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[
"Strong field ionization and gauge dependence of nonlocal potentials",
"Strong field ionization and gauge dependence of nonlocal potentials"
]
| [
"T C Rensink \nUniversity of Maryland\nCollege Park\n",
"T M Antonsen Jr\nUniversity of Maryland\nCollege Park\n"
]
| [
"University of Maryland\nCollege Park",
"University of Maryland\nCollege Park"
]
| []
| Nonlocal potential models have been used in place of the Coulomb potential in the Schrodinger equation as an efficient means of exploring high field laser-atom interaction in previous works. Although these models have found use in modeling phenomena including photo-ionization and ejected electron momentum spectra, they are known to break electromagnetic gauge invariance. This paper examines if there is a preferred gauge for the linear field response and photoionization characteristics of nonlocal atomic binding potentials in the length and velocity gauges. It is found that the length gauge is preferable for a wide range of parameters. | 10.1103/physreva.94.063407 | [
"https://arxiv.org/pdf/1609.02551v1.pdf"
]
| 62,814,765 | 1609.02551 | ce8497ee9cb467940243a7571cd4ddd2a13c58c1 |
Strong field ionization and gauge dependence of nonlocal potentials
T C Rensink
University of Maryland
College Park
T M Antonsen Jr
University of Maryland
College Park
Strong field ionization and gauge dependence of nonlocal potentials
(Dated: 9/6/2016)
Nonlocal potential models have been used in place of the Coulomb potential in the Schrodinger equation as an efficient means of exploring high field laser-atom interaction in previous works. Although these models have found use in modeling phenomena including photo-ionization and ejected electron momentum spectra, they are known to break electromagnetic gauge invariance. This paper examines if there is a preferred gauge for the linear field response and photoionization characteristics of nonlocal atomic binding potentials in the length and velocity gauges. It is found that the length gauge is preferable for a wide range of parameters.
I. INTRODUCTION
Study of strong field ultra-short pulse laser gas interactions, including as THz frequency radiation generation [1][2][3], high harmonic generation [4], and the growing field of attosecond atom-field dynamics [5], relies on numerical modeling of laser-gas interaction. This is often done in two different regimes: A "macroscopic" simulation of laser-pulse evolution over distances of millimeters or centimeters, where the gas is treated as a medium that includes the linear field response, nonlinear field response, including the possibly rotational field response for a diatomic gas, field ionization, and free electron response [6,7]. In the second, "microscopic", regime the interaction of the field with a single atom or molecule is examined in the quantum mechanical picture. This in principle requires solution of the time dependent Schrodinger equation (TDSE) using approximate analytical methods [8], finite-difference time domain (FDTD) numerical solutions [9], or by Floquet expansion schemes [10]. Although attempts have been made to couple Maxwell's Equations with a "microscopic" Schrodinger model [11], these simulations are computationally expensive and remain largely beyond reach at the time of this writing.
Nonlocal binding potentials are a promising tool for efficient solution of the Schrodinger equation, capable of reproducing many the basic quantum mechanical atomic properties efficiently. Despite these successes, it is known that nonlocal models are gauge dependent, while classical electromagnetic theory and the Schrodinger formulation of quantum mechanics are well known to be gauge independent [12]. Breaking this symmetry raises the natural question of how to handle the gauge dependence of these potentials.
This paper examines the gauge dependence of a nonlocal gaussian potential representing the Coulomb potential in a hydrogen-like atom, in the presence of a time varying, spatially uniform electric field. Specifically, we consider the linear polarizability and photoionization rates predicted by the nonlocal models in the length and velocity gauges. The paper is organized as follows: Section II briefly reviews the statement of gauge invariance of the Schrodinger equation for local potentials, section III introduces the nonlocal potential formulation in the length and velocity gauges, section IV reviews some of the basic time-independent properties of the gaussian nonlocal model, and section V examines the static and dynamic atomic polarizability and photoionization characteristics for each gauge. Concluding remarks follow.
II. GAUGE INVARIANCE OF LOCAL POTENTIALS
We briefly examine the gauge invariance of local potential formulations of the time dependent Schrodinger equation. Specifically, we consider the TDSE for the wavefunction of a single electron in the presence of an atomic potential V (x) and a classical electromagnetic field in the dipole approximation with no back-reaction. The time-dependent electric field is represented in the Schrodinger equation via the electromagnetic potential terms, defined through the relation F(t) = −∂ t A(t) − ∇Φ(x, t), noting that, for simplicity we require A(t) depend only on time and that Φ(x, t) be linear in x. The magnetic field is ignored. Atomic units (a.u.) = m e = 1, q e = −1 are used throughout except where noted. The general form of the Schrodinger equation is then:
i∂ t ψ(x, t) = 1 2 (−i∇ + A(t)) 2 − Φ(x, t) − V (x) ψ(x, t).
(1) The choice of A and Φ is not unique; one may define a new set of potentials A , Φ with the addition of a gauge term
A (t) ≡ A(t) + ∇χ(x, t) (2) Φ (x, t) = Φ(x, t) − ∂ t χ(x, t)(3)
that produce the same field F(t), noting that the gauge term takes the form χ(x, t) = x · ∆A(t) for this system. On defining a new wavefunction that is modified by a local phase factor,
ψ (x, t) = exp [−iχ(x, t)] ψ(x, t),(4)
we express the original Schrodinger equation in terms of the primed variables, and operate on the gauge term, i.e. i∂ t ψ = exp(iχ)(i∂ t − ∂ t χ)ψ , and (−i∇ + arXiv:1609.02551v1 [physics.atom-ph] 8 Sep 2016
A(t)) exp(iχ)ψ = exp(iχ)(−i∇ + A(t) + ∇χ)ψ ,
leading to a Schrodinger equation of equivalent form in the transformed variables
i∂ t ψ (x, t) = 1 2 (−i∇ + A (t)) 2 − Φ (x, t) − V (x) ψ (x, t).
(5) Both the original and gauge-transformed Schrodinger equations reproduce the same set of observables and are therefore said to be gauge invariant.
III. GAUGE DEPENDENCE OF NONLOCAL POTENTIALS
If we allow the potential to take the form of an operator acting on the the wavefunction V (x)ψ(x, t) →V ψ(x, t), we may define a nonlocal potential [13][14][15][16] as:
V ψ(x, t) ≡ V 0 u(x)S(t) (6) S(t) ≡ d 3 x u * (x )ψ(x , t) (7) u(x) = σ −3 exp − x 2 /(2σ 2 )(8)
where we have chosen to use a gaussian shape function for u(x). Specifically, the nonlocal potential term is comprised of the function u(x) scaled by the projection of the wavefunction onto u * (x). Projecting onto the complex conjugate ensures the non-local potential remains self-adjoint. Loosely speaking, the positive real valued constant V 0 controls the "strength" of the potential (V 0 > 0 is attractive) and σ, with dimension of length, controls the width of the potential. On performing the same gauge transformation as done in the previous section (and dividing through by an overall phase factor exp(iχ)) the nonlocal potential term appears in the gauge-transformed Schrodinger equation as:
V ψ(x, t) → exp(−iχ)u(x) d 3 x u * (x ) exp(iχ)ψ (x , t)(9)
and it can be seen that the potential term is modified by the phase factor χ.
A form of gauge invariance can be introduced if we treat u(x) as a field that undergoes the same transformation as ψ(x, t), namely u (x, t) ≡ exp(−iχ)u(x); the transformed Schrodinger equation is of the same form as the original and will yield the same observables. However, this implies that u(x) depends on the gauge, and u (x, t), which represents the atomic potential, now depends on the introduced field. This is unphysical, so the question naturally arises: is there a natural gauge for introducing a nonlocal potential? We examine two obvious choices, setting either A = 0 or Φ = 0 in Eq.(1), defining the electric field through a single potential term.
The analysis in the remainder of this paper will be done in the k-space (momentum) representation for con-venience via the Fourier transform definitions,
φ(k) = 1 (2π) 3/2 d 3 x e −ik·x ψ(x ) (10a) ψ(x) = 1 (2π) 3/2 d 3 k e ik ·x φ(k ) (10b)
so that the (canonical) momentum is given by −i∇ → k.
We examine the Schrodinger equation in the so-called length gauge, where A(t) = 0 in Eq.(1), and the velocity gauge, where Φ(x, t) = 0. The momentum-space equations in these two cases are:
i∂ t − 1 2 k 2 + i∂ t A(t) · ∇ k φ L (k, t) = −V φ L (k, t),(11a)
and
i∂ t − 1 2 (k + A(t)) 2 φ V (k, t) = −V φ V (k, t), (11b)
where the subscripts designate length and velocity gauge wavefunctions respectively. The nonlocal potential operator is identical in both equations, specificallŷ
V φ(k, t) ≡ V 0 u(k) d 3 k u * (k )φ(k , t) (12) u(k) ≡ exp(−σ 2 k 2 /2).(13)
We note that the electric potential is written in terms of a single variable A(t) in both equations, where the electric field is defined as
E(t) = −∂ t A(t).
Although we have represented the electric field using a common potential, Eqs.(11a), (11b) are not equivalent. We substitute the explicit expressions for the nonlocal potential in Eqs.(11a) and (11b), introduce integrating factors, and obtain:
φ L (k, t) = iV 0 t −∞ dt exp − i 2 t t dt (k − A(t) + A(t )) 2 . . . u(k − A(t) + A(t ))S L (t ) (14a) φ V (k, t) = iV 0 t −∞ dt exp − i 2 t t dt (k + A(t )) 2 . . . u(k)S V (t ) (14b) where S V,L = d 3 ku * (k)φ V,L ,(15)
which follows from Eq. (7).
IV. FIELD FREE SYSTEM
If A(t) = 0, the length and velocity gauge systems are equivalent. The time independent system is found to have a single bound state which can be represented explicitly in momentum space:
φ 0 (k) = 2V 0 S 0 u(k) |k| 2 + k 2 0 (16) S 0 ≡ d 3 k u * (k )φ 0 (k )(17)
where k 0 ≡ √ 2E 0 is real and positive defined for a state with total energy −|E 0 |. Multiplying both sides of Eq.(16) by u * (k), integrating over all momenta, and dividing both sides by S 0 , we obtain the consistency relation
1 = 2V 0 d 3 k |u(k )| 2 |k | 2 + k 2 0 .(18)
Equation (18) relates E 0 , σ, and V 0 , which can be integrated to give
1 = 4π 3/2 V 0 σ 1 − √ πσk 0 exp σ 2 k 2 0 erfc (σk 0 ) . (19)
Here, erfc is the complimentary error function. For a system with a single nonlocal binding potential term, equation (18) implies that only a single bound state is supported by the nonlocal potential (in contrast to a gaussian local potential [17]); for a chosen value of V 0 and σ, only a single value of E 0 = k 2 0 /2 will satisfy the consistency relation Eq. (18). Figure 1 shows the values of V 0 vs σ for the energies corresponding to the first five states of the hydrogen eigen-spectrum, E n = .5/n 2 . Once the bound state energy is specified, σ is used as a fitting parameter that determines V 0 via Eq. (19). Figure 2 shows a comparison of the nonlocal wavefunction, ψ 0 (x), for various values of σ (E 0 = .5); the hydrogen 1s orbital is provided for comparison.
V. FIELD RESPONSE IN THE LENGTH AND VELOCITY GAUGES
Equations (14a) and (14b) show that the timedependent wavefunction, in the presence of a time varying field, can be recovered if the (gauge dependent) overlap functions S L (t), S V (t) are known. These in turn depend on integrals of the wavefunctions (see Eq. (7)). The advantage of the nonlocal potential model is that these integrals can be carried out analytically, resulting in Volterra (type II) integral equations for the functions S L,V (t).
This method reduces a 3+1 dimensional calculation of ψ(x, t) typically needed to find values of the wavefunction . Further, since the wavefunction has been integrated analytically, the approach is not limited by spatial (or momentum) resolution or extent, which can present difficulties for finite difference solvers. Loosely speaking, the spatial/momentum dependence has been "integrated out" while encoding the wavefunction evolution through the time evolution of the complex variable S(t).
The integral equation for S L,V (t) is found by multiplying Eqs.(14a), (14b) by u(k) and integrating over all momenta. The resulting equation can be written in terms of a kernel function that depends on known quantities:
S L,V (t) = t −∞ dt K L,V (t, t )S L,V (t ).(20)
The kernel K L,V is different in the length and velocity gauges:
K L = iV 0 2π α(t, t ) 3/2 . . . (21a) exp −σ 2 (A 2 + A 2 ) + 1 2α(t, t ) i∆x + σ 2 (A + A ) 2 , K V = iV 0 2π α(t, t ) 3/2 exp i∆x 2α(t, t ) ,(21b)
where
α(t, t ) ≡ 2σ 2 + i(t − t ),(22)
and
∆x ≡ t t A(t )dt = x(t) − x(t ).(23)
The variable ∆x corresponds to the displacement of a classical electron in the presence of A from time t to t (assuming the initial velocity v(t ) = 0). In obtaining (20)-(23), we have absorbed an overall spatially independent phase factor exp( t t dt A 2 (t )) into the definition of the wavefunction, which will not affect any results. The velocity gauge and length gauge kernels differ due to he explicit appearance of the potential, A(t), A(t ) in the length gauge kernel; all the field-dependence in the velocity gauge expression appears through the variable ∆x (as was true for Eqs.(14a), (14b)).
A. Atomic Dipole Moment
The average momentum and time dependent atomic dipole moment are defined as
k ≡ d 3 k φ * (k , t)k φ(k , t)(24)
and
p(t) ≡ − d 3 x ψ * (x , t)xψ(x , t) = −i d 3 k φ * (k , t)∇φ(k , t)(25)
In principle, the nonlinear dipole moment, including the effects of ionization, can be determined from the wavefunction given as the solution of Eqs.(14a), (14b). However, as shown in [16], it is computationally less intensive to solve for the dipole moment using the Ehrenfest relations. These are written as two first-order coupled ODE's with integral expressions for S(t). In both length and velocity gauges:
∂ t k = 2Im V S * (t) t −∞ M(t, t )S(t ) (26) ∂ t p = − k − A(t) + 2Re V S * (t) t −∞ L(t, t )S(t ) (27)
provided we use different definitions for the kernel terms L, M, n,
L L (t, t ) ≡ σ 2 n L (t, t ) − A(t) K L (t, t ) (28) M L (t, t ) ≡ − n L (t, t ) − A(t) K L (t, t ) (29) n L (t, t ) ≡ i∆x(t, t ) + σ 2 A(t) + A(t ) α(t, t )(30)
and
L V (t, t ) ≡ σ 2 n V (t, t )K V (t, t ) (31) M V (t, t ) ≡ −n V (t, t )K V (t, t ) (32) n V (t, t ) ≡ i∆x(t, t ) α(t, t )(33)
where subscript L, V indicate the length and velocity gauges respectively, using previous definitions for ∆x, α, and σ in Eqs. (22), (23). The velocity gauge expressions are again reductions of the length gauge expression where explicit appearances of the potential A(t) and A(t ) are absent.
B. Linear Polarizability
In the low field regime, the (total) dipole moment in Eq.(25) can be characterized by the frequency dependent polarizationp
(ω) = α(ω)F(ω)(34)
where α(ω) is the dynamic polarizability. Although generally a tensor, α(ω) can be represented here by a scalar function because the nonlocal potential is isotropic in k, x and is related to the electric susceptibility tensor χ (1) (ω) through the Clausius-Mossotti relation ( [18]). To obtain the expression for α(ω) for the nonlocal potential model, we define the following:
F(t) =Fe −iωt + c.c., A(t) =F iω e −iωt + c.c., p(t) =pe −iωt + c.c., φ(k, t) → (φ 0 (k) + δφ(k, t)) e iE0t , δφ(k, t) ≡ φ − (k)e −iωt + φ + (k)e iωt ,
S 0 → (S 0 + δS(t)) e iE0t , and
δS(t) ≡ d 3 k u * (k )δφ(k , t) = S − e −iωt + S + e iωt ,
where ω is the frequency of the applied field, and we require F(t), A(t), and p(t) to be real quantities. The expressions above are inserted in a perturbative expansion of the Schrodinger equation (Eq. (11b)) and solved for δφ (discarding all higher order terms). The result is used in Eq.(25) to obtain the first order, frequency dependent dipole moment. For a linearly polarized monochromatic field F(t) = F 0 e −iωtẑ , one obtains the following for the velocity gauge treatment:
φ − = D(−ω) V 0 u(k)S − − φ 0 k z iωF , φ + = D(ω) V 0 u(k)S + + φ 0 k z iωF * , where D(±ω) ≡ E 0 + k 2 /2 ± ω −1 .
With some algebraic manipulation one finds expressions forp ± , e.g:
p − = − d 3 k D(−ω)φ 0 (∂ kz φ 0 ) k z ωF + d 3 k D(ω)φ 0 (∂ kz φ 0 ) k z ωF .
A similar expression can be found for p + . The polarizability is then given by the expression:
α V (ω) = d 3 k [D(ω) − D(−ω)] φ 0 (∂ kz φ 0 ) k z ω ,(35)
and the length gauge polarizability is found by the same method to be
α L (ω) = d 3 k [D(ω) + D(−ω)] (∂ kz φ 0 ) 2 .(36)
Equations (35) 35) and (36). In the limit σ → 0, the nonlocal potential is equivalent to a gaussian potential, and the polarizability is gauge-independent. For positive values of σ, the length-gauge system is more easily polarized by a (DC) applied electric field.
ω → 0 in Fig 3 to show the static (DC) polarizability as a function of the fitting parameter σ. In the limit σ → 0, the gaussian nonlocal potential is equivalent to an attractive delta function potential, and the polarizability is observed to limit to a non-zero gauge-independent value. If σ is increased, the length gauge static polarizability is observed to be much greater than that in the velocity gauge formulation. In Fig.(4), α(ω) is evaluated via Eqs.(35) and (36) (solid lines) and plotted as a function of laser frequency for various values of sigma. The polarization is real for ω < E 0 but complex for ω > E 0 . To evaluate α(ω) for ω ≥ E 0 the laser frequency, previously defined as real, is allowed to become slightly complex, ω → ω + iδ accounting for causality. It should be noted that systems with additional eigenstates would have additional resonances for coupling to excited states, e.g. for hydrogen: ω res = E 0 (1 − 1/n 2 ). These are not present in a single bound state system. The cross marks in Fig.4 represent the α(ω) calculated from numerical simulation via Eqs.(26) and (25). Each cross represents the atomic dipole calculated for a 50 femtosecond low intensity pulse (I max = 1 × 10 10 W/cm 2 ), and the ratio taken of Fourier transform coefficientsp(ω),F(ω). Agreement is observed between the predicted and simulated values. (25)).
C. Ionization
The time dependent bound-electron probability is defined as
ρ(t) ≡ d 3 x ψ * 0 (x )ψ(x , t) 2 = d 3 k φ * 0 (k )φ(k , t) 2 ;
(37) we may use this to define the time dependent ionization rate ν(t) through the relation
ρ(t) = ρ(t 0 ) exp − t t0 ν(t )dt ,(38)
that depends functionally on the field F(t).
In practice, it is often easier to use functions other than ρ(t) that are approximately equal to the bound probability defined by Eq. (37). For the nonlocal potential, we use the quantity
ρ u (t) ≡ S(t) S 0 2 ∝ d 3 k u * (k )φ(k , t) 2(39)
for convenience, noting that the functions u(k) and φ 0 (k) are similar in functional form, and note the limit E 0 → ∞, ρ u (t) → ρ(t) . Although these measures are not identical, any wave density that escapes the nonlocal potential region quickly propagate away from the origin, making ρ u (t) a very good approximation of the bound probability. A comparison of these quantities was rather carefully examined in previous work [16] which demonstrated ρ u (t) and ρ(t) were in agreement in the length gauge formulation. The quantity ρ u (t) does not offer such a straightforward interpretation in the velocity gauge, but can be used as a measure of bound probability for times when A(t) = 0, and can be used for measuring pulse averaged ionization rates. We compare the length and velocity gauge predicted ionization rates using a flat-top laser pulse of form E(t) ≡ F (t) cos(ωt)ẑ with a 15 femtosecond ramp-time (t r ) of the form
F (t) ≡ F 0 sin 2 ( π 2tr t) for 0 ≤ t ≤ t r F 0 for t r < t ≤ t p − t r F 0 cos 2 ( π 2tr (t − t p + t r )) for t p − t r < t ≤ t p(40)
where t r is the ramp time to maximum and t p is the total pulse length, with values of 15 and 90 femtoseconds respectively. This pulse profile was used in place of a gaussian or sin 2 (t) envelope to maximize the time the electric field amplitude was at a fixed value while still maintaining a narrow bandwidth to prevent frequency dependent structure in the ionization rate from being averaged out. The total drop in bound probability ρ u (t f ) (Eq. (39)) is used to calculate a pulse averaged ionization ratẽ
ν = − ln [ρ(t f )] t p − t r .(41)
FIG. 5: The ionization rates predicted by the PMPB model and the nonlocal length/velocity gauge formulations as a function of laser frequency and intensity (100 × 100 data points, interpolated). The laser parameters here span the multiphoton (high frequency, low intensity) and tunnel (low frequency, high intensity) ionization regimes. The length gauge and PMPB ionization rates agree well over the parameter space shown; the velocity gauge ionization rate generally underestimates in the multiphoton regime and overestimates in the tunnel regime. Slices along constant intensity and frequency are shown in Figs.7 and 6 for direct comparison. The ionization rate as a function of intensity for I 0 = 2 × 10 14 W/cm 2 (6a); the values of σ for the length and velocity gauge potentials were calibrated at this intensity, at 800 nm (visible here as the crossing point for all rates). The velocity gauge overestimates the rate towards tunnel regime and underestimates it in the multiphoton regime, while the PMPB and length gauge rates predict similar rates. The rates are also plotted for I 0 = 2 × 10 13 W/cm 2 (6b). Figure 5 shows the ionization rateν landscape as a function of the near infrared to near ultraviolet laser frequency at ionizing intensities, spanning the multiphoton and tunnel ionization regimes. The length gauge and velocity gauge rates are compared with an ionization rate model introduced by Popruzhenko, et. al. in 2008 [8], here referred to as the "PMPB" rate, in reference to the authors' names. The PMPB rate used for comparison here is preferable to Keldysh or ADK models [20][21][22] which are known to underestimate the multiphoton ionization rate by several orders of magnitude; the PMPB model is valid in both the tunneling and multiphoton regimes and was shown to give good agreement with both The PMPB photoionization rate and nonlocal (length gauge) rate also show agreement as a function of intensity for 800 nanometer light; the velocity gauge ionization rate does not (7a). At 400 nm (7b), the velocity gauge ionization rate has the same power dependence as the length gauge and PMPB rates, but strongly underestimates the magnitude for the chosen fitting parameter (σ = 4.785).
Floquet and ab initio TDSE solver simulations [8].
To compare the ionization rate predicted by the nonlocal potential, the tuning parameter σ was fixed by matching the ionization rate of a single run with typical laboratory parameters ω = .057 [a.u.] (800 nm), and F 0 = .01 [a.u.] (Intensity of 2 × 10 14 W/cm 2 ), seen as the crossing point of all rates in Fig.6a. The values σ = 2.482 for the length gauge and σ = 4.785 for the velocity gauge were used in all ionization plots shown.
A glance at Fig. 5 shows that the nonlocal length gauge and PMPB rates share the same general contours across the entire range of intensities and frequencies shown here. Slices taken along lines of constant frequency (7a, 7b) and constant intensity (6b, 6a) give a more direct compari-son and show strong agreement in the PMPB and length gauge ionization rates for all frequencies examined and intensities up to I 0 ∼ 4 × 10 14 W/cm 2 . The deviation above this intensity is only apparent; calculation of the S(t) always leaves residual traces which artificially decreaseν. The PMPB and nonlocal length gauge predict similar ionization rates for all laser parameters shown. It should be stated that the agreement in ionization rate shown in these plots is, in some cases misleading; neither the PMPB nor the nonlocal model here can account for ionization pathways that include intermediate population of excited electron states, [23].
By contrast, the velocity gauge ionization rate does not agree with the PMPB rate; it underestimates ionization below I 0 = 2 × 10 14 W/cm 2 and overestimates it for higher intensities; for this reason, it is unlikely that a different choice of σ could improve the predicted rate in the tunnel and multiphoton regimes (the rate generally changes monotonically with the tuning parameter σ for a specified electric field). This under-prediction at low intensities and over prediction at high-intensities for the velocity gauge formulation is consistent with other work [15] which examined the ionization rate of a similar nonlocal model in the velocity gauge.
VI. CONCLUSION
In this work, we examine the gauge dependence of nonlocal atomic potentials in the time dependent Schrodinger equation. We note that the utility of nonlocal potential models is that the atom-field interaction can be computed in the time domain without having to resolve the spatial or momentum space wavefunction, allowing for rapid evaluation of e.g., the atomic dipole moment and photoionization rate. For this reason, nonlocal potential models are of interest for examining atom-field interactions and for use in Maxwell-Schrodinger laser propagation simulations.
Specifically, we consider the linear dipolar field response and photoionization rate, predicted by a gaussian nonlocal atomic potential, in the length and velocity gauges in a time varying electric field. All examined quantities are found to be gauge dependent. At low intensities (I ∼ 10 10 W/cm 2 and below), both gauge formulations exhibit similar resonant frequency response at photon energies near the ionization threshold, and a static polarizability in the low frequency limit, but differ significantly in magnitude. The photoionization rates predicted in each gauge were compared with the Coulombic photoionization rate model (PMPB) [8], in the frequency (near IR to near UV) and intensity domains ((I ∼ 10 13 − 10 15 W/cm 2 ). It was found that, although gauge formulations demonstrate multiphoton resonance and tunnel features, the velocity gauge formulation generally over estimated the tunnel ionization rate and underestimated the multiphoton ionization rate; the length gauge and PMPB photoionization rates agreed well over
FIG. 1 :
1(Color online) Curves relating V 0 and σ for constant values of E 0 that satisfy Eq.(19). Shown here for the first five hydrogen states, the gaussian nonlocal potential supports a (single) bound state of arbitrary energy.FIG. 2: (Color online) The normalized configuration space wavefunction ψ(|r|) is given by the Fourier transform of Eq.(16) (shown here for E 0 = .5). The variable σ is used as a fitting parameter. and observables of interest to a series of ∼ 2D calculations (the number of operations required to solve the integral equation in time grows like t 2 )
and (36) are evaluated in the limit FIG. 3: (Color online) The static polarizability α(ω = 0) as calculated from Eqs. (
FIG. 4 :
4The dynamic polarizability in the length and velocity gauges, with a single photon resonance at ω = E 0 The solid lines represent the α(ω) given by Eqs.(35) and (36) (ω ≥ E 0 → ω + iδ), and the crosses represent the simulated low field response via the total dipole (Eq.
FIG. 6: The ionization rate as a function of intensity for I 0 = 2 × 10 14 W/cm 2 (6a); the values of σ for the length and velocity gauge potentials were calibrated at this intensity, at 800 nm (visible here as the crossing point for all rates). The velocity gauge overestimates the rate towards tunnel regime and underestimates it in the multiphoton regime, while the PMPB and length gauge rates predict similar rates. The rates are also plotted for I 0 = 2 × 10 13 W/cm 2 (6b).
FIG. 7: The PMPB photoionization rate and nonlocal (length gauge) rate also show agreement as a function of intensity for 800 nanometer light; the velocity gauge ionization rate does not (7a). At 400 nm (7b), the velocity gauge ionization rate has the same power dependence as the length gauge and PMPB rates, but strongly underestimates the magnitude for the chosen fitting parameter (σ = 4.785).
This work was supported by the. U.S. Department of Energy (DoE) and Naval Research Laboratory (NRLThis work was supported by the U.S. Department of Energy (DoE) and Naval Research Laboratory (NRL).
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| []
|
[
"On the reinforcement homogenization in CNT/Metal Matrix Composites during Severe Plastic Deformation",
"On the reinforcement homogenization in CNT/Metal Matrix Composites during Severe Plastic Deformation"
]
| [
"Katherine Aristizabal \nDepartment of Materials Science\nChair of Functional Materials\nSaarland University\nCampus 66123SaarbrückenGermany\n",
"Andreas Katzensteiner \nErich Schmid Institute of Materials Science\nAustrian Academy of Sciences\nJahnstrasse 12A-8700LeobenAustria\n",
"Andrea Bachmaier \nErich Schmid Institute of Materials Science\nAustrian Academy of Sciences\nJahnstrasse 12A-8700LeobenAustria\n",
"Frank Mücklich \nDepartment of Materials Science\nChair of Functional Materials\nSaarland University\nCampus 66123SaarbrückenGermany\n",
"Sebastian Suárez \nDepartment of Materials Science\nChair of Functional Materials\nSaarland University\nCampus 66123SaarbrückenGermany\n"
]
| [
"Department of Materials Science\nChair of Functional Materials\nSaarland University\nCampus 66123SaarbrückenGermany",
"Erich Schmid Institute of Materials Science\nAustrian Academy of Sciences\nJahnstrasse 12A-8700LeobenAustria",
"Erich Schmid Institute of Materials Science\nAustrian Academy of Sciences\nJahnstrasse 12A-8700LeobenAustria",
"Department of Materials Science\nChair of Functional Materials\nSaarland University\nCampus 66123SaarbrückenGermany",
"Department of Materials Science\nChair of Functional Materials\nSaarland University\nCampus 66123SaarbrückenGermany"
]
| []
| Carbon nanotube (CNT)-reinforced nickel matrix composites with different concentrations were processed by high pressure torsion (HPT). We thoroughly characterized the CNT agglomerates' spatial arrangement at different stages of deformation in order to extract information valuable for the optimization of the processing parameters and to elucidate the mechanisms involved during the processing of particle reinforced metal matrix composites by HPT. From the electron micrographs taken on the radial direction with increasing equivalent strains, we observed that CNT agglomerates debond by relative sliding between CNT during HPT, becoming spherical at higher stages of deformation. Furthermore, we introduced a model for the prediction of the minimum strain required for a homogeneous distribution of a second phase during HPT, which can be correlated to the material's three-dimensional structure and agrees well with the experimental data. | 10.1016/j.matchar.2018.01.007 | [
"https://arxiv.org/pdf/2010.14103v1.pdf"
]
| 139,705,594 | 2010.14103 | fc5e2b4bbcbbfe58daec50b44fc0c0e562187de2 |
On the reinforcement homogenization in CNT/Metal Matrix Composites during Severe Plastic Deformation
Katherine Aristizabal
Department of Materials Science
Chair of Functional Materials
Saarland University
Campus 66123SaarbrückenGermany
Andreas Katzensteiner
Erich Schmid Institute of Materials Science
Austrian Academy of Sciences
Jahnstrasse 12A-8700LeobenAustria
Andrea Bachmaier
Erich Schmid Institute of Materials Science
Austrian Academy of Sciences
Jahnstrasse 12A-8700LeobenAustria
Frank Mücklich
Department of Materials Science
Chair of Functional Materials
Saarland University
Campus 66123SaarbrückenGermany
Sebastian Suárez
Department of Materials Science
Chair of Functional Materials
Saarland University
Campus 66123SaarbrückenGermany
On the reinforcement homogenization in CNT/Metal Matrix Composites during Severe Plastic Deformation
10.1016/j.matchar.2018.01.007Carbon nanotubesMetal-matrix compositesSevere Plastic deformationSecond- phase distribution 2
Carbon nanotube (CNT)-reinforced nickel matrix composites with different concentrations were processed by high pressure torsion (HPT). We thoroughly characterized the CNT agglomerates' spatial arrangement at different stages of deformation in order to extract information valuable for the optimization of the processing parameters and to elucidate the mechanisms involved during the processing of particle reinforced metal matrix composites by HPT. From the electron micrographs taken on the radial direction with increasing equivalent strains, we observed that CNT agglomerates debond by relative sliding between CNT during HPT, becoming spherical at higher stages of deformation. Furthermore, we introduced a model for the prediction of the minimum strain required for a homogeneous distribution of a second phase during HPT, which can be correlated to the material's three-dimensional structure and agrees well with the experimental data.
Introduction
The distribution of reinforcing phases in composite materials is of great importance and has a big influence on their mechanical performance. Carbon nanotubes have been widely used as reinforcing phase not only in polymer [1,2] and ceramic matrix composites [3,4] but also in metal matrix composites [5][6][7]. Some authors have explored different strategies for improving the distribution of CNT in MMC such as blending by mixing, nano-scale dispersion, ball milling, cold spraying and molecular-level mixing [6,8,9] However, blending by mixing has been found to deteriorate the mechanical properties of the composites due to poor distribution of the CNT. Although nano-scale dispersion was found to improve significantly the distribution of the CNT in Aluminum matrix, because it consist in utilizing natural rubber in a mixture with the CNT and the metallic powder alternatively stacked in a preform, it needs to be subjected to high temperatures (800°C) in order to burn the rubber and to melt the metal, which would imply the use of even higher temperatures in the case of nickel, which has a melting point of 1455°C, adding further difficulties to the manufacturing process. Ball milling and molecular-level mixing methods also improve significantly the dispersion of the CNT, both producing large agglomerates' sizes of about some microns to some millimeters. Ball milling also results in severe damage to the CNT. A thorough discussion of the advantages and drawbacks of each technique is beyond the scope of this manuscript and can be found in [6] and the references therein. Therefore, given their tendency to form agglomerates due to Van der Waals forces, a homogeneous distribution of CNT is still a challenging task.
High-pressure torsion, a severe plastic deformation process, has shown to be a powerful tool for improving the distribution of particles in MMC. I. Sabirov et al, showed how ceramic particles are reduced in size and dispersed inside the metallic matrix by debonding [10]. During HPT, the sample is placed between two anvils using high pressures (> 2 GPa) and rotated in quasi-constrained conditions a certain number of turns (T). By the action of plastic flow and the generation, mobility and re-arrangement of dislocations, a refinement of the microstructure takes place and the strength of the material increases with increasing strain until the saturation in the microstructural refinement is reached [11].
Nevertheless, these microstructures possess high stored energy in their large grain boundary area. Recently, based on their ability to pin the microstructure [5], CNT have been used as stabilizing phase against grain growth [12,13] in MMC processed by severe plastic deformation. However, in order to efficiently fulfill their stabilization task, CNT should be homogeneously distributed [14], which is also expected to increase the mechanical performance of these MMC. For that purpose, the processing route used in this work started by colloidal mixing of the CNT with the metallic Nickel powder, followed by the cold pressing and hot sintering of the blends and succeeding processing by means of high pressure torsion.
For qualitative assessment of the distribution, directly observation of electron micrographs is sufficient. Previously, a preliminary assessment of the reinforcement homogeneity in CNT/Ni MMC was carried out [15]. In the present study the aim is to extract quantitative information from the electron micrographs on the CNT agglomerate size and spatial distribution in MMC subjected to severe plastic deformation, and correlate it to the mechanisms involved in the deformation of the reinforcement during processing, seeking the optimization of the process parameters and the improvement of the physical properties of the composites. Some quantitative methods on the evaluation of the dispersion and distribution of CNT in MMC have been proposed in the literature [16][17][18][19]. A thorough discussion of the different quantification methods can be found elsewhere [20].
Furthermore, event-to-event (nearest neighbor distance NND), based methods, commonly used in spatial statistics, have been used in unidirectional composites [21]. Moreover, the method of Region Homogeneity HRO has been proposed as an easy way of assessing the distribution homogeneity of second phases in metallic materials [22] and also of CNT agglomerates in MMC [23]. The advantages and limitations of the latter in the evaluation of MMC processed by SPD are discussed here.
In this work, an extensive clustering and distribution homogeneity analysis was performed with the aim of understanding the behavior of the CNT during the deformation process as a function of the accumulated deformation. For the evaluation of the studied composites' homogeneity, a NND clustering analysis-based methodology is used, which can be completely carried out using open source software [24,25]. From the electron micrographs, a CNT agglomerate-debonding mechanism by relative sliding between CNT during HPT is observed, which contributes to the understanding of the reinforcement arrangement after the processing by severe plastic deformation of CNT MMC. Finally, a model for the prediction of the minimum equivalent strain that should be applied during HPT for a homogenous distribution of particles in MMC is proposed, which provides a basis for the optimization of the processing parameters.
Experimental
Manufacturing and HPT processing of CNT/Ni composites
The composites were obtained via powder metallurgy and further processed by HPT. The starting materials were MWCNT (CCVD grown, Graphene Supermarket, USA density 1.84 g/cm 3 , diameter: 50-85 nm, length: 10-15 m, carbon purity: >94%) and dendritic Ni powder (Alfa Aesar, mesh -325). A colloidal mixing process was used to blend the precursor powders by which CNT are dispersed in ethylene glycol EG (CNT/EG concentration ratio at 0.2 mg/ml) and mixed with Ni powder. A thorough description of this process is reported elsewhere [26]. The CNT fractions used were 0.5, 1, 2 and 3 wt. % (2.4, 4.7, 9 and 13 vol. %, respectively). The powders were dried and cold pressed under 990
MPa and subsequently sintered in a hot uniaxial press HUP under vacuum (2 x 10 -6 mbar) at 750 °C for 2.5 h with a 264 MPa axial pressure. Sintered samples were further processed by means of HPT at room temperature using 1, 4, 10 and 20 T under 4 GPa of pressure.
Samples with 1 wt. % CNT were processed 30 T at room temperature RT and at 200ºC, were also analyzed. Fig. 1 shows schematically the HPT set up.
Characterization
The HPT samples were cut in halves, embedded in conductive resin and fine polished using polishing discs with the aid of 6, 3 and 1 m diamond suspensions and finally with OPS colloidal silica. The samples were then characterized by scanning electron microscopy (SEM) using a Helios NanoLab TM 600 dual beam field emission microscope (FEI Company). The analysis of the microstructure was carried out along the middle plane, in order to avoid the effect of the microstructural gradients along the height of the specimens [27]. The images were taken every 1 mm with a resolution of 12.5 nm per pixel along the radial direction, corresponding to increasing equivalent strain values, according to
ε eq = 2πTr t√3
, where T is the number of turns, t is the sample thickness and r the distance from the center of the sample [28]. In the case of HPT samples, the micrographs were acquired at 7 made based on the size of the agglomerates: if the images were to be taken at lower magnifications, information about the smaller agglomerates in the case of the highly deformed samples would be lost due to resolution issues. On the other hand, at higher magnifications the larger agglomerates would be also neglected, since they would hardly fit into the region of interest. Furthermore, the size was kept constant for comparison purposes.
The images were digitally binarised and analyzed using the image processing package FIJI [24]. Micrographs with corresponding binary images are shown exemplarily in Fig. 2. The CNT agglomerates were considered as compact particles and a thorough particle analysis was performed on each image. The particle analysis protocol was as follows: the images were calibrated with the known scale; the threshold was adjusted thoroughly, in order to separate the dark (CNT agglomerates) from the light regions (Ni matrix), without removing pixels from the boundaries (manual segmentation was performed when necessary) and finally, the images were made binary. Different size descriptors (such as the area, the maximum and minimum Feret diameter and the perimeter) and shape factors (such as roundness, circularity, aspect ratio etc.) can be obtained during particle analysis. In this case, the particle area was obtained and the diameter of an equivalent circumference was computed as the agglomerate equivalent diameter DCNT.
Additionally, the position of the agglomerate centers of mass was extracted and used to calculate the NND (nearest neighbor distance). The evolution of the area weighted agglomerate diameter and the mean NND with increasing equivalent strain were studied (Fig. 3).
Quantitative assessment of the distribution homogeneity and clustering behavior
Furthermore, a set of samples was analyzed in terms of homogeneity and distribution using two different methodologies. The quantitative assessment of the homogeneity was carried out following the method proposed by Rossi et al [23]. The agglomerate area fraction (phase amount) and the number of objects were obtained using particle analysis in 20 different regions of interest using ROI manager in FIJI. The region homogeneity HR is a constructed homogeneity obtained by multiplying the object number homogeneity HNO and the phase amount homogeneity HPA. The partial homogeneities H were calculated as the counterpart of the Gini G coefficients G: = 1 − , which is a measure of distribution inequality [22]. The Gini index of each parameter (object number and phase amount as obtained from particle analysis in FIJI) was retrieved using the ineq package in R studio, an integrated development environment for R, which is a programming language for statistical computer and graphics [25].
Information about the agglomerates' centers of mass was used for further point pattern analysis by means of the nearest neighbor distance distribution function (Gfunction) using the spatstat package in R Studio [29], in order to obtain more detailed information about the spatial behavior between the individuals (CNT agglomerates' centers of mass, treated as "points") of the investigated population (studied areas, treated as "point patterns"). This method consists on the comparison of the empirical cumulative distribution function Gobs(r), with the nearest neighbor distribution function Gtheo(r) for complete spatial randomness (CSR). The nearest neighbor distance distribution function for a CSR is described by G theo (r)=1-exp(-N A πr 2 ), where NA is the number of particles per unit area, and r the evaluated distance. According to this, the condition Gobs(r) < Gtheo(r) is inherent of regular patterns and the opposite case, Gobs(r) > Gtheo(r), is interpreted as "clustering" (for example, Fig. a-c) because the NND are smaller than expected for the CSR case [30][31][32].
As there are numerous configurations for CSR, pointwise critical envelops with fixed respective number of points (corresponding to each case), obtained from 100 Monte Carlo simulations of CSR with a significance level of 2/101 = 0.0198 and spatial Kaplan-Meier edge correction, which is performed in order to correct effects arising from the nonvisibility of points lying outside the evaluated field of view during the evaluation of the G-Function [33], were plotted as the theoretical expected behavior. Accordingly, the G-function is a useful tool in spatial statistics that summarizes the "clustering" behavior of points in a point pattern. It displays the empiric cumulative distribution of NND in contrast to that of the CSR with the same number of points within a window of the same size. If a point pattern does not adjust to a Poisson pattern, then it can be either clustered (i.e. when the particles are interacting and tend to come closer together) or regular (i.e. when the points tend to avoid each other). It might be inferred that a homogeneous sample, e.g. with a high HRO value, also does not display a clustering behavior and vice versa. The G-Function can help to confirm or deny this statement. In this case, border corrections are carried out and the size of the window is arbitrary. For this reason, the G-function was chosen as a complementary method.
Results and discussion
By plotting the area weighted mean agglomerate diameter DCNT and the mean NND vs. Eq. Strain (Fig. 3) the evolution of both parameters with increasing strain can be tracked. Results show that both, DCNT and NND decrease significantly in size during the first stages of deformation approximately up to a strain of 20, but do not significantly change afterwards, where both DCNT and NND are between 100 and 400 nm. The deviation of the data also decreases with increasing strain. Fig. 3b also suggests that the agglomerates come closer for higher CNT concentration. Furthermore, a previous study showed that the agglomerate area fraction does not change significantly with increasing strain values [34]. In the latter case, DCNT and NND are also within the same range discussed before.
Figure 3 a) Evolution of agglomerate diameter DCNT, and b) evolution of the mean inter-particle distance (defined as NND) with increasing strain, of samples with different compositions.
Nevertheless, from these results, an explicit relation concerning the evolution of the particle homogeneity during HPT is not feasible. In order to address this matter, analysis of the distribution of the CNT was carried out using the region homogeneity parameter HRO and the nearest neighbor distribution function G.
The parameter region homogeneity HRO, as proposed by Rossi et al. [23] serves as a general evaluation of the distribution of particles lying within a studied area divided in an arbitrary number of regions, by quantifying the similarity of the different regions in terms of number of objects and amount of phase. This method is a practical and straightforward way of assessing the homogeneity distribution, when studying well-dispersed particle composites. Nevertheless, it does not take into account the separation or the interaction of the agglomerates (it is not sensitive to the clustering behavior of particles within the analyzed regions). Furthermore, the analysis is limited to samples (micrographs) of the same dimensions, since this method is area size sensitive.
On the other hand, when studying non-deformed MMC samples, lower magnifications can be used in order to display a higher number of particles without neglecting any significant information since the initial agglomerates are much larger (DCNT > 2m) than in the case of highly deformed samples (DCNT ~100 nm, Fig. 3). cases it is true that HRO increases with increasing strain except after only 1 T. Also, the diameter of the agglomerates and the inter-particle distance decrease significantly compared to the HUP samples (Fig. 3), and this is also an indicative of the improvement of the agglomerate distribution. Fig. 4a to 4c correspond to the G-function of samples with 1 wt. % CNT processed 10 T, which displayed the higher improvement in HRO (Fig. 4d). In all the cases a clustered behavior is present, as the empiric distributions are above the envelope for 100 simulations of the theoretical CSR expectation. In the case of samples with 1 wt. % CNT processed 30 T, HRO increases slightly with increasing strain above HRO = 60 % (Fig. 4d). Fig. 4e to 4h show the G-function for the latter case. It can be seen from Fig. 4e that even for zero equivalent strain the empiric cumulative distribution of NND is closer to the CSR envelope. Nevertheless, for distances > 50 nm, a slight clustering behavior is observable. It can be inferred that for eq. strains 84 < < 100 (Fig. 4c and 4f) the agglomerate NND cumulative distribution starts to behave homogeneously (Fig. 4f to 4h Gobs stays within the envelope for completely random Poisson distributions). processed 30 T at 200ºC. In this case is also true that when = 0, G(r) is closer to the theoretical expectation CRS and for higher , G(r) is within the CRS envelopes, and for distances < 100 nm the agglomerates are more separated than the CRS case. Sabirov [10] proposed the extended Tan and Zhang model (originally conceived for cold rolled MMCs) as an assessment of the equivalent strain values required in HPT to achieve a homogeneous particle distribution in ceramic particle reinforced MMC subjected to extrusion according to ≥
[( π 6f ) 1 3 ]√R γ
where dp is the particle size, dm is the matrix powder size required to attain a homogeneous particle distribution; f the particle volume fraction; R is the extrusion ratio; is the shear strain with γ = √3 . This is based on the assumption that the particles are spherical. They found that SiC and Al2O3 particles behave differently during HPT and that the homogenization occurs according to Tan and Zhang model [35]. Nevertheless, the experimental data did not adjust well to their proposed model, which was attributed to the fact that the ceramic particles are de-clustered through a debonding mechanism without deformation of the particle clusters.
Although, in the samples studied here, the agglomerates cannot be considered spherical throughout the entire process because during the first stages of deformation elongated CNT agglomerates in the shear direction are observed (Fig. 2, = 0).
Nevertheless, at higher applied strains, they start to debond forming more spherical agglomerates (Fig. 2, eq. = 84). Contrary to ceramic particles, which are hard and brittle,
CNT are elastic and CNT agglomerates are bonded by Van der Waals forces that can be overcome during deformation. According to this, CNT agglomerates debond by relative sliding between CNT. Furthermore, sliding between CNT walls may also occur [36].
In the literature, a unique value of inter-particle distance, which only deals with the case of equidistant particles, has been considered for the assessment of a uniform distribution of particles during plastic deformation [10,35]. Nevertheless, as already discussed, the particles can be homogeneously distributed according to different configurations describing a Poisson distribution. Therefore, a model is proposed, starting from the nearest neighbor distribution function in 3D, and its correlation with the CNT agglomerates' spatial behavior. In this case, HPT is considered and the agglomerates are assumed to be spherical.
The nearest neighbor distance NND distribution function in 3D for completely random Poisson distribution is described by equation 1 [37]:
( ) = 1 − exp (− 4 3 3 ) ; ≥ 0 (1)
where NV is the mean number of objects per unit volume. NV for isolated spherical objects of random diameter D is [37] (pg. 78):
= 6 3(2)
Replacing 2 in equation 1 yields:
( ) = 1 − exp (− 8 3 3 )(3)
For point fields it is true that the volume fraction equals the spherical contact distribution V V =H s (r) and from the completely random property of the Poisson field it follows that the NND distribution D(r) and the spherical contact distribution function H s (r)
are identical [37] (pg. 315) H s (r) D(r), thus D(r) = V V . According to this and solving r in equation 2:
= 2 [− ln (1− ) ] 1/3(4)
Also, for uniform distribution of CNT, which are expected to be at the grain boundaries due to the nature of the processing route (powder metallurgy), r should be equal or greater than the grain size of the matrix after HPT GSfHPT:
≥ (5)
Furthermore, for simple shear a volume element, which may be a grain or a second phase in a composite, will be deformed in an ellipsoid with apex ratio ′ . The reduction ratio of an ellipsoid is given by √r ' . For large shear strains ( > 2), r ' = γ 2 , with γ = √3ε eq [28]. In the case of the studied CNT reinforced MMC, it is assumed that the agglomerates are deformed in the same way as the matrix. Accordingly, after HPT the reduction ratio is given by:
= √3(6)
GSiHPT corresponds to the grain size of the matrix before HPT, i.e. the grain size of the composite after the sintering process by HUP GS iHPT = GS fHUP . Thus, replacing in equation 6 and solving GS fHUP :
= √3(7)
and equation 5 can be expressed as: 6 shows the experimental data of the mean grain size measured by EBSD of the studied composites before HPT with the corresponding Zener based model.
2 [− ln (1− ) ] 1/3 ≥ √3(8)
According to equation 9, there is a minimum strain ℎ that should be applied in order to obtain a homogenous distribution of CNT. It can be thus inferred that for increasing CNT volume fraction VCNT a lower strain should be applied in order to obtain a homogeneous distribution of CNT after HPT. This corresponds well with the literature, where, for lower VCNT, inhomogeneity rises due to the presence of reinforcement-depleted regions [23,39]. Table 1 shows the results for ℎ , according to equation 9, for the different CNT compositions used in this work. In the case of the samples with 1wt. %CNT (VCNT = 0.047) ℎ = 96.31, which is within the range 84 < < 100 previously found experimentally, for which these samples were found to be homogenously distributed according to the NND distribution function.
These homogenously distributed samples presented values of region homogeneity HR 60 %. It can thus be inferred, that a HR over 60% is an empirical lower bound, which is necessary but not sufficient to unequivocally identify a homogeneous random distribution of the second phase. It is therefore unavoidable the utilization of a complementary evaluation, as proposed here, which additionally evaluates the distribution of the mean spatial inter-particle distance. Furthermore, this approach provides information valuable for the optimization of the processing parameters by HPT of MMC.
Conclusions
The distribution homogeneity of CNT agglomerates in MMC processed by highpressure torsion is evaluated by the combination of the region homogeneity parameter HRO and the nearest neighbor distribution function G(r). This methodology can be used as a thorough evaluation of the distribution homogeneity of second phases in MMC.
Furthermore, CNT agglomerates debond by relative sliding between CNT during HPT, which differs significantly to the debonding-mechanism in ceramic particles. Finally, a model that predicts the minimum equivalent strain required for a homogenous second phase distribution during HPT is developed, which correlates well with the experimental data and provides a basis for the optimization of the processing parameters of MMC by severe plastic deformation.
Figure 1
1Schematic HPT set up used for sample processing.
Figure 2
2Electron micrographs (upper row) and their respective binarized images (lower row) from a 1 wt.% sample after 10 turns. The equivalent strain increases from left to right. The dark regions correspond to CNT agglomerates.
Fig. 3
3also displays information for a sample with 1 wt. % CNT processed with 30 T at 200 ºC.
Furthermore
, it is possible to keep the size of the analyzed area constant and HRO can be used to compare the homogeneity of different kind of MMC samples. Nevertheless, the inability of using lower magnifications and of keeping the size of the micrographs constant without losing significant information restricts the usefulness of HRO in highly deformed samples.
Fig
. 4d displays the evolution of HRO for the samples with 1 wt. % CNT processed at RT. Even though HRO does not significantly improve during HPT up to 30 T, in most
Figure 4 (
4a-c) G(r) at different equivalent strains for 1wt.% CNT deformed at 10 T. (d) HRO for the selected set of samples. (e-f) G(r) at different equivalent strains for 1wt.%CNT deformed at 30 T. Where ĜObs (r) is the observed value of G(r); Gtheo (r) is the theoretical value of G(r) for complete spatial randomness; Glo (r) and Ghi (r) represent the lower and upper bounds of G(r) from simulations. The deformation was performed at room temperature.
Figure 5 G
5(r) for a 1wt.%CNT sample processed with 30 T at 200ºC, as a function of the equivalent strain. The measured HRO is also displayed.
Figure 6
6Mean grain size of studied composites after HUP as a function of the reinforcement volume fraction (VCNT). Dashed curves represent the empirical Zener pinning model with respective upper and lower bounds (grey envelope).Accordingly, and rearranging equation 8:
Previously, it was shown for consolidated CNT reinforced Nickel matrix composites that, in the presence of CNT a grain growth stagnation takes place during sintering and eventual annealing and the final grain size is related to the CNT volume fraction VV according to the Zener based relationship GS fHUP =0.99±0.07
V V
0.4
[38].
Fig.
Table 1
1Values of equivalent strain required for achieving a homogenous CNT distribution in MMC processed by HPT.VCNT
0.024
126.5
0.047
96.31
0.09
73.71
0.13
63.16
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| []
|
[
"Gradient flows in the normal and Kähler metrics and triple bracket generated metriplectic systems",
"Gradient flows in the normal and Kähler metrics and triple bracket generated metriplectic systems"
]
| [
"Anthony M Bloch ",
"Philip J Morrison ",
"Tudor S Ratiu "
]
| []
| []
| The dynamics of gradient and Hamiltonian flows with particular application to flows on adjoint orbits of a Lie group and the extension of this setting to flows on a loop group are discussed. Different types of gradient flows that arise from different metrics including the so-called normal metric on adjoint orbits of a Lie group and the Kähler metric are compared. It is discussed how a Kähler metric can arise from a complex structure induced by the Hilbert transform. Hybrid and metriplectic flows that arise when one has both Hamiltonian and gradient components are examined. A class of metriplectic systems that is generated by completely antisymmetric triple brackets is described and for finitedimensional systems given a Lie algebraic interpretation. A variety of explicit examples of the several types of flows are given.IntroductionDynamical systems, finite or infinite, that describe physical phenomena typically have parts that are in some sense Hamiltonian and parts that can be recognized as dissipative, with the Hamiltonian part being generated by a Poisson bracket and the dissipative part being some kind of gradient flow. The description of Hamiltonian systems has received much attention over nearly two centuries and, although some forms of dissipation have received general attention, the understanding and classification of dissipative dynamics is a much broader topic and consequently less well developed. Early modern treatments of geometric Hamiltonian mechanics include those of Souriau [1970] andAbraham and Marsden [1978], and the literature on this topic is now immense. A special type of gradient flow that preserves invariants, the double bracket formalism described inBrockett [1991](see, e.g.,Bloch [1990], Bloch [2003), is a formalism that occurs in a variety of contexts (see Ratiu [1994, 1996]) and is welladapted to practical numerical computations (seeVallis, Carnevale and Young [1989]; Flierl and Morrison [2011]). Examples of infinite-dimensional gradient flows include the Cahn-Hilliard systems (see Otto [2001]) and the celebrated Ricci flows (see ; Chow [2004]), which are nonlinear diffusion-like equations. A general form for combined Hamiltonian and gradient flows was described inMorrison [1986], where such flows were termed metriplectic flows (see also Oettinger [2006];Morrison [2009];Liero and Mielke [2012]). Thus, it is evident that there are a variety Hamiltonian and dissipative flows, and the purpose of this paper is to explore the form and geometric structure of such flows in both the ode and pde contexts.We recall the following well-known result(Brockett [1991],Brockett [1994], Bloch, Brockett, and Ratiu [1990], Bloch, Brockett, and Ratiu [1992, Bloch, Flaschka, and , Bloch and Iserles [2005]).(2)is the gradient of the function H(L) = κ(L, N) relative to the normal metric on O. Proof. By the definition of the gradient grad H(L) ∈ T L O ⊂ g u relative to the normal metric, we have for any L ∈ O and δ L ∈ g u , dH(L) · [L, δ L] = grad H(L), [L, δ L] normal (3) where · denotes the natural pairing between 1-forms and tangent vectors and [L, δ L] is an arbitrary tangent vector at L to O. Set grad H(L) = [L, X] = [L, X L ]. Then (3) becomes − [L, δ L], N = [L, X], [L, δ L] normal or, equivalently, Gradient flows and metriplectic systems 3 [L, N], δ L = X L , δ L L = X L , δ L . Since [L, N] ∈ g L u , this implies that X L = [L, N], and hence grad H(L) = [L, [L, N]], as stated. The same computation, for a general function H ∈ C ∞ (g u ), yields grad H(L) = −[L, [L, ∇H(L)]]Proposition 1. The vector field given by the ordinary differential equationwhere ∇H(L) denotes the gradient of the function H relative to the invariant inner product , := −κ( , ), i.e., dH(L) · X = ∇H(L), X for any X ∈ g u . | 10.1007/978-3-0348-0451-6_15 | [
"https://arxiv.org/pdf/1208.6193v1.pdf"
]
| 33,173,127 | 1208.6193 | 0012c2484b9c54e4e979adc1e4c1cf980b823c12 |
Gradient flows in the normal and Kähler metrics and triple bracket generated metriplectic systems
30 Aug 2012
Anthony M Bloch
Philip J Morrison
Tudor S Ratiu
Gradient flows in the normal and Kähler metrics and triple bracket generated metriplectic systems
30 Aug 2012loop groupsadjoint orbitsHamiltonian systemsintegrable systemsgradient flowsmetriplectic systemsthermodynamics
The dynamics of gradient and Hamiltonian flows with particular application to flows on adjoint orbits of a Lie group and the extension of this setting to flows on a loop group are discussed. Different types of gradient flows that arise from different metrics including the so-called normal metric on adjoint orbits of a Lie group and the Kähler metric are compared. It is discussed how a Kähler metric can arise from a complex structure induced by the Hilbert transform. Hybrid and metriplectic flows that arise when one has both Hamiltonian and gradient components are examined. A class of metriplectic systems that is generated by completely antisymmetric triple brackets is described and for finitedimensional systems given a Lie algebraic interpretation. A variety of explicit examples of the several types of flows are given.IntroductionDynamical systems, finite or infinite, that describe physical phenomena typically have parts that are in some sense Hamiltonian and parts that can be recognized as dissipative, with the Hamiltonian part being generated by a Poisson bracket and the dissipative part being some kind of gradient flow. The description of Hamiltonian systems has received much attention over nearly two centuries and, although some forms of dissipation have received general attention, the understanding and classification of dissipative dynamics is a much broader topic and consequently less well developed. Early modern treatments of geometric Hamiltonian mechanics include those of Souriau [1970] andAbraham and Marsden [1978], and the literature on this topic is now immense. A special type of gradient flow that preserves invariants, the double bracket formalism described inBrockett [1991](see, e.g.,Bloch [1990], Bloch [2003), is a formalism that occurs in a variety of contexts (see Ratiu [1994, 1996]) and is welladapted to practical numerical computations (seeVallis, Carnevale and Young [1989]; Flierl and Morrison [2011]). Examples of infinite-dimensional gradient flows include the Cahn-Hilliard systems (see Otto [2001]) and the celebrated Ricci flows (see ; Chow [2004]), which are nonlinear diffusion-like equations. A general form for combined Hamiltonian and gradient flows was described inMorrison [1986], where such flows were termed metriplectic flows (see also Oettinger [2006];Morrison [2009];Liero and Mielke [2012]). Thus, it is evident that there are a variety Hamiltonian and dissipative flows, and the purpose of this paper is to explore the form and geometric structure of such flows in both the ode and pde contexts.We recall the following well-known result(Brockett [1991],Brockett [1994], Bloch, Brockett, and Ratiu [1990], Bloch, Brockett, and Ratiu [1992, Bloch, Flaschka, and , Bloch and Iserles [2005]).(2)is the gradient of the function H(L) = κ(L, N) relative to the normal metric on O. Proof. By the definition of the gradient grad H(L) ∈ T L O ⊂ g u relative to the normal metric, we have for any L ∈ O and δ L ∈ g u , dH(L) · [L, δ L] = grad H(L), [L, δ L] normal (3) where · denotes the natural pairing between 1-forms and tangent vectors and [L, δ L] is an arbitrary tangent vector at L to O. Set grad H(L) = [L, X] = [L, X L ]. Then (3) becomes − [L, δ L], N = [L, X], [L, δ L] normal or, equivalently, Gradient flows and metriplectic systems 3 [L, N], δ L = X L , δ L L = X L , δ L . Since [L, N] ∈ g L u , this implies that X L = [L, N], and hence grad H(L) = [L, [L, N]], as stated. The same computation, for a general function H ∈ C ∞ (g u ), yields grad H(L) = −[L, [L, ∇H(L)]]Proposition 1. The vector field given by the ordinary differential equationwhere ∇H(L) denotes the gradient of the function H relative to the invariant inner product , := −κ( , ), i.e., dH(L) · X = ∇H(L), X for any X ∈ g u .
Specifically, in this paper we discuss the dynamics of gradient and Hamiltonian flows, with particular application to flows on adjoint orbits of a Lie group and the extension of this setting to flows on a loop group. We compare the different types of gradient flows that arise from different metrics, in particular, the so-called normal metric on adjoint orbits of a Lie group and the Kähler metric. We discuss how a Kähler metric can arise from the complex structure induced from the Hilbert transform. We also consider flows that arise when one has both Hamiltonian and gradient structures present. In particular, we discuss metriplectic flows, flows that produce entropy while conserving energy. We consider such flows in both the finite and infinite settings, and discuss a general class of metriplectic flows that arise from completely antisymmetric triple brackets. For finite systems, we show how the triple bracket has a natural Lie algebraic formulation, and for infinite systems we give a procedure for constructing a quite general class of metriplectic pdes. We also consider, hybrid flows, of Hamiltonian and gradient form, that dissipate energy. Several examples of hybrid and metriplectic flows are given, including finite systems such as the Toda lattice on R and metriplectic so(3) brackets. Various infinite-dimensional examples including a 1 + 1 dissipative systems that conserves energy, and hybrid systems such as the KdV with dissipation, the Ott and Sudan [1969] equation that describes Landau damping, and others.
The paper is organized as follows. In section 2 we review material need for latter development. In particular, we discuss metrics on adjoint orbits, Toda flows and the double bracket formulation. Sections 3 and 4 contain the main new results of the paper as described above. In section 3 we discuss metrics on loop groups and related gradient flows, while in section 4 we discuss our results on metriplectic systems, in both finite-and infinite-dimensions, and give examples.
Metrics on adjoint orbits of compact Lie groups and associated dynamical systems
Double bracket systems
Let g u be the compact real form of a complex semisimple Lie algebra g, G u a compact connected real Lie group with Lie algebra g u , and κ the Killing form (on g or g u , depending on the context).
The "normal" metric on the adjoint orbit O of G u through L 0 ∈ g u (see Atiyah [1982], [Besse, 2008, Chapter 8]) is given as follows. Decompose orthogonally g u = g L u ⊕ g uL , relative to to the invariant inner product , := −κ( , ), where g uL := ker ad L is the centralizer of L and g L u = rangead L ; as usual, ad L := [L, ·]. For X ∈ g u denote by X L ∈ g L u and X L ∈ g uL the orthogonal projections of X on g L u and g uL , respectively. Recall that a general vector tangent at L to the adjoint orbit O is necessarily of the form [L, X] for some X ∈ g u . The normal metric on O is the G u -invariant Riemannian metric given by [L, X], [L,Y ]
normal := X L ,Y L(1)
for any X,Y ∈ g u . Fix N ∈ g u and consider the flow on the adjoint orbit O of G u through L 0 ∈ g u given by (2)
The finite Toda system
The double bracket equation (2) is intimately related to the finite non-compact Toda lattice system. This is a Hamiltonian system modeling n particles moving freely on the x-axis and interacting under an exponential potential. Denoting the position of the kth particle by x k , the Hamiltonian is given by
H(x, y) = 1 2 n ∑ k=1 y 2 k + n−1 ∑ k=1 e x k −x k+1
and hence the associated Hamiltonian equations arė
x k = ∂ H ∂ y k = y k ,ẏ k = − ∂ H ∂ x k = e x k−1 −x k − e x k −x k+1 ,(5)
where we use the conventions e x 0 −x 1 = e x n −x n+1 = 0, which corresponds to formally setting x 0 = −∞ and x n+1 = +∞. This system of equations has an extraordinarily rich structure. Part of this is revealed by Flaschka's change of variables (Flaschka [1974]) given by a k = 1 2 e (x k −x k+1 )/2 and b k = − 1 2 y k .
which transform (5) to ȧ k = a k (b k+1 − b k ) , k = 1, . . . , n − 1 , b k = 2(a 2 k − a 2 k−1 ) , k = 1, . . . , n , with the boundary conditions a 0 = a n = 0. This system is equivalent to the Lax equation
d dt L = [B, L] = BL − LB ,(7)
where L = b 1 a 1 0 · · · 0 a 1 b 2 a 2 · · · 0 . . . . . . . . . 0 · · · b n−1 a n−1 0 · · · a n−1 b n
, B =
0 a 1 0 · · · 0 −a 1 0 a 2 · · · 0 . . . . . . . . . 0 · · · 0 a n−1 0 · · · −a n−1 0
.(8)
If O(t) is the orthogonal matrix solving the equation Thus, O −1 LO = L(0), i.e., L(t) is related to L(0) by conjugation with an orthogonal matrix and thus the eigenvalues of L, which are real and distinct, are preserved along the flow. This is enough to show that this system is explicitly solvable or integrable. Equivalently, after fixing the center of mass, i.e., setting b 1 + · · · + b n = 0, the n − 1 integrals in involution whose differentials are linearly independent on an open dense set of phase space {(a 1 , . . . , a n−1 , b 1 , . . . , b n ) | b 1 + · · · + b n = 0} are Tr L 2 , . . . , Tr L n .
Lie algebra integrability of the Toda system
Let us quickly recall the well-known Lie algebraic approach to integrability of the Toda lattice. Let g be a Lie algebra with an invariant non-degenerate bilinear symmetric form , , i.e., [ξ , η], ζ = ξ , [η, ζ ] for all ξ , η, ζ ∈ g and ξ , · = 0 implies ξ = 0. Suppose that k, s ⊂ g are Lie subalgebras and that, as vector spaces, g = k ⊕ s. Let π k : g → k, π s : g → s be the two projections induced by this vector space direct sum decomposition. Since g ∋ ξ ∼ −→ ξ , · ∈ g * is a vector space isomorphism, it naturally induces the isomorphisms k ⊥ ∼ = s * , s ⊥ ∼ = k * . By non-degeneracy of , , we have g = s ⊥ ⊕ k ⊥ ; denote by π k ⊥ : g → k ⊥ , π s ⊥ : g → s ⊥ the two projections induced by this vector space direct sum decomposition. In particular, g, s ⊥ , k ⊥ all carry natural Lie-Poisson structures. The (-)Lie-Poisson bracket of s * ∼ = k ⊥ is given by
{ϕ, ψ}(ξ ) = − ξ , [π s ∇ϕ(ξ ), π s ∇ψ(ξ )] , ξ ∈ k ⊥ ,(9)
where ϕ, ψ : k ⊥ → R are any smooth functions, extended arbitrarily to smooth functions, also denoted by ϕ and ψ, on g and ∇ϕ, ∇ψ are the gradients of these arbitrary extensions relative to , . This formula follows from the fact that the gradient on k ⊥ of ϕ| k ⊥ , which is an element of s due to the isomorphism k ⊥ ∼ = s * , equals π s ∇ϕ. Thus, the Hamiltonian vector field of ψ ∈ C ∞ (k ⊥ ), given byφ = {ϕ, ψ} for any ϕ ∈ C ∞ (k ⊥ ), has the expression
X ψ (ξ ) = −π k ⊥ [π s ∇ψ(ξ ), ξ ] , ξ ∈ k ⊥(10)
with the same conventions as above.
If ψ ∈ C ∞ (g) is invariant, i.e., [∇ψ(ζ ), ζ ] = 0 for all ζ ∈ g, then (10) simplifies to
X ψ (ξ ) = [π k ∇ψ(ξ ), ξ ] = − [π s ∇ψ(ξ ), ξ ] , ξ ∈ k ⊥ .(11)
The Adler-Kostant-Symes Theorem (see Adler [1979], Kostant [1979], Symes [1980a,b], and Ratiu [1980] for many theorems of the same type) states that if ϕ and ψ are both invariant functions on g, then {ϕ, ψ} = 0 on k ⊥ which is equivalent to the commutation of the flows of the Hamiltonian vector fields (11). Suppose that G = KS, where G is a Lie group with Lie algebra g and K, S ⊂ G are closed subgroups with Lie algebras k and s, respectively. The writing G = KS means that each element g ∈ G can be uniquely decomposed as g = ks, where k ∈ K and s ∈ S and that this decomposition defines a smooth diffeomorphism K × S ≈ G. The coadjoint action of S on s * has the following expression, if s * is identified with k ⊥ via , : if s ∈ S, ξ ∈ k ⊥ , then s · ξ = π k ⊥ Ad s ξ , where Ad s ξ is the adjoint action in G of the element s ∈ S ⊂ G on ξ ∈ k ⊥ ⊂ g.
For the Toda lattice (7), this general setup applies in the following way. Let G = GL(n, R), K = SO(n), S = {invertible lower triangular matrices}, G = KS is the Gram-Schmidt orthonormalization process, g = gl(n, R), k = so(n), s = {lower triangular matrices}, ξ , η := Tr(ξ η) for all ξ , η ∈ gl(n, R), k ⊥ = sym(n) the vector space of symmetric matrices, and s ⊥ = n, the nilpotent Lie algebra of strictly lower triangular matrices. The set of matrices L in (8) is a union of S-coadjoint orbits parametrized by the value of the trace; for example, the set of trace zero matrices L of the form (8) equals the S-coadjoint orbit through the symmetric matrix that has everywhere zero entries with the exception of the upper and lower first diagonals where all entries are equal to one. Thus, the Toda lattice is a Poisson system whose restriction to a symplectic leaf is a classical Hamiltonian system with n − 1 degrees of freedom. The Hamiltonian of the Toda lattice is 1 2 Tr L 2 and the f k (L) := 1 k Tr L k , k = 1, . . . , n − 1 are the n − 1 integrals in involution (by the Adler-Kostant-Symes Theorem) and are generically independent.
The Toda system as a double bracket equation
If N is the matrix diag{1, 2, . . . , n}, the Toda equations (7) may be written in the double bracket form (2) for B := [N, L]. This was shown in Bloch [1990]; the consequences of this fact were further analyzed for general compact Lie algebras in Bloch, Brockett, and Ratiu [1990], Bloch, Brockett, and Ratiu [1992], and Bloch, Flaschka, and . As shown in Proposition 1, the double bracket equation, with L replaced by iL and N by iN, restricted to a level set of the integrals described above, i.e., restricted to a generic adjoint orbit of SU(n), is the gradient flow of the function TrLN with respect to the normal metric; see Bloch, Flaschka, and Ratiu [1990] for this approach.
This observation easily implies that the flow tends asymptotically to a diagonal matrix with the eigenvalues of L(0) on the diagonal and ordered according to magnitude, recovering the result of Moser [1975], Symes [1982], and Deift, Nanda, and Tomei [1983].
Riemannian metrics on O
Now, we recall that, in addition to the normal metric on an adjoint orbit, there are other natural G u -invariant metrics: the induced and the group invariant Kähler metrics (as discussed in [Atiyah, 1982, §4], Atiyah and Pressley [1983], and [Besse, 2008, Chapter 8]).
Firstly, there is the induced metric b on O, defined by b := ι * (−κ( , )), where ι : O ֒→ g u is the inclusion and , := −κ( , ) is thought of as a constant Riemannian metric on g u . Therefore,
b(L)([L, X], [L,Y ]) := [L, X], [L,Y ](12)
for any L ∈ O, X,Y ∈ g u . The induced metric on O is also G u -invariant. Secondly, there are the G u -invariant Kähler metrics on O compatible with the natural complex structure (of course, induced by the complex structure of G). These are in bijective correspondence (by the transgression homomorphism) with the set of G u -invariant sections of the trivial vector bundle over O whose fiber at L ∈ O is the center of ker (ad L ) and whose scalar product with all positive roots is positive ( [Besse, 2008, Proposition 8.83]). Among these, there is the G u -invariant Kähler metric b 2 which is compatible with both the natural complex structure on O and has as imaginary part the orbit symplectic structure; b 2 is called the standard Kähler metric on O.
The G u -invariant Riemannian metrics on a maximal dimensional orbit O are completely determined by T -invariant inner products on the direct sum of the two dimensional root spaces of g u , which is the tangent space to O at the point L 0 ∈ t in the interior of the positive Weyl chamber; recall that O intersects the positive Weyl chamber in a unique point. The negative of the Killing form induces on each such 2-dimensional space an inner product. This inner product, left translated at all points of O by elements of G u , yields the normal metric on O. Any other G u -invariant inner product on O is obtained by left translating at all points of O the inner product on this direct sum of 2-dimensional root spaces obtained by multiplying in each 2-dimensional summand the inner product with a positive real constant.
Since L 0 lies in the interior of the positive Weyl chamber (because O is maximal dimensional), α(L 0 ) > 0 for all positive roots α of g u . Then the constant by which the natural inner product on the 2-dimensional root space needs to be multiplied in order to get the standard Kähler metric is α(L 0 ), whereas to get the induced metric, it is α(L 0 ) 2 ( [Atiyah, 1982, Remark 2 in §4]). We can formulate this differently, as in Bloch, Flaschka, and . Since, by (12) and (1),
b(L)([L, X], [L,Y ]) = [L, X], [L,Y ] = [L, X L ], [L,Y L ] = −[L, [L, X L ]],Y L = −[L, [L, X L ]] L ,Y L = − ad 2 L [L, X], [L,Y ] normal we have b(L)([L, X], [L,Y ]) = b 1 (L)(A (L) 2 [L, X], [L,Y ]),(13)
where we denote now by b 1 the normal metric and A (L) := (i ad L ) 2 is the positive square root of (i ad L ) 2 = − ad 2 L = A (L) 2 . The standard Kähler metric on O is then given by
Note that, as opposed to the normal and induced metrics which have explicit expressions, the standard Kähler metric on O requires the spectral decomposition of A (L) at any point L ∈ O. Or, as explained above, one expresses it at the point L 0 in the positive Weyl chamber in terms of the positive roots and then left translates the resulting inner product at any point of O. The normal metric does not depend on the operators A (L), whereas the standard Kähler and induced metrics do.
Gradient flows on the loop group of the circle
In this section we introduce three weak Riemannian metrics on the subgroup of average zero functions of the connected component of the loop group L(S 1 ) of the circle, analogous to the normal, standard Kähler, and induced metrics on adjoint orbits of compact semisimple Lie groups. Of course, we shall not work on adjoint orbits of this group because they degenerate to points, L(S 1 ) being a commutative group. Then we shall compute the gradient flows for these three metrics.
The loop group of S 1
Recall (e.g., Pressley and Segal [1986]) that the loop group L(S 1 ) of the circle S 1 consists of smooth maps of S 1 to S 1 . With pointwise multiplication, L(S 1 ) is a commutative group. Often, elements of L(S 1 ) are written as
e i f , where f ∈ L(R) := {g : [−π, π] → R | g is C ∞ , g(π)
= g(−π) + 2nπ, for some n ∈ Z}; n is the winding number of the closed curve [−π, π] ∋ t → e ig(t) ∈ S 1 about the origin. More precisely, there is an exact sequence of groups If one insists on working with smooth loops, then one can consider L(S 1 ) and L(S 1 ) 0 as Fréchet Lie groups either in the convenient calculus of Kriegl and Michor [1997] or in the tame category of .
0 −→ Z −→ L(R) exp −→ L(S 1 ) −→ Z −→ 0 n −→ 2πn; f −→ e i f −→ f (π)− f (−π)
Alternatively, one can work with loops e i f for f : [−π, π] → R of Sobolev class H s , where s ≥ 1 (or appropriate W s,p or Hölder spaces). By standard theory (see, e.g., Palais [1968] or Adams and Fournier [2003]), it is checked that L(S 1 ) is a Hilbert Lie group (see, e.g., Bourbaki [1971] or Neeb [2004]). We shall not add the index s on L(R) and L(S 1 ); from now on we work exclusively in this category of H s Sobolev class maps and loops. A simple proof of the fact that L(R) is a Hilbert Lie group was given to us by K.-H. Neeb. First, note that L(R) is a closed additive subgroup of the Hilbert space H s (R) := {h : R → R | h of class H s }. Second, L(R) = L(R) 0 × Z as topological groups, where L(R) 0 := {g ∈ L(R) | g(π) = g(−π)} is the closed vector subspace of H s (R) consisting of periodic functions; hence it is an additive Hilbert Lie group. Therefore, there is a unique Hilbert Lie group structure on L(R) for which L(R) 0 is the connected component of the identity. For general criteria that characterize Lie subgroups in infinite dimensions, see [Neeb, 2006, Theorem IV.3.3] (even for certain classes of Lie groups modeled on locally convex spaces). Third, since exp : L(R) → L(S 1 ) maps bijectively each connected component of L(R) to a connected component of L(S 1 ), it induces a Hilbert Lie group structure on L(S 1 ).
The commutative Hilbert Lie algebra of L(S 1 ) is clearly H s (S 1 , R) := {u : S 1 → R | u of class H s }, the space of periodic H s maps, and the exponential map exp : H s (S 1 , R) → L(S 1 ) is given by exp(u)(θ ) = e iu(θ ) , where θ ∈ R/2πZ = S 1 .
The based loop group of S 1
The inner product on the Hilbert space L 2 (S 1 ) of L 2 real valued functions on S 1 is defined by
f , g := 1 2π π −π dθ f (θ )g(θ ) , f , g ∈ L 2 (S 1 ).
Following Pressley [1982] and Atiyah and Pressley [1983], we introduce the closed Hilbert Lie subgroup L(S 1 ) := {ϕ ∈ L(S 1 ) | ϕ(1) = 1} of L(S 1 ) whose closed commutative Hilbert Lie algebra is L(R) := {u ∈ H s (S 1 , R) | u(1) = 0}. The exponential map exp : L(R) ∋ u → e iu ∈ L(S 1 ) is a Lie group isomorphism (with L(R) thought of as a commutative group relative to addition), a fact that will play a very important role later on (see also [Pressley and Segal, 1986, page 151, §8.9]).
There is a natural 2-cocycle ω on L(R), namely
ω(u, v) := 1 2π π −π dθ u ′ (θ )v(θ ) = u ′ , v ,(15)
where u ′ := du/dθ . Therefore, there is a central extension of Lie algebras
0 −→ R −→ L(R) −→ L(R) −→ 0
which, as shown in Segal [1981], integrates to a central extension of Lie groups
1 −→ S 1 −→ L(S 1 ) −→ L(S 1 ) −→ 1.
The "geometric duals" of L(R) and L(R) = R ⊕ L(R) are themselves, relative to the weak L 2 -pairing. It turns out that the coadjoint action of L(S 1 ) on L(R) preserves {1} ⊕ L(R) so that, as usual, the coadjoint action of L(S 1 ) on L(R) is an affine action which, in this case, because the group is commutative, equals
Ad * e i f µ = f ′ f = (log | f |) ′ e i f ∈ L(S 1 ), µ ∈ L(R).
Thus, the orbit of the constant function 0 is L(S 1 )/S 1 (where the denominator is thought of as constant loops), i.e., it equals L(S 1 ). Therefore, every element u ∈ L(R) of its Lie algebra has, in Fourier representation, vanishing zero order Fourier coefficient , i.e., u(0) = 0. Thus, the based loop group is a coadjoint orbit of its natural central extension and, according to §2, has three distinguished weak Riemannian metrics. These were computed explicitly in Pressley [1982], Atiyah and Pressley [1983], Pressley and Segal [1986]; we recall them below.
L(S 1 ) as a weak Kähler manifold
Note that on L(R), the cocycle (15) is weakly non-degenerate. Therefore, left (or right) translating it at every point of the group L(S 1 ) yields a weakly non-degenerate closed two-form, i.e., a symplectic form. Thus, as expected, since it is a coadjoint orbit, the Hilbert Lie group L(S 1 ) carries an invariant symplectic form whose value at the identity element 1 (the constant loop equal to 1) is given by (15).
Now we introduce the Hilbert transform on the circle
H u(θ ) := 1 2π − π −π ds u(s) cot θ − s 2 = 1 2π − π −π ds u(θ − s) cot s 2 := lim ε→0+ 1 π ε≤|s|≤π ds u(θ − s) cot s 2(16)
for any u ∈ L 2 (S 1 ), where − denotes the Cauchy principal value. We adopt here the sign conventions in [King, 2009, Formulas (3.202) and (6.38), Vol. 1]. If u ∈ L 2 (S 1 ), then H u ∈ L 2 (S 1 ) and it is defined for almost every θ ∈ [−π, π] (Lusin's Theorem, [King, 2009, §6.19, Vol. 1]). The Hilbert transform has the following remarkable properties that will be used later on:
• If u(θ ) = ∑ ∞ n=−∞ u(n)e inθ ∈ L 2 (S 1 ), where u(n) := 1 2π π −π dθ u(θ )e −inθ , so u(n) = u(−n) since u is real valued, then H u(θ ) = −i ∞ ∑ n=−∞ u(n) sign(n) e inθ ∈ L 2 (S 1 )(17)
which follows from the identity H f (n) = −i f (n) sign(n) ( [King, 2009, Formulas (6.100) or (6.124), Vol. 1]). Here, sign(n) = 1 if n ∈ N, sign(n) = −1 if n ∈ −N, and sign(0) = 0. Note that H u is also real valued since u(n) sign(n) = − u(−n) sign(−n). The formula above implies that ( [King, 2009, Formula (6.126
), Vol. 1]) π −π ds H u(s) = 0.
• For every u ∈ L 2 (S 1 ), we have the orthogonality property ( [King, 2009, Formula (6.127), Vol. 1]):
u, H u = 0.
• Take the orthonormal Hilbert basis ϕ n (θ ) := e inθ | n ∈ Z of L 2 (S 1 ). Then ( [King, 2009, Formula (6.131
), Vol. 1]): H ϕ n (θ ) = −i sign(n) ϕ n (θ ), for all n ∈ Z.
So, the eigenvalues of H are: −i for all n > 0, i for all n < 0, and 0 if n = 0.
• If u, v ∈ L 2 (S 1 ) then ( [King, 2009, Formula (6.99
), Vol. 1]) u, v = 1 4π 2 π −π ds u(s) π −π ds v(s) + H u, H v
and hence ( [King, 2009, Formula (6.97
), Vol. 1]) u 2 L 2 (S 1 ) = 1 2π π −π ds u(s) 2 + H u 2 L 2 (S 1 )
for any u ∈ L 2 (S 1 ). This shows that H u 2 L 2 (S 1 ) ≤ u 2 L 2 (S 1 ) and the constant 1 is the best possible ( [King, 2009, Formulas (6.167) and (6.168), Vol. 1]). In particular, if the average of u is zero, then H is an isometry of L 2 (S 1 ).
• The Hilbert transform is skew-adjoint relative to the L 2 (S 1 )-inner product, i.e., H * = −H ( [King, 2009, Formula (6.98) or (6.106), Vol. 1]).
• For any u ∈ L 2 (S 1 ) we have ( [King, 2009, Formula (6.34), (6.82), or (6.156), Vol. 1]):
H 2 u(θ ) = −u(θ ) + 1 2π π −π ds u(s) = −u(θ ) + u(0).
• For any u ∈ H s (S 1 ) with s ≥ 0 we have H u ∈ H s (S 1 ); this is an immediate consequence of (17). If s ≥ 1, then
H u ′ = (H u) ′ , i.e., H • d dθ = d dθ • H on H s (S 1 ) with s ≥ 1. Using these properties, if u(θ ) = ∑ ∞ n=−∞ u(n)e inθ ∈ H 1 (S 1 ), then u ′ (θ ) = ∑ ∞ n=−∞ u(n)ine inθ ∈ L 2 (S 1 ) and hence H u ′ (θ ) = (H u) ′ (θ ) = −i ∞ ∑ n=−∞ u(n) sign(n) e inθ ′ = ∞ ∑ n=−∞ |n| u(n)e inθ .(18)On the other hand, if v ∈ H 2 (S 1 ), then − d 2 dθ 2 v(θ ) = ∞ ∑ n=−∞ n 2 v(n)e inθ(19)
and hence if u ∈ H 1 (S 1 ),
− d 2 dθ 2 1 2 u(θ ) = ∞ ∑ n=−∞ |n| u(n)e inθ = (H u ′ )(θ ) = H • d dθ u (θ )(20)
by (18). By the previous properties we have (H • d/dθ) 2 = −d 2 /dθ 2 , as expected; note that the the extra term, which is the zero order Fourier coefficient, does not appear in this case, because the derivative eliminates it.
Now, if ϕ = e i f ∈ L(S 1 ), i.e., ϕ(1) = 1 and f : [−π, π] → R is a periodic function, then f (0) = f (0) = 0.
Similarly, if u ∈ L(R), i.e., u(1) = 0 and we think of u as a periodic function u : [−π, π] → R, then u(0) = u(0) = 0. This, and the properties of the Hilbert transform on the circle, imply: H (L(R)) ⊆ L(R), H is unitary on L(R) (relative to the H s -inner product), H • H = −I on L(R). Concretely, the Hilbert transform on L(R) has the form:
u(θ ) = ∑ n∈Z\{0} u(n)e inθ ∈ L(R) =⇒ H u(θ ) = −i ∑ n∈Z\{0} u(n) sign(n) e inθ ∈ L(R).
Thus, H defines the structure of a complex Hilbert space on L(R), relative to the H s inner product, s ≥ 1. Hence, translating H to any tangent space of L(S 1 ), we obtain an invariant almost complex structure on the Hilbert Lie group L(S 1 ) which is, in fact, a complex structure. For general criteria how to obtain complex structures on real Banach manifolds, see Beltita [2005]; the argument above is a very special case of these general methods.
Finally, L(S 1 ) is a Kähler manifold, as proved in Atiyah and Pressley [1983]. This is immediately seen by noting that
g(1)(u, v) := ω(H u, v) = ∞ ∑ n=−∞ |n| u(n) v(n)(21)
is symmetric and positive definite and so, by translations, defines a weak Riemannian metric on L(S 1 ). Note that this metric is not the H s metric for any s ≥ 1. In fact, the metric g is incomplete, whereas the H s metric is complete. Concluding, (L(S 1 ), ω, g, H ) is a weak Kähler manifold and all structures are group invariant (see Pressley [1982], Atiyah and Pressley [1983], Pressley and Segal [1986]).
Weak Riemannian metrics on L(S 1 )
The three metrics discussed in §2 for L(S 1 ), viewed as a coadjoint orbit of its central extension, have been computed by Pressley [1982]. We recall here relevant formulas.
The induced metric is defined by the natural inner product on L(R), which is the usual L 2 -inner product. Hence, the induced metric is obtained by left (equivalently, right) translation of the inner product
b(1)(u, v) := u, v = 1 2π π −π dt u(t)v(t)(22)
for any two functions u, v ∈ L(R).
Define the following inner products on L(R):
b 2 (1)(u, v) := b(1)(u, H v ′ ) = u, H v ′ , if u, v ∈ H s (S 1 ), s ≥ 1 (23) b 1 (1)(u, v) := b(1)(u ′ , v ′ ) = u ′ , v ′ , if u, v ∈ H s (S 1 ), s ≥ 1.(24)
Bilinearity and symmetry of b 1 (1) and b 2 (1) are obvious.
If u ∈ L(S 1 ), writing u(θ ) = ∑ ∞ n=−∞ u(n)e inθ with u(0) = 0, we have u ′ (θ ) = i ∑ ∞ n=−∞ n u(n)e inθ . Since {e inθ | n ∈ Z} is an orthonormal Hilbert basis of L 2 (S 1 ), we get b 1 (1)(u, u) = ∞ ∑ n=−∞ n 2 | u(n)| 2 ≥ 0.
In addition, b 1 (1)(u, u) = 0 if and only if u(n) = 0 for all n = 0, i.e., u(θ ) = u(0) = 0. This shows that b 1 (1) is indeed an inner product on L(R) which coincides with the H 1 inner product. Hence, if L(R) is endowed with the H s topology for s ≥ 1, this inner product is strong if s = 1 and weak if s > 1. Left translating this inner product to any tangent space of L(S 1 ) (endowed with the H s topology for s ≥ 1), yields a Riemannian metric on L(S 1 ) that is strong for s = 1 and weak for s > 1. This Riemannian metric is the normal metric on L(S 1 ). The inner product b 2 (1) is identical to g(1) by (21), (23), and (15). Thus, translating this inner product to the tangent space at every point of the Hilbert Lie group L(S 1 ), yields the standard Kähler metric b 2 = g on L(S 1 ), endowed with the H s topology for s ≥ 1. Note that if u ∈ L(S 1 ), then
b 2 (1)(u, u) = ∞ ∑ n=−∞ |n|| u(n)| 2
which shows that the Kähler metric b 2 coincides with the H 1/2 metric and is, therefore, a weak metric on L(S 1 ).
There are relations similar to (13) and (14), namely
b(1)(u, v) = b 1 (1)(A 2 u, v), b 2 (1)(u, v) = b 1 (1)(A u, v), where (A 2 u)(θ ) = ∞ ∑ n=−∞ n 2 u(n)e inθ , (A u)(θ ) = ∞ ∑ n=−∞ |n| u(n)e inθ if u(θ ) = ∑ ∞ n=−∞ u(n)e inθ .
However, note that the relation involving A 2 requires that u ∈ H s (S 1 ) with s ≥ 2.
Vector fields on L(S 1 ) and L(R)
Recall that the exponential map exp : L(R) ∋ u → e iu ∈ L(S 1 ) is a Lie group isomorphism ( [Pressley and Segal, 1986, page 151, §8.9]). Here, we identified the Lie algebra of S 1 with R, even though, naturally, it is the imaginary axis, the tangent space at 1 ∈ S 1 to S 1 . This means that care must be taken when carrying out standard Lie group operations with the exponential map, interpreted as the exponential of a purely imaginary number. Since such computations affect our next results, we clarify these statements below.
The tangent space at the identity 1 to S 1 is the imaginary axis. This is the natural Lie algebra of the Lie group S 1 and the exponential map is given by exp : iR ∋ (ix) → e ix ∈ S 1 . Of course, traditionally, one identifies iR with R by dividing by i and thinks of the exponential map as exp : R ∋ x → e ix ∈ S 1 . Unfortunately, this induces some problems. For example, since (left) translation is given by L e ix e iy := e ix e iy , it follows that
T 1 L e ix (iy) := d dε ε=0 L e ix e iεy = d dε ε=0 e ix e iεy = iye ix ,(25)
so the identification of the Lie algebra with R poses no problems and we have, dividing both sides by i,
T 1 L e ix (y) = ye ix .(26)
However, the definition of the exponential map for any Lie group G with Lie algebra g, yields
d dt exp(tξ ) = T e L exp(tξ ) ξ , for all ξ ∈ g.(27)
This formula works perfectly well if the Lie algebra of S 1 is iR. Indeed
d dt e tix = ixe tix
which coincides with (27) in view of (25). On the other hand, if the Lie algebra is thought of as R, i.e., the right hand side needs to be divided by i, then with the definition of exp(tx) = e itx the identity above is no longer valid. What we should get is d dt (26) if exp(tx) = e itx , but the right hand side gives ixe itx , as we saw above. In other words, if the Lie algebra of S 1 is thought of as R, as is traditionally done, then we need a formula for the derivative of the Lie group exponential map in terms of the exponential map of purely imaginary numbers. In view of the previous discussion, this formula is
exp(tx) = x exp(tx) = T 1 L exp(tx) x = xe itx byd dt exp(tx) := 1 i d dt e itx = xe itx .(28)
With these remarks in mind, we shall now compute the push-forward of a vector field on L(R) to L(S 1 ).
Proposition 2. Let X ∈ X(L(R)) be an arbitrary vector field . Then its push-forward to L(S 1 )) has the expression
(exp * X) e iu = X(u)e iu
for any u ∈ L(R).
Proof. By the definition of push forward of vector fields by a diffeomorphism, we have
(exp * X) e iu = T exp •X • exp −1 e iu = T u exp (X(u)) = d dε ε=0 exp (u + εX(u)) = d dε ε=0 exp(u) exp(εX(u)) = d dε ε=0 exp(εX(u)) exp(u) (28) = 1 i d dε ε=0 e iεX(u) e iu = X(u)e iu as stated.
The gradient vector fields in the three metrics of L(S 1 )
We compute now the gradients of a specific function using the three metrics.
Theorem 1. The gradients of the smooth function H : L(S 1 ) → R given by
H e i f = 1 4π π −π dθ f ′ (θ ) 2 are (i) ∇ 1 H e i f = f e i f for the normal metric b 1 ; (ii) ∇H e i f = − f ′′ e i f with respect to the induced metric b for f ∈ H s (S 1 ) with s ≥ 2; (iii) ∇ 2 H e i f = (H f ′ )e i f with respect to the weak Kähler metric b 2 . Proof. (i) Since T 1 L e i f u = ue i f for any u ∈ L(R) and e i f ∈ L(S 1 ), invariance of b 1 yields b 1 (1) e −i f ∇ 1 H e i f , u = b 1 e i f ∇ 1 H e i f , ue i f = dH e i f ue i f = d dt t=0 H e i( f +tu) = d dt t=0 1 4π π −π dθ f ′ (θ ) + tu ′ (θ ) 2 = 1 2π π −π dθ f ′ (θ )u ′ (θ ) = f ′ , u ′ (1) = b 1 (1)( f , u) which shows that ∇ 1 H e i f = f e i f .
(ii) Proceeding as above, using the same notations, and assuming that f ∈ H s (S 1 ) with s ≥ 2, we have
b(1) e −i f ∇H e i f , u = b e i f ∇H e i f , ue i f = dH e i f ue i f = 1 2π π −π dθ f ′ (θ )u ′ (θ ) = − 1 2π π −π dθ f ′′ (θ )u(θ ) = − f ′′ , u (12) = b(1) − f ′′ , u which shows that ∇H e i f = − f ′′ e i f .
(iii) This computation uses the isometry property of H relative to the L 2 inner product. We have,
b 2 (1) e −i f ∇ 2 H e i f , u = b 2 e i f ∇ 2 H e i f , ue i f = dH e i f ue i f = f ′ , u ′ = H f ′ , H u ′ (23) = b 2 (1) H f ′ , u which shows that ∇ 2 H e i f = (H f ′ )e i f .
Since
ω e i f H ∇ 2 H e i f , ue i f (21) = b 2 e i f ∇ 2 H e i f , ue i f = dH e i f ue i f
it follows that the Hamiltonian vector field on L(S 1 ), ω for the function H is X H = H ∇ 2 H. Since H commutes with the tangent lift to group translations, Theorem 1(iii) implies that
X H e i f = H ∇ 2 H e i f = H ∇ 2 H e i f = H H f ′ e i f = − f ′ e i f .
This proves the first part of the following statement.
Corollary 1. The Hamiltonian vector field of H relative to the translation invariant symplectic form ω on L(S 1 ) whose value at the identity element is given by (15) has the expression X H e i f = − f ′ e i f . Its flow is the rotation
F t e i f (θ ) = e −i( f (t+θ )− f (t)) .
Proof. Since L(R) ∋ u −→ e iu ∈ L(S 1 ) is the exponential map and we think of R as the Lie algebra of S 1 (and not the imaginary axis), we write de itu /dt = ue itu without the factor of i in front (see (28)). The verification that F t is indeed the flow of X H is straightforward:
d dt F t e i f (θ ) = d dt e −i( f (t+θ )− f (t)) = −( f ′ (t + θ ) − f ′ (t))e −i( f (t+θ )− f (t)) = X H F t e i f (θ )
as required.
We recover thus [Pressley, 1982, Proposition 3.1] (up to a sign which is due to different conventions calibrating ω, H , and b 2 ).
Applying Proposition 2 to Theorem 1, we get the following result:
Corollary 2. The three gradient vector fields for the smooth function H 1 : L(R) → R given by Since the exponential map is a Lie group isomorphism and the three metrics coincide with the respective inner products at the identity, their left invariance guarantees that the three inner products on L(R) correspond to the three invariant metrics on L(S 1 ).
H 1 (u) = 1 4π π −π dθ (u ′ ) 2 are (i) ∇ 1 H 1 (u) = u
Applying Proposition 2 to Corollary 1, we conclude:
Corollary 3. The Hamiltonian vector field of H 1 relative to the symplectic form ω given by (15) has the expression X H (u) = −u ′ . Its flow is (F t (u)) (θ ) = u(θ − t).
The verification of the statement about the flow is immediate:
d dt (F t (u)) (θ ) = d dt u(θ − t) = −u ′ (θ − t) = (X H (F t (u))) (θ ).
If one is willing to put more stringent hypotheses on the functional, it is possible to obtain a general result.
Theorem 2. Let H : L(S 1 ) → R be a smooth function (with L(S 1 ) endowed, as usual, with the H s topology for s ≥ 1) and assume that the functional derivative δ H/δ u ∈ L(S 1 ) exists. Then the gradient vector fields are (i) ∇H(u) = δ H δ u with respect the weak inner product b(1) defining the induced metric;
(ii) ∇ 1 H(u) (θ ) = − θ 0 dϕ ϕ 0 dψ δ H δ u (ψ) with respect to the (weak) inner product b 1 (1) defining the normal metric, provided both θ 0 dϕ δ H δ u (ϕ) and θ 0 dϕ ϕ 0 dψ δ H δ u (ψ) are periodic; (iii) ∇ 2 H(u) (θ ) = −H θ 0 dϕ δ H δ u (ϕ) with respect to the weak inner product b 2 (1) defining the Kähler metric, provided θ 0 dϕ δ H δ u (ϕ) is periodic. Proof. (i) For the inner product b(1) on L(S 1 ) defining the induced metric, if u, v ∈ L(R), we have by periodicity of u, v, b(1) (∇H(u), v) = DH(u) · v = δ H δ u , v (12) = b(1) δ H δ u , v .
This shows that ∇H(u) = δ H δ u . (ii) For the inner product b 1 (1) on L(S 1 ) defining the normal metric, if u, v ∈ L(R), we have by periodicity of
θ 0 dϕ δ H δ u (ϕ) and θ 0 dϕ ϕ 0 dψ δ H δ u (ψ) , b 1 (1)(∇ 1 H(u), v) = DH(u) · v = δ H δ u , v = 1 2π π −π dθ δ H δ u (θ )v(θ ) = 1 2π θ 0 dϕ δ H δ u (ϕ) v(θ ) π −π − 1 2π π −π dθ θ 0 dϕ δ H δ u (ϕ) v ′ (θ ) = − 1 2π π −π dθ d dθ θ 0 dϕ ϕ 0 dψ δ H δ u (ψ) v ′ (θ ) = − d dθ θ 0 dϕ ϕ 0 dψ δ H δ u (ψ) , v ′ (1) = b 1 − θ 0 dϕ ϕ 0 dψ δ H δ u (ψ) , v which shows that (∇ 1 H(u))(θ ) = − θ 0 dϕ ϕ 0 dψ δ H δ u (ψ) .(
iii) For the inner product b 2 (1) on L(S 1 ) defining the Kähler metric, if u, v ∈ L(R), we have by periodicity of θ 0 dϕ δ H δ u (ϕ) and the isometry property of H ,
b 2 (1)(∇ 2 H(u), v) = DH(u) · v = δ H δ u , v = 1 2π π −π dθ δ H δ u (θ )v(θ ) = 1 2π θ 0 dϕ δ H δ u (ϕ) v(θ ) π −π − 1 2π π −π dθ θ 0 dϕ δ H δ u (ϕ) v ′ (θ ) = − θ 0 dϕ δ H δ u (ϕ), v ′ = − H θ 0 dϕ δ H δ u (ϕ), H v ′ (23) = b 2 (1) −H θ 0 dϕ δ H δ u (ϕ), v which shows that ∇ 2 H(u) (θ ) = −H θ 0 dϕ δ H δ u (ϕ).= θ 0 dϕ δ H δ u (ϕ).(iii)
Of course, using Proposition 2, there are immediate counterparts of Theorem 2 and Corollary 4 on the loop group L(S 1 ), which we shall not spell out explicitly.
The hypotheses guaranteeing the existence of the functional derivative of H relative to the weakly non-degenerate L 2 pairing are quite severe. For example, the theorem can be applied to the functional H 1 in Corollary 2, but one needs additional smoothness. Indeed, the first thing to check is if this functional has a functional derivative. In fact, it does not, unless we assume that u ∈ H s (S 1 ) for s ≥ 2, in which case we have
DH 1 (u) · v = 1 2π π −π ds u ′ (s)v ′ (s) = 1 2π u ′ (s)v(s) π −π − 1 2π π −π ds u ′′ (s)v(s) = −u ′′ , v ,
i.e., δ H/δ u = −u ′′ . With this additional hypothesis, the gradient flow with respect to the weak inner product b(1) defining the induced metric is given by u t = −u ′′ . Therefore, to continue computing the other two gradients of H 1 , we need to assume that u ∈ H s (S 1 ) for s ≥ 2. Provided this holds, to find the gradient relative to the (weak) inner product b 1 (1) defining the normal metric, we have to check that both
θ 0 dϕ δ H δ u (ϕ) = − θ 0 dϕ u ′′ (ϕ) = −u ′ (θ ) + u ′ (0) θ 0 dϕ ϕ 0 dψ δ H δ u (ψ) = − θ 0 dϕ (u ′ (ϕ) − u ′ (0)) = −u(θ ) + u ′ (0)θ
are periodic. While the first one is periodic, the second one is not unless we assume that u ′ (0) = 0. With this additional hypothesis, the gradient is given by u t = u. However, we know from Corollary 2 that neither s ≥ 2, nor u ′ (0) = 0 is needed. In addition, this can also be seen directly, as follows. For any u, v ∈ L(R), we have
b 1 (1)(∇ 1 H(u), v) = DH(u) · v = 1 2π π −π ds u ′ (s)v ′ (s) = u ′ , v ′ (1) = b 1 (u, v)
which shows that ∇ 1 H(u) = u. The same situation occurs in the computation of the third gradient. In the hypotheses of the theorem, we have
∇ 2 H(u) (θ ) = −H θ 0 dϕ δ H δ u (ϕ) = H (u ′ − u ′ (0)) = H u ′
because the Hilbert transform of a constant is zero. Thus, the gradient flow is given in this case by
u t = H u ′ (20) = − d 2 dθ 2 1 2 u.
As before, the same result can be obtained easier and without any additional hypotheses in the following way:
b 2 (1)(∇ 2 H(u), v) = DH(u) · v = u ′ , v ′ = H u ′ , H v ′ (23) = b 2 (1)(H u ′ , v).
Symplectic structure on periodic functions
The form of the periodic Korteweg-de Vries (KdV) equation we shall use is
u t − 6uu θ + u θ θ θ = 0,(29)
where u(t, θ ) is a real valued function of t ∈ R and θ ∈ [−π, π], periodic in θ , and u θ := ∂ u/∂ θ . The KdV equation is, of course, a famous integrable infinite dimensional Hamiltonian system. It is Hamiltonian on the Poisson manifold of all periodic functions relative to the Gardner [1971] bracket
{F, G} = 1 2π π −π dθ δ F δ u d dθ δ G δ u ,(30)
where
F(u) = S 1 dθ f (u, u θ , u θ θ , . . .)
and similarly for G; the functional derivative δ F/δ u is the usual one relative to the L 2 (S 1 ) inner product, i.e.,
δ F δ u = ∂ f ∂ u − d dθ ∂ f ∂ u θ + d 2 dθ 2 ∂ f ∂ u θ θ − · · · .
The Hamiltonian vector field of H(u) = 1 2π π −π dθ h(u, u θ , u θ θ , . . .) has the expression
X H (u) = d dθ δ H δ u .
For the KdV equation one takes
H(u) = 1 2π π −π dθ u 3 + 1 2 u 2 θ .(31)
The Casimir functions of the Gardner bracket are all smooth functionals C for which δC/δ u = c is a constant function, i.e.,
C(u) = c, u = 1 2π π −π dθ cu(θ ) = c u(0).
Thus C −1 (0) is a candidate weak symplectic leaf in the phase space of all periodic functions. The situation in infinite dimensions is not as clear as in finite dimensions, where this would be a conclusion, because there is no general stratification theorem and one cannot expect, in general, more than a weak symplectic form. However, in our case, this actually holds, as shown in Zaharov and Faddeev [1971]. Indeed,
σ (u 1 , u 2 ) : = 1 4π π −π dθ θ 0 dϕ (u 1 (ϕ)u 2 (θ ) − u 2 (ϕ)u 1 (θ )) = 1 2π π −π dθ θ 0 dϕ u 1 (ϕ) u 2 (θ ) = θ 0 dϕ u 1 (ϕ), u 2(32)
defines a weak symplectic form on L(R) whose formal Poisson bracket is (30). This immediately shows that there is a tight relationship with the symplectic form ω of the complex Hilbert space L(R), the Lie algebra of the based loop groups, given by (15), namely
σ d 2 dθ 2 u, v = ω(u, v) for all u, v ∈ L(R) of class H s , s ≥ 2. Defining d dθ −1 u := θ 0 dϕ u(ϕ),
the KdV symplectic form σ has the suggestive expression (see (28))
σ (u 1 , u 2 ) = d dθ −1 u 1 , u 2 ,
which is well defined on H − 1 2 (S 1 , R). On the other hand, the Poisson bracket given by the Kähler symplectic form (15) on L(R) is
{F, G} = 1 2π π −π dθ δ F δ u d dθ −1 δ G δ u ,(33)
which is similarly well defined on H − 1 2 , and the Hamiltonian vector field defined by this bracket is given by Corollary 4, i.e.,
u t = X H (u) = d dθ −1 δ H δ u .(34)
Now, the gradient vector field for the corresponding Kähler metric, as computed in Theorem 2(iii), is written as
u t = −H d dθ −1 δ H δ u .(35)
Metriplectic Systems
In this section we define metriplectic systems and show how to construct general classes of such systems in terms of triple brackets for both finite-and infinite-dimensional theories. We use some of the machinery developed above to address specific examples.
Definition and consequences
A metriplectic system consists of a smooth manifold P, two smooth vector bundle maps π, κ : T * P → T P covering the identity, and two functions H, S ∈ C ∞ (P), the Hamiltonian or total energy and the entropy of the system, such that (i) {F, G} := dF, π(dG) is a Poisson bracket; in particular π * = −π;
(ii) (F, G) := dF, κ(dG) is a positive semidefinite symmetric bracket, i.e., ( , ) is R-bilinear and symmetric, so κ * = κ, and (F, F) ≥ 0 for every F ∈ C ∞ (P);
(iii) {S, F} = 0 and (H, F) = 0 for all F ∈ C ∞ (P) ⇐⇒ π(dS) = κ(dH) = 0.
The metriplectic dynamics of the system is given in terms of the two brackets by
or, equivalently, as an ordinary differential equation, by d dt c(t) = π(c(t))dH(c(t)) + κ(c(t))dS(c(t)).
The Hamiltonian vector field X H := π(dH) ∈ X(P) represents the conservative or Hamiltonian part, whereas Y S := κ(dS) ∈ X(P) the dissipative part of the full metriplectic dynamics (36) or (37).
As far as we know, first attempts to introduce such a structure were given in adjacent papers by Kaufman [1984] and Morrison [1984a]. (See also Kaufman and Morrison [1982].) Kaufman [1984] imposed, instead of (iii), the weaker condition {H, S} = (H, S) = 0, which is enough, as will become apparent below, to deduce the First and Second Laws of Thermodynamics. In the plasma examples presented, he used (iii) for a large class of functions. All three axioms, including the degeneracy condition of (iii), were stated explicitly in Morrison [1984a] and Morrison [1984b]. The former treated the same kinetic example as Kaufman [1984] along with additional formalism, while the latter presented the metriplectic formalism for the compressible Navier-Stokes equations with entropy production. All three axioms were restated in Morrison [1986], where the terminology metriplectic was introduced and a detailed physical motivation for the introduction of (iii) is presented along with other examples such as a dissipative free rigid body equation and the Vlasov-Poisson equation with a collision term that generalizes the Landau and Balescu-Lenard equations. In Grmela andÖttinger [1997], under the name GENERIC (General Equations for Non-Equilibrium Reversible Irreversible Coupling), the same geometric structure was used to analyze many other equations; due to this paper and subsequent work of these authors, the metriplectic formalism has been popularized. For a very interesting modern application of this structure see Mielke [2011] and for further discussion about avenues for generalization see Morrison [2009].
The definition of metriplectic systems has three immediate important consequences. Let c(t) be an integral curve of the system (37).
(1) Energy conservation:
d dt H(c(t)) = {H, H}(c(t)) + (H, S)(c(t)) = 0.(38)
(2) Entropy production:
d dt S(c(t)) = {S, H}(c(t)) + (S, S)(c(t)) ≥ 0.(39)
(3) Maximum entropy principle yields equilibria: Suppose that there are n functions C 1 , . . . ,C n ∈ C ∞ (P) such that {F,C i } = (F,C i ) = 0 for all F ∈ C ∞ (P), i.e., these functions are simultaneously conserved by the conservative and dissipative part of the metriplectic dynamics. Let p 0 ∈ P be a maximum of the entropy S subject to the constraints
H −1 (h) ∩C −1 1 (c 1 ) ∩ . . .C −1 n (c n )
, for given regular values h, c 1 , . . . , c n ∈ R of H,C 1 , . . . ,C n , respectively. By the Lagrange Multiplier Theorem, there exist α, β 1 , . . . , β n ∈ R such that dS(p 0 ) = αdH(p 0 ) + β 1 dC 1 (p 0 ) + · · · + dC n (p 0 ).
But then, assuming that α = 0, for every F ∈ C ∞ (P), we have
{F, H}(p 0 ) + (F, S)(p 0 ) = dF(p 0 ), π(p 0 ) (dH(p 0 )) + dF(p 0 ), κ(p 0 ) (dS(p 0 )) = dF(p 0 ), 1 α π(p 0 ) (dS(p 0 ) − β 1 dC 1 (p 0 ) − · · · − dC n (p 0 )) + dF(p 0 ), κ(p 0 ) (αdH(p 0 ) + β 1 dC 1 (p 0 ) + · · · + dC n (p 0 )) = 1 α {F, S}(p 0 ) − β 1 α {F,C 1 }(p 0 ) − · · · − β n α {F,C n }(p 0 ) + α(F, H)(p 0 ) + β 1 (F,C 1 )(p 0 ) + · · · + β n (F,C n )(p 0 ) = 0
which means that p 0 is an equilibrium of the metriplectic dynamics (36) or (37). This is akin to the free energy extremization of thermodynamics, as noted by Morrison [1984b] and Morrison [1986] where it was suggested that one can build in degeneracies associated with Hamiltonian "dynamical constraints." (See also Mielke [2011].)
Suppose that K ∈ C ∞ (P) is a conserved quantity for the Hamiltonian part of the metriplectic dynamics, i.e., {K, H} = 0. Then, if c(t) is an integral curve of the metriplectic dynamics, we have d dt K(c(t)) = dK(c(t)) (ċ(t)) = dF(c(t)), π(c(t)) (dH(c(t))) + dF(c(t)), κ(c(t)) (dS(c(t))) = {K, H}(c(t)) + (K, S)(c(t)) = (K, S)(c(t)).
As pointed out in Morrison [1986], this immediately implies that a function that is simultaneously conserved for the full metriplectic dynamics and its Hamiltonian part, is necessarily conserved for the dissipative part. Physically, it is advantageous for general metriplectic systems to conserve dynamical constraints, i.e., conserved quantitates of its Hamiltonian part and the examples given in Kaufman [1984], Morrison [1984a], Morrison [1984b], and Morrison [1986] satisfy this condition.
Metriplectic systems based on Lie algebra triple brackets
Associated with any quadratic Lie algebra (i.e., a Lie algebra admitting a bilinear symmetric invariant form) is a natural completely antisymmetric triple bracket. This is used to construct Lie algebra based metriplectic systems. The algebra so(3) is worked out explicitly and examples are given.
General theory
A quadratic Lie algebra is, by definition, a Lie algebra admitting a bilinear symmetric non-degenerate invariant form κ : g×g → R (the letter κ is meant to remind one of the Killing form in a semisimple Lie algebra). Recall that invariance means that κ([ξ , η], ζ ) = κ(ξ , [η, ζ ]) for all ξ , η, ζ ∈ g or, equivalently, that the adjoint operators ad η for all η ∈ g are antisymmetric relative to κ. Non-degeneracy (strong) means that the map g ∋ ξ → κ(ξ , ·) ∈ g * is an isomorphism. Finite dimensional quadratic Lie algebras have been completely classified in Medina and Revoy [1985]. For finite dimensional Lie algebras, non-degeneracy is equivalent to the following statement: κ(ξ , η) = 0 for all η ∈ g if and only if ξ = 0. In infinite dimensions this condition is called weak non-degeneracy and it is implied by non-degeneracy but the converse is, in general, false. For example, let g be an arbitrary finite dimensional Lie algebra. Recall that the Killing form is defined by κ(ξ , η) := Trace(ad ξ • ad η ). If {e i }, i = 1, . . . dim g, is an arbitrary basis of g and c p i j are the structure constants of g, i.e., [e i , e j ] = c p i j e p , then κ(ξ , η) = ξ i c p iq η j c q j p and hence the components of κ in the basis {e i }, i = 1, . . . dim g, are given by
κ i j = κ(e i , e j ) = c p iq c q j p .
The Killing form is bilinear symmetric and invariant; it is non-degenerate if and only if g is semisimple. Moreover, −κ is a positive definite inner product if and only if the Lie algebra g is compact (i.e., it is the Lie algebra of a compact Lie group).
In general, let κ be a bilinear symmetric non-degenerate invariant form and define the completely antisymmetric covariant 3-tensor
c(ξ , η, ζ ) := κ(ξ , [η, ζ ]) = −c(ξ , ζ , η) = −c(η, ξ , ζ ) = −c(ζ , η, ξ ).
In the coordinates given by the basis {e i }, i = 1, . . . dim g, the components of c are
c i jk := κ im c m jk = −c ik j = −c jik = −c k ji .
This construction immediately leads to the triple bracket introduced by Bialynicki-Birula and Morrison [1991] (see also Morrison [1998]
), { · , ·, ·} : C ∞ (g) × C ∞ (g) × C ∞ (g) → C ∞ (g) defined by { f , g, h}(ξ ) := c(∇ f (ξ ), ∇g(ξ ), ∇h(ξ )) := κ (∇ f (ξ ), [∇g(ξ ), ∇h(ξ )]) ,(40)
where the gradient is taken relative to the non-degenerate bilinear form κ, i.e., for any ξ ∈ g we have
κ(∇ f (ξ ), ·) := d f (ξ ) or, in coordinates ∇ i f (ξ ) = κ i j ∂ f ∂ ξ i where [κ i j ] = [κ kl ] −1 , i.e., κ i j κ jk = δ i k .
This triple bracket is trilinear over R, completely antisymmetric, and satisfies the Leibniz rule in any of its variables. In coordinates it is given by
{ f , g, h} = c i jk ∇ i f ∇ j g∇ k h = κ im c m jk κ ip ∂ f ∂ ξ p κ jq ∂ g ∂ ξ q κ kr ∂ h ∂ ξ r = c p jk κ jq κ kr ∂ f ∂ ξ p ∂ g ∂ ξ q ∂ h ∂ ξ r = c pqr ∂ f ∂ ξ p ∂ g ∂ ξ q ∂ h ∂ ξ r ,
where c pqr are the components of the contravariant completely antisymmetric 3-tensorc associated to c by raising its indices with the non-degenerate symmetric bilinear form κ, i.e., for any ξ , η, ζ ∈ g, we havē c (κ(ξ , ·), κ(η, ·), κ(γ, ·)) := c(ξ , η, ζ ).
This construction extends the bracket due to Nambu [1973] to a Lie algebra setting. Nambu considered ordinary vectors in R 3 and defined { f , g, h} Nambu
(Π ) = ∇ f (Π ) · (∇g(Π ) × ∇h(Π)) ,(41)
where '·' and '×' are the ordinary dot and cross products. Thus, the Nambu bracket is a special case of the triple bracket (40) in the case of g = so(3), whose the structure constants are the completely antisymmetric Levi-Civita symbol ε i jk . Such 'modified rigid body brackets' were also described in Bloch and Marsden [1990], Holm and Marsden [1991], and Marsden and Ratiu [1999]. If g is an arbitrary quadratic Lie algebra with bilinear symmetric non-degenerate invariant form κ, the quadratic function
C 2 (ξ ) := 1 2 κ(ξ , ξ )(42)
is a Casimir function for the Lie-Poisson bracket on g, identified with g * via κ, i.e.,
{ f , g} ± (ξ ) = ±κ (ξ , [∇ f (ξ ), ∇g(ξ )]) ,(43)
as an easy verification shows since ∇C 2 (ξ ) = ξ . In view of (43), the following identity is obvious
{ f , g} + = {C 2 , f , g}
(this was first pointed out in Bialynicki-Birula and Morrison [1991]). For example, if g = so (3),
the (-)Lie-Poisson bracket { f , g} so(3) − (Π ) = −{C 2 , f , g} Nambu (Π ) = −Π · (∇ f (Π ) × ∇g(Π))(44)
is the rigid body bracket, i.e., if Note that given any two functions, f , g ∈ C ∞ (g), because the triple bracket satisfies the Leibniz identity in every factor, the map C ∞ (g) ∋ h → {h, f , g} ∈ C ∞ (g) is a derivation and hence defines a vector field on g, denoted by X f ,g : g → g, i.e.,
h(Π ) = 1 2 Π · Ω , where Π i = I i Ω i , I i > 0, i = 1,dh(ξ ), X f ,g (ξ ) = κ ∇h(ξ ), X f ,g (ξ ) = {h, f , g}(ξ ) for all h ∈ C ∞ (g).(45)
Note that X f , f = 0. Thus, for triple brackets, two functions define a vector field, analogous to the Hamiltonian vector field defined by a single function associated to a standard Poisson bracket. From (40) we have the following result.
Proposition 3. The vector field X f ,g on g corresponding to the pair of functions f , g is given by
X f ,g (ξ ) = [∇ f (ξ ), ∇g(ξ )] .(46)
Triple brackets of the form (40) can be used to construct metriplectic systems on a quadratic Lie algebra g in the following manner. Let κ be the bilinear symmetric non-degenerate form on g defining the quadratic structure and fix some h ∈ C ∞ (g). Define the symmetric bracket
( f , g) κ h (ξ ) := −κ X h, f (ξ ), X h,g (ξ ) .(47)
Assume that −κ is a positive definite inner product. Then ( f , f ) ≥ 0. Thus we have the manifold g endowed with the Lie-Poisson bracket (43), the symmetric bracket (47), the Hamiltonian h, and for the entropy S we take any Casimir function of the Lie-Poisson bracket. Then the conditions (i)-(iii) of §4.1 are all satisfied, because (h, g) κ h = −κ(X h,h , X h,g ) = −κ(0, X h,g ) = 0 for any g ∈ C ∞ (g). The equations of motion (36) are in this case given by
d dt f (ξ ) = κ ∇ f (ξ ), d dt ξ = { f , h} ± (ξ ) + ( f , S)(ξ ) = ±κ (ξ , [∇ f (ξ ), ∇h(ξ )]) − κ X h, f (ξ ), X h,S (ξ ) = ∓κ (∇ f (ξ ), [ξ , ∇h(ξ )]) − κ ([∇h(ξ ), ∇ f (ξ )], [∇h(ξ ), ∇S(ξ )])
for any f ∈ C ∞ (g).
This gives the equations of motionξ
= ±[ξ , ∇h(ξ )] + [∇h(ξ ), [∇h(ξ ), ∇S(ξ )]] .(48)
Note that the flow corresponding to S is a generalized double bracket flow. Observe also that this flow reduces to a double bracket flow and is tangent to an orbit of the group if ∇h(ξ ) = ξ . Indeed if h = 1 2 κ(ξ , ξ ) the symmetric bracket (47) reduces to the symmetric bracket induced from the normal metric.
Special case of so(3)
If the quadratic Lie algebra is so(3), we identify it with R 3 with the cross product as Lie bracket via the Lie algebra isomorphismˆ: R 3 → so(3) given byûv := u × v for all u, v ∈ R 3 . Since Ad Aû = Au, for any A ∈ SO(3) and u ∈ R 3 , we conclude that the usual inner product on R 3 is an invariant inner product. In terms of elements of so(3) we have u · v = − 1 2 Trace(ûv). We shall show below that the metriplectic structure on R 3 is precisely the one given in Morrison [1986].
Recall that the Nambu bracket is given for so (3) by (41) and hence the symmetric bracket (47) has the form
κ({Π , h, f }, {Π , h, g}) = ε imn ∂ h ∂ Π m ∂ f ∂ Π n δ i j ε jst ∂ h ∂ Π s ∂ g ∂ Π t = ε imn ε st i ∂ h ∂ Π m ∂ f ∂ Π n ∂ h ∂ Π s ∂ g ∂ Π t = ∇h 2 ∇g · ∇ f − (∇ f · ∇h)(∇g · ∇h)(49)
where in the third equality we have used the identity ε imn ε st i = δ ms δ nt − δ mt δ ns . This coincides with [Morrison, 1986, equation (31)].
With the choice S(Π) = Π 2 /2 and the usual rigid body Hamiltonian, the equations of motion (48) are those for the relaxing rigid body given in Morrison [1986].
Comments.
• In three dimensions any Poisson bracket can be written as
{ f , g} = J i j ∂ f ∂ Π i ∂ g ∂ Π j = ε i j k V k (Π ) ∂ f ∂ Π i ∂ g ∂ Π j(50)
where i, j, k = 1, 2, 3, and V ∈ R 3 . The last equality follows from the identification of 3 × 3 antisymmetric matrices with vectors (the hat map discussed above). Using the well know fact (which is easy to show directly) that brackets of the form of (50) satisfy the Jacobi identity if
V · ∇ × V = 0 ,(51)
we conclude that
{F, G} f = { f , F, G} Nambu(52)
satisfies the Jacobi identity for any smooth function f ; i.e., unlike the general case where the theorem of Bialynicki-Birula and Morrison [1991] requires f to be the quadratic Casimir, one obtains a good Poisson bracket for any f . Thus, for the special case of three dimensions, one can interchange the roles of Hamiltonian and entropy in the metriplectic formalism.
• Thinking in terms of so(3) * , the setting arising from reduction (see e.g. Marsden and Ratiu [1999]), this construction leads to a natural geometric interpretation of a metriplectic system on the manifold P = R 3 . With the Poisson bracket on R 3 of (52), the bundle map π : T * R 3 → T R 3 has the expression since dH(Π ) ⊤ = ∇H(Π ) (dH(Π ) is a row vector and ∇H(Π ) is its transpose, a column vector). Now the triple bracket associated to the equation (48) can be used to generate a symmetric bracket given in Bloch, Krishnaprasad, Marsden, and Ratiu [1994] as follows:
(F, G) BKMR (Π ) = (F, G) κ C = κ({Π ,C, F}, {Π ,C, G}) = (Π × ∇F(Π )) · (Π × ∇G(Π )) .(53)
where now C = ||Π || 2 /2. Hence the bundle map κ : T * R 3 → T R 3 has the expression
κ(x, Π ) = −Π × Π × (·) ⊤ .
Thus, with the freedom to choose any quantity S = f as an entropy, with the assurance that (51) will be satisfied because ∇ × V = ∇ × ∇ f = 0, we can take H = C and have {F, S} f = 0 and (F, H) = 0 for all F ∈ C ∞ (R 3 ). The equations of motion for this metriplectic system arė
Π = −Π × ∇ f (Π ) − Π × (Π × ∇ f (Π )).(54)
The symmetric bracket is the inner product of the two Hamiltonian vector fields on each concentric sphere. As discussed in Bloch, Krishnaprasad, Marsden, and Ratiu [1994], this symmetric bracket can be defined on any compact Lie algebra by taking the normal metric on each coadjoint orbit. • The following set of equations were given in Fish [2005]:
Π = ∇S(Π) × ∇H(Π) − ∇H(Π) × (∇H(Π ) × ∇S(Π)).(55)
Yet, this metriplectic system is identical to that obtained from (48), using (49), viz.
Π = {Π , S, H} + κ ({Π , H, Π }, {Π, H, S}) ,(56)
Replacing H by g in (49) gives
(F, G) g ((Π )) = κ ({(Π ), g, F}, {(Π ), g, G}) = (∇g(Π ) × ∇F(Π )) · (∇g(Π ) × ∇G(Π )).(57)
Thus, the bundle map κ : T * R 3 → T R 3 has the expression
κ g (x, Π ) = −∇g(Π) × ∇Π × (·) ⊤ .
Examples: Two special cases of the equation (55) are of interest.
(i) If we take H = 1 2 Π 2 and S = c · Π, c a constant vector, we obtaiṅ
Π = c × Π − Π × (Π × c).(58)
(ii) If we take S = 1 2 Π 2 and H = c · Π, c a constant, we obtaiṅ
Π = Π × c − c × (c × Π) .(59)
The equations of motion (58) is an instance of double bracket damping, where the damping is due to the normal metric, whereas (59) gives linear damping of the sort arising in quantum systems.
The Toda system revisited
The Toda lattice equation revisited
We note that the Toda lattice equation fits into the metriplectic picture in a degenerate but interesting fashion since it has a dual Hamiltonian and gradient character which may be seen by writing it in the double bracket form (2). . It may be viewed either as the Hamiltonian part or the dissipative part of a metriplectic system with Hamiltonian H = 1 2 Tr L 2 or entropy function S = Tr LN respectively with the Toda lattice equations in the corresponding form (7) or (2), as discussed in Section 2. This observation may be extended to the Toda lattice flow on the normal form of any complex semisimple Lie algebra as can be see in Bloch, Brockett, and Ratiu [1992].
Full Toda with dissipation
It is possible to construct an interesting metriplectic system which incorporates the full Toda dynamics.
We consider the again the flow on the vector space of symmetric matrices k ⊥ = sym(n) but now consider the flow on a generic orbit as discussed in Deift et al. [1992] where it was shown that the flow is integrable. The Hamiltonian is again 1 2 Tr L 2 and the flow on full symmetric matrices is given bẏ
L = [π s L, L](60)
with π s being the projection onto the skew symmetric matrices in the lower triangular skew decomposition of a matrix.
In this setting there are nontrivial Casimir functions of the bracket (9). These are given as follows. For L an n × n symmetric matrix set for 0
≤ k ≤ [ 1 2 n] det(L − λ ) k = n−2k ∑ r−0 E rk (L)λ n−2k−r(61)
where the subscript k denotes the matrix obtained by deleting the first k rows and the last k columns. Then I 1k (L) = E 1k (L)/E 0k (L) are Casimir functions of the generic orbit in sym(n) as shown in Deift et al. [1992].
Thus we obtain the metriplectic systemsL
= [π s L, L] + [L, [L, ∇I 1k ]](62)
where the metric is the normal metric on orbits of su(n) restricted to the symmetric matrices (identified with i times the symmetric matrices) as in Bloch, Brockett, and Ratiu [1992]. Here H = 1 2 Tr L 2 and S = I 1k .
Metriplectic systems for pdes: metriplectic brackets and examples
First we construct a class of metriplectic brackets based on triple brackets for infinite systems, then we consider in detail an example based on Gardner's bracket on S 1 . Lastly, we mention various generalizations.
Symmetric brackets for pdes based on triple brackets
Similar to §4.2 we can construct metriplectic flows for infinite-dimensional systems from completely antisymmetric triple brackets of the form
{E, F, G} = S 1 dθ 1 S 1 dθ 2 S 1 dθ 3 C i jk (θ 1 , θ 2 , θ 3 ) (P i E u )(θ 1 ) (P j F u )(θ 2 ) (P k G u )(θ 3 )(63)
where E, F, and G are smooth functions on S 1 , C i jk is a smooth function on S 1 × S 1 × S 1 which is completely antisymmetric in its arguments, so as to assure complete antisymmetry of {E, F, G}. In addition, we denote E u := δ E/δ u, etc. Let P i , i = 1, 2, 3, be pseudo-differential operators. Evidently, the triple bracket of (63) is trilinear and completely antisymmetric in E, F, G. From (63) and a Hamiltonian H, we construct a symmetric bracket as follows:
(F, G) H = S 1 dθ ′ S 1 dθ ′′ U(θ ′ ), H, F G (θ ′ , θ ′′ ) U(θ ′′ ), H, G ,(64)
where U(θ ) in (64) denotes the functional
U(θ ) : u → S 1 dθ ′ u(θ ′ )δ (θ − θ ′ ).(65)
We shall use this notation in subsequent expressions below. The 'metric' G is assumed to be symmetric and positive semidefinite, i.e., the smooth function G :
S 1 × S 1 → R satisfies G (θ ′ , θ ′′ ) = G (θ ′′ , θ ′ ) and S 1 dθ ′ S 1 dθ ′′ G (θ ′ , θ ′′ ) f (θ ′ ) f (θ ′′ ) ≥ 0(66)
for all functions f ∈ C ∞ (S 1 ). Therefore, by construction, it is clear that (64) satisfies the following:
(i) (F, G) H = (G, F) H for all F, G, (ii) (F, H) H = 0 for all F, and (iii) (F, F) H ≥ 0 for all F.
As a special case suppose P i = P for all i = 1, 2, 3; then (63) becomes
{E, F, G} = S 1 dθ 1 S 1 dθ 2 S 1 dθ 3 C (θ 1 , θ 2 , θ 3 ) P(θ 1 )E u P(θ 2 )F u P(θ 3 )G u .(67)
As a further specialization, suppose C (θ 1 , θ 2 , θ 3 ) is given by
C (θ 1 , θ 2 , θ 3 ) = A(θ 1 , θ 2 ) + A(θ 2 , θ 3 ) + A(θ 3 , θ 1 )(68)
where A is any antisymmetric function, i.e.,
A(θ 1 , θ 2 ) = −A(θ 2 , θ 1 ) .
The form (68), assuming (69), assures complete antisymmetry of C . Finally, a particularly interesting, self-contained, case would be to suppose the A's come from some Poisson bracket, according to A(θ 1 , θ 2 ) = {U(θ 1 ),U(θ 2 )} .
It would be quite natural to choose the entropy, S, to be a Casimir function of this bracket and to choose this bracket as the Hamiltonian part of the metriplectic system with symmetric bracket given by (64). We give an example of this construction in Sec. 4.4.2.
It is evident that one can construct a wide variety of symmetric brackets based on triple brackets. For example, one can choose the pseudo-differential operators from the list {I d , d/dθ , (d/dθ ) −1 , H }, where I d is the identity operator, and the Hamiltonian, H, and entropy (Casimir) C could be one of the following functionals:
H 0 = S 1 dθ u(71)H 2 = S 1 dθ u 2 /2(72)H 1 = S 1 dθ u ′2 /2(73)H KdV = S 1 dθ u 3 + u ′2 /2 .(74)
In the Sec. 4.4.2 we will construct a metriplectic system based on the Gardner bracket (30) of Sec. 3.7. To avoid complications, we choose a simple example, yet one that displays general features of a large class of 1 + 1 energy conserving dissipative system.
Metriplectic systems based on the Gardner bracket
For simplicity we choose P i = I d for all i, and as mentioned above, we suppose A(θ 1 , θ 2 ) is generated from the Gardner bracket (30), i.e.,
A(θ 1 , θ 2 ) := {U(θ 1 ),U(θ 2 )} = S 1 dθ δ (θ − θ 1 ) d dθ δ (θ − θ 2 ) = δ ′ (θ 1 − θ 2 ) ,(75)
where prime denotes differentiation with respect to argument and δ ′ (θ 1 − θ 2 ) is defined by
S 1 dθ 1 S 1 dθ 2 δ ′ (θ 1 − θ 2 ) f (θ 1 )g(θ 2 ) = − S 1 dθ 1 S 1 ds δ ′ (s) f (θ 1 )g(θ 1 − s) = S 1 dθ 1 f (θ 1 )g ′ (θ 1 ) = − S 1 dθ 1 S 1 dθ 2 δ ′ (θ 2 − θ 1 ) f (θ 1 )g(θ 2 )
for any f , g ∈ C ∞ (S 1 ), which shows that δ ′ (θ 2 − θ 1 ) = −δ ′ (θ 1 − θ 2 ). With this choice for A we obtain
C (θ 1 , θ 2 , θ 3 ) = δ ′ (θ 1 − θ 2 ) + δ ′ (θ 2 − θ 3 ) + δ ′ (θ 3 − θ 1 ) ,
and Eq. (67) becomes
{E, F, G} = S 1 dθ 1 S 1 dθ 2 S 1 dθ 3 δ ′ (θ 1 − θ 2 ) + δ ′ (θ 2 − θ 3 ) + δ ′ (θ 3 − θ 1 ) E u (θ 1 ) F u (θ 2 ) G u (θ 3 ) = S 1 dθ G u (θ ) S 1 dθ F u (θ )E ′ u (θ ) + S 1 dθ E u (θ ) S 1 dθ G u (θ )F ′ u (θ ) + S 1 dθ F u (θ ) S 1 dθ E u (θ )G ′ u (θ ).(76)
We shall construct a metriplectic system of the forṁ
F = {H, F, G} + S 1 dθ ′ S 1 dθ ′′ U(θ ′ ), S, F G (θ ′ , θ ′′ ) U(θ ′′ ), S, G ,
using the Gardner bracket (71). Observe if we now set F = H 0 , the Casimir for the Gardner bracket (71), then, since δ H 0 /δ u = 1, we obtain
{F, H 0 , G} = S 1 dθ F u G ′ u(77)
which is precisely the Gardner bracket. To see this, let us compute, for example, the integral in the third term of (76). Changing variables s = θ 3 − θ 1 we get
S 1 dθ 1 S 1 dθ 2 S 1 dθ 3 δ ′ (θ 3 − θ 1 ) E u (θ 1 ) G u (θ 3 ) = − S 1 ds S 1 dθ 3 δ ′ (s) E u (θ 3 − s) G u (θ 3 ) = S 1 dθ 3 E ′ u (θ 3 )G u θ 3 ).
A similar computation shows that the first and second terms vanish. In order to construct the symmetric bracket in (64), we need the following, computed using (76):
{U(θ ), H, G} = − S 1 dθ G u (θ ) H ′ u (θ ) + S 1 dθ G u (θ )H ′ u (θ ) + S 1 dθ H u (θ ) G ′ u (θ ).(78)
Now with the counterpart of (78) for the functional F with U(θ ′ ), a choice for H, and a choice for G , we can construct (F, G) H . We make the following choices:
H 2 (u) = S 1 dθ u 2 2 , S(u) := H 0 (u) = S 1 dθ u (79) G (θ ′ , θ ′′ ) = δ (θ ′ − θ ′′ ) .(80)
Now choose H 2 from (79) and insert it into (78) which gives
{U(θ ), H 2 , G} = − S 1 dθ G u (θ ) u ′ (θ ) + S 1 dθ G u (θ )u ′ (θ ) + SG ′ u (θ )(81)
and to construct the symmetric bracket (64), we need
U(θ ′′ ), H 2 , S = −u ′ (θ ′′ ).(82)
Thus, the equations of motion are
d dt F = {F, H 0 , H 2 } + (F, S) H 2 where F, S H 2 = S 1 dθ ′ S 1 dθ ′′ U(θ ′ ), H 2 , F G (θ ′ , θ ′′ ) U(θ ′′ ), H 2 , S .(83)
This yields
u t − u θ = S u θ θ + Q with Q := S 1 dθ ′ |u θ ′ | 2 .(84)
Equation (84) has several interesting features. For fixed given constant S and Q, it is a linear equation composed of the heat equation with a source and with the inclusion of a linear advection term. One can proceed to solve this equation by the usual method of constructing a temporal Green's function out of the heat kernel and expanding in a Fourier series. After such a solution is constructed, one must enforce the fact that the global quantities S and Q are both time dependent and, importantly, dependent on the solution so constructed. Only after these constraints are enforced would one actually have a solution. Pursuing this construction, although interesting, is outside the scope of the present paper and will be treated elsewhere.
We observe that the equation (84) is metriplectic. Indeed, by construction, we have a Poisson bracket (77) (the Gardner bracket) and a symmetric bracket (83). Since these were constructed out of triple brackets, property (iii) of Definition in Section 4.1 holds. Positive semidefiniteness of the symmetric bracket follows from (81).
The nature of the dissipation of (84) is of particular interest in that it involves the global quantities S and Q. This is reminiscent of collision operators, such as that due to Boltzmann and generalized nonlinear Fokker-Planck operators such as those due to Landau, Lenard-Balescu, and others (see, e.g., Morrison [1986]). The usual dissipation in 1 + 1 systems is local in nature (see Sec. 4.5) and dissipates energy. Thus the metriplectic construction of this section has pointed to a quite natural type of dynamical system that has dynamical versions of both the first and second laws of thermodynamics. The pathway for constructing other systems with nonlinear and dispersive Hamiltonian components, other kinds of dissipation, etc. is now cleared, and some will be considered in future publications.
Some metriplectic generalizations
It is evident that many generalizations are possible. We mention a few.
• Without destroying the symmetries or formal metriplectic bracket properties we could allow one or both of the functions C and G to depend on the field variable u or even contain pseudodifferential operations. In fact, such ideas were used in similar brackets in Flierl and Morrison [2011] to facilitate numerical computation. • It is clear how to generalize (64) to preserve more constraints, say I 1 , I 2 , . . . , in addition to H.
The bracket (F, G) H,I 1 ,I 2 ,... is guaranteed to be symmetric, conserve the invariants, and be positive semidefinite.
• It is of general interest to have metriplectic systems of the forṁ F = {H, F, G} + S 1 dθ ′ S 1 dθ ′′ U(θ ′ ), S, F G (θ ′ , θ ′′ ) U(θ ′′ ), S, G (such as our example of Sec. 4.4.2) for a suitably chosen function G; here H is the Hamiltonian and S is the entropy. Exploring the mathematics of when this is possible is an area to pursue. • The construction here is easily extendable to higher spatial dimensions. For example consider the following triple bracket given in Bialynicki-Birula and Morrison [1991]:
{E, F, G} = D d 6 z E f F f , G f ,(86)
where z = (q, p) is a canonical six-dimensional phase space variable, f (z,t) is a phase space density, as in Vlasov theory, the 'inner' Poisson bracket is defined by
[ f , g] = f q · g p − f p · g q .(87)
We assume that the domain D with boundary conditions enables us to set all surface terms obtained by integrations by parts to zero, thereby assuring complete antisymmetry. Inserting the quadratic Casimir C 2 := D d 6 z f 2 /2 into (86) gives
{F, G} V P = {C 2 , F, G} = D d 6 z f F f , G f ,
the Lie-Poisson bracket for the Vlasov-Poisson system, as given in Morrison [1980]. Thus, this bracket with the quadratic Casimir is formally akin to the construction given in Sec. 4.2.1 (although we note it reduces to a good bracket for any Casimir and in this way is like the case of so(3) of Sec. 4.2.2). The triple bracket of (86) can be used in a generalization of the bracket of (64) to obtain a variety of energy conserving collision operators, with a wide choice of Casimirs as entropies.
Hybrid dissipative structures
Even if a system is not metriplectic, it is of interest to see if it can be obtained from an equation which consists of a Hamiltonian part and a gradient part with respect to a suitable Poisson bracket and metric, respectively. For KdV-like equations, energy (the Hamiltonian) is generally not conserved when dissipation is added to the system. This is common for physical systems, but a more complete model would conserve energy while accounting for heat loss, i.e., entropy production. In the terminology of Morrison [2009], models that lose energy, such as those treated here and those described by the double bracket formalism of §2.1, are incomplete, while those that do represent dynamical models of the laws of thermodynamics, such as metriplectic systems, are termed complete. Although incomplete systems do not conserve energy, they may conserve other invariants, and building this in, represents an advantage of various bracket formulations. Thus, we construct incomplete hybrid Hamiltonian and dissipative dynamics by combining a Hamiltonian and a gradient vector field according to the prescription u t = {u, H} + (u, S)
where u → {u, H} is a Hamiltonian vector field generated by H and u → (u, S) is a gradient vector field generated by S (which could be H). Thus, ( , ) is, up to a sign, an inner product on the space of functions u.
Consider the following examples:
• With the usual KdV Hamiltonian of (31) and the Gardner bracket of (30) describing the Hamiltonian vector field, together with the choice S(u) = H 1 (u) = 1 4π π −π dθ (u θ ) 2 we obtain for the gradients of Corollary 2 (i) u t = {u, H} − ∇ 1 H 1 = −u θ θ θ + 6uu θ − u (ii) u t = {u, H} − ∇H 1 = −u θ θ θ + 6uu θ + u θ θ (iii) u t = {u, H} − ∇ 2 H 1 = −u θ θ θ + 6uu θ − H (u θ ) which is the KdV equation of (29) with the inclusion of a new term that describes dissipation. Case (i) corresponds to simple linear damping, case (ii) to 'viscous' diffusion, and case (iii) to the equation of Ott and Sudan [1969] which adds a term to the KdV equation that describes Landau damping. For these systems the KdV invariant π −π dθ u 2 serves as a Lyapunov function.
• Choosing H = S = H 1 , the Kähler Hamiltonian flow of (34) together with the dissipative flow generated by (21),
yields u t = {u, H 1 } − ∇ 2 H 1 = −u θ − H (u θ )
which describes simple advection with Landau damping. This equation possesses the damped traveling wave solution. • We note that we can derive the heat equation from a symmetric bracket of the form (64), again with G (θ ′ , θ ′′ ) = δ (θ ′ − θ ′′ ). Using this G and noting {U(θ ), H 0 , F} = G ′ u (θ ), we obtain
(F, G) H 0 = S 1 dθ F ′ u G ′ u .(89)
Let us compute, for example,Ḟ(u) = (F, −H 2 ) H 0 (see (72)). Since δ H 2 /δ u = −u, we obtain
S 1 d θ F uu = d dt F(u) = (F, H 2 ) H 0 = − S 1 dθ F ′ u u ′ = S 1 dθ F u u ′′ .
This yields u t = u xx which is the heat equation.
From these examples it is clear how a variety of hybrid Hamiltonian and dissipative flows can be constructed from the machinery we have developed. For example, if we replace the KdV Hamiltonian by H(u) = S 1 dθ 1 2 uH (u θ ) + 1 3 u 3 we obtain the Benjamin-Ono equation with the various dissipative terms. Related ideas apply to fluid dynamics may be found in Gay-Balmaz and Holm [2012].
d dt L(t) = [L(t), [L(t), N]] , L(0) = L 0 ∈ g u .
d dt O = BO , O(0) = Identity,then from(7)we have d dt(O −1 LO) = 0 .
b 2 (L)[L, X], [L,Y ]) = b 1 (A (L)[L, X], [L,Y ]).
2π which shows that ker exp = Z and coker exp = {0}. Thus the connected components of L(S 1 ) are indexed by the winding number. The connected component of the identity L(S 1 ) 0 consists of loops with winding number zero about the origin.
for the weak inner product b 1 ( 1 )
1defining the normal metric; (ii) ∇H 1 (u) = −u ′′ for the weak inner product b(1) defining the induced metric, where for u ∈ H s (R) with s ≥ 2; (iii) ∇ 2 H 1 (u) = H u ′ for the weak inner product b 2 (1) defining the Kähler metric.
Corollary 4 .
4Under the same hypothesis as in Theorem 2(iii), the Hamiltonian vector field of the smooth function H : L(S 1 ) → R relative to the symplectic form ω on L(R) given by (15) has the expression X H (u) = θ 0 dϕ δ H δ u (ϕ)Proof. We have X H (u) = H ∇ 2 H(u)
F
= {F, H + S} + (F, H + S) = {F, H} + (F, S), for all F ∈ C ∞ (P),
2, 3, and I i are the principal moments of inertia of the body, then Hamilton's equations d dt F(Π ) = { f , h} so(3) − (Π ) are equivalent to Euler's equationsΠ = Π × Ω.
One simply first constructs the completely antisymmetric multilinear brackets {E, F, G, H, . . . } paralleling (63), and then, analogous to (64), constructs (F, G) H,I 1 ,I 2 ,... = S 1 dθ ′ S 1 dθ ′′ U(θ ′ ), H, I 1 , I 2 , . . . , F G (θ ′ , θ ′′ ) U(θ ′′ ), H, I 1 , I 2 , . . . , G .
π f (x, Π ) = x, ∇ f (Π ) × (·) ⊤
Acknowledgements AMB was partially supported by NSF grants DMS-090656 and DMS-1207693. PJM was supported by U.S. Department of Energy contract DE-FG05-80ET-53088. TSR was partially supported by Swiss NSF grant 200021-140238, and by the government grant of the Russian Federation for support of research projects implemented by leading scientists, Lomonosov Moscow State University under the agreement No. 11.G34.31.0054.
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"High-Dimensional L 2 -Boosting: Rate of Convergence",
"High-Dimensional L 2 -Boosting: Rate of Convergence"
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| [
"Ye Luo ",
"Kurtluo@hku Hk ",
"Hong Kong ",
"Martin Spindler [email protected] ",
"Jannis Kueck [email protected] ",
"\nInstitute for Statistics\nInstitute for Statistics\nHong Kong University Business School\nThe University of Hong Kong\nUniversity of Hamburg\nGermany\n",
"\nUniversity of Hamburg\n\n"
]
| [
"Institute for Statistics\nInstitute for Statistics\nHong Kong University Business School\nThe University of Hong Kong\nUniversity of Hamburg\nGermany",
"University of Hamburg\n"
]
| [
"Journal of Machine Learning Research"
]
| Boosting is one of the most significant developments in machine learning. This paper studies the rate of convergence of L 2 Boosting, which is tailored for regression, in a highdimensional setting. Moreover, we introduce so-called "post-Boosting". This is a postselection estimator which applies ordinary least squares to the variables selected in the first stage by L 2 Boosting. Another variant is "Orthogonal Boosting"where after each step an orthogonal projection is conducted. We show that both post-L 2 Boosting and the orthogonal boosting achieve the same rate of convergence as LASSO in a sparse, high-dimensional setting. We show that the rate of convergence of the classical L 2 Boosting depends on the design matrix described by a sparse eigenvalue constant. To show the latter results, we derive new approximation results for the pure greedy algorithm, based on analyzing the revisiting behavior of L 2 Boosting. We also introduce feasible rules for early stopping, which can be easily implemented and used in applied work. Our results also allow a direct comparison between LASSO and boosting which has been missing from the literature. Finally, we present simulation studies and applications to illustrate the relevance of our theoretical results and to provide insights into the practical aspects of boosting. In these simulation studies, post-L 2 Boosting clearly outperforms LASSO. | null | [
"https://arxiv.org/pdf/1602.08927v3.pdf"
]
| 15,493,541 | 1602.08927 | 0c608a2c4c3e9ec5e9213629ddad3a62bf1e7b26 |
High-Dimensional L 2 -Boosting: Rate of Convergence
2022
Ye Luo
Kurtluo@hku Hk
Hong Kong
Martin Spindler [email protected]
Jannis Kueck [email protected]
Institute for Statistics
Institute for Statistics
Hong Kong University Business School
The University of Hong Kong
University of Hamburg
Germany
University of Hamburg
High-Dimensional L 2 -Boosting: Rate of Convergence
Journal of Machine Learning Research
12022Submitted XXX; Published XXXGermany Editor: N.N.L 2 Boostingpost-L 2 Boostingorthogonal L 2 Boostinghigh-dimensional modelsLASSOrate of convergencegreedy algorithmapproximation
Boosting is one of the most significant developments in machine learning. This paper studies the rate of convergence of L 2 Boosting, which is tailored for regression, in a highdimensional setting. Moreover, we introduce so-called "post-Boosting". This is a postselection estimator which applies ordinary least squares to the variables selected in the first stage by L 2 Boosting. Another variant is "Orthogonal Boosting"where after each step an orthogonal projection is conducted. We show that both post-L 2 Boosting and the orthogonal boosting achieve the same rate of convergence as LASSO in a sparse, high-dimensional setting. We show that the rate of convergence of the classical L 2 Boosting depends on the design matrix described by a sparse eigenvalue constant. To show the latter results, we derive new approximation results for the pure greedy algorithm, based on analyzing the revisiting behavior of L 2 Boosting. We also introduce feasible rules for early stopping, which can be easily implemented and used in applied work. Our results also allow a direct comparison between LASSO and boosting which has been missing from the literature. Finally, we present simulation studies and applications to illustrate the relevance of our theoretical results and to provide insights into the practical aspects of boosting. In these simulation studies, post-L 2 Boosting clearly outperforms LASSO.
Introduction
In this paper we consider L 2 Boosting algorithms for regression which are coordinatewise greedy algorithms that estimate the target function under L 2 loss. Boosting algorithms represent one of the major advances in machine learning and statistics in recent years. Freund and Schapire's AdaBoost algorithm for classification (Freund and Schapire (1997)) has attracted much attention in the machine learning community as well as in statistics. Many variants of the AdaBoost algorithm have been introduced and proven to be very competitive in terms of prediction accuracy in a variety of applications with a strong resistance to overfitting. Boosting methods were originally proposed as ensemble methods, which rely on the principle of generating multiple predictions and majority voting (averaging) among the individual classifiers (cf Bühlmann and Hothorn (2007)). An important step in the analysis of Boosting algorithms was Breiman's interpretation of Boosting as a gradient descent algorithm in function space, inspired by numerical optimization and statistical estimation (Breiman (1996), Breiman (1998)). Building on this insight, Friedman et al. (2000) and Friedman (2001) embedded Boosting algorithms into the framework of statistical estimation and additive basis expansion. This also enabled the application of boosting for regression analysis. Boosting for regression was proposed by Friedman (2001), and then Bühlmann and Yu (2003) defined and introduced L 2 Boosting. An extensive overview of the development of Boosting and its manifold applications is given in the survey Bühlmann and Hothorn (2007).
In the high-dimensional setting there are two important but unsolved problems on L 2 Boosting. First, the convergence rate of the L 2 Boosting has not been thoroughly analyzed. Second, the pattern of the variables selected at each step of L 2 Boosting is unknown.
In this paper, we show that these two problems are closely related. We establish results on the sequence of variables that are selected by L 2 Boosting. At any step of L 2 Boosting, we call this step "revisiting" if the variable chosen in this step has already been selected in previous steps. We analyze the revisiting behavior of L 2 Boosting, i.e., how often L 2 Boosting revisits. We then utilize these results to derive an upper bound of the rate of convergence of the L 2 Boosting. 1 We show that frequency of revisiting, as well as the convergence speed of L 2 Boosting, depend on the structure of the design matrix, namely on a constant related to the minimal and maximal restricted eigenvalue. Our bounds on convergence rate of L 2 Boosting are in general slower than that of LASSO.
We also introduce in this paper the so-called "post-Boosting", and the orthogonal boosting variant. 2 For orthogonal boosting see also Section 3.1 in Lai and Yuan (2021). We show that both algorithms achieve the same rate of convergence as LASSO in a sparse, high-dimensional setting.
Compared to LASSO, boosting uses a somewhat unusual penalization scheme. The penalization is done by "early stopping" to avoid overfitting in the high-dimensional case. In the low-dimensional case, L 2 Boosting without stopping converges to the ordinary least squares (OLS) solution. In a high-dimensional setting early stopping is key for avoiding overfitting and for the predictive performance of boosting. We give a new stopping rule that is simple to implement and also works very well in practical settings as demonstrated in the simulation studies. We prove that such a stopping rule achieves the best bound obtained in our theoretical results.
In a deterministic setting, which is when there is no noise or error term in the model, boosting methods are also known as greedy algorithms (the pure greedy algorithm (PGA) and the orthogonal greedy algorithm (OGA)). In signal processing, L 2 Boosting is essentially 1. Without analyzing the sequence of variables selected at each step of L2Boosting, only much weaker results on convergence speed of L2Boosting are available based on DeVore and Temlyakov (1996) and Livshitz and Temlyakov (2003). 2. Orthogonal boosting has also similarities with forward step-wise regression. the same as the matching pursuit algorithm of Mallat and Zhang (1993). We will employ the abbreviations post-BA (post-L 2 Boosting algorithm) and oBA (orthogonal L 2 Boosting algorithm) for the stochastic versions we analyze.
The rate of convergence of greedy algorithms has been analyzed in DeVore and Temlyakov (1996) and Livshitz and Temlyakov (2003). Temlyakov (2011) is an excellent survey of recent results on the approximation theory of greedy approximation. To the best of our knowledge, with an additional assumption on the design matrix, we establish the first results on revisiting in the deterministic setting and greatly improve the existing results of DeVore and Temlyakov (1996). These results, being available in the appendix, are essential for our analysis for L 2 Boosting, but might also be of interest in their own right.
As mentioned above, Boosting for regression was introduced by Friedman (2001). L 2 -Boosting was defined in Bühlmann and Yu (2003). Its numerical convergence, consistency, and statistical rates of convergence of boosting with early stopping in a low-dimensional setting were obtained in Zhang and Yu (2005). Consistency in prediction norm of L 2 Boosting in a high-dimensional setting was first proved in Bühlmann (2006). The numerical convergence properties of Boosting in a low-dimensional setting are analyzed in Freund et al. (2016+). The orthogonal Boosting algorithm in a statistical setting under different assumptions is analyzed in Ing and Lai (2011). The rates for the PGA and OGA cases are obtained in Barron et al. (2008).
In this paper we consider linear basis functions. Classification and regression trees, and the widely used neural networks, involve non-linear basis functions. We hope that our results can serve as a starting point for the analysis of non-linear basis functions which is left for future research.
The structure of this paper is as follows: In Section 2 the L 2 Boosting algorithm (BA) is defined together with its modifications, the post-L 2 Boosting algorithm (post-BA) and the orthogonalized version (oBA). In Section 3 we present a new approximation result for the pure greedy algorithm (PGA later) and an analysis of the revisiting behavior of the boosting algorithm. In Section 4 we present the main results of our analysis, namely an analysis of the boosting algorithm and some of its variants. The proofs together with some details of the new approximation theory for PGA are provided in the Appendix. Section 5 contains a simulation study that offers some insights into the methods and also provides some guidance for stopping rules in applications. Section 6 discusses two applications and provides concluding observations. Notation: Let z and y be n-dimensional vectors. We define the empirical L 2 -norm as E n [z] = 1/n n i=1 z i . Define ||z|| to be the Euclidean norm, and ||z|| 2,n := E n [z 2 ]. Define < ·, · > n to be the inner product defined by: < z, y > n = 1/n n i=1 z i y i . For a random variable X, E[X] denotes its expectation. The correlation between the random variables X and Y is denoted by corr(X, Y ).
We use the notation a ∨ b = max{a, b} and a ∧ b = min{a, b}. We also use the notation a b to mean a ≤ cb for some constant c > 0 that does not depend on n; and a P b to mean a = O P (b). For a set U , supp(U ) denotes the set of indices of which the corresponding element in U is not zero. Given a vector β ∈ R p and a set of indices T ⊂ {1, . . . , p}, we denote by β T the vector in which
β T j = β j if j ∈ T , β T j = 0 if j / ∈ T .
L 2 -Boosting with componentwise least squares
To define the boosting algorithm for linear models, we consider the following regression setting:
y i = x i β + ε i , i = 1, . . . , n,(1)
with vector x i = (x i,1 , . . . , x i,pn ) consisting of p n predictor variables, β a p n -dimensional coefficient vector, and a random, mean-zero error term ε i , E[ε i |x i ] = 0. Further assumptions will be employed in the next sections. We allow the dimension of the predictors p n to grow with the sample size n, and is even larger than the sample size, i.e., dim(β) = p n n. But we will impose a sparsity condition. This means that there is a large set of potential variables, but the number of variables which have non-zero coefficients, denoted by s, is small compared to the sample size, i.e. s n. This can be weakened to approximate sparsity, to be defined and explained later. More precise assumptions will also be made later. In the following, we will drop the dependence of p n on the sample size and denote it by p if no confusion will arise.
X denotes the n × p design matrix where the single observations x i form the rows. X j denotes the jth column of design matrix, and x i,j the jth component of the vector x i . We consider a fixed design for the regressors. We assume that the regressors are standardized with mean zero and variance one, i.e., E n [x i,j ] = 0 and E n [x 2 i,j ] = 1 for j = 1, . . . , p, The basic principle of Boosting can be described as follows. We follow the interpretation of Breiman (1998) andFriedman (2001) of Boosting as a functional gradient descent optimization (minimization) method. The goal is to minimize a loss function, e.g., an L 2 -loss or the negative log-likelihood function of a model, by an iterative optimization scheme. In each step the (negative) gradient which is used in every step to update the current solution is modelled and estimated by a parametric or nonparametric statistical model, the so-called base learner. The fitted gradient is used for updating the solution of the optimization problem. A strength of boosting, besides the fact that it can be used for different loss functions, is its flexibility with regard to the base learners. We then repeat this procedure until some stopping criterion is met.
The literature has developed many different forms of boosting algorithms. In this paper we consider L 2 Boosting with componentwise linear least squares, as well as two variants. All three are designed for regression analysis. "L 2 "refers to the loss function, which is the typical sum-of-squares of the residuals Q n (β) = n i=1 (y i − x i β) 2 typical in regression analysis. In this case, the gradient equals the residuals. "Componentwise linear least squares"refers to the base learners. We fit the gradient (i.e. residuals) against each regressor (p univariate regressions) and select the predictor/variable which correlates most highly with the gradient/residual, i.e., decreases the loss function most, and then update the estimator in this direction. We next update the residuals and repeat the procedure until some stopping criterion is met. We consider L 2 Boosting and two modifications: the "classical"one which was introduced in Friedman (2001) and refined in Bühlmann and Yu (2003) for regression analysis, an orthogonal variant and post-L 2 Boosting. As far as we know, post-L 2 Boosting has not yet been defined and analyzed in the literature. In signal processing and approximation theory, the first two methods are known as the pure greedy algorithm (PGA) and the orthogonal greedy algorithm (OGA) in the deterministic setting, i.e. in a setting without stochastic error terms.
L 2 Boosting
For L 2 Boosting with componentwise least squares, the algorithm is given below.
Algorithm 1 (L 2 -Boosting)
1. Start / Initialization: β 0 = 0 (p-dimensional vector), f 0 = 0, set maximum number of iterations m stop and set iteration index m to 0.
At the
(m + 1) th step, calculate the residuals U m i = y i − x i β m .
3. For each predictor variable j = 1, . . . , p calculate the correlation with the residuals:
γ m j := n i=1 U m i x i,j n i=1 x 2 i,j = < U m , x j > n E n [x 2 i,j ]
.
Select the variable j m that is the most correlated with the residuals 3 , i.e., For simplicity, write γ m for the value of γ m j m at the m th step. The act of stopping is crucial for boosting algorithms, as stopping too late or never stopping leads to overfitting and therefore some kind of penalization is required. A suitable solution is to stop early, i.e., before overfitting takes place. "Early stopping" can be interpreted as a form of penalization. Similar to LASSO, early stopping might induce a bias through shrinkage. A potential way to decrease the bias is by "post-Boosting"which is defined in the next section.
In general, during the run of the boosting algorithm, it is possible that the same variable is selected at different steps, which means the variable is revisited. This revisiting behavior is key to the analysis of the rate of convergence of L 2 Boosting. In the next section we will analyze the revisting properties of boosting in more detail.
Post-L 2 Boosting
Post-L 2 Boosting is a post-model selection estimator that applies ordinary least squares (OLS) to the model selected by the first-step, which is L 2 Boosting. To define this estimator formally, we make the following definitions: T := supp(β) andT := supp(β m * ), the support of the true model and the support of the model estimated by L 2 Boosting as described above with stopping at m * . A superscript C denotes the complement of the set with regard to {1, . . . , p}. In the context of LASSO, OLS after model selection was analyzed in Belloni and Chernozhukov (2013). Given the above definitions, the post-model selection estimator or OLS post-L 2 Boosting estimator will take the form β = argmin β∈R p Q n (β) : β j = 0 for each j ∈T C .
(2)
3. Equivalently, which fits the gradient best in a L2-sense.
Comment 2.1 For boosting algorithms it has been recommended -supported by simulation studies -not to update by the full step size x j m but only a small step ν. The parameter ν can be interpreted as a shrinkage parameter, or alternatively, as describing the step size when updating the function estimate along the gradient. Small step sizes (or shrinkage) make the boosting algorithm slower to converge and require a larger number of iterations. But often the additional computational cost in turn results in better out-of-sample prediction performance. By default, ν is usually set to 0.1. Our analysis in the later sections also extends to a restricted step size 0 < ν < 1.
Orthogonal L 2 Boosting
A variant of the Boosting Algorithm is orthogonal Boosting (oBA) or the Orthogonal Greedy Algorithm in its deterministic version. Only the updating step is changed: an orthogonal projection of the response variable is conducted on all the variables which have been selected up to this point. The advantage of this method is that any variable is selected at most once in this procedure, while in the previous version the same variable might be selected at different steps which makes the analysis far more complicated. More formally, the method can be described as follows by modifying
Step (4):
Algorithm 2 (Orthogonal L 2 Boosting) (4 )ŷ m+1 ≡ f m+1 = P m y and U m+1 i = Y i −Ŷ m+1 i ,
where P m denotes the projection of the variable y on the space spanned by first m selected variables (the corresponding regression coefficient is denoted β m o .)
Define X m o as the matrix which consists only of the columns which correspond to the variables selected in the first m steps, i.e. all X j k , k = 0, 1, . . . , m. Then we have:
β m o = (X m o X m o ) −1 X m o y (3) y m+1 = f m+1 o = X m o β m o(4)
Comment 2.2 Orthogonal L 2 Boosting might be interpreted as post-L 2 Boosting where the refit takes place after each step.
Comment 2.3 Both post-Boosting and orthogonal Boosting require, to be well-defined, that the number of selected variables be smaller than the sample size . This is enforced by our stopping rule as we will see later.
New Approximation Results for the Pure Greedy Algorithm
In approximation theory a key question is how fast functions can be approximated by greedy algorithms. Approximation theory is concerned with deterministic settings, i.e., the case without noise. Nevertheless, to derive rates for the L 2 Boosting algorithm in a stochastic setting, the corresponding results for the deterministic part play a key role. For example, the results in Bühlmann (2006) are limited by the result used from approximation theory, namely the rate of convergence of weak relaxed greedy algorithms derived in Temlyakov (2000). For the pure greedy algorithm DeVore and Temlyakov (1996) establish a rate of convergence of m −1/6 in the 2 −norm, where m denotes the number of steps iterated in the PGA. This rate was improved to m −11/62 in Konyagin and Temlyakov (1999), but Livshitz and Temlyakov (2003) establish a lower bound of m −0.27 . The class of functions F which is considered in those papers is determined by general dictionaries D and given by
F = {f ∈ H : f = k∈Λ c k w k , w k ∈ D, |Λ| < ∞ and k∈Λ |c k | ≤ M },
where M is some constant, H denotes a Hilbert space, and the sequence (c k ) are the coefficients with regard to the dictionary D.
In this section we discuss the approximation bound of the pure greedy algorithm where we impose an additional but widely used assumption on the Gram matrix E n [x i x i ] in high dimensional statistics to tighten the bounds. First, the assumptions and an initial result describing the revisiting behavior will be given, then a new approximation result based on the revisiting behavior will be presented which is the core of this section. The proofs for this section and a detailed analysis of the revisiting behavior of the algorithm are moved to Appendix A.
Assumptions
For the analysis of the pure greedy algorithm, the following two assumptions are made, which are standard for high-dimensional models.
A.1 (Exact Sparsity) T = supp(β) and s = |T | n.
Comment 3.1 The exact sparsity assumption can be weakened to an approximate sparsity condition, in particular in the context of the stochastic version of the pure greedy algorithm (L 2 Boosting). This means that the set of relevant regressors is small, and the other variables do not have to be exactly zero but must be negligible compared to the estimation error.
For the second assumption, we make a restricted eigenvalue assumption which is also commonly used in the analysis of LASSO.
Define Σ(s, M ) := {A|dim(A) ≤ s × s, A is any diagonal submatrices of M }, for any square matrix M .
We need the following definition.
Definition 1 The smallest and largest restricted eigenvalues are defined as
φ s (s, M ) := min W ∈Σ(s,M ) φ s (W ), and φ l (s, M ) := max W ∈Σ(s,M ) φ l (W ).
φ s (W ) and φ l (W ) denote the smallest and largest eigenvalue of the matrix W .
A.2 (SE) We assume that there exist constant 0 < c < 1 and C such that
0 < 1 − c ≤ φ s (s , E n [x i x i ]) ≤ φ l (s , E n [x i x i ]) ≤ C < ∞ for any s ≤ M 0 ,
where M 0 is a sequence such that M 0 → ∞ slowly along with n, and M 0 ≥ s.
Comment 3.2
This condition is a variant of the so-called "sparse eigenvalue condition", which is used for the analysis of the Lasso estimator. A detailed discussion of this condition is given in Belloni et al. (2010). Similar conditions, such as the restricted isometry condition or the restricted eigenvalue condition, have been used for the analysis of the Dantzig Selector (Candes and Tao (2007)) or the Lasso estimator (Bickel et al. (2009)). An extensive overview of different conditions on matrices and how they are related is given by van de Geer and Bühlmann (2009). To assume that φ l (m, E n [x i x i ]) > 0 requires that all empirical Gram submatrices formed by any m components of x i are positive definite. It is well-known that Condition SE is fulfilled for many designs of interest.
More restrictive requirements that M 0 should be large enough will be imposed in order to get good convergence rate for the PGA, i.e., L 2 Boosting without a noise term.
Define V m = Xα m as the residual for the PGA. α m is defined as the difference between the true parameter vector β and the approximation at the m th step, β m , α m = β − β m . We would like to explore how fast V m converges to 0. In our notation, ||V m+1 || 2 2,n = ||V m || 2 2,n − (γ m ) 2 , therefore ||V m || 2 2,n is non-increasing in m. As described in Algorithm 1, the sequence of variables selected in the PGA is denoted by
j 0 , j 1 , . . .. Define T m := T ∪ {j 0 , j 1 , . . . , j m−1 }. Define q(m) := |T m | as the cardinality of T m , m = 0, 1, . . .. It is obvious that q(m) ≤ m + s.
It is essential to understand how PGA revisits the set of already selected variables. To analyze the revisiting behavior of the PGA, some definitions are needed to fix ideas.
Definition 2 We say that the PGA is revisiting at the m th step, if and only if j m−1 ∈ T m−1 . We define the sequence of labels A := {A 1 , A 2 , ...} with each entry A i being either labelled as R(revisiting) or N (non-revisiting).
Lemma 1 Assume that assumptions A.1-A.2 hold. Assume that m + k < M 0 . Consider the sequence of steps 1, 2, ..., m.
Denote µ a (c) = [1 − (1 + 1 (1−c) 2 ) − 1 1−c ]
for any c ∈ (0, 1). Then for any δ > 0, the number of Rs in the sequence A at step m, denoted R(m), must satisfy:
|R(m)| ≥ 1 − (1 + δ)µ a (c) 2 − (1 + δ)µ a (c) m − (1 + δ)µ a (c) 2 − (1 + δ)µ a (c) q(0).
The lower bound stated in Lemma 1 has room for improvement, e.g., when c = 0, |R(m)|/m = 1 as it is shown in Lemma 8 in Appendix A, while we get 1/2 in Lemma 1 as lower bounds of |R(m)/m| as m becomes large enough. Deriving tight bounds is an interesting question for future research. More detailed properties of the revisiting behavior of L 2 Boosting are provided in the Appendix A.
Approximation bounds on PGA
With an estimated bound for the proportion of Rs in the sequence A, we are now able to derive an upper bound for ||V m || 2 2,n . By Lemma 1, define n * k := m+µa(c)q(k)
2−µa(c)
which is an upper bound of |q(m + k) − q(k)| up to constant converging to 1 as q(m) goes to infinity. Before we state the main result of this section we present an auxiliary lemma.
Lemma 2 Let λ > 0 be a constant. Let m = λq(k). Consider the steps numbered as
k + 1, ..., k + m. Assume that m + k < M 0 . Define ζ(c, λ) := (1−c)((1−µa(c))λ−µa(c)) 2+λ log( 2+λ 2−µa(c) ) + 1 − c for all λ ≥ µa(c) 1−µa(c)
. Then, for any δ > 0 and q(k) > 0 large enough, the following statement holds:
||V m+k || 2 2,n ≤ ||V k || 2 2,n q(k) q(k) + n * k ζ(c,λ)−δ .
Based on Lemma 2, we are able to develop our main results on approximation theory of pure greedy algorithm under L 2 loss and Assumptions 1 and 2.
Theorem 1 (Approximation Theory of PGA based on revisiting) Define ζ * (c) := max λ≥ µa(c)
1−µa(c)
ζ(c, λ) as a function of c. Then, for any δ > 0 and m < M 0 , there exists a
constant C > 0 so that ||V m || 2 2,n /||V 0 || 2 2,n ≤ C( s m+s ) ζ * (c)−δ for m large enough.
Comment 3.3 Our results stated in Theorem 1 depend on the lower bound of |R(m)|/m, which is the proportion of the Rs in the first m terms in the sequence A. We conjecture that the convergence rate of PGA is close to exponential as c → 0. Denote the actual proportion
of R in the sequence A by ψ(c), i.e., |R(m)| ≥ ψ(c)m−ψ 1 (c)q(0), where ψ(c), ψ 1 (c) are some constants depending on c. If ψ(c) → 1, it is easy to show that ||V m || 2 2,n ||V 0 || 2 2,n s s+m ζ ,
based on the proof of Theorem 1, for any arbitrarily large ζ. In general, further improvements of the convergence convergence rate of PGA can be achieved by improving the lower bounds of |R(m)|/m. Table 1 gives different values of the SE constant c for the corresponding values of ζ * . The convergence rate of PGA and hence of L 2 Boosting is affected by the frequency of revisiting. Because different values of c impose different lower bounds on the frequencies of revisiting, thus different values of c imply a different convergence rate of the process in our framework.
Main Results
In this section we discuss the main results regarding the L 2 Boosting procedure (BA), post-L 2 Boosting (post-BA) and the orthogonal procedure (oBA) in a high-dimensional setting.
We analyze the linear regression model introduced in a high-dimensional setting, which was introduced in Section 2.
L 2 Boosting with Componentwise Least Squares
First, we analyze the classical L 2 Boosting algorithm with componentwise least squares. For this purpose, the approximation results which we derived in the previous section are key. While in the previous section the stochastic component was absent, in this section it is explicitly considered.
The following definitions will be helpful for the analysis: U m denotes the residuals at the m th iteration, U m = Y − Xβ m . β m is the estimator at the m th iteration. We define the difference between the true and the estimated vector as α m := β − β m . The prediction error is given by V m = Xα m .
For the Boosting algorithm in the high-dimensional setting it is essential to determine when to stop, i.e. the stopping criterion. In the low-dimensional case, stopping time is not important: the value of the objective function decreases and converges to the traditional OLS solution exponentially fast, as described in Bühlmann and Yu (2006). In the highdimensional case, such fast convergence rates are usually not available: the residual ε can be explained by n linearly independent variables x j . Thus selecting more terms only leads to overfitting. Early stopping is comparable to the penalization in LASSO, which prevents one from choosing too many variables and hence overfitting. Similarly to LASSO, a sparse structure will be needed for analysis.
At each step, we minimize ||U m || 2 2,n along the "most greedy"variable X j m . The next assumption is on the residual / stochastic error term ε and encompasses many statistical models which are common in applied work.
A.3 With probability greater than or equal 1 − α, we have, sup 1≤j≤p | < X j , ε > n | ≤ 2σ log(2p/α) n := λ n .
Comment 4.1 The previous assumption is, e.g., implied if the error terms are i.i.d. N (0, σ 2 ) random variables. This in turn can be generalized / weakened to cases of non-normality by self-normalized random vector theory (de la Peña et al. (2009)) or the approach introduced in Chernozhukov et al. (2014).
Set σ 2 n := E n [ε 2 ]. Recall that ||U m+1 || 2 2,n = ||U m || 2 2,n −(γ m j ) 2 , where |γ m j | = max 1≤j≤p | < X j , U m > n | = max 1≤j≤p | < X j , V m > n + < X j , ε > n |.
The lemma below establishes the main result of convergence rate of L 2 Boosting.
Lemma 3 Suppose assumptions A.1-A.3 hold and s log(p)
n → 0. Assume M 0 is large enough so that log(M 0 /s) + (ξ + 1 1+ζ * (c) ) log( s log(p) n||V 0 || 2 2,n ) > 0 for some ξ > 0. Write m * + 1 for the first time ||V m || 2,n ≤ η √ m + sλ n , where η is a constant large enough. Then, for any δ > 0, with probability ≥ 1 − α, (1) it holds m * s s log(p) n||V 0 || 2 2,n −1 1+ζ * (c)−δ and m * < M 0 ;(5)
(2) the prediction error ||V m * +1 || satisfies:
||V m * +1 || 2 2,n p ||V 0 || 2 1+ζ * (c)−δ 2,n s log(p) n ζ * (c)−δ 1+ζ * (c)−δ .(6)
Comment 4.2 Lemma 3 shows that the convergence rate of the L 2 Boosting depends on the value of c. For different values of c, the lower bound of the proportion of revisiting ("R") in the sequence A should be different. Such lower bounds on the frequency of revisiting will naturally determine the upper bound for the deterministic component, which affects our results on the rate of convergence of L 2 Boosting. As ζ * (c) → ∞, the statement (2) implies the usual LASSO rate of convergence.
The bound of the approximation error ||V m || 2 2,n stated in inequality (6) is obtained under an infeasible stopping criteria. Below we establish another result which employs the same convergence rate but with a feasible stopping criterion which can be implemented in empirical studies.
Theorem 2 Suppose all conditions stated in Lemma 3 hold. Let c u > 4 be a constant. Let
m * 1 + 1 be the first time such that ||U m || 2 2,n ||U m−1 || 2 2,n > 1 − c u log(p)/n. Then, with probability at least 1 − α, ||V m * 1 || 2 2,n ||V 0 || 2 1+ζ * (c)−δ 2,n ( s log(p) n ) ζ * (c)−δ 1+ζ * (c)−δ .
Comment 4.3 As we have already seen in the deterministic case, the rate of convergence depends on the constant c. In Table 2 we give for different values of c the corresponding rates setting δ equal to zero, so that the rates can be interpreted as upper bounds. Comment 4.4 It is also important to have an estimator for the variance of the error term σ 2 , denoted byσ 2 n,m . A consistent estimation of the variance is given by ||U m || 2 2,n at the stopping time m = m * .
Orthogonal L 2 Boosting in a high-dimensional setting with bounded restricted eigenvalue assumptions
In this section we analyze orthogonal L 2 Boosting. For the orthogonal case, we obtain a faster rate of convergence than with the variant analyzed in the section before. We make use of similar notation as in the previous subsection: U m o denotes the residual and V m o the prediction error, formally defined below. Again, define β m o as the parameter estimate after the m th iteration.
The orthogonal Boosting Algorithm was introduced in the previous section. For completeness we give here the full version with some additional notation which will be required in a later analysis.
Algorithm 3 (Orthogonal L 2 Boosting) 1. Initialization: Set β 0 o = 0, f 0 o = 0, U 0 o = Y and the iteration index m = 0. 2. Define X m
o as the matrix of all X j k , k = 0, 1, . . . , m and P m o as the projection matrix given X m o .
3. Let j m be the maximizer of the following:
max 1≤j≤p ρ 2 (X j , U m o ). Then, f m+1 o = P m o Y with corresponding regression projection coefficient β m+1 o . 4. Calculate the residual U m o = Y − Xβ m o = (I − P m o )Y := M P m o Y and V m o = M P m o Xβ.
5. Increase m by one. If some stopping criterion is reached, then stop; else continue with
Step 2.
It is easy to see that:
||U m+1 o || 2,n ≤ ||U m o || 2,n(7)
The benefit of the oBA method, compared to L 2 Boosting, is that once a variable X j is selected, the procedure will never select this variable again. This means that every variable is selected at most once.
For any square matrix W , we denote by φ s (W ) and φ l (W ) the smallest and largest eigenvalues of W .
Denote by T m the set of variables selected at the m th iteration. Write S m :
= T − T m . We know that |T m | = m by construction. Set S m c = T m − S m .
Lemma 4 (lower bound of the remainder) Suppose assumptions A.1-A.3 hold. For any m such that |T m | < sη, ||V m o || 2,n ≥ c 1 ||Xβ S m || 2,n , for some constant c 1 > 0. If S m = ∅, then ||V m o || 2,n = 0.
The above lemma essentially says that if S m is non-empty, then there is still room for significant improvement in the value ||V m || 2 2,n . The next lemma is key and shows how rapidly the estimates decay. It is obvious that ||U m o || 2,n and ||V m o || 2,n are both decaying sequences. Before we state this lemma, we introduce an additional assumption.
A.4 min j∈T |β j | ≥ J and max j∈T |β j | ≤ J for some constants J > 0 and J < ∞.
Comment 4.5 The first part of assumption A.4 is known as a "beta-min"assumption as it restricts the size of the non-zero coefficients. It can be relaxed so that the coefficients β j are a decreasing sequence in absolute value.
Lemma 5 (upper bound of the remainder) Suppose assumptions A.1-A.4 hold. Assume that √ sλ n → 0. Let m * be the first time that ||U m * || 2 2,n < σ 2 + 2Kσsλ 2 n . Then, m * < Ks and ||V m o || 2 2,n s log(p)/n with probability going to 1.
Although in general L 2 Boosting may have a slower convergence rate than LASSO, oBA reaches the rate of LASSO (under some additional conditions). The same technique used in Lemma 6 also holds for the post-L 2 Boosting case. Basically, we can prove, using similar arguments, that T Ks ⊃ T when K is a large enough constant. Thus, post-L 2 Boosting enjoys the LASSO convergence rate under assumptions A.1-A.4. We state this in the next section formally.
Post-L 2 Boosting with Componentwise Least Squares
In many cases, penalization estimators like LASSO introduce some bias by shrinkage. The idea of "post-estimators" is to estimate the final model by ordinary least squares including all the variables which were selected in the first step. We introduced post-L 2 Boosting in Section 2. Now we give the convergence rate for this procedure. Surprisingly, it improves upon the rate of convergence of L 2 Boosting and reaches the rate of LASSO (under stronger assumptions). The proof of the result follows the idea of the proof for Lemma 5.
Lemma 6 (Post-L 2 Boosting) Suppose assumptions A.1-A.4 hold. Assume that √ sλ n → 0. Let m * = Ks be the stopping time with K a large enough constant. Let S m = T m − T 0 be the set of variables selected at steps 1, 2, ...., m. Then, T 0 ⊂ S m * with probability going to 1, i.e., all variables in T 0 have been revisited, and ||P X T m * Y − Xβ|| 2 2,n ≤ CKsλ 2 n s log(p)/n. Comment 4.6 Our procedure is particularly sensitive to the starting value of the algorithm, at least in the non-asymptotic case. This might be exploited for screening and model selection: the procedure commences from different starting values until stopping. Then the intersection of all selected variables for each run is taken. This procedure might establish a sure screening property.
Simulation Study
In this section we present the results of our simulation study. The goal of this exercise is to illustrate the relevance of our theoretical results for providing insights into the functionality of boosting and the practical aspects of boosting. In particular, we demonstrate that the stopping rules for early stopping we propose work reasonably well in the simulations and give guidance for practical applications. Moreover, the comparison with LASSO might also be of interest. First, we start with an illustrative example, later we present further results, in particular for different designs and settings.
Illustrative Example
The goal of this section is to give an illustration of the different stopping criteria. We employ the following data generating process (dgp): 4 y = 5x 1 + 2x 2 + 1x 3 + 0x 4 + . . . + 0x 10 + ε,
with ε ∼ N (0, 2 2 ), X = (X 1 , . . . , X 10 ) ∼ N 1 0(0, I 10 ), I 10 denoting the identity matrix of size 10. To evaluate the methods and in particular the stopping criteria we conduct an analysis of both in-sample and out-of-sample mean squared error (MSE). For the outof-sample analysis we draw a new observation for evaluation and calculation of the MSE. For the in-sample analysis we also repeat the procedure and form the average over all repetitions. In both cases we employ 60 repetitions. The sample size is n = 20. Hence we have 20 observations to estimate 10 parameters.
The results are presented in Figures 5.1 and 5.2. Both show how MSE depends on the number of steps of the boosting algorithm. We see that MSE first decreases with more steps, reaches its minimum and then starts to increase again due to overfitting. In both graphs the solution of the L 2 Boosting algorithm convergences to the OLS solution. We also indicate the MSE of LASSO estimators as horizontal lines (with cross-validated choice of the penalty parameter and data-driven choice of the penalization parameter.) In order to find a feasible stopping criterion we have to rely on the in-sample analysis. Figure 5.1 reveals that the stopping criterion we introduced in the sections before performs very well and even better than stopping based on a corrected AIC values which has been proposed in the literature as stopping criterion for boosting. The average stopping steps of our criterion and the corrected AIC-based criterion (AICc) are presented by the vertical lines. On average our criterion stops earlier than the AICc based one. As our criterion performs better then the AICc we will not report AICc results in the following subsection. For the post-estimator similar patterns arise and are omitted.
Further Results
In this section we present results for different designs and settings so give a more detailed comparison of the methods.
We consider the linear model
y = p j=1 β j x j + ε,(9)
with ε standard normal distributed and i.i.d. For the coefficient vector β we consider two designs. First, we consider a sparse design, i.e., the first s elements of β are set equal to one, all other components to zero (β = (1, . . . , 1, 0, . . . , 0)). Then we consider a polynomial design in which the jth coefficient given by 1/j, i.e. β = (1, 1/2, 1/3, . . . , 1/p). For the design matrix X we consider two different settings: an "orthogonal" setting and a "correlated" setting. In the former setting the entries of X are drawn as i.i.d. draws from a standard normal distribution. In the correlated design, the x i (rows of X) are distributed according to a multivariate normal distribution where the correlations are given by a Toeplitz matrix with factor 0.5 and alternating signs. We have the following settings:
• X: "orthogonal" or "correlated"
• coefficient vector β: sparse design or polynomial decaying design
• n = 100, 200, 400
• p = 100, 200
• s = 10
• K = 2
• out-of-sample prediction size n 1 = 50
• number of repetitions R = 500
We consider the following estimators: L 2 Boosting with componentwise least squares, orthogonal L 2 Boosting and LASSO. For Boosting and LASSO, we also consider the postselection estimators ("post"). For LASSO we consider a data-driven regressor-dependent choice for the penalization parameter (Belloni et al. (2012)) and cross validation. Although cross validation is very popular, it does not rely on established theoretical results and therefore we prefer a comparison with the formal penalty choice developed in Belloni et al. (2012). For Boosting we consider three stopping rules: "oracle", "Ks", and a "datadependent"stopping criterion which stops if The Ks-rule stops after K × s variables have been selected where K is a constant. As s is unknown, the rule is not directly applicable. The oracle rule stops when the meansquared-error (MSE), defined below, is minimized, which is also not feasible in practical applications.
The simulations were performed in R (R Core Team (2014)). For LASSO estimation the packages Chernozhukov et al. (2015) and Jerome Friedman (2010) (for cross validation) were used. The Boosting procedures were implemented by the authors and the code is available upon request.
To evaluate the performance of the estimators we use the MSE criterion. We estimate the models on the same data sets and use the estimators to predict 50 observations out-ofsample. The (out-of-sample) MSE is defined as
M SE = E[(f (X) − f m (X)) 2 ] = E[(X (β − β m )) 2 ],(10)
where m denotes the iteration at which we stop, depending on the employed stopping rule. The MSE is estimated bŷ
M SE = 1/n 1 n 1 i [(f (x i ) − f m (x i )) 2 ] = 1/n 1 n 1 i [(x i (β − β m )) 2 ](11)
for the out-of-sample predictions.
The results of the simulation study are shown in Tables 3 -10. As expected, the oracle-based estimator dominates in all cases except in the correlated, sparse setting with more parameters than observations. Our stopping criterion gives very good results, on par with the infeasible Ks-rule. Not surprisingly, given our results, both post-Boosting and orthogonal Boosting outperform the standard L 2 Boosting in most cases. A comparison of post-and orthogonal Boosting does not provide a clear answer with advantages on both sides. It is interesting to see that the post-LASSO increases upon LASSO, but there are some exceptions, probably driven by overfitting. Cross validation works very well in many constellations. An important point of the simulation study is to compare Boosting and LASSO. It seems that in the polynomial decaying setting, Boosting (orthogonal Boosting with our stopping rule) dominates post-LASSO. This also seems true in the iid, sparse setting. In the correlated, sparse setting they are on par. Summing up, it seems that Boosting is a serious contender for LASSO.
Comment 5.1 It seems that in very high-dimensional settings, i.e. when the number of signals s is bigger than the sample size n, (e.g. n = 20, p = 50, s = 30) boosting performs quite well and outperforms LASSO which seems to break down. This case is not covered by our setting, but it is an interesting topic for future research and shows one of the advantages of boosting.
Applications
In this section we present applications from different fields to illustrate the boosting algorithms. We present applications to demonstrate how the methods work when applied to real data sets and, then compare these methods to related methods, i.e. LASSO. The focus is on making predictions which is an important task in many instances.
Application: Riboflavin production
Thos application involves genetic data and analyzes the production of riboflavin. First, we describe the data set, then we present the results.
Data set
The data set has been provided by DSM (Kaiserburg, Switzerland) and was made publicly available for academic research in Bühlmann et al. (2014) (Supplemental Material). The real-valued response / dependent variable is the logarithm of the riboflavin production rate. The (co-)variables measure the logarithm of the expression level of 4, 088 genes (p = 4, 088), which are normalized. This means that the covariables are standardized to have variance 1, and the dependent variable and the resources are "de-meaned",which is equivalent to including an unpenalized intercept. The data set consists of n = 71 observations which were hybridized repeatedly during a fed-batch fermentation process in which different engineered strains and strains grown under different fermentation conditions were analyzed. For further details we refer to Bühlmann et al. (2014), their Supplemental Material, and the references therein.
Results
We analyzed a data set about the production of riboflavin (vitamin B 2 ) with B. subtilis. We split the data set randomly into two samples: a training set and a testing set. We estimated models with different methods on the training set and then used the testing set to calculate out-of-sample mean squared errors (MSE) in order to evaluate the predictive accuracy. The size of the training set was 60 and the remaining 11 observations were used (2014)) with the package Chernozhukov et al. (2015) and our own code. Replication files are available upon request.
The results show that, again, post-and orthogonal L 2 Boosting give comparable results. They both outperform LASSO and post-LASSO in this application.
Predicting Test Scores
Data Set
Here the task is to predict the final score in the subjects Mathematics and Portugese in secondary education. This is relevant, e.g., to identify students which need additional support to master the material. The data contains both student grades and demographic, social and school related features and it was collected by using school reports and questionnaires. Two datasets are provided regarding the performance in two distinct subjects: Mathematics and Portuguese. The data set is made available at the UCI Machine Learning Repository and was contributed by Paulo Cortez. The main reference for the data set is Cortez and Silva (2008).
Results
We employed five-fold CV to evaluate the predictive performance of the data set. The results remain stable when choosing a different number of folds. The data sets contain, for both test results, 33 variables, which are used as predictors. The data set for the Mathematics test scores contains 395 observations, the sample size for Portuguese is 649. The results confirm our theoretical derivations that boosting is comparable to Lasso.
Conclusion
Although boosting algorithms are widely used in industry, the analysis of their properties in high-dimensional settings has been quite challenging. In this paper the rate of convergence for the L 2 Boosting algorithm and variants are derived, which has been a long-standing open problem until now.
Appendix A. A new approximation theory for PGA
A.1 New results on approximation theory of PGA
In this section of the appendix we introduce preparatory results for a new approximation theory based on revisiting. These results are useful to prove Lemma 1. The proofs of these lemmas are provided in the next section. For any m 1 ≥ m, define L(m, m 1 ) = ||V m 1 || 2 2,n /||V m || 2 2,n ≤ 1. For any integers q 1 > q, define ∆(q, q 1 ) := Π q 1 −q−1
j=0
(1 − 1−c q+j ) with some constant c. It is easy to see that for any k 1 > k, ∆(k, k 1 )/(k/k 1 ) 1−c > 1 and ∆(k, k 1 )/(k/k 1 ) 1−c → 1 as k → ∞.
First of all, we can establish the following naive bounds on L(m, m 1 ).
Lemma 7 Suppose ||V m || > 0, m + 1, m 1 < M 0 (a) For any m, L(m, m + 1)
≤ 1 − 1−c q(m) . (b) For any m ≥ 0, m 1 > m, L(m, m 1 ) ≤ ∆(q(m), q(m) + m 1 − m).
The bound of L(m, m 1 ) established in Lemma 7 is loose. To obtain better results on the convergence rate of ||V m || 2 , the revisiting behavior of the PGA has to be analyzed in more detail. The revisiting behavior of PGA addresses the question when and how often variables are selected again which have been already selected before. When PGA chooses too many new variables, it leads on average to slower convergence rates and vice versa. The next results primarily focus on analyzing the revisiting behavior of the PGA.
The following lemma summarizes a few basic facts of the sequence of A i , i ≥ 1.
Lemma 8 Suppose m, m 1 < M 0 . Suppose further that conditions A.1 and A.2 are satisfied.
(1) If E n [X i X i ] is diagonal matrix, i.e., c = 0, then there are only Rs in the sequence A.
(2) Define N (m) := {k|A k = N, 1 ≤ k ≤ m}, the index set for the non-revisiting steps, and R(m) := {k|A k = R, 1 ≤ k ≤ m}, the index set for the revisiting steps. Then |R(m)| + |N (m)| = m, q(m) = |N (m)| + q(0), and J N (m) := {j k |k ∈ N (m)} has cardinality equal to |N (m)|.
(
3) L(0, m) ≤ Π |N (m)| i=1 (1 − 1−c q(0)+i−1 ) × (1 − 1−c q(m) ) |R(m)| , i.e.
, the sequence to maximize the upper bound of L(0, m) stated above is N N...N RR...R. Consequently, the sequence {A m+1 , ..., A m 1 } to maximize the upper bound of L(m, m 1 ) for general m 1 > m is also N N...N RR...R.
The proof of this lemma is obvious and hence omitted. Much more involved are the following results, for characterizing the revisiting behavior.
Lemma 9 Suppose conditions A.1 and A.2 are satisfied. Suppose further that there is a consecutive subsequence of N s in the sequence A starting at position m with length k. Assume that m + k < M 0 .Then for any δ > 0, there exists an absolute constant Q(δ) > 0 such that for any q(m) > Q(δ), the length of such a sequence cannot be longer than
((1 + δ) (2−c)(1+c) (2+c)(1−c) ) 1 1−c − 1 q(m).
Lemma 9 establishes some properties of the N sequence. Next, we formulate a lemma which characterizes a lower bound of the proportions of Rs.
Lemma 10 Assume that assumptions A.1-A.2 hold. Assume that m < M 0 . Consider the sequence of steps 1, 2, ..., m. Set µ e (c) = (1 − exp(−1/(1 − c) 2 )) for any c ∈ (0, 1). Then, the number of Rs in the sequence A satisfies:
|R(m)| ≥ 1 − µ e (c) 2 − µ e (c) m − µ e (c) 2 − µ e (c)
q(0).
Lemma 10 illustrates that for any 1 > c > 0, the R spots occupy at least some significant proportion of the sequence A, with the lower bound of the proportion depending on c. In fact, such a result holds for arbitrary consecutive sequence A m , A m+1 , ..., A m+k , as long as m + k < M 0 . In the main text, we further extend results stated in Lemma 10.
A.2 Proofs of Lemmas in Appendix
A.1 Proof [Proof of Lemma 7] By definition, ||V m || 2 = j∈T m α m j < V m , X j >= j∈T m α m j ||V m ||corr(V m , X j ). De- fine ρ j m := |γ m j m |/||V m || = |corr(V m , X j m )|. Therefore, ρ j m | j∈T m α m j | ≥ ||V m ||, i.e., ρ 2 j m | j∈T m α m j | 2 ≥ ||V m || 2 . By the Cauchy-Schwarz inequality, | j∈T m α m j | 2 ≤ q(m)||α m || 2 . Therefore, ρ 2 j m ≥ 1−c q(m) . So ||V m+1 || 2 2,n = ||V m || 2 2,n (1 − ρ 2 j m ) ≥ ||V m || 2 (1 − 1−c q(m) ), i.e., L(m, m + 1) ≤ 1 − 1−c q(m)
. The second statement follows from statement (a) and the fact that q(m + 1) ≤ q(m ) + 1 for any m ≥ 0.
Proof [Proof of Lemma 9]
Denote the length of such a sequence of N s as k. Then by definition of non-revisiting, ||α m+k || 2 = ||α m || 2 + m+k−1 j=m (γ j ) 2 .
We know that ||V m || 2 2,n = ||V m+k || 2 2,n + m+k−1 j=m (γ j ) 2 , therefore, ||V m || 2 2,n = ||V m+k || 2 2,n + ||α m+k || 2 − ||α m || 2 .
Applying Assumption A.2, (2+c)/(1+c)||V m || 2 2,n ≤ (2−c)/(1−c)||V m+k || 2 2,n . Consequently, L(m, m + k) ≥ (2+c)(1−c)
(2−c)(1+c) , with the right-hand side of the inequality being a constant that only depends on c. When c = 0, the constant equals 1, which implies that k has to be 0, i.e., there are no N s in the sequence A.
For any δ > 0, there exists Q > 0 such that for any q(m) > Q,
L(m, m + k) ≤ ∆(q(m), q(m) + k) ≤ (1 + δ)( q(m) q(m)+k ) 1−c . It follows that ( q(m) q(m)+k ) 1−c ≥ 1 1+δ (2+c)(1−c) (2−c)(1+c) , i.e., k ≤ ((1 + δ) (2−c)(1+c) (2+c)(1−c) ) 1 1−c − 1 q(m).
Proof [Proof of Lemma 10] Define N (m) as:
{l : j l / ∈ T 0 , j l is only visited once within steps 1,2,...,m}.
So it is easy to see that N (m) ⊂ N (m) and | N (m)| ≥ 2|N (m)| − m. Therefore, for any j l with l ∈ N (m), α m j l = −γ l . If |R(m)| ≥ m/2, then we already have the results stated in this lemma. Otherwise, N (m) is non-empty. Therefore, ||α m || 2 ≥ l∈ N (m) (γ l−1 ) 2 . By the sparse eigenvalue condition A.2,
1 1 − c ||V m || 2 2,n ≥ ||α m || 2 ≥ l∈ N (m) (γ l−1 ) 2 .(12)
Note that by Lemma 9, (γ l−1 ) 2 = ||V l−1 || 2 2,n − ||V l || ≥ 1−c q(l−1) ||V l−1 || 2 . Therefore, (γ l−1 ) 2 ≥ 1−c q(l−1) ||V m || 2 2,n . Plugging the above inequality back into (12), we get:
1 1 − c ||V m || 2 2,n ≥ (1 − c) l∈ N (m) 1 q(l − 1) ||V m || 2 .
Since these q(l − 1), l ∈ N (m), are different integers with their maximum being less than
or equal to q(m) = q(0) + |N (m)|. Therefore, l∈ N (m) 1 q(l−1) ≥ | N (m)| l=1 1 q(0)+|N (m)|−l ≥ log((q(0) + |N (m)|)/(q(0) + |N (m)| − | N (m)|)).
The above inequality implies that exp (
Appendix B. Proofs for Section 3
Proof [Proof of Lemma 1] First of all, WLOG, we can assume that q(0) exceeds a large enough constant Q(δ). Otherwise, we can consider the true parameter β contains some infinitesimal components such that q(0) > Q(δ).
Let's revisit inequality (12). l∈ N (m) (γ l−1 ) 2 ≥ l∈ N (m) ||V l−1 || 2 1−c q(l−1) . The right-hand side reaches its
q(l−1) ≥ ||V m || 2 2,n (1−c) | N (m)| l=1 1 q(m)−l ×( q(m)−1 q(m)−l−1 ) 1−c ≥ 1 1−c q(m) 1−c ((q(m)−| N (m)|) c−1 −q(m) c−1 )
. Combining the above inequality with (12), we get: 1+δ
1−c ≥ (1 − c)q(m) 1−c ((q(m) − | N (m)|) c−1 − q(m) c−1 ), i.e, | N (m)| ≤ q(m)[1 − (1 + 1+δ (1−c) 2 ) −1 1−c ] ≤ (1 + δ )µ a (c)q(m)
, for some δ > 0, with δ → 0 as δ → 0. The rest of the arguments follow the proof stated for Lemma 10.
Hence, the results stated in Lemma 1 hold.
Proof [Proof of Lemma 2]
Without loss of generality, we can assume that k = 0. We can also assume that ||V 0 || 2 2,n > 0, because otherwise ||V 0 || 2 2,n = ||V m || 2 2,n = 0 so that the conclusion already holds. Set n 0 = |N (m)| ≤ (1 + δ) m+µa(c)q(0)
2−µa(c)
for some δ > 0, when q(0) is large enough.
Then, it easy to see that ||V m || 2 2,n /||V 0 || 2 2,n ≤ Π n 0 i=1 (1 − 1−c q(0)+i−1 )(1 − 1−c q(0)+n 0 ) (m−n 0 ) , and the right hand reaches its maximum, when n 0 = (1 + δ)n * 0 with n * 0 := m+µa(c)q(0) 2−µa(c) . When q(0) is large enough, we know that there exists a δ > 0 such that
Π n * 0 i=1 (1 − 1 − c q(0) + i − 1 ) ≤ (1 + δ)( q(0) q(0) + n * 0 ) 1−c = (1 + δ)( 2 − µ a (c) 2 + λ ) 1−c and (1 − 1 − c q(0) + n * 0 ) m−n * 0 ≤ (1 + δ)(1 − 1 − c q(0) λ+2 2−µa(c) ) q(0) (1−µa(c))λ−µa(c) 2−µa(c) ≤ (1 + δ) exp(− (1 − c)((1 − µ a (c))λ − µ a (c)) 2 + λ ).
Thus, for any δ > 0, and for q(0) large enough,
||V m || 2 2,n /||V 0 || 2 2,n ≤ (1 + δ)( 2 − µ a (c) 2 + λ ) 1−c exp(− (1 − c)((1 − µ a (c))λ − µ a (c)) 2 + λ ).
Notice that the bound on the right-hand side does not depend on q(0) or m.
As defined in the statement of this lemma, ζ(c, λ) =
(1−c)((1−µa(c))λ−µa(c)) 2+λ log( 2+λ 2−µa(c) ) So for any δ > 0, and for q(0) large enough, ||V m || 2 2,n ≤ ||V 0 || 2 2,n (
+ 1 − c, whereq(0) q(0) + n * 0 ) ζ(c,λ)−δ . Proof [Proof of theorem 1]
If q(0) < Q(δ) where Q(δ) is defined in Lemma 1, we can treat β as if there are additional infitestimony coefficients so that q(0) = Q(δ).
Let λ * be the maximizer of ζ(c, λ) given c ∈ (0, 1). For any small δ > 0, define a sequence m 0 , m 1 , . . . according to the following rule:
m 0 = s, m i+1 = m i + λ * n i , i = 1, 2, .
. . , with the sequence n 0 , n 1 , . . . being defined as:
n i+1 = n i + (1 + δ) 1 2 − µ a (c) (m i+1 − m i + µ a (c)n i ) ,
with n 0 = s. It is easy to see that: By Lemma 1, (1). 1 < c λ * < m i+1 /m i ≤ C λ * , for some constant c λ * , C λ * that only depends on λ * and i ≥ I(δ), where I(δ) is a fixed real number depending on δ.
(2). c n ≤ n i /m i ≤ C n , for i ≥ I(δ), with c n , C n being generic constants.
(3). n i ≥ q(m i ), for i ≥ I(δ). And by Lemma 2,
.
||V m i+1 || 2 2,n /||V m i || 2 2,n ≤ ( q(m i ) q(m i ) + 1 2−µa(c) (m i+1 − m i + µ a (c)q(m i )) ) ζ * (c)−δ ≤ ( n i n i + 1 2−µa(c) (m i+1 − m i + µ a (c)n i ) ) ζ * (c)−δ ≤ ( n i n i+1 ) ζ * (c)−δ ,
for all i ≥ I(δ). So, according to statements 1-4, we are able to conclude that:
||V m i || 2 2,n C||V 0 || 2 2,n ( s n i ) ζ * (c)−δ C||V 0 || 2 2,n ( s m i + s ) ζ * (c)−δ ,(13)
for all i ≥ I(δ), with C being a constant. For any m > 0, m < M 0 , since m 0 , m 1 , ... is an increasing sequence of positive integers, there exists i such that m i ≤ m < m i+1 . So m m i ≤ m i+1 m i ≤ C λ * . Also, for m large enough, i must be sufficiently large that i ≥ Q(δ). Therefore, ||V m || 2 2,n ≤ ||V m i || 2 2,n ||V 0 || 2 2,n ( s s+m i ) ζ * (c)−δ ||V 0 || 2 2,n ( s s+m ) ζ * (c)−δ .
Appendix C. Proofs for Section 4
The two lemmas below state several basic properties of the L 2 Boosting algorithm that will be useful in deriving the main results.
Lemma 11 ||U m+1 || 2 2,n = ||U m || 2 2,n − < U m , X jm > 2 n = ||U m || 2 2,n (1 − ρ 2 (U m , X jm )), and
||V m+1 || 2 2,n = ||V m || 2 2,n − 2 < V m , γ m jm X jm > n +(γ m jm ) 2 , where γ m jm =< U m , X jm > n . Moreover, since V m = U m − ε, ||V m+1 || 2 2,n = ||V m || 2 2,n − 2 < U m , X jm > n < ε, X jm > n − < U m , X jm > 2 n = ||V m || 2 2,n − 2γ m jm < ε, X jm > n −(γ m jm ) 2 .
Lemma 12 Assuming that assumptions A.1-A.3 hold, and m ≤ M 0 . Let Z m = ||U m || 2 2,n − ||V m || 2 2,n . Then, with probability ≥ 1−α and uniformly in m, |Z m −σ 2 n | ≤ 2 √ m+s √ 1−c λ n ||V m || 2,n .
Lemma 12 bounds the difference between ||U m || 2 2,n and ||V m || 2 2,n . This difference is
σ 2 (1 − O p (s/n)) if β m = β. Proof [Proof of Lemma 12] From Lemma 1, Z m+1 = Z m − 2γ m j m < ε, X j m > n , and Z m = Z 0 − 2 m−1 k=0 γ k j k < ε, X j k >= Z 0 − 2 < ε, Xβ − V m >. Z 0 = ||y|| 2
2,n − ||Xβ|| 2 2,n = ||ε|| 2 2,n + 2 < ε, Xβ >. Then,
Z m = ||ε|| 2 2,n + 2 < ε, V m > = ||ε|| 2 2,n + 2 < ε, V m ||V m || 2,n > ||V m || 2,n ≤ ||V m || 2,n λ n (||α m || 1 /||V m ||) ≤ ||V m || 2,n √ m + sλ n (||α m ||/||V m ||)
since |supp(V m )| ≤ m + s. By assumption A.2, ||α m ||/||V m || ≤ 1 1−c . Hence, the conclusion holds.
C.1 Proofs for L 2 Boosting
Proof [Proof of Lemma 3] We assume that λ n ≥ max 1≤j≤p | < , X j > n |. This event occurs with probability ≥ 1 − α.
According to our definition, m * + 1 is the first time ||V m || 2,n ≤ η √ m + sλ n , where η is a positive constant. We know that in high-dimensional settings, ||U m || 2,n → 0, so ||V m || 2,n → σ 2 . Thus, by fixing p and n, such an m * must exist.
First, we prove that for any m < m := (m * + 1) ∧ M 0 , we have: ||V m+1 || 2 2,n ≤ ||V m || 2 2,n , i.e., ||V m || 2 2,n is non-increasing with m. By Lemma 1, ||V m+1 || 2 2,n = ||V m || 2 2,n − γ m (γ m − 2 < ε, x j m > n ). To show that ||V m || 2 2,n is non-increasing with m, we only need to prove that γ m and (γ m − 2 < ε, x j m > n ) have the same sign, i.e., |γ m | > 2| < ε, x j > n |. It suffices to prove |γ m | > 2λ n . We know that |γ m | ≥ (1 − c) ||V m || 2,n √ m+s − λ n ≥ λ n (η √ 1 − c − 1). Thus, for any η > 3 √ 1−c , ||V m+1 || 2 2,n ≤ ||V m || 2 2,n for all m < m.
Define q(m) as in Section 3.1, with q(0) = s. For any m < M 0 ∧ (m * + 1), by selecting a variable that is the most correlated with V m , we are able to reduce ||V m || 2 2,n by at least 1−c q(m) ||V m || 2 2,n , and thus ||U m || 2 2,n − ||U m+1 || 2 2,n = ||V m || 2 2,n − ||V m+1 || 2 2,n − 2γ m < X j m , > n ≥ (γ m ) 2 − 2λ n |γ m |.
Define γ m := √ 1−c √ q(m)
||V m || 2,n − λ n . Consider the variable j that is most correlated with V m , and define γ =< X j , V m > n . By Lemma 1, |γ | ≥
√ 1−c √ q(m)
||V m || 2,n . Consequently,
| < X j , U m > n | = |γ + < X j , > n | ≥ γ m . By definition, |γ m | ≥ | < X j , U m > n | ≥ γ m .
Since we assume that ||V m || 2,n > η √ m + sλ n , so γ m :=
√ 1−c √ q(m) ||V m || 2,n − λ n > λ n .
Therefore, |γ m | > λ n , and (γ m ) 2 − 2λ n |γ m | ≥ ( γ m ) 2 − 2λ n γ m . By Lemma 11, ||V m || 2 2,n − ||V m+1 || 2 2,n = ||U m || 2 2,n − ||U m+1 || 2,n − 2γ
m < X j m , > n ≥ |γ m | 2 − 2λ n |γ m | ≥ | γ m | 2 − 2λ n γ m = 1−c q(m) ||V m || 2 2,n − 4 √ 1−c √ q(m) λ n ||V m || 2,n + 3λ 2 n ≥ 1−c q(m) ||V m || 2 2,n − 4 √ 1−c √ q(m) λ n ||V m || 2,n .
Thus,
||V m || 2 2,n − ||V m+1 || 2 2,n ≥ 1 − c q(m) ||V m || 2 2,n − 4 √ 1 − c q(m) λ n ||V m || 2,n .(14)
Plugging in ||V k || 2,n > η √ k + sλ n to inequality (14), for any k > 0, k < M 0 − 1, we obtain that ||V k+1 || 2 2,n ≤ (1 − 1−c q(k) )||V k || 2 2,n + 4
√ 1−c √ q(k) λ n ||V k || 2,n ≤ (1 − 1−c q(k) )||V k || 2 2,n + 4 √ 1−c ηq(k) ||V k || 2 2,n = 1−c−ψ q(k) ||V k || 2 2,n , where ψ = 4 √ 1−c η
can be an arbitrarily small constant when η is large enough.
Similar to the above inequality, recall the definition of N (m), R(m) and N (m). By the argument in Lemma 1, when n is large enough,
1 1−c ||V m || 2 2,n ≥ k∈ N (m) (γ k−1 ) 2 ≥ k∈ N (m) 1−c q(k−1) ||V k−1 || 2 2,n − 2 k∈ N (m) √ 1−c √ q(k−1) λ n ||V k−1 || 2,n ≥ k∈ N (m)
1−c−ψ q(k−1) ||V k−1 || 2 2,n . Thus, following the proof of Lemma 1, we can treat 1 − c − ψ as the constant 1 − c in Lemma 1, and we obtain:
||V m || 2 2,n ||V 0 || 2 2,n ( s m + s ) ζ * (c)−δ−ψ ,(15)
for some small δ > 0, and for all m < m. Define δ = δ + ψ.
On the other hand, ||V m || 2 2,n ≥ (η √ m + sλ n ) 2 for all m < m. Therefore, combining with (15), we get:
s log(p) n ||V 0 || 2 2,n ( s m − 1 + s ) ζ * (c)−δ +1 ,
or equivalently, m s( s log(p)
n||V 0 || 2 2,n ) − 1 1+ζ * (c)−δ .
By assumption, log(M 0 /s) + (ξ + 1 1+ζ * (c) ) log( s log(p) Proof [Proof of Theorem 2] At the (m * 1 + 1) th step, we have:
||U m * 1 +1 || 2 2,n > (1 − c u log(p)/n)||U m * 1 || 2 2,n .
It follows that (γ m * 1 ) 2 < c u log(p)/n||U m * 1 || 2 2,n , while (γ m ) 2 ≥ c u log(p)/n||U m || 2 2,n for all m < m * 1 . Consider the m * defined in Lemma 3 as a reference point. (a) Suppose m * 1 < m * : By the proof of Lemma 3, ||V m || 2 is decreasing when m ≤ m * 1 +1. By Lemma 12, ||U m * 1 || 2 2,n ≤ σ 2 n + 2
√ m * 1 +s √ 1−c λ n ||V m * 1 || 2,n . It follows that (γ m * 1 ) 2 < c u log(p)/n||U m * 1 || 2 2,n < c u λ n + 2c u log(p)/n m * 1 + s √ 1 − c λ n ||V m * 1 || 2,n .(16)
Now we would like to form a lower bound for (γ m * 1 ) 2 .
(γ m * 1 ) 2 = ||U m * 1 || 2 2,n − ||U m * 1 +1 || 2 2,n = ||V m * 1 || 2 2,n − ||V m * 1 +1 || 2 2,n − 2γ m * 1 < X j m * 1 , > n ≥ ||V m * 1 || 2 2,n −||V m * 1 +1 || 2 2,n −2λ n |γ m * 1 |. By inequality (14), ||V m * 1 || 2 2,n −||V m * 1 +1 || 2 2,n ≥ 1−c q(m) ||V m * 1 || 2 2,n − 4 √ 1−c √ q(m) λ n ||V m * 1 || 2,n . So, (γ m * 1 ) 2 ≥ 1−c q(m) ||V m * 1 || 2 2,n − 2λ n |γ m * 1 | − 4 √ 1−c √ q(m) λ n ||V m * 1 || 2,n . Consequently, (γ m * 1 ) 2 ≥ √ 1 − c q(m * 1 ) ||V m * 1 || 2,n − 4λ n(17)
Plugging inequality (17) in inequality (16), it is easy to see that ||V m * 1 || 2 2,n ≤ K(m * 1 + s)λ 2 n (m * ∧ s)λ 2 n for some K > 0. (b) Suppose m * 1 ≥ m * : it follows that (γ m j m ) 2 ≥ c u log(p)/n||U m || 2 2,n for all m < m * 1 . Since ||U m || 2 2,n is a decreasing sequence, for δ small enough, there exists some m 2 such that ||U m 2 || 2 2,n > (1 − δ)σ 2 n for any m ≤ m 2 , and ||U m 2 +1 || 2 2,n ≤ (1 − δ)σ 2 n . For δ small enough and m ≤ m 2 ∧ m * 1 , ||V m+1 || 2 2,n − ||V m || 2 2,n = −(γ m ) 2 − 2γ m < X j m , > n ≤ −(γ m ) 2 +2λ n |γ m |. Since c u > 4, so for δ small enough, |γ m | 2 ≥ c u log(p)/n||U m || 2 2,n ≥ c u (1 − δ)λ 2 n ≥ 4λ 2 n , so −(γ m ) 2 + 2λ n |γ m | < 0. Case (b.2): Suppose m * 1 > m 2 : We show that this leads to a contradiction. First of all, we claim that m 2 ≥ m * + 1. We prove this contradiction: We know that ||U m || 2 2,n = σ 2 n + ||V m || 2 2,n + 2 < V m , > n . Since ||U m 2 +1 || 2 2,n ≤ (1 − δ)σ 2 n , 2 < V m 2 , > n ≤ ||V m 2 || 2 2,n + 2 < V m 2 , > n ≤ −δσ 2 n . So |2 < V m 2 , > n | ≥ δσ 2 n .
Suppose m 2 ≤ m * , it follows that m 2 ≤ m * < M 0 . Since we know that ||V m 2 || 2 2,n is decreasing for all m ≤ m * , we have ||V m 2 || 2 2,n ≥ ||V m * || 2 2,n ≥ η(m * + s)λ 2 n . Equivalently, ||V m * || 2,n ≥ η √ m * + sλ n . Therefore, ||V m 2 || 2 2,n + 2 < V m 2 , > n ≥ ||V m 2 || 2,n (||V m 2 || 2,n − 2 √ m 2 +s √ 1−c λ n ) > 0, which is a contradiction to ||V m 2 || 2 2,n + 2 < V m 2 , > n ≤ −δσ 2 n . So it must hold that m 2 ≥ m * +1. Therefore, ||V m || 2 2,n ≤ ||V m * +1 || 2 2,n ≤ c u (m * +s+1)λ 2 n for any m * + 1 ≤ m ≤ m 2 . We also know that by assumption, (γ m ) 2 ≥ c u (1 − δ)λ 2 n , for any m ≤ m 2 < m * 1 . Since ||V m || 2 2,n − ||V m+1 || 2 2,n = (γ m ) 2 − 2γ m < X j m , > n ≥ c u 1 λ 2 n > 0, for some constant c u 1 > 0 if (1−δ)c u > 2, it follows that ||V m || 2 2,n ≥ ||V m+1 || 2 2,n for m = m * , m * +1, ...., m 2 −1. Consequently, ||V m 2 || 2 2,n ≤ ||V m * || 2 2,n . By assumption, at the (m 2 + 1) th step, we know that ||U m 2 +1 || 2 2,n ≤ (1 − δ)σ 2 n . It follows that:
| < V m 2 , > n | ≥ δ σ 2 n ,
for some positive constant δ > 0. However, ||V m 2 || 2 2,n ≤ ||V m * || 2 2,n , so | < V m 2 , > n | ≤ ||V m 2 || 2,n σ n ≤ ||V m * || 2,n σ n → 0, which contradicts (18).
By collecting all the results in (a), (b).1, (b).2, our conclusion holds.
C.2 Proofs for oBA
Proof [Proof of Lemma 4] By the sparse eigenvalue condition:
||V m o || 2 2,n ≥ (1 − c)||β S m || 2 2 ≥ 1 − c C ||Xβ S m || 2 2,n .
Similarly, for U m o , ||U m o || 2 2,n = ||M P m o ε + M P m o (Xβ)|| 2 2,n = ||M P m ε|| 2 2,n + ||V m o || 2 2,n + 2 < ε, V m o > n ≥ n−m n σ 2 + ||V m o || 2 2,n − 2λ n √ m + s||V m o || 2,n .
Proof [Proof of Lemma 5] At step Ks, if T Ks ⊃ T , then ||V m o || 2 2,n = ||M P m o V m o || 2 2,n = 0. The estimated predictor satisfies: ||xβ − xβ m || 2,n ≤ 2(1 + η)σ s log(p) n with probability going to 1, where η > 0 is a constant.
Let A 1 = {T Ks ⊃ T }. Consider the event A c 1 . Then there exists a j ∈ T which is never picked up in the process at k = 0, 1, . . . , Ks.
At every step we pick a j to maximize | < X j , U m o > 2,n | = | < X j , V m o > 2,n + < X j , ε > 2 |.
Let W m = Xβ S m and W m = Xα m S m c . Then ||V m o || 2 2,n ≥ (1 − c)(||β S m || 2 2 + ||α S m c || 2 2 ) ≥ (1 − c) j∈S m |β S m | 2 .
Also V m o = M P m o Xβ S m , so < V m o , Xβ S m > 2,n = ||M P m o Xβ S m || 2 2,n = ||Xβ S m −X S m c ζ|| 2 2,n ≥ (1 − c)||β S m || 2 2,n , where X S m c ζ = P X S m c (Xβ S m ). Thus, it is easy to see that < V m o , Xβ S m > 2,n = j∈S m β j < V m o , X j > 2,n ≥ (1 − c) j∈S m β 2 j . Thus, there exists some j * such that | < V m o , X j * > | ≥ (1 − c)|β j | ≥ cJ. We know that the optimal j m must satisfy: | < U m o , X jm > 2,n | ≥ | < V m o , X j * > 2,n − < ε, X j * > 2,n | ≥ (1 − c)J − λ n . Thus, | < V m o , X jm > 2,n | > (1 − c)J − 2λ n . Hence, ||V m+1 o || 2 2,n = ||V m o − γ jm X jm || 2 2,n = ||V m o || 2 2,n − 2γ jm < U m o , X jm > +2γ j < , X jm > +γ 2 jm ≤ ||V m o || 2 2,n − γ 2 jm + 2λ n |γ jm | ≤ ||V m o || 2 2,n − ((1 − c)J − λ n ) 2 + 2((1 − c)J − λ n )λ n ≤ ||V m o || 2 2,n − ((1 − c)J) 2 + 4(1 − c)Jλ n . Consequently, ||V m o || 2 2,n ≤ ||V m 0 || 2 2,n − K((1 − c)J) 2 s + 4K(1 − c)Jsλ n . Since ||V 0 o || 2 2,n ≤ CJ 2 s, so let K > (1−c) 2 J 2 CJ 2 and assuming λ n → 0 would lead to a ||V m o || 2 2,n < 0 asymptotically. That said, our assumption that "there exists a j which is never picked up in the process at k = 0, 1, 2, . . . , KS" is incorrect with probability going to 1.
Thus, at time Ks, A 1 must happen with probability going to 1. And therefore, we know that ||U m o || 2 2,n = ||M P Ks o ε|| 2 2,n ≤σ 2 = σ 2 + O p ( 1 √ n ). By definition of m * , m * ≤ Ks. Therefore, ||V m o || 2 2,n ≤ ||U m o || 2 2,n − σ 2 + m * λ n = O p (sλ 2 n ).
C.3 Proofs for post-BA
Proof [proof of Lemma 6] It is sufficient to show that T 0 ⊂ T m * − T 0 for m * = Ks with K being large enough. If there exists a j ∈ T 0 which is never revisited at steps 1, 2, ..., m * , then in each step, we can choose the variable j: By assumption A.2, the optimal step size γ m j :=< U m , X j > n =< , X j > n + < V m , X j > n must satisfy: |γ m j | ≥ p √ 1 − c|β j |−λ n > √ 1 − cJ(1−o(1)). Hence, each step ||U m || 2 2,n must decrease for at least (1 − c)J 2 (1 − δ) 2 for any δ > 0 and n large enough. However, ||U 0 || 2 2,n ≤ (1 + c)s(J ) 2 , which implies that m * = Ks ≤ (1+c)s(J ) 2 (1−c)J 2 (1−δ) 2 , i.e, K ≤ (1+c)(J ) 2 (1−c)J 2 (1−δ) 2 . So for any K > (1+c)(J ) 2 (1−c)J 2 , as n → ∞ and for δ small enough, m * Ks > (1+c)s(J ) 2 (1−c)J 2 (1−δ) 2 , which leads to a contradiction.
Thus, for any K > (1+c)(J ) 2 (1−c)J 2 , all variables in T 0 must be revisited at steps 1, 2, ..., m * with probability going to 1. The rest of the results simply follow T 0 ⊂ T m * − T 0 .
U m , x j )|. 4. Update the estimator: β m+1 := β m + γ m j m e j m where e j m is the j m th index vector and f m+1 := f m + γ m j m x j m 5. Increase m by one. If m < m stop , continue with (2); otherwise stop.
2 :
2Relation between c and ζ *
Figure 1 :
1Figure 5.1 shows the out-of-sample MSE of the L 2 Boosting algorithm depending on the number of steps. The horizontal lines show the MSE of OLS and LASSO estimates.
Figure 2 :
2Figure 5.2 shows the in-sample MSE of the L 2 Boosting algorithm depending on the number of steps. The horizontal lines show the MSE of OLS and LASSO estimates.
>
(1 − C log(p)/n)) for some constant C. This means stopping, if the ratio of the estimated variances does not improve upon a certain amount any more.
1 /( 1
11− c) 2 ) ≥ (q(0) + |N (m)|)/(q(0) + |N (m)| − | N (m)|), i.e., | N (m)| ≤ (1 − exp(−1/(1 − c) 2 ))(q(0) + |N (m)|).Set µ e (c) = (1 − exp(−1/(1 − c) 2 )) ∈ (0, 1) when c ∈ [0, 1). Since we know that | N (m)| ≥ 2|N (m)|−m, we immediately have: |N (m)| ≤ 1 2−µe(c) (m+ µ e (c)q(0)), and |R(m)| ≥ 1−µe(c) 2−µe(c) m − µe(c) 2−µe(c) q(0).
minimum when N (m) = {m−| N (m)|+1, m−| N (m)|+2, ..., m}, and for the step m − | N(m)| + l, q(m − | N (m)| + l − 1) = q(m) − | N (m)| + l − 1, l = 1, 2, ..., | N (m)|.We know that ||V l−1 || 2 2,n /||V m || 2 2,n ≥ 1/L(l−1, m), while L(m−l, m) → ( q(m)−l−1 q(m)−1 ) 1−c as q(m)− m + l → ∞. So for any δ > 0, and q(0) large enough, (1 + δ) l∈ N (m) ||V l−
)
> 0 for some ξ > 0. Thus, asymptotically, m = M 0 ∧ (m * + 1) < M 0 , i.e., m * + 1 < M 0 . Therefore, for δ small enough, m * + 1 < M 0 . Thus, m * s( s log(p) n ) − 1 1+ζ * (c)−δ , and ||V m * +1 || 2 2,n ≤ η √ m * + 1 + sλ n ||V 0 || 2 1+ζ * (c)−δ 2,n s log(p) n ζ * (c)−δ 1+ζ * (c)−δ , for any small δ > 0.
Table
Table
Table 3 :
3Simulation results: sparse, iid design (Boosting)n
p
BA-oracle
BA-Ks
BA-our
p-BA-oracle
p-BA-Ks
p-BA-our
oBA-oracle
oBA-Ks
oBA-our
100
100
0.44
0.69
0.66
0.12
0.58
0.43
0.12
0.82
0.54
100
200
0.48
0.85
1.28
0.14
0.77
1.65
0.12
1.00
0.60
200
100
0.15
0.29
0.26
0.05
0.25
0.21
0.05
0.34
0.20
200
200
0.20
0.41
0.35
0.06
0.31
0.21
0.06
0.41
0.24
400
100
0.07
0.13
0.10
0.03
0.11
0.08
0.03
0.14
0.09
400
200
0.09
0.19
0.16
0.03
0.16
0.12
0.02
0.21
0.14
Table 4 :
4Simulation results: sparse, iid design (Lasso)
n
p
LASSO
p LASSO
Lasso-CV
p-Lasso-CV
100
100
0.88
0.70
0.54
0.94
100
200
1.02
1.30
0.72
1.02
200
100
0.29
0.28
0.22
0.43
200
200
0.37
0.39
0.30
0.44
400
100
0.13
0.11
0.09
0.18
400
200
0.16
0.20
0.14
0.27
Table 5 :
5Simulation results: sparse, correlated design (Boosting)n
p
BA-oracle
BA-Ks
BA-our
p-BA-oracle
p-BA-Ks
p-BA-our
oBA-oracle
oBA-Ks
oBA-our
100
100
1.40
1.70
1.90
0.55
1.02
1.31
0.44
0.96
1.36
100
200
3.02
2.80
2.85
1.65
2.29
2.48
1.25
1.44
1.96
200
100
0.41
0.48
0.54
0.07
0.12
0.16
0.07
0.35
0.24
200
200
0.53
0.60
0.63
0.06
0.15
0.19
0.06
0.42
0.25
400
100
0.16
0.28
0.19
0.02
0.04
0.08
0.02
0.14
0.09
400
200
0.17
0.23
0.21
0.03
0.04
0.10
0.03
0.17
0.10
Table 6 :
6Simulation results: sparse, correlated design (Lasso)n
p
LASSO
p-LASSO
Lasso-CV
p-Lasso-CV
100
100
2.63
1.35
0.97
1.37
100
200
2.96
2.04
1.63
2.38
200
100
1.10
0.23
0.33
0.57
200
200
1.64
0.38
0.49
0.87
400
100
0.38
0.10
0.13
0.23
400
200
0.36
0.15
0.16
0.31
Table 7 :
7Simulation results: polynomial, iid design (Boosting)
n
p
BA-oracle
BA-Ks
BA-our
p-BA-oracle
p-BA-Ks
p-BA-our
oBA-oracle
oBA-Ks
oBA-our
100
100
0.37
0.81
0.58
0.36
1.09
0.64
0.37
1.23
0.73
100
200
0.44
1.04
1.38
0.43
1.32
1.85
0.45
1.39
0.74
200
100
0.26
0.40
0.34
0.26
0.47
0.37
0.26
0.49
0.39
200
200
0.27
0.54
0.39
0.27
0.63
0.42
0.28
0.61
0.44
400
100
0.17
0.19
0.19
0.17
0.21
0.20
0.18
0.22
0.20
400
200
0.18
0.30
0.26
0.18
0.33
0.28
0.17
0.34
0.28
Table 8 :
8Simulation results: polynomial, iid design (Lasso)n
p
LASSO
p-LASSO
Lasso-CV
p-Lasso-CV
100
100
0.43
0.83
0.45
0.84
100
200
0.50
1.06
0.54
0.76
200
100
0.30
0.34
0.26
0.47
200
200
0.34
0.52
0.33
0.52
400
100
0.19
0.19
0.15
0.24
400
200
0.21
0.31
0.20
0.38
Table 9 :
9Simulation results: polynomial, correlated design (Boosting)
n
p
BA-oracle
BA-Ks
BA-our
p-BA-oracle
p-BA-Ks
p-BA-our
oBA-oracle
oBA-Ks
oBA-our
100
100
0.23
0.68
0.46
0.22
0.91
0.49
0.22
1.22
0.51
100
200
0.26
0.87
1.02
0.24
1.10
1.42
0.24
1.46
0.66
200
100
0.19
0.37
0.28
0.15
0.44
0.26
0.14
0.49
0.24
200
200
0.22
0.49
0.35
0.20
0.56
0.34
0.20
0.61
0.34
400
100
0.13
0.20
0.17
0.10
0.20
0.15
0.09
0.20
0.15
400
200
0.14
0.23
0.20
0.11
0.24
0.17
0.11
0.28
0.17
for forecasting. The table below shows the MSE errors for different methods discussed in
the previous sections.
All calculations were peformed in R (R Core Team
Table 10 :
10Simulation results: polynomial, correlated design (Lasso)n
p
LASSO
p-LASSO
Lasso-CV
p-Lasso-CV
100
100
0.33
0.53
0.33
0.55
100
200
0.34
0.93
0.36
0.55
200
100
0.27
0.31
0.23
0.41
200
200
0.28
0.47
0.29
0.46
400
100
0.17
0.18
0.14
0.24
400
200
0.16
0.24
0.15
0.29
Table 11 :
11Results Riboflavin Production -out-of-sample MSE
BA-Ks
BA-our
p-BA-Ks
p-BA-our
oBA-Ks
oBA-our
LASSO
p-LASSO
0.4669
0.3641
0.4385
0.1237
0.4246
0.1080
0.1687
0.1539
Table 12 :
12Prediction of education -out-of-sample MSE
subject
BA
post-BA
oBA
Lasso
post-Lasso
Mathematics
19.1
19.3
19.3
18.4
18.4
Protugese
8.0
7.9
7.9
7.8
7.8
. In order to allow comparability the dgp is adopted fromBühlmann (2006).
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Additive logistic regression: A statistical view of boosting. J H Friedman, T Hastie, R Tibshirani, 10.1214/aos/1016218223The Annals of Statistics. 28J. H. Friedman, T. Hastie, and R. Tibshirani. Additive logistic regression: A statistical view of boosting. The Annals of Statistics, 28:337-407, 2000. doi: 10.1214/aos/1016218223. URL http://projecteuclid.org/euclid.aos/1016218223. with discussion.
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A stepwise regression method and consistent model selection for high-dimensional sparse linear models. Ching-Kang Ing, Tze Leung Lai, Statistica Sinica. 214Ching-Kang Ing and Tze Leung Lai. A stepwise regression method and consistent model selection for high-dimensional sparse linear models. Statistica Sinica, 21(4):1473-1513, 2011.
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| []
|
[
"Non-Archimedean Unitary Operators",
"Non-Archimedean Unitary Operators"
]
| [
"Anatoly N Kochubei [email protected] \nInstitute of Mathematics\nNational Academy of Sciences of Ukraine\nTereshchenkivska 301601KievUkraine\n"
]
| [
"Institute of Mathematics\nNational Academy of Sciences of Ukraine\nTereshchenkivska 301601KievUkraine"
]
| []
| We describe a subclass of the class of normal operators on Banach spaces over non-Archimedean fields (A. N. Kochubei, J. Math. Phys. 51 (2010), article 023526) consisting of operators whose properties resemble those of unitary operators. In particular, an analog of Stone's theorem about one-parameter groups of unitary operators is proved. MSC 2010. Primary: 47S10. | null | [
"https://arxiv.org/pdf/1102.4302v1.pdf"
]
| 118,300,286 | 1102.4302 | 76fbef6bbd2c663eb2d902fdf99fed0eb3e8b49b |
Non-Archimedean Unitary Operators
21 Feb 2011
Anatoly N Kochubei [email protected]
Institute of Mathematics
National Academy of Sciences of Ukraine
Tereshchenkivska 301601KievUkraine
Non-Archimedean Unitary Operators
21 Feb 2011
We describe a subclass of the class of normal operators on Banach spaces over non-Archimedean fields (A. N. Kochubei, J. Math. Phys. 51 (2010), article 023526) consisting of operators whose properties resemble those of unitary operators. In particular, an analog of Stone's theorem about one-parameter groups of unitary operators is proved. MSC 2010. Primary: 47S10.
INTRODUCTION
1.1. In a previous paper [4], we found a class of non-Archimedean normal operators, bounded linear operators on Banach spaces over non-Archimedean fields possessing orthogonal, in the non-Archimedean sense, spectral decompositions. It is a natural problem now to find out what operators in the non-Archimedean setting should be seen as unitary ones. Classically, the correspondence between selfadjoint and unitary operators extends, via the well-known functional calculus, the correspondence λ → e iλ between real numbers and complex numbers from the unit circle.
There is no direct analog of the function e iλ in the non-Archimedean case. However we will see that its natural counterpart in the context of non-Archimedean operator theory is the function λ → z λ where λ runs the ring Z p of p-adic integers, z belongs to the group of principal units of a non-Archimedean field K (in another language, z is a positive element of K [8]). The image of this function also belongs to the group of principal units. This prompts to define a non-Archimedean unitary operator as an operator of the form I + V where I is the unit operator, V is a normal operator in the sense of [4], V < 1. The normality assumption is essential -otherwise I + V can be non-diagonalizable together with V . This shows (see also [11]) that the isometricity is not a substitute of unitarity in the non-Archimedean case. In fact we will use a refined version of the above definition; see Section 2.
In classical operator theory, the main result about unitary operators is Stone's theorem about the representation of a one-parameter unitary group in the form t → e itA where t ∈ R, A is a selfadjoint operator. We find its non-Archimedean analog -a one-parameter group parametrized by the group of principal units of Q p has the form s A where A belongs to the class of normal operators in the sense of [4], A ≤ 1. This result can be reformulated from the setting with the parameter from the group of principal units to the case of the parameter from Z p .
1.2. Let us recall principal notions and results from [4]. We will not explain the basic notions of non-Archimedean analysis; see [5,6,7,8].
Let A be a bounded linear operator on a Banach space B over a complete non-Archimedean valued field K with a nontrivial valuation; | · | will denote the absolute value in K. Denote by L A the commutative Banach algebra generated by the operators A and I. L A is a closure of the algebra K[A] of polynomials in A, with respect to the norm of operators; thus L A is a Banach subalgebra of the algebra L(B) of all bounded linear operators. Elements λ ∈ K are identified with the operators λI.
The spectrum M(L A ) is defined (see [2]) as the set of all bounded multiplicative seminorms on L A . In a natural topology, it is a nonempty Hausdorff compact topological space. If the algebra L A is uniform, that is T 2 = T 2 for any T ∈ L A , and all its characters take their values in K, then An operator A, for which the above picture takes place, is called normal. We will call a normal operator strongly normal, if its spectrum σ(A) is a nonempty totally disconnected compact subset of K, and M(L A ) = σ(A). A strongly normal operator admits the spectral decomposition
A = σ(A) λ E(dλ).
More generally, we get a functional calculus assigning to any K-valued continuous function ϕ the operator
ϕ(A) = σ(A) ϕ(λ) E(dλ), such that ϕ(A) ≤ sup λ∈σ(A) |ϕ(λ)|.
Some sufficient conditions of strong normality were found in [4]. Let dim B < ∞, and B = K n , with the norm (x 1 , . . . , x n ) = max 1≤i≤n |x i |. An operator A is represented, with respect to the standard basis in K n , by a matrix (a ij ) n i,j=1 . Its operator norm coincides with A = max i,j |a ij | (see [10]). It is sufficient to consider the case where A = 1.
Let K be the residue field of the field K. Together with the operator A, we consider its reduction, the operator A on the K-vector space B = K n corresponding to the matrix ( a ij ) where a ij is the image of a ij under the canonical mapping O → K (O is the ring of integers of the field K). An operator A is called nondegenerate, if A = νI for any ν ∈ K.
It was proved in [4] that A is strongly normal, if it is nondegenerate, all its eigenvalues belong to K, and its reduction A is diagonalizable. These conditions are satisfied, for example, if A has n different eigenvalues from K.
In the infinite-dimensional situation, a similar result holds [4] (with the representation of operators by infinite matrices), if we assume in addition that K is algebraically closed, B is the space of sequences tending to zero, A is a bounded operator with a compact spectrum, and the resolvent of A belongs, in a weak sense, to the space of Krasner analytic functions outside the spectrum. For example, if a compact operator (that is a norm limit of a sequence of finite rank operators) is such that its reduction is diagonalizable, then it is strongly normal.
Note that for a strongly normal operator A and any continuous K-valued function ϕ on σ(A), the operator B = ϕ(A) is strongly normal. Indeed, considering the functional model of the algebra L A we see that the spectrum of the operator B coincides with the set f (σ(A)). The Banach algebra L B is a subalgebra of L A , and its functional model coincides with the closure in C(σ(A), K) of the set of functions π • f where π is an arbitrary polynomial. The convergence of the sequence π n •f in C(σ(A), K) is equivalent to the convergence of the sequence of polynomials π n in the space C(σ(B), K). By Kaplansky's theorem (see Theorem 43.3 in [8]), L B is isometrically isomorphic to C(σ(B), K).
2 Unitary Operators 2.1. An operator U on a Banach space B over a complete non-Archimedean valued field K with a nontrivial valuation will be called unitary, if U = I + V where V < 1 and V is strongly normal. A unitary operator admits a spectral decomposition
U = σ(V ) (1 + λ)E V (dλ) = σ(U ) µE U (dµ).
Here E V is the spectral measure of the operator V , the mapping ϕ(λ) = 1 + λ transforms the spectrum of V into that of U, E U (M) = E V (ϕ −1 (M)) for any open-closed subset of σ(U).
Below we assume that the field K is an extension of the field Q p of p-adic numbers, and the absolute value | · | is an extension of the p-adic absolute value.
Denote by U 1 (K) the group of principal units of the field K, that is U 1 (K) = {1 + λ : λ ∈ K, |λ| < 1}. We will consider one-parameter groups U(s), s ∈ U 1 (K), of unitary operators, that is families of unitary operators, continuous with respect to the norm of operators, such that U(s 1 s 2 ) = U(s 1 )U(s 2 ), s 1 , s 2 ∈ U 1 (K).
(1)
A one-parameter group of unitary operators can be constructed as follows. Let A be a strongly normal operator, A ≤ 1, σ(A) ⊆ Z p . Consider the function f s (λ) = (1 + z) λ where s = 1 + z, z ∈ K, |z| < 1, λ ∈ Z p . This function can be defined by its Mahler expansion [6,8]:
f s (λ) = ∞ n=0 z n P n (λ) where P n (λ) = λ(λ − 1) · · · (λ − n + 1) n! , n ≥ 1; P 0 (λ) ≡ 1.
An equivalent definition [6,8] can be made in terms of the approximation of a p-adic integer λ by a sequence of nonnegative integers, for which the function is defined in the straightforward way.
Set
U(s) = (1 + z) A = σ(A) (1 + z) λ E A (dλ), s = 1 + z ∈ U 1 (K).(2)
Due to the non-Archimedean orthonormality of the Mahler basis, we have
(1 + z) λ = 1 + v z (λ), v z (λ) = ∞ n=1 z n P n (λ), sup λ∈Zp ∞ n=1 z n P n (λ) = |s| < 1, so that U(s) is a unitary operator, that is U(s) = I + V (s) where V (s) = σ(A) v z (λ)E A (dλ) = σ(V (s)) µE V (s) (dµ), E V (s) (M) = E A (v −1 z (M)).
It follows from the approximative description of the function f s that U(s) possesses the required group property. Next, let s 1 = 1 + z 1 , s 2 = 1 + z 2 , |z 1 | < 1, |z 2 | < 1. Using (2) we find that
U(s 1 ) − U(s 2 ) = σ(A) ∞ n=1 (z n 1 − z n 2 ) P n (λ) E A (dλ), so that U(s 1 ) − U(s 2 ) ≤ sup λ∈Zp ∞ n=1 (z n 1 − z n 2 ) P n (λ) ≤ sup n≥1 |z n 1 − z n 2 | = |s 1 − s 2 |.
Therefore the function s → U(s) is continuous with respect to the operator norm.
2.2.
In particular, the above construction makes sense for s ∈ U 1 (Q p ), and the formula (2) defines a norm-continuous one-parameter group of unitary operators U 1 (Q p ) ∋ s → U(s). The next result is a converse statement, an analog of Stone's theorem.
Theorem. Let U(s), s ∈ U 1 (Q p ), p = 2, be a norm continuous one-parameter group of unitary operators, such that the spectrum of the strongly normal operator U(1 + p) − I is contained in pZ p . Then there exist such a strongly normal operator A, σ(A) ⊆ Z p , that U(s) = s A , s ∈ U 1 (Q p ).
Proof. Each element s ∈ U 1 (Q p ) can be represented, in a unique way, as
s = (1 + p) ζ , ζ ∈ Z p .(3)
Indeed, set ζ = log s log(1 + p) (see [6,8] regarding properties of the p-adic logarithm). We
have ζ ∈ Q p , | log s| ≤ p −1 , | log(1 + p)| = |p| = p −1 , so that ζ ∈ Z p .
On the other hand, exp(ζ log(1 + p)) = (1 + p) ζ ( [8], Theorem 47.10), which implies (3). Let us write the canonical representation ζ = ζ 0 + ζ 1 p + ζ 2 p 2 + · · · , ζ j ∈ {0, 1, . . . , p − 1}.
The series (1 + p) ζ = ∞ n=0 p n P n (ζ) converges uniformly with respect to ζ ∈ Z p , so that the function ζ → (1 + p) ζ is continuous. Due to the norm continuity of U,
U(s) = lim n→∞ [U(1 + p)] ζ 0 +ζ 1 p+···+ζnp n .(4)
Denote a n (λ) = (1 + p) (ζ 0 +ζ 1 p+···+ζnp n )λ , λ ∈ Z p .
Let us prove that a n (λ) −→ (1 + p) ζλ , as n → ∞,
uniformly with respect to λ. Indeed, we have the estimate a n (λ) − (1 + p) ζλ = (1 + p) (ζ n+1 p n+1 +ζ n+2 p n+2 +··· )λ − 1
= ∞ k=1 p k P k ((ζ n+1 p n+1 + ζ n+2 p n+2 + · · · )λ) ≤ sup k≥1 p −k P k ((ζ n+1 p n+1 + ζ n+2 p n+2 + · · · )λ) where P k ((ζ n+1 p n+1 + ζ n+2 p n+2 + · · · )λ) ≤ p −n−1 |k!| −1 ≤ p −n−1+ k−1 p−1 , so that, uniformly with respect to λ ∈ Z p , a n (λ) − (1 + p) ζλ ≤ p −n−1 sup k≥1 p −k+ k−1 p−1 −→ 0,
as n → ∞. Now we return to the expression (4). By our assumption, U(1 + p) = I + V where V is a strongly normal operator, σ(V ) ⊆ pZ p . We have
U(1 + p) = σ(V ) (1 + λ)E V (dλ)
The Banach algebra L A generated by the operator V contains the strongly normal operator
A = 1 log(1 + p) log(I + V ) = 1 log(1 + p) ∞ k=1 (−1) k−1 k V k = σ(V ) log(1 + λ) log(1 + p) E V (dλ).
Obviously, σ(A) ⊆ Z p and
(1 + p) log(1+λ) log(1+p) = 1 + λ,
so that U(1 + p) = (1 + p) A , and it follows from (4) that
U(s) = lim n→∞ (1 + p) A ζ 0 +ζ 1 p+···+ζnp n .
Switching to the functional model and using (5) and (3) we obtain the required formula for the operators U(s).
Note that the condition regarding the operator U(1 + p) − I is satisfied automatically, if U(1 + p) = I + V , V < 1, and K = Q p .
2.3.
Let W (z), z ∈ Z p , p = 2, be a norm continuous unitary representation of the additive group Z p , and the spectrum of the operator W (p −1 log(1 + p)) − I lies in pZ p . Denote s = e pz , U(s) = W (z). Then s ∈ U 1 (Q p ), and s → U(s) is a one-parameter group satisfying the conditions of the above Theorem. We obtain the expression
W (z) = e pzA , z ∈ Z p ,
where A is a strongly normal operator, σ(A) ⊆ Z p .
Example. Galois Representations
In this section we follow [1,3,9].
Let K be a finite extension of Q p , and ε (n) ∈K (K is an algebraic closure of K) is a sequence of primitive roots of unity of orders p n , such that ε (0) = 1, ε (1) = 1, (ε (n+1) ) p = ε (n) , n = 0, 1, . . . .
Denote K n = K(ε (n) ), K ∞ = ∞ n=0
K n , G K = Gal(K/K). Let µ p n be the set of roots of unity of order p n ; thus ε (n) ∈ µ p n , n ≥ 0. The cyclotomic character χ : G K → Z * p = {x ∈ Z p : |x| = 1} is defined via the equality
σ(ζ) = ζ χ(σ) , for all σ ∈ G K , ζ ∈ µ p ∞ = ∞ n=0 µ p n .
χ is continuous with respect to the standard topology of G K as a profinite group.
The kernel of the cyclotomic character coincides with H K = Gal(K/K ∞ ). Therefore χ identifies Γ K = Gal(K ∞ /K) = G K /H K with an open subgroup of the multiplicative group Z * p .
By definition, a p-adic representation V of the group G K is a finite-dimensional vector space over Q p with a continuous linear action of G K .
Let K ∞ be the p-adic completion of K ∞ . Let us consider the action of Γ K on the K ∞ -vector space C p ⊗ Qp V H K of elements from C p ⊗ Qp V fixed under the action of H K . If d = dim Qp V , then C p ⊗ Qp V H K is a K ∞ -vector space of dimension d. The group Γ K acts on the union D Sen (V ) of finite-dimensional subspaces of C p ⊗ Qp V H K invariant with respect to Γ K , and dim K∞ D Sen (V ) = d. By Sen's theorem [9], there is a unique K ∞ -linear operator Θ V on D Sen (V ), such that for any ω ∈ D Sen (V ), there exists such an open subgroup Γ ω ⊂ Γ K that σ(ω) = [exp (Θ V log χ(σ))] ω (6) for all σ ∈ Γ ω . A representation V is called a Hodge-Tate representation if, for a certain basis e 1 , . . . , e d ∈ D Sen (V ), the operator Θ V is diagonal, with eigenvalues from Z. In this case, we can introduce a norm in D Sen (V ) setting
x 1 e 1 + · · · + x d e d = max(|x 1 |, . . . , |x d |), x 1 , . . . , x d ∈ K ∞ .
Then Θ V is obviously strongly normal, Θ V ≤ 1.
Taking into account the continuity of χ we can choose a so small open subgroup Λ ⊂ Γ K that |χ(σ) − 1| ≤ p −1 for all σ ∈ Λ. For every σ ∈ Λ, the right-hand side of (6) defines a unitary operator on D Sen (V ).
[2] the space M(L A ) is totally disconnected and L A is isomorphic to the algebra C(M(L A ), K) of continuous functions on M(L A ) with values from K. In this case, under the above isomorphism, the characteristic functions η Λ of nonempty open-closed subsets Λ ⊂ M(L A ) correspond to idempotent operators E(Λ) ∈ L A , E(Λ) = 1. These operators form a finitely additive norm-bounded projection-valued measure on the algebra of open-closed sets, with the non-Archimedean orthogonality property f = sup Λ E(Λ)f , f ∈ B.
ACKNOWLEDGEMENTThis work was supported in part by the Ukrainian Foundation for Fundamental Research, Grant 29.1/003.
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Continuous cohomology and p-adic Galois representations. S Sen, Invent. Math. 62S. Sen, Continuous cohomology and p-adic Galois representations, Invent. Math. 62 (1980), 89-116.
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Orthogonal transformations in non-Archimedean spaces. N Shilkret, Arch. Math. 23N. Shilkret, Orthogonal transformations in non-Archimedean spaces, Arch. Math. 23 (1972), 285-291.
| []
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[
"TWISTED NOVIKOV HOMOLOGY AND JUMP LOCI IN FORMAL AND HYPERFORMAL SPACES",
"TWISTED NOVIKOV HOMOLOGY AND JUMP LOCI IN FORMAL AND HYPERFORMAL SPACES"
]
| [
"Toshitake Kohno ",
"Andrei Pajitnov "
]
| []
| []
| Let X be a finite CW complex, and ρ : π 1 (X) → GL(n, C) a representation. Any cohomology class α ∈ H 1 (X, C) gives rise to a deformation γ t of ρ defined by γ t (g) = ρ(g) exp(t α, g ). We show that the cohomology of X with local coefficients γ gen corresponding to the generic point of the curve γ is computable from a spectral sequence starting from H * (X, ρ). We compute the differentials of the spectral sequence in terms of the Massey products. We show that the spectral sequence degenerates in case when X is a Kähler manifold, and ρ is semi-simple.If α ∈ H 1 (X, R) one associates to the triple (X, ρ, α) the twisted Novikov homology (a module over the Novikov ring). We show that the twisted Novikov Betti numbers equal the Betti numbers of X with coefficients in the local system γ gen . We investigate the dependance of these numbers on α and prove that they are constant in the complement to a finite number of integral hyperplanes in H 1 (X, R). | 10.2478/s11533-014-0413-2 | [
"https://arxiv.org/pdf/1302.6785v2.pdf"
]
| 55,415,169 | 1302.6785 | d7f23c91d842e4e8847995ffcd81468ba84b08b7 |
TWISTED NOVIKOV HOMOLOGY AND JUMP LOCI IN FORMAL AND HYPERFORMAL SPACES
27 Feb 2013
Toshitake Kohno
Andrei Pajitnov
TWISTED NOVIKOV HOMOLOGY AND JUMP LOCI IN FORMAL AND HYPERFORMAL SPACES
27 Feb 2013arXiv:1302.6785v1 [math.AT]
Let X be a finite CW complex, and ρ : π 1 (X) → GL(n, C) a representation. Any cohomology class α ∈ H 1 (X, C) gives rise to a deformation γ t of ρ defined by γ t (g) = ρ(g) exp(t α, g ). We show that the cohomology of X with local coefficients γ gen corresponding to the generic point of the curve γ is computable from a spectral sequence starting from H * (X, ρ). We compute the differentials of the spectral sequence in terms of the Massey products. We show that the spectral sequence degenerates in case when X is a Kähler manifold, and ρ is semi-simple.If α ∈ H 1 (X, R) one associates to the triple (X, ρ, α) the twisted Novikov homology (a module over the Novikov ring). We show that the twisted Novikov Betti numbers equal the Betti numbers of X with coefficients in the local system γ gen . We investigate the dependance of these numbers on α and prove that they are constant in the complement to a finite number of integral hyperplanes in H 1 (X, R).
INTRODUCTION
Let X be a finite connected CW-complex; denote its fundamental group by G. Let ρ : G → GL(n, C) be a representation. Any cohomology class α ∈ H 1 (X, C) gives rise to the following deformation of ρ: γ t : G → GL(n, C), γ t (g) = e t α,g ρ(g).
The cohomology groups of X with local coefficients γ t are isomorphic for all t except a subset containing only isolated points. The cohomology group H * (X, γ gen ) corresponding to the generic point of the curve γ t is the first main object of study in the present paper. We prove that there is a spectral sequence F * r starting from the homology of X with coefficients in ρ, converging to H * (X, γ gen ), and the differentials in this spectral sequence are computable in terms of some special higher Massey products with α. The first differential in this spectral sequence is the homomorphism L α of multiplication by α in the ρ-twisted cohomology of X.
This type of spectral sequences appeared in the paper of S.P. Novikov [11] in the de Rham setting. It was generalized to the case of cohomology with coefficients in a field of arbitrary characteristic in the paper [13] of the second author. Our present construction is close to the original ideas of S.P. Novikov. The main technical novelty of our present approach is the systematic use of formal exponential deformations, see § § 3, 4; this allows to avoid the convergency issues for power series.
If ρ is the trivial representation, the differentials in the spectral sequence above are the usual Massey products in the ordinary cohomology with slightly reduced indeterminacy: d r (x) = α, . . . , α, x , see § 3. Thus the spectral sequence degenerates when the space X is formal, by the classical argument of P. Deligne, Ph, Griffiths, J. Morgan, D. Sullivan [3] which applies here as well ( § 3).
In case when ρ is not trivial the situation is more complicated. The spectral sequence degenerates in particular when X is a Kähler manifold and ρ is a semi-simple representation ( §6, Prop. 6.6). The proof uses C. Simpson's theory of Higgs bundles [17], see also the work of H. Kasuya [8]. In this case we have Ker L α Im L α ≈ H * (X, γ gen ).
We introduce a class of hyperformal spaces for which all the spectral sequences F * r corresponding to 1-dimensional representations degenerate in their second term.
If α is a real cohomology class, there is another geometric construction related to ρ and α, namely, the twisted Novikov homology introduced in the works of H. Goda and the second author see [6], [15]. This construction associates to X, ρ and α a module over the Novikov ring L m,α . The rank and torsion numbers of this module are called the twisted Novikov Betti numbers and twisted Novikov torsion numbers; they provide lower bounds for numbers of zeros of any Morse form belonging to the de Rham cohomology class α (see §8 for details). These invariants detect the fibered knots in S 3 as it follows from the recent work of S. Friedl [5]. For a given space X and the representation ρ the Novikov numbers depend on the cohomology class α ∈ H 1 (X, R). The case of the torsion numbers was studied in [14] and [15]. It is proved there that the torsion numbers are constant in the open polyhedral cones formed by finite intersections of certain half-spaces in R n , where n = rkH 1 (X, Z). Similar analysis applies to the Novikov Betti numbers, which are of main interest to us in the present work. We prove in §8 that these numbers do not depend on α in the complement to a finite number of proper vector subspaces. In general the set of all α for which the Novikov Betti number b ρ k (X, α) is greater by q than the generic value (the jump loci for the Novikov numbers) is a union of a finite number of proper vector subspaces, see §8, Prop. 8.6.
It is known that the Novikov homology and the homology with local coefficients are related to each other. This was first observed in the paper [12] of the second author, see also Novikov [11]. Similar result holds also for the twisted Novikov homology. namely we prove (see §8, Prop. 9.2) that for α ∈ H 1 (X, R) we have b ρ k (X, α) = β k (X, γ gen ). This implies several corollaries about both families of numerical invariants. On one hand for given ρ and α the jumping loci for β k (X, ρ, α) are unions of integral hyperplanes. On the other hand the twisted Novikov Betti numbers are computable from the Massey spectral sequence. In the case of degeneracy of this spectral sequence, they equal the dimension of its second term.
EXACT COUPLES
In this section we recall the definition of the spectral sequence of an exact couple (following [9], [7]) and give an equivalent description of the successive terms of the spectral sequence, which will be useful in the sequel.
Let C = (D, E, i, j, k) be an exact couple, so that we have an exact triangle
D i / / D j~⑥⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ E k`❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆
We will usually abbreviate the notation to C = (D, E) and call D and E the first, respectively the second component of the exact couple. Following W. Massey we define the derived exact couple setting
E ′ = Ker (jk)/Im (jk), D ′ = i(D)
and defining j ′ , k ′ suitably. Iterating the process we obtain a sequence of exact couples C r = (D r , E r ), the initial couple being numbered as C 1 ; this sequence is called the spectral sequence associated to the exact couple C .
We will need an alternative description of the groups E r and the maps j r , k r .
Definition 2.1.
1) For r 2 let Z r be the subgroup of all elements x ∈ E, such that k(x) = i r−1 (y) for some y ∈ D. We put Z 1 = E. 2) For r 1 let B r be the subgroup of all elements z ∈ E, such that z = j(y) for some y ∈ D with i r−1 (y) = 0.
The following properties are easy to check:
Z 1 = E ⊃ Z 2 = Ker (jk) ⊃ Z 3 . . . ⊃ Z r ⊃ Z r+1 . . . B 1 = {0} ⊂ B 2 = Im (jk) ⊂ B 3 ⊂ . . . ⊂ B r ⊂ B r+1 ⊂ . . . B i ⊂ Z j for every i, j.
Put
E r = Z r /B r , D r = Im i r .
Define a homomorphism k r : E r → D r setting k r (x) = k(x) for every x ∈ Z r . Define a homomorphism j r : D r → E r as follows: if x ∈ D r and x = i r (y), then put j r (x) = [j(y)]. It is easy to check that these homomorphisms are well-defined and give rise to an exact couple C r = (D r , E r ):
D r i / / D r jr5 ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ E r kr`❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇
The proof of the next is in a usual diagram chasing:
Proposition 2.2.
The exact couples C r and C r are isomorphic for any r.
FORMAL DEFORMATIONS OF DIFFERENTIAL ALGEBRAS AND THEIR
SPECTRAL SEQUENCES
Let
A * = {A k } k∈N = {A 0 d / / A 1 d / / . . .}
be a graded-commutative differential algebra (DGA) over a field K of characteristic zero. Let N * be a graded differential module (DGM) over A * (that is, N * is a graded module over A * endowed with a differential D which satisfies the Leibniz formula with respect to the pairing A * ×N * → N * ). We will use the same symbol d to denote the differentials in both A * and N * , since no confusion is possible. We denote by A * [[t]] the algebra of formal power series over A * endowed with the differential extended from the differential of A * . Let ξ ∈ A 1 be a cocycle.
(1) 0 / / N * [[t]] t / / N * [[t]] π / / N * / / 0
where π is the natural projection t ✤ / / 0. The induced long exact sequence in cohomology can be considered as an exact couple (2) H
* N * [[t]] t / / H * N * [[t]] π * w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ H * (N * ) δ g g ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Proposition 3.1.
The spectral sequence induced by the exact couple (2) depends only on the cohomology class of ξ.
Proof. Let ξ 1 , ξ 2 ∈ A 1 be cohomologous cocycles,
ξ 1 = ξ 2 + df with f ∈ A 0 . Let D t = d + tξ 1 , D ′ t = d + tξ 2 be the correspond- ing differentials. Multiplication by e tf ∈ A 0 [[t]] determines an iso- morphism F : N * [[t]] → N * [[t]]
, commuting with the differentials, namely F (D t ω) = D ′ t (F (ω)). Thus the exact sequences (1) corresponding to ξ and η are isomorphic, as well as the exact couples (2) and their spectral sequences.
Definition 3.2. Put α = [ξ]
. The spectral sequence associated to the exact couple (2) is called deformation spectral sequence and denoted by F * r (N * , α). If N * and α are clear from the context, we write F * r instead of F * r (N * , α). We will compute the differentials in this spectral sequence in terms of special Massey products. Denote by
L α : H * (N * ) → H * (N * )
the multiplication by α. It is clear that
F * 2 = Ker L α /Im L α .
Let a ∈ H * (N * ). An r-chain starting from a is a sequence of elements ω 1 , . . . , ω r ∈ N * such that dω 1 = 0, [ω 1 ] = a, dω 2 = ξω 1 , . . . , dω r = ξω r−1 .
For an r-chain C put ∂C = ξω r ; this is a cocycle in N * . Denote by M Z m (r) the subspace of all a ∈ H * (N * ) such that there exists an r-chain starting from a. Thus
M Z m (1) = H * (N * ), M Z m (2) = Ker L α : H m (N * ) → H m+1 (N * ) . Denote by M B m
(r) the subspace of all β ∈ H * (N * ) such that there exists an (r − 1)-chain C = (ω 1 , . . . , ω r−1 ) with ξω r belonging to β. By definition
M B m (1) = 0, M B m (2) = Im L α : H m−1 (N * ) → H m (N * ) . It is clear that M B m (i) ⊂ M Z m (j) for every i, j. Put M H m (r) = M Z m (r) M B m (r) .
In the next definition we omit the upper indices and write M H (r) , M Z (r) etc. in order to simplify the notation. The correspondence a ✤ / / ξ, a (r+1) gives rise to a well-defined homomorphism of degree 1
∆ r : M H (r) / / M H (r) .
The next proposition is proved by an easy diagram chasing argument.
Proposition 3.5. For any r we have ∆ 2 r = 0, and the cohomology group
H * (M H * (r) , ∆ r ) is isomorphic to M H * (r+1)
. Theorem 3.6. For any r 2 there is an isomorphism
φ : M H * (r) ≈ / / F * r commuting with differentials.
Proof. Recall from Section 2 a spectral sequence F * r isomorphic to F * r . It is formed by exact couples
D r i / / D r~⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ E r δ`❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ where D 1 = H * (N * ), D r = Im t r−1 : D → D , and E r = Z r /B r (the modules Z r , B r are described in the definition 2.1). Lemma 3.7. 1) M Z (r) = Z r , 2) M B (r) = B r .
Proof. We will prove 1), the proof of 2) is similar. Let ζ ∈ H * (N * ) and z be a cocycle belonging to ζ. Then δ(ζ) equals the cohomology
class of ξz ∈ N * [[t]]; and ζ ∈ Z r if and only if there is µ ∈ N * [[t]] such that (3) ξz − D t µ ∈ t r−1 N * [[t]].
This condition is clearly equivalent to the existence of a sequence of elements µ 0 , µ 1 , . . . ∈ N * [[t]] such that (4) ξz = dµ 0 , and dµ i + ξµ i−1 = 0 for every 0 i r − 2.
The condition (4) is in turn equivalent to the existence of an r-chain starting from ζ. The Lemma implies that E * r ≈ M H * (r) and it is not difficult to prove that this isomorphism is compatible with the boundary operators.
In view of the Proposition 3.1 we obtain the next Corollary.
Corollary 3.8. Let ξ ∈ A 1 be a cocycle. The graded groups M H * (r)
defined above depend only on the cohomology class of ξ, which will be denoted by α.
Therefore the differentials in the spectral sequence {F * r } are equal to the higher Massey products with the cohomology class of ξ. Observe that these Massey products, defined above, have smaller indeterminacy than the usual Massey products. The second term of the spectral sequence is described therefore in terms of multiplication by the cocyle [ξ]. It is convenient to give a general definition. Definition 3.9. Let K * be a differential graded algebra, and θ be an element of odd degree s. Denote By L θ : K * → K * +s the homomorphism of multiplication by θ. The quotient Ker L θ Im L θ is a graded module which will be denoted by H * (K * , θ) and called θ-cohomology of K * . The dimension of H k (K * , θ) will be denoted by B k (K * , θ).
We have therefore
F * 2 (N * , α) ≈ H * (H * (N * ), α)
. Let us consider some examples, which will be important for the sequel.
Example 3.10. Let K * = A * . The homomorphism L α is the multiplication by α in the cohomology H * (A * ), and the differentials in the spectral sequence F r are the higher Massey products induced by the ring structure in H * (A * ).
Example 3.11. Let A * be a DGA and η ∈ A 1 be a cocycle. Endow the algebra A * with the differential d η defined by the following formula:
d η (x) = dx + ηx;
we obtain a DGM over A, which we will denote by A * . For an element α ∈ H 1 (A * ) we obtain a spectral sequence F * r with
F * 1 = H * ( A * ); F * 2 = H * (H * ( A * ), α)
. Example 3.12. Let M be a connected C ∞ manifold, and E be an n-dimensional complex flat bundle over M . Denote by ρ : π 1 (M ) → GL(n, C) the monodromy of E. Let A * (M ) be the algebra of complex differential forms on M . The space A * (M, E) of the differential forms with coefficients in E is a DGM over A * (M ); its cohomology is isomorphic to the cohomology H * (M, ρ) with local coefficients with respect to the representation ρ. We obtain a spectral sequence F r with
F * 1 = H * (M, ρ); F * 2 = H * (H * (M, ρ), α)
. Now let us consider some cases when the spectral sequences constructed above, degenerate in its second term. Recall that a differential graded algebra A * is called formal if it has the same minimal model as its cohomology algebra. Here is a useful characterization of minimal formal algebras.
k ; denote by C k ⊂ V k the subspace of the closed generators. The algebra A * is formal if and only if in each V k there is a direct complement N k to C k in V k , such that any cocycle in the ideal generated by ⊕ k (N k ) is cohomologous to zero.
This theorem leads to the proof of the well-known property that in formal algebras all Massey products vanish (see [3]). We will show that a similar result holds for the special Massey products introduced above, for the setting of the Example 3.10.
Theorem 3.14. Let A * be a formal differential algebra, α ∈ H 1 (A * ). Then the spectral sequence F * r (A * , α) degenerates at its second term
F * 2 = H * (H * (A * ), α).
Proof. It suffices to establish the property for the case of formal minimal algebras. Let A * be a formal minimal algebra with the space
of generators V k in dimension k decomposed as V k = C k ⊕ N k , see 3.13. We will prove that M Z (2) = M Z (3) = . . . = M Z (r) and M B (2) = M B (3) = . . . = M B (r) for every r 2.
Choose ξ ∈ A 1 representing α ∈ H 1 (A * ). Let a ∈ Z r , and C = (ω 1 , . . . , ω r ) be an r-chain starting from a. so that
dω 1 = 0, [ω 1 ] = a, dω 2 = ξω 1 , . . . , dω r = ξω r−1 .
Denote by Λ the algebra generated by the space of closed generators C i , so that M = Λ ⊕ I(N * ). Write ω r = ω 0 r + ω 1 r with ω 0 r ∈ Λ, ω 1 r ∈ I(N * ). Then ξω r−1 = dω 1 r , and d(ξω 1 r ) = d(ξω r ) = ξ 2 ω r−1 = 0, so that ξω 1 r is a cocycle belonging to I(N * ). Therefore ξω 1 r = dω r+1 for some ω r+1 ∈ I(N * ) and we obtain an (r + 1)-chain starting from a, so that a ∈ M Z (r+1) .
A similar argument shows that B 2 = B r for every r 2. Therefore the spectral sequence degenerates at the term E 2 .
The next proposition gives a sufficient condition for the degeneracy of the spectral sequence associated with a DGM over a DG-algebra A * . It will be used in Section 6 while studying the case of Kähler manifolds.
Definition 3.15. A DG-module N * over a DG algebra A * is called formal if it is a direct summand of a formal DGA B * over A * , that is,
(5) B * = N * ⊕ K * ,
where both N * and K * are DG-submodules of B * .
Proposition 3.16. Let N * be a formal DG-module over A * , and α ∈ H 1 (A * ). Then the spectral sequence F * r (N * , α) degenerates at its second terms.
Proof. The direct sum decomposition (5) implies that α); the spectral sequence F * r (B * , α) degenerates by the previous proposition and the result follows.
F * r (B * , α) = F * r (N * , α) ⊕ F * r (K * ,
A SPECTRAL SEQUENCE CONVERGING TO THE TWISTED HOMOLOGY
Let X be a connected topological space, G = π 1 (X). Let B be an integral domain and ρ : G → GL(n, B) a representation. The cohomology of the cochain complex C * (X, ρ) = Hom G (C * (X), B n ) is a B-module called the twisted cohomology of X with respect to ρ and denoted by H * (X, ρ). Denote by {B} the fraction field of B. The dimension over {B} of the localization H k (X, ρ) ⊗ {B} will be called the k the Betti number of X with respect to ρ and denoted by β k (X, ρ).
Let us start with a given representation ρ : G → GL(n, C), pick a cohomology class α ∈ H 1 (X, C) and consider the exponential deformation of ρ: γ t : G → GL(n, C), γ t (g) = ρ(g)e t α,g (t ∈ C).
It is not difficult to prove that for a given k the Betti number β k (X, γ t ) does not depend on t in the complement to some subset S k ⊂ C consisting of isolated points. Let Λ = C[[t]] and consider the representation γ of G defined as follows:
γ : G → GL(n, Λ), γ(g) = ρ(g)e t α,g ∈ GL(n, Λ)
(where e t α,g is understood as a formal power series). This representation will be called formal exponential deformation of ρ. It is clear that
β k (X, γ) = β k (X, γ t ) for t / ∈ S k .
Thus the Betti number corresponding to the generic point of the exponential deformation curve equals the Betti number of X with respect to the formal exponential deformation. The exact sequence
0 / / Λ t / / Λ / / C / / 0
gives a short exact sequence of complexes
0 / / C * (X, γ) t / / C * (X, γ) / / C * (X, ρ) / / 0
which in turn induces a long exact sequence in cohomology, that can be interpreted as an exact couple
(6) H * X, γ t / / H * X, γ w w ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ H * (X, ρ) g g ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ Denote by L * r (X, ρ, α) = (U * r , W * r ) the induced spectral sequence. Proposition 4.1. Let X be a finite connected CW-complex. Then dim C W k ∞ = β k (X, γ) = β k (X, γ t ) for generic t. Proof.
The ring Λ is principal, and the Λ-module H * X, γ is a finite direct sum of cyclic modules. Only the free summands survive to E ∞ and the contribution of each such summand to dim C W k ∞ equals 1.
COMPARING THE TWO SPECTRAL SEQUENCES
We are interested in the formal deformations of differential algebras related to topological spaces, mainly manifolds. Let M be a connected C ∞ manifold, and E a flat n-dimensional vector bundle over M . Denote π 1 (M ) by G, and let ρ : G → GL(n, C) be the monodromy representation of E.
In this section we will work with the module N * = A * (M, E) of E-valued C ∞ differential forms on M . This vector space is a DGM over the algebra A * (M ) of the C-valued C ∞ differential forms on M . Let α ∈ H 1 (M ). The corresponding exact couple (7) H
* N * [[t]] t / / H * N * [[t]] π * x x ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ H * (N * ) g g ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆
gives rise to the deformation spectral sequence F * r (N * , α) (see §3). Let γ be the formal exponential deformation of ρ corresponding to the class α: γ(g) = ρ(g)e t α,g ∈ GL(n, Λ), Λ = C[[t]].
We associate to this deformation the spectral sequence L * (M, ρ, α) (see §4).
R 1 = {0 / / A / / T 0 d / / T 1 / / . . .}.
Consider the sheaf Z * of γ-equivariant singular cochains:
Z * (U ) = Hom G S ∞ * (π −1 (U )), C n [[t]] .
The sequence (9)
R 2 = {0 / / A / / Z 0 δ / / Z 1 δ / / . . .}
(where δ is the coboundary operator on singular cochains) is also exact, and we have two soft acyclic resolutions of the sheaf A. The integration map I induces a homomorphism R 1 → R 2 which equals identity on A. The standard sheaf-theoretic result implies that the induced homomorphism in the cohomology groups of the complexes of global sections is an isomorphism (see for example [19], Corollary 3.14). The proof of the Proposition is complete. The isomorphism
I * • Φ * : H * A * (M, E)[[t]] → H * (M, γ)
induces an isomorphism of the exact couple
(10) H * A * (M, E)[[t]] t / / H * A * (M, E)[[t]] π * v v • • • • • • • • • • • • • H * (M, E) h h | | | | | | | | | | | | |
to the exact couple
(11) H * M, γ t / / H * M, γ π * w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ H * (M, E) g g ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖
and therefore the isomorphism of the spectral sequences F * r (N * , α) ≈ L * r (M, ρ, α). Now let us proceed to general topological spaces. The rational homotopy theory of D. Sullivan (see [18], [3], [2]) associates to a connected topological space X a minimal algebra M * (X) over C, well defined up to isomorphism.
Let α ∈ H 1 (X); we obtain a spectral sequence F * r (M * (X), α). It is not difficult to see that for the case when X is a C ∞ manifold, this spectral sequence is isomorphic to the one considered in the previous section. Indeed, we have a homotopy equivalence φ : M * (X) / / A * (X); the induced homomorphism of the spectral sequences
F * r (M * (X), α) → F * r (A * (X), φ * (α))
is an isomorphism in the second term F 2 . Thus the two spectral sequences are isomorphic.
Theorem 5.4. Let X be a finite CW complex. Then α). Proof. Let f : X → M be a homotopy equivalence of X to a C ∞ manifold M (possibly non-compact). Denote by α ′ the (f −1 ) * -image of α. The map f induces a homotopy equivalence F : M * (X) → A * (M ) of DGAs. We obtain isomorphisms of the spectral sequences:
F * r (M * (X), α) ≈ L * r (X,F * r (M * (X), α) ≈ F * r A * (M ), (f −1 ) * (α) (12) L * (X, α) ≈ L * (M ), α).(13)
Now apply Theorem 5.1 and the result follows.
Remark 5.5. Theorem 5.4 can be generalized to the case of local coefficients so as to obtain a result similar to Theorem 5.1 in the setting of general topological spaces.
FORMAL AND HYPERFORMAL SPACES
The next theorem follows immediately from Theorems 5.4 and 3.14.
Theorem 6.1. Let X be a finite connected CW-complex. Assume that X is formal. Then the spectral sequences F * r (M * (X), α) ≈ L * (X, α) degenerate at their second term.
Thus the dimension of the homology of X with coefficients in a generic point of the exponential deformation of the trivial representation can be computed from the multiplicative structure of the ordinary homology. Namely, let α ∈ H 1 (X, C) and let γ t be the exponential deformation of the trivial representation:
γ t (g) = e t α,g .
Denote by b k (X, γ gen ) the Betti number of X with coefficients in a generic point of the curve γ t , and let L α be the operation of multiplication by α in H k (X, C). The theorem above implies that b k (X, γ gen ) = H * (H k (X, C), α).
The case of the spectral sequence L * (X, ρ, α) where ρ : π 1 (X) → GL(n, C) is a non-trivial representation, is more complicated and to guarantee the degeneracy of the spectral sequence a stronger condition is necessary. Let us first consider the case of 1-dimensional representations ρ. Let M be a connected C ∞ manifold. Denote by G the fundamental group of M , let Ch(G) be the group of homomorphisms G → C * = GL(1, C). For a character ρ ∈ Ch(G) denote by E ρ the corresponding flat vector bundle over M . Put
A * (M ) = ρ∈Ch(G) A * (M, E ρ ).
The pairing E ρ ⊗ E η ≈ E ρη induces a natural structure of a differential graded algebra on the vector spaceĀ * (M ). Proof. Let ρ ∈ Ch(π 1 (M )). The flat bundle E ρ has a unique structure of a harmonic Higgs bundle (see [1], [17]); the exterior differential D ρ in the DG-module A * (M, E ρ ) writes therefore as D ρ = D ′ ρ + D ′′ ρ , and the natural homomorphisms of DG-modules
Definition 6.2. A C ∞ manifold M is hyperformal if the algebrā A * (M ) is formal.Ker (D ′ ρ ), D ′′ ρ v v ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ) ) ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ A * (M, E ρ ), D H * DeRham (M, E ρ ), 0
induce isomorphisms in cohomology (see [17], Lemma 2.2 (Formality)). Denote the DG-module (Ker (D ′ ρ ), D ′′ ρ ) by K ρ (M ), and put
K * (M ) = ρ∈Ch(G) K ρ (M ).
The multiplicativity properties of Higgs bundles imply that K * is a DG-algebra and we have the maps of DGAs:
K * (M ) z z t t t t t t t t t t t & & ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ A * (M ) ⊕ ρ H * (M, E ρ )
both inducing isomorphisms in cohomology. The theorem follows. Consider now the case of representations of higher rank.
Proposition 6.6. Let M be a compact Kähler manifold and ρ : π 1 (M ) → GL(n, C) be a semi-simple representation. Then the DG-module A * (M, E ρ ) is formal.
Proof. By [17], Theorem 1 there is a harmonic metric on the bundle E ρ . The tensor powers of this metric provide harmonic metrics on the bundles E ⊗n ρ for any n 1. Put
L * ρ (M ) = ∞ n=0 A * (M, E ⊗n ρ ) (where A * (M, E ⊗0 ρ ) = A * (M ) by convention). Then L * ρ (M )
is a DGalgebra. The same argument as in the proof of Theorem 6.5 implies that this algebra is formal, and it remains to observe that A * (M, E ρ ) is a direct summand of L * ρ (M ).
ON THE BETTI NUMBERS OF CHAIN COMPLEXES OVER LAURENT POLYNOMIAL RINGS
This section contains some preliminary algebraic results. We discuss the Betti numbers of complexes over Laurent polynomial rings in view of further applications to the Novikov Betti numbers and the homology with local coefficients.
Let T be an integral domain and {T } be its fraction field. Let C * be a finite free chain complex over T :
C * = {0 o o C 0 o o . . . ∂ k o o C k ∂ k+1 o o C k+1 o o . . .}
The tensor product of this chain complex with {T } will be denoted by C * and the boundary operator in this complex will be denoted by
∂ * . The Betti number b k (C * ) is equal to dim C k − rk∂ k − rk∂ k+1 = rkC k − rk∂ k − rk∂ k+1
(where rk∂ k stands for the maximal rank of a non-zero minor of the matrix of ∂ k ). Let φ : T → U be a homomorphism to another integral domain. Denote by b k (C * , φ) the k-th Betti number of the tensor product
C * ⊗ φ {U }. Then b k (C * , φ) b k (C * ).
The inequality is strict if the φ-images of all the rk∂ k -minors of ∂ k vanish, or if the φ-images of all the rk∂ k+1 -minors of ∂ k+1 vanish. In general, for q > 0 the condition b k (C * , φ) b k (C * ) + q is equivalent to the existence of a number i, with 0 i q such that all the (rk∂ k − i)-minors of φ(∂ k ) vanish and all the (rk∂ k+1 − (q − i))minors of φ(∂ k+1 ) vanish.
We are interested in the case of the Laurent polynomial rings. Let R be an integral domain and C * be a finite free chain complex over L n = R[t ± 1 , . . . , t ± n ] = R[Z n ]. Let p : Z n → Z m be a group homomorphism, extend it to a ring homomorphism L n → L m which will be denoted by the same letter p.
Denote by Q m is the field of fractions of L m , that is, the field of the rational functions in m variables with coefficients in the fraction field of R.
Form the chain complex C * ⊗ p Q m , and denote by b k (C * , p) the dimension of the vector space H k (C * ⊗ p Q m ) over Q m .
Observe that if p is injective, then b k (C * ) = b k (C * , p). We will now study the dependance of b k (C * , p) on p.
Definition 7.1. A subgroup G ⊂ Z n is called full if it is a direct summand of Z n . We say that a homomorphism p : Z n → Z m is subordinate to a full subgroup G ⊂ Hom(Z n , Z m ) and we write p ⊏ G, if all the coordinates of p are in G.
Remark 7.2. Denote by K the subgroup of Z n dual to G. Then p ⊏ G if and only if p | K = 0. Theorem 7.3. Let C * be a finite free complex over L n . Let k 0, q > 0. Then there is a finite family of proper full subgroups G i ⊂ Hom(Z n , Z) such that for p ∈ Hom(Z n , Z m ) the condition b k (C * , p) b k (C * ) + q is equivalent to the following condition: p ⊏ G i for some i.
Proof. Let us do the case q = 1, the general case is similar. Let F denote the set of all the (rk∂ k )-minors of the matrix ∂ k : C k → C k−1 , and all the (rk∂ k+1 )-minors of the matrix ∂ k+1 : C k+1 k → C k . Let ∆ ∈ F , write ∆ = g∈Z n r g · g.
According to our previous observation it suffices to study the set Σ of all homomorphisms p : Z n → Z m such that p(∆) = 0. Let Γ = supp ∆, this is a finite subset of Z n . Any homomorphism p : Z n → Z m with p(∆) = 0 must be non-injective on Γ. To describe the set of all such homomorphisms let us say that a subdivision
Γ = Γ 1 ⊔ . . . ⊔ Γ N is ∆-fitted, if for any j we have g k ∈Γ j r g = 0.
For any ∆-fitted subdivision S consider the subgroup L(S) ⊂ Hom(Z n , Z) consisting of all homomorphisms h : Z n → Z such that h | Γ i is constant for every i. Then L(S) is a full subgroup of Hom(Z n , Z). Observe that L(S) is a proper subgroup since every Γ i contains at least two elements. It is clear that a homomorphism p : Z n → Z m belongs to Σ if and only if p is constant on each component Γ i of a ∆-fitted subdivision of ∆, that is, p ⊏ L(S).
THE TWISTED NOVIKOV BETTI NUMBERS
Let X be a finite connected CW complex and ρ : π 1 (X) → GL(l, R) be a representation (where R is an integral domain). We will discuss several numerical invariants deduced from ρ. Denote π 1 (X) by G, and rkH 1 (X, Z) by n. The chain complex of the universal cover X has a natural structure of a left G-module; it will be convenient to consider on it a right G-action x · g = g −1 (x). Our first numerical invariant is the twisted Betti number of X with respect to ρ:
b k (X, ρ) = dim {R} H k C * ( X) ⊗ ρ {R} l .
Denote by φ the projection of G onto G/T ors ≈ Z n . We obtain two representations G
φ~⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ρ $ $ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Z n GL(l, R)
Let ρ • : G → GL(l, L n ) denote the tensor product of these two representations (where L n = R[Z n ] is the ring of Laurent polynomials in n variables) and put
b • k (X, ρ) = dim {Ln} H k C * ( X) ⊗ ρ • {L n } l Denote X Z n
/ / X the maximal free abelian covering of X; we will consider b • k (X, ρ) as the Betti number of X with ρ-twisted coefficients.
Let p : Z n → Z m be a homomorphism. Consider the tensor product of representations ρ and p • φ:
ρ p = ρ ⊗ (p • φ) : G → GL(l, L m ) and let b k (X, ρ; p) = dim {Lm} H k C * ( X) ⊗ ρp {L m } l It is not difficult to see that b k (X, ρ; p) b • k (X, ρ) for every p; (14) b k (X, ρ) b • k (X, ρ).(15)
Theorem 7.3 of the previous section implies the following.
Proposition 8.1. Let k 0, q > 0.
Then there is a finite family of proper full subgroups G i ⊂ Hom(Z n , Z) such that for p ∈ Hom(Z n , Z m ) the condition b k (X, ρ; p) b • k (X, ρ) + q is equivalent to the following condition: p ⊏ G i for some i.
Proceeding to the Novikov homology, let us first recall the definition of the Novikov ring. Let η : Z m → R be a group homomorphism. The Novikov completion L m,η of the ring L m = R[Z m ] with respect to η is defined as the set of all series of the form λ = g n g g (where g ∈ G and n g ∈ R) satisfying the following condition of finiteness in the direction corresponding to η:
In general the ring L m,η is rather complicated, however if η is a monomorphism, this ring is Euclidean by a theorem of J.-Cl. Sikorav (see [14], Th. 1.4).
Let ξ : G → R be a homomorphism. We can factor it as follows:
(16) G / / H 1 (X, Z)/T ors ≈ / / Z n / / p ! ! ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ R Z m ξ > > ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥
where p is an epimorphism and ξ is a monomorphism. The Novikov completion of the ring R[Z m ] with respect to ξ will be denoted by L m,ξ ; this ring is principal. Denote by ρ ξ the composition
G ρp / / GL(r, L m ) / / GL(l, L m,ξ ). Definition 8.2. ([6]
, [15]) The L m,ξ -module
H * C * ( X) ⊗ ρp L l m,ξ
is called the ρ-twisted Novikov homology of X with respect to ξ. The rank and the torsion number of its k-th component are denoted respectively by b ρ k (X, ξ) and q ρ k (X, ξ). The geometric reasons to consider these completions of the module C * ( X) are as follows. If X is a compact manifold, and ω is a closed 1-form on M with non-degenerate zeroes, denote by ξ the period homomorphism [ω] = ξ : H 1 (M, Z) → R. The Morse-Novikov theory implies the following lower bounds on the number of zeroes of ω:
m k (ω) 1 r b ρ k (X, ξ) + q ρ k (X, ξ) + q ρ k−1 (X, ξ)
. Compared to the other versions of the Novikov homology, the twisted Novikov homology has the advantage of being computable, and at the same time to keep the information about the non-abelian structure of G and related invariants. In a recent work [5] S. Friedl and S. Vidussi proved that the twisted Novikov homology detects fibredness of knots in S 3 . At present we will need only the simplest part of the twisted Novikov homology invariants, namely the twisted Novikov Betti numbers. is a free finitely generated abelian group; its rank is called irrationality degree of ξ and denoted Irrξ. In particular Irrξ = 1 if and only if ξ is a multiple of a homomorphism G → Z. We say that ξ is maximally irrational if Irrξ = n.
Observe that if ξ is maximally irrational, then the homomorphism p in the diagram ((16)) is an isomorphism and b ρ k (X, ξ) does not depend on ξ.
Definition 8.5. The number b ρ k (X, ξ) where ξ is maximally irrational, will be denoted by b • k (X, ρ).
It is not difficult to prove that
b k (X, ρ) b • k (X, ρ) = b • k (X, ρ),(17)b ρ k (X, ξ) b • k (X, ρ) for every ξ : G → R.(18)
Theorem 7.3 of the previous section implies the following. Proposition 8.6. Let k 0, q > 0. Then there is a finite family of proper full subgroups G i ⊂ Hom(Z n , Z) such that for the condition b ρ k (X, ξ) b • k (X, ρ) + q is equivalent to the following condition: ξ ∈ i G i ⊗ R.
HOMOLOGY WITH LOCAL COEFFICIENTS
Let us proceed now to the homology with local coefficients. Let ρ : G → GL(n, C) be a representation of G and α ∈ H 1 (X, C). The cohomology class α can be considered as a homomorphism α : G →
C[[t]]
is injective by the previous lemma, and therefore can be further extended to a homomorphism Γ : {L m } → C((t)) of the fraction fields. Therefore
dim {Lm} H k C * ( X)⊗ ρp {L m } l = dim C((t)) H k C * ( X)⊗ ρp {L m } l ⊗C((t))
and the point 1) is proved. The point 2) follows from 1) and Proposition 8.3.
The next proposition follows immediately.
Proposition 9.3. Let k 0, q > 0. Then there is a finite family of proper full subgroups G i ⊂ Hom(Z n , Z) such that for the condition β k (X, γ) b • k (X, ρ) + q is equivalent to the following condition: ξ ∈ i G i ⊗ C.
FORMAL AND HYPERFORMAL SPACES II : THE JUMP LOCI
Let X be a manifold, ρ : π 1 (X) → GL(n, C) a representation, and α ∈ H 1 (X, C). In this section we assume that the spectral sequences F * r and L * r associated to the triple (X, ρ, α) degenerate at their second term, so that F * r (X, ρ, α) ≈ L * r (X, ρ, α) ≈ H * H * (X, ρ), α According to Section 6 this holds in particular when X is a Kähler manifold and ρ a semi-simple representation. The following theorem follows from Theorem 5.1, together with Propositions 8.3, 9.2. Theorem 10.1.
1) β k (X, γ) = β k (X, γ gen ) = B k (H * (X, ρ), α). 2) If α ∈ H 1 (X, R) then b ρ k (X, α) = β k (X, γ) = B k (H * (X, ρ), α). We deduce the following result concerning the jump loci of the Betti numbers: is equivalent to the condition α ∈ i G i ⊗ C.
For α ∈ H 1 (X, R) each of the following conditions (i), (ii)
b ρ k (X, ξ) b • k (X, ρ) + q; (22) B k (H * (X, ρ), α) b • k (X, ρ) + q (23)
is equivalent to the condition α ∈ i G i ⊗ R.
ACKNOWLEDGEMENTS
The main part of the work was accomplished during the second author's stay at the Graduate School of Mathematics, Tokyo university, in the fall of 2011. The second author is grateful to the Graduate School of Mathematics for warm hospitality.
The authors thank Hisashi Kasuya for valuable discussions concerning his work.
During the preparation of the paper, the authors became aware of the paper [4] by A. Dimca and S. Papadima, on related subjects. Theorem 6.1 of the present paper is also proved by a completely diffferent method in [4], see Theorem E.
Definition 3 . 3 .
33Let a ∈ H * (N * ), and r 1. We say that the (r + 1)tuple Massey product ξ, . . . , ξ, a is defined, if a ∈ M Z (r) . In this case choose any r-chain (ω 1 , . . . , ω r ) starting from a.The cohomology class of ∂C = ξω r is in M Z (r) (actually it is in M Z (N ) for every N ) and it is not difficult to show that it is well defined modulo M B (r) . The image of δC in M Z (r) /M B (r) is called the (r + 1)-tuple Massey product of ξ and ω: ξ, a (r+1) = ξ, . . . , ξ r , a ∈ M Z (r) M B (r) .
Example 3 . 4 .
34The double Massey product ξ, a (2) equals ξa, the triple Massey product ξ, a (3) equals the cohomology class of ξω 2 where dω 2 = ξω 1 , [ω 1 ] = a, etc.
Theorem 3 .
313. (P. Deligne, Ph, Griffiths, J. Morgan, D. Sullivan,[3], Th. 4.1) Let A * be a minimal algebra over a field of characteristic zero, generated (as a free graded-commutative algebra) in degree k by a vector space V
Theorem 5. 1 .
1The spectral sequences F * r (N * , α) and L * r (M, ρ, α) are isomorphic.Proof. The A * (M )-module A * (M, E)[[t]]can be considered as the vector space of exterior differential forms onM with values in E[[t]]. Let π : M → M be the universal covering of M . The vector bundle E is isomorphic to M × ρ C n , and π * E ≈ M × C n . Denote by T * (M ) the A * (M )-submodule of A * ( M , C n [[t]]), consisting of the differential forms on M which are equivariant with respect to the representation γ. Choose a closed 1-form ξ within the class α. Let F : M → C be a C ∞ function such that π * ξ = dF . The next lemma is obvious.Lemma 5.2. The homomorphism Φ : A * (M, E)[[t]] / / T * (M ); Φ(ω) = e tF π * (ω)is an isomorphism of DG-modules over A * (M ).
a manifold N we denote by S ∞ * (N ) the graded group of all singular C ∞ -chains on N . The integration map determines a homomorphismI : T * (M ) / / Hom G S ∞ * ( M ), C n [[t]] . Here C n [[t]]is endowed with the structure of a G-module via the representation γ : G → GL(n, Λ).Proposition 5.3. The induced map in the cohomology groups I * : H * (T * (M )) / / H * (M, γ) is an isomorphism. Proof. The argument follows the usual sheaf-theoretic proof of the de Rham theorem. Consider the sheaf T k on M whose sections over an open subset U ⊂ M are γ-equivariant k-forms on π −1 (U ) with values in C n [[t]]. Denote by A the sheaf on M , whose sections over U are γ-equivariant locally constant functions π −1 (U ) → C n [[t]]. We have an exact sequence of sheaves(8)
Theorem 6 . 3 .
63Let M be a hyperformal manifold, ρ ∈ Ch(G) and α ∈ H 1 (M, C). Then the spectral sequencesF r (M * (M ), ρ, α) ≈ L * (M, ρ, α) degenerate in their second term. Proof. The DG-module A * (M, E ρ ) is formal; apply Proposition 3.16 and the proof is over. Denote by b k (M, ρ, γ gen ) the k-th Betti number of M with coefficients in a generic point of the curve γ t (g) = ρ(g)e t α,g . Corollary 6.4. Let M be a hyperformal manifold, ρ ∈ Ch(G), and α ∈ H 1 (M, C). Then b k (M, ρ, γ gen ) = B k (H * (M, ρ), α).An big class of examples of hyperformal spaces is formed by Kähler manifolds, as it follows from C. Simpson theory of Higgs bundles[17].
Theorem 6 . 5 .
65Any compact Kähler manifold is hyperformal.
Proposition 8 . 3 .
83Let ξ : G → R be a homomorphism and ρ : G → GL(l, R) a representation. Then for every k we haveb k (X, ρ, p) = b ρ k (X, ξ) (where p is obtained from the diagram(16)).Proof. The twisted Betti Novikov number in question equals the dimension ofH k C * ( X) ⊗ ρp L l m ⊗ { L m,ξ } over the field of fractions { L m,ξ } of the Novikov ring. The Betti number b k (X, ρ; p) is the dimension of the vector space H k C * ( X) ⊗ ρ p L l m ⊗ {L m } over the field of fractions {L m }. The inclusion L m ⊂ L m,ξ extends to an inclusion of fields {L m } ⊂ { L m,ξ } and the result follows. Definition 8.4. For a homomorphism ξ : G → R the subgroup ξ(G)
Proposition 10 . 2 .
102Let k 0, q > 0. Then there is a finite family of proper full subgroups G i ⊂ Hom(Z n , Z) such that each of the following conditions (i), (ii)β k (X, γ) b • k (X, ρ) + q; (20) B k (H * (X, ρ), α) b • k (X, ρ) + q(21)
L m,η = λ ∀ C ∈ R, the set supp λ ∩ η −1 [C, ∞[ is finite .
Proof. If P ∈ C[z 1 , . . . , z m ] is a polynomial such that P (e α1t , . . . , e αmt ) = 0, write P = a I t I where the sum ranges over multiindices I = (k 1 , . . . , k m ) ∈ N m . Denote the string (α 1 , . . . , α m ) by α. The series ζ = P (e α1t , . . . , e αmt ) is then a finite sum of exponential functions of the form a I e t I,α . Observe that I, α = J, α if I = J , since α i are linearly independent over Q. Therefore ζ is a finite linear combination of exponential functions e tβ I with pairwise different β I . Thus ζ = 0 implies a I = 0 for all I.The representation γ factors through Z m as followsLet (e 1 , . . . , e m ) denote the canonical basis in Z m ; then we have Γ(e i ) = e tα i . The extension of Γ to a ring homomorphism Z[Z m ] → † A particular case of this proposition corresponding to the vanishing Novikov Betti numbers was proved by S. Papadima and A. Suciu in[16].
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A Dimca, S Papadima, arXiv:1206.3773Nonabelian cohomology jump loci from an analytic viewpoint. A. Dimca, S. Papadima, Nonabelian cohomology jump loci from an analytic viewpoint, arXiv:1206.3773.
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Twisted Novikov homology and circle-valued Morse theory for knots and links, e-print: math.GT/0312374. H Goda, A Pajitnov, Journal Publication: Osaka Journal of Mathematics. 423H. Goda, A. Pajitnov, Twisted Novikov homology and circle-valued Morse the- ory for knots and links, e-print: math.GT/0312374, Journal Publication: Os- aka Journal of Mathematics, 42 No. 3, 2005, p. 557 -572.
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Exact Couples in Algebraic Topology (Parts I and II). W Massey, Ann. Math. 56W. Massey, Exact Couples in Algebraic Topology (Parts I and II), Ann. Math. 56 (1952), pp. 363-396.
Many-valued functions and functionals. An analogue of Morse theory. S P Novikov, Sov. Math. Dokl. 24S.P. Novikov, Many-valued functions and functionals. An analogue of Morse theory , Sov. Math. Dokl. 24 (1981), 222 -226.
Bloch homology, critical points of functions and closed 1-forms. S P Novikov, Soviet Math. Dokl. 2872S.P. Novikov, Bloch homology, critical points of functions and closed 1-forms, Soviet Math. Dokl. 287 (1986), N • 2.
An analytic proof of the real part of the Novikov inequalities DAN SSSR. A Pajitnov, 293in RussianA. Pajitnov, An analytic proof of the real part of the Novikov inequalities DAN SSSR, 293, 1987 no. 6 (in Russian);
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. Graduate Kavli Ipmu, School, Mathematical, The Sciences, University, Tokyo ; Komaba, Meguro-Ku, TOKYO 153-8914, JAPAN E-mail address: [email protected] IPMU, GRADUATE SCHOOL OF MATHEMATICAL SCIENCES, THE UNIVERSITY OF TOKYO, 3-8-1 KOMABA, MEGURO-KU, TOKYO 153-8914, JAPAN E-mail address: [email protected]
. Jean Laboratoire Mathématiques, Université Leray Umr 6629, Faculté De Nantes, Rue Des Sciences, 2, De La, Houssinière, NANTES. 44072CEDEX E-mail address: [email protected] MATHÉMATIQUES JEAN LERAY UMR 6629, UNIVERSITÉ DE NANTES, FACULTÉ DES SCIENCES, 2, RUE DE LA HOUSSINIÈRE, 44072, NANTES, CEDEX E-mail address: [email protected]
| []
|
[
"Microscopic model of quantum butterfly effect: out-of-time-order correlators and traveling combustion waves",
"Microscopic model of quantum butterfly effect: out-of-time-order correlators and traveling combustion waves"
]
| [
"Igor L Aleiner \nColumbia Physics Department\nDepartment of Physics and Astronomy\nLaboratoire de Physique Theorique et Hautes Energies, CNRS UMR 7589\nUniversites\n530 W 120th Street, Paris 6 et 7, place Jussieu, Cedex 0510027, 75252New York, ParisNYUSA, France\n",
"Lara Faoro \nRutgers University\n136 Frelinghuysen Rd08854PiscatawayNew JerseyUSA\n",
"Lev B Ioffe \nRutgers University\n136 Frelinghuysen Rd08854PiscatawayNew JerseyUSA\n"
]
| [
"Columbia Physics Department\nDepartment of Physics and Astronomy\nLaboratoire de Physique Theorique et Hautes Energies, CNRS UMR 7589\nUniversites\n530 W 120th Street, Paris 6 et 7, place Jussieu, Cedex 0510027, 75252New York, ParisNYUSA, France",
"Rutgers University\n136 Frelinghuysen Rd08854PiscatawayNew JerseyUSA",
"Rutgers University\n136 Frelinghuysen Rd08854PiscatawayNew JerseyUSA"
]
| []
| We extend the Keldysh technique to enable the computation of out-of-time order correlators such as O(t)Õ(0)O(t)Õ(0) . We show that the behavior of these correlators is described by equations that display initially an exponential instability which is followed by a linear propagation of the decoherence between two initially identically copies of the quantum many body systems with interactions. At large times the decoherence propagation (quantum butterfly effect) is described by a diffusion equation with non-linear dissipation known in the theory of combustion waves. The solution of this equation is a propagating non-linear wave moving with constant velocity despite the diffusive character of the underlying dynamics.Our general conclusions are illustrated by the detailed computations for the specific models describing the electrons interacting with bosonic degrees of freedom (phonons, two-level-systems etc.) or with each other. | 10.1016/j.aop.2016.09.006 | [
"https://arxiv.org/pdf/1609.01251v2.pdf"
]
| 118,645,872 | 1609.01251 | a5bc3059ad9f8ac498af4ab94d34196a67b1edd1 |
Microscopic model of quantum butterfly effect: out-of-time-order correlators and traveling combustion waves
Igor L Aleiner
Columbia Physics Department
Department of Physics and Astronomy
Laboratoire de Physique Theorique et Hautes Energies, CNRS UMR 7589
Universites
530 W 120th Street, Paris 6 et 7, place Jussieu, Cedex 0510027, 75252New York, ParisNYUSA, France
Lara Faoro
Rutgers University
136 Frelinghuysen Rd08854PiscatawayNew JerseyUSA
Lev B Ioffe
Rutgers University
136 Frelinghuysen Rd08854PiscatawayNew JerseyUSA
Microscopic model of quantum butterfly effect: out-of-time-order correlators and traveling combustion waves
Quantum butterflydecoherenceout-of-time-order
We extend the Keldysh technique to enable the computation of out-of-time order correlators such as O(t)Õ(0)O(t)Õ(0) . We show that the behavior of these correlators is described by equations that display initially an exponential instability which is followed by a linear propagation of the decoherence between two initially identically copies of the quantum many body systems with interactions. At large times the decoherence propagation (quantum butterfly effect) is described by a diffusion equation with non-linear dissipation known in the theory of combustion waves. The solution of this equation is a propagating non-linear wave moving with constant velocity despite the diffusive character of the underlying dynamics.Our general conclusions are illustrated by the detailed computations for the specific models describing the electrons interacting with bosonic degrees of freedom (phonons, two-level-systems etc.) or with each other.
10 Acknowledgement 37
Motivation
In a chaotic classical system a small perturbation leads to the exponential divergence of trajectories characterized by Lyapunov time, 1/Λ. As a result, the observables in two copies of the system experiencing different perturbations quickly become uncorrelated. In a many body system a local perturbation initially destroys the correlations locally, then the region where the correlations are destroyed quickly grows with time. Killing a butterfly in Ray Bradbury story [1] leads to the spreading perturbation until it reaches the size of the system (Earth in this story). This phenomena is known as butterfly effect.
The concept of butterfly effect can be generalized to a closed chaotic quantum system even though such generic system does not necessarily have a direct analogue of Lyapunov divergence of trajectories because quantum mechanics prohibits the infinitesimal shift of the trajectory. The convenient measure of the butterfly effect is provided by the out-of-time-order correlator (OTOC) that was first introduced by Larkin and Ovchinnokov [2], revived by Kitaev [3,4] and extensively discussed by a number of works recently [5,6,7,8]. OTOC is defined by
A(t) = O(t)Õ(0)O(t)Õ(0) ,(1)
where O(t) andÕ(t) are two local operators in Heisenberg picture. Physically, it describes how much the perturbation introduced byÕ(0) changes the value of the O(t). At large times A(t) goes to a zero, because the state created by the consecutive action of the operators O(t)Õ(0) is incoherent with the state obtained when these operators act in a different order. 1 The anomalous time order in the correlator (1) implies the evolution backward in time, so it is not measurable by direct physical experiments on one copy of the system in the absence of a time machine such as implemented in NMR experiments [9]. One can view the the decrease of the OTOC with time as the consequence of the dephasing between two initially almost identical Worlds evolving with the same Hamiltonian. In this respect it is different from the problems of fidelity [10] and Loschmidt echo ( [9] and references therein) that study evolution forward and backwards with slightly different Hamiltonians. It is also different from a problem of the evolution of a particle along quasiclassically close trajectories appearing in studies of the proximity effects [2] or weak-localization [11] and quantum noise [12]. For physical systems the Hamiltonian is local, so that distant parts of a system are not interacting directly with each other. In this case, one may further distinguish the case when operators O andÕ act far from each other in real space. One expects that the correlator decreases after the significant delay needed for the perturbation to spread over the distance separating these operators. When correlators of this type decayed for any separation between the operators in the real space the coherence is completely lost. The decay of OTOC at long times for all subsystems (i.e. for all separations) for all operators O and O implies complete quantum information scrambling [13]. Note that the separation of the operators in space is equivalent to the separation into subsystems introduced in quantum information works. We are not going to discuss here quantum information implications of OTOC and the exact definition of quantum scrambling; we refer the reader to the literature that discussed its theory [14,15,16,17,18] and the possibility of its experimental measurement [19,20].
The goal of this work is to develop the analytic tools to study OTOC (1) for microscopic models that allow for the solutions for conventional correlators.
The technique that we develop is essentially a straightforward extension of the Keldysh technique. We apply our technique to three models that are basic in condensed matter physics: (i) electrons interacting with localized bosonic degrees of freedom (Einstein phonons or simplified two level systems), (ii) electrons in the disorder potential and (iii) electrons weakly interacting with each other. We find that in models (i) and (iii) the mathematical description of the OTOC is similar to the description of the combustion waves. The small initial perturbation first grows exponentially remaining local and then starts to propagate with a constant velocity and a well defined front, despite the fact that the thermal transport in these models is always diffusive. The velocity of the front propagation is always slower than electron Fermi velocity and it is parametrically slower than it in some models. The apparently slow velocity of the front propagation implies that it does not necessarily saturate Lieb-Robinson bound [21]. This conclusion of the constant velocity of the quantum butterfly propagation agrees with the result obtained in holographical theory of black holes [6,22,23].
The plan of this paper is the following. In Section 2 we introduce the basic elements of our technique: the augmented Keldysh formalism that involves two forward and two backward paths. The state of the system in this formalism is described by the diagonal and off-diagonal Green functions in the augmented space. The diagonal functions describe the quasiparticle distribution function in each "world", the off-diagonal ones describe the coherence between the "worlds". In Section 2.3 we introduce two types of correlators that one can compute in this technique: the observables that can be measured directly in a physical experiment and the computables that one can only compute numerically (or measure given the time machine). In Section 3 we introduce the details of the microscopic models for which the anomalous correlator of type (1) will be computed. In Section 4 we derive the analogue of the kinetic equation for both diagonal and off-diagonal ones. In Section 5 we analyze the stability of the kinetic equations of Section 4 ignoring their spatial structure (i.e. in zero dimensional case) and show that the instability of the off-diagonal functions is described by non-linear ordinary differential equations. Section 6 generalizes these equations for the models with spatial structure for which they become similar to the equations describing the combustion waves. Section 7 describes the formation of the propagating front that follows from the non-linear diffusive equations derived in Section 6. The Section 8 studies the initial time period at which the state of the system is not yet accounted for by the diffusive equations and its match to the evolution at longer times. Finally, the Section 9 gives the summary of the results and discussions of possible extensions.
Augmented Keldysh and Keldysh techniques
Augmented Keldysh technique
Anomalously ordered correlator such as out of time ordered A(t) introduced in the Section 1, see Eq. (1), cannot be computed in conventional techniques that assumes casual time evolution. To circumvent this difficulty we augment the standard Keldysh technique by introducing two forward and two backward evolutions shown in Fig. 1b.
In order to describe the augmented technique we recall the conventional Keldysh technique [24] first. There, the differently ordered correlators are given by
N αβγδ (t) = T C KÔ α (t 2 )Ô β (t 1 )Ô γ (t 2 )Ô δ (t 1 ) exp −i C KĤ int (t)dt ,(2)
where α, β, γ, δ = ± denote the positions of the operators on the traditional Keldysh contour shown in Fig. 1a. Here all observable operatorsÔ and the interaction part of the HamiltonianĤ int are in the interaction representation, the averaging is done with a density matrix (that represents the initial conditions in the past), the symbol T C K denotes ordering of the operators on the Keldysh contour, i.e. the operator referring to the position down the contour is on the left in (2) (for fermion operators the change of order brings in minus sign). One can see that by choosing indexes α, β, γ, δ = ± one can get different order of operators but never the anomalous order required by Eq. (1). Figure 1: The traditional, C K , (a) and the augmented (b) Keldysh contours C aK . Times t 1,2 label the insertion of the operators for observable or computable quantities, see text.
(a) − + t1 t2 (b) d − d + u − u + t1 t2
Operators (fermionic or bosonic) are ordered according their location on the contours C K , C aK .
The augmented contour C aK allows anomalous order of the operators such as in Eq. (1). For this contour the indices α, β, γ, δ can acquire four values, u±, d± (where u stays for the up and d for the down parts of the contour). The expression similar to Eq. (2)
A αβγδ (t) = T C aKÔ α (t 2 )Ô β (t 1 )Ô γ (t 2 )Ô δ (t 1 ) exp −i C aKĤ int (t)dt ,(3)
with the choice α = u+, β = d+,γ = u−, δ = d− becomes out-of-order correlator A(t). Clearly other combinations of indices will produce normal as well abnormal correlators. Equation (3) is the essence of the augmented technique. Unitary evolution on u/d segments of the contour can be viewed as the evolution of the different worlds (we will use this term loosely throughout the paper) with the same Hamiltonian and the same initial conditions ("correlated worlds " initially). The correlator (3) can be viewed as the response at time t to the perturbation (source) at time 0. When the sources are located at the same up/down parts of the contour, the response is directly measurable, we shall refer to these sources as 'physical'. All the other sources will be referred to as 'unphysical'. Our ultimate objective is to describe how these "correlated worlds" become "uncorrelated worlds" provided that a small local perturbation is seeded differently in the two worlds (butterfly effect). 2 In the next few sections we generalize the rules and the results of Keldysh technique for the augmented Keldysh technique. We will see that almost all rules are going through up to the kinetic equation where the correlation function describing not only the occupation numbers but also the measure of the correlation between the different worlds.
Augmented space and Green function
Similarly to usual diagrammatic technique, we introduce the 4 × 4 matrix of Green functions of Fermi or Bose fields. It is convenient to view this four dimensional space as a direct product of 2 × 2 Keldysh and 2 × 2 augmented space. Each operator (fermionic or bosonic) ψ(t) can be placed in four different points of contour at time t, therefore it is enlarged into four dimensional vector:
Ψ(1) = ψ u (1) ψ d (1) a ; ψ i (1) = ψ i,+ (1) ψ i,− (1) K(4)
where (1), (2) are the short hand notations for the coordinates, times (and might be spin) that specifies the single particle state: i ≡ (t i , r i , σ i ). In these notations the 4 × 4 matrix Green function reads
iĜ(1, 2) = T C Ψ(1) ⊗ Ψ † (2) .(5)
As usual, the components of the Green functions are linearly dependent. This redundancy is eliminated by the Keldysh rotation, which is conveniently described by the Pauli matriceŝ
τ · 0 = 1 0 0 1 ;τ · 1 = 0 1 1 0 ;τ · 2 = 0 −i i 0 ;τ · 3 = 1 0 0 −1 ; τ · + = 0 1 0 0 ;τ · − = 0 0 1 0 .(6)
The superscript, · = a, K, describes the space (augmented or Keldysh) in which these matrices act. In terms of matrices (6) the Keldysh rotation is given by 3
Ψ →RΨ,(7a)Ψ † →Ψ = Ψ †R † · τ K 1 ⊗τ a 0 ,(7b)G → RĜR † · τ K 1 ⊗τ a 0 ,(7c)R = exp iπτ K 2 ⊗τ a 0 4 (7d)
After rotation (7), the Green function acquires the form
G = Ĝ uuĜud G duĜdd a , whereĜ uu = G R G K 0 G A K ,Ĝ ud = 0 Γ K 0 0 K G du = 0Γ K 0 0 K ,Ĝ dd = G RGK 0G A K .(8)
In the absence of the non-physical sources 4 , the components diagonal in the augmented space are equal,Ĝ uu =Ĝ dd , and coincide with the conventional Green functions. In particular, the retarded, advanced and Keldysh Green functions, G R,A,K are given by
iG R (1,2) = ψ(1)ψ † (2) ± ψ † (2)ψ(1) θ(t 1 − t 2 ), iG A (1,2) = − ψ(1)ψ † (2) ± ψ † (2)ψ(1) θ(t 2 − t 1 ), iG K (1,2) = ψ(1)ψ † (2) ∓ ψ † (2)ψ(1) ,(9)
(hereinafter, the upper sign corresponds to fermions and lower to bosons unless stated otherwise), whilst the inter-world functions read:
Γ K (1, 2) = −2i ψ(1)ψ † (2) = G K (1, 2) + [G R (1, 2) − G A (1, 2)],(10)Γ K (1, 2) = ±2i ψ † (2)ψ(1) = G K (1, 2) − [G R (1, 2) − G A (1, 2)].
In the absence of non-physical sources, these functions include the information on the single particle spectrum and on the distribution functions of holes (particles), Γ K (Γ K ). We note that even in the presence of non-physical sources the diagonal components are not influenced by the non-diagonal ones. This is because the correlations between the upper and down worlds can not affect the dynamics in each of these worlds. Formally, this means that the structure of the Green functions always retain the form of Eq. (8). Only the relation (10) between diagonal and non-diagonal components in augmented space can be modified by the presence of non-physical sources. In fact, the violation of the relation (10) will be the formal indicator of the quantum butterfly effect.
Observables and computables
We distinguish the correlators (observables) that can be in principle measured by a physics experiment and the ones that can only be studied in the rather artificial system that allows inversion of time directions. Because the latter can be more readily studied by numerical simulations, where the unitary evolution can be formally reversed 5 , we call them computables.
The example of observable is given by the casual correlator
N ρρ (t) = 1 2 T C Ψ (t)(τ K 1 ⊗ τ a 0 )Ψ(t) Ψ (0)(τ K 0 ⊗ τ a 0 )Ψ(0)(11)
that describes the density response at time t to the perturbation at time 0. Indeed the correlator (11) rewritten in terms of the original fields ψ has the form N ρρ (t) = ψ † (t, r)ψ(t, r), ψ † (0, r 0 )ψ(0, r 0 ) , the usual rules of linear response imply that the density induced by the scalar potential applies at point r 0 at time t = 0 is given by −iN ρρ (t). In fact this structure is general for Keldysh technique: the physical perturbation comes with τ K 0 while the observable comes with τ K 1 . In contrast the out-of-time-ordered correlator provides the example of the computable. In this paper we focus on out-of-time-ordered correlators of the form
A γδ αβ (t, r; t r ) = T C ψ α (t, r)ψ † β (t , r) ψ † γ (0, 0)ψ δ (0, 0)(12)
that becomes out-of-time-ordered for many combinations of indices α, β, γ, δ. For instance, for α = u+, β = d+,γ = u−, δ = d− it provides an example of the general correlator (3) discussed in Section 2.1. It is convenient to separate, as we have done here by parenthesis, the 'source' term provided by the product of two operators at time t = 0 and the 'response term' provided by two operators at time t ≈ t > 0. For the fixed γ and δ the correlator (12) can be viewed as the Green function G αβ (t, r; t , r) computed in the states modified by the action of the operators ψ γ (0, 0)ψ † δ (0, 0). In particular, it satisfies the same identities as the Green function:
A γδ u+,d− = A γδ u−,d− = A γδ u−,d+ = A γδ u+,d+ .(13)
After Keldysh rotation in indices α and β the correlator (12) acquires the same general form as the Green function (8).
Because of the identities (13) one can choose many equivalent forms of the out-of-time-ordered correlators that display unusual behavior. It will be more convenient to us to compute the symmetrized correlator defined by
A ρρ (t, t , r) = T C Ψ (t , r)(τ K 1 ⊗ τ a 1 )Ψ(t, r) Ŝ 0 .(14)
The source term,Ŝ 0 , can have many equivalent forms that distinguish upper and down Worlds, we can choose for instancê
S 0 = ψ † u− (0, 0)ψ d− (0, 0)(15)
This term destroys one particle in the down World and creates it back before the evolution in the upper World starts (notice that for the operators at t = 0 ψ † u− (0) = ψ † d+ (0), see Fig. 1). As we shall see below the final results depend very weakly on the particular form of the source term.
The response operator in this correlator is the sum of four termŝ
R tt (r) =Ψ(t , r)(τ K 1 ⊗ τ a 1 )Ψ(t , r) = ψ † u− ψ d− − ψ u− ψ † d− + ψ † u+ ψ d+ − ψ u+ ψ † d+(16)
that measures the product of the distribution functions and the correlations between the worlds. The minus sign in this equation is due to fermionic commutation rules.
As usual, any correlator allows for a pictorial representation to facilitate the basic structure of the theory and to be able to sum up the most important parts of the perturbative expansions up to infinite order. We develop the diagrammatic rules for the technique in the augmented space below in sections 2.6, 2.7, 2.8.
In the absence of the unphysical sources the two worlds remain perfectly correlated. The stability of this solution can be discussed in very general terms without the knowledge of the details of the microscopic model. In fact, the existence of the self-energy, Dyson equation, and the general thermodynamic relations are sufficient to prove that the perfectly correlated solution is unstable. We begin with these general considerations.
Dyson equation.
In any field theory that allows the separation of the Hamiltonian into bare (H 0 ) and interacting (H int ) parts, one can introduce the notion of bare Green function, G 0 , corresponding to Hamiltonian H 0 , the full Green functions (defined above) and the self energies,Σ, that take into account the effects of the interaction on the bare Green functions. In diagramm technique the self-energy can be defined as the sum of all one-particle-irreducible diagrams (see Sections 2.6, 2.7, 2.8). The Green functions and self-energies obey the Dyson equation that can be written in two equivalent forms
(Ĥ 0 τ a 0 −Σ) •Ĝ =1,(17a)G • (Ĥ 0 τ a 0 −Σ) =1,(17b)
where1 is the unit operator in the space-time and the augmented Keldysh space, the symbol • implies the matrix multiplication in the augmented space and the convolution in space-time. The operatorĤ 0 is diagonal in Keldysh and augmented spaces with the diagonal elements defined by equationsĤ 0 G R 0 = 1,
H 0 G A 0 = 1, it is related to the Hamiltonian H 0 byĤ 0 = id/dt − H 0 .
The general structure of the Green functions (8) implies that the parts of the Green function that are retarded and advanced in Keldysh space remain diagonal in the augmented space. Because the bare retarded and advanced Green functions are diagonal in the augmented space, the self energies Σ A,R αβ = δ αβ Σ A,R α remain also diagonal and they are given by the solution of the equations
(Ĥ 0 − Σ R/A α ) • G R/A α = 1 (18a) G R/A α • (Ĥ 0 − Σ R/A α ) = 1, α = u, d.(18b)
As usual, the non-diagonal part of the Green functions in Keldysh space is not entirely determined by the Eqs. (18): it also depends on the initial conditions. Its evolution is described by the homogeneous equations
(Ĥ 0 − Σ R α ) • G K αβ − Σ K αβ • G A β = 0, (19a) G K αβ • (Ĥ 0 − Σ A β ) − G R α • Σ K αβ = 0. (19b)
Notice that both the diagonal and the non-diagonal parts G K αβ are controlled by the initial conditions. We emphasize that the diagonal components Σ A,R α , Σ K αα may depend on the diagonal components of the Green functions in the augmented space, G K αα , but not on the other diagonal (e.g. G A,R β , G K ββ β = α) or the non-diagonal ( G K αβ ) Keldysh components. This observation turns out to be the key of the description of the instability in the evolution of non-diagonal correlations as we see in the next subsection.
Stability and instability.
Let us consider the fermionic Green function for the sake of concreteness. The Keldysh components of the Green function can be conveniently parametrized via
G K αβ = G R α • F αβ − F αβ • G A β.(20)
For α = β this equation reduces to the conventional parametrization of G K in terms of the quantum distribution function, F uu and F dd ; for α = β it gives the parametrization of the new functions Γ K andΓ K in terms of F ud and F du . Substituting Eq.(20) into Eqs. (19) and using Eqs. (18), we find that Eqs. (19) are satisfied for F αβ solving the quantum kinetic equation
H 0 • F αβ − F αβ • H 0 = Σ R α •F αβ − F αβ • Σ A α − Σ K αβ .(21)
In the quasiclassical approximation the two terms in brackets correspond to the outgoing scattering processes (this term taken alone always leads to dissipation) whilst the last term corresponds to the incoming processes (this term taken alone always leads to instability).
In thermal equilibrium the Green functions depend only on the time difference. The diagonal parts of the electron self energies are related by the fluctuation-dissipation theorem (FDT):
Σ K αα ( ) = Σ R α ( ) − Σ A α ( ) n 0 ( ), n 0 ( ) = tanh − µ 2T (22)
where is the frequency conjugated to the time difference, µ is the chemical potential, and T is the temperature: both of them are determined by the initial conditions. For Bosons one should replace n 0 ( ) = tanh(. . .) by p 0 (ω) = coth(. . . ). For phonons (which number is not conserved) the chemical potential µ = 0. In equilibrium the left hand side of Eq. (21) is zero, substituting Eq. (22) into (21) we see that [1 − F uu ( )] /2 has the meaning of the Fermi distribution function.
The FDT also implies that Eq. (21) has a generally stable solution. The only reason for this solution to become unstable is the metastability of the state that might happen on the unstable branch of the phase transition. However, even in this case, the ultimate fate of the system is a different equilibrium characterized by different Σ R,A α and
F uu = F dd = n 0 ( ),(23)
with different µ and T that are found from the number of particle and energy conservation in the new spectrum. That solution would be stable again.
The stability of the thermal solution of Eq. (21) is guaranteed by Boltzmann H-theorem and the global conservation laws (energy and the number of particles). In the framework of Eq. (21) it means that for small deviations of F uu , F dd from the thermal distributions the outgoing terms dominate incoming ones in Eq. (21). This fact is far from trivial because both terms are generically non-linear.
The equation (21) allows for solution similar to Eq. (23) for the off-diagonal components:
F ud = 1 + n 0 , F du = −1 + n 0 .(24)
We will call this solution the "correlated worlds solution". In the Pauli matrix notation given by Eq. (6), Eq. (23) and Eq. (24) can be compactified (for fermions) aŝ
F = n( , p, r, t) (τ a 0 +τ a 1 ) + iτ a 2 ,(25)
where anticipating further applications, we allow the distribution function to depend not only on energy but also on the phase space variable and time. The precise definition of the notion of the semiclassical phase space is given in Sec.
4.
Notice that in contrast to the solution Eq. (21), the stability of the correlated worlds solution is not guaranteed even if the solution Eq. (23) is stable. Indeed, for the diagonal (conventional) distribution function the small deviation from equilibrium results in the non-zero RHS of Eq. (21) in which outgoing terms always dominate incoming ones. Outgoing terms always imply relaxation, which leads to the stability of the equilibrium solution. For non-diagonal distribution function, the small deviation from equilibrium results in the same incoming terms as for diagonal distribution function but smaller outgoing terms. A small deviation of diagonal term leads to two contributions to the outgoing term: one due to the interaction induced change in Σ A,R α , another due to the change in F uu (F dd ) itself. Because Σ A,R α do not depend on the off-diagonal terms F ud (F du ), the former contribution is missing for the off-diagonal terms. This makes deviation of the outgoing term that tries to restore equilibrium smaller for non-diagonal distribution function in the interacting system. Thus, for the offdiagonal distribution function the outgoing terms do not necessarily dominate the incoming ones for small deviations from the equilibrium. This results in a possible instability of the solution of Eqs. (24). The computations for the specific models below show that this instability is indeed present for electronphonon and electron-electron interactions but not for impurity scattering.
The alternative solution (allowed by conservation laws for off-diagonal components) is
F ud = F ud = 0, orF = n( , p, r, t)τ a 0 ,(26)
this solution will be called the "uncorrelated worlds solution". The incoming term is second order (or higher) in F ud (F du ), therefore it vanishes for the small deviation from this solution. In contrast, the outgoing term is always linear in F ud (F du ) and it dominates. Thus, the uncorrelated worlds solution is generally stable and one expects that the correlated worlds solution (24) is not. The only exception is the electron scattering by impurities that conserves the number of particles at each energy separately. In this case both outgoing and incoming terms are linear in all components of F αβ and the previous arguments do not hold.
The meaning of the "uncorrelated worlds solution" (26) is the following. Unlike their diagonal counterparts, F ud (F du ) encode not only the distribution functions but also the overlap of the many-body wave-functions evolving at the upper and lower contour. Any decrease of this correlation diminishes the values of both F ud (F du ). The proposed instability is therefore nothing but the quantum butterfly effect, the decay of F ud (F du ) everywhere in the system results in the loss of the coherence between many body wave functions describing upper and down Worlds. The ultimate solution given in Eq. (26) corresponds to the complete destruction of the coherence between lower and upper contour.
The description of the evolution of the system from the "correlated worlds solution" (25) to the "uncorrelated worlds solution" (26) is the subject of the further sections.
Basic rules of the diagram technique: Green functions
The basic elements of the diagrammatic representation needed to compute the correlators in the augmented space are shown in Fig. 2. Notice that for keeping track of the Keldysh structure putting arrows on the Green function for the real fields (as it is done throughout this paper) is convenient but not necessary. In the absence of interactions the observable (11) and the computable (14) are given by the diagrams shown in Fig. 3. Introduction of the separate notation for the box (see Fig. 3 ) enables one to display the matrix structure of the interaction vertices as well.
= iĜ = iĜ 0 1 2 3 i, j = [iĜ(1, 3) · (τ K i ⊗τ a j ) · [iĜ(3, 2)] 2 1 i, j =n(τ K i ⊗τ a j ) · [iĜ(1, 2)] 1 2 i, j = iĜ(1, 2) · (τ K i ⊗τ a j )n n = ⎡ ⎢ ⎢ ⎣ 1 0 K 1 0 K ⎤ ⎥ ⎥ ⎦ a ; n = 0; 1 K , 0; 1 K a ;
Vertices
In order to develop the perturbation theory one needs to supplement the expression for the Green functions with the expression for the bare vertices. Because the unitary evolution in each sector is formally independent, these vertices do not couple different sectors of the augmented space. In the Keldysh space they have the usual structure.
To illustrate this point, more for the benefit of the readers familiar with the conventional Keldysh technique, let us consider the textbook [27] example of the perturbation theory for the electrons interacting with phonons. The lowest order contribution to the electron self-energy has the form (formal general rules for the diagram techniques will be summed in the next subsection): (11) and (14) in non-interacting problem.
Σ ab uu = iλ 2 γ k aa G a b uu D kk uuγ k b b ,(27a)Σ ab ud = iλ 2 γ k aa G a b ud D kk udγ k b b ,(27b)Σ ab ud = iλ 2 γ k aa G a b ud D kk udγ k b b ,(27c)Σ ab dd = iλ 2 γ k aa G a b dd D kk ddγ k b b , (27d) N ρρ (t) = t 0 1, 0 0, 0 A ρρ (t) = t 0 1, + 0, +
where
γ 1 ij =γ 2 ij = 1 √ 2 τ 0 K ij , γ 2 ij =γ 1 ij = 1 √ 2 τ 1 K ij .
The diagonal components in the augmented space coincides with the ones for the regular technique. The non-diagonal ones are found by using the Wick's theorem and noticing that the vertices by themselves do not mix different sectors of the augmented space. The matrix structure displayed by Eq. (27) can be further compactified by equation
Σ αβ = iΥ γ αα G α β λ 2 D γγ 4 Υ γ β β ,(29)
where 4 × 4 × 4 matrices Υ γ αα are given by
Υ (i ,j ) (i,j),(i ,j ) = l=0,1 m=0,3 n τ K l ⊗ τ a m (i ,j ) τ K l ⊗ τ a m (i,j),(i ,j ) , (30a) Υ (i ,j ) (i,j),(i ,j ) = l=0,1 m=0,3 τ K l ⊗ τ a m n (i ,j ) τ K l ⊗ τ a m (i,j),(i ,j ) . (30b)
Here0 ≡ 1,1 = 0,n = (0, 1; 0, 1), n = (1, 0; 1, 0) T , and the sum over index m = 0, 3 gives the sum τ a 0 ⊗ τ a 0 + τ a 3 ⊗ τ a 3 that is different from zero (and equal 2) only for coinciding indices in the augmented space. Pictorially, this matrix structure can be summarized by Fig. 4 where the basic blocks (boxes and triangles) are again defined in Fig. 2. The appearance of the vector n in the formalism is due to the non-conservation of the number of particles (phonons), see Fig. 2 c).
Note, that the same vertex Υ describes the interaction of the electron with the disorder potential. The only difference is that there is no time dependence of the quenched disorder potential, thus the impurity line connecting different branches of the augmented Keldysh contour never decays. The general structure of the vertices for electron-electron interaction is shown in Fig. 4a. Note that is is described by the same building blocks as the electron-phonon and electronimpurity interaction. Because the number of particles in the electron-electron interaction is conserved the vector n does not appear in this case. The definition of vertices has to be supplied with the remaining bosonic Green functions, defined in Fig. 5
1 2 = − i 4 U (r 1 − r 2 )δ(t 1 − t 2 ) (b)
Diagram technique: summary
We are now prepared to formulate the general rules of the diagram technique that operates with the blocks defined in sections 2.6, 2.7:
In order to compute the correlator (observable or computable) one has: (i) to place the sources and the interaction vertices, (ii) to connect them by the Green function lines, (iii) to trace over the indices in the augmented Keldysh space, (iv) to integrate over positions of the interaction vertices, (v) to multiply the result by (−1) N F L , where N F L is the number of the closed fermionic loops. As usual, in order to derive the physical properties at large scales one introduces the notion of self-energy that is defined as the sum of all one-particle- irreducible diagrams. For example, the self-energies for the electron-phonon, electron-electron and electron-disorder interaction are shown in Fig. 6.
(a) = iλ 2 4D ; = iλ 2 4D 0 ; (b) 1 2 = V (r 1 − r 2 )τ K + ⊗ [τ a 0 +τ a 1 ] ;
Microscopic Models
The instability expected in section 2.5 is of kinetic nature. Its form depends on the detailed form of the kinetic equation and thus on the microscopic model on which the latter is based. In the following we describe the models that allow one to study the development of the instability in detail.
In all these models the main ingredient are mobile electrons that form a Fermi sea. They are described by the quadratic Hamiltonian
H el = p ξ p ψ † p ψ p(31)
and characterized by the the bare Green function
G R 0 = 1 − ξ p + i0 , G A 0 = 1 − ξ p − i0 ,(32)
where the single particle energy ξ p is counted from the Fermi energy F . The condition ξ p = 0 defines the Fermi surface of the electrons. For electrons the operator H 0 introduced in Eqs. (17) acquires the form
H 0 = i ∂ ∂t − ξ p , p = −i ∂ ∂r .(33)
Here we restored the units of for future convenience in developing the quasiclassical approximation later on. The three models for the electron interaction that we formulate below differ by their conservation laws. The primitive model of electron-phonon interaction (section 3.1) preserves the total energy of the system and the number of electrons but not the momentum of the system. The electrons in the impurity potential (section 3.2) is not a translational invariant system, however, the Figure 6: Self-energies for the (a) electron-phonon interaction, (b) electron-electron interaction; and (c) electron in the Gaussian disordered potential. The first order self-energy for the electron electron interaction is discarded as not-related to the collisions but rather renormalizing the self-consistent spectrum for the deterministic motion. The inside lines for the Green function are solid which means that infinite series of the rainbow diagrams is summed. This approximation is known to lead to the quasi-classical Boltzmann equation and can be justified for weak interactions or small enough disorder strength (neglecting localization effects).
scattering by impurities conserves the energy of individual electrons, leading to infinite number of conservation laws in this problem. The electron-electron interaction (section 3.3) preserves both the translational and Galilean invariance, so it conserves the total energy, momentum, and the particle number. 6
Electron-phonon interaction
The simplest interacting model is the one in which the electrons interact with dispersionless phonons with frequency ω 0 (Einstein phonons) with Hamiltonian
H ph = r ω 0 b † r b r ,(34)
To avoid inconsequential consideration of the band structure we simply assume (somewhat artificially) that all the points r are random and dilute, their density per unit volume is n ph . Notice that n ph represents the density of phonon sites, the density of thermally excited phonons is the product of n ph and phonon occupation number. Bosons are interacting with electrons via
H el−ph = r u r ψ † (r)ψ(r),(35)
where u r = λ(b r + b † r ). Correlators of field u are given by the Green functions local in space,
D R 0 = 2ω 0 n ph δ(r 1 − r 2 ) (ω + i0) 2 − ω 2 0 , D A 0 = 2ω 0 n ph δ(r 1 − r 2 ) (ω − i0) 2 − ω 2 0 .(36)
Poles at positive frequencies in these functions correspond to phonon emission and at negative frequencies to phonon absorption (while the physical energies of phonons are of course positive). These Green functions should be used as the basic elements of the diagram technique shown in Fig. 5a. Here and below we adopt the traditional convention in which the phonon frequency is denoted by ω whilst reserving for the electron energy. For phonons the operator H 0 , introduced in Eq. 17, acquires the form
H 0 = − ∂ 2 ∂t 2 − ω 2 0 .(37)
Similarly to the electrons, see Eq. (20), the Keldysh part of the phonon Green function can be parametrized by
D K αβ = D R α • P αβ − P αβ • D A β(38)
.
With this parametrization the form of the kinetic equation (21) for the phonons remains the same;, the only difference is that their equilibrium distribution functions for the correlated world solution iŝ
P = p ph (ω; r, t) (τ a 0 +τ a 1 ) + iτ a 2 ,(39)
and for the uncorrelated world solution:
P = p ph (ω; r, t)τ a 0 ,(40)
In the thermal equilibrium p ph = p 0 (ω).
Electrons in disorder potential
The interaction with quenched disorder potential is described by the Hamiltonian:
H el−imp = r U (r)ψ † r ψ r
After averaging over the disorder potential with correlator U (r)U (r ) = V (r − r ), the translation invariance is restored for averaged correlation functions and the diagrams for the electron correlators become similar to those for electronphonon interaction that carries zero frequency, see Fig. 5.
Electron-electron interaction
The interaction between electrons is given by
H el−el = 1 2 ψ † r ψ † r ψ r ψ r U (r − r )drdr(41)
The rules of the diagram technique are given in Fig. 4. In the discussion of the properties of this model we shall neglect the spin of the electrons. For completeness, we also mention that in the perturbation theory based on Eq. (41) the singular terms proportional to G K (t, t) have to be understood as G K (1, 1) → 2i Ψ † (r 1 )Ψ(r 1 ) , and G R,A (t, t) → 0. Such terms appear only in the Hartree-Fock contributions to the single electron spectrum and not in the collisions interesting for us.
Kinetic equation for normal and augmented Keldysh functions.
Quasiclassical descriptions
The kinetic equation (21) fully determines the evolution of the observables and the computables. However, it is not solvable in a general case. Substantial simplification occurs in the quasiclassical limit in which equations (21) become local in time and phase space. This simplification is possible if the rate of the electron scattering is smaller than the relevant energy scales in the problem: temperature for electron-electron or electron-phonon interactions or Fermi energy for electrons in disorder potential. For the diagonal part of the kinetic equation this is well established and the theory of quantum corrections is well developed, see ref. [29] for a pedestrian introduction. In the following we shall assume that these conditions hold and that the quasiclassical kinetic equation follows for the diagonal terms. Under these conditions similar local equations hold for non diagonal Greem functions despite the fact that these functions do not have classical meaning.
We follow the standard procedure for the derivation of the quasiclassical equations for both diagonal and non-diagonal components. Any function of two coordinates and two times can be represented as a Wigner transformation
W (t 1, r 1 ; t 2 , r 2 ) = w( , p; t, r)e ipr−/ −i t−/ d d d p (2π ) d+1(42)
where t = (t 1 + t 2 )/2, r = (r 1 + r 2 )/2, t − = t 1 − t 2 , r − = r 1 − r 2 . In this section we chose to keep the Planck constant explicitly so that the parameter for semiclassical expansion is always displayed. Using this representation for the Green functions of electrons and employing Eq. (33) we get for the left hand side (LHS) of the quantum kinetic equation (21)
H 0 • F αβ − F αβ • H 0 = i ∂ ∂t + ∂ ∂r dξ dp F αβ ( , p; r, t)(43)
which coincides with the LHS of the classical Boltzmann equation for both diagonal and off-diagonal components of the Green functions. Similar arguments for the phonons lead to the LHS of the kinetic equation
H 0 • P αβ − P αβ • H 0 = i ω ∂ ∂t P αβ (ω, p; r, t).(44)
The LHS of Eqs. (43,44) represent the deterministic (Liouville) evolution corresponding to the unitary quantum dynamics which is identical for both diagonal and non-diagonal components in the augmented space. The right hand side (RHS) of the kinetic equation describes the non-reversible probabilistic parts and it is different for different models. The equations
∂ ∂t + ∂ ∂r dξ dp F αβ ( , p; r, t) = [St ··· el ] αβ (45) ω ∂ ∂t P αβ (ω, p; r, t) = ω St ··· ph αβ(46)
describe the time evolution of the distribution functions. Here [St ··· el ] , St ··· ph denote collision integrals for the particles (electrons or phonons) scattered by other particles (denoted by · · · ). These collisions integrals will be computed in the next section.
Collision integrals for the specific models
The RHS of the kinetic equation allows a number of simplifications in the leading order in . Furthermore, below we shall consider only the leading order term in the interation.
In the leading approximation one can replace
G R − G A = −2πiδ( − ξ p )
and, with the same accuracy,
G K αβ = −2πiδ( − ξ p )F αβ . Analogously for phonons D R − D A = −2πin ph [δ(ω − ω 0 ) − δ(ω + ω 0 )] D K αβ = −2πi [δ(ω − ω 0 )P αβ (ω 0 ) − δ(ω + ω 0 )P αβ (−ω 0 )] .
Note that the fact that D K αα (ω) is an odd function of frequency allows the simultaneous description of phonon emission and absorption processes by a single P αα (ω > 0) > 0 . Substituting these forms into the RHS of the kinetic equation we obtain the collision integrals,
St αβ = 1 Σ R α •F αβ − F αβ • Σ A β − Σ K αβ ,(47)
for different models. In Eq. (47) we kept only the real part of the collision integral, we discuss various approsimation involved in its derivation of in more detail in section 4.4
Electron-phonon scattering.
We calculate the lowest order diagrams shown in Fig. 6. For our model one neglect the correlations between phonons at different space locations, i.e. the blobs for fermionic loop in self-energy shown in Fig. 6 a) correspond to coinciding points (with density n ph ). This implies that the electron self-energy and collision integral contain an extra factor n ph with respect to the phonon ones. In the diagonal sector we obtain (we do not write down the spatial and time coordinates as the semiclassical collision integrals are local in those variables)
St ph el αα = n ph dωdP 1 M (P ; P 1 , ω) (2π)(2π ) (d+1) × − L ph el α (P 1 , ω)F αα (P ) + [P αα (ω)F αα (P 1 ) + 1] ; St el ph αα = 1 2 dP dP 1 M (P ; P 1 , ω) (2πω)((2π ) (d+1) (48) × − L ph el α (P 1 , ω)P αα (ω) + [1 − F αα (P )F αα (P 1 )] ,
where we introduced functions
L ph el α (P 1 , ω) = P αα (ω) + F αα (P 1 ), L el ph α (P 1 , ω) = F αα (P ) − F αα (P 1 ),(49)
that determine outgoing rate. This notation is useful as the same quantity enters the equations for the non-diagonal part. For off-diagonal (α = β) we obtain St ph el αβ = n ph dP 1 dQ 1 M (P ; P 1 , ω) (2π)(2π ) (d+1) −L ph el (P 1 , ω)F αβ (P ) + P αβ (ω)F αβ (P 1 ) ,
St el ph αβ = 1 2 dP dP 1 M (Q; P 1 , ω) (2πω)(2π ) (d+1) −L el ph (P 1 , ω)P αβ (ω) − F αβ (P )F βα (P 1 ) ,(50)
where we introduced the short hand notation
2L ... ... (P 1 , Q 1 ) ≡ [L ... ... ] u (P 1 , Q 1 ) + [L ... ... ] d (P 1 , Q 1 ).(51)
The form factors M include the matrix elements, conservation laws for the electron and phonons colliding with each other, and their spectrum:
M = λ 2 2 (2π ) d+1 δ( − 1 − ω) [2π δ( 1 − ξ(p 1 ))] 2π ± ±δ(ω ∓ ω 0 )
where numerical factor 1/2 includes the difference of F, P from the physical distribution function by a factor of two. We find it is more convenient to keep
dS dt = dP St ph el αα δ[ − ξ(p)] ln 1 + F αα (P ) 1 − F αα (P ) + +n ph dω St el ph αα ± ±δ(ω ∓ ω 0 ) ln 1 + P αα (ω) P αα (ω) − 1 ≥ 0,(54)
which is the microscopic manifestation of Boltzmann H-theorem. Equality is reached only for thermal distribution functions for which it reduces to Eq. (52) and Eq. (53) . These equations allow to prove that the only stable solution of the kinetic equation is given by the thermal distribution functions and all deviations from it decay (generally, exponentially). In contrast, the non-diagonal part does not satisfy any of these properties or conservation laws. As we already mentioned, this absence of conservation laws and H-theorem will be the key to understand the instability of the thermal non-diagonal distributions for correlated worlds (25) and (39) and their subsequent evolution to non-correlated worlds (26), (40). The discussion of the instability will be done in Secs. 5 and 6. In the remainder of this section, we list the properties of the collision integrals for the other physical models of Sec. 3
Electron-electron scattering.
We calculate the lowest order diagram shown on Fig.6. In the diagonal sector we obtain St el el αα = dP 1 dP 2 dP 3 M (P, P 1 ; P 2 P 3 ) (2π ) 3(d+1) − L el el α (P 1 , P 2 , P 3 )F αα (P )
+ [F αα (P 3 ) + F αα (P 2 ) − F αα (P 1 ) − F αα (P 1 )F αα (P 2 )F αα (P 3 )]} ,(55)
where we denoted
L el el α = F αα (P 2 )F αα (P 3 ) − F αα (P 1 )F αα (P 3 ) − F αα (P 1 )F αα (P 2 ) + 1. (56)
As before, the form factors M include the matrix elements, the conservation laws for the electron and phonons colliding with each other, and their spectrum:
M = |U p2−p − U p3−p | 2 8 (2π ) d+1 δ(P + P 1 − P 2 − P 3 ) 3 i=1 [2π δ( i − ξ(p i ))]
where numerical factor 1/8 includes the difference of F, from the physical distribution function by a factor of two, and exchange symmetry of the final state. We find it is more convenient to keep , ω as independent energies connected by δ-functions in M with physical spectrum to have the symmetric form for the conservation laws and use the d + 1 dimensional momentum vectors P = ( , p).
The collision integral (55) satisfies the conditions similar to those for electronphonon scattering dP St el el αα δ( − ξ(p)) = 0, (57)
(particle conservation) dP St el el αα δ( − ξ(p))P = 0,(58)δ( − ξ(p)) ln 1 + F αα (P ) 1 − F αα (P ) ≥ 0,(59)
(entropy growth). For off-diagonal (α = β) we obtain St el el αβ = dP 1 dP 2 dP 3 M (P, P 1 ;
P 2 P 3 ) (2π ) 3(d+1) × −L el el (P 1 , P 2 , P 3 )F(P ) αβ + F αβ (P 2 )F αβ (P 3 )F βα (P 1 ) ,(60)
where once again 2L el el (P 1 , P 2 , P 3 ) ≡ L el el u (P 1 , P 2 , P 3 ) + L el el d (P 1 , P 2 , P 3 ).
Similarly to the electron-phonon interaction, the collision integral (60) for the non-diagonal term is non-linear due to the incoming term. This leads to the instability of the thermal non-diagonal distribution.
Electron-impurity scattering
The collision integral for the electron-impurity scattering is linear in the distribution function
St im el αβ = dp 1 M (p, p 1 ) (2π ) d {−F αβ (p) + F αβ (p 1 )} ,(61)
where
M (p, p 1 ) = 2π |V p−p1 | 2 δ(ξ p − ξ p1 ).
This implies that in the case of the impurity scattering the non-diagonal components of the Keldysh function have the same time evolution as the diagonal ones, so the solution in which it is equal to the thermal equilibrium distribution is stable. Note that electrons in the impurity potential is a chaotic system. In this respect it is not different from the electron-phonon and the electron-electron interaction. Nevertheless, the non-diagonal components are stable, in contrast to the models with electron-phonon and electron-electron interactions. This results in a very different behavior of the out-of-time-ordered correlators in this system.
Additional remarks
It is worthwhile to emphasize that the basic form of the kinetic equation and the forthcoming conclusions are not limited to the lowest order self-energy calculation. In particular, taking into account the commutators of the self-energy with F αβ results in well controllable corrections to the LHS of the kinetic equation and has the meaning of the self-consistent spectrum. The higher order expansion improves the accuracy of the matrix elements in the collision integrals and also produces the real processes involving larger number of particles. Neither of those complications seem to affect the basic relations of the diagonal and non-diagonal evolutions and we will not be dwelling on them in this paper. The imaginary part of the non-diagonal elements of collision integral neglected in Eq. (47) formally appears due to the difference between the distribution functions in upper and down Worlds:
St αβ = 1 Σ R α − Σ R β F αβ
This effect also disappears in the leading quasiclassical approximation and does not affect the instability discussed in the next sections. For instance, for electronphonon model this term becomes
St αβ ∝ dξ D R ( − ξ) (F α (ξ) − F β (ξ)) + G R (ξ) (P α (ξ) − P β (ξ))
It disappears for the two Worlds in equilibrium. Furthermore, it is zero if one World has extra particle density that resulted in the spatially non-unform chemical potential.
Instability of the augmented Keldysh functions in zero dimensional case
In this section we study the instability in the systems in which the spatial dependence of the correlation functions can be neglected.
Instability in electron-phonon model.
The Einstein phonon distribution function is characterized by just two numbers in each sectors of the augmented space that are the values of P αβ (±ω 0 ). In thermal equilibrium the two diagonal components are given by Eq. (39). Because the instability in the non-diagonal sector does not affect the diagonal one (Sec. 2.4), for simplification we assume that the diagonal sector for both electrons and phonons is in equilibrium. This assumption is not essential, and the thermal function can be replaced to its non-equilibrium value without any technical complications. Because the phonon field is real, the non-diagonal sectors are related to each other by the symmetry P ud (ω 0 ) = −P du (−ω 0 ), and the state of the phonons is described by two parameters
P ud (ω 0 ) = θ, P du (ω 0 ) =θ.
The phonon scattering process does not depend on the electron momentum, so we need to keep only the energy, = ξ(p), dependence of the electron distribution function:
F ud ( , p; t) = f ( , t), F du ( , p; t) = −f ( , t).
.
Inserting these definitions in the kinetic equations (45,46,50) and performing integrals over momentum we obtain
τ ∂f ∂t = −L 2T , ω 0 2T f + θf ( − ω 0 ) +θf ( + ω 0 ) , (62a) τ ∂f ∂t = −L 2T , ω 0 2T f + θf ( − ω 0 ) + θf ( + ω 0 ) ,(62b)ητ ∂θ ∂t = −θ + I − ,(62c)ητ ∂θ ∂t = −θ + I + ,(62d)
where we introduced the positive quantities
I ± = 1 2ω 0 d f ( )f ( ± ω 0 );(63)
L(x, y) = 2 coth (y) − tanh (x + y) + tanh (x − y) .
We also replaced ω 0 → ω 0 as the semiclassical expansion is already completed. In deriving Eqs. (62), we neglected the energy dependence of the electron density of states, ν, we denoted
1 τ = 2πνn ph λ 2 ,(65)
and introduced the dimensionless parameter
η = n ph νω o .(66)
Eqs. (62) are further simplified in the limits η 1 and η 1. In the former limit the phonon relaxation is slow compared to electrons, in the latter the electron relaxation is slower. The limit η 1 seems to be the most relevant for physical situations (e.g. to describe electrons interacting with low density TLS) and moreover it will enable to develop intuition for analyzing the more involved kinetics of electron-electron collisions. Thus we focus on this limit only. Because the relaxation of θ is much faster than that of f , we can solve the Eqs (62c,d) for θ,θ in the stationary limit. We obtain
τ ∂f ∂t = −L 2T , ω 0 2T f + [I − f ( − ω 0 ) + I + f ( + ω 0 )] ,(67a)τ ∂f ∂t = −L 2T , ω 0 2T f + I +f ( − ω 0 ) + I −f ( + ω 0 ) . (67b)
Eqs. (67) and Eqs. (63,64) form the complete set of equations describing the time evolution of the non-diagonal components of the distribution function. However, they are still non-linear and nonlocal in energy space.
The further analysis is separated into two regimes: "classical" ω 0 T and "quantum" ω 0 T . The difference between these regimes is expected on the physical grounds: in the "classical" regime a large number of excitations is already present, therefore, one expects (and we will see that that it is indeed the case) that the perturbation results in the evolution that leads to the uncorrelated fixed point, f = 0,f = 0 with the characteristic time of the order of τ. In the quantum regime, one expects that the characteristic time is determined by an exponentially small number of excitations and it becomes infinite at zero temperature.
Generally, one expects that in the gapful systems at T = 0 a small perturbations cannot lead to any instability, in particular, these systems cannot be chaotic, so that the scrambling time is infinite. The exponential growth of the characteristic time at low temperatures in the gapless system found here implies a smooth crossover between the properties of the gapful and gapless systems at T = 0 .
Classical limit (ω 0
T ) At high temperatures we can neglect ω 0 in I ± (63) and in the arguments of f in Eq. (67), we can also approximate L = 2/y in Eq. (64). We see then that the form of f ( ) andf ( ) dependencies is not changed by the evolution. Therefore, we can look for the solution of Eq. (67) in the form
f ( ) = φ(t) [1 + tanh( /2T )] ,(68a)f ( ) = φ(t) [1 − tanh( /2T )] .(68b)
We obtain that the function φ(t) obeys the first order differential equation
τ ∂φ ∂t = − 4T ω 0 φ − φ 3(69)
that has the unstable fixed point at φ = 1 and the stable fixed point at φ = 0. The solution of this equation takes the form
φ(t) = 1 1 + exp [(t − t d )/t cl * ] 1/2 ,(70)
where t cl * = τ ω 0 /8T . This time dependence is typical of the dissipative instabilities. The delay time, t d , depends only logarithmically on the initial conditions: t d = t cl * |ln [1 − φ(0)]| whilst the decay time t cl * coincides with the classical scattering time of the electrons that is inversely proportional to the density of phonon sites, n ph , and the phonon occupation number, T /ω 0 . Note that this classical time, t cl * , is less than the energy relaxation time, in the limit ω 0 T .
5.1.2.
Quantum limit (ω 0 T ) In order to describe the behavior of the solution in the quantum limit, it is instructive to look at the results of the numerical solution of Eq. (67) at low temperatures, see Fig. 7. In contrast to the classical case, the function f ( ) does not preserve the shape with the time evolution. However, one observes that the behavior does not change at negative energies. At positive energies we observe a sequence of peaks at energies n = (n + 1/2)ω 0 that are similar to the first peak at n = 0. This behavior can be qualitative understood as follows.
At low temperatures Eq. (63) implies that I − I + (see also Fig. 7b). If the terms proportional to I + are dropped, Eq. (67a) describes the drift of f ( ) to high energies together with relaxation. Similarly, Eq. (67b) describes the drift to negative energies and relaxation. The L-terms in Eqs. (67) lead to decay, so the frequency regime where these terms dominate cannot contribute to the instability. For frequencies | | > ω 0 the L-term in Eq. (67) is large so the region responsible for the instability is | | < ω 0 . In the absence of I + , the advection term proportional to I − removes perturbations from this region. Because the values of f ( ) at high energies does not feedback on low energies, the instability disappears. In order to see the instability it is therefore essential to keep the I + term in the region | | < ω 0 . From Fig. 7b we see that the main contribution to I ± comes from the vicinity of the frequencies ±ω 0 /2, so the integrals I ± can be approximated by I − ≈ f 2 + and I + ≈ f 2 − where f ± = f (±ω 0 /2) (we drop nonessential numerical factors). For this reason we focus on the time dependence of these two values of f ( ). In the equation for df − /dt we can neglect the ε/Τ contribution of
I + f (−3ω 0 /2) term because f (−3ω 0 /2) ∼ exp(−ω 0 /T )f − .
In the equation for df − /dt we can neglect the contribution of I − f (3ω 0 /2) because it is proportional to I − 1. Then the integral equations (67) reduce to two ordinary differential equations:
τ df + dt = −L 0 f + + f 2 + f − (71) τ df − dt = −L 0 f − + f 2 − f +(72)
where L 0 = 2 exp(−ω 0 /2T ). At low temperatures the linear term in these equations becomes exponentially small, as a result the instability developes exponentially slowly. Solving for the product f + f − we get
f + f − = L 0 1 + exp [(t − t d )/t qu * ] ,(73a)
and
f + + f − = f 0 1 1 + exp [(t − t d )/t qu * ] 1/2 ,(73b)
where t qu * = τ /(2L 0 ). The Eqs. (73) should be compared with the solution (70), we see that they describe similar relaxation but with exponentially smaller rates. Although the behavior of the solution (73) is similar to the one in the high temperature limit, there is an important difference: the relaxation is determined only by a narrow advection region at low energies, whereas the high energy region plays a passive role of a sink. Furthermore, the non-linearity appears first at high energies but it does not affect the fact that the dynamics is determined by the narrow region at low energies that determines the value of I + .
Instability for electron-electron interaction.
As for electron-phonon scattering one can focus only on the energy dependence of the off-diagonal functions
F ud ( , p; t) = f ( , t), F du ( , p; t) = −f ( , t),
and assume that the diagonal functions correspond to the equilibrium. As a result, the time evolution of the functions f,f is described by the equations similar to Eq. (67):
τ F L ∂f ∂t = −L F T f + K( , f,f ), τ F L ∂f ∂t = −L F T f +K( , f,f ),(74)L F (x) = 1 + x 2 /π 2 , where τ −1 F L ∼ T 2 /E * F(75)
is the Fermi liquid relaxation rate, E * F is the parameter built from the electron density of states and the interaction constant. 8 The functional K( , f,f ) is of the second order in f and of the first order inf :
K( , f,f ) = ∞ 0 I − (ω)f ( − ω) dω 2πT + ∞ 0 I + (ω)f ( + ω) dω 2πT , (76a) K( , f,f ) = ∞ 0 I + (ω)f ( − ω) dω 2πT + ∞ 0 I − (ω)f ( + ω) dω 2πT ,(76b)I ± (ω) = 2 d 2πT f ( )f ( ± ω).(76c)
Formally these equations are similar to Eqs. (67) for the electron-phonon scattering, the only difference is that instead of one mode Eqs. (76) contain the integral over frequencies. At large T the functionals K(f,f ) andK(f,f ) are dominated by terms proportional to I − (ω) that describes drift to larger frequencies for f ( ) and to smaller frequencies forf ( ). The feedback that results in dissipation is due to the energies T . The qualitative properties of these equations are thus captured by the simplified equations for two characteristic values of f ± = f (±T ). These equations have exactly the same form as Eqs. (71,72), with the important difference that L 0 ∼ 1. Thus their solution is given by the equations (70) with characteristic decay time t * ∼ τ F L .
It is very important that although the relaxation rate in a Fermi liquid becomes very large at high energies, the processes involving high energy electrons 8 In three dimensional Fermi liquid E * F ∼ E F where E F is the Fermi energy, in two dimensional Fermi liquid it contains additional ln(E F /T ) factors [30] while in 1D models two particle collisions do not lead to dissipation.
do not contribute to the instability of the equations (74) for the non-diagonal parts. Instead, the instability is controlled by the same processes as the physical relaxation and has characteristic time scale of the electron-electron relaxation time at temperature T .
Equations for spatial structure of the instability
As we have seen in section 5 the instability in zero dimensional models is always controlled by equations similar to (62). This equation can be derived and solved analytically in the case of electron-phonon model at high temperatures but it provides the qualitative description in other cases as well. To resolve the spatial structure of the instability we thus begin with the electron-phonon model at high temperatures. The presence of the spacial structure changes the quantum kinetic equations (62) very little (apart from introducing the spacial dependence). Because the phonons in this model are local, the equations for θ andθ contain the fermionic functions taken at the same spatial point. As in section 5.1 in the limit of low phonon density (η 1) the phonon relaxation is fast, so we can solve for local θ andθ:
θ(r) = 1 2ω 0 d f ( , r)f ( − ω 0 , r), θ(r) = 1 2ω 0 d f ( , r)f ( + ω 0 , r),
Performing the standard spatial gradient expansion in the LHS of the kinetic equation (45), we find that df /dt in (62) acquires an additional diffusion term. Parametrizing the solution by the ansatz (68) we obtain the final equation
∂φ ∂t − D * ∇ 2 φ = − 2 φ − φ 3 t * .(77)
This equation is the central result of this paper. As we argue below it holds (with small modifications) for other models as well.
At non-zero temperature the electron-phonon interaction leads to the diffusive motion of electrons characterized by the momentum relaxation time so that the diffusion coefficient D * = v 2 F τ tr /d, where v F is the Fermi velocity and d is the spatial dimensionality. At high temperatures ω 0 T the transport relaxation rate is given by 1/τ tr = λ 2 ν(n ph T / ω 0 ), with (n ph T / ω 0 ) ph having the meaning of the thermal phonon density. In this case the energy relaxation of the electrons becomes parametrically slower than its momentum relaxation: 1/τ e = (ω 0 /T )λ 2 νn ph .
The diffusion approximation used to derive Eq. (77) can be rigorously justified only if the resulting gradient of φ is small on the scale of the mean free path, v F τ tr . This happens only if t cl * τ tr which can occur if the electron diffusion is additionally slowed down by the impurity scattering 1/τ tr = 1/τ (ph) tr
+ 1/τ (imp) tr 1/τ (ph) tr .
Both the diffusion coefficient D * and the time t cl * depend on the local temperature T (r, t) and the electron density n(r, t). Those quantities are described by the standard diffusion and thermal diffusion equations for the diagonal components and their solutions has to be used as entry parameters for Eq. (77). This scheme gives the complete description of the quantum butterfly effect. Notice that depending on the particular model, D * may coincide with the particle or thermal diffusion coefficients or may be different from those by a numerical factor.
A very similar equation can be put forward for the model of the electronelectron interaction. Taking into account that the effective equations for electronelectron interaction is formally the same as that for electron-phonon case, we write
∂ ∂t + v∇ φ − D * ∇ 2 φ = − 2 φ − φ 3 t * .(78)
The only modification here is the appearance of the drift term v∇which is dictated by the Galilean invariance for ξ p = p 2 /2m − F . The macroscopic velocity v(r, t), the local temperature T (r, t), and the electron density n(r, t) are controlled by the usual equations of local hydrodynamics and thermal (entropy) diffusion [31]. It is possible to generalize Eq. (78) for the case of relativistic hydrodynamics. Based on Lorentz invariance one obtains
u i ∂ i + D * ∂ i ∂ i φ = − 2 φ − φ 3 t * ,(79)
where covariant and contravariant component are related by the arbitrary metric tensor and u i is standard four component velocity vector with local constrain u i u i = 1.
To close the section, let us emphasize that the coefficients D * , t * do not affect the diagonal entropy production and do not enter the usual Onsager relations. It is unknown to us whether there is an analogue of the H-theorem that includes the non-diagonal distribution functions as well.
7. Spatial Propagation of the instability: combustion waves.
The equations (77,78,79) for the spatial structure of the instability are well known in the theory of combustion. In particular, Eq. (77) is very similar to Fisher-Kolmogorov-Petrovsky-Piscounov equation (FKPP) [32,33] dy dt − ∇ 2 y = y(1 − y).
All equations of this type possess two stationary solutions y = 0 and y = 1 in case of FKPP, one of them is stable, another is not. In particulr, our Eq. (77), displays the instability of the solution φ(r) = 1 that evolves according to the following scenario. After being seeded at time t = 0 with the small deviation δφ(r) = 1 − φ(r) 1, in a region around 0 (i.e. δφ = 0 for r > R c ) the instability remains localized in the area where it was seeded (r < R c ) for the time t d ∼ ln(1/δφ). After this initial period, the instability starts to grow spatially forming a non-linear wave that moves with a well defined velocity v cw . For Eq. (77) in 1D the solution φ f (x − v cw t) for the front moving with constant velocity v obeys the equation
t * v cw dφ f dx + D * d 2 φ f dx 2 = 2φ f (1 − φ 2 f )(80)
As is established in the theory of combustion [34], the value of the front velocity can be found from the study of the solution of Eq.(80) at x → ∞ where δφ → 0. At δφ 1 the solution of Eq. (80) behaves as δφ ∼ exp(−kx) with k that is
real at v 2 cw ≥ 16D * /t * .(81)
For the initial conditions that correspond to δφ = 0 for r > R c the solution quickly converges to the one moving with the minimal velocity allowed by the constraint (81). The presence of other solutions (with higher velocities) is due to the fact that for the (non-physical) initial conditions that differ from unity everywhere, the instability develops at large r might develop independently of the seed at small r. One concludes that the combustion wave moves with velocity v cw = 4 D * /t * Note that for electron-phonon and electron-electron models D * ∼ t * v 2 F in the absence of electron-impurity and elastic scattering, so that the front velocity v cw ∼ v F . Because no perturbation (even unphysical one) can propagate with velocity larger than v F , v cw v F .
In order to check the conclusions of the semi-quantitative analysis presented above we have studied numerically the front propagation in the dimensionless equation
dφ dt = ∇ 2 φ + 2φ(φ 2 − 1)(82)
and in the similar equation describing evolution of both electrons and phonons
dφ dt = ∇ 2 φ + 2(Θ − 1)φ,(83a)dΘ dt = φ 2 − Θ (83b)
that describes the situation in which the phonon dynamics is of the same order as electron one (i.e. η = 1). We found that in all cases and in all dimensions (d = 1, 2, 3) the front quickly assumes a well defined shape and start to move with the constant velocity. We note that this conclusion for the two component (electron and phonon) systems is not obvious because such equations are known to display more complex behavior in some cases. We now apply the findings of the previous sections, namely, the instability of the off-diagonal part of the Green function to the computation of the out-oftime-ordered correlator (14). In the conventional theory the correlator of two operators at large separations in time or space factorizes
Ŝ 0Rt,t = Ŝ 0 R t,t(84)
The corrections to this factorization are given by irreducible correlator that decreases quickly with distance and time. In the electron models considered here the irreducible part is small in 1/p F r and 1/ F t. Furthermore, in a conventional theory one can evaluate both averages in the RHS of (84) against the background of the unperturbed states. The crucial difference of the two Worlds theory is that the second term in this factorization is unstable. Thus, it is not correct to replace it by its value for the fully correlated, unperturbed state: a small deviation from this value at short distances grows quickly and eventually reduces it to zero in the whole system. Instead one should use for it the results of the solution of the equations for the Green functions discussed in previous sections. In particular, for the response operator in correlator (14) we get R t,t (r) = 2πν f ( , t, r) −f ( , t, r) .
where we emphasized that the augmented distribution functions f andf are generally the functions of the position in the space as well. The space-time dependence of these functions is determined by the equations derived in Sections 5-7. The average of the source operator is a constant factor, for the correlator (14) it is given by the total density of electrons:
Ŝ 0 = n el .(86)
As discussed in Section 5 the time dependence of the augmented distribution functions is simplified in the high temperature regime of the electron-phonon model. In this case the form of the energy dependence of the augmented distribution function does not change with time, the time dependence shows up only in the factor φ(r, t): f ( , t, r) = φ(r, t)f 0 ( ). In this case we can write the final result for the Wigner transform of the out-of-time ordered correlator in the closed formà ρρ ( , t, r) = 2πνn 0 ( )n el φ(t, r),
A ρρ (t , t, r) = (d )e −i (t −t)Ã ρρ ( , t + t 2 , r),(87)
Here φ(t, r) is the solution of the equations (77-79) appropriate for a particular model with the initial conditions
φ(0, r) = 1 − δφ(r)(89)
Here δφ(r) δ (r)/n el δ (r)p −d F
describes the perturbation resulting from the introduction of one extra electron in down World, which serves as a seed of the instability. Hereδ(r) denotes the smeared δ-function that appears because the equations for the distribution function are valid only at the time scales larger than collision time τ tr , so the addition of one particle at time t = 0 in the down World translates in the density spead over distance l tr ∼ v F τ tr for the initial conditions of the Eqs. (77-79). As a result the δ−function in Eq. (90) has to be replaced byδ(r) which is smeared at the distances of mean free path, l tr ∼ v F τ tr . The solutions of the equations (77-79) correspond to the propagation of the front as illustrated by Fig. 8.
Note that the particular symmetric form (16) of the response operator computed here has the property that it vanishes at coinciding times and coordinates. This property disappears for less symmetric form of the correlators, for instance if τ a 1 is replaced by, e.g. τ − = 1 2 (τ 1 − iτ 2 ), in the definition of the response operator (16).Ã
ρρ (t, r) = T C Ψ (t , r)(τ K 1 ⊗ τ a − )Ψ(t, r) Ŝ 0(91)
In this case the first term in (85) disappears and we get after integration over energiesÃ
ρρ (t, r) = 2n 2 el φ(t, r).(92)
At low temperatures for electron-phonon model and for electron-electron interaction at any temperature the energy dependence of the augmented distribution function changes with time as well (Sections 5.1.2, 3.3). In this case the equations for the augmented distribution function are more complicated but the solution remains qualitatively similar.
In all cases, the augmented distribution function that controls the spacial and time dependence of the out-of-time-ordered correlator describes the front propagation, the state of the system before the front has not been affected yet by perturbation whilst the state of the system behind the front is characterized by exponentially vanishing correlations:
A ρρ ( , t, r) = 2πνn 0 ( )n el exp − t−t d −r/vcw 2t * t > r/v cw + t d 1 − exp t−t d −r/vcw t * t < r/v cw + t d(93)
The delay time t d in these equations is controlled by the initial conditions (89) to the Eqs. (77-79) or similar. It depends only logarithmically on the strength of the initial perturbation:
t d = t * |ln [δφ(0)]|(94)
Equation (90) enables us to estimate the strength of the initial perturbation. Indeed, δ(0) 1/(v F τ tr ) d . Thus, we estimate δφ(0) 1/(p F l tr ) d and the delay time
t d = t * d ln(p F v F τ tr )(95)
It is worthwhile to notice that this expression is somewhat similar to the Ehrenfest time [11,35] appearing as the delay time for the quantum correction in quantum chaos for non-interacting system. In this one electron problem the real instability does not occur. In a finite size system of spatial size R the correlator (93) decreases exponentially to zero for all r < R after t scr = t d + R/v cw . The time t scr has the meaning of the time at which the two worlds become completely uncorrelated due to a local perturbation, this is also the time that it takes for the quantum information to be spread over the whole system (scrambling time). We see that although the propagation of the information is controlled by diffusion, it occurs with a constant velocity due to the non-linearity of the equations. The diffusion coefficient controls the velocity of this propagation.
The propagation with constant velocity (87,93) also indicates that in a chaotic many body system the entanglement entropy spreads ballistically despite the diffusive nature of the dynamics. This analytical result confirms the empirical conclusions reached in a number of numerical works.
As we discussed above, the conclusions of the linear propagation of the quantum butterfly effect controlled by the combustion equations is quite general. The details of the equations are sensitive to the microscopic model but the linear propagation similar to combustion front occurs in all of them.
Discussion and conclusions
We developed the technique to study the out-of-time-ordered correlators, such as Eq. (1), based on the extension of Keldysh technique. Similarly to standard Keldysh technique, the augmented technique enables the analytical study of systems in different limits, in particular to obtain the leading result in the quasiclassical approximation and systematic corrections to it. As well as in the Keldysh technique the quasiclassical approximation is valid provided that the particle motion between collisions is quasiclassical whilst collision themselves can be quantum.
We limited ourselves to the leading quasiclassical terms that result in the equations similar to the kinetic equation in traditional statistical mechanics. We found that they describe all (or most of all) non-trivial behavior of the outof-time-ordered correlators. The major difference from the traditional kinetic equation is the appearance of the off-diagonal functions, superficially similar to the conventional distribution function. However, unlike the state occupation probabilities, these new functions also describe the overlap between two copies of the system. The kinetic equation for the off-diagonal functions is dramatically different from that for the diagonal functions: the outgoing term depends on both diagonal and off-diagonal functions whilst the incoming term contains only off-diagonal ones.
The solution with initially unit overlap between two copies becomes unstable when disturbed by a very small perturbation, the phenomenon known as quantum butterfly effect. This instability is described at long times (longer than collision times) by non-linear diffusion equations similar to those appearing in the combustion front propagation. After an initial transient behavior the front of the propagating wave acquires a constant velocity and a shape that does not depend on the initial conditions (Section 7). In the electron models studied in this paper, the velocity of the front is of the order (but less than) the Fermi velocity that serves as natural bound for the propagation speed. In the presence of impurity scattering the velocity of the front can become parametrically slower than Fermi velocity. The microscopic model of electrons interacting with the dilute set of oscillators solved in this work might provide the description of the loss of coherence in the set of two level systems (TLS) that provide both elastic and non-elastic scattering for electrons with the latter becoming small at low temperatures.
Our work suggests a number of exciting developments. First, the quantum butterfly effect studied here can be viewed as the result of the gradual entanglement of the local degrees of freedom with larger and larger part of the surrounding system, and thus is likely to be related to the propagation of entanglement entropy discussed extensively recently [36,37,38,39,7]. Our results would enable us to put these works on the firm ground of an analytical theory if the relation between non-diagonal correlators and entanglement entropy is established. We hope to return to this point in future works.
The quantum butterfly effect can be studied numerically and compared with the analytical theory developed here. Also, the destruction of the coherence between two copies of the system might be a useful tool to study the appearance of the arrow of time in the systems described by the unitary evolution. Finally, the destruction of quantum coherence between two copies of the system is a very important phenomenon for the quantum information protocols that are based on the construction of the initially perfectly entangled states of two (or more) interacting qubit systems because small perturbation to one of these systems would result in a spreading decoherence wave described by our equations.
The spatial and time scales of the effective non-linear diffusive equations that describe the instability of the coherent solution are sensitive to the details of the microscopic theory. Furthermore, their relation to the ones appearing in physical observables is not expected to be universal. Thus they might provide a new tool and the new way of thinking about microscopically different systems that display similar properties such as conductivity.
Our formalism can be extended to the study of many body localization by augmenting the formalism developed in the work [40]. This would provide the analytical approach and qualitative understanding to the problem for which only numerical results are currently available. [41,42,43,44] It might even help to describe the transition itself and even the entanglement propagation in generic glassy systems. Moreover, the question of the propagation of the decoherence front in localized systems is similar to the problem of the decoherence propagation in integrable systems. Note that according to [40,45] in localized and integrable systems the collision integral disappears resulting in the suppression of the chaotic behavior that is responsible for the quantum butterfly effect.
Finally, the microscopic systems studied in this work are described by the combustion equations that display only laminar solution. However, combustion equations for systems with a few components are known to display a large variety of interesting behaviors: Turing instabilities [46], Zhabotinsky cycles [47] to name just a few. It remains to be seen if these solutions are realized in microscopic models as the instabilities of the correlated worlds solution. In particular, they might appear as the solutions against the background of nonequilibrium states such as turbulent hydrodynamics of normal or superfluid liquid.
Acknowledgement
We acknowledge extremely useful discussions with Alexei Kitaev and the hospitality of CTP CSIBS, Daejeon, Korea. Our research was supported by ARO grant W911NF-13-1-0431 and by the Russian Science Foundation grant # 14-42-00044.
2. 5
5Stability and instability. . . . . . . . . . . . . . . . . . . . . . . 10 2.6 Basic rules of the diagram technique: Green functions . . . . . . 12 2.7 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.8 Diagram technique: summary . . . . . . . . . . . . . . . . . . . . 15 3 Microscopic Models 16 3.1 Electron-phonon interaction . . . . . . . . . . . . . . . . . . . . . 17 3.2 Electrons in disorder potential . . . . . . . . . . . . . . . . . . . 18 3.3 Electron-electron interaction . . . . . . . . . . . . . . . . . . . . . 19 4 Kinetic equation for normal and augmented Keldysh functions. 19 4.1 Quasiclassical descriptions . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Collision integrals for the specific models . . . . . . . . . . . . . . 20 4.2.1 Electron-phonon scattering. . . . . . . . . . . . . . . . . 21 4.2.2 Electron-electron scattering. . . . . . . . . . . . . . . . .22 4.3 Electron-impurity scattering . . . . . . . . . . . . . . . . . . . . . 24 4.4 Additional remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5 Instability of the augmented Keldysh functions in zero dimensional case 25 5.1 Instability in electron-phonon model. . . . . . . . . . . . . . . . 25 5.1.1 Classical limit (ω 0 T ) . . . . . . . . . . . . . . . . . . . 26 5.1.2 Quantum limit (ω 0 T ) . . . . . . . . . . . . . . . . . . 27 5.2 Instability for electron-electron interaction. . . . . . . . . . . . . 29 6 Equations for spatial structure of the instability 30 7 Spatial Propagation of the instability: combustion waves. 31 8 Time and spatial dependence of out-of-time-ordered correla-
Figure 2 :
2Definition of the basic elements of the diagrammatic technique. (a) The lines (thick and thin) describe the exact and bare Green functions respectively. (b) The box describes the matrix structure of the vertices. (c) The vertices which do not conserve the number of particles (for example absorption and emission of the phonons or photons).
Figure 3 :
3Diagrammatic expressions for the correlators
Figure 4 :
4Vertex structure for (a) electron-electron interaction; and (b) for the electron-phonon interaction. The outgoing or ingoing vertical arrow in (b) can be any bosonic line including phonons, photons, impurity potential etc. The short dashed lines on panel (b) are the delta function in space and time.
Figure 5 :
5Remaining basic elements of the diagram technique for electron-phonon a) and electron-impurity b) interactions. In (a) we include the interaction constant λinto definition of the propagator to keep the vertices ofFig. 4intact. Correlation function V (r) describes the fluctuations due to the weak random impurities.
, ω as independent energies connected by δ-functions in M with physical spectrum to have the symmetric form for the conservation laws and use the d + 1 dimensional momentum vector P = ( , p).Here comes an important observation. Even though Eq. (48) and Eq. (50) look similar, their properties are very different. Indeed the diagonal part satisfies the electron number conservation law − ξ(p)) + n ph dω St el ph αα ω ± ±δ(ω ∓ ω 0 ) = 0.(53) Moreover, one can explicitly check that the time derivative of the entropy
Figure 7 :
7Left panel, main figure: low temperature evolution of non-diagonal parameter f in the low temperature case, T = 0.1ω 0 for discrete times t/τ = 0.2, 0.4, . . . 4.0. The inset: similar evolution at high temperatures, T = 2ω 0 , for t/τ = 0.2, 0.4, 0.6, . . . shows fast uniform decrease of the non-diagonal parameter. The right panel displays the integrands of I ± (63) at t/τ =2.0 that shows that I ± are dominated by narrow frequency ranges around ±ω 0 /2. This allow us to simplify the equations by considering only the values of f at these points.
Figure 8 :
8Propagation of instability in different dimensions. The upper panel shows the solution of (82) for one component electron system, the lower for the combined system of electron and phonons (83).
8 .
8Time and spatial dependence of out-of-time-ordered correlators.
Here we assume that operators O have zero averages in all states. If not, the irreducible correlators have to be discussed. We also assume that operator O 2 (t) = 0
The two contour formalism of Ref.[25] is not suitable for this purpose: in this formalism the worlds are uncorrelated from the very beginning though the disordered potential acts the same on the both worlds.3 Such parametrization of the Keldysh space was first introduced by Larkin and Ovchinnikov[26].
We distinguish physical sources that can be realized in experiment and the non-physical ones that require time-machine for their implementation
Butterfly effect in this case might appear due to the rounding errors in back and forth evolutions.
We neglect the effects of interference of different processes responsible e.g. for the onset of the localization in disordered systems[28] or renormalization of the electron-phonon interaction strength by disorder. The justification for this omission is that the effects considered in this paper are dramatic already on the level of the kinetic equation approximation.
Note that the momentum is not conserved because of the locality of the phonon correlations even though the averaged system is formally translationally invariant.
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[
"NOISE SENSITIVITY AND NOISE STABILITY FOR MARKOV CHAINS: EXISTENCE RESULTS",
"NOISE SENSITIVITY AND NOISE STABILITY FOR MARKOV CHAINS: EXISTENCE RESULTS"
]
| [
"\nMALIN PALÖ FORSSTRÖM\n\n"
]
| [
"MALIN PALÖ FORSSTRÖM\n"
]
| []
| During the past 15 years, several extensions of the concepts noise sensitivity and noise stability, first coined in [2], has been studied. The purpose in this paper is to give definitions of this concepts in the setting of continuous time Markov chains, which then unifies many of the previously considered generalizations. In addition, a considerable amount of time is spent on proving the existence of sequences of noise stable and nondegenerate functions with respect to various classes of Markov chains, a problem which interestingly will appear to have close connections to the so called localization of eigenvectors, a problem which in the setting of random graphs has recently been given a lot of attention. | null | [
"https://arxiv.org/pdf/1501.01824v1.pdf"
]
| 119,747,672 | 1501.01824 | 437b2c151ce1370e74f592b2b9219b9031b1b353 |
NOISE SENSITIVITY AND NOISE STABILITY FOR MARKOV CHAINS: EXISTENCE RESULTS
8 Jan 2015
MALIN PALÖ FORSSTRÖM
NOISE SENSITIVITY AND NOISE STABILITY FOR MARKOV CHAINS: EXISTENCE RESULTS
8 Jan 2015Date: January 9, 2015.
During the past 15 years, several extensions of the concepts noise sensitivity and noise stability, first coined in [2], has been studied. The purpose in this paper is to give definitions of this concepts in the setting of continuous time Markov chains, which then unifies many of the previously considered generalizations. In addition, a considerable amount of time is spent on proving the existence of sequences of noise stable and nondegenerate functions with respect to various classes of Markov chains, a problem which interestingly will appear to have close connections to the so called localization of eigenvectors, a problem which in the setting of random graphs has recently been given a lot of attention.
Introduction
In [2], Benjamini, Kalai and Schramm coined the term noise sensitivity, looking at how likely the occurance of events where to differ at the starting point and the ending point of sequences of continuous time random walks on Hamming cubes. Since this paper was published, several extensions of this model, as well as similar definitions in slightly different settings, have been studied, including changing the random walk into an exclusion processes on the Hamming cube [3] or into Brownian motion on R n [13,14], as well as trying to understand how the definitions can be applied in the context of functions defined on the leaves of binary trees [15]. Several of the results for the Hamming cube case, which are proven using the theory developed from this research, have also been extended to other settings, such as to the symmetric group or to slices of Hamming cubes [7,8,9,11].
Our main goal of this paper will be to propose a definition of noise sensitivity for general Markov chains and to show that this definition preserves many of the properties from the original setting. Interestingly, we will see that some of the questions that arise will have connections to the so called localization of eigenvectors studied recently in eg. [1], [4], [6] and [10].
Throughout this paper, we will be concerned with sequences (X (n) ) n≥1 of reversible and irreducible continuous time Markov chains X (n) . For each n ≥ 1, let S (n) be the state space, Q n = (q (n) ij ) i,j∈S (n) be the generator and π n be the stationary distribution of X (n) . We write X (n) t to denote the position of X (n) at time t ∈ R + , and will always assume that X (n) 0 has been choosen according to π n .
Next, for all n ≥ 1 and t ≥ 0, let H In other words, H
(n) t
operates on a function f with domain S (n) by
H (n) t f (w) = E[f (X (n) t ) | X (n) 0 = w].
For functions f and g with domain S (n) , we will use the inner product
f, g = f, g πn = E[f (w)g(w)].
As X (n) is assumed to be reversible and irreducible, we can find a set, {ψ (n) j } j of eigenvectors to −Q n , with corresponding eigenvalues
(1) 0 = λ (n) 0 < λ (n) 1 ≤ λ (n) 2 ≤ . . . ≤ λ (n) |S (n) |−1 such that {ψ (n)
j } j is an orthonormal basis with respect to ·, · for the space of real valued functions on S (n) . The eigenvectors {ψ (n) j } j will also be eigenvectors to H j } j is an orthonormal basis, for any f : S (n) → R we can write
f (w) = |S (n) |−1 i=0 f, ψ (n) i ψ (n) i (w).
To simplify notations, we will writef (j) instead of f, ψ (n) j . Note that with this notation, for any function f : S (n) → R,
E[f ] = E[f · 1] = f, 1 = f, ψ (n) 0 =f (0)
and
Var(f n ) = E[f · f ] − E[f ] 2 = f, f −f (0) 2 = |S (n) |−1 i=0f (i)ψ (n) i , |S (n) |−1 j=0f (j)ψ (n) j −f (0) 2 = |S (n) |−1 i=0 |S (n) |−1 j=0f (i)f (j) ψ (n) i , ψ (n) j −f (0) 2 = |S (n) |−1 i=0f (i) 2 −f (0) 2 = |S (n) |−1 i=1f (i) 2
The smallest nonzero eigenvalue, λ (n) 1 , is called the spectral gap of the Markov chain X (n) , and its inverse, t is called the relaxation time. Another characterization of the spectral gap which will be useful for us is
(2) λ (n) 1 = min f : E[f ]=0,f ≡0 −Q (n) f, f f, f ,
where the minimum is attained by the corresponding eigenvector ψ (n) 1 . The right hand side of (2) is called the Rayleigh quotient of −Q (n) .
We will now define the concepts with which we will be concerned in the rest of these notes.
Definition 1.1. Let (X (n) ) n≥1 be a sequence of reversible and irreducible continuous time Markov chains, with state spaces (S (n) ) n≥1 and stationary distributions (π n ) n≥1 , and let t (n) rel be the relaxation time of the nth Markov chain. A sequence (f n ) n≥1 of Boolean functions, f n : S (n) → {0, 1}, is said to be noise sensitive with respect to (X (n) ) n≥1 , if for all α > 0
(3) lim n→∞ Cov f n (X (n) 0 ), f n (X (n) αt (n) rel ) = 0.
Note that as E f n (X (n)
0 ) = E f n (X (n) αt (n) rel
) , we have that
Cov(f n (X (n) 0 ), f n (X (n) αt (n) rel ) = E f n (X (n) 0 )f n (X (n) αt (n) rel ) − E f n (X (n) 0 ) 2 .
In addition to the definition for noise sensitivity given above, we will use the following definition of noise stability, which captures the opposite behaviour. Definition 1.2. Let (X (n) ) n≥1 be a sequence of reversible and irreducible continuous time Markov chains, with state spaces (S (n) ) n≥1 and stationary distributions (π n ) n≥1 , and let t As we will see later, if (X (n) ) n≥1 is a sequence of reversible and irreducible continuous time Markov chains and (f n ) n≥1 is a sequence of Boolean functions with domain S (n) such that lim n→∞ Var(f n ) = 0, then (f n ) n≥1 will be both noise stable and noise sensitive with respect to (X (n) ) n≥1 . For this reason, we are only interested in sequences of Boolean functions with lim n→∞ Var(f n ) > 0.
If this is satisfied for a sequence (f n ) n≥1 of Boolean functions, we say that (f n ) n≥1 is nondegenerate. Remark 1.3. In [2], the concept of noise sensitivity was defined as follows. Given X
(n) 0 = (X (n) 0 (1), . . . , X (n) 0 (n)) ∈ {0, 1} n , letX (n) α = (X (n) α (1), . . . ,X (n)
α (n)) ∈ {0, 1} n be a random perturbation of X (n) 0 , i.e. for every j ∈ {1, . . . , n} independently, setX
(n) α (j) = X (n) 0 (j) with probability 1 − α, andX (n) α (j)) = 1 − X (n) 0 (j) else. A sequence (f n ) n≥1 , f n : {0, 1} n → 0, was said to be asymptotically noise sensitive if for all α > 0, lim n→∞ Cov(f n (X (n)
0 ), f n (X (n) α ) = 0. Now consider the continuous time Markov chain X (n) on {0, 1} n which for each coordinate i, at times that are exponentially distributed with parameter 1, rerandomizes the value at i by setting the value to 1 with probability 1/2 and 0 with probability 1/2. This Markov chain has relaxation time 1, and for any specific coordinate i, the probability that the value at i is not the same at time 0 as at time αt (1−e −α )/2 , why Definition 1.1 in this special case is equivalent with the definition of asymptotic noise sensitivity given in [2].
Comparing Definition 1.1 with the definition of complete graph noise sensitivity given in [3], the two definitions coincide if we consider the continuous time Markov chain corresponding to a random walk on the union of the graphs G n with vertices {w ∈ {0, 1} n : w = k} and an edge between two vertices u and w iff u = w and u − v = 2, and choose π n to be the uniform measure on {0, 1} n . However, note that as these graphs are not connected, the corresponding Markov chains are not irreducible. We will therefore not consider this model directly in this paper, even though some of our results applies to this model as well.
Using the eigenvalues of the generator, we will now give another characterization of noise sensitivity, which generalizes the first part of Theorem 1.9 in [2].
Proposition 1.4. A sequence of Boolean functions (f n ) n≥1 , f n : S (n) → {0, 1}
, is noise sensitive with respect to (X (n) ) n≥1 if and only if for all k > 0,
(5) lim n→∞ i : λ (n) 1 ≤λ (n) i <kλ (n) 1f n (i) 2 = 0. Proof. Fix α > 0. Then E f n (X (n) 0 )f n (X (n) αt (n) rel ) = E f n (X (n) 0 )H (n) αt (n) rel f n (X (n) 0 ) = E |S (n) |−1 i=0f n (i)ψ (n) i (X (n) 0 ) |S (n) |−1 j=0f n (j)H (n) αt (n) rel ψ (n) j (X (n) 0 ) = E |S (n) |−1 i=0f n (i)ψ (n) i (X (n) 0 ) |S (n) |−1 j=0f n (j)e −αt (n) rel ·λ (n) j ψ (n) j (X (n) 0 ) = i,j e −αt (n) rel ·λ (n) jf n (i)f n (j)E ψ (n) i (X (n) 0 )ψ (n) j (X (n) 0 ) = |S (n) |−1 i=0 e −αλ (n) i /λ (n) 1f n (i) 2 .
Here the last equality follows from the fact that {ψ
(n) i } |S (n) |−1 i=0
is an orthonormal set, together with the definition of the relaxation time. As E [f n (w)] 2 =f n (0) 2 , it follows that
Cov(f n (X (n) 0 ), f n (X (n) αt (n) rel )) = E f n (X (n) 0 )f n (X (n) αt (n) rel ) − E f n (X (n) 0 ) 2 = |S (n) |−1 j=1 e −αλ (n) j /λ (n) 1f n (i) 2 .(6)
For any α > 0, it is easy to see that the left hand side of (6) tends to zero as n → ∞ if and only if (5) holds. From this the desired conclusion follows.
Remark 1.5. By the last lines of the proof above it follows that if a sequence of functions satisfies (3) for one α > 0, then it does so for all α > 0, i.e. the proof of Proposition 1.4 in fact shows that a sequence of Boolean functions (f n ) n≥1 is noise sensitive with respect to (X (n) ) n≥1 if and only if lim
n→∞ Cov(f n (X (n) 0 ), f n (X (n) t (n) rel )) = 0.
We can easily obtain a similar characterization of noise stability, which generalizes the second part of Theorem 1.9 in [2]. Proposition 1.6. Let (X (n) ) n≥1 be a sequence of reversible and irreducible continuous time Markov chains. A sequence of Boolean functions (f n ) n≥1 , f n : S (n) → {0, 1}, is noise stable with respect to (X n ) n≥1 if and only if for all δ > 0 there is k ∈ N such that
(7) sup n i : λ (n) i ≥kλ (n) 1f n (i) 2 < δ.
Remark 1.7. Note that the proposition above directly implies that if (X (n) ) n≥1 is a sequence of Markov chains corresponding to random walks on a family of expander graphs, i.e. a sequence of graphs where the correspondinging sequence of spectral gaps is bounded from below, then all sequences of Boolean functions on (S (n) ) n≥1 will be noise stable with respect to (X n ) n≥1 .
Proof. First note that since f n is Boolean, we have that
P f n (X (n) αt (n) rel ) = f n (X (n) 0 ) = E f n (X (n) αt (n) rel )(1 − f n (X (n) 0 )) + E f n (X (n) 0 )(1 − f n (X (n) αt (n) rel )) = 2 E f n (X (n) 0 ) − E f n (X (n) 0 )f n (X (n) αt (n) rel
) .
Using this, as well as the the proof of the previous proposition, we obtain
P f n (X (n) 0 ) = f n (X (n) αt (n) rel ) = 2 E f n (X (n) 0 ) − |S (n) |−1 i=0 e −αλ (n) i /λ (n) 1f n (i) 2 = 2 E f n (X (n) 0 ) 2 − |S (n) |−1 i=0 e −αλ (n) i /λ (n) 1f n (i) 2 = 2 |S (n) |−1 i=0f (i) 2 − |S (n) |−1 i=0 e −αλ (n) i /λ (n) 1f n (i) 2 = 2 |S (n) |−1 i=0 1 − e −αλ (n) i /λ (n) 1 f n (i) 2 .
For the if direction of the proof, suppose that there for any any δ > 0 is k δ ≥ 1 such that
sup n i : λ (n) i ≥k δ λ (n) 1f n (i) 2 < δ.
Then for all δ > 0,
lim α→0 sup n P f n (X (n) αt (n) rel ) = f n (X (n) ) = 2 lim α→0 sup n |S (n) |−1 i=0 1 − e −αλ (n) i /λ (n) 1 f n (i) 2 ≤ 2δ + 2 lim α→0 sup n i : λ (n) i <k δ λ (n) 1 1 − e −αk δ λ (n) 1 /λ (n) 1 f n (i) 2 = 2δ + 2 lim α→0 sup n i : λ (n) i <k δ λ (n) 1 1 − e −αk δ f n (i) 2 ≤ 2δ + 2 lim α→0 1 − e −αk δ = 2δ.
As δ can be chosen to be arbitrarily small, this implies that (f n ) n≥1 is noise stable with respect to (X n ) n≥1 . For the only if direction, suppose that there is δ > 0 such that for all k ≥ 1,
sup n i : λ (n) i ≥kλ (n) 1f n (i) 2 ≥ δ
for all k > 0. Then in particular, this is true for k = α −1 . This implies that
lim α→0 sup n P f n (X (n) 0 ) = f n (X (n) αt (n) rel ) = 2 lim α→0 sup n |S (n) |−1 i=0 1 − e −αλ (n) i /λ (n) 1 f n (i) 2 ≥ 2 lim α→0 sup n i : λ (n) i ≥kλ (n) 1 1 − e −αλ (n) i /λ (n) 1 f n (i) 2 ≥ 2 lim α→0 sup n i : λ (n) i ≥kλ (n) 1 1 − e −αk f n (i) 2 = 2 lim α→0 (1 − e −1 )δ.
In particular, (f n ) n≥1 cannot be noise stable.
Remark 1.8. The proof of Proposition 1.4 shows that Definition 1.1 is sharp in the following sense. Let (T n ) n≥1 be any sequence such that lim n→∞ T n /t (n) rel = ∞. Then for any sequence (f n ) n≥1 of Boolean functions with domain S (n) , and any α > 0,
(8) lim n→∞ Cov(f n (X (n) 0 ), f n (X (n) αTn ) = 0
as we, by the same method as the one used in this proof, can show that
Cov(f n (X (n) 0 ), f n (X (n) αTn ) = |S (n) |−1 i=1 e −αλ (n) i Tnf n (i) 2 ≤ e −αλ (n) 1
Tn
|S (n) |−1 i=1f n (i) 2 ≤ e −αλ (n) 1 Tn f, f = e −αλ (n) 1 Tn E[f 2 ] ≤ e −αλ (n) 1 Tn
and this tends to zero as n → ∞. The sequence of relaxation times is thus the largest sequence of times we can look at and still possibly obtain a nontrivial definition of noise sensitivity. By a similar argument, we find that for any α > 0 and sequence (T n ) n≥1 such that lim n→∞ T n /t
(n) rel = ∞,
then for all sequences (f n ) n≥1 of Boolean functions,
P (f n (X (n) 0 ) = f n (X (n) αTn )) = 2 |S (n) |−1 i=0 1 − e −αλ (n) i Tn f n (i) 2 ≥ 2 1 − e −αλ (n) 1 Tn i∈S (n) : i≥1f n (i) 2 ≥ 2 Var(f n )
for all large enough n. This implies that if we replaced t (n) rel by any sequence (T n ) n≥1 with lim n→∞ T n /t (n) rel = ∞ in the definition of noise stability, then no nondegenerate sequence of Boolean functions would be noise stable with respect to (X (n) ) n≥1 . Remark 1.9. From the fact that Var(f n ) = |S n |−1 i=1f (i) 2 , using Proposition 1.4 and Proposition 1.6 we see that degenerate sequence (f n ) n≥1 of Boolean functions with domains (S (n) ) n≥1 will be both noise stable and noise sensitive with respect to any sequence of reversible and irreducible Markov chains with state spaces (S (n) ) n≥1 .
In the remainder of these notes, we will often be concerned with sequences of Markov chains (X (n) ) n≥1 corresponding to random walks on sequences of connected graphs (G n ) n≥1 . By a random walk on a graph G n we mean a continuous time Markov chain with state space V (G n ) and generator Q n = (q
(n) vw ) v,w∈V (Gn) with q (n) vw = 1/ deg(v) when v and w are neighbours −1 if v = w 0 else where deg(v) is the degree of the vertex v.
Whenever we talk about a sequence of graphs (G n ) n≥1 , we will assume that |V (G n )| → ∞ as n → ∞ and that G n is connected for every n. For a graph G, we will use vol(G) to denote twice the number of (undirected) edges in the graph. For example, for the complete graph on n vertices, K n , we have that vol(K n ) = n(n − 1).
The existence of noise sensitive functions
As our definitions of noise sensitivity and noise stability coincide with the definitions given in [2], we already know that both noise stable and noise sensitive nondegenerate sequences of Boolean functions exist with respect to (X (n) ) n≥1 when X (n) is a random walk on a n-dimensional Hamming cube. A natural question is if such sequences will exist in general, or if the Hamming cube is a very special case. Recall that by Remark 1.7, there are sequences of Markov chains for which no such sequences exists. In this section we will focus on the question about the existence of sequences of noise sensitive functions, and try to give criteria on the Markov chain to guarentee just that.
We begin this section by considering an example of a family of expander graphs, and give a more concrete explanation of why no noise sensitive sequences of functions can exist on these graphs.
Example 2.1. Fix k ≥ 1 and for each n ≥ k, let X (n) be the continuous time Markov chain with state space [n] k := {(w 1 , . . . , w n ) ∈ {0, 1} n : w 1 + · · · + w n = k} which evolves as follows. At times which are exponentially distributed with parameter 1, two indices i, j ∈ [n] are chosen independently at random, and the digits at these positions in the current string are switched. Equivalently, X (n) is an exclusion process on the binary strings of length n and Hamming weight k. For any fixed k, as n → ∞ this Markov chain has relaxation time of order 1 (see e.g. [5]). As a consequence of this, as α → 0, at time α · 1 the probability is very high that none of the balls have moved at all, and thus all sequences of functions are noise stable with respect to this sequence of Markov chains.
Remark 2.2. The simplest of the models in the previous example is given by choosing k = 1, in which case we obtain a random walk on the complete graph on n vertices.
The moral of the previous example is that if for some sequence of random walks on a sequence of graphs, the relaxation time is such that t (n) rel is bounded from above when n tends to infinity, then all functions will be noise stable, the reason being that with high probability, the random walker on the nth graph never moves at all between time 0 and time αt (n) rel . Proposition 1.6 shows that another condition which guarantees that all functions are noise stable is that λ
(n) |S (n) |−1 /λ (n) 1 = O(1)
. However, as for random walks on graphs, λ |S n−1 | ≤ 2, these conditions are equivalent.
In the rest this section, our goal will be to prove the following result, which shows that when with high probability, the position of the random walker at the relaxation time is not the same as the position of the random walker at time zero, there will be at least one nondegenerate sequence of functions which is noise sensitive. Proposition 2.3. Let (X (n) ) n≥1 be a sequence of reversible and irreducible continuous time Markov chains. Then if
(9) lim n→∞ P (X (n) 0 = X (n) t (n) rel ) − i π n (i) 2 = 0
there is at least one nondegenerate sequence of functions (f n ) n≥1 which is noise sensitive with respect to (X (n) ) n≥1 .
Remark 2.4. Note that (9) is equivalent to that
(10) lim n→∞ w∈S (n) π n (w) · P (X (n) t (n) rel = w | X (n) 0 = w) − P (X (n) t (n) rel = w) = 0
i.e. (9) is a measure of how different the probability of being at X (n) 0
at time t (n)
rel is from being in X (n) 0 when in the stationary distribution. Also, (9) can be rewritten as
(11) lim n→∞ w∈S (n) Cov 1 w (X (n) 0 ), 1 w (X (n) t (n) rel ) = 0
i.e. as the sum of the noise sensitivity of the indicator functions of all w ∈ S (n) .
Even though in general, it might be hard to check whether (9) holds, Proposition 2.3 might be useful in some special cases. It is e.g. relatively simple to show that (9) holds for graphs whose minimum degree tends to infinity, such as sequences of hypercubes. On the other hand, the following example, due to Johan Jonasson, shows that there are sequences of graphs on which (9) do not hold, even though the corresponding sequence of relaxation times is unbounded.
Example 2.5. Let G n be the graph obtained by joining 2n stars with n 2 outer vertices byadding all possible edges between their centers (see figure 1). This relaxation time of the random walk on this graph can be at most of the same order as the expected time until one of the edges between the centers is used, which is of order n. As the bottleneck ratio can be at most 1/n, it follows that the relaxation time t Now pick ε > 0 to be very small. Then with probability close to 0.5, given that the random walk started in one of the vertices of the inner complete graph, x, at time εt (n) rel , the random walk is at the same position as where it started. By a similar argument, it follows that given that the random walk started in one of the outer vertices, y, at time εt (n) rel the probability is close to 1/2n 2 that the random walk is in y. Using this, as well as standard results for random walk on discrete graphs, it follows that (9) do not hold. However, as the mixing time of each individual star is of order one, it is relatiively easy to see that any sequence of functions with
lim n→∞ Var i (E[f n (X (n) 0 | X (n) 0
in star number i]) = 0 will be noise sensitive. We now give a proof of Proposition 2.3.
Proof of Proposition 2.3. Fix some n ≥ 1. Let (m n ) n≥1 be a sequence of integers s.t. lim n→∞ m n /|S (n) | ∈ (0, 1). Define f n by choosing exactly m n of the states in S (n) independently at random, and set f n to be one on those vertices and 0 everywhere else. Then
lim n→∞ Var[f n (w)] = lim n→∞ m n (|S (n) | − m n )/|S (n) | 2 ∈ (0, 1)
so that any sequence (f n ) n≥1 made by picking a sequence of functions in this way is nondegenerate.
Define the random vaiable Y m in terms of f n be
Y n = Cov f n (X (n) 0 ), f n (X (n) t (n) rel ) = E f n (X (n) 0 )f n (X (n) t (n) rel ) − E f n (X (n) 0 ) 2 . and let v (n) , w (n) ∈ S (n) , v (n) = w (n) be fixed. Then E[Y n ] = E E f n (X (n) 0 )f n (X (n) t (n) rel ) | f n − E f n (X (n) 0 ) | f n 2 = E fn E f n (X (n) 0 )f n (X (n) t (n) rel ) | f n − E E f n (X (n) ) | f n 2 .(12)
Rewriting the first of these two terms, we obtain
E E f n (X (n) 0 )f n (X (n) t (n) rel ) | f n =E E f n (X (n) 0 )f n (X (n) t (n) rel ) | X (n) 0 , X (n) t (n) rel =E E fn f n (X (n) 0 )f n (X (n) t (n) rel ) | X (n) 0 , X (n) t (n) rel | X (n) 0 = X (n) t (n) rel · P (X (n) 0 = X (n) t (n) rel ) +E E fn f n (X (n) 0 )f n (X (n) t (n) rel ) | X (n) 0 , X (n) t (n) rel | X (n) 0 = X (n) t (n) rel · P (X (n) 0 = X (n) t (n) rel )
.
Here
E E f n (X (n) 0 )f n (X (n) t (n) rel ) | X (n) 0 , X (n) t (n) rel | X (n) 0 = X (n) t (n) rel = E E f n (X (n) 0 ) 2 | X (n) 0 | X (n) 0 = X (n) t (n) rel = E E f n (X (n) 0 ) | X (n) 0 | X (n) 0 = X (n) t (n) rel = E E f n (X (n) 0 ) | X (n) 0 = m n /|S (n) |
and similarly
E E f n (X (n) 0 )f n (X (n) t (n) rel ) | X (n) 0 , X (n) t (n) rel | X (n) 0 = X (n) t (n) rel = m n 2 |S (n) | 2 = m n (m n − 1) |S (n) |(|S (n) | − 1)
.
For the last term in (12) E
[E[f n (X (n) 0 ) | f n ] 2 ] = E P (f n (X (n) 0 ) = 1 | f n ) 2 = |S (n) | m n −1 · |S (n) | − 1 m n − 1 i∈S (n) π n (i) 2 + |S (n) | − 2 m n − 2 i,j∈S (n) : i =j π n (i)π n (j) = m n |S (n) | i∈S (n) π n (i) 2 + m n (m n − 1) |S (n) |(|S (n) | − 1) i,j∈S (n) : i =j π n (i)π n (j) = m n |S (n) | − m n (m n − 1) |S (n) |(|S (n) | − 1) i π n (i) 2 + m n (m n − 1) |S (n) |(|S (n) | − 1) i,j π n (i)π n (j) = m n (|S (n) | − m n ) |S (n) |(|S (n) | − 1) i π n (i) 2 + m n (m n − 1) |S (n) |(|S (n) | − 1) i π n (i) 2 = m n (|S (n) | − m n ) |S (n) |(|S (n) | − 1) i π n (i) 2 + m n (m n − 1) |S (n) |(|S (n) | − 1)
.
Summing up, we obtain
E[Y n ] = m n |S (n) | · P (X (n) 0 = X (n) t (n) rel ) + m n (m n − 1) |S (n) |(|S (n) | − 1) · P (X (n) 0 = X (n) t (n) rel ) − m n (|S (n) | − m n ) |S (n) |(|S (n) | − 1) i π(i) 2 + m n (m n − 1) |S (n) |(|S (n) | − 1) = m n (|S (n) | − m n ) |S (n) |(|S (n) | − 1) · P (X (n) 0 = X (n) t (n) rel ) − i π(i) 2 = |S (n) | |S (n) | − 1 · Var(f n ) · P (X (n) 0 = X (n) t (n) rel ) − i π(i) 2 .
Now for each n, fix a function f * n defined as above for which
Cov f * n (X (n) 0 ), f * n (X (n) t (n) rel ) ≤ E[Y n ].
Then by assumption, (f * n ) n≥1 is noise sensitive with respect to (X (n) ) n≥1 .
The existence of noise stable functions
In analogue with Proposition 2.3, which gives a criteria for when there exist nondegenerate sequences of Boolean functions which are noise sensitive with respect to some Markov chain (X (n) ) n≥1 , the goal of this section is to obtain criteria for when nondegenerate noise stable sequences of Boolean functions exist. In Proposition 3.1, we show that the existence of noise stable functions is tightly connected with the so called delocalization of eigenvectors which has recently been studied for random graphs and random matrices in eg. [1], [4], [6] and [10]. Proposition 3.2 then provides a condition which guarantees the existence of nondegenerate noise stable sequence of Boolean functions whenever (X (n) ) n≥1 is a sequence of transitive, reversible and irreducible Markov chains. In particular, combining the two propositions, we obtain that eigenvectors of transition matrices of transitive Markov chains do not localize.
We now state our main results of this section.
Proposition 3.1. Let (X (n) ) n≥1 be a sequence of reversible and irreducible continuous time Markov chains. Then the following two conditions are equivalent.
(A) there exists a nondegenerate sequence of Boolean functions which is noise stable with respect to (X (n) ) n≥1 (B) there is ε > 0, k > 0 and a sequence (
ψ (n) ) n≥1 , where ψ (n) ∈ Ψ (n) k := Span({ψ (n) i } i : λ (n) 1 ≤λ (n) i ≤kλ (n) 1 ) and ψ (n) , ψ (n) = 1, such that (13) P w (ψ (n) (w) 2 < ε) < 1 − ε for all large enough n.
Although the previous proposition might seem quite technical, we get the following result as a consequence.
Proposition 3.2. For all sequences of transitive, reversible and irreducible continuous time Markov chains (X (n) ) n≥1 there is at least one nondegenerate sequence of Boolean functions which is noise stable with respect to (X (n) ) n≥1 .
By a transitive Markov chain we mean a Markov chain with state space S and generator Q = (q ij ) i,s∈S such that there for any pair of states i, j ∈ S is a bijective function ϕ : S → S such that ϕ(i) = j and q ϕ(i)ϕ(j) = q ij . Let Aut(X n ) denote the set of all such functions ϕ given some Markov chain X n . Note that a random walk on a vertex transitive graph is a transitive Markov chain.
We give a proof of Proposition 3.2 after proving Proposition 3.1.
Remark 3.3. From Proposition 1.6 it is clear that the existence of noise stable nondegenerate sequences of Boolean functions is equivalent to that there must exists some integer k and a sequence of vectors (ψ (n) ) n≥1 with ψ (n) ∈ Ψ (n) k , such that the distance between ψ (n) and f n − E[f n ], for a nondegenerate sequence (f n ) n≥1 of Boolean functions, is asymptotically zero. If this holds then clearly (B) must happen, so one direction of the proposition above is obvious, although we formalize this argument in Lemma 3.9. The idea of the proof in the other direction is that if a noise stable sequence of functions exists, then there should be such a sequence which arises from truncating a sequence of functions (ψ (n) ) n≥1 , where ψ (n) ∈ Ψ (n) k for some k ≥ 1. If this is not possible, we cannot have (B). This idea is very similar to the so called best threshhold cut algorithm which sweeps through all possible truncations of an eigenvector corresponding to the first eigenvalue in order to find a good cut in a graph, used in computer science for e.g. clustering ( [12]). Before moving on to the proof of Proposition 3.1, it should also be noted that there is in fact sequences of graphs such that no nondegenerate sequence of Boolean functions is noise stable with respect to the corresponding random walk. The next example provides such an example.
Example 3.5. Let G n be the graph obtained by attaching a complete graph on n 2 vertices to a complete graph on n vertices by adding a single edge (v, w) between a vertex c in the larger graph and a vertex w in the smaller graph. Consider the random walk X (n) on G n . Let Q (n) = (Q (n) ij ) i,j∈V (Gn) be the corresponding generator and π n be the corresponding stationary distribution.
v w K n 2 K n Write K n and K n 2 for the two subgraphs. Then
vol(G n ) = n 2 (n 2 − 1) + n(n − 1) + 2 π n (K n ) = (n · (n − 1) + 1)/vol(G n ) π n (K n 2 ) = (n 2 · (n 2 − 1)
+ 1)/vol(G n ) π n (v) = n 2 /vol(G n ) π n (w) = n/vol(G n ) q (n) vw = 1/n 2 q (n) wv = 1/n Define a function f n (u) for u ∈ V (G n ) by setting f n (u) = 1 when u ∈ K n 2 −πn(K n 2 ) πn(Kn) when u ∈ K n
Then E[f n (u)] = 0, and clearly f n ≡ 0. Using (2), we obtain
λ (n) 1 = min f : E[f ]=0,f ≡0 1 2 · u1,u2∈V (Gn) |f (u 1 ) − f (u 2 )| 2 π n (u 1 ) q (n) u1u2 f, f ≤ 1 2 · u1,u2∈V (Gn) |f n (u 1 ) − f n (u 2 )| 2 π n (u 1 ) q (n) u1u2 f n , f n = 1 2 · 1 + πn(K n 2 ) πn(Kn) 2 · π n (v) q (n) vw + π n (w) q (n) wv π n (K n 2 ) · 1 2 + π n (K n ) · πn(K n 2 ) πn(Kn) 2
Using that random walks are reversible, and simplifying, we obtain
λ (n) 1 ≤ 1 2 · 1 + πn(K n 2 ) πn(Kn) 2 · π n (v) q (n) vw + π n (w) q (n) wv π n (K n 2 ) · 1 2 + π n (K n ) · πn(K n 2 ) πn(Kn) 2 = 1 + πn(K n 2 ) πn(Kn) 2 · π n (v) q (n) vw π n (K n 2 ) + π n (K n ) · πn(K n 2 ) πn(Kn) 2 = πn(Kn)+πn(K n 2 ) πn(Kn) 2 · π n (v) q (n) vw
π n (K n 2 ) · πn(Kn)+πn(K n 2 ) πn(Kn) = (π n (K n ) + π n (K n 2 )) · π n (v) q (n) vw π n (K n ) · π n (K n 2 ) = 1 · π n (v) q (n) vw π n (K n ) · π n (K n 2 ) = 1 · n 2 / vol(G n ) · 1/n 2 π n (K n ) · π n (K n 2 )
= 1 vol(G n ) · 1 π n (K n ) · π n (K n 2 ) ≍ 1 n 2 .
As π n (K n ) → 0, no nondegenerate sequence of Boolean functions with domains (V (G n )) n≥1 can be noise stable with respect to (X (n) ) n≥1 due to its values on (K n ) n≥1 . As the mixing time of K n 2 is of order 1 for each n, while the calculations above show that the the relaxation time of G n is of order n 2 , this implies that all nondegenerate sequences of Boolean functions are noise sensitive with respect to (X (n) ) n≥1 .
Not much is known about neither the eigenvectors of transition matrices of Markov chains in general, nor of the eigenvectors of transition matrices of random walks on graphs in particular. However, in recent years, conditions similar to (B) have been studied for random graphs. To simplify the notation, consider first the following definition, used in [6] and [10].
Definition 3.6. Let T be a subset of {1, 2, . . . , n} of size L and let δ > 0 be a fixed number. A vector ψ ∈ R n exhibits (T, δ)-localization if
ψ(·)1 ·∈T (·) 2 2 ≥ (1 − δ) ψ 2 2 . A vector ψ is said to be (L, δ)-localized is there is some set T ⊂ {1, 2, . . . , n} with 1 T 2 2 ≤ L such that ψ is (T, δ)-localized.
Note that the definition above is dependent of the chosen basis. If we let · 2 2 = ·, · and note that the opposite of (B) is given by (¬B) for each k > 0 and ε > 0 there is arbitrarily large n such that
P (ψ (n) (w) 2 < ε) ≥ 1 − ε for all ψ (n) ∈ Ψ (n) k with ψ (n) , ψ (n) = 1.
It is now easy to show that (¬B) is equivalent to (¬B) for each k > 0 and ε > 0 there is arbitrarily large n such that all ψ (n) ∈ Ψ (n) k is (εn, ε)localized.
For the usual L 2 -norm on R n , which agrees with ·, · only for Markov chains with uniform stationary distribution π, the following is known. Corollary 3.2 in [4]. For every p > 0 there is δ = δ(p) > 0 and ε(p) ∈ (0, 0.5) such that for almost all G ∼ G(n, p), no eigenvector of the adjency matrix except ψ 0 is (εn, δ)-localized. Theorem 3 in [6]. Fix δ > 0. Let d n = (log n) γ for some γ > 0 and let T n be a deterministic sequence of sets of size
L n = o(2/(exp(d −α n ) − exp(−d −α n ))
) where α ∈ (0, min(1, 1/γ). Then if A n is the adjency matrix of a random regular graph with valency d n , then P (no eigenvector of A n is (T n , δ)-localized) ≥ 1 − o(1/d n ).
Theorem 7.1 in [10]. Let H be an n × n Hermitean random matrix from a Wigner ensamble satisfying two technical conditions (see [10]). Suppose that L ∈ [n] and η > 0 are chosen such that η and ν := L/n is sufficiently small. Then there is a constant c > 0 which do not depend on n such that P (no eigenvector of H is (L, η)-localized) ≥ 1 − e −cn .
Note that none of the results above is exactly what we need. The first and third results are both of the type we need. However, in the first case, the result is for eigenvectors of the adjency matrix instead of for the generator and in addition, the norm is wrong. In the third case, the result is not for transition matrices of Markov chains at all, but is of the correct type. In the second case, the eigenvectors are eigenvectors for the generator as well as the random graphs are regular, and for the same reason the norm is correct as well. However, there are two other problems with this result. Firstly, we need a result for all sequences of sets and not for a beforehand chosen sequence. More problematic for us however, is that the measure of the sets T n will tend to zero.
Given what is currently known however, there seems to be nothing preventing that (B) would hold for a large family of Markov chains, and further results in this direction would be interesting in the light of Proposition 3.1.
We will now give a proof of Proposition 3.1 through a sequence of lemmas. The first of these lemmas is particularly interesting since it provides means by which one can validate that a sequence of truncated real-valued functions is noise stable. Lemma 3.7. Let (X (n) ) n≥1 be a sequence of reversible and irreducible continuous time Markov chains, and let g(ε) be a real valued function such that ε/g(ε) 2 → 0 as ε → 0. Suppose that there is k > 0, δ > 0, a sequence (ψ (n) ) n≥1 where ψ (n) ∈ Ψ (n) k and ψ (n) , ψ (n) = 1, and a sequence (c n ) n≥1 such that for all n ≥ 1,
(i) P (ψ (n) (w) ≥ c n ) > δ (ii) P (ψ (n) (w) < c n ) > δ (iii) lim ε→0 sup n P (ψ (n) (w) ∈ [c n − g(ε)
, c n + g(ε)]) = 0 Then the sequence (1 ψ (n) ≥cn ) n≥1 is nondegenerate and noise stable with respect to (X (n) ) n≥1 .
Proof. Note first that for all functions f with range {0, 1},
(14) P (f (X 0 ) = f (X t )) = E (f (X 0 ) − f (X t )) 2 = 2 |S (n) |−1 i=1 (1 − e −λit )f (i) 2
where the last equality, by the proof of Proposition 1.4 in fact holds for all real valued functions with the same domain as f . Set J n (ε) = [c n − g(ε), c n + g(ε)] and f n = 1 ψ (n) ≥cn . Then (f n ) n≥1 is nondegenerate by (i) and (ii).
To show that (f n ) n≥1 will be noise stable with respect to (X (n) ) n≥1 , note that
P (f n (X (n) 0 ) = f n (X (n) εt (n) rel )) = E (f n (X (n) 0 ) − f n (X (n) εt (n) rel )) 2 ≤ P (ψ (n) (X (n) 0 ) ∈ J n (ε)) + P ψ (n) (X (n) 0 ) − ψ (n) (X (n) εt (n) rel ) ≥ g(ε) | ψ (n) (X (n) 0 ) ∈ J n (ε) ≤ P (ψ (n) (X (n) 0 ) ∈ J n (ε)) + E (ψ (n) (X (n) 0 ) − ψ (n) (X (n) εt (n) rel )) 2 | ψ (n) (X (n) 0 ) ∈ J n (ε) g(ε) 2 ≤ P (ψ (n) (X (n) 0 ) ∈ J n (ε)) + E (ψ (n) (X (n) 0 ) − ψ (n) (X (n) εt (n) rel )) 2 g(ε) 2 P (ψ (n) (X (n) 0 ) ∈ J n (ε)) ≤ P (ψ (n) (X (n) 0 ) ∈ J n (ε)) + 2 i (1 − e −ελit (n) rel )ψ (n) (i) 2 g(ε) 2 P (ψ (n) (X (n) 0 ) ∈ J n (ε)) ≤ P (ψ (n) (X (n) 0 ) ∈ J n (ε)) + 2(1 − e −εk ) iψ (n) (i) 2 g(ε) 2 P (ψ (n) (X (n) 0 ) ∈ J n (ε)) = P (ψ (n) (X (n) 0 ) ∈ J n (ε)) + 2(1 − e −εk ) g(ε) 2 P (ψ (n) (X (n) 0 ) ∈ J n (ε)) ≤ P (ψ (n) (X (n) 0 ) ∈ J n (ε)) + 2εk g(ε) 2 P (ψ (n) (X (n) 0 ) ∈ J n (ε))
.
Using (iii), we obtain
lim ε→0 sup n P (f n (X (n) 0 ) = f n (X (n) εt (n) rel )) = lim ε→0 sup n P (ψ (n) (w) ∈ J n (ε)) + 2εk g(ε) 2 P (ψ (n) (X (n) 0 ) ∈ J n (ε)) = 0
i.e. (f n ) n≥1 is noise stable with respect to (X (n) ) n≥1 .
Lemma 3.8. Let (X (n) ) n≥1 be a sequence of reversible and irreducible continuous time Markov chains. Then either (C) there is k ≥ 1, a sequence (c n ) n≥1 and a sequence (ψ (n) ) n≥1 , where ψ (n) ∈ Ψ (n) k such that (1 ψ (n) (w)>cn ) n≥1 is nondegenerate and noise stable with respect to (X (n) ) n≥1 , or (¬B) for each k > 0 and ε > 0 there is arbitrarily large n such that
P (ψ (n) (w) 2 < ε) ≥ 1 − ε for all ψ (n) ∈ Ψ (n) k
with ψ (n) , ψ (n) = 1. Note in particular that this entails that (B) ⇒ (C) ⇒ (A).
Proof. For each δ > 0, n ≥ 1, k > 0 and ψ (n) ∈ Ψ (n) k , define c δ (ψ (n) ) = sup{c ∈ R : P (ψ (n) (w) ≤ c) < δ} and c 1−δ (ψ (n) ) = inf{c ∈ R : P (ψ (n) (w) ≥ c) < δ}.
and set I δ (ψ (n) ) := [c (n) δ (ψ), c 1−δ (ψ (n) )]
. Note in particular that this implies that (15) P (ψ (n) (w) ∈ I δ (ψ (n) )) ≥ 1 − 2δ.
Suppose that (C) does not hold. Then the assumptions of Lemma 3.7 cannot be satisfied, and in particular, for all k > 0 and δ > 0, there is α = α(k, δ) > 0 such that
lim sup ε ′ →0 sup n inf c∈I δ (ψ (n) ) P (ψ (n) (w) ∈ [c − √ ε ′ , c + √ ε ′ ]) ≥ α for all (ψ (n) ) n≥1 with ψ (n) ∈ Ψ (n) k
and ψ (n) , ψ (n) = 1. By Lemma A.1, this implies that for any ε ′ > 0 there is n(ε ′ , δ) > 0 such that
inf c∈I δ (ψ (n(ε ′ ,δ)) ) P (ψ (n(ε ′ ,δ)) (w) ∈ [c − √ ε ′ , c + √ ε ′ ]) ≥ α/2 for all ψ (n(ε ′ ,δ)) ∈ Ψ (n(ε ′ ,δ)) k
. This in turn implies that for any ψ (n(ε ′ ,δ)) ∈ Ψ (n(ε ′ ,δ)) k and for all c ∈ I δ (ψ (n(ε ′ ,δ)) )
(16) P (ψ (n(ε ′ ,δ)) (w) ∈ [c − √ ε ′ , c + √ ε ′ ]) ≥ α/2.
As the length of I δ (ψ (n(ε ′ ,δ)) ) is c (n(ε ′ ,δ)) 1−δ − c (n(ε ′ ,δ)) δ and any interval of length 2 √ ε ′ with midpoint contained in I δ (ψ (n(ε ′ ,δ)) ) has measure at least α/2 by 16, we obtain 1 ≥ P (ψ (n(ε ′ ,δ)) (w) ∈ I δ (ψ (n(ε ′ ,δ)) )) ≥ c
(n(ε ′ ,δ)) 1−δ − c (n(ε ′ ,δ)) δ 2 √ ε ′ · α 2 = |I δ (ψ (n(ε ′ ,δ)) )| √ ε ′ · α 4 or equivalently, that (17) |I δ (ψ (n(ε ′ ,δ)) )| ≤ 4 α · √ ε ′ .
Now recall that we need to show that (¬B) holds, i.e. that there for any k > 0 and ε > 0 is arbitrarily large n such that
P (ψ (n) (w) 2 < ε) ≥ 1 − ε for all ψ (n) ∈ Ψ (n) k with ψ (n) , ψ (n) = 1.
To this end, fix k > 0 and ε > 0 and pick a sequence (δ j ) j≥1 such that 2δ j /(1 − 2δ j ) 2 < ε/2 for all j ≥ 1, and lim j→∞ δ j = 0. Then by (15),
(18) P (ψ (n) (w) ∈ I δj (ψ (n) )) ≥ 1 − 2δ j > 1 − ε/2 > 1 − ε
for all n ≥ 1 and all sequences (ψ (n) ) n≥1 with ψ (n) ∈ Ψ (n) k . Pick ε ′ (j) such that 4 α(k,δj ) · ε ′ (j) ≤ ε/2. By the derivations above, we can find n(ε ′ (δ j ), δ j ) such that
(19) |I δj (ψ (n(ε ′ (δj ),δj)) )| ≤ 4 α(k, δ j ) · ε ′ (δ j ) = ε/2
for all ψ (n(ε ′ (δj ),δj)) ∈ Ψ (n(ε ′ (δj ),δj )) k . By applying Lemma A.2 to the sets A j = {w ∈ S ((n(ε ′ (δj ),δj))) : ψ ((n(ε ′ (δj ),δj ))) (w) ∈ I δj (ψ ((n(ε ′ (δj ),δj ))) )} we get
E[ψ (n(ε ′ (δj ),δj )) (w) | ψ (n(ε ′ (δj ),δj)) (w) ∈ I n(ε ′ (δj ),δj) ] 2 ≤ 2δ j (1 − 2δ j ) 2 ≤ ε/2.
Combining this with (19), we obtain
I δj (ψ (n(ε ′ (δj ),δj )) ) ⊆ [−ε/2 − ε/2, ε/2 + ε/2] = [−ε, ε].
Summing up, we have that
P (ψ (n(ε ′ (δj ),δj )) (w) 2 < ε) ≥ P (|ψ (n(ε ′ (δj ),δj)) (w)| < ε) ≥ P (|ψ (n) (w)| < ε | ψ (n) (w) ∈ I δ (ψ (n) )) · P (ψ (n) (w) ∈ I δ (ψ (n) )) ≥ 1 · (1 − 2δ) ≥ 1 − ε for all sequences (ψ (n(ε ′ (δj ),δj)) ) j≥1 with ψ (n(ε ′ (δj ),δj )) ∈ Ψ (n(ε ′ (δj ),δj)) k .
The only thing which remains to do to finish the proof is to show that n(ε ′ (δ j ), δ j ) can be chosen arbitrarily large. To do this, it is enough to show that
lim j→∞ n(ε ′ (δ j ), δ j ) = ∞.
To see this, note that if δ < min w∈S (n) π n (w), then ψ (n) ∈ I δ for all w ∈ S (n) , which implies that |I δ | > 0 as ψ (n) , ψ (n) = 1 and E[ψ (n) ] = 0. This contradicts that |I δ | < 4 α · √ ε ′ < ε 2 . As lim j→∞ δ j = 0 by assumption, this shows that
n(ε ′ (δ j ), δ j ) ≥ inf{n ≥ 1 : δ ≥ min w∈S (n) π n (w)}.
This completes the proof. Proof. Suppose that (f n ) n≥1 is a nondegenerate sequence of Boolean functions which is noise stable with respect to (X (n) ) n≥1 . As (f n ) n≥1 is nondegenerate and Boolean, there is δ > 0 such that
δ < Var(f n ) = E[f 2 n ] − E[f n ] 2 = E[f n ] − E[f n ] 2 = E[f n ] · (1 − E[f n ]) for all n ≥ 1. Rewriting this, we obtain E[f n ] − 1 2 < 1 4 − δ which in particular implies that (20) min {|E[f n ] − 0| , |E[f n ] − 1|} > 1 2 − 1 4 − δ
for all n ≥ 1.
As (f n ) n≥1 is noise stable with respect to (X (n) ) n≥1 , by Proposition 1.6, for all α > 0 there is k > 0 such that
(21) i : λi>kλ1f n (i) 2 < α for all n ≥ 1. Fix such a k. For a function f : S (n) → R, let P k f := i≥1 : λi<kλ1f (i)ψ (n) i and let P k,0 f := i≥0 : λi<kλ1f (i)ψ (n)
i . If (¬B) holds, then for all ε > 0 there is arbitrarily large n ≥ 1 such that for all ψ (n) ∈ Ψ (n) k with ψ (n) , ψ (n) = 1,
P (ψ (n) (w) 2 < ε) ≥ 1 − ε.
In particular
P (P k f n (w) 2 < ε) ≥ P (P k f n (w) 2 < ε P k f n , P k f n ) ≥ 1 − ε.
Using (20) we now obtain
i : λi>kλ1f n (i) 2 = f n − P k,0 f n , f n − P k,0 f n = E[(f n − P k,0 f n ) 2 ] ≥ P (P k f n (w) 2 < ε) · E[(f n − P k,0 f n ) 2 | P k f n (w) 2 < ε] ≥ (1 − ε) · 1 2 − 1 4 − δ − √ ε .
If α and ε are both chosen small enough, this contradicts (21). This finishes the proof. We now proceed to the proof of Proposition 3.2.
Proof of Proposition 3.2. We will prove something stronger than what is required to be able to conclude the proposition, namely that there is ε > 0 and a sequence (ψ (n) ) n≥1 , where ψ (n) ∈ Ψ (n) 1
and ψ (n) , ψ (n) = 1 for each n, such that
P w (ψ (n) (w) 2 < ε) < 1 − ε
for all large enough n. Assume for contradiction that this is not true, i.e. assume that for any ε > 0 there is a subsequence (n (ε) k ) k≥1 , such that
(22) P w (ψ (n (ε) k ) (w) 2 < ε) ≥ 1 − ε for all sequences (ψ (n (ε) k ) ) k≥1 with ψ (n (ε) k ) ∈ Ψ (n (ε) k ) 1
. We can without loss of generality assume that {n
(ε) k } k≥1 ⊆ {n (ε ′ )
k } k≥1 whenever ε < ε ′ . To make the notation less cumbersome, we will write n k instead of n and inductively for j = 2, 3, . . . , m (n k ) ,
φ (n k ) j [θ 1 , θ 2 , . . . , θ j ] = φ (n k ) j−1 [θ 1 , . . . , θ j ] sin θ j + χ (n k )
j+1 cos θ j . Then all vectors ψ ∈ Ψ (n k ) 1 of length one can be written as
ψ[θ 1 , . . . , θ m (n k ) −1 ] := φ (n k ) m (n k ) [θ 1 , . . . , θ m (n k ) −1 ] for some tuple (θ 1 , . . . , θ m (n k ) −1 ) ∈ [0, π] × [0, π/2] m (n k ) −2 .
Note that ψ is the parametric description of a m (n k ) -dimensional sphere in R |S (n k ) | . Let µ (n k ) be its surface measure.
Define
Θ (n (ε) k ) w (ε) = (θ 1 , . . . , θ m (n k ) −1 ) ∈ [0, π] × [0, π/2] m (n k ) −2 : φ (n k ) m (n k ) [θ 1 , . . . θ m (n k ) −1 ](w) 2 < ε (22) is then equivalent to that P ({w : (θ 1 , . . . , θ m (n k ) −1 ) ∈ Θ (n (ε) k ) w }) ≥ 1 − ε for each tuple (θ 1 , . . . , θ m (n k ) −1 ). By (22), each ψ ∈ Ψ (n k ) 1 ∩ S |S (n k ) |−1 corresponds to a tuple (θ 1 , . . . , θ m (n k ) −1 ) which lies in Θ (n k ) w (ε) for at least (1 − ε)|S (n k ) | different w ∈ S (n k )
. In terms of probabilities, this means that
P ({w : (θ 1 , . . . , θ m (n k ) −1 ) ∈ Θ (n (ε) k ) w }) ≥ 1 − ε
for each tuple (θ 1 , . . . , θ m (n k ) −1 ). As this holds for all such tuples, we obtain that
P × µ (n k ) ({w : (θ 1 , . . . , θ m (n k ) −1 ) ∈ Θ (n (ε) k ) w }) ≥ 1 − ε
As P a finite probability measure with is uniform on S (nm) , there must be at least one w ∈ S (n k ) such that µ (n k ) (Θ (n k ) w (ε)) ≥ 1 − ε. Fix such an element w ∈ S (n k ) . As µ (n k ) is a surface measure, it is invariant of the choice of basis. In particular, we can assume that the basis is chosen in such a way that
χ (n k ) 1 (w) = . . . = χ (n k ) m (nk) −1 (w) = 0.
Then µ(Θ (n k ) w (ε)) can be rewritten as
µ (n k ) (Θ (n k ) w (ε)) = µ (n k ) (θ 1 , . . . , θ m (n k ) −1 ) : φ (n k ) m (n k ) [θ 1 , . . . , θ m (n k ) −1 ](i) 2 < ε = µ (n k ) (θ 1 , . . . , θ m (n k ) −1 ) : χ (n k ) m (n k ) (w) 2 cos 2 (θ m (n k ) −1 ) < ε = µ (n k ) (θ 1 , . . . , θ m (n k ) −1 ) : cos 2 (θ m (n k ) −1 ) < ε χ (n k ) m (n k ) (w) 2 = µ (n k ) (θ 1 , . . . , θ m (n k ) −1 ) : cos 2 (θ m (n k ) −1 ) < ε M (n k ) w where M (n k ) w = sup θ1,...,θ m (n k ) −1 φ (n k ) m (n k ) [θ 1 , . . . , θ m (n k ) −1 ] 2 .
Since the surface element of a m (n k ) -dimensional sphere is given by
sin(θ 2 ) sin 2 (θ 3 ) · · · sin m (n k ) −2 (θ m (n k ) −1 )dθ 1 dθ 2 . . . dθ m (n k ) −1 we have that µ(Θ (n k ) w (ε)) = π/2 arccos ε/M (n k ) w sin m (n k ) −2 (θ m (n k ) −1 )dθ m (n k ) −1 π/2 0 sin m (n k ) −2 (θ m (n k ) −1 )dθ m (n k ) −1 .
To be able to give bounds on this integral, note first that
arccos ε M (n k ) w = π 2 − ε M (n k ) w + O ε M (n k ) w 3 .
This implies that
µ (n k ) (Θ (n k ′ ) w (ε)) = ε/M (n k ) w +O((ε/M (n k ) w ) 3/2 ) 0 cos m (n k ) −2 (θ) dθ π/2 0 cos m (n k ) −2 (θ) dθ .
For any a ∈ (0, 1),
cos m (n k ) −2 θ = a ⇔ θ = arccos a 1/(m (n k ) −2) = 2(1 − a 1/(m (n k ) −2) ) + O((1 − a 1/(m−2) ) 3/2 ) = −2 log a m (n k ) − 2 + O( log a m (n k ) − 2 ) + O((1 − a 1/(m−2) ) 3/2 ) = −2 log a m (n k ) − 2 + O( log a m (n k ) − 2
).
Using Lemma A.3, we obtain
1 2 · −2 log a m (n k ) −2 +O( log a m (n k ) −2 ) 0 cos m (n k ) −2 (θ) dθ π/2 0 cos m (n k ) −2 (θ) dθ ≤ 1 1 + a .
Since (X (n) ) n≥1 is transitive, Lemma A.4 implies that M (n k ) w = dim Ψ (n k ) = m (n k ) . In particular, this implies that if we choose a = exp(−4ε), then
µ (n k ) (Θ (n k ′ ) w (ε)) = √ ε/M (n k ) +O((ε/M (n k ) ) 3/2 ) 0 cos m (n k ) −2 (θ) dθ π/2 0 cos m (n k ) −2 (θ) dθ = √ ε/m (n k ) +O((ε/m (n k ) ) 3/2 ) 0 cos m (n k ) −2 (θ) dθ π/2 0 cos m (n k ) −2 (θ) dθ ≤ 1 2 · −2 log a m (n k ) −2 +O( log a m (n k ) −2 ) 0 cos m (n k ) −2 (θ) dθ π/2 0 cos m (n k ) −2 (θ) dθ ≤ 1 1 + exp(−4ε)
whenever ε is small enough. In particular, if ε is small enough, then we cannot have that
µ(Θ (n k ′ ) i (ε)) ≥ 1 − ε
for any k ≥ 1. This completes the proof.
Noise stability and the bottleneck ratio
Although Proposition 3.2 tell us that for a relatively large family of sequences of Markov chains (X n ) n≥1 , there is at least one nondegenerate sequence of Boolean functions which is noise stable with respect to (X n ) n≥1 , it does not tell us how to find such a sequence. Lemma 3.7 provides us with means to do so, but it contains many parameters and does not connect to other well known definitions for graphs. We will end this paper by giving a result for reversible and irreducible continuous time Markov chains whose bottleneck ratio and spectral gap is of the same order, which says that in this case, any sequence of sets whose bottleneck ratios approximate the spectral gap well, will be noise stable. When in addition to being reversible and irreducible, the Markov chain is transitive, we show that there is a nondegenerate sequence of sets which minimize the bottleneck ratio. The most interesting thing with this result is that it provides a partial explanation of how noise stable functions can arise.
Recall that for a random walk on a connected graph G with stationary distribution π, the bottleneck ratio Φ * is defined by
(23) Φ * = Φ * (G) := inf A⊂V (G) : π(A)≤1/2 Φ(A) where Φ(A) = |E(A, A c )|/vol(A)
. For a general continuous time Markov chain X with generator Q = (q ij ) and stationary distribution π, we let Φ(A) := ( i∈A, j∈A c π(i)q ij )/π(A) and define the bottleneck ratio as
(24) Φ * = Φ * (X) := inf A⊂S (n) : π(A)≤1/2 Φ(A).
Note that when the rate r = q ii of a random walk equals one,
i∈A, j∈A c π(i)q ij π(A) = i∈A, j∈A c deg(i) vol(G) · 1 deg(i) i∈A deg(i) vol(G) = i∈A, j∈A c 1 i∈A deg(i) = |E(A, A c )| vol(A)
i.e. in this case the two definitions coincide.
The main reason to expect that noise stable sequences of Boolean functions should exist when Φ
(n) * ≍ λ (n) 1
is that heuristically, for a set A ⊆ S (n) , Φ * (A) estimates how large a proportion of the total mass of A that will move to A c during one unit of time. If Φ (n) * ≍ λ (n) 1 , then only about a proportion ε of the total mass in A will move to A c at least once between time zero and time εt (n) rel . If A n minimizes the bottleneck ratio of X (n) , we thus expect that the sequence (1 w∈An ) n≥1 is noise stable, and it only remains to show that we can find such a sequence which is in addition nondegenerate, which is why we want each Markov chain in the sequence (X (n) ) n≥1 to be transitive. In Examples 4.4 and 4.5, we show that even for random walks on connected graphs, we cannot drop the transitivity assumption. However, it is not clear whether this assumption could be replaced with something weaker.
In general, it is known that for any Markov chain, the bottleneck ratio and the spectral gap satisfies Φ 2 * ≤ 2λ 1 ≤ 4Φ * , where the lower bound is attained for a unit rate random walk on Z n and the upper bound is attained for a unit rate random walk on the Hamming cube. We thus know that there are sequences of graphs with Φ (n) * ≍ λ (n) 1 , implying that the result is not void. We are now ready to state our result. Proposition 4.1 will be proven using two lemmas, which we now state and prove. In these lemmas, as well as in the rest of the notes, we will for any Boolean function f :
V (G) → {0, 1} let A f denote the set {w ∈ V (G) : f (w) = 1}.
The first of these lemmas will be proven for general continuous time Markov chains, but due to issues concerning nondegeneracy which will be discussed later, the second lemma will only be proved for transitive Markov chains.
0 ) = f n (X (n) t rel )) ≤ lim α→0 lim n→∞ α t (n) rel · π(A fn ) · Φ(A fn ).
In particular, if Φ(A fn ) ≍ λ (n) 1 , then (f n ) n≥1 is noise stable with respect to (X (n) ) n≥1 . Proof. Fix n ≥ 1 and let α > 0 be arbitrarily chosen. By the definition of the generator Q n , for any h n > 0,
(25) P (X (n) hn = j | X (n) 0 = i) = h n q (n) ij + o(h n ).
We will, to simplify notations, assume that h n is choosen such that αt (n) rel /h n is an integer.
Then
P (f n (X (n) 0 ) = 1, f n (X (n) αt (n) rel ) = 0) ≤ αt (n) rel /hn k=1 P (f n (X (n) (k−1)hn ) = 1, f n (X (n) khn ) = 0) = α t (n) rel h n · P (f n (X (n) 0 ) = 1, f n (X (n) hn ) = 0) (26)
where the last equality follows by stationarity. By definition, the right hand side of (26) equals
(27) α t (n) rel · i∈A fn , j∈A c fn π n (i) · 1 h n P (X (n) hn = j | X (n) 0 = i).
Using (25), (27) can be bounded from above by
α t (n) rel · i∈A fn , j∈A c fn π n (i) · q (n) ij + o(h n ) h n = α t (n) rel · i∈A fn , j∈A c fn π n (i) q (n) ij + α t (n) rel · π(A fn ) · o(h n ) h n = α t (n) rel · π(A fn ) · Φ(A fn ) + α t (n) rel · π(A fn ) · o(h n ) h n .
In particular, we have showed that
(28) P (f n (X (n) 0 ) = 1, f n (X (n) αt (n) rel ) = 0) ≤ α t (n) rel · π(A fn ) · Φ(A fn ) + α t (n) rel · π(A fn ) · o(h n ) h n .
As X (n) is reversible for each n,
P (f n (X (n) 0 ) = f n (X (n) αt rel )) = P (f n (X (n) 0 ) = 1, f n (X (n) αt (n) rel ) = 0) + P (f n (X (n) 0 ) = 0, f n (X (n) αt (n) rel ) = 1) = 2P (f n (X (n) 0 ) = 1, f n (X (n) αt (n) rel ) = 0).
Using the upper bound from (28), we thus obtain
P (f n (X (n) 0 ) = f n (X (n) t rel )) ≤ α t (n) rel · π(A fn ) · Φ(A fn ) + α t (n) rel · π(A fn ) · o(h n ) h n .
As the second term can be made arbitrarily small by choosing h n small, the desired conclusion follows.
Remark 4.3. This directly shows that if G n is the n-dimensional Hamming cube, the sequence (f n ) n≥1 , where f n (w) = w(1), is noise stable.
To finish the proof of Proposition 4.1, it remains to show that given that X (n) is transitive and Φ (n) * ≍ λ (n) 1 , there is at least one nondegenerate sequence (f n ) n≥1 of Boolean functions with Φ(A fn ) ≍ Φ (n) * . It is not obvious that such a sequence exists in a more general setting, and the following two examples show that even in the special case of random walks on graphs, some assumption on the graphs G n , which is stronger than G n being regular, is needed to guarantee the existence of a nondegenerate sequence of Boolean functions which minimizes the bottleneck ratio. In particular, these examples show that Proposition 4.1 is not true for general Markov chains.
Example 4.4. Let, as in Example 3.5, G n be the graph obtained by joining a complete graph on n vertices to a complete graph on n 2 vertices by adding a single edge.
K n 2 K n
We will once more consider the random walks on these graphs. Using the same notation as in Example 3.5, we have |E(K n , K n 2 )| = 1 and vol(K n ) = n(n − 1). This implies that the continuous time random walk on G n with rate 1 has bottleneck ratio
Φ * = Φ(G n ) = |E(K n , K n 2 )| vol(K n ) + 1 = 1 n(n − 1) + 1 ≍ 1 n 2 .
However, π n (K n ) = vol(K n ) + 1 vol(G n ) = n(n − 1) + 1 n(n − 1) + n 2 (n 2 − 1) + 2 ≍ 1 n 2 i.e. this example shows that there are sequences of graphs for which the bottleneck ratio is attained by a proportion of the graphs whose measure tends to zero.
Our next example shows that to assume that graphs in the sequence are regular is not enough to avoid that the bottleneck ratio is attained by an arbitrarily small part of the graph. To sum up, the examples above shows that even if we can find a sequence of sets A n satisfying Φ(A n ) ≍ λ (n) 1 , it is not obvious that we can find such a sequence which is in addition nondegenerate, even given quite strong conditions on the Markov chains, such as it being a random walk on a regular graph. For transitive Markov chains however, the following lemma guareantees the existence of such a sequence. Proof. Suppose that the lemma is false. Then for at least one set A ⊂ S,
(i) π(A) ≤ 1 4 , (ii) Φ(A) = Φ * and (iii) Φ(A ′ ) > Φ(A) for all A ′ ⊆ S with π(A) < π(A ′ ) ≤ 1
2 . Let ϕ ∈ Aut(X) be such that ϕ(A) = A. As π(A) < 1 and X (n) is transitive, at least one such function exists. As and π(A) ≤ 1 4 by (i), we must have that π(A ∪ φ(A)) ≤ 2π(A) ≤ 1 2 . By the definition of Φ,
Φ(A ∪ ϕ(A)) = i∈A∪ϕ(A) j ∈A∪ϕ(A) π(i)q ij π(A ∪ ϕ(A)) = i∈A j ∈A π(i)q ij + i∈ϕ(A) j ∈ϕ(A) π(i)q ij − i∈A j∈A c ∩ϕ(A) π(i)q ij − i∈ϕ(A) j∈A∩ϕ(A) c π(i)q ij − i∈A∩ϕ(A) j∈A c ∩ϕ(A) c π(i)q i π(A) + π(ϕ(A)) − π(A ∩ ϕ(A)) = 2 i∈A j ∈A π(i)q ij − i∈A j∈A c ∩ϕ(A) π(i)q ij − i∈ϕ(A) j∈A∩ϕ(A) c π(i)q ij − i∈A∩ϕ(A) j∈A c ∩ϕ(A) c π(i)q ij 2π(A) − π(A ∩ ϕ(A)) ≤ 2 i∈A j ∈A π(i)q ij − i∈A∩ϕ(A) j∈A c ∩ϕ(A) π(i)q ij − i∈A∩ϕ(A) j∈A∩ϕ(A) c π(i)q ij − i∈A∩ϕ(A) j∈A c ∩ϕ(A) c π(i)q ij 2π(A) − π(A ∩ ϕ(A)) = 2 i∈A j ∈A π(i)q ij − i∈A∩ϕ(A) j ∈A∩ϕ(A) π(i)q ij 2π(A) − π(A ∩ ϕ(A)) = 2π(A)Φ(A) − π(A ∩ ϕ(A))Φ(A ∩ ϕ(A)) 2π(A) − π(A ∩ ϕ(A)) .
As π(A) < π(A ∪ ϕ(A)) ≤ 1 2 , by (iii) we have that Φ(A) < Φ(A ∪ ϕ(A)).
Combining this with the previous equation we obtain
Φ(A) < 2π(A)Φ(A) − π(A ∩ ϕ(A))Φ(A ∩ ϕ(A)) 2π(A) − π(A ∩ ϕ(A)) .
Rearranging, we get Φ(A ∩ ϕ(A)) < Φ(A). This contradicts (ii), why the conclusion of the lemma follows.
Proof of Proposition 4.1. By Lemma 4.6, there exists a nondegenerate sequence of Boolean functions (f n ) n≥1 such that Φ(A fn ) ≍ λ 1 . In the following example we will give an example of a sequence of random walks on graphs which shows that in this setting, a sequence of sets (A n ) n≥1 with Φ * (A n ) ≍ Φ (n) * can be noise stable. Unfortunately, I have yet not found a Markov chain with a sequence of sets (A n ) n≥1 which satisfies that Φ * (A n ) ≍ Ψ (n) * ≍ λ (n) 1 such that (1 An ) n≥1 is nondegenerate but not noise stable, neither with nor without the additional transitivity assumption.
Example 4.7. Let G n be the graph with vertices labeled by the integers 0, 1, . . . , 2n − 1 and with an ende between two vertices if their label diiffer by 1 modulo 2n. The random walk X (n) on this graph has spectral gap λ (n) 1 = 1/n 2 and bottleneck ratio Φ (n) * = 1/n achieved by e.g. the set of vertices A n labeled by {0, 1, . . . , n − 1}. It is easy to show that the sequence (1 An ) n≥1 is noise stable with respect to (X (n) ) n≥1 , even though Φ (n) * ≍ λ (n)) 1 . Except in special cases, the definition of the bottleneck ratio does not give us much direct information about noise stability, as, assuming that A ⊆ S and writing f = 1 A Φ(A) = i∈A, j ∈A π(i)q ij π(A) = 1 2 · i,j∈S π(i)q ij · (f (i) − f (j)) 2
E[f ] = i∈S π(i)f (i) · j∈S q ij · (f (i) − f (j)) E[f 2 ] = f, −Qf f, f = i λ if (i) 2 if (i) 2 = i λ i ·f (i) 2 jf (j) 2 = E f [λ]
where
P f (λ = λ i ) =f (i) 2 jf (j) 2 .
If A (n) * minimizes the bottleneck ratio of X (n) , and we want to show that the sequence (f n ) n≥1 , where f n (w) = 1 w∈A (n) * (w), is not noise stable, we need to show that for all δ > 0 there is k = k δ such that sup n P fn (λ > kλ (n) 1 ) < δ. This is neither implied nor not implied by the fact that E fn [λ] = ω(λ 1 ). Moreover, we cannot even conclude that f is not noise sensitive, as this would require that lim n→∞ P fn (λ < kλ Then there is arbitrarily small x and n(x) ∈ N such that f n(x) (x) ≥ ε/2 for all f n(x) ∈ F n(x) .
Proof. Suppose that the conclusion of the lemma is false. Then for each x 0 > 0 and each n ≥ 1 there is at least one function f n,x0 ∈ F n such that f n,x0 (x 0 ) < ε/2 As each such function is increasing, we must in fact have that f n,x0 (x) < ε/2 for all x < x 0 . This implies that ≤ sup n f n,x0 (x 0 ) < ε/2.
As this contradicts (29), the desired conclusion follows.
Lemma A.2. Let A ⊆ S be a set with 0 < P (A) < 1. Then for any function ψ : S → R with E[ψ 2 (w)] = 1 and E[ψ(w)] = 0,
E[ψ (n) (w) | w ∈ A] 2 ≤ P (w ∈ A) P (w ∈ A) 2 .
Proof. Note first that 0 = E[ψ(w)] = E[ψ(w) | w ∈ A] · P (w ∈ A) + E[ψ(w) | w ∈ A] · P (w ∈ A).
Rewriting this, we obtain
E[ψ (n) (w) | w ∈ A] = − P (w ∈ A) P (w ∈ A) · E[ψ(w) | w ∈ A].
This in turn implies that
1 = E[ψ(w) 2 ] ≥ P (w ∈ A) · E[ψ(w) 2 | w ∈ A] ≥ P (w ∈ A) · E[ψ(w) | w ∈ A] 2 = P (w ∈ A) · P (w ∈ A) P (w ∈ A) · E[ψ(w) | w ∈ A] 2 = P (w ∈ A) 2 P (w ∈ A) · E[ψ (n) (w) | w ∈ A] 2 .
Rearranging, we obtain the desired equation.
Lemma A.3. For any positive decreasing function f : R + → R + and any constants A ≥ a > 0,
a/2 0 f (θ)dθ A 0 f (θ)dθ ≤ f (0) f (0) + f (a) .
i.e. ψ · ϕΨ . From this it follows that the maximum M must be the same for each w ∈ S (n) . This implies that
|S| · M = w∈S χ 1 (w) 2 + . . . + χ dim Ψ (w) 2 = w∈S dim Ψ i=1 χ i (w) 2 = dim Ψ i=1 w∈S χ i (w) 2 = |S| dim Ψ i=1 w∈S π(w)χ i (w) 2 = |S| dim Ψ i=1 χ i , χ i = |S| dim Ψ i=1 1 = |S| · dim Ψ.
Lemma A.5. For any fixed u, the functions {ψ i,u } i : ψi∈Ψ , where ψ i,u (w) = ψ i (w u ), are an orthonormal basis for the eigenspace spanned by {ψ i } i : ψi∈Ψ
Proof. As ψ i,u , ψ j,u = ψ i , ψ j it is immediately clear that {ψ i,u } i : ψi∈Ψ is an orthonormal set, so it remains to show that ψ j,u ∈ Span{ψ i } i : ψi∈Ψ for any j with ψ j ∈ Ψ . To obtain this result, it is enough to show that ψ j,u is an eigenvector of −Q with eigenvalue λ j . This follows as λ j ψ j , ψ k ψ k,u = λ j ψ j,u .
{e −λ (n) j t } j . Since the set {ψ (n)
be the relaxation time of the nth Markov chain. A sequence (f n ) n≥1 of Boolean functions, f n : S (n) → {0, 1}, is said to be noise stable with respect to (X (n) )
= α is (1 − e −α )/2. This implies that the law of X
in fact equals n.
Figure 1 .
1The figure above shows the graph G n , which is obtained by joining n stars with n leaves by adding all possible edges between their centers.
Remark 3. 4 .≡
4If for a sequence (X (n) ) n≥1 of Markov chains, λ R |S (n) | , which implies that (B) holds in this case. In particular, Proposition 3.1 is consistent with Remark 1.7, stating that all sequences of Boolean functions on expander families are noise stable.
Lemma 3 . 9 .
39Let (X (n) ) n≥1 be a sequence of reversible and irreducible continuous time Markov chains. Then if (f n ) n≥1 is a sequence of nondegenerate Boolean functions, both (A) and (¬B) cannot hold. In particular (A) ⇒ (B).
Proof of Proposition 3 . 1 .
31Note first that it is trivial that (C) ⇒ (A). By Lemma 3.9, (A) implies (B). By Lemma 3.8, ¬(C) ⇒ ¬(B), in turn implying that (B) ⇒ (C). Combining these results, we obtain (A) ⇒ (B) ⇒ C ⇒ A, implying that (A) ⇔ (B) which is the desired conclusion.
dependency on ε is clear. Set m (n k ) := dim Ψ
Proposition 4 . 1 .
41Let (X (n) ) n≥1 be a sequence of transitive, reversible and irreducible continuous time Markov chains, with spectral gaps (λ(n) 1 ) n≥1 and bottleneck ratios (Φ (n) * ) n≥1 . If λ ) * , there exist at least one nondegenerate noise stable sequence of Boolean functions on S (n) .
Lemma 4 . 2 .
42Let (X (n) ) n≥1 be a sequence of reversible and irreducible continuous time Markov chains with spectral gaps (λ
Example 4 . 5 .n
45For each n ≥ 1, let G(1) n be the complete graph on n + 1 vertices and G ) ≍ 1/n. Now let G n be the graph obtained by choosing one edge in G(1) n and one edge in G(2) n , removing them both and adding new edges between the ends of the chosen edges in the respective gaphs such that the degree of each vertex is preserved and G n is connected. Then Φ (n) * = 2/(vol(G (1) n ) + 1). Due to the order of Φ * (G / vol(G n ) → 0, this sequence of graphs provides examples of regular graphs where the bottleneck ratio is attained by an arbitrarily small proportion of the graph.
Lemma 4 . 6 .
46Let X be a transitive, reversible and irreducible continuous time Markov chain with finite state space S and stationary distribution π. Then there is a set A ⊂ S with π(A) ∈ 1 4 , 1 2 such that Φ * = Φ(A).
sequence is noise stable. After now having finished the proof of the main result of this section; Propostion 4.1, it is natural to ask what will happen if the assumptions of this does not hold. We have already showed what will happen if we lose the transitivity assumption, so what remains to do is to consider what happens if we drop the assumption that Φ (n) * ≍ λ (n)
there can be no sequence (f n ) n≥1 of Boolean functions in (Span i : λ For each n ∈ N, let F n be a collection of increasing functions and suppose that
) n≥1 . Further, let (f n ) n≥1 , f n : S (n) → {0, 1}, be a sequence of
Proof.a/2 0.Lemma A.4. Let Ψ be an eigenspace of a transitive, reversible and irreducible continuous time Markov chain X. Then for any w ∈ S,Proof. Let χ 1 , . . . , χ dim Ψ be an orthonormal basis for Ψ . Then any ψ ∈ Ψ of length one can be written aswhere (θ 1 , . . . , θ dim Ψ ) are the polar coordinates of ψ.We now claim that, from this representation of ψ ∈ Ψ , it follows that for any w ∈ S, ψ(w) 2 is maximized byTo see this, note first that for dim Ψ = 2, we have thatThis is clearly maximized by χ 2 1 + χ 2 2 when θ 1 = arctan χ 2 /χ 1 . This provides the first step for an argument by induction. To see that this holds in general, define recursivelyfor any choice of θ 1 , . . . , θ m ′ −1 . This finishes the proof of the claim. Now note that for any ψ ∈ Ψ with corresponding eigenvalue λ, and any ϕ ∈ Aut(X), Q • ψ(ϕ(i)) = j∈S q ij ψ(ϕ(j)) = j∈S q ϕ(i)ϕ(j) ψ(ϕ(j)) = j∈S q ϕ(i)j ψ(j) = λψ(ϕ(i))
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[
"Optical Kagomé lattice for ultra-cold atoms with nearest neighbor interactions",
"Optical Kagomé lattice for ultra-cold atoms with nearest neighbor interactions"
]
| [
"J Ruostekoski \nSchool of Mathematics\nUniversity of Southampton\nSO17 1BJSouthamptonUK\n"
]
| [
"School of Mathematics\nUniversity of Southampton\nSO17 1BJSouthamptonUK"
]
| []
| We propose a scheme to implement an optical Kagomé lattice for ultra-cold atoms with controllable s-wave interactions between nearest neighbor sites and a gauge potential. The atoms occupy three different internal atomic levels with electromagnetically-induced coupling between the levels. We show that by appropriately shifting the triangular lattice potentials, experienced by atoms in different levels, the Kagomé lattice can be realized using only two standing waves, generating a highly frustrated quantum system for the atoms. | 10.1103/physrevlett.103.080406 | [
"https://arxiv.org/pdf/0906.3042v1.pdf"
]
| 38,863,927 | 0906.3042 | 5b34f4db7e6b9eabdd53bc55735ad23c52f3c019 |
Optical Kagomé lattice for ultra-cold atoms with nearest neighbor interactions
16 Jun 2009
J Ruostekoski
School of Mathematics
University of Southampton
SO17 1BJSouthamptonUK
Optical Kagomé lattice for ultra-cold atoms with nearest neighbor interactions
16 Jun 2009(Dated: June 16, 2009)arXiv:0906.3042v1 [cond-mat.quant-gas]numbers: 0375Lm0375Mn0375Ss0375Hh
We propose a scheme to implement an optical Kagomé lattice for ultra-cold atoms with controllable s-wave interactions between nearest neighbor sites and a gauge potential. The atoms occupy three different internal atomic levels with electromagnetically-induced coupling between the levels. We show that by appropriately shifting the triangular lattice potentials, experienced by atoms in different levels, the Kagomé lattice can be realized using only two standing waves, generating a highly frustrated quantum system for the atoms.
In the Kagomé lattice the atoms occupy three electronic ground states with a common excited state. The frequencies of the two lattice lasers are tuned between the atomic resonance frequencies in such a way that ω3 > ωα > ω2 > ω β > ω1. The polarization components of the lattice lasers that couple to each individual transition are indicated. Right: Atoms in each of the three internal levels |j (j = 1, 2, 3) experience an equilateral triangular lattice potential. These lattices are shifted with respect to each other, so that the lattice site occupied by the atoms in |2 (|3 ) is at the midpoint of the triangle side of |1 defined by the unit vector k β /k = (êy − √ 3êz)/2 (êz). Each site has four nn sites, so that, e.g., the site of |1 has two nn sites of |2 and two of |3 . dependent lattice system, the Kagomé lattice can be prepared with only two optical standing waves (SWs), as compared to six SWs in previous proposals [14], with the additional advantage of controllable non-local twobody interactions between different sites and the possibility for the creation of Abelian and non-Abelian guage potentials. The strength of the em-driven hopping can be tuned over a wide range of values with respect to tunneling between more distant sites and the nn collisions. Spin-dependent lattices where the different atomic spin components were moved around independently was experimentally created using 87 Rb atoms [15]. Alkalineearth-metal atoms and rare-earth metals with narrow optical resonances (e.g. Sr, Yb) are particularly suitable for realizing spin-dependent optical lattices, because of slow loss rates due to spontaneous emission [16].
We consider ultra-cold (bosonic or fermionic) atoms occupying three internal sublevels |j (j = 1, 2, 3) of the same atom that are coupled by em transitions. The atom dynamics is assumed to be restricted to a 2D layer in the yz plane due to a tightly confined 2D pancake-shaped trap. Such a confinement may be experimentally created, e.g., using magnetic or optical fields [17,18]. Along the 2D trap each species experiences a triangular optical lattice potential, generated by two SWs with wavevectors k α,β = k(ê y ± √ 3ê z )/2. The atom-lattice model is based on an atomic (single-band) Hubbard Hamiltonian [19] where the atoms occupy the lowest mode of each lattice site. We show that by appropriately tuning the frequencies of the lattice lasers, the lattice potentials of the three species can be shifted in a triangular shape to form a Kagomé lattice pattern. Then the four nn sites are occupied by the atoms in the other two sublevels and the hopping of the atoms between adjacent sites, with amplitude κ jk , only occurs as a result of driving by coherent em fields that change the internal atomic level. The hopping to the nearest sites occupied by the atoms in the same sublevel results from tunneling between the sites and, in sufficiently deep lattices, it may be suppressed. Moreover, the specialty of the proposed scheme is that by adjusting the overlap of the nn lattice site wave functions (Wannier functions) we can prepare a lattice system with a non-negligible, controllable s-wave interaction between adjacent sites U nn jk (j = k), with the both limits U nn jk ≫ κ jk and U nn jk ≪ κ jk achievable, providing a frustrated quantum system with nn interactions.
As a realization of a frustrated Kagomé lattice with desired interactions, we consider a tripod four-level scheme; Fig. 1. The atoms in the three electronic ground states are coupled to a common electronically excited level |e by different resonant transition frequencies ω j (ω 3 > ω 2 > ω 1 ). The lattice laser α, with the frequency ω α , is blue-detuned with respect to two of the transitions δ α,j = ω j − ω α < 0, for j = 1, 2, and red-detuned with respect to one δ α,3 > 0. Then the atoms in the states |1 , |2 (|3 ) are attracted towards low-intensity (highintensity) regions of α. The lattice laser β, with the frequency ω β , is blue-detuned from one of the transitions δ β,1 < 0 and red-detuned from the others. For simplicity, we assume |e = |e, m , |1 = |1, m − 1 , |2 = |2, m , and |3 = |3, m + 1 , so that the transitions |1 → |e , |3 → |e , and |2 → |e have dipole matrix elements d e1 = e|d|1 = D e|1, 1 ê * +1 , d e3 = D e| − 1, 3 ê * −1 , and d e2 = D e|0, 2 ê * 0 , coupling to the light with polarizations σ + , σ − , andê 0 , respectively. Here D is the reduced dipole matrix element, e|σg are the Clebsch-Gordan coefficients, and we useê +1 = −(ê x + iê y )/ √ 2, e −1 = (ê x − iê y )/ √ 2, andê 0 =ê z . We assume the polarizations of the two SWs to be orthogonalê α ·ê * β = 0, so that the lattice potentials read
V j = V α j + V β j with V α j =s jα E r sin 2 [k(y + √ 3z)/2 + ϕ j ], V β j =s jβ E r sin 2 [k(y − √ 3z)/2 + η j ] ,(1)ϕ 1 = η 1 = ϕ 3 = 0, ϕ 2 = η 2 = η 3 = π/2. Here s jη ∝ |(d ej ·ê η ) 2 /δ η,j | (η = α, β)
denotes the lattice strength in the lattice photon recoil energy units E r = 2 k 2 /2m, depending on the light polarization, atomic sublevel, and the detuning. To produce three triangular lattices the lasers need to couple simultaneously to all the three transitions, so that s jα and s jβ are non-vanishing for all j. This can be obtained by choosing the polarization vectors for the SWs so thatê η ·ê * ±1,0 = 0 together witĥ e η · k η = 0 (η = α, β) andê α ·ê * β = 0. We will next demonstrate an example to show that such a solution can be found for the polarization vectors and how selecting the lattice light polarization can also be used to control the relative strengths of the three triangular lattices. This can be especially useful if the absolute values of the detunings are very different. We consider the level scheme shown in Fig. 1 and, for simplicity, assume that |δ α, 3
| = |δ α,2 | = 3|δ α,1 | = 3|δ β,3 | = |δ β,2 | = |δ β,1 | (indicating that ω 1 − ω 2 = ω 2 − ω 3
and that the lasers are detuned exactly at the midpoint between the nearest transitions) and that the Clebsch-Gordan coefficients are equal. The polarization vectors (orthogonal to the corresponding wavevectors k j ) for the
SWs areê α = a α (− √ 3ê y +ê z )/2 + b αêx andê β = a β ( √ 3ê y +ê z )/2 + b βêx .
Here we choose the complex coefficients a α,β and b α,β in such a way thatê α ·ê *
β = 0 and |ê * −1 ·ê α | 2 = 3|ê * +1 ·ê α | 2 and |ê * +1 ·ê β | 2 = 3|ê * −1 ·ê β | 2 .
The last two conditions ensure that we have s 1η = s 3η , compensating for the different values of the detunings. A straightforward algebra yields solutions for a α,β and b α,β with s 2η = 4s 3η /5 (η = α, β), resulting in only very small differences between the lattice strengths. We also find that the two SWs in Eq. (1) have equal amplitudes for each level, i.e., s jα = s jβ for all j.
For each species the lattice potential V j , in Eq. (1) is an equilateral triangle with the side of length 2d, where d = π/ √ 3k denotes the nn separation; Fig. 1. The minima of the potential are at the corners of the triangles which for |1 are at (y, z) = [(n + m)π/k, (n − m)d], for |2 at [(n + m + 1/2)π/k, (n − m − 1/2)d], and for |3 at [(n + m + 1)π/k, (n − m)d], with n, m integers. The triangular lattices of |2 and |3 are shifted to coincide with the side midpoints of the triangular lattice of |1 . The combined system of interlaced triangular lattices forms a Kagomé lattice, so that each site has four nn sites which are occupied by the two other species (two of each).
In order to estimate the relative strengths of the different terms in the lattice Hamiltonian we evaluate the corresponding integral representations. The direct tunneling amplitude, where atoms remain in the same hyperfine level during the hopping process, reads
J p b ≃ − dy dz (− 2 2m φ * bn ∇ 2 φ bn ′ + φ * bn V b φ bn ′ ) > 0, where ∇ 2 = ∂ 2
y + ∂ 2 z and V b is given by Eq. (1). The Wannier functions for atoms in |b at site n are φ bn (y, z) and may be approximated by the ground state harmonic oscillator wave function with the trap frequencies ω y ≃ √ 2sE r / and ω z ≃ √ 6sE r / which are obtained by expanding the optical potential at the lattice site minimum [19].
The site n ′ here refers to the nearest site to n occupied by the same atomic hyperfine level along the direction of p, where p takes the values of the three triangle sides: k α,β ,ẑ.
Here J kα b = J k β b by symmetry. For J kα b
and Jẑ b we have φ b,n ′ (y, z) = φ b,n (y + √ 3d, z + d) and φ b,n ′ (y, z) = φ b,n (y, z + 2d), respectively. Due to the anisotropy of the individual lattice site wavefunctions, the hopping amplitudes along the z direction differ slightly from those along the direction of the two SWs. Although a more rigorous calculation of the hopping amplitudes would involve a full band-structure calculation, here it is sufficient to provide order-of-magnitude analytic estimates using the Gaussian approximations to φ bn .
The em field changes the internal level of the atom. Because the lattice site minima of the different sublevels are shifted with respect to each other, the atoms simultaneously also undergo spatial hopping along the lattice. The hopping amplitude reads
κ bc = d 3 r φ * bn R bc φ cn ′ ,(2)
where R bc denotes the effective Rabi frequency for the transition between the levels |b and |c and n ′ refers to the nn site of species |c to the site n occupied by the species |b , with κ bc proportional to the spatial overlap between the atoms in the nn sites. As shown in Fig. 1, κ 12 , κ 23 , and κ 13 represent hopping along the directions of k β , k α , andẑ, respectively. As for the direct tunneling we then have κ 12 = κ 23 , but κ 13 is not exactly equal. Note that κ bc in Eq. (2), unlike the direct tunneling, can take positive, negative, or even complex values. The explicit expression for the Rabi frequency R bc depends on the particular form of the em coupling between the internal levels which can be one or multi-photon transition. For a two-photon transition via an off-resonant intermediate level |e ′ (which for a laser can be electronically excited and for a rf/microwave an electronic ground state) we may adiabatically eliminate |e ′ [20]. We then obtain R bc = E b E * c d e ′ b d * e ′ c /(2 2 ∆) and a contribution to the em-induced level shifts −|E j | 2 |d e ′ j | 2 /(2 2 ∆) for |j . Here d e ′ j ≡ d e ′ j ·ê j and we have assumed that |b is coupled to |e ′ by the em field with the positive fre-
quency component E + b = 1 2ê b E b e ik b
·r e iΩ b t and detuning ∆. Using the Gaussian approximation to φ bn , we obtain κ jk ≃ R jk ǫ jk exp [−(3 + √ 3)π 2 √s jk /48 √ 2], for (j, k) = (1, 2), (2, 3) and κ 13 ≃ R 13 ǫ 13 exp (−π 2 √s 13 /4 √ 6), wheres jk ≡ 4s j s k /( √ s j + √ s k ) 2 and ǫ jk ≡ [s 2 jk /(s j s k )] 1/4 . Here we have assumed that the lattice potentials by the lasers α and β in Eq. (1) have the same amplitude in each level |j , i.e. s jα = s jβ , and we have suppressed in the notation the index referring to the particular laser.
The onsite interaction term may also be obtained from the Wannier functions U jj ≃ g (jj) 2D dy dz |φ jn | 4 . The 2D nonlinearity g (jk)
2D ≃ 2π 2 a jk /ml x √ 2π is given in terms of the scattering length a jk and the 2D trap confinement l x = /mω x where the oscillator frequency per-pendicular to the lattice is ω x . The additional densitydependent contribution to the scattering length in 2D is negligible when √ 2πl x /a jj ≫ ln (8π 3/2 l x n 2D a jj ) [21], where n 2D denotes the 2D atom density. For fermionic atoms a jj = 0 and the onsite interaction term vanishes for a single-species gas if the atoms only occupy the lowest mode of each lattice site, but is non-vanishing for a two-species gas trapped in each lattice site. The nn interaction due to the s-wave scattering between the atoms in different sublevels is always non-vanishing for both bosonic and fermionic atoms. It is generated in the spatial overlapping area of the adjacent sites and may be estimated by U nn jk ≃ 2g
(jk) 2D dy dz |φ jn | 2 |φ kn ′ | 2 . We obtain U jj ≃ a jj 3 1/4 E r √ s j /l x √ π, U nn 13 ≃ 2a 13 3 1/4 E r √s 13 e −π 2 √s 13 /2 √ 6 /l x √ π,
and for other states
U nn jk ≃ 2a jk 3 1/4 E r √s jk e −(3+ √ 3)π 2 √s jk /24 √ 2 /l x √ π.
The Hamiltonian for the atomic system then reads
H = k ǫ k c † k c k + U kk c † k c † k c k c k − jk ′ (J p b c † j c k + H.c.) − jk (κ bc c † j c k + H.c.) + U nn bc c † j c j c † k c k ,(3)
where jk denotes the summation over adjacent lattice sites and jk ′ over the nearest sites that are occupied by the same atomic species. The level shifts and the detunings due to em-induced hoppings are included in ǫ k . Here J p b > 0 but all other coefficients can take either positive or negative values, and κ bc can be complex.
We may compare the different terms in the Hamiltonian. For typical experimental parameters for bosonic atoms we have U jj ≫ U nn jk (j = k). For simplicity, assuming s 1 = s 2 , we obtain for lattice heights s = 25 and 40, U nn 12 /U jj ≃ 10 −3 a 12 /a jj and 10 −4 a 12 /a jj . At the same lattice heights the em-driven hopping amplitudes in terms of the direct tunneling and the nn interactions are given by κ 12 /J kα 1 ≃ 540( R 12 /E r ) and 5100( R 12 /E r ), and κ 12 /U nn 12 ≃ 4.2(l x /a 12 )( R 12 /E r ) and 8.3(l x /a 12 )( R 12 /E r ). In shallow lattices, with weak em-coupling between the sublevels R 12 /E r ≪ 1, the nn hopping κ bc and the direct tunneling J p b between more distant sites may be comparable. In deeper lattices and for stronger κ bc the direct tunneling terms may be ignored. If the transverse confinement of the 2D lattice is weak (l x large) and the Rabi field sufficiently strong R 12 l x /(a 12 E r ) ≫ 1, we may also neglect U nn bc with κ bc ≫ U nn bc and in Eq. (3) we only keep terms proportional to ǫ k , U kk , and κ bc . An especially interesting property of the proposed scheme is that we may also find a wide range of parameter values for which U nn bc κ bc . This limit can always be achieved with sufficiently weak em coupling. If we also simultaneously require that κ bc ≫ J p b , we may, e.g., at s = 50 select R 12 /E r ≃ 5 × 10 −4 and l x ≃ 17a 12 , resulting in U nn bc ≃ 10κ bc . For instance, for 87 Rb the s-wave scattering length between |F = 1, M F = −1 and |2, +1 hyperfine states is about 5.191nm [22], corresponding to the transverse trap frequency of ω x ≃ 2π × 15 kHz, achievable, e.g., by an optical lattice. The interspecies scattering length could be increased by Feshbach resonances, further enhancing the effect of the nn interactions U nn bc , so that one could reach U jj ≫ U nn bc ≫ κ bc ≫ J p b also in shallow lattices. Interspecies Feshbach resonances were observed also in 87 Rb between different hyperfine states [23] and in optical lattices at low occupation numbers the harmful three-body losses are suppressed. Moreover, alkaline-earth-metal and rare-earth-metal atoms with very narrow optical resonances [16] may allow the lattice lasers to be tuned close to the atomic resonance making it easier to produce deep lattices.
The simplest situation is to select the phases of the em fields in Eq. (2) so that all the hopping amplitudes κ bc are real and positive. It is, however, also possible to engineer a non-uniform phase profile for the hopping amplitudes which was in Ref. [24] proposed as a mechanism to construct topologically non-trivial ground states with fractional fermion numbers in 1D. In a 2D lattice the technique can be used to create an effective magnetic field for neutral atoms [25]. Here we may similarly induce a phase for atoms hopping around a closed path in the lattice, mimicing a magnetic flux experienced by charged particles. We write κ bc = |κ bc |e iν bc where the phases ν bc (r) may be constant or spatially varying [24,25]. The hopping around one unit triangle then generates the phase ∆ν = ν 12 + ν 23 − ν 13 for the atoms, corresponding to a magnetic flux Φ ∝ ∆ν through the area enclosed by the triangle. For instance, for spatially constant ν bc with ∆ν = 0, the entire lattice area may be divided into side-sharing triangles where each adjacent triangle experiences the flux with the opposite sign. In the case of atoms occupying more than one sublevel in each lattice site, we may also generate non-Abelian vector potentials similarly to Ref. [26]; see also [27,28].
For strong nn interactions with 1/3 filling, even without a vector potential, our model produces a frustrated ground state where one atom in each triangle of sites is strongly influenced by the atoms in other cornersharing triangles. It is helpful to consider a honeycomb lattice, formed by connecting the centers of triangles of the Kagomé lattice, where the sites are coupled by ring-hopping processes [29]. A lattice system described by an analogous quantum dimer model on a pyrochlore/checkerboard lattice and on a 3D diamond lattice was recently shown to support fractional charges [10]. Similar fractional excitations are expected to exist in the Kagomé lattice system, where they can act as independent, deconfined particles over finite distances at temperatures above the ordering transition driven by quantum fluctuations -in this case by the ring-exchange processes [30]. Atomic states in the prepared lattice system could potentially be detected optically [31].
Our formalism considers one atomic species per lattice site. It is straightforward to generalize it to the situation where a two-species gas is trapped in each site with em field inducing hopping for both species. Then the onsite interaction and the hopping terms can be expressed as an effective Heisenberg spin-1/2 Hamiltonian
H eff ≃ i,j [t z S z i S z j ± t ⊥ (S x i S x j + S y i S y j )]
where the + (−) sign refers to fermionic (bosonic) atoms and S k i denote the spin matrices [32]. The fermionic version has been extensively studied in 2D Kagomé lattices where the ground state of the SU(2) symmetric case (t z = t ⊥ ) still has unsettled questions, e.g., in the existence of spontaneously broken symmetries and finite energy gaps [12].
We are grateful to N. Shannon for explanations of the importance of Kagomé lattices and EPSRC for funding.
FIG. 1: Left: In the Kagomé lattice the atoms occupy three electronic ground states with a common excited state. The frequencies of the two lattice lasers are tuned between the atomic resonance frequencies in such a way that ω3 > ωα > ω2 > ω β > ω1. The polarization components of the lattice lasers that couple to each individual transition are indicated. Right: Atoms in each of the three internal levels |j (j = 1, 2, 3) experience an equilateral triangular lattice potential. These lattices are shifted with respect to each other, so that the lattice site occupied by the atoms in |2 (|3 ) is at the midpoint of the triangle side of |1 defined by the unit vector k β /k = (êy − √ 3êz)/2 (êz). Each site has four nn sites, so that, e.g., the site of |1 has two nn sites of |2 and two of |3 .
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| []
|
[
"Ab initio Investigation on Hybrid Graphite-like Structure Made up of Silicene and Boron Nitride",
"Ab initio Investigation on Hybrid Graphite-like Structure Made up of Silicene and Boron Nitride"
]
| [
"C Kamal \nIndus Synchrotrons Utilization Division\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia\n",
"Aparna Chakrabarti \nIndus Synchrotrons Utilization Division\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia\n\nHomi Bhabha National Institute\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia\n",
"Arup Banerjee \nHomi Bhabha National Institute\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia\n\nBARC Training School\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia\n"
]
| [
"Indus Synchrotrons Utilization Division\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia",
"Indus Synchrotrons Utilization Division\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia",
"Homi Bhabha National Institute\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia",
"Homi Bhabha National Institute\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia",
"BARC Training School\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia"
]
| []
| In this work, we report our results on the geometric and electronic properties of hybrid graphitelike structure made up of silicene and boron nitride (BN) layers. We predict from our calculations that this hybrid bulk system, with alternate layers of honeycomb silicene and BN, possesses physical properties similar to those of bulk graphite. We observe that there exists a weak van der Waals interaction between the layers of this hybrid system in contrast to the strong inter-layer covalent bonds present in multi-layers of silicene. Furthermore, our results for the electronic band structure and the density of states show that it is a semi-metal and the dispersion around the Fermi level (EF ) is parabolic in nature and thus the charge carriers in this system behave as Nearly-Free Particle-Like.These results indicate that the electronic properties of the hybrid bulk system resemble closely those of bulk graphite. Around EF the electronic band structures have contributions only from silicene layers and the BN layer act only as a buffer layer in this hybrid system since it does not contribute to the electronic properties near EF . In case of bi-layers of silicene with a single BN layer kept in between, we observe a linear dispersion around EF similar to that of graphene. However, the characteristic linear dispersion become parabola-like when the system is subjected to a compression along the transverse direction. Our present calculations show that the hybrid system based on silicon and BN can be a possible candidate for two dimensional layered system akin to graphite and multi-layers of graphene. | 10.1016/j.physleta.2014.02.011 | [
"https://arxiv.org/pdf/1312.2772v1.pdf"
]
| 119,276,606 | 1312.2772 | 838336a7627cbce4dcb01590fa2eb8373c514d71 |
Ab initio Investigation on Hybrid Graphite-like Structure Made up of Silicene and Boron Nitride
10 Dec 2013
C Kamal
Indus Synchrotrons Utilization Division
Raja Ramanna Centre for Advanced Technology
452013IndoreIndia
Aparna Chakrabarti
Indus Synchrotrons Utilization Division
Raja Ramanna Centre for Advanced Technology
452013IndoreIndia
Homi Bhabha National Institute
Raja Ramanna Centre for Advanced Technology
452013IndoreIndia
Arup Banerjee
Homi Bhabha National Institute
Raja Ramanna Centre for Advanced Technology
452013IndoreIndia
BARC Training School
Raja Ramanna Centre for Advanced Technology
452013IndoreIndia
Ab initio Investigation on Hybrid Graphite-like Structure Made up of Silicene and Boron Nitride
10 Dec 2013
In this work, we report our results on the geometric and electronic properties of hybrid graphitelike structure made up of silicene and boron nitride (BN) layers. We predict from our calculations that this hybrid bulk system, with alternate layers of honeycomb silicene and BN, possesses physical properties similar to those of bulk graphite. We observe that there exists a weak van der Waals interaction between the layers of this hybrid system in contrast to the strong inter-layer covalent bonds present in multi-layers of silicene. Furthermore, our results for the electronic band structure and the density of states show that it is a semi-metal and the dispersion around the Fermi level (EF ) is parabolic in nature and thus the charge carriers in this system behave as Nearly-Free Particle-Like.These results indicate that the electronic properties of the hybrid bulk system resemble closely those of bulk graphite. Around EF the electronic band structures have contributions only from silicene layers and the BN layer act only as a buffer layer in this hybrid system since it does not contribute to the electronic properties near EF . In case of bi-layers of silicene with a single BN layer kept in between, we observe a linear dispersion around EF similar to that of graphene. However, the characteristic linear dispersion become parabola-like when the system is subjected to a compression along the transverse direction. Our present calculations show that the hybrid system based on silicon and BN can be a possible candidate for two dimensional layered system akin to graphite and multi-layers of graphene.
I. INTRODUCTION
There has been a lot of interest in layered systems, such as two-dimensional (2D) graphene-like honeycomb structure made up of materials other than carbon, because of the novel properties associated with these 2D systems and also due to their potential applications in nanotechnology. The important property associated with the layered systems is that they have a strong in-plane bonding and a weak van der Waals (vdW) bonding in a direction perpendicular to the plane which leads to an anisotropic bonding arrangement in the systems 1 . Among these graphene-like systems, silicene, the graphene analog of silicon, have been extensively studied by both theoreticians and experimentalists in recent past 2-29 . Silicene possesses many physical properties which are similar to those of graphene. For example, silicene is a semi-metal and the charge carriers in this 2D material behave like massless Dirac-Fermions due to the presence of linear dispersion around Fermi level (E F ) at a symmetry point K in the reciprocal lattice 2,3 . Presence of linear dispersion in silicene has been recently confirmed by ARPES measurement 20 . Moreover, it has been observed that a band gap can be opened up and tuned in a monolayer of silicene by applying an external transverse electric field [6][7][8][9] which is, however, not possible in monolayer of graphene.
Though the properties of monolayer of silicene resemble those of monolayer of graphene, the properties of multi-layers of silicene are drastically different from those of multi-layers of graphene. It has been observed from our recent studies 4,5 that the multi-layers of silicene possess strong inter-layer covalent bonds in contrast to the weak van der Waals interaction between the layers of graphene multi-layers and bulk graphite. Presence of inter-layer strong covalent bond influences many properties of multi-layers of silicene. Due to this strong interlayer covalent bonding, the multi-layers of silicene can no longer behave like a layered system. The reason for differences in properties between carbon and silicon based systems is due to the different energetically favourable hybridizations present in these two systems in spite of the fact that these two atoms contain same number of electrons in the valence shell. Most favourable hybridization in silicon system is sp 3 whereas carbon system can exist in sp, sp 2 , as well as sp 3 hybridizations. It is important to note that the difference in the values of energy levels between the two valence sub-shells namely 3s and 3p in Si atom is smaller than the corresponding value between 2s and 2p sub-shells in C atom. This leads to the preference of sp 3 hybridization in Si because 3s can easily mix with all the sub-shells 3p x , 3p y and 3p z .
Due to the above mentioned reason, it is not possible to have bi-and multi-layers of silicene analogous to the biand multi-layers of graphene. Furthermore, the graphitelike layered structure of silicon cannot be constructed by directly stacking one silicene layer over another since they readily form inter-layer covalent bonds 4,5 . However, it is desirable to obtain graphene-like silicon based layered systems possessing similar exciting and novel properties of multi-layer of graphene since the former has an important advantage over carbon based systems because of their compatibility with the existing semiconductor industry. Keeping these points in mind, in the present work, we propose that there is a possibility of creating a graphite-like layered structure of silicon by inserting a buffer layer in between the multi-layers of silicene. The buffer layer prevents the strong inter-layer covalent bonds between the layers of silicene. Moreover, the buffer layer is expected not to alter the electronic properties of the multi-layers of silicene around E F . For this purpose, we consider a hybrid graphite-like layered system made up of alternate layers of honeycomb silicene and honeycomb boron nitride. It has been shown in existing theoretical studies that interaction between monolayer of silicene and boron nitrate substrate is due to weak van der Waals [30][31][32][33] . We study the geometric and electronic structures of the hybrid graphite-like layered system by employing ab initio density functional theory (DFT) 34 based calculations. In this case, BN layer acts as a buffer layer. We want to probe whether it would be possible to have an energetically stable hybrid system with physical properties similar to those of bi-layers of graphene as well as bulk graphite.
The present paper has been arranged in the following manner. We give the details of the computational methods employed in our calculations in the next section. The results of the properties of hybrid graphite-like structure have been discussed in section III, and then followed by the conclusion in section IV. Effect of vdW interaction on the geometric and electronic properties of the hybrid system is studied in the appendix.
II. COMPUTATIONAL DETAILS
Density functional theory (DFT) 34 based calculations have been performed using Vienna ab-initio simulation package (VASP) 35 within the framework of the projector augmented wave (PAW) method. We employ generalized gradient approximation (GGA) given by Perdew-Burke-Ernzerhof (PBE) 36 for exchange-correlation functional. The plane waves are expanded with energy cut of 400 eV. We use Monkhorst-Pack scheme for k-point sampling of Brillouin zone integrations with 10×10×5 and 11×11×1 for bulk and bi-layer systems respectively. The convergence criteria for energy in SCF cycles is chosen to be 10 −6 eV. The geometric structures are optimized by minimizing the forces on individual atoms with the criterion that the total force on each atom is below 10 −2 eV/Å. For bi-layer, we use a super cell geometry with a vacuum of about 15Å in the z-direction (direction perpendicular to the plane of silicene/BN) so that the interaction between two adjacent unit cells in the periodic arrangement is negligible. The geometric structures and charge density distributions are plotted using XCrySDen software 37 . In this section, we start our discussion on the geometric properties of the hybrid graphite-like bulk system (Si 16 B 18 N 18 ) with alternate layers of honeycomb silicene and BN. The space group of the hybrid bulk system is P3m1. The unit cell contains two silicene and two BN (buffer) layers. The super cell of (2×2) of honeycomb silicene is lattice matched with 3×3 of honeycomb BN with the deviation of only about 1.7 %. Each layer of silicene (BN) contains 8 silicon (9 boron and 9 nitrogen) atoms. The calculated optimized geometric structures of the hybrid graphite-like structure are shown in Fig. 1 (a) and (c). Our results on geometric structure obtained by DFT based calculation with PBE XC functional show that the hybrid system has a value of lattice constant about 7.554 A along a axis (which is denoted by A). The calculated values of Si-Si and B-N bond lengths in the basal plane are 2.246 and 1.454Å respectively. We observe that the amount of buckling present in silicene layer in the hybrid system is slightly increased to 0.543Å from its free standing value 4 of 0.457Å. The reason for increase in the buckling length is due to the interaction of silicene layer with the other layers present in the bulk system. Moreover, the increase in the value of buckling leads to a higher contribution of sp 3 -like hybridization in Si atoms in silicene of the hybrid system as compared to that of Si atoms in free standing silicene. Our calculations with PBE XC functional give 14.695Å for the value of lattice constant along c axis (which is denoted by C). Thus, the inter-layer distance, denoted by D, between the silicene and BN layers becomes 3.674Å (in this case, D = C/4, See Fig. 1(a) ). We also perform similar calculations for bi-layer of silicene with a single BN layer kept in between (Si 16 B 9 N 9 ). The optimized geometry of bi-layer is given in Fig. 1 (b) and (d). In this case, the lattice constant 'A' and inter-layer distance of the hybrid bilayer are estimated to be 7.607 and 4.011Å respectively. We observe from these results that for hybrid bi-layer both the values of lattice constant and inter-layer distance slightly increase from the corresponding values in hybrid bulk system. Furthermore, our calculations of cohesive energy of the hybrid systems show that they are energetically stable. We find that the cohesive energy per atom for the hybrid bulk and bi-layer systems are 6.07 and 5.57 eV/atom respectively. These results indicate that the interaction between the layers in bilayer is slightly weaker as compared to that in the hybrid bulk.
In order to get deeper insight into the interaction between the layers of the hybrid systems, we plot the spatial distribution of charge densities for the hybrid graphitelike bulk system (Si 16 B 18 N 18 ) (see (c) and (d)) and bilayer of silicene (Si 16 B 9 N 9 ) (see (e) and (f)) in Fig. 2. These figures clearly show the charge distribution of hybrid systems exhibit characteristic features of layered systems such as (i) presence of more charge densities in basal planes containing covalent bonds between atoms in each layer and (ii) negligible amount of charge densities present between the adjacent layers. We also compare these results with that of bulk graphite (see (a) and (b)) and find that the charge density distributions of the hybrid systems are very similar to that of bulk graphite. Thus, our results clearly indicate that the hybrid sys- tems are similar to those of graphite and multi-layers of graphene.
We also determine the equation of state (EOS)) of the hybrid bulk system (Si 16 B 18 N 18 ). In Fig. 3, we provide the results for variation in total energy of system with the volume of unit cell. We evaluate the bulk modulus of the hybrid system by fitting our results given in Fig. 3 with Birch-Murnaghan formula for EOS. From the fitting, the values of bulk modulus and its derivative with respect to pressure for this system are estimated to be 48.5 GPa and 1.76 respectively. This value of bulk modulus of hybrid bulk system is slightly higher than the corresponding values of layered structures such as bulk graphite (33.8 GPa) 38 and bulk hexagonal BN (25.6 GPa) 39 . However, they are much lower than that of bulk silicon (98 GPa) 40 . This result again corroborates with our earlier results which indicates the layered nature of the bulk hybrid system made up of silicene and BN. Thus, the hybrid system may also be used as a soft material similar to graphite and bulk BN.
B. Electronic Properties
Having analyzed the geometrical properties of hybrid graphite-like structures in the previous section, we now discuss our results on the electronic properties of these hybrid systems. In Fig. 4, we plot the electronic band structures, in two different energy ranges, for both the hybrid bulk and bi-layered systems along the high symmetry points in reciprocal lattice which give dispersions corresponding to the motion of the charge particles along the planar directions. For the sake of comparison, we also include the electronic band structure of bulk graphite in Fig. 4 (a) and (b). We calculate the band structure for graphite super cell 2×2×1 (containing 16 carbon atoms) so that we can directly compare these results with the hy- brid system. Our results on band structure for the hybrid bulk system show that the conduction and valence bands touch each other only at the highly symmetric point K in Brillouin zone. On comparison of the results for the hybrid bulk system, Si 16 B 18 N 18 , (Fig. 4(c) and (d)) with those of bulk graphite (Fig. 4(a) and (b)), we find that the electronic band structures of these two systems are similar. However, we observe that the hybrid system contains additional number of bands which lie above ≈2.0 eV and below ≈-2.0 eV. The contributions for these bands are mainly due to the BN layers. Furthermore, around the Fermi level, we observe a parabola-like dispersion for the hybrid bulk system (see Fig. 4 (d)) as in graphite. Thus, the charge carriers in this system behave nearlyfree-particle-like. It is clearly seen from Fig. 4 (b) and (d) that the band structure of the hybrid bulk system and bulk graphite are rather similar close to the Fermi level. This leads to an important conclusion that the hy- brid bulk system made up of alternate silicene and BN layers can be a possible material for silicon based layered structure similar to that of carbon based bulk graphite.
Interestingly, in case of the hybrid bi-layered system, we observe a linear dispersion around the Fermi level in contrast to the parabola-like dispersion present in bilayer of graphene. Furthermore, this dispersion is also distinctly different from the corresponding dispersion for the pure bi-layer (without BN layer) of silicene 4 where the parabolic dispersions are shifted in both E and k direction in the band structure due to strong inter-layer covalent bonding. A closer look at the bands reveals that two linear dispersions are present in the band structure and they correspond to the two silicene layers of hybrid bi-layer. This suggests that there exists a much weaker interaction between the layers in bi-layer as compared to that between the layers in bulk system. This result again corroborates with our data on the geometric properties where it is observed that the inter-layer distance in bilayer is about 9 % larger as compared to that in the bulk.
In Fig. 5, we present the results of calculated total density of states (DOS) and partial DOS for these two hybrid systems, namely bulk system (Si 16 B 18 N 18 ) and bi-layer (Si 16 B 9 N 9 ). First important observation is that the hybrid systems, both bulk as well as bi-layer, are semi-metallic since the values of DOS at E F are zero. Investigation from the atom projected partial DOS clearly indicates that the contributions of DOS just below and above Fermi levels are mainly due to π and π * orbitals of silicene layers. Moreover, we observe that there is no contribution from either boron or nitrogen (and thus BN layer) around the Fermi level. These atoms contribute to DOS at about -1.8 eV ( and below) and 2.5 eV (and above) compared to the Fermi level respectively for valence and conduction bands.
Variation of Properties with Inter-layer Distance
In order to understand the effect of inter-layer distance on the properties of the hybrid system, we perform the calculations of electronic band structure of the hybrid bi-layer system under various transverse stress. The compression along the direction perpendicular to the plane of silicene sheets (c axis) is simulated by freezing z-components of the positions of the surface atoms during the geometric structure optimization. These surface atoms are denoted by circle with red color in Fig. 6 (a). The strength of the interaction between the layers is varied by changing the inter-layer distance. The stability of this system under the influence of transverse compression is analyzed by the cohesive energy calculation. Our calculations show that the hybrid bi-layered system under the transverse compression is energetically stable. From Fig. 6(b), we observe that the value of cohesive energy per atom decreases monotonically with decrease in the value of inter-layer distance. Investigation on geometric structure indicates that the variation in the value of lattice constant (A) due to the compression is very small and maximum variation is less than 1.5 %. However, the buckling length in silicene layer is largely increased, up to 0.884Å (for the highest compression) from 0.543 A, during the compression. This is due to the increased interaction between the silicene and BN layers. Now, we discuss the results of band structure and charge density distribution for different inter-layer distances which are given in Fig. 7. As the inter-layer distance is decreased the degeneracy of the bands corresponding to the two silicene layer is lifted due to an increased interaction between the layers. It is clearly seen from Fig. 7 that the four bands ( two valence and two conduction bands) of silicene layers become nondegenerate. The characteristics of these four bands decide the electronic properties of bilayer. Fig. 7 (d)-(f) shows that when the value of inter-layer distance is little more than 3Å the characteristic linear dispersion present in the band structure of this hybrid bi-layer is changed to parabola-like dispersion which are very similar to those of bi-layer of graphene [41][42][43][44][45][46][47] . Thus, we infer from these results that the weak interaction between the layers changes the character of charge carriers in this 2D system from a Dirac-Fermion to a Nearly-Free-Particle-Like. Furthermore, we also observe that the lower and upper most bands start to move away from the Fermi level as the inter-layer separation is reduced. The corresponding charge density plots for the system with the inter-layer distance (Fig. 7 (d)-(f)) show that the interaction between the layers is still a weak vdW since there is no significant overlap of charges from the atoms present in the adjacent layers. However, when the value of inter-layer distance is about 3Å and below, we observe opening of a band gap in the band structure of this hybrid system. Thus, the results of our calculation show that there is a semi-metal to semi-conductor transition due to compression along the direction perpendicular to the plane of silicene sheets ( c axis). Up to this point, we observe no deformation in the geometric structure of the bi-layered system. When the inter-layer distance ( Fig. 7(g)-(i)) is further reduced, the interaction between the layer increases which in turn increases the value of induced band gap of the system. We also observe that a significant amount of overlap in the charge density between the layers which indicates the onset of covalent-like bond formation between the atoms in the adjacent layers (See Fig. 7(i)).
On further reduction in the inter-layer distance to about 2.46Å, we see from Fig. 7(j) that there is a noticeable deformation in the geometric structure of the hybrid bi-layer. There is an undulation in the geometry of BN layer. One part of the BN layer is moved upward and another goes downward such that B and N atoms present in these two parts make a covalent-like bond with Si atoms of the adjacent silicene layer. The presence of strong covalent-like bonds is confirmed by the charge density analysis (Fig. 7(j)). Again due to these strong bonds, the electronic behavior of the system becomes completely different from that given in Fig. 7(a)-(f)). In this case, we also observe that the system has a direct band gap of about 824 meV. Furthermore, there is no signature of any parabola-like dispersion around the Fermi level. This clearly indicates that the system no longer behaves like a layered graphite-like material.
IV. CONCLUSION
We have carried out studies on the geometric and electronic properties of hybrid graphite-like structures made up of silicene and boron nitride (BN) layers by employing ab initio DFT based calculations. We observe from the cohesive energy calculations that the hybrid system is energetically stable. Our calculations predict that the hybrid bulk system possesses physical properties similar to those of bulk graphite. The coupling between the layers of silicene and BN of this hybrid system is due to weak van der Waals interaction which is same as that in graphite and multi-layers of graphene. We observe from the results on the electronic band structure and the density of states that the hybrid bulk system is a semimetal and it possesses the dispersion curve, around E F , very similar to that of bulk graphite. Main contributions to the electronic band structure around E F arise only due to silicene layers. Our calculations on bi-layer of silicene with a BN layer show that it possesses the characteristic linear dispersion around E F due to much weaker interaction between the layers. However, the nature of dispersion curve becomes parabola-like when the system is compressed along the direction perpendicular to the plane of silicene sheet. Finally, we also observe an opening up of band gap near the Fermi level when the inter-layer distance is about 3Å and below. These calculations show that the hybrid system based on silicon and BN can be a possible candidate for two dimensional layered soft material akin to graphite and multi-layers of graphene. These results would be important from application point of view since the silicon based systems are more compatible with the existing semiconductor industry as compared to the carbon based systems.
V. APPENDIX
Influence of weak van der Waals Interaction: We would like to mention here that the standard exchangecorrelation (XC) functionals such as local density approximation (LDA) and generalized gradient approximation (GGA) do not take into account weak attractive potential arising due to vdW interaction [48][49][50][51][52][53][54][55][56][57][58] . In order to incorporate the weak vdW interaction between the layers of the hybrid system and study its effect on the geometric and electronic properties of the hybrid system, we perform the electronic structure calculations based on DFT with van der Waals density functional (vdW-DF) 48-50 as implemented in the Quantum-Espresso package 59 . The values of lattice constants obtained by vdW-DF are summarized in Table I. We also include the results of the calculations with PBE XC functional for comparison. We can see clearly from Table I ence between these two values is about 0.2 %. The lattice constant 'A' is determined by the strong intra-layer covalent bonds between the atom in basal plane. Both these functionals accurately describe the covalent bonds present within the layers. However, we observe a large difference of about 7 % in value of lattice constant along c axis obtained by DFT calculations with PBE and vdW-DF XC functionals. The PBE XC functional underestimates the strength of vdW potential between the layers and hence it overestimates the value of lattice constant 'C'. The failure of LDA and GGA is due to fact that for large inter-layer distance 'D', the vdW attractive potential scales as -C 4 /D 4 , where C 4 is vdW interaction coefficient, whereas both LDA and GGA XC functionals predict the wrong trend of exponential decay with the inter-layer distance 51 . It is interesting to note that the inter-layer distance in bulk hybrid system is 3.426 A which is comparable with 3.354Å of bulk graphite. Thus, our calculations on the hybrid system show that vdW interaction play a crucial role in the coupling between the layers of silicene and BN. This vdW interaction is similar to that between the layers of graphite as well as multi-layers of graphene. In order to check the effect of weak vdW interaction on the electronic properties of the hybrid system, we also calculate the electronic band structures for the hybrid bilayer system by employing DFT with vdW-DF XC functional. The results of these calculations are given in Fig. 8. We compare these results with the corresponding one obtained by using PBE XC functional (See Fig. 2 (e)). We observe that the two functionals give similar results for the electronic band structure along Γ-K-M-Γ path in Brillouin zone of the hybrid system and thus the effect of vdW interaction on the electronic properties is negligible.
VI. ACKNOWLEDGMENTS
Authors thank Dr. P. D. Gupta, Dr. S. K. Deb and Dr. P. K. Gupta for encouragement and support. CK thanks Mr. Himanshu Srivastava for useful discussions. The support and help of Mr. P. Thander and the scientific computing group, Computer Centre, RRCAT is acknowledged.
FIG. 1 :
1(color on line) The optimized geometric structures of hybrid graphite-like structures made up of silicene and boron nitride layers. Bulk system (Si16B18N18) : (a) side and (c) top views. Bilayer of silicene with a single boron nitride layer (Si16B9N9) : (b) side and (d) top views.
FIG. 2 :
2(color on line) Spatial charge density distributions of layered systems. Bulk graphite with 2×2×1 super cell ( (a) side and (b) top views) and hybrid graphite-like structures made up of silicene and boron nitride layers: bulk system (Si16B18N18) ((c) side and (d) top views) and bi-layer (Si16B9N9) ((e) side and (f) top views) III. RESULTS AND DISCUSSIONS A. Geometric structures and charge density distributions
FIG. 3 :
3Variation in total energy of hybrid graphite-like bulk system made up of silicene and BN layers (Si16B18N18) with its volume.
FIG. 4 :FIG. 5 :
45Band structures of layered systems in two energy ranges. Bulk graphite with 2x2x1 super cell ( (a) and (b)) and hybrid graphite-like structures made up of silicene and boron nitride layers: bulk system (Si16B18N18) ((c) and (d) ) and bi-layer (Si16B9N9) ((e) and (f) ). The values of energy of bands are with respect to the respective Fermi level. Total and partial density of states for hybrid graphite-like structure made up of silicene and boron nitride layer: (a) bulk system (Si16B18N18) and (b) bi-layer (Si16B9N9). The values of energy are with respect to the Fermi level.
on line) (a) Hybrid bi-layer of silicene with a single BN layer (Si16B9N9) under compression along the transverse direction. The z-components of positions of atoms with circle (Red color) are frozen during the geometric optimization. (b) Variation of cohesive energy per atom with inter-layer separation.
FIG. 7 :
7(color on line) Variation in the electronic band structure and the spatial charge density distribution with different inter-layer separation. The energy of bands are with respect to Fermi level. The value of inter-layer distance (inÅ) is given below each figure.
FIG. 8 :
8Band structures of hybrid graphite-like bi-layered structure (Si16B9N9) obtained by vdW-DF. The values of energy of bands are with respect to the Fermi level.
TABLE I :
IThe results of optimized geometries of hybrid graphite-like structures made up of silicene and boron nitride obtained by DFT with PBE and vdW-DF exchangecorrelation functionals.Lattice Constants (Å)
Inter-layer Distance (Å)
System
A
C
D
PBE vdW-DF PBE vdW-DF PBE
vdW-DF
Bulk
7.554 7.539 14.695 13.704 3.674
3.426
(Si16B18N18)
Bi-layer
7.607 7.646
8.021
7.903 4.011
3.951
(Si16B9N9)
that the lattice constant 'A' obtained by these two XC functionals, namely PBE and vdW-DF, match very well with each other. The differ-
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| []
|
[
"CUTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE",
"CUTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE"
]
| [
"Subhajit Ghosh "
]
| []
| []
| Let {G n } ∞ 1 be a sequence of non-trivial finite groups, and G n denote the set of all non-isomorphic irreducible representations of G n . In this paper, we study the properties of a random walk on the complete monomial group G n ≀ S n generated by the elements of the form (e, . . . , e, g; id) and (e, . . . , e, g −1 , e, . . . , e, g; (i, n)) for g ∈ G n , 1 ≤ i < n. We call this the warp-transpose top with random shuffle on G n ≀ S n . We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is O n log n + 1 2 n log(|G n | − 1) . We show that this shuffle exhibits ℓ 2 -cutoff at n log n + 1 2 n log(|G n | − 1). We prove that this shuffle has total variation cutoff at n log n if |G n | = o(n δ ) for all δ > 0.2010 Mathematics Subject Classification. 60B15, 60J10, 60C05. | null | [
"https://arxiv.org/pdf/2101.00533v2.pdf"
]
| 230,433,780 | 2101.00533 | e22ec8092361fc026ae8ecbb493843ada9469d46 |
CUTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE
29 Mar 2022
Subhajit Ghosh
CUTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE
29 Mar 2022arXiv:2101.00533v2 [math.PR]
Let {G n } ∞ 1 be a sequence of non-trivial finite groups, and G n denote the set of all non-isomorphic irreducible representations of G n . In this paper, we study the properties of a random walk on the complete monomial group G n ≀ S n generated by the elements of the form (e, . . . , e, g; id) and (e, . . . , e, g −1 , e, . . . , e, g; (i, n)) for g ∈ G n , 1 ≤ i < n. We call this the warp-transpose top with random shuffle on G n ≀ S n . We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is O n log n + 1 2 n log(|G n | − 1) . We show that this shuffle exhibits ℓ 2 -cutoff at n log n + 1 2 n log(|G n | − 1). We prove that this shuffle has total variation cutoff at n log n if |G n | = o(n δ ) for all δ > 0.2010 Mathematics Subject Classification. 60B15, 60J10, 60C05.
Introduction
The number of shuffles required to mix up a deck of cards is the main topic of interest for card shuffling problems. It has received considerable attention in the last few decades. The card shuffling problems can be described as random walks on the symmetric group. The generalisation by replacing the symmetric group with other finite groups is also a wellstudied topic in probability theory [17]. A random walk converges to a unique stationary distribution subject to certain natural conditions. For a convergent random walk, the mixing time (number of steps required to reach the stationary distribution up to a given tolerance) of random walks has been intensively studied. It is helpful to know the eigenvalues and eigenvectors of the transition matrix to study the convergence rate of random walks. In general, convergence rate questions for finite Markov chains are useful in many subjects, including statistical physics, computer science, biology and more [15].
In the eighties, Diaconis and Shahshahani introduced the use of non-commutative Fourier analysis techniques in their work on the random transposition shuffle [7]. They proved that this shuffle on n distinct cards has total variation cutoff at 1 2 n log n. After this landmark work, the theory of random walks on finite groups obtained its own independence, its own problems and techniques. Afterwards, some other techniques have come to deal with random walks on finite groups (viz. the coupling argument [1], the strong stationary time approach [2,3]). Our model is mainly inspired by the transpose top with random shuffle on the symmetric group S n [9]. Given a deck of n distinct cards this shuffle chooses a card from the deck uniformly at random and transposes it with the top card. This shuffle exhibits total variation cutoff at n log n [4,5]. The transpose top with random shuffle has been recently generalized by the author to the cards with two orientations known as the flip-transpose top with random shuffle on the hyperoctahedral group B n [10]. The flip-transpose top with random shuffle on B n has total variation cutoff at n log n. In this work, we generalize the flip-transpose top with random shuffle to the complete monomial group G n ≀ S n . As a notable random walk on the complete monomial groups we mention [18], where Schoolfield Jr. generalized the random transposition model to G ≀ S n for a finite group G. However this walk was generated by the probability measure which is constant on the conjugacy classes. In general, it is not easy to study a random walk generated by a probability measure which is not constant on the conjugacy classes (such model is known as the non-conjugacy random walk). Although our model is a non-conjugacy random walk, we study this using the representation theory of the complete monomial group G n ≀ S n . For other random walks on the complete monomial group see [8,12]. Before describing our random walk model, let us first recall the definition of the complete monomial group. Definition 1.1. Let G be a finite group and S n be the symmetric group of permutations of elements of the set [n] := {1, 2, . . . , n}. The complete monomial group is the wreath product of G with S n , is a group denoted by G ≀ S n and can be described as follows: The elements of G ≀ S n are (n + 1)-tuples (g 1 , g 2 , . . . , g n ; π) where g i ∈ G and π ∈ S n . The multiplication in G ≀ S n is given by (g 1 , . . . , g n ; π)(h 1 , . . . , h n ; η) = (g 1 h π −1 (1) , . . . , g n h π −1 (n) ; πη). Therefore (g 1 , . . . , g n ; π) −1 = (g −1 π(1) , . . . , g −1 π(n) ; π −1 ). Now let {G n } ∞ 1 be a sequence of non-trivial finite groups. We consider the complete monomial groups G n := G n ≀ S n for each positive integer n. Let e be the identity of G n and id be the identity of S n . For an element π ∈ S n , let π := (e, . . . , e; π) ∈ G n and for g ∈ G n , let g (i) := (e, . . . , e, g ↑ , e, . . . , e; id) ∈ G n . ith position.
Unless otherwise stated from now on, (e, . . . , e, g −1 , e, . . . , e, g; (i, n)) denotes the element of G n with g −1 in ith position and g in nth position, for g ∈ G n , 1 ≤ i < n. One can check that (g −1 ) (i) g (n) (i, n) is equal to (e, . . . , e, g −1 , e, . . . , e, g; (i, n)) for g ∈ G n , 1 ≤ i < n.
In this work we consider a random walk on the complete monomial group G n driven by a probability measure P , defined as follows:
(1) P (x) = 1 n|Gn| , if x = (e, . . . , e, g; id) for g ∈ G n , 1 n|Gn| , if x = (e, . . . , e, g −1 , e, . . . , e, g; (i, n)) for g ∈ G n , 1 ≤ i < n, 0, otherwise.
We call this the warp-transpose top with random shuffle because at most times the nth component is multiplied by g and the ith component is multiplied by g −1 simultaneously, g ∈ G n , 1 ≤ i < n. We now give a combinatorial description of this model as follows: Let A n (G) denote the set of all arrangements of n coloured cards in a row such that the colours of the cards are indexed by the set G. For example, if Z 2 denotes the additive group of integers modulo 2, then elements of A n (Z 2 ) can be identified with the elements of B n (the hyperoctahedral group). For g, h ∈ G, by saying update the colour g using colour h we mean the colour g is updated to colour g · h. Elements of G n can be identified with the elements of A n (G n ) as follows: The element (g 1 , . . . , g n ; π) ∈ G n is identified with the arrangement in A n (G n ) such that the label of the ith card is π(i), and its colour is g π(i) , for each i ∈ [n]. Given an arrangement of coloured cards in A n (G n ), the warp-transpose top with random shuffle on G n is the following: Choose a positive integer i uniformly from [n]. Also choose a colour g uniformly from G n , independent of the choice of the integer i.
(1) If i = n: update the colour of the nth card using colour g.
(2) If i < n: first transpose the ith and nth cards. Then simultaneously update the colour of the ith card using colour g and update the colour of the nth card using colour g −1 .
123459786 Figure 1. Transitions for the warp-transpose top with random shuffle on Z 3 ≀ S 9 . Z 3 is the additive group of integers modulo 3, consists of the colours red, green and blue such that red represents the identity element. (a) shows transitions when the sixth card is chosen and (b) shows transitions when the last card is chosen.
12345 6 789 − − − → − −− → − − − → 123459786 123459786 123456789 12345678 9 − − − → − −− → − − − → 123456789 123456789 (a) (b)
The flip-transpose top with random shuffle on the hyperoctahedral group serves the case when |G n | = 2 for all n [10]. We now state the main theorems of this paper.
Theorem 1.1. The mixing time for the warp-transpose top with random shuffle on G n is O n log n + 1 2 n log(|G n | − 1) . Theorem 1.2. The warp-transpose top with random shuffle on G n presents ℓ 2 -cutoff at n log n + 1 2 n log(|G n | − 1). Theorem 1.3. The warp-transpose top with random shuffle on G n exhibits total variation cutoff phenomenon with cutoff time n log n if |G n | = o(n δ ) for all δ > 0.
Let us recall some concepts and terminologies from representation theory of finite group and discrete time Markov chains with finite state space to make this paper self contained. Readers from representation theoretic background may skip Subsection 1.1 and from Probabilistic background may skip Subsection 1.2.
1.1. Representation theory of finite group. Let G be a finite group and V be a finite dimensional complex vector space. Also let GL(V ) be the set of all invertible linear operators on V . A linear representation ρ of G is a homomorphism from G to GL(V ). Sometimes this representation is also denoted by the pair (ρ, V ). The dimension of the vector space V is called the dimension d ρ of the representation. V is called the G-module corresponding to the representation ρ in this case. Let C[G] be the group algebra consists of complex linear combinations of elements of G. In particular taking V = C[G], we define the right regular representation R :
G −→ GL(C[G]) of G by R(g) h∈G C h h = h∈G C h hg −1 , where C h ∈ C. A vector subspace W of V is said to be stable ( or 'invariant') under ρ if ρ(g) (W ) ⊂ W for all g in G.
The representation ρ is irreducible if V is non-trivial and V has no non-trivial proper stable subspace. Two representations (ρ 1 , V 1 ) and (ρ 2 , V 2 ) of G are are said to be isomorphic if there exists an invertible linear map T : V 1 → V 2 such that the following diagram commutes for all g ∈ G:
V 1 V 1 V 2 V 2 ρ 1 (g) T T ρ 2 (g)
For each g ∈ G, ρ(g) can also be thought of as an invertible complex matrix of size d ρ ×d ρ . The trace of the matrix ρ(g) is said to be the character value of ρ at g and is denoted by χ ρ (g). It can be easily seen that the character values are constants on conjugacy classes, hence characters are class functions. If χ ρ (g) denote the complex conjugate of χ ρ (g), then one can check that χ ρ (g −1 ) = χ ρ (g) for all g ∈ G. Let C(G) be the complex vector space of class functions of G. Then a 'standard' inner product ·, · on C(G) is defined as follows:
φ, ψ = 1 |G| g∈G φ(g)ψ(g −1 ) for φ, ψ ∈ C(G).
An important theorem in this context is the following [19,Theorem 6]: The characters corresponding to the non-isomorphic irreducible representations of G forms an ·, · -orthonormal basis of C(G).
If V 1 ⊗ V 2 denotes the tensor product of the vector spaces V 1 and V 2 , then the tensor product of two representations ρ 1 : G → GL(V 1 ) and ρ 2 : G → GL(V 2 ) is a representation denoted by (ρ 1 ⊗ ρ 2 , V 1 ⊗ V 2 ) and defined by,
(ρ 1 ⊗ ρ 2 )(g)(v 1 ⊗ v 2 ) = ρ 1 (g)(v 1 ) ⊗ ρ 2 (g)(v 2 ) for v 1 ∈ V 1 , v 2 ∈ V 2 and g ∈ G.
We use some results from representation theory of finite groups without recalling the proof. For details about finite group representation see [14,16,19]
(2) P(X k+1 = y | H k−1 ∩ {X k = x}) = M(x, y).
Equation (2) says that given the present, the future is independent of the past. Let D k denote the distribution after k transitions, i.e. D k is the row (probability) vector (P(X k = x)) x∈Ω . Then D k = D k−1 M for all k ≥ 1, which implies D k = D 0 M k . In particular if the chain starts at x ∈ Ω, then its distribution after k transitions is
D k = δ x M k , i.e. P(X k = y | X 0 = x) = M k (x, y).
Here δ x is defined on Ω as follows:
δ x (u) = 1 if u = x, 0 if u = x.
A Markov chain is said to be irreducible if it is possible for the chain to reach any state starting from any state using only transitions of positive probability. The period of a state x ∈ Ω is defined to be the greatest common divisor of the set of all times when it is possible for the chain to return to the starting state x.
||D k − Π|| 2 := x∈Ω D k (x) Π(x) − 1 2 Π(x) 1 2 .
We now define the total variation distance between two probability measures.
Definition 1.3.
Let µ and ν be two probability measures on Ω. The total variation distance between µ and ν is defined by
||µ − ν|| TV := sup A⊂Ω |µ(A) − ν(A)|.
It can be easily seen that ||µ − ν|| [11,Proposition 4.2]). For an irreducible and aperiodic chain the interesting topic is the minimum number of transitions k required to reach near the stationarity Π up to a certain level of tolerance ε > 0. We first define the maximal ℓ 2 -distance (respectively total variation distance) between the distribution after k transitions and the stationary distribution as follows:
TV = 1 2 x∈Ω |µ(x) − ν(x)| (seed 2 (k) := max x∈Ω M k (x, ·) − Π 2 respectively d TV (k) := max x∈Ω M k (x, ·) − Π TV .
For ε > 0, the ℓ 2 -mixing time (respectively total variation mixing time) with tolerance level ε is defined by
τ mix (ε) := min {k : d 2 (k) ≤ ε} (respectively t mix (ε) := min {k : d TV (k) ≤ ε}).
Most of the notations of this subsection are borrowed from [11].
Non-commutative Fourier analysis and random walks on finite groups.
Let p and q be two probability measures on a finite group G. We define the convolution p * q of p and q by (p * q)(x) := y∈G p(xy −1 )q(y). The Fourier transform p of p at the right regular representation R is defined by the matrix x∈G p(x)R(x). The matrix p(R) can be thought of as the action of the group algebra element g∈G p(g)g −1 on C[G] by multiplication on the right. It can be easily seen that (p * q)(R) = p(R) q(R).
A random walk on a finite group G driven by a probability measure p is a Markov chain with state space G and transition probabilities M p (x, y) = p(x −1 y), x, y ∈ G. It can be easily seen that the transition matrix M p is p(R) and the distribution after kth transition will be p * k (convolution of p with itself k times) i.e., the probability of getting into state y starting at state x after k transitions is p * k (x −1 y). One can easily check that the random walk on G driven by p is irreducible if and only if the support of p generates G [17, Proposition 2.3].
The stationary distribution for an irreducible random walk on G driven by
p, is the uniform distribution U G on G (since x∈G M p (x, y) = x∈G p(x −1 y) = z∈G p(z) = 1, z = x −1 y for all y ∈ G)
. From now on, the uniform distribution on group G will be denoted by U G . For the random walk on G driven by p, it is enough to focus on
p * k − U G 2 and p * k − U G TV because, M k p (x, ·) − U G TV = M k p (y, ·) − U G TV M k p (x, ·) − U G 2 = M k p (y, ·) − U G 2 for any two elements x, y ∈ G.
We now define the cutoff phenomenon for a sequence of random walks on finite groups.
Definition 1.4. Let {G n } ∞
0 be a sequence of finite groups and p n be a probability measure on G n for each n. Consider the sequence of irreducible and aperiodic random walk on G n driven by p n . We say that the ℓ 2 -cutoff phenomenon (respectively total variation cutoff phenomenon) holds for the family {(G n , p n )} ∞ 0 if there exists a sequence {T n } ∞ 0 of positive real numbers tending to infinity as n → ∞, such that the following hold:
(1) For any ǫ ∈ (0, 1) and k n = ⌊(1 + ǫ)T n ⌋,
lim n→∞ p * kn n − U Gn 2 = 0 respectively lim n→∞ p * kn n − U Gn TV = 0 ,
(2) For any ǫ ∈ (0, 1) and
k n = ⌊(1 − ǫ)T n ⌋, lim n→∞ p * kn n − U Gn 2 = ∞ respectively lim n→∞ p * kn n − U Gn TV = 1 .
Here ⌊x⌋ denotes the floor of x (the largest integer less than or equal to x).
Informally, we will say that {(G n , p n )} ∞ 0 has an ℓ 2 -cutoff (respectively total variation cutoff) at time T n . This says that for sufficiently large n the mixing time does not depend on the tolerance level ε(> 0). In other words the distribution after k transitions is very close to the stationary distribution if k = T n but too far from the stationary distribution if k < T n . Roughly the cutoff phenomenon depends on the multiplicity of the second largest eigenvalue of the transition matrix [6]. We now see that the random walk of our concern is irreducible and aperiodic. Proof. The support of P is Γ
= {(g −1 ) (i) g (n) (i, n), g (n) | g ∈ G n , 1 ≤ i < n} and it can be easily seen that {g (k) , (i, n) | g ∈ G n , 1 ≤ k ≤ n, 1 ≤ i < n} is a generating set of G n . (g −1 ) (n) (g −1 ) (i) g (n) (i, n) g (n) = (i, n) for each 1 ≤ i < n and g ∈ G n , (k, n)g (n) (k, n) = g (k) for each 1 ≤ k ≤ n and for all g ∈ G n .(3)
Thus (3) implies Γ generates G n and hence the warp-transpose top with random shuffle on G n is irreducible. Moreover given any π ∈ G n , the set of all times when it is possible for the chain to return to the starting state π contains the integer 1 (as support of P contains the identity element of G n ). Therefore the period of the state π is 1 and hence from irreducibility all the states of this chain have period 1. Thus this chain is aperiodic. Proposition 1.4 says that the warp-transpose top with random shuffle on G n converges to the uniform distribution U Gn as the number of transitions goes to infinity. In Section 2 we will find the spectrum of P (R). We will prove Theorems 1.1 and 1.2 in Section 3. In Section 4, lower bound of P * k − U Gn TV will be discussed and Theorem 1.3 will be proved.
Acknowledgement. I extend sincere thanks to my advisor Arvind Ayyer for all the insightful discussions during the preparation of this paper. I would like to thank the anonymous reviewer of Algebraic Combinatorics for valuable comments, which helped improve the upper bound result and simplify the proof of the lower bound. I would like to acknowledge support in part by a UGC Centre for Advanced Study grant.
Spectrum of the transition matrix
In this section we find the eigenvalues of the transition matrix P (R), the Fourier transform of P at the right regular representation R of G n . To find the eigenvalues of P (R) we will use the representation theory of the wreath product G n of G n with the symmetric group S n . First we briefly discuss the representation theory of G ≀ S n , following the notation from [13].
A partition λ of a positive integer n (denoted λ ⊢ n) is a weakly decreasing finite sequence (λ 1 , · · · , λ r ) of positive integers such that r i=1 λ i = n. The partition λ can be pictorially visualized as a left-justified arrangement of r rows of boxes with λ i boxes in the ith row, 1 ≤ i ≤ r. This pictorial arrangement of boxes is known as the Young diagram of λ.
, we define Y(X) = {µ : µ is a map from X to Y}. For µ ∈ Y(X), define ||µ|| = x∈X |µ(x)|, where |µ(x)| is the number of boxes of the Young diagram µ(x) and define Y n (X) = {µ ∈ Y(X) : ||µ|| = n}.
Let n be a fixed positive integer. Let G denote the (finite) set of all non-isomorphic irreducible representations of G. Given σ ∈ G, we denote by W σ the corresponding irreducible (4) (3, 1) (2, 2) (2, 1, 1) (1, 1, 1, 1) For example let us take n = 10 and G to be Z 10 , the additive group of integers modulo 10. Also let Z 10 := {σ 1 , σ 2 , σ 3 , σ 4 , σ 5 , σ 6 , σ 7 , σ 8 , σ 9 , σ 10 } and µ ∈ Y 10 ( Z 10 ) be such that
µ(σ 1 ) = , µ(σ 2 ) = , µ(σ 8 ) = ,X i (G) = i−1 k=1 g∈G (g −1 ) (k) g (i) (k, i) = i−1 k=1 g∈G (g −1 ) (k) (k, i)g (k) , for all 2 ≤ i ≤ n.
Young-Jucys-Murphy elements generates a maximal commuting subalgebra of C[G ≀ S n ] and act like scalars on the Gelfand-Tsetlin subspaces of irreducible G ≀ S n -modules. We now define Gelfand-Tsetlin subspaces and the Gelfand-Tsetlin decomposition.
Let λ ∈ H n,n (G) and consider the irreducible H n,n (G)-module (the space for the representation of H n,n (G)) V λ . Since the branching is simple [13,Section 4], the decomposition into irreducible H n−1,n (G)-modules is given by
V λ = ⊕ µ V µ ,
where the sum is over all µ ∈ H n−1,n (G), with µ ր λ (i.e there is an edge from µ to λ in the branching multi-graph), is canonical. Iterating this decomposition of V λ into irreducible H 1,n (G)-submodules, i.e.,
(4) V λ = ⊕ T V T ,
where the sum is over all possible chains T = λ 1 ր λ 2 ր · · · ր λ n with λ i ∈ H i,n (G) and λ n = λ. We call (4) the Gelfand-Tsetlin decomposition of V λ and each V T in (4) a Gelfand-Tsetlin subspace of V λ . We note that if 0 = v T ∈ V T , then C[H i,n (G)]v T = V λ i from the definition of V T . From Lemma 6.2 and Theorem 6.4 of [13], we may parametrise the irreducible representations of G ≀ S n by elements of Y n ( G).
Theorem 2.1 ([13, Theorem 6.5]). Let µ ∈ Y n ( G). Then we may index the Gelfand-Tsetlin subspaces of V µ by standard Young G-tableaux of shape µ and write the Gelfand-Tsetlin decomposition as
V µ = ⊕ T ∈tab G (n,µ) V T ,
where each V T is closed under the action of G n and as a G n -module, is isomorphic to the irreducible G n -module W r T (1) ⊗ W r T (2) ⊗ · · · ⊗ W r T (n) .
For i = 1, . . . , n; the eigenvalues of X i (G) on V T are given by |G| dim(W r T (i) ) c(b T (i)).
(i) = µ(σ i ), m i = |µ (i) |, d i = dim(W σ i ) for each 1 ≤ i ≤ t. Then dim(V µ ) = n m 1 , . . . , m t f µ (1) · · · f µ (t) d m 1 1 · · · d mt t .
Here f µ (i) denotes the number of standard Young tableau of shape µ (i) , for each 1 ≤ i ≤ t.
Lemma 2.3.
Let G be a finite group and ρ ∈ G. If W ρ (respectively χ ρ ) denotes the irreducible G-module (respectively character) and d ρ is the dimension of W ρ , then the action of the group algebra element g∈G g on W ρ is given by the following scalar matrix
g∈G g = |G| d ρ χ ρ , χ 1 I dρ .
Here I dρ is the identity matrix of order d ρ × d ρ and 1 be the trivial representation of G.
Proof. It is clear that g∈G g is in the centre of C[G]. Therefore by Schur's lemma ([19, Proposition 4]), we have g∈G g = cI dρ for some c ∈ C. The value of c can be obtained by equating the traces of g∈G g and cI dρ .
Remark 2.4.
Our focus will be on H n,n (G n ) i.e. G n ≀ S n for the sequence of subgroups
H 1,n (G n ) ⊆ · · · ⊆ H i,n (G n ) ⊆ · · · ⊆ H n,n (G n ).
For simplicity we write the Young-Jucys-Murphy elements X i (G n ) of G n ≀ S n (i.e. G n ) as X i for 1 ≤ i ≤ n. Thus Theorems 2.1 and 2.2 are applicable to G n .
Let t := | G n | and G n := {σ 1 , . . . , σ t }, where σ 1 = 1 (the trivial representation of G n ). We write µ ∈ Y n ( G n ) as the tuple (µ (1) , . . . , µ (t) ), where µ (i) := µ(σ i ) for each 1 ≤ i ≤ t. We also denote m i := |µ (i) |, W σ i := the irreducible G n -module corresponding to σ i and d i = dim(W σ i ) for each 1 ≤ i ≤ t. Thus t, σ i , µ (i) , m i , W σ i , and d i depend on G n i.e., on n. To avoid notational complication the dependence of t, σ i , µ (i) , m i , W σ i , and d i on n is suppressed. We note that for T ∈ tab Gn (n, µ) the dimension of V T is d m 1 1 · · · d mt t .
Theorem 2.5. For each
µ = (µ (1) , . . . , µ (t) ) ∈ Y n ( G n ), let P (R) V µ denote the restriction of P (R) to the irreducible G n -module V µ . Then the eigenvalues of P (R) V µ are given by, 1 n dim(W r T (n) ) c(b T (n)) + χ r T (n) , χ 1 , with multiplicity dim(V T ) = d m 1 1 · · · d mt t
for each T ∈ tab Gn (n, µ).
Proof. We first find the eigenvalues of X n + g∈Gn (e, . . . , e, g; id). Let I dim(V T ) denote the identity matrix of order dim(V T ) × dim(V T ). Then from Theorem 2.1 we have
(5) V µ = ⊕ T ∈tab Gn (n,µ) V T and X n V T = |G n | dim(W r T (n) ) c(b T (n))I dim(V T ) .V T = |G n | dim(W r T (n) ) χ r T (n) , χ 1 I dim(V T ) .
We recall P (R) = 1 n|G n | g∈Gn R ((e, . . . , e, g; id)) +
n−1 i=1
R (e, . . . , e, g −1 , e, . . . , e, g; (i, n)) .
Therefore n|G n | P (R) is the action of X n + g∈Gn (e, . . . , e, g; id) on C[G n ] by multiplication on the right. Since dim(V T ) = d m 1 1 · · · d mt t , the theorem follows from (5) and (6).
Upper bound for total variation distance
In this section, we find an upper bounds of P * k − U Gn 2 and P * k − U Gn TV when k ≥ n log n+ 1 2 n log(|G n |−1)+Cn for C > 0. We also prove Theorems 1.1 and 1.2 in this section. Before proving the main results of this section, first we set some notations and prove two useful lemmas. For any positive integer N, we write ξ ⊢ N to denote ξ is a partition of N. Given a partition ξ of the integer N (here we are allowing N to take value 0), throughout this section ξ 1 denotes the largest part of ξ. In particular if ξ ⊢ 0 then f ξ = 1 (as there is a unique Young diagram with zero boxes) and we set ξ 1 = 0.
f 1 (g −1 )f 2 (g) = 1 |G| ρ∈ G d ρ trace f 1 (ρ)f 2 (ρ) ,
where the sum is over all irreducible representations ρ of G and d ρ is the dimension of ρ.
Recall that U G be the uniform distribution on the group G. Then using Lemma 2.3 we have the following
U G (ρ) = 1 if ρ = 1, 0 if ρ = 1, for ρ ∈ G.
Moreover, given any probability measure p on the finite group G, we have p(1) = 1. Therefore setting f 1 = f 2 = p * k − U G , we have the following
(7) p(x) = p(x −1 ) for all x ∈ G =⇒ p * k − U G 2 2 = ρ∈ G\{1} d ρ trace ( p(ρ)) 2k .
We now state the Diaconis-Shahshahani upper bound lemma. The proof follows from the Cauchy-Schwarz inequality and (7).
Lemma 3.2 ([4, Lemma 4.2]). Let p be a probability measure on a finite group G such that
p(x) = p(x −1 ) for all x ∈ G.
Suppose the random walk on G driven by p is irreducible. Then we have the following
p * k − U G 2 TV ≤ 1 4 p * k − U G 2 2 = 1 4 ρ∈ G\{1} d ρ trace ( p(ρ)) 2k ,
where the sum is over all non-trivial irreducible representations ρ of G and d ρ is the dimension of ρ.
Definition 3.1. Let A be a non empty set. Then the indicator function of A is denoted by
Ind A and is defined by
Ind A (x) = 1 if x ∈ A 0 if x / ∈ A.
Lemma 3.3. Let N be a positive integer and s be any non-negative real number. Then we have
λ⊢N (f λ ) 2 λ 1 − s N 2k < e − 2ks N e N 2 e − 2k N .
Proof. For ζ ⊢ (N −λ 1 ), recall that ζ 1 denotes the largest part of ζ. If ζ 1 ≤ λ 1 , then we have
f λ ≤ N λ 1 f ζ . Therefore λ⊢N (f λ ) 2 λ 1 − s N 2k
is less than or equal to
N λ 1 =1 ζ⊢(N −λ 1 ) ζ 1 ≤λ 1 N λ 1 2 (f ζ ) 2 λ 1 − s N 2k ≤ N λ 1 =1 N λ 1 2 λ 1 − s N 2k ζ⊢(N −λ 1 ) (f ζ ) 2 = N −1 u=0 N u 2 1 − u + s N 2k u!.(8)
Equality in (8) is obtained by writing u = N −λ 1 . Using 1 − x ≤ e −x for all x ≥ 0 and N u ≤ N u u! , the expression in the right hand side of (8) is less than or equal to
N −1 u=0 N 2u u! e − 2k N (u+s) < e − 2ks N ∞ u=0 1 u! N 2 e − 2k N u = e − 2ks N e N 2 e − 2k N .
An immediate corollary of Lemma 3.3 follows from the fact
f λ 2 λ 1 − s N 2k = N −s N 2k , if λ = (N) ⊢ N .
Corollary 3.4. Following the notations of Lemma 3.3, we have
λ⊢N λ =(N) (f λ ) 2 λ 1 − s N 2k < e − 2ks N e N 2 e − 2k N − N −s N 2k . Lemma 3.5. Let µ = (µ (1) , . . . , µ (t) ) ∈ Y n ( G n ). Recall that µ (j) 1 (respectively µ (j) ′ 1
) denotes the largest part of µ (j) (respectively its conjugate) for 1 ≤ j ≤ t. Then we have
T ∈tab Gn (n,µ) c(b T (n)) + χ r T (n) , χ 1 n dim(W r T (n) ) 2k < n m 1 , . . . , m t f µ (1) · · · f µ (t) t j=1 M 2k j + M ′2k j Ind (0,∞) (m j ), where M j := µ (j) 1 −1+ χ σ j ,χ 1 nd j and M ′ j := µ (j) ′ 1 −1+ χ σ j ,χ 1 nd j for each 1 ≤ j ≤ t. Proof. Let T i = {(T 1 , . . . , T t ) ∈ tab Gn (n, µ) | b T (n) is in T i } for each 1 ≤ i ≤ t.
Then tab Gn (n, µ) is the disjoint union of the sets T 1 , . . . , T t . Therefore we have
T ∈tab Gn (n,µ) c(b T (n)) + χ r T (n) , χ 1 n dim(W r T (n) ) 2k = t i=1 T ∈T i c(b T (n)) + χ σ i , χ 1 nd i 2k Ind (0,∞) (m i )
and this is equal to,
t i=1 n − 1 m 1 , .., m i − 1, .., m t f µ (1) · · · f µ (t) f µ (i) T i ∈tab(µ (i) ) c(b T i (m i )) + χ σ i , χ 1 nd i 2k Ind (0,∞) (m i ) < t i=1 n m 1 , . . . , m t f µ (1) · · · f µ (t) f µ (i) T i ∈tab(µ (i) ) M 2k i + M ′2k i Ind (0,∞) (m i ).(9)
The inequality in (9) holds because T i ∈ tab(µ (i) ) implies the following:
c(b T i (m i )) + χ σ i , χ 1 nd i 2k ≤ max µ (i) 1 − 1 + χ σ i , χ 1 nd i 2k , µ (i) ′ 1 − 1 − χ σ i , χ 1 nd i 2k ≤ max µ (i) 1 − 1 + χ σ i , χ 1 nd i 2k , µ (i) ′ 1 − 1 + χ σ i , χ 1 nd i 2k , as χ σ i , χ 1 = 0 or 1 < µ (i) 1 − 1 + χ σ i , χ 1 nd i 2k + µ (i) ′ 1 − 1 + χ σ i , χ 1 nd i 2k = M 2k i + M ′2k i .
Therefore the result follows from (9) and
T i ∈tab(µ (i) ) M 2k i + M ′2k i = f µ (i) M 2k i + M ′2k i .
Proposition 3.6.
For the warp-transpose top with random shuffle on G n , we have
4 P * k − U Gn 2 TV ≤ P * k − U Gn 2 2 < 2 e n 2 e − 2k n − 1 + e − 4k n + 2e n 2 e − 2k n e n 2 (|Gn|−1)e − 2k n − 1 + 2(| G n | − 1)n 2 e − 2k n e n 2 e − 2k n 1 n 2 + e n 2 (|Gn|−1)e − 2k n − 1 ,
for all k ≥ max{n, n log n}.
Proof. Let us recall that G n = {σ 1 , . . . , σ t } and σ 1 = 1, the trivial representation of G n . Given µ ∈ Y n ( G n ), throughout this proof we write µ = (µ (1) , . . . , µ (t) ), where µ (i) = µ(σ i ), µ (i) ⊢ m i , and t i=1 m i = n. Now using Lemma 3.2, we have
(10) 4 P * k − U Gn 2 TV ≤ P * k − U Gn 2 2 = µ∈Yn( Gn): µ(1) =(n) dim(V µ ) trace P (R) V µ 2k .
First we partition the set Y n ( G n ) into two disjoint subsets A 1 and A 2 as follows:
A 1 = ∪ 1≤i≤t B i , where B i = {µ ∈ Y n ( G n ) | m i = n, m k = 0 for all k ∈ [t] \ {i}} A 2 = {µ ∈ Y n ( G n ) | t k=1 m k = n, 0 ≤ m k ≤ n − 1}.
It can be easily seen that B i 's are disjoint. Therefore by using Theorem 2.5 and Remark 2.6, the inequality (10) become
4 P * k − U Gn 2 TV ≤ P * k − U Gn 2 2 = µ∈B 1 µ(1) =(n) dim(V µ ) T ∈tab Gn (n,µ) c(b T (n)) + 1 nd 1 2k d n 1 + t i=2 µ∈B i dim(V µ ) T ∈tab Gn (n,µ) c(b T (n)) nd i 2k d n i(11)+ µ∈A 2 dim(V µ ) T ∈tab Gn (n,µ) c(b T (n)) + χ r T (n) , χ 1 n dim(W r T (n) ) 2k d m 1 1 · · · d mt t .
The sum of the first two terms in the right hand side of (11) are equal to
λ⊢n λ 1 =n f λ d n 1 T ∈tab(λ) c(b T (n)) + 1 nd 1 2k d n 1 + t i=2 λ⊢n f λ d n i T ∈tab(λ) c(b T (n)) nd i 2k d n i = λ⊢n λ =(n),(1 n ) f λ T ∈tab(λ) c(b T (n)) + 1 n 2k + n − 2 n 2k + t i=2 d 2n i d 2k i λ⊢n f λ T ∈tab(λ) c(b T (n)) n 2k .(12)
Now recalling λ 1 (respectively λ ′ 1 ) is the largest part of λ (respectively its conjugate), we have the following:
c(b T (n)) + x n 2k ≤ max λ 1 − 1 + x n 2k , λ ′ 1 − 1 − x n 2k , < λ 1 − 1 + x n 2k + λ ′ 1 − 1 + x n 2k
, for T ∈ tab(λ) and x ≥ 0.
This implies
λ⊢n λ =(n),(1 n ) f λ T ∈tab(λ) c(b T (n)) + 1 n 2k < λ⊢n λ =(n),(1 n ) f λ 2 λ 1 n 2k + λ ′ 1 n 2k < 2 λ⊢n λ =(n),(1 n ) f λ 2 λ 1 n 2k and λ⊢n f λ T ∈tab(λ) c(b T (n)) n 2k < λ⊢n f λ 2 λ 1 − 1 n 2k + λ ′ 1 − 1 n 2k < 2 λ⊢n f λ 2 λ 1 − 1 n 2k .
Thus using 1 − x ≤ e x for x ≥ 0, k ≥ n, and d i ≥ 1 for all 1 ≤ i ≤ t, the expression in (12) is bounded above by
2 λ⊢n λ =(n) (f λ ) 2 λ 1 n 2k + 1 − 2 n 2k + 2 t i=2 λ⊢n (f λ ) 2 λ 1 − 1 n 2k < 2 e n 2 e − 2k n − 1 + e − 4k n + 2(t − 1)e − 2k n e n 2 e − 2k n .(13)
The inequality in (13) follows from Corollary 3.4 and Lemma 3.3. Now recalling M j :=
µ (j) 1 −1+ χ σ j ,χ 1 nd j , M ′ j := µ (j) ′ 1 −1+ χ σ j ,χ 1 nd j
, and using Lemma 3.5, the third term in the right hand side of (11) is less than
µ∈A 2 n m 1 , . . . , m t 2 (f µ (1) ) 2 · · · (f µ (t) ) 2 d 2m 1 1 . . . d 2mt t t j=1 M 2k j + M ′2k j Ind (0,∞) (m j ).(14)
We now deal with (14) by considering two separate cases namely j = 1 and 1 < j ≤ t. Now using
µ (1) ⊢m 1 f µ (1) 2 µ (1) ′ 1 nd 1 2k = µ (1) ⊢m 1 f µ (1) 2 µ (1) 1 nd 1 2k
, the partial sum corresponding to j = 1 in (14) is equal to,
n−1 m 1 =1 (m 2 ,...,mt) m k =n−m 1 0≤m k ≤n−1 2 µ (i) ⊢m i 1≤i≤t n m 1 2 n − m 1 m 2 , . . . , m t 2 (f µ (1) ) 2 · · · (f µ (t) ) 2 d 2m 1 1 . . . d 2mt t µ (1) 1 nd 1 2k =2 n−1 m 1 =1 (d 2 2 + · · · + d 2 t ) n−m 1 n m 1 2 (n − m 1 )! 1 d 1 2k−2m 1 m 1 n 2k µ (1) ⊢m 1 (f µ (1) ) 2 µ (1) 1 m 1 2k (15) < 2 n−1 m 1 =1 (d 2 2 + · · · + d 2 t ) n−m 1 n m 1 2 (n − m 1 )! 1 d 1 2k−2m 1 m 1 n 2k e m 2 1 e − 2k m 1 .
The inequality in (15) follows from Lemma 3.3. As n ≥ m 1 , we have
k ≥ m 1 log m 1 + m 1 n k − m 1 log n =⇒ m 2 1 e − 2k m 1 ≤ n 2 e − 2k n .
Thus writing n − m 1 by u, the expression in (15) is less than or equal to
(16) 2e n 2 e − 2k n n−1 u=1 d 2 2 + · · · + d 2 t d 2 1 u 1 d 1 2k−2n n u 2 u! 1 − u n 2k Now using 1 − x ≤ e −µ (j) ⊢m j f µ (j) 2 µ (j) ′ 1 nd j 2k = µ (j) ⊢m j f µ (j) 2 µ (j) 1 nd j 2k
, the partial sum corresponding to 1 < j ≤ t in (14) turns out to be (18) n−1
m j =1 (m 1 ,..., m j ,...,mt) m k =n−m j 0≤m k ≤n−1 2 µ (i) ⊢m i 1≤i≤t n m j 2 n − m j m 1 , . . . , m j , . . . , m t 2 (f µ (1) ) 2 · · · (f µ (t) ) 2 d 2m 1 1 . . . d 2mt t ζ 2k , where ζ = µ (j) 1 −1 nd j .
The expression given in (18) is equal to the following
2 n−1 m j =1 (d 2 1 +· · ·+d 2 t −d 2 j ) n−m j n m j 2 (n−m j )! 1 d j 2k−2m j m j n 2k µ (j) ⊢m j (f µ (j) ) 2 µ (j) 1 − 1 m j 2k (19) < 2 n−1 m j =1 (d 2 1 + · · · + d 2 t − d 2 j ) n−m j n m j 2 (n − m j )! 1 d j 2k−2m j m j n 2k e − 2k m j e m 2 j e − 2k m j .
The inequality in (19) follows from Lemma 3.3. As n ≥ m j , we have
k ≥ m j log m j + m j n k − m j log n =⇒ m 2 j e − 2k m j ≤ n 2 e − 2k n .
Thus writing n − m j by v and using 1 m j ≤ 1, the expression in (19) is less than or equal to 1
(20) 2n 2 e − 2k n e n 2 e − 2k n n−1 v=1 d 2 1 + · · · + d 2 t − d 2 j d 2 j v 1 d j 2k−2n n v 2 v! 1 − v n 2k Now using 1 − x ≤ e −v! n 2 |G n | d 2 j − 1 e − 2k n v < 2n 2 e − 2k n e n 2 e − 2k n e n 2 |Gn| d 2 j −1 e − 2k n − 1 .(21)
Therefore the proposition follows from (11), (13), (17), (21) and 1 d j ≤ 1 for all 1 ≤ j ≤ t.
Lemma 3.7.
For the random walk on G n driven by P we have the following:
(1) Let C > 0. If k ≥ n log n + 1 2 n log(|G n | − 1) + Cn, then
P * k − U Gn TV ≤ 1 2 P * k − U Gn 2 < √ 2 e −2C + 1 e −C + o(1).
(2) For any ǫ ∈ (0, 1), if we set k n = (1 + ǫ) n log n + 1 2 n log (|G n | − 1) , then lim n→∞ P * kn − U Gn 2 = 0.
(3) For any ǫ ∈ (0, 1), if |G n | = o(n δ ) for all δ > 0, then
k ′ n = ⌊(1 + ǫ)n log n⌋ =⇒ lim n→∞ P * k ′ n − U Gn TV = 0.
Proof. Using k ≥ n log n + 1 2 n log(|G n | − 1) + Cn Proposition 3.6 we have the following:
4 P * k − U Gn 2 TV ≤ P * k − U Gn 2 2 < 2 e e −2C |Gn|−1 − 1 + e −4C n 4 (|G n | − 1) 2 + 2e e −2C |Gn|−1 e e −2C − 1 (22) + 2(| G n | − 1) × e −2C |G n | − 1 × e e −2C |Gn|−1 1 n 2 + e e −2C − 1
The sequence {G n } ∞ 1 consists of non-trivial finite groups, thus 1 |Gn|−1 ≤ 1. Also, | Gn|−1 |Gn|−1 ≤ 1. Therefore the expression in the right hand side of (22) is less than
2 + 2e e −2C + 2e −2C e e −2C e e −2C − 1 + e −4C n 4 + 2e −2C e e −2C n 2 Now using e x − 1 < 2x for 0 < x ≤ 1 we have 4 P * k − U Gn 2 TV ≤ P * k − U Gn 2 2 < 4 + 6e −2C + 4e −4C 2e −2C + o(1) =⇒ P * k − U Gn TV ≤ 1 2 P * k − U Gn 2 < √ 2 + 3e −2C + 2e −4C e −C + o(1) < √ 2 e −2C + 1 e −C + o(1).
This proves the first part of the lemma. For any ǫ ∈ (0, 1), setting k n = (1 + ǫ) n log n + 1 2 n log (|G n | − 1) , we have The right hand side of (23) converges to zero as n → ∞ because 1 |Gn|−1 , | Gn|−1 |Gn|−1 ≤ 1. Hence the second part follows.
k n + 1 ≥ (1 + ǫ) n log n + 1 2 n log (|G n | − 1) =⇒ e − kn n ≤ e 1 n n 1+ǫ (|G n | − 1)
Again, for any ǫ ∈ (0, 1), setting k ′ n = ⌊(1 + ǫ)n log n⌋, we have Now using the fact |G n | = o(n δ ) for all δ > 0, the right hand side of (24) converges to zero as n → ∞ because | Gn|−1 |Gn|−1 ≤ 1. Hence the third part follows.
k ′ n + 1 ≥ (1 + ǫ)n log n =⇒ e − k ′ n n ≤ e 1 n n 1+ǫ . Thus Proposition 3.6 implies 0 ≤ 4 P * k ′ n − U Gn 2 TV ≤ P * k ′ n − U Gn
Proof of Theorem 1.1. Let ε > 0 and τ (n) mix (ε) (respectively t (n) mix (ε)) be the ℓ 2 -mixing time (respectively total variation mixing time) with tolerance level ε for the warp-transpose top with random shuffle on G n . We choose C ε > 0 such that √ 2 e −2Cε + 1 e −Cε < ε 4 . Then the first part of Lemma 3.7 ensures the existence of positive integer N such that the following hold for all n ≥ N,
k ≥ n log n + 1 2 n log(|G n | − 1) + C ε n =⇒ P * k − U Gn TV ≤ 1 2 P * k − U Gn 2 < ε 2 =⇒ P * k − U Gn TV < ε and P * k − U Gn 2 < ε.
Finally, using n log n + 1 2 n log(|G n | − 1) + C ε n < 2 n log n + 1 2 n log(|G n | − 1) for all n ≥ N, we can conclude that τ (n) mix (ε) ≤ 2 n log n + 1 2
n log(|G n | − 1) and t (n) mix (ε) ≤ 2 n log n + 1 2 n log(|G n | − 1) .
Thus the theorem follows.
Proposition 3.8. For large n, we have
P * k − U Gn 2 > (n − 2 + n(|G n | − 1))(n − 1) e − k n .
Proof. Recall that the irreducible representations of G n are parameterised by the elements of Y n ( G n ). We now use Theorem 2.5 to compute the eigenvalues of the restriction of P (R) to some irreducible G n -modules. The eigenvalues of the restriction of P (R) to the irreducible G n -module indexed by
n − 1 · · · , φ, . . . , φ ∈ Y n ( G n )(25)
are given below.
Eigenvalues: 1 − 1 n 0 Multiplicities: n − 2 1
The eigenvalues of the restriction of P (R) to the irreducible G n -modules indexed by Young G n -diagram with n boxes of the following form
n − 1 · · · , φ, . . . , φ, ↑ , φ, . . . , φ ∈ Y n ( G n ), for 1 < i ≤ | G n | ith position.(26)
are given below.
Here 'a n ≈ b n ' means 'a n is asymptotic to b n ' i.e. an bn = 1 as n → ∞. Therefore (27) implies P * k − U Gn 2 > (n − 2 + n(|G n | − 1))(n − 1) e − k n for large n.
Proof of Theorem 1.2. For any ǫ ∈ (0, 1), the second part of Lemma 3.7 implies (28) lim
where E k (f) (respectively E k (f 2 )) denotes the expected value of f (respectively f 2 ) with respect to the probability measure P * k . Now recall the expressions for E k (f) and E k (f 2 ) obtained in [10].
E k (f) ≈ 1 + (n − 2)e − k n .
E k (f 2 ) ≈ 2 + 3(n − 2)e − k n + (n 2 − 5n + 5)e − 2k n + (n − 2) 1 + (−1) k n k .
Let us define a homomorphism f from G n onto S n as follows:
(34) f : (g 1 , . . . , g n ; π) → π, for (g 1 , . . . , g n ; π) ∈ G n .
It can be checked that the mapping f defined in (34) is a surjective homomorphism. Moreover, f projects the warp-transpose top with random shuffle on G n to the transpose top with random shuffle on S n i.e., P f −1 = P. We now prove a lemma which will be useful in proving the main result of this section. Proof. We use the first principle of mathematical induction on k. The base case for k = 1 is true by definition. Now assume the induction hypothesis i.e., (P f −1 ) * m = P * m f −1 for some positive integer m > 1. Let π ∈ S n be chosen arbitrarily. Then for the inductive step k = m + 1 we have the following:
Now using the fact that f is a homomorphism, we have the following:
{(ξ ′ , ζ ′ ) ∈ f −1 (ξ) × f −1 (ζ)
: ξ, ζ ∈ S n and ξζ = π} = {(ξ ′ , ζ ′ ) ∈ G n × G n : ξ ′ ζ ′ ∈ f −1 (π)}. Therefore the expression in (35) becomes {ξ ′ ,ζ ′ ∈Gn: ξ ′ ζ ′ ∈f −1 (π)} P (ξ ′ )P * m (ζ ′ ) = π ′ ∈f −1 (π) ξ ′ ,ζ ′ ∈Gn ξ ′ ζ ′ =π ′ P (ξ ′ )P * m (ζ ′ ) = π ′ ∈f −1 (π) P * (m+1) (π ′ ) = P * (m+1) f −1 (π).
Thus the lemma follows from the first principle of mathematical induction.
Lemma 4.2.
For the random walk on G n driven by P we have the following:
(1) For large n, ||P * k − U Gn || TV ≥ 1 − 2 (3 + 3e −c + o(1)(e −2c + e −c + 1)) (1 + (1 + o(1))e −c ) 2 , when k = n log n + cn and c ≪ 0. (2) For any ǫ ∈ (0, 1), lim n→∞ ||P * ⌊(1−ǫ)n log n⌋ − U Gn || TV = 1.
Proof. We know that, given two probability distributions µ and ν on Ω and a mapping ψ : Ω → Λ, we have ||µ−ν|| TV ≥ ||µψ −1 −νψ −1 || TV , where Λ is finite [11,Lemma 7.9]. Therefore we have the following: Now if n is large, c ≪ 0 and k = n log n + cn, then by (37), we have the first part of this lemma.
||P * k − U Gn || TV ≥ ||P * k f −1 − U Gn f −1 || TV = || P f −1 * k − U
Again for any ǫ ∈ (0, 1), from (37), we have (38) 1 ≥ ||P * ⌊(1−ǫ)n log n⌋ − U Gn || TV ≥ 1 − 2 (3 + 3n ǫ + o(1)(n 2ǫ + n ǫ + 1)) (1 + (1 + o(1))n ǫ ) 2 , for large n. Therefore, the second part of this lemma follows from (38) and the fact that lim n→∞ 2 (3 + 3n ǫ + o(1)(n 2ǫ + n ǫ + 1)) (1 + (1 + o(1))n ǫ ) 2 = 0. Theorem 1.3 follows from the third part of Lemma 3.7 and the second part of Lemma 4.2.
Proposition 1. 4 .
4The warp-transpose top with random shuffle on G n is irreducible and aperiodic.
For example there are five partitions of the positive integer 4 viz. . The Young diagrams corresponding to the partitions of 4 are shown in Figure 2.
Definition 2. 1 .
1Let Y denote the set of all Young diagrams (there is a unique Young diagram with zero boxes) and Y n denote the set of all Young diagrams with n boxes. For example, elements of Y 4 are shown inFigure 2. For a finite set X
Figure 2 .Figure 3 .
23Young Standard Young tableaux of shape(3,1).G-module (the space for the corresponding irreducible representation of G). Elements of Y( G) are called Young G-diagrams and elements of Y n ( G) are called Young G-diagrams with n boxes. Given µ ∈ Y( G) and σ ∈ G, we denote by µ ↓ σ the set of all Young G-diagrams obtained from µ by removing one of the inner corners in the Young diagram µ(σ). Let µ ∈ Y. A Young tableau of shape µ is obtained by taking the Young diagram µ and filling its |µ| boxes (bijectively) with the numbers 1, 2, . . . , |µ|. A Young tableau is said to be standard if the numbers in the boxes strictly increase along each row and each column of the Young diagram of µ. The set of all standard Young tableaux of shape µ is denoted by tab(µ). Elements of tab((3, 1)) are listed inFigure 3. Let µ ∈ Y( G). A Young G-tableau of shape µ is obtained by taking the Young G-diagram µ and filling its ||µ|| boxes (bijectively) with the numbers 1, 2, . . . , ||µ||. A Young G-tableau is said to be standard if the numbers in the boxes strictly increase along each row and each column of all Young diagrams occurring in µ. Let tab G (n, µ), where µ ∈ Y n ( G), denote the set of all standard Young G-tableaux of shape µ and let tab G (n) = ∪ µ∈Yn( G) tab G (n, µ). Let T ∈ tab G (n) and i ∈ [n]. If i appear in the Young diagram µ(σ), where µ is the shape of T and σ ∈ G, we write r T (i) = σ. The content of a box in row p and column q of a Young diagram is the integer q − p. Let b T (i) be the box in µ(σ), with the number i resides and c(b T (i)) denote the content of the box b T (i).
, µ(σ 2 )
2µ(σ 10 ) = and µ(σ i ) = φ for all i ∈ {3, 4, 5, 6, 7, 9}, where φ denotes the empty Young diagram (i.e. Young diagram with no boxes). Then for the element T of tab Z 10 (10, µ)
3 8
3, µ(σ 10 )5 and µ(σ i ) φ for i ∈ {3, 4, 5, 6, 7, 9}, we have the following: r T (1) = σ 2 , r T (2) = σ 2 , r T (3) = σ 8 , r T (4) = σ 1 , r T (5) = σ 10 , r T (6) = σ 1 , r T (7) = σ 1 , r T (8) = σ 8 , r T (9) = σ 1 ,r T (10) = σ 1 and c(b T (1)) = 0, c(b T (2)) = −1, c(b T (3)) = 0, c T (4) = 0, c T (5) = 0, c(b T (6)) = 1, c(b T (7)) = −1, c(b T (8)) = −1, c(b T (9)) = 2, c(b T (10)) = 0.Definition 2.2. Let H i,n (G) be the subgroup{(g 1 , . . . , g n , π)∈ G ≀ S n : π(j) = j for i + 1 ≤ j ≤ n} of G ≀ S n for 0 ≤ i ≤ n. In particular H 0,n (G) = H 1,n (G) = G n and H n,n (G) = G ≀ S n .Definition 2.3. The (generalized) Young-Jucys-Murphy elements X 1 (G), . . . , X n (G) of H n,n (G) or C[G ≀ S n ] are given by X 1 (G) = 0 and
Remark 2. 6 .
6In the regular representation of a finite group, each irreducible representation occurs with multiplicity equal to its dimension[19, section 2.4]. Therefore, Theorems 2.2 and 2.5 provide the eigenvalues of P (R).
Theorem 3. 1 (
1Plancherel formula, [4, Theorem 4.1]). Let f 1 and f 2 be two functions on the finite group G. Theng∈G
x for all x ≥ 0 and d 1 = 1 the expression in (16) is less than or Now using the notation m 1 , .., m j , .., m t to denote m 1 , . . . , m j−1 , m j+1 , . . . , m t , and
x for all x ≥ 0 and d 2k−2n j ≥ 1 for all j ∈ {1, . . . , t}, the expression in (20) is less than or equal to 2n 2 e − 2k n e n 2 e
n
2ε (|G n | − 1) 1+ε × e
≈
(n − 1)(n − 2)e − 2k n + n(n − 1)(|G n | − 1)e − 2kn .
Lemma 4. 1 .
1For any positive integer k we have (P f −1 ) * k = P * k f −1 .
P f −1 * (m+1) (π) = P f −1 * P f −1 * m (π) = {ξ,ζ∈Sn: ξζ=π} P f −1 (ξ) P f −1 * m (ζ) = {ξ,ζ∈Sn: ξζ=π} P f −1 (ξ) P * m f −1 (ζ),by the induction hypothesis, = ξ,ζ∈Sn ξζ=πP f −1 (ξ) P * m f −1 (ζ) = ξ,ζ∈Sn ξζ=π ξ ′ ∈f −1 (ξ) ζ ′ ∈f −1 (ζ) P (ξ ′ )P * m (ζ ′ ).
Sn || TV , by Lemma (4.1) and U Gn f −1 = U Sn ,= ||P * k − U Sn || TV , using P f −1 = P.(36)Therefore (31), (32), (33), and (36) implies that||P * k − U Gn || TV ≥1 − 2 3 + 3(n − 2)e − k n − 2(n − 1)e − 2k n + o(1)
. 1.2. Discrete time Markov chain with finite state space. Let Ω be a finite set. A sequence of random variables X 0 , X 1 , . . . is a discrete time Markov chain with state space Ω and transition matrix M if for all x, y ∈ Ω, all k > 1, and all events H k−1 := ∩ {X s = x s } satisfying P(H k−1 ∩ {X k = x}) > 0, we have0≤s<k
Definition 1.2.Let D k denote the distribution after k transitions of an irreducible discrete time Markov chain with finite state space Ω, and Π denote its stationary distribution. Then the ℓ 2 -distance between D k and Π is defined byThe period of all the states of an irreducible
Markov chain are the same [11, Lemma 1.6]. An irreducible Markov chain is said to be
aperiodic if the common period for all its states is 1. A probability distribution Π is said to
be a stationary distribution of the Markov chain if ΠM = Π. Any irreducible Markov chain
possesses a unique stationary distribution Π with Π(x) > 0 for all x ∈ Ω [11, Proposition
1.14]. Moreover if the chain is aperiodic then D k −→ Π as k −→ ∞ [11, Theorem 4.9]. For an
irreducible chain, we first define the ℓ 2 -distance between the distribution after k transitions
and the stationary distribution.
Theorem 2.2 ([13, Theorem 6.7]). Let µ ∈ Y n ( G). Write the elements of G as {σ 1 , . . . , σ t } and set µ
Email address: [email protected]
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[
"Cybersusy Solves the Cosmological Constant Problem",
"Cybersusy Solves the Cosmological Constant Problem"
]
| [
"John A Dixon [email protected] "
]
| []
| []
| Cybersusy is a new mechanism for SUSY breaking. When the auxiliary fields are integrated in any theory like the SSM, certain special new composite superfields ωα arise. Spontaneous breaking of internal symmetry, like SU(2) × U(1) ⇒ U(1), gives rise to a new realization of SUSY for ωα. This realization mixes elementary and composite states. In the resulting effective action, if ωα has mass, then there are SUSY anomalies. Since there are no massless supermultiplets, the SUSY anomalies must be present. They generate a spectrum for SUSY breaking that is consistent with the known particles. Supergravity does not couple to the anomalies because it does not couple to composite states. So unitarity is not violated. There is no cosmological constant generated, because SUSY is not spontaneously broken. 1 | null | [
"https://arxiv.org/pdf/1006.2334v2.pdf"
]
| 118,678,663 | 1006.2334 | f315057ead1bba0fe5ad0711b140857513e81fc8 |
Cybersusy Solves the Cosmological Constant Problem
11 Jun 2010
John A Dixon [email protected]
Cybersusy Solves the Cosmological Constant Problem
11 Jun 2010
Cybersusy is a new mechanism for SUSY breaking. When the auxiliary fields are integrated in any theory like the SSM, certain special new composite superfields ωα arise. Spontaneous breaking of internal symmetry, like SU(2) × U(1) ⇒ U(1), gives rise to a new realization of SUSY for ωα. This realization mixes elementary and composite states. In the resulting effective action, if ωα has mass, then there are SUSY anomalies. Since there are no massless supermultiplets, the SUSY anomalies must be present. They generate a spectrum for SUSY breaking that is consistent with the known particles. Supergravity does not couple to the anomalies because it does not couple to composite states. So unitarity is not violated. There is no cosmological constant generated, because SUSY is not spontaneously broken. 1
Why Cybersusy?
One of the tantalizing features of unbroken supersymmetry (SUSY) is that it solves the 'Cosmological Constant Problem' [1,2,3]. This happens because the effective potential P for unbroken SUSY has a zero Vacuum Expectation Value (VEV) [4,5]:
P = 0(1)
A further tantalizing fact is that this remains true even when the scalar field A i develops a VEV A i = 0 which spontaneously breaks an internal symmetry, so long as SUSY itself is not spontaneously broken [6].
SUSY is spontaneously broken whenever an auxiliary field, such as F i (expressed as a function of A i ), develops a VEV F i (A) = 0. In that case, (1) becomes typically:
P = F i F i ≈ (100 GeV) 4 .(2)
This yields a huge cosmological constant Λ, because of the coupling of P to supergravity:
d 4 x P √ −g → d 4 x P √ −g ⇒ Λ d 4 x √ −g(3)
The Cosmological Constant Problem is that the maximum experimentally possible value [1,2,3] for Λ is about ρ crit ≈ 6 × 10 −47 GeV 4 , whereas (2) yields a value that is 10 54 times too large.
SUSY is certainly not observed, so it must be broken. The usual assumption is that SUSY is spontaneously broken. However, as discussed below, Cybersusy breaks SUSY in a way that is not spontaneous. As a result, the VEV of the potential remains zero as in (1), and the Cosmological Constant Problem does not arise for Cybersusy.
How does Cybersusy arise?
Cybersusy starts with the simplest interacting SUSY action in 3+1 dimensions. This is an action with a set of chiral scalar superfields, interacting with each other through trilinear interactions, with no mass terms. The first step is to integrate the auxiliary fields F i . Although this integration is just a simple Gaussian integration in the path integral, after the completion of a square, it should be remembered that the result is exact and non-perturbative in nature.
Much of the information in the resulting non-linear SUSY theory is encapsulated in the BRST cohomology [7,8,9] of a local nilpotent functional derivative operator δ BRST . This cohomology can be analyzed using spectral sequences [10,11,12,16,17].
The analysis in [13] shows that there are certain composite fields ωα, which behave like chiral dotted spinor superfields. A short way to express this is:
δ BRST ωα = δ SS ωα (4)
where δ SS has the usual 'SuperSpace' form [4,5]:
δ SS = C β Q β + CβQβ(5)
The next step is to reconsider the action with a mass term and an assumption that some scalar superfield develops a VEV which spontaneously breaks some internal symmetry, but without spontaneously breaking SUSY itself [6].
Then there is a new nilpotent realization [13] of SUSY of the form:
δ BRST ωα = δ SS ωα + bm 2Â Cα (6) δ BRSTÂ = δ SSÂ(7)
where the constant b is proportional to the square of the VEV that breaks the internal symmetry, m is a mass parameter, and is one of the chiral superfields in the original theory.
Solutions and Mixing in the SSM
There are algebraic constraints that relate to the construction of the ωα, and to the generation of the algebra (6). For example, in the Leptonic sector of the SSM, these constraints yield composite superfields ωα made of Lepton chiral superfields multiplied by Higgs/Goldstone superfields 2 . The equation (6) applies when SU(2) × U(1) is spontaneously broken down to U(1), and the fields are the corresponding elementary Lepton chiral superfields. Quarks work the same way.
Effective Action and SUSY Anomalies
The hypothesis behind Cybersusy is that the composite superfields ωα correspond to bound states, and that they should be replaced by effective elementary superfields to explore the algebra (6). Effective ωα superfields have dimension m 1 2 , so the new Cybersusy algebra [13] for the effective theory is:
δ CS = δ SS + δ MIX(8)
where δ MIX ωα = b Cα.
It is straightforward to write down a supersymmetric action for the ωα for the case where b = 0 in equation (9). The action for the chiral scalar superfields is, of course, well known [4,5]. The action for the superfield ωα represents a massive supermultiplet with mass m ω including spin 1 2 , spin 1 and spin zero particles, mixed in an unusual way with an unusual propagator [13,15]. The mass term for ωα is:
A Mass ω = m 2 ω d 4 x d 2 θ ωα ωα(10)
When the VEV arises, making b = 0, an attempt to make the effective action invariant under (8) meets an impasse. One needs to compensate for the variation of (10) under (9):
δ MIX A Mass ω = − 2bm 2 ω d 4 x d 2 θ ωα Cα(11)
But this term (11) has exactly the form of the anomalous terms found in the early SUSY cohomology papers [12,16,17]. This means that it is impossible to generate this term by the variation of any local term with (5).
The superspace invariant kinetic term for ωα is of the form:
A Kinetic ω = d 4 x d 4 θ ωα ∂ αα ω α(12)
Terms can be added to (12) so that the result is invariant under (8). This construction works because the relevant variations of (12) with (9) are not in the cohomology space of (5). A simple 3 form for this is:
A CS Kinetic ω = d 4 x d 4 θ ωα + b Aθα ∂ αα ω α + b Aθ α(13)
So the SUSY breaking involves both b and m ω :
1. If b = 0 and m ω = 0, then the action is the superspace invariant action and there are two supermultiplets, one from ωα with mass m ω , and one from with mass m A .
2. If b = 0 and m ω = 0, then SUSY is still unbroken. This is easy to see, because one can change (13) back to (12) by changing variables 4
ωα ⇒ ωα − b Aθα.(14)
3. If b = 0 and m ω = 0, the change of variables (14) is not available because (10) is a chiral integral. This is equivalent to the existence of the SUSY anomaly of the form (11). So one gets a spectrum for broken SUSY.
If m ω = 0, there is a massless supermultiplet of charged Leptons, in sharp contradiction with experiment. The only choice that has a chance to be consistent with experiment is to keep m ω = 0 and b = 0.
In the above, we have oversimplified things a little. In the SSM there are actually Left and Right Superfields, which enables the theory to preserve Lepton number. When these details are added, it is easy to arrange for the Electron to be very light compared to the other members of its broken supermultiplet. The Neutrinos and Quarks work the same way.
Cosmological Constant, Unitarity and Double Counting
The presence of anomalies in a theory normally signals the breakdown of unitarity and consistency [18]. However it is easy to see that this does not happen for Supergravity with SUSY breaking from Cybersusy.
The SUSY anomalies relate to the ωα superfields. But since ωα are composite, we cannot validly couple them directly to supergravity. That would be double counting.
So the correct theory to couple to supergravity is simply the usual SSM with three Higgs superfields H i , J, K i . These spontaneously break SU(2) × U(1) ⇒ U(1), without spontaneous breaking of SUSY, as set out in [13].
At first glance, it seems preposterous to claim that SUSY is broken in this theory. However the situation is simply an extension of ideas that are common in QCD [2] [19] . The hadrons (and the ωα) are not explicit in the SSM action that is coupled to gravity. The hadrons are implicit bound states 5 , and so are the ωα .
It is clear that the Cosmological Constant Problem is not present for this action, because SUSY is not spontaneously broken.
Signature for Cybersusy
One clear signature of Cybersusy is that there should be a Vector Boson Electron at high mass. But this vector boson is not a gauge boson, because it arises from the ωα supermultiplet, which has nothing to do with a gauge multiplet. The same scenario carries through for the Neutrinos and Quarks.
Conclusion
Cybersusy is unavoidable for this H i , J, K i version of the SSM, and it does resolve some problems:
1. The effective ωα superfields have actions which describe massive supermultiplets.
2. The SUSY breaking comes from SUSY anomalies of the ωα superfields.
3. Cybersusy is well adapted to the SSM.
4. The Cybersusy breaking spectrum is consistent with experimental results so far, for Leptons and Quarks at least.
5.
Even though the SUSY breaking is anomalous, no violation of unitarity arises because supergravity does not couple directly to the composite states.
6. Cybersusy breaking of SUSY happens when SU(2) × U(1) spontaneously breaks to U(1), but no spontaneous breaking of SUSY is needed.
7. As a result of item 6 above, the Cosmological Constant Problem does not arise.
But Cybersusy also raises many new questions. Two important and difficult questions are:
1. Is there any way to constrain the magnitude of the breaking parameters b and the mass parameters m ω , to get some phenomenological predictions?
2. How can one add interactions to the quadratic actions for Cybersusy?
Preliminary results indicate that the Hadron, Higgs and Gauge Supermultiplets also get SUSY breaking spectra that do not conflict with known experimental results. This requires detailed, but straightforward, analysis.
See the remarks at[13] for an example of ωα.
This 'broken superspace' form is new, but 'equivalent' to the expression in[13]. The invariance of(13) under(8) can be shown using integration by parts for θ, θ. This construction cannot work for (10), because (10) is a chiral integral.4 This is a dubious transformation because it takes a chiral superfield into one that is not chiral. These results are best established using components as in[13]. But, as usual, superfield notation is easier to follow.
Cybersusy has hadronic supermultiplet towers which break SUSY, but the spectrum is not yet known[14].
This is the classic introduction to the Cosmological Constant Problem , and it also discusses SUSY in this context. Steven Weinberg, Rev. Mod. Phys. 611Steven Weinberg, Rev. Mod. Phys. 61, 1 (1989). This is the classic introduction to the Cosmological Constant Problem , and it also discusses SUSY in this context.
A recent paper on this subject which also refers to various reviews and analysis. J Stanley, Robert Brodsky, Shrock, Standard Model Condensates and the Cosmological Constant': arXiv 0803.2554. hep-thA recent paper on this subject which also refers to various reviews and analysis is Stanley J. Brodsky and Robert Shrock: 'Standard Model Condensates and the Cosmological Constant': arXiv 0803.2554 [hep-th]
Dark Energy and the Accelerating Universe. Joshua A Frieman, Michael S Turner, Dragan Huterer, arXiv:0803.0982Ann. Rev. Astron. Astrophys. 46astro-phA recent review is Joshua A. Frieman, Michael S. Turner, Dragan Huterer, Dark En- ergy and the Accelerating Universe, Ann. Rev. Astron. Astrophys. (2008) 46, also at arXiv:0803.0982[astro-ph].
SUSY and supergravity are collected in Supersymmetry, Vols. 1 and 2. Sergio Ferrara, North HollandWorld ScientificMany of the original papers on SUSY and supergravity are collected in Supersymmetry, Vols. 1 and 2, ed. Sergio Ferrara, North Holland, World Scientific, (1987)
Useful techniques are contained in S. M J T Gates, M Grisaru, W Rocek, Siegel, Superspace. Useful techniques are contained in S. J. Gates, M. T. Grisaru, M. Rocek and W. Siegel, Superspace, Benjamin, 1983.
This is the normal behaviour of pure chiral theories, as was shown by L. O'Raifeartaigh: Nucl. Phys. B96 (1975) 331-352. 1411This paper isThis is the normal behaviour of pure chiral theories, as was shown by L. O'Raifeartaigh: Nucl. Phys. B96 (1975) 331-352. This paper is included in Vol. 1 of [4] at p. 411.
. C Becchi, A Rouet, R Stora, Commun. Math. Phys. 42127C. Becchi, A. Rouet, and R. Stora, Commun. Math. Phys. 42 (1975) 127.
. I V Tyutin, P N Preprint, arXiv:0812.0580Lebedev Physical Institute. 39hep-thI. V. Tyutin, Preprint P.N. Lebedev Physical Institute, No 39, (1975); now at arXiv:0812.0580 [hep-th]
. Friedemann Glenn Barnich, Marc Brandt, Henneaux, Physics Reports. 3385A review of BRST is: Glenn Barnich, Friedemann Brandt, Marc Henneaux, Physics Reports Vol 338, No 5, pp 439-569.
For a modern introduction to spectral sequences in general, see: John McCleary, A User's Guide to Spectral Sequences. Cambridge University PressSecond EditionFor a modern introduction to spectral sequences in general, see: John McCleary, A User's Guide to Spectral Sequences, Cambridge University Press, Second Edition (2001).
. J A Dixon, Commun. Math. Phys. 139J. A. Dixon, Commun. Math. Phys. 139, 495-526 (1991).
. Ibid, Commun. Math. Phys. 140Ibid., Commun. Math. Phys. 140, 169-201 (1991)
Ibid, arXiv:0908.0889AIP Conference Proceedings. George Alverson, Pran Nath, Brent NelsonBoston (Massachusetts1200at P 1085. This short article is a summary of the results in [14Ibid., arXiv:0908.0889 now published in SUSY 2009: AIP Conference Proceedings Vol- ume 1200, Boston (Massachusetts), Eds: George Alverson, Pran Nath, Brent Nelson, at P 1085. This short article is a summary of the results in [14].
An unreported but important piece of progress since this paper is that the left handed Lepton dotspinor should be represented by ω p Liα = g −1L p iψJα + (p −1 ) pq (mk i + K i )ψ P qα − (r −1 ) pq (mh i + H i )ψ Rqα. With this change in equation (4) of [13] , the spectrum described in section 8 of [13] can be derived from the SSM, without the problem mentioned in section 7 of [13] . The Quarks work similarlyAn unreported but impor- tant piece of progress since this paper is that the left handed Lepton dotspinor should be represented by ω p Liα = g −1L p iψJα + (p −1 ) pq (mk i + K i )ψ P qα − (r −1 ) pq (mh i + H i )ψ Rqα . With this change in equation (4) of [13] , the spectrum described in section 8 of [13] can be derived from the SSM, without the problem mentioned in section 7 of [13] . The Quarks work similarly.
These five papers need some revisions and clarifications, but they are largely still relevant. Ibid, arXiv:0908.0889[13The first papers on Cybersusy. hep-th. Related work is in [15Ibid., The first papers on Cybersusy are in arXiv: 0808-0811, 0808-2263, 0808- 2276, 0808-2301, 0808.3749 [hep-th]. A summary of the most crucial material is in arXiv:0908.0889 [13]. These five papers need some revisions and clarifications, but they are largely still relevant. Related work is in [15].
This paper discusses various aspects of chiral dotted spinor superfields. Preliminary results indicate that the Baryons, Hadronic Mesons, Gauge and Higgs/Goldstone sectors get SUSY breaking too, but using multi-dotted spinor chiral superfields likeÂα 1α2 and ωα 1α2α3 . The spectrum here is not yet known. 0911.0199hep-thIbid., 0911.0199 [hep-th]. This paper discusses various aspects of chiral dotted spinor superfields. Preliminary results indicate that the Baryons, Hadronic Mesons, Gauge and Higgs/Goldstone sectors get SUSY breaking too, but using multi-dotted spinor chiral superfields likeÂα 1α2 and ωα 1α2α3 . The spectrum here is not yet known.
. J A Dixon, R Minasian, Commun. Math. Phys. 172J. A. Dixon and R. Minasian, Commun. Math. Phys. 172, 1-11 (1995)
. J A Dixon, R Minasian, J Rahmfeld, Commun. Math. Phys. 171J. A. Dixon, R. Minasian and J. Rahmfeld, Commun. Math. Phys. 171, 459-473 (1995)
Steven See, Weinberg, The Quantum Theory of Fields. Cambridge University Press2See, for example, Steven Weinberg, 'The Quantum Theory of Fields', Cambridge Uni- versity Press, Volume 2 (1998).
A useful discussion of QCD etc. is in J. C. Taylor, Hidden Unity in Nature's Laws. Cambridge University PressA useful discussion of QCD etc. is in J. C. Taylor, Hidden Unity in Nature's Laws, Cambridge University Press (2001).
. L At E X3046, L AT E X3046
| []
|
[
"ON RESTRICTED ANALYTIC GRADIENTS ON ANALYTIC ISOLATED SURFACE SINGULARITIES",
"ON RESTRICTED ANALYTIC GRADIENTS ON ANALYTIC ISOLATED SURFACE SINGULARITIES"
]
| [
"Vincent Grandjean ",
"Fernando Sanz "
]
| []
| []
| Let (X, 0) be a real analytic isolated surface singularity at the origin 0 of a real analytic manifold (R n , 0) equipped with a real analytic metric g. Given a real analytic function f0 : (R n , 0) → (R, 0) singular at 0, we prove that the gradient trajectories for the metric g| X\0 of the restriction (f0|X ) escaping from or ending up at the origin 0 do not oscillate. Such a trajectory is thus a sub-pfaffian set. Moreover, in each connected component of X \ 0 where the restricted gradient does not vanish, there is always a trajectory accumulating at 0 and admitting a formal asymptotic expansion at 0. | 10.1016/j.jde.2013.05.020 | [
"https://arxiv.org/pdf/1105.3888v1.pdf"
]
| 119,571,686 | 1105.3888 | d49628ffd7c1ba5506789e6a894e4de6d20755d6 |
ON RESTRICTED ANALYTIC GRADIENTS ON ANALYTIC ISOLATED SURFACE SINGULARITIES
19 May 2011
Vincent Grandjean
Fernando Sanz
ON RESTRICTED ANALYTIC GRADIENTS ON ANALYTIC ISOLATED SURFACE SINGULARITIES
19 May 2011
Let (X, 0) be a real analytic isolated surface singularity at the origin 0 of a real analytic manifold (R n , 0) equipped with a real analytic metric g. Given a real analytic function f0 : (R n , 0) → (R, 0) singular at 0, we prove that the gradient trajectories for the metric g| X\0 of the restriction (f0|X ) escaping from or ending up at the origin 0 do not oscillate. Such a trajectory is thus a sub-pfaffian set. Moreover, in each connected component of X \ 0 where the restricted gradient does not vanish, there is always a trajectory accumulating at 0 and admitting a formal asymptotic expansion at 0.
Introduction
Let f 0 : (R n , 0) → R be a real analytic function such that 0 is a critical point of f 0 . Let g be a real analytic Riemannian metric defined in a neighborhood of 0. Let γ : [0, +∞[→ R n be a maximal solution of the gradient vector field ∇ g f 0 such that ω(γ) := lim t→∞ γ(t) = 0, and let | γ | ⊂ R n be its image. We are not interested in any particular parameterization and we will simply call γ and | γ | a gradient trajectory. Gradient trajectories γ :] − ∞, 0] → R n escaping from 0 = lim t→−∞ γ(t) will be dealt with in same way in changing the sign of f 0 .
The classical problem of the gradient is to know how, from an analytic point of view, does the solution | γ | go to its limit point 0. For a long time remained undecided Thom's question famously known as Thom's Gradient Conjecture: does the trajectory have a tangent at its limit point, namely does lim t→∞ γ(t) |γ(t)| exist ? (see [20] for an historical account by then). Eventually Kurdyka, Mostowski and Parusiński showed that the length of the radial projection of the curve | γ | onto S n−1 is finite [17], thus proving Thom's Conjecture.
A much more challenging question about the behavior of gradient trajectories at their limit point is to decide whether they oscillate or not. A trajectory γ is (analytically) non-oscillating if given any (semi-)analytic subset H ⊂ R n the intersection | γ | ∩ H has finitely many connected components.
The plane case is well understood. In dimension n ≥ 3, but a few special cases in dimension 3 [24,9,10], the non-oscillation of gradient trajectories is not known.
It is also worth recalling that in the case of real analytic vector fields on a 3-manifold, some very interesting properties of the Hardy field of the real analytic function germs along a given non-oscillating trajectory have been studied in [5], and thus allowing a partial reduction of the singularities result.
In the special case where a real analytic isolated surface singularity is foliated by gradient trajectories, the main result of this paper guarantees, that they do not oscillate. In fact, we will solve the following slightly more complicated problem.
Let X ⊂ R n be a real analytic isolated surface singularity at the origin 0. Each connected component S 0 of the germ at 0 of X \ {0} is a real analytic submanifold of R n . The ambient metric g induces on S 0 an analytic Riemannian metric h := g| S 0 . The gradient vector field ∇ h (f 0 | S 0 ) of the restriction f 0 | S 0 of the function f 0 to S 0 is thus well defined. The vector field ∇ h (f 0 | S 0 ) is called the restricted gradient vector field of f 0 on S 0 and will be shortened as ∇ h (f 0 ).
The main result of this paper is the following:
Theorem 1. Let γ : R ≥0 → S 0 be a trajectory of the restricted gradient vector field ∇ h f 0 accumulating at 0. Then γ is analytically non-oscillating.
A pleasant and cheap consequence of this result is
Corollary 2. The curve |γ| is a sub-pfaffian set.
A natural question is to ask whether there exists a trajectory γ of the restricted gradient accumulating to the origin to apply the main theorem to. Elementary topological arguments and some properties of a gradient vector field show that it is always the case: Proposition 3. There exists a non-stationary trajectory of ∇ h f 0 accumulating to 0 either in positive or in negative time.
It is worth recalling that in the smooth context of an analytic gradient vector field on (R n , 0), there exists furthermore a real analytic curve through 0 invariant for the gradient vector field [20], a real analytic separatrix). For restricted gradients over isolated surface singularities we also prove here there always exists a formal separatrix: If ∇ h f 0 does not vanish S 0 , there exists a trajectory γ : R ≥0 → S 0 of ∇ h f 0 accumulating to 0 which admits a formal asymptotic expansion at the origin such that the associated formal curve Γ is invariant for the restricted gradient vector field.
Structure of the proof
We first recall the case of an analytic Euclidean gradient in R 2 . Trajectories of a real analytic vector field in R 2 accumulating at the origin either "spiral" around the origin or have a tangent. In the latter case, a Rolle's type argument shows that the trajectory is non-oscillating (see [6]). The non-oscillation of a planar analytic gradient trajectory is thus given by the existence of a tangent. Although Thom's Gradient conjecture holds true ( [17]), we sketch the usual simpler proof of the existence of a tangent in the plane case. This will provide a flavor of some of the arguments that makes our proof of Theorem 1 works.
Let (r, ϕ) be the polar coordinates at the origin of R 2 and write f 0 (r cos ϕ, r sin ϕ) = r k [F k (ϕ) + O(r)]
where F k (ϕ) is the restriction to the unit circle of the homogeneous part of f 0 of least degree. The gradient differential equation becomes a differential equation on S 1 × R ≥0 and, after division by r k−1 , writes as (1)ṙ = r(kF k + O(r)) andφ = F ′ k + O(r).
Since F k is not identically zero, when it is constant we divide Equation (1) by r and find that the divided vector field is transverse to C = S 1 × 0 at each point (dicritical case). If F k is not constant, then F ′ k must vanish and change sign along the circle C. This prevents any gradient trajectory from accumulating on the whole bottom circle C (non-monodromic case). In both cases, dicritical and non-monodromic, a plane gradient trajectory does not spiral around its limit point, therefore it has a tangent and thus does not oscillate.
The plane case is enlightening enough to provide us with some of the elements we need to prove Theorem 1. The surface S 0 on which we want to understand the behavior of the restricted gradient trajectories at their limit point 0, is analytically diffeomorphic to a cylinder S 1 ×]0, ε]. We can carry the metric h over this cylinder, so that we have a well defined gradient differential equation. Our concern then becomes: how does a trajectory of this differential equation behave near the bottom circle C = S 1 × 0? There is no canonical way to extend the inverse of the diffeomorphism onto S 1 ×[0, ε], and soà-priori, our differential equation is not well defined on the bottom circle C, if defined at any point of it! Nevertheless, we manage to prove that a limit dynamics exists on the circle C but at finitely many points. We first show that our setting only allows a single possible type of oscillation, that we call spiraling. To keep up with the planar situation, we actually prove that the only possible dynamics of the restricted gradient vector field will either be dicritical-like or non-monodromic-like (see Section 5 for precise definitions). Consequently, trajectories cannot spiral and will therefore be non-oscillating.
In Section 3, Proposition 14 provides a systematic way to parameterize clos(S 0 ) as the surjective image of a continuous mapping defined on S 1 × [0, ε] which induces an analytic diffeomorphism between the open cylinder S 1 ×]0, ε] and S 0 . Such a parameterization is inherited from the resolution of singularities of the analytic surface X and thus comes with some very specific properties on the bottom circle C = S 1 × 0. In Section 4, we use such a parameterization to express the pull-back of the restriction of the function f 0 to S 0 , as well as the corresponding gradient vector field, in polar-like coordinates (ϕ, r) ∈ S 1 × [0, ǫ] as in Equation (1). We obtain a continuous principal part along the bottom circle that will play a similar role to that of the principal part F k in (1). Section 5 deals with the oscillating dynamics of a given real analytic vector field on an isolated surface singularity (such as S 0 ) and vanishing at the tip, which can only be spiraling around this singular point, as we have already suggested. Although of an independent nature, we use the results of the previous sections for the proof. We also describe two local dynamical situations we call "dicritical" and "non-monodromic", generalizing the planar smooth case, and show here that such dynamics are non-oscillating. Our notion of "dicritical-ness": there exists an arc of the bottom circle C such that each point is the ω-limit point of a unique trajectory, is weaker than the usual notion requiring transversality to the exceptional divisor (here the bottom circle). Our notion of "non-monodromic-ness" is also weaker than the notion stated above: the function playing the role of F k in Equation (1), is continuous, not constant but can fail to be differentiable at finitely many points of C. The proof of Theorem 1 is done in Section 6. It uses all the main results of Sections 3, 4 to obtain a differential equation on a cylinder S 1 × [0, ε] which is analytic on S 1 ×]0, ε]. Although there is a slight cost, namely a finite subset of the bottom circle where the differential equation is likely to be not defined, we know enough about it to show that only the dicritical or non-monodromic situations happen. Section 5 then guarantees the nonoscillation of the restricted gradient trajectories. The last section deals with two not-so-unexpected consequences of our main result, Corollary 2 and Theorem 4.
Parameterization of real analytic surfaces
Let X be the germ, at the origin 0 of R n , of a real analytic surface of pure dimension 2. We will not distinguish between the germ of X at 0 and a representative in a sufficiently small neighborhood of 0. Assume that the surface X has an isolated singularity at the origin, that is X \ {0} is a smooth embedded analytic surface of R n .
Let S 0 be a given connected component of the germ at 0 of the regular part X \{0}. The tangent cone of S 0 at 0 ∈ R n is the subset of S n−1 made of the limits of the oriented secant direction p k |p k | taken along sequences of points (p k ) k in S 0 converging to 0. The tangent cone C 0 (S 0 ) is a compact connected subanalytic subset of S n−1 of dimension at most one. We distinguish two cases: -If C 0 (S 0 ) reduces to a single point, we will speak about the cuspidal tangent cone case (CTC for short).
-If C 0 (S 0 ) is a curve we will speak of the open tangent cone case (OTC).
For any ε > 0 sufficiently small, the Local Conic Structure Theorem (see [19,2,27]) states that X is homeomorphic to the cone with vertex 0 over
X ε = X∩S n−1 ε , where S n−1 ε
is the Euclidean sphere of radius ε. Moreover, the surface X is transverse to S n−1 ε so that S 0 ∩S n−1 ε is analytically diffeomorphic to S 1 and S 0 ∩ clos(B(0, ε)) is analytically diffeomorphic to S 1 ×]0, ε].
Definition 5. Assume C 0 (S 0 ) consists of the single oriented direction η ∈ S n−1 . A system of analytic coordinates (x, z) = (x 1 , . . . , x n−1 , z) at 0 is called adapted for S 0 if the half-line R + η is the non-negative z-axis.
Given adapted coordinates (x, z) in the CTC case, taking the height function z instead of the distance function, the proof of the Local Conic Structure's Theorem adapts to obtain the same conclusion: the intersection S 0 ∩ {z = ε} is transverse, thus analytically diffeomorphic to S 1 and
S 0 ∩ {0 < z ≤ ε} is analytically diffeomorphic to S 1 ×]0, ε] for 0 < ε ≤ ε 0 once ε 0 is sufficiently small.
From now on, we fix some ε 0 so that in both cases OTC or CTC, the above properties coming from the locally conic structure are satisfied. We consider a representative of S 0 in {0 < z < ε 0 }, where z stands for the distance to the origin in the OTC case and for the last component of an adapted system of coordinates in the CTC case.
In what follows we will desingularize the surface S 0 . First, it will be convenient for us to open the surface S 0 by means of a single blowing-up-like mapping β. Roughly speaking, we mean that the inverse image of S 0 by β accumulates to a one-dimensional set in the exceptional divisor. In the OTC case, the usual polar blowing-up β : (y, r) → ry, for y ∈ S n−1 and r the distance function, "opens" the surface S 0 , since β −1 (S 0 ) accumulates onto C 0 (S 0 ) ⊂ S n−1 , a subanalytic curve. The CTC case, however, requires more work. Starting with an adapted system of coordinates (x, z), a first and naive candidate mapping to "open" the surface is a "ramified blowing-up" of the form β s : (y, w) → (w s y, w), where y ∈ R n−1 , for a well chosen rational exponent s > 1. Such an exponent s exists when the z-axis is contained in the surface S 0 . However the surface β −1 s (S 0 ) may accumulate to a single point on the divisor β −1 s (z = 0) (or escapes to infinity) whatever the exponent s is. In such a case the surface S 0 cannot be opened with any such ramified blowing-up. In this situation, we consider a given analytic half-branch on S 0 as new non-negative z-axis, and in these new coordinates, a ramified blowing-up as above will open the surface. The next technical lemma will detail such considerations. First, an analytic half-branch at the origin 0 of R n is the germ at 0 of a connected component Γ of Y \ {0}, where Y is a one-dimensional analytic set through 0. When Γ is contained in {z > 0}, it is parametrized as the image of an analytic mapping z → (θ(z), z N ), z > 0, where θ = (θ 1 , . . . , θ n−1 ) :] − ε, ε[→ R n−1 is analytic with θ(0) = 0 and N is a positive integer. Lemma 6. Assume the tangent cone C 0 (S 0 ) is reduced to a point. Let (x, z) be adapted analytic coordinates at 0. (i) There is a unique rational number ν > 1 such that the accumulation set of the mapping S 0 ∋ (x, z) → x z ν ∈ R n−1 is a bounded subset of R n−1 and contains a point that is not (0, . . . , 0). (ii) There exists a unique positive rational number e ≥ ν such that the accumulation set of the mapping S 0 × S 0 ∋ ((x, z), (y, z)) → |x−y| z e ∈ R is a bounded subset of R containing a positive number. (iii) Let Γ : z → (θ(z), z N ) be a real analytic half-branch at 0 such that Γ ⊂ S 0 . Then, the set of accumulation values of the mapping τ e,Γ : S 0 → R n−1 ,
(x, z) → x−θ(z 1/N ) z e
is a connected bounded subanalytic set of dimension 1.
Proof. The uniqueness of ν and e are clear. For (i), let h(z) := sup{|x| for (x, z) ∈ S 0 }. The function h is subanalytic and extends continuously to z = 0 by h(0) = 0. Writing it as a Puiseux's series h(z) = az ν +· · · with a = 0, the exponent ν satisfies the required properties: the cuspidal nature of S 0 and the definition of adapted coordinates imply that ν > 1. We show the existence of e of point (ii) similarly to point (i): We take this time the function h to be defined as h(z) := sup{|x − y| for (x, z), (y, z) ∈ S 0 }. For (iii), let Λ be the set of accumulation values of the mapping τ = τ e,Γ . Since Γ is contained in S 0 , the origin 0 of R n−1 is in Λ. By definition of the exponent e of point (ii), Λ is bounded and contains a point p = 0. The connectedness and subanalyticity of Λ follow from the connectedness of S 0 and the subanalyticity of τ .
Remark 7. The numbers ν, e of Lemma 6 depend on the adapted system of coordinates. Take in R 3 the revolution surface x 2 + y 2 − z 5 = 0, then e = ν = 5/2. Consider now the change of coordinates (x ′ , y ′ , z ′ ) = (x + z 2 , y, z), then e ′ = ν ′ = 2.
The next result synthesizes the discussion about the possible opening of the surface S 0 by a single blowing-up-like mapping. Its proof follows from Lemma 6.
Proposition 8. In the OTC case, let M = S n−1 ⊂ R n with coordinates y = (y 1 , . . . , y n ). In the CTC case, let M = R n−1 with coordinates y = (y 1 , . . . , y n−1 ). Let (x, z) be adapted coordinates for S 0 at 0 and let e ∈ Q >1 be the exponent of point (ii) in Lemma 6 for these adapted coordinates and let z → (θ(z), z N ) be a parametrization of an analytic half-branch Γ in S 0 such that eN ∈ N. Consider the following analytic mapping A mapping β as in (2) is called an opening blow-up of clos(S 0 ). In the CTC case, β depends on the adapted system of coordinates, on the given curve Γ on S 0 and on the number N in the parametrization of Γ. As we will see, the choice of all these parameters will not matter for our purpose, so we do not need the notation β to carry these parameters.
(2) β : M × [0, ε 0 ] → R n (y, z) → zy, OTC case, (z eN y + θ(z), z N ), CTC case
For the rest of this section, assume that we have picked an opening blowup β of the surface S 0 . A key element in our result relies on the construction of an explicit diffeomorphism between S and the open cylinder S 1 ×]0, ε 0 ], which extends to a global parameterization of clos(S) = S ∪ E: a surjective continuous mapping Φ : S 1 × [0, ε 0 ] → clos(S). For this purpose, we first resolve the singularities of the surface clos(S), also providing a resolution of the singularities of clos(S 0 ) (up to ramification). Several formulations are possible. The version we use is stated in the following theorem, an avatar of the general theory on reduction of singularities of real analytic space as found in Hironaka & Al. [13,1] (see also [4]).
Theorem 9 (Reduction of singularities of S). There exists a non-singular real analytic surface S, a normal crossing divisor E ⊂ S and a proper analytic mapping σ :
S → U where U is an open neighborhood U of E in M × R such that: (i) clos(S) ∩ U ⊂ σ( S) and σ −1 (E) ⊂ E, (ii) S ′ = σ −1 (S)
is an open submanifold of S and the restricted mapping
σ| S ′ : S ′ → S ∩ U is an isomorphism, (iii) If E ′ = clos(S ′ ) ∩ E, then E ′ = clos(S ′ ) \ S ′ , it is a compact suban- alytic connected curve of S and σ(E ′ ) = E. (iv) If p ∈ E ′ , there is a fundamental system of neighborhoods {W k } of p
in S such that any connected component of W k \ E is either contained in S ′ or has empty intersection with S ′ .
Proof. Let X 1 = clos(β −1 (X \ {0})) be the strict transform of X by the opening blowing-up β and let Z = (X 1 ∪ D) ∩ U on some open neighborhood U of E in M × R. The general reduction of singularities applied to the real closed analytic set Z states there exists a proper surjective analytic mapping π : M → U , composition of finitely many blowing-ups with closed analytic smooth centers, such that the total transform π −1 (Z) has only normal crossings. Moreover, the smooth centers of blowing-ups are chosen either to be contained in the singular locus of the corresponding strict transform of Z or in the divisors created along the resolution process.
Since sing(Z) ∩ clos(S) ⊂ D, the mapping π induces an isomorphism from π −1 (U \ D) onto U \ D. Let S be the irreducible component of π −1 (Z) containing π −1 (S). Let E = π −1 (D) ∩ S and put σ = π| S . Since S is closed in M and σ is proper, we obtain the first inclusion in point (i). The second inclusion is given by construction. Since S ∩ U ⊂ U \ D and π is an isomorphism on π −1 (U \ D) we get point (ii). To prove point (iii), we first remark that E ′ = clos(S ′ ) \ S ′ as an easy consequence of (i). The properness of σ ensures that E ′ is the Hausdorff limit as ε → 0 of the subanalytic family of compact sets C ε = σ −1 (S ∩ {z = ε}), each analytically diffeomorphic to the circle, and so E ′ is subanalytic, compact and connected. It cannot be reduced to a single point p since, otherwise the curve selection lemma would show that S ′ ∪ {p} ⊂ S is locally open at p and thus p would be isolated in E which cannot be. The properness of σ is used again to prove that σ(E ′ ) = E. Finally, for point (iv), let W be an affine chart at p, isomorphic to R 2 , such that W ∩ E is either one or the two coordinate axis. Let
W k = [−1/k, 1/k] 2 . A connected component of W k \ E is either a half-space or a quadrant. Each contains a single connected component of W k+1 \ E.
If the property described in point (iv) does not hold, there will be points in
W k \ E which belong to the boundary E ′ = clos(S ′ ) \ S ′ of S ′ , thus impos- sible since E ′ ⊂ E.
A triple R = ( S, E, σ) satisfying the properties (i-iv) of Theorem 9 will be called a (total) resolution of singularities of S. The curve E will simply be called the divisor of the resolution R. The surface S ′ = σ −1 (S) and E ′ = clos(S ′ ) ∩ E will be respectively called the strict transform and the strict divisor of the resolution. We will also speak of R ′ = (S ′ , E ′ , σ ′ = σ| S ′ ) as the strict resolution of S (associated to R).
Let R = ( S, E, σ) be a resolution and p be a point of E. Let σ p : S 1 → S be the blowing-up of S at p. This provides a new triple
R p = ( S 1 , σ −1 p ( E), σ• σ p ) which is a new resolution of singularities of S. Definition 10. Let R 1 , R 2 be two resolutions of the surface S = β −1 (S 0 ).
If R 2 is obtained from R 1 by finitely many successive points blowing-ups at points in the successive corresponding divisors, we will say that R 2 dominates R 1 and will write R 2 R 1 .
A resolution dominating a given one will be obtained when we want to "monomialize" one or several functions on S which are restrictions of analytic functions.
Definition 11. Let H = (h 1 , . . . , h k ) be a k-uple of real analytic functions in a neighborhood of E in M × R ≥0 . A resolution R = ( S, E, σ) of S is adapted to H (or briefly a (S, H)-resolution) if, for any j, the composition h j = h j • σ has a monomial representation at any point p ∈ S: There are analytic coordinates (u, v) of S at p such that h j = u a v b G j (u, v), where a, b ∈ N, G j is analytic and G j (0, 0) = 0.
Corollary 12. Let H = (h 1 , . . . , h k ) be as above and suppose that the restriction h j | S has no critical point. Then there exists a resolution R of S such that, for any R 1 R, R 1 is a (S, H)-resolution.
Proof. From classical results in local monomialization of analytic functions in a smooth analytic manifolds (see for instance [3]): just consider a resolution of S and blow-up the points of the divisor where the corresponding total transform of the h j have not yet a monomial representation.
The following terminology is needed to state the principal result of this section. Let N be a real analytic manifold with real analytic smooth boundary ∂N and f : N → R be a continuous map. The function f is ramifiedanalytic at a point p of ∂N , if there are l ∈ N and analytic coordinates (x, z) at p for which N = {z ≥ 0} and ∂N = {z = 0}, such that the mapping (x, z) → f (x, z l ) is analytic at (0, 0). If h : N → M is a continuous mapping into an analytic manifold M , the mapping h will be called ramifiedanalytic at p ∈ ∂N if, in some analytic coordinates of M , its components are ramified-analytic at p.
Remark 13. Let f : N → R be a ramified-analytic function at some point p ∈ ∂N , with analytic coordinates (x, z) at p for which N = {z ≥ 0} and ∂N = {z = 0}. The function z∂ z f extends continuously, in a neighborhood V of p, into a function which is ramified-analytic at p and, moreover, vanishes along the boundary V ∩ ∂N . Proposition 14. Let R = ( S, E, σ) be a (S, z)-resolution and R ′ = (S ′ , E ′ , σ ′ ) be the associated strict resolution. There exist ε > 0 and a continuous mapping Φ : S 1 × [0, ε] → S with the following properties:
(i) It maps S 1 × {r} onto σ −1 (S ∩ {z = r}) for 0 < r ≤ ε and induces an analytic diffeomorphism between S 1 ×]0, ε] and σ −1 (S ∩{0 < z ≤ ε}). (ii) It maps surjectively S 1 × [0, ε] onto clos(S ′ ) = S ′ ∪ E ′ and it maps C = S 1 × {0} onto E ′ . (iii) The set Ω = Ω( Φ) = ( Φ) −1 (E ′ ∩ sing ( E)) ⊂ C is finite and Φ is
uniformly ramified-analytic at any point of C \ Ω: there exists l ∈ N such that (ϕ, r) → Φ(ϕ, r l ) is analytic at every point of C \ Ω.
Using Theorem 9, points (i), (ii) and (iii) are true for Φ := σ • Φ when replacing the strict transforms S ′ and E ′ with the initial subsets Sand E respectively. Namely, Φ maps surjectively S 1 × [0, ε] onto clos(S) = S ∪ E,
C onto E and S 1 ×]0, ε] diffeomorphically onto S, sending S 1 × {r} onto S ∩ {z = r}.
Moreover, Φ is uniformly ramified analytic at every point of C \ Ω. A mapping Φ (or Φ) satisfying points (i) to (iii) of Proposition 14 is called a parameterization associated to the resolution R, and the subset Ω in (iii) and is called the exceptional set of the parameterization Φ (or Φ).
Proof. Let R = ( S, E, σ) be a (S, z)-resolution. We construct a retraction of a neighborhood of E in S onto E by integration of a certain analytic vector field. It is just an avatar of the construction of a Clemens structure on an analytic manifold equipped with a normal crossings divisor (see [8,23]).
Let g be an analytic Riemannian metric on S, whose existence is guaranteed by Grauert's Theorem on the analytic embedding of analytic manifolds in Euclidean spaces [11]. Letz := z • σ : S → R. Let ξ = ∇ g (−z 2 ) be the gradient vector field of −z 2 w.r.t the metric g. Its singular set is exactly the
divisor E = { z = 0}.
Let ε be small enough so that σ induces a diffeomorphism from
σ −1 (S ∩ {0 < z ≤ ε}) to S ∩ {0 < z ≤ ε}.
We can now consider S just as being
S ∩ {0 < z ≤ ε}. For r ∈]0, ε], let C r =z −1 (r) = σ −1 (S ∩ {z = r}). It is an embedded curve in S isomorphic to the circle S 1 . Let ρ : S 1 → C ε , ϕ → ρ(ϕ) be an analytic diffeomorphism.
For p ∈ S ′ , let γ p be the maximal integral curve of ξ with initial data γ p (0) = p. The parameterized curve γ p is defined for times t ≥ 0 and stays in S ′ . Since the function t →z(γ p (t)) strictly decreases to 0 as t goes to infinity γ p cuts (orthogonally) each curve C r for r ∈]0,z(p)] only once. Thanks to Lojasiewicz's Gradient Inequality [18], the omega-limit set ω(γ p ) consists of a single point R(p) ∈ E ′ and the mapping R : S → E ′ is continuous since E is compact. The following mapping is thus well defined:
(3) Φ : S 1 × [0, ε] → S, Φ(ϕ, r) = C r ∩ | γ ρ(ϕ) |, if r = 0; R(ρ(ϕ)), if r = 0, where | γ p |⊂ S is the image set of γ p . The restriction of Φ to the open cylinder S 1 ×]0, ε] is an analytic diffeomorphism onto S ′ , proving point (i).
In order to obtain the continuity of Φ and properties (ii) and (iii), we will show that for
p ∈ E ′ there exists ϕ 0 ∈ S 1 such that Φ(ϕ 0 , 0) = p, Φ is continuous at (ϕ 0 , 0) and ramified-analytic if p ∈ E ′ \ sing( E). Let p ∈ E ′ \sing E. Let (u, v) be analytic coordinates at p such thatz(u, v) = v m with m ≥ 1 and E = {v = 0}. From point (iv) of Theorem 9, there is a neighborhood V of p such that the half-space {v > 0} is contained in S ′ .
The metric writes g = Adu 2 + 2Bdudv + Cdv 2 , and we obtain
ξ = 2(det g) −1 (Bmv 2m−1 ∂ ∂u − Amv 2m−1 ∂ ∂v ).
Since A(p) = 0, the divided vector field ξ ′ := v 1−2m ξ is not singular, transverse to the divisor E at p and generates the same foliation as ξ on {v = 0}. Thus there exists a trajectory | γ | of ξ with ω(γ) = p which extends smoothly and analytically through of p as a trajectory |γ ′ | of ξ ′ . Going backwards in time, | γ | cuts C ε at a point ρ(ϕ 0 ) for some ϕ 0 ∈ S 1 . Thus p = R(ρ(ϕ 0 )) = Φ(ϕ 0 , 0). Let γ ′ q be the trajectory of ξ ′ through a point q ∈ V. Since ξ ′ is not singular in V and transverse to the fibers v = cst, up to shrinking V, the following mapping
H : V×] − δ, δ[→ S, (q, t) → H(q, t) := v −1 (t) ∩ |γ ′ q |, is analytic. Fix v 0 > 0 such that γ cuts v −1 (v 0 ) inside V and denote ψ : S 1 → C v 1/m 0 , ψ(ϕ) = Φ(ϕ, v 1/m 0 )
, an analytic diffeomorphism. By construction the mapping we are looking for satisfies Φ(ϕ, r) = H(ψ(ϕ), r 1/m ) in some neighborhood of (ϕ 0 , 0) and thus is ramified analytic at that point. The number m can be chosen constant for each connected component of E \ sing( E), which are finitely many. Thus there is a uniform ramification index l along C \ Ω. So we get (iii).
Let p ∈ E ′ ∩ sing E. Let (u, v) be analytic coordinates at p such that z(u, v) = u l v m with l, m ≥ 1 and E = {uv = 0}. From point (iv) of Theorem 9 we assume that the first quadrant Q = {u > 0, v > 0} is contained in S ′ for u, v small enough. The metric writes as g = Adu 2 + 2Bdudv + Cdv 2 , and we obtain
ξ = 2(det g) −1 u 2l−1 v 2m−1 [(−lCv + mBu) ∂ ∂u + (lBv − mAu) ∂ ∂v ]
. Since g is positive definite, the divided vector field ξ ′ = u 1−2l v 1−2m ξ has a saddle-type singularity at p: its linear part L p at p has two non-zero eigenvalues with opposite sign. Moreover, each eigen-direction is transverse to the u-axis and v-axis, namely the components of E at p. The only trajectories of ξ ′ with ω-limit point p are the two connected components of W s \ {p}, where W s is the local stable manifold at p. Since ξ and ξ ′ are positively proportional on Q, the separatrix W s ∩ Q ⊂ S ′ is a trajectory |γ q | of ξ and thus ω(γ q ) = p. Going backwards in time, |γ q | cuts C ε at a point ρ(ϕ 0 ) for some ϕ 0 ∈ S 1 and thus Φ(
ϕ 0 , 0) = p. Let H : clos(Q) × [0, δ[→ S, where H(q, t)
is the intersection point of the trajectory of ξ ′ through the point q with the level curve {u l v m = t}. As in the previous case, continuity at p of the mapping Φ will follow from the continuity at p = (0, 0) of the mapping H. This property is easily obtained by explicit computation when the vector field ξ ′ is linear, and we can reduce to this case using Hartman-Grobman Theorem (see for instance [22]).
Definition 15.
Let Ω be a finite subset of C (such as the exceptional set of a parameterization Φ in the proposition above). An analytic mapping F : S 1 ×]0, ε] → N , is called uniformly almost ramified-analytic (with respect to Ω) if there exists some l ∈ N such that (ϕ, r) → F (ϕ, r l ) can be extended as an analytic mapping at any point of C \ Ω. To be shorter, we will either write Ω-u-a-r-a or simply u-a-r-a if the subset Ω is understood.
Part (iv) of Proposition 14 says that Φ (or Φ) is an u-a-r-a mapping with respect to the exceptional set Ω. Since ramified-analyticity at any point of C \ Ω is inherited from the construction of Φ and uniformity comes from the compactness of E, another typical situation example we will come across in the sequel is the following: if h is a continuous function in a
neighborhood of E ⊂ M × R ≥0 which is ramified-analytic along E (with respect to D = M × {0}), the composite mapping h Φ = h • Φ is Ω-u-a-r-a.
Asymptotic expansions of restricted functions
A Q-generalized (real) formal power-series is a formal expansion G(T ) = k 0 a k T α k , where (α k ) k 0 is a strictly increasing sequence of non-negative rational numbers and each coefficient a k is a real number. It is said conver- gent if there exists t 0 > 0 such that the sequence of m-partial sum functions G m : R ≥0 → R, G m (t) = m k=0 a k t α k , converges uniformly in [0, t 0 ],Φ := f S • Φ = f S • σ • Φ : S 1 × [0, ε] → R.
This Section is devoted to prove the following result, establishing an asymptotic expansion of the restricted function f S w.r.t. the height coordinate z : M ×R → R ≥0 (let again (ϕ, r) be the standard coordinates on S 1 ×[0, ε]).
Proposition 16. Assume that f is not identically vanishing on S. One and only one of the following two properties is satisfied: (a) There exists a Q-generalized real formal power-series G(T ) = k 0 a k T α k which is an asymptotic expansion of f S in the following sense: for any positive integer m, there exists a neighborhood V m of E in clos(S) and a bounded function g m : V m → R such that, for any (y, z) ∈ V m with z = 0,
(4) f S (y, z) = m−1 k=0 a k z α k + z αm g m (y, z).
Moreover, the formal power series G(T ) is a convergent Puiseux series and f S (y, z) = G(z) for any (y, z) ∈ S in a neighborhood of E.
(b) Given an initial resolution of S 0 , there exists a dominating resolution R 0 adapted to the function z, a Q-generalized polynomial P (T ) = m k=0 a k T α k and a rational number α > α m such that, for any resolution R R 0 and any associated parameterization Φ, the mapping f Φ : S 1 × [0, ε] → R writes as
(5) f Φ (ϕ, r) = P (r) + r α F (ϕ, r),
where F is a continuous function on S 1 ×[0, ε] and its restriction to C := S 1 × {0} is not constant. Moreover, F is u-a-r-a with respect to the exceptional set Ω of Φ.
The proof will follow from the following lemma.
Lemma 17. With the hypotheses and notations of Proposition 16, we find: (i) There exists a unique α = α(f S ) ∈ Q ≥0 such that the quotient f S /z α is bounded on S and cannot have the value 0 as a single accumulation value as z → 0 + . The number α is called the exponent of the restricted function f S (with respect to E).
(ii) Given an initial resolution of S 0 , there exists a dominating (S, z)resolution R 0 such that, for any other resolution R R 0 and any associated parameterization Φ : S 1 × [0, ε] → S, the quotient function f Φ /r α = (f /z α ) • Φ is well defined and analytic on S 1 ×]0, ε] and extends to a continuous function on S 1 × [0, ε]. Its restriction to the bottom circle C will be denoted by in Φ (f ) and called the initial part of the restricted function f S (relative to Φ).
(iii) An initial part in Φ (f ) like in point (ii) is constant if and only f S /z α has a unique accumulation value as z → 0.
Proof. By definition of a ramified-analytic function along D and since E is a compact subset of D, there exists a positive integer l ∈ N such that the function f : (y, z) → f (y, z l ) is analytic in a neighborhood of E in M × R. If we prove the Lemma for the analytic function (f ) S := f | S , we obtain the exponent α. Then α :=ᾱ/l is the exponent of f S with respect to E and it satisfies (i)-(iii). For the rest of the proof, we suppose that f is analytic in a neighborhood of E in M × R. Proof of (i). The uniqueness of the exponent α is immediate from its definition. Consider the following function µ(t) = max{|f (y, t)| for (y, t) ∈ S}.
It is well defined since S ∩ {z = t} is compact for t > 0. The function µ is subanalytic, continuous and identically zero only if f S is. So assuming that f S does not vanish identically on S, there exists a positive real number a and a non-negative rational number α such that t −α µ(t) → a as t → 0. This proves the claim. Proof of (ii). Assume we are given a first resolution. Let R 0 be a (S, f, z)resolution dominating it. Any other resolution R = ( S, E, σ) dominating R 0 is still a resolution adapted to f and z. Let Φ : S 1 × [0, ε] → S be a parameterization associated with R. The function h = z −α f S is analytic, continuous and bounded on S ∩ {0 < z < ε} for some ε > 0. Let S ′ = σ −1 (S), E ′ = clos(S ′ ) \ S ′ be respectively the strict transform of S and the strict divisor of the resolution R (see the notations of Theorem 9). Let
h ′ = h • σ : S ′ → R. Thus r −α f Φ = h ′ • Φ.
Since Φ is continuous and maps C onto E ′ , there is just to prove that h ′ extends to a continuous function up to E ′ . We also write h
′ = z −α f S where f S = f S • σ and z = z • σ. First, let p ′ ∈ E ′ \ sing( E).
There are analytic coordinates (u, v) of S at p ′ such that E = {v = 0} and {v > 0} ⊂ S ′ using (iv) of Theorem 9. Since R is a resolution adapted to f and z, we write
f S (u, v) = u l 1 v m 1 U 1 (u, v), z = v m 2 U 2 (u, v)
for some integers l 1 , m 1 , m 2 ∈ N and invertible analytic functions U 1 , U 2 with U 2 (0, 0) > 0. For (u, v) close to p ′ = (0, 0) with v > 0, we find
(6) h ′ (u, v) = v m 1 −αm 2 u l 1 U 1 (u, v) U 2 (u, v) α .
Since h ′ is bounded on {v > 0} necessarily m 1 ≥ αm 2 and the right hand term in Equation (6) defines a continuous function on {v ≥ 0}. If clos(S ′ ) ⊂ {v ≥ 0} nearby p ′ , we get the desired conclusion. If instead {v < 0} ⊂ S ′ , necessarily m 2 is even since z is positive on S ′ . In this case, the monomial v m 1 −αm 2 in expression (6) must be read as v m 1 /(v m 2 ) α . The function h ′ turns out to be continuous in a neighborhood of p ′ = (0, 0).
Suppose now that p ′ ∈ E ′ ∩ sing E. There are analytic coordinates (u, v) of S at p ′ with E = {uv = 0} and {u > 0, v > 0} ⊂ S ′ and such that we can write
f S (u, v) = u l 1 v m 1 U 1 (u, v), z = u l 2 v m 2 U 2 (u, v)
for some l 1 , m 1 , l 2 , m 2 ∈ N and analytic functions U 1 , U 2 with U 1 (0, 0) = 0, U 2 (0, 0) > 0. This time, for small and positive u, v, we have
(7) h ′ (u, v) = u l 1 −αl 2 v m 1 −αm 2 U 1 (u, v) U 2 (u, v) α .
Since the function h ′ is bounded in a neighborhood of p ′ in S ′ , l 1 − αl 2 and m 1 − αm 2 are both non-negative. The continuity of h ′ follows by the same arguments as in the previous case.
Proof of (iii). It follows by continuity of f Φ /r α , proved in (ii), the properness of Φ and that Φ maps C onto E = clos(S) ∩ {z = 0}.
Proof of Proposition 16. Let α 0 ∈ Q ≥0 be the exponent of f with respect to E. Let R 0 be a (S, z)-resolution and Φ 0 be an associated parameterization satisfying the properties of (ii) in Lemma 17.
If the initial part in Φ 0 (f ) is not constant then we are in case (b) of the proposition with P = 0 and α = α 0 . Assume now in Φ 0 (f ) ≡ a 0 ∈ R * . The function f 1 := f − a 0 z α 0 is ramifiedanalytic along D. If f 1 | S ≡ 0 then we are in case (a). Otherwise, using Lemma 17, let α 1 ∈ Q ≥0 be the exponent of f 1 w.r.t E. By definition of the exponent, we find α 1 > α 0 . Let R 1 be a (S, z)-resolution with R 1 R 0 and Φ 1 an associated parameterization for which the initial part in Φ 1 (f 1 ) of f 1 exists as in part (ii). If in Φ 1 (f 1 ) is not constant we are in case (b) as above and we are done, otherwise we continue this process. Suppose there exists a sequence of (S, z)-resolutions {R k } k≥0 with R k+1 R k , associated parameterizations Φ k and a Q-generalized power series G(T ) = k≥0 a k T α k such that, for any m ≥ 0, α m is the exponent of the function f m = f − m−1 k=0 a k z α k and the principal part in Φ m (f m ) is a constant function equal to a m = 0. The definition of the exponent α gives directly the asymptotic expansion of f S as in equation (4). Let Γ ⊂ S be an analytic half-branch accumulating to a single point in E, parameterized by the variable z. Let L :]0, ε] → R defined as L(z) = f S (Γ(z)). By (4), we have for any m ≥ 0 and z sufficiently small,
L(z) − m−1 k a k z α k = O(z αm ),
that is, that G(T ) is the asymptotic expansion of L as z → 0 + . Since L is a semi-analytic function, G(T ) is a convergent Puiseux series. Thus L(z) = G(z), where G is considered here as the sum of the expansion G(T ). We define G S : S → R by G S (y, z) = G(z), an analytic function on S which depends only on z. We have shown that the restrictions of f S and G S on Γ coincide. Since Γ can be chosen arbitrarily, f S = G S on the whole surface S. This proves statement (a) of the Proposition. Finally, F = f Φ −P r α is u-a-r-a since both f Φ and P are so.
Remark 18.
Although F depends on the resolution R and on the associated parameterization Φ, we insist it is of the special following form:
F = g • Φ with g := ( f Φ −P r α )
. The function g is continuous in a neighborhood of E in M × R ≥0 , ramifiedanalytic along E, and depends on f and the opening blowing-up β only.
Oscillation vs Spiraling in singular surfaces
Let γ : [0, +∞[→ R n be an analytically parameterized curve such that lim t→+∞ γ(t) = 0 ∈ R n and 0 does not belong to | γ |, the image of γ.
Definition 19.
A parameterized curve γ is said (analytically) non-oscillating if for any semi-analytic subset H of R n , either | γ | is contained in the subset H or the intersection | γ | ∩ H consists at most of finitely many points. If, on the contrary, there exists a semi-analytic set H such that | γ | is not contained in H and the intersection | γ | ∩ H has infinitely many points then we will say that γ is oscillating relatively to H.
The notion of oscillation clearly depends only on the germ at 0 of the image | γ | of the parameterized curve γ, not on any given parameterization.
In dimension 2, the notion of spiraling around a given point is a special case of oscillation for a curve. A convenient definition is found in [6]. We generalize this notion for a curve | γ | contained in an analytic isolated surface singularity X ⊂ R n at the origin 0 and accumulating at 0. Let X be an analytic surface with an isolated singularity at 0 ∈ R n . Let S 0 be a connected component of X \ {0}. Let Γ be an analytic half-branch at 0 contained in S 0 . For a small enough simply connected neighborhood V of (the germ at 0 of) Γ in S 0 , the curve Γ ∩ V separates V \ Γ into two connected components which we call the two local sides of Γ in S 0 . Definition 20. The curve γ : [0, +∞[→ S 0 ⊂ X \ {0} spirals in X if, for any analytic half-branch Γ at 0 in S 0 , there exists an increasing sequence (t k ) k∈N ⊂ R >0 with t k → +∞ such that for each k:
γ([t k , t k+1 [) ∩ Γ = {γ(t k )}, γ(t k − ε k ) ∈ V − and γ(t k + ε k ) ∈ V + ,
for ε k > 0 small and where V − , V + are the local sides of Γ in S 0 .
When γ is a trajectory of a real analytic vector field in a neighborhood of 0 ∈ R 2 , a Rolle-Khovanskii's argument proves that the only oscillating dynamics at 0 is spiraling (see [6]). Proposition 21 below extends this result to analytic isolated surfaces singularities. Let ξ 0 be an analytic vector field on S 0 which extends continuously and subanalytically to the origin by ξ 0 (0) = 0, as a mapping from clos(S 0 ) to T R n | clos(S 0 ) . Proof. If γ spirals then it is oscillating. Suppose that γ does not spiral. There exists an analytic half-branch Γ in S 0 such that either (a) the germ at 0 of the intersection | γ | ∩ Γ is empty, or (b) | γ | ∩ Γ is infinite but γ does not cross Γ from one fixed local side of Γ to the other side at those intersection points. If (b) happens, a Rolle's argument implies that Γ is tangent to ξ 0 at infinitely many points accumulating to 0. The subanalyticity of ξ 0 implies that the half-branch Γ is a trajectory of ξ 0 , contradicting the oscillation of γ relatively to Γ. So (b) is impossible.
Assume we are in case (a). Since the surface S 0 is analytically diffeomorphic to a cylinder, S 0 \ Γ is a simply connected analytic manifold. Using Haefliger's Theorem [12,16,21]), we deduce that any leaf of the real analytic foliation induced by ξ 0 in S 0 \ Γ is a Rolle's leaf. In particular, the curve | γ | ⊂ S 0 \ Γ is a Rolle's leaf and cannot cut infinitely many times any analytic half-branch contained in S 0 \ Γ. Thus γ is non-oscillating.
Despite of the similarities between spiraling in a smooth surface and in an analytic isolated surface singularity, there is however a very important difference. The existence, for a trajectory γ, of a tangent at the origin, that is the limit of secants lim t→∞ γ(t) |γ(t)| exists, prevents, in the smooth surface situation, from spiraling around the origin. For an isolated surface singularity, although in the OTC case this argument is still valid, in the CTC situation, the curve γ will always have a tangent at the origin corresponding to the direction of the tangent cone, regardless if it is spiraling or not A criterion stronger than the existence of tangent to imply non-spiraling is that the lifting of γ by a reduction of singularities of the surface accumulates to a single point on the exceptional divisor. We will use this criterion through its lifting on S 1 × [0, ε] via a parameterization as in section 3.
Criterion for non-spiraling. Let R be a resolution of S = β −1 (S 0 ) where β is an opening blowing-up of S 0 . Let Φ : S 1 × [0, ε] → S be a parameterization associated to R. Assume that | γ | ⊂ S 0 and suppose the ω-limit set ω(γ) of the lifted curve γ = (β • Φ) −1 • γ is such that C \ ω(γ) contains an open non-empty arc. Then γ does not spiral in X.
The proof is easy: the stated property will imply that γ does not intersect a given analytic half-branch Γ on S 1 × [0, ε] through a point p ∈ C \ (ω(γ) ∪ Ω) where Ω is the exceptional set of Φ. Therefore, γ does not intersect the curve Γ = (β • Φ)(Γ) ⊂ S 0 , which is an analytic half-branch by properness of the resolution and the property that Φ is ramified-analytic at p. Thus γ does not spiral in X.
The next result describes, for a vector field ξ 0 on S 0 , two types of dynamics ensuring that none of its trajectories accumulating at the origin is spiraling. These types correspond to either "dicritical" or "non-monodromic" dynamics similar to those in the plane gradient case met in Section 2. (a) Dicritical case: There exist a point p ∈ C \ Ω and a neighborhood U of p in S 1 × [0, ε] disjoint from Ω in which ξ writes as
(8) ṙ = r µ H(r, ϕ) ϕ = r µ−1+η G(r, ϕ)
where µ, η ∈ Q >0 and H, G are continuous on U and ramified-analytic at any point of U ∩ C and such that H is negative on U .
(b) Non-monodromic case: There exist µ ∈ Q ≥0 , u-a-r-a functions G 1 , G 2 : S 1 ×]0, ε] → R so that G 2 vanishes on C \ Ω and an u-a-r-a function H continuous on the whole cylinder S 1 × [0, ε] such that the restricted function H| C is not constant, in such a way that ξ writes in the open cylinder S 1 ×]0, ε] as
(9) ṙ = r µ+1 G 1 ϕ = r µ [ ∂H ∂ϕ + G 2 ].
Then any trajectory γ of ξ 0 accumulating to the origin is non-spiraling and therefore is non-oscillating.
Proof. It suffices to show that any trajectory γ of ξ 0 accumulating to the origin satisfies the non-spiraling criterion above.
In the dicritical situation (a) we prove a slightly stronger result: there exists a non-empty arc I ⊂ U ∩ C such that each point in I is the unique ω-limit point of a trajectory of the transformed vector field ξ. When µ − η + 1 ≥ µ in Equation (8) Assume now that µ − η + 1 < µ in (8). We suppose that U is of the form
U =]ϕ 1 , ϕ 2 [×[0, δ] ∈ S 1 × [0, ε]
for some δ > 0 small enough. Dividing ξ by r µ−1 |H|, our vector field provides the following equations in U :
(10) ṙ = −ṙ ϕ = r η G |H|
Up to shrinking U , and since G is ramified-analytic, we furthermore assume that G does not vanish on U , up to increasing the exponent η. If G(p) = 0 but G| U ∩C ≡ 0, then there are points of C close to p at which G does not vanish. Thus we can also suppose G(p) = 0, for instance that G is positive on U . Up to shrinking U again, we know that
K 1 ≤ G |H| ≤ K 2 on [ϕ 1 , ϕ 2 ] × [0, δ] for some positive constants K 1 , K 2 .
The solution of (10) through a point (ϕ 0 , r 0 ) ∈ [ϕ 1 , ϕ 2 ]×]0, δ], as long as it is in that domain, lies between the solutions through (ϕ 0 , r 0 ) of the systems of equationsṙ = −r andφ = K i r µ for i = 1, 2. These last curves are parameterized by
ϕ → r(ϕ) = [r η 0 − η K i (ϕ − ϕ 0 )] 1/η , i = 1, 2.
We deduce that any point of ]ϕ 1 , ϕ 2 [×0 ⊂ C is the unique accumulation point of a trajectory of the system (10), lying in {r > 0}.
Consider now the non-monodromic situation (b). The hypothesis about H implies its partial derivative ∂ ϕ H is u-a-r-a and continuous along C \ Ω. Let crit * (H| C ) be the critical locus of H| C in C \ Ω, and let Ω ′ = Ω ∪ (H| C ) −1 (H(crit * (H| C )).
Since H| C is not constant, C \ Ω ′ has non empty interior. To show the criterion for non-spiraling for any trajectory γ of ξ 0 , it is enough to check that the limit set ω(γ) of any trajectory γ of ξ accumulating to C is contained in Ω ′ . Assume γ is parameterized by t ∈ R ≥0 and consider the real function H(γ(t)).
h = h γ : R ≥0 → R, t → h(t) =
The function h is C 1 . Let p ∈ C \Ω ′ and let a := H(p). We just have to show that a cannot be an accumulation value of h when t → +∞. The function |∂ ϕ H| is bounded below on the compact set (H| C ) −1 (a): there exists c > 0 such that |∂ ϕ H| ≥ 2c > 0 on a given neighborhood V of (H| C ) −1 (a) in
S 1 × [0, ε].
Since γ(t) = (ϕ(t), r(t)) satisfies Equations (9), if γ(t) ∈ V then, up to shrinking V (taking into account Remark 13), we finḋ
h(t) = ∂H ∂r (γ(t))ṙ(t) + ∂H ∂ϕ (γ(t))φ(t) > c.
For ε ′ > 0 sufficiently small, we assume that H −1 (a) ∩ S 1 × [0, ε ′ ] ⊂ V. Thus, there exists some δ > 0 such that
h(t) ∈]a − δ, a + δ[⇒ γ(t) ∈ V.
From all these properties, for t ∈ h −1 (]a−δ, a+δ[) large enough,ḣ(t) ≥ c/2 > 0. Thus when t → +∞, the value a cannot be an accumulation value of h.
Remark 23. The non-monodromic situation (b) described in Proposition 22
can be generalized as follows: Assume that the given analytic vector field ξ 0 does not vanish in S 0 and that the foliation F on S 1 ×]0, ε] induced by ξ = (β • Φ) * ξ 0 extends continuously to C \ Ω such that C is invariant. Assume there exist, two distinct points q 1 , q 2 of C \ Ω where F is not singular, and two continuous germs of vector fields ξ 1 at q 1 and ξ 2 at q 2 , which are local generators of the foliation F, positively co-linear to ξ in the common domain of definition and "pointing in different directions": if ϕ denotes a global coordinate on C ≃ S 1 , writing ξ i (q i ) = c i ∂ ϕ , then c 1 c 2 < 0. Then any trajectory of ξ 0 accumulating to the origin is non-oscillating.
Proof of the main result
This section is devoted to the proof of the main result of this paper, Theorem 1.
The next Lemma shows that the only case requiring work is when both f 0 | S 0 and ∇ h f 0 do not vanish on S 0 . Lemma 24. If either f 0 | S 0 or ∇ h f 0 vanishes in any neighborhood of 0 in S 0 , then any trajectory of the restricted gradient ∇ h f 0 accumulating to the origin is non-oscillating.
Proof. First, note that ∇ h f 0 extends to a continuous subanalytic mapping ∇ h f 0 : clos(S 0 ) → T R n by ∇ h f 0 (0) = 0, and that f 0 vanishes on any connected component of the zero locus of ∇ h f 0 containing 0 in its closure. The subanalytic Curve Selection Lemma guarantees there exists a subanalytic (thus semi-analytic) curve Γ ⊂ S 0 such that 0 ∈ clos(Γ) and Γ ⊂ f −1 0 (0). Let γ be a non-trivial trajectory of the restricted gradient ∇ h f 0 accumulating to 0. The function t → f 0 (γ(t)) is increasing and tends to 0 as t → ∞. Thus f 0 (γ(t)) < 0 for any t and γ does not cut Γ. Apply now Proposition 21.
Assume from now on that there exists a neighborhood V of 0 in X, such
that f 0 | S 0 and ∇ h f 0 do not vanish in V ∩ S 0 .
The sketch of the proof of Theorem 1 is as follows. We first open the surface S 0 by means of a suitable opening blow-up mapping β : M × R ≥0 → R n as defined in Proposition 8. Then we take a suitable resolution R = ( S, E, σ) of the surface S = β −1 (S 0 ) as in Theorem 9. We then pick a parametrization Φ : S 1 × [0, ε] → S associated to R. Writing Φ = σ • Φ, the mapping β • Φ is a diffeomorphism from the open cylinder S 1 ×]0, ε] onto S 0 . Thus, the pull-back h := (β •Φ) * h of the metric h is an analytic Riemannian metric on the open cylinder. If f Φ denotes the composition f 0 • β • Φ, then the pull-backξ := (β • Φ) * ∇ h f 0 is just the gradient vector field of f Φ with respect to h, that is,ξ
= (β • Φ) * ∇ h f 0 = ∇ h f Φ .
The proof will be finished, using Proposition 22, once we have proved that ξ satisfies one of the two situations described there: either (a), dicritical or (b), non-monodromic.
Our proof will only deal with the metric g on R n be the Euclidean metric. We can reduce to this case using Cartan-Janet's Theorem [15,7]: an analytic Riemannian manifold can be locally isometrically embedded into an Euclidean space as an analytic submanifold equipped with the induced Riemannian structure.
Notation. Let Ω be a finite subset of C (such as for instance the exceptional set of a parameterization as in Proposition 14). In Definition 15 was introduced the notion Ω-u-a-r-a function on the open cylinder S 1 ×]0, ε]. For any rational number ν ≥ 0, let A ≥ν be the real algebra of all the Ω-u-a-r-a functions ψ : S 1 ×]0, ε] → R for which the function r −ν ψ is also an Ω-u-a-r-a function along C. Let A >ν := ∩ µ>ν A ≥µ be the ideal of A ≥ν of the functions ψ such that the function r −ν ψ vanishes identically on C \ Ω. In particular ψ ∈ A >ν means there exists a rational number ν ′ > ν such that ψ ∈ A ≥ν ′ .
We are dealing first with the CTC case in rather detailed fashion. It requires much more work than the OTC case, and this latter will follow from exactly the same arguments as those used in the CTC case.
Cuspidal case. Assume that the tangent cone of S 0 at the origin is reduced to a single point. Take linear coordinates (x 1 , . . . , x n−1 , z) at 0 adapted to S 0 which are also orthonormal coordinates for the Euclidean metric.
We consider an opening blow-up mapping of the form
β : M × R ≥0 → R n , (y, z) → (z eN y + θ(z), z N ),
where M = R n−1 and e, N, θ are defined as in Proposition 8. We recall that z → (θ(z), z) = (θ 1 (z), . . . , θ n−1 (z), z N ) is a parametrization of an analytic half-branch in S 0 . Let m + 1 ∈ N ≥1 be the minimum order at 0 with respect to z of the components θ j . The cuspidal nature of the surface S 0 implies m ≥ N .
We define the following functions on the open cylinder R = eN (y 2 1 + · · · + y 2 n−1 ) • Φ and U = j ((y j • Φ) ϕ ) 2 , where the subscript ϕ stands for partial derivative with respect to the angular variable ϕ. Again R and U depend on the resolution and on the associated parameterization Φ considered, but both are u-a-r-a functions with respect to the resulting exceptional set Ω.
Lemma 25.
There is a non-empty open arc J of C \ Ω with non empty interior in C \ Ω along which the restricted function U | J is positive.
Proof. We just have to show that the function U does not vanish on the whole of C \Ω. If U | C\Ω ≡ 0, by definition of U , each y j •Φ is locally constant when restricted to C \ Ω, and thus constant on C by continuity. Using (iii) of Proposition 14, this would imply the constancy of the coordinates y j along E = σ( Φ(C)), which is impossible since by construction dim E = 1.
Using the coordinates (ϕ, r) in the open cylinder, the metric h writes (11) h = (β • Φ) * g = A(r, ϕ)dr 2 + 2B(r, ϕ)drdϕ + C(r, ϕ)dϕ 2 .
The following lemma describes the coefficients of the metric h. The dominant part of each term of interest is explicit. It is important to remark that the statement neither deals with a fixed resolution R, nor an associated parametrization. It is very useful and necessary to obtain the needed conclusions up to dominating resolutions of a given one as we will see. Lemma 26. There exists a resolution R 0 of S such that, for any other resolution R R 0 and parameterization Φ associated to R, we have the following description: There exist s ∈ Q ≥0 ∪ {+∞} with (12) s ≥ eN + m, an analytic power series ψ(r) with ψ(0) = 0 and an u-a-r-a function H on S 1 ×]0, ε] which extends continuously to the circle C in such a way that H| C is not constant, such that we obtain the following expressions for the coefficients of h in (11):
(13) A = r 2N −2 [N 2 + ψ(r)] + r eN +m−1 A 1 + r 2eN −2 A 2 B = r s H ϕ + r 2eN −1 R ϕ + B C = r 2eN U
where A 1 , A 2 ∈ A ≥0 and B ∈ A >2eN −1 and with the convention that the term r s H ϕ ≡ 0 if s = ∞. Moreover, s and ψ(r) depend neither on R R 0 nor on the parameterization Φ.
Proof. We start with a given resolution R 1 of the surface S and adopt the notations above. Let w = y • Φ = (w 1 , . . . , w n−1 ) and θ(z) = z m+1 θ(z) with θ(0) = 0. Let λ(r) := (1 + m)θ(r) + rθ ′ (r) = r −m θ ′ (r) where the prime denotes the usual derivative. It is an analytic mapping which does not vanish at r = 0. From the expression of β, we deduce
(β • Φ) * dx n = N r N −1 dr (β • Φ) * dx j = [r eN −1 (eN w j + r(w j ) r ) + r m λ j ]dr + r eN (w j ) ϕ dϕ.
Note that each w j is u-a-r-a and extends continuously on the whole bottom circle C. But (w j ) r could even be unbounded. However, by Remark 13, each function r(w j ) r is u-a-r-a and belongs to A >0 . Taking this property into account and since g is the Euclidean metric, we obtain
(14) h = (β • Φ) * (dx 2 1 + · · · + dx 2 n ) = [N 2 r 2N −2 + r 2m j λ 2 j + r eN +m−1 ( j eN λ j w j + · · · ) + r 2eN −2 ( j e 2 N 2 w 2 j + · · · )]dr 2 + 2[r eN +m j λ j (w j ) ϕ + r 2eN −1 ( j eN w j (w j ) ϕ + · · · )]drdϕ + [r 2eN j (w j ) 2 ϕ ]dϕ 2 ,
where · · · stands for an element of A >0 . Let ψ(r) := r 2(m−N )+2 j λ 2 j (r). Since m ≥ N and e > 1 we can define A 1 , A 2 ∈ A ≥0 so that A, the coefficient of dr 2 in (14) writes as in (13). Notice that ψ(r) does not depend on the resolution or the parameterization. On the other hand, the second summand of the coefficient of drdϕ in (14) is given by (14) is just r 2eN U . In order to complete the expressions of (13), let us have a look at the first summand of the coefficient of drdϕ in (14). Consider the function h(y, z) = z eN +m j λ j (z)y j , defined and analytic in a neighborhood of E in M × R. Applying Proposition 16 to h, there exists a resolution R 0 of the surface S so that, given any other resolution R R 0 and any associated parametrization Φ, the composition h Φ = h • Φ on the open cylinder S 1 ×]0, ε] writes (15) h Φ (ϕ, r) = P (r) + r s H(ϕ, r)
r 2eN −1 R ϕ + B where B ∈ A >2eN −1 , while the coefficient of dϕ 2 in
where s ∈ Q ≥0 , P (r) is a Q-generalized polynomial and H is an u-a-r-a function that extends continuously to the bottom circle C such that H| C is either not constant if s < +∞ or, for s = ∞, H is identically zero and P (r) is a convergent Puiseux series. The first summand of the coefficient of drdϕ in (14) is just the partial derivative (h Φ ) ϕ (identically zero if s = ∞), equal to r s H ϕ by (15). Since H ∈ A ≥0 , we get Inequality (12). Moreover, Proposition 16 also ensures that the exponent s does not depend on the given resolution R (or on the parametrization Φ) as long as it dominates R 0 . This completes the proof of the lemma.
Now consider the function f = f 0 • β which is an analytic function in a neighborhood of E in M × R. We can assume that f 0 (0) = 0 so that f | E ≡ 0. Applying Proposition 16 to f , there exists a resolution R of S and an associated parametrization Φ such that expression (5) is valid: We can write either f Φ = f Φ (r) as a convergent Puiseux series only depending on r or else (16) f Φ = a 0 r α 0 + . . . + a m r αm + r α F (ϕ, r) = P (r) + r α F (ϕ, r),
where a j ∈ R \ {0}, 0 ≤ α 0 < · · · < α m < α are non-negative rational numbers and F is u-a-r-a and extends continuously to the whole cylinder and the restriction F | C is not a constant function. Recall moreover that P (r), as well as the exponent α, are independent of the parameterization and of any resolution R ′ which dominates R. Therefore, we can suppose that R and Φ are chosen such that the expressions (13) for the coefficients of the transformed metric in Lemma 26 also hold.
In what follows, we treat the degenerate case when f Φ only depends on r as the case (16) with α as big as we want but without requiring that F | C is not constant.
Up to a multiplication by a function that does not vanish on the open cylinder, the differential equation provided by the transformed vector field ξ writes:
(17) ṙ = [P ′ (r) + (r α F ) r ]C − r α F ϕ Ḃ ϕ = [P ′ (r) + (r α F ) r ]B + r α F ϕ A
We have several cases to deal with.
Case (1): α = 2N − 2 + α < min{s + α 0 − 1, 2eN + α 0 − 2}.
From the expression ofφ in (17) we obtaiṅ
ϕ = r α (N 2 F ϕ + ∆), whith ∆ ∈ A >0 .
On the other hand, from 2eN + α − 1 ≥ 2eN + α 0 − 1 > α + 1 and s + α ≥ s + α 0 > α + 1 we deduceṙ ∈ A > α+1 . Eventuallyξ is in the non-monodromic case (9) of Proposition 22.
Case (2): α = 2N − 2 + α ≥ min{s + α 0 − 1, 2eN + α 0 − 2}. Using (12) in this case, we find α > α 0 and thus P = 0 and a 0 α 0 = 0. We distinguish two sub-cases:
Case (2a): 2eN + α 0 − 2 < s + α 0 − 1. We deduce first thatφ = r 2eN +α 0 −2 (G ϕ + ∆) where (18) G = α 0 a 0 R, if 2eN + α 0 − 2 < α; α 0 a 0 R + N 2 F if 2eN + α 0 − 2 = α.
and ∆ ∈ A >0 . We observe that the function r 2eN +α 0 −2 G is of the form g Φ = g • Φ for some ramified-analytic function g on a neighborhood of E in M × R ≥0 . Thus the function G is continuous on the cylinder and u-a-r-a. On the other hand, the term r α F ϕ B in the expression forṙ in (17) belongs to A >2eN +α 0 −1 . Thusṙ = r 2eN +α 0 −1 (a 0 α 0 U + Υ) for Υ ∈ A >0 . If G| C is not constant, our situation is non-monodromic in the sense of Proposition 22. Otherwise, thanks to Lemma 25, it is the dicritical case of Equation (8) with µ = 2eN + α 0 − 1.
Case (2b): s + α 0 − 1 ≤ 2eN + α 0 − 2.
This case is the most difficult since several of the terms involved in the expression ofφ may be of the same order with respect to r so that all the "initial parts" which are derivatives with respect to ϕ of a function may cancel. From Equation (12) and m ≥ N , we first find (19) α + 2eN − 1 ≥ α + s ≥ 2eN + α 0 + 1.
Using Equations (13) and (19) we get r α F ϕ B ∈ A ≥2eN +α 0 +1 . Thus, in (17), we obtain (20)ṙ = r 2eN +α 0 −1 (a 0 α 0 U + Υ), for Υ ∈ A >0 .
Using again (19) we find the following estimates on the order of some terms in the expression ofφ:
P ′ (r)B ∈ A >2eN +α 0 −2 , (r α F ϕ )r eN +m−1 A 1 ∈ A >2eN +α 0 , (r α F ϕ )r 2eN −2 A 2 ∈ A ≥2eN +α 0 (r α F ) ϕ B ∈ A ≥2eN +α 0 .
This allow us to writeφ as
(21)φ = −P ′ (r)[r s H ϕ + r 2eN −1 R ϕ ] + r α (N 2 + ψ(r))F ϕ + ∆ = G ϕ + ∆,
where ∆ ∈ A >2eN +α 0 −2 and
G = −P ′ (r)[r s H + r 2eN −1 R] + r α (N 2 + ψ(r))F.
Once again G = g Φ = g • Φ for some function g in a neighborhood of E in M × R ≥0 which is ramified analytic along E. From the definition of R and Remark 18 for H and F , the function g depends only on f and β but not on the resolution R or on an associated parameterization Φ. So, up to further finitely many blowing-ups and applying Proposition 16, we can assume that
G(ϕ, r) = Q(r) + r ρ G(ϕ, r)
where Q is a Q-generalized polynomial, ρ ∈ Q >0 and G is an u-a-r-a function which extends continuously to the bottom circle C with, either G | C is not constant or ρ can be chosen as large as we want (we just need ρ > 2eN + α 0 − 2). Two cases are to be considered:
-If ρ ≤ 2eN + α 0 − 2, Equation (21) writesφ = r ρ ( G ϕ + ∆) for ∆ ∈ A >0 .
Combined with (20), we find a non-monodromic situation (9).
-If ρ > 2eN + α 0 − 2 then we are in the dicritical situation (8)
h = (1 + A)dr 2 + 2rBdrdϕ + r 2 U dϕ 2 , where U = i (w i ) 2 ϕ , A = r 2 i (w i ) 2 r and B = r i (w i ) r (w i ) ϕ , since i w 2 i = 1. We note that A, B ∈ A >0 . Writing f Φ = f 0 • β • Φ, the pull-backξ = (β • Φ) * ∇ h f 0 of
the restricted gradient vector field has the following associated system of differential equations (up to the multiplication by the determinant of the metric h):
(22) ṙ = r 2 U (f Φ ) r − rB(f Φ ) φ ϕ = −rB(f Φ ) r + (1 + A)(f Φ ) ϕ .
We consider cases (a) or (b) of Proposition 16 for the function f := f 0 • β.
In case (a) the function f depends only on z and thus (f Φ ) ϕ ≡ 0. Dividing (22) by r(f Φ ) r , which does not vanish on the open cylinder, we obtain the dicritical situation of Proposition 22.
In case (b), we assume that the resolution R is such that
f Φ (ϕ, r) = P (r) + r α F (ϕ, r),
where P (r) = a 0 r α 0 + · · · + a m r αm , α m < α if a 0 = 0, and F extends continuously to the bottom circle C and its restriction F | C is not constant.
If α ≤ 1, Equations (22) becomė r ∈ A ≥α+1 andφ = r α [F ϕ + ∆],
where ∆ ∈ A >0 . We have a non-monodromic situation as in (9) and we are done. If α > 1, we have two sub-cases: -Case α = α 0 . This means that P ≡ 0. Thus r(f Φ ) r = r α (αF + rF r ) and (f Φ ) ϕ = r α F ϕ .
Using Remark 13 and Equation (22), we findφ = r α (F ϕ + ∆) where ∆ ∈ A >0 . We still haveṙ ∈ A ≥α+1 and thus we obtain a non-monodromic situation. Case α > α 0 . We deducė r = r α 0 +1 (a 0 α 0 U + Υ) andφ ∈ A >α 0 with Υ ∈ A >0 . We obtain the dicritical situation (8) with µ = α 0 + 1 thanks to Lemma 25. This finishes all the cases and the proof of the Main Theorem 1.
Consequences
Now we prove Corollary 2 and Theorem 4 as consequences of our main result, Theorem 1. We also sketch a proof of the more elementary Proposition 3.
Proof of Corollary 2. Suppose that | γ | ⊂ S 0 , a connected component of X \ {0}. Let β be an opening blowing-up of S 0 and let R = ( S, E, σ) be a resolution of the surface S = β −1 (S 0 ). Theorem 1 ensures that the lifting L = (β • σ) −1 (| γ |) accumulates at a single point p of E. Thus L is contained in a simply connected semi-analytic open set of the strict transform S ′ = σ −1 (S) where the foliation F has no singularities. Using Haefliger's theorem ( [12,21,25]), we deduce that L is a Rolle's leaf of F and thus a pfaffian set. Its image | γ | = β(σ(L)) is a sub-pfaffian set in R n since σ and β are proper mappings.
Proof of Proposition 3. Let S 0 be a connected component of X \ {0}, homeomorphic to the semi-open cylinder S 1 ×]0, ε] for ε small. Denote by C ε the image of S 1 × {ε} by such homeomorphism. We consider two cases.
Case 1: The function f 0 | S 0 is negative and has no critical point. Let a 0 < 0 be the minimum of the function f 0 restricted to C ε . Consider a point p ∈ S 0 for which a 0 < f 0 (p) < 0 and let γ p be the trajectory of the restricted gradient vector field ∇ h f 0 starting at p. Since t → f 0 (γ p (t)) is increasing, γ p is defined for all positive t and lim t→∞ γ p (t) = 0.
Case 2. Suppose f −1 0 (0) ∩ S 0 = ∅. Up to taking −f 2 0 instead of f 0 ,
we assume that f 0 ≤ 0 on clos(S 0 ) and that
Z 0 = f −1 0 (0) ∩ clos(S 0 )(= crit(f 0 | S 0 )) intersects with S 0 . Let U be a connected component of S 0 \ Z 0 .
Since S 0 is topologically a cylinder and Z 0 consist of finitely many analytic half-branches at 0 (up to taking ε smaller), the component U is simply connected. In fact, we can take a triangle Σ in the plane with sides σ 1 , σ 2 , σ 3 and a continuous map κ : Σ → clos(U ) restricting to a diffeomorphism between Σ \ (σ 1 ∪ σ 2 ) and U , sending each of the sides σ 1 or σ 2 homeomorphically to a half-branch of Z 0 and sending the side σ 3 onto clos(U ) ∩ C ε . Note that, if there are at least two half-branches of Z 0 then κ is a homeomorphism, otherwise clos(U ) = clos(S 0 ) and κ is just a quotient map gluing the two sides σ 1 , σ 2 together.
Since clos(U ) is invariant, we can carry ∇ h f 0 onto Σ via κ (which is singular along σ 1 ∩ σ 2 ). It will be denoted by χ 0 while we will denote g 0 = κ * f 0 and v = σ 1 ∩ σ 2 = κ −1 (0). We just have to prove that there exists a trajectory of χ 0 accumulating to v.
We use the following properties:
(1) Up to taking a smaller ε, each point x ∈ σ 1 ∪ σ 2 \ {v} is the accumulation point of a unique trajectory of χ 0 . (2) No non-stationary trajectory of χ 0 can have its α-limit point and its ω-limit point both in σ 1 ∪ σ 2 \ {v}.
The first property is easy to prove using local coordinates or using the classical Lojasiewicz's retraction map of the gradient (cf. [18]). The second one is a consequence of the fact that g 0 (σ 1 ∪ σ 2 ) = 0 and that g 0 increases along trajectories of χ 0 .
Claim. There exists t ε < 0 such that for t ∈]t ε , 0[, the fiber g −1 0 (t) ⊂ Σ is connected. Proof of the Claim. Each connected component of a (non-empty) fiber g −1 0 (t) with t < 0 is either homeomorphic to a circle or to a closed segment with extremities on σ 3 . Since g 0 has no critical points in the interior of the triangle Σ, the first case cannot occur. On the other hand, the restriction g 0 | σ 3 vanishes only at the extremities. If we take t ε equal to the maximum of the critical values of this restriction, g 0 takes any value t ∈]t ε , 0[ exactly twice along σ 3 and this proves the claim.
For i = 1, 2, choose x i ∈ σ i \ {v} and let γ i be the trajectory of χ 0 accumulating to x i . Take t 0 with t ε < t 0 < 0 such that γ i cuts the fiber g −1 0 (t 0 ), necessarily in a single point y i . Let I be the closed segment in (1) and (2) above, there exists a unique point τ (z) ∈ I such that the trajectory starting at τ (z) accumulates to z for positive infinite time. Moreover, orienting positively I from y 1 to y 2 , we find that τ (z) < τ (w) whenever Proof of Theorem 4. We begin with the definition of formal asymptotic expansion. A formal curve Γ at the origin of R n is a formal Puiseux parameterization Γ(T ) = ( Γ 1 (T ), . . . , Γ n−1 (T ), T N ), where each Γ i is a formal power series in the single indeterminate T with no constant term. A trajectory γ has an asymptotic expansion Γ at the origin if it can be smoothly parameterized as z → γ(z) = (γ 1 (z), . . . , γ n−1 (z), z N ), z > 0 and the component γ i admits Γ i as expansion.
z ∈ [x 1 , v[ and w ∈ [x 2 , v[, or z ∈ [x 1 , v[ and w ∈]z, v[, or w ∈ [x 2 , v[ and z ∈]w, v[.
If the critical locus of f 0 | S 0 is not empty then, each connected component of this locus in a semi-analytic invariant set the restricted gradient, so point (ii) is true.
We assume the restricted gradient of f 0 does not vanish in S 0 . Since any restricted gradient trajectory γ is not oscillating at 0, the function z • γ(t) decreases strictly to 0 as t → +∞. Thus it admits a continuous parameterization z → γ(z) = (γ 1 (z), . . . , γ n−1 (z), z), for z ≥ 0, which is analytic for z > 0.
Let β be an opening blowing-up of S 0 and S = β −1 (S 0 ). Let R = ( S, E, σ) be a resolution of S and let R ′ = (S ′ , E ′ , σ ′ ) be the strict resolution associated to R as in Theorem 9. Let h = (β • σ) * g and f = f 0 • β • σ. The metric h degenerates along the divisor E, so the gradient vector field ∇ h f is defined only on S \ E. However, we can define a one-dimensional analytic foliation F in S whose singular set sing( F ) is a finite subset of E and such that ∇ h f is a local generator of F at any point of S \ E.
The reduction of singularities of an analytic foliation on a smooth surface ( [26]) and the compactness of E ensure we can assume that any singularity p ∈ sing( F ) is simple: a local generator ξ p has a non-nilpotent linear part at p with eigenvalues λ and µ = 0.
Let Σ := E ′ ∩ sing( E) be the finite set of singular points of the strict divisor E ′ . In order to complete the proof, we will show that only situations (1) or (2) below happen and the result holds true in both cases.
(1) Dicritical situation: There is a point p ∈ E ′ \ Σ, not singular for F , and such that E ′ is transverse to F at p. The leave L p of F through p is a nonsingular analytic curve and transverse to E ′ at p. The image β • σ(L p ∩ S ′ ) is an analytic separatrix for the restricted gradient on S 0 accumulating to the origin. In fact through each point q ∈ E ′ in a neighborhood of p, there is a unique analytic separatrix through q.
(2) A non corner singularity: The strict divisor E ′ is an invariant set of F and there is a point p ∈ (E ′ \Σ)∩ sing( F). A local generator ξ p of F at p has a linear part with two real eigenvalues. One eigen-direction is tangent to E ′ and the other one is transverse to E ′ . The theory of local invariant manifolds (see for instance [14]) provides a formal invariant non-singular manifold W at p which is tangent to the transverse eigen-direction 1 . We also get a C ∞ invariant manifold W through p having W as asymptotic expansion at p. The image β • σ(W ∩ S ′ ) is the desired characteristic trajectory γ of the restricted gradient.
In order to find a contradiction, we assume that neither case (1) or (2) above holds. Thus E ′ is invariant for F and sing( F ) ∩ E ′ = Σ. Since any point p ∈ Σ is a simple singularity, the two components of E at p are the two local analytic separatrices of F at p. Let {Q j p } j=1,2,3,4 be the open "quadrants" of U p \ E in a small coordinate neighborhood U p of p in S. Let J(p) ⊂ {1, 2, 3, 4} be the subset of j for which Q j p ⊂ S ′ (see part (iv) of Theorem 9). For j ∈ J(p), the quadrant Q j p is either: -of saddle type, if any trajectory of the restricted gradient ∇ h f through a 1 If the corresponding eigenvalue is non-zero, Briot-Bouquet's theorem guarantees the convergence of W point in Q j limit point which is impossible since the function f 0 increases strictly along this trajectory.
Thanks
This work started during the Thematic Program at the Fields Institute on O-minimal Structures and Real Analytic Geometry January-June 2009. The authors are both very grateful to the Fields Institute and its staff for support, facilities and very good working conditions they found there.
The first author would like to thank the Departamento deÁlgebra, Geometría y Topología of the University of Valladolid for the support and the working conditions provided while visiting to complete this work.
The second author was partially supported by the research projects MTM2007-66262 (Ministerio de Educación y Ciencia) and VA059A07 (Junta de Castilla y León) and by Plan Nacional de Movilidad de RR.HH. 2008/11, Modalidad "José Castillejo".
The authors want to thank also O. LeGal for useful commentaries and remarks.
Theorem 4 .
4Let S 0 be a connected component of X \ {0}.
Then β induces a diffeomorphism from M ×]0, ε 0 ] onto its image. Let S := β −1 (S 0 ), D := {z = 0} ⊂ M × R and E := clos(S) ∩ D. Then E is a closed bounded connected subanalytic curve of D of dimension one.
thus given rise to a continuous function, also denoted G : [0, t 0 ] → R, analytic for t > 0, called the sum of the convergent series. If the exponents α k are in N l for some positive integer l, then G(T ) is called a Puiseux series. If all but finitely many coefficients a k are non-zero then G(T ) is a Q-generalized real polynomial. Let X ⊂ R n be an analytic isolated surface singularity at 0 and let S 0 be a connected component of X \ {0}. Let β : M × R ≥0 → R n be an opening blowing-up of S 0 and denote S = β −1 (S 0 ), D = {z = 0} = M × {0}, E = clos(S) ∩ D as in the previous section. Let f : U → R be a continuous function in U , a neighborhood of E in M × R ≥0 , which is ramified-analytic along D. Let f S : clos(S) → R be the restriction of f to clos(S) = S ∪ E. Given a (S, z)-resolution R = ( S, E, σ) and an associated parameterization Φ : S 1 × [0, ε] → S as in Proposition 14, we denote by f
Proposition 21 .
21Assume that ξ 0 does not vanish in S 0 . Let γ : [0, +∞[→ S 0 be a non-trivial trajectory of ξ 0 accumulating at 0. Then γ is oscillating if and only if it spirals in X.
Proposition 22 .
22Assume that ξ 0 does not vanish in S 0 . Suppose that the transformed vector field ξ = (β •Φ) * ξ 0 on the open cylinder S 1 ×]0, ε] satisfies one of the following non-exclusive situations:
, dividing ξ by r µ , gives a vector field which extends to U ∩ S 1 × [0, ε] as a ramified-analytic vector field transverse to C ∩ U . Thus any point of C ∩ U is the unique accumulation point of a trajectory of ξ living in the open cylinder S 1 ×]0, ε].
with µ = 2eN + α 0 − 1 thanks to Lemma 25. This finishes the proof of the main theorem in the CTC case. Open tangent cone case. Let us have a quick look at the open tangent case (OTC). We first choose orthonormal coordinates x = (x 1 , . . . , x n ). Consider the opening blow-up mapping β : M ×R ≥0 → R n , (y, z) → zy, where M = S n−1 , as in (2), and let S = β −1 (S 0 ). Let R = ( S, E, σ) be a (S, z)-resolution of S as in Theorem 9 and pick an associated parameterization Φ : S 1 × [0, ε] → S satisfying the conditions of Proposition 14. Let h = (β • Φ) * h be the pull-back metric in the open cylinder. With computations similar to those done in the proof of Lemma 26, and using Remark 13, we can write
g − 1 0
1(t 0 ) joining y 1 and y 2 . Consider the domain Λ in Σ enclosed by the piecewise smooth closed curve formed by the segments [x i , v] in σ i , [y i , x i ] in γ i and I. By construction, χ 0 enters Λ only through the segment I and leaves Λ positively invariant. For each z in one of the semi-open sides [x 1 , v[ or [x 2 , v[, thanks to properties
Let a = sup{τ (z)/z ∈ [x 1 , v[} and b = inf{τ (z)/z ∈ [x 2 , v[}.Thus a ≤ b and for every point y ∈ [a, b] in the segment I, the trajectory of χ 0 starting at y accumulates to v.
p escapes from Q j p for positive and negative time; -of node-source type, if any trajectory escapes for positive time but accumulates to p for negative time; -of node-sink type, if each trajectory escapes for negative time but accumulates to p for positive time.We have two possibilities:(a) Each quadrant Q j p is of saddle-type for all p ∈ Σ and all j ∈ J(p) or there are no singularities at all (Σ = ∅). In this case, classical arguments show that the dynamics is "monodromic" in a neighborhood of E ′ in S ′ : there exist an analytic half-branch Λ through a point q ∈ E ′ \Σ contained in S ′ , transverse to F, a neighborhood Λ 0 of q in Λ and a Poincaré first return map P : Λ 0 → Λ such that for x ∈ Λ 0 given, the leaf L x through x cuts again Λ at P (x) after visiting all the quadrants Q j p . Since a gradient vector field cannot have closed orbits, P has no fixed points. Poincaré-Bendixson's type arguments imply there are leaves of F in S ′ accumulating to the whole divisor E ′ , thus producing spiraling trajectories of the restricted gradient, which contradicts Theorem 1.(b) There is a quadrant Q j 0 p 0 ⊂ S ′ of node type for some singularity p 0 ∈ Σ and some j 0 ∈ J(p 0 ). Suppose for instance that it is of node-source type (the case of node-sink type is analogous in reversing time). Consider one of the local analytic separatrices of F at p 0 . There is a connected component of E ′ \ Σ, say E 1 , which meets such a separatrix. Since E ′ is invariant, E 1 is a leaf of F. Let p 1 ∈ Σ be the other accumulation point of E 1 , different from p 0 . The flow-box theorem shows that there is a point q 0 ∈ Q j 0 p 0 such that the trajectory γ 0 issued from the point q 0 visits a point in a quadrant Q j 1 p 1 for some j 1 ∈ J(p 1 ). By definition of a node-source type, the quadrant Q j 1 p 1 cannot be of nodesource type. If Q j 1 p 1 is of saddle-type, we consider the connected component E 2 of E ′ \ Σ meeting the local analytic separatrix at p 1 which is not contained in E 1 and the point p 2 ∈ Σ such that clos(E 2 ) \ E 2 = {p 1 , p 2 }. In this case, choosing q 0 in the initial quadrant Q j 0 p 0 sufficiently close to p 0 , we can suppose that the trajectory γ 0 also visits some quadrant Q j 2 p 2 for some j 2 ∈ J(p 2 ). Continuing this way, if all the visited quadrants are of saddle-type we construct a sequence of singularities p 1 , p 2 , . . . different from p 0 . Since Σ is finite, we create a cycle p l , p l+1 , . . . , p m = p l , with l minimum for this property. Since E m is not equal to E l (otherwise p m−1 = p l−1 against the minimality of l) we find three local analytic separatrices through p l , say E l , E l+1 , E m , which is a contradiction with the fact that p l is a simple singularity.Thus, there exist p k ∈ Σ, for some k ≥ 1 and a quadrant Q j k p k of node-sink type, for some j k ∈ J(p k ), which intersects the trajectory γ 0 . Then p 0 is the α-limit point of γ 0 and p k a ω-limit point. Its image β • σ • γ 0 is a trajectory of the restricted gradient for which the origin is the α and the ω
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E-mail address: [email protected] Permanent Address: F. Sanz, University of Valladolid, Departamento dé Algebra, Geometría y Topología, Facultad de Ciencias, Prado de la Magdalena s/n, E-47006. Toronto, Ontario, M5T 3J1, Canada; Valladolid (Spain). FaxPermanent Address: V. Grandjean, Department of Computer Science, University of Bath, BATH BA2 7AY, EnglandUnited Kingdom) E-mail address: [email protected] Temporary Address: V. Grandjean, Fields Institute, 222 College Street. 34) 983 423788 E-mail address: [email protected] Address: V. Grandjean, Department of Computer Science, Univer- sity of Bath, BATH BA2 7AY, England,(United Kingdom) E-mail address: [email protected] Temporary Address: V. Grandjean, Fields Institute, 222 College Street, Toronto, Ontario, M5T 3J1, Canada E-mail address: [email protected] Permanent Address: F. Sanz, University of Valladolid, Departamento dé Algebra, Geometría y Topología, Facultad de Ciencias, Prado de la Mag- dalena s/n, E-47006, Valladolid (Spain). Fax: (34) 983 423788 E-mail address: [email protected]
| []
|
[
"Semiclassical Trace Formula and Spectral Shift Function for Systems via a Stationary Approach",
"Semiclassical Trace Formula and Spectral Shift Function for Systems via a Stationary Approach"
]
| [
"Marouane Assal ",
"Mouez Dimassi ",
"Setsuro Fujiié "
]
| []
| []
| We establish a semiclassical trace formula in a general framework of microhyperbolic hermitian systems of h-pseudodifferential operators, and apply it to the study of the spectral shift function associated to a pair of selfadjoint Schrödinger operators with matrix-valued potentials. We give Weyl type semiclassical asymptotics with sharp remainder estimate for the spectral shift function, and, under the existence of a scalar escape function, a full asymptotic expansion in the strong sense for its derivative. A time-independent approach enables us to treat certain potentials with energy-level crossings.2010 Mathematics Subject Classification. 81Q10 (47A55, 81Q20, 47N50). | 10.1093/imrn/rnx149 | [
"https://arxiv.org/pdf/1702.07880v1.pdf"
]
| 119,575,994 | 1702.07880 | 285c601ec2e4f0bd704928b0fceab3f046d6163a |
Semiclassical Trace Formula and Spectral Shift Function for Systems via a Stationary Approach
25 Feb 2017
Marouane Assal
Mouez Dimassi
Setsuro Fujiié
Semiclassical Trace Formula and Spectral Shift Function for Systems via a Stationary Approach
25 Feb 2017
We establish a semiclassical trace formula in a general framework of microhyperbolic hermitian systems of h-pseudodifferential operators, and apply it to the study of the spectral shift function associated to a pair of selfadjoint Schrödinger operators with matrix-valued potentials. We give Weyl type semiclassical asymptotics with sharp remainder estimate for the spectral shift function, and, under the existence of a scalar escape function, a full asymptotic expansion in the strong sense for its derivative. A time-independent approach enables us to treat certain potentials with energy-level crossings.2010 Mathematics Subject Classification. 81Q10 (47A55, 81Q20, 47N50).
Introduction
In this paper, we study the spectral shift function (SSF for short) for Schrödinger operators with matrix-valued potentials. Such operators appear in molecular physics in the Born-Oppenheimer approximation. The justification of this approximation and a classification of matrix Schrödinger operators can be found in [5,10,15,17].
More precisely, we are concerned with the SSF for the pair of operators (P 1 , P 0 ) with (1.1)
P 0 := −h 2 ∆ ⊗ I N + V ∞ , P 1 := −h 2 ∆ ⊗ I N + V (x),
where h ∈ (0, 1] is a small positive parameter, I N is the identity N × N matrix, V ∞ is an N × N constant hermitian matrix and V (x) is a smooth hermitian matrix-valued potential which tends rapidly enough to V ∞ at infinity. The SSF associated to (P 1 , P 0 ) denoted s h is defined as distribution (modulo a constant) by the Lifshits-Krein formula
(1.2) s ′ h , f = −tr f (P 1 ) − f (P 0 )
, ∀f ∈ C ∞ 0 (R; R). The SSF is related with the eigenvalue counting function below the level inf σ(P 0 ) and with the scattering determinant above this level (Birman-Krein formula, see [36]). Here σ(P 0 ) stands for the spectrum of P 0 .
The concept of the SSF was introduced in the middle of the previous century by I. M. Lifshits in his investigations in the solid state theory (see [22,23]) and then developed by M. Krein (see [20,21,19]) into a mathematical theory. The work of Krein on the SSF has been described in details in the survey [3]. One can also find detailed account concerning mathematical and historical aspects of the SSF in [2].
In the scalar case N = 1, a lot of works have been devoted to the study of the SSF in different asymptotic regimes (see [28] and the references therein). In particular, a Weyl-type asymptotics of the SSF with a sharp remainder estimate and a complete asymptotic expansion of the derivative of the SSF were studied in high energy regime ( [29]) and in the semiclassical regime ( [30], [31]).
The proofs of these works reduce to the study of (1.3) tr
f (P 1 )F −1 h θ(τ − P 1 ) − f (P 0 )F −1 h θ(τ − P 0 ) ,
where θ is a smooth function of the time t with compact support and F −1 h is the semiclassical Fourier inverse transform defined by (2.2). The method in [30] consists in writing (1.3) as the semiclassical Fourier inverse transform of θ(t) tr f (P 1 )e −itP 1 /h − f (P 0 )e −itP 0 /h , and constructing (modulo O(h ∞ )) the Schwartz' kernel of the evolution operator f (P 1 )e −itP 1 /h . This construction by means of Fourier integral operators is now standard and well known for scalar-valued operators P 1 (see [12,14] for problems concerning the asymptotic distribution of eigenvalues, and [28,29,30] for the SSF). For matrix-valued operators this explicit construction is very complicated (or impossible). To avoid this problem, and to study the counting function of eigenvalues of P 1 , V. Ivrii [14] observed that a rough construction by using the successive approximation method of f (P 1 )e −itP 1 /h for |t| < h 1−δ (with 0 < δ ≤ 1) suffices to get a full asymptotic expansion in powers of h of tr (f (P 1 )F −1 h θ(τ − P 1 )). This beautiful observation is used by the second author and J. Sjöstrand [7] to develop a time-independent approach to get asymptotics of tr (f (P 1 )F −1 h θ(τ − P 1 )) for matrix-valued operator P 1 . The novelty in this approach consists in expressing (1.3) in terms of the resolvent instead of evolution operator, and studying the (almost) analyticity of its trace near the real axis. This method is used in [8] to study the SSF for scalar non semi-bounded operators such as Stark Hamiltonian. The aim of this paper is to develop and apply this stationary approach to the study of the SSF for matrix-valued operators.
In the first part of this work, we consider a general system of h-pseudodifferential operator H w = H w (x, hD x ). For a fixed energy τ 0 such that τ 0 − H(x, ξ) is uniformly microhyperbolic in some direction T (see Definition 2.1), we show that the trace of the operator χ w f (H w )F −1 h θ(τ −H w ) is negligible (= O(h ∞ )) provided that θ is supported in h 1−δ ≤ |t| ≤ κ (for arbitrary positive h-independent κ), see Theorem 2.2. Here χ ∈ C ∞ 0 (R 2n ; R) and f is supported in a small neighborhood of τ 0 . Moreover, under the existence of an escape function associate to H(x, ξ) at τ 0 (see (2.15)), we can take κ = h −ν for arbitrary ν > 0 (see Remark 3.1 and section 4.4). On the other hand, we give a complete asymptotic expansion in powers of h of tr(χ w f (H w )F −1 h θ(τ − H w )) provided that θ is supported in a small h-independent neighborhood of 0 and τ 0 − H(x, ξ) is microhyperbolic at every point (x, ξ) ∈ suppχ, see Theorem 2.4. This is a consequence from the fact that the above trace depends, modulo O(h ∞ ), only on the symbol τ 0 − H(x, ξ) on the support of χ as long as the support of θ is small enough near 0 (Theorem 2.3), and the fact that a symbol τ 0 − H microhyperbolic near a point can be extended to a uniformly microhyperbolic symbol in the whole phase space (Theorem A.3).
To our best knowledge, there are only few works treating the semiclassical asymptotics of the SSF for matrix valued operators (see [4,18] and the references therein). The asymptotics of the SSF for the semi-classical Dirac operator has been studied in [4]. In this case, the classical corresponding Hamiltonian has uniformly distinct eigenvalues, and then the study of the SSF can be reduced to the scalar case by diagonalization. The relation between the spectral shift function and the resonances for Dirac operator with analytic potential has been examined in [18]. In the second part of this paper, we consider the SSF associated to the pair of Schrödinger operators with matrix-valued potentials defined in (1.1), without any condition on the multiplicities of its eigenvalues. First, using Theorem 2.4, we show that (1.3) has a full asymptotic expansion in h when the support of θ is close enough to the origin (Theorem 2.6). This result with a Tauberian argument give the Weyl-type asymptotic formula for the SSF with a sharp remainder estimate (Theorem 2.7). Finally we give a pointwise full asymptotic expansion of the derivative of the SSF near energies τ where there exists a scalar escape function associated to the classical Hamiltonian ξ 2 I N + V (x) (Theorem 2.8). This last theorem is a generalization to the matrix case of the result of [30] at non-trapping energies.
The paper is organised as follows. In section 2, we state our main results and we give an outline of the proofs. The proofs of these results will be given in Sections 3 and 4 respectively. Finally, the appendix A contains some technical lemmas related to the notion of microhyperbolicity used in our proofs.
Notations : For ξ = (ξ 1 , ..., ξ n ) ∈ R n , we use the usual notation ξ := (1 + ξ 2 ) 1/2 , where ξ 2 := ξ 2 1 + ... + ξ 2 n = |ξ| 2 . For z ∈ C, we recall that∂ z := 1 2 (∂ ℜz + i∂ ℑz ). The bracket [a j ] 1 0 stands for the difference a 1 − a 0 . The scalar products in R n and C N will be denoted , and ( , ) respectively. We introduce the following standard asymptotic notations that we shall use through the paper. Given a function f h depending on a small parameter h ∈ (0, 1], the
relation f h = O(h ∞ ) (or f h ≡ 0) means that f h = O(h k ), for all k ∈ N and h small enough. We write f h ∼ j≥0 γ j h j provided that for each k ∈ N, f h − k j=0 γ j h j = O(h k+1 ).
Statement of the results
Let H N be the space of hermitian N × N matrices endowed with the norm · N ×N , where for A ∈ H N , A N ×N := sup {w∈R N ; |w|<1} |Aw|.
Throughout this work we will use the notations of [7] for symbols and h-pseudodifferential operators (see also [14]). In particular, S 0 (R 2n ; H N ) is the class of symbols
S 0 (R 2n ; H N ) := {H ∈ C ∞ (R 2n ; H N ); ∂ α x ∂ β ξ H(x, ξ) N ×N = O α,β (1), ∀α, β}.
We use the standard Weyl quantization of symbols. More precisely, if H ∈ S 0 (R 2n ; H N ) then H w (x, hD x ) is the operator defined by
H w (x, hD x )u(x) = 1 (2πh) n R 2n e i(x−y)·ξ/h H x + y 2 , ξ u(y)dydξ, u ∈ C ∞ 0 (R n ; C N ).
We will occasionally use the shorthand notations Op w h (H) = H w = H w (x, hD x ) when there is no ambiguity.
We recall the following notion of microhyperbolicity which will play an important role in this paper.
Definition 2.1 (Microhyperbolicity). Let H ∈ C ∞ (R 2n ; H N ). We say that H(x, ξ) is micro- hyperbolic at (x 0 , ξ 0 ) in the direction T ∈ R 2n , if there are constants C 0 , C 1 , C 2 > 0 such that (2.1) T, ∇ x,ξ H(x, ξ) w, w ≥ C 0 |w| 2 − C 1 |H(x, ξ)w| 2 ,
for all (x, ξ) ∈ R 2n with |(x, ξ) − (x 0 , ξ 0 )| ≤ 1 C 2 and all w ∈ C N . Here ∇ x,ξ H(x, ξ) = (∂ x H(x, ξ), ∂ ξ H(x, ξ)). If for some constants C 0 , C 1 > 0 the above estimate holds for all (x, ξ) ∈ R 2n , we say that H(x, ξ) is uniformly microhyperbolic on R 2n in the direction T . In the case where H(x, ξ) depends also on an additional parameter, we say that H is uniformly microhyperbolic in the direction T if (2.1) is satisfied with C 0 , C 1 > 0 independent of this parameter.
2.1. Trace formula for systems of h-pseudodifferential operators. Let
θ ∈ C ∞ 0 (] − 1, 1[; R), θ ε (t) := θ(t/ε), where ε > 0 is a positive constant possibly depending on h and (2.2) F −1 h θ ε (τ ) = 1 2πh R e itτ /h θ ε (t)dt,
the semiclassical Fourier inverse operator.
Let A, H ∈ S 0 (R 2n ; H N ), and χ ∈ C ∞ 0 (R 2n ; R). We assume that A w (i + H w ) −k is of trace class for some k ∈ N. Writing A w f (H w ) = A w (i + H w ) −k (i + H w ) k f (H w ) and using the fact that (i+ H w ) k f (H w ) is bounded by the spectral theorem we deduce that A w f (H w ) is of trace class for all f ∈ C ∞ 0 (R; R)
. We recall that χ w is of trace class (with norm trace O(h −n ), see [7,Theorem 9.4]).
Fix τ 0 ∈ R. We denote by O τ 0 the set of open intervals centered at τ 0 , i.e.,
O τ 0 = {]τ 0 − η, τ 0 + η[; η > 0}. Theorem 2.2. Suppose that there exists T ∈ R 2n such that τ 0 − H(x, ξ) is uniformly micro- hyperbolic with respect to (x, ξ) ∈ R 2n in the direction T . If 0 / ∈ supp θ, then there exists I ∈ O τ 0 such that for all f ∈ C ∞ 0 (I; R) and ε ∈ [h 1−δ , κ[ with κ > 0, 0 < δ ≤ 1 independent of h, we have, uniformly for τ ∈ R, (2.3) tr A w f H w )F −1 h θ ε τ − H w ) = O(h ∞ ). Theorem 2.3. Let H 0 , H 1 ∈ S 0 (R 2n , H N ) be such that H 0 = H 1 in a neighborhood of supp χ.
Then there exists ε > 0 small and independent of h such that we have, uniformly for τ ∈ R, (2.4) tr
χ w f (H w j )F −1 h θ ε (τ − H w j ) 1 0 = O(h ∞ ).
The following result is a simple consequence of the above theorems.
Theorem 2.4. Suppose that τ 0 − H(x, ξ) is microhyperbolic at every point (x, ξ) in supp χ. If θ equals 1 near t = 0, then there exist I ∈ O τ 0 and ε > 0 small and independent of h such that for f ∈ C ∞ 0 (I; R), the following full asymptotic expansion in powers of h holds uniformly for τ ∈ R:
(2.5) tr χ w f (H w )F −1 h θ ε (τ − H w ) ∼ (2πh) −n f (τ ) j≥0 γ j (τ )h j as h ց 0.
Remark 2.2. The coefficients τ → γ j (τ ) are smooth, independent of f and θ and can be computed explicitly (see formula (3.25)).
2.2.
Application to Schrödinger operators with matrix-valued potentials. In this section we apply the above trace formula to study the spectral properties of multi-channel semiclassical Schrödinger operators of the form
(2.6) P 1 (h) := −h 2 ∆ ⊗ I N + V (x), P 0 (h) := −h 2 ∆ ⊗ I N + V ∞ , in L 2 (R n ; C N ),
where I N is the N ×N identity matrix and V (x) is a smooth hermitian matrix-valued potential, i.e.,
V (x) = V ij (x) 1≤i,j≤N , V ij (x) = V ji (x).
We assume that the matrix V has a limit V ∞ at infinity and
(2.7) ∃µ > n s.t. ∂ α x (V (x) − V ∞ ) N ×N = O α ( x −µ−|α| )
, ∀α ∈ N n , ∀x ∈ R n . After a linear transformation, we may assume that
V ∞ = e 1,∞ 0 · · · 0 0 e 2,∞ · · · 0 . . . 0 . . . . . . 0 · · · 0 e N,∞ , with e 1,∞ ≤ e 2,∞ ≤ · · · ≤ e N,∞ .
The operator P 0 (h) with domain H 2 (R n ; C N ) is self-adjoint. Its spectrum is [e 1,∞ , +∞[. Since V − V ∞ is ∆-compact, the operator P 1 (h) admits a unique self-adjoint realization in L 2 (R n ; C N ) with domain H 2 (R n ; C N ). Moreover the essential spectra of P 1 (h) and P 0 (h) are the same. The operator P 1 (h) may have discrete eigenvalues in (−∞, e 1,∞ ) and embedded ones in the interval [e 1,∞ , e N,∞ ] contained in the continuous spectrum.
The spectral shift function s h (τ ) associated to (P 1 (h), P 0 (h)) is defined as a real-valued function on R satisfying the Lifshits-Krein formula
(2.8) s ′ h (·), f (·) = −tr f (P 1 (h)) − f (P 0 (h)) , ∀f ∈ C ∞ 0 (R; R)
. The function s h (τ ) is fixed up to an additive constant by the formula (2.8), and we normalize it so that s h (τ ) = 0 for τ < inf(σ(P 1 (h)).
We denote by p 1 (x, ξ) := ξ 2 I N + V (x) and p 0 (x, ξ) := ξ 2 I N + V ∞ , (x, ξ) ∈ R 2n , the classical Hamiltonians associated with the operators P 1 (h) and P 0 (h), respectively. Let e 1 (x) ≤ e 2 (x) ≤ ... ≤ e N (x) be the eigenvalues of V (x) arranged in increasing order.
Theorem 2.5. Assume (2.7) and let f ∈ C ∞ 0 (R; R). There exists a sequence of real numbers (c 2j (f )) j∈N such that
(2.9) s ′ h (·), f (·) ∼ (2πh) −n j≥0 c 2j (f )h 2j as h ց 0, with (2.10) c 0 (f ) = ω n 2 N k=1 R n +∞ 0 f (e k,∞ + τ ) − f (e k (x) + τ ) τ n−2 2 dτ dx,
where ω n is the volume of the unit sphere S n−1 .
For τ 0 ∈ R, set
Σ τ 0 := N k=1 {(x, ξ) ∈ R 2n ; ξ 2 + e k (x) = τ 0 }.
The following theorem is a consequence of Theorem 2.4.
Theorem 2.6 (Weak asymptotics). Let τ 0 ∈ {e 1,∞ , e 2,∞ , · · · , e N,∞ }. Assume (2.7) and τ 0 − p 1 (x, ξ) is microhyperbolic at every point (x, ξ) ∈ Σ τ 0 . Then, if θ is equal to 1 near the origin, there exist I ∈ O τ 0 and ε small enough and independent of h such that for f ∈ C ∞ 0 (I; R), the following asymptotic formula holds uniformly for τ ∈ R:
(2.11) s ′ h (·), F −1 h θ ε (τ − ·)f (·) ∼ (2πh) −n f (τ ) j≥0 γ 2j (τ )h 2j as h ց 0.
The coefficients γ 2j (τ ) are smooth functions of τ , independent of f and θ. In particular,
(2.12) γ 0 (τ ) = ω n 2 N k=1 R n (τ − e k (x)) n−2 2 + − (τ − e k,∞ ) n−2 2 + dx,
where τ + := max (τ, 0). Remark 2.3. According to Definition 2.1, the assumption that τ 0 − p 1 (x, ξ) is microhyperbolic at every point (x, ξ) ∈ Σ τ 0 is equivalent to the following condition: For x 0 with e j (x 0 ) = τ 0 , j = 1, ..., N , there exists T 1 ∈ R n and C > 0 such that
T 1 , ∇ x V (x 0 ) ω, ω ≥ 1 C |ω| 2 , ∀ω ∈ ker(V (x 0 ) − τ 0 I N ).
In particular, if e j (x 0 ) is a simple eigenvalue of V (x 0 ), this is equivalent to ∇e j (x 0 ) = 0.
As a consequence of Theorem 2.6, we get a sharp remainder estimate for the spectral shift function corresponding to the pair (P 1 (h), P 0 (h)).
Theorem 2.7 (Weyl-type asymptotics). Assume that (2.7) holds with V ∞ = 0. Let τ 0 = 0 such that τ 0 − p 1 (x, ξ) is microhyperbolic at every point (x, ξ) ∈ Σ τ 0 . There exists I ∈ O τ 0 such that
(2.13) s h (τ ) = (2πh) −n a 0 (τ ) + O(h −n+1 ) as h ց 0,
uniformly for τ ∈ I, with
(2.14) a 0 (τ ) = ω n n N k=1 R n (τ − e k (x)) n 2 + − τ n 2 + dx.
As indicated in the introduction, in the scalar case a complete asymptotic expansion in powers of h of the derivative of the SSF has been obtained under a non-trapping condition on the classical trajectories corresponding to the energy surface Σ τ 0 (see [30]). In the present matrix-valued case, the treatment is much more complicated. In fact, since the eigenvalues are not enough regular, the usual definition of the Hamilton flow for a matrix-valued Hamiltonian function does not make sense (see [16]). For this reason, we use here the notion of escape function.
More precisely, we suppose that there exists a scalar escape function G ∈ C ∞ (R 2n ; R) associated to p 1 at τ 0 , i.e.,
(2.15) ∃ C > 0, s.t. {p 1 , G}(x, ξ) := ∂G ∂x · ∂p 1 ∂ξ − ∂G ∂ξ · ∂p 1 ∂x ≥ C, ∀(x, ξ) ∈ Σ τ 0 ,
in the sense of hermitian matrices.
In the scalar case N = 1, it is well known that the above assumption is equivalent to the non-trapping condition on the energy τ 0 . In fact, if τ 0 is non-trapping for the classical Hamiltonian p 1 , one can construct an escape function G ∈ C ∞ (R 2n ; R) satisfying (2.15) (see for instance [9], [33], [34], [35]). Conversely, if (2.15) holds then one easily sees that G is strictly increasing along the Hamiltonian flows associated to p 1 in Σ τ 0 which prevents the existence of trapped trajectories at τ 0 . We also point out that (2.15)
implies that τ 0 − p 1 (x, ξ) is microhyperbolic at every point (x, ξ) ∈ Σ τ 0 in the direction of the Hamiltonian vector field (∂ ξ G(x, ξ), −∂ x G(x, ξ)).
Now we can formulate the main result of this paper.
Theorem 2.8 (Strong asymptotics). Fix an energy τ 0 > e N,∞ . Assume that (2.7) and (2.15) are satisfied. Then, there exists I ∈ O τ 0 such that s ′ h (·) has a complete asymptotic expansion of the form
(2.16) s ′ h (τ ) ∼ (2πh) −n j≥0 γ 2j (τ )h 2j as h ց 0,
uniformly for τ ∈ I, where the coefficients γ 2j (τ ) are given in Theorem 2.6.
Examples and further generalizations. First observe that, for
G(x, ξ) = x · ξ, (2.15) is equivalent to (2.17) 2(τ 0 − e k (x)) − x · ∇V (x) ≥ C, ∀x ∈ {x ∈ R n ; τ 0 − e k (x) ≥ 0}, k = 1, · · · N.
Thus, under the assumption (2.7), the asymptotics (2.16) holds near any large τ 0 with
τ 0 > sup x∈R n x · ∇V (x) 2 N ×N + sup x∈R n V (x) N ×N .
Notice that our results extend to the case of potentials depending on h, i.e. V (x; h) = V 0 (x)+hV 1 (x; h). In such a case, we assume (2.7) uniformly with respect to h. In particular, as a simple example, consider the case where V 0 (x) is a diagonal matrix diag (e 1 (x), . . . , e N (x)). If each e j (x) satisfies
2(τ 0 − e j (x)) − x · ∇ x e j (x) ≥ c j > 0, ∀x ∈ {x ∈ R n ; τ 0 − e j (x) ≥ 0},
then (2.17) is satisfied for h small enough and (2.16) holds.
More generally, we can treat the spectral shift function associated to a pair of self-adjoint h-pseudodifferential operators (P 1 (h), P 0 (h)) provided that the SSF is well defined and the existence of a scalar escape function holds.
2.4.
Outline of the proofs. The purpose of this subsection is to provide an outline of the proofs.
As indicated in the introduction, our method is time-independent. The starting point is the functional calculus of h-pseudodifferential operators based on the Helffer-Sjöstrand formula (see [7,Ch. 8]). By this formula, the main object to study will be the integral of the form
(2.18) I(τ, ε; h) = − 1 π C∂f (z)F −1 h θ ε (τ − z)K(z; h)L(dz), τ ∈ R,
where L(dz) = dxdy is the Lebesgue measure on C ∼ R 2 . Here,f ∈ C ∞ 0 (C) denotes an almost analytic extension of f ∈ C ∞ 0 (R; R) (see [7,Ch. 8] and also [11]), i.e., (2.19)f |R = f,
(2.20)∂f (z) = O(|ℑz| ∞ ),
and K, which in fact is the trace of an operator depending on the resolvent, is a complexvalued analytic function defined in a neighborhood of suppf except on the real axis, with an estimate
(2.21) K(z; h) = O h −n |ℑz| −2 .
The right hand side of (2.18) is independent of the particular choice of the almost analytic extensionf . In particular, let ψ L (z) be a function on C defined by
(2.22) ψ L (z) = ψ( ℑz L ), L > 0, C ∞ 0 (R; R) ∋ ψ(t) = 1 (|t| ≤ 1) 0 (|t| ≥ 2).
Thenf ψ L is also an almost analytic extension of f , and we have
(2.23) I(τ, ε; h) = − 1 π C∂ f ψ L (z)F −1 h θ ε (τ − z)K(z; h)L(dz).
From now on, M > 0 is a constant independent of h and we put
(2.24) ζ(h) := h log( 1 h ), L := M ζ(h) ε .
We begin with a general remark on the integral given by the right hand side of (2.23). From (2.20) and the definition of ψ L , we deduce
(2.25)∂ f ψ L (z) = O(h ∞ )ψ L (z) + O( 1 L )f (z)1 [1,2]∪[−2,−1] ℑz L , which together with (2.27) yields I(τ, ε; h) ≡ I + (τ, ε; h) + I − (τ, ε; h), uniformly for 0 < ε ≤ ch −ν (where ν is a fixed constant). Here (2.26) I ± (τ, ε; h) := − 1 π {±ℑz>L}∂ f ψ L (z)F −1 h θ ε (τ − z) K(z; h)L(dz).
We recall that the notation
A ≡ B means A − B = O(h ∞ ). The behavior of the function F −1 h θ ε (τ − z) depends on the support of θ. For general θ with support in ] − 1, 1[, we have (2.27) F −1 h θ ε (τ − z) = O( ε h e ε|ℑz| h ).
In particular, in the support of ψ L , we have
(2.28) F −1 h θ ε (τ − z) = O(εh −2M −1 ). For θ with support only in R + , say in ] 1 2 , 1[, we have (2.29) F −1 h θ ε (τ − z) = O ε h e εℑz h , ℑz > 0, O ε h e εℑz 2h
, ℑz < 0.
This latter estimate implies in particular that
(2.30) I − (τ, ε; h) = O(ε 2 h M 2 −n−2 ), which means I − = O(h ∞ )
if ε is at most of polynomial order in h and M is arbitrary.
Let θ, ε be as in Theorem 2.2 and assume that τ 0 − H(x, ξ) is uniformly microhyperbolic on R 2n in the direction T . Without any loss of generality we may assume that θ ∈ C ∞ 0 (] 1 2 , 1[; R).
According to the Helffer-Sjöstrand formula (see (3.3), (3.4)), the left hand side of (2.3) can be written as (2.18) with
K(z; h) := (λ 0 − z) k−1 tr A w (z − H w ) −1 (λ 0 − H w ) −(k−1) , where λ 0 < inf(σ(H w )) and k ∈ N is large enough so that A w (z − H w ) −1 (λ 0 − H w ) −(k−1) is of trace class.
As explained above, we have
I − = O(h ∞ ). To deal with I + , we conjugate the operator A w (z − H w ) −1 (λ 0 − H w ) −(k−1) with the unitary operator U t := e it h (T 2 ·x−T 1 ·hDx) , t ∈ R. Here T = (T 1 , T 2 ) is the direction of the uniform microhyperbolicity of τ 0 − H(x, ξ). Then the function K t (z; h) := (λ 0 − z) k−1 tr A w t (z − H w t ) −1 (λ 0 − H w t ) −(k−1) , with H w t := U t H w (x, hD x )U −1 t = H w (x + tT 1 , hD x + tT 2 ) etc.
, is invariant with respect to the change of real t and coincides with K(z; h) thanks to the cyclicity of the trace. Now, replacing H, A by their almost analytic extensions (H andÃ), we extend this function to complex t. The extended functionK t (z; h) is defined in {z ∈ C; ℑz > C 0 ℑt} for some positive constant C 0 independent of M ,ε and h. We fix t 0 = iL/C 0 . Then we see that
K t 0 (z; h) is equal to K(z; h) modulo O (h ∞ ) in the domain {ℑz > L}.
The uniform microhyperbolic condition enables us to continueK t 0 (z; h) analytically to the lower half plane with ℑz > −cL for a positive contant c. In fact, the imaginary part of the Weyl symbol of z −H t 0 stays positive definite in such a region, and the sharp Gårding inequality guarantees the invertibility of the operator. Let us now outline the proof of Theorem 2.3. By Helffer-Sjöstrand formula, the left hand side of (2.4) can be written as (2.18)
with K(z; h) = tr (χ w [(z − H w j ) −1 ] 1 0 )
. Using that dist(supp χ, supp (H 0 −H 1 )) > 0, we prove by some exponentially weighted resolvent estimates
K(z; h) = O h M εC −n L 2 ,
uniformly for |ℑz| ≥ L, where C > 0 is a constant independent of h, M, ε. Combining this with (2.28), we get Theorem 2.3 provided that ε > 0 is small enough.
Theorem 2.4 is a consequence of the two previous theorems and the symbolic calculus of h-pseudodifferential operators. Assuming that χ is supported in a small neighborhood of a fixed point (x 0 , ξ 0 ) ∈ R 2n (by a partition of unity there is no loss of generality in doing so) and using the fact that changing H outside the support of χ leads to an error of order
O(h ∞ ) in the trace formula tr(χ w f (H w )F −1 h θ(τ − H w )) (according to Theorem 2.
3), together with Theorem A.3, we may assume that there exists I ∈ O τ 0 such that τ − H is uniformly microhyperbolic with respect to (x, ξ) ∈ R 2n and τ ∈ I. Now, fixε = h 1−δ 0 with δ 0 ∈]0, 1/2[. Applying Theorem 2.2, we obtain (2.31) tr
χ w f (H w )F −1 h θ(τ − H w ) ≡ tr χ w f (H w )F −1 h θε(τ − H w ) .
In fact we can represent the difference θ − θε as a finite sum of functionsθ ε appearing in
Theorem 2.2 (with ε ∈ [ε, 1 C [).
The principal significance of (2.31) is that it allows one to use the standard h-pseudodifferential calculus and get the asymptotic expansion in powers of h given in Theorem 2.4 just by symbolic calculus (see [7,) . To see this, we first recall that for |ℑz| > h δ (with δ ∈]0, 1/2[) the resolvent (z − H w ) −1 is an h-pseudodifferential operator and its corresponding symbol admits an asymptotic expansion in powers of h (see (3.21)). Combining this with the fact
tr χ w f (H w )F −1 h θε(τ − H w ) ≡ − 1 π {|ℑz|>h δ 0 }∂ f ψ( ℑz h δ 0 ) (z)F −1 h θε(τ − z)tr χ w (z − H w ) −1 L(dz),
we see that the left hand side of (2.31) has a complete asymptotic expansion in powers of h, which yields Theorem 2.4.
Turn now to the main ideas in the proofs of the results of subsection 2.2 concerning our application to the SSF. Theorem 2.5 is a simple consequence of the h-pseudodifferential symbolic calculus while Theorems 2.6 and 2.7 are consequences of Theorem 2.4 and standard Tauberian arguments combined with a trick of Robert [27] respectively.
Finally, we sketch the proof of our main result which is Theorem 2.8. According to (2.8) and the Helffer-Sjöstrand formula we have
− s ′ h (·), F −1 h θ ε (τ − ·)f (·) = I(τ, ε; h), with K(z; h) = (z − λ 0 ) q tr (P j (h) − λ 0 ) −q (P j (h) − z) −1 1 0 . Here λ 0 < inf σ(P 1 (h)) and q ∈ N is large enough, see (4.3).
First, suppose that 0 is not contained in the support of θ. Then,
I − (τ, ε; h) = O(h ∞ ) uniformly for ε ∈]h 1−δ , h −ν [ as before.
To deal with I + (τ, ε; h), we adapt an idea from the theory of resonance. More precisely, under the existence of an escape function near τ 0 (assumption (2.15)), we prove by the analytic distortion method that K(z; h) extends analytically from the upper half plane to the lower one with ℑz > −M ζ(h) for all M > 0.
From this, we deduce two important consequences. First,
(2.32) s ′′ h (τ ) = O h −n ζ(h) −2 .
uniformly for τ near τ 0 . Second, the same argument as in the proof of Theorem 2.2 leads to
I + (τ, ε; h) = O(h ∞ ), uniformly for ε ∈]h 1−δ , h −ν [. Hence we obtain (2.33) s ′ h (·), F −1 h θ ε (τ − ·)f (·) = O(h ∞ )
. Now, we assume that θ is equal to one near zero, and let ε be small and independent of h andε = h −ν . As in the proof of (2.31), the formula (2.33) yields
s ′ h (·), F −1 h θ ε (τ − ·)f (·) ≡ s ′ h (·), F −1 h θε(τ − ·)f (·)
. By (2.8) and (2.11), the left hand side of the above equality has an asymptotic expansion in powers of h. The right hand side is written, by Taylor's formula and (2.32),
s ′ h (·), F −1 h θ ε (τ − ·)f (·) = s ′ h (τ )f (τ ) + O(h ν+1−n ζ(h) −2 ).
Since ν is arbitrary, this ends the proof of Theorem 2.8 by taking f = 1 near τ 0 .
Proofs of the results on the semiclassical trace formula
In this section, we prove the results concerning the semiclassical trace formula. Throughout our proofs, when it is not precised, we let C denotes a positive constant that may take different values, but is always independent of ε, h and M .
3.1. Proof of Theorem 2.2. Writing θ = θ 1 + θ 2 , with supp θ 1 ⊂]0, +∞[ and supp θ 2 ⊂ ] − ∞, 0[, we may assume that supp θ ⊂] 1 2 , 1[. For τ ∈ R and ε > 0, we define
(3.1) I(τ, ε; h) := tr A w f H w F −1 h θ ε τ − H w .(3.2) f (H w )g(H w ) = − 1 π C∂f (z) g(z) (z − H w ) −1 L(dz).
Let λ 0 ∈ R be fixed such that λ 0 < inf(σ(H w )) and set, for ℑz = 0,
(3.3) K(z; h) := (λ 0 − z) k−1 tr A w (z − H w ) −1 (λ 0 − H w ) −(k−1) .
Then, using (3.1), (3.2) and (3.
3) with g(z) = (λ 0 − z) k−1 F −1 h θ ε (τ − z), we obtain (3.4) I(τ, ε; h) = − 1 π C∂f (z)F −1 h θ ε (τ − z) K(z; h)L(dz).
Let L and ψ L be defined by (2.24) and (2.22). We write (3.5)
I = I + + I − , I ± := − 1 π {±ℑz>0}∂ (f ψ L )(z)F −1 h θ ε (τ − z) K(z; h)L(dz).
Since the support of θ is included in ] 1 2 , 1[, it follows from (2.30) that (3.6) I − (τ, ε; h) ≡ 0, uniformly for τ ∈ R and ε ∈ [h 1−δ , κ[, for all κ > 0.
Let us now turn to the study of I + (τ, ε; h). By assumption, there exists T = (T 1 , T 2 ) ∈ R 2n and I τ 0 ∈ O τ 0 such that τ − H(x, ξ) is uniformly microhyperbolic in the direction T with respect to (x, ξ) ∈ R 2n and τ ∈ I τ 0 .
For t ∈ R, we define the unitary operator
U t := e it h (T 2 ·x−T 1 ·hDx) . Clearly, we have H w t := U −1 t H w (x, hD x )U t = H w (x, hD x ) + tT = H w x + tT 1 , hD x + tT 2 , A w t := U −1 t A w (x, hD x )U t = A w (x, hD x ) + tT = A w x + tT 1 , hD x + tT 2 .
LetH,Ã be two almost analytic extensions of H and A, respectively, which are bounded together with all theirs derivatives. Put for complex t with small imaginary part
H w t :=H w ((x, hD x ) + tT ) andà w t :=à w ((x, hD x ) + tT ).
By Taylor's formula with respect to ℑt, we have
z −H((x, ξ) + tT ) = z −H (x, ξ) + ℜtT + iℑtT = z − H (x, ξ) + ℜtT − iℑt ∇ x,ξ H((x, ξ) + ℜtT ), T + O(|ℑt| 2 ). (3.7)
Thus, one easily sees by using the Calderón-Vaillancourt theorem (see [7,Theorem 7.11]) that there exists a constant C 0 > 0 (depending only on the L ∞ -norms of a finite numbers of
derivatives of H) such that (z −H w t ) −1 exists for |ℑz| ≥ C 0 |ℑt|. Set K t (z; h) := (λ 0 − z) k−1 tr à w t (z −H w t ) −1 (λ 0 −H w t ) −(k−1) . Using that ∂ tÃt , ∂ tHt = O(|ℑt| ∞ )
, we obtain, uniformly on {z ∈ C; |ℑz| ≥ C 0 |ℑt|},
(3.8)∂ tKt (z; h) = O |ℑt| ∞ |ℑz| 2 .
On the other hand, since U t is unitary for t ∈ R, it follows from the cyclicity of the trace thatK t is independent of ℜt. This implies
(3.9)∂ tKt (z; h) = i 2 ∂ ℑtKt (z; h), andK t (z; h) = K(z; h), ∀t ∈ R.
We have, uniformly for |ℑz| ≥ C 0 |ℑt|,
K(z; h) −K iℑt (z; h) =K ℜt (z; h) −K ℜt+iℑt (z; h) = − ℑt 0 d dsK ℜt+is (z; h)ds = O |ℑt| ∞ |ℑz| 2 .
Fix t 0 = iL/C 0 . By the preceding estimate, we have, uniformly for |ℑz| ≥ L,
(3.10) K(z; h) −K t 0 (z; h) = O(h ∞ ).
In the expression (3.5) of I + , one sees from (2.25) and (2.29) that the restriction of the integral to the domain 0 < ℑz ≤ L is O(h ∞ ). Therefore, by (3.10), we get
I + (τ, ε; h) ≡ − 1 π {ℑz>L}∂ f ψ L (z)F −1 h θ ε (τ − z)K(z; h)L(dz) ≡ − 1 π {ℑz>L}∂ f ψ L (z)F −1 h θ ε (τ − z)K t 0 (z; h)L(dz).
Lemma 3.1. Let t 0 = iL/C 0 = iM C 0 ε ζ(h) be as above. The function z →K t 0 (z; h) extends as a holomorphic function to the zone ℑz ≥ − |t 0 | 2 . Proof. As in (3.7), Taylor's formula yields
z −H t 0 (x, ξ) = z − H(x, ξ) − t 0 T, ∇ x,ξ H(x, ξ) + O(|t 0 | 2 ).
Using the global microhyperbolicity condition, we obtain for small h
(3.11) ℑ(z −H t 0 (x, ξ)) + C|t 0 |(z −H t 0 (x, ξ)) * (z −H t 0 (x, ξ)) ≥ c(|t 0 | + ℑz)I N ,
uniformly on z with ℑz > 0 and ℜz ∈ I (see (A.8)), where C, c > 0 are constants independent of h and M . Here, * stands for the usual complex adjoint of a matrix. Now we pass from the symbolic calculus level to the h-pseudodifferential calculus. The semiclassical version of the sharp Gårding inequality (see [7] Theorem 7.12 and [14, Ch.1] for the matrix case) and (3.11) imply,
(3.12) ℑ(Op w h (z −H t 0 )u, u) + C|t 0 | Op w h (z −H t 0 )u 2 ≥ c(|t 0 | + ℑz) u 2 − O(h) u 2 ≥ c 3 (|t 0 | + ℑz) u 2 ,
for all u ∈ L 2 (R n ; C N ) and h small enough. Here we used the fact that h = o(|t 0 |). Combining (3.12) with the inequality ab ≤ c|t 0 | 6 a 2 + 3
2c|t 0 | b 2 , we obtain c 3 (|t 0 | + ℑz) u 2 ≤ Op w h (z −H t 0 )u u + C|t 0 | Op w h (z −H t 0 )u 2 ≤ c|t 0 | 6 u 2 + ( 3 2c|t 0 | + C|t 0 |) Op w h (z −H t 0 )u 2 , which yields (3.13) c 6 (|t 0 | + ℑz) u 2 ≤ ( 3 2c|t 0 | + C|t 0 |) Op w h (z −H t 0 )u 2 ,
for all u ∈ L 2 (R n ; C N ). We conclude that (z −H w t 0 ) −1 extends as a holomorphic function of z to the zone ℑz ≥ − |t 0 | 2 . This ends the proof of the lemma.
Letψ ∈ C ∞ (R; R) be such thatψ(s) = ψ(s) for s > 0,ψ(s) = 1 for −1/4C 0 < s < 0, and ψ(s) = 0 for s < −1/2C 0 , and defineψ L as in (2.22). Then we have
I + (τ, ε; h) ≡ − 1 π {ℑz>L}∂ f ψ L (z)F −1 h θ ε (τ − z)K t 0 (z; h)L(dz) ≡ − 1 π {ℑz>0}∂ f ψ L (z)F −1 h θ ε (τ − z)K t 0 (z; h)L(dz) ≡ − 1 π {ℑz>0}∂ f ψ LψL (z)F −1 h θ ε (τ − z)K t 0 (z; h)L(dz) ≡ 1 π {ℑz<0}∂ f ψ LψL (z)F −1 h θ ε (τ − z)K t 0 (z; h)L(dz), (3.14)
uniformly for τ ∈ R. Notice that to pass from the first equation to the second we used (2.25), and the last identity follows from the Cauchy formula for analytic functions. Now, with the same argument as for I − , we see that I + = O(ε 2 h M 2 −n−2 ) uniformly for τ ∈ R and ε ∈ [h 1−δ , κ[ for all κ > 0, which gives the result since M > 0 is arbitrary. This ends the proof of Theorem 2.2. To see this, we first see (3.6), since ν is fixed and M is arbitrary. Next, since supp ψ L ⊂ {z ∈ C; |ℑz| ≤ 2 M κ ζ(h))} for all ε ∈ [κ, h −ν [, it follows from the above assumption (with ℓ > 2 M κ ) and the Cauchy formula that
I + (τ, ε; h) = − 1 π {ℑz>0}∂ f ψ L (z)F −1 h θ ε (τ − z)K(z; h)L(dz) = 1 π {ℑz<0}∂ f ψ L (z)F −1 h θ ε (τ − z)K(z; h)L(dz).
Then the same argument as for I − shows I + = O(h ∞ ), and hence (2.3) holds uniformly for τ ∈ R and ε ∈ [κ, h −ν [. Later, in the application to the study of the SSF, we shall show that assumption (2.15) about the existence of an escape function implies that the function z → K(z; h) (defined by (4.1)) satisfies the condition assumed on K(z; h) in this remark in an open complex neighborhood O of τ 0 (see Lemma 4.2). This will be crucial for the proof of the pointwise asymptotics (2.16).
3.2.
Proof of Theorem 2.3. Let ε > 0 be a small constant (independent of h) which will be fixed later. We have
tr χ w f (H w j )F −1 h θ ε (τ − H w j ) 1 0 = I(τ, ε; h) where I(τ, ε; h) is defined by (2.23) with K(z; h) = tr χ w [(z − H w j ) −1 ] 1 0 .
It follows from (2.25) and (2.27) that, uniformly for τ ∈ R,
(3.15) I(τ, ε; h) ≡ − 1 π {L<|ℑz|<2L ∂ f ψ L (z)F −1 h θ ε (τ − z)tr χ w [(z − H w j ) −1 ] 1 0 L(dz).
Let B 0 ∈ C ∞ 0 (R 2n ) be a real-valued function in the phase space such that B 0 = 1 near supp χ and B 0 = 0 near supp (H 1 − H 0 ), and let B = αB 0 for a constant α > 0 that we will choose later. We notice that the symbol b = e B log 1 h is of class S l δ (R 2n ) 1 for l = α B 0 L ∞ (R 2n ) and every δ > 0. By the same notation we also denote the corresponding h-pseudodifferential operator, which is bounded, elliptic and has an inverse operator (e B log 1 h ) −1 with symbol in the same class. Using the h-pseudodifferential calculus (see [7,Ch. 7]) as well as the Calderón-Vaillancourt theorem, it is clear that for some k ∈ N,
(3.16) e B log 1 h (z − H w 1 )(e B log 1 h ) −1 = z − H w 1 + O(αζ(h)) ∇B 0 C k , in operator norm, for h ≤ h(α),
where h(α) > 0 is some continuous function.
It follows that for |ℑz| ≥ Cαζ(h) (where C depends only on H 1 C k and B 0 C k ), the right hand side of (3.16) is invertible and we have
(3.17) e B log 1 h (z − H w 1 ) −1 (e B log 1 h ) −1 = O(|ℑz| −1 ). in operator norm.
On the other hand, since B = α near supp χ and that B = 0 near supp(H 1 − H 0 ), it follows from the h-pseudodifferential calculus again that we have
(e B log 1 h ) −1 (H w 1 − H w 0 ) = (H w 1 − H w 0 ) + O(h ∞ ), χ w e B log 1 h = e α log 1 h χ w + O(h ∞ )
, in operator norm and trace norm respectively. Thus
e α log 1 h tr χ w (z − H w 1 ) −1 (H w 1 − H w 0 )(z − H w 0 ) −1 = tr χ w e B log 1 h (z − H w 1 ) −1 (e B log 1 h ) −1 (H w 1 − H w 0 )(z − H w 0 ) −1 + O h ∞ |ℑz| 2 = tr χ w (z − e B log 1 h H w 1 (e B log 1 h ) −1 ) −1 (H w 1 − H w 0 )(z − H w 0 ) −1 + O h ∞ |ℑz| 2 .
Combining this with (3.16), we deduce that for |ℑz| ≥ Cαζ(h)
(3.18) tr χ w [(z − H w j ) −1 ] 1 0 = O h α−n |ℑz| 2 .
We choose α = M Cε . It follows from (2.28), (3.15) and (3.18)
I(τ, ε; h) = O ε 3 h M ( 1 εC −2)−n−3 log( 1 h ) −2 .
Next we choose ε small enough so that 1 εC > 2. This ends the proof of Theorem 2.3 since M is arbitrary. We recall that C depends only on H 1 C k and B 0 C k .
3.3.
Proof of Theorem 2.4. Without any loss of generality, we may assume that χ is supported in a small neighborhood of a fixed point (x 0 , ξ 0 ). In fact we may replace χ by a finite sum of terms χχ i with i χ i = 1 near the support of χ and χ i has its support in a small neighborhood of a fixed point (x i , ξ i ) ∈ supp χ. Then, choosing the support of χ small enough, we may assume that τ − H(x, ξ) is uniformly microhyperbolic in a fixed direction T for (x, ξ) in suppχ and for τ near τ 0 . Moreover, by modifying H outside suppχ as in Theorem A.3, we may assume that τ − H(x, ξ) is uniformly microhyperbolic in the whole phase space R 2n in the direction T thanks to Theorem 2.3.
Let θ ∈ C ∞ 0 (] − 1; 1[; R) be equal to one near 0, ε > 0 small enough independent of h and D be an integer such that 2 −D ∼ h 1−δ with δ ∈]0, 1 2 [. Putε = 2 −D ε. We write
θ ε (t) − θε(t) = D i=1 Ψ(2 i−1 t),
where Ψ(t) = θ ε (t) − θ ε (2t) ∈ C ∞ 0 (R). Clearly, Ψ(t) = Ψ 1 (t) + Ψ 2 (t) where Ψ 1 and Ψ 2 are equal to 0 near zero, supp Ψ 1 ⊂]0, ε[ and supp Ψ 2 ⊂] − ε, 0[. Now applying Theorem 2.2 (resp. Remark 2.1) to Ψ 1 (2 i−1 t) (resp. Ψ 2 (2 i−1 t)), i = 1, · · · D, we see that there exists I ∈ O τ 0 such that for all f ∈ C ∞ 0 (I; R), we have (3.19) tr
χ w f (H w )F −1 h θ ε (τ − H w ) ≡ tr χ w f (H w )F −1 h θε(τ − H w )
, uniformly for τ ∈ R. As in (2.26), we have
tr χ w f (H w )F −1 h θε(τ − H w ) ≡ (3.20) − 1 π {|ℑz|≥h δ }∂ f ψ h δ (z)F −1 h θε(τ − z)tr χ w (z − H w ) −1 L(dz).
Now in the zone Ω δ := {z ∈ suppf ; |ℑz| ≥ h δ }, with 0 < δ < 1 2 , the resolvent (z − H w ) −1 is an h-pseudodifferential operator. More precisely, according to Proposition 8.6 in [7], there exists a C ∞ matrix-valued function (x, ξ) → G(x, ξ, z; h) such that
∂ α x,ξ G(x, ξ, z; h) ≤ C α h −δ(1+|α|) , ∀α ∈ N 2n , uniformly on z ∈ Ω δ and (3.21) (z − H w ) −1 = Op w h (G(x, ξ, z; h)),
for all z ∈ Ω δ . Moreover x, ξ, z) is a finite sum of terms of the form
(3.22) G(x, ξ, z; h) ∼ G 0 (x, ξ, z) + hG 1 (x, ξ, z) + h 2 G 2 (x, ξ, z) + · · · in S δ δ (R 2n , H N ) where G j ((z−H(x, ξ)) −1 B 1 (x, ξ, z)(z−H(x, ξ)) −1 B 2 (x, ξ, z)(z−H(x, ξ)) −1 · · · B k (x, ξ, z)(z−H(x, ξ)) −1 ,
with k < 2j + 1, B l (x, ξ, z) ∈ S 0 (R 2n ; H N ) holomorphic in z near suppf . Now by classical results on trace class h-pseudodifferential operators (see Theorem II.53 and Proposition II. 56 in [26]), we have for all m ∈ N, (3.23) tr
χ w f (H w )F −1 h θ ε (τ − H w ) = (2πh) −n m j=0 a j (τ ; h)h j + O(h (m+1)(1−2δ)−n ), where (3.24) a j (τ ; h) = − 1 π ∂ f ψ h δ (z)F −1 h θε(τ − z) e j (z) L(dz) with e j (z) := R 2n χ(x, ξ) tr (G j (x, ξ, z)) dxdξ.
Here tr denotes the trace of square matrices. The microhyperbolicity assumption implies that there exists I ∈ O τ 0 such that the function
I ∋ τ → e j (τ ± i0) := lim sց0 e j (τ ± is), is C ∞ (see Proposition A.4). Set (3.25) γ j (τ ) := − 1 2πi e j (τ + i0) − e j (τ − i0) .
Now the following lemma ends the proof of Theorem 2.4.
Lemma 3.2. a j (τ ; h) = f (τ )γ j (τ ) + O(h ∞ ).
Proof. Since z → F −1 h θε(τ −z) e j (z) is holomorphic in the complex domain {z ∈ C; ±ℑz > 0}, it follows from the Green formula that
a j (τ ; h) = − 1 π lim sց0 {ℑz>s}∂ f ψ h δ (z)F −1 h θε(τ − z) e j (z) L(dz) − 1 π lim sց0 {ℑz<−s}∂ f ψ h δ (z)F −1 h θε(τ − z) e j (z) L(dz) = − 1 2πi R f (λ)F −1 h θε(τ − λ) e j (τ + i0) − e j (τ − i0) dλ = R F −1 h θε(τ − λ)f (λ)γ j (λ)dλ = R F −1 1 θ(λ)f (τ − h δ λ)γ j (τ − h δ λ)dλ.
The last equality is obtained by a change of variable. Applying Taylor's formula to the function λ → f (τ − h δ λ)γ j (τ − h δ λ) at λ = 0 and using the fact that F −1 1 θ(λ)(−iλ) k dλ = θ (k) (0) = 0 we obtain the lemma. We recall that θ = 1 near zero. Here F −1
1 is F h −1 with h = 1
Proofs of the results on the SSF
This section is devoted to the proofs of the results of subsection 2.2.
We follow the notations used in section 3. From the assumption (2.7), the operator
(P j (h) − z 0 ) −q (z − P j (h)) −1 1 0
is of trace class for q > n 2 and z 0 ∈ σ(P 1 (h)) ∪ σ(P 0 (h)) which are fixed in what follows. We set
(4.1) K(z; h) := (z − z 0 ) q tr (P j (h) − z 0 ) −q (z − P j (h)) −1 1 0 , ℑz = 0.
As in the proof of (3.4), formulas (2.8) and (3.2) yield, for all f, θ ∈ C ∞ 0 (R; R),
(4.2) s ′ h (·), f (·) = 1 π C∂f (z)K(z; h)L(dz), (4.3) s ′ h (·), F −1 h θ(τ − ·)f (·) = 1 π C∂f (z)F −1 h θ(τ − z)K(z; h)L(dz).
4.1.
Proof of Theorem 2.5. This is a classical result and follows from the functional calculus of h-pseudodifferential operators.
Let δ ∈]0, 1 2 [. The contribution from the domain |Im z| ≤ h δ of the integral in the right hand side of (4.2) is O(h ∞ ).
Next, in the domain |Im z| ≥ h δ , we use the fact that (z − P k (h)) −1 are h-pseudodifferential operators, k = 0, 1 (see (3.21) and (3.22)). This formally yields (2.9) (with q = 0) with
c j (f ) = R 2n 1 π tr C∂f (z) (G j,1 (x, ξ, z) − G j,0 (x, ξ, z)) L(dz) dxdξ.
In particular
c 0 (f ) = R 2n tr f (p 0 (x, ξ)) − f (p 1 (x, ξ)) dxdξ,
and (2.10) trivially follows from this formula. To see that c 2j+1 (f ) = 0, it suffices to notice that h → |2πh| n tr(f (P 1 (h)) − f (P 0 (h)) is an even function. More rigorously for q = 0, one may write K(z; h) as
K(z; h) = (z − z 0 ) q tr (P 1 (h) − z 0 ) −q − (P 0 (h) − z 0 ) −q (z − P 1 (h)) −1 + (z − z 0 ) q tr (P 0 (h) − z 0 ) −q (z − P 1 (h)) −1 − (z − P 0 (h)) −1 ,(4.4)
and use the fact that (P k (h) − z 0 ) −q are h-pseudodifferential operators, k = 0, 1. This ends the proof of (2.9).
4.2.
Proof of Theorem 2.6. The proof of Theorem 2.6 uses (4.3) and is quite similar to that of Theorem 2.4, and we omit the details. The main difference is that Σ τ 0 = {(x, ξ) ∈ R 2n ; det(p 1 (x, ξ) − τ 0 ) = 0} is not a compact set in R 2n . In this case we have to justify that we can cover Σ τ 0 by finite open sets O 1 , O 2 , · · · , O ℓ in which we can constructp 1,k (x, ξ), and T k ∈ R 2n , k = 1, · · · , ℓ, such thatp 1,k (x, ξ) − τ 0 is uniformly microhyperbolic in the direction T k andp 1,k (x, ξ) = p 1 (x, ξ) for all (x, ξ) ∈ O k . To see this, we first notice that Σ τ 0 = Σ τ 0 ∩ {|ξ| ≤ R 0 } (R 0 being large enough), since lim |ξ|→∞ det(p 1 (x, ξ) − τ 0 ) = ∞. Next, fix R 1 large such that inf |x|>R 1 |det(V (x)−τ 0 )| > 0. This is possible since lim |x|→∞ V (x) = V ∞ and τ 0 ∈ σ(V ∞ ) by assumption. On the compact set Σ τ 0 ∩ {|x| ≤ R 1 } we can apply Theorem 2.4 without any modification. On the other hand, we see from the choice of R 1 that∇(|ξ| 2 ) = 0 for all (x, ξ) ∈ Σ τ 0 ∩ {|x| > R 1 } := Σ τ 0 ,R 1 . Thus, we can find finite open covers o 1 , o 2 , · · · o ℓ in R n ,T 1 ,T 2 , · · · ,T ℓ ∈ R n and c 1 , · · · c ℓ > 0 such that {|ξ| ≤ R 0 } ⊂ ℓ j=1 o j and for each j = 1, · · · ℓ, T j , ∇(|ξ| 2 ) ≥ c j , uniformly on ξ ∈ o j ∩ π ξ Σ τ 0 ,R 1 . Now using Theorem A.3, we constructp 1,j (x, ξ), such thatp 1,j (x, ξ) − τ 0 is uniformly microhyperbolic in the direction T j = (0,T j ) andp 1,j (x, ξ) = p 1 (x, ξ) for all (x, ξ) ∈ {|x| > R 1 } × o j . We can now proceed analogously to the proof of Theorem 2.4.
4.3.
Proof of Theorem 2.7. For the proof of Theorem 2.7, assume that τ → s(τ ; h) is monotonic (i.e., s ′ (·; h) is positive or negative in the sense of distributions). In this case Theorem 2.7 is a simple consequence of Theorem 2.6 by standard Tauberian arguments (see [7], [14], [26]).
For the general case, we use a trick due to Robert [27], which consists in writing s(τ ; h) = s 1 (τ ; h) − s 2 (τ ; h) where τ → s i (τ ; h), i = 1, 2 are monotonic. Now, it suffices to apply the above argument to each s i (τ ; h).
Notice that, Robert's trick applies to Schrödinger operators with matrix-valued potential under the assumption (2.7) with scalar matrix V ∞ .
4.4.
Proof of Theorem 2.8. The proof of the following lemma is the same as that of Lemma 2.2 in [8].
s ′ h (τ ) = 1 π ℑK(τ + i0; h) in D ′ (R),
i.e. we have, for all f ∈ C ∞ 0 (R),
s ′ h (·), f = lim κց0 1 π R f (τ )ℑK(τ + iκ; h) dτ.
Let I ∈ O τ 0 such that (2.15) holds on Σ τ for all τ ∈ I. For M ≥ 0, we introduce the following h-dependent set
V (x) − V ∞ ∈ C ∞ 0 (R n ; H N ).
For any M > 0, the function z → K(z; h) has an analytic extension from Γ 0 to Γ M . Moreover, we have, uniformly for z ∈ Γ M ,
(4.6) K (k) (z; h) = O h −n ζ(h) −k−1 , ∀k ∈ N.
In particular, uniformly for τ ∈ I,
(4.7) s (k+1) h (τ ) = O h −n ζ(h) −k−1 , ∀k ∈ N.
Proof. The estimate (4.7) follows immediately from (4.6) and the representation of the SSF given by Lemma 4.1. Hence it is enough to prove (4.6).
Let F : R n → R n be a smooth vector field such that F = 0 in a neighbourhood of supp(V − V ∞ ) and F (x) = x for |x| large enough. For ω ∈ R small enough, we denote U ω : L 2 (R n ; C N ) → L 2 (R n , C N ) the unitary operator defined by
(4.8) U ω φ(x) := |det(1 + ω∇F (x))| 1 2 φ(x + ωF (x)),
and set P j,ω (h) := U ω P j (h)(U ω ) −1 , j = 0, 1.
They are differential operators with analytic coefficients with respect to ω, and can be analytically continued to small enough complex values of ω. It follows from the analytic perturbation theory (see [16]) that for ω 0 small enough, ω ∈] − ω 0 , ω 0 [ → P j,ω (h), j = 0, 1, extends to an analytic type A-family of operators on D(ω 0 ) := {ω ∈ C; |ω| < ω 0 } with domain H 2 (R n ; C N ).
We set, first for real ω and ℑz > 0,
(4.9) K ω (z; h) := (z − z 0 ) q tr (P j,ω (h) − z 0 ) −q (z − P j,ω (h)) −1 1 0 .
Since U ω is unitary for real ω, it follows from the cyclicity of the trace that (4.10)
K(z; h) = K ω (z; h), ∀ω ∈] − ω 0 , ω 0 [, ∀ ℑz > 0.
On the other hand, for ℑz > M 0 ζ(h) for a given h-independent M 0 > 0, the function ω → K ω (z) is analytic in ω ∈ D(2cM 0 ζ(h)) with some c > 0 independent of h and M 0 . Thus, by the uniqueness theorem of analytic continuation, the identity (4.10) remains true for ℑz > M 0 ζ(h) and ω ∈ D(2cM 0 ζ(h)), i.e., (4.11) K(z; h) = K ω (z; h), ∀ω ∈ D 2cM 0 ζ(h)), ∀ ℑz > M 0 ζ(h).
From now on we fix M = cM 0 and set ω 1 = iM ζ(h).
Since τ 0 > e N,∞ , x · ξ is an escape function for p 1 (x, ξ) for |(x, ξ)| large enough. Thus, without any loss of generality, we may assume that G(x, ξ) = x · ξ for |(x, ξ)| large enough. ThenG(x, ξ) := G(x, ξ) − F (x) · ξ has a compact support, and in particular its quanti-zationG w (x, hD x ) is L 2 -bounded by the Calderón-Vaillancourt theorem and the operators e ± M ζ(h) hG w (x,hDx) are well-defined.
Let us defineP
j,ω 1 (h) := e − M ζ(h) hG w (x,hDx) P j,ω 1 (h)e M ζ(h) hG w (x,hDx) , j = 0, 1, (4.12)K ω 1 (z; h) := (z − z 0 ) q tr P j,ω 1 (h) − z 0 −q z −P j,ω 1 (h) −1 1 0 .
From Lemma 4.3 below, z →K ω 1 (z; h) is analytic in Γ cM for some c > 0. Again by the cyclicity of the trace and the uniqueness of the analytic continuation, we conclude
(4.13)K ω 1 (z; h) = K(z; h), ∀z ∈ Γ cM .
This with the resolvent estimate (4.14) leads to
K(z; h) = O(h −n ζ(h) −1 ),
uniformly for z ∈ Γ cM , which yields (4.6) for k = 0. Next, taking the derivative of (4.12) and applying (4.14) we obtain (4.6) for k ≥ 1 (Recall that the trace of semiclassical quantization of a symbol in a suitable class is of O(h −n ), see [7,Theorem 9.4]).
Lemma 4.3. There exists c > 0 such that for all M > 0 the operatorP j,ω 1 (h) − z is invertible for every z ∈ Γ cM . Moreover, one has, uniformly in this domain,
(4.14) z −P j,ω 1 (h) −1 = O ζ(h) −1 .
Proof. We have
(4.15)P j,ω 1 (h) = e − M ζ(h) h adG w P j,ω 1 (h) ∼ ∞ k=0 (−M ζ(h)) k k! 1 h adG w k P j,ω 1 (h), where adG w P j,ω 1 (h) = [G w , P j,ω 1 (h)] = O(h) 2 .
By definition, ζ(h) tends to 0 as h ց 0.
Combining this with the boundedness of h −1 adG w we find that the asymptotic expansion (4.15) makes sense. In particular,
P j,ω 1 (h) = P j,ω 1 (h) − M ζ(h) h G w (x, hD x ), P j,ω 1 (h) + O(M 2 ζ(h) 2 ).
Let p j,ω 1 ,p j,ω 1 be the Weyl symbols of P j,ω 1 ,P j,ω 1 respectively. We obtain from the hpseudodifferential calculus ([7, Ch. 7]),
(4.16)p j,ω 1 = p j,ω 1 − iM ζ(h){p j,ω 1 ,G} + O(M 2 ζ(h) 2 ),
and in particular, using the Taylor expansion of p j,ω 1 with respect to ω 1 ;
p j,ω 1 = p j − iM ζ(h){p j , F (x) · ξ} + O(M 2 ζ(h) 2 ),
we obtain
(4.17) ℑ(p j,ω 1 ) = −M ζ(h){p j ,G + F (x) · ξ} + O(M 2 ζ(h) 2 ). (4.18) ℜ(p j,ω 1 ) = p j + O(M ζ(h)).
Since G(x, ξ) =G(x, ξ) + F (x) · ξ satisfies the assumption (2.15), it follows from (4.17) and (4.18) that there exist C > 0 and I ∈ O τ 0 such that
(4.19) − ℑ(p 1,ω 1 )(x, ξ) ≥ CM ζ(h), ∀(x, ξ) ∈ Σ I := τ ∈I Σ τ ,
Of course, the same estimate holds also for ℑ(p 0,ω 1 )(x, ξ), since (2.15) always holds for p 0 with G = x · ξ for any τ 0 > e N,∞ .
We
writeP j,ω 1 (h) − z = A j,ω 1 (h) − ℜz + i(B j,ω 1 (h) − ℑz) with A j,ω 1 (h) = 1 2 P j,ω 1 (h) + P j,ω 1 (h) * , B j,ω 1 (h) = 1 2i P j,ω 1 (h) − P j,ω 1 (h) * .
Let ψ j,1 , ψ j,2 ∈ C ∞ (R 2n ; R) be such that, for I ′ ⋐ I, ψ 2 j,1 + ψ 2 j,2 = 1, ψ j,1 = 1 on Σ j I ′ , supp(ψ j,1 ) ⊂ Σ j I .
According to Lemma 3.2 in [32], there exist two self-adjoint operators Ψ j,1 and Ψ j,2 with principal symbols respectively ψ j,1 and ψ j,2 such that
(4.20) (Ψ j,1 ) 2 + (Ψ j,2 ) 2 = Id + O(h ∞ ) in L(L 2 (R n )).
We denote by the same letters the operators Ψ j,i := Ψ j,i I N , i = 1, 2. On the support of ψ j,1 , we see from (4.19) that the principal symbol of −B j,ω 1 (h) is estimated from below by CM ζ(h). Then by the Gårding's inequality, we obtain, uniformly for
ℑz > − C 3 M ζ(h), (4.21) (P j,ω 1 (h) − z)Ψ j,1 u · Ψ j,1 u ≥ | (P j,ω 1 (h) − z)Ψ j,1 u, Ψ j,1 u | ≥ | (ℑP j,ω 1 (h) − ℑz)Ψ j,1 u, Ψ j,1 u | = (ℑz − B j,ω 1 (h))Ψ j,1 u, Ψ j,1 u ≥ (ℑz + CM ζ(h) − O(h)) Ψ j,1 u 2 ≥ C 3 M ζ(h) Ψ j,1 u 2 .
On the other hand, since A j,ω 1 (h) − ℜz is uniformly elliptic on the support of ψ j,2 and ℜz ∈ I, the symbolic calculus permits us to construct a parametrix R ∈ S 0 ( ξ −2 ) of A j,ω 1 (h) − ℜz such that, in the symbol sense,
R#(A j,ω 1 (h) − ℜz)ψ j,2 = ψ j,2 + O(h ∞ ),
where # stands for the Weyl composition of symbols. As a consequence, we obtain
(P j,ω 1 (h) − z)Ψ j,2 u ≥ 1 C ′ Ψ j,2 u − O(h ∞ ) u 2 . (4.22)
Furthermore, by means of standard elliptic arguments, one can easily prove the following semiclassical inequality (4.23) [P j,ω 1 (h), Ψ j,i ]u ≤ C 2 h( P j,ω 1 (h)u + u ), ∀u ∈ H 2 (R n ; C N ).
Combining (4.20), (4.21), (4.22), and (4.23) with the estimate
(4.24) (P j,ω 1 (h) − z)u 2 = 2 i=1 Ψ j,i (P j,ω 1 (h) − z)u 2 − O(h ∞ ) (P j,ω 1 (h) − z)u 2 ≥ 1 2 2 i=1 (P j,ω 1 (h) − z)Ψ j,i u 2 − 2 i=1 [P j,ω 1 (h), Ψ j,i ]u 2 − O(h ∞ ) (P j,ω 1 (h) − z)u 2 ,
we deduce, for z ∈ Γ cM (with c > 0 independent of M and h) and sufficiently small h,
(4.25) (P j,ω 1 (h) − z)u ≥ ζ(h) C u .
By the same arguments, we prove an estimate similar to (4.25) for the adjoint operator P j,ω 1 (h) * − z and we conclude thatP j,ω 1 (h) − z is invertible for every z ∈ Γ cM . Moreover (4.25) yields the resolvent estimate (4.14).
End of the proof of Theorem 2.8. Using Lemma 4.2, and applying Theorem 2.2 and Remark 3.1 to the right hand side of (4.3) we obtain the following lemma.
I ∈ O τ 0 such that for f ∈ C ∞ 0 (I; R), we have (4.26) s ′ h (·), F −1 h ϕ ε (τ − ·)f (·) = O(h ∞ ), uniformly for τ ∈ R and ε ∈]κ, h −ν [. Now let θ ∈ C ∞ 0 (] − 1, 1[; R
) be equal to one near 0 and let f ∈ C ∞ 0 (I; R) be as in the above lemma. Suppose ε > 0 is a small enough constant independent of h andε := h −ν with ν ∈ N arbitrary large.
Repeating the same construction as in the proof of Theorem 2.4, we represent the difference θε−θ ε as a finite sum 0≤j≤N (h) ϕ ε j with ϕ ε j as in Lemma 4.4 and N (h) = O(h −ν ). Applying Lemma 4.4 to each term, we get
(4.27) s ′ h (·), F −1 h θ ε (τ − ·)f (·) = s ′ h (·), F −1 h θε(τ − ·)f (·) + O(h ∞ )
, uniformly with respect to τ ∈ R.
Next, by a change of variable we have
s ′ h (·), F −1 h θε(τ − ·)f (·) = R F −1 1 θ(λ)(f s ′ h )(τ − h 1+ν λ)dλ.
Applying Taylor's formula to the function λ → (f s ′ h )(τ − h 1+ν λ) at λ = 0 and using (4.7) with k = 1, we get
(4.28) s ′ h (·), F −1 h θε(τ − ·)f (·) = s ′ h (τ )f (τ ) + O(h ν+1−n ζ(h) −2 )
, uniformly for τ ∈ R since R F −1 1 θ(λ)dλ = θ(0) = 1. From (4.27) and (4.28) we deduce
(4.29) s ′ h (τ )f (τ ) = s ′ h (·), F −1 h θ ǫ (τ − ·)f (·) + O(h ν+1−n ζ(h) −2 )
. By Theorem 2.6, the first term of the right hand side of the above equality has an asymptotic expansion in powers of h. Now, since ν is arbitrary, the asymptotic expansion (2.16) follows from (4.29) by choosing f equal to 1 near τ 0 . This ends the proof of Theorem 2.8 under the assumption V − V ∞ ∈ C ∞ 0 (R n ; H N ). Remark 4.1. Notice that, except for Lemma 4.2, all the steps of the proof of Theorem 2.8 remain valid under the assumptions (2.7) and (2.15) with µ > n. We will now show how to dispense with the assumption on the support of V in Lemma 4.2. According to Proposition 4.2 in [24], if V satisfies (2.7), then for any κ > 0 andμ ∈]0, µ[, we can construct V κ such that V κ can be extended into a holomorphic function of r = |x| in the sector Σ(2κ) = {ℜr ≥ 1; |ℑr| < 2κℜr}, and, for any multi-index α, it satisfies
(4.30) ∂ α x (V κ (x) − V (x)) N ×N = O( x −μ−|α| κ ∞ )
. As in [24], we fix κ = h s with s ∈]0, 1[. We denote by K κ (z; h) the right hand side of (4.1) when we replace V by V κ in P 1 (h). The operator P 1 (h) = −h 2 ∆ + V κ can be distorded analytically intoP 1 (h) = U ν P 1 (h)(U ν ) −1 (see [24]). Now the proof of Lemma 4.2 shows that (4.6) and (4.7) hold for K κ (z; h). On the other hand, using the resolvent identity and we show that
K κ (z; h) − K(z; h) = O(κ ∞ ) = O(h ∞ )
, uniformly on Γ 0 . Consequently, Lemma 4.2 remains true under the assumptions (2.7) and (2.15).
Appendix A. Microhyperbolic functions
In this section, we prove some technical lemmas on the notion of microhyperbolicity used in our proofs. (R 2n ; H N ). The following statements are equivalents (1) H is microhyperbolic at ρ 0 ∈ R 2n in the direction T ∈ R 2n .
Lemma A.1. Let H ∈ C ∞
(2) T, ∇ ρ H(ρ 0 ) | ker H(ρ 0 ) is strictly positive in the sense of hermitian matrices, i.e. there exists C > 0 such that
(A.1) T, ∇ ρ H(ρ 0 ) w, w ≥ C|w| 2 , ∀w ∈ ker H(ρ 0 ).
Proof. Obviously (1) implies (2).
Assume that (2) is satisfied and let us prove (1). Let w = w 1 + w 2 ∈ C N , with w 1 ∈ ker H(ρ 0 ) and w 2 ∈ ker H(ρ 0 ) ⊥ . We have :
T, ∇ ρ H(ρ 0 ) w, w = T, ∇ ρ H(ρ 0 ) w 1 , w 1 + T, ∇ ρ H(ρ 0 ) w 2 , w 2 + 2 i =j=1 T, ∇ ρ H(ρ 0 ) w i , w j =: I 1 + I 2 + I 3 .
By hypothesis, I 1 satisfies
(A.2) |I 1 | ≥ C|w 1 | 2 .
On the other hand, we have
(A.3) |I 2 | ≤ C ′ |w 2 | 2 and |I 3 | ≤ C ′′ |w 1 ||w 2 | < εC ′′ |w 1 | 2 + C ′′ ε |w 2 | 2 ,
for ε > 0 small enough and C ′ , C ′′ > 0. Putting together (A.2), (A.3), we obtain
(A.4) ( T, ∇ ρ H(ρ 0 ) w, w) ≥ C 2 |w| 2 − O 1 ε |w 2 | 2 .
Now, the fact that H(ρ 0 ) : ker H(ρ 0 ) ⊥ → ker H(ρ 0 ) ⊥ is bijective, implies |H(ρ 0 )w 2 | ≥C|w 2 |, ∀w 2 ∈ ker H(ρ 0 ) ⊥ .
Combining this with (A.4), we get
( T, ∇ ρ H(ρ 0 ) w, w) ≥ C 2 |w| 2 − O 1 ε |H(ρ 0 )w 2 | 2 ,
which together with the fact that H(ρ 0 )w 2 = H(ρ 0 )w implies (A.1).
Lemma A.2. Let F ∈ C ∞ (R 2n ; H N −r )
and m(0) ∈ H r invertible, r ≥ 1. Assume that for ρ 0 ∈ R 2n , there exists T ∈ R 2n and C 0 > 0 such that
(A.5) T, ∇ ρ F (ρ 0 ) w, w ≥ C 0 |w| 2 , ∀w ∈ C N −r .
Then
(1) H(ρ) = F (ρ) 0 0 m(0)
is microhyperbolic at ρ 0 in the direction T . Therefore ( T, ∇ ρ H(0) w, w) + ( T, ∇ ρ M (0) w, w) = ( T, ∇ ρ F (0) w, w) ≥ C 0 |w| 2 , ∀w ∈ C N −r .
Since ker (H(0) + M (0)) = ker (H(0)) ⊂ C N −r , it follows from lemma A.1 that H + M is microhyperbolic at ρ 0 = 0 in the direction T . Then, H + M is microhyperbolic near 0 in the direction T .
The main result of this appendix is the following.
Theorem A.3. Let H ∈ C ∞ (R 2n ; H N ). Assume that H is microhyperbolic near ρ 0 ∈ R 2n in the direction T . There existsH ∈ C ∞ (R 2n ; H N ) such thatH = H near ρ 0 andH is uniformly microhyperbolic on R 2n in the direction T . Moreover, we can chooseH bounded together with all its derivatives, i.e.H ∈ S 0 (R 2n ; H N ).
Proof. Without any loss of generality, we may assume that ρ 0 = 0. We know that there exists P such that We recall that ker (H(0)) ⊂ {(w 1 , 0); w 1 ∈ C N −r }, with r = dim Im (m 22 ) (due to (A.6)).
Set
H 0 (ρ) := ∇ ρ m 11 (0)ρ 0 0 m 22 , ρ ∈ R 2n .
It follows from Lemma A.2 and (A.7) that H 0 is microhyperbolic at every point ρ ∈ R 2n in the direction T . Let χ ∈ C ∞ 0 (R 2n ; R) be such that χ(ρ) = 1 for |ρ| ≤ 1 and χ(ρ) = 0 for |ρ| ≥ 2. For δ > 0, set χ δ (ρ) = χ ρ δ . We define H δ (ρ) = χ ρ δ H(ρ) − H 0 (ρ) + H 0 (ρ).
We claim that for δ small enough, H δ is microhyperbolic at every point ρ ∈ R 2n in the direction T . In fact, for |ρ| ≤ δ, H δ (ρ) = H(ρ) is microhyperbolic at ρ 0 = 0 and then at every ρ ∈ R 2n with |ρ| ≤ δ. For |ρ| ≥ 2δ, H δ (ρ) = H 0 (ρ) which is microhyperbolic at every point ρ ∈ R 2n in the direction T . For δ < |ρ| < 2δ, we have
H δ (ρ) = H 0 (ρ) + O(|ρ| 2 ) O(|ρ|) O(|ρ|) O(|ρ|) .
Thus, Lemma A.2 implies that H δ is microhyperbolic in the direction T for δ small enough.
Consequently H δ is microhyperbolic at every point ρ ∈ R 2n in the direction T . To see that we can chooseH ∈ S 0 (R 2n ; H N ), let f ∈ C ∞ (R) such that f (t) = t for |t| < 1, |f (t)| ≥ 1 on |t| ≥ 1 and f (t) is constant at ±∞. PutH(x) = f (H δ (x)). By the functional calculus of self-adjoint operator, it is easy to check thatH satisfies the desired properties.
Proposition A.4. Let H ∈ C ∞ (R 2n ; H N ), χ ∈ C ∞ 0 (R 2n ) and τ 0 ∈ R. Assume that τ 0 − H(ρ) is microhyperbolic at every ρ ∈ suppχ. Let G(ρ, z) be an N × N matrix-valued function (not necessarily Hermitian) smooth with respect to ρ and holomorphic with respect to z in a neighborhood of τ 0 . Set, for ±ℑz > 0 respectively,
F ± (z) = R 2n (z − H(ρ)) −1 G(ρ, z)(z − H(ρ)) −1 χ(ρ)dρ.
Then, for real τ near τ 0 , the limit F ± (τ ± i0) := lim εց0 F ± (τ ± iε) exists and τ → F ± (τ ± i0) is smooth near τ 0 .
Proof. We consider F + . The proof for F − is similar. Decomposing χ into a finite sum of functions χ i with small support, we may assume using Theorem A.3 that τ − H(ρ) is microhyperbolic in the direction T at every point ρ ∈ R 2n and τ near τ 0 . We may also assume that G, H ∈ S 0 (R 2n ; H N ). LetH,G andχ be three almost analytic extensions of H, G and χ respectively, which are bounded together with all their derivatives. Put H(ρ, t) :=H(ρ + itT ),G(ρ, t, z) :=G(ρ + itT, z),χ(ρ, t) :=χ(ρ + itT ), t ∈ R.
We assert that for small enough ℑz ≥ 0, t ≥ 0 with ℑz + t > 0, there exist C, c > 0 such that (A. 8) ℑ((z −H(ρ, t))ω, ω) + Ct|(z −H(ρ, t))ω| 2 ≥ c(t + ℑz)|ω| 2 , ∀ω ∈ C N .
In fact ((z −H(ρ, t))ω, ω) = ((z − H(ρ))ω, ω) − it( T, ∇ ρ H(ρ) ω, ω) + O(t 2 )|ω| 2 and hence the global microhyperbolic condition (see (2.1)) yields, for some c, C 1 , C 2 > 0, H(ρ, t))ω, ω) ≥ (ℑz + ct)|ω| 2 − O(t)|(ℜz − H(ρ))ω| 2 + O(t 2 )|ω| 2 ≥ c(ℑz + t − C 1 (ℑz) 2 − C 2 t 2 )|ω| 2 − O(t)|(z −H(ρ, t))ω| 2 , uniformly on {z ∈ C; ℜz ∈]τ 0 − η, τ 0 + η[, ℑz > 0} for small enough η, and (A.8) follows from this inequality.
ℑ((z −
Applying Cauchy-Schwarz inequality to the first term of (A.8), we easily obtain (A. 9) z −H(ρ, t) N ×N + Ct z −H(ρ, t) 2 N ×N ≥ c(ℑz + t)|ω| 2 , ∀ω ∈ C N . This shows that (z −H(ρ, t)) for t > 0, ℑz ≥ 0.
For simplicity, assume T = (1, 0, · · · , 0). Put ρ = (ρ 1 , ρ ′ ) and fix t 0 > 0. By the Stokes' formula, we have H(ρ, t)) −1G (ρ, z, t)(z −H(ρ, t)) −1χ (ρ, t) dtdρ.
F + (z) = R 2n (z −H(ρ 1 + it 0 , ρ ′ )) −1G (ρ 1 + it 0 , ρ ′ , z)(z −H(ρ 1 + it 0 , ρ ′ )) −1χ (ρ 1 + it 0 , ρ ′ )dρ − R 2n ×[0,t 0 ] 1 2 (∂ ρ 1 + i∂ t ) (z −
Clearly the first term of the right hand side of the above equality extends to a C ∞ function up to ℑz ≥ 0. One sees that the same is true for the second term by using (A.10) and the fact that (∂ ρ 1 + i∂ t )H, (∂ ρ 1 + i∂ t )G, (∂ ρ 1 + i∂ t )χ are all of O(t ∞ ). This ends the proof.
In the integral expression (2.26) of I + , we can replace, modulo O (h ∞ ), K(z; h) byK t 0 (z; h) and then the integral domain ℑz > L by ℑz < −cL by the Cauchy theorem. Thus the estimate I + = O(h ∞ ) is reduced to the same argument as for I − , and we conclude I = O(h ∞ ). This gives Theorem 2.2.
Letf be an almost analytic extension of f satisfying (2.19) and (2.20) with suppf ⊂ {z ∈ C; |ℑz| ≤ 1}. If g is real analytic in a neighborhood of the support off , then we have, by the Helffer-Sjöstrand formula (see [7, Ch. 8]),
Remark 3. 1 .
1Let O be an open bounded subset of C such that suppf ⊂ O, and assume that the function K(z; h) defined by (3.3) in the upper half plane extends as a holomorphic functionK(z; h) to the zone O ℓ (h) := O ∩ {ℑz ≥ −ℓζ(h)} for all ℓ ∈ N and that the estimateK(z; h) = O(h −d(n) ) holds uniformly for z ∈ O ℓ (h) with d(n) depending only on the dimension. Then (2.3) remains true uniformly for ε ∈ [κ, h −ν [ with fixed κ > 0, ν ∈ N.
Lemma 4 . 1 .
41Under the assumption (2.7), we have
:= {z ∈ C; ℜz ∈ I and ℑz > −M ζ(h)} , where we recall that ζ(h) = h log( 1 h ). The idea of the proof of the following lemma is based on the theory of resonance and close to the one of Theorem 1 in[32].
Lemma 4 . 2 .
42In addition to the assumptions (2.7) and (2.15), we assume that
Lemma 4 . 4 .
44Assume that ϕ ∈ C ∞ 0 (] − 1, 1[; R) is 0 in a neighborhood of 0. Let κ be a positive constant independent of h and ν ∈ N. Under the assumptions of Lemma 4.1, there exists
( 2 )
2If (A.5) holds at ρ 0 = 0 and M ∈ C ∞ (R 2n , H N ) with M (ρ) = O(|ρ| 2 ) O(|ρ|) O(|ρ|) O(|ρ|) , then H + M is microhyperbolic near ρ 0 = 0 in the direction T .Proof. Since T, ∇ ρ H(ρ 0 ) = T, ∇ ρ F (ρ ∇ ρ H(0) + T, ∇ ρ M (0) = T, ∇ ρ F (0)
where m 22 is a diagonal and invertible matrix. Replacing H(ρ) by P H(ρ)P −1 , we may assume thatH(ρ) = m 11 (ρ) m 21 (ρ)m 12 (ρ) m 22 (ρ) with m 11 (0) = 0, m 12 (0) = 0, m 21 (0) = 0 and m 22 (0) = m 22 . Since H is microhyperbolic at 0 in the direction T , it follows from Lemma A.1 that (A.7) T, ∇ ρ m 11 (0) T, ∇ ρ m 21 (0) T, ∇ ρ m 12 (0)T, ∇ ρ m 22 , ∇ ρ m 11 (0) w 1 , w 1 ≥ C|w 1 | 2
.10) (z −H(ρ, t)) −1 N ×N = O(
Following[7], S k δ (R 2n ) := {a(·; h) ∈ C ∞ (R 2n ; R); ∀α ∈ N 2n : ∂ α x,ξ a(x, ξ; h) = Oα(h −δ|α|−k )}, for k ∈ R and δ ∈ [0, 1].
We have used the fact G is scalar valued only to prove that [G w , Pj,ω 1 (h)] = O(h).
Acknowledgement. This research was initiated when the first and second authors was visiting the Ritsumeikan University in May 2016; the financial support and kind hospitality are gratefully acknowledged. The first author acknowledges support from JSPS KAKENHI Grant number JP16H03944. The third author was partially supported by the JSPS KAKENHI Grant number JP15K04971.
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| []
|
[
"A POSTERIORI ERROR ESTIMATION FOR FINITE ELEMENT APPROXIMATIONS OF A PDE-CONSTRAINED OPTIMIZATION PROBLEM IN FLUID DYNAMICS *",
"A POSTERIORI ERROR ESTIMATION FOR FINITE ELEMENT APPROXIMATIONS OF A PDE-CONSTRAINED OPTIMIZATION PROBLEM IN FLUID DYNAMICS *"
]
| [
"Alejandro Allendes ",
"Enrique Otárola ",
"Richard Rankin "
]
| []
| []
| We derive globally reliable a posteriori error estimators for a PDE-constrained optimization problem involving linear models in fluid dynamics as state equation; control constraints are also considered. The corresponding local error indicators are locally efficient. The assumptions under which we perform the analysis are such that they can be satisfied for a wide variety of stabilized finite element methods as well as for standard finite element methods. When stabilized methods are considered, no a priori relation between the stabilization terms for the state and adjoint equations is required. If a lower bound for the inf-sup constant is available, a posteriori error estimators that are fully computable and provide guaranteed upper bounds on the norm of the error can be obtained. We illustrate the theory with numerical examples. | null | [
"https://arxiv.org/pdf/1708.00590v1.pdf"
]
| 54,682,412 | 1708.00590 | 50f54ab88c34764eb16d0ced9f4ab7ea592b3015 |
A POSTERIORI ERROR ESTIMATION FOR FINITE ELEMENT APPROXIMATIONS OF A PDE-CONSTRAINED OPTIMIZATION PROBLEM IN FLUID DYNAMICS *
2 Aug 2017
Alejandro Allendes
Enrique Otárola
Richard Rankin
A POSTERIORI ERROR ESTIMATION FOR FINITE ELEMENT APPROXIMATIONS OF A PDE-CONSTRAINED OPTIMIZATION PROBLEM IN FLUID DYNAMICS *
2 Aug 2017AMS subject classifications 49K2049M2565K1065N1565N3065N5065Y20
We derive globally reliable a posteriori error estimators for a PDE-constrained optimization problem involving linear models in fluid dynamics as state equation; control constraints are also considered. The corresponding local error indicators are locally efficient. The assumptions under which we perform the analysis are such that they can be satisfied for a wide variety of stabilized finite element methods as well as for standard finite element methods. When stabilized methods are considered, no a priori relation between the stabilization terms for the state and adjoint equations is required. If a lower bound for the inf-sup constant is available, a posteriori error estimators that are fully computable and provide guaranteed upper bounds on the norm of the error can be obtained. We illustrate the theory with numerical examples.
Introduction.
In this work we shall be interested in the design and analysis of computable a posteriori error estimators for a linear-quadratic optimal control problem involving linear models in fluid dynamics as state equation; control constraints are considered. To make matters precise, let Ω ⊂ R d , with d ∈ {2, 3}, be an open and bounded polytopal domain with Lipschitz boundary ∂Ω and f ∈ L 2 (Ω) d . Given a regularization parameter ϑ > 0 and a desired state y Ω ∈ L 2 (Ω) d , we define
(1.1) J(y, u) = 1 2 y − y Ω 2 L 2 (Ω) d + ϑ 2 u 2 L 2 (Ω) d .
We will be interested in the following PDE-constrained optimization problem: Find with a, b ∈ R d satisfying a < b; the previous vector inequalities being understood componentwise. In (1.3), ε, κ ∈ R and are such that ε > 0 and κ ≥ 0 and c ∈ W 1,∞ (Ω)
is a solenoidal field. The generalized Oseen equations describe the low-Reynoldsnumber flow in porous media in situations where velocity gradients are non-negligible; they provide a unified approach to model flows of viscous fluids in a cavity and a porous media. Our analysis allows for choices of the terms c and κ that yield different flow models:
(1.5)
c = 0, κ = 0 : −ε∆y + ∇p (Stokes), c = 0 : −ε∆y + κy + ∇p (Brinkman), κ = 0 : −ε∆y + (c · ∇) y + ∇p (Oseen).
The design of numerical techniques for approximating the solution to (1.3) has two major difficulties: first, in view of the so-called inf-sup condition [18,19], arbitrary finite element methods are not allowed, and second, considering standard finite element methods produces poor approximation results when convection-dominated regimes are considered [30]. In order to overcome such difficulties, a variety of finite element techniques have been proposed and analyzed in the literature: the family of stabilized finite element methods. We refer the reader to [30] for an extensive overview.
In the PDE-constrained optimization context, a usual alternative for approximating the solution to the optimal control problem (1.2)-(1.4) is based on the so-called optimize-then-discretize approach. This technique discretizes the associated optimality system: the state equations (1.3), the adjoint equations and a variational inequality that characterizes the optimal controlū. Consequently, the difficulties presented in the discretization of (1.3) are also present in the numerical approximation of the solution to (1.2)-(1.4). In addition, (1.2)-(1.4) is intrinsically nonlinear and, if c = 0, presents a crosswind phenomena; the convection field of the adjoint equations is the negative of the one appearing in (1.3). The latter further motives the development of an efficient solution technique that, in convection-dominated regimes, properly treats the oscillatory behaviors that occur when approximatingȳ and its adjoint variablē w and resolves interior or boundary layers exhibited by both variables. Failure to resolve boundary layers can pollute the numerical solution in the entire domain; see [20] for results involving the scalar version of (1.2)-(1.4). However, numerical schemes based only on stabilized techniques are not sufficient to approximate the solution to (1.2)- (1.4): in addition to the efficient resolution of either interior or boundary layers, some possible geometric singularities must be resolved. This motivates the methods that we will use in this work: stabilized adaptive finite element methods.
Adaptive finite element methods (AFEMs) are iterative methods that improve the quality of the finite element approximation to a partial differential equation (PDE) on the basis of an essential ingredient: an a posteriori error estimator. The a posteriori error analysis for the standard finite element approximation of elliptic problems has a solid foundation [2,28,37]. When stabilized approximations are considered, several estimators have been introduced and analyzed in the literature; see, for instance, [1,4,6,35,38]. However, in the PDE-constrained optimization context, the theory has not been fully developed. The main source of difficulty is its inherent nonlinear feature, which appears due to the control constraints. An attempt to unify the available results has been carried out recently in [21] where the authors derive an important relationship between the error in optimal control problems and estimators, that satisfy a set of suitable assumptions, for problems associated with the state and adjoint equations [21,Theorem 3.2].
In the current work, the assumptions under which we perform the analysis are such that they can be satisfied for a wide variety of stabilized finite element methods as well as for standard finite element methods. This includes using a different stabilization method to approximate the state equation from that used to approximate the adjoint equation. We derive a posteriori error estimators that are globally reliable. Moreover, if a lower bound for the inf-sup constant is available, we can obtain a posteriori error estimators that are fully computable and provide guaranteed upper bounds on the norm of the error. Consequently, the estimators can be used as a stopping criterion in adaptive algorithms. The local error indicators that can be used to adaptively refine the mesh are locally efficient. Furthermore, we observe that they can be used to efficiently resolve boundary layers.
The outline of this paper is as follows. In section 2 we introduce some terminology used throughout this work. In section 3 we study the optimal control problem (1.2)-(1.4) and obtain the associated optimality system. In section 4 we give the general form of the finite element methods that we consider for approximating the solution to (1.2)-(1.4). The core of our work is section 5, where we devise a family of a posteriori error estimators. Under suitable assumptions, we obtain abstract reliability results in section 5.1 and local efficiency of the corresponding error indicators in section 5.2. In section 6 we consider the estimators that we can obtain for a particular approximation method in more detail. Finally, in section 7 we present a series of numerical examples to illustrate the theory.
Preliminaries.
2.1. Notation. For a bounded domain A ⊂ R t , t ∈ {1, 2, 3}, L 2 (A) and H 1 (A) denote the standard Lebesgue and Sobolev spaces, respectively; L 2 0 (A) is the subspace of L 2 (A) containing functions with zero mean value on A, and H 1 0 (A) is the subspace of H 1 (A) containing functions whose trace is zero on ∂A. We use bold letters to denote the vector-valued counterparts of the aforementioned spaces and an extra under accent for their matrix-valued counterparts. For instance, for d ∈ {2, 3}, we denote L 2 (A) = L 2 (A) d and L We now proceed to define notation associated with the discretization of the domain. Let T = {K} be a conforming partition ofΩ into simplical elements K [13,18]. We assume that T is a member of a shape regular family of partitions. Let F denote the set of all element edges(2D)/faces(3D) and F I ⊂ F denote the set of interior edges(2D)/faces(3D). For an element K ∈ T , let:
• P n (K) denote the space of polynomials on K of total degree at most n;
• F K ⊂ F denote the set containing the individual edges(2D)/faces(3D) of K;
• h K denote the diameter of K;
• n K γ denote the unit exterior normal vector to the edge(2D)/face(3D) γ ∈ F K . For an edge(2D)/face(3D) γ ∈ F , let:
• P n (γ) denote the space of polynomials on γ of total degree at most n;
• Ω γ = {K ∈ T : γ ∈ F K };
• h γ denote the diameter of the edge(2D)/face(3D) γ. To simplify the exposition of the material, we define V = H 1 0 (Ω) and Q = L 2 0 (Ω) with norms ||| · ||| V,Ω and ||| · ||| Q,Ω defined, for all ξ ∈ V and φ ∈ Q, by The relation a b indicates that there exists a constant C such that a ≤ Cb. The constant C may be different at each occurence but is independent of a, b and the size of the elements in the mesh.
2.2. Inequalities. For K ∈ T and nonnegative integers l, we denote by Π K,l the L 2 (K)-orthogonal projection operator onto P l (K) d . This operator is defined as
(2.3) Π K,l : L 2 (K) → P l (K) d , (t − Π K,l (t), v) L 2 (K) = 0 ∀v ∈ P l (K) d .
Throughout the manuscript we will frequently make use of the following inequalities. First, if K ∈ T and ξ ∈ V, we have the Poincaré inequalities [7,25,29] (2.4)
ξ L 2 (Ω) ≤ C P,Ω ∇ξ L ≈ 2 (Ω) and ξ − Π K,0 (ξ) L 2 (K) ≤ h K π ∇ξ L ≈ 2 (K) , where (2.5) C P,Ω = 1 π d i=1 1 |l i | 2 −1/2
with |l 1 | , . . . , |l d | being the sides of a d-dimensional box containing Ω. We immediately comment that these inequalities imply that, for ξ ∈ V and K ∈ T ,
(2.6) ξ L 2 (Ω) ≤ C Ω |||ξ||| V,Ω and ξ − Π K,0 (ξ) L 2 (K) ≤ C K |||ξ||| V,K , where (2.7) C Ω = CP,Ω √ ε , if κ = 0, min CP,Ω √ ε , 1 √ κ , if κ = 0, and (2.8) C K = hK π √ ε , if κ = 0, min hK π √ ε , 1 √ κ , if κ = 0. We define A : V × V → R, B : V × Q → R and C : V × V → R by (2.9) A(ξ, ζ) := ε(∇ξ, ∇ζ) L ≈ 2 (Ω) + (κξ + (c · ∇) ξ, ζ) L 2 (Ω) , B(ζ, φ) := (φ, ∇ · ζ) L 2 (Ω) , C(ξ, ζ) := ε(∇ξ, ∇ζ) L ≈ 2 (Ω) + (κξ − (c · ∇) ξ, ζ) L 2 (Ω) .
The fact that c is a solenoidal vector field and integration by parts implies that
(2.10) A(ξ, ζ) = C(ζ, ξ) ∀ξ, ζ ∈ V.
Moreover, for all ξ ∈ V,
(2.11) A(ξ, ξ) = C(ξ, ξ) = |||ξ||| 2 V,Ω
and, for all ξ, ζ ∈ V,
(2.12) A(ξ, ζ) ≤ C ct |||ξ||| V,Ω |||ζ||| V,Ω , C(ξ, ζ) ≤ C ct |||ξ||| V,Ω |||ζ||| V,Ω , where (2.13) C ct = 1 + C Ω √ ε |c| L ∞ (Ω) ,
with |c| L ∞ (Ω) being the L ∞ (Ω) norm of |c| and C Ω being given by (2.7). We now recall the standard inf-sup condition [18,19]: there exists a positive constant β such that
(2.14) β φ L 2 (Ω) ≤ sup ξ∈V\{0} B(ξ, φ) ∇ξ L ≈ 2 (Ω) ∀φ ∈ Q.
Notice that, in view of |||ξ||| 2 V,Ω ≤ (ε + κC 2 P,Ω ) ∇ξ 2 L ≈ 2 (Ω) , we have that
(2.15) |||φ||| Q,Ω ≤ C is sup ξ∈V\{0} B(ξ, φ) |||ξ||| V,Ω ∀φ ∈ Q, where (2.16) C is = ε + κC 2 P,Ω β .
3. Optimal control problem: optimize. In this section we briefly analyze the optimal control problem (1.2)-(1.4). To accomplish this task, we begin by introducing the following weak version of the state equations (1.3): Find (y, p) ∈ V × Q such that
(3.1) A(y, ξ) − B(ξ, p) = (f + u, ξ) L 2 (Ω) ∀ ξ ∈ V, B(y, φ) = 0 ∀ φ ∈ Q,
where the bilinear forms A and B are defined by (2.9) and we recall that ε > 0, κ ≥ 0, c ∈ W 1,∞ (Ω) is a solenoidal field, f ∈ L 2 (Ω), V = H 1 0 (Ω) and Q = L 2 0 (Ω). In view of the fact that A satisfies (2.11) and (2.12) and B satisfies the inf-sup conditions (2.14) and (2.15), we conclude the well-posedness of problem (3.1) [18,19]. We also mention that, due to de Rham's Theorem (see Section 4.1.3 and Theorem B73 in [18]), we can consider the following equivalent formulation of problem (3.1): Find y ∈ V 0 such that
(3.2) A(y, ξ) = (f + u, ξ) L 2 (Ω) ∀ξ ∈ V 0 , where V 0 := {v ∈ H 1 0 (Ω) : ∇ · v = 0}.
To analyze our optimal control problem, we follow [22,36] and introduce the socalled control to state map S : L 2 (Ω) → V 0 which, given a control u, associates to it the state y that solves (3.2). In addition, we define, for a, b ∈ R d with a < b, the set the vector inequalities being understood componentwise. The set U ad is a bounded, convex, closed and nonempty subset of L 2 (Ω) and consequently weakly sequentially compact. Thus, in view of the fact that the reduced cost functional
f (u) := 1 2 S(u) − y Ω 2 L 2 (Ω) + ϑ 2 u 2 L 2 (Ω)
is weakly lower semicontinuous and strictly convex (ϑ > 0), we conclude the existence and uniqueness of an optimal controlū and an optimal stateȳ that satisfy (3.2), or equivalently (3.1); see Theorem 2.14 in [36]. The existence ofp such that (ȳ,p) solves (3.1) follows from de Rham's Theorem. In addition, we have thatū satisfies the first-order optimality condition
(3.4) f ′ (ū)(u −ū) ≥ 0 ∀ u ∈ U ad ;
see [36,Lemma 2.21]. To explore this variational inequality, and to obtain optimality conditions, we define, on the basis of the formal Lagrange method (see [16,Section 3.3] and [36, Section 2.10]), the adjoint state (w, q) as the unique solution to the following weak problem: Find (w, q) ∈ V × Q such that
(3.5) C(w, ζ) + B(ζ, q) = (y − y Ω , ζ) L 2 (Ω) ∀ ζ ∈ V, B(w, ψ) = 0 ∀ ψ ∈ Q.
With this adjoint state at hand, the variational inequality (3.4) can be rewritten as
(3.6) (w + ϑū, u −ū) L 2 (Ω) ≥ 0 ∀ u ∈ U ad .
We have thus arrived at the following optimality system: (ȳ,p,ū) ∈ V × Q × U ad is optimal for the PDE-constrained optimization problem (1.
2)-(1.4) if and only if (ȳ,p,w,q,ū) ∈ V × Q × V × Q × U ad solves (3.7) A(ȳ, ξ) − B(ξ,p) = (f +ū, ξ) L 2 (Ω) , ∀ ξ ∈ V, B(ȳ, φ) = 0, ∀ φ ∈ Q, C(w, ζ) + B(ζ,q) = (ȳ − y Ω , ζ) L 2 (Ω) , ∀ ζ ∈ V, B(w, ψ) = 0, ∀ ψ ∈ Q, (w + ϑū, u −ū) L 2 (Ω) ≥ 0, ∀ u ∈ U ad ;
see also [
(3.9) Π [a,b] (ξ) − Π [a,b] (ζ) L 2 (K) ≤ ξ − ζ L 2 (K) ∀ξ, ζ ∈ V.
4. Finite element discretization. We follow the optimize-then-discretize approach and introduce a numerical scheme to approximate the solution to (3.7). The scheme allows for the incorporation of stabilization terms into the standard Galerkin discretizations of the state and adjoint equations; no a priori relation between the stabilized terms is required. We refer the reader to Remark 4.1 below for a discussion regarding the advantages of the proposed approach when solving (1.2)-(1.4).
The stabilized scheme reads as follows:
Find (ȳ T ,p T ,w T ,q T ,ū T ) ∈ V(T ) × Q(T ) × V(T ) × Q(T ) × U ad (T ) such that (4.1) A(ȳ T , ξ) − B(ξ,p T ) + S(ȳ T ,p T , f +ū T ; ξ) = (f +ū T , ξ) L 2 (Ω) , B(ȳ T , φ) + H(ȳ T ,p T , f +ū T ; φ) = 0, C(w T , ζ) + B(ζ,q T ) + Q(w T ,q T ,ȳ T − y Ω ; ζ) = (ȳ T − y Ω , ζ) L 2 (Ω) , B(w T , ψ) + K(w T ,q T ,ȳ T − y Ω ; ψ) = 0, (w T + ϑū T , u −ū T ) L 2 (Ω) ≥ 0, for all (ξ, φ, ζ, ψ, u) ∈ V(T ) × Q(T ) × V(T ) × Q(T ) × U ad (T )
; the bilinear forms A, B and C being defined as in (2.9). We consider the setting where the discrete spaces V(T ) and Q(T ) are subspaces of V and Q, respectively, and the discrete set
U ad (T ) is a subset of U ad . Hence, V(T ) ⊂ V, Q(T ) ⊂ Q and U ad (T ) ⊂ U ad .
The terms S and H, and Q and K in (4.1), correspond to stabilization terms for the state and adjoint equations, respectively. Finally, we assume that V(T ), Q(T ), U ad (T ), S, H, Q and K are such that at least one solution to (4.1) exists.
Remark 4.1 (optimize-then-discretize approach). In this work, we consider the optimize-then-discretize approach because it allows for the incorporation of different stabilization terms into the discrete state and adjoint equations. The purpose of the latter is twofold: first, the use of low-order methods, and second, the efficient resolution of (4.1), by appropriately tuning some associated stabilization parameters, in convection-dominated regimes. The latter is especially important since, as is observed in [20] for the scalar case, the failure to resolve boundary layers exhibited by the solution of (3.7) can pollute the numerical solution in the entire domain. In contrast, the use of the discretize-then-optimize approach imposes a relationship between the stabilization terms. To be precise, for a given stabilization terms S in the state equations, the aforementioned approach imposes that the stabilization term Q is its adjoint counterpart [14]. This could lead to an unnatural stabilization term in the adjoint equations delivering oscillatory solutions and therefore poor approximation results in convection-dominated regimes [14].
Before proceeding with the analysis of our method, it is instructive to comment on those advocated in the literature. Regarding the a priori theory, in the absence of control constraints, the design and analysis of numerical techniques for solving (1.2)-(1.3), with c = 0 and κ = 0, have been investigated in several papers; see [11,32,34] and references therein. To the best of our knowledge, and again, for c = 0 and κ = 0, the first work that incorporates control-constraints and analyzes stabilized schemes for (1.2)-(1.4) is [31]; the optimal control is discretized by using piecewise constant functions. The authors, on the basis of postprocessing techniques, provide a quadratic error estimate for the approximation of the optimal control variable [31, Theorem 2.8]. Subsequently, the authors of [26] extend the results of [31] and analyze nonconforming schemes for the discretization of the state and adjoint equations; in contrast to [31], the vector field is not assumed to be in H 2 (Ω) ∩ W 1,∞ (Ω). In addition, [26] analyzes an anisotropic scheme for approximating the solution to (1.2)-(1.4) when Ω is not convex; a domain with a reentrant edge (d = 3) is considered. We conclude this paragraph by mentioning the reference [17], where the authors investigate numerical techniques for solving a modification of problem (1.2)-(1.4) that, in addition, includes constraints on the state variable.
Regarding the a posteriori error analysis, to the best of our knowledge, the first work to propose an error estimator for (1.2)-(1.4), with c = 0 and κ = 0, is [24]. In this work, the authors follow the discretize-then-optimize approach and obtain a discrete optimality system with no stabilization terms [24, equation (2.9)]. They propose an error estimator in a two-dimensional setting and analyze its reliability properties [24, Theorem 3.1]. However, there is no efficiency analysis. Later, an asymptotically exact ZZ-type a posteriori error estimator was proposed in [23]. The authors derive upper and lower bounds for the error in terms of the proposed estimator [23, Theorem 5.1] that relies on an error non-degeneracy condition [23, inequality (2.24)] and strong regularity assumptions on (ȳ,p): it is assumed to belong to Lemma 4.2]. In [15], the authors propose an a posteriori error estimator for (1.2)-(1.4) but with the state equations (1.3) replaced by a Stokes-Darcy system: they study the reliability and efficiency properties of the proposed estimator. We also mention [27], where a similar PDE-constrained optimization problem has been analyzed but with the control-constraint (1.4) replaced by the state-constraint y L 2 (Ω) ≤ γ, where γ > 0: an error estimator is proposed and its reliability and efficiency properties are investigated. All the aforementioned references consider plain Galerkin discretizations for the state and adjoint equations, i.e., no stabilization terms are considered. We conclude this paragraph by mentioning the so-called dual weighted residual method (DWR) [10] and its applications to the optimal control of flow problems [8,9].
H 3 (Ω) ∩ V × H 1 (Ω) ∩ Q [23,
Recently, the authors of [21] propose and analyze an a posteriori error estimator for problem (1.2)-(1.4) when κ = 0 [21, Section 5]. The associated discrete optimal system incorporates stabilized terms, into the state and adjoint equations, that are based on the streamline-diffusion finite element method (SDFEM). On the basis of proposed and analyzed a posteriori error estimators for the state and adjoint equations, the authors derive an estimator for (1.2)-(1.4). We comment that the obtained upper bound for the error, in terms of the a posteriori error estimator, is not computable.
In this work we analyze a family of a posteriori error estimators in a unifying framework that incorporates a wide variety of standard and stabilized finite element methods.
5.
A posteriori error analysis. In this section we derive and analyze a posteriori error estimators for the solution to the discretization (4.1) of the optimal control problem (3.7).
Reliability analysis.
We begin this section by introducing the following notation. Let e y :=ȳ−ȳ T , e p :=p−p T , e w :=w−w T , e q :=q−q T and e u :=ū−ū T , where (ȳ,p,w,q,ū) ∈ V × Q × V × Q × U ad is the solution to the optimality system (3.7) and (ȳ T ,
p T ,w T ,q T ,ū T ) ∈ V(T ) × Q(T ) × V(T ) × Q(T ) × U ad (T )
is its numerical approximation given as the solution to (4.1). The goal of this section is to obtain an upper bound for
(5.1)
|||(e y , e p , e w , e q , e u )||| 2 Ω := K∈T |||(e y , e p , e w , e q , e u )||| 2 K where |||(e y , e p , e w , e q , e u )|||
2 K := |||e y ||| 2 V,K + ̺|||e p ||| 2 Q,K + |||e w ||| 2 V,K + ̺|||e q ||| 2 Q,K + e u 2 L 2 (K) .
The norms ||| · ||| V,K and ||| · ||| Q,K are defined as in (2.2) and the parameter ̺ is a nonnegative constant that will be arbitrary in the analysis but fixed in the numerical experiments of Section 7. The upper bound for the error (5.1) that we obtain is constructed using upper bounds on the error between the solution to the discretization (4.1) and auxilliary variables that we define in what follows. Let (ŷ,p) ∈ V × Q be the solution to
(5.2) A(ŷ, ξ) − B(ξ,p) = (f +ū T , ξ) L 2 (Ω) ∀ ξ ∈ V, B(ŷ, φ) = 0 ∀ φ ∈ Q.
We notice that, in view of (4.1), we have that (ȳ T ,p T ) ∈ V(T ) × Q(T ) satisfies
(5.3) A(ȳ T , ξ) − B(ξ,p T ) + S(ȳ T ,p T , f +ū T ; ξ) = (f +ū T , ξ) L 2 (Ω) B(ȳ T , φ) + H(ȳ T ,p T , f +ū T ; φ) = 0
for all ξ ∈ V(T ) and φ ∈ Q(T ). Consequently, (ȳ T ,p T ) can be seen as a finite element approximation of the solution to (5.2). We thus make the following assumption: Assumption 1. There exist quantities η y and η p which are such that
(5.4) |||ŷ −ȳ T ||| V,Ω ≤ η y and |||p −p T ||| Q,Ω ≤ η p .
Let (ŵ,q) ∈ V × Q be the solution to
(5.5) C(ŵ, ζ) + B(ζ,q) = (ȳ T − y Ω , ζ) L 2 (Ω) ∀ ζ ∈ V, B(ŵ, ψ) = 0 ∀ ψ ∈ Q.
We notice that, again in view of (4.1),
(w T ,q T ) ∈ V(T ) × Q(T ) satisfies (5.6) C(w T , ζ) + B(ζ,q T ) + Q(w T ,q T ,ȳ T − y Ω ; ζ) = (ȳ T − y Ω , ζ) L 2 (Ω) , B(w T , ψ) + K(w T ,q T ,ȳ T − y Ω ; ψ) = 0,
for all ζ ∈ V(T ) and ψ ∈ Q(T ), and hence (w T ,q T ) corresponds to a finite element approximation of the solution to (5.5). We thus make the following assumption: Assumption 2. There exist quantities η w and η q which are such that
(5.7) |||ŵ −w T ||| V,Ω ≤ η w and |||q −q T ||| Q,Ω ≤ η q .
We introduce the auxiliary control variable
(5.8)ũ = Π [a,b] − 1 ϑw T .
We define the error between this auxilliary control variable andū T as follows:
(5.9) η u := K∈T η 2 ct,K 1/2 , with η u,K := ũ −ū T L 2 (K) .
We also define (5.10) C y = 2 + 2µC 6 Ω + 4(1 + ̺ω)(C 4 Ω + µC 8 Ω + 2µC 12 Ω ), with µ = 4ϑ −2 and ω = C 2 is (1 + C ct ) 2 . We now present the analysis through which we obtain an upper bound for the total error. where (5.14) Υ 2 := C y η 2 y + 2̺η 2 p + C w η 2 w + 2̺η 2 q + C u η 2 u , and C y , C w and C u are defined by (5.10), (5.11) and (5.12), respectively.
Proof. We proceed in 6 steps. Step 1. The goal of this step is to control the term e u L 2 (Ω) . We begin with a simple application of the triangle inequality to write
(5.15) e u 2 L 2 (Ω) ≤ 2 ū −ũ 2 L 2 (Ω) + 2 ũ −ū T 2 L 2 (Ω) = 2 ū −ũ 2 L 2 (Ω) + 2η 2 u , whereũ = Π [a,b] − 1
ϑw T and η u is defined as in (5.9). Let us now bound the first term on the right hand side of (5.15). To accomplish this task we first observe a key property that the auxiliary control variableũ satisfies:
(5.16) (w T + ϑũ, u −ũ) L 2 (Ω) ≥ 0 ∀u ∈ U ad ;
see Lemma 2.26 and Theorem 2.28 in [36]. Set u =ũ in the variational inequality of (3.7) and u =ū in (5.16). We thus obtain that
(w + ϑū,ũ −ū) L 2 (Ω) ≥ 0, (w T + ϑũ,ū −ũ) L 2 (Ω) ≥ 0,
and, consequently, that
(5.17) ϑ ū −ũ 2 L 2 (Ω) ≤ (w −w T ,ũ −ū) L 2 (Ω) .
In order to bound the right hand side of (5.17), we first define (ỹ,p) ∈ V × Q as the solution to
(5.18) A(ỹ, ξ) − B(ξ,p) = (f +ũ, ξ) L 2 (Ω) ∀ ξ ∈ V, B(ỹ, φ) = 0 ∀ φ ∈ Q.
In addition, we define (w,q) ∈ V × Q as the solution to
(5.19) C(w, ζ) + B(ζ,q) = (ỹ − y Ω , ζ) L 2 (Ω) ∀ ζ ∈ V, B(w, ψ) = 0 ∀ ψ ∈ Q.
Utilizing the statesŵ andw defined as the solutions to (5.5) and (5.19), respectively, we arrive at
ϑ ū −ũ 2 L 2 (Ω) ≤ (w −w,ũ −ū) L 2 (Ω) + (w −ŵ,ũ −ū) L 2 (Ω) + (ŵ −w T ,ũ −ū) L 2 (Ω) ≤ (w −w,ũ −ū) L 2 (Ω) + 1 ϑ w −ŵ 2 L 2 (Ω) + 1 ϑ ŵ −w T 2 L 2 (Ω) + ϑ 2 ū −ũ 2 L 2 (Ω)
upon using Cauchy-Schwarz and Young's inequalities. Hence,
(5.20) ū −ũ 2 L 2 (Ω) ≤ 2 ϑ (w −w,ũ −ū) L 2 (Ω) + 2 ϑ 2 w −ŵ 2 L 2 (Ω) + ŵ −w T 2 L 2 (Ω) .
We proceed to bound (w −w,ũ −ū) L 2 (Ω) . To accomplish this task, we first notice that, since (w,q) solves the adjoint problem of the optimality system (3.7) and (w,q) solves (5.19), the fact thatp −p ∈ Q implies that B(w −w,p −p) = 0. Thus, since (ȳ,p) and (ỹ,p) solve (3.7) and (5.18), respectively, we arrive at
(ū −ũ,w −w) L 2 (Ω) = A(ȳ −ỹ,w −w).
We now invoke (2.10) and, again, the fact that (w,q) and (w,q) solve (3.7) and (5.19), respectively, to obtain that
(5.21) (ũ −ū,w −w) L 2 (Ω) = A(ỹ −ȳ,w −w) = C(w −w,ỹ −ȳ) = − ȳ −ỹ 2 L 2 (Ω) ≤ 0,
upon noticing that, since (ȳ,p) solves the state equations of the optimality system (3.7) and (ỹ,p) solves (5.18), the fact thatq −q ∈ Q implies that B(ȳ −ỹ,q −q) = 0. Using the previous estimate in (5.20) we obtain that
(5.22) ū −ũ 2 L 2 (Ω) ≤ 2 ϑ 2 w −ŵ 2 L 2 (Ω) + 2 ϑ 2 ŵ −w T 2 L 2 (Ω) .
The control of the second term on the right hand side of (5.22) follows from (2.6) and Assumption 2:
ŵ −w T 2 L 2 (Ω) ≤ C 2 Ω η 2 w .
We now turn our attention to bounding the term w −ŵ L 2 (Ω) . Applying similar arguments to the ones that lead to (5.21) we obtain that
(5.23) |||w −ŵ||| 2 V,Ω = C(w −ŵ,w −ŵ) = (ỹ −ȳ T ,w −ŵ) L 2 (Ω) ≤ C Ω ỹ −ȳ T L 2 (Ω) |||w −ŵ||| V,Ω ,
where we have also used (2.6). Consequently, w −ŵ 2
L 2 (Ω) ≤ C 4 Ω ỹ −ȳ T 2 L 2 (Ω)
, upon using, again, (2.6). It thus suffices to bound ỹ −ȳ T L 2 (Ω) . We proceed as follows:
ỹ −ȳ T 2 L 2 (Ω) ≤ 2 ỹ −ŷ 2 L 2 (Ω) + 2 ŷ −ȳ T 2 L 2 (Ω) .
To control the second term on the right hand side of the previous expression, we invoke Assumption 1 and (2.6). We thus conclude that
ŷ −ȳ T 2 L 2 (Ω) ≤ C 2 Ω η 2 y .
To bound the first term, we employ that (ŷ,p) and (ỹ,p) solve (5.2) and (5.18), respectively. This, on the basis of ∇ · c = 0 and (2.6), yields
(5.24) |||ỹ −ŷ||| 2 V,Ω = A(ỹ −ŷ,ỹ −ŷ) = (ũ −ū T ,ỹ −ŷ) L 2 (Ω) ≤ C Ω ũ −ū T L 2 (Ω) |||ỹ −ŷ||| V,Ω ,
which allows us to conclude, in view of (5.9) and (2.6), that
ỹ −ŷ 2 L 2 (Ω) ≤ C 4 Ω η 2 u .
On the basis of (5.15) and (5.22), we combine our previous findings and arrive at
e u 2 L 2 (Ω) ≤ 2µC 6 Ω η 2 y + µC 2 Ω η 2 w + 2 + 2µC 8 Ω η 2 u , (5.25) where µ = 4ϑ −2 .
Step 2. The goal of this step is to bound |||e y ||| V,Ω . To accomplish this task, we apply the triangle inequality and invoke Assumption 1. In fact,
(5.26) |||e y ||| 2 V,Ω ≤ 2|||ȳ −ŷ||| 2 V,Ω + 2|||ŷ −ȳ T ||| 2 V,Ω ≤ 2|||ȳ −ŷ||| 2 V,Ω + 2η 2 y .
To control the remaining term we employ similar ideas to the ones that lead to (5.24). These arguments reveal that Step 3. We now bound the term |||e w ||| V,Ω . To accomplish this task, we use, again, the triangle inequality and Assumption 2 to obtain that
(5.27) |||ȳ −ŷ||| 2 V,Ω ≤ C 2 Ω ū −ū T 2 L 2 (Ω) ,(5.29) |||e w ||| 2 V,Ω ≤ 2|||w −ŵ||| 2 V,Ω + 2η 2 w .
To bound |||w −ŵ||| 2 V,Ω we invoke the optimality system (3.7) and (5.5). In fact, the arguments that allow us to obtain (5.23) immediately yield
|||w −ŵ||| 2 V,Ω = C(w −ŵ,w −ŵ) = (ȳ −ȳ T ,w −ŵ) L 2 (Ω) ≤ ȳ −ȳ T L 2 (Ω) w −ŵ L 2 (Ω)
upon using a Cauchy-Schwarz inequality. In view of (2.6), we conclude that
(5.30) |||w −ŵ||| 2 V,Ω ≤ C 4 Ω |||ȳ −ȳ T ||| 2 V,Ω ,
which, combined with the estimates (5.28) and (5.29), yields
(5.31) |||e w ||| 2 V,Ω ≤ 4C 4 Ω 2µC 8 Ω + 1 η 2 y + 2 2µC 8 Ω + 1 η 2 w + 4C 6 Ω 2 + 2µC 8 Ω η 2 u .
Step 4. We now bound |||e p ||| Q,Ω . We start with a simple application of the triangle inequality and Assumption 1:
|||e p ||| 2 Q,Ω ≤ 2|||p −p||| 2 Q,Ω + 2|||p −p T ||| 2 Q,Ω ≤ 2|||p −p||| 2 Q,Ω + 2η 2 p ;
we recall that (ŷ,p) solves (5.2). To control the first term on the right hand side of the previous expression, we utilize the inf-sup condition (2.15):
(5.32) |||p −p||| Q,Ω ≤ C is sup ξ∈V\{0} B(ξ,p −p) |||ξ||| V,Ω .
Since (ȳ,p) and (ŷ,p) solve (3.7) and (5.2), respectively, we conclude that
B(ξ,p −p) = A(ȳ −ŷ, ξ) − (ū −ū T , ξ) L 2 (Ω) ≤ C ct |||ȳ −ŷ||| V,Ω + C Ω ū −ū T L 2 (Ω) |||ξ||| V,Ω ,
upon using (2.6) and (2.12). In view of (5.27) we thus arrive at
B(ξ,p −p) ≤ C Ω (1 + C ct ) ū −ū T L 2 (Ω) |||ξ||| V,Ω .
This and (5.32) imply that |||p −p||| Q,Ω ≤ C is C Ω (1 + C ct ) ū −ū T L 2 (Ω) . Thus,
(5.33) |||e p ||| 2 Q,Ω ≤ 2ωC 2 Ω ū −ū T 2 L 2 (Ω) + 2η 2 p ,
where ω = C 2 is (1 + C ct ) 2 . We conclude the estimate for |||e p ||| (5.34) |||e p ||| 2 Q,Ω ≤4µωC 8 Ω η 2 y + 2µωC 4 Ω η 2 w + 2ωC 2 Ω 2 + 2µC 8 Ω η 2 u + 2η 2 p .
Step 5. We bound |||e q ||| Q,Ω . Similar arguments to the ones employed in the previous step yield
|||e q ||| 2 Q,Ω ≤ 2|||q −q||| 2 Q,Ω + 2|||q −q T ||| 2 Q,Ω ≤ 2|||q −q||| 2 Q,Ω + 2η 2 q and |||q −q||| Q,Ω ≤ C is sup ζ∈V\{0} (ȳ −ȳ T , ζ) L 2 (Ω) − C(w −ŵ, ζ) |||ζ||| V,Ω ≤ C is C 2 Ω |||ȳ −ȳ T ||| V,Ω + C ct |||w −ŵ||| V,Ω .
We finally use (5.30), and conclude that |||q −q||| Q,Ω ≤ C is C 2 Ω (1 + C ct ) |||ȳ −ȳ T ||| V,Ω , and then that
(5.35) |||e q ||| 2 Q,Ω ≤ 2ωC 4 Ω |||ȳ −ȳ T ||| 2 V,Ω + 2η 2 q ,
where, we recall that, ω = C 2 is (1 + C ct ) 2 . Consequently,
(5.36) |||e q ||| 2 Q,Ω ≤ 4ωC 4 Ω 2µC 8 Ω + 1 η 2 y + 4µωC 8 Ω η 2 w + 4ωC 6 Ω 2 + 2µC 8 Ω η 2 u + 2η 2 q .
Step 6. Combining It is important in a posteriori error analysis to have an upper bound for the error that is in terms of local error indicators, so that it can be used to adaptively refine the mesh. Such a bound follows from Theorem 5.1 under the following two assumptions.
Assumption 3. There exist quantities η y,K and η p,K that are such that where (5.40) Υ 2 K := C y η 2 y,K + 2̺η 2 p,K + C w η 2 w,K + 2̺η 2 q,K + C u η 2 u,K , and C y , C w and C u are defined by (5.10) (5.11), and (5.12), respectively.
Proof. In view of Assumptions 3 and 4, the proof follows from a simple application of the result of Theorem 5.1.
Theorem 5.2 can be used to obtain guaranteed upper bounds on the error if the value of a β satisfying (2.14) is known and the quantities η y,K , η p,K , η w,K and η q,K are computable. If this is not the case then Theorem (5.2) can still be used to arrive at an a posteriori error estimator under the following assumption.
Assumption 5. There exist computable quantitiesη y,K ,η p,K ,η w,K andη q,K which are such that η y,K η y,K , η p,K η p,K , η w,K η w,K and η q,K η q,K for all K ∈ T . where (5.42)Υ 2 K :=η 2 y,K +η 2 p,K +η 2 w,K +η 2 q,K +η 2 u,K .
Proof. Upon invoking Assumptions 3, 4 and 5, the estimate (5.41) is a consequence of Theorem 5.2.
Efficiency analysis.
In this section we prove the local efficiency of the a posteriori error indicators Υ K andΥ K defined by (5.40) and (5.42), respectively. In what follows we will assume that Assumptions 3, 4 and 5 are satisfied and that ̺ = 0. In addition, we make two further assumptions. To state them, we first define, for nonnegative integers l, the discrete space
(5.43) P l (T ) = v ∈ L 2 (Ω) : v |K ∈ P l (K) d for all K ∈ T .
Our first additional assumption reads as follows: Assumption 6. The spaces V(T ) and Q(T ) and the set U ad (T ) are such that • V(T ) = V ∩ P lV (T ) for some positive integer l V , • Q(T ) = Q∩P lQ (T ) for some nonnegative integer l Q or Q(T ) = Q∩P lQ (T )∩ H 1 (Ω) for some positive integer l Q , • U ad (T ) = U ad ∩ P lU (T ) for some nonnegative integer l U or U ad (T ) = U ad ∩ P lU (T ) ∩ H 1 (Ω) for some positive integer l U . For K ∈ T , we define the following residuals and oscillation terms:
(5.44) R st K := Π K,m (f) +ū T |K + ε∆y T |K − Π K,m ((c · ∇)ȳ T |K ) − κȳ T |K − ∇p T |K , (5.45) R ad K :=ȳ T |K −Π K,m (y Ω )+ε∆w T |K +Π K,m ((c · ∇)w T |K )−κw T |K +∇q T |K , (5.46) osc st K := f − Π K,m (f) − ((c · ∇)ȳ T |K − Π K,m ((c · ∇)ȳ T |K )), and (5.47) osc ad K := −(y Ω − Π K,m (y Ω )) + ((c · ∇)w T |K − Π K,m ((c · ∇)w T |K )), where m = max {l V , l Q − 1, l U }.
We recall that the operator Π K,m is defined as in (2.3), and notice that, in view of the choice of m, we have the following invariance property: with R ad γ,K := −ε n K γ · ∇ w T |K −q T |K n K γ .
Π K,m (R st K ) = R st K and Π K,m (R ad K ) = R ad K . For γ ∈ F I , we define (5.48) R st γ := K∈Ωγ R st γ,K with R st γ,K := −ε n K γ · ∇ ȳ T |K +p T |K n K γ ,and
We now state our final assumption. Assumption 7. For all K ∈ T , the computable quantitiesη y,K ,η p,Kηw,K , and η q,K , introduced in Assumption 5, are such that
Υ 2 K ∇ ·ȳ T 2 L 2 (K) + ∇ ·w T 2 L 2 (K) + K ′ ∈TK h 2 K R st K ′ 2 L 2 (K ′ ) + R ad K ′ 2 L 2 (K ′ ) + γ∈FK h K R st γ 2 L 2 (γ) + R ad γ 2 L 2 (γ) + K ′ ∈TK h 2 K osc st K ′ 2 L 2 (K ′ ) + osc ad K ′ 2 L 2 (K ′ ) + η 2 u,K (5.50) whereT K ⊂ T andF K ⊂ F I .
Under Assumptions 3, 4, 5, 6 and 7 we present an efficiency analysis. We start by noting that, since Υ K Υ K , we only need to bound terms that appear on the right hand side of (5.50).
We first invoke integration by parts and (3.1) to conclude that
K∈T (R st K , ξ) L 2 (K) + γ∈FI ( R st γ , ξ) L 2 (γ) =A(e y , ξ) + B(ξ, e p ) − (e u , ξ) L 2 (Ω) − K∈T (osc st K , ξ) L 2 (K) ∀ξ ∈ V.
We now apply standard bubble function arguments [2,37] to this equation to obtain
(5.51) R st K 2 L 2 (K) h −2 K |||e y ||| 2 V,K + ̺|||e p ||| 2 Q,K + e u 2 L 2 (K) + osc st K 2 L 2 (K)
for K ∈ T , and that, for γ ∈ F I ,
R st γ 2 L 2 (γ) K ′ ∈Ωγ h −1 K ′ |||e y ||| 2 V,K ′ + ̺|||e p ||| 2 Q,K ′ + h K ′ e u 2 L 2 (K ′ ) + osc st K ′ 2 L 2 (K ′ )
.
(5.52)
On the other hand, using (3.5) and, again, integration by parts we obtain that
K∈T (R ad K , ξ) L 2 (K) + γ∈FI ( R ad γ , ξ) L 2 (γ) =C(e w , ξ) − B(ξ, e q ) − (e y , ξ) L 2 (Ω) − K∈T (osc ad K , ξ) L 2 (K) ∀ξ ∈ V.
Applying standard bubble function arguments, again, to this equation yields
(5.53) R ad K 2 L 2 (K) h −2 K |||e w ||| 2 V,K + ̺|||e q ||| 2 Q,K + e y 2 L 2 (K) + osc ad K 2 L 2 (K)
for K ∈ T , and, for γ ∈ F I ,
R ad γ 2 L 2 (γ) K ′ ∈Ωγ h −1 K ′ |||e w ||| 2 V,K ′ + ̺|||e q ||| 2 Q,K ′ + h K ′ e y 2 L 2 (K ′ ) + osc ad K ′ 2 L 2 (K ′ )
.
(5.54)
We now proceed to bound the terms ∇ ·ȳ T 2 L 2 (K) and ∇ ·w T 2 L 2 (K) in (5.50). To accomplish this task, we notice that ∇ · ξ ∈ Q for all ξ ∈ V. Then, it follows from the second equation of (3.7) that ∇ ·ȳ = 0, and thus that
(5.55) ∇ ·ȳ T 2 L 2 (K) = ∇ · e y 2 L 2 (K) |||e y ||| 2 V,K .
Similarly, it follows from the fourth equation of (3.7) that
(5.56) ∇ ·w T 2 L 2 (K) = ∇ · e w 2 L 2 (K) |||e w ||| 2 V,K .
We conclude with an estimate for the term η u,K defined by (5.9):
η u,K ≤ e u L 2 (K) + Π [a,b] (− 1 ϑw ) − Π [a,b] (− 1 ϑw T ) L 2 (K) ≤ e u L 2 (K) + 1 ϑ e w L 2 (K)
upon invoking the triangle inequality, (3.8), and (3.9). Hence,
(5.57) η 2 u,K e u 2 L 2 (K) + e w 2 L 2 (K) .
The following theorem then follows upon combining (5.50)-(5.57).
Theorem 5.4 (local efficiency). If ̺ = 0 and Assumptions 3, 4, 5, 6 and 7 hold, then
Υ 2 K Υ 2 K e w 2 L 2 (K) + K ′ ∈ΩK
|||(e y , e p , e w , e q , e u )|||
2 K ′ + h 2 K ′ e u 2 L 2 (K ′ ) + e y 2 L 2 (K ′ ) + osc st K ′ 2 L 2 (K ′ ) + osc ad K ′ 2 L 2 (K ′ ) , withΩ K =T K ∪ γ∈FK Ω γ .
The following corollary follows upon using (2.6) and the fact that Ω is bounded.
6.
A particular example. Henceforth, we shall consider a particular case of the approximation scheme (4.1). We set V(T ) = V ∩ P 1 (T ), Q(T ) = Q ∩ P 0 (T ), U ad (T ) = U ad ∩ P 0 (T ),
(6.1) S(ȳ T ,p T , f +ū T ; ξ) = K∈T S K (ȳ T ,p T , f +ū T ; ξ), (6.2) H(ȳ T ,p T , f +ū T ; φ) = τ γ γ∈FI h γ ([p T ], [φ]) L 2 (γ) ,(6.3) Q(w T ,q T ,ȳ T − y Ω ; ζ) = K∈T Q K (w T ,q T ,ȳ T − y Ω ; ζ), and (6.4) K(w T ,q T ,ȳ T − y Ω ; ψ) = −τ γ γ∈FI h γ ([q T ], [ψ]) L 2 (γ) , where S K (ȳ T ,p T , f +ū T ; ξ) = τ K ((c · ∇)ȳ T + κȳ T − (f +ū T ), (c · ∇) ξ) L 2 (K) , Q K (w T ,q T ,ȳ T − y Ω ; ζ) = τ K ((c · ∇)w T − κw T +ȳ T − y Ω , (c · ∇) ζ) L 2 (K)
and
[v] denotes the jumps in v. The stabilization parameters τ γ and τ K are such that τ γ > 0 and 0 < τ K h 2 K . Note that these choices correspond to solving the state equations using a particular case of the method given by [30, equation (3.6)] and are such that Assumption 6 is satisfied.
We note that alternative methods for solving the state equations can be found in [12] but we restrict our attention to the method described above in order to simplify the presentation.
6.1. Fully computable a posteriori error estimators. In this section we obtain a posteriori error estimators that satisfy the assumptions of Section 5 and are fully computable if the value of a β satisfying (2.14) is known. We first define some quantities that the estimators will be defined in terms of.
For ς = st and ς = ad, let the equilibrated fluxes g ς γ,K ∈ P 1 (γ) d be such that
(6.5) g ς γ,K + g ς γ,K ′ = 0, if γ ∈ F K ∩ F K ′ , K, K ′ ∈ T , K = K ′ , (f +ū T , λ) L 2 (K) − ε(∇ȳ T , ∇λ) L ≈ 2 (K) − (κȳ T + (c · ∇)ȳ T , λ) L 2 (K) +(p T , ∇ · λ) L 2 (K) − S K (ȳ T ,p T , f +ū T ; λ) + γ∈FK (g st γ,K , λ) L 2 (γ) = 0
for all λ ∈ P 1 (K) d and all K ∈ P,
(ȳ T − y Ω , λ) L 2 (K) − ε(∇w T , ∇λ) L ≈ 2 (K) − (κw T − (c · ∇)w T , λ) L 2 (K) −(q T , ∇ · λ) L 2 (K) − Q K (w T ,q T ,ȳ T − y Ω ; λ) + γ∈FK (g ad γ,K , λ) L 2 (γ) = 0
for all λ ∈ P 1 (K) d and all K ∈ P, and (6.6)
γ∈FK h K g ς γ,K + R ς γ,K 2 L 2 (γ) K ′ ∈TK h 2 K R ς K ′ 2 L 2 (K ′ ) + γ∈FK h K R ς γ 2 L 2 (γ)
for all K ∈ P, wherê
T K = {K ′ ∈ T : V K ∩ V K ′ = ∅} andF K = γ∈FK {γ ′ ∈ F I : V γ ∩ V γ ′ = ∅}
with V K denoting the set containing the vertices of element K and V γ denoting the set containing the vertices of the edge/face γ. For information that will help with the construction of such g ς γ,K we refer the reader to [2, Chapter 6] and [4,5].
For ς = st and ς = ad, we also define σ ς K ∈ P 2 (K) d×d to be such that
−div σ ς K = R ς K in K, σ ς K n K γ = g ς γ,K + R ς γ,K on γ, ∀ γ ∈ F K ,
and σ ς K L 2 (K) is minimized. We note that the g ς γ,K are such that the data in the above problem are compatible in the sense that σ ς K exists. Moreover, for all K ∈ T ,
(6.7) (σ ς K , ∇ξ) L ≈ 2 (K) = (R ς K , ξ) L 2 (K) + γ∈FK (g ς γ,K + R ς γ,K , ξ) L 2 (γ) ∀ ξ ∈ V and (6.8) σ ς K 2 L 2 (K) h 2 K R ς K ′ 2 L 2 (K) + γ∈FK h K g ς γ,K + R ς γ,K 2 L 2 (γ) .
For information on the construction of such σ ς K we refer the reader to [3,4]. Finally, for ς = st and ς = ad, we define
(6.9) Ψ ς,K = 1 √ ε σ ς K L 2 (K) + C K osc ς K L 2 (K) .
We thus have the following result.
Theorem 6.1. Assumption 3 holds with
(6.10) η 2 y,K = 3Ψ 2 st,K + C 2 is 1 + 2C 2 ct ∇ ·ȳ T 2 L 2 (K)
and (6.11)
η 2 p,K = 2C 2 is 1 + 3C 2 ct Ψ 2 st,K + C 2 is C 2 ct 1 + 2C 2 ct ∇ ·ȳ T 2 L 2 (K) .
Moreover, Assumption 1 holds with Proof. Let E y ∈ V be the solution to
(6.13) ε(∇E y , ∇ξ) L ≈ 2 (Ω) + κ(E y , ξ) L 2 (Ω) = A(ŷ −ȳ T , ξ) − B(ξ,p −p T ) ∀ ξ ∈ V.
Letting φ =p − p T in (2.15) yields that
|||p −p T ||| Q,Ω ≤ C is sup ξ∈V\{0} B(ξ,p −p T ) |||ξ||| V,Ω .
To control the right-hand side of the previous estimate we use (6.13) and obtain that
B(ξ,p −p T ) = A(ŷ −ȳ T , ξ) − ε(∇E y , ∇ξ) L ≈ 2 (Ω) − κ(E y , ξ) L 2 (Ω) ≤ C ct |||ŷ −ȳ T ||| V,Ω |||ξ||| V,Ω + |||E y ||| V,Ω |||ξ||| V,Ω ,
upon using (2.12). Hence, (6.14) |||p −p T ||| Q,Ω ≤ C is |||E y ||| V,Ω + C ct |||ŷ − y T ||| V,Ω .
We now estimate |||ŷ −ȳ T ||| V,Ω . Sincep −p T ∈ Q, by using the second equation of (5.2) we have that
B(ŷ −ȳ T ,p −p T ) = −B(ȳ T ,p −p T ) ≤ ∇ ·ȳ T L 2 (Ω) |||p −p T ||| Q,Ω .
Thus, by using the previous estimate and letting ξ =ŷ −ȳ T in (6.13), we arrive at
|||ŷ −ȳ T ||| 2 V,Ω = ε(∇E y , ∇(ŷ −ȳ T )) L ≈ 2 (Ω) + κ(E y ,ŷ −ȳ T ) L 2 (Ω) + B(ŷ −ȳ T ,p −p T ) ≤ |||E y ||| V,Ω |||ŷ −ȳ T ||| V,Ω + ∇ ·ȳ T L 2 (Ω) |||p −p T ||| Q,Ω .
This, in view of (6.14), then yields that
|||ŷ −ȳ T ||| 2 V,Ω ≤ C is ∇ ·ȳ T L 2 (Ω) |||E y ||| V,Ω + |||E y ||| V,Ω + C is C ct ∇ ·ȳ T L 2 (Ω) |||ŷ −ȳ T ||| V,Ω ≤ C 2 is 2 ∇ ·ȳ T 2 L 2 (Ω) + 1 2 |||E y ||| 2 V,Ω + 1 2 |||E y ||| V,Ω + C is C ct ∇ ·ȳ T L 2 (Ω) 2 + 1 2 |||ŷ −ȳ T ||| 2 V,Ω from which it follows that |||ŷ −ȳ T ||| 2 V,Ω ≤ C 2 is ∇ ·ȳ T 2 L 2 (Ω) + |||E y ||| 2 V,Ω + |||E y ||| V,Ω + C is C ct ∇ ·ȳ T L 2 (Ω) 2 .
Hence, upon observing that
|||E y ||| V,Ω + C is C ct ∇ ·ȳ T L 2 (Ω) 2 ≤ 2|||E y ||| 2 V,Ω + 2C 2 is C 2 ct ∇ ·ȳ T 2 L 2 (Ω) ,
we can arrive at
(6.15) |||ŷ −ȳ T ||| 2 V,Ω ≤ 3|||E y ||| 2 V,Ω + C 2 is 1 + 2C 2 ct ∇ ·ȳ T 2 L 2 (Ω) .
Furthermore, (6.14) allows us to conclude that
|||p −p T ||| 2 Q,Ω ≤ 2C 2 is |||E y ||| 2 V,Ω + C 2 ct |||ŷ −ȳ T ||| 2 V,Ω .
Applying (6.15) then yields that
(6.16) |||p −p T ||| 2 Q,Ω ≤ 2C 2 is 1 + 3C 2 ct |||E y ||| 2 V,Ω + C 2 is C 2 ct 1 + 2C 2 ct ∇ ·ȳ T 2 L 2 (Ω) .
Now, letting ξ = E y in (6.13) yields that
|||E y ||| 2 V,Ω = A(ŷ −ȳ T , E y ) − B(E y ,p −p T ) = K∈T (R st K , E y ) L 2 (K) + γ∈FK (g st γ,K + R st γ,K , E y ) L 2 (γ) + (osc st K , E y ) L 2 (K) by (5.|||E y ||| 2 V,Ω = K∈T (σ st K , ∇E y ) L ≈ 2 (K) + (osc st K , E y − Π K,0 (E y )) L 2 (K) ≤ K∈T Ψ 2 st,K 1/2 |||E y ||| V,Ω
by the Cauchy-Schwarz inequality and (2.4). Consequently,
(6.17) |||E y ||| 2 V,Ω ≤ K∈T Ψ 2 st,K .
The theorem then follows upon combining (6.15), (6.16) and (6.17).
We note that the above theorem is an improvement and adaptation to the case considered in this section of the results from [4]. The below theorem can be proved similarly to how the above theorem was proved.
Theorem 6.2. Assumption 4 holds with
(6.18) η 2 w,K = 3Ψ 2 ad,K + C 2 is 1 + 2C 2 ct ∇ ·w T 2 L 2 (K)
and
(6.19) η 2 q,K = 2C 2 is 1 + 3C 2 ct Ψ 2 ad,K + C 2 is C 2 ct 1 + 2C 2 ct ∇ ·w T 2 L 2 (K) .
Moreover, Assumption 2 holds with
(6.20) η w = K∈P η 2 w,K 1/2 and η q = K∈P η 2 q,K 1/2 .
We note that, if the value of a β satisfying (2.14) is known, then Assumption 5 holds withη y,K = η y,K ,η p,K = η p,K ,η w,K = η w,K andη q,K = η q,K . Furthermore, by (6.6) and (6.8) we have that
η 2 y,K + η 2 p,K ∇ ·ȳ T 2 L 2 (K) + γ∈FK h K R st γ 2 L 2 (γ) + K ′ ∈TK h 2 K R st K ′ 2 L 2 (K ′ ) + osc st K ′ 2 L 2 (K ′ ) (6.21)
and
η 2 w,K + η 2 q,K ∇ ·w T 2 L 2 (K) + γ∈FK h K R ad γ 2 L 2 (γ) + K ′ ∈TK h 2 K R ad K ′ 2 L 2 (K ′ ) + osc ad K ′ 2 L 2 (K ′ ) (6.22)
from which it follows that Assumption 7 is also satisfied. We note that it also follows that
|||ŷ −ȳ T ||| 2 V,Ω + |||p −p T ||| 2 Q,Ω K∈T ∇ ·ȳ T 2 L 2 (K) + γ∈FK h K R st γ 2 L 2 (γ) + h 2 K R st K 2 L 2 (K) + osc st K 2 L 2 (K) (6.23) and |||ŵ −w T ||| 2 V,Ω + |||q −q T ||| 2 Q,Ω K∈T ∇ ·w T 2 L 2 (K) + γ∈FK h K R ad γ 2 L 2 (γ) + h 2 K R ad K 2 L 2 (K) + osc ad K 2 L 2 (K) (6.24)
6.2. Residual-based a posteriori error estimators. From (6.23) and (6.24) the following result follows.
Theorem 6.3. Let η 2 y,K =η 2 p,K = ∇ ·ȳ T 2 L 2 (K) + γ∈FK h K R st γ 2 L 2 (γ) + h 2 K R st K 2 L 2 (K) + osc st K 2 L 2 (K) , (6.25)η 2 w,K =η 2 q,K = ∇ ·w T 2 L 2 (K) + γ∈FK h K R ad γ 2 L 2 (γ) + h 2 K R ad K 2 L 2 (K) + osc ad K 2 L 2 (K) ,(6.
26) η y,K = Cη y,K , η p,K = Cη p,K , η w,K = Cη w,K , η q,K = Cη q,K ,
η y = η p = K∈T η 2 y,K = K∈T η 2 p,K , η w = η q = K∈T η 2 w,K = K∈T η 2 q,K , andT K =F K = {K},
where C is a positive constant that is independent of the size of the elements in the mesh. Then Assumptions 1, 2, 3, 4, 5 and 7 hold.
Numerical examples.
We performed numerical examples using the approximation method described in section 6 with τ K = h 2 K for all K ∈ T and τ γ = 1 for all γ ∈ F I . We considered ϑ = 1 and ̺ = 1. The number of degrees of freedom Ndof = 2dN v + (d + 2)N e , where N v is the number of vertices in the mesh and N e is the number of elements in the mesh.
Two dimensional examples.
We perform two dimensional examples on polygonal domains for which the value of a β satisfying (2.14) is known. After obtaining the approximate solution, the a posteriori error estimator Υ from Theorem 5.1 was computed with the aid of Theorems 6.1 and 6.2. We note that the estimator Υ provides a guaranteed upper bound on |||(e y , e p , e w , e q , e u )||| Ω . The local error indicators Υ K from Theorem 5.2 were also computed, again with the aid of Theorems 6.1 and 6.2. Each mesh T was adaptively refined by marking for refinement the elements K ∈ T that were such that Υ 2 K ≥ N −1 e K ′ ∈T Υ 2 K ′ . In this way a sequence of adaptively refined meshes was generated from the initial meshes shown in Figure 1. Example 1. We consider the square domain Ω = (0, 1) 2 . From [33] we have that (2.14) holds with β = sin(π/8). We took ε = 1, c(x 1 , x 2 ) = (x 2 , −x 1 ), κ = 1, a = (−0.5, −0.5) and b = (0.5, 0.5). The data f and y Ω were chosen to be such that
y(x 1 , x 2 ) = curl (x 1 (1 − x 1 )x 2 (1 − x 2 )) 2 ,p(x 1 , x 2 ) = cos(2πx 1 ) cos(2πx 2 ),
w(x 1 , x 2 ) = curl (sin(2πx 1 ) sin(2πx 2 )) 2 ,q(x 1 , x 2 ) = sin(2πx 1 ) sin(2πx 2 ).
The results are shown in Figure 2. We observe that the error |||(e y , e p , e w , e q , e u )||| Ω and the estimator Υ are decreasing at the optimal rate. Example 2. We consider the triangular domain Ω = {(x 1 , x 2 ) : [33] we have that (2.14) holds with β = sin(π/16). We took ε = 0.01, c = (0, 0), κ = 1, a = (0, 0) and b = (0.1, 0.1). The data f and y Ω were chosen to be such that
x 1 > 0, x 2 > 0, x 1 + x 2 < 1}. Fromy(x 1 , x 2 ) = curl x 1 x 2 2 (1 − x 1 − x 2 ) 2 1 − x 1 − exp(−100x 1 ) − exp(−100) 1 − exp(−100) ,
p(x 1 , x 2 ) = cos(2πx 2 )/1024,
w(x 1 , x 2 ) = curl x 2 1 x 2 (1 − x 1 − x 2 ) 2 1 − x 2 − exp(−100x 2 ) − exp(−100) 1 − exp(−100) , andq (x 1 , x 2 ) = cos(2πx 1 )/1024.
The results are shown in Figure 3. We observe that, once the mesh has been sufficiently refined, the error |||(e y , e p , e w , e q , e u )||| Ω and the estimator Υ decrease at the optimal rate. We also observe that more refinement has been performed in the regions where the solution has boundary layers.
Example 3. We consider the L-shaped domain Ω = (−1, 1) 2 \ ([0, 1) × (−1, 0]). From [33] we have that (2.14) holds with β = 0.1601. We took ε = 1, c = (0, 0), κ = 0, a = (0, 0), b = (1, 1), f = (1, 1) and y Ω (x 1 , x 2 ) = (x 2 , −x 1 ). The results are shown in Figure 4. We observe that the estimator Υ decreases at the optimal rate and that more refinement is being performed in regions close to the reentrant corner. The true solution to this problem is unknown and hence we cannot compute |||(e y , e p , e w , e q , e u )||| Ω . However, from Theorem 5.1 we know that |||(e y , e p , e w , e q , e u )||| Ω ≤ Υ. Example 4. We considered the same problem as in the previous example with the exception that we took the domain to be the T-shaped domain Ω = ((−1.5, 1.5) × (0, 1)) ∪ ((−0.5, 0.5) × (−2, 0]) on which we have that (2.14) holds with β = 0.1076 from [33]. The results are shown in Figure 5. Similar observations to those made about the previous example can be made.
7.2.
Three dimensional examples. Unfortunately, we are not aware of any polyhedral domains for which the value of a β satisfying (2.14) is known. Hence, when the domain is three dimensional, the estimator Υ from Theorem 5.1 and the local error indicators Υ K from Theorem 5.2 are not computable. Consequently, after obtaining the approximate solution, the a posteriori error estimatorΥ and the local error indicatorsΥ K from Theorem 5.3 were computed, with the aid of Theorem 6.3. Each mesh T was adaptively refined by marking for refinement the elements K ∈ T that were such thatΥ 2 K ≥ N −1 e K ′ ∈TΥ 2 K ′ . In this way a sequence of adaptively refined meshes was generated from the initial meshes shown in Figure 6. We note that we have not proved that the estimatorΥ provides a guaranteed upper bound on |||(e y , e p , e w , e q , e u )||| Ω . However, from Theorem 5.3 we know that |||(e y , e p , e w , e q , e u )||| Ω Υ . Example 5. We consider the cuboidal domain Ω = (0, 1) 3 . We took ε = 1, c(x 1 , x 2 , x 3 ) = (x 2 − x 3 , x 3 − x 1 , x 1 − x 2 ), κ = 1, a = (−0.5, −0.5, −0.5) and b = (0.5, 0.5, 0.5). The data f and y Ω were chosen to be such that y(x 1 , x 2 , x 3 ) = curl (x 1 (1 − x 1 )x 2 (1 − x 2 )x 3 (1 − x 3 )) 2 ,p(x 1 , x 2 , x 3 ) = cos(2πx 3 ), w(x 1 , x 2 , x 3 ) = curl (sin(2πx 1 ) sin(2πx 2 ) sin(2πx 3 )) 2 ,q(x 1 , x 2 , x 3 ) = sin(2πx 3 ).
The results are shown in Figure 7. We observe that the error |||(e y , e p , e w , e q , e u )||| Ω and the estimatorΥ are decreasing at the optimal rate. Example 6. We consider the tetrahedral domain Ω = {(x 1 , x 2 , x 3 ) : x 1 > 0, x 2 > 0, x 3 > 0, x 1 + x 2 + x 3 < 1}. We took ε = 0.01, c = (1, 1, 1), κ = 0, a = (0, 0, 0) and b = (0.1, 0.1, 0.1). The data f and y Ω were chosen to be such that andq (x 1 , x 2 , x 3 ) = (sin(2πz) − 3/(2π)) /1024,
where χ = x 2 3 (1 − x 1 − x 2 − x 3 ) 2 .
The results are shown in Figure 8. We observe that, once the mesh has been sufficiently refined, the error |||(e y , e p , e w , e q , e u )||| Ω and the estimatorΥ decrease at the optimal rate. We also observe that more refinement has been performed in the regions where the solution has boundary layers.
+ (c · ∇) y + κy + ∇p = f + u in Ω, ∇ · y = 0in Ω, y = 0 on ∂Ω, and the control constraints(1.4) a ≤ u ≤ b a.e.in Ω,
≈ 2 (
2A) = L 2 (A) d×d .
V
,K := ε ∇ξ 2 L ≈ 2 (K) + κ ξ 2 L 2 (K) and |||φ||| 2 Q,K := φ 2 L 2 (K) .
ad := {v ∈ L 2 (Ω) : a ≤ v ≤ b a.e. in Ω};
Theorem 5.1 (global reliability). If Assumptions 1 and 2 hold, then(5.13)|||(e y , e p , e w , e q , e u )||| 2 Ω ≤ Υ 2
( 5 .
525), (5.28), (5.31), (5.34) and (5.36) allows us to arrive at (5.13).
Assumption 4 .
4There exist quantities η w,K and η q,K that are such thatTheorem 5.2 (global reliability). If Assumptions 3 and 4 hold, then (5.39)|||(e y , e p , e w , e q ,
|||(e y , e p , e w , e q ,
|||(e y , e p , e w , e q , e u
Fig. 1 .
1The initial meshes used for Examples 1, 2, 3 and 4.
2 Fig. 2 .
22e y , e p , e w , e q , e u )||| Ω e y , e p , e w , e q , e u )||| Ω Υ Ndof −1/Example 1: The error |||(ey, ep, ew, eq, eu)||| Ω and estimator Υ (left) and the 19th adaptively refined mesh (right).
2 Fig. 3 .
23e y , e p , e w , e q , e u )||| Ω Υ Ndof −1/Example 2: The error |||(ey, ep, ew, eq, eu)||| Ω and estimator Υ (left) and the 19th adaptively refined mesh (right).
Fig. 4 .
4Example 3: The estimator Υ (left) and the 17th adaptively refined mesh (right).
Fig. 5 .
5Example 4: The estimator Υ (left) and the 15th adaptively refined mesh (right).
Fig. 6 .
6Exterior views of the initial meshes used for Examples 5 and 6.
y(x 1 3 Fig. 7 .
137, x 2 , x 3 ) = curl x 1 x 2 2 χ 1 − x 1 − exp(−100x 1 ) − exp(−100) 1 − exp(−100) , p(x 1 , x 2 , x 3 ) = cos(2πz) − 3/(2π 2 ) /1024, w(x 1 , x 2 , x 3 ) = curl x 2 1 x 2 χ 1 − x 2 − exp(−100x 2 ) − exp(−100) 1 − exp(−100), e y , e p , e w , e q , e u )||| Ω Υ Ndof −1/Example 5: The error |||(ey, ep, ew, eq, eu)||| Ω and estimatorΥ (left) and an exterior view of the 19th adaptively refined mesh (right).
/ 3 Fig. 8 .
38e y , e p , e w , e q , e u )||| Ω e y , e p , e w , e q , e u )||| Ω Υ Ndof −1Example 6: The error |||(ey, ep, ew, eq, eu)||| Ω and estimatorΥ (left) and an exterior view of the 19th adaptively refined mesh (right).
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[
"Omnidirectional excitation of surface waves and super-Klein tunneling at the interface between two different bi-isotropic media",
"Omnidirectional excitation of surface waves and super-Klein tunneling at the interface between two different bi-isotropic media"
]
| [
"Seulong Kim \nDepartment of Energy Systems Research\nDepartment of Physics\nAjou University\n16499SuwonKorea\n",
"Kihong Kim \nDepartment of Energy Systems Research\nDepartment of Physics\nAjou University\n16499SuwonKorea\n"
]
| [
"Department of Energy Systems Research\nDepartment of Physics\nAjou University\n16499SuwonKorea",
"Department of Energy Systems Research\nDepartment of Physics\nAjou University\n16499SuwonKorea"
]
| []
| We study theoretically some unique characteristics of surface electromagnetic waves excited at the interface between two different kinds of general bi-isotropic media, which include Tellegen media and chiral media as special cases. We derive an analytical dispersion relation for those waves, using which we deduce eight different conditions under which they are generated between two Tellegen media and between two chiral media independently of the component of the wave vector along the interface. These make it possible to excite the surface waves for all or a wide range of incident angles in attenuated total reflection experiments on multilayer structures. We generalize the concept of a conjugate matched pair to bi-isotropic media and obtain several conditions under which the omnidirectional total transmission, which we call the super-Klein tunneling, occurs through conjugate matched pairs consisting of Tellegen media and of chiral media. We find that these conditions are closely linked to those for the omnidirectional excitation of surface waves. Using the invariant imbedding method, we perform extensive numerical calculations of the absorptance, the transmittance, and the spatial distribution of the electromagnetic fields for circularly-polarized waves incident on bilayer structures and confirm that the results agree perfectly with the analytical predictions. | 10.1103/physrevb.101.165428 | [
"https://arxiv.org/pdf/2001.01352v2.pdf"
]
| 209,862,170 | 2001.01352 | 0ade1d64abf7738fb67772d92f7b1a5d4cd442f6 |
Omnidirectional excitation of surface waves and super-Klein tunneling at the interface between two different bi-isotropic media
4 Apr 2020
Seulong Kim
Department of Energy Systems Research
Department of Physics
Ajou University
16499SuwonKorea
Kihong Kim
Department of Energy Systems Research
Department of Physics
Ajou University
16499SuwonKorea
Omnidirectional excitation of surface waves and super-Klein tunneling at the interface between two different bi-isotropic media
4 Apr 2020
We study theoretically some unique characteristics of surface electromagnetic waves excited at the interface between two different kinds of general bi-isotropic media, which include Tellegen media and chiral media as special cases. We derive an analytical dispersion relation for those waves, using which we deduce eight different conditions under which they are generated between two Tellegen media and between two chiral media independently of the component of the wave vector along the interface. These make it possible to excite the surface waves for all or a wide range of incident angles in attenuated total reflection experiments on multilayer structures. We generalize the concept of a conjugate matched pair to bi-isotropic media and obtain several conditions under which the omnidirectional total transmission, which we call the super-Klein tunneling, occurs through conjugate matched pairs consisting of Tellegen media and of chiral media. We find that these conditions are closely linked to those for the omnidirectional excitation of surface waves. Using the invariant imbedding method, we perform extensive numerical calculations of the absorptance, the transmittance, and the spatial distribution of the electromagnetic fields for circularly-polarized waves incident on bilayer structures and confirm that the results agree perfectly with the analytical predictions.
I. INTRODUCTION
Surface electromagnetic waves of various kinds have been a focus of intensive research in recent decades [1,2]. It has been well known that when these waves are excited on the surface of a medium or at the interface between two media, the electromagnetic fields close to the surface or the interface are greatly enhanced. The high sensitivity of this enhancement to various parameters and the resulting linear and nonlinear optical effects have been successfully applied in developing efficient photonic devices and sensors [3][4][5]. Strong interests in the physics and applications of these waves have even created a new research area called plasmonics [6,7].
Surface plasma waves or surface plasmons, which are the simplest type of surface electromagnetic waves, can be excited on the surface of metals by external electromagnetic radiation [8]. More recently, there has been a growing interest in the different types of surface waves in other complex media such as Dyakonov waves excited at the interface of anisotropic media [9,10], optical Tamm plasmon polaritons excited on the surface of a photonic crystal [11][12][13], and surface plasmon polaritons associated with chiral media [14,15]. Surface polaritons on the surface of negative index media [16,17], surface waves due to a spatial inhomogeneity near the surface of semiconductors [18,19], surface waves in general bianisotropic media [20], and the influence of optical nonlinearity on surface plasmons [21][22][23] have also attracted some attention.
In a recent paper, we have presented a detailed study of the characteristics of surface waves excited at the interface between a metal and a general bi-isotropic medium * [email protected] [24]. Bi-isotropic media, which include Tellegen media and chiral media as special cases, are the most general form of linear isotropic media, where the electric displacement D and the magnetic induction B are linearly and isotropically related to both the electric field E and the magnetic intensity H [25,26]. In cgs Gaussian units, the constitutive relations for harmonic waves in these media can be expressed as
D = ǫE + aH, B = µH + a * E,(1)
where ǫ is the dielectric permittivity and µ is the magnetic permeability. The magnetoelectric parameter a is written as
a = χ + iγ,(2)
where χ is called the non-reciprocity (or Tellegen) parameter and γ is called the chirality index.
In the present paper, we generalize the theory of Ref. [24] to the case where the surface waves are excited at the interface between two different kinds of general bi-isotropic media. We first derive analytically the generalized dispersion relation for surface waves. Our main focus in this work is to derive the explicit conditions under which surface waves are generated between two Tellegen media and between two chiral media independently of the component of the wave vector along the interface, starting from the dispersion relation. When these conditions are satisfied, it is possible to excite the surface waves for all or a wide range of incident angles in attenuated total reflection (ATR) experiments on multilayer structures. We call this phenomenon the omnidirectional excitation of surface waves. We confirm these predictions by calculating the absorptance and the spatial distribution of the electromagnetic fields for circularly-polarized waves incident on multilayer structures using a generalized version of the invariant imbedding method (IIM) [27][28][29][30][31].
We also find that the omnidirectional excitation of surface waves is intimately related to the phenomenon of omnidirectional total transmission of waves through a conjugate matched bilayer. In Ref. [32], the authors have considered the wave propagation through a pair of slabs with the medium parameters ǫ 1 and µ 1 and ǫ 2 and µ 2 , respectively. It has been demonstrated that when the condition ǫ 1 /ǫ 2 = µ 1 /µ 2 = −1 is precisely satisfied and the thicknesses of the two slabs are the same, waves incident on the bilayer are totally transmitted regardless of the incident angle and the polarization. The pair of slabs satisfying the above conditions has been called the conjugate matched pair. We generalize this concept to bi-isotropic media and show that the omnidirectional total transmission also occurs in such cases in the complete absence of dissipation. If there exists a small dissipation in the same system, however, we find that a finite absorption due to the omnidirectional excitation of surface waves always arises.
Recently, the omnidirectional total transmission of electron waves through a scalar potential barrier has been found to occur in pseudospin-1 Dirac-type materials, when the electron energy is one-half the value of the potential [33][34][35][36][37][38]. In an analogy to the Klein tunneling occurring when electrons are incident normally on a potential barrier of an arbitrary shape in Dirac materials [39][40][41][42], this phenomenon has been termed the super-Klein tunneling. From the viewpoint of the physics of wave propagation, the origins of the the super-Klein tunneling and the omnidirectional total transmission through a conjugate matched pair are quite similar, and therefore we will refer the latter as the super-Klein tunneling as well. This phenomenon is also studied in detail by calculating the transmittance using the IIM.
The rest of this paper is organized as follows. In Sec. II we derive an analytical dispersion relation for surface waves at the interface between two different bi-isotropic media. Using the dispersion relation, we derive eight different conditions for omnidirectional excitation of surface waves in Sec. III. We also specify the conditions under which the omnidirectional total transmission through a conjugate matched pair occurs. In Sec. IV we develop a generalized IIM for wave propagation in stratified biisotropic media. In Sec. V we perform extensive numerical calculations using the IIM and compare the results with the predictions of the dispersion relation. In Sec. VI we comment on the experimental feasibility and conclude the paper.
II. DISPERSION RELATION FOR SURFACE WAVES
We consider a plane interface between two different kinds of general bi-isotropic media located at z = 0 as illustrated in Fig. 1. We seek a surface-wave solution to Maxwell's equations that is propagating in the x direction along the interface but is exponentially damped FIG. 1. Sketch of the surface wave propagating in the x direction along the interface at z = 0 between two different kinds of general bi-isotropic media. The blue and red curves illustrate the field amplitudes for different circular polarizations decaying exponentially away from the interface. The decay rates, κjr and κ jl (j = 1, 2), are different for RCP and LCP modes and are related to the x component of the wave vector q, which is a conserved quantity, by Eq. (6).
away from it. The two bi-isotropic media are characterized by the parameters ǫ 1 , µ 1 , and a 1 (= χ 1 + iγ 1 ) and ǫ 2 , µ 2 , and a 2 (= χ 2 + iγ 2 ), respectively. In a uniform bi-isotropic medium, there exist two eigenmodes of circular polarization and the electromagnetic fields can be decomposed as
E = E r + E l , H = H r + H l ,(3)
where the subscripts r and l denote right-circularly polarized (RCP) and left-circularly polarized (LCP) modes respectively. The effective refractive indices n jr and n jl and the effective impedances η jr and η jl for RCP and LCP modes, respectively, in the bi-isotropic medium j (j = 1, 2) are given by [25,31] n jr = n j + γ j , n jl = n j − γ j ,
η jr = n j + iχ j ǫ j , η jl = n j − iχ j ǫ j ,(4)
where n j is defined by
n j = − ǫ j µ j − χ j 2 , if ǫ j , µ j < 0 and ǫ j µ j > χ j 2 i χ j 2 − ǫ j µ j , if ǫ j µ j < χ j 2 ǫ j µ j − χ j 2 , otherwise .
(5) We note that there are cases where some of the effective refractive indices take negative values, such as in ordinary negative index media with ǫ < 0, µ < 0, and χ = 0.
In the z > 0 region, E r and H r are assumed to depend on x, z, and t as exp(−κ 1r z + iqx − iωt), while E l and H l are as exp(−κ 1l z + iqx − iωt). In the z < 0 region, E r and H r are proportional to exp(κ 2r z + iqx − iωt), while E l and H l are to exp(κ 2l z + iqx − iωt). The imaginary wave-vector components κ jr and κ jl are defined by
κ jr = q 2 − k 0 2 n jr 2 , κ jl = q 2 − k 0 2 n jl 2 ,(6)
where k 0 (= ω/c) is the vacuum wave number. In order to have a pure surface-wave mode, all of these components have to be positive real numbers. The tangential components of the electric and magnetic fields have to be continuous at the interface. In addition, the electric and magnetic fields in bi-isotropic media satisfy the relationships E 1l,y E 2r,y E 2l,y = 0.(10)
To have a non-trivial solution, the determinant of the 4 × 4 coefficient matrix of this equation has to vanish. This condition yields the desired dispersion relation for surface waves at the interface between two different biisotropic media:
(η 1r + η 1l ) (η 2r + η 2l ) κ 1r n 1r + κ 2r n 2r κ 1l n 1l + κ 2l n 2l + (η 1r − η 2r ) (η 1l − η 2l ) κ 1r n 1r + κ 1l n 1l κ 2r n 2r + κ 2l n 2l = 0.(11)
In the special case where both of the two bi-isotropic media are Tellegen media with nonzero χ 1 and χ 2 but with γ 1 = γ 2 = 0, we have n jr = n jl = n j and the dispersion relation can be simplified as
(ǫ 1 κ 2 + ǫ 2 κ 1 ) (µ 1 κ 2 + µ 2 κ 1 ) − (χ 1 κ 2 + χ 2 κ 1 ) 2 = 0,(12)
where
κ j = q 2 − k 0 2 n j 2 .(13)
When both of the two media are ordinary isotropic media with χ 1 = χ 2 = γ 1 = γ 2 = 0, this equation reduces to the well-known dispersion relation
(ǫ 1 κ 2 + ǫ 2 κ 1 ) (µ 1 κ 2 + µ 2 κ 1 ) = 0.(14)
For the discussion of omnidirectional total transmission through a conjugate matched pair in the following sections, it is beneficial to introduce the effective dielectric permittivities ǫ jr and ǫ jl and the effective magnetic permeabilities µ jr and µ jl (j = 1, 2). Starting from Eq. (7) and using the definitions
D r = ǫ r E r , D l = ǫ l E l , B r = µ r H r , B l = µ l H l ,(15)
it is straightforward to derive the expressions
ǫ jr = ǫ j + a j iη jr , ǫ jl = ǫ j − a j iη jl , µ jr = µ j + ia * j η jr , µ jl = µ j − ia * j η jl ,(16)
using which we can rewrite Eq. (4) as
n jr = √ ǫ jr √ µ jr , n jl = √ ǫ jl √ µ jl , η jr = √ µ jr √ ǫ jr , η jl = √ µ jl √ ǫ jl .(17)
III. OMNIDIRECTIONAL EXCITATION OF SURFACE WAVES AND SUPER-KLEIN TUNNELING
In this section, we derive some interesting consequences of the analytical dispersion relation. In particular, we derive the conditions under which the surface waves are excited regardless of the incident angle in ATR experiments on multilayer structures similar to the Kretschmann or Otto configuration.
When both of the two bi-isotropic media are Tellegen media, the dispersion relation [Eq. (11)] is reduced to
(η 1r + η 1l ) (η 2r + η 2l ) κ 1 n 1 + κ 2 n 2 2 + 4 (η 1r − η 2r ) (η 1l − η 2l ) κ 1 n 1 κ 2 n 2 = 0.(18)
If the effective refractive indices of the two media are anti-matched (that is, n 1 = −n 2 and κ 1 = κ 2 ), then this equation becomes
(η 1r − η 2r ) (η 1l − η 2l ) κ 1 n 1 2 = 0.(19)
We notice that it is satisfied for any value of κ 1 if
η 1r = η 2r or η 1l = η 2l .(20)
Since the dependence on the incident angle in ATR experiments occurs only through κ j , we conclude that surface waves will be excited regardless of the incident angle if any of the above conditions is satisfied, as long as the
n1 = −n2 η1r = η2r ǫ1r = −ǫ2r µ1r = −µ2r (1, 0, 1, 0) I n1 = −n2 η 1l = η 2l ǫ 1l = −ǫ 2l µ 1l = −µ 2l (0, 1, 0, 1) II n1 = n2 η1r = −η 2l ǫ1r = −ǫ 2l µ1r = −µ 2l (1, 0, 0, 1) III n1 = n2 η 1l = −η2r ǫ 1l = −ǫ2r µ 1l = −µ2r (0, 1, 1, 0) IV Chiral/Chiral η1 = −η2 n1r = n 2l ǫ1r = −ǫ 2l µ1r = −µ 2l (1, 0, 0, 1) V η1 = −η2 n 1l = n2r ǫ 1l = −ǫ2r µ 1l = −µ2r (0, 1, 1, 0) VI η1 = η2 n1r = −n2r ǫ1r = −ǫ2r µ1r = −µ2r (1, 0, 1, 0) VII η1 = η2 n 1l = −n 2l ǫ 1l = −ǫ 2l µ 1l = −µ 2l (0, 1, 0, 1) VIII
incident angle satisfies the condition that κ j is real. This makes surface waves be excited at all or a continuous range of incident angles, in sharp contrast to the usual case where surface waves are excited at a specific incident angle.
On the other hand, if the effective refractive indices of the two Tellegen media are matched (that is, n 1 = n 2 and κ 1 = κ 2 ), the dispersion relation is reduced to
(η 1r + η 2l ) (η 1l + η 2r ) κ 1 n 1 2 = 0,(21)
which is satisfied for any (real) value of κ 1 if
η 1r = −η 2l or η 1l = −η 2r .(22)
Similarly, when both media are chiral media such that χ 1 = χ 2 = 0 and η jr = η jl ≡ η j , the dispersion relation [Eq. (11)] is reduced to
4η 1 η 2 κ 1r n 1r + κ 2r n 2r κ 1l n 1l + κ 2l n 2l + (η 1 − η 2 ) 2 κ 1r n 1r + κ 1l n 1l κ 2r n 2r + κ 2l n 2l = 0. (23)
If the impedances of the two media are anti-matched (that is, η 1 = −η 2 ), then this equation becomes
η 1 2 κ 1r n 1r − κ 2l n 2l κ 1l n 1l − κ 2r n 2r = 0,(24)
which is satisfied identically if n 1r = n 2l or n 1l = n 2r .
On the other hand, if the impedances of the two media are matched (that is, η 1 = η 2 ), the dispersion relation is reduced to
η 1 2 κ 1r n 1r + κ 2r n 2r κ 1l n 1l + κ 2l n 2l = 0,(26)
which is satisfied identically if n 1r = −n 2r or n 1l = −n 2l .
In Table I we give a summary of the conditions under which the surface waves are excited regardless of the component of the wave vector parallel to the interface, or equivalently, the incident angle in ATR experiments on multilayer structures. We also indicate the polarization of the excited surface-wave mode. For instance, when the conditions n 1 = −n 2 and η 1r = η 2r are satisfied for two Tellegen media, we can easily verify from Eq. (10) that the corresponding surface-wave mode satisfies
(E 1r,y , E 1l,y , E 2r,y , E 2l,y ) ∝ (1, 0, 1, 0).(28)
This implies that the surface wave is RCP in both media 1 and 2. The polarizations in other cases are obtained similarly. In the situation where the surface wave is excited by a wave incident from the side of the medium 1, the circular polarization of the wave reflected from the interface has to match with that of the excited surface-wave mode. An example of this phenomenon will be discussed in Sec. V D.
The conditions II listed in Table I are expressed in terms of the effective dielectric permittivities and magnetic permeabilities and have been derived from the conditions I using Eq. (17). For instance, from n 1 = −n 2 and η 1r = η 2r , we have
√ ǫ 1r √ µ 1r = − √ ǫ 2r √ µ 2r , √ µ 1r √ ǫ 1r = √ µ 2r √ ǫ 2r .(29)
From a simple manipulation of these equations, we obtain
ǫ 1r = −ǫ 2r , µ 1r = −µ 2r .(30)
We notice that the conditions II are closely related to those for the super-Klein tunneling or the omnidirectional total transmission to occur. The generalized definition of a conjugate matched pair is that the two layers have the same thicknesses and satisfy ǫ 1r
ǫ 2r = µ 1r µ 2r = −1 and ǫ 1l ǫ 2l = µ 1l µ 2l = −1(31)
or
ǫ 1r ǫ 2l = µ 1r µ 2l = −1 and ǫ 1l ǫ 2r = µ 1l µ 2r = −1.(32)
FIG. 2. Sketch of the configuration considered in Sec. IV. A plane wave is incident at an angle θ from a bi-isotropic medium with the parameters ǫi, µi, χi, and γi onto a bilayer system consisting of two different kinds of bi-isotropic media with the parameters ǫ1, µ1, χ1, and γ1 and ǫ2, µ2, χ2, and γ2 respectively, and then is transmitted to the bi-isotropic substrate with ǫt, µt, χt, and γt. The wave is evanescent in the regions d2 < z < L and 0 < z < d2. The incident wave tunnels through the top layer, excites the surface wave on the interface at z = d2, and then tunnels through the bottom layer to reach the substrate.
In order for the transmittance to be equal to one for any incident angle and polarization, an additional condition that the incident and transmitted regions are consisted of the same media has to be satisfied. In the special case of a bilayer made of two different chiral media, RCP and LCP waves become completely decoupled, if the incident and transmitted regions and the two chiral media have the same impedance. Then the super-Klein tunneling is achieved separately for RCP and LCP waves when
ǫ 1r ǫ 2r = µ 1r µ 2r = −1, for RCP, ǫ 1l ǫ 2l = µ 1l µ 2l = −1, for LCP.(33)
In dispersive media, the parameters ǫ, µ, χ, and γ depend on the wave frequency. Therefore each of the matching (or anti-matching) conditions listed in Table I can be satisfied only for a specific frequency in dispersive cases. That the dispersion relation is satisfied for any value of q at a certain frequency appears to have a close resemblance to the flat band phenomenon attracting much current interest of researchers [43].
IV. INVARIANT IMBEDDING METHOD
A popular method to excite surface waves experimentally is to perform ATR experiments on multilayer structures similar to the Kretschmann or Otto configuration.
In this paper, we consider a configuration sketched in Fig. 2. A plane wave is assumed to be incident at an angle θ from a uniform bi-isotropic medium with the parameters ǫ i , µ i , and a i (= χ i + iγ i ) onto a bilayer system consisting of two different kinds of bi-isotropic media with the parameters ǫ 1 , µ 1 , and a 1 (= χ 1 + iγ 1 ) and ǫ 2 , µ 2 , and a 2 (= χ 2 + iγ 2 ) respectively, and then is transmitted to the bi-isotropic substrate with ǫ t , µ t , and a t (= χ t + iγ t ). The layers 1 and 2 are assumed to have thicknesses d 1 and d 2 such that d 1 + d 2 = L. The wave is evanescent in the regions d 2 < z < L and 0 < z < d 2 .
The incident wave tunnels through the top layer, excites the surface wave on the interface at z = d 2 , and then tunnels through the bottom layer to reach the substrate. Some examples of the spatial field distribution in the region 0 < z < L will be shown in Figs. 4, 8, and 11.
We will test the predictions of the dispersion relation derived in Secs. II and III by solving Maxwell's equations directly in the configuration of Fig. 2 using the IIM. In this section, we present a generalization of the IIM developed previously in Ref. [31] to the case where the incident and transmitted regions as well as the inhomogeneously stratified medium in between are general bi-isotropic media.
As we have mentioned before, the field components E y and H y can be decomposed as
E y = E r,y + E l,y , H y = H r,y + H l,y = E r,y iη r − E l,y iη l .(34)
We rewrite this as a matrix equation
ψ = N φ,(35)
where
ψ = E y H y , φ = E r,y E l,y , N = 1 1 1 iηr − 1 iη l .(36)
In Ref. [31], we have derived the matrix wave equation for ψ in general bi-isotropic media of the form
E −1 ψ ′ ′ + Dψ = 0,(37)
where
D = k 0 2 M − q 2 E −1 , E = µ −a * −a ǫ , M = ǫ a a * µ .(38)
By substituting Eq. (35) into Eq. (37), we change the basis of the wave function to φ:
E −1 (N φ) ′ ′ + DN φ = 0.(39)
It is necessary to generalize this equation by replacing the vector wave function φ by the 2 × 2 matrix wave function Φ, the jth column vector (Φ 1j , Φ 2j ) T of which represents the wave function when the incident wave consists only of the jth wave (j = 1, 2). We note that here the index j = 1 (j = 2) corresponds to the case where RCP (LCP) waves are incident. We are mainly interested in calculating the 2 × 2 reflection and transmission coefficient matrices r = r(L) and t = t(L), which we consider as functions of L. In our notation, r 21 is the reflection coefficient when the incident wave is RCP and the reflected wave is LCP. Similarly, r 12 is the reflection coefficient when the incident wave is LCP and the reflected wave is RCP. Similar definitions are applied to the transmission coefficients.
We also need to generalize the vector wave function ψ to the 2 × 2 matrix wave function Ψ (= N Φ) in a similar manner. The jth column vector (Ψ 1j , Ψ 2j ) T of Ψ represents the wave function when the incident wave consists only of the jth wave. Ψ 11 (Ψ 12 ) represents the field E y and Ψ 21 (Ψ 22 ) represents the field H y in the inhomogeneous region when the incident wave is RCP (LCP), respectively. The field E y associated with the incident wave is assumed to have a unit amplitude.
We follow the standard procedure to derive the invariant imbedding equations. We introduce 2 × 2 matrix functions
U 1 (z; L) = N Φ, U 2 (z; L) = E −1 (N Φ) ′ ,(40)
which we consider as functions of both z and L. Then the wave equation is transformed to
U 1 U 2 ′ = A U 1 U 2 , A = O E −D O ,(41)
where A is a 4 × 4 matrix and O is the 2 × 2 null matrix. The wave functions in the incident and transmitted regions are expressed in terms of r and t:
U 1 (z; L) = N i e iPi(L−z) I + N i e iPi(z−L) r, z > L N t e −iPtz t, z < 0 ,(42)
where I is the 2 × 2 identity matrix and the matrices N i and N t are the values of N in the incident and transmitted regions, respectively. The diagonal matrices P i and P t are defined by
P i = p ir 0 0 p il , P t = p tr 0 0 p tl ,(43)
where the negative z components of the wave vector in the incident region p ir and p il and those in the transmitted region p tr and p tl are given by
p ir = sgn (n ir ) k 0 2 n ir 2 − q 2 , p il = sgn (n il ) k 0 2 n il 2 − q 2 , p tr = sgn (n tr ) k 0 2 n tr 2 − q 2 , if k 0 2 n tr 2 ≥ q 2 i q 2 − k 0 2 n tr 2 , if k 0 2 n tr 2 < q 2 , p tl = sgn (n tl ) k 0 2 n tl 2 − q 2 , if k 0 2 n tl 2 ≥ q 2 i q 2 − k 0 2 n tl 2 , if k 0 2 n tl 2 < q 2 .(44)
The effective refractive indices n ir , n il , n tr , and n tl are defined similarly as in Eq. (4). In the simpler case where the incident and transmitted regions are ordinary dielectric media, we can simplify Eqs. (43) and (44) as
P i = p i I, P t = p t I, p i = sgn (n i ) k 0 2 n i 2 − q 2 , p t = sgn (n t ) k 0 2 n t 2 − q 2 , if k 0 2 n t 2 ≥ q 2 i q 2 − k 0 2 n t 2 , if k 0 2 n t 2 < q 2 .(45)
At the boundaries of the inhomogeneous medium, we have
U 1 (0; L) = N t t, U 1 (L; L) = N i (r + I) , U 2 (0; L) = −iE t −1 N t P t t = −iE t −1 N t P t N t −1 U 1 (0; L), U 2 (L; L) = iE i −1 N i P i (r − I) = iE i −1 N i P i N i −1 U 1 (L; L) − 2iE i −1 N i P i ,(46)
where E i and E t are the values of E in the incident and transmitted regions respectively. From Eq. (46), we obtain
gŜ + hR = v,(47)whereŜ = U 1 (0; L) U 2 (0; L) ,R = U 1 (L; L) U 2 (L; L) , g = iE t −1 N t P t N t −1 I O O , h = O O iE i −1 N i P i N i −1 −I , v = O 2iE i −1 N i P i .(48)
We define 4 × 4 matrices S and R bŷ
S = Sv = S 11 S 12 S 21 S 22 v,R = Rv = R 11 R 12 R 21 R 22 v,(49)
where S ij and R ij (i, j = 1, 2) are 2 × 2 matrices. The invariant imbedding equations satisfied by R and S have been derived in Ref. [31]:
dR dl = A(l)R(l) − R(l)hA(l)R(l), dS dl = −S(l)hA(l)R(l),(50)
where l is the thickness of the inhomogeneous layer in the z direction. The initial conditions for these matrices are given by
R(0) = S(0) = (g + h) −1 .(51)
From the definitions ofR,Ŝ, R, and S, we obtain
R 12 2iE i −1 N i P i = N i (r + I) , R 22 2iE i −1 N i P i = iE i −1 N i P i (r − I) , S 12 2iE i −1 N i P i = N t t, S 22 2iE i −1 N i P i = −iE t −1 N t P t t.(52)
The expression for S 22 is given for reference, though it is not necessary for the derivation of the invariant imbedding equations.
The invariant imbedding equations satisfied by r and t follow from Eqs. (50) and (52) and take the forms
dr dl = i N i −1 EE i −1 N i P i r + rN i −1 EE i −1 N i P i − i 2 (r + I) N i −1 EE i −1 N i P i − P i −1 N i −1 E i DN i (r + I) , dt dl = itN i −1 EE i −1 N i P i − i 2 t N i −1 EE i −1 N i P i − P i −1 N i −1 E i DN i (r + I) .(53)
The initial conditions for r and t are obtained from Eq. (51):
r(0) = 2N i −1 E i −1 N i P i N i −1 + E t −1 N t P t N t −1 −1 E i −1 N i P i − I, t(0) = 2N t −1 E i −1 N i P i N i −1 + E t −1 N t P t N t −1 −1 E i −1 N i P i .(54)
The invariant imbedding method can also be used in calculating the wave function Ψ(z; L) inside the inhomogeneous medium. It turns out that the equation satisfied by Ψ(z; L) is very similar to that for t and takes the form
∂ ∂l Ψ(z; l) = iΨN i −1 EE i −1 N i P i − i 2 Ψ N i −1 EE i −1 N i P i − P i −1 N i −1 E i DN i (r + I) .(55)
This equation is integrated from l = z to l = L using the initial condition
Ψ(z; z) = N i [r(z) + I] .(56)
The reflectance (transmittance) is defined by the ratio of the energy flux of the reflected (transmitted) wave to that of the incident wave. The energy flux of a wave is given by the z component of the Poynting vector
Re (S z ) = c 4π Re [(E × H * ) z ] = c 4π Re E x H * y − E y H * x .(57)
Starting from this, we obtain the expressions for the components of the 2 × 2 reflectance and transmittance matrices of the form
R 11 = |r 11 | 2 , R 21 = p il n ir p ir n il |r 21 | 2 , R 12 = p ir n il p il n ir |r 12 | 2 , R 22 = |r 22 | 2 , T 11 = p tr n ir µ i ǫ t µ t − χ t 2 p ir n tr µ t ǫ i µ i − χ i 2 |t 11 | 2 , T 21 = p tl n ir µ i ǫ t µ t − χ t 2 p ir n tl µ t ǫ i µ i − χ i 2 |t 21 | 2 , T 12 = p tr n il µ i ǫ t µ t − χ t 2 p il n tr µ t ǫ i µ i − χ i 2 |t 12 | 2 , T 22 = p tl n il µ i ǫ t µ t − χ t 2 p il n tl µ t ǫ i µ i − χ i 2 |t 22 | 2 .(58)
In the case where p tr (p tl ) is imaginary, we have to set T 11 = T 12 = 0 (T 21 = T 22 = 0), since the transmitted wave is evanescent. On the other hand, if both ǫ t and µ t are negative, while ǫ t µ t > χ t 2 , we need to replace ǫ t µ t − χ t 2 by − ǫ t µ t − χ t 2 in the above expressions for the transmittances. In the simpler case where the incident and transmitted regions are ordinary dielectric media, we can simplify Eq. (58) as
R ij = |r ij | 2 , T ij = p t µ i p i µ t |t ij | 2 .(59)
We stress again that the numerical indices 1 and 2 refer to the RCP and LCP waves, respectively, and the first index i of the matrix R ij (T ij ) is used for the reflected (transmitted) wave, while the second index j is used for the incident wave. For example, T 12 represents the transmittance when the incident wave is LCP and the transmitted wave is RCP.
In the absence of dissipation, the law of energy conservation requires that
R 11 + R 21 + T 11 + T 21 = 1, R 22 + R 12 + T 22 + T 12 = 1.(60)
If there is dissipation, then the absorptance is defined by
A 1 = 1 − R 11 − R 21 − T 11 − T 21 , A 2 = 1 − R 22 − R 12 − T 22 − T 12 ,(61)
where A 1 (A 2 ) is the absorptance when the incident wave is RCP (LCP).
V. NUMERICAL RESULTS
The IIM developed in Sec. IV allows us to solve any wave propagation problem in the situation where the when RCP and LCP waves are incident on the same bilayer system considered in Fig. 3. The Ey field associated with the incident wave is assumed to have a unit amplitude. The incident angle is chosen to be θ = 57 • . The interface is located at z = 4Λ and waves are incident from the region where z > 5Λ. medium parameters depend arbitrarily on the coordinate z. In this section, we restrict our attention mainly to the bilayer systems satisfying the matching conditions listed in Table I. Before presenting the numerical results obtained for those cases, we first consider a more generic case where the matching conditions are not satisfied to contrast the result with those in the matched cases.
A. Generic case
In Fig. 3 we plot the absorptances A 1 and A 2 for RCP and LCP waves of frequency ω incident on a bilayer system consisting of a Tellegen medium with ǫ 1 = 2.25 + 0.01i, µ 1 = 1, χ 1 = 4, and γ 1 = 0 and an ordinary dielectric with ǫ 2 = 1.5, µ 2 = 1, and χ 2 = γ 2 = 0 in the configuration shown in Fig. 2 versus incident angle. Plane waves are incident from a dielectric prism with the parameters ǫ i = 9, µ i = 1, and a i = 0. The substrate is assumed to be made of the same material as the prism. The layers 1 and 2 have the thicknesses d 1 = Λ and d 2 = 4Λ, where Λ satisfies ωΛ/c = 0.2π. For the given parameters, we can solve Eq. (12) numerically and find that the surface wave is excited at the incident angle θ ≈ 57.59 • , which corresponds to q ≈ 2.5328. This result agrees perfectly with the numerical result for the absorptances showing narrow sharp peaks at the same incident angle, since the excitation of a surface wave is manifested by a strong absorption of the energy of the incident wave. In the present example, it is interesting to notice that a surface wave is excited even though the values of ǫ 1 , µ 1 , ǫ 2 , and µ 2 are all positive, in contrast to the more conventional cases where ǫ 1 and ǫ 2 (or µ 1 and µ 2 ) have the opposite signs. This is possible because in the medium 1, the real part of the square of the effective refractive index (= −13.75) is negative due to the large value of the Tellegen parameter χ 1 . This makes the medium 1 behave similarly to a metal.
In Fig. 4 we show the spatial distributions of the intensities of the y components of the electric and magnetic fields, |E y | 2 and |H y | 2 , when RCP and LCP waves are incident on the same bilayer system considered in Fig. 3. The E y field associated with the incident wave is assumed to have a unit amplitude. The incident angle is chosen to be θ = 57 • corresponding to the angle at which the absorptances take the peak values. We find that the field intensities are greatly enhanced at the interface between the two media and decay exponentially away from it, as can be expected from a surface wave.
B. Cases I and II
We now move to the cases where the matching conditions listed in Table I are satisfied. In Fig. 5 we consider the excitation of surface waves at the interface between two different Tellegen media (that is, media with χ = 0 and γ = 0) of the same thickness Λ, where the medium parameters are given by ǫ 1 = −3 + νi, µ 1 = −1, and χ 1 = −1 and ǫ 2 = 3, µ 2 = 1, and χ 2 = 1 respectively. We also assume that waves are incident from a dielectric prism with the parameters ǫ i = 4, µ i = 1, and a i = 0 and transmitted to the substrate with the same parameters as the prism. If we ignore the small imaginary part in ǫ 1 (that is, ν), we obtain n 1 2 = ǫ 1 µ 1 − χ 1 2 = 2 and n 2 2 = ǫ 2 µ 2 − χ 2 2 = 2. When both ǫ and µ are negative and ǫµ > χ 2 , we have to choose n = − ǫµ − χ 2 as the effective refractive index. Therefore we have n 1 = −n 2 = − √ 2 in the present case. The effective impedances for RCP and LCP waves are obtained from Eq. (4) and we find that if we ignore ν, η 1r = η 2r = ( √ 2 + i)/3 and η 1l = η 2l = ( √ 2 − i)/3. When these conditions are satisfied, surface waves can be excited for any real value of κ 1 (= κ 2 ), as we have proved in Eq. (20). Since a wave is incident from the region where √ ǫ i µ i = 2, this gives the constraint that κ 1 2 /k 0 2 = 4 sin 2 θ − 2 > 0, that is, θ > 45 • for κ 1 to be real. Therefore surface waves will be excited for any value of θ greater than 45 • for both RCP and LCP incident waves. These predictions are confirmed clearly in Fig. 5, where the absorptances A 1 and A 2 obtained for four different values of ν (= 0, 10 −4 , 10 −5 , and 10 −6 ) when ωΛ/c = 2π and three different values of ωΛ/c (= 1.5π, 2π, and 2.5π) when ν = 10 −5 are plotted versus incident angle. When ν is zero, there is no dissipation in the layers and the absorption of the wave energy does not arise, although even in this case, the surface waves are excited at the interface for θ > 45 • , as will be shown in Fig. 8. When ν is small and nonzero, the absorptances remain zero at θ < 45 • , but become finite at all angles greater than 45 • . In the parameter region where the surface waves are excited, we find that the absorptances depend quite sensitively on the value of ν and ωΛ/c. The broad peaks shown in Fig. 5 are in sharp contrast to the narrow sharp peaks in Fig. 3. In the calculations shown here and in all later calculations, we have assumed a very small value of ν. However, we have checked numerically that qualitatively similar results are obtained for much larger values of ν up to 0.01.
In the case where ν is zero, the bilayer system considered above is an example of a conjugate matched pair. From Eq. (16), we obtain ǫ 2r = −ǫ 1r = 2 − √ 2i, µ 2r = −µ 1r = (2 + √ 2i)/3, ǫ 2l = −ǫ 1l = 2 + √ 2i, and µ 2l = −µ 1l = (2 − √ 2i)/3, which satisfy Eq. (31). In Fig. 6 we plot the transmittances T 11 and T 22 through the bilayer in the same configuration as in Fig. 5. As is expected from a conjugate matched pair, both T 11 and T 22 are identically equal to one for any incident angle when ν is zero. For a small value of ν equal to 10 −5 , however, the transmittances remain equal to one only for θ < 45 • . This occurs because when the surface waves are excited at θ > 45 • , a large amount of absorption arises even in the presence of a very small damping. In all cases where the omnidirectional total transmission occurs, the crosspolarized transmittances T 12 and T 21 obviously vanish. In other cases, however, they are generally nonzero.
In Fig. 7, we make a comparison of the transmittance T 11 obtained for ν = 10 −5 and ωΛ/c = 2π with those for ν = 10 −9 and ωΛ/c = 2π and for ν = 10 −5 and ωΛ/c = 0.8π in the same configuration as in Fig. 5. Although true omnidirectional total transmission occurs only when ν = 0, we find that total transmission is obtained at all angles except for those very close to 90 • , if ν or Λ is sufficiently small. Tuning the slab thickness Λ seems to be a convenient way to observe both the omnidirectional total transmission and the omnidirectional excitation of surface waves experimentally.
In Fig. 8, we show the spatial distributions of |E y | 2 and |H y | 2 , when RCP and LCP waves with a unit amplitude of E y are incident on the same bilayer system considered in Fig. 6 at the incident angle θ = 60 • for two different values of ν (= 0, 10 −5 ). We find that both fields, which are extremely pronounced near the interface between the two media, decay exponentially away from it. We have checked numerically that the decay rates of the fields are approximately consistent with the values obtained from κ 1 and κ 2 for the chosen value of θ. We observe that the surface waves are excited even when ν is zero and the field enhancement in that case is larger than that in the ν = 0 case. We emphasize that these are the solutions in the steady state.
C. Cases III and IV
Next, in Fig. 9 we consider the surface wave between two Tellegen media of the same thickness Λ such that ωΛ/c = π, where ǫ 1 = −3 + νi (ν = 10 −5 ), µ 1 = 1, and χ 1 = 1 and ǫ 2 = 3, µ 2 = −1, and χ 2 = −1 respectively. We also assume that waves are incident from a dielectric prism with the parameters ǫ i = 4, µ i = 1, and a i = 0 and transmitted to the substrate with the same parameters as the prism. If we ignore ν, we obtain n 1 = n 2 = 2i, η 1r = −η 2l = −i, and η 1l = −η 2r = −i/3. When these conditions are satisfied, surface waves can be excited for any real value of κ 1 (= κ 2 ) for both RCP and LCP incident waves, as we have proved in Eq. (22). Since a wave 11. Spatial distributions of the intensities of the y components of the electric and magnetic fields, |Ey| 2 and |Hy| 2 , when RCP and LCP waves with a unit amplitude of Ey are incident on the same bilayer system considered in Fig. 9. The incident angle is chosen to be either 0 • or 45 • . The interface is located at z = Λ and waves are incident from the region where z > 2Λ.
is incident from the region where √ ǫ i µ i = 2, this gives the condition that κ 1 2 /k 0 2 = 4 sin 2 θ + 2 > 0, which is satisfied for all θ. Therefore surface waves will be excited for an arbitrary value of θ including the normal incidence case with θ = 0, which is clearly confirmed in Fig. 9.
We can also show that when ν is zero, the bilayer system considered in Fig. 9 is a conjugate matched pair. From Eq. (16), we obtain ǫ 2r = −ǫ 1l = 6, µ 2r = −µ 1l = −2/3, ǫ 2l = −ǫ 1r = 2, and µ 2l = −µ 1r = −2, which satisfy Eq. (32). In Fig. 10 we plot the transmittances T 11 and T 22 through the bilayer in the same configuration as in Fig. 9. When ν is zero, we confirm that the omnidirectional total transmission of both RCP and LCP waves occurs. When ν is nonzero, the transmittances are substantially reduced at all θ due to the omnidirectional excitation of surface waves. In Fig. 11, we show the spatial distributions of |E y | 2 and |H y | 2 , when RCP and LCP waves with a unit amplitude of E y are incident on the same bilayer system considered in Fig. 9 with ν = 10 −5 at the angles θ = 0 • and 45 • . In all cases including the normal incidence case, we find that surface waves are strongly excited at the interface.
D. Case III or IV
We now consider the third configuration involving two Tellegen media. In Fig. 12(a) we consider the surface wave between two Tellegen media of the same thickness Λ such that ωΛ/c = 0.8π, where ǫ 1 = −3+νi (ν = 10 −5 ), µ 1 = −7, and χ 1 = 5 and ǫ 2 = −3, µ 2 = 1, and χ 2 = 1 respectively. We assume that waves are incident from a dielectric prism with the parameters ǫ i = 4, µ i = 1, and a i = 0 and transmitted to the substrate with the same parameters as the prism. If we ignore ν, we obtain n 1 = n 2 = 2i and the effective impedances for RCP and LCP waves η 1r = −7i/3, η 1l = i, η 2r = −i, and η 2l = −i/3, and therefore we have η 1l = −η 2r but η 1r = −η 2l . In this case, we find that surface waves are excited for an arbitrary value of θ for RCP incident waves. For LCP waves, surfaces waves are also excited at any incident angle greater than 0, but they are not excited at θ = 0 and very weakly excited at small incident angles. These results can be understood from the polarization of the surface-wave mode, (0, 1, 1, 0), corresponding to the fourth case in Table I. This implies that the surface wave in this case is LCP in the medium 1 and RCP in the medium 2. Let us first consider the normal incidence case. When an RCP (LCP) wave is normally incident on the interface between the two media, the transmitted wave will be RCP (LCP) and the reflected wave will be LCP (RCP). Therefore it is not possible to have the correct mode structure when an LCP wave is incident and the surface wave will not be excited in that case. For a nonzero incident angle, the reflected and transmitted waves will have both RCP and LCP components. When θ is sufficiently small, however, the coupling and mixing between RCP and LCP components is weak and the behavior is similar to the normal incidence case. In Fig. 12(b) we have switched the parameters of the media 1 and 2 (except for ν). Then we obtain n 1 = n 2 , η 1r = −η 2l , and η 1l = −η 2r . A similar argument as the above shows that surface waves are excited at all θ for LCP incident waves. For RCP waves, however, surface waves are not excited at θ = 0 and very weakly excited at small incident angles, as is confirmed in Fig. 12(b).
If we ignore ν, we also obtain ǫ 2r = −ǫ 1l = −2, µ 2r = −µ 1l = 2, ǫ 2l = −6, ǫ 1r = −6/7, µ 2l = 2/3, µ 1r = 14/3 from Eq. (16). Therefore the second condition of Eq. (32) is satisfied but the first is not, and therefore this bilayer is not a conjugate matched pair. In Fig. 13 we plot the transmittances T 11 and T 22 through the bilayer in the same configuration as in Fig. 12. We confirm that the omnidirectional total transmission does not occur regardless of the value of ν.
E. Cases V and VI
Next, we consider the excitation of surface waves at the interface between two different chiral media (that is, media with γ = 0 and χ = 0), where η jr = η jl = η j (j = 1, 2). In Fig. 14 we consider the case where ǫ 1 = −3 + νi (ν = 10 −5 ), µ 1 = 1, and γ 1 = 0.1 and ǫ 2 = 3, µ 2 = −1, and γ 2 = −0.1. The two layers have the same thickness Λ such that ωΛ/c = π. If we ignore ν, we obtain η 1 = −η 2 = −i/ √ 3, n 1r = n 2l = 0.1 + √ 3i, and n 1l = n 2r = −0.1+ √ 3i. In this case, we find that surface waves are excited for an arbitrary value of θ for both RCP and LCP incident waves, as we have proved in Eq. (25) and verified in Fig. 14.
We find that when ν is zero, the bilayer considered in Fig. 14
= −ǫ 1l = 3 + 0.1 √ 3i, µ 2r = −µ 1l = −1 − (0.1/ √ 3)i, ǫ 2l = −ǫ 1r = 3 − 0.1 √ 3i, and µ 2l = −µ 1r = −1+(0.1/ √
3)i, which satisfy Eq. (32). In Fig. 15 we plot the transmittances T 11 and T 22 through the bilayer in the same configuration as in Fig. 14 and confirm that the omnidirectional total transmission of both RCP and LCP waves occurs when ν is zero. When ν is nonzero, the transmittances are reduced at all θ due to the omnidirectional excitation of surface waves.
F. Cases VII and VIII
From now on, we will consider the cases where the effective impedances of the media 1 and 2 are the same when we ignore the small imaginary part in ǫ 1 . In those cases, it is more illuminating to have the same impedance value in both the incident region and the substrate as in the media 1 and 2. Then the impedance is matched throughout the whole space and the unwanted wave scattering at the interfaces between the incident region and the medium 1 and between the medium 2 and the substrate is eliminated. In Fig. 16 we consider the case where ǫ 1 = −1.4 + νi (ν = 10 −5 ), µ 1 = −1.4, and γ 1 = 0.1 and ǫ 2 = 1.2, µ 2 = 1.2, and γ 2 = 0.1. The layers 1 and 2 have the same thickness Λ, which satisfies ωΛ/c = 1.5π. In the incident region and the substrate, we choose ǫ i = ǫ t = 2, µ i = µ t = 2, and a i = a t = 0 to have η i = η t = 1. If we ignore ν, we obtain η 1 = η 2 = η i = η t = 1. Ignoring ν, the effective refractive indices for RCP and LCP waves are given by n 1r = −1.3, n 1l = −1.5, n 2r = 1.3, and n 2l = 1.5, and therefore we have n 1r = −n 2r and n 1l = −n 2l . When these conditions are satisfied, surface waves can be excited for any real value of κ 1r (κ 1l ) for incident RCP (LCP) waves, as we have proved in Eq. (27). Since a wave is incident from the region where √ ǫ i µ i = 2, this gives the constraints that κ 1r 2 /k 0 2 = 4 sin 2 θ − 1.3 2 > 0, that is, θ > 40.54 • for RCP waves and κ 1l 2 /k 0 2 = 4 sin 2 θ − 1.5 2 > 0, that is, θ > 48.59 • for LCP waves. That the surface waves are indeed excited in the predicted regions of the incident angle can be seen clearly from Fig. 16.
When ν is zero, the bilayer considered in Fig. 16 is a conjugate matched pair. From Eq. (16), we obtain ǫ 2r = −ǫ 1r = 1.3, µ 2r = −µ 1r = 1.3, ǫ 2l = −ǫ 1l = 1.5, and µ 2l = −µ 1l = 1.5, which satisfy Eq. (31). In Fig. 17 we plot the transmittances T 11 and T 22 through the bilayer in the same configuration as in Fig. 16 and confirm that the omnidirectional total transmission of both RCP and LCP waves occurs when ν is zero. For a small value of ν equal to 10 −5 , the transmittance remains equal to one only at θ < 40.54 • for RCP waves and at θ < 48.59 • for LCP waves. This is because the surface waves are excited at either θ > 40.54 • or θ > 48.59 • depending on the polarization of the incident wave.
In the present example, having the impedance matching in the whole space is not required to have the omnidirectional total transmission, since the two conditions in Eq. (31) are simultaneously satisfied. We have checked numerically that even in the case where ǫ i = ǫ t = 4, µ i = µ t = 1, and a i = a t = 0, the transmittances T 11 and T 22 are identically equal to one at all θ if ν is zero. When ν is small and nonzero, the surface waves are excited at θ > 40.54 • for both RCP and LCP waves and the absorptances are slightly different from those shown in Fig. 16 due to the coupling and mixing of RCP and LCP waves.
G. Case VII or VIII
Finally, we consider the cases which satisfy Eq. (33). In Fig. 18(a) we consider the case where ǫ 1 = −1.4+νi (ν = 10 −5 ), µ 1 = −1.4, and γ 1 = 0.1 and ǫ 2 = 1.2, µ 2 = 1.2, and γ 2 = 0.1. In the incident region and the substrate, the medium parameters are ǫ i = ǫ t = 2, µ i = µ t = 2, and a i = a t = 0. The layers 1 and 2 have the same thickness Λ, which satisfies ωΛ/c = 1.5π. If we ignore ν, we obtain η 1 = η 2 = η i = η t = 1. Ignoring ν, the effective refractive indices for RCP and LCP waves are given by n 1r = −1.3, n 1l = −1.5, n 2r = 1.3, and n 2l = 1.1, and therefore we have n 1r = −n 2r but n 1l = −n 2l . When these conditions are satisfied, surface waves can be excited for any real value of κ 1r only for RCP waves as stated in Eq. (33). Since a wave is incident from the region where √ ǫ i µ i = 2, this gives the constraint that θ > 40.54 • for RCP waves. We confirm that the surface waves are excited in the predicted region of the incident angle only when RCP waves are incident in Fig. 18(a). In Fig. 18(b) we consider the case where ǫ 1 = −1.2 + νi (ν = 10 −5 ), µ 1 = −1.2, and γ 1 = 0.1 and ǫ 2 = 1.4, µ 2 = 1.4, and γ 2 = 0.1. In this case, it is straightforward to verify that the surface waves are excited when θ > 40.54 • only for LCP incident waves.
When ν is zero, the bilayer system considered in Fig. 18(a) is a conjugate matched pair for RCP waves. From Eq. (16), we obtain ǫ 2r = −ǫ 1r = 1.3, µ 2r = −µ 1r = 1.3, ǫ 2l = 1.1, ǫ 1l = −1.5, µ 2l = 1.1, and µ 1l = −1.5, which satisfy the first of Eq. (33). In Fig. 19(a) we plot the transmittances T 11 and T 22 through the bilayer in the same configuration as in Fig. 18(a) and confirm that the omnidirectional total transmission of RCP waves occurs when ν is zero. For a small value of ν equal to 10 −5 , the transmittance remains equal to one only at θ < 40.54 • for RCP waves. There is no omnidirectional total transmission for LCP waves and the transmittance T 22 depends very weakly on ν. In Fig. 19(b) we consider the transmittances in the same configuration as in Fig. 18(b) and find that the omnidirectional total transmission of LCP waves occurs when ν is zero.
In the present case, having the impedance matching in the whole space is a necessary condition to have the omnidirectional total transmission for RCP or LCP waves. In Fig. 20 we consider the same bilayer as in Figs. 18(a) and 19(a) and assume that the medium parameters in the incident region and substrate are ǫ i = ǫ t = 4, µ i = µ t = 1, and a i = a t = 0. Then the impedance matching in the whole space is not achieved and the omnidirectional total transmission does not occur. In Fig. 20(a) we show that the absorptance A 2 is no longer zero due to the coupling between LCP and RCP waves. In Fig. 20(b) we confirm that the transmittance T 11 is smaller than one at all θ even when ν is zero.
VI. CONCLUSION
In this paper, we have investigated the characteristics of the surface waves excited at the interface be-tween two different bi-isotropic media. We have derived an analytical dispersion relation for those waves, using which we have deduced the conditions under which they are excited between two Tellegen media and between two chiral media for all or a wide range of incident angles in ATR experiments on multilayer structures. We have also obtained the conditions under which the omnidirectional total transmission occurs through conjugate matched pairs. We have found that the omnidirectional excitation of surface waves and the omnidirectional total transmission through a conjugate matched pair are intimately related phenomena and discussed the similarities and differences of the respective conditions. We have also pointed out that the omnidirectional total transmission discussed here has basically the same physical origin as the super-Klein tunneling occurring in pseudospin-1 Dirac-type materials. We have confirmed our predictions with detailed numerical calculations of the absorptance, the transmittance, and the spatial distribution of the electromagnetic fields.
The omnidirectional excitation of surface waves and the omnidirectional total transmission through a conjugate matched pair extensively discussed in previous sections require that the chirality index γ or the Tellegen parameter χ takes a fairly large value. Though there exist no natural materials with such properties, there have been many recent theoretical and experimental studies to construct artificial metamaterial structures with a very large and tunable value of γ [44][45][46][47]. It may be possible to test our theory using those metamaterials.
It is more difficult to construct artificial metamaterials with a large value of the Tellegen parameter χ. Recently, it has been pointed out that some topological Dirac materials such as topological insulators and Weyl semimetals can be considered electromagnetically as a kind of Tellegen medium [48][49][50][51][52][53]. The χ value for topological insulators is very small, but it may be possible to obtain strongly enhanced magnetoelectric effects in Weyl semimetals with tilted Dirac cones. Further research in that direction is highly desired.
FIG. 3 .FIG. 4 .
34Absorptances A1 and A2 for RCP and LCP waves of frequency ω incident on a bilayer system consisting of a Tellegen medium with ǫ1 = 2.25 + 0.01i, µ1 = 1, χ1 = 4, and γ1 = 0 and an ordinary dielectric with ǫ2 = 1.5, µ2 = 1, and χ2 = γ2 = 0 in the configuration shown inFig. 2plotted versus incident angle. Waves are incident from a dielectric prism with the parameters ǫi = 9, µi = 1, and ai = 0. The substrate is assumed to be made of the same material as the prism (that is, ǫt = 9, µt = 1, and at = 0). The layers 1 and 2 have the thicknesses d1 = Λ and d2 = 4Λ, where Λ satisfies ωΛ/c = 0.2π. Spatial distributions of the intensities of the y components of the electric and magnetic fields, |Ey| 2 and |Hy| 2 ,
FIG. 5 .FIG. 6 .FIG. 7 .FIG. 8 .
5678Absorptances A1 and A2 for RCP and LCP waves of frequency ω incident on a bilayer system consisting of two different kinds of Tellegen media with ǫ1 = −3 + νi, µ1 = −1, and χ1 = −1 and ǫ2 = 3, µ2 = 1, and χ2 = 1 in the configuration shown inFig. 2plotted versus incident angle. Waves are incident from a prism with the parameters ǫi = 4, µi = 1, and ai = 0 and transmitted to the substrate with the same parameters as the prism. The layers 1 and 2 have the same thickness Λ and the results for (a) four different values of ν (= 0, 10 −4 , 10 −5 , 10 −6 ) when ωΛ/c = 2π and three different values of ωΛ/c (= 1.5π, 2π, 2.5π) when ν = 10 −5 are shown. Transmittances T11 and T22 for RCP and LCP waves of frequency ω in the same configuration as inFig. 5plotted versus incident angle. The layers 1 and 2 have the same thickness Λ such that ωΛ/c = 2π and the results for two different values of ν (= 0, 10 −5 ) are compared. Comparison of T11 obtained for ν = 10 −5 and ωΛ/c = 2π with those for ν = 10 −9 and ωΛ/c = 2π and for ν = 10 −5 and ωΛ/c = 0.8π in the same configuration as inFig. Spatialdistributions of the intensities of the y components of the electric and magnetic fields, |Ey| 2 and |Hy| 2 , when RCP and LCP waves with a unit amplitude of Ey are incident on the same bilayer system considered inFig. 6. The incident angle is chosen to be θ = 60 • and the parameter ν is either 0 or 10 −5 . The interface is located at z = Λ and waves are incident from the region where z > 2Λ.
FIG. 9 .
9Absorptances A1 and A2 for RCP and LCP waves of frequency ω incident on a bilayer system consisting of two different kinds of Tellegen media with ǫ1 = −3 + νi (ν = 10 −5 ), µ1 = 1, and χ1 = 1 and ǫ2 = 3, µ2 = −1, and χ2 = −1 plotted versus incident angle. Waves are incident from a prism with the parameters ǫi = 4, µi = 1, and ai = 0 and transmitted to the substrate with the same parameters as the prism. The layers 1 and 2 have the same thickness Λ, which satisfies ωΛ/c = π.
FIG. 10 .
10Transmittances T11 and T22 for RCP and LCP waves in the same configuration as inFig. 9plotted versus incident angle when ν is 0 and 10 −5 .
FIG. 12 .
12Absorptances A1 and A2 for RCP and LCP waves of frequency ω incident on a bilayer system consisting of two different kinds of Tellegen media (a) with ǫ1 = −3 + νi (ν = 10 −5 ), µ1 = −7, and χ1 = 5 and ǫ2 = −3, µ2 = 1, and χ2 = 1 and (b) with ǫ1 = −3 + νi (ν = 10 −5 ), µ1 = 1, and χ1 = 1 and ǫ2 = −3, µ2 = −7, and χ2 = 5 plotted versus incident angle. Waves are incident from a prism with the parameters ǫi = 4, µi = 1, and ai = 0 and transmitted to the substrate with the same parameters as the prism. The layers 1 and 2 have the same thickness Λ, which satisfies ωΛ/c = 0.8π.
FIG. 13 .
13Transmittances T11 and T22 for RCP and LCP waves in the same configurations as inFig. 12plotted versus incident angle when ν is 0 and 10 −5 .
FIG. 14 .FIG. 15 .
1415Absorptances A1 and A2 for RCP and LCP waves of frequency ω incident on a bilayer system consisting of two different kinds of chiral media with ǫ1 = −3 + νi (ν = 10 −5 ), µ1 = 1, and γ1 = 0.1 and ǫ2 = 3, µ2 = −1, and γ2 = −0.1 in the configuration shown inFig. 2plotted versus incident angle. Waves are incident from a prism with the parameters ǫi = 4, µi = 1, and ai = 0 and transmitted to the substrate with the same parameters as the prism. The layers 1 and 2 have the same thickness Λ, which satisfies ωΛ/c = π. Transmittances T11 and T22 for RCP and LCP waves in the same configuration as inFig. 14plotted versus incident angle when ν is 0 and 10 −5 .
FIG. 16 .
16Absorptances A1 and A2 for RCP and LCP waves of frequency ω incident on a bilayer system consisting of two different kinds of chiral media with ǫ1 = −1.4+νi (ν = 10 −5 ), µ1 = −1.4, and γ1 = 0.1 and ǫ2 = 1.4, µ2 = 1.4, and γ2 = −0.1 plotted versus incident angle. In the incident region and the substrate, the medium parameters are ǫi = ǫt = 2, µi = µt = 2, and ai = at = 0. The layers 1 and 2 have the same thickness Λ, which satisfies ωΛ/c = 1.5π.
FIG. 17 .
17Transmittances T11 and T22 for RCP and LCP waves in the same configuration as inFig. 16plotted versus incident angle when ν is 0 and 10 −5 .
FIG. 18 .
18Absorptances A1 and A2 for RCP and LCP waves of frequency ω incident on a bilayer system consisting of two different kinds of chiral media (a) with ǫ1 = −1.4 + νi (ν = 10 −5 ), µ1 = −1.4, and γ1 = 0.1 and ǫ2 = 1.2, µ2 = 1.2, and γ2 = 0.1 and (b) with ǫ1 = −1.2 + νi (ν = 10 −5 ), µ1 = −1.2, and γ1 = 0.1 and ǫ2 = 1.4, µ2 = 1.4, and γ2 = 0.1 plotted versus incident angle. In the incident region and the substrate, the medium parameters are ǫi = ǫt = 2, µi = µt = 2, and ai = at = 0. The layers 1 and 2 have the same thickness Λ, which satisfies ωΛ/c = 1.5π.
FIG. 19 .
19Transmittances T11 and T22 for RCP and LCP waves in the same configurations as in Fig. 18 plotted versus incident angle when ν is 0 and 10 −5 .FIG. 20. (a) Absorptances A1 and A2 for RCP and LCP waves incident on the same bilayer considered inFig. 18(a) plotted versus incident angle, when the medium parameters in the incident region and the substrate are ǫi = ǫt = 4, µi = µt = 1, and ai = at = 0. (b) Transmittances T11 and T22 for RCP and LCP waves in the same configuration as inFig. 19(a) plotted versus incident angle when ν is 0 and 10 −5 .
TABLE I .
ISummary of the conditions under which the surface waves are excited regardless of the component of the wave vector parallel to the interface, or equivalently, the incident angle in ATR experiments on multilayer structures. The cases where the interface is between two Tellegen media or between two chiral media are considered. The polarization of the excited surface-wave mode is indicated and each case is assigned a case number. The conditions II expressed in terms of the effective dielectric permittivities and magnetic permeabilities are equivalent to the conditions I.Interface
Conditions I
Conditions II
Surface mode
Case no.
Tellegen/Tellegen
is a conjugate matched pair. From Eq. (16), we obtain ǫ 2r
( =10 -5 ) T (a) 1 =-1.4, 1 =-1.4 2 =1.2, 2 =1.2 T 11 ( =0) (b) 1 =-1.2, 1 =-1.2 2 =1.4, 2 =1.4 T 22 ( =0) T 11 ( =10 -5 ) T 22 ( =10 -5 ) T 11 ( =0) T (deg)
ACKNOWLEDGMENTSThis research was supported through a National Research Foundation of Korea Grant (NRF-2020R1A2C1007655) funded by the Korean Government.
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| []
|
[
"Quantum Tomography of Three-Level Atoms",
"Quantum Tomography of Three-Level Atoms"
]
| [
"Andrei B Klimov \nDepartamento de Física\nUniversidad de Guadalajara\n44420Guadalajara, JaliscoMexico\n",
"Hubert De Guise \nDepartment of Physics\nLakehead University\nP7B 5E1Thunder BayOntarioCanada\n"
]
| [
"Departamento de Física\nUniversidad de Guadalajara\n44420Guadalajara, JaliscoMexico",
"Department of Physics\nLakehead University\nP7B 5E1Thunder BayOntarioCanada"
]
| []
| We analyze the possibility of tomographic reconstruction of a system of three-level atoms in both non-degenerate and degenerate cases. In the non-degenerate case (when both transitions can be accessed independently) a complete reconstruction is possible. In the degenerate case (when both transitions are excited simultaneously) the complete reconstruction is achievable only for a single atom in the Ξ configuration. For multiple Ξ atoms, or even a single atom in the Λ configuration, only partial reconstruction is possible. Examples of one and two-atom cases are explicitly considered. | 10.1088/1751-8113/41/2/025303 | [
"https://arxiv.org/pdf/0711.2765v1.pdf"
]
| 16,817,800 | 0711.2765 | 11f4ef4afa67697b3efdd3934cce04c7387cb3a8 |
Quantum Tomography of Three-Level Atoms
17 Nov 2007 November 2007
Andrei B Klimov
Departamento de Física
Universidad de Guadalajara
44420Guadalajara, JaliscoMexico
Hubert De Guise
Department of Physics
Lakehead University
P7B 5E1Thunder BayOntarioCanada
Quantum Tomography of Three-Level Atoms
17 Nov 2007 November 2007
We analyze the possibility of tomographic reconstruction of a system of three-level atoms in both non-degenerate and degenerate cases. In the non-degenerate case (when both transitions can be accessed independently) a complete reconstruction is possible. In the degenerate case (when both transitions are excited simultaneously) the complete reconstruction is achievable only for a single atom in the Ξ configuration. For multiple Ξ atoms, or even a single atom in the Λ configuration, only partial reconstruction is possible. Examples of one and two-atom cases are explicitly considered.
Introduction.
Quantum tomography is now an accepted technique to reconstruct the state of a quantum system [1]. Recent applications include the reconstruction of numerous physical systems, such as a radiation field [2], trapped ions and molecular vibrational states [3]- [5], spin [6] [7] and some other systems [8] [9].
The main idea behind Quantum Tomography is to use population measurements of the "rotated" density matrix of the system. Explicitly, if ρ is the density matrix, a tomogram ω(ψ, κ) is defined by
ω(ψ, κ) ≡ ψ| U (κ) ρ U −1 (κ) |ψ(1)
where
ρ(κ) ≡ U (κ) ρ U −1 (κ)(2)
is the density matrix rotated by the unitary transformation U and κ stands for all the parameters required to uniquely specify U . By varying the input parameters κ, one obtains a complete collection of different observables (called a quorum), from which a characterization of the initial quantum state of the system can be obtained [10].
Every tomographic scheme is centered on the possibility of inverting Eq.(1). This inversion and the corresponding reconstruction of ρ from ω(ψ, κ) can always be mathematically achieved [11] [12] when a Lie group G acts irreducibly in the Hilbert space appropriate for the description of the quantum system under investigation. Experimental success depends on whether or not all the requirements element in G can be practically implemented.
The objective of this paper is to study the possible tomographic reconstruction of the density matrix describing a collection of three-level atoms. Several approaches have already been proposed to reconstruct the state of a general three-level system (qutrit). In [13] [14] [15], the reconstruction of the quantum state of a three-level optical system is implemented for a frequency-and estimating quantum states and measuring fourth-order field moments. The use of non-orthogonal measurements as a way to reconstruct the state of a system (provided those measurements span the Hilbert space) as well as a detailed example of reconstruction for one and two-qutrit systems, is considered in [7].
Even though one technically may require only a finite number of different experimental set-ups for complete tomographic reconstruction of atomic states, we will focus on the so-called redundant reconstruction, which implies a continuous set of measurements "blanketing" all the parameter space. We justify this redundancy on the grounds that the reconstruction of a many-atom system would, in practice, require a large number of such discrete measurements.
Our strategy is to investigate tomographic reconstruction of the atomic state by probing atoms through the application of a carefully selected sequence of dispersive and resonant electromagnetic pulses.
Once an initial pulse of classical light has created a state of collective excitation in an ensemble of cold atoms, another pulse converts the atomic excitations into field excitations generating, for different atomic configurations, photons in a well-defined spatial and temporal mode [16]. The total number of such photons is determined by photoelectric detection so that a probability of detecting a given number of photons can be directly related to a tomogram [17].
In this work, our analysis will focus on two of three fundamentally different atomic configurations. They are the so-called Λ and Ξ configurations, distinguished by the presence of transition degeneracies from the generic nondegenerate configuration.
In Sec.3, we discuss this non-degenerate case, where transition frequencies are essentially different for distinct atomic transitions and where each transition can be independently interrogated by a pulse of the appropriate frequency. In this case, the density matrix can be completely reconstructed.
In Sec.4 and 5, the systems under consideration contain atomic transitions having the same frequency; it is not possible to interrogate every transition individually. We will consider these cases at length and highlight the differences between the (global) symmetries pertinent to the description of these inequivalent degenerate atomic configuration. We will show, for these cases, that the density matrix cannot in general be completely reconstructed and that the partial information extracted from the measurements in the case of Λ and Ξ atoms is essentially different.
u(3), su(3) and three-level atoms
The Hamiltonian governing the evolution of a collection of A three-level atoms in a classical field has the form
H =Ĥ 0 +Ĥ 12 +Ĥ 23(3)
whereĤ 0 is the free atomic Hamiltonian, andĤ ij is the interaction term between levels i and j.
The terms in Eq.(3) are most transparently analyzed by introducing a set {Ŝ ij ; i, j = 1, 2, 3} of collective transition operators that satisfy the standard commutation relations of the u(3) algebra:
[Ŝ ij ,Ŝ kl ] = δ jkŜil − δ ilŜkj .(4)
Thus, the Hilbert space for our systems naturally decomposes into a sum of subspaces invariant under the action of the Lie group U(3). We further assume that the A atoms are indistinguishable and their states are fully symmetric under permutation of the particle indices. Hence, the possible states of our system belong to a single unitariy irreducible representation of U(3) having dimension 1 2 (A + 1)(A + 2) and denoted by (A, 0) in mathematics. The dimension of the Hilbert space is given by the number of ways of distributing A bosons in three modes.
If we introduce the atomic basis {|n 1 n 2 n 3 , n 1 , n 2 , n 3 ≥ 0 , n 1 + n 2 + n 3 = A} ,
where n j denotes the population in the j-th atomic level, the matrix elements ofŜ ij can be easily evaluated using the Schwinger realization:
S ij → a † i a j .(6)
In the one-atom case, this yieldsŜ
ij |j = |i ,(7)
where the following identification
|100 ↔ |1 , |010 ↔ |2 , |001 ↔ |3 ,(8)
has been made. Throughout this work, we will assume the ordering E 1 ≤ E 2 ≤ E 3 of individual atomic levels.
In terms ofŜ ij 's, the Hamiltonian of Eq.(3) takes the form
H 0 = 3 i=1 E iŜii ,(9)H 12 = g 1 e iω1tŜ 12 + e −iω1tŜ 21 ,(10)H 23 = g 2 e iω2tŜ 23 + e −iω2tŜ 32 ,(11)
where ω 1 and ω 2 are frequencies of the external fields and g 1 and g 2 are coupling constants, chosen to be real for simplicity. The operatorN
= 3 i=1Ŝ ii(12)
commutes with all other operators in the u(3) algebra. This operator is proportional to the unit operator when acting on occupational states of the form |n 1 n 2 n 3 ,N |n 1 n 2 n 3 = A|n 1 n 2 n 3 .
Non-Degenerate Case
The non-degeneracy condition is understood to imply that the atomic transition frequencies (E 3 − E 2 ) and (E 2 − E 1 ) are sufficiently distinct to satisfy
(E 3 − E 2 ) − (E 2 − E 1 ) ≫ g 1 , (E 3 − E 2 ) − (E 2 − E 1 ) ≫ g 2 .(14)
In this manner, each transition can be interrogated separately by an external field. A typical non-degenerate system is illustrated in Fig.1. For tomographic purposes, the level pairs (1 ↔ 2), (2 ↔ 3) and their respective transitions, if taken in respective isolation, could be considered as independent two-level subsystems. However the full system must be treated as a three-level system under SU(3) evolution.
In the rotating frame, the Hamiltonian of Eq.(3) takes the form
H int = ∆ 23Ŝ33 − ∆ 12Ŝ11 + g 12 Ŝ 12 +Ŝ 21 + g 23 Ŝ 23 +Ŝ 32 ,(15)
where
∆ 12 ≡ E 2 − E 1 − ω 1 , ∆ 23 ≡ E 3 − E 2 − ω 2 .(16)
Because the frequencies ω 1 and ω 2 of the external field are adjustable parameters, two types of field pulses can be applied to our system. The first type is characterized by ∆ ij ≈ 0 and thus resonant; it stimulates the corresponding atomic transitions. The second type is characterized by ∆ ij ≫ g ij and is thus far-off resonant (dispersive); this kind of pulse leads to some phase shifts.
The corresponding resonant and dispersive evolution operators have the form
U R ij (β ij ) = exp −iβ ij Ŝ ij +Ŝ ji , i = j ,(17)U D 11 (φ 12 ) = exp iφ 12Ŝ11 ,(18)U D 33 (φ 23 ) = exp iφ 23Ŝ33 ,(19)
where β ij = g ij t ij and φ ij = (∆ ij + g 2 ij /∆ ij )τ ij . Here, t ij , τ ij are time intervals, not necessarily equal. It should be noted, that for short interaction times τ ij satisfying g 2 τ ij /∆ ij << 1 and g 2 τ ij << ∆ 2 ij , the second term in the expression for φ ij can be obviously neglected. However, for long interaction times, exp(−i∆ ij τ ij ) becomes strongly oscillating and the measurements should be carried out in stroboscopic times, τ ij = 2πn/∆ ij .
For a complete reconstruction of the density matrix in the absence of degeneracy, it suffices to measure the probability of detecting zero photons (i.e. zero fluorescence condition) in the irradiated field. This corresponds to the detection the atoms in the ground state, |A00 . It is worth noting here that, in the non-redundant scheme when only a finite number of different pulses have to be applied, the measurement of non-zero photons are required [17].
It is possible, by combining operations in Eqs.(17)- (19), to obtain a element of the group U(3) sufficiently general for our purpose. This is best seen by first observing that an element of SU(3) is parameterized by eight real numbers and can be conveniently factorized [18] into a product of SU(2) subgroup transformations:
U (α 1 , β 1 , γ 1 , α 2 , β 2 , α 3 , β 3 , γ 3 ) = R 23 (α 1 , β 1 , γ 1 ) · R 12 (α 2 , β 2 , α 2 ) · R 23 (α 3 , β 3 , γ 3 ) ,(20)
The action of such a group element on basis states of an irreducible representation is given in [18]. The notation n ≡ (n 1 n 2 n 3 ) ,
I = I 23 ≡ 1 2 (n 2 + n 3 ) ,(21)
will be a useful shorthand throughout this paper. In particular, we note that
ν| R 23 (α, β, γ)|n ≡ ν 1 ν 2 ν 3 | R 23 (α, β, γ)|n 1 n 2 n 3 = D I23 Mν ,Mn (Ω) , ν|R 12 (α, β, γ)|n ≡ ν 1 ν 2 ν 3 | R 12 (α, β, γ)|n 1 n 2 n 3 = D I12 mν ,mn (Ω) ,(22)
where D J MM ′ an SU(2) Wigner D-function and
I 12 = 1 2 (n 1 + n 2 ) , M n = 1 2 (n 2 − n 3 ) , m n = 1 2 (n 1 − n 2 ) .(23)
More generally, the definition of states for the irrep (λ, µ) and SU(3) elements between those states can be found in [18]. Specialized results are collected in A.
In particular, some formulae in this section implicitly depend on matrix elements of states in irreps of the type (A, 0), (0, A) and (σ, σ). The notation of SU(3) D functions conforms to that of [18] and uses the pair (λ, µ) with the labels n, I to unambiguously distinguish the SU(3) D functions. For occupational states, n, I are given by Eq. (21). Because the state |A00 is, up to a global phase, unchanged by the action of dispersive pulses and operations of the form R 23 , it is enough to consider a sequence of pulses of the form:
U (ϕ 23 , β 23 , ϕ 12 , β 12 ) = U D 33 (−ϕ 23 ) U R 23 (β 23 ) U D 11 (ϕ 12 ) U R 12 (β 12 ) .(24)
In the single atom case, the evolution operator Eq.(24) is a 3 × 3 matrix explicitly given by
U (ϕ 23 , β 23 , ϕ 12 , β 12 ) = e iϕ12 cos (β 12 ) −i e iϕ12 sin (β 12 ) 0 −i cos (β 23 ) sin (β 12 ) cos (β 12 ) cos (β 23 ) −i sin (β 23 ) −e −iϕ23 sin (β 12 ) sin (β 23 ) −ie −iϕ23 cos (β 12 ) sin (β 23 ) e −iϕ23 cos (β 23 ) .(25)
U (ϕ 23 , β 23 , ϕ 12 , β 12 ) can be more easily analyzed in the factorized form
U (ϕ 23 , β 23 , ϕ 12 , β 12 ) = e − 1 3 i(ϕ12−ϕ23)Ū ,(26)wherē U = 1 0 0 0 −ie i(χ−ϕ12) cos(β 23 ) e i(2χ−ϕ12) sin(β 23 ) 0 −e −i(2χ−ϕ12) sin(β 23 ) ie −i(χ−ϕ12) cos(β 23 ) × e iχ cos(β 12 ) − sin(β 12 ) 0 sin(β 12 ) e −iχ cos(β 12 ) 0 0 0 1 1 0 0 0 ie iχ 0 0 0 −ie −iχ ,(27)
and
χ = 1 3 (2ϕ 12 + ϕ 23 ) .(28)
Clearly, the matrix U (ϕ 23 , β 23 , ϕ 12 ,
β 12 ) of Eq.(26) is an element of U(3) whereas U of Eq.(27) is an SU(3) transformation.
Comparing with the parametrization of [18], we have the correspondences
α 1 → −ϕ 23 − 1 2 π , β 1 → 2β 23 , γ 1 → 3 2 π + 2 3 ϕ 12 + 1 3 ϕ 23 α 2 → − 2 3 ϕ 12 − 1 3 ϕ 23 , β 2 → 2β 12 , α 3 → − 4 3 ϕ 12 − 2 3 ϕ 23 − π , β 3 = 0 , γ 3 = 0 .
(29) Note that, althoughŪ is not the most general SU (3), it can be multiplied on the right by a transformation of the form
R 12 (ᾱ 2 , 0,ᾱ 2 )R 23 (ᾱ 3 ,β 3 ,γ 3 ) = e −iᾱ2 0 0 0 e i 2 i(2ᾱ2−ᾱ3−γ3) cos(β 3 2 ) −e i 2 (2ᾱ2−ᾱ3+γ3) sin(β 3 2 ) 0 e i 2 (ᾱ3−γ3) sin(β 3 2 ) e i 2 (ᾱ3+γ3) cos(β 3 2 ) (30)
without affecting the dynamics of the |100 state. Thus,Ū is equivalent to a general transformation when acting on |100 .
Expanding the density matrix in the occupational basis:
ρ = nν |n ν| ρ n,ν ,(31)
and introducing the shorthand τ = (ϕ 23 , β 23 , ϕ 12 , β 12 ) , we rapidly obtain
ω(τ ) = nν A00|Ū(τ ) |n ν|Ū † (τ ) |A00 ρ n,ν = nν (−1) ν2 D (A,0) (A00)0,nI (τ ) D (0,A) (0AA)0,ν * I ′ (τ )ρ n,ν ,(32)
where D
(λ,µ) n1Ī1,n2I2 (τ ) ≡ (λ, µ)n 1 I 1 | U (τ )|(λ, µ)n 2 I 2(33)
is an SU (3) Wigner-function for the irrep (λ, µ). Further notational details and properties of these functions (in particular Eq.(70)) can be found in A. Products of SU(2) D-functions can be decomposed into sums of D-functions multiplied by products of SU(2) Clebsch-Gordan coefficients. The same holds for products of SU(3) D-functions provided that we use SU(3) Clebsch-Gordan technology. Thus, given that SU (3)
-coupling (A, 0) ⊗ (0, A) decomposes in the direct sum [19] (A, 0) ⊗ (0, A) = (A, A) ⊕ (A − 1, A − 1) ⊕ . . . ⊕ (0, 0) , = A λ=0 (λ, λ) ,(34)
we have
D (A,0) (A00)0, nI (τ ) D (0,A) (0AA)0, ν * I ′ (τ ) = λ,J D (λ,λ) (λλλ)0, N (λ) J (τ ) (A, 0) (A00)0 (0, A) (0AA)0 (λ, λ) (λλλ)0 × (A, 0) nI (0, A) ν * I ′ (λ, λ) N (λ) J ,(35)
where
N (λ) = (n 1 + ν * 1 − (A − λ), n 2 + ν * 2 − (A − λ), n 3 + ν * 3 − (A − λ)) .(36)
and where (A, 0)
n 1 I 1 (0, A) n 2 I 2 (λ, λ) N (λ) I 3(37)
is the SU (3) Clebsch-Gordan coefficient for the coupling of |(A, 0) n 1 I 1 and
|(0, A) n 2 I 2 to |(λ, λ) N (λ) I 3 . The appearance of extra factors of (A − λ) in the construction of N (λ) is discussed in Eq.(86) of B. Inserting Eq.(35) in Eq.(32) yields ω(τ ) = nνλJ (−1) ν2 ρ n,ν D (λ,λ) (λλλ)0, N (λ) J (τ ) × (A, 0) (A00)0 (0, A) (0AA)0 (λ, λ) (λλλ)0 (A, 0) nI (0, A) ν * I ′ (λ, λ) N (λ) J .(38)
After some straightforward manipulations detailed in C, we obtain the final expression
(−1) ν2 ρ n, ν = µ J (µ + 1) 3 1024π 5 (A, 0) (A00)0 (0, A) (0AA)0 (µ, µ) (µµµ) 0 −1 × (A, 0) nI (0, A) ν * I ′ (µ, µ) N (µ) J dΩ D (µ,µ) * (µµµ) 0, N (µ) J (τ ) ω(τ ) .(39)
As there is no restriction on n or ν * , Eq.(39) shows that, in the nondegenerate case, the density matrix can be completely reconstructed.
Degenerate Λ-type atomic systems
Let us turn our attention to the case of a degenerate Λ-type system. A typical Λ atom is schematically illustrated in Fig.2.
In the single-atom case, the allowed transitions are |1 ↔ |3 , |2 ↔ |3 . The degeneracy condition is
E 3 − E 1 = E 3 − E 2 .(40)
In the multi-atom case, the only atomic configuration that can be unambiguously identified by photon counting is when every atom is excited, i.e. when the system of A atoms is in the state |00A .
The evolution
In the rotating frame, the interaction Hamiltonian has the form
H Λ = ∆S 33 + g (S 13 + S 23 ) + g (S 31 + S 32 ) ,(41)
where ∆ = E 3 − E 1 − ω and g 1 = g 2 = g for simplicity.
In the single atom case,Ĥ Λ can be represented as the following 3 × 3 matrix:
H Λ = 0 0 g 0 0 g g g ∆ .(42)
A simple basis transformation
| 1 → |1 = 1 √ 2 (| 1 − | 2 ) , | 2 → |2 = 1 √ 2 (| 1 + | 2 ) , | 3 → |3 ,(43)
given by the constant matrix
T 12 = 1 √ 2 1 √ 2 0 − 1 √ 2 1 √ 2 0 0 0 1 ,(44)
transforms Eq.(42) to the block diagonal form
H Λ →Ĥ T = T −1 12Ĥ Λ T 12 = 0 0 0 0 0 √ 2g 0 √ 2g ∆ .(45)
The effect T 12 on basis states is illustrated in Fig.3; T 12 produces a dark state |1 completely decoupled from the remaining doublet. In view of this we can expect, on general grounds, that a complete reconstruction will not be possible as our HamiltonianĤ Λ cannot possibly probe the dark state |1 .
Using the basis {|1 , |2 , |3 }, the resonant pulses, with ∆ = 0, are of the formŨ
R Λ ( √ 2gt) = 1 0 0 0 cos √ 2gt i sin √ 2gt 0 i sin √ 2gt cos √ 2gt .(46)
In the same basis, the dispersive pulses, with ∆ ≫ g, are described in the stroboscopic approximation by the effective evolution operator
U D Λ 2g 2 t/∆ = 1 0 0 0 e 2ig 2 t/∆ 0 0 0 e −2ig 2 t/∆ .(47)
In the two-dimensional subspace spanned by |2 and |3 , the operatorsŨ R Λ andŨ D Λ correspond to SU(2) rotations about thex andẑ axes, respectively:
U R Λ (α) → R x 23 (α) ,Ũ D Λ (β) → R z 23 (β) ,(48)
in an obvious notation. A sufficiently general sequence of pulses can thus be writtenŨ
Λ (Ω) = R z 23 (α) · R x 23 (β) · R z 23 (γ) , = R z 23 (α + π 2 ) · R y 23 (β) · R z 23 (γ − π 2 ) = R 23 (Ω) .(49)
For the one-atom case, the 3 × 3 matrix representation for this evolution has the formŨ
Λ (Ω) = 1 0 0 0 * * 0 * * (50)
where * indicates a non-zero entry. The block diagonal form ofŨ is explicit. It shows that, for a system containing one or more than one atom, there will always be at least one subspace which cannot be reached in the course of evolution of |00A ; such decoupled subspaces are an obstruction to complete reconstruction.
To illustrate this, we expand the density matrix for an A-atom system in the basis {|ñ 1ñ2ñ3 } of occupation of the states |1 , |2 and |3 :
ρ = ñ1ñ2ñ3ν1ν2ν3 |ñ 1ñ2ñ3 ν 1ν2ν3 |ρ (ñ1ñ2ñ3),(ν1ν2ν3) .(51)
Using the shorthandñ for the tripletñ 1ñ2ñ3 , we can write the tomogram as
ω(Ω) = ñν 00A|R 23 (Ω) |ñ 1ñ2ñ3 ν 1ν2ν3 |R 23 (Ω)|00A ρ (ñ1ñ2ñ3),(ν1ν2ν3) . (52)
As the R 23 (Ω) rotation does not affect the first atomic index and acts irreducibly as an SU(2) rotation in each the subspace spanned by states having a common fixedñ 1 , we write
|ñ 1ñ2ñ3 → |ñ 1 ; Im , |ν 1ν2ν3 → |ν 1 ; Iµ ,(53)
where whereρ (n1n2n3),(ν1ν2ν3) →ρ (n1;Im),(ν1;Iµ) has been used to conform to the notation of Eq.(53), where D I mm ′ is the usual SU(2) D function and where C L M L1 m1, L2 m2 is an SU(2) Clebsch-Gordan coefficient.
I = 1 2 (ñ 2 +ñ 3 ) = 1 2 (ν 2 +ν 3 ) , m = 1 2 (ñ 2 −ñ 3 ) , µ = 1 2 (ν 2 −ν 3 )(54)
Multiplying both sides of (55) by D L ′ * 0,M (Ω), integrating over SU(2) and using orthogonality of the Clebsch-Gordan coefficients rapidly gives the elements of the density matrix that can be reconstructed from the tomographic process as
4.2 Reconstruction for state of one and two Λ-type atoms.
In a system containing a single atom, the Hilbert space is spanned, in the notation of Eq.
Under the evolutionŨ Λ (Ω) = R 23 (Ω), the initial state |0; 1 2 , − 1 2 cannot reach the dark state so it is only possible to reconstruct element of ρ of the form ρ (0; 1 2 m),(0; 1 2 m ′ ) , with m, m ′ = ± 1 2 . The last diagonal element, ρ (1;00),(1;00) can be inferred from the normalization. None of the remaining four matrix elements can be determined by our scheme.
In the two-atom case, an even smaller proportion of matrix elements can be recovered. Using again the notation of Eq.(53), states of the irrep (2, 0) are conveniently given, in the occupational, tensor product and SU(2) basis |ñ 1 ; ℓm , in table 1.
|n 1 n 2 n 3 |1 n1 |2 n2 |3 n3 |ñ 1 ; I, m |002 |3 |3 | 0; 1, −1 |011 1 √ 2 |2 |3 + |3 |2 | 0; 1, 0 |020 |2 |2 | 0; 1, 1 |101 1 √ 2 |1 |3 + |3 |1 | 1; 1 2 , − 1 2 |110 1 √ 2 |1 |2 + |2 |1 | 1; 1 2 , 1 2 |200 |1 |1 | 2; 0, 0
The initial state |002 will not evolve out of the I = 1 subspace, so only matrix elements of the form ρ (0;1m)(0;1m ′ ) can be reconstructed using our scheme. These represent only nine of the possible 36 elements of the density matrix.
We conclude this section by noting that the situation obviously worsens (in the sense that a smaller and smaller proportions of the matrix elements can be recovered) as the number of atoms increases.
Degenerate Ξ-type atomic systems
Finally, we consider the case of the Ξ system. It is illustrated, for a single atom, in Fig.4. For this configuration, the condition E 2 − E 1 = E 3 − E 2 holds.
The evolution
In the rotating frame, the Hamiltonian governing the evolution of a collection of A atoms in the Ξ configuration in an external field has the form (g 1 = g 2 = g)
H Ξ = ∆ Ŝ 33 −Ŝ 11 + g Ŝ 12 +Ŝ 32 +Ŝ 21 +Ŝ 23 ,(58)
where ∆ = (E 3 − E 1 )/2 − ω. Important insight into the nature of this Hamiltonian can be gained by noting that the operatorsŜ 11 −Ŝ 33 andŜ 12 +Ŝ 21 +Ŝ 23 +Ŝ 32 are, in fact, proportional to two of the three generators of the so(3) subalgebra of su (3) :
S 11 −Ŝ 33 →L z ,Ŝ 12 +Ŝ 21 +Ŝ 23 +Ŝ 32 → √ 2L x .(59)
Thus, the possible evolutions are elements of the SO(3) subgroup of SU (3). Clearly, a convenient sequence of pulse is given by
U Ξ (Ω) ≡ R z (α) · R x (β) · R z (γ) , = R z (α + π/2) · R y (β) · R z (γ − π/2) .(60)
Here, the resonant pulses are of the form
R x (β) = exp −iβ (Ŝ 12 +Ŝ 21 +Ŝ 23 +Ŝ 32 )/ √ 2(61)
while the dispersive pulses are generated byŜ 11 −Ŝ 33 . It is important to note that, in the one-atom case, the exponentiation ofĤ Ξ in Eq.(58) produces an evolution that acts irreducibly on the non-degenerate states of the Hilbert space: in contrast with Eq.(49) of the Λ case, the "rotations" R x and R z of the Ξ states are not restricted to a two-dimensional subspace of the whole Hilbert space.
To analyze the many-atom case, we start by observing that the state of the system for which every the atom is completely excited, |00A , is an eigenstate of L z with eigenvalue −A and is annihilated byL − . Here,L − = Ŝ 21 +Ŝ 32 / √ 2 is constructed in the usual way:L − =L x − iL y . Thus, |00A is the unique angular momentum state |L, −L , with L = A:
|00A → |L = A, M = A .(62)
As this state contains the largest possible number of excitations, it can be uniquely identified through photon counting so that the corresponding tomogram is determined from the probability of detecting 2A photons in the irradiated field.
The general correspondence between the occupational basis states |n 1 n 2 n 3 is found in [20] and given by
|LM = 2 L+M 1 2 (A + L) !(L + M )!(L − M )!(2L + 1) 1 2 (A − L) !(A + L + 1)! × (a † 2 ) 2 − 2a † 1 a † 3 1 2 (A−L) p (a † 1 ) p (a † 2 ) L+M−2p (a † 3 ) p−M 2 p p! (p − M )!(L + M − p)! |0 .(63)A L M |LM 1 1 1 a † 1 |0 0 a † 2 |0 −1 a † 3 |0 2 2 2 1 √ 2 (a † 1 ) 2 |0 1 a † 2 a † 1 |0 0 1 √ 3 (a † 2 ) 2 + a † 1 a † 3 |0 −1 a † 2 a † 3 |0 −2 1 √ 2 (a † 3 ) 2 |0 2 0 0 1 √ 6 (a † 2 ) 2 − 2a † 1 a † 3 |0
It is clear that, given an angular momentum state in the irrep (A, 0) of su(3), we can unambiguously write it as a linear combination of occupational states, and vice versa. Thus, we may expand
ρ = L1M1L2M2 |L 1 M 1 L 2 M 2 | ρ L1M1,L2M2 ,(64)
where L 1 , L 2 run from A, A − 2, ..., 1 or 0 depending if A is even or odd. As the evolution is necessarily an element of SO (3), the tomogram takes the general form
ω(Ω) = L1M1L2M2 A, −A|R(Ω)|L 1 M 1 L 2 M 2 |R † (Ω)|A, −A ρ L1M1,L2M2 , = M1M2 D A −A,M1 (Ω)D A * −A,M2 (Ω) ρ AM1,AM2 , = M1M2J (−1) −A−M2 D J 0,M (Ω)C J 0 A −A, A A C J M A M1, A M2 ρ AM1,AM2 ,(65)
with C J M L1 M1, L2 M2 a regular angular momentum Clebsch-Gordan coefficient.
In a manner similar to the previous cases, multiplication by D J ′ * 0,M (Ω), integration over SO(3) and orthogonality of Clebsch-Gordan coefficients yields
(−1) A+M2 ρ AM1AM2 = J 2J + 1 8π 2 C J M A M1, A M2 C J 0 A −A, A A −1 dΩ D J * 0,M (Ω)ω(Ω) .(66)
The result clearly shows that only those linear combinations of occupational state that transform by angular momentum L = A can be reconstructed.
Examples: one and two atom cases
For a single atom, we see from table 2, that the tomogram is constructed from an L = 1 state. There is no other angular momentum multiplet and so the evolution, an element of SO (3), will yield sufficiently many tomograms to guarantee complete reconstruction. The matter is different for the two-atom case. In this case, the tomogram is constructed using an L = 2 state but the Hilbert space also contains an L = 0 subspace, which cannot be reached from L = 2 with our evolution. Thus, if we are limited to measuring a total of 2A photons, it will only be possible to recover ρ 2M,2M ′ and impossible to reconstruct ρ 00,2M , ρ 00,00 , ρ 2M,00 . This is because, in our scheme, it is not possible to extract photons from the L = 0 state, and absence of photon does not pin down a particular state.
It may be possible to measure fewer than 2A photons, but this does not lead to more information. There is only one state with M = A and one state with M = A − 1 (or M = −A and M = −(A − 1)). It is possible to use the L = A, M = A−1 state for the tomogram and measure 2A−1 or 2A−2 photons, but we will recover nothing more than if we had started with L = A, M = A.
There are two states with M = A − 2; they belong different angular momentum multiplet. Thus, if we measure, say, a total of 2(A − 2) photons, it is not possible to know unambiguously if this is the result of a complete cascade within the L = A − 1 multiplet or a partial cascade within the L = A multiplet. This kind of limitation becomes obviously more severe as the number of angular momentum multiplet containing a given M value increases.
Conclusions
We have proposed a physical realization applicable to the reconstruction of the quantum state of three-level atomic systems. The information about atomic states is extracted by measuring the total number of excitations after successive applications of electromagnetic field pulses.
We have shown that, in the non-degenerate case, the complete reconstruction of atomic states is possible. Although the number of independent parameters required for a complete reconstruction is less than needed for the complete parametrization of a generic element of SU(3) group, a complete reconstruction is possible because, in addition to the usual evolution of the system, another tool is available in the reconstruction scheme: the projective measurement.
When degeneracies are present, the possibilities of reconstruction are limited. The origin of these limitations is essentially different for atoms in Λ and Ξ configuration. In both cases, the evolution operator operators belong to a subgroup of the whole SU(3) group, and our work illuminates the subtle distinction between the global properties of SO(3) and SU(2) as subgroups of SU(3).
In the Ξ case, the reconstruction is rooted in an SO(3) symmetry of the physically available evolution operator; this symmetry provides information about a single subspace. In the one-atom case, the Hilbert space contains precisely a single SO(3) subspace, so the density matrix can be completely reconstructed. In the multiple-atom case, only reconstruction in one pre-determined subspace is possible. In this case, our protocol would be to apply the sequence of pulses of Eq.(60) with a subsequent measurement of the number of 2A of photons in the irradiated field, giving us the tomogram appearing in the reconstruction formula of Eq.(66). For completeness, we note here that we did not consider the effective two-photon-like transition in the Ξ system due to extremely narrow width of such transitions (∼ g 2 /∆), which leads to serious experimental difficulties in its detection.
In the case of Λ configuration, the evolution operator generates an SU(2) transformation and, even in the one-atom case, there is always more than a single SU(2) multiplet: a complete reconstruction is impossible because there always exists invariant SU(2) "dark" subspaces, which cannot be uniquely identified by measuring irradiated photons. We stress that the decomposition of the Hilbert space into invariant subspaces occurs as a result of the inability to access independent transitions separately; this to be contrasted with the approach of Ref. [9], wherein SU(2) decomposability arises from considerations of perfectly general polarization states. Note also that although the effective transitions between degenerate levels in the Λ case are not sensitive to the atom-field detunings, they still require long interaction times.
The tomographic protocol for Λ differs from the Ξ. After application of the sequence of pulses Eq.(24) to a Λ-type atom, we have to measure the probability of detecting zero irradiated photons, which leads to the tomogram used in Eq.(39).
Finally, we observe that the tomographic reconstruction process for a collection of non-degenerate three-level atoms is a simple generalization of the familiar process used for two-level quantum systems. In both instances, one uses the whole dynamic symmetry group to carry out the inversion process. In contrast to this, we are restricted to a specific subgroup in the degenerate cases, which essentially reduces our tools and actually limits the possibility of the complete tomographic reconstruction.
We would like to thank Dr. O. Aguilar for his participation in the early stages of this project. The work of A. B. Klimov is partially supported by grants CONACyT 45704. The work of H. de Guise is supported by NSERC of Canada.
A SU(3) basis states and D-functions
In this section, we review some notation useful mostly in section 3. Further details can be found in [18].
The correspondence between the occupational basis and states of the (A, 0) is
|n 1 n 2 n 3 → |(A, 0)nI .(67)
In Eq.(67) and throughout this paper, n is a shorthand for (n 1 n 2 n 3 ). Here, the (A, 0) labels indicate that |n 1 n 2 n 3 can be reached, using theŜ ij operators of Eq.(6), from the state |A00 . This state is killed by the so-called su(3) raising operatorsŜ 12 ,Ŝ 13 andŜ 23 . The eigenvalues of the su(3) diagonal operatorŝ
h 1 =Ŝ 11 −Ŝ 22 ,ĥ 2 =Ŝ 22 −Ŝ 33 ,(68)
acting on |A00 are, respectively, (A, 0). The angular momentum label I is necessary to deal with the general case considered in Ref. [18], where states of more general families of the type (p, q) are constructed. A state |(p, q)nI can be reached from the state |(p, q)(p + q, q, 0) 1 2 p , i.e. with n 1 = p + q, n 2 = q, n 3 = 0 and I = 1 2 p. |(p, q)(p + q, q, 0) 1 2 p is killed by the su(3) raising operators, and the eigenvalues of (ĥ 1 ,ĥ 2 ) are (p, q). When p and q are both non-zero, it is possible to have distinct states in the same (p, q) family that have identical n, so the angular momentum label I is required to distinguish these distinct states.
Some calculations require the evaluation of the matrix elements
Here, D (0,A)
n * I,ν * I ′ (τ ) ≡ (0, A)n * I|Ū(σ)|(0, A)ν * I ′ .(71)
The dimension of (0, A) is the same as the dimension of (A, 0), but the construction of Ref. [18] for basis state of |(0, A)n * I requires twice as many quanta as the basis states of |(A, 0)nI . The relation between n and n * is
n = (n 1 , n 2 , n 3 ) → n * = (A − n 1 , A − n 2 , A − n 3 ) .(72)
Using this and the results from [18], one can verify Eq.(70). Using Eq.(70), one can also verify that the D functions are orthogonal, in the sense that dim(λ, µ)
1024π 5 dΩ D (λ,µ) * nI,νL (Ω)D (λ ′ ,µ ′ ) n ′ I ′ ,ν ′ L ′ (Ω) = δ λλ ′ δ µµ ′ δ nn ′ δ II ′ δ νν ′ δ LL ′ ,(73)
where dΩ = sin β 1 cos 1 2 β 2 sin 1 2 β 2 3 sin β 3 dα 1 dβ 1 dγ 1 dα 2 dβ 2 dα 3 dβ 3 dγ 3
is the invariant measure, which can be found in the usual ways [21]. The normalization follows from the dimensionality formula dim(λ, µ) = 1 2 (λ + 1)(µ + 1)(λ + µ + 2)
for the irrep (λ, µ) and the use the parameter range
0 ≤ α 1 ≤ 4π , 0 ≤ β 1 ≤ π , 0 ≤ γ 1 ≤ 4π , 0 ≤ α 2 ≤ 2π , 0 ≤ β 2 ≤ π , 0 ≤ α 3 ≤ 4π , 0 ≤ β 3 ≤ π , 0 ≤ γ 3 ≤ 4π .(76)
B Reduced SU (3) Clebsch-Gordan coefficients
B.1 Basis states
The construction of states in the irrep (p, q) of su(3) is detailed in [18]. We can summarize this procedure by stating that one requires, at a minimum, a total of p + 2q bosons. These bosons must be of at least two types when q = 0. Thus, if a † ij creates a boson of type j in mode i, we define, quite generally,
S kℓ = a † k1 a ℓ1 + a † k2 a ℓ2 .(77)
containing p + q bosons in mode 1, q boson in mode 2 and none in mode 3, belongs to the (p, q) irrep. It is, in fact, killed by everyĈ kℓ with ℓ > k and is thus the highest weight state of (p, q). Here,
a † 11 a † 12 a † 21 a † 22 = a † 11 a † 22 − a † 12 a † 21(79)
is the determinant of the matrix. Other states in (p, q) are obtained by laddering down from |(p, q)(p + q, q, 0); 1 2 q . This is not the only possibility. One can verify that differs from |(p, q)nI by at most a normalization but contains p+2q+3t bosons.
|(p, q)(p + q + t, q + t, t); 1 2 q) = a † 11 a †
B.2 SU(3) Clebsch-Gordan coefficients
The SU (3)
where |(σ, σ)N (σ) I 3 is the state with 3σ bosons described in [18]. Note that, because the state |(σ, σ)N (σ) I 3 does not contain 3A bosons, we do not have n i + ν * i = N (σ) i etc but rather (3) Clesbsh-Gordan coefficients for (1, 0) ⊗ (0, 1) → (0, 0). does not depend on M i . Tables are provided for A = 2 and A = 1. Table 5: 2-atom case. Reduced SU(3) Clesbsh-Gordan coefficients for (2, 0) ⊗ (0, 2) → (0, 0).
n i + ν * i = N (σ) i + (A − σ) .(86)
In this last expression, the sums over n and ν are not independent but linked by Eq.(36).
To complete the inversion, we use orthogonality of SU(3) CGs:
µJ ′ (A, 0) nI (0, A) ν * I ′ (µ, µ) N (µ) J ′ (A, 0) nĪ (0, A) ν * Ī ′ (µ, µ) N (µ) J ′ = δn n δĪ I δν ν δĪ′ I ′ ,(91)
and rearrange the notation to finally yield Eq.(39).
RemovingN reduces u( 3 )
3to su(3). Thus, the possible evolutions generated by the HamiltonianĤ are, up to an unimportant global phase, finite SU(3) transformations.
Figure 1 :
1A typical level scheme for the non-degenerate case.
Figure 2 :
2Schematic representation of a single atom in a Λ configuration.
Figure 3 :
3The basis states resulting from the transformation T 12 .
Furthermore(− 1 )
1, the tomograms of Eq.(52) must haveñ 1 =ν 1 = 0, so Eq.−I−µ C L 0 I I, I −I C L M I m, I −µ D L 0 ,M (Ω)ρ (0;Im),(0;Iµ) , (55)
×
dΩ ω(Ω) D L * 0,M (Ω).
(53), by states of the form |ñ 1 ; Im , with |1 = |100 , |2 = |0; |3 = |0; 1 2 , − 1 2 .
Figure 4 :
4The Ξ configuration for a single atom.
n 1 n 2 n 3 n
3|Ū (σ)|n 1 n 2 n 3 * = (A, 0)nI|Ū (σ)|(A, 0)νI ′ * , element is related to the matrix element between basis states of the irrep (0, A), which is conjugate to (* I,ν * I ′ (τ ) .
-coupling (A, 0) ⊗ (0, A) can be decomposed in the direct sum[19] (A, 0) ⊗ (0, A) = (A, A) ⊕ (A − 1, A − 1) ⊕ . . .(σ, σ) occurs at most once in the decomposition. To compute SU(3) Clebsch-Gordan coefficients for states in the series of Eq.(83), we must couple states of the form |(A, 0)nI 1 |(0, A)ν * I 2 , (84) which contain a total of 3A bosons of three types. States in the irrep (σ, σ) of the series of Eq.(83) are of the form |(σ, σ)N I σ, σ)N (σ) I 3 ,
, I2 M2 is the usual su(2) coupling coefficient and the reduced Clebsch-Gordan (A
C
Final form of the density matrix for the nondegenerate case In this section we present the technical steps to obtain Eq.(39) from Eq.(38). First, multiply both sides of Eq.(38) by D (µ,µ) * (µµµ)0, N (µ) J ′ (τ ) for fixed N (µ) and fixed J ′ , integrate over the SU(3)-invariant measure of Eq.(74) and rearrange.
Table 1 :
1Basis states for two atoms in the Λ configuration.
Table 2 :
2Angular momentum basis states as linear combinations of occupational number states for one and two atoms in the Ξ configuration.
If |0 denotes state with no boson excitation, the statea †
11
a †
12
a †
21
a †
22
q
(a †
11 ) p |0 ∼ |(p, q)(p + q, q, 0); 1
2 q ,
visibly contains p + 2q + 3t bosons but is equivalent to the state of Eq.(78) More generally, if the usual ket |(p, q)nI denotes a state in (p, q) containing p + 2q bosons, then the (round) ket |(p, q)nI) =12 a †
13
a †
21 a †
22 a †
23
a †
31 a †
32 a †
33
t
a †
11 a †
12
a †
21 a †
22
q
(a †
11 ) p |0 ,
∼
a †
11 a †
12 a †
13
a †
21 a †
22 a †
23
a †
31 a †
32 a †
33
t
|(p, q)(p + q, q, 0); 1
2 q ,
(80)
because the determinant
a †
11 a †
12 a †
13
a †
21 a †
22 a †
23
a †
31 a †
32 a †
33
(81)
is an SU(3) scalar.
a †
11
a †
12 a †
13
a †
21
a †
22 a †
23
a †
31
a †
32 a †
33
t
|(p, q)nI
(82)
Table 3 :
31-atom case. Reduced SU
Table 4 :
41-atom case. Reduced SU(3) Clesbsh-Gordan coefficients for (1, 0) ⊗ (0, 1) → (1, 1).
(n 2 −n 3 ) , M 2 = 1 2 (ν 3 −ν 2 ). The phases of the states |(A, 0)nI 1 , |(0, A)ν * I 2 and |(σ, σ)N (σ) I 3 are those of Ref.[18]. The phase of the Clebsch-Gordan coefficient is determined by forcing
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| []
|
[
"Analysis of level-dependent subdivision schemes near extraordinary vertices and faces",
"Analysis of level-dependent subdivision schemes near extraordinary vertices and faces",
"Analysis of level-dependent subdivision schemes near extraordinary vertices and faces",
"Analysis of level-dependent subdivision schemes near extraordinary vertices and faces"
]
| [
"Costanza Conti ",
"Marco Donatelli ",
"Paola Novara ",
"Lucia Romani ",
"Costanza Conti ",
"Marco Donatelli ",
"Paola Novara ",
"Lucia Romani "
]
| []
| []
| Convergence and smoothness analysis of a bivariate level-dependent (non -stationary) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. In this paper we focus on the problem of analyzing convergence and tangent plane continuity -also known as G 1 -continuity -of non-stationary subdivision schemes. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, we derive new sufficient conditions for establishing G 1 -continuity of any non-stationary subdivision surface at the limit points of extraordinary vertices and/or extraordinary faces.Keywords Non-stationary subdivision · Extraordinary vertex/face · Convergence · Tangent plane (G 1 ) continuity Mathematics Subject Classification (2010) 26A15 · 68U07IntroductionThis paper provides a general procedure to check convergence of level-dependent subdivision schemes in the neighborhood of an extraordinary vertex/face. It also gives sufficient conditions for the limit surface to be tangent plane continuous at the limit point of an extraordinary vertex/face. To the best of our knowledge the only contribution in this domain | null | [
"https://arxiv.org/pdf/1707.01954v3.pdf"
]
| 119,609,198 | 1707.01954 | 9066ce1a7719b722c8f3176a19a8695b2b5e96d2 |
Analysis of level-dependent subdivision schemes near extraordinary vertices and faces
17 Mar 2018
Costanza Conti
Marco Donatelli
Paola Novara
Lucia Romani
Analysis of level-dependent subdivision schemes near extraordinary vertices and faces
17 Mar 2018Received: date / Accepted: dateNon-stationary subdivision · Extraordinary vertex/face · Convergence · Tangent plane (G 1 ) continuity Mathematics Subject Classification (2010) 26A15 · 68U07
Convergence and smoothness analysis of a bivariate level-dependent (non -stationary) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. In this paper we focus on the problem of analyzing convergence and tangent plane continuity -also known as G 1 -continuity -of non-stationary subdivision schemes. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, we derive new sufficient conditions for establishing G 1 -continuity of any non-stationary subdivision surface at the limit points of extraordinary vertices and/or extraordinary faces.Keywords Non-stationary subdivision · Extraordinary vertex/face · Convergence · Tangent plane (G 1 ) continuity Mathematics Subject Classification (2010) 26A15 · 68U07IntroductionThis paper provides a general procedure to check convergence of level-dependent subdivision schemes in the neighborhood of an extraordinary vertex/face. It also gives sufficient conditions for the limit surface to be tangent plane continuous at the limit point of an extraordinary vertex/face. To the best of our knowledge the only contribution in this domain
is the work of Jena et al. in [21] where the behaviour of the limit surfaces generated by a specific non-stationary subdivision scheme in the neighborhood of an extraordinary face is studied. The specific scheme is a generalization of the celebrated Doo-Sabin's proposal [13]. The sufficient conditions we propose are used for the analysis of the family of approximating non-stationary subdivision schemes presented in [17]. The members of the latter family are a generalization of exponential spline surfaces to quadrilateral meshes of arbitrary topology whose tangent plane continuity is conjectured and shown only by numerical evidence in [17,Section 5]. Due to the lack of existing theoretical results for the analysis of level-dependent subdivision schemes, we believe that our contribution could mark a first step forward towards a deeper understanding of non-stationary subdivision with a consequent increase of its use in different fields of application.
Motivation
Non-stationary subdivision schemes were introduced in the last 10 years with the aim of enriching the class of limit functions of stationary schemes and have very different and distinguished properties. Indeed, it is well-known that stationary subdivision schemes are not capable of generating circles, ellipses, or to deal with level-dependent tension parameters that allow to arbitrarily modify the shape of a subdivision limit. Non-stationary schemes generate function spaces that are much richer. For example, in the univariate case, they include exponential B-splines or C ∞ limits with bounded support as the Rvachevtype function (see, e.g., [16]). Since the support size of the subdivision limit is dominated by the effect of the first few refinement steps while the smoothness by the last ones, a narrow support and a high level of derivative continuity can be achieved by using nonstationary schemes. The generation capability of level-dependent schemes (especially the capability of generating exponential-polynomials) is important in several applications, e.g., in biological imaging [1,8,10,12,31], geometric design-approximation [11,23,24,29,34] and in isogeometric analysis [20]. Level-dependent subdivision schemes include Hermite schemes that do not only model curves and surfaces, but also their gradient fields (such schemes are used in geometric modelling and biological imaging, see e.g. [5,6,8,22,28]). Additionally, non-stationary wavelet and frame constructions are level adapted and more flexible [9,14,19,33]. Unfortunately, in practice, the use of subdivision is mostly restricted to the class of stationary subdivision schemes even though the non-stationary ones are equally relatively simple to implement and highly intuitive in use: from an implementation point of view changing coefficients with the levels is not a crucial matter also in consideration of the fact that, in practice, only few subdivision iterations are performed. On the contrary, a crucial limitation to the spread of level-dependent schemes, is a lack of general analysis methods, especially methods for their convergence and regularity analysis. This motivates our study.
Subdivision framework
Subdivision schemes are efficient iterative algorithms to produce smooth surfaces as the limit of a recursive process starting from a given coarse 2-manifold polygon mesh (a polygon mesh is considered to be 2-manifold if it does neither contain non-manifold edges, non-manifold vertices, nor self-intersections, see [18]). Each step of the recursive process produces a finer 2-manifold polygon mesh than the original one, containing many more vertices and polygonal faces. Vertices and faces of a polygon mesh are identified by the so-called vertex valence and face valence, respectively. While the valence of a vertex is the number of edges incident to it, the valence of a face counts the number of edges that delimit it. For a quadrilateral mesh, vertices and faces of valence 4 are called regular. Differently, for a triangular mesh regular vertices are the ones with valence 6 while regular faces have valence 3. A regular mesh is a mesh that contains regular vertices and regular faces only. Non regular vertices and faces are called extraordinary vertices/faces (see Figure 1 for a graphical illustration of these two cases) and, whenever they appear, the mesh is said to be of arbitrary topology. It is evident that to ensure the convergence of a subdivision scheme to a (tangent plane) continuous limit surface, the refinement rules to be applied in the neighborhood of extraordinary vertices/faces have to be different from the ones used in the regular regions, since they must strictly depend on the vertex/face valence (see, e.g., [26]). The refinement rules to be used in the regular regions and in the neighbourhood of extraordinary vertices/faces may change with the refinement level or not. In the latter case the subdivision scheme is called stationary, non-stationary or level-dependent otherwise. A known analysis tool to investigate convergence and smoothness properties of stationary subdivision schemes for regular meshes is the one proposed by Dyn and Levin in [16].
To study convergence and smoothness of a non-stationary subdivision scheme for regular meshes, Dyn and Levin [15] proposed a method based on the comparison with a stationary scheme whose regularity properties are known. In the case of meshes with arbitrary topology, we are currently able to study only convergence and regularity of stationary subdivision schemes near extraordinary vertices/faces, thanks to the results in [26,27,32,35,36]. However, in literature we can find no general results to analyze level-dependent subdivision schemes at extraordinary vertices/faces.To the best of our knowledge, the only contribution in this domain is the work of Jena et al. in [21] where a specific scheme is considered.
Therefore, the goal of our paper is to propose a general procedure to check if a nonstationary subdivision scheme is convergent in the neighborhood of an extraordinary vertex/face. Moreover, it also aims at giving sufficient conditions for the limit surface to be tangent plane continuous at the limit point of an extraordinary vertex/face. The paper is organized as follows. In Section 2 we provide preliminaries on bivariate subdivision schemes. Then, in Section 3, we prove new results concerning non-stationary subdivision schemes. Next, in Section 4 (specifically, Subsection 4.2) sufficient conditions for proving convergence of a non-stationary subdivision scheme in correspondence to extraordinary vertices/faces are considered. Finally, in Subsection 4.3 we give sufficient conditions to verify if the limit surfaces generated by an arbitrary convergent, non-stationary subdivision scheme are tangent plane continuous at the limit points of extraordinary elements. Some application examples of the derived conditions are shown in Section 5.
Preliminaries on bivariate subdivision schemes
A bivariate subdivision scheme is an iterative method capable of producing a smooth surface starting from a given coarse polygonal mesh. Unless explicitly specified, we consider level-dependent subdivision schemes, non-stationary subdivision schemes, alternatively. In the following we denote by S a subdivision scheme for meshes of arbitrary topology. For any given initial mesh M (1) of arbitrary topology, we denote by R (1) and E (1) the submeshes of M (1) that determine the behaviour of the limit surface on the one-ring of a regular vertex and an extraordinary element (vertex or face), respectively. The submesh R (1) is also called the neighborhood of a regular vertex. Similarly, the submesh E (1) is also called the neighborhood of an extraordinary element. The action of S on R (1) , defined by 3D control points having components {f (k) α ∈ R, α ∈ Z 2 }, k ∈ N, can be described either by the componentwise application of the refinement rules f
(k+1) α = ∑ β∈Z 2 c (k) α−2β f (k) β , k ∈ N, α ∈ Z 2 ,(1)
or, equivalently, by the application of the sequence of subdivision operators {S c (k) , k ∈ N},
mapping componentwise the vector f (1) = {f (1) α , α ∈ Z 2 } ∈ ℓ(Z 2 )
with the initial control points into the corresponding vector of level k + 1, i.e., (1) .
f (k+1) = S c (k) S c (k−1) ... S c (1) f
(In the sequel we use k ≥ 1 instead of k ∈ N, omitting the trivial information that the refinement level is always assumed to be an integer). The coefficients in (1) can be conveniently collected in the so-called k-th level subdivision mask
c (k) = {c (k) α , α ∈ Z 2 },
or incorporated in the k-th level subdivision symbol
c (k) (z) = ∑ α∈Z 2 c (k) α z α , z ∈ (C\{0}) 2 .
The notation S c (k) ∞ is for the norm of the operator S c (k) , i.e.
S c (k) ∞ := max ∑ β∈Z 2 |c (k) α−2β | : α ∈ {(0, 0), (0, 1), (1, 0), (1, 1)} .(2)
In conclusion, when applied on R (1) , a subdivision scheme S can be equivalently identified by the sequence of subdivision operators {S c (k) , k ≥ 1}, by the sequence of subdivision masks {c (k) , k ≥ 1} or by the sequence of associated subdivision symbols {c (k) (z), k ≥ 1}.
Instead, in E (1) the subdivision rules relating the vertices of the k-th level mesh with those of the next level k + 1 are encoded in the rows of a local subdivision matrix S k . Thus, in the neighborhood of an extraordinary element the action of the subdivision scheme S is described by a sequence of local subdivision matrices {S k , k ≥ 1}.
Remark 2.1. When the valence of the extraordinary element reduces to the regular value, the local subdivision matrix S k provides another alternative way to represent a subdivision step in the regular case.
In the stationary setting we will use the notationS to refer to a subdivision scheme that is not level-dependent. Hence, it will be identified by a subdivision operator, say S c , a subdivision mask c or an associated subdivision symbol c(z), when applied in R (1) ; a local subdivision matrix S, when applied in E (1) .
Preliminaries for studying convergence of subdivision schemes in regular submeshes
In the following, after recalling some well-known definitions, we present several useful results dealing with the convergence of a non-stationary subdivision scheme in R (1) (see [15] for further details).
Definition 2.1. The non-stationary subdivision scheme S applied to the initial data f (1) ∈ ℓ(Z 2 ) is called convergent if there exists a limit function g f (1) ∈ C(R 2 ) (which is nonzero for at least one initial nonzero sequence f (1) ) such that
lim ℓ→+∞ sup α∈Z 2 |g f (1) (2 −ℓ α) − f (ℓ+1) α | = 0.
We call the subdivision scheme S C r -convergent, r ∈ N 0 , if g f (1) ∈ C r (R 2 ). The limit function is often denoted as g f (1) := lim ℓ→+∞ (S c ) ℓ f (1) .
Definition 2.2. Let δ = {δ 0,α , α ∈ Z 2 }.:= lim ℓ→+∞ (S c ) ℓ δ,(3)
is the basic limit function of the subdivision scheme.
Definition 2.3. Let δ = {δ 0,α , α ∈ Z 2 }. For a convergent, non-stationary subdivision scheme S := {S c (ℓ) , ℓ ≥ 1} the function φ k := lim ℓ→+∞ S c (k+ℓ) S c (k+ℓ−1) . . .S c (k) δ, k ≥ 1,(4)
is the k-th member of the family of basic limit functions {φ k , k ≥ 1}.
Definition 2.4. Let S andS be subdivision schemes defined in R (1) by the subdivision masks {c (k) ∈ ℓ(Z 2 ), k ≥ 1} and c ∈ ℓ(Z 2 ), respectively. If
+∞ ∑ k=1 S c (k) − S c ∞ < +∞,(5)
then S andS are said to be asymptotically equivalent schemes.
Remark 2.2. As observed in [7, page 2], (5) holds if and only if
+∞ ∑ k=1 c (k) − c ∞ < +∞ where c ∞ = sup α∈Z 2 |c α |.
Theorem 2.1. [15, Let S andS be asymptotically equivalent subdivision schemes defined in R (1) by the subdivision masks {c (k) ∈ ℓ(Z 2 ), k ≥ 1} and c ∈ ℓ(Z 2 ), respectively. IfS is convergent, then S is also convergent and
lim k→+∞ sup (u,v)∈R 2 |φ k (u, v) − φ(u, v)| = 0,
where φ is the basic limit function ofS defined in (3) and {φ k , k ≥ 1} the family of basic limit functions of S defined in (4).
Remark 2.3. The condition of asymptotical equivalence in (5), that guarantees convergence, could be relaxed by considering the fulfillment of the weaker condition of asymptotical similarity together with approximate sum rules of order 1, as recently shown in [4].
Preliminaries for studying convergence of subdivision schemes in irregular submeshes
In the neighborhood of an extraordinary vertex/face, each step of a subdivision algorithm can be conveniently encoded in the rows of a local subdivision matrix S k relating the vertices of the k-th level mesh with those of the next level. The matrix S k has a different structure depending on the kind of extraordinary element (face or vertex) appearing in the k-th level mesh. Precisely, if the mesh contains an extraordinary face of valence n, in view of the fact that the valence-n extraordinary face is surrounded by n sectors, each composed by p vertices, the local subdivision matrix S k is of the form
S k = B 0,k B 1,k · · · B n−1,k B n−1,k B 0,k · · · B n−2,k . . . . . . . . . . . . B 1,k · · · B n−1,k B 0,k ,(6)
where B i,k ∈ R p×p , i = 0, . . ., n − 1. Thus S k ∈ R N×N with N = pn has a block-circulant structure. For short we write S k := circ(B 0,k , . . ., B n−1,k ).
Remark 2.4. Due to the structure of S k , it is not difficult to prove that
S k ∞ ≤ n−1 ∑ i=0 B i,k ∞ .
If the k-th level mesh contains an extraordinary vertex of valence n, the refinement rules in its neighborhood involve pn + 1 points instead of pn: p points in each of the n sectors plus the extraordinary vertex. Thus, to construct the local subdivision matrix S k we first build the matrixS
k = α kβ T kβ T k · · ·β T k γ kB0,kB1,k · · ·B n−1,k γ kBn−1,kB0,k · · ·B n−2,k . . . . . . . . . . . . . . . γ kB1,k · · ·B n−1,kB0,k ,(7)
whereα k ∈ R,β k ,γ k ∈ R p andB i,k ∈ R p×p , i = 0, . . . , n − 1. Then, following the method shown in [26,Example 5.14], we transform the matrixS k in a block-circulant matrix S k of the form
S k := circ(B 0,k , . . ., B n−1,k ) with B j,k = α k nβ T k γ k nB j,k , j = 0, . . . , n − 1.(8)
It follows that S k ∈ R N×N , with N = n(p + 1), has a block-circulant structure. Hence, without loss of generality, we can always assume that the local subdivision matrix S k has a block-circulant structure with blocks of dimension m × m, where m = p if the k-th level mesh contains an extraordinary face and m = p + 1 if it contains an extraordinary vertex.
We continue by introducing some important notation from [26,27,32]. We start by assuming that near an isolated extraordinary vertex or face of valence n the regular subdivision surface r is defined on the local domain D n := Ω × Z n (consisting of n copies of Ω ) with
Ω := [0, 2] × [0, 2]
in case of quadrilateral mesh, {(u, v) ∈ R 2 | u, v ≥ 0 and 0 ≤ u + v ≤ 2} in case of triangular mesh, and Z n := Z/nZ. In the case of triangular and quadrilateral meshes, if we apply one step of refinement to the local domain D n , we obtain a new domain with 4n cells: 3n outer ordinary cells and n inner cells that contain the extraordinary element. The restriction r 1 of r to the outer cells is called ring. Denoting byr the inner part of r, that isr := r\r 1 , we can repeat the refinement process only forr to obtain a second ring r 2 and an even smaller inner part. Hence, iterated refinement generates a sequence of rings {r k , k ≥ 1} which covers all of the surface except for the central point (limit of the extraordinary vertex or face), that hereinafter we denote by r c . Precisely, assuming the central point to be placed at 0 and introducing the notatioñ
Ω := [0, 1] × [0, 1] in case of quadrilateral mesh, {(u, v) ∈ R 2 | u, v ≥ 0 and 0 ≤ u + v ≤ 1} in case of triangular mesh,
and
Ω k := 2 1−k (Ω \Ω ), D n,k := Ω k × Z n , k ≥ 1,
we see the ring r k as the restriction of the subdivision surface r : D n → R 3 to the set D n,k , i.e., r k := r| D n,k , and the subdivision surface r as (see Figures 2 and 3)
r = k≥1 r k ∪ {r c }.
Specifically, in the case of quadrilateral meshes, the set Ω k is explicitly given by
Ω k = {(u, v) ∈ R 2 | u, v ≥ 0 and 2 1−k ≤ max{u, v} ≤ 2 2−k },
while in the case of triangular meshes
Ω k = {(u, v) ∈ R 2 | u, v ≥ 0 and 2 1−k ≤ u + v ≤ 2 2−k },
(see Figure 4). As a consequence, both in the case of triangular and quadrilateral meshes, the set Ω k is constituted by the union of 3 cells, say ω [1] k , ω [2] k and ω [3] k , implying that the domain D n,k is indeed made of 3n cells. There follows that the entire surface ring r k is the union of 3n patches, each one denoted by r [ j] k and corresponding to the restriction of the regular subdivision surface r to the single cell ω In the following, we denote by d [ j] k ∈ R P×3 , P < N, the vector of control points of each patch r [ j] k , and with Φ
[ j] k ∈ R P the function vector containing all P shifts of the basic limit function φ k , whose support intersects ω [ j] k . We assume that the functions in Φ [ j] k are ordered as the points in the vector d [ j] k , and thus we call them the associated basic limit functions. Thus, for each j ∈ Z 3n , we have
r [ j] k : ω [ j] k → R 3 (u, v) −→ r [ j] k (u, v) = (d [ j] k ) T Φ [ j] k (u, v).(9)
Denoting byΦ [ j] ∈ R P the vector containing all shifts of the basic limit functionφ , whose support intersects ω
[ j] k , if assumptions of Theorem 2.1 are satisfied, we have that lim k→+∞ sup (u,v)∈ω [ j] k Φ [ j] k (u, v) −Φ [ j] (u, v) ∞ = 0, ∀ j ∈ Z 3n . Remark 2.5. Let x 0 := (1, 1, . . . , 1) T ∈ R P . We observe that, for a convergent stationary subdivision schemeS, we have Φ [ j] (u, v) T x 0 = 1 for all (u, v) ∈ R 2 , j ∈ Z 3n . Instead, for a non-stationary subdivision scheme S, Φ [ j] k (u, v) T x 0 = 1 for all k ≥ 1 and for all (u, v) ∈ R 2 , j ∈ Z 3n
if and only if S has the property of stepwise reproduction of constants (see, e.g., [3] for more details). In general, Φ
[ j] k (u, v) T x 0 = α k with α k ∈ R, for all j ∈ Z 3n . Now, let d 1 ∈ R N×3 be the collection of the vectors of control points d [ j] 1 of all patches r [ j]
1 , j ∈ Z 3n . Denoted by {S k ∈ R N×N , k ≥ 1} the matrix sequence that defines a non-stationary subdivision scheme S in E (1) , we can obtain the entire set of the (k + 1)-th level control points representing the whole ring r k+1 by the matrix multiplication
d k+1 = S k d k = S k S k−1 d k−1 = ... = S (k) d 1 with S (k) := S k S k−1 · · · S 1 , k ≥ 1, I, k = 0.(10)
Moreover, denoting by Φ k+1 the function vector with blocks Φ
[ j] k+1 , j ∈ Z 3n , we can rewrite each patch r [ j] k+1 (u, v) = (d [ j] k+1 ) T Φ [ j] k+1 (u, v), (u, v) ∈ ω [ j]
k+1 , of the surface ring r k+1 as
r [ j] k+1 (u, v) = d T k+1 Φ k+1 (u, v),
(i.e. independently of j) since the function vector Φ k+1 ∈ R N indeed contains only P functions that are non-zero on ω
[ j]
k+1 . Exploiting the given definition of r k+1 , we can now provide the following notion of convergence of a non-stationary subdivision scheme S in E (1) .
Definition 2.5. Let S be a (non-stationary) subdivision scheme whose action in
E (1) is described by the matrix sequence {S k ∈ R N×N , k ≥ 1}. Moreover, let d 1 ∈ R N×3 be the vertices of E (1) . S is said to be convergent in E (1) (i.e.,
in the neighborhood of an extraordinary vertex/face of valence n) if, for all bounded initial data d 1 , there exists a limit point
r c ∈ R 3 such that lim k→+∞ sup (u,v)∈D n,k r k (u, v) − r c ∞ = 0.
We conclude by observing that, if the subdivision scheme S converges, then r = k≥1 r k ∪ {r c } is a surface without gap, i.e. r is a surface which is continuous at all points including r c (which is in fact r(0, 0)). In the following we call r the limit surface of the subdivision scheme S .
3 New results related to the C 1 -continuity analysis of subdivision schemes in regular submeshes
The goal of this section is to prove new preliminary results that allow us to derive a general criterion for verifying if the limit surface r generated by an arbitrary non-stationary subdivision scheme is tangent plane continuous at the limit points of extraordinary elements. Tangent plane continuity of r at the limit point r c is a weaker definition of C 1 -continuity at the limit point r c , named G 1 -continuity. It reads as follows.
(u, v) := ∂ u r k (u,v)∧∂ v r k (u,v) ∂ u r k (u,v)∧∂ v r k (u,v) 2 , (u, v) ∈ D n,k , k ≥ 1}, lim k→+∞ sup (u,v)∈D n,k n k (u, v) − n(r c ) ∞ = 0.
We remark that a scheme S which is tangent plane continuous at each limit point is named
G 1 -continuous.
The preliminary results required in Subsection 4.3 to give sufficient conditions for verifying tangent plane continuity at the limit points of extraordinary elements, deal with new results connected with the C 1 -continuity analysis of subdivision schemes in regular submeshes R (1) . For them we recall the well-known notions of asymptotical equivalence of order 1 and divided-difference scheme, plus related results proven in [15].
Definition 3.2. Let S andS be subdivision schemes defined in R (1) by the subdivision masks {c (k) ∈ ℓ(Z 2 ), k ≥ 1} and c ∈ ℓ(Z 2 ), respectively. If +∞ ∑ k=1 2 k S c (k) − S c ∞ < +∞,
then S andS are said to be asymptotically equivalent schemes of order 1.
(∆ (ℓ) e j f (ℓ) ) α := f (ℓ) α − f (ℓ) α−e j 2 −ℓ , α ∈ Z 2 , j ∈ {1, 2}, ℓ ≥ 1, the e j -directional divided-difference operator. Lemma 3.1. Let j ∈ {1, 2}. If c (ℓ) (z) = 1 2 (1 + z j )b (ℓ) e j (z), then ∆ (ℓ+1) e j f (ℓ+1) (z) = b (ℓ) e j (z) ∆ (ℓ) e j f (ℓ) (z 2 ), and {S b (ℓ) e j , ℓ ≥ 1} is called the e j -directional divided difference scheme of {S c (ℓ) , ℓ ≥ 1}. Proof. Introducing the notation f (ℓ) (z) = ∑ α∈Z 2 f (ℓ) α z α , where the power is intended com- ponentwise, we can write ∆ (ℓ+1) e j f (ℓ+1) (z) = 1 2 −(ℓ+1) (1 − z j ) f (ℓ+1) (z).(11)
Since
f (ℓ+1) (z) = c (ℓ) (z) f (ℓ) (z 2 ), in view of the factorized form of c (ℓ) (z), we obtain ∆ (ℓ+1) e j f (ℓ+1) (z) = 1 2 −ℓ (1 − z j 2 ) b (ℓ) e j (z) f (ℓ) (z 2 ).
Now, taking into account equation (11), the claim follows.
⊓ ⊔ From Lemma 3.1 we have that ∆ (ℓ+1) e j f (ℓ+1) = S b (ℓ) e j ∆ (ℓ) e j f (ℓ) ⇔ ∆ (ℓ+1) e j S c (ℓ) f (ℓ) = S b (ℓ) e j ∆ (ℓ) e j f (ℓ) .(12)
Lemma 3.2. Let S andS be subdivision schemes defined in R (1) by the subdivision symbols {c (k) (z), k ≥ 1} and c(z), respectively. Assume that:
i) S andS are asymptotically equivalent of order 1; ii) the factor (1 + z 1 )(1 + z 2 ) is contained in the symbols c(z) and c (k) (z), for all k ≥ 1.
Then, the divided difference schemes with symbols b e j (z)
:= 2c(z) 1+z j , j ∈ {1, 2} and b (k) e j (z) := 2c (k) (z) 1+z j , j ∈ {1, 2}
, are asymptotically equivalent of order 1.
Proof. We only consider the case corresponding to j = 1, since the case j = 2 can be treated analogously. To simplify the notation we denote b e 1 (z) and b (k) e 1 (z) by b(z) and b (k) (z), respectively. We start by considering the relation
2c(z) = (1 + z 1 )b(z) with c(z) := ∑ α∈[0,N 1 ]×[0,N 2 ] c α z α and b(z) := ∑ α∈[0,N 1 −1]×[0,N 2 ] b α z α .
Comparing the same power of z we easily see that,
c 0,α 2 = 1 2 b 0,α 2 , c N 1 ,α 2 = 1 2 b N 1 −1,α 2 , c α = 1 2 (b α + b α−e 1 ) , α ∈ [1, N 1 − 1] × [0, N 2 ], which means b 0,α 2 = 2c 0,α 2 , b N 1 −1,α 2 = 2c N 1 ,α 2 , b α = 2 α 1 ∑ β 1 =0 (−1) α 1 −β 1 c β 1 ,α 2 , α ∈ [1, N 1 − 2] × [0, N 2 ].
Analogously, working with the relation 2c
(k) (z) = (1 + z 1 )b (k) (z) we get b (k) 0,α 2 = 2c (k) 0,α 2 , b (k) N 1 −1,α 2 = 2c (k) N 1 ,α 2 , b (k) α = 2 α 1 ∑ β 1 =0 (−1) α 1 −β 1 c (k) β 1 ,α 2 , α ∈ [1, N 1 − 2] × [0, N 2 ].
Therefore,
b (k) − b ∞ ≤ 2N 1 c (k) − c ∞ and +∞ ∑ k=1 2 k b (k) − b ∞ ≤ 2N 1 +∞ ∑ k=1 2 k c (k) − c ∞ < ∞.
Thus, in light of Remark 2.2, the result is proven.
⊓ ⊔
The previous Lemma is useful for the next result.
Proposition 3.1. Let S andS be subdivision schemes defined in R (1) . Assume that:
i) S andS are asymptotically equivalent of order 1. ii)S is C 1 -convergent in R (1) , with symbol c(z) that contains the factor (1 + z 1 )(1 + z 2 ); iii) S is defined in R (1) by the subdivision symbols {c (ℓ) (z), ℓ ≥ 1} all containing the factor (1 + z 1 )(1 + z 2 ).
Then, the associated divided difference schemes with symbols b e j (z) := 2c(z) 1+z j and b , ℓ ≥ 1} converges uniformly to the basic
limit function of {S b e j }; b) lim ℓ→+∞ S b (k+ℓ) e j S b (k+ℓ−1) e j . . . S b (k) e j ∆ (k) e j δ = ∂ e j φ k and lim ℓ→+∞ S ℓ+1 b e j ∆(1)e j δ = ∂ e j φ ,
for δ = {δ 0,α , α ∈ Z 2 } and with φ k defined as in (4) and φ as in (3).
Proof. The result in a) is a direct consequence of Lemma 3.2 and Theorem 2.1 (see also [15,Lemma 15]).
To show b) we proceed as follows. In view of the factorization properties of c (ℓ) (z), we can apply Lemma 3.1 to conclude the existence of the e j -directional divided difference scheme of order 1 of {S c (ℓ) , ℓ ≥ 1}. Then, to show convergence of the e j -directional divided difference scheme of order 1, we just recall the result in a). Next, we exploit (12) and
write S b (k) e j ∆ (k) e j δ = ∆ (k+1) e j S c (k) δ, so that S b (k+ℓ) e j S b (k+ℓ−1) e j . . . S b (k) e j ∆ (k) e j δ = ∆ (k+ℓ+1) e j S c (k+ℓ) S c (k+ℓ−1) . . . S c (k) δ.
Moreover, introducing the notation δ (ℓ+1) := S c (k+ℓ) S c (k+ℓ−1) . . . S c (k) δ, we have that
∆ (k+ℓ+1) e j δ (ℓ+1) = δ (ℓ+1) − (δ (ℓ+1) ) ·−e j 2 −k−ℓ−1 , j ∈ {1, 2}. Thus lim ℓ→+∞ S b (k+ℓ) e j S b (k+ℓ−1) e j . . . S b (k) e j ∆ (k) e j δ = lim ℓ→+∞ ∆ (k+ℓ+1) e j S c (k+ℓ) S c (k+ℓ−1) . . . S c (k) δ = lim ℓ→+∞ S c (k+ℓ) S c (k+ℓ−1) ... S c (k) δ− S c (k+ℓ) S c (k+ℓ−1) ... S c (k) δ ·−e j 2 −k−ℓ−1 = ∂ e j φ k , in view of the fact that lim ℓ→+∞ S c (k+ℓ) S c (k+ℓ−1) . . . S c (k) δ = φ k and φ k is C 1 .
The result for the stationary scheme follows by taking S c (ℓ) = S c for all ℓ ≥ 1 and using Theorem 2.1.
⊓ ⊔
As a consequence of the previous Proposition we have
(u,v)∈R 2 |∂ e j φ k (u, v) − ∂ e j φ(u, v)| = 0, j ∈ {1, 2}.
Analysis of (non-stationary) subdivision schemes in irregular submeshes
Before focusing on the sufficient conditions that guarantee the convergence of a nonstationary subdivision scheme in E (1) (Theorem 4.1), we present a few linear algebra results to be used for the subdivision analysis.
Auxiliary linear algebra results
Let M ∈ R N×N . In the following, two simple results based on the Jordan decomposition of M are proven. In the first one we assume d ∈ R N×3 and consider the sequence {M k d, k ≥ 0}. Then, under suitable assumptions on the matrix M, we show its convergence. In the second one (which is a well known result so that we omit its proof) we study the properties of M k , k ≥ 0, again with the help of its Jordan decomposition.
with q T =x T d ∈ R 1×3 ,x T = e T 1 X −1 ∈ R 1×N and e T 1 = (1, 0, ..., 0) ∈ R 1×N . Moreover, x T M =x T .(13)
Proof. Using the Jordan decomposition of M and introducing the notation w := X −1 d, we can write M k d = XJ k w. Hence, recalling that 1 is the unique dominant eigenvalue of J and the associated eigenvector is x = (1, 1, ..., 1) see, e.g., [26,27,35].
T , we have lim k→+∞ M k d = lim k→+∞ XJ k w = X lim k→+∞ J k w = X 1 · · · 0 0 0 0 · · · 0 . . . . . . . . . . . . 0 · · · · · · 0 w = xq T , with q T =x T d andx T = e T 1 X −1 . To conclude we observe thatx T M =x T XJX −1 = e T 1 JX −1 = e T 1 X −1 =x T ,
In the next Proposition we replace the k-th power of the matrix M with the product of k different matrices M k M k−1 · · · M 1 and we successively consider hybrid combinations of the two.
Moreover, in view of the fact that
M j − M M ≤ C M j − M ≤ C C σ j ,
where C is the finite positive constant appearing in the assumption, we can finally arrive at the following result for any arbitrary k ≥ 1:
M (k) ≤ ∏ k j=1 M j M ≤ k ∏ j=1 ( M M + M j − M M ) ≤ k ∏ j=1 1 + C C σ j = e log e ∏ k j=1 1+ C C σ j = e ∑ k j=1 log e 1+ C C σ j ≤ e ∑ k j=1 C C σ j ≤ e C C ∑ +∞ j=1 1 σ j ,
where the last but one inequality follows from the fact that log e (1 + x) ≤ x for all x ≥ 0. Since ∑ +∞
M (k) = M k + k ∑ j=1 M k− j (M j − M)M ( j−1) , for all k ≥ 1. Proof. Assuming ∑ k−1 j=1 M k− j (M j − M)M ( j−1) to be 0 when k = 1, we can write M k + k ∑ j=1 M k− j (M j − M)M ( j−1) = M k + k−1 ∑ j=1 M k− j (M j − M)M ( j−1) + (M k − M)M (k−1) = M k + M (k) + k−1 ∑ j=1 M k− j M ( j) − MM (k−1) − k−1 ∑ j=1 M k− j+1 M ( j−1) = M k + M (k) + k−2 ∑ j=1 M k− j M ( j) − k−2 ∑ j=0 M k− j M ( j) = M k + M (k) − M k = M (k) ,
so concluding the proof. ⊓ ⊔
Convergence analysis in irregular submeshes
In this section we make use of the previous linear algebra results to provide sufficient conditions for establishing the convergence of a non-stationary subdivision scheme S defined in irregular submeshes E (1) by the matrix sequence {S k ∈ R N×N , k ≥ 1}. With the notation previously introduced, let d 1 ∈ R N×3 be the collection of the vectors of control points d 1 , j ∈ Z 3n . According to Definition 2.5, our goal is to study the convergence of the sequence of regular rings {r k+1 , k ≥ 0} whose patches r [ j] k+1 are described by the equation r
[ j] k+1 = (d [ j] k+1 ) T Φ [ j] k+1 = d T k+1 Φ k+1 , j ∈ Z 3n
. According to (10), the entire set of the (k + 1)-th level control points d k+1 representing the whole ring r k+1 is given by the matrix multiplication
d k+1 = S (k) d 1 with S (k) := S k S k−1 · · · S 1 , k ≥ 1, I, k = 0.
The key idea to prove convergence of S is to write the product matrix S (k) in terms of the stationary matrix S k . Indeed, from Proposition 4.4 we write
d k+1 = S k d 1 + y k with y k := k ∑ j=1 S k− j (S j − S)S ( j−1) d 1 ,(14)
and then show our first main result. (i)S is convergent both in R (1) and in E (1) , (ii) S is asymptotically equivalent toS in R (1) , (iii) in E (1) the matrices S k and S satisfy, for all k ≥ 1, S k − S ∞ ≤ C σ k with C some finite positive constant and σ > 1.
Then, for all bounded initial data d 1 ∈ R N×3 , the non-stationary subdivision scheme S is convergent also in E (1) . In particular,
lim k→+∞ sup (u,v)∈Ω k+1 r k+1 (u, v) − (q 0 + β 0 ) ∞ = 0, where -q 0 = d T 1x 0 ∈ R 3 withx 0 such that S Tx 0 =x 0 , -β 0 = (lim k→+∞ y k ) T x 0 x T 0 x 0 ∈ R 3 for y k = k ∑ j=1 S k− j (S j − S)S ( j−1) d 1 and x 0 such that Sx 0 = x 0 .
Proof. The proof follows the line of reasoning of the proof of [15,Theorem 6]. For
d 1 ∈ R N×3 we define u k+1,ℓ := S ℓ S (k) d 1 , ℓ ≥ 0, k ≥ 0.
From assumption (i) we know that lim ℓ→+∞ u k+1,ℓ =: u k+1 , k ≥ 0 namely ∀ε > 0 there exists L ∈ N such that u k+1,ℓ −u k+1 ∞ < ε for all ℓ > L. We continue by proving that the sequence {u k+1 , k ≥ 0} is a Cauchy sequence. Indeed, in view of Proposition 4.2, Proposition 4.3 and assumption (iii) we have
u k+1 − u k ∞ = lim ℓ→+∞ S ℓ (S k − S)S (k−1) d 1 ∞ ≤C S k − S ∞ d 1 ∞ ≤C σ k ,
and thus, for s ≥ 1,
u k+s − u k ∞ ≤ s ∑ j=1 u k+ j − u k+ j−1 ∞ ≤Ĉ σ k s−1 ∑ j=0 1 σ j(15)
withC ,C andĈ finite positive constants. From above we can conclude the existence of the vector u := lim k→+∞ u k and write that ∀ε > 0 there exists N ∈ N such that
u k+1,ℓ − u ∞ ≤ u k+1,ℓ − u k+1 ∞ + u k+1 − u ∞ < ε, ∀k, ℓ > N. Now, using the notation d k,ℓ := S (k+ℓ) d 1 = S k+ℓ · · · S 1 d 1 we estimate d k,ℓ − u ∞ as d k,ℓ − u ∞ ≤ d k,ℓ − u k+1,ℓ ∞ + u k+1,ℓ − u ∞ .
We then write
d k,ℓ − u k+1,ℓ ∞ = S k+ℓ · · · S k+1 − S ℓ S k · · · S 1 d 1 ∞ ≤ C S k+ℓ · · · S k+1 − S ℓ ∞
with C a finite positive constant. In view of Proposition 4.4 (with S k+1 playing the role of M 1 ), using again (iii) we arrive at
d k,ℓ − u k+1,ℓ ∞ ≤C k+ℓ ∑ j=k+1 S k+ℓ− j (S j − S)S ( j−1) d 1 ∞ ≤Ĉ k+ℓ ∑ j=k+1 1 σ j ,(16)
where againC ,Ĉ are finite positive constants. Since the last sum in (16) can be made arbitrarily small, we can conclude that, ∀ε > 0 there exists N ∈ N such that for all k, ℓ > N
d k,ℓ − u ∞ < ε.
As a consequence, ∀ε > 0 there exists N ∈ N such that for all m = k + ℓ > 2N + 1
S (m) d 1 − u ∞ < ε, that is lim m→+∞ S (m) d 1 = u.(17)
We continue by showing that the vector u is an eigenvector of S associated with the eigenvalue 1 (i.e., Su = u). Indeed, observing that Su k+1,ℓ = u k+1,ℓ+1 we write
Su − u ∞ ≤ Su − Su k+1,ℓ ∞ + u k+1,ℓ+1 − d k,ℓ+1 ∞ + d k,ℓ+1 − u ∞ ,
with the right hand side that tends to 0 for k and ℓ going to +∞. In view of assumption (i) and (17), we can thus conclude convergence of the sequence
{y k , k ≥ 0}, with y k := S (k) d 1 − S k d 1 = d k+1 − S k d 1 = k ∑ j=1 S k− j (S j − S)S ( j−1) d 1 .
Moreover, denoting y := lim k→+∞ y k , from the fact that Su = u we can also conclude that Sy = y, which means that y lies in the eigenspace corresponding to the right eigenvector of S associated to the eigenvalue λ 0 = 1. There follows that y is of the form y = x 0 β T 0 with x 0 = (1, 1, ..., 1) T , which implies that β 0 can be written as β 0 = y T x 0
x T 0 x 0 .
From (14) we then write
lim k→+∞ d k+1 = lim k→+∞ S k d 1 + x 0 β T 0 ,(18)
and in view of Proposition 4.1, after replacing (13) in equation (18), we arrive at
lim k→+∞ d k+1 = x 0 (q 0 + β 0 ) T , with q 0 = d T 1x0 .(19)
Then, taking into consideration assumption (ii) and Theorem 2.1, we have that
lim k→+∞ sup (u,v)∈Ω k+1 Φ k+1 (u, v) − Φ(u, v) ∞ = 0.(20)
After recalling that Φ(u, v) T x 0 = 1 for all (u, v) ∈ Ω k+1 (in light of the arguments in Remark 2.5), we continue by writing, for all j ∈ Z 3n , (20)), sup
sup (u,v)∈ω [ j] k+1 r [ j] k+1 (u, v) T − (q 0 + β 0 ) T ∞ = sup (u,v)∈ω [ j] k+1 r [ j] k+1 (u, v) T − Φ(u, v) T x 0 (q 0 + β 0 ) T ∞ = sup (u,v)∈ω [ j] k+1 Φ k+1 (u, v) T d k+1 − Φ(u, v) T x 0 (q 0 + β 0 ) T ∞ = sup (u,v)∈ω [ j] k+1 Φ k+1 (u, v) T d k+1 − Φ k+1 (u, v) T x 0 (q 0 + β 0 ) T + Φ k+1 (u, v) T x 0 (q 0 + β 0 ) T − Φ(u, v) T x 0 (q 0 + β 0 ) T ∞ ≤ sup (u,v)∈ω [ j] k+1 Φ k+1 (u, v) T ∞ d k+1 − x 0 (q 0 + β 0 ) T ∞ + sup (u,v)∈ω [ j] k+1 Φ k+1 (u, v) T − Φ(u, v) T ∞ x 0 (q 0 + β 0 ) T ∞ . Since lim k→+∞ d k+1 − x 0 (q 0 + β 0 ) T ∞ = 0 (in light of (19)), lim k→+∞ sup (u,v)∈Ω k+1 Φ k+1 (u, v) T − Φ(u, v) T ∞ = 0 (in light of(u,v)∈Ω k+1 Φ k+1 (u, v
) T ∞ is uniformly bounded and
x 0 (q 0 + β 0 ) T ∞ is bounded, we finally obtain
lim k→+∞ sup (u,v)∈ω [ j] k+1 r [ j] k+1 (u, v) T − (q 0 + β 0 ) T ∞ = 0, ∀ j ∈ Z 3n ,
which concludes the proof. ⊓ ⊔ Now, following the notation in [26], we denote with λ r , r = 0, . . . , r, 0 ≤ r ≤ N − 1, the r + 1 different eigenvalues of S ∈ R N×N sorted in decreasing order according to their magnitude, i.e. |λ 0 | ≥ |λ 1 | ≥ . . . ≥ |λ r |. Moreover, for ℓ r ≥ 0 we denote by ℓ r + 1, the algebraic multiplicity of λ r . As emphasized in Remark 4.1, it is a known fact that, for a convergent stationary schemeS identified by {S}, all r + 1 eigenvalues have magnitude less than 1, except λ 0 which is required to be exactly 1 and with algebraic and geometric multiplicity 1. It means that 1 = λ 0 > |λ 1 | ≥ . . . ≥ |λ r | and ℓ 0 = 0. Moreover, the eigenvector associated to the unique dominant eigenvalue λ 0 = 1 is required to be x 0 = (1, 1, ..., 1) T ∈ R N (see, e.g., [26,27,35]). Thus, exploiting the Jordan decomposition of S k and the equality S k d 1 = XJ k w with w = X −1 d 1 , for k sufficiently large we can write
S k d 1 = x 0 q T 0 + O(|λ 1 | k ),(21)
where, with a slight abuse of notation, O(|λ 1 | k ) denotes a vector in R N×3 with all its entries behaving as O(|λ 1 | k ). This vector notation is assumed from now on. Equation (21) implies the following convergence rate result for the sequence {y k , k ≥ 0}.
S (k) d 1 = u + O 1 σ k .(22)
Thus
y k = x 0 β T 0 + O 1 σ k .(23)
Proof. We use the notation of the proof of Theorem 4.1 and start proving (22). First we write
u − S (k+ℓ) d 1 = (u − u k+1 ) + (u k+1 − u k+1,ℓ ) + (u k+1,ℓ − S (k+ℓ) d 1 ).
Then, by (15), (16) and (21) we obtain
u − S (k+ℓ) d 1 ∞ ≤ u − u k+1 ∞ + u k+1 − u k+1,ℓ ∞ + u k+1,ℓ − S (k+ℓ) d 1 ∞ ≤ C 1 |λ 1 | ℓ + C 2 σ k ,
with C 1 , C 2 finite positive constants. Since |λ 1 | < 1, we can findL such that |λ 1 | ℓ < 1 σ k for all ℓ >L. Therefore,
u − S (k+L+1) d 1 ∞ ≤ C 3 σ k ,
with C 3 a finite positive constant. Hence, taking the limit to +∞ with respect to k, (22) follows. Similarly, to prove the result in (23) we write
y k+ℓ − x 0 β T 0 = (S (k+ℓ) d 1 − u) − (S k+ℓ d 1 − x 0 q T 0 )
and consider the triangular inequality
y k+ℓ − x 0 β T 0 ∞ ≤ S (k+ℓ) d 1 − u k+1,ℓ ∞ + u k+1,ℓ − u k+1 ∞ + u k+1 − u ∞ + S k+ℓ d 1 − x 0 q T 0 ∞ .
Using again (15), (16) and (21), we obtain the upper bound
y k+ℓ − x 0 β T 0 ∞ ≤C 1 |λ 1 | ℓ +C 2 σ k +C 3 |λ 1 | k+ℓ .
Then, applying the same reasoning as before, (23) is proven. ⊓ ⊔
Tangent plane continuity analysis at the limit point of an extraordinary element
Aim of this section is to provide sufficient conditions to show that a convergent, nonstationary subdivision scheme S produces tangent plane continuous limit surfaces at the limit point. We recall that, if a symmetric stationary subdivision schemeS is tangent plane continuous, the ordered eigenvalues of the associated local subdivision matrix S satisfy
1 = λ 0 > λ 1 > |λ 2 | with λ 1 ∈ R + , ℓ 1 = 1,
i.e. the sub-dominant eigenvalue λ 1 is real, double and with geometric multiplicity equal to algebraic multiplicity. In this case, the eigenvectors associated to λ 1 are linearly independent. In the following we denote by x 0 = (1, 1, ..., 1) T ∈ R N the eigenvector associated to λ 0 = 1, and by x 0 1 , x 1 1 ∈ R N the two linearly independent eigenvectors associated to λ 1 . It is a well-known fact that, if the stationary schemeS produces tangent plane continuous limit surfaces, then its characteristic map is regular (see [26] for details), i.e. the planar ring defined by Ψ (u, v) T
= Φ(u, v) T x 0 1 , x 1 1 ∈ R 1×2 , where Φ(u, v)
∈ R N is the corresponding basic limit function vector, is such that det(J Ψ (u, v) T ) is non zero and of constant sign with
J Ψ (u, v) T := ∂ u Ψ (u, v) T ∂ v Ψ (u, v) T ∈ R 2×2 , (u, v) ∈ Ω 1 .
In the following we provide sufficient conditions to show that a non-stationary subdivision scheme S produces tangent plane continuous limit surfaces at the limit point r c = q 0 + β 0 (see Definition 3.1).
Theorem 4.2. Let S be a non-stationary subdivision scheme whose action in E (1) is described by the matrix sequence {S k , k ≥ 1}. Moreover, letS be a stationary subdivision scheme that in E (1) is identified by {S}. Assume that:
(i)S is C 1 -convergent in R (1) with symbol c(z) containing the factor (1+z 1 )(1+z 2 ), and G 1 -convergent in E (1) ;
(ii) S is defined in R (1) by the symbols {c (k) (z), k ≥ 1} where each c (k) (z) contains the factor (1 + z 1 )(1 + z 2 ); (iii) S is asymptotically equivalent of order 1 toS in R (1) ; (iv) in E (1) the matrices S k and S satisfy, for all k ≥ 1, S k − S ∞ ≤ C σ k with C some finite positive constant, σ > 1 λ 1 > 1 with λ 1 ∈ R + the subdominant eigenvalue of S which is double and non-defective.
Then the subdivision surface generated by S is tangent plane continuous at the limit point r c .
Proof. First we observe that from (i) and Remark 4.1 the matrix S has a simple dominant eigenvalue λ 0 = 1. Also, from Theorem 3.1 we know that S is C 1 -convergent in R (1) and from Theorem 4.1 we also know that S is convergent in E (1) . From Definition 2.5 it follows that, if r k+1 is the regular patches ring near an extraordinary vertex/face and if {d k+1 , k ≥ 0} is the sequence of control points related to r k+1 , then the two sequences {d k+1 , k ≥ 0} and {r k+1 , k ≥ 0} converge. In particular, in view of Proposition 4.4, we know that
d k+1 = S k d 1 + y k with y k = k ∑ j=1 S k− j (S j − S)S ( j−1) d 1 .
To simplify the following analysis, we do not consider the full expression of a sequence of rings, but only the asymptotic behavior of the dominant terms as k tends to infinity. Since the subdivision surface generated by the stationary schemeS is tangent plane continuous at its limit point (assumption (i)), the eigenvalues of S satisfy 1 = λ 0 > λ 1 > |λ i |, i = 2, . . . , r and the sub-dominant eigenvalue λ 1 has geometric multiplicity and algebraic multiplicity two [26]. Thus, exploiting the Jordan decomposition of S k and the equality S k d 1 = XJ k w with w = X −1 d 1 , for k sufficiently large we can write that
S k d 1 = x 0 q T 0 + λ k 1 (x 0 1 (q 0 1 ) T + x 1 1 (q 1 1 ) T ) + o(λ k 1 )
with x 0 1 and x 1 1 denoting the two linearly independent eigenvectors associated to λ 1 and q 0 1 , q 1 1 two vectors in R 3 . Since in view of Corollary 4.1 we also have that
y k = x 0 β T 0 + O 1 σ k , we arrive at d k+1 = x 0 (q T 0 + β T 0 ) + λ k 1 (x 0 1 (q 0 1 ) T + x 1 1 (q 1 1 ) T ) + θ k ,(24)
where θ k denotes a vector in R N×3 with all its entries behaving as o(λ k 1 ) + O 1 σ k . Parameterizing the regular patches ring r k+1 using the basic limit function vector Φ k+1 , we can write (r [ j] k+1 ) T , for each j ∈ Z 3n , as (cf. equation (9)
) (r [ j] k+1 (u, v)) T = Φ T k+1 (u, v)d k+1 , (u, v) ∈ ω [ j] k+1 , j ∈ Z 3n .
Using Remark 2.5 and introducing the notation α k+1 for the value Φ k+1 (u, v) T x 0 ∈ R, thanks to (24), we have
(r [ j] k+1 (u, v)) T = Φ k+1 (u, v) T x 0 (q T 0 + β T 0 ) + λ k 1 (x 0 1 (q 0 1 ) T + x 1 1 (q 1 1 ) T ) + θ k = α k+1 (q T 0 + β T 0 ) + λ k 1 Φ k+1 (u, v) T (x 0 1 (q 0 1 ) T + x 1 1 (q 1 1 ) T ) + Φ k+1 (u, v) T θ k .
To verify the tangent plane continuity of the limit surface at the limit point r c = q 0 + β 0 , we first observe that ∂ u α k+1 = ∂ v α k+1 = 0, and then write
∂ u (r [ j] k+1 (u, v)) T = ∂ u α k+1 (q T 0 + β T 0 ) + λ k 1 Φ k+1 (u, v) T (x 0 1 (q 0 1 ) T + x 1 1 (q 1 1 ) T ) + Φ k+1 (u, v) T θ k = λ k 1 ∂ u Φ k+1 (u, v) T (x 0 1 (q 0 1 ) T + x 1 1 (q 1 1 ) T ) + ∂ u Φ k+1 (u, v) T θ k = λ k 1 ∂ u Φ k+1 (u, v) T (x 0 1 (q 0 1 ) T + x 1 1 (q 1 1 ) T ) + θ k λ k 1 and ∂ v (r [ j] k+1 (u, v)) T = ∂ v α k+1 (q T 0 + β T 0 ) + λ k 1 Φ k+1 (u, v) T (x 0 1 (q 0 1 ) T + x 1 1 (q 1 1 ) T ) + Φ k+1 (u, v) T θ k = λ k 1 ∂ v Φ k+1 (u, v) T (x 0 1 (q 0 1 ) T + x 1 1 (q 1 1 ) T ) + ∂ v Φ k+1 (u, v) T θ k = λ k 1 ∂ v Φ k+1 (u, v) T (x 0 1 (q 0 1 ) T + x 1 1 (q 1 1 ) T ) + θ k λ k 1 .
Therefore, in order to study the evolution of the direction of the normal vectors to the j-th surface patch we write
∂ u (r [ j] k+1 (u, v)) T ∧ ∂ v (r [ j] k+1 (u, v)) T = λ 2k 1 ∂ u Φ k+1 (u, v) T (x 0 1 (q 0 1 ) T + x 1 1 (q 1 1 ) T ) + θ k λ k 1 ∧ ∂ v Φ k+1 (u, v) T (x 0 1 (q 0 1 ) T + x 1 1 (q 1 1 ) T ) + θ k λ k 1
, and, with the help of the formula
(k 1 a + k 2 b) ∧ (h 1 a + h 2 b) = det k 1 k 2 h 1 h 2 (a ∧ b), for k 1 , k 2 , h 1 , h 2 ∈ R and a, b ∈ R 3 ,
we arrive at
∂ u (r [ j] k+1 (u, v)) T ∧ ∂ v (r [ j] k+1 (u, v)) T = λ 2k 1 det(J Ψ k+1 (u, v) T )((q 0 1 ) T ∧ (q 1 1 ) T ) +(∂ u Φ k+1 (u, v) T θ k λ k 1 ) ∧ ∂ v Φ k+1 (u, v) T (x 0 1 (q 0 1 ) T + x 1 1 (q 1 1 ) T ) +∂ u Φ k+1 (u, v) T (x 0 1 (q 0 1 ) T + x 1 1 (q 1 1 ) T ) ∧ (∂ v Φ k+1 (u, v) T θ k λ k 1 ) +(∂ u Φ k+1 (u, v) T θ k λ k 1 ) ∧ (∂ v Φ k+1 (u, v) T θ k λ k 1 ) , where Ψ k+1 (u, v) T := Φ k+1 (u, v) T x 0 1 , x 1 1 and J Ψ k+1 (u, v) T := ∂ u Ψ k+1 (u, v) T ∂ v Ψ k+1 (u, v) T = ∂ u Φ k+1 (u, v) T x 0 1 ∂ u Φ k+1 (u, v) T x 1 1 ∂ v Φ k+1 (u, v) T x 0 1 ∂ v Φ k+1 (u, v) T x 1 1 .
Since in our notation the partial derivatives ∂ u and ∂ v are directional derivatives with respect to the two perpendicular axes directions e 1 and e 2 , using assumptions (i), (ii), (iii) we
have that ∂ u (Φ [ j] k+1 (u, v)) T and ∂ v (Φ [ j]
k+1 (u, v)) T are uniformly bounded in view of Corollary 3.1. Recalling also that θ k denotes a vector in R N×3 with all its entries behaving as o(λ k 1 ) + O 1 σ k and λ 1 σ > 1 (in light of assumption (iv)), this allows us to obtain
∂ u (r [ j] k+1 (u, v)) T ∧ ∂ v (r [ j] k+1 (u, v)) T =λ 2k 1 det(J Ψ k+1 (u, v) T )((q 0 1 ) T ∧ (q 1 1 ) T ) + o(1) , (u, v) ∈ ω [ j] k+1 , j ∈ Z 3n .
Therefore, when computing the sequence {n
[ j] k+1 (u, v), (u, v) ∈ ω [ j]
k+1 , k ≥ 0} of normal vectors to the j-th surface patch we obtain
n [ j] k+1 (u, v) = ∂ u (r [ j] k+1 (u, v)) T ∧ ∂ v (r [ j] k+1 (u, v)) T ∂ u (r [ j] k+1 (u, v)) T ∧ ∂ v (r [ j] k+1 (u, v)) T 2 = det(J Ψ k+1 (u, v) T )((q 0 1 ) T ∧ (q 1 1 ) T ) + o(1) det(J Ψ k+1 (u, v) T )((q 0 1 ) T ∧ (q 1 1 ) T ) + o(1) 2 = sign(det(J Ψ k+1 (u, v) T )) ((q 0 1 ) T ∧ (q 1 1 ) T ) + o(1) ((q 0 1 ) T ∧ (q 1 1 ) T ) + o(1) 2 , (u, v) ∈ ω [ j] k+1 , j ∈ Z 3n .
Again in view of Proposition 3.1 and Corollary 3.1 we can write
lim k→+∞ sup (u,v)∈Ω k+1 ∂ u Φ k+1 (u, v) T − ∂ u Φ(u, v) T ∞ = 0, and lim k→+∞ sup (u,v)∈Ω k+1 ∂ v Φ k+1 (u, v) T − ∂ v Φ(u, v) T ∞ = 0.
From the latter we obtain
lim k→+∞ sup (u,v)∈Ω k+1 J Ψ k+1 (u, v) T − J Ψ (u, v) T ∞ = 0 and also lim k→+∞ sup (u,v)∈Ω k+1 | det(J Ψ k+1 (u, v) T ) − det(J Ψ (u, v) T )| = 0.
Therefore, taking into account that
lim k→+∞ sup (u,v)∈Ω k+1 r k+1 (u, v) − r c ∞ = 0 and n(r c ) := sign(det(J Ψ (0, 0) T ) (q 0 1 ) T ∧ (q 1 1 ) T (q 0 1 ) T ∧ (q 1 1 ) T 2 ,
it has been proven that, for all j ∈ Z 3n ,
lim k→+∞ sup (u,v)∈ω [ j] k+1 n [ j] k+1 (u, v) − n(r c ) ∞ = 0.
Hence the limit surface r obtained by the non-stationary subdivision scheme S is tangent plane continuous at the limit point r c = r(0, 0). ⊓ ⊔
Application examples
In this section we use Theorem 4.1 to study the convergence of two non-stationary subdivision schemes in the neighborhood of extraordinary elements. Also, we use Theorem 4.2 to prove that the limit surfaces obtained by such schemes are tangent plane continuous at the limit points of the corresponding extraordinary elements. This proves a conjecture given in [17,Section 5] where only numerical evidence was shown. Both examples deal with approximating subdivision schemes defined on quadrilateral meshes of arbitrary topology. As a matter of fact, although our theorems apply also to non-stationary interpolatory schemes and to schemes defined on triangular meshes, we are only aware of the above examples of non-stationary subdivision schemes.
Generalized trigonometric spline surfaces of order 3
In [21], the authors presented a non-stationary subdivision scheme which produces tensorproduct trigonometric spline surfaces of order 3 except in the neighborhood of extraordinary faces. This non-stationary scheme can be seen as a generalization of the well-known stationary Doo-Sabin's scheme [13] yielding polynomial spline surfaces of order 3 except in the neighborhood of extraordinary faces.
,(25)
while in E (1) the refinement rules are written in terms of a subdivision matrix S having the structure in (6)
It is a well-known fact that Doo-Sabin's scheme is convergent both in R (1) and in E (1) , and the limit surface is C 1 -continuous in correspondence to regular regions of the mesh and tangent plane continuous at the limit points of extraordinary faces. Moreover, in R (1) the associated subdivision symbol is
c(z 1 , z 2 ) = (z 1 + 1) 3 (z 2 + 1) 3 16 ,
namely it contains the factor (1 + z 1 )(1 + z 2 ). Thus it verifies assumption (i) of Theorem 4.1 and assumption (i) of Theorem 4.2.
The non-stationary scheme in [21], is described in R (1) by the k-level mask (k ≥ 1)
c (k) = c 4,k b k + c 4,k b k + c 4,k c 4,k b k + c 4,k a k + c 4,k a k + c 4,k b k + c 4,k b k + c 4,k a k + c 4,k a k + c 4,k b k + c 4,k c 4,k b k + c 4,k b k + c 4,k c 4,k ,(27)
where
c n,k = 1 4n cos 2 h 2 k cos 2 h 2 k−1 , n ∈ N, n ≥ 4, h ∈ 0, π 3 , k ≥ 1, and a k = 1 4cos 2 h 2 k cos h 2 k−1 + 1 4cos 2 h 2 k , b k = 1 8cos 2 h 2 k cos h 2 k−1 , h ∈ 0, π 3 , k ≥ 1.
The associated subdivision symbol is therefore
c (k) (z 1 , z 2 ) = e i h 2 k−1 (z 1 +1)(z 2 +1)(z 1 +e i h 2 k−1 )(z 1 e i h 2 k−1 +1)(z 2 +e i h 2 k−1 )(z 2 e i h 2 k−1 +1) (e i h 2 k−2 +1) 2 (e i h 2 k−1 +1) 2 ,
which contains the factor (1 + z 1 )(1 + z 2 ), thus satisfying assumption (ii) of Theorem 4.2. Differently, in E (1) the refinement rules are given in terms of the k-level matrix S k having the structure in (6) with blocks
B 0,k = a k + c n,k 0 0 0 a k + c 4,k b k + c 4,k 0 0 a k + c 4,k b k + c 4,k c 4,k b k + c 4,k a k + c 4,k 0 0 b k + c 4,k , B 1,k = b k + c n,
Using (25) and (27), we verify that the stationary and non-stationary subdivision schemes are asymptotically equivalent of order 1. To see it, we use the Lagrange form of the remainder of the Taylor expansion to write
cos(2 −k h) = 1 − h 2 2 2 −2k + h 4 24 2 −4k cos(ξ ), ξ ∈ (0, 2 −k h),
Generalized exponential spline surfaces of order ≥ 3
In [30] a generalization of order-d polynomial spline surfaces to quadrilateral meshes of arbitrary topology has been proposed. For d = 4, the refinement rules of the corresponding scheme are the rules of Catmull-Clark's subdivision scheme [2] which, in the regular regions of the mesh, can be given in terms of the subdivision mask
c = .(29)
Differently, in the neighborhood of an extraordinary vertex of valence n ≥ 5, the subdivision matrix S k of the order-4 scheme is as in (7)
B 0 = ,B 1 = .
It is a well-known fact that Catmull-Clark's scheme is convergent both in R (1) and in E (1) , and the limit surface is C 2 continuous in correspondence to regular regions of the mesh and tangent plane continuous at the limit points of extraordinary vertices. The subdivision symbol associated to the scheme is c(z 1 , z 2 ) = (z 1 + 1) 4 (z 2 + 1) 4 64 , containing the factor (1 + z 1 )(1 + z 2 ). Thus it verifies assumption (i) of Theorem 4.1 and assumption (i) of Theorem 4.2.
The family of approximating subdivision schemes discussed in [17] is a non-stationary extension of the family in [30], and provides a generalization of order-d exponential spline surfaces to quadrilateral meshes of arbitrary topology. The refinement rules defining this family depend on the subdivision level k and, in particular, they are chosen in such a way that the schemes could reproduce particular shapes such as spheres, tori or conical shapes when the initial meshes are suitably selected. In addition, when the initial mesh is regular, these schemes are tensor-product exponential splines (namely tensor-product polynomial splines but also tensor-product trigonometric and hyperbolic splines) [25]. More precisely, the k-level (k ≥ 1) refinement rules characterizing this family of non-stationary subdivision schemes depend on a k-level parameter v k defined as
where a k = (2v k + 1) 2 4(v k + 1) 2 , b k = 4v k + 2 16(v k + 1) 2 , c k = 1 16(v k + 1) 2 ,
d k = 2v k +1 4(v k +1) , e k = 1 8(v k +1) .
Hence, the associated symbol reads as c (k) (z 1 , z 2 ) = (z 1 + 1) 2 (z 2 + 1) 2 (z 1 e , and contains the factor (1 + z 1 )(1 + z 2 ). Differently, the k-level subdivision matrixS k , defined near an extraordinary vertex of valence n ≥ 5, is of the form (7) with The choice of v k specifies the kind of spline surface we get in the limit, in correspondence to the regular regions of the mesh. In fact, if v k < 1 the schemes yield trigonometric splines, if v k = 1 polynomial splines and if v k > 1 hyperbolic splines. In [17], the authors prove that the limit surface obtained by applying the generalized spline schemes of order d to a regular mesh is C d−2 -continuous, while in the neighborhood of extraordinary elements the tangent plane continuity of the limit surface is shown only by numerical evidence. Here we use Theorem 4.1 and Theorem 4.2 to prove convergence and tangent plane continuity of the limit surfaces at the limit points of extraordinary elements. To prove that the non-stationary version of Catmull-Clark's scheme is convergent and produces tangent plane continuous surfaces at the limit points of extraordinary vertices, we first show that the subdivision masks c and c k in (29) and (30) are asymptotically equivalent of order 1. To this purpose we again write cos(2 −k θ ) = 1 − θ 2 2 2 −2k + θ 4 24 2 −4k cos(ξ ), ξ ∈ (0, 2 −k θ ), cos 2 (2 −k θ ) = 1 − θ 2 2 −2k + θ 4 3 2 −4k cos(2ξ ),ξ ∈ (0, 2 −k θ ), and cosh(2 −k θ ) = 1 + θ 2 2 −2k + θ 4 24 2 −4k cosh(η), η ∈ (0, 2 −k θ ), cosh 2 (2 −k θ ) = 1 + 2θ 2 2 −2k + θ 4 3 2 −4k cosh(2η),η ∈ (0, 2 −k θ ), from which we obtain
α k = 1 − 4v k +2 n(v k +1) 2 − 1 n(v k +1) 2 ,β k = 2(2v k + 1) n 2 (v k + 1) 2 ,|a k − 9 16 | ≤ A 4 k , |b k − 3 32 | ≤ B 4 k , |c k − 1 64 | ≤ C 4 k , |d k − 3 8 | ≤ D 4 k , |e k − 1 16 | ≤ E 4 k
with A, B, C , D, E finite positive constants independent of n and k. Thus, we get
S c (k) − S c ∞ = max |a k − 9 16 | + 4|b k − 3 32 | + 4|c k − 1 64 |, 2|d k − 3 8 | + 4|e k − 1 16 | ≤ 1 4 k max {A + 4B + 4C , 2D + 4E} , so that +∞ ∑ k=1 2 k S c (k) − S c ∞ ≤ max {A + 4B + 4C , 2D + 4E} +∞ ∑ k=1 1 2 k < +∞.
As a consequence, assumptions (i)-(iii) of Theorem 4.2 and assumption (ii) of Theorem 4.1 are satisfied. Next, we use formula (8) to transform the matricesS andS k in the block-circulant matrices denoted by S and S k , and verify the existence of a finite positive constant M independent of n and k such that S k − S ∞ ≤ M 4 k for all k ≥ 1, n ≥ 5 and θ ∈ [0, π) ∪ i(0, 2acosh(500)). As before, we first write
Fig. 1
1Example of quadrilateral mesh containing an extraordinary face (left) and of triangular mesh containing an extraordinary vertex (right).
For a convergent, stationary subdivision schemē S := {S c } the functionφ
Fig. 2
2Domains Ω 1 ,Ω 2 ,Ω 3 corresponding to three subdivision steps in the case of a quadrilateral mesh containing an extraordinary vertex.
Fig. 3
3Ring r k in the case of a quadrilateral mesh with an extraordinary vertex (figure taken from[26]).
Fig. 4
4Domain Ω 1 in the case of a triangular (left) and a quadrilateral (right) mesh containing an extraordinary vertex placed at 0.
Definition 3. 1 .
1Let S be a convergent (non-stationary) subdivision scheme and let n(r c ) denote the normal vector at the limit point r c of an extraordinary vertex/face of valence n. The surface r, limit of S , is tangent plane continuous at r c if there exists a unique vector n(r c ) such that, for all sequences of normal vectors {n k
Remark 3. 1 .
1Asymptotical equivalence of order 1 implies asymptotical equivalence.Theorem 3.1. [15, Theorem 8] Let S andS be subdivision schemes defined in R (1) by the subdivision masks {c (k) ∈ ℓ(Z 2 ), k ≥ 1} and c ∈ ℓ(Z 2 ), respectively. If S andS are asymptotically equivalent of order 1, then C 1 -convergence ofS implies C 1 -convergence of S.
Definition 3. 3 .
3For the two perpendicular directions e 1 = (1, 0) T , e 2 = (0, 1) T , we define as
j , j ∈ {1, 2}, satisfy the following properties: a) the sequence of basic limit functions of {S b (ℓ) e j
Corollary 3. 1 .
1Under the assumptions of Proposition 3.1 lim k→+∞ sup
Proposition 4. 1 .
1Let XJX −1 be the Jordan decomposition of M ∈ R N×N . Assume that the unique dominant eigenvalue of J is 1 and the associated eigenvector is x = (1, 1, ..., 1) T . Then, for all bounded d ∈ R N×3 , lim k→+∞ M k d = xq T ,
where the last but one equality is due to the structure of J. ⊓ ⊔ In Propositions 4.2 and 4.3, · refers to any vector norm and its associated induced matrix norm.Proposition 4.2. If the dominant eigenvalue of M is 1 and its algebraic multiplicity is 1, then there exists a finite positive constant C (independent of k) such that M k ≤ C , ∀k ≥ 0.
Remark 4. 1 .
1It is important to remark that the subdivision matrix S defining a convergent, stationary subdivision schemeS in irregular submeshes E (1) , always verifies the assumptions of Propositions 4.1 and 4.2 since a.1) the unique dominant eigenvalue of S is λ 0 = 1, a.2) the algebraic multiplicity of λ 0 is 1, a.3) the eigenvector associated with λ 0 is x 0 = (1, 1, ..., 1) T ,
Proposition 4. 3 .
3Let M (0) := I ∈ R N×N and for all k ≥ 1 let M (k) := M k M k−1 · · · M 1 with M j ∈ R N×N , for all j = 1, ..., k. Let M ∈ R N×N be a nonsingular matrix having 1 as dominant eigenvalue with algebraic multiplicity 1. If, for all k ≥ 1, M k − M ≤ C σ k with σ > 1 and some finite positive constant C (independent of k), then M (k) ≤ C , ∀ k ≥ 1 with C a finite positive constant (independent of k). Proof. The proof takes inspiration from [15, Theorem 5]. Let y ∈ R N . The claim is proven by introducing a new vector norm y M := sup k≥−1 M k+1 y associated to the given nonsingular matrix M ∈ R N×N . In view of Proposition 4.2, our assumption on M implies the existence of a finite positive constant C such that M k ≤ C for all k ≥ 0. Moreover, y ≤ y M since M k+1 y = y when k = −1. There follows that y ≤ y M ≤ C y , with C a finite positive constant, i.e. any standard vector norm and the · M norm are uniformly equivalent. We now consider the induced norm for the matrix M itself, and denote it as · M . Then by exploiting the uniform equivalence of norms to bound M (k) , k ≥ 1. From the definition of induced norm, it holds the submultiplicative property M k M k−1 ...M 1 M ≤ M k M M k−1 M ... M 1 M .
j=1 1 σ
1j < ∞ the claim follows. ⊓ ⊔ We conclude this section with another useful intermediate result relating M (k) with M k .
Proposition 4. 4 .
4Let M ∈ R N×N , M (0) := I ∈ R N×N and for all k ≥ 1 let M (k) := M k · · · M 1 ∈ R N×N . Then,
Corollary 4. 1 .
1LetS be a convergent, stationary scheme identified by {S} and denote by λ 1 the subdominant eigenvalue of S. Under the assumptions of Theorem 4.1 with the additional requirement that σ > 1 |λ 1 | > 1, for u = lim k→+∞ S (k) d 1 we have
, i = 2, . . . , n − 2, B n−1
b k + c 4 ,k c 4
44i = 2, . . . , n − 2,
= 0
06×6 , i = 2, . . . , n − 2,B n−1
∈
(1, 500) if θ ∈ i(0, 2acosh(500)).For the non-stationary approximating scheme of order d = 4 (non-stationary version of Catmull-Clark's scheme), the k-level subdivision mask to be used in the regular regions of the mesh is
1 n 2
2(v k + 1) 2 , 0, 0, 0, 0T , γ k = d k , 1 4 , b k , e k , c k , ,k = 0 6×6 , i = 2, . . . , n − 2,B n−1
Theorem 4.1. Let S be a non-stationary subdivision scheme whose action in E (1) is described by the matrix sequence {S k , k ≥ 1}. Moreover, letS be a stationary subdivision scheme that in E (1) is identified by {S}. Assume that:
In R (1) , Doo-Sabin's scheme is described by the subdivision maskc =
1
16
3
16
3
16
1
16
3
16
9
16
9
16
3
16
3
16
9
16
9
16
3
16
1
16
3
16
3
16
1
16
AcknowledgmentsThis research has been accomplished within RITA (Research ITalian network on Approximation).The previous expression allows us to get the bounds |a k − 1 2 | ≤ A 4 k , |b k − 1 8 | ≤ B 4 k , |c n,k − 1 4n | ≤ n −1 C 4 k , ∀n ≥ 4 with A, B, C finite positive constants independent of n and k. The latter bounds can then be used to show that S c (k) − S c ∞ = a k + c 4,k −916and therefore prove the asymptotical equivalence of order 1. Indeed, using (2), we arrive atSummarizing, assumptions (i) − (iii) of Theorem 4.2 and assumption (ii) of Theorem 4.1 are satisfied. Next, we show that S k − S ∞ ≤ M 4 k for all k ≥ 1, n ≥ 5 and h ∈ 0, π 3 , with M a finite positive constant. Indeed, by(26)and(28)we havefor n ≥ 5 we finally obtain the boundpositive constant independent of n and k. In other words assumption (iii) of Theorem 4.1 and assumption (iv) of Theorem 4.2 are satisfied. Since S has a dominant single eigenvalue λ 0 = 1 and a subdominant eigenvalue 0.5 < λ 1 < 1 with algebraic and geometric multiplicity 2 (i.e. is a double non-defective eigenvalue), all the assumptions of Theorem 4.1 and Theorem 4.2 are verified with σ = 4. Hence, the non-stationary version of Doo-Sabin's scheme is convergent at extraordinary faces and the limit surfaces obtained by such a scheme are tangent plane continuous at the limit points of extraordinary faces.and explicitly compute each norm on the right hand side asMoreover, in view of the boundswe are finally able to bound the norms of the blocks asHence, for all n ≥ 5,with M := M 0 + M 1 + M 2 + M 3 a finite positive constant independent of n and k.The above proves that (iii) of Theorem 4.1 is satisfied. Moreover, since S has a dominant single eigenvalue λ 0 = 1 and a subdominant eigenvalue 0.5 < λ 1 < 1 with algebraic and geometric multiplicity 2 (double non defective eigenvalue), (iv) of Theorem 4.2 is also satisfied with σ = 4. It follows that all the assumptions of Theorem 4.1 and Theorem 4.2 are verified. Thus, this non-stationary version of Catmull-Clark's scheme is convergent at extraordinary vertices and the limit surfaces obtained by such a scheme are tangent plane continuous at the limit points of extraordinary vertices.
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| []
|
[
"Followers Are Not Enough: A Multifaceted Approach to Community Detection in Online Social Networks",
"Followers Are Not Enough: A Multifaceted Approach to Community Detection in Online Social Networks"
]
| [
"David Darmon [email protected] \nDepartment of Mathematics\nUniversity of Maryland\nCollege Park, MarylandUnited States of America\n",
"Elisa Omodei \nLaTTiCe-CNRS\nISC-PIF\nParisFrance\n\nDepartment d'Enginyeria Informatica i Matematiques\nUniversitat Rovira i Virgili\nTarragonaSpain\n",
"3☯Joshua Garland \nDepartment of Computer Science\nUniversity of Colorado\nBoulderColoradoUnited States of America\n"
]
| [
"Department of Mathematics\nUniversity of Maryland\nCollege Park, MarylandUnited States of America",
"LaTTiCe-CNRS\nISC-PIF\nParisFrance",
"Department d'Enginyeria Informatica i Matematiques\nUniversitat Rovira i Virgili\nTarragonaSpain",
"Department of Computer Science\nUniversity of Colorado\nBoulderColoradoUnited States of America"
]
| []
| In online social media networks, individuals often have hundreds or even thousands of connections, which link these users not only to friends, associates, and colleagues, but also to news outlets, celebrities, and organizations. In these complex social networks, a 'community' as studied in the social network literature, can have very different meaning depending on the property of the network under study. Taking into account the multifaceted nature of these networks, we claim that community detection in online social networks should also be multifaceted in order to capture all of the different and valuable viewpoints of 'community.' In this paper we focus on three types of communities beyond follower-based structural communities: activity-based, topic-based, and interaction-based. We analyze a Twitter dataset using three different weightings of the structural network meant to highlight these three community types, and then infer the communities associated with these weightings. We show that interesting insights can be obtained about the complex community structure present in social networks by studying when and how these four community types give rise to similar as well as completely distinct community structure. | 10.1371/journal.pone.0134860 | null | 14,313,481 | 1404.0300 | e2d6393014999bfa3858a320353304c4a68d8fda |
Followers Are Not Enough: A Multifaceted Approach to Community Detection in Online Social Networks
David Darmon [email protected]
Department of Mathematics
University of Maryland
College Park, MarylandUnited States of America
Elisa Omodei
LaTTiCe-CNRS
ISC-PIF
ParisFrance
Department d'Enginyeria Informatica i Matematiques
Universitat Rovira i Virgili
TarragonaSpain
3☯Joshua Garland
Department of Computer Science
University of Colorado
BoulderColoradoUnited States of America
Followers Are Not Enough: A Multifaceted Approach to Community Detection in Online Social Networks
RESEARCH ARTICLE ☯ These authors contributed equally to this work. *
In online social media networks, individuals often have hundreds or even thousands of connections, which link these users not only to friends, associates, and colleagues, but also to news outlets, celebrities, and organizations. In these complex social networks, a 'community' as studied in the social network literature, can have very different meaning depending on the property of the network under study. Taking into account the multifaceted nature of these networks, we claim that community detection in online social networks should also be multifaceted in order to capture all of the different and valuable viewpoints of 'community.' In this paper we focus on three types of communities beyond follower-based structural communities: activity-based, topic-based, and interaction-based. We analyze a Twitter dataset using three different weightings of the structural network meant to highlight these three community types, and then infer the communities associated with these weightings. We show that interesting insights can be obtained about the complex community structure present in social networks by studying when and how these four community types give rise to similar as well as completely distinct community structure.
Introduction
Networks play a central role in online social media services like Twitter, Facebook, and Google-+. These services allow a user to interact with others based on the online social network they curate through a process known as contact filtering [1]. For example, 'friends' on Facebook represent reciprocal links for sharing information, while 'followers' on Twitter allow a single user to broadcast information in a one-to-many fashion. Central to all these interactions is the fact that the structure of the social network influences how information can be broadcast or diffuse through the service.
Because of the importance of structural networks in online social media, a large amount of work in this area has focused on using structural networks for community detection. In this traditional view, a community is defined as a collection of nodes (users) within the network which are more highly connected to each other than to nodes (users) outside of the community [2,3]. For instance, in [4], the authors use a follower network to determine communities within Twitter, and note that conversations tend to occur within these communities. The approach of focusing on the structure of networks makes sense for 'real-world' sociological experiments, where obtaining additional information about user interactions may be expensive and timeconsuming. However, with the prevalence of large, rich data sets from online social networks, additional information beyond the structure alone may be incorporated, and these augmented networks more realistically reflect how users interact with each other on social media services [5,6].
A large body of work exists on methods for automatic detection of communities within networks, e.g., the stochastic blockmodel [7] (and its recent generalization to weighted networks [8]), the Girvan-Newman algorithm [3], clique percolation [9], Infomap [10], Louvain [11], and the recently introduced OSLOM [12]. All these methods begin with a given network, and then attempt to uncover structure present in the network, i.e., they are agnostic to how the network was constructed. As opposed to this agnostic analysis, we propose-and illustrate the importance of-a multifaceted question-focused approach. We believe that in order to understand all of the communities present in a network, it is important to take the following crucial steps:
1. Ask several questions about the communities that may be present in the network, each of which is aimed at revealing a new facet of the community structure present in the network--e.g., which users are tightly connected structurally? which groups of users have similar topical interests?
2. Derive a weighting scheme aimed at answering each of the questions asked in the first step and then perform community detection on each weighted network.
3. Compare the (possibly) different communities that arise from this series of experiments on a micro and macro scale to unveil all the interesting facets of community structure present in the social network. This is especially true for social network analysis. In social networks, a 'community' could refer to several possible structures. The simplest definition of community, as we have seen, might stem from the network of explicit connections between users on a service (friends, followers, etc.). On small time scales, these connections are more or less static. To form a more dynamic picture of community structure, we might instead determine communities based on who is talking to whom or which users talk about similar topics. More abstractly, we could even define communities as groups of people who exhibit similar activity profiles. We can characterize these types of communities based on the questions that motivate them:
• Structure-based: Who are your stated friends? Whom do you follow?
• Activity-based: Who shares similar activity profiles?
• Topic-based: What do you talk about?
• Interaction-based: Whom do you communicate with? This is not meant to be an exhaustive list, but rather a list of some of the more common types of communities observed and studied in social networks. Instead of looking at each of these questions in isolation-which is the standard approach-we propose looking at when and how communities motivated by these different questions overlap and are disjoint, and whether different approaches to asking the question "What community are you in?" leads to different insights about a social network. For example, a user on Twitter might connect mostly with computational social scientists, utilize the service nearly every time a particular user or group of users is active, talk mostly about machine learning, and interact solely with close friends (who may or may not be computational social scientists). For this user, the answers to each question therefore correspond to more or less distinct communities. By contrast, a teenage user may only connect and interact with their friends, will most likely exhibit similar activity patterns dictated by the academic and extracurricular schedule of a student, and will discuss topics typical of their demographic. For this user, the answers to each question map to the same community of users, even though different definitions of community are being used. This highlights an important subtlety of this work: defining communities in different ways does not imply that the collection of users in each community must be different. Said differently, it is not obvious-or always the case-that changing the definition of community will always change the communities that are detected. Studying when this change does and does not occur--as we will show-provides interesting insights into the complex community structure present in these networks.
We now consider in more detail how each of these types of questions describes a unique facet of 'community'. Later in Section 4 we will show how asking all of these questions-instead of focusing on only one-gives a much deeper view of the (possibly) different communities that users belong to.
Naïvely the activity-based approach is motivated by the question of "Which users in a network have similar activity profiles?". With this question in mind, a community can then be thought of as those users who use (or do not use) the service at similar times. However, the measure we use is slightly more subtle that this. In particular, these communities can be thought of as groups of users who possess so-called "activity-based predictive capacity" with one another, i.e., given a user u and follower f, who are members of a community detected with this weighting, there exists a reduction in uncertainty about the activity profile of follower f (tweeting or remaining silent), given the tweet history of user u, ignoring the information present in the tweet history of follower f. To accomplish this, we consider each user on Twitter as an information processing unit, but completely ignore the content of their tweets. We then weight the directed edges (the reported follower-followee relationships) between users with the so-called transfer entropy [13], calculated between the tweet history of follower f and user u. This is discussed in more detail in Section 2.3.
The topic-based and interaction-based approaches, in contrast to the activity-based approach, rely on the content of a user's interactions and ignore their temporal components. The content contains a great deal of information about communication between users. There are two broad approaches to topic-based communities in the literature. [14] used a set of users collected based on their use of a single hashtag, and tracked the formation of follower and friendship links within that set of users. In [15], the authors chose a set of topics to explore, and then seeded a network from a celebrity chosen to exemplify a particular topic. Both approaches thus begin with a particular topic in mind, and perform the data collection accordingly. Other approaches use probabilistic models for the topics and treat community membership as a latent variable [16]. For example, a popular approach to analyzing social media data is to use Latent Dirichlet Allocation (LDA) to infer topics based on the prevalence of words within a status [17,18]. The LDA model can then be used to infer distributions over latent topics, and the similarity of two users with respect to topics may be defined in terms of the distance between their associated topic distributions. Because our focus is not on topic identification, we apply a simpler approach using hashtags as a proxy for topics, similar to the approaches presented in [19,20].
We can then define the similarity of two users in terms of their hashtags, and use this similarity to build a topic-based network. This method is described in detail in Section 2.4.
Finally, the interaction-based approach relies on the meta-data and text of messages to identify whom a user converses with on the social media service. On Twitter, we can use mentions (indicating a directed communication) and retweets (indicating endorsement of another user) to identify conversation. A few works have investigated this type of community. For example, [21] considered both mention and retweet networks in isolation for a collection of users chosen for their political orientation. In [22], the authors construct a dynamic network based on simple time-windowed counts of mentions and retweets, and use the evolution of this network to aid in community detection.
Previous research on communities in social networks focused almost exclusively on different network types in isolation. A notable exception to this is [23], which considered both structure-based and interaction-based communities on Twitter. However, this study focused on data collected based on particular topics (country music, tennis, and basketball), and not on a generic subpopulation of Twitter users. Moreover, it did not explore the differences in community structure resulting from the different network weightings, and focused on aggregate statistics (community size, network statistics, etc.). Another notable exception is [24], where the authors used a tensor representation of user data to incorporate retweet and hashtag information into a study of the social media coverage of the Occupy Movement. The tensor can then be decomposed into factors in a generalization of the singular value decomposition of a matrix, and these factors can be used to determine 'salient' users. However, unlike our work, this approach focused on data for a particular topic (the Occupy Movement) and did not collect users based on a structural network.
Studying how the communities derived from follower-followee, activity-based, topic-based, and interaction-based networks are disjoint as well as overlapping allow for a more complete picture of the implicit networks present in online social media, as opposed to the explicit social network indicated by follower links alone. In this paper, we explore the relation between these various possible networks through their corresponding communities. We begin by describing the methodologies used to generate the various types of networks, and infer their community structure. We then explore how the communities of users differ depending on the type of network used. We conclude by illustrating that this multifaceted question-oriented approach to community detection provides interesting insights into the intricate multifaceted structure of several different users' communities, information that would not be available if only a single form of community detection was performed.
Methodology
In the following sections, we introduce the problem of community detection, and present the data set used for our analyses. We then describe our methodology for constructing the question-specific networks. In particular, we introduce an information theoretic statistic for activity-based communities, a retweet-mention statistic for interaction-based communities, and a hashtag similarity metric for defining topic-based communities. It should be noted that our goal is to choose representative weightings for each community type so that we may compare community structure across types, and not to-aside from the activity-based weightingintroduce novel weighting schemes per se.
Community Detection
As discussed in the introduction, we adopt the standard definition of community: a collection of nodes (users) within a network who are more densely connected to each other than with the rest of the network. Structural community detection is a well studied problem and several different methods and algorithms have been proposed. For a complete review of this subject we refer the reader to [25,26]. In this paper however we focus on a class of networks and communities that is far less studied, in particular we study networks which are both weighted and directed and communities within those weighted directed networks that can (but need not) overlap. When selecting a detection algorithm we propose that all three (weight, direction, and overlap) are important for the following reasons. First, communication on Twitter occurs in a directed manner, with users broadcasting information to their followers. An undirected representation of the network would ignore this fact, and could lead to communities composed of users who do not actually share information. Second, we are interested in not just the structure of links but also in their function, and to capture this we use edge weightings which must be incorporated into the community detection process. Finally, since people can belong to multiple and possibly overlapping social (e.g., college friends, co-workers, family, etc.) and topical (e.g., a user can be interested in both cycling and politics and use the network to discuss the two topics with two different groups of users) communities, we are interested in finding overlapping communities, rather than partitions of the weighted directed network.
This last criterion in particular poses a problem because the majority of community detection algorithms developed so far are built to find partitions of a network and few are aimed at finding overlapping communities [8,9,[27][28][29][30][31][32][33]. Among these methods, even fewer deal with directed or weighted networks. For example, the work of [9] on clique percolation can account for both features, but not at the same time. A recent method proposed in [12], OSLOM (Order Statistics Local Optimization Method), is one of the first methods able to deal with all of these features simultaneously. This method of Lancichinetti et al. relies on a fitness function that measures the statistical significance of clusters with respect to random fluctuations, and attempts to optimize this fitness function across all clusters. Following [12], the significance measure is defined as the probability of finding the cluster in a network without community structure. The null model used is highly similar to the one adopted by Newman and Girvan in [3] to define modularity, i.e., it is a model that generates random graphs with a given degree distribution. The authors tested their algorithm on different benchmarks (LFR and Girvan-Newman) and real networks (such as the US air transportation network and a word association network), and compared its performance against the best algorithms currently available (i.e., the ones mentioned in the introduction), and found excellent results. Moreover, they showed that OSLOM is also able to recognize the absence, and not simply the presence, of community structure, by testing it on random graphs. This feature of the algorithm plays an important role in the analysis of real world networks, since it is not always the case that community structure is indeed present and it is therefore useful to be able to detect its absence too. Therefore, given its versatility and performance with benchmark networks, in this paper we used OSLOM to detect overlapping communities present in our weighted and directed networks.
The Initial Dataset and Network Construction
The dataset for this study consisted of the tweets of 15,000 Twitter users over a 9 week period (from April 25th to June 25th 2011). The network and user activity (including tweets, mentions, and hashtags) were accessed using the Twitter API. The users are embedded in a network collected by performing a breadth-first expansion from a random seed user. In particular, once the seed user was chosen, the network was expanded to include his/her followers, but only included users considered to be 'active' (i.e., users who tweeted at least once per day over the past one hundred days). Network collection continued in this fashion by considering the active followers of the active followers of the seed, and so on until 15,000 users were added to the network. The only biasing steps in this procedure are the selection of the seed node and the filtering out of inactive users. However, because all of the community types we are interested in involve some aspect of the interaction between users, each other, and their content, this procedure provides a reasonable follower-followee network for use with the different question-based analyses.
Since our goal is to explore the functional communities of this network, we filter the network down to the subset of users that actively interact with each other (e.g., via retweets and mentions). We do this by measuring what we call (incoming/outgoing) information events. We define an outgoing information event for a given user u as either a mention made by u of another user in the network, or a retweet of one of u's tweets by another user in the network. The logic for this definition is as follows: if u mentions a user v this can be thought of as u directly sending information to v, and if u is retweeted by v then v received information from u and rebroadcast it to their followers. In either case there was information outgoing from u which affected the network in some way. Analogously, we define the incoming information event for u as either being mentioned by a different user in the network, or as retweeting another user in the network. With (incoming/outgoing) information events defined we filtered the network by eliminating all users with less than a total of 9 outgoing and incoming information events, i.e., less than one information event per type per week on average. We then further restricted our analysis to the strong giant connected component of the network built from the (incoming/outgoing) information filtered set of users. In this study the link is directed from the user to the follower because this is the direction in which information flows. Thus, for a pair of users u and v, an edge a v ! u in the follower-followee network has weight 1 if user u follows v, and 0 otherwise. The final network consists of 6,917 nodes and 1,481,131 edges.
Activity-Based Communities and Transfer Entropy
For the activity-based communities, we consider only the timing of each user's tweets and ignore any additional content. From this starting point, we can view the behavior of a user u on Twitter as a point process, where at any instant t the user has either emitted a tweet (X t (u) = 1) or remained silent (X t (u) = 0). This is the view of a user's dynamics taken in [34] and [35]. Thus, we reduce all of the information generated by a user on Twitter to a time series {X t (u)} where t ranges over the time interval for which we have data (9 weeks in this case). Because status updates are only collected in discrete, 1-second time intervals, it is natural to consider only the discrete times t = 1s, 2s, . . ., relative to a reference time.
Operationally, we expect users to interact with Twitter on a human time scale, and thus the natural one-second time resolution is too fine, since most humans do not write tweets on the time scale of seconds. For this reason, we coarsen each time series by considering nonoverlapping time intervals ten minutes in length. For each time interval, we record a 1 if the user has tweeted during that time interval, and a 0 if he or she has not. Thus, the new coarsened time series now captures whether or not the user has been active on Twitter over any given ten minute time interval in our data set. It should be noted that, as (most) users are not constantly tweeting, these time series are quite sparse. For this particular application, this sparsity appropriately reflects the behavior of the users. However, this sparsity may not be appropriate for all applications; anytime the following measure is used in practice the effects of binning, sparsity, and their role in the activity-based community structure should be considered. We can then compute the flow of information from a user u to a follower f by computing the transfer entropy between their time series X t (u) and X t (f ). See Appendix A for a detailed introduction to transfer entropy and its estimation from data.
For the communities based on transfer entropy, we weighted each edge from a user u to a follower f by the estimated transfer entropy of the user u on f,
w TEðkÞ u!f ¼TE ðkÞ XðuÞ!Xðf Þ :ð1Þ
Transfer entropy is an information theoretic measure of directed information flow. We assume-as is standard-a positive transfer entropy from the tweet history of a user u to X(f) implies a reduction in uncertainty in the activity of the follower, given the information contained in X(u), while removing the information already contained in X(f). This provides an information theoretic framework in which we can capture communities with activity-based predictive capacity between users or users with similar activity profiles. We call this relation "APC" (activity-based predictive capacity).
We computed the transfer entropy on each coarsened time series with lag k ranging from 1 to 6, this corresponds to a lag of ten minutes to an hour. The choice of lag must balance a trade-off between additional information and sparsity of samples: as we increase the lag k, we account for longer range dependencies, but we also decrease the number of samples available to infer a higher dimensional predictive distribution. Interestingly, as we will show later, the underlying communities resulting from the different lags have similar structure. See S1 Text for a discussion of these issues.
It should be noted that according to [35] a positive transfer entropy between X(u) and X(f) indicates that u "influences" f, or that u and f share a common influencer. Similarly, since transfer entropy is a nonlinear generalization of Granger causality [36], it is common to assume a positive transfer entropy between a user u and a follower f implies the relationship is causal in the Granger sense. However, we are skeptical whether this measure truly captures social influence in its entirety, or even causality in the case of online social networks. So instead we simply treat this weighting as a way to explicitly quantify APC between users.
As an aside, while we are skeptical that this measure can capture social influence, later in this paper we show that this information theoretic measure agrees with other concepts of influence such as the Forbes list of "Top 10 Social Media Influencers" and corroborates with the socalled strength of weak ties [37]. Even so, we do not explicitly treat transfer entropy-and we do not believe it should be treated without further investigation-as a quantification of social influence or causality. Instead this measure should be treated merely as a way to quantify APC between users.
To the best of our knowledge, this is the first use of transfer entropy for community detection in online social networks. Various other information theoretic approaches have been used successfully to analyze online social networks, e.g., to gain insight into local user behavior [34], to detect communities based on undirected information flow [38], and to perform network inference and link prediction [35].
Topic-Based Communities and Hashtag Weighting
In contrast to the activity-based approach, the topic-based (or topical) communities, i.e., communities defined by the topics users discuss, rely on the content of a user's interactions and ignore their temporal components. In a topical community, users are defined to be a member of a community if they tweet about similar topics as the other members of the community. In order to detect the topical communities, we weight the edges of the user-follower network through a measure based on the number of common hashtags between pairs of users. Hashtags are a good proxy for a tweet's content as hashtags are explicitly meant to be keywords indicating the topic of the tweet. Moreover they are widely used and straightforward to detect.
To this end, we characterize each user u by a vectorhðuÞ of length equal to the number of unique hashtags in the dataset, and whose elements are defined as
h i ðuÞ ¼ i ðuÞ log N n i ;ð2Þ
where ϕ i (u) is the frequency of hashtag i occuring in user u's tweets, N is the total number of users, and n i is the number of users that have used the hashtag i in their tweets. This adapted term frequency-inverse document frequency (tf-idf) measure [39] captures the importance of a hashtag in the users's tweets through the first factor, but at the same time smooths it through the second factor by giving less importance to hashtags that are too widely used (as N n i approaches one, its logarithm approaches zero).
For the topical communities we weight each directed edge from a user u to a follower f with the cosine similarity of their respective vectorshðuÞ andhðf Þ:
w HT u!f ¼h ðuÞ Áhðf Þ jjhðuÞjj jjhðf Þjj :ð3Þ
This weight captures the similarity between users in terms of the topics discussed in their tweets.
Interaction-Based Communities and Mention / Retweet Weighting
Retweets and mentions are two useful features of Twitter networks which can be used to track information flow through the network. With mentions, users are sending direct information to other users and with retweets users are rebroadcasting information from a user they follow to all of their followers. This type of information flow defines a community in a much different way than transfer entropy. Instead of defining communities by the loss of uncertainty in one user's tweeting history based on another's, we define interaction-based communities by weighting the user-follower network with a measure proportional to the number of mentions and/or retweets between users. We define three weighting schemes. We first consider the number of tweets of a user u that were rebroadcast by a follower f, indicating a flow of information from u to f. Since different users produce different amounts of tweets and retweets, in order to take into account the relative importance of this measure we normalize it by dividing over the total number of retweets made by f,
w R u!f ¼ # retweets of u by f # total retweets made by f ;ð4Þ
We then consider the number of mentions of f made by user u. To account for the fact that some users/accounts are very popular and are often mentioned by so many other users that the flow of information from the latter to the former is flushed out, we normalize this measure by dividing over the total number of mentions received by f.
w M u!f ¼ # mentions of f by u # total mentions of f ;ð5Þ
Finally, we define the mention-retweet proportion as the arithmetic mean of the two measures just defined,
w MR u!f ¼ ðw R u!f þ w M u!f Þ 2 :ð6Þ
Results and Discussion
In this section we show the importance and usefulness of a multifaceted approach to community detection in online social networks. We begin by showing that the communities emerging from the different weightings of the structural network quantitatively differ both at the macroscopic (e.g., number of communities and their size distribution) and microscopic (e.g., specific memberships, comemberships) scale in interesting ways. Finally, in order to provide a practical illustration of the utility of this question-oriented multifaceted approach, we present several concrete examples of multifaceted community memberships discovered with this method. We will show examples both at the community and individual user level.
Comparing Aggregate Statistics of Community Structure
We begin by examining the overall statistics for the communities inferred by OSLOM using the weightings defined in the previous sections. The number of communities by community type is given in Table 1. We see that the topic-and interaction-based networks admit the most communities. The activity-based network admits the fewest communities. One advantage of OSLOM over many other community detection algorithms is that it explicitly accounts for singleton 'communities': those nodes who do not belong to any extant communities. This is especially important when a network is collected via a breadth-first search, as in our network, where we begin from a seed node and then branch out. Such a search, once terminated, will result in a collection of nodes on the periphery of the network that may not belong to any community in the core. We see in Table 1 that the topic-and interaction-based communities have the most singletons. This result for the interaction-based community is partially an artifact of the retweet/ mention weighting: 717 of the users were disconnected from the network by how the weights were defined, resulting in 'orphan' nodes which we have included in the collection of singletons for all of our analyses. However, even after accounting for this artifact, the interaction-based network still has the most non-orphan singletons. This seems to indicate that a large fraction of the 6917 (nearly 25%) do not interact with each other in a concerted way that would mark them as a community under our interaction-based definition. This agrees with a result previously reported in [40] about how most users passively interact with incoming information on Twitter.
Next we consider the distribution of community sizes across the community types. The complementary cumulative distribution of community sizes is given in Fig 1. Note that both axes are plotted on log-scales. Thus, for a fixed community size s, Fig 1 shows the proportion of communities of size greater than s for each community type. We see that the community distributions have longer tails for the non-structural networks, and that the interaction-based network has the longest tail. Note that the activity-based communities, using transfer entropy estimated with varying lags, seem to converge on a similar large-scale community structure around lag 3. That is, the communities based on lag 1 and lag 2 transfer entropies are generally larger, and these communities resolve into smaller communities as the lag increases beyond 2. The distribution of community sizes for lag larger than 2 is insensitive to the lag. Most importantly, we see that the distributions of community sizes differ across the community types, the large-scale community structure is highly dependent on the particular question posed.
Comparing Community Structure with Normalized Mutual Information
In the previous section, we saw that the large scale statistics of the communities were highly dependent on the type of community under consideration. However, macroscale network statistics do not account for differences in community structure that result from operations such as splitting or merging of communities. Moreover, this view does not account for which users belong to which communities, and in particular which users belong to the same communities across community types. To answer this question, we invoke methods for the comparison of clusters: given two different clusterings of nodes into communities, how similar are the two clusters? The standard approach to answering this question is to define a metric on the space of possible partitions. Because we detect coverings rather than partitions, standard cluster comparison metrics like variation of information [41] are not appropriate. Instead, we use a generalization of variation of information first introduced in [31], the normalized mutual information. The normalized mutual information stems from treating clustering as a community identification problem: given that we know a node's community membership(s) in the first covering, how much information do we have about its community membership(s) in the second covering, and vice versa? Consider the two coverings C 1 and C 2 . We think of the community memberships of a randomly chosen node in C 1 as a binary random vector X 2 {0, 1} jC 1 j where the i th entry of the vector is 1 if the node belongs to community i and 0 otherwise. Similarly, Y 2 {0, 1} jC 2 j is a binary random vector indicating the community memberships of the node in C 2 . Then the normalized mutual information is defined as
NMIðC 1 ; C 2 Þ ¼ 1 À 1 2 H½XjY H½X þ H½YjX H½Yð7Þ
where H[Á] denotes a marginal entropy and H[ÁjÁ] denotes a conditional entropy. The normalized mutual information varies from 0 to 1, attaining the value of 1 only when C 1 and C 2 are identical coverings up to a permutation of their labels. See the appendix of [31] for more details.
We considered the normalized mutual information between the communities inferred from the structural network and the networks weighted with lag 1 through 6 transfer entropies, hashtag similarity, and mention, retweet, and mention-retweet activity. The resulting NMI(C i , C j ) are shown in Fig 2. Note the block diagonal structure in Fig 2, which indicates that coverings are most similar within a question category. For example, the activity-based transfer entropy coverings are more similar to each other than to any of the other coverings. Similarly for the interaction-based mention, retweet, and mention-retweet coverings. This point may seem obvious, but the fact that the different weightings within a question category result in similar community structure indicates that each covering is capturing true, latent properties of the social network. Interestingly, the coverings resulting from the different weightings are all more similar to each other than to the structural covering from the unweighted network. Also note that the covering based on the hashtag similarities are different from all of the other weight-based coverings. In agreement with the results reported in the previous section, we see that the activity-based communities share similar structure for lags greater than 2. Because of these two results, in the remainder of the paper, we focus on the activity-based communities inferred using the lag-4 transfer entropy.
Thus, we see that although the activity-based, interaction-based, and topic-based communities relied on the structural network, their community structure differs the most from the community structure of the follower network. This agrees with the results from the previous section, and reinforces that the follower network is a necessary but not sufficient part of detecting communities characterized by properties beyond follower-followee relationships.
Comparing Edges Across Different Community Types
Any covering determined by OSLOM induces a natural partition of the edges in a directed network. In particular, let u and f be two users in the network, and let M u and M f be their community memberships. Then any edge e u ! f can be partitioned into one of three classes by
Tðe u!f Þ ¼ Inter À edge : M u \ M f ¼ ; Intra À edge : M u ¼ M f Mixed À edge : otherwise :ð8Þ8 > < > :
In words, an inter-edge connects two users who share no community memberships, an intraedge connects two users who each belong to the same communities, and a mixed-edge connects two users who share some, but not all, community memberships. Thus, inter-edges cross community boundaries, intra-edges lie within community boundaries, and mixed-edges lie at the borders of community boundaries. See Fig 3 for a schematic of this edge partitioning. Each community type (Structural, Activity-based, Topic-based, Interaction-based) induces a different partition of the edges, and each edge type (Transfer Entropy, Hashtag, Mention-Retweet) induces a different distribution of weights. That is, let W u ! f be the weight on an edge E u ! f chosen uniformly at random from the network, and compute
PðW u!f w u!f jTðE u!f Þ ¼ tÞ;ð9Þ
the empirical distribution over the edge weights conditioned on the edge type t being one of inter, intra, and mixed. The densities associated with these distributions are shown in Fig 4. By considering how the weights vary conditional on the edge type, we can explore whether and how each of the follower-, interaction-, activity-, and topic-based communities define Inter-edges (red dashed) cross community boundaries. Intra-edges (blue solid) remain inside community boundaries. Mixed-edges (purple dotted) both remain in and cross community boundaries due to overlap in community membership. Each column corresponds to the same collection of weights, but partitioned using a different covering. Each row corresponds to the same covering, but for the different weights. functional units within the network. For example, if users interact more-or-less independently of the topics they discuss, then the distributions of the interaction-based / topic-based edge weights should be independent of the edge type determined from the topic-based / interactionbased communities. By contrast, if users tend to interact mostly with those users who discuss similar topics, then the edge weight distributions should vary with the edge type. Each column of Fig 4 demonstrates how the density of weights change with the community type used to induce the partition. For example, the second column shows how the density of hashtag weights change based on the community partition. In general, the densities differ in non-trivial ways across the edge types and coverings. However, we see that the medians of the distributions provide a useful summary statistic. Under each collection of density functions, we list the median value of the inter, intra, and mixed-edges in red, blue, and purple, respectively. Unsurprisingly, we see that the greatest difference between the distributions occurs when we consider matching Community-type / Weight-type pairs, since the communities are determined based on the corresponding weights. However, we also see differences in the distributions when the Community-type / Weight-type pairs do not match. For example, for the covering corresponding to the activity-based communities, the hashtag weights tend to be higher for edges internal to communities than between communities. Similarly, for the covering corresponding to the topic-based communities, the transfer entropy weights tend to be higher on edges within (intra) and between (inter) communities, and lower for edges between members with multiple, non-identical memberships (mixed).
We also note that the distribution of transfer entropy weights always tend to be higher for edges crossing community boundaries (inter-or mixed-edges) compared to those edges within community boundaries (intra-edges). This is a property not exhibited by either the hashtag or mention-retweet weights, independent of the covering used to partition the edges. Moreover, for all but the activity-based covering, the weights of mixed-edges tend to be highest overall. Recall that the transfer entropy TE X(u) ! X(f) quantifies the reduction in uncertainty about a follower f's activity from knowing the activity of a user u. This result therefore implies that, in terms of prediction, it is more useful to know the time series of a user followed outside of the community compared to a user followed inside of the community, and even more useful to know the time series of a user that shares some, but not all, of the same community memberships. Thus, in an information theoretic sense, we see that novel information useful for prediction is more likely to flow across community boundaries than within community boundaries.
Uncovering the Multifacetedness of Community Memberships
In the previous sections we showed that the communities emerging from the different weightings of the structural network quantitatively differ both at the macroscopic and at the microscopic scale. Here, we present some concrete examples of different community membership, in order to provide a practical illustration of the utility of using a multifaceted approach to community detection. We explore this first on a community level and then at the individual level.
In the topic-based communities, we found a single community consisting of 83 users who tweet about environmental issues and frequently use hashtags such as #green, #eco and #sustainability. We also found a different community of 47 users who tweet about small businesses and entrepreneurship, using hashtags such as #smallbiz, #marketing and #enterpreneur. In both cases, almost all members of these topic-based communities are not found in the same community in the other networks e.g., interaction-or activity-based, indicating that while these people talk about the same things and can therefore be considered a community based on their content, they do not strongly interact with each other nor behave the same, and so belong to different social groups with respect to interactions and behavior. This illustrates that the topics discussed by these users only define one facet of the users complex social network.
Another interesting example is a community whose topics tend to focus on Denver and Colorado. These users do not belong to the same community in the interaction-based network, but most of them do belong to the same community in the activity-based network. This indicates that these users react to the same events and issues regarding Colorado and are therefore strongly connected in the topic-based and activity-based networks, but at the same time they do not directly interact with each other and are therefore more loosely connected in the interaction-based networks, where they belong to different communities.
It is interesting to examine the intersection between this topic-based community and the overlapping activity-based communities and explore which users in the intersection of these community types have the highest APC. Users with the highest APC in an activity-based community would provide the highest reduction in uncertainty on average about that communities activity or inactivity on Twitter. Interestingly, among the highest APC users in these communities-those with highest transfer entropy on average-we find @Colorado, which is the state official Twitter account, @ConnectColorado, a page created to connect Coloradans, and the CBS Denver account, a popular news agency. This means that these accounts have high activity-based predictive capacity in terms of their followers' activity on Twitter. This is not surprising as these activity-based communities heavily overlapped with the topic-based communities that discuss Colorado (mentioned above) and these accounts provide information on this topic. This provides a brief proof of concept that the APC measure makes sense and suggests this could be a useful measure for identifying individuals who possess high activity-based predictive capacity for a communities activity on a social media network. However, more work needs to be done to quantify this more fully. At the individual-based level, we also found interesting examples of multifaceted community membership, which show that asking multiple questions leads to finding different kinds of communities a user belongs to. For example, the user @JohnReaves belongs to two different topic communities: one about health care and the other about 'innovation' and 'creativity'. There was a 70% overlap between the innovation and creativity topic community, the interaction-based, and the activity-based community the user is in, whereas there is almost no overlap between the latter two and the healthcare community. This means that the user interacts with and behaves like the people talking about innovation and creativity, but even though he also talks about healthcare he does not interact with (nor behave like) the people talking about that. Another interesting example is given by user @jenajean. She belongs to two different topic communities: she talks about NASA and space but she does not interact nor behave like people belonging to that topic community, whereas she also talks about leadership, and the people she interacts with highly overlap with the people belonging to the topic-based community on leadership for which she belongs to. Lastly, @rajean, @Tekee, @Kimbirly and @mamamonroe are examples of users belonging to the topic community talking about Denver and Colorado. The activity-based community they belong to highly overlaps with the topic community (therefore they behave like the people that talk about Colorado), but they all belong to different interaction-based communities, each of which does not overlap in more than two users with the topic community.
Conclusion and Future Work
In this study, we have demonstrated that the communities observed in online social networks are highly question-dependent. The questions posed about a network a priori have a strong impact on the communities observed. Moreover, using different definitions of community reveal different and interesting relationships between users. More importantly, we have shown that these different views of the network are not revealed by using the structural network or any one weighting scheme alone. By varying the questions we asked about the network and then deriving weighting schema to answer each question, we found that community structure differed across community types on both the macro (e.g., number of communities and their size distribution) and micro (e.g., specific memberships, comemberships) scale in interesting ways.
To verify the validity of these communities we demonstrated that boundaries between communities represent meaningful internal/external divisions. In particular, conversations (e.g., retweets and mentions) and topics (e.g., hashtags) tended to be most highly concentrated within communities. We found this to be the case even when the communities were defined by a different criterion from the edge weights under study.
At first glance the boundaries defined by the activity-based communities derived from the transfer entropy weighting seemed less meaningful. However, upon further investigation our novel use of transfer entropy for the detection of activity-based communities highlighted an important fact about this social network: information transmission tended to be higher across community boundaries than within them. This result echos the 'strength of weak ties' theory from [37], which has found empirical support in online social networks [6]. This means that our use of transfer entropy not only defines boundaries that are meaningful divisions between communities but also illustrates that users who have a strong APC with a community need not be a member of that community.
Our findings may have important implications to a common problem in social network analysis: identification of influential individuals-but further work must be carried out. Many network measures of influence are based on the various types of centrality (degree, betweenness, closeness, eigenvector, etc.) [42]. Most centralities depend explicitly on the structure of the network under consideration. But we have seen in our study that the structural network only highlights one of many possible ways that users interact with online social media. Thus, a naïve application of centrality measures to a structural network may be answering a different question than the one motivating influence detection. For example, a user exhibiting high closeness centrality based on the structural network may be able to make a message visible to most other users in a social network, but this does not mean that those other users are likely to respond to or propagate that message. The importance of correctly framing network measures of influence has been explored previously [43], and our work further highlights its importance.
Based on our preliminary results, we conjecture that weighted generalizations of these centrality measures using transfer entropy might lead to better insights about who is actually influential in an online social network. It is interesting to note for example that two of the Forbes "Top 10 Social Media Influencers", happen to be in our network, viz., Ann Tran and Jessica Northey, and transfer entropy also quantified these two users as having two of the highest APC on average, i.e., there was a high reduction in uncertainty about their followers activity on Twitter given their tweet histories. While we do not believe this information theoretic measure can capture social influence in its entirety, this suggests that this activity-based measure may be useful in finding influential members purely based on their temporal tweet history; even completely ignoring both tweet content and social status. However, a deeper analysis would need to be carried out to verify this finding. In addition to exploring this phenomenon further, we plan to explore a broader selection of choices for both the transfer-entropy lag and tweet history time resolution. We believe that by doing an in-depth analysis of both of these parameters we can discover interesting activity-based communities that occur on much broader time scales.
This work demonstrates the utility of a multifaceted question-oriented approach to community detection. This work shows that crafting several facet-driven weighting schema, doing community detection for each weighting scheme and then comparing the similarities and differences across community types, is an unavoidable process for uncovering the complex-and often hidden-community structure present in online social networks. More generally, this work illustrates that without a clear definition of community-or even only using a single definition of community-many rich and interesting communities present in online social networks remain invisible. Multifaceted question-oriented community detection can bring those hidden communities into the light.
Supporting Information S1 Text. Transfer entropy and its estimation from data. (PDF)
Fig 1 .
1The proportion of communities greater than s in size, across the different community types. Note the logarithmic scale on the horizontal and vertical axes. doi:10.1371/journal.pone.0134860.g001
Fig 2 .
2The normalized mutual information between the coverings inferred from the different community types. Values of normalized mutual information close to 1 indicate similarity in the community structure, while values close to 0 indicate dissimilarity. The normalized mutual information is computed with singletons and orphan nodes included. Note the block-diagonal structure, indicating the strong relationship between question type and community membership.doi:10.1371/journal.pone.0134860.g002
Fig 3 .
3A schematic of how, given a covering, the edges of the network can be partitioned using (8) into inter-edges, intra-edges, and mixed-edges.
doi:10.1371/journal.pone.0134860.g003
Fig 4 .
4The density of edge weights for different community types (rows) and weight types (columns). The red, blue, and purple values below each collection of densities indicate the median of weight on intra-, inter-, and mixed-edges, respectively. doi:10.1371/journal.pone.0134860.g004
Table 1 .
1Number of non-singleton communities and singletons by community type: S(tructural), A (ctivity-based), T(opic-based), and I(nteraction-based).Community Type
# of Communities
# of Singletons
S
201
308
A, Lag 1
101
951
A, Lag 2
99
600
A, Lag 3
106
611
A, Lag 4
105
668
A, Lag 5
107
632
A, Lag 6
106
642
T
289
1064
I
252
2436
doi:10.1371/journal.pone.0134860.t001
PLOS ONE | DOI:10.1371/journal.pone.0134860 August 12, 2015 5 / 20
PLOS ONE | DOI:10.1371/journal.pone.0134860August 12, 2015
PLOS ONE | DOI:10.1371/journal.pone.0134860 August 12, 2015 20 / 20
AcknowledgmentsThanks to Cesar Flores, Luís Seoane, Kevin Stadler, Jody Wright, and Nix Barnett for their contributions to preliminary ideas related to this paper and to Michelle Girvan, William Rand, Aaron Clauset, and Ryan James for their valuable comments and suggestions. Finally, we would like to acknowledge the Santa Fe Institute and its Complex Systems Summer School for providing the intellectually stimulating environment where this project began.Author Contributions
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| []
|
[
"A new analytic method with a convergence- control parameter for solving nonlinear prob- lems",
"A new analytic method with a convergence- control parameter for solving nonlinear prob- lems"
]
| [
"Xiaolong Zhang ",
"Songxin Liang "
]
| []
| []
| In this paper, a new analytic method with a convergencecontrol parameter c is first proposed. The parameter c is used to adjust and control the convergence region and rate of the resulting series solution. It turns out that the convergence region and rate can be greatly enlarged by choosing a proper value of c. Furthermore, a numerical approach for finding the optimal value of the convergence-control parameter is given. At the same time, it is found that the traditional Adomian decomposition method is only a special case of the new method. The effectiveness and applicability of the new technique are demonstrated by several physical models including nonlinear heat transfer problems, nano-electromechanical systems, diffusion and dissipation phenomena, and dispersive waves. Moreover, the ideas proposed in this paper may offer us possibilities to greatly improve current analytic and numerical techniques.Mathematics Subject Classification (2010). 34E10; 35B20; 76M45. | null | [
"https://arxiv.org/pdf/1609.01550v1.pdf"
]
| 119,558,653 | 1609.01550 | e4f05c331d8fb0fefff30160888cc7dda39314bd |
A new analytic method with a convergence- control parameter for solving nonlinear prob- lems
2 Sep 2016
Xiaolong Zhang
Songxin Liang
A new analytic method with a convergence- control parameter for solving nonlinear prob- lems
2 Sep 2016Adomian decomposition methodconvergence-control param- eteracceleration of convergenceoptimal valuenonlinear problems
In this paper, a new analytic method with a convergencecontrol parameter c is first proposed. The parameter c is used to adjust and control the convergence region and rate of the resulting series solution. It turns out that the convergence region and rate can be greatly enlarged by choosing a proper value of c. Furthermore, a numerical approach for finding the optimal value of the convergence-control parameter is given. At the same time, it is found that the traditional Adomian decomposition method is only a special case of the new method. The effectiveness and applicability of the new technique are demonstrated by several physical models including nonlinear heat transfer problems, nano-electromechanical systems, diffusion and dissipation phenomena, and dispersive waves. Moreover, the ideas proposed in this paper may offer us possibilities to greatly improve current analytic and numerical techniques.Mathematics Subject Classification (2010). 34E10; 35B20; 76M45.
Introduction
Nonlinear problems occur in almost every field in science and engineering. The exact solutions are very difficult to obtain for most strong nonlinear differential equations. To solve these problems, several analytic methods such as the traditional perturbation method [1], Lyapunov's artificial small parameter method [2], the δ−expansion method [3] and the Adomian decomposition method (ADM) [4] have been proposed. The Adomian decomposition method, which was first proposed by George Adomian in the 1980s, is a powerful approach for seeking series solutions to linear and nonlinear problems. With the development of computer softwares such as Maple and Mathematica, the advantages of the method for solving complicated nonlinear problems are more and more obvious. It has been successfully applied to solve many problems in physical sciences such as the nonlinear Klein-Gordon equation [5], the Lane-Emden problem [6], the hydromagnetic peristaltic flow [7], the nonlinear fin problem [8] and the plate flow [9]. To make the method more effective, many researchers have done lots of improvements, for example, giving the noise terms [10], comparing with other methods [11,12], discussing the convergence rate [13], analysing the error [14], modifying the recursive scheme [15,16]and indicating the essence [17]. Recently, Duan and his colleagues introduced a convergence parameter into the ADM firstly [18,19,20].
For many real-world problems, one expects larger convergence regions and rates for the resulting series solutions. However, the traditional ADM usually cannot reach the goal, as shown in Section 4.1 below. In the present paper, we introduce a new more general method called Adomian Decomposition Method with a convergence-control Parameter (ADMP), which can be used successfully to achieve the goal and is different with various modifications of the ADM by other researchers. Furthermore, the method offers a family of solutions depending on the convergence-control parameter c, providing a chance to choose the optimal series solution. It turns out that the convergence region and rate of the resulting series solution can be greatly enlarged by choosing a proper value of c.
In Section 2, we present a brief review of the traditional ADM. In Section 3, the ADMP is described explicitly. Moreover, a numerical approach for finding the optimal value of the convergence-control parameter is also given. To demonstrate the effectiveness and applicability, several nontrivial applications including nonlinear ODEs and nonlinear PDEs are discussed in Section 4.
Basic ideas of the traditional ADM
We shall begin to consider the nonlinear differential equation
L[u] + R[u] + N [u] = f (t), t ∈ Ω (2.1)
with some initial/boundary conditions, where u = u(t), L, R, N and f (t) denote some unknown function, an easily invertible linear operator (usually the highest-order linear operator), the remainder of the linear operator, the nonlinear operator and the given source term. The linear operator L is designed for performing the inverse easily, but the choice is not unique. For example, for the Lane-Emden equation u ′′ (t) + p t u ′ (t) + f (u(t)) = 0, one can choose either the highest-order differential operator d 2 dt 2 (·) or t −p d dt (t p d dt (·)) as L. The nonlinear operator N is analytic. Multiplying (2.1) by L −1 , one obtains
L −1 [L(u)] = L −1 [f (t)] − L −1 [R(u)] − L −1 [N (u)]. (2.2)
According to the definition of integral operator, one has
L −1 [L(u)] = u − φ(t),(2.3)
where φ(t) is determined by the initial/boundary conditions mentioned before. Combining Eq. (2.2) with Eq. (2.3), one gets
u = L −1 [f (t)] + φ(t) − L −1 [R(u)] − L −1 [N (u)]. (2.4)
In the traditional ADM, the solution u(t) is expressed as the decomposition series
u(t) = +∞ k=0 u k (t), (2.5)
where u k (t), k ≥ 0 are determined by the recursive scheme
u 0 (t) = L −1 [f (t)] + φ(t), u k (t) = −L −1 [R(u k−1 (t))] − L −1 [A k−1 ], k ≥ 1,(2.6)
and
A k = 1 k! d k dǫ k N +∞ i=0 u i (t)ǫ i ǫ=0 , k ≥ 0 (2.7)
are decomposition polynomials.
To specify the decomposition polynomials in Eq. (2.7), one has
A 0 = N [u 0 (t)], A 1 = N ′ [u 0 (t)]u 1 (t), A 2 = N ′ [u 0 (t)]u 2 (t) + 1 2! N ′′ [u 0 (t)]u 2 1 (t),A 3 = N ′ [u 0 (t)]u 3 (t) + N ′′ [u 0 (t)]u 1 (t)u 2 (t) + 1 3! N ′′′ [u 0 (t)]u 3 1 (t), . . .
It is seen that A k depends on A i , i = 0, 1, · · · , k−1 and the nonlinear operator N . In practice, instead of finding a solution expression (2.5), one can only calculate an nth-order approximation ψ n (t) = n k=0 u k (t), (2.8) of which u k (t), k = 0, 1, · · · , n can be calculated recursively via (2.6).
Description of the new ADMP
Based on the traditional ADM, a convergence-control parameter c and a artificial parameter ǫ are introduced to Eq. (2.1), so a new equation is
L[u] + ǫc + ǫ 2 (1 − c) (R[u] + N [u]) = f (t). (3.1)
One then set
u(t) = +∞ k=0 v k (t, c)ǫ k . (3.2) Multiplying (3.1) by L −1 , one has u(t) = − ǫc + ǫ 2 (1 − c) L −1 (R[u] + N [u]) + L −1 [f (t)] + φ(t). (3.3) Substituting (3.2) into (3.
3) and collecting the coefficients of powers of ǫ, a new recursive scheme is obtained
v 0 (t, c) =L −1 [f (t)] + φ(t), v 1 (t, c) = − cL −1 [R(v 0 (t, c)) + B 0 ] , v k (t, c) = − cL −1 [R(v k−1 (t, c)) + B k−1 ] − (1 − c)L −1 [R(v k−2 (t, c)) + B k−2 ] , k ≥ 2 (3.4)
where the Adomian polynomials are given by
B k = 1 k! d k dǫ k N +∞ i=0 v i (t, c)ǫ i ǫ=0 , k ≥ 0. (3.5)
To specify the polynomials in Eq. (3.5), one has
B 0 = N [v 0 (t, c)] = N [u 0 (t)] = A 0 , B 1 = N ′ [v 0 (t, c)]v 1 (t, c) = cN ′ [u 0 (t)]u 1 (t) = cA 1 , B 2 = N ′ [v 0 (t, c)]v 2 (t, c) + 1 2! N ′′ [v 0 (t, c)]v 2 1 (t, c) = c 2 A 2 + (1 − c)A 1 , B 3 = N ′ [v 0 (t, c)]v 3 (t, c) + N ′′ [v 0 (t, c)]v 1 (t, c)v 2 (t, c) + 1 3! N ′′′ [v 0 (t, c)]v 3 1 (t, c) = c 3 A 3 + 2c(1 − c)A 2 , . . .
The relationship between the new recursive scheme (3.4) and the traditional recursive scheme (2.6) is as follows.
v 0 (t, c) = u 0 (t), v 1 (t, c) = cu 1 (t), v 2 (t, c) = c 2 u 2 (t) + (1 − c)u 1 (t), v 3 (t, c) = c 3 u 3 (t) + 2c(1 − c)u 2 (t),
. . .
When ǫ = 1 in (3.2), the solution u(t) with convergence-control parameter c is presented by v(t, c) = +∞ k=0 v k (t, c).
(3.6) Similar to Eq. (2.8), one can only calculate an nth-order approximation
ψ n (t, c) = n k=0 v k (t, c). (3.7)
Next, an approach for determining the optimal value of the convergencecontrol parameter c for the nth-order approximation is given. The optimal c is determined by solving the equation
∂E(c) ∂c = 0, (3.8)
where the averaged residual error E(c) of the approximation (3.7) on Ω is defined by
E(c) = 1 M M k=1 [(L + R + N ) (ψ n (t k , c)) − f (t k )] 2 ,
and t 1 , t 2 , · · · , t M ∈ Ω are sample points.
Remark 3.1. It is easily seen that the traditional ADM is only a special case of the new approach ADMP when c = 1.
Applications
A nonlinear heat transfer problem
Consider the nonlinear heat transfer problem governed by a nonlinear ordinary differential equation [21] (1 + ǫu(t)) u ′ (t) + u(t) = 0, u(0) = 1,
(4.1)
where ǫ is a physical parameter, and the prime denotes differentiation with respect to the time t. The closed-form solution is unknown, but one can get
u ′ (0) = − 1 1 + ǫ . (4.2)
In the following, the traditional perturbation method, the traditional ADM and the new ADMP are applied in sequence to seek series solutions to u ′ (0), and comparisons between them are made.
Regarding ǫ as a perturbation parameter, one writes u(t) as a perturbation series
u(t) = u 0 (t) + ǫu 1 (t) + ǫ 2 u 2 (t) + ǫ 3 u 3 (t) + · · · .u(t) = e −t + ǫ e −t − e −2t + ǫ 2 1 2 e −t − 2e −2t + 3 2 e −3t + · · · . (4.4)
The derivation of the perturbation solution (4.4) at t = 0 is
u ′ (0) = −1 + ǫ − ǫ 2 + ǫ 3 − ǫ 4 + ǫ 5 − ǫ 6 + ǫ 7 − ǫ 8 + ǫ 9 − ǫ 10 + · · · . (4.5)
Obviously, when ǫ ≥ 1 the series (4.5) is divergent, as shown in Fig. 1. Next, we analyse the problem by the traditional ADM. Following the procedure outlined in Section 2, one obtains u 0 (t) = 1,
u 1 (t) = −t, u 2 (t) = ǫt + 1 2 t 2 , u 3 (t) = −ǫ 2 t − 3 2 ǫ t 2 − 1 6 t 3 ,
. . . The derivation of the ADM solution u(t) at t = 0 is
u ′ (0) = −1 + ǫ − ǫ 2 + ǫ 3 − ǫ 4 + ǫ 5 − ǫ 6 + ǫ 7 − ǫ 8 + ǫ 9 − ǫ 10 + · · · , (4.7)
which is the same as Eq. (4.5) given by the perturbation method, as shown in Fig. 1. Finally, the ADMP is used to solve the problem (4.1). By means of (3.4), one has v 0 (t, c) = 1,
v 1 (t, c) = −ct, v 2 (t, c) = c − 1 + c 2 ǫ t + 1 2 c 2 t 2 , v 3 (t, c) = 2cǫ − 2c 2 ǫ − c 3 ǫ 2 t + c − c 2 − 3 2 c 3 ǫ t 2 − 1 6 c 3 t 3 ,
. . . When c = 1, they are the same as those in (4.6) given by the ADM.
The nth-order approximation of u ′ (0) is
ψ ′ n (0, c) = n k=0 v ′ k (0, c),(4.9)
where v ′ k (0, c), k = 0, 1, · · · , n can be calculated via (4.8) and the prime denotes differentiation with respect to t. For example, when n = 1, 2, 3, · · · , one has It is easy to see that the 1st-order ADMP approximation ψ ′ 1 (0, c) when c = 1 1+ǫ coincides with the exact solution (4.2), whereas the 9th-order perturbation approximation and the 10th-order ADM approximation diverge when ǫ ≥ 1, as shown in Fig. 1.
ψ ′ 1 (0, c) = −c, ψ ′ 2 (0, c) = −1 + c 2 ǫ, ψ ′ 3 (0, c) = −1 + 2c − c 2 ǫ − c 3 ǫ 2 , . . .
A nonlinear electrostatic cantilever NEMS model
Beam-type electrostatic actuators have been widely used to construct nanoelectromechanical system (NEMS). Over the last decades, the fundamental and applied researches as well as engineering developments in NEMS have undergone major improvements [20,22].
A beam-type NEMS actuator is modeled by a beam of length L, width w with a uniform cross section and thickness h, which is suspended over a conductive substrate and separated by a dielectric spacer. In this paper, we discuss the cantilever NEMS. Based on the Euler-Bernoulli beam assumptions, the governing equation for a beam actuator with the first-order fringing field correction may be expressed as
d 4 Y dX 4 = 1 (s − Y ) 2 + 0.65 w(s − Y ) F + 1 EI F k , k = 3, 4,(4.10)
where Y, X, I, E and F k denote the deflection of the beam, the position along the beam as measured from the clamped end, the moment of inertia for the cross-sectional area of the beam, effective material modulus and intermolecular/quantum force per unit length of the beam respectively and F = ǫ 0 wV 2 /2EI with ǫ 0 = 8.854 × 10 −12 C 2 N −1 m −2 the dielectric constant of air, s the original gap between the two electrodes without deflection and V the applied voltage. The effective material modulus will become the plate modulus E/(1 − ν 2 ), when the width satisfies w ≥ 5h. The Van der Waals force [23] is
F 3 = Aw 6π(s − Y ) 3 ,(4.11)
with A the Hamaker constant. The Casimir force [24] is
F 4 = π 2 vw 240(s − Y ) 4 ,d 4 y(x) dx 4 = α k (1 − y(x)) k + α 2 (1 − y(x)) 2 + α 1 1 − y(x) , k = 3, 4, (4.13) where α 4 = π 2 L 4 hvw 240EIs 5 , α 3 = L 4 Aw 6πEIs 4 , α 2 = L 4 ǫ 0 wV 2 2EIs 3 , α 1 = 0.65s w α 2 .
The boundary conditions are
y(0) = 0, y ′ (0) = 0, y ′′ (1) = 0, y ′′′ (1) = 0.
Setting u(x) = 1 − y(x), then Eq. (4.13) becomes
d 4 u(x) dx 4 + α k u(x) k + α 2 u(x) 2 + α 1 u(x)
= 0, (4.14)
with the boundary conditions
u(0) = 1, u ′ (0) = 0, u ′′ (1) = 0, u ′′′ (1) = 0. (4.15)
In this paper k = 3 is considered. It is assumed that α k = 0.2, α 2 = 0.5, α 1 = 0.25. Following the procedure outline in Section 3, one has
L[u(x)] + N [u(x)] = 0, (4.16) where L(·) = d 4 dx 4 (·), N [u(x)] = α 3 u(x) 3 + α 2 u(x) 2 + α 1 u(x) .
The inverse of the linear operator is taken to be
L −1 = x 0 x3 0 x2 1 x1 1 (·)dtdx 1 dx 2 dx 3 .v 0 (x, c) =1, v 1 (x, c) = − c 19x 2 80 − 19x 3 120 + 19x 4 480 , v 2 (x, c) = − 9139c 2 288000 − 19 80 + 19c 80 x 2 + 703 c 2 48000 + 19 120 − 19c 120 x 3 + − 19 480 + 19c 480 x 4 − 703c 2 x 6 576000 + 703c 2 x 7 2016000 − 703c 2 x 8 16128000 ,
. . .
Then the nth-order approximation is
ψ n (x, c) = n k=0 v k (x, c). (4.19)
For example, when n = 1, 2, · · · ,
ψ 1 (x, c) =1 − 19 cx 2 80 + 19 cx 3 120 − 19 cx 4 480 , ψ 2 (x, c) =1 − 9139 c 2 288000 + 19 80 x 2 + 703 c 2 48000 + 19 120 x 3 − 19 x 4 480 − 703 c 2 x 6 576000 + 703 c 2 x 7 2016000 − 703 c 2 x 8 16128000 ,
. . .
The boundary value problem (4.14)-(4.15) does not have a closed-form solution. To determine whether the ADM solution (when c = 1) is optimal or not, one uses the error remainder of the nth-order approximation ψ n (x, c) defined by 20) and the maximal error remainder of ψ n (x, c) defined by
R n (x, c) = L[ψ n (x, c)] + N [ψ n (x, c)],(4.M ER n = max 0≤x≤1 |R n (x, c)|. (4.21)
One takes x j = j/20(j = 1, 2..., 20) as sample points and solves Eq. (3.8). The optimal values of c for n = 1 to n = 10 and the corresponding maximal error remainders are calculated, as shown in the second and third columns of Table 1, and the maximal error remainders of the ADM approximations (when c = 1) are shown in the fourth column. It is seen that the ADM approximations are not optimal and the ADMP approximations are more accurate than the ADM approximations. To further demonstrate the accuracy of the ADMP approximation with the optimal value of c over the ADM approximation, the error remainder R 10 (x, c) of the 10th-order ADMP approximation when c = 1.21135 and the error remainder R 10 (x, 1) of the 10th-order ADM approximation are shown in Fig. 3. It is seen that the error remainder of the ADMP approximation is much smaller than that of the traditional ADM in the whole interval.
A nonlinear Burgers' equation
Burgers' equation is one of the significant tools for describing nonlinear diffusion and dissipation phenomena. It is widely used to model the hydrodynamic motion [25], soil-water flow [26] and nonlinear standing waves in constantcross-sectioned resonators [27]. In this part, a burgers' equation is considered with the form The linear operator is set as L = d dt (·). Following the procedure outlined in Section 3, one has the series solution v(x, t) = +∞ k=0 v k (x, t, c), (4.24) and an nth-order approximation
∂u(x, t) ∂t + u(x, t) ∂u(x, t) ∂x + ∂ 3 u(x, t) ∂x 2 ∂t = 0, u(x, 0) = x.ψ n (x, t, c) = n k=0 v k (x, t, c),(4.25)
where v k (x, t, c), k = 0, · · · , n can be obtained via (3.4). For example, when n = 1, 2, · · · ,
ψ 1 (x, t, c) = (1 − ct) x, ψ 2 (x, t, c) = 1 − t + c 2 t 2 x, ψ 3 (x, t, c) = 1 − t − c 2 t 2 + 2 ct 2 − c 3 t 3 x, ψ 4 (x, t, c) = 1 − t + t 2 + 2 c 3 t 3 − 3 c 2 t 3 + c 4 t 4 x, . . .
Consider the error of the nth-order approximation For n = 10, Fig. 4 shows that the error E 10 (x, t, 1) of the ADM approximation is very large. Therefore, the ADM approximation is far away from the exact solution and it is essential to introduce a convergence-control parameter c to make the results better.
By taking 20 × 20 sample points (x i , t j ) = (i/2, j/20), i, j = 1, 2, . . . , 20 and solving Eq. (3.8), one obtains the optimal value c = 0.8. Fig. 5 shows that the error E 10 (x, t, 0.8) of the ADMP approximation is much smaller than the error E 10 (x, t, 1) of the ADM approximation. To further demonstrate the accuracy of the ADMP approximation, the two approximation methods are contrasted in Fig. 2. Therefore, the ADMP approximation is much more accurate than the ADM approximation.
A nonlinear RLW equation
The RLW equation was first proposed by Peregrine for describing nonlinear dispersive waves. It has been used to model a large class of physics phenomena such as the regularised long wave [28], nonlinear shallow wave and longitudinal dispersive waves in elastic rods. The RLW equation has attracted much attention. Existence and uniqueness of the solution to the RLW equation has been given in [29]. Several numerical method such as finite-difference method, a lumped Galerkin method and septic splines have been introduced in the literature [30]. Consider the RLW equation with this form Following the procedure outlined in Section 3, one obtains the series solution v(x, t) = +∞ k=0 v k (x, t, c), (4.29) and an nth-order approximation ψ n (x, t, c) = n k=0 v k (x, t, c) (4.30) to the problem (4.27). For example, when n = 1, 2, · · · , one has ψ 1 (x, t, c) = x − ct − cxt,
∂u(x, t) ∂t + ∂u(x, t) ∂x + u(x, t) ∂u(x, t) ∂x + ∂ 3 u(x, t) ∂x 2 ∂t = 0, u(x, 0) = x.ψ 2 (x, t, c) = x − t − xt + c 2 xt 2 + c 2 t 2 , ψ 3 (x, t, c) = x − t − xt − c 2 xt 2 − c 2 t 2 + 2 cxt 2 + 2 ct 2 − c 3 xt 3 − c 3 t 3 ,
. . .
Similar to the problem in Section 4.3, one obtains the optimal value c = 0.81 for the 10th-order ADMP approximation. Fig. 6 and Fig. 7 show that the error E 10 (x, t, 0.81) of the ADMP approximation is much smaller that the error E 10 (x, t, 1) of the ADM approximation. Therefore, the ADMP approximation is much more accurate than the ADM approximation.
Conclusion
We have introduced a new convergence-control parameter into the traditional ADM, which can be used to adjust and control the convergence region and rate of the resulting series solution. Furthermore, we have proposed an approach for finding the optimal value of the convergence-control parameter. The traditional ADM is only a special case of the ADMP when c = 1. Usually, the nth-order approximation ψ n (x, t, c) is not optimal when c = 1. The ADMP is significantly helpful for obtaining more accurate approximation.
. (4.3) into Eq. (4.1), one obtains a perturbation solution
Figure 1 .
1Solid line: exact solution (4.2); dash line: 9th-order perturbation approximation; circle line: 10th-order ADM; solid circle line: 1st-order ADMP with c = 1 1+ǫ .
Figure 2 .
2Solutions
055 × 10 −34 Js the Planck's constant and v the speed of light. Substituting Eqs. (4.11), (4.12) and the expressions y = Y /s, x = X/L into Eq. (4.10), yields
Figure 3 .
3LEFT: R 10 (t, 1) for the ADM. RIGHT: R 10 (t, 1.21135) for the ADMP.
E
n (x, t, c) = ψ n (x, t, c) − u(x, t) ex . (4.26)
Figure 4 .
4E 10 (x, t, 1) of the traditional ADM approximation for(4.22).
Figure 5 .
5E 10 (x, t, 0.8) of the ADMP approximation for(4.22).
Figure 6 .
6E 10 (x, t, 1) of the traditional ADM approximation for(4.27).
Figure 7 .
7E 10 (x, t, 0.81) of the ADMP approximation for(4.27).
The M ER n of the ADMP and the ADMn
c
M ER n (ADM P ) M ER n (ADM )(c=1)
1 1.10733
1.01966E-1
2.69754E-1
2 1.21693
2.22708E-2
1.01782E-1
3 1.21761
2.99159E-3
4.26177E-2
4 1.21455
4.61594E-4
1.89559E-2
5 1.21285
7.58058E-5
8.77839E-3
6 1.21203
1.27843E-5
4.18515E-3
7 1.21165
2.17946E-6
2.03977E-3
8 1.21147
3.72997E-7
1.01153E-3
9 1.21139
6.38943E-8
5.08700E-4
10 1.21135
1.09425E-8
2.58802E-4
Table 1.
AcknowledgementThe authors would like to thank Professor Junsheng Duan (College of Science, Shanghai Institute of Technology) for his suggestions.
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[
"Topological asymptotic dimension",
"Topological asymptotic dimension"
]
| [
"Massoud Amini "
]
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| We initiate a study of asymptotic dimension for locally compact groups. This is shown to be finite for residually compact groups. We show that polycyclicby-compact groups and compactly generated, topologically virtually nilpotent groups are residually compact, and show that compactly generated nilpotent groups are polycyclic-by-compact. Along the way, the notion of Hirsch length is extended to topological groups and classical results of Hirsch and Malcev are extended using a topological version of the Poincaré lemma. We prove that for compactly generated, solvable-by-compact groups the asymptotic dimension is majorized by the Hirsch length, and equality holds for polycyclic-by-compact groups. We extend the class of elementary amenable groups beyond the discrete case and show that topologically elementary amenable groups with finite Hirsch length have finite asymptotic dimension. We prove that a topologically elementary amenable group of finite Hirsch length with no nontrivial locally elliptic normal closed subgroup is solvable-by-compact. Finally, we show that a totally disconnected, locally compact, second countable group has finite asymptotic dimension, if all of its discrete quotients are so. | null | [
"https://arxiv.org/pdf/2203.06868v1.pdf"
]
| 247,446,980 | 2203.06868 | c5fbb3a5e0f94621df391c585bb36646f3881111 |
Topological asymptotic dimension
14 Mar 2022
Massoud Amini
Topological asymptotic dimension
14 Mar 2022Received: date / Accepted: datearXiv:2203.06868v1 [math.DS] Noname manuscript No. (will be inserted by the editor)asymptotic dimension · Hirsch length · polycyclic-by-compact · solvable-by-compact · topologically virtually nilpotent · topologically elementary amenable Mathematics Subject Classification (2010) 53C23, 51F30
We initiate a study of asymptotic dimension for locally compact groups. This is shown to be finite for residually compact groups. We show that polycyclicby-compact groups and compactly generated, topologically virtually nilpotent groups are residually compact, and show that compactly generated nilpotent groups are polycyclic-by-compact. Along the way, the notion of Hirsch length is extended to topological groups and classical results of Hirsch and Malcev are extended using a topological version of the Poincaré lemma. We prove that for compactly generated, solvable-by-compact groups the asymptotic dimension is majorized by the Hirsch length, and equality holds for polycyclic-by-compact groups. We extend the class of elementary amenable groups beyond the discrete case and show that topologically elementary amenable groups with finite Hirsch length have finite asymptotic dimension. We prove that a topologically elementary amenable group of finite Hirsch length with no nontrivial locally elliptic normal closed subgroup is solvable-by-compact. Finally, we show that a totally disconnected, locally compact, second countable group has finite asymptotic dimension, if all of its discrete quotients are so.
Introduction
The study of finitely generated discrete groups has witnessed two historical developments. The first is celebrated work of Hirsch on the structure theory of infinite slovable groups in the 30's, where he initiated an invariant (now called the Hirsch length) and showed that many structural arguments could be based on induction M. Amini Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran E-mail: [email protected] on this length, quite like what is done for nilpotent groups using the niloptency class [20]. The second is pioneering work of Gromov in the 90's on asymptotic invariants of infinite groups [17]. These two developments are interestingly related, as the Hirsch length is known to majorize the Gromove asymptotic dimension in many cases [2], [10].
Little is known about both invariants beyond the discrete case. The Hirsch length has technical difficulties when one deals with topological groups. The main source of complication is the failure of the second isomorphism theorem for topological groups which makes the usual definition not well-defined. The Gromov invariant has a better situation, as it is defined as soon as we have a proper, translation invariant metric which generates topology of the group (so-called a plig metric [18]). However, it is not easy to control the invariant not to blow up.
The key tool to maintain such a control is to relate the two invariants, as in the discrete case. This is the main objective of the current paper.
Here instead of finitely generated discrete groups one has to work with compactly generated locally compact groups (usually second countable, to make sure a plig metric exists). There are several topological complications to be addressed, but the outcome is relatively satisfactory: the invariants are well-defined and finite in many interesting cases and interrelate to shed a light on the structure theory of certain natural classes of topological groups.
In Section 2 we introduce box spaces and define asymptotic dimensions of locally compact (second countable) groups and show that the invariant is finite for residually compact groups. In Section 3 we define the Hirsch length for topological groups and prove the Hirsch formula, which is critical for inductive arguments based on the Hirsch length. This section also deals with a natural extension of virtually nilpotent groups in the topological realm, which are shown to be residually compact and so of finite asymptotic dimension. Same is done in Section 4 for topologically elementary amenable groups. A small final epilogue in Section 5 is devoted to finiteness of the asymptotic dimension of totally disconnected, locally compact, second countable groups.
Box spaces and asymptotic dimension
In this section we extend the notion of Box space and its asymptotic dimension from the class of discrete residually finite groups to that of locally compact residually compact groups. This is not a trivial task and we have to overcome several topological technicalities.
Throughout the rest of this section, G is a locally compact, σ-compact, Hausdorff group and H ≤ G is a closed subgroup. We denote both the identity element and trivial subgroup of G by 1. For a complex or real valued function f on G, the left translate of f is defined by ℓ g f (h) = f (g −1 h), for g, h ∈ G. We use the standard notations H g := gHg −1 and g H := {hgh −1 : h ∈ H}, for g ∈ G. We also denote a finite subset E of G by E ⋐ G and a normal subgroup H of G by H G.
Recall that H is called cocompact if there is a compact subset K ⊆ G with G = KH. For a locally compact group G, a closed subgroup H is cocompact iff the quotient space G/H is compact (combine [11, Lemma 2.C.9], with the idea of the proof of [21,Proposition 18]). This is the topological analog of subgroups of finite index.
Definition 1 A residually compact approximation (RC-app.) of G is an decreasing sequence of cocompact closed subgroups G n ≤ G with n≥1 G n = 1. If G has an RC-app., we say that G is residually compact (RC).
We shall introduce an uncountable version of this notion in Section 4 which also applies to non compactly generated groups. The two notions are equivalent for locally compact, σ-compact (or equivalently, compactly generated) groups.
Definition 2 An RC-app. (G n ) of G is called regular if it satisfies any of the following equivalent conditions:
(i) for each g = 1 there is n ≥ 1 with G n g G G n = ∅, (ii) n≥1 g∈G G g n = 1; (iii) for each E ⋐ G there is n ≥ 1 such that the quotient map: G → G/G n is injective on Eg, for each g ∈ G.
The above equivalent conditions trivially hold when each G n is normal. When G is discrete, in a residually finite approximation (RF -app.) (G n ), the subgroups G n are of finite index. In this case, RF -approximations may always be chosen to be regular, as we may replace G n 's with smaller normal subgroups of finite index via the Poincaré lemma: For a discrete group G and subgroup H ≤ G of finite index n := [G : H], there is a normal subgroup N G with N ≤ H and [G : N ]| ≤ n!. The standard proof of Poincaré lemma uses the fact that |Sym(G/H)| = [G : H]! and that for N := ∩ g∈G H g , a copy of G/N embeds in Sym(G/H). This argument breaks down in the topological version, since while the above N is a closed normal subgroup contained in H and G/N continuously embeds in Iso(G/H), the group of isometries of the compact metric space G/H, the copy of G/N fails to be closed in Iso(G/H). For instance, G = SL(2, R) has many cocompact subgroups (cocompact lattices, as well as the upper triangular subgroup), but no proper normal cocompact subgroup. However the situation could be saved in the topological case by relaxing the cocompactness to a local version.
Definition 3 A closed subgroup H is called locally cocompact if G/H has zero asymptotic dimension with respect to the quotient metric induced by an adapted metric on G.
A few comments are in order: First that a closed normal subgroup N is locally cocompact iff the quotient group G/N is locally elliptic (i.e., its closed subgroups generated by compact subsets are compact). Since local ellipticity is indeed a local version of compactness in coarse geometric sense (rather than topological sense), this justifies the term used here. Second, G is throughout assumed to be σ-compact, and so it always has an adapted metric [11,Proposition 4.A.2], and each two such metrics are coarsely equivalent [11,Corollary 4.A.6(2)], and the same holds for the induced metrics on the quotient space, and so the above notion is well defined. Third, the quotient metric is not necessarily G-invariant (though the quotient space G/H is a disjoint union of σ-compact spaces-namely, the cosets of H-and so paracompact [9,Theorem 7.3], the action of G on G/H is not properunless, for instance H is compact-and so results such as [11, Proposition 4.C.8(1)] do not apply).
Lemma 1 (Poincaré) For each cocompact closed subgroup H ≤ G, if there is an adapted metric on G inducing a G-invariant compatible metric on G/H, then there is a locally cocompact closed normal subgroup N G with N ≤ H.
Proof Choose an adapted metric on G with G-invariant quotient metric on G/H. Consider the closed normal subgroup N := ∩ g∈G H g . Then G/H is a compact metric space on which G/N acts from left by
xN · gH = xgH (x, g ∈ G)
This is well defined as N is normal and isometric as the metric on G/H is left translation invariant. In particular, G/N continuously embeds in the compact metric group Iso(G/H). Though the image is not closed, but G/N inherits an adapted metric from the metric group Iso(G/H), which has zero asymptotic dimension (as it is compact), thus asdim(G/N )=0, and so it is locally elliptic by [11,Proposition 4.D.4].
Corollary 1 If G is a residually compact group with an RC-approximation {G n } such that each quotient space G/G n has a G-invariant compatible metric, then G also has a regular RC-approximation.
The assumption that the quotient space G/H has a G-invariant metric could be quite restrictive, for instance, in the above example where G = SL(2, R) and H is a cocompact lattice in G, there is no invariant metric on the orbit space G/H, since otherwise, by the above lemma G must have a locally cocompact normal subgroup N , but then since G = SL(2, R) is compactly generated, and so is G/N [11, Proposition 2.C. 8(4)]. This forces G/N to be compact, i.e., N must be a cocompact normal subgroup, which is not possible, as N is contained in a cocompact lattice.
The problem of existence of a G-invariant compatible (i.e., giving back the topology) metric on a Hausdorff topological group G was independently solved by Birkhoff [5] and Kakutani [24]: a topological group is metrizable if and only if it is Hausdorff and first countable, and in this case, the metric can be taken to be G-invariant (c.f., [29, section 1.22]). If we ask for a G-invariant compatible metric which is also proper (i.e., the balls are relatively compact), the problem is solved by Struble [38]: a locally compact group G has a proper G-invariant compatible metric if and only if it is second countable. The same questions on coset spaces are explored by Delaroche in [1]. Let H be a closed subgroup of a Hausdorff topological group G. When G is metrizable, the quotient space G/H has a quotient metric, however, even if G is a second countable, there does not always exist a G-invariant compatible metric on G/H (see, [1,Example 2.3] A closed subgroup H ≤ G is called a characteristic subgroup if it is invariant under Aut(G) consisting of continuous automorphisms of G. Since inner automorphisms are automatically continuous, each characteristic subgroup is normal (but the converse is not correct even in discrete groups, as for instance G×1 is a normal subgroup of G × G, but is not preserved under the flip automorphism).
One could use the same argument to prove (an extension of) Poincaré lemma for the characteristic subgroup N := σ∈Aut(G) σ(H). We record this result in the next lemma for future use.
Lemma 2 For each cocompact closed subgroup H ≤ G, if there is an adapted metric on G inducing a G-invariant compatible metric on G/H, then there is a locally cocompact closed characteristic subgroup N G with N ≤ H. In particular, if G is abelian, every closed cocompact subgroup H ≤ G contains a locally cocompact closed characteristic subgroup N .
The fact that in the above lemma gives the existence of locally cocompact closed characteristic subgroups for locally compact, σ-compact, abelian groups seems annoying, but in fact it is quite useful, since for virtually nilpotent groups (which are the main objects of interest in this section), one could reduce to the case of abelian groups (using an inductive argument) and use the above result.
the are Let τ = (H n ) and σ = (G n ) be decreasing sequences of cocompact subgroups of G, then we write τ σ if for each n ≥ 1 there is m ≥ 1 with G m ⊆ H n , and write τ ∼ σ if τ σ and σ τ . The set Λ(G) of all RC-apps. of G is upward closed w.r.t. . A maximum element (if any) is called a dominating RC-app. It follows from the above lemma that every compact subgroup H ≤ G is part of a regular RC-app., thus a regular RC-app. (G n ) is dominating iff for each compact subgroup H ≤ G, there is n ≥ 1 with G n ≤ H. Dominating RC-apps. are important as they give the lowest possible asymptotic dimension (among all RCapps.). The corresponding box space is denoted by ✷ s G and is called the standard box space. Note that (as in the discrete case) asdim(✷ σ G) is well defined (c.f. [39,Remark 3.21].)
While each finitely generated RF group has a dominating regular RF -app. (consisting of normal subgroups), this may fail for compactly generated RC groups and regular RC-apps. This is because the former is a consequence of the fact that finitely generated RF groups have at most countably many normal subgroups of finite index, while compactly generated RC groups may have uncountably many normal cocompact subgroups (Of course, for RC groups with at most countably many normal cocompact subgroups, one could always find a dominating regular RC-app.)
Next let G act (from right) by isometries on a metric space (X, d). The action is locally bounded if K · x is bounded for every x ∈ X and every compact set K ⊆ G and it is bounded if every orbit is bounded. The action is (metrically) proper if d(x · g, x) → ∞, for every x ∈ X as g → ∞ in G. Let π : X → X/G be the corresponding quotient map. Then D(π(x), π(y)) := inf g,h∈G d(x · g, y · h) defines a metric on the quotient space X/G. The next lemma is well-known.
Lemma 3 When the action is isometric, properly-discontinuous and cobounded, the metric D induces the quotient topology on X/G.
A length function on G is a map ℓ : G → R + which satisfies (i) ℓ(gh) ≤ ℓ(g) + ℓ(h); (ii) ℓ(g −1 ) = ℓ(g); (iii) ℓ(g) = 0 ⇔ g = 1, for each g, h ∈ G.
If G admits a locally bounded action by isometries on some metric space (X, d), then for every x ∈ X the map g → d(x · g, x) is a length function on G, which is bounded on compact subsets. It is known that G admits a proper length function (bounded on compact subsets) if and only if G is σ-compact [8]. Also every locally bounded action of G by isometries is either bounded or proper, iff every length function on G is either bounded or proper [8] (This is called property P L by Cornulier, see also [40]).
Length functions also naturally induce proper, left translation invariant metrics which generate the topology on G. Such a metric is called a plig metric by Haagerup and Przybyszewska [18]. It is easy to show that if a locally compact group G admits a plig metric, then G is second countable. Moreover, any two plig metrics on a second countable group G are coarsely equivalent, in the sense of Roe [36]. Lubotzky, Moser and Ragunathan showed that every compactly generated second countable group has a plig metric [26] (same holds for countable discrete groups by a result of Tu [41]). In the most general case, Haagerup and Przybyszewska showed that every locally compact, second countable group admits a plig metric (in a way that the balls have exponential growth w.r.t. the Haar measure). They also showed that the existence of a plig metric on G implies that G has bounded geometry.
Next let G be a second-countable RC group with a plig metric d G and for an RC-app. σ = (G n ), let d n be the corresponding quotient metric on G/G n . We define the box space of the RC-app. σ by
✷ σ G := n≥1 (G/G n , d n ),
with box metric d B which restricts to d n on G/G n and d B (G/G n , G/G m ) → ∞ as n + m → ∞. When d G is induced by a length function ℓ on G, one such box metric is given by d B (gG n , hG m ) := ℓ(g) + ℓ(h) + n + m, for g, h ∈ G and n, m ≥ 1.
The following lemma is proved as in the discrete case [39,Lemma 3.13].
Lemma 4 (Basic Lemma) Let G be RC with a given proper right invariant metric d and left Haar measures λ. Let G act on itself by right multiplication. For d ≥ 0 and a given RC-app. σ = (G n ), the following are equivalent: (i) σ is regular and asdim(✷ σ G) ≤ d;
(ii) for each R > 0 there is n ≥ 1 and a uniformly bounded cover U = {U 0 , · · · , U d } of G consisting of mutually disjoint G n -invariant sets, with Lebesgue number at most R;
(iii) for each ε > 0 and each E ⋐ G there is n ≥ 1 and functions
f i : G → [0, 1] with compact support K i , with K i ∩K i ·g = ∅, for g = 1, satisfying f i −ℓ g f i ∞ ≤ ε and d i=0 G n ℓ g f i dλ = 1 (g ∈ G), for 0 ≤ i ≤ d.
The family {f i } d i=0 is called a system of decay for σ. The next lemma follows immediately from definition.
Lemma 5 Let H ≤ G be a closed subgroup. For each regular RC-app. (G n ) of G, (H ∩ G n ) is a regular RC-app. of H.
Proposition 1 Given a regular RC-app. σ = (G n ) of G, (i) asdim(G) ≤ asdim(✷ σ G); (ii) for any closed subgroup H ≤ G and regular RC-app. τ = (H ∩ G n ) of H, asdim(✷ τ H) ≤ asdim(✷ σ G).
(iii) for any closed cocompact subgroup H ≤ G, asdim(✷ s H) = asdim(✷ s G).
Proof Parts (i), (ii) and the easy inequality in part (iii) follow from Lemmas 4 and 5. We just need to verify that asdim(✷ s H) ≥ asdim(✷ s G). Let d H be a plig metric on H. Take a compact set F H of representatives of G/H such that
F H × H → G; (x, h) → x · h
is bijective (Use the properness of the quotient map and take F H := q −1 (G/H)). Choose R > 0 large enough that F H ⊆ B R (e). Use Basic Lemma to find n ≥ 1 and collections W (0) , · · · , W (s) , with s = asdim(✷ s H, forming a uniformly bounded cover of H with Lebesgue number at most 8R such that each W (j) consists of (H ∩ G n )-right translation invariant disjoint sets, for
0 ≤ j ≤ s. Choose n large enough that G n ⊆ H. For 0 ≤ j ≤ s, put V (j) = {B −4R (W ) : W ∈ W (j) }, where B −4R (W ) := {h ∈ H : B 4R (h) ∩ H ⊆ W },
and observe that V (j) consists of G n -right translation invariant sets with minimal distance 4R. By the above bound on the Lebesgue number, any 4R-ball in H is contained in a member of W (j) 's, and so V (j) 's form a cover of H. Now U (j) := {B 2R (e)V : V ∈ V (j) } consists of disjoint G n -right translation invariant sets, and the Lebesgue number of the cover
U (0) ∪ · · · ∪ U (s) is at most R (for each g ∈ G there is x ∈ F H and h ∈ H such that g = xh and there is V ∈ V (0) ∪ · · · ∪ V (s) with h ∈ V . Since F H ⊆ B R (e), g ∈ B R (e)V and so B R (g) = B R (e)g ⊆ B 2R (e)V ∈ U (0) ∪ · · · ∪ U (s) , as claimed)
. This finishes the proof of the reverse inequality.
Topologically virtually nilpotent and polycyclic-by-compact groups
Recall that a group G is (r-step) nilpotent if there is a finite series
1 = G 0 ✂ G 1 ✂ · · · ✂ G r = G
of normal subgroups with [G, G i ] ≤ G i−1 , for i = 1, · · · , r. We say that G is topologically virtually nilpotent (TVN) if it has a nilpotent cocompact subgroup. Hirsch proved that every finitely generated (discrete) nilpotent group is polycyclic (i.e., has a finite series with cyclic factors) [20]. He also showed that polycyclicby-finite groups are residually finite (RF). We use the following series of lemmas which extend a classical result of Malcev on polycyclic-by-finite groups [28], to show the analog of Hirsch result for polycyclic-by-compact groups, i.e., groups with a polycyclic cocompact subgroup. For this purpose, we need the notion of Hirsch length for topological groups. Let G be a topological group and define the Hirsch length h(G) of G as the maximum number n (possibly infinity) of infinite cyclic factors in all possible normal series
1 = G 0 ✂ G 1 ✂ · · · ✂ G m = G
of closed subgroups of G. This is clearly well-defined. When G is discrete, this is the same as the number of infinite cyclic factors in any series
1 = G 0 ✂ G 1 ✂ · · · ✂ G n = G
with cyclic or finite factors, by the Jordan-Hölder-Schreier Theorem [3]. Unfortunately this could not be adapted to the topological case (with finite replaced by compact), as the second isomorphism theorem is not valid for topological groups (except under some extra conditions; c.f., [19,Theorem 5.3]). The argument in the algebraic case has two parts: each refinement of a series as above has the same number of infinite cyclic factors as the original series, and that any two such series have isomorphic refinements by the Jordan-Hölder-Schreier Theorem. It is indeed the second part which fails in the topological case, as the next lemma shows.
Lemma 6 The number of infinite cyclic factors in any series
1 = G 0 ✂ G 1 ✂ · · · ✂ G n = G
with cyclic or compact factors remains unchanged after refinement with closed subgroups.
Proof If we insert normal closed subgroups in the series as
G i+1 ✂ A ✂ B ✂ G i
to get a proper refinement, then either G i+1 is cocompact in G i , in which case A/G i+1 is compact as a closed subgroup of a compact group, and so are G i /B and B/A, as a quotient or a closed subquotient of G i /G i+1 , and nothing is added to the number of non-compact cyclic factors; or G i+1 is not cocompact in G i , in which case G i /G i+1 is an infinite cyclic group and has no proper infinite quotient, thus as we have started with a proper refinement, A has to be of finite index in G i and so is B. This forces G i+1 to be of infinite index in A and again the number of non-compact cyclic factors is not effected. This shows that each refinement of the above series has the same number of non-compact cyclic factors.
Let us see what goes wrong with refinements: take another series
1 = H 0 ✂ H 1 ✂ · · · ✂ H n = G,
and observe that, for each i,
1 = G i ∩ H 0 ✂ G i ∩ H 1 ✂ · · · ✂ G i ∩ H n = G i
is an ascending series of closed subgroups. This gives subnormal series
G i+1 = (G i ∩ H 0 )G i+1 ✂ (G i ∩ H 1 )G i+1 ✂ · · · ✂ (G i ∩ H n )G i+1 = G i ,
and repeating this for i = 0, · · · , m, we get a refinement of the first series with mn terms. Similarly, we could get a refinement of the second series with nm terms by inserting subnormal series
H j+1 = (H j ∩ G 0 )H j+1 ✂ (H j ∩ G 1 )H j+1 ✂ · · · ✂ (H j ∩ G m )H j+1 = H j ,
for j = 0, · · · , n. These refinements are algebraically isomorphic, since for subgroups Q, N, and L of G with L normal subgroup of Q satisfying qN = N q, for every q ∈ Q, we have QN/LN ≃ Q/L(Q ∩ N ) [3, Lemma 1]). This applies in the first refinement to Q := G i ∩ H j , N := G i+l , and L := G i ∩ H j+1 , and to the second refinement with interchanging the roles of the G i 's and H j 's. However, since one could not guarantee that QN/LN is locally compact, there is no way to show that it is also homeomorphic to Q/L(Q ∩ N ) (which is clearly locally compact).
On the other hand, there are situations where Hirsch length is well-behaved (i.e., it satisfies the Hirsch formula). This is the content of next lemma.
Lemma 7 If G is a topological group and H ✂ G is a closed normal subgroup, then h(G) ≥ h(H) + h(G/H).
If moreover, H is compact, cocompact, or open, then the equality holds.
Proof Let us first assume that h(H) = n and h(G/H) = m are both finite. Then there are normal series
1 = H 0 ✂ H 1 ✂ · · · ✂ H k = H
of closed subgroups of H with n infinite cyclic factors and normal series
H/H ✂ G 1 /H ✂ · · · ✂ G ℓ /H = G/H
of closed subgroups of G/H with m infinite cyclic factors. Since the quotient map:
G → G/H is open, each G i is a closed subgroup of G containing H. Now we get the subnormal series 1 = H 0 ✂ H 1 ✂ · · · ✂ H k−1 ✂ H ✂ G 1 ✂ · · · ✂ G ℓ = G with n + m infinite cyclic factors, thus h(G) ≥ n + m.
If one of the terms in right hand side of the inequality is infinite, say that of H, then again, for each n, there is a normal series of closed subgroups of H with n infinite cyclic factor, and by the above argument, h(G) ≥ n + m, for each n, thus h(G) = ∞, and we again have the inequality.
Next let us assume that N is compact. Then given a subnormal series
1 = G 0 ✂ G 1 ✂ · · · ✂ G n = G we get a series 1 ✂ G 1 N/N ✂ · · · ✂ G n−1 N/N ✂ G/N
where each G i N is a closed subgroup of G (since G i is closed and N is compact) and so locally compact. It is known that in this case, the second isomorphism theorem holds, that's is,
G i N/N ≃ G i /(G i ∩ N ), as topological groups [19, Theorem 5.3]. Also, G i+1 N/G i N ≃ G i+1 /G i (G i+1 ∩ N ). Now there is a group epimorphism: G i+1 /G i → G i+1 /G i (G i+1 ∩ N ) with kernel G i+1 ∩ N .
In particular, if G i+1 /G i is the infinite cyclic group, then G i+1 ∩ N has to be isomorphic to kZ, for some integer k, but this intersection is a compact group, so k = 0 and G i+1 N/G i N is the infinite cyclic group. This shows that h(G/N ) ≥ h(G), and since h(N ) = 0, equality (Hirsch formula) holds.
When N is cocompact, we get a series
1 ✂ G 1 ∩ N ✂ · · · ✂ G n−1 ∩ N ✂ N
and there is a group monomorphism:
(G i+1 ∩ N )/(G i ∩ N ) → G i+1 /G i , so if G i+1 /G i is the infinite cyclic group, (G i+1 ∩ N )/(G i ∩ N ) is either isomorphic to the infinite cyclic group, or is trivial. But if it is trivial, then G i+1 ∩ N = G i ∩ N . In this case, since N is cocompact in G, G i+1 ∩ N is cocompact in G i+1 , thus in the series of closed subgroups G i ∩ N ≤ G i ≤ G i+1
the first is cocompact in the last, and so is the second in the last, i.e., G i+1 /G i has to be compact, which contradicts our assumption. Thus (
G i+1 ∩ N )/(G i ∩ N ) is infinite cyclic, whenever G i+1 /G i is so, that is, h(N ) ≥ h(G)
, and since h(G/N ) = 0, again the equality holds. Finally, if N is open, then G/N is discrete, and given a subnormal series
1 = G 0 ✂ G 1 ✂ · · · ✂ G n = G in the series 1 ✂ G 1 N/N ✂ · · · ✂ G n−1 N/N ✂ G/N all factors G i N/N is discrete and G i N/N ≃ G i /(G i ∩ N ). Also G i N is open and G i+1 N/G i N ≃ G i+1 /G i (G i+1 ∩ N ), with both sides discrete. Again there is a group epimorphism: G i+1 /G i → G i+1 /G i (G i+1 ∩ N ) with kernel G i+1 ∩ N . If G i+1 /G i is the infinite cyclic group, then G i+1 ∩ N has to be isomorphic to kZ,
for some integer k. But then G i ∩ N also has to be of the form ℓZ, for some integer ℓ (deviding k). Since there is a group monomorphism:
(G i+1 ∩ N )/(G i ∩ N ) → G i+1 /G i , and G i+1 /G i
has no non-trivial finite subgroup, both k and ℓ cannot be non-zero. If k = 0, then ℓ = 0 and we get no infinite factor in the subnormal series of N at the i th position, but get one such factor in the subnormal series of G/N at the i th position. If k = 0, then ℓ = 0 and we get one infinite factor in the subnormal series of N at the i th position. This shows that
h(N ) + h(G/N ) ≥ h(G),
which finishes the proof. Now since the polycyclic groups have finite Hirsch length we immediately get the following result from the cocompact case of the above lemma. One could extend the notion of polycyclic groups by considering a family X of locally compact groups and call G a poly-X group if G has a finite subnormal series
1 = G 0 ✂ G 1 ✂ · · · ✂ G n = G
whose factors belong to X. The next lemma extends and is proved similar to [33, 10.2.4].
Lemma 8 If X a family of locally compact groups, stable under taking closed subgroups (resp. under taking quotients by closed normal subgroups), then so is the class of poly-X groups.
Lemma 9 Each poly-{cyclic, compact} group has a characteristic closed cocompact subgroup that is poly-{infinite cyclic}.
Proof We adapt the proof of [33, 10.2.5]. Consider the situation that N ✂ L is compact with L/N infinite cyclic. If L = N, x , for some x ∈ L, then some power x n of x with n ≥ 1 centralizes both N and x. Therefore, x n is central in L, and in particular, x n is a normal infinite cyclic cocompact closed subgroup of L. Now take a poly-{cyclic, compact} group G and take a finite subnormal series
1 = G 0 ✂ G 1 ✂ · · · ✂ G n = G
with cyclic or compact quotients. We argue by induction on i to show that each G i has a characteristic closed cocompact subgroup H i that is poly-{infinite cyclic}. This is trivial for i = 0, and if it holds for i, then since
G i ✂ G i+1 and H i is charac- teristic in G i , then H i ✂ G i+1 . Let us first observe that G i+1 has a normal closed cocompact poly-{infinite cyclic} subgroup. This is clear if G i+1 /G i is compact, thus we may assume that G i+1 /G i is infinite cyclic. In this case, L := G i+1 /H i has a compact normal subgroup N = G i /H i with L/N = G i+1 /G i infinite cyclic.
By the observation of the previous paragraph, L has a normal infinite cyclic cocompact closed subgroup. The inverse image M in G i+1 of this group under the quotient map is then a normal poly-{infinite cyclic} closed cocompact subgroup of G i+1 . In particular, M is finitely generated and so countable. Since M is locally compact in the relative topology of G, it has to be discrete. By the classical Poincare Lemma, M contains a closed cocompact characteristic subgroup of G i+1 (c.f., [37,Theorem 7.1.7]). Let us take H i+1 to be this subgroup, which is poly-{infinite cyclic} by Lemma 8. This finishes the argument of the inductive step, and the induction stops at G = G n .
Lemma 10 Each noncompact polycyclic-by-compact group has a closed nontrivial discrete finitely generated torsion-free abelian characteristic subgroup.
Proof Let G be a noncompact polycyclic-by-compact group. By Lemma 9, G has a closed cocompact poly-{infinite cyclic} characteristic subgroup H. Since G is not compact, H = 1. By definition, H is discrete, finitely generated and solvable. Take the derived series
H = H 0 ☎ H 1 ☎ · · · ☎ H n+1 = 1
for H, where n is minimal with H n+1 = 1, then H n is a nontrivial abelian characteristic subgroup of G. Finally, H n ≤ H is torsion free and hence infinite.
We have already used an inductive argument (based on the indices in subnormal series) in the proof of Lemma 9. Another instance of inductive argument, this time based on the Hirsch length, is used in the proof of the following extension of a classical result of Malcev for polycyclic-by-finite groups [28].
Lemma 11 (Malcev) If G is polycyclic-by-compact, then each closed subgroup H ≤ G is the intersection of all closed cocompact subgroups of G containing H.
Proof LetH be the intersection of all closed cocompact subgroups of G containing H. To show that K = H, we proceed by induction on the Hirsch length h(G). The result is trivial when G is compact, that is, h(G) = 0. When G is not compact, it has a non trivial discrete finitely generated torsion-free abelian normal subgroup A. Since A is abelian and torsion-free, the map x → x r is an a group monomorphism. Let us denote its range by A r . Then A r is a discrete (and so closed) normal subgroup of G. Also, since A is finitely generated and torsion-free, it is free abelian, that is, A is isomorphic to Z n , for some n (which is then non-zero, as A is not trivial). In particular, r≥1 A r = 1 and h(A) > 0. But then, since A and A r are isomorphic, h(A r ) > 0. It follows from Lemma 7 that h(G/A r ) < h(G). By the induction hypothesis applied to G/A r we have,
H ≤H ≤ r≥1 HA r ≤ HA, henceH ≤ H(A ∩ r≥1 HA r ) = H r≥1 (H ∩ A)A r ) = H(H ∩ A) = H,
which finishes the proof.
Lemma 12
Let G be polycyclic-by-compact and H be a closed subgroup. Then In particular, A has to be locally finite by [11, 4.D.2], which in turn contradicts [11, 2.E.17(3a)], as A has no non-trivial finite subgroup (since it is torsion-free). This proves the claim, which implies that h(H/B) = h(G/B) < h(G). Now by the induction hypothesis, we have H/B ≤ G/B is cocompact and so is H ≤ G.
h(H) = h(G) if H is cocompact. Conversely, if H a closed normal subgroup with h(H) = h(G),
Proposition 2 (Hirsch) Polycyclic-by-compact groups are residually compact (RC) and always have a regular RC-approximation.
Proof Take any cocompact closed subgroup L ≤ G and take N := g∈G L g which is the largest closed normal subgroup of G contained in L. Choose a compact subset F L ⊆ G of the representatives of the transversal decomposition G = x∈F L Lx into right L-cosets and observe that N = x∈F L L x is cocompact. Take any closed subgroup H ≤ G, and observe that in the notations of the Malcev lemma, Lemma 12 could be used to give an alternative proof of the above proposition as follows: Again we argue by induction on the Hirsch length h(G). We may assume that G is noncompact. Let A be a normal torsion-free infinite abelian subgroup of G given by Lemma 10. Then the set A r consisting of r-th powers of elements of A is a characteristic subgroup of A with finite index and r≥1 A r = 1. On the other hand, h(G/A r ) < h(G), for each r, and by the inductive assumption, G/A r is RC. Now since r≥1 A r = 1, it follows that G is also RC.
Next we want to show that compactly generated topologically virtually nilpotent groups are residually compact. We first need a modification of result due to Hirsch, who showed that finitely generated nilpotent groups are polycyclic [20].
Lemma 13 (Hirsch) compactly generated nilpotent groups are polycyclic-by-compact.
Proof We adapt the inductive argument of Hirsch. Let G be nilpotent of class r with a compact set S of generators. If r = 1 then G is a compactly generated abelian group, which is then polycyclic-by-compact by [30,Proposition 33]. If r > 1, then let G 1 be the last non-trivial term in the upper central series of G. Without loss of generality we may assume that G 1 is closed (otherwise use its closure instead) and so compactly generated by [11,Proposition 5.A.7]. Then G 1 is abelian and compactly generated and so polycyclic-by-compact, by the argument in case of r = 1. Let A ≤ G 1 be a polycyclic cocompact subgroup. By the argument of [30,Proposition 33], we may also take A to be closed. Use the same argument as in the proof of Lemma 9 to choose a cocompact closed characteristic subgroup B of G 1 contained in A. Then since A is polycyclic, so is B. Choose a compact subset K ⊆ G 1 with G 1 = KB. Since G/G 1 is compactly generated and nilpotent of class r − 1, by induction hypothesis it is polycyclic-by-compact. Choose a polycyclic cocompact subgroup N/G 1 and compact subset E ⊆ G/G 1 with G/G 1 = E(N/G 1 ). By [11, Lemma 2.C.9], we may choose a compact subset L ⊆ G with LN/G 1 = E. Thus G = LN . Finally, since N/G 1 is polycyclic, it is finitely generated, hence there is a finitely generated subgroup C of N with N = CG 1 . But C is also nilpotent as a subgroup of G, thus it is polycyclic-byfinite [20], that is C = F D, for a polycyclic group D and a finite subset F . Now we have
G = LN = CLG 1 = LF DG 1 = LF G 1 D = LF KBD,
where the fourth equality follows from the fact that G 1 is normal in N . Since B ≤ G 1 is a characteristic subgroup and G 1 ✂ G, we have D ✂ G, and so BD is a subgroup of G. But BD/D ≃ B/(B ∩ D) algebraically, and the right hand side is polycyclic as a quotient of a polycyclic group. Thus BD is polycyclic as an extension of a polycyclic group by another polycyclic group [37, 7.1.13(c)]. On the other hand, by the continuity of the product map, LF K is compact in G, and so BD is a cocompact subgroup of G. Therefore, G is polycyclic-by-compact and the inductive argument is complete.
Lemma 14
Each compactly generated topologically virtually nilpotent group contains a finitely generated torsion-free nilpotent cocompact subgroup.
Proof Let G be compactly generated and TVN. By definition, G has a closed nilpotent cocompact subgroup H. Then H is compactly generated [11, 2.C.8(3)] and so polycyclic-by-compact by Lemma 13. Choose a polycyclic cocompact subgroup A of H and observe that A is torsion-free-by-finite [43,Corollary 2.7]. The finite index torsion-free subgroup B of A would then be cocompact in G, finitely generated (as it is a finite index subgroup of the finitely generated group A), and nilpotent (as it is a subgroup of the nilpotent group H).
Note that in the above proposition we could not guarantee that G has a cocompact closed polycyclic subgroup, since though the closure of a cocompact subgroup is again cocompact, the closure of a polycyclic subgroup may fail to be polycyclic (e.g. take the dense polycyclic subgroup Z + αZ of R, for α irrational).
Lemma 13 and Proposition 2 prove the following extension of a classical result due to Hirsch (c.f. [43, 2.10, 2.13]) for topological groups.
Proposition 3 (Hirsch)
Topologically virtually nilpotent (TVN) compactly generated groups are residually compact (RC) and always have a regular RC-approximation.
Combining Propositions 1, 2, and 3, we get the first main result of this section.
Theorem 1 Topologically virtually nilpotent groups and polycyclic-by compact groups have standard box space.
Finally, using Lemma 14, we get the second main result of the section.
Theorem 2 Topologically virtually nilpotent compactly generated groups have finite asymptotic dimension.
Proof Let G be compactly generated and TVN. Choose a finitely generated torsionfree nilpotent cocompact subgroup A in G by Lemma 14. By [23,Theorem 5.2], there is an embedding of A as a subgroup into the unitary group U n (Z), for some n ≥ 2. But asdim ✷ s U n (Z) = n(n − 1)/2, for n ≥ 2 [39,Theorem 4.9]. Hence A has finite asymptotic dimension. Now, since A is a cocompact subgroup, G has finite asymptotic dimension by Proposition 1(iii).
In Section 4 we prove more general results of the above nature (Theorem 3). For that, we would need the following result.
Proposition 4 If G is a compactly generated, abelian-by-compact, locally compact group then asdim(G) = h(G).
Proof First assume that G is abelian. By [30,Proposition 33], there is a non negative integer n, and a cocompact subgroup A of G, topologically isomorphic to Z n . Since A is cocompact in G, we have asdim(G)=asdim(A) = n, by Lemma 15. On the other hand, by definition, h(G/A) = 0, and so h(G) = h(A) = n, by the Hirsch formula in the cocompact case.
Next consider the case where G is abelian-by-compact. Then exactly by the same argument as above, G has the same asymptotic dimension and Hirsch length as its cocompact abelian subgroup (which could clearly assumed to be closed and so locally compact) and the result follow from the previous case.
This plus Lemma 15 (proved in Section 4) shows the following more general result.
Corollary 3 If G is a compactly generated, solvable-by-compact, locally compact group then asdim(G) ≤ h(G).
We also have the following sharp estimate for case of polycyclic-by-compact groups.
Proposition 5 If G is polycyclic-by-compact, asdim(G) = h(G).
Proof Both the asymptotic dimension and Hirsch length are stable under passing to a cocompact subgroup, so we may assume that G is polycyclic, but the result in this case is already known [10, Theorem 3.5].
Elementary amenable groups
A discrete group is elementary amenable (EA) if it can be constructed from finite and abelian groups through the processes of direct unions and extensions. The class was first considered by von Neumann in the study of the Banach-Tarski paradox [42]. Although the class [EA] already contains non discrete groups (including all locally compact abelian ones), it is quite natural to extend the notion by adding some more topological flavor. We could not trace the following natural topological version in the literature: the class of topologically elementary amenable (TEA) groups is the smallest family of locally compact Hausdorff groups containing all compact and all locally compact abelian Hausdorff groups, and is closed under taking closed subgroups, quotients by closed normal subgroups, topological group extensions and increasing unions. Note that starting with a locally compact Hausdorff group and a closed (and normal if needed) subgroup, we end up a locally compact Hausdorff group in the first three cases, and the same holds in the fourth one if we take care of topology as follows: given an increasing family of locally compact Hausdorff groups indexed by a net such that each group sits in a group with larger index as an open subgroup, topologize the union by letting a subset of the union to be open iff its intersection with each of the groups in the family is open. This way we get a locally compact Hausdorff group again.
Let C, D be classes of locally compact groups (which are also assumed to be Hausdorff from now on, without further notice). Then the classes LC and CD are the class of "locally C" groups (that is, locally compact groups with all compactly generated locally compact subgroups in C) and that of C-by-D groups (that is, groups with a compact or cocompact normal subgroup in C whose quotient group is in D). Then the class [TEA] could be constructed by transfinite induction starting with X 0 = {1}, X 1 be the class of abelian-by-compact groups, and X α+1 := (LX α )X 1 , and for a limit ordinal β, X β := ∪ α<β X α . Then adapting an argument of Hillman and Linnel in [22], we have [TEA] is nothing but the union of all X α 's. Now, following the idea of Hillman and Linnel, we define the Hirsch length on the class X 1 of abelian-by-compact groups as in the previous section 1 , on groups in G ∈ LX α by h(G) := sup{h(H) : H ∈ X α }, and on groups G ∈ X α+1 having a compact or cocompact normal subgroup N ∈ X α with G/N ∈ X 1 , by h(G) = h(N ) + h(G/N ). Again the inductive argument of Let us compare and contrast the class [TEA] with other known classes of topological groups. Our reference for the properties of various classes discussed here is [32], from which we borrow class notations as well.
[TEA] clearly con- 1 We do not follow the footsteps of [12] here, where they define the Hirsch length of an abelian-by-finite group A as the module dimension dim Q (A ⊗ Q), as in [10]. This causes no problem as long as A is countable, but even for an innocent looking group like R, the module dimension is ∞, where as the Hirsch length-based on our definition in previous section-is 1 (c.f. [10,Remark 3]). The same of hesitation is needed when one defines different notions of asymptotic dimension. For instance the asymptotic dimension of R with respect to the coarse structure consisting of subsets E ⊆ R × R with {x − y : (x, y) ∈ E} finite, is ∞ [10, Remark 2], where as it is 1, if we replace "finite" by "relatively compact" in the above coarse structure. The main idea of this paper is that that such pathologies disappears if one take the topology into account when working with classes such as solvable-by-compact groups. tains the classes [Z] of central groups (groups whose quotient over center is compact) which is included in the larger class of extensions of abelian groups by compact groups. It does not contains the class [MAP] of maximally almost periodic groups (groups with enough finite dimensional irreducible representations to separate the points, or equivalently, enough continuous almost periodic functions to separate the points), as these are subgroup of compact groups and so contains many non amenable groups (this justifies our assumption that the class [ [32,Example 7]; also there are compactly generated solvable non Type I groups, c.f., [32,Example 17]; and connected solvable non Type I groups, c.f., [32,Example 19]). Another structural result of Grosser and Moskowitz [15,Theorem 3.16], states that for a group in [FC] − (topologically finite conjugacy class groups, that is groups relatively compact conjugacy classes) the quotient by the intersection of all compact normal subgroups is a direct product of a vector group and a discrete torsion free abelian group. This shows that In this section we adapt the argument in [12] to show that for second countable topologically elementary amenable groups, the asymptotic dimension of any box space is bounded above by the Hirsch length. Since we need a plig metric (see section 2) in order to make sense of asymptotic dimension, one first needs to restrict to second countable locally compact groups. Such a group is then automatically σcompact, and so compactly generated by [11, 2.C.6]. For the general case, we define asdim(G) as the supremum of asdim(K), over all compactly generated subgroups K ≤ G (or equivalently, over all compactly generated closed subgroups by [11, 2.C.2(5)]). This is then the same as the asymptotic dimension of G as a topological group endowed with its canonical left coarse structure [2,Theorem 2.8].
A locally compact group G is called locally elliptic if every compact subset of G is contained in a compact open subgroup. These are exactly those locally compact groups with zero asymptotic dimension [11, 4.D.4].
Lemma 15 Let G be a locally compact, compactly generated group. For a short exact sequence
1 → N → G → G/N → 1
with N compact or cocompact, we have asdim(G) ≤ asdim(N ) + asdim(G/N ), and equality holds if the exact sequence is split and N is locally elliptic.
Proof Since G is compactly generated, so are N and G/N by [11, 2.C.8]. Take any plig metric on G (no matter which one, as they are all coarsely equivalent) and observe that the quotient map q : G → G/N is an isometry (and so Lipschitz) with respect to the canonical quotient metric on G/N , and that the coarse fibres are translates of a neighborhood of N (and so coarsely equivalent to N ). Since G is coarsely equivalent to a geodesic metric space [11, 4.B.9], the desired inequality follows from [4,Theorem 29].
If N is locally elliptic, then asdim(N ) = 0 and asdim(G) ≤ asdim(G/N ). On the other hand, since the exact sequence is split (in the category of topological groups), G/N is isomorphic (as a topological group) to a closed subgroup of G, and so asdim(G) ≥ asdim(G/N ), and we get the equality.
The first main result of this section reads as follows.
Theorem 3 Let G be topologically elementary amenable (TEA), then asdim(G) ≤ h(G). In particular groups in [TEA] with finite Hirsch length have finite asymptotic dimension.
Proof Assume first that G is compactly generated. By Proposition 4 we have the the desired inequality (even equality) when G is in class X 1 . By lemma 7, the Hirsch length satisfies h(G) = h(N ) + h(G/N ), for any compact or cocompact normal subgroup N , whereas the asymptotic dimension satisfies asdim(G) ≤ asdim(N )+ asdim (G/N ), by Lemma 15. The result now follows by a transfinite induction on the ordinal α where G belongs to X α .
Next in the general case, we could calculate both the asymptotic dimension and Hirsch length by taking supremum of the same invariant over all compactly generated closed subgroups. Since being (TEA) passes to closed subgroups, the result follows from the previous case.
The second statement in the above theorem has a more structural version (Corollary 4), which extends [22,Corollary 1]. First we need the following adaptation of [22, Main Theorem]. We use the notation Λ(G) to denote the maximal locally elliptic closed normal subgroup of G, which exist by Zorn lemma (but might be trivial), is unique [11, 4.D.7(7)], and is called the locally elliptic radical of G.
Recall that an action of G on a topological space is called effective (or faithful) if for every element g = e in G, there exists x ∈ X with g · x = x.
Lemma 16
For each G in [TEA] of finite Hirsch length, G/Λ(G) has a maximal solvable normal closed cocompact subgroup.
Proof Assume first that G is compactly generated. Arguing by induction, let the result be true for such groups with Hirsch length at most n and let G be in [T EA] of Hirsch length n + 1. Then by transfinite induction, one could show that G has a normal closed subgroup K with G/K infinite and abelian-by-compact. Modding out the maximal compact normal subgroup, we may assume that G/K is abelian, but not compact. Since G/K is compactly generated [11, 2.C.8(4)], it is Z r -bycompact, for some r > 0 [30,Proposition 30]. In particular, h(G/K) = r > 0. Choose a subgroup H of G containing K such that H/K is a maximal abelian normal closed subgroup of G/K, then H is cocompact in G. Then the action of G/H on H/K (by conjugation) is effective, and r = h(G/K) < h(G) = n + 1. Since h(K) ≤ h(G) − r < n, by the inductive hypothesis, K has a normal cocompact closed subgroup L containing Λ(K) with L/Λ(K) a maximal solvable normal subgroup of K/Λ(K) say with derived length d. As both Λ(K) and L are characteristic in K, they are also normal in G. Moreover, Λ(K) = K ∩ Λ(G). Now the centralizer of K/L in H/L is a normal solvable closed cocompact subgroup of G/L with derived length at most 2. Finally, G/A(G) has a maximal solvable normal closed cocompact subgroup of derived length at most 2 + d, since it contains the pre-image of the centralizer of K/L in H/L. This completes the inductive argument in the cocompact case.
In general, let {G i } i∈I be the set of compactly generated subgroups of G. By the above argument, each G i has a normal closed subgroup H i containing Λ(G i ) such that H i /Λ(G i ) is the maximal solvable normal closed cocompact subgroup of G i /Λ(G i ) with derived length at most 2 + d, for d as in the previous case. Clearly H i is the maximal closed normal locally elliptic-by-solvable subgroup of G i . For any index j ∈ I, H i ∩ G j is a normal locally elliptic-by-solvable subgroup of G j , and so
H i ∩ G j ≤ H j . Also if G j ≤ G i , then H j ≤ H i .
Let H be the union of all H i 's and for x, y ∈ H and g ∈ G, choose indices i, j, k ∈ I with x ∈ H i , y ∈ H j , and g ∈ G k . Choose ℓ ∈ J such that G ℓ contains G i ∪ G j ∪ G k . Then x, y ∈ H ℓ (by the above argument) and so are xy −l and gxg −1 , that is, H is a normal (not necessarily closed) cocompact subgroup of G. Let D i be the 2+d derived subgroup of H i . Then D i is a locally elliptic normal subgroup of G i , and by an argument similar to the above, D := i∈I D i is a locally elliptic (not necessarily closed) normal subgroup of G. But the 2 + d derived subgroup of H is contained in D (as each iterated commutator involves only finitely many elements of H), thus by the algebraic isomorphism, HΛ(G)/Λ(G) ≃ H/(H ∩ Λ(G)), the left hand side is solvable of derived length at most 2 + d, and the inductive argument is complete in the general case.
As a result we get the following corollary which extends [ Proof This is proved by induction on n. If n = 0, then G is compact and so in X 1 . In general, if G is compactly generated and non compact, by the argument used in the proof of the above lemma, it has normal closed subgroups K ≤ H ≤ G such that H is cocompact in G and H/K is a free abelian of positive rank, h(K) < h(G)−1 and h(G/K) < h(G). By the induction hypothesis, K ∈ LX n−2 and G/K ∈ LX n−1 . Then H ∈ (LX n−2 )X 1 = X n−1 . Thus G ∈ X n−1 X 1 ⊆ X n . If G is not compactly generated, then all of its compactly generated subgroups are in X n by the above argument, and so G ∈ LX n .
Note that there are finitely generated, torsion free, elementary amenable groups which are not virtually solvable [20, page 162]. An argument based on [12, Lemma 2.2] proves the next lemma (c.f. the proof of [12,Proposition 3.7]).
Lemma 17
Let G be a locally compact group and σ be a family of cocompact closed subgroups of G. Then asdim(✷ σ G) = sup{asdim(✷ τ G) : τ ⊆ σ countable}.
Note that, unless σ is countable, the box space is not metrizable [36,Theorem 2.55], and the left hand side of the equality in the above lemma should be understood in the general context of asymptotic dimension of (uncountable) families of metric spaces (the so called, total box space). But this does not lead to a different asymptotic dimension (by the subspace and union permanence of asymptotic dimension [17,Theorems 6.2,6.3]; c.f. the argument in the proof of [12,Proposition 3.7]).
A family σ of cocompact subgroups of G is called separating if for any nonempty relatively compact subset F ⊆ G\{1}, there is G α ∈ σ with F ⊆ G\G α . A group G is residually compact (RC) if it has a separating family of cocompact subgroups, or equivalently, the family of all cocompact subgroups of G is separating. This is the same as the notion introduced in section 2 for compactly generated groups. The family σ is called semiconjugacy-separating if for any non-empty relatively compact subset F ⊆ G, there is G α ∈ σ with G α ∩ g G G α empty for every g ∈ F \{1}, where g G is the conjugacy class of g. As in [12,Lemma 3.12], this is equivalent to the condition that for any non-empty relatively compact subset F ⊆ G, there is G α ∈ σ with the quotient map q α : G → G/G α injective on F g, for each g; which in turn is equivalent to the condition that for any nonempty relatively compact subset F ⊆ G, there is G α ∈ σ with N (G α ) ∩ F either empty or {1}, where N (H) := ∪ g∈G H g , for a subgroup H. In particular, every separating family of cocompact normal subgroups is semi-conjugacy-separating. Also any family σ of cocompact subgroups is semi-conjugacy-separating whenever there exists a separating family τ of cocompact normal subgroups such that σ is a refinement of τ (that is, for any G β ∈ τ there is G α ∈ σ with G α ≤ G β ). Now a locally compact group G is residually compact (RC) if and only if it has a semiconjugacy-separating family of cocompact subgroups.
The next result extends [12,Proposition 3.16].
Lemma 18 Let a residually compact (RC) group G with a a plig metric d act on itself by right multiplication. Let σ be a semi-conjugacy-separating family of cocompact subgroups of G. Then asdim(G) ≤ asdim(✷ σ G).
Proof As mentioned in the paragraph after Lemma 17, we work with the total box space (which is always metrizable) if needed, but use the same notation for the sake of simplicity. Assume that asdim(✷ σ G) ≤ n for some n ∈ Z + and consider the metric on ✷ σ G that restricts to the quotient metric of on quotients of G by elements of σ. Given R > 0, exists S ≥ R and S-bounded cover U of ✷ σ G with multiplicity at most n + 1 and Lebesgue number at least R. Choose G α ∈ σ with q α : G → G/G α injective on B 3S (1)g = B 3S (g), for any g ∈ G. By [12,Lemma 3.15], q α is isometric on B S (g) for any g. Let U α consist of intersections of elements of U with G/G α and lift it to a G α -invariant cover V of G of multiplicity at most n + 1 and Lebesgue number at least R, exactly as in the proof of [12,Proposition 3.16]. This means that asdim(G) ≤ n and we are done.
In the next lemma, for R > 0 and a subset Y of a metric space (X, d), we use the notation P (Y ; R; d) := {y ∈ X : d(Y, y) ≤ R}.
Lemma 19 Let 1 → N ι − → G π − → K → 1
be a short exact sequence of locally compact groups with ι (identified with) an inclusion. Let H be a closed (but not necessarily normal) subgroup of G and let q : G → G/H and p : K → K/π(H) be the corresponding quotient maps. Then (i) for the G-action induced by π on K/π(H), there is a unique G-equivariant map ρ :
G/H → K/π(H) with ρ • q = p • π,
(ii) for any k ∈ K, ρ −1 (p(k)) = q(gN ), for any g ∈ π −1 (k), and this N -
invariant, (iii) H is cocompact in G iff π(H) is cocompact in K and N ∩ H is cocompact in N ,
Furthermore, for any plig metric d on G, (iv) the quotient metric d ′ on K/π(H) is the same as the metric induced by ρ from the quotient metric of G/H, (v) for any R > 0, k ∈ K and g ∈ π −1 (k), ρ −1 (P (p(k); R; d ′ )) = q(P (N ; R; d)g) contains the R-net q(N g) isometric to N/(N ∩ H g ) under the corresponding quotient metric.
Proof (i). The map ρ(gH) := p(π(g)) is well defined and G-equivariant, and unique by surjectivity of q.
(ii). The first statement is clear and the second follows by normality of N .
(iii). This follows from (i) and (ii) and continuity and properness of the action of G on G/H by left multiplication.
(iv). This follows from (i).
(v) This follows from normality of N , the fact that d is a plig metric and that the metrics on K and K/π(H) are quotient metrics (c.f., the proof of part (5) in [12,Lemma 4.1]).
The next lemma extends [12,Proposition 4.3].
Lemma 20 Let 1 → N ι − → G π − → K → 1
be a short exact sequence of locally compact groups, where G has a plig metric. Let σ be a family of cocompact subgroups of G. Consider the corresponding families σ 1 := {N ∩ G g α : G α ∈ σ, g ∈ G} and σ 2 := {π(G α ) : G α ∈ σ} of cocompact subgroups of N and K, respectively. Then
asdim(✷ σ G) ≤ asdim(✷ σ 1 N ) + asdim(✷ σ 2 K).
Proof The box families for which the asymptotic dimensions are calculated are quotients of the group by the cocompact subgroups in the family with the quotient metric. By the previous lemma, there are 1-Lipschitz maps ρ α : G/G α → K/π(G α ), and for any R > 0 and y ∈ K/π(G α ), the coarse fibre ρ −1 (B R (y) contains an R-net isometric to N/(N ∩ G g α ), for g ∈ (p • π) −1 (y). This means that we have a uniform coarse equivalence as in [12,Theorem 3.8], which gives the desired inequality.
The following special case is of great importance.
Lemma 21 Let 1 → N ι − → G π − → K → 1
be a short exact sequence of locally compact groups, where G has a plig metric. Let max be the family of all cocompact (normal) subgroups of G, and use the same notation for N and K. Then asdim(✷ max G) ≤ asdim(✷ max N ) + asdim(✷ max K).
Proof As in the above lemma, max induces families max 1 of (normal) subgroups of N and max 2 of (normal) subgroups of K. Since asdim(✷ max 1 N ) ≤ asdim(✷ max N ), and the same for K and max 2 , the inequality follows from Lemma 20. Now we are ready to prove the second main result of this section.
Theorem 4 If G is topologically elementary amenable (TEA) and residually compact (RC), then for any semi-conjugacy-separating family σ of cocompact subgroups of G,
asdim(G) ≤ asdim(✷ σ G) ≤ asdim(✷ max G) ≤ h(G).
Proof The first inequality is proved in Proposition 1, and the second is trivial. For the last inequality, note that for group extensions by compact or cocompact normal subgroups, we have inequality of Lemma 21 for asymptotic dimensions, and equality h(G) = h(N ) + h(K) for the Hirsch length. Also we have the desired inequality when G is in class X 1 , by Proposition 4. Now the result follows by transfinite induction.
Corollary 5 In any of the following cases we have the equalities asdim(G) = asdim(✷ σ G) = h(G), for any semi-conjugacy-separating family σ of cocompact subgroups of G:
(i) G is residually compact and a semidirect product of a locally elliptic group by a polycyclic-by-compact group, (ii) G is the wreath product of a compact abelian group with a polycyclic-bycompact group.
Proof Part (i) follows from Lemma 15 and Proposition 5. For part (ii), observe that by an argument similar to the proof of [16,Theorem 3.2], G is residually compact in this case. Now apply part (i).
Totally disconnected groups
The class of totally disconnected locally compact second countable (t.d.l.c.s.c.) groups are recently subject of study as an important subclass of Polish groups (separable and completely metrizable topological groups). The second countability assumption is not that essential, as every t.d.l.c. group is a directed union of open subgroups which are s.c. modulo a compact normal subgroup [19, 8.7]. The basic advantage of being t.d. is that by a classical result of van Dantzig a t.d.l.c. group admits a local basis at identity of compact open subgroups [19, 7.7] (showing that compact t.d.l.c. groups are profinite and so are compact open subgroups of t.d.l.c. groups). Platonov has shown that a t.d.l.c.s.c. group, in which every finitely generated subgroup is relatively compact, may be written as a countable increasing union of profinite groups [34]. Also a residually discrete t.d.l.c.s.c. group may be written as a countable increasing union of [SIN]-groups [7]. These two results make the subclass of t.d.l.c.s.c. groups built from profinite and discrete groups specially interesting. One should ad to these evidences the fact that kernel of the adjoint representation of a p-adic Lie group belongs to this subclass [13].
Let us define the above subclass more rigorously. The class of elementary groups is the smallest class E of t.d.l.c.s.c. groups which contains all s.c. profinite groups and countable discrete groups, and is closed under taking topological group extensions by s.c. profinite groups and countable discrete groups and increasing countable unions (of open subgroups) [44, 1.1]. This class is known to be closed under group extensions, taking closed subgroups and quotients by closed normal subgroups, inverse limits, quasi-products, and local direct products, as long as the resulting group remains t.d.l.c.s.c. Also it passes from a cocompact subgroup to a t.d.l.c.s.c. group [44, 1.3].
Each t.d.l.c.s.c. group G has a unique maximal (resp. minimal) closed normal subgroup Rad E (G) (resp. Res E (G)) whose quotient group is elementary [44, 1.5]. This is called the elementary radical (resp. residual) of G. Also, a t.d.l.c.s.c. group with an open solvable subgroup, is known to be elementary [44, 1.11]. Moreover, the structural results of Wesolek reduces the study of compactly generated t.d.l.c.s.c. groups to that of elementary groups and topologically characteristically simple non-elementary groups [44, 1.6, 1.7].
Since the topological version of the second isomorphism theorem is valid for t.d.l.c.s.c. groups [44, 2.2] (c.f., [19, 5.33]), the intersection of two closed cocompact subgroups of a t.d.l.c.s.c. group is again cocompact. In particular, the family of all closed cocompact subgroups is filtering in the sense of Caprace and Monod, and it follows from [7, 2.5], that if a compactly generated t.d.l.c.s.c. group is residually compact (RC), then each RC-approximation of G includes a subgroup which is compact-by-discrete (c.f., [44, 2.6]).
In a t.d.l.c.s.c. group G, the quasi-center QZ(G) of G consists of elements with open centralizer. This is a (not necessarily closed) characteristic subgroup [6]. The locally elliptic radical Λ(G) of Platonov also plays an important role in t.d.l.c. groups and could be characterized as the C-core , for the the collection C of all compact subgroups. Also, when G is t.d.l.c.s.c., it has a closed characteristic subgroup SIN(G) containing QZ(G), which is an increasing union of compactly generated relatively open [SIN]-subgroups.
The class E of elementary t.d.l.c. groups could be constructed by a transfinite inductive process: let E 0 be the trivial class and E 1 be the class of profinite or discrete groups. Suppose that E α is defined and put E α+1 = (LE α )E 1 (using the notation of the previous section). Finally, for a limit ordinal β, put E β := ∪ α<β E α . Then E = ∪ α<ω 1 E α [44, page 1396]. We define the construction rank of G ∈ E by rk(G) := min{α : G ∈ E α }. If rk(G) > 1 and G is compactly generated, then G is a group extension of either a profinite group or a discrete group by an elementary group of strictly lower rank. Also if G ∈ E , then so is any open subgroup H and rk(H) ≤ rk(G). Similarly, if G ∈ E , then so is the quotient by any compact subgroup K and rk(G/K) ≤ rk(G) (note that the quotient G/K remains elementary, even if K is not compact [44, 3.12]). If G is t.d.l.c.s.c. and H is a normal subgroup with H, G/H ∈ E , then so is G and rk(G) ≤ rk(H)+ rk(G/H). Finally if H is a closed subgroup, rk(H) ≤ 3rk(G) [44, 3.2-3.8] (note that in [44], the right hand sides of the last two inequalities are added with 1 and 3, but we avoid this by shifting the construction rank by 1). If G is a locally solvable t.d.l.c.s.c. group, then G is elementary with rk(G) ≤ 4 n , where n is the solvable rank of G (i.e., the minimum of the derived length of solvable compact open subgroups of G) [44, 8.1]. In particular, non-discrete, compactly generated, t.d.l.c. topologically simple group are not locally solvable (c.f., [46, Theorem 2.2]). As a concrete case of rank calculation, let G be residually discrete, Res(G) = {1}, and G is a countable increasing union of open [SIN]-groups. A [SIN]-group is compact-by-discrete and so is elementary with rank at most 2, thus rk(G) is at most 3.
For a class G of t.d.l.c.s.c. groups, the elementary closure E G of G is the smallest class of t.d.l.c.s.c. groups containing G and all s.c. profinite groups and countable discrete groups, which is closed under group extensions of s.c. profinite groups, countable discrete groups, and groups in G and closed under countable increasing unions [44, 3.19]. An example is the class of all l.c.s.c. p-adic Lie groups for all primes p [45, Theorem 6.3].
As far as we know, there is no result on the asymptotic dimension of t.d.l.c.s.c. groups or elementary groups. In this section we give some results in this direction.
A subquotient of a topological group is a closed subgroup of a quotient of G by a closed normal subgroup. This includes both closed subgroups and quotients by closed normal subgroups at the same time. When the group has a plig metric, any subquotient has a canonical induced plig metric. If the asymptotic dimension of the group is finite, so is that of any subquotient, but a group with infinite asymptotic dimension may have many subquotients with finite asymptotic dimension. The key fact, we are going to show, is that for t.d.l.c.s.c. groups, the asymptotic dimension of the group is controlled by the asymptotic dimension of its discrete subquotients.
As we already mentioned, locally solvable groups are elementary with finite construction rank. It is known (and indeed shown by von Neumann himself in [42]) that locally solvable groups are amenable. As these are not necessarily elementary amenable, the bounds in previous section could not be applied. Unfortunately, the above result also does not effectively apply in this case, as a locally solvable group is not finitely generated unless it is solvable. However, for a locally solvable group, the Hirsch length dominates the asymptotic dimension, but the former (which is the supremum of the Hirsch length of finitely generated solvable subgroups) may blow up.
In the compactly generated case, we have the following characterization of groups with finite asymptotic dimension, which the main result of this section.
Theorem 5 Let G be a t.d.l.c.s.c. compactly generated group and consider the following conditions:
(i) G has finite asymptotic dimension, (ii) G has a discrete quotient with finite asymptotic dimension, (iii) every discrete quotient of G has finite asymptotic dimension. Then (i) ⇒ (ii) and (iii) ⇒ (i). If moreover, all open subgroups of G are compact, then these conditions are equivalent. When G is also a [SIN]-group, in parts (ii) and (iii) quotients could be taken to be finitely generated quotient groups (i.e., quotients by open normal subgroups).
Proof We employ van Dantzig theorem (ensuring that G always have open compact subgroups; c.f., [19, 7.7]) and a topological version of Milnor-Švarc lemma: If X pseudo-metric space and a locally compact group G acts on X with orbit map ι x : G → X; g → gx, so that the action is geometric, namely it is isometric, cobounded (i.e., X could be covered with translate of a subset with finite diameter), locally bounded (i.e., for a bounded subset B of X, each g ∈ G has a neighborhood V with V B bounded in X) and metrically proper (i.e., ι −1
x (B r (x)) is relatively compact in G, for each x ∈ X and each r > 0), then G is σ-compact and ι x is a quasi-isometry, for each x ∈ X. If moreover, X is coarsely-connected (i.e., for some c > 0, between each two points in X one could insert finitely many points with distance of consecutive ones at most c), then G is also compactly generated (c.f., [11, 4.C.5]). Now if H ≤ G is a compact open subgroup, then the quotient group G/H is paracompact. In particular, the translation action of G on G/H is geometric (c.f., [11,4.C.8]), and so by Milnor-Švarc lemma, asdim(G) ≤ asdim(G/H). Indeed, in this case we have asdim(G) = asdim(G/H), as the orbit map is just the quotient map, which is surjective. This shows that (i) implies (ii) and (iii) implies (i). If all open subgroups of G are compact, then all discrete quotients of G have the form G/H, for H open and compact, thus (i) implies (iii). Finally, when G is a t.d.l.c.s.c. compactly generated [SIN]-group, it has a local basis consisting of compact open normal subgroups, and in the above argument, H could be taken to be normal as well. The rest works as above.
Corollary 6 Let G be a t.d.l.c.s.c. compactly generated group all of whose discrete quotients are finite. Then asdim(G) = asdimRes(G) < ∞.
Proof If all discrete quotients of G are finite, then by implication (iii) ⇒ (i) of Theorem 5, G has finite asymptotic dimension. Also in this case, Res(G) is a cocompact, compactly generated subgroup of G [27], [7, Theorem F], thus asdim(G) = asdimRes(G).
Note that by a deep result of Caprace and Monod, for a compactly generated totally disconnected locally compact group all of whose discrete quotients are finite, the discrete residual admits no non-trivial discrete or compact quotient [7,Corollary G]. However, this condition on all discrete quotients is a rather strong condition, for instance in this case if G is moreover residually discrete, then it is forced to be compact. A less restrictive situation is when G is a t.d.l.c.s.c. compactly generated [SIN]-group. In this case, G is residually discrete, i.e., Res(G) = {1} [7, Corollary 4.1], however, one could only deduce that G is compact-by-discrete [15,Theorem 2.13], so it may have infinite asymptotic dimension.
Corollary 2
2Polycyclic-by-compact groups have finite Hirsch length.
then H is cocompact. Proof First assume that H is cocompact and N be the cocompact closed normal subgroup of G contained in H, constructed in the proof of Poincaré lemma. Then clearly h(G/N ) = h(H/N ) = 0, and so h(H) = h(N ) = h(G), by the Hirsch formula in the cocompact case. Conversely, if H is normal, then h(G) ≥ h(H) + h(G/H) = h(G) + h(G/H), thus h(G/H) = 0. Since G/H is also polycyclic-by-compact, it could only have zero Hirsch length if it is compact. Therefore, H is cocompact. Note that in the discrete case, the above result could be proved without normality assumption by an inductive argument on the Hirsch length [33, 10.2.10]. Let us see what goes wrong in the topological case. To use induction on h(G), let us first consider the case h(G) = 0, then G is compact and the claim is trivial. If G is noncompact, one could use Lemma 10 to choose a normal torsion-free infinite discrete abelian subgroup A of G. If we knew that the Hirsch formula h(H) = h(H ∩ A) + h(H/H ∩ A) holds, the rest would follow: the right hand side of the above equality is less than or equal h(A) + h(G/A) which in turn is at most h(G), and since h(H) = h(G), it follows that, h(H ∩ A) = h(A), which yields h(A/H ∩ A) = 0, showing that H ∩ A ≤ A is cocompact. Now choose a closed locally cocompact characteristic subgroup B of A contained in H ∩ A by Lemma 2. Since A ✂ G and B ≤ A is characteristic, we get B✂G. Let us observe that h(B) > 0, indeed, if h(B) = 0, then B is compact, but then both B and A/B are locally elliptic, and so is A [11, 4.D.6].
H
:= {L : H ≤ L ≤ G, L is closed and cocompact} := {HN : N ✂ G, N is closed and cocompact}. Applying this to H = 1 we getH = 1, i.e., {N : N ✂ G, N is closed and cocompact} = 1, that is, G is RC, with a regular RC-approximation.
[ 22 ,
22Page 238] could be adapted to show that the Hirsch length is well-defined for G in [TEA], and satisfies the natural properties: h(H) ≤ h(G), for each closed subgroup H, h(G) = h(N ) + h(G/N ), for each compact or cocompact normal subgroup N , and h(G) = sup{h(H) : H ≤ G closed and compactly generated} (c.f. [20, Theorem 1]). This plus the Hirsch formula in the previous section show that the notion defined here coincides with the notion defined in previous section for polycyclic-by-compact groups.The class [EA] is known to include all virtually solvable groups (and so all virtually nilpotent and all polycyclic-by-finite groups), but as we assume that [TEA] is stable only under taking "closed" subgroups, we would not have all topologically virtually nilpotent, or even all polycyclic-by-compact groups in the class[TEA]. The choice of assuming stability under taking arbitrary subgroups has its own price to pay: the class [TEA] would then include the class of maximally almost periodic groups, and in particular the non amenable free groups would be in[TEA] (c.f.,[32, Example 12]).
[FC] − is included in [TEA] (note that for compactly generated groups, [FC] − = [FD] − (c.f., [15, Theorem 3.20]), however there is a semidirect product of R by Z 2 (which is then in [TEA]) not included in [FC] − (c.f., [32, Example 11]).
Corollary 4
4A topologically elementary amenable (TEA) group of finite Hirsch length with no nontrivial locally elliptic normal closed subgroup is solvable-bycompact. The next result extends the first assertion of [20, Theorem 2].Proposition 6 If G is topologically elementary amenable (TEA) of finite Hirsch length n, then G is in LX n+1 .
). When H is an open subgroup of a second countable group G, a G-invariant compatible metric exists on G/H iff H is almost normal (i.e., for every finite subset F of G there exists a finite subset E of G such that HF ⊆ EH) [1, Theorem 2.15], but this is not sufficient for closed subgroups [1, Example 2.14].
TEA] is stable only under taking closed subgroups). [MAP] includes the class [Moore] of Moore groups (groups whose irreducible representations are finite dimensional) by Gelfand-Raikov theorem. Note that a Lie group belongs to [Moore] if and only if it is a finite extension of a central group, so [TEA] contains Lie groups in the class [Moore]. For compactly generated groups, [MAP] is included in [SIN] (small invariant neighborhood groups, so the latter is not included in [TEA], indeed [SIN] includes all discrete groups (more generally, [SIN]-groups are discrete extensions of direct products of vector groups and compact groups (c.f., [15, Theorem 2.13]). However connected [SIN] groups are in [TEA] (as they are in [Z], by Freudenthal-Weil theorem), though there are connected groups in [TEA] included neither in [MAP] nor in [SIN] (c.f., [32, Example 8]) as well as compactly generated totally disconnected groups in [TEA] included neither in [MAP] nor in [SIN] (c.f., [32, Example 9]). More generally, connected [IN]-groups are extension of compact groups by vector groups, and so are in [TEA] [15, Theorem 2.9, Corollary 2.8]. A less trivial inclusion follows by a result of Grosser and Moskowitz [15, Proposition 4.5], stating that a compactly generated [FIA] − group (topologically finite inner automorphism group, that is a group with relatively compact inner automorphisms group) is an extension of a direct product of some vector group R n with a compact group by some Z d , showing that [TEA] contains compactly generated [FIA] − groups. Since [SIN]∩[FD] − ⊆ [FIA] − [15, Theorem 4.6], same is true for compactly generated [SIN]-groups which are in [FD] − (topologically finite derived subgroup, that is a groups with relatively compact commutator or derived subgroup). This class is quite interesting as it contains non Type I groups, showing that [TEA] is not included in [Type I]: there is a central extension of T by Z 2 with derived subgroup T, which is a compactly generated group in [SIN]∩[FD] − , with a non Type I, Z 2 multiplier (and so non Type I) (c.f.,
For the discrete residual Res(G) of G, defined as the intersection of all open normal subgroups, the quotient G/Res(G) is a countable increasing union of [SIN]-groups [44, 2.7]. Elementary groups contain t.d.l.c.s.c. which are either [SIN]-groups, solvable, locally elliptic t.d.l.c.s.c., or contain a compact open subgroup that has a dense quasi-center.
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Zur allgemeine theorie der masse. J Neumann, Fund. Math. 13J. von Neumann, Zur allgemeine theorie der masse, Fund. Math. 13 (1929), 73-116.
B A F Wehrfritz, Group and ring theoretic properties of polycyclic groups. LondonSpringer-VerlagB.A.F. Wehrfritz, Group and ring theoretic properties of polycyclic groups, Springer- Verlag, London, 2009.
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"HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inference"
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"Gianluca Detommaso \nAmazon.com\n\n",
"Ullrich Köthe \nHeidelberg University\n\n",
"Robert Scheichl \nHeidelberg University\n\n"
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| Many recent invertible neural architectures are based on coupling block designs where variables are divided in two subsets which serve as inputs of an easily invertible (usually affine) triangular transformation. While such a transformation is invertible, its Jacobian is very sparse and thus may lack expressiveness. This work presents a simple remedy by noting that subdivision and (affine) coupling can be repeated recursively within the resulting subsets, leading to an efficiently invertible block with dense, triangular Jacobian. By formulating our recursive coupling scheme via a hierarchical architecture, HINT allows sampling from a joint distribution p(y, x) and the corresponding posterior p(x | y) using a single invertible network. We evaluate our method on some standard data sets and benchmark its full power for density estimation and Bayesian inference on a novel data set of 2D shapes in Fourier parameterization, which enables consistent visualization of samples for different dimensionalities. * equal contribution. | null | [
"https://arxiv.org/pdf/1905.10687v4.pdf"
]
| 216,634,548 | 1905.10687 | e6ed6106ba720cf4cf451b10677ca15a19eeeafa |
HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inference
Jakob Kruse [email protected]
Heidelberg University
Gianluca Detommaso
Amazon.com
Ullrich Köthe
Heidelberg University
Robert Scheichl
Heidelberg University
HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inference
Many recent invertible neural architectures are based on coupling block designs where variables are divided in two subsets which serve as inputs of an easily invertible (usually affine) triangular transformation. While such a transformation is invertible, its Jacobian is very sparse and thus may lack expressiveness. This work presents a simple remedy by noting that subdivision and (affine) coupling can be repeated recursively within the resulting subsets, leading to an efficiently invertible block with dense, triangular Jacobian. By formulating our recursive coupling scheme via a hierarchical architecture, HINT allows sampling from a joint distribution p(y, x) and the corresponding posterior p(x | y) using a single invertible network. We evaluate our method on some standard data sets and benchmark its full power for density estimation and Bayesian inference on a novel data set of 2D shapes in Fourier parameterization, which enables consistent visualization of samples for different dimensionalities. * equal contribution.
Introduction
Invertible neural networks based on the normalizing flow principle have recently gained increasing attention for generative modeling, in particular networks built on a coupling block design (Dinh, Sohl-Dickstein, and Bengio 2017). Their success is due to a number of useful properties: (a) they can tractably model complex high-dimensional probability densities without suffering from the curse-ofdimensionality, (b) training via the maximum likelihood objective is generally very stable, (c) their latent space opens up opportunities for model interpretation and manipulation, and (d) the same trained model can be used for both efficient data generation and efficient density calculation.
While autoregressive models can also be trained as normalizing flows and share properties (a) and (b), they sacrifice efficient invertibility for expressive power and thus lose properties (c) and (d). In contrast, lack of expressive power of a single invertible block is a core limitation of invertible networks, which needs to be compensated by extremely deep models with dozens or hundreds of blocks, e.g., the GLOW architecture (Kingma and Dhariwal 2018). While invertibility allows to back-propagate through very Figure 1: Sparse (left) and dense (right) triangular Jacobian of a standard coupling block and of our recursive design, respectively. Nonzero parts of the Jacobian in gray.
x ∇ • f C (x) x ∇ • f R (x)
deep networks with minimal memory footprint (Gomez et al. 2017), more expressive invertible building blocks are still of great interest. The superior performance of autoregressive approaches such as ( Van den Oord et al. 2016) is due to the stronger interaction between variables, reflected in a dense triangular Jacobian matrix, at the expense of cheap inversion. The theory of transport maps (Villani 2008) provides certain guarantees of universality for triangular maps, which do not hold for the standard coupling block design with a comparatively sparse Jacobian (figure 1, left).
Here, we propose an extension to the coupling block design that recursively fills in the previously unused portions of the Jacobian using smaller coupling blocks. This allows for dense triangular maps (figure 1, right), or any intermediate design if the recursion is stopped before, while retaining the advantages of the original coupling block architecture. Furthermore, the recursive structure of this mapping can be used for efficient conditional sampling and Bayesian inference. Splitting the variables of interest into two subsets x and y, a single normalizing flow model can be built that allows efficient sampling from both the joint distribution p(x, y) and the conditional p(x | y). It should be noted that our extension would also work for convolutional architectures like GLOW.
Finally, we introduce a new family of data sets based on Fourier parameterizations of two-dimensional curves. In the normalizing flow literature, there is an abundance of twodimensional toy densities that provide an easy visual check for correctness of the model output. However, the sparsity of the basic coupling block only becomes an issue beyond two dimensions where it is challenging to visualize the distribution or individual samples. Pixel-based image data sets, on the other hand, quickly are too high dimensional for a mean-ingful assessment of the quality of the estimated densities.
A step towards visualizable data sets of intermediate size has been made in ), but their fourdimensional problems are still too simple to demonstrate the advantages of the recursive coupling approach described above. To fill the gap, we describe a way to generate data sets of arbitrary dimension, where each data point parameterizes a closed curve in 2D space that is easy to visualize. Increasing the input data dimension allows the representation of distributions of more and more complex curves.
To summarize, the contributions of this paper are: (a) a simple, efficiently invertible flow model with dense, triangular Jacobian; (b) a hierarchical architecture to model joint as well as conditional distributions; (c) a novel family of data sets allowing easy visualization for arbitrary dimensions.
The remainder of this work consists of a literature review, some mathematical background, a description of our method and supporting numerical experiments, followed by closing remarks.
Related Work
Normalizing flows were popularized in the context of deep learning chiefly by the work of (Rezende and Mohamed 2015) and (Dinh, Krueger, and Bengio 2015). By now, a large variety of architectures exist to realize normalizing flows. The majority falls into one of two groups: coupling block architectures and autoregressive models. For a comprehensive overview and background information on invertible neural networks and normalizing flows see (Kobyzev, Prince, and Brubaker 2019) or .
Additive and then affine coupling blocks were first introduced by (Dinh, Krueger, and Bengio 2015;Dinh, Sohl-Dickstein, and Bengio 2017), while (Kingma and Dhariwal 2018) went on to generalize the permutation of variables between blocks by learning the corresponding matrices, besides demonstrating the power of flow networks as generators. Subsequent works have focused on replacing the (componentwise) affine transformation at the heart of such networks, which limits expressiveness, e.g., by replacing affine couplings with more expressive monotonous splines (Durkan et al. 2019), albeit at the cost of evaluation speed.
On the other hand, there is lso a rich body of work on autoregressive (flow) networks (Huang et al. 2018;Kingma et al. 2016;Van den Oord et al. 2016;Van den Oord, Kalchbrenner, and Kavukcuoglu 2016;Papamakarios, Pavlakou, and Murray 2017). More recently, (Jaini, Selby, and Yu 2019) applied second-order polynomials to improve expressive power over typical autoregressive models and proved that their model is a universal density approximator. While such models provide excellent density estimation compared to coupling architectures (Liao, He, and Shu 2019;Ma et al. 2019), generating samples is often not a priority and can be prohibitively slow.
There are other approaches, outside those two subfields, that also seek a favorable trade-off between expressive power and efficient invertibility. Residual Flows Chen et al. 2019) impose Lipschitz constraints on a standard residual block, which guarantees invertibility with a full Jacobian and enables approximate maximumlikelihood training but requires an iterative procedure for sampling. Similarly, (Song, Meng, and Ermon 2019) uses lower triangular weight matrices that can be inverted via fixed-point iteration. The normalizing flow principle is formulated continuously as a differential equation (DE) by , which allows free-form Jacobians but requires integrating a DE for each network pass. (Karami et al. 2019) introduce another method with dense Jacobian, based on invertible convolutions in the Fourier domain.
In terms of modeling conditional densities with invertible neural networks, (Ardizzone et al. 2019a) proposed an approach that divides the network output into conditioning variables and a latent vector, training the flow part with a maximum mean discrepancy objective (MMD, Gretton et al. 2012) instead of maximum likelihood. Later (Ardizzone et al. 2019b) introduced a simple conditional coupling block to construct a conditional normalizing flow.
Mathematical Background
For an input vector x ∈ R N , a standard, invertible coupling block is abstractly defined by
x = f C (x) = x 1 C x 2 | x 1 = x 1 x 2 ,(1)
where x 1 = x 0: N/2 and x 2 = x N/2 :N are the first and second half of the input vector and x 2 = C(x 2 | x 1 ) is an easily invertible transform of x 2 conditioned on x 1 . Its inverse is then simply given by
x = f 1 C (x ) = x 1 C 1 x 2 | x 1 .(2)
For affine coupling blocks (Dinh, Sohl-Dickstein, and Bengio 2017), C takes the form C(u | v) = u exp s(v) + t(v) with s and t unconstrained feed-forward networks. The logarithm of the Jacobian determinant of such a block can be computed very efficiently as
log det J fC (x) = log det ∂f C (x) ∂x = sum(s(x 1 )). (3)
To ensure that all entries of x are transformed and interact with each other, a pipeline that alternates between coupling blocks and random orthogonal matrices Q is constructed, where the orthogonal block x = f Q (x) = Qx can trivially be inverted as
x = f 1 Q (x ) = Q x with log-determinant log det J f Q (x) = 0.
Normalizing Flows and Transport Maps
To create a normalizing flow, this 'pipeline'
T = f C1 • f Q1 • f C2 • f Q2 • . . . is trained via maximum likelihood loss L(x) = 1 2 T (x) 2 2 − log |J T (x)|(4)
to transport the data distribution p X to a standard normal latent distribution p Z = N (0, I). The map T can then be used to sample from p X by drawing a sample z (i) from p Z in the latent space and by passing it through the inverse model S = T 1 to obtain x (i) = S(z (i) ).
Using the change-of-variables formula, the density at a given data point x can also be calculated as p X (x) = p Z (T (x)) · |det J T (x)|. The mathematical basis of this procedure is the theory of transport maps (Villani 2008), which are employed in exactly the same way to push a reference density (e.g. Gaussian) to a target density (e.g. the data distribution, (Marzouk et al. 2016)). In fact, up to a constant, namely the (typically inaccessible) fixed entropy H(p X ) of the data distribution, the expected value of the objective in equation (4) is the Kullback-Leibler (KL) divergence between the data distribution p X and the push-forward of the latent density S # p Z :
D KL (p X S # p Z ) = p X (x) log p X (x) S # p Z (x) dx = E x∼p X [L(x)] + H(p X ).
(5) Normalizing flows represent one parametrised family of maps over which equation (4) can be minimized. Other examples include polynomial (Marzouk et al. 2016), kernelbased (Liu and Wang 2016) or low-rank tensor (Dolgov et al. 2020) approximations.
Note also that each pair f Ci • f Qi in T is a composition of an orthogonal transformation and a triangular map, where the latter is better known in the field of transport maps as a Knothe-Rosenblatt rearrangement (Marzouk et al. 2016). This can be interpreted as a non-linear generalization of the classic QR decomposition (Stoer and Bulirsch 2013). Whereas the triangular part encodes the possibility to represent non-linear transformations, the orthogonal part reshuffles variables to foster dependence of each part of the input to the final output, thereby drastically increasing the representational power of the map T .
Bayesian Inference with Conditional Flows
Inverse problems arise when one possesses a wellunderstood model for the forward mapping x → y from hidden parameters x to observable outcomes y, e.g. in the form of an explicit likelihood p(y | x) or a Monte-Carlo simulation. However, the actual object of interest is the inverse mapping y → x from observations to parameters. According to Bayes' theorem, this requires estimation of the posterior conditional density p(x | y). Such Bayesian inference problems arise frequently in the sciences and are generally very hard.
Normalizing flows can be used in several ways to estimate conditional densities. The approach in this paper is inspired by (Marzouk et al. 2016) and exploits the link to Knothe-Rosenblatt maps highlighted above. As described below, see figure 3 (left), it suffices to constrain the possible rearrangements of variables in the coupling blocks, i.e. the choice of the orthogonal blocks f Qi , to enable conditional sampling. This was first noted in (Detommaso et al. 2019).
Independently, (Ardizzone et al. 2019b) and (Winkler et al. 2019) introduced conditional coupling blocks that allow an entire normalizing flow to be conditioned on external variables. By conditioning the transport T between p X (x) and p Z (z) on the corresponding values of y as z = T (x | y), its inverse T 1 (z | y) can be used to turn the latent distribution p Z (z) into an approximation of the posterior p(x | y).
f R x Q + Q · f R (x) s(x1) t(x1) fR(x1)
fR(x2) Figure 2: A recursive affine coupling block. The inner functions f R (x i ) take again the form of the outer gray block, repeated until the maximum hierarchy depth is reached. Each such coupling block in itself has a triangular Jacobian.
Method
We extend the basic coupling block design in two ways.
The Recursive Coupling Block
As visualized in figure 1 (left), the Jacobian J f of a simple coupling block is very sparse, i.e. many possible interactions between variables are not modelled. However the efficient determinant computation in equation (3) works for any lower triangular J f , and indeed theorem 1 of (Hyvärinen and Pajunen 1999) states that a single triangular transformation can, in theory, already represent arbitrary distributions.
The following recursive coupling scheme f R makes use of this potential and fills the empty areas below the diagonal: Given x ∈ R N and a hierarchy depth K ∈ N, we define recursively, for k = K, K − 1, . . . , 1:
x = f R,k (x) = f C (x), if N k ≤ 3, f R,k 1 (x 1 ) C k f R,k 1 x 2 x 1 , else,(6)
where for each k, x 1 = x 0: N k /2 and x 2 = x N k /2 :N k , N k is the size of the current input vector and N K = N . Note that each sub-coupling has its own coupling function C k with independent parameters. The inverse transform is
x = f 1 R,k (x ) = f 1 C (x ), if N k ≤ 3, f 1 R,k 1 (x 1 ) f 1 R,k 1 C 1 k x 2 f 1 R,k 1 (x 1 )
, else.
For K = log 2 N , this procedure leads to the dense lower triangular Jacobian visualized in figure 1 (right), the log-determinant of which is simply the sum of the logdeterminants of all sub-couplings C k . A visual representation of the architecture can be seen in figure 2. However, since the (sub-)coupling blocks are affine, f R,K can only represent an approximation of the exact Knothe-Rosenblatt map and it is still necessary, as for standard coupling blocks, to create a normalizing flow by composing several recursive coupling blocks interspersed with orthogonal transformations f Q . Thus, in practice, it is also more economical to limit the depth of the hierarchy to 2 or 3. This already increases the amount of interaction between individual variables considerably, while limiting the computational overhead, but it allows to use much shallower networks. The trade-off between number of blocks and hierarchy depth will be studied for the Fourier shapes data set in this work.
y Qy Q y · f H,y (y) fR(y) + s(y) t(y) x Qx Q x · f H,x (x | y) fR(x) y x ∇• fH(y) ∇• fH(x)
y zy zy
x zx ∼ N (zx; 0, I |x| ) Figure 3: Top: Single HINT block with recursive coupling, and its Jacobian matrix. Transformation of x is influenced by y, but not vice-versa, imposing a hierarchy on variables. Bottom: Using HINT flows for conditional sampling/Bayesian inference.
Hierarchical Invertible Neural Transport
While the recursive coupling block defined above is motivated by the search for a more expressive architecture, it is also ideally suited for estimating conditional flows and thus for Bayesian inference. Specifically, in a setting with paired data (x i , y i ), where subsequently we want a sampler for x conditioned on y, we can provide both variables as input to the flow, separating them in the first hierarchy level for further transformation at the next recursion level. Crucially, x and y variables are never permuted between lanes, thus only feeding forward information from the y-lane to the x-lane as shown in figure 3 (top left). Instead of one large permutation operation over all variables, as in the hierarchical coupling block design in figure 2, we apply individual permutations Q y and Q x to each respective lane at the beginning of the block. A normalizing flow model constructed in this way performs hierarchical invertible neural transport, or HINT for short.
The output of a HINT model is a latent code with two components, z = [z y , z x ] = T (y, x), but the training objective stays the same as in equation (4):
L(y, x) = 1 2 T (y, x) 2 2 − log |J T (y, x)|(8)
As with a standard normalizing flow, the joint density of input variables is the pull-back of the latent density via T :
p T (y, x) = S # p Z (z) = S # N (0, I |y|+|x| ),(9)
where S = T −1 . But because the y-lane in HINT can be evaluated independently of the x-lane, we can determine the partial latent code z y for a given y and hold it fixed (figure 3, bottom left), while drawing z x from the x-part of the latent distribution (bottom right). This yields samples from the conditional density:
x = S x ([z y , z x ]) ∼ p T (x | y) with z y = T y (y),(10)
where superscripts x and y respectively denote xand ylanes of the transformations. This means HINT gives access to both the joint density of x and y, as well as the conditional density of x given y, e.g. for Bayesian inference.
Computational Complexity
The number of couplings doubles in every recursion level, whereas the workload per coupling decreases exponentially, so that the total order of complexity of HINT and RealNVP is the same. All sub-networks s and t within one level are independent of each other and can be processed in a single parallel pass on the GPU (see appendix). Only the final affine transformations and some bookkeeping operations must be executed sequentially, but at negligible cost compared to the other tensor operations. Our first, non-parallel implementation of HINT is 2-10 times slower than RealNVP.
Experiments
We perform experiments on classical UCI data sets (Dua and Graff 2017) and a new data set called "Fourier shapes". We introduce this new data set to balance four conflicting goals:
1. The dimension of the data should be high enough for HINT's hierarchical decomposition to make a difference.
2. The dimension should be low enough to allow for accurate quantitative evaluation and comparison of results.
3. Learning the joint distribution should be challenging due to complex interactions between variables.
4. Visualizations should allow intuitive qualitative comparison of the differences between alternative approaches.
Our new data set represents families of 2-dimensional contours in terms of the probability density of their Fourier coefficients and fulfills the above requirements: The dimension of the problem can be easily adjusted by controlling the complexity of the shapes under consideration and the number of Fourier coefficients (1,2). Shapes with sharp corners and long-range symmetries require accurate alignment of many Fourier coefficients (specifically, of their phases, 3 -see appendix). Humans can readily recognize the quality of a shape representation in a picture (4). Our experiments show considerable improvements of HINT over RealNVP. To clearly demonstrate these advantages, we heavily restrict the networks' parameter budgetslarger networks would be much more accurate, but exhibit less meaningful differences. Models were trained on an RTX 2080 Ti GPU. Hyper-parameters are listed in the appendix. Table 1: Normal and recursive coupling compared on UCI benchmarks in terms of average log-likelihood (mean ± std over 3 training runs; higher is better ↑).
UCI Density Estimation Benchmarks
The tabular UCI data sets (Dua and Graff 2017) are popular for comparing density models. Using public code for preprocessing 2 , we compare several flow models with REAL-NVP and RECURSIVE coupling blocks in terms of the average log-likelihood on the test set. To tease out shortcomings, each model is "handicapped" to a budget of 500k (POWER, GAS) or 250k (MINIBOONE) trainable parameters. Table 1 shows that the recursive design achieves similar or better test likelihood in all cases, even when the low dimensionality (DIM) of the data allows little recursion.
Fourier Shapes Data Set
A curve g(t) ∈ R 2 , parameterized by 2M + 1 complex 2d Fourier coefficients a m ∈ C 2 , can be traced as
g(t) = M m=−M a m · e 2π·i·m·t(11)
with parameter t running from 0 to 1. This parameterization will always yield a closed, possibly self-intersecting curve (McGarva and Mullineux 1993). Vice-versa, we can calculate the Fourier coefficients
a m = 1 L L−1 l=0 p l · e −2π·i·m·l/L , for m ∈ [−M, M ],(12)
to approximately fit a curve through a sequence of L points p l ∈ R 2 , l = 0, . . . , L − 1. By increasing M , higher order terms are added to the parameterization in equation (11) and the shape is approximated in greater detail. An example of this effect for a natural shape is shown in figure 4 (right). Note that the actual dimensionality of the parameterization in our data set is |x| = 4 · (2M + 1), as each complex 2d coefficient a m is represented by four real numbers.
We perform experiments on two specific data sets, first using curves of order M = 2, i.e. |x| = 20, to represent a distribution of simple shapes that arise from the intersection of two randomly placed circles with a fixed ratio of radii and a fixed distance. The resulting Lens shapes can be seen in figure 5 (left), together with the highly structured correlation matrix of Fourier parameters x i that yield such shapes.
The second data set uses M = 12, i.e. |x| = 100, to represent Cross shapes which are generated by crossing two 2 https://github.com/LukasRinder/normalizing-flows (12). Right: Tracing the curve g(t) according to equation (11), for different numbers M of Fourier terms a m .
bars of random length, width and lateral shift at a right angle, oriented randomly, but positioned close to the origin. This results in a variation of Xs, Ls and Ts, some of which are shown in figure 5 (right) together with the even more complicated (100 × 100) parameter correlation matrix.
Density estimation. For density estimation, a single-block and a two-block network are trained on the Lens-shapes data, once with standard coupling blocks and once with the new recursive design. All networks have the same total parameter budget -details in the appendix.
Samples from the two-block models and absolute differences to the true parameter correlation matrices are shown in figure 6 (left). Qualitatively, samples from the recursive model are visually more faithful and have smaller errors in the correlation matrices. Quantitatively, we compare over three training runs per model using the following metrics:
• Maximum mean discrepancy (MMD, Gretton et al. 2012) measures the dissimilarity of two distributions using only samples from both. Following (Ardizzone et al. 2019a), we use MMD with an inverse multi-quadratic kernel and average the results over 100 batches from the data prior and from each trained model. Lower is better.
• Average log-likelihood (LL) of the test data under the model, i.e. − 1 2 T (x) 2 + log|J T (x)| − log(2π N 2 ) where N is the data dimensionality. Higher is better.
• Average intersection-over-union (IOU) between generated shapes and the best fitting shape that follows the construction rules of the data set. See appendix for details on the fitting procedure. Higher is better.
• Average Hausdorff distance (H-DIST) between the contours of the generated shapes and those of the best fitting shapes, as above. Lower is better. Table 2 shows how recursive coupling blocks outperform the conventional design. The difference is especially striking for the single-block network, as a non-recursive coupling block leaves half the variables untouched and is thus inherently unable to model the data distribution properly.
We also trained standard and recursive networks with 4 and 8 coupling blocks on the larger Cross-shapes data set. Representative samples from the 4-block models are shown in figure 6 (right) together with the best fitting, actual Cross shape. Here, it is even more clearly visible that the recursive Table 2: Comparing REAL-NVP and RECURSIVE coupling for sampling and density estimation for the Lens-shapes data (plotting mean ± standard deviation over 3 training runs). model produces samples with better geometry, i.e right angles, straight lines and symmetries in the expected locations.
A quantitative comparison in terms of LL, IOU and H-DIST is presented in table 3, with recursive coupling consistently outperforming standard REAL-NVP blocks.
Bayesian Inference on Fourier Shapes
To set-up Bayesian inference tasks, we formulate forward mappings x → y from x to observable features y. Since these features are incomplete shape descriptors, the inverse y → x is ambiguous, and p(x | y) is learned with HINT. Given a Lens shape, our forward mapping locates its two tips and returns their horizontal and vertical distances d h and d v . The tips' absolute positions and the side of the lens' "bulge" remain undetermined by these features.
The forward mapping for Cross shapes returns four geometrical features. These are the 2d coordinates of the center, i.e where the bars cross, plus the angle of and thickness ratio between the two bars. What remains free, are the absolute thickness, as well as the length and lateral shift of the bars. Finally, noise σ ∼ N (0, 1 20 I) is added to the output of each forward mapping to obtain observed data vectors y.
We trained a conditional flow model (cINN) and HINT with 1, 2, 4 and 8 blocks for Bayesian inference on the Lens shapes. A quantitative comparison, in terms of MMD, IOU and H-DIST, is given in table 4. Here, however, MMD does not compare to samples from the prior p X (x), but to samples from an estimate of the true posterior p(x | y), estimated via Approximate Bayesian Computation (ABC, Csilléry et al. 2010), as in (Ardizzone et al. 2019a), see appendix.
In table 4 we see that HINT consistently produces better shapes (measured by IOU and H-DIST), especially in the case of a single block, and it exhibits better conditioning (as evidenced by MMD) for all but the 8-block model, most likely due to the limited parameter budget, which leaves some of the sub-networks underparameterized. A similar effect can be observed in the LL (not shown in table 4), when only measuring the log-likelihood of the x-lane in HINT and simply excluding contributions from the y-lane.
Qualitative results in figures 7 and 8, for the Lens and for the Cross shapes, respectively, confirm the superior performance of HINT also visually, in particular producing significantly better angles and symmetries for the Cross shapes. This is quantitatively supported in the metrics in table 5.
Recursion Depth vs. Number of Blocks
Seeing that recursive coupling blocks outperform standard ones, we also tested if using more standard coupling blocks closes the gap. We looked at several combinations of number of blocks, recursion depth and parameter budget on the Cross-shapes data. In summary, the first two recursion levels improve performance more than additional coupling blocks. Beyond that, we see diminishing returns, as the limited parameter budget gets distributed over too many subnetworks. Full results are in the appendix.
Conclusion
We presented recursive coupling blocks and HINT, a new invertible architecture for normalizing flow, improving on the traditional coupling block in terms of expressive power by densifying the triangular Jacobian, while keeping the advantages of an accessible latent space. This keeps the efficient sampling and density estimation of RealNVP, which is often compromised by other approaches to denser Jacobians, e.g. auto-regressive flows. To evaluate the model, we introduced a versatile family of data sets based on Fourier decompositions of simple 2D shapes that can be visualized easily, independent of the chosen dimension. In terms of future improvements, we expect that our formulation can be made more computationally efficient through the use of e.g. masking operations, enabling more advanced parallelization.
-APPENDIX -
Details on Lens Shapes Data Set
The shapes in the Lens data set arise from the intersection of two circles. The first has a uniformly random radius r within [1, 2], the second has exactly double that radius. They are positioned at a uniformly random angle, with their centers set apart by a distance of 2.4r. Their intersection is centered at the origin and offset by a random distance from N (0, 1 2 ) in either dimension.
Details on Cross Shapes Data Set
The shapes for the larger Cross data set are generated by taking the union of two oblong rectangles crossing each other at a right angle. For both rectangles, the longer side length is drawn uniformly from [ 3, 5 ] and the other from [ 1 2 , 2 ]. We shift both rectangles along their longer side by a uniformly random distance drawn from [ − 3 2 , 3 2 ]. Then we form the union and insert equally spaced points along the resulting polygon's sides such that no line segment is longer than 1 5 . This is necessary to obtain dense point sequences which are approximated more faithfully by Fourier curves.
Finally, we center the shape at the origin, rotate it by a random angle and again shift it within the plane by a distance drawn from N (0, 1 2 ).
Training Hyper-Parameters
For all experiments, we used the Adam optimizer with betas (0.9, 0.95) for 50 epochs with L2 weight regularization strength 1.86 · 10 −5 . Unless otherwise stated, we used exponential learning rate decay, starting at 0.01 and ending at 0.0001. We employ a hundredfold reduced learning rate for the first three epochs, as we find this helps set training of invertible networks on a stable track. Where we use recursive coupling blocks, the number of neurons per hidden layer in the subnetworks is decreased by half with each level of recursion, down to a minimum of 1/8 of the top level network size. Each subnetwork consists of two hidden layers and ReLU activations. Network parameters are initialized from a Gaussian distribution with σ = 0.005.
UCI Data
For the POWER data set, models were scaled to a budget of 5 · 10 5 trainable parameters and we used a batch size of 1660 (i.e. 1000 batches/epoch).
For the GAS data set, models were scaled to a budget of 5 · 10 5 trainable parameters and we used a batch size of 853 (i.e. 1000 batches/epoch).
For the MINIBOONE data set, models were scaled to a budget of 2.5 · 10 5 trainable parameters and we used a batch size of 300 (i.e. 100 batches/epoch).
Lens Shapes
We instantiate this data set with 10 6 training and 10 5 test samples and use a batch size of 10 4 . All unconditional models were scaled to a budget of 1 · 10 5 trainable parameters, all conditional models to a budget of 4 · 10 5 . Figure 9: Configurations for initializing the shape fitting algorithm on Cross shapes, with a given angle.
Cross Shapes
We instantiate this data set with 10 6 training and 10 5 test samples and use a batch size of 10 4 . All unconditional models were scaled to a budget of 2 · 10 6 trainable parameters, all conditional models to a budget of 4 · 10 6 . For some of the deeper models shown in the trade-off section later on (16 and 32 blocks), we initialize the learning rate to 0.001 to stabilize training.
Shape Fitting Algorithm
In order to judge the quality of individual generated shapes, we compare to the best match among shapes from the ground truth distribution. This means we have to find the best match in an automated fashion. The way we do this differs slightly for the two types of shapes we used.
The algorithm for fitting a proper Lens shapeŜparametrized by a tuple θ consisting of angle, scale and center -to a generated one S works as follows: 1. find the two points P 0 , P 1 ∈ S which are furthest apart 2. α = the angle of the line − −− → P 0 P 1 3. initialize shape parameters θ to α, 2, 1 n n j=1 S j 4. use gradient descent to minimize L Ŝ (θ), S , where shape, and vice versa, starting from a strong heuristic guess for the correct angle. Since the angle has such a strong effect on the fit and our initial guess should be reasonably good, we set a lower learning rate for the angle parameter during gradient descent.
L Ŝ (θ), S = 1 m m i=1 min j D ij + 1 n n j=1 min i D ij with D ∈ R m×n and D ij = Ŝ i − S j
The equivalent procedure for the more complex Cross shapes data set requires a few changes to work well.
• The shapeŜ is parameterized by a tuple θ consisting of width, length and lateral shift of each of the two bars, as well as an angle α and the center of their crossing. • The initial angle α is taken from the dominant straight line within S, as determined by RANSAC (Fischler and Bolles 1981).
• A general clean Cross shapeŜ is a twelve-sided polygon. We adapt the loss L from above by changing the first term to the average distance from each corner of the polygon S to the closest point in S, and the second term to the average distance from each point in S to the closest line segment ofŜ. • Because our Cross shapes can have very diverse layouts, initializations with wrong lateral shift often get stuck in a bad local minimum. To counteract this, we repeat the optimization for each of the nine archetypes shown in figure 9 and keep the result where L is lowest.
With these changes we consistently get a good fitŜ, provided the generated S actually resembles a Cross shape.
Samples from Deeper Models
In figure 10 we show qualitative results for deeper nonrecursive models trained on the Cross shapes data set, all under identical training conditions and at the same total parameter budget of 2M . Both visually and in terms of the x correlation matrix, recursive coupling with only 4 blocks is on par or better than non-recursive coupling with the same parameter budget spread across up to 32 blocks.
Sensitivity of the Fourier Curve Parameterization
To demonstrate how quickly Fourier shapes deteriorate as soon as a single parameter goes "out of sync", we present figure 11. In each row, the same original shape is manipulated by adding an increasing offset (0.1 per column, left to right) to one arbitrary Fourier parameter x i . It is clear that even minor mistakes in a single variable affect the entire shape. The quality of generated samples thus relies heavily on modelling correlations between the variables correctly.
Recursion Depth vs. Number of Blocks
We trained a number of models with varying recursion depth and number of blocks on the unconditional Cross data set to investigate the trade-off between these hyperparameters. The results are presented in tables 6 to 8. A recursion depth of 0 corresponds to standard REAL-NVP coupling. Note that for the latter two tables, we lift the total parameter budget and instead only set a fixed size for the coupling networks within each single block.
We find that in all cases, the best results both statistically and in terms of shape quality are achieved by networks with one or two levels of recursion. Beyond that, the advantages of recursive coupling start to be outweighed by the parameter trade-off or the challenges of training more complex networks. We also find that the deeper networks (i.e. 16 and 32 blocks) exhibit less stable training results, which we consider an argument in favor of using fewer, more expressive coupling blocks.
Proof of Concept: Bayesian filtering using HINT
In the following, we compare HINT and a conditional adaptation of Real-NVP (Ardizzone et al. (2019a), called INN for short) on a Bayesian filtering test case. Filtering is a recursive Bayesian estimation task on hidden Markov models which requires to estimate the posterior of unobserved variables. The posterior is then used as a prior for the next step, and subsequently updated by newly arriving observations. Standard sampling algorithms for filtering require a Monte Carlo approximation of the analytic expression of the prior, which may dramatically deteriorate performance due to large variance. HINT, on the other hand, does not need a functional form of the prior but only samples from it, which makes it an interesting candidate method for filtering.
As a task, we use the general predator-prey model, also termed competitive Lotka-Volterra equations, which typically describes the interaction of d species in a biological system over time:
∂ ∂t x i = β i x i 1 − d j=1 α ij x j (13)
The undisturbed growth rate of species i is given by β i (can be < or > 0, growing or shrinking naturally), and is further affected by the other species in a positive way (α ij > 0, predator), or in a negative way (α ij < 0, prey). The solutions to this system of equations can not be expressed analytically, which makes their prediction a challenging task. Additionally, we make the process stochastic, by adding random noise at each time step. Therefore, at each time step, noisy measurements of x 1 , x 2 , x 3 are observed, with the task to predict the remaining population x 4 , given the current and past observations. We compare this to an implementation using a standard coupling block INN with the training procedure proposed by Ardizzone et al. (2019a). The results for a single example time-series are shown in figure 12. We find that the standard INN does not make meaningful predictions in the shortor mid-term, only correctly predicting the final dying off of population x 4 . HINT on the other hand is able to correctly model the entire sequence. Importantly, the modeled posterior distribution at each step correctly contracts over time, as more measurements are accumulated. The experimental details are as follows: Parameters α and β for the 4d competitive Lotka-Volterra model in equation (13) Figure 13 shows the resulting development of the four populations over ten unit time steps, including noise on the observed values for x 1 , x 2 and x 3 .
Both the INN and the HINT network we trained consist of ten (non-recursive) coupling blocks with a total of 10 6 trainable weights. We used Adam to train both for 50 epochs per time step with 64000 training samples and a batch size of 500. The learning rate started at 10 2 and exponentially decayed to 10 3 (HINT) and 10 4 (INN), respectively. Inference on x 4 begins with an initial guess at t = 0 drawn from N (1, 1 10 ).
Figure 4 :
4Left: A 2d polygon obtained from the segmentation of a natural image. Middle: The vertices p l of the polygon forming the basis for computing the Fourier coefficients in equation
Figure 5 :
5Left: Samples from the Lens shapes data set, and true correlation matrix of Fourier coefficients for a large batch of Lens shapes. Right: The same for Cross shapes. Red lines behind the Fourier curves show the original geometry they approximate.
Figure 6 :
6Samples from REAL-NVP (top) and RECURSIVE (bottom) coupling block networks, trained on the Lens shapes (left) and the Cross shapes (right) from figure 5. The closest fitting shapes from the true distributions are depicted in red for reference. Here and in subsequent figures, the fifth panel shows the absolute differences between true and sampling correlation matrices. BLOCKS REAL-NVP RECURSIVE MMD ↓ 0.370 ± 0.000 0.140 ± 0.002 1 LL ↑ 2.021 ± 0.001 2.861 ± 0.006 IOU ↑ 0.449 ± 0.010 0.688 ± 0.014 H-DIST ↓ 0.519 ± 0.005 0.109 ± 0.005 MMD ↓ 0.012 ± 0.001 0.009 ± 0.004 2 LL ↑ 3.141 ± 0.036 3.219 ± 0.005 IOU ↑ 0.789 ± 0.024 0.819 ± 0.006 H-DIST ↓ 0.063 ± 0.007 0.057 ± 0.000
Figure 7 :
7Samples from a conditional coupling net (CINN, left) and from HINT (right), trained Lens shapes. Green dotted lines mark the largest diameter of each shape; red lines show how it should look according to the data y. Both models do well with 4 blocks (bottom), but only HINT generates reasonable samples with a single coupling block (top). Last panels as infigure 6.
Figure 8 : 6
86Samples from a conditional coupling net (CINN, left) and from HINT (right) with 4 (top) or 8 (bottom) blocks, trained on Cross shapes. The expected center, angle and thickness ratio of the Cross according to the data y are shown in green, the best fitting Cross shape in red. Visually, HINT reproduces shapes from the data set much better. Last panels as in figure
Figure 10 :
10for initialization with angle α = α + π (i.e. the "flipped" lens) and keep the result with lower L In other words, we minimize the average distance from each point in the fitted shape to the nearest point in the generated Samples from deeper REAL-NVP models vs. ours, visualized in the same style as in the main paper.
Figure 11 :
11Sensitivity of Fourier shapes to changes in a single parameter.
Figure 12 :Figure 13 :
1213Contracting Population development in competitive Lotka-Volterra model. x 1 , x 2 and x 3 are only accessible via noisy measurements (small crosses), the colored bands show the standard deviation of this noise. The task is to predict x 4 as the sequence develops.
MINIBOONE 42 −16.625 ± 0.119 −14.117 ± 0.1631
BLOCKS
DIM
REAL-NVP
RECURSIVE
POWER
6
−0.054 ± 0.017
−0.027 ± 0.018
4
GAS
8
7.620 ± 0.136
7.662 ± 0.094
MINIBOONE 42 −19.296 ± 0.395 −14.547 ± 0.164
POWER
6
0.093 ± 0.002
0.080 ± 0.007
8
GAS
8
8.062 ± 0.177
8.137 ± 0.055
Table 3 :
3Comparison of flow models for Cross shapes with normal (RNVP) and recursive coupling (REC), respectively.
Table 4 :
4Conditional (CINN) vs. hierarchical (HINT) cou-
pling for Bayesian inference on Lens shapes (mean ± stan-
dard deviation over 3 training runs). See text for details.
Table 5 :
5Fidelity of Cross shapes generated by conditional
(CINN) and hierarchical (HINT) flows, respectively.
were picked randomly as β = 0.78382338 1.26614888 1.25787652 0.80904295 1.00470891 0.32719451 1.34501072 1.29758381 1.28599724 0.39362355 0.89977679 1.00063551 1.12163602 1.08672758 1.39634746 0.53592833 and the true initial population values x *
1.14143055
0.64270729
1.42981209
0.90620443
and
α =
,
0 set to
x *
0 =
1.00471435
0.98809024
1.01432707
0.99687348
.
Table 6 :
6Comparing the effect of more coupling blocks versus more levels of recursion within the blocks, while keeping the network parameter budget fixed at 2 · 10 6 .RECURSION DEPTH
0
1
2
3
4 BLOCKS
LL ↑
3.419
3.680
3.663
3.666
IOU ↑
0.314
0.857
0.823
0.835
H-DIST ↓
0.236
0.066
0.077
0.071
CORR ↓
0.0454
0.0022
0.0027
0.0026
8 BLOCKS
LL ↑
3.329
3.592
3.593
IOU ↑
0.588
0.802
0.817
H-DIST ↓
0.138
0.085
0.083
CORR ↓
0.0064
0.0030
0.0028
16 BLOCKS
LL ↑
3.530
3.623
IOU ↑
0.778
0.841
H-DIST ↓
0.101
0.073
CORR ↓
0.0035
0.0026
32 BLOCKS
LL ↑
3.578
IOU ↑
0.836
H-DIST ↓
0.082
CORR ↓
0.0029
Table 7 :
7Comparing the effect of more coupling blocks versus more levels of recursion within the blocks, while affording internal networks a width of 128 neurons, halved at every recursion level.RECURSION DEPTH
0
1
2
3
4 BLOCKS
LL ↑
3.496
3.595
3.612
3.614
IOU ↑
0.713
0.806
0.813
0.804
H-DIST ↓
0.115
0.090
0.085
0.082
CORR ↓
0.0032
0.0025
0.0024
0.0023
8 BLOCKS
LL ↑
3.623
3.664
3.651
IOU ↑
0.822
0.864
0.860
H-DIST ↓
0.081
0.065
0.067
CORR ↓
0.0023
0.0018
0.0018
16 BLOCKS
LL ↑
3.631
3.613
IOU ↑
0.839
0.819
H-DIST ↓
0.074
0.080
CORR ↓
0.0020
0.0022
32 BLOCKS
LL ↑
3.612
IOU ↑
0.814
H-DIST ↓
0.086
CORR ↓
0.7490
Table 8 :
8Comparing the effect of more coupling blocks versus more levels of recursion within the blocks, while affording internal networks a width of 512 neurons, halved at every recursion level.RECURSION DEPTH
0
1
2
3
4 BLOCKS
LL ↑
3.620
3.685
3.685
3.681
IOU ↑
0.846
0.837
0.825
0.810
H-DIST ↓
0.082
0.069
0.068
0.071
CORR ↓
0.0023
0.0025
0.0023
0.0025
8 BLOCKS
LL ↑
3.669
3.716
3.722
IOU ↑
0.863
0.883
0.888
H-DIST ↓
0.064
0.053
0.052
CORR ↓
0.0020
0.0018
0.0017
16 BLOCKS
LL ↑
3.702
3.685
IOU ↑
0.882
0.878
H-DIST ↓
0.055
0.060
CORR ↓
0.0017
0.0017
32 BLOCKS
LL ↑
3.654
IOU ↑
0.813
H-DIST ↓
0.072
CORR ↓
0.0021
Code and data at https://github.com/VLL-HD/HINT.
Acknowledgments Jakob Kruse was supported by Informatics for Life funded by the Klaus Tschira Foundation. Gianluca Detommaso was supported by the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (EP/L015684/1).
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"Strong Successive Refinability and Rate-Distortion-Complexity Tradeoff",
"Strong Successive Refinability and Rate-Distortion-Complexity Tradeoff"
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"Albert No [email protected] \nDepartment of Electrical Engineering\nStanford University\n\n",
"Amir Ingber [email protected] \nDepartment of Electrical Engineering\nStanford University\n\n",
"Tsachy Weissman [email protected] \nDepartment of Electrical Engineering\nStanford University\n\n"
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"Department of Electrical Engineering\nStanford University\n",
"Department of Electrical Engineering\nStanford University\n",
"Department of Electrical Engineering\nStanford University\n"
]
| []
| We investigate the second order asymptotics (source dispersion) of the successive refinement problem. Similarly to the classical definition of a successively refinable source, we say that a source is strongly successively refinable if successive refinement coding can achieve the second order optimum rate (including the dispersion terms) at both decoders. We establish a sufficient condition for strong successive refinability. We show that any discrete source under Hamming distortion and the Gaussian source under quadratic distortion are strongly successively refinable.We also demonstrate how successive refinement ideas can be used in point-to-point lossy compression problems in order to reduce complexity. We give two examples, the binary-Hamming and Gaussian-quadratic cases, in which a layered code construction results in a low complexity scheme that attains optimal performance. For example, when the number of layers grows with the block length n, we show how to design an O(n log(n) ) algorithm that asymptotically achieves the rate-distortion bound.Index TermsComplexity, layered code, rate-distortion, refined strong covering lemma, source dispersion, strong successive refinability, successive refinement. | 10.1109/tit.2016.2549540 | [
"https://arxiv.org/pdf/1506.03407v2.pdf"
]
| 1,044,352 | 1506.03407 | 679ebfa29408351333168ec520e8c2e8770b4679 |
Strong Successive Refinability and Rate-Distortion-Complexity Tradeoff
15 Mar 2016
Albert No [email protected]
Department of Electrical Engineering
Stanford University
Amir Ingber [email protected]
Department of Electrical Engineering
Stanford University
Tsachy Weissman [email protected]
Department of Electrical Engineering
Stanford University
Strong Successive Refinability and Rate-Distortion-Complexity Tradeoff
15 Mar 2016arXiv:1506.03407v2 [cs.IT] 1 † Yahoo! LabsIndex Terms Complexitylayered coderate-distortionrefined strong covering lemmasource dispersionstrong successive refinabilitysuccessive refinement
We investigate the second order asymptotics (source dispersion) of the successive refinement problem. Similarly to the classical definition of a successively refinable source, we say that a source is strongly successively refinable if successive refinement coding can achieve the second order optimum rate (including the dispersion terms) at both decoders. We establish a sufficient condition for strong successive refinability. We show that any discrete source under Hamming distortion and the Gaussian source under quadratic distortion are strongly successively refinable.We also demonstrate how successive refinement ideas can be used in point-to-point lossy compression problems in order to reduce complexity. We give two examples, the binary-Hamming and Gaussian-quadratic cases, in which a layered code construction results in a low complexity scheme that attains optimal performance. For example, when the number of layers grows with the block length n, we show how to design an O(n log(n) ) algorithm that asymptotically achieves the rate-distortion bound.Index TermsComplexity, layered code, rate-distortion, refined strong covering lemma, source dispersion, strong successive refinability, successive refinement.
I. INTRODUCTION
In the successive refinement problem, an encoder wishes to send a source to two decoders with different target distortions. Instead of designing separate coding schemes, the successive refinement encoder uses a code for the first decoder which has a weaker link and sends extra information to the second decoder on top of the message of the first decoder. In general, the performance of a successive refinement coding scheme is worse than separate coding for each decoder. However, for some cases, we can simultaneously achieve the optimum rates for both The material in this paper has been presented in part at the 2013 51st Annual Allerton Conference on Communication, Control, and Computing decoders as if the optimum codes were used separately. In this case, we say the source is successively refinable.
Necessary and sufficient conditions for successive refinement were independently proposed by Koshélev [1], [2] and Equitz and Cover [3]. Rimoldi [4] found the full rate-distortion region of the successive refinement problem including non-successively refinable sources. Kanlis and Narayan [5] extended the result to the error exponent that quantifies "how fast the excess distortion probability decays". Tuncel [6] characterized the entire region of ratedistortion-exponents with separate handling of the two error events. Both lines of work considered error exponents in the spirit of Marton [7], which characterized the error exponent for the point-to-point case.
For the point-to-point source coding problem, Ingber and Kochman [8] and Kostina and Verdù [9] independently proposed an asymptotic analysis that complements the error exponent analysis. In this setting, the figure of merit is the minimum achievable rate when the excess distortion probability ǫ and the block length n are fixed. This can be quantified by the source dispersion. For an i.i.d. source with law P , the minimum rate can be approximated by R(P, D) + V (P, D)/nQ −1 (ǫ), where R(P, D) and V (P, D) are, respectively, the rate-distortion function and dispersion of a source P at distortion level D. We can consider this rate as a "second order" optimum rate (where the classical rate-distortion function is the first order result).
With this stronger notion of optimality, it is natural to ask whether successive refinement schemes can achieve the second order optimum rates at both decoders simultaneously. An obvious necessary condition for the existence of such schemes is that the source be successively refinable, so we refer to such a source as "strongly successively refinable" (formal definitions follow in the sequel). In this paper, we present a second order achievability result for the successive refinement problem. As a corollary, we derive a sufficient condition for strong successive refinability and show that a source P is strongly successively refinable if all sourcesP in the neighborhood of P are successively refinable.
In the second part of the paper, we show that successive refinement codes can be useful in the point-to-point source coding problem when we want to achieve lower encoding complexity. The idea is that finding the best representing codeword in a successive manner is often easier than finding a codeword from the set of all codewords, which normally has exponential complexity. Moreover, storing exponentially many codewords is often prohibitive, while successive refinement encoding can reduce the size of codebooks. Our findings here contribute to the recent line of work on reducing the complexity of rate-distortion codes, cf. [10]- [12] and references therein.
We aim to study the general approach of using successive encoding to reduce complexity. We denote this approach by "layered coding", a family that includes all coding schemes that can be implemented in a successive manner.
Basically, the layered coding scheme is searching for an appropriate codeword over a tree structure where the number of decoders corresponds to the level of the tree. The larger the tree, the faster the codeword can be found, and therefore the lower decoding complexity. In order to reduce the encoding complexity significantly, we generalize the result to the case where the number of decoders is increasing with block length n. This is different from the classical successive refinability where only a fixed number of decoders are considered. On the other hand, the larger tree structure restricts the class of coding schemes, and therefore too many decoders may cause a rate loss. Our result for this setting characterizes an achievable trade-off between encoding complexity (how fast can we find the codeword) and performance (how much do we end up compressing). Note that SPARC [12] and CROM [13] are manifestations of the layered coding approach that attain good performance.
The rest of the paper is organized as follows. In Section II, we revisit the known results about successive refinement and source dispersion. Section III provides the problem setting. We present our main results in Section IV, where proof details are given in Section V. Section VI is dedicated to a layered coding scheme, and we conclude in Section VII.
Notation: X n and X denotes an n-dimensional random vector (X 1 , X 2 , . . . , X n ) while x n and x denotes a specific realization of it. When we have two random vectors, we use the notation such asX n 1 = (X 1,1 ,X 1,2 , . . . ,X 1,n ) andX n 2 = (X 2,1 ,X 2,2 , . . . ,X 2,n ).
II. PRELIMINARIES
A. Source Dispersion
Consider an i.i.d. source X n with law P where the source alphabet is X and the reconstruction alphabet iŝ X . Let d : X ×X → [0, ∞) be a distortion measure where d(x n ,x n ) = (1/n) n i=1 d(x i ,x i ). It is well known that the rate-distortion function R(P, D) is the optimal asymptotic compression rate for which distortion D can be achieved. However, this first order optimum rate can be achieved only when the block length n goes to infinity.
Beyond the first order rate, we can consider two 1 asymptotic behaviors which are excess distortion exponent [7] and the source dispersion [9], [16]. The former considers how fast the excess distortion probability Pr d(X n ,X n ) > D is decaying, while the latter considers how fast the minimum number of codewords converges to R(P, D) when excess distortion probability ǫ and block length n are given. It was shown that the difference between the minimum rate for fixed n and R(P, D) is inversely proportional to square root of n. More formally, let R P,D,ǫ (n) be the minimum compression rate for which the excess distortion probability is smaller than ǫ. The result is given by: Theorem 1 ( [16]): Suppose R(P, D) is twice differentiable 2 with respect to D and the elements of P in some neighborhood of (P, D). Then
R P,D,ǫ (n) = R(P, D) + V (P, D) n Q −1 (ǫ) + O log n n(1)
where V (P, D) is the source dispersion, given by
V (P, D) VAR [R ′ (X, D)] (2) = x∈X P (x)(R ′ (x, D)) 2 − x∈X P (x)R ′ (x, D) 2(3)
1 These asymptotic approaches analyze the excess distortion probability. Other approaches exist which analyze the average achievable distortion [14], [15]. 2 We say R(P, D) is differentiable at P if there is an extensionR(·, D) : R m → R which is differentiable. Under this definition, R ′ (x, D)
and V (P, D) are well and uniquely defined. Details are given in Appendix A. and R ′ (x, D) denotes the derivative of R(P, D) with respect to the probability P (x):
R ′ (x, D) ∂R(Q, D) ∂Q(x) Q=P .(4)
We have a similar result for the Gaussian source under quadratic distortion:
Theorem 2 ( [8]): Consider an i.i.d. Gaussian source X n distributed according to N (0, σ 2 ), and quadratic dis-
tortion, i.e., d(x n ,x n ) = (1/n) n i=1 (x i −x i ) 2 . Then R P,D,ǫ (n) = 1 2 log σ 2 D + 1 2n Q −1 (ǫ) + O log n n .(5)
Note that the dispersion of the Gaussian source is V (P, D) = 1/2 nats 2 /source symbol for all D ≤ σ 2 .
B. Successive Refinement
The successive refinement problem with two decoders can be formulated as follows. Again, let X n be i.i.d. with law P . The encoder sends a pair of messages (m 1 ,
m 2 ) where 1 ≤ m i ≤ M i for i ∈ {1, 2}.
The first decoder takes m 1 and reconstructsX n 1 (m 1 ) ∈X n 1 where the second decoder takes (m 1 , m 2 ) and reconstructsX n 2 (m 1 , m 2 ) ∈X n 2 . Note thatX 1 andX 2 denote the respective reconstruction alphabets of the decoders. The i-th decoder employs the distortion measure d i (·, ·) : X ×X i → [0, ∞) and wants to recover the source x n with distortion D i , i.e.,
d i (x n ,X n i ) ≤ D i for i ∈ {1, 2}.(6)
The rates of the code are defined as
R 1 = 1 n log M 1 (7) R 2 = 1 n log M 1 M 2 .(8)
An (n, R 1 , R 2 , D 1 , D 2 , ǫ)-successive refinement code is a coding scheme with block length n and excess distortion probability ǫ where rates are (R 1 , R 2 ) and target distortions are (D 1 , D 2 ). Since we have two decoders, the excess distortion probability is defined by Pr d i (X n ,X n i ) > D i for some i .
Definition 1: A rate-distortion tuple (R 1 , R 2 , D 1 , D 2 ) is achievable, if there is a family of (n, R (n) 1 , R (n) 2 , D 1 , D 2 , ǫ (n) )-successive refinement codes where lim n→∞ R (n) i = R i for i ∈ {1, 2},(9)lim n→∞ ǫ (n) = 0.(10)
The achievable rate-distortion region is known:
Theorem 3 ([4]):
Consider a discrete memoryless source X n with law P . The rate-distortion tuple (R 1 ,
R 2 , D 1 , D 2 )
is achievable if and only if there is a joint law P X,X1,X2 of random variables (X,X 1 ,X 2 ) (where X is distributed according to P ) such that
I(X;X 1 ) ≤R 1(11)I(X;X 1 ,X 2 ) ≤R 2 (12) E d i (X,X i ) ≤D i for i ∈ {1, 2}.(13)
In some cases, we can achieve the optimum rates at both decoders simultaneously:
Definition 2: For i ∈ {1, 2}, let R i (P, D i ) denote the rate-distortion function of the source P when the distortion measure is d i (·, ·) and the distortion level is D i . If (R 1 (P, D 1 ), R 2 (P, D 2 ), D 1 , D 2 ) is achievable, then we say the source is successively refinable at (D 1 , D 2 ). Furthermore, if the source is successively refinable at (D 1 , D 2 ) for all non-degenerate D 1 , D 2 (i.e., for which R 1 (P, D 1 ) < R 2 (P, D 2 )), then we say the source is successively refinable.
A necessary and sufficient condition for successive refinability is known. [3]): A source P is successively refinable at (D 1 , D 2 ) if and only if there exists a conditional distribution PX 1 ,X2|X such that X −X 2 −X 1 forms a Markov chain and
Theorem 4 ([1],R i (P, D i ) = I(X;X i )(14)E d i (X,X i ) ≤ D i(15)
for i ∈ {1, 2}.
The condition in the theorem holds for the cases of a Gaussian source under quadratic distortion and for any discrete memoryless sources under Hamming distortion. Note that the successive refinability is not shared by all sources and distortion measures. For instance, symmetric Gaussian mixtures under quadratic distortion are not successively refinable [17]. The above results of successive refinability can be generalized to the case of k decoders.
Note that we can also define successive refinability using R(P, D 1 , D 2 ) where R(P, D 1 , D 2 ) is the minimum rate R 2 such that (R 1 (P, D 1 ), R 2 , D 1 , D 2 ) is achievable. Using Theorem 3, we can characterize R(P, D 1 , D 2 ),
R(P, D 1 , D 2 ) = inf PX 1 ,X 2 |X : E[d1(X,X1)]≤D1, E[d2(X,X2)]≤D2, I(X;X1)≤R1(P,D1) I(X;X 1 ,X 2 ).(16)
Definition 2 implies that the source is successively refinable at (D 1 , D 2 ) if and only if R(P, D 1 , D 2 ) = R 2 (P, D 2 ).
III. PROBLEM SETTING
We consider the successive refinement problem with two decoders. Let X n = (X 1 , · · · , X n ) be i.i.d. with law
P , where the source alphabet is X . An encoder f (n) = f (n) 1 , f (n) 2
maps a source sequence to a pair of messages,
f (n) 1 : X n → {1, · · · , M 1 } (17) f (n) 2 : X n → {1, · · · , M 2 }.(18)
The first decoder receives only the output of f 1 (X n ), and therefore we say that its rate is R 1 = (1/n) log M 1 . The second decoder receives the output of both functions, so its rate is
R 2 = (1/n) log M 1 M 2 .
Decoder 1 employs a decoder g
d i (x n ,x n i ) = 1 n n j=1 d i (x j ,x i,j )(19)
for all i ∈ {1, 2}, x n ∈ X n ,x n 1 ∈X n 1 andx n 2 ∈X n 2 . The setting is described in Figure 1.
1 (f (n) 1 (X n ))) > D 1 ≤ǫ 1(20)
Pr d 2 (X n , g
(n) 2 (f (n) 1 (X n ), f (n) 2 (X n ))) > D 2 ≤ǫ 2 ,(21)
and such a code is called a (n, M 1 , M 2 , D 1 , D 2 , ǫ 1 , ǫ 2 )-code.
Note that we consider the two error events separately, unlike in the definition of a successive refinement code in Section II-B. Our goal is to characterize the achievable (n, M 1 , M 2 , D 1 , D 2 , ǫ 1 , ǫ 2 ) region in general. Motivated by successive refinability, we define strong successive refinability as follows.
Definition 4:
The source is strongly successively refinable at (D 1 ,
D 2 , ǫ 1 , ǫ 2 ) if (n, M 1 , M 2 , D 1 , D 2 , ǫ 1 , ǫ 2 ) is achievable for some M 1 , M 2 satisfying 1 n log M 1 = R 1 (P, D 1 ) + V 1 (P, D 1 ) n Q −1 (ǫ 1 ) + o 1 √ n (22) 1 n log M 1 M 2 = R 2 (P, D 2 ) + V 2 (P, D 2 ) n Q −1 (ǫ 2 ) + o 1 √ n(23)
where R i (P, D i ) and V i (P, D i ) are the point-to-point rate-distortion function and the source dispersion for the i-th decoder. Furthermore, if the source is strongly successively refinable at (D 1 , D 2 , ǫ 1 , ǫ 2 ) for all non-degenerate D 1 , D 2 , ǫ 1 , ǫ 2 (i.e., R P,D1,ǫ (n) < R P,D2,ǫ (n)), then we say the source is strongly successively refinable.
While standard successive refinability implies that the successive refinement structure does not cause any loss in the compression rate (asymptotically), strong successive refinability implies that we also do not lose from the dispersion point of view.
Note that in order to verify that a source is strongly successively refinable, it is sufficient to find an achievability scheme since the converse will follow from the converse in point-to-point source coding.
IV. MAIN RESULTS
Our results in this section pertain to discrete memoryless sources under general distortion, as well as Gaussian sources under quadratic distortion. The results are given here, with proofs in Section V.
A. Discrete Memoryless Source
Let X n be i.i.d. with distribution P and the distortion measures be d 1 :
X ×X 1 → [0, ∞) and d 2 : X ×X 2 → [0, ∞).
We assume that the alphabets X ,X 1 andX 2 are finite, and therefore distortion measures d 1 and d 2 are bounded by some constant d M . We further assume that P (x) > 0 for all x ∈ X since one can remove the source symbol from X that has zero probability. Then, the following theorem provides the achievable rates including the second order term:
Theorem 5 (Achievability for Discrete Memoryless Source): Assume that both R 1 (P, D 1 ) and R(P, D 1 , D 2 ) are continuously twice differentiable with respect to D 1 , D 2 , and the elements of P in some neighborhood of (P, D 1 , D 2 ).
Then, there exists an (n, M 1 , M 2 , D 1 , D 2 , ǫ 1 , ǫ 2 )-code such that
1 n log M 1 = R 1 (P, D 1 ) + V 1 (P, D 1 ) n Q −1 (ǫ 1 ) + O log n n (24) 1 n log M 1 M 2 = R(P, D 1 , D 2 ) + V (P, D 1 , D 2 ) n Q −1 (ǫ 2 ) + O log n n (25) where V 1 (P, D 1 ) VAR [R ′ 1 (X, D 1 )] (26) = x∈X P (x)(R ′ 1 (x, D 1 )) 2 − x∈X P (x)R ′ 1 (x, D 1 ) 2 (27) V (P, D 1 , D 2 ) VAR [R ′ (X, D 1 , D 2 )] (28) = x∈X P (x)(R ′ (x, D 1 , D 2 )) 2 − x∈X P (x)R ′ (x, D 1 , D 2 ) 2 .(29)
Similarly to Theorem 1, R ′ 1 (x, D 1 ) is the derivative of R 1 (P, D 1 ) with respect to the probability P (x) and R ′ (x, D 1 , D 2 ) is the derivative 3 of R(P, D 1 , D 2 ) with respect to the probability P (x):
R ′ 1 (x, D 1 ) ∂R 1 (Q, D 1 ) ∂Q(x) Q=P (30) R ′ (x, D 1 , D 2 ) ∂R(Q, D 1 , D 2 ) ∂Q(x) Q=P .(31)
By applying the above theorem to the special case where R(P , D 1 , D 2 ) = R 2 (P , D 2 ) for allP in some neighborhood of P , we get the following corollary. 3 Similar to the definition of R ′ (x, D), we can define R ′ (x, D 1 , D 2 ) using an extension. Then, R ′ (x, D 1 , D 2 ) and V (P, D 1 , D 2 ) are well and uniquely-defined as well, where details are given in Appendix A.
Corollary 6: Suppose R i (P, D i ) is continuously twice differentiable with respect to D i and the elements of P in some neighborhood of (P, D i ) for i ∈ {1, 2}. If all sourcesP in some neighborhood of P are successively refinable at D 1 , D 2 , then there exists an (n, M 1 , M 2 , D 1 , D 2 , ǫ 1 , ǫ 2 ) code such that
1 n log M 1 = R 1 (P, D 1 ) + V 1 (P, D 1 ) n Q −1 (ǫ 1 ) + O log n n (32) 1 n log M 1 M 2 = R 2 (P, D 2 ) + V 2 (P, D 2 ) n Q −1 (ǫ 2 ) + O log n n ,(33)
i.e., the source P is strongly successively refinable at (D 1 , D 2 , ǫ 1 , ǫ 2 ).
The corollary is because R(P , D 1 , D 2 ) = R 2 (P , D 2 ) for allP in some neighborhood of P implies that their derivatives at (P, D 1 , D 2 ) coincide, i.e.,
∂R 2 (Q, D 2 ) ∂Q(x) Q=P = ∂R(Q, D 1 , D 2 ) ∂Q(x) Q=P .(34)
Since the source dispersion is the variance of the derivatives, we have V (P, D 1 , D 2 ) = V 2 (P, D 2 ).
Remark 1: Any discrete memoryless source under Hamming distortion measure is successively refinable. Therefore, Corollary 6 implies that any discrete memoryless source under Hamming distortion is also strongly successively refinable provided R(P, D) is appropriately differentiable. Note that the size of the set {D :
R(P, D) is not differentiable} is at most |X | [18].
B. Gaussian Memoryless Source
Let X n be an i.i.d. N (0, σ 2 ) source, and suppose the distortion measure is quadratic (at both decoders).
Theorem 7 (Achievability for Gaussian Memoryless Source):
The memoryless Gaussian source under quadratic distortion is strongly successively refinable, i.e., for σ 2 > D 1 > D 2 , there exists an (n,
M 1 , M 2 , D 1 , D 2 , ǫ 1 , ǫ 2 ) code such that 1 n log M 1 = 1 2 log σ 2 D 1 + 1 2n Q −1 (ǫ 1 ) + O log n n (35) 1 n log M 1 M 2 = 1 2 log σ 2 D 2 + 1 2n Q −1 (ǫ 2 ) + O log n n .(36)
V. PROOF
A. Method of Types
Our proofs for finite alphabet sources rely heavily on the method of types [19]. In this section, we briefly review its notation and results that we use. Without loss of generality we assume X = {1, 2, . . . , r x }. For any sequence
x n ∈ X n , let N (a|x n ) be the number of symbol a ∈ X in the sequence x n . Let the type of a sequence x n be an r x dimensional vector P x n = (N (1|x n )/n, N (2|x n )/n, . . . , N (r x |x n )/n). Then, denote P n (X ) be the set of all types on X n , i.e., P n (X ) = {P x n | x n ∈ X n }. The size of the set P n (X ) is at most polynomial in n, more precisely,
|P n (X )| ≤ (n + 1) rx .(37)
For given type P , define type class of P by
T P = {x n ∈ X n | P x n = P }.(38)
We can also define type class T x n = {x n ∈ X n | Pxn = P x n } using a sequence x n ∈ X n . We can bound the size of type class.
1 (n + 1) rx exp (nH(P )) ≤ |T P | ≤ exp (nH(P ))(39)
where H(P ) denote an entropy of random variable with law P .
We further consider the conditional types. Let Y be a set of alphabet where we also assume Y = {1, 2, . . . , r y } to be finite. Consider a stochastic kernel W : X → Y. We say that y n ∈ Y n has conditional type W given
x n ∈ X n if N (a, b|x n , y n ) = N (a|x n )W (b|a).(40)
Then, we can define conditional type class of W given x n ∈ X n by
T W (x n ) = {y n ∈ Y n | y n has conditional type W given x n }.(41)
We can also bound a size of conditional type class. For sequence x n ∈ X n with type P , and for conditional type W , we have
1 (n + 1) rxry exp (nH(P |W )) ≤ |T W (x n )| ≤ exp (nH(P |W )) .(42)
H(P |W ) denotes a conditional entropy of U given V where (U, V ) are random variables with a joint law P × W .
B. Proof of Theorem 5
A key tool used in the proof is a refined version of the type covering lemma [19]. We say a set B is D-covering a set A if for all a ∈ A, there exists an element b ∈ B such that d(a, b) ≤ D. In the successive refinement setting, we need to cover a set in a successive manner.
Definition 5: Let d 1 : A × B → [0, ∞) and d 2 : A × C → [0, ∞) be distortion measures. Consider sets A ⊂ A, B ⊂ B and C b ⊂ C for all b ∈ B. We say B and {C b } b∈B successively (D 1 , D 2 )-cover a set A, if for all a ∈ A, there exist b ∈ B and c ∈ C b such that d 1 (a, b) ≤ D 1 and d 2 (a, c) ≤ D 2 .
The following lemma provides an upper bound of minimum size of sets that successively (D 1 , D 2 )-cover a type class T P .
Lemma 8 (Refined Covering Lemma):
For fixed n, let P ∈ P n (X ) be a type on X where P (x) > 3/n for all
x ∈ X . Suppose ∇R(P, D 1 , D 2 ) is bounded in some neighborhood of (D 1 , D 2 ) where ∇R(P, D 1 , D 2 ) = ∂ ∂D 1 R(P, D 1 , D 2 ), ∂ ∂D 2 R(P, D 1 , D 2 ) .(43)
Then for D 1 , D 2 ∈ (0, d M ), there exist sets B 1 ⊂X n 1 and B 2 (x n 1 ) ⊂X n 2 for eachx n 1 ∈ B 1 where B 1 and {B 2 (x n 1 )}xn 1 ∈B1 successively (D 1 , D 2 )-cover T P with following properties:
• The size of B 1 is upper bounded:
1 n log |B 1 | ≤R 1 (P, D 1 ) + k 1 log n n . (44) • For allx n 1 ∈ B 1 , the size of B 2 (x n 1 ) is also bounded: 1 n log (|B 1 | · |B 2 (x n 1 )|) ≤R(P, D 1 , D 2 ) + k 2 log n n ,(45)
where k 1 and k 2 are universal constants, i.e., do not depend on the distribution P or n.
The proof of Lemma 8 is given in Appendix B. The following corollary provides a successive refinement scheme using B 1 and {B 2 (x n 1 )}xn 1 ∈B1 from Lemma 8. Corollary 9: For length of sequence n and type Q ∈ P n (X ), letR satisfyR ≥ R 1 (Q, D) + k 1 log n/n. Then, there exists a coding scheme for T Q such that
• Encoding functions are f Q,1 : T Q → {1, . . . , M Q,1 } and f Q,2 : T Q → {1, . . . , M Q,2 }.
• Decoder 1 and Decoder 2 employ
g Q,1 : {1, . . . , M Q,1 } →X n 1 (46) g Q,2 : {1, . . . , M Q,1 } × {1, . . . , M Q,2 } →X n 2 (47)
respectively.
• For all x n ∈ T Q , encoding and decoding functions satisfy
d 1 (x n , g Q,1 (f Q,1 (x n ))) ≤D 1 (48) d 2 (x n , g Q,2 (f Q,1 (x n ), f Q,2 (x n ))) ≤D 2 .(49)
• The number of messages are bounded:
R ≤ 1 n log M Q,1 ≤R + log n n (50) 1 n log M Q,1 M Q,2 ≤R(Q, D 1 , D 2 ) + (k 2 + 1) log n n .(51)
The proof of Corollary 9 is given in Appendix C.
Let us now describe the achievability scheme. Similar to the idea from [6], we will consider the four cases according to the type Q of the input sequence x n . For each case, the encoding will be done in a different manner.
Before specifying four cases, we need to define ∆R 1 and ∆R 2 . Let ∆R 1 be the infimal value such that the probability of {R 1 (P X n , D 1 ) > R 1 (P, D 1 ) + ∆R 1 } is smaller than ǫ 1 , and ∆R 2 be the infimal value such that the probability of {R(P X n , D 1 , D 2 ) > R(P, D 1 , D 2 ) + ∆R 2 } is smaller than ǫ 2 . Recall that P X n denotes the type of X n . The error occurs at decoder 1 if and only if R 1 (P X n , D 1 ) > R 1 (P, D 1 ) + ∆R 1 , and therefore probability of error at decoder 1 is less than ǫ 1 . Similarly, the error occurs at decoder 2 if and only if R(P X n , D 1 , D 2 ) > R(P, D 1 , D 2 ) + ∆R 2 , and therefore probability of error at decoder 2 is less than ǫ 2 . The following lemma bounds ∆R 1 and ∆R 2 .
Lemma 10:
∆R 1 = V 1 (P, D 1 ) n Q −1 (ǫ 1 ) + O log n n (52) ∆R 2 = V (P, D 1 , D 2 ) n Q −1 (ǫ 2 ) + O log n n .(53)
The proof follows directly from [16, Lemma 3].
We are ready to define four cases based on the type of the source sequence as well as corresponding encoding schemes.
1) Q ∈ A (0,0) {Q ∈ P n (X ) : R 1 (Q, D 1 ) − R 1 (P, D 1 ) ≤ ∆R 1 , R(Q, D 1 , D 2 ) − R(P, D 1 , D 2 ) ≤ ∆R 2 }.
In this case, both decoders decode successfully. Since R(Q, D 1 ) ≤ R(P, D 1 ) + ∆R 1 , by Corollary 9, there exist encoding and decoding functions f Q,1 , f Q,2 , g Q,1 , g Q,2 such that
d 1 (x n , g Q,1 (f Q,1 (x n ))) ≤D 1 (54) d 2 (x n , g Q,2 (f Q,1 (x n ), f Q,2 (x n ))) ≤D 2(55)
for all x n ∈ T Q and
R 1 (P, D 1 ) + ∆R 1 + k 1 log n n ≤ 1 n log M (0,0) Q,1 ≤R 1 (P, D 1 ) + ∆R 1 + (k 1 + 1) log n n (56) 1 n log M (0,0) Q,1 M (0,0) Q,2 ≤R(Q, D 1 , D 2 ) + (k 2 + 1) log n n .(57)
We emphasize that we have
R 1 (P, D 1 ) instead of R 1 (Q, D 1 ) in (56)
. This is because we need to aggregate the codewords at the end of the proof. More precisely, we have to fix the number of codewords for decoder 1 in order to bound the number of codewords only for decoder 2.
2) Q ∈ A (0,1) {Q ∈ P n (X ) :
R 1 (Q, D 1 ) − R 1 (P, D 1 ) ≤ ∆R 1 , R(Q, D 1 , D 2 ) − R(P, D 1 , D 2 ) > ∆R 2 }.
For those Q, the encoder only D 1 covers T Q . Thus, decoder 1 will decode successfully and decoder 2 will declare an error. In this case, we do not need a message for decoder 2 and we can think of M } →X n 1 such that
d 1 (x n , g (0,1) (f (0,1) (x n ))) ≤ D 1 (58) for all Q ∈ A (0,1) and x n ∈ T Q where 1 n log M (0,1) 1 = R 1 (P, D 1 ) + ∆R 1 + O log n n . (59) 3) Q ∈ A (1,0) {Q ∈ P n (X ) : R 1 (Q, D 1 ) − R 1 (P, D 1 ) > ∆R 1 , R(Q, D 1 , D 2 ) − R(P, D 1 , D 2 ) ≤ ∆R 2 }.
In this case, the encoder only D 2 covers T Q . Thus, decoder 2 will decode successfully and decoder 1 will declare an error. In this case, we do not need a message for decoder 1. However, because of the structure of successive refinement code, we need to reformulate the point-to-point code for the second decoder into the form of successive refinement code. More precisely, we can find functionsf
(1,0) Q : X n → {1, . . . ,M (1,0) Q,2 } andg (1,0) Q : {1, . . . ,M (1,0) Q,2 } →X n 2 such that d 2 (x n ,g (1,0) Q (f (1,0) Q (x n ))) ≤ D 2 (60) for all x n ∈ T Q where 1 n logM (1,0) Q,2 ≤ R 2 (Q, D 2 ) + k 1 log n n . (61) Let M (1,0) Q,1 and M (1,0) Q,2 be R 1 (P, D 1 ) + ∆R 1 + k 1 log n n ≤ 1 n log M (1,0) Q,1 ≤R 1 (P, D 1 ) + ∆R 1 + (k 1 + 1) log n n (62) 1 n log M (1,0) Q,1 M (1,0) Q,2 ≤ 1 n logM (1,0) Q,2 + log n n .(63)
For simplicity, we neglect the fact that the number of messages are integers since it will increase the rate by at most log n/n bits/symbol. Let h be a one to one mapping from {1, . . . , M
(1,0) Q,1 } × {1, . . . , M (1,0) Q,2 } to {1, . . . ,M (1,0)
Q,2 }. Then, we can define encoding and decoding functions f
(1,0) Q,1 : X n → {1, . . . , M (1,0) Q,1 }, f (1,0) Q,2 : X n → {1, . . . , M (1,0) Q,2 }, and g (1,0) Q,2 : {1, . . . , M (1,0) Q,1 } × {1, . . . , M (1,0) Q,2 } →X n 2 where f (1,0) Q,1 (x n ), f (1,0) Q,1 (x n ) =h −1 f (1,0) Q (x n ) (64) g (1,0) Q,2 (m 1 , m 2 ) =g (1,0) Q (h(m 1 , m 2 )).(65)
Note that the first message is useless for decoder 1, but we do not care since it will declare an error anyway.
On the other hand, decoder 2 will decode both m 1 and m 2 successfully.
4) Q ∈ A (1,1) {Q ∈ P n (X ) : R 1 (Q, D 1 ) − R 1 (P, D 1 ) > ∆R 1 , R(Q, D 1 , D 2 ) − R(P, D 1 , D 2 ) > ∆R 2 }.
The encoder sends nothing and the both decoder will declare errors. We can assume M
(1,1) 1 = M (1,1) 2 = 1.
Finally, we merge all encoding functions together. Given source sequence x n , the encoder describes a type of sequence as a part of the first message using |X | log(n + 1) bits. This affects at most O(log n/n) bits/symbol in rates. Based on the type of sequence, it employs an encoding function accordingly, as described above. Since the decoder also knows the type of the sequence, it can employ the corresponding decoding function. Since all
M (0,0) Q,1 , M (0,1) 1 , M (1,0) Q,1 have the same upper bound, we can bound M 1 : 1 n log M 1 ≤R 1 (P, D 1 ) + ∆R 1 + (k 1 + 1) log n n + |X | log(n + 1) n (66) ≤R 1 (P, D 1 ) + V 1 (P, D 1 ) n Q −1 (ǫ 1 ) + O log n n .(67)
Similarly, we can show that
1 n log M 1 M 2 ≤ R(P, D 1 , D 2 ) + V (P, D 1 , D 2 ) n Q −1 (ǫ 2 ) + O log n n .(68)
This concludes the proof.
C. Proof of Theorem 7
Instead of type covering arguments that we used in the previous section, we use the result of sphere covering for Gaussian sources.
Theorem 11 ([20]): There is an absolute constant k s such that, if R > 1 and n ≥ 9, any n-dimensional spheres of radius R can be covered by less than k s n 5/2 R n spheres of radius 1.
For simplicity, we refer to the sphere of radius r by r-ball and denote by B(x n , r) = {x n : d(x n ,x n ) ≤ r 2 }, the set of points in the sphere centered at x n with radius r. The above theorem immediately implies the following corollary.
Corollary 12: For n ≥ 9 and R 1 > R 2 > 0, we can find a set C ⊂ R n of size M that satisfies:
• For all x n ∈ B(0, R 1 ), there is an elementx n ∈ C such that x n ∈ B(x n , R 2 ).
• The size of the set M is upper bounded by
1 n log M ≤ 1 2 log R 1 R 2 + 5 2 log n n + O 1 n .(69)
Let r 1 and r 2 be radius of the balls such that Pr X 2 1 + · · · + X 2 n > r 2 1 = ǫ 1 and Pr X 2 1 + · · · + X 2 n > r 2 2 = ǫ 2 . First, consider the case ǫ 1 < ǫ 2 . It is clear that Q −1 (ǫ 1 ) > Q −1 (ǫ 2 ) and r 1 > r 2 . We can further divide this case into the following three cases, 1) X n ∈ B(0, r 2 ), i.e., X 2 1 + · · · + X 2 n ≤ r 2 2 . In this case, we design a code such that both decoders can decode successfully.
Let C (0,0) 1 ⊂ R n be the set that satisfies: can be found similarly to the proof of Theorem 2. Since Q −1 (ǫ 1 ) > Q −1 (ǫ 2 ), it is clear that
• C (0,0) 1 = M (0,0) 1 • B(0, r 2 ) ⊂ x n ∈C (0,0) 1 B(x n , √ nD 1 ) • 1 n log M (0,0) 1 ≤ 1 2 log σ 2 D1 + 1 2n Q −1 (ǫ 2 ) + O log1 n log M (0,0) 1 ≤ 1 2 log σ 2 D 1 + 1 2n Q −1 (ǫ 1 ) + O log n n .(70)
Similarly, we can cover a √ nD 1 -ball with M ⊂ R n that satisfies: Thus, if x n ∈ B(0, r 2 ), then we can findx n 1 ∈ C (0,0) 1
• C (0,0) 2 = M (0,0) 2 • B(0, √ nD 1 ) ⊂ x n ∈C (0,0) 2 B(x n , √ nD 2 ) • 1 n log M (0,0) 2 ≤ 1 2 log D1 D2 + O log
such that x n ∈ B(x n 1 , √ nD 1 ) which implies
(1/n) x n −x n 1 2 2 ≤ D 1 . Furthermore, since x n −x n 1 ∈ B(0, √ nD 1 ), we can findx n ∈ C (0,0) 2
such that
x n −x n 1 ∈ B(x n , √ nD 2 ) which implies (1/n) x n −x n 1 −x n 2 2 ≤ D 2 .
Finally, we can takex n 2 =x n 1 +x n , and we get (1/n) x n −x n 2 2 2 ≤ D 2 .
2) X n ∈ B(0, r 1 ) but X n / ∈ B(0, r 2 ), i.e., r 2 2 < X 2 1 + · · · + X 2 n ≤ r 2 1 . We will only send a message to decoder 1, and decoder 2 will declare an error. We can cover r 1 -ball with
M (0,1) 1 number of √ nD 1 -balls where 1 n log M (0,1) 1 ≤ 1 2 log σ 2 D 1 + 1 2n Q −1 (ǫ 1 ) + O log n n .(71)
Therefore, there exists C (0,1) 1 that satisfies:
• C (0,1) 1 = M (0,1) 1 • B(0, r 2 ) ⊂ x n ∈C (0,1) 1 B(x n , √ nD 1 ) • 1 n log M (0,1) 1 ≤ 1 2 log σ 2 D1 + 1 2n Q −1 (ǫ 1 ) + O log n n .
We can think M (0,1) 2 to be one.
3) X n / ∈ B(0, r 1 ) and X n / ∈ B(0, r 2 ), i.e., r 2 1 < X 2 1 + · · · + X 2 n . The encoder does not send any messages and both decoder will declare an error. We can think both M Then, we can see that
1 n log M 1 ≤ 1 2 log σ 2 D 1 + 1 2n Q −1 (ǫ 1 ) + O log n n (72) 1 n log M 1 M 2 ≤ 1 2 log σ 2 D 2 + 1 2n Q −1 (ǫ 2 ) + O log n n .(73)
Similarly, we can consider the case ǫ 1 ≥ ǫ 2 . In this case, it is clear that Q −1 (ǫ 1 ) ≤ Q −1 (ǫ 2 ) and r 1 ≤ r 2 . We can further divide the case into the following three cases, 1) X n ∈ B(0, r 1 ), i.e., X 2 1 + · · · + X 2 n ≤ r 2 1 . In this case, both decoders can decode successfully. We can find M (1,0) 1 number of √ nD 1 -balls that covers r 1 -ball where
1 n log M (1,0) 1 ≤ 1 2 log σ 2 D 1 + 1 2n Q −1 (ǫ 1 ) + O log n n .(74)
Similar to previous cases, we can define C (1,0) 1 to be a set of the ball centers.
Also, we can cover
√ nD 1 -ball with M (1,0) 2 number of √ nD 2 -balls where 1 n log M (1,0) 2 ≤ 1 2 log D 1 D 2 + O log n n .(75)
Since
Q −1 (ǫ 1 ) ≤ Q −1 (ǫ 2 ), it is clear that 1 n log M (1,0) 1 M (1,0) 2 ≤ 1 2 log σ 2 D 2 + 1 2n Q −1 (ǫ 2 ) + O log n n .(76)
2) X n ∈ B(0, r 2 ) but X n / ∈ B(0, r 1 ), i.e., r 2 1 < X 2 1 + · · · + X 2 n ≤ r 2 2 . We will only send a message to decoder 2, and decoder 1 will declare an error. We can cover r 2 -ball with
M (1,1) number of √ nD 2 -balls where 1 n logM (1,1) ≤ 1 2 log σ 2 D 2 + 1 2n Q −1 (ǫ 2 ) + O log n n .(77)=M (1,1) (78) 1 n log M (1,1) 1 ≤ 1 2 log σ 2 D 1 + 1 2n Q −1 (ǫ 2 ) + O log n n (79) 1 n log M (1,1) 2 ≤ 1 2 log D 1 D 2 + O log n n .(80)
Recall that the decoder 1 does not care about the reconstruction of the source, where, on the other hand, decoder 2 will get both M and will be able to reconstruct the source based onM (1,1) .
3) X n / ∈ B(0, r 1 ) and X n / ∈ B(0, r 2 ), i.e., r 2 2 < X 2 1 + · · · + X 2 n . We will not send any messages and both decoder will declare an error. We can think both M to be one.
Similar to the case of ǫ 1 < ǫ 2 , we can combine the codebooks and get
1 n log M 1 ≤ 1 2 log σ 2 D 1 + 1 2n Q −1 (ǫ 1 ) + O log n n (81) 1 n log M 1 M 2 ≤ 1 2 log σ 2 D 2 + 1 2n Q −1 (ǫ 2 ) + O log n n .(82)
This concludes the proof.
Remark 2:
If we have ǫ 1 = ǫ 2 = ǫ, radius r 1 and r 2 are the same and the proof can be simplified. In this case, an error will occur at both decoders if and only if X 2 1 +· · ·+X 2 n > r 2 where r = r 1 = r 2 . Since both decoders share the same error events, encoding can be done successively in a simple manner and we do not have to consider the case of message splitting. More precisely, given codebook {(X n 1 (i),X n 2 (j)) : 1 ≤ i ≤ M 1 , 1 ≤ j ≤ M 2 }, the encoder finds i such that (1/n) X n −X n 1 (i) 2 2 ≤ D 1 and then finds j such that (1/n) X n −X n 1 (i) −X n 2 (j)
2 2 ≤ D 2 .
This is the key idea of Section VI where we use the successive refinement technique to construct a point-to-point source coding scheme with low complexity.
VI. LAYERED CODES
We considered the successive refinement problem with two decoders so far. In this section, we show that the idea of successive refinement is also useful for point to point lossy compression where we have one encoder and one decoder. The intuition is that successive refinement coding provides a tree structure for a coding scheme which allows low encoding complexity. More precisely, if the source is successively refinable, we can add L − 1 virtual mid-stage decoders and employ a successive refinement scheme for L decoders without any (asymptotic) performance loss. For fixed L, this is a simple extension of successive refinement, however, we also provide a result for L = L n growing with n. Since the number of decoders L corresponds to the level of tree and larger L leads to lower complexity of the scheme, we have a great advantage in terms of complexity by taking growing L = L n .
Note that the tree structured vector quantization (TVSQ) has been extensively studied, and also has a successive approximation property. For example, in [21], Effros et al. combined pruned TVSQ with a universal noiseless coder which enables progressive transmission of sources. While this approach guarantees optimality at zero distortion, it cannot achieve the rate-distortion function in general.
The precise problem description is the following. Let n be the block length of the coding scheme. The codebook consists of L sub-codebooks (C (n) 1 , C (n) 2 , · · · , C (n) L ) and each sub-codebook consists of M i codewords for 1 ≤ i ≤ L. We consider the following encoding scheme which we call layered coding:
• Find c 1 ∈ C (n) 1
that minimizes some function ψ 1 (x n , c 1 ).
• For i ≥ 2, given c 1 , · · · , c i−1 , find c i ∈ C (n) i that minimizes ψ i (x n , c 1 , c 2 , · · · , c i−1 ),
where ψ 1 , ..., ψ L are simple functions that depend on the specific implementation of the scheme. One can think of (c 1 , . . . , c i ) as messages for an i-th (virtual) decoder. The compressed representation of the source consists of a length L vector (m 1 , · · · , m L ) which indicates the index of codeword from each sub-codebook. Note that the total number of codewords is M 1 × · · · × M L and the rate of the scheme is R = L i=1 1 n log M i . Once the decoder receives the message, it reconstructsX n = φ(m 1 , · · · , m L ) with some function φ. Note that the definition of the layered code is exactly equal to that of the successive refinement code except the fact that the layered coding scheme only considers the distortion at the last decoder.
A. Layered Coding Schemes
We show the existence of layered coding schemes for a Gaussian source under quadratic distortion and for a binary source under Hamming distortion. For fixed L, it is easy to have a layered coding scheme, since sources are successively refinable in both cases and we can apply the successive refinement schemes. In this section, we generalize the result even further in two aspects. First, we consider how fast the coding rate can converge to the rate-distortion function, and provide an achievable rate including a dispersion term. Then, we allow L to be a function of block length n, and provide a layered coding scheme for L = L n growing with n. Our next theorem shows an existence of a rate-distortion achieving layered coding scheme for given n and L.
L i=1 1 n log M i ≤ R(D) + V (D) n Q −1 (ǫ) + Lk log n n + O log n n(83)
for some constantk where the O (log n/n) term does not depend on D or L.
The proof and discussion of Theorem 13 are given in Section VI-A1 and Section VI-A2. Note that Lk log n/n is also in the class of O(log n/n) for constant L, however, we will also consider the case where L = L n grows with n. We would like to point out that the last O(log n/n) remains the same even when L = L n increases as n grows.
1) Gaussian source under quadratic distortion: For Gaussian source under quadratic distortion, we can generalize
Theorem 7 to the case of multiple decoders. As we mentioned in Remark 2, we choose all ǫ i to be equal to ǫ.
1 n log M 1 ≤ 1 2 log σ 2 D 1 + 1 2n Q −1 (ǫ) + O log n n (84) 1 n log M i ≤ 1 2 log D i−1 D i + 3 log n n for 2 ≤ i ≤ L(85)
for any D 1 > D 2 > · · · > D L = D where the O (log n/n) term depends on ǫ but not on L or the D i values.
The choice of ψ i and φ will be specified in the proof. The fact that the O (log n/n) term is not dependent on the specific choice of D i 's and L is important in cases we consider later where L and D i vary with n.
Proof: Consider the successive refinement problem with target distortions D 1 > · · · > D L = D and target excess distortion probabilities ǫ 1 = · · · = ǫ L = ǫ. Given sub-codebooks C 1 , . . . , C L , the basic idea of the scheme is as as shown in Algorithm 1. Note that the input of the algorithm is a given sequence x n and the set of sub-codebooks C 1 , . . . , C L where the output is the collection of sub-codewords c m1 , . . . , c mL .
Algorithm 1 Encoding Scheme.
Set D 1 > D 2 > · · · > D L = D, and let x (0) = x n .
for i = 1 to L do Find a codeword c mi ∈ C i such that x (i−1) − c mi 2 2 ≤ nD i . If there is no such codeword, declare an error. Let x (i) = x (i−1) − c mi .
end for
We construct sub-codebooks based on Corollary 12. Let r be a radius such that Pr X 2 1 + · · · + X 2 n > r 2 = ǫ. Similar to the proof of Theorem 7, we can find M 1 number of √ nD 1 -balls that covers the r-ball where
1 n log M 1 ≤ 1 2 log σ 2 D 1 + 1 2n Q −1 (ǫ) + O log n n .(86)
Again, the term O (log n/n) only depends on ǫ where we provide the details in Appendix D. Then, for i ≥ 2, we can cover
nD i−1 -ball with M i number of √ nD i -balls where 1 n log M i ≤ 1 2 log D i−1 D i + 3 log n n .(87)
The i-th sub-codebook C i is a set of centers of √ nD i -balls, and therefore |C i | = M i .
Suppose the encoder found c m1 , · · · , c mi−1 successfully, which implies c m1 + · · · + c mi−1 − x n 2 2 ≤ nD i−1 . In other words, x n is in the ball with radius nD i−1 where the center of the ball is at c m1 + · · · + c mi−1 . Then, by construction, we can always find c mi ∈ C i such that ψ i (x n , c m1 , . . . , c mi ) = c m1 + · · · + c mi − x n 2 2 ≤ nD i . We can repeat the same procedure L times and find (m 1 , m 2 , . . . , m L ).
The error occurs if and only if the event X 2 1 + · · · + X 2 n > r 2 happens at the beginning, and therefore the excess distortion probability is ǫ. The reconstruction at the decoder will be φ(c m1 , . . . , c mL ) = c m1 + · · · + c mL .
The overall rate of Lemma 14 can be bounded by Proof: Similar to the proof of Lemma 14, we can consider the successive refinement problem with target distortions D 1 > · · · > D L = D and target excess distortion probabilities ǫ 1 = · · · = ǫ L = ǫ. The basic idea of coding is very similar to the Gaussian case. The difference is that we use Hamming instead of l 2 balls, and therefore we need Lemma 8 instead of Corollary 12. A Hamming ball with radius r is defined by
L i=1 1 n log M i (88) ≤ 1 2 log σ 2 D 1 + 1 2n Q −1 (ǫ) + O log n n + L i=2 1 2 log D i−1 D i + 3 log n n (89) = 1 2 log σ 2 D + 1 2n Q −1 (ǫ) + 3(L − 1) log n n + O log n n .(901 n log M 1 ≤h 2 (p) − h 2 (D 1 ) + V (p, D) n Q −1 (ǫ) + O log n n (91) 1 n log M i ≤h 2 (D i−1 ) − h 2 (D i ) + k 3 log n n , for 2 ≤ i ≤ L(B H (r) ∆ = {y n ∈ {0, 1} n : n i=1 y i ≤ r}.(93)
Given sub-codebooks C 1 , . . . , C L , the basic idea of the achievability scheme is the following:
Algorithm 2 Enoding Scheme.
Set D 1 > D 2 > · · · > D L = D, and let x (0) = x n .
for i = 1 to L do Find the codeword c mi ∈ C i such that d(x (i−1) , c mi ) ≤ D i .
If there is no such codeword, declare an error.
Let
x (i) = x (i−1) ⊕ c mi .
end for
Similar to Algorithm 1, the input of the algorithm is a given sequence x n and the set of sub-codebooks C 1 , . . . , C L where the output is the collection of sub-codewords c m1 , . . . , c mL .
In the first stage, similar to [16, Theorem 1], we can find a sub-codebook C 1 with size M 1 such that the excess distortion probability is smaller than ǫ and
1 n log M 1 ≤ h(p) − h(D 1 ) + V (p, D 1 ) n Q −1 (ǫ) + O log n n .(94)
Similar to the Gaussian case, the term O (log n/n) only depends on ǫ, where the detail is provided in Appendix E.
For i ≥ 2 and the given type Q, Lemma 8 implies that there is M Q,i Hamming balls with radius nD i that covers all sequences of type Q where
1 n log M Q,i ≤R(Q, D i ) + k 1 log n n (95) =h(Q(1)) − h(D i ) + k 1 log n n .(96)
Let C Q,i be a set of centers of Hamming balls with radius nD i , and therefore |C Q,i | = M Q,i . The i-th sub-codebook
C i is union of C Q,i 's for all type Q ∈ T (D i−1 , D i ) {Q ∈ P n (X ) : D i < Q(1) ≤ D i−1 } and zero codeword
(0, 0, 0, · · · , 0), i.e.,
C i = {(0, . . . , 0)} ∪ Q∈T (Di−1,Di) C Q,i .(97)
Then, we have
1 n log M i = 1 n log |C i | (98) ≤ 1 n log 1 + Q∈T (Di−1,Di) |C Q,i | (99) ≤ 1 n log 1 + (n + 1) max Q∈T (Di−1,Di) M Q,i (100) ≤ h(D i−1 ) − h(D i ) + (k 1 + 1) log n n (101) where (100) is because |T (D i−1 , D i )| ≤ nD i−1 − nD i + 1. We can set k 3 ∆ = k 1 + 1.
Suppose the encoder could find c m1 , · · · , c mi−1 successfully which implies d(c m1 ⊕ · · · ⊕ c mi−1 , x n ) ≤ nD i−1 .
In other words, x n is in the Hamming ball with radius nD i−1 where the center of ball is at c m1 ⊕ · · · ⊕ c mi−1 .
Then, by construction, we can always find c mi ∈ C i such that ψ i (c m1 , . . . , c mi ) = d(c m1 ⊕ · · · ⊕ c mi , x n ) ≤ nD i .
We can repeat the same procedure L times and find (m 1 , m 2 , . . . , m L ).
The error occurs if and only if the first sub-codebook fails to cover the source x n at the beginning, and therefore the excess distortion probability is ǫ. The reconstruction at the decoder will be φ(c m1 , . . . , c mL ) = c m1 ⊕ · · · ⊕ c mL .
Remark 3:
We would like to point out that Lemma 15 is limited to memoryless binary sources while Theorem 5 holds for any discrete memoryless sources. The main difference is the operation between source symbols. More precisely, in Lemma 15, the source is encoded and then the "error" sequence (modulo 2 difference) is encoded again. Note that Hamming distortion is closely related to this operation. However, It is hard to generalize this idea to non-binary sources because there are no corresponding differences when the distortion measure is arbitrary. The modulo |X | difference could work, but it is complex to analyse even when the distortion measure is still Hamming distortion.
The overall rate of Lemma 15 can be bounded by
L i=1 1 n log M i ≤h 2 (p) − h 2 (D 1 ) + V (p, D 1 ) n Q −1 (ǫ) + O log n n + L i=2 h 2 (D i−1 ) − h 2 (D i ) + k 3 log n n (102) =h 2 (p) − h 2 (D) + V (p, D 1 ) n Q −1 (ǫ) + k 3 (L − 1) log n n + O log n n .(103)
B. Discussion 1) Rate-Distortion Trade-Off: In both (90) and (103), it is obvious that the choice of L has an important role.
For simplicity, we only consider the case where M 1 = M 2 = · · · = M L = M , and we neglect the fact that the number of messages M is an integer. We can find M and D 1 > D 2 > · · · > D L which satisfy (84) and (85) (or (91) and (92)) with equality. For example, in the Gaussian case, we can find M and D 1 , · · · , D L sequentially:
1 n log M = 1 2 log σ 2 D 1 + 1 2n Q −1 (ǫ) + O log n n (104) 1 n log M = 1 2 log D i−1 D i + 3 log n n for 2 ≤ i ≤ L,(105)
Clearly, the number of possible reconstructions is M L = e nR and the rate of the scheme is R = (1/n)L log M .
On the other hand, the complexity is of order M × L since the encoder is searching a right codeword over M sub-codewords at each stage. Thus, for fixed rate R, we can say that the coding complexity (or size of codebooks) scales with L exp (nR/L) which is a decreasing function of L. This shows that larger L provides a lower complexity of the scheme. It is worth emphasizing that we can set L = L n to be increasing with n. This is because the bounds in both corollaries hold uniformly for all L.
On the other hand, in both corollaries, the overall rate can be bounded by
R(D) + V (D) n Q −1 (ǫ) +kL n log n n + O log n n(106)
for some constantk, where we denote by R(D) and V (D) the rate-distortion function and the source dispersion.
However, the optimum rate is given by
R(D) + V (D) n Q −1 (ǫ) + O log n n .(107)
We can see that there is a penalty termkL n (log n/n) because of using layered coding. If L n is growing too fast with n in order to achieve low-complexity of the scheme, then the rate penalty term L n (log n/n) can be too large and we may lose (second-order) rate optimality. This shows the trade-off between the rate and complexity of the scheme. Consider the following two examples, which are valid for both the Gaussian and binary cases.
• If L n = L is constant, then the scheme achieves the rate-distortion and the dispersion as well, but the complexity is exponential (albeit with a smaller exponent).
• If L n (log n/n) → 0 as n → ∞, we can achieve the rate-distortion function. For example, if L n = n/log 2 n+1, then the achieved rate is
R = R(D) + O 1 log n ,(108)
i.e., the scheme achieves the rate-distortion function as n increases, while the coding complexity is of order n log 2 n n R log n . Note that the excess distortion probability ǫ is fixed. We would like to point out that the rate is near polynomial in n.
• If L n (log n/ √ n) → 0 as n → ∞, we can achieve the source dispersion. For example, if L n = √ n/log 2 n + 1, then the achieved rate is
R =R(D) + V (D) n Q −1 (ǫ) + O 1 √ n log n .(109)Note that R − R(D) is inversely proportional to √ n with coefficient V (D)Q −1 (ǫ)
, in other words, layered coding can achieve the second order optimum rate. On the other hand, coding complexity is of order ( √ n/log 2 n)n √ nR log n which is better than the original exponential complexity.
2) Generalized Successive Refinability:
We would like to emphasize another interesting feature of layered coding.
Layered coding can be viewed as a successive refinement scheme with L decoders. Since our result allows L = L n to be increasing with n, this can be viewed as another generalized version of successive refinement. If the source is either binary or Gaussian and lim n→∞ L n (log n/n) = 0, the source is successively refinable with infinitely many decoders, where the rate increment is negligible. For comparison, in the classical successive refinement result, the number of decoders is not increasing and the rate increment between neighboring decoders is strictly positive. In [13], this property is termed infinitesimal successive refinability, and the results here establish that Gaussian and binary sources are infinitesimally successively refinable sources (under the relevant distortion criteria). Moreover, if we further assume lim n→∞ L n log n √ n = 0, each decoder can achieve the optimum distortion including dispersion term. In this case, we can say that the binary and Gaussian sources are strongly infinitesimally successively refinable sources.
In [13], the authors also pointed out that infinitesimal successive refinability yields another interesting property called ratelessness. Consider a binary or Gaussian source with lim n→∞ L log n n = 0, where the decoder received the first few fraction of messages, i.e., (m 1 , m 2 , . . . , m αL ) for some 0 < α < 1. Based on the proof of Lemma 14 and Lemma 14, the decoder will still be able to reconstruct the source sequence with distortion D(αR) which is the minimum achievable distortion at rate αR. If we have lim n→∞ L n log n √ n = 0, an even stronger ratelessness property can be established. In this case, the decoder can achieve the optimum distortion including dispersion terms.
VII. CONCLUSIONS
We have considered the problem of successive refinement with a focus on the optimal rate including the second order dispersion term. We have proposed the concept of "strong successive refinability" of the source and obtained a sufficient condition for it. In particular, any discrete memoryless source under Hamming distortion, or the Gaussian source under quadratic distortion are strongly successively refinable. We also show that the complexity of point-topoint source coding can be reduced using the idea of successive refinement. For binary and Gaussian sources, we characterize an achievable trade-off between rate and complexity of the scheme. We establish, for these cases, the existence of schemes which are infinitesimally successively refinable, rateless, achieve optimum dispersion, with sub-exponential complexity. Alternatively, essentially polynomial complexity is attainable if one is willing to back off from attaining the dispersion term.
APPENDIX
A. Derivative of Rate-Distortion Function
For fixed D > 0, the rate-distortion function is a mapping between C m to R where C m = {(x 1 , . . . , x m ) :
x i ≥ 0, ∀i, m i=1 x i = 1} ⊂ R m . Note that the tangent space of C m is (m − 1)-dimensional hyperplane that contains C m itself. We say R(·, D) is differentiable at P = (p 1 , . . . , p m ) ∈ C m if there is an extensionR(·, D) : R m →R
which is differentiable at P . The derivative of R(·, D) is defined by a derivative of its extension, i.e.,
R ′ (P, D) ∆ =R ′ (P, D) = ∂R(P, D) ∂p 1 , ∂R(P, D) ∂p 2 , . . . , ∂R(P, D) ∂p m T ∈ R m(110)
Since C m is smooth, the derivative R ′ (P, D) is well-defined in the following sense [22, 4p]. LetR 1 (·, D) : R m →R be another extension of R(·, D), then for any Q ∈ C m , we have
R ′ 1 (P, D), Q − P = R ′ (P, D), Q − P .(111)
This implies that the derivative along its tangent plane is the same regardless of the choice of extension. This is enough to use Taylor series since R(Q, D) = R(P, D) + R ′ (P, D), Q − P + high order terms.
Now, consider the well-definedness of V (P, D). For an extensionR(·, D) : R m →R, the source dispersion is defined by
V (P, D) = VAR R ′ 1 (P, D) = m i=1 ∂R 1 (P, D) ∂p i 2 p i − m i=1 ∂R 1 (P, D) ∂p i p i 2 .(113)
SupposeR 1 (·, D) is another extension of R(·, D), then (111) implies that
R ′ 1 (P, D) =R ′ (P, D) + α1 m(114)
for some α ∈ R where 1 m = (1, 1, . . . , 1) T ∈ R m . Then, we have
VAR R ′ 1 (P, D) = m i=1 ∂R 1 (P, D) ∂p i 2 p i − m i=1 ∂R 1 (P, D) ∂p i p i 2 (115) = m i=1 ∂R 1 (P, D) ∂p i − m j=1 ∂R 1 (P, D) ∂p j p j 2 p i (116) = m i=1 ∂R(P, D) ∂p i − m j=1 ∂R(P, D) ∂p j p j 2 p i (117) =VAR R ′ (P, D) .(118)
Therefore, VAR [R ′ (P, D)] does not depend on the particular choice of extension.
The same argument holds for R ′ (P, D 1 , D 2 ) and V (P, D 1 , D 2 ) as well. More precisely, for any Q ∈ C m and extensionsR(P, D 1 , D 2 ) andR 1 (P, D 1 , D 2 ), we have
R ′ (P, D 1 , D 2 ), Q − P = R ′ 1 (P, D 1 , D 2 ), Q − P .(119)
Also, VAR [R ′ (P, D 1 , D 2 )] does not depend on the particular choice of extension.
B. Proof of Refined Covering Lemma for Successive Refinement
The proof is similar to the proof of [5,Lemma 1], however, we have to consider vanishing terms more carefully in order to deal with source dispersions. Given type class T P , we want to construct sets B 1 ⊂X n 1 and B 2 (x n 1 ) ⊂X n 2 for allx n 1 ∈ B 1 such that
T P ⊂ x n 1 ∈B1 B 1 (x n 1 , D 1 ),(120)B(x n 1 , D 1 ) ⊂ x n 2 ∈B2(x n 1 ) B 2 (x n 2 , D 2 ) for allx n 1 ∈ B 1 ,(121)
where B i (x n i , D) = {x n ∈ X n : d i (x n ,x n i ) ≤ D} for i ∈ {1, 2}. We construct such sets using conditional types. Let
D ⋆ 1 =D 1 − 1 n |X | · X 1 · d M (122) D ⋆ 2 =D 2 − 1 n |X | · X 1 · X 2 + 1 d M .(123)
Then, there exist probability kernels W 1 : X →X 1 and W 2 : X ×X 1 →X 2 such that
I(X;X 1 ) =R 1 (P, D ⋆ 1 )(124)I(X;X 1 ,X 2 ) =R(P, D ⋆ 1 , D ⋆ 2 )(125)
where the joint law of (X,X 1 ,
X 2 ) is P × W 1 × W 2 and E d 1 (X,X 1 ) = x,x1 P (x)W 1 (x 1 |x)d 1 (x,x 1 ) ≤ D ⋆ 1 (126) E d 2 (X,X 2 ) = x,x1,x2 P (x)W 1 (x 1 |x)W 2 (x 2 |x,x 1 )d 2 (x,x 2 ) ≤ D ⋆ 2 .(127)
The structure of kernels are described in Figure 2.
|[W 1 ](x 1 |x) − W 1 (x 1 |x)| ≤ 1 nP (x) (128) X n W 1 : X →X 1 W 2 : X ×X 1 →X 2X n 1 X n 2 Fig. 2. Structure of Kernels |[W 2 ](x 2 |x,x 1 ) − W 2 (x 2 |x,x 1 )| ≤ 1 n[W 1 ](x 1 |x)P (x)
.
Let T [W1] (x n ) be the conditional type class of [W 1 ] given x n , and T [W2] (x n ,x n 1 ) be the conditional type class of [W 2 ] given (x n ,x n 1 ). Then, following lemma shows that x n ,x n 1 andx n 2 from those type classes satisfy distortion constraints.
Lemma 16:
For any x n ∈ T P ,x n
1 ∈ T [W1] (x n ) andx n 2 ∈ T [W2] (x n ,x n 1 ), we have d 1 (x n ,x n 1 ) ≤D 1 (130) d 2 (x n ,x n 2 ) ≤D 2 .(131)
The proof of Lemma 16 is given in Appendix F.
To construct the codebook, we further let [Q] be a marginalized type ofX 1 and [V 2 ] be a marginalized kernel fromX 1 toX 2 . More precisely,
[Q](x 1 ) = x∈X [W 1 ](x 1 |x)P (x) (132) [V 2 ](x 2 |x 1 ) = 1 [Q](x 1 ) x∈X [W 2 ](x 2 |x,x 1 )[W 1 ](x 1 |x)P (x).(133)
We further let
G 1 = T [Q] ,G 1 (x n ) = T [W1] (x n ), G 2 (x n 1 ) = T [V2] (x n 1 ) andG 2 (x n ,x n 1 ) = T [W2] (x n ,x n 1 )
for all x n ∈ T P ,x n 1 ∈ G 1 . It is clear thatG 1 (x n ) ⊂ G 1 andG 2 (x n ,x n 1 ) ⊂ G 2 (x n 1 ). We generate codebook randomly based on these sets.
Let Z M = (Z 1 , · · · , Z M ) be a randomly generated codebook where Z 1 , . . . , Z M ∈X n 1 are i.i.d. random variables that has uniform distribution over G 1 . Also, for given Z i = z i , let Ξ N i = (Ξ i,1 , · · · , Ξ i,N ) ⊂X n 2 be i.i.d. random variables uniformly distributed over G 2 (z i ). The size of codebook M and N will be specified later. We denote U 1 (Z M ) the set of source words that are not covered by the codebook Z M , i.e.,
U 1 (Z M ) ={x n ∈ T P : d 1 (x n , Z i ) > D 1 , for all 1 ≤ i ≤ M }.(134)
Also, for each 1 ≤ i ≤ M , let U 2 (Ξ N i ) be the set of source words that are covered by Z i but not covered by the codebook Ξ N i , i.e.,
U 2 (Ξ N i ) ={x n ∈ T P : d 1 (x n , Z i ) ≤ D 1 , d 2 (x n , Ξ i,j )) > D 2 , for all 1 ≤ j ≤ N }.(135)
If we can show that E U(Z m ) ∪ ∪ M i=1 U 2 (Ξ N i ) < 1, then we can say that there exist sets B 1 and B 2 (x n 1 ) that satisfy (120) and (121). This is because the random variable only gets integer values, and the fact that its expectation is less than one implies that there exists an event of the variable being equal to zero with non-zero probability, as required. We will show that the expectation can be made to be less than one, by taking M and N to be large enough, but not too large so that (44) and (45) are satisfied. Note that this argument is similar to that of [19,Chapter 9].
We begin with union bound.
E U 1 (Z M ) m i=1 U 2 (Ξ N i ) = x n ∈TP Pr x n ∈ U 1 (Z M ) M i=1 U 2 (Ξ N i ) (136) ≤ x n ∈TP Pr x n ∈ U 1 (Z M ) + x n ∈TP M i=1 Pr x n ∈ U 2 (Ξ N i ) .(137)
We can bound the first term using type counting lemma.
x n ∈TP Pr [x n ∈ U 1 (Z m )] = x n ∈TP (1 − Pr [d 1 (x n , Z 1 ) ≤ D 1 ]) M (138) ≤ x n ∈TP 1 − G 1 (x n ) |G 1 | M (139) ≤ x n ∈TP exp − G 1 (x n ) |G 1 | M (140)
where the joint law of (X, [
X 1 ], [X 2 ]) is P × [W 1 ] × [W 2 ].
Note that (141) is because of (39) and (42), while (143) is due to (37).
We can bound the second term using a similar technique.
Pr x n ∈ U 2 (Ξ N i ) = Pr [d 1 (x n , Z i ) ≤ D 1 , d 2 (x n , Ξ i,j ) > D 2 , ∀j](144)= 1 |G 1 | x n 1 ∈G1 Pr [d 1 (x n ,x n 1 ) ≤ D 1 , d 2 (x n , Ξ i,j ) > D 2 , ∀j | Z i =x n 1 ](145)= 1 |G 1 | x n 1 ∈G1 d1(x n ,x n 1 )≤D1 Pr [d 2 (x n , Ξ i,1 ) > D 2 | Z i =x n 1 ] N (146) = 1 |G 1 | x n 1 ∈G1 d1(x n ,x n 1 )≤D1 exp −N G 2 (x n ,x n 1 ) |G 2 (x n 1 )| (147) = 1 |G 1 | x n 1 ∈G1 d1(x n ,x n 1 )≤D1 exp −N (n + 1) −|X |·|X1|·|X2| exp(−n(H([X 2 ]|[X 1 ]) − H([X 2 ]|X, [X 1 ]))) .(148)
Finally, we get
x n ∈TP M i=1 P x n ∈ U 2 (Ξ N i ) ≤ M |T P | exp −N (n + 1) −|X |·|X1|·|X2| exp(−n(H([X 2 ]|[X 1 ]) − H([X 2 ]|X, [X 1 ]))) (149) ≤ M |T P | exp −N (n + 1) −|X |·|X1|·|X2| exp(−nI(X; [X 2 ]|[X 1 ])) .(150)
We choose M and N that satisfy
(n + 1) |X |·|X1|+2 exp(nI(X; [X 1 ])) ≤ M ≤ (n + 1) |X |·|X1|+4 exp(nI(X; [X 1 ]))(151)(n + 1) |X |·|X1|·|X2|+2 exp(nI(X; [X 2 ]|[X 1 ])) ≤ N ≤ (n + 1) |X |·|X1|·|X2|+4 exp(nI(X; [X 2 ]|[X 1 ])).(152)
If we apply such M and N to (137), (143) and (150), it automatically gives E |U(Z m ) ∪ ∪ M i=1 U 2 (Ξ N i ) | < 1 for n > |X | · X 1 + 4 + H(P ) + I(X; [X 1 ]). Therefore, there exists sets B 1 and B 2 (x n 1 ) that satisfies (120) and (121) where
1 n log |B 1 | ≤I(X; [X 1 ]) + 2 · |X | · X 1 + 8 n log n (153) 1 n log (|B 1 | · |B 2 (x n 1 )|) ≤I(X; [X 1 ], [X 2 ]) + 2 · |X | · X 1 · X 2 + 2 · |X | · X 1 + 16 n log n(154)
for allx n 1 ∈ B 1 . Note that we bound log(n + 1) by 2 log n. Then, the following lemma bounds the gap between I(X;X 1 ) and I(X; [X 1 ]) (also for I(X;X 1 ,X 2 ) and I(X; [X 1 ], [X 2 ])) where the proof is given in Appendix G.
Lemma 17:
I(X;X 1 ) − I(X; [X 1 ]) ≤ 2 |X | · X 1 n log n(155)I(X;X 1 ,X 2 ) − I(X; [X 1 ], [X 2 ]) ≤ 4 |X | · X 1 · X 2 n log n.(156)
With (153) and (154), we can bound the size of B 1 and B 2 (x n 1 )'s by
1 n log |B 1 | ≤I(X;X 1 ) + 4 · |X | · X 1 + 8 n log n(157)
1 n log (|B 1 | · |B 2 (x n 1 )|) ≤I(X;X 1 ,X 2 ) + 6 · |X | · X 1 · X 2 + 2 · |X | · X 1 + 16
n log n(158)
Recall that we setX 1 that satisfies I(X;X 1 ) = R(P, D ⋆ 1 ). Thus, the final step of the proof should be bounding the difference between R(P, D 1 ) and R(P, D * 1 ), and also between R(P, D 1 , D 2 ) and R(P, D ⋆ 1 , D ⋆ 2 ). Lemma 18: For large enough n, we have
R 1 (P, D ⋆ 1 ) ≤R 1 (P, D 1 ) + log n n (159) R(P, D * 1 , D * 2 ) ≤R(P, D 1 , D 2 ) + log n n .(160)
The proof is given in Appendix H Finally, we have 1 n log |B 1 | ≤R 1 (P, D 1 ) + (4 · |X | · X 1 + 9) log n n (161) log (|B 1 | · |B 2 (x n 1 )|) ≤R(P, D 1 , D 2 ) + (6 · |X | · X 1 · X 2 + 2 · |X | · X 1 + 17) log n n .
We can see that the coefficients of the log n/n terms are
k 1 =4 · |X | · X 1 + 9 (163) k 2 =6 · |X | · X 1 · X 2 + 2 · |X | · X 1 + 17(164)
which are independent of the distribution P and block length n. This concludes the proof of the lemma.
C. Proof of Corollary 9
By Lemma 8, there exist B 1 ,
{B 2 (x n 1 )}xn 1 ∈B1 that successively (D 1 , D 2 )-cover T Q where 1 n log |B 1 | ≤R 1 (Q, D 1 ) + k 1 log n n (165) 1 n log (|B 1 | · |B 2 (x n 1 )|) ≤R(Q, D 1 , D 2 ) + k 2 log n n for allx n 1 ∈ B 1 .(166)
For simplicity, we neglect the fact that the number of messages and the size of sets are integers. Let M Q,1 = e nR and let M Q,2 that satisfies M Q,1 M Q,2 = |B 1 | · maxxn 1 ∈B1 |B 2 (x n 1 )|. Then, (50) and (51) hold by definition. Then, we can find an one to one function h :
such thatx n 1 can be uniquely recovered based only on m 1 where (m 1 , m 2 ) = h 1 (x n 1 ,x n 2 ), i.e., there exists a functioñ h such thatx n 1 =h(m 1 ). This is because |B 1 | ≤ M Q,1 . For all x n ∈ T Q , there existsx n 1 ∈ B 1 andx n 2 ∈ B 2 (x n 1 ) such that d 1 (x n ,x n 1 ) ≤ D 1 and d 2 (x n ,x n 2 ) ≤ D 2 . Let f Q,1 (x n ) and f Q,2 (x n ) be the first argument and the second argument of h(x n 1 ,x n 2 ), respectively. Further let g Q,1 (m 1 ) =h(m 1 ) and g Q,2 (m 1 , m 2 ) be an inverse function of h(·, ·). By construction of B 1 and {B 2 (x n 1 )}xn 1 ∈B1 , encoder and decoder satisfies (48) and (49).
Note that M Q,1 has to be an integer, and may not be exactly equal to e nR . However, we can set (1/n) log M Q,1 to be close toR, i.e.,R
≤ 1 n log M Q,1 ≤R + log n n .(168)sup x |F n (x) − Φ(x)| ≤ Cρ σ 3 √ n .(169)
In [24], Shevtsova showed the optimum C is smaller than 1 2 . Let X n be i.i.d. Gaussian random variables with zero mean and variance σ 2 . Then, for r 2 > nσ 2 , we have
Pr n i=1 X 2 i > r 2 =Pr n i=1 (X 2 i −σ 2 ) √ 2nσ 2 > r 2 −nσ 2 √ 2nσ 2 (170) ≤Q r 2 − nσ 2 √ 2nσ 2 + 1 2 15σ 6 2 √ 2nσ 6(171)
where we want this probability to be smaller than ǫ. Thus, we can set r such that
r 2 = nσ 2 + √ 2nσ 2 Q −1 ǫ − 15 4 √ 2n .(172)
where O (1/n) term does not depend on L or D 1 .
E. Bound O (log n/n) term for binary case
Let X n be i.i.d. Bernoulli(p) where p < 1/2. Then, for 1/2 > q > p, we have
Pr [ n i=1 X i > q] =Pr n i=1 (Xi−p) √ np(1−p) > (q − p) n p(1−p) (178) ≤Q (q − p) n p(1 − p) + 1 2 p p 3/2 (1 − p) 3/2 √ n(179)
where we want this probability to be smaller than ǫ. Thus, we set q such that q = p + p(1 − p) n Q −1 ǫ − 1 2 np(1 − p) 3 .
By Lemma 8, we can cover T Q with M 1 number of √ nD 1 -balls where 1 n log M 1 ≤h(q) − h(D 1 ) + k 1 log n n (181) ≤h(p) + (q − p)h ′ (p) − h(D 1 ) + k 1 log n n (182)
≤h(p) + p(1 − p) n Q −1 ǫ − 1 2 np(1 − p) 3 log 1 − p p − h(D 1 ) + k 1 log n n (183) =h(p) − h(D 1 ) + V (p, D 1 ) n Q −1 ǫ − 1 2 np(1 − p) 3 + k 1 log n n .(184)
Using Taylor's expansion, one can bound Q −1 ǫ − 1/(2 np(1 − p) 3 ) by Q −1 (ǫ) + O (1/ √ n). Finally, we have
1 n log M 1 ≤h(p) − h(D 1 ) + 1 √ n Q −1 (ǫ) + k 1 log n n + O 1 n ,(185)
where O (1/n) term does not depend on L or D 1 .
F. Proof of Lemma 16
For any x n ∈ T P ,x n 1 ∈ T [W1] (x n ) andx n 2 ∈ T [W2] (x n ,x n 1 ), we have
d 1 (x n ,x n 1 ) = x,x1 P (x)[W 1 ](x 1 |x)d 1 (x,x 1 ) (186) ≤ x,x1 P (x)W 1 (x 1 |x)d 1 (x,x 1 ) + 1 n |X | · X 1 d M(187)≤D ⋆ 1 + 1 n |X | · X 1 d M (188) =D 1 .(189)
Similarly, we have d 2 (x n ,x n 2 ) = P (x)W 1 (x 1 |x)W 2 (x 2 |x,x 1 )d 2 (x,x 1 ) +
x,x1,x2 1 n W 2 (x 2 |x,x 1 )d 2 (x,x 1 )
+ 1 n |X | · X 1 · X 2 d M(192)≤D ⋆ 2 + 1 n |X | · X 1 d M + 1 n |X | · X 1 · X 2 d M (193) =D 2 .(194)
G. Proof of Lemma 17
Let Q be Q(x 1 ) = x∈X P (x)W 1 (x 1 |x).
|H(X 1 ) − H([X 1 ])| ≤ − |X | · X 1 n log |X | n (199) ≤ |X | · X 1 n log n.(200)
Using τ (x) = −x log x, we can also bound the difference between conditional entropies:
|H(X 1 |X) − H([X 1 ]|X)| ≤ x∈X P (x) x1∈X1 τ (W 1 (x 1 |x)) − τ ([W 1 ](x 1 |x)) (201) ≤ x∈X P (x) x1∈X1 τ (|W 1 (x 1 |x) − [W 1 ](x 1 |x)|) (202) ≤ x∈X P (x) x1∈X1 τ 1 nP (x)(203)
Similarly, we can bound the difference between I(X;X 1 ,X 2 ) and I(X
Q (x 1 ,x 2 ) − [Q](x 1 ,x 2 ) ≤ x P (x) |W 1 (x 1 |x)W 2 (x 2 |x,x 1 ) − [W 1 ](x 1 |x)[W 2 ](x 2 |x,x 1 )| (211) ≤ x P (x) |W 1 (x 1 |x)W 2 (x 2 |x,x 1 ) − [W 1 ](x 1 |x)W 2 (x 2 |x,x 1 )| + x P (x) |[W 1 ](x 1 |x)W 2 (x 2 |x,x 1 ) − [W 1 ](x 1 |x)[W 2 ](x 2 |x,x 1 )| (212) ≤ x 1 n W 2 (x 2 |x,x 1 ) + x 1 n (213) ≤ 2 |X | n |(214)≤ 2 |X | · X 1 · X 2 n log n.(216)
Note that
≤ 1 nP (x) W 2 (x 2 |x,x 1 ) + 1 nP (x) (218) ≤ 2 nP (x)(219)
Since we assumed that nP (x) > 3, we have
H(X 1 ,X 2 |X) − H([X 1 ], [X 2 ]|X) ≤ x P (x) x1,x2 τ (W 1 (x 1 |x)W 2 (x 2 |x,x 1 )) − τ ([W 1 ](x 1 |x)[W 2 ](x 2 |x,x 1 )) (220) ≤ x P (x) x1,x2 τ (|W 1 (x 1 |x)W 2 (x 2 |x,x 1 ) − [W 1 ](x 1 |x)[W 2 ](x 2 |x,x 1 )|) (221) ≤ x P (x) x1,x2 τ 2 nP (x) (222) ≤ x 2 X 1 · X 2 n log nP (x) 2(223)
(
Allerton), and at the 2014 International Symposium on Information Theory. This work was supported by the NSF Center for Science of Information under Grant Agreement CCF-0939370.
:
{1, · · · , M 1 } × {1, · · · , M 2 } →X n 2 , whereX 1 andX 2 are the reconstruction alphabets for each decoder. Decoder i has its own 6 distortion measure d i : X ×X i → [0, ∞) with a target distortion D i . Both d 1 and d 2 are symbol by symbol distortion measures which induce block distortion measures by
Definition 3 :
3We say that (n, M 1 , M 2 , D 1 , D 2 , ǫ 1 , ǫ 2 ) is achievable if there exists an encoder-decoder pair that satisfies Pr d 1 (X n , g
Theorem 1, we can find encoding and decoding functions f (0,1) : X n → {1, . . .
nD 1 -balls that covers the r 2 -ball. Upper bound on M
same for C 2 where |C 1 | = M 1 and |C 2 | = M 2 .
Definition 6 :
6An (n, L, {M 1 , · · · , M L }, D, ǫ)-layered code is a coding scheme with L sub-codebooks where the size of the i-th sub-codebook is M i , and the probability of excess distortion Pr d(X n ,X n ) > D is at most ǫ.
Theorem 13 :
13For i.i.d. Gaussian sources under quadratic distortion and i.i.d. binary sources under Hamming distortion, there exists a (n, L, {M 1 , . . . , M L }, D, ǫ)-layered code such that
Lemma 14 :
14Let a source be i.i.d. Gaussian N (0, σ 2 ) under quadratic distortion. For all L, there exists a (n, L, {M 1 , · · · , M L }, D, ǫ)-layered code such that
) 2 )
2Binary source under Hamming distortion: The next lemma provides a similar result for a binary source under Hamming distortion. Lemma 15: Let the source be i.i.d. Bern(p) and the distortion be measured by Hamming distortion function, where the target distortion is D. For large enough n, there is a (n, L, {M 1 , · · · , M L }, D, ǫ)-layered code for all L and D 1 > D 2 > · · · > D L = D such that
92)where O (log n/n) only depends on ǫ, we denote dispersion of Bern(p) source with V (p, D) = p(1 − p) log 2 ((1 − p)/p), and a binary entropy function with h 2 (p) = −p log p − (1 − p) log(1 − p) and k 3 is a constant that does not depend on any of the variables.
exp −(n + 1) −|X |·|X1| exp(n(H([X 1 ]|X) − H([X 1 ])))M (141) = |T P | exp −(n + 1) −|X |·|X1| exp(n(H([X 1 ]|X) − H([X 1 ])))M (142) ≤ exp(nH(P )) exp −(n + 1) −|X |·|X1| exp(−nI(X; [X 1 ]))M
(
{x n 1 } × B 2 (x n 1 )) → {1, . . . , M Q,1 } × {1, . . . , M Q,2 }
because nP (x) > 3 for all x. Equation (204) is because |τ (x) − τ (y)| ≤ τ (|x − y|) if |x − y| ≤ 1/2.Finally, we getI(X;X 1 ) − I(X; [X 1 ]) ≤ H(X 1 ) − H([X 1 ]) + H(X 1 |X) − H([X 1 ]|X)
|W 1
1(x 1 |x)W 2 (x 2 |x,x 1 ) − [W 1 ](x 1 |x)[W 2 ](x 2 |x,x 1 )| ≤ |W 1 (x 1 |x)W 2 (x 2 |x,x 1 ) − [W 1 ](x 1 |x)W 2 (x 2 |x,x 1 )| + |[W 1 ](x 1 |x)W 2 (x 2 |x,x 1 ) − [W 1 ](x 1 |x)[W 2 ](x 2 |x,x 1 )|(217)
28 D .
DBound O log n n term for Gaussian case Theorem 19 (Berry-Esseen Theorem[23]): Let Z n be i.i.d. random variables with E [Z i ] = 0, E Z 2 i = σ 2 and E |Z i | 3 = ρ < ∞.Let F n be the cumulative distribution function of ( n i=1 Z i )/(σ √ n) and Φ be the cumulative distribution function of the standard normal distribution. Then, for all n,
By Corollary 12, we can cover r-ball with M 1 number of √ nD 1 -balls where Using Taylor's expansion, one can bound Q −1 ǫ − 15/(4 √ 2n) by Q −1 (ǫ) + O (1/ √ n). Finally, we have1
n
log M 1 ≤
1
2
log
r 2
nD 1
+
5
2
log n
n
+
1
n
log k s
(173)
=
1
2
log
nσ 2 +
√
2nσ 2 Q −1 ǫ − 15
4
√
2n
nD 1
+
5
2
log n
n
+
1
n
log k s
(174)
=
1
2
log
σ 2
D 1
+
1
2
log 1 +
2
n
Q −1 ǫ −
15
4
√
2n
+
5
2
log n
n
+
1
n
log k s
(175)
≤
1
2
log
σ 2
D 1
+
1
√
2n
Q −1 ǫ −
15
4
√
2n
+
5
2
log n
n
+
1
n
log k s .
(176)
1
n
log M 1 ≤
1
2
log
σ 2
D 1
+
1
√
2n
Q −1 (ǫ) +
5
2
log n
n
+ O
1
n
,
x,x1,x2P (x)[W 1 ](x 1 |x)[W 2 ](x 2 |x,x 1 )d 2 (x,x 2 ) (190) P (x)[W 1 ](x 1 |x)W 2 (x 2 |x,x 1 )d 2 (x,x 1 ) + 1 n |X | · X 1 · X 2 d M(191)≤
x,x1,x2
≤
x,x1x2
Therefore, we have|Q(x 1 ) − [Q](x 1 )| = x∈X P (x)(W 1 (x 1 |x) − [W 1 ](x 1 |x))(196)which implies Q − [Q] 1 ≤ |X | · X 1 /n. By [19, Lemma 2.7], we can bound the difference between entropies:≤
x∈X
P (x) |W 1 (x 1 |x) − [W 1 ](x 1 |x)|
(197)
≤
x∈X
1
n
=
|X |
n
(198)
; [X 1 ], [X 2 ]). Recall that (X,X 1 ,X 2 ) has a joint law P × W 1 × W 2 and (X, [X 1 ], [X 2 ]) has a joint law P × [W 1 ] × [W 2 ]. [W 1 ](x 1 |x)[W 2 ](x 2 |x,x 1 )P (x).Then,Q and [Q] should be similar:LetQ and [Q] beQ
(x 1 ,x 2 ) =
x
W 1 (x 1 |x)W 2 (x 2 |x,x 1 )P (x)
(209)
[Q](x 1 ,x 2 ) =
x
(210)
which implies Q − [Q] ≤ 2 |X | · X 1 · X 2 /n.By [19, Lemma 2.7], we can bound the difference between entropiesH(X 1 ,X 2 ) − H([X 1 ], [X 2 ]) ≤ − 2 |X | · X 1 · X 21
n
log
2 |X |
n
(215)
Using(216)and(224), we can bound the gap between mutual informations:H. Proof of Lemma 18We know that D ⋆ 1 = D 1 − |X | · |X 1 |d M /n. Using the convexity and monotonicity properties of the rate-distortion function, we find an upper bound on the difference between R(P, D ⋆ 1 ) and R 1 (P, D 1 ):Therefore, we can bound R 1 (P, D ⋆ 1 ) using R 1 (P, D 1 ):for large enough n. Similarly, by the mean value theorem, there exists a c such that for large enough n,where D ′ 1 = cD 1 + (1 − c)D * 1 , D ′ 2 = cD 2 + (1 − c)D * 2 .
Hierarchical coding of discrete sources. V Koshelev, Problemy peredachi informatsii. 163V. Koshelev, "Hierarchical coding of discrete sources," Problemy peredachi informatsii, vol. 16, no. 3, pp. 31-49, 1980.
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Fixed-length lossy compression in the finite blocklength regime. V Kostina, S Verdú, IEEE Transactions on. 586Information TheoryV. Kostina and S. Verdú, "Fixed-length lossy compression in the finite blocklength regime," Information Theory, IEEE Transactions on, vol. 58, no. 6, pp. 3309-3338, 2012.
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Polar codes are optimal for lossy source coding. S B Korada, R L Urbanke, Information Theory, IEEE Transactions on. 56S. B. Korada and R. L. Urbanke, "Polar codes are optimal for lossy source coding," Information Theory, IEEE Transactions on, vol. 56, no. 4, pp. 1751-1768, 2010.
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Rateless lossy compression via the extremes. A No, T Weissman, arXiv:1406.6730arXiv preprintA. No and T. Weissman, "Rateless lossy compression via the extremes," arXiv preprint arXiv:1406.6730, 2014.
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| []
|
[
"Production at LHC of composite particles from strongly interacting elementary fermions via four-fermion operators of Einstein-Cartan type",
"Production at LHC of composite particles from strongly interacting elementary fermions via four-fermion operators of Einstein-Cartan type"
]
| [
"R Leonardi \nINFN\nSezione di Perugia\nVia A. PascoliI-06123PerugiaItaly\n",
"F Romeo \nDepartment of Physics and Astronomy\nVanderbilt University\n37235NashvilleTNUSA\n",
"† H Sun \nInstitute of Theoretical Physics\nSchool of Physics\nDalian University of Technology\nNo.2 Linggong Road116024Dalian, LiaoningP.R.China\n",
"A Gurrola \nDepartment of Physics and Astronomy\nVanderbilt University\n37235NashvilleTNUSA\n",
"§ O Panella \nINFN\nSezione di Perugia\nVia A. PascoliI-06123PerugiaItaly\n",
"S.-S Xue \nICRANet\nPiazzale della Repubblica10-65122PescaraItaly\n\nPhysics Department\nSapienza University of Rome\nPiazzale Aldo Moro 500185RomaItaly\n"
]
| [
"INFN\nSezione di Perugia\nVia A. PascoliI-06123PerugiaItaly",
"Department of Physics and Astronomy\nVanderbilt University\n37235NashvilleTNUSA",
"Institute of Theoretical Physics\nSchool of Physics\nDalian University of Technology\nNo.2 Linggong Road116024Dalian, LiaoningP.R.China",
"Department of Physics and Astronomy\nVanderbilt University\n37235NashvilleTNUSA",
"INFN\nSezione di Perugia\nVia A. PascoliI-06123PerugiaItaly",
"ICRANet\nPiazzale della Repubblica10-65122PescaraItaly",
"Physics Department\nSapienza University of Rome\nPiazzale Aldo Moro 500185RomaItaly"
]
| []
| A new physics scenario shows that four-fermion operators have a strong-coupling UV fixed point, where composite fermions F (bosons Π) form as bound states of three (two) SM elementary fermions and they couple to their constituents via effective contact interactions at the composite scale Λ ≈ O(TeV). We present a phenomenological study to investigate such composite particles at the LHC. Using these contact interactions, we compute the production cross sections and decay widths of composite fermions in the context of the relevant experiments at LHC with pp collisions at √ s = 13 TeV and √ s = 14 TeV. In particular, we focus on the resonant channel pp → e + F → e + e − qq , whose cross section has been recently limited by the CMS Collaboration. By a simple recasting of this result, we obtain a constraint on the model parameters such that composite fermions of mass mF below 4.25 TeV are excluded for Λ = mF . We further compute 5σ contour plots of the statistical significance and highlight the region of parameter space where F can manifest using 3 ab −1 , expected by the High-Luminosity LHC. It turns out that there is a large portion of the parameter space where F can be discovered and that deserve a dedicated investigation. In addition, we also study the composite boson state Π0 with the estimation of branching ratios into two quarks (two jets) B(Π0 → qq ) and into two boosted gauge bosons B(Π0 →GG ), from which we obtain the branching ratios of composite-fermion decay into an electron and two boosted gauge bosons B(F → eGG ). Moreover we briefly discuss the possible final states of four jets or one jet with two gauge bosons in LHC pp collision. | 10.1140/epjc/s10052-020-7822-0 | [
"https://arxiv.org/pdf/1810.11420v2.pdf"
]
| 53,371,332 | 1810.11420 | a1bf4a129e7dc04d8adc529de35b0a5b619b0144 |
Production at LHC of composite particles from strongly interacting elementary fermions via four-fermion operators of Einstein-Cartan type
(Dated: October 29, 2018)
R Leonardi
INFN
Sezione di Perugia
Via A. PascoliI-06123PerugiaItaly
F Romeo
Department of Physics and Astronomy
Vanderbilt University
37235NashvilleTNUSA
† H Sun
Institute of Theoretical Physics
School of Physics
Dalian University of Technology
No.2 Linggong Road116024Dalian, LiaoningP.R.China
A Gurrola
Department of Physics and Astronomy
Vanderbilt University
37235NashvilleTNUSA
§ O Panella
INFN
Sezione di Perugia
Via A. PascoliI-06123PerugiaItaly
S.-S Xue
ICRANet
Piazzale della Repubblica10-65122PescaraItaly
Physics Department
Sapienza University of Rome
Piazzale Aldo Moro 500185RomaItaly
Production at LHC of composite particles from strongly interacting elementary fermions via four-fermion operators of Einstein-Cartan type
(Dated: October 29, 2018)
A new physics scenario shows that four-fermion operators have a strong-coupling UV fixed point, where composite fermions F (bosons Π) form as bound states of three (two) SM elementary fermions and they couple to their constituents via effective contact interactions at the composite scale Λ ≈ O(TeV). We present a phenomenological study to investigate such composite particles at the LHC. Using these contact interactions, we compute the production cross sections and decay widths of composite fermions in the context of the relevant experiments at LHC with pp collisions at √ s = 13 TeV and √ s = 14 TeV. In particular, we focus on the resonant channel pp → e + F → e + e − qq , whose cross section has been recently limited by the CMS Collaboration. By a simple recasting of this result, we obtain a constraint on the model parameters such that composite fermions of mass mF below 4.25 TeV are excluded for Λ = mF . We further compute 5σ contour plots of the statistical significance and highlight the region of parameter space where F can manifest using 3 ab −1 , expected by the High-Luminosity LHC. It turns out that there is a large portion of the parameter space where F can be discovered and that deserve a dedicated investigation. In addition, we also study the composite boson state Π0 with the estimation of branching ratios into two quarks (two jets) B(Π0 → qq ) and into two boosted gauge bosons B(Π0 →GG ), from which we obtain the branching ratios of composite-fermion decay into an electron and two boosted gauge bosons B(F → eGG ). Moreover we briefly discuss the possible final states of four jets or one jet with two gauge bosons in LHC pp collision.
I. INTRODUCTION
The parity-violating gauge symmetries and spontaneous/explicit breaking of these symmetries for the hierarchy pattern of fermion masses have been at the center of a conceptual elaboration that has played a major role in donating to mankind the beauty of the Standard Model (SM) and possible scenarios beyond SM for fundamental particle physics. A simple description is provided on the one hand by the composite Higgs-boson model or the Nambu-Jona-Lasinio (NJL) model [1] with effective four-fermion operators, and on the other by the phenomenological model of the elementary Higgs boson [2]. These two models are effectively equivalent for the SM at low energies. After a great experimental effort for many years, using pp collision data at √ s = 7, 8 TeV at the Large Hadron Collider (LHC), the ATLAS [3] and CMS [4] collaborations have shown the first observations of a 125 GeV scalar particle in the search for the SM Higgs * [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] * * [email protected] ; corresponding author boson [5,6]. This far-reaching result begins to shed light on this most elusive and fascinating arena.
Recently, in the Run 2 of the upgraded LHC, studies on √ s = 13 TeV pp collision data are performed by ATLAS and CMS to search for new (beyond the SM) resonant and/or non-resonant phenomena [7][8][9][10][11]. These studies are continuously pushing up exclusion bounds on the parameter spaces of many possible scenarios beyond SM [12][13][14][15].
Composite-fermion scenarios have offered a possible solution to the hierarchy pattern of fermion masses [16,17]. In this context [18][19][20][21][22], SM quarks "q" and leptons " " are assumed to be bound states of some not yet observed fundamental constituents generically referred as preons and to have an internal substructure and heavy excited states F of masses m * F that should manifest themselves at the high energy compositeness scale Λ. Exchanging preons and/or binding quanta of unknown interactions between them result in effective contact interactions of SM fermions and heavy excited states. While different heavy excited states have been considered in literature [23][24][25], in this article we take as a reference the case of a heavy composite Majorana neutrino, N l , for which the interaction lagrangian would be (g * /Λ) 2q L γ µ q LNL γ µ L . Its theoretical studies and numerical analysis have been carefully elaborated in [26,27]. Moreover, an experimental analysis of √ s = 13 TeV pp collisions at LHC of the pro-cess pp → N l → qq of the dilepton (dielectrons or dimuons) plus diquark final states has been carried out by the CMS collaboration [28] excluding the existence of N l for masses up to 4.60 (4.70) TeV at 95% confidence level, assuming m N l = Λ. Motivated by the theoretical inconsistency [29] of SM bilinear Lagrangian of chiral gauged fermions and quantum-gravity natural regularization, as well as by quadrilinear four-fermion operators of Einstein-Cartan type [30], an alternative physics scenario had been proposed [31,32] on the basis of SM gauge symmetric fourfermion operators of SM left-and right-handed fermions (ψ L , ψ R ) in the charge sector "Q i " and flavor family "f "
f =1,2,3 G ψ f L ψ f Rψ f R ψ f L Qi=0,−1,2/3,−1/3 .(1)
These effective operators are attributed to the new physics at the cutoff Λ cut , and reduce to the NJL-type operator for the top-quark channel. The effective coupling G (1) has two fixed points: the weak-coupling infrared (IR) fixed point and the strong-coupling ultraviolet (UV) fixed point. In the scaling domain of IR fixed point of the weak four-fermion coupling G at the electroweak scale v ≈ 239.5 MeV, effective Operators (1) give rise to SM physics with tightly composite Higgs particle via NJL mechanism, and also offer possible solution to the hierarchy pattern of fermion masses [31,33]. In the scaling domain of UV fixed point of the strong four-fermion coupling G at the composite scale Λ ≈ O (TeV), composite fermions (bosons) form as bound states of three (two) SM elementary fermions and they couple to their constituents via effective contact interactions [32,34]. In the two previous scenarios, two model-independent properties are experimentally relevant for the study presented below: (i) the existence of contact interactions, in addition to SM gauge interactions, which represents an effective approach for describing the effects of the unknown internal dynamics of compositeness; (ii) the existence of composite fermions or excited states of SM fermions. For more details about the former scenario see Refs. [16][17][18][19][20][21][22][23][24][25].
In this article we study the latter scenario, focusing on the composite particles arising from four-fermion operators of Einstein-Cartan type, with massive (m F )
composite fermions F f R ∼ ψ f R (ψ f R ψ f L ) (bound states of three SM fermions) and massive (m Π ) composite bosons Π f ∼ (ψ f R ψ f L ) (
bound states of two SM fermions) forming in the scaling domain of a UV fixed point of the strong four-fermion coupling G at the composite scale Λ 5.14 TeV and Λ m F m Π [32,35]. The effective coupling between the composite fermion (boson) and its constituents is given by the following contact interaction, which describes composite particle F f (Π f ) production and decay:
(g * /Λ) 2ψf L (ψ f L ψ f R )F f R + h.c.,(2)(F Π /Λ) 2 (ψ f L ψ f R )Π f + h.c.,(3)
where (g * /Λ) 2 is a phenomenological parameter, and one can chose g 2 * = 4π so that 4π/Λ 2 is a geometric crosssection in the order of magnitude of inelastic processes forming composite fermions ( Fig. 1). Whereas, (F Π /Λ) 2 is the Yukawa coupling between composite boson (Fig. 2) and two fermionic constituents, and (g * /F Π ) 2 relates to the form factor of composite boson. The composite fermion is in fact a bound state of a SM fermion and composite boson, namely F f R ∼ ψ f R Π f . The composite scale Λ > F Π can only be experimentally determined like the electroweak scale v. The composite-fermion (-boson) mass m F , m Π ∝ Λ and the proportionality is of the order of unity.
In the follows we will consider the model in Eq. (1) with contact interactions of Eqs. (2) and (3) to study composite fermion production and decay at LHC and rely on the aforementioned heavy composite Majorana neutrino experimental studies [26,27] for what concerns the determination of constraints on the model parameters. We further compute 5σ contour plots of the statistical significance and highlight the region of parameter space where F can manifest using 3 ab −1 .
The article is arranged as follow. We discuss in Sec. II composite fermions' constituents and effective contact interactions among them. Focusing on the e + e − qq final state in Sec. III, the production cross sections and decay widths of these composite fermions are calculated in Sec. IV. In Sec. V we present the branching ratios of composite fermions decays in terms of selected parameters: composite scale Λ, composite particle masses m F and m Π , constrained by the recast of the upper limit on σ(pp → eeqq ) [28] in Sec. VI. In Sec. VI we further investigate the region of the paramete space where we expect composite fermion to appear with 3 ab −1 , which is the statistics expected in the High-Luminosity (HL) LHC. We find out that there is a wide region of model phase space where the composite fermions can be discovered in future searches. We also discuss other channels of composite fermions in Sec. VII and, in particular, we foresee a new full hadronic final state that, to the best of our knowledge, has not been investigated at the LHC. Finally, we summarize the work with some closing remarks in Sec. VIII.
II. QUARK-LEPTON OPERATORS AND CONTACT INTERACTIONS
A. Composite fermions F
To be relevant for possible final states with leptons and quarks in ongoing high-energy experimental searches in pp collisions, we first consider, among four-fermion operators (1), the following SM gauge-symmetric and fermionnumber conserving four-fermion operators, being the SM doublet i L = (ν e L , e L ) and singlet e R with an additional right-handed neutrino ν e R for leptons; ψ Lia = (u La , d La ) and u a R , d a R for quarks, where the color a and SU L (2)-isospin i indexes are summed over. Eq. (4) is for the first family only, as a representative of the three fermion families. The SM left-and right-handed fermions are mass eigenstates, their masses are negligible in TeV-energy regime and small mixing among three families encoded in G is also neglected [31].
G (¯ i L e R )(d a R ψ Lia ) + (¯ i L ν e R )(ū a R ψ Lia ) + h.c.,(4)
In Eq. (4), each four-fermion operator has the two possibilities to form composite fermions, listed in Table I. Up to a form factor, E (N ) indicates a composite fermion made of an electron (a neutrino) and a colorsinglet quark pair, and its superscript for electric charge. There are four independent composite fields F :
E 0 R , N − R , E − R , N 0 R and their Hermitian conjugates:Ē 0 L = (E 0 R ) † γ 0 , N + L = (N − R ) † γ 0 ,Ē + L = (E − R ) † γ 0 ,N 0 L = (N 0 R ) † γ 0 . They carry SM quantum numbers t i 3L , Y , and Q i = Y + t i 3L
, which are the sum of SM quantum numbers (t i 3L , Y, Q i ) of their constituents, i.e., the elementary leptons and quarks in the same SM family [32], listed in Table II, so that the contact interactions in Eq. (2) are SM gauge symmetric.
The contact interactions for the production and decay of a composite fermions F are:
L F CI = V F + V † F ,(5)
where
VĒ0 = g 2 * Λ 2 (Ē 0 L e R )(d a R u La ), pp or ep →Ē 0 L e R ,(6)VN+ = g 2 * Λ 2 (N + L ν e R )(ū a R d La ), pp or ep →N + L ν e R ,(7)VĒ+ = g 2 * Λ 2 (Ē + L e R )(d a R d La ), pp or ep →Ē + L e R , (8) VN0 = g 2 * Λ 2 (N 0 L ν e R )(ū a R u La ), pp or ep →N 0 L ν e R ,(9)
and
V † E 0 = g 2 * Λ 2 (ē L E 0 R )(ū a R d aL ), E 0 R →ē L (ū a R d aL ) (10) V † N − = g 2 * Λ 2 (ν e L N − R )(d a R u aL ), N − R →ν e L (d a R u aL ) (11) V † E − = g 2 * Λ 2 (ē L E − R )(d a L d Ra ), E − R →ē L (d a L d Ra ) (12) V † N 0 = g 2 * Λ 2 (ν e L N 0 R )(ū a L u Ra ), N 0 R →ν e L (ū a L u Ra ).(13)
These are relevant contact interactions for phenomenological studies of possible inelastic channels of compositefermion production and decay in pp or ep collisions.
B. Composite bosons Π 0,±
From the four-fermion interaction in Eq. (4), it is possible to form composite bosons
Π + = (g * /F Π ) 2 (d a R u La ), Π − = (Π + ) † (14) Π 0 d = (g * /F Π ) 2 (d a R d La ),(15)Π 0 u = (g * /F Π ) 2 (ū a R u La ),(16)
and their Hermitian conjugates. Such normalized composite boson field has the same dimension [energy] of elementary boson field. The composite boson carries the quantum numbers that are the sum over SM quantum numbers of its two constituents, see Table III. These are pseudo composite bosons Π 0,± , analogous to charged and neutral pions π 0,± in the low-energy QCD. As shown in Fig. 2, the effective coupling between composite boson and its two constituents can be written as an effective contact interaction,
L Π ± CI = g Y (d a R u La )Π − + h.c.,(17)L Π 0 d CI = g Y (d a R d La )Π 0 d + h.c.(18)L Π 0 u CI = g Y (ū a R u Ra )Π 0 u + h.c.(19)
where g Y = (F Π /Λ) 2 . Appropriate normalizing the composite boson Π with the form factor (g * /F Π ) 2 in Eqs. (14)(15)(16), the effective contact interaction in Eqs. (17)(18)(19) can be expressed as a dimensionless Yukawa coupling g Y , whose value, corresponding to F Π value, can be different for composite bosons in Eqs. (14)(15)(16), but we do not consider such difference here.
Operator Composite fermion FR Composite fermionFL Composite boson Π (ν e L eR)(d a R uLa) E 0 R ∼ eR(d a R uLa)Ē 0 L ∼ēL(ū a R dLa) Π + ∼ (d a R uLa) (ēLν e R )(ū a R dLa) N − R ∼ ν e R (ū a R dLa)N + L ∼ν e L (d a R uLa) Π − ∼ (ū a R dLa) (ēLeR)(d a R dLa) E − R ∼ eR(d a R dLa)Ē + L ∼ēL(d a L dRa) Π 0 d ∼ (d a R dLa) (ν e L ν e R )(ū a R uLa) N 0 R ∼ ν e R (ū a R uLa)N 0 L ∼ν e L (ū a L uRa) Π 0 u ∼ (ū a R uLa)composite fermions FR constituents charge Qi = Y + t i 3L SUL(2) 3-isospin t i 3L UY (1)-hypercharge Y E 0 R eR(d a R uLa) 0 1/2 −1/2 N − R ν e R (ū a R dLa) −1 −1/2 −1/2 E − R eR(d a R dLa) −1 −1/2 −1/2 N 0 R ν e R (ū a R uLa) 0 1/2 −1/2
C. Contact interaction of composite fermion and boson
In the view of the composite fermion being a bound state of a composite boson and a SM fermion, using composite-boson fields in Eqs. (14)(15)(16), we rewrite V † in Eqs. (10-13) as follow,
V † E 0 = g Y (ē L E 0 R )Π − , E 0 R →ē L Π − (20) V † N − = g Y (ν e L N − R )Π + , N − R →ν e L Π +(21)V † E − = g Y (ē L E − R )Π 0 d , E − R →ē L Π 0 d (22) V † N 0 = g Y (ν e L N 0 R )Π 0 u , N 0 R →ν e L Π 0 u ,(23)
and their Hermitian conjugates V in Eqs. (6)(7)(8)(9), as shown in Fig.3. These contact interactions in Eqs. (20)(21)(22)(23) imply that composite fermions F :
E 0 R , N − R , E − R , N 0 R
can decay into composite bosons Π ± and Π 0 , which decay then to SM fermions, following the contact interactions in Eqs. (17)(18)(19) at the leading order of tree level. However, we shall consider other decay channels at the next leading order, such as neutral composite boson decay to two SM gauge bosons Π 0 u,d →G +G .
D. Contact interaction of Π 0 composite boson and gauge bosons
Analogously to π 0 → γγ, the massive Π 0 u,d composite boson can also decay into two gauge bosons [32] : via the contact interaction
Π 0 u,d → γγ,(24)Π 0 u,d → γZ 0 ,(25)Π 0 u,d → Z 0 Z 0 ,(26)Π 0 u,d → W + W − ,(27)L Π 0 GG = i=u,d gg N c 4π 2 F Π µνρσ (∂ ρ A µ )(∂ σ A ν )Π 0 i(28)
where g and g represent the couplings of gauge bosons A µ and A ν to the SM quarks u and d with different
SU L (2)-isospin i = u, d.
Actually, this effective contact interaction (28) is an axial anomaly vertex, as a result of a triangle quark loop and standard renormalization procedure in SM.
III. e + e − qq FINAL STATE IN pp COLLISIONS
In this section we study the processes giving the e + e − qq final state, which we can use to set bounds on the parameters of the model by using the recast of the experimental upper limit on σ(pp → eeqq ) published in [28]. For this purpose, we consider only the case of composite fermions F = E 0 ,Ē 0 , E + , E − . The detailed analysis of composite fermions F = N 0 ,N 0 , N + , N − , giving the ννqq final states, will be considered in future. If the energy √ s in the parton center of mass frame is larger than composite fermion masses, the resonant processes described below can occurs. The virtual processes of composite fermions are not considered here. The kinematics of final states is simple in the center of mass frame of pp collisions. In pp collisions at LHC, the e + e − qq final state with this model can be obtained via the production of the composite fermions E 0 ,Ē 0 , E − , E + in association with an electron or a positron and the subsequent decay of the composite fermion to a positron or an electron and two quarks:
composite bosons Π constituents charge Qi = Y + t i 3L SUL(2) 3-isospin t i 3L UY (1)-hypercharge Y Π + (d a R uLa) +1 1/2 1/2 Π − (ū a R dLa) −1 −1/2 −1/2 Π 0 d (d a R dLa) 0 −1/2 1/2 Π 0 u (ū a R uLa) 0 1/2 −1/2pp → e + E 0 → e + e − qq ,(29)pp → e −Ē 0 → e − e + qq ,(30)pp → e + E − → e + e − qq ,(31)pp → e − E + → e − e + qq ,(32)
The quark-family mixing is neglected, so at parton level the previous equations are:
ud → e + E 0 → e + e − ud,(33)ud → e −Ē 0 → e − e +ū d,(34)dd → e + E − → e + e − dd,(35)dd → e − E + → e − e + dd.(36)
The decay of the composite fermion to a lepton and two quarks can happen directly, via the interactions in Eq. (10,12), or with the decay of the composite fermion to a lepton and the composite boson, via the interactions in Eq. (20,22), and the subsequent decay of the composite boson to two quarks, via the interactions in Eq. (17,18,19):
E 0 → e − Π + → e − ud,(37)E 0 → e + Π − → e +ū d,(38)E − → e − Π 0 d → e − dd,(39)E + → e + Π 0 d → e + dd.(40)
The cross sections of these processes are:
σ(pp → eF → e + e − qq ) = σ(pp → eF ) × B(F → eqq ), (41) where B(F → eqq ) = Γ 3−body (F → eqq ) + Γ(F → eΠ)B(Π → qq ) Γ tot (F )(42)
and
Γ tot (F ) = Γ(F → eΠ) + Γ 3−body (F → eqq ).(43)
The total cross section of the e + e − qq channel from the model in pp collisions is approximately given by
σ(pp → e + e − qq ) ≈ σ(pp → e + E 0 ) × B(E 0 → e +ū d) + σ(pp → e −Ē0 ) × B(Ē 0 → e − ud) + σ(pp → e +Ē− ) × B(Ē − → e − dd) + σ(pp → e − E + ) × B(E + → e + dd). (44)
The calculation of these quantities will be given in the next sections.
IV. CROSS SECTIONS AND DECAY WIDTHS
The partonic cross section of qq → eF is calculated by standard methods via the contact interaction in Eqs. (5-9) (all of them give the same result),
σ(ŝ, m F ) = 1 3 × 64π g 2 * Λ 2 2 (ŝ − m 2 F ) 2 m 2 F ,(45)
where √ŝ stands for the parton center-mass-energy of pp collisions in LHC experiments.
We consider the production cross sections for the composite fermions F in pp collisions expected at the CERN LHC collider according to Feynman's parton model. The QCD factorization theorem allows to obtain any hadronic cross section (e.g. in pp collisions) in terms of a convolution of the hard partonic cross sectionsσ, evaluated at the parton center of mass energy √ŝ = √ τ s, with the universal parton distribution functions f a (x,Q) which depend on the parton longitudinal momentum fractions x, and on the factorization scaleQ:
σ = ij 1 m 2 F s dτ 1 τ dx x f i (x,Q 2 )f j ( τ x ,Q 2 )σ(τ s) .(46)
The factorization and renormalization scale Q is generally fixed at the value of the mass that is being produced. The parametrization of the parton distribution function is NNPDF3.0 [36] and the factorization scale has been chosen asQ = m F . The right panel of Fig. 4 shows the agreement between analytical calculations based on Eqs. (45) and (46), for the case of the fermion E 0 , and the results of simulations with CalcHEP where the model with four-fermion interactions has been implemented. We remark the quite 45) and (46) and the filled circles (black) represent the results from our implementation of the model in CalcHEP. We find good agreement. On the right panel, we plot the decay width of composite fermion F as a function of its mass mF for the case Λ = mF . Again, we observe a good agreement between the expectation from a CalcHEP simulation and the analytical result based on Eq. (47). In both the plots we used the particular case F = E 0 , but the results are the same for the other choises of F . good agreement that validates our model implementation in CalcHEP.
Analytical calculations, in the similar way as the first term in Eq. (5) of Ref. [26], yield the width of composite fermion decay to its quark and lepton constituents
Γ 3−body (F → eqq ) = g 2 * Λ 2 2 m 5 F 4 × (8π) 3 .(47)
Note that at TeV energy scales, composite fermions are massive (m F ) Dirac fermions, whereas all SM elementary fermions are treated as massless Dirac fermions of four spinor components, consisting of right-and left-handed Weyl fermions of two spinor components. Alternatively, the decay width Γ F has also been evaluated via CalcHEP, and numerical results are completely in agreement with analytical one in Eq. (47), see left panel of Figure 4. The decay width of the composite fermion to a lepton and a composite boson Π can easily be computed from the effective contact lagrangian in Eqs. (20) and (22):
Γ(F → eΠ) = 1 32π F 2 Π Λ 2 2 m F 1 − m 2 Π m 2 F 2 .(48)
The decay width of the Π boson to two quarks is simply calculated by using the effective contact Lagrangian in Eq. (17) and (18),
Γ(Π → qq ) = 3 16π F Π Λ 4 m Π .(49)
For the Π + and Π − composite bosons this is the only decay channel, therefore composite fermions E 0 andĒ 0 have B(F → eqq ) = 1. The Π 0 d composite boson, instead, can also decay to two gauge bosonsGG , according to the contact interaction (28), the corresponding decay widths are [32] :
Γ Π 0 d →γγ = 5 9 2 Γ,(50)Γ Π 0 d →γZ 0 = 1 sin 2 2θ W 1 2 − 5 9 sin 2 θ W 2 Γ,(51)Γ Π 0 d →Z 0 Z 0 = 1/2−sin 2 θ W +(5/9) sin 4 θ W sin 2 2θ W 2 Γ,(52)Γ Π 0 d →W + W − = 1 8 sin 2 θ W 2 Γ,(53)
where θ W is the Weinberg angle,
Γ = αN c 3πF Π 2 m 3 Π 0 d 64π ,(54)
and the number of colors N c = 3. Total decay rate Γ tot (Π 0 d →GG ) is the sum over all contributions from Eqs. (50-53). The total Π 0 d -decay rate reads
Γ tot (Π 0 d ) = Γ(Π 0 d → qq ) + Γ tot (Π 0 d →GG ),(55)
where Γ(Π 0 d → qq ) is given by Eq. (49). Based on these results, we calculate the branching ratios of different channels in next section.
V. PARAMETERS AND BRANCHING RATIOS
In order to present the branching ratios of different possible channels in terms of parameters of the model, we are bound to discuss physically sensible parameters to explore. This model has four parameters that can be rearranged to three dimensionless parameters for a given Λ value:
(Λ, m F , F Π , m Π ) → (m F /Λ, m Π /m F , F Π /m Π ).
The ratio m F /Λ < 1 (m Π /Λ < 1) of the composite fermion (boson) mass and the basic composite scale Λ gives us an insight into the dynamics of composite fermion (boson) formation. On the other hand, a composite fermion F is composed by a composite boson and an elementary SM fermion, to represent this feature, we adopt the ratio m Π /m F < 1 as a parameter. In addition, considering the parameters m Π and F Π represent the same dynamics of composite boson formation, we approximately adopt F Π ≈ m Π so as to reduce the numbers of free parameters at this preliminary stage. As a result, for given √ s and Λ values, we have two parameters (m Π /m F , m F /Λ) to represent the results of cross sections, decay rates and branching ratios of various decay channels E → eqq , E → eΠ → eqq , and Π →GG . Figure 5 shows the branching ratios of the composite fermion E decay to eqq , i.e., B(E → eqq ) of Eqs. (42, 43), and E decay to e Π,
B(E → eΠ) = Γ(E → eΠ)/Γ tot (E),(56)
where Γ(E → eΠ) is given by Eq. (48). The results show the direct decay channel E → eqq is dominant over the decay channel E → eΠ. Note that these branching ratios are independent from m F /Λ for the parameterization F Π = m Π . Figure 6 shows for F Π = m Π and two selected m F /Λ values, the branching ratios of the Π 0 d decay to two quarks qq ,
B(Π 0 d → qq ) = Γ(Π 0 d → qq )/Γ tot (Π 0 d )(57)
and the Π 0 d decay to two gauge bosonsGG ,
B(Π 0 d →GG ) = Γ tot (Π 0 d →GG )/Γ tot (Π 0 d ).(58)
The results show that the decay of Π 0 →GG is not negligible only for small values of both m F /Λ and m Π /m F , see Figure 6 left panel. Figure 7 shows that the branching ratios of the direct E decay to a charged lepton and two quarks E → eqq ,
B(E → eqq , direct) = Γ 3−body (E → eqq )/Γ tot (E),(59)
and indirect E decay E → eΠ → eqq ,
B(E → eΠ → eqq ) = Γ(E → eΠ) Γ tot (E) B(Π → qq ),(60)
and the sum of these two branching ratios gives the total branching ratio B(E → eqq ) of E decay to eqq . In addition, it is also shown in Figure 7 that the branching ratio of the decay channel E → eΠ 0 → eGG ,
B(E → eΠ 0 d → eGG ) = Γ(E → eΠ 0 d ) Γ tot (E) B(Π 0 d →GG ).
(61) Despite this decay channel would be peculiar, having a final state signature not typical of the standard model processes, with highly energetically boosted gauge bosons plus an electron, the results show that the branching ratio B(E ± → eqq ) is much larger than B(E ± → eΠ 0 → eGG ) in the parameter space we have explored with the aforementioned parameter assumptions.
VI. BOUNDS ON THE MODEL
In this section we provide a discussion of the bounds on this model by recasting the 95% confidence level (C.L.) experimental upper limit on σ(pp → eeqq ) using a recent analysis [28] We also performed a study about the potential of a dedicated analysis in the High Luminosity LHC (HL-LHC) conditions (center of mass energy of 14 TeV and luminosity of 3 ab −1 ). We used CalcHEP to generate the processes and DELPHES [37] to simulate the detector effects. In order to separate the signal from the background, we selected events with pt e1 ≥ 180 GeV, pt e2 ≥ 80 GeV, pt j1 ≥ 210 GeV, m ee ≥ 300 GeV (pt is the transverse momentum, e 1 the leading electron, e 2 the subleading electron, j 1 the leading jet and m ee the invariant mass of the two electrons). Then we evaluated the reconstruction and selection efficiencies for signal ( s ) and background ( b ) as the ratio of the selected and the total generated events. From these efficiencies, the signal and background cross sections (σ s , σ b ) and the integrated luminosity (L), it is possible to evaluate the expected number of events for the signal (N s ) and the SM background (N b ) and finally the statistical significance (S):
m Π /m F m F /Λ=0.2 F Π =m Π E → e q q - E → e q q -Direct E → e Π → e q q - E → e Π → e GG'm Π /m F m F /Λ=0.8 F Π =m Π E → e q q - E → e q q -Direct E → e Π → e q q - E → e Π → e GG'N s = Lσ s s , N b = Lσ b b , S = N s √ N b .(62)
The S = 5 contour curve is shown by the upper (solid) line in Figure 9. It can be used to get indications about the potential for discovery or exclusion with the experiments at the HL-LHC, showing that there is a wide region of the model phase space where the existence of the composite fermions can be investigated; for the case Λ = m F we can reach masses up to ≈ 6.2 TeV.
VII. OTHER CHANNELS OF COMPOSITE FERMIONS
In this article, we have carried out the analysis of composite fermions F = E 0 ,Ē 0 , E + , E − produced in LHC pp collisions for the final states eeqq or eeGG . However, the exact same analysis can be done for composite fermions F = N 0 ,N 0 , N + , N − for the final state ννqq or ννGG ,
pp → νN → ννqq , or ννGG(63)
where νν stands for the SM left-handed neutrino ν e L and/or sterile right-handed neutrino ν e R . The latter is a candidate of dark-matter particles, represented by missing energy and momentum in the final states. Substituting e + e − by ν e L ν e R in above calculations, we obtain the same results at this preliminary level without turning on SM gauge interactions. For example, analogously to Eqs. (59,60,61), the branching ratios of the direct N decay to a neutrino and two quarks N → νqq ,
B(N → νqq , direct) = Γ 3−body (N → νqq )/Γ tot (N ),(64)
and indirect N decay N → νΠ 0 u → νqq ,
B(N → νΠ 0 u → νqq ) = Γ(N → νΠ 0 u ) Γ tot (N ) B(Π 0 u → qq ),(65)
and the sum of these two branching ratios gives the total branching ratio B(N → νqq ) of N decay to νqq . The branching ratio of the decay channel N → νΠ 0 u → νGG ,
B(N → νΠ 0 u → νGG ) = Γ(N → νΠ 0 u ) Γ tot (N ) B(Π 0 u →GG ).
(66) The same numerical results can be found in Fig. (7).
In fact, both composite bosons (Π) and fermions (F ) have definite SM quantum numbers, so that the Feynman diagrammatic representations of SM perturbative gauge interactions can be easily implemented, see Eqs. (4.8)-(4.11) in Ref. [32]. However, at the leading order of contact interactions discussed in this article, all gauge interactions are neglected, except the effective contact interaction (28) of the triangle anomaly, which couples to two SM gauge bosonsGG . It should be mentioned that these gauge bosonsGG in Eq. (28) can be two gluons that possibly fuse to a Higgs particle in the final states.
A. Composite fermions Q and four-jet final states
We further consider, among the variants of Eq. (1), the following SM gauge-symmetric and fermion-number conserving four-fermion operators of the quark sector, choosing as representative the first family [31,32],
G (ψ bi L d Rb )(d a R ψ Lia ) + (ψ bi L u Rb )(ū a R ψ Lia ) + h.c..(67)
Each four-fermion operator has the two possibilities to form composite fermions, listed in Table IV. Up to a certain form factor, D (U ) indicates a composite fermion made of a down quark d (an up quark u) and a color-singlet quark pair, and its superscript for electric charge. There are four independent composite fields Q: D 2/3
Ra , U −1/3 Ra , D −1/3 Ra , U 2/3
Ra and their Hermitian conjugates:D
−2/3 La = (D 2/3 Ra ) † γ 0 ,Ū 1/3 La = (U −1/3 Ra ) † γ 0 ,D 1/3 La = (D −1/3 Ra ) † γ 0 ,Ū −2/3 La = (U 2/3
Ra ) † γ 0 . They carry SM quantum numbers t i 3L , Y , and Q i = Y + t i 3L , which are the sum of SM quantum numbers (t i 3L , Y, Q i ) of their constituents, i.e., the elementary quarks in the same SM family [32], listed in Table V. These composite fermions D and U are analogous to the composite fermions E and N that have been previously analyzed.
The contact interactions for the production and decay of a composite fermions Q are, where
L Q CI = V Q + V † Q ,(68)∼d Lb (ū a R dLa) Π + ∼ (d a R uLa) (d Lb u Rb )(ū a R dLa) U −1/3 Rb ∼ u Rb (ū a R dLa)Ū 1/3 Lb ∼ū Lb (d a R uLa) Π − ∼ (ū a R dLa) (d Lb d Rb )(d a R dLa) D −1/3 Rb ∼ d Rb (d a R dLa)D 1/3 Lb ∼d Lb (d a L dRa) Π 0 d ∼ (d a R dLa) (ū Lb u Rb )(ū a R uLa) U 2/3 Rb ∼ u Rb (ū a R uLa)Ū −2/3 Lb ∼ū Lb (ū a L uRa) Π 0 u ∼ (ū a R uLa)charge Qi = Y + t i 3L SUL(2) 3-isospin t i 3L UY (1)-hypercharge Y D 2/3 Rb d Rb (d a R uLa) 2/3 1/2 1/6 U −1/3 Rb u Rb (ū a R dLa) −1/3 −1/2 1/6 D −1/3 Rb d Rb (d a R dLa) −1/3 −1/2 1/6 U 2/3 Ra u Rb (ū a R uLa) 2/3 1/2 1/6VD−2/3 = g 2 * Λ 2 (D −2/3 Lb d Rb )(d a R u La ); pp →D −2/3 La d Ra ,(69)VŪ1/3 = g 2 * Λ 2 (Ū 1/3 Lb u Rb )(ū a R d La ); pp →Ū 1/3 La u Ra , (70) VD1/3 = g 2 * Λ 2 (D 1/3 Lb d Rb )(d a R d La ); pp →D 1/3 La d Ra , (71) VŪ−2/3 = g 2 * Λ 2 (Ū −2/3 Lb u Rb )(ū a R u La ); pp →Ū −2/3 Lb u Rb ,(72)
and
V † D 2/3 = g 2 * Λ 2 (d Lb D 2/3 Rb )(ū a R d La ); D 2/3 Rb →d Lb (ū a R d La )(73) V † U −1/3 = g 2 * Λ 2 (ū Lb U −1/3 Rb )(d a R u La ); U −1/3 Rb →ū Lb (d a R u La )(74)V † D −1/3 = g 2 * Λ 2 (d Lb D −1/3 Rb )(d a L d Ra ); D −1/3 Rb →d Lb (d a L d Ra ) (75) V † U 2/3 = g 2 * Λ 2 (ū Lb U 2/3
Rb )(ū a L u Ra ); U The pp or ep collisions produce a composite fermion Q and a quark q = u, d, i.e., the production process pp → Qq via the contact interactions in Eq. V Q (69-72). The composite fermions Q decay to a quark and a pair of quarks, Q →qqq via the contact interactions V † Q (73-76).
A composite fermion Q = D 2/3 Ra , U −1/3 Ra , D −1/3
Ra , U 2/3 Ra appears in the s-channel as a resonance.
The following final states are foreseen in pp collisions at LHC:
pp (ud) →dD 2/3 →d + d + two jets (ud),(77)
pp (dd) →dD −1/3 → d +d + two jets (dd); (78) pp (dū) →ūU −1/3 →ū + u + two jets (dū), (79) pp (uū) →ūU 2/3 →ū + u + two jets (uū).
These four-jet events pp → jj + jj have a simple, but peculiar kinematic distribution that may be easily identified from background. At the present tree-level approximation of contact interactions, the cross sections, decay rates, kinematics and parameters (Λ, m F ) are the same as those of pp → † + jj processes. Analogously to the branching ratios (61) and (66) of the decay channels E → e Π 0 d → eGG and N → ν Π 0 u → νGG , we can obtain the branching ratios of composite fermions D −1/3 and U 2/3 decay into a quark (jet) and two boosted gauge bosons,
B(D −1/3 → d Π 0 d → dGG ) = Γ(D −1/3 → d Π 0 d ) Γ tot (D −1/3 ) × B(Π 0 d →GG );(81)B(U 2/3 → u Π 0 u → uGG ) = Γ(U 2/3 → u Π 0 u ) Γ tot (U 2/3 ) × B(Π 0 u →GG ).
(82) At the present tree-level approximation of contact interactions, the cross sections, decay rates, branching ratios, kinematics and parameters (Λ, m F ) of the composite fermions D −1/3 and U 2/3 production and decay into a jet and two boosted gauge bosons are the same as those of the composite fermions E ± and N production and decay into a lepton and two boosted gauge bosons discussed in Eqs. (61) and (66).
Here again we neglect the small contributions from perturbative SM gauge interactions, and only consider the dominant tree-level contributions of the first-family contact interactions (68) without any flavor mixing. Other possible channels with final states of gauge and Higgs bosons, as well as heavy quarks [32,35] are expected to have much smaller branching ratios and will be duly discussed and analyzed in future.
VIII. SUMMARY AND REMARKS
In the weak coupling regime the effective four-fermion operators of NJL-type possess an IR-fixed point, rendering the elegant Higgs mechanism of the SM of particle physics at low energies. In the strong coupling regime, on the other end, these operators could possess an UVfixed point, giving rise to composite fermions/bosons composed by SM fermions and their relevant contact interactions with SM fermions at high energies O(TeV). Using the first SM family, we study the spectra of composite particles and contact interactions in quark-lepton and quark-quark sectors. The cross sections and decay rates of composite particles are calculated to study their phenomenologies based on the LHC physics from pp collision at high energy TeV scale. In particular, the processes giving e + e − qq final state are analyzed by using the recast of the experimental upper limit on σ(pp → eeqq ) to set bounds on the parameters of composite particles and their contact interactions. We determine that a composite fermion, (F ), of mass m F below 4.25 TeV can be excluded for Λ = m F . At the same time, we compute 3σ and 5σ contour plots of the statistical significance and highlight the phase space in which F can manifest using 3 ab −1 , foreseen at the high luminosity LHC (HL-LHC). This result shows that there is a vast range of model parameters to which a dedicated search can be sensitive to F composite fermions and we thus encourage such efforts in future investigations at the LHC. Moreover, we further consider other decay channels of composite fermions and, in particular, we find that the case of Q-resonances can lead to a four jets final state (a triplet of jets produced in association with another jet). This signature, to the best of our knowledge, has escaped the realm of the searches at the LHC and can offer a new possibility to search for composite fermions and physics beyond the SM. The detailed phenomenology of the Q-resonances is the subject of an ongoing work. The phenomenology of two boosted SM gauge bosons Eqs. (58,61,66,81,82), and two gluons fusing into a Higgs boson in final states of composite boson and fermion decays will be further studied.
It is an interesting question to see how these phenomenologies can possibly account for some recent results obtained in both space and underground laboratories. The cosmic rays pp collisions produce composite particles E that decay into electrons and positrons. This may explain an excess of cosmic ray electrons and positrons around TeV scale [38,39]. In addition, recent AMS-02 results [40] show that at TeV scale the energydependent proton flux changes its power-law index. This implies that there would be "excess" TeV protons whose origin could be also explained by the resonance of composite fermions N due to the interactions of dark-matter and normal-matter particles. These composite fermions should appear as resonances by high-energy sterile neutrinos inelastic collisions with nucleons (xenon) at the largest cross-section, then resonances decay and produce some other detectable SM particles in underground laboratories [41]. Similarly, in the ICECUBE experiment [42], we expect events where the neutrinos change their directions (lower their energies) by their inelastic collisions to form the resonances of composite fermions N at a high energy scale (≈ TeV). Similarly to the analogy between the Higgs mechanism and BCS superconductivity, the composite-particle counterparts in condensed matter physics have been recently discussed [43].
IX. ACKNOWLEDGEMENTS
The work of Alfredo Gurrola and Francesco Romeo is supported in part by NSF Award PHY-1806612. The work of Hao Sun is supported by the National Natural Science Foundation of China (Grant No.11675033).
FIG. 1 .
1A lepton , two quarks q (u-type) andq (d-type) form a composite fermion F via the contact interaction (dark blob) PL,R(g 2 * /Λ 2 ), where PL,R = (1 ∓ γ5)/2. The thin solid line represents an SM elementary fermion, and the thick double line represents a composite fermion F . By a crossing symmetry applied to the lepton line → † (dashed line) the same diagram describes a 2 → 2 production process qq → † F .
FIG. 2 .
2We show the Feynman diagrammatic representation for the contact interaction between the composite boson and its constituent quarks, where the thin solid line represents an SM elementary fermion, double wave line represents a composite boson, and the blob represents an interacting vertex (FΠ/Λ) 2 PL,R.
FIG. 3 .
3We show the Feynman diagrammatic representation for the contact interaction between the composite fermion and boson, where the thin solid line represents a SM elementary fermion, the double solid line is a composite fermion and the double wave line represents a composite boson and the blob represents an interacting vertex (FΠ/Λ) 2 PL,R.
FIG. 4 .
4(Color Online). On the left panel, we show the production cross section of pp → F as a function of mF for the case Λ = mF and at a center of mass energy √ s = 13 TeV. The solid red line represents the results of an analytical and numerical calculation based on Eqs. (
Online). The branching ratios of the composite fermion F decay to eqq and to e Π. The branching ratio F → eqq is much larger than the branching ratio F → eΠ, except in a small regime 0.4 mΠ/mF 0.9, where the branching ratio F → e Π is not completely negligible.
of 2.3 fb −1 data from the 2015 Run II of the LHC by the CMS collaboration with respect to the predictions of the model of composite fermions discussed in this article. Note that both electrons and positrons are collected in the final states of eeqq , electrons and positrons are not distinguished in the data analysis. For the case m F = Λ one obtains that the composite fermions of this model are excluded up to masses m ex F ≈ 4.25 TeV. This result is shown in Figure 8, together with the exclusion limits m ex F ≈ 3.3, 2.4, 1.5 TeV for Λ fixed at 6, 9 and 12 TeV. Figure 9 shows the exclusion curve, lower (dashed) line, in the 2-dimensional parameter space (Λ, m F ) for our model obtained via the recasting of the analysis [28] of 2.3 fb −1 data from the 2015 Run II of the LHC by the CMS collaboration. Here the regions of the parameter space below the curves are excluded.
Online). The branching ratios of the Π 0 decays to qq and toGG , for the case mF /Λ = 0.2 on the left panel and for the case mF /Λ = 0.8 on the right panel.
FIG. 7 .
7(Color Online). The branching ratios for the composite fermion decays: (1) directly decay to eqq; (2) decay to eqq via intermediate composite boson Π; (3) decay to two gauge bosons in the case that the intermediate state is a neutral composite boson Π 0 d . The E → eqq decay is the sum of the direct E → eqq decay and the E → eΠ → eqq decay. The case mF /Λ = 0.2 on the left panel and the case mF /Λ = 0.8 on the right panel.
Online). Recast of the experimental upper limit on σ(pp → eeqq ) published in[28] against the model of composite fermions studied in this article. The dotted line (solid green line) is the 95% C.L. observed (expected) upper limit on σ(pp → eeqq ) as reported in[28]. The solid line (red) is the theoretical expectation from the model described in this work as given by Eq. (44) for the case Λ = mF , the dashed lines (orange) are the theoretical expectation from the model for the cases Λ = 6, 9, 12 TeV. If Λ = mF one obtains that the composite fermions of this model are excluded up to masses m ex F ≈ 4.25 TeV. Online). Recast of the experimental upper limit from[28] (dashed line) and the predicted contour curve at a 5-level statistical significance (solid line) in the 2-dimensional parameter space (Λ, mF ). The shaded region denotes unphysical values of the parameters (Λ < mF ).
Rb →ū Lb (ū a L u Ra ). (76) These are the relevant contact interactions for phenomenological studies of possible inelastic channels of composite-fermion Q = (D, U ) production and decay in pp collision. Without considering SM gauge interactions and family mixings, the phenomenological analysis and results are the same as those of composite fermions (E, N ), apart from different final states. B. Possible Q-resonances and four jets final states in pp collisions
TABLE I .
IFour-fermion operators in Eq. (4) and possible composite fermions F and composite bosons Π. The color a index is summed.
TABLE II .
IIComposite fermions FR, their constituents and SM quantum numbers.
TABLE III .
IIIComposite bosons Π 0,± , their constituents and SM quantum numbers.
TABLE IV .
IVFour-fermion operators (67) and possible composite fermions Q. The color a index is summed.composite fermions QR
constituents
TABLE V .
VComposite fermions QR, their constituents and SM quantum numbers.
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| []
|
[
"A SPATIALLY RESOLVED INNER HOLE IN THE DISK AROUND GM AURIGAE",
"A SPATIALLY RESOLVED INNER HOLE IN THE DISK AROUND GM AURIGAE"
]
| [
"A Meredith Hughes ",
"Sean M Andrews ",
"Catherine Espaillat ",
"David J Wilner ",
"Nuria Calvet ",
"Paola D'alessio ",
"Chunhua Qi ",
"Jonathan P Williams ",
"Michiel R Hogerheijde "
]
| []
| []
| We present 0. ′′ 3 resolution observations of the disk around GM Aurigae with the Submillimeter Array (SMA) at a wavelength of 860 µm and with the Plateau de Bure Interferometer at a wavelength of 1.3 mm. These observations probe the distribution of disk material on spatial scales commensurate with the size of the inner hole predicted by models of the spectral energy distribution. The data clearly indicate a sharp decrease in millimeter optical depth at the disk center, consistent with a deficit of material at distances less than ∼20 AU from the star. We refine the accretion disk model ofCalvet et al. (2005)based on the unresolved spectral energy distribution (SED) and demonstrate that it reproduces well the spatially resolved millimeter continuum data at both available wavelengths. We also present complementary SMA observations of CO J=3−2 and J=2−1 emission from the disk at 2 ′′ resolution. The observed CO morphology is consistent with the continuum model prediction, with two significant deviations: (1) the emission displays a larger CO J=3−2/J=2−1 line ratio than predicted, which may indicate additional heating of gas in the upper disk layers; and (2) the position angle of the kinematic rotation pattern differs by 11 • ± 2 • from that measured at smaller scales from the dust continuum, which may indicate the presence of a warp. We note that photoevaporation, grain growth, and binarity are unlikely mechanisms for inducing the observed sharp decrease in opacity or surface density at the disk center. The inner hole plausibly results from the dynamical influence of a planet on the disk material. Warping induced by a planet could also potentially explain the difference in position angle between the continuum and CO data sets. | 10.1088/0004-637x/698/1/131 | [
"https://arxiv.org/pdf/0903.4455v1.pdf"
]
| 10,053,453 | 0903.4455 | fa27eb2103fdeae37c418915762b1fc1a742dedb |
A SPATIALLY RESOLVED INNER HOLE IN THE DISK AROUND GM AURIGAE
25 Mar 2009
A Meredith Hughes
Sean M Andrews
Catherine Espaillat
David J Wilner
Nuria Calvet
Paola D'alessio
Chunhua Qi
Jonathan P Williams
Michiel R Hogerheijde
A SPATIALLY RESOLVED INNER HOLE IN THE DISK AROUND GM AURIGAE
25 Mar 2009Accepted for publication in ApJ: March 25, 2009 Accepted for publication in ApJ: March 25, 2009Preprint typeset using L A T E X style emulateapj v. 10/09/06Subject headings: circumstellar matter -planetary systems: protoplanetary disks -stars: individual (GM Aurigae)
We present 0. ′′ 3 resolution observations of the disk around GM Aurigae with the Submillimeter Array (SMA) at a wavelength of 860 µm and with the Plateau de Bure Interferometer at a wavelength of 1.3 mm. These observations probe the distribution of disk material on spatial scales commensurate with the size of the inner hole predicted by models of the spectral energy distribution. The data clearly indicate a sharp decrease in millimeter optical depth at the disk center, consistent with a deficit of material at distances less than ∼20 AU from the star. We refine the accretion disk model ofCalvet et al. (2005)based on the unresolved spectral energy distribution (SED) and demonstrate that it reproduces well the spatially resolved millimeter continuum data at both available wavelengths. We also present complementary SMA observations of CO J=3−2 and J=2−1 emission from the disk at 2 ′′ resolution. The observed CO morphology is consistent with the continuum model prediction, with two significant deviations: (1) the emission displays a larger CO J=3−2/J=2−1 line ratio than predicted, which may indicate additional heating of gas in the upper disk layers; and (2) the position angle of the kinematic rotation pattern differs by 11 • ± 2 • from that measured at smaller scales from the dust continuum, which may indicate the presence of a warp. We note that photoevaporation, grain growth, and binarity are unlikely mechanisms for inducing the observed sharp decrease in opacity or surface density at the disk center. The inner hole plausibly results from the dynamical influence of a planet on the disk material. Warping induced by a planet could also potentially explain the difference in position angle between the continuum and CO data sets.
INTRODUCTION
Understanding of the planet formation process is intimately tied to knowledge of the structure and evolution of protoplanetary disks. Of particular importance is how and when in the lifetime of the disk its constituent material is cleared, which provides clues to how and when planets may be assembled. While observations suggest that the inner and outer dust disk disperse nearly simultaneously (e.g. Skrutskie et al. 1990;Wolk & Walter 1996;Andrews & Williams 2005), it is not clear which physical mechanism(s) drives this process, or the details of how it progresses. Possible dispersal mechanisms, of which several may come into play over the lifetime of a disk, include a drop in dust opacity due to grain growth (e.g. Strom et al. 1989;Dullemond & Dominik 2005), photoevaporation of material by energetic stellar radiation (e.g. Clarke et al. 2001), photophoretic effects of gas on dust grains (Krauss & Wurm 2005), inside-out evacuation via the magnetorotational insta-bility (Chiang & Murray-Clay 2007), and the dynamical interaction of giant planets with natal disk material (e.g. Lin & Papaloizou 1986;Bryden et al. 1999). Observing the distribution of gas and dust in disks allows us to evaluate the roles of these disk clearing mechanisms.
One particular class of systems, those with "transitional" disks (e.g. Strom et al. 1989;Skrutskie et al. 1990), have become central to our understanding of disk clearing. These disks exhibit a spectral energy distribution (SED) morphology with a deficit in the nearto mid-infrared excess over the photosphere consistent with a depletion of warm dust near the star. The advent of the Spitzer Space Telescope has allowed detailed measurement of mid-infrared spectra with unprecedented quality and quantity. Combined with simultaneous advances in disk modeling that can now reproduce in detail the SED features (e.g. D' Alessio et al. 1999Alessio et al. , 2001Dullemond et al. 2002;D'Alessio et al. 2006), these observations have revolutionized the study of disk structure. However, such studies rely entirely on SED deficits whose interpretations are not unique, since effects of geometry and opacity can mimic the signature of disk clearing (Boss & Yorke 1996;Chiang & Goldreich 1999).
Spatially resolved observations are crucial for confirming the structures inferred from disk SEDs. High resolution imaging at millimeter wavelengths is especially important because dust opacities are low, and the disk mass distribution can be determined in a straightforward way for an assumed opacity. Millimeter observations also avoid many of the complications present at shorter wavelengths, including large optical depths, spectral features, and contrast with the central star. Sev-eral recent millimeter studies have resolved inner emission cavities for disks with infrared SED deficits through direct imaging observations, e.g. TW Hya (Calvet et al. 2002;Hughes et al. 2007), LkHα 330 (Brown et al. , 2008, and LkCa 15 (Piétu et al. 2007;Espaillat et al. 2008). These observations unambiguously associate infrared SED deficits with a sharp drop in millimeter optical depth in the disk center. More information is needed to determine whether the low optical depth is a result of decreased surface density or opacity.
GM Aurigae is a prototypical example of a star host to a "transitional" disk. The ∼1-5 Myr old T Tauri star (Simon & Prato 1995;Gullbring et al. 1998) of spectral type K5 is located at a distance of 140 pc in the Taurus-Auriga molecular complex (Bertout & Genova 2006), and its brightness and relative isolation from intervening cloud material have enabled a suite of observational studies of its disk properties. The presence of circumstellar dust emitting at millimeter wavelengths was first inferred by Weintraub et al. (1989), and the disk structure was subsequently resolved in the 13 CO J=2-1 transition by Koerner et al. (1993). Their arcsecond-resolution mapping of the gas disk revealed gaseous material in rotation about the central star. Assuming a Keplerian rotation pattern allowed a determination of the dynamical mass for the central star of 0.8 M ⊙ . Further modeling of the structure and dynamics of the disk was carried out by Dutrey et al. (1998), using higher-resolution 12 CO J=2-1 observations. Scattered light images revealed a dust disk inclined by 50-56 • extending to radii ∼ 300 AU from the star (Stapelfeldt & The WFPC2 Science Team 1997;Schneider et al. 2003).
Efforts to model the SED of GM Aurigae have long indicated the presence of an inner hole, and estimates of its size have grown over the years as the quality of data and models have improved. In the early 1990s, the low 12 µm flux led to ∼ 0.5 AU estimates of the inner disk radius (Marsh & Mahoney 1992;Koerner et al. 1993). That value was later increased to 4.8 AU by Chiang & Goldreich (1999) in the context of hydrostatic radiative equilibrium models, and a putative planet at a distance of 2.5 AU from the star was shown to be capable of clearing an inner hole of this extent using simulations of the relevant hydrodynamics (Rice et al. 2003). With the aid of a ground-based mid-IR spectrum, Bergin et al. (2004) increased the gap size estimate to 6.5 AU, and subsequently Calvet et al. (2005) inferred an inner hole radius of 24 AU using a Spitzer IRS spectrum in combination with sophisticated disk structure models. Recently, Dutrey et al. (2008) have argued for a 19±4 AU inner hole in the gas distribution, using combined observations of several different molecular line tracers. Like the SED-based measurements, their method is indirect: they use a model of the disk in Keplerian rotation to associate a lack of high-velocity molecular gas with a deficit of material in the inner disk.
We present interferometric observations at 860 µm from the Submillimeter Array 7 and 1.3 mm from the Plateau de Bure Interferometer 8 that probe disk material 7 The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academica Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academica Sinica.
8 Based on observations carried out with the IRAM Plateau on scales commensurate with the 24 AU inner disk radius inferred from the SED. These data allow us to directly resolve the inner hole in the GM Aur disk for the first time. We describe the observations in §2 and present the dual-wavelength continuum data in §3.1. We also present observations of the molecular gas disk in the CO J=3-2 and J=2-1 lines that allow us to study disk kinematics in §3.2. We use these data to investigate disk structure in the context of the SED-based models of Calvet et al. (2005), described in §4. Implications for the disk structure and evolutionary status are discussed in §5.
OBSERVATIONS AND DATA REDUCTION
The GM Aur disk was observed with the 8-element (each with a 6 m diameter) Submillimeter Array (SMA; Ho et al. 2004) in the very extended (68-509 m baselines) and compact (16-70 m baselines) configurations on 2005 November 5 and 26, respectively. Observing conditions on both nights were excellent, with ∼1 mm of precipitable water vapor and good phase stability. Double sideband receivers were tuned to a central frequency of 349.935 GHz (857 µm), with each 2 GHz-wide sideband centered ±5 GHz from that value. The SMA correlator was configured to observe the CO J=3−2 (345.796 GHz) and HCN J=4−3 (354.505 GHz) transitions with a velocity resolution of 0.18 km s −1 . No HCN was detected, with a 3σ upper limit of 0.9 Jy beam −1 in the 2. ′′ 2×1. ′′ 9 synthesized beam. The observing sequence alternated between GM Aur and the two gain calibrators 3C 84 and 3C 111. The data were edited and calibrated using the MIR software package. 9 The passband response was calibrated using observations of Saturn (compact configuration) or the bright quasars 3C 273 and 3C 454.3 (very extended configuration). The amplitude scale was determined by bootstrapping observations of Uranus and these bright quasars, and is expected to be accurate at the ∼10% level. Antenna-based gain calibration was conducted using 3C 111, while the 3C 84 observations were used to check on the quality of the phase transfer. We infer that the "seeing" induced on the very extended observations by phase noise and small baseline errors is small, 0. ′′ 1. Wideband continuum channels from both sidebands and configurations were combined. The derived 870 µm flux of GM Aur is 640 ± 60 mJy.
Additional SMA observations in the extended (28-226 m) and sub-compact (6-69 m baselines) configurations were conducted on 2006 December 10 and 2007 September 14, respectively, with a central frequency of 224.702 GHz (1335 µm). While the sub-compact observations were conducted in typical weather conditions for this band (2.5 mm of water vapor), the extended data were obtained in better conditions similar to those for the higher frequency observations described above. The correlator was configured to simultaneously cover the J=2−1 transitions of CO (230.538 GHz), 13 CO (220.399 GHz), and C 18 O (219.560 GHz) with a velocity resolution of ∼0.28 km s −1 . The calibrations were performed as above.
GM Aurigae was also observed with the 6-element (each with a 15 m diameter) Plateau de Bure Interferde Bure Interferometer. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). 9 See http://cfa-www.harvard.edu/$\sim$cqi/mircook.html. ometer (PdBI) in the A configuration (up to 750 m baselines) on 2006 January 15. Observing conditions were excellent, with atmospheric phase noise generating a seeing disk of 0.2 ′′ . The PdBI dual-receiver system was set to observe the 110.201 GHz (2.7 mm) and 230.538 GHz (1.3 mm) continuum simultaneously. As with the SMA data, observations alternated between GM Aur and two gain calibrators, 3C 111 and J0528+134. The data were edited and calibrated using the GILDAS package (Pety 2005). The passband responses and amplitude scales were calibrated with observations of 3C 454.3 and MWC 349, respectively. The derived 1.3 and 2.7 mm fluxes of GM Aur are 180 ± 20 and 21 ± 2 mJy.
The standard tasks of Fourier inverting the visibilities, deconvolution with the CLEAN algorithm, and restoration with a synthesized beam were conducted with the MIRIAD software package. A high spatial resolution image of the 860 µm continuum emission from the SMA data was created with a Briggs robust = 1.0 weighting scheme for the visibilities, excluding projected baselines ≤ 70 kλ, resulting in a synthesized beam FWHM of 0. ′′ 30 × 0. ′′ 24 at a position angle of 34 • . A similar image of the 1.3 mm continuum emission with a synthesized beam FWHM of 0. ′′ 43 × 0. ′′ 30 at a position angle of 35 • was generated from the PdBI data using natural weighting (robust = 2.0). Table 1 summarizes the line and continuum observational parameters. Figure 1 shows the results of the SMA and PdBI continuum observations in both the image and Fourier domains. The presence of an inner hole in the GM Aur disk, as predicted by models of the SED, is clearly indicated both by the double-peaked emission structure in the image and by the null in the visibility data. The doublepeaked emission structure points to a deficit of flux near the disk center; the null in the visibility function, or the location at which the real part of the visibilities change sign, similarly reflects a decrease in flux at small angular scales. The resolution of the 2.7 mm data from the PdBI was insufficient to provide about the inner hole.
RESULTS
Millimeter Continuum Emission
The maps in the left panel of Fig. 1 show a doublepeaked brightness distribution at both wavelengths, with peak flux densities of 59 ± 4 mJy beam −1 at 860 µm and 16.6 ± 0.3 mJy beam −1 at 1.3 mm. For all but the most edge-on viewing geometries (e.g. Wolf et al. 2008), a continuous density distribution extending in to the dust destruction radius (∼0.05-0.1 AU; Isella et al. 2006) would be expected to result in a centrally-peaked brightness distribution. In the case of GM Aurigae, the double-peaked emission structure is a geometric effect due to the truncation of disk material at a much larger radius, viewed at an intermediate inclination of 50-56 • (Dutrey et al. 1998(Dutrey et al. , 2008: the region of highest density is near the inner disk edge, with a large column density of optically thin material in this ring effectively generating limb brightening at the inner edge of the outer disk, at two points along the disk major axis.
The size of the inner hole can be roughly estimated by the separation of the emission peaks, although the peak separation will also depend on the brightness of the directly-illuminated inner edge of the outer disk relative to the extended disk component (Hughes et al. 2007). The separation of the peaks in the 860 µm image is 0. ′′ 38± 0. ′′ 03, corresponding to a physical diameter of 53 ± 4 AU (radius 27 ± 2 AU) at a distance of 140 pc. A position angle of 66 • is estimated by the orientation of a line that bisects the two peaks, although a more robust value of 64 • ± 2 • is derived in §4.1 below. Since the peaks are not distinctly separated in the 1.3 mm image, the same estimate cannot be made, but the position angle is clearly consistent with that derived from the 860 µm visibilities and indicated by the perpendicular dashed lines in Fig. 1. The presence of an inner hole is also evident from the visibilities displayed in the right panel of Fig. 1. The real part of the complex visibilities have been averaged in concentric annuli of deprojected (u, v) distance from the disk center. For details of the deprojection process, see Lay et al. (1997). As discussed in the appendix of Hughes et al. (2007), the presence of a null in the visibility function indicates a sharp decrease in flux at a radius corresponding roughly to the angular scale of the null position. The precise position of the null depends primarily on the angular size of the inner hole, but also on the radial gradients of the surface density and temperature distribution and the relative brightness of the directly illuminated wall at the inner edge of the outer disk. In a standard power-law parameterization, the disk temperature T and surface density Σ vary inversely with radius as Σ ∝ R −p and T ∝ R −q . Neglecting the emission from the wall and assuming standard values of p = 1.0 and q = 0.5, expected for a typical viscous disk with constant α and consistent with previous studies of the GM Aur disk (Dutrey et al. 1998;Andrews & Williams 2007;Hughes et al. 2008), we may obtain a rough estimate of the size of the inner hole using the observed null position and Eq. A9 from Hughes et al. (2007): R null (kλ) = (1 AU/R hole )(D source /100 pc)[2618 + 1059(p + q)]. A polynomial curve fit to the visibilities yields a null position of 190 kλ at 860 µm and 224 kλ at 1.3 mm, which correspond to inner hole radii of 31 and 26 AU, respectively. However, these estimates are uncertain to within ∼30%, as the data are consistent with a broad range of null positions. We therefore turn to a more sophisticated modeling procedure described in §4.1 below. and first (color) moments of the data: these are the velocity-integrated intensity and intensity-weighted velocities, respectively. The peak flux density is 6.7 ± 0.3 Jy beam −1 in the CO J=3-2 line and 2.4 ± 0.1 Jy beam −1 in the CO J=2-1 line, with integrated fluxes of 9.4 Jy km s −1 and 21.2 Jy km s −1 , respectively (although emission from extended ambient cloud material is likely to increase the CO J=2-1 integrated flux over that originating from the disk alone). The channel and moment maps are broadly consistent with the expected kinematic pattern for material in Keplerian rotation about the central star, substantially inclined to our line of sight (as in Dutrey et al. 1998;Simon et al. 2000).
CO Channel and Moment Maps
The short-baseline spatial frequencies in the (u, v) plane provided by the subcompact configuration of the SMA during our observations of the J=2−1 transition are sensitive to emission on the largest spatial scales. These short antenna spacings reveal the severity of the cloud contamination to an extent not possible with previous data. The contamination is evident as an extended halo around the disk emission in the central channels of the J=2−1 channel maps near LSR velocities of 5-6 km s −1 (Fig. 3). It is also evident in the moment map (Fig. 4) as an elongation of emission near the systemic velocity (green-yellow) to the northwest along the disk minor axis. This contamination indicates that caution must be exercised when deriving kinematic information from the CO lines, particularly the central channels. Spatial filtering by the interferometer does not ameliorate cloud contamination in an abundant, easily-excited, high-optical depth tracer like COJ=2-1. The J=3−2 line appears less contaminated than J=2−1 (Figs. 2 and 4), although similarly short antenna spacings (8-43 m) are not present in this data set. Nevertheless, we expect less cloud contamination in the J=3−2 transition, since the temperature of the cloud will be lower than that of the disk and will therefore populate the upper rotational levels of the CO molecule less efficiently. The cloud contamination prevents detection of self-absorption in the central channels of the CO J=2-1 channel maps along the near (northwest) edge of the disk (as determined by scattered light observations; see Schneider et al. 2003). Dutrey et al. (1998) report selfabsorption along the southeast edge, but our observations suggest that this brightness asymmetry may be due to cloud contamination. It is also possible that the contamination is due to a residual envelope, although we are unable to determine the large-scale structure of the extended line emission with our interferometric data.
In all figures, the disk orientation based on the position angle of 64 • derived from the continuum emission ( Fig. 1 and §4.1) is plotted over the CO emission as a set of crossed dashed lines, with the relative extent of the major and minor axes (based on the inclination angle of 55 • ) indicated by the length of the perpendicular lines. The position angle of 51 • derived by Dutrey et al. (1998) from fitting the CO J=2−1 emission, consistent with our own J=3−2 and J=2−1 observations, is illustrated by the solid line. Note that the position angle of the CO emission differs slightly from the position angle of the continuum emission, by 11 • ± 2 • (see §4.1). The trend is clear for both transitions, but more obvious in the less-contaminated J=3−2 transition. Note that the position angle for the CO emission is derived entirely from the rotation pattern (evident in the isovelocity contours) and not from the geometry of the integrated CO emission: the integrated emission appears to match the position angle from the continuum emission reasonably well. We do not observe the isophote twisting in integrated CO emission seen by Dutrey et al. (1998). The cloud contamination and differences in antenna spacings may play a role.
DISK STRUCTURE MODELS
Updated SED Model
Here we revisit the broadband SED modeling of GM Aur presented by Calvet et al. (2005). Taking into consideration new observational constraints at submillimeter and millimeter wavelengths, we use the irradiated accretion disk models of D' Alessio et al. (2005Alessio et al. ( , 2006 to re-derive the properties of the outer disk of GM Aur and its inner, truncated edge or "wall." Our grainsize distribution follows a power-law of a −3.5 , where a is the grain radius. We assume ISM-sized grains in the upper layers of the disk and accordingly adopt a min =0.005 µm and a max =0.25 µm (Draine & Lee 1984). Closer to the disk midplane grains have a maximum size of 1 mm. Input parameters for the outer disk include the stellar properties, the mass accretion rate, the viscosity parameter (α), and the settling parameter (ǫ) which measures the dust-to-gas mass ratio in the upper layers of the disk relative to the standard dustto-gas mass ratio. Following Calvet et al. (2005), we adopt the same extinction, distance, inclination, dust grain opacities, and stellar properties (i.e. luminosity, radius, and temperature; see Table 2). We use a mass accretion rate of 7.2×10 −9 M ⊙ yr −1 which was derived using HST STIS spectra by Ingleby & Calvet (2009), in contrast to the value of 10 −8 M ⊙ yr −1 derived from veiling measurements in Calvet et al. (2005). We assume an outer disk radius of 300 AU, which matches the observed extent of scattered light from the dust disk (Schneider et al. 2003) and previous fits to the continuum emission (Hughes et al. 2008), as well as the short- baseline data presented here.
In order to reproduce the outer disk component of the SED, we vary ǫ and α ( Figure 5). As described in , α effectively determines the mass surface density distribution and therefore the disk mass, which is best reflected by the longest-wavelength SED points. The value of ǫ has the greatest effect on the slope of the SED beyond 100 µm. With the new millimeter data we find ǫ=0.5, indicating less settling than reported previously. We also find α=0.002 and a more massive outer disk of 0.16 M ⊙ . This mass is significantly larger than an estimate based on the 860 µm and 1.3 mm flux measurements using opacities from Beckwith et al. (1990), which yields ∼0.04 M ⊙ , and is only marginally Toomre stable at 300 AU (Q∼1.1). The outer disk model uses an opacity of ∼0.1 cm 2 g −1 at 1 mm (D'Alessio et al. 2001) which is about four times lower than that derived from the Beckwith et al. (1990) opacities, accounting for the discrepancy in mass. Within the inner disk hole, there are 1.1×10 −11 M ⊙ of optically thin small dust grains, which account for the 10 µm emission and the near-IR excess. The mass in solids could be much larger than this mass if pebbles, rocks, or even planetesimals have grown in the inner disk, since they would have a negligible opacity in the near-IR. We note that Calvet et al. (2005) reports the mass of the dust as 7×10 −10 M ⊙ ; this is actually the mass of the gas within the hole, assuming the standard dust to gas mass ratio. The gas mass could be significantly larger, depending on the total amount of solids and the actual ratio, but these are poorly constrained by existing data.
We vary the temperature of the wall to best reproduce the data. The radius of the wall is set by the temperature and dust composition, and the wall's height is set by the disk scale height. We assume that the wall is axisymmetric and composed of relatively small grains, as well as vertically flat in order to reproduce the rapid rise of the mid-IR excess at wavelengths beyond 10 µm. We adopt the dust composition used in D' Alessio et al. (2005) and Calvet et al. (2005). The maximum grain size is adjusted from ISM sizes to reproduce the shape of the IRS spectrum as necessary. At short wavelengths, larger grains have smaller opacities than ISM-sized grains. Therefore, at a given temperature large grains will be at smaller radii than ISM-sized grains as per Eqn. 12 of D' Alessio et al. (2005). The derived size of the inner hole varies somewhat depending on whether the SED or the resolved millimeter visibilities are included. Fitting only the broadband SED and neglecting the resolved millimeter-wavelength data, the wall is located at 26 AU and has a temperature of 130K and a height of ∼2 AU with maximum grain size a max =0.25 µm (Fig. 5, left panel). The radius of the wall differs by ∼2 AU from Calvet et al. (2005), since here we take L acc ∼ GMṀ/R assuming magnetospheric accretion while Calvet et al. (2005) uses L acc ∼ GMṀ/2R as per the boundary layer model. We also adopt a different mass accretion rate.
In order to compare the SED model with the resolved continuum data, it is necessary to fix the disk geometry. As listed in Table 2, we adopt an inclination of 55 • , in order to maintain consistency with Calvet et al. (2005). However, the position angle is poorly reproduced by the value of 53.4 • ± 0.9 • that is the weighted average of fits to the CO emission (Dutrey et al. 1998(Dutrey et al. , 2008, see Fig. 1). To derive a more appropriate position angle, we generate a sky-projected image from the disk model and use the MIRIAD task uvmodel to sample the image at the same spatial frequencies as the data. We compare these model visibilities with the observed 860 µm visibilities (which have the finest resolution). We repeat this process for a range of position angles and calculate a χ 2 value comparing each set of model visibilities with the data. Using this method, we fit a position angle of 64 • ± 2 • , which differs by 11 • ± 2 • from the position angle of the CO disk derived by Dutrey et al. (1998Dutrey et al. ( , 2008.
When considering the resolved millimeter-wavelength visibilities, a disk with a 20 AU hole reproduces the emission much better (Fig. 5, right panel, and Fig. 1, center panels). Using the same χ 2 comparison of visibilities as described in the previous paragraph, the 20 AU model represents a 3σ improvement over the 26 AU model, which significantly underpredicts the amount of flux produced close to the star. This 20 AU hole has a wall with a temperature of 120 K, a height of 1.4 AU, and maximum grain size a max =5 µm. For neither the 20 AU nor the 26 AU model does the wall contribute significant continuum emission at the wavelengths and spatial scales probed by our data. The main discrepancy between the fits to the SED and the millimeter visibilities occurs between wavelengths of ∼20-40 µm where the 20 AU hole model overpredicts the flux. However, the SED morphology in this region is likely sensitive to the properties of the wall at the inner disk edge, which are not well known and are not constrained by our data. It is also possible that the composition of the grains, particularly whether the silicate and graphite form composite grains or are separated, can affect the temperature and therefore the mid-IR morphology of the wall component of the SED (D'Alessio et al. 2009, in prep). Since our focus is on the interferometric millimeter-wavelength data, we adopt the model with a 20 AU inner hole for the remainder of the analysis. Figure 1 compares this model with the data in the image plane (center panel) and in the visibility domain (red line in the right panel). The agreement is excellent, and the residuals are less than 3σ within the 2 ′′ box shown.
The flux density of the eastern peak of the 860 µm image is 50 mJy beam −1 , while that of the western peak is 59 mJy beam −1 . The corresponding peaks in the model images are 49 and 50 mJy beam −1 , respectively. Given the rms noise of 3.5 mJy beam −1 , these values are consistent with no flux difference and hence axially symmetric emission from the inner disk edge. The positional accuracy of the data and knowledge of the stellar proper motion are insufficient to determine whether or not the emission peaks are equally offset from the star. This result may be contrasted with the strong asymmetries observed by Brown et al. (2008) in their observations of the inner hole in LkHα 330, although these data are missing short antenna spacings present in the GM Aur data that may dilute asymmetries. However, as in the case of LkHα 330, we find that the GM Aur continuum presents a sharp contrast in brightness between the inner and outer disk, reflected by the null in the visibility function and the strong agreement between the data and the model containing an inner hole. The 1.1 × 10 −11 M ⊙ of dust within the central hole in the model implies a reduction in the mass surface density of small grains of at least 6 orders of magnitude at 1 AU relative to a continuous model of the dust disk, indicating that the data are consistent with an inner disk region that is essentially completely evacuated of small grains.
Comparison with CO Observations
In order to compare the gas and dust properties of the GM Aur disk, we used the SED-based model described above to generate predicted CO J=3−2 and J=2−1 emission. We assume that gas and dust are well mixed, with a uniform gas-to-dust mass ratio of 100 (neglecting the complication of dust settling) and a constant CO abundance relative to H 2 of 10 −6 , which is required to reproduce the peak CO J=2−1 flux. We also add microturbulence with a FWHM of 0.17 km s −1 throughout the outer disk, as derived by Dutrey et al. (1998). This is comparable to the 0.18 km s −1 spectral resolution of the data and does not affect our determination of the disk geometry. Due to the position angle differences ev- ident between the continuum emission in Fig. 1 and the central channels in Fig. 2, we also adjust the position angle to 51 • (as in Dutrey et al. 1998). Finally, we note that with an outer radius of 300 AU, the continuum model severely underpredicts the CO emission at large radii, as expected for a model with a sharp cutoff at its outer edge (Hughes et al. 2008). We therefore extrapolate the model to 525 AU to match the spatial extent of the CO emission (Dutrey et al. 1998). While this larger CO model no longer matches perfectly the continuum emission for the shortest baselines, based on the prediction assuming a constant gas-to-dust mass ratio, it retains the kinematic and thermal structure of the smallscale continuum model. In order to consistently solve for the level populations and generate sky-projected images in the CO lines, we use the Monte Carlo radiative transfer code RATRAN (Hogerheijde & van der Tak 2000). We then use the MIRIAD task uvmodel to sample the model image at identical spatial frequencies to those present in our interferometric CO data set. Figure 6 compares the predicted CO emission from the extended SED model (right) with the observed emission from the GM Aur disk (left) for the J=2−1 (top) and J=3−2 transitions. It is clear that the velocity pattern in the disk is consistent with Keplerian rotation (as previously noted by Koerner et al. 1993;Dutrey et al. 1998), and that the SED-based model is capable of reproducing the basic morphology of the CO emission.
The primary difference between data and model is the CO J=3-2/J=2-1 line ratio: the disk structure model that reproduces the peak flux density of the J=2−1 transition underpredicts the peak J=3−2 flux by 30%. This difference may be attributed to a ∼10 K difference in temperature between the gas and dust in the upper layers of the GM Aur disk that are probed by these optically thick CO lines. While the vertical temperature gradient of the dust in the model is fixed by the SED, a relative increase in gas temperature would populate the upper rotational transition of the molecule more efficiently and produce more J=3−2 emission relative to J=2−1. The temperature and the CO abundance are also somewhat interdependent, since the CO abundance sets the vertical location, and therefore the temperature, . Open pentagons represent millimeter observations obtained from Andrews & Williams (2005); Beckwith & Sargent (1991); Dutrey et al. (1998); Kitamura et al. (2002); Koerner et al. (1993); Looney et al. (2000); Rodmann et al. (2006); Weintraub et al. (1989). Closed pentagons are from this work. The final model (solid line) includes the following components: stellar photosphere (dotted line), optically thin dust region (long-dashed line), disk wall (short-long dashed line), outer disk (dot-dashed line). The peak at ∼1µm from the wall emission is due to scattered light. While the 20 AU model does not fit the IRS spectrum as well between ∼20-40 µm as the 26 AU model, it reproduces the millimeter continuum emission very well at both 860 µm and 1.3 mm (Fig. 1).
of the τ =1 surface from which most of the line emission originates. An increase in temperature would therefore also vary the anomalously low CO/H 2 ratio necessary to reproduce the J=2−1 flux. Such line ratio differences have been previously observed in the disk around TW Hya (Qi et al. 2004(Qi et al. , 2006, and may be due to additional heating of gas in the upper disk by such processes as xray and UV irradiation, dissociative or mechanical heating (e.g. Glassgold et al. 2004;Kamp & Dullemond 2004;Nomura et al. 2007) Nevertheless, while the flux levels vary between the data and model prediction, the similarity in morphology makes it clear that the overall disk structure is consistent between the molecular gas traced by CO and the model based on dust traced by continuum emission and the SED. The only other significant difference between the two is in the position angle of the emission, which differs by ∼11 • . The implications of this result are discussed in §5.2 below.
DISCUSSION
Inner Disk Clearing
The resolved millimeter continuum observations of the GM Aur system are consistent with the prediction from the SED model. Models of the observed 860 µm and 1.3 mm maps in conjunction with the SED and Spitzer IRS spectrum, give a value of ∼20 AU for the extent of this inner cleared region. The inference of an inner hole of this size from the SED and resolved millimeter visibilities is consistent with recent millimeter-wave observations of rotational transitions of CO isotopologues from the GM Aur disk that provide spectroscopic evidence for a diminished density of cold CO within 20 AU (Dutrey et al. 2008). However, other observations indicate that this region cannot be entirely devoid of gas. Salyk et al. (2007) detect CO rovibrational emission originating from hot gas at radii near ∼ 0.5 AU, from which they infer a total gas mass in the inner disk of ∼ 0.3 M ⊕ . Measurements of the Hα linewidth imply an accretion rate of ∼ 10 −8 M ⊙ yr −1 (White & Ghez 2001;Ingleby & Calvet 2009); accretion at this rate requires a steady supply of gas from the inner disk. The SED model also requires 3 × 10 −4 lunar masses of dust in the inner disk, to account for the 10 µm silicate feature and slight near-to mid-IR excess .
A wide variety of mechanisms has been invoked to explain the low optical depth of the central regions of transition disks (see e.g. Najita et al. 2007, and references therein), each with different implications for planet formation and the process of evolution between the primordial and debris disk stages. The available measurements of properties of the inner hole in the GM Aur disk allow us to evaluate the plausibility of each mechanism as the driver of disk clearing in this system.
Grain Growth -The agglomeration of dust into larger particles should proceed faster in central regions where relative velocities of particles are faster and surface densities are higher. This would produce a drop in opacities associated only with the inefficiency of emission of large grains at the observed wavelengths (e.g. Strom et al. 1989;Dullemond & Dominik 2005). However, this process is inconsistent with the clearing of CO from the central region observed by Dutrey et al. (2008), as grain growth should proceed without diminishing the gas density. Grain growth is also somewhat inconsistent with the steep submillimeter slope observed by Rodmann et al. (2006) for the GM Aur system. The value inferred for the millimeter wavelength slope α of 3.2 is the steepest in their sample of ten T Tauri stars, and is typical of a grain population that has undergone little growth, with grain The outer radius of the model has been extended to 525 AU to reproduce the extent of the molecular gas emission (see §4.2 for details). The CO morphology is consistent with the SED-based model, with the exception of the line ratio: the model that best reproduces the peak flux of the CO J=2-1 line underpredicts the CO J=3-2 brightness by 30%.
size a max ≤ 1 mm. Furthermore, the original SED model and the submillimeter visibilities both independently indicate a sharp decrease in surface density or opacity near 24 AU, while grain growth and dust settling are predicted to be a continuous process and so should display a more gradual transition between the inner and outer disk (Weidenschilling et al. 1997;Dullemond & Dominik 2005).
Photoevaporation -Another proposed process to generate inside-out clearing of protoplanetary disks is photoevaporation via the "UV switch" mechanism (Clarke et al. 2001). In this scenario, high-energy photons from the star heat the upper disk layers, allowing material to escape the system at a rate that gradually diminishes the disk mass, while most of the disk mass drains onto the star via viscous accretion (e.g. Hartmann et al. 1998). Once the photoevaporation rate matches the accretion rate near 1 AU and prevents resupply of material from the outer disk, the inner disk will decouple and drain onto the star within a viscous timescale, leaving an evacuated central region surrounded by a lowmass outer disk that will then rapidly disperse. As noted by Alexander & Armitage (2007), the properties of the GM Aur system are inconsistent with a photoevaporative scenario because the large mass of the outer disk should still be sufficient to provide a substantial accretion rate to counteract the photoevaporative wind. Furthermore, the measured accretion rate is high enough that within the framework of the photoevaporation scenario, it would only be observed during the brief period of time when the inner disk was draining onto the star. Photoevaporation may yet play a role in clearing the outer disk of its remaining gas and dust, but it cannot explain the current lack of inner disk material.
Inside-Out MRI Clearing -The magnetorotational instability operating on the inner disk edge may also drive accretion and central clearing, although it should be noted that this is purely an evacuation mechanism: it can only take hold after the generation of a gap by some other means. Nevertheless, given the creation of a gap, MRI clearing is predicted to operate in systems like GM Aur whose outer disks are still too massive for photoevaporation to dominate (Chiang & Murray-Clay 2007). The observed depletion of CO interior to 20 AU radius (Dutrey et al. 2008) is consistent with this theory, which predicts a total gas mass depletion of order 1000× interior to the rim radius relative to the extrapolated value from the outer disk power law fit, normalizing to the total disk mass of 0.16M ⊙ . This theory is consistent with the substantial accretion rate of the GM Aur system, yielding a value of α of 0.005, only slightly greater than the derived value of 0.002 from the model. Salyk et al. (2007) estimate a gas-to-dust ratio of ∼ 1000 in the inner disk, roughly 10 times greater than that of the outer disk, which is consistent with the prediction of the inside-out MRI evaporation scenario that flux from the star should promote blowout of small dust grains by radiation pressure, substantially clearing the inner disk of dust even as the gas continues to accrete onto the star. However, it is difficult to reconcile this with the substantial population of µm-size grains that must be present in the inner disk to account for the 10 µm silicate feature in the IRS spectrum. It is also important to consider the source of the requisite initial gap in the disk.
Binarity -The dynamical influence of an unseen stellar or substellar companion would also cause clearing of the inner disk. A notable example is the recent result by Ireland & Kraus (2008) demonstrating that the inner hole in the transition disk around CoKu Tau/4 is caused by a previously unobserved companion. There are relatively few constraints on the multiplicity of GM Aur at the < 20 AU separations relevant for the inner hole. Radial velocity studies with km s −1 precision do not note variability (Bouvier et al. 1986;Hartmann et al. 1986), ruling out a close massive companion. As Dutrey et al. (2008) discuss, the stellar temperature and dynamical mass from the disk rotation combined with the H-band flux place an upper limit of ∼0.3 M ⊙ on the mass of a companion. Interferometric aperture-masking observations with NIRC2 that take advantage of adaptive optics on the Keck II telescope place an upper limit of ∼40 times the mass of Jupiter on companions with separations between 1.5 and 35 AU from the primary (A. Kraus and M. Ireland, private communication). The presence of hot CO in the central 1 AU of the system (Salyk et al. 2007) and the high accretion rate, undiminished relative to the Taurus median, also argue against the presence of a massive close companion. A stellar companion is therefore an unlikely origin for the central clearing in the GM Aur system.
Planet-Disk Interaction -Perhaps the most compelling mechanism for producing a transition disk is the dynamic clearing of material by a giant planet a few times the mass of Jupiter. The opening of gaps and holes in circumstellar disks has long been predicted as a consequence of giant planet formation (e.g. Lin & Papaloizou 1986;Bryden et al. 1999). Some simulations have shown that inner holes may in fact be a more common outcome than gaps as angular momentum transfer mediated by spiral density waves can clear the inner disk faster than the viscous timescale (Varnière et al. 2006;Lubow & D'Angelo 2006). The planet-induced clearing scenario was considered in detail for GM Aur by Rice et al. (2003) and found to be globally consistent with the observed properties of the system (although their estimate of the inner hole radius is based on pre-Spitzer SED information). This mechanism naturally explains the diminished but persistent accretion rates and presence of small dust grains through two predictions of models of planet-disk interaction: (1) filtration of dust grains according to size is expected at the inner disk edge, leading to a dominant population of small grains in the inner disk (Rice et al. 2006); and (2) a sustained reduction in accretion rate to ∼ 10% of that through the outer disk is predicted as the giant planet begins to intercept most of the accreting material (Lubow & D'Angelo 2006). These effects combined may also explain the enhanced gas-to-dust ratio in the inner disk. A planet-induced gap could also serve as a catalyst for inside-out MRI clearing Chiang & Murray-Clay (2007).
Given the observed 20 AU inner disk radius and the scenario of clearing via dynamical interaction with a giant planet, it is possible to make a simple estimate of the distance of the planet from the star. The width of a gap opened by a planet is approximately 2 √ 3 Roche radii (Artymowicz 1987), and simulations show that the minimum mass necessary to open a gap is of order 1 Jupiter mass (e.g. Lin & Papaloizou 1993;Edgar et al. 2007). If the outer edge of the planet-induced gap coincides with the 20 AU inner disk radius (with the portion of the disk interior to the planet cleared via spiral density waves or the MRI), then a companion between 1 and 40 times the mass of Jupiter would be located between 11 and 16 AU from the star. The influence of a planet carving out an inner cavity in the dust distribution is therefore a plausible scenario, bolstered by recent results demonstrating that a planet is responsible for dynamical sculpting of dust in the much older Fomalhaut system (Kalas et al. 2008).
Evidence for a Warp?
While the model comparison in §4 above shows that CO emission from the disk is globally consistent with Keplerian rotation, the 11 • difference in position angle between the continuum data and the two CO data sets is significant at the ∼ 5σ level, and may indicate some kinematic deviation from pure Keplerian rotation in a single plane. Changes in position angle with physical scale are commonly interpreted as warps in the context of studies of galaxy dynamics (e.g. Rogstad et al. 1974); it may be that the change in position angle in the GM Aur disk indicates a kinematic warp.
The possibility of a warp or other deviation from Keplerian rotation was discussed by Dutrey et al. (1998), although their discussion was based on possible isophote twisting observed in integrated CO J=2−1 contours. We observe no such isophote twisting in the integrated CO J=2-1 or J=3-2 emission presented here (Fig. 4), although this determination may be influenced by the differing baseline lengths and beam shapes in the respective interferometric data sets. Instead, we observe deviations from the expected position angle only in the rotation pattern of the resolved CO emission, which is reflected in the isovelocity contours of Fig. 4. This position angle change does not appear to be related to the cloud contamination, as it is more clear in the less-contaminated CO J=3−2 data set. In order to test whether the position angle of the true brightness distribution might have been altered by incomplete sampling of the data in the Fourier domain, we generated a model of the disk at a position angle of 64 • , consistent with that measured independently for the two continuum data sets. We then fit the position angle by χ 2 minimization as in §4.1 above. With this method, after sampling with the response at the spatial frequencies in the CO J=3-2 data set, we recover the position angle to within less than a degree of the input model. This is to be expected, since the χ 2 fitting procedure takes into account the interferometer response when fitting for the position angle. The position angle change is therefore robust independent of beam convolution effects.
In order to cause a change in position angle on physical scales between those probed by the continuum (∼ 30 AU) and the CO (∼ 200 AU), a warp would have to occur at a size scale of order 100 AU. The most natural explanations for the presence of a warp in a gas-rich circumstellar disk include flybys and perturbations by a planet or substellar companion. A simple estimate of the timescale of flyby interactions is τ = 1/(N πb 2 σ), where N is the number density of stars, b is the approach distance, and σ is the velocity dispersion. Assuming typical values for Taurus, including a stellar density of ∼10 pc −3 (e.g. Gomez et al. 1993) and velocity dispersion of 0.2 km s −1 (Kraus & Hillenbrand 2008), the timescale for interactions at distances of ∼1000 AU, sufficient to cause significant perturbations at Oort Cloud radii (Scholl et al. 1982), is of order 1 Gyr. Since the results of a one-time perturbation would likely damp in a few orbital periods (10 3 yr at a distance of 100 AU), such an interaction is statistically unlikely. However, it should be noted that a recent interaction might have been capable of producing an extended feature like the "blue ribbon" observed in scattered light by Schneider et al. (2003).
The influence of a massive planet or substellar companion has been investigated as the origin of warps observed in gas-depleted debris disks, including β Pic (Mouillet et al. 1997) and HD 100546 (Quillen 2006). However, there is a dearth of theoretical investigation into the plausibility of warps caused by planetary systems in gas-rich disks more closely analogous to the GM Aur system. Since the warp in the GM Aur disk must occur between the Hill sphere of the putative planet and the ∼200 AU resolution of the CO line observations, it is plausible that the warp could be due to the gravitational influence of the same body responsible for evacuating the inner disk. A theoretical inquiry into this possibility would be useful, but is beyond the scope of this paper.
CONCLUSIONS
Spatially resolved observations in millimeter continuum emission, obtained using the SMA at 860 µm and PdBI at 1.3 mm, reveal a sharp decrease in optical depth near the center of the GM Aur disk. Simple estimates of the extent of this region, based on the separation of peaks in the continuum images and the position of the null in the visibility functions in Fig. 1, are consistent with the inner hole radius of 24 AU derived by Calvet et al. (2005) using disk structure models to fit the SED. No significant azimuthal asymmetry is detected in the continuum emission.
Refined versions of the SED-based model of Calvet et al. (2005) show that the data are very well reproduced by a disk model with an inner hole of radius 20 AU. This model overpredicts the broadband SED flux in the 20-40 µm wavelength regime, but this region of the spectrum likely depends on the properties of the wall at the inner disk edge, which are poorly constrained by available data.
CO emission in the J=3−2 and J=2−1 transitions confirms the presence of a disk with kinematics consistent with Keplerian rotation about the central star, but at a position angle offset from the continuum by ∼11 • . The morphology of the CO emission is broadly consistent with the SED model, but with a larger CO J=3-2/J=2-1 line ratio than predicted for the SED model. This is a likely indication of additional gas heating relative to dust in the upper disk atmosphere.
Given the observed properties of the GM Aur system, photoevaporation, grain growth, and binarity are unlikely physical mechanisms for inducing a sharp decrease in opacity or surface density at the disk center. The inner hole plausibly results from the dynamical influence of a planet on the disk material, with the inner disk possibly cleared by spiral density waves or the MRI. While a recent flyby is statistically unlikely, warping induced by a planet could also explain the difference in position angle between the continuum and CO data sets.
Fig. 1 .
1-Continuum emission from the disk around GM Aur at wavelengths of 860 µm observed with the SMA (top) and 1.3 mm observed with PdBI (bottom). The data are displayed in both the image (a) and Fourier (b) domains. In the image domain (a), the observed brightness distribution at each wavelength (left) is compared with the model prediction (center; see §4.1 for model details), and the residuals are also shown (right). In the data and model frames, the contours are[3, 6, 9, ...]× the rms noise (3.5 mJy beam −1 at 860 µm and 0.75 mJy beam −1 at 1.3 mm). In the residual frame, the contours start at 2σ and are never greater than 3σ. The synthesized beam sizes and orientations for the two maps are, respectively, 0.′′ 30 × 0. ′′ 24 at a position angle of 34 • and 0. ′′ 43 × 0. ′′ 30 at a position angle of 35 • . Two sets of axes are shown: the dotted line indicates the position angle of the double-peaked continuum emission, while the solid line indicates the best-fit position angle of the CO emission (see §3.2 for details). In the Fourier domain (b), the visibilities are averaged in bins of deprojected u-v distance from the disk center, and compared with the model prediction (red line). The inner hole in the GM Aur disk is clearly observed at both wavelengths, as a double-peaked emission structure in the image domain or as a null in the visibility function in the Fourier domain.
Figures 2-4 display the new SMA observations of CO emission from the GM Aur disk. Figures 2 and 3 show channel maps with contours starting at twice the rms noise level and increasing by factors of √ 2, while Figure 4 displays the zeroeth (contours)
Fig. 2 .
2-Channel maps of CO J=3−2 emission from the GM Aur disk. Contour levels start at 0.61 Jy (2 times the rms noise) and increase by factors of √ 2. LSR velocity is indicated by color and quoted in the upper right of each panel. The synthesized beam (2. ′′ 2×1. ′′ 9 at a PA of 14 • ) and physical scale are indicated in the lower left panel. Two sets of axes are shown: the dotted line indicates the position angle of the double-peaked continuum emission, while the solid line indicates the best-fit position angle of the CO emission.
Fig. 3 .
3-Channel maps of CO J=2−1 emission from the GM Aur disk. Contour levels start at 0.17 Jy (2 times the rms noise) and increase by factors of √ 2. LSR velocity is indicated by color and quoted in the upper right of each panel. The synthesized beam (2. ′′ 1×1. ′′ 4 at a PA of 56 • ) and physical scale are indicated in the lower left panel. Two sets of axes are shown: the dotted line indicates the position angle of the double-peaked continuum emission, while the solid line indicates the best-fit position angle of the CO emission. Cloud contamination is evident in at least the central four channels.
Fig. 4 .
4-Zeroeth (contours) and first (colors) moment map of the CO J=3−2 (top) and J=2−1 (bottom) data in Figs. 2 and 3. The dotted line indicates the position angle of the double-peaked continuum emission, while the solid line indicates the best-fit position angle of the CO emission. The zeroeth moment contours are well aligned with the latter, while the isovelocity contours of the first moment map are more consistent with the former. Cloud contamination is evident in the CO J=2−1 map in the northwest region along the disk minor axis.
Fig. 5 .
5-Model of the SED of GM Aur using the method of D'Alessio et al. (2005, 2006). The final model of the SED alone has an inner disk hole of 26 AU (left), while the model that best reproduces the resolved millimeter-wavelength visibilities has a hole of radius 20 AU (right). See §4 for model details. We show optical (open circles; Kenyon & Hartmann 1995), 2MASS (closed circles), IRAC (open squares; Hartmann et al. 2005), and IRAS (closed squares; Weaver & Jones 1992) data and a Spitzer IRS spectrum
Fig. 6 .
6-Position-velocity diagram comparing the molecular line observations (left) with the predicted (right) CO J=2-1 (top) and CO J=3-2 (bottom) emission from the GM Aur disk, assuming a standard gas-to-dust mass ratio of 100. The plots show the brightness as a function of distance along the disk major axis, assuming a position angle of 51 • . Contours are [2,4,6,...] times the rms flux density in each map (0.17 and 0.61 Jy beam −1 , respectively). The dotted line shows the expected Keplerian rotation curve for a star of mass 0.84 M ⊙ .
TABLE 1
1Observational parameters for GM AurContinuum
TABLE 2
2Stellar and Model Properties Star 1 L * (L ⊙ ).................. 1.1 R * (R ⊙ ).................. 1.5 T * (K).................. 4730 M (M ⊙ yr −1 ).................. 7.9 × 10 −9 Distance (pc).................. 140 A V .................. 1.2 Inclination (deg).................. 55 Optically Thick Wall 2 R wall (AU ).................. 20 (26) a min (µm) 1 .................. 0.005 amax (µm) 3 .................. wall (K) 3 .................. 120 (130) z wall (AU ) 3,4 .................. AU ) 1 .................. 300 ǫ 3 .................. 0.5 α 3 .................. 0.002 M d (M ⊙ ).................. 0.165 (0.25)
The authors would like to thank the IRAM staff, particularly Roberto Neri, for their help with the observations and data reduction. We thank Lee Hartmann for helpful discussions in the early stages of this project. Partial support for this work was provided by NASA Origins of Solar Systems Program Grant NAG5-11777. A. M. H. acknowledges support from a National Science Foundation Graduate Research Fellowship. Support for S. M. A. was provided by NASA through Hubble Fellowship grant #HF-01203-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. N. C. acknowledges support from NASA Origins Grant NNG05GI26G and JPL grant AR50406. P. D. acknowledges grants from CONACyT, México. J. P. W. acknowledges support from NSF grant AST-0808144.
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| []
|
[
"Generation For Adaption: A GAN-Based Approach for Unsupervised Domain Adaption with 3D Point Cloud Data",
"Generation For Adaption: A GAN-Based Approach for Unsupervised Domain Adaption with 3D Point Cloud Data"
]
| [
"Junxuan Huang [email protected] \nState University of New York at Buffalo\n\n",
"Junsong Yuan [email protected] \nState University of New York at Buffalo\n\n",
"Chunming Qiao [email protected] \nState University of New York at Buffalo\n\n"
]
| [
"State University of New York at Buffalo\n",
"State University of New York at Buffalo\n",
"State University of New York at Buffalo\n"
]
| []
| Recent deep networks have achieved good performance on a variety of 3d points classification tasks. However, these models often face challenges in "wild tasks" where there are considerable differences between the labeled training/source data collected by one Lidar and unseen test/target data collected by a different Lidar. Unsupervised domain adaptation (UDA) seeks to overcome such a problem without target domain labels. Instead of aligning features between source data and target data, we propose a method that uses a Generative Adversarial Network (GAN) to generate synthetic data from the source domain so that the output is close to the target domain. Experiments show that our approach performs better than other state-of-theart UDA methods in three popular 3D object/scene datasets (i.e., ModelNet, ShapeNet and ScanNet) for cross-domain 3D object classification. | null | [
"https://arxiv.org/pdf/2102.07373v4.pdf"
]
| 232,074,645 | 2102.07373 | 044fd3ebc340ef3d5548daf4e7faab56819fec58 |
Generation For Adaption: A GAN-Based Approach for Unsupervised Domain Adaption with 3D Point Cloud Data
Junxuan Huang [email protected]
State University of New York at Buffalo
Junsong Yuan [email protected]
State University of New York at Buffalo
Chunming Qiao [email protected]
State University of New York at Buffalo
Generation For Adaption: A GAN-Based Approach for Unsupervised Domain Adaption with 3D Point Cloud Data
Recent deep networks have achieved good performance on a variety of 3d points classification tasks. However, these models often face challenges in "wild tasks" where there are considerable differences between the labeled training/source data collected by one Lidar and unseen test/target data collected by a different Lidar. Unsupervised domain adaptation (UDA) seeks to overcome such a problem without target domain labels. Instead of aligning features between source data and target data, we propose a method that uses a Generative Adversarial Network (GAN) to generate synthetic data from the source domain so that the output is close to the target domain. Experiments show that our approach performs better than other state-of-theart UDA methods in three popular 3D object/scene datasets (i.e., ModelNet, ShapeNet and ScanNet) for cross-domain 3D object classification.
Introduction
Deep learning with 3D point cloud data has achieved significant outcomes in different tasks. Classification is the most fundamental task [26] and plays a central role in a number of modern applications, e.g., robots, self-driving cars or virtual reality. Despite their impressive success, a Deep Neural Network (DNN) requires a large amount of labeled point cloud data for training. Point clouds data can be captured with different sensors (e.g. 64 beam lidar, 128 beam lidar and RGB-D camera) which produces different 3D sampling patterns. In addition, in real-world where objects are scanned from LiDAR, it also happens that some parts are lost or occluded (e.g.chairs lose legs) while some datasets are generated form 3D polygonal models with a smooth and uniform surface. As Fig. 1 shows, apparently, those datasets have different geometric feature and data distribuations. Which may leads to a performance drop when we adapting a classification model. This issue is known as domain gap. To solve the performance dropping issue, labeling every 3D object in the unseen dataset is the most straightforward solution. But different with 2D object dataset. During labeling, people need to rotate several times and look through different angles to identify an object. It is timeconsuming and expensive. The objective of domain adaptation (DA) is to leverage rich annotations in a source domain to achieve a good performance in a target domain having few annotations. Some of the existing DA methods have focused on mapping features into a shared subspace or minimizing instance-level distances such as MMD [3], CORAL [35]. Other adversarial-training DA methods, like DANN [11], ADDA [37], MCD [30] have tried to use adversarialtraining to select domain invariant features during training so that the trained model could perform better in the target domain. The main idea of domain adversarial training is that if the feature representation is domain invariant, a classifier trained on the source domain's features will operate on the target domain as well. However, the assumption about the presence of domain invariant features may not always be true. As an alternative, PointDan [28] designed a Self-Adaptive Node Construction for aligning 3D local fea- Figure 2. The distribution of points cloud in target object (right) looks much uniform than source object (left), and our synthetic output keep the shape of the source domain object, but the distribution of points are more similar to the target domain object tures with points cloud data. A recent work [42] built a DA method that upsampled the source domain data into a canonical domain and trained a semantic segmentation network over the canonical domain.
In this paper, we tackle the problem of unsupervised domain adaptation (UDA), where the target domain is entirely unlabelled. Our approach chooses to generate synthetic data from a source domain according to the target domain data pattern. Unlike other adversarial-training DA methods, we fully utilize the advantage of adversarial-training by generating synthetic data while keeping the label of the source domain. As Fig. 2 shows, our model generates a synthetic sofa with a very similar shape to the sofa from the source dataset and its point distribution looks much closer to the target dataset object compared with the sofa from the source dataset. Our model architecture consists of four parts: generator,latent reconstruct module, discriminator, and feature encoder/decoder. We have two inputs for the generator, in addition to the encoded feature of input points cloud, we also have a latent code z sampled from standard Gaussian distribution as a condition input. Because of that, we can generate multi possible outputs from a single input, and this can improve the quality of output object from the generator as shown in [39]. Latent reconstruct module recovered the latent code z during training for calculating latent reconstruct loss. For the encoder, we use Point transformer structure [16] and decoder uses a 3-layer MLP (multilayer perceptron). Similar to D2GAN [25], we have an additional discriminator, a classifier to restrict the output from generator. Extensive experiments that involve the use of different pairs of a dataset to perform cross-domain 3D object classification tasks show that our approach's predicting accuracy is 2.6% higher than PointDan on average, 2.8% higher than MCD, and 4.6% higher than ADDA.
We summarize our contributions as follows:
(1) A new UDA model for 3D object classification which different from other methods in this task. We generate synthetic data for training classification models instead of searching common feature space or aligning features. Based on that, we can visualize the effect of domain adaption and indicate which part of the object is transferred
(2) A novel design that adding a latent space reconstruct module and using an additional discriminator to restrict the generator's output which further improves the adaptation.
(3) We perform extensive experimental tests on various 3D classification datasets and show that the proposed approach can achieve competitive results compared to the state-of-the-art methods.
Related Work
3D Vision Understanding
3D vision has various data representations: multi-view, voxel grid, 3D mesh and point cloud data. Various Deep networks have been designed to deal with the above different formats of 3D data [33,24,43,9]. Point cloud can be directly obtained by LiDAR sensors, and contain a lot of 3D spatial information. PointNet [26], which takes advantage of a symmetric function to process the unordered point sets in 3D, is the first deep neural networks to directly deal with point clouds. Later research [27] proposed to stack Point-Nets hierarchically to model neighborhood information and increase model capacity. Several subsequent works considered how to perform convolution operations like 2D images on point clouds.
One main approach is to convert a point cloud into a regular voxel array to allow convolution operations. Tchapmi et al. [36] proposed SEGCloud for pointwise segmentation. It maps convolution features of 3D voxels to point clouds using trilinear interpolation and keeps global consistency through fully connected conditional random fields. Atzmon et al [2] presented the PCNN framework with extension and restriction operators to map between point-based representation and voxel-based representation by performing volumetric convolution on voxels to extract point features. MCCNN by Hermosilla et al. [17] allows non-uniformly sampled point clouds by treating convolution as a Monte Carlo integration problem. Similarly, in PointConv proposed by Wu et al. [40], 3D convolution was performed through Monte Carlo estimation and importance sampling. Liu et al. [21] took advantages of volumetric 3D convolution and point-based method by proposing point-voxel convolution.
Unsupervised Domain Adaptation (UDA)
Early work on UDA mainly performed reweighing [34,44] or resampling [13] of the source-domain examples to match the target distribution. Besides, there has been a prolific body of works on learning domain-invariant representations. For instance, Fernando et al [10] tried to use subspace alignment and Gong et al [14] performed interpolation. Adversarial training [12,37,4,32,18] tried to use a discriminator to select domain-invariant features. Other Figure 3. A 3D classification model trained on source dataset Scannet (left) make a wrong prediction on object from target dataset Modelnet (right) due to domain gap between two datasets methods such as Correlation Alignment (CORAL) [35], Maximum Mean Discrepancy (MMD) [3,22], or Geodesic distance [15] tried to minimize the intra-class distance in a subspace simultaneously. However, there have been only a few works on domain adaptation for 3D point clouds. Among them. Rist et al. [29] proposed that dense 3D voxels are preferable to point clouds for sensor-invariant processing of LiDAR point clouds. Salah et al. [31] proposed a CycleGAN approach to the adaptation of 2D bird's eye view images of LiDAR between synthetic and real domains. Wu et al. [38] used geodesic correlation alignment between real and synthetic data. Qin et al. [28] designed a Self-Adaptive Node Construction for aligning 3D local features with points cloud data.
Proposed GAN-based DA method
As Fig. 3 shows, when we directly use a classifier trained on the source dataset over the target dataset, some objects may be misclassified due to the domain gap. A sofa from Modelnet may be classified as a bookshelf if we use a classifier trained on Scannet without any domain adaption methods. To overcome the domain gap between different point cloud datasets, we propose a generative model, which takes point cloud from the source dataset and target dataset as input first and then tries to maintain the shape of an object while keeping the label information when generating the synthetic object according to the target domain dataset's distribution. This adaption process's objective is that classifier could learn the distribution of point cloud in the target domain and perform better in the test of the target dataset.
As Fig. 5 shows in the training source domain object and target domain object will go through a shared encoder. Like most GANs, the encoded features will be sent to a discriminator and, the discriminator tries to distinguish features from source domain or target domain. Following the idea in [39] of adding multimodal information to the model, we also have Gaussian samples z for latent condition input to the generator. To force the generator to use the Gaussian samples z , we introduce a VAE encoder to recover z from the synthetic output. In addition, inspired by [20], in order to enhance the quality of output object from G, we have an additional discriminator, a classifier C in training the model.Below, we explain our model in more details.
Definition and Notations
We consider an unsupervised domainadaptation (UDA) setting, A point cloud from a source domain is represented
as (X s {x s i } i=1...N , y s ), where x s i ∈ R 3
is a 3D point and N is the number of points in the point cloud; y ∈ {1, 2, ..., k} is the ground-truth label, where k is the number of classes. In UDA, we have access to a set of labeled LiDAR point clouds, from the source domain and a set of unlabeled LiDAR point clouds a set of unlabeled LiDAR point clouds
(X t {x t i } i=1...N ) in a target domain.
It is assumed that two domains are sampled from the distributions P s (X s ) and P t (X t ) while the distribution P s = P t . We use classification as the task of domain adaption and denote the classification network as C θ (·)
{C θ,j |j = 1...k}, whose input is a point cloud X and output is a probability vector C θ (X). our approach tend to generate synthetic object X s from X s and for the generative generative adversarial network we have encoder E, decoder D, generator G and discriminator F. The process is denoted as X = D(G(E(X)))
Learn mapping of point cloud to latent space
We obtain feature vector X s of input point cloud by training an autoencoder. As Fig. 4 shows the encoder E consists of 4 self-attention layer [16]. The object X s from the source dataset is encoded as feature vector X s and decoder D reconstructs objectX from the latent feature vector X s . The encoder and decoder are trained with the reconstruction loss, and we choose Earth Mover's Distance (EMD) to measure the distance between reconstructed objectX and input object X s .
L recon = d EMD (X s , D(E(X s ))),(1)
As for the object X t from the target dataset, instead of training another autoencoder for the target domain, we directly feed X t to E because [7] showed that doing this would achieve a better performance in subsequent adversarial training.
Latent space reconstruction
Like most of the generative adversarial networks, we set a min-max game between generator and discriminator. The generator is trained to fool the discriminator so that the discriminator fails to tell if the latent vector comes from the source domain X s or target domain X t . Use Earth Mover's Distance(EMD) to restrict the synthetic outputs can make it close to the input's shape, but we do not want the synthetic object to having the exact shape of input. Because we are building a synthetic dataset which means the variety is also significant. So we bring a random sampled variable z into Formally, the latent representation of the source domain input x s = E AE (X s ), along with random sampled variable z from a standard Gaussian distribution N (0, I). Thus, a latent representationx t = G(x s , z) will generated by generator. Then discriminator will try to distinguish betweenx t and x t = E AE (X t ). The mode encoder E z which is the encoder part of VAE will encode the synthetic outputX t , which is decoded from the latent representa-tionX t = D AE (x t ), to reconstruct the conditional input
z = E z (X t ).
Train generator with classification loss
The job of discriminator F is to only distinguish the latent representation from the source domain or the target domain, and there is no guarantee that the synthetic output will be the same class as input object, though we use a reconstruction loss during training to make sure the output point set will be close to the input point set in Earth Mover's Distance (EMD). So we add a discriminator to utilize label information from the source dataset fully. The classifier C will predict the class of output point set and compare it with the ground truth label in the source dataset. The loss will be backward to the generator to encourage it generates synthetic objects in the same class as input objects from the source dataset.
Overall loss function and training
To optimize GAN's output object quality, we set a minmax game between the generator, the discriminator, and the classifier. Given training examples of source domain object X s , and Gaussian samples z, we seek to optimize the following training losses over the generator G, the discrim-inator F, the encoder E z and the classifier C:
Adversarial loss. For trianing of the generator and discriminator we add the an adversarial loss and we implement least square GAN [23] for stabilizing the training. Hence, the adversarial losses will be minimized for the generator and the discriminator are defined as:
L GAN F = E X t ∼p(X t ) [F (E AE (X t )) − 1] 2 + E X s ∼p(X s ) [F (E AE (X s )] 2 (2) L GAN G = E X s ∼p(X s ),z∼p(z) [F (G(E AE (X s ), z)) − 1] 2 ,(3)
where X t ∼ p(X t ), X s ∼ p(X s ) and z ∼ p(z) denotes samples drawn from the set of complete point sets, the set of partial point sets, and N (0, I).
Reconstruction loss. To make the output object similar to the input object in shape, we add a reconstruction loss to encourage the generator to reconstruct the input so that the output object is more likely to be considered the same class as the input object. Here we use Earth Mover's Distance (EMD) to measure the distance between the reconstructed objectX and the input object X s .
L recon G = E X s ∼p(X s ),z∼p(z) d EMD (X s , D AE (G(E AE (X s ), z)) ),(4)
Latent space reconstruction. A reconstruction loss on the z latent space is also added to force G to use the conditional mode vector z in generate output object:
L latent G,Ez = E X s ∼p(X s ),z∼p(z) [ z, E z (D AE (G(E AE (X s ), z))) 1 ],(5)
Classification loss. To restrict the class of output object, we added the classification loss to the classifier and the generator.
L Cls C = E X s ∼p(X s ),z∼p(z) [− K k=1 l k log (G(E AE (X s ), z))](6)L GAN G = E X s ∼p(X s ),z∼p(z) [− K k=1 l k log (G(E AE (X s ), z))](7)
where l k is the kth label among all classes in source dataset The full objective function for training the domain transfer network is described as:
argmin (G,Ez,C) argmax F L GAN F +L GAN G +αL recon G +βL latent G,Ez +γL Cls G ,(8)
where α, β and γ are importance weights for the reconstruction loss, the latent space reconstruction loss and classification loss respectively.
In choosing the importance weights, α controls how similar the shape of synthetic output object with the shape of the input object in 3D space, β determines how close the synthetic output object with input object in latent space, and γ influences how much probability the synthetic object will be considered as same class as input object under classifier.
Network implementation
In the experiments, each point cloud object is set to 1024 points the VAE follows [1,6]: using PointNet [26] as the encoder and a 3-layer MLP as the decoder. The autoencoder encodes a point set into a latent vector of fixed dimension |x| = 256. Similar to [16], we use the 4-layer self attention layers with MLP as encoder and 3-layer MLP for both generator G and discriminator F . Classifier is also using Pointnet structure. To train the VAE, we use the Adam optimizer [19] with an initial learning rate 0.0005, β 1 = 0.9 and train 2000 epochs with a batch size of 200. To train the autoencoder we use the Adam optimizer with an initial learning rate 0.0005, β 1 = 0.5 and train for a maximum of 1000 epochs with a batch size of 32. The parameters of the pre-trained autoencoder and variational autoencoder are fixed during GAN training. To train the GAN, we use the Adam optimizer with an initial learning rate 0.0005, β 1 = 0.5 and train for a maximum of 1000 epochs with a batch size of 50. The classifier used to train the GAN was pre-trained on the source dataset with the Adam optimizer in an initial learning rate 0.0001 for 200 epochs.
Experiments
Datasets
We verify our domain adaption model on three public point cloud datasets: shapenet [5], scannet [8] and modelnet [41]. Following the same setting as PointDA-10 [28], we choose ten common classes among three datasets. The results are shown in Table 2 Table Total M Train 106 515 572 200 889 124 465 240 680 392 4, 183 Test 50 100 100 86 100 20 100 100 100 100 856 S Train 599 167 310 1, 076 4, 612 1, 620 762 158 2, 198 5, 876 17, 378 Test 85 23 50 126 662 232 112 30 330 842 2, 492 S* Train 98 329 464 650 2, 578 161 210 88 495 1, 037 6, 110 Test 26 85 146 149 801 41 61 25 134 301 1, 769 to fully cover the object. To extract overlapped classes, we selected classes in ModelNet-10.
ShapeNet-10 (S): ShapeNet contains 3D CAD models of 55 categories gathered from online repositories: Trimble 3D Warehouse and Yobi3D2. ShapeNet contains more samples and its objects have larger variance in structure compared with ModelNet. Uniform sampling is applied to collect the points of ShapeNet on surface. So may lose some marginal points compared with ModelNet. Same as ModelNet-10 we collect 10 common classes from ShapeNet.
ScanNet-10 (S*): Different with ModelNet and ShapeNet, ScanNet contains scanned and reconstructed real-world indoor scenes. We isolate 10 common classes instances contained in annotated bounding boxes for classification. The objects are scanned by RGB-D camera, they often lose some parts and get occluded by surroundings. ScanNet is a realistic domain but more challenging for DNN based model.
Experiments Setup
We choose the PointNet [26] as the backbone of the classifier in evaluation. The learning rate is set to 0.0001 under the weight decay 0.0005 and α, β, γ follow the Table 1. All models have been trained for 200 epochs of batch size 64 in both source domain and synthetic datasets. Baselines: In our experiments, we evaluate the performance when the model is trained only by source training samples (w/o Adapt). We also compare our model with five Figure 6. Visualization for source domain object, target domain object and synthetic object general-purpose UDA methods including: Maximum Mean Discrepancy (MMD) [22], Adversarial Discriminative Domain Adaptation (ADDA) [37], Domain Adversarial Neural Network (DANN) [11], Maximum Classifier Discrepancy (MCD) [30] and PointDAN [28] in the same training policy. For further performance comparison, we also evaluated the performance of an ablated version of our model that removes the classifier and only kept the generator and discriminator. Finally, we show the performance of a full supervised method (Supervised). As expected, supervised performs better than all unsupervised methods.
As shown from table 3, our approach can improve over all five existing UDA models in all six scenarios. In particular, for S* → S, our approach achieves 65.7% pre-dicting accuracy, which represents about 10.8% improvement over the best result from the existing approaches. For S*→M, our model also has over 5.7% improvement. Note that, among the three different datasets, Scannet (S*) is the most challenging dataset because Scannet's point cloud objects are scanned from the real world, while the other two datasets are generated from 3D polygonal models. This implies that the difference between S* and S, and between S* and M are more significant from the observation and our approach is better at generating synthetic objects by minimizing the gap in point distributions between the source and target datasets.
To better understand how does our DA method improve the classifier over multi-class classification task. Confusion matrix over ten classes is shown in Fig.7. Since each class has a different amount of objects, we normalize the amount to ratio. The higher value in the confusion matrix, the darker color it is. In Scannet(S*) to Modelnet(M) scenario, we can see a large amount of bathtub, bed, and bookshelf objects misclassified as a sofa. After domain adaption by our method, most of them are correctly classified. What's more, in Modelnet(M) to Scannet(S*) scenario, which is a very challenging scenario. From table 3 we find it has only 22.3% accuracy without domain adaption, most of the objects are misclassified as chair or monitor. After applying our method to this scenario, the amount of objects that are misclassified as a monitor is significantly decreased, and we have more correctly classified objects in the bathtub and bed. AE means use autoencoder with reconstruction loss in model ,L denotes latent space reconstruction with VAE , C represents the additional discriminator, a classifier.
Ablation Study Setup:
To study the effect of latent reconstruction module and additional discriminator, we first construct a model that only keeps encoder/decoder and generator. From table 4 we can see this ablated model could only slightly improve the classifier over the none-adaption case. After applying the latent space reconstruction module to this ablated model, the classifier's performance significantly increases in all six scenarios due to the high fidelity, quality synthetic output. Even the ablated model can outperform the existing models most of the time, but adding classifier C to our model as an additional discriminator will further improve the result. Fig.8 shows the visualized result of synthetic output generated by each ablated model from the same object in the source domain. We can find the model that only uses Au- Figure 7. Confusion matrix toencoder(AE) with reconstruction loss, can only roughly reconstruct the shape of the object. Nevertheless, after applying latent reconstruction loss with Variational Autoencoders(VAE), the quality of synthetic objects is significantly increased. In the last, using the classifier as an additional discriminator provides the synthetic objects more fidelity because it will encourage the generator to ingratiate the classifier so that the synthetic is more likely to be considered as the same class as the source domain object.
Conclusion
We have proposed a novel approach to unsupervised domain adaptation in the 3D classification task in this work. The basic idea is to transfer training data into the target domain rather than selecting domain invariant feature or implementing feature alignment. In our approach, a Generative Adversarial Network (GAN) is constructed for domain transfer in 3D point-clouds to perform unsupervised domain adaptation in 3D classification. Our model generates synthetic objects from the source domain objects such that the output will have the same shape and label as the source domain objects but are constructed according to the pattern of the target domain. In this way, classifier trained using the synthetic dataset will perform better in the target domain dataset. To encourage the model to generate an object in the same class as input, we designed a GAN-based framework that takes the classifier as an addition discriminator.
Using extensive experiments involving three datasets and comparing them with five existing DA methods, we have demonstrated our approach's superiority over the state-ofthe-art domain adaptation methods.
Figure 1 .
1Point cloud generate from different source or sensors may have different geometric feature. Even the same car scanned by two different lidar have different points density.
Figure 4 .
4The structure of encoder our model and a Variational Autoencoder(VAE) is trained to encode synthetic objects to recover latent input vector, encouraging the use of conditional mode input z.
Figure 5 .
5Illustration of training and inference .
, where M, S and S* represent subset of Modelnet, Shapenet and Scannet respectively. We consider six types of adaptation scenarios which are M → S, M → S*, S → M, S → S*, S* → M and S* → S. ModelNet-10 (M): ModelNet40 contains clean 3D CAD models of 40 categories. Those downloaded 3D CAD models from 3D Warehouse, and Yobi3D search engine indexing 261 CAD model websites. After getting the CAD model, they sample points on the surface
Figure 8 .
8visualized result of different ablated models
Table 1 .
1Importance weights used in loss function
Table 2 .
2Number of samples in proposed datasets.Dataset
Bathtub Bed Bookshelf Cabinet Chair Lamp Monitor Plant
Sofa
Table 3 .
3M means ModelNet and S denotes ShapeNet while S* represents ScanNet.Quantitative classification results (%) on PointDA-10
Dataset[28].
M→S M→S* S→M S→S* S*→M S*→S Avg
w/o Adapt
42.5
22.3
39.9
23.5
34.2
46.9
34.9
MMD[22]
57.5
27.9
40.7
26.7
47.3
54.8
42.5
DANN[11]
58.7
29.4
42.3
30.5
48.1
56.7
44.2
ADDA[37]
61.0
30.5
40.4
29.3
48.9
51.1
43.5
MCD[30]
62.0
31.0
41.4
31.3
46.8
59.3
45.3
PointDAN[28]
62.5
31.2
41.5
31.5
46.9
59.3
45.5
Ours
62.8
36.5
41.9
31.6
50.4
65.7
48.1
Supervised
90.5
53.2
86.2
53.2
86.2
90.5
76.6
Table 4 .
4Ablation analysis AE L C M→S M→S* S→M S→S* S*→M S*→S Avgw/o Adapt
42.5
22.3
39.9
23.5
34.2
46.9
34.9
only AE
√
59.5
33.5
34.2
16.1
43.3
55.4
40.3
AE+L
√ √
62.6
34.1
40.4
29.1
49.6
64.3
46.7
GFA
√ √ √
62.8
36.5
41.9
31.6
50.4
65.7
48.1
Supervised
90.5
53.2
86.2
53.2
86.2
90.5
76.6
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"Emulation of stochastic simulators using generalized lambda models"
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"Bruno Sudret \nChair of Risk, Safety and Uncertainty Quantification\nETH Zürich\nStefano-Franscini-Platz 58093ZürichSwitzerland\n"
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| Stochastic simulators are ubiquitous in many fields of applied sciences and engineering. In the context of uncertainty quantification and optimization, a large number of simulations is usually necessary, which becomes intractable for high-fidelity models. Thus surrogate models of stochastic simulators have been intensively investigated in the last decade. In this paper, we present a novel approach to surrogating the response distribution of a stochastic simulator which uses generalized lambda distributions, whose parameters are represented by polynomial chaos expansions of the model inputs. As opposed to most existing approaches, this new method does not require replicated runs of the simulator at each point of the experimental design. We propose a new fitting procedure which combines maximum conditional likelihood estimation with (modified) feasible generalized least-squares. We compare our method with state-of-the-art nonparametric kernel estimation on four different applications stemming from mathematical finance and epidemiology. Its performance is illustrated in terms of the accuracy of both the mean/variance of the stochastic simulator and the response distribution. As the proposed approach can also be used with experimental designs containing replications, we carry out a comparison on two of the examples, showing that replications do not necessarily help to get a better overall accuracy and may even worsen the results (at a fixed total number of runs of the simulator). | 10.1137/20m1337302 | [
"https://arxiv.org/pdf/2007.00996v3.pdf"
]
| 220,302,557 | 2007.00996 | 95c554142399a7ceb52381c184ccc401c4983378 |
Emulation of stochastic simulators using generalized lambda models
February 9, 2022
Xujia Zhu
Chair of Risk, Safety and Uncertainty Quantification
ETH Zürich
Stefano-Franscini-Platz 58093ZürichSwitzerland
Bruno Sudret
Chair of Risk, Safety and Uncertainty Quantification
ETH Zürich
Stefano-Franscini-Platz 58093ZürichSwitzerland
Emulation of stochastic simulators using generalized lambda models
February 9, 2022
Stochastic simulators are ubiquitous in many fields of applied sciences and engineering. In the context of uncertainty quantification and optimization, a large number of simulations is usually necessary, which becomes intractable for high-fidelity models. Thus surrogate models of stochastic simulators have been intensively investigated in the last decade. In this paper, we present a novel approach to surrogating the response distribution of a stochastic simulator which uses generalized lambda distributions, whose parameters are represented by polynomial chaos expansions of the model inputs. As opposed to most existing approaches, this new method does not require replicated runs of the simulator at each point of the experimental design. We propose a new fitting procedure which combines maximum conditional likelihood estimation with (modified) feasible generalized least-squares. We compare our method with state-of-the-art nonparametric kernel estimation on four different applications stemming from mathematical finance and epidemiology. Its performance is illustrated in terms of the accuracy of both the mean/variance of the stochastic simulator and the response distribution. As the proposed approach can also be used with experimental designs containing replications, we carry out a comparison on two of the examples, showing that replications do not necessarily help to get a better overall accuracy and may even worsen the results (at a fixed total number of runs of the simulator).
system already at the design phase. Classical simulators are usually deterministic because they implement solvers for the governing equation of the system. Thus, repeated model evaluations with the same input parameters consistently result in the same value of the output quantities of interest (QoIs). In contrast, stochastic simulators contain intrinsic randomness, which leads to the QoI being a random variable conditioned on the given set of input parameters. In other words, each model evaluation with the same input values generates a realization of the response random variable that follows an unknown distribution. Formally, a stochastic simulator M s can be expressed as
M s : D X × Ω → R (x, ω) → M s (x, ω),(1)
where x is the input vector that belongs to the input space D X , and Ω denotes the sample space of the probability space {Ω, F, P} that represents the internal source of randomness.
Stochastic simulators are widely used in modern engineering, finance, and medical sciences.
Typical examples include evaluating the performance of a wind turbine under stochastic loads [1], predicting the price of an option in financial markets [2], and the spread of a disease in epidemiology [3].
Due to the random nature of stochastic simulators, repeated model evaluations with the same input parameters, called hereinafter replications, are necessary to fully characterize the probability distribution of the corresponding QoI. In addition, uncertainty quantification and optimization problems typically require model evaluations for various sets of input parameters. Altogether, it is necessary to have a large number of model runs, which becomes intractable for costly models.
To alleviate the computational burden, surrogate models, a.k.a. emulators, can be used to replace the original model. Such a model emulates the input-output relation of the simulator and is easy and cheap to evaluate.
Among several options for constructing surrogate models, this paper focuses on the so-called nonintrusive approaches. More precisely, the computational model is considered as a "black box" and is only required to be evaluated on a limited number of input values, called the experimental design (ED).
Three classes of methods can be found in the literature for emulating the entire response distribution of a stochastic code in a nonintrusive manner. The first one is the random field approach, which approximates the stochastic simulator by a random field. The definition in Eq. (1) implies that a stochastic simulator can be regarded as a random field indexed by its input variables. Controlling the intrinsic randomness allows one to get access to different trajectories of the simulator, which are deterministic functions of the input variables. In practice, this is achieved by fixing the random seed inside the simulator. Evaluations of the trajectories over the experimental design can then be extended to continuous trajectories, either by classical surrogate methods [4] or through Karhunen-Loève expansions [5]. Since this approach requires the effective access to the random seed, it is only applicable to data generated in a specific way.
Another class of methods is the replication-based approach, which relies on using replications at all points of the experimental design to represent the response distribution through a suitable parametrization. The estimated distribution parameters are then treated as (noisy) outputs of a deterministic simulator. Then, conventional surrogate modeling methods, such as Gaussian processes [6] and polynomial chaos expansions (PCEs) [7], can emulate these parameters as a function of the model input [8,9]. Because this approach employs two separate steps, the surrogate quality depends on the accuracy of the distribution estimation from replicates in the first step [10]. Therefore, many replications are necessary, especially when nonparametric estimators are used for the local inference [8,9].
A third class of methods, known as the statistical approach, does not require replications or controlling the random seed. If the response distribution belongs to the exponential family, generalized linear models [11] and generalized additive models [12] can be efficiently applied.
When the QoI for a given set of input parameters follows an arbitrary distribution, nonparametric estimators can be considered, notably kernel density estimators [13,14] and projection estimators [15]. However, it is well known that nonparametric estimators suffer from the curse of dimensionality [16], meaning that the necessary amount of data increases drastically with increasing input dimensionality.
In a recent paper [10], we proposed a novel stochastic emulator called the generalized lambda model (GLaM). Such a surrogate model uses generalized lambda distributions (GLDs) to represent the response probability density function (PDF). The dependence of the distribution parameters on the input is modeled by PCEs. However, the methods developed in [10] rely on replications. In the present contribution, we propose a new statistical approach combining feasible generalized leastsquares with maximum conditional likelihood estimations to get rid of the need for replications. Therefore, the proposed method is much more versatile in the sense that replications and seed controls are no longer necessary.
The paper is organized as follows. In Sections 2 and 3, we briefly review GLDs and PCEs, which are the two main elements constituting the GLaM. In Section 4, we recap the GLaM framework and introduce the maximum conditional likelihood estimator. Then, we present the algorithm developed to find an appropriate starting point to optimize the likelihood, and to design ad hoc truncation schemes for the PCEs of distribution parameters. In Section 5, we validate the proposed method on two analytical examples and two case studies in mathematical finance and epidemiology, respectively, to showcase its capability to tackle real problems. Finally, we summarize the main findings of the paper and provide an outlook for future research in Section 6.
Generalized lambda distributions
Formulation
The generalized lambda distribution (GLD) is a flexible probability distribution family. It is able to approximate most of the well-known parametric distributions [17,18], e.g., uniform, normal, Weibull, and Student's t distributions. The definition of a GLD relies on a parametrization of the quantile function Q(u), which is a nondecreasing function defined on [0, 1]. In this paper, we consider the GLD of the Freimer-Kollia-Mudholkar-Lin family [17], which is defined by
Q(u; λ) = λ 1 + 1 λ 2 u λ 3 − 1 λ 3 − (1 − u) λ 4 − 1 λ 4 ,(2)
where λ = {λ l : l = 1, . . . , 4} are the four distribution parameters. More precisely, λ 1 is the location parameter, λ 2 is the scaling parameter, and λ 3 and λ 4 are the shape parameters. To ensure valid quantile functions (i.e., Q being nondecreasing on u ∈ [0, 1]), it is required that λ 2 be positive. Based on the quantile function, the PDF f W (w; λ) of a random variable W following a GLD can be derived as
f W (w; λ) = 1 Q (u; λ) = λ 2 u λ 3 −1 + (1 − u) λ 4 −1 1 [0,1] (u), with u = Q −1 (w; λ),(3)
where Q (u; λ) is the derivative of Q with respect to u, and 1 [0,1] is the indicator function. A closed-form expression of Q −1 , and therefore of f W , is in general not available, and thus the PDF is evaluated by solving the nonlinear equation Eq. (3) numerically.
Properties
GLDs cover a wide range of unimodal shapes, including bell-shaped, U-shaped, S-shaped and bounded-mode distributions, which is determined by λ 3 and λ 4 , as illustrated in Figure 1 [10]. For instance, λ 3 = λ 4 produces symmetric PDFs, and λ 3 , λ 4 < 1 leads to bell-shaped distributions. Moreover, λ 3 and λ 4 are closely linked to the support and the tail properties of the corresponding PDF. λ 3 > 0 implies that the PDF support is left-bounded and λ 4 > 0 corresponds to right-bounded PDFs. Conversely, the distribution has lower infinite support for λ 3 ≤ 0 and upper infinite support for λ 4 ≤ 0. More precisely, the support of the PDF denoted by supp (f W (w; λ)) = [B l , B u ] is given by
B l (λ) = −∞, λ 3 ≤ 0, λ 1 − 1 λ 2 λ 3 , λ 3 > 0, B u (λ) = +∞, λ 4 ≤ 0, λ 1 + 1 λ 2 λ 4 , λ 4 > 0.(4)
Importantly, for λ 3 < 0 (λ 4 < 0), the left (resp., right) tail decays asymptotically as a power law, and thus the GLD family can also provide fat-tailed distributions. Due to this power law decay, for λ 3 ≤ − 1 k or λ 4 ≤ − 1 k , moments of order greater than k do not exist. For λ 3 , λ 4 > −0.5, the mean and variance exist and are given by
µ = E [W ] = λ 1 − 1 λ 2 1 λ 3 + 1 − 1 λ 4 + 1 , v = Var [W ] = (d 2 − d 2 1 ) λ 2 2 ,(5)
where the two auxiliary variables d 1 and d 2 are defined by
d 1 = 1 λ 3 B(λ 3 + 1, 1) − 1 λ 4 B(1, λ 4 + 1), d 2 = 1 λ 2 3 B(2λ 3 + 1, 1) − 2 λ 3 λ 4 B(λ 3 + 1, λ 4 + 1) + 1 λ 2 4 B(1, 2λ 4 + 1),(6)
with B denoting the beta function.
Moments of order k do not exist
Moments of order k do not exist and λ 4 . The values of λ 1 and λ 2 are set to 0 and 1, respectively. The blue points indicate that the PDF has infinite support in the marked direction. In contrast, both the red and green points denote the boundary points of the PDF support. More precisely, the PDF f W (w) = 0 on the red dots, whereas f W (w) = 1 on the green ones.
Consider a deterministic computational model M d (x) that maps a set of input parameters
x = (x 1 , x 2 , . . . , x M ) T ∈ D X ⊂ R M to the system response z ∈ R.
In the context of uncertainty quantification, the input variables are affected by uncertainty due to lack of knowledge or intrinsic variability (also called aleatory uncertainty). Therefore, they are modeled by random variables and grouped into a random vector X characterized by a joint PDF f X . The uncertainty in the input variables propagates through the the model M d to the output, which becomes a random variable denoted by Z = M d (X).
Remark. f X is the joint PDF for the input variables, which is needed to define orthogonal polynomials as described below. It should not be confused with the stochasticity of the simulator addressed in the next sections.
Provided that the output random variable Z has finite variance, M d belongs to the Hilbert space H of square-integrable functions associated with the inner product
u, v H def = E [u(X)v(X)] = D X u(x)v(x)f X (x)dx.(7)
If the joint PDF f X fulfills certain conditions [19], the space spanned by multivariate polynomials is dense in H. In other words, H is a separable Hilbert space admitting a polynomial basis.
In this study, we assume that X has mutually independent components, and thus the joint distribution f X is expressed as
f X (x) = M j=1 f X j (x j ).(8)
Let {φ (j) k : k ∈ N} be the orthogonal polynomial basis with respect to the marginal distribution of f X j , i.e.,
E φ (j) k (X j ) φ (j) l (X j ) = δ kl ,(9)
with δ being the Kronecker symbol defined by δ kl = 1 if k = l and δ kl = 0 otherwise. Then, the multivariate orthogonal polynomial basis can be obtained as the tensor product of univariate polynomials [20]:
ψ α (x) = M j=1 φ (j) α j (x j ),(10)
where α = (α 1 , . . . , α M ) ∈ R M denotes the multi-index of degrees. Each component α j indicates the polynomial degree of φ α j and thus of ψ α in the jth variable x j . For some classical distributions, e.g., normal, uniform, exponential, the associated univariate orthogonal polynomials are well known as Hermite, Legendre, and Laguerre polynomials [21]. For arbitrary marginal distributions, such a basis can be computed numerically through the Stieltjes procedure [22].
Following the construction defined in Eq. (10), ψ α (·), α ∈ N M forms an orthogonal basis for H. Thus, the random output Z can be represented by
Z = M d (X) = α∈N M c α ψ α (X),(11)
where c α is the coefficient associated with the basis function ψ α . The spectral representation in Eq. (11) is a series with infinitely many terms. In practice, it is necessary to adopt truncation schemes to approximate M d (x) with a finite series defined by a finite subset A ⊂ N M of multi-indices. A typical scheme is the hyperbolic (q-norm) truncation scheme [23]:
A p,q,M = α ∈ N M , α q = M i=1 |α i | q 1 q ≤ p ,(12)
where p is the maximum total degree of polynomials, and q ≤ 1 defines the quasi-norm · q .
Note that with q = 1, we obtain the so-called full basis of total degree less than p.
For an arbitrary distribution f X with dependent components of X, the usual practice is to transform X into an auxiliary vector ξ with independent components (e.g., a standard normal vector) using the Nataf or Rosenblatt transform [24]. Alternatively, polynomials orthogonal to the joint distribution may be computed on the fly using a numerical Gram-Schmidt orthogonalization [25].
Generalized lambda models (GLaM)
Introduction
Because of their flexibility, we assume that the response random variable of a stochastic simulator for a given input vector x follows a GLD. Hence, the distribution parameters λ are functions of the input variables:
Y (x) ∼ GLD (λ 1 (x), λ 2 (x), λ 3 (x), λ 4 (x)) .(13)
Under appropriate conditions discussed in Section 3, each component of λ(x) admits a spectral representation in terms of orthogonal polynomials. Recall that λ 2 (x) is required to be positive (see Section 2). Thus, we choose to build the associated PCE on the natural logarithm transform log (λ 2 (x)). This results in the following approximations: The generalized lambda model presented above is a statistical model. It involves two approximations. First, the response distribution of a stochastic simulator is approximated by GLDs.
λ l (x) ≈ λ PC l (x; c) = α∈A l c l,α ψ α (x), l = 1, 3, 4,(14)λ 2 (x) ≈ λ PC 2 (x; c) = exp α∈A 2 c 2,α ψ α (x) ,(15)
As illustrated in Figure 1, GLDs cover a wide range of unimodal shapes but cannot produce multimodal distributions. Thus, the GLD representation is appropriate when the response distribution stays unimodal. In this case, the flexibility of GLDs allows capturing the possible shape variation of the response distribution within a single parametric family. Second, the distribution parameters λ(x) seen as functions of x are represented by truncated polynomial chaos expansions. So they must belong to the Hilbert space of square-integrable functions with respect to f X (x) dx.
Estimation of the model parameters
Given the truncation sets A, the coefficients c need to be estimated from data to build the surrogate model. In this paper, as opposed to [10] and the vast majority of the literature on stochastic simulators, the simulator is required to be evaluated only once on the experimental design (1) , . . . , x (N ) , and the associated model responses are collected in Y = y (1) , . . . , y (N ) .
X = x
To develop surrogate models in a nonintrusive manner, we propose using the maximum conditional
likelihood estimator:ĉ = arg max c∈C L (c) ,(16)
where
L (c) = N i=1 log f GLD y (i) ; λ PC x (i) ; c .(17)
Here, f GLD denotes the PDF of the GLD defined in Eq. (3), and C is the search space for c. The estimator introduced in Eq. (17) can be derived from minimizing the Kullback-Leibler divergence between the surrogate PDF and the underlying true response PDF over D X ; see details in [10].
(i) P X is absolutely continuous with respect to the Lebesgue measure of R M , i.e., the joint
PDF f X (x) is Lebesgue-measurable.
(ii) f X has a compact support D X .
(iii) C is compact, and c 0 ∈ C.
(iv) There exists a set A ⊂ D X with P X (X ∈ A) > 0 such that ∀x ∈ A, Y (x) does not follow a uniform distribution.
Most of the assumptions in the Theorem 1 are realistic, except the one that the true model can be exactly represented by a GLaM, which is rather technical to guarantee the consistency. In practice, we do not require the QoI for any input parameters following a GLD but assume that the response distribution can be well approximated by GLDs.
It is worth remarking that since a GLD can have very fat tails (see Section 2.2), solving the optimization problem may produce response PDFs with unexpected infinite moments when the model is trained on a small data set. To prevent too-fat tails (if no prior knowledge suggests it), we apply the threshold λ PC
3 (x) = max λ PC 3 (x;ĉ), −0.3 and λ PC 4 (x) = max λ PC 4 (x;ĉ), −0.
3 , which indicates that we enforce the surrogate PDFs to have finite moments up to order 3 (higher order moments may exist depending onĉ). Thresholds larger than −0.3 (e.g., from −0.1 to 0) can be used if the response PDF is known to be light-tailed. Note that when enough data are available, these operations are unnecessary because the resulting model does not exceed the threshold. Although the thresholdings could have been imposed in the model definition in Eq. (14), they change the regularity of the optimization problem, and do not generally improve the performance according to our experience. Therefore, we only use them for postprocessing.
Remark 1.
While we consider the simulator to be evaluated only once for each point of the experimental design in this paper, the estimator defined in Eq. (16) is not limited to this type of data. When replications are available, the objective function can be reformulated to
L (c) = N i=1 1 R (i) R (i) r=1 log f GLD y (i,r) ; λ PC x (i) ; c ,(19)
where R (i) denotes the number of replications at point x (i) , and y (i,r) is the model response for x (i) at the rth replication. In addition, if R (i) is constant for all points x (i) ∈ X , Eq. (19) provides the same estimator as in our previous work [10].
Fitting procedure
In practice, the evaluation of L(c) is not straightforward because the PDF of GLDs does not have an explicit form as shown in Eq. (3). Details about the evaluation procedure are given in [10]. Note that the optimization problem Eq. (16) is subject to complex inequality constraints due to the dependence of the PDF support on λ (see Eq. (4)). Given a starting point, we follow the optimization strategy developed in [10]: We first apply the derivative-based trustregion optimization algorithm [26] without constraints. If none of the inequality constraints is activated at the optimum, we keep the results as the final estimates. Otherwise, the constrained (1+1)-CMA-ES algorithm [27] available in the software UQLab [28] is used instead.
Because L(c) is highly nonlinear, a good starting point is necessary to guarantee the convergence of the optimization algorithm. In this section, we introduce a robust method to find a suitable starting point.
According to Eq. (5), the mean µ(x) and the variance function v(x) of a GLaM satisfy
µ(x) = λ PC 1 (x) + 1 λ PC 2 (x) g λ PC 3 (x), λ PC 4 (x) , log (v(x)) = −2 log λ PC 2 (x) + h λ PC 3 (x), λ PC 4 (x) ,(20)
where we group the dependence of µ and log(v) on λ 3 and λ 4 into g and h, respectively, for the purpose of simplicity. If λ PC 3 (x) and λ PC 4 (x) do not vary strongly on D X , we observe that the variations of the mean and the variance function are mostly dominated by the location parameter λ PC 1 (x) and the scale parameter λ PC 2 (x).
Recall that the spectral approximation for λ 2 (x) is on its logarithmic transform. If a PCE can be constructed for µ(x) and − 1 2 log (v(x)), the associated coefficients can be used as a preliminary guess for the coefficients of λ PC 1 (x) and λ PC 2 (x), respectively. As a result, we first focus on estimating the mean and the variance function as follows:
µ(x) = α∈Aµ c µ,α ψ α (x), v(x) = exp α∈Av c v,α ψ α (x) ,
where the form of the variance function implies a multiplicative heteroskedastic effect (see [29]).
The mean estimation is a classical regression problem. However, since the variance function is also unknown and needs to be estimated, the heteroskedastic effect should be taken into account.
Many methods have been developed in statistics and applied science to tackle heteroskedastic regression problems. They can be classified into two groups: one class of methods relies on repeated measurements at given input values [30][31][32] (replication-based), whereas a second class of methods jointly estimates both quantities by optimizing certain functions without the need for replications [33][34][35][36]. Some studies [34,36] have shown higher efficiency of the second class of methods over the former. This finding supports our pursuit for a replication-free approach. In particular, we opt for feasible generalized least-squares (FGLS) [37], which iteratively fits the mean and variance functions in an alternative way.
The details are described in Algorithm 1. In this algorithm, OLS denotes the use of ordinary least-squares, and WLS is weighted least-squares.v corresponds to the set of estimated variances on the design points in X which are then used as weights in WLS to re-estimate c µ .
Algorithm 1 Feasible generalized least-squares (FGLS)
1:ĉ µ ← OLS (X , Y) 2: for i ← 1, . . . , N FGLS do 3:μ ← α∈Aµ c µ,α ψ α (X ) 4:r ← 2 log (|Y −μ|) 5:ĉ v ← OLS (X ,r) 6:v = exp α∈Av c v,α ψ α (X ) 7:ĉ µ ← WLS (X , Y,v) 8: end for 9: Output:ĉ µ ,ĉ v
After obtainingĉ µ andĉ v from FGLS, we perform two rounds of the optimization procedure described at the beginning of this section to build the GLaM surrogate. First, we set the starting points as c 1
= c µ , c 2 = − 1 2 c v , and λ PC 3 (x) = λ PC 4 (x) = 0.13,
Truncation schemes
Provided that the bases of λ PC (x) are given, we have presented a procedure to construct GLaMs from data in the previous section. However, there is generally no prior knowledge that would help select the truncation sets A l 's ab initio. In this section, we develop a method to determine a suitable hyperbolic truncation scheme A p,q,M presented in Eq. (12) for each component of
λ PC (x).
As discussed in Section 2, λ PC 3 (x) and λ PC 4 (x) control the shape variations of the response PDF. We assume that the shape does not vary in a strongly nonlinear way. Hence, the associated p can be set to a small value, e.g., p = 1, in practice. In contrast, λ PC 1 (x) and λ PC 2 (x) require possibly larger degree p since their behavior is associated with the mean and the variance function, which might vary nonlinearly over D X . To this end, we modify Algorithm 1 to adaptively find appropriate truncation schemes for µ(x) and v(x), which are then used for λ 1 (x) and λ 2 (x), respectively.
Algorithm 2 presents the modified FGLS. Instead of using OLS, we apply the adaptive ordinary least-squares with degree and q-norm adaptivity (referred to as AOLS) [38]. This algorithm builds a series of PCEs, each of which is obtained by applying OLS with the truncation set A p,q,M defined by a particular combination of p ∈ p and q ∈ q. Then, it selects the truncation scheme for which the associated PCE has the lowest leave-one-out error. In the modified FGLS, the truncation set A µ for µ(x) is selected only once (before the loop), whereas several truncation Algorithm 2 Modified feasible generalized least-squares
1: Input: (X , Y), p 1 , q 1 , p 2 , q 2 2: A µ ,ĉ µ ← AOLS (X , Y, p 1 , q 1 ) 3: for i ← 1, . . . , N FGLS do 4:μ ← α∈Aµ c m,α ψ α (X ) 5:r ← 2 log (|Y −μ|) 6: A i v ,ĉ i v , ε i LOO ← AOLS (X ,r, p 2 , q 2 ) 7:v ← exp α∈Av c v,α ψ α (X ) 8:ĉ µ ← WLS (X , Y, A µ ,v) 9: end for 10: i * = arg min ε i LOO : i = 1, . . . , N FGLS 11: Output: A µ ,ĉ i * µ , A i * v ,ĉ i * v schemes A i v : i = 1, . . . , N FGLS are obtained.
We select the one corresponding to the smallest leave-one-out error on the expansion of the variance as the truncation set A v for v(x). After running Algorithm 2, we apply the two-round optimization strategy described in the previous section to build the GLaM corresponding to the selected truncation schemes.
There are several parameters to be determined in Algorithm 2. In the following examples and applications, we set the candidate degrees p 1
= {0, . . . , 10} for λ PC 1 (x), and p 2 = {0, . . . , 5} for λ PC 2 (x)
. p 1 contains high degrees to approximate possibly highly nonlinear mean functions, the accuracy of which is crucial for basis selections for λ 2 (x) in Algorithm 2. p 2 is set to have degrees up to 5, allowing relatively complex variations. The lists of q-norms are q 1 = q 2 = {0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}, which contains the full basis. The total number of FGLS iterations is set to N FGLS = 10 which, according to our experience, is enough to find an appropriate truncated set for λ PC 2 (x).
Application examples
In this section, we validate the proposed algorithm on two analytical examples and two case studies in mathematical finance and epidemiology. In the four cases, the response distributions do not belong to a single parametric family, so as to test the flexibility of the proposed method.
In addition, we compare the performance of GLaMs with the nonparametric kernel conditional density estimator from the package np [39] implemented in R. The latter performs a thorough leave-one-out cross-validation with a multistart strategy to choose the bandwidths [14], which is one of the state-of-the-art kernel estimation methods. The surrogate model built by this method is referred to as the kernel conditional density estimator (KCDE).
Alongside GLaM and KCDE, another surrogate model, the heteroskedastic Gaussian process (denoted by GP), is also considered. This model assumes that the response distribution is Gaussian, and the mean and variance functions are represented by Gaussian processes. We apply the method proposed by Binois et. al. [40] which adopts a sequential design strategy to actively balance the trade-off between replications and explorations. The algorithm is available in the package hetGP in R. However, due to the sequential design (the new points are added one by one), building such a surrogate can be very time-consuming (cf. Section 5.2 for details).
Consequently, we present the comparisons with hetGP only for the first two examples.
Moreover, for comparison purposes, we consider another "Gaussian" surrogate model where we represent the response distribution with a normal distribution. The associated mean and variance, which are functions of the input x, are not fitted to data but set to the true values of the simulator. In other words, this surrogate model should represent the "oracle" of Gaussian-type mean-variance surrogate models, such as the ones presented in [36,41].
We use Latin hypercube sampling [42] to generate the experimental design for GLaM and KCDE. The stochastic simulator is only evaluated once for each vector of input parameters.
The associated QoI values are used to construct surrogate models with the proposed estimation procedure in Section 4.3. In contrast, the construction of the GP relies on a sequential design strategy which adaptively find new points to evaluate [40]. Hence, we use Latin hypercube sampling of 20% of the total number of model runs to initiate the process. Then, the algorithm proceeds by iteratively looking for points to evaluate and updating the surrogate.
To quantitatively assess the performance of the surrogate model, we define an error measure between the underlying model and the emulator by
ε = E d Y (X),Ŷ (X) ,(21)
where Y (X) is the model response,Ŷ (X) corresponds to that of the surrogate, d (Y 1 , Y 2 ) denotes the contrast measure between the probability distributions of Y 1 and Y 2 , and the expectation is taken with respect to X. In this study, we use the normalized Wasserstein distance, defined by
d (Y 1 , Y 2 ) = d WS (Y 1 , Y 2 ) σ (Y 1 ) ,(22)
where d WS is the Wasserstein distance of order two [43] defined by
d WS (Y 1 , Y 2 ) def = Q 1 − Q 2 2 = 1 0 (Q 1 (u) − Q 2 (u)) 2 du ,(23)
where Q 1 and Q 2 are the quantile functions of Y 1 and Y 2 , respectively. As a summary, by combining Eq. (21) and Eq. (23) the global error reads
ε = D X 1 0 Q Y (x) (u) − QŶ (x) (u) 2 du f X (x) Var [Y (x)] dx(24)
Following this definition, the standard deviation σ Y 1 can be seen as the Wasserstein distance between the distribution of Y 1 and a degenerate distribution concentrated at the mean value µ Y 1 .
As a result, the Wasserstein distance normalized by the standard deviation can be interpreted as the ratio of the error related to emulating the distribution of Y 1 by that of Y 2 , and to using the mean value µ Y 1 as a proxy of Y 1 .
Because d WS is invariant under translation, the normalized Wasserstein distance is invariant under both translation and scaling; that is,
∀a ∈ R \ 0, b ∈ R d WS (a Y 1 + b, a Y 2 + b) σ(a Y 1 + b) = d WS (Y 1 , Y 2 ) σ(Y 1 ) .(25)
To calculate the expectation in Eq. (21)
ε b = Ntest i=1 b (i) S −b (i) 2 Ntest i=1 b(i) −b 2 , withb = 1 N test Ntest i=1b (i) ,(26)
where b
Example 1: a two-dimensional simulator
The first example is the Black-Scholes model used for stock prices [44]:
dS t = x 1 S t dt + x 2 S t dW t ,(27)
where x = (x 1 , x 2 ) T are the input parameters, corresponding to the expected return rate and volatility of a stock, respectively. W t is a standard Wiener process, which represents the source of stochasticity. Equation (27) is a stochastic differential equation whose solution S t (x) is a stochastic process for given parameters x. Note that we explicitly express x in S t (x) to emphasize that x are input parameters, but the stochastic equation is defined with respect to time. Without loss of generality, we set the initial condition to S 0 (x) = 1.
In this example, we are interested in Y (x) = S 1 (x), which corresponds to the stock value in one
year i.e., t = 1. We set X 1 ∼ U(0, 0.1) and X 2 ∼ U(0.1, 0.4) to represent the input uncertainty,
where the ranges are selected based on parameters calibrated from real data [45].
The solution to Eq. (27) can be derived using Itô calculus [2]: Y (x) follows a lognormal distribution defined by
Y (x) ∼ LN x 1 − x 2 2 2 , x 2 .(28)
As the distribution of Y (x) is known, it is not necessary to simulate the whole process S t (x)
with time integration to evaluate S 1 (x). Instead, we can directly generate samples from the distribution defined in Eq. (28). In terms of the average error, GLaMs outperform KCDEs for all sizes of experimental design.
Furthermore, GLaMs yield an average error near 0.1 for N = 1,000, which can be hardly achieved by KCDEs even with four times more model runs.
Example 2: a five-dimensional simulator
The second example is given by
Y (x) = M s (x, ω) = µ(x) + σ(x) · Z(ω),(29)
where X ∼ U [0, 1] 5 are the input variables, and Z ∼ N (0, 1) is the latent variable that introduces the stochasticity. The simulator has an input dimension of M = 5, which is used to show the performance of the proposed method in a moderate-dimensional problem. By definition,
Y (x) is a Gaussian random variable with mean µ(x) and standard deviation σ(x) which are defined by
µ(x) = 3 − 5 j=1 j x j + 1 5 5 j=1 j x 3 j + 1 15 5 j=1 j log (x 2 j + x 4 j ) + x 1 x 2 2 − x 5 x 3 + x 2 x 4 , σ(x) = exp 1 10 5 j=1 j x j ,(30)
Thus, this example has a nonlinear mean function and a strong heteroskedastic effect: the variance varies between 1 and 20. The results show that the GLaM provides more accurate estimates for both functions. Similar to the first example, we perform a convergence study for N ∈ {250; 500; 1,000; 2,000; 4,000}, the results of which are shown in Fig. 9. The underlying response distribution is Gaussian, and thus the oracle normal approximation has ε = 0, which is not reported in the figure.
Surprisingly, GP gives the worst results. This may be understood as follows: the updating criterion of the sequential design targets at minimizing the integrated mean-squared error. The latter mainly focuses on improving the mean estimation (as illustrated in Figs. 7 and 8), yet both the mean and variance contribute to the Wasserstein distance Eq. (23). Also, this example is a five-dimensional problem, which results in more parameters to estimate for GP. In the case of small N , namely N = 250, both the GLaMs and KCDEs perform poorly, with the GLaMs showing a similar average error but higher variability. This is explained as follows. Because of the use of AOLS in the modified FGLS procedure, we observe that the total number of coefficients of GLaMs to be estimated varies between 19 to 39 for N = 250. Since the GLD is very flexible, a relatively large data set is necessary to provide enough evidence of the underlying PDF shape.
Consequently, a small N can lead to overfitting for high-dimensional c, but good surrogates can be obtained for more parsimonious models. In contrast, KCDE always performs a thorough leave-one-out cross-validation strategy to select the bandwidths. Therefore, KCDEs show a slightly more stable estimate for N = 250. With N increasing, however, GLaMs converge much faster and outperform KCDEs for N ≥ 500 both in terms of the mean and median of the errors.
For N ≥ 1,000, the average performance of GLaM is even better than the best KCDE model among the 50 repetitions.
In this example of moderate dimensionality, building a GP with sequential design is surprisingly
Effect of replications
As pointed in Remark 1, the proposed method can also work with a data set containing replicates.
The latter are simply treated as separate points in the ED. In this section, we analyze the effect
Example 3: Asian options
In this third example, we apply the proposed method to a financial case study, namely an Asian option [46]. Such an option, a.k.a. average value option, is a derivative contract, the payoff of which is contingent on the average price of the underlying asset over a certain fixed time period.
Due to the path-dependent nature, an Asian option has complex behavior, and its valuation is not straightforward, as opposed to European options.
Recall the Black-Scholes model defined in Eq. (27) that represents the evolution of a stock price S t (x). Instead of relying on the stock price on the maturity date t = T , the payoff of an Asian call option reads
C(x) = max {A T (x) − K, 0} , with A t (x) = 1 t t 0 S u (x)du.(31)
where A t (x) is called the continuous average process, and K denotes the strike price. Because
S t+∆t (x) | S t (x) ∼ LN log (S t (x)) + x 1 − x 2 2 2 ∆t, x 2 √ ∆t .
Finally, the continuous average defined in Eq. (31) is approximated by the arithmetic mean, that is,
A 1 (x) = 1,000 k=1 S k∆t (x)
1,000 Figure 12 shows two response PDFs predicted by the two surrogate models constructed on an experimental design of N = 500. The reference histograms are calculated from 10 4 repeated runs of the simulator for each set of input parameters. We observe that the KCDE exhibits slight fluctuations at the right tail for high volatility (in Figure 12a) and does not well approximate the bulk of the response distribution for low volatility (in Figure 12b). In comparison, the GLaM can well represent the PDF shape in both cases and also more accurately approximates the tails. Figures 13 and 14 KCDEs. The oracle Gaussian approximation in this case study has a similar error to GLaMs built on 1,000 model runs. For N ≥ 2,000, GLaMs fitted from data are much more accurate than the best possible Gaussian-type mean-variance model. not only is important for making investment decisions but also has a very similar form to the option price [46]. The definition Eq. (31) implies that the payoff C(x) is a mixed random variable, which has a probability mass at 0 and a continuous PDF on the positive line depending on the strike price K. In the following analysis, K is set to 1.
For GLaMs, µ C (x) can be calculated by
µ C (x) = λ 1 − 1 λ 2 λ 3 + 1 λ 2 λ 4 − K (1 − u K ) + 1 λ 2 1 − u λ 3 +1 K λ 3 (λ 3 + 1) − (1 − u K ) λ 4 +1 λ 4 (λ 4 + 1)(32)
where λ's are the distribution parameters at x, and u K is the solution of the nonlinear equation
Q(u K ; λ) = K.(33)
with Q being the quantile function defined in Eq. (2). Figure 16 shows the convergence of estimations of µ C (x) in terms of the error defined in Eq. (26).
The difference between the performance of GLaMs and KCDEs is not as significant as for the distribution estimation of A 1 (x) in Figure 15. For relatively small data sets, namely N ≤ 500, both models work poorly: they are only able to explain on average no more than 70% of the variance of µ C (X). In addition, GLaMs demonstrate a higher variability of the errors. For larger experimental designs N ≥ 2,000, however, the performance of GLaMs improves significantly more than that of KCDEs. For N = 4,000, the average error of GLaMs is twice smaller than that of KCDEs, and the smallest error achieved by GLaMs is one order of magnitude smaller than the best KCDE.
Example 4: Stochastic SIR model
In this fourth example, we apply the proposed method to a stochastic susceptible-infected-recovered (SIR) model in epidemiology [3]. This model simulates the spread of an infectious disease, which can help find appropriate epidemiological interventions to minimize social and ethical impacts during the outbreak.
According to the standard SIR model, at time t a population of size P t contains three groups of individuals: susceptible, infected, and recovered, the counts of which are denoted by S t , I t ,
and R t , respectively. These three quantities fully characterize a population configuration at time t. Among the three groups, only susceptible individuals can get infected due to close contact with infected individuals, whereas an infected person can recover and becomes immune to future infections. We consider a fixed population without newborns and deaths, i.e., the total population size is constant, P t = P . As a result, S t , I t , and R t satisfy the constraint S t + I t + R t = P , and only the time evolution of (S t , I t ) is necessary to characterize the spread of a disease.
To account for random recoveries and interactions among individuals, stochastic SIR models are usually preferred to represent the epidemic evolution. Without going into details, the model dynamics is briefly summarized as follows. The pair (I t , S t ) evolves as a continuous-time Markov process following mutual transition rates β and γ, which denote the contact rate and recovery rate, respectively. The epidemic stops at time t = T where I T = 0, indicating that no further infections can occur. The evolution process is simulated by the Gillespie algorithm [47]. The reader is referred to [3] for a more detailed presentation of stochastic SIR models.
In this case study, we set the total population equal to P = 2,000 and β = γ = 0.5 as in [41].
The initial configuration x = (S 0 , I 0 ) is the vector of input parameters. To account for different scenarios, the input variables X are modeled as X 1 ∼ U(1200, 1800) (initial number of susceptible individuals) and We observe that the PDF shape varies: it changes from symmetric to slightly right-skewed distributions depending on the input variables.
The GLaM is able to accurately capture this shape variation, while KCDE exhibits relatively poor shape representations. Figures 18 and 19 illustrate the mean and variance function. Because the analytical results are unknown for this simulator, we use 1,000 replications to estimate these quantities for plotting. In epidemiological management, the expected value µ(x) = E [Y (x)] is crucial for decision making [48]. Therefore, we investigate the accuracy of µ(x) estimations, and the results are in Figure 21.
First, both GLaM and KCDE can explain more than 90% of the variance in µ(X) for N = 250, which implies an overall accurate approximation to the mean function. With increasing N , GLaM shows a more rapid decay of the error. Furthermore, GLaMs built on N = 1,000 have a similar (or even slightly better) performance to KCDEs with N = 4,000.
Conclusions
This paper presents an efficient and accurate nonintrusive surrogate modeling method for stochastic simulators that does not require replicated runs of the latter. We follow the setting of Zhu and Sudret [10], where the generalized lambda distribution is used to flexibly approximate the response probability density function. The distribution parameters, as functions of the input variables, are approximated by polynomial chaos expansions. In this paper, however, we do not require replicated runs of the stochastic simulator, which provides a more general and versatile approach. We propose the maximum conditional likelihood estimator to construct such a model for given basis functions. This estimation method is shown to be consistent and applicable to data with or without replications. In addition, we modify the feasible generalized least-squares algorithm to select suitable truncation schemes for the distribution parameters, which also provides a good starting point for the subsequent optimization of the likelihood function.
The performance of the new method is illustrated on analytical examples and case studies in mathematical finance and epidemics. The results show that with a reasonable number of model runs, the developed algorithm can produce surrogate models that accurately approximate the response probability density function and capture the shape variations of the latter with
x. Considering the normalized Wasserstein distance as an error metric, generalized lambda models always show a better convergence rate than the nonparametric kernel conditional density estimator with adaptive bandwidth selections (from the package np in R). Furthermore, the proposed method generally yields more reliable estimates of certain important quantities.
Quantifying the uncertainty of surrogate models that emulate the entire response distribution of a stochastic simulator remains to be developed in future work, especially when no or only a few replications are available. One possibility is to use cross-validation to calculate the expected loss.
However, when the log-likelihood is used as the loss function such as Eq. (17), the resulting score is not intuitive and is difficult to interpret. Alternatively, with a given basis for λ(x) in GLaMs, one can use bootstrap [49] to assess the uncertainty in the estimation of the coefficients. Figure 22 illustrates the PDF predictions of 100 bootstrapping GLaMs of a data set with N = 500 of Possible interesting applications of the proposed method to be investigated in future studies include reliability analysis and sensitivity analysis [50]. To improve the performance of the generalized lambda surrogate model for small data sets, we plan to develop algorithms that select only important basis functions based on appropriate model selection criteria. Finally, since the generalized lambda distribution cannot represent multimodal distributions, potential extensions to mixtures of generalized lambda distributions may provide a more flexible surrogate for simulators with multimodal response distribution [51].
family for a particular set of coefficients c 0 , i.e., q 0 = q c 0 and p 0 = p c 0 . We denote the probability measure of the probability space of (X, Y ) by P 0 and the Lebesgue measure by µ.
The maximum likelihood estimation defined in Eq. (16) belongs to the generalized method of moments (GMM) [52] for which we define the loss function by
c (x, y) = − log (q c (x, y)) 1 q 0 (x,y)>0 (x, y).(34)
It holds that
c 0 = arg min c l(c), where l(c) = E [ c (X, Y )] .
The maximum likelihood estimator is then defined bŷ
c = arg min c l n (c), where l n (c) = 1 n n i=1 c X (i) , Y (i) ,
where l n is the empirical version of l.
To prove the consistency of a GMM estimator, the uniform law of large numbers is usually used. In the case of a maximum likelihood estimator for the generalized lambda model, classical methods [53] to prove the uniform law of large numbers cannot be applied directly, due to the fact that the support of q c can depend on the model parameters c, as shown in Eq. (4). To circumvent this problem, we use the techniques suggested by [54] for the proof. (ii) Continuity: ∀c ∈ C, the map c → q c is continuous atc for µ-almost all (x, y) ∈ D x × R.
Proof. (i) As the conditions of Theorem 1 indicate that D X and C are compact, the two sets are bounded according to the Heine-Borel theorem. Hence, the value of λ PC (x; c) is also bounded.
We denote respectively C i , i = 1, . . . , 4 and {C i , i = 1, . . . , 4} as the upper and lower bounds for each component of λ:
C i ≤ λ i ≤ C i ∀i = 1, . . . , 4.(35)
In addition, Eq. (15) guarantees that λ PC 2 (x; c) is bounded away from 0, i.e., C 2 > 0. Consider now Eq.
λ 2 u λ 3 −1 + (1 − u) λ 4 −1 ≤ C 2 u k + (1 − u) k ,(36)
where
k = max C 3 − 1, C 4 − 1 .
Define the function m(u) = u k + (1 − u) k , which corresponds to the denominator of Eq. (36).
λ 2 u λ 3 −1 + (1 − u) λ 4 −1 ≤ C 2 C m = C q .(37)
Therefore, sup c∈C q c (x, y) ≤ C q .
(ii) Next, we prove the continuity. For anyc ∈ C, we classify the points (x, y) ∈ D x × R into three groups based on their corresponding latent variableũ: (1)ũ ∈ (0, 1), (2)ũ does not exist within [0, 1], and (3)ũ = 0 or 1.
For (x, y) in the first class, y is an interior point of the support of the conditional distribution qc(x, ·). Thereby, the following equation holds:
y = Q(ũ;λ) =λ 1 + 1 λ 2 ũλ 3 − 1 λ 3 − (1 −ũ)λ 4 − 1 λ 4 ,(38)
where the distribution parametersλ are obtained by evaluating λ PC (x;c). The partial derivatives of Q(u; λ) with respect to all the relevant parameters are
∂Q ∂u = 1 λ 2 u λ 3 −1 + (1 − u) λ 4 −1 ,(39)∂Q ∂λ 1 = 1,(40)∂Q ∂λ 2 = − 1 λ 2 2 u λ 3 − 1 λ 3 − (1 − u) λ 4 − 1 λ 4 ,(41)∂Q ∂λ 3 = 1 λ 2 λ 2 3 u λ 3 ln(u)λ 3 − u λ 3 − 1 ,(42)∂Q ∂λ 4 = 1 λ 2 λ 2 4 (1 − u) λ 4 − 1 − (1 − u) λ 4 ln(1 − u)λ 4 .(43)
It can be easily observed that Eq. (39) and Eq. (40) are continuous functions of u ∈ (0, 1) and λ. Although Eq. (41) is undefined for λ 3 = 0 and λ 4 = 0, the limit exists according to l'Hôpital's rule. The same holds for Eq. (42) and Eq. (43). As a result, we can extend Eqs. (41) to (43) by continuity, and thus they become continuous functions of u ∈ (0, 1) and λ. Therefore, Now consider a point (x, y) in the second class, which implies that y is outside the support of qc(x, ·), say, y is smaller than the lower bound of the support of qc(x, ·). In this case, qc(x, y) = 0.
Q(u,
According to Eq. (4), if the lower bound is finite, it is a continuous function of λ and thus continuous atc. As a result, for c within a certain neighborhood ofc, the lower bound is larger than y, which implies q x (x, y) = 0 for c in this neighborhood. Thereby, q c (x, y) is continuous at c. Analogous reasoning holds for the case where y is bigger than the upper bound of the support.
The last class corresponds to the case where y is located on the endpoint of the support of qc(x, ·).
By takingũ = 0 and 1 in Eq. (38) or considering directly Eq. (4), we obtain two associated deterministic functions between x and y. As a result, points of the third class can be represented by two curves in D x × R, whose Lebesgue measure is zero. This closes the proof of continuity.
Lemma 2.
The class G defined below satisfies the uniform strong law of large numbers:
G = g c = log q c + q 0 2q 0 1 q 0 >0 : c ∈ C .(44)
Proof. According to the continuity property in Lemma 1, it is obvious that for allc ∈ C, the map c → g c is continuous atc for µ-almost all (x, y) ∈ D × R. By assumption, the probability measure P 0 is absolutely continuous with respect to µ, and thus g c is continuous for P 0 -almost all (x, y) ∈ D × R.
Define G as the envelope function of the class G, i.e., G(x, y) = sup c∈C |g c (x, y)|. Let us prove that G ∈ L 1 (P 0 ), where L 1 (P 0 ) denotes the set of absolutely integrable functions with respect to P 0 .
Taking the boundedness property in Lemma 1 into account, we obtain g c (x, y) ≤ log 2C q q 0 (x, y) = log(2C q ) − log(q 0 (x, y)).
Obviously, g c (x, y) ≥ − log(2). Therefore, |g c (x, y)| ≤ max {log(2), |log(2C q )| + |log(q 0 (x, y))|} ≤ log(2) + |log(C q )| + |log(q 0 (x, y))|.
Because the inequality is independent of c, we have G(x, y) ≤ log(2) + |log(C q )| + |log(q 0 (x, y))|,
E [G(X, Y )] ≤ log(2) + |log(C q )| + E [|log(q 0 (X, Y )|] .(47)
Now consider the last term in Eq. (47):
E [|log(q 0 (X, Y )|] = Dx×R |log (q 0 (x, y))|p 0 (x, y)dxdy = Dx R |log (q 0 (x, y))|q 0 (x, y)dy f X (x)dx.
Through a change of variables, the integral within the parenthesis of Eq. (48) can be calculated as B(x) = R |log (q 0 (x, y))|q 0 (x, y)dy = 1 0 log λ 2 u λ 3 −1 + (1 − u) λ 4 −1 du, (49) where λ = λ PC (x; c 0 ). According to Eq. (35), we have
B(x) ≤ 1 0 |log(λ 2 )| + log u λ 3 −1 + (1 − u) λ 4 −1 du ≤ k 2 + 1 0 max log u k + (1 − u) k , log u k + (1 − u) k du,(50)
where
k 2 = max log C 2 , |log (C 2 )| , k = min {C 3 − 1, C 4 − 1} , k = max C 3 − 1, C 4 − 1 .
Using the symmetry of the integrand, we get
B(x) ≤ k 2 + 2 · max 1 2 0 log u k + (1 − u) k du, 1 2 0 log u k + (1 − u) k du ≤ k 2 + 2 · 1 2 0 log u k + (1 − u) k du + 1 2 0 log u k + (1 − u) k du .(51)
Without loss of generality, we now study the property of the integral
1 2 0 log u k + (1 − u) k du.(52)
For k = 0, Eq. (52) is equal to 1 2 log(2). For k > 0, we have u k ≤ (1 − u) k , and thus (2)) .
Through similar calculation, for k < 0, we have
As a result, Eq. (52) is finite. More precisely, 1 2 0 log u k + (1 − u) k du ≤ 1 2 log(2) + |k| 2 (log(2) + 1) .
Equation (55) implies B(x) ≤ k 2 + log(2) + |k| + k (log(2) + 1) = C B .
By inserting Eq. (56) into Eq. (48), we obtain
E [|log(q 0 (X, Y )|] ≤ C B .(57)
Then, according to Eq. (47), the envelope function G fulfills E [G(X, Y )] ≤ log(2) + |log(C q )| + E [|log (q 0 (X, Y ))|] = log(2) + |log(C q )| + C B < +∞.
(58)
Since G is always positive according to its definition, Eq. (58) means G ∈ L 1 (P 0 ). The continuity and the property of the envelope function G shown above allow applying [54, Lemma 3.10], which guarantees that G satisfies the uniform weak law of large numbers:
sup c∈C 1 n n i=1 g c X (i) , Y (i) − E [g c (X, Y )] P − −−−− → n→+∞ 0.(59)h 2 (qĉ, q 0 | x) f X (x)dx ≤ 8 N i=1 gĉ X (i) , Y (i) − E [gĉ (X, Y )] ,(60)
where the Hellinger distance is given by
h 2 (qĉ, q 0 | x) = 1 2 R qĉ(x, y) − q 0 (x, y) 2 dy.
According to Lemma 2, Eq. (60) implies
which is called the Hellinger consistency.
We define the function
R(c) = Dx h 2 (q c , q 0 | x) f X (x)dx.(62)
According to Lemma 1, ∀c ∈ C, the map c → √ q c − √ q 0 2 is continuous atc for all x ∈ D x and almost all y ∈ R. Since √ q c − √ q 0 2 ≤ q c + q 0 , and R (q c + q 0 ) dy = 2 < +∞, the map c → h 2 (q c , q 0 | x) is continuous for all x ∈ D x , which is guaranteed by the generalized Lebesgue dominated convergence theorem. Similarly, the map c → R(c) is also continuous.
Without going into lengthy discussions, it can be shown that the GLD is not identifiable only
Figure 1 :
1A graphical illustration of the PDF of the FKML family of GLD as a function of λ 3
where A = {A l : l = 1, . . . , 4} are the truncation sets defining the basis functions, and c = {c l,α : l = 1, . . . , 4, α ∈ A l } are coefficients associated to the bases. For the purpose of clarity, we explicitly express c in the spectral approximations as in λ PC (x; c) to emphasize that c are the model parameters.
The advantages of this estimation method are twofold. On the one hand, it removes the need for replications in the experimental design. On the other hand, if a GLaM for a certain choice of c can exactly represent the stochastic simulator, the proposed estimator is consistent under mild conditions, as shown in Theorem 1 (see Appendix A.1 for a detailed proof).
Theorem 1 .
1Let (X(1) , Y(1) ), . . . , (X (N ) , Y (N ) ) be independent and identically distributed random variables following X ∼ P X and Y (x) ∼ GLD λ PC (x; c 0 ) . If the following conditions are fulfilled, the estimator defined in Eq.(16) is consistent, that is,c a.s. − − → c 0 .
which corresponds to a normal-like shape. Then, we fit a GLaM with λ PC 3 (x) λ PC 4 (x) being only constant; i.e., the coefficients of nonconstant basis functions are kept as zeros during the fitting. Finally, we use the resulting estimates as a starting point and construct a final GLaM with all the considered basis functions by solving Eq. (17).
S
is the value predicted by the surrogate for x (i) ∈ X test , andb (i) denotes the quantity estimated from 10 4 replicated runs of the original stochastic simulator for x(i) . The error ε b defined in Eq.(26) indicates how much of the variance of b(X) cannot be explained by b S (X) estimated from surrogate model. Experimental designs of various size N ∈ {250; 500; 1,000; 2,000; 4,000} are investigated to study the convergence of the proposed method. Each scenario is run 50 times with independent experimental designs to account for statistical uncertainty in the random design for GLaM and KCDE. For GP, N corresponds to the total number of model runs. We repeat 10 times for each value of N (i.e., 10 heteroskedastic Gaussian processes are built using the same number of model runs). As a consequence, error estimates for each N are represented by box plots.
(a ) Figure 2 :
)2PDF for x = (0.03, 0.33) T (b) PDF for x = (0.07, 0.11) T Example 1 -Comparisons of the emulated PDF, N = 500.
Figure 2 Figure 3 :Figure 4 :
234shows two PDFs predicted by a GLaM and a KCDE built on an experimental design of size N = 500. We observe that with 500 model runs, the KCDE yields PDFs with spurious oscillations and demonstrates relatively poor representation of the bulk. In contrast, the GLaM can better approximate the underlying response PDF in terms of both magnitude and shape variations.Figures 3 and 4compare the mean and variance function predicted by the GLaM, KCDE, and GP. The analytical mean function following Eq. (28) is exp(x 1 ), which only depends on the first variable. The GLaM gives an accurate estimate of the mean function, whereas the KCDE captures a wrong dependence, and GP produces a rather complex structure. For the variance function, the GLaM yields a more detailed trend than the KCDE and GP.For quantitative comparisons,Figure 5summarizes the error measure Eq. (21) with respect to the size of experimental design. The accuracy of the oracle normal approximation is also reported (black dashed line). This error is only due to model misspecifications because we use the true mean and variance (however, the true response distribution is lognormal). The GP approach performs rather poorly and converges to the oracle normal approximation when the number of points in the experimental design increases. This means that it can accurately estimate the Example 1 -Comparisons of the mean function estimation, N Example 1 -Comparisons of the variance function estimation, N = 500. mean and variance functions for large data sets. However, due to the limitation of the Gaussian assumption, GP cannot further decrease the error. The average error of GLaMs built on N = 500 model runs are smaller than that of the normal approximation. For N > 500, GLaMs clearly provide more accurate results. KCDEs show a slow rate of convergence even in this example of dimension two. In contrast, GLaMs reveal high efficiency with a faster decrease of the errors.
Figure 5 :
5Example 1 -Comparison of the convergence between GLaMs and KCDEs in terms of the normalized Wasserstein distance as a function of the size of the experimental design. The dashed lines denote the average value over 50 repetitions of the full analysis. The green box plots and associated dashed lines correspond to the errors of the heteroskedastic Gaussian Process with sequential design (10 repetitions for each size of the experimental design). The black dash-dotted line represents the error of the model assuming that the response distribution is normal with the true mean and variance.
Figure 6 :
6Example 2 -Comparisons of the emulated PDF, N = 1,000. Variance values 1.35, 3.32, 8.17, 14.88 from (a) to (d) Figure 6 compares the model response PDFs (with different variances) for four input values with those predicted by a GLaM and a KCDE built upon 1,000 model runs. The results show that the GLaM correctly identifies the shape of the underlying normal distribution among all possibleshapes of the GLD. Moreover, it yields a better approximation to the reference PDF, whereas KCDE tends to "wiggle" inFigure 6d(high variance) and overestimate the spread inFigure 6a
(
low variance). Figures 7 and 8 illustrate the mean and variance function predicted by the GLaM, KCDE, and GP in the x 4 − x 5 plan with all the other variables fixed at their expected value.
Figure 7 :
7Example 2 -Comparisons of the mean function estimation in the plan x 4 − x 5 with all the other input fixed at their expected value. The surrogate models are fitted to an ED with N = 1,000.
Figure 8 :
8Example 2 -Comparisons of the variance function estimation in the plan x 4 − x 5with all the other input fixed at their expected value. The surrogate models are fitted to an ED with N = 1,000.
Figure 9 :
9Example 2 -Comparison of the convergence between GLaMs and KCDEs in terms of the normalized Wasserstein distance as a function of the size of the experimental design. The dashed lines denote the average value over 50 repetitions of the full analysis. The green box plots and associated dashed lines correspond to the errors of the heteroskedastic Gaussian Process with sequential design (10 repetitions for each size of the experimental design). The "oracle" normal model has an error ε = 0 that is not plotted here.
time-consuming, especially for large experimental designs. This is probably due to the sequential design of experiments, which adds new points one by one and updates the surrogate after each enrichment. The associated simulations were performed on the ETH Euler cluster, and the average CPU time varied from 463 seconds for N = 250 to over 9 days for N = 4,000 to build a single GP. For KCDE, it took about 20 CPU seconds for N = 250 up to 30 minutes for N = 4,000 on a standard laptop. In comparison, constructing a GLaM is always on the order of seconds: around 8 seconds for both N = 250 and N = 4,000 on a standard laptop.
of replications using the previous two analytical examples. To this end, we generate data by replicating R ∈ {5; 10; 25; 50} for each set of input parameters in the ED. We keep the total number of simulations the same as nonreplicated cases by reducing the size of the ED accordingly.For instance, a data set of total N = 1,000 model evaluations with 10 replications consists of 100 different sets of input parameters, each of which is simulated 10 times.For quantitative comparisons, we investigate a convergence study similar to Sections 5.1 and 5.2: the total number of runs N varies in {250; 500; 1,000; 2,000; 4,000}, and each scenario is repeated 50 times.Figures 10 and 11 summarize the error defined in Eq. (21) averaged over the 50 repetitions for each R ∈ {5; 10; 25; 50}. In the first example, replications do not have a strong effect for R ∈ {5; 10; 25}. This is because the expansions for λ(x) contain only a few terms. Therefore, as long as we have enough ED points, exploring the input space and performing replications bring similar improvements to the surrogate accuracy. However, a large number of replications, i.e., R = 50, gives too few ED points for small values of N , which yields GLaMs of poor performance. In the second example, we observe a clear negative effect of replications: for the same total amount of model runs, the surrogate quality deteriorates when increasing the number of replications / decreasing the size of the experimental design.In summary, homogeneous replications (i.e., those with the same number of replicates for each point of the experimental design) do not necessarily bring additional accuracy and may even lead to a "waste" of computational budget for the proposed GLaM method. Nevertheless, this does not imply that replications are always useless. On the one hand, for methods that explore the usage of replications, there is a trade-off between replications and exploration[40]. On the other hand, an adaptive selection of different numbers of replications for each point in the experimental design could possibly improve the performance of the proposed method. However, unlike the heteroskedastic GP, GLaM not only estimates the mean and the variance but also produces the whole PDF. As a result, sequential design strategies for building GLaMs remain to be developed in future study and are outside the scope of the paper.
Figure 10 :
10Example 1 -Comparison of the GLaMs built on data with different number of replications. The curves corresponds to the mean error over the 50 repetitions.
Figure 11 :
11Example 2 -Comparison of the GLaMs built on data with different number of replications. The curves corresponds to the mean error over the 50 repetitions.
A
T (x) plays an important role in the Asian option modeling Eq. (31), the PDF of A T (x) is of interest in this case study. As in Section 5.1, we set T = 1, which corresponds to a one-year inspection period. We choose X 1 ∼ U(0, 0.1) and X 2 ∼ U(0.1, 0.4) for the two input random variables. Unlike S 1 (x), the distribution of A 1 (x) cannot be derived analytically. It is necessary to simulate the trajectory of S t (x) to compute A 1 (x). Based on the Markovian and lognormal properties of S t (x), we apply the following recursive equations for the path simulation with a time step ∆t = 0.001: S 0 (x) = 1,
Figure 12 :Figure 13 :
1213shows the mean and variance function, where the reference values can be obtained by applying Itô's calculus. For the experimental design of N = 500, the GLaM more accurately predicts the two functions. Finally, quantitative comparisons in Figure 15 confirm the superiority of GLaMs to KCDEs: GLaMs yield smaller average error for all N ∈ {250; 500; 1,000; 2,000; 4,000} and demonstrate a better convergence rate. Moreover, for large experimental designs (N ≥ 2,000), the average error of GLaMs is nearly half of that of (a) PDF for x = (0.03, 0.33) T (b) PDF for x = (0.07, 0.11) T Asian option -Comparisons of the emulated PDF, Asian option -Comparisons of the mean function estimation, N = 500.
Figure 14 :
14Asian option -Comparisons of the variance function estimation, N = 500.
Figure 15 :
15Asian option, average process A 1 (x) at T = 1 year -Comparison of the convergence of GLaMs and KCDEs in terms of the normalized Wasserstein distance as a function of the size of the experimental design. The dashed lines denote the average value over 50 repetitions of the full analysis. The black dash-dotted line represents the error of the model assuming that the response distribution is normal with the true mean and variance As a second quantity of interest, we consider the expected payoff µ C (x) = E [C(x)]. This quantity
Figure 16 :
16Asian option, expected payoff estimations -Comparison of the convergence of GLaMs and KCDEs in terms of the normalized mean squared error as a function of the size of the experimental design. The dashed lines denote the average value over 50 repetitions of the full analysis.
X 2 ∼Figure 17 :Figure 18 :
21718U(20, 200) (initial number of infected individuals). The QoI is the total number of newly infected individuals during the outbreak, i.e., Y (x) = S T − S 0 . (a) PDF for x = (1714, 165) T (b) PDF for x = (1364, 61) T SIR model -Comparisons of the emulated PDF, SIR model -Comparisons of the mean function estimation in the plan N = 500.
Figure 17
17compares two response PDFs estimated by a GLaM and by a KCDE for two sets of initial configurations, using an experimental design of size N = 500. The reference histograms are obtained by 10 4 repeated model runs for each x.
Figure 19 :
19SIR model -Comparisons of the variance function estimation, N = 500.We observe that both functions vary nonlinearly in the input space. Compared with the KCDE, the GLaM is able to capture the trend of the two functions and provides more accurate estimates.More detailed comparisons of the surrogate models are shown inFigure 20. The error of the oracle Gaussian approximation is quite small. This implies that the response distribution for most of the input parameters in the input space is close to a Gaussian distribution. Nevertheless, GLaMs built on N = 4,000 model runs still demonstrate better average behavior. For all sizes of experimental design, GLaMs clearly outperform KCDEs. For N ≥ 500, the biggest error of GLaMs is smaller than the smallest error of KCDEs among the 50 repetitions. Finally, to achieve the same accuracy as GLaMs, KCDEs require around 7 times more model runs.
Figure 20 :
20SIR model -Comparison of the convergence between GLaMs and KCDEs in terms of the normalized Wasserstein distance as a function of the size of the experimental design. The dashed line denotes the average value over 50 repetitions of the full analysis. The black dash-dotted line represents the error of the model assuming that the response distribution is normal with the true mean and variance
Figure 21 :
21SIR model, mean value estimations -Comparison of the convergence between GLaMs and KCDEs in terms of the normalized mean-squared error as a function of the size of the experimental design. The dashed line denotes the average value over 50 repetitions of the full analysis.
Example 1 .Figure 22 :
122Note that the associated theoretical aspects remain to be developed: it is necessary to prove the bootstrap consistency, which is usually achieved by showing the asymptotic normality of the estimator. As a result, the asymptotic properties of the maximum likelihood estimator in Eq. (17) need to be further investigated.(a) PDF for x = (0.03, 0.33) T (b) PDF for x = (0.07, 0.11) T Example 1 -Uncertainty on the PDF predicted by GLaM for two values of the input parameters, using an experimental design of N = 500. The blue line is the PDF predicted by GLaM from the 500 data points. The grey lines correspond to 100 PDFs generated by GLaM using bootstrapped experimental designs.
Lemma 1 .
1Under the conditions described in Theorem 1, we have the following: (i) Boundedness: sup c∈C q c (x, y) < +∞.
(3) to evaluate the PDF of GLDs. If u in Eq. (3) does not exist in [0, 1], q c = 0 and thus bounded. For u ∈ [0, 1], we have
For k = 0
0and 1, m(u) is a constant function equal to 2 and 1, respectively. If k = 0, 1, the derivative m (u) = k u k−1 − (1 − u) k−1 is equal to 0 only at u = 0.5 in [0, 1]. As a result, min m(u) = min {m(0), m(0.5), m(1)}. For k < 0, min m(u) = m(0.5) = 2 1−k . While for k > 0, min m(u) = min {m(0), m(0.5), m(1)} = min 1, 2 1−k . Hence, we have min m(u) ≥ min 1, 2 1−k = C m . Taking this property into account, Eq. (36) becomes
Dx h 2
2(qĉ, q 0 | x) f X (x)dx a.s. − − → 0,
for λ 3
3= λ 4 = 1 and λ 3 = λ 4 = 2. In other words, by excluding two points in the λ 3 − λ 4 plane, different values of λ lead to different distributions. Note that the two exceptions are the only two cases where the corresponding distributions are uniform distributions. As a result, the last condition in Theorem 1 excludes the nonidentifiable cases. Furthermore, λ PC (x; c) are polynomials in x and linear in c. Therefore, for c =c, λ PC (x; c) and λ PC (x;c) are not identical for µ-almost all x ∈ R M , and thus for P X -almost all x ∈ D X . Hence, there exists a set Ω x with P X (Ω x ) > 0 such that as long as c = c 0 , h (q c , q 0 | x) > 0 ∀x ∈ Ω x , which implies the uniqueness. Finally, combining Eq. (61) with the continuity and uniqueness of R(c), we haveĉ a.s. − − → c 0 .
, we use Latin hypercube sampling to generate a test set X test of size N test = 1,000 in the input space. The normalized Wasserstein distance is calculated for each x ∈ X test and then averaged by N test .For the last two case studies, the analytical response distribution of Y (x) is unknown. To
characterize it, we repeatedly evaluate the model 10 4 times for x. In addition, we also compare
some summarizing statistical quantity b(x) of the model response Y (x), such as the mean
E [Y (x)] or variance Var [Y (x)], depending on the focus of the application. Note that b(x) is a
deterministic function of input variables, and we define the normalized mean-squared error by
λ) is continuously differentiable. In addition, Eq. (39) is bounded away from 0. These two properties allow one to apply the implicit function theorem, and thus u is a continuous function of λ in a neighborhood ofλ, which implies that u is continuous atλ. According to Eq.(3), the PDF is a continuous function of both u and λ. Hence, using the continuity shown before, f Y (y; λ) is continuous atλ. Furthermore, λ PC (x; c) are C ∞ functions of c, and thus λ PC (x; c) is continuous atc. Combining both the continuity of f Y (y; λ) and λ PC (x; c), we have that q c (x, y) is continuous atc for the point (x, y).
Finally,[55, Theorem 22] extends the convergence to almost surely, which is the uniform strong law of large numbers. Now, we have all the ingredients to prove Theorem 1. Proof. Following [54, Lemma 4.1, 4.2], it can be easily shown that 0 ≤Dx
q c (x, y) = f Y |X y λ PC (x; c) , p c (x, y) = f X,Y (x, y) = f X (x)q c (x, y),where q c denotes the conditional PDF with model parameters c, and p c corresponds to the associated joint PDF. Under this setting, we assume that the true distribution q 0 belongs to the
AcknowledgmentsThis paper is a part of the project "Surrogate Modeling for Stochastic Simulators (SAMOS)" funded by the Swiss National Science Foundation (Grant #200021_ 175524), whose support is gratefully acknowledged.A AppendixA.1 Consistency of the maximum likelihood estimatorIn this section, we prove the consistency of the maximum likelihood estimator, as described in Theorem 1. For the ease of derivation, we introduce the following notation:
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| []
|
[
"Tunneling interstitial impurity in iron-chalcogenide based superconductors",
"Tunneling interstitial impurity in iron-chalcogenide based superconductors"
]
| [
"Huaixiang Huang \nDepartment of Physics\nShanghai Key Laboratory of High Temperature Superconductors\nShanghai University\n200444ShanghaiChina\n\nTexas Center for Superconductivity\nDepartment of Physics\nUniversity of Houston\n77204HoustonTexasUSA\n",
"Degang Zhang \nTexas Center for Superconductivity\nDepartment of Physics\nUniversity of Houston\n77204HoustonTexasUSA\n\nCollege of Physics and Electronic Engineering\nSichuan Normal University\n610101ChengduChina\n",
"Yi Gao \nDepartment of Physics\ninstitute of Theoretical Physics\nNanjing Normal University\n210023NanjingJiangsuChina\n",
"Wei Ren \nDepartment of Physics\nShanghai Key Laboratory of High Temperature Superconductors\nShanghai University\n200444ShanghaiChina\n\nInternational Centre for Quantum and Molecular Structures\nMaterials Genome Institute\nShanghai University\n200444ShanghaiChina\n",
"C S Ting \nTexas Center for Superconductivity\nDepartment of Physics\nUniversity of Houston\n77204HoustonTexasUSA\n"
]
| [
"Department of Physics\nShanghai Key Laboratory of High Temperature Superconductors\nShanghai University\n200444ShanghaiChina",
"Texas Center for Superconductivity\nDepartment of Physics\nUniversity of Houston\n77204HoustonTexasUSA",
"Texas Center for Superconductivity\nDepartment of Physics\nUniversity of Houston\n77204HoustonTexasUSA",
"College of Physics and Electronic Engineering\nSichuan Normal University\n610101ChengduChina",
"Department of Physics\ninstitute of Theoretical Physics\nNanjing Normal University\n210023NanjingJiangsuChina",
"Department of Physics\nShanghai Key Laboratory of High Temperature Superconductors\nShanghai University\n200444ShanghaiChina",
"International Centre for Quantum and Molecular Structures\nMaterials Genome Institute\nShanghai University\n200444ShanghaiChina",
"Texas Center for Superconductivity\nDepartment of Physics\nUniversity of Houston\n77204HoustonTexasUSA"
]
| []
| A pronounced local in-gap zero-energy bound state (ZBS) has been observed by recent scanning tunneling microscopy (STM) experiments on the interstitial Fe impurity (IFI) and its nearestneighboring (nn) sites in FeTe0.5Se0.5 superconducting (SC) compound. By introducing a new impurity mechanism, the so-called tunneling impurity, and based on the Bogoliubove-de Gennes (BDG) equations we investigated the low-lying energy states of the IFI and the underlying Feplane. We found the peak of ZBS does not shift or split in a magnetic field as long as the tunneling parameter between IFI and the Fe-plane is sufficiently small and the Fe-plane is deep in the SC state. Our results are in good agreement with the experiments. We also predicted that modulation of spin density wave (SDW), or charge density wave (CDW) will suppress the intensity of the ZBS. | 10.1103/physrevb.93.064519 | [
"https://arxiv.org/pdf/1507.06358v1.pdf"
]
| 118,370,253 | 1507.06358 | 9d789e519ec05438d87b4e70759486e4a3c06cde |
Tunneling interstitial impurity in iron-chalcogenide based superconductors
22 Jul 2015
Huaixiang Huang
Department of Physics
Shanghai Key Laboratory of High Temperature Superconductors
Shanghai University
200444ShanghaiChina
Texas Center for Superconductivity
Department of Physics
University of Houston
77204HoustonTexasUSA
Degang Zhang
Texas Center for Superconductivity
Department of Physics
University of Houston
77204HoustonTexasUSA
College of Physics and Electronic Engineering
Sichuan Normal University
610101ChengduChina
Yi Gao
Department of Physics
institute of Theoretical Physics
Nanjing Normal University
210023NanjingJiangsuChina
Wei Ren
Department of Physics
Shanghai Key Laboratory of High Temperature Superconductors
Shanghai University
200444ShanghaiChina
International Centre for Quantum and Molecular Structures
Materials Genome Institute
Shanghai University
200444ShanghaiChina
C S Ting
Texas Center for Superconductivity
Department of Physics
University of Houston
77204HoustonTexasUSA
Tunneling interstitial impurity in iron-chalcogenide based superconductors
22 Jul 2015(Dated: July 24, 2015)arXiv:1507.06358v1 [cond-mat.supr-con]
A pronounced local in-gap zero-energy bound state (ZBS) has been observed by recent scanning tunneling microscopy (STM) experiments on the interstitial Fe impurity (IFI) and its nearestneighboring (nn) sites in FeTe0.5Se0.5 superconducting (SC) compound. By introducing a new impurity mechanism, the so-called tunneling impurity, and based on the Bogoliubove-de Gennes (BDG) equations we investigated the low-lying energy states of the IFI and the underlying Feplane. We found the peak of ZBS does not shift or split in a magnetic field as long as the tunneling parameter between IFI and the Fe-plane is sufficiently small and the Fe-plane is deep in the SC state. Our results are in good agreement with the experiments. We also predicted that modulation of spin density wave (SDW), or charge density wave (CDW) will suppress the intensity of the ZBS.
Since the discovery of iron-based superconductor, [1] new compounds continue to be found. The 11 type iron chalcogenides has attracted much attention due to the simplicity of its crystal structure. Angle-resolved photoemission spectroscopy, STM, and transport experiments [2][3][4][5] have been performed to investigate the electronic structure and SC gap. By substituting Se for Te in FeTe compound, superconductivity appears and presents a variety of phenomena for different mixing ratios of Te and Se [6]. The as-grown Fe 1+y Te x Se 1−x single crystals contain a large amount of excess Fe that are randomly situated at the interstitial sites in the crystal. Recent STM experiments investigated the electronic state near the interstitial Fe in FeSe 0.5 Te 0.5 superconductors. [7] A robust ZBS is observed in the tunneling spectrum taken at the center of the IFI and its nn sites with the intensity decaying quickly. Interestingly, the peak does not split or shift in a magnetic field which is drastically inconsistent with the magnetic or non-magnetic impurity effects in either d-wave or s-wave superconductors. [8][9][10] The interplay of the IFI with the in-plane Fe and the opposite Se (Te) are complex. The valence of excess iron does not equal to that of the in-plane iron, the effects of the IFI to the physical property are sensitive to the stoichiometry. [3,[11][12][13][14]. Although the magnetic moment of the access Fe has finite value, for SC state Fe 1+y Te 0.5 Se 0.5 , we assume that the IFI is coupled to the underlying Fe-plane by a hopping term. Taking the IFI as a tunneling impurity [15] without scattering potential, and based on the BDG equations in the presence as well as in the absence of the magnetic field, we systematically investigated the low-lying energy states of the IFI and the in-plane Fe. We found the IFI-induced in-gap ZBS is a common feature for iron-based superconductors as long as the hopping term between the IFI and the underlying Fe-plane is sufficiently small and Fe-plane is deep in the SC state. The ZBS is isotropic and localized, the intensity of the induced ZBS decays to zero at a distance of three or four lattice constants away from the IFI.
In the optimally doped regime, and when the external magnetic field is not very large, beside the appearance of the in-gap resonance peaks induced by the magnetic field [16,17], the IFI-induced no-splitting no-shifting ZBS shows up at its nn sites no matter where we put the IFI. This phenomenon is contrary to the Zeeman effect of single electron since the IFI-induced ZBS results from Cooper pairs. Intensity of the ZBS at nn sites has the maximum value as the IFI is located above the vortex core center. Considerably large magnetic field may induce charge oscillation, and breaks the process of reforming Cooper pairs when electron tunnels back to the Feplane from the IFI. Therefore larger magnetic field will suppress the ZBS at the nn sites of the IFI. We believe that the tunneling impurity is a new scenario to understand the experimental finding of IFI effect.
Since the electronic structure and Fermi-surface topology of the 11 system is very similar to those of the other iron-based superconductors, [3] we use a minimal twoorbital four-band tight binding model [18] to investigate Fe 1+y Te 0.5 Se 0.5 . Different from the co-planar CuO 2 , superconducting Fe-plane is sandwiched between two adjacent Se(Te) layers, the next-nearest-neighbor (nnn) hoppings are mediated by up and down Se (Te) and are not equal to each other. This asymmetry should be correct when one investigates the surface properties as in STM experiments, because the bonds between up Te (Se) ions and Fe ions are broken after cleavage. Theoretical results [16,17,[19][20][21][22][23] based on this model are qualitatively consistent with experiments. [24][25][26] The projected up and down Se (Te) atoms stay at the centers of the Fe plaquette alternatively, and the IFI is located at the opposite side of the down Se (Te) above the Fe-plane[see Fig.1]. The total Hamiltonian is expressed as H = H 0 + H IFI , H 0 is the mean-field phenomenological model Hamiltonian without excess Fe, reads
H 0 = − iµjνσ (t iµjν c † iµσ c jνσ + h.c.) −μ iµσ c † iµσ c iµσ + U ′ i,ν,σ n i n iνσ + iµjνσ (∆ iµjν c † iµσ c † jνσ + h.c.) ,(1)
where n iµσ is the electron concentration at site i, orbital µ for spin σ,μ is the chemical potential, determined by the average electron filling on Fe site n i = 2 + x, x denotes electron doping concentration. H 0 can be separated to three parts. The first line is tight binding part, with the hopping integral t 1−4 = 1, 0.4, −2, 0.04 [18] as depicted in Fig.1. The first term in the second line is the mean field expression of Coulomb and Hund's interaction H 0,int in the SC state without SDW or CDW. It will be more complicated when SDW or CDW exists
H 0,int = i,ν ′ =ν,σ U n iνσ n iνσ + (U − 3J H ) n iν ′ σ n iνσ + (U − 2J H ) n iν ′σ n iνσ .(2)
Here we take U = 3.4,
J H = 1.3 and U ′ = 3U−5JH 4
. [19][20][21][22][23] The last term in the second line is the phenomenological superconducting pairing part. Although there are controversies about the symmetry of the SC pairing, s ± -wave [4,27,28] is suggested for Fe 1+y Te 1−x Se x . In real space it is intraorbital pairing ∆ iµjν = V 2 c iµ↑ c jµ↓ − c iµ↓ c jµ↑ , where j is the nnn of i site. The self-consistent mean field are ∆ iµjµ and n iµ .
The role of the IFI is to provide two tunneling channels corresponding to the two orbitals of Fe. Electron tunneling onto IFI belongs to a Cooper pair since the Fe-plane is in the SC state, when it tunnels back to the Fe-plane a new Cooper pair reforms. Because the effective magnetic moment of the cooper pair is zero, there is no magnetic interaction between the IFI and the Fe-plane. The effective coupling of the IFI to the Fe-plane reads
H IFI = −t IFI iµσ c † µσ c iµσ ,(3)
wherec † is the creation operator of the IFI, means the summation up to its four nn sites in the Fe-plane. Tunneling magnitude t IFI is a tuning parameter in our calculations.
Magnetic field contains a dynamical term for electrons, in the mixed state, the effect of the magnetic field is included through the Peierls phase factor. For a perpendicular magnetic field, the hopping integral in Eq. (1) and (3) should be changed to t ′ which can be expressed as
t ′ iµjν = t iµjν exp [iϕ ij ], where ϕ ij = π Φ0
i j A(r) · dr, and Φ 0 = hc/2e is the superconducting flux quantum. We assume the applied magnetic field B to be uniform and the vector potential is A = (−By, 0, 0) in the Landau gauge. Periodic boundary condition should be consistent with the requirement that the total phase factor along each small plaquette given by ϕ ij = πBa 2 Φ0 . The presence of Peierls phase makes the usual translation operator change to magnetic translation operator. To ensure them commutable with each other and the Hamiltonian, each magnetic unit cell has to contain 2Φ 0 flux [29,30]. In this case the supercell technique can be used to calculate the local density of states (LDOS) in the presence of B. We self-consistently solve the BDG equations in real space, linear dimension of a unit cell is N x × N y = 32 × 32 for B = 0, while for finite B we adopt N x = 2N y . The LDOS is calculated according to
ρ i (ω) = nµk [|u n iµσk | 2 δ(E n,k − ω) + |v n iµσk | 2 δ(E n,k + ω)],
where the delta function δ(x) is Γ/π(x 2 + Γ 2 ), with the quasiparticle damping Γ = 0.005. u n iµk and v n iµk are the eigenstate component of the energy E n,k at site i orbital µ for wave vector k, and they correspond to the particlelike component and hole-like component respectively. The number of unit cell is taken as M x × M y = 20 × 20 for B = 0 and 10 × 20 when external B is applied, with k α = − π Nα + j2π NαMα , α = x, y. Throughout the paper, temperature is zero K, the energy and length are measured in units of t 1 and the nearest Fe-Fe distance a, respectively.
Although the experiments was performed on half-filled SC compounds [7], we found ZBS is a common feature for SC state as long as t IFI is small. Fig.2 shows LDOS ρ for different t IFI with x = 0.0 and x = 0.12 for the Hamiltonian of Eq.1. For x = 0.0 and t IFI less than 0.04, LDOS at the IFI has a single sharp ZBS peak which can be seen in Fig.2(a), and the peak height is decreased with the increasing of t IFI .
At its four nn sites ρ is isotropic. Compared to the pure sample, a pronounced in-gap ZBS appears and the height of the peak is enhanced with t IFI increasing from 0.01 to 0.03 as seen in Fig.2(b). At the sites two lattice constants away from the nn sites, the intensity of the in-gap ZBS decays almost to zero as shown in Fig.2(c). For different doping cases, we expect the ZBS still exist whenever Fe-plane is in the SC state. Fig.2(d)-(f) plot ρ at the IFI and its vicinity for x = 0.12. The curves are very similar to those of the x = 0.0 cases with the corresponding peak of the ZBS a little higher.
Small and large t IFI give significantly distinct results. For relatively larger t IFI , the results deviate from the experimental observation. When x = 0.12, t IFI = 0.05, ρ at the IFI splits into two asymmetrical peaks shown by the violet dotted line in Fig.2(d). Due to the proximity effect the LDOS at the nn sites of the IFI also shows double peaks with heavy weight on the positive energy and can be seen in Fig.2(e). Therefore, sufficiently small t IFI is an important condition for the appearance of ZBS. As temperature increases, the phonon mediated layer tunneling t IFI is also increased, thus ZBS will not appear for higher temperature cases.
We have addressed the cases when Fe-plane does not have magnetic order defined as m i = (n i↑ − n i↓ )/2. In fact, as magnetic order coexists with the SC order which will happen in underdoped cases, the LDOS at the IFI and its nn sites will still have the in-gap ZBS. In the following we will discuss the more complicated cases with H 0,int taking the form of Eq.2. The formation of the ZBS at the nn sites is similar to the Andreev reflection and we expect the homogeneous magnetic order itself does not suppress ZBS as long as the sample is in the SC state. We can see from Fig.3(a) that the ZBS still appears at nn sites of the IFI for underdoped case x = 0.06 with t IFI = 0.02. While an external B could drive SDW modulation [see Fig.3(c)] and CDW in this case, the competition between the oscillated SDW and SC order ruins the proximity effect. Therefore as SDW exists, an external magnetic field will suppress the ZBS as well as other ingap states which is depicted in Fig.3(b). In Fig.3(b), the IFI is located above the vortex core center with the ZBS almost vanished; as the above IFI moves away from the center, the ZBS does not appear.
In the optimally doped regime, we investigated perturbation of IFI on the mixed state. ρ is plotted in Fig.4 at IFI as well as at its nn sites for different B with fixed t IFI = 0.02, x = 0.12. A unit cell of linear dimension 64 × 32 corresponds to a relatively small magnetic field B ≈ 13T and 40 × 20 corresponds to relatively large field B ≈ 32T. In all cases of small t IFI , ρ on the IFI has a sharp ZBS regardless of the strength of B and the position of the IFI. The height of the peak becomes lower as the IFI moves close to the vortex core center which can be seen in Fig.4(a). Now we look at the relatively small magnetic field B ≈ 13T. In the absence of the IFI, two in-gap resonance peaks induced by B are located at negative energies in the vortex core. At sites away from the vortex, resonance peaks are suppressed and move to the gap edge, finally evolve into its bulk feature [see Fig.4(b)]. When introducing IFI into the system, the above characteristics do not change except for the additional appearance of ZBS induced by the IFI which is depicted in Fig.4(c). We can see that the in-gap ZBS shows up at nn sites no matter where we put the IFI. Compared with the cases without B [Fig.2], the height of the ZBS at nn sites of the IFI is obviously enhanced, especially when the IFI is located above the vortex core center and has its maximum value. As the IFI moves away from the vortex center, the peak on nn sites is decreased. When the projected point of IFI on Fe-plane is 16 lattice constants away from vortex core center along x-axis, the height of the ZBS of nn sites evolves back to the value of B = 0 case. We also notice that a lower height of ZBS on the IFI [see Fig.4(a)] corresponds to a higher peak of ZBS on its nn sites [see Fig.4(c)]. The more of quasiparticle tunneling onto nn site, the less part is remained on the IFI. Results of B ≈ 22T (48 × 24 unit cell) are very similar to those of B ≈ 13T and we do not show them here.
Without the IFI, n i is enhanced and the pairing parameter ∆ i = j ∆ ij /4 is almost vanished at the vortex core center, with ∆ i and n i being symmetric with respect to the two vortex cores. Introducing of the IFI breaks the symmetry. For smaller B ≈ 13T and projected IFI at vortex core center (16.5, 16.5), although n i on the two vortex core are different, it increases to bulk value at the scale of coherence length. Out of vortex core, n i is almost homogeneous which can be seen in the Fig.4(e). While larger B leads to oscillation of n i , Fig.4(f) displays the inhomogeneous n i for B ≈ 32T with IFI located above vortex core center (29.5, 15.5). The difference of n i on the two vortex cores is 0.01, much larger than B = 0 cases in which ∆n i induced by IFI is at the order of 10 −3 . Obviously, magnetic field significantly enhanced the impact of IFI to the Fe-plane. The oscillated CDW ruins the Cooper pair tunneling and reforming, hence suppresses the ZBS on nn sites. For large B ≈ 32T, Fig.4(d) shows that when IFI locates above vertex core center, ZBS appears at its nn sites with the peak much lower than that of small B case. As the above IFI moves away from vortex core center, it still have impact on nn sites, however the induced state is not ZBS and almost invisible in Fig.4(d).
In summary, motivated by the recent STM experiments [7] on iron-based superconductor FeTe 0.5 Se 0.5 , we investigated the impact of IFI on the s ± -wave iron-based superconductors. Taking IFI as a tunneling impurity, [15] we calculated LDOS on the IFI as well as on Fe-plane in the absence and in the presence of magnetic field. In all cases with and without external magnetic field, LDOS at the IFI has a single sharp ZBS as long as hopping parameter t IFI is sufficiently small and Fe-plane is in SC state. The ZBS always appears on its nn sites due to the proximity effect in the absence of B. When a magnetic field is applied, our results are in good agreement with the experiments. When Fe-plane is deep in SC state, the IFI-induced ZBS shows up at nn sites no matter where we put the IFI. It does not split or shift and the intensity is enhanced. The effect of the IFI is isotropic and localized in all cases. However when Fe-plane has oscillated SDW or CDW, the intensity of ZBS on nn sites will be suppressed. In underdoped cases, a magnetic field can lead to modulation of SDW. And when the applied magnetic field is considerably large, it may induce CDW in the IFI system.
Tunneling impurity only provide tunneling channels in the system. Due to the superconducting state of Fe-plane, the effective tunneling is Cooper pair tunneling with zero magnetic moment thus no Zeeman splitting. Although experiments and density functional theory study verified the access Fe is spin polarized [14], we do not consider exchange interaction between excess Fe and superconducting Fe-plane. The origin of IFI-induced ZBS on nn sites is related to Andreev reflection. As far as we know, a convinced explanation for the STM experiments is still lacking. Although it may associated with non-trivial topological or other factors, our simple tunneling IFI gives a natural explanation for the experiments and provides a new interpretation for interstitial impurity.
We thank Jiaxin Yin for helpful discussions. This work was supported by the Texas Center for Superconductivity at the University of Houston and the
online)Schematic plot of the tight-binding model. The right picture shows the relative position of IFI to the Fe-plane.
FIG
. 2: (color online) LDOS at IFI and its nn sites as well as at the nearby nn + 2 site for different tIFI = 0.01t1, 0.02t1, 0.03t1. Panel (a)-(c) are for x = 0.0 while panel (d)-(f) for x = 0.12. In (d) and (e) the dotted volet line are for tIFI = 0.05.
FIG. 3 :
3(color online) For tIFI = 0.02, x = 0.06, (a)LDOS at nn sites without B. (b)LDOS at nn site for B ≈ 22T with IFI projected at vortex core center(12.5, 12.5). (c)Imagine of modulated |mi| for B ≈ 22T with IFI above vortex core center.
online)(a)LDOS at different IFI for B ≈ 13T.(b)Without IFI, LDOS at vortex core center and at sites away from vortex for B ≈ 13T.(c)With the IFI located at different place, LDOS at the corresponding nn site for B ≈ 13T. (d)Similar to (c) but for B ≈ 32T. From up to down the curves correspond to the projected IFI at vortex core center, and L lattice constants away from the center along horizontal axis. The gray dashed line indicate the position of zero energy. (e)Image of ni for B ≈ 13T with the projected IFI at (16.5, 16.5). (f)Image of ni for B ≈ 32T with the projected IFI at (29.5, 15.5). The curves of (b)(c)and (d) are displaced vertically by 0.5 unit for clarify.
Robert A. Welch Foundation under the Grant No. E-1146, National Key Basic Research Program of China (Grant No. 2015CB921600), QiMingXing Project (No. 14QA1402000) of Shanghai Municipal Science and Technology Commission, Eastern Scholar Program and Shuguang Program (No. 12SG34) from Shanghai Municipal Education Commission, NSF of Shanghai(Grant No. 13ZR1415400), Shanghai Key Lab for Astrophysics (Grant No. SKLA1303), NSFC (Grant Nos. 11204138 and 11274222) and NSF of Jiangsu Province of China
BK2012450), the Sichuan Normal University and the "thousand talent program. Sichuan Province, ChinaGrant No. BK2012450), the Sichuan Normal University and the "thousand talent program" of Sichuan Province, China.
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| []
|
[
"Optimal Placement of Distributed Energy Storage in Power Networks",
"Optimal Placement of Distributed Energy Storage in Power Networks"
]
| [
"Student Member, IEEEChristos Thrampoulidis ",
"Student Member, IEEESubhonmesh Bose ",
"Fellow, IEEE.Babak Hassibi "
]
| []
| []
| Large-scale storage is a promising emerging technology to realize a reliable smart-grid since it can enhance sustainability, reliability and asset utilization. At fast time-scales, storage can be used to mitigate fluctuations and time-variations on renewable generation and demand. On slower time scales, it can be used for load shifting and reducing the generation costs. This paper deals with the latter case and studies the following important decision problem: Given a network and an available storage budget, how should we optimally place, size and control the energy storage units across the network. When the generation cost is a nondecreasing convex function, our main result states that it is it is always optimal to allocate zero storage capacity at generator buses that connect to the rest of the power grid via single links, regardless of demand profiles and other network parameters. This provides a sound investment strategy for network planners, especially for distribution systems and isolated transmission networks.For specific network topologies, we also characterize the dependence of the optimal production cost on the available storage resources, generation capacities and flow constraints. | 10.1109/tac.2015.2437527 | [
"https://arxiv.org/pdf/1303.5805v2.pdf"
]
| 52,319 | 1303.5805 | a73f14567bb2488c97a5370f00121f03756027db |
Optimal Placement of Distributed Energy Storage in Power Networks
23 Mar 2013
Student Member, IEEEChristos Thrampoulidis
Student Member, IEEESubhonmesh Bose
Fellow, IEEE.Babak Hassibi
Optimal Placement of Distributed Energy Storage in Power Networks
23 Mar 2013arXiv:1303.5805v1 [math.OC]
Large-scale storage is a promising emerging technology to realize a reliable smart-grid since it can enhance sustainability, reliability and asset utilization. At fast time-scales, storage can be used to mitigate fluctuations and time-variations on renewable generation and demand. On slower time scales, it can be used for load shifting and reducing the generation costs. This paper deals with the latter case and studies the following important decision problem: Given a network and an available storage budget, how should we optimally place, size and control the energy storage units across the network. When the generation cost is a nondecreasing convex function, our main result states that it is it is always optimal to allocate zero storage capacity at generator buses that connect to the rest of the power grid via single links, regardless of demand profiles and other network parameters. This provides a sound investment strategy for network planners, especially for distribution systems and isolated transmission networks.For specific network topologies, we also characterize the dependence of the optimal production cost on the available storage resources, generation capacities and flow constraints.
I. INTRODUCTION
Emerging energy storage technologies in the electricity grid can be potentially used for various services, e.g., to enhance sustainability, reliability, efficiency and asset utilization [1]- [4]. The economic value [5]- [7] and national impact [8], [9] of these services has been emphasized in various recent studies. Over the last decade or so, these technologies have shown remarkable Emails: (cthrampo, boses, hassibi)@caltech.edu. This work was supported in part by NSF under grants CCF-0729203, CNS-0932428 and CCF-1018927, NetSE grant CNS 0911041, Office of Naval Research MURI grant N00014-08-1-0747, Caltech Lee Center for Adv. Net., ARPA-E grant de-ar0000226, Southern Cali. Edison, Nat. Sc. Council of Taiwan, R.O.C. grant NSC 101-3113-P-008-001, Resnick Inst., Okawa Foundation and Andreas Mentzelopoulos Scholarships for the Univ. of Patras. technical improvements and their cost of production has dropped significantly [10]- [12]. Also, the obvious challenges in integrating intermittent renewable generation has spurred the interest in large-scale storage to deal with their variability [13]- [16] . In summary, the authors in [8] identify the use of energy storage as "critical to achieving national energy policy objectives and creating a modern and secure electric grid system." For this promising new technology, we study an efficient investment strategy of placing and sizing such units in a network to minimize generation cost.
Applications of energy storage can be classified according to time scale of operation [3], [5], [17]. At fast-time scales (seconds to minutes), storage is mainly used to mitigate variability of renewable generation and demand and reduce the role of costly ancillary services to balance demand and supply, e.g., [18]- [21]. On a slower time scale (over hours), bulk storage devices aim at load shifting, i.e., generate when it is cheap and supply the residual load using storage dynamics [5], [10]. In the present work, we only deal with the latter. Our results are therefore complementary to those of [18]- [21]. The short-time scale strategies developed there can be implemented " on top " of the long-term strategies developed here. Now, we provide a brief overview of the relevant literature. Optimal control policy for storage units has been extensively studied. While the authors in [20], [22] examine the control of a single storage device without a network, [21], [23], [24] explicitly model the role of the networks in the operation of distributed storage resources. The engineering constraints are designed based on one of the two popular power flow models [25]- [27], namely, AC power flow and DC power flow. Storage resources at each node in the network are assumed to be known a priori in these settings.
For the emerging smart grid technology, the investment decision problem of selecting, sizing and placement of distributed energy resources is gaining importance. As noted in [28], it is critical to "assess the technical and economic attributes of energy storage specifically reflecting the operational demands and opportunities presented by the smart grid environment. Without the ability to analyze network features of transmission and distribution, storage system technologies and their efficiencies, along with their cost benefits for various value streams, there is no ability for the utility to make comparative business decisions that will enable the optimal siting of energy storage." Several researchers have analyzed aspects of this problem, e.g., [29], [30] using purely economic arguments, without explicitly considering the network constraints of the physical system. Authors in [20], [31] have looked at optimal sizing of storage devices in singlebus power systems, while Kanoria et al. [21] compute the effect of sizing of distributed storage resources on generation cost for specific networks. Gayme et. al [23] study a similar problem in IEEE benchmark systems [32]. Recently, a more general framework to study the optimal storage placement problem in generic networks has been formulated and studied through simulations in [33], [34] using tools similar to optimal power flow (OPF) . The resulting problem is non-convex and hence Sjödin et al. in [33] use the approximate DC-OPF [26], [27] while Bose et al. in [34] use the relaxation of AC-OPF [35]- [38] based on semidefinite programming [39], [40].
In this paper, we consider the investment decision problem of how much storage to place on each node in a network given an available storage budget. As a by-product, we also derive the optimal control policy for the storage units. Our main result shows that when minimizing a convex and nondecreasing generation cost with any fixed available storage budget, there always exists an optimal storage allocation that assigns zero storage at those generator nodes that connect via single lines to the network, for arbitrary demand profiles and other network parameters. It suggests that in most distribution networks and isolated transmission networks, it is always optimal to allocate the entire available storage budget among demand buses. Also, this work provides an analytic justification to the conjecture that the optimal allocation of storage resources mainly depends on the network structure as opposed to the total available storage budget as noted through simulations in [23], [33], [34]. For generator buses with multiple line connections, however, we show that the optimal storage placement may not in general place zero storage capacity at such nodes. Furthermore, we study the dependence of optimal production cost on the available storage budget, line flow and generation capacities for specific network topologies. The physical network has been modeled using DC power flow. Preliminary versions of our result have appeared in [41] and [42].
The paper is organized as follows. We formulate the optimal storage placement problem in Section II. The main result is proven and discussed in Section III. More detailed analysis of the problem on specific network topologies is discussed in Section IV. Concluding remarks and directions for future work are outlined in Section V. Technical details appear in the Appendix.
II. PROBLEM FORMULATION
Consider a power network that is defined by an undirected connected graph G on n nodes (or buses) N = {1, 2, . . . , n}. For two nodes k and l in N , let k ∼ l denote that k is connected to l in G by a transmission line. For the power flow model, we use the linearized DC approximation; see [26] for a detailed survey. In this approximation, the network is assumed to be lossless, the voltage magnitudes are assumed to be at their nominal values at all buses and the voltage phase angle differences are small.
Time is discrete and is indexed by t. Now consider the following notation.
• d k (t) is the known real power demand at bus k ∈ N at time t. Demand profiles often show diurnal variations [43], i.e., they exhibit cyclic behavior. Let T time-steps denote the cycle length of the variation. In particular, for all k ∈ N , t ≥ 0, assume
d k (t + T ) = d k (t).
• g k (t) is the real power generation at bus k ∈ N at time t and it satisfies
0 ≤ g k (t) ≤ g k ,(1)
where, g k is the generation capacity at bus k.
• c k (g k ) denotes the cost of generating power g k at bus k ∈ N . The cost of generation is assumed to be independent of time t and depends only on the generation technology at bus k. Also, suppose that the function c k : R + → R + is non-decreasing, differentiable and convex. A cost function commonly found in the literature [23], [25], [37], [38] that satisfies the above assumptions is:
c k (g k ) = γ k,2 g 2 k + γ k,1 g k + γ k,0 ,
where γ k,2 , γ k,1 , γ k,0 are known nonnegative coefficients.
• θ k (t) is the voltage phase angle at bus k ∈ N at time t.
• For two nodes k ∼ l, let p kl (t) be the power flowing from bus k to bus l at time t. It satisfies
p kl (t) = y kl [θ k (t) − θ l (t)] ,(2)
where y kl is the admittance of the line joining buses k and l. Also, the power delivered over this line is limited by thermal effects and stability constraints and hence, we have
|p kl (t)| ≤ f kl ,(3)
where f kl is the capacity of the corresponding line. • γ k (t) and δ k (t) are the average charging and discharging powers of the storage unit at bus k ∈ N at time t, respectively. The energy transacted over a time-step is converted to power units by dividing it by the length of the time-step. This transformation conveniently allows us to formulate the problem in units of power [34]. Let 0 < α γ , α δ ≤ 1 denote the charging and discharging efficiencies, respectively of the storage technology used, i.e., the power flowing in and out of the storage device at node k ∈ N at time t is α γ γ k (t) and 1 α δ δ k (t), respectively [20], [44]. The roundtrip efficiency of this storage technology is α = α γ α δ ≤ 1.
storage device ∼ · · · g k (t) d k (t) γ k (t) α γ γ k (t) δ k (t) 1 α δ δ k (t) p kl (t)
• s k (t) denotes the storage level at node k ∈ N at time t and s 0 k is the storage level at node k at time t = 0. From the definitions above, we have that
s k (t) = s 0 k + t τ =1 α γ γ k (τ ) − 1 α δ δ k (τ ) .(4)
For each k ∈ N , assume s 0 k = 0, so that the storage units are empty at installation time.
• b k ≥ 0 is the storage capacity at bus k. Thus, s k (t) for all t satisfies the following:
0 ≤ s k (t) ≤ b k .(5)
• h is the available storage budget and denotes the total amount of storage capacity that can be installed in the network. Our optimization algorithm decides the allocation of storage capacity b k at each node k ∈ N and thus, we have
k∈N b k ≤ h.(6)
• The charging and discharging rates of each storage device are assumed to be upper-bounded by ramp limits. Suppose these limits are proportional to the capacity of the corresponding device, i.e., for all k ∈ N ,
0 ≤ γ k (t) ≤ ǫ γ b k , (7a) 0 ≤ δ k (t) ≤ ǫ δ b k ,(7b)
where ǫ γ ∈ (0, 1 αγ ] and ǫ δ ∈ (0, α δ ] are fixed constants. Balancing power that flows in and out of bus k ∈ N at time t, as shown in Figure 1, we have:
g k (t) − d k (t) − γ k (t) + δ k (t) = l∼k p kl (t).(8)
Also, optimally placing storage over an infinite horizon is equivalent to solving this problem over a singe cycle, provided the state of the storage levels at the end of a cycle is the same as its initial condition [34]. Thus, for each k ∈ N , we have
T t=1 α γ γ k (t) − 1 α δ δ k (t) = 0.(9)
For convenience, denote [T ] := {1, 2, . . . , T }. Using the above notation, we define the following optimization problem.
Storage placement problem P : Restrict attention to network topologies where each bus either has generation or load but not both. Partition the set of buses N into two groups N G and N D where they represent the generation-only and load-only buses respectively and assume N G and N D are non-empty. For any subset K of N G , define the following optimization problem.
minimize k∈N T t=1 c k (g k (t)) over (g k (t), γ k (t), δ k (t), θ k (t), p kl (t), b k ), k ∈ N , k ∼ l, t ∈ [T ],
Restricted storage placement problem Π K :
minimize k∈N T t=1 c k (g k (t)) over (g k (t), γ k (t), δ k (t), θ k (t), p kl (t), b k ), k ∈ N , k ∼ l, t ∈ [T ], subject to (1), (2), (3), (4), (5), (6), (7), (8), (9), b i = 0, i ∈ K.
Problem Π K corresponds to placing no storage at the (generation) buses of the network in subset K. We study the relation between the problems P and Π K in the rest of the paper.
We say bus k ∈ N has a single connection if it has exactly one neighboring node l ∼ k.
Similarly, a bus k ∈ N has multiple connections if it has more than one neighboring node in G. We illustrate the notation using the network in Figure
III. MAIN RESULT
For a subset K ⊆ N G , let p * and π * K be the optimal values for problems P and Π K , respectively. Now, we are ready to present the main result of this paper. Theorem 1. Suppose K ⊆ N G and each node i ∈ K has a single connection. If P is feasible,
then Π K is feasible and p * = π K * .
The theorem states that, for any available storage budget, there always exists an optimal allocation of storage capacities that places no storage at any subset of generation buses with single connections. The result holds, regardless of the demand profiles and other network parameters, such as line flow constraints and impedances of transmission lines. In our model, the demand profiles of the load buses are deterministic. However, since Theorem 1 holds for arbitrary demand profiles, our result extends to the case where the load is stochastic.
Note that we restrict attention to generator buses in K that have single connections only. This is applicable in many practical scenarios, as discussed in Section III-B. The result is not true, in general, if K includes generator buses with multiple connections. We provide an example in Section III-C to illustrate this.
A. Proof of Theorem 1
We only prove for the case where the round-trip efficiency is α < 1, but the result holds for α = 1 as well. Assume P is feasible throughout. For any variable z in problem P , let z * be the value of the corresponding variable at the optimum. In our proof, we use the following technical result.
Lemma 1.
Suppose φ : R → R is convex and differentiable. Then, for any x 1 < x 2 and
0 ≤ η ≤ (x 2 − x 1 ): φ(x 1 + η) + φ(x 2 − η) ≤ φ(x 1 ) + φ(x 2 ).
Proof: We only prove the result for 0 < η ≤ (x 2 − x 1 )/2. The other cases can be shown similarly. The function φ(·) is continuous and differentiable. From mean-value Theorem [39], it follows that there exists
ξ 1 ∈ (x 1 , x 1 + η) and ξ 2 ∈ (x 2 − η, x 2 ), such that [φ(x 1 + η) − φ(x 1 )] − [φ(x 2 ) − φ(x 2 − η)] = φ ′ (ξ 1 )η − φ ′ (ξ 2 )η. Also, ξ 1 < ξ 2 and hence φ ′ (ξ 1 ) ≤ φ ′ (ξ 2 ) since φ is
convex. This completes the proof.
Consider node i ∈ K and j ∼ i. Node j is uniquely defined as i has a single connection.
It can be shown that problem P , in general, has multiple optima. In the following result, we characterize only a subset of these optima.
Lemma 2. There exists an optimal solution of P such that for all t ∈ [T ] and all
i ∈ K, j ∼ i, (a) g * i (t)γ * i (t)δ * i (t) = 0, (b) g * i (t) ≤ f ij .
The first part of Lemma 2 essentially says that for some optimum solution of P , the storage units should not charge and discharge at the same time step if there is positive generation at the same bus at that time step. This is expected since the round-trip efficiency of the storage devices α = α γ α δ is less than one and since the generation cost is a nondecreasing function.
The second part can be interpreted as follows. Power that flows from bus i to bus j at each
t ∈ [T ] is p ij (t) = g i (t) − γ i (t) + δ i (t) and we have p ij (t) ≤ f ij . But Lemma 2(b) states that there exists an optimum for which, g * i (t), t ∈ [T ]
itself defines a feasible flow over this line. Proof: The feasible set of problem P is a bounded 1 polytope and the objective function is a continuous convex function. Hence the set of the optima of P is a convex and compact set [39]. Now, with every point in the set of optimal solutions of P , consider the function i∈K,t∈[T ] (γ i (t) + δ i (t)). This is a linear and hence continuous function on the compact set of optima of P and hence attains a minimum. Consider the optimum of P where this minimum is attained. We show that for this optimum, g * i (t)γ * i (t)δ * i (t) = 0 and g * i (t) ≤ f ij for all t ∈ [T ] and i ∈ K, j ∼ i.
(a) Suppose, on the contrary, we have g *
i (t 0 ) > 0, γ * i (t 0 ) > 0 and δ * i (t 0 ) > 0 for some t 0 ∈ [T ]. Define ∆g ′ := min (1 − α)γ * i (t 0 ) , 1 − α α δ * i (t 0 ) , g * i (t 0 ) .
Note that ∆g ′ > 0. Now, for bus i, construct modified generation, charging and discharging
profilesg i (t),δ i (t),γ i (t), t ∈ [T ] that differ from g * i (t), δ * i (t), γ * i (t)
only at t 0 as follows:
g i (t 0 ) := g * i (t 0 ) − ∆g ′ , γ i (t 0 ) := γ * i (t 0 ) − 1 1 − α ∆g ′ , δ i (t 0 ) := δ * i (t 0 ) − α 1 − α ∆g ′ .
Note that, for all t ∈ [T ], the storage level s i (t) and the power p ij (t) flowing from bus i to bus j remain unchanged throughout. It can be checked that the modified profiles define a feasible point of P . Since c i (·) is non-decreasing, we have c i (g i (t 0 )) ≤ c i (g * i (t 0 )) and hence the additivity of the objective in P over i and t implies that this feasible point has an objective function value of at most p * . It follows that this feasible point defines an optimal point of P . However, we haveγ i (t 0 ) +δ i (t 0 ) < γ * i (t 0 ) + δ * i (t 0 ) and thus, this optimum of P has a strictly lower i∈K,t∈[T ] (γ i (t) + δ i (t)), contradicting our hypothesis. This completes
the proof of g * i (t 0 )γ * i (t 0 )δ * i (t 0 ) = 0. (b) If g * i (t) = 0 for all t ∈ [T ], then g * i (t) ≤ f ij clearly holds. Henceforth, assume max t∈[T ] g * i (t) > 0, and consider any t 0 ∈ [T ], such that g * i (t 0 ) = max t∈[T ] g * (t). If γ * i (t 0 ) = 0, then, max t∈[T ] g * i (t) = g * i (t 0 ) = p * ij (t 0 ) ≤f ij + γ * i (t 0 ) =0 − δ * i (t 0 ) ≥0 ≤ f ij .(10)
and Lemma 2(b) holds.
Suppose now that γ * i (t 0 ) > 0 and hence δ * i (t 0 ) = 0 from Lemma 2(a). First, we show that the storage device discharges at some point after t 0 .
s * i (t 0 ) = s * i (t 0 − 1) ≥0 + α γ γ * i (t 0 ) >0 > 0.
We also have s * i (T ) = s 0 i = 0 by hypothesis. Thus the storage device at node i needs to discharge in [t 0 + 1, T ] and hence α γ γ *
i (t) − 1 α δ δ * i (t) < 0 for some t ∈ [t 0 + 1, T ].
Let t 1 be the first time instant after t 0 when the storage device at bus i is discharged, i.e.
t 1 := min t ∈ [t 0 + 1, T ] | α γ γ * i (t) − 1 α δ δ * i (t) < 0 .(11)Thus, δ * i (t 1 ) > 0. Define ∆g := min γ * i (t 0 ) , 1 α δ * i (t 1 ) , g * i (t 0 ) .(12)
Then ∆g > 0. Now, consider the case where:
g * i (t 1 ) > 0, and g * i (t 0 ) ≤ g * i (t 1 ) + α∆g.(13)
Since g * i (t 1 ) > 0 and δ * i (t 1 ) > 0, then γ * i (t 1 ) = 0, by Lemma 2(a). In that case, g * i (t 1 ) + δ * i (t 1 ) = p * ij (t 1 ) is the power that flows from bus i to bus j at time t 1 . Combining (12) and (13), we have
max t∈[T ] g * i (t) = g * i (t 0 ) ≤ g * i (t 1 ) + α∆g ≤ g * i (t 1 ) + δ * i (t 1 ) = p * ij (t 1 ) ≤ f ij .
Hence, Lemma 2(b) holds when (13) is satisfied. Next, we show that if (13) does not hold, then we can construct an optimum of P with a lower i∈K,t∈[T ] (γ i (t) + δ i (t)) and this contradicts our hypothesis.
Suppose (13) does not hold. If g * i (t 1 ) = 0, then we have
g * i (t 0 ) ≥ ∆g > α∆g = g * i (t 1 ) + α∆g.
Thus, it suffices to only consider the following case:
g * i (t 0 ) > g * i (t 1 ) + α∆g.(14)
Construct the modified generation, charging and discharging profiles at node i,g i (t),δ i (t),γ i (t)
using (12), that differ from g * i (t), δ * i (t), γ * i (t) only at t 0 and t 1 as follows:
g i (t 0 ) = g * i (t 0 ) − ∆g,g i (t 1 ) = g * i (t 1 ) + α∆g, γ i (t 0 ) = γ * i (t 0 ) − ∆g,γ i (t 1 ) = γ * i (t 1 ), δ i (t 0 ) = δ * i (t 0 ) = 0,δ i (t 1 ) = δ * i (t 1 ) − α∆g.
Also, define the modified storage levels i (t) usingγ i (t) andδ i (t). To provide intuition to the above modification, we essentially generate and store less at time t 0 by an amount ∆g. This means at a future time t 1 , we can discharge α∆g less from the storage device and hence have to generate α∆g more to compensate. To check feasibility, it follows from (12), that for t = t 0 , t 1 , we have
0 ≤g i (t) ≤ g i , 0 ≤γ i (t) ≤ ǫ γ b * i , 0 ≤δ i (t) ≤ ǫ δ b * i .
Also, the line flows p ij (t) remain unchanged. For the storage levels, it can be checked that the following holds:
0 ≤ s * i (t 0 − 1) ≤s i (t) ≤ s * i (t) ≤ b * i , for t ∈ [t 0 , t 1 − 1], s i (t) = s * i (t), otherwise.
This proves that the modified profiles define a feasible point for P . The cost satisfies
c i (g i (t 0 )) + c i (g i (t 1 )) ≤ c i (g * i (t 0 ) − α∆g) + c i (g * i (t 1 ) + α∆g) (15a) ≤ c i (g * i (t 0 )) + c i (g * i (t 1 )) .(15b)
Equation (15a) follows from the non-decreasing nature of c i (·) and equation (15b) follows from using (14) and Lemma 1. Thus the modified profilesg i (t),δ i (t),γ i (t) define a feasible point of P with a cost at most p * and, hence, are optimal for P . However, we also havẽ
γ i (t 0 ) +γ i (t 1 ) +δ i (t 0 ) +δ i (t 1 ) = γ * i (t 0 ) + γ * i (t 1 ) + δ * i (t 0 ) + δ * i (t 1 ) − (1 + α)∆g >0 .
Thus, the modified profiles define an optimum of P with a lower i∈K,t∈[T ] (γ i (t) + δ i (t)).
This is a contradiction and completes the proof of the Lemma.
To prove Theorem 1, consider the optimal solution of P that satisfies Lemma 2(b). For all i ∈ K, g * i (t) itself defines a feasible flow over the line joining buses i and j, where j is the unique neighboring node of i. Now the proof idea is as follows. For i ∈ K, transfer all storage capacities b * i and the associated charging/ discharging profiles (γ * i (t), δ * i (t)), to the neighboring node j. In particular, consider the point g *
k (t),γ k (t),δ k (t),θ k (t),p kl (t),b k , k ∈ N , k ∼ l, t ∈ [T ] defined as follows.γ i (t) = 0,γ j (t) = γ * i (t) + γ * j (t), γ k (t) = γ * k (t), k ∈ N \ {i, j}, δ i (t) = 0,δ j (t) = δ * i (t) + δ * j (t), δ k (t) = δ * k (t), k ∈ N \ {i, j}, θ i (t) = θ * i (t) + 1 y ij (γ * i (t) − δ * i (t)), θ k (t) = θ * k (t), k ∈ N \ {i}, b i = 0,b j = b * i + b * j , b k = b * k , k ∈ N \ {i, j}, p ij (t) = p * ij (t) + γ * i (t) − δ * i (t), p kl (t) = p * kl (t), k ∼ l, (k, l) = (i, j).
We do this successively for each i ∈ K to obtain a feasible point of Π K . Since the generation profiles remained invariant, the resulting point is optimal for Π K . This completes the proof of Theorem 1.
B. Discussion
First, we explore a few practical power networks, where Theorem 1 applies, i.e., network topologies with generator buses that have single connections. In particular, consider the networks shown in Figure 3. The single generator single load case in Figure 3a models topologies where generators and loads are geographically separated and are connected by a transmission line, e.g., see [45]. This is common where the inputs for the generation technology (like coal or natural gas) are available far away from where the loads are located in a network. Figure 3b is an example of a radial network, i.e., an acyclic graph. Most distribution networks conform to this topology, e.g., see [38], [46]. Also, isolated transmission networks, e.g., the power network in
Catalina island [13] are radial in nature. Theorem 1 also applies to generic mesh networks that have generator buses with single connections, e.g., the network in Figure 2.
In all these examples, Theorem 1 implies that for any available storage budget h, it is always optimal to place no storage on generator buses that have single connections. Problem P , in general, has multiple optimal solutions, but Theorem 1 proves the existence of an optimum with the property that b * i = 0 for all i in K. Our result is robust to changes in demand profiles, generation capacities, line flow capacities and admittances in the entire network, i.e., it remains optimal not to place any storage at buses in set K under any changes to the above-mentioned parameters. The optimal sizing and the operation of the storage devices, however, might vary with changes in these parameters. To illustrate the efficacy of this robustness, consider the example in Figure 3a. Suppose the line flow capacity is larger than the peak value of the demand profile, i.e., f 12 ≥ max t∈[T ] d 2 (t). It can be checked that placing all the available storage at the generator bus is an optimal solution.
If at a later time during the operation of the network, the demand increases such that the peak demand surpasses the line capacity, this placement of storage no longer remains optimal and requires new infrastructure for storage to be built on the demand side to avoid load shedding.
If, however, we use the optimum as suggested by the problem Π K and place all storage on the demand side from the beginning, then this placement not only can accommodate the change in the demand, but, it also, remains optimal under the available storage budget. To explore a different direction, suppose another generator is built to supply the load in Figure 3a. Then, Theorem 1 implies that the optimal placement still has no storage at bus 1 and thus is robust to such extensions of the network. In summary, Theorem 1 suggests that the optimal storage placement problem has an interesting underlying structure and the solution to this problem is robust to changes in a wide class of system parameters and hence useful for network planners.
The network planner uses Theorem 1 to solve the investment decision strategy and the optimal control policy for the generation and storage devices for a power network as follows. Let K be the set of all generator buses with single connections. With the demand profiles and network parameters as input, solve the problem Π K optimally. The optimal storage capacities b * k , k ∈ N \K define the investment decision strategy for sizing storage units at different buses. The optimal generation profiles g * k (t), k ∈ N G define the economic dispatch of the various generators and the optimal charging/ discharging profiles γ * k (t), δ * k (t), k ∈ N \ K define the optimal control of the installed storage units.
We end this discussion with a simple remark regarding the incorporation of renewables to the power network. Renewable generators with marginal cost of production can be treated as negative loads (and the corresponding buses as load buses). The result of Theorem 1 still applies to all conventional generator buses that have single links. However, buses with single connections with a renewable energy source differ from the corresponding ones with conventional sources and thus are not guaranteed to have zero storage in the optimal placement.
C. On generators with multiple connections
Generator buses with multiple connections may not always have zero storage capacity in the optimal allocation. In this section, we illustrate this fact through a simple example. Consider a 3-node network as shown in Figure 4. All quantities are in per units. Let the cost of generation at node 1 be c 1 (g 1 ) = g 2 1 . Let T = 4 and the demand profiles at nodes 2 and 3 be d 2 = (9, 10, 0, 10) and d 3 = (0, 10, 10, 10).
Also, suppose that the line capacities are f 12 = f 13 = 9.5 and the available storage budget is h = 5. Finally, assume no losses and ignore the ramp constraints in the charging and discharging processes, i.e. α = 1 and ǫ γ = ǫ δ = 1. It can be checked that p * = 877 < π
IV. RESULTS ON SPECIFIC NETWORK TOPOLOGIES
In problems P and Π K , we solve for the optimal placement and control of storage in a powernetwork, given the demand profiles d k (t), t ∈ [T ], the storage budget h, the capacities of the generators g k , k ∈ N G and other network parameters such as the line flow limits f kl , k ∼ l. In this section, we explore the behavior of the optimal cost of production as a function of these parameters. This provides valuable insights on various design issues, e.g., how much savings in terms of generation cost do we achieve by investing in an extra unit of storage. We explore such questions for specific network topologies.
We make a few simplifying assumptions in this section. Let c k (·), k ∈ N G be strictly convex and let α = 1 and ǫ γ = ǫ δ = 1. The proofs of the results are included in the Appendix.
A. Single Generator Single Load Network
Consider the single generator single load network shown in Figure 5. Generator at bus 1 is connected to a load (or demand) at bus 2 using a single line, i.e., K = N G = {1} and N D = {2}. For this network, placing all the available storage resources at the load bus is always optimal.
∼ 1 2 r 2 (t) r 1 (t) b 1 b 2 |p 12 (t)| ≤ f 12 g 1 (t) ≤ g 1 d 2 (t)
This is an immediate consequence of Theorem 1. In this section, for any fixed demand profile
d 2 (t), t ∈ [T ]
of the load bus, we analyze the behavior of the optimal cost of production as a function of the generation capacity g 1 , the line flow capacity f 12 and the available storage budget h; in particular, let the parameterized storage placement problem be P (g 1 , f 12 , h) and its optimal cost be p * (g 1 , f 12 , h). Similarly define, Π {1} (g 1 , f 12 , h) and π
{1} * (g 1 , f 12 , h). At the optimum of P (g 1 , f 12 , h), we have g * 1 (t) ≤ f 12 , t ∈ [T ] from Lemma 2. Also, it satisfies g * 1 (t) ≤ g 1 , t ∈ [T ]
. Thus, to characterize the optimal point of P (g 1 , f 12 , h), it is equivalent to consider the constraint g 1 (t) ≤ min {g 1 , f 12 } , t ∈ [T ].
Proposition 1. For any
h ≥ 0, problem P (g 1 , f 12 , h) is feasible iff min {g 1 , f 12 } ≥ f min , where f min = max max 1≤t≤T t τ =1 d 2 (τ ) t , max 1≤t 1 <t 2 ≤T t 2 τ =t 1 +1 d 2 (τ ) − h t 2 − t 1 .(16)
Moreover, if min {g 1 , f 12 } ≥ f min , then p * (g 1 , f 12 , h) = p * (f min , f min , h).
We interpret this result as follows. If either the line flow limit f 12 < f min or the generation capacity g 1 < f min , the load cannot be satisfied. Notice that f min for h > 0 is no more than f min for h = 0. Thus, storage can be used to reduce the cost of operation avoiding transmission upgrades and generation capacity expansion [30]. Interestingly, for f 12 ≥ f min and g 1 ≥ f min , the optimal cost of operation does not depend on the specific values of f 12 and g 1 . From transmission or distribution planning perspective, investment in line and generation capacities over f min do not reduce the cost of operation. We provide an illustrative example at the end of this section.
Next, we characterize the behavior of P (g 1 , f 12 , h) and its optimal cost p * (g 1 , f 12 , h) as a func-tion of h. For a given f 12 and g 1 , the minimum required storage budget to serve the load depends on the demand profile d 2 (t), t ∈ [T ]. This may or may not be zero, depending on d 2 (t), t ∈ [T ], f 12 and g 1 . We calculate this minimum required storage budget, (say h min ) in Proposition 2.
Also, it is easy to observe that as we allow larger storage budget, the generation cost does not reduce beyond a point, i.e., there exists h sat such that p * (g 1 , f 12 , h) = p * (g 1 , f 12 , h sat ) for all h ≥ h sat . We also calculate h sat in Proposition 2. First, we introduce some notation. Construct the sequence {τ m } M m=0 as follows. Let τ 0 = 0. Define τ m iteratively:
τ m = arg max τ m−1 +1≤t≤T t τ =τ m−1 +1 d 2 (τ ) t − τ m−1 ,(17)(a) If min {g 1 , f 12 } < max t∈[T ] t τ =1 d 2 (τ ) t , then P (g 1 , f 12 , h) is infeasible for all h ≥ 0. (b) Suppose, min {g 1 , f 12 } ≥ max t∈[T ] t τ =1 d 2 (τ ) t . Then, P (g 1 , f 12 , h) is feasible iff h ≥ h min and p * (g 1 , f 12 , h) is convex and non-increasing in h, where h min = max 0≤t 1 ≤t 2 ≤T t 2 τ =t 1 +1 (d 2 (τ ) − min {g 1 , f 12 }) + .(18)Furthermore, p * (g 1 , f 12 , h) is constant for all h ≥ h sat , where h sat = max 1≤m≤M max τ m−1 +1≤t≤τm τm τ =τ m−1 +1 d 2 (τ ) t − τ m−1 τ m − τ m−1 − t τ =τ m−1 +1 d 2 (τ ) .(19)
The condition min
{g 1 , f 12 } ≥ max t∈[T ] t τ =1 d 2 (τ ) t
implies that there is some h > 0 for which P (g 1 , f 12 , h) is feasible. If this condition is violated, the problem remains infeasible no matter how large the storage budget h is. More the storage budget, lesser is the generation cost and hence p * (g 1 , f 12 , h) is decreasing in h. The convexity, however, implies that there is diminishing marginal returns on the investment on storage, i.e., the benefit of the first unit installed is more than that from the second unit. As a final note, observe that h sat is a function of only the demand profile and is independent of the generation and line flow capacities. In Section III-C we showed that placing zero storage at the generator bus of a star network is not optimal, i.e., in general, p * (h) = π * (h). In Figure 8, we plot p * (h) and π * (h) for the 3-node star network shown in Figure 4 over a range of values of the total storage budget h. Observe that p * (h) < π {1} * (h) for some values of h but they coincide at:
• Minimum value of h for which P (h) and Π {1} (h) are feasible.
• Large enough values of h.
We formally state this for a general n-node star network in the following. Assume g = ∞.
V. CONCLUSIONS AND FUTURE WORK
In this paper, we have derived analytic results on the optimal storage placement and sizing of storage units in the power grid. This provides valuable information for transmission and distribution system planners to find a suitable investment strategy in large-scale storage for the future smart grid. This work suggests that the storage placement problem has an interesting underlying structure and can be exploited to make sound planning decisions for sizing storage, that are robust to changes in a wide class of network parameters. As a result, it provides a framework to find the optimal storage capacity allocations in a network and in addition finds the optimal control policy for the generators and storage devices.
A natural direction for future work is to study the same problem with performance metrics other than the production cost such as how storage can be used to defer transmission line upgrades or generation capacity expansion. This is a first step to "assess the technical and economic attributes" [28] of this emerging smart grid technology. Another interesting direction is characterizing the optimal allocation of storage with stochastic demand and generation. In particular, we believe that storage placement on generators with multiple connections with realistic stochastic demand models would provide us with valuable insights.
VI. ACKNOWLEDGEMENTS
Fig. 1 :
1Power balance at node k ∈ N .
constraints imposed on the charging/discharging control policy of the energy storage devices, (8) represents the power balance constraints at each bus of the network and (6) represents the constraint on the sum of the capacities of all storage devices being no greater than the available storage budget.
Fig. 2 :
22. N G = {1, 2, 7} and N D = {3, 4, 5, 6}. Buses 1 and 2 have single connections and all other buses in the network have A sample network.
Fig. 3 :
3Examples of power networks (a) Single generator single load system (b) A radial network.
Fig. 4 :
4A network with a generator that has multiple connections.
Fig. 5 :
5Single generator single load network. Available storage budget is h ≥ b 1 + b 2 .
Fig. 6 :
6Plots to illustrate propositions 1 and 2. (a) Typical hourly load profile and optimal generation portfolio for line flow capacity f 12 = 0.85, generation capacity g 1 = 1 and storage budget h = 1 (b) p * (g 1 = 1, f 12 = 0.85, h). (c) p * (g 1 = 1, f 12 , h = 1).Example: Now we explain propositions 1 and 2 with an example. All quantities are in per units.Consider an hourly load profile d 2 (t), t ∈ [T ] as shown inFigure 6a. The optimal generation profile g * 1 (t), t ∈ [T ] for P (g 1 = 1, f 12 = 0.85, h = 1) has been plotted in the sameFigure.
Notice that max t∈[T ] g * 1 (t) ≤ f 12 as stated in Lemma 2. Consider the plots inFigures 6b and 6c. We plot p * (g 1 = 1, f = 0.85, h) for h in [0, 3] inFigure 6b. Notice that f 12 ≤ max t∈[T ] d 2 (t), i.e., the problem is infeasible in the absence of storage. We calculate h min = 0.226 and h sat = 2.598 from proposition 2. In Figure 6c, we plot p * (g 1 = 1, f 12 , h = 1) for f 12 in [0, 2]. As in proposition 1, the problem is infeasible for f 12 < f min = 0.683 and the optimal cost remains constant for f 12 ≥ f min . B. Star Network Consider a star network on n ≥ 2 nodes as shown in Figure 7. N G = {1} and N D = {2, 3, . . . , n}. For fixed demand profiles d k (t), t ∈ [T ], k ∈ N D , line flow capacities f 1k , k ∈ N D and capacity of the generator g 1 , let P (h) and Π {1} (h) denote the storage placement problem and its restricted version as functions of the available storage budget h. Also, let p * (h) and π {1} * (h) be their optimal costs respectively.
Fig. 8 :
8P (h) and Π {1} (h) for the simple 3-node star network inFigure 4.
Proposition 3 .
3Suppose f 1k ≥ max t∈[T ] t τ =1 d k (τ ) t for all k ∈ N D . Then, P (h) and Π {1} (h) are feasible iff h ≥ h min , where ) p * (h min ) = π {1} * (h min ), (b) There exists h o ≥ h min such that p * (h) = π {1} * (h) for all h ≥ h o .
for 1 ≤ m ≤ M, where M is the smallest integer for which τ M = T . Note that the sequence depends only on the demand profile d 2 (t), t ∈ [T ]. For any x ∈ R, let [x] + := max(x, 0). Proposition 2. Problem P (g 1 , f 12 , h) satisfies:
Without loss of generality, let bus 1 be the slack bus and hence θ1(t) = 0 for all t ∈ [T ]. Boundedness of the set of feasible solutions of P then follows from the relations in (1), (2), (3), (6) and(7),.
The authors gratefully acknowledge Prof. K. Mani Chandy at California Institute of Technology and Mr. Paul DeMartini from Resnick Institute for their helpful comments.A. Proofs for Single Generator Single Load NetworksWe drop subscripts from the variables d 2 (t), g 1 (t), t ∈ [T ], f 12 , g 1 , c 1 (·) and the superscript from Π {1} (·), π {1} * (·) for ease of notation throughout this section.Moreover, if min {g, f } ≥ f min , then p * (g, f, h) = p * (f min , f min , h).Proof: From Theorem 1, it suffices to show the claim for Π(g, f, h) and π * (g, f, h). First, we show that if Π(g, f, h) is feasible, then min {g, f } ≥ f min . Fix any h ≥ 0 and let g(t), t ∈ [T ] be a feasible generation profile. Since t τ =1 r 2 (τ ) = s 2 (t) ≥ 0, we have for any t ∈ [T ]Furthermore, for any 1 ≤ t 1 < t 2 ≤ T , the power extracted from the storage device between time instants t 1 and t 2 cannot exceed the total storage budget h and hence we haveSince g(t), t ∈ [T ] is feasible, g(t) ≤ min {g, f } for all t ∈ [T ]. Hence, combining(22)and(23), we getNext, we show that min {g, f } ≥ f min is sufficient for Π(g, f, h) to be feasible. Consider the optimal generation profile g * (t), t ∈ [T ] for the relaxed problem Π(+∞, +∞, h). Suppose itThen g * (t), t ∈ [T ] is also feasible and optimal for problem Πholds. Consider the following notation.In the above definition g * (0) := 0 for convenience. If g * (t max ) = 0, then (24) clearly holds.Henceforth, assume g * (t max ) > 0. Then, g * (t), t ∈ [T ] satisfies:Now, suppose the following holds:If(26)holds, it follows from(25):and hence (24) is satisfied. Next, we show that(26)indeed holds to complete the proof. First we prove that s * 2 (t max ) = 0, i.e., the storage device at node 2 fully discharges at time t max . Suppose s * 2 (t max ) > 0. As in Lemma 2, we construct a modified generation profile and storage control policy that is feasible and has an objective function value no greater than π * (+∞, +∞, h). But, the optimal generation profile g * (t), t ∈ [T ] is unique since the cost function c(·) is assumed to be strictly convex. Hence we derive a contradiction. By hypothesis, s * 2 (t max ) > 0 and hence the storage device at bus 2 discharges for some t > t max . Let t 1 be the first such time instant and defineNotice that ∆ 1 > 0. Consider the modified generation profileg(t) and control policyr 2 (t), that differ from g * (t) and r * 2 (t) only at t max and t 1 as follows:Using Lemma 1, we have c(g(t max )) + c(g(t 1 )) ≤ c(g * (t max )) + c(g * (t 1 )).It can be checked that the modified profiles are feasible for Π(+∞, +∞, h). The details are omitted for brevity. This is a contradiction and hence s * 2 (t max ) = 0. Next, we characterize s * 2 (t less ). If t less = 0, then s * 2 (t less ) = s 0 2 = 0. If t less > 0, we prove that s * 2 (t less ) = h, i.e., the storage device at node 2 is fully charged at time t less . Suppose s * 2 (t less ) < h. As above, we construct a modified generation profileg(t) and storage control policỹ r 2 (t) that achieves an objective value no greater than π * (+∞, +∞, h) to derive a contradiction.In particular, defineConsiderg(t) andr 2 (t), that differ from g * (t) and r * 2 (t) only at t less and t less + 1 as follows:As above, this defines a feasible point for Π(+∞, +∞, h) and achieves an objective value strictly less than π * (+∞, +∞, h). This is a contradiction and hence s * 2 (t less ) = h for t less > 0.Proof: From Theorem 1, it suffices to prove the claim for Π(g, f, h) and π * (g, f, h).(a) To the contrary of the statement of the Proposition suppose that minThen, it follows directly from Proposition 1 that, contradicting our hypothesis.Also, h ≥ 0 and hence:can be equivalently written as follows:Also, by hypothesis, we haveCombining(29)and(30), we get min {g, f } ≥ f min . Then, Proposition 1 implies that Π(g, f, h) is feasible. Convexity and non-decreasing nature of p * (g, f, h) as a function of h follows from linear parametric optimization theory[39].Finally, we prove that p * (g, f, h) is constant for all h ≥ h sat , where h sat is as defined in(28). The proof idea here is as follows. We construct the optimal generation profile g * (t), t ∈ [T ] for the problem Π(+∞, +∞, +∞) and show that it is feasible and hence optimal for the problem Π(g, f,holds.Problem Π(+∞, +∞, +∞) can be re-written as follows.Let the Lagrange multipliers in equations (31a)-(31b) be λ(t), ℓ(t), t ∈ [T ] and ν, respectively. It can be checked that the following primal-dual pair satisfies the Karush-Kuhn-Tucker conditions and hence is optimal for the convex program Π(+∞, +∞, +∞) and itsLagrangian dual[39]. We omit the details for brevity.. . , τ m and m = 1, 2, . . . , M,, and ν * = −c ′ (g * (T )).The above profile g * (t), t ∈ [T ] of Π(+∞, +∞, +∞) satisfies:and hence is feasible and optimal for Π(g, f, +∞). Note that τmMaximizing the above relation over all t ∈ [T ] we get max t∈[T ] s * 2 (t) = h sat . Therefore, g * (t), t ∈ [T ] is feasible and optimal for Π(g, f, h) provided that h ≥ h sat .B. Proofs for Star NetworksMoreover:Proof: First we show that h ≥ h min is necessary for P (h) to be feasible. Consider any feasible solution of P (h). For any k ∈ N D and 0 ≤ t 1 < t 2 ≤ T , we have t 2 τ =t 1 +1 r k (τ ) ≥ −b k , since the power extracted from the storage device at node k cannot exceed the corresponding storage capacity b k . Also, for any k ∈ N D the power flow on the line joining buses 1 andCombining the above relations andthen P (h) is also feasible and henceh ≥ h min is necessary for both problems to be feasible. Now we prove that it is also sufficient.In particular, we show that for h = h min , Π {1} (h) is feasible. For convenience, definẽThen h min = k∈N Dh k . Rearranging(35), we getAlso, by hypothesis, we haveCombining equations(36)and(37), we haveFor each k ∈ N D , consider a single generator single load system as follows. Let the demand profile be d k (t), the capacity of the transmission line be f 1k and the total available storage budget beh k . For this system, the right hand side in(38)coincides with the definition of f min in(16). From Proposition 1, it follows that there is a feasible generation profile (say g (k) (t)) and a storage control policy r k (t) that define a feasible flow over this single generator single load system and meet the demand. Now, for the star network, construct the generation profile g 1 (t)and operate the storage units at each node k ∈ N D with the control policy r k (t) defined above.Also, r 1 (t) = 0 for all t ∈ [T ]. It can be checked that this defines a feasible point for Π {1} (h min ).Next, we prove that p * (h min ) = π {1} * (h min ). Let b * k , k ∈ N be optimal storage capacities for problem P (h min ). Then the optimal storage capacities satisfy the following relations:where the first one follows from (34) and the second one follows from the constraint on the total available storage capacities. Rearranging the above equations, we get b * 1 = 0 and hence p * (h min ) = π {1} * (h min ). This completes the proof of part (a). To prove part (b) of Proposition 3, we start by showing thatAssume P (∞) is feasible. For h = ∞, we drop the variables b k , k ∈ N , and consider the problems P (∞) and Π {1} (∞) over the variables g 1 (t), r k (t), k ∈ N . The variables p 1k (t) ands k (t) are defined accordingly for all k ∈ N . We argue that the optimal points of P (∞) lie in a bounded set. Note that |p 1k (t)| = |d k (t) + r k (t)| ≤ f 1k and thus the control policies r k (t) are bounded for all k ∈ N D . Also, the cost function c 1 (·) is convex and hence its sub-level sets[39]are bounded. From the above arguments and the power-balance at bus 1, the optimal policy r 1 (t) is also bounded. Thus, the set of optimal solutions of P (∞) is a bounded set. Furthermore, this set is also closed since the objective function and the constraints are continuous functions.As in the proof of Lemma 2, associate the function t∈[T ] |r 1 (t)| with every point in the set of optimal solutions of P (∞). This is a continuous function on a compact set and hence attains a minimum. Consider the optimum of P (∞) where this minimum is attained. We prove(39)byshowing that r * 1 (t) = 0 for all t ∈ [T ] at this optimum. Assume to the contrary, that r * 1 (t) is non-zero for some t ∈ [T ]. DefineAlso, define ∆ := min {r * 1 (t 0 ), −r * 1 (t 1 )} and notice that ∆ > 0. Case 1: g * 1 (t 0 ) > g * 1 (t 1 ) + ∆: Construct the modified generation and charging/ discharging profilesg 1 (t),r 1 (t) that differ from g * 1 (t), r * 1 (t) only at t 0 and t 1 as follows:where ∆g := min {∆, g * 1 (t 0 )} > 0. As in the proof of Lemma 2, this is feasible for P (∞). Also, by Lemma 1:The details are omitted for brevity. This feasible point satisfiesand hence defines an optimal point of P (∞) with a strictly lower value of the function t∈[T ] |r 1 (t)|.This is a contradiction.Case 2: g * 1 (t 0 ) ≤ g * 1 (t 1 ) + ∆: As above we construct modified storage control policiesr k (t) for all k ∈ N , keeping the generation profile constant to define an optimal point of P (∞) with a lower value of t∈[T ] |r 1 (t)| to derive a contradiction.Let the modified control policy at bus 1 be as follows:Instead, we distribute this to storage devices at k ∈ N D , as follows:for some ψ k ≥ 0, k ∈ N D and k∈N D ψ k = ∆. To ensure feasibility of the modified profiles it suffices to check that the line flow constraints are satisfied at t 0 and t 1 . In other words, weshow that there exists ψ k , k ∈ N D such that for all k ∈ N D ,Equivalently, we prove thatRecall that p * 1k (t 0 ) and p * 1k (t 1 ) are feasible for P (∞). Thus p * 1k (t 0 ) ≤ f 1k and p * 1k (t 1 ) ≥ −f 1k . Also, g * 1 (t) − r * 1 (t) = k∈N D p * 1k (t) at t = t 0 and t = t 1 . Thus, we havewhere the last inequality follows from the hypothesis g * 1 (t 0 ) ≤ g * 1 (t 1 ) + ∆. The modified profiles satisfy |r 1 (t 0 )| + |r 1 (t 1 )| < |r * 1 (t 0 )| + |r * 1 (t 1 )| as in(40). As argued above this is a contradiction and hence (39) holds.For P (∞), s * k (t), k ∈ N , t ∈ [T ] is finite. Define h o := k∈N D max t∈[T ] s * k (t). Then, note that (g * 1 (t), r * k (t), t ∈ [T ] k ∈ N ) are also feasible for Π {1} (h) and P (h) for all h ≥ h o . This completes the proof of Proposition 3.
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APPENDIX Here, we present the proofs of the results presented in Section IV. For the single generator single node and the star network. IEEE Distribution Test Feeders. we drop the voltage angles θ k (t), k ∈ N , t ∈ [TIEEE Distribution Test Feeders. [Online]. Available: http://www.ewh.ieee.org/soc/pes/dsacom/testfeeders/index.html APPENDIX Here, we present the proofs of the results presented in Section IV. For the single generator single node and the star network, we drop the voltage angles θ k (t), k ∈ N , t ∈ [T ].
. Furthermore, since α = 1, define r k (t) := γ k (t)−δ k (t) be the power that flows into the storageFurthermore, since α = 1, define r k (t) := γ k (t)−δ k (t) be the power that flows into the storage
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"Bill Kay \nErskine College\nHarvard University\nUniversity of South Carolina\nRose-Hulman Institute of Technology\n\n",
"Emma E Snively \nErskine College\nHarvard University\nUniversity of South Carolina\nRose-Hulman Institute of Technology\n\n"
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"Erskine College\nHarvard University\nUniversity of South Carolina\nRose-Hulman Institute of Technology\n",
"Erskine College\nHarvard University\nUniversity of South Carolina\nRose-Hulman Institute of Technology\n",
"Erskine College\nHarvard University\nUniversity of South Carolina\nRose-Hulman Institute of Technology\n"
]
| []
| Let D be a simple digraph without loops or digons (i.e. if (u, v) ∈ E(D), then (v, u) ∈ E(D)). For any v ∈ V (D) let N 1 (v) be the set of all vertices at out-distance 1 from v and let N 2 (v) be the set of all vertices at out-distance 2. We provide sufficient conditions under which there must exist some v ∈ V (D) such that |N 1 (v)| ≤ |N 2 (v)|, as well as examine properties of a minimal graph which does not have such a vertex. We show that if one such graph exists, then there exist infinitely many strongly-connected graphs having no such vertex. arXiv:0808.0946v3 [math.CO] | 10.2140/involve.2009.2.387 | [
"https://arxiv.org/pdf/0808.0946v3.pdf"
]
| 6,206,110 | 0808.0946 | 4c05e36c1e04dea298f3e2744f58c84d075251e5 |
Contributions to Seymour's Second Neighborhood Conjecture
August 19, 2008
James N Brantner
Erskine College
Harvard University
University of South Carolina
Rose-Hulman Institute of Technology
Greg Brockman
Erskine College
Harvard University
University of South Carolina
Rose-Hulman Institute of Technology
Bill Kay
Erskine College
Harvard University
University of South Carolina
Rose-Hulman Institute of Technology
Emma E Snively
Erskine College
Harvard University
University of South Carolina
Rose-Hulman Institute of Technology
Contributions to Seymour's Second Neighborhood Conjecture
August 19, 2008
Let D be a simple digraph without loops or digons (i.e. if (u, v) ∈ E(D), then (v, u) ∈ E(D)). For any v ∈ V (D) let N 1 (v) be the set of all vertices at out-distance 1 from v and let N 2 (v) be the set of all vertices at out-distance 2. We provide sufficient conditions under which there must exist some v ∈ V (D) such that |N 1 (v)| ≤ |N 2 (v)|, as well as examine properties of a minimal graph which does not have such a vertex. We show that if one such graph exists, then there exist infinitely many strongly-connected graphs having no such vertex. arXiv:0808.0946v3 [math.CO]
Introduction
For the purposes of this article, we consider only simple nonempty digraphs (those containing no loops or multiple edges and having a nonempty vertex set), unless stated otherwise. We also require that our digraphs contain no digons, that is, if D is a digraph then (u, v) ∈ E(D) ⇒ (v, u) / ∈ E(D). If i is a positive integer, we denote the i th neighborhood of a vertex u in D by N i,D (u) = {v ∈ V (D)|dist D (u, v) = i}, where dist D (u, v) is the length of the shortest directed path from u to v in D (if there is no directed path from u to v, we set dist D (u, v) = ∞). If D is clear from context, we simply write N i (u) and dist (u, v). We also may wish to consider the i th in-neighborhood of a vertex N −i (u) = {v ∈ V (D)|dist(v, u) = i}. In addition, if V ⊆ V (D), we let D[V ] be the subgraph of D induced by V .
Graph theorists will be familiar with the following conjecture due to Seymour (see [3]), now more than a decade old: In 1995, Dean [3] conjectured this to be true when D is a tournament. Dean's Conjecture was subsequently proven by Fisher [5] in 1996. Further, in their 2001 paper Kaneko and Locke [6] showed Conjecture 1.1 to be true if the minimum outdegree of vertices in D is less than 7, and Cohn, Wright, and Godbole [2] showed that it holds for random graphs almost always. And finally, in 2007 Fidler and Yuster [4] proved that Conjecture 1.1 holds for graphs with minimum out-degree |V (D)|−2, tournaments minus a star, and tournaments minus a sub-tournament. While over the years there have been several attempts at a proof of Conjecture 1.1, none of these have yet been successful.
For completeness, we introduce the related Caccetta-Häggkvist conjecture [1], which was posed in 1978:
Conjecture 1.2 (Caccetta-Häggkvist Conjecture)
. If D is a directed graph with minimum outdegree at least |V (D)|/k, then D has a directed cycle of length at most k. We do not seek to prove Conjecture 1.1 in this paper. Rather, we prove the conjecture for various classes of graphs. We then take a different tack and provide conditions that must be satisfied by any appropriately-defined minimal counterexample to Seymour's Second Neighborhood Conjecture. This provides tools with which the conjecture can be approached; in one direction it may aid in showing the nonexistence of such a graph, while in the other direction we restrict the search space of possible counterexamples.
Definitions
We begin our investigation by defining some useful terms.
Definition 2.1. Suppose that D is digraph and u ∈ V (D). We say that u is satisfactory if |N 1 (u)| ≤ |N 2 (u)|. Also, u is a sink if |N 1 (u)| = 0. Note that a sink is trivially satisfactory. Also define A s,D (u) = |N 1 (u)| − |N 2 (u)| to be the anti-satisfaction of u. As usual, if G is clear from context, we simply write A s (v). Notice that u is satisfactory if and only if A s (u) ≤ 0.
Definition 2.4. Again let D be a directed graph. Recall that a transitive triangle T is a directed graph on three nodes a, b, c such that (a, b), (a, c), (b, c) ∈ E(T ). If (u, v) ∈ E(D), we say that edge (u, v) is the base of a transitive triangle if u and v share a common first neighbor; that is,
|N 1 (u) ∩ N 1 (v)| ≥ 1.
Base Figure 1 -Demonstration of an edge that is the base of a transitive triangle We now have the tools to delve into our results.
If, for distinct t, u, v, w ∈ V (D), we have that (t, u), (u, w), (t, v), (v, w) ∈ E then we call {(t, u), (u, w), (t, v), (v, w)} a 2-directed diamond.
Directed cycles and underlying girth
In this section we show that certain classes of graphs satisfy Seymour's Second Neighborhood Conjecture. The following theorem shows that directed cycles are necessary for a graph to be a counterexample to the conjecture. Proof. Let D be a directed graph, and suppose that D contains no satisfactory vertices. Then D has no sink, as noted in Definition 2.1. It is a well-known fact that a graph with no sinks has a directed cycle. We include the standard proof, however, since the same technique will be useful to us later:
Because D has sink, for v ∈ V (D), |N 1 (v)| > 0. Pick an arbitrary vertex v 0 ∈ V (D), and consider the sequence {v i } |V (D)| i=0 defined recursively by v i+1 ∈ N 1 (v i ) for i ≥ 0. By the Pigeonhole principle, there exist some r = s such that v r = v s . Then we note that the sequence of edges (v r , v r+1 ), (v r+1 , v r+2 ), . . . , (v s−1 , v s = v r ) defines a dicycle in D, thus completing our proof.
The following theorem provides another sufficient condition for a graph to contain a satisfactory vertex: Proof. Let v 0 ∈ V (D) have the minimal out-degree in D. If |N 1 (v 0 )| = 0, then v 0 is a sink and hence a satisfactory vertex. Otherwise, let v 1 ∈ N 1 (v 0 ). By construction, we have that
|N 1 (v 1 )| ≥ |N 1 (v 0 )|. Furthermore, D contains no transitive triangles, so |N 1 (v 0 )∩N 1 (v 1 )| = 0. Thus, |N 2 (v 0 )| ≥ |N 1 (v 1 )| ≥ |N 1 (v 0 )|, and by definition v 0 is satisfactory.
Remark. Recall that the girth of a undirected graph is the length of its shortest cycle. Theorem 3.2 shows that any counterexample to Conjecture 1.1 must have underlying girth of exactly 3.
Minimal Criminals
To this point, we have been showing that classes of graphs satisfy Conjecture 1.1. In this section we reverse course and explore necessary properties of the minimal criminal graphs of A from Definition 2.2. If Seymour's Second Neighborhood Conjecture is true, then our goal should be to derive such strong constraints on the graphs of A that a contradiction is obtained. On the other hand, if the conjecture is false, then our goal is to find necessary or sufficient conditions for a graph to be in A ; we provide a number of necessary conditions here. Proof of 3: We see that |N 2,Z (u)| ≥ |N 2,M (u)|, since otherwise A s,Z (u) ≤ 0 and u is not satisfactory in Z, a contradiction. Consider now X = N 2,Z (u) \ N 2,M (u). We note that X ⊆ {v}, since v is the only vertex that could have been added to u's second neighborhood in Z (Case 1 in Figure 3). Thus we see that (u, y)). If |N 2,M (u) \ N 2,Z (u)| = 0, then for z ∈ Z, we therefore have a path of length 2 from u to z in Z, and considering this path in M yields a path from u to z avoiding e. And finally, if |N 2,M (u) \ N 2,Z (u)| = 1, then we have a path of length 2 from u to z in Z for all but 1 vertex in Z, and as before we have a corresponding path from u to z avoiding e. But in this case, there is a path of length 2 from u to v avoiding e, and hence we have obtained the desired result.
Proof of 4: Paths of length 1 from u to v ∈ N 1 (v) yield transitive triangles with e as the base, and paths of length 2 from u to v ∈ {v} ∪ N 1 (v) yield 2-directed diamonds with e as one of the bases. By part 3, at least one of these structures exists, and hence we are done.
Proof of 5:
Since N 1 (v) \ (N 1 (u) ∩ N 1 (v)) ⊆ N 2 (u), we have that |N 2 (u)| ≥ |N 1 (v)| − |N 1 (u) ∩ N 1 (v)|. But since M contains no satisfactory vertices, we have that |N 2 (u)| < |N 1 (u)|. By transitivity, we obtain |N 1 (v)| − |N 1 (v) ∩ N 1 (u)| < |N 1 (u)|. It then follows that |N 1 (v)| − |N 1 (u)| < |N 1 (v) ∩ N 1 (u)|, but |N 1 (v) ∩ N 1 (u)
| is the number of transitive triangles having base e, so we have proved the first half of part 5.
To prove the second half of this part, we consider the following cases: Case 1 : Suppose there exists a vertex u such that (u, u ), (u , v) ∈ E(M). By part 3, we know that u must be connected to at least |N 1 (v)| − 1 elements of N 1 (v) via a path of length 1 or 2 avoiding e. But we see that u is adjacent to at most |N 1 (u) − 2| vertices in N 1 (v). Subtracting, we see that u is connected via a path of length 2 avoiding e to at least
|N 1 (v)| − 1 − (|N 1 (u)| − 2) = (|N 1 (v)| − |N 1 (u)|) + 1 vertices in N 1 (v)
; each of which yields a 2-directed diamond of which e is the base, which is the desired result.
Case 2 : Suppose there is no such u . Then again applying part 3, it must be that there exists a path of length 1 or 2 avoiding e to each vertex in N 1 (v). But u is adjacent to at most |N 1 (u)| − 1 of these vertices, and as before we count that there is a path of length 2 avoiding e from u to at least |N 1 (v)| − (|N 1 (u)| − 1) = |N 1 (v)| − |N 1 (u)| + 1 vertices in |N 1 (v)|. Since each of these paths yield a 2-directed diamond with e as the base, we are done.
Proof of 6: In M, pick an arbitrary vertex u. Delete this vertex (and all edges incident with it) and label the resulting directed graph Z. Then in a similar manner to before, one of the vertices in N −1,M (u) must be satisfactory in Z by vertex minimality of M. Label this vertex t. Since |N 1,Z (t)| = |N 1,M (t)| − 1, t is satisfactory, and |N 2,Z (t)| ⊆ |N 2,M (t)| (note that in contrast to deleting an edge, deleting a vertex does not allow any vertices to add vertices to their second neighborhoods), we see that we must have |N 2,Z (t)| = |N 2,M (t)|. It is then necessary that A s,M (t) = 1. Since u was arbitrary, we have obtained the desired result.
Proof of 7: We apply the same technique as we used Theorem 3.1. We present a brief sketch of our proof: by part 5, each vertex in M has an in-neighbor having anti-satisfaction of exactly 1. If we begin at an arbitrary vertex and choose one of its in-neighbors having anti-satisfaction of exactly 1, do the same for the resulting vertex, and iterate this process, at some point we must arrive back at a vertex we have already visited, thus constructing a directed cycle of vertices having anti-satisfaction exactly 1.
Finally, we show that there is not a finite nonzero number of strongly-connected counterexamples to the conjecture. That is, either the conjecture is true, or there are an infinite number of (non-isomorphic) strongly-connected graphs that violate Conjecture 1. Proof. Suppose that Seymour's Second Neighborhood Conjecture is false, and suppose that digraph D is any strongly-connected counterexample to Seymour's Second Neighborhood Conjecture. (By Part 1 of Theorem 4.1, such a D must exist.) Let H be any digraph satisfying the condition ∀v ∈ V (H), A s (v) ≥ 0; that is, all of H's vertices have nonnegative anti-satifaction. Note that any dicycle satisfies the relevant condition, and hence there exists a choice of H on any number n of vertices, n ≥ 3.
We now construct a graph D on |V (D)|·|V (H)| vertices such that D is a counterexample to Seymour's Second Neighborhood Conjecture, thus proving our theorem. We define our graph D as follows: defines a directed path in D from (d 1 , h 1 ) to (d 2 , h 2 ). If d 1 = d 2 , let d 3 ∈ N 1,D (d 1 ); we know that (d 1 , h 1 ), (d 3 , h 2 ) are adjacent in D , and since d 2 = d 3 there is a path from (d 3 , h 2 ) to (d 2 , h 2 ) in D , the existence of a path from (d 1 , h 1 ) to (d 2 , h 2 ) follows.
• V (D ) = V (D) × V (H) • If u = (d 1 , h 1 ), v = (d 2 , h 2 ) ∈ V (D ),
By definition, we then have that D is a strongly-connected counterexample to Seymour's Second Neighborhood Conjecture.
Conjecture 1 . 1 (
11Seymour's Second Neighborhood Conjecture). Let D be a directed graph. Then there exists a vertex v 0 ∈ V (D) such that |N 1 (v 0 )| ≤ |N 2 (v 0 )|.
Definition 2 . 2 .
22Let A = {D|D is a simple directed graph with no satisfactory vertices} be the set of counterexamples to Seymour's Second Neighborhood Conjecture. Let A = {D | |E(D)| = min H∈A |E(H)|} be the set of graphs in A with the fewest number of edges. Finally, let A = {D | |V (D)| = min H∈A |V (H)|} be the set of graphs in A with the fewest number of vertices. We will refer to any element of A as a minimal criminal. Note that A is empty if and only if Conjecture 1.1 is true.
Definition 2. 3 .
3Let D be a digraph. Suppose that u ∈ V (D). We define W D (u) = {v|dist(u, v) = ∞} to be the walkable neighborhood of u with respect to D. If D is clear from context, we simply write W (u).
Figure 2 -
2We say the edges (t, u), (t, v) are the bases of the 2Demonstration of the bases of a 2-directed diamond
Theorem 3 . 1 .
31If a digraph contains no directed cycles, then it must have a satisfactory vertex.
Theorem 3 . 2 .
32Let D be a directed graph containing no transitive triangles. Then D contains a satisfactory vertex.
Theorem 4. 1 .Figure 3 -
13If M ∈ A , we have the following: 1. M is strongly connected. 2. For each u ∈ V (M), A s (u) ∈ {1, 2}. 3. For every edge e = (u, v) ∈ E(M), there exists a path of length 1 or 2 avoiding e from u to all but at most 1 element of {v} ∪ N 1 (v). 4. Every edge of M is the base of either a transitive triangle or a 2-directed diamond. 5. Suppose that e = (u, v) ∈ E(M) and |N 1 (u)| ≤ |N 1 (v)|. Then e must be the base of at least |N 1 (v)|−|N 1 (u)|+1 transitive triangles and the base of at least |N 1 (v)|−|N 1 (u)|+1 2-directed diamonds. 6. For any vertex u ∈ V (M), there exists a vertex v ∈ N −1 (u) such that A s (v) = 1. 7. There exists a directed cycle in M such that every vertex on the cycle has antisatisfaction of exactly 1. Proof. Proof of 1: Recall that a directed graph is strongly connected if there exists a directed path between any two of its vertices. Pick an arbitrary vertex u from the vertex set of M. Now consider M = M[W (u)]. We now pick an arbitrary vertex v ∈ W (u). Clearly, N 1,M (v) ⊆ W (u) and N 2,M (v) ⊆ W (u). But this implies that A s,M = |N 1,M (v)| − |N 2,M (v)| = |N 1,M (v)| − |N 2,M (v)| = A s,M , and hence v is satisfactory in M if and only if v is satisfactory in M. Since by definition M contains no satisfactory vertices, v cannot be satisfactory in M . Thus M contains no satisfactory vertices. But M is a subgraph of M, and so by minimality of M we have that M = M . Two possible cases resulting from deleting an edge from M. In Case 1, there is a length 2 path from u to v, while in Case 2 no such path exists. Note that it is possible that deleting e will increase the size of u's second neighborhood, as shown in Case 1. Proof of 2: Fix u and pick an arbitrary edge e = (u, v) ∈ E(M). Consider the directed graph Z obtained by deleting e from M. Since Z has fewer edges than M, we have that Z contains a satisfactory vertex. For each vertex w ∈ V (M), we note that |N 1,Z (w)| = |N 1,M (w)| unless w = u, in which case |N 1,Z (u)| = |N 1,M (u)| − 1. Furthermore, we have that |N 2,Z (w)| ≤ |N 2,M (w)|, except if w = u, in which case we have that |N 2,Z (u)| ≤ |N 2,M (u)|+1. (See Figure 3.) Thus, we obtain that in Z for w = u ∈ V (Z), A s,Z (w) ≥ A s,M (w), and hence all vertices in Z besides u are not satisfactory. Thus by process of elimination we have that u is satisfactory in Z. Thus 0 ≥ A s,Z (u) = |N 1,Z (u)| − |N 2,Z (u)| ≥ (|N 1,M (u)| − 1) − (|N 2,M (u)| + 1), and hence we have that 0 < A s,M (u) = |N 1,M (u)| − |N 2,M (u)| ≤ 2. Result 2 follows immediately.
|N 2,M (u) \ N 2,Z (u)| ≤ 1, with equality only if v ∈ N 2,Z (u). Note that N 1,M (v) ⊆ N 1,M (u)∪N 2,M (u). Let Y = N 1,M (u)∩N 1,M (v) and Z = N 2,M (u)∩ N 1,M (v). For y ∈ Y , we clearly have a path of length 1 from u to y avoiding e (namely the edge
Figure 4 -
4A partial representation of the graph D , given D and H. We can think about D as being made by replacing each vertex of D with a copy of H. Note that for clarity we replace only one vertex in the above picture.is especially interesting in light of Part 1 of Theorem 4.1, which shows that all minimal criminals are strongly connected.
Theorem 4 . 2 .
42If Seymour's Second Neighborhood Conjecture is false, there are infinitely many non-isomorphic strongly-connected counterexamples to Seymour's Second Neighborhood Conjecture.
then (u, v) ∈ E(D ) if and only if either 1. d 1 = d 2 and (h 1 , h 2 ) ∈ E(H), or 2. d 1 = d 2 and (d 1 , d 2 ) ∈ E(D).
For any vertex v = (d, h) ∈ V (D ), we calculate that
|N 1 ,
1D (v)| = |N 1,H (h)| + |V (H)| · |N 1,D (d)|, by construction. Furthermore, we have that|N 2,D (v)| = |N 2,H (h)| + |V (H)| · |N 2,D (d)|.We then calculate thatA s,D (v) = |N 1,D (v)| − |N 2,D (v)| = (|N 1,H (h)| − |N 2,H (h)|) + |V (H)|(|N 1,D (d)| − |N 2,D (d)|).But by our choice of H, we have that|N 1,H (h)| − |N 2,H (h)| ≥ 0, and by our choice of D we have that |N 1,D (d)| − |N 2,D (d)| > 0. Hence we obtain A s,D (v) > 0, thus implying that every vertex in D has positive anti-satisfaction. Furthermore, D is strongly connected: fix (d 1 , h 1 ), (d 2 , h 2 ) ∈ V (D ). If d 1 = d 2 , let d 1 , δ 1 , . . . , δ i , d 2 define a directed path in D from d 1 to d 2 . Then (d 1 , h 1 ), (δ 1 , h 2 ), . . . , (δ i , h 2 ), (d 2 , h 2 )
Conjecture 1.1 would imply the k = 3 case of Conjecture 1.2. Much work has been done on Conjecture 1.2, including an entire workshop in 2006 sponsored by AIM and the NSF, yet Conjectures 1.1 and 1.2 both remain open.
Bill Kay Emma Snively University of South Carolina Rose-Hulman Institute of Technology Columbia, SC Terre Haute, IN United States United States [email protected] [email protected]
AcknowledgementsThis work was done at the East Tennessee State University REU, NSF grant 0552730, under the supervision of Dr. Anant Godbole.
On minimal digraphs with given girth. L Caccetta, R Häggkvist, Congressus Numerantium. 21L. Caccetta and R. Häggkvist, On minimal digraphs with given girth, Congressus Nu- merantium 21 (1978), 181-187.
Probabilistic versions of Seymour's distance two conjecture. Z Cohn, E Wright, A Godbole, PreprintZ. Cohn, E. Wright, and A. Godbole, Probabilistic versions of Seymour's distance two conjecture, Preprint.
Squaring a tournament-an open problem. N Dean, B Latka, Congressus Numerantium. 109N. Dean and B. Latka, Squaring a tournament-an open problem, Congressus Numeran- tium 109 (1995), 73-80.
Remarks on the second neighborhood problem. D Fidler, R Yuster, Journal of Graph Theory. 55D. Fidler and R. Yuster, Remarks on the second neighborhood problem, Journal of Graph Theory 55 (2007), 208-220.
Squaring a tournament: a proof of Dean's conjecture. C David, Fisher, Journal of Graph Theory. 231David C. Fisher, Squaring a tournament: a proof of Dean's conjecture, Journal of Graph Theory 23 (1996), no. 1, 15-20.
The minimum degree approach for Paul Seymour's distance 2 conjecture. Yoshihiro Kaneko, Stephen C Locke, Congressus Numerantium. 148Yoshihiro Kaneko and Stephen C. Locke, The minimum degree approach for Paul Sey- mour's distance 2 conjecture, Congressus Numerantium 148 (2001), 201-206.
. N James, Brantner Greg, SC Cambridge, MA United StatesBrockman Erskine College Harvard University Due WestUnited States [email protected] [email protected] N. Brantner Greg Brockman Erskine College Harvard University Due West, SC Cambridge, MA United States United States [email protected] [email protected]
| []
|
[
"Conformation of single-stranded RNA in a virus capsid: implications of dimensional reduction",
"Conformation of single-stranded RNA in a virus capsid: implications of dimensional reduction"
]
| [
"Rouzbeh Ghafouri \nDepartment of Physics and Astronomy\nUCLA\n90095-1547Los AngelesCA\n",
"Joseph Rudnick \nDepartment of Physics and Astronomy\nUCLA\n90095-1547Los AngelesCA\n",
"Robijn Bruinsma \nDepartment of Physics and Astronomy\nUCLA\n90095-1547Los AngelesCA\n"
]
| [
"Department of Physics and Astronomy\nUCLA\n90095-1547Los AngelesCA",
"Department of Physics and Astronomy\nUCLA\n90095-1547Los AngelesCA",
"Department of Physics and Astronomy\nUCLA\n90095-1547Los AngelesCA"
]
| []
| The statistical mechanics of a treelike polymer in a confining volume is relevant to the packaging of the genome in RNA viruses. Making use of the mapping of the grand partition function of this system onto the statistical mechanics of a hard-core gas in two fewer spatial dimensions and of techniques developed for the evaluation of the equilibrium properties of a one-dimensional hard rod gas, we show how it is possible to determine the density and other key properties of a collection of rooted excluded-volume tress confined between two walls, both in the absence and in the presence of a onedimensional external potential. We find, somewhat surprisingly, that in the case of key quantities, the statistical mechanics of the excluded volume, randomly branched polymer map exactly into corresponding problems for an unrestricted linear polymer. | null | [
"https://arxiv.org/pdf/0804.1347v1.pdf"
]
| 16,336,490 | 0804.1347 | 64105c09f696a04ec035033ea77a606a8e53a241 |
Conformation of single-stranded RNA in a virus capsid: implications of dimensional reduction
Rouzbeh Ghafouri
Department of Physics and Astronomy
UCLA
90095-1547Los AngelesCA
Joseph Rudnick
Department of Physics and Astronomy
UCLA
90095-1547Los AngelesCA
Robijn Bruinsma
Department of Physics and Astronomy
UCLA
90095-1547Los AngelesCA
Conformation of single-stranded RNA in a virus capsid: implications of dimensional reduction
(Dated: January 5, 2014)numbers: 3620Ey8239Pj8714Gg0520y8715Cc
The statistical mechanics of a treelike polymer in a confining volume is relevant to the packaging of the genome in RNA viruses. Making use of the mapping of the grand partition function of this system onto the statistical mechanics of a hard-core gas in two fewer spatial dimensions and of techniques developed for the evaluation of the equilibrium properties of a one-dimensional hard rod gas, we show how it is possible to determine the density and other key properties of a collection of rooted excluded-volume tress confined between two walls, both in the absence and in the presence of a onedimensional external potential. We find, somewhat surprisingly, that in the case of key quantities, the statistical mechanics of the excluded volume, randomly branched polymer map exactly into corresponding problems for an unrestricted linear polymer.
I. INTRODUCTION AND MOTIVATION
Unlike the structure of the protein envelope of viruses, which is well-studied and precisely characterized [1,2,3] important aspects of the precise physical organization of the packaged genome are as yet undetermined. This is true in the case of both DNA [4] and RNA [5] viruses. However, it has also been determined that a single strand of RNA will organize into a tree-like secondary structure [6,7]. Such secondary RNA structure occurs in general, and tree-like configurations are known to characterize the genomic conformation of certain RNA viruses [8,9]. In light of those facts, one can hope to construct a reasonably accurate theoretical model of the statistical and mechanical properties of long segments of single-stranded (ss) RNA-and in particular of the genomic matter in ss RNA viruses-if one can properly evaluate the statistical mechanics of a tree-like polymer in the presence of an external potential energy. The potential energy plays two roles in the context of the genomic conformation in RNA viruses. First, one naturally posits an energetic barrier that serves to confine the polymer to a particular region in space. Second, given the known interaction between RNA and the protein shell [10,11], it is reasonable to assume an attractive potential in the vicinity of the boundaries of that region.
The key challenge in this problem is taking into account the effects of excluded volume. That excluded volume effects are of central importance has been known since the late 1940's when Zimm and Stockmayer [12] showed that the radius of gyration R(N ) of an ideal branching polymer scales with the number of monomers N as N 1/4 . This scaling relation means that nonexcluded-volume branching polymers are highly con- * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] densed; the monomer density grows linearly with distance from the center of the polymer coil, a situation that cannot be sustained in light of excluded volume constraints. A field theory for excluded-volume interactions of branched polymers has been constructed by Lubensky and Isaacson in the form of a 6 − expansion [13]. More recently, Parisi and Sourlas [14] have utilized supersymmetry techniques to argue that the exponents of a d-dimensional branched polymer with excluded-volume interaction described by this field theory can be obtained by a mapping to the Yang-Lee edge singularity [15,16] of a d − 2 dimensional Ising model. For d = 3, this leads to the scaling relation R(N ) ∝ N 1/2 , with a density profile that now decreases inversely proportional to distance. The supersymmetry method is a demanding formalism, but Brydges and Imbrie [17,18] showed that it could be reformulated as a relation between the conformational statistics of a branched polymer with excluded volume effects in d dimensions and the statistical mechanics of a hard-core liquid in d − 2 dimensions. See also Cardy [19] for a particularly accessible exposition.
A previous paper by the present authors [20] contains an account of the utilization of dimensional reduction methods introduced by Brydges and Imbrie to determine the conformational properties, in particular the density, of rooted trees confined to a finite region and subject to an external potential. Although dimensional reduction holds in a curved geometry (see appendix E), such as the interior of a sphere-the geometry most relevant to RNA encapsulation in a viral capsid-it does not appear to lead to the kind of fundamental simplification that allows for the analysis of the effects of interactions between the branched polymer and the surrounding walls. Consequently, our attention was focused on the simpler, but still relevant, problem of a branched polymer confined between two walls parallel in 3 dimensions. We were able to allow for interactions between the polymer and the walls, as well as any other one-dimensional external potential energy. This paper provides background to that shorter work by filling in important calculational details.
It also extends the results reported there. In particular, we build on the central-and somewhat surprisingoutcome that the statistical mechanics of the excludedvolume branched polymer maps onto the statistics of an unrestricted chain polymer to investigate both the density profile of the branched polymer and the interaction between bounding surfaces mediated by it. Given the substantial history of research on the statistics of unrestricted chain polymers in the literature(see, for instance, [21]), the results we present here are not entirely new. However, we hope that they will prove stimulating in further investigations of the structure and assembly process of spherical viruses.
An outline of the paper is as follows. In Section II we review the consequences of the connection between the statistics of an excluded volume randomly branched polymer and the statistical mechanics of a gas of hard rods in two fewer dimensions. We focus on the simplest case of a one-dimensional hard rod gas, namely a gas in which the external potential is equal to zero. This is in order to develop key formulas and, additionally, to build some mathematical intuition with regard to the behavior of the more general system in which the external potential is not constant. Section III addresses the means of solving for the density and partition function of the one-dimensional gas when the external potential varies spatially. The method utilized is based on an integral equation for the density of the gas derived to Percus [22]. We exploit the reformulation of that method by Vanderlick, et. al. [23], which we recast into a form suitable for a "lattice gas" of one-dimensional hard rods. In Section IV the approach is further developed, so that the density of the gas, and hence the generating function for branched polymer statistics, follows from the solution of linear recursion relations. The connection between those recursion relations and the Schrödinger equation is developed in Section V. This connection provides the justification for the close relationship between the statistics of the excluded volume, randomly branched polymer and the unrestricted linear polymer chain. The principal result that we will utilize relates the density, n(x, z) of a hard-core gas in d − 2 dimensions to the generating function, Σ(x, z) of rooted, branched polymers in d dimensions. Here, we assume translational symmetry in all but one dimension of the two systems, and x is the one coordinate in the direction along which there is any spatial variation. The link between the two quantities is expressed in the two relationships [17,18,19]
ρ d−2 (x, z) = ∞ N =1 z(−z/π) N −1 N Z N (x) (2.1) Σ d (x, z) = ∞ N =1 z N Z N (x) (2.2)
which tells us that the quantity Z N contains information concerning the number of configurations of an Nmonomer branched polymer in d dimensions that is rooted at the position x (through (2.2)) or, alternatively, concerning the density at x of an N -particle gas with hard core repulsion in d − 2 dimensions (through (2.1)). According to the above equations, the behavior of the polymer system with positive fugacity, z, is obtained by investigating the mathematical structure of the gas in the grand canonical ensemble, but with negative fugacity.
As an example of the application of Eqs (2.1) and (2.2), and to establish some points of reference for the discussion to follow, we will review the statistical mechanics of a gas of rods in a very large one-dimensional interval subject to a constant potential energy.
A. One-dimensional rods in an extended region under the influence of a constant external potential
The constant external potential can be set equal to zero. The grand partition function of the onedimensional gas is given by
Q(L) = ∞ N =0 (L − aN ) N a N N ! z N N < La (2.3)
where a is the hard-core "radius." The factor a N in the denominator guarantees that each term in the summand is dimensionless. In the limit of very large L, the sum will be dominated by the N value for which the summand is maximum. To locate that term we first exponentiate the summand and then re-express the exponent in terms of the density ρ = N/L. The function he h on the left hand side of (2.9) is graphed in Fig. 1 W (z), where W is the Lambert W function [24]. Figure 2 shows what the solution looks like, as the fugacity varies from a negative to positive values. Note the onset of imaginary components to the solution. The departure from a purely real solution follows from the minimum in the function he h , as displayed in Fig. 1. When z lies below this minimum, there is no purely real solution to (2.9). The behavior of the solutions in the vicinity of the point at which the imaginary part emerges yields information about both the location of this "critical point" and about its implications for the statistics of selfavoiding rooted trees in three dimensions. The minimum in he h is at h = h c = −1, corresponding to z = z c = −1/e in (2.9 This tells us that, in the immediate vicinity of z = z c ,
h = −1 ± √ 2eδz (2.12) or, from (2.7) ρ → ∓ 1 a 1 √ 2eδz (2.13)
When z > z c continuity of the solution with z > 0 requires that we take the upper sign in (2.11)-(2.13). However, when z < z c , the choice of sign is controlled by the choice of location with respect to a branch cut in the complex z plane, starting at the branch cut at z = z c and extending to z = −∞ along the negative z axis.
The fact that the (uniform) density of the onedimensional hard rod gas possess a singularity going as (z − z c ) −1/2 , where z c = −1/e, allows us to extract the leading behavior of the large N coefficients in a power series expansion of this function of the fugacity [25]. Making use of the general result that if a function f (z) can be written as a power series about z = 0, so that
f (z) = ∞ N =0 F N z N , then F N = 1 2πi f (z) z N +1 dz (2.14)
where the contour over which the integral in (2.14) is performed encircles the origin in the complex z plane and does not enclose any singularities in the function f (z). Expanding the contour so that it impinges on the branch point and, ultimately surrounds the branch cut (see Fig. 3), we end up with the following result for the coefficient of z N in the power series expansion of ρ(z):
1 πa √ 2e ∞ 0 (−1) N +1 (+|z c | + δz) N +1 δz −1/2 dδz = 1 πa √ 2e (−1) N +1 |z c | N +1 ∞ 0 e −(N +1) ln(1+δz/|zc|) δz −1/2 dδz → 1 πa √ 2e (−1) N +1 |z c | N +1 ∞ 0 e −(N +1)δz/|zc| δz −1/2 dδz ∝ (−1) N +1 |z c | N +1/2 N −1/2 (N 1) (2.15)
According to (2.1) and (2.2), we see that the number of conformations of three dimensional trees with N monomers should grow geometrically as 1/(πz c ) N , with the additional power law modification N −3/2 .
Re(z)
Im(z) z c FIG. 3:
The original contour in (2.14) (small circle surrounding the origin) and the distorted contour that bounds the branch cut originating at the branch point, zc.
Note that in the above discussion, we have neglected the possibility of any other non-analyticity in ρ(z). As one can readily verify, the leading contributions to the integral in (2.14) will, for large N , be controlled by the singularities in ρ(z) that lie closest to the origin. The possible existence of other poles, branch points or essential singularities in ρ(z), all of which will occur at |z| > |z c |, is irrelevant to the results obtained here.
The fact that the statistics of the hard-core gas-and by extension randomly configured trees-are controlled by non-analyticity in the grand canonical ensemble that is closest to the origin in the complex z plane can be exploited to infer the emergence of a bound state generated by the presence of an attractive external potential. The argument is as follows [19].
We start with the density at a particular point of the gas of hard rods. In the grand canonical ensemble, this density equal to the sum over configurations in which the a rod is at that point, divided by the sum over all allowable configurations. If the system is confined to the one dimensional region between x = 0 and x = L, then this leads to
ρ(x, z) = Q(x, z)zQ(L − x − a) Q(L, z) (2.16)
with Q the grand partition function as before. If, now, there is a non-zero, delta function potential energy, V at the position x 1 , then the grand partition function contains an additional contribution from the associated Boltzmann factor weighting configurations in which a gas particle occupies that particular location. If we write
e −βV = 1 + λ (2.17)
where an attractive potential (negative V ) generates a λ > 0, then the new partition function is
Q (L, z) = Q(L, z) + Q(x 1 , z)λzQ(L − x 1 , z) = Q(L, z)(1 + λρ(x 1 , z)) (2.18)
The new density at x 1 is, then, given by
ρ (x 1 , z) = Q(x 1 , z)zQ(L − x 1 , z) Q (L, z) = Q(x 1 , z)zQ(L − x 1 , z) Q(L, z)(1 + λρ(x 1 , z)) (2.19)
In the case of a very long interval and a large value of x 1 , the density ρ(x 1 , z) will be independent of x 1 , and is given by (2.5)-(2.9). Figure 4 illustrates the behavior of the density as a function of z for z > z c = −1/e. The fact that ρ(z) is negative for z < 0 and that it goes to −∞ as z → z + c (see also (2.13)) ensures that, for any positive value of λ, there will be a zero in Q (L.z) and hence a pole in ρ (x 1 , z) for a (negative) value of z closer to the origin than z c . If we associate this singularity in the density with a bound state, we are led to conclude that an arbitrarily weak one-dimensional potential well in the interior of an infinitely extended system will "bind" a three dimensional random tree. Further analysis of this situation requires a more searching exploration of the spatial structure of this putative bound state. The problem of a one dimensional gas of hard core rods was considered by Percus [22], who derived an integral equation for the equilibrium density of that system, from which quantity the partition function and all interesting equilibrium correlation functions can be obtained. The integral equation, which has been reduced to a very useful and tractable form by Vanderlick, et. al. [23], serves as a starting point for the exploration of the equilibrium statistics of randomly branched polymers in a one-dimensional environment, given the connections established by Brydges and Imbrie [17,18]. The two equations leading to the calculation of the density in this one-dimensional system are [22,23]
h(x) = ρ(x) 1 − x+σ x ρ(t)dt (3.1) h(x) = e β(µ−u(x)) exp − x x−σ h(t)dt (3.2)
where (3.1) can be taken as a definition of the function h(x). Note the strong similarity between (3.2) and (2.9). In fact, if we set u(x) = 0, replace e βµ by z, take σ = 1, and assume an x-independent h(x), then (3.2) reproduces (2.9), while, if we also assume an x-independent density, ρ, and set a = 1, then (3.1) reduces to (2.6).
For our purposes, it proves more useful to focus our investigations on a discrete version of the one-dimensional gas of rods-a one-dimensional, hard core lattice gas. As it turns out, the equations leading to a solution for the density of this system have already been worked out in a different context [26]. To maintain a self-contained exposition, the derivation of the equations governing the density in this discrete system are also presented below. This derivation closely parallels the arguments of Percus [22].
We assume rods that can sit on specific points on a line. As shown in rods are at sites labeled by the integer n. The actual locations of the sites are indicated by the vertical lines in the figure. The boundaries between different regions of the gas lie between the sites at the points indicated by the ×'s. These boundaries are utilized in the definitions of "partial" grand partition functions. For instance the grand partition function Ξ(−∞, n) corresponds to a system in which the rods can occupy the region to the left of the × at n in Fig. 5. The full grand partition function of the infinite system is Ξ(−∞, ∞) ≡ Ξ. Now, assume a chemical potential µ and site-dependent local potentials u n . We locate a rod by indicating the site at which its far right portion sits. For instance, Fig. 6 shows a rod with a length equal to two that is located at the site n. Given this, it is straightforward to show that the density n n+1 n-1 of rods at the site n is given by
ρ n = Ξ(−∞, n − σ)Ξ(n, ∞)e β(µ−un) Ξ (3.3)
The next step is to establish a relation for the product of two partial grand partition functions. In particular, we are interested in Ξ(−∞, n)Ξ(n, ∞). This product is almost the entire partition function. In fact, it only omits configurations in which a rod overlaps the boundary at n. In the case of the rod pictured in Fig. 6, there is precisely one such configuration: the one in which the right hand side of the rod lies on the site n + 1. In general there are σ − 1 such configurations. Let us look at the one corresponding to the location of the rod with length σ lying on the site n + 1. The contribution to Ξ of that particular configuration is Ξ(−∞, n + 1 − σ)Ξ(n + 1, ∞)e β(µ−un+1) = Ξρ n+1 (3.4) Making the appropriate corrections for all missing contributions to Ξ, we see that
Ξ(−∞, n)Ξ(n, ∞) = Ξ × 1 − σ−1 j=1 ρ n+j (3.5)
Now, we construct a recursion relation for the partial grand partition function. By inspection, one sees that the difference between the partial function Ξ(−∞, n) and the function Ξ(−∞, n − 1) lies in configurations in which there is a rod at n. In other words
Ξ(−∞, n) = Ξ(−∞, n − 1) + e β(µ−un) Ξ(−∞, n − σ) = Ξ(−∞, n − 1) + ρ n Ξ Ξ(n, ∞) = Ξ(−∞, n − 1) + ρ n Ξ(−∞, n) 1 − σ−1 j=1 ρ n+j (3.6)
The second line of (3.6) follows from (3.3) and the last line from (3.5). Thus,
Ξ(−∞, n − 1) = Ξ(−∞, n) 1 − ρ n 1 − σ−1 j=1 ρ n+j ≡ Ξ(−∞, n)(1 − h n ) (3.7)
where the last line of (3.7) serves as a definition of the quantity h n , given as
h n = ρ n 1 − σ−1 j=1 ρ n+j (3.8)
Returning to the expression on the right hand side of (3.3), we note that, given (3.7),
ρ n = e β(µ−un) Ξ(−∞, n)(1 − h n )(1 − h n−1 ) · · · (1 − h n−σ+1 )Ξ(n, ∞) Ξ = e β(µ−un) (1 − σ−1 j=1 ρ n+j ) σ m=1 (1 − h n−m+1 ) (3.9)
We then divide both sides of (3.9) by the term 1 − σ−1 j=1 ρ n+j . Then, making use of the definition of h k implicit in the last line of (3.7), and singling out the first term in the product on the right hand side of (3.9) we have
h n = e β(µ−un) (1 − h n ) σ m=2 (1 − h n−m+1 ) (3.10)
Solving for h n , we are left with
h n = e β(µ−un) σ m=2 (1 − h n−m+1 ) 1 + e β(µ−un) σ m=2 (1 − h n−m+1 ) (3.11)
If we define
H n = σ m=2 (1 − h n−m+1 ) (3.12)
then the equation (3.11) becomes
h n = H n e β(µ−un) 1 + H n e β(µ−un) (3.13)
The quantity H n can be thought of in terms of a local modification of the fugacity arising from the excluded volume constraint. Note that the quantity h n is determined by h n 's to the left of it. This means that the system of equations (3.11) can be solved by iteration to the right. Furthermore, given the explicit definition of h n in equation (3.8), we see that the density, ρ n is expressed in terms of h n and ρ m 's with m > n, so having solved for h n , we obtain the ρ n 's by iterating to the left.
To highlight the precise points of reference between the discrete equations above and the integral equations developed by Percus [22] and refined by Vanderlick, et. al. [23], we note that (3.8) corresponds to (3.1) while (3.11) is the analogue in the discrete system of (3.2).
Another version of the set of equations that is a bit simpler to iterate replaces the variables h n by
g n = 1 − h n (3.14)
Then,
H n = σ m=2 g n−m+1 (3.15)
and the equation for g n becomes
g n = 1 1 + H n e β(µ−un) (3.16) with g n = 1 − ρ n 1 − σ−1 j=1 ρ n+j (3.17)
The equations above constitute discrete version of the method of Vanderlick, et. al. [23].
A. The case σ = 1
It is worthwhile to ask what happens when σ = 1. In this case, the rods are "point particles," and excluded volume does not play a role. The solution of the equations is greatly simplified. Given (3.9) and (3.8) with σ = 1, one arrives at the following end-result.
ρ n = e β(µ−un) 1 + e β(µ−un) (3.18)
The lattice gas in this case is, in fact, well-described by an appropriate version of the one-dimensional Ising model. This system maps onto trees in which the "branches" have zero extension, and in which, furthermore, the radius of the hard core at the vertices between branches is vanishingly small.
B. The lattice gas with σ = 2
The simplest non-trivial version of this lattice gas as it applies to the conformational statistics or random trees has σ = 2. Then (3.11) and (3.8) become
h n = e β(µ−un) (1 − h n−1 ) 1 + e β(µ−un) (1 − h n−1 ) (3.19) h n = ρ n 1 − ρ n+1 (3.20)
For the time being, we will focus on a uniform gas confined to a finite region. This means that we are going to set u n = 0, supplemented by boundary conditions to be developed below. Replacing e βµ by the fugacity, z, the equation for h n becomes
h n = z(1 − h n−1 ) 1 + z(1 − h n−1 ) (3.21)
In terms of g n , as defined in (3.14), (3.21) becomes
g n = 1 1 + zg n−1 (3.22)
This equation leads to a solution for g n in terms of g 1 in terms of continued fractions:
g n = 1 1 + z 1+ z 1+··· (3.23)
We can rewrite the solution to the equation for g n in terms of an iterated matrix equation [27]. Given the fact that the continued fraction is repeated, the iteration of the matrix equation is relatively straightforward. Suppose we have the solution for g m in terms of g m−l as follows:
g m = A l + B l g m−l C l + D l g m−l (3.24)
Then, given
g m−l = 1 1 + zg m−l−1 (3.25)
we have
g l = A l+1 + B l+1 g m−l−1 C l+1 + D l+1 g m−l−1 = A l + B l 1 1+zg m−l−1 C l + D k l 1 1+zg m−l−1 (3.26)
Rationalizing the last line of (3.26), we end up with the following recursion relations
A l+1 = A l + B l (3.27) B l+1 = zA l (3.28) C l+1 = C l + D l (3.29) D l+1 = zB l (3.30)
Equations (3.27) and (3.28) can be written as follows
A l+1 B l+1 = 1 1 z 0 A l B l (3.31)
with a similar equations for C and D. Given that we have the initial conditions A 0 = 0, B 0 = 1, C 0 = 1 and D 0 = 0, we have
A l B l = 1 1 z 0 l 0 1 (3.32)
Our task now is to take the l th power of the matrix in (3.31).
C. A digression: the uniform case
Before performing the requisite calculations, we will consider the solution (3.22) under the assumption that the g's are independent of location. The equation that results from that model is
zg 2 + g − 1 = 0 (3.33)
The solution is
g = 1 2z √ 1 + 4z − 1 (3.34)
There is a branch point at z = −1/4. Figure 7 shows what the real and imaginary parts of g look like as a function of z. Close to the critical point, the general form of the plot is qualitatively identical to the result for h(z) displayed in Fig. 2. We can reconstruct the (constant) density, ρ, from (3.20), with h = 1 − g, where g is given by (3.34). From (3.20) with ρ, h and g constant, we have
ρ = h 1 + h = 1 − g 2 − g = 2z + 1 − √ 1 + 4z 1 + 4z − √ 1 + 4z (3.35) If z = −1/4 + δ, then ρ = 1/2 + 2δ − 2 √ δ 4δ − 2 √ δ → − 1 4 √ δ (3.36)
The solution (3.34) suggests a reparameterization that is useful in the vicinity of the "critical point."
z = − 1 4 cosh 2 k (3.37)
Then, (3.34) becomes
g = 2 cosh ke −k (3.38)
When z passes through -1/4, (3.37) requires that we replace the real k by an imaginary quantity.
D. Completion of the calculation
The results we need follow from the eigenvectors and eigenvalues of the matrix 1 1 z 0 . Solving the relevant equations and making use of the reparameterization (3.37), we find that (3.32) reduces to
A n B n = sinh kn (2 cosh k) n−1 sinh k − sinh(n−1)k (2 cosh k) n sinh k (3.39)
With a similar set of steps, we end up with
C n D n = sinh(n+1)k sinh k(2 cosh k) n − sinh kn sinh k(2 cosh k) n+1 (3.40)
We assume that the initial g is equal to zero. This is consistent with assumptions in the case of the ordinary one-dimensional excluded-volume gas. We will assign this g the index 0. Then,
g n (z) = A n C n = 2 sinh nk cosh k sinh(n + 1)k = 2 cosh 2 k − sinh 2k coth(n + 1)k (3.41)
If k is real, the n → ∞ limit of the above expression is consistent with (3.35). However, if k is imaginary, then the second term on the last line of (3.41) behaves like the cotangent function, and is periodic with a set of poles. See Fig. 8. The graph in this figure was generated by numerical solution of the recursion relation (3.22). The key property of this plotted solution for g n , and in the analytical form in (3.41), is the appearance of poles. In fact, as will be demonstrated below, the passage of a pole in g n through the far boundary of the interval of interest is directly associated with the appearance of a singularity in the dependence of the density on the fugacity, z. A fuller discussion of this point will follow the development of an alternative approach to the solution of the recursion relation for g n (z).
IV. MODIFIED APPROACH TO THE SOLUTION OF THE RECURSION RELATIONS
We now describe an alternate route to the analysis of this system. Recall Eq. (3.22). Suppose we write
1 + zg n (z) = ψ n+1 (z) ψ n (z) (4.1)
Then, (3.22) is transformed to the following recursion relation
ψ n+1 (z) ψ n (z) = 1 + z ψ n (z)/ψ n−1 (z) (4.2)
which is manipulated to
ψ n+1 (z) = ψ n (z) + zψ n−1 (z) (4.3) Now, let ψ n (z) = (−z) n/2 φ n (z) = 1 2 cosh k n φ n (z) (4.4)
where in the last line of (4.4) use has been made of (3.37). Substituting from (4.4) into (4.3), we end up with the recursion relation
φ n+1 (z) − 2 cosh kφ n (z) + φ n−1 (z) = 0 (4.5)
The solution to this equation is
φ n (z) = Ae kn + Be −kn (4.6)
In the case that k is imaginary, corresponding to z < −1/4, the solutions are complex exponential-or, replacing k by ik, φ n (z) = A cos kn + B sin kn (4.7)
A. Boundary conditions
We require that the density be zero at the two ends of the interval. We set the leftmost end at n = 0 and the rightmost one at n = L + 1. Making use of (3.8) with h n expressed in terms of g n , we have
ρ n = (1 − g n )(1 − ρ n+1 ) (4.8)
We ensure that ρ 0 = 0 by requiring g 0 = 1. As for ρ L+1 , we simply set it equal to zero. The pole in the solution for the density will be due to a pole in g L , which will, according to (4.1) follow from ψ L (z) passing through zero. The boundary condition at n = 0 will be obtained if we set
ψ 0 (z) = ψ 1 (z) = 1 (4.9)
Making use of the above equation and Eq. (4.1) we see g 0 = 0 The pole at n = L follows from the requirement that
ψ L (z) = φ L (z) = 0 (4.10)
Given that k will be small in most of the cases of interest to us, our solution in the uniform case will be generated from φ n = sin(lπ(n + 1)/(L + 1)) (4.11)
with l an integer. Here is how the argument goes. We start with φ n of the form sin(kn+θ). Then, given that the ratio ψ 1 /ψ 0 is equal to one, making use of the relationship (4.4) between ψ n and φ n , we have sin(k + θ) 2 cos k sin θ = 1 (4.12)
Anticipating that both k and θ will be small, we expand the sine and cosine functions in terms of their arguments, and we end up with
k + θ 2θ = 1 (4.13)
The solution to this equation is θ = k. Requiring that kL + θ = lπ, we end up with k = lπ/(L + 1) (4.14)
Substituting the results into the argument of the sine function, we end up with (4.11).
B. Reconstruction of the density: general formulas
The next step will be to reconstruct the density from the function g n (z). We can do this formally, by noting that the general form of the equation (3.20) is
ρ n+1 = α n + β n ρ n (4.15)
We can solve this equation by analogy with the solution of a linear differential equation. We start by introducing a new variable
R n = ρ n n−1 m=m0 β −1 m (4.16)
Making use of this new variable, (4.15) is recast into the recursion relation
R n+1 = R n + α n n m=m0 β −1 m ≡ R n + γ n (4.17)
The solution of this very simple recursion relation is
R n = n−1 l=l0 γ l (4.18)
In the case of the product in (4.16), the understanding is that if n − 1 < m 0 , then the product is replaced by m0 m=n−1 β m and similarly for the product in (4.17). As for the sum in (4.18), if n − 1 < l 0 then the sum is replaced by − l0 l=n γ l . Finally, if n − 1 = m 0 in (4.16), then the product is replaced by β n−1=m0 , and similarly for the product in (4.17). Note that all considerations are simplified if we take m 0 = l 0 = 0.
Ultimately, we recover the density ρ n through the inverse of (4.16), with one proviso. Taking m 0 = l 0 = 0, we note that a full solution to (4.15) is of the form
ρ n = C + n−1 l=0 γ l n−1 m=0 β m (4.19)
with C an arbitrary constant. In the case of interest, the equation corresponding to (4.15), as given by (3.20), is
ρ n+1 = 1 − h −1 n ρ n (4.20)
so here,
α n = 1 (4.21) β n = −h −1 n (4.22)
C. Specific relations for the reconstruction of the density Given the relationships (4.1) and (4.4), we can write
1 + zg n (z) = 1 2 cosh k φ n+1 (z) φ n (z) (4.23)
Solving for g n (z), we have
g n (z) = −4 cosh 2 k 1 2 cosh k φ n+1 (z) φ n (z) − 1 (4.24)
Now, we use (4.11), with k general and assume trigonometric functions (k → ik).Then, φ n (z) = sin k(n + 1) (4.25)
Where, recall, z = −1/4 cos 2 k. Substituting this into (4.24), we have
h n (z) = 1 − g n (z) = 1 − 4 cos 2 k + 2 cos k sin k(n + 2) sin k(n + 1) = − sin k(n − 1) sin k(n + 1) (4.26)
As our next step, we will implement the recursion relation repeatedly. Starting with a particular value of n, which we will set equal to L, and assuming ρ L+1 = 0, we have
ρ L = sin k(L − 1) sin k(L + 1) (ρ L+1 − 1) = − sin k(L − 1) sin k(L + 1) (4.27)
If we apply this recursion relation down to ρ L−3 , we end up with
ρ L−3 = − sin k(L − 4) sin k(L − 2) − sin k(L − 4) sin k(L − 3) sin k(L − 2) sin k(L − 1) − sin k(L − 4) sin k(L − 3) sin k(L − 1) sin k(L) − sin k(L − 4) sin k(L − 3) sin k(L) sin k(L + 1) (4.28)
There will clearly be a pole in the right hand side of (4.28) when sin k(L+1) = 0. However, the denominators in other terms that expression also pass through zero, at different values of k. As we will see in our discussion of the more general case, those denominators do not give rise to additional poles.
D. The general case
We now turn to the way in which the reconstruction of the density works in the most general case. That is, we look into what happens when the fugacity varies from position to position. In that case, the recursion relation can be written in the form
g n+1 = 1 1 + z n g n (4.29)
If we write 1 + z n g n = ψ n+1 ψ n (4.30)
we obtain the following relationship:
ψ n+2 = ψ n+1 + z n+1 ψ n (4.31)
Then,
h n = 1 − g n = z n−1 ψ n−2 ψ n (4.32)
We can reconstruct the density in the same way as we did in the case of no potential. Carrying out the same procedure as above, we find
ρ L−3 = z L−4 ψ L−5 ψ L−3 − z L−4 z L−3 ψ L−5 ψ L−4 ψ L−3 ψ L−2 +z L−4 z L−3 z L−2 ψ L−5 ψ L−4 ψ L−2 ψ L−1 −z L−4 z L−3 z L−2 z L−1 ψ L−5 ψ L−4 ψ L−1 ψ L (4.33)
Once again, the pole in the density results from a zero in ψ L . To see that no other zeros lead to a pole, we consider the case of ψ L−2 . The terms in (4.33) in which that function appears combine as follows.
−z L−4 z L−3 ψ L−5 ψ L−4 1 ψ L−2 1 ψ L−3 − z L−2 ψ L−1 = −z L−4 z L−3 ψ L−5 ψ L−4 1 ψ L−2 ψ L−1 − z L−2 ψ L−3 ψ L−1 ψ L−3 = −z L−4 z L−3 ψ L−5 ψ L−4 1 ψ L−2 ψ L−2 ψ L−1 ψ L−3 (4.34)
where the last line of (4.34) follows from (4.31) with a suitable adjustment of n. We can alter the equations for ψ m and ρ m by introducing a modification of the function φ n . Our new and generalized version is via the following alteration of (4.4)
ψ n = φ n n m=1 (−z n ) 1/2 (4.35)
Then, (4.31) becomes
(−z n+2 ) 1/2 φ n+2 = φ n+1 − (−z n+1 ) 1/2 φ n (4.36)
Furthermore, the last term in (4.34), the term in which there is a pole resulting from a zero of ψ L -and hence φ L -reduces to
− (−z L−4 ) 1/2 (−z L ) 1/2 φ L−5 φ L−4 φ L−1 φ L (4.37) V.
RELATION TO THE SCHRÖDINGER EQUATION
The recursion relation (4.5), and particularly the solutions (4.6) and (4.7), strongly suggest a relationship between the equations that we solve for the quantity g n (z) and thence the density ρ n (z) and the Schrödinger equation for a free particle. In fact, when the variable k is small and imaginary, Eq. (4.5) becomes
φ n+1 (z) − 2φ n (z)(1 − k 2 /2) + φ n−1 (z) = 0 (5.1)
which reduces, under the assumption of slowly-varying φ n (z), to
− d 2 φ n (z) dn 2 = k 2 φ n (z) (5.2)
Now, consider (4.36). We write
z n = − 1 4 cos 2 k e −βun (5.3)
Expanding in k and assuming that both φ n (z) and u n are slowly varying functions of n, this recursion relation becomes
− d 2 φ n (z) dn 2 + e βun − 1 φ n (z) = k 2 φ n (z) (5.4)
If βu n is small, the second term on the left hand side of (5.4) is just the standard potential energy contribution to the Schrödinger equation-with the multiplicative factor β. Under the conditions described above, the recursion relations and, more particularly, the density through (4.37), are obtained via the solution to the Schrödinger equation. In fact, again, if βu n is small, the residue of the pole in the density as a function of z is given by
− φ n (z) 2 Res(1/φ L−1 (z)φ L (z)) (5.5)
In fact, the residue function can be directly related to the normalization of the solutions to the equation (5.4). This fact can be established by appealing to a standard result for the normalization of the solutions to the onedimensional Schrödinger equation [28]. This argument is contained in Appendix A. For an outline of a demonstration based on the discrete equations, see Appendix B.
VI. ALTERNATE DERIVATION OF THE SCHRÖDINGER EQUATION FORMALISM FROM A GRADIENT EXPANSION
The fact that we are led by the developments in Sections IV and V from a discrete version of the onedimensional lattice gas problem as formulated by Percus and Vanderlick to the continuous Schrödinger equation suggests the possibility of a direct route from the constitutive equations of the continuous hard rod gas to the Schrödinger equation approach. In fact, guided by what has been described previously in this article, one can outline just such a development, based on a gradient expansion of (3.1) and (3.2). We start with an analysis of the second of those two equations
h(x) = e β(µ−u(x)) exp − x x−σ h(t)dt ≡ z(x) exp − x x−σ h(t)dt (6.1) where z(x) = ze −βu(x) (6.2)
When the system is uniform, the singularity occurs on the vicinity of z(x) = z 0 = −1/σe, h(x) = h 0 = −1/σ. We rewrite the (6.1) as follows
σh(x)e −σh(x) = σ(−1/σ + δh(x))e σ(−1/σ+δh(x)) = −(1 − σδh(x))e −1 e σδh(x) = −e −1 (1 − σδh(x))(1 + σδh(x) + 1 2 σ 2 δh(x) 2 + · · · ) = −e −1 (1 + 1 2 σ 2 δh(x) 2 + · · · ) = σ(−1/σe + δz)e −βu(x) exp − x x−σ h(t)dt − σh(x) = −e −1 (1 − σeδz)e −βu(x) e σ 2 dδh(x)/dx/2+··· = −e −1 (1 − σeδz)(1 − βu(x) + · · · ) ×(1 + 1 2 σ 2 dδh(x) dx + · · · ) (6.3)
Equating the fifth and eighth line of (6.3), and expanding to second order in σ and first order in βu(x), we end up with the following equation
1 − 1 2 σ 2 δh(x) 2 = 1 − σeδz + 1 2 σ 2 dδh(x) dx − βu(x) (6.4) or − dδh(x) dx − δh(x) 2 + 2 σ 2 βu(x) = − 2e σ δz (6.5) If we set δh(x) = 1 2 W (x) (6.6) then (6.5) becomes − dW (x) dx − 1 2 W (x) 2 + 4 σ 2 βu(x) = − 4e σ δz (6.7) Let W (x) = 2 ψ (x) ψ(x) = d dx ln ψ(x) 2 (6.8)
Then,
dW (x) dx + 1 2 W (x) 2 = 2 d 2 ψ(x)/dx 2 ψ(x) (6.9)
and (6.7) becomes
− d 2 ψ(x) dx 2 + 2 σ 2 βu(x)ψ(x) = − 2e σ δz ψ(x) (6.10)
We next turn to the other constitutive equation, (3.1). To analyze this relation, we multiply both sides by σ and perform the same kind of expansion as we did on (6.1). To facilitate this expansion, we replace ρ(x) by −ρ(x), anticipating that the density becomes negative in the regime of interest. In fact, it will become large and negative, as we are interested in the behavior in the vicinity of a pole. Performing the gradient expansion utilized above, (3.1) becomes
σ(−1/σ + δh(x))(1 + σρ(x) + 1 2 σ 2 dρ(x) dx + · · · ) = −(1 − σδh(x))(1 + σρ(x) + 1 2 σ 2 dρ(x) dx + · · · ) = −(1 + σρ(x) − σ 2 ρ(x) δh(x) + 1 2 σ 2 dρ(x) dx −σδh(x) + · · · ) = −σρ(x) (6.11)
The neglected terms in (6.11) are higher order in σ.
Equating the last line in (6.11) to the next-to-last line, we end up with the equation
1 2 σ 2 dρ(x) dx − σ 2 ρ(x)δh(x) = σδh(x) − 1 (6.12)
Making use of (6.6), this equation becomes
dρ(x) dx − W (x)ρ(x) = 2 σ 2 (σδh(x) − 1) (6.13)
Note that the solution to the homogeneous version of (6.13) is
ρ(x) = exp x x0 W (x )dx = exp ln(ψ(x) 2 − ψ(x 0 ) 2 ) = Aψ(x) 2 (6.14)
where we have made use of (6.8).
To further analyze (6.13), we replace δh(x) on the right hand side by ψ (x)/ψ(x), as mandated by (6.6) and (6.8).
If we further set
ρ(x) = ψ(x) 2 r(x) (6.15)
Eq. (6.13) becomes
ψ(x) 2 dr(x) dx = 2 σ 2 σ dψ(x)/dx ψ(x) − 1 (6.16)
In keeping with our gradient expansion approach, we focus our attention on second term in parentheses on the right hand side of (6.16). The equation that results from ignoring the first term is
r(x) = − 2 σ 2 x x0 dx ψ(x ) 2 (6.17)
and we obtain the following expression for ρ(x)
ρ(x) = − 2 σ 2 ψ(x) 2 x x0 dx ψ(x ) 2 (6.18)
We now note that the function
φ(x) = ψ(x) x x0 dx ψ(x ) 2 (6.19)
is also a solution to Eq. (6.10), and that it has a Wronskian of one with the function ψ(x), in that φ (x)ψ(x) − ψ (x)φ(x) = 1 [29]. Both of these properties can be verified by direct substitution. Note that they are independent of the lower bound of integration, x 0 . A general version of (6.19) is
ρ(x) = − 2 σ ψ(x)(φ(x) + Aψ(x)) (6.20)
We will assume that ψ(x) satisfies the boundary condition ψ(0) = 0, which yields a ρ(x) that is also zero at that interval boundary. To insure that ρ(L) = 0, we set the constant A equal to −φ(L)/ψ(L). There is then a pole in the density when ψ(L) = 0. To determine the residue at that pole, we note the following
2 σ 2 ψ(x) 2 φ(L) ψ(L) = 2 σ 2 ψ(x) 2 φ(L) ψ(L) ψ (L) ψ (L) = 2 σ 2 ψ(x) 2 φ (L)ψ(L) − 1 ψ(L)ψ (L) (6.21)
The last line of (6.21) follows from the Wronskian relation between φ(x) and ψ(x). The pole term on the right hand side is the one going as (ψ(L)ψ (L)) −1 . Following the reasoning in Appendix A (see especially (A4)) we find that this generates the factor −σ/(2e)1/((δz − δz 0 ) L 0 ψ(x) 2 dx), where δz 0 is the value (actually, one of the values) of δz at which ψ(L) = 0. Thus, the residues at the poles incorporate the normalization of the eigenfunctions there, as was the case in the discrete version of the model.
VII. APPLICATION OF THE SCHRÖDINGER EQUATION FOMALISM: THE CASE OF A DELTA-FUNCTION POTENTIAL
In a case of particular interest to us the potential energy, u n , is non-zero at one site, n 0 . Here, it is not correct to treat the potential energy as either always small or slowly-varying. However, as we will see, the connection between solutions to the Schrödinger equation and the density as constructed from discrete recursion relation still holds. The recursion relation to which this potential energy leads is
g n0+1 = 1 1 + ze −βu g n0 ≡ 1 1 + z g n0 (7.1)
Rewriting the g's in terms of φ's, we end up with the relation
φ n0+2 = 2 cos kφ n0+1 − φ n0 +φ n0 1 − z z 2 cos kφ n0 − φ n0+1 2 cos k(1 − z /z)φ n0 + φ n0+1 z /z = φ 0 n0+2 +φ n0 1 − z z φ n0−1 2 cos k(1 − z /z)φ n0 + φ n0+1 z /z (7.2)
The quantity φ 0 n0+2 is the value of φ n0+2 in the absence of the delta function potential. If that potential is small, then (1 − z /z) will be small, and the right hand side of (7.2) becomes
φ 0 n0+2 + φ n0 z z − 1 φ n0−1 /φ n0+1 (7.3)
Now, we once again take the ratio with which we started. This leads us to the equation
φ n0+2 φ n0+1 = φ 0 n0+2 φ n0+1 + z z − 1 φ n0 φ n0−1 φ 2 n0+1 (7.4)
We will consider two cases: extended states and the possibility of a bound state. The latter solution exists if the potential is attractive and exceeds a threshold value.
Extended state solutions
The assumption that we now work with is that the solutions to the equations have the following form φ n = a sin(k(n + 1)) n ≤ n 0 + 1 b sin(k(n + 1) + θ) n ≥ n 0 + 1 (7.5) Note that there are two possible forms for of φ n at n = n 0 +1. That is, a sin(k(n 0 +2)) = b sin(k(n 0 +2)+θ). The quantity φ 0 n0+2 is, according to (7.5), equal to a sin(k(n+ 3)). Given all this, (7.4) becomes sin(k(n 0 + 3) + θ) sin(k(n 0 + 2) + θ) = sin(k(n 0 + 3)) sin(k(n 0 + 2)) + z z − 1 sin(k(n 0 + 1)) sin(kn 0 ) sin(k(n 0 + 2)) 2 (7.6) We now consider the most general version of (7.6) as a condition on the phase shift θ. That is, we look at the equation
sin(k(n 0 + 3) + θ) sin(k(n 0 + 2) + θ) = A (7.7)
Expanding the sine functions and solving for θ, we end up with the following result:
tan θ = sin(k(n 0 + 3)) − A sin(k(n 0 + 2)) A cos(k(n 0 + 2)) − cos(k(n 0 + 3)) (7.8)
Inserting the right hand side of (7.6) into (7.8) as a substitute for the quantity A, we end up with the result for tan θ:
tan θ = − (z/z − 1) sin(k(n 0 + 1)) sin(kn 0 ) sin k + (z/z − 1) sin(k(n 0 + 1)) sin(kn 0 ) cos(k(n 0 + 2))/ sin(k(n 0 + 2)) (7.9)
We are now in a position to work out the allowed values of the quantity k. Given the boundary condition (4.10), the requirement on k is sin(k(L + 1) + θ(k)) = 0 (7.10)
The bound state
Here, we assume hyperbolic functions, corresponding to a form for k that places the fugacity, z, closer to the origin than −1/4. The solution to the recursion relations for φ n (z) will then be φ n (z) = a sinh(k(n + 1)) n ≤ n 0 + 1 be −kn n ≥ n 0 + 1 (7.11) This solution is appropriate to a system in which the length of the region to which the rods are confined is arbitrarily great. Applying (7.4) to this conjectured so-lution, we end up with the relationship e −k = sinh(k(n 0 + 3)) sinh(k(n 0 + 2)) + z z − 1 sinh(k(n 0 + 1)) sinh(kn 0 ) sin 2 (k(n 0 + 2)) (7.12)
which can be manipulated to sinh k sinh(k(n 0 + 2)) = 1 − z z sinh(kn 0 ) sinh(k(n 0 + 1))e −k(n0+2) (7.13)
Limiting cases
Two limits are of interest to us. First, if n 0 1, then (7.13) reduces to 1 − z z = 2 sinh ke 3k (7.14)
Given 1 − z/z = 1 − e βu , we note that (7.14) has a solution of the type desired only if u < 0. If we expand the left hand side of (7.14) in u and, assuming negative u, replace the left hand side of (7.14) by −|βu|, we obtain the following equation for k e 4k − e 2k − |βu| = 0 (7.15)
The solution to this equation is k = 1 2 ln 1 2 1 + 1 + 4|βu| (7.16)
When k is small-the other limit of interest-the right hand side of (7.16) can be expanded, and we find k = |βu| (7.17)
The case of small k
Continuing in our investigation of the small-k regime, we note that when k 1 we can ignore the difference between sinh(kn 0 ) and sinh(k(n 0 +1)) and sinh(k(n 0 +2)) in (7.13), and we can also replace sinh k by k. Then, the condition on k is
k = 1 − z z sinh(kn 0 )e −k (7.18)
This limit can also be applied to the calculation of the properties of bound states. For example, Eq. (7.9) reduces to tan θ = −(z/z − 1) sin 2 (kn 0 )/k 1 + (z/z − 1) sin(kn 0 ) cos(kn 0 )/k (7.19)
As will be verified in Section VIII, Eqs. (7.17)-(7.19) are consistent with the equations at which one arrives in the case of a delta function potential in the corresponding Schrödinger equation. This is especially the case if we expand the factor (1 − z/z ) to first order in βu, which corresponds to the magnitude of the Dirac delta function potential.
VIII. DELTA-FUNCTION POTENTIAL WELL IN THE VICINITY OF A SURFACE
Henceforth, we will assume that the functions, φ n (z) can be rewritten in the form φ(x, z) where x is now a continuous variable, and that those function can be determined by solving the appropriate version of a Schrödinger equation. We then assume an attractive delta function potential in the vicinity of the bounding surface at x = 0, and we rederive the equations satisfied by the bound state and the phase shift in extended states. We start with the standard "matching condition"
φ (x) φ(x) x=x1− − φ (x) φ(x) x=x1+ = V (8.1)
where V is the strength of the attractive potential at the point x 1 , which we use as a simpler substitute for the combination (z/z − 1). Note that the dependence of φ(x, z) on z has been suppressed. This practice will be followed throughout this section.
A. The bound state
The unnormalized bound state is given by
φ b (x) = sinh κx x < x 1 sinh κx 1 e −κ(x−x1) x > x 1 (8.2)
Applying the matching condition (8.1) to the solution (8.2), we end up with the equation for the quantity κ
coth κx 1 + 1 = V κ (8.3)
One readily establishes, by looking at small-κ limits, that there is now a minimum value of V required to sustain a bound state. In order for this to be possible, we must have V > 1/x 1 . An alternate, but equivalent, version of (8.3) is
κ − V e −κx1 sinh κx 1 = 0 (8.4)
This relation is to be compared with (7.18).
The normalization of the bound state is obtained by taking the integral
∞ 0 φ b (x) 2 dx = 1 2κ [sinh κx 1 e κx1 − κx 1 ] (8.5)
B. Extended states
Here, the states are of the form φ(x) = sin kx sin kx1 sin(kx 1 + θ(k)) x < x 1 sin(kx + θ(k)) x > x 1 (8.6) Note that the form of the extended eigenfunctions is consistent with the normalization considerations laid out in Appendix C. The equation satisfied by θ(k) is
k cot kx 1 − k cot(kx 1 + θ(k)) = V (8.7)
After manipulations like those in Section VII, we end up with the equation for θ(k)
tan θ(k) = V k sin 2 kx 1 1 − V k cos kx 1 sin kx 1 (8.8)
This relation effectively replicates (7.19).
C. The calculation of the generating function
We start by noting that the following holds in the vicinity of the critical point.
z = − 1 4 1 cos 2 k → − 1 4 (1 + k 2 ) (8.9)
Given the values that k takes in a system with constant potential and large system size, L, (see (4.14)), the poles in the density as a function of fugacity, z, will be closelyspaced on the negative z-axis as indicated in Fig. 9.
Re(z)
Im(z) z c We will start with the generating function for the gas of rods deep in the interior of a constant potential system. We use the Mittag-Leffler theorem [30] to reconstruct the density from its poles. In the case at hand, we can ignore the spatial structure of the modes, so that the residues can be assumed to be constant. The relevant expression
is 1 L l 1 z − z l (8.10)
where quantities z l are the poles and 1/L encapsulates the normalization of the modes. Given (8.9), this sum reduces to
1 2π|z c | ∞ −∞ dk (z + |z c |)/|z c | + k 2 = 1 |z c | 1 2 |z c | + z (8.11
) As previously, z c = −1/4. Obtaining the coefficient of z N in the expansion of the right hand side of (8.11) is a pretty straightforward exercise. I will utilize a method that is, at least initially, a bit more complicated. We start with the standard contour integral-based expression for the coefficient of z N in the expansion of a function of z. As applied to the function at issue here, it is
1 2 |z c | 1 2πi 1 z N +1 1 |z c | + z dz (8.12)
where the closed contour encircles the origin. We distort the contour so that it wraps around the branch cut that starts at z = z c and extends to −∞. If we write z = −|z c | − ζ then the integral will look like this, to within overall multiplicative constants.
∞ 0 1 (−|z c | − ζ) N +1 1 √ ζ dζ = (−1) N +1 ∞ 0 1 (|z c | + ζ) N +1 1 √ ζ dζ (8.13)
Now, we introduce the following identity
1 (|z c | + ζ) N +1 = 1 N ! ∞ 0 t N e −t(zz+ζ) dt (8.14)
The double integral we now have to perform is
(−1) N +1 1 N ! ∞ 0 dt ∞ 0 dζt N e −t(|zc|+ζ) √ ζ (8.15)
We will perform the integral over ζ first. It is
∞ 0 e −tζ √ ζ dζ = 2 ∞ 0 e −ty 2 dy = ∞ −∞ e −ty 2 dy = π t (8.16)
The remaining integral is
(−1) N +1 √ π N ! ∞ 0 t N −1/2 e −|zc|t dt = (−1) N +1 √ π N ! ∞ 0 exp [(N − 1/2) ln t − t|z c |] dt (8.17)
When N 1, we can evaluate the integral by looking for its maximum. The extremum equation that determines the maximizing value of t is
N − 1/2 t − |z c | = 0 (8.18)
The solution to this equation is
t = N − 1/2 |z c | (8.19)
Substituting this back into the integrand we find for the result of the integral
(−1) N +1 √ π 1 N ! exp (N − 1/2) ln N − 1/2 |z c | − (N − 1/2) = (−1) N +1 √ π 1 N ! exp [(N − 1/2) ln |z c | + (N − 1/2) ln N − 1/2 − (N − 1/2)] = (−1) N +1 √ π 1 N ! |z c | −(N −1/2) exp [(N − 1/2) ln N − N ] = (−1) N +1 √ π|z c | −(N −1/2) exp [(N − 1/2) ln N − N − (N ln N − N )] = (−1) N +1 √ π|z c | −(N −1/2) exp [−(1/2) ln N ] = (−1) N +1 √ π|z c | −(N −1/2) N −1/2 (8.20)
In the fourth line of (8.20), Stirling's formula for N ! was used.
The principal result, that the coefficient of z N goes as |z c | −N N −1/2 , could have been derived considerably more easily. However, the method can be generalized. For example, consider the case of the generating function near a boundary. Here, we have for the generating function
2 π ∞ 0 sin 2 kx |z c | + z + k 2 dk = 1 π ∞ 0 1 − cos 2kx |z c | + z + k 2 dk = 1 2π ∞ −∞ 1 − cos 2kx |z c | + z + k 2 dk = 1 − e −2 √ |zc|+z x 2 |z c | + z (8.21)
We know how to extract the coefficient of z N in part of the expression on the last line of (8.21). For the additional part, the double integral corresponding to (8.15) is
(−1) N +1 1 N ! ∞ 0 dt ∞ 0 dζ t N e −t(|zc|+ζ) cos(2 √ ζ x) √ ζ (8.22
) Again, we perform the integral over ζ first, changing integration variables as in (8.16). We end up with the integral
2 ∞ 0 e −ty 2 cos(2yx)dy = ∞ −∞ e −ty 2 +2iyx dy = π t e −x 2 /t (8.23)
The final integration to perform is
(−1) N +1 √ π N ! ∞ 0 exp (N − 1/2) ln t − t|z c | − x 2 /t (8.24) The new extremum equation is N − 1/2 t − |z c | + x 2 t 2 = 0 (8.25)
The analysis can be short-circuited if we take into account the following facts:
1. The correction to the solution of the equation due to the last term will be small.
2. The effect of the correction on the first two terms in the exponent in (8.24) will also be very small, as t has already been adjusted so that those terms are at an extremum.
This all means that the result of the integration in (8.24) will to be the same as in (8.17), except that there is the additional term
− x 2 t = − x 2 |z c | N − 1/2 (8.26)
Combining this with the term we have already evaluated we have for the coefficient of z N in the case of the hardrod gas near an end-wall
(−1) N +1 √ π|z c | −(N −1/2) N −1/2 1 − e −|zc|x 2 /(N −1/2) (8.27)
The difference between N and N + 1/2 can be neglected in the denominator in the exponent in (8.27).
D. Density in a finite interval
We can also utilize the Mittag-Leffler method to reconstruct the density in the case of a finite interval. Here, the reconstructed density is, to within an overall multiplicative factor
ρ(n) = 1 L ∞ m=1 1 4 + (πm) 2 4L 2 −N sin 2 (πmn/L) = 4 N L ∞ m=1 e −N ln(1+π 2 m 2 /L 2 ) sin 2 (πmn/L) → 4 N L ∞ m=1
e −π 2 m 2 N/L 2 sin 2 (πmn/L) (8.28) where the explicit value of z c is used. There are two different limits to consider, based on the ratio N/L 2 . If N L 2 , then the sum is dominated by the first term, and the density is
ρ(n) = 4 N L e −π 2 N/L 2 sin 2 (πn/L) (8.29)
On the other hand, if N L 2 , then the sum is as given by (8.27). Figure 10 shows how the density as given by (8.28) behaves as a function of n when L = 10, 000 for various values of N . The function is multiplied by 4 −N √ N so that the various curves tend to the same value in the interior of the interval when N is small enough. The density as given by the last line of (8.28), for a gas of hard rods, normalized as in Fig. 10. In this case the values of N are 50, 5,000, 500,000, 5,000,000 and 50,000,000. Again, the highest curves correspond to the smallest values of N .
E. The attractive potential
We can make use of the results above to perform the integral needed to reconstruct the density as a function of the fugacity. An important precursor to the calculation of the generating function is the reconstruction of the sum l φ l (x) 2 . The details of this calculation are contained in Appendix D. Making use of the results of the results of that appendix, we find that the modification of the generating function for the density due to the presence of the delta function potential near the boundary is
− 1 π ∞ −∞ sin 2 kx 1 Re e 2ikx V k − V sin kx 1 e ikx1 × dk (z + |z c |) + k 2 + sin 2 κx 1 e −2κ(x−x1) 1 2κ [sinh κx 1 e κx1 − κx 1 ] = sinh 2 ( z + |z c | x 1 ) z + |z c | × e −2 √ z+|zc| x V z + |z c | − V sinh( z + |z c | x 1 )e − √ z+|zc| x1 (8.30)
In the above equation, the quantity κ satisfies (8.3) or, equivalently, (8.4). This contribution to the generating function is, recall, in addition to the contribution that one derives in the absence of the attractive potential. The results in (8.30) are relevant to the case x > x 1 . Now, the extraction of the actual density at fixed monomer number, N , from (8.30) entails the kind of contour integral described in Section II. The result of that integration depends on the value of V . If the potential strength is sufficiently great that there is a bound state, then one can show straightforwardly that there is a pole in the last line of (8.30) with a residue that yields the normalized bound state with a prefactor going as z −N * , where
|z c | + z * = κ (8.31)
The quantity κ, again, satisfies the equivalent equations (8.3) or, equivalently (8.4), for the bound state. Note that the absolute value of z * is less than |z c |, in that z * = κ 2 − |z c |. We are assuming a κ that is not too large. If V does not exceed the threshold for a bound state, then things are a bit different. There is no pole in (8.30), but rather a branch cut. A detailed calculation, under the assumption x ∼ √ N and
x 1 √ N , yields ρ(x) ∝ |z c | −N N −1/2 1 − e −|zc|x 2 /N 1 − 2V |z c |x 2 1 x/ √ N 1 − V x 1 (8.32)
Note the denominator in the last term in brackets on the left hand side of (8.32), a signature of an impending ground state.
F. Attractive potential at the boundaries of a finite interval
Some numerical results serve to illustrate the effects of attractive potentials on the density of branched polymers confined to a finite interval. For example, Fig. 12 shows what the density looks like for various values of the number of monomers, N , when there is an attractive potential a distance n = 40 from the boundaries of a region with extension L = 10, 000. Here, we take N = 50, 500, 5,000 and 50,000. The figure graphs the density close in to one of the bounding surfaces. In each case, the density is multiplied by z N * , where z * is the value of z corresponding to the pole associated with the lowest energy bound state. In the case of Fig. 12, the attractive potentials are sufficiently strong to ensure two bound states. Figure 13 displays the same set of modified densities, this time over the entire interval. Finally, we consider the case of an attractive potential that is not quite strong enough to generate bound states. Here, one might expect to see the influence of a "precursor effect." Figure 14 graphs the density, normalized as in Fig. 10. The density, normalized as in Fig. 10, in the vicinity of one of the boundaries of a system with length, L = 10, 000, for N equal to 50, 500, 5,000 and 50,000. The attractive potential, which is not quite strong enough to induce a bound state, is at a distance 40 from the boundary.
on N in this case is to be contrasted with the graphs of ρ(n) in Fig. 10, where at N = 5, 000, the normalized density is greatly suppressed with respect to the density at lower values of N .
IX. THE PRESSURE OF ROOTED TREES
Recall that the generating function for the number of trees with roots at the position x is derivable from the density of a gas of hard-core particles. As we have seen, the form of this density is
k ψ k (x) 2 z k + z (9.1)
The functions ψ k (x) are solutions to a Schrödinger-like equation that are, furthermore, normalized. If we sum over all possible locations of the roots, we end up with a generating function directly derivable from the sum for all rooted trees in the interval. This generating function is
k 1 z k + z (9.2)
We are interested in the coefficient of z N in this sum, which leads us to the following analogue of the partition function for rooted trees in the interval:
k z −N k (9.3)
There are additional combinatorial factors, but, as we will be looking at large N and taking a log, they turn out to be unimportant. From the expression above, we obtain the following result for the effective free energy:
− k B T ln k |z k | −N (9.4)
If N is large enough that the z k closest to the origin dominates, then the free energy reduces to
N k B T ln z 1 (9.5)
where z 1 is the location of the singularity that lies closest to the origin.
A. Pressure in the absence of a potential energy: scaling formulas
In the case of an interval with no potential, we can write
z k = −z c − w kπ L 2 (9.6)
where z c = w = 1/4. Then, the free energy has the form
−k B T ln k z c + w kπ L 2 −N = N k B T ln z c − k B T ln k 1 + w z c kπ L 2 −N → N k B T ln z c − k B T ln k e −N w zc (kπ/L) 2 (9.7)
The last line of (9.7) follows if L is sufficiently large. We assume this to be the case and proceed. The pressure is the negative of the derivative of the free energy with respect to L. Making use of (9.7) we find
P = 2k B T N w z c k k 2 π 2 L 3 e −N w zc (kπ/L) 2 k e −N w zc (kπ/L) 2 (9.8)
Keeping the full expression for the free energy, we obtain a general result for the force between the two walls. Figure 15 illustrates the dependence on L of the force divided by k B T , with w = z c = 1 and N = 100. One can easily show that at sufficiently large L the pressure will go as 1/L. In fact, a straightforward analysis tells us that the pressure will, in this case have the general form P (L, N )/k B T = P(L/ √ N )/L (9.9) Figure 16 is a graph of the function on the right hand side of (9.9). The large L behavior of the quantity P is evident from the figure, in that it approaches a limiting value as its argument goes to infinity.
There is also the question of the behavior of the pressure at intermediate values of L √ N . In the regime in which one singularity in z dominates, one can show that the pressure goes as N/L 3 . This is consistent with P ∝ N/L 2 . Figure 17 is a plot of (L/ √ N ) 2 × P. Note that this combination is effectively a constant between L/ √ N = 0 and L/ √ N = 3. From this we can infer two different regimes for the pressure. In the first, when L √ N , the force goes as 1/L, independent of N . In the second in which L/ √ N 1, the force goes as N/L 3 .
X. THE INFLUENCE OF ATTRACTIVE POTENTIALS
Once again, we assume that there are attractive potentials near the two bounding surfaces, as indicated in Fig. 18. There will be two types of eigenfunction in this configuration: even and odd parity about the center. In the region 0 < y < L/2, the even functions have the form ψ e (y) = sin ky y < x cos(k(L/2 − y)) x < y < L/2 (10.1)
The odd parity functions are of the form ψ o (y) = sin ky y < x sin(k(L/2 − y)) x < y < L/2 (10.
2)
The equation for the eigenvalues, expressed through the quantity k, is where V is the strength of the delta function potential, x − is just to the left of x and x + is just to the right. Making use of the two forms in (10.1) and (10.2), we have for the equations satisfied by k in the case of the even and odd eigenfunctions, respectively
ψ (x − ) ψ(x − ) − ψ (x + ) ψ(x + ) = V(k cot(kx) + k tan k x − L 2 = V (10.4) k cot(kx) − k cot k x − L 2 = V (10.5)
For sufficiently large values of V , there may also be solutions at imaginary k = iκ. The equations that are satisfied in the even and odd parity cases are, respectively,
κ coth(κx) − κ tanh κ x − L 2 = V (10.6) κ coth(κx) − κ coth κ x − L 2 = V (10.7)
The threshold values of V for which there are solutions to (10.6) and (10.7) are, respectively
V even = 1 x (10.8) V odd = 1 x + 1 L/2 − x (10.9)
Given these solutions, we are able construct the expression for the force generated by the rooted branched polymers. As previously, we start with the result (9.4) for the effective free energy. The general result for the force associated with this free energy, corresponding to (9.8), is
P k B T = ∂ ∂L ln l |z l | −N = −N l |z l | −N −1 ∂|z l |/∂L l |z l | −N (10.10)
To assess the derivative of the singularities with respect to L, we make use of the general relationship |z l | = 1 4 cos 2 k l (10.11) Then,
∂|z l | ∂L = − sin k l 2 cos 3 k l ∂k l ∂L (10.12)
There will be corresponding equations for the up to two bound states, in which an analytic continuation has been performed from the variable k to the variable iκ. Equations (10.10)-(10.12) allow us to calculate results for the force exerted by the branched polymer between two walls. Figure 19 summarizes results for the pressure for various values of the attractive potential. In the case of interest here, there is no bound state unless V > 10. Note that for sufficiently small separa- tions and sufficiently attractive potentials the pressure becomes negative, corresponding to an attraction between the walls. This follows from the fact that the bound state free energy decreases as the attractive wells approach each other, in analogy to the simplest version of the chemical bond [31]. Figures 20 and 21 are two interesting and contrasting plots. In both figures, the attractive potential is a distance 0.1 from the edges of the system. The threshold for a bound state is V = 10. In the first plot, V = 10.00025, which is just sufficiently strong that there is a bound state In the second plot, V = 9.9997, and the attractive delta function does not quite suffice to produce such a solution to the Schrödinger-like equation. The plots are for a range of values of N , as indicated in the caption. In the case of the first plot, Fig. 20, the larger the value of N , the lower, or more negative, the pressure. In the case of the second plot, Fig. 21, the greater N , the higher the pressure.
What this tells us is a very small change in the attractive potential suffices to give rise to a considerable change in the force between the walls. It also tells us that an attractive potential can be "mediated" by the branched polymer in such a way as to facilitate the assembly of a capsid.
Finally, note in the case of Fig. 20 that the force is concave downward when L is small enough. This leads to a mechanical instability in that range, in which the walls will continue to collapse, assuming a constant countervailing force. This means that the walls will be pulled together until other mechanisms, such as excluded volume effects, intervene.
ne n=n b φ n (k ) e −βun+1/2 φ n+1 (k) + e −βun/2 φ n−1 (k) − ne n=n b φ n (k) e −βun+1/2 φ n+1 (k ) + e −βun/2 φ n−1 (k ) = ne n=n b φ n (k )e −βun+1/2 φ n+1 (k) − φ n (k)e −βun/2 φ n−1 (k ) − ne n=n b φ n (k)e −βun+1/2 φ n+1 (k ) − φ n (k )e −βun/2 φ n−1 (k) = φ ne (k )e −βun e +1/2 φ ne+1 (k) − φ n b (k)e −βun b /2 φ n b −1 (k ) −φ ne (k)e −βun e+1 /2 φ ne+1 (k ) + φ n b (k )e −βun b /2 φ n b −1 (k) = 2(cos k − cos k ) ne n=n b φ n (k )φ n (k) (B2)
We will take the solutions of the equations in the equation above to satisfy the boundary conditions in Section IV. Furthermore, we take the end point of the summation to be n b = 1, n e = L − 1. Furthermore, we will assume that the potential energy is zero at and near the boundaries. Then, φ n b (k) = φ n b −1 (k) and similarly for k . Additionally, we will assume that the value k is consistent with the boundary condition φ ne=L (k) = 0. Then, (B2) reduces to
− φ L−1 (k)φ L (k ) = 2(cos k − cos k ) L−1 n=1 φ n (k)φ n (k )
(B3) From here on, the analysis follows that in Appendix A.
APPENDIX C: NOTE ON THE NORMALIZATION OF EXTENDED EIGENSTATES
As an essential step in the calculation of generating functions, we establish the proper way to normalize the eigenfunctions we deal with in the case of very large systems. In particular, we are interested in the case of an eigenfunction in a long interval that goes as φ(x) ∝ sin(kx+θ(k)) towards the left end of the interval. The boundary condition will be φ(x 0 ) = 0, which tells us that
k l x 0 + θ(k l ) = lπ (C1)
Recall (A6). For a given value of k l , we have ∆x(x 0 + θ (k l )) + x 0 ∆k = 0 (C2) This tells us that
dx 0 dk = − x 0 + θ (k) k (C3)
Given that, in our version of the Schrödinger equation, E = k 2 , we can rewrite (C3) as
dx 0 dE = 1 2k dx 0 dk = − x 0 + θ (k) 2k 2 (C4)
From (A6) and the above, we have
x0 0 φ(x) 2 dx = dφ(x) dx 2 x=x0 x 0 + θ (k) 2k 2 (C5)
We now return to (C1). If the integer l increments by one, then k will change as follows ∆k(x 0 + θ (k)) = π (C6) This tells us that ∆k (x 0 + θ (k)) π = 1 (C7) and, we then have the following result for the eigenfunction sum leading to the density:
l φ l (x) 2 x0 0 φ l (x) 2 dx = l φ l (x) 2 2k 2 l (dφ(x)/dx) 2 x=x0 (x 0 + θ (k l )) ∆k(x 0 + θ (k l )) π → 2 π k 2 (dφ(x)/dx) 2 x=x0 φ(x) 2 dk (C8)
If we choose φ(x) precisely equal to sin(kx + θ(k)) at x near x 0 , then the last line in (C8) reduces to 2 π φ(x) 2 dk (C9) defining the partition function of a super gas. We show those two are equal using cluster expansion and the properties of gaussian integration. Finally we put the partition function of a super gas into the form of the partition function of a solution of branched polymers. This last step is done using the Taylor expansion of functions of Grassman variables.
Classical Hardcore Gas
Consider the classical gas in a D dimensional space. We assume the interaction potential energy of two gas particles, U (r 2 ij ), depends only on the distance between those two particles: r 2 ij . The labels, i and j, indicate the gas particles. We also include an external potential, V (r 2 i ). This potential depends on the radial distance of the gas particle from the origin: r 2 i . The partition functions of the hardcore gas is
G HG (z) = Z HG (N )z N (E2) Z HG (N ) = 1 N ! i d D r i e − P ij U (r 2 ij )− P i V (r 2 i )(E3)
If we define:
g ij = g r 2 ij = exp(U (r 2 ij )) − 1 and h i = h r 2 i = exp(V (r 2 i ))
then it is well known that the grand partition function can also be represented in the form of cluster expansion [32].
G HG (z) = exp( k b k z k ) (E4) b k = 1 k! CG k i d D r i h i ij g ij (E5) (E6)
The label CG stands for 'Connected Graphs'. b k is the sum of all connected graphs (clusters) with k particles divided by k!. in the next section we define the partition function of a super symmetric gas.
supersymmetric gas
Consider the supersymmetric gas in a super-space of d = D + 2 real plus 2 Grassman coordinates. As in Cardy's exposition [19] the distance between two particles and also a particle from the origin is defined as:
R 2 ij = (r i − r j ) 2 + (θ i −θ j )(θ i − θ j ) (E7) R 2 i = r 2 i +θ i θ i (E8)
The quantities r ij and r i are the usual real-valued distances in d dimensional real space, while θ i andθ i are the Grassman coordinates of the particles. As in the case of the classical hardcore gas, it is natural to define the partition function of a supersymmetric gas as ('SSG' stands for 'Super Symmetric Gas'):
G SSG (z) = Z HG (N )z N (E9) Z SSG (N ) = 1 N ! i d d r i dθ i dθ i e − P ij U (R 2 ij )− P i V (R 2 i ) (E10)
Here, the cluster expansion works as well. The only modification is that wherever we have r 2 ij or r 2 i in the classical case, we use their counterparts for the supersymmetric case, i.e. R 2 ij and R 2 i . In the next section we demonstrate that, given a supersymmetric gas in super-space of d = D + 2 real and two Grassman coordinates and a hardcore classical gas in a D dimensional space with the same inter-particle and external potentials, we have G SSG (z) = G HG (z) (E11)
Super Symmetric Gas versus Classical Gas
As in [19], we define the functions p(µ) and q(ν) as follows:
g(r 2 ij ) = ∞ 0 dµ ij p(µ ij )e −µij r 2 ij h(r 2 i ) = ∞ 0 dν i q(ν i )e −νir 2 i
Notice that in the above expressions, R 2 ij can also be used instead of r 2 ij . Consider the contribution of of a connected graph of k supersymmetric particles, CG. We use the above relations to transform the functions g and h into the functions p and q. Regardless of integrations over the parameters µ ij and ν i , we are left with k i=1 d d r i dθ i θ i exp(− 1 2 jl r j A CG jl r l +θ j A CG ij θ l
The quantity A CG is a k×k matrix dependent on the connected graph CG. The gaussian integration over Grassman numbers evaluates to
(− 1 2 Λ) k det(A CG )
We have used the following convention in this evaluation:
dθdθθθ = Λ
On the other hand the integration over the real coordinates contribute a factor of (2π) kd/2 det(A CG ) −d/2
If we choose Λ = − 1 π , the final expression evaluates to (2π) k(d−2)/2 det(A CG ) (d−2)/2
The two Grassman coordinates have canceled the effect of two real coordinates. It is evident that if we start with a classical hardcore gas in D = d − 2 dimensions, we arrive at the same expression as the above. This completes the proof for the equation (E11).
In the next section, we expand the partition function of a supersymmetric gas. However, instead of cluster expansion, we use a Taylor expansion. It turns out that the super-symmetric gas is related to the generating function of branched polymers.
Super Symmetric Gas versus Branched Polymers
We begin this section by defining four new functions:
P (r 2 ij ) = e −U (r 2 ij ) (E12) S(r 2 i ) = e −V (r 2 i )(E13)
Notice that if we Taylor-expand P (R 2 ij ) and R(R 2 i ) around r 2 ij and r 2 i , because of the properties of Grassman numbers, only the first two terms survive P (R 2 ij ) = P (r 2 ij ) +θ ij θijQ(r 2 ij ) (E14) S(R 2 r ) = P (r 2 i ) +θ i θ i T (r 2 i ) (E15)
In the above equations, Q and T are the first derivatives of the functions P and S respectively. Using the above equations, we can expand the grand partition function of a supersymmetric gas. We can construct a graphical expression for each term. We indicate the term Q(r 2 ij ) by a pair of connected particles (monomers) at r i ad r j . The term T (r 2 i ) is represented by a solid black circle at point r i representing a root monomer. For each non-root monomer, represented by a hollow circle, located at r i we multiply a factor of S(r 2 i ). For any pair of monomers, located at r i and r j , which are not connected we multiply a factor of P (r 2 ij ). For each monomer we also have a factor of z. The supersymmetric grand partition function is the sum of all these terms (graphs). Each term has a product of its connected graphs. It can be shown that any connected graph which has a loop or no root or even more than one root is zero (See figure 23).
Thus, only those graphs which are products of connected, loopless and single rooted graphs, ie: branched polymers, contribute. As the result of Grassman integration, for each monomer a factor of −1/π is generated. In order to go further we need to specify the potentials U (r 2 ij ) and V (r 2 i ). We choose a hardcore repulsive potential for U , a being the dimameter of monomers. Imagine a spherical container for the particles of radius R. The external potential is zero inside and infinitely large outside. This results in step functions for P and S: U (r 2 ij ) = ∞ r 2 ij < a 2 0 r 2 ij > a 2 ⇒ P (r 2 ij ) = 0 r 2 ij < a 2 1 r 2 ij > a 2 V (r 2 i ) = 0 r 2 i < R 2 ∞ r 2 i > R 2 ⇒ S(r 2 i ) = 1 r 2 i < R 2 ∞ r 2 i > R 2 Because of this, Q and T take the form of delta functions Q(r 2 ij ) = δ(r 2 ij − a 2 ) = 1 2a δ(r ij − a) T (r 2 i ) = δ(r 2 i − R 2 ) = − 1 2R δ(r i − R) With the above choice of functions, any pair of connected monomers have a fixed separation of a, the diameter of monomers. Also, any root monomer, because the delta function is constrained to stay on the surface of the sphere. For any root monomer we have a factor of −1/2R and for all other monomers a factor of 1/2a. Recall that there is also a factor of −z/π for any monomer. Therefore, z/2πR appears as the fugacity of the roots and −z/2πa as the fugacity of other monomers. Putting everything together we obtain
G SSG (z) = G BP (− z 2πa , z 2πR ) (E16)
Comparing this with our previous equation (E11) provides the main result:
G HG (z) = G BP (− z 2πa
, z 2πR ) (E17)
Discussion
As we see the fugacities in G BP appear with opposite signs in the equation (E1). However, the physical region of the partition function of G BP is where both of its fugacity arguments are positive. Consequently, one cannot explore the physical region of the branched polymer solution using this equation.
II. DIMENSIONAL REDUCTION: THE MAP FROM THE TREE-LIKE POLYMER IN d DIMENSIONS TO THE HARD-ROD GAS IN d − 2 DIMENSIONS
FIG. 4 :
4The function ρ(z), as given by (2.5)-(2.9), for z > zc, the location of which is indicated by the vertical dashed line.
Fig. 5 FIG. 5 :
55The setup for the one dimensional hard rod gas.
FIG. 6 :
6A rod with a lenth σ = 2 located at the site n.
FIG. 7 :
7The real (solid curve) and imaginary (dashed curves) parts of h(z) = 1 − g(z), where g(z) is as given by(3.34).
FIG. 8 :
8The function gn when z is slightly to the right of the critical point at -1/4. The dashed line at gn = 2 indicates the value to which gn tends when z = −1/4 exactly.
FIG. 9 :
9The locations of the poles in the density as a function of the fugacity, z. The location of zc = −1/4, is indicated.
FIG. 10 :
10The density, as given by the last line of (8.28), as a function of n, multiplied by 4 −N √ N , with L = 10, 000, for the following value of N : 50, 500, 5,000, 50,000. The highest curves correspond to the smallest values of N . The plot is restricted to the region adjacent to the boundary.
Figure 11
11displays the density profile over the entire interval for a different set of values of N . The dominance of the single Schrödinger equation eigenfunction, sin(π(n + 1)/L), is evident in N = 50, 000, 000 curve.
FIG. 12 :
12The density, ρ(n), for the case of attractive potentials near the bounding surfaces of of a region of length L = 10, 000. The curves correspond to N =50, 500, 5,000 and 50,000. The heights of the density curves decrease with increasing N .
FIG. 13 :
13The same set of densities graphed inFig. 12, this time over the entire interval.
The dependence on number of particles, N , which is not as simple and monotonic as in the previous cases, is indicated in the figure. The dependence of ρ(
FIG. 15 :
15The pressure as a function of L when N = 100. Here, w = zc = 1.
FIG. 16 :
16The quantity LP (L, N )/kBT = P(L/ √ N ).
FIG. 17 :
17The quantity (L.√ N ) 2 P, plotted against L/ √ N .
FIG. 18 :
18The potential configuration with which we will work. There are two attractive delta function potentials, each a distance x from the bounding walls of a region with an extent L.
FIG. 19 :
19The pressure as a function of distance, L, between the walls for x = 0.1 and the following values of the attractive potential strength, V : 9.9, 10.01, 10.02, 10.05. The smaller V the higher the pressure.
20: The pressure divided by kBT for V = 10.00025 and the following values of N : 300,000, 100,000, 30,000, 10,000, 3,000. The larger N the more negative (or less positive) the pressure. . 21: The pressure divided by kBT for V = 9.9997 and the following values of N : 300,000, 100,000, 30,000, 10,000, 3,000. Here, the larger N the greater (more positive) the pressure.
FIG. 22 :
22Solution of branched polymers. Roots are depicted by solid black circles.
FIG. 23 :
23Some of the vanishing terms in the expansion of partirion function
Then the summand is of the formexp L ρ ln
1 − aρ
aρ
+ ρ + ρ ln z
(2.4)
where we have made use of Stirling's formula: ln N ≈
N ln(N/e). The extremum equation that follows from an
attempt to maximize (2.4) with respect to N , and hence
ρ, is
ln
1 − aρ
aρ
−
aρ
1 − aρ
+ ln z = 0
(2.5)
We introduce the new variable
h =
aρ
1 − aρ
(2.6)
Then,
ρ =
1
a
h
1 + h
(2.7)
and (2.5) becomes
− ln h − h + ln z = 0
(2.8)
or
he h = z
(2.9)
FIG. 1: The function, he h , on the left hand side of (2.9), graphed as a function of h.. The solution to Eq. (2.9) is, formally, h =
-3
-2
-1
1
2
1
2
3
4
h
he h
). Expanding both sides of that equation about those critical values we have (h c + δh)e hc+δh = −1
e
+
δh 2
2e
+ O(δh 3 )
= z c + δz
= −
1
e
+ δz
(2.10)
Solving to lowest order for δh:
δh = ±
√
2eδz
(2.11)
-2
-1
1
2
-1
1
z
h(z)
FIG. 2: The real (solid curve) and imaginary (dashed curves)
parts of solutions to (2.9).
AcknowledgmentsThe authors of the paper gratefully acknowledge useful conversations with David Schwab. J.R. is especially indebted to Christian Rose for collaborations on a project that led directly to some of the key elements reported in this paper. This research was supported by the National Science Foundation through DMR Grant 04-04507.APPENDIX A: NORMALIZATION OF EIGENFUNCTIONSWe start with the two equationsWe will assume that φ 1 (x) has zeros at x = 0 and x = x 0 . As for φ 2 (x), it is also zero at x = 0, and it has another zero close to x 0 . In fact, we assume that as ∆E → 0, φ 2 (x) → φ 1 (x). Subtracting (A1) from (A2) and integrating from x = 0 to x = x 0 , we findAs for the next to last line in (A3), because of boundary conditions all contributions are equal to zero exceptUnder the assumption that ∆E is small, the last line reduces to ∆EGiven that the zero of φ 2 (x) is close to x 0 , we can writeThis can be established graphically. Combining the results, we haveNow, let us look at the denominator φ n φ n−1 in (4.37). In the vicinity of the zero of φ n , we can writeOn the other handIf we divide by this, we have a contribution to the residue of the pole going exactly like 1/( n 0 φ 2 n dn), to within a multiplicative constant. In fact, a careful analysis of the relationship between the variables z in our system and the energy, E, in the Schródinger equation leads to the conclusion that the multiplicative constant is precisely 1/4. This means that, to within the multiplicative factor of 1/4, the residue effectively normalizes the contributions to the density.APPENDIX B: NORMALIZATION IN THE DISCRETE SYSTEMWe start with the following version of the equation for the quantity φ n e −βun+1/2 φ n+1 (k) + e −βun/2 φ n−1 (k) = 2 cos k φ n (k) (B1) We consider two versions of this equation, one just like (B1) and one with the parameter k slightly different. Denoting that new value of k as k , we have, multiplying (B1) by φ n (k ) and the corresponding equation for k = k by φ n (k), subtracting the results and summing over n,We focus on the case x < x 1 and begin with the extended states, and we will seek the difference between this sum and the sum in the absence of the attractive potential. That is, our task is to find the value of the sum.After expanding sin(kx 1 + θ) and carrying out some simple algebra and trigonometry, the above expression reduces toGiven the trigonometric identityThe summand becomeswhere (8.8) has been used. The integral to be performed iswhere the removal of a factor of two in front of the integral (see (C9)) is compensated for by the fact that the range of integration has been extended by a factor of two. In fact, the result of the integration will be real as a matter of course, so we can remove the "Re" function from the expression. The sole contribution to the integration is from a pole on the upper imaginary axis, at a value of k = iκ such that the denominator k − V sin kx 1 e ikx1 → i(κ − V sinh κx 1 e −κx1 ) = 0. This equation for the pole is the same as the requirement(8.4)for the bound state. The residue at that pole isFocusing on the denominator in (D6), we haveThe third line of (D7) follows from(8.4). Inserting this result into (D6) we end up with the result for the integrationAgain, we have utilized(8.4) to obtain the right hand side of the equation above. Referring to Section VIII A, we see that this exactly cancels the contribution of the bound state to the left of the delta function potential. We have recovered the standard result for a complete orthonormal set of eigenfunctions, and that is that the sum is independent of the set chosen. The corresponding integration when x > x 1 isThe same sort of contour integration yields a result that precisely cancels the contribution of the bound state in that regime.APPENDIX E: DIMENSIONAL REDUCTION FOR A SPHERICAL GEOMETRYA prime motivation for the work reported here is the packing of complex RNA into a viral capsid. The closest approximation to the geometry of the packing environment entails spherical symmetry. It is natural to ask whether dimensional reduction will prove useful in this case. In this Appendix, we explore the consequences of the results of Brydges and Imbrie[17,18]when spherical symmetry holds.The two physical systems related by dimensional reduction are a hard-core classical gas in D dimensions and a solution of rooted branched polymers in d = D + 2 dimensions. The particles have a hard core repulsive interaction with diameter a, and they are confined in a container with radius R. The equation relating these two systems is:where G HG on the left side of the final equation is the grand partition function of the gas, z being the fugacity of the particles. The quantity G BP (z, w), on the right side of the equation, is the grand partition function of the solution of annealed branched polymers. The first argument, z, is the fugacity of the monomers and the second argument, w, is the fugacity of the roots. The roots are constrained to be on the surface of the container (see figure(22)). We provide the proof of the equation (E1) in the following sections. The reasoning is parallel to the one suggested by Cardy[19]. We start from the partition function of hardcore classical gas in. Then we continue with
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| []
|
[
"Generalized common index jump theorem with applications to closed characteristics on star-shaped hypersurfaces and beyond",
"Generalized common index jump theorem with applications to closed characteristics on star-shaped hypersurfaces and beyond"
]
| [
"Huagui Duan [email protected]. \nSchool of Mathematical Sciences and LPMC\nNankai University\n300071Tianjin\n",
"Hui Liu \nSchool of Mathematics and Statistics\nWuhan University\n430072Wuhan, Hubei\n",
"Yiming Long [email protected]. \nChern Institute of Mathematics and LPMC\nNankai University\n300071Tianjin\n",
"Wei Wang [email protected] \nSchool of Mathematical Sciences\nKey Laboratory of Pure and Applied Mathematics\nPeking University\n100871BeijingThe People's Republic of China\n"
]
| [
"School of Mathematical Sciences and LPMC\nNankai University\n300071Tianjin",
"School of Mathematics and Statistics\nWuhan University\n430072Wuhan, Hubei",
"Chern Institute of Mathematics and LPMC\nNankai University\n300071Tianjin",
"School of Mathematical Sciences\nKey Laboratory of Pure and Applied Mathematics\nPeking University\n100871BeijingThe People's Republic of China"
]
| []
| In this paper, we first generalize the common index jump theorem of Long-Zhu in 2002 and Duan-Long-Wang in 2016 to the case where the mean indices of symplectic paths are not required to be all positive. As applications, we study closed characteristics on compact star-shaped hypersurfaces in R 2n , when both positive and negative mean indices may appear simultaneously.Specially we establish the existence of at least n geometrically distinct closed characteristics on every compact non-degenerate perfect star-shaped hypersurface Σ in R 2n provided that every prime closed characteristic possesses nonzero mean index. Furthermore, in the case of R 6 we remove the nonzero mean index condition by showing that the existence of only finitely many geometrically distinct closed characteristics implies that each of them must possess nonzero mean index. We also generalize the above results about closed characteristics on non-degenerate starshaped hypersurfaces to closed Reeb orbits of non-degenerate contact forms on a broad class of prequantization bundles. | null | [
"https://arxiv.org/pdf/2205.07082v1.pdf"
]
| 248,811,101 | 2205.07082 | f4b34f546b44b2e30e0d4f98b0e7485253fa8042 |
Generalized common index jump theorem with applications to closed characteristics on star-shaped hypersurfaces and beyond
14 May 2022 May 17, 2022
Huagui Duan [email protected].
School of Mathematical Sciences and LPMC
Nankai University
300071Tianjin
Hui Liu
School of Mathematics and Statistics
Wuhan University
430072Wuhan, Hubei
Yiming Long [email protected].
Chern Institute of Mathematics and LPMC
Nankai University
300071Tianjin
Wei Wang [email protected]
School of Mathematical Sciences
Key Laboratory of Pure and Applied Mathematics
Peking University
100871BeijingThe People's Republic of China
Generalized common index jump theorem with applications to closed characteristics on star-shaped hypersurfaces and beyond
14 May 2022 May 17, 20221Closed characteristicstar-shaped hypersurfaceMaslov-type indexcommon index jump theorem 2010 Mathematics Subject Classification: 58E0537J4534C25
In this paper, we first generalize the common index jump theorem of Long-Zhu in 2002 and Duan-Long-Wang in 2016 to the case where the mean indices of symplectic paths are not required to be all positive. As applications, we study closed characteristics on compact star-shaped hypersurfaces in R 2n , when both positive and negative mean indices may appear simultaneously.Specially we establish the existence of at least n geometrically distinct closed characteristics on every compact non-degenerate perfect star-shaped hypersurface Σ in R 2n provided that every prime closed characteristic possesses nonzero mean index. Furthermore, in the case of R 6 we remove the nonzero mean index condition by showing that the existence of only finitely many geometrically distinct closed characteristics implies that each of them must possess nonzero mean index. We also generalize the above results about closed characteristics on non-degenerate starshaped hypersurfaces to closed Reeb orbits of non-degenerate contact forms on a broad class of prequantization bundles.
Introduction and main results
There are three goals in this paper. The first one is to generalize the common index jump theorem (CIJT for short below). This theorem is the first result which exhibits certain common index property of iterates of more than one but finitely many symplectic matrix paths. This theorem was discovered and proved by Y. Long and C. Zhu in 2002 (Theorems 4.1 and 4.3 of [LoZ], cf. also Theorems 11.1.1 and 11.2.1 of [Lon4]), and yields a breakthrough in the study of the multiplicity and stability of closed characteristics on compact convex hypersurfaces in R 2n . In the last twenty years, this theorem has been applied to study various problems, provides an important method in the studies of the multiplicity and stability of closed characteristics in symplectic and contact dynamics. Specially it is one of the few methods which work effectively for such studies when the dimension of the symplectic manifold is not only 4. In [Wan2] and [Wan3] of 2016, W. Wang found certain useful symmetric property in CIJT. In 2016 also, an important extension of this CIJT was established by H. Duan, Y. Long and W. Wang (Theorem 3.5 of [DLW]), which gives an enhanced version of the CIJT by giving precise indices for iterates of related symplectic paths near the carefully chosen iterates in the study. This enhanced CIJT has been used to establish sharp estimates on the multiplicity and stability of prime closed geodesics on compact simply-connected bumpy Finsler manifolds whose loop spaces possess bounded Betti number sequences, provided the number of prime closed geodesics is finite and each of them possesses non-zero Morse index, which implies the positivity of their mean indices in [DLW]. This enhanced CIJT has also been applied to the studies of closed characteristics on star-shaped hypersurfaces in R 2n by H. Duan, H. Liu, Y. Long and W. Wang in [DLLW] and on other contact manifolds by V. Ginzburg, B. Z. Gürel and L. Macarini in [GGM] for example, under the assumption that all the prime closed characteristics possess positive (or negative) mean indices. Note that in Theorems 5.1 and 5.2 as well as Corollaries 5.3 and 5.4 of [GGu], V. Ginzburg and B. Z. Gürel extended the enhanced CIJT of [DLW] to the case admitting the mean indices of all symplectic paths being nonzero via a new index recurrence arguments. Subsequently, based on [GGu], V. Ginzburg, B. Z. Gürel and L. Macarini in Theorem 4.1 of [GGM] further studied the enhanced CIJT of [DLW] for strongly non-degenerate symplectic paths with positive mean indices. Note that based on the theorems these authors gave interesting results about Reeb orbits on contact manifolds. But their extensions of the enhanced CIJT missed the precise values of indices of some crucial iterates as those listed in Theorem 3.6 below, and missed the symmetric property of the CIJT produced by the vertices in the cube [0, 1] l in the proof of CIJT discovered first in [LoZ] and then in [DLW] as those in the Step 2 of the proof of Theorem 1.2 below where the two opposite vertices are used. Note that such missing might be due to the index recurrence arguments. Note also that these missing properties are very crucial in our study in the current paper and for the future study, more precisely in order to get sharp estimates on multiplicities of periodic orbits, we do need to compute Morse type number quantities accurately and to apply the mentioned symmetric property of CIJM. Thus the first goal of this paper is to further extend this enhanced CIJT of [DLW] to the case that there exist prime symplectic paths possessing positive as well as negative mean indices simultaneously and the above mentioned information can be reserved at the same time by generalizing the method of [DLW].
The main idea in the proof of the CIJT in [LoZ] is to show that there exist large suitable iterate of each path among the given finitely many prime symplectic paths such that the corresponding enlarged index intervals at their these iterates possess a non-empty common intersection interval which is sufficiently large to contain certain integers, and the number of these integers yields a lower bound estimate for the number of these prime symplectic paths, provided all of these paths possess positive mean indices. Note that when prime closed characteristics on a compact star-shaped hypersurface in R 2n are considered, in general some of them may possess positive mean indices and the others possess negative mean indices. Then the iterated index sequences of them may tend to positive infinity as well as negative infinity simultaneously, and consequently it seems impossible to apply the CIJT to get a common intersection interval of the iterated enlarged index intervals of all the prime symplectic paths. To overcome this difficulty, suggested by the resonance identity for a star-shaped hypersurface in R 2n with only finitely many prime closed characteristics which was proved by H. Liu, Y. Long and W. Wang in [LLW] of 2014 (cf. Theorem 2.3 below), we can consider iterates of all these prime symplectic paths together by adding some −1 parameter to the negative mean indices, and we have improved the common index jump theorem (Theorems 4.1 and 4.3 of [LoZ] as well as Theorems 11.1.1 and 11.2.1 of [Lon4]) to the new enhanced common index jump Theorems 3.4 and 3.6 below to deal with mixed mean indices. This improvement allows our Theorem 1.2 below to deal with closed characteristics on non-degenerate star-shaped hypersurfaces when their mean indices are non-zero.
The second goal of this paper is to apply this generalized enhanced common index jump Theorem to study the multiplicity and stability of closed characteristics on compact star-shaped hypersurfaces in R 2n . In this paper, we let Σ be always a C 3 compact hypersurface in R 2n strictly star-shaped with respect to the origin, i.e., the tangent hyperplane at any x ∈ Σ does not intersect the origin.
We denote the set of all such hypersurfaces by H st (2n), and denote by H con (2n) the subset of H st (2n) which consists of all strictly convex hypersurfaces. We consider closed characteristics (τ, y) on Σ, which are solutions of the following problem ẏ = JN Σ (y),
y(τ ) = y(0),(1.1)
where J = 0 −I n I n 0 , I n is the identity matrix in R n , τ > 0, N Σ (y) is the outward normal vector of Σ at y normalized by the condition N Σ (y) · y = 1. Here a · b denotes the standard inner product of a, b ∈ R 2n . A closed characteristic (τ, y) is prime, if τ is the minimal period of y. Two closed characteristics (τ, y) and (σ, z) are geometrically distinct, if y(R) = z(R). We denote by T (Σ) the set of geometrically distinct closed characteristics (τ, y) on Σ ∈ H st (2n The study on closed characteristics in the global sense started in 1978, when the existence of at least one closed characteristic was first established on any Σ ∈ H st (2n) by P. Rabinowitz in [Rab] and on any Σ ∈ H con (2n) by A. Weinstein in [Wei] independently. Since then the existence of multiple closed characteristics on Σ ∈ H con (2n) has been deeply studied by many mathematicians, for example, studies in [EkL], [EkH], [HWZ1], [Szu], [LoZ], [LLZ], [Wan2], [Wan3], [WHL] as well as [Lon4] and references therein.
For the star-shaped hypersurfaces, We are only aware of a few papers about the multiplicity of closed characteristics. In [Gir] of 1984 and [BLMR] of 1985, # T (Σ) ≥ n for Σ ∈ H st (2n) was proved under some pinching conditions. In [Vit2] of 1989, C. Viterbo proved a generic existence result for infinitely many closed characteristics on star-shaped hypersurfaces. In [HuL] of 2002, X.
Hu and Y. Long proved that # T (Σ) ≥ 2 for non-degenerate Σ ∈ H st (2n). In [HWZ2] of 2003, H.
Hofer, K. Wysocki, and E. Zehnder proved that # T (Σ) = 2 or ∞ holds for every non-degenerate Σ ∈ H st (4) provided that all stable and unstable manifolds of the hyperbolic closed characteristics on Σ intersect transversally. Furthermore, recently this alternative result was proved to be true for every non-denernerate Σ ∈ H st (4) without the above transversal condition by D. Cristofaro-Gardiner, M. Hutchings and D. Pomerleano in [CGHP]. In [CGH] of 2016, D. Cristofaro-Gardiner and M. Hutchings proved that # T (Σ) ≥ 2 for every contact manifold Σ of dimension three. Later various proofs of this result for star-shaped hypersurfaces have been given in [GHHM], [LLo1] and [GiG]. There are also some multiplicity results for closed orbits of dynamically convex Reeb flows, cf., [GuK] and [AbM].
On the stability problem, we refer the readers to [Eke], [DDE], [Lon1]- [Lon3], [LoZ], [WHL], [Wan1] for convex hypersurfaces and [LiL], [LLo2], [CGHHL] for star-shaped hypersurfaces. The following index perfect condition was first introduced by H. Duan, H. Liu, Y. Long and W. Wang in [DLLW] of 2018, for the star-shaped hypersurfaces in R 2n which is much weaker than the dynamically convexity condition introduced by H. Hofer, K. Wysocki, and E. Zehnder in [HWZ1] of 1998 (cf. also [HWZ2] in 2003).
Definition 1.1. We call a compact star-shaped hypersurface Σ in R 2n perfect, if for every prime closed characteristic (τ, y) on Σ, the Maslov-type index of each good m-th iterate (mτ, y) of (τ, y) with some m ∈ N satisfies i(y, m) = −1 if n is even, or i(y, m) ∈ {−2, −1, 0} if n is odd.
Here an iterate (mτ, y) of a prime closed characteristic (τ, y) on Σ with m ∈ N is called good, if its Maslov-type index has the same parity as that of (τ, y), otherwise it is called bad. Note that for a bad closed characteristic (mτ, y), the element x m corresponding to (mτ, y) satisfies β(x m ) = −1 in Lemma 2.2 below, and consequently the equivariant critical module of the functional F a,K at S 1 · x m must be trivial, i.e., x m is homologically invisible and thus can be ignored in the Morse theory study. This property was used first in Definition 4.8 and Remark 4.9 via Proposition 4.2 of [LLW] to compute the Euler characteristicχ(y). Then this property was used in Section 3 of [GGM] to compute the local equivariant symplectic homology. (cf. also the condition (A) below Theorem 1.2). Here in the current paper, we shall use this property in the computations of the Morse type numbers in Section 4 below.
In [DLLW], the authors proved specially the existence of at least n closed characteristics on every non-degenerate perfect star-shaped hypersurfaces in R 2n when every closed characteristic possesses positive mean index. Then V. Ginzburg, B.Z. Gürel and L. Macarini in [GGM] obtained the same multiplicity result of closed Reeb orbits on contact manifolds under non-degenerate condition and the index perfect condition introduced in [DLLW] (i.e., the perfect condition given in Definition 1.1), provided the contact form α is index-positive (or index-negative), i.e., all contractible periodic orbit γ of α possess positive (or negative) mean index. Most recently V. Ginzburg and L. Macarini in [GM] obtained some optimal multiplicity results of closed Reeb orbits on symmetric contact spheres under the so called strong dynamical convexity which extended results of C. Liu, Y. Long and C. Zhu in [LLZ] of 2002.
When we consider prime closed characteristics on a compact star-shaped hypersurface in R 2n , a priori the mean indices of some of them can be non-positive. By the resonance identity for a star-shaped hypersurface in R 2n with only finitely many prime closed characteristics proved by H.
Liu, Y. Long and W. Wang in [LLW] of 2014 (cf. Theorem 2.3 below), at least one of these prime closed characteristics must possess positive mean index, but some of the others may possess zero or negative mean indices. Thus even if we assume that every prime closed characteristic possesses nonzero mean index, their iterated index sequences may still tend to positive infinity as well as negative infinity simultaneously as we mentioned before. To overcome this difficulty, by Theorem 2.3 below, we notice that the existence of some prime closed characteristics possessing positive mean index can be crucially used to construct actually the mentioned common index intersection interval, and that the prime closed characteristics with negative mean indices make no contributions to it and thus can be ignored in some sense. This understanding is rigorously realized by applying our new generalized enhanced common index jump Theorem 3.6 together with the mentioned existence of prime closed characteristics possessing positive mean indices so that we can deal with positive and negative mean indices simultaneously, and establish the following more general existence result, provided the mean indices of all the prime closed characteristics are non-zero. Note that another key observation in our proof is that the Morse inequalities still hold under non-zero mean index condition when the degrees of Morse-type numbers and Betti numbers are carefully chosen (cf.
(2.16) and (2.17) below).
Theorem 1.2. Let Σ be a compact non-degenerate perfect star-shaped hypersurface in R 2n .
If every prime closed characteristic on Σ possesses nonzero mean index, then there exist at least n geometrically distinct closed characteristics. Furthermore, if the total number of prime closed characteristics on Σ is finite, then Σ carries at least n non-hyperbolic closed characteristics with even Maslov-type indices when n is even, and at least n closed characteristics with odd Maslov-type indices and at least (n − 1) of them are non-hyperbolic when n is odd.
Based on Theorem 3.6 below, we can generalize Theorem 1.2 about closed characteristics on star-shaped hypersurfaces to closed Reeb orbits of contact forms on a broad class of prequantization bundles, which improves Theorem 2.1 and Theorem 2.10 of [GGM]. To clarify it, we review some terminologies from contact geometry, following Section 2 of [GGM].
Let (M 2n+1 , ξ) be a closed contact manifold satisfying c 1 (ξ)| π 2 (M ) = 0 and α be the contact form supporting the contact structure ξ. A non-degenerate periodic orbit γ is called good if its Conley-Zehnder index µ(γ) has the same parity as that of the underlying simple closed orbit. Note that the Maslov-type index of γ equals to µ(γ) − 1. In the following, let (M 2n+1 , ξ) be a prequantization circle bundle over closed integral symplectic manifolds (B 2n , ω), i.e., the first Chern class of the
principle bundle M → B is −[ω]
. Denote by χ(B) the Euler characteristic of B and by
c B := inf{k ∈ N | ∃S ∈ π 2 (B) with c 1 (T B), S = k}
its minimal Chern number. We impose the following condition which is weaker than the condition (F) in Section 2 of [GGM]:
(A) (i) The manifold (M 2n+1 , ξ) admits a strong symplectic filling (W, Ω) which is symplectically aspherical, i.e., Ω| π 2 (W ) = 0 and c 1 (T W )| π 2 (W ) = 0, and the map π 1 (M ) → π 1 (W ) induced by the inclusion is injective.
(ii) The contact form α is non-degenerate, the mean indexμ(γ) is nonzero for every contractible periodic orbit γ of α and has no contractible good periodic orbits γ such that µ(γ) = 0 if n is odd or µ(γ) ∈ {0, ±1} if n is even. (2) Note that we are unable to weaken the condition (NF) in Section 2 of [GGM] as (A), since the proof in [GGM] relies on the machinery of positive equivariant symplectic homology.
A remarkable observation by F. Bourgeois and A. Oancea in Section 4.1.2 of [BO] is that under suitable additional assumptions that all closed contractible Reeb orbits on M are non-degenerate and have Conley-Zehnder index strictly greater than 3 − n, the positive equivariant symplectic homology is defined and well-defined even when M does not have a symplectic filling. But the existence of closed contractible Reeb orbit with negative mean index will destroy this assumption and then the positive equivariant symplectic homology is not well defined, thus we are unable to weaken the index-positive condition in (NF) of [GGM] to the nonzero mean index condition.
An important result of V. Bangert and W. Klingenberg in [BaK] implies that if c is a closed geodesic on a compact Riemannian (or Finsler) manifold M such that it possesses zero mean index and c is neither homologically invisible nor an absolute minimum of the energy functional, then there exist infinitely many closed geodesics on M . In fact, we tend to believe that when the number of prime closed geodesics on a compact Finsler manifold or prime closed characteristics on a compact star-shaped hypersurface in R 2n is finite, then each one of them must be homologically visible as well as variationally visible (cf. [BaK] and [Lon4] for definitions). Motivated by the result in [BaK] and the well-known weakly non-resonant ellipsoids, we tend to believe that the following conjecture for closed characteristics should hold:
Conjecture 1.5. If there exist only finitely many geometrically distinct closed characteristics on a compact star-shaped hypersurface Σ in R 2n , then no one of them possesses zero mean index.
Our third goal of this paper is to give a positive answer to this conjecture below in the case of n = 3 for non-degenerate star-shaped hypersurfaces. But up to our knowledge, this conjecture seems to be rather challenging in its full generality.
Theorem 1.6. If there exist only finitely many geometrically distinct closed characteristics on a compact non-degenerate star-shaped hypersurface Σ in R 6 , then every prime closed characteristic possesses nonzero mean index.
Using Theorem 1.6, we can remove the nonzero mean index condition in Theorem 1.2 in the case of n = 3:
Corollary 1.7. If Σ is a compact non-degenerate perfect star-shaped hypersurface in R 6 , then there exist at least three geometrically distinct closed characteristics. Furthermore, if the total number of prime closed characteristics on Σ is finite, then Σ carries at least three geometrically distinct closed characteristics with odd Maslov-type indices and at least two of them are non-hyperbolic.
Note that one may generalize our Theorem 1.6 to contact manifolds by the idea of our proof of Theorem 1.6, especially the key observation in our Lemma 5.1 that the Viterbo index i(y m ) always equals to −4 for every closed characteristic (τ, y) with zero mean index and all m ∈ N.
In this paper, let N, N 0 , Z, Q, R, C and R + denote the sets of natural integers, non-negative integers, integers, rational numbers, real numbers, complex numbers and positive real numbers respectively. We define the functions
[a] = max {k ∈ Z | k ≤ a}, {a} = a − [a], E(a) = min {k ∈ Z | k ≥ a}, ϕ(a) = E(a) − [a]. (1.3)
Denote by a · b and |a| the standard inner product and norm in R 2n . Denote by ·, · and · the standard L 2 inner product and L 2 norm. For an S 1 -space X, we denote by X S 1 the homotopy
quotient of X by S 1 , i.e., X S 1 = S ∞ × S 1 X, where S ∞ is the unit sphere in an infinite dimensional
complex Hilbert space. In this paper we use Q coefficients for all homological and cohomological modules. By t → a + , we mean t > a and t → a.
2 Mean index identities for closed characteristics on compact starshaped hypersurfaces in R 2n
In this section, we briefly review the mean index identities for closed characteristics on Σ ∈ H st (2n) developed in [LLW] which will be needed in Section 4. All the details of proofs can be found in [LLW]. Now we fix a Σ ∈ H st (2n) and assume the following condition on T (Σ):
(F) There exist only finitely many geometrically distinct prime closed characteristics
{(τ j , y j )} 1≤j≤k on Σ.
Let j : R 2n → R be the gauge function of Σ, i.e., j(λx) = λ for x ∈ Σ and λ ≥ 0, then
j ∈ C 3 (R 2n \ {0}, R) ∩ C 0 (R 2n
, R) and Σ = j −1 (1). Letτ = inf 1≤j≤k τ j and T be a fixed positive constant. Then following [Vit1] and Section 2 of [LLW], for any a >τ T , we can construct a function
ϕ a ∈ C ∞ (R, R + ) which has 0 as its unique critical point in [0, +∞). Moreover, ϕ ′ a (t) t is strictly decreasing for t > 0 together with ϕ a (0) = 0 = ϕ ′ a (0) and ϕ ′′ a (0) = 1 = lim t→0 + ϕ ′ a (t)
t . The precise definition of ϕ a and the dependence of ϕ a on a are given in Lemma 2.2 and Remark 2.3 of [LLW] respectively. As in [LLW], we define a Hamiltonian function
H a ∈ C 3 (R 2n \ {0}, R) ∩ C 1 (R 2n , R) satisfying H a (x) = aϕ a (j(x)) on U A = {x | aϕ a (j(x)) ≤ A} for some large A, and H a (x) = 1
2 ǫ a |x| 2 outside some even larger ball with ǫ a > 0 small enough such that outside U A both ∇H a (x) = 0 and H ′′ a (x) < ǫ a hold as in Lemmas 2.2, 2.4 and Proposition 2.5 of [LLW]. We consider the following fixed period probleṁ
x(t) = JH ′ a (x(t)), x(0) = x(T ). (2.1) Then solutions of (2.1) are x ≡ 0 and x = ρy(τ t/T ) with ϕ ′ a (ρ) ρ = τ aT , where (τ, y)
is a solution of (1.1). In particular, non-zero solutions of (2.1) are in one to one correspondence with solutions of (1.1) with period τ < aT .
For any a >τ T , we can choose some large constant K = K(a) such that
H a,K (x) = H a (x) + 1 2 K|x| 2 (2.2)
is a strictly convex function, that is,
(∇H a,K (x) − ∇H a,K (y), x − y) ≥ ǫ 2 |x − y| 2 , (2.3)
for all x, y ∈ R 2n , and some positive ǫ. Let H * a,K be the Fenchel dual of H a,K defined by
H * a,K (y) = sup{x · y − H a,K (x) | x ∈ R 2n }.
The dual action functional on
X = W 1,2 (R/T Z, R 2n ) is defined by F a,K (x) = T 0 1 2 (Jẋ − Kx, x) + H * a,K (−Jẋ + Kx) dt. (2.4)
Then F a,K ∈ C 1,1 (X, R) and for KT ∈ 2πZ, F a,K satisfies the Palais-Smale condition and x is a critical point of F a,K if and only if it is a solution of (2.1). Moreover, F a,K (x a ) < 0 and it is independent of K for every critical point x a = 0 of F a,K .
When KT / ∈ 2πZ, the map x → −Jẋ + Kx is a Hilbert space isomorphism between X = W 1,2 (R/(T Z); R 2n ) and E = L 2 (R/(T Z), R 2n ). We denote its inverse by M K and the functional
Ψ a,K (u) = T 0 − 1 2 (M K u, u) + H * a,K (u) dt, ∀ u ∈ E. (2.5) Then x ∈ X is a critical point of F a,K if and only if u = −Jẋ + Kx is a critical point of Ψ a,K .
Suppose u is a nonzero critical point of Ψ a,K . Then the formal Hessian of Ψ a,K at u is defined by
Q a,K (v) = T 0 (−M K v · v + H * ′′ a,K (u)v · v)dt, (2.6) which defines an orthogonal splitting E = E − ⊕ E 0 ⊕ E + of E intoi K (u) = dim E − and ν K (u) = dim E 0
respectively. Similarly, we define the index and nullity of x = M K u for F a,K , we denote them by i K (x) and ν K (x). Then we have
i K (u) = i K (x), ν K (u) = ν K (x), (2.7)
which follow from the definitions (2.4) and (2.5). The following important formula was proved in Lemma 6.4 of [Vit2]:
i K (x) = 2n([KT /2π] + 1) + i v (x) ≡ d(K) + i v (x), (2.8) where the Viterbo index i v (x) does not depend on K, but only on H a .
By the proof of Proposition 2 of [Vit1], we have that v ∈ E belongs to the null space of Q a,K if and only if z = M K v is a solution of the linearized systeṁ
z(t) = JH ′′ a (x(t))z(t). (2.9)
Thus the nullity in (2.7) is independent of K, which we denote by
ν v (x) ≡ ν K (u) = ν K (x).
By Proposition 2.11 of [LLW], the index i v (x) and nullity ν v (x) coincide with those defined for the Hamiltonian H(x) = j(x) α for all x ∈ R 2n and some α ∈ (1, 2). Especially 1 ≤ ν v (x) ≤ 2n − 1 always holds.
For every closed characteristic (τ, y) on Σ, let aT > τ and choose ϕ a as above. Determine ρ
uniquely by ϕ ′ a (ρ) ρ = τ aT . Let x = ρy( τ t T )
. Then we define the index i(τ, y) and nullity ν(τ, y) of (τ, y) by
i(τ, y) = i v (x), ν(τ, y) = ν v (x).
Then the mean index of (τ, y) is defined bŷ
i(τ, y) = lim m→∞ i(mτ, y) m .
Note that by Proposition 2.11 of [LLW], the index and nullity are well defined and are independent of the choice of a. For a closed characteristic (τ, y) on Σ, we simply denote by y m ≡ (mτ, y) the m-th iteration of y for m ∈ N.
We have a natural S 1 -action on X or E defined by
θ · u(t) = u(θ + t), ∀ θ ∈ S 1 , t ∈ R.
Clearly both of F a,K and Ψ a,K are S 1 -invariant. For any κ ∈ R, we denote by
Λ κ a,K = {u ∈ L 2 (R/(T Z); R 2n ) | Ψ a,K (u) ≤ κ}, X κ a,K = {x ∈ W 1,2 (R/(T Z), R 2n ) | F a,K (x) ≤ κ}.
For a critical point u of Ψ a,K and the corresponding
x = M K u of F a,K , let Λ a,K (u) = Λ Ψ a,K (u) a,K = {w ∈ L 2 (R/(T Z), R 2n ) | Ψ a,K (w) ≤ Ψ a,K (u)}, X a,K (x) = X F a,K (x) a,K = {y ∈ W 1,2 (R/(T Z), R 2n ) | F a,K (y) ≤ F a,K (x)}.
Clearly, both sets are S 1 -invariant. Denote by crit(Ψ a,K ) the set of critical points of Ψ a,K . Because Ψ a,K is S 1 -invariant, S 1 · u becomes a critical orbit if u ∈ crit(Ψ a,K ). Note that by the condition (F), the number of critical orbits of Ψ a,K is finite. Hence as usual we can make the following definition.
Definition 2.1. Suppose u is a nonzero critical point of Ψ a,K , and N is an
S 1 -invariant open neighborhood of S 1 · u such that crit(Ψ a,K ) ∩ (Λ a,K (u) ∩ N ) = S 1 · u. Then the S 1 -critical module of S 1 · u is defined by C S 1 , q (Ψ a,K , S 1 · u) = H q ((Λ a,K (u) ∩ N ) S 1 , ((Λ a,K (u) \ S 1 · u) ∩ N ) S 1 ).
Similarly, we define the S 1 -critical module C S 1 , q (F a,K , S 1 · x) of S 1 · x for F a,K .
We fix a and let u K = 0 be a critical point of Ψ a,K with multiplicity mul(u K ) = m, that is, u K corresponds to a closed characteristic (τ, y) ⊂ Σ with (τ, y) being m-iteration of some prime closed characteristic. Precisely, we have u K = −Jẋ + Kx with x being a solution of (2.1) and x = ρy( τ t T ) with ϕ ′ a (ρ) ρ = τ aT . Moreover, (τ, y) is a closed characteristic on Σ with minimal period τ m . For any p ∈ N satisfying pτ < aT , we choose K such that pK / ∈ 2π T Z, then the pth iteration u p pK of u K is given by −Jẋ p + pKx p , where x p is the unique solution of (2.1) corresponding to (pτ, y) and is a critical point of F a,pK , that is, u p pK is the critical point of Ψ a,pK corresponding to x p .
Lemma 2.2. (cf. Proposition 4.2 and Remark 4.4 of [LLW] ) If u p pK is non-degenerate, i.e., ν pK (u p pK ) = 1, let β(
x p ) = (−1) i pK (u p pK )−i K (u K ) = (−1) i v (x p )−i v (x) , then C S 1 ,q−d(pK)+d(K) (F a,K , S 1 · x p ) = C S 1 ,q (F a,pK , S 1 · x p ) = C S 1 ,q (Ψ a,pK , S 1 · u p pK ) = Q, if q = i pK (u p pK ) and β(x p ) = 1, 0,
otherwise.
(2.10) Theorem 2.3. (cf. Theorem 1.1 of [LLW] and Theorem 1.2 of [Vit2]) Suppose that Σ ∈ H st (2n)
satisfying # T (Σ) < +∞. Denote by {(τ j , y j )} 1≤j≤k
all the geometrically distinct prime closed characteristics. Then the following identities hold
1≤j≤k i(y j )>0χ (y j ) i(y j ) = 1 2 , 1≤j≤k i(y j )<0χ (y j ) i(y j ) = 0, (2.11)
whereχ(y) ∈ Q is the average Euler characteristic given by Definition 4.8 and Remark 4.9 of [LLW].
In particular, if all y m 's are non-degenerate for m ≥ 1, then
χ(y) = (−1) i(y) , if i(y 2 ) − i(y) ∈ 2Z, (−1) i(y) 2 ,
otherwise.
(2.12)
Let F a,K be a functional defined by (2.4) for some a, K ∈ R large enough and let ǫ > 0 be small
enough such that [−ǫ, 0) contains no critical values of F a,K . For b large enough, The normalized Morse series of F a,K in X −ǫ \ X −b is defined, as usual, by M a (t) = q≥0, 1≤j≤p dim C S 1 , q (F a,K , S 1 · v j )t q−d(K) , (2.13)
where we denote by {S 1 · v 1 , . . . , S 1 · v p } the critical orbits of F a,K with critical values less than −ǫ.
The Poincaré series of H S 1 , * (X, X −ǫ ) is t d(K) Q a (t), according to Theorem 5.1 of [LLW], if we set Q a (t) = k∈Z q k t k , then
q k = 0 ∀ k ∈I,
where I is an interval of Z such that I ∩ [i(τ, y), i(τ, y) + ν(τ, y) − 1] = ∅ for all closed characteristics (τ, y) on Σ with τ ≥ aT . Then by Section 6 of [LLW], we have
M a (t) − 1 1 − t 2 + Q a (t) = (1 + t)U a (t),
where U a (t) = i∈Z u i t i is a Laurent series with nonnegative coefficients. If there is no closed characteristic withî = 0, then
M (t) − 1 1 − t 2 = (1 + t)U (t),
(2.14)
where M (t) = p∈Z M p t p denotes M a (t) as a tends to infinity. In addition, we also denote by b p the coefficient of t p of 1 1−t 2 = p∈Z b p t p , i.e. there holds b p = 1, for all p ∈ 2N 0 , and b p = 0 for all p ∈ 2N 0 .
For any two positive integers n 1 and n 2 , it follows from (2.14) that
2n 2 +1 p=−2n 1 +1 M p t p − 2n 2 +1 p=−2n 1 +1 b p t p = (1 + t) 2n 2 +1 p=−2n 1 u p t p − u 2n 2 +1 t 2n 2 +2 − u −2n 1 t −2n 1 ,(2.15)
which, through letting t = −1, yields the following Morse inequality As in [Lon3], denote by
2n 2 +1 p=−2n 1 +1 (−1) p M p ≤ 2n 2 +1 p=−2n 1 +1 (−1) p b p . (2.16) Similarly we have 2n 2 p=−2n 1 (−1) p M p ≥ 2n 2 p=−2n 1 (−1) p b p .N 1 (λ, b) = λ b 0 λ , for λ = ±1, b ∈ R, (3.1) D(λ) = λ 0 0 λ −1 , for λ ∈ R \ {0, ±1}, (3.2) R(θ) = cos θ − sin θ sin θ cos θ , for θ ∈ (0, π) ∪ (π, 2π), (3.3) N 2 (e θ √ −1 , B) = R(θ) B 0 R(θ) ,
for θ ∈ (0, π) ∪ (π, 2π) and
B = b 1 b 2 b 3 b 4 with b j ∈ R, and b 2 = b 3 . (3.4) Here N 2 (e θ √ −1 , B) is non-trivial if (b 2 − b 3 ) sin θ < 0, and trivial if (b 2 − b 3 ) sin θ > 0.
As in [Lon3], the ⋄-sum (direct sum) of any two real matrices is defined by Lemma 3.1. (cf. [Lon3], Lemma 9.1.5 and List 9.1.12 of [Lon4]) For M ∈ Sp(2n) and ω ∈ U, the splitting number S ± M (ω) (cf. Definition 9.1.4 of [Lon4]) satisfies
A 1 B 1 C 1 D 1 2i×2i ⋄ A 2 B 2 C 2 D 2 2j×2j = A 1 0 B 1 0 0 A 2 0 B 2 C 1 0 D 1 0 0 C 2 0 D 2 .ForS ± M (ω) = 0, if ω ∈ σ(M ). (3.5) S + N 1 (1,a) (1) = 1, if a ≥ 0, 0, if a < 0. (3.6)
For any M i ∈ Sp(2n i ) with i = 0 and 1, there holds
S ± M 0 ⋄M 1 (ω) = S ± M 0 (ω) + S ± M 1 (ω), ∀ ω ∈ U.
(3.7)
We have the following decomposition theorem
f (1) = M 1 ⋄ · · · ⋄ M k , (3.8) where each M i is a basic normal form listed in (3.1)-(3.4) for 1 ≤ i ≤ k. For every γ ∈ P τ (2n) ≡ {γ ∈ C([0, τ ], Sp(2n)) | γ(0) = I 2n }, we extend γ(t) to t ∈ [0, mτ ] for every m ∈ N by γ m (t) = γ(t − jτ )γ(τ ) j
∀ jτ ≤ t ≤ (j + 1)τ and j = 0, 1, . . . , m − 1, (3.9) as in p.114 of [Lon2]. As in [LoZ] and [Lon4], we denote the Maslov-type indices of γ m by (i(γ, m), ν(γ, m)).
Then the following iteration formula from [LoZ] and [Lon4] can be obtained.
Theorem 3.3. (cf. Theorem 9.3.1 of [Lon4]) For any path γ ∈ P τ (2n), let M = γ(τ ) and
C(M ) = 0<θ<2π S − M (e √ −1θ )
. We extend γ to [0, +∞) by its iterates. Then for any m ∈ N we Theorem 3.4. Fix an integer q > 0. Let µ i ≥ 0 and β i be integers for all i = 1, · · · , q. Let α i,j be positive numbers for j = 1, · · · , µ i and i = 1, · · · , q. Let δ ∈ (0, 1 2 ) satisfying δ max
have i(γ, m) = m(i(γ, 1) + S + M (1) − C(M )) +2 θ∈(0,2π) E mθ 2π S − M (e √ −1θ ) − (S + M (1) + C(M )),(3.1≤i≤q µ i < 1 2 . Suppose D i ≡ β i + µ i j=1
α i,j = 0 for i = 1, · · · , q. Then there exist infinitely many (N, m 1 , · · · , m q ) ∈ N q+1 such that
m i β i + µ i j=1 E(m i α i,j ) = ̺ i N + ∆ i , ∀ 1 ≤ i ≤ q. (3.12) min{{m i α i,j }, 1 − {m i α i,j }} < δ, ∀ j = 1, · · · , µ i , 1 ≤ i ≤ q, (3.13) m i α i,j ∈ N, if α i,j ∈ Q,(3.
14)
where
̺ i = 1, if D i > 0, −1, if D i < 0, ∆ i = 0<{m i α i,j }<δ 1, ∀ 1 ≤ i ≤ q. (3.15)
Remark 3.5. When D i > 0 for all 1 ≤ i ≤ q, this is precisely the Theorem 4.1 of [LoZ] (also cf. Theorem 11.1.1 of [Lon4]).
Proof of Theorem 3.4. By assumption D i = 0, ∀ 1 ≤ i ≤ q, we further assume that there
exists some integer 0 ≤ q 0 ≤ q with D i < 0 for 0 ≤ i ≤ q 0 and D i > 0 for q 0 + 1 ≤ i ≤ q.
Next we will do with both of these two cases simultaneously. In fact we only need to use ̺ i N and m i = ̺ i N M D i + χ i M to replace the corresponding N and m i s in the proof of Theorem 4.1 of [LoZ] (cf. Theorem 11.1.1 of [Lon4]). For reader's convenience and because the proof is almost self-contained, in the following we only give some different points and details.
In order to get (3.12), we consider
m i D i = ̺ i N M D i M D i − ̺ i N M D i M D i + χ i M D i = ̺ i N + χ i − ̺ i N M D i M D i , ∀ 1 ≤ i ≤ q,(3.16)
where, to get (3.14), we require M ∈ N to satisfy M α i,j ∈ N when α i,j ∈ Q for j = 1, · · · , µ i , and χ i ∈ {0, 1} will be determined later.
Set
m i = ̺ i N M D i + χ i M. (3.17)
Then by (1.3) and (3.16), following the proofs from (4.11) and (4.13) of [LoZ] (or (11.1.11) to (11.1.13) of [Lon4]) and using ∆ i and δ defined there, it yields
m i β i + µ i j=1 E(m i α i,j ) = m i D i + µ i j=1 (ϕ(m i α i,j ) − {m i α i,j }) = ̺ i N + χ i − ̺ i N M D i M D i + µ i j=1 (ϕ(m i α i,j ) − {m i α i,j }) = ̺ i N + χ i − ̺ i N M D i M D i + ∆ i − 0<{m i α i,j }<δ {m i α i,j } + 0<1−{m i α i,j }<δ (1 − {m i α i,j }),(3.18)
which, together with requiring (3.16) and (3.18) simultaneously, implies that
m i β i + µ i j=1 E(m i α i,j ) − ̺ i N − ∆ i ≤ ̺ i N M D k − χ i M |D i | + µ i δ, 1 ≤ i ≤ q. (3.19)
Notice that δ max 1≤i≤q µ i < 1 2 holds by assumption. So by (3.19), in order to obtain (3.12) we need to choose M, N ∈ N and χ i s such that the following estimate holds
̺ i N M D i − χ i M |D i | < 1 2 .
(3.20)
On the other hand, by the choice (3.17) of m i , we have
{m i α i,j } = ̺ i N M D i + χ i M α i,j = ̺ i N α i,j D i + χ i − ̺ i N M D i M α i,j ≡ {A i,j (̺ i N ) + B i,j (̺ i N )}, j = 1, · · · , µ i , 1 ≤ i ≤ q, (3.21) where A i,j (̺ i N ) = ̺ i N α i,j D i − χ i,j , B i,j (̺ i N ) = χ i − ̺ i N M D i M α i,j ,(3.22)
and χ i,j ∈ {0, 1} will be determined later.
Following the arguments between (4.18) and (4.20) of [LoZ], it can be easily shown that {m i α i,j } must be close enough to 0 or 1, i.e., satisfying (3.13), if
max {|A i,j (̺ i N )|, |B i,j (̺ i N )|} < δ 1 3 , for 0 < δ 1 < δ. (3.23)
By (3.20) and (3.23), in order to get (3.12)-(3.14) we only need to choose integers χ i , χ i,j ∈ {0, 1} and infinitely many integers N ∈ N such that all the quantities
̺ i N α i,j D i − χ i,j , ̺ i N M D i − χ i (3.24)
can be made simultaneously to be small enough, which can be reduced to a dynamical problem on a torus (cf. pages 233-234 of [Lon4]). Here we omit rest of details in [Lon4].
In 2002, Y. Long and C. Zhu [LoZ] has established the common index jump theorem for symplectic paths, which has become one of the main tools to study the periodic orbit problem in Hamiltonian and symplectic dynamics. In [DLW] of 2016, H. Duan, Y. Long and W. Wang further improved this theorem to an enhanced version which gives more precise index properties of γ 2m k k and γ 2m k ±m k with 1 ≤ m ≤m for any fixedm. Under the help of Theorem 3.4, following the proofs of Theorem 3.5 in [DLW], next we further generalize this theorem to the case of admitting the existence of symplectic paths with negative mean indices.
Theorem 3.6. (Generalized common index jump theorem for symplectic paths) Let γ i ∈ P τ i (2n) for i = 1, · · · , q be a finite collection of symplectic paths with nonzero mean indiceŝ
i(γ i , 1). Let M i = γ i (τ i ).
We extend γ i to [0, +∞) by (3.9) inductively.
Then for any fixedm ∈ N, there exist infinitely many (q + 1)-tuples (N, m 1 , · · · , m q ) ∈ N q+1 such that the following hold for all 1 ≤ i ≤ q and 1 ≤ m ≤m,
ν(γ i , 2m i − m) = ν(γ i , 2m i + m) = ν(γ i , m), (3.25) i(γ i , 2m i + m) = 2̺ i N + i(γ i , m), (3.26) i(γ i , 2m i − m) = 2̺ i N − i(γ i , m) − 2(S + M i (1) + Q i (m)), (3.27) i(γ i , 2m i ) = 2̺ i N − (S + M i (1) + C(M i ) − 2∆ i ), (3.28) where ̺ i = 1, ifî(γ i , 1) > 0, −1, ifî(γ i , 1) < 0, ∆ i = 0<{m i θ/π}<δ S − M i (e √ −1θ ), Q i (m) = e √ −1θ ∈σ(M i ), { m i θ π }={ mθ 2π }=0 S − M i (e √ −1θ ).
(3.29)
More precisely, by (3.17) and (4.40), (4.41) in [LoZ] , we have
m i = N M |î(γ i , 1)| + χ i M, ∀ 1 ≤ i ≤ q,(3.
30)
where χ i = 0 or 1 for 1 ≤ i ≤ q and M θ π ∈ Z whenever e √ −1θ ∈ σ(M i ) and θ π ∈ Q for some 1 ≤ i ≤ q. Furthermore, by (3.24), for any ǫ > 0, we can choose N and {χ i } 1≤i≤q such that
N M |î(γ i , 1)| − χ i < ǫ, ∀ 1 ≤ i ≤ q. (3.31) Proof. For 1 ≤ i ≤ q, let µ i = 0<θ<2π S − M i (e √ −1θ ), α i,j = θ j π where e √ −1θ j ∈ σ(M i ) for 1 ≤ j ≤ µ i , and D i = i(γ i , 1) + S + M i (1) − C(M i ) + θ∈(0,2π) θ π S − M i (e √ −1θ
). Then Theorem 3.6 can be proved by Theorem 3.4 and using ̺ k N and m i = ̺ i N M D i + χ i M to replace the corresponding N and m i s in the proof of Theorem 3.5 of [DLW]. Here we omit all details.
Remark 3.7. Let l = q + q k=1 µ k , and v = 1 M |î(γ 1 , 1)| , · · · , 1 M |î(γ 1 , 1)| , α 1,1 |î(γ 1 , 1)| , · · · , α 1,µ 1 |î(γ 1 , 1)| , · · · , α q,1 |î(γ q , 1)| , · · · , α q,µq |î(γ q , 1)| ∈ R l .
Theorem 3.6 also shows that for any given small ǫ > 0 one can find a vertex χ = (χ 1 , · · · , χ q , χ 1,1 , · · · , χ 1,µ 1 , · · · , χ q,1 , · · · , χ q,µq ) of the cube [0, 1] l and infinitely many N ∈ N such that |{N v} − χ| < ǫ.
Theorem 3.8. (cf. Theorem 2.1 of [HuL] and Theorem 6.1 of [LLo2]) Suppose Σ ∈ H st (2n) and (τ, y) ∈ T (Σ). Then we have i(y m ) ≡ i(mτ, y) = i(y, m) − n, ν(y m ) ≡ ν(mτ, y) = ν(y, m), ∀m ∈ N, (3.32)
where i(y m ) and ν(y m ) are the index and nullity of (mτ, y) defined in Section 2, i(y, m) and ν(y, m)
are the Maslov-type index and nullity of (mτ, y) (cf. Section 5.4 of [Lon3]). In particular, we havê i(τ, y) =î(y, 1), whereî(τ, y) is given in Section 2,î(y, 1) is the mean Maslov-type index (cf.
Definition 8.1 of [Lon4]). Hence we denote it simply byî(y).
Proof of Theorem 1.2
In order to prove Theorem 1.2, let Σ ∈ H st (2n) be a non-degenerate perfect star-shaped hypersurface which possesses only finitely many prime closed characteristics {(τ k , y k )} q k=1 withî(y k , 1) = 0. Note that there exist at least one closed characteristic on Σ with positive mean index by the first identity of (2.11) in Theorem 2.3. So without loss of generality, the following mixed mean index condition holds:
(MMI) There exists an integer q 0 ∈ [1, q] such thatî(y k , 1) > 0 for 1 ≤ k ≤ q 0 andî(y k , 1) < 0
for q 0 + 1 ≤ k ≤ q.
Denote by γ k ≡ γ y k the associated symplectic path of (τ k , y k ) for 1 ≤ k ≤ q. Then by Lemma 3.3 of [HuL] and Lemma 3.2 of [Lon1], there exists P k ∈ Sp(2n) and U k ∈ Sp(2n − 2) such that
M k ≡ γ k (τ k ) = P −1 k (N 1 (1, 1)⋄U k )P k , ∀ 1 ≤ k ≤ q,(4.1)
where every U k has the following form:
R(θ 1 ) ⋄ · · · ⋄ R(θ r ) ⋄ D(±2) ⋄s ⋄ N 2 (e α 1 √ −1 , A 1 ) ⋄ · · · ⋄ N 2 (e αr * √ −1 , A r * ) ⋄ N 2 (e β 1 √ −1 , B 1 ) ⋄ · · · ⋄ N 2 (e βr 0 √ −1 , B r 0 ), where θ j 2π ∈ Q for 1 ≤ j ≤ r; α j 2π ∈ Q for 1 ≤ j ≤ r * ;
β j 2π ∈ Q for 1 ≤ j ≤ r 0 and r + s + 2r * + 2r 0 = n − 1.
(4.2)
Proof of Theorem 1.2.
We prove Theorem 1.2 in two cases:
Case 1. n is even.
We continue the proof in three steps.
Step 1. The first set of iterates for the choice of the vertex χ in the cube [0, 1] l .
By (MMI), we haveî(y k ) =î(y k , 1) > 0 for 1 ≤ k ≤ q 0 andî(y k ) =î(y k , 1) < 0 for q 0 +1 ≤ k ≤ q, which implies that i(y k , m) → +∞ for 1 ≤ k ≤ q 0 and i(y k , m) → −∞ for q 0 + 1 ≤ k ≤ q as m → +∞. So the positive integerm defined bȳ
m 1 = max 1≤k≤q 0 {min{m 0 ∈ N | i(y k , m + l) ≥ i(y k , l) + n + 1, ∀ l ≥ 1, m ≥ m 0 }} m 2 = max q 0 +1≤k≤q {min{m 0 ∈ N | i(y k , m + l) ≤ i(y k , l) − n − 1, ∀ l ≥ 1, m ≥ m 0 }} m = max{m 1 ,m 2 } (4.3)
is well-defined and finite.
For the integerm defined in (4.3), it follows from Theorem 3.6 and Remark 3.7 that there exist a vertex χ of [0, 1] l and infinitely many (q + 1)-tuples (N, m 1 , · · · , m q ) ∈ N q+1 such that for any
1 ≤ k ≤ q, there holdsm + 2 ≤ min{2m k , 1 ≤ k ≤ q}, (4.4) i(y k , 2m k − m) = 2̺ k N − 2 − i(y k , m), ∀ 1 ≤ m ≤m, (4.5) i(y k , 2m k ) = 2̺ k N − 1 − C(M k ) + 2∆ k , (4.6) i(y k , 2m k + m) = 2̺ k N + i(y k , m), ∀ 1 ≤ m ≤m,(4.7)
where note that S + M k (1) = 1, Q k (m) = 0, ∀ m ≥ 1 by (4.1)-(4.2). By the definition (4.3) ofm and (4.6), for any m ≥m + 1, we obtain
i(y k , 2m k − m) ≤ i(y k , 2m k ) − n − 1 = 2N − n − 2 + 2∆ k − C(M k ) ≤ 2N − 3, 1 ≤ k ≤ q 0 , (4.8) i(y k , 2m k − m) ≥ i(y k , 2m k ) + n + 1 = −2N + n + 2∆ k − C(M k ) ≥ −2N + 1, q 0 + 1 ≤ k ≤ q,(4.9)
i(y k , 2m k + m) ≥ i(y k , 2m k ) + n + 1
= 2N + n − C(M k ) + 2∆ k ≥ 2N + 1, 1 ≤ k ≤ q 0 , (4.10) i(y k , 2m k + m) ≤ i(y k , 2m k ) − n − 1 = −2N − n − 2 − C(M k ) + 2∆ k ≤ −2N − 3, q 0 + 1 ≤ k ≤ q, (4.11)
where we use the facts 2∆ k − C(M k ) ≤ n − 1 and C(M k ) ≤ n − 1.
Then by (4.5)-(4.11) and Theorem 3.8, we obtain In fact, let ǫ <
i(y 2m k −m k ) ≤ 2N − n − 3, ∀m + 1 ≤ m ≤ 2m k − 1, 1 ≤ k ≤ q 0 , (4.12) i(y 2m k −m k ) ≥ −2N − n + 1, ∀m + 1 ≤ m ≤ 2m k − 1, q 0 + 1 ≤ k ≤ q, (4.13) i(y 2m k −m k ) = 2̺ k N − 2n − 2 − i(y m k ), ∀ 1 ≤ m ≤m, (4.14) i(y 2m k k ) = 2̺ k N − C(M k ) + 2∆ k − n − 1, (4.15) i(y 2m k +m k ) = 2̺ k N + i(y m k ), ∀ 1 ≤ m ≤m, (4.16) i(y 2m k +m k ) ≥ 2N − n + 1, ∀ m ≥m + 1, 1 ≤ k ≤ q 0 , (4.17) i(y 2m k +m k ) ≤ −2N − n − 3, ∀ m ≥m + 1, q 0 + 1 ≤ k ≤ q.
1+2M
1≤k≤q |χ(y k )| , by Theorem 2.3 and (MMI) we have
q k=1χ (y k ) |î(y k )| = î (y k )>0χ (y k ) i(y k ) − î (y k )<0χ (y k ) i(y k ) = 1 2 ,
which, together with (3.30)-(3.31), yields
N − q k=1 2m kχ (y k ) = q k=1 2Nχ(y k ) |î(y k )| − q k=1 2χ(y k ) N M |î(y k )| + χ k M ≤ 2M q k=1 |χ(y k )| N M |î(y k )| − χ k . < 2M ǫ q k=1 |χ(y k )| < 1.
(4.20)
It proves Claim 1 since each 2m kχ (y k ) is an integer.
Now by Lemma 2.2, it yields
2m k m=1 (−1) d(K)+i(y m k ) dim C S 1 ,d(K)+i(y m k ) (F a,K , S 1 · x m k ) = 2m k m=1 (−1) i(y m k ) dim C S 1 ,d(K)+i(y m k ) (F a,K , S 1 · x m k ) = m k −1 i=0 2i+2 m=2i+1 (−1) i(y m k ) dim C S 1 ,d(K)+i(y m k ) (F a,K , S 1 · x m k ) = m k −1 i=0 2 m=1 (−1) i(y m k ) dim C S 1 ,d(K)+i(y m k ) (F a,K , S 1 · x m k ) = m k 2 m=1 (−1) i(y m k ) dim C S 1 ,d(K)+i(y m k ) (F a,K , S 1 · x m k ) = 2m kχ (y k ), ∀ 1 ≤ k ≤ q, (4.21)
where x k is the critical point of F a,K corresponding to y k , and we choose large enough K such that Since i(y m k ) = −n − 1 when (mτ k , y k ) is good, by (MMI), Definition 1.1 and Theorem 3.8, we set
d(K) = 2n([KT /2π] + 1) ≥ −i(y m k ) for 1 ≤ m ≤ 2m k and 1 ≤ k ≤ q. For 1 ≤ k ≤ q,M e + (k) = # {1 ≤ m ≤m | i(y m k ) ≤ −n − 2, i(y 2m k +m k ) ∈ 2Z, i(y k ) ∈ 2Z}, if 1 ≤ k ≤ q 0 , # {1 ≤ m ≤m | i(y m k ) ≥ −n, i(y 2m k +m k ) ∈ 2Z, i(y k ) ∈ 2Z}, if q 0 + 1 ≤ k ≤ q, M o + (k) = # {1 ≤ m ≤m | i(y m k ) ≤ −n − 2, i(y 2m k +m k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1}, if 1 ≤ k ≤ q 0 , # {1 ≤ m ≤m | i(y m k ) ≥ −n, i(y 2m k +m k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1}, if q 0 + 1 ≤ k ≤ q, M e − (k) = # {1 ≤ m ≤m | i(y m k ) ≤ −n − 2, i(y 2m k −m k ) ∈ 2Z, i(y k ) ∈ 2Z}, if 1 ≤ k ≤ q 0 , # {1 ≤ m ≤m | i(y m k ) ≥ −n, i(y 2m k −m k ) ∈ 2Z, i(y k ) ∈ 2Z}, if q 0 + 1 ≤ k ≤ q, M o − (k) = # {1 ≤ m ≤m | i(y m k ) ≤ −n − 2, i(y 2m k −m k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1}, if 1 ≤ k ≤ q 0 , # {1 ≤ m ≤m | i(y m k ) ≥ −n, i(y 2m k −m k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1}, if q 0 + 1 ≤ k ≤ q,
which, together with i(y 2m k +m k ) − i(y 2m k −m k ) ∈ 2Z by (4.14) and (4.16), yields
M e + (k) = M e − (k), M o + (k) = M o − (k), ∀ 1 ≤ k ≤ q. (4.22)
For 1 ≤ k ≤ q 0 and 1 ≤ m ≤m satisfying i(y m k ) ≤ −n − 2, and for q 0 + 1 ≤ k ≤ q and 1 ≤ m ≤m satisfying i(y m k ) ≥ −n, by (4.14) and (4.16) it yields
i(y 2m k −m k ) ≥ 2N − n, i(y 2m k +m k ) ≤ 2N − n − 2, ∀ 1 ≤ k ≤ q 0 , (4.23) i(y 2m k −m k ) ≤ −2N − n − 2, i(y 2m k +m k ) ≥ −2N − n, ∀ q 0 + 1 ≤ k ≤ q.
(4.24)
So, for 1 ≤ m ≤m, by (4.23)-(4.24) and Lemma 2.2, we know that all y 2m k +m k 's with i(y m k ) ≤ −n − 2 and 1 ≤ k ≤ q 0 , or i(y m k ) ≥ −n and q 0 + 1 ≤ k ≤ q, only have contribution to the alternative sum 2N −n−1 p=−2N −n−1 (−1) p M p , and all y 2m k −m k 's with i(y m k ) ≤ −n − 2 and 1 ≤ k ≤ q 0 , or i(y m k ) ≥ −n and q 0 + 1 ≤ k ≤ q, have no contribution to 2N −n−1 p=−2N −n−1 (−1) p M p . Thus for the Morse-type numbers M p 's in (2.14), by (4.21)-(4.24) we have
2N −n−1 p=−2N −n−1 (−1) p M p = q k=1 1≤m≤2m k +m −2N−n−1≤i(y m k )≤2N−n−1 (−1) d(K)+i(y m k ) dim C S 1 ,d(K)+i(y m k ) (F a,K , S 1 · x m k ) = q k=1 2m k m=1 (−1) d(K)+i(y m k ) dim C S 1 ,d(K)+i(y m k ) (F a,K , S 1 · x m k ) + q k=1 M e + (k) − M o + (k) − q k=1 M e − (k) − M o − (k) − 1≤k≤q 0 i(y 2m k k )≥2N−n (−1) i(y 2m k k ) dim C S 1 ,d(K)+i(y 2m k k ) (F a,K , S 1 · x 2m k k ) − q 0 +1≤k≤q i(y 2m k k )≤−2N−n−2 (−1) i(y 2m k k ) dim C S 1 ,d(K)+i(y 2m k k ) (F a,K , S 1 · x 2m k k ) = q k=1 2m kχ (y k ) − 1≤k≤q 0 i(y 2m k k )≥2N−n (−1) i(y 2m k k ) dim C S 1 ,d(K)+i(y 2m k k ) (F a,K , S 1 · x 2m k k ) − q 0 +1≤k≤q i(y 2m k k )≤−2N−n−2 (−1) i(y 2m k k ) dim C S 1 ,d(K)+i(y 2m k k ) (F a,K , S 1 · x 2m k k ). (4.25)
In order to exactly know whether the iterate y 2m k k of y k has contribution to the alternative sum
2N −n−1 p=−2N −n−1 (−1) p M p , 1 ≤ k ≤ q, we set N e + = # {q 0 + 1 ≤ k ≤ q | i(y 2m k k ) ≤ −2N − n − 2, i(y 2m k k ) ∈ 2Z, i(y k ) ∈ 2Z} + # {1 ≤ k ≤ q 0 | i(y 2m k k ) ≥ 2N − n, i(y 2m k k ) ∈ 2Z, i(y k ) ∈ 2Z}, (4.26) N o + = # {q 0 + 1 ≤ k ≤ q | i(y 2m k k ) ≤ −2N − n − 2, i(y 2m k k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1} + # {1 ≤ k ≤ q 0 | i(y 2m k k ) ≥ 2N − n, i(y 2m k k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1}, (4.27) N e − = # {q 0 + 1 ≤ k ≤ q | i(y 2m k k ) ≥ −2N − n, i(y 2m k k ) ∈ 2Z, i(y k ) ∈ 2Z} + # {1 ≤ k ≤ q 0 | i(y 2m k k ) ≤ 2N − n − 2, i(y 2m k k ) ∈ 2Z, i(y k ) ∈ 2Z}, (4.28) N o − = # {q 0 + 1 ≤ k ≤ q | i(y 2m k k ) ≥ −2N − n, i(y 2m k k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1} + # {1 ≤ k ≤ q 0 | i(y 2m k k ) ≤ 2N − n − 2, i(y 2m k k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1}. (4.29)
Thus by Claim 1, (4.25), the definitions of N e + and N o + and (2.16), we have
N + N o + − N e + = q k=1 2m kχ (y k ) + N o + − N e + = 2N −n−1 p=−2N −n−1 (−1) p M p ≤ 2N −n−1 p=−2N −n−1 (−1) p b p = 2N −n−2 p=0 b p = N − n 2 , (4.30)
where the first equality holds by Claim 1, the second equality follows from (4.25) and the definitions of N e + and N o + , and the last equality follows from b 2j = 1 and b 2j−1 = 0 for 0 ≤ j ≤ N − n−2 2 by (2.16) where n is even.
So (4.30) give the following estimate
N e + ≥ n 2 .
(4.31)
Step 2. The second set of iterates for the choice of the dual vertexχ = 1 − χ in the cube [0, 1] l . Similar to (4.12)-(4.18), forχ =1 − χ of the cube [0, 1] l with χ chosen below (4.3) wherê 1 = (1, · · · , 1), it follows from Theorem 3.6 (also cf. Theorem 2.8 of [HaW] and Theorem 4.2 of [LoZ]) and Remark 3.7 that there exist also infinitely many (q + 1)-tuples (N ′ , m ′ 1 , · · · , m ′ q ) ∈ N q+1 such that for any 1 ≤ k ≤ q, there holds
i(y 2m ′ k −m k ) ≤ 2N ′ − n − 3, ∀m + 1 ≤ m ≤ 2m ′ k − 1, 1 ≤ k ≤ q 0 , (4.32) i(y 2m ′ k −m k ) ≥ −2N ′ − n + 1, ∀m + 1 ≤ m ≤ 2m ′ k − 1, q 0 + 1 ≤ k ≤ q, (4.33) i(y 2m ′ k −m k ) = 2̺ k N ′ − 2n − 2 − i(y m k ), ∀ 1 ≤ m ≤m, (4.34) i(y 2m ′ k k ) = 2̺ k N ′ − C(M k ) + 2∆ ′ k − n − 1, (4.35) i(y 2m ′ k +m k ) = 2̺ k N ′ + i(y m k ), ∀ 1 ≤ m ≤m, (4.36) i(y 2m ′ k +m k ) ≥ 2N ′ − n + 1, ∀ m ≥m + 1, 1 ≤ k ≤ q 0 , (4.37) i(y 2m ′ k +m k ) ≤ −2N ′ − n − 3, ∀ m ≥m + 1, q 0 + 1 ≤ k ≤ q,(4.38)
where, furthermore, ∆ k and ∆ ′ k satisfy the following relationship
∆ ′ k + ∆ k = C(M k ), ∀ 1 ≤ k ≤ q, (4.39)
by the factχ =1 − χ and the proof of Claim 4 in the proof of Theorem 1.1 of [DLW] or (42) in Theorem 2.8 of [HaW].
Similarly, we define
N ′ e + = # {q 0 + 1 ≤ k ≤ q | i(y 2m ′ k k ) ≤ −2N ′ − n − 2, i(y 2m ′ k k ) ∈ 2Z, i(y k ) ∈ 2Z} + # {1 ≤ k ≤ q 0 | i(y 2m ′ k k ) ≥ 2N ′ − n, i(y 2m ′ k k ) ∈ 2Z, i(y k ) ∈ 2Z}, (4.40) N ′ o + = # {q 0 + 1 ≤ k ≤ q | i(y 2m ′ k k ) ≤ −2N ′ − n − 2, i(y 2m ′ k k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1} + # {1 ≤ k ≤ q 0 | i(y 2m ′ k k ) ≥ 2N ′ − n, i(y 2m ′ k k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1}, (4.41) N ′ e − = # {q 0 + 1 ≤ k ≤ q | i(y 2m ′ k k ) ≥ −2N ′ − n, i(y 2m ′ k k ) ∈ 2Z, i(y k ) ∈ 2Z} + # {1 ≤ k ≤ q 0 | i(y 2m ′ k k ) ≤ 2N ′ − n − 2, i(y 2m ′ k k ) ∈ 2Z, i(y k ) ∈ 2Z}, (4.42) N ′ o − = # {q 0 + 1 ≤ k ≤ q | i(y 2m ′ k k ) ≥ −2N ′ − n, i(y 2m ′ k k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1} + # {1 ≤ k ≤ q 0 | i(y 2m ′ k k ) ≤ 2N ′ − n − 2, i(y 2m ′ k k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1}. (4.43)
So by (4.35) and (4.39) it yields
i(y 2m ′ k k ) = 2̺ k N ′ − C(M k ) + 2(C(M k ) − ∆ k ) − n − 1 = 2̺ k N ′ + C(M k ) − 2∆ k − n − 1.N ′ + N ′ o + − N ′ e + = q k=1 2m ′ kχ (y k ) + N ′ o + − N ′ e + = 2N ′ −n−1 p=−2N ′ −n−1 (−1) p M p ≤ 2N ′ −n−1 p=−2N ′ −n−1 (−1) p b p = 2N ′ −n−2 p=0 b p = N ′ − n 2 ,(4.46)
which, together with (4.45), implies
N e − = N ′ e + ≥ n 2 .
(4.47)
Step 3. The summary.
So by (4.31) and (4.47) it yields
q ≥ N e + + N e − ≥ n. (4.48)
In addition, any hyperbolic closed characteristic y k must have i(y 2m k k ) = 2̺ k N − n − 1 since there holds C(M k ) = 0 in the hyperbolic case. However, by (4.26) and (4.28), there exist at least (N e + + N e − ) closed characteristics with even indices i(y 2m k k ). So all these (N e + + N e − ) closed characteristics are non-hyperbolic. Then (4.48) shows that there exist at least n distinct nonhyperbolic closed characteristics. Now (4.26), (4.28) and (4.48) show that all these non-hyperbolic closed characteristics and their iterations have even Maslov-type indices. This completes the proof of Case 1.
Case 2. n is odd.
In this case, (MMI) still holds. Here the arguments are similar to those in the proof of Case 1. So we only give some different parts in the proof and omit other details.
Claim 2: There exist at least (n−1) geometrically distinct non-hyperbolic closed characteristics denoted by {y k } n−1 k=1 with odd Maslov-type indices on such hypersurface Σ. Here one crucial and different point from the proof of Case 1 is that we need to consider the (4.25)). This difference is mainly due to the different parity of n. Since the method is similar to that in proof of Case 1, we only list some necessary parts.
alternative sum 2N −n p=−2N −n (−1) p M p (cf. (4.52)) instead of 2N −n−1 p=−2N −n−1 (−1) p M p (cf.
At first, there holds i(y m k ) / ∈ {−n − 2, −n − 1, −n} when (mτ k , y k ) is good, by (MMI), Definition 1.1 and Theorem 3.8, we set
M e + (k) = # {1 ≤ m ≤m | i(y m k ) ≤ −n − 3, i(y 2m k +m k ) ∈ 2Z, i(y k ) ∈ 2Z}, if 1 ≤ k ≤ q 0 , # {1 ≤ m ≤m | i(y m k ) ≥ −n + 1, i(y 2m k +m k ) ∈ 2Z, i(y k ) ∈ 2Z}, if q 0 + 1 ≤ k ≤ q, M o + (k) = # {1 ≤ m ≤m | i(y m k ) ≤ −n − 3, i(y 2m k +m k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1}, if 1 ≤ k ≤ q 0 , # {1 ≤ m ≤m | i(y m k ) ≥ −n + 1, i(y 2m k +m k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1}, if q 0 + 1 ≤ k ≤ q, M e − (k) = # {1 ≤ m ≤m | i(y m k ) ≤ −n − 3, i(y 2m k −m k ) ∈ 2Z, i(y k ) ∈ 2Z}, if 1 ≤ k ≤ q 0 , # {1 ≤ m ≤m | i(y m k ) ≥ −n + 1, i(y 2m k −m k ) ∈ 2Z, i(y k ) ∈ 2Z}, if q 0 + 1 ≤ k ≤ q, M o − (k) = # {1 ≤ m ≤m | i(y m k ) ≤ −n − 3, i(y 2m k −m k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1}, if 1 ≤ k ≤ q 0 , # {1 ≤ m ≤m | i(y m k ) ≥ −n + 1, i(y 2m k −m k ) ∈ 2Z − 1, i(y k ) ∈ 2Z − 1}, if q 0 + 1 ≤ k ≤ q,
which, together with i(y 2m k +m k ) − i(y 2m k −m k ) ∈ 2Z by (4.14) and (4.16), yields
M e + (k) =M e − (k),M o + (k) =M o − (k), ∀ 1 ≤ k ≤ q.
(4.49)
For 1 ≤ k ≤ q 0 and 1 ≤ m ≤m satisfying i(y m k ) ≤ −n − 3, and for q 0 + 1 ≤ k ≤ q and 1 ≤ m ≤m satisfying i(y m k ) ≥ −n + 1, by (4.14) and (4.16) it yields
i(y 2m k −m k ) ≥ 2N − n + 1, i(y 2m k +m k ) ≤ 2N − n − 3, ∀ 1 ≤ k ≤ q 0 , (4.50) i(y 2m k −m k ) ≤ −2N − n − 3, i(y 2m k +m k ) ≥ −2N − n + 1, ∀ q 0 + 1 ≤ k ≤ q, (4.51)
Then, similarly to the equation (4.25), we have This completes the proof of Claim 2.
2N −n p=−2N −n (−1) p M p = q k=1 2m kχ (y k ) − 1≤k≤q 0 i(y 2m k k )≥2N−n+1 (−1) i(y 2m k k ) dim C S 1 ,d(K)+i(y 2m k k ) (F a,K , S 1 · x 2m k k ) − q 0 +1≤k≤q i(y 2m k k )≤−2N−n−3 (−1) i(y 2m k k ) dim C S 1 ,d(K)+i(y 2m k k ) (F a,K , S 1 · x 2m k kN + H o + − H e + = q k=1 2m kχ (y k ) + H o + − H e + = 2N −n p=−2N −n (−1) p M p ≤ 2N −n p=−2N −n (−1) p b p = 2N −n−1 p=0 b p = 2N − n − 1 2 + 1 = N − n − 1 2 ,(4.
Claim 3: There exist at least another geometrically distinct closed characteristic different from those found in Claim 2 with odd Maslov-type indices on such hypersurface Σ.
In fact, for those (n − 1) distinct closed characteristics {y k } n−1 k=1 found in Claim 2, there holds i(y 2m k k ) = ±2N − n − 1 by the definitions of H e + and H e − , which, together with (4.12)-(4.18) and (MMI), yields
i(y m k ) = 2N − n − 1, ∀ m ≥ 1, k = 1, · · · , n − 1. (4.58)
Then by Lemma 2.2 it yields
1≤k≤n−1 m≥1 dim C S 1 ,d(K)+2N −n−1 (F a,K , S 1 · x m k ) = 0. (4.59)
By (4.12)-(4.14), (4.16)-(4.17) and (MMI), it yields i(y m k ) = 2N − n − 1 for any m = 2m k and 1 ≤ k ≤ q. Therefore, by (4.59) and (2.14) we obtain
n≤k≤q dim C S 1 ,d(K)+2N −n−1 (F a,K , S 1 · x 2m k k ) = n≤k≤q, m≥1 dim C S 1 ,d(K)+2N −n−1 (F a,K , S 1 · x m k ) = 1≤k≤q, m≥1 dim C S 1 ,d(K)+2N −n−1 (F a,K , S 1 · x m k ) = M 2N −n−1 ≥ b 2N −n−1 = 1. (4.60)
Now by (4.60) and Lemma 2.2, it yields that there exist at least another closed characteristic 5 Proof of Theorem 1.6
In this section, we prove Conjecture 1.5 for the case of n = 3, i.e., Theorem 1.6, by contradiction.
We assume first the following condition (C):
(C) There are finitely many prime closed characteristics {(τ k , y k )} 1≤k≤q on Σ ∈ H st (6), and i(y k ) = 0 for 1 ≤ k ≤ q 0 with some integers q 0 ∈ [1, q] and q ∈ N.
Let P Σ = {mτ k | 1 ≤ k ≤ q, m ∈ N} be the period set of all closed characteristics on Σ.
Denote by γ k ≡ γ y k the associated symplectic path of (τ k , y k ) for 1 ≤ k ≤ q. Then by Lemma 3.3 of [HuL] and Lemma 3.2 of [Lon1], there exists P k ∈ Sp(6) and U k ∈ Sp(4) such that
M k ≡ γ k (τ k ) = P −1 k (N 1 (1, 1)⋄U k )P k , ∀ 1 ≤ k ≤ q, (5.1)
where every U k has the following form by Theorem 3.2:
R(θ 1 ) ⋄ · · · ⋄ R(θ r ) ⋄ D(±2) ⋄s ⋄ N 2 (e α 1 √ −1 , A 1 ) ⋄ · · · ⋄ N 2 (e αr * √ −1 , A r * ) ⋄ N 2 (e β 1 √ −1 , B 1 ) ⋄ · · · ⋄ N 2 (e βr 0 √ −1 , B r 0 ), where θ j 2π ∈ Q for 1 ≤ j ≤ r; α j 2π ∈ Q for 1 ≤ j ≤ r * ; β j 2π ∈ Q for 1 ≤ j ≤ r 0 and r + s + 2r * + 2r 0 = 2. (5.2)
Hence by (5.1), Theorem 3.8 and the precise index iteration formulae for symplectic paths due to Y. Long (cf. [Lon3] or Chapter 8 of [Lon4]), we have Proof. By (5.4) we have
i(y m k ) = m(i(y k ) + n + 1 − r) + 2 r j=1 mθ j 2π + r − 1 − n = m(i(y k ) + 4 − r) + 2 r j=1 mθ j 2π + r − 4, ∀ m ≥ 1, 1 ≤ k ≤ q,(5.i(y k ) + 4 − r + r j=1 θ j π = 0, ∀ 1 ≤ k ≤ q 0 . (5.5)
Note that θ j π / ∈ Q and 0 ≤ r ≤ 2 by (5.2), and then it yields r = 0 or 2.
If r = 0, then i(y k ) = −4 by (5.5). Together with (5.3), it yields i(y m k ) = −4 for all m ∈ N. If r = 2, note that 2 j=1 θ j π ∈ (0, 4) and i(y k ) ∈ 2Z by (8. The proof of Lemma 5.1 is finished.
Remark 5.2. In our proof of Theorem 1.6 below, we shall apply results in Section 7 of [Vit2] frequently. Note that in the Theorem 7.1 of [Vit2] and its proof, the two end points of the index interval I were carefully avoided (cf. Theorem 7.1 on p.637, (7.24) and (7.25) in p.647, and the top part on p.648 of [Vit2]), which are due to the effect of the S 1 -action on the homologies with two adjacent dimensions (cf. Corollary in Appendix 1 on p.653 of [Vit2]). In our proof of Theorem 1.6 we are dealing with only non-degenerate critical orbits. Note that because in our case the critical orbit S 1 · y m k studied below is always orientable on the star-shaped hypersurface Σ, by the homological correspondence of the Theorem I.7.5 and the comments below it on pp.78-79 of [Cha], only the homology with dimension to be precisely equal to the Morse index of the critical orbit survives. Therefore in our case, all the results in (7.24) and (7.25) of [Vit2] work for all k ∈ I, not only for k ∈ I o . Specially, by (5.7) below, whenever the dimension of the related homology is d(K) − 3 or d(K) − 5, i.e., the Viterbo index of the corresponding closed characteristic is −3 or −5, results in (7.24) and (7.25) of [Vit2] can be applied. Such arguments have no requirement on those Viterbo index near and is not −3 and −5 of the related closed characteristics, and thus can be applied to the case in the current paper. This understanding is applied below, whenever we apply Theorem 7.1 of [Vit2]. We refer readers also to Theorem 5.1 and its proof in [LLW] which generalized Theorem 7.1 of [Vit2] to the degenerate case.
Proof of Theorem 1.6.
Based on Lemma 5.1, we carry out the proof of Theorem 1.6 in several steps below.
Step 1. On one hand, by Lemma 5.1, there always holds i(y m k ) = −4 for any 1 ≤ k ≤ q 0 and m ∈ N. On the other hand, note thatî(y k ) > 0 (respectively < 0) implies i(y m k ) → +∞ (respectively −∞) as m → +∞. Thus iterates y m k of every y k for q 0 + 1 ≤ k ≤ q have indices satisfying i(y m k ) = −3 and −5 for any large enough m ∈ N. Therefore for large enough a, all the closed characteristics y m k for 1 ≤ k ≤ q with period larger than aT , which implies that the iterate number m is very large, will have their Viterbo indices: either (i) equal to − 4, whenî(y k ) = 0, or (ii) different from − 3, −4 and − 5, whenî(y k ) = 0.
(5.7)
Step 2. For a ∈ R, let X − (a, K) = {x ∈ X | F a,K (x) < 0} with K = K(a) as defined in the above (2.2) as well as in Section 7 of [Vit2]. Note also that the origin 0 of X is not contained in X − (a, K) by definition. Because the Hamiltonian function H a,K is quadratic homogeneous as assumed at the beginning of Section 7 of [Vit2] due to the study there being near the origin, the functional F a,K is homogeneous too.
For any large enough positive a < a ′ , we fix the same constant K > 0 in (2.2) to be sufficiently large such that the Hamiltonian function H t,K (x) is strictly convex for every t ∈ [a, a ′ ]. Now let A = X − (a, K) and A ′ = X − (a ′ , K). Because the period set P Σ defined at the beginning of this section is discrete, we choose the above constants a and a ′ carefully such that aT and a ′ T do not belong to P Σ . Note that for t ∈ [a, a ′ ] because every critical orbit S 1 · x of the functional F t,K always possesses the critical value F t,K (S 1 · x) = 0 as mentioned in p.639 of [Vit2] and by (2.7) of [LLW], the boundary sets of A and A ′ , i.e., {x | F t,K (x) = 0} with t = a or a ′ , possess no critical orbits, and specially the origin 0 of X is not contained in A and A ′ . Therefore by the homogeneity mentioned above we have
H S 1 ,d(K)+i (A ′ , A) = H S 1 ,d(K)+i (A ′ ∩ S(X), A ∩ S(X)), ∀ i ∈ Z,(5.8)
where S(X) is the unit sphere of X.
Because each critical orbit S 1 · x of the functional F t,K for some t ∈ [a, a ′ ] corresponds to an iterate y m k for some 1 ≤ k ≤ q and m ∈ N, we denote this critical orbit by S 1 · x t,k,m . Thus by the definition of F t,K we have the period T t,k,m of x t,k,m satisfies T t,k,m = tT . Consequently every
critical orbit S 1 · x t,k,m of F t,K for some t ∈ [a, a ′ ] contained in (A ′ \ A) ∩ S(X) must satisfy aT ≤ T t,k,m ≤ a ′ T.
(5.9)
Thus the period of the corresponding y m k is also T t,k,m and satisfies (5.9) too. Consequently the total number of such critical orbits contained in (A ′ \ A) ∩ S(X) is finite, which is denoted byĵ.
That is, there exist preciselyĵ times of t ∈ (a, a ′ ) which we denote by t j with 1 ≤ j ≤ĵ satisfying a < t 1 < . . . tĵ < a ′ , such that F t j ,K with 1 ≤ j ≤ĵ possesses critical orbits in (A ′ \ A) ∩ S(X), and any other F t,K with t ∈ [a, a ′ ] \ {t j | 1 ≤ j ≤ĵ} possesses no any critical orbit in (A ′ \ A) ∩ S(X).
In order to compute the homology in (5.8), we introduce below a new functionalt, which is motivated by the proof of Proposition 3 in Appendix 1 of [Vit2].
Claim 4. The partial derivative ∂ ∂t F t,K (x) of F t,K (x) with respect to t ∈ [a, a ′ ] satisfies ∂ ∂t F t,K (x) < 0 for all (x, t) ∈ S(X) × [a, a ′ ].
In fact, by the definition of H t (x) in Section 2, it is strictly increasing in t when x = 0, and then so is H t,K (x). Then the Fenchel dual function H * t,K (y) is strictly decreasing in t when y = 0. Consequently F t,K (x) is strictly decreasing in t too. Thus Claim 4 is proved. Now based on Claim 4 and the well-known implicity function theorem, there exists a unique smooth functiont : S(X) → R given by the equation
Ft (x),K (x) = 0, ∀ x ∈ S(X). (5.10) It further implies ∂ ∂t Ft (x),K (x)t ′ (x) + F ′ t(x),K (x) = 0, ∀ x ∈ S(X). (5.11)
Then for (x 0 , t 0 ) ∈ S(X) × R, we have that x 0 is a critical point oft with critical value t 0 if and only if F ′ t 0 ,K (x 0 ) = 0 by (5.11). Note thatt is S 1 -invariant since so is F t,K .
Claim 5. At any critical point x 0 oft with critical value t 0 , we have
C S 1 , * (t, x 0 ) ∼ = C S 1 , * (F t 0 ,K | S(X) , x 0 ). (5.12)
In fact, let U be a small enough S 1 -invariant open neighborhood of S 1 · x 0 in S(X). Since ∂F t,K (x 0 )/∂t < 0 by Claim 4, we obtain thatt(x) ≤ t 0 for x ∈ U if and only if F t 0 ,K (x) ≤ 0 by (5.10). Thus we obtain
{x ∈ U |t(x) ≤ t 0 } = {x ∈ U | F t 0 ,K (x) ≤ 0}, {x ∈ U |t(x) ≤ t 0 } \ {S 1 · x 0 } = {x ∈ U | F t 0 ,K (x) ≤ 0} \ {S 1 · x 0 }. (5.13)
Then by the definition of S 1 -critical module in Sections I.4 and I.7 of [Cha], the two S 1 -critical modules in (5.12) are isomorphic to each other at every dimension. Thus Claim 5 holds.
Remark 5.3. Note that the isomorphic identity (5.12) holds without further showing that functionalt(x) is C 2 and its Morse index and nullity at its critical point x 0 with critical value t 0 are the same as those of the functional F t 0 ,K (x 0 ) at its critical point x 0 , although these can be proved by using the implicity function theorem and (5.11). Here the Hessian matrices of these two functionals differ by only a positive constant which can be obtained by differentiating both sides of (5.11) with respect to x, and then evaluating at the critical points respectively.
By Claim 4 and (5.10), we then obtain A ′ ∩ S(X) = {x ∈ S(X) | F a ′ ,K (x) < 0} = {x ∈ S(X) |t(x) < a ′ },
A ∩ S(X) = {x ∈ S(X) | F a,K (x) < 0} = {x ∈ S(X) |t(x) < a}.
Note that both a and a ′ are regular values oft since aT and a ′ T do not belong to P Σ . Then for small enough ǫ > 0, A ′ ∩ S(X) and A ∩ S(X) are S 1 -homotopy equivalent witht a ′ −ǫ andt a+ǫ respectively, wheret κ denotes the level sett κ = {x ∈ S(X) |t(x) ≤ κ}. Thus by the homotopy invariance of the homology, we obtain H S 1 ,d(K)+i (A ′ ∩ S(X), A ∩ S(X)) = H S 1 ,d(K)+i (t a ′ −ǫ ,t a+ǫ ), ∀ i ∈ Z.
(5.14)
Step 3. Now for the chosen large enough a and a ′ with a < a ′ , by (5.7) there exists no any closed characteristic whose period locates between aT and a ′ T possessing Viterbo index −3 or −5.
Therefore by the discussion in pp.78-79 of [Cha], we obtain C S 1 ,d(K)−3 (F t j ,K | S(X) , S 1 · x t j ,k,m ) = C S 1 ,d(K)−5 (F t j ,K | S(X) , S 1 · x t j ,k,m ) = 0.
which together with (5.12) yields C S 1 ,d(K)−3 (t, S 1 · x t j ,k,m ) = C S 1 ,d(K)−5 (t, S 1 · x t j ,k,m ) = 0. Step 4. Now we consider the following exact sequence of the triple (X, A ′ , A) Now we fix the above chosen a ′ > 0 and choose another large enough a ′′ > a ′ , and enlarge the constant K in (2.2) chosen above (5.8) so that the conclusions between (5.7) and (5.8) hold when we replace (a, a ′ ) by (a ′ , a ′′ ). Then repeating the above proof with the long exact sequence of the triple (X, A ′′ , A ′ ) instead of (X, A ′ , A) in the above arguments with A ′′ = X − (a ′′ , K), similarly we Step 5. When a increases, we always meet infinitely many closed characteristics with Viterbo index −4 due to the existence of y k withî(y k ) = 0 for 1 ≤ k ≤ q 0 by Lemma 5.1. For the above chosen large enough a < a ′ , there exist only finitely many closed characteristics among {y m k | 1 ≤ k ≤ q 0 , m ≥ 1} such that their periods locate between aT and a ′ T . Therefore for the corresponding critical orbits S 1 · x t j ,k,m of F t j ,K , all of them possess Morse index d(K) − 4. Then by the equivariant version of Theorem I.4.2 as well as the discussions there on pp.78-79 of [Cha], when the condition (C) holds, i.e., q 0 ≥ 1 here, we obtain H S 1 ,d(K)−4 (A ′ , A) = H S 1 ,d(K)−4 (A ′ ∩ S(X), A ∩ S(X)) = H S 1 ,d(K)−4 (t a ′ −ǫ ,t a+ǫ ) = aT ≤T t j ,k,m ≤a ′ T 1≤k≤q 0 C S 1 ,d(K)−4 (t, S 1 · x t j ,k,m ) = aT ≤T t j ,k,m ≤a ′ T 1≤k≤q 0 C S 1 ,d(K)−4 (F t j ,K | S(X) , S 1 · x t j ,k,m ) = aT ≤T t j ,k,m ≤a ′ T where the first equality follows from (5.8), the second equality follows from (5.14), the fourth equality follows from (5.12), and for the third equality we give more explanations as follows:
· · · −→ H S 1 ,d(K)−3 (A ′ , A) i 3 * −→ H S 1 ,d(K)−3 (X, A) j 3 * −→ H S 1 ,d(K)−3 (X, A ′ ) ∂ 3 * −→ H S 1 ,d(K)−4 (A ′ , A) i 4 * −→ H S 1 ,d(K)−4 (X, A) j 4 * −→ H S 1 ,d(K)−4 (X, A ′ ) ∂ 4 * −→ H S 1 ,d(K)−5 (A ′ , A) −→ · · · .
Denote by M q (a, a ′ ) = aT ≤T t j ,k,m ≤a ′ T 1≤k≤q rank C S 1 ,q (t, S 1 · x t j ,k,m ), β q (a, a ′ ) = rank H S 1 ,q (t a ′ −ǫ ,t a+ǫ ).
Then M d(K)−3 (a, a ′ ) = M d(K)−5 (a, a ′ ) = 0 and β d(K)−3 (a, a ′ ) = β d(K)−5 (a, a ′ ) = 0 hold by (5.15) and (5.16) respectively, which together with an equivariant version of Theorem I.4.3 of [Cha] yield M d(K)−4 (a, a ′ ) = β d(K)−4 (a, a ′ ). Then the third equality in (5.22) holds.
Step 6. By the exactness of the sequence (5.21) and (5.22), we obtain H S 1 ,d(K)−4 (X, A) = H S 1 ,d(K)−4 (A ′ , A) H S 1 ,d(K)−4 (X, A ′ ) = 0.
Then, by our choice of a, a ′ and a ′′ , and replacing (X, A ′ , A) by (X, A ′′ , A ′ ) in the above arguments, similarly we obtain H S 1 ,d(K)−4 (X, A ′ ) = 0. (5.23)
Now on one hand, if j 4 * in (5.21) is a trivial homomorphism, then by the exactness of the sequence (5.21) it yields H S 1 ,d(K)−4 (X, A ′ ) = Ker(∂ 4 * ) = Im(j 4 * ) = 0, which contradicts to (5.23). Therefore j 4 * in (5.21) is a non-trivial homomorphism.
However, on the other hand, by (7.4) of [Vit2], j 4 * in (5.21) is a zero homomorphism. This contradiction completes the proof of Theorem 1.6.
Theorem 1. 3 .
3Let (M 2n+1 , ξ) be a prequantization S 1 -bundle of a closed symplectic manifold (B, ω) such that ω| π 2 (B) = 0 and c B > n/2 and, furthermore, H k (B; Q) = 0 for every odd k or c B > n. Let α be a contact form supporting ξ and assume that M and α satisfy condition(A). Then α carries at least r B geometrically distinct contractible periodic orbits. Furthermore, if there exist finitely many geometrically distinct contractible closed orbits, Then α carries at least r non−hyp B geometrically distinct contractible non-hyperbolic periodic orbits, where r non−hyp B := r B − dimH n (B; Q) and r B := χ(B) + 2 dimH n (B; Q), if n is odd, χ(B) + 4 dimH n−1 (B; Q), if n is even. (1.2) Remark 1.4. (1) The proof of Theorem 1.3 follows the proofs of Theorems 2.1 and 2.10 of [GGM] via replacing the enhanced common jump theorem of H. Duan, Y. Long and W. Wang by our Theorem 3.6 and the proof of Theorem 1.2 below. We should emphasize that under nonzero mean index condition, there are similar Morse inequalities in the setting of equivariant symplectic homology to (2.16)-(2.17) below, cf., (68) in p.221 of[HM].
negative, zero and positive subspaces. The index and nullity of u are defined by
generalized common index jump theorem for symplectic paths In [Lon2] of 1999, Y. Long established the basic normal form decomposition of symplectic matrices. Based on this result he further established the precise iteration formulae of indices of symplectic paths in [Lon3] of 2000.
every M ∈ Sp(2n), the homotopy set Ω(M ) of M in Sp(2n) is defined by Ω(M ) = {N ∈ Sp(2n) | σ(N ) ∩ U = σ(M ) ∩ U ≡ Γ and ν ω (N ) = ν ω (M ) ∀ω ∈ Γ}, where σ(M ) denotes the spectrum of M , ν ω (M ) ≡ dim C ker C (M − ωI) for ω ∈ U. The component Ω 0 (M ) of P in Sp(2n) is defined by the path connected component of Ω(M ) containing M .
Theorem 3. 2 .
2(cf. [Lon3] and Theorem 1.8.10 of [Lon4]) For any M ∈ Sp(2n), there is a path f : [0, 1] → Ω 0 (M ) such that f (0) = M and
10) andî (γ, 1) = i(γ, 1) + S + M (1) − C(M ) + θ∈(0,2π)
Claim 1 :
1For N ∈ N in Theorem 3.6 satisfying (4.12)-(4.18), we have q k=1 2m kχ (y k ) = N.(4.19)
by (4.17)-(4.18) and Lemma 2.2, we know that all y 2m k +m k 's with m ≥m + 1 have no contribution to the alternative sum 2N −n−1 p=−2N −n−1 (−1) p M p , where the Morse-type number M p is defined in (2.14). Similarly again by Lemma 2.2 and (4.12)-(4.13), ally 2m k −m k 's withm + 1 ≤ m ≤ 2m k − 1 only have contribution to 2N −n−1 p=−2N −n−1 (−1) p M p .For 1 ≤ m ≤m, by (4.16) and Lemma 2.2, we know that all y 2m k +m k 's with −n ≤ i(y m k ) for 1 ≤ k ≤ q 0 , or i(y m k ) ≤ −n − 2 for q 0 + 1 ≤ k ≤ q, have no contribution to the alternative sum2N −n−1 p=−2N −n−1 (−1) p M p . Similarly again by Lemma 2.2 and (4.14), for 1 ≤ m ≤m, all y 2m k −m k 's with −n ≤ i(y m k ) and 1 ≤ k ≤ q 0 , or i(y m k ) ≤ −n − 2 for q 0 + 1 ≤ k ≤ q, only have contribution to 2N −n−1 p=−2N −n−1 (−1) p M p .
y n with i(y 2mn n
2mn) = 2N − n − 1 and i(y 2mn n ) − i(y n ) ∈ 2Z. Thus y n and its iterations have odd Maslov-type indices. This completes the proof of Claim 3. Now for Case 2, Theorem 1.2 follows from Claim 2 and Claim 3. The proof of Theorem 1.2 is finished.
. 1 .
1(5.3), we have used E(a) = [a] + 1 for a ∈ R \ Z. Thuŝ i(y k ) = i(y k ) + 4 − r + Whenî(y k ) = 0 with 1 ≤ k ≤ q 0 , we have i(y m k ) = −4 for any m ∈ N.
8.1.7 of [Lon4], then by (5.5) we have 2 j=1 θ j π = 2 and i(y k ) = −4. Since 2 j=1 mθ j 2π = m implies that 2 j=1 [ mθ j 2π ] = m − 1, so by (5.3) we have i(y m k ) = −2m + 2 r j=1 mθ j 2π − 2 = −2m + 2(m − 1) − 2 = −4, ∀ m ≥ 1. (5.6)
H
with an equivariant version of Theorem I.4.3 of[Cha], i.e., the Morse inequality, we obtainH S 1 ,d(K)−3 (t a ′ −ǫ ,t a+ǫ ) = H S 1 ,d(K)−5 (t a ′ −ǫ ,t a+ǫ ) S 1 ,d(K)−3 (A ′ , A) = H S 1 ,d(K)−5 (A ′ , A) = 0.(5.17)
from (7.4) on p.639 of [Vit2] that the homomorphisms j 3 * in (5.18) is a zero map. Thus (5.17) and (5.18) yield H S 1 ,d(K)−3 (X, A) = Ker(j 3 * ) = Im(i 3 * ) = H S 1 ,d(K)−3 (A ′ , A) = 0. (5.19)
). A closed characteristic (τ, y) is non-degenerate if 1 is a Floquet multiplier of y of precisely algebraic multiplicity 2; hyperbolic if 1 is a double Floquet multiplier of it and all the other Floquet multipliers are not on U = {z ∈ C | |z| = 1}, i.e., the unit circle in the complex plane; elliptic if all the Floquet multipliers of y are on U. We call a Σ ∈ H st (2n) non-degenerate if all the closed characteristics on Σ, together with all of their iterations, are non-degenerate.
±2N ′ − n and ±2N ′ − n − 2 are replaced by ±2N ′ − n + 1 and ±2N ′ − n − 3, respectively, and these numbers satisfy the following relationship53)
which yields
H e
+ ≥ H e
+ − H o
+ ≥
n − 1
2
.
(4.54)
Similarly, denote by H
′ e
+ , H
′ o
+ , H
′ e
− , H
′ o
− the numbers similarly defined by (4.40)-(4.43) where
H e
± = H
′ e
∓ ,
H o
± = H
′ o
∓ .
(4.55)
Similarly to the inequality (4.46), by the same arguments above and (4.55) we can obtain
H e
− = H
′ e
+ ≥ H
′ e
+ − H
′ o
+ ≥
n − 1
2
.
(4.56)
Therefore it follows from (4.54) and (4.56) that
q ≥ H e
+ + H e
− ≥
n − 1
2
+
n − 1
2
= n − 1.
(4.57)
By the same arguments in the proof of Case 1, it follows from the definitions of H e
+ and H e
−
that these (n − 1) distinct closed geodesics are non-hyperbolic, and the Viterbo indices of them and
their iterations are even, and thus the Maslov-type indices of them and their iterations are odd.
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[]
| [
"Román Orús \nDept. d'Estructura i Constituents de la Matèria\nUniversality of Entanglement and Quantum Computation Complexity\nUniv. Barcelona\n08028BarcelonaSpain\n",
"José I Latorre \nDept. d'Estructura i Constituents de la Matèria\nUniversality of Entanglement and Quantum Computation Complexity\nUniv. Barcelona\n08028BarcelonaSpain\n"
]
| [
"Dept. d'Estructura i Constituents de la Matèria\nUniversality of Entanglement and Quantum Computation Complexity\nUniv. Barcelona\n08028BarcelonaSpain",
"Dept. d'Estructura i Constituents de la Matèria\nUniversality of Entanglement and Quantum Computation Complexity\nUniv. Barcelona\n08028BarcelonaSpain"
]
| []
| We study the universality of scaling of entanglement in Shor's factoring algorithm and in adiabatic quantum algorithms across a quantum phase transition for both the NP-complete Exact Cover problem as well as the Grover's problem. The analytic result for Shor's algorithm shows a linear scaling of the entropy in terms of the number of qubits, therefore difficulting the possibility of an efficient classical simulation protocol. A similar result is obtained numerically for the quantum adiabatic evolution Exact Cover algorithm, which also shows universality of the quantum phase transition the system evolves nearby. On the other hand, entanglement in Grover's adiabatic algorithm remains a bounded quantity even at the critical point. A classification of scaling of entanglement appears as a natural grading of the computational complexity of simulating quantum phase transitions. | 10.1103/physreva.69.052308 | [
"https://arxiv.org/pdf/quant-ph/0311017v3.pdf"
]
| 14,552,768 | quant-ph/0311017 | 11104a948aee4d0d9170f69423dc854cd79f92a8 |
2 Apr 2004 March 10, 2021
Román Orús
Dept. d'Estructura i Constituents de la Matèria
Universality of Entanglement and Quantum Computation Complexity
Univ. Barcelona
08028BarcelonaSpain
José I Latorre
Dept. d'Estructura i Constituents de la Matèria
Universality of Entanglement and Quantum Computation Complexity
Univ. Barcelona
08028BarcelonaSpain
2 Apr 2004 March 10, 2021numbers: 0367-a0365Ud0367Hk
We study the universality of scaling of entanglement in Shor's factoring algorithm and in adiabatic quantum algorithms across a quantum phase transition for both the NP-complete Exact Cover problem as well as the Grover's problem. The analytic result for Shor's algorithm shows a linear scaling of the entropy in terms of the number of qubits, therefore difficulting the possibility of an efficient classical simulation protocol. A similar result is obtained numerically for the quantum adiabatic evolution Exact Cover algorithm, which also shows universality of the quantum phase transition the system evolves nearby. On the other hand, entanglement in Grover's adiabatic algorithm remains a bounded quantity even at the critical point. A classification of scaling of entanglement appears as a natural grading of the computational complexity of simulating quantum phase transitions.
Introduction
One of the main theoretical challenges in quantum computation theory is quantum algorithm design. Some attempts to uncover underlying principles common to all known efficient quantum algorithms have already been explored though not definite and satisfactory answer has been found yet. On the one hand, it has been seen that majorization theory seems to play an important role in the efficiency of quantum algorithms [1][2][3]. All known efficient quantum algorithms verify a step by step majorization of the probability distribution associated to the quantum register in the measurement basis. Therefore, efficient quantum algorithms drive the system towards the final solution by carefully reordering probability amplitudes in such a way that a majorization arrow is always present. On the other hand, the most relevant ingredient is likely the role entanglement plays in quantum computational speedup. Regarding this topic several results have recently been found [4][5][6][7][8][9] which suggest that entanglement is at the heart of the power of quantum computers.
An important result has been obtained by Vidal [8], who proved that large entanglement of the quantum register is a necessary condition for exponential speedup in quantum computation. To be concrete, a quantum register such that the maximum Schmidt number of any bipartition is bounded at most by a polynomial in the size of the system can be simulated efficiently by classical means. The figure of merit χ proposed in [8] is the maximum Schmidt number of any bi-partitioning of the quantum state or, in other words, the maximum rank of the reduced density matrices for any possible splitting. It can be proved that χ ≥ 2 E(ρ) , where the Von Neumann entropy E(ρ) refers to the reduced density matrix of any of the two partitions. If χ = O(poly(n)) at every step of the computation in a quantum algorithm, then it can be efficiently classically simulated. Exponential speed-up over classical computation is only possible if at some step along the computation χ ∼ exp(n a ), or E(ρ) ∼ n b , being a and b positive constants. In order to exponentially accelerate the performance of classical computers any quantum algorithm must necessarily create an exponentially large amount of χ at some point.
Another topic of intense research concerns the behavior of entanglement in systems undergoing a quantum phase transition [10]. Quantum correlations in critical systems have been analyzed in many situations and using a wide range of entanglement measurements [9,[11][12][13][14][15][16][17][18]. In particular, it has been noted [13,14,16,17] that some of these measurements have important connections to well-known results arising from conformal field theory [19][20][21][22]. More generally, when a splitting of a d-dimensional spin system is made, the Von Neumann entropy for the reduced density matrix of one of the subsystems E(ρ) = −tr(ρ log 2 ρ) at the critical point should display a universal leading scaling behavior determined by the area of the region partitioning the whole system. This result depends on the connectivity of the Hamiltonian and applies as is to theories with a Gaussian continuum limit. For example, when separating the system in the interior and the exterior of a sphere of radius R and assuming an ultraviolet cutoff x 0 , the entropy of e.g. the interior
is E = c 1 R x 0 d−1(1)
where c 1 corresponds to a known heat-kernel coefficient [21]. In terms of the number of spins present in the system, this leading universal scaling behavior can be written as
E ∼ n d−1 d(2)
(which reduces to a logarithmic law for d = 1). This explicit dependence of entanglement with dimensionality throws new light into some well established results from quantum computation. A similar situation is present in quantum adiabatic algorithms, initially introduced by Farhi et. al. [23], where the Hamiltonian of the system depends on a control parameter s which in turn has a given time dependence. The Hamiltonians related to adiabatic quantum computation for solving some NP-complete problems (such as 3-SAT or Exact Cover) can be directly mapped to interactive non-local spin systems, and therefore we can extend the study of entanglement to include this kind of Hamiltonians. This point of view has the additional interest of being directly connected to the possibility of efficient classical simulations of the quantum algorithm, by means of the protocol proposed in ref. [8].
In this paper we analyze the scaling of the entropy of entanglement in several quantum algorithms. More concretely, we focus on Shor's quantum factoring algorithm [24] and on a quantum algorithm by adiabatic evolution solving the NP-complete problem Exact Cover [25], finding for both of them evidence of a quantum exponential speedup with linear scaling of quantum correlations, which difficults the possibility of an efficient classical simulation. We furthermore study the adiabatic implementation of Grover's quantum search algorithm [29][30][31], in which entanglement is a bounded quantity even at the critical point, regardless of the size of the system.
We have structured the paper as follows: in Sec. 2 we analytically address the study of quantum entanglement present in Shor's factoring algorithm. We consider the problem of universal scaling of entanglement at the critical point of an adiabatic quantum algorithm solving the NP-complete problem Exact Cover in Sec. 3, where we present numerical results for systems up to 20 qubits. In Sec. 4 we focus on the adiabatic implementation of Grover's quantum searching algorithm, and derive analytical expressions for the study of entanglement in the system. Finally, in Sec. 5 we collect the conclusions of our work.
Scaling of entanglement in Shor's factoring algorithm
It is believed that the reason why Shor's quantum algorithm for factorization [24] beats so clearly its classical rivals is rooted in the clever use it makes of quantum entanglement. Several attempts have been made in order to understand the behavior of the quantum correlations present along the computation [6,7]. In our case, we will concentrate in the study of the scaling behavior for the entanglement entropy of the system. We shall first remember both Shor's original [24] and phase-estimation [32] proposals of the factoring algorithm and afterwards we shall move to the analytical analysis of their quantum correlations.
The factoring algorithm
The interested reader is addressed to [24,[32][33][34] for precise details. Given an odd integer N to factorize, we pick up a random number a ∈ [1, N ]. We make the assumption that a and N are Co-primes (otherwise the greatest common divisor of a and N would already be a non-trivial factor of N ). There exists a smaller integer r ∈ [1, N ], called the order of the modular exponentiation a x mod N , such that a r mod N = 1. Let us assume that the a we have chosen is such that r is even and a r/2 mod N = −1, which happens with very high probability (bigger than or equal to 1/(2 log 2 N )). This is the case of interest because then the greatest common divisor of N and a r/2 ± 1 is a non-trivial factor of N . Therefore, the factoring problem has been reduced to the order-finding problem of the modular exponentiation function a x mod N , and it is at this point where quantum mechanics comes at work. The procedure can be casted in two different ways:
Shor's proposal for order-finding
We make use of two quantum registers: a source register of k qubits (such that 2 k ∈ [N 2 , 2N 2 ]) and a target register of n = ⌈log 2 N ⌉ qubits. The performance of the quantum algorithm is shown in Fig. 1, where we are making use of the Hadamard gate initially acting over the k qubits of the source, the unitary implementation of the modular exponentiation function
U f |q |x = |q |(x + a q ) mod N(3)
(where |q and |x respectively belong to the source and target registers), and the Quantum Fourier Transform operator
QF T |q = 1 2 k/2 2 k −1 m=0 e 2πiqm/2 k |m .(4)
All these operations can be efficiently implemented by means of one and twoqubit gates. Finally, a suitable classical treatment of the final measurement of this quantum algorithm provides us with r in few steps, and therefore the prime factorization of N in a time O((log 2 N ) 3 ). 00 11 00 00 11 11 0 k 0 n 00 00 00 11 11 11 00 00 00 11 11 11 H U U f QFT (n) (k) Figure 1: quantum circuit for the order-finding algorithm for the modular exponentiation function.
Phase-estimation proposal for order-finding
We refer the interested reader to [32] for more details. The quantum circuit is similar to the one shown in the previous section but slightly modified, as is shown in Fig. 2. The unitary operator V f to which the phase-estimation procedure is applied is defined as
V f |x = |(a x) mod N(5)
(appreciate the difference between expressions (5) and (3)), being diagonalized by eigenvectors
|v s = 1 r 1/2 r−1 p=0 e −2πisp/r |a p mod N(6)
such that V f |v s = e 2πis/r |v s ,
and satisfying the relation 1
r 1/2 r−1 s=0 |v s = |1
. The operator is applied over the target register being controlled on the qubits of the source in such a way that
Λ(V f )|j |x = |j V j f |x ,(8)
where by Λ(V f ) we understand the full controlled operation acting over the whole system, which can be efficiently implemented in terms of one and two-qubit gates. As in the previous case, the information provided by a final measurement of the quantum computer enables us to get the factors of N in a time O((log 2 N) 3 ).
Analytical results
We choose to study the amount of entanglement between the source and the target register in the two proposed quantum circuits, right after the modular exponentiation operation U f (Fig. 1) or the controlled V f operation (Fig. 2), and before the Quantum Fourier Transform in both cases. At this step of the computation, the pure quantum state of the quantum computer is easily seen to be exactly the same for both quantum circuits, and is given by
|ψ = 1 2 k/2 2 k −1 q=0 |q |a q mod N ,(9)
and therefore the density matrix of the whole system is
|ψ ψ| = 1 2 k 2 k −1 q,q ′ =0 |q q ′ | |a q mod N a q ′ mod N | .(10)
Tracing out the quantum bits corresponding to the source, we get the density matrix of the target register, which reads
ρ target = tr source (|ψ ψ|) = 1 2 k 2 k −1 p,q,q ′ =0 p|q q ′ |p |a q modN a q ′ modN | ,(11)
that is,
ρ target = 1 2 k 2 k −1 p=0 |a p mod N a p mod N | ∼ 1 r r−1 p=0 |a p modN a p modN | .(12)
The last step comes from the fact that a r mod N = 1, being r ∈ [1, N ] the order of the modular exponentiation. If 2 k were a multiple of r there would not be any approximation and the last equation would be exact. This is not necessarily the case, but the corrections to this expression go like O(1/2 k ), thus being exponentially small in the size of the system. It follows from expression (12) that the rank of the reduced density matrix of the target register at this point of the computation is rank(ρ target ) = r .
Because r ∈ [1, N ], this rank is usually O(N ). If this were not the case, for example if r were O(log 2 N ), then the order-finding problem could be efficiently solved by a classical naive algorithm and it would not be considered as classically hard. Because N is exponentially big in the number of qubits, we have found a particular bipartition of the system (namely, the bipartition between the source register and the target register) and a step in the quantum algorithm in which the entanglement, as measured by the rank of the reduced density matrix of one of the subsystems, is exponentially big. This implies in turn that Shor's quantum factoring algorithm can not be efficiently classically simulated by any protocol in ref. [8] owing to the fact that at this step χ = O(N ), therefore constituting an inherent exponential quantum speed-up based on an exponentially big amount of entanglement. It is worth noticing that the purpose of the entanglement between the two registers consists on leaving the source in the right periodic state to be processed by the Quantum Fourier Transform. Measuring the register right after the entangling gate disentangles the two registers while leaving the source in a periodic state, and this effect can only be accomplished by previously entangling source and target. These conclusions apply both to Shor's original proposal (circuit of Fig. 1) and to the phase-estimation version (circuit of Fig. 2). The behavior of the rank of the system involves that the entropy of entanglement of the reduced density matrix at this point will mainly scale linearly with the number of qubits, E ∼ log 2 r ∼ log 2 N ∼ n, which is the hardest of all the possible scaling laws. We will find again this strong behavior for the entropy in Sec. 3.
Scaling of entanglement in an NP-complete problem
We now turn to analyze how entanglement scales for a quantum algorithm based on adiabatic evolution [23], designed to solve the NP-complete problem Exact Cover [25]. We first briefly review the proposal and, then, we consider the study of the properties of the system, in particular the behavior of the entanglement entropy for a given bipartition of the ground state.
Adiabatic quantum computation
The adiabatic model of quantum computation deals with the problem of finding the ground state of a given system represented by its Hamiltonian. Many relevant computational problems (such as 3-SAT) can be mapped to this situation. The method is briefly summarized as follows: we start from a time dependent Hamiltonian of the form
H(s(t)) = (1 − s(t))H 0 + s(t)H p ,(14)
where H 0 and H p are the initial and problem Hamiltonian respectively, and s(t) is a time-dependent function satisfying the boundary conditions s(0) = 0 and s(T ) = 1 for a given T . The desired solution to a certain problem is codified in the ground state of H p . The gap between the ground and the first excited state of the instantaneous Hamiltonian at time t will be called g(t). Let us define g min as the global minimum of g(t) for t in the interval [0, T ]. If at time T the ground state is given by the state |E 0 ; T , the adiabatic theorem states that if we prepare the system in its ground state at t = 0 (which is assumed to be easy to prepare) and let it evolve under this Hamiltonian, then
| E 0 ; T |ψ(T ) | 2 ≥ 1 − ǫ 2 (15) provided that max| dH 1,0 dt | g 2 min ≤ ǫ(16)
where H 1,0 is the Hamiltonian matrix element between the ground and first excited state, ǫ << 1, and the maximization is taken over the whole time interval [0, T ]. Because the problem Hamiltonian codifies the solution to the problem in its ground state, we get the desired solution with high probability after a time T . A closer look to the adiabatic theorem tells us that T dramatically depends on the scaling of the inverse of g 2 min with the size of the system. More concretely, if the gap is only polynomially small in the number of qubits (that is to say, it scales as O(1/poly(n)), the computational time is O(poly(n)), whereas if the gap is exponentially small (O(2 −n )) the algorithm makes use of an exponentially big time to reach the solution.
The explicit functional dependence of the parameter s(t) on time can be very diverse. The point of view we adopt in the present paper is such that this time dependence is not taken into account, as we study the properties of the system as a function of s, which will be understood as the Hamiltonian parameter. We will in particular analyze the entanglement properties of the ground state of H(s), as adiabatic quantum computation assumes that the quantum state remains always close to the instantaneous ground state of the Hamiltonian all along the computation. Note that we are dealing with a system which is suitable to undergo a quantum phase transition at some critical value of the Hamiltonian parameter, and therefore we expect to achieve the biggest quantum correlations at this point. The question is how this big quantum correlations scale with the size of the system when dealing with interesting problems. This is the starting point for the next two sections.
Exact Cover
The NP-complete problem Exact Cover is a particular case of the 3-SAT problem, and is defined as follows: given the n boolean variables
{x i } i=1,...n , x i = 0, 1 ∀ i,
where i is regarded as the bit index, we define a clause of Exact Cover involving the three qubits i, j and k (say, clause "C") by the equation
x i + x j + x k = 1.
There are only three assignments of the set of variables {x i , x j , x k } that satisfy this equation, namely, {1, 0, 0}, {0, 1, 0} and {0, 0, 1}. The clause can be more specifically expressed in terms of a boolean function in Conjunctive Normal Form (CNF) as
φ C (x i , x j , x k ) = (x i ∨ x j ∨ x k ) ∧ (¬x i ∨ ¬x j ∨ ¬x k ) ∧ (¬x i ∨ ¬x j ∨ x k ) ∧(¬x i ∨ x j ∨ ¬x k ) ∧ (x i ∨ ¬x j ∨ ¬x k ) ,(17)
so φ C (x i , x j , x k ) = 1 as long as the clause is properly satisfied. An instance of Exact Cover is a collection of clauses which involves different groups of three qubits. The problem is to find a string of bits {x 1 , x 2 . . . , x n } which satisfies all the clauses. This problem can be mapped into finding the ground state of a Hamiltonian H p in the following way: given a clause C define the Hamiltonian associated to this clause as
H C = 1 2 (1 + σ z i ) 1 2 (1 + σ z j ) 1 2 (1 + σ z k ) + 1 2 (1 − σ z i ) 1 2 (1 − σ z j ) 1 2 (1 − σ z k ) + 1 2 (1 − σ z i ) 1 2 (1 − σ z j ) 1 2 (1 + σ z k ) + 1 2 (1 − σ z i ) 1 2 (1 + σ z j ) 1 2 (1 − σ z k ) + 1 2 (1 + σ z i ) 1 2 (1 − σ z j ) 1 2 (1 − σ z k ) ,(18)
where we have defined σ z |0 = |0 , σ z |1 = −|1 . Note the parallelism between equations (17) and (18). The quantum states of the computational basis that are eigenstates of H C with zero eigenvalue (ground states) are the ones that correspond to the bit string which satisfies C, whereas the rest of the computational states are penalized with an energy equal to one. Now, we construct the problem Hamiltonian as the sum of all the Hamiltonians corresponding to all the clauses in our particular instance, that is to say,
H p = C ∈ instance H C ,(19)
so the ground state of this Hamiltonian corresponds to the quantum state whose bit string satisfies all the clauses. We have reduced the original problem stated in terms of boolean logic to the hard task of finding the ground state of a two and three body interactive spin Hamiltonian with local magnetic fields. Observe that the couplings depend on the particular instance we are dealing with, and that the spin system has not an a priori well defined dimensionality neither a well defined lattice topology, in contrast with some usual simple spin models. We now define our s-dependent Hamiltonian H(s) as a linear interpolation between an initial Hamiltonian H 0 and H p :
H(s) = (1 − s)H 0 + sH p(20)
where we take the initial Hamiltonian H 0 to be basically a magnetic field in the x direction, more concretely,
H 0 = n i=1 d i 2 (1 − σ x i ) ,(21)
where d i is the number of clauses in which qubit i appears, and σ x |+ = |+ , with |+ = 1 √ 2 (|0 + |1 ), so the ground state of H 0 is an equal superposition of all the possible computational states. Observe that H(s) is, apart from a constant factor, a sum of terms involving local magnetic fields in the x and z direction, together with two and three-body interaction coupling terms in the z component. This system is suitable to undergo a quantum phase transition (in the limit of infinite n) as s is shifted from 0 to 1. The study of this phenomena is the aim of the following section.
Numerical results up to 20 qubits
We have randomly generated instances for Exact Cover with only one possible satisfying assignment and have constructed the corresponding problem Hamiltonians. Instances are produced by adding clauses at random until there is exactly one satisfying assignment, starting over if we end up with no satisfying assignments. According to [25], these are believed to be the most difficult instances for the adiabatic algorithm. Our analysis proceeds as follows:
Appearance of a quantum phase transition We have generated 300 Exact Cover instances (300 random Hamiltonians with a non-degenerated ground state) and have calculated the ground state for 10, 12 and 14 qubits for different values of the parameter s in steps of 0.01. We then consider a particular bipartition of the system into two blocks of n/2 qubits, namely, the first n/2 qubits versus the rest, and have calculated the entanglement entropy between the two blocks. For each of the randomly generated Hamiltonians we observe a peak in the entanglement entropy around a critical value of the parameter s c ∼ 0.7. We have averaged the obtained curves over the 300 instances and have obtained the plot from Fig. 3. Figure 3: evolution of the entanglement entropy between the two blocks of size n/2 when a bipartition of the system is made, on average over 300 different instances with one satisfying assignment. A peak in the correlations appears for s c ∼ 0.7 in the three cases.
The point at which the entropy of entanglement reaches its maximum value is identified as the one corresponding to the critical point of a quantum phase transition in the system (in the limit of infinite size). This interpretation is reinforced by the observation of the typical energy eigenvalues of the system. For a typical instance of 10 qubits we observe that the energy gap between the ground state and the first excited state reaches a minimum precisely for a value of the parameter s c ∼ 0.7 (see Fig. 4).
We observe from Fig. 3 that the peak in the entropy is highly asymmetric with respect to the parameter s. A detailed study of the way this peak seems to diverge near the critical region seems to indicate that the growth of entanglement is slower at the beginning of the evolution and fits remarkably well a curve of the type E ∼ log | log (s − s c )|, whereas the falling down of the peak is better parameterized by a power law E ∼ |s − s c | −α with α ∼ 2.3, being α a certain critical exponent. These laws governing the critical region fit better and better the data as the number of qubits is increased.
Analysis of different bi-partitions of the system Explicit numerical analysis for 10 qubits tells us that all possible bi-partitions for each one of the instances produce entropies at the critical point of the same order of magnitude -as expected from the non-locality of the interactions-. This is represented in Fig. 5, where we plot the minimum and maximum entanglement obtained from all the possible partitions of the system for each one of the generated instances (points are sorted such that the minimum entropy monotonically increases).
Similar conclusions derive from the data plotted in Fig. 6, where we have considered again the same quantities but looking at 64 partitions of the ground state for 10 different instances of 16 qubits. According to these results we restrict ourselves in what follows to the analysis of a particular bipartition of the system, namely the first n/2 qubits versus the rest.
It is worth emphasizing that the existence of a single partition with exponentially large entanglement makes the algorithm not amenable to classical simulation. The above result is stronger and shows that essentially all partitions are highly entangled. The system is definitely hard to simulate by classical means. Figure 6: minimum and maximum entropy over 64 bi-partitions of a 16-qubit system for 10 randomly generated instances of Exact Cover. Instances are sorted such that the minimum entanglement monotonically increases.
Scaling laws for the minimum energy gap and the entanglement entropy
To characterize the finite-size behavior of the quantum phase transition, we have generated 300 random instances of Exact Cover with only one satisfying assignment from 6 to 20 qubits, and studied the maximum Von Neumann entropy for a bipartition of the system as well as the minimum gap, both in the worst case and in the mean case over all the randomly generated instances. We must point out that the scaling laws found in this section are limited to the small systems we can handle with in our computers. Increasing the number of qubits may lead to corrections in the numerical results, which should be of particular importance for a more precise time-complexity analysis of the adiabatic algorithm. Fig. 7 represents the behavior of the gap in the worst and mean cases. From Fig. 8 it is noticed that the gap seems to obey a scaling law of the style O(1/n), being n the number of qubits, which would assure a polynomial-time quantum computation. This law is in agreement with the results in [25], and are in concordance with the idea that the energy gap typically vanishes as the inverse of the volume in condensed matter systems (here the volume is the number of qubits). Error bars in the two plots give 95 per cent of confidence level in the numerically calculated mean.
We have considered as well the scaling behavior of the entanglement entropy for an equally sized bipartition of the system also in the worst and in the mean case. The obtained data from our simulations are plotted in Fig. 9 -where error bars give 95 per cent of confidence level in the mean-and seem to be in agreement with a strongly linear scaling of entanglement as a function of the size of the number of qubits. More concretely, a numerical linear fit for the mean entanglement entropy gives us the law E ∼ .1 n. Observe that the entropy of entanglement does not get saturated in its maximum allowed value (which would be E = n/2 for n qubits), so we can say that only a twenty percent of all the possible potential available entanglement appears in the quantum algorithm. Linearity in the scaling law would imply that this quantum computation by adiabatic evolution, after a suitable discretization of the continuous time dependence, could not be classically simulated by the protocol of ref. [8]. Given that the scaling of the gap seems to indicate that the quantum computation runs in a polynomial time in the size of the system, our conclusion is that apparently we are in front of an exponentially fast quantum computation that seems extremely difficult (if not impossible) to be efficiently simulated by classical means. This could be an inherent quantum mechanical exponential speedup that can be understood in terms of the linear scaling of the entropy of entanglement. Note also the parallelism with the behavior of the entanglement found in Shor's algorithm in Sec. 2. As a remark, our numerical analysis shows that the quantum algorithm is difficult to be simulated classically in an efficient way, which does not necessarily imply that the quantum computer runs exponentially faster than the classical one, as our time-complexity analysis is limited to 20 qubits. Figure 9: scaling of the entanglement entropy for an equally sized bipartition of the system, both in the worst case and in the mean case over all the randomly generated instances. Error bars give 95 per cent of confidence level for the mean. The data are consistent with a linear scaling.
The linear behavior for the entropy with respect to the size of the system could in principle be expected according to the following qualitative reasoning. Naively, the entropy was expected to scale as the area of the boundary of the splitting, according to some considerations taken from conformal field theory (see [13,14,[19][20][21]). This area-law is in some sense natural: because the entropy value is the same for both density matrices arising from the two subsystems, it can only be a function of their shared properties, and these are geometrically encoded in the area of the common boundary. For a system of n qubits, this implies a scaling law for the entropy like E ∼ n (d−1)/d (which reduces to a logarithm for d = 1). Our system does not have a well defined dimensionality, but owing to the fact that there are many random two and three body interactions, the effective (fractal) dimensionality of the system should be very large. Therefore, we expect a linear (or almost linear) scaling, which is what we have numerically obtained. The data seems to indicate that such an effective dimensionality is around d ∼ n, thus diverging as n goes to infinity.
It is possible to compare our apparently linear scaling of the mean entropy of entanglement with the known results obtained by averaging this quantity over the entire manifold of n-qubit pure states, with respect to the natural Fubini-Study measure. According to the results conjectured by Page [26] and later proved in [27], the average entropy for an equally-sized bipartition of a random n-qubit pure state in the large n limit can be approximated by E ∼ (n/2) − 1/(2 ln 2) (in our notation), therefore displaying as well a linear scaling law (but different from ours). In fact, this is an indicator that most of the n-qubit pure states are highly entangled, and that adiabatic quantum computation naturally brings the system close to these highly entangled regions of the pure state manifold (more information about the average entanglement of an n-qubit system can be found in [28]).
The entanglement-gap plane
The plots in Fig. 10 and Fig. 11 show the behavior of the peak in the entanglement versus the gap, both again in the average and the worst case for all the generated instances. Clearly, as the gap becomes smaller the production of entanglement in the algorithm increases. A compression of the energy levels correlates with high quantum correlations in the system.
Convergence of the critical points
The critical point s c seems to be bounded by the values of s associated to the minimum gap and to the maximum entropy. Actually, the critical point corresponding to the minimum size of the energy gap is systematically slightly bigger than the critical point corresponding to the peak in the entropy. By increasing the size of the system these two points converge towards the same value, which would correspond to the true critical point of a system of infinite size. This effect is neatly observed in Fig. 12, which displays the values of s associated to the mean critical points both for the gap and for the entropy as a function of n. Figure 12: mean critical point for the energy gap and for the entropy. Error bars give 95 per cent of confidence level for the means. Note that they tend to approach as the size of the system is increased.
Universality
All the above results suggest that the system comes close to a quantum phase transition. The characterization we have presented based on the study of averages over instances reconstructs its universal behavior. Results do not depend on particular microscopic details of the Hamiltonian, such as the interactions shared by the spins or the strength of local magnetic fields. Any adiabatic algorithm solving a k-sat problem and built in the same way we have done for Exact Cover should display on average exactly the same properties we have found regardless of the value of k, which follows from universality (the case k = 2, though not being NPcomplete, should display also this property as its hamiltonian would consist as well of local interactions in a big-dimensional lattice; k = 1 is a particular case, as its hamiltonian is non-interacting). Linear scaling of entanglement should therefore be a universal law for these kind of quantum algorithms. The specific coefficients of the scaling law for the entropy should be a function only of the connectivity of the system, that is on the type of clauses defining the instances.
We have explicitly checked this assertion by numerical simulations for clauses of Exact Cover but involving 4 qubits (x i + x j + x k + x l = 1), which is a particular case of 4-sat. In Fig. 13 we plot the behavior of the entropy of entanglement for a 10-qubit system for these type of clauses and compare it to the same quantity calculated previously for the clauses involving 3 qubits (the usual Exact Cover Hamiltonian). We observe again the appearance of a peak in the entropy, which means that the system is evolving close to a quantum phase transition. Figure 15: scaling of the entanglement entropy for an equally sized bipartition of the system, both in the worst and in the mean cases over all the randomly generated instances of clauses involving 4 qubits, up to n = 16. Error bars give 95 per cent of confidence level for the mean. The data are consistent with a linear scaling. Figures 14 and 15 respectively show the scaling of the energy gap in the mean and worst case and the scaling of the peak in the entropy in the mean and worst case as well, up to 16 qubits. Error bars give again 95 per cent of confidence level for the means. The behavior is similar to the one already found for the instances of Exact Cover involving 3 qubits (figures 8 and 9), which supports the idea of the universality of the results. The minimum energy gap seems to scale in this case as ∼ 1 n 3 (n being the number of qubits), which would guarantee again a polynomial-time quantum adiabatic evolution.
Scaling of entanglement in adiabatic Grover's algorithm
Let us now consider the adiabatic implementation of Grover's quantum searching algorithm in terms of a Hamiltonian evolution [29][30][31] and study its properties as a function of the number of qubits and the parameter s. For this problem, it is possible to compute all the results analytically, so we shall get a closed expression for the scaling of entanglement. As a side remark, it is worth noting that the treatment made in [8] is not valid for oracular problems as it is assumed that all quantum gates are known in advanced. Independently of this issue, we shall see that the system remains little entangled between calls to the oracle.
Implementation of Grover's searching algorithm with adiabatic quantum computation
Grover's searching algorithm [29] can be implemented in adiabatic quantum computation by means of the s-dependent Hamiltonian
H(s) = (1 − s)(I − |s s|) + s(I − |x 0 x 0 |) ,(22)
where |s ≡ 1 2 n/2 2 n −1 x=0 |x , n is the number of qubits, and |x 0 is the marked state. The computation takes the quantum state from an equal superposition of all computational states directly to the state |x 0 , as long as the evolution remains adiabatic. The time the algorithm takes to succeed depends on how we choose the parameterization of s in terms of time. Our aim is to compute the amount of entanglement present in the register and need not deal with the explicit dependence of the parameter s on time and its consequences (see [30,31] for further information about this topic).
It is straightforward to check that the Hamiltonian (22) has its minimum gap between the ground and first excited states at s = 0.5, which goes to zero exponentially fast as the number of qubits in the system is increased. Therefore, this Hamiltonian apparently seems to undergo a quantum phase transition in the limit of infinite size at s = 0.5. Quantum correlations approach their maximum for this value of s (for more on Grover's problem as a quantum phase transition, see [37]).
Analytical results
It can be seen (see for example [36]) that the ground state energy of the Hamiltonian given in equation (22) corresponds to the expression
E − (s) = 1 2 1 − (1 − 2s) 2 + 4 2 n s(1 − s) ,(23)
being s is the Hamiltonian parameter. The corresponding normalized ground state eigenvector is given by
|E − (s) = a|x 0 + b x =x 0 |x ,(24)
where we have defined the quantities
a ≡ α b b 2 ≡ 1 2 n − 1 + α 2 α ≡ 2 n − 1 2 n − 1 − 2 n 1−s E − (s) .(25)
In all the forthcoming analysis we will assume that the marked state corresponds to |x 0 = |0 , which will not alter our results. The corresponding density matrix for the ground state of the whole system of n qubits is then given by
ρ n = b 2 (α 2 − 2α + 1)|0 0| + b 2 |φ φ| + b 2 (α − 1)(|φ 0| + |0 φ|) ,(26)
where we have defined |φ as the the unnormalized sum of all the computational quantum states (including the marked one), |φ ≡ 2 n −1 x=0 |x . Taking the partial trace over half of the qubits, regardless of what n/2 qubits we choose, we find the reduced density matrix
ρ n/2 = b 2 (α 2 − 2α + 1)|0 ′ 0 ′ | + 2 n/2 b 2 |φ ′ φ ′ | + b 2 (α − 1)(|φ ′ 0 ′ | + |0 ′ φ ′ |) ,(27)
where we understand that |0 ′ is the remaining marked state for the subsystem of n/2 qubits and |φ ′ ≡ 2 n/2 −1 x=0 |x is the remaining unnormalized equally superposition of all the possible computational states for the subsystem. Defining the quantities A ≡
α 2 + 2 n/2 − 1 α 2 + 2 n − 1 B ≡ α + 2 n/2 − 1 α 2 + 2 n − 1 C ≡ 2 n/2 α 2 + 2 n − 1(28)
(note that A + (2 n/2 − 1)C = 1), the density operator for the reduced system of n/2 qubits can be expressed in matrix notation as
ρ n/2 = A B · · · B B C · · · C . . . . . . . . . . . . B C · · · C ,(29)
where its dimensions are 2 n/2 × 2 n/2 . We clearly see that the density matrix has rank equal to 2. Therefore, because rank(ρ) ≥ 2 E(ρ) ∀ρ (where E(ρ) is the Von Neumann entropy of the density matrix ρ) we conclude that E(ρ n/2 ), which corresponds to our entanglement measure between the two blocks of qubits, is always ≤ 1. This holds true even for non symmetric bi-partitions of the complete system. Regardless of the number of qubits, entanglement in Grover's adiabatic algorithm is always a bounded quantity for any s, in contrast with the results obtained in the previous sections for Shor's factoring algorithm and for the Exact Cover problem. Grover's adiabatic quantum algorithm essentially makes use of very little entanglement, but even this bounded quantity of quantum correlations is enough to give a square root speedup. We have explicitly calculated the Von Neumann entropy for ρ n/2 . Because the rank of the reduced density matrix is two, there are only two non-vanishing eigenvalues that contribute in the calculation which are
λ ± = 1 2 1 ± 1 − 4(2 n/2 − 1)(AC − B 2 ) .(30)
We analyze the limit n → ∞ for s = 0.5 and s = 0.5 separately:
(i) s = 0.5
In the limit of very high n we can approximate the ground state energy given in equation (23) by
E − (s) ∼ 1 2 1 − 1 − 4s(1 − s) .(31)
Therefore, the quantity
α ∼ 1 1 − E − (s) 1−s(32)
diverges at s = 0.5, which implies that this limit can not be correct for that value of the parameter. The closer we are to s = 0.5, the bigger is α. In this limit we find that
A ∼ α 2 + 2 n/2 α 2 + 2 n(33)
B ∼ α + 2 n/2 α 2 + 2 n (34)
C ∼ 2 n/2 α 2 + 2 n ,(35)
where all these quantities tend to zero as n → ∞. It is important to note that the convergence of the limit depends on the value of α or, in other words, how close to s = 0.5 we are. The closer we are to s = 0.5, the slower is the convergence, and therefore any quantity depending on these parameters (such as the entropy) will converge slower to its assimptotical value. For the eigenvalues of the reduced density matrix we then find that when n → ∞
λ ± → 1 2 (1 ± 1) ,(36)
so λ + ∼ 1 and λ − ∼ 0, and therefore the assimptotical entropy is
E(s = 0.5, n → ∞) = −λ + log 2 λ + − λ − log 2 λ − = 0 .(37)
The convergence of this quantity is slower as we move towards s = 0.5.
(ii) s = 0.5
We begin our analysis by evaluating the quantities at s = 0.5 and then taking the limit of big size of the system. We have that α(s = 0.5) = 2 n −1 2 n/2 −1 ∼ 2 n/2 . From here it is easy to get the approximations Figure 16: Von Neumann entropy for the reduced system as a function of s for 10, 12 and 14 qubits. As the size of the system increases the entropy tends to zero at all points, except at s = 0.5 in which tends to 1.
A ∼ 1 2 B ∼ 1 2 n/2 C ∼ 1 2 n/2+1 ,(38)
and therefore λ ± ∼ 1 2 1 ± 1 − 4 2 n/2 1 4
1 2 n/2 − 1 2 n = 1 2 ± 1 2 n/4 ,(39)
so λ ± → 1 2 and E(s = 0.5, n → ∞) = 1. According with (39) we can evaluate the finite size corrections to this behavior and find the scaling of the entropy with the size of the system for very large n. The final result for the entropy at the critical point reads
E(s = 0.5, n >>) ∼ 1 − 4 ln 2 2 −n/2 .(40)
Note that the entropy remains bounded and tends to 1 for s = 0.5 as an square root in the exponential of the size of the system, which is the typical factor in Grover's quantum algorithm.
We have represented the evolution of the entanglement entropy as a function of s for different sizes of the system in Fig. 16 and have plotted in Fig. 17 the maximum value of the entropy along the computation as a function of the size of the system according to the expression given in equation (40). We can now compare the two plots with Fig. 3 and Fig. 9 in the previous section. The behavior for the entropy in Grover's adiabatic algorithm is dramatically different to the one observed in the NP-complete problem. Entanglement gets saturated in Grover's adiabatic algorithm even at the critical point, which is reminiscent of short ranged quantum correlations in quantum spin chains 1 . Let us note that, in the limit of infinite size, the quantum state in Grover's algorithm is separable with respect to any bipartition of the system (and therefore not entangled, as it is a pure state) for any s except for s = 0.5. All the entanglement along the algorithm is concentrated at this point, but this entanglement is still a bounded quantity and actually equal to 1. Consequently, a small amount of entanglement appears essentially only at one point when the size of the system is big, whereas the rest of the algorithm needs to handle just separable states. We point out that these results apply as well to the traditional discrete-time implementation of Grover's searching algorithm, as the states between iterations are the same as in the adiabatic version for discrete s values.
Conclusions
In this paper we have studied the scaling of the entanglement entropy in several quantum algorithms. In particular, we have analytically proven that Shor's factoring algorithm makes use of an exponentially large amount of entanglement between the target register and the source register after the modular exponentiation operation, which in turn implies the impossibility of an efficient classical simulation by means of the protocol of ref. [8]. Furthermore, we have provided numerical evidence for a universal linear scaling of the entropy with the size of the system together with a polynomially small gap in a quantum algorithm by adiabatic evolution devised to solve the NP-complete problem Exact Cover, therefore obtaining a polynomial-time quantum algorithm which would involve exponential resources if simulated classically, in analogy to Shor's algorithm. Universality of this result follows from the fact that the quantum adiabatic algorithm evolves close to a quantum phase transition and the properties at the critical region do not depend on particular details of the microscopic Hamiltonian (instance) such as interactions among the spins or local magnetic fields. We have also proven that the Von Neumann entropy remains a bounded quantity in Grover's adiabatic algorithm regardless of the size of the system even at the critical point. More concretely, the maximum entropy approaches to one as an square root in the size of the system, which is the typical Grover's scaling factor.
Our results show that studying the scaling of the entropy is a useful way of analyzing entanglement production in quantum computers. Results from other fields of physics [19][20][21] can be directly applied to bring further insight into the analysis of quantum correlations. Different entanglement scaling laws follow from different situations according to the amount of correlations involved, as can be seen in Table 1. A quantum algorithm can be understood as the simulation of a system evolving close to a quantum phase transition. The amount of entanglement involved depends on the effective dimensionality of the system, which in turn governs the possibilities of certain efficient classical simulation protocols.
These scaling laws provide a new way of understanding some aspects from one-way quantum computation. It is known that the so-called cluster state of the one-way quantum computer can be generated by using Ising-like interactions on a planar two-dimensional lattice [38][39][40]. This fact can be related to the linear (in the size of a box) behavior of entropy for spin systems in two dimensions. Onedimensional models seem not to be able to efficiently create the highly-entangled cluster state [41]. Again, this fact can be traced to the logarithmic scaling law of the entropy in spin chains which is insufficient to handle the large amount of entanglement to carry out e.g. Shor's algorithm. Note also that d ≥ 3 dimensional systems bring unnecessarily large entanglement.
Quantum phase transitions stand as the more demanding systems in terms of entanglement. They are very hard to simulate classically. It is then reasonable to try to bring NP-complete problems to a quantum phase transition setup, which quantum mechanics handles naturally. 11053, PB98-0685 and BFM2000-1320-C02-01. Part of this work was done at the Benasque Center for Science.
Figure 2 :
2phase-estimation version of the quantum circuit for the orderfinding algorithm. The controlled operation is Λ(V f ).
Figure 4 :
4energies of the ground state and first excited state for a typical instance with one satisfying assignment of Exact Cover in the case of 10 qubits (in dimensionless units). The energy gap approaches its minimum at s c ∼ 0.7.
Figure 5 :
5minimum and maximum entropy over all possible bi-partitions of a 10-qubit system for each of the 300 randomly generated instances of Exact Cover. Instances are sorted such that the minimum entanglement monotonically increases.
Figure 7 :
7scaling of the minimum energy gap (in dimensionless units) with the size of the system, both in the worst case and in the mean case over all the randomly generated instances. Error bars give 95 per cent of confidence level for the mean.
Figure 8 :
8minimum energy gap (in dimensionless units) versus the inverse size of the system, both in the worst case and in the mean case over all the randomly generated instances. Error bars give 95 per cent of confidence level for the mean. The behavior is apparently linear.
Figure 10 :
10mean entropy of entanglement versus mean size of the energy gap (in dimensionless units). Error bars give 95 per cent of confidence level for the means. Each point corresponds to a fixed number of qubits.
Figure 11 :
11maximum entropy of entanglement versus minimum size of the energy gap (in dimensionless units). Each point corresponds to a fixed number of qubits.
Figure 13 :Figure 14 :
1314entanglement as a function of the Hamiltonian parameter for clauses of Exact Cover involving 3 (k = 3) and 4 (k = 4) qubits, for a 10-qubit system, averaged over all the randomly generated instances. minimum energy gap (in dimensionless units) versus 1/(n 3 ), both in the worst and in the mean cases over all the randomly generated instances of clauses involving 4 qubits, up to n = 16. Error bars give 95 per cent of confidence level for the mean. The behavior seems to be linear.
Figure 17 :
17Von Neumann entropy for the reduced system at s = 0.5 as a function of n. For infinite size of the system there is a saturation at 1.
Table 1 :
1entanglement scaling laws in different problems, in decreasing complexity order.
A somehow similar situation is present in one-dimensional quantum spin chains outside of the critical region, where the entanglement entropy also reaches a saturation when increasing the size of the system[14]. Saturation does not appear in higher dimensional systems.
Vidal for discussions about the content of this paper. We acknowledge financial support from the projects MCYT FPA2001-3598, GC2001SGR-00065, IST-1999-
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| []
|
[
"Scale-invariant perturbations in ekpyrotic cosmologies without fine-tuning of initial conditions",
"Scale-invariant perturbations in ekpyrotic cosmologies without fine-tuning of initial conditions"
]
| [
"Aaron M Levy ",
"Anna Ijjas ",
"Paul J Steinhardt ",
"\nDepartment of Physics\nPrinceton Center for Theoretical Science\nPrinceton University\n08544PrincetonNJUSA\n",
"\nDepartment of Physics\nPrinceton University\n08544PrincetonNJUSA\n",
"\nPrinceton Center for Theoretical Science\nPrinceton University\n08544PrincetonNJUSA\n",
"\nPrinceton University\n08544PrincetonNJUSA\n"
]
| [
"Department of Physics\nPrinceton Center for Theoretical Science\nPrinceton University\n08544PrincetonNJUSA",
"Department of Physics\nPrinceton University\n08544PrincetonNJUSA",
"Princeton Center for Theoretical Science\nPrinceton University\n08544PrincetonNJUSA",
"Princeton University\n08544PrincetonNJUSA"
]
| []
| Ekpyrotic bouncing cosmologies have been proposed as alternatives to inflation. In these scenarios, the universe is smoothed and flattened during a period of slow contraction preceding the bounce while quantum fluctuations generate nearly scale-invariant super-horizon perturbations that seed structure in the post-bounce universe. An analysis byTolley and Wesley (2007)showed that, for a wide range of ekpyrotic models, generating a scale-invariant spectrum of adiabatic or entropic fluctuations is only possible if the cosmological background is unstable, in which case the scenario is highly sensitive to initial conditions. In this paper, we analyze an important counterexample: a simple action that generates a Gaussian, scale-invariant spectrum of entropic perturbations during ekpyrotic contraction without requiring fine-tuned initial conditions. Based on this example, we discuss some generalizations. | 10.1103/physrevd.92.063524 | [
"https://arxiv.org/pdf/1506.01011v1.pdf"
]
| 12,220,224 | 1506.01011 | 78b362d1628c54eefcfe12a22d96fd4fcae3db2a |
Scale-invariant perturbations in ekpyrotic cosmologies without fine-tuning of initial conditions
Aaron M Levy
Anna Ijjas
Paul J Steinhardt
Department of Physics
Princeton Center for Theoretical Science
Princeton University
08544PrincetonNJUSA
Department of Physics
Princeton University
08544PrincetonNJUSA
Princeton Center for Theoretical Science
Princeton University
08544PrincetonNJUSA
Princeton University
08544PrincetonNJUSA
Scale-invariant perturbations in ekpyrotic cosmologies without fine-tuning of initial conditions
(Dated: June 3, 2015)cyclic/ekpyrotic cosmologyentropic mechanismattractor solutions
Ekpyrotic bouncing cosmologies have been proposed as alternatives to inflation. In these scenarios, the universe is smoothed and flattened during a period of slow contraction preceding the bounce while quantum fluctuations generate nearly scale-invariant super-horizon perturbations that seed structure in the post-bounce universe. An analysis byTolley and Wesley (2007)showed that, for a wide range of ekpyrotic models, generating a scale-invariant spectrum of adiabatic or entropic fluctuations is only possible if the cosmological background is unstable, in which case the scenario is highly sensitive to initial conditions. In this paper, we analyze an important counterexample: a simple action that generates a Gaussian, scale-invariant spectrum of entropic perturbations during ekpyrotic contraction without requiring fine-tuned initial conditions. Based on this example, we discuss some generalizations.
I. INTRODUCTION
Observations of the cosmic microwave background from the Wilkinson Microwave Anisotropy Probe (WMAP) [1], the Planck satellite [2,3], the Atacama Cosmology Telescope (ACT) [4] and other experiments have shown that the primordial scalar (density) fluctuation spectrum is adiabatic and nearly scale-invariant with nearly Gaussian statistics. Inflation [5][6][7] has been suggested as a mechanism for generating perturbations with these properties, though, it is known that to do so it requires rare initial conditions [8,9] and results in a multiverse of outcomes [10][11][12][13][14].
Bouncing cosmologies with a period of ultra-slow (ekpyrotic) contraction have been proposed as alternatives. In these theories, smoothing contraction occurs because the energy density of a scalar field with equationof-state > 3 (where ≡ 3(p + ρ)/ρ with p being the pressure and ρ the energy density) grows to dominate all other forms of energy, including inhomogeneities, anisotropy and spatial curvature [15]. A key advantage compared to inflation is that the ekpyrotic mechanism does not lead to a multiverse.
The currently best understood way to produce density fluctuations in the ekpyrotic theory involves two scalar fields that generate a scale-invariant spectrum of entropy perturbations. After the ekpyrotic smoothing phase, these perturbations convert into a scale-invariant spectrum of adiabatic perturbations [16][17][18][19]. In the first * [email protected] examples discussed in the literature, the background cosmological solution describing the evolution of the two fields along the potential energy surface is unstable, which means finely-tuned initial conditions are required to begin the ekpyrotic phase [20][21][22][23]. Tolley and Wesley [24] analyzed the dynamics of a wide class of contracting cosmological models that generate scale-invariant adiabatic or entropic perturbations and suggested that the problem may be generic. More specifically, they showed that the cosmological background solutions are not attracted to a fixed point and, from this, concluded that the models are highly sensitive to initial conditions.
In this paper, we present simple ekpyrotic models that generate a scale-invariant, nearly Gaussian spectrum of density perturbations but do not require fine-tuning of initial conditions. Although these models belong in the class considered by Tolley and Wesley, we show that the background solutions are attracted to a fixed-curve along which scale-invariant fluctuations are generated. The existence of such a fixed-curve is sufficient to ensure that the observational predictions are insensitive to the choice of initial conditions. In other words, being attracted to a fixed-point is not necessary to avoid fine-tuning.
In Section II, we summarize the general argument that suggests the need for finely-tuned initial conditions. In Section III, we review a simple ekpyrotic model for which we find a fixed-curved but no fixed-point attractor. In Section IV, we describe how to construct more general examples that also avoid the need for fine-tuning of initial conditions. Finally, we discuss the implications for cosmology.
II. SCALING SOLUTIONS, SCALE-INVARIANCE AND INSTABILITY
Scaling solutions are solutions to the equations of motion for which there exists a set of field variables such that all contributions (treating the kinetic and potential energy densities as distinct) to the total energy density scale identically with time, keeping their fractional contributions constant. Scaling background solutions are particularly important because they are exactly solvable and can yield a scale-invariant spectrum of perturbations.
In Ref. [24], Tolley and Wesley presented an instability argument that applies to contracting, scaling solutions derived from two-derivative, two-field actions
S = d 4 x √ −g 1 2 R (1) − d 4 x √ −g 1 2 G ab (Φ)g µν ∂ µ Φ a ∂ ν Φ b − V (Φ)
that possess a continuous symmetry generated by a parameter κ such that
dΦ a dκ = ξ a (Φ), g µν → e κ g µν , S → e κ S.(2)
Here g µν is the spacetime metric with (− + ++) signature convention, R is the Ricci scalar, ξ a (Φ) is a function of the two scalar fields Φ = {Φ a } where a = 1, 2, G ab is the metric on field space, and V (Φ) is the potential energy density; reduced Planck units (8πG N = 1 where G N is Newton's gravitational constant) are used throughout. This symmetry guarantees the existence of a set of field variables (Φ 1 , Φ 2 ) → (φ, σ) such that the Lagrangian density can be rewritten as
L = 1 2 R − 1 2 (∂σ) 2 − 1 2 f (σ)(∂φ) 2 − V 0 e −cφ h(σ),(3)
where c is a real constant and V 0 < 0 (that is, the ekpyrotic potential V (φ, σ) is negative). Along the background solution, f (0) = h(0) = 1 and σ = 0. It proves useful to introduce the dynamical variables
(w, x, y, z) ≡ f (σ)φ √ 6H , σ √ 6H , − a −V 0 h(σ)e − c 2 φ √ 3H , σ (4)
where τ < 0 is conformal time running from large negative to small negative values, a prime denotes a derivative with respect to conformal time, a is the scale factor and H ≡ a /a is the conformal Hubble parameter. The four variables are dimensionless using reduced Planck units. With these variables, the Friedmann-Robertson-Walker (FRW) equations of motion become Here we introduced the dimensionless time variable N ≡ ln a that denotes the number of e-folds of ekpyrotic contraction and runs from large positive to small positive values. We eliminated y using the Friedmann constraint
w, N = 3(w 2 + x 2 − 1) w − c 6f (z) − 3 2 f, z f (z) xw,(5) x, N = 3(w 2 + x 2 − 1) x + 1 √ 6 h, z h(z) + 3 2 f, z f (z) w 2 ,(6)z, N = √ 6x.(7)w 2 + x 2 − y 2 = 1.(8)
The equation of state takes the simple form
= 1 − H H 2 = 3(w 2 + x 2 ).(9)
From the Friedmann constraint, we also see that the square of each variable w, x, and y is a fractional contribution to the total energy density: w 2 is the φ-kinetic energy; x 2 is the σ-kinetic energy; and y 2 is the potential energy. The equation-of-state parameter is the sum of kinetic energies. As can be verified by direct substitution, the background Eqs. (5-7) admit a fixed-point solution,
(w, x, y, z) = c √ 6 , 0, c 2 6 − 1, 0 ,(10)
in which σ = 0 with and only φ is changing, provided the constraint h, σ (0) = −c 2 f, σ (0)/(c 2 − 6). Obviously, this is a scaling solution since any fractional contribution to the total energy density, w 2 , x 2 , and y 2 , is constant. The cosmological background solutions correspond to a field-space trajectory like the one shown in Fig. 1. Perturbations of this trajectory can be decomposed into those along the red curve (adiabatic perturbations) and perpendicular to it (entropic perturbations).
The instability argument connects the stability of the solution in Eq. (10) with the spectral indices of its perturbations. The basic idea makes use of the fact that, since z ≡ σ = 0 in the background solution, the second order action and, hence, the perturbation spectra derived from it are determined by a few parameters {c ; f, σ (0); f, σσ (0); h, σσ (0)}. 11) that results in a scale-invariant spectrum of perturbations (entropic or adiabatic) renders the background solution dynamically unstable to perturbations in the sense that the matrix associated with the linearized system has at least one negative eigenvalue. In a contracting universe, a negative eigenvalue means a dynamically unstable direction in (w-x-z)-space. An example is given by the Lagrangian density
L = 1 2 R − 1 2 (∂Φ 1 ) 2 + 1 2 (∂Φ 2 ) 2 −Ṽ 0 e −c1Φ1 −Ṽ 0 e −c2Φ2 ,(12)
where c 1 , c 2 are positive-definite constants andṼ 0 < 0. The corresponding FRW equations of motion admit a scaling solution with Φ i = A i ln |τ | + B i and c 1 A 1 = c 2 A 2 that has been shown to generate a scale-invariant spectrum of entropic perturbations [17,18]. According to the instability argument, it should have an unstable background which in this simple case (G ab = δ ab ) can be depicted as in Fig. 2.
For the purpose of illustration, we briefly outline how the instability emerges from a negative eigenvalue of the linearized system. The change of variables,
φ = c 2 Φ 1 + c 1 Φ 2 c 2 1 + c 2 2 ,(13)σ = c 1 Φ 1 − c 2 Φ 2 c 2 1 + c 2 2 + σ 0(14)
with σ 0 = 2 ln(c 2 /c 1 )/(c 2 1 + c 2 2 ), brings the Lagrangian density in Eq. (12) to the form of Eq. (3). The coupling function to the kinetic energy of φ, f (σ), and the coupling function to the potential energy of φ, h(σ), are given by
f (σ) = 1,(15)h(σ) = 1 + c 2 2 σ 2 + O(σ 3 ),(16)
and the parameters, c and V 0 are defined such that
1 c 2 = 1 c 2 1 + 1 c 2 2 ,(17)V 0 = c 2 c 1 2c 2 1 c 2 1 +c 2 2 + c 1 c 2 2c 2 2 c 2 1 +c 2 2 Ṽ 0 .(18)
Linearizing the background Eqs.
(5-7) about the fixed- point in Eq. (10), the perturbations (δw, δx, δz) ≡ (w − c √ 6 , x, z) satisfy δw, N δx, N δz, N = M · δw δx δz (19) with M defined as M ≡ 1 2 c 2 − 6 0 0 0 1 2 c 2 − 6 c 2 2 √ 6 c 2 − 6 0 √ 6 0 . (20)
The eigenvalues of M are c 2 − 6 /2 and (c 2 − 6) ± √ 9c 4 − 60c 2 + 36 /4. Note that the smallest eigenvalue is negative for ekpyrosis, as is clear from substituting Eq. (10) into the equation of state, Eq. (9): > 3 requires c > √ 6. Therefore, as the universe contracts, N decreases, and perturbations along the eigenvector corresponding to the negative eigenvalue grow so that the system is carried away from the fixed-point solution in Eq. (10). In this case, the negative eigenvalue means that the initial conditions for the fields must be fine-tuned to lie close to the trajectory or else the fields will evolve far-off course as illustrated in Fig. 2.
III. EKPYROSIS AND SCALE-INVARIANCE WITHOUT FINE TUNING
In this section, we describe the case where the negative eigenvalue exists but is physically irrelevant. As we will show, the occurrence of the negative eigenvalue only means that the attractor is a fixed-curve rather than a fixed point.
We consider the Lagrangian density first discussed by Li in Ref. [25],
L = 1 2 R − 1 2 (∂ψ) 2 − 1 2 e −λψ (∂χ) 2 −Ṽ 0 e −λψ ,(21)Forỹ = 0, (w,x,z) = ψ /( √ 6H), e − λ 2 ψ χ /( √ 6H), e − λ 2 ψ (χ − χ0) .
The green curve, (w,x,z) = (1, 0,z), is the fixed-curve attractor. The red arrow points to the fixed-point (w,x,z) = (1, 0, 0). where λ is a positive andṼ 0 < 0. The model involves an ekpyrotic field, ψ, with a negative potential, similar to ordinary, single-field ekpyrosis. The novel feature is the non-canonical, exponential coupling to the massless spectator field, χ. We begin with the simple case where V (ψ) = 0. This corresponds to the borderline ekpyrotic equation of state = 3. Then, we generalize to V (ψ) = 0 ( > 3) and provide a full, analytic treatment.
FIG. 3.b. The trajectories in (φ, σ) variables: For y = 0, (w, x, z) = f (σ) φ /( √ 6 H), σ /( √ 6 H), σ .A. V (ψ) = 0
The basic idea is captured in Fig. 3a, which illustrates background trajectories corresponding to different initial conditions in the space (w,x,
¡ ¡ ! 0 y,z) = ψ √ 6H , e − λ 2 ψ χ √ 6H , $ $ $ $ $ $ $ $ X 0 −a −Ṽ 0 e − λ 2 ψ √ 3H , e − λ 2 ψ (χ − χ 0 ) .(22)
We use the superscript ∼ to distinguish quantities expressed in (ψ, χ) variables from those expressed in the special (φ, σ) variables that were used to derive the original instability argument.
Any set of initial conditions for a, ψ, ψ , χ, and χ corresponds to a particular point on the cylinder. The background solution follows the blue arrow originating at this point. There are two special initial conditions at the points (w,x,z) = (±1, 0, 0); the one with the + sign corresponds to the red arrow in Fig. 3a. These two points are special because the blue arrows vanish here, i.e., if the background solution starts here, it stays here. Hence, these are fixed-point solutions.
For any other initial conditions, (i.e., any other point on the cylinder), the blue arrows carry the background solution toward the green curve. The green curve is therefore a strong attractor for generic initial conditions. Henceforth, we call such an attractor curve a "fixed-curve attractor." Solutions evolving along this curve are scaling solutions sincew,x, andỹ are constant. They describe a universe dominated by the kinetic energy of the adiabatic field, ψ; the entropic field, χ, is constant. For the case (λ = √ 6) depicted in Fig. 3, these solutions can be shown to generate a scale-invariant spectrum of perturbations in the entropic field, χ.
As for the negative eigenvalue associated with the linearized equations of motion, in the cases discussed in this paper, it only indicates the existence of a fixed-curve attractor instead of a fixed-point attractor. The existence of a fixed-curve attractor means there is no need for finetuning of initial conditions.
One can describe the same dynamics in the special (φ, σ) variables (see Fig. 3b). Since the Lagrangian density in Eq. (21) has the shift symmetry of Eq. (2) assumed by the instability argument, it can be put into the form of Eq. (3) through a variable transformation (ψ, χ) → (φ, σ),
ψ = φ + 2 λ ln sech λσ 2 + ψ 0 ,(23)χ = 2 λ e λ 2 (φ+ψ0) tanh λσ 2 + χ 0 ,(24)
where ψ 0 is a real constant. Substituting this transformation into Eq. (21) yields the Lagrangian density in Eq.
(3) with f (σ) = h(σ) = cosh 2 (λσ/2) and c = λ. The two Lagrangian densities describe the same theory in different field variables: the cylinder on the right is a twisted version of the one on the left. It is clear from the blue arrows that the green curve in Fig. 3b is an attractor, just like the green curve in Fig. 3a. Solutions along the green-curve attractor generate a scale-invariant spectrum of entropic perturbations.
B. V (ψ) = 0
We extend our analysis to the more general form of the Lagrangian density in Eq. (21) that applies both to V (ψ) = 0 and to V (ψ) = 0.
Extremizing the action with respect to variations of the fields (ψ, χ) yields the field equations ψ + 2Hψ − λṼ 0 e −λψ a 2 + λ 2 e −λψ χ 2 = 0, (25) χ + 2Hχ − λψ χ = 0.
The Friedmann constraint is
H 2 = 1 6 ψ 2 + e −λψ χ 2 + 2a 2Ṽ 0 e −λψ .(27)
Using the variables defined in Eq. (22), these equations can be recast as the autonomous, dynamical system
w, N = 3(w 2 +x 2 − 1) w − λ √ 6 − 3 2 λx 2 , (28) x, N = 3x(w 2 +x 2 − 1) + 3 2 λwx,(29)z, N = − 3 2 λwz + √ 6x.(30)
If V = 0, this system admits three fixed-point solutions at (w,x,ỹ,z) = (−1, 0, 0, 0) ,
(+1, 0, 0, 0) ,(31)λ √ 6 , 0, λ 2 6 − 1, 0 ,(32)
all of which are unstable (i.e., associated with a negative eigenvalue). The third fixed-point solution given by Eq. (33) bisects two fixed-curve solutions
(w,x,ỹ,z) = λ √ 6 , 0, λ 2 6 − 1, ±Z(34)
withZ ∝ e − λ 2 2 N that generate a scale-invariant spectrum of entropic perturbations, as shown in Ref. [25].
If V =ỹ = 0, λ 2 must be 6 in order for the fixed-point in Eq. (33) to be a solution. For λ = √ 6, this coincides with the fixed-point in Eq. (32) and corresponds to the red arrow shown in Fig. 3a; Eq. (34) parameterizes the vertical, green, fixed-curve attractor.
Changing variables (ψ, χ) → (φ, σ) as defined in Eqs. (23) and (24)
w, N = 3(w 2 + x 2 − 1) w − λ √ 6 sech (λz) − √ 6c tanh(λz) xw,(35)
x, N = 3(w 2 + x 2 − 1) x + 2 3 λ tanh(λz)
+ √ 6c tanh(λz) w 2 ,(36)z, N = √ 6x.(37)
The variable transformations as defined in Eqs. (23) and (24) imply the following relationship between the variables (w, x, y, z) → (w,x,ỹ,z):
w = sech λ 2 z w − tanh λ 2 z x, (38) x = tanh λ 2 z w + sech λ 2 z x,(39)y = y,(40)z = 2 λ sinh λ 2 z .(41)
If V = 0, these transformations quantify how to "twist" Fig. 3b to generate Fig. 3a.
The fixed-point solutions in Eqs. (31-33) are given in the new variables (w, x, y, z) as
(−1, 0, 0, 0) ,(42)(+1, 0, 0, 0) ,(43)λ √ 6 , 0, λ 2 6 − 1, 0 ,(44)
and the fixed-curve solutions in Eq. (34) are
λ √ 6 sech λ 2 z , − λ √ 6 tanh λ 2 z , λ 2 6 − 1, ±Z(45)
with Z = (2/λ) sinh −1 λZ/2 . These fixed-curves lie on the surface of the cylinder w 2 + x 2 = λ 2 /6. For V = 0 and λ = √ 6, Eq. (45) corresponds to the twisted green curve that is confined to the surface of the unit cylinder in Fig. 3b.
Direct substitution verifies that the curves in Eq. (34) and Eq. (45) are solutions to the background equations given in Eqs. (28)(29)(30) and Eqs. (35-37) for both V = 0 with λ = √ 6 and for V = 0. To show the existence of a negative eigenvalue, we linearize the equations of motion about the fixed-points, in Eq. (33) and (44), respectively, for the two sets of variables. Linearizing Eqs. (28)(29)(30) about Eq. (33) yields a matrix equation like that given in Eq. (19) with (δw, δx, δz) = w − λ/ √ 6,x,z and
M ≡ λ 2 2 − 3 0 0 0 λ 2 − 3 0 0 √ 6 − λ 2 2 .(46)M ≡ λ 2 2 − 3 0 0 0 λ 2 2 − 3 λ 2 (λ 2 −3) 2 √ 6 0 √ 6 0 .(47)
BothM and M have eigenvalues −λ 2 /2, λ 2 /2 − 3, λ 2 − 3 , the first of which is negative.
In the first set of variables, the eigenvector corresponding to the eigenvalue −λ 2 /2 is parallel to the unit vector in thez-direction, which is tangent to the fixed-curve solution in Eq. (34). In the second set of variables, the eigenvector associated with the eigenvalue −λ 2 /2 is parallel to a linear combination of unit vectorsx,ẑ, namelyẑ − λ 2 /(2 √ 6)x, that is tangent to the fixed-curve solution in Eq. (45).
For the case, V = 0 and λ = √ 6, these eigenvectors are tangent to the green fixed-curves in Fig. 3 at the red arrows. The existence of a negative eigenvalue in this model is harmless, since it only means that the system is attracted to a fixed-curve solution (instead of a fixedpoint solution) that generates a scale-invariant spectrum of entropic perturbations.
C. Further Generalizations
Although the remainder of this work will consider actions with the shift symmetry in Eq. (2), our results can be generalized to cases without shift symmetry. For example, the field contribution to the shift-symmetric Lagrangian density in Eq. (21) is a special case of the more general Lagrangian density
L = 1 2 R− 1 2 (∂ψ) 2 − 1 2 e −λψ (∂χ) 2 −(1 + r(χ)) V 0 e −µψ +q(χ)
(48) with µ = λ and r(χ) = q(χ) = 0. The addition of q(χ) and r(χ) breaks the shift symmetry since V = (1 + r(χ)) V 0 e −µψ + q(χ)
→ 1 + r(e λ 2µ κ χ) V 0 e −µψ−κ + r(e λ 2µ κ χ) = e −κ V.(49)
If µ = λ, the ekpyrotic Lagrangian density in Eq. (48) admits a scaling solution that is a fixed-curve attractor with χ = 0 and that generates a scale-invariant spectrum of entropic perturbations. This is due to the fact that, as χ → 0, r(χ) and q(χ) approach constants r(χ 0 ) and q(χ 0 ). The first, r(χ 0 ), can be reabsorbed into V 0 , and the second, q(χ 0 ), is negligible along the fixed-curved attractor.
IV. CONSTRUCTING NEW MODELS
In this section, we derive the most general ekpyrotic, two-field Lagrangian density with shift symmetry that admits scaling solutions which are either fixed-point or fixed-curved attractors and generate a scale-invariant spectrum of entropy perturbations.
First, we consider the Lagrangian density in Eq. (3) with arbitrary parameters and couplings,
{V 0 < 0, c ∈ R, h(σ) > 0, f (σ) > 0}.(50)
In the Appendix, we show that the combined conditions of shift symmetry, scaling solution, fixed-curved attractor and scale-invariant spectrum of perturbations imply the following properties:
P1: lim |σ|→∞ f (σ) = ∞ monotonically; P2: lim |σ|→∞ h(σ) ∝ e −µσ ; P3: lim |σ|→∞ (w, x, y, z) = 0, µ √ 6 , µ 2 6 − 1, −sgn(µ)∞ ; P4: |µ| > √ 6.
Property P1 says that the coupling f (σ) must grow without bound. Property P2 constrains the form of the potential energy density to be exponential at late times. Property P3 defines the scaling solution. It implies w = 0 so that the φ field is fixed; furthermore, since the background equations in Eqs. (5-7) depend explicitly on σ which will vary, it also implies that the scaling solution is a fixed-curve (rather than a fixed-point) in (w-x-z)space. Property P4 is necessary for ekpyrosis ( > 3) as follows from substituting this solution into the equation of state, Eq. (9). The example from the last section has these four properties. At late times, the fixed-curve attractor in Eq. (45) goes to z ≡ σ → ±∞. In this limit, f (σ) = h(σ) = cosh 2 (λσ/2) is dominated by the single exponential e |λσ| /4. For example, if λ > 0 and σ → −∞, choosing µ = λ in the solution in property P3 reproduces Eq. (45) at late times. Similar arguments apply for the different combinations of sgn(λ) and sgn(σ).
Assuming properties P1 thru P4, the only remaining degrees of freedom are the parameters, c, V 0 , and the latetime behavior of f (σ), modulo property P1. We show now that, given these four properties, it is possible to obtain a scale-invariant spectrum for the entropic modes but not for the adiabatic modes.
We perturb Einstein's equations about the fixed-curve solution specified by Property P3, working in the longitudinal gauge [26,27] where the metric takes the form
ds 2 = a 2 −(1 + 2Φ)dτ 2 + (1 − 2Φ)d x 2 .(51)
Since φ = 0 along the background solution in property P3, the quantity Q s ≡ f (σ)δφ is automatically gauge-invariant and represents the entropy perturbation; the Mukhanov-Sasaki variable Q σ ≡ δσ + (σ /H) Φ is also gauge-invariant and represents the adiabatic perturbation [28,29]. Property P3 implies that the equation of state = µ 2 /2 and, therefore, the conformal Hubble parameter is H −1 = (1 − )(−τ ) < 0. Then, the mode functions u σ ≡ a Q σ and u s ≡ a Q s can be shown to satisfy
u σ + k 2 − θ σ (−τ ) 2 u σ = α (−τ ) 2 u s + β (−τ ) u s ,(52)u s + k 2 − θ s (−τ ) 2 u s = γ (−τ ) 2 u σ + δ (−τ ) u σ ,(53)
where the the background-dependent quantities can be derived, for example, from the expressions in Ref. [30]:
θ σ = −2 µ 2 − 4 (µ 2 − 2) 2 − µ 2 − 6 2 µ 2 (µ 2 − 2) 2 c 2 f ,(54)θ s = −2 µ 2 − 4 (µ 2 − 2) 2 + 3 µ 2 − 6 µ 2 (µ 2 − 2) c 2 f − µ(µ 2 − 6) (µ 2 − 2) 2 f, σ f − µ 2 (µ 2 − 2) 2 f, σ f u s = 0.(70)
For the mode function u s , we find the solution
u s (τ ) = f (τ ) c 1 (k) τ −1/k dτ f (τ ) + c 2 (k) ,(71)
where c 1 (k) and c 2 (k) are constants of integration. Choosing c 1 (k) and c 2 (k) so that u s and u s match the Bunch-Davies solution, (1/ √ 2k)e −ikτ , at horizon crossing, −τ = 1/k, yields
c 1 (k) = e i (f (−1/k) + 2ikf (−1/k)) 2 2kf (−1/k) ,(72)c 2 (k) = e i 2kf (−1/k) .(73)
For fast-growing f , at late times the integral in Eq. (71) is very closely approximated by (1/k)(1/f (−1/k)). With Eqs. (72) and (73), the entropic mode function is given by
u s = Σ(k) f (τ ) (74) where Σ(k) = e i (−f (−1/k) + (2 − 2i)kf (−1/k)) 2 √ 2k 3/2 (f (−1/k)) 3/2 .(75)
Substituting this result into the right side of the adiabatic equation, Eq. (52), we find that u σ satisfies
u σ + 2 µ 2 − 4 (µ 2 − 2) 2 1 (−τ ) 2 u σ = 4 µ 2 − 6 µ (µ 2 − 2) 2 c Σ(k) (−τ ) 2 (76) with solution u σ = c 3 (k)(−τ ) µ 2 −4 µ 2 −2 + c 4 (k)(−τ ) 2 µ 2 −2 + 2 µ 2 − 6 µ (µ 2 − 4) cΣ(k),(77)
where c 3 (k) and c 4 (k) are constants. The first two terms vanish as τ → 0 by property P4. This shows that our assumption in Eq. (69) is justified.
From the solutions for the mode functions in Eqs. (71) and (77), it is clear that both the adiabatic and the entropic spectra are proportional to k 3 |Σ(k)| 2 . Scale invariance is obtained if and only if
|Σ(k)| 2 = ξk −3 ,(78)
for some constant ξ that is independent of k. Eq. (78) is a first order differential equation for the coupling function f , namely
f f 2 − 4 (−τ ) f f = 8ξ f − 8 (−τ ) 2 ;(79)
its solution is given by
f (τ ) = sec 2 [ln(τ /τ 0 )] 2ξ(−τ ) 2 ,(80)
where τ 0 is an integration constant. This is clearly not monotonic as τ → 0 and, therefore, violates property P1. Hence, we conclude scale-invariance is impossible for fastgrowing f . If |f, σ /f | 1, then |f /f | 1/(−τ ). To leading order, both mode functions satisfy
u + 2 µ 2 − 4 (µ 2 − 2) 2 1 (−τ ) 2 u = 0,(81)
as is clear from Eqs. (52), (53) and (61-66). In this case, both the adiabatic and the entropic spectra are given by
n S = 4 − µ 2 − 6 µ 2 − 2 ,(82)
which is blue by property P4. Hence, we conclude scaleinvariance is impossible for slow-growing f .
C. "Just-so" growth: |f,σ /f | ∼ 1
If |f, σ /f | ∼ 1, f (σ) = e −λσ for some λ ∈ R such that sgn(λ) = sgn(µ). The coupling functions α, β, γ, δ are all proportional to 1/ √ f so the right sides of Eqs. (52) and (53) can be neglected. The mode functions effectively decouple and the adiabatic spectral index is again given by Eq. (82); the entropic spectral index is
n S = 4 − 2 λ µ − 2 µ 2 − 2 + 1 ,(83)
which is scale-invariant when λ = µ. For a given µ > √ 6, any n S < (3µ 2 − 2)/(µ 2 − 2) can be achieved by choosing λ = (n S − 1)/µ − µ (n S − 3) /2.
Note that since "just-so" growth implies f (σ) = e −λσ (and property P2 requires h(σ) = e −µσ ), the Lagrangian density is given by
L = 1 2 R − 1 2 (∂σ) 2 − 1 2 e −λσ (∂φ) 2 − V 0 e −cφ e −µσ ,(84)
which is equivalent to the Lagrangian density in Eq. (48) with the identifications σ ←→ ψ, φ ←→ χ.
V. DISCUSSION
In this paper, we have presented explicit examples of ekpyrotic models, with and without shift symmetry, that have fixed-curve attractor background cosmological solutions generating a scale-invariant spectrum of entropic perturbations. The existence of a negative eigenvalue associated with the linearized dynamical equations, as identified by Tolley and Wesley, can indicate a true instability requiring fine-tuning of initial conditions for some actions. But, for actions of the type discussed here, the negative eigenvalue only indicates that the attractor solution is a curve rather than a point. Fine-tuning of initial conditions is thus avoided. Furthermore, as in all ekpyrotic models, this class of actions avoids the multiverse and the problem that all cosmological outcomes are possible. Hence, the predictions are "generic," the same on average for any Hubble-sized patch.
One might be concerned if, in order to avoid fine-tuning of initial conditions, the models had to be made more complicated. However, the opposite is the case here. The actions impose less stringent constraints on the equationof-state, , during the contracting phase and, hence, less fine-tuning of parameters. For such a choice of parameters, a scalar spectrum of density perturbations with the observed spectral tilt can be generated. As in all ekpyrotic models, though, it is not possible to generate a detectable spectrum of primordial gravitational waves (the ratio of the tensor-perturbation amplitude to the scalar-perturbation amplitude, r ≈ 0 ), consistent with current limits [15,31]. The same class of actions has also been shown to generate zero non-Gaussianity during the ekpyrotic contraction phase; a small amount of local non-Gaussianity may be generated during the bounce, but at a level well within current observational bounds on f NL [32,33].
It is notable that current cosmological observations are constraining ekpyrotic models to be in a class that is the simplest, as measured by parameters, degrees of freedom, and initial conditions. By contrast, the same observations are pointing away from the simplest models of inflation [34].
We thank D. Wesley and A. Tolley for useful comments. This research was supported in part by the U.S. Department of Energy under grant number DE-FG02-91ER40671.
FIG. 1 .
1A schematic trajectory of a solution through fieldspace: The field variables (φ, σ) were constructed so that, along the field-space trajectory in Eq. (10), only φ varies and σ = 0. Perturbations δφ are tangent to the trajectory and are adiabatic; perturbations δσ are orthogonal to the trajectory and are entropic.
FIG. 2 .
2The instability of the scaling solution corresponding to the Lagrangian density in Eq. (12): The background cosmological solution corresponds to a trajectory along the ridge of the potential as indicated by the red arrow and red curve. Quantum fluctuations along the trajectory produce a blue spectrum of adiabatic perturbations and fluctuations normal to the trajectory produce a scale-invariant spectrum of entropic perturbations. After the ekpyrotic phase, the entropic perturbations convert into adiabatic perturbations due to the bending of the trajectory curve (not shown here). The fact that the background trajectory is a ridge means that it is unstable if the initial conditions are sufficiently far from the ridge. Linearized around the fixed-point in Eq. (10), the background Eqs. (5-7) reduce to a matrix equation, where the parameters in Eq. (11) determine the eigenvalues of the matrix. Tolley and Wesley's analysis showed that any combination of the parameters in Eq. (
FIG. 3 .
3The Lagrangian density in Eq. (21) with V = 0 and λ = √ 6: Green curves and red arrows map onto each other according to the transformations in Eqs. (38-41). Solutions follow the blue arrows. FIG. 3.a. The trajectories in (ψ, χ) variables:
The green curve, (w, x, z) = (sech(λ z/2), − tanh(λ z/2), z), is the fixed-curve attractor. The red arrow points to the fixed-point (w, x, z) = (1, 0, 0).
, the Lagrangian density in Eq.(21) takes the form of Eq.(3) with f (σ) = h(σ) = cosh 2 (λσ/2) , c = λ, and V 0 =Ṽ 0 e −λψ0 .Repeating the same analysis in the new variables defined in Eq. (4), the equations of motion become
B. Slow growth: |f,σ /f | 1
These variables are generally time-dependent because f = f (σ) is a function of τ . Substituting the expression for H −1 into the definition of x given in Eq.(4), we findUsing this expression, Eqs. (54-59) can be rewritten as:Eqs. (52) and (53) are a coupled, linear system of differential equations which must be solved as τ → 0 − to find the spectra. Depending on the growth rate of the coupling function f , different terms in Eqs. (61-66) come to dominate in this regime. For example, the first term in θ σ always dominates over the second since f → ∞ as τ → 0. By contrast, the relative sizes of the different terms in θ s depend on the magnitude of f /f . For clarity, we define the symbol " " to meanSimilarly, we define "∼" to mean(68) For example, 1/(−τ ) 2 1/(−τ ) and 3/(−τ ) ∼ 2/(−τ ).There are three qualitatively different cases to consider: fast growth, |f, σ /f | 1, slow growth, |f, σ /f | 1, and "just-so" growth, |f, σ /f | ∼ 1.so that to leading order, the entropic mode evolves independently:Here we derive the four properties listed at the beginning of Sec. IV using a series of lemmas:abatic spectrum without fine-tuning.the Lagrangian density can be recast asEq. (91) has the shift-symmetric form of Eq. (3) with the solution (x, z) = (0, z 0 ) at (x,z) = (0, 0) (withz ≡σ) for which Tolley and Wesley proved that any solutions of interest are unstable. For fixed-point solutions like this (as opposed to fixed-curve solutions) instability implies the need for finely-tuned initial conditions. Thus, we conclude |σ| → ∞; we are forced to fixed-curve solutions.Lemma 2: f → const cannot generate a scale-invariant adiabatic or entropic spectrum without fine-tuning.When f = f 0 = const, the non-canonical coupling becomes canonical. Then, the background Eqs.(5)and(6)give (w, x) = (w 0 , x 0 ) with x 0 = 0 (cf. Lemma 1). Then h = e − √ 6x0σ . In such canonically-coupled theories, the adiabatic perturbation decouples from the entropic perturbation and has a blue tilt. More precisely, the equations of motion to the Lagrangian densitylinearized around (w 0 , x 0 ) = (d/ √ 6, c/ √ 6f 0 ) yield the same equation for both the adiabatic and the entropic mode functionsso that the spectral index is given byAvoiding fine-tuning, i.e., negative eigenvalues in the linearized equations of motion, requires c 2 /f 0 + d 2 > 6. In this regime the spectral indices are blue for both the adiabatic and entropic spectra.Lemma 3: If w = 0 then f → (∞ or const) at late times.For scaling solutions, the background equation for w, Eq. (5), can be recast asx > 0 and z decreases. Hence, sgn(f ) = −sgn(f, z ). From the right side of Eq. (95), we conclude that, if B < 0, f → 0 at late times and that, if B > 0, f → (B 2 or ∞) at late times. With Lemma 2, we only have to consider the case f → ∞.Lemma 4: If f → ∞, then w = 0 at late times.If f → ∞, the background equation for w in Eq. (5) becomesAssume w = 0. Then f, z /f = √ 6(x 2 + w 2 − 1)/x and, hence, f (σ) = e −λσ with λ = − √ 6(x 2 + w 2 − 1)/x. In particular, λ and x must have opposite sign. From z, N = √ 6x, we see, though, that x and z must have different sign (since N decreases as the system evolves, z will, for example, decrease when z, N > 0). Therefore, λ and z must have the same sign. But then f = e −λz → 0 at late times, which contradicts our assumption, f → ∞.To this point, we have shown that any ekpyrotic, scaling attractor that generates a scale-invariant spectrum of either adiabatic or entropic perturbations lies at w = 0 with f → ∞ and x = 0. Defining µ ≡ √ 6x, Eq. (6) implies h, z /h = −µ, from which we conclude, given that z ≡ σ, h(σ) ∝ e −µσ (property P2).Lemma 5:The scaling solution must correspond to the limit given in property P3.With h(σ) = e −µσ , the only scaling solutions of Eqs.(5)and(6)are (w, x) = (0, ±1), (0, µ/ √ 6) if f, σ /f = const. For the first two solutions, = 3. Since ekpyrosis corresponds to > 3, we only consider the third solution (property P3).Note, if f, σ /f = const ≡ −λ, there is another scaling solution at (w, x) = ± µ(λ + µ) − 6 |λ + µ| ,At late times, this solution is equivalent to one in Ref.[19]that was shown never to be an attractor.Lemma 6:The coupling f (σ) must grow monotonically.Since the solution in property P3 is a fixed-curve, it can be parameterized by one variable, z. For any finite |z|, f will be finite so that even if the system lies at (w, x) = (0, µ/ √ 6), the kinetic energy of the ekpyrotic field evolves as w, N = −3 c (µ 2 /6 − 1) √ f (cf. Eq.(5)). Thus, in any interval over which f shrinks, |w, N | grows. For this reason, we only consider solutions for which f grows monotonically (property P2). Linearizing Eqs.(5)and(6)about the background in property P3 yields δw, N = 3where δw ≡ w and δx ≡ x − µ/ √ 6. The eigenvalues of this system are (µ 2 − 6)/2, µ 2 − 6 − µf, z /f /2 . Since µf, z /f < 0, this reduced 2 × 2 system is stable, i.e., has positive eigenvalues, if |µ| > √ 6 (property P4).
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| []
|
[
"Modelling approach of a near-far-field model for bubble formation and transport",
"Modelling approach of a near-far-field model for bubble formation and transport"
]
| [
"Jürgen Geiser [email protected] \nRuhr University of Bochum\nThe Institute of Theoretical Electrical Engineering\nUniversitätsstrasse 150D-44801BochumGermany\n",
"Paul Mertin \nRuhr University of Bochum\nThe Institute of Theoretical Electrical Engineering\nUniversitätsstrasse 150D-44801BochumGermany\n"
]
| [
"Ruhr University of Bochum\nThe Institute of Theoretical Electrical Engineering\nUniversitätsstrasse 150D-44801BochumGermany",
"Ruhr University of Bochum\nThe Institute of Theoretical Electrical Engineering\nUniversitätsstrasse 150D-44801BochumGermany"
]
| []
| In this paper, we present a model based on a near-far-field bubble formation. We simulate the formation of a gas-bubble in a liquid, e.g., water and the transportation of such a gas-bubble in the liquid. The modelling approach is based on coupling the near-field model, which is done by the Young-Laplace equation, with the far-field model, which is done with a convection-diffusion equation. We decouple the small and large time-and space scales with respect to each adapted model. Such a decoupling allows to apply the optimal solvers for each near-or farfield model. We discuss the underlying solvers and present the numerical results for the near-far-field bubble formation and transport model. | null | [
"https://arxiv.org/pdf/1809.00899v2.pdf"
]
| 119,677,260 | 1809.00899 | 30e789fc2e7547cca921faa23edec88ab3985974 |
Modelling approach of a near-far-field model for bubble formation and transport
Jürgen Geiser [email protected]
Ruhr University of Bochum
The Institute of Theoretical Electrical Engineering
Universitätsstrasse 150D-44801BochumGermany
Paul Mertin
Ruhr University of Bochum
The Institute of Theoretical Electrical Engineering
Universitätsstrasse 150D-44801BochumGermany
Modelling approach of a near-far-field model for bubble formation and transport
Near-far-field approachYoung-Laplace equationConvection- diffusion equationLevel-set methodcoupling analysis ACM subject classifications F21G17G18J2 AMS subject classifications 35K2034L3065M0676M20
In this paper, we present a model based on a near-far-field bubble formation. We simulate the formation of a gas-bubble in a liquid, e.g., water and the transportation of such a gas-bubble in the liquid. The modelling approach is based on coupling the near-field model, which is done by the Young-Laplace equation, with the far-field model, which is done with a convection-diffusion equation. We decouple the small and large time-and space scales with respect to each adapted model. Such a decoupling allows to apply the optimal solvers for each near-or farfield model. We discuss the underlying solvers and present the numerical results for the near-far-field bubble formation and transport model.
Introduction
We are motivated to model bubble formation and transport in liquid, which are applied in controlled production of gas bubbles in chemical-, petro-chemical-, plasma-or biomedical-processes, see [4], [8], [12] and [5].
We consider to decompose the formation process of bubbles, we call it nearfield approach, and the transport process of bubbles, we call it far-field approach. Such a decomposition allows to separate the large scale-dependencies of the bubble formation, which has smaller time and space scales as the transport of the bubbles, which applies larger time and space scales, see [3]. For such a decomposition, we assume that the bubble is formated from an orifice in a solid surface and submerged in a liquid (viscous Newtonian liquid), see [10]. Therefore, the first process (formation) has to be finalized, when we start with the second process (transport). Such a decomposition allows to choose the optimal discretization and solver methods, i.e., we apply fast ODE-solvers for the near-field model and level-set methods for the far-field model. The paper is organized as following: The modelling problems and their solvers are presented in Section 2. The coupling of the models are discussed in Section 3 The numerical experiments are presented in Section 4. In the contents, that are given in Section 5, we summarize our results.
Mathematical Model
The mathematical model is based on a real-life experiment, where gas-bubbles are formed in a liquid and are transported after the formation process, see the plasma-experiment in [5]. The experiment is given as a thin capillary, where the gas-bubbles are streamed in an homogeneous form and transported in a tube, which is filled with liquid, see the We consider the profile of the tube and deal with the simplified approach of the experiment, which is given in Figure 2.
Based on the decoupling of formation and transport, while we assume, that the formation process is not influenced by the transport, see [1], we deal with two different decoupled models:
-Near-field approach based on a Young-Laplace equation, see [11], where we have a static shape after the formation of the bubbles. -Far-field approach based on a convection-diffusion equation, see [3], where we have a rewriting into a level-set equation, such that we could transport the static bubble-shapes, see [9].
In the following, we discuss the different models.
Periodic sources (stable bubble sources) u 1 u u 2 3 ...
Tube with water and periodical inflow−sources
u i t t 0 ∆ t ∆ t 1 2 Fig. 2.
Periodically inflow of the stable bubble sources.
Far-field approach
The first modelling approach is given with a convection-diffusion equation in cylindrical coordinate as:
∂u ∂t = −v ∂u ∂z + D L ∂ 2 u ∂z 2 + D t r ∂ ∂r (r ∂u ∂r ), (r, z, t) ∈ Ω × [0, T ],(1)u(x, z, 0) = u near (x, z), (r, z, t) ∈ Ω,(2)
where we assume u near is the solution of the bubble-formation in the near-field and we assume to have Dirichlet-boundary conditions. Here, we have the benefit and drawbacks of the modelling approach:
-Benefits:
• The model is simple and fast to compute.
• The model also allows to discuss a dynamical shape.
-Drawbacks:
• The shape of the bubble is not preserved, while we assume a static shape.
• The influence of the speed of motion in the outer normal direction is not possible.
Level-Set method
We apply an improved model, that allows to follow the shapes of the bubble, see [9]. The convection-diffusion equation is reformulated in the notation of a levelset equation, which is given as
∂u ∂t = −v · ∇u − F 0 |∇u|, (x, t) ∈ Ω × [0, T ],(3)u(x, 0) = u 0 (x),(4)u(x, t) = 0.0, (x, t) ∈ ∂Ω × [0, T ],(5)
where v is the convection vector and F 0 is the speed of motion in the outer normal direction. Further Ω is the computational domain and T is the end time.
The initialisation u(x, 0) is the results of the near-field computations. Such equations are wel-known as level-set equations and can be solved like convection-diffusion equations, see [9].
In the following, we apply the explicit different discretization methods in space, while we apply the level-set equation with the explicit time-discretisation and apply upwind methods for the advection and outer normal direction term only in the x-direction, the same is also done with the y-direction.
We have the following terms:
D − x u i,j = u i,j − u i−1,j ∆x ,(6)D + x u i,j = u i+1,j − u i,j ∆x ,(7)|D + x u i,j | = (max(D − x u i,j , 0)) 2 + (min(D + x u i,j , 0)) 2 1/2 ,(8)u n+1 i = u n i − ∆t v x u n i − u n i−1 ∆x − ∆t v y u n i,j − u n i,j−1 ∆x + ∆t F o |D + x u i,j |,(9)
where we assume v x , v y , F 0 ≥ 0. We discretize the Level-set equation with the explicit time-discretisation and apply upwind methods for the advection and outer normal direction term in the x-and y-direction and have the following terms:
D − x u i,j = u i,j − u i−1,j ∆x ,(10)D + x u i,j = u i+1,j − u i,j ∆x ,(11)D − y u i,j = u i,j − u i,j−1 ∆y ,(12)D + y u i,j = u i,j+1 − u i,j ∆y ,(13)|D + x u i,j | = (max(D − x u i,j , 0)) 2 + (min(D + x u i,j , 0)) 2(14)+(max(D − y u i,j , 0)) 2 + (min(D + y u i,j , 0)) 2 1/2 . (15) u n+1 i = u n i − ∆t v x u n i − u n i−1 ∆x − ∆t v y u n i,j − u n i,j−1 ∆x + ∆t F o |D + x u i,j |,(16)
where we assume v x , v y , F 0 ≥ 0.
Remark 1. An alternative approach of the shape transport can be done with the volume of fluid (VOF) method. Such a method is based on a free-surface modelling technique, while the method is tracking and locating the free surface, see also [6].
In the following, we discuss the near-field approach.
Near field model
The near-field model is discussed with respect to the formation of a drop or bubble, see [11] and [10]. The basic modelling idea is based on the so called Young-Laplace equation, see [2] and deals with the following simplied shape of the bubble, see Figure 3.
We deal with the following parameterisation, see [11]:
β = −ρgR 2 t /σ,(17)
where β is the Bond number, σ is the surface tension, ρ is the liquid density, g is the gravity and R t is the curvature of the drop. The near-field equations are given as: where s is the arc length along the curve and θ the angle of elevation for its slope and α is the mono-layer surface tension.
dr ds = cos(θ),(18)dz ds = sin(θ),(19)dθ ds = 2 + βz − sin(θ) r ,(20)
We have the conditions:
r = a, z = 0, at s = 0, (21) dz ds = dz dr = 0, at s = L or (r = 0),(22)
where L is the arc length of the bubble which is a-priori unknown, so here we apply r = a = 0.
Remark 2. The ODE system can be solved with numerical methods, e.g., with the MATLAB function ode45. Based on the value of R t , means the possible curvature of the bubble, we solve the half shape of the bubble and measure the different diameters.
Coupling Near-Field and Far-Field
The modelling assumes, that we could decouple the near and far-field, while we neglect the coalescence or ruptures of the bubbles, e.g., in the flow-field, see [7]. We assume that in terms of the bubble-density function:
f b (r, z, x, y, t) = u(x, y, t)δ((r − R(x, y, t)), (z − Z(x, y, t))),(23)
where u is the concentration of the bubble and R and Z are obtained with the bubble-formation equations, while r and z are the cylinder coordinates of the density function, that we do not have an influence means r ≈ R and z ≈ Z for the formation process. We discuss the following different coupling ideas:
-Parameters of the ellipse are computed in the near-field and initialise the far-field bubble. -The near-field computation is directly implemented into the far-field.
Decoupled computation of Near-and Far-Field
The near-field bubble is computed with the ODE's given in (18)-(20).
We estimate the characteristic parameters of the ellipse in the Figure 4. Based on the estimation of the elliptic-parameters, we obtain the curvature of the ellipse:
(x − x a ) 2 a 2 + (y − y b ) 2 b 2 = 1.(24)
The ellipse is the curvature of the far-field, which is computed with the levelset method.
Remark 3. The transformation of the elliptic parameters of the near-field model allows to simplify the construction of the shape in the far-field. We only apply the ellipses in the far-field transport modell.
Numerical Experiments
In the following, we apply the bubble-formation based on the simplified model, i.e., Young-Laplace equation, and the bubble-transport model, based on the level-set equations.
Bubble-Formation: Experiment 1
The near-field equations are given as:
dr ds = cos(θ),(25)dz ds = sin(θ),(26)dθ ds + sin(θ) r = ∆p α ,(27)
where s is the arc length along the curve and θ the angle of elevation for its slope and α is the mono-layer surface tension. We have the conditions:
r = a, z = 0, at s = 0, (28) dz ds = dr ds = 0, at s = L or (r = 0),(29)
where L is the arc length of the bubble which is a-priori unknown, so here we apply r = 0. We deal with the following domain:
(r, z) ∈ [0, b] × [0, d], where a = 1, b = 3, d = 3. Further α = 0.1, ρ = 0.1, g = 9.81, p tube = 0.001.
The numerical results of bubble-formation is given in Figure 5.
Bubble-Formation: Experiment 2
In the following, we couple the near-field and far-field computations.
We have the following setting, see Figure 6. The numerical results of far-field bubble-transport is given in Figure 10. We apply the following parameters:
-Input-parameters of the near-field bubble computation: r 0 = 10, z 0 = 0, θ 0 = 0, α = 0.2, ρ = 0.1, g = 9.81, ∆p = 0.9. The numerical results of the near-far-field coupled bubble-transport code, which is given in Figure 11. Remark 5. The coupling of the formation and transport of the bubbles are done with ordinary and partial differential equations. Based on decoupling such systems of mixed ordinary and partial differential equations, we could compute each separate part with the optimal numerical solvers.
Bubble-Formation: Multiple Bubble Experiment (2 Bubbles)
In the following, we couple the near-field and far-field computations with multiple bubbles.
We have the following setting, see Figure 9.
The numerical results of far-field bubble-transport is given in Figure 10. We apply the following parameters:
-Computation of the near-field bubbles (a representing bubble is computed)
• Input-parameters of the near-field bubbles computation: a bubble2 = 0.4720, b bubble2 = 1.2147.
• Ellipse: (x − x bubblei ) 2 + ((y − y bubblei ) * a bubble /b bubble ) 2 − a 2
bubble , where (x bubblei , y bubblei ) is the origin of the i-th bubble.
-Computation of the far-field bubbles (level-set initialisation):
• Parameterisation of the level-set initial-function, e.g., two bubbles:
φ 0 (x, y) = (x − x bubble1 ) 2 + ((y − y bubble1 ) a bubble 1 b bubble 1 ) 2 − a 2 bubble1 , a x ≤ x ≤ 50, a y ≤ x ≤ b y , (x − x bubble2 ) 2 + ((y − y bubble2 ) a bubble 2 b bubble 2 ) 2 − a 2 bubble2 , 50 ≤ x ≤ b x , a y ≤ x ≤ b y ,(30)
where (x bubble1 , y bubble1 ) = (20, 50), (x bubble2 , y bubble2 ) = (80, 50) with the coordinates of the grid (a x , a y ) = (0, 0) and b x , b y ) = (100, 200).
The numerical results of the near-far-field coupled bubble-transport code, which is given in Figure 11. Remark 6. The level-set method allows to deal with different level-set functions, such that we could transport multiple bubbles.
Bubble-Formation: Multiple Bubble Experiment (10 Bubbles)
In the following, we extend the near-field and far-field computations with 10 bubbles. We also apply the decomposition of near-field and far-field computations as given in Figure 9.
We apply the following parameters: -Computation of the near-field bubbles (a representing bubble is computed)
• Input-parameters of the near-field bubbles computation: * Bubble 1: r 0 = 0.1, z 0 = 0, θ 0 = 0, α = 0.2, ρ = 0.1, g = 9.81, σ = 0.2, r t = 0.05. r 0 = 0.1, z 0 = 0, θ 0 = 0, α = 0.2, ρ = 0.1, g = 9.81, σ = 0.2, r t = 0.14.
Conclusion
We present a bubble model, which is a coupled model based on a bubble formation and bubble transport model. The decoupling into near-and far-field models allows to apply optimal solver and discretization methods. We apply different numerical experiments, which shows the benefit of such a treatment. In future, we consider the fully coupled problem, while we deal with bubble density functions and the coupling between the formation and transport process. Such an extension allows to see the ruptures of the bubbles.
Figure 1 .
1
Fig. 1 .
1Sketch of the real-life experiment (capillary with gaseous outflow into a tube filled with water).
Fig. 3 .
3Near field parameters of the bubble shape.
Fig. 4 .
4Final bubble based on the near-field computation and estimation of the bubble-parameters (we assume an elliptic curve).
Fig. 5 .
5Bubble-formation, left figure with parameters a = 1, b = 3, d = 3, α = 0.1, ρ = 0.1, g = 9.81, p tube = 0.001 and right figure with parameters a = 10, b = 3, d = 3, α = 0.2, ρ = 0.1, g = 9.81, p tube = 0.4.
Remark 4 .
4The Young-Laplace equation allows to formulate the bubble-formation such that we could obtain the radii of the different bubbles based on the various pressure parameters.
Fig. 6 .
6Left figure: Bubble-formation with the ODEs and right figure: Bubbletransport with the PDEs (level-set equations).
-
Output-parameters of the near-field bubble computation: a bubble = 0.1825, b bubble = 0.2216. -Ellipse: (x − 50) 2 + ((y − 50) * a bubble /b bubble ) 2 − a 2 bubble .
Fig. 7 .
7Left figure with bubble formation (near-field) and right figure with the bubble transport (far-field).
Fig. 8 .
8Upper figures: Transport of the first bubble with the level-set function, lower figures: Transport of the second bubble with the level-set function.
*Fig. 9 .
9Bubble 1: r 0 = 0.1, z 0 = 0, θ 0 = 0, α = 0.2, ρ = 0.1, g = 9.81, σ = 0.2, r t = 0.05. Left figure: Bubble-formation with the ODEs and right figure: Bubbletransport with the PDEs (level-set equations). * Bubble 2: r 0 = 0.1, z 0 = 0, θ 0 = 0, α = 0.2, ρ = 0.1, g = 9.81, σ = 0.2, r t = 0.2. • Output-parameters of the near-field bubble computation: * Bubble 1: a bubble1 = 0.5051, b bubble1 = 0.9909. * Bubble 2:
Fig. 10 .
10Left figure with bubble formation (near-field) and right figure with the bubble transport (far-field).
Fig. 11 .
11Multi-bubble transport with the level-set function.
*
Bubble 2: r 0 = 0.1, z 0 = 0, θ 0 = 0, α = 0.2, ρ = 0.1, g = 9.81, σ = 0.2, r t = 0.06. * Bubble 3: r 0 = 0.1, z 0 = 0, θ 0 = 0, α = 0.2, ρ = 0.1, g = 9.81, σ = 0.2, r t = 0.07. * Bubble 4: r 0 = 0.1, z 0 = 0, θ 0 = 0, α = 0.2, ρ = 0.1, g = 9.81, σ = 0.2, r t = 0.08. * Bubble 5: r 0 = 0.1, z 0 = 0, θ 0 = 0, α = 0.2, ρ = 0.1, g = 9.81, σ = 0.2, r t = 0.1. * Bubble 6: r 0 = 0.1, z 0 = 0, θ 0 = 0, α = 0.2, ρ = 0.1, g = 9.81, σ = 0.2, r t = 0.12. * Bubble 7:
Fig. 12 .
12Transport of 10 bubbles with the level-set function.
Remark 7. In the experiment, we deal with at least 10 bubbles, which are different formated and transported via the level-set method. The numerical experiments allows to accelerate the formation and transport of such processes.
* Bubble 8: r 0 = 0.1, z 0 = 0, θ 0 = 0, α = 0.2, ρ = 0.1, g = 9.81, σ = 0.2, r t = 0.16. * Bubble 9: r 0 = 0.1, z 0 = 0, θ 0 = 0, α = 0.2, ρ = 0.1, g = 9.81, σ = 0.2, r t = 0.18. * Bubble 10: • Parameterisation of the level-set initial-function, e.g., two bubbles:where (x bubble1 , y bubble1 ) = (20, 50), (x bubble2 , y bubble2 ) = (80, 50) with the coordinates of the grid (a x , a y ) = (0, 0) and b x , b y ) = (100, 200).The numerical results of the near-far-field coupled bubble-transport code, which is given inFigure 12.
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Volume of fluid (VOF) method for the dynamics of free boundaries. C W Hirt, B D Nichols, Journal of Computational Physics. 391C.W. Hirt and B.D. Nichols. Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics, 39 (1):201-225, 1981.
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The formation of a bubble from a submerged orifice. J A Simmons, J E Sprittles, Y D Shikhmurzaev, European Journal of Mechanics -B/Fluids. 53Supplement CJ.A. Simmons and J.E. Sprittles and Y.D. Shikhmurzaev. The formation of a bubble from a submerged orifice. European Journal of Mechanics -B/Fluids, 53, Supplement C, 24-36, 2015.
The coalescence speed of a pendent and a sessile drop. S T Thoroddsen, K Takehara, T G Etoh, Journal of Fluid Mechanics. 527Cambridge University PressS.T. Thoroddsen, K. Takehara, and T.G. Etoh. The coalescence speed of a pendent and a sessile drop. Journal of Fluid Mechanics, Cambridge University Press, 527: 85-114, 2005.
Bubble formation and dynamics in gasliquidsolid fluidization -A review. G Q Yang, B Du, L S Fan, Chemical Engineering Science. 62G.Q. Yang, B. Du, and L.S. Fan. Bubble formation and dynamics in gasliquidsolid fluidization -A review. Chemical Engineering Science 62, 2-27, 2007.
| []
|
[
"BUBBLE TREE COMPACTIFICATION OF MODULI SPACES OF VECTOR BUNDLES ON SURFACES",
"BUBBLE TREE COMPACTIFICATION OF MODULI SPACES OF VECTOR BUNDLES ON SURFACES"
]
| [
"D Markushevich ",
"A S Tikhomirov ",
"G Trautmann "
]
| []
| []
| In this article we announce some results on compactifying moduli spaces of rank-2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example the compactification of the space of stable rank-2 vector bundles with Chern classes c 1 = 0, c 1 = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers. | 10.2478/s11533-012-0072-0 | [
"https://arxiv.org/pdf/1110.6525v1.pdf"
]
| 119,283,150 | 1110.6525 | 852a88939d42280c58e389bf579e462309ac5704 |
BUBBLE TREE COMPACTIFICATION OF MODULI SPACES OF VECTOR BUNDLES ON SURFACES
29 Oct 2011
D Markushevich
A S Tikhomirov
G Trautmann
BUBBLE TREE COMPACTIFICATION OF MODULI SPACES OF VECTOR BUNDLES ON SURFACES
29 Oct 2011AMS 2010 Subject Classification: 14J6014D2014D21
In this article we announce some results on compactifying moduli spaces of rank-2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example the compactification of the space of stable rank-2 vector bundles with Chern classes c 1 = 0, c 1 = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.
Introduction
In this article, we describe a conceptual scheme of a new construction of a compactification of moduli spaces of stable bundles on surfaces whose boundary consists of vector bundles on trees of surfaces, replacing the torsion free semistable sheaves appearing in the Gieseker-Maruyama compactification. In this description the long proofs of completeness, separatedness, versality, properness are replaced by brief sketches, and the complete versions will appear in full detail in subsequent papers. In conclusion, we produce a concrete example of the compactification of the moduli space of stable bundles on the projective plane P 2 with second Chern class c 2 = 2. In this example, we provide an alternative explicit construction of the same compactification and give a complete description of its boundary.
To some extent, the replacement of limit sheaves in a compactification by vector bundles on trees of bubbles is very natural. The bubbling phenomenon appeared in eighties and nineties in the description of degeneration processes in several conformally invariant problems of geometric analysis: minimal surfaces (Sacks-Uhlenbeck), harmonic maps (Parker), pseudoholomorphic curves in symplectic varieties (Parker-Wolfson, Rugang Ye), and Yang-Mills fields on 4-manifolds (Feehan [Fe], Taubes [Ta], Uhlenbeck [U]).
Donaldson-Uhlenbeck constructed a (partial) compactification Y M n of the moduli space Y M n of instantons of charge n on a 4-manifold S, where the instantons are defined as the ASD Yang-Mills connections on a vector bundle over a 4-manifold S with Chern classes c 1 = 0, c 2 = n: Y M n ⊂ Y M n ∪ (Y M n−1 × X) ∪ (Y M n−2 × S 2 X) ∪ · · · ∪ S n X,
In [MT1] and [MT2] it was shown that in the case of S = S 4 , Y M n has a real semi-algebraic structure.
The boundary of Y M n consists of "ideal instantons", that is, singular connections whose curvature is a sum of a smooth part and of several delta-functions. A degenerating family of bundles and connections on them encloses more information than is kept by an "ideal instanton". The bubble-tree compactification of Feehan-Taubes-Uhlenbeck (FTU) is a kind of blowup of the boundary of Y M n , encoding the way an ASD connection degenerates into an ideal instanton by a connection over a tree of surfaces, obtained from S by successive gluings of spheres S 4 at a finite set of points.
When S is a complex projective surface, Donaldson proved that the Kobayashi-Hitchin correspondence identifies Y M n with the moduli space of µ-stable vector bundles M µ-s (0, n) with c 1 = 0, c 2 = n on S ( [D1]; see also [LT] for further developments). Thus the natural question can be asked, whether the Uhlenbeck-Donaldson and the FTU compactifications also have an algebro-geometric interpretation. For the first one, the answer is known: Jun Li [JL] endowed Y M n with a structure of a quasi-projective scheme and defined a birational morphism M µ-s (0, n) → Y M n , where the closure of M µ-s (0, n) is taken in the Gieseker-Maruyama moduli space of semistable sheaves on S. See also [HL], Sect. 8.2, where a similar compactification, called moduli space of µ-semistable sheaves, is constructed for the sheaves with arbitrary determinant.
The main motivation of our work is to find an algebro-geometric analog of the FTU compactification, which would be a kind of blowup of M µ-s (0, n). By analogy with the topological bubbles which are 4spheres, we can introduce the notion of an algebraic bubble. Given a complex surface S and its blowup S in a point p with the exceptional line L, an algebraic bubble is a complex projective plane P = P 2 (C), attached to S along L. Algebraic bubble trees are obtained by iterating this construction. They appear as fibers of the semi-universal family over the compactified configuration spaces of Fulton-MacPherson [FM].
In the realm of algebraic geometry, the degenerations are described in terms of flat families. It turns out that using flat deformations, one can "replace" singular sheaves F on S by bundles on the trees of surfaces S T , and that the thus obtained tree bundles together with bundles on the original surface S fit into a separated algebraic space of finite type. This is the main result of the paper, stated as Theorem 7.2. In some particular cases, for example, if S = P 2 , we can assert that this algebraic space is proper (Theorem 7.3).
This result still does not contain an answer to the question which served its motivation, by the following reasons. First, our construction does not provide any morphism between our moduli space and the Gieseker-Maruyama compactification in any direction. Second, though one can associate a topological bubble tree with 4-spheres as bubbles to any algebraic bubble tree by contracting to points the intersection lines of the algebraic bubbles, this does not lead to a correspondence between the bundles on algebraic bubble trees and the ASD connections on the associated spherical bubble trees. The problem is that the stock of treelike bundles we admit in our compactification contains bundles which are nontrivial on the intersection lines of bubbles, so there is no way to push them forward along the topological contraction in order to get a bundle on a spherical bubble tree. This is certainly a drawback of our approach based on the Serre construction for flat families of rank 2 bundles over a curve. It is a challenging problem to find another approach which would bring us to a stock of boundary bundles, trivial on the intersections of components.
We will briefly mention some related work. Tree-like bundles were used with the opposite goal by D.Gieseker in [G2] in order to construct bundles on S by deforming bundles on trees to bundles on S. Over curves, Nagaraj and Seshadri [NS] considered bundles on a degenerating family of curves and compactified them by bundles on reducible curves, pasting in trees of rational components. Buchdahl [B1], [B2] studied, by differential geometric means, a compactification of degenerating bundles on S by bundles on a blowup of points in S (which is an irreducible surface, unlike our bubble trees).
In this article, S is always a smooth complex projective surface, endowed with a very ample polarization class h, and M S,h (2; N , n) denotes the Gieseker-Maruyama moduli space of semistable torsion free sheaves E of rank 2 on S with fixed Chern classes c 1 (E) = N and c 2 (E) = n ≥ (N 2 )/4, where "semistable" means "Gieseker semistable with respect to h". The space we are compactifying is the open subspace M s S,h (2; N , n) of stable vector bundles, assumed nonempty. If not otherwise stated, all the schemes are locally of finite type over C, and the base of any family we consider is always assumed to be a scheme.
Trees of surfaces
2.1. Trees. A tree T in this article is a finite graph, oriented by a partial order < and satisfying:
• there is a unique minimal vertex α ∈ T , the root of T ;
• for any a ∈ T, a = α, there is a unique maximal vertex b < a, the predecessor of a, denoted by a − ; • By a + := {b ∈ T | b − = a} we denote the set of direct successors of a ∈ T . We let T top denote the vertices of T without successor.
A weighted tree is a pair (T, c) of a tree T with a map c which assigns to each vertex a ∈ T an integer n a , called the weight or charge of the vertex, subject to the conditions
(1) n a ≥ 0 if a = α,(2)
#a + ≥ 2 if n a = 0 and a = α,
(3) n α ≥ C,
where C ≤ 0 is some constant depending on S, h and N , as specified below in formula (10). The total weight or total charge of a weighted tree is the sum Σ a∈T n a = n of all the weights. We denote by T n the set of all trees which admit a weighting of total charge n. It is obviously finite.
2.2. Trees of surfaces. Let S be a smooth complex projective surface with an ample invertible sheaf O S (h). Our trees of surfaces are reduced surfaces S T , defined for any tree T and whose components are indexed by the vertices of the tree T . They are constructed by the following data.
(i) For each vertex a ∈ T top ∪{α}, let P a be a copy of P 2 (C) together with a line l a ⊂ P a and a finite subset Z a = {x b ∈ P a l a | b ∈ a + }, and let S a σa − → P a be the blowup of P a along Z a . We will denote the exceptional lines in S a byl b , b ∈ a + , and the inverse image of l a in S a by the same symbol l a . (ii) For a = α, we set S α σα − → S to be the blowup of S at a finite set Z α = {x a ∈ S| a ∈ α + }.
(iii) For a ∈ T top , S a is a copy of P 2 (C).
(iv) For each a > α, we fix some isomorphisml a φa − → l a .
A tree-like surface of type T over S or a T -surface is now defined as the result S T of gluing the above surfaces S a along the isomorphisms φ a into a reduced connected normal crossing surface with components S a . We write S T = ∪ a∈T S a and identifyl a with l a for all a ∈ T . After this identification the lines l a are the intersections l a = S a ∩ S a − .
By the construction of S T , all or a part of its components can be contracted. In particular, there is the morphism
(4) S T σ − → S
which contracts all the components except S α to the points of the finite set Z α .
Note that: 1) There are no intersections of the components other than the lines l a .
2) If a ∈ T top then S a is a plane P 2 (C).
3) If T = {α} is trivial, then S T = S. 4) After contracting the lines l a topologically (over C), one obtains a tree of 4-sphere bubbles.
2.3. Line bundles on T -surfaces. Let now L be a line bundle on a T -surface S T , and let the inverse image of the divisor class h on S α also be denoted by h. Then the restrictions of L to the components can be written as
L|S α = O Sα (mh − Σ a∈α + m a l a )
and
L|S a = O Sa (m a l a − Σ b∈a + m b l b ).
L is called ample if each L|S a is ample for all a ∈ T . This means that m, m a > 0 and m 2 > Σ a∈α + m 2 a and m 2 a > Σ b∈a + m 2 b . for all a ∈ T. Let m, r be positive integers. We will say that L is of type (r, m, h) if its restriction to the root surface is given by L|S α = O Sα (rmh − rΣ a∈α + m a l a ) for some m a (a = α).
We define the multitype of a line bundle on a T-surface S T to be the sequence
m T := (m a ) a∈T ,
where m α = m The above inequalities imply 2.4. Lemma.
1) For any m, r > 0 and any weighted tree (T, c), T ∈ T n , there are at most finitely many ample line bundles of type (r, m, h). 2) Given n, there is an integer m 0 such that for any m ≥ m 0 and any (T, c), T ∈ T n , there is an ample line bundle on S T of type (1, m, h).
Tree-like bundles
3.1. Definition. Let S T = ∪ a∈T S a be a T -surface with tree T ∈ T n . A vector bundle E = E T on S T or the pair (E T , S T ) is called a tree of vector bundles or a T -bundle if the restrictions E a = E|S a , a ∈ T , satisfy the following conditions:
(i) E a has rank 2, c 1 (E a ) = 0 for a = α, c 1 (E α ) = σ * α N , and the second Chern classes c 2 (E a ) = n a , a ∈ T, define a weighting of T in the sense of definition 2.1, (2). (ii) If T = {α}, then E a is admissible as defined below for any a ∈ T.
(iii) If T = {α}, then [E] ∈ M S,h (2; N , n).
3.2. Definition. Let S T = ∪ a∈T S a be a T -surface with tree T ∈ T n . Assume that T = {α}. Let a ∈ T , and let E a be a rank-2 vector bundle on S a . We will say that E a is admissible if one of the following conditions is satisfied:
(i) In case a ∈ T top , with S a a plane P a , E a is an extension of type
(5) 0 → O Pa → E a → I x,Pa → 0, x = {pt} ∈ l a , c 2 (E a ) = 1,
(ii) or in case a ∈ T top , with S a a plane P a , E a is an extension of type
(6) 0 → O Pa (−1) → E a → I Z,Pa (1) → 0, where dim Z = 0, Z ∩ l a = ∅, c 2 (E a ) = length(Z) − 1 ≥ 2. (iii) In case a ∈ T top , a = α, |a + | ≥ 2, E a is a non-split extension of type (7) 0 → O Sa (−l a + Σ b∈a + l b ) → E Sa → I Z,Sa (l a − Σ b∈a + l b ) → 0, where dim Z ≤ 0, Z ⊂ S a {( ∪ b∈a + l b ) ∪ l a }, c 2 (E Sa ) = length(Z) + |a + | − 1 ≥ 1,(
iv) or a ∈ T top , a = α, and E a is a non-split extension of the type
(8) 0 → O Sa ( Σ b∈a + l b ) → E a → O Sa (− Σ b∈a + l b ) → 0, c 2 (E a ) = |a + | ≥ 1, (v) or a ∈ T top , a = α, and E a = 2O Sa .
(vi) In case a = α, E α is a non-split extension of type
(9) 0 → O Sα (−qh + Σ b∈α + 0 l b ) → E α → I Z,Sα (qh − Σ b∈α + 0 l b + σ * α N ) → 0,(10)c 2 (E α ) ≥ C := −q 2 0 (h 2 ) − q 0 |(h · N )|, for some subset α + 0 of α + , where 0 ≤ q ≤ q 0 for some integer q 0 depending on S, and where dim Z ≤ 0, Z ⊂ S α ( ∪ b∈α + l b ), length(Z) ≤ n + q 2 0 (h 2 ) + q 0 |(h · N )|.
Remark. The above definitions single out a possibly redundant class of vector bundles including all the bundles which may occur in degenerations. The conditions (i)-(vi) guarantee the boundedness of the family of tree-like bundles and replace the (semi)stability conditions, which are not obvious for tree-like bundles. Those T -bundles which indeed occur in degenerations will be called limit T -bundles, see Definition 4.5.
Notation. Let E = E T be a T -bundle on S T for a tree T ∈ T n . We also write E T = # a∈T E a and define its total second Chern class to be
c 2 (E T ) := Σ a∈T c 2 (E a ) = Σ a∈T n a = n.
Two T -bundles (E T , S T ) and (E ′ T , S ′ T ) are called isomorphic if there exists an isomorphism φ :
S T → S ′ T over S such that E T = φ * E ′ T .
In the following we need formulas for the Euler characteristics of line bundles and T -bundles. These follow by standard computations on the components of the trees.
Lemma.
For any T -bundle (E T , S T ), T ∈ T n , and any ample line bundle L on S T of type (1, m, h), the following Euler characteristics are independent of the tree and are given by the formulas
(11) χ(E ⊗ L) = m 2 (h 2 ) + m(h · (2N − K S )) + 1 2 (N · (N − K S ))+ 2χ(O S ) − n =: N m (12) χ(L) = m 2 2 (h 2 ) − m 2 (h · K S ) + χ(O S )
Later we will consider the embeddings of S T into the Grassmannian G = Grass(N m , 2) of 2-dimensional quotient spaces of the space C Nm , defined by the global sections of an appropriate twist of tree-like bundles E T on S T . Here N m denotes the integer given by (11). The universal rank-2 quotient bundle on G will be denoted by Q. We have the following boundedness result.
3.4. Proposition (Boundedness). For any n there is an integer m 0 > 0 such that, for any m ≥ m 0 and any T ∈ T n , there is an ample line bundle L of type (1, m, h) on S T such that for any T -bundle E T on S T ,
(i) h i (E T ⊗ L) = 0 for i > 0 and h 0 (E T ⊗ L) = χ(E T ⊗ L) = N m ; (ii) the evaluation map C Nm ⊗ O S T ։ E T ⊗ L is surjective; (iii) the induced map S T i G − → G = Gr(C Nm , 2) is a closed embedding such that i * G Q ≃ E T ⊗ L , i * G O G (1) ≃ L ⊗2 ⊗ O S T (σ * N ); (iv) for any j, q > 0, h j (i * G O G (q)) = 0, and the Hilbert polynomial P G (q) := χ(i * G O G (q)) is given by the formula (13) P G (q) = 2q 2 m 2 (h 2 ) + 2q 2 m(h · N ) − qm(h · K S ) + 1 2 q 2 (N 2 ) − 1 2 q(N · K S ) + χ(O S (N )).
Proof. This follows from Serre's theorems A and B, the boundedness of the family of all admissible bundles of given type on each S a , which follows easily from the definition, and from the boundedness of the family of h-semistable vector bundles [Sim] with c 1 = N and c 2 ≤ n on S. (ii) follows from the above and [Tr,lemma 5.13]. Formula (13) follows directly from 3.3.
Families of tree-like bundles
In this section we fix the definition of families of T -surfaces S T and T -bundles (E T , S T ) for trees T ∈ T n with fixed total charge n.
4.1. Definition (T n -families of surfaces). 1) Let X π → Y be a flat family of trees of surfaces over a scheme Y locally of finite type, whose trees belong to T n . Such a family is called a family of trees of surfaces of type T n , or simply a T n -family, if there exists a morphism σ : X → S × Y such that the following holds:
(i) π = pr 2 • σ. (ii) For each closed point y ∈ Y the morphism σ y = σ|S y : S y → S × {y} ≃ S, where S y = π −1 (y), is the standard contraction (4), (iii) There is a union of irreducible components X b of X such that the restriction σ|X b is a birational morphism X b → S × Y.
The morphism σ : X → S×Y will also be called a standard contraction.
2) A T n -family of surfaces X π → Y is called a good T n -family of surfaces if σ is birational on the whole of X.
3) A T n -family of surfaces
X π → Y is called trivial if σ is an isomor- phism. 4) Two T n -families X π → Y and X ′ π ′ → Y over the same base Y are called isomorphic if there exists an isomorphism X φ − → X ′ such that (14) π ′ • φ = π and σ ′ • φ = σ, where X σ → S × Y and X ′ σ ′ → S × Y are the standard contractions. 4.2. Examples. 1) The standard contraction X = S T → S of a single T -surface for a tree T ∈ T n is a T n -family over a point with X b = S α , σ|X b = σ α : S α → S. This T n -family is good only if T = {α}.
2) Let C be a smooth curve and let X → S × C be the blowup of a point (s, c). Then X → C is a good T 1 -family.
3) Let X = S T × Y be the product of a T -surface with some scheme Y locally of finite type, T ∈ T n , T = {α}. Then X → Y is a T n -family with X b = S α × Y which is not good.
For families over smooth curves we have the following 4.3. Theorem. Let X → C be a good T n -family of surfaces over a smooth curve C. Then X has at most A k -singularities, analytically locally trivial along the lines of intersection of components in the fibres of X.
The A k -singularities in T n -families over curves really appear in the construction of limit bundles in the proof of the Completeness Theorem 4.9 as a result of certain contractions of tree-like surfaces.
Definition (T n -families of bundles). (a) For a fixed total second
Chern class n, a T n -family of tree-like bundles is given as a triple (E/X/Y ), where X/Y denotes a T n -family X π → Y of surfaces and E is a vector bundle on X, such that its restriction to all the components of all the fibers of π over the closed points of Y are admissible.
(b) Let now X π → Y be a good T n -family of surfaces. By Definition 4.1, (ii), there exists a maximal dense open subset U of Y with the property that the fibers X y = π −1 (y) are isomorphic to S for all closed points y ∈ U. A triple (E/X/Y ) as in (a) will be called a good T n -family of (tree-like) bundles if the restrictions of E to the fibers X y over the closed points y ∈ U are stable vector bundles from M s S,h (2; N , n). There is an obvious notion of isomorphy and equivalence for T nfamilies. Two T n -families of bundles (E/X/Y ) and (
E ′ /X ′ /Y ) are called isomorphic, (E/X/Y ) ≃ (E ′ /X ′ /Y ), if there is an isomorphism X φ − → X ′ of T n -families of surfaces such that E ≃ φ * E ′ . The families are called equivalent if there is an isomorphism X φ − → X ′ and an invertible sheaf L on Y such that (E ⊗ π * L/X/Y ) and (E ′ /X ′ /Y ) are isomorphic.
Definition (Limit bundles).
A T -bundle (E T , S T ) of total charge n is called a limit T -bundle if there exists a (germ of a) smooth pointed curve (C, 0) and a good T n -family of bundles (E/X/C) as defined in
4.4 (b), such that X|C {0} σ − → S × (C {0}) is an isomorphism, E|C {0}
is a family of stable vector bundles from M s S,h (2; N , n) and E T ≃ E|X 0 on the fibre of X over 0 ∈ C.
A limit T -bundle is called sss-limit T -bundle if there exists a (germ of a) smooth pointed curve (C, 0) and a T n -family of bundles (
E/X/C) as defined in 4.4 (a), such that X|C {0} σ − → S × (C {0}) is an isomorphism, E|C {0} is a family of strictly semi-stable vector bundles from M S,h (2; N , n) and E T ≃ E|X 0 on the fibre of X over 0 ∈ C.
Let M n (pt) denote the set of all limit T -bundles with T ∈ T n and M n (pt) the set of their isomorphism classes. More generally, we give the following definition. 4.6. Definition (Moduli stack and moduli functor). For any Y as above, we denote by M n (Y ) the set of all T n -families (E/X/Y ) of limit bundles. Given a morphism f : Y ′ → Y , it is obvious how to define the pullback f * (E/X/Y ) of a family (E/X/Y ) and it is easy to verify that this is again a T n -family of limit bundles. Thus M n is a pseudofunctor M n : (Sch/C) op → (Sets) in the language of stacks, where in our setting (Sch/C) denotes the category of complex schemes locally of finite type over C. By definition, M n (pt) can be identified with M n (Spec(C)). To get a functor, we define M n (Y ) := M n (Y )/ ∼, where ∼ denotes equivalence.
Open sub-pseudo-functors of M n .
Consider the open subpseudo-functors M g n , M t n , M 0 n and M s n of the pseudo-functor M n : (Sch/C) op → (Sets), defined as follows:
M g n (Y ) = {(E/X/Y ) ∈ M n (Y ) | ∀ y ∈ Y, E y is not an sss-limit bundle}, M t n (Y ) = {(E/X/Y ) ∈ M n (Y ) | X → Y is a trivial T n -family}, M 0 n (Y ) = M g n (Y ) ∪ M t n (Y ), M s n (Y ) = {(E/X/Y ) ∈ M n (Y ) | X → Y is a trivial T n -family, and ∀ y ∈ Y, E y ∈ M s S,h (2; N , n)} (Y ∈ Sch/C).
Obviously, for any Y ∈ Sch/C,
(15) M s n (Y ) = M g n (Y ) ∩ M t n (Y ).
The pseudofunctors M g n , M t n , M 0 n and M s n and their associated functors M g n , . . . will be used in the construction of the moduli spaces M g n , M 0 n , see Section 7.
4.8. Lift of families. In the proofs of the main results one mostly has to deal with good families of tree-like bundles over curves. We use the standard notation (C, 0) for a curve C with a marked point 0 ∈ C and denote by C * the punctured curve C {0}. By a finite covering of curves τ : (C, 0) → (C, 0) we understand a finite morphism τ :C → C such that τ (0) = 0.
For a given curve (C, 0) and a given family of tree-like bundles F = (E/X/C) ∈ M n (C) over C we denote the restriction of F onto C * by
F * = (E * /X * /C * ) ∈ M n (C * ), where X * = C * × C X, π * = π|X * , σ * = σ|X * , E * = E|X * . Respectively, we denote the lift of F = (E/X/C) ∈ M n (C) toC bỹ F =C × C F = (Ẽ/X/C) ∈ M n (C),
whereX =C × C X, and X λ ←Xπ →C denote the natural projections withẼ = λ * E.
Moreover, the notation for the liftFφ − →F ′ toC of an isomorphism F φ − → F ′ over C should be selfexplaining.
The proofs of properness and separatedness of the moduli space we are going to construct are reduced to the respective properties of the pseudofunctor M n over the smooth curves, basing upon the valuative criteria for properness and separatedness. Thus the following completeness and separatedness theorems for the pseudofunctor M n are key results for the existence of a proper moduli space of tree-like bundles. The proofs can only roughly be indicated in this note.
4.9. Completeness Theorem. The pseudo-functor M n is complete in the following sense. For any smooth pointed curve (C, 0) and any family of tree-like bundles (E/C * × S/C * ) ∈ M n (C * ), there is a finite covering
( C, 0) τ − → (C, 0)
and a family ( E/ X/ C) ∈ M n ( C) of tree-like bundles together with an isomorphism ϕ,
C × S X o o ? _ X * ϕ ≈ / / C * × S { { w w w w w w w w w C C o o ? _ C * such that E| X * ∼ = ϕ * (τ × id S ) * E.
Sketch of the proof.
Step 1: Let X = C × S. We use the description of reflexive rank-2 sheaves F on X via the Serre construction 1 in the relative situation:
0 → O X (−qh) s − → F → I Z (qh + N ) → 0, where Z is finite over C.
Consider the closure M s S,h (2; N , n) of the moduli space of stable vector bundles M s S,h (2; N , n) in the projective scheme of Gieseker-Maruyama of χ-semistable torsion-free sheaves. Using its construction and projectivity, we can replace C by a finite covering and assume that there is a reflexive sheaf E over the whole of X such that:
• E|C * × S is the given bundle;
• there is a number q with an exact sequence
0 → O X (−qh) → E → I Z (qh + N ) → 0 ;
• Z is reduced, smooth over C * , and Z → C is flat and finite.
Step 2:
Smoothing Z by blowups of points of Z ∩ ({0} × S) in the 3-fold X = C × S.
Step 3: Separating the components of Z by lifting the families over finite base changes and by further blowing up, thus getting "bubbles" and separating the corresponding extension sequences of the Serre construction.
Step 4: Contracting superfluous bubbles in the 3-fold obtained in Step 3, thereby producing a 1-parameter family of tree-like surfaces whose total space is a three-dimensional variety having at worst curves of A k singularities.
Step 5: The case of a T n -family (E/X * /C * ) such that all the fibres of (X * /C * ) are reducible. Since all the bundles E y , y ∈ C, are limit bundles, after a possible shrinking of C * , there exists a surface Y 0 containing C * and a T n -bundle
(E 0 /X Y 0 /Y 0 ) ∈ M n (Y 0 ) such that, for Y * = Y 0 C * one has X Y 0 × Y 0 Y * ≃ S × Y * . Extend Y 0 to a surface Y containing C and, respectively, extend (E 0 |S × Y * /S × Y * /Y * ) to a family of Gieseker semistable sheaves (E Y /S × Y /Y ). As in
Step 1, represent E Y as a sheaf obtained by the Serre construction, applied to some subscheme Z Y of S × Y , finite over Y . Then take Z = Z Y × Y C and apply Steps 2-4 to the family (E Y |S × C/S × C/C). As a result, possibly after a base change over C, we obtain a T n -family ( E/ X/C) extending (E/X * /C * ).
4.10. Separatedness Theorem. The pseudofunctor M g n is separated in the following sense: Let (C, 0) be a smooth pointed curve and let F = (E/X/C), F ′ = (E ′ /X ′ /C) ∈ M g n (C) be two families of tree-like bundles. Suppose that there is an isomorphism
X * = C * × C X σ 1 ( ( P P P P P P P P P P P P φ 0 ∼ / / C * × C X ′ = X ′ * σ 2
v v n n n n n n n n n n n n C * × S such that the restrictions F * = (E * /X * /C * ), F ′ * = (E ′ * /X ′ * /C * ) ∈ M g n (C * ) of F, F ′ over C * are isomorphic via φ 0 , i.e. there exists an isomorphism of vector bundles ψ 0 : E * −→ ∼ φ * 0 E ′ * . Then φ 0 extends to an isomorphism φ : X−→ ∼ X ′ , and there exists an invertible sheaf L on C with an isomorphism ρ :
L|C * −→ ∼ O C * such that the isomorphism ψ 0 ⊗ π * 0 (ρ) : E * ⊗ π * 0 (L|C * )−→ ∼ φ * 0 E ′ * extends to an isomorphism E ⊗ π * L−→ ∼ φ * E ′ , which provides an isomorphism of the T n -families of bundles (E ⊗ π * L/X/C) ≃ (E ′ /X ′ /C).
The proof is rather elaborate, and what follows gives a brief idea of it. The first step is blowing up X, X ′ to obtain a modelX, smooth over C and dominating both X, X ′ . There is an isomorphism of the lifted vector bundles µ 0 :Ẽ * −→ ∼Ẽ ′ * over C * . We can twist E by π * (I k ) for some integer k, where I is the ideal sheaf of 0 ∈ C, so that µ 0 extends to a sheaf morphism µ :Ẽ →Ẽ ′ . We assume that k is chosen to be minimal with this property, so that the restriction of µ to the fiber over 0 ∈ C is a nonzero morphism of sheaves µ 0 :Ẽ 0 →Ẽ ′ 0 .
Our definitions imply that the determinants of E, E ′ are lifts of line bundles from C × S, and det µ can be viewed as a section of (detẼ) −1 ⊗ detẼ ′ ≃π * L, where L is a line bundle on C. This implies that the support of coker µ is a simple normal crossing surface, a union of components S a ofX 0 . A combinatorial argument shows that det µ cannot vanish only on a part of components, so if µ is not an isomorphism, then µ 0 is degenerate on every component ofX 0 . It is quite obvious then that the image of µ 0 , restricted to S a , is a rank 1 sheaf over every component S a ofX 0 .
The second step is the proof of a fact from commutative algebra, which, stated in geometric terms, reads as follows:
4.11. Lemma. Let φ : E → E ′ be an injective morphism of rank 2 locally free sheaves on a smooth irreducible threefold X, and assume that D = Supp coker φ is an effective divisor on X having smooth irreducible components D i , i ∈ A, such that L i := coker φ|D i is a rank 1 sheaf for each i ∈ A. Then L i is a line bundle on D i , i ∈ A. In particular, for each i ∈ A the bundle E ′ |D i has a quotient line bundle L i .
As follows from Definition 3.2 (i-ii), there is always a component inX 0 , namely, the inverse image of any top component of X ′ , on which the restriction ofẼ ′ has no invertible quotient. This proves that µ has to be an isomorphism.
The last step of the proof is an argument, showing that there is only one way to contract some of the components inX, on whichẼ is trivial, in order that the result of the contraction might satisfy the condition (2).
4.12. Remark. The separatedness fails for the full functor M n as there may be non-isomorphic S-equivalent vector bundles on S which are limits of the same family of stable ones.
Fulton-MacPherson configuration spaces
The construction of the moduli space, up to technical details, follows the standard pattern. First we use Hilbert schemes and embeddings into Grassmannians for constructing a space H, which parametrizes all the objects we want to include in our moduli space, and then we quotient H by a group action. First, to construct the parameter space H, we invoke the results of W. Fulton and R. MacPherson from their paper [FM,, where they introduce the configuration spaces S[n] for any natural number n.
5.1. Notation. Let n := max{|T | − 1 | T ∈ T n }, and let Y F M = S[n] be the Fulton-MacPherson configuration space with the semi-universal family S[n] + over S[n] of the so-called "n-pointed stable configurations" over Y F M . Let
(16) π F M : X F M := S[n] + σ F M −→ S × Y F M pr 2 → Y F M .
denote the standard morphisms, where σ F M is a birational morphism which decomposes into a sequence of blowups with explicitly described smooth centers. The family (16) has a number of nice properties. In particular, both X F M and Y F M are smooth projective varieties and the following "versality" property holds:
5.2. Proposition (Versality). Let X → S × Y → Yphism V (y) f − → Y F M such that X V → S × V (y) → V (y) is the pull back of X F M → S × Y F M → Y F M under f .
In particular, for any T ∈ T n and any tree of surfaces S T there exists a point (not unique) y ∈ Y F M such that
(17) S y := π −1 F M (y) ≃ S T .
Note that Y F M represents a functor of configuration families, [FM,Theorem 4], but there is no obvious way of deriving the above versality from that. A proof can be done by a detailed analysis of the Kodaira-Spencer map related to this deformation problem.
The birational morphism σ F M : X F M → S × Y F M is by its construction decomposed into a sequence σ F M = σ 1 • ... • σ R of blowups with smooth centers, say, Z i ⊂ (σ 1 • ... • σ i−1 ) −1 (S 0 × Y F M ), i = 1, ..., R. By this construction, the divisors D i := (σ i • ... • σ R ) −1 (Z i ) satisfy the property that, for any y ∈ Y F M , D i ∩ S y is a subtree of the tree S y = π −1 F M (y). For a sequence of positive integers m 0 , n 1 , ..., n R we define the invertible sheaves
M 0 := O S (m 0 h) ⊠ O Y F M on S × Y F M and M := σ * F M (M 0 ) ⊗ O X F M (−Σn i D i ) on X F M .
Using the above property of the divisors and Serre's theorems A and B, one can derive the following Lemma.
Lemma.
For the given number n there is a sequence of positive integers m 0 , n 1 , . . . , n R such that (i) M r 0 is pr 2 -very ample for any r ≥ 1 and pr 2 * M r 0 is locally free. (ii) M r is π F M -very ample for any r ≥ 1 and π F M * M r is locally free.
Remark.
(i) Note that, since Y F M is a projective variety, then for each T ∈ T n and each a ∈ T the number ǫ(T, a) := max{ǫ ∈ N| there exists y ∈ Y F M such that S y has T as its graph and m a (M|S y ) = ǫ} is clearly finite, where, as above, S y = π −1 F M (y) and we use the notation from 2.3.
(ii) From the definition of the line bundle M it follows that, if y, y ′ ∈ Y F M are two points such that the fibres as trees of surfaces S y and S y ′ have the same graph T , then the line bundles M|S y and M|S y ′ have the same multitype: m T (M|S y ) = m T (M|S y ′ ). In particular,
S y ≃ S y ′ implies M|S y ≃ M|S y ′ .
We thus are led to the following notation for an arbitrary tree of surfaces S T with T ∈ T n :
(18) m T (M) := m T (M|S y )
for any isomorphism S T ∼ → S y , y ∈ Y F M . This notation is coherent, for the right hand side of (18) does not depend on the choice of y.
Next, since the set of all pairs (E T , S T ) ∈ M n (C) is bounded by Proposition 3.4, we may strengthen the result of Lemma 5.3 in the following way.
5.5. Proposition. One can choose the numbers m 0 , n 1 , ..., n R in Lemma 5.3 in such a way that, for any (E T , S T ) ∈ M n (C) and any isomorphism φ y : S T ∼ −→ S y , y ∈ Y F M , the following holds:
(i) h j (S y , M|S y ) = 0 for any j > 0 and the number r 0 := h 0 (S y , M|S y ) is independent of y with the above property.
(ii) Put m := r 0 m 0 . Then the line bundle L := φ * y (M r 0 ) on S T has type (1, m, h), h j (E T ⊗ L) = 0, j > 0, and N m := h 0 (E T ⊗ L) = χ(E T ⊗ L) is given by (11).
(iii) For any isomorphism θ y : C Nm −→ ∼ H 0 (E T ⊗L), the induced map
(19) θ(y) : C Nm ⊗ O S T θy⊗id −→ H 0 (E T ⊗ L) ⊗ O S T ev −→ E T ⊗ L
is surjective, and the induced morphism i θ(y) : S T → G := Grass(N m , 2) to the Grassmannian of 2-dimensional quotients of C Nm is an embedding such that
(20) i * θ(y) Q = E T ⊗ L and i * θ(y) O G (1) ≃ L 2 ⊗ O S T (σ * N ),
where Q is the universal rank 2 quotient sheaf on G.
(iv) h j (i * θ(y) O G (q)) = 0 for all j, q > 0 and (21)
χ(i * θ(y) O G (q)) = 2q 2 m 2 (h 2 ) + 2q 2 m(h · N ) − qm(h · K S ) + 1 2 q 2 (N 2 ) − 1 2 q(N · K S ) + χ(O S (N ))
(the same value as in (13)).
Proof. (i) follows by a standard argument and Lemma 5.3, (ii) and (iv) follow directly from Lemma 5.3 and Riemann-Roch. (iii) follows from a lemma on embeddings into Grassmannians in [Tr,lemma 5.13], and boundedness, Proposition 3.4.
Note that from (20) it follows immediately that
(22) m T (i * θ(y) O G (1) ⊗ O S T (−σ * N )) = 2m T (φ * y M r 0 F M )
. 5.6. A functorial line bundle. By the above the sheaf π F M * M is locally free on Y F M of rank r 0 . We consider the line bundle
L c F M := M r 0 ⊗ π * F M (detπ F M * M) −1 on X F M .
Using this line bundle, one can construct for any T n -family (E/X/Y ) a line bundle L Y on X such that these line bundles are compatible with base change in the sense of J.Kollár, [Ko, Definition 2.3], see 6.2 and Section 7. This defines a descent of the line bundles L Y to a line bundle L Mn on the moduli space. One of the possible approaches to the proof of the projectivity of our moduli space would be to show that L Mn is ample. To this end, one might verify the weak positivity property [V] for the bundles L Y , but this seems to be difficult for the T n -families (E/X/Y ) that are not good.
The Hilbert scheme construction
The parameter space for T n -bundles will be an open part of the Hilbert scheme Hilb P H (S × G) consisting of T n -surfaces, where as above G is the Grassmannian Gr(N m , 2), and P H will be determined below.
6.1. Definition. Let m = r 0 m 0 be as in 5.5. An embedded T n -surface is defined to be a closed embedding S T i ֒→ S × G such that
(i) the composition S T i G ֒→ G is a closed embedding with i * G O G (1) ⊗ O S T (−σ * N ) ample of type (2, m, h).
(ii) the composition S T → S is a standard contraction as defined in (4), which is an isomorphism if T is a single vertex.
Let O S×G (1) := O S (mh − N ) ⊠ O G
(1) be the chosen very ample polarization of S × G and, for an embedded tree S T i ֒→ S × G, let P H (q) = χ(i * (O S×G (q))) be the corresponding Hilbert polynomial. From Definition 6.1 it follows immediately that i * (O S×G (1)) is a very ample line bundle of type (3, m, h) on S T . Hence, P H (q) is given by the formula (12) with qm substituted for m:
(23) P H (q) = 9 2 q 2 m 2 (h 2 ) − 3 2 qm(h · K S ) + χ(O S ).
Consider now the Hilbert scheme Hilb P H (S × G) and let
H ′ ⊂ Hilb P H (S × G)H = {(S T ֒→ S×G) ∈ H s | m T (O S (−N )⊠O G (1)|S T ) = 2m T (M r 0 ) and (i * (O S (−mh) ⊠ Q)/S T / Spec(C)) ∈ M n (Spec(C))}.
Since H is a locally closed subscheme of the Hilbert scheme, there is the semi-universal family X H of embedded T n -surfaces with diagram
(24) X H π H & & L L L L L L L L L L L i H / / S × G × H pr 3
H.
Lemma.
There is a line bundle L H on X H such that for any fibre X H,z , z = (S T ֒→ S × G), and any isomorphism φ :
S T ∼ − → S y (y ∈ Y F M ),
L H |X H,z ≃ L c F M ≃ M r 0 |S y and that the Hilbert polynomials of the fibres X H,z are formed with respect to the line bundle
L H ⊗ i * H (O S (−N ) ⊠ O G (1) ⊠ O H ).
The bundle L H plays also the role of a functorial polarization in the sense of J.Kollár. It can be obtained as follows. By the versality of
Y F M there is an open covering (H i ) of H with morphisms H i f i − → Y F M such that X H |H i is the pullback of X F M under f i .E H := O S ⊠ Q ⊠ O H |X H ⊗ (L H ) −1 ,
Then (E H , X H , H) is a T n -family and belongs to M n (H).
6.3. Remark (Boundedness). It follows from the results of the next section that any T n -bundle occurs in the family (E H /X H /H), proving that the functor M n is bounded.
The coarse moduli space
Given an arbitrary family (E/X/Y ) ∈ M n (Y ), one can construct a line bundle L Y on X as in the case of the family over H using the versality of Y F M . This bundle is fibrewise isomorphic to M r 0 and has type (1, m, h). If there is a morphism ρ : Y ′ → Y , then the constructions of the line bundles L Y are compatible, so that L Y ′ ≃ ρ * L Y . By Proposition 5.5, π * (E ⊗ L Y ) is locally free of rank N m , and we can consider the principal GL(N m )-bundlẽ
Y := Isom(k Nm ⊗ O Y , π * (E ⊗ L Y )) ρ → Y over Y with the Cartesian diagram (26)X π ρ / / X π Y ρ / / Y
Denote the lifts of the bundles byẼ =ρ * E, LỸ =ρ * L Y . In view of Proposition 5.5, we have onX a universal epimorphism
(27) Θ : C Nm ⊗ OX ∼ →π * π * (Ẽ ⊗ LỸ ) ev ։Ẽ ⊗ LỸ ,
which induces an embedding i Θ :X ֒→ S × G ×Ỹ in the commutative diagram (28)Xπ
i Θ / / S × G ×Ỹ pr 3 Ỹ such that (29)Ẽ ⊗ LỸ = i * Θ (O S ⊠ Q ⊠ OỸ ) Let now LỸ ⊗ i * Θ (O S ⊠ O G (1) ⊠ OỸ )
serve as the polarization for computing the Hilbert polynomial in S × G ×Ỹ . The restriction of LỸ to each fibre ofX over a pointỹ ∈Ỹ is by its construction isomorphic to some
M r 0 y ⊗ O S T ⊗ O G (1) or to some O S T (mh) ⊗ O G (1).
Hence the fibres ofX all have the Euler characteristic P H (q), and the conditions of the definition of H are satisfied for the fibers ofX. By the universal property of the Hilbert scheme, there is a morphism φ in the following Cartesian diagram such thatX is the pull back of X H ,
(30)Xπ φ / / X H π H Y φ / / H and such thatẼ ⊗ LỸ ≃ i * Θ (O S ⊠ Q ⊠ OỸ ) ≃ i * Θ (id × φ) * (O S ⊠ Q ⊠ O H ) ≃φ * i * H (O S ⊠ Q ⊠ O H ) ≃φ * (E H ⊗ L H ).
By functoriality LỸ ≃φ * L H , and henceẼ ≃φ * E H .
The group SL(N m ) acts naturally on the Grassmannian G = Grass(N m , 2) and induces an action on Hilb P H (S × G), under which H is invariant. In order to find an algebraic structure on the quotient by the action of SL(N m ), we have to shrink H. Consider the open SL(N m )-invariant subsets H g and H t of H defined as follows:
H g := {(S T i ֒→ S × G) ∈ H | (i * (O S (−mh) ⊠ Q))/S T /C) ∈ M g n (C)}, H t := {(S T i ֒→ S × G) ∈ H | (i * (O S (−mh) ⊠ Q))/S T /C) ∈ M t n (C)}.
Note that
(31) H s = H g ∩ H t .
It is well known that Since M S,h (2; N , n) is projective, M t n is quasi-projective. In particular, M t n is separated. 7.1. Proposition. The action SL(N m ) × H g → H g is proper.
For the proof we use the properness criterion via families over curves and Theorem 4.10 on separatedness. Now we invoke the following result of Kollár [Ko1,Theorem 1.5].
Theorem. Fix an excellent base scheme Λ. Let G be an affine algebraic group scheme of finite type over Λ and X a separated algebraic space of finite type over Λ. Let m : G × X → X be a proper G-action on X. Assume that one of the following conditions is satisfied:
(1) G is a reductive group scheme over Λ.
(2) Λ is the spectrum of a field of positive characteristic.
Then a geometric quotient p X : X → X//G exists and X//G is a separated algebraic space of finite type over Λ.
Applying now the case (1) of this theorem to our situation with Λ = Spec(C), G = SL(N m ), X = H g , we obtain from Proposition 7.1 that M g n := H g //SL(N m ) is a separated algebraic space of finite type over C and p :
H g → H g //SL(N m ) = M g n is a geometric quotient. Note that M s S,h (2; N , n) = M g n ∩ M t n
is open in both M g n , M t n , so we can glue them together along M s n into an algebraic space M 0 n = M g n ∪ M t n , which is of finite type, separated but not necessarily complete.
For an arbitrary T n -family (E/X/Y ) one has an SL(N m )-equivariant diagram (30). Assume in addition that the family belongs to M g (Y ). Then there is the diagram
(32)Ỹ ρ φ / / H g p H Y M g n .
As ρ is a principal bundle map, it follows that there exists a morphism (of algebraic spaces) f : Y → M g n which extends (32) to a commutative diagram
(33)Ỹ ρ φ / / H g p H Y f / / M g n .
The existence of this modular morphism f : Y → M g n and the existence of the semi-universal family (E g H /X g H /H g ) means that M g n corepresents the functor M g n , i.e. it is the wanted moduli space. The same argument applies to M t n and M 0 n . Thus we have the following 7.2. Theorem. There exists a separated algebraic space M g n (resp. M 0 n ) of finite type corepresenting the functor M g n (resp. M 0 n ).
There is a particular case in which we can say more. This is the case when M g n = M n . This happens when M S,h (2; N , n) contains no strictly semistable locally free sheaves, for example, when S = P 2 . Then we can state: 7.3. Theorem. Let S = P 2 . Then M g n = M n , and there exists a proper 2 algebraic space M n corepresenting the functor M n .
In the general case, we can only suggest a conjecture.
7.4. Conjecture. Let S be any smooth projective surface. Then there exists a proper algebraic space corepresenting M n .
It is not clear whether one should expect M n to be projective. However, in some examples one can present an explicit construction of M n as a projective variety. One such example is treated in the next section.
8. The bubble-tree compactification of M P 2 (2; 0, 2) Let M(0, 2) = M P 2 (2; 0, 2) be the moduli space of semistable sheaves with Chern classes c 1 = 0, c 2 = 2 and rank 2. It is well known that M(0, 2) is isomorphic to the P 5 of conics in the dual plane, the isomorphism being given by [F ] ↔ C(F ), where C(F ) is the conic of jumping lines of [F ] in the dual plane, see [Ba], [Ma], [OSS]. For a more explicit description we use the following notation.
Beilinson resolutions.
In the sequel S will be the projective plane: S = P (V ) = P 2 , where V is a fixed 3-dimensional vector space. We will write mF for C m ⊗ F , where F is a sheaf.
It is well known that any sheaf F from M(0, 2) has two Beilinson resolutions 0 → 2 Ω 2 S (2)
A −→ 2 Ω 1 S (1) → F → 0 0 → 2 O S (−2) B −→ 4 O S (−1) → F → 0,
where the matrices A (of vectors in V ) and B (of vectors in V * ) are related by the exact sequence
0 → C 2 A −→ C 2 ⊗ V B −→ C 4 → 0.
Recall that Hom(Ω 2 S (2), Ω 1 S (1)) is canonically isomorphic to V with v ∈ V acting by contraction. The matrix product BA is zero, where the elements of the two matrices are multiplied by the rule that for v ∈ V and f ∈ V * , the product f v is f (v).
The matrices A and B are determined by F uniquely up to isomorphisms of the above resolutions. The first resolution implies that det(A) ∈ S 2 V is non-zero. It is the equation of the conic C(F ). The sheaf F is locally free if and only if C(F ) is smooth, or if and only if F is stable. If C(F ) decomposes into a pair of lines, then A is equivalent to a matrix of the form x 0 z y , and F is an extension 0 → I x → F → I y → 0.
In this case F is S-equivalent to I x ⊕ I y .
We will first present the T 2 -bundles appearing in the compactificaton as limits of 1-parameter degenerations. We will start from the following explicit description of the blowup of A 1 × S at a point.
A special blowup.
Let (e 0 , e 1 , e 2 ) be a basis of V and (x 0 , x 1 , x 2 ) the corresponding homogeneous coordinates on S. The blowup
X σ − → A 1 × S
of the point p = (0, e 0 ) is the subvariety of A 1 × S × P 2 given by the equations
tx 0 u 1 − x 1 u 0 = 0 tx 0 u 2 − x 2 u 0 = 0 x 1 u 2 − x 2 u 1 = 0,
where the u ν are the coordinates of the third factor P 2 . We consider the following divisors on X:
1)S, the proper transform of {0} × S, isomorphic to the blowup of S at p; 2) S 1 , the exceptional divisor of σ; 3) H, the lift of A 1 × h, where h is a general line in S; 4) L, the divisor defined by O X (L) = pr * 3 O P 2 (1). We haveS ∼ H − L. We also let x ν resp. u ν denote the sections of O X (H) resp. O X (L) lifting the above coodinates. Using the equations of X, we see that the canonical section s of O X (S) fits into the diagrams
(34) O X s / / (t 0 ,x 1 ,x 2 ) % % K K K K K K K K K K O X (H − L) (u 0 ,u 1 ,u 2 ) O X (H).
8.3. Example. Using the notation from 8.2, we choose A = e 0 0 0 e 0 , representing F 0 = I 0 ⊕ I 0 , where I 0 is the ideal sheaf of the point e 0 ∈ P (V ). Denote C = A 1 (C) and let F be the family of M(0, 2)sheaves over C × S defined as the cokernel of the matrix
A(t) = e 0 −ta −tb e 0
with a = α 1 e 1 + α 2 e 2 and b = β 1 e 1 + β 2 e 2 . Then F |{t} × S is locally free for t = 0. The second Beilinson resolution of F (see 8.1) is then
0 → 2 O C ⊠ O S (−2) B(t) −→ 4 O C ⊠ O S (−1) → F → 0.
with B(t) given by
B(t) = x 1 x 2 α 1 tx 0 α 2 tx 0 β 1 tx 0 β 2 tx 0 x 1 x 2 .
Let now X σ − → C × S be the blowup of C × S at p = (0, e 0 ). We have the following commutative diagram with exact rows and columns:
0 0 2 O S 1 (−1) 0 / / 2 O X (−2H) σ * B(t) / / s 4 O X (−H) / / σ * F / / 0 0 / / 2 O X (−H − L) B X / / 4 O X (−H) / / F / / 0 2 O S 1 (−1) 0 0
By (34),
B X = u 1 u 2 α 1 u 0 α 2 u 0 β 1 u 0 β 2 u 0 u 1 u 2 .
Here 2 O S 1 (−1) is the torsion of σ * F , supported on the exceptional divisor, and F is its locally free quotient. Finally, let E := F(−S). One can verify that the restriction of E toS is trivial, ES ≃ 2 OS, and that the restriction E S 1 belongs to M s P 2 (2; 0, 2). Thus E|S ∪ S 1 is a T 2 -bundle with weighted tree of type (II), see (35), and is a flat degeneration of bundles in M(0, 2). Note that the vectors a, b used in the construction of this degeneration correspond to two points on the double line C(F 0 ) = {e 2 0 = 0} and thus determine a "complete conic" in classical terminology. Choosing different such pairs will lead to non-isomorphic T 2 -bundles. 8.4. Example. Let now A = e 0 0 0 e 2 and F 0 = I 0 ⊕ I 2 , where I 0 resp. I 2 are the ideal sheaves of the points p 0 = e 0 resp. p 2 = e 2 in S = P (V ). Here we consider the blowup X σ − → C × S of C × S at the two points p 0 , p 2 . LetS = S 0 again denote the proper transform of {0} × S, and let S 1 and S 2 denote the exceptional divisors of σ. We embed X into C × S × P 2 × P 2 with equations
tx 0 u 1 − x 1 u 0 = 0 tx 0 u 2 − x 2 u 0 = 0 x 1 u 2 − x 2 u 1 = 0 tx 2 v 0 − x 0 v 2 = 0 tx 2 v 1 − x 1 v 2 = 0 x 1 v 0 − x 0 v 1 = 0,
where u ν and v ν are the coordinates of the third and fourth factors P 2 respectively. Define the sheaf F over C × S as the cokernel of the matrix
A(t) = e 0 −t 2 e 1 −t 2 e 1 e 2 .
The corresponding matrix B(t) is then given by
B(t) = x 1 x 2 0 t 2 x 0 t 2 x 2 0 x 0 x 1 .
In order to find the limit T 2 -bundle on the T 2 -surfaceS ∪ S 1 ∪ S 2 , we proceed as in Example 8.3. Let x ν , u ν , v ν denote the sections of O X (H), O X (L 1 ), O X (L 2 ) obtained by lifting the respective homogeneous coordinates, and let s 1 ∈ ΓO X (S 1 ) and s 2 ∈ ΓO X (S 2 ) be the canonical sections with divisors S 1 ∼ H − L 1 and S 2 ∼ H − L 2 . We obtain a torsion free sheaf F on X as the quotient in the exact triple
0 → O S 1 (−1) ⊕ O S 2 (−1) → σ * F → F → 0 with resolution 0 → O X (−H − L 1 ) ⊕ O X (−H − L 2 ) B X − − → 4 O X (−H) → F → 0, where B X = u 1 u 2 0 tu 0 tv 2 0 v 0 v 1 .
The restrictions of F toS, S 1 , S 2 are now
FS ≃ OS(−ℓ 1 ) ⊕ OS(−ℓ 2 ),
where ℓ i =S ∩ S i (i = 1, 2) are the two exceptional curves onS, and
F S 1 ≃ O S 1 ⊕ I q 1 (1), F S 2 ≃ O S 2 ⊕ I q 2 (1),
where q 1 resp. q 2 , are the points {u 1 = u 2 = 0} resp. {v 0 = v 1 = 0}.
Next, let E ′ be the elementary transform given by the exact sequence
0 → E ′ → F → O S 1 ⊕ O S 2 → 0.
This sheaf turns out to be locally free on X. Then the sheaf E := E ′ (−S) has the restrictions
ES ≃ 2 OS, E S 1 , E S 2 ,
where the Chern classes of E S 1 , E S 2 are c 1 = 0, c 2 = 1, and there are non-split extensions of the form
0 → O S 1 → E S 1 → I q 1 → 0, 0 → O S 2 → E S 2 → I q 2 → 0.
Thus E|S ∪ S 1 ∪ S 2 is a T 2 -bundle with weighted tree of type (III), see (35). It is a again a flat degeneration of vector bundles from M(0, 2).
The third example with weighted tree of type (IV) can be constructed by a similar but slightly more complicated procedure. The final result about the moduli space M 2 is the following theorem.
8.5. Theorem. Let P (S 2 V ) denote the blowup of P (S 2 V ) along the Veronese surface.
(1) The moduli space M 2 , defined in Theorem 7.3 with n = 2, is isomorphic to P (S 2 V ).
(2) The isomorphism classes [E T , S T ], T ∈ T 2 , are in 1:1 correspondence with the points of P (S 2 V ). (3) The weighted trees associated to the pairs [E T , S T ] that occur in M 2 are of one of the following four types:
(35) GFED @ABC 1 A A A A A A GFED @ABC 1 } } } } } } GFED @ABC 2 GFED @ABC 1 B B B B B B GFED @ABC 1 | | | | | | GFED @ABC 0 α : _ _ GFED @ABC 2 _ _ _ _ GFED @ABC 0 _ _ _ _ _ _ GFED @ABC 0 _ _ _ _ _ _ _ _ _ GFED @ABC 0 _ _ _ (I) (II) (III)(IV)
The four types of weighted trees define a stratification of P (S 2 V ) in locally closed subsets in the following way. Let Σ 0 be the exceptional divisor of P (S 2 V ) and Σ 1 the proper transform of the cubic hypersurface of decomposable conics in P (S 2 V ). Then
• the points of P (S 2 V ) Σ 0 ∪ Σ 1 represent the bundles of type (I) on the original surface S; • the bundles in Σ 0 Σ 1 are of type (II); • the bundles in Σ 1 Σ 0 are of type (III); • the bundles in Σ 1 ∩ Σ 0 are of type (IV).
There is a construction of a complete family (E/X/F ) of T 2 -bundles which contains all types of such bundles. This will be sketched next.
8.6. Semi-universal family for M(0, 2). From now on H will denote the vector space C 2 and G will be the Grassmannian Grass(2, H ⊗ V ) = G(2, 6) of 2-dimenional subspaces of H ⊗ V . Let U denote the universal subbundle on G. For any subspace C 2 y ֒→ H ⊗ V there is the
determinant homomorphism C 2 y / / ∧ 2 H ⊗ S 2 V ≃ S 2 V.
We denote by G ss the open subset defined by ∧ 2 y = 0. By the description in 8.1, this open subset parametrizes all the sheaves in M(0, 2), and there is a semi-universal family F on G ss × P with resolution
0 → U ⊠ Ω 2 S (2) → H ⊗ O G ss ⊠ Ω 1 S (1) → F → 0,
where U is the universal subbundle on G, restricted to G ss . The map y → ∧ 2 y defines a morphism G ss → P (S 2 V ) which is the modular morphism of the family and at the same time is a good quotient by the natural action of SL(H) on G ss :
G ss //SL(H) ≃ P (S 2 V ) ≃ M(0, 2).
Consider the subvarieties ∆ 0 , ∆ ′ 1 , ∆ ′′ 1 , ∆ ′′′ 1 of G ss defined as follows. For each y ∈ G ss , let l y ⊂ P (H ⊗ V ) be the corresponding line, and let S be the image of the Segre embedding P (H) × P (V ) ֒→ P (H ⊗ V ). Then: ∆ 0 := {y ∈ G ss | l y ⊂ S}, ∆ ′ 1 := {y ∈ G ss | l y ∩ S consists of two simple points}, ∆ ′′ 1 := {y ∈ G ss | l y ∩ S is a double point}, ∆ ′′′ 1 := {y ∈ G ss | l y ∩ S is a simple point}.
One finds the following normal forms for the matrices A defining the inclusions C 2 A ֒→ H ⊗ V that represent the points y ∈ G ss :
-y ∈ ∆ 0 if and only if y is represented by a matrix A = ( x 0 0 x ) , -y ∈ ∆ ′ 1 if and only if y is represented by a matrix A = ( x 0 0 x ′ ) with independent vectors x and x ′ , -y ∈ ∆ ′′ 1 if and only if y is represented by a matrix A = ( x 0 z x ) with independent vectors x and z, -y ∈ ∆ ′′′ 1 if and only if y is represented by a matrix A = ( x 0 z x ′ ), where (x, x ′ , z) is a basis ov V.
8.7. The Kirwan blowup. LetGσ − → G be the blowup of G along ∆ 0 followed by the blowup along the proper transform∆ 1 of ∆ 1 := ∆ ′ 1 ∪∆ ′′ 1 . ThenG is also acted on by SL(H) and there is a suitable linearization of this action such thatG s =G ss . The geometric quotientG s //SL(H) is isomorphic to P (S 2 V ), and we have the following commutative diagram:
(36)G s π σ / / G ss π P (S 2 V ) σ / / P (S 2 V ).
We have the following exceptional divisors inG s . Let D 0 :=σ −1 (∆ 0 ), and let D 1 be the inverse image of∆ 1 inG s . Then where Σ 0 and Σ 1 are the divisors in P (S 2 V ) introduced in Theorem 8.5.
8.8. Univesal family of Serre constructions. We parametrized the first Beilinson resolutions 0 → 2O S (−1) → 2Ω 1 S (1) → F → 0, of the sheaves from M(0, 2) by the two-dimensional subspaces U of H ⊗ V = H 0 (S, H ⊗ Ω 1 S (2)). Now we want to parametrize all the Serre constructions of these sheaves. A Serre construction for F is determined by a section s of F (1) with finitely many zeros. The global sections of F (1), up to proportionality, are in one-to-one correspondence with three dimensional subspaces W of H ⊗ V such that U ⊂ W . Here Z is a zero-dimensional subscheme of S of length 3. The second sequence represents the Serre construction for F . For a given F , we can always find W such that Z is either the union of three distinct points, or an isotropic fat point Spec O S /m 2 x (x ∈ S) of length 3. Let now relativize this construction over the whole ofG s . Let There are natural projectionsG s γ ← F δ → F 2,3 . We have the semiuniversal family of Beilinson resolutions of the polystable sheaves from M(0, 2) overG s . Lifting it to F , we obtain the exact sequence 0 → U ⊠ Ω 2 S (2) → H ⊗ O F ⊠ Ω 1 S (1) → F → 0. Shrinking F to an appropriate open subset, mapped surjectively ontõ G s , we obtain a semi-universal family of Serre constructions over F ,
0 → W/U ⊠ Ω 2 S (2) → F → I Z,F ×S ⊗ O F ⊠ O S(37)
(1) → 0, where U and W are the lifts of the tautological subbundles from F 2,3 , and where Z is a flat family of zero-dimensional subschemes of S of length 3 over F . We can shrink further F in such a way that γ remains surjective, so that the only singularities of Z are the quasi-transversal intersections of three smooth branches over the points of D 0 . 8.9. Semi-universal family for M 2 . Let D 0 := γ −1 (D s 0 ) and D 1 := γ −1 (D s 1 ) be the lifted divisors in F , and let B 0 := Z ∩ (D 0 × S) be the codimension 3 intersection. This is the singular locus of Z, where three branches intersect. Besides B 0 , the singular locus Sing F of F contains the points (f, x i ) ∈ Z such that f ∈ D 1 , x i ∈ S (i = 1, 2), x 1 = x 2 and F| f ×S ≃ I x 1 ,S ⊕ I x 2 ,S . At these points Z is smooth, but the local extension class of the Serre sequence degenerates. We are to resolve both types of singularities.
Let
X ′ σ 0 − → F × S
be the blowup of B 0 . Let further B 1 be the closure of σ −1 0 (Sing F ∪ ((F − D 0 ) × S)).
Let X σ 1 − → X ′ be the blowup of B 1 . Consider the composed morphism
p : X σ 1 •σ 0 − −− → F × S pr 2 −→ F,
and let E 0 , E 1 denote the exceptional divisors of the last two blowups. Then X p − → F is a family of T 2 -surfaces which includes the above examples. Moreover, the proper transformZ of Z in X is smooth. In order to replace it by a vector bundle, we consider the Serre construction on X lifting (37). A computation shows that the local extension class of (37), when lifted to a local section of the invertible sheaf ǫ ∈ Ext 1 IZ ,X ⊗ pr * S O S (1), p * (W/U ⊠ Ω 2 S (2)) , has a simple pole along D 0 ∩ Z and is regular and nonvanishing everywhere else.
To transform the simple pole into a double one, we make the base changeX → X, which is a double covering branched at D 0 + D 1 . It is defined locally over X. We add˜to mark the lifts toX of all the objects defined on X. Thenǫ acquires a double pole alongD 0 ∩Z and has no other singularities. Hence it defines a regular nowhere vanishing section of the invertible sheaf Ext 1 IZ ,X ⊗ pr * S O S (1),p * (W/U ⊠ Ω 2 S (2))(2D 0 ) . This implies that in the extension 0 →p * (W/U ⊠ Ω 2 S (2))(D 0 ) → E → IZ ,X ⊗ pr * S O S (1)(−D 0 ) → 0 defined byǫ, the middle term E is a vector bundle. This is the wanted family of T 2 -bundles. It is defined overX, which can be thought of as a DM stack with stabilizers of order ≤ 2 whose associated coarse moduli space is X. The statements of Theorem 8.5 follow by considering the classifying map of our functor M 2 on this family towards the moduli space M 2 .
be a flat deformation of a standard contraction S T → S with base point y ∈ Y. Then there is an open neighbourhood V (y) ⊂ Y of the point y and a mor-
be the open subscheme of all embedded T n -surfaces in the sense of Definition 6.1, and letH s ⊂ H ′ be the open part of those S T i ֒→ S × G for which S T ≃ S and (i * (O S (−mh) ⊠ Q)/S T / Spec(C)) ∈ M s n (Spec(C)).Finally, let H s be the closure of H s in H ′ and define H := {(S T i ֒→ S × G) ∈ H s | i * (O S (−N ) ⊠ O G (1)) = φ * M 2r 0 for an isomorphism φ : S T ∼ → S y for some point y ∈ Y F M and (i * (O S (−mh) ⊠ Q)/S T / Spec(C)) ∈ M n (Spec(C))}.Here the existence of such a point y ∈ Y F M follows from the versality of Y F M and the property that i * (O S (−N ) ⊠O G (1)) ≃ φ * M 2r 0 does not depend on the choice of the point y in view of (5.4), (ii). Equivalently,
(i) the action SL(N m ) × H s → H s is proper and H s //SL(N m ) is isomorphic to M s S,h (2; N , n); (ii) M t n := H t //SL(N m ) is isomorphic to an open subscheme of the Gieseker-Maruyama moduli space M S,h (2; N , n) containing M s S,h (2; N , n)as a dense open subscheme. In fact, the embeddings into S × G used in our construction are equivalent to the embeddings into G used by Gieseker, in the case when the underlying tree-like surface is irreducible, hence the Gieseker (semi)stability condition on the vector bundles from H t coincides with the Mumford (semi)stability under the action of SL(N m ). This proves (ii). From Propositions 3.1, 3.2 of[G1], we conclude that the points of H t //SL(N m ) represent exactly the S-equivalence classes of semistable vector bundles on S.
= D i ∩G s =π −1 (Σ i ), i = 1, 2,
Such a subspace provides an extension of the injection in the Beilinson reso-lution 2O S (−1) = U ⊗ O S (−1) ֒→ 2Ω 1 S (1) = H ⊗ Ω 1 S(1) to an injection W ⊗ O S (−1) ֒→ H ⊗ Ω S (1), determined up to the action of GL(W ), and this extension defines two exact sequences 0 → W ⊗ O S (−1) → H ⊗ Ω 1 S (1) → I Z,S (1)
F 2 , 3
23:= {(U, W )| U ⊂ W ⊂ H ⊗ V },be the flag variety of 2-and 3-dimensional subspaces of H ⊗ V , andF = F 2,3 × GG s .
For S = P 2 one might use monads.
I. e. complete, separated and of finite type
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A construction of stable bundles on an algebraic surface. D Gieseker, J. Differential Geom. 27Gieseker D. A construction of stable bundles on an algebraic surface, J. Differential Geom. 27 (1988), 137-154.
The Geometry of Moduli Spaces of Sheaves. D Huybrechts, M Lehn, Aspects of Mathematics E. 31ViewegHuybrechts D., Lehn M. The Geometry of Moduli Spaces of Sheaves, Aspects of Mathematics E 31, Braunschweig: Vieweg 1997.
Partial desingularisations of quotients of nonsingular varieties and their Betti numbers. F Kirwan, Ann of Math. 122Kirwan F. Partial desingularisations of quotients of nonsingular va- rieties and their Betti numbers, Ann of Math. 122 (1985), 41-85.
Projectivity of complete moduli. J Kollár, J. Differential Geometry. Kollár J. Projectivity of complete moduli, J. Differential Geometry, 32 (1990), 235-268.
Quotient spaces modulo algebraic groups. J Kollár, Annals of Math. 145Kollár J. Quotient spaces modulo algebraic groups, Annals of Math. 145 (1997), 33-79.
Algebraic geometric interpretation of Donaldson's polynomial invariants. Jun Li, J. Diff. Geom. 37Li, Jun: Algebraic geometric interpretation of Donaldson's polynomial invariants, J. Diff. Geom. 37, 417-466, 1993
The Kobayashi-Hitchin correspondence. M Lübke, A Teleman, World ScientificRiver EdgeLübke M., Teleman, A.: The Kobayashi-Hitchin correspondence, World Scientific, River Edge, 1995.
Singularities of the curves of jumping lines of a vector bundle of rank 2 on P 2 . Algebraic Geometry. M Maruyama, Proc.of Japan-France Conf., Tokyo and Kyoto. .of Japan-France Conf., Tokyo and KyotoSpringerMaruyama M. Singularities of the curves of jumping lines of a vector bundle of rank 2 on P 2 . Algebraic Geometry, Proc.of Japan-France Conf., Tokyo and Kyoto, 1982, Lect. Notes in Math., 1016, Springer, 1983, 370-411.
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M Maruyama, G Trautmann, Limits of instantons. 3Maruyama M., Trautmann G.: Limits of instantons, Intern. Jour- nal of Math. 3, (1992), 213-276.
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. Proc. Indian Acad. Sci. Math. Sci. 109IIII, Proc. Indian Acad. Sci. Math. Sci. 109, 165-201 (1999).
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Quasi-projective moduli for polarized manifolds. E Viehweg, SpringerViehweg E. Quasi-projective moduli for polarized manifolds, Springer, 1995.
Removable singularities in Yang-Mills fields. K K Uhlenbeck, Comm. Math. Phys. 83Uhlenbeck K. K. Removable singularities in Yang-Mills fields, Comm. Math. Phys. 83, 11-29 (1982).
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"Differentiable perturbations of Ornstein-Uhlenbeck operators",
"Differentiable perturbations of Ornstein-Uhlenbeck operators"
]
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"Luigi Manca [email protected] \nDipartimento di Matematica P. e A\nUniversità di Padova\nVia Trieste 6335121PadovaItaly\n"
]
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"Dipartimento di Matematica P. e A\nUniversità di Padova\nVia Trieste 6335121PadovaItaly"
]
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| We prove an extension theorem for a small perturbation of the Ornstein-Uhlenbeck operator (L, D(L)) in the space of all uniformly continuous and bounded functions f : H → R, where H is a separable Hilbert space. We consider a perturbation of the form N 0 ϕ = Lϕ + Dϕ, F where F : H → H is bounded and Fréchet differentiable with uniformly continuous and bounded differential. Hence, we prove that N 0 is m-dissipative and its closure in C b (H) coincides with the infinitesimal generator of a diffusion semigroup associated to a stochastic differential equation in H. | null | null | 13,900,717 | 0705.3126 | 2d1cff649622e1bc614ffa469ee2b0986b3d669e |
Differentiable perturbations of Ornstein-Uhlenbeck operators
22 May 2007 February 1, 2008
Luigi Manca [email protected]
Dipartimento di Matematica P. e A
Università di Padova
Via Trieste 6335121PadovaItaly
Differentiable perturbations of Ornstein-Uhlenbeck operators
22 May 2007 February 1, 2008Ornstein-Uhlenbeck semigroupm-dissipative operatorLumer-Phillips theorem MSC 2000: 47B38, 47A55,
We prove an extension theorem for a small perturbation of the Ornstein-Uhlenbeck operator (L, D(L)) in the space of all uniformly continuous and bounded functions f : H → R, where H is a separable Hilbert space. We consider a perturbation of the form N 0 ϕ = Lϕ + Dϕ, F where F : H → H is bounded and Fréchet differentiable with uniformly continuous and bounded differential. Hence, we prove that N 0 is m-dissipative and its closure in C b (H) coincides with the infinitesimal generator of a diffusion semigroup associated to a stochastic differential equation in H.
Introduction and setting of the problem
Let H be a separable Hilbert space endowed with scalar product ·, · and norm | · |. We shall always identify H with its topological dual space H * . L(H) is the Banach space of all the linear and continuous maps in H, endowed with the usual norm · L(H) . With C b (H) (resp. C b (H; H)) we denote the Banach space of all uniformly continuous and bounded functions f : H → R (resp. f : H → H), endowed with the supremum norm · (resp. · 0 ). We also denote by C 1 b (H) (resp. C 1 b (H; H)) the space of all f ∈ C b (H) (resp. f : H → H) that are Fréchet differentiable with differential in C b (H; H) (resp. with uniformly continuous and bounded differential Df : H → L(H)). We assume the following (iii) For any t > 0 the linear operator Q t , defined as
Q t x = t 0 e sA Qe sA * xds, x ∈ H, t ≥ 0,(2)
is of trace class.
(iv) F ∈ C 1 b (H; H), and K = sup x∈H DF (x) L(H) .
It is well known (see, for instance, [44,4]) that thanks to conditions (i)-(iii) it is possible to define the so called Ornstein-Uhlenbeck (OU) semigroup
(R t ) t≥0 in C b (H) by the formula R t ϕ(x) = 0 ϕ(e tA x + y)N Qt (dy), x ∈ H,(3)
where N Qt is the Gaussian measure on H of mean 0 and covariance operator Q t (see [44,4]). It turns out that the OU semigroup is not a strongly continuous semigroup in C b (H) but it is a weakly continuous semigroup (see [11,1]) and a π-semigroup (see [77,7]). However, it is possible to define its infinitesimal generator in the weaker sense
D(L) = ϕ ∈ C b (H) : ∃g ∈ C b (H), lim t→0 + R t ϕ(x) − ϕ(x) t = g(x), x ∈ H, sup t∈(0,1) R t ϕ − ϕ t < ∞ Lϕ(x) = lim t→0 + R t ϕ(x) − ϕ(x) t , ϕ ∈ D(L), x ∈ H.(4)
We are interested in the operator (N 0 , D(N 0 )) defined by
N 0 ϕ = Lϕ + F ϕ, ϕ ∈ D(N 0 ) = D(L) ∩ C 1 b (H), where F ϕ(x) = Dϕ(x), F (x) .
Now let us consider the stochastic differential equation in H
dX(t) = AX(t) + F (X(t)) dt + Q 1/2 dW (t) t > 0, X(0) = x x ∈ H,(5)
where (W (t) t≥0 is a cylindrical Wiener process defined on a stochastic basis (Ω, F , (F t ) t≥0 , P). Since F ∈ C 1 b (H; H), problem (5) has a unique mild solution X(t, x) t≥0,x∈H (see [44,4]), that is for any x ∈ H the process (X(·, x)) t≥0 is adapted to the filtration (F t ) t≥0 and it is continuous in mean square, i.e. lim
t→s E |X(t, x) − X(s, x)| 2 = 0, ∀s ≥ 0.
This allows us to define a transition semigroup (P t ) t≥0 in C b (H), by setting
P t ϕ(x) = E ϕ(X(t, x)) , t ≥ 0, ϕ ∈ C b (H), x ∈ H.
The semigroup (P t ) t≥0 is not strongly continuous in C b (H). However, it is a π-semigroup, and we can define its infinitesimal generator (N, D(N)) in the same way as for the OU semigroup
D(N) = ϕ ∈ C b (H) : ∃g ∈ C b (H), lim t→0 + P t ϕ(x) − ϕ(x) t = g(x), x ∈ H, sup t∈(0,1) P t ϕ − ϕ t < ∞ Nϕ(x) = lim t→0 + P t ϕ(x) − ϕ(x) t , ϕ ∈ D(N), x ∈ H.
The main result of this paper is the following Theorem 1.2. Let us assume that Hypothesis 1.1 holds. Then, the operator
(N 0 , D(N 0 ), defined by D(N 0 ) = D(L) ∩ C 1 b (H) and N 0 ϕ = Lϕ + F ϕ, ∀ϕ ∈ D(N 0 ), is m-dissipative in C b (H)
and its closure is the operator (N, D(N)).
In [22,2], it is proved that Theorem 1.2 holds with F ∈ C 1,1 b (H; H), that is F is Fréchet differentiable and its differential DF : H → L(H) is Lipschitz continuous.
Perturbations of OU operators as been the object of several papers (see, for instance, [22,233,355,588,8]). Frequently, additional assumptions are taken on the OU operator in order to have D(L) ⊂ C 1 b (H). In order to prove Theorem 1.2 we develope a technique introduced in [22, 2]. The idea is the following: since F ∈ C 1 b (H; H), there exists a unique solution η(·, x) of the abstract Cauchy problem
d dε η(ε, x) = F (η(ε, x)), ε > 0, η(0, x) = x, x ∈ H.
Then, for any ε > 0 we define the operators F ε :
C b (H) → C b (H) and N ε : D(N ε ) ⊂ C b (H) → C b (H) by setting F ε ϕ(x) = 1 ε ϕ(η(ε, x)) − ϕ(x) , D(N ε ) = D(L) ∩ C 1 b (H), N ε ϕ = Lϕ + F ε ϕ, ϕ ∈ D(N ε ).
By an approximation argument, we are able to prove that the operator
(N 0 , D(N 0 )) is m-dissipative in C b (H).
Then, by the Lumer-Phillips theorem, it will follow that the closure of (N 0 , D(N 0 )) coincides with the operator (N, D(N)).
Properties of F ε
The following lemma collects some useful properties of η.
Lemma 1.3. The following estimates hold
|η(t, x)| ≤ e F 0 t |x|; (6) |η(t, x) − η(t, y)| ≤ e Kt |x − y|; (7) |η(t, x) − x| ≤ c F 0 t (8) η x (t, x) L(H) ≤ e Kt (9) η x (t, x) − η x (t, y) L(H) ≤ e Kt θ DF (e Kt |x − y|),(10)
where θ DF : R + × R + → R + is the modulus of continuity of DF .
Proof. (6), (8), (9) have been proved in [22; 2, Lemma 2.1]. (7). We have
|η(t, x) − η(t, y)| ≤ |x − y| + t 0 F (η(s, x)) − F (η(s, y))|ds ≤ K t 0 |η(s, x) − η(s, y)|ds.
Then (7) follows by Gronwall's Lemma.
(10). Let x, y, h ∈ H and set
r h (t) = η x (t, x) · h − η x (t, y) · h = p h (t, x) − p h (t, y), where P h (t, x) = η x (t, x) · h and p h (t, y) = η x (t, y) · h.
Then r h (t) fulfills the following equation
d dt r h (t) = DF (η(t, x))r h (t) + DF (η(t, x)) − DF (η(t, y)) p h (t, x), t > 0 r h (0) = 0. Since |DF (η(t, x))r h (t)| ≤ K|r h (t)| it follows that r h (t) is bounded by |r h (t)| ≤lim ε→0 + F ε ϕ = F ϕ in C b (H). (11) F ε ϕ ≤ Dϕ 0 F 0 .(12)
Proof. For all ϕ ∈ C 1 b (H) we have
F ε ϕ(x) − F ϕ(x) = 1 ε ε 0
Dϕ(η(s, x)) − Dϕ(x), F (η(s, x)) ds
+ 1 ε ε 0 Dϕ(x), F (η(s, x)) − F (x) ds.
Then by (8) we have
|F ε ϕ(x) − F ϕ(x)| ≤ ≤ 1 ε ε 0 (|θ Dϕ (|η(s, x) − x|) F 0 + Dϕ 0 K|η(s, x) − x|)) ds ≤ 1 ε ε 0 (θ Dϕ ( F 0 s|) F 0 + Dϕ 0 K F 0 s) ds ≤ (θ Dϕ ( F 0 ε|) + Dϕ 0 Kε) F 0
where θ Dϕ , is the modulus of continuity of Dϕ. This yields (11). Moreover, we have
F ε ϕ(x) = 1 ε ε 0
Dϕ(η(s, x)), F (η(s, x)) ds that implies (12).
m-dissipativity of N .
Given ε > 0 we introduce the following approximating operator
N ε = L + F ε , D(N ε ) = D(L) ∩ C 1 b (H). We have Proposition 1.5. N ε is a m-dissipative operator in C b (H) for any ε > 0.
Moreover, for any f ∈ C 1 b (H) and any λ > ω + (e εK − 1)/ε the operator
R(λ, N ε ) = 1 − T λ ) −1 R λ + 1 ε , L , where T λ : C b (H) → C b (H) is defined by T λ ψ(x) = R λ + 1 ε , L 1 ε ψ(η(ε, x)) , x ∈ H, ψ ∈ C b (H) (13) maps C 1 b (H) into D(L) ∩ C 1 b (H) and DR(λ, N ε )f 0 ≤ 1 λ − ω − e Kε −1 ε Df 0 .(14)
Proof. Let ε > 0, λ > 0, f ∈ C b (H). The equation
λϕ ε − Lϕ ε − F (ϕ ε ) = f is equivalent to λ + 1 ε ϕ ε − Lϕ ε − F (ϕ ε ) = f + 1 ε ϕ ε (η(ε, ·))
and to
ϕ ε = R λ + 1 ε , L f + T λ ϕ ε .(15)
Since, as we can easily see, for any λ > 0
T λ ψ ≤ 1 1 + λε ψ , ∀ψ ∈ C b (H),(16)
the operator T λ is a contraction in C b (H) and so equation (15) has a unique solution ϕ ε ∈ C b (H) done by ϕ ε = R(λ, N ε )f . Moreover, by (13), (16) it holds
ϕ ε ≤ 1 λ + 1 ε f + 1 ε ϕ ε .
Consequently,
ϕ ε ≤ 1 λ f .
Then, N ε is m-dissipative. Now let f ∈ C 1 b (H). We recall that for any λ > 0,
ψ ∈ C b (H) R(λ, L)ψ(x) = ∞ 0 e −λt R t ψ(x)dt(17)
and that DR t ψ(x) = H e tA * Dψ(e tA x + y)N Qt (dy).
Hence, for any λ > ω
DR(λ, L)ψ(x) = ∞ 0 H e −λt e tA * Dψ(e tA x + y)N Qt (dy)dt(18)
and so
DR(λ, L)ψ 0 ≤ 1 λ − ω Dψ 0(19)
Moreover, as it can be easily seen by (18), DR(λ, L)ψ is uniformly continuous. Then R(λ, L) :
C 1 b (H) → C 1 b (H). Now, in order to prove that T λ : C 1 b (H) → C 1 b (H)
it is sufficient to show that ψ(η(ε, x)) ∈ C 1 b (H), for any ψ ∈ C 1 b (H). Indeed, by a standard computation, we have Dψ(η(ε, ·))(x) = η * x (ε, x)Dψ(η(ε, x)), x ∈ H. Consequently, by (7), (10) we have
|Dψ(η(ε, ·))(x) − Dψ(η(ε, ·))(x)| ≤ η * x (ε, x) − η * x (ε, x) L(H) |Dψ(η(ε, x))| + η * x (ε, x) L(H) |Dψ(η(ε, x)) − Dψ(η(ε, x))| ≤ e εK θ DF (e εK |x − x|) Dψ 0 + e εK θ Dψ (|η(ε, x) − η(ε, x)|) ≤ e εK θ DF (e εK |x − x|) Dψ 0 + e εK θ Dψ (e εK |x − x|),
for any x, x ∈ H. So, DT λ ψ(·) is uniformly continuous. Now we prove that T λ is a contraction in C 1 b (H). By (13), (17) we have
T λ ψ(x) = 1 ε ∞ 0 e −(λ+ 1 ε )t R t ψ(η(ε, ·))(x)dt = 1 ε ∞ 0 H e −(λ+ 1 ε )t ψ(η(ε, e tA x + y))N Qt (dy)dt Then DT λ ψ(x) = = 1 ε ∞ 0 H e −(λ+ 1 ε )t e tA * η *
x (ε, e tA x + y)Dψ(η(ε, e tA x + y))N Qt (dy)dt By (9) it follows
|DT λ ψ(x)| ≤ 1 ε ∞ 0 e −(λ+ 1 ε −ω)t e εK Dψ 0 dt = e εK 1 + ε(λ − ω) Dψ 0 .
Therefore, for any λ > ω + (e εK − 1)/ε the linear operator T λ is a contraction in C 1 b (H) and its resolvent satisfies
(1 − T λ ) −1 (C 1 b (H)) ⊂ C 1 b (H), D(1 − T λ ) −1 ψ 0 ≤ 1 1 − e εK 1+ε(λ−ω) Dψ 0 .(20)
This implies
R(λ, N ε )(C 1 b (H)) = (1 − T λ ) −1 R λ + 1 ε , L (C 1 b (H)) ⊂ C 1 b (H).
Then, N ε is m-dissipative. Finally, (14) follows by (19) and (20).
Lemma 1.6. The operator N 0 is dissipative in C b (H).
Proof. We have to prove that λϕ−N 0 ϕ ≥ λ ϕ for any ϕ ∈ D(N 0 ), λ > 0. So, if ϕ ∈ D(L) ∩ C 1 b (H) and λ > 0 we set
λϕ − Lϕ − F ϕ = f.
then for any ε > 0 we have
λϕ − N ε ϕ = f + F ϕ − F ε ϕ. It follows ϕ = R(λ, N ε )(f + F ϕ − F ε ϕ) and ϕ ≤ 1 λ ( f + F ϕ − F ε ϕ )
Then by (11) it follows
ϕ ≤ 1 λ f .
Since N 0 is dissipative, its closure N 0 is still dissipative (maybe it is multivalued). By the following theorem follows Theorem 1.2.
Theorem 1.7. N 0 is m-dissipative.
Proof. Let f ∈ C 1 b (H), ε ∈ (0, 1) and λ > ω + e K − 1. We denote by ϕ ε the solution of λϕ ε − N ε ϕ ε = f.
By Proposition (1.5) we have ϕ ε ∈ D(L) ∩ C 1 b (H) = D(N 0 ), then ϕ ε is solution of λϕ ε − N 0 ϕ ε = f + F ε ϕ ε − F ϕ ε . We claim that F ε ϕ ε − F ϕ ε → 0 in C b (H) as ε → 0 + . Indeed it holds F ε ϕ ε (x) − F ϕ ε (x) = 1 ε ε 0 ( Dϕ ε (η(s, x)), F (η(s, x)) + Dϕ ε (x), F (x) ) ds = 1 ε ε 0 ( Dϕ ε (η(s, x)) − Dϕ ε (x), F (η(s, x)) + Dϕ ε (x), F (η(s, x)) − F (x) ) ds.
Hence
|F ε ϕ ε (x) − F ϕ ε (x)| ≤ ≤ 1 ε ε 0 (|Dϕ ε (η(s, x)) − Dϕ ε (x)| F 0 + Dϕ ε 0 |F (η(s, x)) − F (x)|) ds By (8) we have |F (η(s, x)) − F (x)| ≤ K|η(s, x) − x| ≤ K F 0 s ≤ K F 0 ε.
Notice now that since ϕ ε = R(λ, N ε )f and ε ∈ (0, 1), by (14) it follows
Dϕ ε 0 ≤ c 1 Df 0 , for all ε ∈ (0, 1), where c 1 = (λ − ω − Ke K ) −1 . This also implies |Dϕ ε (η(s, x)) − Dϕ ε (x) 0 ≤ c 1 Df (η(s, x) + ·) − Df (x + ·) 0 ≤ c 1 |θ Df (|η(s, x) − x|) ≤ c 1 θ Df ( F 0 ε),
where θ Df : R + → R + is the modulus of continuity of Df . So we find
|F ε ϕ ε (x) − F ϕ ε (x)| ≤ ≤ c 1 F 0 θ Df ( F 0 ε) + c 1 Df 0 K F 0 ε.
Then F ε ϕ ε − F ϕ ε → 0 in C b (H), as ε → 0 + . Finally, we have obtained lim ε→0 + λϕ ε − N 0 ϕ ε ] = f in C b (H). Therefore the closure of the range of λ − N includes C 1 b (H), which is dense in C b (H). So, since N 0 is dissipative, by the Lumer-Phillips theorem the closure N 0 of N 0 is m-dissipative.
Proof of Theorem 1.2
By Theorem 1.7 the operator N 0 is m-dissipative in C b (H). It is also known that if ϕ ∈ D(L) ∩ C 1 b (H), then Nϕ = Lϕ + F ϕ (see, for instance, [66, 6]) and therefore (N, D(N)) is an extension of (N 0 , D(N 0 )). Finally, since the operator (N, D(N)) is closed (see Proposition 3.4 in [77,7]), by the Lumer-Phillips theorem it follows that the closure of (N 0 , D(N 0 )) in C b (H) coincides with (N, D(N)).
Hypothesis 1. 1 .
1(i) A : D(A) ⊂ H → H is the infinitesimal generator of a strongly continuous semigroup (e tA ) t≥0 of type G(1, ω), i.e. there exists ω ∈ R such that e tA L(H) ≤ e ωt , t ≥ 0; (1) (ii) Q ∈ L(H) is self adjoint and positive;
t 0 e 0 e
00K(t−s) DF (η(s, x)) − DF (η(s, y)) L(H) |p h (s, x)|ds. By taking into account that DF : H → L(H; H) is uniformly continuous and bounded, we denote by θ DF the modulus of continuity of DF . Hence, by (7), (9) we have |r h (t)| ≤ t Ks θ DF (|η(s, x) − η(s, y)|)ds|h| ≤ e Kt θ DF (e Kt |x − y|)|h| Proposition 1.4. For any ϕ ∈ C 1 b (H) we have
A Hille-Yosida theorem for weakly continuous semigroups. Semigroup Forum. cite.Cerrai1Sandra Cerrai493cite.Cerrai1Sandra Cerrai, A Hille-Yosida theorem for weakly continuous semigroups, Semigroup Forum 49 (1994), no. 3, 349-367.
DP032Giuseppe Da Prato, Perturbations of Ornstein-Uhlenbeck operators: an analytic approach, and economics. Progr. Nonlinear Differential Equations Appl. 55cite.DP032Giuseppe Da Prato, Perturbations of Ornstein-Uhlenbeck operators: an an- alytic approach, and economics (Levico Terme, 2000), Progr. Nonlinear Differential Equations Appl., vol. 55, Birkhäuser, Basel, 2003, pp. 127-139.
Kolmogorov equations for stochastic PDEs, Advanced Courses in Mathematics. CRM Barcelona. BaselBirkhäuser Verlagcite.DP043 , Kolmogorov equations for stochastic PDEs, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2004.
Da Prato, Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications. CambridgeCambridge University Press44cite.DPZ924Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992.
Manca065Luigi Manca, On a class of stochastic semilinear PDEs. Stoch. Anal. Appl. 242cite.Manca065Luigi Manca, On a class of stochastic semilinear PDEs, Stoch. Anal. Appl. 24 (2006), no. 2, 399-426.
Kolmogorov equations for measures. Preprintcite.Manca076 , Kolmogorov equations for measures, 2007. Preprint.
On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Studia Math. 1363Priola7Enrico Priolacite.Priola7Enrico Priola, On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions, Studia Math. 136 (1999), no. 3, 271-295.
Zambotti008Lorenzo Zambotti, An analytic approach to existence and uniqueness for martingale problems in infinite dimensions. Probab. Theory Related Fields. 1182cite.Zambotti008Lorenzo Zambotti, An analytic approach to existence and uniqueness for martingale problems in infinite dimensions, Probab. Theory Related Fields 118 (2000), no. 2, 147-168.
| []
|
[
"Who discovered the expanding universe?",
"Who discovered the expanding universe?"
]
| [
"Harry Nussbaumer ",
"Lydia Bieri "
]
| []
| []
| Does it really matter who discovered the expanding universe? Great discoveries are anyway never done single-handedly. This is a valid attitude. However, those interested in the evolution of our scientific culture are eager to know the intricate patterns that lead to new insights. As the expanding universe is one of the most important discoveries ever made, it is not astonishing that the question of how it happened is still widely discussed.The debate on this topic has flared up again due to an article in Nature View, where Eugenie Reich highlighted two contributions by Sidney van den Bergh [12] and David Block [1]. Their effect was to reanimate the discussion whether Hubble or Lemaître discovered the expanding universe, or whether it was simply a nearly predictable outcome of the normal scientific activity of those days. We have investigated this question in our book Discovering the Expanding Universe [10], where we reconstructed the discovery from original documents.The story exemplifies two different paths of scientific progress. The Hubble-myth is that of a chance discovery, the story of Lemaître is a deliberate search by an individual scientist for the solution of a long standing problem. We draw attention to this aspect, and we also want to summarise the main facts for those too busy to do their own research, or to read our whole book[10], where also the detailed references can be found.During the whole story we should not forget that neither Lemaître nor Hubble worked in isolation, and that a discovery also feeds on direct and indirect contributions from others. In this case the discovery was imbedded in the many contributions to general relativity, the search for an adequate interpretation of the enigmatic large nebular redshifts, and the challenge of measuring distances to extragalactic objects.The facts are: | null | [
"https://arxiv.org/pdf/1107.2281v3.pdf"
]
| 119,218,170 | 1107.2281 | 02fedccb3b4061c2225673c2fdcba6b42651e26a |
Who discovered the expanding universe?
16 Jan 2012
Harry Nussbaumer
Lydia Bieri
Who discovered the expanding universe?
16 Jan 2012which he had been observing at Flagstaff since 1912. De Sitter's universe also contained the cosmological constant Λ, but no other matter. 3. In 1922 Friedmann found that Einstein's equations allowed a dynamic universe. He did not connect this finding to astronomical observations, and he did not spot the 1
Does it really matter who discovered the expanding universe? Great discoveries are anyway never done single-handedly. This is a valid attitude. However, those interested in the evolution of our scientific culture are eager to know the intricate patterns that lead to new insights. As the expanding universe is one of the most important discoveries ever made, it is not astonishing that the question of how it happened is still widely discussed.The debate on this topic has flared up again due to an article in Nature View, where Eugenie Reich highlighted two contributions by Sidney van den Bergh [12] and David Block [1]. Their effect was to reanimate the discussion whether Hubble or Lemaître discovered the expanding universe, or whether it was simply a nearly predictable outcome of the normal scientific activity of those days. We have investigated this question in our book Discovering the Expanding Universe [10], where we reconstructed the discovery from original documents.The story exemplifies two different paths of scientific progress. The Hubble-myth is that of a chance discovery, the story of Lemaître is a deliberate search by an individual scientist for the solution of a long standing problem. We draw attention to this aspect, and we also want to summarise the main facts for those too busy to do their own research, or to read our whole book[10], where also the detailed references can be found.During the whole story we should not forget that neither Lemaître nor Hubble worked in isolation, and that a discovery also feeds on direct and indirect contributions from others. In this case the discovery was imbedded in the many contributions to general relativity, the search for an adequate interpretation of the enigmatic large nebular redshifts, and the challenge of measuring distances to extragalactic objects.The facts are:
1. In 1917 Einstein found from his fundamental equations of general relativity a static model of the universe. More than two thousand years of astronomical observations showed the universe to be stable and practically immutable in space as well as in time. Thus, very naturally, Einstein was looking for a static world. To achieve his aim he introduced the cosmological constant Λ.
2. In 1917, a few months after Einstein, de Sitter found another solution which seemed to contain the explanation of Slipher's then already well known nebular redshifts, which he had been observing at Flagstaff since 1912. De Sitter's universe also contained the cosmological constant Λ, but no other matter. 3. In 1922 Friedmann found that Einstein's equations allowed a dynamic universe. He did not connect this finding to astronomical observations, and he did not spot the flaw in de Sitter's model. Except for Einstein, who did not think that dynamic solutions were physically relevant, no one took note of Friedmann.
4. There were other theoreticians, who in the 1920ies derived dynamical universes from the Einstein equations. But none of them linked their results to observations, nor did they propose an expanding universe. Lanczos in 1922 derived a formal solution of a spatially closed dynamical universe. In the same paper, he commented on publications by Hermann Weyl 1918 and 1919, which discussed redshifts in de Sitter's model. These papers attest to the confusion generated by de Sitter's empty universe. And as the mathematical tools had not yet been as developed as nowadays, interpretations often took intricate paths. An important contribution came from Weyl with his concept of a "causally connected world". It is also instructive to follow the Einstein-Weyl postcard exchange about the cosmological constant. More details about this topic and other players in the game can be found in our book [10].
5. During the early twenties not only the theoreticians but also the observers tried to make sense of de Sitter's universe, and to determine its radius of curvature. In addition, the community still debated whether nebulae were extragalactic or not. In the course of these investigations Carl Wirtz found in 1924 for spiral nebulae a relationship between their apparent photographic diameters and the radial velocities, and in the same year Knut Lundmark published in Monthly Notices the first distancevelocity diagram, distances being given in units of the distance to Andromeda. In the discussion on the nature of spiral nebulaeÖpik, by an ingenious method, had already in 1922 found a distance of 450 kpc to Andromeda, much closer to the real value than Hubble's later distance of 285 kpc. But it was Hubble's paper, read at the January 1, 1925 meeting of the American Astronomical Society, which cleared the sky for extragalactic nebulae (now called galaxies) as building blocks of the universe: the island universes hypothesised by Kant and Laplace were accepted as reality. For further details go to our book [10]. 6. In 1927 Lemaître [4] . He was satisfied that the observations did not contradict his theoretical conclusions: the universe itself is expanding. But he was also well aware that there was an enormous scatter in the observations, and that further observations would have to confirm the linear relationship.
Lemaître was fully aware of the significance of his discovery. It is all the more astonishing that he did not try to place it in one of the prestigious astronomical journals, but published it in French in the Annales de la Société scientifique de Bruxelles.
7. In 1929 Hubble [2] set out to study the motion of the sun against the background of the extragalactic nebulae, for which he tabulated, as others had done before, distances and velocities of extragalactic nebulae: v = rK +X cos α cos δ+Y sin α cos δ+ Z sin δ. In the course of that investigation he found with his improved distances that "The data in the table indicate a linear correlation between distances and velocities, whether the latter are used directly or corrected for solar motion, according to the older solutions". Having realised that fact, he turned away from the solar problem, concentrating on the linear relationship. Depending on how he grouped the galaxies, he found K = 473, 513 or 530(km/s)/Mpc, but opted for K = 500(km/s)/Mpc as his favourite value. To derive the numerical value of K, Hubble worked with his own distances. For the redshifts he mainly took those of Slipher, as tabulated in Eddington's 1923 The Mathematical Theory of Relativity (second edition 1924), without however, giving references. Hubble refrained from interpreting his observational discovery, he concluded "The outstanding feature, however, is the possibility that the velocity-distance relation may represent the de Sitter effect, and hence that numerical data may be introduced into discussions of the general curvature of space". In a later letter to de Sitter, Hubble wrote that he would leave the interpretation of his observations to those "competent to discuss the matter with authority". In none of the seven pages of Hubble's paper is there a single word about an expanding universe, actually Hubble never believed in such a thing. Hubble's observations confirmed Lemaître's predictions. In our book we also show how they re-ignited the cosmological debate, as exemplified by the crucial de Sitter-Eddington discussion of Friday, 10 January 1930 [11]. Livio). In his answer to Smart of 13 February 1931 he specified:"I did not find it advisable to reprint the provisional discussion of radial velocities which is clearly of no actual interest, ...". (For "actual" he certainly had the French meaning of "current" in mind.) He cut his derivation of the Hubble constant, which at that time was simply called the coefficient of expansion, and he cut the discussion of the astronomical data from which he had derived it. He replaced it with the sentence: "From a discussion of the available data, we adopt", after which he gives his numerical result for R ′ R . However, he adds a reference list containing what for him must have been the crucial "available data", namely the 1930 series of observationally based papers by de Sitter. He certainly meant no offence to Hubble by not mentioning his 1929 publication, which in 1931 was outdated by de Sitter's thorough investigations. -Lemaître was a modest man. When his discoveries began to be attributed to Hubble, he refrained from a campaign to defend his priorities. However, in 1950, he reminded his readers that he had already determined the Hubble constant in 1927 [7]. 9. It is occasionally stated that the mathematical prediction of the expanding universe was also made by Robertson. This is a misunderstanding which we also discuss in our book. In 1928 Robertson submitted to the Philosophical Magazine an article, in which he wanted to replace de Sitter's line element by "a mathematically equivalent solution in which many of the apparent paradoxes inherent in [de Sitter's solution] were eliminated". He also arrived at the formula which in Lemaître's hand had become the distance-velocity relation. However, he wrote this as v = c · (l/R), where l is the distance of the nebula and R the radius of curvature of the universe, for which he was looking within a static solution.
Robertson then took practically the same set of observations as had been taken by Lemaître one year before and would be taken by Hubble one year later. From this he calculated R = 2 · 10 27 cm. His c/R corresponds to H = 463(km/s)/Mpc; but this he did not calculate. Robertson placed an important milestone in our understanding of cosmological solutions of the Einstein equations. Solutions of Einstein's equations, in general, do not obey special symmetries. Yet, to describe the large scale structure of a spatially homogeneous universe, the four-dimensional spacetime is usually separated into a spatial and a time component. Moreover, to 'treat every point in this world equally' -the content of the Copernican principle -and to implement the observational constraints into our models, leads to the hypothesis of universal homogeneity and isotropy. These premises imply symmetries in the solutions of Einstein's equations. Robertson was the first to search in detail for all the mathematical universes that satisfy these physical requirements.
10. It is occasionally claimed that it was Hubble who converted Einstein to the expanding universe. This is very unlikely. Although there is no written report about the moment when Einstein was converted, it is highly probable that it happened, when Eddington showed him that his static solution was unstable. We discuss the circumstances further in our book.
11. Neither Hubble nor Lemaître rested on their laurels. With the help of the world's most powerful telescopes Hubble and Humason began measuring nebular redshifts on Mount Wilson. Their data -later continued by Sandage -would become one of the cornerstones of observational cosmology. Lemaître had another impact, when in 1931 [6] he suggested in a one-column letter to Nature, what would become the Big Bang, and in 1933, in a paper read before the American National Academy of Sciences, Lemaître suggested vacuum energy as the deeper meaning of the cosmological constant Λ. These exploits have also been highlighted in detail by Jean-Pierre Luminet [9].
12. If Hubble was not the discoverer of the expanding universe, why is he still often venerated as such. Kragh and Smith [3] have looked into the evolution of the 'Hubble-myth'. They find that not until the 1950ies did the notion of 'Hubble's law' and 'Hubble as the astronomer who had discovered the expanding universe' become common in the scientific literature, where Hubble's role was gradually elevated at the expense of everyone else's. They conclude: the label 'Hubble's law' is an example of what has been called Stigler's law of eponymy, namely, 'No scientific discovery is named after its original discoverer'.
The discovery of the expanding universe is a picture book example of an individual scientist who was aware of a burning scientific issue and solved it. It did not happen in a vacuum. Lemaître had benefitted from Eddington's insights into general relativity. In his 1927 paper he also cites Lanczos and Weyl, and he stood, of course, on the shoulders of Einstein and de Sitter. But similar arguments could be held against Newton and Einstein. However, if we apply our usual standards of attributing scientific discoveries, we should recall the situation of 1927. Einstein's static universe could not explain Slipher's redshifts, de Sitter's theory which provided redshifts was incomprehensible. Lemaître spotted the problem in de Sitter's work, one of the great figures of astronomy in the first half of the twentieth century. Before Lemaître only Friedmann had been sufficiently reckless to seriously follow up the idea of a truly dynamical universe. Now, in 1927, Lemaître derived from Einstein's fundamental equations the solution of a dynamical universe. To create a link to observations, he looked for the effect that his model would have on spectra of distant sources. This gave him the linear velocity-distance relationship v = H · d, where a redshift signified an expanding universe, blueshifted spectra would have meant a shrinking universe. He then collected the available redshifts and distances to derive the missing factor of proportionality, which could not be derived from theory. The observations assured him that we live in an expanding universe. This was one of the most fascinating discoveries ever made.
The full story is much richer and more colourful than what can be summarised on a few pages, and the following very incomplete list of references is much extended in our book [10].
8 .
8Lemaître's article of 1927 appeared in French in the Annales de la Société scientifique de Bruxelles. It was translated into English and published in 1931 in Monthly Notices[5]. However, there was a historically momentous omission. His derivation of the numerical value of H was cut out by a deliberate act. Until recently it was an unsolved puzzle why this was done, and who was responsible. Thus the public, who read the English version, was left with the impression that Hubble had been the first to derive H. Hubble was even accused of having instigated the cuts in the translation. However, in 2011, two letters resolved the riddle.The first letter was a request to Lemaître by Dr. Smart on behalf of the Royal
Astronomical Society (RAS) for a translation of his 1927 article for publication
in the Monthly Notices (article [1] by Block). The secretary of the RAS stresses
that "This request of the Council is almost unique in the Society's annals and
it shows you how much the Society would appreciate the honour of giving your
paper a greater publicity amongst English speaking scientists". Smart adds "... if
you have any further additions etc on the subject, we would glad[ly] print these
too. I suppose that if there were additions a note could be inserted to the effect
that § §1-72 are substantially from the Brussels paper + the remainder is new (or
something more elegant)". Lemaître obliged (article [8] by
. D Block, arxiv.org/abs/1106.3928D. Block. arxiv.org/abs/1106.3928 (2011).
A relation between distance and radial velocity among extra-galactic nebulae. E Hubble, PNAS15E. Hubble. A relation between distance and radial velocity among extra-galactic neb- ulae, PNAS, 15, 168-173. (1929).
Who discovered the expanding universe?. H Kragh, R W Smith, History of Science. 41H. Kragh and R.W. Smith. Who discovered the expanding universe?, History of Sci- ence, Vol. 41, 141-162. (2003).
Un univers homogène de masse constante et de rayon croissant, rendant compte de la vitesse radiale des nébuleuses extra-galctiques, Annales de la Société scientifique de Bruxelles. G Lemaître, Série A. 47G. Lemaître. Un univers homogène de masse constante et de rayon croissant, rendant compte de la vitesse radiale des nébuleuses extra-galctiques, Annales de la Société scientifique de Bruxelles, Série A, 47, 49-59. (1927).
A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae. G Lemaître, MNRAS. 91G. Lemaître. A homogeneous universe of constant mass and increasing radius ac- counting for the radial velocity of extra-galactic nebulae, MNRAS, 91, 483-490. (1931a).
The beginning of the world from the point of view of quantum theory. G Lemaître, Nature. 127706G. Lemaître. The beginning of the world from the point of view of quantum theory, Nature, 127, 706. (1931c).
L'expansion de l'Univers, par Paul Couderc. G Lemaître, Annales d'Astrophysique. 13G. Lemaître. L'expansion de l'Univers, par Paul Couderc. Annales d'Astrophysique, 13, 344-345. (1950).
Lost in translation: Mystery of the missing text solved. M Livio, Nature. 479Livio, M. Lost in translation: Mystery of the missing text solved. Nature 479, 171-173. (2011).
. J.-P Luminet, arxiv.org/abs/1105.6271v1J.-P. Luminet. arxiv.org/abs/1105.6271v1 (2011).
Discovering the Expanding Universe. H Nussbaumer, L Bieri, 978-0-521-51484-2Cambridge University PressCambridge, UK.H. Nussbaumer and L. Bieri. Discovering the Expanding Universe, Cambridge Uni- versity Press, Cambridge, UK. (2009). ISBN 978-0-521-51484-2.
The Observatory. A S Proceeding Of The R, 53Proceeding of the R.A.S., The Observatory, Vol. 53, 37-39. (1930).
. S Van Den, Bergh, arxiv.org/abs/1106.1195S. van den Bergh. arxiv.org/abs/1106.1195 (2011).
| []
|
[
"Universality in dissipative Landau-Zener transitions",
"Universality in dissipative Landau-Zener transitions"
]
| [
"Peter P Orth \nDepartment of Physics\nYale University\n06520New HavenConnecticutUSA\n",
"Adilet Imambekov \nDepartment of Physics and Astronomy\nRice University\n77005HoustonTexasUSA\n",
"Karyn Le Hur \nDepartment of Physics\nYale University\n06520New HavenConnecticutUSA\n"
]
| [
"Department of Physics\nYale University\n06520New HavenConnecticutUSA",
"Department of Physics and Astronomy\nRice University\n77005HoustonTexasUSA",
"Department of Physics\nYale University\n06520New HavenConnecticutUSA"
]
| []
| We introduce a random variable approach to investigate the dynamics of a dissipative two-state system. Based on an exact functional integral description, our method reformulates the problem as that of the time evolution of a quantum state vector subject to a Hamiltonian containing random noise fields. This numerically exact, non-perturbative formalism is particularly well suited in the context of time-dependent Hamiltonians, both at zero and finite temperature. As an important example, we consider the renowned Landau-Zener problem in the presence of an Ohmic environment with a large cutoff frequency at finite temperature. We investigate the 'scaling' limit of the problem at intermediate times, where the decay of the upper spin state population is universal. Such a dissipative situation may be implemented using a cold-atom bosonic setup. | 10.1103/physreva.82.032118 | [
"https://arxiv.org/pdf/0912.3531v4.pdf"
]
| 119,251,180 | 0912.3531 | 16fcc16be7c9cc08bdc61ae24c5d93c52db05ec4 |
Universality in dissipative Landau-Zener transitions
Peter P Orth
Department of Physics
Yale University
06520New HavenConnecticutUSA
Adilet Imambekov
Department of Physics and Astronomy
Rice University
77005HoustonTexasUSA
Karyn Le Hur
Department of Physics
Yale University
06520New HavenConnecticutUSA
Universality in dissipative Landau-Zener transitions
(Dated: October 5, 2010)numbers: 0365Xp0365Yz3380Be7450+r
We introduce a random variable approach to investigate the dynamics of a dissipative two-state system. Based on an exact functional integral description, our method reformulates the problem as that of the time evolution of a quantum state vector subject to a Hamiltonian containing random noise fields. This numerically exact, non-perturbative formalism is particularly well suited in the context of time-dependent Hamiltonians, both at zero and finite temperature. As an important example, we consider the renowned Landau-Zener problem in the presence of an Ohmic environment with a large cutoff frequency at finite temperature. We investigate the 'scaling' limit of the problem at intermediate times, where the decay of the upper spin state population is universal. Such a dissipative situation may be implemented using a cold-atom bosonic setup.
I. INTRODUCTION
A two-level system is never completely isolated resulting in dissipation, decoherence and entanglement [1].Therefore, one primary task for experimentalists is to manipulate and read out the internal state of the dissipative two-level system (qubit) with a high fidelity. Often, this can be achieved by sweeping the two energy levels through an avoided crossing, a situation that occurs in a variety of physical areas such as molecular collisions [2], chemical reaction dynamics [3], molecular nanomagnets [4], quantum information and metrology [5][6][7][8]. For a constant crossing speed v this is known as the Landau-Zener problem [9][10][11][12] which can be solved exactly in the absence of dissipation. Naturally, it is important to know the effect of the dissipative universe on the probability p(t) for the spin to remain in its initial state at time t [13][14][15][16]. Exact results [17,18] are only available at zero temperature and in the limit t → +∞, where the energy difference of the two spin states is much larger than the bandwidth ω c of the environmental bath. Typically however, ω c is much larger than the tunneling coupling between the two states ∆. Here, we rather focus on the experimentally relevant "scaling" regime at intermediate times, where the spin energies have not completely traversed the bath's energy band: ∆ < = vt < ω c with v > 0. To resolve the dissipative spin dynamics, we develop a powerful numerically exact stochastic Schrödinger equation formalism (SSE). Compared to earlier SSE approaches [19][20][21][22], our method allows easier exact consideration of initial spin-bath correlations, which are crucial in the Landau-Zener context. It may also be applied to other many-body environments that can be represented in the form of a Coulomb gas such as the Kondo model. [23,24] We prove that p(t) exhibits a universal decay in the intermediate (scaling) regime due to phonon assisted spin transitions. The size of the jump at the level crossing decreases for increasing dissipation and p(t) converges to the infinite time value only when t ∼ ω c /v. We also derive an approximate analytical decay formula valid for slow sweeps at zero temperature, which agrees well with our numerical results.
II. MODEL AND NOTATIONS
Specifically, we study a two-level system coupled to a bath of harmonic oscillators (the spin-boson Hamiltonian) [25,26]
H = ∆ 2 σ x + 2 σ z + σ z 2 k λ k (b † k + b k ) + k ω k b † k b k . (1)
Here, σ x,z are the Pauli matrices, ∆ is the bare tunneling coupling and the detuning. The bosonic oscillator operators have frequencies ω k and coupling constants λ k . We express the components of the reduced spin density matrix ρ(t) using functional integrals [25,26] ρ
(σ f , σ f ; t) = Dσ(·) Dσ (·)A[σ]A * [σ ]F [σ, σ ] ,(2)
where A[σ] is the amplitude for the spin to follow the path σ(t) in the absence of the bath, and F [σ, σ ] is the real-time influence functional of the bath
F [σ, σ ] = exp − 1 π t t0 ds s t0 ds {−iL 1 (s − s )ξ(s)η(s ) + L 2 (s − s )ξ(s)ξ(s )} ,(3)
written in terms of symmetric and antisymmetric spin paths η(s) = 1 2 [σ(s) + σ (s)] and ξ(s) = The kernel functions L 1 (t) = ∞ 0 dωJ(ω) sin ωt and L 2 (t) = ∞ 0 dωJ(ω) cos ωt coth ω/2k B T are determined by the bath spectral function J(ω) = π k λ 2 k δ(ω − ω k ) and the temperature T .
At time t 0 → −∞, the spin-bath interaction is first turned on, but the spin is held fixed in position σ i for arXiv:0912.3531v4 [cond-mat.other] 1 Oct 2010 t 0 < t ≤ 0. The spin paths {σ(t), σ (t)} in Eq. (2) are constrained to σ(t) = σ (t) = σ i for t ≤ 0 and to σ(t f ) = σ f , σ (t f ) = σ f . At t = 0, the bath is in the shifted canonical equilibrium state. For positive times, the spin jumps between the states {|↑ , |↓ } and the spin double path occurring in Eq. (2) can thus be regarded as a single path between the four states {|↑↑ , |↑↓ , |↓↑ , |↓↓ }. If the path starts and ends in a diagonal ("sojourn") state {| ↑↑ , | ↓↓ } and makes 2n transitions at times t 1 < t 2 < . . . < t 2n along the way, it can be parametrized as
ξ(t) = 2n j=1 Ξ j Θ(t − t j ) and η(t) = 2n j=0 Υ j Θ(t − t j ).
The variables {Ξ 1 , . . . , Ξ 2n } = {ξ 1 , −ξ 1 , . . . , −ξ n } embody the n off-diagonal ("blip") parts of the path between the times t 2m−1 and t 2m (m = 1, . . . , n), and characterize the time spent by the path in the states {|↑↓ , |↓↑ } such that ξ(t) = ±1, η(t) = 0. The variables {Υ 0 , . . . , Υ 2n } = {η 0 , −η 0 , . . . , η n } describe the n + 1 diagonal (sojourn) parts in the time period (t 2m , t 2m+1 ) during which η(t) = ±1, ξ(t) = 0 (here, we have m = 0, . . . , n and t 2n+1 ≡ t f ). The path's boundary conditions then specify η 0 and η n .
Inserting this general spin path ξ(t), η(t) into Eq. (3) and performing the time integrations yields
F n [Ξ j , Υ j , t j ] = Q 1 Q 2 with Q 1 = exp i π 2n j>k≥0 Ξ j Υ k Q 1 (t j − t k ) ,(4)Q 2 = exp 1 π 2n j>k≥1 Ξ j Ξ k Q 2 (t j − t k ) ,(5)
where Q 1,2 are the second integrals of L 1,2 . The free spin-path amplitudes A[σ]A * [σ ] give a factor iξη∆/2 to switch from a sojourn state η to a blip state ξ (and vice versa) as well as a bias-dependent phase factor
H n = exp[i 2n j=1 Ξ j s(t j )] with s(t) = t 0 dt (t ).
Altogether, the probability p(t) = ρ(| ↑ , | ↑ ; t) to find the system in state | ↑ at time t takes the form,
p(t) = 1 + ∞ n=1 i∆ 2 2n t 0 dt 2n · · · t2 0 dt 1 {ξj ,ηj } F n H n .(6)
III. RANDOM VARIABLES
We now proceed and decouple the terms bilinear in the blip and sojourn variables by Hubbard-Stratonovich transformations. Such a decoupling is useful since Eq. (6) has the Coulomb gas structure. [23]. Our formalism may thus be applied to other models which allow a Coulomb gas representation such as the Kondo model [24]. The resulting expression then suggests that p(t) can be obtained as a statistical average of a stochastic Schrödinger equation [19][20][21][22][23]27].
For definiteness, we will now focus on the case of an Ohmic bath with spectral function J(ω) = ηω exp(−ω/ω c ). It contains the viscosity coefficient η and a high-frequency cutoff ω c , and we also introduce the dimensionless dissipation parameter α = η/2π . We like to emphasize that our method is able to solve for the system's dynamics at any temperature T . The bath correlation functions read Q 1 (t) = η tan −1 (ω c t) and [25,26]. In fact, to apply a Hubbard-Stratonovich transformation to Eq. (5), we need to write
Q 2 (t) = η 2 ln(1 + ω 2 c t 2 ) + η ln πk B T t sinh πk B T tQ 2 (t j −t k ) in a factorized form Q 2 (t j − t k ) = η 2 [G 0 + mmax m=1 G m Ψ m (t j )Ψ m (t k )].
Since the kernel is translationally invariant, this can be achieved by a Fourier series expansion. To obtain only negative Fourier coefficients, we rather expandQ 2
(τ ) = Q 2 (τ ) − Q 2 (2) = η 2 [g 0 + mmax/2 m=1 g m cos mπτ 2 ]
, where we introduced the rescaled time τ = t/t max , with t max being the final time of our numerical simulation. Thus, the coefficients are
G 0 = g 0 + 2 η Q 2 (2), G 2k−1 =G 2k =g k < 0,Q 2 = e −nα[ 2 η Q2(2)+G] dS exp i 2n j=1 Ξ j h(τ j ) ,(7)
where the sum G= −αG m Ψ m (τ ). We can proceed similarly with Q 1 after separating it into a symmetric Q 1 (|t|) and an antisymmetric part Q 1 (t) in order to extend the sum to j ≤ k. On the other hand, for zero detuning = 0 and α < 1/2 [25,26], one can safely approximate Q 1 (t) ≈ ηπ/2. This approximation becomes exact for ∆/ω c → 0 since the main contribution to the functional integral of Eq. (6) stems from spin flips with time separations larger than ω −1 c . The finite bias case = 0 requires more consideration of the first sojourn as it accounts for the spin-bath preparation, which affects the long-time behavior of p(t) [18] (see below). For = 0, Eq. (6) reads
mmax/2 m=0 g m is equal to [− 2 η Q 2 (2)] for m max →∞,p(τ ) = 1 + dS ∞ n=1 i∆t max e − α 2 [ 2 η Q2(2)+G] 2 2n τ 0 dτ 2n × · · · τ2 0 dτ 1 {ξj ,ηj } exp[iπα n−1 k=0 η k ξ k+1 ] 2n j=1 exp[iΞ j h(τ j )] .(8)
Without the summation over blip and sojourn variables {ξ j , η j }, this expression has the form of a timeordered exponential, averaged over the random variables {s m }. This summation, however, can be incorporated into a product of matrices in the vector space of states
V = A 0 e −ih(τ ) −e ih(τ ) 0 e iπα e ih(τ ) 0 0 −e −iπα e ih(τ ) −e −iπα e −ih(τ ) 0 0 e iπα e −ih(τ ) 0 −e −ih(τ ) e ih(τ ) 0 ,(9)i ∂ ∂τ | Φ(τ ) = V (τ )| Φ(τ ) ,(10)
with initial and final conditions | Φ i,f = (1, 0, 0, 0) T for N different realizations of the noise variables {s m }. Averaging the results gives p(τ ) = 1
N N k=1 Φ (k) 1 (τ ), where Φ 1 (τ ) is the first component of | Φ(τ ) .
Other components of the density matrix (2) can be obtained using different initial and final conditions. In fact, the differential equations obey the additional symmetries ImΦ 1 = 0, Φ * 3 = Φ 2 and Φ 4 = 1 − Φ 1 , such that only three real-variables are independent. Since the evolution is unitary (for = 0) and Φ 2 1 + 2|Φ 2 | 2 = 1 is an integral of motion, we can introduce a classical unitlength spin S = ( √ 2ReΦ 2 , √ 2ImΦ 2 , Φ 1 ) that evolves according to dS/dτ = H × S in a random magnetic field H = (cos h(τ ), sin h(τ ), 0), and from which we find p(τ ) = 1 2 (1 + S z (τ ) ). Hence, the time-evolution of a dissipative quantum spin can be formulated as that of a classical spin in a random magnetic field. The quantum nature of the problem is hidden in the fact that spin rotations about different axes do not commute and through the averaging over random field configurations.
IV. APPLICATIONS
A. Spin dynamics at zero detuning
To prove the feasibility of our method, we have computed the spin dynamics for zero detuning in the range 0 < α < 1/2 for different temperatures T . We express T in units of ∆ (hereafter we set = k B = 1). Results for P (t) = 2p(t) − 1 in Fig. 1 exhibit damped oscillations with the correct renormalized tunneling frequency of order ∆ r = ∆( ∆ ωc ) α/(1−α) for T ∆ r . The quality factor of the oscillations agrees with predictions from the Non Interacting Blip Approximation (NIBA) [25], field theory [28] and from the time-dependent numerical renormalization group (TD-NRG) [30].
For intermediate values of alpha we are able to access the asymptotic long-time behavior of P (t), where |P (t)| 1, within our numerical approach. At T = 0, we find that the system exhibits exponentially damped coherent oscillations as predicted in Ref. [28] and in agreement with recent TD-NRG calculations [30]. Note that the statistical error of our method scales like N −1/2 , where N is the number of different realizations of the noise. We thus cannot resolve the existence of a small incoherent part at weak-coupling α ln ωc ∆ 1, which was predicted in Ref. 31 using a rigorous Born approximation.
For increasing temperature, the coherence of oscillations gets lost more rapidly, and finally for T ∆ r we observe incoherent decay P (t) = exp[−t/τ ] with rate [25]. Note that our method gives reliable results over the full range of temperatures.
τ −1 = √ πΓ(α) 2Γ(α+ 1 2 ) ∆ 2 r T πT ∆r 2α
B. Dissipative Landau-Zener transition
Next, we turn to the case of a Landau-Zener sweep of the detuning (t) = vt with v > 0. We examine the survival probability p(t) that the spin remains in its initial state if swept across the resonance. Neglecting the bath, this problem can be solved exactly [9][10][11][12] and one finds that p(t) converges toward the celebrated Landau-Zener formula p lz = exp[−π∆ 2 /2v] for t ∆/v. A fundamental question is thus how this result is modified in the presence of dissipation. Surprisingly, at zero temperature the bath does not affect the final transition probability p lz in the limit t → +∞, if the spin couples longitudinally to the reservoir via its σ z component [17]. This limit, however, corresponds to very large times t ω c /v where the separation of the spin energies is larger than the bosonic bandwidth. In contrast, we explore the so-called scaling regime, where one first takes the limit ω c → ∞, holding ∆ r t = y fixed, and only then considers y → ∞. This limit is important because it allows the spin-boson model to exhibit universal behavior [25,26]. For large but finite ω c the scaling regime corresponds to an intermediate time regime where the spin energy separation is smaller than ω c but possibly much larger than ∆: ∆ vt ω c . Phonon assisted spin transitions therefore still occur even though ∆, and the probability p(t) converges toward its final value p lz only for times of the order t ∼ ω c /v. Note that this is in stark contrast to the non-dissipative (perfectly isolated) case where this convergence happens much faster for t ∼ ∆/v.
In the context of Landau-Zener transitions, the bath preparation affects the long-time result of p(t) [18]. Thus, it is important to consider the contribution of the initial sojourn exactly, as it accounts for the fact that the bath starts out from a shifted equilibrium state. It is given by the k = 0, 1 terms in Q 1 (Eq. (4)). We can incorporate this term by adding it to the height function
h(τ, τ 1 ) = vt 2 max 2 (τ 2 − 2τ c τ ) + mmax m=1 s m −αG m Ψ m (τ ) − 2α tan −1 [ω c t max (τ − τ 1 )] .(11)
Here, t max determines the time interval length of our simulation, and [0, τ c ] ([τ c , 1]) correspond to times before (after) the level crossing. The fact that the height function now contains τ 1 forces us to explicitly perform the τ 1 -integration in Eq. (8). We thus randomly pick a uniformly distributed τ 1 ∈ [0, 1], which determines h(τ, τ 1 ) as well as the initial state | Φ τ1 = −i(0, e ih(τ1,τ1) , −e −ih(τ1,τ1) , 0) T . We then propagate this initial state in the interval [τ 1 , 1] according to Eq. (10) and calculate the survival probability as p(τ ) = 1 + Φ 1 (τ ) , where the average is over N choices of τ 1 and random variables {s m }. Here we set | Φ(τ < τ 1 ) = 0 in an individual run since Φ 1 (τ ) only accounts for the contribution of paths with at least one spin jump. In Eq. (10), the evolution is not unitary.
In Fig. 2 (a), we check that p(t) converges toward p lz at long times t ω c /v for T = 0. For the large sweeping speed v/∆ 2 = 10 in Fig. 2 (a), we find that this also holds for T /∆ = {1, 5}, since thermal effects only occur during the short period where |vt| T [16]. In Fig. 2 (b), we show results for slow sweep velocity v/∆ 2 = 0.5, and observe clearly that the size of the jump in p(t) at the crossing reduces with enhancing dissipation and temperature. Following the jump, we observe a decay of p(t) in the intermediate time regime up to t ∼ ω c /v due to bath mediated spin transitions. The final probability increases with temperature due to thermalization.
We now derive an analytical formula describing the universal decay in the scaling regime at T = 0, which holds for slow sweeping speeds only. For large static detuning ∆ r (but still ω c ) the NIBA can be justified [25] and predicts an overdamped exponential relaxation with a decay rate Γ = π∆r 2Γ(2α) ( /∆ r ) 2α−1 . Inserting (t) = vt and integrating dp/dt = −Γp(t) for α < 1/2 yields
p[ (t)] = C exp −π∆ 2 r 4αΓ(2α)v ∆ r 2α .(12)
If we except the integration constant C, this formula contains only scaling variables, which shows that the decay is universal. It reduces to p lz in the limit α → 0 (with C = 1) and breaks down for times of the order t ∼ ω c /v, where it becomes a function of the bare ∆ again: Fig. 2 (c), we show that the decay is indeed described by this formula for T = 0, v/∆ 2 = 0.5 and α = 0.05. Note that this decay does not occur at α = 1/2 [32]. We like to emphasize that our numerical method gives reliable results over the whole range of sweep velocities and temperatures.
p[ = ω c ] = C exp[− π∆ 2 4αΓ(2α)v ]. In
C. Realization with cold-atom quantum dot setup
The intermediate (scaling) time regime ∆ vt ω c might be accessed using the cold-atom geometry of Refs. [33][34][35]. It comprises a bosonic mixture of atoms in two hyperfine ground states a and b, subject to state-selective traps. One species forms a onedimensional Bose-Einstein Condensate (BEC), representing the Ohmic reservoir, and the other species is trapped in a tight harmonic potential, operated in the collisional blockade limit, representing the "spin". Coupling the different species by Raman lasers, the system is described by Eq. (1) with ∆ and being proportional to the laser intensity and frequency, respectively. Using the parameters of Ref. [36], we estimate α = 1 4K (−1 + g ab /g aa ) 2 ≈ 0.06; K ∼ ρ a /g aa is the Luttinger parameter of the BEC, g αβ = 2 ω ⊥ a αβ are the scattering amplitudes containing the transverse trapping frequency ω ⊥ = 2π × 67kHz and the scattering length a aa = 5.2nm. The value of a ab must be tuned such that g ab g aa using optical Feshbach resonances [37]. Choosing ∆ ≈ 100Hz and v ≈ 1kHz/s, the intermediate time (scaling) regime occurs between 0.1s < t < 10s.
V. CONCLUSIONS
To summarize, we have developed a stochastic Schrödinger method to investigate the dissipative Landau-Zener problem in the scaling limit ∆/ω c 1 at finite temperature. Assuming α < 1/2, we have shed light on an experimentally relevant intermediate timeregime where p(t) shows universal decay due to bath mediated spin transitions. Our results are relevant in quantum information, where fast quantum processes are more useful. Our method can also be extended to other many-body environments.
VI. ACKNOWLEDGMENTS
1 2
1[σ(s) − σ (s)], respectively.
the integration over the Gaussian distributed Hubbard-Stratonovich variables reads dS =
FIG
. 1. (Color online): (a) P (t) as a function of t for various values of α, ∆ = 1, ωc = 100, = 0 and T = 0. We checked that for a given α curves corresponding to different ωc/∆ 1 scale on top of each other in units of the renormalized tunneling rate ∆r = ∆( ∆ ωc ) α/(1−α) . Quality factor Ω/γ of damped oscillations agrees with prediction Ω/γ = cot πα 2(1−α) from Refs. [25, 26, 28]. Results are obtained with mmax = 3000, N = 5 · 10 4 . (b) P (t) for different temperatures T (here, = kB = 1), dissipation strength α = 0.1, and other parameters as in (a).{| ↑↑ , | ↑↓ , | ↓↑ , | ↓↓ }, which have the form[29]
with A = 1 2 (∆t max e −(α/2)[(2/η)Q2(2)+G] ). Then, Eq. (8) becomes p(τ ) = dS Φ f |T e −i τ 0 dsV (s) |Φ i which can be calculated by solving the stochastic Schrödinger equation
FIG. 2 .
2(Color online) (a) p(t) for a fast sweep with v/∆ 2 = 10, ωc/∆ = 200, α = {0.05, 0.1} and T /∆ = {0, 1, 5} (here, = kB = 1). We choose mmax = 4000, N = 4 · 10 6 . (b) Slow sweep with v/∆ 2 = 0.5. Other parameters as in (a). (c) Fit of universal decay of p[ (t)] using Eq. (12) with α = 0.05 and single fit parameter C = 0.59.
and the trigonometric functions read Ψ 2k−1 =cos kπτ 2 , Ψ 2k =sin kπτ 2 , where k=1, . . . , m max /2. Decoupling the blip variables by m max Hubbard-Stratonovich transformations then results in
We acknowledge discussions with L. Glazman, W. Hofstetter, and D. Roosen. This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG02-08ER46541 and by the Yale Center for Quantum Information Physics through the grant NSF DMR-0653377.
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| []
|
[
"Towards Exascale Simulations of the ICM Dynamo with WENO-Wombat",
"Towards Exascale Simulations of the ICM Dynamo with WENO-Wombat"
]
| [
"Julius Donnert \nINAF Istituto di Radioastronomia\nvia P. Gobetti 101I-40129BolognaItaly\n\nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMNUSA\n",
"Hanbyul Jang \nDepartment of Physics\nSchool of Natural Sciences\nUlsan National Institute of Science and Technology\nUNIST-gil 5044919UlsanKorea\n",
"Peter Mendygral \nCray Inc\n55425BloomingtonMNUSA\n",
"Gianfranco Brunetti [email protected] \nINAF Istituto di Radioastronomia\nvia P. Gobetti 101I-40129BolognaItaly\n",
"Dongsu Ryu [email protected]. \nDepartment of Physics\nSchool of Natural Sciences\nUlsan National Institute of Science and Technology\nUNIST-gil 5044919UlsanKorea\n",
"Thomas Jones \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMNUSA\n"
]
| [
"INAF Istituto di Radioastronomia\nvia P. Gobetti 101I-40129BolognaItaly",
"School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMNUSA",
"Department of Physics\nSchool of Natural Sciences\nUlsan National Institute of Science and Technology\nUNIST-gil 5044919UlsanKorea",
"Cray Inc\n55425BloomingtonMNUSA",
"INAF Istituto di Radioastronomia\nvia P. Gobetti 101I-40129BolognaItaly",
"Department of Physics\nSchool of Natural Sciences\nUlsan National Institute of Science and Technology\nUNIST-gil 5044919UlsanKorea",
"School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMNUSA"
]
| []
| In galaxy clusters, modern radio interferometers observe non-thermal radio sources with unprecedented spatial and spectral resolution. For the first time, the new data allows to infer the structure of the intra-cluster magnetic fields on small scales via Faraday tomography. This leap forward demands new numerical models for the amplification of magnetic fields in cosmic structure formation-the cosmological magnetic dynamo. Here we present a novel numerical approach to astrophyiscal MHD simulations aimed to resolve this small-scale dynamo in future cosmological simulations. As a first step, we implement a fifth order WENO scheme in the new code WOMBAT. We show that this scheme doubles the effective resolution of the simulation and is thus less expensive than common second order schemes. WOMBAT uses a novel approach to parallelization and load balancing developed in collaboration with performance engineers at Cray Inc. This will allow us scale simulation to the exaflop regime and achieve kpc resolution in future cosmological simulations of galaxy clusters. Here we demonstrate the excellent scaling properties of the code and argue that resolved simulations of the cosmological small scale dynamo within the whole virial radius are possible in the next years. | 10.3390/galaxies6040104 | [
"https://www.mdpi.com/2075-4434/6/4/104/pdf"
]
| 54,035,551 | 1808.10633 | e41198ace280a5d08ae36a8e4cb98fc255d3ded6 |
Towards Exascale Simulations of the ICM Dynamo with WENO-Wombat
Published: 29 September 2018
Julius Donnert
INAF Istituto di Radioastronomia
via P. Gobetti 101I-40129BolognaItaly
School of Physics and Astronomy
University of Minnesota
55455MinneapolisMNUSA
Hanbyul Jang
Department of Physics
School of Natural Sciences
Ulsan National Institute of Science and Technology
UNIST-gil 5044919UlsanKorea
Peter Mendygral
Cray Inc
55425BloomingtonMNUSA
Gianfranco Brunetti [email protected]
INAF Istituto di Radioastronomia
via P. Gobetti 101I-40129BolognaItaly
Dongsu Ryu [email protected].
Department of Physics
School of Natural Sciences
Ulsan National Institute of Science and Technology
UNIST-gil 5044919UlsanKorea
Thomas Jones
School of Physics and Astronomy
University of Minnesota
55455MinneapolisMNUSA
Towards Exascale Simulations of the ICM Dynamo with WENO-Wombat
Published: 29 September 201810.3390/galaxies6040104Received: 28 August 2018; Accepted: 27 September 2018;galaxies Article * Correspondence: [email protected]; Tel.: +39-051-639-9364galaxy clustersmagnetic fieldsnumerical methodsmagneto-hydrodynamics
In galaxy clusters, modern radio interferometers observe non-thermal radio sources with unprecedented spatial and spectral resolution. For the first time, the new data allows to infer the structure of the intra-cluster magnetic fields on small scales via Faraday tomography. This leap forward demands new numerical models for the amplification of magnetic fields in cosmic structure formation-the cosmological magnetic dynamo. Here we present a novel numerical approach to astrophyiscal MHD simulations aimed to resolve this small-scale dynamo in future cosmological simulations. As a first step, we implement a fifth order WENO scheme in the new code WOMBAT. We show that this scheme doubles the effective resolution of the simulation and is thus less expensive than common second order schemes. WOMBAT uses a novel approach to parallelization and load balancing developed in collaboration with performance engineers at Cray Inc. This will allow us scale simulation to the exaflop regime and achieve kpc resolution in future cosmological simulations of galaxy clusters. Here we demonstrate the excellent scaling properties of the code and argue that resolved simulations of the cosmological small scale dynamo within the whole virial radius are possible in the next years.
Introduction
Most of the Baryonic matter in our Universe is in the form of magnetized plasma. Hence, astronomers observe the signature of astrophysical magnetic fields from the solar system to the large scale structure. In galaxy clusters, radio telescopes detect the synchrotron radiation (50 MHz-30 GHz) emitted by relativistic electrons (γ > 1000) gyrating in the magnetic field (B ∼ 1 µG) of the intra-cluster-medium (ICM), a hot and underdense plasma (T ∼ 10 8 K, n th ∼ 10 −3 cm −3 ). The next generation of radio interferometers will infer the three dimensional structure of the field through Faraday tomography on kpc scales. This represents a first serious probe of the small scale properties of the whole intra-cluster-medium that demands detailed predictions to interpret the new data. As radio brightness is not strongly correlated with thermal density, upcoming studies will probe the whole virial volume of a cluster.
The ICM itself is a weakly collisional plasma, whose micro-physical properties are set by turbulence and electromagnetic interactions (plasma-waves), not particle Coulomb scattering [1]. Thus the magnetic field plays a crucial role in making the medium "collisional" on large scales, i.e., behave like a magnetised fluid [2]. In the currently favoured model of the ICM, the evolution of the magnetic field is governed by a turbulent small-scale dynamo that grows small seed fields at high redshift (B ∼ 10 −13 G) into µG fields via an inverse cascade at the Alfvén scale of the medium [3,4]. In merging galaxy clusters the Alfvén scale 1 reaches a few kpc, thus it is now in range of next-generation radio interferometers for Faraday tomography studies.
The strength and geometry of the magnetic field is set by the local seeding and turbulence history of the gas parcel under consideration, thus the new data demands numerical simulations to compare with expectations from dynamo theory. However, the nature of the small scale dynamo has made it very difficult to obtain accurate numerical models for the ICM magnetic field [6]. The crucial time scale of magnetic field growth is set by the smallest length scale available in the turbulent system, where the eddy turnover time is smallest. In nature, this can be far below pc scale, in simulations this is at best the resolution scale. Current state-of-the-art Eulerian simulations start seeing numerical effects below scales of 10 kpc, Lagrangian simulations reach better resolution in the cluster center, but do not come close at the cluster outskirts due to density adaptivity (see Donnert et al., subm. to SSRv for a review). Thus resolving the Alfvén scale at 3 kpc in the whole cluster volume and faithfully evolving the magnetic field through structure formation is not possible with current community codes.
In the preferred Eulerian approach, such simulations would require ∼4096 3 zones inside the virial radius, run with a highly accurate finite volume or finite difference scheme. This translates into 50-100 TBytes of memory and would generate 1-4 PByte of data. This is well in range of current Petascale and upcoming Exascale supercomputers, but requires near ideal weak scaling of the simulation code to 5-10 thousand compute nodes. Current state-of-the-art simulations typically run on a few thousand nodes, to maximize parallel efficiency [7]. Hence, it stands to reason that in practice resolutions close the Alfvén scale in the ICM will be challenging to achieve with current codes.
Here we present a performance-aware implementation of the a fifth order constrained transport weighted essentially non-oscillatory (WENO) scheme in the scalable WOMBAT code 2 . This implementation represents a first step towards simulations of the small-scale dynamo in the ICM in a cosmological framework that resolve the Alfvén scale. We will show that WENO doubles the effective resolution of the simulation, but is only a factor 10 more computationally expensive than commonly used 2nd order schemes at the same resolution. Hence it is a more efficent algorithm. WOMBAT itself is an on-going research effort of performance engineers at Cray Inc. (Seattle, WA, USA) to maximize computational efficiency on upcoming exascale systems [8]. We will show that the new code indeed achieves excellent performance on large supercomputers.
WENO-Wombat
WENO Algorithm
The Weighted Essentially Non Oscillatory schemes [9] are an improvement of ENO schemes presented in Harten et al. [10], Shu and Osher [11]. ENO schemes chose one out of several stencils around a zone i based on a mathematical smoothness criterion to avoid spurious oscillations close to flow discontinuities (shocks). WENO schemes combine a weighted average of the stencils, so that the scheme is high order away from shocks, but still avoids Gibbs phenoma adjacent to them see [12,13] for a review. WENO-Wombat implements the classical scheme from Jiang and Shu [14], Jiang and Wu [15], which combines three stencils to achieve formal fifth order trunctation error in space. The algorithm uses a Roe-type Riemann solver to decouple the system of 8 partial differential equations into 1 The scale where magnetic and turbulent pressure are comparable, i.e., where the Lorentz force becomes important [5]. 2 wombatcode.org independent advection equations [16]. The WENO interpolated fluxes on the faces of zone i in the decoupled system are given by F s
i+ 1 2 = 1 12 −F s i−1 + 7F s i + 7F s i+1 − F s i+2 − ϕ N ∆F s+ i− 3 2 , ∆F s+ i− 1 2 , ∆F s+ i+ 1 2 , ∆F s+ i+ 3 2 + ϕ N ∆F s− i+ 5 2 , ∆F s− i+ 3 2 , ∆F s− i+ 1 2 , ∆F s− i− 1 2 ,(1)
where ∆F s− i the flux on the cell boundaries obtained from a simple Lax-Friedrichs splitting.
The WENO5 interpolant ϕ N (a, b, c, d) is defined as ϕ N (a, b, c, d) = 1 3 ω 0 (a − 2b + c) + 1 6 ω 2 − 1 2 (b − 2c + d) .(2)
The non-linear weights are given by:
ω 0 = α 0 α 0 + α 1 + α 2 , ω 2 = α 2 α 0 + α 1 + α 2 (3) α 0 = 1 ( + IC 0 ) 2 , α 1 = 6 ( + IC 1 ) 2 , α 2 = 3 ( + IC 2 ) 2 ,(4)
with = 10 −6 and
IS 0 = 13(a − b) 2 + 3(a − 3b) 2 ,(5)IS 1 = 13(b − c) 2 + 3(b + c) 2 ,(6)IS 2 = 13(c − d) 2 + 3(3c − d) 2 .(7)
Time integration is realized with a fourth order four-stage Runge Kutta integrator. It has been shown that the resulting scheme is only third order accurate close to critical points (extrema) of the flow. In real world applications this is actually beneficial, because the scheme becomes more robust than a full fifth order scheme like WENO-Z [17] that would fall back to protection fluxes instead.
Magnetic fields are treated in a constrained transport staggered mesh approach following Ryu et al. [18], which is formally only second order accurate. However, using the high order fluxes the scheme conserves magnetic energy density to fifth order (Jang et al., in prep.), which is of crucial importance for the small dynamo. For the complete description of the algorithm including eigenvectors we refer the reader to the full method papers Jang et al., in prep. and Donnert et al., in prep.
Wombat Implementation
WOMBAT is a hybrid parallel MPI/OpenMP code written in object oriented Fortran 2008 [8]. It is developed in collaboration with the programming environment group at Cray Inc., a major manufacturer of super computers.
The code initially divides the computational domain Ω into rectangular sub-domains (domains) residing each on a separate MPI rank. Each sub-domain is further divided into independent pieces of work, rectangular patches, usually of size 18 D -32 D zones, where D is the number of dimensions. Patches and domains are implemented using Fortran 2008 objects, which makes the code highly modular. Patches carry ghost zones, domains carry ghost domains, which overlap with neighbouring MPI ranks and facilitate communication. Their size is tunable. If a patch is exported into a ghost domain, it is communicated to the according MPI rank owning the domain. This facilitates load-balancing among ranks. Ghost/boundary zones of patches are overlapping zones between neighbouring patches that need to be communicated for the patch to be computed. Once a patch has received all its boundary zones, it can be computed (resolved) independently for the rest of the world grid. Depending on the solver, this may happen many times per time step.
WOMBAT implements communication of ghost zones and domains using fully one-sided thread-asynchronous MPI-RMA with MPI_THREAD_MULTIPLE. The scheme is continuously improved and an active research topic for the exa-scale by performance engineers are Cray Inc. To minimize OpenMP overhead the code uses a single OpenMP parallel region, in which all threads perform work and communication independently and asynchronously from each other. This requires the MPI library to support OpenMP lock-free communication. Patches internal to a rank are resolved in memory and can be immediately computed. Patches with boundary zones overlapping with another rank are communicated: The boundary zones are copied ("packed") into mailboxes by the neighbouring rank, communicated with MPI_Get, unpacked on the local rank and eventually the patch is resolved. The status of the mailbox (empty, packed) is communicated with 8 Byte signals (the "heartbeat"). The heartbeat signals are always issued and facilitate a weak form of synchronization among ranks. Every step of the communication (packing, heartbeat signal, unpacking, resolution) can be done by any thread on the rank at any time. Thus the scheme achieves computation/communication overlap at the thread level and can react to imbalance from work decomposition or network contention on the machine.
We have implemented the WENO scheme into the WOMBAT framework as a separate solver module. At the beginning, the state vector grid of the resolved patch is copied into thread local memory. In multiple dimensions, it is flattened into a one dimensional array to increase vector length. This introduces memory overhead of about 25 Megabytes per thread and rank for 18 3 zone patches. The whole algorithm operates on single index arrays, thus all loops are SIMD vectorized by the compiler. This is necessary to achieve a significant fraction of peak double precision performance on modern CPUs and greatly simplifies future GPU ports of the algorithm. In multiple dimensions, the grid has to be swept along the y-direction and z-direction. We implement the sweeps by re-ordering the data arrays, which corresponds to rotations of the grid in three dimensions. The resulting fluxes are then rotated back into the original layout, so that final updates can be performed. At the end of a sub-step, the grid is saved into the multi-index arrays in global memory.
Per time step, the boundaries of 3 zones are communicated 8 times, i.e., the patches is passed 8 times until it is resolved. The WENO and CT scheme require one pass each per sub-step.
Results
Fidelity
We demonstrate the fidelity of the WENO MHD algorithm with several test cases. In Figure 1, we show the convergence test involving advecting small (10 −5 ) perturbations in the four MHD waves across a one dimensional domain, following Gardiner and Stone [19]. All waves converge at fifth order down to L1 errors below 10 −11 . For the compressive fast and slow modes, wave steepening prevents further convergence in this test. Entropy and Alfén mode converge to 10 −13 , where the roundoff error from 8 byte floating point precision prevents further convergence. A comparison with results from ATHENA [19] shows that the WENO5 scheme improves on the CTU+CT by more than a factor of two in effective resolution, i.e., ATHENA reaches L1 errors of 10 −11 at 512 zones, WENO5 reaches theis L1 error at 64 zones.
In Figure 2 we show the density of the 2 dimensional Orszang-Tang vortex test from Orzang and Tang [20], Tóth [21], Stone et al. [22]. The left panel shows the result from a run with WOMBAT's new WENO5 implementation with 128 2 zones. On the right the result from the run with 2nd order TVD+CTU implementation at 256 2 zones. No difference is visible by eye. In Figure 3 we show pressure slices of the test at t = 0.5. Again WENO5 with 128 2 zones resolves the complex pressure topology equally or better than the TVD + CTU result with 256 2 zones. In particular, the pressure blip at y = −0.1875, x = −0.45. This demonstrates that the higher fidelity of the WENO5 algorithm effectively doubles the resolution of a MHD simulation. At the same time, a WENO5 run in 3D uses only an 8th of the memory and 75% of the runtime of a TVD + CTU run at double its resolution. Thus we argue that fifth order WENO5 represents an optimal compromise between algorithmic fidelity and computational expense. This result is in line with previous work on WENO3, WENO5 and WENO9: Zhang et al. [23] found that every increase in order effectively doubles the resolution of the scheme. In Figure 4, we show a 2D Kelvin Helmholtz instability test following the ICs of Lecoanet et al. [24] who use a sinusoidal perturbation (A = 0.01, P = 10, a = 0.05, σ = 0.2, z 1 = 0.5, z 2 = 1.5, u flow = 1, ∆ρ = 1) at t = 2. We note that the with these parameters the instability is not resolved in both runs. The left panel shows is the WENO5 solution with 64 × 128 zones. The right panel shows the TVD+CTU solution with twice the resolution. In the WENO5 simulation, the instability grows as expected and develops the famous vortices as well as fluctuations away from the shearing layer. In contrast, the second order run, does not show well developed vortices and substantially more diffusion. The growth of the instability is significantly slowed. This test shows that WENO5 resolves instabilities close to the resolution scale significantly better than a lower order code at twice the WENO5 resolution.
In Figure 1 right, we show the time evolution of magnetic energy in the field loop advection test from Stone et al. [22]. Runs at 32 2 , 64 2 , 128 2 zones in red, blue, green respectively are shown with WENO5 (solid lines) and with TVD+CTU (dashed lines). Magnetic energy is conserved much better in the WENO5 case, due to the fifth order fluxes in the scheme. A comparison with the ATHENA results presented in Gardiner and Stone [19] again show that WENO5 roughly doubles the effective resolution, e.g., at t = 1 ATHENA with 128 zones conserves 95% of magnetic energy (Figure 1 in [17]), just as WENO5 with 64 zones. We argued above that magnetic field growth close to the resolution scale is the crucial mechanism for magnetic field amplification in galaxy clusters. Thus this result demonstrates the advantages of using a highly accurate CT scheme in cluster MHD simulations.
Efficiency
On a single Broadwell core, the WENO5 implementation is found to perform at about 4.9 GFlops or 20% of double precision peak, due to the high degree of SIMD vectorization of the code. On a single Broadwell node, the code performs at 7.2% of peak, or 1.3 GFlops per core.
We showcase the performance of the implementation on multiple nodes, by running a cache blocking study on a Cray Inc. development system. To this end, we vary the patch size of a problem at roughly 1024 3 zones on 27 nodes with Aries interconnect and 2 ×18 core Intel Broadwell CPUs per node. We vary the patch size between 4 3 and 72 3 zones and the number of MPI ranks per node between 2 and 36, which corresponds to 18 and 1 OpenMP threads. The resulting throughput in Million zones per second per node is shown in Figure 5 left. The optimal patch size is found to be about 18 3 zones, performing at 0.6 Million zones per second per node. At this patch size, most of the patch fits the level 3 cache of the Broadwell CPU, but not the level 2 cache. We note that WOMBAT's TVD+CTU scheme has an optimal patch size of 32 3 on the same system and a strong drop in throughput towards larger patch sizes. Here our implementation approaches optimal performance again. We speculate that because TVD+CTU does not implement array flattening in the algorithm, the hardware pre-fetching is not efficiently keeping data in the cache hierarchy, leading to the drop in performance. In contrast, the flattened array in the WENO5 implementation lead to stride 1 array accesses throughout the algorithm, which allows the pre-fetcher to keep data in the cache hierarchy efficiently for large patch sizes. Nonetheless, smaller patches are highly desirable for load balancing and thus 16 3 -20 3 is the optimal patch size for WENO5 applications. In weak scaling tests, the problem size is increased alongside the machine size to expose the degree of non-parallelizeable overhead in a program [25]. We chose a small problem size with a runtime of 2.4 s per step to clearly expose communication overhead at scale. On the right of Figure 5, we show the time per update/weak scaling of the WENO5 implementation on the Cray XC40 supercomputer "Hazel Hen" at the HLRS in Stuttgart. The system was not dedicated to the test, thus the interconnect is subject to the usual contention seen on large shared systems in practice. From 4 nodes/96 cores onwards, the step time is found relatively constant between 2.4 and 2.5 s per step with a 10% scatter among time steps, represented by the error bars. This scatter is typical for Haswell system of this size. We note that smallest data point corresponds to workstation size machine with 48 cores. The largest run used 4096 nodes/96.000 cores, which is more than half of the machine. With a problem size of 4.5 Billion zones such a run would resolve the Alfvén scale in a triple zoomed cosmological simulation of a galaxy cluster.
Conclusions
We have presented a new implementation of a fifth order WENO5 scheme for constrained transport magneto-hydrodynamics in the WOMBAT framework. Our code is aimed at the simulation of cosmic magnetic fields in galaxy clusters, in particular the turbulent small-scale dynamo in the ICM and its Faraday tomography signal. We have motivated the need for new community codes for this particular problem as supercomputers enter the exasflops regime. We have given a concise overview of the WENO5 algorithm and the implementation in WOMBAT. Finally we shown a few code tests with the new code and argued that the algorithm represents an efficiency optimum as it doubles the effective resolution compared to second order codes common in the field today. We have also shown that given the same resolution, WENO5 resolves instabilities better than second order TVD+CTU and improves on magnetic energy conservation. Finally we have shown cache optimization tests and demonstrated excellent weak scaling of the code up to realistic problem sizes of 4.5 billion zones on a current Cray XC40 supercomputer. Thus we are confident that accurate predictions of the magnetic field distribution in galaxy clusters from the small scale dynamo with resolved Alfvén scale are within reach in the next two years. Funding: This research has received funding from the People Programme (Marie Sklodowska Curie Actions) of the European Unions Eighth Framework Programme H2020 under REA grant agreement no 658912, "Cosmo Plasmas". TWJ acknowledges support from the US NSF through grant AST1714205.
Figure 1 .Figure 2 .
12Left: L1 error of the wave convergence test from in one dimension Gardiner and Stone[19]. Alfvén mode in red, entropy mode in blue, fast mode in the green and slow mode in purple.Right: Evolution of magnetic energy in the advection of a field loop test. Density in the 2D Orszag-Tang Vortex test. Left: WENO5 solution with 128 2 zones; Right: TVD solution with 256 2 zones. −0.5−0.4−0.3−0.2−0.1 0 0.1 0.2 0.3 0.
Figure 3 .
3Pressure slices in the 2D Orszag-Tang Vortex test at y = −0.1875 (left) and y = −0.073 (right). TVD solution with 256 2 zones (red line), WENO5 solution with 128 2 zones (black squares).
Figure 4 .
4Density in the 2D Kelvin Helmholtz test at time t = 2. Left: WENO5 solution with 64 × 128 zones; Right: TVD solution with 128 × 256 zones.
Figure 5 .
5Left: Performance in Millions of zones per second per node for different patch sizes on 27 nodes with Intel Broadwell CPUs for a problem of approximate 1024 3 zones. Right: Weak scaling on the Cray XC40 "Hazel Hen" at HLRS Stuttgart, Germany. Shown is the mean time per update of 100 updates over the number of nodes/cores. Error bars indicate the largest and smallest timestep of 100, variations are due to node variablity of the shared system.
Author
Contributions: Software, J.D., P.M., H.J.; supervision, T.J., D.R.; project administration, G.B.; writing, J.D.
Intel Broadwell, 27 nodes, ≈ 1024 3 zones Zones / dim / patch 10 6 zones / sec / node Number of Nodes Time per update [sec] WENO-Wombat on Cray XC40 "Hazel Hen" ideal, 2.45 sec0
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Acknowledgments:We would like to thank the two referees for constructive criticism that improved the paper significantly. Access to the 'Hazel Hen' at HLRS has been granted through PRACE preparatory access project "PRACE 4477". All graphs in the work have been produced using PGF and using the Julia 3 programming language.Conflicts of Interest:The authors declare no conflict of interest.
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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution. CC BY) license. New York, NY, USA; Basel, SwitzerlandACMc 2018 by the authors. Licensee MDPIACM: New York, NY, USA, 1967; pp. 483-485. c 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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"Some Intensive and Extensive Quantities in High-Energy Collisions *",
"Some Intensive and Extensive Quantities in High-Energy Collisions *"
]
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"A Tawfik \nEgyptian Center for Theoretical Physics (ECTP)\nEgypt and World Laboratory for Cosmology And Particle Physics (WLCAPP)\nMTI University\n11571Cairo, CairoEgypt\n"
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"Egyptian Center for Theoretical Physics (ECTP)\nEgypt and World Laboratory for Cosmology And Particle Physics (WLCAPP)\nMTI University\n11571Cairo, CairoEgypt"
]
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| We review the evolution of some statistical and thermodynamical quantities measured in difference sizes of high-energy collisions at different energies. We differentiate between intensive and extensive quantities and discuss the importance of their distinguishability in characterizing possible critical phenomena of nuclear collisions at various energies with different initial conditions. | 10.5506/aphyspolbsupp.7.17 | [
"https://arxiv.org/pdf/1310.0718v1.pdf"
]
| 118,475,345 | 1310.0718 | 71929e4940ef275f83710ca0607cf768f6013edf |
Some Intensive and Extensive Quantities in High-Energy Collisions *
2 Oct 2013
A Tawfik
Egyptian Center for Theoretical Physics (ECTP)
Egypt and World Laboratory for Cosmology And Particle Physics (WLCAPP)
MTI University
11571Cairo, CairoEgypt
Some Intensive and Extensive Quantities in High-Energy Collisions *
2 Oct 2013numbers: 2575Gz2575Dw0540-a0570Fh
We review the evolution of some statistical and thermodynamical quantities measured in difference sizes of high-energy collisions at different energies. We differentiate between intensive and extensive quantities and discuss the importance of their distinguishability in characterizing possible critical phenomena of nuclear collisions at various energies with different initial conditions.
Introduction
The terminology "intensive and extensive quantity" was introduced by Richard C. Tolman [1] in order to distinguish between different thermodynamical parameters, properties, variables, etc. Therefore, the defining of such quantities as intensive or extensive may depend on the way in which subsystems are arranged [1]. In order to characterize possible critical phenomena of the nuclear collisions, which likely become complex at ultra high energy, various signatures have been proposed [2]. It is obvious that the critical phenomena of intensive or extensive variables [3] should be differentiated. The extensive variables, like total charge multiplicity, obtain about equal contributions from the initial (due to fluctuations in spectators) and final stage (resonances). The intensive variables, like particle ratios, are well described by resonances at the freeze-out [4,5,6,7]. In the present work, we show how the distinguishability between extensive and intensive quantities behaves at various energies and with different initial conditions. The implication of statistical-thermal models on high-energy physics dates back to about six decades [8]. Koppe introduced an almost-complete recipe for the statistical description of particle production [9]. The particle abundances in Fermi model [10] are treated by means of statistical weights. Furthermore, Fermi model [10] gives a generalization of the "statistical model", in which one starts with a general cross-section formula and inserts into it a simplifying assumption about the matrix element of the process, which reflects many features of the high-energy reactions dominated by the density in phase space of the final states. In 1951, Pomeranchuk [11] came up with the conjecture that a finite hadron size would imply a critical density above which the hadronic matter cannot be in the compound state, known as hadrons. Using all tools of statistical physics, Hagedorn introduced in 1965 the mass spectrum to describe the abundant formation of resonances with increasing masses and rotational degrees of freedom [12] which relate the number of hadronic resonances to their masses as an exponential. Accordingly, Hagedorn formulated the concept of limiting temperature based on the statistical bootstrap model.
The statistical and thermodynamical variables, properties and parameters can be classified into intensive, extensive, normalized intensive and extensive, process and conjugate. There are physical properties which neither intensive nor extensive, e.g. electric resistance, invariant mass and special relativity. The intensivity is apparently additive and therefore a state variable. The intensive (bulk) properties do not depend on the system size or the amount of existing material. Therefore, it is scale invariant. The extensivity is field and point variable but not additive. The extensive properties are additive for independent and non-interacting subsystems. They are directly proportional to the amount of existing material. Normalized intensive and extensive quantities are densities. They are not additive. The process depends on past history of the system. Therefore, they are differentiable, inexactly. The conjugates are intensive and extensive pairs, like temperature and entropy. For example, in grand canonical ensemble, strongly intensive quantities have been suggested as fluctuation measures not depending on the system volume and its fluctuations [13]. The charge distribution is inclusive, while isotropically resolved particle observation is an exclusive property. We review the evolution of some statistical and thermodynamical quantities measured in difference sizes of the high-energy collisions at different energies.
The present paper is organized as follows. The intensivity and extensivity of statistical properties are shortly reviewed in section 2. The dissipative properties are elaborated in section 3. The energy dependence of temperature shall be estimated in section 4. Section 5 is devoted to the conclusions and outlook.
2. Statistical properties: multiplicities and particle ratios (GeV) NN s 1 10 100 1000 Only two independent intensive variables are needed in order to fully specify the entire state of the system of interest. Other intensive properties can be derived from these known ones. An exclusive property implies that energy and momentum, for instance, of all products are measured. The intensivity means that some quantities of the products are left unmeasured.
〉 part N 〈 ) / η /d ch (dN 0 1 2 3 4 5 Central AA ALICE ATLAS CMS RHIC SPS AGS FOPI ) NSD p pp (p ALICE CDF CMS RHIC UA1 UA5 ) INEL p pp (p ISR UA5 PHOBOS ALICE pPb ALICE dAu PHOBOS pAu NA35 0.15 NN s ∝ 0.11 NN s ∝ 0.10 NN s ∝
An extensive comparison between the particle multiplicity dN ch /dη per participating nucleon at mid-rapidity in central heavy-ion collisions [14,15,16,17,18,19,20,21,22,23,24] and the corresponding results from p+p(p) [25,26,27,28,29,30,31,32,33] and p(d)+A collisions [34,35,14] is presented in Fig. 1. It is obvious that the energy dependence of the total multiplicity is distinguishable. In order words, the initial state plays an essential role. The extenstivity can be related to canonical ensemble,
Z(N, T, V ) = Tr N exp − H T ,(1)
where H is the Hamiltonian, while grand canonical ensemble is related to intensivity,
Z(µ, T, V ) = Tr N exp − H − µ N T ,(2)
where N stands for the degrees of freedom. With Dirac delta function and when the chemical potential µ is Wick rotated, then extenstivity can be related to intensivity In Fig. 2, the results ofp/p calculated in HRG are represented by solid line, which seems to be a kind of a universal curve. In heavy-ion collisions, the proton ratio varies strongly with the center-of-mass energy √ s. The HRG models describes very well the heavy-ion results. Also, AL-ICE pp results are reproduced by means of HRG model. The ratios from pp-and AA-collisions runs very close to unity implying almost vanishing matter-antimatter asymmetry. On the other hand, it can also be concluded that the statistical-thermal models including HRG seem to excellently describe the hadronization at very large energies and the condition deriving the chemical freeze-out at the final state of hadronization, the constant degrees of freedom or S( √ s, T ) = 7(4/π 2 )V T 3 , seems to be valid at all center-of-mass-energies spanning between AGS and LHC. So far, we conclude that the distinguishability between proton ratios in pp-collisions and that in AA-collisions disappears with increasing √ s.
Z(N, T, V ) = 1 2 π 2 π 0 Z(iT θ, T, V ) exp(−iN θ) dθ.(3)
Dissipative properties: elliptic flow
The azimuthal distribution with respect to the reaction plane reads
d N d(φ i − Ψ n ) ∼ 1 + 2 n=1 v n cos [n (φ i − Ψ n )] .(4)
The reaction plane angle Ψ n is not directly measurable, but can be determined from particle azimuthal distributions. There are various possible sources of azimuthal correlations like, jet formation, resonances exist, which do not depend on the reaction plane (non-flow correlations). The Fourier coefficient v n , which refers to the correlation in n particle emission with respect to the reaction plane, is given by will set on. At larger energies, the behavior can be described by in-plane elliptic flow due to pressure gradient. the elliptic flow shows a rich structure; a transition from in-plane to out-of-plane and back to in-plane emission. Apparently, it is sensitive to the properties of the medium created in heavy-ion collisions. There are evidences that the elliptic flow of charged and identified particles indicates a strong rise of the expansion velocity of the medium (radial fow) at RHIC vs LHC.
v n = cos [n (φ i − Ψ n )] .(5)
On the other hand, it was assumed that there are no correlations due to elliptic flow in pp collisions at RHIC energy [40]. The methods of measuring elliptic flow can hardly be employed with the currently available number of recorded pp interactions of ALICE at the LHC. Furthermore, none of available microscopic Monte Carlo (MC) models describes the development of anisotropic flow in elementary hadron-hadron interactions yet [40]. Particular non-perturbative approach was suggested as a mechanism of anisotropic flow might be a leading one in hadron collisions, since those have smaller geometrical extension and the probability of hydrodynamical generation of elliptic flow is lower compared to the collisions of nuclei [41].
pp collisions simulated by PYTHIA, PHOJET and EPOS at 900 and 7000 GeV are analyzed by two-particle correlation methods. The integrated v 2 coefficients reconstructed by the methods are found to vary from 10% − 15%. These values are attributed solely to the non-flow correlations [40].
Hagedorn temperature: energy and system size dependence
The transverse mass spectra of well-identified particles have been studied at various energies, for instance [45]. Accordingly, Stefan-Boltzmann approximation results is
1 m T dN dm T dy = a exp − m T T ,(6)
where m T = p 2 T + m 2 is the dispersion relation and a is a fitting parameter. Fig. 4 presents the energy dependence of the inverse slope parameter T of the transverse mass spectra of K + (left panel) and K − mesons (right panel) produced in central Pb+Pb and Au+Au collisions. There is a plateau at SPS energies [45] which is preceded by a steep rise of T measured at the AGS [42] and followed by an indication of a further increase of the RHIC data [43]. Although the scatter of data points is large, T appears to increase smoothly in p + p(p) collisions [44], left panel of Fig. 4
The dependence on system size is illustrated in left panel of Fig. 4. The Hagedorn temperature in pp collisions seems to be smaller than that in AA collisions. Its variation with the center-of-mass energy is apparently weaker than the variation in AA collisions. A much more systematic measurement would help in proving or disproving such a conclusion.
Conclusions and outlook
The ultimate goal of the physics program of high-energy collisions is the study of properties of strongly interacting matter under extreme conditions of temperature and/or compression. The particle multiplicities and their fluctuations and correlations are experimental tools to analyse the nature, composition, and size of the medium, from which they are originating. Of particular interest is the extent to which the measured particle yields are showing equilibration. Based on analysing the particle abundances or momentum spectra, the degree of equilibrium of the produced particles can be estimated. The particle abundances can help to establish the chemical composition of the system. The momentum spectra can give additional information on the dynamical evolution and the collective flow.
In order to characterize possible critical phenomena, signatures based on particle multiplicities and their fluctuations and correlations have been proposed. Intensive or extensive quantities should be separated, systematically. Extensivity obtains about equal contributions from the initial and final stage. Intensivity is well described by produced particles in final state. The present work introduces the importance of distinguishability between extensive and intensive quantities at various energies and in different system sizes.
Fig. 1 .
1A comparison of dN ch /dη per participating nucleon at mid-rapidity in central heavy-ion collisions to corresponding results from p+p(p) and p(d)+A collisions. The quantities are given in physical units. Graph taken from Ref.[36].
Fig. 2 .
2np/n p ratios depicted in whole available range of √ s. Open symbols stand for the results from various pp experiments (labeled). The solid symbols give the heavy-ion results from AGS, SPS and RHIC, respectively. The fitting of pp results according to Regge model is given by the dashed curve[37]. The solid curve is the HRG results. Contrary to the dashed curve, the solid line is not a fitting to experimental data. The graph taken from Ref.[38].
Fig. 3 .
3Integrated elliptic flow measured in central heavy-ion collisions (20 − 30%) is given in dependence on Nucleus-Nucleus center-of-mass energy. The graph taken from Ref.[39].
Fig. 3
3shows data collected over about four decades spanning from GSI, AGS, SPS, RHIC to LHC facilities. The integrated elliptic flow measured in relative central heavy-ion collisions (20 − 30%) is given in dependence on Nucleus-Nucleus center-of-mass energy inFig. 3. For the comparison, the integrated elliptic flow is corrected for p t cutoff of 0.2 GeV/c. The estimated magnitude of this correction is 12 ± 5% based on calculations with Therminator. The figure shows that there is a continuous increase in the magnitude of elliptic flow for this centrality region from RHIC to LHC energies. In comparison to the elliptic flow measurements in Au-Au collisions at √ s N N = 200 GeV we observe about a 30% increase in the magnitude of v 2 at √ s N N = 2.76 TeV. The rapid decrease of v 2 at very low energy, FOPI data, refers to bounce-off. Increasing √ s N N , a squeeze-out
Fig. 4 .
4Energy dependence of T related to the transverse mass spectra of K + (left panel) and K − mesons (right panel) produced in central Pb+Pb and Au+Au collisions. The graphs taken from Ref.[45].
. The dependence of T on the system size is obvious. For completeness, we recall that the direct thermal photons have been used to estimate the Hagedorn temperature,T =
304 ± 51 MeV ALICE
[46]
221 ± 19 MeV PHENIX [47]
AcknowledgementThis work is financially supported by the World Laboratory for Cosmology And Particle Physics (WLCAPP), http://wlcapp.net/.
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| []
|
[
"Digital Resistance during COVID-19: A Workflow Management System of Contactless Purchasing and Its Empirical Study of Customer Acceptance Digital Resistance during COVID-19: A Workflow Management System of Contactless Purchasing and Its Empirical Study of Customer Acceptance",
"Digital Resistance during COVID-19: A Workflow Management System of Contactless Purchasing and Its Empirical Study of Customer Acceptance Digital Resistance during COVID-19: A Workflow Management System of Contactless Purchasing and Its Empirical Study of Customer Acceptance"
]
| [
"Yang Lu \nInformation Systems and Operations Management\nCollege of Business\nUniversity of Central\n\n",
"Edmond Oklahoma \nInformation Systems and Operations Management\nCollege of Business\nUniversity of Central\n\n",
"O K 73034 \nInformation Systems and Operations Management\nCollege of Business\nUniversity of Central\n\n"
]
| [
"Information Systems and Operations Management\nCollege of Business\nUniversity of Central\n",
"Information Systems and Operations Management\nCollege of Business\nUniversity of Central\n",
"Information Systems and Operations Management\nCollege of Business\nUniversity of Central\n"
]
| []
| The COVID-19 pandemic has stimulated the shift of work and life from the physical to a more digital format. To survive and thrive, companies have integrated more digital-enabled elements into their businesses to facilitate resilience, by avoiding potential close physical contact.Following Design Science Research Methodology (DSRM), this paper builds a workflow management system for contactless digital resilience when customers are purchasing in a store.The findings show that response costs have a positively significant effect on customers' behavioral intention to adopt digital resilience, while self-efficacy plays a negative role on customers' behavioral intention. The findings reveal that, during the COVID-19 pandemic, customers are more concerned about health issues and put more effort into the deployment of digital resilience to mitigate the consequences of the virus. These results indicate that, even beyond the performance of technology itself, another factor (the health issue) can play the key role in customers' acceptance of digital resilience. | null | [
"https://arxiv.org/pdf/2105.07838v2.pdf"
]
| 234,741,713 | 2105.07838 | 8f17a4f83f85e01352428d46c069cbd9d63f61cb |
Digital Resistance during COVID-19: A Workflow Management System of Contactless Purchasing and Its Empirical Study of Customer Acceptance Digital Resistance during COVID-19: A Workflow Management System of Contactless Purchasing and Its Empirical Study of Customer Acceptance
Yang Lu
Information Systems and Operations Management
College of Business
University of Central
Edmond Oklahoma
Information Systems and Operations Management
College of Business
University of Central
O K 73034
Information Systems and Operations Management
College of Business
University of Central
Digital Resistance during COVID-19: A Workflow Management System of Contactless Purchasing and Its Empirical Study of Customer Acceptance Digital Resistance during COVID-19: A Workflow Management System of Contactless Purchasing and Its Empirical Study of Customer Acceptance
Digital resilienceCOVID-19Design Science Research Methodology (DSRM)workflow management systemSIR (Susceptible-Infectious-Removed) model
The COVID-19 pandemic has stimulated the shift of work and life from the physical to a more digital format. To survive and thrive, companies have integrated more digital-enabled elements into their businesses to facilitate resilience, by avoiding potential close physical contact.Following Design Science Research Methodology (DSRM), this paper builds a workflow management system for contactless digital resilience when customers are purchasing in a store.The findings show that response costs have a positively significant effect on customers' behavioral intention to adopt digital resilience, while self-efficacy plays a negative role on customers' behavioral intention. The findings reveal that, during the COVID-19 pandemic, customers are more concerned about health issues and put more effort into the deployment of digital resilience to mitigate the consequences of the virus. These results indicate that, even beyond the performance of technology itself, another factor (the health issue) can play the key role in customers' acceptance of digital resilience.
INTRODUCTION
Until the end of September 2020, the cumulative number of confirmed cases of worldwide, stood at 34.1 million. The number of deaths, at that point in time, was 1.02 million 1 .
The COVID-19 pandemic has caused dramatic damage and has thoroughly changed organizations' operating modes, as well as people's lives and habits. Close physical contact is the major reason given for the fast spread and infection of COVID-19 Wu & McGoogan, 2020;Sohrabi et al., 2020). As one effective measure, digital resilience is being deployed, and it is quickly being improved to its highest level, in order to mitigate the influence of the pandemic. Both companies and individuals hold virtual meetings instead of face-to-face ones. Since the inception of COVID-19, the popular virtual communication and conference software called ZOOM's stock price has skyrocketed to the price of $559 on October 16, 2020, up from around $70 in January 2020.
Close physical contact 2 happens among people every day; contact is unavoidable. Walmart provides three types of grocery shopping: purchasing at a local store (conventional), ordering online with pickup (blended), and ordering online with delivery (e-commerce). It is important to continue to offer the conventional purchasing style, since many customers still prefer to select their food themselves or because certain categories of food are not available when using the other two purchasing styles. Because of these, there is an urgent need for businesses to assist their customers in avoiding potential close physical contact. This study focuses on the first purchasing style, conventional purchasing, by designing a digital resilience workflow management system that helps customers avoid potential close physical contacts when they are purchasing in a store. This robust and applicable digital infrastructure will enhance companies' resilience in fighting against the virus and will assist in making more profitable businesses, as well.
As the groundwork of information systems, digital resilience describes an organization's capability to deal with unexpected disruptions, in order to continue doing business and to be successful after the emergency. Digital resilience has the potential to change not only an organization's operating modes, but also people's behavior and habits. When they are trying to mitigate the influences of the COVID-19 pandemic, companies, for their businesses to benefit, need to deploy more resiliency-related digital techniques; individual customers, for their health, need to cope with these resiliency-related digital strategies to avoid potential infection.
Information systems is an important discipline that can be used to explore insights that can help to resolve the many issues caused by the unexpected COVID-19 disruptions. Digital resilience can be achieved through information systems' integration of high technology with advanced devices. A well-designed digital resilience workflow management system can sustain businesses' continuity and can mitigate the impact of COVID-19. Our research lies mainly in helping information systems to find a way to accelerate digital resilience during the COVID-19 period.
This study articulates three objectives:
First, it should be noted that design science is a distinguished and classical methodology among the IS disciplines. This paper uses the framework of Design Science Research Methodology (DSRM) not only to identify the critical problem of potential close physical contact in the COVID world, but to define the objectives of the proposed workflow management system by flowchart using the epidemiological SIR model, to design and to illustrate the digital resilience flow of contactless purchasing by using the Petri net workflow management system, and to assess the feasibility of the workflow management system by using behavioral theories associated with empirical study. DSRM is considered an interdisciplinary methodology composed of design science and empirical research, in this paper.
Second, the workflow management system described herein is built to embed digital resilience, in order to allow businesses, the chance to help their customers avoid potential close physical contact when they are making a purchase in a store. Digital resilience is a must-have tool for a business' continuity and performance, especially in the event of an emergency. The proposed workflow system offers good guidance that a company can follow, so that it can continue to be successful despite unexpected disruptions -having previously invested in digital resilience and having facilitated digital resilience into its enterprise management system.
Last but not least, depending on the workflow management system of contactless purchasing, the feasibility of the proposed workflow management system should be considered. This workflow depicts the human behaviors of considering, recognizing, coping, behaving, and using digital resilience to prevent potential close physical contacts under the COVID-19 pandemic. If customers are reluctant (or are not able) to adopt digital resilience measures when making a purchase in a store, the company can still effectively change and successfully implement digital resilience to keep its customers away from potential infection. This paper describes an empirical examination that considers what factors most relate to customers' intention to cope with digital resilience in a store, and we found two factors that have different impacts on customers' intention. The response cost (facilitating conditions) is positively associated with customers' digital resilience adoption, and self-efficacy (facilitating conditions) is negatively associated with customers' digital resilience adoption. COVID-19 has changed people's behavior and habits, not only when making a purchase in a store but also when accomplishing many other daily activities, e.g., wearing masks and hand sanitizing.
The overall infrastructure of this study (Appendix Figure A1) follows the main steps of Design Science Research Methodology (DSRM) (Hevner et al., 2004;Peffers et al., 2007;Vandenbosch & Higgins, 1995;Carvalho, 2020): (1) problem identification and motivation, (2) definition of the objectives for a solution, (3) design and development of the model, (4) demonstration of the model, and (5) evaluation of the model. We use DSRM to propose a Petri net workflow management system that guides customers away from potential close physical contact when they are making a purchase in a store.
PROBLEM IDENTIFICATION AND MOTIVATION OF DIGITAL RESILIENCE
The first step of DSRM is to identify problem and motivation. The Novel COVID-19 is a coronavirus that is similar to SARS-CoV and MERS-CoV (Lai et al., 2020;Shereen et al., 2020;He, Deng, & Li, 2020). The difference is that COVID-19 spread all across the globe, as a pandemic, within a short six-month period, causing huge impacts on people's lives and on society Chan et al., 2020). In this paper, the epidemiological SIR model is employed to explore the reason why so many people have become infected. The main reason appears to be the close physical contact between infected (symptomatic and asymptomatic) and susceptible people.
The SIR Model
In general, the SIR model consists of three compartments (Figure 1.), susceptible (S), infectious (I), and removed (R). S describes the people who are susceptible to the disease. At the beginning of the pandemic, S equals to the total population in a certain area. I describes the people who are infectious. The infectious people have the disease, and they can infect others. R (or removed) describes the people who have caught the disease and who have now either recovered from it or have died. These recovered people are immune to the disease. Thus, the removed people are those who are not infectious anymore (Chen et al., 2020;Lin et al., 2020).
In the SIR model, several assumptions are used to simplify the real-world phenomenon of COVID-19. Explanations of the variables of SIR model are addressed in Appendix 2. (Table A1).
(1) The total population (TP) remains constant during the pandemic. It means that the rate of change of the susceptible population plus the rate of change of the infectious population plus the rate of the removed population must be zero. The total population (TP) is given by (S+I+R). This will be the same constant value for all the possible values of time. The initial value will be the starting point: the value of the total population at the beginning of the pandemic. As time progresses, it will not change. It will always equal the initial value.
(2) The transmission rate (γ) is proportional to the contact between the susceptible and the infectious people. And γ occurs at a constant rate. The transmission rate (γ) will decrease as more people become infectious.
(3) The removed rate (α) is a constant rate. It could be a death rate or a recovery rate, or it could be the composite of the death and recovery rates.
(4) The contact ratio (q) is the fraction of the population that comes into contact with an infected individual during the period when they are infectious, q = γ / α.
(5) The basic reproductive ratio ( 0 ) is the reciprocal of the contact ratio (q), 0 = α / γ. This ratio indicates that there will be an epidemic if 0 > 1 (6) The initial number of susceptible people is 0 , the initial number of infectious people is 0 , and the initial value of removed people is 0.
The rate of change of the number of susceptible people over time:
dS/dt = -γ * I * S(1)
The rate of change of the number of infectious people over time:
dI/dt = γ * I * S -α * I (2)
The rate of change of the number of removed people over time:
dR/dt = α * I(3)
These three differential equations are for the three compartments of people of the population.
Equation (1) indicates that the number of susceptible people is going to change according to the number of contacts between susceptible and infectious people. Equation (2) indicates that the number of infectious will increase because of the contact between people who have either recovered or died as a result of the disease spread. Equation (3) indicates that the rate of removed people is going to increase at the constant rate, depending on how many infectious people there are.
The SIR model assumes that susceptible people will transfer to other states with a certain probability of infection, according to the development pattern of COVID-19. The dynamic model of "susceptible-infectious-removed" can predict the trend of COVID-19 within a certain range, geographical area, or time segment.
Evaluating the Importance of Contact Ratio (q)
The initial number of susceptible people is 0 , the initial number of infectious people is 0 , and the initial value of removed people is 0. The following equation is the initial point of COVID-19.
S+I+R = 0 + 0(4)
Next, we investigate and discuss three important issues of COVID-19 based on the SIR model:
the severe spread, the potential maximum number of infectious people, and the potential number of infected people by the end of the pandemic. All three problems are related to the contact ratio (q).
The Severe Spread of COVID-19
The initial number of infectious people at the beginning of the outbreak is given by 0 . The question is whether or not the number of infectious people will grow. If the number of infectious people starts to grow, the disease will spread throughout the population. Here, we focus on Equation (2), the rate of change of infectious people over time. S is smaller than its initial value (S ≤ 0 ). In the context of the disease, at the beginning of the outbreak, everyone in the total population theoretically was susceptible to the disease, especially since it was a Novel Coronavirus, i.e., one that had never been seen before.
Since S ≤ 0 , we have
dI/dt < I (γ * 0 -α)(5)
An epidemic will occur if the size of I increases from the initial value of infectious people ( 0 ).
In the very real situation of COVID-19, it became clear that the number of infectious people was increasing very quickly. For the other part of Equation (γ 0 -a), if this term is positive, there will be a spread of the disease. It means,
0 > α / γ(6)
The basic reproductive ratio 0 = α / γ. This ratio indicates that there will be an epidemic if 0 > 1. This ratio represents the secondary infections in the population caused by one initial primary infection. In other words, if one person has the disease, 0 will show how many infections, on average, that person is likely to cause. This current coronavirus is an ongoing outbreak that we have never seen before. The reproductive ratio, as described in the research, is estimated to be more likely 2 to 4 (Chen et al., 2020;Li et al., 2020). COVID-19 is an epidemic that spreads quickly. Therefore, avoiding potential close physical contact is an effective way to reduce the contact ratio and to decrease the number of infected people.
The Potential Maximum Number of Infectious of COVID-19
There is a known lack of appropriate and effective approaches to detection and diagnosis in the early stages of any disease outbreak, especially in unknown epidemics like COVID-19.
Knowledge of the precise estimate of the number of people infected is essential, in order to be able to judge the severity of the epidemic and to make corresponding decisions. A common method used is to estimate the number of infections based on the proportion of outflowing people in a certain area. The early report from Northeastern University (Chinazzi et al., 2020) made a similar relevant analysis.
Knowing the number of infected people is very helpful when it comes to planning how to distribute health resources and how to implement anti-COVID measures. In Equations (1) and (2),
dI/dS = (γIS -aI)/(-γ IS) = -1 + a/ γ s (7)
The contact ratio q = γ / a, we have
I + S -1/q * lnS = 0 + 0 -1/q * ln 0 (8)
The maximum will occur, when S = 1/q. Substituting this value into the equation (8),
= 0 + 0 -1/q (1 + ln(q 0 )) (9)
The maximum number of infectious people ( ) is the maximum number of people that will have the disease at a given time. The term (1/q (1 + ln(q 0 ))) depends on the parameter q, the contact ratio. In the outbreak of COVID-19, the value of q is high; the disease is very easy to transmit. Many susceptible people are becoming infected when encountering potential close physical contact with infectious people, especially since COVID-19 has a relatively long incubation period, during which its symptoms might not yet have appeared. Avoiding potential close physical contact separates the susceptible from the infectious people, in order to reduce the quantity of overall infectious (both symptomatic and asymptomatic) people.
The Potential Number of Infected People by the End
How can we know that the pandemic is at its end? The number of infectious people will go down to zero. This, in the future, will signal the end of the outbreak. Let us rearrange to find the size of the removed people (R), those who have either recovered or died, at the end of the pandemic.
The total number of people who have caught the disease by the end is,
R(end) = 0 + 0 -S(end)(10)
Based on Equation (8), the removed people or the size of the removed population at the end of the epidemic is,
S(end) -1/q * ln(S(end)) = 0 + 0 -1/q * ln( 0 )(11)
If the value of q is sufficiently large, most of the population will not catch the disease. In the case of COVID-19, if there is a large value of q, the potential maximum number of infectious people at any given time is almost equal to the whole population, in theory.
In summary, the contact ratio (q) appears in the answers to all three key questions. It is impossible to stop the spread of COVID-19 that has already occurred; what we can do is reduce the number of people who will get infected ( ). It is practical to isolate the susceptible people from the infectious people. This is exactly why we need to avoid potential close physical contact.
In reality, grocery shopping has become one of the major channels to explore, during the COVID-19 era. Our study depicts a workflow management system to solve the issue of potential close physical contact when a customer is making a purchase in a store. The proposed workflow management system will contribute to the IS community's fight against the COVID-19
pandemic.
DEFINITION OF THE OBJECTIVES OF DIGITAL RESILIENCE MEASURES
The second step of DSRM is to interpret the objectives of a solution. Administrative authorities have suggested several policies to be taken against COVID-19, such as staying at home, avoiding gatherings or parties, closing stores and places to shop, etc. However, people cannot escape their need for groceries. There are different groups of personnel at grocery stores; this leads to a complicated COVID-19 infection network fraught with potential close physical contacts. To mitigate infection, a store can deploy anti-COVID measures; digital resilience is one of the most effective ways to avoid potential physical contact.
Our goal is to mitigate the influence of the COVID-19 pandemic, specifically by focusing on avoiding close physical contacts between a business and its customers, by the use of digital N W C CW C O C resilience. A flowchart (Figure 2.) illustrates how a customer can avoid potential physical contact when making a purchase in a store. The detailed procedure and the relevant activities in this digital resilience system are described and explained in the next section.
DESIGN OF DIGITAL RESILIENCE WORKFLOW MANAGEMENT SYSTEM
Workflow Management System of Avoiding Contact
A Petri net workflow (Salimifard & Wright, 2001;Xu et al., 2009) Each role player is represented by a labeled Petri net (LPN) model, and all LPN models are combined as the complete workflow management system. The system includes five interactive transactions: the interactions between C and SC in EP, between C and PM in PuP, between C and PA in PaP, between C and DA in DP, and between C and CS in CSP. Within the entire process, many digital resilience-enabled devices and sensors are available to assist customers.
From a customer behavioral perspective, companies can recognize which factors are likely to impact their customers' intention and can adjust accordingly.
Labeled Petri Net Workflow Management System
The proposed labeled Petri net model is constructed based on previous studies (Van der Aalst, workflow net (LWN). LPN represents each role player, and LWN represents the complete system. Transitions are divided into three categories: In, Out, and Inner transitions. The In Transition refers to "receiving a message from a partner via network"; the Out Transition refers to "sending a message to a partner via network"; and the Inner Transition "contains all inner activities" ). In the proposed workflow management system, customer and the five assistant systems are partners, and all messages and relevant activities are interacted between these six role players throughout the system. All messages and activities are recorded in the system for further analysis.
Definition 1. A labeled Petri net (LPN) is composed of 7 tuples,
LPN = (P, T, F, 0 , , , ).
Criteria:
(1) P is a finite set of places.
(2) T is a finite set of transitions. T = U U . The three categories (In, Out, and Inner) are mutually exclusive in a workflow system.
(3) F ⊆ (P×T) ∪(T×P), which refers to a set of directed arcs (relations) connecting Places to
Transitions and Transitions to Places.
(4) (P, T, F) represents a Petri net.
(5) M: P → {0,1} is a marking function. 0 is the initiation marking.
(6) is the set of messages between customers and business. Each message is defined as the form of [(msg, Sender, Receiver)]; msg is the name of a specific message or task.
(7) (M, ) is a state of LPN. ( 0 , 0 ) is an initial state, where 0 is a non-empty set.
(8) is a finite set of activity labels, e.g., Greek or Arabic.
(9) : T→ is defined as a labeling or weight function.
Definition 2. LWN = (P, T, F, 0 , , , ) = LPN.
Labeled workflow net (LWN) is an LPN, if and only if
(1) P consists of a source place i, which is a non-empty set.
(2) P consists of outcome places , which is a non-empty set.
DEMONSTRATION OF DIGITAL RESILIENCE WORKFLOW MANAGEMENT
SYSTEM
In the proposed LWN, P ( 1 , 2 , … , 20 ) is place that is expressed by a circle. The Specifically, M ( 1 ) =1 indicates that a customer's access to a store has been denied because the customer has failed a physical temperature check. M ( 2 ) =1 indicates that a customer's access to a store has been denied because the customer has refused to wear a mask. M ( 3 ) =1 indicates that a customer has successfully finished a purchasing process in a store with the assistance of the digital resilience workflow management system, which has provided store access check (the store's customer capacity, the customer's temperature, and the wearing of a mask); purchasing process assistance (crowd density, one-way direction); self-payment system (cash, card, or App Pay); delivery assistance (a self-delivery system); and customer service (a self-customer service system). Messages are exchanged between the customers and the business. (Access, C, B) means that a store receives an access request from a customer; Out (N_Tem, SC, C) means that the Sensor Checking System sends a message of a customer's temperature fail from SC to C. More detailed explanations of the messages are shown in Appendix (Table A2).
In the proposed workflow management system, there are three terminal goals: 1 , 2 , and 3 .
Only 3 consists of all the possible digital resilience-enabled purchasing processes. Both 1 and 2 deny access to a store because of a temperature check failure or no mask wearing, respectively. Let us have a detailed look at 3 from the starting point i. The complete workflow system (Figure 3.) includes the five procedures mentioned above.
In the first procedure, Access, the interaction between C and SC in EP involves several sensors and protocols. Since the COVID-19 pandemic is severe, every customer is required to follow In the second procedure, Purchasing, the interaction between C and PM in PuP, digital resilience measures will assist and warn customers ([(Pur, PM, C)]), all throughout the store, if a certain area has a dense crowd or if the customer has not followed the correct direction during shopping.
Customers can also install the related App to track and to instantly obtain useful information.
In the third procedure, Payment, the interaction between C and PA in PaP, there is no personal assistant. What the customer ([(Pay, C, PA)]/ [(Pay, PA, C)]) needs to do is adopt a self-assistant system to scan and pay for his/her items by cash or by card. Another potential digital resilience measure is that a customer can use his/her own cell phone to scan and pay through a payment
App.
In the fourth procedure, Delivery, the interaction between C and DA in DP, a customer ([(Y_Deli, C, DA)]) can use a digital device to process a delivery if the customer needs some of the items to be delivered. If there is no request from the customer ([(N_Deli, C, DA)]) to deliver anything, the customer will be directed to the final step: Customer Service.
In the fifth procedure, Customer Service, the interaction between C and CS in CSP, many types of contactless service can be implemented, such as a voice assistant, a virtual assistant, an App assistant, a message assistant, etc. If the customer ([(Y_Ser, C, CS)]) needs customer service, the system will assist him/her. If not, the system will finish all of its possible assisting and the customer's purchasing will end at 3 .
THEORETICAL CONTRIBUTIONS
This study is a good attempt to present a workflow management system of digital resilience to mitigate consequences of the COVID-19 pandemic by integrating the two major research paradigms of information systems: design science and behavioral research. First, the structure and the context of this paper are based on DSRM (Design Science Research Methodology). That methodology is suitable to use in identifying a practical problem (the potential close physical contacts of customers during purchasing in a store), in building a workflow management system to help businesses' customers avoid potential close physical contacts, and in empirically evaluating the feasibility of the system (whether or not customers will adopt digital resilience when making a purchase, and what factors impact customers' behavioral intention). DSRM is an appropriate measure, both theoretical and indirect, to use in mitigating the influence of the unexpected disruptions.
LIMITATIONS AND FUTURE RESEARCH
While our paper focuses on the ways in which customers can accept digital resilience measures taken to counter the risks of COVID-19, another important angle to consider is the employees' prospects for digital resilience. A real-life example happened at Walmart's "Order Online & Pickup." Walmart had already made a digital resilience effort; the App indicated that when a customer arrived to make a pickup, he or she should "Roll Up" the vehicle's windows to protect the driver and the employee from potential infection. However, as happens often, neither the employees nor the customers obeyed this principle, because it was easier to open the window for communication between employees and customers. Nevertheless, during the COVID-19 pandemic, we must learn to tolerate the inconvenience of avoiding potential close physical contact. Employees need to be trained and must follow the policy of digital resilience in order to avoid any potential close physical contact. Businesses must learn how to monitor and manage their employees' behavior regarding digital resilience. Otherwise, digital resilience may not perform well in mitigating COVID-19 influences or other unexpected disruptions.
Previous studies have explored the behavioral model's relationship to moderating effects, such as age, gender, education level, work experience (Compeau & Higgins, 1995;Venkatesh et al., 2003). But different groups of people have addressed the COVID-19 pandemic with different thinking, recognition, and behaviors. It will be valuable to investigate the way in which people behave heterogeneously; then, companies can accompany that information as they work to improve the performance of their workflow management system in satisfying their customers'
requests. The relationship between effort expectancy and behavioral intention is vague. The moderating effects may strengthen the relationship between effort expectancy and behavioral intention.
The evaluation of DSRM used in this study was to assess the feasibility of the proposed digital resilience workflow management system by employing empirical tests on factors that impact customers' intention to adopt digital resilience, not on the productivity of the workflow management system. Future research could assess the effectiveness and productivity of ways to improve the entire workflow management system.
MANAGERIAL IMPLICATIONS
This study seeks to construct reliable measures for organizations as they implement digital resilience and as they work to prevent their customers' contracting COVID-19 by helping the customers to avoid potential close physical contact. On the basis of the epidemiological SIR model, it is clear that close physical contact is a major reason why so many people became infected; in fact, it is the major reason why COVID-19 spread all over the world so rapidly. The avoidance of any potential close physical contact is an effective way to protect the susceptible from the infectious people. Although the authorities closed many local stores, grocery stores remain open for necessary daily needs. Digital resilience is the key measure that can assist a local store in implementing anti-COVID measures by setting up a contactless purchasing environment. In this way, potential close physical contact can be greatly reduced, and the store's customers will be safer.
Second, our study presents a workflow management system that solves a real problem: the avoidance of potential close physical contact in stores. The system could be a good example for companies that are seeking to facilitate their own digital resilience measures in order to mitigate the influences of COVID-19, especially in places with the potential for many people to gather, e.g., schools, hospitals, etc. The proposed workflow management system could easily be adjusted to fulfill the various standards and requirements of both organizations and individuals.
Third, the proposed workflow management system is a foundational framework, since it is clear that emerging technologies will be employed to improve organizations' digital capability. Many will look to implement the proposed workflow management system on a broad IoT (Internet of Things) platform integrated with both AI (artificial intelligence) and blockchain technology. IoT offers the potential to implement digital resilience to all of the devices within the system for information sharing, data storage, and performance estimation (Xu, He, & Li, 2014). AI could improve digital resilience by making the workflow management system more intelligent and automatic (Lu, 2019a), and blockchain technology offers a strong, decentralized platform that can provide security and privacy-preserving auditing for processing digital resilience (Lu, 2019b).
CONCLUSIONS
A digital transformation has never been more urgently needed than it is now, following the unexpected disruptions from COVID-19. For a company to succeed in this world of unprecedented constraints upon its customers, it needs to empower enterprise information systems, to optimize operational activities, to foster the new culture of a hybrid work environment, and to engage its customers in new ways, intelligently and virtually transforming products and services with new business models. Digital resilience has the potential to help companies maintain their business performance and continuity in the COVID-19 world.
Customers can adopt digital resilience to protect themselves from the threat of potential infection while completing necessary daily tasks. This study shows that customers are more willing to adopt digital resilience that is implemented by companies (e.g., grocery stores).
This study designs a digital resilience workflow management system that specifically focuses on protecting a business' customers from the infection of COVID-19 by assuring their avoidance of potential close physical contact with other shoppers. Another critical point is the customers' acceptance of digital resilience. Our findings demonstrate that the COVID-19 pandemic has forced customers to form new grocery shopping habits by using the digital resilience-enabled contactless method of grocery shopping. The institution of appropriate digital resilience-enabled measures is necessary a) to reduce the contact ratio (q) of COVID-19 and b) to keep customers both healthy and safe. It is expected that the more digital resilience-enabled companies will offer more competitive advantages that will both prevent the further dissemination of COVID-19 and will attract more customers for them, during the pandemic.
Zheng, Y. Y., Ma, Y. T., Zhang, J. Y., and Xie, X. 2020. "COVID-19 and the Cardiovascular System," Nature Reviews Cardiology (17:5), pp. 259-260 (doi.org/10.1038/s41569-020-0360-5). [(N_Cap, B, C)] N_Cap represents a store isn't full.
APPENDIX SUPPLEMENTAL TABLES
In (N_Cap, B, C) means a store sends a message of it is not full to customer. [(Y_Cap, B, C)] Y_Cap represents a store is full.
In (Y_Cap, B, C)
TP = S + I + R = 0 + 0 d/dt (S+I+R) = (-γ * I * S) + (γ * I * S -α * I) + (α * I) = 0
is built to help customers avoid close potential physical contact in a store. The proposed workflow management system consists of five major procedures: the Entering Procedure (EP), the Purchasing Procedure (PuP), the Payment Procedure (PaP), the Delivery Procedure (DP), and the Customer Service Procedure (CSP), as well as six role players, the Customer (C), the Sensor Checking System (SC), the Purchasing Monitoring System (PM), the Payment Assistant System (PA), the Delivery Assistant System (DA), and the Customer Service System (CS).
and transitions are represented by rectangles with exchanged messages. The transition is represented by a solid rectangle. The terminal goal is G = {M ( 1 ) =1; M ( 2 ) =1; M ( 3 ) =1}.
three access checks: the store capacity check ([(Cap, SC, C)]), the body temperature check ([(Temp, SC, C)]), and the mouth and nose mask check ([(Mask, SC, C)]). Here is the order: first, a customer requests access to a grocery store ([(Access, C, B)]). If that store already has its maximum number of customers inside, as a safety issue, the customer ([(Y_Cap, B, C)]) is told to wait to enter until another customer finishes shopping. If a store does not have its maximum number of customers, the customer ([(N_Cap, B, C)]) will be allowed to enter if the customer satisfies the temperature ([(Y_Tem, SC, C)]) and mask wearing ([(Y_Mas, SC, C)])requirements. Any customer will be denied entry to a store if the customer is reluctant either a) to check his/her body temperature or b) to wear a mask ([(N_Mas, SC, C)]). If a customer has a temperature check and shows a temperature that is above the normal range, the customer ([(N_Tem, SC, C)]) will be denied access to the store. All three activities would be monitored and controlled by digital devices, with notice and instructions sent to the customer. It is voluntary that customers complete extra anti-infection measures, such as hand sanitizing, cart cleaning, and wearing gloves, etc.
Table A1 .
A1Main Variables and ExplanationsThe initial value of infectious C CustomerThe maximal number of infectious F A set of directed arcsThe initial value of removal A labeling or weight function S Susceptible P A finite set of places.The initial value of susceptive T A finite set of transitionsSIR Model
Workflow Management System
Variable
Description
Variable
Description
q
The contact ratio
i
The starting place
I
Infectious
B
Business/Company
0
R
Removed
A terminal goal ( 1 , 2 , 3 )
0
0
TR
Total Population
0
The start marking
α
The removal rate
A finite set of activity labels
γ
The rate of increase in the infectious
φ
A set of messages
Table A2 .
A2Explanations of Messages between Customer and Business MessageNotification Explanation [(Access, C, B)] Access represents access.In (Access, C, B) means a store receives request of access from a customer.[(Cap, B, C)]Cap represents store capacity check.Out (Cap, B, C) means a store sends capacity check to a customer.[(Temp, SC, C)] Temp represents customer body temperature check.Out (Temp, SC, C) means Sensor Checking System sends temperature check to a customer.Out (Mask, SC, C) means Sensor Checking System sends mask check message to a customer.[(Mask, SC, C)] Mask represents
customer wearing
mask check.
means a sore receives a message of it is full. [(N_Tem, SC, C)] In (N_Tem, SC, C) means Sensor Checking System receives a message of temperature fails. Out (N_Tem, SC, C) means Sensor Checking System sends a message of temperature fails to a customer. [(Y_Tem, SC, C)] In (Y_Tem, SC, C) means Sensor Checking System receives a message of temperature passes from a store. [(N_Mas, SC, C)] N_Mas represents no wearing mask. In (N_Mas, SC, C) means Sensor Checking System receives a message of mask check fails. Out (N_Mas, SC, C) means Sensor Checking System sends a message of maks check fails to a customer. [(Y_Mas, SC, C)] In (Y_Mas, SC, C) means Sensor Checking System receives a message of mask check passes. [(Pur, C, PM)] Pur represents purchasing procedure. In (Pur, C, PM) means Purchasing Monitoring System receives a message of purchasing from a customer. [(Pur, PM, C)] Pur represents purchasing procedure. Out (Pur, PM, C) means Purchasing Monitoring System sends a message of purchasing to a customer. [(Pay, C, PA)] Pay represents payment procedure. In (Pay, C, PA) means Payment Assistant System receives a message of payment from a customer. [(Pay, PA, C)] Pay represents payment procedure. Out (Pay, PA, C) means Payment Assistant System sends a message of payment to a customer. [(N_Deli, C, DA)] N_Deli represents no delivery request. Out (N_Deli, C, DA) means a customer sends a message of no delivery to Delivery Assistant System.N_Tem represents
temperature check
fails.
Y_Tem represents
temperature check
passes.
Y_Mas represents
wearing mask.
Source: (WHO) https://covid19.who.int, (CDC) https://www.cdc.gov/coronavirus/2019-ncov/index.html, and (Johns Hopkins Coronavirus Resource Center) https://coronavirus.jhu.edu/map.html. Accessed on September 30, 2020. 2 Close physical contacts refer to contacts that are within 2 meters (6 feet) for over 15 minutes.
[(Y_Deli, C, DA)] Y_Deli represents requesting delivery.In[(Y_Deli, C, DA)In [(Y_Ser, C, CS)] means Customer Service System receives a message of customer service from a customer. Notes:1. The format of an exchanged message is: (Msg, Sender, Receiver). 2. Msg is the key message, Sender or Receiver is one of the six role players. 3. In represents a receiving message from sender to receiver, Out represents a sending message from sender to receiver.APPENDIX 3. DSRM PROCESS MODEL
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|
[
"Bures metric over thermal state manifolds and quantum criticality",
"Bures metric over thermal state manifolds and quantum criticality"
]
| [
"Paolo Zanardi \nDepartment of Physics and Astronomy\nUniversity of Southern California\n90089-0484Los AngelesCAUSA\n\nInstitute for Scientific Interchange\nVilla Gualino, Viale Settimio Severo 65I-10133TorinoItaly\n",
"Lorenzo Campos Venuti \nInstitute for Scientific Interchange\nVilla Gualino, Viale Settimio Severo 65I-10133TorinoItaly\n",
"Paolo Giorda \nInstitute for Scientific Interchange\nVilla Gualino, Viale Settimio Severo 65I-10133TorinoItaly\n"
]
| [
"Department of Physics and Astronomy\nUniversity of Southern California\n90089-0484Los AngelesCAUSA",
"Institute for Scientific Interchange\nVilla Gualino, Viale Settimio Severo 65I-10133TorinoItaly",
"Institute for Scientific Interchange\nVilla Gualino, Viale Settimio Severo 65I-10133TorinoItaly",
"Institute for Scientific Interchange\nVilla Gualino, Viale Settimio Severo 65I-10133TorinoItaly"
]
| []
| We analyze the Bures metric over the manifold of thermal density matrices for systems featuring a zero temperature quantum phase transition. We show that the quantum critical region can be characterized in terms of the temperature scaling behavior of the metric tensor itself. Furthermore, the analysis of the metric tensor when both temperature and an external field are varied, allows to complement the understanding of the phase diagram including cross-over regions which are not characterized by any singular behavior. These results provide a further extension of the scope of the metric approach to quantum criticality. | 10.1103/physreva.76.062318 | [
"https://arxiv.org/pdf/0707.2772v2.pdf"
]
| 119,145,026 | 0707.2772 | 9396dfa5596c0fb1c714add311c2aae2b3f58e07 |
Bures metric over thermal state manifolds and quantum criticality
Jul 2007 (Dated: February 1, 2008)
Paolo Zanardi
Department of Physics and Astronomy
University of Southern California
90089-0484Los AngelesCAUSA
Institute for Scientific Interchange
Villa Gualino, Viale Settimio Severo 65I-10133TorinoItaly
Lorenzo Campos Venuti
Institute for Scientific Interchange
Villa Gualino, Viale Settimio Severo 65I-10133TorinoItaly
Paolo Giorda
Institute for Scientific Interchange
Villa Gualino, Viale Settimio Severo 65I-10133TorinoItaly
Bures metric over thermal state manifolds and quantum criticality
Jul 2007 (Dated: February 1, 2008)arXiv:0707.2772v2 [quant-ph] 19numbers: 0365Ud0570Jk0545Mt
We analyze the Bures metric over the manifold of thermal density matrices for systems featuring a zero temperature quantum phase transition. We show that the quantum critical region can be characterized in terms of the temperature scaling behavior of the metric tensor itself. Furthermore, the analysis of the metric tensor when both temperature and an external field are varied, allows to complement the understanding of the phase diagram including cross-over regions which are not characterized by any singular behavior. These results provide a further extension of the scope of the metric approach to quantum criticality.
I. INTRODUCTION
These years are witnessing an increasing research effort at the intersection of Quantum Information Science [1] and more established fields like theoretical condensed matter [2]. It belongs to this class the approach to quantum phase transitions (QPT) [3] based on the information-geometry of quantum states that has been recently proposed in Ref [5] and [6]. Further developments, for specific, yet important, class of quantum states have been then reported in [7,8,9,10,11,12,13,14]. The underlying idea is deceptively simple: the major structural change in the ground state (GS) properties at the QPT should reveal itself by some sort of singular behavior in the distance function between the GSs corresponding to slightly different values of the coupling constants. This intuition can me made more quantitative by analyzing the leading order terms in the expansion of the quantum fidelity between close GSs.
A general differential-geometric framework encompassing all of these result has been offered in Ref [15]. There it has been shown that these leading order terms do correspond to a Riemannian metric g over the parameter manifold. This metric g is nothing but the pull-back of the natural metric over the projective Hilbert space via the map associating the Hamiltonian parameters with the corresponding GS. In the thermodynamical limit the singularities of g correspond to QPTs. In Ref. [16] the nature of this correspondence has been further investigated and it has been shown that both the metric approach to QPT and the one based on geometrical phases [17,18] can be understood in terms of the critical scaling behavior of the quantum geometric tensor [19].
The conceptually appealing and potentially practically relevant feature of this strategy consists of the fact that its viability does not rely on any a priori knowledge of the physics of the model e.g., order parameters, symmetry breaking patterns,... but just on a universal geometrical structure (basically the Hilbert scalar product). Very much in the spirit of Quantum Information the metric approach is fully based on quantum states rather than Hamiltonians (that might be even unknown), once these are given the machinery can be applied.
In this paper we further extend the scope of this metric approach by considering the manifold of thermal states of a family of Hamiltonians featuring a zero-temperature PT. In [20] it was shown that by studying the mixed-state fidelity [21] between Gibbs states associated with slightly different Hamiltonians one could detect the influence of the zero-temperature quantum criticality over a finite range of temperatures. Here we will refine that analysis and make it more quantitative by resorting to the concept of Bures metric between mixed quantum states. This metric provides the natural finite-temperature extension of the metric tensor g studied in the GS case and corresponds again to the leading order in the expansion of the (mixed-state) fidelity between close states i.e., associated with infinitesimally close parameters. By analyzing the case of the Quantum Ising model we shall show how the quantum-critical region above the zero-temperature QPT can be remarkably characterized in terms of the scaling behavior of the Bures metric tensor.
The paper is organized as follows: in sect. II we introduce the basic concepts about mixed-state metrics and in Sect III we specialize them to the case of thermal (Gibbs) states. In sect. IV we provide generalities about quasifree fermion systems and in Sect VI we analyze in detail the Bures metric tensor for the quantum Ising model. Finally in sect V conclusions and outlook are given.
II. PRELIMINARIES
The Bures distance between two mixed-states ρ and σ is given in terms of the Uhlmann fidelity [21] F (ρ, σ) = tr ρ 1/2 σρ 1/2 (1)
by d B (ρ, σ) = 2 [1 − F(ρ, σ)].
The starting point of our analysis is provided by the following expression for the Bures distance between two infinitesimally close density matrices (see e.g. [22] for a derivation)
ds 2 (dρ) := d 2 B (ρ, ρ + dρ) = 1 2 n,m | m|dρ|n | 2 p m + p n ,(2)
where |n is the eigenbasis of ρ with eigenvalues p n i.e., ρ = n p n |n n|. Even though in the sum in (2) p n and p m cannot be simultaneously in the kernel of ρ, since |n , |m ∈ Ker(ρ) ⇒ n|dρ|m = 0, one can formally extend the sum to all possible pairs by setting to zero the unwanted terms. For ρ pure i.e., ρ = |ψ ψ| one has dρ = |dψ ψ| + |ψ dψ| from which one sees that the diagonal matrix elements of dρ are vanishing and one is left with ds 2 B = m∈Ker(ρ) | dψ|m | 2 = dψ|(1 − |ψ ψ|)|dψ . This expression coincides with the Riemannian metric considered in [15]. The bures metric (2) is tightly connected to the so-called quantum Fisher information and it appears in the quantum version of the celebrated Cramer-Rao bound [23]. This suggests the possible relevance of the results that we are going to present in this paper to the field of quantum estimation [24].
To begin with we would like to cast Eq. (2) in a from suitable for future elaborations. Let us first differentiate the density matrix dρ = n (dp n |n n| + p n |dn n| + p n |n dn|) and consider to begin the matrix element (dρ) ij . We observe that i|j = δ i,j ⇒ di|j = − i|dj ; whence i|dρ|j = δ i,j dp i + i|dj (p j − p i ). Putting this expression back into (2) one obtains
ds 2 = 1 4 n dp 2 n p n + 1 2 n =m | n|dm | 2 (p n − p m ) 2 p n + p m .(3)
This relation is quite interesting since it tells apart the classical and the quantum contributions. Indeed the first term in (3) is nothing but the Fisher-Rao distance between the probability distributions {p n } n and {p n + dp n } n whereas the second term takes into account the generic non-commutativity of ρ and ρ ′ := ρ + dρ. We will refer to these two terms as the classical and nonclassical one respectively. When [ρ ′ , ρ] = 0 the problem gets effectively classical and the Bures metric collapses to the Fisher-Rao one; this latter being in general just a lower bound [23,25].
Before moving to the analysis of the metric (2) we would like to comment about the connection with the recently established quantum Chernoff bound [26]. This latter, denoted by ξ QCB , is the quantum analogue of the Chernoff bound in classical information theory; it quantifies the rate of exponential decay of the probability of error in discriminating two quantum states ρ and σ when a large number n of them is provided and collective measurements are allowed i.e., P err ∼ exp(−nξ QCB ). The Chernoff bound naturally induces a distance function over the manifold of quantum states with a well defined operational meaning (the bigger the distance between the states the smaller the asymptotic error probability in telling one from the other). In [26] it has been proven that exp(−ξ QCB ) = min 0≤s≤1 tr ρ s σ 1−s ≤ F(ρ, σ) and that for infinitesimally close states i.e., σ = ρ + dρ, one has
ds 2 QCB := 1 − exp(−ξ QCB ) = 1 2 n,m | m|dρ|n | 2 ( √ p m + √ p n ) 2 . (4)
From this expression we see that the distinguishability metric associated with the quantum Chernoff bound has the same form of the Bures one (2) but the denominators p n + p m are replaced by (
√ p m + √ p n ) 2 .
Using the inequalities ( √ p m + √ p n ) 2 ≥ p n + p m and 2(p n + p m ) ≥ ( √ p m + √ p n ) 2 one immediately sees that
ds 2 2 ≤ ds 2 QCB ≤ ds 2 .(5)
This relation shows that, as far as divergent behavior is concerned, the Bures and the Chernoff bound metric are equivalent i.e., one metric diverges iff the other does.
On the other hand in the metric approach to QPTs the identification of divergences of the rescaled metric tensor and their study plays the central role [15]. Therefore one expects the two distinguishability measures to convey equivalent information about the location of the QPTs. Though most of the calculations that are reported in this paper could be easily extended to the Chernoff bound metric, here we will limit ourselves to the analysis of the Bures metric (2).
III. THERMAL STATES
From now on we specialize our analysis to the case of thermal states. If the Hamiltonian smoothly depends on a set of parameters, denoted by λ, living in same manifold M one has the smooth map (λ, β) → ρ(β, λ) := Z −1 e −βH(λ) , (Z = tre −βH ). What we are going to study in this paper is basically the pull-back onto the (λ, β) plane of the Bures metric through this map. This is the obvious finite-temperature extension of the ground-state approach of Ref. [15].
We start by studying the Bures distance when T = 0 is fixed and for infinitesimal variations of the Hamiltonian's parameters λ. Notice first that ρ = Z −1 n e −βEn |n n| where E n and |n are the eigenvalues and eigenvectors of the Hamiltonian operator H. With a standard reasoning, by differentiating the Hamiltonian eigenvalue equation one finds that i|dj = i|dH|j /(E i − E j ). Moreover one easily sees that dp i = d(e −βEi /Z) = −βp i (dE i −( j dE j p j )), therefore the first term in equation (3) can be written as
β 2 /4 i p i (dE 2 i − dE ) 2 where dE β := j dE j p j )
. This means that the Fisher-Rao distance is expressed by the thermal variance of the diagonal observable dH d := j dE j |j j| times the square of the inverse temperature. Summarizing
ds 2 B = β 2 4 ( dH 2 d β − dH d 2 β ) + 1 2 n =m n|dH|m E n − E m 2 (e −βEn − e −βEm ) 2 Z(e −βEn + e −βEm ) .(6)
The two terms correspond to the first and second term of (3) respectively and they depend on β and on the other parameters of the Hamiltonian. For example, when a single parameter h is considered, the Bures distance defines a simple metric that can be expressed in term of the classical and non-classical part
g hh (h, β) = g c hh (h, β) + g nc hh (h, β)(7)
such that ds 2
B = g hh (h, β)dh 2 .
Let us now explore the behavior of the Bures distance in presence of infinitesimal variations of both the temperature (β variations) and a field h in the Hamiltonian. It is easy to see that the variation of β only affects the Fisher Rao classical term in (3). In fact the variation dH in (6), or analogously the variations |dm in (3), are taken with respect to h only. The calculations can be summarized as follows. We first have to expand the dp n as dp n = (∂ β p n )dβ + (∂ h p n )dh. We have that
(∂ β p n )dβ = p n [ H − E n ] dβ and (∂ h p n )dh = β p n [ ∂ h H d − ∂ h E n ] dh where E n = E n (h).
The complete classical term of the Bures distance can be written expanding (dp n ) 2 = (∂ h p n dh) 2 + (∂ β p n dβ) 2 + 2∂ β p n ∂ h p n dβdh, and summing over n. We thus have three different contributions:
1 4 n (dp n ) 2 p n = 1 4 H 2 − H 2 dβ 2 +β 2 [(∂ h H) d ] 2 − (∂ h H) d 2 dh 2 +2β [ H(∂ h H) d − (∂ h H) d H ] dβdh(8)
These terms correspond to the elements of the metric g β,β , g h,β , g c h,h respectively. The full metric can be written once one calculates the non classical term in (3). The infinitesimal Bures distance can then be written in terms of the 2 × 2 metric tensor g as :
ds 2 = (dh, dβ) g dh dβ , g = g hh g hβ g hβ g ββ .(9)
where again g hh (h, β) = g c hh (h, β) + g nc hh (h, β). It is at this point interesting to check whether, for β → ∞, one recovers the known results for groundstate (pure) fidelity and metric tensor. In order to do that we will consider separately the classical and non-classical term in (3). In fact ρ(β) − ρ(∞) 1 = (1 − p 0 ) + n>0 p n ≤ 2 n>0 e −β(En−E0) ; from which one sees that, for finite-dimensional systems, the thermal density matrix converges (in trace norm) exponentially fast to the projector over the ground state |0 . Then it follows that all the expectations values will converge exponentially fast to their zero-temperature limits: |tr(Aρ(β)) − tr(Aρ(∞))| ≤ A ρ(β) − ρ(∞) 1 , this in turn guarantees that the covariances of diagonal operators in the Fisher-Rao term (8) are vanishing (since e.g., dH d is diagonal dH 2 ∞ = dH 2 ∞ .) in the zerotemperature limit. In the infinite dimensional case the convergence to zero of this term will typically be only algebraic in the region where the smallest excitation gap is small compared to the temperature, whereas it will be exponential elsewhere. The overall convergence behavior of the classical term for β → ∞ depends now on the detailed interplay between the decay of covariances we just discussed in (8) and the divergence of the powers of β in front of them. An analysis of the zero-temperature limit of these terms will be provided later for the quantum Ising model. We will see that all the classical terms vanish in the zero temperature limit but at the critical value of the parameter. As far as the second non-classical term in (3) is concerned one has just to notice that from lim β→∞ p n (β) = δ n,0 it follows that the only contributions will come from the elements involving the groundstate i.e., 0|dj = 0|dH|j /(E j − E 0 ). This completes the remark.
Before moving to the next sections, where we will specialize the previous results to the particular case of the Quantum Ising model, we would like to notice that the variation of the Bures distance with temperature only, given by the element g ββ of the metric, is precisely proportional to the specific heat c v [15], i.e.
ds 2 B = dβ 2 4 ( H 2 β − H 2 β ) = dβ 2 4 T 2 c v .
This simple fact was already observed in [15] and [10] and provides, we believe, a neat connection between quantum-information theoretic concept, geometry and thermodynamics.
IV. QUASI FREE FERMIONS
In this section we specialize the study of the behavior of the Bures metric to systems of quasi-free fermions when one has the variation of one parameter h of the Hamiltonian and of the temperature T . The results that we present here are a finite-temperature generalization of those given in Refs [7] and [8] and directly related to the mixed-state fidelity ones reported in [20].
The quasi-free Hamiltonians we consider are given, after performing a suitable Bogoliubov transformation, by
H = ν Λ ν η † ν η ν ,(10)
where Λ ν > 0 and η ν denote the quasi-particle energies and annihilation operator respectively. One has that ν is a suitable quasi-particle label, that for translationally invariant systems amounts to a linear momentum; the ground state is the vacuum of the η ν operators i.e., η ν |GS = 0, ∀ ν. The dependence on the parameter h is both through the Λ ν 's and the η ν 's. We now derive the explicit general form of the Bures distance (2) starting from the classical part (8). We observe that the (many-body) Hamiltonian eigenvalues are given by E j = ν n ν Λ ν where the n ν 's are fermion occupation numbers i.e., n ν = 0, 1. Therefore we have that dE j = ν n ν dΛ ν and dE j β = ν n ν β dΛ ν where the averages are easy to compute since the probability distribution of the dE j factorizes over the ν's. Furthermore, n µ n ν β − n µ β n ν β = δ µν n ν β (1 − n ν β ) and we can thus write:
1 4 n (dp n ) 2 p n = 1 4 k n k (1 − n k ) × Λ 2 k dβ 2 +β 2 (∂ h Λ k ) 2 dh 2 +2βΛ k ∂ h Λ k dβ dh} .(11)
The term in dh 2 is the classical term due to the infinitesimal variations of the parameters of the Hamiltonian at fixed T and it corresponds to the variance, see (6),
var(H d ) = ν n ν β (1 − n ν β )dΛ 2 ν .
Since we are dealing with independent free-fermions one has that n ν β = (exp(βΛ ν ) + 1) −1 , whence
ds 2 c = β 2 16 ν (∂ h Λ k ) 2 cosh 2 (βΛ ν /2) dh 2(12)
In order to compute the non-classical part of Eq. (3), one has to explicitly consider the eigenvectors of (10). Following the notation of Ref. [7] one has |m =
{α ν , α −ν } ν>0 = ⊗ ν>0 |α ν , α −ν where |0 ν 0 ν = cos(θ ν /2)|00 ν,−ν − sin(θ ν /2)|11 ν,−ν , |0 ν 1 −ν = |01 ν,−ν , |1 ν 0 −ν = |10 ν,−ν , |1 ν 1 ν = cos(θ ν /2)|11 ν,−ν + sin(θ ν /2)|00 ν,−ν .
We assume now that parameter dependence is only in the angles θ ν 's (this assumption holds true for all the translationally invariant systems). It is easy to see from the above factorized form that the only non vanishing matrix elements n|dm are given by 0 ν 0 −ν |d|1 ν 1 −ν = dθ ν /2 and that the thermal factor (p n − p m ) 2 /(p n + p m ) has the form sinh 2 (βΛ ν )/[(cosh(βΛ ν ) + 1)(cosh(βΛ ν )] = (cosh(βΛ ν ) − 1)/ cosh(βΛ ν ). Putting all together one finds
ds 2 nc = 1 4 ν>0 cosh(βΛ ν ) − 1 cosh(βΛ ν ) (∂ h θ ν ) 2 dh 2(13)
We finally note that the two elements (12) and (13) define the metric element (7).The results of this section can be applied to any quasi-free fermionic model (10).
V. QUANTUM ISING MODEL
We are now going to discuss in some detail the behavior of the metric tensor for a paradigmatic example in the class of quasi-free fermionic models, the Ising model in transverse field. The model is defined by the Hamiltonian
H = − j σ x j σ x j+1 + hσ z j .(14)
At T = 0 this system undergoes a quantum phase transition for h = 1 . For h < 1 the system is in an ordered phase as the correlator σ x 1 σ x r T =0 tends to a non zero value: lim r→∞ σ x 1 σ x r T =0 = 1 − h 2 1/4 . The excitations in this region are domain walls in the σ x direction. Instead for h > 1 the magnetic field dominates, and excitations are given by spin flip over a paramagnetic ground state. The transition point h = 1 is described by a c = 1/2 conformal field theory,which implies that means that the dynamical exponent z = 1; the correlation function exponent is ν = 1. As is well known [3], a signature of the ground state phase diagram remains at positive temperature. In the quasi classical region T ≪ ∆, where ∆ = |1 − h| is the lowest excitation gap, the system can be described by a diluted gas of thermally excited quasi-particles, even if the nature of the quasi-particles is different at the different sides of the transition. Instead in the quantum critical region T ≫ ∆ the mean inter-particle distance becomes of the order of the quasi-particle de Broglie wavelength and thus quantum critical effects dominate and no semiclassical theory is available. In each of the above described regions of the (h, T ) plane the system displays very different dynamical as well thermodynamical properties. For example, in the quantum critical region the specific heat approaches zero linearly with the temperature (this is in fact a general feature of all conformal field theories), whereas in the quasi-classical regions the approach is exponentially fast.
A. Bures metric tensor in the (h, T ) plane
We now investigate whether the signature of the physically different regions can be revealed by analyzing the elements of the metric tensor defined by the Bures distance. We begin by studying the temperature dependence of the metric tensor when only the external field is varied i.e., the term g hh (h, T ), see Eq. (7). The Hamiltonian (14) is equivalent to a quasi-free fermionic model, and following our previous notation one has ǫ k = cos (k) − h, ∆ k = sin (k), Λ k = ǫ 2 k + ∆ 2 k and tan (ϑ k ) = ∆ k /ǫ k . Using formulae (12) and (13) it is straightforward to write (7). After rescaling g → g/L and passing to the thermodynamic limit we obtain
g c hh = β 2 16π π −π 1 cosh (βΛ k ) + 1 ǫ 2 k Λ 2 k dk g nc hh = 1 8π π −π cosh (βΛ k ) − 1 cosh (βΛ k ) ∆ 2 k Λ 4 k dk
The integrals are better evaluated by transforming momentum integration to energy integration in a standard way. As previously noticed, on general grounds, the classical term g c hh vanishes when the temperature goes to zero. In the quantum-critical region β∆ ≈ 0, and one obtains the following low temperature expansion:
g c hh = π 96h 2 T + O T 2 .
Instead in the quasi-classical region where β∆ ≫ 1 the fall-off to zero is exponential. With a saddle point approximation one obtains
g c hh = ∆ 32πh
T −3/2 e −∆/T + lower order.
We now analyze the scaling behavior of the nonclassical term of the metric g. From the results of [5,7] it is known that the geometric tensor at zero temperature diverges as ∆ −1 when ∆ → 0. The non classical term matches this ground state behavior from positive temperature. Indeed, in the quantum-critical region the integral is well approximated by
g nc hh ≈ 1 8πh 2 2β 0 cosh (x) − 1 cosh (x) 4β 2 − x 2 x 2 dx.
For large β (low temperatures) this expression can be Laurent expanded and the resulting integrals can be resummed using residuum theorem, giving
g nc hh = 1 h 2 C π 2 T −1 − 1 16 + O (T ) ,(15)
where C is Catalan's constant C = 0.915966 . . .. We would like to point out that the behavior of the metric tensor in the quasi-critical region, can be inferred from dimensional scaling analysis in much the same spirit as was done in [16] for the zero temperature metric tensor. From Eq. (6) we see that the scaling dimension of g nc hh is ∆ nc = 2∆ V − 2z − d, where ∆ V is the scaling dimension of the operator dH, z is the dynamical exponent and d is the spatial dimensionality. Following [16] (β now plays the role of the length) we obtain
g nc hh ∼ T ∆nc/z(16)
In the present case, z = d = ∆ V = 1 (the scaling dimension of σ z i -a free fermionic field-is one) which agrees with (15).
We now pass to analyze the behavior of g nc hh in the quasi-classical region i.e. when β∆ ≫ 1. In this case the "temperature" part of the integral is never effective, i.e. one has cosh(βΛ)−1 cosh(βΛ) ≈ 1, so it is quite clear that, in first approximation, one recovers the zero temperature result first given in [5] which we re-write here as an energy integral
g nc hh = 1 8πh 2 |1+h| |1−h| (h − 1) 2 − ω 2 ω 2 − (1 + h) 2 ω 3 dω,(17)
and we assumed h > 0. For small values of the gap,hence we are in a situation where we consider first the limit T → 0 and then ∆ → 0-we observe the following divergence
g nc hh (T = 0, ∆ → 0) ≈ 1 16∆ ,
which is a result also reported in [5,7]. Instead when the gap is large, -so that we are necessary on the h > 1 side-we can approximate the radical in Eq. (17) with an ellipse centered at (h, 0) with semiaxes r x = 1 and r y = 2h, that amounts to write
(h − 1) 2 − ω 2 ω 2 − (1 + h) 2 ≈ 2h 1 − (ω − h) 2 .
In this case the integral gives
g nc hh (T = 0, ∆ ≫ 1) ≈ 1 8h 5/2 (h − 1) 3/2 ≈ 1 8∆ 4 .
Again, by doing a saddle point approximation one realizes that the zero temperature results are approached exponentially fast with the temperature, more precisely one has g nc hh (β∆ ≫ 1) = g nc hh (T = 0) − const. × T 3/2 e −∆/T .
We now extend our analysis to the other terms of metric tensor (9). When we consider the case in which both the temperature and the field h are varied two new matrix elements come into play:
g T T = β 4 16π π −π Λ 2 k cosh (βΛ k ) + 1 dk g hT = β 3 16π π −π ǫ k cosh (βΛ k ) + 1 dk.
Let us first comment on the behavior observed at very low temperature. In the quasi-classical region (∆ ≫ T ) all matrix elements of g tend to zero except for g nc hh . This is a general feature and is due to the fact that these terms are absent in the zero temperature expression. As previously stated the fall-off to zero is exponential, and in particular, for the model in exam, we have that
g T h ≈ T −5/2 e −∆/T , T ≪ ∆ g T T ≈ T −7/2 e −∆/T .(18)
Let us now look at the quantum critical region T ≫ ∆, small temperature. The mixed term tends to a constant:
g hT = π 48 + O T 2 .(19)
Instead g T T must diverge at zero temperature, as it has to match with the diverging behavior observed in the ground state [5]. For the diagonal term g T T one has
g T T = T −2 4 c v = π 24 1 T + O (T ) , T ∆.(20)
We note in passing that this result agrees with the one for the specific heat obtained for general conformal theories [27] c v = (πcT ) / (3v) , as T → 0, since in our case the velocity v is one and the conformal charge c is one-half.
We thus see that, in the present case, both g hh and g T T diverge as T −1 . This has not to be the case in general, indeed at any quantum critical point described by a conformal field theory, g T T will diverge as T −1 whereas the behavior of g nc hh is dictated by Eq. (16). In this section we have analyzed the behavior of all the elements of the geometric tensor g. The result of this analysis allows one to conclude that indeed, at least for the specific model studied, the quantum critical and quasi classical regions can be clearly identified in terms of the markedly different temperature behavior of the geometric tensor g.
B. Directions of maximal distinguishability
The analysis carried out in the previous section can be further deepened by studying some useful quantities that can be derived by the analysis of the metric tensor g. Indeed, we will see that these quantities allow to give a finer description of the behavior of the system in the plane (h, T ) and to reveal new unexpected features. We first start by noticing that at each point (h, T ) the eigenvectors of the metric tensor g define the directions of maximal and minimal growth of the line element ds 2 B . Hence the vector field v M (h, T ) given by the eigenvector of g related to the highest eigenvalue λ M , defines at each point of the (h, T ) plane the direction along which the fidelity decreases most rapidly: the latter represents the direction of highest distinguishability between two nearby Gibb's states.
We now focus our analysis on the study of the vector field v M (h, T ) in the specific case of the quantum Ising model, see Fig. 1. We first observe that there clearly are some interesting features for small temperatures, that reflect the analysis previously carried out on the metric elements of g. On one hand, in the quasi-classical region, when h ≃ 0 we have that the direction of highest fidelity drop, is parallel to the h axis. This reflects the fact that, in this region, all the elements of g tend to zero except for the term g nc hh . On the other hand, in the quasi critical region, the direction of highest fidelity drop is parallel to the T axis. Again, this feature can be linked to the previous analysis of the terms (15) and (20). Indeed both g hh and g T T diverge as T −1 , but, as C/π 2 / (π/24) = 0.70 . . . < 1, g T T eventually becomes bigger and thus the direction of highest distinguishability turns parallel to the T axis. We proceed in our description of the phase diagram through the introduced vector field by examining what happens at the h = 0 axis. Here one has the impression that a kind of singular point appear around T ≈ 1. The reason for this is that at h = 0 the system becomes the purely classical Ising model, which possess only classical behavior at any temperature. This implies that the quantum-critical region cannot extend over this line. As the dispersion Λ k is flat, it is straightforward to write down the metric tensor on the h = 0 line. It turns out that g is completely diagonal meaning that eigenvectors are parallel to the (h, T ) axes. One sees that for 0.852 < T < ∞, g hh > g T T then for 0.101 < T < 0.852, g T T > g hh and then finally, at very small temperature, quantum fluctuations dominates and for 0 < T < 0.101, g hh > g T T . The appearance of the purely classical Ising line at h = 0, which forbids the quantum critical phase extend over this line, is related to the fact that the model (14) is invariant under the Z 2 symmetry h → −h. This in turns implies that the phase diagram is mirror symmetric around the h = 0 line and that there is another quantum critical point at h = −1. The physical consequence is that the semiclassical ordered region is much smaller than one would think and the actual phase diagram is much like in Figure 2.
Finally we now discuss another feature that can observed by studying v M (h, T ). As one can see in Fig. 1, along the line T = h 1 the vector field becomes parallel to the vector w = (−1, 1). It turns out that this feature can be understood analytically by studying the behavior of the metric tensor g when |h| ≫ 1. Indeed, by evaluating the dominant part of the various metric elements on the line T = h = t ≫ 1, one sees that all 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 0000000000 0000000000 0000000000 0000000000 0000000000 1111111111 1111111111 1111111111 1111111111 the Fisher-Rao terms decay as t −2 while g nc hh ∼ t −4 , and, what is most surprising, all matrix elements tend to have the same value in magnitude. This feature can be understood by simply observing that when |h| ≫ 1 it is only the classical term proportional to the external magnetic field of the Quantum Ising Hamiltonian that survives i.e., H ≃ h i σ z i . The density matrix of the system can be written as ρ(h, T ) = exp (−h i σ z i /T )/Z; in this approximation the only non zero terms of the metric are the Fisher-Rao ones and all the covariances that define these terms, see (8), coincide with var(H). Thus, in the limit |h| ≫ 1 the Bures distance reads
ds 2 B = var(H)[dT 2 /T 2 − hdT dh/T 3 + h 2 dh 2 /T 4 ]
. If now one chooses the particular case T = h = t and evaluates the density g/L one finds that
g (t, t) = t −2 16 cosh 2 (1/2) 1 −1 −1 1 + O t −3 .
Thus, one has that on the line T = h ≫ 1 the only non zero eigenvalue is 2 var(H)/(Lt 2 ) and it corresponds to the eigenvector w = (−1, 1). In this approximation, that amounts to neglecting the term g c hh ∼ t −4 , when moving along the line T = h ≫ 1 i.e., along the direction defined by w ⊥ , no changes in the state of the system occur.
C. Crossover and metric tensor g
We finally present some preliminary results related to the intriguing possibility of determining the crossover lines between the quasi-classical and quasi-critical region (14) through the analysis of the elements of metric tensor g and the induced Gaussian curvature [28] in the plane (h, T ). The capability of the highest (in modulus) eigenvalue of the g and of the Gaussian curvature induced by the metric to capture, in terms of divergencies or discontinuities, the existence of QPTs has been already tested in [8] and [15]. Here we would like to test whether these quantities are able to identify the crossover between the quasi-classical and quantum-critical region. Notice that the curvature of the Bures metric in the case of squeezed states has been studied in [29] and an operational interpretation attempted. It is also worthwhile to stress that the so-called thermodynamical curvature plays a central role in the geometrical theory of classical phase transition developed by Ruppeiner and coworkers [30].
As already pointed out, at each point (h, T ) the vector field v M (h, T ) defines the direction of highest distinguishability between two nearby Gibb's states. The degree of distinguishability along this direction is quantified by the maximal eigenvalue λ M (h, T ). Since the quasi-classical and quantum-critical regions are characterized by significantly different physical properties, it is natural to investigate whether the change of the latter, in spite of not involving a phase transition, could be revealed by our measures of statistical distinguishability and by the related functionals.
We now give a descriptive analysis of the raw data. In figure 3, we have plotted the contour plot of the maximal eigenvalue of g. The main feature is the presence for T > 0 of two patterns of high distinguishability (white) that separate the regions (h < 1, T 0.25) and (h > 1, T 0.25) from the rest of the diagram. Thus, the first information that can be drawn is that a change of parameters inside these regions implies a small change in the statistical properties of the corresponding ground states. On the contrary, if one varies h and T and moves from these regions towards the center of the diagram, for example moving along the integral lines of v M (h, T ) , the statistical properties of the state necessarily have to significantly change. One can see that, the "transition" lines between the different regions can be extrapolated numerically by tracing the "ridge" lines of the two patterns of high distinguishability. It turns out that the same result can be achieved by looking at the lines where the Gaussian curvature of g changes sign, see figure 4. For example, when h > 1, one can see that along the de- termined transition line, T has a linear dependence on h − 1. This preliminary descriptive analysis seems thus to indicate that a neat distinction between the quasi-classical regions (characterized by a negative curvature) and the quantum-critical (characterized by a positive curvature) can be made on the basis of study of the metric g. This is indeed the first time that the use of the fidelity, and of the related functionals, allows to identify the crossover between two distinct phases.
VI. CONCLUSIONS
In this paper we have analyzed the relation between quantum criticality, finite temperature and the differential-geometry of the manifold of mixed quantum states. We studied the Bures metric over the set of thermal quantum states associated with Hamiltonians featuring a zero-temperature quantum phase transition i.e., quasi-free fermionic systems. In particular we focused on the study of the quantum Ising model for which we provided a fully analytical characterization of the Bures metric tensor g. Quantum critical and semiclassical regions in the temperature, magnetic field plane can be easily identified in terms of different scaling behavior of the components of g as a function of the temperature. Cross-over lines between the different regions can be found just by looking at the shape of the graph of the largest eigenvalue of the metric as a function of temperature and magnetic field. Remarkably these cross-over lines seem to be associated also with the change of sign of the Gaussian curvature of the metric g.
The results presented in this paper provide further support to the validity of the statistical-metric approach to phase transitions [15] and clearly show that the scope of this geometrical method can be extended to finite temperatures. The physical significance of the curvature of the metric as well as the study of the thermal states geometry associated with other distinguishability distances e.g., the quantum Chernoff bound metric, are topics deserving further investigations.
Figure 1 :
1Vector field of the eigenvector associated to the highest eigenvalue of g, in the plane (h, T ) for the Ising model in transverse field.
Figure 2 :
2Phase diagram of the Ising model in transverse field taking into account both critical points at h = ±1, and the purely classical Ising line h = 0. The arrow indicates the direction of highest fidelity decrease (the direction of the arrows is conventional, but fixed once for all)
Figure 3 :
3Contour plot of the highest eigenvalue of g, in the plane (h, T ) for the Ising model in transverse field.
Figure 4 :
4Contour plot of the Gaussian curvature of g, in the plane (h, T ) for the Ising model in transverse field. The arrows indicate the zero curvature lines.
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 h T
AcknowledgmentsThe authors would like to thank for useful discussions R. Ionicioiu and M. Paris.
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| []
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[
"Pb chains on reconstructed Si(335) surface",
"Pb chains on reconstructed Si(335) surface"
]
| [
"Mariusz Krawiec \nInstitute of Physics\nM. Curie-Sk lodowska University\nPl. M. Curie-Sk lodowskiej 120-031LublinPoland\n"
]
| [
"Institute of Physics\nM. Curie-Sk lodowska University\nPl. M. Curie-Sk lodowskiej 120-031LublinPoland"
]
| []
| The structural and electronic properties of Si(335)-Au surface decorated with Pb atoms are studied by means of density-functional theory. The resulting structural model features Pb atoms bonded to neighboring Si and Au surface atoms, forming monoatomic chain located 0.2 nm above the surface. The presence of Pb chain leads to a strong rebonding of Si atoms at the step edge. The fact that Pb atoms occupy positions in the middle of terrace is consistent with STM data, and also confirmed by simulated STM images. The calculated band structure clearly shows one-dimensional metallic character. The calculated electronic bands remain in very good agreement with photoemission data. PACS numbers: 73.20.At, 71.15.Mb, 79.60.Jv, 68.47.Fg I. | 10.1103/physrevb.79.155438 | [
"https://arxiv.org/pdf/0904.0399v1.pdf"
]
| 119,290,859 | 0904.0399 | 0145830c7667db34b1c040d53d4f48b74581d23c |
Pb chains on reconstructed Si(335) surface
2 Apr 2009
Mariusz Krawiec
Institute of Physics
M. Curie-Sk lodowska University
Pl. M. Curie-Sk lodowskiej 120-031LublinPoland
Pb chains on reconstructed Si(335) surface
2 Apr 2009(Dated: April 2, 2009)numbers: 7320At7115Mb7960Jv6835B-6847Fg
The structural and electronic properties of Si(335)-Au surface decorated with Pb atoms are studied by means of density-functional theory. The resulting structural model features Pb atoms bonded to neighboring Si and Au surface atoms, forming monoatomic chain located 0.2 nm above the surface. The presence of Pb chain leads to a strong rebonding of Si atoms at the step edge. The fact that Pb atoms occupy positions in the middle of terrace is consistent with STM data, and also confirmed by simulated STM images. The calculated band structure clearly shows one-dimensional metallic character. The calculated electronic bands remain in very good agreement with photoemission data. PACS numbers: 73.20.At, 71.15.Mb, 79.60.Jv, 68.47.Fg I.
INTRODUCTION
One-dimensional (1D) atomic chains created on semiconductor templates have recently attracted much attention due to new phenomena characteristic for a reduced dimensionality [1]. The most spectacular examples include a breakdown of Fermi liquid theory [2,3] and Peierls metal-insulator transition [4]. A specially attractive route to create such 1D structures is a process of selforganization of atoms into very regular arrays of parallel metallic chains on stepped semiconducting or insulating substrates [5,6]. In this case electrons near the Fermi energy are completely decoupled from the substrate due to band gap in electronic spectrum of the substrate. The chain atoms are bounded to the surface by low energy states which do not contribute to electronic properties of the system.
The Si(335)-Au surface is one of the simplest examples of high-index surfaces which stabilize the onedimensional structures, and has been studied by number of techniques, including: reflection high energy electron diffraction (RHEED) [7], scanning tunneling microscopy (STM) [8,9], angle-resolved photoemission spectroscopy (ARPES) [8,10] and first principles density functional theory (DFT) [11]. In particular, the STM topography data show regular arrays of monoatomic chains separated by width of Si(335) terrace and a few nanometers long [8,9]. The ARPES spectra taken in the direction parallel to the steps show two highly dispersive bands crossing the Fermi energy (E F ), thus indicating one-dimensional metallic nature of the system [8]. The structural model of the Si(335)-Au surface has also been proposed [8] and confirmed later by the DFT calculations [11], which well describes all the available experimental data.
Recently, other materials (Na, Pb, In) deposited on Au decorated Si(335) surface have also been studied [12,13,14,15,16,17]. Perhaps lead is the most intensively studied among them. Unlike Si(335)-Au surface, in * Electronic address: [email protected] which gold substitutes Si atoms in the surface layer, the Si(335)-Au/Pb is a representative of new class of systems in which the deposited material forms one-dimensional structures adsorbed on-top of the surface. The lead deposited on flat Si(111)-Au(6×6) surface forms well ordered monoatomic layers [18,19,20], and is known to weakly interact with the Si substrate [21]. Thus, in the case of Si(335)-Au/Pb structure, one can expect a more pronounced one-dimensional character of chains as compared to clean Si(335)-Au reconstruction. Indeed, the deposition of 0.28 ML of Pb on reconstructed Si(335) surface (0.28 ML gives exactly 1 Pb atom per Si(335) unit cell) also leads to one-dimensional objects on the surface. The STM topography shows a few nanometers long chains placed in between Si chains of original Si(335)-Au surface [14]. The photoemission spectra taken in the [110] direction (i.e. parallel to the steps), show a highly dispersive band crossing the Fermi energy and quite flat bands in the [112] direction (perpendicular to the steps), thus indicating clear one-dimensional character of the reconstruction. A more detailed analysis of morphology of the surface and of the ARPES data is difficult or even impossible without appealing to DFT calculations. It is the purpose of the present work to propose a structural model of the Si(335)-Au/Pb surface and calculate corresponding band structure.
Here I present a structural model of the Pb chains on Si(335)-Au surface, derived from total energy DFT calculations. It features single Pb atom per Si(335) unit cell placed near the Au chain. The Pb atoms are bonded to neighboring Si and Au atoms, forming monoatomic chain located ∼ 0.2 nm above the surface. This picture is consistent with the STM topography data of Ref. [14], and also confirmed by simulated STM topography images. On the other hand, the calculated band structure for the present model clearly shows one-dimensional character, i.e. a strong dispersion in the direction parallel to the steps and their lack in the direction perpendicular to them, and remains in good agreement with the ARPES spectra of Ref. [14]. The rest of the paper is organized as follows. In Sec. II the details of calculations are presented. The structural and electronic properties of the clean Si(335)-Au surface are briefly discussed in Sec. III. The structural model of Pb chains, simulated STM topography images and the band structure are presented and discussed in Sec. IV, V, VI, respectively. The influence of step-edge buckling on electronic properties is discussed in Sec. VII. Finally, Sec. VIII contains some conclusions.
II. DETAILS OF CALCULATIONS
The calculations have been performed using standard pseudopotential density functional theory and linear combination of numerical atomic orbitals as a basis set, as implemented in the SIESTA code [22,23,24,25,26]. The local density approximation (LDA) to DFT [27], and Troullier-Martins norm-conserving pseudopotentials [28] have been used. In the case of Pb and Au pseudopotentials, the semicore 5d states were included. A double-ζ polarized (DZP) basis set was used for all the atomic species [23,24]. The DZP utilizes two radial functions for each angular momentum and additional polarization shell. The radii of the orbitals for different species were following (in Bohrs): Si -5.13 (3s), 6.59 (3p) and 5.96 (3d), Au -4.39 (5d), 6.24 (6s) and 5.79 (6p), Pb -3.56 (5d), 4.68 (5f ), 5.30 (6s) and 6.48 (6p), and H -5.08 (1s) and 4.48 (2p). A Brillouin zone sampling of 24 nonequivalent k points, and a real-space grid equivalent to a planewave cutoff 100 Ry (up to 82 k points and 300 Ry in the convergence tests) have been employed. This guarantees the convergence of the total energy within ∼ 2 meV per atom in the supercell.
The Si(335)-Au/Pb system has been modeled by four silicon double layers and a vacuum region of 18Å. All the atomic positions were relaxed except the bottom layer. The Si atoms in the bottom layer were fixed at their bulk ideal positions and saturated with hydrogen. To avoid artificial stresses, the lattice constant of Si was fixed at the calculated value, 5.41Å. The atomic positions were relaxed until the maximum force in any direction was less than 0.01 eV/Å.
III. CLEAN SI (335)-AU SURFACE
The deposition of 0.28 ML of gold on Si(335) surface forms well ordered arrays of chain structure. The surface consists of (111) terraces which have a width 3 2 3 × a [112] (1.26 nm) [7]. Each terrace contains a single row of gold atoms running parallel to the step edge, i.e. in the [110] direction. The gold chain is formed by substitution of Si atoms in the middle of terrace. The step edge Si atoms form a 'honeycomb' substructure [29,30,31], which is a common feature of all the Au-induced vicinal Si surfaces [8]. The structural model of the Si(335)-Au surface is shown in Fig. 1, where the Si surface atoms (Si 1 -Si 6 ) are labeled by numbers 1-6, and the gold atom by Au. Originally this model has been proposed as a sim- ple truncation of Si(557)-Au reconstruction [8], and later confirmed by DFT calculations [11]. The STM topography data [8,9] show a single chain per terrace, which is associated with the step edge Si atoms rather than with the Au chain. The gold substitutes some of top layer Si atoms in the middle of terrace, and is not visible to STM. This has also been confirmed by DFT calculations [11]. See also simulated STM topography images presented in Fig. 3. The ARPES spectra show two highly dispersive bands crossing the Fermi energy in the direction parallel to the steps, and quite flat bands in the direction perpendicular to them [8], clearly indicating one-dimensional metallic nature of the system. The DFT calculations indicate that one of the bands crossing E F is associated with the step edge Si atoms, having unsaturated dangling bonds, while the other one originates from hybridization of the Au and neighboring Si atoms in the middle of terrace in the surface layer [11].
IV. STRUCTURAL MODEL OF PB CHAINS ON SI(335)-AU
One-dimensional structures of Pb on Si(335)-Au reconstruction one obtains assuming single Pb atom per Si(335)-Au unit cell, which corresponds to the experimental Pb coverage -0.28 ML [14]. The total energy calculations show that Pb atoms prefer to adsorb on the surface. The substitution of Pb atoms into top Si layer is energetically less favorable, as the surface energy is by 0.3-0.6 eV (per unit cell) higher in comparison to clean Si(335)-Au surface and Pb atom in the bulk fcc structure. Similarly, the substitution into the second Si Si3
-Au-Au −0.41 2 Si1 (subst) −0.31 3 Si5-Si6-Si1 −0.25 4 Au-Si1-Si1 −0.06 5 Si4-Au −0.02
layer is not preferred. In this case the energy cost is 1 eV. The exception is the substitution of Pb at the step edge, where the energy gain is 0.31 eV. More than 40 structural models have been investigated, and only five of them lead to stable structures. Corresponding surface energies, with respect to clean Si(335)-Au surface and bulk Pb atom, are shown in Table I. The differences in energy are rather small, however, as it will be argued later, the model with the lowest energy, i.e. Si 3 -Au-Au model, is the best candidate for a true model of Pb chains on Si(335)-Au surface. Note the nomenclature used here reflects the bonding of lead with corresponding surface atoms. For labeling see Fig. 2. The structural model with the lowest surface energy features Pb atoms located 2Å above the surface near the gold chain, and is shown in bounded to two Au atoms and one Si 3 atom, but not to Si 4 atom (Fig. 2). The Pb-Si 3 bond length is equal 2.75 A, and Pb-Au -2.92Å. Both bonds are shorter than the Pb bonds in the bulk fcc structure (3.46Å). The presence of Pb atoms on-top of the surface also modifies positions of underneath Au and Si atoms, leading to change in their bond lengths. Thus the Au-Si 4 bond is equal 2.52Å, and Au-Si 5 -2.47Å, to compare with 2.43 A and 2.38Å, respectively, in the clean Si(335)-Au surface. What is more important, presence of Pb atoms also leads to a strong rebonding at the step edge. In the clean Si(335)-Au surface, the Si atoms near the step edge form a sort of honeycomb chain (see Fig. 1), which is common feature of all the vicinal Si surfaces [8]. Main feature of this substructure is a true double bond between Si 2 and Si 3 atoms, which is responsible for the stability of the honeycomb chain [29]. In the presence of Pb atoms on Si(335)-Au surface, there is no longer double bonds between Si 2 and Si 3 atoms. The atoms are just single bonded, as in common silicon structures. The Si 3 atom is now bonded with the lead, and Si 2 with underneath Si atom in the second layer (compare Fig. 1 and Fig. 2).
At this point I would like to comment on main structural features of the other models, listed in Table I, which have slightly higher relative surface energies. The next 'best' structural model features the step edge Si atoms substituted by lead (model 2 in Table I). Model 3, having relative surface energy -0.25 eV, accounts for Pb atom bonded to Si 5 and Si 6 atoms within the same terrace and Si 1 atom at the step edge of the neighboring terrace (see Fig. 2 for labeling). In model 4, with the energy -0.06 eV, Pb is located between two Si 1 atoms at the step edge and one gold atom on neighboring terrace, while in model 5 the Pb takes bridge position between Si 4 and Au atoms on the same terrace. However those models can be ruled out, according to arguments given below.
The present model, i.e. the model with the lowest relative surface energy, is a good candidate for structural model of Si(335)-Au/Pb reconstruction. There are few strong arguments supporting this model. First one is that the model has the lowest energy. The second one is that the model explains the STM data of Ref. [14]. In STM images of Si(335)-Au/Pb surface, the Pb chains are located in between the chain structure of original Si(335)-Au reconstruction (see Fig. 1 of Ref. [14]). According to the STM data, the Pb chains should be located in the middle of terraces, above the surface. This is exactly what the present model shows, and STM simulations, which will be presented in the next section, confirm this. The other models, listed in Table I remain in disagreement with the STM data. Model 1 leads to a single chain in STM topography, as the step edge Si atoms, which formed the monoatomic chains in clean Si(335)-Au surface are now replaced by the lead atoms. Similarly, models 2, 3 and 4 also lead to single chain within terrace, as the Pb atoms saturate the step edge Si dangling bonds. In fact, the STM topography of model 4 shows a zig-zag chain, associated with Pb atoms and less visible step edge Si atoms. Only the last model, i.e. model 5, remains in reasonably good agreement with STM topography data. The next argument concerns the band structure, as the only present model reproduces the photoemission spectra of Ref. [14] very well. The other models, which have slightly higher relative surface energies disagree with the ARPES data, in particular, they do not give correct band structure near the Fermi energy. This will be further discussed in Sec. VI. The next argument, albeit intuitive, is based on the analogy with the growth of Pb on flat Si(111) surface. It is well known, that lead grows in very regular fashion, i.e. layer by layer, on Si(111)-Au(6×6) surface from very beginning [18,19,20], contrary to the growth on Si(111) with 7×7 reconstruction, where it forms amorphous wetting layer [18,32,33]. This indicates that Pb prefers to bond with Au rather than with Si atoms. The present model of Si(335)-Au/Pb surface also reflects this fact, as the Pb atom is bonded with one Si and two Au atoms. The last argument comes from the conditions at which the experiment of Ref. [14] has been performed. Namely, the temperature of deposition of Pb on Si(335)-Au surface was 260 K. This temperature is far to low to substitute the Si atom by the lead, and the model with Si 1 atoms replaced by Pb can be ruled out. All the above arguments show that the model shown in Fig. 2 is very good candidate to be a true model of the Pb chains on Si(335)-Au surface.
V. STM SIMULATIONS
The STM topography data of clean Si(335)-Au surface shows one-dimensional structures which are interpreted as the step edge Si atoms [8,11]. The deposition of 0.28 ML of Pb leads to monoatomic Pb chains located in the middle of the Si(335)-Au terraces, i.e. between the Si chains of original Si(335)-Au surface. The structural model discussed in previous section supports this scenario. To further check the validity of the structural model, I have performed STM simulations within the Tersoff-Hamann approach [34]. The results of constant current topography for different bias voltages are shown in Fig. 3. Top panels represent simulated STM topography of 3 × 3 nm 2 of the same area of the Si(335)-Au/Pb surface for sample bias U = −1 V (a) and U = −0.1 V (b). For comparison, corresponding images of the same area of clean Si(335)-Au surface are shown in the bottom panels. As it was discussed previously (see Sec. III), the STM topography of clean Si(335)-Au surface features monoatomic chains, which are associated with the step edge Si atoms (see Fig. 3c) and d)). The Pb atoms deposited on this surface form monoatomic chains, located in the middle of terraces, i.e. between the chains of clean Si(335)-Au surface. This is evident, if one compares panels a) and c) or b) and d) of Fig. 3.
As one can read off from Fig. 3, the Pb atoms are more pronounced than the step edge Si atoms, especially at high sample bias. Such a behavior can be explained gies −0.7 eV and +1.5 eV (see solid line in Fig.4), while PDOS of the step edge Si atoms is larger near the Fermi energy (dashed line). In first approximation, the STM current is proportional to integrated density of states be-tween the Fermi energy and applied bias voltage (eU). This contributes to the fact that the Pb chain is more pronounced at higher sample bias. At very low sample bias (U = −0.1) V, the step edge Si atoms have larger values of PDOS, and as a result both chains have comparable topography amplitude (see panel b) of Fig. 3). However, the Pb chain is still more pronounced than the Si one due to the fact that it sticks out above the surface. The STM topography along the Pb chain shows very small changes of the amplitude at low voltages (see panel b) of Fig. 3), indicating strong overlapping of the 6p states. This is also reflected in the band structure, where 6p band of lead, crossing the Fermi energy, is very dispersive in the direction along the chain (see Fig. 4). On the other hand, there is no such an effect in the STM topography of the step edge Si atoms due to more localized character of 3p states of silicon. As a result the step edge Si band is rather flat, i.e. is less dispersive than 6p band of lead (see Fig. 4). All this shows that the Pb chain has more metallic character than the step edge Si chain, which makes the lead a good candidate to look for exotic phenomena characteristic for the systems of reduced dimensionality.
VI. BAND STRUCTURE
The calculated band structure for present structural model, along the high symmetry lines of two-dimensional Brillouin zone (BZ), is shown in Fig. 5. The direction in To be more precise, those bands originate from the hybridization of the Au chain with neighboring Si 4 and Si 5 atoms (see Fig. 2). All the bands discussed above have also been identified in the clean Si(335)-Au surface [11] and similar bands have been found in the Si(557)-Au reconstruction [35,36]. In fact, the band marked with filled squares has also Pb character. This band reflects the hybridization of gold with Pb and Si 5 atoms. Finally, the band shown as filled triangles is associated with 6p states of lead. All the above bands do not have (or have very weak) dispersion in the direction perpendicular to the steps (Γ-M), indicating one-dimensional character of the structure. Since Pb features a strong spin-orbit (SO) interaction [21,37], and the low-dimensionality of the system can increase it [38], it is worthwhile to comment on this effect in the Si(335)-Au/Pb system. Although the spin-orbit interaction was not included in present calculations, one can draw some conclusions appealing to the Si(557)-Au reconstruction. The measured band structure of Si(557)-Au surface shows two Au induced proximal bands crossing the E F near the K point (of 2 × 1) zone with ∼ 300 meV splitting [10,39,40,41]. An explanation of the splitting was given in terms of SO interaction [42]. Since both Si(335)-Au and Si(557)-Au surfaces belong to the same family of vicinal surfaces (Si(335)-Au reconstruction may be considered as a simple truncation of Si(557)-Au surface), one can expect SO interaction to play similar role here. In fact, Crain et al. [8] observed small splitting of the band crossing the E F . This was interpreted as two different bands, one coming from Au-Si hybridization and the other one from the step edge Si atoms [11]. It is also possible that the splitting has its origin in the SO interaction, similar like in Si(557)-Au surface. On the other hand, if Pb is deposited onto Si(335)-Au surface, one would expect SO effect to be more pronounced. However, the ARPES data [14] shows a single metallic band and thus no evidence of SO splitting. So one can conclude that if SO effect is really present in this system, certainly the splitting of the band is smaller than the energy resolution of the ARPES apparatus, which is 50 meV in this case. This is rather an intriguing result, however it can be true, as recent spin-polarized photoemission measurements report values of SO splitting as small as 15 meV in Pb deposited on Si(111) surface [43].
A comparison of the calculated band structure with the photoemission spectra of Ref. [14] is shown in Fig. 6. As one can see, the present DFT calculations agree very well with the experimental data. In particular, the band near the Fermi energy, which has high photoemission intensity, is the 6p band of lead. One can also identify other bands, marked with symbols in Fig. 5, except the band coming from the step edge Si atoms. The reason that this band is not observed in ARPES can be twofold. First, the band is to close to the Fermi energy which makes it very difficult to verify experimentally. Second, there may be some mechanism which leads to the saturation of the Si bonds, accounting for various imperfections or impurities. It is more likely that the band associated with the unsaturated step edge bonds (open circles in Fig. 5) is eliminated in the real surface by some reconstruction and passivation. The main effects of the reconstruction can be achieved by saturating the dangling bonds with hydrogen [35]. This process removes extra band from the gap region (open circles in Fig. 5), and is energetically very favorable, as the energy gain is ∼ 1.30 eV with respect to the unsaturated surface and H 2 molecule. The comparison of the calculated band structure with saturated Si 1 bonds and the ARPES spectra of Ref. [14] is shown in Fig. 7. Clearly, the band structure is in bet- ter agreement with the ARPES experiment now. This suggests that the step edge Si bonds are really saturated by some surface reconstruction, however this cannot be accounted for without multiplying the unit cell of the Si(335)-Au surface.
VII. STEP-EDGE BUCKLING
The Si(557)-Au reconstruction is known to undergo a buckling of the step edge at low temperatures [36,42,44]. The step edge Si atoms alternate between up and down positions, with distortion in z direction ∆z = 0.65Å. In general, all the step edge atoms occupy equivalent positions and feature half-filled dangling bonds, which give rise to a flat band near the Fermi energy. This is referred as high-temperature phase. However this situation is energetically unfavorable at low temperature and the system tends to lower its energy by the buckling of the step edge. The energy gain is 130.9 meV. The dangling bonds of the up-edge Si atoms became fully occupied, while those of the down-edge are empty. As a result the band structure features the fully occupied and empty step-edge electron bands. The buckling of the step edge influences the properties of the band structure near the Fermi energy through changes in the Au-Si-Au bond angles. Those changes are responsible for the band gap that opens in dispersive Au-Si band crossing the E F , and thus for the metal-insulator transition at low temperature [44].
Doubling the unit cell of Si(335)-Au/Pb system in direction [110] one can account for similar effect associated with the buckling of the step edge. In the present system the distortion ∆z is slightly smaller than in Si(557)-Au system, and equal to 0.57Å. This smaller value of the ∆z can be related to the fact that Si(557)-Au supercell cell consists of two non-equivalent ×1 unit cells with extra Si adatom in one of them, which in turn can increase ∆z. The ×2 Si(335)-Au/Pb supercell is constructed from two equivalent ×1 cells. The buckling of the step edge lowers the total energy of the system by 117.5 meV, to be compared with 130.9 meV in the case of Si(557)-Au surface, and leads to a splitting of the step edge Si band, originally pinning the Fermi energy. As a result one observes two bands, a fully occupied flat band (with a bandwidth W ≈ 0.09 eV) at energy 0.29 eV below E F , and empty more dispersive band (W ≈ 0.38 eV) at energy 0.25 eV above the E F . Those bands are associated with up-edge and down-edge Si atoms, respectively. Similar bands have also been observed in the Si(557)-Au surface [44]. The step edge buckling is also reflected in different bond angles between Au and Si 5 atoms (see Fig. 2). In situation when all the step edge Si atoms occupy equivalent positions (∆z = 0), these angles are equal 101.64 • , to be compared with 111.6 • and 103.7 • for Si(557)-Au surface. Again, different values of the Si(557)-Au angles come from non-equivalent ×1 unit cells, and the buckling can change them by ±10 • [44]. On the other hand, the buckling in the present system changes the Si 5 -Au-Si 5 angles to 101.75 • (up-edge) and 101.29 • (down-edge ×1 cell). Moreover, the angles between Au and Pb atoms are also slightly changed, and equal 81.78 • (81.54 • ) in up-edge (down-edge) ×1 cells, to be compared with equilibrium value 81.97 • . All this is reflected in the band structure, where very small energy gap develops in dispersive Au-Pb-Si band (filled squares in Fig. 5), when the step-edge buckling takes place. However, this small energy gap is slightly below the Fermi level. Similar effect, although with slightly larger energy gap, has been found in Si(557)-Au surface [44], and has been assigned to the shortcomings of the LDA functionals, which fail to describe excitation spectra. So one can expect that better description of the exchange and correlation effects (like GW approximation) will move the empty states to higher energies, thus driving the system into insulating state [45,46]. However, it is also possible, that present LDA results are correct, and the system stays in metallic phase. The step-edge buckling may not be able to open a true gap in Au-Pb-Si band, as it gives much smaller values of Au-Si angles than in case of the Si(557)-Au surface. This last scenario seems to be supported by experimental data of Ref. [14], shown in Fig. 8 together with calculated surface band structure for the step-edge buckled Si(335)-Au/Pb reconstruction. As one can see the agree- [14]) and the calculated surface band structure for the model with the buckling of the step edge. ment between measured and calculated band structure is very good. In particular there is no band pinned at the Fermi energy.
To calculate the surface band structure shown in Fig. 8, following procedure has been applied. Since the bulk periodicity in direction [110] is 1×a [110] , and calculations have been performed in ×2 unit cell in this direction, the resulting band structure features twice as many bands as in ×1 unit cell, folded back into first BZ of ×2 structure. This of course is not observed in experiment, as the ARPES is surface sensitive technique. To get rid of these bulk electron bands, only the surface atoms have been taken into account in calculations. The positions of these atoms have been obtained in full slab calculations, as described in Sec. II. To mimic rest of the Si layers, the surface Si atoms were passivated by hydrogen. The H atoms were placed in a way that the bond lengths and bond angles between surface and next from the surface Si layers were preserved. In this way calculated band structure is shown in Fig. 8. Note that differences between in this way calculated surface band structure and that obtained from full slab calculations are negligible.
VIII. CONCLUSIONS
In conclusion, the structural and the electronic properties of the monoatomic Pb chains on Si(335)-Au surface have been studied within the density functional theory. The obtained structural model features Pb chain on-top of the surface in the middle of terrace. As a result, two monoatomic chains are observed on single terrace, one made of lead, and the other one associated with the step edge Si atoms. Both chains are visible in STM experimental data. The STM topography is very well reproduced within the present model. The calculated band structure shows clear one-dimensional character of the structure and remains in almost perfect agreement with the ARPES data. Finally, the buckling of the step edge Si atoms has been found, and unlike for the Si(557)-Au surface, does not drive the system into insulating phase.
FIG. 1 :
1(Color online) Structural model of Si(335)-Au surface. Top panel shows side view of the structure, and bottom panel shows top view with marked surface unit cell. Labels 1-6 stand for silicon surface atoms (Si1-Si6), while Au denotes gold atom. The dashed lines in bottom panel indicate step edges.
Fig. 2 .FIG
2As one can read off from the figure, . 2: (Color online) Structural model of Si(335)-Au/Pb surface. Top panel -side view, and bottom panel -top view of the model. Labels 1-6 stand for silicon surface atoms (Si1-Si6), while Au denotes gold and Pb -lead atom. Surface unit cell is also indicated in the bottom panel.
FIG. 3 :FIG. 4 :
34(Color online) STM simulations of 3 × 3 nm 2 area of Si(335)-Au/Pb surface (top panels) and Si(335)-Au (bottom panels) for sample bias U = −1 V (a and c) and U = −0.1 V (b and d). All the images show the same area of the surface.by the combination of structural end electronic effects. The Pb chain sticks out (by 2Å ) above the surface, and this mainly contribute to the discussed effect. On the other hand, the electronic properties also play significant role. The calculated projected density of states (PDOS) of the Pb atoms and the step edge Si atoms is shown inFig. 4. The Pb PDOS features larger values at ener-(Color online) Projected density of states of the Pb (solid line) and the step edge Si atom (dashed line).
FIG. 5 :
5(Color online) Calculated band structure along high symmetry lines in two-dimensional Brillouin zone (shown on the right). The [110] direction is parallel to the steps, while [112] -perpendicular to them. The bands associated with Pb chain is marked with filled triangles, the step edge Si atom band -open circles, while the bands originating from the Au chain hybridizing with Si4 and Si5 atoms are marked with squares. The Fermi level is taken as the zero of energy. the 2D Brillouin zone, defined by the points Γ, K and M', is parallel to the steps of the Si(335)-Au surface, while the Γ-M is perpendicular to them (see right panel of Fig. 5). A few important surface bands are marked with symbols in Fig. 5. The band marked with open circles pinning the Fermi energy, originates from unsaturated bonds of the Si atoms at the step edge. crossing the E F (filled squares) are associated with the Au chain.
FIG. 6 :
6(Color online) Comparison of the measured ARPES intensity along Γ-K-M' line of the 2D Brillouin zone (Ref.[14]) and the calculated band structure. The high intensity is shown light.
FIG. 7 :
7(Color online) Comparison of the measured ARPES intensity (Ref.[14]) and the calculated band structure for the model with saturated step edge Si bonds.
FIG. 8 :
8(Color online) Comparison of the measured ARPES intensity (Ref.
TABLE I :
IThe relative surface energies of most stable structural models of Si(335)-Au/Pb structure. The energies are referred to clean Si(335)-Au surface and Pb atom in the bulk fcc structure. model position of Pb surface energy (eV)1
AcknowledgmentsI would like to thank Prof. M. Ja lochowski for valuable discussions. This work has been supported by the Polish Ministry of Education and Science under Grant No. N202 081 31/0372.
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| []
|
[
"Phase Transitions in Neutron Stars and Gravitational Wave Emission",
"Phase Transitions in Neutron Stars and Gravitational Wave Emission"
]
| [
"G F Marranghello \nInstituto de Física\nObservatoire de la Côte d'Azur\nUniversidade Federal do Rio Grande do Sul\nPorto Alegre, NiceBrazil., France\n",
"Cesar A Z Vasconcellos \nInstituto de Física\nObservatoire de la Côte d'Azur\nUniversidade Federal do Rio Grande do Sul\nPorto Alegre, NiceBrazil., France\n",
"J A De Freitas Pacheco \nInstituto de Física\nObservatoire de la Côte d'Azur\nUniversidade Federal do Rio Grande do Sul\nPorto Alegre, NiceBrazil., France\n"
]
| [
"Instituto de Física\nObservatoire de la Côte d'Azur\nUniversidade Federal do Rio Grande do Sul\nPorto Alegre, NiceBrazil., France",
"Instituto de Física\nObservatoire de la Côte d'Azur\nUniversidade Federal do Rio Grande do Sul\nPorto Alegre, NiceBrazil., France",
"Instituto de Física\nObservatoire de la Côte d'Azur\nUniversidade Federal do Rio Grande do Sul\nPorto Alegre, NiceBrazil., France"
]
| []
| We review the detectability of gravitational waves generated by oscillations excited during a phase transition from hadronic matter to deconfined quark-gluon matter in the core of a neutron star. Neutron star properties were computed using a Boguta and Bodmer's based model and the MIT bag model. The maximum energy available to excite mechanical oscillations into the star is estimated by energy difference between the configurations with and without a quark-gluon matter core. On basis of the planned sensitivity of present laser interferometers (VIRGO or LIGO I) and those of the next generation (LIGO II), the maximum volume to be probed by these experiments is determined. These results are used as an indication of the potential detectability of neutron stars as sources of gravitational waves. Our results indicate that the maximum distance probed by the detectors of the first generation is well beyond M31, whereas the second generation detectors will probably see phase transition events at distances two times longer, but certainly not yet attaining the Virgo cluster. | 10.1103/physrevd.66.064027 | [
"https://arxiv.org/pdf/astro-ph/0208456v1.pdf"
]
| 37,259,125 | astro-ph/0208456 | 7a5f8e6369858ecc40411b067fff3b43815449b8 |
Phase Transitions in Neutron Stars and Gravitational Wave Emission
26 Aug 2002
G F Marranghello
Instituto de Física
Observatoire de la Côte d'Azur
Universidade Federal do Rio Grande do Sul
Porto Alegre, NiceBrazil., France
Cesar A Z Vasconcellos
Instituto de Física
Observatoire de la Côte d'Azur
Universidade Federal do Rio Grande do Sul
Porto Alegre, NiceBrazil., France
J A De Freitas Pacheco
Instituto de Física
Observatoire de la Côte d'Azur
Universidade Federal do Rio Grande do Sul
Porto Alegre, NiceBrazil., France
Phase Transitions in Neutron Stars and Gravitational Wave Emission
26 Aug 2002(Dated: October 24, 2018)numbers: 0430Db1239Ba2660+c2160-n
We review the detectability of gravitational waves generated by oscillations excited during a phase transition from hadronic matter to deconfined quark-gluon matter in the core of a neutron star. Neutron star properties were computed using a Boguta and Bodmer's based model and the MIT bag model. The maximum energy available to excite mechanical oscillations into the star is estimated by energy difference between the configurations with and without a quark-gluon matter core. On basis of the planned sensitivity of present laser interferometers (VIRGO or LIGO I) and those of the next generation (LIGO II), the maximum volume to be probed by these experiments is determined. These results are used as an indication of the potential detectability of neutron stars as sources of gravitational waves. Our results indicate that the maximum distance probed by the detectors of the first generation is well beyond M31, whereas the second generation detectors will probably see phase transition events at distances two times longer, but certainly not yet attaining the Virgo cluster.
I. INTRODUCTION
The first generation of large gravitational interferometric detectors as the French-Italian VIRGO and the American LIGO, should be fully operational within one or two years. The best signal-to-noise (S/N) ratio that can be achieved from these detectors implies the use of matchedfilter techniques, that require a priori the knowledge of the signal waveform. Thus, the identification of possible sources having a well defined signal is a relevant problem in the detection strategy.
Neutron stars are certainly one of the most popular potential sources of gravitational waves (GWs), since they can emit by different mechanisms having a well known waveform. Rotating neutron stars may have a timevarying quadrupole moment and hence radiate GWs, by either having a triaxial shape or a misalignment between the symmetry and the spin axes, which produces a precessional motion [1,2,3]. Moreover, fast rotating protoneutron stars may develop different instabilities such as the so-called Chandrasekhar-Friedman-Schutz (CFS) instability [4,5], responsible for the excitation of density waves traveling around the star in the sense opposite to its rotation, or undergo a transition from axi-symmetric to triaxial shapes through the dynamical "bar-mode" instability [6]. All these mechanisms are potentially able to generate large amounts of energy in the form of GWs. Nonradial oscillations are also a possible mechanism for neutron stars emit GWs and this possibility was al- * Electronic address: [email protected]; † Electronic address: [email protected] ‡ Electronic address: [email protected] ready discussed in the late sixties [7]. One of the difficulties with this mechanism concerns the energy source necessary to excite the oscillations. The elastic energy stored in the crust and released by tectonic activity was recently considered [8], but the maximum available energy is likely to be of the order of 10 44−45 erg. Thus, even if all this energy could be converted into nonradial modes, the maximum distance that a signal could be seen by a laser interferometer like VIRGO is only about 3.0 kpc [8]. This result depends, of course, on the adopted equation of state, which fixes the mode frequencies and damping timescales. A considerable amount of energy would be available if the neutron star undergo a phase transition in the core. For instance, if quark deconfinement occurs, then the star will suffer a micro-collapse since the equation of state of the quark matter is softer than that of the hadronic matter and the new equilibrium configuration will be more compact, having a larger binding energy. The energy difference will partially cover the cost of the phase transition and will partially be used to excite mechanical modes, whose energy will be dissipated by different channels. The physical conditions of matter at densities above the saturation density (ρ 0 = 0.15f m −3 ) is badly known and many first or second order phase transitions have been speculated upon. The possibility of a quark and hadron mixed phase was considered in ref. [9], where the conservation of the electric and the baryonic charges are satisfied not locally but within a volume including many droplets of quarks and of hadrons. This mixed phase does not include local surface and Coulomb energies of quarks and nuclear matter and, as a consequence of these terms, if the interface tension between quark and nuclear matter is too large, the mixed phase is not favored energetically [10,11]. In this case, the neutron star will then have a core of pure quark matter and a mantle of hadrons surrounding it. In the absence of dynamical models able to follow the evolution of the star during the phase transition, we assume as in ref. [12], that the structural rearrangement suffered by a "cold" star occurs on a dynamical timescale, i.e., of the order of milliseconds, shorter than the smooth global stellar readjustment suggested in refs. [13,14] and the gradual transition expected to occur in hot proto-neutron stars [15]. Radial modes excited by the micro-collapse do not radiate GWs, but an important coupling with rotation exists [16] and, if the star rotates, the oscillations will be damped not only by dissipation of the mechanical energy into heat but also by the emission of GWs.
In the present work, the detectability of GWs generated by oscillations excited during a phase transition in the core of a neutron star is reviewed [17]. Neutron star properties were computed using a description of the hadronic matter based on the work of Boguta and Bodmer [18], including the fundamental baryon octet, the isovector meson ̺ and lepton degrees of freedom (see references [15] for details). Hybrid models, including a quark-gluon core, with the same baryonic number as the pure hadron configuration were also computed and the energy difference between both states was used as an estimate of the maximum energy available to either excite mechanical oscillations or to be converted into heat. Then, using the planned sensitivity of present laser interferometers like VIRGO (or LIGO I) and those of the next generation (LIGO II), the maximum volume of space that can be probed by these experiments is calculated, since this indicates the potential detectability of these sources. The plan of this paper is the following: in section 2 the star models are reviewed, in section 3 the gravitational wave emission is discussed and finally, in section 4, conclusions are given.
II. THE NEUTRON STAR MODEL
The present model for the nuclear matter was already discussed by Marranghello et al. [19], who have investigated the effects of a finite temperature in the equation of state. Here, the same description for dense matter is adopted but only models with T=0 will be considered. Moreover, the coupling constants are slightly modified with respect to those considered in that work, in order to allow the maximum mass of the configuration to be compatible with recent results on the binary X-ray system Vela X-1 [20]. For the sake of completeness, we recall here the main points of the model.
The lagrangian density describing the nuclear matter is
L = Bψ B [(iγ µ (∂ µ − g ωB ω µ ) − (M B − g σB σ)]ψ B − Bψ B [ 1 2 g ̺B τ · ̺ µ ]ψ B + bM 3 σ 3 + c 4 σ 4 + 1 2 (∂ µ σ∂ µ σ − m 2 σ σ 2 ) − 1 4 ω µν ω µν + 1 2 m 2 ω ω µ ω µ − 1 4 ̺ µν · ̺ µν + 1 2 m 2 ̺ ̺ µ · ̺ µ + lψ l [iγ µ ∂ µ − M l ]ψ l .(1)
This equation represents nuclear matter as composed by a mixture of the fundamental baryon octet (p, n, Λ, Σ + , Σ 0 , Σ − , Ξ − , Ξ 0 ) coupled to three mesons (σ, ω, ̺) and leptons (for the details see [15]). The scalar and vector coupling constants, g σ , g ω and the coefficients b, c were determined by imposing that the model bulk properties should be able to reproduce the binding energy E b (= -16.3 MeV), the compression modulus K (= 240 MeV) and the nucleon effective mass M * = M − g σσ (= 732 MeV) at the saturation density ρ 0 (=0.153 fm −3 ). Additionally, the isovector coupling constant g ̺ is determined from the coefficient for the symmetry energy in nuclear matter, a 4 (= 32.5 MeV). We have used the universal hyperon-nucleon coupling ratios χ i = g Hi /g i , with i = σ, ̺, ω. The resulting coefficients used in our computations, are given in table 1. Figure 1 shows the equation of state derived from our model and in figure 2 it is shown the energy density profile inside the star for two configurations having gravitational masses equal to 1.2 and 1.6 M ⊙ respectively. Hybrid models including a quark-gluon core were also computed. The quark matter was described by the MIT figure 3 for the case of a pure hadronic configuration (dashed line), a hybrid star with a quark core (solid line) and, for comparison, a case where a mixed phase exist in the center, compute in the same way as ref. [9].
It should be emphasized that our equation of state is quite steep ( dlogP dlog̺ ≈ 2.6) near saturation and, as a consequence, the deconfinement transition occurs at densities just above the saturation value (ρ ∼ 1.7ρ 0 ), producing hybrid stars with very extended quark-gluon cores. Here we take the opportunity to reiterate that, from the actual status of theoretical predictions to the EOS, with so many parameters to be adjusted, even in the quark phase and specially in the hadron phase, we believe that only qualitative results can be obtained with some insights on the quantitative results. The maximum stable mass of a pure hadronic configuration is about M max ≈ 2.1 M ⊙ while for hybrid stars this limit is reduced to 1.73 M ⊙ . Thus, the present calculations exclude the possibility that Vela X-1 has a quark core. Notice that our models obey the Seidov criterium [22], namely, that the hybrid star will be stable only if the energy jump across the transition surface satisfies the condition
ǫ q ǫ H < 3 2 (1 + P ǫ H )(2)
where P, e q , e H are respectively the pressure, the energy density of quarks and hadrons at the transition point.
One of the goals of this work is the determination of an upper limit for the energy able to excite the different oscillation modes of the star, when a modification in its internal structure occurs. In this sense, the simplest approach is to compute the energy difference between two configurations having the same baryonic number; the first constituted of pure hadrons (H), the second having a core of deconfined matter (HQ), and use such a difference as an indication of the maximum available energy. Table 3 gives the parameters for five models defined by a given baryonic number [23]
N = R 0 4πr 2 1 − 2Gm(r) r 2 −1/2 ρ B (r)dr ,(3)
or the baryonic mass of the star (M bar = N M B ); the expected gravitational mass of both configurations (pure hadron and hybrid),
M g = m(R) = R 0 4πr 2 ǫ(r)dr ;(4)
expected radii, masses of the quark-gluon core and the energy difference obtained from the passage of configuration H to HQ, the binding energy.
E g = M g − M bar .(5)
III. THE GRAVITATIONAL WAVE EMISSION
The transition from configuration H to HQ may occur through the formation of a mestastable core, built up by an increasing central density. The increase in the central density may be a consequence of a continuous spindown or other different mechanisms the star could suffer. This transition releases energy, exciting mainly the radial modes of the star [17]. These modes do not emit GWs, unless when coupled with rotation [16], a situation which will be assumed here.
In order to simplify our analysis, we will consider that most of the mechanical energy is in the fundamental model. In this case, the gravitational strain amplitude can be written as
h(t) = h 0 e −(t/τgw−ıω0t)(6)
where h 0 is the initial amplitude, ω 0 is the angular frequency of the mode and τ gw is the corresponding damping timescale. The initial amplitude is related to the total energy E g dissipated under the form of GWs by the relation [24] h 0 = 4 ω 0 r GE g τ gw c 3
1/2 (7)
where G is the gravitational constant, c is the velocity of light and r is the distance to the source. Relativistic calculations of radial oscillations of neutron star with a quark core were recently performed by Sahu et al. [25]. However, the relativistic models computed by those authors do not have a surface of discontinuity where an energy jump occurs. Instead a mixing region was considered, where the charges (electric and baryonic) are conserved globally but not locally [9]. Oscillations of star models including an abrupt transition between the mantle and the core were considered by Haensel [26] and Miniutti [27]. However a Newtonian treatment was adopted and the equation of state used in the calculations does not correspond to any specific nuclear interaction model. In spite of these simplifications, these hybrid models suggest that rapid phase transitions, as that resulting from the formation of a pion condensate, proceed at the rate of strong interactions and affect substantially the mode frequencies. However, the situation is quite different for slow phase transitions (the present case), where the mode frequencies are quite similar to those of a one-phase star [26]. In this case, scaling the results of [26], the frequency of the fundamental mode (uncorrected for gravitational redshift) is given approximately by
ν 0 ≈ 63.8 (M/M ⊙ ) R 3 1/2 kHz (8)
where the mass is given in solar units and the radius in km.
Once the transition to quark-gluon matter occurs, the weak interaction processes for the quarks u, d and s
u + s → d + u(9)
and
d + u → u + s(10)
will take place. Since these reactions are relatively slow, they are not balanced while the oscillations last and thus, they dissipate mechanical energy into heat [28]. According to calculations of reference [21], the dissipation timescale can be estimated by the relation
τ d ≈ 0.01 150M eV m s 4 M ⊙ M c s(11)
where m s is the mass of the s-quark in MeV and M c is the mass of the deconfined core in solar masses. This equation is valid for temperatures in the range 10 8 − 10 9 K. On the other hand, according to reference [16] the damping timescale by GW emission is
τ gw = 1.8 M ⊙ M P 4 ms R 2 s(12)
where again the stellar mass is in solar units, the radius is in km and the rotation period P is in milliseconds. In a first approximation, the fraction of the mechanical energy which will be dissipated under the form of GWs is f g = 1 (1+τgw/τ d ) . Notice that the damping timescale by GW emission depends strongly on the rotation period. Therefore one should expect that slow rotators will dissipate mostly of the mechanical energy into heat. In table 4 is given for each star model the expected frequency of the fundamental mode (corrected for the gravitational redshift) , the critical rotation period (in ms) for having f g = 0.50, the GW damping for this critical period and the quality factor of the oscillation, Q =πν 0 τ gw .
After filtering the signal, the expected signal-to-noise ratio is
(S/N ) 2 = 4 ∞ 0 |h(ν) | S n (ν) dν(13)
whereh(ν) is the Fourier transform of the signal and S n (ν) is the noise power spectrum of the detector. Performing the required calculations, the S/N ratio can be written as In the equation above, the angle average on the beam factors of the detector were already performed [29]. From eqs. ( 4) and (11), once the energy and the S/N ratio are fixed, one can estimate the maximum distance D max to the source probed by the detector. In the last two columns of table 4 are given distances D max derived for a signal-to-noise ratio S/N = 2.0 and the sensitivity curve of the laser beam interferometers VIRGO and LIGO II. In both cases, it was assumed that neutron stars underwent the transition having a rotation period equal to the critical value.
(S/N ) 2 = 4 5 h 2 0 τ gw S n (ν gw ) Q 2 1 + 4Q 2(14)
We emphasize again that our calculations are based on the assumption that the deconfinement transition occurs in a dynamical timescale [12]. In the scenario developed in ref. [13], a mixed quark-hadron phase appears and the complete deconfinement of the core occurs according to a sequence of quasi-equilibrium states. The star contracts slowly, decreasing its inertia moment and increasing the its angular velocity until the final state be reached in a timescale of the order of 10 5 yr [13]. Clearly, in this scenario no gravitational waves will be emitted and this could be a possibility to discriminate both evolutionary paths.
IV. CONCLUSIONS
The structure of pure hadronic configurations and that of hybrid stars having a quark core, with the same bary-onic number, were computed using a new equation of state [19]. The maximum stable mass for pure hadronic configurations satisfies the requirement of being higher than the mass of Vela X-1 ( M = 1.86±0.16 M ⊙ ), determined recently [20] from a new study of its orbital motion.
As one should expect, the masses of quark cores increase with the baryon number of the configuration, as well as the difference between the binding energies between hybrid and single phase objects. On the basis of this result and from the point of view of the GW emission, it would be natural to expect that phase transitions occurring in more massive stars would be more easy to detect. In fact, this is not the case because, in the one hand the damping timescale due to the emission of GWs is inversely proportional to the mass and hence massive stars have low oscillation quality factors and, on the other hand the detectors have sensitivities optimized at frequencies around 200 Hz. In comparison with non-radial oscillations (for instance m= ℓ = 2 modes), these have comparable frequencies but damping timescales one order of magnitude higher than those resulting from radial modes coupled to rotation. In this case, higher quality factors Q may be obtained but most of the mechanical energy will be dissipated into heat. In order to be an efficient source of GWs by the mechanism here considered, the star must be a fast rotator, i.e., to have rotation periods of the order of few milliseconds. Even in the most favorable case (model 2, corresponding to a baryonic mass equal to 1.32 M ⊙ ), if the rotation period is 4.0 ms, only 1% of the available energy will be emitted as GWs.
Inspection of table 4 indicates that the maximum distance probed by detectors of first generation (VIRGO, LIGO I) is about 6.4 Mpc, well beyond M31, whereas the second generation (LIGO II) will probably see phase transition events at distances two times longer, but certainly not yet attaining the Virgo cluster. The small probed volume and the rapid rotation required for this mechanism be efficient imply in a low event rate, imposing severe limitations on the detectability of such a signal.
FIG. 1 :
1Equation of state for hadronic matter (solid line) and for the quark-gluon matter (dotted line).
FIG. 2 :
2Neutron star energy density-radius relation for M=1.6M⊙ (solid line) and M=1.2M⊙ (dotted line) on which the quark-gluon core extends up to 8.6 km and 7 km, respectively. FIG. 3: Neutron star mass as a function of central energy density for hybrid star with constant pressure transition (solid line), hybrid (dotted line) and pure hadronis star (dashed line). configurations Lagrangian[21] and the physical conditions at the deconfinement transition were estimated from the Gibbs criteria, namely, by the equality of the chemical potential and pressure of both phases, under conservation of the baryon number and electrical charge. The physical parameters at the transition point are given in table 2, for a bag constant B 1/4 = 180M eV and a strange quark mass m s = 150 MeV. The gravitational mass of the star as a function of the central energy density is shown in
TABLE I :
I(gσ/mσ) 2
(gω/mω) 2
(g̺/m̺) 2
b(×100)
c(×100)
9.927
4.820
4.791
0.8659
-0.2421
TABLE II :
IIPhysical conditions at the phase transition: energy densities and pressure are given in GeV fm−3 ǫH
ǫq
P
0.236 0.349 0.0163
TABLE III :
IIIParameters of the stellar models: masses are in solar units and radii in km; H and HQ mean pure hadron and hybrid configurations respectivelyM bar Mg (H) Mg (HQ) Mg (core) R (H) R(HQ) log (E) erg
1.0749 1.0003 0.9625
0.1150
12.82 12.47 52.83
1.3202 1.2009 1.0953
0.3608
13.04 12.16 53.28
1.5724 1.4008 1.2468
0.6013
13.13 11.98 53.44
1.8342 1.6006 1.4164
0.8626
13.01 11.82 53.52
2.1045 1.8002 1.6141
1.1766
12.69 11.50 53.52
TABLE IV :
IVOscillation parameters: the damping timescale
τgw is given for the critical period; maximum distances for
VIRGO (V) and LIGO II (L) are in Mpc
ν0
Pcrit τgw Q Dmax
Dmax
(kHz) (ms) (ms) -
(VIRGO) (LIGO II)
1.62 1.64 87.0 442 4.9
10.2
1.83 1.25 27.0 155 6.4
13.5
2.06 1.13 17.0 110 6.0
12.8
2.32 1.06 11.5 84 5.1
11.1
2.72 1.00 8.4 72 3.6
5.7
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[]
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"S A Barannikov ",
"François Laudenbach "
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| Given a Morse function f on a closed manifold M with distinct critical values, and given a field F, there is a canonical complex, called the Morse-Barannikov complex, which is equivalent to any Morse complex associated with f and whose form is simple. In particular, the homology of M with coefficients in F is immediately readable on this complex. The bifurcation theory of this complex in a generic one-parameter family of functions will be investigated. Applications to the boundary manifolds will be given. | null | [
"https://arxiv.org/pdf/1509.03490v2.pdf"
]
| 89,614,080 | 1509.03490 | 014e1e4454168bb7d6396b1b4e607bbf5d377526 |
S A Barannikov
François Laudenbach On an article by
Given a Morse function f on a closed manifold M with distinct critical values, and given a field F, there is a canonical complex, called the Morse-Barannikov complex, which is equivalent to any Morse complex associated with f and whose form is simple. In particular, the homology of M with coefficients in F is immediately readable on this complex. The bifurcation theory of this complex in a generic one-parameter family of functions will be investigated. Applications to the boundary manifolds will be given.
The Morse-Barannikov complex
We adopt a presentation which is a mix of the presentation given by S. Barannikov in [1] and a more abstract one given by C. Viterbo in his joint work with D. Le Peutrec and F. Nier [7] which we slightly simplify.
We are given a closed manifold M , a Morse function f : M → R whose critical values are distinct, and a field F. For each integer k the critical points of index k are numbered in the increasing order of the critical values: f (p 1 ) < f (p 2 ) < . . . (the function is just generic and it is not assumed to be ordered). We shall often identify the set of critical points and the set of critical values.
For defining the Morse complex it is necessary to have two extra data: -A (decreasing) pseudo-gradient, that is, a vector field on M which satifies X · f < 0 out of the critical points, and some non-degeneracy condition for the vanishing of X at each critical point; therefore, the zeroes of X are hyperbolic. As a consequence, each critical point p has a stable manifold W s (p) and an unstable manifold W u (p). This pseudogradient is chosen Morse-Smale, a generic property meaning that the stable manifolds are transverse to the unstable manifolds. -An orientation of the unstable manifolds.
The Morse complex C * (f, X) is made as follows. In degree k the module C k (f, X) is the free Z-module generated by the critical points of index k and the differential ∂ k : C k (f, X) → C k−1 (f, X) counts the signed number of connecting orbits. Observe that, for (p, q), a pair of critical points of respective index k and k − 1, W u (p) ∩ W s (q) is made of a finite number of connecting orbits. Since W s (q) is co-oriented by the orientation of W u (q), each orbit descending from p to q gets a sign. Define a q p to be the signed number of the connecting orbits and define the Morse differential C k (f, X) by ∂ k (p) = a q p q .
Definition 1.2. A chain complex C * with coeffcients in F is said to be F-equivalent to C * (f, X)
if it has the same generators and if its differential δ is made from ∂ by conjugating in each degree by an invertible upper triangular matrix T with coefficients in F:
C k+1 (f, X) ⊗ F T ∂ / / C k (f, X) ⊗ F ∂ / / T C k−1 (f, X) ⊗ F T C k+1 δ / / C k δ / / C k−1
Here, C k (f, X) ⊗ F is a vector space equipped with its canonical ordered basis. Remark 1.3. When changing the pseudo-gradient or the orientations of the unstable manifolds the Morse complex is changed by Z-equivalence. Conversely, if dim M > 1 and if the level sets of f are connected (or f has one local minimum and one local maximum only, i.e. f is polar in Morse's terminology) every Z-equivalence is realizable by such changes. When the coefficients are in a field an F-equivalence has no longer such a geometrical meaning in general. But, an F-equivalence keeps the memory of the filtration by the sub-level sets of the function f . This fact will be used in the last step of the proof of Barannikov's theorem.
Theorem 1.4. (Barannikov [1]) The Morse complex C * (f, X) is F-equivalent to a simple complex (C * , ∂ B )
, that is, for every generator p, ∂ B (p) is 0 or a generator and ∂ B (p) = ∂ B (p ) if p = p and ∂ B (p) = 0. Moreover, (C * , ∂ B ) is unique and depends only on C * (f, X) ⊗ F for any pseudo-gradient X.
This complex is called the Morse-Barannikov complex associated with the Morse function f ; it depends on the field F. Corollary 1.5. The homology H * (M ; F) is graded isomorphic to the sub-space generated by the critical points having the homological type, in the sense given below: ∂ B (p) = 0 and p / ∈ Im ∂ B .
One important point in the statement is the coupling of some critical points, the unpaired generators being "isolated" in the complex. This fact plays a deep rôle in the work by Y. Chekanov & P. Pushkar [4]. When the Morse complex is concentrated in two degrees, the statement amounts to the fact that the double coset GL(n, Z)/T (n) × T (n) is isomophic to the symmetric group S n , a fact which was important in Cerf's work on pseudo-isotopy (here, T (n) denotes the sub-group of invertible upper triangular matrices) [3]. In Barannikov's paper the proof of existence follows more or less from Gauss' algorithm. The proof of uniqueness remains mysterious. It is clarified by C. Viterbo in [7].
Viterbo's important remark is that the critical points of the given Morse function f are divided in three types: upper, lower and homological, depending of the place of a zero map in the diagram below of F-vector spaces and F-linear maps, in which c denotes a critical value of index k + 1 and, for brevity, c + ε stands for the sub-level set f c+ε := f −1 (−∞, c + ε] :
F 0 / / H k+1 (c − ε) / / H k+1 (c + ε) J / / H k+1 (c + ε, c − ε) ∼ = O O ∆ / / I H k (c − ε) / / H k (c + ε) / / 0 H k+1 (+∞, c − ε) ∆ 6 6
The horizontal line is an exact sequence. The critical point p such that f (p) = c is said to be of upper type when J = 0, implying ∆ injective. It is said to be of lower type when J is surjective and I = 0. It is said of homological type when I is injective and ∆ = 0. Clearly, since F is a field, all possibilities are covered, making a partition of the critical points. The type of a critical point p of f is readable on the Morse complex with coefficients in F, that is C * (f, X) ⊗ F. For instance, p is of lower type if there is an F-linear combination of critical points higher than p whose boundary is p ; this property does not depend on the chosen pseudo-gradient X.
The coupling of critical points. If p is a critical point of index k + 1, the local unstable manifold W u loc (p) is unique up to isotopy and orientation. Set where σ runs among the k-cycles of the sub-level set f c−ε representing ∆([p]); it is a critical value λ(p) = f (q) where q is a critical point of index k. By identifying critical point and critical value, we set q := λ(p).
Lemma 1.6. The critical point q := λ(p) is of lower type. Proof. Denote ∆ q ([p]) the class of ∆([p]) modulo the sub-level set f (q) − ε in H k f (p) − ε, f (q) − ε .
By definition of the minimax, this class is not zero and we have
∆ q ([p]) = αI p q [q] , α ∈ F, α = 0, where I p q : H k (f (q) + ε, f (q) − ε) → H k (f (p) − ε, f (q) − ε) is induced by the inclusion. Thus, W u loc (q) is the boundary of 1 α W u (p) in the pair (+∞, f (q) − ε). Hence I([q]) = 0. The Barannikov differential. Set ∂ B (p) = λ(p) if p isProof. 1) Injectivity. Let q = ∂ B (p) = ∂ B (p ) with f (p) > f (p ).
We are using the same notation as in Lemma 1.6. We have
∆ q ([p]) = αI p q ([q]) = 0 in H k f (p) − ε, f (q) − ε and ∆ q ([p ]) = α I p q ([q]) = 0 in H k f (p ) − ε, f (q) − ε . By construction I p q ∆ q ([p ]) = 0. Thus, we have: ∆ q ([p]) = ∆ q ([p]) − α α I p q ∆ q ([p ]) = αI p q ([q]) − α α α I p q ([q]) = 0 .
Therefore, f (q) is not the minimax value associated with p.
2) Surjectivity. Let q be a critical point of lower type and index k. Set
µ(q) = inf σ max(f |σ)
where σ runs among the relative chains of (+∞, f (q) − ε) whose boundary is a relative cycle representing the class [q]. This µ(q) is a critical value of index k + 1 with µ(q) = f (p) for some critical point p. A chain σ approximating the infimum has a non vanishing class in
H k+1 f (p) + ε, f (p) − ε ; hence, [σ] = β[p] in the pair (f (p) + ε, f (p) − ε) with β ∈ F, β = 0. We have ∆ q [p] = I p q ( 1 β [q]
). If this element is zero, this means that there is another relative chain bounded by W u (q) under the level of p, contradicting the definition of µ(q). Then, ∆ q ([p]) = 0. A fortiori, ∆([p]) = 0 and p is of upper type.
We have also to show that λ(p) = q. By the above σ, ∆([p]) is homologous to [q] up a non zero scalar. If it is homologous to [q ] with f (q ) < f (q), then I p q ([q]) = 0, and this is not the case.
At this point we have the uniqueness part in Barannikov's theorem.
Lemma 1.8. The Morse-Barannikov complex is F-equivalent to the Morse complex. In particular, its homology is isomorphic to H * (M, F).
Proof. Suppose we have a chain complex (C * , ∂), F-equivalent to the Morse complex and which is simple until the degree k. Then, ∂ (C k+1 ) is orthogonal to the critical points of upper type (with respect to the canonical scalar product of a based vector space). If not, ∂ • ∂ = 0.
Let p 1 , . . . , p m be the critical points of index k + 1, with f (p 1 ) < . . . < f (p m ). Let q 1 , . . . , q r be the critical points of index k whose type is lower or homological; f (q 1 ) < . . . < f (q r ). We assume that, for some j ≤ r, we have ∂p i = 0 or ∂p i = q k(i) for every i < j, the map i → k(i) being injective. We have
∂p j = i<j,∂p i =0 α i q k(i) + 0 or β k 0 q k 0 + k<k 0 ,k =k(i) β k q k , with β k 0 = 0 .
In the first case we use the following upper triangular matrix in degree k + 1:
T (p j ) = p j − i<j,∂p i =0 α i p i T (p ) = p if = j
and we set∂ = ∂ • T . We get∂(p j ) = 0 and∂(p i ) = ∂(p i ) for every i < j; so we have improved the simpleness of the differential.
In the second case, we use an upper triangular matrix in both degree k + 1 and k:
T (p j ) = 1 β k 0 p j − i<j, ∂p i =0 α i p i , T (p ) = p if = j, T (q k 0 ) = q k 0 + k<k 0 ,k =k(i) β k β k 0 q k T (q k ) = q k if k = k 0 . We set∂ k+1 = T −1 • ∂ k+1 • T and∂ k = ∂ k • T . We observe that∂ k = 0 on the k-cycles as T keeps this set invariant. We have∂(p j ) = q k 0 and∂(p i ) = ∂(p i ) if i < j.
Thus,∂ has the simple form for 1 ≤ i ≤ j. Arguing this way recursively, we get a simple complex which is F-equivalent to the Morse complex.
Since the equivalence relation involves upper triangular matrices only, ∂(p) remains the class of ∆([p]) in H * f (p) − ε; F as it is in the Morse complex. Therefore, when the F-equivalent complex is simple, the type of each critical point can be easily derived and this complex is the Morse-Barannikov complex. The proof of Theorem 1.4 is completed.
Bifurcations
Bifurcations occur in a path of functions. It follows from Thom's transversality theorems, as it is explained by J. Cerf in the beginning of [3], that the space F of real smooth functions on M has a natural stratification whose strata of codimension ≤ 1 are the following:
0) The stratum F 0 , an open dense set in F, is formed by Morse functions whose critical values are all simple. The next two strata are of codimension one. 1) The stratum F 1 is formed by the functions whose critical values are simple and whose critical points are all non-degenerate but one where the Hessian has a kernel of dimension one.
2) The stratum F 2 is formed by the Morse functions whose critical values are all simple but one which has multiplicity 2.
3) The complement R in F of the preceding strata. A generic path has its end points in F 0 , avoids R which in turn is said to be of codimension greater than 1, and crosses F 1 ∪ F 2 transversely in a finite number of points.
Since the Morse-Barannikov complex is well defined for functions in F 0 only, it is necessary to study the bifurcation when crossing F 1 and F 2 . In that aim, it is convenient to introduce the Barannikov diagram and the Cerf diagram, which are defined as follows.
The Barannikov diagram deals with a generic Morse function f ∈ F 0 . Let n = dim M . The vertical lines D k , k = 0, 1, . . . , n, are drawn in the plane, the x-coordinate of D k being n − k.
The critical values of f of index k are marked on D k . When the pair of critical points (p, q) are coupled in the Barannikov complex a segment is drawn from f (p) to f (q); clearly, the slope of this segment is negative (Fig. 1). The critical values, of homological type, in particular max f and min f , remain non-connected to another one.
[0, 1] × R of {t} × f t (critf t ).
It is made of finitely many smooth arcs transverse to the verticals {t} × R, ending at cusp points or in {0, 1} × R, crossing one another transversely.
Bifurcation at birth times.
This event is the crossing of the stratum F 1 , which is co-oriented: the crossing in the positive direction corresponds to the birth of a pair of critical points of index k and k + 1 respectively. The crossing in the opposite direction corresponds to the cancellation of a pair of critical points.
The birth is modeled by the following formula:
f (x, y) = c + Q(y) + x 3 − (t − t 0 )x,
where t 0 is the birth time, c is the critical value of the birth point, (x, y) ∈ R × R n−1 are local coordinates at the birth point p 0 and Q is a non-degenerate quadratic form of index k on R n−1 .
In the Cerf diagram, there is a cusp of coordinates (t 0 , c). For 0 < t 0 − t small, there are no critical points in the vicinity of p 0 . For 0 < t − t 0 small, there are two critical points p t , q t of index k + 1 and k close to p 0 (Fig. 2).
Figure 2
It follows from the model that p t is of upper type and q t is of lower type and this pair is coupled in the Barannikov diagram. The other critical points keep their type and coupling when crossing the birth time.
Bifurcation at double critical value time.
This event is the crossing of the stratum F 2 , say at time t 0 . Since the functions f t are Morse for t close to t 0 , the critical points can be followed continuously on (t 0 −η, t 0 +η). Generically, the pseudo-gradient is Morse-Smale at the time t 0 . Therefore, the Morse complex remains the same on a small interval. But, as the order of the critical values is modified the Barannikov complex could change.
Denote by (p 1 t , p 2 t ) the pair of critical points whose values cross at time t 0 ; say f t (
p 1 t ) < f t (p 2 t ) when t < t 0 ; hence, f t (p 1 t ) > f t (p 2 t ) when t > t 0 .
The question is how the types and coupling of critical points are changing when t crosses the time t 0 .
With Barannikov we limit ourselves to the case when M is the n-sphere S n and the crossing does not involve the extremal values, this latter question being left as an exercise to the reader. Since on a sphere the only critical points of homological type are the extrema, the crossing deals with critical values of upper/lower type.
We shall say that there is no bifurcation if the crossing keeps all critical points with their initial types and coupling (remember that, near a crossing time all the functions of the considered path are Morse and the critical points can be followed smoothly in time due to the implicit function theorem).
One checks by hand that there is no bifurcation if p 1 t and p 2 t have distinct indices. Now, we are reduced to the case where the two crossing critical values have the same index, say k.
We are going to prove in the next three propositions that, in our restrictive setting, there are only three types of bifurcations which are shown with their Barannikov diagrams before and after crossing (Fig. 3).
2.3.
Notation. Before stating the bifurcation propositions, it is useful to introduce some notation. Denote by c 0 the double critical value:
c 0 = f t 0 (p 1 t 0 ) = f t 0 (p 2 t 0 )
. We have two (k − 1)-spheres Σ 1 and Σ 2 traced in the level set f t 0 = c 0 − ε by the respective unstable manifolds of p 1 t 0 and p 2 t 0 . When t runs in a small interval around t 0 , these objects move by a small isotopy:
Σ 1 t , Σ 2 t ⊂ (f t = c 0 − ε)
. Finally, if α is a non-zero (k − 1)-homology class is the sub-level set c 0 − ε, we denote by λ(α) the critical value which is the infimum of c such that α vanishes in the relative homology H k−1 (c 0 − ε, c; F) of the pair of the denoted sub-level sets. Proposition 2.4. We assume that, for t < t 0 , p 1 t and p 2 t have not the same type upper/lower. Then, the following holds true. 1) A bifurcation can occur only in case (A left), that is: p 1 t is of upper type and below p 2 t which is of lower type. 2) Assuming the above necessary condition, a bifurcation occurs if and only if [Σ 1 ] = [Σ 2 ] in H k−1 (c 0 ; F) up to a scalar. In that case, for t > t 0 , p 1 t becomes of lower type and p 2 t becomes of upper type; the new coupling is shown on (A right).
Proof. 1) Assume p 1 t and p 2 t are respectively of lower and upper type for t < t 0 . When t is close to t 0 , t < t 0 , by assumption, Σ 1 f t = f t (p 1 Morse inequalities have been formulated and proved for manifolds with non-empty boundary. Notice that generically a function F : W → R is a Morse function whose critical points lie in the interior of W and whose restriction to the boundary is Morse.
Actually, the problem considered by Barannikov in [1] was more ambitious, that is, given a generic germf along the boundary M , to give a bound from below of the number of critical points of any generic extension F : W → R off , as acute as possible. We focus on M = S n , the n-sphere, and W = D n+1 . Moreover, we limit ourselves to answer the question of knowing when this bound is positive. This problem was completely solved by S. Blank & F. Laudenbach [2] for n = 1 and by C. Curley [5] for n = 2.
3.1. The framing. Here we use Barannikov's terminology. Given a Morse function f : M → R a framing of f is the data of one vertical arrow at each critical value of f . A generic germ f : M × [0, ε) → R along the boundary determines a framing according the following rule: for the critical p of f the arrow at f (p) points up (resp. down) if < df (p), n(p) > is positive (resp. negative), where n(p) is a tangent vector at p pointing inwards. Conversely, the framing classifies the germf up to isotopy fixing the boundary.
This yields some information about the non-existence of an extension without critical points F : W → R of the germf . For instance, if the framing points up at the maximum, the maximum principle tells us that any extension must have at least one critical point in the interior. Barannikov's discussion will be of course more subtle. The framing may be attached to a Morse function on M , an F-equivalence class of Morse complexes or the associated Morse-Barannikov complex C B (f ) as well. We will speak of the framed Barannikov diagram. We are going to look at the framed Barannikov complex and get some information in relation to the extension problem.
From now on, we restrict ourselves to M = S n and W = D n+1 . A framed function f : S n → R is said to be standard if f has two critical points only, the framing pointing down at the maximum and pointing up at the minimum. In this case, there is a standard extension to the (n + 1)-ball without critical points. Now, start with a generic germf along S n and assume there is an extension without critical points F : D n+1 → R. Then, there is a one-parameter family of spheres S t ⊂ D n+1 , t ∈ [0, 1], such that S 0 = M , S t lies inside S t when t > t and the germ of F along S 1 is standard. Set f t := F |S t ; it is a function thought of as defined on M = S n , equipped with a framing due to the knowledge of S t for t = t + ε, ε > 0. The framing says the direction of moving of the spheres near a critical point p of f t (up to isotopy, F may be locally thought of as the height function in R 3 ).
Generically, for such a family of spheres, the map t → f t is a generic path of functions in the sense of Section 2. Therefore, there is a sequence of bifurcations starting from a given germ and ending at the standard germ.
There is no bifurcation of framing. At a birth time the pair (p t , q t ) which has born with f t (p t ) > f t (q t ) has the standard framing: the arrows are both up or both down; such a pair will be said to be a standard pair. At a crossing time t 0 involving the pair of critical values f t (p 1 t ) < f t (p 2 t ), with the same indices, each critical point keeps its framing during the crossing. But, the rule of moving added to the rule of numbering implies that, for t < t 0 , the arrow of p 1 t is up and the arrow of p 2 t is down (Compare Fig. 4). At each generic time, we have a well-defined Morse-Barannikov framed complex and the bifurcations are those allowed by Propositions 2.4 to 2.6. Of course, in each case the bifurcation occurs only when the required homological condition is satisfied. If it is not, the crossing yields no bifurcation. Figure 4 3.2. Barannikov idea's. Not any sequence of allowed bifurcations of the framed Barannikov's complex is realizable by a sequence of bifurcations of framed Morse complex; at the level of framed Barannikov diagram there is no longer homological condition. Therefore, there is a formal problem which is the following: given a framed Barannikov complex, does there exist a sequence of allowed bifurcations connecting it to the standard Barannikov complex (i.e. one maximum down and one minimum up)? If there is no solution to the formal problem, a fortiori there is no solution to the extension problem without critical point. Now, the question is whether it is possible to answer the formal problem in finite time. This question is solved by the last theorem in Barannikov's article.
Theorem 3.3. Given a framed Barannikov complex C 0 , if it is connected to the standard Barannikov complex C st , then C 0 is connected to C st without any birth.
In particular, the formal problem reduces to a finite combinatorics. There are two types of such inverted pairs: in type I (resp. type II) the upper point is equipped with an arrow pointing up (resp. down).
The index of a coupled pair (inverted or standard) will be the index of the upper point.
One checks on the list of bifurcations that such a pair could not disappear alone. At best, it is possible to shift the indices of the involved critical points (use the bifurcation of Fig. 4 one pair being inverted and the other being standard in the sense of the birth bifurcation). But, two inverted pairs involving of a bifurcation as shown on Fig.4 become both standard and then, each of them can be cancelled.
As a consequence, an obvious obstruction to extending without critical points is the parity of the number of inverted pairs in the initial framed Morse function. The obstruction which follows from Theorem 3.3 is more subtle as shown in the next example. The proof of Theorem 3.3 is based on the fact that there is a very short list of bifurcations involving at least one inverted pair; moreover, the role of the two types, I and II, are completely different. Here is the list of these bifurcations:
(1) a pair of type I and a pair of type II of the same indices yield two standard pairs;
(2) two pairs of type I whose indices differ by 1 yield two standard pairs whose indices differ by 1; (3) two standard pairs having the same index k yield a pair of type I and a pair of type II both having the index k; (4) two standard pairs whose indices are k and k + 1 yield two pairs of type II whose indices are still k and k + 1; (5) a pair of type I and index k and a standard pair of index k ± 1 yield a pair of type II and index k ± 1 and a standard pair of index k.
In particular, it is impossible to change the index of a pair of type II. For proving Theorem 3.3, without losing generality we may assume that the initial complex C 0 has no standard pairs. All the coupled pairs are inverted and we are facing the problem of canceling them, maybe with the help of introducing standard pairs (births) whose bifurcations could create new inverted pairs in good positions for a total cancellation. One checks that this event cannot happen.
According to the previous list, for canceling one pair A of type II it is required to have one pair B of type I and of the same index is the right position allowing the bifurcation (1). If B comes from births followed by the bifurcation (3), then B is born with another pair of type II still with the same index. So the price to pay the cancellation of A with B is the appearance of C which is almost in the same position as A in the Barannikov diagram, up to a shift of the heights in the direction of the arrows of the framing. So, nothing is gained. The complete proof follows the same line. We refer to Barannikov's article for the details.
Theorem 1.1. (Milnor [6], Th. 7.2). The Morse complex is a chain complex: ∂ • ∂ = 0. Moreover, H (C * (f, X)) ∼ = H * (M ; Z). A fortiori, H (C * (f, X); F) ∼ = H * (M ; F).
[p] := [W u loc (p)] ∈ H k+1 (c+ε, c−ε) where c = f (p). If p is of upper type, one defines λ(p) := inf σ max(f |σ)
of upper type and ∂ B (p) = 0 in the two other cases. According to the previous lemma, ∂ B • ∂ B = 0. Lemma 1.7. The map λ defines a bijection from the set of critical points of upper type onto the set of critical points of lower type.
Figure 1 The
1Cerf diagram deals with a generic path γ = (f t ) t∈[0,1] . Its Cerf diagram is the union in
Figure 3
3Figure 3
Definition 3. 4 .
4Let f be a framed Morse function. A coupled pair of critical points is said to be inverted when one of its two arrows points up and the other points down.
Figure 5
5Figure 5
t is null-homologous in its sub-level set while Σ 2 t is nonhomologous to 0. Since these spheres change with t by an isotopy, this property persists up to t = t 0 and still a little further, up to some t > t 0 . Therefore the types are unchanged after crossing and it is easy to check that the coupling is also unchanged. Hence, no bifurcation.2) Then, we assume (A left). Again, for t < t 0 and close to t 0 , certainly[Σ 1 t ] is not homologous to 0 in H k−1 ((f t = c 0 − ε); F). The unstable manifold W u (p 2 t ) traces in f t = f t (p 2 t ) − ε , ε > 0, a sphere which is homologous to 0 in its sub-level set if f t (p 2 t ) − ε > f t (p 1 t ). Thus, for Σ 2 t in f t = c 0 − ε, there are two possibilities: (i) [Σ 2 t ]=0 or (ii) [Σ 2 t ] is a non-zero multiple of [Σ 1t ] since it should vanish when passing in a sub-level set containing p 1 t . In both cases, this property persists up to t 0 and up to some t > t 0 . One checks easily that in case (i) there is no bifurcation.Consider case (ii). The point p 2 t is of upper type, by the very definition. Now, with ε > 0 small enough so that f t (p 2 t ) < f t (p 1 t ) − ε , we see that the trace of W u (p 1 t ) in the level set
Proposition 2.6. In this situation, there is a bifurcation (exchange of coupling) if and only if µ([e 1 ]) = µ([e 2 ]).In that case f t (q 1 t ) > f t (q 2 t ) when t < t 0 as shown on (C left).3. The non-empty boundary caseFollowing S. Barannikov, we discuss in this section the problem of extending without critical points a germ of function given along the boundary M of a compact (n+1)-dimensional manifold W , M = ∂W . This setting was already considered in 1934 by Morse-van Schaack[8] where the
t ) − ε is homologous to 0 in its sub-level set. Therefore, p 1 t is of lower type and the types of the two critical points exchange. One checks that the coupling is as shown on (A right).Assume now p 1 t and p 2 t are both of upper type. By the same homological argument as before, the type cannot change. But the coupling could change. Denote q 1 t and q 2 t the points associated with p 1 t and p 2 t respectively before t 0 . Proposition 2.5. In this situation, there is a bifurcation (exchange of coupling) if and only if. But this vanishing holds true for every t close t 0 . This proves that the coupling of p 1 t remains unchanged; hence, no bifurcation.What happens in case of equality? First, we look at t < t 0 .Since the function f t , restricted to the sub-level set c 0 − ε, moves by isotopy when t runs inand, hence, f t (q 2 t ) < f t (q 1 t ) as shown on (B left).Second, we look at t > t 0 . Now, λ([Σ 2 t ]) varies continuously on [t 0 , t 0 + η); thus, this value is f t (q 1 t ) due to the above equality at time t 0 . Thus p 2 t is coupled with q 1 t . Therefore, p 1 t must be coupled with q 1 t ; there is, indeed, no other free place!The last case to consider is when p 1 t and p 2 t are both of lower type. For homological reasons, there is no change of types. Let q 1 t and q 2 t be the points of index k + 1 with which they are coupled respectively when t < t 0 .At time t 0 , denotes by e i , i = 1, 2, the k-cell traced by the unstable manifold of p i t 0 in the level set f = c 0 − ε. The map µ that we introduced in Lemma 1.7 is still defined: µ([e i ]) is the infimum of c such that the class of e i is 0 in the pair (c, c 0 − ε) with coefficients in F. Arguing similarly as in the previous proposition, one proves the following.
The framed Morse complex and its invariants. S Barannikov, Advances in Soviet Math. 21S. Barannikov, The framed Morse complex and its invariants, Advances in Soviet Math. 21 (1994), 93-115.
Extension à une variété de dimension 2 d'un germe de fonction donné au bord. S Blank, F Laudenbach, C.R. Acad.Sci. Paris. 270S. Blank, F. Laudenbach, Extension à une variété de dimension 2 d'un germe de fonction donné au bord, C.R. Acad.Sci. Paris 270 (1970), 1663-1665.
La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudoisotopie. J Cerf, Inst. Hautes Études Sci. Publ. Math. 39J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo- isotopie, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 5-173.
Combinatorics of fronts of Legendrian links and the Arnol'd 4-conjectures Russian Math surveys. Y Chekanov, P Pushkar, 60Y. Chekanov, P. Pushkar, Combinatorics of fronts of Legendrian links and the Arnol'd 4-conjectures Russian Math surveys, 60 n• 1 (2005), 95-149.
Non-singular extension of Morse functions. C Curley, Topology. 16C. Curley, Non-singular extension of Morse functions, Topology 16 (1977), 89-97.
Lectures on the h-cobordism theorem. J Milnor, L. Siebenmann & J. SondowPrinceton Univ. PressJ. Milnor, Lectures on the h-cobordism theorem, Notes by L. Siebenmann & J. Sondow, Princeton Univ. Press, 1965.
Precise Arrhenius law for p-forms, The Witten Laplacian and Morse-Barannikov complex. D Le Peutrec, F Nier, C Viterbo, Annales Henri Poincaré. onlineD. Le Peutrec, F. Nier, C. Viterbo, Precise Arrhenius law for p-forms, The Witten Laplacian and Morse- Barannikov complex, Annales Henri Poincaré (August 2012), online.
The critical point theory under general boundary conditions. M F Morse, G B Van Schaack, Annals of Math. 353M.F. Morse, G.B. Van Schaack, The critical point theory under general boundary conditions, Annals of Math. 35 n • 3 (1934), 545-571.
UMR 6629 du CNRS, Faculté des Sciences et Techniques. Laboratoire De Mathématiques, Jean Leray, 2Université de Nantesrue de la Houssinière, F-44322 Nantes cedex 3, France. E-mail address: [email protected] de mathématiques Jean Leray, UMR 6629 du CNRS, Faculté des Sciences et Techniques, Université de Nantes, 2, rue de la Houssinière, F-44322 Nantes cedex 3, France. E-mail address: [email protected]
| []
|
[
"THE EFFECTS OF INVASIVE EPIBIONTS ON CRAB-MUSSEL COMMUNITIES: A THEORETICAL APPROACH TO UNDERSTAND MUSSEL POPULATION DECLINE",
"THE EFFECTS OF INVASIVE EPIBIONTS ON CRAB-MUSSEL COMMUNITIES: A THEORETICAL APPROACH TO UNDERSTAND MUSSEL POPULATION DECLINE"
]
| [
"Jingjing Lyu \nDepartment of Mathematics\nClarkson University\n13699PotsdamNew YorkUSA\n",
"Linda A Auker \nDepartment of Biology\nSt. Lawrence University\n13617CantonNew YorkUSA\n",
"Anupam Priyadarshi \nDepartment of Mathematics\nInstitute of Science\nBanaras Hindu University\n221005VaranasiIndia\n",
"Rana D Parshad \nDepartment of Mathematics\nIowa State University\n50011AmesIowaUSA\n"
]
| [
"Department of Mathematics\nClarkson University\n13699PotsdamNew YorkUSA",
"Department of Biology\nSt. Lawrence University\n13617CantonNew YorkUSA",
"Department of Mathematics\nInstitute of Science\nBanaras Hindu University\n221005VaranasiIndia",
"Department of Mathematics\nIowa State University\n50011AmesIowaUSA"
]
| []
| Blue mussels (Mytilus edulis) are an important keystone species that have been declining in the Gulf of Maine. This could be attributed to a variety of complex factors such as indirect effects due to invasion by epibionts, which remains unexplored mathematically. Based on classical optimal foraging theory and anti-fouling defense mechanisms of mussels, we derive an ODE model for crab-mussel interactions in the presence of an invasive epibiont, Didemnum vexillum. The dynamical analysis leads to results on stability, global boundedness and bifurcations of the model. Next, via optimal control methods we predict various ecological outcomes. Our results have key implications for preserving mussel populations in the advent of invasion by non-native epibionts. In particular they help us understand the changing dynamics of local predator-prey communities, due to indirect effects that epibionts confer.1991 Mathematics Subject Classification. Primary: 34C11, 34C23, 49J15; Secondary: 92D25, 92D40. | 10.1142/s0218339020500060 | [
"https://arxiv.org/pdf/1810.04256v1.pdf"
]
| 52,955,435 | 1810.04256 | 8fd75f06aaf700e9b1f4cc7e33e8f4cd0249a52b |
THE EFFECTS OF INVASIVE EPIBIONTS ON CRAB-MUSSEL COMMUNITIES: A THEORETICAL APPROACH TO UNDERSTAND MUSSEL POPULATION DECLINE
Jingjing Lyu
Department of Mathematics
Clarkson University
13699PotsdamNew YorkUSA
Linda A Auker
Department of Biology
St. Lawrence University
13617CantonNew YorkUSA
Anupam Priyadarshi
Department of Mathematics
Institute of Science
Banaras Hindu University
221005VaranasiIndia
Rana D Parshad
Department of Mathematics
Iowa State University
50011AmesIowaUSA
THE EFFECTS OF INVASIVE EPIBIONTS ON CRAB-MUSSEL COMMUNITIES: A THEORETICAL APPROACH TO UNDERSTAND MUSSEL POPULATION DECLINE
Blue mussels (Mytilus edulis) are an important keystone species that have been declining in the Gulf of Maine. This could be attributed to a variety of complex factors such as indirect effects due to invasion by epibionts, which remains unexplored mathematically. Based on classical optimal foraging theory and anti-fouling defense mechanisms of mussels, we derive an ODE model for crab-mussel interactions in the presence of an invasive epibiont, Didemnum vexillum. The dynamical analysis leads to results on stability, global boundedness and bifurcations of the model. Next, via optimal control methods we predict various ecological outcomes. Our results have key implications for preserving mussel populations in the advent of invasion by non-native epibionts. In particular they help us understand the changing dynamics of local predator-prey communities, due to indirect effects that epibionts confer.1991 Mathematics Subject Classification. Primary: 34C11, 34C23, 49J15; Secondary: 92D25, 92D40.
1. Introduction 1.1. Background. Blue mussels (Mytilus edulis) are an ecologically and economically important species [1,2,3]. They play several roles in marine ecosystems: as important prey for many species, such as crabs, shorebirds, sea stars, and gastropod molluscs [3,4,5]; as nutrient recyclers and pollution indicators [6]; and as a keystone species, serving as habitat for benthic infaunal organisms [3,7]. However, M. edulis has declined in the Gulf of Maine by over 60% since the 1970s [8]. Mussel post-larval settlement, consistent with this observation, has also declined [11]. The reasons for this decline are unclear, but are almost certainly complex. Thus a clearer understanding of the ecological factors that influence mussel populations is needed.
A primary cause of a species population decline is predation. Invasive predators, like the green crab (Carcinus maenas) and the Asian shore crab (Hemigrapsus sanguineus), readily prey on the blue mussel [12,13,15]. However, mussel size limits crab predation, with crabs consuming mussel prey below 70 mm in shell length [16]. Furthermore, mussels have also adapted to crab predation by thickening their shells in response to novel predator presence, in extremely short time periods [15]. Furthermore, substrate complexity reduces predation on mussels as increasingly complex habitats provide refuge from crab predation [17]. Thus while predation has put considerable pressure on mussel populations, rapidly evolving defense mechanisms, escape from predation via growth, and physical refuges have counteracted predator impacts.
Though mussels do have the aforementioned protections against predation, they are still in decline. Curiously, in the 1970s, an introduced ascidian species Didemnum vexillum, arrived in the Gulf of Maine [18]. D. vexillum is a colonial ascidian that is dominant as a competitor for substratum, prolifically laying down mat-like structures on any hard substrate [22]. Consequently, it acts as an epibiont (fouling organism) on M. edulis [11,12]. Epibionts impact predator-prey communities indirectly by affecting predation rates on basibionts. D. vexillum in particular has chemical anti-predatory defenses. If a crab predator attempts to break off pieces of the D. vexillum colony to reach the mussel, D. vexillum releases secondary metabolites and sulfuric acid that deters the crab [22]. This mechanism by which the epibiont protects the mussel from crab predation is known as associational resistance [24]. While it appears to protect mussels from crab predation, D. vexillum also negatively affects mussel fecundity and fitness, resulting in fewer progeny [11]. Thus D. vexillum has both a positive and negative impact on mussel populations.
Given this complex relationship, we ask, • Could the introduced epibiont D. vexillum change the predator-prey dynamics in an established local crab-mussel community? • Could the net effect of positive and negative impacts from D. vexillum epibiosis, lead to mussel population decline? Our analysis is (to the best of our knowledge) the first mathematical investigation of predator-prey dynamics under pressure of fouling from epibionts in a crab-mussel community. Herein,
• we derive a predator-prey model for crab-mussel interactions, given that clearly a certain size of mussel is preferred or "optimal" for the crab, • we consider the effects of an invasive epibiont by meshing OFT and antfouling defense of the mussels, • we model the effects of associational resistance and reduced fecundity, due to the epibiont, and • we next investigate dynamical aspects of the model, and use optimal control theory to predict various outcomes. In adult mussels the protective periostracum, which inhibits epibiont settlement when present [26] tends to wear off due to age, decay and abrasion. Consequently, the periostracum is more prevalent on newer regions of the shell, while absent on older regions [27]. This means juvenile mussels are less likely to be overgrown with epibionts than are adult mussels. When crabs forage for mussels, they typically prefer a medium sized adult. But this preferred size tends to get easily overgrown by epibionts. Epibionts can alter the prey size choice of predators, including crabs, in experiments [22,28,29,30], though this has not yet been tested with D. vexillum. We hypothesize in the current manuscript that D. vexillum can change the feeding preference of crabs away from mid-size adult mussels (that are easily overgrown and therefore less likely to be eaten) towards juvenile mussels (which are less likely to be overgrown and so are easier targets), even though the latter are not the crab's preferred food source. To elucidate our approach we survey some classical results from OFT.
1.2. Optimal Foraging Theory. Optimal foraging theory (OFT) predicts how animals behave when they forage for food. It is well known that predators optimize feeding strategies to maximize energy intake [31]. Essentially, predators try to gain the most energy from their prey by expending the least amount of energy in the hunting process. For a crab foraging for mussels this amounts to maximizing (1) e h = energy gained from mussel intake handling time of mussel .
This translates to a medium-sized adult mussel as the optimally preferred prey by adult crabs. While large mussels have a potentially high source of energy, they take a much longer time to open and consume than smaller mussels. Small mussels, conversely, take a short time to consume, but they offer very little reward. Even so, juvenile mussels are readily consumed by many species, including green crabs and dogwhelk [12]. We refer the reader to the mathematical treatment by Krivan [32,33], who describes, via a three species ODE model, in which a predator hunts (optimally) for two prey species u and v, where u is favored to v. The term u 1 denotes the attack rate with which prey u is hunted, and u 2 denotes the attack rate with which prey v is hunted. The analysis presented in [33] draws from classic results in OFT and shows that, in order to maximize e h , the optimal pair of (u 1 , u 2 ) is given by u 1 = 1, u 2 = 0 if u > u * or u 1 = 1, u 2 = 1 if u < u * , or u 1 = 1, 0 < u 2 < 1 if u = u * , where u * is the critical density for switching.
1.2.1. OFT in the presence of epibionts. Predators are known to switch prey if preferred prey drop below a threshold density [35]. For example, fish species have been shown to switch habitats if foraging in one habitat becomes less fruitful than in another habitat [36]. Theoretical studies also support this [32]. In our context, If e1 h1 > e2 h2 , the adult mussels are preferred to juveniles as the optimal prey for crabs. If e1 h1 < e2 h2 , juveniles mussels are optimal to hunt. Conjecture 1. An increase in epibiont density will cause crabs to switch from adult mussels to juveniles even though the juvenile is less preferred and the adult mussel density remains high.
Essentially, if one considers a predator-prey model with these species (crabmussel-epibiont), there are two limiting cases • There is no epibiont (e = 0), in this setting e1 h1 > e2 h2 , so u 1 = 1, u 2 = 0; • The epibiont achieves its carrying capacity K, that is, e = K, in this setting, high epibiont density causes the handling time h 1 1, thus e1 h1 < e2 h2 , and so the crab switches to juveniles, and now u 1 = 0, u 2 = 1.
To this end we first split the mussel class into adults and juveniles (we assume the juveniles have the protective periostracum whereas the adults do not). A crab species preying on two separate classes of adult and juveniles mussels (with adults being the preferred food type), places this in a classic one predator-two prey setting [32]. Our hypotheses for OFT are as follows:
(1) In the absence of epibionts, crab-mussel interactions follow classical OFT.
That is, adult mussels will be attacked with rate 1, whilst the less preferred juveniles will not be attacked (u 1 = 1, u 2 = 0). We claim this is the only optimal strategy for the crab, as long as d 1 > e2 h2 , where d 1 is the death rate of prey type 1.
(2) There is a change to the classical case, under pressure of epibiosis from D.
vexillum. (3) If a crab were presented with a preferred adult mussel overgrown with D.
vexillum, it would switch to a prey of a less optimal size, even if the overall adult mussel density was high (assuming their was uniform overgrowing of all adult mussels). (4) The switch would be to juvenile mussels, which we know are almost never overgrown because of their intact protective peristrocum. That is in the (e = K) case, we have (u 1 = 0, u 2 = 1). We claim this is the only optimal strategy for the crab, as long as d 1 > e1 h1 . (5) This will in turn directly affect the feedback loop to the adult mussel population, given that juveniles are transitioning to adults.
The above is rigorously shown in appendix section 8.1. mussels size crab preference best size mussels size crab preference Figure 1. Classic crab preference (without epibiont) shown in left panel. Crabs were presented with clean and overgrown mussels [11,12], and the handling times of the overgrown ones were up to 6 times greater than the clean ones (seen in center panel). These experimental results [11,12] lead us to conjecture that crabs will switch to the smaller uncovered juvenile mussels under pressure of epibiosis, shown in right panel.
Mathematical Formulation
Our goal is to derive a mathematical model that best captures our hypotheses. To this end, we make the following assumptions, (1) An epibiont has invaded into a local crab-mussel community, and is growing logistically. It will eventually reach a critical carrying capacity.
(2) We model the pressure from epibiosis, in terms of the attack rates u 1 , u 2 . That is we assume these are dependent on the epibiont density. As a simple first approach we assume
u 1 (e) = K − e K , u 2 (e) = e K .
Thus, without any epibiont presence (e = 0), the adult mussel is the only prey eaten and the juvenile is not eaten at all because there is not enough energy gain for the effort involved, so u 1 = 1, u 2 = 0, in line with classical theory [33]. However, this starts to change as the epibiont starts to overgrow the mussels. When the epibiont is at carrying capacity, e = K, we assume the adult mussels are completely overgrown, and thus is not consumed at all. The crab switches completely to juveniles, so that u 1 = 0, u 2 = 1.
(3) We model the decreasing fecundity in mussels due to epibiont cover by considering a growth rate a = a(e). We consider a = a(e) = a
K− e 2 K .
Hence as epibionts get to carrying capacity e = K, this growth rate is cut in half and becomes a/2. (4) We assume the intraspecies competition is present only in adult mussels, and not juveniles [49]. (5) We assume the search rates λ 1 , λ 2 to be the same, and normalized to 1, so λ 1 = λ 2 = 1. Based on the above assumptions, we have the following system of differential equations,
dC dt = − d 1 C + e 1 u 1 (e) M A 1 + h 1 u 1 (e)M A + h 2 u 2 (e)M J C + e 2 u 2 (e) M J 1 + h 1 u 1 (e)M A + h 2 u 2 (e)M J C,(2)(3) dM A dt = bM J − δ 1 M 2 A − u 1 (e) M A 1 + h 1 u 1 (e)M A + h 2 u 2 (e)M J C, (4) dM J dt = a(e)M A − bM J − u 2 (e) M J 1 + h 1 u 1 (e)M A + h 2 u 2 (e)M J C,(5)de dt = b 1 e(1 − e K ).
where Here C, M A , M J are the densities of crabs, adult mussels and juvenile mussels population at a given time t respectively. The population density of D. vexillum is e, while d 1 is the mortality rate of the crab, e 1 , e 2 is the energy gain to the crab from preying on the adult mussel and juvenile mussel respectively, h 1 , h 2 are the handling time of the adult mussel and juvenile mussel respectively, b is the rate at which juveniles leave the juvenile class and become adults, a is the rate at which juveniles are produced, δ 1 measures the intraspecific competition among adult mussels, b 1 is the intrinsic rate of growth of the epibiont population, and K is its carrying capacity.
(6) u 1 (e) = K − e K , u 2 (e) = e K , a(e) = a K − e 2 K .
Dynamical Analysis
3.1. Boundedness. The equation for e is bounded trivially by K. Addition of (129)-(131), and given the fact that e 1 < 1 and e 2 < 1, yields:
(7) d(C + M A + M J ) dt ≤ −d 1 C + aM A − δ 1 M 2 A ≤ aM A − δ 1 M 2 A .
Thus, comparison with a logistic ODE yields:
(8) C + M A + M J ≤ a δ 1 ,
and thus we can state the following theorem:
Theorem 3.1. Consider the crab-mussel system (129)-(132). The solutions (C, M A , M J , e), satisfy the following uniform bounds
(9) ||C|| ∞ ≤ K 1 , ||M A || ∞ ≤ K 1 , ||M J || ∞ ≤ K 1 , ||e|| ∞ ≤ K 1 , for any initial conditions (C(0), M A (0), M J (0), e(0)) ∈ L ∞ , where K 1 = max(K, a δ1
). 3.2. Equilibrium and Local Stability with no epibiont. We now consider the existence and stability of the equilibrium for the system when there is no epibiont(e = 0). The system is simplified as
(10) dC dt = −d 1 C + e 1 M A 1 + h 1 M A C,(11)dM A dt = bM J − δ 1 M 2 A − M A 1 + h 1 M A C,(12)dM J dt = aM A − bM J .
Two equilibria, (0, 0, 0) and (0, a δ1 , a 2 δ1b ), on the boundary and one interior equilibrium (C * , M * A , M * J ). It is easy to see (0, 0, 0) is unstable.
And (0, a δ1 , a 2 δ1b ) is globally stable if 0 < e 1 − d 1 h 1 < d1δ1 a and unstable if e 1 − d 1 h 1 > d1δ1 a .
The interior equilibrium is given by
(13) C * = e 1 (a(e 1 − d 1 h 1 ) − d 1 δ 1 ) (e 1 − d 1 h 1 ) 2 ,(14)M * A = d 1 e 1 − d 1 h 1 ,(15)M * J = ad 1 b(e 1 − d 1 h 1 )
.
Note that C * > 0, M * A > 0 andM * J > 0 if (16) e 1 − d 1 h 1 > d 1 δ 1 a .
We next state the following theorem Theorem 3.2. Consider the crab-mussel system (129)-(132), without the presence of an epibiont, that is when e = 0. There exists an interior steady state (C * , M * A , M * J ), which is locally asymptotically stable under the following criteria,
(17) d 1 δ 1 a < e 1 − d 1 h 1 < d 1 δ 1 a + d 1 δ 1 e 1 ah 1 , d 1 > e 2 h 2 .
The proof is relegated to the appendix section 8.2.
Remark 1. Note, the second condition d 1 > e2 h2 is not a result of the Routh-Hurwitz criterion, rather it follows from lemma 8.1 in appendix section 8.1. We enforce it so that the attack rates should be as predicted via classical OFT.
Equilibrim and Local
Stability Analysis with Epibiont. The system (129)-(132) has five possible equilibria. There is one in the interior of the positive octant (C * , M * A , M * J , e * ), and four on the boundary, (0, 0, 0, 0), (0, 0, 0, K), (0, a δ1 , a 2 bδ1 , 0) and (0, a 2δ1 , a 2 4bδ1 , K). It is easy to check that the equilibria with no epiboint, (0, 0, 0, 0) and (0, a δ1 , a 2 bδ1 , 0), are unstable. Furthermore, (0, a 2δ1 ,
a 2 4bδ1 , K) is stable if 0 < e 2 − d 1 h 2 < 4bd1δ1 a 2 and unstable if e 2 − d 1 h 2 > 4bd1δ1 a 2 .
In fact, it is common that prey exist in a stable state in the absence of the predator. Finally, (0, 0, 0, K) is also unstable. We will focus on the interior equilibrium.
Consider the interior equilibrium, i.e. (C * , M * A , M * J , e * ). It is easy to see e * = K in the equilibrium state.Then we have u 1 (e * ) = 0, u 2 (e * ) = 1, a(e * ) = a 2 . To get (C * , M * A , M * J , e * ) explicitly, it is equivalent to solve the following equations:
(18) − d 1 C + e 2 M J 1 + h 2 M J C = 0, (19) bM J − δ 1 M 2 A = 0, (20) a 2 M A − bM J − M J 1 + h 2 M J C = 0, (21) e = K.
Thus the interior equilibrium is given by
(22) C * = 1 2 ae 2 bd1 (e2−d1h2)δ1 d 1 − e 2 b e 2 − d 1 h 2 ,(23)M * A = bd 1 (e 2 − d 1 h 2 )δ 1 ,(24)M * J = d 1 e 2 − d 1 h 2 ,(25)e * = K. Note that M * A > 0 and M * J > 0 if e 2 − d 1 h 2 > 0. And C * > 0 if e 2 − d 1 h 2 > 4bd1δ1 a 2 .
Therefore, the feasibility criteria for this system is
(26) e 2 − d 1 h 2 > 4bd 1 δ 1 a 2 .
We next state the following theorem Theorem 3.3. Consider the crab-mussel system (129)-(132), when the epibiont has reached equilibrium, that is e = K. There exists an interior steady state (C * , M * A , M * J , K), which is locally asymptotically stable under the following criteria,
(27) 4bd 1 δ 1 a 2 < e 2 − d 1 h 2 < 16bd 1 δ 1 a 2 , d 1 > e 1 h 1 .
The proof relies on the Routh-Hurwitz criterion [37], and is relegated to the appendix section 8.3.
Remark 2.
Note that the condition d 1 > e1 h1 is not a result of the Routh-Hurwitz criterion, rather it follows from lemma 8.1 in appendix section 8.1. In a sense we enforce it so that the attack rates should be as predicted via classical OFT.
3.4. Global stability. We now derive some results on the global stability of the internal equilibrium.
Theorem 3.4. Consider the model (10)- (12). There exists an > 0, s.t. the internal equilbrium point, (C * , M * A , M * J ), is globally asymptotically stable under the following parametric restriction
(28) d 1 δ 1 a < e 1 − d 1 h 1 < d 1 , 0 < < a δ 1 , 1 2 < e 1 < 1.
Proof. Consider M A = M A + , Under this transformation we have the following transformed system
(29) dC dt = −d 1 C + e 1 M A + 1 + h 1 (M A + ) C,(30)dM A dt = bM J − δ 1 (M A + ) 2 − M A + 1 + h 1 (M A + ) C,(31)dM J dt = a(M A + ) − bM J .
It is enough to show the new system (29)-(31) is globally asymptotically stable. The equilibrium of (29)-(31) is given by
C * = e 1 a d 1 (M A * + ) − e 1 δ 1 d 1 (M A * + ) 2 , M A * = d 1 e 1 − d 1 h 1 − , M * J = ad 1 b(e 1 − d 1 h 1 ) .(32)
Note that solutions to this new system are feasible if
(33) d 1 δ 1 a < e 1 − d 1 h 1 < d 1 , 0 < < a δ 1 .
We define the following Lyapunov function,
(34) V (C, M * A , M J ) = C + M * A + M J . Note that V ≥ 0, because of the positivity of the solutions. Furthermore, V is radially unbounded. Now consider dV dt = dC dt + dM A dt + dM J dt = −d 1 C + (e 1 − 1) M A + 1 + h 1 (M A + ) C − δ 1 (M A + ) 2 + a(M A + ) < −d 1 C − δ 1 (M A + ) 2 + a(M A + ) = −d 1 C − δ 1 (M A ) 2 − 2δ 1 M A − δ 1 2 + aM A + a = −d 1 C − δ 1 (M A − a 2δ 1 ) 2 − δ 1 ( − a 2δ 1 ) 2 + a 2 2δ 1 − 2δ 1 M A = −δ 1 [(M A − a 2δ 1 ) 2 + ( − a 2δ 1 ) 2 ] + a 2 2δ 1 − 2δ 1 M A − d 1 C.(35)
We
hope a 2 2δ1 < δ 1 [(M A − a 2δ1 ) 2 + ( − a 2δ1 ) 2 ] + 2δ 1 M A + d 1 C. Since 2[(M A − a 2δ1 ) 2 + ( − a 2δ1 ) 2 ] ≥ (M A − ) 2 , it is enough to show =⇒ a 2 δ 2 1 < (M A − ) 2 + 2 M A + 2d 1 C δ 1 = (M A ) 2 + 2 + 2d 1 C δ 1 .(36)
Since M A is bounded by a δ1 − from the feasibility conditions, we show
(37) =⇒ a 2 δ 2 1 − 2 < ( a δ 1 − ) 2 + 2d 1 C δ 1 ,(38)=⇒ a δ 1 < + d 1 C δ 1 . Since + d1C δ1 ≥ 2 d1C δ1 , it is enough to show a δ1 < 2 d1C δ1 . Due to C * = e1a d1 (M A * + ) − e1δ1 d1 (M A * + ) 2 = e1a d1 M * A − e1δ1 d1 (M * A ) 2 , it is enough to show a 2 δ 1 < 4d 1 C = 4e 1 aM A − 4e 1 δ 1 M 2 A = −4e 1 δ 1 (M A − a 2δ 1 ) 2 + 2e 1 a 2 δ 1 .(39)
Therefore we only need to show
(40) a 2 δ 1 < max(−4e 1 δ 1 (M A − a 2δ 1 ) 2 + 2e 1 a 2 δ 1 ) = 2e 1 a 2 δ 1 ,
and this requires we have 1 2 < e 1 < 1.Then the system (10)- (12) is globally stable if
(41) d 1 δ 1 a < e 1 − d 1 h 1 < d 1 , 0 < < a δ 1 , 1 2 < e 1 < 1.
Hopf Bifurcation
Now we will investigate the Hopf bifurcation for the system in terms of parameter a. In this paper, we will follow the method developed by Liu [38]. Firstly, let us consider the system (129)-(131), without the presence of an epibiont (e = 0), that is when e = 0. The Hopf bifurcation at a = a * can occur if A 2 (a * ), A 0 (a * ), and φ(a * ) = A 2 (a * )A 1 (a * ) − A 0 (a * ) are smooth functions of a in an open interval of a * ∈ R such that:
(1) A 1 (a * ) > 0, A 0 (a * ) > 0, and φ(a * ) = A 2 (a * )A 1 (a * ) − A 0 (a * ) = 0. (2) dφ(a)
da | a=a * = 0. We check the above, in appendix section 8.4, to state the following theorem
Optimal Control
In this section our goal is to investigate mechanisms in our crab-mussel-epibiont system, that, if controlled, could lead to optimal levels of crab or mussel densities. We assume that the attack rates u 1 , u 2 are not known a priori and enter the system as time-dependent controls. They no longer depend on the epibiont density. Instead we assume that the handling time depends on the epibiont density e in the following way, where
(42) h 1 (e) = 1 + e K , a(e) = a K − e 2 K .
These responses are for the range 0 ≤ e ≤ K. Increase in epibiont density still negatively effects mussel fecundity and the handling time for adult mussels increases with increasing epibiont density. This has a twofold advantage. We can visualise the system from the crab's point of view. That is, the crab can "optimally" control its attack rate, to reach the best possible population density. Also we can visualise the system from the mussel's point of view. That is, the mussel can induce defenses or other mechanisms, that would alter the attack rate of the crab, thus enabling the mussel population density to reach optimum levels. Our model takes the following form,
C = − d 1 C + e 1 u 1 (t) M A 1 + h 1 (e)u 1 (t)M A + h 2 u 2 (t)M J C + e 2 u 2 (t) M J 1 + h 1 (e)u 1 (t)M A + h 2 u 2 (t)M J C,(43)(44) M A = bM J − δ 1 M 2 A − u 1 (t) M A 1 + h 1 (e)u 1 (t)M A + h 2 u 2 (t)M J C,(45)M J = a(e)M A − bM J − u 2 (t) M J 1 + h 1 (e)u 1 (t)M A + h 2 u 2 (t)M J C,(46)e = b 1 e(1 − e K ).
We next derive optimal strategies for three objective functions, where we maximize both crab and mussel populations. To simplify the calculation, we will consider the case when e = K, which is when the epibiont achieves carrying capacity. In this case our system reduces to
C = − d 1 C + e 1 u 1 (t) M A 1 + 2u 1 (t)M A + h 2 u 2 (t)M J C + e 2 u 2 (t) M J 1 + 2u 1 (t)M A + h 2 u 2 (t)M J C,(47)(48) M A = bM J − δ 1 M 2 A − u 1 (t) M A 1 + 2u 1 (t)M A + h 2 u 2 (t)M J C,(49)M J = a 2 M A − bM J − u 2 (t) M J 1 + 2u 1 (t)M A + h 2 u 2 (t)M J C.
5.1.
Maximizing crab denisty w.r.t. attacking rates. To maximize density of the crab, the density of juvenile mussels (thus leading to more adult mussels, its favored food) should be maximized. Crab attack rates should be miminized on the juvenile mussels, as they are less preferred by the crab. Thus we choose the following objective functional,
(50) J 1 (u 1 , u 2 ) = T 0 (C + M J − 1 2 u 2 2 )dt, s.t. (47)-(49) and C(t 0 ) = C 0 , M A (t 0 ) = M A0 , M J (t 0 ) = M J0
. and we search for the optimal controls in the set U where
(51) U = {(u 1 , u 2 )|u i measurable, 0 ≤ u 1 ≤ 1, 0 ≤ u 2 ≤ 1, t ∈ [0, T ], ∀T }.
The goal is to seek an optimal (u * 1 , u * 2 ) s.t.,
(52) J 1 (u * 1 , u * 2 ) = max (u1,u2) T 0 (C + M J − 1 2 u 2 2 )dt.
We can state the following existence theorem, Theorem 5.1. Consider the optimal control problem (47)- (49).
There exists (u * 1 , u * 2 ) ∈ U s.t. (53) J 1 (u * 1 , u * 2 ) = max (u1,u2)∈U T 0 (C + M J − 1 2 u 2 2 )dt.
Proof. The compactness (closed and bounded in ODE case) of the functional J 1 follows from the global boundedness of the state variables via theorem 3.3, and the boundedness assumption on the controls. Also the functional J 1 is concave in the argument u 2 . This is easily verified via standard application [39]. These in conjunction give the existence of an optimal control via application of classical one predator-two prey theory [40].
In order to derive necessary conditions on the optimal control, we use Pontryagin's maximum principle (PMP). The Hamiltonian for our problem is given by
(54) H = C + M J − 1 2 u 2 2 + λ 1 C + λ 2 M A + λ 3 M J .
We use the Hamiltonian to find a differential equation of the adjoint λ i , i = 1, 2, 3.
λ 1 (t) = − λ 1 −d 1 + M A e 1 u 1 + M J e 2 u 2 M A h 1 u 1 + M J h 2 u 2 + 1 + λ 2 u 1 M A M A h 1 u 1 + M J h 2 u 2 + 1 + λ 3 u 2 M J M A h 1 u 1 + M J h 2 u 2 + 1 − 1, λ 2 (t) = − λ 1 e 1 u 1 C M A h 1 u 1 + M J h 2 u 2 + 1 − (M A e 1 u 1 + M J e 2 u 2 ) Ch 1 u 1 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 − λ 2 −2 δ 1 M A − u 1 C M A h 1 u 1 + M J h 2 u 2 + 1 + u 1 2 M A Ch 1 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 − λ 3 a 2 + u 2 M J Ch 1 u 1 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 , λ 3 (t) = − λ 1 e 2 u 2 C M A h 1 u 1 + M J h 2 u 2 + 1 − (M A e 1 u 1 + M J e 2 u 2 ) Ch 2 u 2 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 − λ 2 b + u 1 M A Ch 2 u 2 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 − λ 3 −b − u 2 C M A h 1 u 1 + M J h 2 u 2 + 1 + u 2 2 M J Ch 2 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 − 1,(55)
with the transversality condition given as
(56) λ 1 (T ) = λ 2 (T ) = λ 3 (T ) = 0.
Considering the optimality conditions, the Hamiltonian function is differentiated with respect to control variables u 1 and u 2 resulting in
∂H ∂u 1 =λ 1 M A e 1 C M A h 1 u 1 + M J h 2 u 2 + 1 − (M A e 1 u 1 + M J e 2 u 2 ) CM A h 1 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 + λ 2 − M A C M A h 1 u 1 + M J h 2 u 2 + 1 + u 1 M A 2 Ch 1 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 + λ 3 u 2 M J CM A h 1 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 , ∂H ∂u 2 =λ 1 M J e 2 C h 1 u 1 M A + h 2 u 2 M J + 1 − (e 1 u 1 M A + e 2 u 2 M J ) CM J h 2 (h 1 u 1 M A + h 2 u 2 M J + 1) 2 + λ 2 u 1 M A CM J h 2 (h 1 u 1 M A + h 2 u 2 M J + 1) 2 + λ 3 − M J C h 1 u 1 M A + h 2 u 2 M J + 1 + u 2 M J 2 Ch 2 (h 1 u 1 M A + h 2 u 2 M J + 1) 2 − u 2 .(57)
We find a characterization of u * 1 by considering three cases:
∂H ∂u 1 < 0 ⇒ u * 1 = 0, ∂H ∂u 1 = 0 ⇒ u * 1 = u 11 s.t. ∂H ∂u 1 u1 1 = 0, ∂H ∂u 1 > 0 ⇒ u * 1 = 1.(58)
When the control is at the upper bound,u 11 is strictly greater than 1. When the control is at the lower bound, the solution of u 11 is strictly less than 0. Similarly for u * 2 . Thus a compact way of writing the optimal control is
u * 1 = min(1, max(0, u 11 )), u * 2 = min(1, max(0, u 21 )),(59)
where u 11 and u 21 are given by
u 11 = w 1 w 2 , u 21 = −e 1 λ 1 + λ 2 M J (e 1 h 2 λ 1 − e 2 h 1 λ 1 + h 1 λ 3 − h 2 λ 2 ) .(60)
with
w 1 =CM J 2 e 1 3 h 2 3 λ 1 3 − 3 CM J 2 e 1 2 e 2 h 1 h 2 2 λ 1 3 + 3 CM J 2 e 1 e 2 2 h 1 2 h 2 λ 1 3 − CM J 2 e 2 3 h 1 3 λ 1 3 + 3 CM J 2 e 1 2 h 1 h 2 2 λ 1 2 λ 3 − 3 CM J 2 e 1 2 h 2 3 λ 1 2 λ 2 − 6 CM J 2 e 1 e 2 h 1 2 h 2 λ 1 2 λ 3 + 6 CM J 2 e 1 e 2 h 1 h 2 2 λ 1 2 λ 2 + 3 CM J 2 e 2 2 h 1 3 λ 1 2 λ 3 − 3 CM J 2 e 2 2 h 1 2 h 2 λ 1 2 λ 2 + 3 CM J 2 e 1 h 1 2 h 2 λ 1 λ 3 2 − 6 CM J 2 e 1 h 1 h 2 2 λ 1 λ 2 λ 3 + 3 CM J 2 e 1 h 2 3 λ 1 λ 2 2 − 3 CM J 2 e 2 h 1 3 λ 1 λ 3 2 + 6 CM J 2 e 2 h 1 2 h 2 λ 1 λ 2 λ 3 − 3 CM J 2 e 2 h 1 h 2 2 λ 1 λ 2 2 + CM J 2 h 1 3 λ 3 3 − 3 CM J 2 h 1 2 h 2 λ 2 λ 3 2 + 3 CM J 2 h 1 h 2 2 λ 2 2 λ 3 − CM J 2 h 2 3 λ 2 3 + e 1 e 2 h 1 2 λ 1 2 − e 1 h 1 2 λ 1 λ 3 − e 2 h 1 2 λ 1 λ 2 + h 1 2 λ 2 λ 3 , w 2 =M A h 1 2 (e 1 2 h 2 λ 1 2 − e 1 e 2 h 1 λ 1 2 + e 1 h 1 λ 1 λ 3 − 2 e 1 h 2 λ 1 λ 2 + e 2 h 1 λ 1 λ 2 − h 1 λ 2 λ 3 + h 2 λ 2 2 ).(61)
We can thus state the following theorem, Theorem 5.2. An optimal control (u * 1 , u * 2 ) ∈ U for the system (47)-(49) that maximises the objective functional J 1 is characterised by (59).
5.2.
Maximizing mussel density w.r.t. attacking rates. To maximinize mussel density, the attack rate on the adult mussels should be minimized. We choose the following objective function,
(62) J 2 (u 1 , u 2 ) = T 0 (M A + M J − 1 2 u 2 1 )dt,(63) U = {(u 1 , u 2 )|u i measurable, 0 ≤ u 1 ≤ 1, 0 ≤ u 2 ≤ 1, t ∈ [0, T ], ∀T }.
We can state the following existence theorem, Theorem 5.3. Consider the optimal control problem (47)- (49). There exists (u * 1 , u * 2 ) ∈ U s.t.
(64) J 2 (u * 1 , u * 2 ) = max (u1,u2)∈U T 0 (M A + M J − 1 2 u 2 1 )dt.
The proof is similar to theorem 5.1. We can next state Theorem 5.4. An optimal control (u * 1 , u * 2 ) ∈ U for the system (47)-(49) that maximizes the objective function J 2 is characterised by u * 1 = min(1, max(0, u 12 )), u * 2 = min(1, max(0, u 22 )).
For the details of the proof of the above necessary conditions and forms of u 12 , u 22 the reader is refered to the appendix section 8.5.
5.3.
Maximizing mussel density w.r.t. intraspecific competition rate. In this approach we view the competition coefficient as a control. To reach certain optimal population densities, the mussels would maximise the densities of both adult and juvenile groups whilst minimising intraspecific competition. To this end our system reduces to
(66) C = −d 1 C + e 1 u 1 M A 1 + 2u 1 (t)M A + h 2 u 2 (t)M J C + e 2 u 2 M J 1 + 2u 1 M A + h 2 u 2 M J C,(67)M A = bM J − δ 1 (t)M 2 A − u 1 M A 1 + 2u 1 M A + h 2 u 2 M J C,(68)M J = a 2 M A − bM J − u 2 M J 1 + 2u 1 M A + h 2 u 2 M J C.
We choose the following objective function,
(69) J 3 (δ 1 ) = T 0 (M A + M J − 1 2 δ 2 1 )dt, s.t. (66)-(68) and C(t 0 ) = C 0 , M A (t 0 ) = M A0 , M J (t 0 ) = M J0
. and we search for the optimal controls in the set U 1 . Where,
(70) U 1 = {δ 1 |δ 1 measurable, 0 ≤ δ 1 ≤ ∞, t ∈ [0, T ], ∀T }.
We can state the following existence theorem, Theorem 5.5. Consider the optimal control problem (66)-(68). There exist (u * 1 , u * 2 ) ∈ U s.t.
(71)
J 3 (δ 1 ) = max (u1,u2)∈U T 0 (M A + M J − 1 2 δ 2 1 )dt.
The proof is similar to theorem 5.1. We can next state Theorem 5.6. An optimal control (u * 1 , u * 2 ) ∈ U for the system (66)-(68) that maximises the objective function J 3 is characterised by
(72) δ * 1 = max(0, −M 2 A λ 2 )
. For the details of the proof of the above necessary conditions and forms of u 12 , u 22 the reader is refered to the appendix section 8.6. We set h 1 = 2 since h 1 = 1 + e K , however, if we just assume h 1 as a constant and increase h 1 and keep other parameters the same, we found the optimal control u 2 always to be 0, and u 1 decreases and gradually become stable. In fact, when h 1 achieves to some critical value, u 1 begins to increase slightly. Based on e = K, as for system (129)-(132) , the optimal foraging strategies are u 1 = 0, u 2 = 1. However, for the control system (43)-(46) , J 1 (u 1 , u 2 ) will be maximized when u 1 = 0.4343, u 2 = 0; J 2 (u 1 , u 2 ) will be maximized when u 1 = 0, u 2 = 0.9121 and J 3 (δ 1 ) will be maximized when δ 1 = 0 with data set we mentioned.
Discussion and Conclusion
Epibiotic invasive species often have anti-predator defenses that are behavioral, chemical, or mechanical [41,42], giving them a survival advantage in a novel habitat because potential predators avoid using them as a food source [43,44]. While this provides a benefit to the basibiont, it impacts other members of the community, including predators of the basibiont as they may show lower preference for basibionts that are overgrown by invasive epibionts [12,45]. However, the effects of epibionts on basibionts are not always positive. Many times the epibiont may attract predators resulting in consumption of the epibiont, which automatically leads to consumption of the basibiont. This is refered to in the literature as "shared doom" [24]. Epibionts can also negatively affect basibiont fecundity and fitness, resulting in fewer offspring [11]. In essence, invasion of predator-prey communities by epibionts is complex, and warrants a thorough mathematical investigation of their impact on predator-prey interactions and populations.
Population cycles are common in predator-prey communities, and although these are possible in our model without epibionts, extensive numerical simulations indicate that at carrying capacity e = K, a Hopf bifurcation is not possible. This points to the epibiont having a stabilizing influence in that it can eliminate population oscillations. A rigorous proof of this is an interesting future direction. Within our study, theorem 3.4 tells us that if the energy gain from the adult mussel is in a certain critical region 0 < e 1 < 1, then one has global stability; even very large perturbations would still allow the system to return to its base state.
Our central question focuses on the effect of the introduced epibiont on the population densities of the local crab-mussel communities. Could high epibiont density lead to lower mussel populations (and so subsequently lower crab populations)? To answer this we compare the equilibrium levels of the juvenile mussel population, "no epibiont" case versus "epibiont reaches carrying capacity" case. If the epibionts do have an adverse effect then we would have M * J | (e=K) < M * J | (e=0) . Comparing these yields,
(74) e 1 + h 2 a < e 2 + h 1 b.
Although we know e 1 > e 2 , under high epibiont density (e = K), we have h 1 >> h 2 , thus even if a < b, (74) could easily hold meaning that there is an adverse effect on the juvenile mussel density via epibiont presence, leading to fewer adults subsequently, and so epibionts could clearly be a factor in mussel population declines as seen via data from the Gulf of Maine [8]. Such decline could eventually lead to crab population decline as well, if the crab species is a specialist on mussels. However, the effects of epibionts on mussel fecundity could also be a cause of predator decline. In order to understand the effects of epibiosis, we investigate the equilibrium density of the crab populations for the "no epibiont" case versus the "epibiont reaches carrying capacity" case. What we note is
(75) C * J | (e=0) = e 1 (a(e 1 − d 1 h 1 ) − d 1 δ 1 ) (e 1 − d 1 h 1 ) 2 , C * J | (e=K) = 1 2 ae 2 bd1 (e2−d1h2)δ1 d 1 − e 2 b e 2 − d 1 h 2 .
Clearly, as epibiont cover reduces mussel fecundity from a, to a/2, this directly affects the crab population. In the (e = K) case there is an increase by a factor of only a 2 , as opposed to a factor of a in the (e = 0) case. Thus reduced fecundity in mussels due to epibiont cover, can also reduce crab populations as well.
We assume logistic growth in the epibiont density. Although in the Gulf of Maine epibiont density fluctuates seasonally, our model could be a useful predictive tool in periods where logistic growth is seen. In locations such as Japan and New Zealand, the epibiont D. vexillum grows logistically [11] due to water temperatures staying above the threshold for D. vexillum viability.
We also use optimal control theory to visualise various optimal scenarios to maximize each crab and mussel densities. Herein, we change the problem slightly, and assume the attack rates u 1 , u 2 are not known a priori, but are time-dependent. Our objective is to explore various scenarios that a species of crab or mussel might attempt to optimise, by manipulating the attack rates. Epibionts are assumed to be present, and their effect is modeled via increasing the handling time h 1 of adult mussels, as epibiont density increase.
Simulations suggest (Fig. 4a) that even under high epibiont density (which in this scenario amounts to doubled h 1 ), the crab should not attack juvenile mussels, but attempt to attack adults. Fig. 4b demonstrates, that what is optimal for the mussel is if u 1 = 0, and so the adult mussel must induce defenses to reduce u 1 , even if it realistically cannot drive that rate to zero. This confirms experimental results of rapid shell thickening by mussels, seen via [15]. Fig. 5a looks at the attack rate on adut mussels u 1 , as h 1 changes. Here, we are trying to maximize mussel populations, and u 1 decreases as h 1 increases, as expected. However, u 1 is approximately 0.17; that is, it does not change significantly if handling times become very large. Curiously, it goes up ever so slightly as shown in Fig. 5b. This likely corresponds to the adult mussel thickening its shell just enough to increase handling time by the crab. Fig. 8 looks at the attack rate on juvenile mussels u 2 , as h 1 changes. Here again we are trying to maximize mussel populations. When handling time on adults is low, juvenile mussels are protected from crab predation due to the predators preference for larger mussels. However, when handling time on adults is high (greater than 2.1 in this simulation), it is likely that crabs would switch to the juvenile; therefore, juveniles must be able to disperse or seek refuge in order to bring attack rates on the juvenile mussels to zero (in turn maximizing their population size). Young mussels drift in the water column until they reach a size of approximately 2.5mm, then they settle on a filamentous algal substrate [3]. Some mussel species settle on algal substrate until they are 30 mm in length [48]. This substrate acts as refuge and must be available for juveniles in order to maximize mussel populations. However, with degradation of suitable habitat, the opportunities for escape from crab populations becomes diminished. Major disturbance events, either natural or anthropogenic, in conjunction with invasion by substrate-smothering colonial species and voracious predators, are likely to decrease opportunities for escape. Endeavors to model predator-prey systems incorporating prey refuge may yield surprising results on stability [23], [25]. Thus it would be very interesting to model refuge effects for the juvenile mussels herein.
As a future direction in modelling the crab-mussel-epibiont interaction, we would also like to examine interference effects [19,21,20]. This effect has often seen to be stabilising [34], and thus modeling interference among the crab population, at high epibiont density is also realistic. Note, Theorem 5.6 suggests that eliminating intraspecific competition among mussels is optimal from their point of view, and yields a maximum density, if there was no competition present. Future modelling endeavors may also investigate if high epibiont cover promotes cannibalism in crabs. That is, under high epibiont cover of adult mussels, would a crab prefer to cannibalise its own conspecifics [46,47], rather than switching to juvenile mussels? Another interesting future direction is to look at the foraging of crabs as they move in and out of patches containing mussels, some of which might be protected by mussel farmers, akin to marine protected areas [9,10]. A spatially explicit approach to this end, modeling a changing habitat based on mussel density, would also be interesting [14].
The empirical literature shows that while epibionts alter the prey choice of predators, including crabs and sea stars [28,29,30,22], there are no prey switching experiments using D. vexillum. Our goal is to provide firm modeling grounds for the scope of such experiments in the future. Thus a logical next step for empirical studies is to conduct experiments with D. vexillum to confirm our switching hypothesis, as well as look at switching scenarios under varying levels of overgrowth (with both living and artificial epibionts such as in [28]). An interesting research question therein would be to ask if one sees the inverted parabola shaped curve, typical of OFT scenarios when measuring crab size versus mussel preference. If our switching hypothesis is confirmed, this should not be the case, as smaller size juvenile mussels should be preferred to adults under heavy epibiont cover. All in all we hope our results will help devise suitable strategies and measures that will enable a boost in dwindling mussel populations, particularly as new complexities arise in ecosystems, driven by rapid increase in invasions.
Acknowledgements
JL and RP would like to acknowledge valuable support from the NSF via DMS-1715377 and DMS-1839993.
Appendix
8.1. Optimal Strategy in our setting. Here, we give a rigorous reasoning for our switching hypothesis. If we following standard OFT, we can consider a fitness function
(76) R(u 1 , u 2 ) = e 1 u 1 M A 1 + h 1 u 1 M A + h 2 u 2 M J + e 1 u 1 M J 1 + h 1 u 1 M A + h 2 u 2 M J .
We endeavor to maximize R(u 1 , u 2 ), the net rate of energy intake during foraging. The optimal strategy for a crab (according to classical OFT) relies on the density of mussels. That is for each (M A , M J ) , we get a set of optimal controls S(M A , M J ) known as the strategy map.
(77) S(M A , M J ) = {(u 1 , u 2 )|R(u 1 , u 2 ) = max 0≤p1,p2≤1 R(p 1 , p 2 )}.
This is (129)-(132), which is actually a control system with controls (u 1 , u 2 ) relying on the state of the system. Now we look for controls belonging to the strategy map S(M A , M J ). Then we calculate the derivatives of S(M A , M J ) to investigate the maximizing controls u 1 and u 2 .
(78) ∂R ∂u 1 = M A e 1 + M A M J u 2 (e 1 h 2 − e 2 h 1 ) (1 + h 1 u 1 M A + h 2 u 2 M J ) 2 ,(79)∂R ∂u 2 = M J (e 2 − M A u 1 (e 1 h 2 − e 2 h 1 )) (1 + h 1 u 1 M A + h 2 u 2 M J ) 2 .
The sign of ∂R ∂u1 and ∂R ∂u2 depend on the e 1 h 2 − e 2 h 1 . A tricky point here is that attack rates depend critically on the density of adult and juvenile mussels. That is of (u 1 = 1, u 2 = 0), or (u 1 = 0, u 2 = 1) are feasible as attack rates if the mussel densities are above a certain density. However if M A , or M J fall below a certain critical level, theory predicts that the less preferred prey should also be attacked, and one might have a situation of (u 1 = 1, u 2 = 1). What we show next, is that if certain parametric restrictions are met, (u 1 = 1, u 2 = 0), or (u 1 = 0, u 2 = 1) are the only optimal choices for the crab, irrespective of mussel density. Proof. If e1 h1 > e2 h2 , ∂R ∂u1 > 0, the maximum of R(u 1 , u 2 ) is thusly achieved for u 1 = 1. And since the sign of ∂R ∂u2 does not depend on u 2 it follows if ∂R ∂u2 = 0, R(u 1 , u 2 ) will be maximized either with u 2 = 0 or u 2 = 1.Then we get the strategy map
(80) S(M A , M J ) = (1, 1) if M A < e2 e1h2−e2h1 , (1, 0) if M A > e2 e1h2−e2h1 , (1, u 2 ), 0 ≤ u 2 ≤ 1 if M A = e2 e1h2−e2h1 . Now M * A = d1
e1−d1h1 from the earlier stability calculations. We note,
(81) M * A = d 1 e 1 − d 1 h 1 > e 2 e 1 h 2 − e 2 h 1 ,
as long as d 1 > e2 h2 , and if this is enforced (u 1 = 1, u 2 = 0) is the only optimal strategy for the crab.
If e1 h1 < e2 h2 in order to maximize R(u 1 , u 2 ), we need u 2 = 1. The strategy map will switch to
(82) S(M A , M J ) = (1, 1) if M J < e1 e2h1−e1h2 , (0, 1) if M J > e1 e2h1−e1h2 , (u 1 , 1), 0 ≤ u 1 ≤ 1 if M J = e2 e1h2−e2h1 . Now M * J = d1
e2−d1h2 from the earlier stability calculations. We note,
(83) M * J = d 1 e 2 − d 1 h 2 > e 1 e 2 h 1 − e 1 h 2 as long as d 1 > e1
h1 , and if this is enforced (u 1 = 0, u 2 = 1) is again, the only optimal strategy for the crab.
8.2. Proof of theorem 3.2. The Jacobian matrix about (C * , M * A , M * J ) of system (10)- (12), without epiboint, is given by
(84) J = 0 J 12 0 J 21 J 22 J 23 0 J 32 J 33 where (85) J 12 = a(e 1 − d 1 h 1 ) − d 1 δ 1 ,(86)J 21 = − d 1 e 1 ,(87)J 22 = − a(e 1 − d 1 h 1 ) 2 + d 2 1 δ 1 h 1 + d 1 δ 1 e 1 e 1 (e 1 − d 1 h 1 ) ,(88)J 23 = b,(89)J 32 = a,(90)J 33 = −b.
The characteristic equation is
(91) λ 3 + A 2 λ 2 + A 1 λ + A 0 = 0, with (92) A 2 = −J 33 − J 22 ,(93)A 1 = −J 12 J 21 + J 22 J 33 − J 23 J 32 ,
and (94)
A 0 = J 33 J 21 J 12 .
It follows from the Routh-Hurwitz stability criteria that all eigenvalues have negative real part if
(95) A 2 > 0, A 0 > 0, A 2 A 1 > A 0 .
It is obvious that the first two conditions are always satisfied under feasibility condition (16). Furthermore,
A 2 A 1 − A 0 > 0 if J 23 J 32 − J 22 J 33 < 0. (96) J 23 J 32 − J 22 J 33 = − bd 1 (ad 1 h 2 1 − ae 1 h 1 + d 1 δ 1 h 1 + δ 1 e 1 ) e(e 1 − d 1 h 1 ) < 0.
It is enough to solve bd 1 (ad 1 h 2 1 − ae 1 h 1 + d 1 δ 1 h 1 + δ 1 e 1 ) > 0, that is, e 1 − d 1 h 1 < d1δ1 a + d1δ1e1 ah1 . Therefore, the system (10)- (12) is asymptotically stable if
(97) d 1 δ 1 a < e 1 − d 1 h 1 < d 1 δ 1 a + d 1 δ 1 e 1 ah 1 .
8.3. Proof of Theorem 3.3. The equilibrium state, of the system (129)-(132), for the epiboint is e = K. At the interior equilibrium state, the parameters u 1 = 0, u 2 = 1 and a(e) = a 2 . Since e will not effect the solution of C, M A and M J once u 1 , u 2 and a(e) are determined, then it is enough to nvestigate the following
three dimension system with the equilibrium (C * , M * A , M * J ) = ( 1 2 ae2 bd 1 (e 2 −d 1 h 2 )δ 1 d1 − e2b e2−d1h2 , bd1 (e2−d1h2)δ1 , d1 e2−d1h2 ) when e = K. (98) dC dt = −d 1 C + e 2 M J 1 + h 2 M J C, (99) dM A dt = bM J − δ 1 M 2 A , (100) dM J dt = aM A − bM J − M J 1 + h 2 M J C.
The Jacobian matrix about (C * , M * A , M * J ) is
(101) J = 0 0 J 13 0 J 22 J 23 J 31 J 32 J 33 where (102) J 13 = 1 2 [a(e 2 − d 1 h 2 ) bd1 (e2−d1h2)δ1 − 2bd 1 ](e 2 − d 1 h 2 ) d 1 ,(103)J 22 = −2δ 1 bd 1 (e 2 − d 1 h 2 )δ 1 ,(104)J 23 = b,(105)J 31 = − d 1 e 2 ,(106)J 32 = a 2 ,(107)J 33 = − 1 2 a(e 2 − d 1 h 2 ) 2 bd1 (e2−d1h2)δ1 + 2bd 2 1 h 2 e 2 d 1 .
Since all the parameters are positive, it is obvious that J 22 < 0, J 23 > 0, J 31 < 0, J 32 > 0, and J 33 < 0. Under the feasibility condition (26), J 13 > 0. And the characteristic equation is given by
(108) λ 3 + B 2 λ 2 + B 1 λ + B 0 = 0, where (109) B 2 = −J 33 − J 22 ,(110)B 1 = −J 13 J 31 + J 22 J 33 − J 23 J 32 ,(111)B 0 = J 13 J 22 J 31 ,
By Routh Hurwitz stability criteria, all eigenvalues have negative real part if
(112) B 0 > 0, B 1 > 0, B 2 > 0, B 2 B 1 − B 0 > 0.
It is easy to check B 2 > 0 and B 0 > 0 under the feasility criterion (26). And
B 1 > 0 if J 22 J 33 − J 23 J 32 > 0. J 22 J 33 − J 23 J 32 = (4 bd1 δ1(e2−d1h2) d 1 δ 1 h 2 − 2ad 1 h 2 + ae 2 )b 2e 2 d 1 = (4 bd1 δ1(e2−d1h2) d 1 δ 1 h 2 − ad 1 h 2 − ad 1 h 2 + ae 2 )b 2e 2 d 1 = (4 bd1 δ1(e2−d1h2) d 1 δ 1 h 2 − ad 1 h 2 + a(e 2 − d 1 h 2 ))b 2e 2 d 1 .(113)
To make J 22 J 33 −J 23 J 32 > 0, it is enough to show 4 bd1 δ1(e2−d1h2) d 1 δ 1 h 2 −ad 1 h 2 > 0, which gives us e 2 − d 1 h 2 < 16bd1δ1 a 2 . Furthermore,
B 2 B 1 − B 0 = J 13 J 31 J 33 − J 2 22 J 33 + J 22 J 23 J 32 − J 22 J 2 33 + J 23 J 32 J 33 = J 13 J 31 J 33 + J 22 (J 23 J 32 − J 22 J 33 ) + J 33 (J 23 J 32 − J 22 J 33 ).(114)
Since J 22 < 0 and J 33 < 0,
J 22 J 33 − J 23 J 32 > 0 implies B 2 B 1 − B 0 > 0. Thus, the system (129)-(132) is asymptotically stable if (115) 4bd 1 δ1 a 2 < e 2 − d 1 h 2 < 16bd 1 δ 1 a 2 .
8.4. Proof of theorem 4.1. Now let a, the growth rate of juvenile mussels, as the bifurcation parameter. Therefore, if condition (16) holds, A 0 (a * ) are always positive.
A 2 (a * ) > 0 if e 1 − d 1 h 1 < d1δ1 a + d1δ1e1 ah1 . And φ(a * ) = A 2 (a * )A 1 (a * ) − A 0 (a * ) = 0 if (116) a * = f 1 f 2 , where f 1 = 4bδ 2 1 h 5 1 (M * A ) 7 + (2b 2 δ 1 h 5 1 + 20bδ 2 1 h 4 1 (M * A ) 6 + (10b 2 δ 1 h 4 1 + 40bδ 2 1 h 3 1 )(M * A ) 5 + (4C * bδ 1 h 3 1 + 20b 2 δ 1 h 3 1 + 2C * δ 1 e 1 h 2 1 + 40bδ 2 1 h 2 1 )(M * A ) 4 + (C * b 2 h 3 1 + 12C * bδ 1 h 2 1 + 20b 2 δ 1 h 2 1 + 4C * δ 1 e 1 h 1 + 20bδ 2 1 h 1 )(M * A ) 3 + (3C * b 2 h 2 1 + 12C * bδ 1 h 1 + 10b 2 δ 1 h 1 + 2C * δ 1 e 1 + 4bδ 2 1 )(M * A ) 2 + ((C * ) 2 bh 1 + 3C * b 2 h 1 + (C * ) 2 e 1 + 4C * bδ 1 + 2b 2 δ 1 )M * A + (C * ) 2 b + C * b 2 , f 2 = b(M * A h 1 + 1) 3 (2(M * A ) 3 δ 1 h 2 1 + b(M * A ) 2 h 2 1 + 4(M * A ) 2 δ 1 h 1 + 2bM * A h 1 + 2M * A δ 1 + C * + b),(117)
and C * , M * A are given by (13) and (14). Furthermore, it is easy to verify that
dφ(a) da | a=a * = − (b(M * A ) 3 h 3 1 + 3b(M * A ) 2 h 2 1 + 3bM * A h 1 + b)(2(M * A ) 2 δ 1 h 2 1 (M * A + 1) 5 − (bh 2 1 + 4δ 1 h 1 )(M * A ) 2 + (2bh 1 + 2δ 1 )M * A + C * + b) (M * A + 1) 5 < 0 = 0.(118)
8.5. Proof of theorem 5.4. The Hamiltonian of the system is given by
(119) H = M A + M J − 1 2 u 2 1 + λ 1 C + λ 2 M A + λ 3 M J .
We use the Hamiltonian to find a differential equation of the adjoint λ i , i = 1, 2, 3.
λ 1 (t) = −λ 1 −d 1 + M A e 1 u 1 + M J e 2 u 2 M A h 1 u 1 + M J h 2 u 2 + 1 + λ 2 u 1 M A M A h 1 u 1 + M J h 2 u 2 + 1 + λ 3 u 2 M J M A h 1 u 1 + M J h 2 u 2 + 1 , λ 2 (t) = −λ 1 e 1 u 1 C M A h 1 u 1 + M J h 2 u 2 + 1 − (M A e 1 u 1 + M J e 2 u 2 ) Ch 1 u 1 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 − λ 2 −2 δ 1 M A − u 1 C M A h 1 u 1 + M J h 2 u 2 + 1 + u 1 2 M A Ch 1 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 − λ 3 a/2 + u 2 M J Ch 1 u 1 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 − 1, λ 3 (t) = −λ 1 e 2 u 2 C M A h 1 u 1 + M J h 2 u 2 + 1 − (M A e 1 u 1 + M J e 2 u 2 ) Ch 2 u 2 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 − λ 2 b + u 1 M A Ch 2 u 2 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 − λ 3 −b − u 2 C M A h 1 u 1 + M J h 2 u 2 + 1 + u 2 2 M J Ch 2 (M A h 1 u 1 + M J h 2 u 2 + 1) 2 − 1,(120)
with the transversality condition gives as (121) λ 1 (T ) = λ 2 (T ) = λ 3 (T ) = 0 8.6. Proof of theorem 5.6. The Hamiltonian of our problem is given by
(126) H = M A + M J − 1 2 δ 2 1 + λ 1 C + λ 2 M A + λ 3 M J .
The differential equations for λ 1 (t), λ 2 (t), λ 3 (t), are standard and are derived as in Theorem 5.4. the transversality condition is
(127) λ 1 (T ) = λ 2 (T ) = λ 3 (T ) = 0.
Considering ∂H ∂δ1 = −M A 2 λ 2 − δ 1 , we derive the optimal control for J 3 (δ 1 ) optimal control change with increasing handling time of adult mussels Figure 8. In this simulation we look at how u 2 changes w.r.t. h 1 .
(128) δ * 1 = max(0, −M 2 A λ 2 ).
Here h 2 = 1. We want to see the change in the control u 2 as h 1 increases. The control u 1 = 0 no matter how large h 1 is. What we notice is that u 2 suddenly goes down to 0 from 1, at a critical value h * 1 = 2.1.
Numerical Explorations of Bifurcations of alternate models
In the case e = K, we do not see a Hopf bifurcation numerically. It is worthwhile considering certain alternate models for the epibiont dynamics as future work. We motivate this via considering the following model, The only change here to crab-mussel system (129)-(132) is that we assume the carrying capacity of the epibiont is density dependent, and depends primarily on the adult mussel density that is K = K = k 1 M A + k 2 . Here k 2 represents alternate substrate that the epibiont can grow on. The four dimensional system has 11 parameters with four dependent variables. The following parameters are used in numerical simulations: The system evolve the stable limit cycles for parameter set 134. Time series for all species is shown in the figure 9a, while limit cycles in 2-D phase space are shown in fig 9b, 10a and 10b. To observe the more qualitative behavior of the model, one-parameter bifurcation diagram is drawn with respect to parameter d 1 and parameter a in the figures . A supercritical Hopf bifurcation occurs at d 1 = 0.3567 which emanates stable limit cycles. There is another supercritical hopf bifurcation at d 1 = 0.444. Between these two Hopf bifurcation, model has periodic solutions. After second Hopf bifurcation point model has stable solutions but crab populations are going to extinct. The dynamics is shown in one-parameter bifurcation diagram fig 11a. The qualitative dynamics has been also obtained for range of parameter a drawn in the fig 11b. Initially, for low parameter value a < 1.265 the crab population is too low but as parameter a increases, model exhibits stable coexistence. Further, it undergoes through supercritical Hopf bifurcation at parameter a = 3.386 which emanates stable limit cycles (green filled circle).
The parameter region has been obtained by drawing two-parameter (a, d 1 ) bifurcation diagram in the fig 12a. The parameter region for which one species goes extinction is shown by shaded region (extreme left), region for which stable coexistence is possible shown by red and region for which perioidc solution is possible shown by blue color in the diagram fig 12a. Another two-parameter (d 1 , δ) bifurcation diagram is drawn in the fig 12b. The parameter region for which stable coexistence occurs and region for which perioidc solution is possible is depicted in the diagram fig 12b. As shown by these bifurcation graphs, model has periodic solutions for biologically feasible choice of parameters and one can find the Hopf bifurctaion point for each of the parameters used in the model.
These results show that a Hopf bifurcation is possible, if one considers a density dependent carrying capacity for the epibiont. These results are robust in nature as (a) The biomasses of all species exhibited the periodic coexistence against the Time series is shown for parameter set 134. (b) A stable limit cycle in the Twodimensional phase space E, Mj for parameter set 134 Figure 10. Stable limit cycle in the 2-D pahase space plot different sets of parameters will yield the same qualitative behavior. The periodicity in the system is beneficial for harvesting and coexistence of all the species involved. (b) One-parameter bifurcation diagram with respect to parameter a Figure 11. One-parameter bifurcation diagrams to depict stablilty, Hopf Bifurcation point and periodic solutions with respect to parameter set 134.
(a) Two-parameters (a, d1) bifurcation diagram with respect to parameter set 134 (b) Two-parameters (d1, δ) bifurcation diagram with respect to parameter set 134 Figure 12. Two-parameter bifurcation diagrams to depict parameter region for the stable coexistence and periodic coexistence with respect to parameter set 134.
with positive initial conditions C(0) = C 0 , M A (0) = M A0 , M J (0) = M J0 , e(0) = e 0 . These responses are for the range 0 ≤ e ≤ K.
Figure 2 .
2The above figures verify theorem 3.4. We consider the parameters e 1 = 0.9, e 2 = 0.01, d 1 = 0.2, b = 1, h 1 = 0.2, h 2 = 0.1, δ 1 = 0.6, a = 0.8, = 0.2.(A) The initial condition (C 0 , M A0 , M J0 )=(0.5 0.5 0.5) (B) The initial condition (C 0 , M A0 , M J0 )=(500 500 500) (zooming in time scale). They both reach a stable level (0.9312 0.0326 0.2394).
Remark 3 .
3Note, although we prove global stability (under certain parametric restrictions) for the case without epibiont (e = 0), it is easily proven using the same approach as above for the (e = K) case by just replacing M A = M A + and defining a new Lyapunov function as V (C, M A , M J , e) = C + M A + M J + e.
Theorem 4. 1 .Figure 3 .
13Under the condition (16), there is a simple Hopf bifurcation of the positive equilibrium point (C * , M * A , M * J ) of model system (10)-(12) at some critical value of parameter a * given by (116) and (117). Here we demonstrate the species density change with time. We see in (A), the population of the species are stable when a = 0.235, while in (B) occurence of a Hopf bifurcation has lead to population cycles.
s.t. (47)-(49) and C(t 0 ) = C 0 , M A (t 0 ) = M A0 , M J (t 0 ) = M J0 . and we search for the optimal controls in the set U . Where,
5. 4 .Figure 4 .
44Numerical Simulations. In this subsection, we investigate via numerical simulation and compare the species' population of the control system (43)-(46) and the classical system (129)-(132) under the epibiont achieving the carrying capacity (e = K). Since the solutions of the states and adjoint equations are a prior bounded and concavity in the controls holds, the optimal controls exist by using a result from Fleming and Rishel[Chap III, Theorem 2.1, pp 63][40]. Forward-Backward Sweep iteration algorithms are used for numerical simulations. The following parameter set is chosen (73) d 1 = 0.1, e 1 = 0.9, e 2 = 0.5, h 1 = 2.0, h 2 = 1.0, b = 0.2, δ 1 = 0.1, (A) Solid curves are the density change for each species of the system (129)-(132) under e = K and the dashed line are the optimal state of the control system (43)-(46) for the objective function J 1 (u 1 , u 2 ) (B) Optimal controls of J 1 (u 1 , u 2 ) with the above parameter set (73).
Figure 5 .Figure 6 .Figure 7 .
567control change with increasing handling time of adult mussels (A) Optimal control u 1 changes with increasing h 1 (B) u 1 increases slightly with large h 1 (A) Solid curves are the density change for each species of the system (129)-(132) under e = K and the dashed line are the optimal state of the control system (43)-(46) for the objective function J 2 (u 1 , u 2 ) (B) Optimal controls forJ 2 (u 1 , u 2 ) with the ablove parameter set shown in (73). (A) Solid curves are the density change for each species of the system (129)-(132) under e = K and the dashed line are the optimal state of the control system (43)-(46) for the objective function J 3 (δ 1 ) (B) The optimal control for the objective function J 3 is awalys to be δ 1 = 0.
Lemma 8 . 1 .
81Consider (129)-(132). If e = 0, and d 1 > e2 h2 then (u 1 = 1, u 2 = 0) is the only optimal choices for the crab. Whereas if e = K, and d 1 > e1 h1 (u 1 = 0, u 2 = 1) is the only optimal choices for the crab.
dC dt = − d 1 C + e 1 u 1 (e) M A 1 + h 1 u 1 (e)M A + h 2 u 2 (e)M J C + e 2 u 2 (e) M J 1 + h 1 u 1 (e)M A + h 2 u 2 (e)M J C,(129)(130) dM A dt = bM J − δ 1 M 2 A − u 1 (e) M A 1 + h 1 u 1 (e)M A + h 2 u 2 (e)M J C,(131)dM J dt = a(e)M A − bM J − u 2 (e) M J 1 + h 1 u 1 (e)M A + h 2 u 2 (e= k 1 M A + k 2 .with positive initial conditions C(0) = C 0 , M A (0) = M A0 , M J (0) = M J0 , e(0) = e 0 . These responses are for the range 0 ≤ e ≤ K.
134 Figure 9 .
1349) A stable limit cycle in the Twodimensional phase space MA, MJ for parameter set Time-series and limit cycle in the 2-) A stable limit cycle in the Twodimensional phase space C, MJ for pa-
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| []
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[
"On The Spherical Clothoid",
"On The Spherical Clothoid"
]
| [
"Alexandru Ionut "
]
| []
| []
| We revisit a nonlinear spline primitive for 3-space first studied by Even Mehlum. It is the spherical clothoid, the spherical curve with geodesic curvature a linear function of arc length. We present its Cartesian coordinate functions using confluent hypergeometric functions (the Kummer functions) and its stereographic projection onto the complex plane. New Humbert series results are also presented along with generating function formulas related to the associated Meixner-Pollaczek polynomials. arXiv:2203.07963v2 [math.DG] | null | [
"https://arxiv.org/pdf/2203.07963v2.pdf"
]
| 247,450,845 | 2203.07963 | 99d8aba6a03ae6ad48ade58e96ef9aefd0cf4614 |
On The Spherical Clothoid
March 23, 2022 22 Mar 2022
Alexandru Ionut
On The Spherical Clothoid
March 23, 2022 22 Mar 2022
We revisit a nonlinear spline primitive for 3-space first studied by Even Mehlum. It is the spherical clothoid, the spherical curve with geodesic curvature a linear function of arc length. We present its Cartesian coordinate functions using confluent hypergeometric functions (the Kummer functions) and its stereographic projection onto the complex plane. New Humbert series results are also presented along with generating function formulas related to the associated Meixner-Pollaczek polynomials. arXiv:2203.07963v2 [math.DG]
Notation
The Pochhammer symbol:
(q) n = q(q + 1) . . . (q + n − 1) = Γ(q + n) Γ(q)
Gauss' hypergeometric series [1]:
2 F 1 (a, b, c; z) = ∞ m=0 (a) m (b) m (c) m z m m!
The confluent hypergeometric functions (the Kummer function) [2]:
1 F 1 (a, b; z) = ∞ m=0 (a) m (b) m z m m!
Humbert hypergeometric series in 2 variables [3,4]: The associated Meixner-Pollaczek polynomials Q λ n (x; φ, c) [6,7] defined by the recurrence relation:
φ0 = (n + c + 1)Q λ n+1 (x; φ, c) − 2[(n + λ + c) cos φ + x sin φ]Q λ n (x; φ, c) + (n + 2λ + c − 1)Q λ n−1 (x; φ, c)
where n = 0, 1, 2... and Q λ −1 (x; φ, c) = 0, Q λ 0 (x; φ, c) = 1
Summary of prior work
In mathematics, the word spline has come to mean a function defined piecewise by polynomials with certain continuity constraints. We are interested in a more general concept i.e. nonlinear splines meaning the curve primitive, the building block of the spline, can be an analytic function. In English, The term spline goes back centuries referring to the flexible strip of wood used by draftsmen to draw smooth curves. Lead weights called ducks or whales held the spline in place at certain points and the strip would take the shape that minimized the bending energy yielding a smooth curve. This shape arising in the physical world corresponds to the minimum energy curve (MEC) i.e. the curve that minimizes the L 2 norm of curvature (see [8] for a more comprehensive treatise). We can also examine the MEC spline element in 3D space. First we recall that the intrinsic quantities needed to describe the shape of a space curve are its curvature κ and torsion τ as functions of arc length s. Mehlum derived the relationship between curvature and torsion for this curve and a differential equation characterizing the curvature (1.4 and 1.3 in [9]):
κ 2 τ = C (1) (κ 2 ) 2 + κ 2 (κ 2 − 2D) 2 − 4ψ 2 + 4C 2 = 0(2)
C, D, and ψ are constants. The calculus of variations problem is solved without the approximation that leads to the ubiquitous cubic spline. However, working with the MEC in practical applications can have its own drawbacks. Mehlum goes on to study a modified version of the differential equation for curvature. (1.5 and 1.6 in [9]) Assuming κ 2 2D yields:
(κ 2 ) 2 − 4α 2 κ 2 + 4C 2 = 0 (3) α = ψ 2 − D 2 is a constant.
We have a simple solution:
κ 2 = α 2 s 2 + C 2 α 2(4)
When C = 0 the resulting planar curve is the clothoid i.e. curvature is a linear function of arc length. The clothoid has since been heavily investigated in the theory of nonlinear planar splines [8,10,11]. For all other C, Mehlum discovered that this curve lies on a sphere and that R = α C (2.16 in [9]). Mehlum and Wimp bring up a known necessary and sufficient condition for a space curve to be spherical (2.8 in [12]):
τ κ − κ τ κ 2 = 0(5)
Mehlum and Resch [9] go on to describe this curve with a geometric/kinematic construction by rolling a sphere without slipping or twisting on a planar clothoid. They thereby reveal that this is the spherical curve with geodesic curvature a linear function of arc length (see [13], Theorem 2.3):
κ g = αs(6)
It is fitting to call it the sherical clothoid, a term first coined byÜlo Lumiste [14] in work unrelated to the theory of nonlinear splines. Mehlum manages to express the cartesian coordinate functions of the spherical clothoid using Humbert series φ 1 , φ 2 and some more common functions.
Novel results
Simplified hypergeometric Representation
The main purpose of this work is to present a simpler hypergeometric representation of the coordinate functions. Without loss of generality, we will restrict our attention to the case R = 1 since all other curves can be obtained with simple scaling. Let r(s) = (x(s), y(s), z(s)) be the parametric equation of our space curve. We consider the differential equation and initial conditions first examined by Mehlum (1.13, 2.17, 2.18, 2.19 and 2.21 in [9]):
r (s) + (α 2 s 2 + 1)r (s) + 3α 2 sr (s) = 0 r(0) = (0, 0, 0) r (0) = (1, 0, 0) r (0) = (0, 1, 0) r (0) = (−1, 0, α)(7)
The following solution holds:
x(s) = s 1 F 1 − i 8 α , 1 2 ; − i αs 2 2 1 F 1 i 8 α + 1 2 , 3 2 ; i αs 2 2 y(s) = −2 1 F 1 i 8 α , 1 2 ; i αs 2 2 2 + 2 z(s) = s 1 F 1 − i 8 α , 1 2 ; − i αs 2 2 1 F 1 i 8 α + 1 2 , 3 2 ; i αs 2 2(8)
x, y and z can also be expressed equivalently using conjugation instead of using real and imaginary parts and the absolute value. Consult Appendix A for a computer-assisted proof using Mathematica.
Derivation
We begin by investigating a loose end from the kinematic construction. The following system of equations describes the rolling without slipping and twisting of a unit sphere on a planar clothoid with scale factor α (see 3 in [15] and 29a in [16]) :
d ds x ỹ z = 0 0 − sin αs 2 2 0 0 cos αs 2 2 sin αs 2 2 − cos αs 2 2 0 x ỹ z (9)
We introduce two new complex variables (see 10 in [15], 8 in [16]):
x = ab * + ba * ,ỹ = i (ab * − ba * ) ,z = aa * − bb * .(10)
This tool was first presented by Feynman, Vernon and Hellwarth to geometrically represent the Schrödinger equation of a two-level quantum system [17]. a and b must satsfy the following system in order forx,ỹ andz to satisfy (9):
i d ds
a b = − 1 2 0 e −iαs 2 /2 e iαs 2 /2 0 a b(11)
We are now left with a system that can be solved directly in a computer algebra system. Consult Appendix B for a solution using Mathematica. This system arises in the Landau-Zener problem (4, [18]). In fact, Zener first solved this system analytically using parabolic cylinder functions in this quantum physics context. We then obtain the solution to Mehlum's equation using (10) again and a rigid motion to satisfy the initial conditions. Stereographic projection is a mapping that projects the sphere onto the plane. Circles on the sphere are mapped to circles on the plane (as long as they do not pass through the point of projection) and loxodromes are mapped to logarithmic spirals. We shift the sphere to center it on the origin and consider the stereographic projection of our curve onto the complex plane:
Stereographic projection
ζ(s) = X(s) + iY (s) = x(s) + iz(s) 1 − (y(s) − 1) = s 1 F 1 i 8 α + 1 2 , 3 2 ; i αs 2 2 2 1 F 1 i 8 α , 1 2 ; i αs 2 2(12)
We also have an alternative representation using a quotient of odd and even parabolic cylinder functions:
ζ(s) = e −iπ
Humbert series corollaries
Let us revisit Mehlum and Wimp's work on the spherical clothoid. We begin with Mehlum's hypergeometric expression for y restricted to the unit sphere (2.54 in [9]):
y(s) = 1 − φ 2 i 4 α , − i 4 α , 1 2 ; i αs 2 2 , − i αs 2 2(14)
Combining this work with our new representation yields a novel reduction formula for a special form of φ 2 :
φ 2 a, −a, 1 2 ; x, −x = 2 1 F 1 a 2 , 1 2 ; x 1 F 1 − a 2 , 1 2 ; −x − 1(15)
We can use some suggestions in Mehlum's work to obtain an expression for x+iz using φ 1 special forms and less mysterious functions (see Appendix B for the derivation):
u = √ π cos(π i 4 α )Γ − i 4 α − 1 2 i 4 α Γ − i 8 α 2 v = − √ π cos(π i 4 α )Γ i 4 α − 1 2 2Γ i 8 α + 1 2 2 φ 1 = φ 1 i 4 α + 1, i 4 α , i 4 α + 3 2 ; 1 2 , i αs 2 2 φ * 1 = φ 1 − i 4 α + 1, − i 4 α , − i 4 α + 3 2 ; 1 2 , − i αs 2 2 x(s) + i z(s) = s u e − i αs 2 2 φ 1 + v e i αs 2 2 φ * 1(16)
Once again combining this with our new representation yields a new identity:
√ π cos(πa)Γ −a − 1 2 aΓ − a 2 2 e −x φ 1 a + 1, a, a + 3 2 ; 1 2 , x − √ π cos(πa)Γ a − 1 2 2Γ a+1 2 2 e x φ 1 −a + 1, −a, −a + 3 2 ; 1 2 , −x = 1 F 1 − a 2 , 1 2 ; −x 1 F 1 a 2 + 1 2 , 3 2 ; x(17)
It is interesting to note that these reductions demystify some quadratic relationships for special functions studied by Mehlum and Wimp (section 5 in [12]).
Generating function formulas related to the associated Meixner-Pollaczek polynomials
Mehlum found expressions for the parameter functions of the spherical clothoid involving associated Meixner-Pollaczek polynomials (2.38, 2.39 and 2.40 in [9]):
x(s) = ∞ j=0 Q 0 j ( 1 4α ; π 2 , 1 2 ) −α 2 j s 2j+1 (1) j (18) y(s) = 1 2 ∞ j=0 Q 0 j ( 1 4α ; π 2 , 1) −α 2 j s 2j+2 3 2 j (19) z(s) = α 6 ∞ j=0 Q 0 j ( 1 4α ; π 2 , 3 2 ) −α 2 j s 2j+3 (2) j(20)
Generating functions related to this family of orthogonal polynomials arise in recent research [19][20][21]. Combining these expressions with our new parameter functions, we obtain the following generating function results:
∞ j=0 Q 0 j (x; π 2 , 1 2 ) t j j! = 1 2 1 F 1 − ix 2 , 1 2 ; it 1 F 1 ix 2 + 1 2 , 3 2 ; −it + 1 F 1 ix 2 , 1 2 ; −it 1 F 1 − ix 2 + 1 2 , 3 2 ; it (21) ∞ j=0 Q 0 j (x; π 2 , 1) t j 3 2 j = 1 F 1 ix 2 , 1 2 ; −it 1 F 1 − ix 2 , 1 2 ; it − 1 2xt (22) ∞ j=0 Q 0 j (x; π 2 , 3 2 ) t j (j + 1)! = 3i 2t 1 F 1 − ix 2 , 1 2 ; it 1 F 1 ix 2 + 1 2 , 3 2 ; −it − 1 F 1 ix 2 , 1 2 ; −it 1 F 1 − ix 2 + 1 2 , 3 2 ; it(23)
Conclusion
We have found simple expressions for the Cartesian coordinates of the spherical clothoid and its projection with some special function results as a bonus. It is remarkable that tools from quantum physics elucidate a classical problem. Suddenly this curve studied by Mehlum does not seem so exotic, heightening its potential in the world of computer aided geometric design. The computation of the special functions presented in this work presents an avenue for future research.
Acknowledgments
I would like to thank Dr. Zurab Silagadze for discussions that sparked the idea behind this hypergeometric reduction tied to the spherical clothoid. I am grateful to Dr. Khalid Ahbli, Dr. James Hateley, Max Kölbl and Dr. Robert Lewis for their attention and support during my research.
Appendix B Complex differential equation system
Here we take an experimental approach and consider the following initial conditions: a(0) = 1, b(0) = 0 (implyingx(0) = 0,ỹ(0) =,z(0) = 1)
Mathematica Session
Let's examine the special forms of 2 F 1 that arise. We first recall some identities derived by Mitra (2.68 and 2.69 in [9]): 2 F 1 a, a + 1, a + (28) Examining the first few terms in the Maclaurin series of s (u Ξ 1 + v Ξ * 1 ) and our initial conditions, we can now find expressions for u and v explicitly in terms of the gamma function and more common functions. There are no terms with even powers of s in our Maclaurin series and that satisfies the first and third initial conditions. We examine the second and fourth:
Mathematica Session
Figure 1 :
1Plot of the spherical clothoid for α = 1 and s ∈ [−5, 5]
Figure 2 :
2Plot of the spherical clothoid and its stereographic projection for α = 1 and s ∈ [0, 5]
Using the contiguous relations of 2 F 1 we can deduce another relation: 2 F 1 a, a + 1, a +1
2
;
1
2
=
√
πΓ a +
1
2
1
Γ a
2 + 1
2
2 +
1
Γ a
2 + 1 Γ a
2
(26)
2 F 1 a, a + 1, a +
3
2
;
1
2
= 2
√
πΓ a +
3
2
1
Γ a
2 + 1
2
2 −
1
Γ a
2 + 1 Γ a
2
(27)
5
2
;
1
2
=
4
3
√
πΓ a +
5
2
1 − 4a
Γ a
2 + 1
2
2 +
1 + 4a
Γ a
2 + 1 Γ a
2
Appendix A ODE solution verificationMathematica SessionAppendix C Mehlum's x + i z Combining Mehlum's expression for x and z (2.65 and 2.67 in[9]) and a Humbert series transform (2.61 and 2.62 in[9]), it is possible to obtain expressions of this form:u, v do not depend on s and these are the special forms of Ξ 1 we are playing with:
. Hypergeometric1f1, I/(8 α. I α sˆ2)/2Hypergeometric1F1[I/(8 α),1/2,(I α sˆ2)/2]
3/2,(I α sˆ2)/2]−1/(2 I) s Hypergeometric1F1[I/(8 α),1/2. Hypergeometric1F1[1/2+I/(8 α. I α sˆ2)/2] Hypergeometric1F1[1/2−I/(8 α),3/2,−((I α sˆ2)/2)Hypergeometric1F1[1/2+I/(8 α),3/2,(I α sˆ2)/2]−1/(2 I) s Hypergeometric1F1[I/(8 α),1/2,(I α sˆ2)/2] Hypergeometric1F1[1/2−I/(8 α),3/2,−((I α sˆ2)/2)]
. = {0, 10= {0,1,0}
. = Fullsimplify, r '''[0In[8]:= FullSimplify [ r '''[0]]
. = Fullsimplify, r ''''[ s]+(αˆ2 sˆ2 + 1) r''[s] + 3 αˆ2 s r'[sIn[9]:= FullSimplify [ r ''''[ s]+(αˆ2 sˆ2 + 1) r''[s] + 3 αˆ2 s r'[s ]]
. = {0, 00= {0,0,0}
== 1/2 I b[s] Eˆ(1/2 (−I) α sˆ2), b'[s] == 1/2 I a. In[1]:=DSolve[{a'[s. s] Eˆ(1/2 I α sˆ2), a[0] == 1, b[0] == 0}, {a, b}, sIn[1]:=DSolve[{a'[s ] == 1/2 I b[s] Eˆ(1/2 (−I) α sˆ2), b'[s] == 1/2 I a[s] Eˆ(1/2 I α sˆ2), a[0] == 1, b[0] == 0}, {a, b}, s ]
. ={{a −> Function, {s}, Eˆ(−(1/2) I sˆ2 α) Hypergeometric1F1[−((−I − 4 α)/(8 α)={{a −> Function[{s}, Eˆ(−(1/2) I sˆ2 α) Hypergeometric1F1[−((−I − 4 α)/(8 α)),
−> Function[{s}, 1/2 s (I Hypergeometric1F1[1 − (−I − 4 α)/(8 α), 3/2, 1/2 I sˆ2 α] + 4 α Hypergeometric1F1[1 − (−I − 4 α). /2, 1/2 I sˆ2 α. 8 α), 3/2, 1/2 I sˆ2 α] − 4 α Hypergeometric1F1[−((−I − 4 α)/(8 α)) 1/2, 1/2 I sˆ2 α])]}}/2, 1/2 I sˆ2 α]], b −> Function[{s}, 1/2 s (I Hypergeometric1F1[1 − (−I − 4 α)/(8 α), 3/2, 1/2 I sˆ2 α] + 4 α Hypergeometric1F1[1 − (−I − 4 α)/(8 α), 3/2, 1/2 I sˆ2 α] − 4 α Hypergeometric1F1[−((−I − 4 α)/(8 α)) 1/2, 1/2 I sˆ2 α])]}}
/2) I sˆ2 α) Hypergeometric1F1[−((−I − 4 α). = FullSimplify [Eˆ(−8 α)), 1/2, 1/2 I sˆ2 αIn[2]:= FullSimplify [Eˆ(−(1/2) I sˆ2 α) Hypergeometric1F1[−((−I − 4 α)/(8 α)), 1/2, 1/2 I sˆ2 α]]
/2 s ( I Hypergeometric1F1[1 − (−I − 4 α)/(8 α), 3/2, 1/2 I sˆ2 α] + 4 α Hypergeometric1F1[1 − (−I − 4 α). = Fullsimplify, 8 α), 3/2, 1/2 I sˆ2 α] − 4 α Hypergeometric1F1[−((−I − 4 α)/(8 α)), 1/2, 1/2 I sˆ2 α])In[3]:= FullSimplify [1/2 s ( I Hypergeometric1F1[1 − (−I − 4 α)/(8 α), 3/2, 1/2 I sˆ2 α] + 4 α Hypergeometric1F1[1 − (−I − 4 α)/(8 α), 3/2, 1/2 I sˆ2 α] − 4 α Hypergeometric1F1[−((−I − 4 α)/(8 α)), 1/2, 1/2 I sˆ2 α])]
Gamma[−I/(4 α)−1/2]/(I/(4 α)Gamma[−I/(8α)]ˆ2). = v=−Sqrt[Pi]Cos[Pi I/In[1]:= u=Sqrt[Pi]Cos[Pi I/(4 α). 4 α)] Gamma[I/(4 α)−1/2]/(2Gamma[I/(8α)+1/2]ˆ2) In[3]:= b=2 Sqrt[Pi] Gamma[I/(4 α)+3/2](1/Gamma[I/(8α)+1/2]ˆ2−1/(Gamma[I/(8α) +1]Gamma[I/(8α)In[1]:= u=Sqrt[Pi]Cos[Pi I/(4 α)] Gamma[−I/(4 α)−1/2]/(I/(4 α)Gamma[−I/(8α)]ˆ2) In[2]:= v=−Sqrt[Pi]Cos[Pi I/(4 α)] Gamma[I/(4 α)−1/2]/(2Gamma[I/(8α)+1/2]ˆ2) In[3]:= b=2 Sqrt[Pi] Gamma[I/(4 α)+3/2](1/Gamma[I/(8α)+1/2]ˆ2−1/(Gamma[I/(8α) +1]Gamma[I/(8α)]))
(8α)+1/2]ˆ2+(1−I/α)/(Gamma[−I/(8α)+1]Gamma[−I/(8α)])) (I α/2) In[8]:= FullSimplify. /2]((1+I/α)/ Gamma[−I/Factorial [3/2]((1+I/α)/ Gamma[−I/(8α)+1/2]ˆ2+(1−I/α)/(Gamma[−I/(8α)+1]Gamma[−I/(8α)])) (I α/2) In[8]:= FullSimplify [ Factorial [3
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When physics helps mathematics: Calculation of the sophisticated multiple integral. Arkady Kholodenko, Zurab Silagadze, Physics of Particles and Nuclei. 43Arkady Kholodenko and Zurab Silagadze. When physics helps mathemat- ics: Calculation of the sophisticated multiple integral. Physics of Particles and Nuclei, 43, 01 2012.
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Certain results on generating functions related to the associated meixner-pollaczek polynomials. Min-Jie Luo, Ravinder Krishna Raina, Shu-Han Zhao, Integral Transforms and Special Functions. 00Min-Jie Luo, Ravinder Krishna Raina, and Shu-Han Zhao. Certain re- sults on generating functions related to the associated meixner-pollaczek polynomials. Integral Transforms and Special Functions, 0(0):1-17, 2021.
A generating function and formulae defining the first-associated meixner-pollaczek polynomials. Khalid Ahbli, Zouhair Mouayn, Integral Transforms and Special Functions. 29Khalid Ahbli and Zouhair Mouayn. A generating function and formulae defining the first-associated meixner-pollaczek polynomials. Integral Trans- forms and Special Functions, 29, 02 2018.
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. E- , mail address: [email protected] address: [email protected]
| []
|
[
"Spin temperature concept verified by optical magnetometry of nuclear spins",
"Spin temperature concept verified by optical magnetometry of nuclear spins"
]
| [
"M Vladimirova \nLaboratoire Charles Coulomb\nUMR 5221\nCNRS\nUniversité de Montpellier\nF-34095MontpellierFrance\n",
"S Cronenberger \nLaboratoire Charles Coulomb\nUMR 5221\nCNRS\nUniversité de Montpellier\nF-34095MontpellierFrance\n",
"D Scalbert \nLaboratoire Charles Coulomb\nUMR 5221\nCNRS\nUniversité de Montpellier\nF-34095MontpellierFrance\n",
"I I Ryzhov \nSpin Optics Laboratory\nSt. Petersburg State University\n1 Ul'anovskaya198504Peterhof, PetersburgStRussia\n",
"V S Zapasskii \nSpin Optics Laboratory\nSt. Petersburg State University\n1 Ul'anovskaya198504Peterhof, PetersburgStRussia\n",
"G G Kozlov \nLaboratoire Charles Coulomb\nUMR 5221\nCNRS\nUniversité de Montpellier\nF-34095MontpellierFrance\n",
"A Lemaître \nCentre de Nanosciences\nCNRS\nUniversité Paris-Saclay -Université Paris-Sud\nRoute de Nozay91460MarcoussisFrance\n",
"K V Kavokin \nSpin Optics Laboratory\nSt. Petersburg State University\n1 Ul'anovskaya198504Peterhof, PetersburgStRussia\n\nIoffe Physico-Technical Institute of the RAS\n194021St.PetersburgRussia\n"
]
| [
"Laboratoire Charles Coulomb\nUMR 5221\nCNRS\nUniversité de Montpellier\nF-34095MontpellierFrance",
"Laboratoire Charles Coulomb\nUMR 5221\nCNRS\nUniversité de Montpellier\nF-34095MontpellierFrance",
"Laboratoire Charles Coulomb\nUMR 5221\nCNRS\nUniversité de Montpellier\nF-34095MontpellierFrance",
"Spin Optics Laboratory\nSt. Petersburg State University\n1 Ul'anovskaya198504Peterhof, PetersburgStRussia",
"Spin Optics Laboratory\nSt. Petersburg State University\n1 Ul'anovskaya198504Peterhof, PetersburgStRussia",
"Laboratoire Charles Coulomb\nUMR 5221\nCNRS\nUniversité de Montpellier\nF-34095MontpellierFrance",
"Centre de Nanosciences\nCNRS\nUniversité Paris-Saclay -Université Paris-Sud\nRoute de Nozay91460MarcoussisFrance",
"Spin Optics Laboratory\nSt. Petersburg State University\n1 Ul'anovskaya198504Peterhof, PetersburgStRussia",
"Ioffe Physico-Technical Institute of the RAS\n194021St.PetersburgRussia"
]
| []
| We develop a method of non-perturbative optical control over adiabatic remagnetisation of the nuclear spin system and apply it to verify the spin temperature concept in GaAs microcavities. The nuclear spin system is shown to exactly follow the predictions of the spin-temperature theory, despite the quadrupole interaction that was earlier reported to disrupt nuclear spin thermalisation. These findings open a way to deep cooling of nuclear spins in semiconductor structures, with a prospect of realisation of nuclear spin-ordered states for high fidelity spin-photon interfaces.The concept of nuclear spin temperature is one of the cornerstones of the nuclear magnetism in solids 1,2 . It has made possible realisation of the cryogenic cooling into the microKelvin range 3 and observation of nuclear spin ordering in metals and insulators 4,5 . Such degree of control of the nuclear spin system (NSS) in semiconductor heterostructures would allow enhancing the efficiency of spin-based information storage and processing 6-9 . However, proving the validity of the spin temperature concept for semiconductor nano-and microstructures is challenging due to the lack of techniques capable of precise sensing of weak nuclear magnetisation in a small volume. In addition, recent experiments showed that in quantum dots, where strong quadrupole-induced local fields have been reported, nuclear spin temperature failed to establish 10 . In this context, NSS thermalisation sensing in semiconductor heterostructures is one the central issues for both fundamental questions related to the realisation of nuclear spin-ordered states, and for potential applications, such as high fidelity spin-photon interfaces 6-9 .The basic postulates of the spin temperature theory are illustrated inFig. 1(a). It is assumed that during the characteristic time T 2 determined by spin-spin interactions the NSS reaches the internal equilibrium. This means that properties of the NSS are governed by a single parameter, the spin temperature Θ N . When this temperature is made different from the lattice temperature Θ L (e.g. by the optical pumping), the thermalisation of the NSS with the crystal lattice usually requires a much longer characteristic time T 1 .Fig.1(b)illustrates one of the main predictions of the spin temperature theory: if the NSS is subjected to a slowly varying magnetic field, such that dB/dt < B L /T 2 , then Θ N and the nuclear spin polarisation P N change obeying universal expressions:(1) Here γ N is the gyromagnetic ratio of the nuclear spin I, angular brackets denote the averaging over all nuclear species, k B is the Boltzman constant, and Θ N i is the spin temperature at strong magnetic field B i >> B L , where B L is the local field induced by the fluctuating nuclear spins. These generic relations are based on the principle of entropy conservation in a thermodynamic system during adiabatic process. They constitute the basis for the nuclear spin cooling by adiabatic demagnetisation, a widely used cryogenic technique 4,11-13 . The nuclear spin temperature may take either positive or negative values, in the latter case the magnetisation being anti-parallel to the applied field.Various optical and magnetic techniques have been employed to measure nuclear spin temperature, mostly by the magnetisation measurement at a fixed value of the external magnetic field 4,11,14-16 . On the other hand, a direct measurement of the nuclear magnetisation as a function of slowly varying magnetic field is extremely challenging and has never been realised to the best of our knowledge. Such an experiment is required to check rigorously the validity of the concept of spin temperature as applied to a specific system.In this Letter we report on realisation of such a proofof-concept experiment in microcavities, semiconductor microstructures with enhanced light-matter coupling 17 . The principle of our experiment is sketched inFig. 1(c). Prior to the measurement, the NSS of the n-GaAs layer embedded in a microcavity is polarised by optical pumping in the presence of the longitudinal magnetic field. Nuclear spin polarisation is probed by linearly polarized cavity mode photons with the photon energy in the transparency band of GaAs. Polarisation of the light beam transmitted through the cavity is sensitive to the Overhauser field, an effective magnetic field created by NSS and acting on electron spins 18 . Two methods of detection of nuclear spin polarisation are used: (i) the Faraday effect induced by the Overhauser field 19,20 and (ii) the spin noise spectroscopy of resident electrons subject to the Overhauser field 21-23 . The main features of the behaviour of the optically cooled NSS under varying external magnetic fields are demonstrated in the experi-arXiv:1706.02528v1 [cond-mat.other] | 10.1103/physrevb.97.041301 | [
"https://arxiv.org/pdf/1706.02528v1.pdf"
]
| 118,977,620 | 1706.02528 | 65fb4d5d4ca90d35f37a51c71c2ca404a77c539b |
Spin temperature concept verified by optical magnetometry of nuclear spins
8 Jun 2017
M Vladimirova
Laboratoire Charles Coulomb
UMR 5221
CNRS
Université de Montpellier
F-34095MontpellierFrance
S Cronenberger
Laboratoire Charles Coulomb
UMR 5221
CNRS
Université de Montpellier
F-34095MontpellierFrance
D Scalbert
Laboratoire Charles Coulomb
UMR 5221
CNRS
Université de Montpellier
F-34095MontpellierFrance
I I Ryzhov
Spin Optics Laboratory
St. Petersburg State University
1 Ul'anovskaya198504Peterhof, PetersburgStRussia
V S Zapasskii
Spin Optics Laboratory
St. Petersburg State University
1 Ul'anovskaya198504Peterhof, PetersburgStRussia
G G Kozlov
Laboratoire Charles Coulomb
UMR 5221
CNRS
Université de Montpellier
F-34095MontpellierFrance
A Lemaître
Centre de Nanosciences
CNRS
Université Paris-Saclay -Université Paris-Sud
Route de Nozay91460MarcoussisFrance
K V Kavokin
Spin Optics Laboratory
St. Petersburg State University
1 Ul'anovskaya198504Peterhof, PetersburgStRussia
Ioffe Physico-Technical Institute of the RAS
194021St.PetersburgRussia
Spin temperature concept verified by optical magnetometry of nuclear spins
8 Jun 2017
We develop a method of non-perturbative optical control over adiabatic remagnetisation of the nuclear spin system and apply it to verify the spin temperature concept in GaAs microcavities. The nuclear spin system is shown to exactly follow the predictions of the spin-temperature theory, despite the quadrupole interaction that was earlier reported to disrupt nuclear spin thermalisation. These findings open a way to deep cooling of nuclear spins in semiconductor structures, with a prospect of realisation of nuclear spin-ordered states for high fidelity spin-photon interfaces.The concept of nuclear spin temperature is one of the cornerstones of the nuclear magnetism in solids 1,2 . It has made possible realisation of the cryogenic cooling into the microKelvin range 3 and observation of nuclear spin ordering in metals and insulators 4,5 . Such degree of control of the nuclear spin system (NSS) in semiconductor heterostructures would allow enhancing the efficiency of spin-based information storage and processing 6-9 . However, proving the validity of the spin temperature concept for semiconductor nano-and microstructures is challenging due to the lack of techniques capable of precise sensing of weak nuclear magnetisation in a small volume. In addition, recent experiments showed that in quantum dots, where strong quadrupole-induced local fields have been reported, nuclear spin temperature failed to establish 10 . In this context, NSS thermalisation sensing in semiconductor heterostructures is one the central issues for both fundamental questions related to the realisation of nuclear spin-ordered states, and for potential applications, such as high fidelity spin-photon interfaces 6-9 .The basic postulates of the spin temperature theory are illustrated inFig. 1(a). It is assumed that during the characteristic time T 2 determined by spin-spin interactions the NSS reaches the internal equilibrium. This means that properties of the NSS are governed by a single parameter, the spin temperature Θ N . When this temperature is made different from the lattice temperature Θ L (e.g. by the optical pumping), the thermalisation of the NSS with the crystal lattice usually requires a much longer characteristic time T 1 .Fig.1(b)illustrates one of the main predictions of the spin temperature theory: if the NSS is subjected to a slowly varying magnetic field, such that dB/dt < B L /T 2 , then Θ N and the nuclear spin polarisation P N change obeying universal expressions:(1) Here γ N is the gyromagnetic ratio of the nuclear spin I, angular brackets denote the averaging over all nuclear species, k B is the Boltzman constant, and Θ N i is the spin temperature at strong magnetic field B i >> B L , where B L is the local field induced by the fluctuating nuclear spins. These generic relations are based on the principle of entropy conservation in a thermodynamic system during adiabatic process. They constitute the basis for the nuclear spin cooling by adiabatic demagnetisation, a widely used cryogenic technique 4,11-13 . The nuclear spin temperature may take either positive or negative values, in the latter case the magnetisation being anti-parallel to the applied field.Various optical and magnetic techniques have been employed to measure nuclear spin temperature, mostly by the magnetisation measurement at a fixed value of the external magnetic field 4,11,14-16 . On the other hand, a direct measurement of the nuclear magnetisation as a function of slowly varying magnetic field is extremely challenging and has never been realised to the best of our knowledge. Such an experiment is required to check rigorously the validity of the concept of spin temperature as applied to a specific system.In this Letter we report on realisation of such a proofof-concept experiment in microcavities, semiconductor microstructures with enhanced light-matter coupling 17 . The principle of our experiment is sketched inFig. 1(c). Prior to the measurement, the NSS of the n-GaAs layer embedded in a microcavity is polarised by optical pumping in the presence of the longitudinal magnetic field. Nuclear spin polarisation is probed by linearly polarized cavity mode photons with the photon energy in the transparency band of GaAs. Polarisation of the light beam transmitted through the cavity is sensitive to the Overhauser field, an effective magnetic field created by NSS and acting on electron spins 18 . Two methods of detection of nuclear spin polarisation are used: (i) the Faraday effect induced by the Overhauser field 19,20 and (ii) the spin noise spectroscopy of resident electrons subject to the Overhauser field 21-23 . The main features of the behaviour of the optically cooled NSS under varying external magnetic fields are demonstrated in the experi-arXiv:1706.02528v1 [cond-mat.other]
We develop a method of non-perturbative optical control over adiabatic remagnetisation of the nuclear spin system and apply it to verify the spin temperature concept in GaAs microcavities. The nuclear spin system is shown to exactly follow the predictions of the spin-temperature theory, despite the quadrupole interaction that was earlier reported to disrupt nuclear spin thermalisation. These findings open a way to deep cooling of nuclear spins in semiconductor structures, with a prospect of realisation of nuclear spin-ordered states for high fidelity spin-photon interfaces.
The concept of nuclear spin temperature is one of the cornerstones of the nuclear magnetism in solids 1,2 . It has made possible realisation of the cryogenic cooling into the microKelvin range 3 and observation of nuclear spin ordering in metals and insulators 4,5 . Such degree of control of the nuclear spin system (NSS) in semiconductor heterostructures would allow enhancing the efficiency of spin-based information storage and processing [6][7][8][9] . However, proving the validity of the spin temperature concept for semiconductor nano-and microstructures is challenging due to the lack of techniques capable of precise sensing of weak nuclear magnetisation in a small volume. In addition, recent experiments showed that in quantum dots, where strong quadrupole-induced local fields have been reported, nuclear spin temperature failed to establish 10 . In this context, NSS thermalisation sensing in semiconductor heterostructures is one the central issues for both fundamental questions related to the realisation of nuclear spin-ordered states, and for potential applications, such as high fidelity spin-photon interfaces [6][7][8][9] .
The basic postulates of the spin temperature theory are illustrated in Fig. 1(a). It is assumed that during the characteristic time T 2 determined by spin-spin interactions the NSS reaches the internal equilibrium. This means that properties of the NSS are governed by a single parameter, the spin temperature Θ N . When this temperature is made different from the lattice temperature Θ L (e.g. by the optical pumping), the thermalisation of the NSS with the crystal lattice usually requires a much longer characteristic time T 1 . Fig.1(b) illustrates one of the main predictions of the spin temperature theory: if the NSS is subjected to a slowly varying magnetic field, such that dB/dt < B L /T 2 , then Θ N and the nuclear spin polarisation P N change obeying universal expressions:
Θ N / B 2 + B 2 L = Θ N i /B i ; P N = B 3k B Θ Nh γ N (I+1) .
(1) Here γ N is the gyromagnetic ratio of the nuclear spin I, angular brackets denote the averaging over all nuclear species, k B is the Boltzman constant, and Θ N i is the spin temperature at strong magnetic field B i >> B L , where B L is the local field induced by the fluctuating nuclear spins. These generic relations are based on the principle of entropy conservation in a thermodynamic system during adiabatic process. They constitute the basis for the nuclear spin cooling by adiabatic demagnetisation, a widely used cryogenic technique 4,[11][12][13] . The nuclear spin temperature may take either positive or negative values, in the latter case the magnetisation being anti-parallel to the applied field.
Various optical and magnetic techniques have been employed to measure nuclear spin temperature, mostly by the magnetisation measurement at a fixed value of the external magnetic field 4,11,[14][15][16] . On the other hand, a direct measurement of the nuclear magnetisation as a function of slowly varying magnetic field is extremely challenging and has never been realised to the best of our knowledge. Such an experiment is required to check rigorously the validity of the concept of spin temperature as applied to a specific system.
In this Letter we report on realisation of such a proofof-concept experiment in microcavities, semiconductor microstructures with enhanced light-matter coupling 17 . The principle of our experiment is sketched in Fig. 1(c). Prior to the measurement, the NSS of the n-GaAs layer embedded in a microcavity is polarised by optical pumping in the presence of the longitudinal magnetic field. Nuclear spin polarisation is probed by linearly polarized cavity mode photons with the photon energy in the transparency band of GaAs. Polarisation of the light beam transmitted through the cavity is sensitive to the Overhauser field, an effective magnetic field created by NSS and acting on electron spins 18 1. (a) Sketch of the two heat reservoirs, the atomic lattice at temperature ΘL, and the nuclear spin system (NSS) at temperature ΘN . The equilibrium within the NSS is established during the spin-spin relaxation time T2 << T1, the spin-lattice relaxation time. (b) Evolution of the nuclear spin temperature (dashed lines) and polarisation (solid lines) in the adiabatic de(re)-magnetisation process starting from either positive (red lines) or negative initial spin temperature ΘNi under magnetic field Bi, as described by equation (1). The lowest nuclear spin temperature ΘN0 that can be reached in the adiabatic demagnetisation procedure is determined by the initial temperature of the nuclei ΘNi in the strong magnetic field Bi and the local field BL. (c) Schematic view of the sample and the detection stage of Faraday rotation and spin noise experiments. NSS in the cavity probed using two optical technics, that allow us to trace the evolution of the initially prepared nuclear spin polarization PN and temperature ΘN along the demagnetisation process. Spin noise spectrum is obtained as Fourier transformation of the stochastic Faraday rotation. The spectral peak frequency is directly related to the Overhauser field acting on electrons in the presence of the in-plane magnetic field. ment where the Faraday rotation angle is measured while ramping the longitudinal magnetic field across zero (Fig. 2). The experiment is conducted in two steps: preparation and measurement ( Fig. 2 (a)). The measured signal ( Fig. 2(b)) contains two contributions: Faraday rotation directly induced by the external field (shown by solid lines, it remains unchanged for all the scans), and the Faraday rotation induced by the Overhauser field φ N ( shown separately in Fig. 2(e) for the first scan), which is proportional to the nuclear spin polarisation. In each consecutive scan, φ N diminishes due to the nuclear spinlattice relaxation, but the behaviour of nuclear polarisation is described by Eqs. (1): the polarisation is an odd function of the applied field, there is no remanent magnetisation at B = 0, and B L = 8 ± 2 G. We have performed this analysis for two samples with different concentrations of Si donors n d : an insulating sample with n d = 2·10 15 cm −3 (Sample A) and a sample characterised by a metallic conductivity (n d = 2 · 10 16 cm −3 , Sample B), for NSS prepared either at positive, or at negative spin temperature. The value of B L obtained for both samples is the same within our experimental accuracy.
We complemented these results by spin-noise measurements of nuclear remagnetisation under magnetic field perpendicular to the light and the structure axis (Fig. 3). Color maps in Figs. 3b,c show the evolution of the electron spin noise spectra under varying magnetic fields. The narrow peak in the spectra appears at the frequency ν of the electron Larmor precession in the total effective magnetic field acting upon the electron spins. This field is given by the sum of the external and the Overhauser field, which allows us to extract the nuclear spin polar- isation. The asymmetry of the recorded sets of spectra with respect to zero magnetic field is due to nuclear spinlattice relaxation. We have taken it into account when fitting equation (1) to the data (black dashed lines in Fig. 3(b-e)). For both samples and both signs of the nuclear spin temperature, the value of the local field was found to be B L = 12 ± 2 G. Thus, the NSS does obey the prediction of the thermodynamic theory expressed by Eq. (1), but value of the local field is surprisingly large, B L ≈ 10 G. Indeed, the spin-spin interactions in GaAs are dominated by magnetic dipole-dipole coupling, which yields a much weaker local field B dd = 1.5 G 24 .
To elucidate the origin of this striking discrepancy, we performed spin noise measurements with the bulk GaAs layer without a microcavity, Sample C (Fig. 3(fg)). Although the signal is much weaker, the best fit using Eqs. (1) and taking into account spin-lattice relaxation during the measurement yields B L = 2 G and Θ N 0 = ±4 µK 25 . This comparison shows unambiguously the enhanced value of local field in the microcavities, compared to that in the bulk GaAs. Within the thermodynamic description of the NSS, the local field which enters Eqs. (1) is defined as 1 :
B 2 L = T r(H 2 S )/T r(M 2 B ),(2)
where H S is the Hamiltonian of all nuclear spin interactions, excluding Zeeman part (typically it includes the magnetic dipole-dipole interactions, and the indirect exchange), and M B is the parallel to the magnetic field component of the nuclear magnetic moment. In n-GaAs, magnetic dipole-dipole interaction is well-studied, and B L = 2 G measured in bulk GaAs agrees well with the previous estimations for B dd 24 . The only plausible explanation for the unexpectedly strong local field detected in microcavities is the quadrupole splitting hν Q of the nuclear spin states induced by an uniaxial strain. In Eq.(2) it can be accounted for by introducing H S = H dd + H Q , where H Q is the Hamiltonian of the quadrupole interaction
H Q = 3 i=1 hν i Q 2 (Î 2 z − I(I + 1) 3 ).(3)
Here the index i stands for the summation over the three isotopes ( 69 Ga, 71 Ga, 75 As), andÎ z is the projection on the nuclear spin operator on the growth (strain) axis. Using equation (2) and the parameters of strain-induced quadrupole splittings in GaAs 26 , one can estimate that the strain as weak as 0.01% induces the local field B L = 10 G in GaAs 27 . Because B L >> B dd , it is the quadrupole interaction that determines the capacity of the NSS to store the energy in the internal degrees of freedom. But in contrast with dipole-dipole interaction, the quadrupole interaction does not provide any coupling between the spins, and can not establish the thermodynamic equilibrium within the NSS. Indeed, in quantum dots, where strong quadrupole-induced local fields have been reported, nuclear spin temperature failed to establish 10 . From our data we can estimate the lower limit of 50 G for the mixing field B m , at which Zeeman and internal energy reservoirs come to equilibrium between each other, so that the NSS can be described by the unique spin temperature 4 (see Supplemental material).
The question remains, how can the thermodynamic equilibrium be established under magnetic field B m > 50 G, much larger than the characteristic field of the dipole-dipole interaction B dd = 1.5 G? We suggest that this is made possible by the multi-isotope nature of the NSS in GaAs. The difference in the quadrupole splittings and gyromagnetic ratios between the three isotopes yields a rich variety of possible inter-isotope flip-flop transitions. These transitions frequencies ν N are illustrated in Fig. 4 as functions of the magnetic field in the absence (Fig. 4a) and in the presence of the quadrupole splitting of the nuclear spin states along z-axis (Fig. 4 (c, e)). The spin flipflop transitions involving different isotopes ensure the energy transfer between the Zeeman and quadrupole energy reservoirs, with total energy conservation of the NSS. These transitions are broadened by dipole-dipole interactions. It is usually assumed 4 that the efficient equilibration of energy reservoirs is ensured at detuning from the resonance less than δν N = 5B dd /(2π γ N = 8 kHz. One can see in Fig. 3d,f, that for both orientations of the magnetic field, the transitions involving such a small detuning are available at B < 50 G, and the mixing remains as efficient as in the absence of the quadrupole splitting ( Fig. 3(b)).
Our results show that the strain-induced nuclear quadruple splittings in semiconductor microcavity do not hinder the establishment of the thermodynamic equilibrium within the nuclear spin system. The quadrupole effects result in the increase of the local field, indicating that the heat capacity of the NSS is dominated by the quadrupole energy reservoir. The energy transfer between the Zeeman and quadrupole reservoirs during adiabatic demagnetisation is made possible by dipoledipole interaction via spin flip-flop transitions involving different isotopes. Thus, deep cooling of the NSS down to microKelvin temperature range via adiabatic demagnetisation is possible in photonic microstructures. This paves the way towards realisation of nuclear magnetically ordered states and their applications, including spin-photon interfaces with reduced thermal noise.
Supplemental Material
I. SAMPLES
The studied microcavity structures consist of Si-doped GaAs 3λ/2-cavity with electron concentrations n e = 2 × 10 15 cm −3 (Sample A) and n e = 4 × 10 16 cm −3 (Sample B). The front (back) mirrors are distributed Bragg reflectors composed of 25 (30) pairs of AlAs/Al 0.1 Ga 0.9 As layers, grown on a 400 µm thick GaAs substrate. Due to multiple round trips in the cavity, the Faraday rotation (FR) is amplified by a factor of N ∼ 1000, corresponding to the interaction length L = 0.7 mm (quality factor Q ∼ 20000 was measured by interferometric techniques) Sample C is the bulk 20 µm-thick GaAs layer grown by liquid-phase epitaxy, with Si donor concentration of n d = 4 × 10 15 cm 3 . All these samples have been studied previously 20,21,28,29 .
II. EXPERIMENTAL TECHNIQUES
Both the spin-noise (SN) and FR techniques have been used previously for studies of the NSS 20,21,29 and are described in in detail in Ref. 29. They have an advantage of being virtually non-perturbative for the NSS, because pumping and measurement stages are separated in time, and cooled NSS is optically probed via the polarisation rotation of the light beam with photon energy tuned below (here 20 meV) the band gap of the studied GaAs layer. In a typical measurement, the sample is placed in a cold finger cryostat at T = 5 K, B z = 180 G. At the first stage, it is optically pumped during 3-15 minutes by the circularly polarised laser diode with photon energy 1.57 eV and power P = 10 mW, focused on 1 mm spot on the sample surface. In the case of SN experiments, the transverse field B ⊥ is also applied during pumping. After the pumping stage, we wait for t w = 1 minute before lowering down B z (Fig. 2(a), 3(a)), to be sure that nuclear spins situated under the orbits of the donors and characterised by the relatively short T 1 do not contribute to the signal 20,21,23 . At the last preparation step, B z is lowering down to the value from which the measurement stage starts (B z = B i z = 50 G for FR and B z = 0 for SN). FR and SN experiments mainly differ by the measurement stage. In FR experiment, the rotation of the linearly polarised probe beam is detected in the presence of the slowly varying longitudinal magnetic field B z . In the SN experiment, the spin noise of the resident electrons is measured in the presence of the slowly varying transverse magnetic field B ⊥ via the fluctuation spectrum of the Faraday rotation angle. The probe beam has the photon energy 20 meV below GaAs band gap, power of 0.5 mW, and is focused on 30 µm spot on the sample surface.
II.1. Faraday rotation
To extract nuclear spin temperature and the local field from the Faraday rotation angle measured as a function of the slowly varying magnetic field B z (the duration of each scan is ∼ 10 s so that dB/dt = 10 G/s), we proceed as follows. First, we subtract the external field contribution from the total signal. This contribution to the signal remains unchanged for all consecutive scans and depends linearly on the magnetic field. The remaining part of the FR is induced by the Overhauser field B N , which is proportional to the nuclear spin polarisation P N .
φ N = B N V N L = b N P N V N L,(S1)
where b N = 5.3 T is the Overhauser field produced by the fully polarised nuclear spins 18 , V N is the nuclear Verdet constant, L is the effective optical length of the sample accounting for by multiple round trips of light in the cavity 20 . Therefore, from Eqs. (1) we get:
φ N = φ N i B/ B 2 + B 2 L (S2)
where φ N i is the Faraday rotation angle at the saturation field B i z . Fitting φ N to equation (S2) we determine B L = 8 ± 2 G. Using the values of V N = 0.1 mrad/G/cm and L = 0.7 mm determined in our previous work 20 we also extract Θ N 0 = −6 µK in sample B (Fig. 2 (e) in the main text) from Eqs. (S1), (S2) and (1). The values of B L and Θ N 0 extracted from FR measurement are averaged over the crystal volume, since the signal is given by the electron band spin splitting 20 .
II.2. Spin noise spectroscopy
The electron spin noise spectrum exhibits a pronounced peak at the electron Larmor frequency ν corresponding to the total (B ⊥ and B N ) field , so that: 21,23
ν = γ e (B ⊥ + B N ) = γ e (B ⊥ + b N P N ),(S3)
where γ e = 0.64 MHz/G is the gyromagnetic ratio of the electrons in the conduction band of GaAs 21 . Thus, by measuring ν as a function of B ⊥ and fitting equations (1) and (S3) to the data we obtain the values of Θ N 0 and B L . Because each field scan takes 100 s (10 times longer than in the case of the Faraday rotation measurements), the spin-lattice relaxation of the NSS is not negligible on this time-scale. It manifests itself in the asymmetry of the recorded sets of spectra with respect to zero magnetic field. For the quantitative comparison with the theory predictions given by Eqs.
(1) and (S3) we measured the magnetic field-dependent relaxation times in an independent set of experiments 29 . Note that in metallic samples the SN signal is mediated by the electron gas, and is therefore contributed by all the nuclei. In the insulating samples, only the nuclei situated under the donor orbits can be detected. However, the polarisation of the nuclear spins situated in the core of the donor orbit decays rapidly, and vanishes during t w . Thus the SN signal comes from the nuclei situated in the periphery of the donor orbits, so that the extracted Θ N 0 and B L are close to those of the bulk nuclei.
III. ESTIMATION OF THE MIXING FIELD
The mixing field B m , is the field at which Zeeman (for each of three isotopes) and internal energy reservoirs come to equilibrium between each other 4 . Only the Zeeman reservoirs can be cooled down via dynamic nuclear polarisation at strong field. By measuring nuclear polarisation, we get access to the average energy of the Zeeman reservoirs E Z = BP N hγ N . During adiabatic demagnetisation, energy transfer and the thermalisation between Zeeman and internal energy reservoirs is achieved at B = B m . At this field, a part of Zeeman energy is transferred to the internal energy reservoir, which results in the modification of the nuclear polarisation. The nonadiabaticity of this process would lead to a deviation from Eqs. (1), quantified by the nonadiabaticity factor f na = B m / B 2 + B 2 m 4 . Comparing the magnetisation measured at B = 50 G before and after the passage through zero field, we have not observed any difference within the experimental precision of 2%. This yields f na > 0.98, and therefore B m > 50 G.
. Two methods of detection of nuclear spin polarisation are used: (i) the Faraday effect induced by the Overhauser field 19,20 and (ii) the spin noise spectroscopy of resident electrons subject to the Overhauser field 21-23 . The main features of the behaviour of the optically cooled NSS under varying external magnetic fields are demonstrated in the experi-arXiv:1706.02528v1 [cond-mat.other] 8 Jun 2017 FIG.
magnetometry by Faraday rotation. (a) Timeline of the experiment. The preparation (blue area) consists in pumping under longitudinal magnetic field B p z , waiting for eventual nuclear relaxation in the vicinity of the localised electrons during tw and fast demagnetisation down to B i z . Faraday rotation of the probe beam is measured during successive scans of the magnetic field across zero (pink area, only first scan is shown). (b) Raw measurements of the Faraday rotation in Sample B (circles). NSS is prepared at ΘN < 0. During nine successive scans of the magnetic field (tm = 5 s, direction shown by arrows) conventional Faraday rotation remains constant, this contribution is shown by solid lines. The remaining contribution to the signal is due to the nuclear spin polarisation. It is shown separately in (e) for the first scan. (c-d) Faraday rotation induced in Sample A by nuclear spin prepared either at negative (c) or at positive (d) temperature (circles). Lines in (c-e) are calculated from Eq. (1), assuming different values of the local field, see Supplemental Material (SM).
FIG. 3 .
3Nuclear spin magnetometry by spin noise spectroscopy. (a) Timeline of the experiment. The preparation (blue area) consists in pumping under oblique magnetic field, waiting during the time tw required for nuclear relaxation in the vicinity of localised electrons and fast demagnetisation down to B i ⊥ . Spin noise spectra of the probe beam are measured while scanning B ⊥ across zero (pink area). (b-g) Color maps of the spin noise spectra during adiabatic demagnetisation procedure at positive (c, e and g) and negative (b, d and f) spin temperature (measured in the signal to shot noise ratio units) for two microcavity samples A ( b-c), B (d-e) and a bulk sample C (f-g). Black lines in (b-e) and red line in (f-g) are fits to Eqs. 1, that determine the values of BL and ΘN0 indicated on the figure (see also SM). Red lines in (b-e) illustrate how the the value BL = 2 G fails to describe the experiment.
FIG. 4 .
4(a) Nuclear spin flip transition frequencies νN for three GaAs isotopes, and (b) the differences ∆νN between them as functions of magnetic field in the absence of the quadrupole splitting. (c, d) Same as (a) and (b), respectively, but in the presence of the quadrupole spitting in z-direction. (d, f), Same as (c) and (d), respectively, but the magnetic field is applied in the plane of the structure. The blue area νN < 8 KHz in (a, c, d) shows the extent of splittings in the local magneto-dipole field and indicates the range of external fields where mixing is possible within each individual isotope. Similarly, the yellow area in (b, d, f) indicates the range of magnetic fields where mixing becomes possible if assisted by inter-isotope transitions. ACKNOWLEDGMENTS This work was supported by the joint grant of the Russian Foundation for Basic Research (RFBR, Grant No. 16-52-150008) and National Center for Scientific Research (CNRS, PRC SPINCOOL No. 148362), as well as French National Research Agency (Grant OBELIX, No. ANR-15-CE30-0020-02). IIR, VSZ and GGK acknowledge Russian Foundation for Basic Research (grant No. 17-12-01124) for the financial support of their experimental work.
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See details in Supplemental Material. See details in Supplemental Material.
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| []
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[
"PERTURBATIVE RESULTS WITHOUT DIAGRAMS",
"PERTURBATIVE RESULTS WITHOUT DIAGRAMS"
]
| [
"R Rosenfelder [email protected] \nParticle Theory Group\nLaboratory for Particle Physics Paul Scherrer Institut\nCH-5232Villigen PSISwitzerland\n"
]
| [
"Particle Theory Group\nLaboratory for Particle Physics Paul Scherrer Institut\nCH-5232Villigen PSISwitzerland"
]
| []
| Higher-order perturbative calculations in Quantum (Field) Theory suffer from the factorial increase of the number of individual diagrams. Here I describe an approach which evaluates the total contribution numerically for finite temperature from the cumulant expansion of the corresponding observable followed by an extrapolation to zero temperature. This method (originally proposed by Bogolyubov and Plechko) is applied to the calculation of higher-order terms for the ground-state energy of the polaron. Using state-of-the-art multidimensional integration routines 2 new coefficients are obtained corresponding to a 4-and 5-loop calculation. | 10.1142/9789812837271_0042 | [
"https://arxiv.org/pdf/0711.3989v1.pdf"
]
| 14,508,358 | 0711.3989 | 0b0be70289a0c219842f8a14e003d28170f97b12 |
PERTURBATIVE RESULTS WITHOUT DIAGRAMS
26 Nov 2007 February 2, 2008
R Rosenfelder [email protected]
Particle Theory Group
Laboratory for Particle Physics Paul Scherrer Institut
CH-5232Villigen PSISwitzerland
PERTURBATIVE RESULTS WITHOUT DIAGRAMS
26 Nov 2007 February 2, 200810:21 WSPC -Proceedings Trim Size: 9in x 6in pi˙dresden 1high-order perturbative calculationscumulant expansionMonte- Carlo integration
Higher-order perturbative calculations in Quantum (Field) Theory suffer from the factorial increase of the number of individual diagrams. Here I describe an approach which evaluates the total contribution numerically for finite temperature from the cumulant expansion of the corresponding observable followed by an extrapolation to zero temperature. This method (originally proposed by Bogolyubov and Plechko) is applied to the calculation of higher-order terms for the ground-state energy of the polaron. Using state-of-the-art multidimensional integration routines 2 new coefficients are obtained corresponding to a 4-and 5-loop calculation.
Introduction
Highly accurate measurements require precise theoretical calculations which perturbation theory can yield if the coupling constant is small. However, in Quantum Field Theory (QFT) the number of diagrams grows factorially with the order of perturbation theory and they become more and more complicated. The prime example is the anomalous magnetic moment of the electron where new experiments 1 need high-order quantum-electrodynamical calculations but the number of diagrams for them "explodes" as shown by the generating function 2 Γ(α) = 1 + α + 7 α 2 + 72 α 3 + 891 α 4 + 12672 α 5 + 202770 α 6 + . . . (1) There are ongoing efforts 3 to calculate all 12672 diagrams in O(α 5 ) -a huge, heroic effort considering the complexity of individual diagrams and the large cancellations among them.
Obviously new and more efficient methods would be most welcome for a cross-check or further progress.
2.
A new method (applied to the polaron g.s. energy)
Here I present a "new" method which -as I learned during the conference -was already proposed 20 years by Bogolyubov (Jr.) and Plechko (BP) 4 . However, to my knowledge it has been never applied numerically which turned out to be quite a challenging task.
The BP method is formulated for the polaron problem, a non-relativistic (but non-trivial) field theory describing an electron slowly moving through a polarizable crystal. Due to medium effects its energy is changed and it acquires an effective mass : E p = E 0 + p 2 /(2m ⋆ ) + . . .. The aim is to calculate the power series expansion for the g.s. energy
e 3 = −0.00080607(3)
there has been no progress towards higher-order terms. This will be remedied by the first numerical application of the BP method. For this purpose the path integral formulation of the polaron problem will be used where the phonons have been integrated out exactly 7 . For large Euclidean times β this gives the following effective action
S eff [x] = β 0 dt 1 2ẋ 2 − α √ 2 β 0 dt t 0 dt ′ e −(t−t ′ ) d 3 k 2π 2 exp [ik · (x(t) − x(t ′ ))] k 2(
4) which will be split into a free part S 0 and an interaction term S 1 . The g.s. energy may be obtained from the partition function
Z(β) = D 3 x e −S eff [x] β→∞ −→ e −βE0(5)
at asymptotic values of β, i.e. zero temperature. The central idea is to use the cumulant expansion of the partition function
Z(β) = Z 0 exp n=1 (−) n n! λ n (β)(6)
where the λ n (β)'s are the cumulants w.r.t. S 1 . These are obtained from the moments
m n : = N D 3 x ( S 1 [x] ) n e −S0[x] , m 0 = 1(7)
by the recursion relation (see, e.g. Eq. (51) in Ref. 8)
λ n+1 = m n+1 − n−1 k=0 n k λ k+1 m n−k .(8)
Explicitly the first cumulants read
λ 1 = m 1 , λ 2 = m 2 − m 2 1 , λ 3 = m 3 − 3 m 2 m 1 + 2 m 3 1 λ 4 = m 4 − 4 m 3 m 1 − 3 m 2 2 + 12 m 2 m 2 1 − 6 m 4 1 (9) λ 5 = m 5 − 5 m 4 m 1 − 10 m 3 m 2 + 20 m 3 m 2 1 + 30 m 2 2 m 1 − 60 m 2 m 3 1 + 24 m 5 1
By construction m n ∝ α n and Eq. (8) shows that the cumulants share this property. Thus we immediately obtain
e n = lim β→∞ 1 β (−) n+1 α n n! λ n (β) .(10)
The functional integral for the moments can be done since it is Gaussian. The integrals over the phonon momenta k m , m = 1 . . . n can also be performed if the m th propagator is written as
1 k 2 m = 1 2 ∞ 0 du m exp − 1 2 k 2 m u m .(11)
Then one obtains
m n = (−) n α n (4π) n/2 n m=1 β 0 dt m tm 0 dt ′ m ∞ 0 du m exp − n m=1 (t m − t ′ m ) · [ det A (t 1 . . . t n , t ′ 1 . . . t ′ n ; u 1 . . . u n ) ] −3/2 .(12)
Here the (n × n)-matrix A
A ij = 1 2 −|t i − t j | + |t i − t ′ j | + |t ′ i − t j | − |t ′ i − t ′ j | + u i δ ij .(13)
is non-analytic in the times t i , t ′ i , but analytic in the auxiliary variables u i .
Numerical procedures and results
The task is now to perform the (3n)-dimensional integral over t i , t ′ i , u i for large enough β in the expression for the cumulants/moments. It is clear that any reduction in the dimensionality of the integral will greatly help in obtaining reliable numerical results in affordable CPU-time. A closer inspection of the structure of the integrand reveals that 2 integrations over the auxiliary variables (say u n , u n−1 ) can always be done analytically. Furthermore, we do not use Eq. (10) to extract the energy coefficient e n but e n = (−) n+1 α n n! lim β→∞ ∂λ n (β) ∂β = : lim β→∞ e n (β) .
This "kills two birds with one stone": first the derivative w.r.t. β takes away one further integration over a time (see Eq. (12) where β appears as upper limit) requiring that only a (3n − 3)-dimensional integral has to be done numerically. Second, it vastly improves the convergence to e n ≡ e n (β = ∞) because now
e n (β) β→∞ −→ ∂ ∂β β · e n +const− a n √ β e −β +. . . = e n + a n √ β e −β +. . . . (15)
In other words : we obtain an exponential convergence to the value e n whereas previously the approach would be very slow, like const/β. This exponential convergence of the derivative version has been demonstrated analytically for n = 1, 2 and numerically for n = 3 (see below). In the following we will assume that it holds for all n. After mapping to the hypercube [0, 1] the remaining (3n − 3)-dimensional integral can be evaluated by Monte-Carlo techniques utilizing the classic VEGAS program 9 or the more modern programs from the CUBA library 10 . We first have tested this approach by comparing with the analytical result given in Eq. (3). Fig. 1 shows e 3 (β) and the best fit to the data However, when extending these calculations to the case n = 4 a very slow convergence of the numerical result with the number of function calls n tot is observed at fixed β. Fortunately, a solution was found by performing the remaining (n − 2) u i -integrations not by stochastic (Monte-Carlo) methods but by deterministic quadrature rules. This is possible since the u i -dependence of the integrand is analytic (see Eq. (13)). We have used the very efficient "tanhsinh-integration" method 11 but Gaussian quadrature is nearly as good. A dramatic improvement in stability results together with a reduction of n tot needed for the much smaller values of |e n | , n > 3. This allows a reliable evaluation of e 4 (see Fig. 2 a) and also makes the determination of e 5 feasible as shown in Fig. 2 The best fit values for e 4 and e 5 displayed in Figs. 2 a, b are still preliminary as a more detailed error analysis has to be made. Also for the n = 5 case the Monte-Carlo statistics should be improved. Note that each high-statistic point in Fig. 2 b took about 30 days runtime on a Xeon 3.0 GHz machine.
b. (a) (b)
Summary and Outlook
• Two additional perturbative coefficients e 4 , e 5 for the polaron g.s. energy have been determined by the method of Bogolyubov and Plechkov (rediscovered independently). This amounts to performing a 4-loop and 5-loop calculation in Quantum Field Theory. • The method is based on a combination of Monte-Carlo integration techniques and deterministic quadrature rules for finite β (temperature) and on a judicious extrapolation to β → ∞ (zero temperature). As a check the value of e 3 calculated analytically by Smondyrev has been reproduced with high accuracy. • The cancellation in n th order is not among many individual diagrams but among the much fewer terms in the integrand of the (3n − 3)-dimensional integral (see Eq. (9)). • The method can be simply extended to the calculation of higherorder terms in the small-coupling expansion of the effective mass m ⋆ (α) for a moving polaron. • Generalizing this approach to relativistic QFT in the worldline representation 12 and calculation of higher-order terms for the anomalous magnetic moment of the electron is under investigation. New challenges arise from the divergences which now occur and the need for renormalization.
of the dimensionless electron-phonon coupling constant α. The lowest-order coefficients are well-known 5 ( e 1 = −1 , e 2 = −0.01591962 ) but since Smondyrev's calculation 6 in 1986
Fig. 1 .
1(color online). Monte-Carlo results for the derivative of the 3 rd cumulant as function of the Euclidean time β. The total number of function calls is denoted by ntot and the full (open) circles are the points used (not used) in the fit. assuming the β-dependence (15). Since the asymptotic behaviour is not valid for low values of β we have eliminated small-β points successively until the resulting χ 2 /dof of the fit reaches a minimum. Excellent agreement with Smondyrev's result (3) is found. If one allows for a different power of β in the prefactor of Eq. (15) then the fit gives an exponent −0.55(3) instead of −0.5 assumed before.
Fig. 2 .
2(a) Same as Fig. 1 but for the 4 th cumulant. (b) Data for the derivative of the 5 th cumulant. Open triangles denote results (not used in the fit) which have a χ 2 > 1.5 indicating that successive Monte-Carlo iterations are not consistent with each other.
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[
"The Vervaat transform of Brownian bridges and Brownian motion",
"The Vervaat transform of Brownian bridges and Brownian motion"
]
| [
"Titus Lupu ",
"Jim Pitman ",
"Wenpin Tang "
]
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| []
| For a continuous function f ∈ C([0, 1]), define the Vervaat transform V (f )(t) := f (τ (f ) + t mod 1) + f (1)1 {t+τ (f )≥1} − f (τ (f )), where τ (f ) corresponds to the first time at which the minimum of f is attained. Motivated by recent study of quantile transforms of random walks and Brownian motion, we study the Vervaat transform of Brownian motion and Brownian bridges with arbitary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decomposition and study its semimartingale property. The expectation and variance of the Vervaat transform of Brownian motion are also derived. 1 0 1 Bs≤a ds > t} is the quantile function of occupation measure (see Dassios [16], Embrechts et al [19] for general background).The key result of [3] is to identify the distribution of the somewhat mysterious Q(w) with that of the Vervaat transform V (w) defined as:where τ n := argmin i∈ [1,n]where A is the a.s. arcsine split (1 − A := argmin t∈[0,1] B t , see Karatzas and Shreve [28]).As a result, to understand quantile transform of B, it is equivalent to study its substitute, the Vervaat transform V (B). Historically, Vervaat[41]showed that if B is conditioned to both start and end at 0, then V (B) is a Brownian excursion:is a Brownian bridge of length 1 starting at 0 and ending at 0. Biane [7] proved a converse theorem to Vervaat's result, i.e. recover standard Brownian bridges from Brownian excursion by uniform sampling: Theorem 1.2 [7] Let B ex be a standard Brownian excursion and U a uniformly distributed random variable independent of B ex . Then the shifted process θ(B ex , U ) defined by θ(B ex , U ) t := B ex U +t − B ex U for 0 ≤ t ≤ 1 − U B ex U +t−1 − B ex U for 1 − U ≤ t ≤ 1, is a standard Brownian bridge. | 10.1214/ejp.v20-3744 | [
"https://arxiv.org/pdf/1310.3889v3.pdf"
]
| 17,537,465 | 1310.3889 | 481e8d09192fa4f02c81d7522e7ec18f6d92c5e2 |
The Vervaat transform of Brownian bridges and Brownian motion
October 16, 2013
Titus Lupu
Jim Pitman
Wenpin Tang
The Vervaat transform of Brownian bridges and Brownian motion
October 16, 2013AMS 2010 Mathematics Subject Classification: 60C0560J6060J65 Keywords: Brownian quartetBessel processesMarkov propertypath decomposi- tionsemimartingale propertyVervaat transform
For a continuous function f ∈ C([0, 1]), define the Vervaat transform V (f )(t) := f (τ (f ) + t mod 1) + f (1)1 {t+τ (f )≥1} − f (τ (f )), where τ (f ) corresponds to the first time at which the minimum of f is attained. Motivated by recent study of quantile transforms of random walks and Brownian motion, we study the Vervaat transform of Brownian motion and Brownian bridges with arbitary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decomposition and study its semimartingale property. The expectation and variance of the Vervaat transform of Brownian motion are also derived. 1 0 1 Bs≤a ds > t} is the quantile function of occupation measure (see Dassios [16], Embrechts et al [19] for general background).The key result of [3] is to identify the distribution of the somewhat mysterious Q(w) with that of the Vervaat transform V (w) defined as:where τ n := argmin i∈ [1,n]where A is the a.s. arcsine split (1 − A := argmin t∈[0,1] B t , see Karatzas and Shreve [28]).As a result, to understand quantile transform of B, it is equivalent to study its substitute, the Vervaat transform V (B). Historically, Vervaat[41]showed that if B is conditioned to both start and end at 0, then V (B) is a Brownian excursion:is a Brownian bridge of length 1 starting at 0 and ending at 0. Biane [7] proved a converse theorem to Vervaat's result, i.e. recover standard Brownian bridges from Brownian excursion by uniform sampling: Theorem 1.2 [7] Let B ex be a standard Brownian excursion and U a uniformly distributed random variable independent of B ex . Then the shifted process θ(B ex , U ) defined by θ(B ex , U ) t := B ex U +t − B ex U for 0 ≤ t ≤ 1 − U B ex U +t−1 − B ex U for 1 − U ≤ t ≤ 1, is a standard Brownian bridge.
Introduction
In a recent work of Assaf et al [3], a novel path transform, called the quantile transform Q has been studied both in discrete and continuous time settings. Inspired by previous work in fluctuation theory (see e.g. and Wendel [42] and Port [36]), the quantile transform for simple random walks is defined as follows. For w a simple walk of length n, with increments of ±1 the quantile transform associated to w is defined by:
∀j ∈ [1, n], Q(w) j := j i=1 w(φ w (i)) − w(φ w (i) − 1),
where φ w is the quantile permutation on [1, n] defined by lexicographic ordering on pairs (w(j − 1), j), that is w(φ w (i) − 1) < w(φ w (j) − 1) or w(φ w (i) − 1) = w(φ w (j) − 1), φ w (i) ≤ φ w (j) if and only if i ≤ j.
As shown in [3], the scaling limit of this transformation of simple random walks is the quantile transform in the continuous case of Brownian motion B := (B t ; 0 ≤ t ≤ 1):
∀t ∈ [0, 1], Q(B) t := 1 2 L a(t) 1 + (a(t)) + − (a(t) − B 1 ) + ,
where L a 1 is the local time of B at level a up to time 1 and a(t) := inf{a;
Chaumont [13] extended partly the result to stable cases, Chassaing and Jason [12] to the reflected Brownian bridges case, Miermont [31] to the spectrally positive case, Fourati [23] to the general Lévy case under some mild hypotheses, Le Gall and Weill [29] to the Brownian tree case and more recently, Lupu [30] to the diffusion case. However, as far as we are aware, there has not been previous study of the Vervaat transform of an unconditioned Brownian motion B or of the Brownian bridges B λ,br ending at λ = 0.
The contribution of the current paper is to give some path decomposition result of Vervaat transform of Brownian bridges (for simplicity, call them Vervaat bridges) with non-zero endpoints. In the case of a Vervaat bridge with negative endpoint V (B λ,br ) where λ < 0, the key idea is to decompose it into two pieces, the first piece a Brownian excursion and the second piece a first passage bridge. The main result is stated as follows: Theorem 1.3 Let λ < 0. Given Z λ the first return to 0 of V (B λ,br ), whose density is given by
f Z λ (t) = |λ| 2πt(1 − t) 3 exp − λ 2 t 2(1 − t) ,(1)
the path is decomposed into two (conditionally) independent pieces:
• (V (B λ,br ) u ; 0 ≤ u ≤ Z λ ) is a Brownian excursion of length Z λ ;
• (V (B λ,br ) u ; Z λ ≤ u ≤ 1) is a first passage bridge through level λ of length 1 − Z λ . Note that Theorem 1.1 [41] is recovered as a weak limit λ → 0 of the previous theorem. The parametric density family (f Z λ ) λ<0 appears earlier in the work of Aldous and Pitman [2], Corollary 5 when they studied the standard additive coalescent. Precisely, Z λ d =
B 2 1 λ 2 +B 2 1
where B 1 is normal distributed with mean 0 and variance 1. We also refer readers to Pitman [33], Chapter 4 for some discussion therein.
For the Vervaat bridges V (B λ,br ) which ends up with some positive value, it is easy to see that we have the following duality relation:
V (B λ,br ) t ; 0 ≤ t ≤ 1 d = V (B −λ,br ) 1−t + λ; 0 ≤ t ≤ 1 for λ > 0.(2)
In other words, looking backwards, we have a first piece of excursion above level λ followed by a first passage bridge. Note in addition that a first passage bridge form λ > 0 to 0 has the same distribution as a three dimensional Bessel bridge from λ to 0 (see Biane and Yor [8]). We have the following decomposition of Vervaat bridges with negative endpoints:
Corollary 1.4 Let λ > 0. Given Z λ the time of last hit of λ by V (B λ,br ) strictly before 1, whose density is given by f Z λ (t) = f Z −λ (1 − t) as in (1), the path is decomposed into two (conditionally) independent pieces:
• (V (B λ,br ) u ; 0 ≤ u ≤ Z λ ) is a three dimensional Bessel bridge of length Z λ starting from 0 and ending at λ;
• (V (B λ,br ) u ; Z λ ≤ u ≤ 1) is a Brownian excursion above level λ of length 1 − Z λ . The rest of the paper is organized as follows. In Section 2, we provide two different proofs of Theorem 1.3, one via weak limit approach which is based on some bijection lemma proved in Assaf et al [3] and the other with resort to excursion theory that relies on results in Pitman and Yor [35].
In Section 3, we give a thorough study of V (B λ,br ) where λ = 0 using Theorem 1.3 and Corollary 1.4. We prove that such processes are not Markov (Section 3.2). However, they are semimartingales and an explicit decomposition is given (Section 3.3). We also relate these processes to some simpler ones (Section 3.1, 3.4) and study the convex minorant of such processes (Section 3.5).
In Section 4, we focus on studying the Vervaat transform of Brownian motion. We first prove that V (B) is not Markov as well (Section 4.1). Nevertheless, we show that it is a semimartingale and the semimartingale decomposition is given (Section 4.2, 4.3). Finally, we provide explicit formulae for the first two moments of the Vervaat transform of Brownian motion (Section 4.4).
Path decomposition for Vervaat bridges
The whole section is devoted to proving Theorem 1.3. First, we use a discrete approximation argument to obtain the path decomposition of V (B λ,br ) where λ < 0. Also we obtain an analog to Theorem 1.2 as a by-product. In the second part, we recover the same result via excursion theory.
Path decomposition via weak limit approach
Discrete case analysis
We begin with the discrete time analysis of random walk cases which is based on combinatorial principles. For a simple random walk w of length n with increments ±1, we would like to describe the law of V (w a ) := (V (w)|w(n) = a) where a < 0 having the same parity as n.
Denote τ V (w) = min{j ∈ [0, n]; w(j) ≤ w(i), ∀i ∈ [0, n]} (the first global minimum of the path) and K(w) = n − τ V (w) (distance from the first global minimum to the end of the path). Following from Theorem 7.3 in Assaf et al [3], the mapping w → (V (w), K(w)) is a bijection between walk(n), the set of simple random walks of length n and the set
{(v, k); v ∈ walk(n), v(j) ≥ 0 for 0 ≤ j ≤ k and v(j) > v(n) for k ≤ j < n},
where k, called a helper variable, records the splitting position in the original path.
The following result turns out to be a direct consequence of this theorem related to Vervaat bridges. Lemma 2.1 w a → (V (w a ), K(w a )) forms a bijection between {w ∈ walk(n) : w(n) = a} (simple random walk bridges which end at a < 0) and the set
{(v, k); v ∈ walk(n), v(j) ≥ 0 for 0 ≤ j ≤ k, v(j) > a for k ≤ j < n and v(n) = a}.
Observe that, to each pair (v, k) in the above set, one can associate a unique triple
(Z a , f br,1 Z a , f br,2 Z a ) where
• Z a is the first time that the path hits level −1,
• f br,1 Z a is the sample path of a first passage bridge of length Z a through level −1,
• f br,2 Z a is that of a first passage bridge of length n − Z a starting at −1 through a.
Remark that to different pairs (v, k), one may have the same triple (Z a , f br,1 Z a , f br,2 Z a ). We now focus on calculating explicitly the distribution of Z a by counting paths. By Lemma 2.1, the total number of the Vervaat transform paths (counting with multiplicity) is (see Chapter III of Feller [20]). Therefore, the total number of the Vervaat transform configurations (counting with multiplicity ) is
|a| − 1 l(n − l) l+1 2 l n−l+|a|−1 2 n − l .
Also note that every Vervaat transform configuration is counted exactly l times (by bijection lemma 2.1). Hence,
P(Z a = l) = |a| − 1 n − l l+1 2 l n−l+|a|−1 2 n−l n+|a| 2 n .(3)
Combining the above discussions, we get the following path decomposition result for discrete Vervaat bridges with negative endpoint: Theorem 2.2 Let a < 0 and have the same parity as n. Given Z a := min{j > 0; V (w a ) j = −1} (distributed as (3)), the path is decomposed into two (conditionally) independent pieces:
• V (w a )| [0,Z a ] is a random walk first passage bridge of length Z a through level −1, • V (w a )| [Z a ,
n] is a random walk first passage bridge starting at −1 through level a of length n − Z a . The theorem provides a path decomposition of Vervaat bridges into two pieces of first passage bridges, one through level −1 and the other from −1 to a. Note that it is also possible to decompose the path slightly differently by a first piece of excursion and the second a first passage bridge through level −a. However, the distribution of the splitting position is much less explicit and thus does not make the proof any easier when passing to the scaling limit.
Continuous case: passage to weak limit
We now turn to the continuous case by appealing to invariance principles. We derive the path decomposition result from Theorem 2.2.
For λ < 0 and 0 < t < 1, let λ n ∼ λ √ n and have the same parity as n and t n := 2[ tn 2 ] + 1 be two fixed sequences. Let S λn be simple random walks of length n with increments ±1 which end at λ n , V (S λn ) be the associated discrete Vervaat bridge and Z λn := inf{j > 0 : V (S λ n ) j = −1}. Define V (S λn )(u); 0 ≤ u ≤ n to be the linear interpolation of the discrete Vervaat bridge V (S λn ).
Recall some invariance principle results on metric space C[0, 1] (continuous functions on [0, 1]). For general background on weak convergence in C[0, 1], we refer the readers to Chapter 2, Billingsley [11].
Lemma 2.3 (a).( 1 √ n V (S λn )(nu); 0 ≤ u ≤ 1) converges in C[0, 1] to (V (B λ,br ) u ; 0 ≤ u ≤ 1). (b).Given Z λn = t n , ( 1 √ n V (S λn )(nu); 0 ≤ u ≤ t) converges in C[0, 1] to a Brownian excursion of length t and ( 1 √ n V (S λn )(nu); t ≤ u ≤ 1) converges in C[0, 1
] to a first passage bridge through level λ of length 1 − t, (conditionally) independent of the excursion.
Proof: The assertion (a) can be viewed as a variant of the results proved in Vervaat [41]. According to Theorem 2.2, given Z λn = t n , the path of V (S λn ) is split into two (conditionally) independent pieces of discrete first passage bridges. Following Bertoin et al [6] and Iglehart [25], the scaled first passage bridge through level −1 converges weakly to a Brownian excursion and the scaled first passage bridge from −1 to λ n converges weakly to a first passage bridge through level λ. This proves (b).
To prove Theorem 1.3, we need to compute the limiting distribution of Z λn = t n as n → ∞. Precisely,
nP(Z λn = t n ) = n|λ n | n − t n tn+1 2 tn n−tn+|λn|−1 2 n−tn n+|λn| 2 n .(4)
Using Stirling's formula, we see:
tn+1 2 t n ∼ 2 πnt 2 nt ; n+|λn| 2 n ∼ 2 πn 2 n exp − λ 2 2 ;
and n−tn+|λn|−1 2
n − t n ∼ 2 πn(1 − t) 2 n(1−t) exp − λ 2 2(1 − t)
.
Injecting these terms in (4), we deduce the limiting distribution as n → ∞ given by (1) . By a local limit argument (see Billingsley [10], Exercise 25.10), we conclude that Z λ has density f Z λ given in (1).
The next theorem is a direct consequence of Theorem 1.3 and should be called a corollary at best. Because of its importance, however, we give it status of a theorem. Theorem 2.4 Given Z λ the length of first excursion of (V (B λ,br ) t ; 0 ≤ t ≤ 1) where λ < 0, the split position A λ := 1 − argmin t∈[0,1] B λ,br t (distance from the minimum of the original bridge path to the end) is (conditionally) independent of V (B λ,br ) and uniformly distributed on [0, Z λ ], In particular, its density is
f A λ (a) = 1 a f Z λ (t) t dt,
where f Z λ is given by (1).
Proof: Note that in the discrete case, given a Vervaat bridge path, the helper variable k takes values exactly in {0, ..., Z λn } where Z λn is the first time that the path returns to 0. This implies that given Z λn , the minimum position of the original bridge is uniformly distributed on [0, Z λn ]. We then obtain the results in the theorem by passing to the scaling limit. Remark: The above corollary holds true for λ ≤ 0 and the case λ = 0, i.e. Theorem 1.2 [7] is recovered as a weak limit λ → 0:
Z λ d → 1 and A λ d → Uniform[0, 1].
Path decomposition via excursion theory
In the current section, we provide an alternative proof of Theorem 1.3 using excursion theory. The proof relies on the decomposition of bridges at their minimum, similar to the decomposition at the maximum that appears in Pitman and Yor [35]. We begin with some notations.
Let p t (x, y) be the heat kernel:
p t (x, y) = 1 √ 2πt exp − (y − x) 2 2t ,
and P T 0,λ be the law of the Brownian bridge from 0 to λ of length T and P Ty x be the law of the Brownian path starting from x until the first time it hits y for y < x.
Given a distribution Q on paths of finite length, denote Q ∧ its image by time reversal. Given Q and Q two distributions on paths of finite length, Q • Q will be the distribution obtained by concatenating two independent paths, one following the distribution Q and the other the distribution Q . According to Corollary 3 in Pitman and Yor [35]: Observe that P Ty 0 can be decomposed as:
P Ty 0 = P T y−λ 0 • P Ty y−λ . Therefore, +∞ 0 dT p T (0, λ)P T 0,λ = 2 λ −∞ dy P T y−λ 0 • P Ty y−λ • P Ty∧ λ .(5)
We extend the definition of Vervaat's transform to continuous path with finite but arbitrary life-time: given a continuous function f on [0, T ] and τ (f ) the first time it attains its minimum, define
V T (f )(t) := f (τ (f ) + t mod T ) + f (T )1 {t+τ (f )≥T } − f (τ (f )).
Apply the Vervaat's operator V T to (5), we get:
+∞ 0 dT p T (0, λ)V T (P T 0,λ ) = 2 λ −∞ dy P T 0 ∧ λ−y • P T 0 λ−y • P T λ 0 = 2 +∞ 0 dy P T 0 ∧ y • P T 0 y • P T λ 0 .(6)
Take λ = 0 in (6), we obtain:
+∞ 0 dT p T (0, 0)V T (P T 0,0 ) = 2 +∞ 0 dy P T 0 ∧ y • P T 0 y .
Let Q T 0,0 be the law of positive Brownian excursion of length T (three dimensional Bessel bridge from 0 to 0). According to Vervaat's result [41], V T (P T 0,0 ) = Q T 0,0 . Thus,
2 +∞ 0 dy P T 0 ∧ y • P T 0 y = +∞ 0 dT p T (0, 0)Q T 0,0 .
Injecting the above identity in (6), we get:
+∞ 0 dT p T (0, λ)V T (P T 0,λ ) = +∞ 0 ds p s (0, 0)Q s 0,0 • P T λ 0 .(7)
By disintegrating (7) with respect to the life-time of paths, we see that V T (P T 0,λ ) is a concatenation of an excursion and a first passage bridge.
Let g t (λ) be the density of the first hit of λ by Brownian motion starting from 0:
g t (λ) = |λ| √ 2πt 3 exp − λ 2 2t .(8)
It follows from (7) that the density of the splitting point Z λ between the excursion and the first passage bridge in V (P 1 0,λ ) is:
p t (0, 0)g 1−t (λ) p 1 (0, λ) = f Z λ (t) as in (1).
Remark: P.Fitzsimmons points out that the decomposition result is also a consequence of a local Williams decomposition, which can be found in the section 6 of [22].
Study of Vervaat bridges
In this section, we will study thoroughly the Vervaat bridges with non-zero endpoint. First, we give an alternative construction of V (B λ,br ) using length-biased sampling techniques. Next we show that such processes are not Markov with respect to their induced filtrations. Despite lack of markovianity, they are semimartingales with respect to their own filtrations and explicit decomposition formulae are given in both positive and negative endpoint cases. Moreover, we relate Vervaat bridges to drifting excursion by additional conditioning. To close the section, we study some properties of convex minorant of V (B λ,br ) where λ < 0.
Construction of Vervaat bridges via Brownian bridges
In the current part, we try to provide an alternative construction of the Vervaat bridges with negative endpoint via standard Brownian bridges (which end at 0). It is obvious that the Vervaat bridges with positive endpoint can be treated similarly by time reversal.
Let λ < 0. As seen in the last section, conditioned on Z λ the first return to 0, the process is split into B ex,Z λ an excursion of length Z λ followed by F λ,1−Z λ a first passage bridge through λ of length 1 − Z λ , independent of each other. Formally, V (B λ,br ) looks much like a standard first passage bridge (of length 1) except that it has an excursion piece placed first. Therefore, it is interesting to ask whether this process can be derived from standard first passage bridge via some simple operations.
Recall that a standard first passage bridge can be constructed via standard Brownian bridge by conditioning on its local time. Denote (F λ t ; 0 ≤ t ≤ 1) for a standard first passage bridge through λ < 0. Following from Bertoin et al [6],
(F λ t ; 0 ≤ t ≤ 1) d = (|B 0,br t | − L 0 t (B 0,br ); 0 ≤ t ≤ 1|L 0 1 (B 0,br ) = |λ|),(9)
where L 0 t is the local time (of a Brownian bridge) at level 0 up to time t. In light of the above construction, the following theorem tells how to construct the Vervaat bridges with negative terminal value by standard Brownian bridges.
Theorem 3.1 Let U be uniformly distributed on (0, 1) independent of X := (B 0,br (t); 0 ≤ t ≤ 1|L 0 1 (B 0,br ) = |λ|) where λ < 0 and (G U , D U )
be the signed excursion interval which contains U . LetX be the process by exchanging the position of the excursion of X straddling time U and the path along [0, G U ], namely:
X t = X t+G U for 0 ≤ t ≤ D U − G U X t−D U +G U for D U − G U ≤ t ≤ D U X t for D U ≤ t ≤ 1.
Then we have the following identity in law: Proof: According to Theorem 1.3, the law of V (B λ,br ) is uniquely determined by that of the triple (Z λ , B ex,Z λ , F λ,1−Z λ ). It suffices to prove that the law of the process on the left hand side of (10) is entirely characterized by the same triple. Following Theorem 3.1 in Perman et al [32] and the discussion below Lemma 4.10 of Pitman [33],
(|X(t)| − L 0 t (X); 0 ≤ t ≤ 1|L 0 1 (X) = |λ|) d = (V (B λ,br ) t ; 0 ≤ t ≤ 1).(10)conditioned on ∆ := D U − G U , (X t ; 0 ≤ t ≤ ∆) and (X t ; ∆ ≤ t ≤ 1)
are independent and ∆ corresponds to the length of first excursion of X via length-biased sampling:
f ∆ (t) = f Z λ (t) as in (1). Thus, ∆ d = Z λ . Finally, since L 0 t (X) = 0 on (0, D U − G U ), we have that (|X(t)| − L 0 t (X); 0 ≤ t ≤ D U − G U |L 0 1 (X) = |λ|) is a Brownian excursion of length ∆ (conditionally) independent of (|X(t)| − L 0 t (X); D U − G U ≤ t ≤ 1|L 0 1 (X) = |λ|)
, which is a first passage bridge through level λ of length 1 − ∆ by construction (9).
Remark:
The process X defined in the above theorem is a Brownian bridge conditioned on its local time, see Chassaing and Janson [12] for detail discussions. In addition, the proof of Theorem 3.1 in Perman et al [32] is extensively based on the concept of Palm distribution, which can be read from Fitzsimmons et al [21].
Vervaat bridges are not Markov
It is natural to ask whether the Vervaat bridges are Markov (with respect to their induced filtrations). In the case of negative endpoints, it is equivalent to ask whether the entrance law after the excursion piece is nice enough for the first passage bridge to produce Markov property. The following result gives a negative answer.
Proposition 3.2 (V (B λ,br ) t ; 0 ≤ t ≤ 1)) where λ < 0 is not Markov with respect to its induced filtration.
Before proving the proposition, we introduce some notations that we use in the current section and rest of the paper. For x, y > 0, denotẽ
q t (x, y) := 1 xy √ 2πt exp − (x − y) 2 2t − exp − (x + y) 2 2t .
Note thatq t (x, y)y 2 dy is the transition kernel of three dimensional Bessel process and
q t (0, y) = lim x→0 +q t (x, y) = 2 √ 2πt 3 exp − y 2 2t = 2 y g t (y), q t (0, 0) = 2 √ 2πt 3 ,
where g t (y) is the density of the first hitting at level y for Brownian motion given in (8).
Proof of Proposition 3.2: Fix t 0 ∈ (0, 1) and
x 0 > 0. Let T t 0 be the first return of V (B λ,br ) to 0 after time t 0 . Consider the distribution of T t 0 given V (B λ,br )t 0 2 = 0 and V (B λ,br ) t 0 = x 0 . According to Theorem 1.3, given T t 0 , (V (B λ,br ) t ; t 0 ≤ t ≤ T t 0 ) and (V (B λ,br ) t ; T t 0 ≤ t ≤ 1)
are two independent first passage bridges from x 0 > 0 to 0 respectively from 0 to λ < 0. Therefore, its density is given by
f 1 (t) = g t−t 0 (x 0 )g 1−t (|λ|) g 1−t 0 (1 + |λ|) 1 t>t 0 = C 1 (t 0 , x 0 , λ) (t − t 0 ) 3 (1 − t) 3 exp − x 2 0 2(t − t 0 ) − λ 2 2(1 − t) 1 t>t 0 ,(11)
for some C 1 (t 0 , x 0 , λ) > 0. Next we consider the distribution of T t 0 given that ∀u ∈ (0, t 0 ), V (B λ,br ) u > 0 and V (B λ,br ) t 0 = x 0 , whose density can be computed using Bayes recipe:
f 2 (t) = C 2 (t 0 , x 0 , λ)q t 0 (0, x 0 )q t−t 0 (x 0 , 0) q t (0, 0) x 2 0 f Z λ (t)1 t>t 0 = C 2 (t 0 , x 0 , λ) t (t − t 0 ) 3 (1 − t) 3 exp − x 2 0 2(t − t 0 ) − λ 2 2(1 − t) 1 t>t 0 .(12)
for some C 2 (t 0 , x 0 , λ) > 0 and C 2 (t 0 , x 0 , λ) > 0. Comparing (11) to (12), we have that f 2 (t) = C 1,2 (t 0 , x 0 , λ)tf 1 (t) for some C 1,2 (t 0 , x 0 , λ) > 0. The two conditional densities of T t 0 fail to be equal and we conclude that the Vervaat bridges with negative endpoint are not Markov.
Remark: The counter-example provided in the proof of Proposition 3.2 indicates that the main reason that makes Vervaat bridges with negative endpoint non-Markov is the lack of information on Z. Indeed, for s ≤ t ≤ 1, V (B λ,br ) t depends not only on V (B λ,br ) s but also on the event {Z ≤ s}.
It is well-known that the time reversal of any Markov process is still Markov. This result leads to the following corollary saying that the Vervaat bridges with positive endpoint is not Markov as well.
Corollary 3.3 (V (B λ,br ) t ; 0 ≤ t ≤ 1)) where λ > 0 is not Markov with respect to its induced filtration.
Now we know that the Vervaat bridges of non-zero endpoint are not Markov. Thus it is natural to ask how bad they may behave so that Markov property cannot be produced. This leads to the question that whether they are semimartingales and what is the semimartingale decomposition. The following section provides some positive answers to this question.
Semimartingale decomposition of the Vervaat bridges
This section is devoted to the semimartingale decomposition of Vervaat's bridges with both negative and positive endpoint. However, the treatments in two cases are different. The reason is that the split position in the case of negative endpoint (i.e. first return to 0) is a stopping time while the split position of the Vervaat bridges with positive endpoint (i.e. the last hit of λ > 0 strictly before 1) is not.
Semimartingale decomposition of Vervaat bridges with negative endpoints
Let λ < 0. Recall from Theroem 1.3 that the transformed bridge (V (B λ,br ) t ; 0 ≤ t ≤ 1) is split into a Brownian excursion and a first passage bridge from 0 to λ. The density Z λ := inf{t > 0; V (B λ,br ) t = 0} is given by (1).
We start by studying the semimartingale decomposition of (V (B λ,br ) t ; Z λ ≤ t ≤ 1).
Proposition 3.4 Let λ < 0 and V λ := V (B λ,br ). Conditional on the value of Z λ , V λ t − t Z λ 1 V λ s + λ − V λ s + λ 1 − Z λ − s ds Z λ ≤t≤1
is a Brownian motion.
Proof: This is an easy consequence of Theorem 1.3 and the following identity due to Biane and Yor [8]
F λ,l d = BES(3) |λ|→0,l + λ,
where F λ,l is the first passage bridge of length l from 0 to λ and BES(3) |λ|→0,l is a three dimensional Bessel bridge from |λ| to 0.
We now deal with the semimartingale decomposition of (V (B λ,br ) t∧Z λ ; 0 ≤ t ≤ 1). The process is a Brownian excursion of length Z λ , absorbed at 0 after Z λ with density given as (1).
Proposition 3.5 Let λ < 0 and J λ t (y) := 1 tq s−t (0, y) q s (0, 0) f Z λ (s)ds, J λ t (y) := 1 t 1 √ s − tq s−t (0, y) q s (0, 0) f Z λ (s)ds. Let V λ := V (B λ,br ), then (Y t ) 0≤t≤1 := V λ t∧Z λ − t∧Z λ 0 ds V λ s + t∧Z λ 0 V λ sJ s (V λ s ) 1 + J s (V λ s ) ds 0≤t≤1
is a Brownian motion with respect the filtration of V (B λ,br ), stopped at time Z λ .
Proof: Let ε ∈ (0, 1). We introduce (B λ,ε t ; t ≥ 0) a Brownian motion with the starting point B λ,ε 0 having the same distribution as V (B λ,br ) ε∧Z λ . Let µ λ ε be the density of this distribution, we have for x > 0,
µ λ ε (x) =q ε (0, x)x 2 J λ ε (x); and µ λ ε (0) = ε 0 f Z λ (s)ds. Let T λ,ε 0 be the first time B λ,ε hits 0. For any t and ε, the law of (V (B λ,br ) s∧Z λ ; ε ≤ s ≤ t) is absolutely continuous with respect the law (B λ,ε (s−ε)∧T ε 0 ; ε ≤ s ≤ t).
In fact, conditional on Z λ > t and the value of V (B λ,br ) t , the path (V (B λ,br ) s∧Z λ ; ε ≤ s ≤ t) has the same distribution as a three dimensional Bessel bridge. It is the same for (B λ,ε
(s−ε)∧T λ,ε 0 ; ε ≤ s ≤ t)
given T λ,ε 0 > t − ε and the value of B λ,ε t−ε . The corresponding density is:
D λ,ε t = 1 B λ,ε 0 =0 + 1 B λ,ε 0 >0,T λ,ε 0 ≤t−ε + 1 T λ,ε 0 >t−εq ε (0, B λ,ε 0 )(B λ,ε 0 ) 2q t−ε (B λ,ε 0 , B λ,ε t−ε )(B λ,ε t−ε ) 2 J λ t (B λ,ε t−ε ) µ λ ε (B λ,ε 0 )B λ,ε 0 B λ,ε t−εqt−ε (B λ,ε 0 , B λ,ε t−ε ) = 1 B λ,ε 0 =0 + 1 B λ,ε 0 >0,T λ,ε 0 ≤t−ε + 1 T λ,ε 0 >t−ε B λ,ε t−ε J λ t (B λ,ε t−ε ) B λ,ε 0 J λ ε (B λ,ε 0 )
.
Since for t ∈ (0, 1), ∂J λ t (y) ∂y = −yJ λ t (y).
Apply Girsanov's theorem, we obtain that (Y t ) t≥ε is a continuous martingale relative to the filtration of (
V (B λ,br ) t ; ε ≤ t ≤ 1) with quadratic variation (t − ε) ∧ (Z λ − ε) + .
Since this holds for all ε sufficiently small, this proves the proposition.
The above results provide the semimartingale decomposition of V (B λ,br ) for λ < 0.
Theorem 3.6 Let λ < 0 and V λ := V (B λ,br ). Then
V λ t − t∧Z λ 0 ds V λ s + t∧Z λ 0 V λ sJ s (V λ s ) 1 + J s (V λ s ) ds − t Z λ 1 V λ s + λ − V λ s + λ 1 − Z λ − s ds 0≤t≤1
is a Brownian motion.
Semimartingale decomposition of Vervaat bridges with positive endpoints
Let λ > 0. Recall from Corollary 1.4 that the transformed bridge V (B λ,br ) can be decomposed into a three dimensional Bessel bridge from 0 to λ and a positive excursion above λ. The density of the split position Z λ is given by
f Z λ (t) = f Z λ (1 − t),
given as (1).
For x, y ≥ 0 let Q t x,y be the law of the bridge of three dimensional Bessel bridge from x to y of length t. Let (R t ) t≥0 be a three dimensional Bessel process starting from 0.
The key idea is to show that for any t ∈ [0, 1), the law of (V (B λ,br ) s ) 0≤s≤t is absolutely continuous with respect to the law of (R s ; 0 ≤ s ≤ t), identify the corresponding density D λ t and deduce by applying Girsanov's theorem the semi-martingale decomposition of (V (B λ,br ) t ; 0 ≤ t ≤ 1).
Let θ λ t := sup{s ∈ [0, t]|R s ≤ λ}. Note that if R t ≤ λ then θ λ t = t.
We begin with a lemma computing the joint distribution of (R t , θ t ) in the case where the last hit at level λ has not yet been attained.
Lemma 3.7 On the event R t > λ, the joint distribution of (R t , θ λ t ) is:
q t (0, y) g t−s (y − λ)g s (λ) g t (y) 1 0<s<t ds 1 y>λ y 2 dy Conditionally on R t > λ, the value of R t and of θ λ t , (R s ; 0 ≤ s ≤ θ λ t ) and (R θ λ t −s − λ; 0 ≤ s ≤ t − θ λ t ) are independent and follow the law Q θ λ t 0,λ respectively Q t−θ λ t 0,Rt−λ .
Proof: Let y > λ. Conditionally on R t = y, (R t−s ; 0 ≤ s ≤ t) is a Brownian first passage bridge from y to 0 and t − θ λ t is the first time it hits λ. Thus conditionally on R t = y, t − θ λ t is distributed according to
g s (y − λ)g t−s (λ) g t (y) 1 0<s<t ds.
Moreover, conditionally on R t = y and on the value of θ λ t , (R t−s ; 0 ≤ s ≤ t − θ λ t ) and (R θ λ t −s ; 0 ≤ s ≤ θ λ t ) are two independent Brownian first passage bridges, from y to λ and from λ to 0.
The next proposition proves that the law of (V (B λ,br ) s ; 0 ≤ s ≤ t) is absolutely continuous with respect to the law of (R s ; 0 ≤ s ≤ t). We express the density D λ t as a deterministic function of t, R t and θ λ t .
Proposition 3.8 For any t ∈ [0, 1), the law of (V (B λ,br ) s ; 0 ≤ s ≤ t) is absolutely continuous with respect to the law of (R s ; 0 ≤ s ≤ t). The corresponding density is:
D λ t = 1 tq s−t (R t , λ) q s (0, λ) f Z λ (s) ds + 1 Rt>λ (1 − θ λ t )(R t − λ) (1 − t) 3 R t exp λ 2 2 exp − (R t − λ) 2 2(1 − t) := Φ λ (t, R t , θ λ t ).
Proof: Observe that as a stochastic process, (D λ t ; 0 ≤ t < 1) is continuous and in particular there is no discontinuity as R t crosses the level λ. Let t ∈ (0, 1). We decompose the density D λ t as sum of two parts: D λ t = D 1,λ t + D 2,λ t , D 1,λ t accounting for the situation Z λ > t and D 2,λ t for the situation Z λ < t. On the event R t < λ, we have D λ t = D 1,λ t . Conditionally on Z λ > t and on the position of V (B λ,br ) t , the paths (V (B λ,br ) s ; 0 ≤ s ≤ t) is a three dimensional Bessel bridge from 0 to V (B λ,br ) t , i.e. these are the same conditional laws as the laws of (R s ; 0 ≤ s ≤ t) conditioned on the value of R t . Conditionally on Z λ > t and on the value of Z λ , the distribution of V (B λ,br ) t is:
q t (0, y)q Z λ −t (y, λ) q Z λ (0, λ) 1 y>0 y 2 dy.
Therefore,
D 1,λ t := 1 tq s−t (R t , λ) q s (0, λ) f Z λ (s) ds.
Next we consider the case Z λ < t. Conditionally on Z λ < t and the position of Z λ and V (B λ,br ) t , the paths (V (B λ,br ) s ; 0 ≤ s ≤ Z λ ) and (V (B λ,br ) Z λ +s − λ; 0 ≤ s ≤ t − Z λ ) are independent and follow the law Q Z λ 0,λ respectively Q t− Z λ 0,V (B λ,br )t−λ . These are the same conditional laws as in Lemma 3.7. On the event Z λ < t, the joint distribution of (V (B λ,br ) t , Z λ ) is:
f Z λ (s)q t−s (0, y − λ)q 1−t (y − λ, 0) q 1−s (0, 0) 1 y>λ (y − λ) 2 dy 1 0<s<t ds.
We have then,
D 2,λ t =1 Rt>λ f Z λ (θ λ t )q t−θ λ t (0, R t − λ)q 1−t (R t − λ, 0) q 1−θ λ t (0, 0) (R t − λ) 2 q t (0, R t ) g t−θ λ t (R t − λ)g θ λ t (λ) g t (R t ) R 2 t =1 Rt>λ (1 − θ λ t )(R t − λ) (1 − t) 3 R t exp λ 2 2 exp − (R t − λ) 2 2(1 − t) .
Lemma 3.9 For any t ∈ (0, 1) and a ≥ 0:
1 t ds (1 − s)(s − t) exp − a s − t = √ π +∞ a 1−t e −u du √ u .
Proof: By change of variables z := 1 − s 1 − t , we get:
1 t ds (1 − s)(s − t) exp − a s − t = 1 0 dz z(1 − z) exp − a (1 − t)z := ϕ( a 1 − t ),
Note that
ϕ(x) := 1 0 dz z(1 − z) exp − x z v=z −1 = +∞ 1 dv v √ v − 1 e −xv .
Differentiating with respect to x, we obtain:
ϕ (x) = − +∞ 1 dv √ v − 1 e −xv = − e −x √ x +∞ 0 dv √ v e −v = − √ π e −x √ x .
Moreover, ϕ satisfies the boundary condition ϕ(+∞) = 0. Thus,
ϕ(x) = √ π +∞ x e −u du √ u . Let Φ 1,λ (t, y) := 1 2 √ 2y exp λ 2 2 (y+λ) 2 2(1−t) (y−λ) 2 2(1−t) e −u du √ u ; Φ 2,λ (t, y) := (y − λ) (1 − t) 3 y exp λ 2 2 exp − (y − λ) 2 2(1 − t) .
According to Lemma 3.9:
Φ λ (t, y, θ) = Φ 1,λ (t, y) + (1 − θ)(0 ∨ Φ 2,λ (t, y)).(13)
Observe that Φ 2,λ is C 1 . Φ 1,λ and the partial derivative ∂ 1 Φ 1,λ are continuous as functions in (t, y). However, ∂ 2 Φ 1,λ (t, y) is not defined at y = λ:
∂ 2 Φ 1,λ (t, λ + ) − ∂ 2 Φ 1,λ (t, λ − ) = − 1 √ 1 − tλ exp λ 2 2 , ∂ 2 Φ λ (t, λ + , θ) − ∂ 2 Φ λ (t, λ − , θ) = ∂ 2 Φ 1,λ (t, λ + ) − ∂ 2 Φ 1,λ (t, λ − ) + (1 − θ)∂ 2 Φ 2,λ (t, λ) = (t − θ) (1 − t) 3 λ exp λ 2 2 .
For t > 0, let
W t := R t − t 0 ds R s ,
where (W t ; t ≥ 0) is a standard Brownian motion starting from 0, predictable with respect the filtration of (R t ; t ≥ 0).
Lemma 3.10
For all t ∈ [0, 1) and λ > 0,
D λ t = 1 + t 0 ∂ 2 Φ λ (s, R s , θ λ s ) dW s .
Proof: Remark that we cannot just apply directly Itô's formula to Φ λ (t, R t , θ λ t ) since Φ λ is not regular enough. It is easy to check that Φ 2,λ and Φ 1,λ outside {y = λ} satisfy the PDE:
1 2 ∂ 2,2 Φ(t, y) + 1 y ∂ 2 Φ(t, y) + ∂ 1 Φ(t, y) = 0.
Let (L λ t (R); t ≥ 0) be the local time at level λ of (R t ; t ≥ 0). Apply Itô-Tanaka's formula, and take into account the discontinuity of partial derivatives ∂ 2 at level y = λ, we get:
Φ 1,λ (t, R t ) = 1 + t 0 ∂ 2 Φ 1,λ (s, R s ) dW s − 1 λ exp λ 2 2 t 0 1 √ 1 − s dL λ s (R). 0 ∨ Φ 2,λ (t, R t ) = t 0 1 Rs>λ ∂ 2 Φ 2,λ (s, R s ) dW s + 1 λ exp λ 2 2 t 0 1 (1 − s) 3 dL λ s (R).
(1−θ λ t ) is constant on the intervals of time where 0∨Φ 2,λ (t, R t ) is positive. From Theorem 4.2, Chapter VI of Revuz and Yor [38], follows that:
(1 − θ λ t )(0 ∨ Φ 2,λ (t, R t )) = t 0 1 Rs>λ (1 − θ λ s )∂ 2 Φ 2,λ (s, R s ) dW s + 1 λ exp λ 2 2 t 0 (1 − θ λ s ) (1 − s) 3 dL λ s (R) = t 0 1 Rs>λ (1 − θ λ s )∂ 2 Φ 2,λ (s, R s ) dW s + 1 λ exp λ 2 2 t 0 1 √ 1 − s dL λ s (R).
on the support of dL λ s (R), (1 − θ λ s ) being equal to 1 − s. Finally
Φ 1,λ (t, R t ) + (1 − θ λ t )(0 ∨ Φ 2,λ (t, R t )) = 1 + t 0 (∂ 2 Φ 1,λ (s, R s ) + (1 − θ λ s )∂ 2 Φ 2,λ (s, R s ))1 Rs>λ dW s .
which finishes the proof.
The next theorem gives the semimartingale decomposition of V (B λ,br ) where λ > 0. Let V λ := V (B λ,br ), then
V λ t − t 0 ds V λ s − t 0 ∂ 2 Φ λ Φ λ (s, V λ s ,θ λ,br s ) ds 0≤t≤1
is a standard Brownian motion.
Proof: For t ∈ [0, 1), let
X t := V (B λ,br ) t − t 0 ds V (B λ,br ) s .
The law of (X s ; 0 ≤ s ≤ t) is absolutely continuous with respect to the law of (W s ; 0 ≤ s ≤ t), with density D λ t . From Lemma 3.10 follows that
[log(D λ ), W ] t = t 0 ∂ 2 Φ λ Φ λ (s, R s , θ λ s ) ds.
From Girsanov's theorem follows that the process:
X t − t 0 ∂ 2 Φ λ Φ λ (s, V (B λ,br ) s ,θ λ,br s ) ds
is a Brownian motion.
Relation with drifting excursion
In Bertoin [5], he studied a fragmentation process by considering the excursion dragged down by drift λ < 0: B ex,λ↓ t := B ex t + λt, for 0 ≤ t ≤ 1.
Notice that V (B λ,br ) (with λ < 0) also looks similar to this process except that the former always stays above the line t → λt while the latter doesn't share this property. A natural way to relate these two processes is to see whether conditioned on staying above the dragging line, the Vervaat bridge is absolutely continuous with respect to drifting excursion. First we need to justify that the conditioning event has positive probability. The next proposition provides a positive answer with an explicit formula. Proposition 3.12 ∀λ < 0,
P(∀t ∈ (0, 1), V (B λ,br ) t > λt) = 1 − |λ| exp λ 2 2 ∞ |λ| exp − t 2 2 dt.
Proof: Following from Proposition 15 of Schweinsberg [40], fix x ∈ [λ, 0] we know the probability of a first passage bridge through level λ to stay above the dragging line tying x to λ:
P(∀t ∈ [0, l], F λ,l (t) > x − (x − λ)t) = |x| |λ| .(14)
Therefore,
P(∀t ∈ (0, 1), V (B λ,br ) t > λt) = 1 0 P ∀s ∈ (t, 1), V (B λ,br ) s > λs|Z λ = t f Z λ (t)dt = 1 0 t |λ| 2πt(1 − t) 3 exp − λ 2 t 2(1 − t) dt = EZ λ .
where the first equality follows from the fact that the excursion piece is always above the dragging line and the second equality is a direct consequence of (14). Following the notations of discussion below Lemma 4.10 in Pitman [33],
EZ = h −2 (λ)EB 2 1 = 1 − λ exp λ 2 2 ∞ λ exp − t 2 2 dt,
where h −2 is the Hermite function of index −2.
Now we know that the Vervaat bridge (with negative endpoint) conditioned to stay above the dragging line is well-defined. In addition, the law of its first return to 0 is given by:
f Z λ (t) = t 1 − |λ| exp λ 2 2 ∞ |λ| exp − t 2 2 dt f Z λ (t).(15)
The next theorem provides a path decomposition result of Vervaat's bridge conditioned to stay above the dragging line and establishes connection to drifting excursion. Theorem 3.13 Let λ < 0. Given Z λ the length of first excursion of (V (B λ,br ) t ; 0 ≤ t ≤ 1|∀t ∈ (0, 1), V (B λ,br ) t > λt) (whose distribution density is given by (15)), the path is decomposed into two (conditionally) independent pieces:
• V (B λ,br ) u ; 0 ≤ u ≤ Z λ |∀t ∈ (0, 1), V (B λ,br ) t > λt is an excursion of length Z λ ;
• V (B λ,br ) u ; Z λ ≤ u ≤ 1|∀t ∈ (0, 1), V (B λ,br ) t > λt is a first passage bridge of length 1 − Z λ conditioned to stay above t → λ(t + Z λ ) for t ∈ (0, 1 − Z λ ).
In addition, (V (B λ,br ) t ; 0 ≤ t ≤ 1|∀t ∈ (0, 1), V (B λ,br ) t > λt) is absolutely continuous with respect to (B ex,λ↓ t ; 0 ≤ t ≤ 1). The corresponding density is:
H 1 − |λ| exp λ 2 2 ∞ |λ| exp − t 2 2 dt , where H := inf{t > 0; B ex,λ↓ t < 0}.
Proof: According to Proposition 11 of Bertoin [5], H is distributed as (1). Following Theorem 2.6 of Chassaing and Jason [12], conditioned on H, (B ex,λ↓ t ; 0 ≤ t ≤ H) is a Brownian excursion of length H. In addition, Proposition 4 of Schweinsberg [40] states that given H, (B ex,λ↓ t ; H ≤ t ≤ 1) is a first passage bridge of length 1 − H conditioned to stay above the line t → λ(t + H) for t ∈ (0, 1 − H), (conditionally) independent of the excursion piece. By change of measures, we obtain the same triple characterization in law.
Convex minorant of Vervaat bridges
In this part, we will study some properties of convex minorant of Vervaat bridge V (B λ,br ) where λ < 0. The convex minorant of a real-valued function (X t ; t ∈ [0, 1]) is the maximal convex function (C t ; t ∈ [0, 1]) such that ∀t ∈ [0, 1], C t ≤ X t . We refer to the points where the convex minorant equals the process as vertices. Note that these points are also the endpoints of the linear segments. See Pitman and Ross [34] and Abramson et al [1] for general background.
Similar to the computation in Proposition 3.12 , we have the explicit formula for the distribution of the last segment's slopes.
Corollary 3.14 Denote s l the slope of the last segment of the convex minorant for
(V (B λ,br ) t ; 0 ≤ t ≤ 1)). ∀a ∈ [λ, 0], we have P(s l ∈ [λ, a]) = 1 + a exp λ 2 2 ∞ |λ| exp − t 2 2 dt.
As discussed in Pitman and Ross [34], a standard first passage bridge can only have accumulations of linear segments at its start point (while Brownian motion has accumulations at two endpoints). However, seen in the beginning of the section, the greatest difference between the Vervaat bridges and the standard first passage bridges is the first excursion piece for the former. Then we can expect that the Vervaat bridges have almost surely a finite number of segments. Proof: We adopt a sample paths argument. Consider a sample path of Brownian bridge B λ,br where λ < 0and 1−A λ := argmin B λ,br (which is a.s. unique). Note that V (B λ,br ) t > 0 for t ∈ (0, A]. Consequently, the first vertex of the Vervaat bridge α 1 > A a.s. According to Pitman and Ross [34], there can be only a finite number of segments on [α 1 , 1] since accumulations can only happen at 0 on the restricted path B λ,br | [0,1−A] . Thus, the number of segments of the Vervaat bridges is a.s. finite.
However, we expect a stronger result regarding the number of segments:
The Vervaat transform of Brownian motion
In this section, we are devoted to studying the Vervaat transform of Brownian motion. We first prove that the process is not Markov with respect to its induced filtration. Next, (V (B) t ; 0 ≤ t ≤ 1) is shown to be a semimartingale with explicit formula. The computation is essentially based on the results in Section 3.3. Finally, we provide the mean and the variance of this process. Proof: According to the above discussion,
V (B) is not Markov
P(V (B) 1 > 0|V (B)1 2 > 0, V (B)1 4 = 0) = 0;(16)
since once it hits 0 on its path, V (B) has to end negatively. On the other hand,
P(V (B) 1 > 0|∀t ∈ (0, 1 2 ], V (B) t > 0) = P(V (B) 1 > 0 and ∀t ∈ (0, 1 2 ], V (B) t > 0) P(∀t ∈ (0, 1 2 ], V (B) t > 0) ≥ P(∀t ∈ ([0, 1], V (B) t > 0) = P(B 1 > 0) = 1 2 .(17)
By comparing (16) and (17), we see that these two conditional probabilities fail to be equal, which implies that (V (B) t ; 0 ≤ t ≤ 1) is not Markov. Formally this means that we obtain the information at time 1 from some prior time, which violates the Markov property.
V (B) is a semimartingale -a conceptual approach
In general, when a process is Markov (with state space in R d ), we know sufficient and necessary conditions for it to be a semimartingale, see Cinlar et al [15]. However, we have seen in the preceding subsection that V (B) is not Markov. Therefore, whether V (B) is a semimartingale or not cannot be judged by classical Markov-semimartingale procedures.
In this section, we provide a soft argument to prove that V (B) is indeed a semimartingale with respect to its induced filtration using Denisov's decomposition for Brownian motion as well as Bichteler-Dellacherie's characterization for semimartingales.
We first recall some paths decomposition result for standard Brownian motion, which permits a characterization for the Vervaat transform. Following the notations in the introduction, A is the a.s. arcsine split (1−A := argmin t∈[0,1] B t ) for a standard Brownian motion. The following theorem is due to Denisov [18]: a) ), the path is decomposed into two independent pieces:
Theorem 4.2 Denisov's decomposition [18] Given A (which is arcsine distributed, i.e. f A (a) = 1 π √ a(1−• 1 √ A (B 1−A+uA − B 1−A ); 0 ≤ u ≤ 1 is a standard Brownian meander; • 1 √ 1−A (B (1−A)(1−u) − B 1−A ); 0 ≤ u ≤ 1 is a standard Brownian meander.
Remark: The theorem simply says that given its a.s. minimum A, a standard Brownian motion is split into two conditional independent meanders of length A and 1 − A joint back to back. Therefore, (V (B) t ; 0 ≤ t ≤ 1) can be viewed as mixing of two independent joint back-to-back Brownian meanders with respect to arcsine distribution. Now we turn to some results in the classical semimartingale theory. Given a filtration (F t ) t , a process H is said to be simple predictable if H has a representation
∀t ∈ [0, 1], H t = H 0 1 {0} (t) + n−1 i=1 H i 1 (t i ,t i+1 ] (t) where H i ∈ F t i and |H i | < ∞ a.s. for 0 = t 1 ≤ t 2 ≤ ... ≤ t n ≤ ∞.
Denote S the collection of simple predictable processes and B = {H ∈ S : |H| ≤ 1} (the unit ball in S). For a given process, we define a linear mapping I X : S → L 0 by
I X (H t ) = H 0 X 0 + n−1 i=1 H i (X t i − X t i−1 ),
for H ∈ S. In fact, I X is defined as stochastic integral with respect to X for simple predictable processes.
The following theorem, proved independently by Bichteler [9] and Dellacherie [17] provides a useful characterization for semimartingales. We refer the readers to Jacod [27], Protter [37] and Rogers and Williams [39] for more details. Remark: Fundamentally, this theorem tells that the notion of semimartingale is equivalent to the notion of "good stochastic integrator" and it depends only on the law of the processes.
We now state the main theorem of the section: Theorem 4.4 (V (B) t ; 0 ≤ t ≤ 1) is semimartingale with respect to its induced filtration.
Proof: Fix H ∈ B and η > 0,
P(|I V (B )(H)| > η) = 1 0 P(|I V (B)|A=a (H)| > η) 1 π a(1 − a) da.(18)
Note that (V (B)|A = 1) is a standard Brownian meander and B me d = R 0→ρ where R x→y is a three dimensional Bessel bridge from x to y and ρ is Rayleigh distributed
P(ρ ∈ dx) = x exp(− x 2 2 )dx.
Girsanov's change of measure theorem guarantees that (V (B)|A = 1) is a semimartingale and so is (V (B)|A = 0) (see e.g. Imhof [26] and Azéma-Yor [4]). Thus, by Theorem 4.3, Proof: Observe that I V (B)|A=a (H) = I 1 + I 2 , where
I 1 := i H i (V (B) τ i+1 ∧a − V (B) τ i+1 ∧a ) and I 2 := i H i (V (B) τ i+1 ∨a − V (B) τ i+1 ∨a ).
We have then,
P(|I V (B)|A=a (H)| > η) ≤ P(|I 1 | > η 2 ) + P(|I 2 | > η 2 )
.
DenoteĨ 1 = I 1 √
a and note that
I 1 = i H i V (B) τ i+1 ∧a − V (B) τ i ∧a √ a = i H i (Ṽ (B)τ i+1 ∧a a −Ṽ (B)τ i ∧a a ).
where (Ṽ (B) t ; 0 ≤ t ≤ 1) is a standard Brownian meander and H i is FṼ τ i ∧a a -adapted ∀i. We have then
P(|I 1 | > η 2 ) ≤ P(|Ĩ 1 | > η 2 ) ≤ sup H∈B P(|I V (B)|A=1 (H)| > η 2 ).
Similarly, by independence of two decomposed meanders,
P(|I 2 | > η 2 ) ≤ P(|Ĩ 2 | > η 2 ) = sup H∈B P(|I V (B)|A=0 (H)| > η 2 ).
whereĨ 2 is the stochastic integral associated to reversed Brownian meander. The following corollary states that the Vervaat bridges are also semimartingales, which provides an alternative proof of the semimartingale property for Vervaat bridges obtained in Section 3.3.
Corollary 4.6
For each fixed λ ∈ R, (V (B λ,br ) t ; 0 ≤ t ≤ 1) is a semimartingale with respect to its induced filtration.
Proof: Fix H ∈ B and η > 0,
P(I V (B) (H) > η) = R P(I V (B λ,br ) (H) > η) 1 √ 2π exp − λ 2 2 dλ.(19)
Note that V (B 0,br ) is Brownian excursion, thus a semimartingale. It suffices then to prove V (B λ,br ) for λ = 0 is a semimartingale. If not the case, ∃ > 0, ∀K > 0 ∃η > K such that sup
H∈B P(I V (B λ,br ) (H) > η) > .
Note in addition that (H, λ) → P(I V (B λ,br ) (H) > η) is jointly continuous in B × (R \ {0}) (concatenation of continuity that is left for readers to check). Thus ∃H λ, ∈ B and θ ∈ (0, |λ|) such that ∀λ ∈ (λ − θ, λ + θ),
P(I V (Bλ ,br ) (H) > η) > 2 .(20)
Injecting (20) into (19), we obtain:
P(I V (B) (H) > η) > 2 λ+θ λ−θ 1 √ 2π exp − λ 2 2 dλ.
which violates that fact that V (B) is a semimartingale.
However, one can hardly derive an explicit decomposition formula using Bichteler-Dellacherie's approach. Let's explain why: a generic approach for the proof of Bichteler-Dellacherie's theorem is to find Q equivalent to P such that X is Q−quasimartingale (see e.g. Protter [37] for definition). By Rao's theorem, X is Q−semimartingale, which is also P−semimartingale by Girsanov's theorem. Note that Rao's theorem is based on Doob-Meyer's decomposition theorem, which in general does not give an explicit expression for two decomposed terms (in fact they are defined as some limiting processes).
Nevertheless, in the next subsection, we do provide an explicit semimartingale decomposition of V (B). The method is similar to that in Section 3.3.
Semimartingale decomposition of the Vervaat transform of Brownian motion
In this part, we will use extensively the notations defined in the section 3.3. We consider V (B) the Vervaat's transform of a Brownian motion on [0, 1]. By definition, V (B) 1 = B 1 a.s., there is ε > 0 such that for all t ∈ (0, ε), V (B) t > 0. Let
T 0 := inf{t ∈ (0, 1]|V (B) t = 0}
Then P( T 0 ≤ 1) = 1 2 and more precisely
{ T 0 ≤ 1} = {V (B) 1 ≤ 0}. Conditional on T 0 ≤ 1,
T 0 follows the arcsine law 1 0<t<1 dt π t(1 − t)
. Conditionally on T 0 ≤ 1 and on the value of T 0 , (V (B) t ; 0 ≤ t ≤ T 0 ) has the law Q T 0 0,0 and is independent from (V (B) t ; T 0 ≤ t ≤ 1). The joint law of (V (B) 1 , T 0 ) on the event T 0 ≤ 1 is:
1 λ<0 dλ √ 2π exp − λ 2 2 |λ| 2πt(1 − t) 3 exp − λ 2 t 2(1 − t) 1 0<t<1 dt.
Thus the law of V (B) 1 conditionally on T 0 is:
|λ| 1 − T 0 exp − λ 2 2(1 − T 0 ) 1 λ<0 dλ.(21)
For the semi-martingale decomposition of (V (B), we will split the task in two: the decomposition of (V (B) t ; 0 ≤ t ≤ T 0 ) and the decomposition of (V (B) t ; T 0 ≤ t ≤ 1). We will start with the latter. Let ( M t ; t ≥ 0) be the process:
M t := min [0,t] V (B).V t + t T 0 V s − M s 1 − s ds T 0 ≤t≤1
is a Brownian motion
Proof: The value of T 0 is considered as fixed. Let (B t ) t≥0 be a Brownian motion starting from 0 and M t := min
[0,t] B .
For any t ∈ [ T 0 , 1), the law of (V (B) s ; T 0 ≤ s ≤ t) is absolutely continuous with respect to the law of (B s ; 0 ≤ s ≤ t − T 0 ). The corresponding density is:
M t− T 0 −∞ g 1−t (B t− T 0 − λ) g 1− T 0 (|λ|) |λ| 1 − T 0 exp − λ 2 2(1 − T 0 ) dλ.(22)
In the above expression, we integrate with respect to the density (21) the function:
1 λ<M t− T 0 g 1−t (B t− T 0 − λ) g 1− T 0 (|λ|) ,
which is the density corresponding to a Brownian first passage bridge from 0 to λ of length 1 − T 0 . (22) rewrites as:
1 − T 0 (1 − t) 3 M t− T 0 −∞ (B t− T 0 − λ) exp − (B t− T 0 − λ) 2 2(1 − t) dλ = 1 − T 0 1 − t exp − (B t− T 0 − M t− T 0 ) 2 2(1 − t) .
Apply Girsanov's theorem, we get the result of the lemma.
Next we deal with the semi-martingale decomposition of (V (B) t∧ T 0 ; 0 ≤ t ≤ 1). As an auxiliary problem we will study first the semi-martingale decomposition of a process (ξ t ; t ≥ 0) defined as follows: with probability 1 2 , ξ is a Bessel 3 process starting from 0. For t ∈ (0, 1), with infinitesimal probability
dt 2π √ t(1−t)
, ξ is a positive excursion of length t, absorbed at 0 after time t. For any t ∈ (0, 1), the law of (V (B) s∧ T 0 ; 0 ≤ s ≤ t) is absolutely continuous with respect the law of (ξ s ; 0 ≤ s ≤ t). The following lemma is a variant of Proposition 3.5.
Proposition 4.8 Let
T ξ 0 := inf{t > 0|ξ t = 0}. Let J t (y) := t≤s≤1 ds π s(1 − s)q s−t (0, y) q s (0, 0) . J t (y) := t≤s≤1 ds π(s − t) s(1 − s)q s−t (0, y) q s (0, 0) .
The process
(Y t ) t≥0 := ξ t − t∧T ξ 0 0 ds ξ s + t∧T ξ 0 0 ξ sJs (ξ s ) 1 + J s (ξ s ) ds t≥0
is a Brownian motion with respect the filtration of ξ, stopped at time T ξ 0 .
Proof: Let ε ∈ (0, 1). We introduce (B ε t ; t ≥ 0) a Brownian motion with the starting point B ε 0 having the same distribution as ξ ε∧T ξ 0 . Let µ ε be the density of this distribution on (0, +∞) (total mass < 1).
µ ε (x) =q ε (0, x)x 2 2 (1 + J ε (x)) .
Let T ε 0 be the first time B ε hits 0. For any t and ε, the law of (ξ s ; ε ≤ s ≤ t) is absolutely continuous with respect the law (B ε
(s−ε)∧T ε 0 ; ε ≤ s ≤ t).
The corresponding density is:
D ε t =1 B ε 0 =0 +q ε (0, B ε 0 )B ε2 0q T ε 0 (0, B ε 0 ) q T ε 0 +ε (0, 0)2π T ε 0 (1 − T ε 0 )µ ε (B ε 0 )g T ε 0 (B ε 0 ) 1 T ε 0 ≤t−ε,B ε 0 >0 + 1 T ε 0 >t−ε µ ε (B ε 0 )B ε 0 B ε t−εqt−ε (B ε 0 , B ε t−ε ) ×q ε (0, B ε 0 )B ε2 0q t−ε (B ε 0 , B ε t−ε )B ε2 t−ε 2 × 1 + 1 t ds π s(1 − s)q s−t (0, B ε t−ε ) q s (0, 0) =1 B ε 0 =0 +q ε (0, B ε 0 )B ε 0 µ ε (B ε 0 )π T ε 0 (1 − T ε 0 )q T ε 0 +ε (0, 0) 1 T ε 0 ≤t−ε,B ε 0 >0 + 1 T ε 0 >t−εqε (0, B ε 0 )B ε 0 B ε t−ε 2µ ε (B ε 0 ) × 1 + J t (B ε t−ε ) .
(D ε t ; t ≥ 0) seen as a time-dependent process is continuous. In particular, there is no discontinuity at T ε 0 + ε. This follows from the fact that as y tends to 0, the convolution kernel y 2q u (0, y) 1 u>0 du is an approximation to the delta function. Since for t ∈ (0, 1),
∂J t (y) ∂y = −yJ t (y).
Apply Girsanov's theorem we get that (Y t ; t ≥ ε) is a continuous martingale relative to the filtration of (ξ t ; t ≥ ε) with quadratic variation (t − ε) ∧ (T ξ 0 − ε) + . Since this holds for all ε sufficiently small, this implies the lemma.
We introduce the functionals Φ(t, γ) andΦ(t, γ) where t is a time and γ a continuous path:
Φ(t, γ) := 2 √ 2π +∞ 0 Φ λ (t, γ(t), sup{s ∈ [0, t]|γ(s) ≤ λ}) exp − λ 2 2 dλ, Φ(t, γ) := 2 √ 2π +∞ 0 ∂ 2 Φ λ (t, γ(t), sup{s ∈ [0, t]|γ(s) ≤ λ}) exp − λ 2 2 dλ,
where Φ λ is defined as (13).
For any t ∈ (0, 1), the law of (V (B) s∧ T 0 ; 0 ≤ s ≤ t) is absolutely continuous with respect to the law of (ξ s ; 0 ≤ s ≤ t) with density
D t = 1 T ξ 0 ≤t + Φ(t, ξ) + J t (ξ t ) 1 + J t (ξ t ) 1 T ξ 0 >t .(23)
Lemma 4.9 There are positive functions c 1 (t) and c 2 (t) bounded on intervals of form [0, 1 − ε], such that for all λ > 0, y > 0, θ ≤ t ∈ [0, 1):
Φ λ (t, y, θ) ≤ c 1 (t) exp λ 2 2 exp − (y − λ) 2 2(1 − t)
and
|∂ 2 Φ λ (t, y, θ)| ≤ c 2 (t)(1 + λ 2 ) y + 1 y exp λ 2 2 exp − (y − λ) 2 2(1 − t) Proof: Note that Φ λ (t, y, θ) ≤ Φ 1,λ (t, y) + 1 y>λ Φ 2,λ (t, y). For y > λ, Φ 2,λ (t, y) ≤ 1 (1 − t) 3 exp λ 2 2 exp − (y − λ) 2 2(1 − t) and Φ 1,λ (t, y) ≤ 1 2 √ 2y exp λ 2 2 exp − (y − λ) 2 2(1 − t) (y+λ) 2 2(1−t) (y−λ) 2 2(1−t) du √ u = 1 √ 1 − t min(y, λ) y exp λ 2 2 exp − (y − λ) 2 2(1 − t) .
In addition, for y > λ we obtain,
|∂ 2 Φ 2,λ (t, y)| = 1 (1 − t) 3 λ y 2 − (y − λ) 2 (1 − t)y exp λ 2 2 exp − (y − λ) 2 2(1 − t) ≤ 1 (1 − t) 3 1 y + y 2 + λ 2 (1 − t)y exp λ 2 2 exp − (y − λ) 2 2(1 − t)
and
|∂ 2 Φ 1,λ (t, y)| ≤ 1 2 √ 2 exp λ 2 2 1 y 2 (y+λ) 2 2(1−t) (y−λ) 2 2(1−t) e −u √ u du + √ 2 y √ 1 − t exp − (y − λ) 2 2(1 − t) − exp − (y − λ) 2 2(1 − t) ≤ 1 + 2 √ 2 2 √ 2y √ 1 − t exp λ 2 2 exp − (y − λ) 2 2(1 − t) ,
which permits to have the desired estimation. Proof: It is clear that the quadratic variation [Φ(·, ξ), ξ] t does not increase for t ≥ T ξ 0 . We need only to show that for a Bessel 3 process (R t ; t ≥ 0)
[Φ(·, R), R] t = t 0Φ (s, R) ds.(24)
Indeed, given any T ∈ (0, 1) and t ∈ [0, T ), the law of (ξ s ; 0 ≤ s ≤ t) on the event T ξ 0 > T is absolutely continuous with respect the law of (R s ; 0 ≤ s ≤ t). For any λ > 0 (Φ λ (t, R t , θ λ t ); 0 ≤ t < 1) is a positive martingale with mean 1. Apply Fubini's theorem, we obtain that (Φ(t, R); 0 ≤ t < 1) is a positive martingale with mean 1. Let (W t ; t ≥ 0) be the Brownian motion martingale part of (R t ; t ≥ 0). To prove (24) we need only to show that the process
Φ(t, R)W t − t 0Φ (s, R) ds 0≤t<1(25)
is a (true) martingale. Lemma 3.10 ensures that for any λ > 0 the process
Φ λ (t, R t , θ λ t )W t − t 0 ∂ 2 Φ λ (s, R s , θ λ s ) ds 0≤t<1(26)
is a local martingale. Next we show that (26) is a (true) martingale. It suffices to bound the expectation of its supreme and dominated convergence theorem permits to conclude. According to Burkholder-Davis-Gundy inequality, ∃C > 0 such that
E sup 0≤s≤t Φ λ (t, R t , θ λ t )W t − t 0 ∂ 2 Φ λ (s, R s , θ λ s ) ds 2 ≤ CE t 0 ∂ 2 Φ λ (s, R s , θ λ s ) 2 W 2 s + Φ λ (s, R s , θ λ s ) 2 ds = C t 0 E ∂ 2 Φ λ (s, R s , θ λ s ) 2 W 2 s + E Φ λ (s, R s , θ λ s ) 2 ds.
From Lemma 3.10 and the bound of Lemma 4.9 follow that (Φ λ (t, R t , θ λ t ); 0 ≤ t < 1) is a square integrable martingale and
1 + E s 0 ∂ 2 Φ λ (u, R u , θ λ u ) 2 du =E Φ λ (s, R s , θ λ s ) 2 ≤c 1 (s) 2 exp λ 2 E exp − (R s − λ) 2 (1 − s) ,
which is integrable on (0, t) for any 0 ≤ t < 1. Moreover, by Cauchy-Schwarz's inequality,
E[∂ 2 Φ λ (s, R s , θ λ s ) 2 W 2 s ] ≤ E[∂ 2 Φ λ (s, R s , θ λ s ) 4 ] 1 2 E[W 4 s ] 1 2 .
The problem of integrability may only occur at 0. However, by the bound of ∂ 2 Φ λ in Lemma 4.9, we know that E[∂ 2 Φ λ (s, R s , θ λ s ) 4 ] = O( 1 s 2 ) as s → 0. Thus the above term is also integrable on (0, t) for 0 ≤ t ≤ 1. We have proved that (26) is a (true) martingale for any λ > 0. Again by Cauchy-Schwarz's inequality,
E Φ λ (t, R t , θ λ t )W t − t 0 ∂ 2 Φ λ (s, R s , θ λ s ) ds ≤ 2 √ tE t 0 ∂ 2 Φ λ (s, R s , θ λ s ) 2 ds 1 2 ≤ 2 √ tc 1 (t) exp λ 2 2 E exp − (R t − λ) 2 (1 − t) 1 2 < ∞.
It follows that the expectation of the absolute value of the martingale (26) Then
V t − t∧ T 0 0 ds V s + t∧ T 0 0Φ (s, V ) + V sJs (V s ) Φ(s, V ) + J s (V s ) ds + t T 0 V s − M s 1 − s ds 0≤t≤1
is a Brownian motion.
Proof: The density process (D t ) 0≤t≤1 given by (23) is time-continuous. In particular it follows from lemma 4.9 that on the event T ξ 0 < 1, as t converges to T ξ 0 from below and ξ t converges to 0, Φ(t, ξ) remains bounded. Besides J t (ξ t ) tends to +∞ at T ξ 0 . Hence
lim t→T ξ 0 Φ(t, ξ) + J t (ξ t ) 1 + J t (ξ t ) = 1,
and D t is continuous as T ξ 0 . Using the semimartingale decomposition of (ξ t ; t ≥ 0) given by Lemma 4.8 and applying Girsanov's theorem together with Lemma 4.10 we get that the process
V (B) t∧ T 0 − t∧ T 0 0 ds V (B) s + t∧ T 0 0Φ (s, V (B)) + V (B) sJs (V (B) s ) Φ(s, V (B)) + J s (V (B) s ) ds
Expectation and variance for V (B)
In the current subsection, we provide the formulae for the first two moments of the Vervaat transform of Brownian motion.
Proposition 4.12 ∀t ∈ [0, 1], we have: The computation is based on Theorem 4.2, which is stated in the Section 4.2 as well as the following identities for standard Brownian meander, whose proof will be reported to the Appendix: Proposition 4.13 Let (B me t , t ∈ [0, 1]) be standard Brownian meander. We have:
EV (B) t = 8 π ( √ t + √ 1 − t − 1);(27)E(V (B) 2 t ) = 3t + 4 − 8t π arcsin √ t − 4 π t(1 − t).(28)EB me t = 2 π ( t(1 − t) + arcsin √ t).(29)E(B me t ) 2 = 3t − t 2 .(30)
EB me t B me = 2 √ t.
Remark: One can also think of computing the expectation and the variance of the Vervaat bridges. However, we are not able to derive some explicit formulae for them except in the case of zero endpoint (correspond to Brownian excursion). Also note that the expectation as well as the variance of the Vervaat transform of Brownian motion can be obtained by discrete approximation.
Expectation for V (B)
Let's compute the expectation of (V (B) t , 0 ≤ t ≤ 1). Recall that A is the a.s. arcsine split (1 − A := argmin t∈[0,1] B t ), we have:
EV (B) t = E(V (B) t 1 A>t ) + E(V (B) t 1 A≤t ). Lemma 4.14 ∀t ∈ [0, 1], E(V (B) t 1 A>t ) = 2 π ( √ 1 − t + 2 √ t − t − 1).(32)
Proof: According to the formula for EB me (t), t ∈ [0, 1], we have:
E(V (B) t 1 A>t ) = 1 t √
aEB me ( t a ) π a(1 − a) da (29) = √ 2
α 2 = −2 √ 1 − a arcsin t a 1 t + 2 1 t √ 1 − a arcsin t a da = π √ 1 − t − √ t 1 t √ 1 − a a √
a − t da.
Therefore,
α 1 + α 2 = π √ 1 − t + √ t 1 t √ a − t a √ 1 − a − √ 1 − a a √ a − t da = π √ 1 − t + 2 √ t 1 t da √ 1 − a √ a − t − √ t(t + 1) 1 t da a √ 1 − a √ a − t .(33)
By change of variables, we obtain: Observe that
1 t da √ 1 − a √ a − t a=t+(1−t)x = 1 0 dx x(1 − x) = π.(34)1 1−t 1 − t a (1 − 1 − t a )da = 2 (a − 1 + t)(1 − t) − 2(1 − t) arcsin a − 1 + t a 1 1−t = 2 t(1 − t) − 2(1 − t) arcsin √ t.(44)
and
1 1−t arcsin 1 − t a da = a arcsin 1 − t a 1 1−t − 1 1−t a arcsin 1 − t a da = arcsin √ 1 − t − π 2 (1 − t) + √ 1 − t 2 1 1−t da √ a − 1 + t = arcsin √ 1 − t − π 2 (1 − t) + t(1 − t).(45)
We have by (44) and (45):
t 0 γ 1 da π a(1 − a) = t 2 + 2t − 3 π arcsin √ t + 3 π t(1 − t).(46)
In addition,
Combining (42), (43), (46) and (47), we obtain (41).
Finally, by adding (40) to (41), we get (28).
Appendix: Computations for Brownian meander
Let (B me t , t ∈ [0, 1]) be standard Brownian meander. From Chung [14], we derive easily the density for meander along the paths as well as its joint distribution with terminal value:
P(B me t ∈ dx) = t − 3 2 x exp − x 2 2t erf x 2(1 − t) dx.(48)
P(B me t ∈ dx, B me 1 ∈ dy) = t − 3 2 x exp − x 2 2t (p 1−t (x, y) − p 1−t (x, −y)) dydx.
where erf is the error function for standard normal distribution: erf(x) = 2
√ π
x −∞ exp(−t 2 )dt and p is the transition kernel associated to Brownian motion: p t (x, y) = 1 √ 2πt exp − (x−y) 2 2t .
Proof of Proposition 4.13: (a). We compute the expectation of standard Brownian meander along the path, which relies on the following identity found in Gradshteyn and Ryzhik [24]: .
∀a > 0,
By change of variables, we obtain:
EB me t (48) = ∞ 0 t − 3 2 y 2 exp − y 2 2t erf y 2(1 − t) dy x= y √ 2(1−t) = √ 8 1 − t t 3 2 ∞ 0 x 2 exp − 1 − t t x 2 erf(x)dx (50) = 2 π t(1 − t) + arcsin √ t .
(b). We next calculate meander's second moment along the paths with the following identity also found in Gradshteyn and Ryzhik [24]:
∀a > 0,
By change of variables, we get:
E(B me t ) 2 (48) = ∞ 0 t − 3 2 y 3 exp − y 2 2t erf y 2(1 − t) dy x= y √ 2(1−t) = 4t − 3 2 (1 − t) 2 ∞ 0 x 3 exp − 1 − t t x 2 erf(x)dx (51) = 3t − t 2 .
(c). Finally we will compute EB me t B me 1 for 0 ≤ t ≤ 1.
EB me t B me 1 (49)) = ∞ 0 ∞ 0 y(p 1−t (x, y) − p 1−t (x, −y))dy t − 3 2 x 2 exp − x 2 2t dx.
Remark that t 0 yp t (x, y)dy = t 2π e − x 2 2t + x erf(− x 2 √ t ), we have:
∞ 0 y(p 1−t (x, y) − p 1−t (x, −y))dy = x erf(− x 2 √ 1 − t ) + erf( x 2 √ 1 − t ) = x.
Since it's well-known that for a > 0, ∞ 0 x 3 e −ax 2 dx = 1 2a 2 , we get:
EB me t B me 1 = t − 3 2 ∞ 0 x 3 e − x 2 2t dx = 2 √ t Remark:
The result in (c), i.e. the identity (31) can be directly derived from Imhof's relation [26] (between Brownian meanders and three dimensional Bessel processes).
Fig 1 .
1Vervaat bridge = Excursion + First passage bridge.
Fig 2 .
2Vervaat bridge=Bessel bridge+Excursion.
the mapping w → V (w) is not injective. Moreover, the number of first passage bridges through level −1 of odd length l is 1 l l+1 2 l and the number of first passage bridges starting at −1 through level a of length n − l is |a|
Fig 3 .
3Discrete Vervaat bridge=First passage bridge+First passage bridge.
Corollary 2. 5
5Let V (B λ,br ) be the Vervaat transform of a Brownian bridge ending at λ < 0. Given Z λ the first return to 0 of V (B λ,br ), let A λ be uniformly distributed on [0, Z λ ]. Then the shifted process θ(V (B λ,br ), A λ ) as defined in Theorem 1.2 is a Brownian bridge ending at λ which attains its minimum at 1 − A λ .
Fig 4 .
4Length-biased pick for a Brownian bridge conditioned on its local time.
Fig 5 .
5Paths which make the Vervaat bridge non-Markov, t 0 = 0.5 and x 0 = 1.
Theorem 3. 11
11Let λ > 0 and for t ∈ (0, 1), θ λ,br t := sup{s ∈ [0, t]|V (B λ,br ) s ≤ λ}.
Proposition 3 .
315 The number of segments of the convex minorant of V (B λ,br ) for λ < 0 is a.s. finite.
Conjecture 3 .
316 The expected number of segments of the Vervvat bridges is finite.
An important property of V (B) is that it has the same terminal value as B: V (B) 1 = B 1 . We then encounter two cases: If B 1 > 0, then V (B) never returns to 0 along the path. Otherwise B 1 ≤ 0, then V (B) 1 ≤ 0. By path continuity, V (B) has to hit 0 somewhere on its path.
Proposition 4.1 (V (B) t ; 0 ≤ t ≤ 1)is not Markov with respect to its induced filtration.
Fig 6 .
6Paths make Vervaat's transform of BM non-Markov. Remark: If we denote T := inf{t > 0; V (B) t = 0}, we have {T ≤ 1} = {V (B) 1 ≤ 0}.
Theorem 4. 3
3Bichteler-Dellacherie's theorem[9],[17] An adapted, càdlàg process X is a semimartingale iff I X (B) is bounded in probability, which means lim η→∞ sup H∈B P(|I X (H)| ≥ η) = 0.
|I V (B)|A=0 (H)| ≥ η) = 0. From (18), to prove sup H∈B P(|I V (B )(H)| ≥ η) → 0, we need some uniform control for sup H∈B P(|I V (B)|A=a (H)| > η). The following result permits to estimate for all a ∈ [0, 1], sup H∈B P|I V (B)|A=a (H)| > η) in terms of sup H∈B P(|I V (B)|A=1 (H)| > η) and sup H∈B P(|I V (B)|A=0 (H)| > η).
to 0 as η → ∞.
Lemma 4. 7
7Let V := V (B).Conditionally on the value of T 0 ,
Theorem 4 .
411 LetM t := min [0,t] V (B),T 0 := inf{t > 0; V (B) t = 0} and V := V (B).
is a martingale with quadratic variation t ∧ T 0 . Lemma 4.7 describes the semi-martingale decomposition of V (B) after the stopping time T 0 .
Fig 7 .
7Expectation and variance for V (B).
da. Using integration by parts, we get:
1 − x)[t + (1 − t)x]
Acknowledgement:The second and the third authors would like to express their gratitude to N.Forman for helpful discussion and suggestions throughout the preparation of this work. The authors would also like to thank P.Fitzsimmons for his remarks on the path decomposition result.Observe that θ → 1 √ a(a+1) arctan( a a+1 tan x) is a primitive θ → 1 sin 2 θ+a for a > 0. Take a = t 1−t , we have from(35):and consequently,Combining(33),(34)and(36), we get(32).where the first term can be derived from (32) by change of variables:The second one can be computed by independence of two meanders:Then we get easily (37) by adding(38)and(39).Remark:The result can also be obtained by observing the dualityFrom Lemma 4.14 and Lemma 4.15 follows easily(27).Variance for V (B)We now turn to calculate EV (B) 2 t for 0 ≤ t ≤ 1. We haveProof: According to the formula for E(B me ) 2 (t), t ∈ [0, 1], we have:Proof:We haveDenote β 1 , β 2 and β 3 the three terms on the right hand side of the above equation. Note that β 1 can be easily computed from Lemma 4.16 by change of variables:According to Denisov's decomposition, for 0 ≤ a ≤ t, we have:andwhere
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| []
|
[
"Transition Matrix Monte Carlo Reweighting and Dynamics",
"Transition Matrix Monte Carlo Reweighting and Dynamics"
]
| [
"Jian-Sheng Wang \nDepartment of Computational Science\nNational University of Singapore\n119260SingaporeRepublic of Singapore\n",
"Tien Kiat Tay \nDepartment of Computational Science\nNational University of Singapore\n119260SingaporeRepublic of Singapore\n",
"Robert H Swendsen \nDepartment of Physics\nCarnegie Mellon University\n15213PittsburghPA\n"
]
| [
"Department of Computational Science\nNational University of Singapore\n119260SingaporeRepublic of Singapore",
"Department of Computational Science\nNational University of Singapore\n119260SingaporeRepublic of Singapore",
"Department of Physics\nCarnegie Mellon University\n15213PittsburghPA"
]
| []
| We study an induced dynamics in the space of energy of single-spin-flip Monte Carlo algorithm. The method gives an efficient reweighting technique. This dynamics is shown to have relaxation times proportional to the specific heat. Thus, it is plausible for a logarithmic factor in the correlation time of the standard 2D Ising local dynamics. 05.50.+q, 02.70Lq, 64.60.Ht.The investigation of new Monte Carlo algorithms has been actively pursued due to their importance for efficient computer simulation. Cluster algorithms [1-3], generalized ensemble methods [4], together with advanced techniques of analysis, such as the histogram methods[5][6][7][8], greatly enhanced our ability to do efficient computer simulations.Recently, Swendsen and Li [9] proposed a novel algorithm named transition matrix Monte Carlo (TMMC). The method first determines a transition matrix W (E|E ′ ), to be defined below, by some sampling technique. The invariant state of this matrix gives the canonical probability distribution of energy P eq (E) ∝ n(E) exp(−E/k B T ), where n(E) is the density of states. This method has connections with the broad histogram method [7], the projection dynamics [10], and the canonical transition probability method [11], but its starting point and considerations are quite different.There are three important aspects in this approach: (1) The computation of the matrix can be efficiently implemented with the N -fold way [12]; (2) the method gives a new reweighting technique, efficient and simple in comparison with multiple histogram method [6];(3) the artificial dynamics is of considerable theoretical interest-it could imply a logarithmic correction in the two-dimensional Ising model critical relaxation, indirectly supporting Domany's conjecture[13].The rest of the paper is organized as follows: we first define the transition matrix Monte Carlo (TMMC) dynamics. We discuss the reweighting method under TMMC. We then describe the steps leading to an exact result for W (E|E ′ ) for the one-dimensional Ising model. We give a differential equation obeyed in the large-size limit and indicate its consequence. We give arguments for relaxation time in one dimension and relate the relaxation time to specific heat. We present simulation data to verify the general results. | 10.1103/physrevlett.82.476 | [
"https://arxiv.org/pdf/cond-mat/9809181v1.pdf"
]
| 118,911,736 | cond-mat/9809181 | 1081d712608bf9b80d8c85ba09d86c4e071dc03a |
Transition Matrix Monte Carlo Reweighting and Dynamics
12 Sep 1998 (10 September 1998)
Jian-Sheng Wang
Department of Computational Science
National University of Singapore
119260SingaporeRepublic of Singapore
Tien Kiat Tay
Department of Computational Science
National University of Singapore
119260SingaporeRepublic of Singapore
Robert H Swendsen
Department of Physics
Carnegie Mellon University
15213PittsburghPA
Transition Matrix Monte Carlo Reweighting and Dynamics
12 Sep 1998 (10 September 1998)
We study an induced dynamics in the space of energy of single-spin-flip Monte Carlo algorithm. The method gives an efficient reweighting technique. This dynamics is shown to have relaxation times proportional to the specific heat. Thus, it is plausible for a logarithmic factor in the correlation time of the standard 2D Ising local dynamics. 05.50.+q, 02.70Lq, 64.60.Ht.The investigation of new Monte Carlo algorithms has been actively pursued due to their importance for efficient computer simulation. Cluster algorithms [1-3], generalized ensemble methods [4], together with advanced techniques of analysis, such as the histogram methods[5][6][7][8], greatly enhanced our ability to do efficient computer simulations.Recently, Swendsen and Li [9] proposed a novel algorithm named transition matrix Monte Carlo (TMMC). The method first determines a transition matrix W (E|E ′ ), to be defined below, by some sampling technique. The invariant state of this matrix gives the canonical probability distribution of energy P eq (E) ∝ n(E) exp(−E/k B T ), where n(E) is the density of states. This method has connections with the broad histogram method [7], the projection dynamics [10], and the canonical transition probability method [11], but its starting point and considerations are quite different.There are three important aspects in this approach: (1) The computation of the matrix can be efficiently implemented with the N -fold way [12]; (2) the method gives a new reweighting technique, efficient and simple in comparison with multiple histogram method [6];(3) the artificial dynamics is of considerable theoretical interest-it could imply a logarithmic correction in the two-dimensional Ising model critical relaxation, indirectly supporting Domany's conjecture[13].The rest of the paper is organized as follows: we first define the transition matrix Monte Carlo (TMMC) dynamics. We discuss the reweighting method under TMMC. We then describe the steps leading to an exact result for W (E|E ′ ) for the one-dimensional Ising model. We give a differential equation obeyed in the large-size limit and indicate its consequence. We give arguments for relaxation time in one dimension and relate the relaxation time to specific heat. We present simulation data to verify the general results.
Let us consider a single-spin-flip Glauber dynamics [14] of the Ising model which is described in continuous time as
∂P (σ, t) ∂t = {σ ′ } Γ(σ, σ ′ )P (σ ′ , t) = N i=1 −w i (σ i ) + w i (−σ i )F i P (σ, t),(1)
where N is the total number of spins, and
w i (σ i ) = 1 2 1 − σ i tanh K nn of i σ k , K = J k B T ,(2)
is the flip rate of site i, which depends on the spin value at the site i as well as the values of its nearest neighbor spins σ k . And F i is a flip operator such that F i P (. . . , σ i , . . .) = P (. . . , −σ i , . . .). The transition matrix Monte Carlo dynamics is defined, based on the above dynamics, by
∂P (E, t) ∂t = E ′ W (E|E ′ )P (E ′ , t),(3)
where P (E, t) is the probability of having energy E at time t, and
W (E|E ′ ) = 1 n(E ′ ) H(σ)=E H(σ ′ )=E ′ Γ(σ, σ ′ ).(4)
Although Eq. (1) and (3) give the same equilibrium distribution of energy, they have totally different dynamics. We note that the transition matrix is banded along diagonal. The matrix elements are not independent-the column sum is zero due to the conservation of total probability, and the row sum satisfies E W (E ′ |E)P eq (E) = 0 due to the fact that the equilibrium distribution is a stationary distribution. The transition matrix also satisfies detailed balance condition, W (E|E ′ )P eq (E ′ ) = W (E ′ |E)P eq (E), inherited from the detailed balance in the original dynamics of spins. The detailed balance conditions put strong constraint on the matrix elements. For example, for any three energies with nonzero transition rates among them, we have
W (E|E ′′ )W (E ′′ |E ′ )W (E ′ |E) = W (E|E ′ )W (E ′ |E ′′ )W (E ′′ |E).(5)
The matrix elements can be computed by Monte Carlo sampling as follows. For a configuration σ, we consider the number N (σ, ∆E) of cases that energy is changed by ∆E, for the N possible single-spin flips. Then for ∆E = 0,
W (E + ∆E|E) = w(∆E) N (σ, ∆E) H(σ)=E ,(6)
where the average is over all configurations having energy E, and w(∆E) = 1 2 1 − tanh(∆E/(2k B T ) for the Glauber dynamics. Since the quantity in the angular brackets of Eq. (6) is independent of temperature (or flip rates), once it is determined, we can use it at any temperature (or with any flip rates) for W (E|E ′ ) and consequently the equilibrium distribution at any temperature.
The microcanonical averages can be computed in any ensembles which have the property that equal energy states have equal probability. We used canonical simulations at a selection of temperatures, so that the total histogram H(E) is approximately flat. Of course, the total H(E) obtained by adding results at different T is not meaningful. However, it is perfectly correct to add statistics to N (σ, ∆E) E from equilibrium simulations at different temperatures.
The single histogram method [5] uses H(E) in a canonical simulation as an estimate to the equilibrium energy distribution P (E). Multiple histogram method attempts to overcome the problem of narrow window in the distribution [6]. In the present method, adding simulations at different T is handled rather naturally. Such a "multiple histogram" TMMC is extremely simple and effective. In connection with the broad histogram method [7], we note that when T is set to ∞, the detailed balance conditions reduce to the broad histogram equations [8]. Unfortunately, the dynamics proposed in Ref. [7] is incorrect [15].
There are a number of different ways of determining the canonical distributions from W (E|E ′ ). We can solve the linear equation,
E ′ W (E|E ′ )P (E ′ ) = 0, or use the detailed balance condition W (E|E ′ )P (E ′ ) = W (E ′ |E)P (E)
. The latter seems numerically more stable-the equations give us a set of recursion relations for P (E). In Fig. 1 we present a calculation of the Ising model heat capacity on a 64 by 64 lattice with 25 simulations at selected temperatures. The errors are small for the whole temperature range.
The one-dimensional dynamics of spins, Eq. (1), can be solved analytically [14]. In particular, the full set of eigenvalues of Γ is known [16]. We show that the matrix W (E|E ′ ) can be obtained in closed form in one dimension. To begin with, the density of states is n(E) = 2C L 2k , where k = 0, 1, 2, . . . , ⌊L/2⌋, assuming periodic boundary condition. The number of possible states with energy E/J = −L + 4k is twice the number of ways (due to overall up-down symmetry) of putting 2k unsatisfied bonds among L possible positions.
The summation over spin states with energy E and E ′ can be expressed in terms of a one-dimensional lattice gas problem. In one dimension, the flip rate can be written as
w i (σ i ) = (1/2) 1 − γσ i (σ i−1 + σ i+1 )/2 where γ = tanh 2K.
Notice that for each final configuration σ associated with E, contributions to the transition from E + = E + 4J to E come only from each pair of nearest neighbors satisfied bonds in configuration σ. Thus we can write
∆ + k = n(E + )W (E|E + ) 1 + γ = ni=L−2k i n i n i+1 ,(7)
where n i = 0 and 1 for unsatisfied and satisfied bond, respectively. Similar expressions can be written down associated with the transition from E − = E − 4J to E. The restricted sums are obtained through a partition function in a field as a generating function. We find ∆ + k = LC L−2 2k and ∆ − k = LC L−2 L−2k−2 and the transition matrix elements are thus
W k,k+1 = (k + 1)(2k + 1) L − 1 (1 + γ),(8)W k+1,k = (L − 2k)(L − 2k − 1) 2(L − 1) (1 − γ).(9)
The diagonal term is computed from the relation
W k−1,k + W k,k + W k+1,k = 0,(10)
and the rest of the elements W k,k ′ = 0 if |k − k ′ | > 1.
The matrix can not be diagonalized analytically in general. Nevertheless, at zero temperature, γ = 1, the eigenvalues can be obtained explicitly as λ k = −2(k + 1)(2k + 1)/(L−1), which give us relaxation times as −1/λ k , with the longest relaxation time τ = (L − 1)/2.
The TMMC for large system follows a simple and interesting dynamics. It can be shown rigorously, with the method of Ω-expansion for master equation [17], that in the large-size scaling limit, the process is described by the equation
∂P (x, t ′ ) ∂t ′ = ∂ ∂x ∂P (x, t ′ ) ∂x + xP (x, t ′ ) ,(11)
where t ′ and x are properly scaled time and energy deviation from equilibrium.
x = E − u 0 N (N c ′ ) 1/2 , u 0 N =Ē,(12)
and
t ′ = bt with b = lim N →∞ 1 2c ′ N E W (Ē|E)(E −Ē) 2 ,(13)
where u 0 is the average energy per spin and c ′ = k B T 2 c is the reduced specific heat per spin. The equation is obtained by replacing E by x and expanding all the terms in small parameter 1/ √ N . Details will be presented elsewhere.
The continuum limit equation describes a constrained random walk. There are two competing effects in the current j = −∂P/∂x − xP ; while the first term is the usual diffusion, the second term keeps the walker at the origin. Equilibrium is obtained when j = 0, giving the well-known Gaussian distribution P eq (x) ∝ e −x 2 /2 .
Equation (11) can be transformed into a onedimensional quantum harmonic oscillator equation with the change of variable P (x, t ′ ) = e −x 2 /4 φ(x, t ′ ). The relaxational spectrum is discrete and equally spaced (in 1/τ ).
In particular, the relaxation modes are e −nt ′ −x 2 /2 H n (x/ √ 2), where H n is Hermite polynomials. Translating back to the original time t, the relaxation times are 1/(nb) ∝ c. Applying the general result, Eq. (13), to the one-dimensional Ising model, we have
τ n = 1 2n cosh 2K, n = 0, 1, 2, · · · .(14)
It is instructive to have an intuitive picture of the asymptotic dynamics, which we argue as follows.
The TMMC is equivalent to the following steps (Algorithm A).
1. Do perfect microcanonical simulation, i.e., pick a state at random among the degenerate states of current energy E.
2. Do one canonical Monte Carlo move, chosen a site at random.
Repeat step (1) and (2) N times, where N is the number of spins in the system. This is one TMMC step (sweep).
Consider the dynamics at very low temperatures. Then only two energies E 0 (ground state) and E 1 = E 0 + 4J (first excited states) dominate. Consider K such that correlation length (ξ ∝ exp(2K)) is about the size of the system.
P (E 0 ) ∝ exp(−E 0 /k B T ), P (E 1 ) ∝ L 2 exp (−E 0 − 4J)/k B T , and P (E 1 )/P (E 0 ) ≈ L 2 exp(−4K) ≈ 1.
We now consider the time scales (from the point of view of TMMC) that a transition is made E 0 → E 1 , and
E 1 → E 0 .
Let's assume that the system is in its ground state E 0 .
Step (1) actually does nothing; in step (2), a kink pair (unsatisfied bonds) is excited with probability exp(−4K). Thus, we need exp(4K) moves to produce a kink pair. So the time scale in units of TMMC steps is τ ∝ exp(4K)/L ∝ L. We have used the fact that correlation length in one dimension is proportional to exp(2K) and that correlation length is comparable to system size L. The reverse process has a similar scale. This argument is consistent with the exact calculation.
A more general argument [18] relating the relaxation time to the specific heat can be given, as follows: the variance of the energy distribution P (E) is related to the specific heat as δE 2 = c N k B T 2 . Transition matrix Monte Carlo moves are random walks in energy confined in a region of δE. Thus the typical time for energy varying over δE is proportional to the square of distance in energy. Therefore, time is proportional to δE 2 , when we measure in steps of moves. To get τ in sweeps, we must divide by N . Let a be the typical time for one step of walk in energy, the equation should be
τ ∝ a(δE 2 /N ) = a c k B T 2 = a c ′ ,(15)
We can give a a precise meaning as 1/(c ′ b), where b is given in Eq. (13). In one dimension, the unit time a diverges. However, this does not happen in dimensions d > 1 as T c > 0, and a will be some finite value asymptotically independent of N . We check the above results in two dimensions by extensive Monte Carlo simulation using two different methods: (1) the microcanonical/canonical algorithm A described above, by computing the time correlation functions; and (2) by direct computation of W (E|E ′ ) with a Wolff cluster algorithm [3]. The eigenvalues are computed with standard package [19]. While the first method is restricted to very small sizes, the second method can apply to large systems. In Fig. 2 we plot only the inverse eigenvalues. Very good confirmation of the log L behavior for the relaxation time is observed. This result has serious implication. In the standard single-spin-flip dynamics, if we add between canonical Monte Carlo moves microcanonical moves, the resulting dynamics is TMMC and still has a residue slowing down τ ∼ log L at T c . It is thus difficult to imagine how this logarithmic factor can be canceled exactly in the original dynamics. In fact, such logarithmic factor is conjectured [13], while many numerical computations [20] did not seem to find it explicitly. We think that the Domany conjecture is still an open question.
One of the prediction of Eq. (11) is that the eigenvalues of W (E|E ′ ) in the large-size limit are equally spaced, this is indeed the case for large systems. The eigenfunctions associated with these eigen modes are compared with numerical results. In Fig. 3, we plot the analytic results (curves) together with numerical values (symbols) from exact diagonalization of matrix W (E|E ′ ) for a threedimensional Ising model of 16 3 at k B T /J = 6.0. There are no adjustable parameters in the comparison except an overall normalization.
In conclusion, the transition matrix Monte Carlo shows a novel dynamical behavior. We find an unusual critical slowing down in one dimension. For two dimensions and higher, we conclude that correlation time is proportional to the specific heat. This is in contrast with cluster algorithms where Li and Sokal [21] showed that the specific heat is only a lower bound to the correlation time. While the TMMC artificial dynamics is of theoretical interest, the reweighting technique is very useful in practice.
FIG. 1 .
1The specific heat of the two-dimensional Ising model on a 64 2 lattice by the transition matrix Monte Carlo reweighting method. The insert shows the relative error with respect to the exact result (obtained numerically based on[22]). The simulations are done at 25 temperatures, each with 10 6 Monte Carlo steps with a single-spin-flip dynamics.
FIG. 2 .
2Relaxation time for two-dimensional Ising model at Tc of the transition matrix Monte Carlo dynamics computed from the inverse of the greatest non-zero eigenvalues of matrix W (E|E ′ ). The matrices are obtained by a Wolff Monte Carlo cluster algorithm with 10 7 to 10 9 cluster flips.
FIG. 3 .
3The first four eigenmodes of the transition matrix Monte Carlo dynamics. The continuous curves are analytic results ∝ e −x 2 /2 Hn(x/ √ 2). Symbols are from exact diagonalizations of matrix W (E|E ′ ) for a three-dimensional Ising model on a 16 3 system at kBT /J = 6.0. For clarity, only every fifth data points are plotted.
Han Tang for an important suggestion on large-size limit. He also thanks Hong Guo and Donghui Zhang for discussions. This work is supported in part by. J S Thanks Lei, Academic Research Grant No. RP950601J. S. W thanks Lei-Han Tang for an important sugges- tion on large-size limit. He also thanks Hong Guo and Donghui Zhang for discussions. This work is supported in part by Academic Research Grant No. RP950601.
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[
"Fermi I particle acceleration in converging flows mediated by magnetic reconnection (Research Note)",
"Fermi I particle acceleration in converging flows mediated by magnetic reconnection (Research Note)"
]
| [
"V Bosch-Ramon [email protected] \nDublin Institute for Advanced Studies\n31 Fitzwilliam PlaceDublin\n\nIreland\n"
]
| [
"Dublin Institute for Advanced Studies\n31 Fitzwilliam PlaceDublin",
"Ireland"
]
| []
| Context. Converging flows with strong magnetic fields of different polarity can accelerate particles through magnetic reconnection. If the particle mean free path is longer than the reconnection layer is thick, but much shorter than the entire reconnection structure, the particle will mostly interact with the incoming flows potentially with a very low escape probability. Aims. We explore, in general and also in some specific scenarios, the possibility of particles to be accelerated in a magnetic reconnection layer by interacting only with the incoming flows. Methods. We characterize converging flows that undergo magnetic reconnection, and derive analytical estimates for the particle energy distribution, acceleration rate, and maximum energies achievable in these flows. We also discuss a scenario, based on jets dominated by magnetic fields of changing polarity, in which this mechanism may operate. Results. The proposed acceleration mechanism operates if the reconnection layer is much thinner than its transversal characteristic size, and the magnetic field has a disordered component. Synchrotron losses may prevent electrons from entering in this acceleration regime. The acceleration rate should be faster, and the energy distribution of particles harder than in standard diffusive shock acceleration. The interaction of obstacles with the innermost region of jets in active galactic nuclei and microquasars may be suitable sites for particle acceleration in converging flows. | 10.1051/0004-6361/201219231 | [
"https://arxiv.org/pdf/1205.3450v2.pdf"
]
| 59,126,700 | 1205.3450 | 4c6615e32f8dc2e08f432a4ddcc8ea4df77ea6f4 |
Fermi I particle acceleration in converging flows mediated by magnetic reconnection (Research Note)
4 Jun 2012 May 1, 2014
V Bosch-Ramon [email protected]
Dublin Institute for Advanced Studies
31 Fitzwilliam PlaceDublin
Ireland
Fermi I particle acceleration in converging flows mediated by magnetic reconnection (Research Note)
4 Jun 2012 May 1, 2014Received ¡date¿ / Accepted ¡date¿Astronomy & Astrophysics manuscript no. text c ESO 2014Magnetic reconnection -Acceleration of particles -Radiation mechanisms: non-thermal
Context. Converging flows with strong magnetic fields of different polarity can accelerate particles through magnetic reconnection. If the particle mean free path is longer than the reconnection layer is thick, but much shorter than the entire reconnection structure, the particle will mostly interact with the incoming flows potentially with a very low escape probability. Aims. We explore, in general and also in some specific scenarios, the possibility of particles to be accelerated in a magnetic reconnection layer by interacting only with the incoming flows. Methods. We characterize converging flows that undergo magnetic reconnection, and derive analytical estimates for the particle energy distribution, acceleration rate, and maximum energies achievable in these flows. We also discuss a scenario, based on jets dominated by magnetic fields of changing polarity, in which this mechanism may operate. Results. The proposed acceleration mechanism operates if the reconnection layer is much thinner than its transversal characteristic size, and the magnetic field has a disordered component. Synchrotron losses may prevent electrons from entering in this acceleration regime. The acceleration rate should be faster, and the energy distribution of particles harder than in standard diffusive shock acceleration. The interaction of obstacles with the innermost region of jets in active galactic nuclei and microquasars may be suitable sites for particle acceleration in converging flows.
Introduction
The interaction of outflows with themselves or their environment usually leads to particle acceleration (e.g. Rieger et al. 2007), and the most common acceleration mechanism in these circumstances is thought to be diffusive shock (Fermi I) acceleration (e.g. Krymskii 1977;Bell 1978a,b;Blandford & Ostriker 1978;Drury 1983). This process works well when a plasma with a weak disordered magnetic field (B) suffers a strong shock. The Fermi I acceleration efficiency depends on the energy gain when particles cross the shock, and on the escape probability downstream the shock, which is related to the shock compression ratio R = v/v ′ , v ′ (v) being the post(pre)-shock velocity.
In strong non-relativistic adiabatic shocks in the testparticle approximation R equals 4, and the resulting energy distribution is Q(E) ∝ 1/E 2 (e.g. Drury 1983). In radiative shocks, R ≫ 4, but densities become very high and energy losses and damping of magnetic irregularities may limit the efficiency of the acceleration process (e.g. Drury et al. 1996, Reville et al. 2007; see nevertheless Sect. 2.2). For plasmas that carry a strong perpendicular B-component, R is ∼ 1, and Fermi I acceleration is suppressed. However, if the Bfield has reversals, energy can be dissipated via magnetic reconnection in regions where the charge density is not sufficiently high to sustain different polarity B-lines (e.g.
Send offprint requests to: V. Bosch-Ramon, e-mail: [email protected] Parker 1957;Petschek 1964;Speiser 1965;Sonnerup 1971;Zenitani & Hoshino 2001), allowing for R ≫ 1 (e.g. Drury 2012).
Studies show that magnetic reconnection can accelerate the bulk of particles up to the pre-reconnection Alfvénic speed, with some particles reaching even higher energies (e.g. Zenitani & Hoshino 2001;Lyubarsky & Liverts 2008;Kowal et al. 2011). Magnetic reconnection may also lead to the formation of strong shocks, accelerating particles through the standard Fermi I mechanism in an otherwise weakly compressive flow (e.g . Forbes 1988;Blackman & Field 1994;Sironi & Spitkovsky 2011). Fermi I particle acceleration may operate in the reconnection region itself (e.g. de Gouveia dal Pino & Lazarian 2005;Giannios 2010;Drury 2012), since particles can bounce back and forth between the layer and the converging flows. The mean free path λ of these particles can eventually overcome the layer thickness, ∆Y , and then particles will not interact significantly with the reconnection layer, but with the perpendicular B-lines of the converging flows (Giannios 2010). In this case, the advection escape probability becomes formally zero because the flows do not move away from the reconnection layer. This is true as long as the mean free path of the particle parallel to B, λ , is much shorter than the characteristic size of the entire reconnection region, ∆X (to order the magnetic field reversal scale). Escape in the opposite direction to the incoming flow can be neglected unless the region extension in this direction is ≪ ∆X. A schematic picture of this scenario is presented in Fig. 1, which shows two cases: the interaction of a flow carrying different polarity B-lines with an obstacle (described in Sect. 3); and the interaction of two flows with different polarity B-lines. As shown in the figure, particles can spiral back and forth between the two sides of the reconnection layer until they diffuse away from the region.
In this note, we analytically explore the Fermi I mechanism when particles interact with converging flows in the null-escape probability regime. We assume that particles can effectively diffuse through the incoming flows. We focus on the case when magnetic reconnection is the mechanism that leads to R ≫ 1. We derive in Sect. 2 analytical estimates for the acceleration rate, the maximum energy and energy distribution of the accelerated particles, and present in Sect. 3 an illustrative case in the context of galactic and extragalactic jets. For simplicity, the discussions are restricted to Newtonian flows, although our conclusions qualitatively apply to the relativistic case as well. For some magnitudes, we will adopt the convention Q x = (Q/10 x cgs).
Particle acceleration in converging flows
In converging flows, particles slowly diffuse perpendicularly to the B-lines, with λ ⊥ ∼ r g /χ, where χ, with value ≥ 1, is the total to the disordered magnetic energy density ratio. Remarkably, a disordered B-component can enhance the reconnection rate and the reconnecting flow velocity v, as shown for instance in Lazarian & Vishniac (1999). The following limit for Fermi I to operate between the converging flows can be imposed: in the current sheet, λ ⊥ > ∆Y , i.e. E min ∼ χ q B r ∆Y , where B r is the B-field in there. Although one can expect that B r ≪ B close to the center of the current sheet, we conservatively assume that over the whole reconnection layer B r ∼ B. On the other hand, the B-lines become more and more entangled close to the re-connection layer, so one can set there χ = 1. The previous condition becomes
E > E min ∼ 0.3 B 0 ∆Y 9 TeV.
(1)
In addition, to be efficiently accelerated, particles in the incoming flows should not drift too early along the B-lines, parallel to the layer, out of the reconnection region (crossfield diffusive escape will be slower by 1/χ 2 ). This is a sort of Hillas limit: λ ∼ χ r g < ∆X, or
E H max ∼ q B∆X/χ ≈ 3 B 0 ∆X 10 χ −1 TeV.(2)
This gives the maximum dynamical range
E H max /E min ∼ 10 χ −1 (∆X 10 /∆Y 9 ) .(3)
Giannios (2010) accounted for the electric field between the sides of the reconnection layer: ǫ ∼ v B. The ǫ-field accelerates the particles perpendicularly to B and v, with
E max ∼ (v/c)qB∆X.
The fraction of time spent by particles within the reconnection layer during an acceleration cycle is ∆Y /c t cycle , and thus this effect will be important at E ∼ E min or for very ordered fields. From Eqs. (2-3) one can see that the quantities χ and ∆X/∆Y are crucial for the efficiency of the discussed acceleration mechanism. The derivation of χ requires a detailed treatment of the plasma, but as mentioned, given the dramatic change of the B-geometry, close to the reconnection layer χ ∼ 1. On scales of ∆X, and given the high plasma characteristic speeds, the disordered B-field generated in the reconnection layer may propagate to the converging flows, although such a B-component may also have an external origin. To estimate ∆X/∆Y , we briefly discuss the flow dynamics below.
Flow dynamics
The plasma that enters, flows along, and eventually leaves the reconnection layer, as shown in Fig. 2, can be described by the particle and energy flux conservation equations:
Aρ v = A ′ ρ ′ v ′ and (4) A v B 2 /8π = A ′ ρ ′ v ′3 /2 ,(5)
where unprimed/primed quantities correspond to the incoming/outgoing flows, ρ is the density, v the velocity, and A ∼ ∆X 2 and A ′ ∼ 4∆X∆Y are the in-and out-flow areas, respectively. The flow pressure is assumed to be negligible in the converging flows because of strong B-dominance. When leaving the region, under negligible external pressure and reasonable geometries, the flow kinetic energy becomes dominant and pressure can be neglected as well.
Given the symmetry of the problem, there is no momentum conservation equation, and the conditions deep within the reconnection layer are hard to determine. Equations 4 and 5 yield v ′ ≈ v A = B 2 /4πρ and thus ρ ′ ≈ ρ (A/A ′ )(v/v A ), but one has to make additional assumptions to obtain ∆X/∆Y . An estimate may be derived assuming one-dimensional adiabatic compression of B r , the remaining field after reconnection in the layer, until it balances the incoming field pressure. This gives ρ ′ /ρ ∼ (B/B r ) and therefore an area ratio A ′ /A = (B r /B)(v/v A ), or ∆X/∆Y ∼ 4(B/B r )(v A /v). An upper limit for ∆X/∆Y would come from ∆Y ∼< r g >, where < r g > is the gyroradius of the average particle in the reconnection layer. A lower limit may be derived assuming equilibrium between the upstream magnetic pressure and the thermal pressure inside the layer of thickness ∆Y , implying (∆X/∆Y ) ∼ 4(v A /v). One can therefore conclude that the dynamical range may span several orders of magnitude in energy, but it needs an efficient magnetic-to-kinetic energy transfer (which may be reasonable, as suggested in Drury 2012).
Acceleration and radiation processes
Like in standard diffusive shock acceleration, particles gain energy by moving back and forth between the incoming flows as ∆E/E ∼ v/c per cycle. In addition, since there is no advection directed outwards, the probability to cross back to the other side of the reconnection layer becomes 1 after covering few λ ⊥ in each incoming flow, i.e. t cycle ∼ (k/χ) r g /c, with k ∼ 10. This t cycle gives an acceleration ratė
E acc ∼ ∆E/t cycle = (χ/k) (v/c) q B c ,(6)
and the zero advection escape probability renders a particle energy distribution Q(E) ∝ 1/E up to energies a few times lower than E H max , at which the distribution drops very quickly. Note that this mechanism is faster inĖ acc ((v/c) vs (v/c) 2 ) and yields a harder Q(E) (1/E vs 1/E 2 ) than standard Fermi I. Such a hard distribution of accelerated particles can become dominant in pressure. This should smooth the v profile and affect Q(E) (as in non-linear Fermi I acceleration; e.g. Drury 1983).
Radiation losses can stop the acceleration at E < E H max . Typically, for protons these losses are fairly inefficient, whereas for electrons synchrotron cooling (e.g. Blumenthal & Gould 1970) tends to be important, which renders a maximum energy E sy max ≈ 60 (χ/k) 1/2 (v/c) 1/2 B −1/2 0 TeV.
In environments with very dense matter or photon fields, other cooling processes might be relevant, such as relativistic Bremsstrahlung and inverse Compton (IC) for electrons (e.g. Blumenthal & Gould 1970), and protonproton collisions (pp), synchrotron and photomeson production for protons (e.g. Kelner et al. 2006;Aharonian 2000;Kelner & Aharonian 2008). Diffusive escape parallel to B can also be more restrictive than the Hillas limit:
E di max ≈ (3v/2c k) 1/2 q B∆X ≈ 4 (v/c k) 1/2 B 0 ∆X 10 TeV.(8) The corresponding dynamical ranges are E sy max /E min ∼ 200 (χ/k) 1/2 (v/c) 1/2 B −3/2 0 ∆Y −1 9 and (9) E di max /E min ∼ 10 (v/c k) 1/2 ∆X 10 ∆Y −1 9 .(10)
In general, to accelerate particles to TeV energies, at least mildly relativistic v A -values will be required. In addition, for electrons, if converging flow acceleration is to be efficient, ∆Y 9 B −3/2 1.5 . This will typically require in astrophysical sources either very high compression ratios, or very low magnetic fields, the latter implying a very large structure (e.g. magnetized turbulent ISM) if the process is to have observable radiative effects. For protons, the Hillas and diffusive limits lead to the more reasonable conditions ∆X > χ∆Y and > (v/c k) −1/2 ∆Y , respectively. Note nevertheless that electrons remaining within the reconnection layer may still be accelerated by electric fields generated in the reconnection process, resulting in energies E min . In this case, the accelerated energy distribution could also be Q(E) ∝ 1/E (e.g. Zenitani & Hoshino 2001).
Synchrotron emission from electrons injected with Q(E) ∝ 1/E and cooled through this process would have a spectral energy distribution ν L ν ∝ ν 1/2 . On the other hand, proton radiation would show ν L ν ∝ ν for synchrotron (or ∝ ν 1/2 in saturation) and for π 0 -decay from pp and photomeson production (above threshold). The last two processes also generate secondary e ± pairs from π ±decay with spectra and luminosities similar to those of the photons from π 0 -decay. These pairs will predominantly cool through synchrotron, like primary electrons, also with ν L ν ∝ ν 1/2 but peaking at much higher energies.
Before exploring the case of magnetized jets, we note that our results may also be applied to weakly magnetized flows with radiative shocks. Particle acceleration between the upstream and the cooled downstream would require sufficiently energetic particles to cross the adiabatic region of the post-shock flow and reach the cooled downstream medium. Another condition would be that the particle cooling/escape timescales were longer everywhere than the acceleration one. In this context, young stellar object jet shocks, or the dense winds of two massive stars colliding, may be sources worthy of being investigated.
The case of magnetized jets
Powerful astrophysical outflows, such as pulsar winds, and microquasar and AGN jets, are expected to be magnetically dominated at their formation and acceleration zones (e.g . Coroniti 1990;Bogovalov & Tsinganos 1999;Beskin & Nokhrina 2006;Komissarov et al. 2007). In pulsar winds, magnetic reconnection may take place naturally in a current sheet that originated at the wind equatorial region (Coroniti 1990;Lyubarsky & Kirk 2001). In relativistic jets, a B-field with reversals could come from the accretion disc (e.g. Barkov & Baushev 2011), but magnetic dissipation may need to be triggered through jet acceleration, which could drive magnetic reconnection through the Kruskal-Schwarzschild instability (see Lyubarsky 2010). A similar though more extreme effect can be expected if an obstacle is entrained by the jet (e.g. Hubbard & Blackman 2006;Araudo et al. 2009Araudo et al. , 2010Barkov et al. 2010;Bosch-Ramon et al. 2012;Barkov et al. 2012) with a dominant B-field with polarity reversals of size ∆X. Under the ongoing magnetic reconnection, the incoming jet material and the obstacle itself can play the role of converging flows with a high ∆X/∆Y ratio. To avoid the suppression of the acceleration process, the shocked obstacle must fulfil the following conditions: high inertia, to avoid quick dragging by the jet; not too high density, to avoid fast radiation cooling; and a mean free path much shorter than the obstacle size (D o ), to avoid fast particle escape.
For a jet/obstacle interaction at z j ∼ 100 R Sch from the central object, where R Sch ≈ 3×10 13 (M/10 8 M ⊙ ) cm, a jet width D j ∼ 0.1 z j ∼ 3 × 10 14 (M/10 8 M ⊙ ) cm and magnetic field B ∼ 10 3 L 1/2 j,44 (M/10 8 M ⊙ ) −1 G, and D o ∆X, the luminosity budget available for dissipation will be L d ∼ (D o /D j ) 2 L j ≤ L j , where L j is the jet power. Assuming ∆X ∼ R Sch , the electron and proton maximum energies will be E e max ∼ 1 (M/10 8 M ⊙ ) 1/2 L −1/4 j,44 TeV, and E p max ∼ 10 7 L 1/2 j,44 TeV. Given the strong magnetic fields, the condition ∆Y 9 B −3/2 1.5 may not hold; if so, electrons could only be accelerated within the reconnection layer. Thus, in AGN jet/obstacle interactions, magnetic reconnection itself could potentially accelerate electrons up to TeV energies. Protons, on the other hand, could be accelerated in the converging flows 1 to ultra high energies or even beyond depending on ∆X, L j , and D o (see Giannios 2010, for similar results). Electrons would yield hard synchrotron radiation peaking around 10 MeV. Proton synchrotron emission, of hard spectrum and peaking around 100 GeV, may be efficient as well under these conditions (see, e.g., Barkov et al. 2012). Emission through pp inside the obstacle could be also significant (Barkov et al. 2010), as well as photomeson production in very bright AGN. Secondary very energetic e ± pairs from π ± -decay would radiate ultra high-energy photons through synchrotron emission that could pair-create in the jet radio fields on pc-scales, being reprocessed as synchrotron and IC photons of lower energies.
In microquasar jets, where, say, M ∼ 10 M ⊙ and L j ∼ 10 36 erg s −1 , magnetic reconnection in the jet/obstacle boundary at z j ∼ 100 R Sch could lead to synchrotron MeV flares produced by GeV electrons. A minor synchrotron self-Compton component in GeV may be also expected. PeV protons may be accelerated in the converging flows, and if they met a nearby thick target, for instance the obstacle itself or a dense field of energetic photons, they could produce hard emission peaking at ∼ 100 TeV through pp or photomeson production, plus hard TeV synchrotron radiation from the subsequent e ± pairs. In compact and powerful objects photon-photon absorption would be severe, radiation being re-emitted as soft gamma rays via synchrotron emission. It is not clear which kind of obstacle may be present in the innermost regions of a microquasar jet, although in the (unlikely) event that the jet remained strongly magnetized up to the binary scales (≫ 100 R Sch ), a clumpy stellar wind could provide those obstacles (Araudo et al. 2009).
In both microquasars and AGN, the reconnection process can be efficient as long as the obstacle has a velocity different from that of the flow. In this process, for D o ∼ D j , a significant fraction of L j may be released, with potential fluxes at high energies F ∼ 10 −10 erg cm −2 s −1 ×L d,44 (d/100 Mpc) −2 for AGN, and ×L d,36 (d/10 kpc) −2 for microquasars. This radiation would be roughly isotropic, but for obstacles small and light enough to be accelerated by the jet, the emission would become progressively Doppler-boosted and enhanced for specific viewing angles. The obstacle may expand, enhancing the radiation, and possibly fragment while accelerated. All this should lead to complex spectral and variability patterns (e.g. Barkov et al. 2010Barkov et al. , 2012. The typical timescale of the whole interaction process will be the time the obstacle remains as such (see, e.g., Bosch-Ramon et al. 2012), whereas the reconnection events will have a variability timescale ∼ min[D o , ∆X]/v, where v c.
Fig. 1 .
1Sketch of the considered scenario: a) the interaction of one flow carrying different polarity B-lines with an obstacle (see Sect. 3); b) the interaction of two flows with different polarity B-lines.
Fig. 2 .
2Sketch of the converging and the outgoing flows in the reconnection region.
Even if jets were formed only by e ± pairs, protons may be entrained from the environment, e.g. the obstacle itself.
Acknowledgements. We thank an anonymous referee for constructive and useful comments and suggestions. We are grateful to Maxim Barkov, Luke Drury and Dmitry Khangulyan for fruitful discussions.
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| []
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[
"Exploring the Gamma Ray Horizon with the next generation of Gamma Ray Telescopes. Part 3: Optimizing the observation schedule of γ-ray sources for the extraction of cosmological parameters",
"Exploring the Gamma Ray Horizon with the next generation of Gamma Ray Telescopes. Part 3: Optimizing the observation schedule of γ-ray sources for the extraction of cosmological parameters"
]
| [
"O Blanch \nIFAE\nBarcelonaSpain\n",
"M Martinez \nIFAE\nBarcelonaSpain\n"
]
| [
"IFAE\nBarcelonaSpain",
"IFAE\nBarcelonaSpain"
]
| []
| The optimization of the observation schedule of γ-ray emitters by the new generation of Cherenkov Telescopes to extract cosmological parameters from the measurement of the Gamma Ray Horizon at different redshifts is discussed. It is shown that improvements over 30% in the expected cosmological parameter uncertainties can be achieved if instead of equal-observation time, dedicated observation schedules are applied. | 10.1016/j.astropartphys.2005.03.010 | [
"https://arxiv.org/pdf/astro-ph/0409591v1.pdf"
]
| 11,549,801 | astro-ph/0409591 | 42b7bd0a821b12e119d554cf01f6d66fdd4a6067 |
Exploring the Gamma Ray Horizon with the next generation of Gamma Ray Telescopes. Part 3: Optimizing the observation schedule of γ-ray sources for the extraction of cosmological parameters
24 Sep 2004 November 22, 2018
O Blanch
IFAE
BarcelonaSpain
M Martinez
IFAE
BarcelonaSpain
Exploring the Gamma Ray Horizon with the next generation of Gamma Ray Telescopes. Part 3: Optimizing the observation schedule of γ-ray sources for the extraction of cosmological parameters
24 Sep 2004 November 22, 2018
The optimization of the observation schedule of γ-ray emitters by the new generation of Cherenkov Telescopes to extract cosmological parameters from the measurement of the Gamma Ray Horizon at different redshifts is discussed. It is shown that improvements over 30% in the expected cosmological parameter uncertainties can be achieved if instead of equal-observation time, dedicated observation schedules are applied.
Introduction
Imaging AirČerenkov Telescopes (IACT) have proven to be the most successful tool developed so far to explore the γ-ray sky at energies above few hundred GeV. A pioneering generation of installations has been able to detect a handful of sources and to start a whole program of very exciting physics studies. Nowadays a second generation of more sophisticated Telescopes is starting to provide new observations. One of the main characteristics of some of the new Telescopes is the potential ability to reduce the gamma ray energy threshold below ∼ 30 GeV [1].
In the framework of the Standard Model of particle interactions, high energy gamma rays traversing cosmological distances are expected to be absorbed through their interaction with the diffuse background radiation fields, or Extragalactic Background Field (EBL), producing e + e − pairs. Then the flux is attenuated as a function of the gamma energy E and the redshift z q of the gamma ray source. This flux reduction can be parameterized by the optical depth τ (E, z q ), which is defined as the number of e-fold reductions of the observed flux as compared with the initial flux at z q . This means that the optical depth introduces an attenuation factor exp[−τ (E, z q )] modifying the gamma ray source energy spectrum.
The optical depth can be written with its explicit redshift and energy dependence [2] as:
τ (E, z) = z 0 dz ′ dl dz ′ 2 0 dx x 2 ∞ 2m 2 c 4 Ex(1+z ′ ) 2 dǫ · n(ǫ, z ′ ) · σ[2xEǫ(1 + z ′ ) 2 ](1)
where x ≡ 1 − cos θ being θ the angle between the photon directions, ǫ is the energy of the EBL photon and n(ǫ, z ′ ) is the spectral density at the given z'.
For any given gamma ray energy, the Gamma Ray Horizon (GRH) is defined as the source redshift z for which the optical depth is τ (E, z) = 1.
In a previous work [3], we discussed different theoretical aspects of the calculation of the Gamma Ray Horizon, such as the effects of different EBL models and the sensitivity of the GRH to the assumed cosmological parameters.
Later, on [4] we estimated with a realistic simulation the accuracy in the determination of the GRH that can be expected from an equal-time obser-vation of a selection of extragalactic sources. The results obtained in that previous study assumed an observation schedule of equal observing time per source which was set to a canonical value of 50 hours (rather standard assumption in IACTs). Although the actual observing time per source might have a lot of constraints (such as significance of the observation, physics interest of the source, competition in time for the observation of other sources, etc...), in this work we want to explore, taking into account just the unavoidable observability constraints, which time scheduling would optimize the power of this method to extract cosmological parameters and by how much the measurement of these parameters could improve.
One must also take into account that the determination of the cosmological parameters extensively discussed in [4] is based on the observations of Active Galactic Nuclei (AGN), which are intrinsically variable. Therefore one of the main parameters to decide which AGN is observed at any time will be their flaring state. For some AGN, it is possible to estimate its activity from observations in other wavelength, for instance using the X-ray data [5] provided by the All-Sky Monitor (ASM) [6] onboard the Rossi X-ray Timing Explorer. Unfortunately, there are a lot of AGNs for which there is not online data that would allow to infer the flaring state. Actually in the current catalogue of sources that are monitored by ASM only 3 of the 22 used on these studies appear. So that, here we'll present an observational scheduling for the AGNs in table 1, which does not care about the activity of the source, and just optimizes the observation time to get the best precision on the cosmological parameter measurements.
The work is organized as follows: in section 2, the expected improvement in the precision of the GRH determination as a function of the observation time is discussed. Section 3 deals with the observational constraints in the optimization procedure. In section 4 we describe the optimization technique employed and describe the actual algorithm used. Section 5 presents the results of the optimization procedure in different scenarios considered and finally in section 6 we summarize the conclusions of this study.
Gamma Ray Horizon energy precision
To optimize the observation time, the first step is the study of the GRH precision as a function of the time that is dedicated for each source. The estimated precision on the GRH comes from the extrapolation of the detected spectra of each source by MAGIC (see [4]). In there, the observation time enters as a multiplicative term to get the number of γs.
In figure 1, the expected σ of the GRH (σ grh ) using several observation times is shown. In these plots only the statistical error from the fitting parameter is shown. That error comes from the error bars in the extrapolated spectra. On the one hand there is the uncertainty on the flux (Φ) that is modeled as "n" times the square-root of Φ, which is proportional to N γ being N γ the number of detected γs. On the other hand, the error on the determination of the energy improves also with N γ if one assumes a gaussian statistic. Therefore, one expects that the σ grh decreases with the square-root of time. Actually, the extrapolated errors show a good agreement with the blue line that is a fit to:
σ grh = k/ √ time (2)
This latter expression would mean that the σ grh can be as small as desired if enough observation time is used. But it does not represent the reality. One should also take into account the systematics, which become more and more important when reducing the statistical error. As it has already been discussed in [4] the main systematic errors in the GRH determination from the simulated experimental data are due to the uncertainty in the global energy scale and to some approximations used to fit the data. The former is a global systematic, which is absolutely independent and uncorrelated to the observation time, and hence it is not considered here. Instead, the latter should have an impact in the precision of the GRH determination as a function of the observation time that may be different for each source. The main effect of those approximations is that the value of the GRH differs slightly from the one that has been introduced. This difference is added quadratically to the statistical error to account for the systematic difference. Then the figure 1 has been repeated and the result for the same 4 sources are shown in figure 2. In that scenario the curve is fitted to :
σ grh = a + k/ √ time (3)
where a is the contribution coming from the systematic, which does not decrease with the amount of observation time and therefore becomes impor- tant when the time is large. In fact the parameterization that have been finally used is :
σ grh = a + GRH(50h) * 50 hours time(hour)(4)
where GRH(50h) are the statistic errors for the GRH using 50 hours of observation time and a is the constant term of the above mentioned fit (table 1).
Constraints
The aim of this work is the optimization of the time dedicated to each source, to understand which improvement can be obtained in the cosmological measurements. Nevertheless, that time should make sense in the frame of the possible observations performed by an IACT such as, for instance, MAGIC. Therefore some constraints should be set. The first constraint is the total amount of time used for those observations. For that, we used 1000 hours to compare it with the "50 h per source" configuration. On that naive configuration 50 hours were chosen since it is a reasonable time to spend in a single source and it was already a criteria to do the list of the best MAGIC targets in [7]. Since 20 sources were used (see reference [4]), it accounts for 1 000 hours. Moreover, taking into account thať Cerenkov Telescopes have typically observations times of about 1 200 hours per year, the limit used could be reached even in one single year. And it is more than acceptable for 2-3 years, since AGNs are one of the main targets of the new generation of IACTs.
One of the singularities of the astrophysics field respect to other physics disciplines is that it studies phenomena that cannot be generated by the humans in a laboratory. Therefore one has to use what nature provides. In this sense, IACTs cannot observe one given source for an infinite time during a year, not even those 1200 hours of observation time, since each source is only visible during some months every year. Based on that fact, we have computed the amount of time that each source is visible below 45 degrees zenith angle form the MAGIC location. In table 1, one can see that time for each of the used AGNs, actually it holds for the year 2005 and it may change a few percent from year to year due to the full moon periods. To compute the optimal distribution of observation times, the constraint "MaxTime" used for each source is :
MaxT ime < T (1 year) * Y ears * F(5)
where T (1 year) are the number of hours stated in table 1. "Y ears" is the number of years during which data would be collected and it is set to 3. And F is the fraction of the available time during which data would be taken. It is set to 0.25 and it accounts for bad weather conditions, off data needed for the standard "On-Off" analysis and time dedicated to other sources or targets of opportunity.
Time Optimization
In reference [4] the capability of the new IACTs to measure cosmological constants has been discussed as well as the systematics on these measurements. There, the main emphasis was put in the 68% contour in the Ω m − Ω λ plane, and it has been shown that it is competitive taking into account the systematics (15% of energy scale, fit approximation and unknown Extragalactic Background Light (EBL)) if a 15% external constraint on the Ultra Violet Table 1: Parameters used for the optimization of the time observation dedicated to each source. The parameter a is the time independent term contributing to the σ grh (equation 4). And "T (1year)" is the time that the source is below 45 degrees zenith angle during 2005 at the MAGIC location.
(UV) background is used. Under this scenario and scheduling 50 hours to each source, there is also the possibility to fit Ω m and Ω λ . Now, we would like to optimize the distribution of the observation time among the used sources to get the best precision on the measurement of the cosmological densities. In order to optimize the distribution of the observation time by requiring a minimum error in some given parameter, a technique based upon a multidimensional constrained minimization using the Fisher Information Matrix has been used.
The Fisher Information Matrix, is defined as
F ij = −d 2 logL dθ i dθ j (6)
where L is the likelihood function of the measurements, θ i and θ j are fitting parameters and < ... > denotes expected value. In "normal" conditions, it is the inverse of the error matrix for the parameters i, j. For large samples, a good estimator of F is simply the function
F ij = −d 2 logL dθ i dθ j(7)
evaluated at θ =θ namely, at the best fit parameter values.
In case L could be simply approximated by a gaussian centered at the measured values, then
F ij = k,l (df k /dθ i )(V −1 kl )(df l /dθ j )(8)
evaluated at θ =θ. The i, j indices run over all the fitting parameters (the cosmological parameters in our case) and the k, l run over all the measurements (the GRH measurements for different redshift in our case). V is the error matrix of the measurements and f k (θ) is the theoretical prediction for measurement k.
External constraints on the parameters are included by adding their corresponding Fisher Information Matrix. For instance, if Ω λ corresponds to parameter i = 3, and we want to include the constraint due to a measurement Ω CM B λ ± ∆Ω CM B λ we just have to add to F ij a matrix F ′ ij with
F ′ 33 = Ω CM B λ ∆Ω CM B λ 2(9)
and zero in all the other matrix elements. This way, one can compute the expected fit parabolic error for any parameter without having actually to perform the fit. The expected error in Ω λ for instance would simply be (F −1 ) 33 . Now one must minimize this quantity (or any desired function of the fitting parameters) with respect to the observation time expended in each source (which enters in the evaluation of V and hence, on F ) with the relevant physical boundaries and constraints. For that we use the mathematical approach implemented in the code "DONLP2" developed by M.Spelucci [8]. The mathematical algorithm evaluates the function to be minimized only at points that are feasible with respect to the bounds. This allows to solve a smooth nonlinear multidimensional real function subject to a set of inequality and equality constraints. In our particular case:
• Problem function: It depends on the variable that we want to minimize but it is always a combination of the elements of the Fisher Information Matrix of the four dimensional fit in terms of H 0 , Ω λ , Ω M and the amount of UV background as described in [4].
• Equality constraints: The global amount of observation time, which is set to 1000 hours.
• Inequality constraints: The maximum available time for each source.
The result of this procedure is an array providing the optimal distribution of the observation times assuming parabolic errors, though we have explicitly checked that for the optimal time distribution the obtained precision on the fit parameters does not depend sizably on the assumption of parabolic errors.
Results
After the optimization to get the minimum error on Ω m or Ω λ , this precision is improved by about 35% (see table 2). It is worth to notice that the obtained uncertainties for σ Ωm and σ Ω λ do not significantly differ while optimizing for one or the other. Even optimizing for the area of 68% contour in Ω m − Ω λ plane, which is done assuming that the contour is an ellipse, the precision obtained is at the same level. This effect is mainly due to the correlation between Ω m and Ω λ . Therefore, we will refer as the optimum time the one that minimizes the area of the Ω m − Ω λ contour. This optimum time for each source is shown in table 3. One should notice that in this table only some of the initial 20 extragalactic considered sources remain. For the others, the optimization suggests that it is less interesting to observe them in terms of cosmological measurements. Moreover, the remaining sources are the ones at lowest and highest redshifts as well as the ones with smaller errors. Both were expected to survive since the former give the capacity to disentangle the cosmological parameters (see reference [3]) and the latter give larger constraints with less dedicated time. The improvement for the 68% contour can be seen in figure 3.
It has already been mentioned in [4] that a different approach to extract information from the GRH can be done: one can use the present constraints of the cosmological parameters to get information on the EBL. In reference [4], the complexity of such analysis is discussed but a simple first step can be done within the scenario of the 4 dimensional fit used for these studies. If one uses the current measurements of the cosmological parameters ( H o = 72 ± 4, Ω m = 0.29 ± 0.07 and Ω λ = 0.72 ± 0.09 [9][10]) as external constraints and then optimizes the time distribution to get the minimum error on the fourth parameter which gives a scale factor for the UV background, one can reach a precision of 13.5%. It is worth to notice that , despite the distribution of time among them is different, the sources that are still used are roughly the same than the ones for the Ω m − Ω λ optimization.
Conclusions
In our previous works on this subject [3,4], it was shown that the precision reached to measure the GRH for 50 hours of observation time is not the same for each of the 20 considered extragalactic sources. Moreover, it was also clear that the sensitivity to Ω m and Ω λ was larger at high redshift and that the capability to disentangle the cosmological parameters is based on having measurements at low and high redshift. Therefore, it is clear that a cleverer distribution of the observation time would lead to better results. The optimization of that time distribution pointed out the need of having low redshift measurements (3EG J1426+428 at z = 0.129) as well as others at high redshift (3EG J1635+3813 at z = 1.814). Together with these extreme sources, the dedication of time at sources that reach the best precision of the GRH (3EG J0340-0201) would also help to improve the results. In this work, the optimal distribution of the observation time, taking into account scheduling constraints, has been studied by applying a technique based upon a multidimensional constrained minimization using the Fisher Information Matrix. The results obtained show that a proper scheduling optimized for the determination of the cosmological parameters could allow to reduce by 35% the error on the determination of Ω m and Ω λ and a notable reduction of the 68% contour in the Ω m − Ω λ plane.
Figure 1 :
1Evolution of the statistic precision of the GRH determination as a function of observation time for four of the used AGNs (3EG J1426+428, 3EG J1255-0549, 3EG J0340-0201 and 3EG J1635+3813). The blue line is the fit to one over square-root of time.
Figure 2 :
2Evolution of the precision of the GRH determination , adding the systematics due to the approximation in the fit of the spectra, as a function of observation time for four of the used AGNs (3EG J1426+428, 3EG J1255-0549, 3EG J0340-0201 and 3EG J1635+3813). The blue line is the fit to equation 3.
Figure 3 :
3Improvement on the 68% contour in the Ω m − Ω λ plane. The red solid line is the 68% contour, taking into account the systematics and imposing a 15% constraint on the UV background, when 50 hours for each of the 20 used sources are scheduled. The blue dashed line is the 68% contour under the same conditions but with the optimized time distribution.
Table 2 :
2Error for the cosmological densities observing for a total of 1000 hours the considered 20 AGNs. In each column the distribution of these 1000 hours is done following different criteria. The first column is a distribution of 50 hours each source. Second and third are times optimized to minimize the uncertainty on Ω m and Ω λ . The last column optimizes the area of the 68% contour in the Ω m − Ω λ plane.Source Name
z
Tarea(hour) TUV (hour)
W Comae , 3EG J1222+2841 0.102
60
278
3C 279 , 3EG J1255-0549
0.538
78
21
3EG J0852-1216
0.566
7
-
CTA026 , 3EG J0340-0201
0.852
167
3
3C454.3 , 3EG J2254+1601
0.859
14
7
3EG J0450+1105
1.207
-
13
3EG J1323+2200
1.400
10
278
3EG J1635+3813
1.814
351
351
1ES J1426+428
0.129
312
49
Table 3 :
3Time scheduled for each source. First column optimizes the area of the 68% contour and the second one the determination of the scale factor for the UV background.
AcknowledgmentsWe are indebted to R.Miquel for his advice with the use the Fisher Information Matrix and for providing us with M.Spelucci's DONLP2 code. We want to thank our colleagues of the MAGIC collaboration for their comments and support.
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| []
|
[
"Textural Approach for Mass Abnormality Segmentation in Mammographic Images",
"Textural Approach for Mass Abnormality Segmentation in Mammographic Images"
]
| [
"Khamsa Djaroudib \nFaculty of Science\nComputer Science Department\nLaStic Laboratory\nBatna University\n) Algeria\n",
"Abdelmalik Taleb Ahmed \nLAMIH Laboratory\nValenciennes University\nUVHC\nFrance\n",
"Abdelmadjid Zidani \nFaculty of Science\nComputer Science Department\nLaStic Laboratory\nBatna University\n) Algeria\n"
]
| [
"Faculty of Science\nComputer Science Department\nLaStic Laboratory\nBatna University\n) Algeria",
"LAMIH Laboratory\nValenciennes University\nUVHC\nFrance",
"Faculty of Science\nComputer Science Department\nLaStic Laboratory\nBatna University\n) Algeria"
]
| []
| Mass abnormality segmentation is a vital step for the medical diagnostic process and is attracting more and more the interest of many research groups. Currently, most of the works achieved in this area have used the Gray Level Co-occurrence Matrix (GLCM) as texture features with a region-based approach. These features come in previous phase for segmentation stage or are using as inputs to classification stage. The work discussed in this paper attempts to experiment the GLCM method under a contour-based approach. Besides, we experiment the proposed approach on various tissues densities to bring more significant results. At this end, we explored some challenging breast images from BIRADS medical Data Base. Our first experimentations showed promising results with regard to the edges mass segmentation methods. This paper discusses first the main works achieved in this area. Sections 2 and 3 present materials and our methodology. The main results are showed and evaluated before concluding our paper. | null | [
"https://arxiv.org/pdf/1412.1506v1.pdf"
]
| 653,327 | 1412.1506 | 4061198377029d7b211ec568692951b645b84810 |
Textural Approach for Mass Abnormality Segmentation in Mammographic Images
Khamsa Djaroudib
Faculty of Science
Computer Science Department
LaStic Laboratory
Batna University
) Algeria
Abdelmalik Taleb Ahmed
LAMIH Laboratory
Valenciennes University
UVHC
France
Abdelmadjid Zidani
Faculty of Science
Computer Science Department
LaStic Laboratory
Batna University
) Algeria
Textural Approach for Mass Abnormality Segmentation in Mammographic Images
Textural approachmammographyMass segmentationcontourtissuesGray Level co-occurrence Matrix (GLCM)
Mass abnormality segmentation is a vital step for the medical diagnostic process and is attracting more and more the interest of many research groups. Currently, most of the works achieved in this area have used the Gray Level Co-occurrence Matrix (GLCM) as texture features with a region-based approach. These features come in previous phase for segmentation stage or are using as inputs to classification stage. The work discussed in this paper attempts to experiment the GLCM method under a contour-based approach. Besides, we experiment the proposed approach on various tissues densities to bring more significant results. At this end, we explored some challenging breast images from BIRADS medical Data Base. Our first experimentations showed promising results with regard to the edges mass segmentation methods. This paper discusses first the main works achieved in this area. Sections 2 and 3 present materials and our methodology. The main results are showed and evaluated before concluding our paper.
Introduction
Diagnosis and cure of breast cancer depend strongly of on early detection and treatment of abnormalities (mass or micro-calcification) of breast. And breast cancer is the first cause of death by cancer at the women. Unfortunately, due to the large variability of size, shape and margin, and its confusion with the mammary tissue, mass abnormality detection and/or segmentation is still a very difficult task for the researchers more than micro-calcification abnormalities detection. Computer Aided Diagnosis (CAD) being an effective tool for radiologist [1][2] [3], for giving a second and more reliable opinion in diagnosis. The detection and/or segmentation is the first and key stage in the complete process of CAD [4] [5].
Masses are characterized by their location, size, shape and margin [6] [7] and the large variation in size and shape in which masse can appear, make mass segmentation a challenging task for researchers. In additional, at the most of cases, mammograms exhibit poor image contrast tissue density (fatty, dense or glandular), then tissue can overlap with breast tumor region [8] as the mass abnormality [9]. According to these problems, many mass segmentation and/or detection methods are developed. We can see review and recent advance of them in [10] [4], [11], [12]. For example, pixel based methods [13] [14] [15], such as region growing and its extensions; region based methods [16] [17], e.g., filter based methods; and simple edges based methods [18], e.g., the gradient filters, are employed widely in the early stage for mass segmentation. Though these types of methods are easily to implement, it is still difficult to acquire satisfied segmentation results for masses of ambiguous boundaries. This is because simple feature cannot handle the complex density distributions and topologies of the masses and normal breast tissue. To find more accurate boundaries of masses, some researchers use active contour methods [19][20] [21], the efficiency of depends for adjusting parameters. Many methods cited above use texture information, because textures features are more rich information in segmentation process [22] [23] and specially in medical images [24], these have been proven to be useful in differentiating mass and normal breasts tissues. The authors in [26] show that the area of a tumor exhibit typically low texture compared to normal parenchyma, and the authors in [8] concluded that the texture features demonstrate more prominent differences between tumor and normal tissues than the intensity feature. In this idea, most methods include textures features use GLCM in segmentation or classification stage of CAD [27] and most of them use GLCM in region approaches, in order to extract texture features in previous stage for mass segmentation or classification stages, we cite some examples in [28] [29][30] [31]. However, there is no significant work which used these matrices in an edges approach. In this paper, we contribute and propose a mass segmentation method by edges detection approach, based on GLCM in order to extract textures images representing textures parameters. Our idea is based on the fact that variance or contrast parameter can detect the spatial change between mass and non mass tissue in region border. Then texture descriptor as the contrast extract from GLCM is compute in each pixel in ROI (Region of Interest) image give an important information to detect edges mass contours.
Our approach split in two stages. At first, we applied smoothing (denoising) and enhancing method to enhance breast image [32]. Respectively, an anisotropic filter diffusion SRAD (Speckle Reducing Anisotropic Diffusion) [33] and Contrast-limited Adaptive Histogram Equalization (CLAHE) are used. Second, for each pixel in a ROI, a contrast descriptor is computed from the cooccurrence matrix of the pixels, and the contrast image is obtained. Mass contour is identified. We applied the proposed algorithm to some challenging breast images in BIRADS database including poor contrast tissue density (fatty, dense or granular) and the segmented mass done by our algorithm is compared to segmentation carried by an expert radiologist by measuring Dice coefficient, Fmeasure and area under the curve (Az).
Materials and Data description
Our method was applied on the mini-MIAS dataset (http://peipa.essex.ac.uk/info/mias.html). It is available online freely for scientific purposes. The films were digitized and the corresponding images were annotated according to their breast density by expert radiologists, using three distinct classes: Fatty (F), Fatty-Glandular (G) and Dense-Glandular (D). Any abnormalities were also detected and described, including calcifications, welldefined, spiculated or ill-defined masses, architectural distortion or asymmetry. Each pair of images in the database is annotated as Symmetric or Asymmetric. The severity of each abnormality is also provided, i.e., benignancy or malignancy.
Methodology: GLCM for edges mass detection and segmentation
The main steps of proposed methodology are summarized in Fig. 1: Fig. 1 Steps for the mass segmentation methodology
Enhancement Images
The performance of methods based on texture information is highly dependent on the pre-processing (enhancement) of the input image [26], so many researchers focus in this stage of CAD. For our approach, this stage is our key to have the best results for the mass segmentation stage. Most mammogram images have low intensity contrast, then we applied smoothing (denoising) and enhancing method to enhance breast image [32]. We suggested applying respectively, the anisotropic filter diffusion SRAD (Speckle Reducing Anisotropic Diffusion) [33] and Contrast-limited Adaptive Histogram Equalization (CLAHE) for enhancing image.
Instead of most studies, in our approach and in the aim to perform texture information, denoising and enhancing steps are applied in whole breast image and then, we extract suspicious ROI image. So, our SRAD algorithm can take speckle for every image independently of another one which makes this approach is more efficiency for image speckle reducing. Fig.2, show an example for input image and enhancing image with delimited ROI, and then zoom of ROI extraction image.
Images in
We used the YU scripts for SRAD [33] and results of this step are shown at Fig.2. In this figure, the image of enhancement show clearly more regions in the breast image. The clear regions are even clearer, which can correspond to a region of the masses tissue, and the dark regions are darker, what can correspond to the regions of the normal tissue (without mass). Fig.2, we showed an image mdb004 which represents the most difficult case for the detection of the mass in the clear normal tissue, which is the dense tissue.
Besides in
For other cases, the images are even more contrasted to improve the next stage of our methodology which is the computing of the images of texture.
Edges mass detection by computing GLCM
In ROI image, we compute GLCM according to three important parameters: direction (angle), a neighbourhood size and texture descriptor of Harralick [34].
Compute direction:
Compute one angle 0°, 45°, 90°, 135° do not give closed outlines, then we compute all directions and calculate their sum, see Fig.3. Fig. 3, is an example of Brodatz image. We show images of texture which is contrast descriptor of Harralick [34], in direction 0°, 45°, 90°, 135° and image representing the sum of these four images. In the image sum, we see clearly more closed outlines. Fig.4, is our mammographic ROI image of breast mdb004. We confirm so the remark on the Brodatz image of Fig.3.
Compute texture descriptor:
Instead of taking the most known four descriptors extracted from GLCM (1.entropy, 2.contrast, 3.second angular moment and 4.inverse differential moment), we take only the contrast descriptor, which measures the heterogeneity of an image and detect spatial variations of grey level intensity in image. Besides, it can summarize all the information of texture we needs.
To extract texture images, and according to these previous parameters (direction, neighborhood size), we compute contrast descriptor of Harralick [34]. Each pixel of ROI image is replaced by this descriptor.
In this work, we use "MatlabR2008b" formulation of contrast descriptor, Eq. (1):
2 , ( , ) ij contrast i j p i j -(1)
This equation returns a measure of the intensity contrast between a pixel p and its neighbourhood in the size of 7x7. Then our algorithm computes this descriptor over the whole ROI image.
Experimental results
Edges mass detection in different densities of tissue
We applied the proposed approach to BIRADS database of breast images, on Mini-MIAS dataset. Tests are done in images with different densities of tissue, fatty, dense or glandular. For each image, contrast descriptor of Harralick is computed. We obtain the contrast texture images where the mass is identified by its borders.
In Fig.6, the examples of three images mdb004, mdb005, mdb019 which represent respectively: in first an breast image with a dense tissue, regions are of clear white color on the images of mammography; in second an image with a fatty tissue, regions are of dark grey color on the images; and finally an image with a glandular tissue, regions are of mixed color, clear white time and grey dark on the mammographic images. This information is given according to the annotations of the MIAS database.
Quantitative evaluation with Dice Coefficient:
The segmented mass is compared to segmentation carried by an expert radiologist by measuring Dice coefficient.
Our approach was applied and tested in the challenging images of the mini-MIAS dataset. We show here the most representative and speaking cases. We quote, the cases where the tissue is dense (e.g. Fig.7), the cases where the tissue is fatty (e.g. Fig.8) and the cases where the tissue is glandular (e.g. Fig.10). Fig. 7 shows the case where tissue surrounding region of mass is dense. A good Percentage of resemblance with expert radiologist segmentation is compute by Dice Coefficient 93.39%. This will certainly help the expert to interpret better the shape of the mass such as detected. Fig. 8 shows the case where tissue surrounding region of mass is Fatty. Here, other borders inside the region of mass delimited by the expert are detected by our method. If we follow the expert, the coefficient of resemblance will be very good 97.74%, otherwise it will not be satisfactory 69.35% in Fig. 9. Fig. 9 shows the case where we take the mask detected inside mass region, with Dice coeff=69.35%. Fig. 9 From left to right: ROI image, ROI image with borders (whites) of the mass region by the expert radiologist, image ROI of the texture with mass detection, image mass segmentation by applied mask, Dice Coeff. = 69.35%. Fig.10 and Fig.11 show the case where tissue surrounding region of mass is Glandular. The same comment as Fig 8: other borders inside the region of mass delimited by the expert are detected by our method, here the rate of resemblance with the demarcations of the expert is 86.18%, see Fig.11. But if we take the borders following the borders delimited by expert, the rate of resemblance remains good 96.67%.
Quantitative evaluation with F-measure:
We compute another evaluation in order to compare with other methods of detection and/or segmentation of mass abnormality, by area under the curve Az.
We quantify TP, FP, TN and FN as: Then, we compared with the works of Arnau Oliver in [4], who summarized recently all the methods of mass detection and/or segmentation and who gives the obtained better results, applied also on MIAS Database. These results calculated by area under de curve Az and is between Az=0.751 and Az=0780. For our method, area under the curve Az=0.81.
Discussion and Conclusion
The stage of the detection and/or the segmentation of cancerous anomalies in the mammographic image is the key step to determine performances of CAD systems. However, the low contrast of the mammographic images and the breast tissues complexity (as well as visually and quantitatively), made that until now, most hard task is really the discrimination between the mammary tissue and the abnormality. The more the tissue is white in mammographic image (dense), the more the confusion increases.
In this study, we contributed to discriminate between normal and mass abnormality tissues by using a texture descriptor (contrast descriptor) given by the GLCM with contour-based approach, while the majority of the similar works used the texture features in a region-based approach.
We also contributed to clear up the cases for three types of tissues: dense, fatty and glandular, which are most often present in the breast image.
Our results given by the Dice coefficient, F-Measure and area under the curve (Az) are good by comparing to another recent similar works and they are promising for future researches.
Detecting and/or segmenting mass edges may help radiologist experts to find size, shape and margin of mass, which are very important for decision process that leads to classify a mass as benign or malign cancer. For this, the proposed approach is especially easy and fast in terms of response time for the radiologists. The contrast descriptor allows the mass margins detecting and so her shape.
We also concluded that the enhancement stage may also be considered as a key stage in our approach. By using SRAD and CLAHE in a different way, or other enhancement methods, would give less successful and different results.
Finally, we can say that texture information is necessary for removing the ambiguity between the regions of the anomalies and the healthy regions. However, till now, the texture is hard to express under a mathematical formalism and this area of research is still open to contributions.
Fig. 2
2Breast image mdb004 input (in left), mdb004 enhancement (center) show ROI (red color) and mass (green color), image zoom of ROI (in right)
Fig. 3
3From left to right : Brodatz D75 image, Image Contrast in 0°, Image Contrast in 45°, Image Contrast in 90°, Image Contrast in 135° and Image Contrast Sum (0°+ 45°+ 90°+135°).
Fig. 4
4Top: in left, mdb004 image input, in right, the image sum of four directions contrast image which shows closely contours. Bottom: Four images of contrast , from left to right, respectively in 0°, 45°, 90°, 135°3.2.2 Compute a neighbourhood size:For synthetic Brodatz images, we can see onFig.5that in mask 3x3, edges are more smooth than mask 7x7 and 9x9. The detected edges are more fuzzy if the neighbourhood size is big. But in reality, the choice of the neighbourhood size depends on textures of objects in image. For the images of mammography, neighbourhood in mask size of 7x7 give better smooth edges than mask size of 3x3 and finer outlines than mask size of 9x9.
Fig. 5
5From left to right: Brodatz D75 image, Image Contrast in mask 3x3, Image Contrast in mask 7x7 and Image Contrast in mask 9x9.
Fig
Fig. 6 From left to right: mdb004 breast, mdb005 breast, mdb019 breast. Blue color: masses, red color: dense tissue, green: fatty tissue, yellow color: Glandular tissue
Fig. 7
7In dense tissue: From left to right: ROI image, ROI image with borders (whites) of the mass region by the expert radiologist, image ROI of the texture with mass detection, image mass segmentation by applied mask, Dice Coeff = 93.39%
Fig. 8
8In Fatty tissue: From left to right: ROI image, ROI image with borders (whites) of the mass region by the expert radiologist, image ROI of the texture with mass detection, image mass segmentation by applied mask, Dice Coeff. = 97.74%
Fig. 10 In
10Glandular tissue: From left to right: ROI image, ROI image with borders (whites) of the mass region by the expert radiologist, image ROI of the texture with mass detection, image mass segmentation by applied mask, Dice Coeff. = 96.67%.
Fig. 11 From
11left to right: ROI image, ROI image with borders (whites) of the mass region by the expert radiologist, image ROI of the texture with mass detection, image mass segmentation by applied mask, DiceCoeff. = 86.18%.
. 6 From left to right: mdb004 breast, mdb005 breast, mdb019 breast. Blue color: masses, red color: dense tissue, green: fatty tissue,yellow color: Glandular tissue
4.2 Quantitative evaluation of mass edges detection
For evaluating edge detection, we select identified mass
contours, according to expert image and we compute Dice
Coefficient at first. Second we compute Precision, Recall
and F-measure in order to calculate the area under the
curve (Az). Then a comparison with an example of the
recent results in the literature is given. It is results obtained
for various methods of detection and/or segmentation of
masses.
The area under the curve Az was computed with Sensitivity Eq. (5) and Specificity Eq. (6).Tab. 1 Quantitative evaluation of the segmentation of breast mass, in training images mdb004, mdb005 and mdb019.TP: True Positive, means region segmented as
mass that proved to be mass.
FP: False Positive, means region segmented as
mass that proved to be not mass.
FN: False Negative, means region segmented as
not mass that proved to be mass.
TN: True Negative, means region segmented as
not mass that proved to be not mass.
TPR: True Positive Rate, Eq. (2)
FPR: False Positive Rate, Eq. (3)
TP
TPR
TP FN
(2)
FP
FPR
FP TN
(3)
Tab. 1 shows the Precision Eq. (4), Recall Eq. (5) and F-
measure Eq. (7) for the three cases of images cited above.
Pr
TP
ecision
TP FP
(4)
Re call Sensitivity TPR
(5)
1
Specificity
FPR
(6)
2
1 / Pr
1 / Re
F measure
ecision
call
(7)
Image
Precision
Recall
F-
measure
mdb004, mass in
dense tissue
0.9978
0.9933
0.9956
mdb005, mass in
fatty tissue
0.9983
0.9781
0.9881
mdb0019, mass in
Glandular tissue
0.9997
0.9375
0.9676
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| []
|
[
"CARLEMAN ESTIMATES FOR GEODESIC X-RAY TRANSFORMS",
"CARLEMAN ESTIMATES FOR GEODESIC X-RAY TRANSFORMS"
]
| [
"Gabriel P Paternain ",
"Mikko Salo "
]
| []
| []
| In this article we introduce an approach for studying the geodesic X-ray transform and related geometric inverse problems by using Carleman estimates. The main result states that on compact negatively curved manifolds (resp. nonpositively curved simple or Anosov manifolds), the geodesic vector field satisfies a Carleman estimate with logarithmic weights (resp. linear weights) on the frequency side. As a particular consequence, on negatively curved simple manifolds the geodesic Xray transform with attenuation given by a general connection and Higgs field is invertible modulo natural obstructions. The proof is based on showing that the Pestov energy identity for the geodesic vector field completely localizes in frequency. Our approach works in all dimensions ≥ 2, on negatively curved manifolds with or without boundary, and for tensor fields of any order. | null | [
"https://arxiv.org/pdf/1805.02163v2.pdf"
]
| 119,628,472 | 1805.02163 | d9c6eb482acb7186f3adae3bc72870e7e116cf3e |
CARLEMAN ESTIMATES FOR GEODESIC X-RAY TRANSFORMS
25 Nov 2021
Gabriel P Paternain
Mikko Salo
CARLEMAN ESTIMATES FOR GEODESIC X-RAY TRANSFORMS
25 Nov 2021
In this article we introduce an approach for studying the geodesic X-ray transform and related geometric inverse problems by using Carleman estimates. The main result states that on compact negatively curved manifolds (resp. nonpositively curved simple or Anosov manifolds), the geodesic vector field satisfies a Carleman estimate with logarithmic weights (resp. linear weights) on the frequency side. As a particular consequence, on negatively curved simple manifolds the geodesic Xray transform with attenuation given by a general connection and Higgs field is invertible modulo natural obstructions. The proof is based on showing that the Pestov energy identity for the geodesic vector field completely localizes in frequency. Our approach works in all dimensions ≥ 2, on negatively curved manifolds with or without boundary, and for tensor fields of any order.
Introduction
Motivation. The geodesic X-ray transform is a central object in geometric inverse problems. In Euclidean space it reduces to the standard X-ray transform, which encodes the integrals of a function over straight lines and provides the mathematical model for imaging methods such as X-ray CT and PET [Na01]. If the underlying medium is not Euclidean, one needs to consider more general curve families. On a Riemannian manifold the geodesic curves provide a natural candidate, and the geodesic X-ray transform encodes the integrals of a function over geodesics. This transform plays an important role (often via linearization or pseudo-linearization arguments) in inverse problems such as • boundary and lens rigidity on manifolds with boundary [PU05,Gu17,SUV21]; • spectral rigidity on closed manifolds with Anosov geodesic flow [PSU14b]; • inverse boundary value problems for hyperbolic equations [SY18];
• inverse boundary value problems for elliptic equations (Calderón problem) [DKSU09,DKLS16].
Several approaches have been introduced for studying the X-ray transform:
• direct methods such as Fourier analysis in symmetric geometries [He99];
• microlocal methods based on interpreting the transform or its normal operator as a Fourier integral or pseudodifferential operator [SU05,SU09,UV16]; • reductions to PDE and energy methods [Sh94,PSU13,GPSU16].
In particular, the works [PSU13, GPSU16,UV16] involve various L 2 estimates with exponential weights. This suggests that the Carleman estimate methodology, which relies on exponentially weighted L 2 estimates and provides a very powerful general approach to uniqueness results for linear PDE [Hö85, Chapter XXVIII], could have consequences for the geodesic X-ray transform as well. However, as explained in Section 3, the PDE related to the X-ray transform are nonstandard and thus the existing theory of Carleman estimates cannot be directly applied.
This paper presents the first steps toward a Carleman estimate approach for the geodesic X-ray transform. We prove two Carleman estimates for the geodesic vector field, one with logarithmic and another with linear Carleman weights, that have the right form in order to be applied to the geodesic X-ray transform. As a consequence we obtain invertibility results for certain weighted X-ray transforms that correspond to large lower order perturbations in the related PDE, and uniqueness results in related problems such as scattering rigidity and transparent connections. Our approach works in all dimensions ≥ 2, on manifolds with or without boundary, and for tensor fields of any order. However, the main Carleman estimate requires that the manifold has negative curvature, and the Carleman weights are purely on the frequency side (one can think of them as pseudodifferential weights) and thus the estimates do not currently lead to local results for the X-ray transform on the space side as in [UV16].
Main estimates. Let (M, g) be a compact oriented Riemannian manifold with or without boundary, and let X be the geodesic vector field regarded as a first order differential operator X : C ∞ (SM) → C ∞ (SM) acting on functions on the unit sphere bundle SM. Our first result is a new energy estimate for X when g is negatively curved. The new estimate involves polynomial weights on the Fourier side and it can be regarded as a Carleman estimate for the transport equation Xu = f (see below for analogies with elliptic operators). To describe it we recall some basic harmonic analysis on SM. By considering the vertical Laplacian ∆ on each fibre S x M of SM we have a natural L 2 -decomposition L 2 (SM) = ⊕ m≥0 H m (SM) into vertical spherical harmonics. We set Ω m := H m (SM) ∩ C ∞ (SM). Then a function u belongs to Ω m if and only if −∆u = m(m+d−2)u where d = dim(M) (for details see [GK80b,DS11]).
If u ∈ L 2 (SM), this decomposition will be written as
u = ∞ m=0 u m , u m ∈ H m (SM).
We say that u has finite degree if the above sum is finite.
Here is our first main result (the norms are L 2 (SM) norms). We have a similar estimate for compact manifolds with nonpositive curvature, provided that they are simple (simply connected with strictly convex boundary and no conjugate points) or Anosov (no boundary, and the geodesic flow satisfies the Anosov property). Both the simple and Anosov conditions can be seen as slightly strengthened forms of the condition that no geodesic has conjugate points (see [PSU14b]). However, if the curvature is only nonpositive, the logarithmic weights above need to be replaced by stronger linear weights. We remark that both logarithmic and linear weights have been prominent in the theory of Carleman estimates, see e.g. [JK85,KRS87].
Theorem 1.2. Let (M, g) be a simple/Anosov Riemannian manifold having nonpositive sectional curvature. There exist m 0 , τ 0 , κ > 0 such that for any τ ≥ τ 0 , for any m ≥ m 0 and for any u ∈ C ∞ (SM) of finite degree (with u| ∂(SM ) = 0 in the boundary case) one has ∞ ℓ=m e 2τ φ ℓ u ℓ 2 ≤ 24 κe 2τ ∞ ℓ=m+1 e 2τ φ ℓ (Xu) ℓ 2 where φ ℓ = ℓ.
Both theorems above are based on weighted frequency localized versions of the Pestov energy identity, which has been the main tool in energy methods for X-ray transforms (see e.g. [Sh94,PSU13,PSU15]). The Pestov identity has been used in many forms. A powerful recent variant is the inequality proved in [PSU15],
(1.1) ∇ SM u ≤ C v ∇Xu ,
which is valid on any compact simple/Anosov manifold for any u ∈ C ∞ (SM) (with u| ∂(SM ) = 0 in the boundary case). Here v ∇ is the vertical gradient (see Section 2). So far the Pestov identity has been expressed in terms of L 2 norms, and it has not been known if it is possible to shift the related estimates to different Sobolev scales.
However, Theorem 1.1 is actually a shifted version of the Pestov identity with respect to the mixed norms
u L 2 x H s v = ∞ ℓ=0 ℓ 2s u ℓ 2 1/2
where s ∈ R and ℓ = (1 + ℓ 2 ) 1/2 . Similar mixed norms for ∇ SM u and v ∇Xu are defined in Section 6. Then, the inequality (1.1) can be expressed as
∇ SM u L 2 x H 0 v ≤ C v ∇Xu L 2 x H 0 v .
The next result is a shifted version of this inequality.
Theorem 1.3. Let (M, g) be a compact Riemannian manifold with sectional curvature ≤ −κ where κ > 0. For any s > −1/2 there is C = C d,s,κ > 0 such that ∇ SM u L 2 x H s v ≤ C v ∇Xu L 2 x H s v for any u ∈ C ∞ (SM) (with u| ∂(SM ) = 0 in the boundary case).
Applications.
Uniqueness results for the transport equation. The significance of Theorem 1.1 is best illustrated by considering the transport equation
Xu = −f in SM , u| ∂(SM ) = 0,
where f has finite degree m (i.e. f ℓ = 0 for ℓ ≥ m+1). Then Theorem 1.1 implies that any solution u ∈ C ∞ (SM) must satisfy u ℓ = 0 for ℓ ≥ m (u has finite degree m − 1) at least when m ≥ 1. This fact provides a full solution to the tensor tomography problem in negative curvature (see [PS88,CS98] and [PSU14a] for a recent survey). The case m = 2 is at the heart of the proof that closed negatively curved manifolds are spectrally rigid [CS98,GK80a]. Given an isospectral deformation g ε of the metric g it was shown in [GK80a] that the symmetric 2-tensor f := ∂gε ∂ε | ε=0 must have the property that it integrates to zero along every closed geodesic of g. Since the geodesic flow of g is Anosov an application of the Livsic theorem [dMM86] shows that there is a smooth function u such that Xu =f ∈ Ω 0 ⊕ Ω 2 , wheref is the natural function induced on SM by the symmetric 2-tensor f . The Carleman estimate in Theorem 1.1 implies that u = u 1 and this uniquely determines a vector field V on M. Doing this for every ε produces a parameter dependent vector field V ε whose flow makes the deformation g ε trivial, thus showing spectral rigidity.
However, the estimate in Theorem 1.1 is considerably more powerful since we can in principle adjust the parameter τ at will as long as we know that u ∈ C ∞ (SM).
Here is the typical application we have in mind. Suppose that Φ ∈ C ∞ (M, C n×n ) is a smooth n × n matrix-valued potential in M (we call Φ a general Higgs field ). Consider the attenuated transport equation
(1.2) (X + Φ)u = −f 0 ∈ Ω 0 , u| ∂(SM ) = 0 where u ∈ C ∞ (SM, C n ).
The objective is to show that f 0 = 0; this is precisely the uniqueness problem that arises when trying to prove injectivity of the attenuated geodesic X-ray transform where the attenuation is given by the matrix Φ (see below). Since (Φu) ℓ = Φu ℓ , we see using (1.2) that
(Xu) ℓ ≤ C u ℓ , ℓ ≥ 1, where C = Φ L ∞ (M )
. If we input this information into Theorem 1.1 and we take τ sufficiently large (depending on Φ L ∞ (M ) ) we deduce that u = u 0 ∈ Ω 0 . Since Xu 0 ∈ Ω 1 , using (1.2) we obtain f 0 = 0 as desired. However, the Carleman estimate can also deal with other matrix attenuations like connections. For our purposes these are simply n × n matrices A of complex valued 1-forms, i.e. A ∈ C ∞ (M, (Λ 1 M) n×n ). We call such geometric objects general connections on M, and they can be naturally considered as smooth functions A : SM → C n×n such that A ∈ Ω 1 . More generally, the Carleman estimate even makes it possible to include nonlinear attenuation terms in the transport equation.
Here is a general result in this direction.
Theorem 1.4. Let (M, g) be a compact Riemannian manifold with sectional curvature ≤ −κ where κ > 0. Let ℓ 0 ≥ 2, and suppose that A : C ∞ (SM, C n ) → C ∞ (SM, C n ) is a map such that (A(u)) ℓ ≤ C( u ℓ−1 + u ℓ + u ℓ+1 ), ℓ ≥ ℓ 0 .
If f ∈ C ∞ (SM, C n ) has finite degree and if u ∈ C ∞ (SM, C n ) satisfies
Xu + A(u) = −f in SM, u| ∂(SM ) = 0,
then u has finite degree.
The previous result can be interpreted as a fundamental uniqueness, or vanishing, theorem for the attenuated transport equation. In [GPSU16], such a result was proved on negatively curved manifolds in the special (linear) case
A(u) = (A + Φ)u
where A and Φ are a skew-Hermitian connection and Higgs field (when A is skew-Hermitian, the connection is also referred to as unitary because the underlying structure group is the unitary group). The skew-Hermitian condition implies that the transport operator X + A + Φ is skew-symmetric, which led to improved energy estimates in [GPSU16]. In our case, where A and Φ can be general matrices, there is a loss of symmetry and large error terms appear in the energy estimates. The Carleman estimate in Theorem 1.1 is required to deal with these large errors (see Section 3 for a more detailed discussion).
In the more general case where (M, g) is a compact simple/Anosov manifold with nonpositive sectional curvature, one could use Theorem 1.2 to prove an analogue of Theorem 1.4 with two additional restrictions: the map A should satisfy (A(u)) ℓ ≤ C u ℓ , ℓ ≥ ℓ 0 , and the solution u ∈ C ∞ (SM, C n ) should additionally satisfy
(1.3) lim ℓ→∞ e τ ℓ X − u ℓ = 0,
where X − is the part of X that maps Ω m to Ω m−1 (see Section 2) and τ > 0 is sufficiently large (depending on A). The condition for A is satisfied if A is a general Higgs field, but (1.3) should be viewed as an additional regularity condition (it states that the Fourier coefficients of X − u decay exponentially, which corresponds to realanalyticity of X − u in the v variable). Thus, the Carleman estimate in Theorem 1.2 shows that the injectivity result for a general Higgs field reduces to the regularity result (1.3). However, it is not at all clear how to establish (1.3). For simple surfaces, it is possible to bypass these difficulties using a loop group factorization result and reducing the problem to the skew-hermitian case, see [PS20]. Theorem 1.4 easily implies uniqueness results for X-ray transforms, whenever one has a regularity result showing that the vanishing of the X-ray transform yields a C ∞ solution u of a transport equation. Let us next discuss this in more detail.
The attenuated X-ray transform. Let (M, g) be a compact Riemannian manifold with smooth boundary. We assume that (M, g) is nontrapping, i.e. for any (
x, v) ∈ SM the geodesic γ x,v (t) through (x, v) exits M at some finite time τ (x, v) ≥ 0.
Given a ∈ C ∞ (M) (scalar attenuation), the attenuated geodesic X-ray transform of f is the map
I a f : ∂ + (SM) → C, I a f (x, v) = τ (x,v) 0 f (γ x,v (t)) e t 0 a(γx,v (s)) ds dt, where ∂ ± (SM) = {(x, v) ∈ SM ;
x ∈ ∂M, ± v, ν ≤ 0} are the inward and outward pointing boundaries of SM (with ν being the outer unit normal to ∂M). It is easy to see that I a f can be equivalently defined as
I a f = u| ∂ + (SM )
where u is the solution of
(X + a)u = −f in SM, u| ∂ − (SM ) = 0.
More generally, we can consider the vector-valued case where f ∈ C ∞ (SM, C n ), and the attenuation is given by a general connection A ∈ C ∞ (M, (Λ 1 M) n×n ) and a general Higgs field Φ ∈ C ∞ (M, C n×n ). The matrix of 1-forms A can be identified with the connection ∇ = d + A on the trivial bundle E = M × C n . In this case, the attenuated X-ray transform of f is defined as
I A+Φ f = u| ∂ + (SM ) ,
where u is the solution of
(X + A + Φ)u = −f in SM , u| ∂ − (SM ) = 0.
The basic inverse problem for 0-tensors (resp. m-tensors) is the following: given f ∈ C ∞ (M, C n ) (resp. f ∈ C ∞ (SM, C n ) with degree m) such that I A,Φ f = 0, is it true that f ≡ 0 (resp. f = −(X + A + Φ)u for some u ∈ C ∞ (SM, C n ) of degree m − 1 with u| ∂(SM ) = 0)? The following theorem gives an affirmative answer on nontrapping negatively curved manifolds.
Theorem 1.5. Let (M, g) be a compact Riemannian nontrapping negatively curved manifold with C ∞ boundary, let A be a general connection, and let Φ be a general Higgs field in M. Let f ∈ C ∞ (SM, C n ), and assume either that M has strictly convex boundary, or that f is supported in the interior of SM. If f has degree m ≥ 0 and if the attenuated X-ray transform of f vanishes (i.e. I A+Φ f = 0), then
f = −(X + A + Φ)u for some u ∈ C ∞ (SM, C n ) with degree m − 1 such that u| ∂(SM ) = 0.
This kind of result was proved for skew-Hermitian A and Φ in [GPSU16] when M is negatively curved with strictly convex boundary (in this case there is a stronger regularity result based on the work [Gu17], showing that one can drop the nontrapping assumption). Also, for strictly convex manifolds with dim(M) ≥ 3 a stronger result was proved in [PSUZ16], based on the method of [UV16]. Genericity results for simple manifolds are established in [Z17] and reconstruction formulas for simple surfaces (up to a Fredholm error) are given in [MP18]. The result for nonconvex boundaries is similar to [Da06,GMT17] which consider the unattenuated case. Theorem 1.5 is new if the boundary is nonconvex.
Scattering rigidity and transparent pairs. Finally, we briefly discuss results for nonlinear geometric inverse problems that can be reduced to Theorems 1.4 and 1.5 via pseudo-linearization arguments. Given a general connection A and a Higgs field Φ on the trivial bundle M × C n , the scattering data for (A, Φ) is the map
C A,Φ : ∂ − (SM) → GL(n, C), C A,Φ = U| ∂ − (SM ) ,
where U : SM → GL(n, C) is the unique matrix solution of the transport equation
(X + A + Φ)U = 0 in SM, U| ∂ + (SM ) = Id.
The data C A,Φ corresponds to boundary measurements for parallel transport if (M, g) is known (it measures how vectors are transformed when they are parallel transported with respect to (A, Φ) along geodesics between two boundary points). The map (A, Φ) → C A,Φ is sometimes called the non-abelian Radon transform, or the X-ray transform for a non-abelian connection and Higgs field (see [PSU12,No18]). The data C A,Φ has a natural gauge invariance under a change of basis, i.e.,
C Q −1 (X+A)Q,Q −1 ΦQ = C A,Φ if Q ∈ C ∞ (M, GL(n, C)) satisfies Q| ∂M = Id.
It follows that from the knowledge of C A,Φ one can only expect to recover A and Φ up to a gauge transformation via Q which satisfies Q| ∂M = Id. Our next result gives the following positive answer: Theorem 1.6. Let (M, g) be a compact Riemannian negatively curved nontrapping manifold, let A and B be two general connections, and let Φ and Ψ be two general Higgs fields. Assume either that M has strictly convex boundary, or that the pairs (A, Φ), (B, Ψ) are supported in the interior of M. Then C A,Φ = C B,Ψ implies that there exists Q ∈ C ∞ (M, GL(n, C)) such that Q| ∂M = Id and B = Q −1 dQ + Q −1 AQ, Ψ = Q −1 ΦQ.
Again, when M has strictly convex boundary, this was proved for skew-Hermitian connections and Higgs fields in [GPSU16] (also without the non-trapping assumption, based on the regularity theory of [Gu17]), and for a more general class of manifolds with dim(M) ≥ 3 in [PSUZ16]. In the Euclidean case the problem has been extensively studied, cf. [Es04,FU01,No02,No18].
An analogous result can be formulated on closed negatively curved manifolds without boundary. In this case, we show that if A is a general connection and Φ is a general Higgs field in M, and if the parallel transport for (A, Φ) along periodic geodesics is the identity map, then (A, Φ) is gauge equivalent to the trivial pair (0, 0) (i.e. there are no transparent pairs) unless there is an obstruction given by twisted conformal Killing tensors. We refer to Section 9 for more details.
Methods and analogies. The starting point of the Carleman estimate is a wellknown L 2 energy identity on SM called Pestov identity. If u ∈ C ∞ (SM), there is a splitting (induced by the Sasaki metric and the Levi-Civita connection) where R(x, v) : {v} ⊥ → {v} ⊥ is the operator determined by the Riemann curvature tensor R of (M, g) by R(x, v)w = R x (w, v)v. An important observation of the present paper is that (1.4) localizes in frequency. By this we mean that it is possible to recover (1.4) by summing in ℓ the identity (1.4) applied to functions u ∈ Ω ℓ . This was observed in [PSU15] for d = 2, but here we prove it for all dimensions. This fact paves the way for the proof of the Carleman estimates: by multiplying the frequency localized estimates by suitable weights, adding them up and using negative curvature to absorb errors we are able to derive the desired inequality. The localization in frequency has recently been applied in [PS21] to give a sharp stability estimate for tensor tomography in non-positive curvature.
∇ SM u = (Xu)X + h ∇u x-derivatives + v ∇u v-derivatives
There is an interesting analogy between the proof of Theorem 1.5, which is related to the X-ray transform, and the absence of embedded eigenvalues for Schrödinger operators. If H = −∆ + V − λ is a Schrödinger operator in R d where V is a shortrange potential (i.e. |V (x)| ≤ C(1 + |x|) −1−ε for some ε > 0 and for a.e. x ∈ R d ) and λ > 0 is a positive energy, the absence of embedded eigenvalues states that any solution u ∈ L 2 (R d ) of Hu = 0 must satisfy u ≡ 0 (see e.g. [Hö85, Section 14.7]). A standard proof of this fact proceeds in three steps:
1. Rapid decay: any solution u decays rapidly at infinity.
2.
Unique continuation at infinity: any solution u that decays rapidly at infinity must vanish outside a compact set. 3. Weak unique continuation: any solution u that vanishes outside a compact set must be identically zero.
Our proof of Theorem 1.5 follows the exact same pattern, even though the equation is very different. The rapid decay result is replaced by a regularity result from [PSU12], unique continuation at infinity is provided by Theorem 1.1, and weak unique continuation is replaced by the absence of twisted conformal Killing tensors proved in [GPSU16]. As a final remark, the uniqueness problems studied in this article are actually closer to long-range scattering or the Landis conjecture rather than short-range scattering. For instance, a Higgs field Φ can be interpreted as a "potential" V : u → Φu in the equation (X + Φ)u = −f . In terms of Fourier coefficients, one has
(V(u)) ℓ ≤ C u ℓ .
This means that the potential V is bounded but has no decay as ℓ → ∞. The analogue of a short-range condition would be (V(u)) ℓ ≤ ℓ −1−ε u ℓ , and an analogue of a long-range condition would be (V(u)) ℓ ≤ C ℓ −ε u ℓ for some ε > 0 (and a condition for derivatives, see [Hö85, Chapter XXX]). The fact that general connections and Higgs fields lead to potentials having no decay as ℓ → ∞, as in the Landis conjecture [KSW15], makes the uniqueness questions studied in this article challenging.
Organization of the paper. Section 2 contains geometric preliminaries on the unit sphere bundle and vertical spherical harmonics. Section 3 discusses in more detail certain useful analogies between the geodesic vector field X and the Laplace operator that have in part motivated this paper. Section 4 describes a general Pestov identity with connection and Section 5 contains the key result on frequency localization mentioned above. Section 6 establishes the Carleman estimate in Theorem 1.1, and contains a proof Theorem 1.3 which interprets the Carleman estimate as a version of the Pestov identity (1.4) that has been shifted to a different regularity scale.
Section 7 proves the Carleman estimate in nonpositive curvature, Theorem 1.2. Section 8 contains a regularity result for the transport equation in the case when the boundary is not strictly convex. Finally, Section 9 contains all the applications and the proofs of Theorems 1.4-1.6.
Acknowledgements. The authors would like to thank Colin Guillarmou for helpful remarks related to Theorem 1.3, and the referee for helpful comments. GPP was partially supported by EPSRC grant EP/R001898/1.
Geometric preliminaries
Unit sphere bundle. To begin, we need to recall certain notions related to the geometry of the unit sphere bundle. We follow the setup and notation of [PSU15]; for other approaches and background information see [GK80b,Sh94,Pa99,Kn02,DS11].
Let (M, g) be a d-dimensional compact Riemannian manifold with or without boundary, having unit sphere bundle π : SM → M, and let X be the geodesic vector field. We equip SM with the Sasaki metric. If V denotes the vertical subbundle given by V = Ker dπ, then there is an orthogonal splitting with respect to the Sasaki metric:
(2.1) T SM = RX ⊕ H ⊕ V.
The subbundle H is called the horizontal subbundle. Elements in H(x, v) and V(x, v) are canonically identified with elements in the codimension one subspace
{v} ⊥ ⊂ T x M by the isomorphisms dπ x,v : H(x, v) → {v} ⊥ , K x,v : V(x, v) → {v} ⊥ , here K (x,v)
is the connection map coming from Levi-Civita connection. We will use these identifications freely below. We shall denote by Z the set of smooth functions Z :
SM → T M such that Z(x, v) ∈ T x M and Z(x, v), v = 0 for all (x, v) ∈ SM.
Another way to describe the elements of Z is a follows. Consider the pull-back bundle π * T M over SM. Let N denote the subbundle of π * T M whose fiber over (x, v) is given by N (x,v) = {v} ⊥ (which we can also identify with T v S x M). Then Z coincides with the smooth sections of the bundle N. Notice that N carries a natural scalar product and thus an L 2 -inner product (using the Liouville measure on SM for integration).
Given a smooth function u ∈ C ∞ (SM) we can consider its gradient ∇u with respect to the Sasaki metric. Using the splitting above we may write uniquely in the decomposition (2.1)
∇u = ((Xu)X, h ∇u, v ∇u).
The derivatives h ∇u ∈ Z and v ∇u ∈ Z are called horizontal and vertical derivatives respectively. Note that this differs from the definitions in [Kn02,Sh94] since here all objects are defined on SM as opposed to T M.
Observe that X acts on Z as follows:
(2.2) XZ(x, v) := DZ(ϕ t (x, v)) dt | t=0
where D/dt is the covariant derivative with respect to Levi-Civita connection and ϕ t is the geodesic flow. With respect to the L 2 -product on N, the formal adjoints
of v ∇ : C ∞ (SM) → Z and h ∇ : C ∞ (SM) → Z are denoted by − v div and −
h div respectively. Note that since X leaves invariant the volume form of the Sasaki metric we have X * = −X for both actions of X on C ∞ (SM) and Z. In what follows, we will need to work with the complexified version of N with its natural inherited Hermitian product. This will be clear from the context and we shall employ the same letter N to denote the complexified bundle and also Z for its sections.
Let
R(x, v) : {v} ⊥ → {v} ⊥ be the operator determined by the Riemann curvature tensor by R(x, v)w = R x (w, v)v, and let d = dim(M).
Spherical harmonics decomposition. Recall the spherical harmonics decomposi-
tion with respect to the vertical Laplacian ∆ = v div v ∇ (cf. [PSU15, Section 3]): L 2 (SM, C n ) = ∞ m=0 H m (SM, C n ),
so that any f ∈ L 2 (SM, C n ) has the orthogonal decomposition
f = ∞ m=0 f m . We write Ω m = H m (SM, C n ) ∩ C ∞ (SM, C n ). Then −∆u = λ m u for u ∈ Ω m , where we set λ m := m(m + d − 2)
. Decomposition of X. The geodesic vector field behaves nicely with respect to the decomposition into fibrewise spherical harmonics: it maps Ω m into Ω m−1 ⊕ Ω m+1 [GK80b, Proposition 3.2]. Hence on Ω m we can write
X = X − + X +
where X − : Ω m → Ω m−1 and X + : Ω m → Ω m+1 . By [GK80b, Proposition 3.7] the operator X + is overdetermined elliptic (i.e. it has injective principal symbol). One can gain insight into the decomposition X = X − + X + as follows. Fix x ∈ M and consider local coordinates which are geodesic at x (i.e. all Christoffel symbols vanish at x). Then Xu(x, v) = v i ∂u ∂x i . We now use the following basic fact about spherical harmonics: the product of a spherical harmonic of degree m with a spherical harmonic of degree one decomposes as the sum of spherical harmonics of degree m−1 and m+1.
Useful analogies
Analogies with elliptic operators. It is instructive to compare our approach to X-ray transforms based on energy methods and related well known energy methods for elliptic operators in R n . Let Ω ⊂ R n be a bounded domain, and let P = −∆ + V be the Schrödinger operator in Ω where V ∈ L ∞ (Ω). We consider the uniqueness problem for solutions of the equation
(3.1) P u = 0 in Ω, u| ∂Ω = 0.
It is well known that under a positivity condition for the potential V , any solution u of (3.1) in C 2 (Ω) (say) must be zero. In fact, if we assume that V ≥ 0, this follows from the simple energy estimate where we integrate the equation P u = 0 against the test function u. This implies that, in terms of L 2 (Ω) inner products, Here we integrated by parts using that u| ∂Ω = 0, and used that V ≥ 0. Thus ∇u = 0, showing that u is constant, and the boundary condition u| ∂Ω = 0 implies that u ≡ 0.
Uniqueness for solutions of (3.1) still holds under the weaker condition V > −λ 1 , where λ 1 > 0 is the first Dirichlet eigenvalue of −∆ in Ω. Then the Poincaré inequality implies that ∇w 2 ≥ λ 1 w 2 whenever w| ∂Ω = 0. The same argument as above shows that
0 = (P u, u) = (−∆u, u) + (V u, u) = ∇u 2 + (V u, u) ≥ ((λ 1 + V )u, u). If λ 1 + V ≥ c a.e.
in Ω for some c > 0, it follows that u ≡ 0. (Combining this argument with the unique continuation principle, it would be enough to assume that
λ 1 + V | U > 0 in some set U of positive measure.)
If the potential V is very negative, uniqueness for solutions of (3.1) may fail. For example, if V = −λ 1 where λ 1 is the first Dirichlet eigenvalue, then the corresponding Dirichlet eigenfunction is a nontrivial solution of (3.1). However, uniqueness will be true if we assume more vanishing. For instance, any u ∈ C 2 (Ω) satisfying
(3.2) P u = 0 in Ω, u| ∂Ω = ∂ ν u| Ω = 0
must be identically zero. This follows from the unique continuation principle, which is typically established by using Carleman estimates (i.e. exponentially weighted L 2 estimates for P ). Carleman estimates themselves correspond to a version of energy methods (more precisely, positive commutator methods), which makes use of the gauge invariance of the problem: writing u = e ϕ w for some ϕ ∈ C 2 (Ω), the problem (3.2) is equivalent with
P ϕ w = 0 in Ω, w| ∂Ω = ∂ ν w| Ω = 0
where P ϕ = e −ϕ P e ϕ is the operator P conjugated with an exponential weight. For some choices of ϕ (for which the commutator [P * ϕ , P ϕ ] is positive), the operator P ϕ is "more positive" than P , and an energy estimate for P ϕ w = 0 (where one integrates against the test function P ϕ w) implies that w ≡ 0.
Let us now return from the case of the Schrödinger operator back to X-ray transforms. As explained in [PSU13,PSU14b], in this case we are considering the operator P = v ∇X on SM, and we wish to show (say) that any u ∈ C ∞ (SM) satisfying
P u = 0 in SM , u| ∂(SM ) = 0
must be identically zero. The basic energy identity for the operator P is (1.4), where the operator −X 2 − R formally corresponds to the operator −∆ + V in the above examples. In particular, −R (curvature) plays a similar role as V (potential). For instance, the condition K ≤ 0 where K denotes sectional curvature corresponds to V ≥ 0. Moreover, the simple/Anosov condition for (M, g) ensures that −X 2 − R > 0 which corresponds to V > −λ 1 (i.e. −∆ + V > 0). Let us formulate these analogies in the following table:
P = −∆ + V P = v ∇X potential V curvature −R V ≥ 0 K ≤ 0 V > −λ 1 simple/Anosov Carleman estimates ?
An analogue of the Carleman estimates approach for uniqueness in elliptic equations has so far been missing in the case of X-ray transforms. Theorem 1.1 represents the first progress in this direction.
Properties of P . Let us consider in more detail the operator P = v ∇X on SM that is fundamental in the energy method for geodesic X-ray transforms. The operator P is a second order differential operator, scalar if d = 2 and vector-valued if d ≥ 3, on the compact (2d − 1)-dimensional manifold SM with boundary. However, even in the case d = 2 where P = V X is the product of two vector fields, the equation P u = 0 does not seem to fall into any known class of PDEs for which there would be a uniqueness theory.
One major issue is that P is not of principal type. To see this, let d = 2 and write P = P 1 P 2 where P 1 = V and P 2 = X. Then P 1 and P 2 are smooth vector fields on SM. Following [PSU13, Section 2] (and writing x 3 = θ) we consider local coordinates (x 1 , x 2 , x 3 ) on SM so that g = e 2λ(x 1 ,x 2 ) (dx 2 1 + dx 2 2 ). In this notation, the principal symbols of P 1 and P 2 are
p 1 = ξ 3 , p 2 = e −λ (cos(x 3 )ξ 1 + sin(x 3 )ξ 2 + [−∂ x 1 λ sin(x 3 ) + ∂ x 2 λ cos(x 3 )]ξ 3 ).
Thus P has real principal symbol p = p 1 p 2 , and the characteristic set p −1 (0) is the union of p −1 1 (0) and p −1 2 (0). However, the intersection
p −1 1 (0) ∩ p −1 2 (0) = {(x, ξ) ∈ T * (SM) ; ξ 3 = 0, cos(x 3 )ξ 1 + sin(x 3 )ξ 2 = 0}
is nontrivial and in this set dp = p 1 dp 2 + p 2 dp 1 vanishes. This means that any null bicharacteristic curve through p −1 1 (0) ∩ p −1 2 (0) reduces to a point, and in particular P is not of principal type (see [Hö85, Chapter XXVI] for more on principal type operators).
The fact that P is not of principal type means that its properties may depend on lower order terms. Indeed, if d = 2 one can find first order operators W on SM so that P + W has nontrivial compactly supported solutions. To see this, note that if W = X ⊥ then P + W = XV (see [PSU13]), and any u ∈ C ∞ c (M int ) satisfies XV u = 0 in SM. Moreover, the counterexample in [Bo10] implies that one may even find such an operator W arising from a weighted X-ray transform: when (M, g) is the Euclidean unit disc, there is A ∈ C ∞ (SM) such that V (X + A)u = 0 for some nontrivial u ∈ C ∞ c (SM int ). These observations indicate that the structure of lower order terms is crucial for the uniqueness problem, and principal symbol computations will not be sufficient.
However, in spite of the above issues it is possible to obtain uniqueness results for P via energy methods. Based on the Pestov energy identity, in [PSU15] it was proved that if (M, g) is a compact simple/Anosov manifold, one has the inequality
u − (u) SM H 1 (SM ) ≤ C P u L 2 (SM ) valid for all u ∈ H 2 (SM) (with u| ∂(SM ) = 0 in the boundary case). Here (u) SM = 1 Vol(SM ) SM u. Such an inequality immediately implies that if P u = 0 in SM and u| ∂(SM ) = 0, then u ≡ 0.
Various generalizations of the geodesic X-ray transform lead to estimates of the form ∇u) in the Pestov identity is also of H 1 type, and spatial localization corresponds to commuting P with a spatial cutoff function which again introduces an H 1 error.
The above statements indicate that unitary connections and higher order tensors lead to H 1/2 errors in the energy estimate (3.3). This is a weaker norm than the H 1 norm appearing on the left hand side of (3.3), which suggests that such perturbations might be manageable (indeed, the high frequency components in the error term can be absorbed, which in particular implies that there is a finite dimensional kernel). However, general connections, curvature or spatial localization correspond to H 1 error terms which are as strong as the H 1 term on the left, suggesting that these perturbations may be challenging to handle by energy methods (it seems that so far only the method in [UV16] can really deal with such perturbations). In this article, we introduce energy methods that are able to deal with H 1 perturbations arising from general connections.
Pestov identity with general connection
In this section we will prove a version of the Pestov identity that involves a general connection, extending the version for unitary connections proved in [GPSU16]. We also give an equivalent version stated in terms of the X ± operators.
We emphasize that for the applications we will eventually only use the identity for A = 0. However, the general setup here highlights the fact that dealing with general connections results in a lack of symmetry in the energy estimates. This lack of symmetry indicates that general connections are indeed stronger perturbations than unitary connections, as explained in Section 3, and it forces one to use other methods such as the Carleman estimates in this article. For simplicity of presentation we will work on the trivial bundle M × C n , but straightforward modifications would lead to analogous results on general Hermitian bundles as in [GPSU16]. We will use the notation from Section 2.
Let (M, g) be a compact manifold with or without boundary, with d = dim(M). Let A be an n × n matrix of smooth complex 1-forms on M, or equivalently a smooth function A : SM → C n×n so that A(x, v) is linear in v. Then A defines a connection on the trivial bundle M × C n . We define the following operators on C ∞ (SM, C n ),
X A := X + A, h ∇ A := h ∇ + ( v ∇A)
.
Here A and ( v ∇A) act by multiplication. The horizontal divergence h div A is defined for Z ∈ Z n by h div A Z := h div Z + v ∇A, Z .
Finally, we define F A ∈ C ∞ (SM, N ⊗ C n×n ) by
F A := X( v ∇A) − h ∇A + [A, v ∇A].
The element F A acts on functions u ∈ C ∞ (SM, C n ) by matrix multiplication and thus it induces and operator of order zero. In the natural L 2 inner product, one has
(X A ) * = −X −A * , ( h ∇ A ) * = − h div −A * .
Note that if A is a unitary connection, meaning that A * = −A, then of course
(X A ) * = −X A , ( h ∇ A ) * = − h div A ,
and F A is the curvature operator in [GPSU16]. With the above conventions, one can check by direct computations that the basic commutator formulas given in [PSU15] for A = 0 and in [GPSU16] for unitary A remain true for a general connection A. Recall that R is the curvature operator defined before and acts on C ∞ (SM, N ⊗ C n ) as RId n×n .
Lemma 4.1. One has the following commutator identities on C ∞ (SM, C n ),
[X A , v ∇] = − h ∇ A , [X A , h ∇ A ] = R v ∇ + F A , h div A v ∇ − v div h ∇ A = (d − 1)X A ,
and by duality one gets the following identity on Z n ,
[X A , v div] = − h div A .
The paper [GPSU16] gave a version of the Pestov identity with unitary connection in any dimension. If the connection is not unitary, one loses symmetry in the Pestov identity but the following form of this identity remains valid.
Lemma 4.2. If A is a general connection, one has the identity
( v ∇X A u, v ∇X −A * u) = (X A v ∇u, X −A * v ∇u) − (R v ∇u, v ∇u) − (F A u, v ∇u) + (d − 1)(X A u, X −A * u)
for any u ∈ C ∞ (SM, C n ) with u| ∂(SM ) = 0 in the boundary case.
Proof. Observe that
( v ∇X A u, v ∇X −A * u) − (X A v ∇u, X −A * v ∇u) = ((X A v div v ∇X A − v divX A X A v ∇)u, u).
The commutator identities above imply that
X A v div v ∇X A − v divX A X A v ∇ = −(d − 1)X A X A + v div(R v ∇ + F A ).
The result follows.
If A is a general connection, the same argument as in Section 2 shows that X A maps Ω m to Ω m−1 ⊕ Ω m+1 . Thus we have the decomposition
X A = X A + + X A − , X A ±
: Ω m → Ω m±1 . We will next rewrite the Pestov identity in terms of X A + and X A − . To do this, we need some notation: for a polynomially bounded sequence α = (α ℓ ) ∞ ℓ=0 of real numbers, we define a corresponding "inner product"
(u, w) α = ∞ ℓ=0 α ℓ (u ℓ , w ℓ ) L 2 (SM ) , u, w ∈ C ∞ (SM, C n ).
We also write u 2 α = ∞ ℓ=0 α ℓ u ℓ 2 . (If each α ℓ is positive one gets an actual inner product and norm, but it is notationally convenient to allow zero or negative α ℓ .)
The Pestov identity can then be written in the following form.
(X A − u, X −A * − u) α − (R v ∇u, v ∇u) − (F A u, v ∇u) + (Z A (u), Z −A * (u)) = (X A + u, X −A * + u) β where Z A (u) is the v div-free part of h ∇ A u, and α ℓ = d − 1, ℓ = 0, (2ℓ + d − 2) 1 + 1 ℓ+d−2 , ℓ ≥ 1, β ℓ = 0, ℓ = 0, 1, (2ℓ + d − 2) 1 − 1 ℓ , ℓ ≥ 2.
Proof. Note that u ∈ C ∞ (SM, C n ), implies rapid decay, namely, u ℓ = O((1 + |ℓ|) −∞ ). We will use Lemma 4.2 in the form
( v ∇X A u, v ∇X −A * u)−(d−1)(X A u, X −A * u) = (X A v ∇u, X −A * v ∇u)−(R v ∇u, v ∇u)−(F A u, v ∇u).
Note that Lemma 4.1 gives (4.1)
X A v ∇u = v ∇X A u − h ∇ A u.
We also have the commutator formula
[X A , − v div v ∇] = 2 v div h ∇ A + (d − 1)X A by∇ A u = v ∇ ∞ ℓ=1 1 ℓ X A + u ℓ−1 − 1 ℓ + d − 2 X A − u ℓ+1 + Z A (u)
where Z A (u) ∈ Z n satisfies v div Z A (u) = 0. Thus (4.1) and (4.2) yield that
X A v ∇u = v ∇ ∞ ℓ=1 1 − 1 ℓ X A + u ℓ−1 + 1 + 1 ℓ + d − 2 X A − u ℓ+1 − Z A (u).
Applying this for A and −A * gives (recall that
λ ℓ = ℓ(ℓ + d − 2)) (X A v ∇u, X −A * v ∇u) = ∞ ℓ=1 λ ℓ 1 − 1 ℓ X A + u ℓ−1 + 1 + 1 ℓ + d − 2 X A − u ℓ+1 , 1 − 1 ℓ X −A * + u ℓ−1 + 1 + 1 ℓ + d − 2 X −A * − u ℓ+1 + (ZA(u), Z−A * (u)) = ∞ ℓ=1 λ ℓ 1 − 1 ℓ 2 (X A + u ℓ−1 , X −A * + u ℓ−1 ) + 1 + 1 ℓ + d − 2 2 (X A − u ℓ+1 , X −A * − u ℓ+1 ) + ∞ ℓ=1 λ ℓ 1 − 1 ℓ 1 + 1 ℓ + d − 2 (X A + u ℓ−1 , X −A * − u ℓ+1 ) + (X A − u ℓ+1 , X −A * + u ℓ−1 ) + (ZA(u), Z−A * (u)).
On the other hand, one has
( v ∇X A u, v ∇X −A * u) − (d − 1)(X A u, X −A * u) = −(d − 1)(X A − u 1 , X −A * − u 1 ) + ∞ ℓ=1 (λ l − (d − 1))(X A + u ℓ−1 + X A − u ℓ+1 , X −A * + u ℓ−1 + X −A * − u ℓ+1 ) = −(d − 1)(X A − u 1 , X −A * − u 1 ) + ∞ ℓ=1 (λ ℓ − (d − 1)) (X A + u ℓ−1 , X −A * + u ℓ−1 ) + (X A − u ℓ+1 , X −A * − u ℓ+1 ) + ∞ ℓ=1 (λ ℓ − (d − 1)) (X A + u ℓ−1 , X −A * − u ℓ+1 ) + (X A − u ℓ+1 , X −A * + u ℓ−1 ) .
Somewhat miraculously, we observe that
λ ℓ 1 − 1 ℓ 1 + 1 ℓ + d − 2 = λ ℓ − (d − 1).
This means that the two sums above involving (X
A + u ℓ−1 , X −A * − u ℓ+1 ) + (X A − u ℓ+1 , X −A * + u ℓ−1 )
terms are equal. The Pestov identity in the beginning of the proof now yields
−(d−1)(X A − u 1 , X −A * − u 1 )+ ∞ ℓ=1 (λ ℓ −(d−1)) (X A + u ℓ−1 , X −A * + u ℓ−1 ) + (X A − u ℓ+1 , X −A * − u ℓ+1 ) = ∞ ℓ=1 λ ℓ 1 − 1 ℓ 2 (X A + u ℓ−1 , X −A * + u ℓ−1 ) + 1 + 1 ℓ + d − 2 2 (X A − u ℓ+1 , X −A * − u ℓ+1 ) − (R v ∇u, v ∇u) − (F A u, v ∇u) + (Z A (u), Z −A * (u)).
We rewrite this as
∞ ℓ=0 α ℓ (X A − u ℓ+1 , X −A * − u ℓ+1 ) − (R v ∇u, v ∇u) − (F A u, v ∇u) + (Z A (u), Z −A * (u)) = ∞ ℓ=1 β ℓ (X A + u ℓ−1 , X −A * + u ℓ−1 ) where α ℓ = λ ℓ 1 + 1 ℓ + d − 2 2 − 1 + (d − 1), β ℓ = λ ℓ 1 − 1 − 1 ℓ 2 − (d − 1).
The result follows after simplifying the expressions for α ℓ and β ℓ .
We will use the previous identity only in the case where the connection A is unitary, or when A = 0. In these cases the inner products become squares of L 2 norms, and we obtain the following energy identity which is equivalent with the Pestov identity with unitary connection given in [GPSU16].
Proposition 4.4 (Pestov identity in terms of X A ± ). Let (M, g) be a compact manifold with or without boundary, and let A be a unitary connection. Then
X A − u 2 α − (R v ∇u, v ∇u) − (F A u, v ∇u) + Z A (u) 2 = X A + u 2 β
for any u ∈ C ∞ (SM, C n ) with u| ∂(SM ) = 0 in the boundary case.
Frequency localization
Recall that any u ∈ C ∞ (SM, C n ) admits an L 2 -orthogonal decomposition
u = ∞ ℓ=0 u ℓ , u ℓ ∈ Ω ℓ ,
where Ω ℓ corresponds to the set of vertical spherical harmonics of degree ℓ. Since X A ± maps Ω ℓ to Ω ℓ±1 , it is immediate that the Pestov identity with unitary connection (Proposition 4.4) reduces to the following identity when applied to functions in Ω ℓ .
Proposition 5.1 (Pestov identity on Ω ℓ ). Let (M, g) be a compact manifold with or without boundary, let A be a unitary connection, and let ℓ ≥ 0. One has If dim(M) = 2, the Pestov identity on Ω ℓ is the same as the Guillemin-Kazhdan energy identity [GK80a]. In [PSU15, Appendix B] it was observed that in two dimensions the Guillemin-Kazhdan identity is actually equivalent with the Pestov identity, in the sense that summing that Guillemin-Kazhdan identity over all ℓ gives back the Pestov identity.
α ℓ−1 X A − u 2 − (R v ∇u, v ∇u) − (F A u, v ∇u) + Z A (u) 2 = β ℓ+1 X A + u 2 , u ∈ Ω ℓ ,
We will now show that the same is true in any dimension: the Pestov identity is equivalent with the frequency localized identities of Proposition 5.1. This means that the Pestov identity localizes completely with respect to vertical Fourier decompositions. This will be a very important observation in what follows.
Lemma 5.2. The Pestov identity on Ω ℓ is equivalent with the Pestov identity with unitary connection in the following sense: for any u ∈ C ∞ (SM, C n ) with u| ∂(SM ) = 0 in the boundary case, one has
∞ ℓ=0 α ℓ−1 X A − u ℓ 2 − (R v ∇u ℓ , v ∇u ℓ ) − (F A u ℓ , v ∇u ℓ ) + Z A (u ℓ ) 2 − β ℓ+1 X A + u ℓ 2 = X A − u 2 α − (R v ∇u, v ∇u) − (F A u, v ∇u) + Z A (u) 2 − X A + u 2 β .
The result will follow if we can show that the curvature and Z A terms localise. Thus Lemma 5.2 is a corollary of the next result.
Lemma 5.3. If (M, g) is a Riemannian manifold and A is a unitary connection, then
(R v ∇u, v ∇w) = 0, (F A u, v ∇w) = 0, (Z A (u), Z A (w)) = 0,
whenever u ∈ Ω m , w ∈ Ω ℓ and m = ℓ.
To prove this lemma, we first prove two auxiliary lemmas. For the auxiliary lemmas we work in R n with the standard inner product and let S n−1 be the unit sphere. We denote by Ω m ⊂ C ∞ (S n−1 ) the space of spherical harmonics of degree m; let ∇ denote the gradient in S n−1 with the canonical metric induced by R n .
Lemma 5.4. Let α be any anti-symmetric 2-form in R n . Given u ∈ Ω m , the function
S n−1 ∋ x → α(x, ∇u(x)) belongs to Ω m .
Proof. It suffices to prove the claim for elements dx i ∧ dx j of the standard basis of Λ 2 (R n ) * . Let f : R n \ {0} → S n−1 be the map f (x) = x/|x| and introduce the vector fields on S n−1 given by
Y j (x) := df x (∂/∂x j ), for x ∈ S n−1 . Since df x (v) = v − x, v x for x ∈ S n−1 and v ∈ R n we have (5.1) Y j (x) = ∂ ∂x j − x j x.
One easily checks that
∇u = Y j (u) ∂ ∂x j = ∇ R n (f * u)| S n−1 .
Using (5.1) we see that
x i Y j − x j Y i = x i ∂ ∂x j − x j ∂ ∂x i
and since the latter is a Killing field of S n−1 we have
[∆, x i Y j − x j Y i ] = 0.
Thus
dx i ∧ dx j (x, ∇u) = x i Y j (u) − x j Y i (u) ∈ Ω m as desired.
Given α, β ∈ Λ 2 ((R n ) * ) recall that the symmetric product of α and β as a 4-tensor is given by (α ⊙ β)(x, y, z, w) = α(x, y)β(z, w) + α(z, w)β(x, y). This gives α⊙β ∈ S 2 (Λ 2 (R n ) * ) and moreover, elements of this form span S 2 (Λ 2 (R n ) * ).
Lemma 5.5. Let R ∈ S 2 (Λ 2 (R n ) * ). Then S n−1 R(∇u, x, x, ∇w) dx = 0 for any u ∈ Ω m and w ∈ Ω k with m = k.
Proof. Given that elements of the form α ⊙ β span S 2 (Λ 2 (R n ) * ) it suffices to show the claim for such 4-tensors. Using the definition of the symmetric product it is enough to prove that
S n−1 α(∇u, x)β(x, ∇w) dx = 0
for any u ∈ Ω m and w ∈ Ω k with m = k. This follows right away from the previous lemma.
Proof of Lemma 5.3. The proof reduces to a statement for fixed x and the first claim involving the Riemann curvature tensor follows directly from Lemma 5.5.
For the result involving F A , we first use [GPSU16, Lemma 3.1] and observe that given u = (u 1 , . . . , u n ) ∈ C n and Z = (Z 1 , . . . , Z n ) with Z γ , v = 0 for 1 ≤ γ ≤ n, one has that the pointwise inner product is given by
F A u, Z = γ,δ (F A ) γδ u γ , Z δ = γ,δ (f A ) γδ (v, u γ Z δ ) where f A = dA + A ∧ A is an n × n matrix of complex 2-forms. It follows that (F A u, v ∇w) = SxM (f A ) γδ (v, u γ v ∇w δ ) d(S x M).
Since (f A ) γδ is a 2-form for all γ and δ we may use Lemma 5.4 to deduce that
(F A u, v ∇w) = 0 whenever m = ℓ.(5.2) (F A u, v ∇w) = (F A w, v ∇u).
Now observe that Proposition 4.4, the formula (5.2), and the polarization identity imply that
(Z A (u), Z A (w)) = (X A + u, X A + w) β − (X A − u, X A − w) α + (R v ∇u, v ∇w) + (F A u, v ∇w).
The statements proved above imply that (Z A (u), Z A (w)) = 0 when m = ℓ.
Carleman estimates in negative curvature
We will now give the first Carleman estimate for X-ray transforms, valid for negative sectional curvature. Throughout this section we will assume that (M, g) is a compact Riemannian manifold with or without boundary, with d = dim(M) ≥ 2. The following theorem is the main result of this section.
Theorem 6.1. Assume that that (M, g) has sectional curvature ≤ −κ where κ > 0. For any s > −1/2, for any m ≥ 1 and for any u ∈ C ∞ (SM, C n ) (with u| ∂(SM ) = 0 in the boundary case) one has
m+1 ℓ=m ℓ 2s+1 X − u ℓ 2 + (2s + 1) ∞ ℓ=m+2 (ℓ − 1) 2s X − u ℓ 2 + κ ∞ ℓ=m ℓ 2s+2 u ℓ 2 + ∞ ℓ=m ℓ 2s Z(u ℓ ) 2 ≤ C 2s + 1 ∞ ℓ=m+1 ℓ 2s+2 (Xu) ℓ 2 where C = C(d) > 0.
If s is large, the previous result easily implies the Carleman estimate stated in the introduction.
Theorem 6.2. Assume that that (M, g) has sectional curvature ≤ −κ where κ > 0. For any τ ≥ 1 and m ≥ 1, one has
∞ ℓ=m e 2τ log(ℓ) u ℓ 2 ≤ C κτ ∞ ℓ=m+1 e 2τ log(ℓ) (Xu) ℓ 2 whenever u ∈ C ∞ (SM, C n ) (with u| ∂(SM ) = 0 in the boundary case).
Proof. It is enough to choose τ = s + 1 with s ≥ 0 in Theorem 6.1.
As discussed in the introduction, the Carleman estimate in Theorem 6.1 can also be understood as a version of the Pestov identity that has been shifted to a different regularity scale. To explain this, introduce the mixed norms
u 2 L 2 x H s v = ∞ ℓ=0 ℓ 2s u ℓ 2 , v ∇u 2 L 2 x H s v = ∞ ℓ=0 ℓ 2s v ∇u ℓ 2 , ∇ SM u 2 L 2 x H s v = ∞ ℓ=0 ℓ 2s ( X − u ℓ+1 2 + X + u ℓ−1 2 + Z(u ℓ ) 2 + v ∇u ℓ 2 )
where ℓ = (1 + ℓ 2 ) 1/2 , v ∇u ℓ 2 = λ ℓ u ℓ 2 , and u −1 = 0.
Clearly u L 2 x H 0 v = u L 2 (SM ) and v ∇u L 2 x H 0 v = v ∇u L 2 (SM ) . We also have that ∇ SM u L 2 x H 0 v ∼ ∇ SM u L 2 (SM ) by the formula (see [LRS18, Lemma 5.1]) ∇ SM u 2 = Xu 2 + h ∇u 2 + v ∇u 2 ∼ X − u 2 + X + u 2 + Z(u) 2 + v ∇u 2 ,
and using that Z(u) 2 = ∞ ℓ=0 Z(u ℓ ) 2 by Lemma 5.3. Theorem 6.1 can now be restated as a shifted Pestov identity. Theorem 6.3. Assume that that (M, g) has sectional curvature ≤ −κ where κ > 0. For any s > −1/2 there is C = C d,s,κ > 0 such that
∇ SM u L 2 x H s v ≤ C v ∇Xu L 2 x H s v for any u ∈ C ∞ (SM, C n ) (with u| ∂(SM ) = 0 in the boundary case).
Proof. Theorem 6.1 with the choice m = 1 yields that
X − u 1 2 + X − u 2 2 + (2s + 1) ∞ ℓ=2 ℓ 2s (X − u) ℓ 2 + κ ∞ ℓ=1 ℓ 2s+2 u ℓ 2 + ∞ ℓ=1 ℓ 2s Z(u ℓ ) 2 ≤ C 2s + 1 ∞ ℓ=2 ℓ 2s+2 (Xu) ℓ 2 ≤ C 2s + 1 v ∇Xu 2 L 2 x H s v .
Note also that Z(u 0 ) = 0, which follows from Proposition 5.1 with ℓ = 0 (recall that
β 1 = 0). Thus we have ∞ ℓ=0 ℓ 2s ( X − u ℓ+1 2 + Z(u ℓ ) 2 + v ∇u ℓ 2 ) ≤ C 2s + 1 v ∇Xu 2 L 2 x H s v .
Finally, note that
X + u 2 L 2 x H s v = ∞ ℓ=1 ℓ 2s (Xu) ℓ − X − u ℓ+1 2 ≤ C( v ∇Xu 2 L 2 x H s v + X − u 2 L 2 x H s v ).
The result follows upon combining the last two inequalities.
We now begin the proof of Theorem 6.1. The first step is to observe that the localized Pestov identity in Proposition 5.1 gains a positive term in negative sectional curvature.
Lemma 6.4. Assume that (M, g) has sectional curvature ≤ −κ where κ > 0, and let ℓ ≥ 0. One has Proof. This follows from Proposition 5.1 upon choosing A = 0 and noting that
α ℓ−1 X − u 2 + κλ ℓ u 2 + Z(u) 2 ≤ β ℓ+1 X + u 2 , u ∈ Ω ℓ ,−(R v ∇u, v ∇u) ≥ κ v ∇u 2 = κ(− v div v ∇u, u) = κλ ℓ u 2 .
Remark 6.5. The Z(u) 2 term can be simplified when d = 2. In this case, one has in the notation of [PSU15, Appendix B]
Z(u) = −(X ⊥ u) 0 iv = (i(η + u −1 − η − u 1 ))iv,
which shows in particular that Z(u k ) = 0 unless k = ±1. To prove the above claim, assume that d = 2 and note that, in the notation of [PSU15,Appendix B],
h ∇u = −(X ⊥ u)iv, v ∇a = (V a)iv.
We may write X ⊥ u as
X ⊥ u = (X ⊥ u) 0 + V a, a = k =0 1 ik (X ⊥ u) k .
It follows that
h ∇u = v ∇(−a) − (X ⊥ u) 0 iv.
Since Z(u) was defined as the v div-free part of h ∇u, we must have Z(u) = −(X ⊥ u) 0 iv.
The Carleman estimate in Theorem 6.1 will follow after multiplying the localized estimates by suitable weights and adding them together. We first give an estimate with rather general weights. In what follows, C ∞ F (SM) denotes the set of smooth functions in SM with finite degree.
Proposition 6.6. Assume that (M, g) has sectional curvature ≤ −κ where κ > 0. If m ≥ 1 and if (γ ℓ ) ∞ ℓ=0 is any sequence of positive real numbers satisfying (6.1)
α ℓ γ 2 ℓ+1 > β ℓ γ 2 ℓ−1 , ℓ ≥ m + 1, and if (δ ℓ ) ∞ ℓ=0 is any sequence where δ ℓ ∈ (0, 1], then one has m+1 ℓ=m α ℓ−1 γ 2 ℓ X − u ℓ 2 + ∞ ℓ=m+2 (1 − δ ℓ−1 )(α ℓ−1 γ 2 ℓ − β ℓ−1 γ 2 ℓ−2 ) X − u ℓ 2 + κ ∞ ℓ=m λ ℓ γ 2 ℓ u ℓ 2 + ∞ ℓ=m γ 2 ℓ Z(u ℓ ) 2 ≤ ∞ ℓ=m+1 1 + 1 − δ ℓ δ ℓ β ℓ γ 2 ℓ−1 α ℓ γ 2 ℓ+1 α ℓ γ 2 ℓ+1 β ℓ γ 2 ℓ−1 α ℓ γ 2 ℓ+1 − β ℓ γ 2 ℓ−1 (Xu) ℓ 2 .
whenever u ∈ C ∞ F (SM, C n ) (with u| ∂(SM ) = 0 in the boundary case). Remark 6.7. The sequence (γ ℓ ) corresponds to the weights in the Carleman estimate, and the parameters (δ ℓ ) fine-tune the weights for the X − terms on the left. Later we will essentially choose γ ℓ = ℓ s for s > −1/2 and δ ℓ ≡ 1/2. An even simpler choice, which would also be sufficient for most of our purposes, would be to take δ ℓ ≡ 1. Then the above estimate becomes (after dropping the X − u ℓ and Z(u ℓ ) terms)
κ ∞ ℓ=m λ ℓ γ 2 ℓ u ℓ 2 ≤ ∞ ℓ=m+1 α ℓ γ 2 ℓ+1 β ℓ γ 2 ℓ−1 α ℓ γ 2 ℓ+1 − β ℓ γ 2 ℓ−1 (Xu) ℓ 2
for any u ∈ C ∞ F (SM) (with u| ∂(SM ) = 0 in the boundary case) Proof of Proposition 6.6. Let u ∈ C ∞ F (SM, C n ), and let (γ ℓ ) ∞ ℓ=0 be a sequence of positive real numbers satisfying (6.1). Lemma 6.4 implies that for each ℓ ≥ 0,
α ℓ−1 γ 2 ℓ X − u ℓ 2 + κλ ℓ γ 2 ℓ u ℓ 2 + γ 2 ℓ Z(u ℓ ) 2 ≤ β ℓ+1 γ 2 ℓ X + u ℓ 2 .
Let now ε ℓ > 0 be positive numbers, and observe that
X + u ℓ 2 = (Xu) ℓ+1 − X − u ℓ+2 2 = (Xu) ℓ+1 2 − 2 Re((Xu) ℓ+1 , X − u ℓ+2 ) + X − u ℓ+2 2 ≤ 1 + 1 ε ℓ (Xu) ℓ+1 2 + (1 + ε ℓ ) X − u ℓ+2 2 .
(6.2)
Combining the last two inequalities gives the estimate
α ℓ−1 γ 2 ℓ X − u ℓ 2 + κλ ℓ γ 2 ℓ u ℓ 2 + γ 2 ℓ Z(u ℓ ) 2 ≤ β ℓ+1 γ 2 ℓ 1 + 1 ε ℓ (Xu) ℓ+1 2 + β ℓ+1 γ 2 ℓ (1 + ε ℓ ) X − u ℓ+2 2 .
Adding up these estimates for m ≤ ℓ ≤ N, where m ≥ 1, yields
(6.3) N ℓ=m α ℓ−1 γ 2 ℓ X − u ℓ 2 + N ℓ=m κλ ℓ γ 2 ℓ u ℓ 2 + N ℓ=m γ 2 ℓ Z(u ℓ ) 2 ≤ N +1 ℓ=m+1 β ℓ γ 2 ℓ−1 1 + 1 ε ℓ−1 (Xu) ℓ 2 + N +2 ℓ=m+2 β ℓ−1 γ 2 ℓ−2 (1 + ε ℓ−2 ) X − u ℓ 2 .
We would like to choose (γ ℓ ) and (ε ℓ ) so that a large part of the last term on the right can be absorbed in the first term on the left. The minimal requirement is that
β ℓ−1 γ 2 ℓ−2 (1 + ε ℓ−2 ) ≤ α ℓ−1 γ 2 ℓ , m + 2 ≤ ℓ ≤ N.
We will choose (ε ℓ ) so that
(6.4) ε ℓ−2 = δ ℓ−1 α ℓ−1 β ℓ−1 γ 2 ℓ γ 2 ℓ−2 − 1 , ℓ ≥ m + 2,
where δ ℓ−1 ∈ (0, 1]. In order for ε ℓ to be positive, we need to have α ℓ−1 γ 2 ℓ > β ℓ−1 γ 2 ℓ−2 for ℓ ≥ m + 2, which follows from the assumption (6.1). Using (6.4), we may express the weights on the right hand side of (6.3) as β ℓ−1 γ 2 ℓ−2 (1 + ε ℓ−2 ) = β ℓ−1 γ 2 ℓ−2 + δ ℓ−1 (α ℓ−1 γ 2 ℓ − β ℓ−1 γ 2 ℓ−2 ) and
β ℓ γ 2 ℓ−1 1 + 1 ε ℓ−1 = β ℓ γ 2 ℓ−1 1 + 1 δ ℓ β ℓ γ 2 ℓ−1 α ℓ γ 2 ℓ+1 − β ℓ γ 2 ℓ−1 = α ℓ γ 2 ℓ+1 β ℓ γ 2 ℓ−1 α ℓ γ 2 ℓ+1 − β ℓ γ 2 ℓ−1 1 + 1 − δ ℓ δ ℓ β ℓ γ 2 ℓ−1 α ℓ γ 2 ℓ+1 .
Inserting these expressions in (6.3), it follows that
m+1 ℓ=m α ℓ−1 γ 2 ℓ X − u ℓ 2 + N ℓ=m+2 (1 − δ ℓ−1 )(α ℓ−1 γ 2 ℓ − β ℓ−1 γ 2 ℓ−2 ) X − u ℓ 2 + κ N ℓ=m λ ℓ γ 2 ℓ u ℓ 2 + N ℓ=m γ 2 ℓ Z(u ℓ ) 2 ≤ N +1 ℓ=m+1 1 + 1 − δ ℓ δ ℓ β ℓ γ 2 ℓ−1 α ℓ γ 2 ℓ+1 α ℓ γ 2 ℓ+1 β ℓ γ 2 ℓ−1 α ℓ γ 2 ℓ+1 − β ℓ γ 2 ℓ−1 (Xu) ℓ 2 + N +2 ℓ=N +1 β ℓ−1 γ 2 ℓ−2 + δ ℓ−1 (α ℓ−1 γ 2 ℓ − β ℓ−1 γ 2 ℓ−2 ) X − u ℓ 2 .
Since u has finite degree we can take the limit as N → ∞, which proves the result.
We will next prove Theorem 6.1 by choosing suitable weights in Proposition 6.6. The proof will involve two elementary lemmas.
Lemma 6.8. If s ≥ 0, then
(ℓ + 1) s − (ℓ − 1) s ≥ sℓ s−1 , ℓ ≥ 1.
Proof. Writing x = 1/ℓ and using that log(1 − x) ≤ −x and log(1 + x) ≥ x log(2) for x ∈ (0, 1), we obtain
(1 + x) s − (1 − x) s ≥ 2 sinh(sx log(2)) ≥ 2 log(2)sx.
Since 2 log(2) ≥ 1, the result follows.
Remark 6.9. By using the generalized binomial theorem, one can show that the sharp constant on the right hand side of Lemma 6.8 is min(2s, 2 s ).
Lemma 6.10. For any ℓ ≥ 2 one has
1 4 √ ℓ ≤ [ ℓ−2 2 ] j=0 ℓ − 1 − 2j ℓ − 2j ≤ 4 √ ℓ .
Proof. We do the proof for ℓ = 2N even (the odd case is similar). Note that
2N − 1 2N 2N − 3 2N − 2 · · · 1 2 = (2N)! 2 2N (N!) 2 . The Stirling approximation √ 2πN N e N ≤ N! ≤ √ 2πN N e N e 1 12N yields that e − 1 6N √ 2π · 2N ( √ 2πN ) 2 ≤ (2N)! 2 2N (N!) 2 ≤ √ 2π · 2N ( √ 2πN) 2 e 1 24N .
This proves the result.
Proof of Theorem 6.1. Define a sequence (γ ℓ ) by
γ 1 = γ 2 = 1, γ 2 ℓ+1 = β ℓ α ℓ γ 2 ℓ−1 for ℓ ≥ 2.
Choose γ ℓ = γ ℓ ℓ σ/2 in Proposition 6.6, where σ > 0. By Lemma 6.8 we have
(6.5) α ℓ γ 2 ℓ+1 − β ℓ γ 2 ℓ−1 = β ℓ γ 2 ℓ−1 [(ℓ + 1) σ − (ℓ − 1) σ ] ≥ β ℓ γ 2 ℓ−1 σℓ σ−1 and (6.6) α ℓ γ 2 ℓ+1 β ℓ γ 2 ℓ−1 α ℓ γ 2 ℓ+1 − β ℓ γ 2 ℓ−1 ≤ (β ℓ γ 2 ℓ−1 ) 2 (ℓ 2 − 1) σ β ℓ γ 2 ℓ−1 [σℓ σ−1 ] ≤ β ℓ γ 2 ℓ−1 ℓ σ+1 σ .
We will next estimate γ 2 ℓ−1 . By definition we have
γ 2 ℓ+1 = β ℓ α ℓ γ 2 ℓ−1 = . . . = [ ℓ−2 2 ] j=0 β ℓ−2j α ℓ−2j .
Using the definition of α ℓ and β ℓ we further have
β ℓ α ℓ = (ℓ − 1)(ℓ + d − 2) ℓ(ℓ + d − 1) . Writing g ℓ = [ ℓ−2 2 ] j=0 ℓ − 1 − 2j ℓ − 2j ,
it follows that
γ 2 ℓ+1 = g ℓ g ℓ+d−2 gd whered = d − 1 if d is even, andd = d if d is odd.
Using Lemma 6.10 we obtain
(6.7) c ℓ ≤ γ 2 ℓ ≤ C ℓ , ℓ ≥ 1,
where c and C only depend on d.
We now choose δ ℓ = 1/2 in Proposition 6.6 and observe, using (6.5)-(6.7), that for ℓ ≥ 1 one has
α ℓ−1 γ 2 ℓ ≥ cℓ σ , α ℓ−1 γ 2 ℓ − β ℓ−1 γ 2 ℓ−2 ≥ cσ(ℓ − 1) σ−1 , λ ℓ γ 2 ℓ ≥ cℓ σ+1 , γ 2 ℓ ≥ cℓ σ−1 , and β ℓ γ 2 ℓ−1 α ℓ γ 2 ℓ+1 ≤ 1, α ℓ γ 2 ℓ+1 β ℓ γ 2 ℓ−1 α ℓ γ 2 ℓ+1 − β ℓ γ 2 ℓ−1 ≤ C ℓ σ+1 σ
for C, c > 0 depending on d. The theorem follows upon choosing σ = 2s + 1.
Carleman estimates in nonpositive curvature
In this section we discuss Carleman estimates in the case of nonpositive sectional curvature instead of strictly negative sectional curvature. The extra positive term in Lemma 6.4 becomes zero in the case, but one gains some positivity from the fact that there are no conjugate points (in a suitable strong sense). Thus, in this section (M, g) will be a compact Riemannian manifold with or without boundary, and we will assume that (M, g) is simple/Anosov with nonpositive sectional curvature.
The main result is the following Carleman estimate with linear instead of logarithmic weights.
Theorem 7.1. Let (M, g) be a simple/Anosov manifold having nonpositive sectional curvature. There exist m 0 , τ 0 , κ > 0 such that for any τ ≥ τ 0 , for any m ≥ m 0 and for any u ∈ C ∞ F (SM, C n ) (with u| ∂(SM ) = 0 in the boundary case) one has
∞ ℓ=m e 2τ φ ℓ u ℓ 2 ≤ 24 κe 2τ ∞ ℓ=m+1 e 2τ φ ℓ (Xu) ℓ 2 where φ ℓ = ℓ.
The constant κ > 0 in the above theorem comes from the fact that compact/simple manifolds with nonpositive curvature have no conjugate points. It is the same constant that appears in the following lemma proved in [PSU15, Theorem 7.2 and Lemma 11.2]:
Lemma 7.2. If (M, g) is a compact simple/Anosov manifold, there exist α ∈ (0, 1] and κ > 0 so that XZ 2 − (RZ, Z) ≥ α XZ 2 + κ Z 2 for any Z ∈ Z n (with Z| ∂(SM ) = 0 in the boundary case).
We begin the proof of Theorem 7.1 with the following result, which is a counterpart of Lemma 6.4. Lemma 7.3. Let (M, g) be a simple/Anosov manifold having nonpositive sectional curvature. There is κ > 0 such that for any ℓ ≥ 1 and for any σ ℓ with 0 ≤ σ ℓ ≤ 1 one has
α ℓ−1 − σ ℓ λ ℓ−1 1 + 1 ℓ + d − 3 2 X − u 2 + σ ℓ κλ ℓ u 2 ≤ β ℓ+1 + σ ℓ λ ℓ+1 1 − 1 ℓ + 1 2 X + u 2 , u ∈ Ω ℓ ,
if additionally u| ∂(SM ) = 0 in the boundary case.
Proof. We use the standard Pestov identity (Lemma 4.2) with A = 0:
X v ∇u 2 − (R v ∇u, v ∇u) + (d − 1) Xu 2 = v ∇Xu 2 .
The nonpositive curvature assumption implies that
X v ∇u 2 − (R v ∇u, v ∇u) ≥ X v ∇u 2 .
On the other hand, the simple/Anosov assumption together with Lemma 7.2 implies that there is κ > 0 (we omit the α X v ∇u 2 term) such that
X v ∇u 2 − (R v ∇u, v ∇u) ≥ κ v ∇u 2 .
Interpolating the last two inequalities yields that for any σ ℓ with 0 ≤ σ ℓ ≤ 1,
X v ∇u 2 − (R v ∇u, v ∇u) ≥ (1 − σ ℓ ) X v ∇u 2 + σ ℓ κ v ∇u 2 .
Inserting this estimate in the Pestov identity implies that
(7.1) (1 − σ ℓ ) X v ∇u 2 + σ ℓ κ v ∇u 2 ≤ v ∇Xu 2 − (d − 1) Xu 2 .
We now assume that u = u ℓ ∈ Ω ℓ and simplify (7.1) as in the proof of Lemma 4.3. This gives
α ℓ−1 X − u ℓ 2 + Z(u ℓ ) 2 + σ ℓ κ v ∇u 2 ≤ β ℓ+1 X + u ℓ 2 + σ ℓ X v ∇u 2
Again as in the proof of Lemma 4.3, one has
X v ∇u ℓ 2 = v ∇ 1 − 1 ℓ + 1 X + u ℓ + 1 + 1 ℓ + d − 3 X − u ℓ − Z(u ℓ ) 2 = λ ℓ+1 1 − 1 ℓ + 1 2 X + u ℓ 2 + λ ℓ−1 1 + 1 ℓ + d − 3 2 X − u ℓ 2 + Z(u ℓ ) 2 .
The result follows by combining the last two formulas and using the trivial estimate
(1 − σ ℓ ) Z(u ℓ ) 2 ≥ 0.
The estimate in Lemma 7.3 is of course only useful when the coefficient in front of X − u 2 is nonnegative. For large ℓ one has
α ℓ−1 − σ ℓ λ ℓ−1 1 + 1 ℓ + d − 3 2 ∼ 2ℓ − σ ℓ ℓ 2 ,
so one needs to have σ ℓ ≤ 2 ℓ for ℓ large. (If one keeps the α X v ∇u 2 that was omitted in the proof of Lemma 7.3, the previous condition becomes σ ℓ ≤ 2
(1−α)ℓ so one still needs σ ℓ ℓ −1 for ℓ large.)
We next give the counterpart of Proposition 6.6 (for δ ℓ ≡ 1), which corresponds to a Carleman estimate with general weights.
Proposition 7.4. Assume that (M, g) is a simple/Anosov manifold of nonpositive sectional curvature. If (γ ℓ ) and (σ ℓ ) are sequences satisfying γ ℓ > 0, 0 < σ ℓ < 1, and
(7.2) α ℓ − σ ℓ+1 λ ℓ 1 + 1 ℓ + d − 2 2 γ 2 ℓ+1 > β ℓ + σ ℓ−1 λ ℓ 1 − 1 ℓ 2 γ 2 ℓ−1 then one has κ ∞ ℓ=m σ ℓ λ ℓ γ 2 ℓ u ℓ 2 ≤ ∞ ℓ=m+1 (α ℓ − σ ℓ+1 λ ℓ (1 + 1 ℓ+d−2 ) 2 )γ 2 ℓ+1 (β ℓ + σ ℓ−1 λ ℓ (1 − 1 ℓ ) 2 )γ 2 ℓ−1 (α ℓ − σ ℓ+1 λ ℓ (1 + 1 ℓ+d−2 ) 2 )γ 2 ℓ+1 − (β ℓ + σ ℓ−1 λ ℓ (1 − 1 ℓ ) 2 )γ 2 ℓ−1 (Xu) ℓ 2
whenever m ≥ 1 and u ∈ C ∞ F (SM, C n ) (with u| ∂(SM ) = 0 in the boundary case).
Proof. Multiplying the estimate in Lemma 7.3 by γ 2 ℓ and using (6.2), we obtain
α ℓ−1 − σ ℓ λ ℓ−1 1 + 1 ℓ + d − 3 2 γ 2 ℓ X − u ℓ 2 + σ ℓ κλ ℓ γ 2 ℓ u ℓ 2 ≤ β ℓ+1 + σ ℓ λ ℓ+1 1 − 1 ℓ + 1 2 γ 2 ℓ 1 + 1 ε ℓ (Xu) ℓ+1 2 + (1 + ε ℓ ) X − u ℓ+2 2 .
Summing up these estimates for ℓ ≥ m, where m ≥ 1, yields
∞ ℓ=m α ℓ−1 − σ ℓ λ ℓ−1 1 + 1 ℓ + d − 3 2 γ 2 ℓ X − u ℓ 2 + ∞ ℓ=m σ ℓ κλ ℓ γ 2 ℓ u ℓ 2 ≤ ∞ ℓ=m+1 β ℓ + σ ℓ−1 λ ℓ 1 − 1 ℓ 2 γ 2 ℓ−1 1 + 1 ε ℓ−1 (Xu) ℓ 2 + ∞ ℓ=m+2 β ℓ−1 + σ ℓ−2 λ ℓ−1 1 − 1 ℓ − 1 2 γ 2 ℓ−2 (1 + ε ℓ−2 ) X − u ℓ 2 .
We now choose ε ℓ so that for ℓ ≥ m + 2,
1 + ε ℓ−2 = (α ℓ−1 − σ ℓ λ ℓ−1 (1 + 1 ℓ+d−3 ) 2 )γ 2 ℓ (β ℓ−1 + σ ℓ−2 λ ℓ−1 (1 − 1 ℓ−1 ) 2 )γ 2 ℓ−2 .
It follows that
1 ε ℓ−1 = (β ℓ + σ ℓ−1 λ ℓ (1 − 1 ℓ ) 2 )γ 2 ℓ−1 (α ℓ − σ ℓ+1 λ ℓ (1 + 1 ℓ+d−2 ) 2 )γ 2 ℓ+1 − (β ℓ + σ ℓ−1 λ ℓ (1 − 1 ℓ ) 2 )γ 2 ℓ−1 .
The result follows.
We will now prove the Carleman estimate by using suitable choices for the weights σ ℓ and γ ℓ .
Proof of Theorem 7.1. Fix a constant µ > 1, choose γ ℓ so that ℓγ 2 ℓ = µ ℓ , and choose σ ℓ = 2δ ℓ where 0 < δ < 1 is fixed. The estimate in Proposition 7.4 implies that
2κδ ∞ ℓ=m µ ℓ u ℓ 2 = κ ∞ ℓ=m σ ℓ ℓµ ℓ u ℓ 2 ≤ κ ∞ ℓ=m σ ℓ λ ℓ ℓ µ ℓ u ℓ 2 = κ ∞ ℓ=m σ ℓ λ ℓ γ 2 ℓ u ℓ 2 ≤ ∞ ℓ=m+1 (α ℓ − σ ℓ+1 λ ℓ (1 + 1 ℓ+d−2 ) 2 )γ 2 ℓ+1 (β ℓ + σ ℓ−1 λ ℓ (1 − 1 ℓ ) 2 )γ 2 ℓ−1 (α ℓ − σ ℓ+1 λ ℓ (1 + 1 ℓ+d−2 ) 2 )γ 2 ℓ+1 − (β ℓ + σ ℓ−1 λ ℓ (1 − 1 ℓ ) 2 )γ 2 ℓ−1 (Xu) ℓ 2 .
Using the formulas in Lemma 4.3 and the fact that σ ℓ = 2δ ℓ , we have
α ℓ − σ ℓ+1 λ ℓ 1 + 1 ℓ + d − 2 2 = (2ℓ + d − 2) 1 + 1 ℓ + d − 2 − 2δ ℓ + 1 λ ℓ 1 + 1 ℓ + d − 2 2 = 2(1 − δ)ℓ + O(1) and β ℓ + σ ℓ−1 λ ℓ 1 − 1 ℓ 2 = (2ℓ + d − 2) 1 − 1 ℓ + 2δ ℓ − 1 λ ℓ 1 − 1 ℓ 2 = 2(1 + δ)ℓ + O(1)
when ℓ is large. These estimates also imply that
α ℓ − σ ℓ+1 λ ℓ 1 + 1 ℓ + d − 2 2 γ 2 ℓ+1 = (2(1 − δ) + O(ℓ −1 ))µ ℓ+1 , β ℓ + σ ℓ−1 λ ℓ 1 − 1 ℓ 2 γ 2 ℓ−1 = (2(1 + δ) + O(ℓ −1 ))µ ℓ−1
for ℓ large.
Using the above bounds and choosing m large enough, we obtain the estimate
2κδ ∞ ℓ=m µ ℓ u ℓ 2 ≤ ∞ ℓ=m+1 (2(1 − δ) + O(ℓ −1 ))(2(1 + δ) + O(ℓ −1 )) (2(1 − δ) + O(ℓ −1 )) − (2(1 + δ) + O(ℓ −1 ))µ −2 µ ℓ−1 (Xu) ℓ 2 ≤ 8(1 − δ)(1 + δ) (1 − δ) − 4(1 + δ)µ −2 ∞ ℓ=m+1 µ ℓ−1 (Xu) ℓ 2 .
This estimate makes sense if 1 − δ > 4(1 + δ)µ −2 , i.e. µ 2 > 4(1+δ) 1−δ . Writing µ 2 = 4(1+δ) 1−δ a where a > 1, the above estimate becomes
2κδ ∞ ℓ=m µ ℓ u ℓ 2 ≤ 8(1 + δ) 1 − 1/a ∞ ℓ=m+1 µ ℓ−1 (Xu) ℓ 2 .
We will now make the choices
δ = 1 2 , a = 2, µ = e 2τ
where τ ≥ τ 0 with τ 0 chosen so that e 4τ 0 > 12. Then one has µ 2 > 4(1+δ) 1−δ , and the estimate takes the required form
κ ∞ ℓ=m e 2τ ℓ u ℓ 2 ≤ 24 e 2τ ∞ ℓ=m+1 e 2τ ℓ (Xu) ℓ 2 .
Remark 7.5. So far, we have considered the case where (M, g) has nonpositive curvature. One could ask if it is possible to obtain weighted estimates whenever (M, g) is a general compact simple/Anosov manifold. In this case the sectional curvatures may be positive, and the most efficient way seems to be to write the Pestov identity as
X v ∇u 2 − (R v ∇u, v ∇u) + (d − 1) Xu 2 = v ∇Xu 2
and use the fact that, by Lemma 7.2,
X v ∇u 2 − (R v ∇u, v ∇u) ≥ α X v ∇u 2 + κ v ∇u 2
for some α, κ > 0. If one does this and runs the argument as in the negative curvature case, one arrives at the following estimate for any u ∈ C ∞ (SM) (with u| ∂(SM ) = 0 in the boundary case):
α ∞ ℓ=0 α ℓ X − u ℓ+1 2 + κ ∞ ℓ=1 λ ℓ u ℓ 2 ≤ (1 − α) ∞ ℓ=2 (λ ℓ − (d − 1)) (Xu) ℓ 2 + α ∞ ℓ=2 β ℓ X + u ℓ−1 2 .
The X + terms on the right can be absorbed in the X − terms on the left, even with weights, just as in the negative curvature case. However, the new (Xu) ℓ terms on the right present problems. If one considers u with u ℓ = 0 for ℓ < m, which is the case relevant for the X-ray transform on m-tensors, then the (Xu) m−1 = X − u m and (Xu) m = X − u m+1 terms on the right cannot be absorbed to the left if m ≥ 2 is large. Moreover, adding weights (i.e. replacing u ℓ by γ ℓ u ℓ ) does not help, since the X − u m term on the right is multiplied by the same γ 2 m as the X − u m and u m terms on the left. Thus this method does not seem to shed new light to the tensor tomography problem of simple/Anosov manifolds (which is still open when dim(M) ≥ 3 and m ≥ 2).
One could still ask if one gets new weighted estimates in the cases m = 0, 1. In these cases the (Xu) 0 and (Xu) 1 terms pose no problem (they are not present on the right). However, replacing u by its weighted version γ ℓ u ℓ in the (Xu) ℓ terms on the right will generate new X − and X + terms, and it seems that absorbing the new terms from right to left requires weights that grow at most mildly. This is not sufficient for dealing with large general connections or Higgs fields, but could lead to a shifted version of the Pestov identity as in Theorem 6.3 but where s has to be in the range −1/2 < s ≤ 1/2. We omit the details.
A regularity result for the transport equation
Let (M, g) be a compact Riemannian manifold with boundary whose geodesic flow is nontrapping, and let A : SM → C n×n be an arbitrary smooth attenuation. If the boundary of M is strictly convex, it was proved in [PSU12, Proposition 5.2] that any solution to Xu + Au = −f in SM with u| ∂(SM ) = 0 is smooth in SM whenever f is smooth. We would like to prove an analogue of this statement, but without assuming that the boundary of M is strictly convex. We will do so at the cost of assuming that the function f is supported in the interior of SM.
Given a smooth f : SM → C n , denote by u f the unique solution to
Xu + Au = −f, u| ∂ − (SM ) = 0.
The function u f may fail to be differentiable due to the non-smoothness of τ . In fact, τ might not even be continuous. Proof. We consider (M, g) isometrically embedded in a closed manifold (N, g) and we extend f by zero. We also extend A smoothly to N. The idea is to use the argument in [Da06, Lemma 2.3] and replace τ locally by other suitable smooth functions. Let ρ(x, v) be any time such that γ x,v (ρ(x, v)) / ∈ M. Consider the function w(t, x, v) defined by the ordinary differential equation
d dt w(t, x, v) + A(ϕ t (x, v))w(t, x, v) = −f (ϕ t (x, v)), w(ρ(x, v), x, v) = 0. We claim that for (x, v) ∈ SM (8.1) u f (x, v) = w(0, x, v).
To prove (8.1) observe that the function u(t, x, v) := u f (ϕ t (x, v)) solves the differential equation
d dt u(t, x, v) + A(ϕ t (x, v))u(t, x, v) = −f (ϕ t (x, v)), u(τ (x, v), x, v) = 0.
Next note that the interval [τ, ρ] can be decomposed into two types of intervals: the first type consists of intervals where the geodesic γ x,v is a maximal geodesic segment in M with boundary points on ∂M, and the second type consists of intervals where γ x,v runs outside M. Recall that outside M, f = 0 and while inside M, I A (f ) = 0. Thus w(τ (x, v), x, v) = 0. Hence (8.1) follows from uniqueness of solutions of ordinary differential equations.
Since γ x,v (ρ(x, v)) / ∈ M, we can take a small hypersurface Σ in N transversally intersecting γ x,v at the point γ x,v (ρ(x, v)) and disjoint from M. Then there is a neighbourhood of (x, v) in SN such that for every θ in this neighbourhood the geodesic γ θ will hit Σ at the time ρ(θ) smoothly depending on θ. If we now use this local function ρ(θ) to define w, using (8.1) we see that u f is smooth around (x, v) since w(0, θ) is.
Remark 8.2. All that is needed is that f has vanishing jet at the boundary. It might be possible to prove that if f = f (x) and I A (f ) = 0, then f has vanishing jet at the boundary following the ideas in [GMT17] or [SU09], but we do not pursue this matter here.
Applications
Finally, we will employ the Carleman estimates proved in this article in various applications as described in the introduction.
Uniqueness results for the transport equation and X-ray transforms. We start with the proof of Theorem 1.4.
Proof of Theorem 1.4. Suppose that f has degree m 0 ≥ 0, and let m ≥ max{ℓ 0 − 1, m 0 }. Then (Xu) ℓ = −(A(u)) ℓ for ℓ ≥ m + 1 and
(Xu) ℓ ≤ R( u ℓ−1 + u ℓ + u ℓ+1 ), ℓ ≥ m + 1.
Inserting this estimate in Theorem 1.1 yields that, for τ ≥ 1,
∞ ℓ=m ℓ 2τ u ℓ 2 ≤ CR τ ∞ ℓ=m+1 ℓ 2τ ( u ℓ−1 2 + u ℓ 2 + u ℓ+1 2 ) ≤ CR τ ∞ ℓ=m (ℓ + 1) 2τ u ℓ 2
where C = C d,κ . If we additionally assume that m ≥ 2τ , then
(ℓ + 1) 2τ = 1 + 1 ℓ 2τ ℓ 2τ ≤ eℓ 2τ , ℓ ≥ m. Thus if m ≥ max{ℓ 0 − 1, m 0 , 2τ } we have ∞ ℓ=m ℓ 2τ u ℓ 2 ≤ CR τ ∞ ℓ=m ℓ 2τ u ℓ 2 .
Now fix τ = τ 0 = max{2CR, 1} and let m = max{ℓ 0 − 1, m 0 , 2τ 0 }. Then the right hand side of the last inequality can be absorbed to the left, and we obtain ∞ ℓ=m ℓ 2τ u ℓ 2 ≤ 0.
Thus u ℓ = 0 for ℓ ≥ m, so u has degree ≤ m − 1 where m = max{ℓ 0 − 1, m 0 , 4CR} where C = C d,κ .
We next show Theorem 1.5.
Proof of Theorem 1.5. Let u = u f be the solution of (X + A + Φ)u = −f in SM , u| ∂ − (SM ) = 0.
Since the attenuated X-ray transform of f vanishes, we have u| ∂(SM ) = 0. This implies that u ∈ C ∞ (SM, C n ); if M has strictly convex boundary this follows from [PSU12,Proposition 5.2], and alternatively if f is supported in the interior of SM we may use Proposition 8.1. We are now exactly in the setting of Theorem 1.4 with A(u) = Au + Φu and l 0 = max{m − 1, 2}, and since f has finite degree it follows that also u has finite degree, i.e. u l = 0 for l ≥ m 0 + 1 for some m 0 . We still need to show that m 0 ≤ m. If this is not the case, we use the transport equation Xu + Au + Φu = −f again to derive X A + u m 0 = 0. Since u m 0 | ∂(SM ) = 0, [GPSU16, Theorem 5.2] implies that u m 0 = 0 and arguing in the same way with u m 0 −1 and so on, we deduce that all u l = 0 for l ≥ m. This shows that f = −(X + A + Φ)u where u has degree m − 1 and u| ∂(SM ) = 0.
Scattering rigidity for general connections in negative curvature. Let us move to our next application, concerning an inverse problem with scattering data. On a nontrapping compact manifold (M, g) with strictly convex boundary, the scattering relation α = α g : ∂ + (SM) → ∂ − (SM) maps a starting point and direction of a geodesic to the end point and direction. If (M, g) is simple, then knowing α g is equivalent to knowing the boundary distance function d g which encodes the distances between any pair of boundary points [Mi81]. On two dimensional simple manifolds, the boundary distance function d g determines the metric g up to an isometry which fixes the boundary [PU05].
Given a connection A and a potential Φ on the bundle M ×C n , there is an additional piece of scattering data. Consider the unique matrix solution U : SM → GL(n, C) to the transport equation XU + AU + ΦU = 0 in SM, U| ∂ + (SM ) = Id.
As discussed in the introduction, the scattering data (or relation) corresponding to a matrix attenuation pair (A, Φ) in (M, g) is the map C A,Φ : ∂ − (SM) → GL(n, C), C A,Φ := U| ∂ − (SM ) .
Theorem 1.6 in the introduction proves that knowledge of C A,Φ uniquely determines the pair (A, Φ) up to gauge. The theorem can be proved by introducing a pseudolinearization that reduces the nonlinear problem to the linear one in Theorem 1.5. The argument is identical to the one used in [PSU12] and [PSUZ16], and we do not need to repeat it here.
Transparent pairs. We next discuss the problem of when the parallel transport associated with a pair (A, Φ) determines the pair up to gauge equivalence in the case of closed manifolds. This problem is discussed in detail in [GPSU16,Pa09,Pa11,Pa12,Pa13], but the results are only for pairs taking values in the Lie algebra of the unitary group. The extension to the group GL(n, R) is non-trivial on two counts. It requires a recent extension of the Livsic theorem to arbitrary matrix groups [Ka11] and our new Carleman estimate for the geodesic vector field in negative curvature.
Since there is no boundary, we need to consider the parallel transport of a pair along closed geodesics. We shall consider a simplified version of the problem, which is interesting in its own right. We will attempt to understand those pairs (A, Φ) with the property that the parallel transport along closed geodesics is the identity. These pairs will be called transparent as they are invisible from the point of view of the closed geodesics of the Riemannian metric.
Let (M, g) be a closed Riemannian manifold, A a connection taking values in the set of real n × n matrices and Φ a potential also taking values in R n×n . The pair (A, Φ) naturally induces a GL(n, R)-cocycle over the geodesic flow ϕ t of the metric g acting on the unit sphere bundle SM with projection π : SM → M. The cocycle is defined by d dt C(x, v, t) = −(A(ϕ t (x, v)) + Φ(π • ϕ t (x, v)))C(x, v, t), C(x, v, 0) = Id.
The cocycle C is said to be cohomologically trivial if there exists a smooth function u : SM → GL(n, R) such that
C(x, v, t) = u(ϕ t (x, v))u −1 (x, v)
for all (x, v) ∈ SM and t ∈ R. We call u a trivializing function and note that two trivializing functions u 1 and u 2 (for the same cocycle) are related by u 2 w = u 1 where w : SM → GL(n, R) is constant along the orbits of the geodesic flow. In particular, if ϕ t is transitive (i.e. there is a dense orbit) there is a unique trivializing function up to right multiplication by a constant matrix in GL(n, R).
Definition 9.1. We will say that a pair (A, Φ) is cohomologically trivial if C is cohomologically trivial. The pair (A, Φ) is said to be transparent if C(x, v, T ) = Id every time that ϕ T (x, v) = (x, v).
Observe that the gauge group given by the set of smooth maps r : M → GL(n, R) acts on pairs as follows:
(A, Φ) → (r −1 dr + r −1 Ar, r −1 Φr).
This action leaves invariant the set of cohomologically trivial pairs: indeed, if u trivializes the cocycle C of a pair (A, Φ), then it is easy to check that r −1 u trivializes the cocycle of the pair (r −1 dr + r −1 Ar, r −1 Φr).
Obviously a cohomologically trivial pair is transparent. There is one important situation in which both notions agree. If ϕ t is Anosov, then the Livsic theorem for GL(n, R) cocycles due to Kalinin [Ka11] (extending the work of Livsic for the case of a cocycle taking values in a compact Lie group [Li71,Li72]) together with the regularity results in [NT98, Theorem 2.4] imply that a transparent pair is also cohomologically trivial. We already pointed out that the Anosov property is satisfied, if for example (M, g) has negative curvature.
Given a cohomologically trivial pair (A, Φ), a trivializing function u satisfies (9.1) (X + A + Φ)u = 0.
If we assume now that (M, g) is negatively curved and the kernel of X A + on Ω m is trivial for m ≥ 1 (i.e. there are no nontrivial twisted conformal Killing tensors (CKTs), see [GPSU16]), then the proof of Theorem 1.5 implies that u = u 0 . If we split equation (9.1) in degrees zero and one we obtain Φu 0 = 0 and du + Au = 0. Equivalently, Φ = 0 and A is gauge equivalent to the trivial connection. Hence we have proved Theorem 9.2. Let (M, g) be a closed negatively curved manifold and (A, Φ) a transparent pair. If there are no nontrivial twisted CKTs, then A is gauge equivalent to the trivial connection and Φ = 0.
Theorem 1. 1 .e
1Let (M, g) be a compact Riemannian manifold with sectional curvature ≤ −κ where κ > 0. Let also φ ℓ = log(ℓ). For any τ ≥ 1 and m ≥ 1, one has 2τ φ ℓ (Xu) ℓ 2 whenever u ∈ C ∞ (SM) (with u| ∂(SM ) = 0 in the boundary case), where C is a positive constant depending only on the dimension of M.
− 1 )
1Xu 2
MS was partly supported by the Academy of Finland (Finnish Centre of Excellence in Inverse Problems Research, grant numbers 284715 and 309963) and by the European Research Council under FP7/2007-2013 (ERC StG 307023) and Horizon 2020 (ERC CoG 770924).
− (u) SM H 1 (SM ) ≤ C P u L 2 (SM ) these statements, note that a unitary connection A introduces the term −(F A u, v ∇u) in the Pestov identity (cf. Lemma 4.2 below), and that when d = 2 the argument leading to [PSU13, equation (6)] shows that higher order tensors introduce a term of the form ∞ k=−∞ |k|(Kv k , v k ) (this extends to any dimension), where K is the Gaussian curvature and the expansion is taken with respect to the vertical Fourier decomposition of functions in the unit circle bundle. Both cases correspond to H 1/2 error terms in (3.3). However, a general (non-unitary) connection A can be introduced to the Pestov identity by writing P as P + v ∇A− v ∇A and using the triangle inequality, leading to an H 1 error term v ∇(Au) . The curvature term −
Lemma 4. 3 .
3If u ∈ C ∞ (SM, C n ) with u| ∂(SM ) = 0 in the boundary case, then
if additionally u| ∂(SM ) = 0 in the boundary case. (We define α −1 = 0.)
Finally, to prove to statement involving Z A we argue as follows. The definition of F A and Lemma 4.) = 0 it suffices to note that if a, b are scalar 1-formssince ∆a = −(d − 1)a and ∆b = −(d − 1)b. Thus invoking the fact that A is unitary
if additionally u| ∂(SM ) = 0 in the boundary case. (We define α −1 = 0.)
Proposition 8. 1 .
1Let f : SM → C n be smooth, supported in the interior of SM with I A (f ) = u f | ∂ + (SM ) = 0. Then u f : SM → C n is smooth.
= (P u, u) = (−∆u, u) + (V u, u) ≥ ∇u 2 .
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| []
|
[
"DERIVED CATEGORIES OF COHERENT SHEAVES AND MOTIVES",
"DERIVED CATEGORIES OF COHERENT SHEAVES AND MOTIVES"
]
| [
"Dmitri Orlov "
]
| []
| []
| The bounded derived category of coherent sheaves D b (X) is a natural triangulated category which can be associated with an algebraic variety X. It happens sometimes that two different varieties have equivalent derived categories of coherent sheaves D b (X) ≃ D b (Y ). There arises a natural question: can one say anything about motives of X and Y in that case? The first such example (see[4]) -abelian variety A and its dual A -shows us that the motives of such varieties are not necessary isomorphic. However, it seems that the motives with rational coefficients are isomorphic Recall a definition of the category of effective Chow motives CH eff (k) over a field k. The category CH eff (k) can be obtained as the pseudo-abelian envelope (i.e. as formal adding of cokernels of all projectors) of a category, whose objects are smooth projective schemes over k, and the group of morphisms from X to Y is the sum ⊕ X i A m (X i × Y ) (over all connected components X i ) of the groups of cycles of codimension m = dim Y on X i ×Y modulo rational equivalence (see[3,1]). In [7] Voevodsky introduced a triangulated category of geometric | 10.1070/rm2005v060n06abeh004292 | [
"https://arxiv.org/pdf/math/0512620v2.pdf"
]
| 11,484,447 | math/0512620 | 2837b0ac767e1b2d581a7fcde6909a5ef544ac90 |
DERIVED CATEGORIES OF COHERENT SHEAVES AND MOTIVES
5 Jul 2011
Dmitri Orlov
DERIVED CATEGORIES OF COHERENT SHEAVES AND MOTIVES
5 Jul 2011
The bounded derived category of coherent sheaves D b (X) is a natural triangulated category which can be associated with an algebraic variety X. It happens sometimes that two different varieties have equivalent derived categories of coherent sheaves D b (X) ≃ D b (Y ). There arises a natural question: can one say anything about motives of X and Y in that case? The first such example (see[4]) -abelian variety A and its dual A -shows us that the motives of such varieties are not necessary isomorphic. However, it seems that the motives with rational coefficients are isomorphic Recall a definition of the category of effective Chow motives CH eff (k) over a field k. The category CH eff (k) can be obtained as the pseudo-abelian envelope (i.e. as formal adding of cokernels of all projectors) of a category, whose objects are smooth projective schemes over k, and the group of morphisms from X to Y is the sum ⊕ X i A m (X i × Y ) (over all connected components X i ) of the groups of cycles of codimension m = dim Y on X i ×Y modulo rational equivalence (see[3,1]). In [7] Voevodsky introduced a triangulated category of geometric
motives DM eff gm (k). He started with an additive category SmCor(k), objects of which are smooth schemes of finite type over k, and the group of morphisms from X to Y is the free abelian group generated by integral closed subschemes Z ⊂ X ×Y such that the projection on
[U ∩V ] → [U]⊕[V ] → [X]
for any open covering U ∪V = X. Triangulated category DM eff gm (k) is defined as the pseudo-abelian envelope of the quotient category H b (SmCor(k))/T (see [7,1] . Therefore, we can work in the category DM gm (k). Moreover (see [7]), for any smooth projective varieties X, Y and for any integer i there is an isomorphism
Hom DMgm(k) (M(X), M(Y )(i)[2i]) ∼ = A m+i (X × Y ), where m = dim Y.
Conjecture 2. Let X and Y be smooth projective varieties and let F :
D b (X) → D b (Y ) be a fully faithful functor. Then the motive M(X) Q (k)[2k] is a direct summand of the motive M(Y ) Q for some integer k ∈ Z.
Suppose, one has a fully faithful functor F : D b (X) → D b (Y ) between derived categories of coherent sheaves of two smooth projective varieties X and Y of dimension n and m respectively. Any such functor has a right adjoint F * by [2], and by Theorem 2.2 from [5] (see also [6, 3.2.1]) the functor F can be represented by an object on the product X × Y, i.e.
F ∼ = Φ A , where Φ A = Rp 2 * (p * 1 (−) L ⊗ A) for some A ∈ D b (X × Y ). With any functor of the form Φ A : D b (X) → D b (Y ) one can associate an element a ∈ A * (X × Y, Q) by the following rule (1) a = p * 1 td X · ch(A) · p * 2 td Y ,
where td X and td Y are Todd classes of the varieties X and Y. The cycle a has a mixed type. Let us consider its decomposition into components a = a 0 + · · · + a n+m , where index is the codimension of a cycle on X × Y. Each component a q induces a map of motives If, in addition, the functor F is an equivalence, then the motives M(X) Q and M(Y ) Q are isomorphic.
Assume now that dim X = dim Y = n and, moreover, suppose that the support of the object A also has the dimension n. Therefore, a q = 0 when q = 0, . . . , n−1, i.e. a = a n +· · ·+a 2n .
It is easily to see that in this case b = b n + · · · + b 2n as well. This implies that the composition β · α : M(X) Q → M(X) Q , which is the identity, coincides with β n · α n . Hence, M(X) Q is a direct summand of M(Y ) Q . Furthermore, since the cycles a n and b n are integral in this case we get the same result for integral motives, i.e. the integral motive M(X) is a direct summand of the motive M(Y ) as well. Thus, we obtain Theorem 1. Let X and Y be smooth projective varieties of dimension n, and let F : Examples of such functors are known, they come from birational geometry (see e.g. [6]). In these examples one of the connected components of supp(A) gives a birational map X Y.
D b (X) → D b (Y ) be
Blow ups and antiflips induce fully faithful functors, and flops induce equivalences. Note that an isomorphism of motives implies an isomomorphism of any realization (singular cohomologies, l-adic cohomologies, Hodge structures and so on).
For arbitrary equivalence Φ
A : D b (X) → D b (Y ) the map of motives α n : M(X) Q → M(Y ) Q ,
induced by the cycle a n ∈ A n (X × Y, Q), is not necessary an isomorphism (e.g. Poincare line bundle P on the product of abelian variety A and its dual A ). However, the following conjecture, which specifies Conjecture 1, may be true. I am grateful to Yu. I. Manin for very useful discussions.
X
is finite and surjective onto a connected component of X. There is a natural embedding [−] : Sm(k) → SmCor(k) of the category Sm(k) of smooth schemes of finete type over k. The category SmCor(k) is additive and one has [X Y ] = [X] ⊕ [Y ]. Further, he considered the quotient of the homotopy category H b (SmCor(k)) of bounded complexes by minimal thick triangulated subcategory T, which contains all objects of the form [X × A 1 ] → [X] and
motive of a variety X by M(X), and its motive in the category of motives with rational coefficients DM eff gm (k) ⊗ Q (and in CH eff (k) ⊗ Q ) by M(X) Q .Conjecture 1. Let X and Y be smooth projective varieties, and let D b (X)≃D b (Y ). Then the motives M(X) Q and M(Y ) Q are isomorphic in CH eff (k) ⊗ Q (and in DM eff gm (k) ⊗ Q ) The category DM eff gm (k) has a tensor structure, and M(X) ⊗ M(Y ) = M(X × Y ). One defines the Tate object Z(1) to be the image of the complex [P 1 ] → [Spec(k)] placed in degree 2 and 3 and put M(p) = M ⊗ Z(1) ⊗p for any motive M ∈ DM eff gm (k) and p ∈ N. The triangulated category of geometric motives DM gm (k) is defined by formally inverting the functor − ⊗ Z(1) on DM eff gm (k). The important and nontrivial fact here is the statement that the canonical functor DM eff gm (k) → DM gm (k) is a full embedding [7, 4.3.1]
α q : M(X) Q → M(Y ) Q (q − m)[2(q − m)]. Thus the total cycle a gives a map α : M(X) Q → n i=−m M(Y ) Q (i)[2i]. Now consider the object B ∈ D b (X × Y ) which represents the (left) adjoint functor F * , i.e. F * ∼ = Ψ B , where Ψ B = Rp 1 * (p * 2 (−) L ⊗ B). One attaches to the object B a cycle b = b 0 + · · · + b n+m defined by the same formula (1). The cycle b induces a map β : n i=−m M(Y ) Q (i)[2i] → M(X) Q . Since the functor Φ A is fully faithful, the composition Ψ B • Φ A is isomorphic to the identity functor. Applying the Riemann-Roch-Grothendieck theorem, we obtain that identity map, i.e. M(X) Q is a direct summand of n i=−m M(Y ) Q (i)[2i]. Denote by M(X) Q the infinite direct sum ∞ i=−∞ M(X) Q (i)[2i]. The total cycle a defined above gives a map α : M(X) Q → M(Y ) Q whose component from M(X) Q (k)[2k] to M(Y ) Q (q− m + k)[2(q − m + k)] coincides with α q for any k. By the same way the cycle b induces a map β from M(X) Q to M(Y ) Q . The above consideration give us that the composition β · α is the identity map. Thus we obtain the following proposition. Proposition 1. Let X and Y be smooth projective varieties and let F : D b (X) → D b (Y ) be a fully faithful functor. Then the motive M(X) Q is a direct summand of the motive M(Y ) Q .
a fully faithful functor such that the dimension of the support of an object A on X × Y, which represents F, is equal to n. Then the motive M(X) is a direct summand of the motive M(Y ). If, in addition, the functor F is an equivalence, then the motives M(X) and M(Y ) are isomorphic.
Conjecture 3 .
3Let A be an object of D b (X × Y ), for which Φ A : D b (X) → D b (Y ) isan equivalence. Then there exist line bundles L and M on X and on Y respectively such that the component a ′ n of the object A ′ := p * 1 L ⊗ A ⊗ p * 2 M gives an isomorphism between motives M(X) Q and M(Y ) Q .
) .
)There exists a canonical functor CH eff (k) → DM eff gm (k), which is a full embedding if k admits resolution of singularities ([7, 4.2.6]). Thus, it doesn't matter in which category (in CH eff (k) or in DM eff gm (k) ) motives of smooth projective varieties are considered. Denote theThis work was done with a partial financial support from grant RFFI 05-01-01034, from the President of RF
young Russian scientists award MD-2731.2004.1, from grant CRDF Award No RUM1-2661-MO-05, and from
the Russian Science Support Foundation.
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Triangulated categories of motives over a field. V Voevodsky, Cycles, transfers, and motivic homology theories. 143Voevodsky, V. Triangulated categories of motives over a field. In Cycles, transfers, and motivic homology theories, vol. 143 of Ann. of Math.Stud, (2000), 188-238.
Steklov Mathematical Institute RAN E-mail address: [email protected]. Steklov Mathematical Institute RAN E-mail address: [email protected]
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[
"Fixed Point Theorems for Hypersequences and the Foundation of Generalized Differential Geometry I: The Simplified Algebra",
"Fixed Point Theorems for Hypersequences and the Foundation of Generalized Differential Geometry I: The Simplified Algebra"
]
| [
"S O Juriaans ",
"J Oliveira ",
"J F Colombeau ",
"E E Rosinger ",
"J Jelínek ",
"M Kun-Ziger ",
"J "
]
| []
| []
| Fixed point theorems are one of the many tools used to prove existence and uniqueness of differential equations. When the data involved contains products of distributions, some of these tools may not be useful. Thus rises the necessity to develop new environments and tools capable of handling such situations. | 10.13140/rg.2.2.32732.05761 | [
"https://arxiv.org/pdf/2205.00114v1.pdf"
]
| 248,496,177 | 2205.00114 | ff4435c68ecaa2e3d1c259ef4a85afde1a9bf96b |
Fixed Point Theorems for Hypersequences and the Foundation of Generalized Differential Geometry I: The Simplified Algebra
30 Apr 2022
S O Juriaans
J Oliveira
J F Colombeau
E E Rosinger
J Jelínek
M Kun-Ziger
J
Fixed Point Theorems for Hypersequences and the Foundation of Generalized Differential Geometry I: The Simplified Algebra
30 Apr 2022
Fixed point theorems are one of the many tools used to prove existence and uniqueness of differential equations. When the data involved contains products of distributions, some of these tools may not be useful. Thus rises the necessity to develop new environments and tools capable of handling such situations.
Introduction
The Theory of Colombeau Generalized Functions is a nonlinear Theory of Generalized Functions which includes Schwartz' Theory of Linear Generalized Functions, i.e., Schwartz Distribution Theory. Colombeau's Theory is well documented by now. Excellent textbooks and articles exist that are pitstops to appreciate and understand the theory and the wide spectra of applications. We refer the reader to some of these excellent text, [1,2,14,16,17,18,19,32,37,38,39,40,43,44,45,46,48,49,50,51,53,61,55,56,57,58,59,62,63,64,65], to get the zest of the basics and relish advanced parts of this theory.
In the Colombeau environments, proving existence for differential equations involving products of distributions in their data has always leaned on classical results to guarantee existence. To achieve this, classical existence results are used, proceeding to prove moderateness and conclude existence of solutions in the environments of Colombeau Algebras. This can be highly nontrivial. One of the setbacks is that most tools used are not intrinsic to these environments. The development of Generalized Differential Calculus (see [3,9]) envisaged the buildout of tools, intrinsic to the generalized environments, making it possible to pin less faith on the classical ones.
Let M be an n−dimensional manifold. The idea of linking a generalized objected M c to M was first employed in [41] where a blueprint was given how to use these objects to solve important problems in General Relativity. Based on this pivotal idea, in [9], the notion of a generalized manifold was introduced. The definition is exactly the same as the classical one the difference being that local charts take values in open subsets of R n and differentiability is checked using the Generalized Differential Calculus. In [3], more details of Generalized Differential Calculus were worked out, showing that it extends and behaves very similar to classical Calculus and an example of a generalized manifold, different from M c , was also given (actually it is a subset of M c ). As far as we know, other examples were not given yet and it remained unclear whether M c was a generalized manifold and whether there existed other examples. A pursue in another direction was the construction of an diffeomorphism invariant Colombeau algebra. This was early undertaken in [20,34] and was settled in the definite in [40] well afore Generalized Differential Calculus was proposed. These are top-notch papers which show that the main obstruction to the construction of such an algebra has a topological nature: Colombeau algebras are ultrametric spaces which naturally mismatch with the classically used topologies. This translate into a highly non-trivial endeavor the creation of an algebra that can be attached to classical manifolds. The last example given in [40] shows that having such an algebra does not necessarily make things much easier when applying the theory to obtain existence of solutions of differential equations having products of distributions in their data. It is essential to observe that an algebra of generalized functions that can be attached to a manifold was also achieved in [60]. Amazingly enough, in this case, technicalities are not that involved.
In [9], all necessary machinery of Generalized Differential Calculus (such as the Inverse Mapping Theorem, The Implicit Function Theorem and others) were proved so that a consistent basis could be laid for a Generalized Differential Geometry. At first, definitions given and results obtained are exactly the same as the classical ones but extend the latter in a non-trivial way. Howbeit, much has yet to be accomplished before this Generalized Differential Geometry unveils its smoldering potential. Generalized Differential Calculus is based on key ideas developed over the years by all prominent researchers in the field but the decisive ideas are due to Kuzinger-Oberguggenberger ( [38]) and 61]). The topology in use (see [4,5,6]) is a slide modification of the sharp topology introduced by Biagioni-Scarpelézos (see [14,61]), yet equivalent to it, is more natural and in harmony with the algebraic structure (see for example [6,7]) of the Colombeau algebras. An interesting fact is that, in the sharp topology, R n embeds as a discrete subset of R n and yet the Generalized Differential
Calculus is a near perfect extension of the Newtonian Differential Calculus (see the Embedding Theorem in [9]). In particular, Classical Space-Time becomes a grid of equidistant points in Generalized Space-Time, a possibility that was raised along time by many physicists and more recently also in [68]. The common distance betwixt grid points could well be glossed as Planck's constant. Can this discontinuity in classical space-time be perceived experimentally? Or at least, can one be convinced that we do have an issue in this direction? This is where the notion of hypersequences steps into the picture. Since classical space-time is a grid of equidistant points it is impossible for classical sequences to converge in this new environment. In particular, it is no longer true that the sequence ( 1 n ) n∈N converges in the ring of Colombeau generalized numbers. But the hypersequence it generates is of the form ( 1 n ) n∈ N and does converge to zero in the Colombeau environment. From the point of view of someone living in generalized space-time, classical convergence of a sequence (x n ) is equivalent to the existence of a n 0 ∈ N such that x n − x m ∈ V 1 (0) if n, m > n 0 . So classically we only measure upto scale α = [ε −→ ε], which is the reason for calling it our natural gauge, the latter being first introduced in [10]. A similar problem occurs when proving existence of differential equations using classical tools to prove moderateness and existence in environments that are like chalk and cheese, in the topological sense, compared to the environment where these tools come from. It takes much more to be convincing.
The paper is structured as follows. In the next section we recall the necessary machinery needed to understand the context and prove subsequent results. In the third section we prove a fixed point theorem for hypersequences, prove that association is a topological and not an algebraic concept and that D ′ (Ω) is discretely embedded in G(Ω) thus proving that classical functions are extremely rare. In the fourth section we prove that M c is a generalized manifold and devote the last section to examples and the enumeration of some results in this new generalized geometry. This is the first of two papers. The second paper is in the context of the full algebra of Colombeau Generalized Functions (see [35]) thus completing the proposal of this new Generalized Differential Geometry as a roundabout route to define generalized functions on manifolds.
The notation K, for the ring of Colombeau generalized numbers, was introduced by Colombeau. However, developments overtime show that it is more reasonable to use the notation K to denote this ring. Here we will still be using Colombeau's original notation to be consistent with the notation in [3,4,5,9]. This paper was written while the second author held a pos-doc position at IME-USP, the University of São Paulo-Brazil. The dimension invariance theorem of section four and some of the examples of the last section are part of his Ph.D. thesis ( [52]) written under the supervision of the first author.
Preliminaries
We shall mainly work over the field R of real numbers but all results also hold for C. This is the reason why sometimes we use K to denote either of these fields. One could rightfully ask "why not consider the field Q ?" The answer is simples: The Colombeau Theory constructed using R is the same as the Colombeau Theory constructed using Q since real numbers can be seen as nets of rational numbers. The reason why we end up with a bigger structure, which is not a field but never the less very interesting, is because we mod out some, but not all, nets converging to zero. These surviving nets, converging to zero, are the infinitesimals which inhabit the halos of the elements of the newly formed environments.
Set I =]0, 1], I η =]0, η], for η ∈]0, 1[ and let α be the identity map α : I −→ R, α(ε) = ǫ. We shall denote, once in a while, α n = α n and call α the standard or natural gauge. Nearly all results in this paper can be proved for other gauges using the already existing results for these gauges (see [54,66]).
A map (also called a net) x : I −→ R is moderate if |x| < α r , for some r ∈ R, i.e. |x(ε)| < ε r , ∀ε ∈ I η = ]0, η], ∃ η < 1. Denote the set of moderate maps by E M and by I = {x : x is moderate and |x| < α n , ∀n ∈ N}. For x ∈ E M , denote by V (x) = Sup{r ∈ R : |x| < α r } and set x = e −V (x) . Then I is a radical ideal of the ring E M and setting, R := EM I , we have that (R, ) is an ultrametric partially ordered topological ring whose group of units, Inv(R), is open and dense (see [10]). The latter property is essential in developing the Generalized Differential Calculus ( [3,9]). This topological ring, (R, ), is called the ring of Colombeau Generalized (real) numbers. A generalized number x ∈ R is a unit if and only if |x| > α r for some r ∈ R and it is a non-unit if and only if there exist a nontrivial idempotent e ∈ B(K) such that e · x = 0 (see [10,12]). In particular, a generalized number is either a unit or a zero divisor. The ring R, contains R as a discrete subfield. Actually, R is a grid of equidistant points in R. The latter is a partially ordered ring whose maximal ideals and idempotents have been completely determined (see [9,10,12,64]). The partial order is not intrinsic but stems from the order of R. This is maybe the only definition that is not, yet, intrinsic. Distance emerges from this order and that is why it is important to understand order. In the references we just mentioned, one finds the following facts: the Jacobson Radical of R is trivial, its ideals are convex, its Krull dimension is infinite and it has a minimal prime which is also a maximal ideal. Its Boolean algebra, B(R), consists of {0, 1} and positive elements each of which is a characteristic functions of a subset S ⊂ I, such that 0 ∈ S ∩ S c , where the last two bars stand for topological closure in R. The set of these subsets is denoted by S and was defined in [10]. Ultrafilters of S partially parametrize prime and maximal ideals of K. It also holds that B(R) = B(C) (see [12]). In particular, the Heaviside function H / ∈ B(R), i.e., H 2 = H (see [25]).
H. Biagioni and Scarpelézos were the first to suggest the topology, defined above, for R. It came to be known as the sharp topology turning R into a complete ultrametric algebra and hence, its topology is generated by balls. In [4,5] it was shown that this topology was also generated by the sets V r [x] = {y ∈ R : |y − x| < α r }, balls, with generalized numbers as radii , compatible with the ring structure. It is easily seen that
B 2r (0) ⊂ V r [0] ⊂ B r/2 (0) if r > 0.
Let Ω ⊂ R n be an open subset with an exhaustion by relatively compact subset Ω m ⊂ Ω. Consider nets p = (p ε ), p ε ∈ Ω m(p) ∀ε ∈ I η , m(p) ∈ N, such that the net ( p ε ) ∈ E M . Factoring out nets p for which also ( p ε ) ∈ I, give rise to a subset of R n which is denote by Ω c (see [37,38]). The notation Ω is used if one does not require the existence of m(p). The algebra of Colombeau generalized functions G(Ω) (see [2,16,17,37] for the original definition), defined on the open subset Ω ⊂ R n , can be viewed as C ∞ −functions defined on Ω c ⊂ R n and taking values in R (see [37,38,3,9]). In [9] (see also [3]) the foundation of Colombeau Generalized Calculus is laid and shown that G(Ω) can be embedded into C ∞ ( Ω c , R). In particular, Schwartz space of linear distributions, D ′ (Ω), can be seen as infinitely differentiable functions where differentiability is defined a la Newton. So we have come full circle from seeing elements of D ′ (Ω) as linear maps, and hence not undergoing variation, to seeing them as functions undergoing variation (note that, classically, derivation in D ′ (Ω) is defined without the use of variation). (An interesting fact is that in the proof of the differentiability of objects in the construction of an invariant algebra in [40], Schwartz distributions are treated as linear maps. In the proof of their embedding in this newly constructed algebra the linearity of a Schwartz distribution is not used. When a Differential Calculus for these invariant algebras is ready, distribution will have both the classical and generalized meaning). More on this, in the next section (see also the remark after [37, Proposition 1.6.3, 1.7.28] and the remark after the latter). For example, the Delta Dirac function δ is the derivate of H, the Heaviside function, and the calculation is just the ordinary calculation from classical Calculus. Another interesting point is that, in the presence of moderateness, negligibility only has to be checked at level 0. This is mentioned and proved in several references. See, for example, [40, paragraphs after I.Theorem 7.13]. Generalized Differential Calculus allows to give an easy proof of this fact. In fact, in the presence of moderateness negligibility at level 0 is a statement about point values: Iff (ε, x) is moderate, then it defines an element f ∈ C ∞ ( Ω c , R). Given x ∈ Ω c there exists a compact subset K ⊂ Ω containing a representative of x. Moderateness at level 0 implies that (f ) |K ∞ = 0 (note that this is exactly the uniformity on K). It follows that f (x) = 0, proving that f = 0 in Ω c . Generalized Differential Calculus gives that ∂ β f = 0, ∀ β ∈ N n . Since the Embedding Theorem [9, Theorem 4.1] tells us that derivations commutes with the embedding, it follows that f = 0 in G(Ω). So there is no need to check other levels. The same proof holds for the full algebra and should also work for the invariant algebra once we have at hand a Generalized Differential Calculus for the latter. Note however that one must have negligibility at level 0 and not just point values of elements of Ω being zero. To see why, consider f = xδ ∈ G(R), where δ = [(ρ ε )], ρ a mollifier.
We have that f (x) = 0, ∀x ∈ R. Also, for
x 0 ∈ R, f (x 0 α) = x 0 · ρ(x 0 ), for ϕ ∈ D(Ω) we have that R f (x)ϕ(x)dx = [( R (xϕ)(x)ρ ε )] and hence, since (ρ ε ) is a delta-net, R f (x)ϕ(x)dx = ( R xϕ(x)ρ ε (x)dx) ≈ (xϕ)(0) = 0.
This proves that f ≈ 0 but f = 0. This example also shows an interesting phenomena: f (0) = 0 and thus for x ∈ V r (0), a small enough sharp neighborhood of 0, f (x) ∈ V 1 (0). For classical mathematics (and hence measurements) f (x) = 0 and thus seemingly does not interfere with physical reality. But for histories of the form x = x 0 α, x 0 ∈ R we have that f (x) = x 0 · ρ(x 0 ) ∈ R and thus interfere with physical reality. These "waves" of appearing and disappearing from physical reality are the source of the turbulence effects we see in physical reality. And it can be worse. Consider g(x) = f k , k ∈ N. Then g(0) = 0, g(α) = (ρ(1)) k . So if ρ(1) > 1 and k is large then these "waves" coming from V r (0) which we cannot measure, can effect in a non-trivial way physical reality. Note also that
δ(0) = α −1 · ρ(0)
is an infinity we can not measure but it is cancelled out on histories, x 0 α near 0, giving us a real number, f (x 0 α) = x 0 · ρ(x 0 ) ∈ R, that we can measure. Even though the history x 0 α is near 0, the position in physical reality where we observe the effect can be faraway from 0 (in this case at x 0 ∈ R) and the result of the measurement x 0 ρ(x 0 ) becomes small as x 0 goes to infinity. So turbulence should be the interaction of elements of B 1 (0) and infinities, i.e., elements of norm greater then 1, producing a measurable but not predictable effect on physical reality. The nonpredictability stems from the fact that spheres in Colombeau environments are clopen sets and classical space-time is a grid of equidistant points. Jumps from one sphere to another sphere occur multiplying with the α r , r ∈ R, which form a discrete chain of quanta. The construction carried out above with Ω ⊂ R n can also be carried out with any subset X ⊂ R n . In fact, consider X with the induced topology, consider an exhaustion (X n ) of X by relatively compact subsets and proceed as before. The set obtained in R n will be denoted by X c . We embed X into X c using constant
nets p = (p ε ), p ε = x ∈ X, ε ∈ I η .
It is clear that, in the sharp topology, X is a discrete grid of equidistant points contained in X c . The remarkable thing is that, in case of a submanifold M of R n , Generalized Differential Calculus on M c will be a generalization of the Classical Differential Calculus on M , although M is discretely embedded in M c . Let q be any norm on R n . Extending it in the obvious way to R n , we define for x = (x 1 , · · · , x n ) ∈ R n , x q = q(x) ∈ R and q x ∈ R to be the norm of q(x) as an element of R. If q(x) = x 2 1 + · · · + x 2 n then we write x q = x 2 (see [3,12]). Since all norms on R n are equivalent, it is easily seen that q x does not depend on the norm q and thus we shall write it as x .
The positive cone, R + , of R is not an open subset. In fact, let e be an idempotent and set x n = e − (1 − e) · α n . Then |x n | = e + (1 − e) · α n . We clearly have that x n is not in the positive cone but x n −→ e. However if we let Inv(R) + = Inv(R) ∩ R + then we have:
Lemma 1. Let Inv(R) + = Inv(R) ∩ R + . 1. Inv(R) + is an open subgroup of R. 2. Let t ∈ [0, 1] and x, y ∈ Inv(R) + . Then tx + (1 − t)y ∈ Inv(R) + . Proof. The fact that Inv(R) + is a subgroup of R is clear. So let x ∈ Inv(R) ∩ R + . Since x is invertible, there exists α r such that x > α r . If y ∈ V r (x) then |y − x| < α r and thus 0 < x − α r < y.
On the other hand, since Inv(R) is open and the α t 's form a totally ordered set, we may take r such that V r (x) ⊂ Inv(R).
To prove the second part, take m > 0 such that α m < min{x, y}. Then if follows that tx The lemma shows that there can not exist a continuous curve whose initial value is negative, its final value is positive and at all other instants its values are comparable with 0. This is exactly what is needed to define the notion of orientation on generalized manifolds.
+ (1 − t)y ≥ tα m + (1 − t)α m = α m and thus, tx + (1 − t)y ∈ Inv(R).
The ideas to consider nets of point in R n was introduced in [32,37,38]. This was used in [3] to define the notion of membranes and histories in R n .
Subsequently, in [46], the notions of internal and strong internal sets (internal sets are generalization of membranes) were introduced, inspired also by concepts of nonstandard analysis. In this same paper, ( [46]), very strong and relevant properties involving these notions were proved. For example, it is proved that strongly internal sets are open subsets of R n where as internal sets are closed subsets of the same space. We will be using freely the results contained in these references. Given a net (A ε ) in R n we shall write it also as A α , being α = [ε → ε] our natural or standard gauge. For an idempotent e ∈ B(K) we define eα = e(ε)ε, meaning that when e(ε) = 0 this index will be omitted. We also write eA α for the net A eα , ∂A α for the net (∂A ε ) (the boundary) and int(U ) for the set of interior point of a subset U ⊂ R n . Given a net A α ⊂ R n of subset of R n , we denote the membrane, or internal set, it originates by [A α ] and the strongly internal set it originates by A α . As mentioned above, internal sets are closed in the sharp topology while strongly internal sets are open in the sharp topology. We say that (A α ) is regular if there exists k ∈ N such that at each boundary point A ε we can inscribe balls of radius ε k tangent to ∂A ε and contained in int(A ε ) and another ball of the same radius tangent at the same point but contained in (int(A ε )) c . As a result, the volume of a regular net is a unit since [3]). For example, this is the case if the boundaries are compact hyper surfaces whose encompassing volumes are not shrinking too fast.
vol(A α ) ≥ vol(V k (0)) = πα 2k ∈ Inv(K) (see
Lemma 2. Let (A α ) be a net in R n and U = A α its strong internal set. Then
∂ A α = [∂A α ]. Proof. For z ∈ ∂ A α , there exist sequences (z n ) ⊂ A α and (p n ) ⊂ ( A α ) c both converging to z in K n . Let w = dist(z, [∂A α ]) be the distance of z to the membrane [∂A α ]. If w = 0 then z ∈ [∂A α ].
If not, then there exist e ∈ B(K) and t >> k such that e · w > e · α t . In particular, dist(z ε , ∂A ε ) > ε t = 0 if e(ε) = 1. Hence, for ε such that e(ε) = 1, we have that either z ε ∈ int(A ε ) or z ε ∈ int((A ε ) c ). Consequently, we may write e = e 1 + e 2 , a sum of orthogonal idempotents, such that e 1 · z ∈ e 1 (int(A α )) and e 2 · z ∈ e 2 (int(A α ) c ). On the other hand, since p n → z , there exists n 0 such that if n > n 0 we have that dist(e 1 · p n , e 1 · [(∂A α )]) > e 1 · α 7t . But since e 1 · z ∈ e 1 · ∂ A α this implies that e 1 · p n ∈ e 1 · A α , a contradiction, unless e 1 = 0. If so, then revers the roles of z n and p n obtaining another contradiction. Thus we have that w = 0 and the result is proved.
In case A α consists of intervals J ε =]a ε , b ε [⊂ R, uniformly bounded, we have that ∂ A α = Interleaven{a, b}, where a = [(a ε )] and b = [(b ε )].
The notion of interleaven is defined in [46] which is as follows: the interleaved of a set
X ⊂ R n is the set of all finite sums m i=1 e i · x i , with x i ∈ X and {e 1 , · · · , e m } a
complete set of mutually orthogonal idempotents in R, i.e., e i · e j = 0 if i = j and m i=1 e i = 1. Note that the latter definition is not exactly the one used in Algebra. We stick to this one since in K there are no primitive idempotents. We extend the definition of interleaving allowing that the number of idempotents involved in the sum of the interleaving is countable and not necessarily finite. Interleavings can also be done with hypersequences and elements of C ∞ (Ω) (see the next section). The expressions entanglement or entertwine express the same idea, since several points are connected in the same net that cannot be undone since it is a point in generalized space-time. This is actually the way that new points in generalized space-time are created. Observing a point x corresponds to the creation of the point x · α n , where α n is defined in the second paragraph of the next section. Hence, observing is seeing a part of the interleaving x. An observation does not change the part of the point that it observes if and only if α n is an idempotent. Consider again f (x) = xδ. The halo(0) = B 1 (0), or the halo of any other point, contains information of any subset of R n via the histories R n · α r , r > 0 or, in general R n · y, y ∈ B 1 (0) and their interleavings. In the same way, information of any subset of R n is contained in the complement of B 1 (0) via histories R n · α r , r < 0. This can also be seen using the homeomorphisms of K n like those whose existence are proved in [10, Theorem 3.3, Theorem 3.4]. The same holds for subsets of K. We can entertwine the history of points x 0 = x 1 ∈ R using the notion of interleaving: x = (x 0 e 1 + x 1 e 2 )α, e 1 + e 2 = 1, the latter being idempotents. As a result, the measurement, f (x), is also an
entertwine: f (x) = x 0 ρ(x 0 )e 1 + x 1 ρ(x 1 )e 2 which
is what is measured in physical reality. We proceed to give an interpretation of this measurement in probabilistic terms. Consider an entertwine
i e i · x i . For each idempotent e ∈ B(R) involved
in the sum, there exists a set S e ∈ S, (see [10, Definition 4.1]), such that e = χ Se is the characteristic function of S e (see [10,12]). If there exists η 0 > 0
such that ]0, η 0 ] ∩ S e is measurable, define µ(e) := lim η→0 1 η η 0 χ Se dµ . Since,
in an interleaving, the idempotents involved form a complete set of mutually orthogonal idempotents, it follows that i µ(e i ) = 1. Hence the µ(e i )'s can be seen as probabilities and, being countable in number, there exists i 0 such that µ(e i0 ) > 0. The interpretation is that whenever µ(e i ) > 0 the measurement at the corresponding point x i is more likely to be obtained because f (x i ) will appear with the same probability in the resulting measurement (see the example in the previous paragraph). We say that the entanglement is a complete
entertwine if µ(e i ) > 0, ∀ i and is a perfect entertwine if µ(e i ) = µ(e j ), ∀ i, j.
In the latter case the number of idempotents involved must be finite. Since measurements involve the same generalized function f , the whole history of measurement is determent and cannot be changed unless the entanglement is undone. One can let the x i 's in an interleaving take values in disjoint subsets X i ⊂ R n letting the probabilities relate to the sets X i which, for example, can be regions in physical reality. In this case, an interleaving can be seen as a function from x : I −→ i X i . For example, rolling a dice produces an
entertwine x = 6 i=1 e i · i, with X i = {i},
being perfect only if the dice is honest.
A Generalized Fixed Point Theorem
In this section we prove a fixed point theorem which is one more piece of the Generalized Differential Calculus whose development started in [3,9]. All the features of this calculus have been extended in [30] to the context of Robinson-Colombeau rings of generalized numbers which includes the fields K/M, where M ✁ K is maximal (see [10,64]). In [30] a differential calculus is also developed for the latter rings and the idea of membrane extended.
In the sequel, ideas contained in [29,42] will be used. Let N ⊂ R be the set of generalized numbers with a representative in N I and N ∪ {∞} elements of the form e · +(1 − e) · ∞, e 2 = 1. Another way to view these elements is to consider E M (N) and factor by the ideal I defined in the previous section. The elements of N are called hyper natural numbers. In the same way one can define the ring of hyper integers Z. We extend the notation α n , n ∈ N, introduced in [10], to the case when n ∈ N ∪ {∞}: if n = [(n ε )] then α n = [(ε nε )]. We can extend this definition to n ∈ Z as long as the set {n ε < 0} is bounded, the reason to require this being obvious. With this notation, idempotents are also of the form α n , where, in this case, n consists of a string of 0's and ∞'s.
A hypersequence is a map x : N −→ G(Ω) and is denoted by (x n ). If x( N ) ⊂ K, we say that (x n ) converges to L ∈ K if given r > 0, there exists n 0 ∈ N such that if n > n 0 then L − x n ∈ V r (0).
Since we are in a Hausdorff space, limits are unique whenever they exist. Such a hypersequence (x n ) is a Cauchy hypersequence if given r > 0 there exists n 0 ∈ N such that if m, n > n 0 then x m − x n ∈ V r (0). K being a complete metric space, we have:
Lemma 3. Let x = (x n ) be
a hypersequence. Then x is a convergent sequence if and only if it is a Cauchy sequence if and only if for each
r ∈ R * + there exists n 0 ∈ N such that n > m > n 0 implies that x n − x m ∈ V r (0).
The sequence x = (x n = 1 n ), n ∈ N does not converge in K because its elements form a grid of equidistant points. However, the hypersequence x = (x n = 1 n ), n ∈ N, converges to 0 ∈ K. In fact, given r > 0, take α −r ≤ n 0 = [(⌊ε −r +1.5⌋)] ≤ α −(r+1) . If n > n 0 , we have that 1 n ∈ V r (0). We also know that n∈N 1 n diverges. However if one sums over a countabel subset of N containing a finite number of elements of norm one then the sum converges. Recall that in this setting a series a n converges if and only if a n −→ 0. Let r ∈ ]0, 1[ be fixed and suppose that we want r n ∈ V t (0), where n = [(n ε )]. For this to occur one must have r nε < ε t . From this it follows that
n ε > ( −t | ln(r)| ) · ln(ε)
. Hence we may take n ε = 2 · −t | ln(r)| · ln(ε)) . Note that n < α −1 and hence is moderate (actually its norm equals 1). Clearly, for any m > n we have r m ∈ V r (0). This proves that the hypersequence (r n ) converges to 0. Any sequencex 0 : N −→ K defines a mapx : N −→ K I in the obvious way:x n = (ε −→x 0nε ). Ifx is moderate then it defines a hypersequence x. If there exists n 0 ∈ N and L ∈ K as = K + K 0 (see [10]) such that n > n 0 implies x(n) − x(n 0 ) ∈ V 1 (0), then the sequence (x 0n ) n∈N converges to L 0 ∈ K, with L 0 ≈ L, in the classical sense and the whole history of measuring this convergence is contained in the sentence "n > n 0 implies
x(n) − x(n 0 ) ∈ V 1 (0)".
Conversely, if (x 0n ) n∈N converges to the real number L 0 and for each ε one choses n ε minimal such that n > n ε implies |x n − L| < ε then n = [(n ε )] must be an element of N if x were to converge. This shows that a sequence of measurements can have precision α k0 , for some k 0 , but not for all k ∈ N. It might be possible to infer from this if classical space-time is discontinuous (unless we declare it continuous and stop measuring beyond V 1 (0)). Since it is not at all clear that the convergence ofx 0 implies de convergence of x, one might question the definition of the notion of integral given in [3,9,37]. We shall prove that, at least in this case, limits do exists and are equal. It is important that this is true if we want that classical theories also hold in the Colombeau environment. If the classical sequence (x 0n ) converges to L 0 and the hypersequence it induces converges to L ≈ L 0 , but L = L 0 , then this might be the reason for deviations or strange behavior of systems where L 0 is used. We shall come back to this again further in this section when we consider the notion of association in G(Ω). As we shall see, the same caution should be taken when dealing with differential equations.
Looking at classical continuity of a function f at a point x 0 , its history of measurements produces a function δ(ε) such that
x − x 0 ∈ V [δ(ε)] (0) implies that f (x) − f (x 0 ) ∈ V 1 (0) with V [δ(ε)]
(0) defined accordingly and noting that continuity implies that we can take the class [δ(ε)] ∈ K 0 . If [δ(ε)] ∈ Inv(K) then there exists r > 0 such that V r (0) ⊂ V [δ(ε)] (0). Once again, classical theory stops measuring at V 1 (0) and one declares a classical function continuous in
x 0 once f (x) − f (x 0 ) ∈ V 1 (0)
. Again, scales less then α are not considered because they are not "detected" by the definitions of classical analysis. Generalized Differential Calculus, links all scales providing a commonage, generalized spacetime.
Let f ∈ G(Ω) and let [K ε ] be a membrane (see [3]). To keep things simple we suppose that K ε = K is compact for all ε and contained in an open and relatively compact subset Ω m ⊂ Ω. Note however that the result obtained also holds for a general membrane. Given a fixed dV ∈ { 1 n , α r : r ∈ I, n ∈ N }, consider the partition P of norm dV of K contained in Ω m , i.e., for each ε ∈ I and dV < 2 µ(Ωm) , P ε has norm dV ε ∈ { 1 nε , ε r }, where µ(Ω m ) denotes the Lebesgue measure of Ω m . Since K is compact, there exists x 0 , x 1
∈ K such that m = f (x 0 ) ≤ f (x) ≤ M = f (x 1 ), ∀x ∈ K. Let s(f, dV ), S(f, dV )
and S(f, dV, * ) be, respectively the lower and upper Riemann sum and any other starred Riemann sum with this dV as the norm of the partition. Then [37,3,9]. From this it follows that the classical and generalized limits are the same and if r = 1 then we already have that s(f, dV ), and S(f, dV ), S(f, dV, * ) are all associated to If ϕ ∈ D(Ω) is non-negative, then
s(f, n) ≤ S(f, dV, * ) ≤ S(f, dV ) and |S(f, dV ) − s(f, dV )| ≤ α −N · µ(Ω m ) · dV , where N > 0 is such that (∇f ) |K ∞ ≤ α −N . Choosing dV < α N +r+1 (this is possible because the hypersequence ( 1 n ) converges to zero), we have that |S(f, dV ) − s(f, dV )| ∈ V r+1 (0) and thus {s(f, dV ), S(f, dV ), S(f, dV, * )} ⊂ V r ( K f (x)dx), where K f (x)dx) is as defined inK f ϕ(x)dx) = f (c)· K ϕ(x)dx, for some c ∈ K.
In particular, if ϕ ∈ A 0 (Ω) (see [2]) is positive, then
K f ϕ(x)dx) = f (c)
. This is useful when looking at the notion of association in G(Ω).
Fix k ∈ R * + and, in E M (N), definex(n)(ε) = χ S(n) · ε k , where χ S(n) is the characteristic function of the set S(n) = {(n ε ) −1 : ε ∈ I}. If n ∈ N then S(n) = {n −1 } andx(n) would be non-zero only when ε = 1 n , having value 1 n k . If these were measurements or observations, and since we cannot make infinitely many measurements or observations, the result would be points of the sequence In the introduction of [33] there are two examples that are worth rewriting into this context. The first is that of a point mass of weight 1 at a point x 0 on the real axis. Consider the membrane M = V 1 (x 0 ). Then the corresponding functionals can be written as
L(ϕ) = 1 vol(M ) M ϕ(x)dx = 2α −1 M ϕ(x)dx By [3, Proposition 3] we have that L(ϕ) = ϕ(c), for some c ∈ M . Since ϕ ∈ D(R), we have that ϕ(M ) ⊂ B 1 (ϕ(x 0 )) = halo(ϕ(x 0 )
) and hence ϕ(c) ≈ ϕ(x 0 ), the latter being what measurements give us, but ϕ(c) ∈ R being the actual value. We can extend this to ϕ ∈ G(R) as (which will be defined yet in this section) by taking
M = V k (x 0 ) such that ϕ(M ) ⊂ B 1 (ϕ(x 0 )).
The other example is that of a dipole at 0 with moment 1. In this case, the functionals can be written as
M (ϕ) = α −1 · (ϕ(α) − ϕ(0))
This amounts to ϕ(α) = ϕ(0) + M (ϕ) · α. Expanding around 0 gives us that M (ϕ) − ϕ ′ (0) ∈ V 1 (0), i.e., M (ϕ) ≈ ϕ ′ (0), the latter being what we measure, the actual value being M (ϕ) ∈ R. Again, this can be extended to G(R). In both cases, our measurements are the only "real" point in the halo of the actual values. Howbeit, classically we cannot differentiate among points in a halo. From the classical point of view, each halo contains only one point but, as we already saw, they do interfere in physical reality. In both examples, the measurable effect in physical reality is a product of an infinity, 2α −1 respectively α −1 , and an infinitesimal, M ϕ(x)dx respectively ϕ(α) − ϕ(0), both accredited and coexisting in harmony in this generalized milieu.
In the generalized environments the history of measurements and of convergence is captured and not each measurement separately (however arbitrarily). If the problem lies in the classical mathematical tools that we use and not in space-time its self, then we hope that these examples help to convince that the Colombeau environments, in particular generalized space-time, are environments that maybe should be considered. See also [37, Section 1.6] and the references mentioned there.
For the reader's sake, we recall the basics of the sharp topology (see [61,14,4,5,6]). Let (Ω m ) be an exhaustion of relatively compact and open subsets of Ω ⊂ R n . Given f ∈ G(Ω) and (m, p) ∈ N 2 , define V mp := Sup{a ∈ R | ∀β ∈ N n , |β| ≤ p, ∂ β f (ε, · ) Ωm = o(ε a ), for ε small } and D mp (f, g) := exp(−V mp (f −ĝ)), whereˆstand for representative. The latter are pseudo-ultrametrics defining the Biagioni-Scarpalézos sharp topology on G(Ω) (see [1, Definition 1.9, Proposition 1.10 and 1.11]). This topology is proved to be equivalent to the topologies given in [4,5,6]. As observed in these references, this topology is metrizible and, with the notation given above, an ultrametric in G(Ω) is given by
D(f, g) = sup 2 · D mm (f, g) 1 + D mm (f, g) : m ∈ N .
With the notation of [4,5,6]
, W k m,r (0) = {f ∈ G(Ω) | |∂ β f (x)| ∈ V r (0), ∀ |β| ≤ k, ∀ x ∈ Ω m }, V r (0) = {x ∈ K : |x| < α r } and ∂ β f β,m := [ε −→ (∂ β f ε ) | Ωm ∞ ].
By [5,Theorem 3.6], the sets W k m,r (0) form a filtered basis of the sharp topology of G(Ω).
If [37]). In the latter case, one says that f and g are test equivalent and in the former case one says that they are associated. In [10], K 0 = {x ∈ K : x ≈ 0} was first formally given a notation as was K as = K + K 0 (see [ [25]). Confusion should not arise between G(Ω) 0 and G 0 (Ω), the original Colombeau algebra (see [37]). Association in Colombeau algebras has been presented as an algebraic notion substituting equality in some sense. We proceed to prove that it is in fact a topological notion and use this to prove that, in a topological sense, Schwartz generalized functions, and hence classical solutions of differential equations, are scarce.
f, g ∈ G(Ω) then f ≈ g iff Ω (f − g)ϕdx ∈ K 0 and f ∼ g iff Ω (f − g)ϕdx = 0 ∈ K, ∀ϕ ∈ D(Ω) (see
Proposition 1. Let f, g ∈ G(Ω). Then the following hold.
1. If g ∈ halo(f ) then g ≈ f . 2. B 1 (0) = halo(0) ⊂ G(Ω) 0 . 3. If f m ∈ K 0 , ∀ m, then f ∈ G(Ω) 0 . 4. If Im(f ) ⊂ K 0 then f ∈ G(Ω) 0 . 5. G(Ω) as = D ′ (Ω) + G(Ω) 0 .
Proof. To prove the first two items, by hypothesis, h := f − g ∈ B 1 (0). Since D(h, 0) < 1, it follows that D mm (h, 0) < 1, ∀m ∈ N and hence there exists a decreasing sequence ((r m ) ,
r m > 0, ∀ m), such that V mm (h) = r m > 0, ∀m ∈ N. Consequently, h ∈ m∈N, W m m,rm . In particular, h(x) ∈ V rm (0), ∀x ∈ Ω m , i.e.,
|h(x)| < α rm , ∀x ∈ Ω m . Let ϕ ∈ D(Ω) and m ∈ N be such that supp(ϕ) ⊂ Ω m ; then |
Ω hϕdx| = | Ωm hϕdx| ≤ α rm · ϕ ∞ · µ(Ω m ). Hence Ω hϕdx ∈ K 0 . If ϕ ∈ D(Ω) then there exists m such that supp(ϕ) ⊂ Ω m . Hence | Ω f (x)ϕ(x)dx| = | Ωm f (x)ϕ(x)dx| ≤ f m · ϕ ∞ · µ(Ω) ∈ K 0 , proving the third item.
To prove the fourth item, use an appropriate Riemann sum, as was shown to exist in the examples preceding the proposition, and the fact that K 0 is a ring. It also follows by the previous item. The fifth item is an obvious statement.
If f ≈ 0 then for each x 0 ∈ Ω and each B r (x 0 ) ⊂ Ω we have that there exists c r ∈ B r (x 0 ) such that f (c r ) ∈ K 0 . This follows from the observation at the end of the paragraph about Riemann sums.
In [9] it was proved that R n is a discrete subset of R n and that if r ∈ K * and x ∈ K then rx = x . Our next results state that the same is true for D ′ (Ω) and C ∞ (Ω). The first part of the next corollary is in fact nothing more than a topological interpretation of [37, Proposition 1. Since g ≈ f if g ∈ halo(f ), it follows that uniqueness of solutions and association can be seen as statements about halos. That is, the weaker notion of equality called association is a topological notion! Equality is an algebraic notion! Elements of G(Ω) that are associated are indistinguishable one from another from the classical point of view (See also [37,Section 1.6]). An element of G(Ω), not in the halo of any point of D ′ (Ω), is called a vampier in [37]: it has no distributional shadow. Note however that it can be that, multiplying it with an e ∈ B(R) it has a shadow, even infinitely many and thus, it can flaunt omnipresence (see the notion of entertwine or of support in the next section). This shows that the construction of Ω c starting from Ω is the same as the construction of G(Ω) starting from D(Ω). All that is classical becomes discrete in the generalized environments. Once again, no sequence converging in D ′ (Ω) can converge in G(Ω) and, classically, measuring convergence is stopped at the sets W m p,1 ! Again, it appears that one has to call upon hypersequences to fix this.
That being so, even in the Colombeau Algebras there is just one notion of equality, i.e., the classical one! Association is not equality in any sense but a topological statement and test equivalent looks more like classical equality but is not. In fact, once again, let f = xδ, q ∈ N and let the delta-net be induced by the mollifier ρ. Then
Ω f ε ϕdx = ε · Ω zρ(z)ϕ(εz)dz = ε · Ω zρ(z)[ϕ(εz) − ϕ(0)]dz = ε q (q−1)! · Ω zρ(z)ϕ(θ · εz)dz = o(ε q ).
This proves that Ω f ϕdx = 0 ∈ K and thus, f ∼ 0 but f = 0. For this f , f (x) = 0, ∀ x ∈ R but it is not true that it is zero uniformly on compact subset of R, i.e., it is not negligible at level zero and even more, f / ∈ W 0 m,r (0) with r > 0. Showing that, classical mathematical tools are incomplete not being able to measure the full physical reality of space-time.
There should be no strangeness in the fact that f (x) = xδ is identically zero on R but is not in R c . Recall that R forms a grid of equidistant points in R and thus f = 0 on a discrete grid with no accumulation points. However, f ′ (x) = δ + xδ ′ is not necessarily identically zero in R because f ′ (0) = ρ(0) · α −1 . For comparison, g(x) = sin(πx), x ∈ Z, is identically zero but g ′ = πcos(πx), x ∈ Z, is not. The most puissant differential calculus developed in Z will not be able to detect that g is not identically zero but that its derivatives effect reality in Z.
In case of K, nets of K are its building blocks and one knows that K embeds as a grid of equidistant point into K. In case of G(Ω), the building blocks are nets of elements of C ∞ (Ω) (see also [2,14,17,37] about how to construct intrinsically D ′ (Ω) starting with the embedding of C ∞ (Ω)). Hence the following results should not come as a surprise (see also [12,Theorem 5.8]).
Corollary 2. The elements of C ∞ (Ω) form a grid of equidistant points in G(Ω). In particular, G(Ω) is a fractal.
Proof. Since C ∞ (Ω) is diagonally embedded it follows that V mp = 0, ∀m, p. By [10,Lemma 3.6], it follows that the inductive dimension dim(G(Ω)) = ∞ and, being an ultrametric space, Ind(G(Ω)) = 0. Consequently, G(Ω) is a fractal.
Generalized Differential Calculus aims at relying soly on intrinsic results and not on classical results to prove existence and uniqueness of differential equations, among other things. Should it be that results proved using classical tools to prove moderateness and establishing solutions for differential equations should be considered incomplete? Our next endeavor is to show how to sidestep this, at least in the direction that we are going to investigate, in such a way that it extends naturally all classical results in this direction. Might it be the way to complete results obtained until now using classical tools?
Given a net of maps (T ε ) and n = [(n ε )] ∈ N, we denote by T n the net (T nε ε ) acting on nets f = (f ε ) as T n (f ) = (T nε ε (f ε )). Suppose that T n is well defined in G(Ω), with Ω ⊂ R n an open subset, and denote by T the net when n = 1. Let A ⊂ G(Ω) be a closed subspace and suppose that the restriction T |A :
A −→ A. The map T : A −→ A is said to be a contraction if there exist L ∈ R * + , λ ∈]0, 1[ such that L < λ and |T (f ) − T (g)|(x) ≤ L · |f − g|(x). It easily follows that |T n (f ) − T n (g)|(x) ≤ L n · |f − g|(x) ≤ λ n · |f − g|(x).
Our interest is when the hypersequence (T n (f )) converges in G(Ω). We look at some examples inspired by the classical analog.
Let Ω ⊂ R n be an open subset, L ∈ R + and x 0 ∈ Ω c . Define A = {f ∈ G(Ω) : |f (x) − x 0 | ≤ L, x ∈ Ω c } and consider it with the induced sharp topology. Then A is a closed subset of G(Ω). In fact, let (f n ) be a Cauchy sequence in A. Since G(Ω) is complete (see [4,5,6]), f n −→ f , for some f , and thus x n := |f n (x) − x 0 | −→ |f (x) − x 0 | =: a. Since x n converges to a, for each r > 0 there exists n 0 such that if n > n 0 then a − x n ∈ V r (0) and thus |a − x n | < α r . Hence a < x n + α r ≤ L + α r . It follows that a ≤ L, i.e., f ∈ A, thus proving that A is a closed subset of G(Ω). We did not use the fact that L ∈ R but the reason why it appears here is that when considering some differential equations, compositions of generalized maps is necessary. In [9] it was shown that this results in the classical composition of maps. Since domains of generalized functions are Ω c , the real bound L will guarantee that compositions of maps are allowed.
Consequently, if f ∈ A then |f (x)| ≤ L + |x 0 | proving that f ≤ 1, i.e., A ⊂ B 1 (0).
Suppose that for each ε ∈ I we have a map T ε :
A ε −→ A ε , with A ε = {h ∈ C(Ω, R) : |h(x) − x 0ε | < L},
which is a contraction with Lipschitz constant K ε < λ ∈ ]0, 1[, the latter being independent of ε. It is clear that with these settings T n , with n ∈ N , is a well defined and continuous map in G(Ω) and T n+m = T n • T m , n, m ∈ N . In particular, T n+1 = T • T n , observing that 1 must be seen as an element of N .
Another classical situation is the following. The set Ω ⊂ R n is open and
relatively compact, for each ε ∈ I, T ε (x)(t) = x 0ε + t t0ε h(s, x(s))ds, t ∈ Ω,
h ∈ C ∞ (R n+1 ) and A ε = C(Ω, R n ) (see [22,23] and [24,Theorem 3.1]). In this case, there exists n 0ε ∈ N, such that T n0ε ε is a contraction with Lipschitz constant K ε ≤ λ < 1, the latter being real and fixed. If n 0 := [(n 0ε )], then define T = [(T n0ε ε )] and thus reducing it to the case of the previous paragraph, a classical argument.
Theorem 1 (The Generalized Fixed point Theorem).
Let Ω ⊂ R N , A ⊂ G(Ω) ∩ B 1 (0) a close subset. For each ε ∈ I, let A ε = C ∞ (Ω) ∩ A, initial conditions taken for that specific ε, and (T ε ) a net of functionals from A ε to its self. If there exists k ∈ N such that each T kε ε is Lipschitz with Lipschitz constant K ε < λ ∈ ]0, 1[, then T = (T kε ε ) is well defined, continuous and has a unique fixed pointx ∈ A.
Proof. The proof uses what was already discussed and also freely facts about the topology.
Choose any x ∈ A and consider the hypersequence (T n (x)). Given r > 0 and m ∈ N, choose n 0 ∈ N such that λ n0 · α −1 ∈ V 4 mN r (0) = V 4 N r1 (0), where r 1 = 4 N (m−1) r. If n, s > n 0 then, writing n = n 0 + k, s = n 0 + l, we have that
|T n (x) − T s (x)|(t) ≤ λ n0 · |T k (x) − T l (x)|(t) ≤ λ n0 · α −1 ∈ V 4 N r1 (0). Setting F (t) = (T k (x) − T l (x))(t), with t ∈ Ω m , we have that F ∈ B 1 (0) ∩ W 0
m,4 N r1 (0) and hence |F ′′ (t)| ≤ α −0.5r1 . Without loss of generality, we may considering
N = 1, obtaining, since Ω m is open, F (t+α 2r1 )−F (t) = F ′ (t)·α 2r1 + F ′′ (c) 2 ·α 4r1 and thus |F ′ (t)| = F (t+α 2r 1 )−F (t) α 2r 1 + F ′′ (c) −2 · α 2r1 ≤ 2α 4r 1 α 2r 1 + α −0.5r 1 2 · α 2r1 < α r1 .
This proves that F ∈ W 1 m,r1 (0) = W 1 m,4 m−1 r (0). By induction, we have that F ∈ W m m,r (0). This proves that (T n (x)) is a Cauchy hypersequence and hence converges. Since T n+1 = T • T n and T is continuous, the limit is a fixed point of T .
In case compositions of maps are not involved, the theorem still holds if A ⊂ B R (0), R > 1 (It always holds since estimates are made in Ω m ). Together with the other tools, it makes the Generalized Differential Calculus developed in [3,9] (see also [26]) a useful tool to generalize most classical results. For example, one can prove the local, and hence the global, existence of geodesics in M c (see the next section). If the solution, given by the theorem, is not a classical solution but it has a distributional shadow, it is a non-vampire, this might lead one to change = in the equation by ≈. Actually, this is often done but maybe it should be rethought. Why not solve the equation as it is in the generalized environment and then see if it has a distributional shadow? The distributional shadow will not be the solution but will behave like the nonvampire. Having observed this, there exist classical equations which do not have a classical solution but do have a generalized solution which is a nonvampire or omnipresent. There are eidolons roaming under and over classical radars, verbalized by the ancients and Dirac as atoms and infinities, which, at rendezvous, tellingly influence physical reality.
Let's formalize the argument used in the last part of the proof of the theorem since it is a useful tool to be considered in Generalized Analysis.
Definition 1. The Down Sequencing Argument
Let f ∈ G(Ω), with Ω ⊂ R n . If f ∈ W 0 m,r (0) with r > 0 and p 0 ∈ N n then f ∈ W |p0| m,s (0), where s = 4 −n|p0| · r, i.e., W 0 m,r (0) ⊂ W |p0| m,s (0).
Using the DSA, another proof of a fact already mentioned can be given: If f is moderate and negligible at level 0, then f is negligible. This can be considered a statement about rigidity. In fact, such an f ∈ W 0 m,r , ∀ m, r > 0 and hence DSA gives what was claimed. In other words r>0 W 0 m,r = {0}, showing that these sets serve as a basis for the sharp topology. Let's consider the example that inspired the Generalized Fixed Point Theorem. Consider the following equation from [39,40].
x(t) = f (x(t))δ(t) + h(t) x(−1) = x 0 x(−1) =ẋ 0
with h, f ∈ C ∞ (R) and δ the Dirac function. This is a typical differential equations from Physics and Engineering having a product of distributions in its data and does not allow the use of classical tools to obtain a solution. In the references just mentioned, one has to prove moderateness and other nontrivial steps involving non trivial classical results. Would this make such proofs incomplete? We show that proving moderateness is not needed and thus suggesting a possible remedy. Ω c ,R)) n we have that A ⊂ (G(Ω)) n and the latter is a complete metric space (see [4,5,6]). From what we already discussed, it follows that A is a closed subspace of a complete algebra. Define A ε accordingly and let T ε be defined in A ε by
= {x = (x 1 , · · · , x n ) ∈ C ∞ ( Ω c ,R n ) : x i ∈ G(Ω) and |x(t) − x 0 | ≤ L}. Since C ∞ ( Ω c ,R n ) ∼ = (C ∞ (T ε (x)(t) = x 0ε +ẋ 0 (t + 1) + t −1 s −1 f (x(r))ρ ε (r)drds + 1 −1 s −1 h(r)drds. We have that |T ε (x)(t) − x 0ε | = |ẋ 0ε (t + 1) + t −1 s −1 f (x(r))ρ ε (r)drds + t −1 s −1 h(r)drds| ≤ |ẋ 0ε |·(|t|+1) + t −1 s −1 |f (x(r))ρ ε (r)|drds + t −1 s −1 |h(r)|drds ≤ |ẋ 0ε |·(2+a)+(2+ a)· f ·C +M = |ẋ 0ε | +M +(1+a)(|ẋ 0ε |+ f ·C) ≤ |ẋ 0ε | +M +b+ f ·C) = L. It is also Lipschitz: |T ε (x)(t)− T ε (x)(t)| = | t −1 s −1 (f (x(r))− f (x(t)))ρ ε (t)drds| ≤ f ·KC ·(2+a)·|x(t)−x(t)| ≤ 2 f C ·(1+a)·K|x(t)−x(t)| ≤ K|x(t)−x(t)|.
It follows that the T they define is Lipschitz and hence well defined and continuous in G(Ω). The first part shows that T maps A into itself. Hence we are in the setting of the Generalized Fixed Point Theorem. We have a Lipschitz map and the corresponding hypersequence has a fixed point which is a solution for the system. Once the Generalized Fixed Point Theorem is applied, there is no need to prove moderateness. The result follows forthwith! From the equation it readily allows thatẍ ≈ h + f (x(0)) · δ, i.e., it is a non-vampier.
Supposing h = 0, expand δ at collision time t = 0 and chose the mollifier such that ρ(0) = 1. The equation to be solved is
x(t) = f (x(t))α −1 x(−1) = x 0 x(−1) =ẋ 0
Its solution is obtained just as in the classical case but using Generalized Differential Calculus. Albeit, the solution might not be a Colombeau generalized function since it might only be defined on a membrane. This happens because the Theory of Generalized Differential Calculus strictly contains the Theory of Colombeau Generalized Functions (see [3] for an explicit example). There exists a non-constant functions with zero derivate (see [9]) behaving like a quanta. For applications, a solution in this more generalized environment does provide a solution since ε small is enough; one does not need an explicit solution in the Colombeau algebra. The environment of the full algebra is treated in [35] relying on results from ( [4,5,7,21,44]). In the case of the invariant algebra, the notion of point value is undertaken in [44]. As for the latter, as already explained, it will depend whether this algebra is or not complete. If it is not complete, then what comes next might be the Umweg, the roundabout route. The original full Colombeau algebra might be all that is needed to have a Generalized Differential Geometry able to handle differential equations on manifolds having products of distributions in their data. This is settled in [35].To get the hang of it, we first look at the case of the simplified algebra.
Generalized Manifolds
In this section, α will stand for an index and not for our standard gauge defined in the previous section. In [9] the definition of a generalized manifold was given and proved that each such manifold had a maximal G−atlas. These manifolds were denoted short by G−manifolds and the atlas by G−atlas. In case the underlying field is R, we have a real generalized manifold and in case the underlying field is C we have a complex generalized manifold. The topology in R n is the sharp topology and differentiability is in the sense of Generalized Differential Calculus ( [3,9]). Other notions such as diffeomorphism and continuity are those defined in [3,4,5,9]. One could ask the reason for pursuing this path. First of all, our aim is to establish the foundations of a Generalized Differential Geometry based on the Generalized Differential Calculus developed in [3,9]. The latter has shown to be a natural extension of Newton's and Schwartz's Calculus. Second, although there exists an invariant Colombeau algebra (see [40]), it is not clear yet if this is a topological algebra and, if it were, how its topology interacts with the topology of a classical manifold. It is also not clear if this algebra is complete since continuity of the parameters involved in its construction is imposed (necessary). Our proposal is that having the Colombeau full algebra at hand, might be what is needed to settle questions and problems on manifolds whose data involves products of distributions. For the sake of completeness, we recall the definition of a generalized manifold in case the underlying field is R.
∅ = U λ ⊂ M onto the open subset ϕ λ (U λ ) ⊂ R n . 2. M = λ∈Λ U λ 3. For every pair α, β ∈ Λ, with U α,β = U α ∩ U β = ∅, the subsets ϕ α (U α,β ) and ϕ β (U α,β ) are open contained in R n such that ϕ β • ϕ −1 α : ϕ α (U αβ ) −→ ϕ β (U α,β ) is a diffeomorphism of class C ∞ .
• The pair (U λ , ϕ λ ) is denominated a local chart (or coordinate system) of A.
• If U ⊂ M and ϕ :
U −→ u(U) is a homeomorphism of U, where ϕ(U)
is an open set of R n , the pair (U, ϕ) is said to be compatible with A if for
each pair (U λ , ϕ λ ) ∈ A with W λ = U ∩ U λ = ∅ we have that ϕ • ϕ −1 λ : ϕ λ (W λ ) −→ ϕ(W λ ) is a diffeomorphism of class C ∞ , where ϕ λ (W λ ) and ϕ(W λ ) are open subsets of R n .
By Zorn's Lemma, it follows that, for a given G-atlas A of dimension n on M , there exists an unique maximal G-atlas A * of dimension n on M defined by:
(U, ϕ) ∈ A * , if and only if, (U, ϕ) ∈ A or (U, ϕ) is compatible with A.
A Generalized Manifold, or G−manifold, is a set M with a G−atlas defined on it. A G−manifold M with a maximal G−atlas is called a G−differential structure of M . If clear from the context, the prefix G will be omitted. The topology on the G−manifold is the one that makes all charts simultaneously homeomorphisms. Our first step in setting the foundations of the Generalized Differential Geometry is settling the invariance of the dimension of a G−manifold. [52]] Let (M, A) be a G-manifold. Then, the dimension of a G-atlas A is constant in each connected component of M .
Theorem 2. [Dimension Invariance
Proof. Suppose there are two intersecting local charts (U α , φ α ) and (U β , φ β ) belonging to the G-atlas
A = {(U α , φ α )} α∈Λ , such that φ α (U α ) ⊂ R m , and φ β (U β ) ⊂ R n . If U α,β = U α ∩ U β = ∅, then we have that φ β • φ −1 α : φ α (U αβ ) −→ φ β (U αβ )
is a diffeomorphism and therefore its differential in a point p ∈ φ α (U αβ ),
D(φ β • φ −1 α )(p) : R m −→ R n , is a R−isomorphism of R−modules.
Since R is a commutative ring with unity, it follows from a result of [13] that n = m.
Of course, we can define such manifolds for any differential class C k , k ≥ 2. The definition is the same, only substituting C ∞ by C k . Only one explicit, but rather trivial, example of a G−manifold was given in [9]. We shall prove that any classical n−dimensional manifold M can be discretely embedded in a G−manifold of the same dimension thus proving that Generalized Differential Geometry is a natural extension of Classical Differential Geometry with the advantage that problems involving irregular data can be considered translating them into the G−context.
Let A = {(U α , φ α ), α ∈Λ}, be an altlas of a C ∞ n-dimensional connected submanifold M of R N . Suppose that for each α ∈ Λ we have that φ α (U α ) = Ω 0 = B r (0) ⊂ R n ,
the open ball of fixed radius r > 0 centered at the origin.
Denote by M c the subset of R N c ⊂ R N constructed from M (see the previous sections). We saw that M is discretely embedded in M c as constants nets. Recall from [9] that R N c ⊂ B 1 (0), the ball of radius 1 centered at the origin and that the image of R N under this map is a grid of equidistant points. We denote by Λ the set of maps from I =]0, 1] into Λ and, for λ ∈ Λ, we denote by
U λ = the strongly internal set U λ(ε) contained in R N c φ λ = (φ λ(ε) ) ε∈I φ λ : U λ −→ R n c , defined by φ λ ([(p ε )]) = [(φ λ(ε) (p ε ))]
For p = [(p ε )] ∈ M c , consider the set {q ∈ M : ∃ ε n → 0, p εn → q}. Algebraically this can be written as: Given q 0 ∈ R n , we have that q 0 ∈ {q ∈ M : ∃ ε n → 0, p εn → q} if and only if there exists e ∈ B(R) such that e · p ≈ e · q 0 (extending the notion of association to K n in the obvious way).
This is a compact subset of M to which we shall refer as the support of the point p and denote it by supp(p). It follows that there exists a complete set of orthogonal idempotents (e λ ) such that
p = x λ ∈supp(p) e λ · p (e λ · p ≈ e λ · x λ ) For example, if p = [(p ε )] = [(sin( 1 ε ))] then supp(x) = [−1, 1].
On the other hand, if p ∈ B 1 (0) then supp(p) = {0}. Although supp(p) might be uncountable, nevertheless the sum above is well defined and can be thought of as a history of events: multiplying by idempotents one sees its behavior along a specific path.
It generalizes the concept of interleaving. The support of elements belonging to a halo of a point in R n consists of only that single point.
Our next result gives conditions under which the objects defined above turn M c into a G−manifold of dimension n. This will enable us to look at Generalized Differential Geometry as a natural extension of Classical Differential Geometry. The latter having as basis Newton's Differential Calculus and the former having as basis the Generalized Differential Calculus defined and developed in [3,9].
Proposition 2.
Suppose that for each α ∈ Λ the map α and its inverse are Lipschitz functions with respect to the norms of R N and R n . Then the following hold.
:= U λ1 ∩U λ2 = ∅ then φ λ2 •φ −1 λ1 is a C ∞ diffeomorphism on its domain φ λ1 (U λ1,λ2 ).
Proof. Let p = [(p ε )] ∈ M c . Since supp(p) is compact, there exist a finite number of α ∈ Λ such that supp(p) is contained in the union of the corresponding U α 's. Let δ be a Lebesgue number of this covering and choose a finite number of q i ∈ supp(p) such that supp(p) ⊂ i B δ1 (q i ), with δ 1 = δ 2 . Starting with q 1 , define λ(ε) = α q1 ∈ Λ, where α q1 is such that B δ1 (q 1 ) ⊂ U αq 1 and p ε ∈ B δ1 (q 1 ).
For λ(ε) not yet defined, continue defining λ(ε) = α q2 ∈ Λ, where α q2 is such that B δ1 (q 2 ) ⊂ U αq 2 and p ε ∈ B δ1 (q 2 ). Since there are a finite number of q i 's, this process ends in a finite number of steps. If λ is not defined on I then there exists a sequence (ε n ), converging to 0, with p εn → q ∈ supp(p). Since the balls B δ1 (q i ) cover A p , there exists a n 0 such that n > n 0 implies that p εn is in the ball B δ (q i0 ), say. But this is a contradiction, since, for these ε's, λ(ε) was already defined. Hence, we defined a λ ∈ Λ such that p ε ∈ U λ(ε) , ε ∈ I and the distance to the boundary is bigger than δ 2 . It follows that p ∈ U λ , with λ being of finite range.
For each λ(ε) there exists an open subset
U λ(ε) ⊂ R N such that U λ(ε) = M ∩ U λ(ε) . Setting U λ = U λ(ε) , we have that U λ = M c ∩ U λ , with U λ an open subset of R N ,
proving that M c has the induced topology. The fact that strongly internal sets are open can be found in [46]. This settles the proof of the first three and the fifth items.
To finish the proof, we prove the forth and sixth items. We first prove that φ λ is well defined. In fact, since local charts are Lipschitz, and the λ's are of finite range, it follows easily that p − q = φ λ (p) − φ λ (q) , i.e., the φ α 's are isometries considered as maps from a subset of R N to a subset of R n . From this it follows that if dist(p, (U λ ) c ) is an invertible then dist(φ α (p), ( Ω 0 ) c ) is an invertible, thus proving that φ λ is well defined and its image is in U λ . Surjectivity, injectivity and continuity are now obvious. The map resulting from a change of coordinates, is a homeomorphism that stems from a net whose elements are all infinitely differentiable taking value in a bounded subset of R n . Hence it originates a diffeomorphism.
The condition that all charts have the same image is not necessary. We could just suppose that there exists r > 0 such that B r (0) ⊂ R n contains all of them. The (U λ , φ λ ) are called local charts of M c . If λ is constant then U λ is called a principal chart. For λ ∈ Λ and x ∈ B 1 (0) define xλ as xλ(ε) = λ(x(ε)). For an idempotent e ∈ B(K) we define eλ = e(ε)λ(ε), meaning that when e(ε) = 1 this index will be omitted. With this notation fixed, the proof of the proposition gives us the following corollary.
Corollary 3.
Let U λ be a local chart with λ of finite range. There exist principal charts U λi , i = 1, k and a complete set of mutually orthogonal idempotents e 1 , · · · , e k such that U λ ⊂ U λi and U λ = U eiλi
The corollary shows the importance of principal charts since they can serve as tools to use sheaf like arguments in the proofs. For results to hold, they just have to be proved for principle charts. Proof. Only the last part of the theorem needs to be proved. To see this, we use Lemma A.1 of [66] which state that for each compact subset of K ⊂ M its Riemannian metric satisfies a p − q ≤ dist M (p, q) ≤ C · p − q , for some C > 0, p, q ∈ K and dist M the Riemannian metric of M . This implies that local charts are isometries.
Using Whitney's Embedding Theorem, the above theorem extends to abstract manifolds. The construction of an atlas for M c shows that if M is a classical manifold then M c can be covered with a countable number of local charts. Another important fact in the construction of Λ is that since its elements have finite range and we can work with local charts (U, φ) of M such that U is relatively compact in M , it follows that for λ 1 , λ 2 ∈ Λ we have that det(D(φ λ2 • φ −1 λ1 )(p)) ∈ Inv(R), for each point p in its domain. If we define orientabillity as is done the classically, this shows that if M is orientable then so is M c . In particular, complex G−manifolds will be orientable. It is important to note that the construction of the differential structure does not depends on the Riemannian metric of the manifold M (see [41]).
Using the Generalized Fixed Point Theorem, one can prove the global existence of geodesics without using the usual stratagem: proving existence depending on ε and then proving moderateness. There is no need to expand on this since everything works just as in the classical case. The environments of Classical Differential Geometry are discretely embedded in the the environments of Generalized Differential Geometry.
Differential Functions on G−manifold
A one dimensional G−manifold shall be referred to as a curve. This definition agrees with the notion of a history given in [9], only now we parametrize the history. If M is an n−dimensional G−manifold and p ∈ M then the tangent vectors and space at p are defined just as in the classical way (see [52]). The tangent space we shall also denote by T p M . Furthermore, it is easily proved that T p M is a free R−module which is R−isomorphic to R n .
As in the classical case, one defines the tangent bundle of M and denote it by T M . It is easily seen that if M is a classical n-dimensional manifold then (T M ) c = T M c is a G−manifold of dimension n 2 .
Given another G−manifold N , define a differential function between M and N using the same classical definition. In case N = R we call a differentiable function a scaler field and if N = R n , with n > 1, then we call such a map a vector valued map, being a vector field if also the dimension of M equals n.
The notions of an immersion and an embedding are defined completely analogous as in classical geometry. Using the Chain rule of Colombeau Generalized Calculus it follows easily that composition of differentiable maps between G−manifolds also satisfy the Chain Rule (see [52]). We recall a Linear Algebra results (see [32]). We sum up, without proofs, some of the most classical theorems that also hold for G−manifolds. Some of the proofs rely on the previous lemma and the fact that Inv(R) is open (see [52]). Theorem (see [3]) to prove that at every point M is locally a graph over a subset of Ω c . The standard classical argument still holds to complete the assertion.
An example of such a function is f (x) = x 2 2 and the value in question is a = 1. In this case we have that ∇f (x) 2 2 = 4 x 2 2 = 4 ∈ Inv(R). 6. The halo of any point in K n is a generalized manifold which is not classical. In [9] an example of a function f ≡ 0 is given such that f ′ ≡ 0.
This function is f (x) = α −2 ln( x ) which is not a Colombeau generalized function. This f is constant on spheres S R (0) and the only point where it is not locally constant is x 0 = 0. This f is easily modified such that it is of class C k . Since the origin is the only points where such spheres accumulate, and spheres are clopen, we have that Graf (f ) − { 0} is a generalized manifold but Graf (f ) is not a generalized manifold.
The negative cone Inv(R) + = Inv(R) ∩ R − is also an open subset of R. It follows that Inv(R) has two connected component which are both open subsets of R. Moreover, Inv(R) + ∩ Inv(R) − = ∅ and 0 is in the topological closure of both Inv(R) + and Inv(R) − . Both are closed under addition and interleaven (see the end of this section or [46]).
Kf
(x)dx and thus, in the classical sense, they are all equal. In case we take dV = 1 n we have a hypersequence and we just proved its convergence. For each s(f, dV ), S(f, dV ), S(f, dV, * ) and K f (x)dx) there exists an element c ∈ K, for each one of them there exists one such element, so that it is equal to f (c)·µ(K).
( 1 n k ) n∈N and hence converge to 0. However, if we look at the hypersequencex then x(n) = e n · α k , where e n = [χ S(n) ] ∈ B(K). Hence x(n) = 0, ∀ n ∈ N and x(n) − x(m) = e −k · e n − e m ∈ {0, e −k }. Consequently, this is not a converging hypersequence. So it might be that measurements in classical space-time are incomplete.
37 ,
37Definition 1.6.5] where they were first defined). Here, we introduce the notation G(Ω) 0 = {f ∈ G(Ω) : f ≈ 0} and G(Ω) as = {f ∈ G(Ω) : ∃ T ∈ D ′ (Ω), such that f ≈ T }. Clearly, K 0 ⊂ G(Ω) 0 .Define the halo of f ∈ G(Ω) as halo(f ) := f + B 1 (0) = B 1 (f ), where B 1 (0) is the ball {f ∈ G(Ω) : D(f, 0) < 1} ⊂ G(Ω) (see also
6.3] and[37, Proposition 1.7.28]. The second part looks at the building blocks of G(Ω) and equate them with the building blocks of K.Corollary 1. Let Ω ⊂ R n be an open subset. The embedding of D ′ (Ω) in G(Ω) is a discrete embedding. Moreover, if h ∈ C ∞ (Ω) * and f ∈ G(Ω) then hf = f .Proof. Since the embedding is linear, we just have to prove discreteness at the origin. Let f ∈ B 1 (0)∩D ′ (Ω). By the previous proposition, f ≈ 0 and hence, by [37, Proposition 1.6.3], f = 0. The proof of the second part is straightforward.
Let b > 0, C > 0 a positive constant limiting the L 1 norm of the delta−net, M r)|drds, L := b + M + |ẋ 0 | + f · C and 1 + a = min{ b C· f +|ẋ0| , 1 2CK , 2}, where K is a Lipschitz constant of f on a compact subset of R containing Ω = ] − 1 − a 2 , a 2 [. The norm f is also taken over the same compact subset. Let A
Definition 2 .
2Let M be a non-void set. A G-atlas of dimension n and class C ∞ of M is a family A = {(U λ , ϕ λ )} λ∈Λ verifying the following conditions: 1. For every λ ∈ Λ the map ϕ λ : U λ −→ R n is a bijection of the open subset
Theorem 3 .
3Let M be a submanifold of R N of dimension n, and suppose that its local charts and their inverses are Lipschitz functions with respect to the norms of R N and R n . Then ( M c , A), A = {(U λ , φ λ ) : λ ∈ Λ of finite range} is a generalized submanifold of R N of codimension N − n containing M as a discrete subset. Moreover, each local chart is an isometry in the sharp topologies and the geometry of M c extends in a natural way the geometry of M .
Lemma 4 .
4Let A : R n −→ R n be a R−linear map. Then A is injective if and only if it is surjective if and only if det(A) ∈ Inv(R).
Theorem 4 .
4Let M 1 , M 2 and M 3 be G-manifolds. If f : M 1 −→ M 2 and g : M 2 −→ M 3 are differentiable applications at p ∈ M 1 and f (p) ∈ M 2 , respectively, then g • f : M 1 −→ M 3 is differentiable at p and D(g • f ) p = (Dg) f (p) • Df p . Theorem 5. Let M 1 and M 2 be n-dimensional G-manifolds and f : M 1 −→ M 2 a map of class C ∞ , such that for p 0 ∈ M 1 we have that Df p0 : T p0 M 1 −→ T f (p0) M 2 , is an isomorphism. Then f is a local diffeomorphism of class C ∞ . Theorem 6. Let f : M −→ N be an immersion at p of class C ∞ , where M and N are G-manifolds of dimension m and n, respectively. Then there exist local coordinate systems around p and f (p), such that f (x 1 , ..., x m ) = (x 1 , ..., x m , 0, ..., 0). Theorem 7. If f : M −→ N is an generalized embedding, then f (M ) is an G-submanifold of N .
Definition 3 .
3Let f : Ω c ⊂ R n −→ R m be a differentiable map, where Ω is an open subset of R n . A point a ∈ R m is called a regular value of f if for each x ∈ f −1 (a) the derivative f ′ (x) : R n → R m is surjective.
1 .
1The topology of M c is induced by the topology of R n .2. If λ ∈ Λ has finite range thenU λ is an open subset of M c . 3. Ω 0 is an open subset of R n c 4. If λ ∈ Λ has finite range then φ λ : U λ −→ Ω 0 ⊂ R n is a isometry w.r.tto the sharp topologies of R n and R N .5. M c = λ∈ Λ U λ , with λ of finite range.6. If λ 1 , λ 2 ∈ Λ are of finite range and U λ1,λ2
AcknowledgementsThe second author is indebted to the University of São Paulo for its warm hospitality and to the Federal University of Roraima, for providing the opportunity to do this research.Theorem 8.Let Ω be an open subset of R m × R n and f : Ω c −→ R n be a application of class C ∞ , where Ω c ⊂ R m × R n . If a ∈ Im(f ) is a regular value of f , then:Let's look at some examples of G−manifolds.Example 1. 4. The sphere S = S 1 (0) contained in R n is a generalized manifold whose local charts do not come from subsets of R n . In fact, S is an open subset of R n because given x ∈ and y ∈ B 1 (0) we have that x + y = max{ x , y } = 1, since x = 1 > y . Consequently, we can take local charts to be the identity map with domain B 1 (x). Since these balls are either equal or disjoint, it follows that they form an atlas for S.This proves that the conditions of Proposition 2 are satisfied and thusLet f ∈ G(Ω)with Ω ⊂ R n . We know (see[9]) that f can be viewed as a differentiable map Ω c −→ R and its differential at each point is a R−linear map from R n to R. A value a ∈ Im(f ) is said to be a regular value of f if for each x ∈ f −1 (a) we have that Df is surjective. This only happens if, writing ∇f (x) = (z 1 , · · · , z n ), the ideal in R generated by z 1 , · · · , z n equals R. In particular, this is the case if ∇f (x) 2 2 ∈ Inv(R). If a is a regular value of f set M = f −1 (a). We assert that M is a submanifold of R n . In fact, just like in the classical case, we can use the Implicit Function
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| []
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[]
| [
"S P Baranov \nP.N. Lebedev Physics Institute\n119991MoscowRussia\n",
"H Jung \nDeutsches Elektronen-Synchrotron\nNotkestrasse 85HamburgGermany\n",
"A V Lipatov \nSkobeltsyn Institute of Nuclear Physics\nMoscow State University\n119991MoscowRussia\n\nJoint Institute for Nuclear Research\n141980Dubna, Moscow regionRussia\n",
"M A Malyshev \nSkobeltsyn Institute of Nuclear Physics\nMoscow State University\n119991MoscowRussia\n"
]
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"P.N. Lebedev Physics Institute\n119991MoscowRussia",
"Deutsches Elektronen-Synchrotron\nNotkestrasse 85HamburgGermany",
"Skobeltsyn Institute of Nuclear Physics\nMoscow State University\n119991MoscowRussia",
"Joint Institute for Nuclear Research\n141980Dubna, Moscow regionRussia",
"Skobeltsyn Institute of Nuclear Physics\nMoscow State University\n119991MoscowRussia"
]
| []
| We consider the production of Z bosons associated with beauty quarks at the LHC using a combined k T + collinear QCD factorization approach, that interpolates between small x and large x physics. Our consideration is based on the off-shell gluon-gluon fusion subprocess g * g * → ZQQ at the leading order O(αα 2 s ) (where the Z boson further decays into a lepton pair), calculated in the k T -factorization approach, and several subleading O(αα 2 s ) and O(αα 3 s ) subprocesses involving quark-antiquark and quark-gluon interactions, taken into account in conventional (collinear) QCD factorization. The contributions from double parton scattering are discussed as well. The transverse momentum dependent (or unintegrated) gluon densities in a proton are derived from Catani-Ciafaloni-Fiorani-Marchesini (CCFM) evolution equation. We achieve reasonably good agreement with the latest data taken by CMS and ATLAS Collaborations. The comparison of our results with next-to-leading-order pQCD predictions, obtained in the collinear QCD factorization, is presented. We discuss the uncertainties of our calculations and demonstrate the importance of subleading quark involving contributions in describing the LHC data in the whole kinematic region. | 10.1140/epjc/s10052-017-5369-5 | [
"https://arxiv.org/pdf/1708.07079v2.pdf"
]
| 119,417,146 | 1708.07079 | 02cbb1dceeff98142bbf76de79f1c86ccf143391 |
19 Nov 2017 March 27, 2018
S P Baranov
P.N. Lebedev Physics Institute
119991MoscowRussia
H Jung
Deutsches Elektronen-Synchrotron
Notkestrasse 85HamburgGermany
A V Lipatov
Skobeltsyn Institute of Nuclear Physics
Moscow State University
119991MoscowRussia
Joint Institute for Nuclear Research
141980Dubna, Moscow regionRussia
M A Malyshev
Skobeltsyn Institute of Nuclear Physics
Moscow State University
119991MoscowRussia
19 Nov 2017 March 27, 2018Associated production of Z bosons and b-jets at the LHC in the combined k T + collinear QCD factorization approach
We consider the production of Z bosons associated with beauty quarks at the LHC using a combined k T + collinear QCD factorization approach, that interpolates between small x and large x physics. Our consideration is based on the off-shell gluon-gluon fusion subprocess g * g * → ZQQ at the leading order O(αα 2 s ) (where the Z boson further decays into a lepton pair), calculated in the k T -factorization approach, and several subleading O(αα 2 s ) and O(αα 3 s ) subprocesses involving quark-antiquark and quark-gluon interactions, taken into account in conventional (collinear) QCD factorization. The contributions from double parton scattering are discussed as well. The transverse momentum dependent (or unintegrated) gluon densities in a proton are derived from Catani-Ciafaloni-Fiorani-Marchesini (CCFM) evolution equation. We achieve reasonably good agreement with the latest data taken by CMS and ATLAS Collaborations. The comparison of our results with next-to-leading-order pQCD predictions, obtained in the collinear QCD factorization, is presented. We discuss the uncertainties of our calculations and demonstrate the importance of subleading quark involving contributions in describing the LHC data in the whole kinematic region.
Introduction
With the LHC in operation, one can access a number of "rare" processes which could have never been systematically studied at previous accelerators. In this article we draw attention to the associated production of Z bosons and b-jets. This process involves both weak and strong interactions and therefore serves as a complex test of Standard Model, perturbative QCD (pQCD) and our knowledge of parton distribution functions in a proton. Similarly to the W + c and W + b processes considered earlier [1,2] it probably provides an arena for double parton scattering (DPS), now widely discussed in the literature. We wish to clarify this point in our paper. Besides that, this process constitutes a substantial background in studying the associated production of Higgs and Z bosons, where the Higgs boson is identified via its decay into a bb pair [3][4][5]. A number of physics scenarios beyond Standard Model also refer to final states containing Z bosons and beauty quarks [6][7][8], thus making the related studies important and topical.
Our present study is greatly stimulated by the recent ATLAS measurements [9] of the total and differential production cross sections of Z bosons associated with beauty quark jets at √ s = 7 TeV accompanied by the CMS measurements [10] of kinematic correlations between Z bosons and b-hadrons at √ s = 7 TeV. We investigate these processes in the framework of a combined QCD approach, based on the k T -factorization formalism [11,12] in the small-x domain and conventional (collinear) QCD factorization at large Bjorken x. Doing so, we employ the k T -factorization approach to calculate the leading contributions from the off-shell gluon-gluon fusion g * g * → ZQQ and, to extend the consideration to the whole kinematic range, take into account several subleading quark-involved subprocesses using collinear QCD factorization. The k T -factorization approach has certain technical advantages in the ease of including higher-order radiative corrections that can be taken into account in the form of transverse momentum dependent (TMD) parton distributions 1 . This approach has become a widely exploited tool and it is of interest and importance to test it in as many cases as possible. Closely related to this is the selection of TMD parton densities best suited to describe the data. These tasks form the major goal of our article. The outline of the paper is the following. In Section 2 we describe our approach and parameter setting. In Section 3 we present the results of our calculations and confront them with the available data. Our conclusions are summarised in Section 4.
The model
Let us start from a short review of calculation steps. The leading contribution comes from the O(αα 2 s ) off-shell gluon-gluon fusion subprocess:
g * (k 1 ) + g * (k 2 ) → Z(p) + b(p 1 ) +b(p 2 )(1)
where the four-momenta of all particles are given in the parentheses. The corresponding gauge-invariant off-shell amplitude was calculated earlier [14,15] and implemented into the Monte-Carlo event generator cascade [16]. All the details of these calculations have been explained [14,15], we only mention here that the standard QCD Feynman rules were employed with the only exception that the initial off-shell gluon spin density matrix was determined according to the k T -factorization prescription [11,12]:
ǫ µ (k i )ǫ * ν (k i ) = k µ iT k ν iT k 2 iT(2)
with k i T being the component of the gluon momentum k i (with i = 1 or 2) perpendicular to the beam axis (k 2 i = −k 2 iT = 0). In the collinear limit k 2 iT → 0 this expression converges to the ordinary one after averaging on the azimuthal angle.
In order to fully reproduce the experimental setup [9,10], we simulate the subsequent decay Z → l + l − incorporated with the production step at the amplitude level. Then, the Z boson propagator is parametrised in Breit-Wigner form with mass m Z = 91.1876 GeV and total decay width Γ Z = 2.4952 GeV [17]. The role of virtual photons in the Z boson resonance region is found to be small: it makes not more than a 2% or 3% correction (including the Z/γ * interference effects). This is much less than the scale uncertainty of the main subprocess (see Section 3), and, therefore, is neglected in our analysis.
In addition to off-shell gluon-gluon fusion, we take into account several subprocesses involving quarks in the initial state. These are the flavor excitation at O(αα 2 s ):
q(k 1 ) + b(k 2 ) → Z(p) + q(p 1 ) + b(p 2 );(3)
the quark-antiquark annihilation at O(αα 2 s ):
q(k 1 ) +q(k 2 ) → Z(p) + b(p 1 ) +b(p 2 );(4)
and the quark-gluon scattering at O(αα 3 s ):
q(k 1 ) + g(k 2 ) → Z(p) + q(p 1 ) + b(p 2 ) +b(p 3 ).(5)
Quark densities are typically much lower than the gluon density at the LHC conditions; however, these processes may become important at very large transverse momenta (or, respectively, at large parton longitudinal momentum fraction x, which is needed to produce large p T events) where the quarks are less suppressed or can even dominate over the gluon density. Here we find it reasonable to rely upon collinear Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) factorization scheme [18], which provides better theoretical grounds in the large-x region. So, we consider a combination of two techniques with each of them being used at the kinematic conditions where it is best suitable (gluoninduced subprocess (1) at small x and quark-induced subprocesses (3) -(5) at large x values). For the flavor excitation and the quark-antiquark annihilation we apply the on-shell limit of formulas obtained earlier [19] supplementing them by the Z boson decays. The amplitude of quark-gluon scattering subprocess can be easily derived from the gluon-gluon fusion one. As usual, to calculate the contributions of quark-induced subprocesses (3) -(5) one has to convolute the corresponding partonic cross sections dσ ab with the conventional parton distribution functions f a (x, µ 2 ) in a proton:
σ = dx 1 dx 2 dσ ab (x 1 , x 2 , µ 2 )f a (x 1 , µ 2 )f b (x 2 , µ 2 ),(6)
where indices a and b denote quark and/or gluon, x 1 and x 2 are the fractions of longitudinal momenta of colliding protons and µ 2 is the hard scale. In the case of off-shell gluon-gluon fusion we employ the k T -factorization formula:
σ = dx 1 dx 2 dk 2 1T dk 2 2T dσ * gg (x 1 , x 2 , k 2 1T , k 2 2T , µ 2 )f g (x 1 , k 2 1T , µ 2 )f g (x 2 , k 2 2T , µ 2 ), (7)
where f g (x, k 2 T , µ 2 ) is the TMD gluon density in a proton. To obtain the latter we use a numerical solution of the CCFM equation [20]. It provides a suitable tool as it smoothly interpolates between the small-x Balitsky-Fadin-Kuraev-Lipatov (BFKL) [21] gluon dynamics and large-x DGLAP one. We adopt the latest JH'2013 parametrization [22], taking JH'2013 set 2 as the default choice. The corresponding TMD gluon density has been fitted to high-precision DIS data on the proton structure functions F 2 (x, Q 2 ) and F c 2 (x, Q 2 ). The fit was based on TMD matrix elements and involves two-loop strong coupling constant, kinematic consistency constraint [23,24] and non-singular terms in the CCFM gluon splitting function [25]. For the conventional quark and gluon densities we use the MSTW'2008 (LO) set [26].
Throughout this paper, all calculations are based on the following parameter setting. In the collinear QCD factorization case we use one-loop running strong and electroweak coupling constants with n f = 4 massless quark flavors and Λ QCD = 200 MeV; the factorization and renormalization scales are both set equal to the Z boson transverse mass, so that we have α s (m 2 Z ) = 0.1232 and α(m 2 Z ) = 1/128. In the k T -factorization case we use a two-loop expression for the strong coupling constant (as it was originally done in the fit [22]) and define the factorization scale as µ 2 F =ŝ + Q 2 T withŝ and Q 2 T being the subprocess invariant energy and the net transverse momentum of the initial off-shell gluon pair, respectively. The latter definition of µ F is unusual and is dictated by the CCFM evolution algorithm [22]. The b-quark mass and Weinberg mixing angle were set to m b = 4.75 GeV and sin 2 θ W = 0.2312 [17]. When necessary, b-quarks were converted into b-hadrons using Peterson fragmentation function [27] with ǫ b = 0.006.
We close our consideration with DPS contributions where we apply a simple factorization formula (for details see the reviews [28][29][30] and references therein):
σ DPS (Z + b +b) = σ(Z) σ(b +b) σ eff ,(8)
where σ eff is a normalization constant which incorporates all "DPS unknowns" into a single phenomenological parameter. A numerical value of σ eff ≃ 15 mb was earlier obtained from fits to pp and pp data. This will be taken as the default value throughout the paper. The calculation of inclusive cross sections σ(b +b) and σ(Z) is straightforward and needs no special explanations. Here we strictly follow the approach described earlier [31][32][33]. The multidimensional phase space integration was performed by means of the Monte Carlo technique, using the routine vegas [34]. In the next section we confront our predictions with the latest LHC data.
Numerical results
This section presents a detailed comparison between theoretical calculations and recent LHC data. The essential measurements have been carried out by the ATLAS [9] and CMS [10] Collaborations and refer to the following categories: Z bosons produced in association with one beauty jet, Z bosons produced in association with two beauty jets and Z bosons produced in association with explicitly reconstructed b-hadrons. In addition to the above, the ATLAS Collaboration has presented [9] inclusive cross sections for Z bosons associated with any number of b-jets. We do not analyse events of this kind in the present study and only concentrate on the production of Z bosons with one or two b-jets.
Production of Z bosons in association with one b-jet
The ATLAS Collaboration has collected the data [9] at √ s = 7 TeV. Both leptons originating from the Z boson decay are required to have p l T > 20 GeV and |η l | < 2.4, the lepton pair invariant mass lies in the interval 76 < M ll < 106 GeV, the beauty jets are required to have p b T > 20 GeV and |η b | < 2.4. We confront our predictions with the available data in Figs. 1 and 2. To estimate the theoretical uncertainties in the quark-involving subprocesses (3) -(5), calculated using the collinear QCD factorization, we have varied the scales µ R and µ F by a factor of 2 around their default values. In the k T -factorization approach, employed for off-shell gluon-gluon fusion subprocess (1), the scale uncertainties were estimated by using the gluon densities JH'2013 set 2+ and JH'2013 set 2− instead of default density JH'2013 set 2. These two sets refer to the varied hard scales in the strong coupling constant α s in the off-shell amplitude: JH'2013 set 2+ stands for 2µ R , while JH'2013 set 2− refers to µ R /2 (see [22] for more information). The estimated scale uncertainties are shown as shaded bands. As one can see, we achieve reasonably good agreement with the ATLAS data [9] within the experimental and theoretical uncertainties, although we observe some underestimation of these data at high p Z T and a slight overestimation at small transverse momenta. The slight overestimation of the data at low p Z T can probably be attributed to the TMD gluon density used, since the region p Z T < 100 GeV is fully dominated by off-shell gluon-gluon fusion, as it is demonstrated in Fig. 2. The rapidity distribution is well described practically everywhere. The NLO pQCD calculations 2 , performed using mcfm routine [35], tend to slightly overestimate our predictions and better decribe the data at large transverse momenta.
To investigate the importance of k T -factorization, we have repeated the calculation using collinear QCD factorization for all considered subprocesses (dash-dotted histograms in Fig. 1). We find that these effects are significant at low and moderate p Z T (up to p Z T ∼ 100 GeV), where the off-shell gluon-gluon fusion dominates. The effect of using k Tfactorization for gluon-dominated processes is clearly demonstrated in Fig. 2. The quarkinitiated subprocesses (3) -(5) become important only at high transverse momenta, where the typical x values are large, and that supports using of the DGLAP quark and gluon dynamics for these subprocesses (see Fig. 2). The subprocesses (3) -(5) are important to achieve an adequate description of the data in the whole p Z T region. The estimated DPS contributions are found to be small in the considered kinematic region. Some reasonable variations in σ eff ≃ 15 ± 5 mb would affect DPS predictions, though without changing our basic conclusion. We note also that scale uncertainties of the CCFM-based predictions are comparable with the ones of NLO pQCD calculations.
Production of Z bosons in association with two b-jets
The data provided by the ATLAS [9] Collaboration refer to the same energies and kinematic restrictions as in the previous subsection. The observables shown by the ATLAS Collaboration are the Z boson transverse momentum p Z T and rapidity y Z , invariant mass of the b-jet pair M bb and angular separation in η − φ plane between the jets ∆R bb . The latter is useful to identify the contributions where scattering amplitudes are dominated by terms involving gluon splitting g → Q +Q.
The results of our calculations are shown in Fig. 3 in comparison with the ATLAS data [9]. As one can see, our results describe the data reasonably well within the experimental and theoretical uncertainties, although some tendency to slightly underestimate the data at high transverse momentum p Z T and large M bb can be seen. The role of offshell gluon-gluon fusion subprocess is a bit enhanced here compared to the case of Z + b production because the quark-antiquark annihilation subprocess (4) gives a negligible contribution and gluon splitting subprocess (5) populates mainly at low η − φ distances ∆R bb . This subprocess is complementary to the one [36] where quark-gluon scattering q + g * was dominant. The estimated DPS contribution is small and can play a role at low p Z T only. The NLO pQCD calculations, performed using mcfm program 3 , tend to slightly underestimate the ATLAS data at low ∆R bb and M bb , although provide better description of the data at large transverse momentum p Z T and invariant mass M bb .
Production of Z bosons in association with two b-hadrons
In the measurements reported by CMS Collaboration [10], both b-hadrons have been identified explicitly by their full decay reconstruction. This data sample allows to study the production properties of a Zbb system even in the region of small angular seperation between the b quarks (where the usual jet analysis is not possible as the jets would overlap). In a specific subsample, an additional cut on the Z boson transverse momentum is applied, p Z T > 50 GeV. The CMS Collaboration described the angular configuration of the Zbb system in terms of spatial (in η − φ plane) and azimuthal separation between the b-hadrons ∆R bb and ∆φ bb , spatial separation min ∆R Zb between the Z boson and closest b-hadron and the asymmetry in the Zbb system defined as
A Zbb = max ∆R Zb − min ∆R Zb max ∆R Zb + min ∆R Zb ,(9)
where max ∆R Zb is the distance between the Z boson and remote b-hadron. The correlation observables are useful to identify the different production mechanisms (or specific higher-order corrections). For example, low min ∆R Zb identifies Z bosons in the vicinity of one of the b-hadron (Z bosons promptly radiated from b-quarks), small ∆φ bb indicates gluon to quark splitting g → Q+Q. Moreover, while the configurations where the two b-hadrons are emitted symmetrically with respect to the Z directions leads to a zero value of A Zbb assymetry, the additional final-state gluon radiation results in a non-zero one, that provides us with the possibility to test the high-order pQCD corrections. Our predictions are shown in Figs. 4 and 5 in comparison with the CMS data [10]. As one can see, our results with default b-quark fragmentation parameters reasonably well describe the data within the theoretical and experimental uncertainties. To estimate an additional uncertainty coming from the b-quark fragmentation, we repeated our calculations with varied shape parameter ǫ b = 0.003 (not shown), which is often used in NLO pQCD calculations. We find that the predicted cross sections (in the considered p T region) are larger for smaller ǫ b values. However, the typical dependence of numerical predictions on the fragmentation scheme is much smaller than the scale uncertainties of our calculations. The NLO pQCD predictions, obtained using the amc@nlo [37] event generator 4 , are rather close to our results.
Conclusions
We have considered the associated Z boson and beauty quark production at the LHC conditions. The calculations were done in a "combined" scheme employing both the k Tfactorization and collinear factorization in QCD, with each of them used in the kinematic conditions of its best reliability. The dominant contribution is represented by the gluongluon fusion subprocess g * g * → Zbb with Z boson further decaying into a lepton pair. This subprocess is entirely (for the first time) calculated in the k T -factorization approach. A number of subleading subprocesses contributing at O(αα 2 s ) and O(αα 3 s ) have been considered in the conventional collinear scheme.
Using the TMD gluon densities derived from the CCFM evolution equation, we have achieved reasonably good agreement between our theoretical predictions and latest CMS and ATLAS experimental data collected at √ s = 7. We find that the (formally subleading) quark-involving subprocesses become especially important at high transverse momenta and are necessary to describe the data in the whole kinematic range. Our estimations of the double parton scattering show that the latter is unimportant. This conclusion is also confirmed by the fact that our single parton scattering calculations show no room for additional contributions when compared to the ATLAS and CMS data. The dash-dotted histograms correspond to the collinear limit of our calculations. The estimated DPS contributions and mcfm [35] predictions (taken from [9]) are shown additionally. The data are from ATLAS [9]. calculated as a function of the Z boson transverse momentum, rapidity, invariant mass of the b-jet pair and angular separation between the jets. Notation of the histograms is the same as in Fig. 1. The data are from ATLAS [9]. The mcfm [35] predictions are taken from [9]. Figure 4: Associated production of a Z boson and two b-hadrons at √ s = 7 TeV. Notation of the histograms is the same as in Fig. 1. The data are from CMS [10]. The amc@nlo [37] predictions are taken from [10]. additional kinematical cut on the Z boson transverse momentum p Z T > 50 GeV. Notation of the histograms is the same as in Fig. 1. The data are from CMS [10]. The amc@nlo [37] predictions are taken from [10].
Figure 1 :
1Associated Z + b production cross section at √ s = 7 TeV presented as a function of the Z boson transverse momentum (left panel) or rapidity (right panel). The solid histograms show our predictions at the default scale while shaded bands correspond to scale variations described in the text.
Figure 2 :
2The off-shell gluon-gluon fusion contribution to the associated Z + b production at √ s = 7 TeV. The on-shell limit of our calculations is shown additionally. The data are from ATLAS[9].
Figure 3 :
3Associated production of a Z boson with two beauty jets at √ s = 7 TeV
Figure 5 :
5Associated production of a Z boson and two b-hadrons at √ s = 7 TeV under
See reviews[13] for more information.
We take them from ATLAS publication[9].
We take them from ATLAS publication[9].4 We take them from CMS publication[10].
AcknowledgementsWe thank F. Hautmann
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| []
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[]
| [
"I B Khriplovich \nBudker Institute of Nuclear Physics\nNovosibirsk University\n630090NovosibirskRussia\n",
"G Yu Ruban \nBudker Institute of Nuclear Physics\nNovosibirsk University\n630090NovosibirskRussia\n"
]
| [
"Budker Institute of Nuclear Physics\nNovosibirsk University\n630090NovosibirskRussia",
"Budker Institute of Nuclear Physics\nNovosibirsk University\n630090NovosibirskRussia"
]
| []
| We derive in a straightforward way the spectrum of a hydrogen atom in a strong magnetic field. | null | [
"https://arxiv.org/pdf/quant-ph/0309014v2.pdf"
]
| 872,421 | quant-ph/0309014 | e304de7ff34f4b5a2dcc6eff76c51c6a38a7d7b2 |
Dec 2003
I B Khriplovich
Budker Institute of Nuclear Physics
Novosibirsk University
630090NovosibirskRussia
G Yu Ruban
Budker Institute of Nuclear Physics
Novosibirsk University
630090NovosibirskRussia
Dec 2003arXiv:quant-ph/0309014v2 3 Hydrogen atom in strong magnetic field revisited
We derive in a straightforward way the spectrum of a hydrogen atom in a strong magnetic field.
u ′′ + − 1 4 + ν z u = 0.(1)
We have introduced in it the usual dimensionless variable:
2z aν → z;
here a =h 2 /m e e 2 is the Bohr radius, m e is the electron mass, ν is the effective quantum number, related to the electron energy as
E ν = − m e e 4 2h 2 ν 2 .(2)
Equation (1) coincides exactly with the radial equation for the s-wave in the threedimensional Coulomb potential −e 2 /r, and has therefore the common hydrogen spectrum
E − n = −
m e e 4 2h 2 n 2 n = 1, 2, 3... ,
and the set of solutions
u − n (z) = exp (−z/2) z F (−n, 2; z) , ν = n ,(4)
where F is the confluent hypergeometric function (here and below we are not interested in the normalization factors). These solutions vanish at the origin and are trivially continued to z < 0. Thus obtained solutions on the whole z axis are odd under z → −z (as reflected by the superscripts "minus" in (3), (4)).
There is however an essential difference between the present problem and the s-wave Coulomb one. In the last case (4) is the only solution. The reason is well-known. Naively the radial wave equation for R(r)(= u(r)/r) has two independent solutions, which behave for r → 0 as R ∼ const (u ∼ r) and R ∼ 1/r (u ∼ const), respectively. However, in fact R ∼ 1/r is no solution at all for the homogeneous wave equation if the point r = 0 is included, since △(1/r) = −4πδ(r). As to our problem, equation (1) does not describe really the vicinity of z = 0 since therein we have to consider seriously the magnetic field itself. Therefore, there are no reasons to discard those solutions of (1) which tend to a constant for small z (and of course decrease exponentially for z → ∞).
Such solutions are presented in a convenient form in [2] (Mathematical Appendices, § d, (d.17)). To our purpose they can be written for z > 0 as
u + ν (z) = exp (−z/2) 1 − νz ln z F (1 − ν, 2; z) + ∞ k=0 Γ(1 − ν + k) [ψ(1 − ν + k) − ψ(k + 2) − ψ(k + 1)] Γ(1 − ν) k! (k + 1)! z k ;(5)
here ψ(α) denotes the logarithmic derivative of the gamma function:
ψ(α) = Γ ′ (α)/Γ(α).
Being trivially continued to z < 0, thus obtained solutions on the whole z axis are even under z → −z (as reflected by the superscript "plus" in (5) and in the corresponding eigenvalues below). Under any reasonable regularization of the logarithmic singularity at z → 0, the even solutions should have vanishing first derivative at the origin. In this way we obtain the following equation for the eigenvalues of ν:
ln a a H = 1 2ν + ψ(1 − ν);(6)
here a H = hc/eH is the typical scale for the radius of electron orbits in the magnetic field H. We are working in the logarithmic approximation, i.e. assume that
λ = ln a/a H ≫ 1.(7)
This allows us to use a crude cut-off at a H for the formal logarithmic divergence at z → 0, as well as to simplify somewhat this equation. The smallest root of equation (6) is
ν + 0 = 1 2λ ,(8)
which gives the ground state energy
E + 0 = − m e e 4 2h 2 ln −2 h 3 H 2 m 2 e e 3 c .(9)
Other roots of equation (6) are
ν + n = n + 1 λ n = 1, 2, 3... ,(10)
with the corresponding energies
E − n = − m e e 4 2h 2 n 2 1 − 2 n ln −1 h 3 H 2 m 2 e e 3 c .(11)
Let us mention that the one-dimensional Coulomb problem was considered in [3 -5] with various regularizations of the singularity at z → 0, but without any relation to the problem of the hydrogen atom in a strong magnetic field.
But let us come back to our problem. The resulting spectrum of the hydrogen atom in a strong magnetic field looks as follows. Each Landau level in this field serves as an upper limit to the sequence of discrete levels of the Coulomb problem in the z direction. This discrete spectrum consists of a singlet ground state with the energy given by formula (9), and close doublets of odd and even states of energies given by formulae (3) and (11). There is also a continuous spectrum of the motion along z above each Landau level.
This picture is valid for sufficiently low Landau levels, as long as the radius of a magnetic orbit is much less than the Bohr radius. Obviously, in a strong magnetic field this description fails for large magnetic quantum numbers, i.e., in the semiclassical region.
Here we can estimate the orbit radius directly from the well-known spectrum of electron in a magnetic field (see, e.g., [2] ( §112, Problem 1)):
E =h eH m e c (N + 1/2), N = n ρ + m + |m| 2 ;
(12) here n ρ is the radial quantum number in the xy plane, and m is the angular momentum projection onto the z axis. Now the semiclassical estimate for the magnetic radius is
a H (N) ≈ hc eH N = a H √ N .
Thus the present picture of levels holds as long as
λ = ln a a H ≫ ln N.(13)
At last, let us consider the correspondence between the obtained system of levels in a strong magnetic field and the hydrogen spectrum in a vanishing field. A beautiful solution of this problem was given in [6] (and is quoted in [1]). We would like to present the solution here as well (having in mind in particular that our note can be considered as a mini-review).
The crucial observation made in [6] is as follows. While changing the magnetic field from vanishingly small to a very strong one, the number of nodal surfaces of a given wave function remains the same. A hydrogen wave function (in zero magnetic field) with quantum numbers n, l, m has n r = n − l − 1 nodal spheres and l − |m| nodal cones with a z axis. With the increase of the magnetic field, the nodal spheres become ellipsoids of rotation, more and more prolate, tending to cylinders in the limit of infinite field. The correspondence between n r and n ρ (n ρ being the radial quantum number in the xy plane in a strong magnetic field) gets obvious from this picture:
n ρ = n r .(14)
The evolution of the hydrogen nodal cones is less obvious. However, due both to equation (14) and to the conservation of the total number of nodal surfaces, the number n z of the nodes of a solution of equation (1) should coincide with the number of nodes of the corresponding spherical function,
n z = l − |m|.(15)
In other words, l − |m| nodal cones of a hydrogen wave function evolve into n z planes of constant z corresponding to the nodes of an eigenfunction of equation (14). And at last, let us note that during the whole evolution of the magnetic field, m remains constant.
Let us consider now, for instance, the ground state in the magnetic field, with N = 0 (see (12)). Obviously, it is degenerate, and its corresponding magnetic wave functions have n ρ = 0 and m = 0, −1, −2 ... . Let us confine further to its lowest sublevel corresponding to the ground state solution of equation (1), with n z = 0. According to the above arguments, the hydrogen ancestors of those wave functions should have n r = 0 and l = |m|. In other words, these ancestors are:
1s; 2p, m = −1; 3d, m = −2; and so on.We are grateful to V.A. Novikov, V.V. Sokolov, M.I. Vysotsky, and A.V. Zolotaryuk for useful discussions. We acknowledge the support by the Russian Foundation for Basic Research through grant No. 03-02-17612.
. R J Elliott, R Loudon, J. Phys. Phys. Chem. Solids. 15196R.J. Elliott and R. Loudon, J. Phys. Phys. Chem. Solids. 15 (1960) 196
L D Landau, E M Lifshitz, Quantum Mechanics. Nauka, MoscowL.D. Landau and E.M. Lifshitz, Quantum Mechanics (1989, Nauka, Moscow)
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. W H Kleiner, Lincoln Lab, Progr. Rep. W.H. Kleiner, Lincoln Lab. Progr. Rep. (Feb. 1960)
| []
|
[
"Photonic-crystal-reflector nano-resonators for Kerr-frequency combs",
"Photonic-crystal-reflector nano-resonators for Kerr-frequency combs"
]
| [
"Su-Peng Yu [email protected] ",
"Hojoong Jung ",
"Travis C Briles ",
"Kartik Srinivasan ",
"Scott B Papp ",
"\nFrequency Division\n‡Department of Physics\nNIST\nBoulderColoradoUSA\n",
"\n¶Present address: Center for Quantum Information\nUniversity of Colorado\nBoulderColoradoUSA\n",
"\n§Microsystems and Nanotechnology Division\nKorea Institute of Science and Technology\nSeoulSouth Korea\n",
"\nNIST\nGaithersburgMarylandUSA\n"
]
| [
"Frequency Division\n‡Department of Physics\nNIST\nBoulderColoradoUSA",
"¶Present address: Center for Quantum Information\nUniversity of Colorado\nBoulderColoradoUSA",
"§Microsystems and Nanotechnology Division\nKorea Institute of Science and Technology\nSeoulSouth Korea",
"NIST\nGaithersburgMarylandUSA"
]
| []
| We demonstrate Kerr-frequency-comb generation with nanofabricated Fabry-Perot resonators with photonic-crystal-reflector (PCR) end mirrors.The PCR group-velocitydispersion (GVD) is engineered to counteract the strong normal GVD of a rectangular waveguide fabricated on a thin, 450 nm silicon nitride device layer. The reflectors provide the resonators with both the high optical quality factor and anomalous GVD required for Kerrcomb generation. We report design, fabrication, and characterization of devices in the 1550 nm wavelengths bands, including the GVD spectrum and quality factor. Kerr-comb generation is achieved by exciting the devices with a continuous-wave (CW) laser. The versatility of PCRs enables a general design principle and a material-independent device infrastructure for Kerr-nonlinear-resonator processes, opening new possibilities for manipulation of light. Visible and multi-spectral-band resonators appear to be natural extensions of the PCR approach. arXiv:1904.07289v2 [physics.optics] | 10.1021/acsphotonics.9b00578 | [
"https://arxiv.org/pdf/1904.07289v2.pdf"
]
| 119,113,596 | 1904.07289 | db5887502c7548c6843f356e71b06f29c6767095 |
Photonic-crystal-reflector nano-resonators for Kerr-frequency combs
Su-Peng Yu [email protected]
Hojoong Jung
Travis C Briles
Kartik Srinivasan
Scott B Papp
Frequency Division
‡Department of Physics
NIST
BoulderColoradoUSA
¶Present address: Center for Quantum Information
University of Colorado
BoulderColoradoUSA
§Microsystems and Nanotechnology Division
Korea Institute of Science and Technology
SeoulSouth Korea
NIST
GaithersburgMarylandUSA
Photonic-crystal-reflector nano-resonators for Kerr-frequency combs
We demonstrate Kerr-frequency-comb generation with nanofabricated Fabry-Perot resonators with photonic-crystal-reflector (PCR) end mirrors.The PCR group-velocitydispersion (GVD) is engineered to counteract the strong normal GVD of a rectangular waveguide fabricated on a thin, 450 nm silicon nitride device layer. The reflectors provide the resonators with both the high optical quality factor and anomalous GVD required for Kerrcomb generation. We report design, fabrication, and characterization of devices in the 1550 nm wavelengths bands, including the GVD spectrum and quality factor. Kerr-comb generation is achieved by exciting the devices with a continuous-wave (CW) laser. The versatility of PCRs enables a general design principle and a material-independent device infrastructure for Kerr-nonlinear-resonator processes, opening new possibilities for manipulation of light. Visible and multi-spectral-band resonators appear to be natural extensions of the PCR approach. arXiv:1904.07289v2 [physics.optics]
Introduction
Optical-frequency combs provide revolutionary technologies for applications and research ranging from optical spectroscopy 1 and metrology 2 to optical communication. 3 In particular, nanofabricated frequency-comb systems would allow for wide-spread application of frequencycomb techniques, enabled by scalable fabrication at low cost. Kerr-microresonator frequency combs-microcombs -realized with integrated photonics are also likely to be a potent propelling force for emerging technologies such as optical waveform generation, 4,5 ranging, 6 inertial navigation, 7 and LIDAR applications. 8 A variety of microcomb systems, with diverse materials and methods, are under active study.
Silicon nitride (Si 3 N 4 , hereafter SiN) is a thoroughly studied material for micro-and nanophotonic applications due to its low optical loss and compatibility with standard semiconductor manufacturing processes. SiN material is particularly advantageous for Kerr frequency comb generation in ring resonators due to high third-order nonlinearity, 9,10 and has seen rapid development in recent years. 11 The challenge for Kerr combs in the ring resonator geometry lies in the control of group-velocity dispersion (GVD, or simply dispersion). Typi-cally, photonic waveguides with sub-wavelength cross-section demonstrate strong normal GVD associated with index guiding, and this inhibits phase-matching of four-wave mixing processes that are necessary for Kerr-comb generation. To maintain anomalous dispersion, a thick SiN device layer is often utilized, leading to challenges in fabrication due to etching thick layers and high materials stresses that lead to cracking and low device yield. 12 Additionally, the bulk SiN material demonstrates strong normal dispersion at shorter wavelength ranges, 13 resulting in challenges for Kerr-comb generation with visible light. Methods to alleviate such constraints are under active development, including exotic waveguide geometries 14,15 and novel materials. 16 In this article, we leverage sophisticated nanofabrication capabilities applicable to SiN photonic devices to create Fabry-Perot-type microresonators with photonic-crystal reflectors (PCR). Photonic crystals are periodic dielectric structures demonstrating unprecedented capabilities to engineer optical properties including group velocity 17 and mode profile. 18,19 The capability the PCR devices offer is that targeted optical characteristics are generated based on nanophotonic patterning independent of device layer thickness. A wider dynamic range of dispersion can be reached using this method to enable compensation of intrinsic dispersion contributions with a wider range of materials, hence enabling realization of Kerr comb generation in a broad range of integrated photonics platforms. For application in SiN devices proposed here, the requirement for a thick device layer can be alleviated, and the optical properties of such micro-cavities can be completely controlled by single-layer patterning using lithography. Here we demonstrate high optical quality factor resonators with engineered anomalousdispersion capable of generating Kerr frequency combs, based on patterned SiN with a 450 nm thickness device layer.
The article will be presented as follows: first, we provide an intuitive concept for the PCR resonator Kerr combs, with numerical design methods and measurement plan; then, we report the fabrication and characterization, fol-lowed by comb generation experiment; finally, we propose possibilities made available by the PCR technology, including designs for a 532 nm pumped visible-band comb. Figure 1(a) presents a conceptual picture of our PCR resonators and their use for Kerrfrequency-comb generation. The resonators are composed of two PCRs, and light is guided between them by a waveguide. The dispersion manifests in resonators in the form of frequency dependence of the free-spectral range (FSR) between adjacent resonances. The free-spectral range has the form FSR = c/2 n g L in a Fabry-Perot resonator, where n g is the group index. In a Fabry-Perot-type resonator formed with subwavelength cross-section photonic waveguides, strong normal geometric dispersion arises from gradual concentration of optical field into the high-index waveguide region as the wavelength is reduced, therefore increasing n g of the guided optical mode. In such a normal-dispersion condition, FSR decreases with increasing optical frequency. To counteract decreasing FSR, the frequency-dependent effective cavity length L can also be utilized as a engineerable parameter. We design PCRs to effectively reduce the cavity length L with increasing optical frequency. Specifically, a PCR can be designed to reflect light at varying depth, with the highest optical frequencies reflecting at the shallowest depth. This is achieved by varying the local bandgap structures on the PCR (gray shaded areas in fig. 1(a)). The total GVD of a PCR resonator is a combination of the PCR contribution and the waveguide contribution, which scales directly with the waveguide length. Therefore, we may adjust the total dispersion simply by changing the length of the resonator. Figure 1(b) presents the experimental setup used to characterize both passive device properties, such as broadband on-resosance transmission (c) and observing Kerr comb formation (d). In particular, broadband transmission measurements offer a rapid testing procedure to rapidly explore the parameter space for PCR designs. We use an unseeded semiconductor optical amplifier as a broad-band light source, and a polarization controller and lensed fibers are used to launch light onto the silicon chip. The transmitted light is directly fed into an optical spectrum analyzer (OSA). The PCR resonators transmit light only on resonance, producing a low-background signal on the OSA even for high-Q devices with linewidth smaller than the resolution bandwidth of the OSA. Fitting the peak centers of the broad-band spectra allows for rapid extraction of device dispersion information. We are able to efficiently screen through devices to find ones with the desired reflection band placement and dispersion profile. Selected PCR resonators are tested in detail with tunable laser sweep for accurate determination of Q and coupling condition, shown in the inset of 1(c). We experiment with devices with proper dispersion and high Q factors to create an optical frequency comb from the PCR cavities. The tunable laser is amplified with an erbium-doped fiber amplifier (EDFA) to provide sufficient power to bring the PCR cavities above the comb generation threshold. An example PCR Kerr comb, analyzed by the OSA, is shown in Fig. 1(d).
Design Concepts
Optical Characterization
PCR design through finite-Element Simulations
Here we present the design procedure for PCR resonators. There are two primary requirements to be satisfied by the PCRs. First, the PCR must show sufficiently high reflectivity in the design wavelength ranges, characterized by the finesse F = F SR F W HM π 1−R , where R is the reflectivity of the PhC reflector. Second, the PCR must generate anomalous group-delay dispersion (GDD) in the target wavelength ranges. For ease of comparison, the GDD will be converted to an effective GVD over the in-resonator waveguide by GVD = GDD/L. The PCRs are composed of sub-wavelength unit cells, whose geometry determines the local reflection band of the crystal, shown in 2(a). A PCR is then built from a stack of continuously varying cells. Finite-element method (FEM) tools are employed to explore the parameter space of unitcell geometry, and the reflection and transmission characteristics of a stack of multiple cells. Fig. 2 presents our design process for PCRs to satisfy the requirements for Kerr-comb generation. A typical unit cell dispersion curve is plotted for the 1D Brillouin zone in Fig. 2(a), with its geometry and electric field patterns shown in panel (i-iii). Strong reflection is achieved by creating a photonic bandgap, a frequency range where no guided propagation mode exists. The bandwidth and center of the bandgap can be rapidly simulated with a FEM eigenfrequency solver in full 3D, since the simulation volume contains only one unit-cell. We rapidly map the bandgap placement as function of unit cell geometries using this method. The unit cells parameters are then used as components to design reflectors made from a stack of varying cells. A typical reflector begins as an unpatterned waveguide, the photonic crystal is introduced by adiabatic onset of modulation to the local width of the waveguide. For simplicity, we use a simple linear tapering scheme for the adiabatic onset, where the parameters such as the waveguide width, modulation amplitude, and lattice constant are varied linearly with the tapering parameter η:
v(η) = v 0 + η( v 1 − v 0 )
where v represents the list of unit cell geometry parameters. The local bandgap of such a taper is shown in Fig. 2(b), where the unit cell continuously transforms from a waveguide with width = 750 nm and lattice constant = 430 nm (η = 0) to the nominal cell of minimum width = 200 nm, lattice constant = 480 nm, and modulation peak amplitude of 875 nm (η = 1). The geometry is designed so that high optical frequencies reflect at shallower depth in the taper, and also to ensure the absence of localized resonance modes inside the reflector itself. The latter meaning no frequencies pass through two disjoint sections of bandgap on the tapering profile. We also design PCRs with chirped nominal parameters. The scale invariance of Maxwell's equations indicate that a change in reflection wavelength range λ → ξλ can be achieved by scaling all geometry parameters by v → ξ v. In practice, this is carried out for all parameters except the device layer thickness. This chirping scheme enables sweeping of bandgaps over a larger total bandwidth. The local bandgap of this chirped mirror is plotted in Fig. 2(c). The desired anomalous dispersion engineering can be achieved by either using the adiabatic tapering or chirping. We were able to achieve higher finesse with the former, while broader bandwidth with the later. With a design in hand, the sweeping trajectory in the parameter space is realized with a finite number of sites, and the resulting stack geometry is simulated in 3D to extract the complex transmission and reflection coefficients of the PCR. With sufficiently slow sweeping of geometry, the scattering loss due to group index and mode profile mismatch can be adiabatically suppressed. 20 A typical simulated finesse curve and an expected transmission spectrum assuming a Fabry-Perot-like 1D resonator formed from such PCRs are plotted in Fig. 2(d), showing a series of resonances with strong suppression of off-resonance transmission in the target bandwidth from 190 to 200 THz.
PCR-resonator nanofabrication
We fabricate PCR resonators by the process flow in 3(a). We acquire silicon wafers with thermally grown silicon dioxide for a cladding layer, and the silicon nitride device layer is grown with low-pressure chemical vapor deposition (LPCVD). We choose a device layer thickness of 450 nm to specifically allow for conventional, single-step LPCVD processing and subsequent nanofabrication without the need for stress-relieving patterns, 12,21 annealing steps between multiple LPCVD runs, 13 or other steps necessary to process thick SiN films. The devices are patterned using electron-beam lithography (EBL), and transferred into the device layer using a fluorine-based reactive ion etching (RIE) process. The patterns for the EBL step are adjusted empirically to compensate for dimension changes during pattern transfer, enabling us to fabricate devices in good agreement with the design geometries. We apply a top cladding layer of silicon dioxide, using a plasma-enhanced chemical vapor deposition (PECVD) process, as fully oxide-clad devices enable more efficient input power coupling using lens fibers. Top cladding also makes the devices more robust against contamination. We perform a high-temperature annealing step of 900 • C for three hours to improve the optical quality of the PECVD oxide after deposition, and we observed up to a factor of three reduction of the optical absorption. To separate chips from the fabrication wafer, we use deep reactive ion etching, which also enable us to fabricate tapering bus waveguides terminating at the chip facets for high fiber-to-chip coupling efficiency. Overall, we fabricate chips with hundreds of PCR resonator devices, and lensed fiber coupling to the chip devices offers 3.5 dB insertion loss per facet.
PCR-resonator GVD design and characterization
We have carried out a detailed set of simulations to demonstrate GVD engineering in our PCR resonators. With the thin SiN device layer in this study, the total effective GVD is a competition between the normal dispersion of the waveguide and the anomalous effective dispersion of the reflectors. As the length of the resonator is increased, the total effective dispersion continuously varies from strongly anomalous from the reflectors to normal dispersion of the waveguides. Here we engineer the anomalous PCR dispersion using the aforementioned tapering and chirping methods. Following the convention mentioned previously, the GDD of the PCR will be averaged over the in-resonator waveguide length for the following discussion. A set of simulated traces demonstrating such dispersion change is plotted in Fig. 4(a). Here, the design goal is to form a pocket of controlled anomalous dispersion around the Telecom C band. The resonator length is swept to demonstrate the dispersion balancing, where a nearzero dispersion is predicted at a length of 700 µm.
We can systematically vary the dispersion setting by the tapering condition-with a longer taper resulting in larger anomalous dispersion. Our simulations indicate that a range of dis-persion is achievable by this method, although constraints exist from the desire to also achieve high resonator finesse and efficient resonator coupling. Specifically, a taper that is too short in length is no longer adiabatic and causes scattering loss and reduces finesse, while one that is too long results in a reflector with insufficient transmission, hence under-coupling the resonator. For future designs, the restriction associated with insufficient PCR transmission can potentially be alleviated by evanescent coupling the bus waveguide directly to the in-resonator waveguide in a manner similar to the ring resonator case. The strength of normal dispersion of the waveguide section depends on the waveguide width, which can also be varied as a design parameter.
To put PCR-resonator dispersion engineering into practice, we fabricated a series of devices that feature a systematic GVD variation; see Fig. 4(b,c). In particular, we design sweeps in device length to demonstrate the counterbalancing of waveguide and reflector dispersion. Transmission measurements using a broadband source, here a supercontinuum source 22 for better measurement bandwidth, allow us to characterize the GVD in detail. The reflecting PCR resonators only transmit light on resonance, therefore the transmission as measured by the OSA is comprised of a set of narrow peaks; the resulting spectra are shown in Fig. 4(b). We identify the frequency of the resonances by finding the peaks of the transmitted power, then calculate the resonator dispersion profiles D int , shown in 4(c). The dispersion profiles demonstrate the predicted gradual transition from strong anomalous dispersion for the L = 100 µm to normal dispersion for the longest L = 1000 µm case, with the zero-dispersion occurring near L = 700 µm.
PCR-resonator Kerr-comb generation
A demonstration of Kerr-frequency combs generation with PCR resonators is shown in Fig. 5. We selected a device length of L c = 250 µm, which is within the range described in the previous section. The loaded quality factor of the device is ≥ 10 5 . The specific dispersion parameters of the device are characterized by D 2 = −c/n·β 2 ·F SR 2 ≈ 0.45 GHz/mode, where β 2 is the second-order GVD, and the integrated dispersion
D int = D 2 2 · m 2 + O(m 3 ),
where m is the mode index; see Fig. 5(a). We amplify the CW pump laser in an EDFA and couple up to 550 mW onto the chip incident on the first PCR. As the pump laser is tuned into the cavity resonance starting on the blue detuned side, we observe characteristic thermo-optic bistability and self-locking of the resonator onto the pump laser allowing for a large buildup of intracavity power. Given sufficient power, the parameteric oscillation threshold is reached, and we observe the formation of a Kerr comb in the modulation instability regime excited from vacuum. Here the device reflection band is chosen to be off-center from the pump to achieve desired coupling condition, leading to the asymmetric shape in D int . (b) The onset of primary comb, and (c) the filling to a full comb with increasing pump detuning. We note that traces in (b) and (c) were measured from two different devices. Fig. 5(b,c) show the evolution of comb generation, operationally as we increase in-resonator power by tuning the CW laser onto resonance. We observe Turing-pattern generationso-called primary comb -in modes ±2 FSRs away from the pump mode. Harmonics of the primary comb lines become populated with increasing in-resonator power, shown in Fig. 5(b). We also observed filling-in of the comb lines to form a full 1-FSR (268 GHz) comb, given sufficient power, shown in Fig. 5(c). The elevated background noise level in this figure is the amplified spontaneous emission noise from the amplifier used with the CW laser. The comb generation unambiguously verifies the creation of anomalous dispersion in the PCR resonators. The kinds of Turing patterns that we observe are the fundamental process of any Kerr-microresonator. Our experiments show the opportunity to systematically vary the optical-mode spacing of Turing patterns, which offers functionality to generate millimeter-wave electronic signals through photodetection.
Moreover, access to Turing patterns offer the possibility for opticalparametric amplification processes of external fields injected to the PCR resonator and related optical-wavelength translation processes. In future device fabrication iterations, we plan to target higher Q factors that can both lead to lower threshold and potentially access to Kerrsoliton generation. We would like to emphasize that a significant advantage of PCR-based resonators is the decoupling of device layer configuration from dispersion engineering. To demonstrate this capability, we utilize the same design principle as the 1550 nm band devices described above to create anomalous dispersion for multiple wavelength bands. An interesting case is shown in Fig. 6(a,b), where anomalous dispersion is created around 532 nm and 1064 nm wavelength ranges, using a shared SiN device layer thickness of 450 nm. The strong anomalous dispersion of the PCR is sufficient to balance the typically strong normal dispersion of materials in the visible ranges. The capability to engineer dispersion profiles independent of device layer enables integration of multiple wavelength band devices onto a same chip. Taking this concept further, we point out that the PCR is transparent to frequencies below its bandgap, making it possible to build cascaded reflectors like the case shown in 6(c). Multifrequency cascaded resonators as such can potentially provide enhancement for second-or third-harmonics generation, or to facilitate interlocked combs between different wavelength ranges. 14 The PCR resonators provide flexibility in dispersion control beyond conventional ring-and disc-resonators, and makes available dispersion parameter ranges previously difficult to reach.
Future Prospects
Conclusion
We have presented on-chip Fabry-Perot-type resonators with tailored dispersion based on PCRs, providing a pathway toward integrated photonics frequency comb generation. Conceptually, the PCRs are the microscopic analogy of chirped dielectric stack mirrors, capable of producing anomalous dispersion based on their geometry, independent of the device layer configuration. We designed and fabricated PCR resonators based on a SiN device layer and standard fabrication techniques. Using a broadband light source and an OSA, we were able to directly verify the dispersion profile of such resonators. Anomalous dispersion of varying strength is created in a 450 nm thickness device layer, which is challenging to achieve with conventional ring resonators. We also demonstrated frequency comb generation in these devices. The PCR cavities provide versatile ca-pabilities in dispersion engineering, and should enable construction of on-chip frequency comb sources for a wide range of design wavelengths.
Graphical TOC Entry
Figure 2 :
2(a) Band structure of the 1D photonic crystal, where a is the lattice constant. At higher frequencies, the upper band (red) couples to free-propagating continuum and is no longer guided. The profiles of (i) nominal unit cell, (ii) upper and (iii) lower band electric field are also shown. (b) Local band structures of an adiabatic taper from ordinary waveguide to photonic crystal reflector. (c) Local band structure of a chirped photonic crystal reflector. (d) Calculated cavity finesse and simulated transmission using an 1D model with reflection coefficient from FEM results and a designated waveguide length.
Figure 3 :
3(a) Fabrication process steps include e-beam lithography, pattern transfer by RIE, cladding using PECVD oxide, and chip separation with deep RIE. (b) A stitched-together full view of a resonator, and (c) zoom-in image of a photonic crystal reflector using SEM.
Figure 4 :
4(a) Calculated D int and total dispersion for cavities with chirped photonic crystal reflector and 500 nm width waveguide of varying length. (b) Measured broad-band transmission spectra of chirped reflector and 750 nm width waveguides of varying length, where the FSR are 70, 98, 184, and 484 GHz from panel one (top) through four (bottom), and (c) calculated D int from resonance frequencies demonstrating transition from anomalous to normal dispersion.
Figure 5 :
5(a) The calculated integrated dispersion D int relative to the pumped mode at 1550 nm.
Figure 6 :
6(a) Total dispersion for photonic resonators optimized for anomalous dispersion at 532 nm wavelength, and (b) 1064 nm wavelength. (c) Illustration and simulated electric field profiles for 1064 nm to 1550 nm cascaded photonic crystal reflectors.
Acknowledgement Funding provided by DARPA DODOS, Air Force Office of Scientific Research (FA9550-16-1-0016), NIST, and UMD/NIST-CNST Cooperative Agreement (70NANB10H193). We acknowledge the Boulder Microfabrication Facility, where the devices were fabricated. We thank Nima Nader and Jeff Chiles for helpful suggestions, and Tara E. Drake and Jizhao Zang for a careful reading of the manuscript. This work is a contribution of the U.S. Government and is not subject to copyright. Mention of specific companies or trade names is for scientific communication only, and does not constitute an endorsement by NIST.
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[
"Star formation history and environment of the dwarf galaxy UGCA 92 ⋆",
"Star formation history and environment of the dwarf galaxy UGCA 92 ⋆"
]
| [
"Lidia Makarova \nSpecial Astrophysical Observatory\nNizhniy Arkhyz369167Karachai-CherkessiaRussia\n\nNewton Institute of Chile\nSAO Branch\nRussia\n",
"Dmitry Makarov \nSpecial Astrophysical Observatory\nNizhniy Arkhyz369167Karachai-CherkessiaRussia\n\nUniversité Lyon 1\nF-69622VilleurbanneFrance\n\nCRAL\nObservatoire de Lyon\n\nSt. Genis Laval\nF-69561France\n",
"Sergey Savchenko \nNewton Institute of Chile\nSAO Branch\nRussia\n\nAstronomical Institute, Saint-Petersburg State University\nSaint-PetersburgRussia\n"
]
| [
"Special Astrophysical Observatory\nNizhniy Arkhyz369167Karachai-CherkessiaRussia",
"Newton Institute of Chile\nSAO Branch\nRussia",
"Special Astrophysical Observatory\nNizhniy Arkhyz369167Karachai-CherkessiaRussia",
"Université Lyon 1\nF-69622VilleurbanneFrance",
"CRAL\nObservatoire de Lyon",
"St. Genis Laval\nF-69561France",
"Newton Institute of Chile\nSAO Branch\nRussia",
"Astronomical Institute, Saint-Petersburg State University\nSaint-PetersburgRussia"
]
| [
"Mon. Not. R. Astron. Soc"
]
| We present a quantitative star formation history of the nearby dwarf galaxy UGCA 92. This irregular dwarf is situated in the vicinity of the Local Group of galaxies in a zone of strong Galactic extinction (IC 342 group of galaxies). The galaxy was resolved into stars with HST/ACS including old red giant branch. We have constructed a model of the resolved stellar populations and measured the star formation rate and metallicity as function of time. The main star formation activity period occurred about 8 -14 Gyr ago. These stars are mostly metal-poor, with a mean metallicity [Fe/H] ∼ −1.5 -−2.0 dex. About 84 per cent of the total stellar mass was formed during this event. There are also indications of recent star formation starting about 1.5 Gyr ago and continuing to the present. The star formation in this event shows moderate enhancement from ∼ 200 Myr to 300 Myr ago. It is very likely that the ongoing star formation period has higher metallicity of about −0.6 -−0.3 dex. UGCA 92 is often considered to be the companion to the starburst galaxy NGC 1569. Comparing our star formation history of UGCA 92 with that of NGC 1569 reveals no causal or temporal connection between recent star formation events in these two galaxies. We suggest that the starburst phenomenon in NGC 1569 is not related to the galaxy's closest dwarf neighbours and does not affect their star formation history. | 10.1111/j.1365-2966.2012.20872.x | [
"https://arxiv.org/pdf/1203.1397v1.pdf"
]
| 118,479,246 | 1203.1397 | 20d8047000ce89a773f2833a67d4febecbee5c75 |
Star formation history and environment of the dwarf galaxy UGCA 92 ⋆
May 2014
Lidia Makarova
Special Astrophysical Observatory
Nizhniy Arkhyz369167Karachai-CherkessiaRussia
Newton Institute of Chile
SAO Branch
Russia
Dmitry Makarov
Special Astrophysical Observatory
Nizhniy Arkhyz369167Karachai-CherkessiaRussia
Université Lyon 1
F-69622VilleurbanneFrance
CRAL
Observatoire de Lyon
St. Genis Laval
F-69561France
Sergey Savchenko
Newton Institute of Chile
SAO Branch
Russia
Astronomical Institute, Saint-Petersburg State University
Saint-PetersburgRussia
Star formation history and environment of the dwarf galaxy UGCA 92 ⋆
Mon. Not. R. Astron. Soc
000May 2014Accepted XXX. Received XXX; in original form XXX(MN L A T E X style file v2.2)galaxies: dwarf -galaxies: formation -galaxies: evolution -galaxies: stellar content -galaxies: individual: UGCA 92
We present a quantitative star formation history of the nearby dwarf galaxy UGCA 92. This irregular dwarf is situated in the vicinity of the Local Group of galaxies in a zone of strong Galactic extinction (IC 342 group of galaxies). The galaxy was resolved into stars with HST/ACS including old red giant branch. We have constructed a model of the resolved stellar populations and measured the star formation rate and metallicity as function of time. The main star formation activity period occurred about 8 -14 Gyr ago. These stars are mostly metal-poor, with a mean metallicity [Fe/H] ∼ −1.5 -−2.0 dex. About 84 per cent of the total stellar mass was formed during this event. There are also indications of recent star formation starting about 1.5 Gyr ago and continuing to the present. The star formation in this event shows moderate enhancement from ∼ 200 Myr to 300 Myr ago. It is very likely that the ongoing star formation period has higher metallicity of about −0.6 -−0.3 dex. UGCA 92 is often considered to be the companion to the starburst galaxy NGC 1569. Comparing our star formation history of UGCA 92 with that of NGC 1569 reveals no causal or temporal connection between recent star formation events in these two galaxies. We suggest that the starburst phenomenon in NGC 1569 is not related to the galaxy's closest dwarf neighbours and does not affect their star formation history.
INTRODUCTION
Dwarf galaxies are most numerous objects in the Universe. The question of star formation in dwarf galaxies is extremely important for understanding of their origin and evolution. The local Universe ( 10 Mpc) is particularly important and convenient for studying dwarf galaxies. Nearby galaxies are resolved into individual stars, which allow us to study stellar population of these galaxies directly. In the last decades, significant progress has been achieved in the study of resolved stellar populations due to the Hubble Space Telescope (HST ) and new large ground-based telescopes.
About 50 per cent of nearby galaxies are situated in groups and clouds (Makarov & Karachentsev 2011). Taking ⋆ Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. † E-mail: [email protected] into account loose associations of dwarf galaxies ), most of the galaxies within 3 Mpc are not isolated objects. The nearby group around IC 342 is obscured by strong Galactic extinction (see Fig. 1). The IC 342-Maffei complex should have significant impact on dynamic and evolution of the nearby Universe. Unfortunately, the 'Zone of Avoidance' hides this region from us and a determination of main properties of galaxies behind it is a challenge.
In the framework of the study of the structure of the nearby Universe (HST project 9771) we have obtained the images of dwarf galaxies within IC 342/Maffei complex. In the present work we have considered the star formation history of the dwarf irregular galaxy UGCA 92, which has for a long time been considered to be the closest neighbour of the nearest starburst galaxy NGC 1569 (Karachentsev et al. 1994b;Makarova & Karachentsev 2003).
The small irregular galaxy UGCA 92 was discovered by Nilson (1974) and independently catalogued as a possible planetary nebula by Ellis et al. (1984). CCD observations of Hoessel et al. (1988) partially resolving it into individ- ual stars in the g and r passband images first showed that UGCA 92 (EGB 0427+63) is a dwarf irregular galaxy. Hodge & Miller (1995) detected 25 H ii regions within UGCA 92, concentrated in two well-separated regions of the galaxy. The reddening, measured from the emission-line spectra for bright H ii regions is E(B − V ) = 0.90 ± 0.08 and the mean oxygen abundance is about 13 per cent solar, with an uncertainty of 50 per cent. Karachentsev & Kaisin (2010) carried out Hα flux measurement for UGCA 92 and derived a current star formation rate of log(SFR) = −1.51 M⊙/yr.
The general parameters of UGCA 92 are presented in Table 1. The total magnitudes, colours and central surface brightnesses, µ(0), are not corrected for Galactic extinction.
THE DATASET
The dwarf irregular galaxy UGCA 92 was observed aboard HST using Advanced Camera for Surveys (ACS) at March 28, 2004 (SNAP 9771, PI I. Karachentsev). Two exposures were made with the filters F606W (1200 s) and F814W (900 s). An ACS image of the dwarf galaxy is shown in Fig. 2. The photometry of resolved stars in the galaxy was performed with the ACS module of the dolphot package 1 for crowded field photometry (Dolphin 2002) using the recommended recipe and parameters. Only stars with photometry of good quality were included in the final list, following recommendations given in the dolphot User's Guide. We have selected stars with signal-to-noise (S/N) 5 in both filters, χ 2 2.5 and |sharp| 0.3. The resulting colour-magnitude diagram contains 32699 stars (Fig. 3).
Artificial star tests were performed using the same reduction procedures to estimate photometric errors, crowdedness and blending effects in the most accurate way. A large library of artificial stars was generated spanning the necessary range of stellar magnitudes and colours so that the distribution of the recovered photometry is adequately sampled. The photometric errors and completeness are pre- sented in Fig. 4. The 1 σ photometric precision is 0.07 at F814W = 25 and 0.19 at F814W = 27 mag. Malmquist bias becomes notable for stars with F814W > 26.8 mag. In F606W the 1 σ photometric precision is 0.06 at 26 and 0.18 at 28 mag.
COLOUR-MAGNITUDE DIAGRAM
All resolved stars are significantly shifted to redder colours due to high extinction in the Zone of Avoidance of the Milky Way. The diagram is typical for dwarf irregular galaxies. A pronounced upper main sequence (MS) and probable helium-burning blue loop stars are found at F606W − In the dense parts of the diagram the colour represents the density in the Hess diagram, while individual stars are represented where they can be individually distinguished. The magnitudes are not corrected for Galactic extinction. Padova isochrones (Girardi et al. 2000) corresponding to the mean age and metallicity of detected star formation episodes are shown: '1' is Z=0.008,t=10 Myr; '2' -Z=0.001,t=50 Myr; '3' -Z=0.0004,t=150 Myr; '4' -Z=0.0004,t=13 Gyr.
F814W < 1.2 mag. The red supergiant branch (RSG) and the intermediate age asymptotic giant branch (AGB) are also well populated. The most abundant feature in the CMD is the red giant branch (RGB) (see Fig. 3).
Extinction
The dwarf galaxy UGCA 92 is situated at the Galactic latitude l = +10.5 • . Galactic extinction in the field of UGCA 92 needs to be addressed in more detail because reddening estimation at low Galactic latitude in the Zone of Avoidance is highly uncertain (see Schlegel et al. 1998, appendix C). Schlegel et al. (1998) give a colour excess of E(B − V ) = 0.79 ± 0.13 using IRAS/DIRBE maps of infrared dust emission. We apply this value to the colour-magnitude diagram and estimate a mean colour of the upper MS as a first test of the feasibility of the colour excess value. The unreddened V − I colour of upper MS stars is about zero (see Kenyon & Hartmann 1995, for example). We have selected main sequence stars in the appropriate magnitude and colour range, |(V −I)0| 0.5 and 19.5 I0 24.0. According to our measurements, the mean MS colour is (V − I)0 = 0.04 mag. The colour spread of the MS ∆(V − I)0 = 0.20 mag is rather large. This spread, as well as the slightly reddish mean MS colour, could indicate a presence of a blue loop population. Thereby, the extinction value given by Schlegel et al. (1998) seems reasonable, and we also do not expect a significant value of internal extinction in this faint dwarf galaxy. The reddening in UGCA 92 was also determined by Hodge & Miller (1995) from the spectroscopy of 25 bright H ii regions within the galaxy. Their reddening value E(B − V ) = 0.90 ± 0.08 is in agreement with the value of Schlegel et al. (1998) within uncertainties. The resolution in the IRAS/DIRBE maps is 6.1 arcmin (Schlegel et al. 1998). It is considerably larger than the ACS field of view. Therefore, we cannot get an information about possible variations in extinction across the field of view from these maps. However, a simple test was made using a colour-magnitude diagram of UGCA 92. A value of the tip of the red giant branch was measured in four different subframes along y axis of the whole ACS frame. We have found, that the TRGB value has no variations within 1.5 σ of the TRGB uncertainty. Consequently, we suggest no variations of external extinction within the ACS field of UGCA 92.
Therefore, we use the colour excess E(B − V ) = 0.79 ± 0.13 from Schlegel et al. (1998) in all our measurements in the present paper.
Foreground contamination
The colour-magnitude diagram is highly contaminated by Milky Way (MW) stars. To account for this contamination we need to construct colour-magnitude diagram of the MW in the direction of UGCA 92. This information can be obtained from images of neighbouring 'empty' fields. Three fields nearby UGCA 92 were found in the HST data archive exposed with WFPC2 in the parallel mode (coordinates of the centre are 4 h 29 m 53 s , +64 • 42 ′ 34 ′′ ). However, the photometric limit of these images is about 3 mag higher than photometric limit of the working UGCA 92 images. To account for the MW contamination in the region of faint stars, we have used the trilegal program (Girardi et al. 2005), which computes synthetic colour-magnitude diagrams for the specified coordinates in the sky and given parameters of the Milky Way models. Consequently, we use neighbouring field stars to account for the Milky Way contribution to the UGCA 92 CMD brighter than F814W = 24 mag and the trilegal synthetic data for the fainter part of the CMD. Thirty synthetic CMDs were constructed with trilegal and then averaged to avoid stochastic errors in synthetic CMDs. Random and systematic photometric uncertainties and completeness measured from artificial star experiments were applied to the synthetic CMDs. Resulting contamination by the MW stars was determined to be 202 stars over the CMD of UGCA 92.
DISTANCE DETERMINATION
The distance of UGCA 92 was first estimated by Karachentsev et al. (1994a) using the brightest blue and red supergiants as distance indicators. The estimated distance mod- ulus (m − M )0 = 26.72 mag suffered from large photometric uncertainties associated with crowded stellar fields and uncertain reddening. Later Karachentsev et al. (1997) estimated a distance to UGCA 92 using the same distance indicators and better quality data. The resulting distance modulus is (m−M )0 = 26.25 mag. Recently, more precise distance determination was made using HST /ACS data . A deep colour-magnitude diagram permitted the use of the TRGB distance indicator, resulting in a distance modulus of (m − M )0 = 27.39 mag.
However, the TRGB method has recently been considerably improved. We have determined the photometric TRGB distance with our trgbtool program, which uses a maximum-likelihood algorithm to determine the magnitude of the tip of the red giant branch from the stellar luminosity function . The estimated value of TRGB is equal to F814W = 24.84 ± 0.02 mag in the ACS instrumental system. The calibration of the TRGB distance indicator has also recently been improved (Rizzi et al. 2007), where the colour dependence of the absolute magnitude of the TRGB and zero-point issues in HST /ACS and WFPC2 have been addressed. Using this calibration, we have obtained the true distance modulus to UGCA 92 of (m − M )0 = 27.41 ± 0.25 mag and a distance of D = 3.03±0.35 Mpc. Bear in mind that the given small error of the TRGB measurement (0.02 mag) and the high precision of the calibration (0.02 mag), the resulting accuracy is entirely determined by uncertainty of foreground extinction of 0.25 mag in the direction of UGCA 92. The colourmagnitude diagram with the fitted RGB luminosity function and the resulting TRGB value is shown in Fig. 5.
STAR FORMATION HISTORY
The star-formation and metal-enrichment history of UGCA 92 has been determined from its CMD using our StarProbe package. This program adjusts the observed photometric distribution of stars in the colour-magnitude diagram against a positive linear combination of synthetic diagrams of single stellar populations (SSPs, single age and single metallicity). Our approach is described in more detail in Makarov & Makarova (2004) and Makarova et al. (2010).
The observed data were binned into Hess diagrams, giving the number of stars in cells of the CMD (two-dimensional histogram). The size of the cell is 0.05 mag in each passband. The synthetic Hess diagrams were constructed from theoretical stellar isochrones and initial mass function (IMF). Each artificial diagram is a map of probabilities to find a star in a cell for given age and metallicity. We used the Padova2000 set of theoretical isochrones (Girardi et al. 2000), and a Salpeter (1955) IMF. The distance is adopted from the present paper (see above) and the Galactic extinction from Schlegel et al. (1998). The synthetic diagrams were altered by the same incompleteness and crowding effects, and the photometric systematics, as those determined for the observations using artificial stars experiments. We also have taken into account the presence of unresolved binary stars (binary fraction). Following Barmina et al. (2002), the binary fraction was taken to be 30 per cent. The mass distribution for the secondary was taken to be flat in the range 0.7 to 1.0 of the primary mass. The best-fitting combination of synthetic CMDs is a maximum-likelihood solution taking into account the Poisson noise of star counts in the cells of the Hess diagram. The resulting star formation history (SFH) is shown in the Fig. 6. The 1 σ error of each SSP is derived from an analysis of likelihood function.
According to our measurements, the main star formation event occurred in the period 12 -14 Gyr ago with a rather high mean star formation rate (SFR) of 1.2 ± 0.1 × 10 −1 M⊙ yr −1 . This is the total SFR over the whole galaxy. The metallicity range for the most stars formed during this event is [Fe/H] = [−2.0 : −1.5] dex. This initial burst accounts for about 84±7 per cent of the total mass of formed stars.
The quiescence period has appearing from about 6 to 12 Gyr ago. However, the absence of star formation activity in this period could be due to tight packing of the different age stars in the upper part of the RGB. At the distance of 3 Mpc, fainter stellar populations, like a horizontal branch and a lower part of the main sequence, are hardly resolved at the HST. Without these details it is difficult to resolve an agemetallicity-SFR relation for the oldest ( > 6 ÷ 8 Gyr) star formation events, due to tight packing of the correspondent isochrones for the brightest part of the CMD.
There are signs of marginal (insignificant) star formation 4 -6 Gyr ago. A metallicity of these stars is similar to the metal abundance of the oldest stellar population.
There are also indications of recent star formation starting about 1.5 -2 Gyr ago and continuing to the present. We have measured the SFH in short age periods in the last 500 Myr to give more detail for the recent and ongoing star formation. A mean SFR of the stars formed in the last 50 Myr is 3.9 ± 0.6 × 10 −2 M⊙ yr −1 . This ongoing star formation rate is in good agreement with the independent estimation by Karachentsev & Kaisin (2010) from Hα flux measurements (3.2 ± 0.3 × 10 −2 M⊙ yr −1 ). The recent star formation is showing moderate enhancement from ∼ 200 Myr to 300 Myr. A mean star formation rate in the last 500 Myr is 4.3 ± 0.6 × 10 −2 M⊙ yr −1 . A mass portion of stars formed in the last 500 Myr is 7.6±0.7 per cent of the total stellar mass. A metallicity of the recent star formation event is determining with large uncertainty due to relatively poor statistic in comparison with sufficiently more numerous old stars. However, the measurements show that a significant part of the young stars is evidently metal enriched. It is very likely that the ongoing star formation has a metallicity of −0.6 -−0.3 dex.
Spatial distribution of stellar populations
We have selected few stellar populations depending on their position in the CM diagram. The result of this selection is presented in Fig. 7. The upper main sequence stars (upper left panel) has the unreddened colour (V − I)0 0, their age range is about 10-100 Myr. These youngest stars of the galaxy are mainly concentrated in few knots which highlight regions of ongoing star formation and form actual irregular structure of UGCA 92.
In the upper right panel of the figure we show supposed blue-loop (BL) stars selected by a colour 0.0 (V − I)0 0.5. Their age range is ∼10-200 Myr. Spatial distribution of these stars is considerably more smooth and regular in comparison with MS stars though they are well-concentrated to the central parts of the galaxy.
A selection criterion for red supergiant stars (RS), with ages in range ∼10 Myr -1 Gyr, (lower left panel) was quite conservative (1.0 (V − I)0 1.5 and I0 21.5) to avoid possible contamination of the spatial structure by young and intermediate age AGB stars (which are redder) and RGB (which are fainter). As a result, the selected population is not numerous and smoothly distributed in the central part of the galaxy nearly along the major axis. A most numerous population is the red giant branch (>1 Gyr) observed nearly (V − I)0 0.5 and I0 23.0. The stars are widely distributed in the image with apparent concentration to the centre of UGCA 92 and nearly the major axis of the object. The density of the stars is smoothly decayed to the edge of the galaxy, which probably extends beyond the image boundary.
STAR FORMATION AND ENVIRONMENT
Maffei -IC 342 is highly obscured nearby galaxy complex. These galaxies are gathered in two groups. One group is around the giant face-on galaxy IC 342 and another one is around the pair of E+S galaxies Maffei 1 and Maffei 2. According to Karachentsev (2005), the groups contain eight members each. One should keep in mind, that the distances to the group members could be highly uncertain due to the high uncertainty in the Galactic extinction in this direction. The IC 342 group structure is shown in the Fig. 8. There are 19 galaxies in the plot which situated at distances 1.8 Mpc from IC 342. The distances were taken from the Catalog of Neighboring Galaxies (Karachentsev et al. 2004). The data on the distances were updated recently (Karachentsev, private communications). The two galaxy groups, around IC 342 and around Maffei 1 and Maffei 2 could be seen in the figure. Most of the complex members are dwarf galaxies. The morphological types of the objects are coded by a colour according to de Vaucouleurs et al. (1991) and Karachentsev et al. (2004) from the early types (red) to the late types (dark blue). It is interesting to note that all the dwarf satellites of IC 342, including the galaxy under study UGCA 92 are irregulars. An absence of dwarf spheroidal satellites subsystem in the group can imply that dSphs still were not discovered, because their lower surface brightness and high Galactic extinction put serious observational constrains.
The dwarf irregular galaxy under study UGCA 92 is situated at a linear distance D = 440 kpc from the gravitational centre of the group IC 342. The three nearest to UGCA 92 galaxies are UGCA 86 (a linear distance D = 260 kpc), UGCA 105 (D = 300 kpc) and NGC 1569 (D = 360 kpc). UGCA 92 is often considered to be the companion to the starburst galaxy NGC 1569. They have been known to have close radial velocities, VLG = 93 km s −1 for UGCA 92 (Begum et al. 2008) and VLG = 106 km s −1 for NGC 1569 (Walter et al. 2008). Indeed, except for the giant spiral IC 342 (D = 320 kpc from NGC 1569) and a very small irregular galaxy Cam B (D = 190 kpc from NGC 1569), UGCA 92 is the spatially closest companion to NGC 1569.
A median radial velocity of galaxies within IC 342 group is VLG = 244 km s −1 with the velocity dispersion of 79 km s −1 . The pair of galaxies NGC 1569 -UGCA 92 has a maximal peculiar velocity within the group. The question arises whether these galaxies form a real subsystem within IC 342 group. Indeed, the uncertainty in UGCA 92 distance is 350 kpc (see Tab. 1), which is pretty close to derived linear separation between NGC 1569 and UGCA 92. The main source of this uncertainty is a huge Galactic absorption. Taking into account the close radial velocities and close positions on the sky, these galaxies could form a tight physical pair. In this case we could expect to find a correlation in their star formation. From the other hand, the estimation of the linear distance between the galaxies is reasonable. Therefore, the similar velocities and positions on the sky are just coincidence in a virialized system. NGC 1569 is the well-known nearest strong starburst galaxy. Vallenari & Bomans (1996) have found evidence of a recent burst of star formation from about 15 to 4 Myr ago. One more distinct burst was found not older than 150 Myr ago. Later Angeretti et al. (2005) have found three recent distinct starbursts using HST/NICMOS data: 13 -37 Myr, 40 -150 Myr and ∼ 1 Gyr ago (for 2.2 Mpc distance). A presence of bright H ii regions also indicate substantial ongoing star formation. The galaxy contains two extremely luminous super-star clusters (Hunter et al. 2000). In the last work 45 other young clusters also have been identified, the most of them have an age < 30 Myr. The distance to this galaxy was uncertain for the long time. The accurate TRGB distance (3.36±0.20 Mpc) was measured by Grocholski et al. (2008) using HST /ACS images. Stellar content of the NGC 1569 halo was studied in great detail recently by Ryś et al. (2011). Judging from these two papers, the old (about 10 Gyr) RGB stars, including the outer halo in this starburst galaxy should be more metal reach ([Fe/H] ≃ −1) than we measured for the old RGB stars of UGCA 92. One of the interesting conclusions of Ryś et al. (2011) is that the outer stellar halo of the starburst galaxy NGC 1569 is not tidally truncated and it is not outward extension of the inner disk, but instead it is the distinct stellar halo with no evident age/metallicity gradient, i.e. the starburst phenomenon is highly centrally concentrated. Probably, such a morphology could be the result of past interaction/merging. The question arises whether we can find any signs of past interactions for the dwarf galaxies using the information on star formation of these objects.
Besides UGCA 92, in the subgroup of the closest neighbours NGC 1569-UGCA 92-UGCA 86-UGCA 105 only UGCA 86 has observations deep enough (HST /ACS images within the project number 9771, PI I. Karachentsev) to judge about overall star formation from the photometry of resolved stars. The colour-magnitude diagram of this galaxy is presented in the Fig. 9. The both UGCA 92 (D = 3.03 Mpc) and UGCA 86 (D = 2.96 Mpc, Karachentsev et al. (2006)) are at nearly the same distance from us. However, UGCA 86 has the brighter absolute stellar magnitude (MB = −17.95) and larger angular size. We have measured more resolved stars in the ACS field. An upper main sequence and helium-burning blue loop stars are found at F606W − F814W < 1.6 mag. The well-populated RSG and the huge AGB are situated above the TRGB at F606W − F814W > 1.6 mag and F814W < 25.14 mag. Galactic extinction for UGCA 86 is even higher (E(B −V ) = 0.94 mag according to Schlegel et al. (1998)). A photometric limit and the large colour excess seriously affect the RGB zone in the CMD (see Fig. 9). This fact reduces seriously a reliability of a computational modelling of the star formation older than 1 Gyr. Therefore, we only fitted a number of theoretical isochrones to the young stellar population of UGCA 86. The age and metallicity of these populations are similar in UGCA 86 and UGCA 92.
H i and Hα observations of NGC 1569, UGCA 92 and UGCA 86 were performed by different authors. Numerous Hα knots were detected in all the three galaxies, tracking the ongoing star formation events in these objects (Hodge & Miller 1995;Kingsburgh & McCall 1998;Karachentsev & Kaisin 2010). High-sensitivity H i maps of NGC 1569 show an evidence for a companion H i cloud connected with the galaxy by a low surface brightness H i bridge. At the edge of NGC 1569 it coincides with H i arcs (Stil & Israel 1998). The hydrogen cloud is apparently not correlated with any optical satellites/counterparts. Stil & Israel (2002) argued, that about 10 per cent by mass of all H i in NGC 1569 have unusually high velocities. Some of this H i may be associated with the mass outflow evident from Hα measurements, but some may also be associated with the NGC 1569's H i companion and H i bridge, in which case, infall rather than outflow might be the cause of the discrepant velocities. H i observations by Stil et al. (2005) show a complex structure of UGCA 86, with two separate components: a rotating disk and a highly elongated spur that is kinematically disjunct from the disk.
Hereby, summarizing the cited results, we could conclude, that hydrogen at the outskirts of the starburst galaxy NGC 1569 is highly disturbed with the signs of infall, but no similar features were detected in the neighbouring dwarfs UGCA 92 and UGCA 86.
It should be note, that all the data known to date could not give us the particular age, when the series of the distinct intensive star bursts in NGC 1569 was started. Angeretti et al. (2005) mention that the last starburst should occur 8 -27 Myr ago if the distance to the galaxy is 2.9 Mpc instead 13 -37 Myr assuming the distance of 2.2 Mpc. The revised accurate distance to NGC 1569 is 3.36 Mpc according to Figure 8. The 3D structure of the IC 342 group. The size of the data cube is 1.8 × 1.8 × 1.8 Mpc. The giant spiral IC 342 is placed in the centre of the cube. Grocholski et al. (2008). It is possible that other two starburst periods occurred 40 -150 Myr ago and about 1 Gyr ago will be shifted to the younger ages assuming the new distance. Our data on recent star formation in the companion UGCA 92 galaxy show substantial and continuous star formation in the last 500 Myr with the enhancement at about 200 -300 Myr. Therefore, our results do not indicate a direct connection between the recent star formation in the two galaxies.
CONCLUSIONS
We have derived a quantitative star formation history of the dwarf irregular galaxy UGCA 92 situated in the highly obscured nearby galaxy group IC 342. Due to a low Galactic latitude (l = +10.5 • ) the extinction is very high (E(B − V ) = 0.79 ± 0.13 according to Schlegel et al. (1998)) in this direction. The star formation history was reconstructed using HST /ACS images of the galaxy and a resolved stellar population modelling. According to our measurements, 84 per cent of the total stellar mass were formed during the star formation occurred about 12 -14 Gyr ago. A metallicity range of these stars is [Fe/H] = [−2.0 : −1.5] dex. There are also signs of marginal star formation 4 -6 Gyr ago. A metallicity of these stars is similar to the metal abundance of the oldest stellar population.
UGCA 92 has a typical morphology of an irregular dwarf with apparent associations of bright blue stars in its body. Numerous H ii knots were also detected in the galaxy, tracking the ongoing star formation. According to our measurements, recent star formation was started about 1.5 -2 Gyr ago and continuing to the present. We modelled the star formation history with good time resolution for the recent star formation event. The continuous star formation in this period shows moderate enhancement from about 200 Myr to 300 Myr ago. A mean star formation rate in the last 500 Myr is 4.3 ± 0.6 × 10 −2 M⊙ yr −1 and 3.9 ± 0.6 × 10 −2 M⊙ yr −1 in the last 50 Myr. The mass portion of the stars formed in the last 500 Myr is 7.6 per cent of the total mass of formed stars. A metallicity of the recent star formation event is determining with a large uncertainty due to relatively poor statistic in comparison with sufficiently more numerous old stars. However, the measurements show that a significant part of the young stars is evidently metal enriched. It is very likely that the ongoing star formation has a metallicity of −0.6 -−0.3 dex. UGCA 92 is often considered to be the companion to the starburst galaxy NGC 1569. They have been known to have close radial velocities, VLG = 93 km s −1 for UGCA 92 (Begum et al. 2008) and VLG = 106 km s −1 for NGC 1569 (Walter et al. 2008). The linear distance between the galaxies is D = 360 kpc. These two objects are evidently could be considered as a close pair of galaxies.
Another dwarf galaxy close to UGCA 92 is UGCA 86 with the linear distance D = 260 kpc. Our HST/ACS data allow to judge about overall star formation from the photometry of resolved stars. Theoretical isochrone fitting shows an apparent similarity of the resolved stellar populations in the two galaxies.
It is worth to note, that the mean metallicity of the old RGB stars measured by us in UGCA 92 is lower, than the known metallicity of the halo RGB stars in NGC 1569.
Comparing our star formation history of UGCA 92 with that of NGC 1569 reveals no causal or temporal connection between recent star formation events in these two galaxies. We suggest, that the starburst phenomenon in NGC 1569 has not related to the closest dwarf neighbours and does not affect their star formation history.
Probably, detailed N-body modelling of the group within 300 kpc from IC 342 is necessary to clarify a reason of recent star bursts in NGC 1569.
Figure 1 .
1Location of the IC 342/Maffei complex of galaxies in Galactic coordinates. Colours denote galaxy type. The IRAS extinction map is shown in grey scale.
Figure 2 .
2HST /ACS image of UGCA 92 in F606W filter. The image size is 3.4 × 3.4 arcmin.
Figure 4 .
4Photometric errors and completeness for UGCA 92. The top panels show the completeness, i.e. the fraction of artificial stars recovered within the photometric reduction procedure, as a function of the F606W and F814W magnitudes. The bottom panels give the difference between the measured and the true input magnitude (∆mag = measure − input). The error bars are 1 σ residuals.
Figure 3 .
3The (F606W − F814W), F814W CMD of the dwarf galaxy UGCA 92.
Figure 5 .
5The colour-magnitude diagram (left panel) and the TRGB calculation results for UGCA 92. The upper right panel shows the completeness, photometric errors and dispersion in errors (vertical bars) versus the HST /ACS F814W filter magnitude. The lower right panel gives a histogram of the F814W luminosity function. The resulting model LF convolved with photometric errors and incompleteness is displayed as a bold solid line with a jump at the position of the TRGB.
Figure 6 .
6The star formation history of the dwarf irregular galaxy UGCA 92. Top panel shows the star formation rate (SFR) (M ⊙ /yr) against the age of the stellar populations. The bottom panel represents the metallicity of stellar content as function of age. The colour corresponds to the strength of SFR for given age and metallicity.
Figure 7 .
7Spatial distribution of the stellar populations in UGCA 92. The main sequence stars is bounded by (V − I) 0 0. The boundaries of the shown blue loop are 0.0 (V − I) 0 0.5. RS stars correspond to the range 1.0 (V − I) 0 1.5 and I 0 21.5. The red giant population bounded by (V − I) 0 0.5 and I 0 23.0. Stellar density contours of the populations are shown with the solid lines.
Figure 9 .
9The (F606W − F814W), F814W CMD of the dwarf galaxy UGCA 86. The magnitudes are not corrected for Galactic extinction. The same Padova theoretical isochrones(Girardi et al. 2000) as for UGCA 92 CMD were shown.
Table 1 .
1General parameters of UGCA 92.R.A.(J2000)
04 h 32 m 03.5 s
[2]
Dec (J2000)
+63 • 36 ′ 58 ′′
[2]
size, arcmin
2.0 × 1.0
[3]
Linear diameter, kpc
3.1
[1]
(m − M ) 0 , mag
27.41 ± 0.25
[1]
Distance, Mpc
3.03 ± 0.35
[1]
B T , mag
15.22
[4]
(B − V ) T , mag
1.34
[4]
V 3 ′ , mag
14.55
[5]
I 3 ′ , mag
12.87
[5]
J T , mag
12.38
[6]
Ks T , mag
11.12
[6]
µ(0) B , mag arcsec −2
25.1
[4]
µ(0) V , mag arcsec −2
24.18 ± 0.01
[5]
µ(0) I , mag arcsec −2
22.61 ± 0.01
[5]
E(B − V ), mag
0.79 ± 0.13
[7]
A I , mag
1.54 ± 0.25
[7]
V LG , km s −1
93
[8]
M B , mag
−15.61
[1]
M HI , M ⊙
1.56 × 10 8
[8]
M HI /L B
0.55
[8]
Fraction of old stars (12-14 Gyr), %
84 ± 7
[1]
Metallicity of old stars, [Fe/H], dex
−2.0 -−1.5
[1]
Fraction of young stars (500 Myr), %
7.6 ± 0.7
[1]
SFR , 12 -14 Gyr ago, M ⊙ yr −1
1.2 ± 0.1 × 10 −1
[1]
SFR , last 500 Myr, M ⊙ yr −1
4.3 ± 0.6 × 10 −2
[1]
[1] this work; [2] LEDA; [3] Karachentsev et al. (2004);
[4] Karachentseva et al. (1996); [5] Sharina et al. (2008);
[6] Vaduvescu et al. (2005); [7] Schlegel et al. (1998); [8] Be-
gum et al. (2008)
http://purcell.as.arizona.edu/dolphot/
ACKNOWLEDGEMENTSThe work was supported by the Russian Foundation for Basic Research (RFBR) grant 11-02-00639 and Russian-Ukrainian RFBR grant 11-02-90449. We acknowledge the support of the Ministry of Education and Science of the Russian Federation, the contract 14.740.11.0901. This work was partially supported by Physical Sciences Department program of Russian Academy of Sciences. We are thankful to the anonymous referee for the very useful comments to the paper and to Scott Trager for his kind help with the text preparation. We acknowledge the usage of the HyperLEDA database (http://leda.univ-lyon1.fr).
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| []
|
[
"arXiv:cs/0505035v1 [cs.CC] 12 May 2005 Beyond Hypertree Width: Decomposition Methods Without Decompositions",
"arXiv:cs/0505035v1 [cs.CC] 12 May 2005 Beyond Hypertree Width: Decomposition Methods Without Decompositions"
]
| [
"Hubie Chen [email protected] \nDepartament de Tecnologia\nUniversitat Pompeu Fabra Barcelona\nSpain\n",
"Víctor Dalmau [email protected] \nDepartament de Tecnologia\nUniversitat Pompeu Fabra Barcelona\nSpain\n"
]
| [
"Departament de Tecnologia\nUniversitat Pompeu Fabra Barcelona\nSpain",
"Departament de Tecnologia\nUniversitat Pompeu Fabra Barcelona\nSpain"
]
| []
| The general intractability of the constraint satisfaction problem has motivated the study of restrictions on this problem that permit polynomial-time solvability. One major line of work has focused on structural restrictions, which arise from restricting the interaction among constraint scopes. In this paper, we engage in a mathematical investigation of generalized hypertree width, a structural measure that has up to recently eluded study. We obtain a number of computational results, including a simple proof of the tractability of CSP instances having bounded generalized hypertree width. | 10.1007/11564751_15 | [
"https://arxiv.org/pdf/cs/0505035v1.pdf"
]
| 6,604,750 | cs/0505035 | b8b36539fe00b09812e15affe64889129b67dc72 |
arXiv:cs/0505035v1 [cs.CC] 12 May 2005 Beyond Hypertree Width: Decomposition Methods Without Decompositions
Hubie Chen [email protected]
Departament de Tecnologia
Universitat Pompeu Fabra Barcelona
Spain
Víctor Dalmau [email protected]
Departament de Tecnologia
Universitat Pompeu Fabra Barcelona
Spain
arXiv:cs/0505035v1 [cs.CC] 12 May 2005 Beyond Hypertree Width: Decomposition Methods Without Decompositions
The general intractability of the constraint satisfaction problem has motivated the study of restrictions on this problem that permit polynomial-time solvability. One major line of work has focused on structural restrictions, which arise from restricting the interaction among constraint scopes. In this paper, we engage in a mathematical investigation of generalized hypertree width, a structural measure that has up to recently eluded study. We obtain a number of computational results, including a simple proof of the tractability of CSP instances having bounded generalized hypertree width.
Introduction
The constraint satisfaction problem (CSP) is widely acknowledged as a convenient framework for modelling search problems. Instances of the CSP arise in a variety of domains, including artificial intelligence, database theory, algebra, propositional logic, and graph theory. An instance of the CSP consists of a set of constraints on a set of variables; the question is to determine if there is an assignment to the variables satisfying all of the constraints. Alternatively, the CSP can be cast as the fundamental algebraic problem of deciding, given two relational structures A and B, whether or not there is a homomorphism from A to B. In this formalization, each relation of A contains the tuples of variables that are constrained together, which are often called the constraint scopes, and the corresponding relation of B contains the allowable tuples of values that the variable tuples may take.
It is well-known that the CSP, in its general formulation, is NP-complete; this general intractability has motivated a large and rich body of research aimed at identifying and understanding restricted cases of the CSP that are polynomialtime tractable. The restrictions that have been studied can, by and large, be placed into one of two categories, which-due to the homomorphism formulation of the CSP-have become known as left-hand side restrictions and right-hand side restrictions. From a high level view, left-hand side restrictions, also known as structural restrictions, arise from prespecifying a class of relational structures A from which the left-hand side structure A must come, while right-hand side restrictions arise from prespecifying a class of relational structures B from which the right-hand side structure B must come. As this paper is concerned principally with structural restrictions, we will not say more about right-hand side restrictions than that their systematic study has origins in a classic theorem of Schaefer [21], and that recent years have seen some extremely exciting results on them (for instance [4,5]).
The structural restrictions studied in the literature can all be phrased as restrictions on the hypergraph H(A) naturally arising from the left-hand side relational structure A, namely, the hypergraph H(A) with an edge {a 1 , . . . , a k } for each tuple (a 1 , . . . , a k ) of A. Let us briefly review some of the relevant results that have been obtained on structural tractability. The tractability of left-hand side relational structures having bounded treewidth was shown in the constraint satisfaction literature by Dechter and Pearl [9] and Freuder [10]. 1 Later, Dalmau et al. [8] building on ideas of Kolaitis and Vardi [19,20] gave a consistency-style algorithm for deciding the bounded treewidth CSP. For our present purposes, it is worth highlighting that although the notion of bounded treewidth is defined in terms of tree decompositions, which can be computed efficiently (under bounded treewidth), the algorithm given by Dalmau et al. [8] does not compute any form of tree decomposition. Dalmau et al. also identified a natural expansion of structures having bounded treewidth that is tractable-namely, the structures homomorphically equivalent to those having bounded treewidth. The optimality of this latter result, in the case of bounded arity, was demonstrated by Grohe [15], who proved-roughly speaking-that if the tuples of A are of bounded arity and A gives rise to a tractable case of the CSP, then it must fall into the natural expansion identified by Dalmau et al. [8].
A number of papers, including [17,16,13,14,11,7], have studied restrictions that can be applied to relational structures of unbounded arity. (Note that any class of relational structures of unbounded arity cannot have bounded treewidth.) In a survey [13], Gottlob et al. show that the restriction of bounded hypertree width [11] is the most powerful structural restriction for the CSP in that every other structural restriction studied in the literature is subsumed by it. Since this work [11,13], whether or not there is a more general structural restriction than bounded hypertree width that ensures tractability, has been a tantalizing open question.
In this paper, we study generalized hypertree width, a structural measure for hypergraphs defined in [12] that is a natural variation of hypertree width; we call this measure coverwidth. Coverwidth is trivially upper-bounded by hypertree width, and so any class of hypergraphs having bounded hypertree width has bounded coverwidth. We define a combinatorial pebble game that can be played on any CSP instance, and demonstrate that this game is intimately linked to coverwidth (Theorem 13). Our study of coverwidth is conceptually simple, mathematically elegant, and relatively compact; we believe that this hints that coverwidth is in fact a natural and robust mathematical concept that may find further applications. Overall, the investigation we perform takes significant inspiration from methods, concepts, and ideas developed by Kolaitis, Vardi, and coauthors [19,20,8,2] that link together CSP consistency algorithms, the existential k-pebble games of Kolaitis and Vardi [18], and bounded treewidth.
Using the pebble game perspective, we are able to derive a number of computational results. One is that the structural restriction of bounded coverwidth implies polynomial-time tractability; this result generalizes the tractability of bounded hypertree width. It has been independently shown by Adler et al. that the hypertree width of a hypergraph is linearly related to the coverwidth [1]. This result can be used in conjunction with the tractability of bounded hypertree width to derive the tractability of bounded coverwidth. However, we believe our proof of bounded coverwidth tractability to be considerably simpler than the known proof of bounded hypertree width tractability [11], even though our proof is of a more general result.
To describe our results in greater detail, it will be useful to identify two computational problems that every form of structural restriction gives rise to: a promise problem, and a no-promise problem. In both problems, the goal is to identify all CSP instances obeying the structural restriction as either satisfiable or unsatisfiable. In the promise problem, the input is a CSP instance that is guaranteed to obey the structural restriction, whereas in the no-promise problem, the input is an arbitrary CSP instance, and an algorithm may, on an instance not obeying the structural restriction, decline to identify the instance as satisfiable or unsatisfiable. Of course, CSPs arising in practice do not come with guarantees that they obey structural restrictions, and hence an algorithm solving the no-promise problem is clearly the more desirable. Notice that, for any structural restriction having a polynomial-time solvable promise problem, if it is possible to solve the identification problem of deciding whether or not an instance obeys the restriction, in polynomial time, then the no-promise problem is also polynomial-time solvable. For bounded hypertree width, both the identification problem and the no-promise problem are polynomial-time solvable. In fact, the survey by Gottlob et al. [13] only considers structural restrictions for which the identification problem is polynomial-time solvable, and thus only considers structural restrictions for which the no-promise problem is polynomial-time solvable.
One of our main theorems (Theorem 20) is that the promise problem for bounded coverwidth is polynomial-time tractable, via a general consistency-like algorithm. In particular, we show that, on an instance having bounded coverwidth, our algorithm detects an inconsistency if and only if the instance is unsatisfiable. Our algorithm, like the consistency algorithm of Dalmau et al. [8] for bounded treewidth, can be applied to any CSP instance to obtain a more constrained instance; our algorithm does not need nor compute any form of decomposition, even though the notion of coverwidth is defined in terms of decompositions! Intriguingly, we are then able to give a simple algorithm for the no-promise problem for bounded coverwidth (Theorem 21) that employs the consistency-like algorithm for the promise problem. The behavior of this algorithm is reminiscent of self-reducibility arguments in computational complexity theory, and on an instance of bounded coverwidth, the algorithm is guaranteed to either report a satisfying assignment or that the instance is unsatisfiable. We believe that our results offer a direct challenge to the view of structural tractability advanced in the Gottlob et al. survey [13], since we are able to give a polynomial-time algorithm for the bounded coverwidth no-promise problem without explicitly showing that there is a polynomial-time algorithm for the bounded coverwidth identification problem.
Returning to the promise problem, we then show that the tractability of bounded coverwidth structures can be generalized to yield the tractability of structures homomorphically equivalent to those having bounded coverwidth (Theorem 22). This expansion of bounded coverwidth tractability is analogous to the expansion of bounded treewidth tractability carried out in [8].
In the last section of this paper, we use the developed theory as well as ideas in [6] to define a tractable class of quantified constraint satisfaction problems based on coverwidth.
We emphasize that none of the algorithms in this paper need or compute any type of decomposition, even though all of the structural restrictions that they address are defined in terms of decompositions.
Definitions. In this paper, we formalize the CSP as a relational homomorphism problem. We review the relevant definitions that will be used. A relational signature is a finite set of relation symbols, each of which has an associated arity. A relational structure A (over signature σ) consists of a universe A and a relation R A over A for each relation symbol R (of σ), such that the arity of R A matches the arity associated to R. We refer to the elements of the universe of a relational structure A as A-elements. When A is a relational structure over σ and R is any relation symbol of σ, the elements of R A are called A-tuples. Throughout this paper, we assume that all relational structures under discussion have a finite universe. We use boldface letters A, B, . . . to denote relational structures.
A homomorphism from a relational structure A to another relational structure B is a mapping h from the universe of A to the universe of B such that for every relation symbol R and every tuple (a 1 , . . . , a k ) ∈ R A , it holds that (h(a 1 ), . . . , h(a k )) ∈ R B . (Here, k denotes the arity of R.) The constraint satisfaction problem (CSP) is to decide, given an ordered pair A, B of relational structures, whether or not there is a homomorphism from the first structure, A, to the second, B. A homomorphism from A to B in an instance A, B of the CSP is also called a satisfying assignment, and when a satisfying assignment exists, we will say that the instance is satisfiable.
This section defines the structural measure of hypergraph complexity that we call coverwidth. As we have mentioned, coverwidth is equal to generalized hypertree width, which was defined in [12]. We begin by defining the notion of hypergraph.
Definition 1. A hypergraph is an ordered pair (V, E) consisting of a vertex set V and a hyperedge set E. The elements of E are called hyperedges; each hyperedge is a subset of V .
Basic to the measure of coverwidth is the notion of a tree decomposition.
Definition 2. A tree decomposition of a hypergraph (V, E) is a pair (T = (I, F ), {X i } i∈I ) where -T = (I, F ) is a tree, and -each X i (with i ∈ I) is called a bag and is a subset of V ,
such that the following conditions hold:
1. V = ∪ i∈I X i . 2. For all hyperedges e ∈ E, there exists i ∈ I with e ⊆ X i . 3. For all v ∈ V , the vertices T v = {i ∈ I : v ∈ X i } form a connected subtree of T .
Tree decompositions are generally applied to graphs, and in the context of graphs, the measure of treewidth has been heavily studied. The treewidth of a graph G is the minimum of the quantity max i∈I |X i | − 1 over all tree decompositions of G. In other words, a tree decomposition is measured based on its largest bag, and the treewidth is then defined based on the "lowest cost" tree decomposition.
The measure of coverwidth is also based on the notion of tree decomposition. In coverwidth, a tree decomposition is also measured based on its "largest" bag; however, the measure applied to a bag is the number of hyperedges needed to cover it, called here the weight.
Definition 3. A k-union over a hypergraph H (with k ≥ 0) is the union e 1 ∪ . . . ∪ e k of k edges e 1 , . . . , e k of H.
The empty set is considered to be the unique 0-union over a hypergraph.
Definition 4. Let H = (V, E) be a hypergraph. The weight of a subset X ⊆ V is the smallest integer k ≥ 0 such that X ∩ (∪ e∈E e) is contained in a k-union over H.
We measure a tree decomposition according to its heaviest bag, and define the coverwidth of a hypergraph according to the lightest-weight tree decomposition.
Definition 5. The weight of a tree decomposition of H is the maximum weight over all of its bags.
Definition 6. The coverwidth of a hypergraph H is the minimum weight over all tree decompositions of H.
It is straightforward to verify that the coverwidth of a hypergraph is equal to the generalized hypertree width of a hypergraph [12]. Since the generalized hypertree width of a hypergraph is always less than or equal to its hypertree width, coverwidth is at least as strong as hypertree width in that results on bounded coverwidth imply results on bounded hypertree width.
There is another formulation of tree decompositions that is often wieldy, see for instance [3].
Definition 7. A scheme of a hypergraph H = (V, E) is a graph (V, F ) such that -(V, F ) has a perfect elimination ordering, that is, an ordering v 1 , . . . , v n of its vertices such that for all i < j < k, if (v i , v k ) and (v j , v k ) are edges in F , then (v i , v j ) is also an edge in F , and -the vertices of every hyperedge of E induce a clique in (V, F ).
It is well known that the property of having a perfect elimination ordering is equivalent to being chordal. The following proposition is also well-known. Let us define the weight of a scheme (of a hypergraph H) to be the maximum weight (with respect to H) over all of its cliques. The following proposition is immediate from Proposition 8 and the definition of coverwidth, and can be taken as an alternative definition of coverwidth.
Proposition 9. The coverwidth of a hypergraph H is equal to the minimum weight over all schemes of H.
We now define the hypergraph associated to a relational structure. Roughly speaking, this hypergraph is obtained by "forgetting" the ordering of the Atuples.
Definition 10. Let A be a relational structure. The hypergraph associated to A is denoted by H(A); the vertex set of H(A) is the universe of A, and for each A-tuple (a 1 , . . . , a k ), there is an edge {a 1 , . . . , a k } in H(A).
We will often implicitly pass from a relational structure to its associated hypergraph, that is, we simply write A in place of H(A). In particular, we will speak of k-unions over a relational structure A.
We now define a class of pebble games for studying the measure of coverwidth. These games are essentially equivalent to the existential k-pebble games defined by Kolaitis and Vardi and used to study constraint satisfaction [18,20]. The pebble game that we use is defined as follows. The game is played between two players, the Spoiler and the Duplicator, on a pair of relational structures A, B that are defined over the same signature. Game play proceeds in rounds, and in each round one of the following occurs:
1. The Spoiler places a pebble on an A-element a. In this case, the Duplicator must respond by placing a corresponding pebble, denoted by h(a), on a B-element. 2. The Spoiler removes a pebble from an A-element a. In this case, the corresponding pebble h(a) on B is removed.
When game play begins, there are no pebbles on any A-elements, nor on any B-elements, and so the first round is of the first type. We assume that the Spoiler never places two pebbles on the same A-element, so that h is a partial function (as opposed to a relation). The Duplicator wins the game if he can always ensure that h is a projective homomorphism from A to B; otherwise, the Spoiler wins. A projective homomorphism (from A to B) is a partial function h from the universe of A to the universe of B such that for any relation symbol R and any tuple (
a 1 , . . . , a k ) ∈ R A of A, there exists a tuple (b 1 , . . . , b k ) ∈ R B where h(a i ) = b i for all a i on which h is defined.
As we mentioned, the definition of this game is based on the existential kpebble game introduced by Kolaitis and Vardi [18,20]. In the existential k-pebble game, the number of pebbles that the Spoiler may use is bounded by k, and the Duplicator need only must ensure that h is a partial homomorphism. A close relationship between this game and bounded treewidth has been identified [2]. Theorem 11. [2] Let A and B be relational structures. For all k ≥ 2, the following are equivalent.
-There is a winning strategy for the Duplicator in the existential k-pebble
game on A, B. -For all relational structures T of treewidth < k, if there is a homomorphism from T to A, then there is a homomorphism from T to B.
To relate the game that we have defined with coverwidth, we are interested in parameterized versions of the game where the weight of the pebbles that the Spoiler has in play, is bounded by a constant k. (Here, by "weight" we are using Definition 4.) That is, the weight of the A-elements that have pebbles, is bounded by the constant k. We call this the existential k-cover game. We now formalize the notion of a winning strategy for the Duplicator in the existential k-cover game. Note that when h is a partial function, we use dom(h) to denote the domain of h. Theorem 13 can be easily applied to show that in an instance A, B of the CSP, if the left-hand side structure has coverwidth bounded by k, then deciding if there is a homomorphism from A to B is equivalent to deciding the existence of a Duplicator winning strategy in the existential k-cover game.
Theorem 14. Let A be a relational structure having coverwidth ≤ k, and let B be an arbitrary relational structure. There is a winning strategy for the Duplicator in the k-cover game on A, B if and only if there is a homomorphism from A to B.
We will use this theorem in the next section to develop tractability results. Although we use Theorem 13 to derive this theorem, we would like to emphasize that the full power of Theorem 13 is not needed to derive it, as pointed out in the proof.
Proof. If there is a homomorphism from A to B, the Duplicator can win by always setting pebbles according the homomorphism. The other direction is immediate from Theorem 13 (note that we only need the forward implication and T = A).
⊓ ⊔
The Algorithmic Viewpoint
The previous section introduced the existential k-cover game. We showed that deciding a CSP instance of bounded coverwidth is equivalent to deciding if the Duplicator has a winning strategy in the existential k-cover game. In this section, we show that the latter property-the existence of a Duplicator winning strategycan be decided algorithmically in polynomial time. To this end, it will be helpful to introduce the notion of a compact winning strategy.
∈ H with dom(h ′ ) = U such that for every v ∈ dom(h) ∩ dom(h ′ ), h(v) = h ′ (v).
Proposition 16. In the existential k-cover game on a pair of relational structures A, B, the Duplicator has a winning strategy if and only if the Duplicator has a compact winning strategy.
Proof. Suppose that the Duplicator has a winning strategy H. Let C be the set containing all functions h ∈ H such that dom(h) is a k-union. We claim that C is a compact winning strategy. Clearly C satisfies the first property of a compact winning strategy, so we show that it satisfies the second property. Suppose h ∈ C and let U be a k-union. By the subfunction property of a winning strategy, the restriction r of h to dom(h) ∩ U is in H. By repeated application of the forth property, there is an extension e of r that is in H and has domain U , which serves as the desired h ′ . Now suppose that the Duplicator has a compact winning strategy C. Let H be the closure of C under subfunctions. We claim that H is a winning strategy. It suffices to show that H has the forth property. Let h ∈ H and suppose that a is an A-element where dom(h) ∪ {a} has weight ≤ k. Let U be a k-union such that dom(h) ∪ {a} ⊆ U . By definition of H, there is a function e ∈ C extending h. Apply the second property of a compact winning strategy to e and U to obtain an e ′ ∈ C with domain U such that for every v ∈ dom(e) ∩ dom(e ′ ), e(v) = e ′ (v). Notice that dom(h) ⊆ dom(e) ∩ dom(e ′ ). Thus, the restriction of e ′ to dom(h) ∪ {a} is in H and extends h.
⊓ ⊔
We have just shown that deciding if there is a winning strategy, in an instance of the existential k-cover game, is equivalent to deciding if there is a compact winning strategy. We now use this equivalence to give a polynomial-time algorithm for deciding if there is a winning strategy.
Theorem 17. For all k ≥ 1, there exists a polynomial-time algorithm that, given a pair of relational structures A, B, decides whether or not there is a winning strategy for the Duplicator in the existential k-cover game on A, B.
Proof. By Proposition 16, it suffices to give a polynomial-time algorithm that decides if there is a compact winning strategy. It is straightforward to develop such an algorithm based on the definition of compact winning strategy. Let H be the set of all functions h such that dom(h) is a k-union (over A) and such that h is a projective homomorphism from A to B. Iteratively perform the following until no changes can be made to H: for every function h ∈ H and every kunion U , check to see if there is h ′ ∈ H such that the second property (of compact winning strategy) is satisfied; if not, remove h from H. Throughout the algorithm, we have maintained the invariant that any compact winning strategy must be a subset of H. Hence, if when the algorithm terminates H is empty, then there is no compact winning strategy. And if H is non-empty when the algorithm terminates, H is clearly a compact winning strategy.
The number of k-unions (over A) is polynomial in the number of tuples in A. Also, for each k-union U , the number of projective homomorphisms h with dom(h) = U from A to B is polynomial in the number of tuples in B. Hence, the size of the original set H is polynomial in the original instance. Since in each iteration an element is removed from H, the algorithm terminates in polynomial time.
⊓ ⊔
The algorithm we have just described in the proof of Theorem 17 may appear to be quite specialized. However, we now show that essentially that algorithm can be viewed as a general inference procedure for CSP instances in the vein of existing consistency algorithms. In particular, we give a general algorithm called projective k-consistency for CSP instances that, given a CSP instance, performs inference and outputs a more constrained CSP instance having exactly the same satisfying assignments as the original. On a CSP instance A, B, the algorithm might detect an inconsistency, by which we mean that it detects that there is no homomorphism from A to B. If it does not, then it is guaranteed that there is a winning strategy for the Duplicator. Proof. The first property is straightforward to verify. For the second property, observe that, each time a tuple is removed from B ′ , the set of satisfying assignments is preserved. For the third property, observe that, associating B ′ -tuples to functions as in Definition 18, the behavior of the projective k-consistency algorithm is identical to the behavior of the algorithm in the proof of Proposition 16.
⊓ ⊔
By using the results presented in this section thus far, it is easy to show that CSP instances of bounded coverwidth are tractable. Define the coverwidth of a CSP instance A, B to be the coverwidth of A. Let CSP[coverwidth ≤ k] be the restriction of the CSP to all instances of coverwidth less than or equal to k. Proof. Immediate from Theorem 14 and the third property of Theorem 19. ⊓ ⊔ Note that we can derive the tractability of CSP instances having bounded hypertree width immediately from Theorem 20. Now, given a CSP instance that is promised to have bounded coverwidth, we can use projective k-consistency to decide the instance (Theorem 20). This tractability result can in fact be pushed further: we can show that there is a generic polynomial-time that, given an arbitrary CSP instance, is guaranteed to decide instances of bounded coverwidth. Moreover, whenever an instance is decided to be a "yes" instance by the algorithm, a satisfying assignment is constructed.
Theorem 21. For all k ≥ 1, there exists a polynomial-time algorithm that, given any CSP instance A, B,
outputs a satisfying assignment for A, B, 2. correctly reports that A, B is unsatisfiable, or 3. reports "I don't know".
The algorithm always performs (1)
or (2) on an instance of CSP[coverwidth ≤ k].
Proof. The algorithm is a simple extension of the projective k-consistency algorithm. First, the algorithm applies the projective k-consistency algorithm; if an inconsistency is detected, then the algorithm terminates and reports that A, B is unsatisfiable. Otherwise, it initializes V to be the universe A of A, and does the following:
-If V is empty, terminate and identify the mapping taking each a ∈ A to the B-element in R B a , as a satisfying assignment.
-Pick any variable v ∈ V . -Expand the signature of A, B to include another symbol R v with R A v = {(v)}.
-Try to find a B-element b such that when R B v is set to {(b)}, no inconsistency is detected by the projective k-consistency algorithm on the expanded instance.
• If there is no such B-element, terminate and report "I don't know".
• Otherwise, set R B v to such a B-element, remove v from V , and repeat from the first step using the expanded instance.
If the procedure terminates from V being empty in the first step, the mapping that is output is straightforwardly verified to be a satisfying assignment.
Suppose that the algorithm is given an instance of CSP[coverwidth ≤ k]. If it is unsatisfiable, then the algorithm reports that the instance is unsatisfiable by Theorem 20. So suppose that the instance is satisfiable. We claim that each iteration preserves the satisfiability of the instance. Let A, B denote the CSP instance at the beginning of an arbitrary iteration of the algorithm. If no inconsistency is detected after adding a new relation symbol
R v with R A v = {(v)} and R B v = {(b)},
there must be a satisfying assignment mapping v to b by Theorem 20. Note that adding unary relation symbols to a CSP instance does not change the coverwidth of the instance.
⊓ ⊔
We now expand the tractability result of Theorem 20, and show the tractability of CSP instances that are homomorphically equivalent to instances of bounded coverwidth. Formally, let us say that A and A ′ are homomorphically equivalent if there is a homomorphism from A to A ′ as well as a homomorphism from A ′ to A. Let CSP[H(coverwidth ≤ k)] denote the restriction of the CSP to instances A, B where A is homomorphically equivalent to a relational structure of coverwidth less than or equal to k. Proof. Let A, B be a CSP instance where A is homomorphically equivalent to a relational structure A ′ of coverwidth ≤ k. The following conditions are equivalent; after stating each condition, we indicate how to show equivalence with the previous condition.
-There is a homomorphism from A to B. -There is a homomorphism from A ′ to B (straightforward).
-The Duplicator has a winning strategy in the existential k-cover game on A ′ , B (Theorem 14). -The Duplicator has a winning strategy in the existential k-cover game on A, B (Theorem 13). -The projective k-consistency algorithm does not report an inconsistency on A, B (Theorem 19). ⊓ ⊔
Quantified Constraint Satisfaction
We now sketch how the ideas given in this paper on constraint satisfaction can be combined with the ideas in [6] to yield results on quantified constraint satisfaction. Specifically, we define a notion of coverwidth for QCSPs, and show that bounded coverwidth QCSPs are tractable.
Definitions. We first briefly define the QCSP and relevant associated notions. A quantified relational structure is a pair (p, A) where A is a relational structure and p is a quantifier prefix, an expression of the form Q 1 v 1 . . . Q n v n where each Q i is a quantifier (either ∃ or ∀) and v 1 , . . . , v n are exactly the elements of the universe of A. The quantified constraint formula φ (p,A) associated to a quantified relational structure (p, A) (where A is over signature σ) is defined to be the formula pC A , where C A is the conjunction of all atomic formulas in the set {R(a 1 , . . . , a k ) : R ∈ σ, (a 1 , . . . , a k ) ∈ R A }. We say that there is a homomorphism from (p, A) to B if B |= φ (p,A) . We define the QCSP as the problem of deciding, given a quantified relational structure (p, A) and a relational structure B, if there is a homomorphism from (p, A) to B. A quantifier prefix p = Q 1 v 1 . . . Q n v n can be viewed as the concatenation of quantifier blocks where quantifiers in each block are the same, and consecutive quantifier blocks have different quantifiers. For example, the quantifier prefix ∀v 1 ∀v 2 ∃v 3 ∀v 4 ∀v 5 ∃v 6 ∃v 7 ∃v 8 , consists of four quantifier blocks: ∀v 1 ∀v 2 , ∃v 3 , ∀v 4 ∀v 5 , and ∃v 6 ∃v 7 ∃v 8 . We say that a variable v j comes after a variable v i in p if they are in the same quantifier block, or v j is in a quantifier block following the quantifier block of v i . Equivalently, the variable v j comes after the variable v i in p if one of the following conditions holds: (1) j ≥ i, or (2) j < i and all of the quantifiers Q j , . . . , Q i are of the same type.
Coverwidth. We now define a notion of coverwidth for quantified relational structures. This can be viewed as a generalization of the definition of coverwidth in terms of schemes, given in Proposition 9.
Definition 23. A scheme of a quantified relational structure (p, A) is a scheme (V, F ) of the hypergraph H(A) (in the sense of Definition 9) such that (V, F ) has a perfect elimination ordering v 1 , . . . , v n respecting the quantifier prefix p in that if i < j, then v j comes after v i in p.
Definition 24. The coverwidth of a quantified relational structure (p, A) is equal to the minimum weight (with respect to A) over all schemes of (p, A).
The quantified k-cover game. We can naturally extend the k-cover game, making use of ideas from [6], to define the quantified k-cover game. We describe the quantified k-cover game by defining the notion of a winning strategy for the Duplicator.
Definition 25. A winning strategy for the Duplicator in the quantified k-cover game on (p, A) and B is a non-empty set H of projective homomorphisms (from A to B) having the following properties. The algorithm for Theorem 27 is similar to projective k-consistency, but in addition to removing tuples that are not projective homomorphisms, it removes further tuples, as follows. Let A 1 . . . A m denote the quantifier blocks of the prefix p, and let A i be an existential quantifier block. Take a tuple from the right-hand side structure, view it as a mapping h : U → B, and consider its restriction h ′ to A 1 ∪ . . . ∪ A i . Let Y be the set of all universally quantified variables in U ∩ (A i+1 ∪ . . . ∪ A m ). If there exists any extension of h ′ to dom(h ′ ) ∪ Y that is not a projective homomorphism, then the tuple is removed. After the procedure terminates, if no inconsistency is detected, then the projective homomorphisms h of the new instance where the weight of dom(h) is ≤ k, is a winning strategy for the Duplicator in the quantified k-cover game on the original instance.
We can expand the result of Theorem 27 by Q-homomorphic equivalence, defined in [6]. Let QCSP[H(coverwidth ≤ k)] denote the restriction of the QCSP to all instances (p, A), B where (p, A) is Q-homomorphically equivalent to a quantified relational structure that has coverwidth less than or equal to k.
Theorem 28. For all k ≥ 1, the problem QCSP[H(coverwidth ≤ k)] is decidable in polynomial time.
Proof. (⇒) Let H be a winning strategy for the Duplicator in the k-cover game on A and B, let T be any structure of coverwidth ≤ k, let f be any homomorphism from T to A, let G = (T, F ) be a scheme for T of weight ≤ k, and let v 1 , . . . , v n be a perfect elimination ordering of G.
We shall construct a sequence of partial mappings g 0 , . . . , g n from T to B such that for each i: We define g 0 to be the partial function with empty domain. For every i ≥ 0, the partial mapping g i+1 is obtained by extending g i in the following way. As v 1 , . . . , v n is a perfect elimination ordering, the set
L = {v i+1 } ∪ {v j : j < i + 1, (v j , v i+1 ) ∈ F } is a clique of G. Define L ′ as L \ {v i+1 }. By the induction hypothesis, there exists h ∈ H such that for every v ∈ L ′ , h(f (v)) = g i (v). Let us consider two cases. If f (v i+1 ) = f (v j ) for some v j ∈ L ′ then we set g i+1 (v i+1 ) to be g i (v j )
. Note that in this case property (2) is satisfied, as every clique in G containing v i+1 is contained in L and h serves as a certificate. (For any clique not containing v i+1 , we use the induction hypothesis.)
Otherwise, that is, if f (v i+1 ) = f (v j ) for all v j ∈ L ′ , we do the following. First, since the weight of L is bounded above by k and f defines an homomorphism from T to A then the weight of f (L) is also bounded by k. Observe that f (L) = dom(h) ∪ {f (v i+1 )}. By the forth property of winning strategy there exists an extension h ′ ∈ H of h that is defined over v i+1 . We set g i+1 (v i+1 ) to be h ′ (f (v i+1 )). Note that h ′ certifies that property (2) is satisfied for very clique containing v i+1 ; again, any clique not containing v i+1 is covered by the induction hypothesis.
Finally, let us prove that g n indeed defines an homomorphism from T to B. Let R be any relation symbol and let (t 1 , . . . , t l ) be any relation in R T . We want to show that (g n (t 1 ), . . . , g n (t l )) belongs to R B . Since G is an scheme for T, {t 1 , . . . , t l } constitutes a clique of G. By property (2) there exists h ∈ H such that h(f (t i )) = g(t i ) for all i. Observing that as f is an homomorphism from T to A, we can have that (f (t 1 ), . . . , f (t l )) belongs to R A . Finally, as h is a projective homomorphism from A to B, the tuple (h(f (t 1 )), . . . , h(f (t l ))) must be in B.
(⇐) We shall construct a winning strategy H for the Duplicator. We need a few definitons. Fix a sequence a 1 , . . . , a m of elements of A. A valid tuple for a 1 , . . . , a m is any tuple (T, G, v 1 , . . . , v m , f ) where T is a relational structure, G is an scheme of weight k for T, {v 1 , . . . , v m } is a clique of G, and f is an homomorphism from T, v 1 , . . . , v m to A, a 1 , . . . , a m . (By a homomorphism from T, v 1 , . . . , v m to A, a 1 , . . . , a m , we mean a homomorphism from T to A that maps v i to a i for all i.) By S(T, G, v 1 , . . . , v m , f ) we denote the set of all mappings h with domain {a 1 , . . . , a m } such that there is an homomorphism from T, v 1 , . . . , v m to B, h(a 1 ), . . . , h(a m ). We are now in a situation to define H. H contains for every subset a 1 , . . . , a m of weight at most k, every partial mapping h that is contained in all S(T, G, v 1 , . . . , v m , f ) where (T, G, v 1 , . . . , v m , f ) is a valid tuple for a 1 , . . . , a m .
Let us show that H is indeed a winning strategy. First, observe that H is nonempty, as it contains the partial function with empty domain. Second, let us show that H contains only projective homomorphisms. Indeed, let h be any mapping in H with domain a 1 , . . . , a m , let R be any relation symbol and let (c 1 , . . . , c l ) be any tuple in R A . Let us define T to be the substructure (not necessarily induced) of A with universe {a 1 , . . . , a k , c 1 , . . . , c l } containing only the tuple (c 1 , . . . , c l ) in R T . It is easy to verify that the graph G = ({a 1 , . . . , a k , c 1 , . . . , c l }, F ) where F = {(a i , a j ) : i = j} ∪ {(c i , c j ) : i = j} is an scheme of T of weight ≤ k. Consequently, (T, G, a 1 , . . . , a m , id) is a valid tuple for a 1 , . . . , a m and therefore there exists an homomorphism g from T to B, and hence satisfying (g(c 1 ), . . . , g(c l )) ∈ R B , such that g(a i ) = h(a i ) for all i = 1, . . . k.
To show that H is closed under subfunctions is rather easy. Indeed, let h ′ be any mapping in H with domain a 1 . . . , a m . We shall see that the restriction h of h ′ to {a 1 , . . . , a m−1 } is also in H. Let (T, G, v 1 , . . . , v m−1 , f ) be any valid tuple for a 1 , . . . , a k−1 . We construct a valid tuple (T ′ , G ′ , v 1 , . . . , v m , f ′ ) for a 1 , . . . , a m in the following way: v m is a new (not in the universe of T) element, T ′ is the structure obtained from T by adding v m to the universe of T and keeping the same relations, f ′ is the extension of f in which v m is map to a m , and G ′ is the scheme of T obtained by adding to G an edge (v j , v m ) for every j = 1, . . . , m− 1. Since (T ′ , G ′ , v 1 , . . . , v m , f ′ ) is a valid tuple for a 1 , . . . , a m and h ′ ∈ H, there exists an homomorphism g ′ from T ′ , v 1 , . . . , v m to B, h ′ (a 1 ), . . . , h ′ (a m ). Observe then that the restriction g of g ′ to {a 1 , . . . , a m−1 } defines then an homomorphism from T, v 1 , . . . , v m−1 to B, h(a 1 ), . . . , h(m 1 ).
Finally, we shall show that H has the forth property. The proof relies in the following easy properties of the valid tuples. Let a 1 , . . . , a m be elements of A and let (T 1 , G 1 , v 1 , . . . , v m , f 1 ) and let (T 2 , G 2 , v 1 , . . . , v m , f 2 ) be valid tuples for a 1 , . . . , a m such that T 1 ∩ T 2 = {v 1 , . . . , v m }, let T be T 1 ∪ T 2 (that is, the structure T whose universe is the union of the universes of T 1 and T 2 , and in which R T = R T1 ∪ R T2 for all relation symbols R), G = G 1 ∪ G 2 and let f be the mapping from the universe T of T to B that sets a to f 1 (a) if a ∈ T 1 and to f 2 (a) if a ∈ T 2 (observe that f 1 and f 2 coincide over {v 1 , . . . , v m }). Then (T, G, v 1 , . . . , v m , f ) is a valid tuple for a 1 , . . . , a m . We call (T, G, v 1 , . . . , v m , f ), the union of (T 1 , G 1 , v 1 , . . . , v m , f 1 ) and (T 2 , G 2 , v 1 , . . . , v m , f 2 ). Furthermore, S(T, G, v 1 , . . . , v m , f ) ⊆ S(T 1 , G 1 , v 1 , . . . , v m , f 1 ) ∩ S(T 2 , G 2 , v 1 , . . . , v m , f 2 ) (in fact, S(T, G, v 1 , . . . , v m , f ) = S(T 1 , G 1 , v 1 , . . . , v m , f 1 )∩S(T 2 , G 2 , v 1 , . . . , v m , f 2 ), although we do not need the equality in our proof).
Proposition 8 .
8Let H be a hypergraph. For every tree decomposition of H, there exists a scheme such that each clique of the scheme is contained in a bag of the tree decomposition. Likewise, for every scheme of H, there exists a tree decomposition such that each bag of the tree decomposition is contained in a clique of the scheme.
Definition 18 .
18The projective k-consistency algorithm takes as input a CSP instance A, B, and consists of the following steps.-Create a new CSP instance A ′ , B ′ as follows. Let the universe of A ′ be the universe of A, and the universe of B ′ be the universe of B. Let the signature of A ′ and B ′ contain a relation symbol R U for each k-union U over A. For each k-union U , the relation R A ′ U is defined as (u 1 , . . . , u k ), where u 1 , . . . , u k are exactly the elements of U in some order; and R B ′ U is defined as the set of all tuples (b 1 , . . . , b k ) such that the mapping taking u i → b i is a projective homomorphism from A to B.-Iteratively perform the following until no changes can be made: remove any B ′ -tuple (b 1 , . . . , b k ) that is not a projective homomorphism. We say that aB ′ -tuple (b 1 , . . . , b k ) ∈ R B ′ U is a projective homomorphism if, letting (u 1 , . . . , u k ) denote the unique element of R A ′ U , the function taking u i → b i is a projective homomorphism from A ′ to B ′ .-Report an inconsistency if there are no B ′ -tuples remaining.Theorem 19. For each k ≥ 1, the projective k-consistency algorithm, given as input a CSP instance A, B:runs in polynomial time, -outputs a CSP instance A ′ , B ′ that has the same satisfying assignments as A, B, and reports an inconsistency if and only if the Duplicator does not have a winning strategy in the existential k-cover game on A, B.
Theorem 20 .
20For all k ≥ 1, the problem CSP[coverwidth ≤ k] is decidable in polynomial time by the projective k-consistency algorithm. In particular, on an instance of CSP[coverwidth ≤ k], the projective k-consistency algorithm reports an inconsistency if and only if the instance is not satisfiable.
Theorem 22 .
22For all k ≥ 1, the problem CSP[H(coverwidth ≤ k)] is decidable in polynomial time by the projective k-consistency algorithm. In particular, on an instance of CSP[H(coverwidth ≤ k)], the projective k-consistency algorithm reports an inconsistency if and only if the instance is not satisfiable.
1. dom(g i ) = {v 1 , . . . , v i }, and 2. for every clique L ⊆ {v 1 , . . . , v i } in G, there exists a projective homomorphism h ∈ H with domain f (L) in the winning strategy of the Duplicator, such that for every v ∈ L, h(f (v)) = g i (v).
Definition 12. A winning strategy for the Duplicator in the existential k-cover game on relational structures A, B is a non-empty set H of projective homomorphisms (from A to B) having the following two properties. 1. (the "forth" property) For every h ∈ H and A-element a / ∈ dom(h), if dom(h) ∪ {a} has weight ≤ k, then there exists a projective homomorphism h ′ ∈ H extending h with dom(h ′ ) = dom(h) ∪ {a}. 2. The set H is closed under subfunctions, that is, if h ∈ H and h extends h ′ , then h ′ ∈ H. For all relational structures T of coverwidth ≤ k, if there is a homomorphism from T to A, then there is a homomorphism from T to B.We have the following analog of Theorem 11.
Theorem 13. Let A and B be relational structures. For all k ≥ 1, the following
are equivalent.
-There is a winning strategy for the Duplicator in the k-cover game on A, B.
-
Definition 15. A compact winning strategy for the Duplicator in the existential k-cover game on relational structures A, B is a non-empty set H of projective homomorphisms (from A to B) having the following properties.1. For all h ∈ H, dom(h) is a k-union (over A).
2. For every h ∈ H and for every k-union U (over A), there exists h ′
1 .
1For every h ∈ H and every existentially quantified A-element a / ∈ dom(h) coming after all elements of dom(h), if dom(h) ∪ {a} has weight ≤ k, then there exists a projective homomorphism h ′ ∈ H extending h with dom(h ′ ) = dom(h) ∪ {a}. 2. For every h ∈ H, every B-element b, and every universally quantified Athen there exists a projective homomorphism h ′ ∈ H extending h with dom(h ′ ) = dom(h) ∪ {a} and h ′ (a) = b. 3. The set H is closed under subfunctions, that is, if h ∈ H and h extends h ′ , then h ′ ∈ H. Theorem 27. For all k ≥ 1, the problem QCSP[coverwidth ≤ k] is decidable in polynomial time.element a /
∈ dom(h) coming after all elements of dom(h), if dom(h)∪{a} has
weight ≤ k, We have the following analog of Theorem 14.
Theorem 26. Let (p, A) be a quantified relational structure having coverwidth
≤ k, and let B be an arbitrary relational structure. There is a winning strategy
for the Duplicator in the quantified k-cover game on (p, A), B if and only if there
is a homomorphism from (p, A) to B.
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Let us assume, towards a contradiction, that there is not extension h ′ of h in H. Then there exists a finite collection {(T i , G i , v 1 , . . . , v m , f i ) : i ∈ I} of valid tuples for a 1 , . . . , a m such that the intersection i∈I S(T i , G i , v 1 , . . . , v m , f i ) does not contain any extension of h. We can rename the elements of the universes so that for every different i, j ∈ I we have that T i ∩ T j = {v 1. Thomas J Schaefer, Let h be any mapping in H, let {a 1 , . . . , a m−1 } be its domain, and let a m be any element in the universe of A such that {a 1 , . . . , a m } has weight ≤ k. v m }. Let (T, G, v 1 , . . . ,T i , G i , v 1 , . . . , v m , f i ), i ∈ Iv m , f ) be the union of. Since S(T, G, v 1 , . . . , v m , f ) ⊆ i∈I S(T i , G i , v 1 , . . . , v m , f iThomas J. Schaefer. The complexity of satisfiability problems. In Proceedings of the ACM Symposium on Theory of Computing (STOC), pages 216-226, 1978. Let h be any mapping in H, let {a 1 , . . . , a m−1 } be its domain, and let a m be any element in the universe of A such that {a 1 , . . . , a m } has weight ≤ k. Let us assume, towards a contradiction, that there is not extension h ′ of h in H. Then there exists a finite collection {(T i , G i , v 1 , . . . , v m , f i ) : i ∈ I} of valid tuples for a 1 , . . . , a m such that the intersection i∈I S(T i , G i , v 1 , . . . , v m , f i ) does not contain any extension of h. We can rename the elements of the universes so that for every different i, j ∈ I we have that T i ∩ T j = {v 1 , . . . , v m }. Let (T, G, v 1 , . . . , v m , f ) be the union of (T i , G i , v 1 , . . . , v m , f i ), i ∈ I, which is a valid tuple for a 1 , . . . , a m . Since S(T, G, v 1 , . . . , v m , f ) ⊆ i∈I S(T i , G i , v 1 , . . . , v m , f i )
f ) does not contain any extension of h. We are almost at home. It is only necessary to observe that. T , G , . . , ; T , G , . . , S(T, G, v 1, v m , f ) does not contain any extension of h. v m−1 , f ) cannot contain h, in contradiction with h ∈ H. ⊓ ⊔we can conclude that S(T, G, v 1 , . . . , v m , f ) does not contain any extension of h. We are almost at home. It is only necessary to observe that (T, G, v 1 , . . . , v m−1 , f ) is a valid tuple for a 1 , . . . , a m−1 and since S(T, G, v 1 , . . . , v m , f ) does not contain any extension of h, S(T, G, v 1 , . . . , v m−1 , f ) cannot contain h, in contradiction with h ∈ H. ⊓ ⊔
| []
|
[
"ON A STEKLOV SPECTRUM IN ELECTROMAGNETICS",
"ON A STEKLOV SPECTRUM IN ELECTROMAGNETICS"
]
| [
"Francesco Ferraresso ",
"Pier Domenico Lamberti ",
"AND IOANNISG Stratis "
]
| []
| []
| After presenting various concepts and results concerning the classical Steklov eigenproblem, we focus on analogous problems for time-harmonic Maxwell's equations in a cavity. In this direction we discuss recent rigorous results concerning natural Steklov boundary value problems for the curlcurl operator. Moreover, we explicitly compute eigenvalues and eigenfunctions in the unit ball of the three dimensional Euclidean space by using classical vector spherical harmonics.Classification: 35P15, 35Q61, 78M22. | null | [
"https://arxiv.org/pdf/2206.00505v1.pdf"
]
| 249,240,532 | 2206.00505 | d0d86eb3abcaff27c03118bd653eb11c62741142 |
ON A STEKLOV SPECTRUM IN ELECTROMAGNETICS
Francesco Ferraresso
Pier Domenico Lamberti
AND IOANNISG Stratis
ON A STEKLOV SPECTRUM IN ELECTROMAGNETICS
Steklov boundary conditionsimpedance problemMaxwell's equationseigenvalue analysisinterior Calderón problem
After presenting various concepts and results concerning the classical Steklov eigenproblem, we focus on analogous problems for time-harmonic Maxwell's equations in a cavity. In this direction we discuss recent rigorous results concerning natural Steklov boundary value problems for the curlcurl operator. Moreover, we explicitly compute eigenvalues and eigenfunctions in the unit ball of the three dimensional Euclidean space by using classical vector spherical harmonics.Classification: 35P15, 35Q61, 78M22.
Introduction
At the macroscopic level, Maxwell's equations read (1). The constitutive relations for an electromagnetic medium reflect the physics that govern the phenomena and are expected to comply with the fundamental physical laws, which play the role of physical hypotheses, or postulates, concerning the properties of the material inside the considered domain. Some constitutive equations are simply phenomenological; others are derived from first principles.
1.1. Time-harmonic Maxwell's equations in a linear homogeneous isotropic conductive medium. We now consider electromagnetic wave propagation in a linear homogeneous isotropic medium in R 3 , with electric permittivity ε, magnetic permeability µ, and electric conductivity σ. By our assumptions on the medium, all these three parameters are constant. For such a medium the constitutive relations are
D = ε E , B = µ H.(2)
The electromagnetic wave with angular frequency ω > 0 will be described by the electric and magnetic field
E(x, t) = ε + i σ ω − 1/2 E(x) e −iωt ,(3)H(x, t) = µ − 1/2 H(x) e −iωt .(4)
We additionally consider that the space dependent part J(x) of J (x, t) satisfies
J(x) = σE(x) (Ohm's law).(5)
From Maxwell's equations (1), we obtain that the space dependent parts E and H satisfy the time-harmonic Maxwell equations
curl E(x) − i ωµ H(x) = 0 , curl H(x) + i ωε E(x) = 0.(6)
Let us note that since ε and µ are constant we have that both the fields E and H are divergence-free: div E = div H = 0.
Let Ω be a bounded domain (i.e. a bounded connected open set) in R 3 with smooth boundary Γ. We consider the following classical boundary value problem that involves the "perfect conductor" condition on Γ curl E − iωµH = 0 , curl H + iωεE = 0,
in Ω,
ν × E = m, on Γ,(8)
where ν denotes the unit outer normal to Γ. By eliminating H we obtain
curl curl E − k 2 E = 0, in Ω, ν × E = m, on Γ,(9)
where k 2 := ω 2 εµ, and as usual we assume that Imk ≥ 0. Instead of the standard interior Calderón operator (see e.g., [8], [26]), in what follows we consider its variant defined by m → (ν × H) × ν, i.e.
ν × E → − i ωµ (ν × curl E) × ν .(10)
Calderón operators are also called Poincaré-Steklov, or impedance, or admittance, or capacity operators.
1.2. The classical Steklov eigenvalue problem. We recall that the classical Steklov 1 eigenvalue problem on a bounded domain Ω of R n , n ≥ 2, is the problem ∆u = 0, in Ω, ∂u ∂ν = λu, on ∂Ω (11) in the unknowns u (the eigenfunction) and λ (the eigenvalue). The domain Ω is assumed to be sufficiently regular (usually one requires that the boundary Γ is at least Lipschitz continuous) and the (scalar) harmonic function u is required to belong to the standard Sobolev space H 1 (Ω). This problem can be considered as the eigenvalue problem for the celebrated Dirichlet-to-Neumann map defined as follows. Given the solution u ∈ H 1 (Ω) to the the Dirichlet problem
∆u = 0, in Ω, u = f, on ∂Ω (12)
with datum f ∈ H 1/2 (Γ), one can consider the normal derivative ∂u ∂ν of u as an element of H −1/2 (Γ), where H 1/2 (Γ) is the standard Sobolev space defined on Γ and H −1/2 (Γ) its dual. This allows to define the map D from
H 1/2 (Γ) to H −1/2 (Γ) by setting Df = ∂u ∂ν .
The map D is called Dirichlet-to-Neumann map and its eigenpairs (f, λ) correspond to the eigenpairs (u, λ) of problem (11), f being the trace of u on Γ.
1.3. The electromagnetic Steklov eigenvalue problem. The natural analogue in electromagnetics of the classical Steklov problem (11) in R 3 can be defined as the eigenvalue problem for the (rescaled) interior Calderón operator defined above, namely the map
ν × E → −(ν × curl E) × ν.(13)
Therefore, one looks for values λ such that (ν × curl E) × ν = −λν × E, or, equivalently (by taking another cross product by ν)
ν × curl E = λE T ,(14)
where E satisfies the equation curl curl E − k 2 E = 0 and E T := (ν × E) × ν is the tangential component of E. In conclusion, the Steklov eigenvalue problem for Maxwell's equations is
curl curl E − k 2 E = 0, in Ω, ν × curl E = λE T , on Γ.(15)
To the best of our knowledge problem (15) was first introduced for k > 0 by J. Camanõ, C. Lackner and P. Monk in [6] where it was pointed out that the spectrum of this problem is not discrete. In particular, for the 1 Vladimir Andreevich Steklov (1864 -1926) was not only an outstanding mathematician, who made many important contributions to Applied Mathematics, but also had an unusually bright personality. The Mathematical Institute of the Russian Academy of Sciences in Moscow bears his name. On his life and work see the very interesting paper [27].
case of the unit ball in R 3 it turns out that the eigenvalues consist of two infinite sequences, one of which is divergent and the other other is converging to zero. To overcome this issue, in that paper a modified problem, having discrete spectrum, is considered and then used to study an inverse scattering problem.
On the other hand, in [32], two of the authors of the present paper have analyzed problem (15) only for tangential vector fields E in which case the problem can be written in the form
curl curl E − k 2 E = 0, in Ω, ν × curl E = λE, on Γ.(16)
Note that the boundary condition in (16) automatically implies that E is tangential, that is E·ν = 0 on Γ. Because of this restriction, the null sequence of eigenvalues disappears and the spectrum turns out to be discrete.
Steklov eigenproblems have been and are extensively studied for a variety of differential operators, linear and nonlinear, mostly in the scalar case. On the contrary, there are not so many publications devoted to analogous problems for Maxwell's equations: apart the two papers mentioned above, we are aware of the ones by F. Cakoni, S. Cogar and P. Monk [5], by S. Cogar [11], [12], [13], by S. Cogar, D. Colton and P. Monk [14], by S. Cogar, P. Monk [15], and by M. Halla [21], [22].
In this paper, after presenting several comments and results for the classical Steklov eigenproblem (see Section 2), we discuss the approach introduced in [32] (see Section 3) and we include explicit computations of the eigenvalues of the related eigenvalue problems in the case of the unit ball in R 3 (see Section 4).
2.
On the classical Steklov eigenvalue problem 2.1. Some indicative applications. Problem (11) has a long history which goes back to the paper [39] written by Steklov himself. It is not easy to give a complete account of all possible applications of problem and its variants. Here we briefly mention three main fields of investigation.
• The sloshing problem. It consists in the study of small oscillations of a liquid in finite basin which can be thought as bounded container (a tank, a mug, a snifter etc.). The basin is represented by a bounded domain Ω in R 3 with boundary Γ = Γ 1 ∪ Γ 2 , where Γ 1 is a two dimensional domain representing the horizontal (free) surface of the liquid at rest, and Γ 2 represents the bottom of the basin. In this case the Steklov boundary condition ∂u ∂ν = λu is imposed only on Γ 1 , while the Neumann condition ∂u ∂ν is imposed on Γ 2 . The gradients grad u(x, y, z) of solutions u represent the (stationary) velocity fields of the oscillations and √ λ the corresponding frequency. In particular u(x, y, 0) is proportional to the elevation of the free surface and the so-called "high-spots" ' correspond to its maxima. We refer to [7], [9], [27], [34] for more details on classical and more recent aspects of the problem.
• Electrical prospection The study of the Dirichlet-to-Neumann map received a big impulse from the seminal paper [4] by Calderón which poses the inverse problem of recovering the electric conductivity γ of an electric body Ω from the knowledge of (the energy form associated with) the voltage-to-current map D γ defined in the same way as
D γ f = γ ∂u γ ∂ν ,
where u γ is the solution to the problem div(γ grad u γ ) = 0, in Ω,
u γ = f, on Γ.(17)
The problem of Calderón was solved in the fundamental paper [38], where it is proved that the map
f → Γ f D γ f dσ = Ω γ| grad u γ | 2 dx
uniquely identifies γ (in the class of conductivities γ of class C ∞ (Ω)). • Vibrating membranes Problem (11) can be used in linear elasticity to model the vibrations of a free membrane Ω in R 2 with mass concentrated at the boundary. Recall that the normal modes of a free membrane with mass density ρ are the solutions to the Neumann eigenvalue problem −∆u = λρu, in Ω, ∂u ∂ν = 0, on Γ.
The total mass of the membrane is given by M = Ω ρ(x) dx. If we consider a family of mass densities ρ for > 0, such that the support of ρ is contained in a neighborhood of the boundary Γ of radius (with ρ constant therein) and such that the total mass M = M does not depend on , then the solutions of problem (18) converge to the solutions of
∆u = 0, in Ω, ∂u ∂ν = λρu, on Γ,(19)
where ρ = M/|Γ| and |Γ| is the perimeter of Γ. Thus problem (19) can be considered as a limiting/critical case of a family of Neumann eigenvalue problems. We refer to [18,29,30] for further details, in particular for an asymptotic analysis.
Finally, we mention that the Steklov problem has been recently used in [23] for a mathematical model related to the study of information transmission in the neural network of the human brain.
2.1.1. Details on the formulation and its connections to trace theory. For any n ≥ 2 the weak (variational) formulation of problem (11) is easily obtained by multiplying the equation ∆u = 0 by a test function ϕ and integrating by parts over Ω. This simple computation leads to the equality Ω grad u · grad ϕ dx = λ Γ uϕ dσ (20) which needs to be satisfied for all functions ϕ ∈ H 1 (Ω), and can be taken as the formulation of the classical problem (11) in H 1 (Ω). The advantage of formulation (20) is evident since it easily allows to apply standard tools from functional analysis and calculus of variations to prove existence of solutions. Indeed, one can prove that the spectrum of problem (20) is discrete and consists of a divergent sequence of eigenvalues
0 = λ 1 ≤ λ 2 ≤ · · · ≤ λ k ≤ . . .
where it is assumed that each eigenvalue is repeated as many times as its multiplicity, which is finite. The corresponding eigenfunctions ϕ j define a complete orthogonal system for the subspace H(Ω) of harmonic functions in H 1 (Ω). Note that H(Ω) can be described as
H(Ω) = {u ∈ H 1 (Ω)
:
Ω grad u·grad ϕ dx = 0, ∀ϕ ∈ C ∞ c (Ω)}.
In fact, the following decomposition holds
H 1 (Ω) = H 1 0 (Ω) ⊕ H(Ω) .(21)
We describe one straightforward way to prove these results since this method will be applied to the case of Maxwell's equations.
By adding the term Ω uϕ dσ to both sides of the equation (20) and setting µ = λ + 1, one gets the equation
Ω grad u · grad ϕ dx + Γ uϕ dσ = µ Γ uϕ dσ
where the quadratic form associated to the left-hand side, namely
Q(u) := Ω | grad u| 2 dx + Γ |u| 2 dσ
is coercive in H 1 (Ω) and in particular defines a norm Q(u) 1/2 equivalent to the Sobolev norm of H 1 (Ω). Thus, the operator L from H 1 (Ω) to its dual defined by the pairing
Lu, ϕ = Ω grad u · grad ϕ dx + Γ uϕ dσ is invertible. Then we can consider the operator T from H 1 (Ω) to itself defined by T = L −1 • J where J is the operator from H 1 (Ω) to its dual defined by the pairing Ju, ϕ = Γ uϕ dσ.
It is simple to see that T is a selfadjoint operator with respect to the scalar product associated with the quadratic form Q above (see e.g., [28] for an analogous Steklov problem). Moreover, since the standard trace operator T r from H 1 (Ω) to L 2 (Γ) is compact, it follows that T is also compact, hence its spectrum consists of zero and a decreasing divergent sequence of positive eigenvalues µ j . Thus the eigenvalues λ j above can be defined by the equality µ j = (λ j + 1) −1 . Moreover, since the kernel of T is exactly H 1 0 (Ω) and its orthogonal is H(Ω), the decomposition (21) immediately follows.
If Ω is the ball of radius R centred at zero, the eigenvalues are given by all numbers of the form
l j = j R , j ∈ N 0 .
The corresponding eigenfunctions are the homogeneous polynomials of degree j and can be written in spherical coordinates in the form
u(r, ξ) = r j Y j (ξ) for r = |x| ≥ 0 and ξ = x/|x| ∈ S n−1 (the (n − 1)-dimensional unit sphere),
where Y j is any spherical harmonic of degree j. In particular, the multiplicity of l j is (2j + n − 2)(j + n − 3)!/(j!(n − 2)!), and only l 0 is simple, the corresponding eigenfunctions being the constant functions. Note that the enumeration l j , j ∈ N 0 is different from the enumeration λ j , j ∈ N discussed above since it does not take into account the multiplicity of the eigenvalues. We note en passant that the eigenvalues of the Laplace-Beltrami operator on the (n−1)-dimensional sphere of radius R are given by the formula σ l = l(l+ n−2)/R 2 and coincide with the squares of the l j for n = 2 (the corresponding eigenfunctions are given by the restrictions of the corresponding Steklov eigenfunctions). We refer to [36] for further discussions. We would like now to describe the method of Auchmuty [2] for the spectral representation of the trace space H 1/2 (Γ), since the same method will be used in the vectorial case (in which case the Steklov eigenvectors for Maxwell's equations will be used).
By exploiting an argument similar to the one discussed above (with an operator analogous to T defined on L 2 (Γ) rather than on H 1 (Ω)), one can actually see that the traces of the eigenfunctions ϕ j on Γ define a complete orthogonal system for L 2 (Γ). Here for simplicity we also write ϕ j instead of T r(ϕ j ). Assume that those eigenfunctions are normalized in L 2 (Γ), that is Γ |ϕ j | 2 dσ = 1. Then, by equation (20) it follows that ϕ j / λ j + 1 is normalized in H 1 (Ω), that is Q(ϕ j / λ j + 1) = 1. Thus, H(Ω) can be described as follows
H(Ω) = ∞ j=1 c j ϕ j λ j + 1 : ∞ j=1 |c j | 2 < ∞ .(22)
Recall that the trace space T r(H 1 (Ω)) coincides with the standard fractional Sobolev space H 1/2 (Γ) and note that T r(H 1 (Ω)) = T r(H(Ω)) by (21). This, combined with (22), yields
H 1/2 (Γ) = T r(H(Ω)) = ∞ j=1 c j T r(ϕ j ) λ j + 1 : ∞ j=1 |c j | 2 < ∞ = ∞ j=1 c j T r(ϕ j ) : ∞ j=1 (λ j + 1)|c j | 2 < ∞ .(23)
If Ω is sufficiently smooth then Weyl's law, describing the asymptotic behavior of the eigenvalues, is
λ j ∼ cj 1 n−1 , as j → ∞
where c is an explicitly known constant. It follows that the space H 1/2 (Γ) can be described as folllows
H 1/2 (Γ) = ∞ j=1 c j ϕ j : ∞ j=1 j 1 n−1 |c j | 2 < ∞ .
Thus the condition on the Fourier coefficients c j is that the sequence
j 1 2(n−1) c j , j ∈ N(24)
belongs to the space 2 of square summable sequences. We note that the appearance of the factor 1/2 at the exponent in (24) is not artificial and corresponds to the exponent of the space H 1/2 (Γ). In fact an analogous representation was found in [31] for the space H 3/2 (Γ) where the exponent 3/2 naturally appears by using the Weyl asymptotic for a biharmonic Steklov eigenvalue problem.
On the electromagnetic Steklov eigenproblem
In this section we briefly present some of the results in [32] for problem (16). In the sequel Ω will denote a bounded domain in R 3 with sufficiently smooth boundary, say of class C 1,1 (see e.g., [33, Definition 1]). As done in [16] for analogous problems, we introduce a penalty term θ grad div u in the equation, where θ can be any positive number, in order to guarantee the coercivity of the quadratic form associated with the corresponding differential operator. Namely, we consider the eigenvalue problem
curl curl E − k 2 E − θ grad div E = 0, in Ω, ν × curl E = λE, on Γ, E · ν = 0, on Γ(25)
where E is the unknown vector field. Here we allow k 2 ∈ R to be not necessarily positive. Recall that the second boundary condition above is in fact embodied in the first one but we prefer to emphasize it since we need to include it in the definition of the energy space.
By L 2 (Ω), H 1 (Ω), H 1 0 (Ω), L 2 (Γ), H 1/2 (Γ), H −1/2 (Γ)
, we denote the standard Lebesgue and Sobolev spaces. We also employ the following spaces:
• H(curl, Ω) = {u ∈ (L 2 (Ω)) 3 : curl u ∈ (L 2 (Ω)) 3 } , with norm: u H(curl,Ω) = u 2 (L 2 (Ω)) 3 + curl u 2 (L 2 (Ω)) 3 1/2 • H(div, Ω) = {u ∈ (L 2 (Ω)) 3 : div u ∈ L 2 (Ω)} , with norm: u H(div,Ω) = u 2 (L 2 (Ω)) 3 + div u 2 L 2 (Ω) 1/2 • H 0 (div, Ω) = {u ∈ H(div, Ω) : ν · u = 0 on Γ} • X T (Ω) = H(curl, Ω) ∩ H 0 (div, Ω) , with norm: u X T (Ω) = u 2 (L 2 (Ω)) 3 + curl u 2 (L 2 (Ω)) 3 + div u 2 L 2 (Ω) 1/2
We refer to [1], [8], [19], [20], [24], [35], [37] for details. It is important to note that since we have assumed Ω to be of class C 1,1 , the space X T (Ω) is continuously embedded in (H 1 (Ω)) 3 and there exists c > 0 such that the Gaffney inequality
u (H 1 (Ω)) 3 ≤ c u L 2 (Ω) 3 + curl u L 2 (Ω) 3 + div u L 2 (Ω) ,
holds for all u ∈ X T (Ω).
Problem (25) has to be interpreted in the weak sense as follows: find
E ∈ X T (Ω) such that Ω curl E · curl ϕ dx − k 2 Ω E · ϕ dx + θ Ω div E div ϕ dx = −λ Γ E · ϕ dσ ,(26)
for all ϕ ∈ X T (Ω).
The above formulation is obtained from (25) by a standard procedure: for a smooth solution E of (25), we multiply both sides of the first equation in (25) by ϕ ∈ X T (Ω), integrate by parts and use the following standard Green-type formula
Ω curl E · curl ϕ dx = Ω curl curl E · ϕ dx − Γ (ν × curl E) · ϕ dσ . (27)
Conversely, by the Fundamental Lemma of the Calculus of Variations (see e.g., [3]), one can see that if E is a smooth solution of (26) then it is also a solution of (25) in the classical sense.
Note that the weak formulation allows to avoid assuming additional regularity assumptions on Γ, see e.g., [40].
In order to study our eigenvalue problem, we need to assume that k 2 does not coincide with an eigenvalue A of the problem
curl curl E − θ grad div E = AE, in Ω, ν × E = 0, on Γ, E · ν = 0, on Γ.(28)
Clearly the two boundary conditions above are equivalent to the Dirichlet condition E = 0 on Γ. We note that (28) has a discrete spectrum which consists of a sequence A n , n ∈ N of positive eigenvalues of finite multiplicity, the first one being
A 1 = min ϕ∈(H 1 0 (Ω)) 3 ϕ =0 Ω | curl ϕ| 2 dx + θ Ω | div ϕ| 2 dx Ω |ϕ| 2 dx > 0.(29)
For the sake of brevity, we assume in the sequel that k 2 < A 1 . For details on the more general case A n < k 2 < A n+1 (30) we refer to [32].
The key result in the considered case is the following. Theorem 1. Let k 2 < A 1 and θ > 0. The eigenvalues of problem (25) are real, have finite multiplicity and can be represented by a sequence λ n , n ∈ N, divergent to −∞. Moreover, the following min-max representation holds:
λ n = − min V ⊂X T (Ω) dimV =n max ϕ∈V \(H 1 0 (Ω)) 3 Ω | curl ϕ| 2 − k 2 |ϕ| 2 + θ| div ϕ| 2 dx Γ |ϕ| 2 dx .(31)
To prove this result we follow the strategy described in Section 2.1.1. Namely, by adding the term η Γ E · ϕ dσ to both sides of equation (26) we obtain (32) where γ = −λ + η. Under our assumptions, it is proved in [32,Thm. 3.1] that if η is big enough then the quadratic form associated with the left-hand side of equation (32), that is
Ω curl E · curl ϕ dx − k 2 Ω E · ϕ dx + θ Ω div E div ϕ dx + η Γ E · ϕ dσ = γ Γ E · ϕ dσQ(E) := Ω | curl E| 2 dx − k 2 Ω |E| 2 dx + θ Ω | div E| 2 dx + η Γ |E| 2 dσ,
is coercive in X T (Ω), hence (Q(E)) 1/2 defines a norm equivalent to that of X T (Ω).
Thus, the operator L η from X T (Ω) to its dual defined by the pairing
L η E, ϕ := Ω curl E · curl ϕ dx − k 2 Ω E · ϕ dx + θ Ω div E div ϕ dx + η Γ E · ϕ dσ is invertible.
Then we can consider the operator T from X T (Ω) to itself, defined by
T = (L η ) −1 • J
where J is the operator from X T (Ω) to its dual defined by the pairing
J E, ϕ = Γ E · ϕ dσ
for all E, ϕ ∈ X T (Ω). As in the case described in Section 2.1.1, it is not difficult to prove that T is a selfadjoint operator with respect to the scalar product associated with the quadratic form Q above. Again, since the trace operator is compact, it follows that T is also compact, hence its spectrum consists of zero and a decreasing divergent sequence of positive eigenvalues γ j . Thus the eigenvalues γ j above can be defined by the equality γ j = (−λ j + η) −1 . Then the characterization in (31) follows by the classical Min-Max Principle applied to the operator T . It follows by the previous results that the space X T (Ω) can be decomposed as an orthogonal sum with respect to the scalar product associated with the form Q, namely
X T (Ω) = KerT ⊕ (KerT ) ⊥ = (H 1 0 (Ω)) 3 ⊕ H(Ω) .
where H(Ω) := (KerT ) ⊥ = E ∈ X T (Ω) :
Ω curl E · curl ϕ dx −k 2 Ω E · ϕ dx + θ Ω div E div ϕ dx = 0, ∀ϕ ∈ (H 1 0 (Ω)) 3 .(33)
Note that E ∈ H(Ω) if and only if E is a weak solution in (H 1 (Ω)) 3 of the problem
curl curl E − k 2 E − θ grad div E = 0, in Ω, ν · E = 0, on Γ.(34)
Solutions to problem (34) play the same role as the harmonic functions in H(Ω) used in Section 2.1.1 and, similarly, the eigenfunctions associated with the eigenvalues γ n define a complete orthonormal system of H(Ω).
Let Σ = {λ n : n ∈ N}. It is important to known whether 0 ∈ Σ. This condition can be clarified as follows. We consider two auxiliary eigenproblems. The first is the classical eigenvalue problem for the Neumann Laplacian
−∆φ = λφ, in Ω, ∂φ ∂ν = 0, on ∂Ω ,(35)
which admits a divergent sequence λ N n , n ∈ N, of non-negative eigenvalues of finite multiplicity, with λ N 1 = 0. The second is the eigenproblem
curl curl ψ = λψ,
in Ω, div ψ = 0
in Ω, ν × curl ψ = 0, on Γ, ψ · ν = 0, on Γ
which admits a divergent sequence λ M n , n ∈ N, of non-negative eigenvalues of finite multiplicity.
In the following result from [32, Thm. 3.10], one has actually to better assume the condition k = 0. On the other hand it is straightforward that if Ω is simply connected and k = 0 then 0 / ∈ Σ because the corresponding eigenfunctions would have zero div and zero curl (see (31)) hence, being tangential, they would be identically zero, see [19,Prop. 2,p. 219] for more information; see also the more recent paper [10].
Theorem 2. Assume that k = 0 and θ > 0. We have that 0 ∈ Σ if and only if k 2 ∈ {θλ N n : n ∈ N} ∪ {λ M n : n ∈ N}. 3.1. Remarks on trace problems and Steklov expansions. We denote by E Ω n , n ∈ N, an orthonormal sequence of eigenvectors associated with the eigenvalues λ n of problem (25), where it is understood that they are normalized with respect to the quadratic from Q.
Let π T denote the trace operator from X T (Ω) to T L 2 (Γ) where T L 2 (Γ) = {u ∈ (L 2 (Γ)) 3 : ν · u = 0 on Γ}.
By setting E Γ n := |λ n − η| π T E Ω n
one can prove, in the spirit of Section 2.1.1, that E Γ n , n ∈ N, is an orthonormal basis of T L 2 (Γ).
These bases can be used to represent the solutions of the following problem
curl curl U − k 2 U − θ grad div U = 0, in Ω, ν × curl U = f , on Γ,(38)
where f ∈ T L 2 (Γ). Let f have the following representation
f = ∞ n=1 c n E Γ n ,
with (c n ) n∈N ∈ 2 . It is proved in [32,Thm. 4.1] that if 0 / ∈ Σ then the solution U of (38) can be expanded in terms of the above basis as follows
U = ∞ n=1 |λ n − η| λ n c n E Ω n .(39)
Finally, under our assumptions we can represent the trace space of X T (Ω) as follows
π T (X T (Ω)) = π T (H(Ω)) = ∞ j=1 c j E Γ j : ∞ j=1 |λ j − η||c j | 2 < ∞ (40)
which is the counterpart of the representation (23) for our problem.
The case where Ω is the unit ball
In this section we consider problem (25) in Ω = B, where B is the unit ball in R 3 centred at zero and we compute explicitly its eigenvalues and eigenvectors. In particular, we shall prove that there exist two families of eigenvectors, one of which is not divergence-free.
We proceed to define the vector spherical harmonics, following the notation of [25]. Recall that the (scalar) spherical harmonics are given by
Y σml (θ, φ) = ε m 2π (2l + 1)(l − m)! 2(l + m)! P m l (cos θ)F σ (φ) = C lm P m l (cos θ)F σ (φ) (41) where θ ∈ [0, π], φ ∈ [0, 2π), σ ∈ {e, o}, l ∈ N ∪ {0}, m ∈ N ∪ {0}, m ≤ l, P m
l is the Legendre function associated with the Legendre polynomial P l , ε m = 2 − δ m0 , and F e = cos mφ, F o = sin mφ.
According to [25, p. 627] we will use the multi-index notation
Y n := Y σml ,l(l + 1) grad ξ Y n (ξ) × ξ (42) A 2n (ξ) = 1 l(l + 1) grad ξ Y n (ξ) (43) A 3n (ξ) = Y n (ξ)ξ(44)
for all ξ ∈ ∂B, where Y n = Y σml is defined as in (41), see [25, p. 350]. We extend this definition to points x ∈ B \ {0} by setting
A 1n (x) = |x| l(l + 1) grad x Y n x |x| × x |x| (45) A 2n (x) = |x| l(l + 1) grad x Y n x |x| (46) A 3n (x) = Y n x |x| x |x| (47)
By definition A 1σ00 = A 2σ00 = 0. The family {A τ n : τ ∈ {1, 2, 3}, n ∈ O} is a complete orthonormal system in L 2 (∂B) 3 . Therefore, we can expand any vector field E ∈ L 2 (B) 3 as follows:
E(r, ξ) = n∈O E 1 n (r)A 1n (ξ) + E 2 n (r)A 2n (ξ) + E 3 n (r)A 3n (ξ) .
We note immediately that since we are interested in solutions of (25), the boundary condition E · ν = 0 is equivalent to E 3 n (1) = 0, since one easily checks that A 2n (ξ) · ξ = 0 and A 1n (ξ) · ξ = 0. Recall that the vector Laplacian acts in the following way
∆(f (r)V (ξ)) = 1 r 2 ∂ ∂r r 2 ∂f (r) ∂r V (ξ) + f (r)∆V (ξ).(48)
We have the following formulae for the vector Laplacian of the vector spherical harmonics A n
∆A 1n = − 1 r 2 l(l + 1)A 1n ∆A 2n = 2 r 2 (l(l + 1))A 3n − 1 r 2 l(l + 1)A 2n ∆A 3n = − 1 r 2 (2 + l(l + 1))A 3n + 2 l(l + 1) r 2 A 2n .(49)
Moreover, one can compute (see also (C.14) in [25])
div E(r, ξ) = n∈O ∂E 3 n (r) ∂r + 2 r E 3 n (r) − l(l + 1) r E 2 n (r) Y n (ξ).(50)
Let us define
Φ(r) := ∂E 3 n (r) ∂r + 2 r E 3 n (r) − l(l + 1) r E 2 n (r) .
Note that div E = 0 is equivalent to Φ = 0.
Remark 3. In [6], the authors find two families of solutions to the equation curl curl u − k 2 u = 0: one is given by M n = curl(xj l (k|x|)Y lm (x/|x|)), the other one by N n = curl M n . Here j l is the spherical Bessel function of the first kind of order l, namely j l (z) = π/(2z)J l+1/2 (z). These two families are divergence-free by definition. Indeed, the solutions M n have E j = 0, j = 2, 3, while the solutions N n satisfy Φ(r) = 0 for E 2 , E 3 not identically zero. The function N n = curl M n (x) in [6], for n ∈ O, are given by
curl M n (x) = curl j l (k|x|) grad x Y n x |x| × x |x| = j l (k|x|)l(l + 1) |x| A 3n + l(l + 1) j l (k|x|)k + j l (k|x|) 1 |x| A 2n .
This is consistent with the fact that div curl M n = 0.
From (50) it is easy to compute
grad(div E)(r, ξ) = n∈O Φ (r)A 3n (ξ) + n∈O Φ(r) r l(l + 1)A 2n (ξ). (51)
Due to (48), (49), (51) we then have that
− ∆E + (1 − θ) grad div E = n∈O − 1 r 2 ∂ ∂r r 2 ∂E 1 n ∂r − E 1 n l(l+1) r 2 − E 3 n 2 √ l(l+1) r 2 − 1 r 2 ∂ ∂r r 2 ∂E 2 n ∂r + E 2 n l(l+1) r 2 + (1 − θ) l(l + 1) Φ r − 1 r 2 ∂ ∂r r 2 ∂E 3 n ∂r + (2+l(l+1))E 3 n r 2 − E 2 n 2 √ l(l+1) r 2 + (1 − θ)Φ · A 1n A 2n A 3n (52) Consider now −∆E + (1 − θ) grad div E − k 2 E = 0 in B.
It is clear that the first equation provided by the first row in (52) above can be solved independently of the other two, by means of separation of variables and classical Sturm-Liouville theory, yielding
E 1 n (r) = E 1 l (r) = j l (kr)(53)
where j is the spherical Bessel function of the first kind of order l as above. The other two equations form a coupled system of Sturm-Liouville equations. For the sake of clarity, we first solve the system in the case θ = 1.
Case θ = 1. We need to solve the two-parameter family of ODE systems
− 1 r 2 ∂ ∂r r 2 ∂E 2 n ∂r + l(l+1)E 2 n r 2 − 2 √ l(l+1)E 3 n r 2 − k 2 E 2 n = 0, in (0, 1), − 1 r 2 ∂ ∂r r 2 ∂E 3 n ∂r + (2+l(l+1))E 3 n r 2 − 2 √ l(l+1)E 2 n r 2 − k 2 E 3 n = 0, in (0, 1), E 3 n (1) = 0.(54)
Recall the spherical Bessel equation ∂ ∂r r 2 ∂ ∂r f + (k 2 r 2 − l(l + 1))f = 0.
Define the spherical Bessel operator of indices k ∈ R and l ∈ Z as follows L k,l (f ) = ∂ ∂r r 2 ∂ ∂r f + (k 2 r 2 − l(l + 1))f.
We will require f ∈ H 2 ([0, 1]), which is equivalent to imposing that the solution f to (55) is not singular in zero. System (54) can be then rewritten in the simpler form
L k,l (E 2 ) = −2 l(l + 1)E 3 , L k,l (E 3 ) = 2E 3 − 2 l(l + 1)E 2 , E 3 (1) = 0.(56)
where we have omitted the dependence on n in E 2 , E 3 .
Remark 4. Note that, upon replacing E 2 in the first equation, by means of the equality
2E 3 −L k,l (E 3 ) 2 √ l(l+1)
= E 2 , we obtain the fourth-order equation
(L k,l ) 2 (E 3 ) − 2L k,l (E 3 ) = 4l(l + 1)E 3 ,
with the boundary condition E 3 (1) = 0.
In view of Remark 3 a solution to the differential equation in (56) is given by
E 2 (r) = l(l + 1) j l (kr)k + j l (kr) r , E 3 (r) = j l (kr)l(l + 1) r ;(57)
note however that the condition E 3 (1) = 0 is not satisfied for general k. To overcome this problem, it is convenient to first find another explicit solution of (56). We claim that the functions E 2 (r) := l(l + 1) j l (kr) r , E 3 (r) := kj l (kr),
are solutions of the system of differential equations (56). In the sequel we use the abbreviated notation L := L k,l . Let us compute
L(k j l (k r)) = k 3 r 2 j(3)
l (k r) + 2k 2 r j l (k r) + k (k 2 r 2 − l(l + 1)) j l (k r) (59) Recall that j l (·) satisfies (55). Differentiation of (55) with f (r) = j l (kr) gives
k 3 r 2 j (3)
l (k r) + 4k 2 r j l (k r) + k(k 2 r 2 + 2 − l(l + 1)) j l (kr) + 2k 2 r j l (k r) = 0 (60) Equation (60) implies that we can rewrite (59) as follows L(kj l (kr)) = −4k 2 rj l (kr) − k(k 2 r 2 + 2 − l(l + 1))j l (kr) − 2k 2 rj l (kr) + 2k 2 rj l (kr) + k(k 2 r 2 − l(l + 1))j l (kr)
= −2k 2 rj l (kr) − 2kj l (kr) − 2k 2 rj l (kr) (61)
The spherical Bessel equation L(j l ) = 0, see (55), again implies that k 2 r j l (kr) = −2kj l (kr) − k 2 r 2 − l(l + 1) j l (kr) r hence (61) can be rewritten as L (k j l (kr)) = 2 2kj l (kr) + k 2 r 2 − l(l + 1)
j l (kr) r − 2kj l (kr) − 2k 2 r j l (kr) = 2kj l (kr) − 2l(l + 1) r j l (kr) = 2E 3 − 2 l(l + 1)E 2 ,
which implies the identity L(E 3 ) = 2E 3 − 2 l(l + 1)E 2 . We note that the functions E 2 , E 3 defined in (57) satisfy the equalities
E 2 = E 3 / l(l + 1), and L(E 3 ) = 2E 3 − 2 l(l + 1)E 2 .
Then it is immediate to check that L(E 2 ) = −2 l(l + 1)E 3 which implies that the couple (E 2 , E 3 ) t solves the two equations in (56), as claimed.
Note that the divergence of the function
E(r, ξ) = n∈O E 2 n (r)A 2n (ξ) + E 3 n (r)A 3n (ξ) is non-trivial unless k = 0. Indeed, div E(r, ξ) = n∈O ∂E 3 (r) ∂r + 2 r E 3 (r) − l(l + 1) r E 2 (r) Y n (ξ) = n∈O j l (kr)k 2 + 2 r j l (kr)k − l(l + 1) r j l (kr) r Y n (ξ) = −k 2 n∈O j l (kr)Y n (ξ) = 0,(62)
where in the last equality we used again (55). We note en passant that the previous formula for the divergence, namely
∂E 3 (r) ∂r + 2 r E 3 (r) − l(l + 1) r E 2 (r) = −k 2 j l (kr)
gives the equality E 2 (r) = r l(l + 1)
1 r 2 ∂(r 2 E 3 (r)) ∂r + k 2 j l (kr) .(63)
A similar computation for the functions E 2 , E 3 defined in (57) gives
E 2 (r) = 1 l(l + 1) r ∂(r 2 E 3 (r)) ∂r(64)
in agreement with Formula (7.4) in [25]. For the boundary condition to be satisfied we then choose a linear combination of (E 2 , E 3 ) t and (E 2 , E 3 ) t . If j l (k) = 0 then (E 2 , E 3 ) t is already a solution of (56) with non-trivial divergence. Otherwise, we just set F 2 (r) := − (l(l + 1)) 3/2 j l (k) kj l (k) j l (kr) r + l(l + 1) j l (kr) r + kj l (kr)
F 3 (r) := −l(l + 1) j l (k) j l (k) j l (kr) − j l (kr) r ,(65)
and then F := (F 2 , F 3 ) t solves (56) with the right boundary condition. For subsequent use, note that
F 2 (1) = l(l + 1) 1 − l(l + 1)j l (k) kj l (k) j l (k) + l(l + 1)kj l (k) = l(l + 1) j l (k)j l (k)k − l(l + 1)j l (k) 2 + k 2 (j l (k)) 2 kj l (k)
for all l ≥ 1.
Let us also compute
1 l(l + 1) d dr F 2 (r) = k j l (kr) r − j l (kr) r 2 1 − l(l + 1)j l (k) kj l (k) + k 2 j l (kr) (55) = j l (kr) r 2 l(l + 1)j l (k) kj l (k) − k 2 + l(l + 1) − 1 r 2 − k j l (kr) r 1 + l(l + 1)j l (k) kj l (k) therefore d dr F 1 (r) | r=1 = l(l + 1) − (k 2 + 1)j l (k) − kj l (k) + j l (k) 2 l(l + 1) kj l (k) Note that the solution E of curl curl E − k 2 E = 0, in B, ν × curl E − λ(ν × E × ν) = 0, on ∂B, ν · E = 0, on ∂B,
can be found in the form
E = n∈O a n (F 2 n A 2n + F 3 n A 3n ) + b n E 1 l A 1n
where F 2 n , F 3 n are defined in (65), E 1 n is defined in (53), and a n , b n ∈ C. For the following computations it is useful to recall the formulae (see also [25, p. 633])
curl(f (r)A 1n ) = l(l + 1) r f (r) A 3n + 1 r ∂ ∂r (rf (r)) A 2n , curl(f (r)A 2n ) = − 1 r ∂ ∂r (rf (r)) A 1n , curl(f (r)A 3n ) = l(l + 1) r f (r) A 1n .
In particular, we have that
(ξ × curl(F 2 n A 2n + F 3 n A 3n ))| r=1 = − ξ × ((F 2 n ) (1) + F 2 n (1))A 1n + ξ × ( l(l + 1)F 3 n (1)A 1n ) = −((F 2 n ) (1) + F 2 n (1))A 2n
Imposing the Steklov condition and taking into account that ν · E = 0 give
0 = (ν × curl E − λ(ν × E × ν))| r=1 = n∈O −a n ((F 2 n ) (1) + (1 + λ)F 2 n (1))A 2n n∈O −b n j l (k) + j l (k)k + λj l (k) A 1n .
Hence, the eigenvalues λ n of (25) are given by two families. The first one is obtained by imposing
0 = ((F 2 n ) (1) + (1 + λ)F 2 n (1)) ⇒ λ (1) l = k 2 k j l (k) j l (k) kj l (k)j l (k) − l(l + 1)(j l (k)) 2 + k 2 (j l (k)) 2
for all l ≥ 1. The second one is obtained by imposing that the coefficient of A 1n vanishes for some n ∈ O. We then obtain
λ (2) l = − j l (k) + j l (k)k j l (k)(66)
It is not difficult to check that the eigenvalues λ l , we note that, due to the recurrence formulae for the derivatives of Bessel functions,
kj l (k)j l (k) − l(l + 1)(j l (k)) 2 + k 2 (j l (k)) 2 k 2 = −j l−1 (k)j l+1 (k) hence, λ (1) l = − kj l (k)j l (k) j l+1 (k)j l−1 (k)(67)
for l ≥ 1. Now, recalling that j l (z) = π 2z J l+1/2 (z), where J is the Bessel function of the first kind, and that we have the following large index asymptotic formula:
J s (z) s→+∞ ∼ 1 √ 2πs
ez 2s s for all z ∈ C, we deduce that
j l (z) l→∞ ∼ e l+1/2 z l 2 l+3/2 (l + 1/2) l+1(68)
for l ≥ 1. By using formulae (67) and (68) General case 0 < θ < +∞. Since we are interested in solutions of
curl curl E − θ grad div E − k 2 E = 0(69)
with non-trivial divergence, it is convenient to find an equation for div E.
The following discussion applies to a general domain Ω (not necessarily a ball). Let ϕ ∈ H 2 (Ω) be an arbitrary function. Multiply (69) by grad ϕ and integrate in Ω to obtain
Ω (curl curl E · grad ϕ − θ grad div E · grad ϕ − k 2 E · grad ϕ) dx = 0.
Due to the Steklov boundary condition, the identity curl grad ϕ = 0, and standard integration by parts we have
Ω curl curl E · grad ϕ dx = Γ (ν × curl E) · grad Γ ϕ dσ = λ Γ E · grad ϕ dσ = λ Γ (div Γ E) ϕ dσ,
where grad Γ and div Γ denote the standard tangential gradient and tangential divergence respectively. Moreover,
−θ Ω grad div E · grad ϕ dx = θ Ω ∆ div Eϕ dx − θ Γ ∂ ∂ν (div E)ϕ dσ,
as well as
−k 2 Ω E · grad ϕ dx = k 2 Ω div Eϕ dx − k 2 Γ (E · ν) = 0 ϕ dσ, hence Ω (θ∆ div E + k 2 div E)ϕ dx = Γ λ div Γ E + θ ∂ ∂ν (div E) ϕ dσ,
which implies that div E satisfies the following boundary value problem
−∆ div E − k 2 θ div E = 0, in Ω, ∂ ∂ν div E = − λ θ div Γ E, on Γ.(70)
Returning to the case of the ball, the equation in Ω = B has the explicit solutions div E n (r, ξ) = aj l k √ θ r Y n (ξ), n ∈ O, a ∈ C.
Therefore, arguing as in (62) we have the equality
∂E 3 (r) ∂r + 2 r E 3 (r) − l(l + 1) r E 2 (r) = aj l k √ θ r
which leads to the ansatz (see also (63))
E 2 (r) = r l(l + 1) 1 r 2 ∂(r 2 E 3 (r)) ∂r − aj l k √ θ r .(71)
As in the case θ = 1 we consider the linear combination F 2
n F 3 n := a E 2 n E 3 n + b E 2 n E 3
n and we impose the boundary condition F 3 n (1) = 0, corresponding to E · ν = 0 on ∂B. We obtain
a = −b j l k √ θ k √ θ j l (k)l(l + 1) ,
hence, up to a constant factor, j l (kr) r + j l (kr)k + l(l + 1)
j l k √ θ r r (73) F 3 n (r) := − j l k √ θ k √ θ j l (k) j l (kr) r + j l k √ θ r k √ θ(74)
As in the case θ = 1, we deduce that there are two families of eigenvalues diverging to −∞: the first one coincides with λ (2) n defined in (66), and it is associated with eigenfunctions in the form E n (r, ξ) = j l (kr)A 1n (ξ); the second one is obtained as in the case θ = 1 by imposing the Steklov boundary conditions, and they are given explicitly as solutions of the equation
λ (1) l = − F 2 l (1) + (F 2 l ) (1) F 2 l (1)
.
We proceed therefore to computing F 2 l (1) and (F 2 l ) (1). To simplify the notation we will not write the index l. We easily obtain
l = − F 2 l (1) + (F 2 l ) (1) F 2 l (1) , n ∈ O E n = E 1 n A 1n , n ∈ O div E n = 0.(76)
for all l ∈ N, where A τ n , τ = 1, 2, 3, n ∈ O are the vector spherical harmonics defined in (42), the functions F 2 n , F 3 n are defined respectively in (73), (74) and E 1 n is defined in (53). Remark 6. The squared values of k = 0 for which the denominators in (75) and (76) vanish are the eigenvalues A of the Dirichlet problem (28). Indeed, our computations in this section show that the solutions of the partial differential equation in (25), which are tangential at the boundary of B, are given by the two families of vector fields F n = F 2 n A 2n + F 3 n A 3n and E n = E 1 n A 1n . Note that E n is tangential because A 1n is tangential, while F n is tangential because F 3 n (1) = 0. Consider now the condition
E 1 n (1) = 0 .(77)
The values of k for which (77) is satisfied are exactly the values of k for which the denominator in (76) vanishes: in this case the vector field E n vanishes at the boundary of B, hence E n is a solution of (28) with A = k 2 . Assume now that the denominator in (76) does not vanish. In this case, the function F 2 n is well-defined, hence the solution F n is well-defined as well. Under this assumption consider the condition
The values of k for which (78) is satisfied are exactly the values of k for which the denominator in (75) vanishes: in this case the vector field F n vanishes at the boundary of B, hence F n is a solution of (28) with A = k 2 . Thus, assuming that the denominators of (75) and (76) do not vanish corresponds to our assumption (30), which is at the base of the analysis carried out in [32].
We note en passant that the standard eigenvalues of Maxwell's equations in a cavity are given by two families of positive numbers, one of which is the family of the squares of the zeros of the equation j l (k) = 0, see [17] , or [33,Appendix], for more details.
curl H(x, t) = ∂ ∂t D(x, t) + J (x, t) (Ampère's law), curl E(x, t) = − ∂ ∂t B(x, t) (Faraday's law of induction), div D(x, t) = (x, t) (Gauss's law), div B(x, t) = 0 (Gauss's law for magnetism),(1)where E, H are the electric and the magnetic fields, D, B are the electric and the magnetic flux densities, J is the electric current density, and is the density of the (externally impressed) electric charge. Constitutive relations, i.e., relations of the form D = D( E, H), B = B( E, H), must accompany
and we shall denote by O the set of triple indices σml where σ ∈ {e, o}, l ∈ N ∪ {0} and m ∈ {0, . . . , l }. For n ∈ O, let A 1n (ξ) = 1
eigenvalues of the first family are diverging to −∞ as l → +∞ (as expected).
Figure 1 .l
1Eigenvalues , l = 1, . . . , 10, as a function of k 2 ∈ (−100, 100).
θ
kj (k) + j(k)(k 2 + 1) − j k √ θ j(k)l(l + 1) j(k) l(l + 1)
F 2 n
2(1) = 0 .
Acknowledgements The authors are thankful to the Departments of Mathematics of the University of Padova and of the National and Kapodistrian University of Athens for the kind hospitality. In addition, they acknowledge financial support from the research project BIRD191739/19 "Sensitivity analysis of partial differential equations in the mathematical theory of electromagnetism" of the University of Padova. The first named author (FF) is grateful for the received support to the UK Engineering and Physical Sciences Research Council through grant EP/T000902/1. The second named author (PDL) is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).According to the previous computations and keeping in mind the case θ = 1, in view of (58) we claim that E 2 (r) = l(l + 1)verify (71) for a = −k 2 /θ and they are solutions of the system (54). Let us show that (71) is satisfied. Note thatwhere the last equality is deduced by (55) with k replaced by k/ √ θ, so the right-hand side of (71) becomesfor a = −k 2 /θ, as claimed. With similar computations one can check that the couple (E 2 , E 3 ) satisfies the systemwhich replaces system (54) used in the case θ = 1. Clearly, another solution (which is divergence-free) is given by (E 2 n , E 3 n ) as defined in (57).
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| []
|
[
"Water flow under rectangular dam *",
"Water flow under rectangular dam *"
]
| [
"A B Bogatyrëv ",
"O A Grigor'ev "
]
| []
| []
| A new analytical method for calculation the characteristics of the flow in porous medium bounded by a rectangular polygonal path is proposed. The method is based on the usage of high genus Riemann theta functions.MSC2010: 30C20, 14H42, 30C30Numerical conformal mappings are commonplace in 2D filtration theory[2]. The majority of model problems being considered deal with flows of a complex kinematic structure, with a lot of singularities (e.g., points where velocity reaches zero or pressure tends to infinity). Therefore, the boundaries of hodograph areas are rich in spikes and right angles. A typical example of such an area is a special case of a circular polygon, namely, a rectangular polygon on a polar grid.A recent trend in filtration theory is to develop numerical techniques which deal specifically with said types of areas[3,4]. These is mostly due to computational problems all the general techniques are ridden with, the most important of those being the crowding phenomenon. However, there already exists an approach addressing precisely these problems[11,12]. The cases treated in[3]are essentially special cases of this general approach.To illustrate the power of this technique, however, it's sufficient to consider an even simpler case -an underground flow in porous media bounded by a watertight surface from below. Despite such a design may seem unrealistic (as a rule, even the simplest models take groundwater afflux into account, optionally along with a watertight structure covering a finite area), our results may have some practical use when rivers with rock formations underneath are dealt with. This toy model also allows a researcher to solve some geometric design problems, which we illustrate in this paper. Other problems, e.g. free boundary problems and finding a depression curve are yet to be addressed. | 10.4213/tm4145 | [
"https://arxiv.org/pdf/1805.03542v1.pdf"
]
| 21,737,542 | 1805.03542 | 2cf235d2acab0225a9c77a659b87dcc6dd1e2fb3 |
Water flow under rectangular dam *
A B Bogatyrëv
O A Grigor'ev
Water flow under rectangular dam *
rectangular polygonsSchwarz-Christoffel integraltheta functionsconformal mappingJaco- bian
A new analytical method for calculation the characteristics of the flow in porous medium bounded by a rectangular polygonal path is proposed. The method is based on the usage of high genus Riemann theta functions.MSC2010: 30C20, 14H42, 30C30Numerical conformal mappings are commonplace in 2D filtration theory[2]. The majority of model problems being considered deal with flows of a complex kinematic structure, with a lot of singularities (e.g., points where velocity reaches zero or pressure tends to infinity). Therefore, the boundaries of hodograph areas are rich in spikes and right angles. A typical example of such an area is a special case of a circular polygon, namely, a rectangular polygon on a polar grid.A recent trend in filtration theory is to develop numerical techniques which deal specifically with said types of areas[3,4]. These is mostly due to computational problems all the general techniques are ridden with, the most important of those being the crowding phenomenon. However, there already exists an approach addressing precisely these problems[11,12]. The cases treated in[3]are essentially special cases of this general approach.To illustrate the power of this technique, however, it's sufficient to consider an even simpler case -an underground flow in porous media bounded by a watertight surface from below. Despite such a design may seem unrealistic (as a rule, even the simplest models take groundwater afflux into account, optionally along with a watertight structure covering a finite area), our results may have some practical use when rivers with rock formations underneath are dealt with. This toy model also allows a researcher to solve some geometric design problems, which we illustrate in this paper. Other problems, e.g. free boundary problems and finding a depression curve are yet to be addressed.
Consider an analytic function f (w) := p + iq with q(w) being harmonic conjugate function to p(w) which is locally constant at the surface of the water resistant surfaces. This function conformally maps the porous layer to a rectangle whose aspect ratio is related to the drop of the water level behind the dam and to the integral water flow Q through it. The latter value is one of the main characteristics of the hydraulic structure:
Q := 0 −∞ −κ dp dy dx = κ 0 −∞ dq dx dx = κ(q(0) − q(−∞)),(1)
which is proportional to the height of the rectangle (provided zero is the horizontal endpoint of the upper basin).
The width of the rectangle is proportional to the drop of the water level behind the dam minus the difference in the levels of the bottom itself. The function f (w) maps stream lines of the flow to the horizontal lines in the rectangle.
The domain filled by the porous medium will be mapped to the rectangle in two steps. First we map it to the upper half plane H := {x ∈ C : Im x > 0}. The second step consists in mapping the half plane to the rectangle with four marked points on the boundary (where the type of the boundary conditions breaks) being mapped to the vertexes of the rectangle. The second map is an elliptic integral, its calculation is a standard procedure -see e.g. [1,5] 3 Space of octagonal domains
First we consider a relatively simple case when the porous medium fills out an octagonal domain shown in Fig. 1 on the right. After we explain the key features of this approach, it will be shown how to increase the complexity of the domain by adding more rectangular steps and/or vertical or horizontal cuts.
We enumerate the vertexes of the octagon in the counterclockwise direction starting from the endpoint of the channel pointing in the eastward direction: +∞ := w+, w1, w2, . . . , w6, w− := −∞. Exactly three of its right angles are intruding, that is equal to 3 2 π if measured from inside the domain. We parametrize the space P of polygons shown in the right Fig. 1 with the lengths Hs of their sides, to which we ascribe signs for technical reasons:
i s Hs := ws+1 − ws, s = 1, 2, . . . , 5.(2)
Obviously, H1 and H4 are negative, the rest of the values Hs are positive. We also consider the positive widths H± of two channels as in Fig. 1
H+ + H1 − H3 > 0,(3)
which eliminate the degenerate polygons having e.g. self intersection of the boundary or vanishing isthmi.
Hyperelliptic curves with six real branch points
The conformal mapping of the upper half plane to any octagon from the space P may be represented by the Schwarz-Christoffel integral:
w(x) := A x (t − x2)(t − x4)(t − x5)dt (t − x+)(t − x−)y(t) , A > 0,(4)
where the real points x * are apriori unknown pre-images of the vertexes w * of the octagon and y 2 (t) = 6 j=1 (t − xj). This integral is naturally defined on a hyperelliptic curve with six real branch points and will be eventually expressed in terms of other function theoretic objects. In this section we list essential facts about these curves.
The survey is a short version of Sects. 2, 4 of [11] and does not contain any proofs, more details may be found in the textbooks like [9,16].
Algebraic model
The double cover of the sphere with six real branch points x1 < x2 < · · · < x5 < x6 is a compact genus two Riemann surface X, the coordinates of points in its affine part satisfying
y 2 = 6 s=1 (x − xs), (x, y) ∈ C 2 .(5)
Cycles, differentials, periods
We fix a special basis in the 1-homology space of the curve X intrinsically related to the latter. The first and second real ovals give us two 1-cycles, a1 and a2 respectively. Both cycles are oriented (up to simultaneous change of sign) as the boundary of a pair of pants obtained by removing all real ovals from the surface. Two remaining cycles b1 and b2 are represented by coreal ovals of the curve oriented so that the intersection matrix takes the canonical form -see Fig. 2.
The reflection of the surface acts on the introduced basis as follows Jas = as,Jbs = −bs, s = 1, 2.
Holomorphic differentials on the curve X take the form
du * = (C1 * x + C2 * )y −1 dx,(7)
with constant values C1 * , C2 * . The basis of differentials dual to the basis of cycles
as duj := δsj; s, j = 1, 2,(8)
determines Riemann period matrix Π with the elements Πsj := bs duj; s, j = 1, 2.
It is a classical result that Π is symmetric and has positive definite imaginary part [9].
From the symmetry properties (6) of the chosen basic cycles it readily follows that:
• Normalized differentials are real, i.e.Jdus = dus, in other words the coefficients C * in the representation (7) are real.
• Period matrix is purely imaginary, therefore we can introduce the symmetric and positive definite real matrix Ω := Im(Π),
Jacobian and Abel-Jacobi map
Definition 1 Given a Riemann period matrix Π, we define the full rank period lattice
L(Π) = ΠZ 2 + Z 2 = H 1 (X,Z) du, du := (du1, du2) t ,
in C 2 and the 4-torus Jac(X) := C 2 /L(Π) known as the Jacobian of the curve X.
This definition depends on the choice of the basis in the lattice H1(X, Z), other choices bring us to isomorphic tori.
It is convenient to represent the points u ∈ C 2 as a theta characteristic [ , ], i.e. a couple of real 2-vectors (columns) 1 , :
u = 1 2 (Π + ).(10)
The points of Jacobian Jac(X) in this notation correspond to two vectors with real entries modulo 2. Second order points of Jacobian are represented as 2 × 2 matrices with binary entries 0, 1.
Definition 2 Abel-Jacobi (briefly: AJ) map u(p) := p p 1 du mod L(Π), p1 := (x1, 0); du := (du1, du2) t ,(11)
is a correctly defined mapping from the surface X to its Jacobian.
From Riemann-Roch formula [9] it easily follows that Abel-Jacobi map is a holomorphic embedding of the curve into its Jacobian. In Sect. 4.4 we give an explicit equation for the image of the genus two curve in its Jacobian. Meanwhile, let us compute the images of the branching points ps = (xs, 0), s = 1, . . . , 6, of the curve X:
p u(p) mod L(Π) [ , ](u(p)) p2 Π 1 /2 10 00 p3 (Π 1 + E 1 )/2 11 00 p4 (Π 2 + E 1 )/2 01 10 p5 (Π 2 + E 1 + E 2 )/2 01 11 p6 (E 1 + E 2 )/2 01 01
where Π s and E s are the s-th columns of the period and identity matrix respectively. One can notice that vector (u(p)) is constant along the real ovals and (u(p)) is constant along the coreal ovals.
Theta functions on genus two curves
Here we give a crash course on the theory of Riemann theta functions adapted to genus two surfaces. Three problems related to conformal mappings will be effectively solved in terms of Riemann theta functions:
• Localization of the curve inside its Jacobian;
• Representation of the 2-sheeted projection of the curve to the sphere;
• Evaluation of the normalized abelian integrals of the second and the third kinds which appear in the canonical decomposition of CS-integral into elementary ones.
Definition 3
Let u ∈ C 2 and Π ∈ C 2×2 be a Riemann matrix, i.e. Π = Π t and Im Π > 0. The theta function of those two arguments is the following Fourier series
θ(u, Π) := m∈Z 2 exp(2πim t u + πim t Πm).
Also considered are theta functions with characteristics obtained by a slight modification of the above.
θ[2 , 2 ](u, Π) := m∈Z 2 exp(2πi(m + ) t (u + ) + πi(m + ) t Π(m + )) = exp(iπ t Π + 2iπ t (u + ))θ(u + Π + , Π), , ∈ R 2 .
The matrix argument Π of theta function is usually omitted when it is clear which matrix is used. Omitted vector argument u is supposed to be zero and the appropriate function of Π is called the theta constant: The convergence of these series follows from Im Π being positive-definite. The series has high convergence rate with well controlled accuracy [10]. Theta function has the following easily checked quasi-periodicity properties with respect to the lattice L(Π) := ΠZ 2 + Z 2 :
θ(u + Πm + m , Π) = exp(−iπm t Πm − 2iπm t u)θ(u, Π), m, m ∈ Z 2 .(12)
Quasi-periodicity of theta with characteristics is easily deduced from the last formula.
Image of Abel-Jacobi map
The locus of a genus 2 curve embedded to its Jacobian may be reconstructed by solving a single equation.
For the localization of the image of AJ embedding for higher genus curves see [13].
Projection to the sphere
Any meromorphic function on the curve may be efficiently calculated via the Riemann theta functions once we know the positions of its zeros and poles (the divisor) [9,8]. Take for instance the degree 2 function x(p) on the hyperelliptic curve (5). The normalization x(p+) := 0; x(p1) := 1; x(p−) := ∞ leads us to a simple expression for the two sheeted projection to the sphere:
x(p) := Const ± θ[35](u ± u + ) ± θ[35](u ± u − ) , u = u(p), u ± := u(p±),(14)
here
Const := θ 2 [35](u − ) θ 2 [35](u + ) .
Third kind abelian integrals
On any Riemann surface there exists a unique abelian differential of the third kind dηpq with two simple poles at the prescribed points p, q only, residues +1, −1 respectively and trivial periods along all a− cycles. There is a closed formula for the appropriate abelian integral, due to Riemann:
ηpq(s) := s * dηpq = log θ[ , ](u(s) − u(p)) θ[ , ](u(s) − u(q)) , p, q, s ∈ X;(15)
with any odd theta characteristic [ , ], say those indicated in the third and the fifth rows of the table in Sect 4.3.
The algorithm of conformal mapping
The SC integral (4) conformally mapping the upper half-plane to an octagon from the space P contains the Schwarz-Christoffel differential of the kind
dw := A (x − x2)(x − x4)(x − x5)dx (x − x+)(x − x−)y , A > 0.(16)
The latter differential can be canonically decomposed into a sum of a-normalized elementary ones:
dw = H− π dηp − Jp − − H+ π dηp + Jp + + C1 · du1 + C2 · du2,(17)
where dηpq is an abelian differential of the third kind with poles at p and q only and residues ±1; du1 and du2 are holomorphic differentials. All the differentials here are real, so are all the coefficients in the decomposition.
In particular, the residues of dw at its poles are related to the widths of appropriate channels -this is where the coefficients H± in formula (17) originate from. It remains to get the efficient evaluation of the SC integral itself and all the auxiliary parameters. Using Riemann's formula (15), we represent the integral of (17) in terms of the Jacobian variables:
w(u) = H− π log θ[3](u − − u) θ[3](u − + u) − H+ π log θ[3](u + − u) θ[3](u + + u) + C1u1 + C2u2.(18)
where u := (u1, u2) t = u(p); u ± := u(p±) = (u ± 1 , u ± 2 ) and the periods matrix for theta is iΩ. As such, we found an efficiently computable expression for the SC integral depending on nine real auxiliary parameters contained
in Ω, u ± and C1, C2.
Mapping octagon to the half plane and back
Here we describe (somewhat sketchy, more details may be found in [11]) the algorithm of the conformal mapping of a fixed octagon from the space P to the upper half-plane and back. Suppose we know the values of all the auxiliary parameters (we postpone their calculation until section 5.2) and a point w * lies in the octagon normalized by the condition w1 = 0, we consider a system of two complex equations
w(u * ) = w * , θ[35](u * , iΩ) = 0,(19)
with respect to the unknown complex 2-vector u * . Following arguments in [11], we assert that system (19) has the unique solution u * with theta characteristic from the block from the reflection principle for the conformal mappings) that the set of two transcendental equations (19) may have many solutions. We use theta characteristic to single out the unique one.
Substituting the solution u * to the right hand side of the expression (14) we get the evaluation at the point w * of the conformal mapping x(w) of the octagon to the half plane with normalization w+, w1, w− → 0, 1, ∞.
In particular, the aspect ratio of the rectangle relating the fall of the water level behind the dam to the total water influx to the upper basin (namely, κ := ∆q/∆p) is the ratio of complete elliptic integrals for the curve with the Weierstrass module
1 < λ := x(p6) = θ 2 [135](u − )θ 2 [356](u + ) θ 2 [135](u + )θ 2 [356](u − ) .
This value may be eventually expressed via the hypergeometric functions F (a, b, c|z) -see e.g. [17] -as
κ = λ − 1 2 F ( 1 2 , 1 2 , 1|1/λ) F ( 1 2 , 1 2 , 1|1 − λ)
.
Conversely, given a point x * in the upper half plane we solve a system of two equations (14) and (13) with respect to a complex 2-vector u * with characteristics from the block we mentioned above. Then substitute this solution to the formula (18) for SC integral in the Jacobian variables to get the image w * of the point x * in the octagon.
Auxiliary parameters
Given the dimensions H * of the octagon, we have to find nine real parameters: the imaginary part Ω of the coefficients C1, C2. Those parameters give a solution to a system of nine real equations we describe below.
dθ[35](u, iΩ) ∧ dw(u) = 0, when u = u(pj), j = 2, 4, 5;
(20)
which means that CS differential dw has (double) zeros at three points pj;
θ[35](u ± , iΩ) = 0,(21)
which means that the points u ± lie on the AJ image of the curve in the Jacobian and finally −2H2 = C1;
2H4 = C2; −2H1 = 2H−(2u − 1 − 1) − 2H+(2u + 1 − 1) + C1Ω11 + C2Ω12; 2H5 = 4H−u − 2 − 4H+u + 2 + C1Ω12 + C2Ω22;(22)
Last four equations specify the side lengths of the polygon and can be deduced by taking the SC integral along the a-and b-cycles with subsequent use of the Riemann reciprocity laws [9,16] for two latter cases. We note that the set of equations (22) immediately gives the unknown coefficients C1, C2 and depends linearly on the remaining unknowns. We also chose one in the stock of these polygons (|w3 − w2| = 2, |w9 − w8| = 4) to illustrate the global behaviour of stream lines under the dam (Fig. 4)
Conclusion
Today there are many techniques for the realistic numerical simulation of sophisticated 3D filtration problems.
However 2D problems make up an essential part of the latter as main underground stream occurs in the direction across the dam. Coarse characteristic of the flow may be found in the framework of planar models.
Figure 1 :
1Left: Cross section of a rectangular dam; Right: Porous layer is a domain of the solution 2 Problem setting We consider a dam of rectangular cross section built on a layer of porous medium (sand or clay) with underlying rocky foundation as in Fig. 1. The body of the dam may have intruding riblets. In accordance with the Darcy law the velocity of the stationary underground water flow is proportional to the gradient of pressure: V = −κ∇p. The continuity condition ∇ · V = 0 implies that the pressure field is harmonic ∆p(w) = 0 in the medium. Equipped with natural boundary conditions, namely the pressure p(w) being locally constant on the bottom of upper/lower basin while its normal derivative being equal to zero on the watertight surfaces of the dam and the rocky foundation, this brings us to the mixed boundary value problem for the pressure p(w).
Figure 2 :
2Canonical basis in homologies of the curve X This curve admits a conformal involution J(x, y) = (x, −y) with six stationary points ps := (xs, 0) and an anticonformal involution (or reflection)J(x, y) = (x,ȳ). The stationary points set of the latter has three components known as real ovals of the curve. Each real oval is an embedded circle and doubly covers exactly one of the segments [x2, x3], [x4, x5], [x6, x1] ∞ of the extended real lineR := R ∪ ∞. The upper half plane H := {Im x > 0} -the image of the octagonal domain -may be lifted to the Riemann surface (5) in exactly two ways. We choose the one with the positive value of the branch y(x) for x ∈ [x6, ∞) on the boundary of the half plane. The pre-images x± of the endpoints of two channels w± are lifted respectively to the points p± on the boundary of embedded upper half plane and lie on the third real oval of the surface (5).
θ[ , ] := θ[ , ](0, Π).
Remark 1
1(i) Theta function with integer characteristics[2 , 2 ] is either even or odd depending on the parity of the inner product 4 t · . In particular, all odd theta constants are zeros.(ii) It is convenient to represent integer theta characteristics as the sums of AJ images of the branch points, keeping only the indices of these points: [sk..l] means the sum modulo 2 of the theta characteristics of points u(ps), u(p k ), . . . , u(p l ), e.g.[35] corresponds to the characteristics10 11 .
Theorem 1 (
1Riemann) A point u of Jacobian lies in the image u(X) of Abel-Jacobi map if and only if θ[35](u) = 0.
Lemma 1 Figure 3 :
13Let the side lengths H±, H1, H2 . . . , H5 satisfy the restrictions (3), then the system of nine equations (20), (21), (22) has a unique real solution with 0 < u + 1 < u − 1 < 1/2 and Ω in the trihedral cone 0 < Ω12 < min(Ω11, Ω22). Sketch of the proof. The existence and the uniqueness of the solution to (20) -(22) in the specified domain follows from the existence and the uniqueness of the conformal mapping of a given octagon to the upper half plane and explicit description of the image of the periods map [11]. The strategy for the solution of the auxiliary set of equations (20)-(22) by Newton method with parametric continuation is discussed in [11]. 6 Applications One can enlarge the stock of polygonal domains at the cost of minor complication of the algorithm. Suppose three equations (20) are not satisfied which means the zeros of the SC differential have moved from the branch points of the curve to the neighbouring real or coreal ovals. Then the composite function w(u(x)) maps the upper half-plane to an octagon with vertical or horizontal cuts emanating from the intruding right angles. The spaces of the 11-gons of this kind -with six right angles, three full angles (cuts) and two zero angles at infinity have dimension nine. The corresponding conformal mappings to the half-plane use the moduli space of real genus two curves with five marked points on their (co)real ovals. The parametric representation of the mapping itself is the same as above but it implicitly depends on extra auxiliary parameters. Now we have three more unknown points in the Jacobian corresponding to zeros of CS differential and described by six real values and six Example of a ribbed channel corresponding to a curve of genus g = 3 more real equations. Three of those equations mean the same as (21): additional points live on the AJ images of (co)real ovals of the curve. Another three equations specify the lengths of three cuts. Note that six additional variables do not explicitly participate in the final parametric representation of the conformal mapping, however they influence the values of the involved auxiliary parameters. We hope that an interested reader will easily reconstruct the aforementioned set of 15 equations (many of those being linear) relating 15 unknowns. Adding new rectangular steps to the domain is more painful for our approach as it rises the genus of the curve bearing the SC integral. Starting from g = 3 we have to characterize the period matrices of hyperelliptic curves, which is the simplest version of the notorious Schottky problem. For the hyperelliptic case the latter has an effective solution [8]: certain even theta constants vanish at the hyperelliptic locus of the Siegel upper half space. Below we present an example of a construction with two riblets (see fig. 3) along with certain numerical results. First of all, by changing the lengths of the cuts and leaving the rest of the dimensions unchanged (namely, H− = 13, H+ = 11, |w10 − w9| = 10, |w7 − w9| = 3, |w6 − w7| = |w5 − w6| = 2, |w4 − w5| = 4, |w4 − w3| = |w1 − w3| = 2), we obtain different values for the ratio κ of the rectangles' sides we map the porous area to. Those values are summarized in Tab. 1.
Figure 4 :
4Global behaviour of the stream lines under the dam with additional riblets. The approach we use reduces finding main characteristics of the underground flow to the solution of a compact set of transcendental and linear equations. This may be applied e.g. to the effective solution of the problems of optimization of the geometry of the dam.
(right). Those seven values satisfy a single linear equation and two more inequalities H+ + H1 − H3 + H5 = H−; H+ + H1 > 0;
Table 1: Aspect ratio κ as a function of the lengths of additional edges.P
P
P
P
P
P
P
P
P
P
P
P
|w 3 − w 2 |
|w 9 − w 8 |
1
2
3
4
1
4.604358822 4.866760353 5.152737177 5.452678199
2
4.647433395 4.909830312 5.195805421 5.495745840
3
4.710043569 4.972431551 5.258403366 5.558342600
4
4.809684090 5.072053634 5.358018625 5.657955418
Our notation of theta characteristic as two column vectors is not universally accepted: sometimes the transposed matrix is used.
× 2 period matrix, the AJ image u ± := u(p±) of two marked points in the Jacobian of the curve, and 2 real
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E T Whittaker, G N Watson, A Course of Modern Analysis. Russia, Moscow GSP-1, ulCambridge University Press8Russian Academy of Sciences [email protected], [email protected]. Whittaker, G.N. Watson, A Course of Modern Analysis -Cambridge University Press, 1996 119991 Russia, Moscow GSP-1, ul. Gubkina 8, Institute for Numerical Mathematics, Russian Academy of Sciences [email protected], [email protected]
| []
|
[
"Morphological Constraints for Phrase Pivot Statistical Machine Translation",
"Morphological Constraints for Phrase Pivot Statistical Machine Translation"
]
| [
"Ahmed El Kholy \nCenter for Computational Learning Systems\nColumbia University\n\n",
"Nizar Habash [email protected] \nCenter for Computational Learning Systems\nColumbia University\n\n"
]
| [
"Center for Computational Learning Systems\nColumbia University\n",
"Center for Computational Learning Systems\nColumbia University\n"
]
| []
| The lack of parallel data for many language pairs is an important challenge to statistical machine translation (SMT). One common solution is to pivot through a third language for which there exist parallel corpora with the source and target languages. Although pivoting is a robust technique, it introduces some low quality translations especially when a poor morphology language is used as the pivot between rich morphology languages. In this paper, we examine the use of synchronous morphology constraint features to improve the quality of phrase pivot SMT. We compare hand-crafted constraints to those learned from limited parallel data between source and target languages. The learned morphology constraints are based on projected alignments between the source and target phrases in the pivot phrase table. We show positive results on Hebrew-Arabic SMT (pivoting on English). We get 1.5 BLEU points over a phrase pivot baseline and 0.8 BLEU points over a system combination baseline with a direct model built from parallel data. | null | [
"https://www.aclanthology.org/2015.mtsummit-papers.9.pdf"
]
| 18,467,827 | 1609.03376 | fd0e0a9999448bc893c21a5e02598bdcad03b3f4 |
Morphological Constraints for Phrase Pivot Statistical Machine Translation
Ahmed El Kholy
Center for Computational Learning Systems
Columbia University
Nizar Habash [email protected]
Center for Computational Learning Systems
Columbia University
Morphological Constraints for Phrase Pivot Statistical Machine Translation
Computer Science, New York University Abu Dhabi
The lack of parallel data for many language pairs is an important challenge to statistical machine translation (SMT). One common solution is to pivot through a third language for which there exist parallel corpora with the source and target languages. Although pivoting is a robust technique, it introduces some low quality translations especially when a poor morphology language is used as the pivot between rich morphology languages. In this paper, we examine the use of synchronous morphology constraint features to improve the quality of phrase pivot SMT. We compare hand-crafted constraints to those learned from limited parallel data between source and target languages. The learned morphology constraints are based on projected alignments between the source and target phrases in the pivot phrase table. We show positive results on Hebrew-Arabic SMT (pivoting on English). We get 1.5 BLEU points over a phrase pivot baseline and 0.8 BLEU points over a system combination baseline with a direct model built from parallel data.
Introduction
One of the main challenges in statistical machine translation (SMT) is the scarcity of parallel data for many language pairs especially when the source and target languages are morphologically rich. A common SMT solution to the lack of parallel data is to pivot the translation through a third language (called pivot or bridge language) for which there exist abundant parallel corpora with the source and target languages. The literature covers many pivoting techniques. One of the best performing techniques, phrase pivoting (Utiyama and Isahara, 2007), builds an induced new phrase table between the source and target. One of the main issues of this technique is that the size of the newly created pivot phrase table is very large. Moreover, many of the produced phrase pairs are of low quality which affects the translation choices during decoding and the overall translation quality.
In this paper, we focus on improving phrase pivoting. We introduce morphology constraint scores which are added to the log linear space of features in order to determine the quality of the pivot phrase pairs. We compare two methods of generating the morphology constraints. One method is based on hand-crafted rules relying on the authors knowledge of the source and target languages; while in the other method, the morphology constraints are induced from available parallel data between the source and target languages which we also use to build a direct translation model. We then combine both the pivot and direct models to achieve better coverage and overall translation quality. We show positive results on Hebrew-Arabic SMT. We get 1.5 BLEU points over a phrase-pivot baseline and 0.8 BLEU points over a system combination baseline with a direct model built from given parallel data.
Next, we briefly discuss some related work. In Section 3, we review the best performing pivoting strategy and how we use it. In Section 4, we discuss the linguistic differences among Hebrew, Arabic, and the pivot language, English. This is followed by our approach to using morphology constraints in Section 5. We finally present our experimental results in Section 6 and a case study in Section 7.
Related Work
Many researchers have investigated the use of pivoting (or bridging) approaches to solve the data scarcity issue (Utiyama and Isahara, 2007;Wu and Wang, 2009;Khalilov et al., 2008;Bertoldi et al., 2008;Habash and Hu, 2009). The main idea is to introduce a pivot language, for which there exist large source-pivot and pivot-target bilingual corpora. Pivoting has been explored for closely related languages (Hajič et al., 2000) as well as unrelated languages (Koehn et al., 2009;Habash and Hu, 2009). Many different pivot strategies have been presented in the literature. The following three are the most common. The first strategy is the sentence translation technique in which we first translate the source sentence to the pivot language, and then translate the pivot language sentence to the target language (Khalilov et al., 2008). The second strategy is based on phrase pivoting (Utiyama and Isahara, 2007;Cohn and Lapata, 2007;Wu and Wang, 2009). In phrase pivoting, a new source-target phrase table (translation model) is induced from sourcepivot and pivot-target phrase tables. Lexical weights and translation probabilities are computed from the two translation models. The third strategy is to create a synthetic source-target corpus by translating the pivot side of source-pivot corpus to the target language using an existing pivot-target model (Bertoldi et al., 2008). In this paper, we use the phrase pivoting approach, which has been shown to be the best with comparable settings (Utiyama and Isahara, 2007).
There has been recent efforts in improving phrase pivoting. One effort focused on improving alignment symmetrization targeting pivot phrase systems (El Kholy and Habash, 2014). In another recent effort, Multi-Synchronous Context-free Grammar (MSCFG) is leveraged to triangulate source-pivot and pivot-target synchronous Context-free Grammar (SCFG) rule tables into a source-target-pivot MSCFG rule table that helps in remembering the pivot during decoding. Also, pivot LMs are used to assess the naturalness of the derivation (Miura et al., 2015).
In our own previous work, we demonstrated quality improvement using connectivity strength features between the source and target phrase pairs in the pivot phrase table (El Kholy et al., 2013). These features provide quality scores based on the number of alignment links between words in the source phrase to words of the target phrase. In this work, we extend on the connectivity scores with morphological constraints through which we provide quality scores based on the morphological compatibility between the connected/aligned source and target words.
Since both Hebrew and Arabic are morphologically rich, we should mention that there has been a lot of work on translation to and from morphologically rich languages (Yeniterzi and Oflazer, 2010;Elming and Habash, 2009;Habash and Sadat, 2006;Kathol and Zheng, 2008). Most of these efforts are focused on syntactic and morphological processing to improve the quality of translation.
Until recently, there has not been much parallel Hebrew-English and Hebrew-Arabic data (Tsvetkov and Wintner, 2010), and consequently little work on Hebrew-English and Hebrew-
Training Corpora
Phrase Arabic SMT. Lavie et al. (2004) built a transfer-based translation system for Hebrew-English and so did Shilon et al. (2012) for translation between Hebrew and Arabic. Our previous work discussed above (El Kholy et al., 2013) was demonstrated on Hebrew-Arabic with English pivoting.
Phrase Pivoting
In this section, we review the phrase pivoting strategy in detail as we describe how we built our baseline for Arabic-Hebrew via pivoting on English. We also discuss how we overcome the large expansion of source-to-target phrase pairs in the process of creating a pivot phrase table. In phrase pivoting (which is sometimes called triangulation or phrase table multiplication), we train a Hebrew-Arabic and an English-Arabic translation models, such as those used in the sentence pivoting technique. Based on these two models, we induce a new Hebrew-Arabic translation model. Since our models are based on a Moses phrase-based SMT system , we use the standard set of phrase-based translation probability distributions. 1 We follow Utiyama and Isahara (2007) in computing the pivot phrase pair probabilities. The following are the set of equations used to compute the lexical probabilities (p w ) and the phrase translation probabilities (φ):
φ(h|a) = e φ(h|e)φ(e|a) φ(a|h) = e φ(a|e)φ(e|h) p w (h|a) = e p w (h|e)p w (e|a) p w (a|h) = e p w (a|e)p w (e|h)
Above, h is the Hebrew source phrase; e is the English pivot phrase that is common in both Hebrew-English translation model and English-Arabic translation model; and a is the Arabic target phrase. We also build a Hebrew-Arabic reordering table using the same technique but we compute the reordering probabilities in a similar manner to Henriquez et al. (2010).
Filtering for Phrase Pivoting As discussed earlier, the induced Hebrew-Arabic phrase and reordering tables are very large. first rank all the candidates based on the log linear scores computed from the phrase translation probabilities and lexical weights multiplied by the optimized decoding weights then we pick the top [n] pairs. In our experiments, we pick the top 1000 pairs for pivoting.
Linguistic Comparison
In this section we present the challenges of preprocessing Arabic, Hebrew, and English, and how we address them. Both Arabic and Hebrew are morphologically complex languages. One aspect of Arabic's complexity is its various attachable clitics and numerous morphological features (Habash, 2010). Clitics include conjunction proclitics, e.g., + w+ 3 'and', prepositional proclitics, e.g., + l+ 'to/for', the definite article + Al+ 'the', and the class of pronominal enclitics, e.g., + +hm 'their/them'. All of these clitics are separate words in English. Beyond the clitics, Arabic words inflect for person, gender, number, aspect, mood, voice, state and case. Additionally, Arabic orthography uses optional diacritics for short vowels and consonant doubling. This, together with Arabic's morphological richness, leads to a high degree of ambiguity: about 12 analyses per word, typically corresponding to two lemmas on average (Habash, 2010). We follow El Kholy and Habash (2010) and use the PATB tokenization scheme (Maamouri et al., 2004) in our experiments. The PATB scheme separates all clitics except for the determiner clitic Al+(DET). We use MADA v3.1 (Habash and Rambow, 2005;) to tokenize the Arabic text. We only evaluate on detokenized and orthographically correct (enriched) output following the work of El Kholy and Habash (2010). Similar to Arabic, Hebrew poses computational processing challenges typical of Semitic languages (Itai and Wintner, 2008;Shilon et al., 2012). Hebrew orthography also uses optional diacritics and its morphology inflects for gender, number, person, state, tense and definiteness. Furthermore, Similar to Arabic, Hebrew has a set of attachable clitics, e.g., conjunctions (such as + ו w+ 4 'and'), prepositions (such as + ב b+ 'in'), the definite article (+ ה h+ 'the'), or pronouns (such as +ה +hm 'their'). These issues contribute to a high degree of ambiguity that is a challenge to translation from Hebrew to English or to any other language. We follow Singh and Habash (2012)'s best preprocessing setup which utilized a Hebrew tagger (Adler, 2007) and produced a tokenization scheme that separated all clitics.
English, our pivot language, is quite different from both Arabic and Hebrew. English is poor in morphology and barely inflects for number and tense, and for person in a limited context. English preprocessing simply includes down-casing, separating punctuation and splitting off "'s".
Approach
One of the main challenges in phrase pivoting is the very large size of the induced phrase table. It becomes even more challenging if either the source or target language is morphologically rich. The number of translation candidates (fanout) increases due to ambiguity and richness which in return increases the number of combinations between source and target phrases. Since the only criteria of matching between the source and target phrase is through a pivot phrase, many of the induced phrase pairs are of low quality. These phrase pairs unnecessarily increase the search space and hurt the overall quality of translation. A basic solution to the combinatorial expansion is to filter the phrase pairs used in pivoting based on log-linear scores as discussed in Section 3, however, this doesn't solve the low quality problem.
Similar to factored translation models where linguistic (morphology) features are augmented to the translation model to improve the translation quality, our approach to address the quality problem is based on constructing a list of synchronous morphology constraints between the source and target languages. These constraints are used to generate scores to determine the quality of pivot phrase pairs. However, unlike factored models, we do not use the morphology in generation and the morphology information comes completely from external resources. In addition, since we work in the pivoting space, we only apply the morphology constraints to the connected words between the source and target languages through the pivot language. This guarantees a fundamental level of semantic equivalence before applying the morphology constraints especially if there is distortion between source and target phrases.
We build on our approach in El Kholy et al. (2013) where we introduced connectivity strength features between the source and target phrase pairs in the pivot phrase table. These features provide quality scores based on the number of alignment links between words in the source phrase and words in the target phrase. The alignment links are generated by projecting the alignments of the source-pivot phrase pairs and the pivot-target phrase pairs used in pivoting. We use the same concept but instead of using the lexical mapping between source and target words, we compute quality scores based on the morphological compatibility between the connected source and target words.
To choose which morphological features to work with, we performed an automatic error analysis on the output of the phrase-pivot baseline system. We did the analysis using AMEANA (El Kholy and Habash, 2011), an open-source error analysis tool for natural language processing tasks targeting morphologically rich languages. We found that the most problematic morphological features in the Arabic output are gender (GEN), number (NUM) and determiner (DET). We focus on those features in addition to (POS) in our experiments.
Next, we present our approach to generating the morphology constraint features using hand-crafted rules and compare this approach with inducing these constraints from Hebrew-Arabic parallel data.
Rule-based Morphology Constraints
Our rule-based morphology constraint features are basically a list of hand-crafted mappings of the different morphological features between Hebrew and Arabic. Since both languages are morphologically rich as explained in Section 4, it is straightforward to produce these mappings for GEN, NUM and DET. Note, however, that we also account for ambiguous cases; e.g., feminine gender in Arabic can map to words with ambiguous gender in Hebrew. We additionally use different POS tag sets for Arabic (47 tags) and Hebrew (25 tags) and in many cases one Hebrew tag can map to more than one Arabic tag; for example, three Arabic noun tags abbrev, noun and noun prop map to two Hebrew tags feminine, masculine noun. 5 Table 2 shows a sample of the morphological mappings between Arabic and Hebrew.
After building the morphological features mappings, we use them to judge the quality of a given phrase pair in the phrase pivot model. We add two scores W s and W t to the log linear space. Given a source-target phrase pairs,t and a word projected alignment a between the source word positions i = 1, ..., n and the target word positions j = 1, ..., m, W s and W t are defined in equations 1 and 2. F is the set of morphological features (we focus on GEN, NUM, DET and POS). M f is the hand-crafted rules mapping between Arabic and Hebrew feature values of feature f ∈ F . In case of ambiguity for a given feature; for example, a word's gender being masculine or feminine, we use the maximum likelihood value of this feature given the word. M LE f (i) is the maximum likelihood feature value of feature f for the source word at position i, and M LE f (j) is the maximum likelihood feature value of feature f for the target word at position j. The maximum likelihood feature values for Hebrew were computed from the Hebrew side of the training data. As for Arabic, the maximum likelihood feature values were computed from the Arabic side of the training data in addition to Arabic Gigaword corpus, which was used in creating the language model (more details in Section 6.1).
Ws = 1 |F | ∀f ∈F ∀(i,j)∈a 1 n [(M LE f (i), M LE f (j)) ∈ M f ] (1) Wt = 1 |F | ∀f ∈F ∀(i,j)∈a 1 m [(M LE f (i), M LE f (j)) ∈ M f ](2)
Induced Morphology Constraints
In this section, we explain our approach in generating morphology constraint features from a given parallel data between source and target languages. Unlike the rule-based approach we build a translation model between the source and target morphological features and we use the morphology translation probabilities as metric to judge a given phrase pair in the pivot phrase table. For the automatically induced constraints, we jointly model mapping between conjunctions of features attached to aligned words rather than tallying each feature match independently. Writing good manual rules for such feature conjunction mappings would be more difficult. Table 3 shows some examples of mapping (GEN), number (NUM) and determiner (DET) in Hebrew to their equivalent in Arabic and their respective bi-directional scores. 0.0047 0.5000 Table 3: Examples of induced morphology constraints for (GEN), number (NUM) and determiner (DET) and their respective scores.
Hebrew (H) Arabic (A)
P F C (A|H) P F C(
As in rule-based approach, we add two scores W s and W t to the log linear space which are defined in equations 3 and 4. P F C is the conditional morphology probability of a given feature combination (F C) value. Similar to rule-based morphology constraints, we resort to the maximum likelihood value of a feature combination when the values are ambiguous. M LE F C (i) is the maximum likelihood feature combination (F C) value for the source word at position i while M LE F C (j) is the maximum likelihood feature combination (F C) value for the target word at position j.
Ws = 1 n ∀(i,j)∈a PF C (M LEF C (i)|M LEF C (j)) (3) Wt = 1 m ∀(i,j)∈a PF C (M LEF C (j)|M LEF C (i))(4)
Model Combinations
Since we use parallel data to induce the morphology constraints, it would make sense to measure the effect of combining (a) the pivot model with added morphology constraints, and (b) the direct model trained on the parallel data used to induce the morphology constraints. We perform the combination using Moses' phrase table combination techniques. Translation options are collected from one table, and additional options are collected from the other tables. If the same translation option (in terms of identical input phrase and output phrase) is found in multiple tables, separate translation options are created for each occurrence, but with different scores (Koehn and Schroeder, 2007). We show results over a learning curve in Section 6.5.
Experiments
In this section, we present a set of experiments comparing the use of rule-based versus induced morphology constraint features in phrase-pivot SMT as well as model combination to improve Hebrew-Arabic pivot translation quality.
Experimental Setup
In our pivoting experiments, we build two SMT models; one model to translate from Hebrew to English, and another model to translate from English to Arabic. The English-Arabic parallel corpus is about (≈ 60M words) and is available from LDC 6 and GALE 7 constrained data. The Hebrew-English corpus is about (≈ 1M words) and is available from sentence-aligned corpus produced by Tsvetkov and Wintner (2010). For the direct Hebrew-Arabic SMT model, we use a TED parallel corpus of about (≈ 2M words) (Cettolo et al., 2012). Word alignment is done using GIZA++ (Och and Ney, 2003). For Arabic language modeling, we use 200M words from the Arabic Gigaword Corpus (Graff, 2007) together with the Arabic side of our training data. We use 5-grams for all language models (LMs) implemented using the SRILM toolkit (Stolcke, 2002).
All experiments are conducted using the Moses phrase-based SMT system . We use MERT (Och, 2003) for decoding weight optimization. Weights are optimized using a tuning set of 517 sentences developed by Shilon et al. (2010).
We use a maximum phrase length of size 8 across all models. We report results on a Hebrew-Arabic development set (Dev) of 500 sentence with a single reference and an evaluation set (Test) of 300 sentences with three references developed by Shilon et al. (2010). We evaluate using BLEU-4 (Papineni et al., 2002).
Baselines
We compare the performance of adding the connectivity strength features (+Conn) to the phrase pivoting SMT model (Phrase Pivot) and building a direct SMT model using all parallel He-Ar corpus available. The results are presented in Table 4. Consistently with our previous effort (El Kholy et al., 2013), the performance of the phrase-pivot model improves with the connectivity strength features. While the direct system is better than the phrase pivot model in general, the combination of both models leads to a high performance gain of 1.7/4.4 BLEU points in Dev/Test over the best performers of both the direct and phrase-pivot models. Table 5: Morphology constraints results. The "Single" columns show the results of a single model of either the direct model or the phrase pivoting models with additional morphological constraints features. The "Combined" show the results of system combination between the direct model and the different phrase pivoting models. In the first row, the "Combined" results are not applicable for the direct model. (*) marks a statistically significant result against both the direct and phrase-pivot baseline.
Model
In this experiment, we show the performance of adding hand-crafted morphology constraints (+Morph Rules) to determine the quality of a given phrase pair in the phrase-pivot translation model. The third row in Table 5 shows that although the rules are based on a one-toone mapping between the different morphological features, the translation quality is improved over the baseline phrase-pivot model by 0.5/0.8 BLEU points in Dev/Test sets.
As expected, the system combination of the pivot model with the direct model improves the overall performance but the gain we get from the morphology constraints only appears in the Dev set with 0.8 BLEU points, and not much in the Test set.
Induced Morphology Constraints
In this experiment, we measure the effect of using induced morphology constraints (+Morph Auto) on MT quality. The last row in Table 5 shows that the induced morphology constraints improve the results over the baseline phrase-pivot model by 0.5/1.5 BLEU points in Dev/Test sets and over the Rule-based morphology constraints by 0.7 BLEU points in the Test set. Similar to the Rule-based constraints, the performance did not improve compared to the direct model in the Dev set; but, again, the Test set showed a great improvement of 1.5 and 1.2 BLEU points over the pivot and direct models, respectively. Also the system combination of the pivot model with the direct model improves the overall performance. The model using induced morphological features is the best performer with an increase in the performance gain by 1.0/0.8 BLEU points in Dev/Test sets. This shows that the benefit we get from the induced morphology constraints were not diluted when we do the model combination given the fact that the constraints were induced from the parallel data to start with.
It is important to note here that the induced morphology constraints outperformed the rule-based constraints across all settings. This shows that the complex morphology constraints extracted from the parallel data provide knowledge that can not be covered by simple linguistic rules. However, the simple rule-based approach comes in handy when there is no data between the source and target languages.
Learning Curve
In this experiment, we examine the effect of using less data in inducing morphology constraints rules and the overall performance when we combine systems. Table 6 shows the results on a learning curve of 100% (2M words), 25% (500K words) and 6.25% (125K words) of the parallel Hebrew-Arabic corpus.
As expected, The system combination between the direct translation models and the phrase-pivot translation model leads to an improvement in the translation quality across the learning curve even when there is small amount of parallel corpora. Despite the weak performance (2.7 BLEU) of the direct system built on 6.25% of the parallel Hebrew-Arabic corpus, the system combination leads to 1.4 BLEU points gain.
An interesting observation from the results is that we always get a performance gain from the induced morphology constrains across all settings. This shows that the system combination helps in adding more lexical translation choices while the constraints help in a different dimension, which is selecting the best phrase pairs from the pivot system.
Case Study
In this section we consider an example from our Dev set that captures many of the patterns and themes in the evaluation. Table 7 shows a Hebrew source sentence and its Arabic reference. This is followed by the output from the pivot system, the direct system, the Phrase Pivot+Conn+Morph Auto system and the combined system. Two particular aspects should be noted. First is the complementary lexical coverage of the direct and pivot systems. This is seen in how one of each covers half of the phrase middlemen and traders. The combined system captures both. Secondly, the gender, number and tense of the main verb prove challenging in many ways (and this is an issue for a majority of the sentences in the Dev set). The Hebrew verb in the present tense is masculine and plural; and naturally follows the subject. The Arabic reference verb appears at the beginning of the sentence, in which location it only agrees with the subject in gender (while number is singular). Arabic Verbs in SVO order agree in gender and number. All the MT systems we compare leave the verb after the subject. The direct, Phrase Pivot+Conn+Morph Auto, and combination systems get the number and gender correctly; however, the direct and combined system make the verb tense past. The Phrase Pivot+Conn+Morph Auto example highlights the value of morphology constraints; but the example points out that they sometimes are hard to evaluate automatically, since there are morphosyntactically allowable forms that do not match the translation references.
Conclusion and Future Work
In this paper, we presented the use of synchronous morphology constraint features based on hand-crafted rules compared to rules induced from parallel data to improve the quality of phrase-pivot based SMT. We show that the two approaches lead to an improvement in the translation quality. The induced morphology constraints approach is a better performer, however, it relies on the fact there is a parallel corpus between source and target languages. We show positive results on Hebrew-Arabic SMT. We get 1.5 BLEU points over phrase-pivot baseline and 0.8 BLEU points over system combination baseline with direct model built from given parallel data.
In the future, we plan to work on reranking experiments as a post-translation step based on morphosyntactic information between source and target languages. We also plan to work on word reordering between morphologically rich language to maintain the relationship between the word order and the morphosyntactic agreement in the context of phrase pivoting.
6
LDC Catalog IDs: LDC2005E83, LDC2006E24, LDC2006E34, LDC2006E85, LDC2006E92, LDC2006G05, LDC2007E06, LDC2007E101, LDC2007E103, LDC2007E46, LDC2007E86, LDC2008E40, LDC2008E56, LDC2008G05, LDC2009E16, LDC2009G01. 7 Global Autonomous Language Exploitation, or GALE, was a DARPA-funded research project.
Table Translation Table 1 :
Translation1Translation Models Phrase Table comparison in terms of number of lines and sizes.Model
Size
# Phrase Pairs
Size
Hebrew-English
≈1M words
3,002,887
327MB
English-Arabic
≈60M words
111,702,225
14GB
Pivot Hebrew-Arabic
N/A
> 30 Billion
≈2.5TB
Table 1
1shows the amount of parallel corpora used to train
Table 2 :
2Rule-based mapping between Arabic and Hebrew morphological features. Each feature value in Arabic can map to more than one feature value in Hebrew.
Table 4 :
4Comparing phrase pivoting SMT with connectivity strength features, direct SMT and the model combination. The results show that the best performer is the model combination in Dev and Test sets.6.3 Rule-based Morphology Constraints
Model
Dev
Test
Single Combined Single Combined
Direct
9.7
n/a
20.4
n/a
Phrase Pivot+Conn
9.1
11.4
20.1
24.5
Phrase Pivot+Conn+Morph Rules
9.6
12.2*
20.9*
24.6
Phrase Pivot+Conn+Morph Auto
9.6
12.4*
21.6*
25.3*
Table 6 :
6Learning curve results of 100% (2M words), 25% (500K words) and 6.25% (125K
words) of the parallel Hebrew-Arabic corpus.
Hebrew Source .
Sourceהמחירי על בפומבי לדבר מסרבי והסוחרי המתווכי the+middlemen and+the+traders refuse[m.p.] to+speak publicly about the+prices Arabic Reference refuse[m.s.] the+middlemen and+the+traders the+speaking publicly about the+prices Phrase Pivot+Conn והסוחרי middlemen והסוחרי refuse[m.s.] the+speaking publicly about the+prices Direct המתווכי המתווכי and+the+traders refused[m.p.] the+speaking upon the+public about the+prices Phrase Pivot+Conn+ והסוחרי Morph Auto the+middlemen והסוחרי refuse[m.p.] the+speaking publicly about the+prices Direct+Phrase Pivot+ Conn+Morph Auto middlemen and+the+traders refused[m.p.] the+speaking publicly about the+prices
Table 7 :
7Translation examples.
Arabic transliteration throughout the paper is presented in the Habash-Soudi-Buckwalter scheme(Habash et al., 2007).4 The following Hebrew 1-to-1 transliteration is used(in Hebrew alphabetical order): abgdhwzxTiklmns'pcqršt. All examples are undiacritized and final forms are not distinguished from non-final forms. Proceedings of MT Summit XV, vol.1: MT Researchers' Track Miami, Oct 30 -Nov 3, 2015 | p. 107
Please refer to) for a complete set of Arabic POS tag set and(Adler, 2007) for Hebrew POS tag set. Proceedings of MT Summit XV, vol.1: MT Researchers' Track Miami, Oct 30 -Nov 3, 2015 | p. 108
Proceedings of MT Summit XV, vol.1: MT Researchers' Track Miami, Oct 30 -Nov 3, 2015 | p. 109
Proceedings of MT Summit XV, vol.1: MT Researchers' Track Miami, Oct 30 -Nov 3, 2015 | p. 111
Proceedings of MT Summit XV, vol.1: MT Researchers' Track Miami, Oct 30 -Nov 3, 2015 | p. 112
Proceedings of MT Summit XV, vol.1: MT Researchers' Track Miami, Oct 30 -Nov 3, 2015 | p. 113
Proceedings of MT Summit XV, vol.1: MT Researchers' Track Miami, Oct 30 -Nov 3, 2015 | p. 114
AcknowledgmentsThe work presented in this paper was possible thanks to a generous Google Research Award. We would like to thank Reshef Shilon and Shuly Winter for helpful discussions and support with processing Hebrew. We also thank the anonymous reviewers for their insightful comments.
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"Periods of tropical K3 hypersurfaces",
"Periods of tropical K3 hypersurfaces",
"Periods of tropical K3 hypersurfaces",
"Periods of tropical K3 hypersurfaces"
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"Yuto Yamamoto ",
"Yuto Yamamoto "
]
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| Let ∆ be a smooth reflexive polytope in dimension 3 and f be a tropical polynomial whose Newton polytope is the polar dual of ∆. One can construct a 2-sphere B equipped with an integral affine structure with singularities by contracting the tropical K3 hypersurface defined by f . We write the complement of the singularity as ι : B 0 ֒→ B, and the local system of integral tangent vectors on B 0 as T Z . Let further Y be an anti-canonical hypersurface of the toric variety associated with the normal fan of ∆, and Pic(Y ) amb be the sublattice of the Picard group of Y coming from the ambient space. In this article, we give a primitive embedding Pic(Y ) amb ֒→ H 1 (B, ι * T Z ) that preserves the pairing, and compute the radiance obstruction of B, which sits in the subspace generated by the image of Pic(Y ) amb . | null | [
"https://arxiv.org/pdf/1806.04239v1.pdf"
]
| 119,619,684 | 1806.04239 | ac8f97c583bb66aefb35d422841c9653dfdb6fb2 |
Periods of tropical K3 hypersurfaces
11 Jun 2018
Yuto Yamamoto
Periods of tropical K3 hypersurfaces
11 Jun 2018arXiv:1806.04239v1 [math.AG]
Let ∆ be a smooth reflexive polytope in dimension 3 and f be a tropical polynomial whose Newton polytope is the polar dual of ∆. One can construct a 2-sphere B equipped with an integral affine structure with singularities by contracting the tropical K3 hypersurface defined by f . We write the complement of the singularity as ι : B 0 ֒→ B, and the local system of integral tangent vectors on B 0 as T Z . Let further Y be an anti-canonical hypersurface of the toric variety associated with the normal fan of ∆, and Pic(Y ) amb be the sublattice of the Picard group of Y coming from the ambient space. In this article, we give a primitive embedding Pic(Y ) amb ֒→ H 1 (B, ι * T Z ) that preserves the pairing, and compute the radiance obstruction of B, which sits in the subspace generated by the image of Pic(Y ) amb .
Introduction
Let M be a free Z-module of rank 3 and N := Hom(M, Z) be the dual lattice. We set M R := M ⊗ Z R and N R := N ⊗ Z R = Hom(M, R). Let ∆ ⊂ M R be a smooth reflexive polytope of dimension 3, and∆ ⊂ N R be the polar polytope of ∆. Let further Σ andΣ be the normal fans to ∆ and∆ respectively. We choose a refinementΣ ′ ⊂ M R ofΣ such that the primitive generator of any 1-dimensional cone inΣ ′ is contained in ∆ ∩ M.
Let A ⊂ N denote the subset consisting of all vertices of∆ and 0 ∈ N. We consider a tropical Laurent polynomial f (x) = max n∈A {a(n) + n 1 x 1 + n 2 x 2 + n 3 x 3 } ,
(1.1) such that the function A → R, n → a(n) (1.2) induces a central subdivision of∆, i.e., every maximal dimensional simplex of the subdivision has the origin 0 ∈ N as its vertex. Let V (f ) be the tropical hypersurface defined by f in the tropical toric variety associated withΣ ′ . See Section 3.1 for the definition of tropical toric varieties. We can construct a 2-sphere B equipped with an integral affine structure with singularities by contracting V (f ). See Section 4 for details about the construction. The same construction has already been performed in Gross-Siebert program [Gro05], [GS06]. There is also another construction by Haase and Zharkov, which was discovered independently [HZ02]. It is also known that maximally degenerating families of complex K3 surfaces with Ricci-flat Kähler metrics converge to 2-spheres with integral affine structures with singularities in the Gromov-Hausdorff limit [GW00].
Let ι : B 0 ֒→ B denote the complement of singularities of B. Let further T Z be the local system on B 0 of integral tangent vectors. The cohomology group H 1 (B, ι * T Z ) has the cup product
∪ : H 1 (B, ι * T Z ) ⊗ H 1 (B, ι * T Z ) → H 2 (B, ι * ∧ 2 T Z ) ∼ = Z (1.3)
induced by the wedge product, where the last isomorphism is determined by choosing an orientation of B. Let Y be an anti-canonical hypersurface of the complex toric variety X Σ associated with Σ, and Pic(Y ) amb := Im (Pic(X Σ ) ֒→ Pic(Y )) (1.4) be the sublattice of Pic(Y ) coming from the Picard group of the ambient space. The first main result of this paper is the following: that preserves the pairing.
We also compute the radiance obstruction of B. Radiance obstructions are invariants of the integral affine manifolds, which were introduced in [GH84]. See Section 2 for its definition. Let Σ(1) be the set of 1-dimensional cones of Σ and D ρ be the restriction to Y of the toric divisor on X Σ corresponding to ρ ∈ Σ(1). There is a one-to-one correspondence between Σ(1) and the set of vertices of∆. We write the vertex of∆ corresponding to ρ ∈ Σ(1) as n ρ ∈∆∩N. (1.6)
It is known that the valuation of the j-invariant of an elliptic curve over a non-archimedean valuation field coincides with the cycle length of the tropical elliptic curve obtained by tropicalization [KMM08], [KMM09]. Theorem 1.1 and Theorem 1.2 are a generalization of this to the case of K3 hypersurfaces.
The definition of periods for general tropical curves was given in [MZ08]. It was also shown in [Iwa10] that the leading term of the period map of a degenerating family of Riemann surfaces is given by the period of the tropical curve obtained by tropicalization.
The period map is approximated by Schmid's nilpotent orbit [Sch73] in the limit to the degeneration point. The leading term of the nilpotent orbit is determined by the monodromy around the degeneration point. It was also shown in [GS10] that the wedge product of the radiance obstruction corresponds to the monodromy operator around the degeneration point in the case of Calabi-Yau varieties. Hence, we can see that the radiance obstruction gives the leading term of the period map of the corresponding family of Calabi-Yau varieties.
The organization of this paper is as follows: In Section 2, we recall the definitions of integral affine manifolds and their radiance obstructions. We also recall the definition of integral affine manifolds with singularities and we define their radiance obstructions. In Section 3, we recall some notions of tropical geometry, such as tropical toric varieties and tropical modifications. In Section 4, we explain the details about how to construct integral affine spheres with singularities from tropical K3 hypersurfaces. In Section 5, we discuss how dispersing singular points piling at one point affects the cohomology group H 1 (B, ι * T Z ) and the radiance obstruction c B . The results in Section 5 will be used for proofs of the main theorems. In Section 6, we give proofs of Theorem 1.1 and Theorem 1.2. In Section 7, we discuss the relation with asymptotic behaviors of period maps of complex K3 hypersurfaces.
Acknowledgment: I am most grateful to my advisor Kazushi Ueda for his encouragement and helpful advice. In particular, the problem of computing radiance obstructions of tropical K3 hypersurfaces was suggested by him. Most parts of this work was done during a visit to Yale University. I thank Sam Payne for his encouragement on this project, valuable comments on an earlier draft of this paper, and the financial support for the visit. I also stayed at University of Geneva and attended the exchange student program "Master Class in Geometry, Topology and Physics" by NCCR SwissMAP in the academic year 2016/2017. During the stay, I learned a lot about tropical geometry from Grigory Mikhalkin. I thank him for many helpful discussions and sharing his insights and intuitions. Many ideas of Section 4 of this article stem from discussions with him. I also thank NCCR SwissMAP for the financial support and the opportunity to study at University of Geneva. I am also grateful to Mark Gross for letting me know the work by Haase and Zharkov [HZ02]. This research was supported by Grant-in-Aid for JSPS Research Fellow (18J11281) and the Program for Leading Graduate Schools, MEXT, Japan. Let B be an integral affine manifold. We give an affine bundle structure to the tangent bundle T B of B as follows: For each U i and x ∈ U i , we set an affine isomorphism
θ i,x : T x B → M R , v → ψ i (x) + dψ i (x)v, (2.1)
and define an affine trivializations by
θ i : T U i → U i × M R , (x, v) → (x, θ i,x (v)), (2.2)
where v ∈ T x B. This gives an affine bundle structure to T B. We write T B with this affine bundle structure as T aff B. Let T Z be the local system on B of integral tangent vectors. We set T := T Z ⊗ Z R.
Definition 2.2. We choose an sufficiently fine open covering U := {U i } i of B so that there is a flat section s i ∈ Γ(U i , T aff B) for each U i . When we set c B ((U i , U j )) := s j − s i for each 1simplex (U i , U j ) of U, the element c B becomes aČech 1-cocycle for T . We call c B ∈ H 1 (B, T ) the radiance obstruction of B.
Definition 2.3. An integral affine manifold with singularities is a topological manifold B with an integral affine structure on B 0 := B \ Γ, where Γ ⊂ B is a locally finite union of locally closed submanifolds of codimension greater than 2. We call Γ the singular locus of B.
In this article, we assume that integral affine manifolds with singularities satisfy the following condition. This was mentioned in [KS06, Section 3.1] as the fixed point property.
Condition 2.4. For any x ∈ Γ, there is a small neighborhood U such that for each connected component (U \ Γ) i of U \ Γ, the monodromy representation π 1 ((U \ Γ) i ) → Aff(M R ) has a fixed vector.
Let B be an integral affine manifold with singularities satisfying the above condition. We write the complement of the singular locus as ι : B 0 ֒→ B. Let further T Z be the local system on B 0 of integral tangent vectors. We set T := T Z ⊗ Z R again.
Definition 2.5. We choose a sufficiently fine covering {U i } i of B so that there is a flat section s i ∈ Γ(U i ∩ B 0 , T aff B 0 ) for each U i . This is possible as long as we assume Condition 2.4. When we set c B ((U i , U j )) := s j − s i , the element c B becomes aČech 1-cocycle for ι * T . We call c B ∈ H 1 (B, ι * T ) the radiance obstruction of B. (3.1)
For each cone σ ∈ F , we set
σ ∨ := {m ∈ M R | m, n ≥ 0 for all n ∈ σ} , (3.2) σ ⊥ := {m ∈ M R | m, n = 0 for all n ∈ σ} . (3.3)
Let Σ be a fan in N R . We define the toric variety X Σ (T) associated with Σ over T as follows: For each cone σ ∈ Σ, we define X σ as the set of monoid homomorphisms σ ∨ ∩ M → (T, ·)
X σ (T) := Hom(σ ∨ ∩ M, T) (3.4)
with the compact open topology. For cones σ, τ ∈ Σ such that σ ≺ τ , we have a natural immersion,
X σ (T) → X τ (T), (v : σ ∨ ∩ M → T) → (τ ∨ ∩ M ⊂ σ ∨ ∩ M v − → T), (3.5)
where σ ≺ τ means that σ is a face of τ . By gluing {X σ (T)} σ∈Σ together, we obtain the tropical toric variety X Σ (T) associated with Σ,
X Σ (T) := σ∈Σ X σ ∼ . (3.6)
We also define the torus orbit O σ corresponding to σ by
O σ (T) := Hom(σ ⊥ ∩ M, R). (3.7)
There is a projection map to the torus orbit
p σ : X σ (T) → O σ (T), (w : σ ∨ ∩ M → T) → (σ ⊥ ∩ M ⊂ σ ∨ ∩ M w − → T). (3.8)
See [Pay09] or [Kaj08] for more details about tropical toric varieties.
Tropical modifications
Tropical modifications are first introduced in [Mik06]. We briefly recall the idea of it. Let (T, +, ·) be the tropical semifield, where T := R ∪ {−∞} and
a + b := max {a, b} , (3.9) a · b := a + b,
(3.10)
and (T, ⊕, ⊙) be the tropical hyperfield, where
a ⊕ b := max {a, b} , a = b, {t ∈ T | t ≤ a} , a = b, (3.11) a ⊙ b := a + b, (3.12)
where the additions + in right hand sides of (3.10) and (3.12) mean the usual addition. In this subsection, all additions + and multiplications · mean max and + respectively in the following unless otherwise mentioned. Let A ⊂ Z n be a finite subset. We consider a tropical polynomial
f (x) = n∈A a n ⊙ x n , (3.13)
which is a multi-valued function on T n defined by f (x) := n∈A a n x n = a n 0 x n 0 , when ∃n 0 ∈ A s.t. a n 0 x n 0 > a n x n (∀n = n 0 ), t ∈ T t ≤ n∈A a n x n , otherwise.
(3.14)
We consider the graph Γ f ⊂ T n+1 of the function f
Γ f := (x, y) ∈ T n+1 y ∈ f (x) . (3.15)
This coincides with the bend locus of f ′ (x, y) := y + n∈A a n x n (3.16)
in T n+1 and has a natural balanced polyhedral structure. Let δ f : Γ f → T n be the projection forgetting the last component.
Definition 3.1. We call the balanced polyhedral complex Γ f the tropical modification of T n with respect to f . We also call the map δ f : Γ f → T n the contraction with respect to f .
The graph of n∈A a n x n is isomorphic to T n as sets. Hence, we can think that associating Γ f with T n corresponds to replacing + and · of n∈A a n x n with ⊕ and ⊙ respectively.
We can also define tropical modifications of general tropical varieties in affine spaces. We define tropical modifications of tropical manifolds that are not necessarily embedded in ambient spaces as maps between tropical manifolds which locally coincide with a tropical modification of an affine tropical variety. For a tropical manifold X, we regard a tropical manifold X ′ which relates to X by tropical modifications as a tropical manifold that is equivalent to X. We refer the reader to [Sha15] or [Kal15] for details.
Contractions of tropical hypersurfaces
In this section, all additions + and multiplications · mean max and + respectively unless otherwise mentioned. We also let M, N denote a free Z-module of rank 3 and its dual lattice respectively.
A construction of focus-focus singularities
We fix basis vectors e 1 , e 2 , e 3 of N. Consider the cone σ k generated by ke 1 + e 2 , e 2 ∈ N in N R , where k is some positive integer. Then the dual cone σ ∨ k is generated by e * 1 , −e * 1 + ke * 2 , ±e * 3 and we have
X σ k (T) := Hom(σ ∨ k ∩ M, T) (4.1) = (x, y, z, w) ∈ T 3 × R xy = z k . (4.2)
We define the space X k,l by X k,l := (x, y, z, w) ∈ X σ k (T) z = 0 + w l ,
(4.3)
where l is also some positive integer. This space consists of two 2-dimensional faces F + := (x, y, z, w) ∈ T 3 × R xy = w kl , z = w l , w > 0 , (4.4)
F − := (x, y, z, w) ∈ T 3 × R xy = z = 0 > w , (4.5)
and a 1-dimensional face
L := (x, y, z, w) ∈ T 3 × R xy = z = w = 0 . (4.6)
Each of these faces has an integral affine structure induced from the ambient space T 3 × R. We extend them and construct an integral affine structure with a singular point on X k,l as follows: First, we choose a point p = (x 0 , y 0 , 0, 0) ∈ L. We set
U x := X k,l \ {(x, y, z, w) ∈ L | x ≥ x 0 } , (4.7) U y := X k,l \ {(x, y, z, w) ∈ L | x ≤ x 0 } .
(4.8)
These give a covering of X k,l \ {p}. Consider projections p x : U x → R 2 , (x, y, z, w) → (x, w), (4.9) p y : U y → R 2 , (x, y, z, w) → (y, w).
(4.10)
The restrictions of p x and p y to F ± are integral affine isomorphisms onto their images. Hence, we can extend the integral affine structures on F ± to U x and U y so that projections p x and p y are integral affine isomorphisms onto their images. Here we have U x ∩ U y = F + ∪ F − and the integral affine structures on U x and U y coincide on F + and F − with each other. Hence, we can extend the integral affine structures on U x and U y to an integral affine structure on X k,l \ {p}. We can easily calculate the monodromy of the integral affine structure around p.
Lemma 4.1. Consider a loop around the point p, which starts from a point in U x , passes through F − , U y , and F + in this order, and comes back to the original point. The monodromy along this loop is given by the matrix
1 kl 0 1 , (4.11)
under the basis e x , e w corresponding to the coordinate (x, w) of U x . The point p is a concentration of kl focus-focus singularities at one point.
Proof. A point (x, w) = (x 0 , w 0 ) of U x is shown as (x, y, z, w) = (x 0 , x −1 0 , 0, w 0 ) in F − . If we see this in U y , we have (y, w) = (x −1 0 , w 0 )
. This is shown as (x, y, z, w) = (x 0 w kl , x −1 0 , w l 0 , w 0 ) in F + . If we see this in U x again, we get (x, w) = (x 0 w kl 0 , w 0 ). Hence, the monodromy transformation is given by e x → e x , e w → (kl)e x + e w . Remark 4.4. When k = l = 1, the space X 1,1 is an integral affine surface with a focus-focus singularity. Here we have X 1,1 ∼ = (x, y, w) ∈ T 2 × R xy = 0 + w , (x, y, z, w) → (x, y, w).
(4.12)
In [KS06, Section 8], a non-archimedean torus fibration corresponding to a surface containing a focus-focus singularity is constructed by using the algebraic surface defined by (αβ −1)γ = 1. Here, the subtraction and multiplication mean the usual ones. When we set α = x, β = y, γ = w −1 , the tropicalizaion of it coincides with xy = 0 + w, the equation defining X 1,1 .
A tropical modification of focus-focus singularities
Consider replacing the right hand side 0 + w l of the equation of (4.3) with 0 ⊕ w l . Then the solution of the equation z = 0 ⊕ w l in X σ k (T) is the union of X k,l and the additional face
F 0 := (x, y, z, w) ∈ T 3 × R xy = z k , z < 0, w = 0 ,(4.13)
and this solution set coincides with the tropical hypersurface V (f ) defined by f = 0 + z + w l in X σ k (T). The multiplicity of the facet F 0 is l and those of others are 1. The point (−∞, −∞, −∞, 0) ∈ F 0 is a singular point of multiplicity k.
We can think that the surface X k,l is obtained by contracting the tropical hypersurfaces V (f ) to x-direction and y-direction at the same time. We choose a point p = (x 0 , y 0 , 0, 0) ∈ L and define a contraction map δ f,p :
V (f ) → X k,l by (x, y, z, w) → (x, y, z, w) (x, y, z, w) ∈ X k,l (x, x −1 , 0, 0)
x ≥ x 0 (y −1 , y, 0, 0) y ≥ y 0 p = (x 0 , y 0 , 0, 0) otherwise.
(4.14)
The face F 0 is contracted to the line L by this map. The tropical hypersurface V (f ) and the contraction δ f,p are shown in Associating V (f ) with X k,l is similar to tropical modifications which we recalled in Section 3.2 in the sense that we replace operations + contained in a function with hyperoperations ⊕. In this article, we call associating the tropical hypersurface V (f ) with X k,l a tropical modification with respect to 0 + w l , and the map δ f,p : V (f ) → X k,l the contraction with respect to 0 + w l .
Blowing up and focus-focus singularities
We fix basis vectors e 1 , e 2 , e 3 of N again. Consider a fan Σ consisting of all faces of a cone σ k 1 generated by k 1 e 1 + e 2 , e 2 ∈ N and a cone σ k 2 generated by k 1 e 1 + e 2 , (k 1 + k 2 )e 2 ∈ N, where k 1 and k 2 are some positive integers. This fan Σ is a refinement of σ k where k 1 + k 2 = k. The intersection τ of σ k 1 and σ k 2 is the 1-dimensional cone generated by k 1 e 1 + e 2 . The tropical affine space corresponding to the cone τ is
X τ (T) := Hom(τ ∨ ∩ M, T) ∼ = R × T × R.
(4.15)
The tropical toric variety X Σ (T) associated with this fan Σ is obtained by gluing the tropical affine spaces
X σ k 1 (T) := Hom(σ ∨ k 1 ∩ M, T) = (x, y, z, w) ∈ T 3 × R xy = z k 1 , (4.16) X σ k 2 (T) := Hom(σ ∨ k 2 ∩ M, T) = (x ′ , y ′ , z ′ , w ′ ) ∈ T 3 × R x ′ y ′ = z ′k 2 , (4.17)
by the maps
X τ (T) ֒→ X σ k 1 (T), (y, z, w) → (y −1 z k 1 , y, z, w), (4.18) X τ (T) ֒→ X σ k 2 (T), (y, z, w) → (yz k 2 , y −1 , z, w). (4.19)
The coordinate transformation between (x, y, z, w) and (x ′ , y ′ , z ′ , w ′ ) is given by
x ′ = x −1 z k 1 +k 2 , y ′ = y −1 , z ′ = z, w ′ = w.
(4.20)
We consider the space X k 1 ,k 2 ,l defined by
X k 1 ,k 2 ,l := (x, y, z, w) ∈ X Σ (T) z = 0 + w l , (4.21)
where l is some positive integer. This space consists of two 2-dimensional faces
F + := (x, y, z, w) ∈ X Σ (T) xy = w k 1 l , z = w l , w > 0 , (4.22) F − := {(x, y, z, w) ∈ X Σ (T) | xy = z = 0 > w} , (4.23)
and a 1-dimensional face
L := {(x, y, z, w) ∈ X Σ (T) | xy = z = w = 0} . (4.24)
We construct an integral affine structure with two singular points on X k 1 ,k 2 ,l as follows:
We choose two points p 1 = (x 1 , y 1 , 0, 0) ∈ L in the coordinate (x, y, z, w) and
p 2 = (x ′ 2 , y ′ 2 , 0, 0) ∈ L in the coordinate (x ′ , y ′ , z ′ , w ′ ) such that y 1 < y −1 2 . We set U x := X k 1 ,k 2 ,l \ {(x, y, z, w) ∈ L | y ≥ y 1 } , (4.25) U y := X k 1 ,k 2 ,l \ {(x, y, z, w) ∈ L | y ≤ y 1 or y ≥ y 2 } , (4.26) U x ′ := X k 1 ,k 2 ,l \ {(x, y, z, w) ∈ L | y ≤ y 2 } .
(4.27)
These form a covering of X k 1 ,k 2 ,l \ {p 1 , p 2 }. Consider projections
p x : U x → R 2 , (x, y, z, w) → (x, w), (4.28) p y : U y → R 2 , (x, y, z, w) → (y, w), (4.29) p x ′ : U x ′ → R 2 , (x ′ , y ′ , z ′ , w ′ ) → (x ′ , w).
(4.30)
As in the case of Section 4.1, we can extend the integral affine structures on F + and F − to an integral affine structure on X k 1 ,k 2 ,l \ {p 1 , p 2 } by using these projections. Note that the integral affine structure on U y induced by p y is the same as the one induced by the projection
p y ′ : U y → R 2 , (x ′ , y ′ , z ′ , w ′ ) → (y ′ , w). (4.31)
As in Lemma 4.1, we can easily see that singular points p 1 and p 2 are concentrations of k 1 l and k 2 l focus-focus singularities respectively, and both of the monodromy invariant subspaces of p 1 and p 2 are the tangent space of L. Blowing up the ambient space by taking a refinement of the cone σ k corresponds to dispersing the concentration of focus-focus singularities which are piled at one point to the monodromy invariant direction.
We can also think that the space X k 1 ,k 2 ,l can be obtained by contracting a tropical hypersurface as in Section 4.2. Consider replacing the right hand side 0 + w l of the equation of (4.21) with 0 ⊕ w l again. Then the solution of the equation z = 0 ⊕ w l in X Σ (T) is the union of X k 1 ,k 2 ,l and the additional face
F 0 := (x, y, z, w) ∈ X Σ (T) xy = z k 1 , z < 0, w = 0 , (4.32)
and this solution set coincides with the tropical hypersurface V (f ) defined by f = 0 + z + w l in X Σ (T). The multiplicity of the facet F 0 is l and those of others are 1. The points
(−∞, −∞, −∞, 0) ∈ X σ k 1 and (−∞, −∞, −∞, 0) ∈ X σ k 2 of V (f )
are singular points of multiplicity k 1 and k 2 respectively. We define a contraction map δ f,p 1 ,
p 2 : V (f ) → X k 1 ,k 2 ,l by setting (x, y, z, w) → (x, y, z, w), (x, y, z, w) ∈ X k 1 ,k 2 ,l , (x, x −1 , 0, 0), (x, y, z, w) / ∈ X k 1 ,k 2 ,l , x ≥ x 1 , (y −1 , y, 0, 0), (x, y, z, w) / ∈ X k 1 ,k 2 ,l , y 1 ≤ y ≤ y ′−1 2 , p 1 = (x 1 , y 1 , 0, 0), otherwise (4.33) for (x, y, z, w) ∈ X σ k 1 such that y ≤ y −1 2 , and (x ′ , y ′ , z ′ , w ′ ) → (x ′ , y ′ , z ′ , w), (x ′ , y ′ , z ′ , w ′ ) ∈ X k 1 ,k 2 ,l , (x ′ , x ′−1 , 0, 0), (x ′ , y ′ , z ′ , w ′ ) / ∈ X k 1 ,k 2 ,l , x ′ ≥ x ′ 2 , (y ′−1 , y ′ , 0, 0), (x ′ , y ′ , z ′ , w ′ ) / ∈ X k 1 ,k 2 ,l , y ′ 2 ≤ y ′ ≤ y −1 1 , p 2 = (x ′ 2 , y ′ 2 , 0, 0), otherwise (4.34) for (x ′ , y ′ , z ′ , w ′ ) ∈ X σ k 2 such that y ′ ≤ y −1 1 .
Note that contractions (4.33) and (4.34) coincide with each other on the intersection. The face F 0 is contracted to the line L by this map.
Example 4.5. The space X 1,1,1 is an integral affine surface with two focus-focus singularities p 1 and p 2 whose monodromy invariant subspaces are tangent spaces of the line passing through p 1 and p 2 . A contraction of the face F 0 to obtain X 1,1,1 is shown in Remark 4.6. By taking a finer refinement of the cone σ k and the same procedure, we can construct an integral affine surface X k 1 ,··· ,km,l with m singular points p 1 , · · · , p m , where k 1 , · · · k m are positive integers such that k 1 + · · · + k m = k. All singular points p 1 , · · · , p m are on the same line L, and the i-th singular point p i is a concentration of k i l focus-focus singularities. The monodromy invariant subspace of any singular point is the tangent space of L. We can also construct a contraction map δ f,p 1 ,··· ,pm : V (f ) → X k 1 ,··· ,km,l as we did in (4.33), (4.34).
Contractions of tropical toric K3 hypersurfaces
Let ∆ ⊂ M R := M ⊗ Z R be a reflexive polytope of dimension 3, which is not necessarily smooth. We write the polar polytope of ∆ as∆ ⊂ N R := N ⊗ Z R, and the normal fans of ∆,∆ as Σ ⊂ N R ,Σ ⊂ M R respectively. We choose a refinementΣ ′ ⊂ M R ofΣ such that the primitive generator of any 1-dimensional cone inΣ ′ is contained in ∆ ∩ M. This gives rise to a crepant resolution of the toric variety associated withΣ.
Let A ⊂ N denote the subset consisting of all vertices of∆ and 0 ∈ N. We consider a tropical Laurent polynomial
f (x) = n∈A a(n)x n , (4.35)
such that the function
A → R, n → a(n) (4.36)
induces a central subdivision of∆. Let V (f ) be the tropical hypersurface defined by f in the tropical toric variety XΣ′(T) associated withΣ ′ . The tropical hypersurface V (f ) intersects with the toric boundary as follows: Let ρ ∈Σ ′ be a 1-dimensional cone, and F (ρ) be the face of ∆ which contains the primitive generator of ρ in its interior. Recall that there is a one-to-one correspondence between k-dimensional faces of ∆ and (2 − k)-dimensional faces of∆. LetF (ρ) be the face of∆ corresponding to F (ρ).
On the torus orbit O ρ (T) ⊂ XΣ′(T), the tropical hypersurface V (f ) is defined by n∈A∩F (ρ) a(n)x n .
(4.37)
The number of elements of A ∩F (ρ) is greater than or equal to 2 if and only if F (ρ) is a vertex or an edge. Hence, the tropical hypersurface V (f ) intersects with the torus orbit O ρ (T) if and only if F (ρ) is a vertex or an edge. Let σ ∈Σ ′ be a cone of dimension greater than 1, and
{ρ i } m i=1 ⊂Σ ′ be the set of 1-dimensional faces of σ. On the torus orbit O σ (T) ⊂ XΣ′(T), the tropical hypersurface V (f ) is defined by n∈ m i=1 A∩F (ρ i ) a(n)x n . (4.38)
The number of elements of l i=1 A ∩F (ρ i ) is greater than or equal to 2 if and only if the dimension of σ is 2 and the primitive generators of ρ 1 , ρ 2 are contained in a common edge of ∆. This is when the tropical hypersurface V (f ) intersects with the torus orbit O σ (T).
We write the union of cells of V (f ) that do not intersect with the toric boundary as B. This is topologically a 2-sphere. In the following, we contract the tropical hypersurface V (f ) to the 2-sphere B, and equip B with an integral affine structure with singularities.
First, we choose positions of singular points. Let τ be a 1-dimensional cell of B. Recall that there is a one-to-one correspondence between k-dimensional cells of B and k-dimensional faces of ∆. Let {ρ i } m+1 i=1 be the set of 1-dimensional cones inΣ ′ whose primitive generators are contained in the edge of ∆ corresponding to τ . We write the primitive generator of ρ i as v i ∈ M. We renumber cones ρ i (1 ≤ i ≤ m + 1) so that v 1 , v m+1 are vertices of ∆, and v i is nearer to v 1 than v i+1 for any 1 ≤ i ≤ l. We choose m distinct points p(τ ) i (1 ≤ i ≤ m) on the interior of τ so that the point p(τ ) i is nearer to the vertex v 1 than p(τ ) i+1 for any 1 ≤ i ≤ m − 1. These points will be singular points of the integral affine structure of B. For each 1-dimensional cell τ of B, we choose points p(τ ) i in this way and fix them.
For each 1-dimensional cell τ of B, we take an open set U τ containing all of these points p(τ ) i (1 ≤ i ≤ m). We also take an open neighborhood U v for each vertex v of B. Here, we take these open sets so that they do not contain any other singular points or any other vertices of B, and all of these open sets U τ , U v and interiors of all facets of B form a covering of B. We contract the tropical hypersurface V (f ) to B as follows:
• Around U v
Let ρ ∈Σ ′ be the cone whose primitive generator is the vertex of ∆ corresponding to v, and X ρ (T) ⊂ XΣ′(T) be the tropical affine toric variety corresponding to ρ. Let further V (f ) v be the star of v in V (f ). We consider the projection
p ρ : X ρ (T) → O ρ (T), (w : ρ ∨ ∩ M → T) → (ρ ⊥ ∩ M ⊂ ρ ∨ ∩ M w − → T).
(4.39)
We setŨ v := p −1 ρ (p ρ (U v )) ∩ V (f ) v and defined the map δ v :Ũ v pρ − → U v as δ v :Ũ v pρ − → p ρ (U v ) ∼ = U v , (4.40) where p ρ (U v ) ∼ = U v is the inverse map of the bijection p ρ : U v → p ρ (U v ). We equip U v with the integral affine structure induced by the integral affine structure of p ρ (U v ) ⊂ O ρ ∼ = R 2 .
The dominant part of f at v is given by
a(0) + n∈A∩F (ρ) a(n)x n . (4.41)
By taking an appropriate coordinate, the function (4.41) can be rewritten as a function of the form y + f v , where f v is a function on O ρ . The map δ v coincides with a restriction of the contraction of the hypersurface defined by y + f v with respect to the function f v , which we considered in Section 3.2.
• Around U τ Let {ρ i } m+1 i=1
be the set of all 1-dimensional cones inΣ ′ whose primitive generators are contained in the edge of ∆ corresponding to τ . We write the 2-dimensional cone whose 1-dimensional faces are ρ i and ρ i+1 as
σ i ∈Σ ′ (1 ≤ i ≤ m). Let furtherΣ ′ τ be the subfan ofΣ consisting of σ i (1 ≤ i ≤ m) and all of their faces. The star V (f ) τ of τ in V (f ) is contained in the subvariety XΣ′ τ (T) ⊂ XΣ′(T).
The tropical toric variety XΣ′ τ (T) is also the ambient space that we used when we construct X k 1 ,··· ,km,l of Remark 4.6, where k i is the integral distance between the primitive generators v i , v i+1 of ρ i , ρ i+1 . On the other hand, the dominant part of f at τ is given by
a(0) + n∈A∩F (ρ 1 )∩F (ρ l+1 ) a(n)x n .
(4.42)
The set A∩F (ρ 1 )∩F (ρ l+1 ) consists of two elements. By taking an appropriate coordinate, the function (4.42) can be rewritten as a function f τ of the form 0 + z + w l , where l is the integral distance between the elements of A ∩F (ρ 1 ) ∩F (ρ l+1 ). Hence, we can
embed V (f ) τ into the tropical hypersurface V (f τ ) defined by f τ in XΣ′ τ (T). The open set U τ ⊂ V (f ) τ is embedded into X k 1 ,··· ,km,l by it. We defineŨ τ ⊂ V (f ) τ as the inverse image of U τ by the map V (f ) τ ֒→ V (f τ ) δ fτ ,p(τ ) 1 ,··· ,p(τ )m − −−−−−−−−− → X k 1 ,··· ,km,l ⊃ U τ , (4.43)
where δ fτ ,p(τ ) 1 ,··· ,p(τ )m is the contraction map of Remark 4.6. We also define δ τ :Ũ τ → U τ as the restriction of (4.43) toŨ τ . We equip U τ with the integral affine structure induced from the integral affine structure of X k 1 ,··· ,km,l ⊃ U τ .
• Around facets
We consider the identity map from the interior of each facet of B to itself. The interior of each facet has an integral affine structure induced from the ambient space
O {0} (T) ∼ = R 3 .
The open setsŨ v ,Ũ τ and interiors of all facets of B form a covering of V (f ). By gluing the above maps δ v , δ τ , and identity maps of interiors of facets together, we obtain a contraction map δ : V (f ) → B and an integral affine surface B with singular points on each edge. We regard B as a tropical K3 surface equivalent to the tropical K3 hypersurface V (f ).
f (x, y, z) = 1 + x 3 y −1 z −1 + x −1 y 3 z −1 + x −1 y −1 z 3 + x −1 y −1 z −1 .
(4.44)
The Newton polytope∆ ⊂ N R of f is the simplex whose one side is 4. In this case, there are no further crepant refinements ofΣ. We choose a point on the interior of each edge of B, which will be a singular point. Let ρ 1 and ρ 2 be the 1-dimensional cones in the normal fanΣ of∆ generated by (1, 0, 0) and (0, 1, 0) respectively. Let further v 1 and v 2 be the vertices of B that correspond to ρ 1 and ρ 2 respectively, and τ be the edge of B connecting v 1 and v 2 .
Around v 1 , the tropical hypersurface V (f ) is locally defined by Around v 2 , the tropical hypersurface V (f ) is locally defined by
1 + x −1 y 3 z −1 + x −1 y −1 z 3 + x −1 y −1 z −1 ,(4.1 + x 3 y −1 z −1 + x −1 y −1 z 3 + x −1 y −1 z −1 , (4.47)
and the contraction δ v 2 coincides with a restriction of the contraction with respect to the function f v 2 on O ρ 2 (T) defined by
f v 2 (x, z) := x 3 z −1 + x −1 z 3 + x −1 z −1 . (4.48)
Around τ , the tropical hypersurface V (f ) is locally defined by
1 + x −1 y −1 z 3 + x −1 y −1 z −1 . (4.49)
When we set x ′ := 1x, y ′ := yz, z ′ := 1xyz, w ′ := z, it is locally defined by
f τ (z ′ , w ′ ) := 0 + z ′ + w ′ 4 .
(4.50)
The contraction δ τ coincides with a restriction of the contraction δ fτ ,p of (4.14) (k = 1, l = 4). The open set U τ is equipped with the integral affine structure of X 1,4 . The singular point p which is in the interior of τ is a concentration of 4 focus-focus singularities. These contractions are shown in Figure 4.3. Black points are chosen points as singular points of integral affine structures. The red region shows the contraction δ v 1 and the blue region shows the contraction δ v 2 . The green region shows the contraction δ τ . These contractions coincide with each other on their intersections.
Dispersions of focus-focus singularities
Let S be an integral affine surface with some singular points. We suppose that the monodromy around one of the singular points p of S is given by the matrix
1 k 0 1 , (5.1)
under a local coordinate system (x, y) near p, where k is a non-zero integer. Here, the tangent vector e x corresponding to the coordinate x is monodromy invariant, and the coordinate y is globally well-defined on a sufficiently small open neighborhood U of p. We write the line defined by y = 0 on U as L. We can construct another integral affine structure with singularities on U, which has just two singular points p 1 and p 2 on L whose monodromies are given by
1 k i 0 1 , (5.2)
under the same coordinate system (x, y) respectively, where k 1 , k 2 are non-zero integers such that k 1 + k 2 = k. By replacing the original integral affine structure with singularities on U with this new one, we can obtain another integral affine surface S ′ with singularities, since monodromies of both integral affine structures with respect to the loop along the boundary of U are the same. Assume that the determinant of the monodromy matrix around any singular point of S is 1. Then we have ι * ∧ 2 T Z ∼ = Z for both S and S ′ , where ι is the inclusion of the complement of singularities. The cohomology groups H 1 (S, ι * T Z ) and H 1 (S ′ , ι * T Z ) have the cup product (1.3) induced by the wedge product. We also write the radiance obstructions of S and S ′ as c S and c S ′ respectively.
Let U ′ = {U j } j∈J ′ be a sufficiently fine acyclic covering of S ′ for ι * T Z such that each open set have one singular point at most and each singular point is contained by only one open set. Let U jα , U j β ∈ U ′ be the open sets containing p 1 and p 2 respectively. We set
U jγ := U jα ∪ U j β , J • := J ′ \ {j α , j β }, and J := J • ∪ {j γ }. We replace U ′ if necessary so that U j 1 ∩U j 2 does not intersect with U jα ∩U j β for any j 1 , j 2 ∈ J • . The set of open sets U := {U j } j∈J is an acyclic covering of S for ι * T Z .
We define a map f :
H 1 (S, ι * T Z ) → H 1 (S ′ , ι * T Z ) by setting f (φ) ((U j 1 , U j 2 )) := φ U ′ j 1 , U ′ j 2 U j 1 ∩U j 2 (5.3)
for each φ ∈ Z 1 (U, ι * T Z ) and j 1 , j 2 ∈ J ′ , where
U ′ j := U j , j ∈ J • , U jγ , j ∈ {j α , j β } . (5.4) Lemma 5.1. The map f : H 1 (S, ι * T Z ) → H 1 (S ′ , ι * T Z ) is well-defined.
Proof. Since we have
δ(f (φ)) (((U j 1 , U j 2 , U j 3 ))) = f (φ) ((U j 2 , U j 3 )) − f (φ) ((U j 1 , U j 3 )) + f (φ) ((U j 1 , U j 2 )) (5.5) = φ U ′ j 2 , U ′ j 3 − φ U ′ j 1 , U ′ j 3 + φ U ′ j 1 , U ′ j 2 (5.6) = (δφ) U ′ j 1 , U ′ j 2 , U ′ j 3 = 0, (5.7) f (φ) is a cocyle. For any element θ ∈ C 0 (U, ι * T Z ), we take the element θ ′ ∈ C 0 (U ′ , ι * T Z ) defined by θ ′ (U j ) := θ(U ′ j ) U j . Then we have f (δθ) ((U j 1 , U j 2 )) = δθ U ′ j 1 , U ′ j 2 U j 1 ∩U j 2 = θ(U ′ j 2 ) U j 1 ∩U j 2 − θ(U ′ j 1 ) U j 1 ∩U j 2 , (5.8) δθ ′ ((U j 1 , U j 2 )) = θ ′ (U j 2 )| U j 1 ∩U j 2 − θ ′ (U j 1 )| U j 1 ∩U j 2 = θ(U ′ j 2 ) U j 1 ∩U j 2 − θ(U ′ j 1 ) U j 1 ∩U j 2 . (5.9)
Hence, we obtain f (δθ) = δθ ′ .
Proposition 5.2. The map f : H 1 (S, ι * T Z ) → H 1 (S ′ , ι * T Z ) is a primitive embedding that preserves the pairing.
Proof. First, we check that the map f is injective. Suppose there exists θ ′ ∈ C 0 (U ′ , ι * T Z ) such that δ(θ ′ ) = f (φ). We will construct an element θ ∈ C 0 (U, ι * T Z ) such that δ(θ) = φ. Here since we have
θ ′ (U j β ) − θ ′ (U jα ) = (δθ ′ ) U jα , U j β = f (φ) U jα , U j β = φ U jγ , U jγ = 0, (5.10)
there is a section s ∈ Γ(U jγ , ι * T Z ) such that s| U jα = θ ′ (U jα ) and s| U j β = θ ′ (U j β ). We define
θ ∈ C 0 (U, ι * T Z ) by setting θ((U j )) := θ ′ ((U j )), j ∈ J • , s, j = j γ . (5.11)
Then when j 1 , j 2 ∈ J • , we have
δθ ((U j 1 , U j 2 )) = θ ′ (U j 2 ) − θ ′ (U j 1 ) = δθ ′ ((U j 1 , U j 2 )) = f (φ) ((U j 1 , U j 2 )) = φ ((U j 1 , U j 2 )) . (5.12) When j 1 = j γ , j 2 ∈ J • , we have δθ U jγ , U j 2 U jα ∩U j 2 = θ ′ (U j 2 )| U jα ∩U j 2 − s | U jα ∩U j 2 (5.13) = θ ′ (U j 2 )| U jα ∩U j 2 − θ ′ (U jα )| U jα ∩U j 2 (5.14) = δθ ′ ((U jα , U j 2 )) = f (φ) ((U jα , U j 2 )) = φ U jγ , U j 2 U jα ∩U j 2 . (5.15)
Since we can also get δθ U jγ , U j 2 U j β ∩U j 2 = φ U jγ , U j 2 U j β ∩U j 2 in the same way, we obtain δθ U jγ , U j 2 = φ U jγ , U j 2 .
(5.16) Therefore, we obtain δ(θ) = φ. Next, we check that the map f preserves the pairing. We take a total orders of J • . By adding j γ to J • as the minimum element, we obtain a total order of J. We also consider the total order of J ′ obtained by adding j α , j β to J • as the minimum and the second minimum elements respectively. For any φ 1 , φ 2 ∈Ȟ 1 (U, ι * T Z ), we can calculate as follows:
φ 1 ∪ φ 2 − f (φ 1 ) ∪ f (φ 2 ) = j 1 <j 2 U jγ ∩U j 1 ∩U j 2 =∅ φ 1 ((U jγ , U j 1 )) ∧ φ 2 ((U j 1 , U j 2 )) (5.17) − j 1 <j 2 U jα ∩U j 1 ∩U j 2 =∅ f (φ 1 )((U jα , U j 1 )) ∧ f (φ 2 )((U j 1 , U j 2 )) (5.18) − j 1 <j 2 U j β ∩U j 1 ∩U j 2 =∅ f (φ 1 )((U j β , U j 1 )) ∧ f (φ 2 )((U j 1 , U j 2 )) (5.19) − j∈J • U jα ∩U j β ∩U j =∅ f (φ 1 )((U jα , U j β )) ∧ f (φ 2 )((U j β , U j )) (5.20) =0,(5.21)
where j 1 , j 2 ∈ J • . Lastly, we show that the map f is primitive. Consider the map
f ⊗ id R : H 1 (S, ι * T Z ) ⊗ Z R → H 1 (S ′ , ι * T Z ) ⊗ Z R, φ ⊗ t → f (φ) ⊗ t, (5.22)
which we will also write as f . Assume that there exists an element θ ′ ∈ C 0 (U ′ , ι * T ) such that δ(θ ′ ) + f (φ ⊗ t) ∈ C 1 (U ′ , ι * T Z ). We will construct an element θ ∈ C 0 (U, ι * T ) such that δ(θ)+φ⊗t ∈ C 1 (U, ι * T Z ). Since the monodromy invariant directions of p 1 and p 2 are the same, we can extend sections θ ′ (U jα ), θ ′ (U j β ) to U jγ . Let s α , s β ∈ Γ(U jγ , ι * T Z ) denote the extensions of θ ′ (U jα ), θ ′ (U j β ) respectively. Since we have
(δ(θ ′ ) + f (φ ⊗ t)) ((U jα , U j β )) = θ ′ (U j β ) U jα ∩U j β − θ ′ (U jα )| U jα ∩U j β ∈ ι * T Z (U jα ∩ U j β ), (5.23)
we can see s α − s β ∈ ι * T Z (U jγ ). We define θ ∈ C 0 (U, ι * T Z ) by setting
θ((U j )) := θ ′ ((U j )), j ∈ J • , s α , j = j γ . (5.24)
Then, in the case where j 1 , j 2 ∈ J • , we have
(δθ + φ ⊗ t) ((U j 1 , U j 2 )) = θ ′ (U j 2 ) − θ ′ (U j 1 ) + (φ ⊗ t) ((U j 1 , U j 2 )) (5.25) = δθ ′ ((U j 1 , U j 2 )) + f (φ ⊗ t) ((U j 1 , U j 2 )) (5.26) = (δθ ′ + f (φ ⊗ t)) ((U j 1 , U j 2 )) ∈ ι * T Z (U j 1 ∩ U j 2 ). (5.27) In the case where j 1 = j γ , j 2 ∈ J • , U jα ∩ U j 2 = ∅ or U j β ∩ U j 2 = ∅. When U jα ∩ U j 2 = ∅, we have (δθ + φ ⊗ t) U jγ , U j 2 U jα ∩U j 2 = θ ′ (U j 2 )| U jα ∩U j 2 − s α | U jα ∩U j 2 + (φ ⊗ t) U jγ , U j 2 U jα ∩U j 2 (5.28) = θ ′ (U j 2 )| U jα ∩U j 2 − θ ′ (U jα )| U jα ∩U j 2 + f (φ ⊗ t) ((U jα , U j 2 )) (5.29) = (δθ ′ + f (φ ⊗ t)) ((U jα , U j 2 )) ∈ ι * T Z (U jα ∩ U j 2 ). (5.30) Hence, we obtain (δθ + φ ⊗ t) U jγ , U j 2 ∈ ι * T Z (U jγ ∩ U j 2 ). When U j β ∩ U j 2 = ∅, we have (δθ + φ ⊗ t) U jγ , U j 2 U j β ∩U j 2 = θ ′ (U j 2 )| U j β ∩U j 2 − s α | U j β ∩U j 2 + (φ ⊗ t) U jγ , U j 2 U j β ∩U j 2 (5.31) = θ ′ (U j 2 )| U j β ∩U j 2 − s β | U j β ∩U j 2 − (s α − s β ) | U j β ∩U j 2 +f (φ ⊗ t) U j β , U j 2 (5.32) = θ ′ (U j 2 )| U j β ∩U j 2 − θ ′ (U j β ) U j β ∩U j 2 − (s α − s β ) | U j β ∩U j 2 +f (φ ⊗ t) U j β , U j 2 (5.33) =(δθ ′ + f (φ ⊗ t)) U j β , U j 2 − (s α − s β ) | U j β ∩U j 2 (5.34) ∈ι * T Z (U j β ∩ U j 2 ). (5.35)
Hence, we obtain (δθ + φ ⊗ t) U jγ , U j 2 ∈ ι * T Z (U jγ ∩ U j 2 ). Therefore, we have δ(θ) + φ ⊗ t ∈ C 1 (U, ι * T Z ).
Proposition 5.3. One has f (c S ) = c S ′ .
Proof. When we take a set of sections {s j ∈ Γ(U j , ι * T )} j∈J , the radiance obstruction c S of S is given by
c S ((U j 1 , U j 2 )) = s j 2 | U j 1 ∩U j 2 − s j 1 | U j 1 ∩U j 2 ∈ Γ(U j 1 ∩ U j 2 , ι * T ) (5.36)
for any j 1 , j 2 ∈ J. We set s jα := s jγ | U jα and s j β := s jγ | U j β . We have
f (c S )((U j 1 , U j 2 )) = c S ((U ′ j 1 , U ′ j 2 )) U j 1 ∩U j 2 = s j 2 | U j 1 ∩U j 2 − s j 1 | U j 1 ∩U j 2 ∈ Γ(U j 1 ∩ U j 2 , ι * T ) (5.37)
for any j 1 , j 2 ∈ J ′ . This is just the radiance obstruction c S ′ of S ′ constructed from the set of sections {s j ∈ Γ(U j , ι * T )} j∈J ′ . In the subsequent subsections, we give proofs of Theorem 1.1 and Theorem 1.2 in this setting. Note that the statements of Theorem 1.1 and Theorem 1.2 do not depend on the the choices of positions of singular points. See Remark 4.7. In Section 4.3, we also saw that blowing up the ambient toric variety corresponds to dispersing concentrations of focus-focus singularities. Proposition 5.2 and Proposition 5.3 ensure that Theorem 1.1 and Theorem 1.2 hold also when we replaceΣ with a refinementΣ ′ ⊂ M R ofΣ such that the primitive generator of any 1-dimensional cone inΣ ′ is contained in ∆ ∩ M, if we could prove the theorems in the above setting.
Proof of Theorem 1.1
We consider the toric variety X Σ associated with Σ. We write the group of toric divisors on X Σ as
Div T (X Σ ) := ρ∈Σ(1) Z · D ρ , (6.3)
where Σ(1) is the set of 1-dimensional cones in Σ, and D ρ is the toric divisor corresponding to ρ ∈ Σ(1). We take the barycentric subdivision of P and let U := {U τ } τ ∈P be the covering of B, where U τ is the open star of the barycenter of τ ∈ P. The covering U is acyclic for ι * T Z and ι * T , andȞ
1 (U, ι * T Z ) = H 1 (B, ι * T Z ),Ȟ 1 (U, ι * T ) = H 1 (B, ι * T ). (6.4)
There is a one-to-one correspondence between the facets of B, vertices of∆, and Σ(1). We write the facet of B and the vertex of∆ that correspond to ρ ∈ Σ(1) as σ(ρ) ∈ P and n ρ = (n 1 (ρ), n 2 (ρ), n 3 (ρ)) ∈ N respectively. Let v ∈ P be a vertex, and {σ(ρ i )} 3 i=1 be the set of facets containing v. The equation defining the plane containing σ(ρ i ) is given by
a(n ρ i ) + n 1 (ρ i )x 1 + n 2 (ρ i )x 2 + n 3 (ρ i )x 3 = a(0). (6.5)
For a divisor D = ρ∈Σ(1) k ρ D ρ ∈ Div T (X Σ ) and the vertex v ∈ P, let v(D) denote the element of M defined by for each 1-simplex (U τ 0 , U τ 1 ) of U. We will check that this map ψ gives the map of Theorem 1.1 in the following lemmas, from Lemma 6.1 to Lemma 6.5.
n 1 (ρ i )x 1 + n 2 (ρ i )x 2 + n 3 (ρ i )x 3 = −k ρ i , 1 ≤ i ≤ 3.
Lemma 6.1. The map ψ is a well-defined group homomorphism.
Proof. First, we check that ψ(D) ((U τ 0 , U τ 1 )) is certainly a section of ι * T Z over U τ 0 ∩ U τ 1 . Assume that we choose vertices v 0 , v 1 for τ 0 , τ 1 when we determine τ 0 (D), τ 1 (D). Then we have
ψ(D) ((U τ 0 , U τ 1 )) = v 1 (D) − v 0 (D). (6.9)
Consider the case where either of τ 0 or τ 1 is a facet. Assume that τ 1 is a facet and the 1-dimensional cone ρ ∈ Σ(1) corresponds to it. Here, we have τ 0 ≺ τ 1 . Since points v 0 (D) and v 1 (D) are contained in the plane defined by n 1 (ρ)x 1 + n 2 (ρ)x 2 + n 3 (ρ)x 3 = −k ρ , (6.10) the vector v 1 (D) − v 0 (D) is contained in the plane defined by n 1 (ρ)x 1 + n 2 (ρ)x 2 + n 3 (ρ)x 3 = 0. (6.11)
On the other hand, the tangent space at U τ 0 ∩ U τ 1 is also this subspace. Hence, we have
ψ(D) ((U τ 0 , U τ 1 )) ∈ ι * T Z (U τ 0 ∩ U τ 1 ).
In the case where neither of τ 0 nor τ 1 is a facet, one of these is a vertex and the other is an edge. Assume that τ 0 is a vertex and τ 1 is an edge. Here, the edge τ 1 contains τ 0 = v 0 as its endpoint. Let σ(ρ 1 ) and σ(ρ 2 ) be the facets containing τ 1 as their face. Since the points v 0 (D), v 1 (D) are contained in the 1-dimensional space defined by
n 1 (ρ i )x 1 + n 2 (ρ i )x 2 + n 3 (ρ i )x 3 = −k ρ i , i = 1, 2, (6.12) the vector v 1 (D) − v 0 (D)
is contained in the 1-dimensional subspace defined by
n 1 (ρ i )x 1 + n 2 (ρ i )x 2 + n 3 (ρ i )x 3 = 0, i = 1, 2. (6.13)
On the other hand, the tangent space at U τ 0 ∩ U τ 1 contains this subspace. Hence, we have
ψ(D) ((U τ 0 , U τ 1 )) ∈ ι * T Z (U τ 0 ∩ U τ 1 ).
Next, we show that ψ(D) is a cocycle. For a given 2-simplex (U τ 0 , U τ 1 , U τ 2 ), assume that we choose vertices v 0 , v 1 , v 2 for τ 0 , τ 1 , τ 2 when we determine τ 0 (D), τ 1 (D), τ 2 (D). Then we have
δ (ψ(D)) ((U τ 0 , U τ 1 , U τ 2 )) = {v 2 (D) − v 1 (D)} − {v 2 (D) − v 0 (D)} + {v 1 (D) − v 0 (D)} = 0. (6.14)
Lastly, we show that the map ψ is a group homomorphism. We will show ψ(D
+ D ′ ) = ψ(D) + ψ(D ′ ) for any D = ρ k ρ D ρ , D ′ = ρ k ′ ρ D ρ ∈ Div T (X Σ ).
Let v ∈ P be a vertex, and {σ(ρ i )} 3 i=1 be the set of facets containing v. Then the point v(D + D ′ ) is defined by
n 1 (ρ i )x 1 + n 2 (ρ i )x 2 + n 3 (ρ i )x 3 = −k ρ i − k ′ ρ i , 1 ≤ i ≤ 3. (6.15)
The point v(D) is defined by (6.16) and the point v(D ′ ) is defined by
n 1 (ρ i )x 1 + n 2 (ρ i )x 2 + n 3 (ρ i )x 3 = −k ρ i , 1 ≤ i ≤ 3,n 1 (ρ i )x 1 + n 2 (ρ i )x 2 + n 3 (ρ i )x 3 = −k ′ ρ i , 1 ≤ i ≤ 3. (6.17) Hence, we have v(D + D ′ ) = v(D) + v(D ′ ). If we choose vertices v 0 , v 1 for τ 0 , τ 1 when we determine τ 0 (D), τ 1 (D), we have ψ(D + D ′ )((U τ 0 , U τ 1 )) = v 1 (D + D ′ ) − v 0 (D + D ′ ) = {v 1 (D) + v 1 (D ′ )} − {v 0 (D) + v 0 (D ′ )} = {v 1 (D) − v 0 (D)} + {v 1 (D ′ ) − v 0 (D ′ )} (6.18) = ψ(D)((U τ 0 , U τ 1 )) + ψ(D ′ )((U τ 0 , U τ 1 )) (6.19)
for any 1-simplex (U τ 0 , U τ 1 ) of U.
Lemma 6.2. The map ψ is independent of the choice of the vertex that we choose for each τ ∈ P when we determine τ (D).
Proof. Let P := {τ i } i be the set of all cells of B. Let further ψ be the map which we obtain when we choose v i for each τ i , and ψ ′ be the map which we obtain when we choose v ′ i for each τ i . We show that ψ(D) = ψ ′ (D) for any D ∈ Div T (X Σ ).
For each D = ρ∈Σ(1) k ρ D ρ ∈ Div T (X Σ ), we define φ(D) ∈ C 0 (U, ι * T Z ) by setting
φ(D)((U τ i )) := v ′ i (D) − v i (D) (6.20)
for each 0-simplex (U τ i ) of U. We will show that the coboundary of φ(D) coincides with ψ ′ (D) − ψ(D). First, we check that φ(D) is certainly an element of C 0 (U, ι * T Z ).
When τ i is a vertex, v i = v ′ i = τ and we have φ(D)(U τ i ) = 0. Hence, φ(D)(U τ i ) ∈ ι * T Z (U τ i ).
When τ i is an edge and is contained in facets σ(ρ i 1 ) and σ(ρ i 2 ), vertices v i (D), v ′ i (D) are contained in the 1-dimensional space defined by
n 1 (ρ i j )x 1 + n 2 (ρ i j )x 2 + n 3 (ρ i j )x 3 = −k ρ i j , j = 1, 2.
(6.21)
Hence, the vector v ′ i (D) − v i (D)
is contained in the 1-dimensional subspace defined by
n 1 (ρ i j )x 1 + n 2 (ρ i j )x 2 + n 3 (ρ i j )x 3 = 0, j = 1, 2. (6.22)
The section ι * T Z (U τ i ) is the lattice of integral tangent vectors that are invariant under the monodromy transformation around the singular point on τ i . That is the lattice contained in the subspace defined by (6.22). Hence, we have φ(D)
((U τ i )) ∈ ι * T Z (U τ i ).
When τ i is a facet, vertices v i (D) and v ′ i (D) are contained in the plane defined by
n 1 (ρ i )x 1 + n 2 (ρ i )x 2 + n 3 (ρ i )x 3 = −k ρ i , (6.23)
where ρ i is the 1-dimensional cone corresponding to τ i . Hence, the vector v ′ i (D) − v i (D) is contained in the plane defined by n 1 (ρ i )x 1 + n 2 (ρ i )x 2 + n 3 (ρ i )x 3 = 0.
(6.24)
On the other hand, the section ι * T Z (U τ i ) is the lattice of integral tangent vectors on U τ i . That is the lattice contained in the subspace defined by (6.24). Hence, we have φ(D) ((U τ i )) ∈ ι * T Z (U τ i ). Therefore, we have φ(D) ∈ C 0 (U, ι * T Z ). For any 1-simplex (U τ i , U τ j ) of U, one can get
ψ ′ (D)((U τ i , U τ j )) − ψ(D)((U τ i , U τ j )) = v ′ j (D) − v ′ i (D) − {v j (D) − v i (D)} = v ′ j (D) − v j (D) − {v ′ i (D) − v i (D)} (6.25) = (δφ(D))((U τ i , U τ j )). (6.26) Hence, we have ψ(D) = ψ ′ (D) inȞ 1 (U, ι * T Z ).
Recall that we have the exact sequence
M → Div T (X Σ ) → Pic(X Σ ) → 0, (6.27)
where the map M → Div T (X Σ ) is given by m → div(χ m ).
Lemma 6.3. The map ψ induces an injection
Pic(X Σ ) ֒→Ȟ 1 (U, ι * T Z ). (6.28)
Proof. First, we check that ψ(div(χ m )) = 0 for any m = (m 1 , m 2 , m 3 ) ∈ M. The facet σ(ρ) of B corresponding to the 1-dimensional cone ρ ∈ Σ is defined by a(n ρ ) + n 1 (ρ)x 1 + n 2 (ρ)x 2 + n 3 (ρ)x 3 = a(0). (6.29)
The primitive generator of ρ is n ρ = (n 1 (ρ), n 2 (ρ), n 3 (ρ)) ∈ N. Hence, we have div(χ m ) = ρ∈Σ(1) m, n ρ D ρ . (6.30)
Let v ∈ P be a vertex of B, and {σ(ρ i )} 3 i=1 be the set of facets containing v. The element v(div(χ m )) satisfies
n 1 (ρ i )x 1 + n 2 (ρ i )x 2 + n 3 (ρ i )x 3 = − m, n ρ i = − 3 j=1 m j n j (ρ i ) , 1 ≤ i ≤ 3.
(6.31) Therefore, we have v(div(χ m )) = −(m 1 , m 2 , m 3 ) ∈ M for any vertex v ∈ P. From the definition of ψ, we can see that ψ(div(χ m )) = 0. Next, we show that the induced map Pic(X Σ ) →Ȟ 1 (U, ι * T Z ) is injective. Assume that ψ(D) = δ(φ) for some D ∈ Div T (X Σ ) and φ ∈ C 0 (U, ι * T Z ). We show that D comes from M. Let τ ∈ P be a 1-dimensional cell, and v 0 , v 1 be its endpoints. Suppose that we choose v 1 when we determine τ (D), i.e., τ (D) = v 1 (D). Then we have
ψ(D)((U v 0 , U τ )) = v 1 (D) − v 0 (D) = φ((U τ )) − φ((U v 0 )), (6.32) ψ(D)((U v 1 , U τ )) = 0 = φ((U τ )) − φ((U v 1 )).
(6.33)
Here, ψ(D)((U v 0 , U τ )) = v 1 (D) − v 0 (D)
and φ((U τ )) are parallel to the direction which is invariant under the monodromy around the singular point on τ . Hence, from (6.32), (6.33), it turns out that φ((U v 0 )) and φ((U v 1 )) also have to be parallel to this direction. Let τ ′ ∈ P be another 1-dimensional cell that has v 0 as its vertex. By the same argument, we can see that φ((U v 0 )) has to be parallel also to the direction which is invariant under the monodromy around the singular point on τ ′ . Since these two monodromy invariant directions are linearly independent, φ((U v 0 )) has to be zero. Similarly, we get φ((U v 1 )) = 0. Hence, by (6.32), (6.33), we obtain φ((U τ )) = 0, (6.34) v 1 (D) = v 0 (D).
(6.35)
Since there is a sequence of edges of B connecting arbitrary two vertices of B, we can conclude that the element v(D) ∈ M is the same for any vertex v. We write this point as m(D) ∈ M. We set D =: ρ∈Σ(1) k ρ D ρ and div(χ −m(D) ) =: ρ∈Σ(1) k ′ ρ D ρ . For any ρ ∈ Σ(1), we take a vertex v ∈ P contained in the facet σ(ρ) ∈ P corresponding to ρ. The element v(D) satisfies n 1 (ρ)x 1 + n 2 (ρ)x 2 + n 3 (ρ)x 3 = −k ρ , (6.36) and the element v(div(χ −m(D) )) = m(D) satisfies
n 1 (ρ)x 1 + n 2 (ρ)x 2 + n 3 (ρ)x 3 = −k ′ ρ . (6.37)
Since v(D) = m(D), we have k ρ = k ′ ρ . Hence, we obtain D = div(χ −m(D) ). Therefore, the induced map Pic(X Σ ) →Ȟ 1 (U, ι * T Z ) is injective. This map Pic(X Σ ) → H 1 (B, ι * T Z ) will also be denoted by ψ.
Lemma 6.4. The embedding ψ : Pic(X Σ ) → H 1 (B, ι * T Z ) is primitive, i.e., the image of the map
ψ : Pic(X Σ ) → H 1 (B, ι * T Z ) → H 1 (B, ι * T ) = H 1 (B, ι * T Z ) ⊗ R (6.38) coincides with Im(ψ) ⊗ Z R ∩ H 1 (B, ι * T Z ).
Proof. For the proof, we use another covering U ′ := {U ′ τ } τ ∈P defined as follows: For each vertex v of B, we choose two distinct 1-dimensional cells τ 1 , τ 2 ∈ P that contain v as their endpoint. We set U ′ v := U v ∪ U τ 1 ∪ U τ 2 for each vertex v ∈ P, and U ′ τ = U τ for each 1 or 2-dimensional cell τ ∈ P. We can check that H i (U ′ v , T Z ) = 0 for i ≥ 0 and the covering U ′ is also acyclic for ι * T Z and ι * T . The covering U is a refinement of U ′ and we have an isomorphism γ :Ȟ 1 (U ′ , ι * T ) →Ȟ 1 (U, ι * T ). We also consider the map
ψ ′ : Div T (X Σ ) ⊗ Z R →Ȟ 1 (U ′ , ι * T ), (6.39)
which is defined as follows: For an element D ⊗ t = ρ∈Σ(1) k ρ D ρ ⊗ t ∈ Div T (X Σ ) ⊗ Z R and a vertex v ∈ P, let {σ(ρ i )} 3 i=1 be the set of facets containing v, and we define v(D ⊗ t) as the point of M R := M ⊗ Z R determined by
n 1 (ρ i )x 1 + n 2 (ρ i )x 2 + n 3 (ρ i )x 3 = −tk ρ i , 1 ≤ i ≤ 3. (6.40) Here, we have v(D ⊗ t) = t · v(D)
. For each cell τ ∈ P, we choose an arbitrary vertex v of τ and set τ (D ⊗ t) := v(D ⊗ t). We define a map ψ ′ by setting
ψ ′ (D ⊗ t) (U ′ τ 0 , U ′ τ 1 ) := τ 1 (D ⊗ t) − τ 0 (D ⊗ t) (6.41)
for each 1-simplex (U ′ τ 0 , U ′ τ 1 ) of U ′ . The arguments of Lemma 6.1 and Lemma 6.2 are applicable also to ψ ′ and we have γ • ψ ′ = ψ ⊗ id. This induces an injection
ψ ′ : Pic(X Σ ) ⊗ Z R = Div T (X Σ ) ⊗ Z R/M R →Ȟ 1 (U ′ , ι * T ).
(6.42)
We will show that there exists D ∈ Div T (X Σ ) such that ψ ′ (D) = λ for any λ ∈ Im(ψ ′ ) ∩ H 1 (U ′ , ι * T Z ). There exists D ′ ⊗ t ∈ Div T (X Σ ) ⊗ Z R such that ψ ′ (D ′ ⊗ t) = λ. By adding some element div(χ m ) ⊗ t ′ (t ′ ∈ R, m ∈ M) to D ′ ⊗ t, we can assume that there is a vertex v 0 ∈ P such that v 0 (D) ∈ M, where D := D ′ ⊗ t + div(χ m ) ⊗ t ′ . Note that ψ ′ (D) = ψ ′ (D ′ ⊗ t) = λ. Since λ is an element ofȞ 1 (U ′ , ι * T Z ), there exists an element φ ∈ C 0 (U ′ , ι * T ) such that
δ(φ) + ψ ′ (D) ∈ C 1 (U ′ , ι * T Z ).
(6.43)
Let τ ∈ P be any 1-dimensional cell and v 1 , v 2 be its endpoints. We choose v 2 when we determine τ (D), i.e., τ (D) = v 2 (D). Since Recall that the cohomology group H 1 (B, ι * T Z ) has the cup product (1.3) induced by the wedge product. Let Y be an anti-canonical hypersurface of the complex toric variety X Σ associated with Σ, and Pic(Y ) amb := Im (Pic(X Σ ) ֒→ Pic(Y )) (6.46) be the sublattice of Pic(Y ) coming from the Picard group of the ambient space.
ι * T (U ′ v 1 ) = ι * T (U ′ v 2 ) = 0, we have {δ(φ) + ψ ′ (D)} ((U ′ v 1 , U ′ τ )) = φ(U ′ τ ) + (v 2 (D) − v 1 (D)) , (6.44) {δ(φ) + ψ ′ (D)} ((U ′ v 2 , U ′ τ )) = φ(U ′ τ ).
Lemma 6.5. The embedding ψ : Pic(X Σ ) ֒→ H 1 (B, ι * T Z ) preserves the pairing.
Proof. We show that D ρ i 1 ·D ρ i 2 = ψ(D ρ i 1 )∪ψ(D ρ i 2 ) for any ρ i 1 , ρ i 2 ∈ Σ(1). First, we show this in the case where D ρ i 1 = D ρ i 2 . A hypersurface in X Σ defined by a polynomial whose Newton polytope is ∆ is an anti-canonical hypersuface.
D ρ i 1 ·D ρ i 2 is equal to the number of intersection points Y ∩ D ρ i 1 ∩ D ρ i 2 . When σ(ρ i 1 ) ∩ σ(ρ i 2 ) is empty, the intersection Y ∩ D ρ i 1 ∩ D ρ i 2 is also empty. When σ(ρ i 1 ) ∩ σ(ρ i 2 )
is not empty, the number of intersection points Y ∩ D ρ i 1 ∩ D ρ i 2 is the integral length of the edge σ(ρ i 1 ) ∩ σ(ρ i 2 ). We write the length of σ(ρ i 1 ) ∩ σ(ρ i 2 ) as l.
We will show that ψ(D ρ i 1 ) ∪ ψ(D ρ i 2 ) = l. When we determine τ (D ρ i j ) for each τ ∈ P in order to obtain ψ(D ρ i j ) where j = 1, 2, we choose a vertex in the following way: For τ (D ρ i j ), if τ is not contained in σ(ρ i j ) and has a vertex outside σ(ρ i j ), we choose one of such vertices. If τ is contained in σ(ρ i j ) and all its vertices are in σ(ρ i j ), we choose one of them arbitrarily.
Here, v(D ρ ) = 0 when v is not a vertex of σ(ρ). Hence, we have ψ(D ρ i j )((U τ 0 , U τ 1 )) = 0 if both τ 0 and τ 1 are not in σ(ρ i j ). For each 2-simplex (U τ 0 , U τ 1 , U τ 2 ) of U, one of τ 0 , τ 1 , τ 2 is a vertex of B. Assume τ i is that vertex. When ψ(D ρ i 1 ) ∪ ψ(D ρ i 2 )((U τ 0 , U τ 1 , U τ 2 )) = 0, either τ 0 or τ 1 is in σ(ρ i 1 ), and either τ 1 or τ 2 has to be in σ(ρ i 2 ). Then the vertex τ i is contained in both σ(ρ i 1 ) and σ(ρ i 2 ). When σ(ρ i 1 ) ∩ σ(ρ i 2 ) is empty, this never happens. Therefore, we get ψ(D ρ i 1 ) ∪ ψ(D ρ i 2 ) = 0 for ρ i 1 , ρ i 2 ∈ Σ(1) such that σ(ρ i 1 ) ∩ σ(ρ i 2 ) = ∅. In the following, we assume σ(ρ i 1 ) ∩ σ(ρ i 2 ) = ∅, and directly compute ψ(D ρ i 1 ) ∪ ψ(D ρ i 2 )((U τ 0 , U τ 1 , U τ 2 )) for 2simplexes (U τ 0 , U τ 1 , U τ 2 ) such that the vertex τ i is contained in the face σ(ρ i 1 ) ∩ σ(ρ i 2 ). Those simplexes are shown in Figure 6.1. ψ(D ρ i 1 ) ∪ ψ(D ρ i 2 ) can be nonzero for them. Let v 0 , · · · , v 10 denote the vertices of these 2-simplexes as shown in Figure 6.1. We use the ascending order of index numbers as the total order of the vertices. Let further τ j (0 ≤ j ≤ 10) be the cell of B whose barycenter is v j . For instance, τ 6 = σ(ρ i 1 ), τ 4 = σ(ρ i 2 ), and τ 5 = σ(ρ i 1 ) ∩ σ(ρ i 2 ). Let e i (1 ≤ i ≤ 3) be the integral tangent vectors at v 8 contained in
R >0 · − − → v 8 v 9 , R >0 · − − → v 8 v 7 , R >0 · − − → v 8 v 5 respectively.
Let further e i (i = 4, 5) be the integral tangent vectors at v 2 contained in R >0 · − − → v 2 v 3 , R >0 · − − → v 2 v 1 respectively. We can assume that e 4 = e 1 + se 3 , e 5 = e 2 + te 3 , (6.47)
where s, t ∈ Z.
Consider the monodromy with respect to a loop that starts at v 8 , passes through v 4 , v 2 , v 6 in this order, and comes back to v 8 . From Lemma 4.1, we can see that it is given by
1 −l 0 1 , (6.48)
under the basis (e 3 , e 1 ), where l is the length of σ(ρ i 1 ) ∩ σ(ρ i 2 ). On the other hand, we can also calculate the monodromy of e 1 as follows: We have e 1 = −e 2 − e 3 in U v 8 . Since we have e 2 = e 5 − te 3 , e 1 becomes (t − 1)e 3 − e 5 when it arrives at v 2 . We also have e 5 = e 3 − e 4 in U v 2 , and e 4 = e 1 + se 3 . Therefore, e 1 becomes (s + t − 2)e 3 + e 1 when it is back to v 2 . Hence we obtain
s + t − 2 = −l. (6.49)
In order to determine each τ j (D ρ i 1 ), we choose a vertex as follows: For v 0 , v 1 , v 4 , we choose the endpoint of τ 1 which is not v 2 . For v 3 , v 5 , v 6 , we choose v 2 . For v 7 , v 10 , we choose the endpoint of τ 7 which is not v 8 . For v 9 , we choose v 8 . In order to determine τ i (D ρ i 2 ), we choose a vertex for each v i as follows: For v 0 , v 3 , v 6 , we choose the endpoint of τ 3 which is not v 2 . For v 1 , v 4 , v 5 , we choose v 2 . For v 7 , we choose v 8 . For v 9 , v 10 , we choose the endpoint of τ 9 which is not v 8 . Then we have
ψ(D ρ i 1 )((U j 1 , U j 2 )) = −e 5
(j 1 , j 2 ) = (0, 2), (0, 3), (1, 2), (4, 5), e 5 (j 1 , j 2 ) = (2, 4), −e 2 (j 1 , j 2 ) = (4, 8), (7, 8), e 5 − e 2 (j 1 , j 2 ) = (5, 8), (6, 8), (6, 9), e 2 (j 1 , j 2 ) = (8, 10), (9, 10), 0 otherwise,
(6.50) ψ(D ρ i 2 )((U j 1 , U j 2 )) =
−e 4 (j 1 , j 2 ) = (0, 1), (0, 2), e 4 (j 1 , j 2 ) = (2, 3), (2, 6), (5, 6), e 4 − e 1 (j 1 , j 2 ) = (4, 7), (4, 8), (5, 8), −e 1 (j 1 , j 2 ) = (6, 8), e 1 (j 1 , j 2 ) = (7, 10), (8, 9), (8, 10), 0 otherwise (6.51) for 0 ≤ j 1 < j 2 ≤ 10. We obtain
ψ(D ρ i 1 ) ∪ ψ(D ρ i 2 )((U j 1 , U j 2 , U j 3 )) =
−e 5 ∧ e 4 (j 1 , j 2 , j 3 ) = (0, 2, 3), −e 5 ∧ (e 4 − e 1 ) (j 1 , j 2 , j 3 ) = (4, 5, 8), (e 5 − e 2 ) ∧ e 1 (j 1 , j 2 , j 3 ) = (6, 8, 9), −e 2 ∧ e 1 (j 1 , j 2 , j 3 ) = (7, 8, 10), 0 otherwise, (6.52)
∼
(2 − s − t)e 1 ∧ e 2 (j 1 , j 2 , j 3 ) = (7, 8, 10), 0 otherwise (6.53) for 0 ≤ j 1 < j 2 < j 3 ≤ 10. We choose the orientation of B so that the element in C 2 (U, Z) taking e 1 ∧ e 2 for the 2-simplex (U 7 , U 8 , U 10 ) and 0 for any other 2-simplexes represents 1 ∈ Z ∼ = H 2 (B, Z). Then we get ψ(
D ρ i 1 ) ∪ ψ(D ρ i 2 ) = 2 − s − t = l.
Lastly, we show that D ρ 0 · D ρ 0 = ψ(D ρ 0 ) ∪ ψ(D ρ 0 ) for any ρ 0 ∈ Σ. Since there exists a primitive element m ∈ M such that div(χ m ) = D ρ 0 + ρ =ρ 0 a ρ D ρ (a ρ ∈ Z), we have
D ρ 0 ∼ − ρ =ρ 0 a ρ D ρ .
(6.54)
Hence, we obtain For each cell τ ∈ P, we choose an arbitrary vertex v(τ ) ∈ M R of τ . Here, if τ is a vertex, then v(τ ) = τ . By enlarging U τ slightly, we assume that U τ contains v(τ ). We take each chart ψ τ : U τ → M R so that ψ τ (v(τ )) = 0 ∈ M R . In order to specify the radiance obstruction c B of B, we choose the zero section 0 ∈ Γ(U τ ∩ B 0 , T aff B 0 ) for each U τ . Then the radiance obstruction c B is represented by the element of C 1 (U, ι * T ) determined by setting
D ρ 0 · D ρ 0 = D ρ 0 · − ρ =ρ 0 a ρ D ρ = − ρ =ρ 0 a ρ D ρ 0 · D ρ (6.55) = − ρ =ρ 0 a ρ ψ(D ρ 0 ) ∪ ψ(D ρ ) = ψ(D ρ 0 ) ∪ ψ − ρ =ρ 0 a ρ D ρ (6.56) = ψ(D ρ 0 ) ∪ ψ(D ρ 0c B ((U τ 0 , U τ 1 )) := v(τ 1 ) − v(τ 0 ) (6.58) for each 1-simplex (U τ 0 , U τ 1 ) of U.
Let v ∈ P be a vertex, and {σ(ρ i )} 3 i=1 be the set of facets of B containing v. Then the vertex v is determined by We consider a Laurent polynomial F = n∈A k n x n ∈ K[x ± 1 , x ± 2 , x ± 3 ] over K whose Newton polytope is∆, where A ⊂ N denotes the subset consisting of all vertices of∆ and 0 ∈ N. We assume that the function where H is the upper half plane. We set
n 1 (ρ i )x 1 + n 2 (ρ i )x 2 + n 3 (ρ i )x 3 = −a(n ρ i ) + a(0), 1 ≤ i ≤ 3.A → R, n → val(k n ) (7.2H R := {z ∈ H | Im z > R} , (7.5)
where R is a positive real number such that e( √ 1R) is smaller than the radius of convergence of k n for any n ∈ A. For each element z ∈ H R , we consider the polynomial f z ∈ C[x ±
1 , x ± 2 , x ± 3 ] obtained by substituting e(z) to t in F . Let V z be the complex hypersurface defined by f z in the complex toric variety XΣ′ associated withΣ ′ . This is a quasi-smooth K3 hypersurface.
We describe the asymptotic behavior of the period of V z in the limit R → ∞ by using the radiance obstruction c B . In the following, we assume k 0 = 1 by multiplying an element of K to F . Some parts of the following are borrowed from [Ued14, Section 7].
For a given element α = (a n ) n∈A\{0} ∈ (C × ) A\{0} , we associate the polynomial W α (x) = 1 + n∈A\{0} a n x n . (7.6)
We write the toric hypersurface defined by W α in the complex toric variety XΣ′ as Y α . Let (C × )
A\{0} reg be the set of α ∈ (C × ) A\{0} such that Y α isΣ ′ -regular, i.e., the intersection of Y α with any torus orbit of XΣ′ is a smooth subvariety of codimension one. We consider the family ofΣ ′ -regular hypersurfaces given by the second projection
ϕ : Y := (x, α) ∈ XΣ′ × (C × ) A\{0} reg W α (x) = 0 → (C × ) A\{0} reg ,
(7.7) and the action of M ⊗ Z C × to this family given by t · (x, α) := t −1 x, (t n a n ) n∈A\{0} , When R is sufficiently large, the hypersurface V z isΣ ′ -regular for any z ∈ H R . We have a map l given by
l : H R → M reg , z → [(k n (e(z))) n∈A\{0} ], (7.9)
where k n (e(z)) is the complex number obtained by substituting e(z) to t in k n . We define a holomorphic form on V z by
Ω z := dx 1 x 1 ∧ dx 2 x 2 ∧ dx 3 x 3 df z . (7.10)
This defines a section of l * H B over H R . Let Y be an anti-canonical hypersurface of the complex toric variety X Σ associated with Σ, and ι : Y ֒→ X Σ be the inclusion. Choose an integral basis {p i } r i=1 of Pic X Σ such that each p i is nef. This determines a coordinate q = (q 1 , · · · , q r ) on M := Pic X Σ ⊗ Z C × = Z A\{0} /M ⊗ Z C × ⊃ M reg and a coordinate τ = (τ 1 , · · · , τ r ) on H 2 amb (Y, C). Let u i ∈ H 2 (X Σ , Z) be the Poincaré dual of the toric divisor D ρ i corresponding to the one-dimensional cone ρ i ∈ Σ, and v = u 1 + · · · + u m be the anticanonical class. Givental's I-function is defined as the series The mirror map ς : M → H 2 amb (Y, C) is a multi-valued map defined by ι * G(q) F (q) .
(7.12)
The residual B-model VHS is isomorphic to the ambient A-model VHS (H A , ∇ A , F • A , Q A ) [Iri11, Definition 6.2] via the mirror map ς [Iri11, Theorem 6.9]. Hence, the section of l * H B over H R defined by Ω z can also be regarded as a section of (ς • l) * H A via the mirror isomorphism.
Here we replace the real number R with a larger one if necessary.
We also choose elements p 0 ∈ H 0 amb (Y ; Q) and p r+1 ∈ H 4 amb (Y ; Q) so that we have p 0 ∪ p r+1 = 1, and define sectionsp i := exp(−τ ) ∪ p i , 0 ≤ i ≤ r + 1, (7.13) of H A . These are flat sections with respect to the Dubrovin connection ∇ A . Note that the quantum cup product coincides with the ordinary cup product, since Y is a K3 surface. The sections {p i } r+1 i=0 form an integral structure of H A,C := Ker ∇ A . This is related to the integral structure H amb A,Z defined in [Iri11, Definition 6.3] by a linear transformation. We consider a map φ : l * H A,C (H R ) → (U ⊕ Pic(Y ) amb ) ⊗ Z C defined bỹ
p 0 → 2π √ −1e,p r+1 → 2π √ −1f,p i → −2πp i (1 ≤ i ≤ r), (7.14)
where U denotes the hyperbolic plane and (e, f ) is its standard basis. This is an isomorphism preserving the pairing. We obtain the period map Here we have an isomorphism D ′ ∼ = D of complex manifolds given by
k 1 e + k 2 f + σ → − √ −1 k 1 σ,(7.18)
where k 1 , k 2 ∈ C and σ ∈ Pic(Y ) amb ⊗ Z C. By this isomorphism, we obtain the period map P : H R → D.
Corollary 7.1. The leading term of the period map P in the limit R → ∞ is given by −2π √ −1z · ψ −1 (c B ), (7.19)
where ψ denotes the map ψ ⊗ Z id R : Pic(Y ) amb ⊗ Z R ֒→ H 1 (B, ι * T ).
Proof. From Theorem 1.2, we can see that the radiance obstruction c B of B is given by
c B = n∈A\{0}
val(k n )ψ(D ρn ), (7.20)
where ρ n ∈ Σ(1) is the cone corresponding to the vertex n ∈ A \ {0} of∆. The holomorphic form Ω z corresponds to F (q) · 1 ∈ H 0 amb (Y, C) under the mirror isomorphism [Iri11, Theorem 6.9], where F (q) is the first term of the Givental's I-function (7.11). It turns out that the period map P : H R → D is given by z → ς(l(z)). The leading term of this is given by r i=1 p i log q i (l(z)) .
(7.21)
Suppose D ρn = r i=1 b n,i p i in Pic(X Σ ), where b n,i ∈ Z. Then we have q i (l(z)) = n∈A\{0} k n (e(z)) b n,i . Corollary 7.1 implies that the radiance obstruction ψ −1 (c B ) ∈ Pic(Y ) amb ⊗ Z R can be regarded as the period of the tropical K3 hypersurface defined by trop(F ). We can also obtain (ψ −1 (c B ), ψ −1 (c B )) ≥ 0 (7.28) from Corollary 7.1 and the inequality (Re τ, Re τ ) > 0 of (7.17). The following inequality (7.29) can be regarded as a tropical version of the Hodge-Riemann bilinear relation for K3 surfaces appearing in (7.16).
Corollary 7.2. One has (ψ −1 (c B ), ψ −1 (c B )) > 0. (7.29)
Proof. From Theorem 1.2 and the assumption that the function A → R, n → val(k n ) induces a central subdivision of∆, we can see that there exists a set of negative real numbers {b(n)} n∈A\{0,n 0 } such that We can infer that we should think that a tropical K3 surface which is obtained by moving singular points to monodromy invariant directions is "equivalent" to the original one.
Remark 7.4. The space
{σ ∈ Pic(Y ) amb ⊗ Z R | (σ, σ) > 0} (7.36)
is the period domain of tropical K3 hypersurfaces. This is the numerator of the moduli space of lattice polarized tropical K3 surfaces [HU18, Section 5]. In [OO18], [OOon], they construct Gromov-Hausdorff compactifications of polarized complex K3 surfaces by adding moduli spaces of lattice polarized tropical K3 surfaces to their boundaries.
Theorem 1. 1 .
1There is a primitive embeddingψ : Pic(Y ) amb ֒→ H 1 (B, ι * T Z ),(1.5)
Theorem 1 . 2 .
12The radiance obstruction c B of B is given byc B = ρ∈Σ(1) {a(n ρ ) − a(0)} ψ(D ρ ).
2
Integral affine structures with singularities and radiance obstructions Let M be a free Z-module of rank n and N := Hom Z (M, Z) be the dual lattice of M. We set M R := M ⊗ Z R, N R := N ⊗ Z R = Hom Z (M, R), and Aff(M R ) := M R ⋊ GL(M). Definition 2.1. An integral affine manifold is a real topological manifold B with an atlas of coordinate charts ψ i : U i → M R such that all transition functions ψ i • ψ −1 j are contained in Aff(M R ).
Remark 2 . 6 .
26The inclusion ι : B 0 ֒→ B induces a map ι * : H 1 (B, ι * T ) ֒→ H 1 (B 0 , T ). Then we can see ι * (c B ) = c B 0 from the definitions.
Let M be a free Z-module of rank n and N := Hom Z (M, Z) be the dual lattice of M. We set M R := M ⊗ Z R and N R := N ⊗ Z R = Hom Z (M, R). We have a canonical R-bilinear pairing −, − : M R × N R → R.
Remark 4 . 2 .
42The monodromy invariant subspace of p coincides with the tangent space of L.
Remark 4 . 3 .
43In the above construction of X k,l , there is an ambiguity in the choice of the position of p ∈ L.
Figure 4
4
Figure 4 . 1 :
41The tropical hypersurface V (f ) and the contraction of the face F 0
Figure 4
4
Figure 4 .
42: A contraction to obtain X 1,1,1
Remark 4 . 7 .
47There is an ambiguity in the choice of the position of each singular point p(τ ) i . However, neither the cohomology group H 1 (B, ι * T Z ) nor the radiance obstruction c B of B does not depend on this choice. We will reconsider this point in Remark 7.3.
Remark 4 . 8 .
48Kontsevich and Soibelman constructed a 2-sphere with an integral affine structure with singularities by contracting a Clemens polytope of a degenerating family of K3 surfaces [KS06, Section 4.2.5]. Their contraction is quite similar to the above contraction of tropical hypersurfaces. Compare the local contraction given in [KS06, Section 4.2.5] to the contraction given in (4.14) of this article.
Example 4. 9 .
9Consider the polynomial
45) and the contraction δ v 1 coincides with a restriction of the contraction with respect to the function f v 1 on O ρ 1 (T) defined by f v 1 (y, z) := y 3 z −1 + y −1 z 3 + y −1 z −1 .(4.46)
Figure 4 .
43: A contraction of the tropical hypersurface V (f )
Let M be a free Z-module of rank 3 and N := Hom(M, Z) be the dual lattice. We set M R := M ⊗ Z R and N R := N ⊗ Z R = Hom(M, R). Let ∆ ⊂ M R be a smooth reflexive polytope of dimension 3, and∆ ⊂ N R be the polar polytope of ∆. Let further Σ andΣ be the normal fans to ∆ and∆ respectively.Let A ⊂ N denote the subset consisting of all vertices of∆ and 0 ∈ N. We consider a tropical Laurent polynomialf (x) = max n∈A {a(n) + n 1 x 1 + n 2 x 2 + n 3 x 3 } , (6.1)such that the functionA → R, n → a(n) (6.2)induces a central subdivision of∆, i.e., every maximal dimensional simplex of the subdivision has the origin 0 ∈ N as its vertex. Let V (f ) be the tropical hypersurface defined by f in the tropical toric variety XΣ(T) associated withΣ. Let further B be the 2-sphere with an integral affine structure with singularities obtained by contracting V (f ) in the way of Section 4.4, and P be the natural polyhedral structure of it. For each 1-dimensional cell of B, we choose the barycenter of it as a position of the singular point that should be on it. We write the complement of singularities of B as ι : B 0 ֒→ B. Let T Z be the local system on B 0 of integral tangent vectors. We set T := T Z ⊗ Z R.
element v(D) always uniquely exists. For each cell τ ∈ P, we choose an arbitrary vertex v of τ and set τ (D) := v(D). Here, when τ is a vertex v of B, we assume τ (D) = v(D). We define a map ψ : Div T (X Σ ) →Ȟ 1 (U, ι * T Z ), D → ψ(D), D) ((U τ 0 , U τ 1 )) := τ 1 (D) − τ 0 (D) (6.8)
sections of ι * T Z . Hence, the vector (v 2 (D) − v 1 (D)) is contained in M. Since there is a sequence of edges of B connecting any vertex of B and v 0 , we have v(D) ∈ M for any other vertex v ∈ P. This happens only when D ∈ Div T (X Σ ).
Figure 6 .
61: 2-simplexes where ψ(D ρ i 1 ) ∪ ψ(D ρ i 2 ) can be nonzero
behaviors of period maps Let K := C (t) be the convergent Puiseux series field, equipped with the standard nonarchimedean valuationval : K −→ Q ∪ {∞}, k = j∈Z c j t j → − min {j ∈ Q | c j = 0} .(7.1) Let M be a free Z-module of rank 3 and N := Hom(M, Z) be the dual lattice. We set M R := M ⊗ Z R and N R := N ⊗ Z R = Hom(M, R). Let ∆ ⊂ M R be a smooth reflexive polytope of dimension 3, and∆ ⊂ N R be the polar polytope of ∆. Let further Σ ⊂ N R ,Σ ⊂ M R be the normal fans of ∆,∆ respectively. We choose a refinementΣ ′ ⊂ M R ofΣ which gives rise to a projective crepant resolution XΣ′ → XΣ of toric varieties associated withΣ.
) induces a central subdivision of∆. Let trop(F ) be the tropicalization of F , which is defined as the tropical polynomial defined by trop(F )(x) := max n∈A {val(k n ) + n 1 x 1 + n 2 x 2 + n 3 x 3 } . (7.3) Let further B be the 2-sphere with an integral affine structure with singularities obtained by contracting the tropical hypersurface defined by trop(F ) in the tropical toric variety XΣ′(T) associated withΣ ′ in the way of Section 4.4. We write the radiance obstruction of B as c B ∈ H 1 (B, ι * T ). Let D ⊂ C be the open unit disk. We consider the universal covering of D \ {0} e : H → D \ {0} , z → exp(2π √ −1z), (7.4)
∈ M ⊗ Z C × . We write the quotient by this action asφ :Ỹ → M reg , where M reg := (C × ) A\{0} reg / (M ⊗ Z C × ). The space M reg can be regarded as a parameter space ofΣ ′ -regular hypersurfaces whose Newton polytopes are∆. Let (H B , ∇ B , H B,Q , F • B , Q B ) be the residual B-model VHS of the familyφ :Ỹ → M reg [Iri11, Definition 6.5].
I
X Σ ,Y (q, z) = e p log qa multi-valued map from an open subset of M × C × to the classical cohomology ring H • (X Σ , C[z −1 ]). Here, Eff(X Σ ) denotes the set of effective toric divisors on X Σ . We write I X Σ ,Y (q, z) = F (q) + G(q) z + H(q) z 2 + O(z −3 ). (7.11)
H
R → P((U ⊕ Pic(Y ) amb ) ⊗ Z C) (7.15)determined by Ω z via the mirror isomorphism and the map φ. The image of this map is contained inD ′ := {[σ] ∈ P((U ⊕ Pic(Y ) amb ) ⊗ Z C) | (σ, σ) = 0, (σ, σ) > 0} .(7.16)We also set D := {σ ∈ Pic(Y ) amb ⊗ Z C | (Re τ, Re τ ) > 0} . (7.17)
. 3 .
3n 0 ∈ A \ {0}, and hence ψ −1 (c B ), ψ −1 (c B As we saw in Remark 4.3 and Remark 4.7, there are ambiguities in the choices of positions of singular points when we contract tropical toric hypersurfaces, and the radiance obstruction does not depend on these choices. This means that moving singular points to monodromy invariant directions does not change the period of the tropical K3 surface B.
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| []
|
[
"Compact radio emission in Ultraluminous X-ray sources",
"Compact radio emission in Ultraluminous X-ray sources"
]
| [
"Astron ",
"Nachr "
]
| []
| []
| We present results from our studies of radio emission from selected Ultraluminous X-ray (ULX) sources, using archival Giant Metrewave Radio Telescope (GMRT) data and new European VLBI Network (EVN) observations. The GMRT data are used to find possible faint radio emission from ULX sources located in late-type galaxies in the Chandra Deep Fields. No detections are found at 235, 325 and 610 MHz, and upper limits on the radio flux densities at these frequencies are given. The EVN observations target milliarcsecond-scale structures in three ULXs with known radio counterparts (N4449-X1, N4088-X1, and N4861-X2). We confirm an earlier identification of the ULX N4449-X1 with a supernova remnant and obtain the most accurate estimates of its size and age. We detect compact radio emission for the ULX N4088-X1, which could harbour an intermediate mass black hole (IMBH) of 10 5 M⊙ accreting at a sub-Eddington rate. We detect a compact radio component in the ULX N4861-X2, with a brightness temperature > 10 6 K and an indication for possible extended emission. If the extended structure is confirmed, this ULX could be an HII region with a diameter of 8.6 pc and surface brightness temperature ≥ 10 5 K. The compact radio emission may be coming from a ∼ 10 5 M⊙ black hole accreting at 0.1Ṁ Edd . | 10.1002/asna.201011504 | [
"https://arxiv.org/pdf/1011.0946v2.pdf"
]
| 118,516,628 | 1011.0946 | 4f43c224577c9f71599990aab14890ebbcf92b8d |
Compact radio emission in Ultraluminous X-ray sources
2011
Astron
Nachr
Compact radio emission in Ultraluminous X-ray sources
999992011/ DOI please set DOI! The dates of receipt and acceptance should be inserted laterGalaxies: general -X-rays: general -ISM: HII regions
We present results from our studies of radio emission from selected Ultraluminous X-ray (ULX) sources, using archival Giant Metrewave Radio Telescope (GMRT) data and new European VLBI Network (EVN) observations. The GMRT data are used to find possible faint radio emission from ULX sources located in late-type galaxies in the Chandra Deep Fields. No detections are found at 235, 325 and 610 MHz, and upper limits on the radio flux densities at these frequencies are given. The EVN observations target milliarcsecond-scale structures in three ULXs with known radio counterparts (N4449-X1, N4088-X1, and N4861-X2). We confirm an earlier identification of the ULX N4449-X1 with a supernova remnant and obtain the most accurate estimates of its size and age. We detect compact radio emission for the ULX N4088-X1, which could harbour an intermediate mass black hole (IMBH) of 10 5 M⊙ accreting at a sub-Eddington rate. We detect a compact radio component in the ULX N4861-X2, with a brightness temperature > 10 6 K and an indication for possible extended emission. If the extended structure is confirmed, this ULX could be an HII region with a diameter of 8.6 pc and surface brightness temperature ≥ 10 5 K. The compact radio emission may be coming from a ∼ 10 5 M⊙ black hole accreting at 0.1Ṁ Edd .
Introduction
Several scenarios have been proposed to explain the high luminosities (L X > 10 39 erg/s) of Ultraluminous X-ray sources (ULXs), but none of them is able to reveal the physical nature of all ULXs. If ULXs are powered by accretion at the Eddington rate, this would imply accreting compact objects of masses 10 2 -10 5 M ⊙ . Such intermediate compact objects can only be black holes (Colbert & Mushotzky 1999) and they would be the missing link between stellar mass black holes and supermassive black holes in the nuclei of galaxies. These Intermediate Mass Black Holes (IMBHs) could form from the death of very massive and hot stars or from multiple stellar interactions in dense stellar clusters (Portegies Zwart 2003). It has also been suggested that ULX objects may harbour secondary nuclear black holes in postmerger galaxies (Lobanov 2008), with masses in excess of 10 5 M ⊙ and accreting at sub-Eddington rates. Alternatively, ULXs could be neutron stars or stellar mass BHs apparently radiating at super-Eddington luminosities (Begelman 2002).
Radio observations of ULXs bear an excellent potential for uncovering the nature of these objects, by detecting and possibly resolving their compact radio emission, measuring its brightness temperature and spectral properties, and assessing the physical mechanism for its production. Few ULXs have been studied in the radio domain (Kaaret et al. 2003;Körding et al. 2005) and a small sample of ⋆ e-mail: [email protected] Member of the International Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the Universities of Bonn and Cologne. ULXs has been cross-identified in the existing radio catalogs (Sánchez-Sutil et al. 2006).
An increase of the number of radio detections and subsequent Very Long Baseline Interferometry (VLBI) studies of detected radio counterparts could potentially help to clarify the nature of ULX sources. With this aim, we 1) analyze archive images of the Chandra Deep Fields taken with the Giant Meterwave Radio Telescope (GMRT) looking for faint radio counterparts of the ULX sources located in this fields; and 2) initiate an European VLBI Network (EVN) program to detect and study milliarcsecond-scale emission in ULX objects with known radio counterparts. In Section 2, we present the two samples of ULX objects studied. The observations and data reduction are explained in Section 3. The results obtained are shown in Section 4, leading to a discussion and final summary presented in Section 5.
Throughout this paper we assume a Λ cold dark matter (CDM) cosmology with parameters H 0 = 73 km s −1 M pc −1 , Ω Λ = 0.73 and Ω m = 0.27.
Observing targets
We use archival GMRT data to search for radio counterparts of 24 ULX objects identified in the Chandra Deep Field North (CDFN), Chandra Deep Field South (CDFS), and Extended CDFS (Lehmer et al. 2006). All these ULXs have luminosities L X ≥ 10 39 erg/s in the 0.5-2.0 keV band, and are located in optically bright irregular and late-spiral galaxies. Ten of the 24 X-ray sources appear to be coincident with optical knots of emission, with optical properties that are consistent with those of giant HII regions in the local universe, suggesting that these ULX sources trace distant star formation (Lehmer et al. 2006).
The objects targeted in our EVN observations are selected from a sample of 11 ULX objects with radio counterparts (Sánchez-Sutil et al. 2006) in the VLA FIRST catalog (Becker et al. 1995). We select three ULX which are brighter than 1 mJy and clearly away from the nuclear region in their respective host galaxies. The first target, N4449-X4, is identified as the most luminous and distant member of the class of oxygen-rich Supernova Remnants (SNRs) (Blair et al. 1983) and classified as an ULX source by Liu & Bregman 2005. A detailed study of this source is described in Mezcua & Lobanov (in prep.). The second target, N4088-X1, is an ULX located at a distance of 13.0 Mpc in the asymmetric spiral galaxy NGC 4088. This ULX is located within the extended emission of a spiral arm and is coincident with a conspicous maximum of radio emission of 1.87 mJy (at 1.4 GHz) with an offset of 3.62" to the Xray peak (Sánchez-Sutil et al. 2006). The ULX has a X-ray luminosity in the 0.3-8.0 keV band of 5.86 x 10 39 erg/s (Liu & Bregman 2005, who don't rule out the possibility of it being an HII region). The third target, N4861-X2, is located in the spiral galaxy NGC 4861 and has an X-ray luminosity of 8.4 x 10 39 erg/s (Liu & Bregman 2005). Its radio counterpart is offset from the X-ray position by 1.97". This ULX has been suggested to coincide with an HII region powered by massive early OB type stars (Pakull & Mirioni 2002). The observations were carried out using the standard GMRT phase calibration mode and spectral line mode. The data were analyzed with the NRAO Astronomical Image Processing System (AIPS). In order to detect very faint sources, we first image the entire primary beam area and extract all strong point-like objects. Then the cleaned field of strong sources is subtracted from the visibility data, and deeper imaging is performed. Finally selfcalibration is applied to increase the dynamic range of the resulting image. At 610 MHz, 3 pointings of the same field (CDFS) are observed.
Observations and data reduction
The EVN observations (project codes EM072A & EM072B) have been made on June 1st & 2nd 2009, in two separate blocks of 12 hours in duration, using 9 antennas (Ef, Jb-1, Cm, Wb, On, Mc, Nt, Tr, Sh) at the wavelength of 18 cm. In order to detect weak emission from the ULX objects, we use the phase-referencing technique, calibrating the phases of the target objects with nearby strong and point-like calibrators. The data reduction was performed in the standard way using AIPS. Uniform weighting was used in the imag- ing, and tapering of the longest baselines was applied for N4088-X1 and N4861-X2 to improve the detection of extended emission.
Results
ULX sources in the Chandra Deep Fields
We show the final GMRT images obtained at 235 MHz and 325 MHz in Fig. 1. The primary beam sizes are 114 arcmin at 235 MHz, 81 arcmin at 325 MHz, and 43 arcmin at 610 MHz. The respective rms noise in the maps at each frequency are 1.4, 0.6, and 0.8 mJy/beam. No radio counterparts of the ULX sources located in the CDFs are detected in a circle radius for each ULX position of 28 arcsec, which is more than 10 times the Chandra positional error circle. Upper limits on their flux densities at each frequency are given in Table 1 (for ULX located in the CDFN) and Table 2 (ULXs in the CDFS), obtained by estimating the local rms at the ULX locations. These upper limits range between 2-4.6 mJy at 235MHz, 1-2.5 mJy at 332 MHz, and 0.5-2 mJy at 610MHz. The position of only three ULX sources fall in the images at 610MHz, thus only 3 upper limits are given at this frequency.
Compact radio emission in ULX sources
In Fig. 2 we show the final images of N4088-X1 (left) and N4861-X2 (right). The noise levels achieved are 26µJy/beam for N4088-X1 and 3µJy/beam for N4861-X2, and the restoring beams are 31 x 29 mas and 11 x 5 mas, respectively. For N4088-X1, we identify a compact component of flux density 0.1 mJy at a 5σ level. The component is centered at RA(J2000) = 12 h 05 m 31.7110 s ± 0.0003 s , DEC(J2000) = 50 • 32'46.729" ± 0.002". For this component, we estimate a brightness temperature of T B > 7 x 10 4 K and an upper limit of 34 x 26 mas for the size. Adopting a distance of 13.0 Mpc yields, for N4088-X1, an integrated 1.6 GHz radio luminosity of 3.8 x 10 34 erg/s. The ULX N4861-X2 (Fig 2, right) has a compact component A centered at RA(J2000) = 12 h 59 m 00.3563037 s ± 0.0000008 s , DEC(J2000) = 34 • 50'42.87500" ± 0.00002". It has a flux density of ∼80µJy (for which we derive a radio luminosity L 1.6GHz = 3.3 x 10 34 erg/s assuming a distance to the host galaxy of 14.80 Mpc) and a size upper limit of 9.8 x 3.8 mas, corresponding to a brightness temperature T B > 1.1 x 10 6 K. Two additional components (B and C), with a total flux density of ∼70µJy are detected, but cannot be firmly localized with the present data. If this extention were confirmed, the whole structure (including component A, B
& C) would have a total flux density of 0.18 mJy, a luminosity of L 1.6GHz = 7.7 x 10 34 erg/s and diameter D∼120 mas. This diameter corresponds to 8.6 pc at the distance of the host galaxy, and it is in agreement with the typical size of HII regions found in our Galaxy, like G18.2-0.3 (Fürst et al. 1987), which has a size of 200 mas, a luminosity of L 1.4GHz = 1.1 x 10 33 erg/s and is formed by several discrete sources.
Discussion
We use the upper limits obtained from the GMRT data on the radio flux densities of the ULX objects to locate them in the fundamental plane of sub-Eddington accreting black holes (cf., Corbel et al. 2003;Gallo et al. 2003;Merloni et al. 2003;Falcke et al. 2004) as defined by a correlation between radio core (L R ) and X-ray (L X ) luminosity and black hole mass, M BH , log L R = 0.6 log L X + 0.78 log M BH + 7.33. For our calculations, we assume a radio spectral index α R ≃ 0.15 and a X-ray spectral index α X ≃ −0.6 adopted previously by Falcke et al. (2004). The resulting radio and X-ray luminosities of the ULX objects in our sample are compared in Fig. 3 to the results of Corbel et al. (2003) and Merloni et al. (2003). The resulting high upper limits on the BH masses do not provide strong constraints on the nature of these ULX objects.
A similar relation is shown in Fig. 4 for the most compact components in the EVN images of N4088-X1 and N4861-X2. Using our radio luminosity at 1.6 GHz appropriately scaled to 5 GHz, the X-ray luminosity scaled to the 2-10 www.an-journal.org (-2, 2, 3, 4, 5) x 26 µJy/beam, the rms noise off-source. For the ULX on the right, the contours are (-32, 2, 3, 4, 5, 6, 7, 10, 20, 30, 40, 50) x 3 µJy/beam. These radio counterparts are offset from the X-ray peak position by 3.62" for N4088-X1, and by 1.97" for N4861-X2. These offsets lie within the X-ray positional error. The compact component of N4861-X2 is indicated with an A. Possible extended emission might be detected in regions B and C. keV band, and assuming a sub-Eddington accretion regime, we derive a black hole mass of 10 5.1 M ⊙ and of 10 4.9 M ⊙ for N4088-X1 and N4861-X2 (component A), respectively. These masses are in agreement with the IMBH scenario for both objects.
Higher sensitivity observations are needed, and observational time has already been guaranteed, to try to detect and/or confirm possible extended structure for both N4088-X1 and N4861-X2.
Summary
Radio observations of ULX sources can help to unveil the nature of these objects. Analysis of archival GMRT data of the Chandra Deep Fields at 235, 325 & 610 MHz have not yielded any radio counterparts for these sources but yielded upper limits on their flux densities. These ULXs are too weak for deep field radio observations so higher sensitivity is needed in order to detect any faint radio emission.
New EVN observations of three ULXs with known radio counterparts yielded first milliarcsecond-scale images of all three objects. The EVN observations have confirmed the earlier identification of the ULX N4449-X1 with a SNR and obtained the most accurate estimates of its size and age (Mezcua & Lobanov in prep.). For the two other ULXs studied, N4088-X1 and N4861-X2, the EVN measurements have provided improved estimates of the compact radio flux, yielding better localizations of these objects in the L X -L radio diagram. The suggested nature of these objects can be best verified with more sentitive observations at 5 GHz aimed at both improving the brightness temperature estimates and obtaining spectral index information. The success of the EVN observations also calls for expanding this study to more ULX objects. new (this work, red squares) location of the ULX sources N4088-X1 and N4861-X2 in the fundamental plane of sub-Eddington accreting black holes. The parallel lines correspond to the labeled black hole mass relative to that of the Sun. We show for comparison the Corbel et al. (2003) data for the X-ray binary GX 339-4 (filled circles), and the Merloni et al. 2003 www.an-journal.org
We use archival GMRT data of the Hubble Deep Field North (overlapping with our CDFN region of interest) at 235 MHz (experiments 01NIK04 & 11TMA01) observed on 2001 December 17, 2002 January 4 and 2002 January 26, and of the CDFS at 325 & 610 MHz (experiments 03JAA01 & 11RNA01), observed on 2003 February 13-17 and 2007 February 11-12, respectively.
Fig. 1
1GMRT images of the HDFN at 235 MHz (left) and the CDFS at 325 MHz (right). The restoring beam sizes are 20.1 x 16.3 arcsec at 235 MHz and 13.7 x 11.8 arcsec at 325 MHz. The best rms sensitivities achieved are 1.4, and 0.6 mJy/beam, respectively. The ULXs positions are marked with squares.
Fig
CONT: N4088-X1 IPOL 1650.561 MHZ N4088-LOW.ICL001.1 Cont peak flux = 1.3259E-04 JY/BEAM Levs = 2.599E-05 * (31.718 31.716 31.714 31.712 31.710 31.708 31.706 31.. 2 1.6 GHz EVN images of the ULX sources N4088-X1 (left) and N4861-X2 (right). The restoring beam sizes are 31 x 29 mas for N4088-X1 and 11 x 5 mas for N4861-X2, with the major axis of the beam oriented along a position angle of 21.99 • for N4088-X1 and 11.36 • for N4861-X2. The contours for the left image are
Fig. 3 Fig. 4
34Location of the 24 ULXs (red arrows) in the fundamental plane of sub-Eddington accreting black holes. The parallel lines correspond to the labeled black hole mass relative to that of the Sun. We show for comparison theCorbel et al. (2003) data for the X-ray binary GX 339-4 (filled circles), and theMerloni et al. (2003) data for the Low Luminosity AGN (LLAGN) NGC 2787, NGC 3147, NGC 3169, NGC 3226, and NGC 4143 (inverted triangles). Old (Sánchez-Sutil et al. 2006, grey squares) and
Table 1
1ULX in the CDFN at 235.5MHz.Name
S
[mJy/beam]
CXOHDFN J123631.66+620907.3
< 3.63
CXOHDFN J123632.55+621039.5
< 2.76
CXOHDFN J123637.18+621135.0
< 2.88
CXOHDFN J123641.81+621132.1
< 3.20
CXOHDFN J123701.47+621845.9
< 2.35
CXOHDFN J123701.99+621122.1
< 3.12
CXOHDFN J123706.12+621711.9
< 2.06
CXOHDFN J123715.94+621158.3
< 4.58
CXOHDFN J123721.60+621246.8
< 3.47
CXOHDFN J123723.45+621047.9
< 4.58
CXOHDFN J123727.71+621034.3
< 4.21
CXOHDFN J123730.60+620943.1
< 2.11
Table 2
2ULX in the CDFS and ECDFS at 332MHz and 610MHz.Name
S332MHz
S610MHz
[mJy/beam] [mJy/beam]
CXOECDFS J033122.00-273620.1
< 1.17
...
CXOECDFS J033128.84-275904.8
< 1.38
...
CXOECDFS J033139.05-280221.1
< 1.38
< 1.62
CXOECDFS J033143.46-275527.8
< 1.45
...
CXOECDFS J033143.48-275103.0
< 1.63
...
CXOCDFS J033219.10-274445.6
< 1.85
...
CXOCDFS J033221.91-275427.2
< 2.39
...
CXOCDFS J033230.01-274404.0
< 1.63
...
CXOCDFS J033234.73-275533.8
< 1.28
...
CXOECDFS J033249.26-273610.6
< 2.98
...
CXOECDFS J033316.29-275040.7
< 0.99
< 0.66
CXOECDFS J033322.97-273430.7
< 1.08
< 0.80
data for the Low Luminosity AGN (LLAGN) NGC 2787, NGC 3147, NGC 3169, NGC 3226, and NGC 4143 (inverted triangles).
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.an-journal.org
Acknowledgements. The authors are grateful to M. López-Corredoira and M. W. Pakull for their valuable comments. M. Mezcua was supported for this research through a stipend from the International Max Planck Research School (IMPRS) for Radio and Infrared Astronomy at the Universities of Bonn and Cologne.
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| []
|
[
"Subatomic systems need not be subatomic",
"Subatomic systems need not be subatomic"
]
| [
"Luca Roversi [email protected] \nDipartimento di Informatica\nUniversit di Torino C.so Svizzera 185\n10149TorinoITALY\n"
]
| [
"Dipartimento di Informatica\nUniversit di Torino C.so Svizzera 185\n10149TorinoITALY"
]
| []
| Subatomic systems were recently introduced to identify the structural principles underpinning the normalization of proofs. "Subatomic" means that we can reformulate logical systems in accordance with two principles. Their atomic formulas become instances of sub-atoms, i.e. of non-commutative self-dual relations among logical constants, and their rules are derivable by means of a unique deductive scheme, the medial shape. One of the neat results is that the cutelimination of subatomic systems implies the cut-elimination of every standard system we can represent sub-atomically.We here introduce Subatomic systems-1.1. They relax and widen the properties that the sub-atoms of Subatomic systems can satisfy while maintaining the use of the medial shape as their only inference principle. Since sub-atoms can operate directly on variables we introduce P. The cut-elimination of P is a corollary of the cut-elimination that we prove for Subatomic systems-1.1. Moreover, P is sound and complete with respect to the clone at the top of Post's Lattice. I.e. P proves all and only the tautologies that contain conjunctions, disjunctions and projections. So, P extends Propositional logic without any encoding of its atoms as sub-atoms of P.This shows that the logical principles underpinning Subatomic systems also apply outside the sub-atomic level which they are conceived to work at. We reinforce this point of view by introducing the set of medial shapes R 23 . The formulas that the rules in R 23 deal with belong to the union of two disjoint clones of Post's Lattice. The SAT-problem of the first clone is in P-Time. The SAT-problem of the other is NP-Time complete. So, R 23 and the proof technology of Subatomic systems could help to identify proof-theoretical properties that highlight the phase transition from P-Time to NP-Time complete satisfiability. | null | [
"https://arxiv.org/pdf/1804.08105v1.pdf"
]
| 5,079,047 | 1804.08105 | 26149940c9acf899d5c7ae3f214c92d653cb72d6 |
Subatomic systems need not be subatomic
22 Apr 2018 April 24, 2018
Luca Roversi [email protected]
Dipartimento di Informatica
Universit di Torino C.so Svizzera 185
10149TorinoITALY
Subatomic systems need not be subatomic
22 Apr 2018 April 24, 2018arXiv:1804.08105v1 [cs.LO]
Subatomic systems were recently introduced to identify the structural principles underpinning the normalization of proofs. "Subatomic" means that we can reformulate logical systems in accordance with two principles. Their atomic formulas become instances of sub-atoms, i.e. of non-commutative self-dual relations among logical constants, and their rules are derivable by means of a unique deductive scheme, the medial shape. One of the neat results is that the cutelimination of subatomic systems implies the cut-elimination of every standard system we can represent sub-atomically.We here introduce Subatomic systems-1.1. They relax and widen the properties that the sub-atoms of Subatomic systems can satisfy while maintaining the use of the medial shape as their only inference principle. Since sub-atoms can operate directly on variables we introduce P. The cut-elimination of P is a corollary of the cut-elimination that we prove for Subatomic systems-1.1. Moreover, P is sound and complete with respect to the clone at the top of Post's Lattice. I.e. P proves all and only the tautologies that contain conjunctions, disjunctions and projections. So, P extends Propositional logic without any encoding of its atoms as sub-atoms of P.This shows that the logical principles underpinning Subatomic systems also apply outside the sub-atomic level which they are conceived to work at. We reinforce this point of view by introducing the set of medial shapes R 23 . The formulas that the rules in R 23 deal with belong to the union of two disjoint clones of Post's Lattice. The SAT-problem of the first clone is in P-Time. The SAT-problem of the other is NP-Time complete. So, R 23 and the proof technology of Subatomic systems could help to identify proof-theoretical properties that highlight the phase transition from P-Time to NP-Time complete satisfiability.
Introduction
Subatomic systems were recently introduced to identify the structural principles underpinning the normalization of proofs. "Subatomic" means that we can reformulate logical systems in accordance with two principles. The atomic constituents of the formulas become instances of sub-atoms, i.e. of non-commutative self-dual relations among logical constants, and the rules are derivable by means of a unique deductive scheme, the medial shape. In its not full, but general enough, form it is:
(A α B) β (C γ D) (A β C) α (B δ D)
where A, B, C, D are formulas and α, β, γ, δ relations. For example, let us focus on propositional logic. The sub-atomic rule (A ∨ B) ∧ (C ∨ D)
(A ∧ C) ∨ (B ∨ D)
stands for the introduction to the right of the conjunction. It is a rule in deep inference which we can read as follows. Let A∨ B and C ∨D be two given disjunctions where B is the premise that allows to derive A and D the one for deriving C. Then, the rule derives A ∧ C from the premise B ∨ D. The sub-atomic rule (f ∨ t) a (t ∨ f) (f a t) ∨ (t a f) represents the excluded-middle t a ∨ a . The sub-atoms (f a t) and (t a f) stand for the atoms a and a, respectively, where a is a self-dual non commutative relation which obeys the equivalence (f ∨ t) a (t ∨ f) = t a t = t. Instead, the rule (f a t) ∨ (f a t)
(f ∨ f) a (t ∨ t) corresponds to the contraction a ∨ a a . Under the same representation of a as before, the conclusion represents a up to the standard equivalences f ∨ f = f and t ∨ t = t. One reason why Subatomic systems are a deep inference formalism is that they target the representation of a class of logical systems as wide as possible which may well include self-dual non-commutative logical operators and we know that there cannot be analytic and complete Gentzen (linear) proof systems with self-dual non-commutative connectives in them [10]. Another reason is that, by means of the uniform representation they allow, Subatomic systems help to identify sufficient conditions to characterize proof systems that enjoy decomposition, i.e. the reorganization of contractions inside a proof, and cut-elimination. This is possible because Subatomic systems abstract at the right level the proofs of decomposition and of cut-elimination that the literature contains in relation to deep inference logical systems for classical, modal, linear and sub-structural logics.
Very briefly, deep inference looks at deductive processes as rewriting procedures where rules apply to an arbitrary depth in the syntax tree of formulas. This is equivalent to saying that deep inference logical systems compose derivations and formulas exactly with the same set of logical connectives. Subatomic systems witness how effective the reduction can be of syntactic bureaucracy that follows from the deep inference approach to proof theory to get closer to the semantic nature of proof and proof normalization. An informative survey about deep inference is [3]. An up-do-date information about its literature is [4].
This paper introduces Subatomic systems-1.1 (Section 2), a slight generalization of the original Subatomic system in [11,12] that we dub as version 1.0, for easiness of reference. Version 1.1 relaxes and widen the properties that the sub-atoms of version 1.0 can satisfy while maintaining the use of the medial shape as the only inference principle. As effect of the generalization, the formulas of Subatomic systems 1.1 build also on variables. Hence, we can introduce P (Section 3.) We show that P is sound and complete with respect to the clone at the top of Post's Lattice (Section 6.) I.e. P proves all and only the tautologies that contain conjunctions, disjunctions and the self-dual projections π 0 and π 1 . So, P extends Propositional logic without any encoding of its atoms as sub-atoms of P. We also prove that the cut and other rules are admissible for a specific fragment of P (Section 5.) The proof is a corollary of the same property that we prove for version 1.1 and which extends the one for version 1.0 (Section 4.)
The existence of P shows that the logical principles underpinning Subatomic systems also apply outside the subatomic level which they are conceived to work at. We reinforce this idea by introducing the set R 23 of medial shapes (Section 7.) The formulas that occur in the rules of R 23 belong to the union of the two clones C 2 and C 3 of Post's Lattice [7]. Both C 2 and C 3 are two of the five maximal clones strictly contained in C 1 . The logical operators that build the formulas of C 2 and C 3 are strongly interrelated but the satisfiability problem for C 2 is in P-Time while the one for C 3 is NP-Time complete. That R 23 can be a Subatomic system-1.1 is still an open question. The conjecture is that we need a further extension of Subatomic systems to prove a cut-elimination for a system with R 23 as its core. The relevance of R 23 is twofold. On one side, it can help focusing on proof-theoretical properties that highlight how and when the phase transition from the satisfiability in P-Time to the satisfiability in the class of NP-Time complete problems occurs. On the other, the way we obtain R 23 strongly suggests that Subatomic systems can be viewed as a framework where looking for grammars that follow a very regular pattern able to generate possibly interesting logical systems, so contributing to the so called systematic proof theory [1]. The side effect would be that the larger will be the class of interesting logical systems that we can generate by means of Subatomic system, the clearer the reason could be why the medial scheme rule is so pervasive, something that, so far, has no a priory convincing explanation.
2 Subatomic systems-1.1
We generalize Subatomic systems-1.0 [11,12] to Subatomic systems-1.1.
Definition 1 (Subatomic systems-1.1). Let U be a denumerable set of constants t, u, v, . . .. Let V be a denumerable set of variables x, y, w, . . .. Let R be a denumerable set of symbol relations α, β, . . . and let ≺ ⊆ R 2 be a partial order among the symbols in R. Let F ::= U | V | F R F generate formulas A, B, C, . . .. Let ( ) : (U → U) ∪ (V → V) ∪ (R → R) be an involutive negation:
A = u if A = u and u ∈ U x if A = x and x ∈ V B α C if A = B α C and B α C ∈ F .
Fixed n ∈ N, let = ⊆ F 2 be the least congruence on F generated by any subset E 1 = F 1 , . . . , E n = F n of axioms taken among following:
(A α B) α C = A α (B α C) (A, B, C ∈ F , α ∈ R) (1) A α B = B α A (A, B ∈ F , α ∈ R) (2) A α u α = A = u α α A (A ∈ F , u α ∈ U, α ∈ R) (3) A α B = A (A, B ∈ F , α ∈ R) (4) B α A = A (A, B ∈ F , α ∈ R) (5) t α u = v (t, u, v ∈ U, α ∈ R) (6) x α y = z (x, y, z ∈ V, α ∈ R) (7) x α y = u (x, y ∈ V, u ∈ U, α ∈ R) (8) t = u (t, u ∈ U) .(9)
A Subatomic system-1.1 S on F , R, ≺ and = has all and only the instances of the following schemes: [11,12], the role of Definition 1 is to delineate the formal framework we are going to work in. The constraints on the framework are very lax. It should not surprise how simple is to think of semantically meaningless instances of Subatomic systems where, for example, the two propositional constants T (true) and F (false) exist and are equated by an instance of (9).
Down-rules Up-rules Splitting (A β B) α (C β D) (α ≺ β) (A α C) β (B α D) (A α C) β (B α D) (α ≻ β) (A β B) α (C β D) Contractive (A α B) β (C α D) (α ≺ β) (A β C) α (B β D) (A β C) α (B β D) (α ≻ β) (A α B) β (C α D) Equational E i F i E i F i • Like in
The framework we delineate is slightly more general than the one in [11,12]. The language F contains variables of V and the set of axioms is extended in two directions. Axioms (4), (5) admit the existence of relations that erase structure. Axioms (7) and (8) allow the existence of relations among constants and variables. This extends the proof theoretical technology of Subatomic systems-1.0 outside its intrinsic sub-atomic nature.
Notation and terminology. Let S be a Subatomic system-1.1 with formulas F built on the symbols in R. Let ≺ be the order relation on R 2 . A context S { } is a formula A ∈ F with any of its sub-formulas, possibly A itself, erased. In the last
case S { } is { }. A relation α of S is unitary if it enjoys axiom (3). A relation α is a right weakening if it enjoys (4) and is a left weakening if (5) holds for it. A relation α ∈ R is strong if no β ∈ R exists such that β ≺ α. A relation α ∈ R is weak if no β ∈ R exists such that α ≺ β. The map ( ) is ≺-consistent if a strong α ∈ R implies that α is weak, and vice versa. A derivation A D S
B of S from A to B is any obvious concatenation of rules instances of S.
• Remark 1. Strong relations are defined as minimal elements of the partial order ≺ ⊆ R 2 . Dually, weak relations are maximal elements. We share this terminological choice with [12]. The justification is semantical. A relation is strong if its truth implies the truth of a weaker one. For example, the classical conjunction is strong and the classical disjunction weak. • Proposition 1 (Excluded middle). Let S be a Subatomic system-1.1 with = as its equational theory. Let α ∈ R be strong. Let the following instances of (6) and (8) hold in S:
v α v = u α (∀v ∈ U)(10)u α γ u α = u α (∀γ ∈ R.γ ≺ α)(11)
x
α x = u α (∀x ∈ V)(12)
where u α is a single and distinguished element of U. The rule
u α A α A is derivable, for every A ∈ F .
Proof. The proof is by induction on the structure of A. The two base cases with A = x or A = v hold because (12) and (10) hold in the given S. Let A be A 0 γ A 1 where, we underline, γ can also be α itself. Moreover, α strong implies that α weak.
Then
u α (11) u α γ u α inductive hypothesys (A 0 α A 0 ) γ (A 1 α A 1 ) (γ ≺ α) (A 0 γ A 1 ) α (A 0 γ A 1 ) (A 0 γ A 1 ) α (A 0 γ A 1 )
. Proposition 1 justifies the following:
Definition 2 (Unit). The constant u α ∈ U is a unit if it enjoys axioms (10), (11) and (12).
• Proposition 2 (Contraction). Let S be a Subatomic system-1.1 with = as its equational theory. Let β ∈ R be weak. Let the following instances of (1), (2), (6) and (7) hold in S:
(A β B) β C = A β (B β C) (∀A, B, C ∈ F ) (13) A β B = B β A (∀A, B ∈ F ) (14) v β v = v (∀v ∈ U) (15) x β x = x (∀x ∈ V) . (16) The rule A β A A is derivable, for every A ∈ F .
Proof. The proof is by induction on the structure of A. The base cases A = x and A = v holds because (15) and (16) hold in the given S. Let A be A 0 γ A 1 , for any γ ≺ β. Then the following derivation
(A 0 γ A 1 ) β (A 0 γ A 1 ) (γ ≺ β) (A 0 β A 0 ) γ (A 1 β A 1 ) inductive hypothesys A 0 γ A 1 exists. Finally, let A be A 0 β A 1 . Then (A 0 β A 1 ) β (A 0 β A 1 ) (13),(14) (A 0 β A 0 ) β (A 1 β A 1 ) inductive hypothesys A 0 β A 1 exists.
Propositions 1 and 2 say that the medial shape is an invariant of two inference mechanisms. One is "Splitting" or, dually, "annihilation". It distributes negation. So, the proofs of a Subatomic system-1.1 can start from units which split into a pair of structures that annihilate each other. The other is "Contraction" or, dually, "sharing". It distributes subformulas with the goal of identifying two occurrences of the same formula into a single one. This is a consequence of a step-wise deductive process that reduces the global identification to the identification on constants or variables only. We introduce the instance P of Subatomic systems-1.1 which we could not see how to obtain as an instance of Subatomic systems-1.0 [11,12].
Definition 3 (Formulas of P). Let F P be the language of formulas generated by:
A, B ::= T | F | V P | V P | A ∧ B | A π 0 B | A π 1 B | A ∨ B .
The set V P contains the variables x, y, z, . . . and V P their negations. Both π 0 and π 1 stand for the self-dual projections on first or second argument, respectively.
• Definition 4 (Order relation among the relations of P). The operator ∧ is strong, ∨ is weak and every π i is in between.
i.e. A ∧ B ≺ P A π 0 B, A π 1 B ≺ P A ∨ B. •
The order relation of Definition 4 originates from the following lattice which pointwise sorts the boolean functions it contains under the assumption that F is smaller than T:
B A π 0 B F T A F F F T T T B A ∧ B F T A F F F T F T B A ∨ B F T A F F T T T T B A π 1 B F T A F F T T F T
Definition 5 (Negation among formulas of P). For every x, A, B ∈ F P , let ( ) be the following involutive and ≺ P -consistent negation:
T = F (17) F = T x = x (∀x ∈ V) A ∨ B = A ∧ B (∀A, B ∈ F ) (18) A ∧ B = A ∨ B (∀A, B ∈ F ) A π i B = A π i B (∀A, B ∈ F and i ∈ {0, 1}) .(19)
• Axiom (19) sets π 0 and π 1 to be self-dual operators like the boolean functions they represent.
Definition 6 (Congruence on formulas of P). Let F be the unit u ∨ of ∨ and T the unit u ∧ of ∧. For every A, B and C in F P , let = P be the congruence that the following axioms induce:
(A α B) α C = A α (B α C) (∀A, B, C ∈ F and α ∈ {π 0 , π 1 , ∨, ∧}) (20) A α B = B α A (∀A, B ∈ F and α ∈ {∨, ∧}) (21) A ∨ F = A (∀A ∈ F ) (22) A π 0 B = A (∀A, B ∈ F ) (23) A π 1 B = B (∀A, B ∈ F ) (24) u ∧ F = F (u ∈ {F, T}) (25) u ∨ u = T (u ∈ {F, T}) (26) x ∨ x = T (∀x ∈ V P ) (27) u ∨ u = u (u ∈ {F, T}) (28) x ∨ x = x (∀x ∈ V P ) .(29)
• Definition 6 gives the least set of axioms. The missing ones can be derived by negation.
Definition 7 (System P). P contains the rules:
(A ∨ B) π j (C ∨ D) ai j ↓ j ∈ {0, 1} (A π j C) ∨ (B π j D) (A π j C) ∧ (B π j D) ai j ↑ j ∈ {0, 1} (A ∧ B) π j (C ∧ D) (A ∨ B) ∧ (C ∨ D) s↓ (A ∧ C) ∨ (B ∨ D) (A ∨ C) ∧ (B ∧ D) s↑ (A ∧ B) ∨ (C ∧ D) (A ∧ B) ∨ (C ∧ D) m↓ (A ∨ C) ∧ (B ∨ D) (A ∧ C) ∨ (B ∧ D) m↑ (A ∨ B) ∧ (C ∨ D) (A π j B) ∨ (C π j D) c j ↓ j ∈ {0, 1} (A ∨ C) π j (B ∨ D) (A ∧ C) π j (B ∧ D) c j ↑ j ∈ {0, 1} (A π j B) ∧ (C π j D)
with formulas of F P (Definition 3) taken up to both = P (Definition 6) and the negation in Definition 5, with ∧, π 0 , π 1 and ∨ ordered under ≺ P (Definition 4.) • So, P is a Subatomic system-1.1 because its formalization fits in the framework of Definition 1. Hence, Proposition 1, axioms (20), (21), (28) and (29), and the rules ai 0 ↓, ai 1 ↓, s↓ imply:
Corollary 1 (Excluded middle in P). For every A ∈ F P , the rule T A ∨ A is derivable.
Moreover, Proposition 2, axiom (22), (23), (24), (26) and 27, and the rules m↓, c 0 ↓, c 1 ↓ imply:
Corollary 2 (Idempotence in P). For every A ∈ F P , the rule A ∨ A A is derivable.
Remark 2. As far as we can see, P cannot be a Subatomic system-1.0, in accordance with Definition 2.5 in [11, page 10] and [12, page 6]. The axiom scheme (3) of those two works classifies every unitary relation α as one for which we have:
A α u = A = u α A .(30)
However, the natural behavior of the relations π 0 and π 1 of P is given by (23) and (24), instances of (4) and (5). So, they cannot satisfy (30). We will see that the weaker behavior of π 0 and π 1 as compared to the one of a unitary relation requires to generalize the Splitting theorem (Section 4.) •
Remark 3.
There is an aspect of P for which we have no convincing a priori justification. For every j, the rule c j ↓ is ai j ↓ flipped up side down. Currently, we limit to observe that this is harmless. Both rules are semantically sound, i.e. the truth of the premise implies the one of the conclusion. • 4 Cut-elimination in Subatomic systems-1.1
We here adapt the proof of the cut-elimination for Subatomic systems-1.0 [11,12] 2. For every weak relation β in S↓ with unit u β ∈ U the following axioms hold:
u β α u β = u β ∀α ∈ R.α ≺ β (31) (A β B) β C = A β (B β C) ∀A, B, C ∈ F (32) A β B = B β A ∀A, B ∈ F (33) u β u = u β ∀u ∈ U (34) A β u β = A ∀A ∈ F (35) x β x = u β ∀x ∈ V ;(36)
3. S↓ contains all and only the splitting and equational down-rules, as in Definition 1. So, it does not contain any contractive down-rule. • Like in [11,12], axioms (31), (32) and (33) are strongly linked to the way that splitting works. Once decomposed a proof into independent subproofs, they can be composed back into a new proof exactly because the here above axioms hold. Also (34) is in [11,12]. Instead, both (35)
u β u β (35) (u β u β ) β u β P S↓ (u β u β ) β (u β B) s↓ (u β u) β (u β β B) (34) u β β (u β β B) (33),(35) B , while D ′ is x (35) x β u β (35) (x β u β ) β u β P ′ S↓ (x β u β ) β (x β B) s↓ (x β x) β (u β β B) (36) u β β (u β β B) (33),(35) B .
The following theorem strictly generalizes the namesake one in [12].
Theorem 1 (Shallow splitting). Let β be a weak relation with unit u β in a Splittable Subatomic system-1.1 S↓. For every
α ≺ β, let u β P S↓ (A 0 α A 1 ) β B
be given.
1. If α is a right weakening, then
K 0 α K 1 D S↓ B exists such that u β Q 0 S↓ A 0 β K 0
exists as well and |Q 0 | ≤ |P|. If α is a left weakening, replace 1 for 0.
If α is unitary, then
K 0 α K 1 D S↓ B exists such that, for every i ∈ {0, 1}, u β Q i S↓ A i β K i
exists as well and |Q 0 | + |Q 1 | ≤ |P|.
Proof. We prove both points simultaneously, proceeding by induction on |P|. The value of |P| is at least 1 because α ≺ β and α is not weak. Necessarily, an occurrence of (31) exists in P which generates a formula out of u β with α in it.
• The base case is with |P| = 1 and (31) occurs in P. So, P is composed by the three derivations -Let α be unitary. For every i ∈ {0, 1}, the proof Q i is
u β P ′ S↓ u β β B ′ (u β α u β ) β B ′u β u β β u β P i S↓ A i β u β . Moreover, |Q 0 | + |Q 1 | = |P 0 | + |P 1 | < |P| = 1
because none among Q 0 , Q 1 , P 0 and P 1 contain axioms that count 1.
-If α is a right or a left weakening we proceed as here above, but focusing only on one of the two proofs Q 0 and Q 1 .
• The inductive case has |P| > 1. We only develop the details of the relevant cases. The first relevant case is a refinement of point (3) in the original proof of Shallow splitting of [11,12]. The refinement requires to consider the possibilities that we introduce a constant by distinguishing among unitary relations, right weakening and left weakening.
-Let α and γ be right weakening such that P is
u β P ′ S↓ (((A 0 α A 1 ) β B 0 ) γ C) β B 1 (4) (A 0 α A 1 ) β B 0 β B 1 .
Because |P ′ | < |P|, by the inductive hypothesis
K l γ K r D ′ S↓ B 1 exists such that u β Q ′ S↓ (A 0 α A 1 ) β B 0 β K l exists as well with |Q ′ | ≤ |P ′ | < |P|. So,
the inductive hypothesis holds on Q ′ and a derivation
K 0 α K 1 D ′′ S↓ B 0 β K l exists such that u β Q ′′ S↓ A 0 β K 0
exists as well with |Q ′′ | ≤ |Q ′ | ≤ |P ′ | < |P|. The proof we are looking for is Q ′′ . The derivation is
K 0 α K 1 D ′′ S↓ B 0 β K l (4) B 0 β (K l γ K r ) D ′ S↓ B 0 β B 1 .
-Let α be unitary and let γ be right weakening such that P is
P ′ S↓ (((A 0 α A 1 ) β B 0 ) γ C) β B 1 (4) (A 0 α A 1 ) β B 0 β B 1 . Because |P ′ | < |P|,
by the inductive hypothesis
K l γ K r D ′ S↓ B 1 exists such that u β Q ′ S↓ (A 0 α A 1 ) β B 0 β K l exists as well with |Q ′ | ≤ |P ′ | < |P|.
So, the inductive hypothesis holds on Q ′ and a derivation
K 0 α K 1 D ′′ S↓ B 0 β K l exists such that, for every i ∈ {0, 1}, the proof u β Q i S↓ A i β K i
exists as well and |Q 0 | + |Q 1 | ≤ |Q ′ | ≤ |P ′ | < |P|. The proofs we are looking for are Q 0 and Q 1 .
The derivation D is
K 0 α K 1 D ′′ S↓ B 0 β K l (4) B 0 β (K l γ K r ) D ′ S↓ B 0 β B 1 .
-The cases with both α and γ left weakening or with α unitary and γ left weakening are symmetric.
The further relevant cases come from points (13) and (14) in the original proof of Shallow splitting of [11,12]. In our case, point (13) requires to focus also on a right weakening α in a proof P with form
u β P ′ S↓ A 0 β B (4) (A 0 α A 1 ) β B . From (4) we get that D is B α K (4) B
. So, the proof Q is simply P ′ . For the analogous of (14) with a left weakening it is enough to proceed as just done her above.
Definition 11 (Provable context). Let β be a weak relation with unit u β in some Subatomic system-1. 1
. A context H is
provable if H u β = u β . •
Theorem 1 implies that Context reduction holds exactly as formulated and proved in [11,12]:
Theoremu β P S↓ S {(A γ B) α (C γ D)} ρ↑ S {(A α C) γ (B α D)} , with ρ↑ in S↑, then u β P ′ S↓ S {(A α C) γ (B α D)}
which means that ρ↑ is admissible in S↓.
Proof. We develop a case specific to version 1.1 where γ and, hence γ, is a right weakening. Theorem 2 on P im-
plies H {{ } β K} D S↓ S { } and u β Q S↓ ((A γ B) α (C γ D)) β K
with H provable. Theorem 1 on Q implies
Q 1 β Q 2 D ′ S↓ K and u β Q 1 S↓ (A γ B) β Q 1 and u β Q 2 S↓ (C γ D) β Q 2
. Theorem 1 on Q 1 implies
Q A γ Q B D 1 S↓ Q 1 and u β Q A S↓ A β Q A
. Theorem 1 on Q 2 implies
Q C γ Q D D 2 S↓ Q 2 and u β Q C S↓ C β Q C . So H u β H u β α u β S↓ H{(A β Q A ) α (C β Q C )} H{(A α C) β (Q A β Q C )} H{((A α C) β (Q A β Q C )) γ ((B α D) β (Q B β Q D ))} H{((A α C) γ (B α D)) β ((Q A β Q C ) γ (Q B β Q D ))} H{((A γ B) α (C γ D)) β ((Q A β Q C ) γ (Q B β Q D ))} H{((A γ B) α (C γ D)) β ((Q A γ Q B ) β (Q C γ Q D ))} S↓ H{((A γ B) α (C γ D)) β (Q 1 β Q 2 )} S↓ H{((A γ B) α (C γ D)) β K} S↓ S {(A γ B) α (C γ D)} .
The splittable fragment P↓ in P
In this section we take advantage of having identified the properties that a Subatomic system-1.1 must meet to enjoy the cut-elimination property. From Definitions 8, 9 and 7 it follows:
Fact 2. The Splittable down-fragment P↓ of P contains the down-rules:
(A ∨ B) π 0 (C ∨ D) ai 0 ↓ (A π 0 C) ∨ (B π 0 D) (A ∨ B) π 1 (C ∨ D) ai 1 ↓ (A π 1 C) ∨ (B π 1 D) (A ∨ B) ∧ (C ∨ D) s↓ (A ∧ C) ∨ (B ∨ D)
while the Splittable up-fragment P↑ of P contains the up-rules:
(A π 0 B) ∧ (C π 0 D) ai 0 ↑ (A ∧ C) π 0 (B ∧ D) (A π 1 B) ∧ (C π 1 D) ai 1 ↑ (A ∧ C) π 1 (B ∧ D) (A ∨ B) ∧ (C ∧ D) s↑ (A ∧ C) ∨ (B ∧ D)
Theorem 3 holds on P, hence on the subset of rules of P↓ and P↑. So, we get:
Corollary 3. Every up-rule of P↑ is admissible in P↓. •
The system P and Post's Lattice
We show that P is related to Post's Lattice [7]. It follows that P extends Propositional logic without relying on any representations of the atoms of P in terms of sub-atoms, i.e. in terms of some encoding which is based on self-dual non-commutative relations.
Definition 12 (Clones [7]). Let B be a set of boolean operators. A clone [B]
is the least set of boolean operators of any arity, closed under composition that contains: (i) propositional variables x, y, z, . . .; (ii) projections of every finite arity,
π 1 1 (x) = x included; (iii) f ∈ B applied to propositional variables. •
The class of all clones is Post's Lattice which is infinite and complete [7]. The top of the lattice is C 1 = [∨, T, ] which strictly contains five pairwise incomparable maximal clones:
C 2 = [∧, →, ←, ∨, T] C 3 = [F, ∧, ←, →, ∨] L 1 = [F, ∧, ∨, T] A 1 = [F, ⇔, ⊕, T, ] D 3 = [(x ∧ y) ∨ (y ∧ z) ∨ (x ∧ z) min(x,y,z) , (x ∧ y) ∨ (y ∧ z) ∨ (x ∧ z)
min(x,y,z) ] whose names come from [7,6].
Proposition 3 (Soundness of P). The Subatomic system-1.1 P is sound for C 1 . I.e., let A, B 1 , . . . , B n ∈ F P be such that B ∈ C 1 exists and A is equivalent to B, up to De Morgan equivalences. Given
T P P A ∨ B 1 ∨ . . . ∨ B n , if B 1 ∧ . . . ∧ B n is true, then A is true.
Proof. P only contains rules of P. By definition, the conclusion of every rule in P is true whenever its premise is true. Since the formula on top of P is T also A ∨ B 1 ∨ . . . ∨ B n = B 1 ∧ . . . ∧ B n ∨ A = B 1 ∧ . . . ∧ B n ⇒ A must be true. Forcefully, the truth of B 1 ∧ . . . ∧ B n implies the one of A.
The proof of completeness follows a standard technique.
Definition 13. Let A[x 1 , . . . , x n ] denote any formula of F P such that x 1 , . . . , x n are all and only its variables. Let T A be the following truth table of A[x 1 , . . . , x n ]:
x 1 x 2 . . . x n A[x 1 , . . . , x n ] F F . . . F χ 0 F F . . . T χ 1 . . . . . . . . . . . . T T . . . T χ 2 n −1 where χ l ∈ {Fτ(l, x i ) = x i if T A (l, x i ) = T τ(l, x i ) = x i if T A (l, x i ) = F τ(l, A[x 1 , . . . , x n ]) = A[x 1 , . . . , x n ] if T A (l, A[x 1 , . . . , x n ]) = T τ(l, A[x 1 , . . . , x n ]) = A[x 1 , . . . , x n ] if T A (l, A[x 1 , . . . , x n ]) = F .x i ∨ x i τ(l, x i ) ∨ τ(l, x i ) . If τ(l, x i ) = x i , then P is T (27) x i ∨ x i τ(l, x i ) ∨ τ(l, x i )
. Let
A[x 1 , . . . , x n ] be π i (x 1 , . . . , x n ). If τ(l, π i (x 1 , . . . , x n )) = π i (x 1 , . . . , x n ), then P is
T x i ∨ x i π i (. . . , x i , . . .) ∨ π i (. . . , x i , . . .) (19) π i (. . . , x i , . . .) ∨ π i (. . . , x i , . . .) Fact 3 τ(l, π i (x 1 , . . . , x n )) ∨ τ(l, π i (x 1 , . . . , x n ))
.
If A ∈ {A 0 ∧ A 1 , A 0 ∨ A 1 } itT P l P τ(l, X) ∨ τ(l, x) ∨ τ(l, y)
for every 1 ≤ l ≤ 2 2 , where X shortens (A ∨ B 1 ∨ . . . ∨ B n )[x, y] and 2 2 is the number of lines that all the combinations of the literals x, x, y and y generate in the truth table of τ(l, X). In fact, for every 1 ≤ l ≤ 4, the proof P l has form
T P l P X ∨ τ(l, x) ∨ τ(l, y)
because X, i.e. A ∨ B 1 ∨ . . . ∨ B n , is a tautology. So D we are looking for is
T T ∧ T ∧ T ∧ T ((X ∨ x ∨ y) ∧ (X ∨ x ∨ y)) ∧ ((X ∨ x ∨ y) ∧ (X ∨ x ∨ y)) s↓,s↓ ((x ∧ x) ∨ (X ∨ X ∨ y ∨ y)) ∧ ((x ∧ x) ∨ (X ∨ X ∨ y ∨ y)) (F ∨ (X ∨ X ∨ y ∨ y)) ∧ (F ∨ (X ∨ X ∨ y ∨ y)) (X ∨ X ∨ y ∨ y) ∧ (X ∨ X ∨ y ∨ y) (X ∨ y) ∧ (X ∨ y) s↓ (y ∧ y) ∨ X ∨ X F ∨ X ∨ X X ∨ X X .
Conclusion and developments
This work highlights how much effective the work in [11,12] that aims at identifying the core mechanism of cutelimination is. The notion of Subatomic system allows to prove modularly, generally and once the cut-elimination of interesting deep inference systems. We show that the original notion of subatomic system can be slightly generalized. This allows to identify new logical systems without any need to encode their constants sub-atomically and without loosing splitting, i.e. cut-elimination. The Subatomic system-1.1 P, sound and complete for the tautologies of Post's clone C 1 , is a witness of how that is possible. Of course, the introduction of P is not breathtaking, but logical systems that smoothly incorporate self-dual operators -in the case of P they are operators as natural as projections -and which keep maintaining good logical properties are not so common [5,8,9]. On going work aims at using the framework of subatomic systems, may be upgraded to some version x.y -this work introduces release 1.1 -, for systematically identifying logical systems with good properties and, possibly, of some relevance. Saying it in another way, the idea is to use the pattern that subatomic systems suggest for contributing to systematic proof theory [1]. The following list of medial shapes should show to what extent this idea can be concrete and potentially interesting:
(A → B) π 0 (C → D) ai 0 ↓ (A π 0 C) → (B π 0 D) (A π 0 C) → (B π 0 D) ai 0 ↑ (A → B) π 0 (C → D) (A → B) π 1 (C → D) ai 1 ↓ (A π 1 C) → (B π 1 D) (A π 1 C) → (B π 1 D) ai 1 ↑ (A → B) π 1 (C → D) (A → B) ← (C → D) s↓ (A ← C) → (B ← D) (A ← C) → (B ← D) s↑ (A → B) ← (C → D) (A π 0 B) → (C π 0 D) c 0 ↓ (A → C) π 0 (B → D) (A → C) π 0 (B → D) c 0 ↑ (A π 0 B) → (C π 0 D) (A ← B) π 0 (C ← D) ai 0 ↓ (A π 0 C) ← (B π 0 D) (A π 0 C) → (B π 0 D) ai 0 ↑ (A → B) π 0 (C → D) (A ← B) π 1 (C ← D) ai 1 ↓ (A π 1 C) ← (B π 1 D) (A π 1 C) → (B π 1 D) ai 1 ↑ (A → B) π 1 (C → D) (A ← B) → (C ← D) s↓ (A → C) ← (B → D) (A → C) → (B → D) s↑ (A → B) → (C → D) (A π 0 B) ← (C π 0 D) c 0 ↓ (A ← C) π 0 (B ← D) (A → C) π 0 (B → D) c 0 ↑ (A π 0 B) → (C π 0 D) (A π 1 B) → (C π 1 D) c 1 ↓ (A → C) π 1 (B → D) (A → C) π 1 (B → D) c 1 ↑ (A π 1 B) → (C π 1 D) (A π 1 B) ← (C π 1 D) c 1 ↓ (A ← C) π 1 (B ← D) (A → C) π 1 (B → D) c 1 ↑ (A π 1 B) → (C π 1 D)
The whole list is candidate to become a subatomic system R 23 . Endowed it with the right equational theory among the propositional logic formulas that the rules infer, R 23 should derive tautologies of C 2 ∪ C 3 we recall in Section 6. Lewis shows that the satisfiability of formulas of C 2 belongs to P-Time problems while the satisfiability of formulas in C 3 is NP-Time complete [6]. So, R 23 would be a logical system where looking for proof-theoretical properties that highlight the phase transition from P-Time to NP-Time complete satisfiability.
The rules of R 23 come from the following complete sub-lattice:
B ← F T A F F T T F F B → ≡ f 4 F T A F F F T T F B π 0 F T A F T T T F F B π 1 F T A F F T T F T B F F T A F F F T F F A T F T B F T T T T T B π 0 F T A F F F T T T B π 1 F T A F T F T T F B → F T A F T T T F T B ← F T A F T F T T T
which is inside the complete lattice of binary boolean functions pointwise ordered in accordance with the convention that F is smaller than T. The lattice shows that it is natural to work with more than one weak relation in the same system. Both ← and → are weak and play the same role as that played by ∧ in the lattice that drives the definition of P in Section 3. Two weak relations are required because the negation ← of ← is the least upper bound of π 0 and π 1 and not of π 0 and π 1 of which ← is greatest lower bound. Of course, symmetrically, the same observation holds for →. The lattice here above should immediately suggests that the search of subatomic systems need not be confined to the set of sixteen two-valued boolean functions. For any k ≥ 3, the use of k-valued operators as relations for subatomic systems is perfectly viable. For example, the subatomic system that corresponds to the paradigmatic deep inference system BV [5] can be seen as a system that uses 3-valued operators that define Coherence Spaces [2]. Considered the huge number of k-valued operators, as k grows, subatomic systems look like grammars that generate specific languages, i.e. logical systems, with good proof theoretical properties, of possible unexpected interest, as consequence of the consistent use of non standard logical operators. This should definitely make it evident the contribution that the introduction of Subatomic systems-1.1 can give to Systematic Proof Theory.
Fact 1 (
1Equation derivations). Let D be a derivation that only contains equation rules of a given Subatomic system-1.1 S. We can obtain derivations of S from D in two steps: (i) negating every of formula of D, (ii) flipping D up-side down. • 3 The Subatomic system-1.1 P
every i ∈ {0, 1}, where |P ′ | = |P ′′ | = |P 0 | = |P 1 | = 0. Lemma 1 holds on P ′ . So,
,
T}, for every 0 ≤ l ≤ 2 n − 1. For every 0 ≤ l ≤ 2 n − 1 and every B ∈ {x 1 , . . . , x, A[x 1 , . . . , x n ]} let T A (l, B) the entry of T A at line l and column B. By definition, let τ be the following map:
Fact 3 (
3Arbitrary projections in P). For every x i ∈ V P , the derivation x i π i (. . . , x i , . . .) exists by applying suitable combinations of the axioms (23) and (24). The same holds for every x i ∈ V P • Proposition 4 (Compactness of P). Let A[x 1 , . . . , x n ] ∈ F P be given. Then, for every 0 ≤ l ≤ 2 n − 1, the proof T P P τ(l, A[x 1 , . . . , x n ]) ∨ τ(l, x 0 ) ∨ . . . ∨ τ(l, x n ) exists. Proof. We proceed by induction on the structure of A[x 1 , . . . , x n ]. Let A[x 1 , . . . , x n ] be x i . If τ(l, x i ) = x i , then P is T (27)
to version 1.1. Definition 8 (Splittable down-fragment). Let S be a Subatomic system-1.1. Then, S↓ is the Splittable down-fragment of S if: 1. S↓ contains at least one weak relation;
2 (Context reduction). Let S be a Subatomic system-1.1 whose fragment S↓ is splittable. Let β a weak relation in S with unit u β . For every A ∈ F and context S , if Theorem 3 (Splittable up-fragment is admissible). Let S be a Subatomic system-1.1 with splittable S↓ and S↑ in it. Let A, B, C, D ∈ F and S be a context. Let α ∈ R be strong. For every γ ∈ R such that α ≺ γ, ifu β
P S↓
S {A}
, then there is K ∈ F and a provable context H such that
H {{ } β K}
P S↓
S { }
and
u β
P S↓
A β B
.
•
is enough to standardly apply the inductive hypothesis.Proof. The assumption saying that the truth of B 1 ∧. . .∧B n implies the truth of A is equivalent to saying that A∨B 1 ∨. . .∨B n is a tautology. To keep the proof readable we assume that x, y are all and only the free variables of A ∨ B 1 ∨ . . . ∨ B n .Of course, what we are going to do, works for any finite set of variables in A ∨ B 1 ∨ . . . ∨ B n . Proposition 4 assures the existence ofTheorem 4 (Completeness of P). The Subatomic system-1.1 P is complete for C 1 . I.e., let A, B 1 , . . . , B n ∈ F P be such that
B ∈ C 1 exists and A is equivalent to B, up to De Morgan equivalences. Let us also assume that the truth of B 1 ∧ . . . ∧ B n
implies the truth of A. Then
T
D P
A ∨ B 1 ∨ . . . ∨ B n
exists.
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[
"Tensor pomeron, vector odderon and diffractive production of meson and baryon pairs in proton-proton collisions",
"Tensor pomeron, vector odderon and diffractive production of meson and baryon pairs in proton-proton collisions"
]
| [
"Piotr Lebiedowicz \nInstitute of Nuclear Physics Polish Academy of Sciences\nul. Radzikowskiego 15231-342KrakówPLPoland\n",
"Otto Nachtmann \nInstitut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany\n",
"⋆⋆Antoni Szczurek \nInstitute of Nuclear Physics Polish Academy of Sciences\nul. Radzikowskiego 15231-342KrakówPLPoland\n"
]
| [
"Institute of Nuclear Physics Polish Academy of Sciences\nul. Radzikowskiego 15231-342KrakówPLPoland",
"Institut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg 16D-69120HeidelbergGermany",
"Institute of Nuclear Physics Polish Academy of Sciences\nul. Radzikowskiego 15231-342KrakówPLPoland"
]
| []
| We review some selected results of the tensor-pomeron and vector-odderon model of soft high-energy proton-proton scattering and central exclusive production of meson and baryon pairs in proton-proton collisions. We discuss the theoretical aspects of this approach and consider the phenomenological implications in a variety of processes at high energies, comparing to existing experimental data. We consider the diffractive dipion and dikaon production including the continuum and the dominant scalar and tensor resonance contributions as well as the photoproduction processes. The theoretical results are compared with existing CDF experimental data and predictions for planned or current LHC experiments, ALICE, ATLAS, CMS, LHCb are presented. ⋆ | 10.1051/epjconf/201920606005 | [
"https://arxiv.org/pdf/1812.00138v1.pdf"
]
| 119,188,478 | 1812.00138 | 7145f788e2c464599091a75327791ae3e875f4c8 |
Tensor pomeron, vector odderon and diffractive production of meson and baryon pairs in proton-proton collisions
1 Dec 2018
Piotr Lebiedowicz
Institute of Nuclear Physics Polish Academy of Sciences
ul. Radzikowskiego 15231-342KrakówPLPoland
Otto Nachtmann
Institut für Theoretische Physik
Universität Heidelberg
Philosophenweg 16D-69120HeidelbergGermany
⋆⋆Antoni Szczurek
Institute of Nuclear Physics Polish Academy of Sciences
ul. Radzikowskiego 15231-342KrakówPLPoland
Tensor pomeron, vector odderon and diffractive production of meson and baryon pairs in proton-proton collisions
1 Dec 2018
We review some selected results of the tensor-pomeron and vector-odderon model of soft high-energy proton-proton scattering and central exclusive production of meson and baryon pairs in proton-proton collisions. We discuss the theoretical aspects of this approach and consider the phenomenological implications in a variety of processes at high energies, comparing to existing experimental data. We consider the diffractive dipion and dikaon production including the continuum and the dominant scalar and tensor resonance contributions as well as the photoproduction processes. The theoretical results are compared with existing CDF experimental data and predictions for planned or current LHC experiments, ALICE, ATLAS, CMS, LHCb are presented. ⋆
Introduction
There is a growing experimental and theoretical interest in understanding diffractive processes at high energy in proton-(anti)proton collisions. A particularly interesting are central exclusive production (CEP) processes where all centrally produced particles are detected. In the CDF [1] and the CMS [2] experiments only large rapidity gaps around the centrally produced dimeson system are checked but the forward and backward going protons are not detected. Preliminary results of similar CEP studies were presented by the ALICE [3] and LHCb [4] Collaborations at the LHC and by the STAR [5] Collaboration at RHIC. Although such results will have diffractive nature, further efforts are needed to ensure their exclusivity. These measurements are important in the context of resonance production, in particular, in searches for the gluon bound states (glueballs). Future experiments at the LHC will be able to detect all particles produced in CEP, including the forward and backward going protons. Feasibility studies for the pp → ppπ + π − process with tagging of the scattered protons as carried out for the ATLAS and ALFA detectors are shown in [6]. Similar possibilities exist with the CMS and TOTEM detectors; see, e.g. [7].
It was known for a long time that the frequently used vector-pomeron model has problems from the point of view of field theory. Taken literally it gives opposite signs for pp andpp total cross sections.
A way how to solve these problems was discussed in [8] where the pomeron was described as a coherent superposition of exchanges with spin 2+4+6+.... The same idea is realised in the tensor-pomeron model formulated in [9] where the soft pomeron exchange can effectively be treated as the exchange of a rank-2 symmetric tensor. In [10] it was shown that the tensor-pomeron model is consistent with the experimental data on the helicity structure of proton-proton elastic scattering at √ s = 200 GeV and small |t| from the STAR experiment [11]. In Ref. [12] the tensor-pomeron model was applied to the diffractive production of several scalar and pseudoscalar mesons in the reaction pp → ppM and it was shown that this model does quite well in reproducing the data when available. In [13] an extensive study of the photoproduction reaction γp → π + π − p in the framework of the tensor-pomeron model was presented. The resonant (ρ 0 → π + π − ) and non-resonant (Drell-Söding) photon-pomeron/reggeon π + π − production in pp collisions was studied in [14]. The central exclusive diffractive production of π + π − continuum together with the dominant scalar f 0 (500), f 0 (980), and tensor f 2 (1270) resonances was studied by us in [15]. The experimental data on central exclusive π + π − production measured at Fermilab [1], CERN [2], and RHIC [5] all show visible structures in the π + π − invariant mass. As we discussed in Ref. [15] the pattern of these structures has mainly resonant origin and is very sensitive to the cuts used in a particular experiment (usually these cuts are different for different experiments). The ρ 0 meson production associated with a very forward/backward πN system in the pp → ppρ 0 π 0 and pp → pnρ 0 π + processes was discussed in [16]. In [17], the exclusive diffractive production of the K + K − in the continuum and via the dominant scalar f 0 (980), f 0 (1500), f 0 (1710), and tensor f 2 (1270), f ′ 2 (1525) resonances, as well as the K + K − photoproduction contributions, were discussed in detail. In [18] the pp → pppp reaction was studied. Also the central exclusive π + π − π + π − production via the intermediate σσ and ρ 0 ρ 0 states in pp collisions was studied in [19].
Formalism
We discuss central exclusive production of π + π − pairs in proton-proton collisions at high energies
p (p a ) + p (p b ) → p (p 1 ) + π + (p 3 ) + π − (p 4 ) + p (p 2 ) ,(1)
where p i , indicated in brackets, denote the 4-momenta of the particles. The generic diagrams for diffractive exclusive pp → ppπ + π − reaction are shown in Fig. 1. At high energies the exchange objects to be considered are the photon γ, the pomeron P, the odderon O, and the reggeons R. Their charge conjugation and G-parity quantum numbers are listed in Table I of [15]. We treat the C = +1 pomeron and the reggeons R + = f 2R , a 2R as effective tensor exchanges while the C = −1 odderon and the reggeons R − = ω R , ρ R are treated as effective vector exchanges. Note that G-parity invariance forbids the vertices a 2R ππ, ω R ππ and Oππ. The total amplitude for the pp → ppπ + π − reaction is a coherent sum of continuum amplitudes and the amplitudes with the s-channel scalar (J PC = 0 ++ ) and tensor (J PC = 2 ++ ) resonances. The photoproduction contributions, discussed in detail in [14], must be also added coherently at the amplitude level and in principle could interfere.
For example, the PP-exchange "Born-level" amplitude (without absorption effects) for π + π − production through a tensor resonance f 2 → π + π − can be written as
M (PP→ f 2 →π + π − ) λ a λ b →λ 1 λ 2 π + π − = (−i)ū(p 1 , λ 1 )iΓ (Ppp) µ 1 ν 1 (p 1 , p a )u(p a , λ a ) i∆ (P) µ 1 ν 1 ,α 1 β 1 (s 1 , t 1 ) × iΓ (PP f 2 ) α 1 β 1 ,α 2 β 2 ,ρσ (q 1 , q 2 ) i∆ ( f 2 ) ρσ,αβ (p 34 ) iΓ ( f 2 ππ) αβ (p 3 , p 4 )
× i∆ (P) α 2 β 2 ,µ 2 ν 2 (s 2 , t 2 )ū(p 2 , λ 2 )iΓ (Ppp)
µ 2 ν 2 (p 2 , p b )u(p b , λ b ) ,(2)
where λ i ∈ {+1/2, −1/2} denote the helicities of the nucleons, t 1 = q 2 1 = (p 1 −p a ) 2 , t 2 = q 2 2 = (p 2 −p b ) 2 , s 1 = (p a +q 2 ) 2 = (p 1 + p 34 ) 2 , s 2 = (p b +q 1 ) 2 = (p 2 + p 34 ) 2 , p 34 = p 3 + p 4 . The amplitude for diffractive Figure 1. The diagram (a) shows the double-pomeron/reggeon and photon mediated central exclusive I G J PC = 0 + 0 ++ and 0 + 2 ++ resonances production and their subsequent decays into π + π − in proton-proton collisions. Two diagrams in panel (b) show exclusive continuum π + π − production. exclusive π + π − continuum production can be written as the sum M (PP→π + π − ) = M (t) + M (û) ,
(a) γ, IP, IR γ, IP, IR f 0 , f 2 p (pa) p (p1) p (p b ) p (p 2 ) π + (p3) π − (p4) (b) γ, IP, IR γ, IP, IR π + (p3) π − (p4) p (pa) p (p1) p (pb) p (p2) t t1 t2 γ, IP, IR γ, IP, IR π − (p4) π + (p3) p (pa) p (p1) p (pb) p (p2) u t1 t2M (t) λ a λ b →λ 1 λ 2 π + π − = (−i)ū(p 1 , λ 1 )iΓ (Ppp) µ 1 ν 1 (p 1 , p a )u(p a , λ a ) i∆ (P) µ 1 ν 1 ,α 1 β 1 (s 13 , t 1 ) iΓ (Pππ) α 1 β 1 (p t , −p 3 ) i∆ (π) (p t ) × iΓ (Pππ) α 2 β 2 (p 4 ,p t ) i∆ (P) α 2 β 2 ,µ 2 ν 2 (s 24 , t 2 )ū(p 2 , λ 2 )iΓ (Ppp) µ 2 ν 2 (p 2 , p b )u(p b , λ b ) ,(3)M (û) λ a λ b →λ 1 λ 2 π + π − = (−i)ū(p 1 , λ 1 )iΓ (Ppp) µ 1 ν 1 (p 1 , p a )u(p a , λ a ) i∆ (P) µ 1 ν 1 ,α 1 β 1 (s 14 , t 1 ) iΓ (Pππ) α 1 β 1 (p 4 ,p u ) i∆ (π) (p u ) × iΓ (Pππ) α 2 β 2 (p u , −p 3 ) i∆ (P) α 2 β 2 ,µ 2 ν 2 (s 23 , t 2 )ū(p 2 , λ 2 )iΓ (Ppp) µ 2 ν 2 (p 2 , p b )u(p b , λ b ) ,(4)
wherep t = p a − p 1 − p 3 andp u = p 4 − p a + p 1 , s i j = (p i + p j ) 2 , ∆ (π) (p) = (p 2 − m 2 π ) −1 . For extensive discussions we refer to [9]. The pomeron-proton vertex function, supplemented by a vertex form factor, taken here to be the Dirac electromagnetic form factor of the proton for simplicity, has the form
iΓ (Ppp) µν (p ′ , p) = iΓ (Ppp) µν (p ′ , p) = −i3β PNN F 1 (p ′ − p) 2 1 2 γ µ (p ′ + p) ν + γ ν (p ′ + p) µ − 1 4 g µν (p / ′ + p /) ,(5)with β PNN = 1.87 GeV −1 . For the Pππ vertex we have iΓ (Pππ) µν (k ′ , k) = −i2β Pππ (k ′ + k) µ (k ′ + k) ν − 1 4 g µν (k ′ + k) 2 F M ((k ′ − k) 2 ) ,(6)
with β Pππ = 1.76 GeV −1 and we use the pion electromagnetic form factor in a simple parametrization
F M (p 2 ) = (1 −p 2 /Λ 2 0 ) −1 , Λ 2 0 = 0.5 GeV 2 .
The off-shellness of the intermediate pions in (3) and (4) is taken into account by the inclusion of form factors. The form factors are parametrised in the monopole formF π (
p 2 ) = (Λ 2 o f f,M − m 2 π )/(Λ 2 o f f,M −p 2 ) or, alternatively, in the exponential form F π (p 2 ) = exp (p 2 − m 2 π )/Λ 2 o f f,E , where Λ o f f,M (or Λ o f f,E )
could be adjusted to experimental data. Here the normalisation conditionF π (m 2 π ) = 1 is clearly satisfied. Our effective pomeron propagator reads
i∆ (P)
µν,κλ (s, t) = 1 4s g µκ g νλ + g µλ g νκ − 1 2 g µν g κλ (−isα ′ P ) α P (t)−1 (7) and fulfills the following relations:
∆ (P) µν,κλ (s, t) = ∆ (P) νµ,κλ (s, t) = ∆ (P) µν,λκ (s, t) = ∆ (P) κλ,µν (s, t) , g µν ∆ (P) µν,κλ (s, t) = 0, g κλ ∆ (P) µν,κλ (s, t) = 0 .(8)
Here, the pomeron trajectory α P (t) is assumed to be of standard linear form, see e.g. [20],
α P (t) = α P (0) + α ′ P t , α P (0) = 1.0808 , α ′ P = 0.25 GeV −2 .(9)
The PP f 2 vertex can be written as
iΓ (PP f 2 ) µν,κλ,ρσ (q 1 , q 2 ) = iΓ (PP f 2 )(1) µν,κλ,ρσ | bare + 7 j=2 iΓ (PP f 2 )( j) µν,κλ,ρσ (q 1 , q 2 ) | bare F (PP f 2 ) (q 2 1 , q 2 2 , p 2 34 ) .(10)
A possible choice for the PP f 2 couplings, denoted by j = 1, ..., 7 terms, is given in Appendix A of [15]. Our attempts to determine the parameters of pomeron-pomeron-meson couplings as far as possible from experimental data have been presented in [12,15,17]. Other details as form of form factors, the tensor-meson propagator ∆ ( f 2 ) and the f 2 ππ vertex are given in [9,15,17].
The ansatz for the C = +1 reggeons R + = f 2R , a 2R is similar to (5) - (9). The f 2R -and a 2Rproton vertex functions are obtained from (5) with the replacements
(M 0 = 1 GeV) 3β PNN → g f 2R pp M 0 , g f 2R pp = 11.04, and 3β PNN → g a 2R pp M 0 , g a 2R pp = 1.68, respectively. The f 2R -pion vertex function is obtained from (6) with the replacement 2β Pππ → g f 2R ππ 2M 0 , g f 2R ππ = 9.30. The R + propagator is obtained from (7) with the replacements α P (t) → α R + (t) = α R + (0) + α ′ R + t, α R + (0) = 0.5475, α ′ R + = 0.9 GeV −2 . The R − -proton vertex (for the C = −1 reggeons R − = ω R , ρ R ) reads iΓ (R − pp) µ (p ′ , p) = −iΓ (R −pp ) µ (p ′ , p) = −ig R − pp F 1 (p ′ − p) 2 γ µ ,(11)
with g ω R pp = 8.65 and g ρ R pp = 2.02. For the ρ R -pion vertex we write
iΓ (ρ R π + π + ) µ (k ′ , k) = −iΓ (ρ R π − π − ) µ (k ′ , k) = − i 2 g ρ R ππ F M (k ′ − k) 2 (k ′ + k) µ ,(12)
with ρ R ππ = 15.63. We assume an effective vector propagator
i∆ (R − ) µν (s, t) = ig µν 1 M 2 − (−isα ′ R − ) α R− (t)−1 ,(13)with α R − (t) = α R − (0) + α ′ R − t, α R − (0) = 0.5475, α ′ R − = 0.9i∆ (O) µν (s, t) = −ig µν η O M 2 0 (−isα ′ O ) α O (t)−1 ,(14)
iΓ
(Opp) µ (p ′ , p) = −iΓ (Opp) µ (p ′ , p) = −i3β Opp M 0 F 1 (p ′ − p) 2 γ µ .(15)
We take here what we think are representative values for the odderon parameters in light of the recent TOTEM results [21],
η O = −1, α O (t) = α O (0) + α ′ O t, α O (0) = 1.05, α ′ O = 0.25GeV −2 , β ONN = 0.2 GeV −1 .
All numbers for the parameters listed above should be considered as default values to be checked and -if necessary -adjusted using relevant experimental data. Some estimates of the present uncertainties of the parameters are discussed in Sec. 3 of Ref. [9].
In reality the Born approximation, e.g. for the amplitude (2), is not sufficient and absorption corrections (rescattering effects) must be taken into account, see [22,23]. A Monte Carlo generator containing a various processes and including detector effects (acceptance, efficiency) would be useful in theory-data comparison. The GenEx Monte Carlo generator [24] could be used in this context.
Selected results
pp andpp elastic scattering
In [10] we confronted three hypotheses for the soft pomeron, tensor, vector, and scalar, with current experimental data on polarised pp elastic scattering [11]. For the vector-pomeron case a big problem arise if we consider pp andpp scattering. Taken literally it gives opposite signs for pp andpp total cross sections. Thus, we shall not consider a vector pomeron further. In order to discriminate between the tensor and scalar cases we turn to the experiment [11]. There a good measurement of the ratio of single-flip to non-flip amplitudes at √ s = 200 GeV and for 0.003 |t| 0.035 GeV 2 was performed. The relevant quantity is r 5 (s, t) = 3.2 pp → ppπ + π − and pp → ppK + K − reactions In Fig. 3 we present the dipion/dikaon invariant mass distributions imposing experimental cuts. The short-dashed lines represent the purely diffractive continuum term including both pomeron and reggeon exchanges, discussed in [15,17]. Exclusive production of light mesons both in the pp → ppπ + π − and pp → ppK + K − reactions are measurable at RHIC and LHC. The pattern of visible structures in the invariant mass distributions is related to the scalar and tensor isoscalar mesons and it depends on experimental kinematics. For example, for the pp → ppK + K − reaction presented in Fig. 3, the solid lines represent the coherent sum of the diffractive continuum, and the scalar f 0 (980), f 0 (1500), f 0 (1710), and tensor f 2 (1270), f ′ 2 (1525) resonances. The lower red lines show the photoproduction term including the dominant φ(1020) → K + K − and the continuum (Drell-Söding) contributions. The narrow φ resonance is visible above the continuum term. It may, in principle, be visible on top of the broader f 0 (980) resonance. The coupling parameters of the tensor pomeron to the φ meson was fixed based on the HERA experimental data for the γp → φp reaction.
One can expect, with our default choice of parameters, that the scalar f 0 (980), f 0 (1500), f 0 (1710) and the tensor f 2 (1270), f ′ 2 (1525) mesons will be easily identified experimentally in CEP processes. The absorption effects lead to a huge damping of the cross section for the purely diffractive contribution and a relatively small reduction of the cross section for the φ(1020) photoproduction contribution. Therefore we expect that one could observe the φ resonance term, especially when no restrictions on the leading protons are included. We note that central exclusive production of φ offers also the possibility to search for effects of the elusive odderon, as was pointed out in [25]. Figure 3. The invariant mass distributions for centrally produced π + π − (the black top lines) and K + K − (the blue bottom lines) pairs with the relevant experimental kinematical cuts specified in the legend. Theoretical results including both the non-resonant continuum and resonances are represented by the solid lines, respectively. The short-dashed lines represent the purely diffractive continuum term alone. The CDF experimental data from [1] in the left panel for the pp → ppπ + π − reaction are presented. For the pp → ppK + K − reaction the solid and longdashed blue lines correspond to the results for φ f 0 (980) = 0 and π/2 in the coupling constant g f 0 (980)K + K − e iφ f 0 (980) , respectively. The lower red line represents the φ(1020) meson plus continuum photoproduction contribution. Absorption effects were taken into account effectively by the gap survival factors, S 2 = 0.1 for the purely diffractive contributions and S 2 = 0.9 for the photoproduction contributions.
In Fig. 4 we present distributions in a special "glueball filter variable" dP t [26] defined by the difference of the transverse momentum vectors dP t = |dP t |, dP t = q t,1 − q t,2 = p t,2 − p t,1 . Results for the ALICE kinematics and for two M K + K − regions are shown. No absorption effects were taken into account here. We see that the maximum for the qq state f ′ 2 (1525) is around dP t = 0.6 GeV. On the other hand, for the scalar glueball candidates f 0 (1500) and f 0 (1710) the maximum is around dP t = 0.25 GeV, that is, at a lower value than for the f ′ 2 (1525) resonance. This is also in accord with the discussion in Refs. [27,28].
pp → pppp reaction
In Fig. 5, we show the invariant mass distributions for centrally produced π + π − , K + K − and pp systems (the left panel) and the distributions in rapidity difference y di f f = y 3 − y 4 (the right panel) at √ s = 13 TeV. In our calculations we include both the tensor-pomeron and the reggeon R + and R − exchanges. We predict a dip in the rapidity difference between the antiproton and proton for y di f f = 0. This novel effect is inherently related to the spin 1/2 of the produced hadrons. We have checked that for the pp production thet-andû-channel diagrams interfere destructively for (C 1 , C 2 ) = (1, 1) and (−1, −1) exchanges and constructively for (1, −1) and (−1, 1) exchanges. For the π + π − production, we get the opposite interference effects between thet-andû-channel diagrams. In our calculations we have included both pomeron and reggeon exchanges. The reggeon exchange contributions lead to an enhancement of the cross section mostly at large rapidities of the centrally produced hadrons. For the production of the dipion continuum, the cross section is concentrated along the diagonal η 3 = η 4 . For the production of pp pairs, the dip extends over the whole diagonal in (η 3 , η 4 ) space, see the left panel in Fig. 6. In the right panel of Fig. 6 we show the asymmetry defined for two pseudorapidities η and η ′
A (2) (η, η ′ ) = d 2 σ dη 3 dη 4 (η, η ′ ) − d 2 σ dη 3 dη 4 (η ′ , η) d 2 σ dη 3 dη 4 (η, η ′ ) + d 2 σ dη 3 dη 4 (η ′ , η) .(16)
For the investigated pseudorapidity range the asymmetries due to pomeron plus reggeon exchanges show a positive sign for |η| > |η ′ | and negative sign for |η| < |η ′ |. That is, antiprotons are predicted to come out typically with a higher absolute value of the (pseudo)rapidity than protons. The asymmetry is caused by interference effects of the dominant (P, P) with the subdominant (R − , P + R + ) and (P + R + , R − ) exchanges. We have checked that in the limited range of pseudorapidities corresponding to the ATLAS and LHCb experiments the effects of the secondary reggeons are predicted to be in the ranges of 2 -11 % and 5 -26 %, respectively. The addition of an odderon with the parameters of (14) et seq. has only an effect of less than 0.5 %. In [18] we have discussed a first qualitative attempt to "reproduce" the experimentally observed behavior of the pp invariant mass (M 34 ) spectra observed in [29,30]. Our calculation shows that the diffractive production of pp through the s-channel f 0 (2100) resonance leads to an enhancement at low M 34 and that the resonance contribution is concentrated at |y di f f | < 1. In general, more resonances can contribute, e.g., f 0 (2020), f 0 (2200), f 0 (2300), f 2 (1950). Also, the subthreshold m R < 2m p resonances that would effectively generate a continuum pp contribution should be taken into account; see [31]. Interference effects between the continuum and resonant mechanisms certainly will occur. We leave this interesting issue for future studies.
of elementary exchanges this should be viewed as a coherent sum of exchanges of spin 2 + 4 + 6 + .... This is the structure obtained in [8] within nonperturbative QCD investigations. Investigations of the pomeron using the models of AdS/QCD also prefer a tensor nature for pomeron exchange [32].
We have given a consistent treatment of continuum and resonance production for different processes in central exclusive pp and pp collisions in an effective field-theoretic approach. We have analysed the central exclusive production of π + π − and K + K − pairs at high energies. We have taken into account purely diffractive and diffractive photoproduction mechanisms. For the purely diffractive mechanism we have included the continuum and the dominant scalar f 0 (980), f 0 (1500), f 0 (1710) and tensor f 2 (1270), f ′ 2 (1525) resonances. The amplitudes have been calculated using Feynman rules within the tensor-pomeron model [9]. The effective Lagrangians and the vertices for PP fusion into the scalar and tensor mesons were discussed in [12,15,17]. Some model parameters (PPM couplings, the off-shell dependence of form factors) have been roughly adjusted to CDF data [1] and then used for predictions for the STAR, ALICE, ATLAS, CMS and LHCb experiments. The distributions, in the so-called glueball filter variable dP t , show different behavior in the K + K − invariant mass windows around glueball candidates with masses ∼ 1.5 GeV and ∼ 1.7 GeV than in other regions. The dP t distribution may help to interpret the relative rates between the f 0 and f 2 resonances.
The photoproduction and purely diffractive contributions have different dependences on the proton transverse momenta. Furthermore, the absorptive corrections for the photoproduction processes lead to a much smaller reduction of the cross section than for the diffractive ones. It can therefore be expected that the ρ-and φ-photoproduction will be seen in experiments requiring only a very small deflection angle for at least one of the outgoing protons. However, we must keep in mind that other processes can contribute in experimental studies of exclusive photoproduction where only large rapidity gaps around the centrally produced vector meson are checked and the forward and backward going protons are not detected. Experimental results for this kind of processes were published by the CDF [1] and CMS [2] collaborations. We refer the reader to Ref. [16] in which ρ 0 production in pp collisions was studied with one proton undergoing diffractive excitation to a πN system.
Recently, in [18] we discussed exclusive production of pp and ΛΛ pairs in proton-proton collisions. At the present stage, we have taken into account mainly the diffractive production of the pp continuum. For our predictions for the LHC we have used the off-shell proton form factor parameter in the range 0.8 GeV < Λ o f f,E < 1 GeV. The invariant mass distribution for pp pairs is predicted to extend to larger dihadron invariant masses than for the production of π + π − or K + K − . Especially interesting is the distribution in the rapidity difference between centrally produced antiproton and proton. For continuum pp production, we predict a dip at y di f f = 0, in contrast to π + π − and K + K − production in which a maximum of the cross section occurs at y di f f = 0. The dip is caused by a good separation of t andû contributions in (η 3 , η 4 ) space and destructive interference of them along the diagonal η 3 = η 4 characteristic for our Feynman diagrammatic calculation with correct treatment of spins. Any experimentally observed distortions from our continuum-pp predictions may therefore signal the presence of resonances. This could give new interesting information for meson spectroscopy.
To describe the relatively low-energy ISR and WA102 data [29,30] for the pp → pppp process we find that we must include also subleading reggeon exchanges in addition to the two-pomeron exchange. Then we made predictions for the LHC energy. The reggeon exchange contributions lead to enhancements at large absolute values of the p andp (pseudo)rapidities, see Fig. 6. A similar effect was predicted for the pp → ppπ + π − reaction in [33]. We have predicted asymmetries in the (pseudo)rapidity distributions of the centrally produced antiproton and proton. The asymmetry should be much more visible for the LHCb experiment which covers a region of larger pseudorapidities where the reggeon exchanges become more relevant. Also the odderon will contribute to such asymmetries. However, we find for typical odderon parameters allowed by recent pp elastic data [21] only very small effects, roughly a factor 10 smaller than the effects due to secondary reggeons.
GeV − 2
2, and M − = 1.41 GeV. Our ansatz for the odderon follows (3.16), (3.17) and (3.68), (3.69) of [9]:
Figure 2 .
2Im[φ 1 (s,t)+φ 3 (s,t)] . The complex r 5 parameter is only weakly dependent on t. Therefore, we can approximately set t = 0 and get r P T 5 (s, 0) = (−0.28 − i2.20) × 10 −5 and r P S 5 (s, 0) = −0.064 − i0.500 for the scalar-and tensor-pomeron model, respectively. InFig. 2we show the STAR experimental result[11] together with the scalar-pomeron and tensor-pomeron results. Clearly, the tensor-pomeron result is perfectly compatible with the experiment whereas the scalar-pomeron result is far outside the experimental error ellipse. The experimental results for r 5 at √ s = 200 GeV fromFig. 5of[11] together with the results for the tensor and the scalar pomeron from[10].
Figure 4 .
4The differential cross sections dσ/d(dP t ) as a function of the dP t "glueball filter" variable for the pp → ppK + K − reaction. Calculations were done for √ s = 13 TeV, |η K | < 1, p t,K > 0.1 GeV, and in two dikaoninvariant mass regions: (a) M 34 ∈ (1.45, 1.60) GeV and (b) M 34 ∈ (1.65, 1.75) GeV.
Figure 5 .
5In the left panel we show the invariant mass distributions for centrally produced π + π − , K + K − and pp systems for a typical LHC experimental conditions at √ s = 13 TeV. Results for the combined tensor-pomeron and reggeon exchanges and Λ o f f ,E = 1 GeV are presented. For the pp production we show results also for Λ o f f ,E = 0.8 GeV. In the right panel we show the distributions in the rapidity difference between the centrally produced hadrons. No absorption effects were included here.
Figure 6 .
6The two-dimensional distribution in (η 3 , η 4 ) for the diffractive continuum pp production for the full phase space and the asymmetry A (2) (η, η ′ ) [see Eq. (16)] at √ s = 13 TeV. In addition, regions of the coverage for the ALICE, ATLAS and LHCb experiments are shown.
ConclusionsWhen considering pp andpp elastic scattering and the ratio of helicity-flip to non-flip amplitudes we found that only the tensor pomeron, introduced in[9], is a viable option for the soft pomeron. In terms
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"The Shape and Topology of the Universe",
"The Shape and Topology of the Universe"
]
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"Jean-Pierre Luminet [email protected] \nLaboratoire Univers et Théories (LUTH)\nObservatoire de Paris\nCNRS\nUniversité Paris Diderot\n5 place Jules Janssen92190MeudonFrance\n"
]
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"Laboratoire Univers et Théories (LUTH)\nObservatoire de Paris\nCNRS\nUniversité Paris Diderot\n5 place Jules Janssen92190MeudonFrance"
]
| []
| What is the shape of the Universe? Is it curved or flat, finite or infinite ? Is space "wrapped around" to create ghost images of faraway cosmic sources? We review how tessellations allow to build multiply-connected 3D Riemannian spaces useful for cosmology. We discuss more particularly the proposal of a finite, positively curved, dodecahedral space for explaining some puzzling features of the cosmic microwave background radiation, as revealed by the | null | [
"https://arxiv.org/pdf/0802.2236v1.pdf"
]
| 17,045,208 | 0802.2236 | ef5b02771b8689ed07b74164549620fe8fb23835 |
The Shape and Topology of the Universe
15 Feb 2008
Jean-Pierre Luminet [email protected]
Laboratoire Univers et Théories (LUTH)
Observatoire de Paris
CNRS
Université Paris Diderot
5 place Jules Janssen92190MeudonFrance
The Shape and Topology of the Universe
15 Feb 2008
What is the shape of the Universe? Is it curved or flat, finite or infinite ? Is space "wrapped around" to create ghost images of faraway cosmic sources? We review how tessellations allow to build multiply-connected 3D Riemannian spaces useful for cosmology. We discuss more particularly the proposal of a finite, positively curved, dodecahedral space for explaining some puzzling features of the cosmic microwave background radiation, as revealed by the
The Hall of Mirrors
Imagine a room paneled with mirrors on all four vertical walls, and place ourselves somewhere within the room: a kaleidoscopic effect will be produced in the closest corner. Moreover, the repeated reflections of each pair of opposing mirrors ceaselessly reproduce the effect, creating the illusion of an infinite network extending in a plane. This paving of an infinite plane by a repeating design is called a tessellation (tessella being the name for a mosaic tile) of the Euclidean plane.
Let us now consider a room paneled with mirrors on all six surfaces (including the floor and the ceiling). If we go into the room, the interplay of multiple reflections will immediately cause us to have the impression of seeing infinitely far in every direction. Cosmic space, which is seemingly gigantic, might be lulling us with a similar illusion. Of course, it possesses neither walls nor mirrors, and the ghost images would be created not by the reflection of light from the surface of the Universe, but by a multiplication of the light ray trajectories following the folds of a wraparound universe. We could live in a physical space which is closed, small and multiply-connected, yet have the illusion that the observed space is greater, as a part of a tessellation built on repetitions of a fundamental cell. Treating such global aspects of space requires a mixture of advanced mathematics and subtle cosmological observations : Cosmic Topology [1]. Fig. 1. A very simple, so-called toric universe in two dimensions, (3), shows how an observer situated in galaxy (b) can see multiple images of galaxy (a). This model of a "wraparound" universe is constructed by starting with a square (1), whose opposite borders have been glued together (2): everything which leaves on one side reappears immediately on the opposite side, at the corresponding point. The light from galaxy (a) reaches the galaxy (b) by several distinct trajectories, because of which the observer in galaxy (b) sees numerous images of the galaxy (a), spread in all directions across the sky. Although the space of the torus is finite, a being who lives there has the illusion of seeing a space that, if not infinite (in practice, there are horizons which limit the view), at least seems larger than it really is. This fictional space (4) looks like a network tessellated from a fundamental cell, which endlessly repeats each of the objects within the cell.
Tessellations and Topology
There are two complementary aspects of geometry as the science of space: the metric part deals with the properties of distance, while the topological part studies the global properties, without introducing any measurements. The topological properties are those which remain insensitive to deformations, provided that these are continuous.
Let us take the Euclidean plane : its local geometry is determined by the metric, i.e. the way in which lengths are measured. Here, it is sufficient to apply the Pythagorean theorem for a system of two rectilinear coordinates covering the plane : ds 2 = dx 2 + dy 2 . This is a local measurement which says nothing about the finite or infinite character of space. Now let us change the topology. To do so, we cut a strip of infinite length in one direction and finite width in the other, then we glue the two sides of the strip: we obtain a cylinder. In this operation, the metric has not changed : the Pythagorean theorem still holds for the surface of the cylinder. Nevertheless, the cylinder has a different topology : its most remarkable characteristic is the existence of an infinite number of "straight lines" which join two arbitrary distinct points (viewed in three dimensions, they are helices with constant spacing). Now take a rectangle and glue its opposite edges two by two. We obtain a flat torus, a surface whose global properties are identical to those of a ring but whose curvature is everywhere zero. The metric (local geometry) of the flat torus is still given by the Pythagorean theorem, just like that of the plane and the cylinder. But the global shape is radically different, since space is now of finite extent.
Through simple cutting and re-gluing of parts of the plane, we have thus defined two surfaces with different topologies than the plane: the cylinder and the flat torus, which however belong to the same family, the locally Euclidean surfaces. The gluing method becomes extremely fruitful when the surfaces are more complicated. Let us take two tori and glue them to form a "double torus". As far as its topological properties are concerned, this new surface with two holes can be represented as an eight-sided polygon (an octagon), which can be understood intuitively by the fact that each torus was represented by a quadrilateral. But this surface is not capable of tessellating the Euclidean plane, for an obvious reason: if one tries to add a flat octagon to each of its edges, the eight octagons will overlap each other. One must curve in the sides and narrow the angles, in other words pass to a hyperbolic space: only there does one succeed in fitting eight octagons around the central octagon, and starting from each of the new octagons one can construct eight others, ad infinitum. By this process one tessellates an infinite space : the Lobachevsky hyperbolic plane (Fig. 2).
A fascinating representation of a hyperbolic tessellation was given by Poincaré. A conformal change of coordinates allows us to bring infinity to a finite distance, with the result that the entire Lobachevsky space is contained in the interior of the unit disk. The famous Dutch graphic artist Maurits Cornelis Escher created a series of prints entitled Circle Limit, in which he used Poincaré's representation (see Fig. 3).
More generally, a torus with n holes, T n , can be constructed as the connected sum of n simple tori. It is topologically equivalent to a 4n-gon where all the vertices are identical with each other and the sides are suitably identified by pairs. The n-torus (n ≥ 2) is a compact surface of negative curvature. This type of surface is most commonly seen at bakeries, in the form of pretzels. They all have the same local geometry, of hyperbolic type ; however, they do not have the same topology, which depends on the number of holes.
Thus it is possible to represent any surface whatsoever with a polygon whose sides one identifies, two by two. The polygon is called a fundamental Paving the hyperbolic plane with octagons. It is impossible to tessellate the Euclidean plane with octagons, which implies that the double torus is not a Euclidean surface. On the other hand, the hyperbolic plane can be paved by octagons cut from the hollow of a saddle. The hyperbolic plane is thus the universal covering space for the double torus. The eight corners of the octagon must all be identified as a single point; this is the reason why one must use a negatively curved octagon with angles of 45 • (8 × 45 = 360), in place of a flat octagon, whose angles are each 135 • .
domain (hereafter FD). The FD distinctly characterizes a certain aspect of the topology. But this is not enough; we must also specify the geometric transformations which identify the points. Indeed, starting from a square, one could identify the points diametrically opposite with respect to the center of symmetry of the square, and the surface obtained will no longer be a flat torus; it will no longer even be Euclidean, but spherical, a surface called the projective plane. The mathematical transformations used to identify points form a group of symmetries, called the holonomy group. If the holonomy group is trivial, the space is simply connected. If not, it is multiply connected.
The holonomy group is discrete, i.e. there is a non zero shortest distance between any two homologous points, and the generators of the group (except the identity) have no fixed point. This last property is very restrictive (it excludes for instance the rotations) and allows the classification of all possible holonomy groups. Due to the fact that the holonomy group is discrete, the FD is always convex and has a finite number of faces. In two dimensions, it is a surface whose boundary is constituted by lines, thus a polygon. In three dimensions, it is a volume bounded by faces, thus a polyhedron.
Starting from the fundamental domain and acting with the transformations of the holonomy group on each point, one creates a number of replicas of the FD ; we produce a tessellation of a larger space, called the universal covering space (hereafter UC) M * . By construction, M * is locally indistinguishable from M. But its topological properties can be quite different. The UC is necessarily simply connected. When M is multiply connected, each point of M generates replicas of points in M * . The universal covering space can be thought of as an unwrapping of the original manifold. For instance, Fig. 3. Upper : Poincaré's representation of the hyperbolic plane. By acting with the holonomies on each point of the fundamental octagon, and repeating the process again and again, one creates a tessellation of the hyperbolic plane by regular and identical octagons. Poincaré demonstrated that the hyperbolic plane, normally infinite, could be represented entirely within the interior of a disk, whose edge represents infinity. Poincaré's model deforms distances and shapes, which explains why the octagons seem irregular and increasingly tiny as we approach the boundary of the disk. All of the lines in the figure represent straight lines of the hyperbolic plane, and meet the boundary at a right angle. Lower : In this 1959 woodcutting entitled Circle Limit III, Escher has used the representation given by Poincaré to tessellate the hyperbolic plane using fish.
the UC of the flat torus is the Euclidean plane E 2 , which indeed reflects the fact that the flat torus is a locally Euclidean surface.
The shape of a homogeneous space is entirely specified if one is given a fundamental domain; a particular group of symmetries, the holonomies (fixed point-free discrete subgroup of isometries), which identify the edges of the domain two by two; and a universal covering space that is tessellated by fundamental domains. Classifying the possible shapes thus reduces, in part, to classifying symmetries.
Species of Spaces
Cosmological solutions of general relativity focus mainly on locally homogeneous and isotropic spaces, namely those admitting one of the three geometries of constant curvature. Any compact 3-manifold M with constant curvature k can thus be expressed as the quotient M = M * /G where the universal covering space M * is either :
• the Euclidean space E 3 if k = 0 • the hypersphere S 3 if k > 0 • the hyperbolic 3-space H 3 if k < 0
and the holonomy group G is a subgroup of isometries of M * acting freely and discontinuously.
Given the recent observational constraints on the curvature of cosmic space (see below), in the remaining of this short review we focus our attention to Euclidean and spherical spaces only.
The multiply connected Euclidean spaces are characterized by their fundamental polyhedra and their holonomy groups. The fundamental polyhedra are either a finite or infinite parallelepiped, or a prism with a hexagonal base, corresponding to the two ways of tessellating Euclidean space. The various combinations generate seventeen multiply connected Euclidean spaces, as shown in Fig. 4 (for an exhaustive study, see [2]). Seven of these spaces (called slabs and chimneys) are of infinite volume. The ten other are of finite volume, six of them being orientable hypertori. The latter present a particular interest for cosmology, since they could perfectly model the spatial part of the so-called "flat" universe models.
For spherical spaces, the simply-connected hypersphere S 3 can be viewed as composed of two spherical balls embedded in Euclidean space, glued along their boundaries in such a way that each point on the boundary of one ball is the same as the corresponding point on the other ball. The full isometry group of S 3 is SO(4). The holonomies that preserve the metric of the hypersphere, i.e. the admissible subgroups G of SO(4) without fixed point, acting freely and discontinuously on S 3 , belong to three categories :
• the cyclic groups of order p, Z p (p ≥ 2), made up of rotations by an angle 2π/p around a given axis, where p is an arbitrary integer ; • the dihedral groups of order 2m, D m (m > 2), which are the symmetry groups of a regular plane polygons of m sides; • the binary polyhedral groups, which preserve the shapes of the regular polyhedra. The group T * preserves the tetrahedron (4 vertices, 6 edges, 4 faces), of order 24 ; the group O * preserves the octahedron (6 vertices, 12 edges, 8 faces), of order 48 ; the group I * preserves the icosahedron (12 vertices, 30 edges, 20 faces), of order 120. There are only three distinct polyhedral groups for the five polyhedra, because the cube and the octahedron on the one hand, the icosahedron and the dodecahedron on the other hand are duals, so that their symmetry groups are the same.
If one identifies the points of the hypersphere by holonomies belonging to one of these groups, the resulting space is spherical and multiply connected. For an exhaustive classification, see [3]. There is a countable infinity of these, because of the integers p and m which parametrize the cyclic and dihedral groups.
Since the universal covering S 3 is compact, all the multiply connected spherical spaces are also compact. As the volume of S 3 is 2π 2 R 3 , the volume
of M = S 3 /G is simply vol(M) = 2π 2 R 3 /G where G is the order of the group G.
For topologically complicated spherical 3-manifolds, G becomes large and vol(M) is small. There is no lower bound since G can have an arbitrarily large number of elements (for lens and prism spaces, the larger p and m are, the By analogy with the two-dimensional case, the three-dimensional hypertorus T 3 is obtained by identifying the opposite faces of a parallelepiped. The resulting volume is finite. Let us imagine a light source at our position, immersed in such a structure. Light emitted backwards crosses the face of the parallelepiped behind us and reappears on the opposite face in front of us; therefore, looking forward we can see our back. Similarly, we see in our right our left profile, or upwards the bottom of our feet. In fact, for light emitted isotropically, and for an arbitrarily large time to wait, we could observe ghost images of any object (here the Earth) viewed arbitrarily close to any angle. The resulting visual effect would be comparable (although not identical) to what could be seen from inside a parallelepiped of which the internal faces are covered with mirrors. Thus one would have the visual impression of infinite space, although the real space is closed (courtesy Jeff Weeks).
smaller the volume of the corresponding spaces). Hence 0 < vol(M) ≤ 2π 2 R 3 . In contrast, the diameter, i.e., the maximum distance between two points in the space, is bounded below by 0.326R, corresponding to the dodecahedral space.
Let us now concentrate on the properties of the Poincaré Dodecahedral Space S 3 /I * (hereafter PDS), obtained by identifying the opposite pentagonal faces of a regular spherical dodecahedron after rotating by 36 • in the clockwise direction around the axis orthogonal to the face (Fig. 6). This configuration involves 120 successive operations and gives some idea of the extreme complication of such multiply connected topologies. Its volume is 120 times smaller than that of the hypersphere with the same radius of curvature, and it is of particular interest for cosmology, giving rise to fascinating topological mirages (Fig. 7). 6. Left : Poincaré Dodecahedral Space can be described as the interior of a spherical dodecahedron such that when one goes out from a pentagonal face, one comes back immediately inside the space from the opposite face, after a 36 • rotation. Such a space is finite, although without edges or boundaries, so that one can indefinitely travel within it. Right : View from inside PDS perpendicularly to one pentagonal face. In such a direction, ten dodecahedra tile together with a 1/10 th turn to tessellate the universal covering space S 3 . Since the dodecahedron has 12 faces, 120 dodecahedra are necessary to tessellate the full hypersphere. Thus, an observer has the illusion to live in a space 120 times vaster, made of tiled dodecahedra which duplicate like in a mirror hall (courtesy Jeff Weeks).
The tessellation of S 3 by 120 copies of the PDS is not obvious to visualize, see Fig. 8. It involves 9 successive layers. Start with the original cell (layer 1). It has 12 pentagonal faces, on each of them one builds a spherical dodecahedron, thus we get 12 dodecahedra on layer 2. Then we go further on and we get 20 dodecahedra in layer 3, 12 in layer 4, 30 in layer 5, 12 in layer 6, 20 in layer 7, 12 in layer 8 and 1 in layer 9 (it can be checked that the sum is 120). Layers 6 through 9 are of course symmetrical to layers 4 through 1. Note that the cells in layer 5 sit "vertically" with respect to the equatorial hyperplane (i.e. they are orthogonal to the equatorial hyperplane), which is why they appear flat in the image (each dark blue hexagon is the 2D shadow of a 3D cell when it is projected from 4D space to 3D space).
Topology and Cosmology
It is presently believed that our Universe is correctly described at large scale by a Friedmann-Lemaître (hereafter FL) model. The FL models are homogeneous and isotropic solutions of Einstein's equations, of which the spatial sections have constant curvature. The FL models fall into 3 general classes, according to the sign of their spatial curvature k = −1, 0, +1. The spacetime manifold is described by the metric ds 2 = c 2 dt 2 − R 2 (t)dσ 2 , where dσ 2 = dχ 2 + S k 2 (χ)(dθ 2 + sin 2 θdϕ 2 ) is the metric of a 3-dimensional homogeneous manifold, flat [k = 0] or with curvature [k ± 1]. The function
S k (χ) is defined as sinh(χ) if k = −1, χ if k = 0, sin(χ) if k = 1; R(t)
is the scale factor, chosen equal to the spatial curvature radius for non flat models.
In most studies, the spatial topology is assumed to be that of the corresponding simply connected space: the hypersphere, Euclidean space or the 3D-hyperboloid, the first being finite and the other two infinite. However, there is no particular reason for space to have a simply connected topology. In any case, general relativity says nothing on this subject; it is only the strict application of the cosmological principle, added to the theory, which encourages a generalization of locally observed properties to the totality of the Universe. However, to the metric element given above there are several, if not an infinite number, of possible topologies, and thus of possible models for the physical Universe. For example, the hypertorus and the familiar Euclidean space are locally identical, and relativistic cosmological models describe them with the same FL equations, even though the former is finite and the latter infinite. Only the boundary conditions on the spatial coordinates are changed. Thus the multiply connected cosmological models share exactly the same kinematics and dynamics as the corresponding simply connected ones (for instance, the time evolutions of the scale factor R(t) are identical).
In FL models, the curvature of physical space depends on the way the total energy density of the Universe may counterbalance the kinetic energy of the expanding space. The normalized density parameter Ω 0 , defined as the ratio of the actual energy density to the critical value that an Euclidean space would require, characterizes the present-day contents (matter, radiation and all forms of energy) of the Universe. If Ω 0 is greater than 1, then space curvature is positive and geometry is spherical; if Ω 0 is smaller than 1 the curvature is negative and geometry is hyperbolic; eventually Ω 0 is strictly equal to 1 and space is locally Euclidean.
The next question about the shape of the Universe is to know whether space is finite or infinite -equivalent to know whether space contains a finite or an infinite amount of matter-energy, since the usual assumption of homogeneity implies a uniform distribution of matter and energy through space. From a purely geometrical point of view, all positively curved spaces are finite whatever their topology, but the converse is not true : flat or negatively curved spaces can have finite or infinite volumes, depending on their degree of connectedness (see e.g. [4], [1]).
From an astronomical point of view, it is necessary to distinguish between the "observable universe", which is the interior of a sphere centered on the observer and whose radius is that of the cosmological horizon (roughly the radius of the last scattering surface), and the physical space. There are only three logical possiblities. First, the physical space is infinite -like for instance the simply connected Euclidean space. In this case, the observable universe is an infinitesimal patch of the full universe and, although it has long been the preferred model of many cosmologists, this is not a testable hypothesis. Second, physical space is finite (e.g. an hypersphere or a closed multiply connected space), but greater than the observable space. In that case, one easily figures out that if physical space is much greater that the observable one, no signature of its finitude will show in the observable data. But if space is not too large, or if space is not globally homogeneous (as is permitted in many space models with multiply connected topology) and if the observer occupies a special position, some imprints of the space finitude could be observable. Third, physical space is smaller than the observable universe. Such an apparently odd possibility is due to the fact that space can be multiply connected and have a small volume. There is a lot of possibilites, whatever the curvature of space. Small universe models may generate multiple images of light sources, in such a way that the hypothesis can be tested by astronomical observations. The smaller the fundamental domain, the easier it is to observe the multiple topological imaging. How do the present observational data constrain the possible multi-connectedness of the universe and, more generally, what kinds of tests are conceivable ? (see [5] for a non-technical book about all the aspects of topology and its applications to cosmology).
If the Universe was finite and small enough, we should be able to see "all around" it, because the photons might have crossed it once or more times. In such a case, any observer might identify multiple images of a same light source, although distributed in different directions of the sky and at various redshifts, or to detect specific statistical properties in the apparent distribution of faraway sources such as galaxy clusters. To do this, methods of cosmic crystallography have been devised (see e.g. [6], [7]). The main limitation of cosmic crystallography is that the presently available catalogs of observed sources at high redshift are not complete enough to perform convincing tests.
Fortunately, the topology of a small Universe may also be detected through its effects on such a Rosetta stone of cosmology as is the cosmic microwave background (hereafter CMB) fossil radiation (for a review, [8]). If you sprinkle fine sand uniformly over a drumhead and then make it vibrate, the grains of sand will collect in characteristic spots and figures, called Chladni patterns. These patterns reveal much information about the size and the shape of the drum and the elasticity of its membrane. In particular, the distribution of spots depends not only on the way the drum vibrated initially but also on the global shape of the drum, because the waves will be reflected differently according to whether the edge of the drumhead is a circle, an ellipse, a square, or some other shape. In cosmology, the early Universe was crossed by real acoustic waves generated soon after the big bang. Such vibrations left their imprints 380 000 years later as tiny density fluctuations in the primordial plasma. Hot and cold spots in the present-day 2.7 K CMB radiation reveal those density fluctuations. Thus the CMB temperature fluctuations look like Chladni patterns resulting from a complicated three-dimensional drumhead that vibrated for 380 000 years. They yield a wealth of information about the physical conditions that prevailed in the early Universe, as well as present geometrical properties like space curvature and topology. More precisely, density fluctuations may be expressed as combinations of the vibrational modes of space, just as the vibration of a drumhead may be expressed as a combination of the drumhead's harmonics. The shape of space can be heard in a unique way. Lehoucq et al. [9] calculated the harmonics (the so-called "eigenmodes of the Laplace operator") for most of the spherical topologies, and Riazuelo et al. [2] did the same for all 18 Euclidean spaces. Then, starting from a set of initial conditions fixing how the universe originally vibrated (the so-called Harrison-Zeldovich spectrum), it is possible to evolve the harmonics forward in time to simulate realistic CMB maps for a number of flat and spherical topologies [10].
The "concordance model" of cosmology describes the Universe as a flat infinite space in eternal expansion, accelerated under the effect of a repulsive dark energy. The data collected by the NASA satellite WMAP [11] have produced a high resolution map of the CMB which showed the seeds of galaxies and galaxy clusters and allowed to check the validity of the dynamic part of the expansion model. However, combined with other astronomical data [12], they suggest a value of the density parameter Ω 0 = 1.02 ± 0.02 at the 1σ level. The result is marginally compatible with strictly flat space sections. Improved measurements could indeed lower the value of Ω 0 closer to the critical value 1, or even below to the hyperbolic case. Presently however, taken at their face value, WMAP data favor a positively curved space, necessarily of finite volume since all spherical spaceforms possess this property.
CMB temperature anisotropies essentially result from density fluctuations of the primordial Universe : a photon coming from a denser region will loose a fraction of its energy to compete against gravity, and will reach us cooler. On the contrary, photons emitted from less dense regions will be received hotter.
The density fluctuations result from the superposition of acoustic waves which propagated in the primordial plasma. They can be decomposed into a sum of spherical harmonics, much like the sound produced by a music instrument may be decomposed into ordinary harmonics. The "fundamental" fixes the height of the note, whereas the relative amplitudes of each harmonics determine the tone quality. Concerning the relic radiation, the relative amplitudes of each spherical harmonics determine the power spectrum, which is a signature of the space geometry and of the physical conditions which prevailed at the time of CMB emission.
The power spectrum depicts the minute temperature differences on the last scattering surface, depending on the angle of view. It exhibits a set of peaks when anisotropy is measured on small and mean scales (i.e. concerning regions of the sky of relatively modest size). These peaks are remarkably consistent with the infinite flat space hypothesis. At large angular scales, the first observable harmonics is the quadrupole, whose wavenumber is = 2 (i.e. concerning CMB spots typically separated by 90 • ). The concordance model predicts that the power spectrum should follow the so-called "Sachs-Wolfe plateau". However, WMAP measurements fall well below the plateau : the measured value of the quadrupole is 7 times weaker than expected in a flat infinite Universe. The probability that such a discrepancy occurs by chance has been estimated to 0.2% only. The octopole (whose wavenumber is = 3) is also weaker, but still compatible with the error bar, which is larger in this range of wavenumbers due to cosmic variance. For larger wavenumbers up to = 900 (which correspond to temperature fluctuations at small angular scales), observations are remarkably consistent with the concordance model.
The unusually low quadrupole value means that long wavelengths are missing. Some cosmologists have proposed to explain the anomaly by still unknown physical laws of the early universe [13]. A more natural explanation may be because space is not big enough to sustain long wavelengths. Such a situation may be compared to a vibrating string fixed at its two extremities, for which the maximum wavelength of an oscillation is twice the string length. On the contrary, in an infinite flat space, all the wavelengths are allowed, and fluctuations must be present at all scales. Thus this geometrical explanation relies on a model of finite space whose size smaller than the observable universe constrains the observable wavelengths below a maximum value.
Weeks et al. [14] showed that some finite multiconnected topologies do lower the large-scale fluctuations whereas others may elevate them. In fact, the long wavelengths modes tend to be relatively lowered only in a special family of closed multiconnected spaces called "well-proportioned". Generally, among spaces whose characteristic lengths are comparable with the radius of the last scattering surface R lss (a necessary condition for the topology to have an observable influence on the power spectrum), spaces with all dimensions of similar magnitude lower the quadrupole more heavily than the rest of the power spectrum. As soon as one of the characteristic lengths becomes significantly smaller or greater than the other two, the quadrupole is boosted in a way not compatible with WMAP data. In the case of flat tori, a cubic torus lowers the quadrupole whereas an oblate or a prolate torus increase the quadrupole; for spherical spaces, polyhedric spaces suppress the quadrupole whereas high order lens spaces (strongly anisotropic) boost the quadrupole. Thus, well-proportioned spaces match the WMAP data much better than the infinite flat space model.
The "football" universe
Among the family of well-proportioned spaces, the best fit to the observed power spectrum is the already mentioned Poincaré Dodecahedral Space [15]. Recall that this space is positively curved, and is a multiply connected variant of the simply connected hypersphere S 3 , with a volume 120 times smaller for the same curvature radius.
The associated power spectrum, namely the repartition of fluctuations as a function of their wavelengths corresponding to PDS, strongly depends on the value of the mass-energy density parameter. Luminet et al. [15] computed the CMB multipoles for = 2, 3, 4 and fitted the overall normalization factor to match the WMAP data at = 4, and then examined their prediction for the quadrupole and the octopole as a function of Ω 0 . There is a small interval of values within which the spectral fit is excellent, and in agreement with the value of the total density parameter deduced from WMAP data (1.02 ± 0.02). The best fit was obtained for Ω 0 = 1.016 (with the matter component Ω m = 0.28).
Since then, the properties of PDS have been investigated in more details by various authors. [16] found an analytical expression of the eigenmodes of PDS, whereas [17] and [18] computed numerically the power spectrum up to the = 15 mode (corresponding to the calculation of 10,521 eigenmodes) and showed that the fit with WMAP was obtained for 1.016 < Ω 0 < 1.020. More recently, Caillerie et al. [19] computed the power spectrum of PDS until = 35 (involving the calculation of 1.7 × 10 9 eigenmodes) and confirmed the fit (Fig. 9).
The result is quite remarkable because the Poincaré space has no degree of freedom. By contrast, a 3-dimensional torus, constructed by gluing together the opposite faces of a cube and which constitutes a possible topology for a finite Euclidean space, may be deformed into any parallelepiped : therefore its geometrical construction depends on 6 degrees of freedom.
The values of the matter density Ω m , of the dark energy density Ω Λ and of the expansion rate H 0 fix the radius of the last scattering surface R lss as well as the curvature radius of space R c , thus dictate the possibility to detect the topology or not. For Ω m = 0.28, Ω 0 = 1.016 and H 0 = 62km/s/M pc, R lss = 53Gpc and R c = 2.63R lss . It is to be noticed that the curvature radius R c is the same for the simply-connected universal covering space S 3 and for the multiply connected PDS. Incidently, the numbers above show that, Fig. 9. Comparative power spectra as a function of the multipoles for WMAP3 (errorbars), the concordance model (dotdashed curve) and PDS (solid curve) for Ω0 = 1.018, Ωm = 0.27 and h = 0.70 Here we calculate the modes up to k = 3000 using the conjecture of [17] proved by [18].
contrary to a current opinion, a cosmological model with Ω 0 1.02 is far from being "flat" (i.e. with R c = ∞) ! For the same curvature radius than the simply-connected hypersphere S 3 , the smallest dimension of the fundamental dodecahedron is only 43 Gpc, and its volume about 80% the volume of the observable universe (namely the volume of the last scattering surface). This implies that some points of the last scattering surface will have several copies. Such a lens effect is purely attributable to topology and can be precisely calculated in the framework of the PDS model. It provides a definite signature of PDS topology, whereas the shape of the power spectrum gives only a hint for a small, well-proportioned universe model.
To be confirmed, the PDS model (sometimes popularized as the "football" universe model) must satisfy two experimental tests :
• New data from the future European satellite Planck Surveyor (scheduled 2008) could be able to determine the value of the energy density parameter with a precision of 1%. A value lower than 1.009 would discard the PDS as a model for cosmic space, in the sense that the size of the corresponding dodecahedron would become greater than the observable universe and would not leave any observable imprint on the CMB, whereas a value greater than 1.01 would strengthen its cosmological pertinence.
• If space has a non trivial topology, there must be particular correlations in the CMB, namely pairs of "matched circles" along which temperature fluctuations should be the same [20]. The PDS model [15] predicts 6 pairs of antipodal circles with an angular radius comprised between 5 • and 55 • (sensitively depending on the cosmological parameters). Fig. 10. A multiply connected topology translates into the fact that any object in space may possess several copies of itself in the observable Universe. For an extended object like the region of emission of the CMB radiation we observe (the so-called last scattering surface) it can happen that it intersects with itself along pairs of circles. In this case, this is equivalent to say that an observer (located at the center of the last scattering surface) will see the same region of the Universe from different directions. As a consequence, the temperature fluctuations will match along the intersection of the last scattering surface with itself, as illustrated in the above figure. This CMB map is simulated for a multiconnected flat space -namely a cubic hypertorus whose length is 3.17 times smaller than the diameter of the last scattering surface. Only two duplicates are depicted.
Such circles have been searched in WMAP data by several teams, using various statistical indicators and massive computer calculations. First, Cornish et al. [21] claimed to have found no matched circles on angular sizes greater than 25 • , and thus rejected the PDS hypothesis. Next, Roukema et al. [22] performed the same analysis for smaller circles, and found six pairs of matched circles distributed in a dodecahedral pattern, each circle on an angular size about 11 • . This implies Ω 0 = 1.010 ± 0.001 for Ω m = 0.28 ± 0.02, values which are perfectly consistent with the PDS model. Finally, Aurich et al. [23] performed a very careful search for matched circles and found that the putative topological signal in the WMAP data was considerably degraded by various effects [24], so that the dodecahedral space model could be neither confirmed nor rejected...
The controversy still went up a tone when Key et al. [25] claimed that their negative analysis was not disputable, and that accordingly, not only the dodecahedral hypothesis was excluded, but also any multiply-connected topology on a scale smaller than the horizon radius. Since such an argument of authority, a fair portion of the academic community believes the WMAP data has ruled out multiply-connected universe models. However, at least the second part of the claim is wrong. The reason is that they searched only for antipodal or nearly-antipodal matched circles. But Riazuelo et al. [10] have shown that for generic multiply-connected topologies (including the well-proportioned ones, which are good candidates for explaining the WMAP power spectrum), the matched circles are generally not antipodal; moreover, the positions of the matched circles in the sky depend on the observer's position in the fundamental polyhedron. The corresponding larger number of degrees of freedom for the circles search in the WMAP data generates a dramatic increase of the computer time, up to values which are out-of-reach of the present facilities. It follows that the debate about the pertinence of a multiply connected well-proportioned space model to reproduce CMB observations is fully open. The new release of WMAP data [26], integrating two additional years of observation with reduced uncertainty, strengthened the evidence for an abnormally low quadrupole and other features which do not match with the infinite flat space model. Besides the quadrupole suppression, an anomalous alignment between the quadrupole and the octopole was put in evidence along a so-called "axis of evil" [27]. Thus the question arose to know whether, since non-trivial spatial topology can explain the weakness of the low-modes, might it also explain the quadrupole-octupole alignment? Until then no multiply-connected space model, either flat [28] or spherical [30], [29] was proved to exhibit the alignment observed in the CMB sky. This is not a strong argument against such models, since the "axis of evil" is generally interpreted as due to local effects and foreground contaminations [31].
As a provisory conclusion, since some power spectrum anomalies are one of the possible signatures of a finite and multiply-connected universe, there is sill a continued interest in the Poincaré dodecahedral space and related finite universe models. And even if the particular dodecahedral space is eventually ruled out by future experiments, all of the other models of well-proportioned spaces will not be eliminated as such. In addition, even if the size of a multiplyconnected space is larger (not too much) than that of the observable universe, we could all the same discover an imprint in the fossil radiation, even while no pair of circles, much less ghost galaxy images, would remain [32]. The topology of the universe could therefore provide information on what happens outside of the cosmological horizon! But this is search for the next decade . . .
Fig. 2 .
2Fig. 2. Paving the hyperbolic plane with octagons. It is impossible to tessellate the Euclidean plane with octagons, which implies that the double torus is not a Euclidean surface. On the other hand, the hyperbolic plane can be paved by octagons cut from the hollow of a saddle. The hyperbolic plane is thus the universal covering space for the double torus. The eight corners of the octagon must all be identified as a single point; this is the reason why one must use a negatively curved octagon with angles of 45 • (8 × 45 = 360), in place of a flat octagon, whose angles are each 135 • .
Fig. 4 .
4The Seventeen Multiply-Connected Euclidean Spaces. The orientation of the doors indicates how the corresponding walls must be glued together (courtesy Adam Weeks Marano).
Fig. 5 .
5Fig. 5. By analogy with the two-dimensional case, the three-dimensional hypertorus T 3 is obtained by identifying the opposite faces of a parallelepiped. The resulting volume is finite. Let us imagine a light source at our position, immersed in such a structure. Light emitted backwards crosses the face of the parallelepiped behind us and reappears on the opposite face in front of us; therefore, looking forward we can see our back. Similarly, we see in our right our left profile, or upwards the bottom of our feet. In fact, for light emitted isotropically, and for an arbitrarily large time to wait, we could observe ghost images of any object (here the Earth) viewed arbitrarily close to any angle. The resulting visual effect would be comparable (although not identical) to what could be seen from inside a parallelepiped of which the internal faces are covered with mirrors. Thus one would have the visual impression of infinite space, although the real space is closed (courtesy Jeff Weeks).
Fig.
Fig. 6. Left : Poincaré Dodecahedral Space can be described as the interior of a spherical dodecahedron such that when one goes out from a pentagonal face, one comes back immediately inside the space from the opposite face, after a 36 • rotation. Such a space is finite, although without edges or boundaries, so that one can indefinitely travel within it. Right : View from inside PDS perpendicularly to one pentagonal face. In such a direction, ten dodecahedra tile together with a 1/10 th turn to tessellate the universal covering space S 3 . Since the dodecahedron has 12 faces, 120 dodecahedra are necessary to tessellate the full hypersphere. Thus, an observer has the illusion to live in a space 120 times vaster, made of tiled dodecahedra which duplicate like in a mirror hall (courtesy Jeff Weeks).
Fig. 7 .
7View from inside PDS calculated by the CurvedSpaces program (courtesy Jeff Weeks).
Fig. 8 .
8The first 5 layers of PDS (courtesy Jeff Weeks).
Fig. 11 .
11The last scattering surface seen from outside in the universal covering space of the Poincaré dodecahedral space with Ω0 = 1.02, Ωm = 0.27 and h = 0.70 (using modes up to a resolution of 6 • ). Since the volume of the physical space is about 80% of the volume of the last scattering surface, the latter intersects itself along six pairs of matching circles.
Fig. 12 .
12A pocket dodecahedral space, to be cut and glued by the reader (courtesy R. Lehoucq).
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| []
|
[
"DROPPING THE INDEPENDENCE: SINGULAR VALUES FOR PRODUCTS OF TWO COUPLED RANDOM MATRICES",
"DROPPING THE INDEPENDENCE: SINGULAR VALUES FOR PRODUCTS OF TWO COUPLED RANDOM MATRICES"
]
| [
"Gernot Akemann ",
"Eugene Strahov "
]
| []
| []
| We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination of two independent, complex Gaussian random matrices, which is equivalent to a coupling of the two factors via an Itzykson-Zuber term. We prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability. The parameter dependence allows us to interpolate between the singular value statistics of the Laguerre ensemble and that of the product of two independent complex Ginibre ensembles which are both known. We give exact formulae for the correlation kernel in terms of a complex double contour integral, suitable for the subsequent asymptotic analysis. In particular, we derive a Christoffel-Darboux type formula for the correlation kernel, based on a five term recurrence relation for our biorthogonal functions. It enables us to find its scaling limit at the origin representing a hard edge. The resulting limiting kernel coincides with the universal Meijer G-kernel found by several authors in different ensembles. We show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly. | 10.1007/s00220-016-2653-4 | [
"https://arxiv.org/pdf/1504.02047v3.pdf"
]
| 118,761,310 | 1504.02047 | 0f1157e1044223271a70c3b56c24dd68dd935876 |
DROPPING THE INDEPENDENCE: SINGULAR VALUES FOR PRODUCTS OF TWO COUPLED RANDOM MATRICES
13 Apr 2015
Gernot Akemann
Eugene Strahov
DROPPING THE INDEPENDENCE: SINGULAR VALUES FOR PRODUCTS OF TWO COUPLED RANDOM MATRICES
13 Apr 2015
We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination of two independent, complex Gaussian random matrices, which is equivalent to a coupling of the two factors via an Itzykson-Zuber term. We prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability. The parameter dependence allows us to interpolate between the singular value statistics of the Laguerre ensemble and that of the product of two independent complex Ginibre ensembles which are both known. We give exact formulae for the correlation kernel in terms of a complex double contour integral, suitable for the subsequent asymptotic analysis. In particular, we derive a Christoffel-Darboux type formula for the correlation kernel, based on a five term recurrence relation for our biorthogonal functions. It enables us to find its scaling limit at the origin representing a hard edge. The resulting limiting kernel coincides with the universal Meijer G-kernel found by several authors in different ensembles. We show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly.
Introduction
A remarkable feature of products of independent complex Gaussian matrices, i.e. independent matrices with i.i.d. standard complex Gaussian entries, is the exact solvability of the statistical properties of their eigenvalues and singular values. Indeed, it was shown in [2] that the eigenvalues of such products form a determinantal point processes in C. The behaviour of singular values for such products was studied in [6,4], where it was observed that its (squared) singular values also form a determinantal point process in R ≥0 . The correlation kernels of these two different determinantal point processes can be written explicitly in terms of Meijer G-functions, with suitable choices of parameters.
These results have opened the possibility to investigate products of independent complex Gaussian matrices on the same level as the well-known classical ensembles of Random Matrix Theory, such as the Ginibre ensemble and the Laguerre ensemble. We refer the reader to the books by Anderson, Guionnet and Zeitouni [9], and by Forrester [19] for an introduction to Random Matrix Theory, as well as to [1] for a compilation of its most recent applications.
The study of products of random matrices goes back to Furstenberg and Kesten [24] who were interested in its Lyapunov exponents that characterise dynamical systems. Many statistical mechanics applications have been summarised in the book by Crisanti, Paladin, and Vulpiani [16], and most recent examples for applications include telecommunications [39] and combinatorics [42]. A very particular case of the product of two coupled matrices was applied to Quantum Chromodynamics (QCD) with chemical potential in [41], where the complex eigenvalue spectrum was determined. This example will be important for our paper, due to the coupling of the matrices.
The recent rapid development on products of matrices is summarised in the review [3], to where we refer for details and references. In particular, in the work by Kuijlaars and Zhang [36] a new class of so-called Meijer G-kernels was found near the origin, representing a hard edge. The name alludes to the appearance of the Meijer G-function. This kernel generalises the Bessel kernel and contains the kernels of Borodin [13], as pointed out by Kuijlaars and Stivigny [35]. It is universal as it remains unchanged when multiplying by an additional independent inverse complex Gaussian matrices as shown by Forrester [20] or by an additional truncated unitary matrix as shown by Kuijlaars and Stivigny [35]. Furthermore, it appears in the Cauchy twomatrix model [11] and its multi-matrix extension [12] of Bertola and coworkers. Because the Cauchy two-matrix model was used recently to solve the (Laplace transform) of a matrix model with Bures measure by Forrester and Kieburg [21], this kernel enjoys applications to quantum density matrices. And we will also find this limiting Meijer G-kernel for two independent matrices, starting from two coupled random matrices. It was shown in Kuijlaars and Zhang [36] that the class of kernels is integrable in the sense of Its, Isergin, Korepin, and Slavnov [30]. This enabled the description of the squared singular values by Hamiltonian equations [44]. For a survey on integrable operators see Deift [17]. Furthermore, contact was made to questions from Gaussian analytic functions in [3,5] by studying the asymptotics of gap and overcrowding probabilities. For very recent results on determinantal point processes related to products of independent complex Gaussian matrices we refer the reader to Kuijlaars [34], Forrester and Wang [23], and Forrester and Liu [22].
Two questions arise naturally: What happens when the assumption of a Gaussian distribution of matrix elements is dropped? When the matrices in the product are independent, but not necessarily Gaussian, a number of results for the statistics of eigenvalues and of singular values in the global asymptotic regime is available. The paper by O'Rourke and Soshnikov [43] gives an analogue of the circular law for the product of a finite number of non-Hermitian random matrices, generalising the result by Burda, Janik, and Waclaw [15]. For a description of the statistics of singular values of products of independent matrices, and, in particular, for the Central Limit Theorem for the squared singular values we refer the reader to the papers by Götze, Tikhomirov and their co-workers [8,26,25]. Results on the local statistics for products of independent matrices with non-Gaussian entries are still not available, to the best of our knowledge. This is no doubt due to the lack of integrability in the non-Gaussian case.
The second question is whether some of the above results can be extended to those of coupled random matrices. Here we consider a product of two dependent matrices, and concentrate on the statistics of the squared singular values. Such random matrices appeared first in the work by Osborn [41] in the context of QCD with a baryon chemical potential µ as follows:
(1.1) D = 0 iA + µB iA * + µB * 0 .
Here A and B are rectangular independent matrices with i.i.d standard complex Gaussian entries, and µ ∈ [0, 1] is a dimensionless parameter. The motivation to consider (1.1) comes from the observation that the QCD Dirac operator D has this off-diagonal block form in the so-called chiral basis. For the random matrix application to QCD we refer to the review by Verbaarschot and Wettig [46], see also chapter 32 in [1] by Verbaarschot. In [41] the correlations of complex eigenvalues of D were determined, which is equivalent to determining the eigenvalues of the product matrix Y = X 1 X 2 , with X 1 = (iA + µB) and X 2 = (iA * + µB * ). The change of variables from matrices A, B to X 1 , X 2 reveals that the latter are coupled by an Itzykson-Zuber term, in addition to their Gaussian weight. Very recently it has been suggested in [32,33] to study the singular values of the Dirac operator in QCD and QCD-like theories instead, in order to better understand the high-density regime. This is one of the motivations for us to study the (squared) singular values of the product matrix Y . Apart from this physical interpretation the parameter µ allows to interpolate between the classical Laguerre ensemble at µ = 0 solved by orthogonal Laguerre polynomials and the recent solution of the product of two independent Gaussian random matrices at µ = 1 given in terms of biorthogonal functions. This paper is organised as follows. In Section 2 we define the notion of µ-dependent Gaussian complex random matrices, making this notion of interpolation and of its limits more precise. We state our main results in Section 3. In particular we demonstrate the exact solvability of the statistical properties of the singular values of the product matrix Y for arbitrary parameter values µ: the joint probability density function of the squared singular values is a determinantal process on R ≥0 and can be computed explicitly in terms of modified Bessel functions of first and second kind. This determinantal point process is a biorthogonal ensemble in the sense of Borodin [13]. For this parameter dependent ensemble we derive different formulae for the correlation kernel including a Christoffel-Darboux type formula and a double complex contour integral representation. We compute the hard edge scaling limit at the origin, and we obtain a Central Limit Theorem for fluctuations of linear statistics. Sections 4-11 and Appendix A contain the proofs of our statements. Acknowledgements. We are grateful to Percy Deift for discussions, and to Jonathan Breuer for a clear explanation of the results in Ref. [14] to us. One of us (G.A.) would like to thank the LPTMS Orsay for hospitality where part of these results were finalised.
Parameter dependent Gaussian complex matrices
Before we present our results we will define a family of parameter dependent coupled Gaussian random variables, and the corresponding notion for random matrices. By this we mean the following. Definition 2.1. Let µ ∈ (0, 1), α(µ) = 1+µ 2µ , and δ(µ) = 1−µ 2µ . We will refer to two complex random variables, z and ξ as to µ-dependent Gaussian complex variables if the joint density of these variables is given by
ρ(z, ξ) = 1 π 2 µ exp −α(µ)(zz + ξξ) + δ(µ)(zξ +zξ) . Definition 2.2. Let X 1 = X (1) 1,1 . . . X (1) 1,M . . . X (1) N,1 . . . X (1) N,M , X 2 = X (2) 1,1 . . . X (2) 1,N . . . X (2) M,1 . . . X (2) M,N
be two matrices whose complex random entries are defined by the following conditions
• X (1) i,j , 1 ≤ i ≤ N, 1 ≤ j ≤ M are independent; • X (2) i,j , 1 ≤ i ≤ M, 1 ≤ j ≤ N are independent; • For each 1 ≤ i ≤ N, and for each 1 ≤ j ≤ M the pair (X (1) i,j , X(2)
j,i ) is a pair of µ-dependent Gaussian complex random variables.
We will refer to such random matrices X 1 and X 2 as to µ-dependent Gaussian complex random matrices.
Alternatively, we can define µ-dependent Gaussian complex random matrices as follows. Let Mat(C, N × M) denote the space of N × M complex matrices X 1 , and Mat(C, M × N) denote the space of M × N complex random matrices X 2 . We consider the probability distribution P N,M (X 1 , X 2 )dX 1 dX 2 on the Cartesian product of Mat(C, N × M) and Mat(C, M × N)
P N,M (X 1 , X 2 )dX 1 dX 2 =c · exp [−α(µ) Tr (X 1 X * 1 + X * 2 X 2 ) + δ(µ) Tr (X 1 X 2 + X * 2 X * 1 )] × N i=1 M j=1 dX (1) i,j R dX (1) i,j I M i=1 N j=1 dX (2) i,j R dX (2) i,j I , (2.1) where X (1) i,j = X (1) i,j R + iX (1) i,j I , X (1) i,j = X (1) i,j R + iX (1) i,j I
denote the sums of the real and imaginary parts of the matrix entries X (1) i,j and X (2) i,j , and c is a normalising constant. The second term in the exponent proportional to δ(µ) is nothing else than the Itzykson-Zuber term (for nonhermitian matrices) coupling the two matrices 1 . We have
Tr(X 1 X * 1 ) = N i=1 M j=1 X (1) i,j X (1) i,j , Tr(X * 2 X 2 ) = N i=1 M j=1 X (2) j,i X (2) j,i , and Tr(X 1 X 2 ) = N i=1 M j=1 X (1) i,j X (2) j,i , Tr(X * 2 X * 1 ) = N i=1 M j=1 X (1) i,j X(2)
j,i .
1 However, because we will be interested in the singular values of the product matrix X 1 X 2 , we will not use their integration formula [28,31] for this term. Therefore the formula for P N,M (X 1 , X 2 )dX 1 dX 2 can be rewritten as
P N,M (X 1 , X 2 )dX 1 dX 2 =c · N i=1 M j=1 e −α(µ) X (1) i,j X (1) i,j +X (2) j,i X (2) j,i +δ(µ) X (1) i,j X (2) j,i +X (1) i,j X (2) j,i × dX (1) i,j R dX (1) i,j I dX (2) j,i R dX (2) j,i I . (2.2)
It is clear from the formula just written above that P N,M (X 1 , X 2 ) is indeed the probability distribution of the µ-dependent Gaussian complex matrices X 1 and X 2 . In addition, note that the normalising constant c is equal to
c = 1 (π 2 µ) N M .
Proposition 2.3. Let A, B be two independent matrices of size N × M with i.i.d standard complex Gaussian entries. Define the random matrices X 1 and X 2 as
(2.3) X 1 = 1 √ 2 (A − i √ µB) , X 2 = 1 √ 2 (A * − i √ µB * ) .
Then the matrices X 1 and X 2 are µ-dependent Gaussian complex random matrices.
Proof. This can be checked by direct calculation.
Statement of results
3.1. The joint probability density function. Our first result is an explicit formula for the joint probability density function for the squared singular values of the random matrix X 1 X 2 .
Recall that the modified Bessel function of the first kind I κ (z) is defined by
(3.1) I κ (z) = ∞ m=0 1 m!Γ(κ + m + 1) z 2 2m+κ
, and the modified Bessel function of the second kind K κ (z) can be defined by the integral formula
(3.2) K κ (z) = Γ κ + 1 2 (2z) κ √ π ∞ 0 cos(t)dt (t 2 + z 2 ) κ 2 + 1 2 ,
see, for example, Gradshteyn and Ryzhik [27]. Then the joint probability density function for the squared singular values y 1 , . . ., y N of the matrix Y = X 1 X 2 is given by
P (y 1 , . . . , y N ) = 1 Z N det y j−1 2 i I j−1 (2δ(µ) √ y i ) N i,j=1 det y j+ν−1 2 i K j+ν−1 (2α(µ) √ y i ) N i,j=1 , (3.3) where (3.4) Z N = N! α(µ) N ν+ N(N−1) 2 δ(µ) N(N−1) 2 2 N (α(µ) 2 − δ(µ) 2 ) N ν+N 2 N j=1 Γ(j)Γ(j + ν).
Let us regard µ as a deformation parameter, and consider two interesting limits of the joint probability density function P (y 1 , . . . , y N ). In the first limiting case the two Gaussian matrices become independent, corresponding to µ → 1, with δ(µ) → 0 and α(µ) → 1. This fact is obvious from the very definition of two µ-dependent Gaussian complex matrices. It can also be seen directly from the explicit formula for the joint probability density function P (y 1 , . . . , y N ), equation (3.3), as shown in Appendix A:
(3.5) lim µ→1 P (y 1 , . . . , y N ) = det y j−1 i N i,j=1 N! N j=1 Γ(j) 2 Γ(j + ν) det G 2,0 0,2 − 0, j + ν − 1 y i N i,j=1
.
Here we have introduced the Meijer G-function (see e.g. Luke [38])
(3.6) G m,n p,q a 1 , a 2 , . . . , a p b 1 , b 2 , . . . , b q z = 1 2πi C m j=1 Γ(b j − s) n j=1 Γ(1 − a j + s) q j=m+1 Γ(1 − b j + s) p j=n+1 Γ(a j − s) z s ds.
An empty product is interpreted as unity, for the indices m ≥ q, n ≥ p. The contour of integration C depends on the location of the poles of the Gamma functions, and we refer to the NIST handbook [40] for details on the different possibilities. In particular the following formula holds, see 9.34.3 in [27],
(3.7) G 2,0 0,2 − 0, l y = 2y l 2 K l (2 √ y).
The right-hand side of equation (3.5) agrees with the joint probability density function of squared singular values of two independent rectangular complex Ginibre matrices, see Akemann, Ipsen and Kieburg [4], formulae (18) and (21). Here we only consider the special case that the matrix Y = X 1 X 2 is square, with ν = ν 1 and ν 2 = 0 compared to there. We will need Y to be square for the group integrals that we encounter in the derivation for general µ ∈ (0, 1). The second interesting limit is that of µ → 0. In this limit δ(µ) and α(µ) diverge, and we obtain the joint density equivalent to the classical Laguerre ensemble. To find the limit of the joint probability density function as µ → 0 we use formula (3.3), and replace the modified Bessel functions inside the determinants by their large argument asymptotic expressions. A short calculation in Appendix A yields Changing variables in equation (3.8),
(3.9) y i → v i = 2y 1 2
i , we obtain the joint probability density function of the classical Laguerre ensemble
1 N! N j=1 Γ(j)Γ(j + ν) det v j−1 i N i,j=1 2 N i=1 v ν i e −v i ,
see Forrester [19], Chapter 7. The change of variables is necessary because we started from the singular values of Y = X 1 X * 1 in this limit, rather than of X 1 which is the single matrix with Gaussian distribution left in this limit.
We conclude that the product X 1 X 2 of two µ-dependent Gaussian complex matrices represents an interpolating biorthogonal ensemble. It interpolates between the ensemble of two independent complex Gaussian matrices, and the Laguerre ensemble of a single complex Gaussian matrix.
3.2.
Exact formulae for the correlation kernel. Theorem 3.1 implies that the squared singular values y 1 , . . ., y N of the product X 1 X 2 of two µ-dependent Gaussian complex matrices form a determinantal point process, (3.10) P (y 1 , . . . , y N ) = det [K N (y i , y j )] N i,j=1 . Here we present exact formulae for the correlation kernel of this process.
Theorem 3.2. The correlation kernel K N (x, y) of the determinantal point process formed by the squared singular values of X 1 X 2 is given by
(3.11) K N (x, y) = N −1 n=0 P n (x)Q n (y),
where the functions P 0 (x), P 1 (x), . . . are defined by
(3.12) P n (x) = (−1) n (ν + n)!n! n k=0 (α(µ) 2 − δ(µ) 2 ) k+ 1 2 δ(µ) k (−n) k (ν + k)!k! x k 2 I k (2δ(µ) √ x),
and the functions Q 0 (y), Q 1 (y), . . . are defined by
(3.13) Q n (y) = (−1) n 2 (n!) 2 n l=0 (α(µ) 2 − δ(µ) 2 ) l+ν+ 1 2 α(µ) l+ν (−n) l (ν + l)!l! y l+ν 2 K l+ν (2α(µ) √ y).
Using explicit formulae for the functions P n (x) and Q n (x) (equations (3.12) and (3.13)) we derive the following formula for the correlation kernel K N (x, y) Theorem 3.3. The correlation kernel K N (x, y) can written as
K N (x, y) = 2 N −1 k,l=0 l i=0 (−1) i+k (ν + N + i)! (N − 1 − k)!(ν + k)!i!(l − i)!k!(ν + i)!(ν + k + i + 1) × (α(µ) 2 − δ(µ) 2 ) k+l+ν+1 α(µ) ν+l δ(µ) k x k 2 y l+ν 2 I k (2δ(µ) √ x)K l+ν (2α(µ) √ y) .
(3.14)
Theorem 3.3 enables us to compare our biorthogonal ensemble with the family of the Laguerre-type biorthogonal ensembles introduced and studied in Borodin [13], Section 4. Also, Theorem 3.3 can be used to investigate the transition of our biorthogonal ensemble to a Laguerre-type ensemble as µ approaches zero. Consider the Laguerre-type ensemble defined by the right-hand side of equation (3.8). Using the same argument as in Borodin [13], Section 4, Theorem 4.1 we can write the correlation kernel K Lag N (x, y) of the ensemble from [13] with θ = 1 as
(3.15) K Lag N (x, y) = e −x 1 2 −y 1 2 x 1 4 y 1 4 N −1 k,l=0 l i=0 (−1) i+k (ν + N + i)!2 k+l+ν (N − 1 − k)!(ν + k)!i!(l − i)!k!(ν + i)! x k 2 y l+ν 2 (ν + k + i + 1) .
It is not hard to check using the asymptotic expressions for the modified Bessel functions of large arguments (see eq. (A.1)) that the kernel K N (x, y) turns into a kernel equivalent to
K Lag N (x, y) as µ → 0. 2
The subsequent asymptotic analysis requires a detailed investigation of the properties of the functions P n (x) and Q n (y) determining the correlation kernel K N (x, y). In particular, we show that these functions satisfy the following biorthogonality condition.
Proposition 3.4. The functions P n (x), Q n (x) defined by equations (3.12) and (3.13) correspondingly satisfy the biorthogonality condition:
∞ 0 P n (x)Q m (x)dx = δ n,m , n, m = 0, 1, 2, . . . .
Moreover, both P n (x) and Q n (y) satisfy five term recurrence relations, and can be represented as contour integrals. Namely, the following Proposition holds true. Proposition 3.5. (a) For the functions P n (x) we have the following five term recurrence relation
(3.16) xP n (x) = a 2,n P n+2 (x) + a 1,n P n+1 (x) + a 0,n P n (x) + a −1,n P n−1 (x) + a −2,n P n−2 (x),
where the coefficients a 2,n , a 1,n , a 0,n , a −1,n , and a −2,n are given explicitly by
a 2,n = 1 (n + 2)(n + 1) δ(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 , (3.17) a 1,n = 1 α(µ) 2 − δ(µ) 2 + 2(2n + ν + 2) (n + 1) δ(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 , (3.18) a 0,n = 3n 2 + 2νn + 3n + ν + 1 α(µ) 2 − δ(µ) 2 + (6n 2 + 6nν + ν 2 + 6n + 3ν + 2) δ(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 , (3.19) a −1,n = n 2 (n + ν)(3n + ν) α(µ) 2 − δ(µ) 2 + 2n 2 (ν + n)(2n + ν) δ(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 , (3.20) a −2,n = (ν + n)(ν + n − 1)n 2 (n − 1) 2 α(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 . (3.21) (b)
For the functions Q n (y) we have the following five term recurrence relation
(3.22) yQ n (y) = b 2,n Q n+2 (y) + b 1,n Q n+1 (y) + b 0,n Q n (y) + b −1,n Q n−1 (y) + b −2,n Q n−2 (y),
where the coefficients b 2,n , b 1,n , b 0,n , b −1,n , and b −2,n are given explicitly by
b 2,n = (ν + n + 2)(ν + n + 1)(n + 2) 2 (n + 1) 2 α(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 , (3.23) b 1,n = − (n + 1) 2 (n + ν + 1) 2 α(µ) 2 − δ(µ) 2 + 2(2n + ν + 2)(n + ν + 1)(n + 1) 2 α(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 , (3.24) b 0,n = − (n + ν) 2 + 2(n + 1)(n + ν) + n + 1 α(µ) 2 − δ(µ) 2 (3.25) + ((n + ν)(5n + ν + 3) + n(n + 3) + 2) α(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 , b −1,n = − (3n + 2ν) n 1 α(µ) 2 − δ(µ) 2 + 2(2n + ν) n α(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 , (3.26) b −2,n = 1 n(n − 1) δ(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 . (3.27)
Note that the recurrence coefficients are related as
(3.28) a 2,n = b −2,n+2 , a 1,n = b −1,n+1 , a 0,n = b 0,n , a −1,n = b 1,n−1 , a −2,n = b 2,n−2 .
This follows from the biorthogonality of the functions P n (x) and Q n (y), see Proposition 3.4. Equation (3.28) can be checked directly as well using the formulas in Proposition 3.5 for the recurrence coefficients.
Using the recurrence relations stated in Proposition 3.5 we derive the following Christoffel-Darboux type formula for the correlation kernel K N (x, y).
Theorem 3.6. The Christoffel-Darboux type formula for the correlation kernel K N (x, y) valid for N ≥ 2 and x = y is given by
K N (x, y) = − a −2,N P N −2 (x)Q N (y) + a −2,N +1 P N −1 (x)Q N +1 (y) + a −1,N P N −1 (x)Q N (y) x − y + a 1,N −1 P N (x)Q N −1 (y) + a 2,N −2 P N (x)Q N −2 (y) + a 2,N −1 P N +1 (x)Q N −1 (y) x − y , (3.29)
where the coefficients a −2,N , a −1,N , a 1,N and a 2,N are given by Proposition 3.5.
The next Proposition gives contour integral representations for the functions P n (x) and Q n (y).
Proposition 3.7. (a)
The following contour integral representation for the function P n (x) holds:
P n (x) = 1 2πi (ν + n)!(n!) 2 α(µ) 2 − δ(µ) 2 1 2 × Σ Γ(t − n) (α(µ) 2 − δ(µ) 2 ) t x t (Γ(t + 1)) 2 Γ(t + ν + 1) 0 F 1 − t + 1 δ(µ) 2 x dt, (3.30)
where Σ is a closed contour that encircles 0, 1, . . ., n once in positive direction, n = 0, 1, . . ., and x > 0. (b) The following contour integral representation for the function Q n (y) is true:
Q n (y) = 1 2πi(n!) 2 (n + ν)! 1 − δ(µ) 2 α(µ) 2 ν α(µ) 2 − δ(µ) 2 1 2 × c+i∞ c−i∞ Γ 2 (s)Γ(s + ν) Γ(s − n) 2 F 1 −n, ν + s s − n δ(µ) 2 α(µ) 2 α(µ) 2 y −s ds, (3.31)
where c > 0, n = 0, 1, . . ., and y > 0.
Finally, we state that as a consequence the correlation kernel K N (x, y) admits a double contour integral representation.
Theorem 3.8. The correlation kernel, K N (x, y), can be written as
(3.32) K N (x, y) = N −1 k=0 K (k) N (x, y) δ(µ) α(µ) k , where K (k) N (x, y) = k m=0 (−1) m (2πi) 2 N m Σ dt c+i∞ c−i∞ ds Γ 2 (s)Γ(s + ν + k)Γ(s − t + m − 1)Γ(t − N + 1) Γ 2 (t + 1)Γ(t + ν + 1)Γ(s − N + m)Γ(s − t + k) × 1 − δ(µ) 2 α(µ) 2 ν α(µ) 2 − δ(µ) 2 t+1 x t 0 F 1 − t + 1 δ(µ) 2 x α(µ) 2 y −s . (3.33)
The contour Σ is chosen in the same way as in Proposition 3.7, and c > 0.
As µ → 1, the biorthogonal ensemble defined by equation (3.3) turns into that for the squared singular values of the product of two matrices with independent complex Gaussian entries, see equation (3.5). The biorthogonal ensemble for the squared singular values of products of M matrices with independent complex Gaussian entries was studied in [6,4]. As µ → 1, the functions P n (x) and Q n (y) defined by equations (3.12) and (3.13) turn into the biorthogonal polynomials and their normalised dual functions there, see equations (43) and (47) in [4], resepctively. Furthermore, our Propositions 3.5 and 3.7 are extensions of the results obtained by Kuijlaars and Zhang, see their Proposition 3.2 and formula (3.6), and the recurrence relations in Section 4 of Kuijlaars and Zhang [36]. As µ → 1, the formulae for the correlation kernel K N (x, y) given in Theorem 3.8 turn into the double integral formula of Proposition 5.1 in Kuijlaars and Zhang [36].
3.3. The hard edge scaling limit of the correlation kernel. We use the Christoffel-Darboux type formula for the correlation kernel K N (x, y) given by Theorem 3.6, and the contour integral representations for the functions P n (x) and Q n (y) of Proposition 3.7 to find the scaling limit of K N (x, y) near the origin (hard edge). Theorem 3.9. Let ν and µ be fixed. For x and y in a compact subset of the positive real axis,
K ν (x, y) = lim N →∞ 1 N (α(µ) 2 − δ(µ) 2 ) K N x N (α(µ) 2 − δ(µ) 2 ) , y N (α(µ) 2 − δ(µ) 2 ) ,
where the limiting Meijer G-kernel K ν (x, y) is given by
K ν (x, y) = 1 0 G 1,0 0,3 − 0, −ν, 0 ux G 2,0 0,3 − ν, 0, 0 uy du.
Here The resulting limiting kernel K ν (x, y) coincides with the scaling limit found by Bertola, Gekhtman, and Szmigielski in the Cauchy-Laguerre two-matrix model [11], with the scaling limit for the product of two independent complex Gaussian matrices found by Kuijlaars and Zhang [36], and with the limiting kernel for the product of two independent complex Gaussian matrices times a fixed arbitrary number of inverses of such matrices found by Forrester [20]. This confirms that the family of new limiting so-called Meijer G-kernels obtained in Kuijlaars and Zhang [36] in the context of products of independent matrices represents a new universality class.
3.4. The Central Limit Theorem. Proposition 3.5 gives explicitly the recurrence coefficients for the functions P n (x) and Q n (y) determining the correlation kernel of the biorthogonal ensemble defined by equation (3.3). This enables us to derive a Central Limit Theorem for the linear statistics of singular values of X 1 X 2 , and to give the limiting variance explicitly.
Here, instead of the probability distribution P N,M (X 1 , X 2 )dX 1 dX 2 (defined by equation (2.1)) we consider the probability distribution P N,M (X 1 , X 2 )dX 1 dX 2 on the Cartesian product of Mat(C, N × M) and Mat(C, M × N) defined by
P N,M (X 1 , X 2 )dX 1 dX 2 =c · exp [−Nα(µ) Tr(X 1 X * 1 + X * 2 X 2 ) + Nδ(µ) Tr(X 1 X 2 + X * 2 X * 1 )] × N i=1 M j=1 dX (1) i,j R dX (1) i,j I M i=1 N j=1 dX (2) i,j R dX (2) i,j I , (3.34) where X (1) i,j = X (1) i,j R + iX (1) i,j I and X (2) i,j = X (2) i,j R + iX (2) i,j
I denote the sums of the real and imaginary parts of the matrix entries X
i,j and X (2) i,j , and c is a normalising constant. Equation (3.34) is obtained from Equation (2.1) by a simple rescaling of the matrix elements by √ N . Let y 1 , . . ., y N be the squared singular values of the matrix X 1 X 2 , and define the linear statistics of y 1 , . . ., y N by the formula
Y (N ) f = N i=1 f (y i ).
Theorem 3.10. Let f be a polynomial with real coefficients. Then we have
Y (N ) f − EY (N ) f → N 0, ∞ k=1 kf kf−k in distribution, wheref k = 1 2πi |w|=1 f (s(w; µ)) w k dw w , s(w; µ) = 1 4w 2 (w + 1) 3 w(1 − µ) 2 + (1 + µ) 2 . (3.35)
Remark 3.11. Since f is a polynomial with real coefficients,f k is real. Furthermore, the Central Limit Theorems for the limiting cases µ → 0 and µ → 1 can be immediately read off by taking the limits on the Laurent polynomial s(w; µ). For the product of two independent complex Gaussian matrices we obtain lim µ→1 s(w; µ) = (w + 1) 3 /w 2 . This agrees with the results following from the recursion coefficients by Kuijlaars and Zhang, by specifying to two matrices there. In the opposite limit we obtain lim µ→0 s(w; µ) = (w + 1) 4 /(4w 2 ). It is not difficult using Laguerre polynomials of square root arguments to directly show that this is the correct limit for the ensemble in eq. (3.8) -which is not the standard Laguerre ensemble due to the change of variables in equation (3.9).
The proof of Theorem 3.10 uses the results for biorthogonal ensembles obtained in Breuer and Duits [14]. They showed that whenever the asymptotic of recurrence coefficients is available, a Central Limit Theorem for the linear statistics can be derived. In our case, Proposition 3.5 gives the recurrence coefficients explicitly. Considering the rescaled probability distribution P N,M (X 1 , X 2 )dX 1 dX 2 we obtain recurrence coefficients that have finite limits as N → ∞, which gives Theorem 3.10.
Proof of Theorem 3.1
First we show that the computation of the joint probability density function of (squared) singular values for products of rectangular matrices can be reduced to that for products of square matrices of the same size. Namely, the following Lemma holds true.
G 0 X 0 , where G 0 ∈ Mat(C, N × N), X 0 ∈ Mat(C, N × N)
, and the joint distribution of G 0 , X 0 is given by
P (N,M ) (G 0 , X 0 )dG 0 dX 0 = const · det (X * 0 X 0 ) M −N × e −α(µ) Tr(G 0 G * 0 +X * 0 X 0 )+δ(µ) Tr(G 0 X 0 +X * 0 G * 0 ) dG 0 dX 0 . (4.1)
Here and below the computation of the µ-dependent constants is suppressed until the last part of the proof of Theorem 3.1.
Proof. If M = N, then the statement of the Lemma follows immediately. Consider the case when M > N. Recall that the matrices G, X are distributed in accordance with P (N,M ) (G, X)dGdX = const ·e −α(µ) Tr(GG * +X * X)+δ(µ) Tr(GX+X * G * ) dGdX.
Consider the following decomposition of the matrix X
X = U X 0 O M −N,N ,
where U is an M × M unitary matrix, X 0 is an N × N complex matrix, and O M −N,N is a complex matrix of size (M − N) × N with zero entries. 3 We have
P (N,M ) (G, X)dGdX = const · det (X * 0 X 0 ) M −N e −α(µ) Tr(GG * +X * 0 X 0 ) × exp δ(µ) Tr GU X 0 O M −N,N + Tr X * 0 O N,M −N U * G * dGdUdX 0 ,
where we have used the results of Section 2 in Fischmann, Bruzda, Khoruzhenko, Sommers, and Zyczkowski [18] (see also the discussion in Ipsen and Kieburg [29], Section III, A). Here dU denotes the Haar measure. IfĜ = GU, then the equation above can be rewritten as
P (N,M ) (G, X)dGdX = const · det (X * 0 X 0 ) M −N e −α(µ) Tr(ĜĜ * +X * 0 X 0 ) × exp δ(µ) Tr Ĝ X 0 O M −N,N + Tr X * 0 O N,M −N Ĝ * dĜdUdX 0 ,P (N,M ) (G, X)dGdX = const · det (X * 0 X 0 ) M −N e −α(µ)(Tr(G 0 G * 0 )+Tr(ĜN,M−NĜ * N,M −N )+Tr(X * 0 X 0)) × e δ(µ) Tr(G 0 X 0 +X * 0 G * 0 ) dĜ 0 dĜ N,M −N dUdX 0 .
The formula just written above implies that the joint distribution of G 0 , X 0 is given by equation (4.1). Moreover, by construction the squared singular values of GX coincide with those of G 0 X 0 .
Let us turn to the proof of Theorem 3.1. We use Lemma 4.1, and assume that both matrices X 1 , X 2 are taken from Mat (C, N × N), and that the joint distribution of X 1 , X 2 is given by
P (N,M ) (X 1 , X 2 )dX 1 dX 2 = const · det (X * 2 X 2 ) M −N × e −α(µ) Tr(X 1 X * 1 +X * 2 X 2 )+δ(µ) Tr(X 1 X 2 +X * 2 X * 1 ) dX 1 dX 2 ,(4.4)
where M ≥ N. In fact we need that X 1 , X 2 ∈ Gl(N, C). Because the set of invertible matrices is dense in Mat (C, N × N) this will not change the joint distribution. Consider the change of variables
X 1 → Y 2 = X 1 X 2 , X 2 → Y 1 = X 2 .
It is known that this transformation has a Jacobian det (Y * 1 Y 1 ) −N . Therefore we can write
P (N,M ) (X 1 , X 2 )dX 1 dX 2 = const · det (Y * 1 Y 1 ) M −2N × e −α(µ) Tr Y * 2 Y 2 Y −1 1 (Y * 1 ) −1 +Tr(Y * 1 Y 1 ) +δ(µ) Tr(Y 2 +Y * 2 ) dY 2 dY 1 .
(4.5)
Next we use the singular value decomposition for both Y 2 and Y 1
Y 1 = V 1 Λ 1 U 1 , Λ 1 = λ (1) 1 0 . . . 0 λ (1) N , Y 2 = V 2 Λ 2 U 2 , Λ 2 = λ (2) 1 0 . . . 0 λ (2) N ,
where Λ 1 , Λ 2 are diagonal matrices with the singular values along the diagonals, and V 1 , V 2 , U 1 and U 2 are unitary N × N matrices. It is known that
dY 1 = const ·△ Λ 2 1 2 N j=1 λ (1) j dλ (1) j dU 1 dV 1 , dY 2 = const ·△ Λ 2 2 2 N j=1 λ (2) j dλ (2) j dU 2 dV 2 ,
where we have introduced the Vandermonde determinant
△ Λ 2 1 = N ≥j>k≥1 λ (1) j 2 − λ (1) k 2 , △ Λ 2 2 = N ≥j>k≥1 λ (2) j 2 − λ (2) k 2 ,
and where dU 1 , dV 1 , dU 2 , and dV 2 are the corresponding Haar measures on the unitary group U(N). Combining these formulae we obtain a probability measure
P (N,M ) (X 1 , X 2 )dX 1 dX 2 = const ·e −α(µ)(Tr(Λ 2 1 )+Tr(U 1 U * 2 Λ 2 2 U 2 U * 1 Λ −2 1 ))+δ(µ)(Tr(V2Λ2U2)+Tr(U * 2 Λ 2 V * 2 )) × △ Λ 2 1 2 △ Λ 2 2 2 det M −2N Λ 2 1 N j=1 λ (1) j dλ (1) j N j=1 λ (2) j dλ (2) j dU 1 dU 2 dV 1 dV 2 .
Using the invariance of the Haar measures under the subsequent shifts U 1 → U 1 U 2 , and U 2 → U 2 V * 2 , and integrating over V 1 and V 2 we obtain
P (N,M ) (X 1 , X 2 )dX 1 dX 2 = const ·e −α(µ)(Tr(Λ 2 1 )+Tr(U 1 Λ 2 2 U * 1 Λ −2 1 ))+δ(µ)(Tr(Λ2U2)+Tr(U * 2 Λ 2)) × △ Λ 2 1 2 △ Λ 2 2 2 det M −2N Λ 2 1 N j=1 λ (1) j dλ (1) j N j=1 λ (2) j dλ (2) j dU 1 dU 2 . (4.6)
The integration over U 1 can be performed using the Harish-Chandra-Itzykson-Zuber integration formula [28,31] (4.7)
U (N ) e − Tr(U 1 Λ 2 2 U * 1 Λ −2 1 ) dU 1 = const · det exp − λ (2) j 2 λ (1) i −2 N i,j=1 △(Λ 2 2 )△(Λ −2 1 ) ,
where the constant does not depend on Λ 1 and Λ 2 , and we have used the transformation Λ 1 → α(µ) 1 2 Λ 1 . In addition, we apply the fact that the Vandermonde determinant of inverse powers is proportional to the Vandermonde determinant with positive powers, namely
(4.8) △ Λ −2 1 = const · △(Λ 2 1 ) det N −1 [Λ 2 1 ]
.
As a result of application of formulae (4.7), (4.8) to probability measure (4.6) we have
P (N,M ) (X 1 , X 2 )dX 1 dX 2 = const ·e −α(µ) 2 Tr(Λ 2 1 )+δ(µ)(Tr(Λ 2 U 2 )+Tr(U * 2 Λ 2)) △ Λ 2 1 △ Λ 2 2 × det exp − λ (2) j 2 λ (1) i −2 N i,j=1 det M −N −1 Λ 2 1 N j=1 λ (1) j dλ (1) j N j=1 λ (2) j dλ (2) j dU 2 .
(4.9)
Now our task is to perform the integration over U 2 . This can be done exploiting the following Leutwyler-Smilga integral formula [37], see e.g. [10] for a derivation based on group characters,
(4.10) U (N ) e δ(µ) Tr(Λ 2( U 2 +U * 2 )) dU 2 = const · det λ (2) j i−1 I i−1 2δ(µ)λ (2) j N i,j=1 △(Λ 2 2 )
.
Here I k (x) denotes the modified Bessel function of the first kind. After the integration over U 2 we obtain the following probability distribution
P (N,M ) (X 1 , X 2 )dX 1 dX 2 = const ·e −α 2 (µ) Tr(Λ 2 1 ) △ Λ 2 1 det λ (2) j i−1 I i−1 2δ(µ)λ (2) j N i,j=1 × det exp − λ (2) j 2 λ (1) i −2 N i,j=1 det M −N −1 Λ 2 1 N j=1 λ (1) j dλ (1) j N j=1 λ (2) j dλ (2) j . (4.11)
To get the induced probability distribution of the singular values λ
1 ,. . .,λ
N of the matrix Y 2 = X 1 X 2 we only need to integrate the probability distribution (4.11) over the variables λ
I = det λ (1) i 2(j−1) N i,j=1 det exp − λ (2) j 2 λ (1) i −2 N i,j=1 × N j=1 e −α(µ) 2 λ (1) j 2 λ (1) j 2ν−1 dλ(1)
j .
(4.12)
Applying the Andréief integral identity valid for a set of integrable functions,
det [ϕ i (x j )] N i,j=1 det [ψ i (x j )] N i,j=1 N j=1 dµ(x j ) = N! det ϕ i (x)ψ j (x)dµ(x) N i,j=1 , (4.13) to ϕ i (x) = x 2(i−1) , ψ i (x) = e − λ (2) i 2 x −2
, and dµ(x) = e −α(µ) 2 x 2 x 2ν−1 dx on R + , we obtain that integral (4.12) is equal
I = N! det ∞ 0 e −α(µ) 2 x 2 − λ (2) j 2 x −2 x 2(i+ν)−3 dx N i,j=1 .
To compute the integral inside the determinant above we use the formula [27]
8.432.6 ∞ 0 x ν−1 exp −x − ρ 2 4x dx = 2 ρ 2 ν K −ν (ρ),
where K −ν (ρ) = K +ν (ρ) is the modified Bessel function of the second kind. The result is that integral (4.12) is proportional to
I = const · det λ (2) j i+ν−1 K i+ν−1 2α(µ)λ (2) j N i,j=1
.
We conclude that the joint density of the singular values of the matrix X 1 X 2 with ν = M − N s given by
P (N,M ) (X 1 , X 2 )dX 1 dX 2 = const · det λ (2) i j−1 I j−1 2δ(µ)λ (2) i N i,j=1 × det λ (2) i j+ν−1 K i+ν−1 2α(µ)λ (2) i N i,j=1 N j=1 λ (2) j dλ (2)
j .
(4.14)
Changing to squared singular values, λ In order to compute this constant we can apply again the Andréief identity equation (4.13), interpreting the left hand side as a probability measure, with the following choice of functions for the squared singular values y j of X 1 X 2 :
(4.15) ψ j (x) = x j 2 I j 2δ(µ) √ x , and (4.16) ϕ j (x) = x j+ν 2 K j+ν 2α(µ) √ x .
After applying the integral identity the requirement that this probability measure is normalised reads as follows,
1 = Z −1 N N! det ∞ 0 y i+j+ν 2 I i (2δ(µ) √ y) K j+ν (2α(µ) √ y) dy N −1 i,j=0
.
I i (2δ(µ) √ y) K j+ν (2α(µ) √ y) dy = 1 2 α(µ) j+ν δ(µ) i α(µ) 2 − δ(µ) 2 −j−ν−i−1 Γ(i + j + ν + 1).
(4.17)
Taking into account the formula known from the normalisation of the Laguerre ensemble,
det [Γ(i + j + ν + 1)] N −1 i,j=0 = N j=1 Γ(j)Γ(j + ν),
we obtain the normalising constant (3.4) in the formula for P (y 1 , . . . , y N ). The statement of Theorem 3.1 follows immediately.
Proof of Theorem 3.2
To derive an explicit formula for the correlation kernel of the biorthogonal ensemble under considerations we need the following Proposition.
(5.1) K N (x, y) = N −1 k,l=0 c k,l ψ k (x)ϕ l (y),
where the matrix C = (c k,l ) N −1 k,l=0 is defined by
(5.2) C = G −1 , G = (g k,l ) N −1 k,l=0 , g k,l = ∞ 0 ψ l (x)ϕ k (x)dx.
Proof. See Borodin [13], Section 2.
The matrix entries of G can be computed explicitly. Using equation (4.17) the result is
(5.3) g k,l = 1 2 α(µ) k+ν δ(µ) l α(µ) 2 − δ(µ) 2 −k−ν−l−1 (k + l + ν)! .
This yields
(5.4) c k,l = 2 (α(µ) 2 − δ(µ) 2 ) k+l+ν+1 α(µ) ν+l δ(µ) k a k,l ,
where (a k,l ) N −1 k,l=0 is the inverse of the Hankel matrix (5.5)
H N −1 = (h k+l ) N −1 k,l=0 , h k = (k + ν)! .
Thus the problem of the computation of the correlation kernel is reduced to that of finding the inverse of the Hankel matrix H N −1 defined by equation (5.5). A general method to find the inverse of a Hankel matrix can be described as follows.
Assume that there exists a probability measure dµ(x) on R such that all moments exist:
h k = x k dµ(x), k = 0, 1, . . . .
Construct the corresponding system {P k } of orthonormal polynomials,
P k (x)P l (x)dµ(x) = δ k,l , k, l ≥ 0.
Consider the Christoffel-Darboux kernel K n (x, y),
K n (x, y) = n k=0 P k (x)P k (y),
rewrite this kernel in the form
K n (x, y) = n i=0 n j=0 q (n) i,j x i y j , and set Q n = q (n) i,j n i,j=0
. Proposition 5.2. We have H n Q n = I n , where H n = (h i+j ) n i,j=0 , and I n is the unit matrix of order n + 1. Proof. Using the reproducing property of the Christoffel-Darboux kernels we obtain
x k K n (x, y)dµ(x) = y k , 0 ≤ k ≤ n.
This can be rewritten as After splitting factors accordingly among the functions P n (x) and Q n (x), including a factor of unity (−1) n+n , formulae (5.1), (5.4), and (5.6) give us the expression for the correlation kernel stated in Theorem 3.2. Here we have also used that (−n) k = 0 for k > n > 0.
Proof of Theorem 3.3
In this Section we derive the formula for the correlation kernel K N (x, y) stated in Theorem 3.3 (equation 3.14). To obtain equation (3.14) from equations (3.11)-(3.13) of Theorem 3.2 we use the following combinatorial fact. Proposition 6.1. Define S(α; k, r, N) by
(6.1) S(α; k, r, N) = N −1 n=0 n! (n − k)!(n − r)! Γ(α + n + 1),
where N = 1, 2, . . . ; k, r are two integers such that 0 ≤ k, r ≤ N − 1, and α > −1. We have
(6.2) S(α; k, r, N) = (−1) r Γ(α + r + 1)r! (N − 1 − k)! r i=0 Γ(N + i + α + 1) Γ(i + α + 1) (−1) i i!(r − i)!(α + k + i + 1) .
Proof. We will prove the equivalence of expressions (6.1) and (6.2) by induction with respect to r. Namely, we will check that the equivalence of expressions (6.1) and (6.2) takes place for r = 0, then we will assume that equation (6.2) is valid for an arbitrary r, and then we will show that this identity remains to be valid when we replace r by r + 1. Γ(α + k + n + 1) Γ(n + 1) .
Using the formula Γ(a + s)
Γ(s) − Γ(a + s + 1) Γ(s + 1) = −a Γ(a + s) Γ(1 + s) ,
it is not hard to see that for a > 0
L n=0 Γ(a + n) Γ(1 + n) = Γ(a + L + 1) aΓ(L + 1) .
Replacing L by N − k − 1, and a by α + k + 1 > 0, we obtain S(α; k, r = 0, N) = Γ(N + α + 1) (α + k + 1)Γ (N − k) .
On the other hand, if r = 0, then the right-hand side of equation (6.2) can be rewritten as
Γ(α + 1) (N − 1 − k)! Γ(N + α + 1) Γ(α + 1)(α + k + 1) = Γ(N + α + 1) (α + k + 1)Γ(N − k) .
So the Proposition is proved for r = 0.
Using formula (6.1) we can obtain a recurrence relation for S(α; k, r, N), namely (6.3) S(α; k, r + 1, N) = S(α + 1; k, r, N) − (α + r + 1)S(α; k, r, N).
Now assume that formula (6.2) holds true for a certain r ∈ N. In order to see that it remains to be valid for r + 1 it is enough to show that the right-hand side of equation (6.2) satisfies equation (6.3). To see this, note that the right-hand side of equation (6.3) (with S(α; k, r, N) given by equation (6.2)) can be explicitly rewritten as
(−1) r Γ(α + r + 2)r! (N − 1 − k)! r i=0 Γ(N + i + α + 2) Γ(i + α + 2) (−1) i i!(r − i)!(α + k + i + 2) − (−1) r Γ(α + r + 2)r! (N − 1 − k)! r i=0 Γ(N + i + α + 1) Γ(i + α + 1) (−1) i i!(r − i)!(α + k + i + 1)
.
(6.4)
Changing the index of summation in the first sum by one, i → j = i + 1, we can rewrite expression (6.4) as
(−1) r+1 Γ(α + r + 2)r! (N − 1 − k)! r+1 j=1 Γ(N + j + α + 1) Γ(j + α + 1) (−1) j j j!(r + 1 − j)!(α + k + j + 1) + (−1) r+1 Γ(α + r + 2)r! (N − 1 − k)! r j=0 Γ(N + j + α + 1) Γ(j + α + 1) (−1) j (r + 1 − j) j!(r + 1 − j)!(α + k + j + 1) .
Clearly, the sum of the two terms just written above can be represented as
(−1) r+1 Γ(α + r + 2)(r + 1)! (N − 1 − k)! r+1 j=0 Γ(N + j + α + 1) Γ(j + α + 1) (−1) j j!(r + 1 − j)!(α + k + j + 1) ,
which is S(α; k, r + 1, N) as given by equation (6.2). Thus we have seen that the right-hand side of equation (6.2) satisfies equation (6.3). The Proposition is proved.
Setting α = M − N and r = l in Proposition 6.1 and multiplying with (−1) k+l we obtain the following Corollary 6.2. The following identity holds true
N −1 p=0 (M − N + p)! p! (−p) k (−p) l = (M − N + l)!l! (N − 1 − k)! l i=0 (i + M)! (M − N + i)! (−1) i+k i!(l − i)!(M − N + k + i + 1)
,
(6.5)
where M ≥ N.
To get equation (3.14) for the correlation kernel K N (x, y) use formula (6.5), and equations (3.11)-(3.13) of Theorem 3.2.
Proof of Proposition 3.4
In this Section we begin to investigate the properties of the functions P n (x) and Q n (x) defined by equations (3.12) and (3.13). In particular, we show that P n (x) and Q n (x) are biorthogonal functions. To see this define two matrices, V = (v k,p ) N −1 k,p=0 and W = (w p,l ) N −1 p,l=0 , by the formulae
(7.1) v k,p = (−1) p (ν + p)!p!(−p) k (ν + k)!k! (α(µ) 2 − δ(µ) 2 ) k+ 1 2 δ(µ) k , and (7.2) w p,l = (−1) p 2(−p) l (p!) 2 (ν + l)!l! (α(µ) 2 − δ(µ) 2 ) l+ν+ 1 2 α(µ) l+ν .
In addition, introduce two column vectors, Ψ(x), and Φ(y),
Ψ(x) = ψ 0 (x) ψ 1 (x) . . . ψ N −1 (x) , Φ(y) = ϕ 0 (y) ϕ 1 (y) . . . ϕ N −1 (y) ,
where ψ j (x) and ϕ j (y) are defined by equations (4.15) and (4.16). Set
(7.3) P(x) = V T Ψ(x), Q(y) = W Φ(y).
By elementary Linear Algebra calculations, the correlation kernel K N (x, y) equation (3.11) can be written as
(7.4) K N (x, y) = P T (x)Q(y).
Observe that the matrix G (defined by equation (5.2)) can be written as
G = ∞ 0 Φ(x)Ψ T (x)dx.
The notation above means that we integrate each matrix element of the N × N matrix Φ(x)Ψ T (x) from 0 to ∞. The matrix C = (c k,l ) N −1 k,l=0 (whose matrix elements are given explicitly by equations (5.4) and (5.6)) is the inverse of the matrix G. Therefore we can write
C −1 = ∞ 0 Φ(x)Ψ T (x)dx.
The key observation is that
C = V W,
as it follows from equations (5.4), (5.6), (7.1), and (7.2). Since C is invertible, both matrices V , W are invertible, and we have
(V W ) −1 = ∞ 0 Φ(x)Ψ T (x)dx, or I = ∞ 0 Φ(x)Ψ T (x)dx V W.
Multiplying both sides of the equation just written above by W from the left, and using the definitions of the vectors P(x), Q(y) (see equation (7.3) solved for Φ(x) and Ψ T (x)) we obtain
W = ∞ 0 Q(x)P T (x)dx W.
Since the matrix W is invertible, we conclude that
∞ 0 Q(x)P T (x)dx = I.
In other words, P n (x) and Q n (x) are biorthogonal functions. Proposition 3.4 is proved.
Proof of Proposition 3.5 and Theorem 3.6
In this Section we derive the recurrence relations for the functions P n (x) and Q n (y) stated in Proposition 3.5. Using these recurrence relations we derive the Christoffel-Darboux type formula for the correlation kernel K N (x, y), and prove Theorem 3.6. First, let us obtain equations (3.16)-(3.21). Setting
(8.1) I k (x) = k!x k 2 δ(µ) k I k (2δ(µ) √ x), k = 0, 1, . . . ,
the following recurrence relation holds true:
(8.2) x I k (x) = I k+1 (x) + δ(µ) 2 (k + 1)(k + 2) I k+2 (x), k = 0, 1, . . .
To see this, use the recurrence relation for the Bessel functions, namely zI ν (z) = 2(ν + 1)I ν+1 (z) + zI ν+2 (z).
Introduce the vectors
I(x) = I 0 (x) I 1 (x) . . . , P(x) = P 0 (x) P 1 (x) . . . .
The recurrence relations for the functions I k (equation(8.2)) can be rewritten as
(8.3) x I(x) = E I(x),
where the matrix E is defined by the formula
(8.4) E k,m = δ k+1,m + δ(µ) 2 (k + 1)(k + 2) δ k+2,m ; k, m = 0, 1, . . . .
Moreover, set
(8.5) V p,k = (−1) p (ν + p)!p!(−p) k (ν + k)!(k!) 2 α(µ) 2 − δ(µ) 2 k+ 1 2 , p, k = 0, 1, . . . .
Then we have
(8.6) P(x) = V I(x),
where V = (V p,k ) 0≤p,k≤∞ . From equations (8.3) and (8.6) we immediately obtain
(8.7) xP(x) = VE I(x).
Introduce the matrix R P by the formula
(8.8) xP(x) = R P P(x).
The matrix R P is defining the recurrence relation for the functions P 0 (x), P 1 (x), . . . In the explicit calculations of the matrix R P below (and in the derivation of the recurrence relations) we will exploit the following Lemma.
Lemma 8.1. For any non-negative integers i,j the following formulae hold true:
(8.10) ∞ m=0 (−1) m+i (i − m)!(m − j)! = δ i,j , ∞ m=0 (−1) m+i (ν + m + 1)(m + 1) 2 (i − m)!(m + 1 − j)! = (ν + i + 1)(i + 1) 2 δ i+1,j + i 2 + 2i(ν + i) + ν + 3i + 1 δ i,j + (ν + 3i)δ i−1,j + δ i−2,j ,(8.
11)
∞ m=0 (−1) m+i (ν + m + 1) 2 (m + 1) (i − m)!(m + 1 − j)! = (i + 1)(i + ν + 1) 2 δ i+1,j + (ν + i) 2 + 2(i + 1)(i + ν) + i + 1 δ i,j + (2ν + 3i)δ i−1,j + δ i−2,j ,
(8.12) ∞ m=0 (−1) m+i (m + 1)(m + 2)(ν + m + 1)(m + 2) (i − m)!(m + 2 − j)! = (ν + i + 2)(ν + i + 1)(i + 2)(i + 1)δ i+2,j + 2(i + 1)(ν + 2i + 2)(ν + i + 1)δ i+1,j + ((ν + i)(ν + 5i + 3) + i(i + 3) + 2) δ i,j + 2(ν + 2i)δ i−1,j + δ i−2,j . (8.13)
Proof. Equation (8.10) is a reformulation of the fact that (x − 1) k−m x=1 = δ k,m . Equations (8.11)-(8.13) can be derived using straightforward calculations. For example equation (8.11) is obtained by differentiating ∂ x (x ν+1 ∂ x (x∂ x (x j (x − 1) i−j+1 ))) at x = 1, after normalising by (i − j + 1)!. Here j = 0, 1 have to be treated separately. The remaining equations follow in a similar fashion.
Proposition 8.2. The matrix V is invertible, and its inverse is given by
(8.14) V −1 k,l = (k!) 2 (ν + k)! (k − l)!(l!) 2 (ν + l)! 1 (α(µ) 2 − δ(µ) 2 ) k+ 1 2 k, l = 0, 1, . . . .
Proof. This can be checked by direct calculations using formula (8.10).
Equation (8.8) says that it is enough to compute the matrix R P explicitly to obtain the recurrence for P 0 (x), P 1 (x), . . . (equations (3.16)- (3.21)). This can be done exploiting formula (8.9), the formula for the matrix elements of V (equation (8.5)), and that for the matrix elements of E (equation (8.4)). In the computations we use formulae (8.11), (8.13) to express the sums involved in terms of the Kronecker symbols. Now we turn to derivation of the recurrence relation for Q 0 (y), Q 1 (y), . . . (equations (3.22)-(3.27)). Set
K l (y) = (l + ν)!y l+ν 2 α(µ) l+ν K l+ν (2α(µ) √ y).
We have
(8.15) y K l (y) = − K l+1 (y) + α(µ) 2 (l + ν + 2)(l + ν + 1) K l+2 (y).
To see that equation (8.15) holds true use the recurrence relations zK ν (z) = −2(ν + 1)K ν+1 (z) + zK ν+2 (z).
Introduce the vectors
K(y) = K 0 (y) K 1 (y) . . . , Q(y) = Q 0 (y) Q 1 (y) . . . .
Then the recurrence relation for the functions K k (y) (equation (8.15)) can be rewritten as
(8.16) y K(y) = E K(y),
where the matrix E is defined by the formula
(8.17) E k,m = −δ k+1,m + α(µ) 2 (k + ν + 1)(k + ν + 2)
δ k+2,m , k, m = 0, 1, . . . .
Moreover, set
W p,k = 2(−1) p (−p) k (p!) 2 k!((ν + k)!) 2 α(µ) 2 − δ(µ) 2 ν+k+ 1 2 , p, k = 0, 1, . . . .
We have
Q(y) = W K(y).
By the same argument as in the derivation of the recurrence relation for the functions P p (x) we find that the recurrence matrix R Q for the functions Q p (y) is given by
R Q = W EW −1 . Proposition 8.3. We have W −1 k,l = ((ν + k)!) 2 l!k! 2(k − l)! 1 (α(µ) 2 − δ(µ) 2 ) k+ν+ 1 2 .
Proof. The formula for (W −1 ) k,l can be obtained by direct calculations using formula (8.10).
The subsequent computation leading to the recurrence relation for the functions Q 0 (y), Q 1 (y), . . . is very similar to that leading to the recurrence relation for the functions P 0 (x), P 1 (x), . . ., where in the evaluation of the matrix R Q we use equations (8.12) and (8.13). Proposition 3.5 is proved. Now let us prove Theorem 3.6. Setting P −n (x) = 0 = Q −n (x) for n = 1, 2 we can apply the recurrence from Proposition 3.5 as follows: (x − y)P n (x)Q n (y) = a −2,n P n−2 (x)Q n (y) − a −2,n+2 P n (x)Q n+2 (y) + a −1,n P n−1 (x)Q n (y) − a −1,n+1 P n (x)Q n+1 (y) + a 1,n P n+1 (x)Q n (y) − a 1,n−1 P n (x)Q n−1 (y) + a 2,n P n+2 (x)Q n (y) − a 2,n−2 P n (x)Q n−2 (y) , for n = 0, 1, . . .
Here we have already used the relation between the coefficients a k,n and b k,n in equation (3.28). Summing up the right-hand side and the left-hand side of the equation above from n = 0 to (8.18) after shifting several summation indices. Cancelling all terms and dividing by (x − y) = 0 we obtain formula (3.29) for the correlation kernel K N (x, y).
N − 1 we obtain (x − y)K N (x, y) = N −1 n=0 (a 2,n P n+2 (x) + a 1,n P n+1 (x))Q n (y) + N −2 n=0 a −1,n+1 P n (x)Q n+1 (y) + N −3 n=0 a −2,n+2 P n (x)Q n+2 (y) − N −3 n=0 a 2,n P n+2 (x)Q n (y) − N −2 n=0 a 1,n P n+1 (x)Q n (y) − N −1 n=0 P n (x)(a −1,n+1 Q n+1 (y) + a −2,n+2 Q n+2 (y)),
Proof of Proposition 3.7 and Theorem 3.8
Let us first obtain the contour integral representation for the functions P 0 (x), P 1 (x), . . . as given in equation (3.30). Recall that P n (x) is given explicitly by equation (3.12). We express the Bessel function in equation (3.12) as an infinite sum,
I k 2δ(µ)x 1 2 = ∞ l=0 1 l!(k + l)! δ(µ)x 1 2 k+2l .
Next we rewrite the formula for P n (x) as
P n (x) = (−1) n (ν + n)!n! ν! α(µ) 2 − δ(µ) 2 1 2 ∞ l=0 x l δ(µ) 2l l! n k=0 (−n) k (α(µ) 2 − δ(µ) 2 ) k (ν + 1) k k!(k + l)! x k .
The expression in the bracket on the right-hand side of the equation for P n (x) above can be written as a generalised hypergeometric series, so we have
P n (x) =(−1) n (ν + n)!n! ν! α(µ) 2 − δ(µ) 2 1 2 × ∞ l=0 x l δ(µ) 2l (l!) 2 1 F 2 −n ν + 1, l + 1 α(µ) 2 − δ(µ) 2 x . (9.1)
The following contour integral representation can be obtained from residue calculus
1 F M −n 1 + ν 1 , . . . , 1 + ν M x = (−1) n M j=1 Γ(ν j + 1)n! 2πi Σ Γ(t + 1)Γ(t − n) M j=1 Γ(t + ν j + 1) x t dt,
where Σ is a closed contour that encircles 0, 1, . . . , n once in the positive direction. In particular,
1 F 2 −n ν + 1, l + 1 α(µ) 2 − δ(µ) 2 x = (−1) n Γ(ν + 1)Γ(l + 1)n! 2πi Σ Γ(t − n) ((α(µ) 2 − δ(µ) 2 ) x) t Γ(t + 1)Γ(t + ν + 1)Γ(t + l + 1) dt.
Inserting the above formula into equation (3.12), we obtain the desired expression for P n (x), equation (3.30), after writing the remaining sum as another hypergeometric function. Now we derive the contour integral representation for the functions Q 0 (y), Q 1 (y), . . . (equation (3.31)). We start from the formula (3.13), and use the relation (3.7). This enables us to rewrite equation (3.13) as
Q n (y) = (−1) n (n!) 2 ν! α(µ) 2 − δ(µ) 2 1 2 × n l=0 α(µ) 2 − δ(µ) 2 α(µ) 2 l+ν (−n) l (ν + 1) l l! G 2,0 0,2 − 0, l + ν α(µ) 2 y . (9.2)
Following the definition (3.6) a contour integral representation for the Meijer G-function in the formula above holds:
(9.3) G 2,0 0,2 − 0, l + ν α(µ) 2 y = 1 2πi c+i∞ c−i∞ Γ(s + l + ν)Γ(s) α(µ) 2 y −s ds,
with c > 0. Formulae (9.2) and (9.3) result in the following expression for the function Q n (y)
Q n (y) = (−1) n (n!) 2 ν! 1 − δ(µ) 2 α(µ) 2 ν α(µ) 2 − δ(µ) 2 1 2 × 1 2πi c+i∞ c−i∞ Γ(s)Γ(s + ν) 2 F 1 −n, ν + s 1 + ν 1 − δ(µ) 2 α(µ) 2 (α(µ) 2 y) −s ds. (9.4)
The (Gauss) hypergeometric function inside the integral above can be written as follows using [27] 9.131.2
2 F 1 −n, ν + s 1 + ν 1 − δ(µ) 2 α(µ) 2 = Γ(1 + ν)Γ(1 − s + n) Γ(1 + ν + n)Γ(1 − s) 2 F 1 −n, n + s s − n δ(µ) 2 α(µ) 2 . (9.5)
Applying this formula, and the fact that
(9.6) Γ(1 − s + n) Γ(1 − s) = (−1) n Γ(s) Γ(s − n) ,
which can be shown using [27] 8.334.3, we obtain equation (3.31). Proposition 3.7 is proved.
To obtain the contour integral representation for the correlation kernel K N (x, y) given in Theorem 3.8 we need the following Lemma.
Lemma 9.1. We have N −1 n=0 Γ(t − n) Γ(s − n) (−n) k (s − n) k =k! Γ(t − N + 1) Γ(s − t + k) k m=0 (−1) m N m Γ(s − t + m − 1) Γ(s + m − N) − Γ(t + 1)Γ(s − t − 1)k! Γ(s)Γ(s − t + k) ,(9.7)
where k = 0, 1, . . . , N − 1.
Proof. Denote by S N (t, s; k) the sum on the left hand side of equation (9.7),
S N (t, s; k) = N −1 n=0 Γ(t − n) Γ(s − n) (−n) k (s − n) k . Also, set S N (t, s; k) = N −1 n=0 Γ(t − n) Γ(s − n) n! (n − k)! .
These sums are related to each other according to the formula
(9.8) S N (t, s; k) = (−1) k S N (t, s + k; k).
Thus it is enough to find a closed formula for S N (t, s; k). Using the elementary property xΓ(x) = Γ(x + 1) is is easy to check that the following identity holds true
Γ(t − n − 1) Γ(s − n − 1) (n + 1)! (n + 1 − k)! − Γ(t − n) Γ(s − n) n! (n − k)! = k Γ(t − n − 1) Γ(s − n − 1) n! (n − k + 1)! − (t − s) Γ(t − n − 1) Γ(s − n) n! (n − k)! . (9.9)
This identity implies the following recurrence relation
(s − t − 1) S N (t, s; k) + k S N (t, s − 1; k − 1) = Γ(t − N + 1) Γ(s − N) N! (N − k)! ,
starting from k = 1, . . . , N − 1. The recurrence relation above can be solved, and a formula for S N (t, s; k) can be obtained. Namely, beginning with k = 0,
S N (t, s; k = 0) = N −1 n=0 Γ(t − n) Γ(s − n) = Γ(t − N + 1) (s − t − 1)Γ(s − N) − Γ(t + 1) (s − t − 1)Γ(s) ,
which can be easily seen by induction in N we find
S N (t, s; k) = k l=0 Γ(t − N + 1)(−k) l (s − t − 1)(s − t − 2) . . . (s − t − l − 1)Γ(s − N − l) N! (N − k + l)! + (−1) k−1 Γ(t + 1)k! (s − t − 1)(s − t − 2) . . . (s − t − 1 − k)Γ(s − k)
.
(9.10) Formulae (9.8), (9.10) imply
S N (t, s; k) = k l=0 (−1) k−l Γ(t − N + 1)Γ(s − t + k − l − 1)k! Γ(s − t + k)(k − l)!Γ(s + k − l − N) N! (N − k + l)! − Γ(t + 1)k!Γ(s − t − 1) Γ(s − t + k)Γ(s) .
Then, after setting m = k − l we get formula (9.7).
Recall that the correlation kernel K N (x, y) can be represented as the sum of the biorthogonal functions P n (x) and Q n (y), see equation (3.11). We insert the integral representations for the functions P n (x) and Q n (y) (see Proposition 3.7) into equation (3.11). We write the hypergeometric function 2 F 1 as a finite sum up to N − 1. Then we interchange the finite sum and the double contour integral, use the combinatorial identity (9.7), and observe that the second term on the right-hand side of equation (9.7) does not contribute to the double contour integral. The result of these calculations is the formula for the correlation kernel in the statement of Theorem 3.8.
Proof of Theorem 3.9
We use the contour integral representations for the functions P n (x) and Q n (y) obtained in Proposition 3.7 together with the Christoffel-Darboux type formula for the correlation kernel K N (x, y), see Theorem 3.6. Namely, we insert the contour integrals representing P n (x) and Q n (y) into formula (3.29). In the numerator of the right-hand side of equation (3.29) we obtain a double contour integral. Let us write this contour integral representation of the correlation kernel explicitly. We have
K N (x, y) = α(µ) 2 − δ(µ) 2 (2πi) 2 (x − y) 1 − δ(µ) 2 α(µ) 2 ν × c+i∞ c−i∞ ds Σ dt Γ 2 (s)Γ(s + ν)(α(µ) 2 − δ(µ) 2 ) t x t 0 F 1 − t + 1 δ(µ) 2 x (Γ(t + 1)) 2 Γ(t + ν + 1) (α(µ) 2 y) −s × − α(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 Γ(t − N + 2) Γ(s − N ) 2 F 1 −N, ν + s s − N δ(µ) 2 α(µ) 2 − α(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 Γ(t − N + 1) Γ(s − N − 1) 2 F 1 −N − 1, ν + s s − N − 1 δ(µ) 2 α(µ) 2 − 3N + ν α(µ) 2 − δ(µ) 2 + 2(2N + ν)δ(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 Γ(t − N + 1) Γ(s − N ) 2 F 1 −N, ν + s s − N δ(µ) 2 α(µ) 2 + N 2 (N + ν) α(µ) 2 − δ(µ) 2 + 2N (N + ν)(2N + ν)δ(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 Γ(t − N ) Γ(s − N + 1) 2 F 1 −N + 1, ν + s s − N + 1 δ(µ) 2 α(µ) 2 + N (N − 1)(N + ν)(N + ν − 1)δ(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 Γ(t − N ) Γ(s − N + 2) 2 F 1 −N + 2, ν + s s − N + 2 δ(µ) 2 α(µ) 2 + N (N + 1)(N + ν)(N + ν + 1)δ(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 Γ(t − N − 1) Γ(s − N + 1) 2 F 1 −N + 1, ν + s s − N + 1 δ(µ) 2 α(µ) 2 . (10.1)
We note that as N → ∞, we have the following ratio asymptotic of Gamma functions
Γ(t − N) Γ(s − N) = sin(πs) sin(πt) Γ(1 − s + N) Γ(1 − t + N) ≃ sin(πs) sin(πt) N t−s 1 + O(N −1 ) ,
upon using equation (9.6), Γ(1 − x)Γ(x) = π/ sin(πx), and the standard asymptotic expansion of the Gamma-function. Moreover, we have [45]
2 F 1 −N, ν + s s − N δ(µ) 2 α(µ) 2 ≃ 1 1 − δ(µ) 2 α(µ) 2 ν+s 1 + O(N −1 ) ,
as N → ∞. Using the asymptotic formulae just written above, we find
1 N (α(µ) 2 − δ(µ) 2 ) K N x N (α(µ) 2 − δ(µ) 2 ) , y N (α(µ) 2 − δ(µ) 2 ) = 1 (2πi) 2 (x − y) c+i∞ c−i∞ ds Σ dt Γ 2 (s)Γ(s + ν) (Γ(t + 1)) 2 Γ(t + ν + 1) sin πs sin πt x t y s × A(s, t; N) + δ(µ) 2 α(µ) 2 − δ(µ) 2 B(s, t; N) (1 + O(N −1 )) , (10.2)
where we used that the hypergeometric function 0 F 1 of rescaled argument tends to unity. The functions A(s, t; N) and B(s, t; N) are given by
A(s, t; N) = N 2 (N + ν) s − N − (t − N)(s + t + N + ν), B(s, t; N) = N(N + 1)(N + ν)(N + ν + 1) (t − N − 1)(s − N) + N(N − 1)(N + ν)(N + ν − 1) (s − N + 1)(s − N) + 2N(N + ν)(2N + ν) s − N − (t − N)(t + s + 2N + 2ν).
Note that the additional factor in front of the kernel compensates the rescaling of the arguments of the factor 1/(x − y). Computations show that Now we take the limit N → ∞ from both sides of equation (10.2), and interchange the limit and integrals in the right-hand side. The fact that we are allowed to take the limit inside the integrals can be justified as in the proof of Theorem 5.3 in Kuijlaars and Zhang [36] using the dominated convergence theorem and the asymptotic properties of Gamma functions. Thus we obtain the limiting relation
lim N →∞ 1 N (α(µ) 2 − δ(µ) 2 ) K N x N (α(µ) 2 − δ(µ) 2 ) , y N (α(µ) 2 − δ(µ) 2 ) = −1 (2πi) 2 (x − y) c+i∞ c−i∞ ds Σ dt
Γ 2 (s)Γ(s + ν) (Γ(t + 1)) 2 Γ(t + ν + 1) sin πs sin πt x t y s (s(s + ν) + t(t + ν) + st) . Since Γ(s) sin πs
Γ(t + 1) sin πt = − Γ(−t) Γ(1 − s) ,
we can rewrite the equation above as
lim N →∞ 1 N (α(µ) 2 − δ(µ) 2 ) K N x N (α(µ) 2 − δ(µ) 2 ) , y N (α(µ) 2 − δ(µ) 2 ) = 1 (2πi) 2 (x − y) c+i∞ c−i∞ ds Σ dt Γ(−t)Γ(s)Γ(s + ν) Γ(t + 1)Γ(t + ν + 1)Γ(1 − s)
x t y s (s(s + ν) + t(t + ν) + st) .
(10.4)
It follows from the definition (3.6) that x − y +
1 2πi Σ dt Γ(−t) Γ(t + 1)Γ(t + ν + 1) x t = −G 1,0 0,3 − 0, −ν,x d dx f (x) −νg(y) + y d dy g(y) x − y − (x d dx ) 2 f (x)g(y) x − y , (10.5) where (10.6) f (x) = G 1,0 0,3 − 0, −ν, 0 x , g(y) = G 2,0 0,3 − ν, 0, 0 y .
Expression (10.5) (with the functions f (x), g(y) defined by equation (10.6)) gives the limiting kernel for the product of two matrices with independent complex Gaussian entries, see Proposition 5.4 in Kuijlaars and Zhang [36]. As it is shown in Kuijlaars and Zhang [36] (see the proof of Theorem 5.3) such limiting kernel can be also written as Theorem 3.9 is proved.
Proof of Theorem 3.10
We use the following result for biorthogonal ensembles obtained by Breuer and Duits [14]. Assume we are given a biorthogonal ensemble on R ≥0 defined by the joint probability density function P N (x 1 , . . . , x N ). Assume further that the correlation kernel of this ensemble, K N (x, y), is given by
K N (x, y) = N −1 p=0 ψ (N ) p (x)φ (N ) p (y),
where the functions ψ
(N ) p , φ (N ) k are orthonormal, ∞ 0 ψ (N ) p (x)φ (N ) k (x)dx = δ p,k .
Suppose we know that the functions ψ f (s(w))w k dw w .
Proof. This statement is a corollary of a more general result for biorthogonal ensembles obtained by Breuer and Duits [14], see Theorem 2.1 and Corollary 2.2 therein.
Note that since f is a polynomial with real coefficients,f k is real. Now, let us consider the N-dependent probability distribution P N,M (X 1 , X 2 ) on the Cartesian product of Mat(C, N × M) and Mat(C, M × N) defined by equation (3.34). Let y 1 , . . ., y N be the squared singular values of the random matrix X 1 X 2 , with its linear statistics given by
Y (N ) f = N i=1 f (y i ).
By Theorem 3.1 the squared singular values y 1 , . . ., y N of the random matrix X 1 X 2 form a biorthogonal ensemble on R ≥0 . The correlation kernel of this ensemble, K N (x, y), can be written as
K N (x, y) = N −1 n=0 P ′ n (x)Q ′ n (y).
The new functions, P ′ n (x) and Q ′ n (y), are defined in terms of P n (x) and Q n (y) as P ′ n (x) = 1 n!(n + ν)! P n (x), Q ′ n (y) = n!(n + ν)!Q n (y),
where P n (x) and Q n (y) are defined as previously by equations (3.12), (3.13), with α(µ) replaced by Nα(µ), and δ(µ) replaced by Nδ(µ). Clearly, the functions P ′ p (x) and Q ′ m (x) are orthonormal,
∞ 0 P ′ p (x)Q ′ m (x)dx = δ p,m .
Moreover, the 5 term recurrence relation (m = 2 here) for the functions P ′ n (x) can be written as (11.1) xP ′ n (x) = a ′ 2,n P ′ n+2 (x) + a ′ 1,n P ′ n+1 (x) + a ′ 0,n P ′ n (x) + a ′ −1,n P ′ n−1 (x) + a ′ −2,n P ′ n−2 (x).
The coefficients a ′ 2,n , a ′ 1,n , a ′ 0,n , a ′ −1,n , and a ′ −2,n easily follow from Proposition 3.5 and are given explicitly by a ′ 2,n = δ(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 (n + ν + 1)(n + ν + 2) N 2 , (11.2) a ′ 1,n = 1 α(µ) 2 − δ(µ) 2 (n + 1)(n + ν + 1) N 2 + δ(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 2(2n + ν + 2)(n + ν + 1) N 2 , (11.3) a ′ 0,n = 1 α(µ) 2 − δ(µ) 2 3n 2 + 2νn + 3n + ν + 1 N 2 + δ(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 6n 2 + 6nν + ν 2 + 6n + 3ν + 2 N 2 , (11.4) Thus, Proposition 11.1 can be applied, and the relevant Laurent polynomial can be computed explicitly. The result follows.
a ′ −1,n = 1 α(µ) 2 − δ(µ) 2 n(3n + ν) N 2 + δ(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 2n(ν + 2n) N 2 , (11.5) a ′ −2,n = 1 α(µ) 2 − δ(µ) 2 n(n − 1) N 2 + δ(µ) 2 (α(µ) 2 − δ(µ) 2 ) 2 n(n − 1) N 2 .
Appendix A. Limits of the joint probability density function
In this appendix we derive the two limits µ → 1 and µ → 0 of the joint probability density function P (y 1 , . . . , y N ) equation (3.3), as given in equations (3.5) and (3.8), respectively.
For the first limit µ → 1 leading to two independent Gaussian complex matrices we have δ(µ) → 0, α(µ) → 1. From the series representation of the function I κ (z), equation .
The limit of the remainder of the pre-factor Z N and of the modified Bessel function of the second kind K κ (2α(µ) √ y) is trivial, and after expressing the latter in terms of the Meijer G-function from equation (3.7) the limiting joint probability density function lim µ→1 P (y 1 , . . . , y N ) in equation (3.5) follows.
In the second limit µ → 0 both δ(µ) and α(µ) diverge. Hence in equation (3.3) we have to replace the modified Bessel functions inside the determinants by their large argument asymptotic expressions. Namely, we use the formulae .
(A.1) I κ (z) ≃ e z √ 2πz , K κ (z) ≃ π 2z e −z ,
Putting all these results together we obtain equation (3.8).
Theorem 3. 1 .
1Let X 1 ∈ Mat (C, N × M) and X 2 ∈ Mat (C, M × N) be two µ-dependent Gaussian complex matrices. Assume that M ≥ N, and set ν = M − N.
P
(y 1 , . . . , y N ) = 2 N (M −1)
are Meijer G-functions with a suitable choice of parameters.
Lemma 4 . 1 .
41Let X ∈ Mat(C, M × N) and G ∈ Mat(C, N × M) be two µ-dependent Gaussian complex matrices. Assume that M ≥ N. Then the squared singular values of the matrix GX are distributed in the same way as the squared singular values of the matrix
have used the invariance of the corresponding Lebesgue measure dG under unitary transformations. Now, set (4.3)Ĝ = G 0ĜN,M −N . This is a block decomposition of the rectangular matrixĜ of size N × M (M > N) such that G 0 is the square matrix of size N × N whose entries are those of the first N columns ofĜ, andĜ N,M −N is the remaining rectangular matrix of size N × (M − N). Inserting (4.3) into equation (4.2), we obtain
N
. The integral over these variables is
=
y i , we obtain equation (3.3) up to a normalisation constant const = 1/Z N .
Proposition 5 . 1 .
51Let ψ j (x), ϕ j (x) be defined by equations (4.15) and (4.16), where j = 0, 1, . . . , N − 1. The correlation kernel K N (x, y) of the biorthogonal ensemble defined by equation (3.10) can be written as
j h k+i y j = y k , 0 ≤ k ≤ n. j = δ k,j , and the statement of the Proposition follows.
Proof. Use Proposition 5.2, and observe that the relevant family of orthogonal polynomials is that of the classical Laguerre polynomials {L(ν+k) n (x)}. Then use the explicit formulae for {L (ν+k) n (x)} (see, for example, [27] 8.970.1).
P = VEV −1 .
A
(s, t; N) = −s(s + ν) − t(t + ν) − st,
Now we can rewrite the right-hand side of equation (10.4) as f (x) νy d dy g(y)
.
= 0, 1, . . ., and m is independent of N. Here we define that ψ In other words, there exists a banded matrix J (N ) ,N = α −m .In this situation we associate with J (N ) a Laurent polynomial s(w) Let f be a polynomial with real coefficients, and define the linear statistics of the biorthogonal ensemble by the formulaX (N ) f = N i=1 f (x i ),where x 1 , . . ., x N are the points of the biorthogonal ensemble under considerations.
Γ
Moreover, as µ → 0, the normalising constant Z N in equation ((j)Γ(j + ν)
Two kernels K(x, y) andK ′ (x, y) are called equivalent if det [K(x i , x j ]) m i,j=1 = det [K ′ (x i , x j )] mi,j=1 , for any m = 1, 2, . . ., for example K ′ (x, y) = (f (x)/f (y))K(x, y). Thus two equivalent kernels define the same correlation functions.
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| []
|
[
"Global aspects of the renormalization group and the Hierarchy problem",
"Global aspects of the renormalization group and the Hierarchy problem"
]
| [
"Andrei T Patrascu \nDepartment of Physics and Astronomy\nUniversity College London\nWC1E 6BTLondonUK\n"
]
| [
"Department of Physics and Astronomy\nUniversity College London\nWC1E 6BTLondonUK"
]
| []
| The discovery of the Higgs boson by the ATLAS and CMS collaborations allowed us to precisely determine its mass being 125.09 ± 0.24GeV. This value is intriguing as it lies at the frontier between the regions of stability and meta-stability of the standard model vacuum. It is known that the hierarchy problem can be interpreted in terms of the near criticality between the two phases. The coefficient of the Higgs bilinear in the scalar potential, m 2 , is pushed by quantum corrections away from zero, towards the extremes of the interval [−M 2 P l , M 2 P l ] where M P l is the Planck mass. In this article, I show that demanding topological invariance for the renormalisation group allows us to extend the beta functions such that the particular value of the Higgs mass parameter observed in our universe regains naturalness. In holographic terms, invariance to changes of topology in the bulk is dual to a natural large hierarchy in the boundary quantum field theory. The demand of invariance to topology changes in the bulk appears to be strongly tied to the invariance of string theory to T-duality in the presence of H-fluxes. | 10.1016/j.physletb.2017.09.010 | [
"https://arxiv.org/pdf/1709.10359v1.pdf"
]
| 119,503,606 | 1709.10359 | 29bc9c1cf261d6e38d9b56e27b2812b824ca40b5 |
Global aspects of the renormalization group and the Hierarchy problem
28 Sep 2017
Andrei T Patrascu
Department of Physics and Astronomy
University College London
WC1E 6BTLondonUK
Global aspects of the renormalization group and the Hierarchy problem
28 Sep 2017
The discovery of the Higgs boson by the ATLAS and CMS collaborations allowed us to precisely determine its mass being 125.09 ± 0.24GeV. This value is intriguing as it lies at the frontier between the regions of stability and meta-stability of the standard model vacuum. It is known that the hierarchy problem can be interpreted in terms of the near criticality between the two phases. The coefficient of the Higgs bilinear in the scalar potential, m 2 , is pushed by quantum corrections away from zero, towards the extremes of the interval [−M 2 P l , M 2 P l ] where M P l is the Planck mass. In this article, I show that demanding topological invariance for the renormalisation group allows us to extend the beta functions such that the particular value of the Higgs mass parameter observed in our universe regains naturalness. In holographic terms, invariance to changes of topology in the bulk is dual to a natural large hierarchy in the boundary quantum field theory. The demand of invariance to topology changes in the bulk appears to be strongly tied to the invariance of string theory to T-duality in the presence of H-fluxes.
INTRODUCTION
The discovery of the Higgs boson and the measurement of its mass as lying at the very edge between EW stability and metastability regions, not accompanied by the discovery of supersymmetric particles, already casts shadows of doubt upon a potential supersymmetric solution to the hierarchy problem. This state of affairs asks for the search of new principles of nature that may allow us to understand the origins of the standard model parameters without having to rely on symmetries and without having to give up calculability as is the case with multiverse interpretations. As recent experimental results suggest, the standard model vacuum appears to be metastable. Understanding the parameters of the standard model that lead to such a conclusion would therefore reveal information about the future fate of our universe. In this article, I show that naturalness can be restored if one imposes invariance of the physical theory with respect to topology changes associated to the renormalisation group. Let BG be the classifying space of the (in this case renormalisation) group G constructed as the quotient EG/G, where EG is a contractible space on which G acts freely.
By means of the isomorphism in cohomology
H n (BG, Z) = H n (G, Z)(1)
it is clear that the topological properties (as described by cohomology) of the renormalisation group G, are equivalent to the topological properties of the space on which G acts. According to the duality between renormalisation group equations of the low energy theory and the evolution equations along the new directions in a higher dimensional effective theory (e.g. 5-d Supergravity) [1][2][3] the cohomology of the renormalisation group is to be associated to the cohomology of the subspace induced by the evolution equations along the additional directions in a higher energy effective theory. Therefore, imposing invariance with respect to the change in the renormalisation group topology is equivalent to imposing invariance with respect to topology changes of the high energy effective theory. This prescription is the low energy "remnant" of the string theoretical T-duality. Therefore, understanding how the principle of topological invariance, observed initially in string theory, affects low energy effective field theories like the standard model or 5-d supergravity could herald new ways of connecting low energy physics to the physics determined by extended fundamental objects expected to arise at or near the Planck scale. While the observation that the string theoretical T-duality may be related to the hierarchy problem is not new [1], the fact that the topological invariance implied by T-duality is the key to restoring naturalness, is.
The beta functions (generators of the renormalisation group) allow us to probe high energies starting from our low energy effective field theory. However, at some point they must enter the regime where string geometry effects become relevant. In order to connect lower energy scales to string geometry effects involving topology changing dualities, such beta functions must not discern topologies connected by means of dualities. This requirement alters the topological structure of the renormalisation group such that it compensates for the topology changes that may occur at the string level. The construction of topology invariant theories with a focus on group quantisation was performed in [4]. The methods introduced there are particularly relevant because they deal with group topology.
The creation of explicitly T-duality invariant string theories resolved the questions regarding the high energy domain. Representative articles are [27], [33]. Transferring such invariance to the renormalisation group is the focus of this work. Due to the duality between RG flux and extra dimensional equations of motion, the goal of this article is to explore the field theoretical, boundary effects of the invariance of the theory to topology changes in the bulk. Of course, after the methodology is established, new insights upon the cosmological con-stant, the vacuum stability, or the nature of dark matter may be obtained. I will only discuss here applications to the hierarchy problem, leaving the other subjects to a future, cosmology oriented article. Of course the standard model hierarchy problem and the cosmological constant problem are most likely related.
In quantum field theory, energy scales are regarded as linearly ordered, from the lowest to the highest. Moving from high energy scales to low energy scales leads to decoupling of the heavy states and to effective theories. The domain where such quantum field theories are valid is also where the renormalisation group is useful, allowing us to move between different energy scales. As we move towards higher energies, in supergravity, we may identify renormalisation group flows with equations of motion on the additional directions. In string theory however, distinguishing the scales is not perfectly defined. T-duality itself must arise by means of a "conspiracy" between physics at all scales simultaneously [5]. T-duality in string theory, together with Buscher's rules for the target space [6,7], is to be associated with the low energy demand for invariance of physics under a change of topology of the renormalisation group. The plan of this article is as follows. First, I introduce the renormalisation group from an algebraic perspective, connecting the form of the generators (beta functions) with the global topological structure of the group. Next, I impose the invariance of physical results with respect to group topology changes by suitably altering the group laws. In order to take this new principle into account a modification of the beta function will be required. Finally, I will show on a simple quantum field theoretical model how these modifications explain the naturalness without the need of additional symmetries or particles. Of course, the presence of new symmetries at higher scales is not excluded by these results. Supersymmetry may still be a symmetry of nature at higher energies but as experiments suggest, it may not be able to explain naturalness on its own.
BULK SPACE AND THE RENORMALISATION GROUP
Given holography and, particularly, the AdS/CFT duality, quantum gravity can be written in terms of a nongravitational quantum field theory on a boundary, allowing the application of Wilson's renormalisation group. In this context, T-duality is seen as a duality in the bulk space involving string theory. Invariance with respect to T-duality has been made explicit at this level, for example by using a type of space that looks locally like a Riemannian manifold but which is glued together from these local patches not just by diffeomorphisms but also by T-duality transformations along some torus fibres (Tfolds). Such a T-fold would be a target space for a string sigma-model that is only locally a Riemannian manifold but globally would look like a more general geometry. Tfolds have a description in terms of spaces that locally are fibre products of a torus fibre bundle with its T-dual [34]. Among the first to create an explicitly T-duality invariant formulation for the string world sheet action was Tseytlin [27]. Here, I show that by holographic duality, a demand for topological invariance resulting from T-duality on the bulk, results in the restoration of naturalness in the hierarchy of the boundary quantum field theory.
The gauge/gravity duality can be interpreted, according to [20] as a geometrisation of the quantum field theoretical renormalisation group. While the connection between a higher dimensional equation of motion and the lower dimensional renormalisation group flow was known already in [1], it was [20] who reformulated this connection in modern, holographic terms. The exact renormalisation group equations can be expressed as Hamilton equations for the radial evolution in a model space-time with one higher dimension.
The jet-bundle structure corresponding to the quantum field theory is employed by [20] in order to show how the geometry corresponding to the RG equations emerges. These equations are being interpreted as higher spin equations of motion. In this article I emphasise mostly the topological aspects of the RG and how dualities connecting topologically distinct string geometries manifest themselves at lower energies as invariances to topological changes of the renormalisation group. To understand such a topological approach to the bulk-space structure, it is important to understand how the generators of the renormalisation group, the beta functions, emerge as geometrical objects. The scale transformations in the field theory correspond to movement in the additional radial dimension [20] and the renormalisation group trajectories are associated to specific geometries. If the RG flow begins or ends near a fixed point, the corresponding geometries are anti-de Sitter and connections between gauge/gravity dualities and RG flows are known [21].
The picture that emerges is that we should formulate the holographic duals of exact RG equations in terms of connections and sections of certain bundles over a model space-time. Indeed, once such a formulation has been constructed the next step will be to establish the topological invariance at low energies that results from the string theoretical T-duality. Following ref. [20] in the first step of the Wilsonian RG process, we integrate out a shell of fast modes and analyse the effect on the sources. Lowering M → λM , where λ = 1 − ǫ is interpreted as an integration of the fast modes, we obtain the variation of the sources
δ ǫ B = −ǫM d dM B δU = −ǫM d dM U(2)
We demand that for the overall partition function
M M dM Z = Z −1 0 [dφdφ * ]{(M d dM e iS0 e iS1 + e iS0 (M M dM e iS1 ) − Z −1 0 e iS0+iS1 M d dM Z 0 } = 0 (3)
where the action has been regulated and split into S = S 0 + S 1 with S 1 containing the B(x, y) sources. Following the calculations of [20] we obtain
δ ǫ B = −ǫM d dM B = ǫB∆ B B δ ǫ U = −ǫM d dM U = −iǫN T r(∆ B )B(4)
The second step of the renormalisation group approach is to perform a transformation that brings the cutoff back while changing the conformal factor of the metric
L(x, y) = δ d (x, y) + ǫzW (0) z (x, y)(5)
and hence we obtain, again, making use of [20] B
(z + ǫz) = B(z) − ǫ[W (0) z , B] + ǫB∆ B B U (z + ǫz) = U (z) − iǫN T r(∆ B )B(6)
We may redefine
∆ B = M z d dM (D 2 (0) ) −1(7)
and this allows us to extend the definition of the sources to the whole of the RG bulk space. The connection between the holographic renormalisation group in the bulk and the Wilsonian renormalisation group in the dual field theory on the boundary has been discussed in [30] where the Wilson RG transformation is mapped by the AdS/CFT duality and takes a holographic form.
At the string scale the formalism works as for the Tduality invariant doubled string formalism. The UVdivergences in the doubled formalism were studied via background field expansion in [39]. Their effects on the beta functions have also been studied and it was noticed that in the doubled interpretation, the high energy beta functionals are to be analysed in a target space that has been dimensionally reduced so that only the base coordinates remain [40]. This shows that the high energy beta functions have been made topologically invariant by means of the doubling string interpretation, and there are no additional UV instabilities produced at that level. The IR region can be reached by changing the coefficient structure in cohomology from one allowing extended objects to one allowing only points. Demanding that these two situations are dual according to the universal coefficient theorem and hence demanding for the associated groups to be isomorphic both as original groups and as extensions leads to an additional co-boundary term in the group law. Such a term represents a unification of low and high energy scales and, according to the universal coefficient theorem, by using the Ext group, leads to a unified interpretation of both IR and UV domains. Therefore this cobordism stabilises also the IR region (see next section).
Scale transformations in the field theory correspond to movement in the extra dimension and therefore the specific RG trajectories correspond to particular bulk geometries. On the quantum field theoretical side, the RG transformations in a perturbative context are regarded as deformations away from a free RG fixed point [20]. In holography, simple geometric constructions in the bulk correspond to strongly coupled dynamics in the dual field theory and hence the free fixed point RG discussion on the field theoretical side may seem incompatible with the standard holographic image. This point of view is however too superficial. In analysing the conjectured duality between free vector models in d = 2 + 1 dimensions and higher spin theories on AdS 4 it was noted in [31] and [32] that the field theory side is well understood while the bulk theory was highly non-linear, containing arbitrary high spins. An example used in [20] was the connection between three dimensional Chern-Simons theories known to be dual to two dimensional Wess-Zumino-Witten models. The theory in the bulk is now topological, meaning that it does not depend on the bulk metric and diffeomorphism invariance is broken only on the boundary by terms explicitly containing the boundary metric [20]. From studying the exact RG equations, [20] shows that the holographic duals of vector models can be expressed in terms of connections and sections of certain bundles over a model spacetime. The goal of this paper is to further generalise this to a situation where topology transformations in the bulk, induced by the action of a T-duality transformation, are supposed to provide the same physics. In this case, insensibility to bulk topology changes is understood by means of the methods discussed in [20] as modifications of the beta functions in the field theory sector that are capable of restoring naturalness of the hierarchy. Otherwise stated, invariance to topology changes in the bulk is dual to a natural hierarchy in the quantum field theory. Such topology changes are of the type induced by T-duality in the presence of non-trivial H-fluxes. Of course, imposing topological invariance is al-ready a non-perturbative criterion imposed on the bulk, hence the method has already from the construction nonperturbative stability.
When we parametrise the infinitesimal RG transformation according to [20] by writing it as
L = 1 + ǫ · z · W (0) z (8)
the second step changes z → λ −1 z and hence the RG flow becomes parametrised by z instead of M which becomes just an auxiliary parameter in the cutoff function. Here, z represents a conformal factor in the background metric η µν → z −2 η µν while the sources B and W are transformed as B old = z d+2 B new and W old = z d W new . The effective renormalisation scale will therefore be µ = M z and the renormalisation group flow will be parametrised by z. This will then be interpreted as a bulk coordinate. The infinitesimal piece of L is W (0) z which can be thought as the z component of the connection. W (0) z is regarded as a bookkeeping device, keeping track of the gauge transformations along the RG flow. In order to use the notation of [20] one may reabsorb the tensorial components of W into B and to redefine B as [20]. In this context, where a free bosonic vector model is considered to be perturbed away from the fixed point by the bi-local source B, and where the fields B and W (0) are extended into the entire RG mapping space (the RG bulk space), we may interpret and re-organise the renormalisation group equations as covariant equations with the beta function playing the role of a curvature. Let the bulk extension of B be called B and the bulk extension of W (0) be called W (0) . In ref. [20] it is shown that the renormalisation group equations emerge as gaugecovariant equations in the bulk. With W (0) µ flat in the transverse directions by construction we can write
B = B−{Ŵ µ , D (0) µ }−Ŵ µ ·Ŵ µ where D 0 µ is the covariant derivative defined inF (0) = dW (0) + W (0) ∧ W (0) = 0(9)
where d = dx µ ·[∂ µ , ·]+dz ·∂ z · is the bulk exterior derivative. This being a boundary operator it satisfies d 2 = 0. For the extension of the B field we have
∂ z B + [W (0) z , B] = β (B)(10)
The
z component of the 1-form β (B) = β (B) µ dx µ +β (B) dz is given by β (B) = B · ∆ B · B (11) with ∆ B = M z d dM (D (0)−2 µ ) where D (0)−2 µ
is the covariant derivative for a background gauge symmetry as defined in [20]. The beta function is therefore interpreted as a curvature. A series of topological invariants arise as integrals over curvatures. It is interesting to note that in the context of [22] and [23], there is no measurement in general relativity that can unambiguously detect the presence of a generic wormhole geometry. This statement is equivalent, due to ER-EPR with the statement that entanglement is not detectable. Such an observation is in agreement with the fact that (co)homology with different choices of coefficients may detect certain aspects of a topological space while masking others and therefore, that physics must remain invariant with respect to topology changes, due for example to string theoretical T-duality transformations. Now that a geometrical interpretation of the beta function has been established [20], it remains to be seen how the invariance with respect to topology changes affects the beta functions. Many topological invariants can be expressed in terms of integrals over various differential geometrical functions like curvature, Gauss curvature, etc. In order to make the low energy results invariant to topological changes, the integrals over the beta functions must not change when topological changes due to T-duality are invoked. To see how this works it is important to first consider the string theoretical duality symmetry.
TOPOLOGICAL ASPECTS OF THE RENORMALISATION GROUP
Let g(l) be the renormalisation group transformation such that, when it acts on a parameter of the theory it obeys the group law g(l)×g(λ) = g(l+λ). Given a parameter of the theory m, we have g(l)m = m ′ = G(l, m). G is a continuous function of the two parameters satisfying the normalisation condition G(0, m) = m corresponding to the transformation g(0) = Id. We can of course transform the group law from this algebraic form to a functional form writing G(l, G(λ, m)) = G(l + λ, m), and in the infinitesimal form we obtain the well known expression ∂G(l,m) ∂l = β(G(l, m)). We can clearly see that the group generator is β(m) = ∂G(ǫ,m) ∂ǫ at ǫ = 0. Let there be a physical quantity F (Q 2 , m) calculated under a certain renormalisation prescription as F ( Q 2 µ 2 , m µ ), m µ being the renormalised parameter (coupling) at some renormalisation point Q = µ. Demanding that F does not depend on the energy scale reads dF dµ = 0, or, making use of the parameters on which F depends
[x ∂ ∂x − β(m) ∂ ∂m ]F (x, m) = 0(12)
where x = Q 2 µ 2 , and m = m µ =m(µ 2 ) is called the effective coupling, β being the group generator defined above. As they stay, the renormalisation group equations represent simply the group laws and do not contain any physics. The change in the parameters of the theory and the energy scale where the measurements are performed do obey such a group law. However, when the energy scale becomes high enough for string geometry to play a role, we need to take into account the possibility of topology change and to make the observable results independent of such a change. One way in which this can be taken into account is by means of cohomology with non-trivial coefficients and the use of the universal coefficient theorem.
Non-trivial coefficients in cohomology represent the topological and geometrical structure associated to a point of the manifold upon which the group acts. When the coefficients are trivial the point has no additional structure. When the coefficients are non-trivial the point may get the structure of various extended objects (strings, branes). The fact that the coefficient structure in cohomology encodes the algebraic structure added to a point on the manifold, giving it the properties of extended objects has been discussed in [38]. In this sense, the universal coefficient theorem, by allowing us to move from one coefficient structure to another, can allow us to move from the trivial structure of a point (classical geometry) to an extended structure with various topologies (stringy geometry). This property is of utmost importance. Properly analysed, it may reveal the origin of the holographic principle and of AdS/CFT. Otherwise stated, this exact sequence in homological algebra potentially encodes several classes of dualities between string theoretical constructions and field theoretical ones. Locality on the boundary is associated to coefficients in co-homology encoding ordinary points. Universal coefficient theorems may define exact sequences connecting such local quantum field theories with non-local theories in the bulk, associated to coefficients in cohomology encoding "generalised" points i.e. algebraic curves, bundles, etc. Understanding the maps arising in the universal coefficient theorem may therefore reveal new dualities and new ways in which certain structures may be seen from a dual perspective. Using the universal coefficient theorem to define new dualities, beyond holography or the ER-EPR conjecture is the subject of future articles. Here, I will use it to reinterpret the hierarchy problem from a different perspective. While a classical point is restricted in probing the topology of a manifold, a string will be able to detect non-trivial topology due to its ability of wrapping around a non-trivial cycle. There is however another way of probing topology, even at lower energies, by means of quantum field theories defined via path integrals. Indeed quantum field theories integrate all available paths which therefore will have access to the global structure of the manifold. A discussion on this aspect of path integral quantisation may be found in [41].
Suppose we have already reached the scale where extended structures become relevant. For group cohomology, the universal coefficient theorem is
0 → Ext(H p−1 (G; M 1 ), M 2 ) → H p (G; M 2 ) h − → Hom(H p (G; M 1 ), M 2 ) → 0(13)
where G is our group, H q represents the q-th order homology while H q represents the q-th order cohomology, and M 1 , resp. M 2 represent the coefficients that make the trivial resp. non-trivial topology of the target space manifest. The sequence above is exact. Were it not for the Ext group on the left, the arrow h would have been an isomorphism. Clearly, the Ext group is what will signal to our beta function the distinction given by different topologies. In order to make our renormalisation group equation insensible to changes in topology and therefore to be able to take into account T -duality with non-trivial H-fluxes I will show what group laws are permitted. Let by notation call H p−1 (G, M 1 ) = G 1 . Then we are interested in Ext(G 1 , M 2 ). G 1 controls the topological aspects of our renormalisation group G as seen by means of homology with coefficients in M 1 . The extension will henceforth be calledG. Its group elementsg ∈G arẽ g = (m, g), with m ∈ M 2 . The group law ofG is
(m ′ , g ′ )(m, g) = (m ′ + m + ξ(g ′ , g), g ′ g)(14)
Let there be two extensionsG 1 andG 2 associated to two potential universal coefficient sequences. Starting for example with the trivial coefficient structure Z which would encode ordinary points and therefore local quantum field theory, the two universal coefficient theorems would expose the maps that would lead to target manifolds of different topologies (for example spheric and toroidal). Let them be
1 → K i1 − →G 1 π1 −→ G → 1 1 → K i2 − →G 2 π2 −→ G → 1(15)
with the two extensions related by a mapf =G 1 → G 2 such that i 2 =f • i 1 and π 1 = π 2 •f . We have therefore two group laws, associated to two different twococycles defining the group extensionsG 1 andG 2 . I will distinguish the two by using round brackets for the first group law and square brackets for the second group law (basically, using the notation and derivation given in [8])
(m ′ , g ′ )(m, g) = (m ′ + m + ξ 1 (g ′ , g), g ′ g), [m ′ , g ′ ][m, g] = [m ′ + m + ξ 2 (g ′ , g), g ′ g](16)
If we assume there exists an isomorphismf :G 1 →G 2 then, since (m, g) = (m, e)(0, g) and sincef is a homomorphism, it is clear thatf is fully determined once the images of the elements (m, e) and (0, g) are given. But then a commutative diagram arises when the two exact sequences above are glued together at the left and right ends and the isomorphismf is employed to connect the two extensions in the middle. For the diagram to be commutative, a set of conditions will appear onf
f • i 1 = i 2 ⇒f (m, e) = [m, e] π 2 •f = π ⇒f (0, g) = [η(g), g](17)
This implies thatf must have a generic formf (m, g) = [m + η(g), g]. When we know η(g) we have definedf . But we do know thatf is a homomorphism, hencẽ
f (m ′ +m+ξ 1 (g ′ , g), g ′ g) = [m ′ +m+ξ 1 (g ′ , g)+η(g ′ g), g ′ g] (18) must be equal tõ f (m ′ , g ′ )f (m, g) = [m ′ + η(g ′ ), g ′ ][m + η(g), g] = = [m ′ + m + ξ 2 (g ′ , g) + η(g ′ ) + η(g), g ′ g](19)
and hence η(g) must satisfy the equality
ξ 1 (g ′ , g) = ξ 2 (g ′ , g) + η(g ′ ) + η(g) − η(g ′ g) = = ξ 2 (g ′ , g) + ξ cob (g ′ , g)(20)
where ξ cob (g ′ , g) is the two-coboundary generated by η(g). Therefore, when the diagram is commutative ξ 1 and ξ 2 define the same extension. When ξ 1 and ξ 2 are proportional i.e. ξ 2 = λξ 1 then the last equality cannot be satisfied for all g ′ and g and therefore they define isomorphic groupsG 1 andG 2 that are different as extensions. Otherwise stated, they define different elements of the second cohomology group associated to the original group. This implies that for the group laws (m ′ , g ′ )(m, g) = (m ′ + m + ξ 1 (g ′ , g), g ′ g), the correction to a simple addition m ′ + m, namely ξ 1 , must satisfy the equality ξ 1 (g ′ , g) = ξ 2 (g ′ , g) + η(g ′ ) + η(g) − η(g ′ g) = ξ 2 (g ′ , g) + ξ cob (g ′ , g). This translates into an additional freedom in defining the beta functions and the introduction of additional terms.
In the context of the renormalisation group, the group laws defined above are basically the renormalisation group equations governing the flows of the parameters of an effective field theory. The demand that the renormalisation group equations and the respective flows be independent of the topology changes emerging from stringy features at high energies, results in the requirement that the groups obtained above are not distinguishable as extensions. In figure 1 it is explained how different topologies in the bulk (the upper level), when connected by means of a cobordism may lead to boundary effective field theories (lower level in the diagrams of Figure 1) that would not distinguish the underlying topological structure. While the string theoretical topologies are connected by T-duality, the freedom of adding a cobordism in the low energy beta function results in a form of invariance to renormalisation group changes in topology.
T-DUALITY
As a generalisation of the R → 1 R invariance of string theory compactified on a circle of radius R, T-duality transformations in the case of the low energy effective field theory are given by the Buscher rules [6][7]. However, the application of such rules is restricted in the topologically non-trivial case due to their validity only over local spacetime patches. In the case of the topologically non-trivial Neveu-Schwarz 3-form H-flux, several T-duals have been found and it has been noticed that T-duality not only changes the H-flux but also the spacetime topology [9][10][11]. It is known that low energy supersymmetry solves the hierarchy problem elegantly. However, from the point of view of the low energy effective actions it may happen that the original and T-dual theories do not have the same number of spacetime supersymmetries [12]. In certain extreme cases, supersymmetry may completely disappear after performing a T-duality operation. We may look at the T-duality transformation as to a change of variables and therefore the symmetries of the original theory are expected to be preserved, although when those symmetries do not commute with the duality, they may be realised only non-locally. That perturbative string theoretical calculations may not detect supersymmetry which is however visible in M-theory is a well known fact that led the authors of [13] to discuss the so called phenomenon of "supersymmetry without supersymmetry". Strings, unlike point particles, probe the target spacetime differently, leading to various effects that may not have analogues in theories based on dimensionless objects. Among these effects, the non-localisation of the extended worldsheet and associated target space supersymmetry under a T-duality transformation appears to be particularly important. A given conformal field theory may appear differently when analysed from different target space perspectives related by T-duality. If we formulate T-duality transformations in the form of canonical transformations of the worldsheet theory, the coordinate with respect to which duality is performed (say η) and the corresponding coordinate in the dual theory (sayη) are non-local functions of each other. When we integrate over the string length parameter which appears in the relation between η andη the non-locality emerges. Any η dependent quantity in one theory becomes a non-local function of the corresponding coordinate in the dual theory. To this effect one adds also the interchange between momentum modes (local) and winding modes (non-local) which is also exclusively due to the extended nature of the string. When the worldsheet theory has an extended supersymmetry, if a complex structure associated to such an extended supersymmetry does not have a dependence on the coordinate η, then in the dual model the extended supersymmetry is realised in the usual way. However, there exist situations when supersymmetry is not preserved by T-duality as noticed above. In all these situations, the complex structure associated with the supersymmetry depends on the coordinate η and in the dual theory the complex structure is replaced by a non-local object. In these cases, the extended supersymmetry in the dual theory, albeit still present, is realised only nonlocally. The relation between supersymmetry and target space geometry is modified [14]. However, non-local effects may be eliminated by changes in the underlying structure associated via coefficients in cohomology to every point of the space. Such modifications are allowed exclusively by freedoms given by the stringy nature of the high energy domain but in the low energy domain they allow for corrections in the beta functions and the RG flow equations identified as transformation group laws.
DUALITY SYMMETRIC STRING WORLD SHEETS
Having explained how the renormalisation group flows can be interpreted in geometric terms and, following [20], how the beta function can be interpreted as a geometric curvature, it remains to be seen how string theory can be formulated in a T-duality symmetric way so that we can easily transfer its topological invariance to low energy effective theories. String theory implies the existence of a target space in which the dynamics of strings manifests itself. This target space is usually associated with spacetime and allows the construction of S-matrix like observables that connect with low energy local effective field theories. However, the idea that spacetime itself may be emergent and that the dynamics and interaction of strings may determine its geometrical properties seems to be gaining some momentum [25], [26]. Moreover, at low energies, string theory possesses duality symmetries which relate apparently different supergravity backgrounds. The string spectrum is known to contain winding modes which are capable of probing spacetime in a different way. T-duality is defined in terms of exchanging the winding and momentum modes while changing the gravity background to render physics invariant. The mere existence of T-duality shows that strings experience geometry differently compared to point particles. It therefore makes sense to keep both the winding and momentum modes on equal footing. This has been done [27], [28] by reformulating string theory in a manifest duality symmetric way. As a consequence the number of dimensions of the target spacetime doubled. The extended nature of such a target space allowed the symmetric inclusion of both a geometry and its T-dual. The background equations of motion for the doubled target space are given by the so called Double Field Theory (DFT). This realises T-duality as a manifest symmetry and incorporates both the momentum modes and the winding modes [27]. This method doubles the spacetime dimension with the doubled dimensions being related by T-duality. Additional degrees of freedom will then appear which will have to be reduced by means of chirality constraints. From a spacetime perspective Tduality is a solution generating symmetry of the low energy equations of motion. From a world sheet point of view, T-duality is a non-perturbative symmetry [36]. The presence of T-duality allows for the construction of nongeometric manifolds where locally geometric regions are patched together by means of T-duality transformations. An analysis of the beta functional of the string in the doubled field formalism in order to determine the background field equations for the doubled space has been done in [36]. There it has been established that the background field equations arising from the one loop beta functional for the doubled formalism are the same as for the usual string. The doubled formalism can be seen as another description of string theory on target spaces described in the notation of [36] as locally T n bundles, with fibre coordinates X i , over the base space N with coordinates Y a . The fibre of the bundle is then doubled to T 2n with the coordinates denoted X A . The resulting sigma model Lagrangian will then be
L = 1 4 H(Y )dX A ∧ * dX B + L(Y ) + L top (X)(21)
where L(Y ) is the string Lagrangian on the base, H(Y ) is the metric on the fibre, and L top is a purely topological term. Explicitly we have
H = h − bh −1 b bh −1 −h −1 b h −1
where h and b are the target space metric and the Bfield on the fibre of the undoubled space. We define X = (X i ,X i ) with {X i } being the coordinates on the T-dual torus. Reference [37] establishes the quantum equivalence of the doubled formalism to the usual sigma model formalism for worldsheets of arbitrary genus, provided the topological term mentioned above is added to the action. The topological term does not affect the classical theory but instead introduces relative signs in the sum over topological sectors. It depends only on the winding numbers around homology cycles in the worldsheet and hence does not affect the classical theory. The periodicities of the direct and dual coordinates are such that the T d torus parametrised by the dual coordinates is dual to the one parametrised by the direct coordinates. Then the term in the action S top = 1 2πα ′ L top becomes a sum of terms involving products of winding numbers for a conjugate pair of cycles, with a sum over 1-cycles. This leads to an alternating sign sum contribution to the functional integral given as a sum over winding numbers. This alternating sum involves the extra-dimensions arising due to the doubling and would not appear in an effective field theory. They will however influence the form of the corrections to the beta functions appearing in the low energy domain.
NATURALNESS AS A LOW ENERGY EFFECT OF T-DUALITY
It is important to notice that the hierarchy problem is strongly related to the cosmological constant problem. It is known [15] that T-duality does not keep the cosmological constant, defined as the asymptotic value of the scalar curvature, invariant. First this has been noted in [16] where for an WZW model with a groupSL(2, R) a discrete subgroup has been gauged and relativity of the cosmological constant was first observed. Such a space has negative cosmological constant and under T-duality it is mapped into an asymptotically flat space. From a string theoretical perspective one can ask, together with the authors of [15] whether the standard low energy effective field theoretical definition of a cosmological constant makes sense. One can indeed wonder to what extent the cosmological constant can be seen as an unambiguous string observable. Ref. [15] notes that the change of the cosmological constant under T-duality is generic in the sense that it remains valid in higher dimensions as well. Starting with a ten dimensional type IIA theory with cosmological constant, a T-duality symmetry, if required to be a good symmetry, will force us to equate the cosmological constant to zero. Is it therefore possible that the network of string dualities will finally impose constraints on the cosmological constant, strong enough to explain its small value? A discussion of how making all scales of the matter sector functionals of the 4-volume element of the universe can remove the vacuum energy contributions from the field equations has been analysed in [24]. Such a discussion is compatible with large hierarchies between the Planck scale, electroweak scale, and vacuum curvature scale.
While the hierarchy problem and naturalness are related to the problem of the small cosmological constant, and may actually be one and the same problem, I will focus here on the naturalness issues. T-duality relates different geometries and/or topologies as seen from the perspective of extended objects (strings). As introducing such extended objects is equivalent to modifying a cohomology theory such that the coefficient structure replaces the normal geometric points with segments of algebraic curves, passing from a string to an effective description and back should be described simply in terms of modifying the coefficient structure of the cohomology theory in a controlled fashion. Duality symmetries are important in determining properties of the low energy effective theories and prove extremely useful in answering questions related to supersymmetry breaking.
To see how T-duality affects the low energy renormalisation group equations I consider a cutoff regularisation scheme where the relation between the bare and renormalised Green's function is
Γ bare ({p 2 }, 1/ǫ, {g bare }) = Z −1 Γ (1/ǫ, {g µ })Γ({p 2 }, µ, {g µ })(22)
where g bare = Z g ((1/ǫ), {g µ })g. Multiplication by the constant Z g obeys the group property and after eliminating the divergencies, such multiplications with finite constants are equivalent with making different choices of renormalisation schemes. Such schemes are therefore related by operations belonging to a Lie group. Given an arbitrary Green function Γ obeying the normalisation condition Γ({p 2 }, µ 2 , 0) = 1 and looking for the variation of the Green's function with respect to the energy scale parameter µ one obtains
µ 2 d dµ 2 Γ = (µ 2 ∂ ∂µ 2 + µ 2 ∂g ∂µ 2 ∂ ∂g )Γ = µ 2 dlnZ Γ dµ 2 Z Γ Γ bare(23)
or, using the standard notation, we obtain the renormalisation group equation in partial derivatives
(µ 2 ∂ ∂µ 2 + β(g) ∂ ∂g + γ Γ )Γ({p 2 }, µ 2 , g µ ) = 0(24)
The beta function and the anomaly dimensions are well known
β = µ 2 dg dµ 2 | g bare γ Γ = −µ 2 dlnZΓ dµ 2 | g bare(25)
By using characteristics we can write the general form of the solution of this equation as Γ(e t {p 2 } µ 2 ,ḡ(t, g))e t 0 γΓ(ḡ(t,g))dt (26) where the characteristic equation is
d dtḡ (t, g) = β(ḡ),ḡ(0, g) = g(27)
We callḡ(t, g) the effective coupling. One usually includes the vertex function and defines the whole product together as effective coupling
gΓ( {p 2 } µ 2 , g)(28)
If our Green function is an n-point function we can write the renormalisation of the coupling g as g bare = Z Γ Z −n/2 2 g and the the product is renormalised as gΓ = Z n/2 2 g bare Γ bare . One can construct a renormalisation group invariant quantity called the invariant charge ξ by multiplying the product by the corresponding propagators
ξ = gΓ( {p 2 } µ 2 , g) n i D 1/2 ( p 2 i µ 2 , g)(29)
The characteristic solution of the renormalisation group equation allows us to sum up an infinite series of logs coming from Feynman diagrams in both the IR (t → −∞) or UV (t → ∞) regions and induces nonperturbative corrections to the otherwise perturbative expansion. As an example, the invariant charge in a massless theory with only one coupling constant is
ξ( p 2 µ 2 , g) = g(1 + b · g · ln p 2 µ 2 + ...)(30)
The beta function for a one loop approximation is β(g) = b · g 2 . The beta function can be written as a derivative of the invariant charge with respect to the log of the momentum β(g) = p 2 d dp 2 ξ(
p 2 µ 2 , g)| p 2 =µ 2(31)
the RG-improved formula for the invariant charge is ξ RG ( p 2 µ , g) = ξ P T (1,ḡ( p 2 µ 2 , g)) =ḡ(
p 2 µ 2 , g)(32)
where I replaced t = ln p 2 µ 2 . The effective coupling is a solution of the characteristic equation d dtḡ (t, g) = bḡ 2 ,ḡ(0, g) = g, t = ln p 2
µ 2 (33) namelyḡ (t, g) = g 1 − b · g · t(34)
When expanding this in terms of t the geometrical progression above reproduces the expansion of the invariant charge with the difference that in this final expression, we also have an infinite series of terms g n t n called the leading log approximation. For the next order in the logs one considers the next term in the expansion of the beta function and sums up the next series of terms in the form g n t n−1 and so on. The behaviour of the effective couplings is determined by the beta function. When the minimal subtraction scheme is employed, the renormalisation of the mass occurs in the same way as that of the couplings
m bare = Z m m(35)
The mass renormalisation constant Z m is independent of the mass parameters and depends only on dimensionless couplings. The effective mass is then given by
Up to one loop one has β(α) = b · α 2 and γ m (α) = c · α with the solution
m(t) = m 0 ( α(t) α 0 ) c/b(38)
The mass is running due to radiative corrections. In the minimal subtraction scheme one may stop the running at p 2 = m 2 and identify m 2 =m 2 (m 2 ). A better way to define the mass is by identifying the physical mass with the pole mass. This does not depend on the scale and is scheme independent. The pole mass can be expressed by means of the running mass at the scale of a mass with finite and calculable corrections. If one calculates radiative corrections to the mass of the Higgs boson based entirely on the Standard Model, one obtains a loop integral of the form
d 4 p[ 1 ( / p − m f )( / p + / k − m f ) ](39)
where k is the Higgs momentum. This diverges quadratically for large p, independent of k and hence generates a correction δm 2 ∼ = Λ 2 where Λ is the scale beyond which the low energy theory can no longer be applied. Let µ 2 be the scale at which the breaking of the SU (2)×U (1) takes place. We assume that the standard model is the low energy theory of a more fundamental theory that rises at scale µ 1 . Using the fundamental theory one may imagine to be able to calculate the mass of the Higgs boson. The result of such a calculation would be a scale dependent mass parameter evaluated at the fundamental scale µ 1 . The important quantity for the low energy theory is the running mass calculated at the scale µ 2 . The relation between these two masses is given by
m 2 H (µ 2 ) = m 2 H (µ 1 ) + Cg 2 µ 2 1 µ 2 2 dk 2 + Rg 2 + O(g 4 ) (40)
where g is a coupling constant, C is dimensionless, and R grows as a logarithm function with respect to µ 1 as µ 1 → ∞. The term proportional to C diverges quadratically when µ 1 → ∞. Usually, in order for m 2 H (µ 2 ) ≪ µ 2 1 one has to fine-tune the parameter m 2 H (µ 1 ) to cancel the second term in the equation above which is of order µ 2 1 . The "natural" value for m 2 H (µ 2 ) is a number of the order of µ 2
1 . Therefore what is the mechanism behind the fact that µ 2 ≪ µ 1 ? Given the experimental exclusion of various supersymmetric models at LHC, it appears implausible for supersymmetry to still be able to solve such a hierarchy problem. However, one may ask how the invariance of the high energy physical theory to topology changes induced by, say, T-duality may affect the low energy calculations given by the last equation?
The requirement of topological invariance as demanded by T-duality indeed has some non-trivial effects on the renormalisation group equations. As said in the introduction, the beta functions are generators of the renormalisation group, while the group laws are given by the respective renormalisation group equations. To demand topological invariance of the theory with respect to high energy changes in topology amounts to modifications of the group laws which implicitly induce modifications in the form of the renormalisation group equations at low energies. Indeed, as will be shown further on, such modifications have precisely the form required to restore naturalness, i.e. they re-create terms similar to the terms obtained by adding supersymmetric partners, only this time, they originate simply from topological restrictions in the high energy end of the theory. The physical mass of the Higgs boson m H at tree level is proportional to the square root of the Higgs self-interaction coupling λ. We know that the observed value of m H is in the range that corresponds to vacuum metastability if there is no new physics between the electroweak scale and the Plank scale. Results for a full 2-loop calculation of the Higgs boson pole mass m H having theM S Lagrangian parameters v, λ, y t , g, g ′ , g 3 with the leading 3-loop corrections in the limit (g 3 , y t ) >> (λ, g, g ′ ) have been presented in [17]. In order to compute the Higgs boson physical mass m H , the self energy function consisting of the sum of all 1-particle-irreducible 2-point Feynman diagrams needs to be calculated
Π(s) = 1 16π 2 Π (1) (s) + 1 (16π 2 ) 2 Π (2) (s) + ...(41)
The complex pole squared mass is the solution of
m 2 H − iΓ H M H = m 2 B + 3λ B v 2 B + 1 16π 2 Π (1) (s pole ) + 1 (16π 2 ) 2 Π (2) (s pole )(42)
In a Wilsonian RG the hierarchy problem appears as a radiative mixing of multiple relevant operators, caused by logarithmic divergences. Thus, because in Wilsonian RG, quadratic divergences determine the position of the critical surface in the theory space, and the scaling behavior of RG flows around the critical surface is determined only by the logarithmic divergences, we can subtract the quadratic divergences easily. The subtraction can therefore be interpreted as a choice of parametrization in the theory space [18]. The required fine tuning appears therefore to be the distance of the bare parameters from the critical surface. This obviously means that quadratic divergences are not the main problem. However, Wilsonian RG radiative mixing implies that the lower mass scale is affected by higher scales through RG transfor-mations. It is therefore important to understand what are the relevant high energy effects that can explain the non-naturalness. Consider the Higgs self coupling λ introduced in the potential of V (0) = m 2 H † 2 H 2 which we treat as a running low energy effective parameter. The RG improved Higgs mass can be written as
(m 2 H 0 ) RG = λv 2 0(43)
where the running quartic Higgs self coupling evaluated at the scale s = m 2 H 0 is
λ = dtβ λ(44)
using the β function at two-loop order β λ we obtain the next to leading log radiative corrections to the Higgs mass summed to all orders in perturbation theory [19]. This formula is defining for the self-coupling. However [20] showed that the beta function can be interpreted as a RG bulk space geometric curvature. When integrating over a curvature two form, the result may depend on the topological contributions to the beta function. Certain quantities constructed from the curvature on a differentiable principal bundle are inherent to the bundle and do not dependent on the specific curvature used for their definition. These are characteristic to the bundle, preserved by bundle diffeomorphisms and define topological invariants associated with the bundle. For details I refer the reader to [29] for further details on the construction of such invariants.
In the most restricted case, the integral of the curvature can be interpreted as a topological invariant. An example is the generalisation of the Gauss-Bonnet theorem for higher dimensional spaces. In this case, denoting the curvature by Ω we may write M P f (Ω) = (2π) n χ(M ) with P f (Ω) the Pfaffian. For a four dimensional oriented manifold the Gauss-Bonnet theorem reads
1 32π 2 M (|R m | 2 − 4|R c | 2 + R 2 )dµ = χ(M )(45)
where R m is the Riemann curvature tensor, R c is the Ricci curvature tensor, and R is the scalar curvature. In the general case we have the so called Chern-Weil theory which is capable of generating various other invariants. In the situation at hand, the beta function has been shown in [20] to be interpreted as a curvature and its integral is known to contribute to the running quartic Higgs self coupling. What is important is that a topological invariant arises in the form of the running quartic Higgs self coupling. This is particularly important because in order to impose topology invariance in agreement with the string geometric T-duality one needs to add a series of terms that would counter the effects of these topological invariants in order to make the theory independent of topology changes as induced by T-duality. Indeed, such terms can be added when the groups associated to different coefficient structures are equivalent as extensions and hence we cannot distinguish topological differences. In fact, this is precisely our requirement in agreement with the topology changing T-duality. Therefore, the renormalisation group equations will be changed due to high energy T-duality effects. These effects can be thought of as mixing between scales as visible in the Wilson renormalisation group approach. Particularly demanding that the renormalisation group operations are not sensitive to topological changes is equivalent to demanding that the low energy groups differ only by terms that do not distinguish the respective extensions to the analysed topologies. For group laws of the form (m ′ , g ′ )(m, g) = (m ′ + m + ξ 1 (g ′ , g), g ′ g) the correction must be ξ 1 (g ′ , g) = ξ 2 (g ′ , g) + ξ cob (g ′ , g). Translated into the RG equation language this provides us with topologi-cally covariant terms that take into account the fact that the high energy theory must keep the universal coefficient theorem that leads to changes in the coefficients which would switch between detecting and not detecting topological features related by T-duality, trivial. Such shifts in the low energy theory will have strictly no physical effects (i.e. no particle detection whatsoever) because they are fundamentally string-geometrical effects that have no analogues for point-particle effective theories. However, they will have detectable effects on the radiative corrections to scalar masses due to the changes of the renormalisation group flows. Indeed the radiative corrections give a modification of the Higgs mass in the form of
m * = m 2 − λ 2 M 2 32π 2 [1 − γ E + ln M 2 4πµ 2 ](46)
but demanding topological insensitivity at the string level leads, at low energies, according to the Verlinde duality [1], [2], to a redefinition of the renormalisation group laws namely to a modification of the renormalisation group equations. Considering that the renormalisation group equations are dual to equations of motion on the extra dimensions and the beta functions represent bulk space curvatures, after expressing the string theory in the bulk in a manifestly T-dual form, we double the string coordinates. Extra dimensional string coordinates correspond to generators of the renormalisation group and hence to beta functions. In modern language, this can be understood in terms of double field theory as resulting from incorporating string theoretical T-duality as a symmetry of a field theory defined on a double configuration space. T-duality however requires extended objects because it is based essentially on the existence of winding modes associated to wrappings of such extended objects around non-contractible cycles. A T-duality symmetric field theory will have to take the winding modes information into account. If we compactify the strings on a torus we obtain compact momentum modes, dual to compact coordinates x i , i = 1, ..., n, as well as string winding modes. A new set of coordinates,x i dual to windings must also be considered for the compactified sector in the field theory.
There is a vast literature on this field, including reviews like [35] on which I based my discussion up to now. In what concerns the low energy effective field theory, this doubling amounts to re-writing the RG equation as (µ ∂ ∂µ + β ⊥ (g) ∂ ∂g − β (g) ∂ ∂g + 1 2 nγ(g))Γ (n) (p i ; g, µ) = 0 (47) where the beta function corresponding to the curvature has been split over the two terms, one over the original fibre bundle and one on the T-dual bundle used to Tsymmetrise the bulk theory. These are of course only low energy remnants of the high energy T-duality symmetrised theory. Their existence in the low energy domain however implies cancellation of terms that would aid the naturalness of the effective theory. This can be rewritten by adding a term ξ(g ′ , g) which corrects the perturbative formulation of the beta function, a term corresponding to the one derived as the cobordism that restores topological insensitivity by demanding that the two coefficient structures in cohomology generate extensions within the universal coefficient theorem that do not single out any topological change. Here g ′ and g are nothing but the parameters defined at the two different scales. I rewrite therefore, heuristically (µ ∂ ∂µ + (β(g) + ξ(m, M )) ∂ ∂g + 1 2 nγ(g))Γ (n) (p i ; g, µ) = 0 (48) Interestingly enough, while keeping the prescription that the modifications brought to the low energy effective theory must obey the laws of topological insensitivity imposed by the universal coefficient theorem's extensions, such terms induce additional scale effects that, because of their dependence on the topology at a specific order (considering for example the large N topological expansion) will contribute with different signs. For example torus terms will annihilate sphere terms, due to the requirement of T-duality, etc. Therefore we have
m * = m 2 − λ 2 M 2 32π 2 [1−γ E +ln M 2 4πµ 2 + i (−1) i ζ(m 2 , M 2 )]
(49) which will introduce the corrections to the extreme scale dependence. The Wilsonian physical mixing effect is accounted for by the scale mixing between stringy effects and effective field theoretical effects.
CONCLUSIONS
This article provides a heuristic solution to the hierarchy problem by explaining it in terms of a non-trivial mixing of UV and IR effects. In modern terminology, we may understand the naturalness of the standard model hierarchy as a dual phenomenon to the existence of high energy T-duality and particularly of bulk-space invariance to topology changes when explicit T-duality invariant string theory is employed. As there is no direct reason for such a requirement in the low energy domain where objects are point-like it comes as no surprise that the hierarchy problem remained so long with no clear resolution.
FIG. 1 :
1The cobordism modifying the group operation connects two distinct high energy topologies, making them indistinguishable in the low energy theory. The doubling method assures indistinguishability in the explicitly T-dual formulation of string theory. The lower part of the figures represent the low energy, the upper part represents string theoretical topology changing effects.
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| []
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[
"Multisoliton solutions of the vector nonlinear Schrödinger equation (Kulish-Sklyanin model) and the vector mKdV equation",
"Multisoliton solutions of the vector nonlinear Schrödinger equation (Kulish-Sklyanin model) and the vector mKdV equation"
]
| [
"Takayuki Tsuchida "
]
| []
| []
| There exist two natural vector generalizations of the completely integrable nonlinear Schrödinger (NLS) equation in 1 + 1 dimensions: the well-known Manakov model and the lesser-known Kulish-Sklyanin model. In this paper, we propose a binary Darboux (or Zakharov-Shabat dressing) transformation that can be directly applied to the Kulish-Sklyanin model. By deriving a simple closed expression for iterations of the binary Darboux transformation, we obtain an explicit formula for the N -soliton solution of the Kulish-Sklyanin model under vanishing boundary conditions. Because the third-order symmetry of the vector NLS equation can be reduced to a vector generalization of the modified KdV (mKdV) equation, we can also obtain multisoliton (or multi-breather) solutions of the vector mKdV equation in closed form. | null | [
"https://arxiv.org/pdf/1512.01840v2.pdf"
]
| 67,826,296 | 1512.01840 | bbb393e89128cd8ad3c3531f5f5c93e06f70757e |
Multisoliton solutions of the vector nonlinear Schrödinger equation (Kulish-Sklyanin model) and the vector mKdV equation
21 Dec 2015 December 22, 2015
Takayuki Tsuchida
Multisoliton solutions of the vector nonlinear Schrödinger equation (Kulish-Sklyanin model) and the vector mKdV equation
21 Dec 2015 December 22, 2015
There exist two natural vector generalizations of the completely integrable nonlinear Schrödinger (NLS) equation in 1 + 1 dimensions: the well-known Manakov model and the lesser-known Kulish-Sklyanin model. In this paper, we propose a binary Darboux (or Zakharov-Shabat dressing) transformation that can be directly applied to the Kulish-Sklyanin model. By deriving a simple closed expression for iterations of the binary Darboux transformation, we obtain an explicit formula for the N -soliton solution of the Kulish-Sklyanin model under vanishing boundary conditions. Because the third-order symmetry of the vector NLS equation can be reduced to a vector generalization of the modified KdV (mKdV) equation, we can also obtain multisoliton (or multi-breather) solutions of the vector mKdV equation in closed form.
Introduction
The cubic nonlinear Schrödinger (NLS) equation [1,2] iq t + q xx + 2σ|q| 2 q = 0, σ = +1 or −1,
(1.1)
is a representative integrable system in 1 + 1 dimensions. The case σ = +1 and the case σ = −1 correspond to the self-focusing and self-defocusing NLS equation, respectively. The NLS equation can be generalized to a single vector equation involving the standard scalar product · , · in two distinct ways while preserving the integrability [3]; that is, the Manakov model [4] iq t + q xx + 2 q, q * q = 0 (1.2) and the Kulish-Sklyanin model [5] iq t + q xx + 4 q, q * q − 2 q, q q * = 0.
(1.3)
Here, q is a vector dependent variable and the asterisk denotes the complex conjugation. For brevity, we write down only the self-focusing case here, but it is straightforward to extend these models to the self-defocusing or a mixed focusing-defocusing case [6][7][8][9][10][11]. Note that these models often appear in some disguised forms; any invertible linear transformation can be applied to the vector q, which mixes its components. The Kulish-Sklyanin model (1.3) can be reduced to the Manakov model (1.2) by setting q, q = 0, up to a trivial rescaling; this can be realized by restricting the components of q as, e.g., q = (q 1 , ±iq 1 , q 3 , ±iq 3 , . . . , q 2m−1 , ±iq 2m−1 ) .
This simple observation demonstrates that the explicit formula for the Nsoliton solution of the Kulish-Sklyanin model (1.3) and the vector soliton interactions thereof are highly nontrivial and more complicated than those for the Manakov model (1.2) reported in [4,[12][13][14].
Clearly, the Manakov model (1.2) is obtained from the (generally rectangular) matrix generalization of the scalar NLS equation, i.e., the matrix NLS equation [15]:
iQ t + Q xx + 2QQ † Q = O,(1.Q = q 1 I + 2m−1 j=1 q j+1 e j .
Here, I is the identity matrix; {e 1 , e 2 , . . . , e 2m−1 } are skew-Hermitian (e † j = −e j ) matrices that form generators of the Clifford algebra, i.e., they satisfy the anticommutation relations:
{e j , e k } + := e j e k + e k e j = −2δ jk I, (1.5) where δ jk is the Kronecker delta. We require that {I, e 1 , e 2 , . . . , e 2m−1 } are linearly independent. Because the ancestor model, the matrix NLS equation (1.4), can be solved using the inverse scattering method and the N-soliton solution can be written down explicitly, it is straightforward to obtain the N-soliton solution of the Kulish-Sklyanin model (1.3) through the reduction. However, the obtained expression is "non-classical" in the sense that it involves the generators of the Clifford algebra {e 1 , e 2 , . . . , e 2m−1 } explicitly in a rather complicated manner; it is a highly nontrivial task to translate such a "non-classical" expression into a more user-friendly "classical" expression not involving {e 1 , e 2 , . . . , e 2m−1 }, using the anticommutation relations (1.5). Indeed, this can be achieved for the one-and two-soliton solutions, but not for the general N-soliton solution in practice.
The main objective of this paper is to derive a simple closed expression for the general N-soliton solution of the Kulish-Sklyanin model (1.3) without recourse to the N-soliton solution of the matrix NLS equation (1.4). To this end, we consider a nonstandard Lax representation for the Kulish-Sklyanin model (1.3) [16], which does not involve the generators of the Clifford algebra {e 1 , e 2 , . . . , e 2m−1 }, and apply a binary Darboux (or Zakharov-Shabat dressing) transformation [17][18][19][20][21]. A peculiar structure of the binary Darboux transformation allows us to express an arbitrary number of its iterations in simple explicit form. Thus, by applying the N-fold binary Darboux transformation to the trivial zero solution, we obtain the bright N-soliton solution of the Kulish-Sklyanin model (1.3) in closed form. Actually, the binary Darboux transformation can be applied to all the isospectral flows that belong to the same integrable hierarchy as the Kulish-Sklyanin model (1.3). Among the higher flows of this integrable hierarchy, the third-order flow is particularly interesting because it simplifies to a vector analog of the modified KdV (mKdV) equation [22,23]:
q y + q xxx + 6 q, q q x = 0 (1.6)
under the reduction q = q * . Thus, with a minor tune-up of the multifold binary Darboux transformation, we can obtain multisoliton solutions, multibreather solutions and their mixtures of the vector mKdV equation (1.6). This paper is organized as follows. In section 2, we summarize two different Lax representations for the Kulish-Sklyanin model (1.3) and make some remarks on its soliton solutions. In section 3, we propose the binary Darboux transformation and apply its N-fold version to the Kulish-Sklyanin model (1.3) to obtain its general N-soliton solution in simple explicit form. We also discuss how to obtain exact solutions such as the N-soliton solution of the vector mKdV equation (1.6); the obtained N-soliton forumula is different from the multisoliton formula proposed by Iwao and Hirota [24] using the Hirota bilinear method [25], and our formula has its own advantages. Section 4 is devoted to concluding remarks.
Lax representations
We start with the matrix generalization of the nonreduced NLS system [26,27] proposed by Zakharov and Shabat as early as 1974 [15]:
iQ t + Q xx − 2QRQ = O, iR t − R xx + 2RQR = O. (2.1)
Here, Q and R are l 1 × l 2 and l 2 × l 1 (generally rectangular) matrices. Some relevant information and references on the matrix NLS system (2.1) can be found in [28]. The Lax representation [29] for the matrix NLS system (2.1) is given by the following overdetermined linear system [30,31]:
Ψ 1 Ψ 2 x = −iζI l 1 Q R iζI l 2 Ψ 1 Ψ 2 , (2.2) Ψ 1 Ψ 2 t = −2iζ 2 I l 1 − iQR 2ζQ + iQ x 2ζR − iR x 2iζ 2 I l 2 + iRQ Ψ 1 Ψ 2 . (2.3)
Here, ζ is a spectral parameter independent of x and t, and I l 1 and I l 2 are the l 1 × l 1 and l 2 × l 2 identity matrices, respectively; in this paper, we usually consider the l 1 = l 2 case and omit the index of the identity matrix. The matrix NLS system (2.1) is a positive flow in the integrable hierarchy associated with the spectral problem (2.2). The next higher flow in the integrable hierarchy is a matrix analog [30,31] of the nonreduced complex mKdV equation [26,27,32], i.e.
Q y + Q xxx − 3Q x RQ − 3QRQ x = O, R y + R xxx − 3R x QR − 3RQR x = O. (2.4)
To reduce the matrix NLS system (2.1) to the Kulish-Sklyanin model (1.3) or, more generally, the matrix NLS hierarchy to the Kulish-Sklyanin hierarchy, we introduce 2 m−1 × 2 m−1 skew-Hermitian matrices {e 1 , e 2 , . . . , e 2m−1 } that satisfy the anticommutation relations (1.5). Then, we set
Q = q 1 I + 2m−1 j=1 q j+1 e j , R = r 1 I − 2m−1 j=1 r j+1 e j .
(2.5)
The matrices {I, e 1 , e 2 , . . . , e 2m−1 } are assumed to be linearly independent. Lax representations involving the generators of the Clifford algebra (or quaternions in the m = 2 case) can be traced back to the references [5,33,34]. As a natural extension of the complex conjugate, we define "Clifford conjugate" denoted as , which acts on the linear span of {I, e 1 , e 2 , . . . , e 2m−1 } to reverse the sign of the coefficients of {e 1 , e 2 , . . . , e 2m−1 }. For instance,
Q = q 1 I − 2m−1 j=1 q j+1 e j , R = r 1 I + 2m−1 j=1 r j+1 e j .
Note that Q = Q. Because of the anticommutation relations (1.5), we have useful relations such as Q Q = QQ = q, q I, QR + R Q = Q R + RQ = 2 q, r I,
QRQ = QR + R Q Q − R QQ = 2 q, r Q − q, q R, (2.6) RQR = R QR + R Q − R R Q = 2 q, r R − r, r Q,(2.7)
(I − µQR) I − µ R Q = 1 − 2µ q, r + µ 2 q, q r, r I. Here, q = (q 1 , q 2 , . . . , q 2m ) and r = (r 1 , r 2 , . . . , r 2m ) are 2m-component row vectors; · , · denotes the standard scalar product, e.g., q, r = 2m j=1 q j r j , etc.
Owing to (2.6) and (2.7), the reduction (2.5) simplifies the matrix NLS system (2.1) to the nonreduced Kulish-Sklyanin model: iq t + q xx − 4 q, r q + 2 q, q r = 0, ir t − r xx + 4 q, r r − 2 r, r q = 0.
(2.9)
Note that q, r and q j r k − q k r j are conserved densities for (2.9). By further imposing the general complex conjugation reduction r j = σ j q * j , σ j = ±1, j = 1, 2, . . . , 2m, we obtain the Kulish-Sklyanin model with a mixed focusing-defocusing nonlinearity: iq t + q xx − 4 q, q * Σ q + 2 q, q q * Σ = 0.
(2.10)
Here, Σ := diag(σ 1 , σ 2 , . . . , σ 2m ) is a diagonal matrix with each entry σ j equal to +1 or −1. In the following, we mainly consider the Kulish-Sklyanin model in the self-focusing case:
iq t + q xx + 4 q, q * q − 2 q, q q * = 0. (2.11)
The third-order symmetry of the nonreduced Kulish-Sklyanin model (2.9) is obtained by imposing the reduction (2.5) on the matrix complex mKdV system (2.4) and noting the identities
Q x RQ + QRQ x = (QRQ) x − QR x Q, R x QR + RQR x = (RQR) x − RQ x R
in view of (2.6) and (2.7); by further setting r = −q (and thus R = − Q in (2.5)), (2.4) reduces to the vector mKdV equation [22,23]:
q y + q xxx + 6 q, q q x = 0. (2.12)
The matrix NLS hierarchy can be solved using the inverse scattering method based on the spectral problem (2.2), so the exact solutions such as the N-soliton solution of the matrix NLS system (2.1), as well as the thirdorder symmetry (2.4), can be obtained explicitly in closed form. Thus, the exact solutions of the Kulish-Sklyanin model (2.11), as well as the vector mKdV equation (2.12), can also be obtained by imposing the corresponding reduction conditions on the scattering data involved in the solution. However, this approach is useful only if the number of components or solitons is small enough. Indeed, the obtained formula for the N-soliton solution of the 2mcomponent Kulish-Sklyanin model (2.11) involves the inverse of an N × N block matrix, where each block is a 2 m−1 × 2 m−1 matrix taking values in the linear span of {I, e 1 , e 2 , . . . , e 2m−1 }. The formula is too bulky and not a mathematically tractable object for 2m > 4 and N > 2.
In the four-component case (2m = 4), the reduction (2.5) is no longer a restriction. Indeed, one can employ 2 × 2 Pauli's matrices multiplied by the imaginary unit i as a matrix representation for {e 1 , e 2 , e 3 }:
e 1 = 0 i i 0 , e 2 = 0 1 −1 0 , e 3 = i 0 0 −i .
These matrices together with the identity matrix form a basis, i.e., any 2 × 2 complex matrix can be expressed as a linear combination of {I, e 1 , e 2 , e 3 }; thus, (2.5) is merely a linear transformation mixing the elements in the 2 × 2 matrices Q and R. In the self-focusing case, this linear transformation reads
Q = q 1 + iq 4 iq 2 + q 3 iq 2 − q 3 q 1 − iq 4 , R = −Q † = −q * 1 + iq * 4 iq * 2 + q * 3 iq * 2 − q * 3 −q * 1 − iq * 4 ,
where Q satisfies the matrix NLS equation (1.4). Clearly, the N-soliton solution of the Kulish-Sklyanin model (2.11) for a four-component vector q can be directly obtained from the N-soliton solution of the matrix NLS equation (1.4) for a 2 × 2 matrix Q by applying this linear transformation. The Kulish-Sklyanin model (2.11) for a three-component vector q is obtained by setting one component, say q 3 , in the four-component case as identically zero. The reduction q 3 = 0 corresponds to the restriction of Q to a symmetric matrix [35]; the corresponding reduction of the N-soliton solution from the four-component case to the three-component case is straightforward [36,37].
It is clear by setting q 2 = q 3 = 0 in the above representation that the Kulish-Sklyanin model (2.11) in the two-component case, say q = (q 1 , q 4 ) can be decoupled into two scalar NLS equations in the variables q 1 ± iq 4 [38]. Thus, any solution of the two-component Kulish-Sklyanin model can be written as a linear combination of two solutions of the scalar NLS equation; in this sense, the two-component case is trivial and less interesting. The rank-1 one-soliton solution in the two-component case is
q(x, t) = 2η sech [2η(x + 4ξt) + α] e −2iξx−4i(ξ 2 −η 2 )t+iϕ 1 2 , ± i 2 ,
and the rank-2 one-soliton solution is
q(x, t) = 2η sech [2η(x + 4ξt) + α 1 ] e −2iξx−4i(ξ 2 −η 2 )t+iϕ 1 1 2 , − i 2 + 2η sech [2η(x + 4ξt) + α 2 ] e −2iξx−4i(ξ 2 −η 2 )t+iϕ 2 1 2 , i 2 .
Here, η > 0 and the other parameters are real constants. This implies that the rank-1 one-soliton solution in the general component case is
q(x, t) = 2η sech [2η(x + 4ξt) + α] e −2iξx−4i(ξ 2 −η 2 )t u (2.13)
where u, u = 0 and u, u * = 1 2 , and the rank-2 one-soliton solution is
q(x, t) = 2η sech [2η(x + 4ξt) + α 1 ] e −2iξx−4i(ξ 2 −η 2 )t+iϕ u + 2η sech [2η(x + 4ξt) + α 2 ] e −2iξx−4i(ξ 2 −η 2 )t+iϕ u * (2.14)
where u, u = 0 and u, u * = 1 2 . In the two-component case, the Kulish-Sklyanin model with a mixed focusing-defocusing nonlinearity is more interesting than the model with a simple focusing (or defocusing) nonlinearity. Indeed, (2.10) with q = (q 1 , q 4 ) and Σ = diag(−1, 1) [39]:
iq 1,t + q 1,xx + 2 |q 1 | 2 − 2|q 4 | 2 q 1 − 2q 2 4 q * 1 = 0, iq 4,t + q 4,xx + 2 2|q 1 | 2 − |q 4 | 2 q 4 + 2q 2 1 q * 4 = 0, (2.15)
is obtained from the matrix NLS system (2.1) through the reduction
Q = q 1 + iq 4 0 0 q 1 − iq 4 , R = −q * 1 − iq * 4 0 0 −q * 1 + iq * 4 .
Thus, the two-component Kulish-Sklyanin model with a mixed focusingdefocusing nonlinearity (2.15) is equivalent to the nonreduced scalar NLS system [26,27]:
iq t + q xx − 2q 2 r = 0, ir t − r xx + 2r 2 q = 0, through the linear change of variables q = q 1 + iq 4 , r = −q * 1 − iq * 4 (or q = q 1 − iq 4 , r = −q * 1 + iq * 4 )
. In this paper, we aim to obtain a compact and tractable expression for the N-soliton solution of the Kulish-Sklyanin model (2.11), which is valid for an arbitrary number of components and does not involve the generators of the Clifford algebra. To derive such a "classical" expression, we first rewrite the spectral problem (2.2) under the reduction (2.5) to a more convenient form. We consider a linear eigenfunction the first component of which is an invertible matrix; then, the spectral problem (2.2) can be rewritten in terms of P := Ψ 2 Ψ −1 1 as a matrix Riccati equation (see [40][41][42] for the scalar case and [43,44] for the vector case):
P x = R + 2iζP − P QP. (2.16)
Thus, under the reduction (2.5) and appropriate boundary conditions, we can confine P to the linear span of {I, e 1 , e 2 , . . . , e 2m−1 }. By setting
P = p 1 I − 2m−1 j=1 p j+1 e j , p = (p 1 , p 2 , . . . , p 2m )
and noting the relation (2.6), we can simplify (2.16) to a vector Riccati equation:
p x = r + 2iζp − 2 p, q p + p, p q. (2.17)
We introduce the scalar denominator f and the vector numerator g as
p = g f , (2.18a) and set g, g = f h. (2.18b)
Noting the freedom to multiply f and g by any common factor, we can linearize the vector Riccati equation (2.17) as
f g T h x = −2iζ 2q 0 r T O q T 0 2r 2iζ f g T h ,
where the superscript T denotes the matrix transpose. This spectral problem is the spatial part of a nonstandard Lax representation for the nonreduced Kulish-Sklyanin model (2.9) [16]; this kind of nonstandard spectral problem first appeared in [7,27] through the investigation of the squared eigenfunctions associated with the scalar NLS hierarchy and a certain vector generalization was studied in [45,46]. The corresponding time part of the Lax representation can, in principle, be derived from (2.3) in an analogous manner, but it is easier to obtain the temporal Lax matrix from the compatibility condition as a truncated power series in the spectral parameter ζ [7,26,27]. For later convenience, we rescale q, r and g T by a factor of 1/ √ 2 and set 2ζ =: λ to reformulate the nonstandard Lax representation in a more symmetric and concise form. iq t + q xx − 2 q, r q + q, q r = 0,
ir t − r xx + 2 q, r r − r, r q = 0, (2.19)
is equivalent to the compatibility condition for the overdetermined linear system [16]:
ψ 1 ψ 2 ψ 3 x = −iλ q 0 r T O q T 0 r iλ ψ 1 ψ 2 ψ 3 , (2.20) ψ 1 ψ 2 ψ 3 t = −iλ 2 − i q, r λq + iq x 0 λr T − ir T x ir T q − iq T r λq T + iq T x 0 λr − ir x iλ 2 + i q, r ψ 1 ψ 2 ψ 3 .
(2.21)
Here, λ is a constant spectral parameter, q and r are row vectors and ψ 2 is a column vector.
By rewriting the spectral problem (2.20) as the adjoint problem 22) or noting the identity
ψ 3 −ψ T 2 ψ 1 x = − ψ 3 −ψ T 2 ψ 1 −iλ q 0 r T O q T 0 r iλ ,(2.ψ 3 −ψ T 2 ψ 1 ψ 1 ψ 2 ψ 3 x = ψ 3 −ψ T 2 ψ 1 −iλ q 0 r T O q T 0 r iλ ψ 1 ψ 2 ψ 3 = 0,
and similar for t-differentiation, we notice that the quantity 2ψ 1 ψ 3 − ψ 2 , ψ 2 is a constant. In fact, the derivation from the standard Lax representation, (2.2) and (2.3), through the reduction (2.5) implies that we only need to consider linear eigenfunctions satisfying the condition 2ψ 1 ψ 3 = ψ 2 , ψ 2 (cf. (2.18b)). In this paper, we use the notation · , · to denote the scalar product of two row vectors as well as column vectors.
q y + q xxx − 3 q x , r q − 3 q, r q x + 3 q, q x r = 0, r y + r xxx − 3 q, r x r − 3 q, r r x + 3 r, r x q = 0. (2.23)
The reduction r = −q simplifies (2.23) to the vector mKdV equation [22,23]:
q y + q xxx + 3 q, q q x = 0, (2.24)
which is obtained as the compatibility condition for the overdetermined linear system:
ψ 1 ψ 2 ψ 3 x = −iλ q 0 −q T O q T 0 −q iλ ψ 1 ψ 2 ψ 3 , (2.25) ψ 1 ψ 2 ψ 3 y = −iλ 3 + iλ q, q λ 2 q + iλq x − α 0 −λ 2 q T + iλq T x + α T −2q T x q + 2q T q x λ 2 q T + iλq T x − α T 0 −λ 2 q + iλq x + α iλ 3 − iλ q, q ψ 1 ψ 2 ψ 3 ,(2.
26)
with α := q xx + q, q q. Let Λ be the block anti-diagonal matrix:
Λ := 1 −I 1 , Λ T = Λ, Λ 2 = I,
and denote a column-vector eigenfunction of the spectral problem (2.20) at λ = µ and its matrix transpose (i.e., row vector) as
|µ := ψ 1 ψ 2 ψ 3 λ=µ , µ| := ψ 1 ψ T 2 ψ 3 λ=µ ,
which satisfy the condition
µ | Λ | µ = 2ψ 1 ψ 3 − ψ 2 , ψ 2 | λ=µ = 0.
In the same manner, we introduce a column-vector eigenfunction |ν of the spectral problem (2.20) at λ = ν and its matrix transpose ν|.
ψ 1 ψ 2 ψ 3 ∝ I + ν − µ λ − ν |µ ν| Λ ν | Λ | µ + µ − ν λ − µ |ν µ| Λ µ | Λ | ν ψ 1 ψ 2 ψ 3 ,(3.
1)
up to an overall constant, where µ | Λ | µ = ν | Λ | ν = 0 and the transformed potentials q and r are given by
q = q + i (µ − ν) (|µ ν| − |ν µ|) 12 ν | Λ | µ , (3.2a) r = r + i (ν − µ) (|µ ν| − |ν µ|) 32 ν | Λ | µ . (3.2b)
Here, the subscripts 12 and 32 denote the (1, 2) and (3, 2) sub-matrices (row vectors in this case) in the 3 × 3 block matrix.
In (3.1), |µ ν | Λ | µ , |ν µ | Λ | ν
provide linear eigenfunctions of the transformed spectral problem at λ = ν and λ = µ, respectively; for a suitable choice of |µ and |ν , these correspond to bound states generated by the binary Darboux transformation. Note that overall factors of |µ and |ν play no role in the definition of the binary Darboux transformation.
If |µ and |ν satisfy not only the spectral problem (2.20) but also the isospectral evolution equation (2.21) at λ = µ and λ = ν, respectively, the binary Darboux transformation (3.1) preserves the Lax representation, (2.20) and (2.21), form-invariant with the potentials transformed as q → q and r → r. This is also true for other flows of the integrable hierarchy. Thus, (3.2) can be used to generate a new nontrivial solution of the Kulish-Sklyanin hierarchy from its trivial solution.
Similar results on the Darboux transformations have been obtained by Mikhailov and coworkers (see, in particular, the pioneering paper [18] and the recent papers [49,50]).
The Darboux matrix defined in (3.1):
D µ,ν = I + ν − µ λ − ν |µ ν| Λ ν | Λ | µ + µ − ν λ − µ |ν µ| Λ µ | Λ | ν
has the important invariance property:
D T µ,ν ΛD µ,ν = Λ,
which implies det D µ,ν = 1 and
D −1 µ,ν = ΛD T µ,ν Λ.
Thus, the constant quantity 2ψ 1 ψ 3 − ψ 2 , ψ 2 for any linear eigenfunction is invariant under the binary Darboux transformation, i.e.
2 ψ 1 ψ 3 − ψ 2 , ψ 2 = 2ψ 1 ψ 3 − ψ 2 , ψ 2 .
In particular, if we start from a linear eigenfunction satisfying the condition 2ψ 1 ψ 3 = ψ 2 , ψ 2 , any linear eigenfunction generated by iterations of the binary Darboux transformation also satisfies the same condition.
We can consider an arbitrary number of iterations of the binary Darboux transformation (3.1) with different values of µ and ν in each step. For instance, the twofold binary Darboux transformation can be represented by the Darboux matrix:
D µ 2 ,ν 2 D µ 1 ,ν 1 = I + ν 2 − µ 2 λ − ν 2 | µ 2 ν 2 | Λ ν 2 | Λ | µ 2 + µ 2 − ν 2 λ − µ 2 | ν 2 µ 2 | Λ µ 2 | Λ | ν 2 × I + ν 1 − µ 1 λ − ν 1 |µ 1 ν 1 | Λ ν 1 | Λ | µ 1 + µ 1 − ν 1 λ − µ 1 |ν 1 µ 1 | Λ µ 1 | Λ | ν 1 , where | µ 2 = |µ 2 + ν 1 − µ 1 µ 2 − ν 1 ν 1 | Λ | µ 2 ν 1 | Λ | µ 1 |µ 1 + µ 1 − ν 1 µ 2 − µ 1 µ 1 | Λ | µ 2 µ 1 | Λ | ν 1 |ν 1 , | ν 2 = |ν 2 + ν 1 − µ 1 ν 2 − ν 1 ν 1 | Λ | ν 2 ν 1 | Λ | µ 1 |µ 1 + µ 1 − ν 1 ν 2 − µ 1 µ 1 | Λ | ν 2 µ 1 | Λ | ν 1 |ν 1 , and µ j | Λ | µ j = ν j | Λ | ν j = 0 (j = 1, 2).
Noting that a multifold binary Darboux transformation can be defined as the order-independent composition of binary Darboux transformations, we can assume that the N-fold binary Darboux transformation takes the following form (cf. [18]):
D λ 1 ,λ 2 ,...,λ 2N = I + 2N k=1 1 λ − λ k 2N j=1 g jk |λ j λ k | Λ. (3.3)
Here, {λ 1 , λ 2 , . . . , λ 2N } are pairwise distinct constants, g jk is a scalar function to be determined, |λ j is a nonzero column-vector eigenfunction of the spectral problem (2.20) at λ = λ j , and |λ j and its matrix transpose (i.e., row vector) λ j | satisfy the condition λ j | Λ | λ j = 0. Then, substituting (3.3) into the invariance property:
D T λ 1 ,λ 2 ,...,λ 2N ΛD λ 1 ,λ 2 ,...,λ 2N = Λ, and noting that this is an identity in λ, we obtain the relations: 2N. (3.4) Thus, g jk + g kj = 0 and (3.4) can be written as a 2N × 2N matrix equation:
g kk = 0, k = 1, 2, . . . , 2N, (λ k − λ j ) g jk + 2N i=1 2N l=1 g ik g lj λ i | Λ | λ l = 0, j, k = 1, 2, . . . ,GA − AG − GLG = O,
which is equivalent to
AG −1 − G −1 A = L. (3.5)
Here, A is a diagonal matrix, G is a skew-symmetric matrix and L is a symmetric matrix, defined as
A := diag (λ 1 , λ 2 , . . . , λ 2N ) , G := (g jk ) j,k=1,2,...,2N , L := ( λ l | Λ | λ i ) l,i=1,2,...,2N .
By solving the linear equation (3.5) for G −1 , we find that off-diagonal entries of the skew-symmetric matrix G −1 are given by
G −1 jk = λ j | Λ | λ k λ j − λ k , j = k. (3.6)
In view of (2.20) at λ = λ k and (2.22) at λ = λ j , we can compute the xderivative of G −1 as
∂ x G −1 jk = − G −1 G x G −1 jk = λ j | Λ diag (i, 0, . . . , 0, −i) | λ k . (3.7)
With the aid of (3.4) and (3.7), we can prove a multifold generalization of Proposition 3.1 by a direct calculation.
ψ 1 ψ 2 ψ 3 ∝ I + 2N k=1 1 λ − λ k 2N j=1 g jk |λ j λ k | Λ ψ 1 ψ 2 ψ 3 ,
up to an overall constant, where |λ j is a linear eigenfunction of the original spectral problem (2.20) at λ = λ j satisfying λ j | Λ | λ j = 0 and g jk is the (j, k) element of the inverse of the skew-symmetric matrix G −1 determined by (3.6). The transformed potentials q and r are given by
q = q − i 1≤j<k≤2N g jk (|λ j λ k | − |λ k λ j |) 12 , r = r + i 1≤j<k≤2N g jk (|λ j λ k | − |λ k λ j |) 32 ,
where the subscripts 12 and 32 denote the (1, 2) and (3, 2) sub-matrices (row vectors in this case) in the 3 × 3 block matrix.
Note that overall factors of |λ 1 , |λ 2 , . . . , |λ 2N are irrelevant to the definition of the N-fold binary Darboux transformation.
Multisoliton solutions
We first notice that the spectral problem (2.20) under the complex conjugation reduction r = −q * has the following symmetry property (a kind of
involution): if ψ 1 ψ 2 ψ 3 is a linear eigenfunction at λ = µ, then ψ * 3 −ψ * 2 ψ * 1 = Λ ψ * 1 ψ * 2 ψ * 3
is a linear eigenfunction at λ = µ * . By applying Proposition 3.1 using these two linear eigenfunctions as |µ and |ν , we obtain new potentials q and r, which also satisfy the same relation r = − q * . To obtain the bright N-soliton solution of the Kulish-Sklyanin model ((2.19) under the reduction r = −q * ): iq t + q xx + 2 q, q * q − q, q q * = 0, (3.8)
we start with the trivial zero solution q = r = 0 and apply Proposition 3.2.
In view of the above symmetry property, we consider a set of 2N eigenvalues {λ 1 , λ 2 , . . . , λ 2N } that consist of N complex conjugate pairs. The ordering of the 2N eigenvalues is irrelevant to the definition of the N-fold binary Darboux transformation, so it can be altered depending on one's preference; in this paper, we number the 2N eigenvalues as λ N +j = λ * j , j = 1, 2, . . . , N, and choose a column-vector eigenfunction |λ j of the linear problem (2.20) and (2.21) at λ = λ j as
|λ j = e −iλ j x−iλ 2 j t c T j 1 2 c j , c j e iλ j x+iλ 2 j t ∝ 1 c T j e iλ j x+iλ 2 j t 1 2 c j , c j e 2iλ j x+2iλ 2 j t , j = 1, 2, . . . , N,
(3.9a) and
|λ N +j = 1 2 c * j , c * j e −iλ * j x−iλ * 2 j t −c † j e iλ * j x+iλ * 2 j t ∝ 1 2 c * j , c * j e −2iλ * j x−2iλ * 2 j t −c † j e −iλ * j x−iλ * 2 j t 1 , j = 1, 2, . . . , N,
(3.9b) where c j is a constant row vector. Note that these linear eigenfunctions indeed satisfy the condition λ j | Λ | λ j = 0, j = 1, 2, . . . , 2N.
Recalling that overall factors of |λ 1 , |λ 2 , . . . , |λ 2N are irrelevant in the N-fold binary Darboux transformation, we can rescale these eigenfunctions as in (3.9) and translate the skew-symmetric matrix G −1 determined by (3.6) into a slightly simpler skew-symmetric matrix:
G −1 → U V −V T W , U T = −U, W T = −W,
where the entries of the N × N matrices U := (u jk ) j,k=1,2,...,N , V := (v jk ) j,k=1,2,...,N and W := (w jk ) j,k=1,2,...,N are defined as
u jk := 1 2 c j , c j e 2iλ j x+2iλ 2 j t + 1 2 c k , c k e 2iλ k x+2iλ 2 k t − c j , c k e i(λ j +λ k )x+i(λ 2 j +λ 2 k )t λ j − λ k = c j e iλ j x+iλ 2 j t − c k e iλ k x+iλ 2 k t , c j e iλ j x+iλ 2 j t − c k e iλ k x+iλ 2 k t 2 (λ j − λ k ) , j < k, (3.10a) v jk := 1 + c j , c * k e i(λ j −λ * k )x+i(λ 2 j −λ * 2 k )t + 1 4 c j , c j c * k , c * k e 2i(λ j −λ * k )x+2i(λ 2 j −λ * 2 k )t λ j − λ * k , (3.10b) w jk := 1 2 c * j , c * j e −2iλ * j x−2iλ * 2 j t + 1 2 c * k , c * k e −2iλ * k x−2iλ * 2 k t − c * j , c * k e −i(λ * j +λ * k )x−i(λ * 2 j +λ * 2 k )t λ * j − λ * k , j < k.
Note that u jj = w jj = 0 and u jk and w jk for j > k are given by −u kj and −w kj , respectively. Moreover, we have w jk = u * jk and v * jk = −v kj , so W = U * and V † = −V . Now, by applying Proposition 3.2, we obtain 11) and r = − q * , so the complex conjugation reduction is realized. It only remains to compute the entries of the inverse matrix in (3.11). While the inverse of a general square matrix is given in terms of the determinant and cofactors [51], the inverse of a skew-symmetric matrix can be expressed more simply in terms of the Pfaffian and cofactors [25]. The Pfaffian [25,51] is a square root of the determinant of a skew-symmetric matrix of even dimension (see https://en.wikipedia.org/wiki/Pfaffian). For example, the inverse of a 4 × 4 skew-symmetric matrix is given as
q = −i 1≤j<k≤N U V V * U * −1 jk c k e iλ k x+iλ 2 k t − c j e iλ j x+iλ 2 j t − i 1≤j,k≤N U V V * U * −1 j,N +k −c * k e −iλ * k x−iλ * 2 k t − 1 2 c * k , c * k c j e i(λ j −2λ * k )x+i(λ 2 j −2λ * 2 k )t − i 1≤j<k≤N U V V * U * −1 N +j,N +k − 1 2 c * j , c * j c * k e −i(2λ * j +λ * k )x−i(2λ * 2 j +λ * 2 k )t + 1 2 c * k , c * k c * j e −i(λ * j +2λ * k )x−i(λ * 2 j +2λ * 2 k )t ,(3. 0 d 12 d 13 d 14 −d 12 0 d 23 d 24 −d 13 −d 23 0 d 34 −d 14 −d 24 −d 34 0 −1 = 1 d 12 d 34 − d 13 d 24 + d 14 d 23 0 −d 34 d 24 −d 23 d 34 0 −d 14 d 13 −d 24 d 14 0 −d 12 d 23 −d 13 d 12 0 .
Following the notation and definition in Hirota's book [25], we write the Pfaffian of the 2N × 2N skew-symmetric matrix
U V V * U * , U T = −U, V † = −V (3.12)
as (1, 2, . . . , 2N) and denote cofactors as
Γ(j, k) = (−1) j+k−1 (1, 2, . . . , j − 1, j + 1, . . . , k − 1, k + 1, . . . , 2N) , 1 ≤ j < k ≤ 2N. (3.13)
Then, we have
U V V * U * −1 = 1 (1, 2, . . . , 2N) 0 −Γ(1, 2) −Γ(1, 3) · · · −Γ(1, 2N) Γ(1, 2) 0 −Γ(2, 3) · · · −Γ(2, 2N) Γ(1, 3) Γ(2, 3) 0 · · · −Γ(3, 2N) . . . . . . . . . . . . . . . Γ(1, 2N) Γ(2, 2N) Γ(3, 2N) · · · 0 .
Using this formula in (3.11) and omitting the tilde of q, we arrive at the main result of this paper.
q = i (1, 2, . . . , 2N) 1≤j<k≤N Γ(j, k) c k e iλ k x+iλ 2 k t − c j e iλ j x+iλ 2 j t + 1≤j,k≤N Γ(j, N + k) −c * k e −iλ * k x−iλ * 2 k t − 1 2 c * k , c * k c j e i(λ j −2λ * k )x+i(λ 2 j −2λ * 2 k )t + 1≤j<k≤N Γ(N + j, N + k) − 1 2 c * j , c * j c * k e −i(2λ * j +λ * k )x−i(2λ * 2 j +λ * 2 k )t + 1 2 c * k , c * k c * j e −i(λ * j +2λ * k )x−i(λ * 2 j +2λ * 2 k )t ,
(3.14)
where the Pfaffian (1, 2, . . . , 2N) and the cofactors Γ(j, k) are defined from the skew-symmetric matrix in (3.12) with the entries of U and V given by (3.10).
By extending the definition of the cofactors in (3.13) as Γ(k, j) = −Γ(j, k) and Γ(j, j) = 0 [25], we can rewrite (3.14) more concisely as
q = − i (1, 2, . . . , 2N) N j=1 N k=1 Γ(j, k) + Γ(j, N + k) 1 2 c * k , c * k e −2iλ * k x−2iλ * 2 k t c j e iλ j x+iλ 2 j t + N j=1 N k=1 Γ(j, N + k) + Γ(N + j, N + k) 1 2 c * j , c * j e −2iλ * j x−2iλ * 2 j t c * k e −iλ * k x−iλ * 2 k t = − i (1, 2, . . . , 2N) N j=1 (1, 2, . . . , j − 1, β, j + 1, . . . , 2N) c j e iλ j x+iλ 2 j t − N k=1 (1, 2, . . . , N + k − 1, β, N + k + 1, . . . , 2N) c * k e −iλ * k x−iλ * 2 k t , with (β, k) = 1, (β, N + k) = 1 2 c * k , c * k e −2iλ * k x−2iλ * 2
k t for k = 1, 2, . . . , N. By setting N = 1, we obtain the one-soliton solution of the Kulish-Sklyanin model (3.8) as
q = −i (λ 1 − λ * 1 ) c * 1 e −iλ * 1 x−iλ * 2 1 t + 1 2 c * 1 , c * 1 c 1 e i(λ 1 −2λ * 1 )x+i(λ 2 1 −2λ * 2 1 )t 1 + c 1 , c * 1 e i(λ 1 −λ * 1 )x+i(λ 2 1 −λ * 2 1 )t + 1 4 c 1 , c 1 c * 1 , c * 1 e 2i(λ 1 −λ * 1 )x+2i(λ 2 1 −λ * 2 1 )t .
The case c 1 , c 1 = 0 and the case c 1 , c 1 = 0 correspond to the rank-1 one-soliton solution (2.13) and the rank-2 one-soliton solution (2.14) respectively, up to a rescaling of q. The one-and two-soliton solutions of the Kulish-Sklyanin model (3.8) (up to a linear transformation mixing the components) have been studied in detail in [28,36,37,52]. Note that the Nsoliton solution (3.14) is a linear combination of the 2N constant vectors c 1 , . . . , c N , c * 1 , . . . , c * N , so it is mathematically redundant to consider the case where the number of the components of q is more than 2N.
Let us move on to the solutions of the vector mKdV equation (2.24). We first notice that the spectral problem (2.25), i.e., (2.20) under the reduction r = −q, has the following symmetry property (a kind of involution): if
ψ 1 ψ 2 ψ 3 is a linear eigenfunction at λ = µ, then ψ 3 −ψ 2 ψ 1 = Λ ψ 1 ψ 2 ψ 3
is a linear eigenfunction at λ = −µ. By applying Proposition 3.1 using these two linear eigenfunctions as |µ and |ν , we obtain new potentials q and r, which also satisfy the same relation r = − q.
To obtain the multisoliton (or multi-breather) solutions of the vector mKdV equation (2.24), we start with the trivial zero solution q = r = 0 in the spectral problem (2.20) and apply Proposition 3.2. In view of the above symmetry property, we consider the case where the 2N eigenvalues {λ 1 , λ 2 , . . . , λ 2N } occur in plus-minus pairs. The ordering of the 2N eigenvalues is irrelevant to the definition of the N-fold binary Darboux transformation and can be altered; in this paper, we number the 2N eigenvalues as λ N +j = −λ j , j = 1, 2, . . . , N, and choose a column-vector eigenfunction |λ j of the linear problem (2.25) and (2.26) at λ = λ j as
|λ j = e −iλ j x−iλ 3 j y c T j 1 2 c j , c j e iλ j x+iλ 3 j y ∝ 1 c T j e iλ j x+iλ 3 j y 1 2 c j , c j e 2iλ j x+2iλ 3 j y , j = 1, 2, . . . , N,
(3.15a) and
|λ N +j = Λ |λ j ∝ 1 2 c j , c j e 2iλ j x+2iλ 3 j y −c T j e iλ j x+iλ 3 j y 1 , j = 1, 2, . . . , N, (3.15b)
where c j is a constant row vector. Note that these linear eigenfunctions indeed satisfy the condition λ j | Λ | λ j = 0, j = 1, 2, . . . , 2N.
Recalling that overall factors of |λ 1 , |λ 2 , . . . , |λ 2N are irrelevant in the N-fold binary Darboux transformation, we can rescale these eigenfunctions as in (3.15) and translate the skew-symmetric matrix G −1 determined by (3.6) into a slightly simpler skew-symmetric matrix:
G −1 → U V −V T −U , U T = −U,
where the entries of the N × N matrices U := (u jk ) j,k=1,2,...,N and V := (v jk ) j,k=1,2,...,N are defined as
u jk := 1 2 c j , c j e 2iλ j x+2iλ 3 j y + 1 2 c k , c k e 2iλ k x+2iλ 3 k y − c j , c k e i(λ j +λ k )x+i(λ 3 j +λ 3 k )y λ j − λ k = c j e iλ j x+iλ 3 j y − c k e iλ k x+iλ 3 k y , c j e iλ j x+iλ 3 j y − c k e iλ k x+iλ 3 k y 2 (λ j − λ k ) , j < k, (3.16a) v jk := 1 + c j , c k e i(λ j +λ k )x+i(λ 3 j +λ 3 k )y + 1 4 c j , c j c k , c k e 2i(λ j +λ k )x+2i(λ 3 j +λ 3 k )y λ j + λ k . (3.16b) Note that v jk = v kj , so V T = V .
Now, by applying Proposition 3.2, we obtain
q = −i 1≤j<k≤N U V −V −U −1 jk c k e iλ k x+iλ 3 k y − c j e iλ j x+iλ 3 j y − i 1≤j,k≤N U V −V −U −1 j,N +k −c k e iλ k x+iλ 3 k y − 1 2 c k , c k c j e i(λ j +2λ k )x+i(λ 3 j +2λ 3 k )y − i 1≤j<k≤N U V −V −U −1 N +j,N +k − 1 2 c j , c j c k e i(2λ j +λ k )x+i(2λ 3 j +λ 3 k )y + 1 2 c k , c k c j e i(λ j +2λ k )x+i(λ 3 j +2λ 3 k )y ,(3.17)
and r = − q, so the required reduction is indeed realized. Because
O I I O U V −V −U −1 + U V −V −U −1 O I I O = O,
the inverse matrix should take the form:
U V −V −U −1 = X Y −Y −X , X T = −X, Y T = Y,
which is skew-symmetric. Thus, we can rewrite (3.17) as
q = −i 1≤j<k≤N U V −V −U −1 jk −c j e iλ j x+iλ 3 j y + 1 2 c j , c j c k e i(2λ j +λ k )x+i(2λ 3 j +λ 3 k )y + c k e iλ k x+iλ 3 k y − 1 2 c k , c k c j e i(λ j +2λ k )x+i(λ 3 j +2λ 3 k )y − i 1≤j<k≤N U V −V −U −1 j,N +k −c j e iλ j x+iλ 3 j y − 1 2 c j , c j c k e i(2λ j +λ k )x+i(2λ 3 j +λ 3 k )y − c k e iλ k x+iλ 3 k y − 1 2 c k , c k c j e i(λ j +2λ k )x+i(λ 3 j +2λ 3 k )y − i N j=1 U V −V −U −1 j,N +j −c j e iλ j x+iλ 3 j y − 1 2 c j , c j c j e 3iλ j x+3iλ 3 j y ,(3.18)
where the tilde of q is omitted. This is a fairly general complex-valued solution of the vector mKdV equation (2.24). To turn it into real-valued solutions, we first set λ j = iη j , j = 1, 2, . . . , N, (3.19) and rewrite the solution (3.18) as follows. is given by 20) where (1, 2, . . . , 2N) is the Pfaffian (a square root of the determinant) of the skew-symmetric matrix with the entries
q = 1 (1, 2, . . . , 2N) 1≤j<k≤N Γ(j, k) c j e −η j x+η 3 j y − 1 2 c j , c j c k e −(2η j +η k )x+(2η 3 j +η 3 k )y − c k e −η k x+η 3 k y + 1 2 c k , c k c j e −(η j +2η k )x+(η 3 j +2η 3 k )y + 1≤j<k≤N Γ(j, N + k) c j e −η j x+η 3 j y + 1 2 c j , c j c k e −(2η j +η k )x+(2η 3 j +η 3 k )y + c k e −η k x+η 3 k y + 1 2 c k , c k c j e −(η j +2η k )x+(η 3 j +2η 3 k )y + N j=1 Γ(j, N + j) c j e −η j x+η 3 j y + 1 2 c j , c j c j e −3η j x+3η 3 j y ,(3.(j, k) = − (N + j, N + k) = c j e −η j x+η 3 j y − c k e −η k x+η 3 k y , c j e −η j x+η 3 j y − c k e −η k x+η 3 k y 2 (η j − η k ) , 1 ≤ j < k ≤ N, (j, N + k) = 1 + c j , c k e −(η j +η k )x+(η 3 j +η 3 k )y + 1 4 c j , c j c k , c k e −2(η j +η k )x+2(η 3 j +η 3 k )y η j + η k , 1 ≤ j, k ≤ N,
and the cofactors Γ(j, k) for 1 ≤ j < k ≤ 2N are defined as in (3.13).
Note that (3.20) is of the form:
q = N l=1 G l c l e −c 1 c 1 , c 1 , c 2 c 2 , c 2 , . . . , c N c N , c N .
We can consider a generalization of the vector mKdV equation (2.24) as considered by Iwao and Hirota [24]:
q y + q xxx + 3 qB, q q x = 0. (3.22)
Here, B = (b jk ) is a constant square matrix, which can be assumed to be symmetric (b jk = b kj ) without loss of generality. Then, an N-soliton solution of this generalized vector mKdV equation (3.22) is given by (3.20) with the involved scalar products generalized as c j , c k → c j B, c k , 1 ≤ j, k ≤ N. This formula generalizes the multisoliton formula proposed by Iwao and Hirota [24] using the Hirota bilinear method [25] (also see the relevant results in [53][54][55]) and appears to be more efficient. In fact, (3.20) with positive η 1 , . . . , η N and real c 1 , . . . , c N is only a special N-soliton solution of the vector mKdV equation (2.24), which does not exhibit any oscillating behavior in each component of the vector variable q. In particular, it cannot reproduce the one-soliton solution of the complex mKdV equation [7,27,32], involving a complex carrier wave, which is equivalent to the two-component vector mKdV equation ((2.24) with a real two-component vector q), up to a linear transformation.
To obtain the general real-valued multisoliton solutions of the vector mKdV equation (2.24), we require that {η 1 , η 2 , . . . , η N } in (3.19) and (3.20), as well as the corresponding linear eigenfunctions in (3.15a), are either real or occur in complex conjugate pairs [49,56]. That is, up to a re-ordering, we assume (a) η M +j = η * j (Re η j > 0), c M +j = c * j , j = 1, 2, . . . , M, (b) η j > 0, c * j = c j , j = 2M + 1, . . . , 2M + L(= N). Thus, the original 2N eigenvalues {λ 1 , λ 2 , . . . , λ 2N } occur in (a) plus-minus and complex-conjugate quartets or (b) plus-minus pairs. Under these conditions, we can easily show from Proposition 3.2 that the new solution generated by the N-fold binary Darboux transformation is indeed real-valued.
q = 2 (η 1 + η 2 ) c 1 η 1 e −η 1 x+η 3 1 y − c 2 η 2 e −η 2 x+η 3 2 y η 1 − η 2 − 4η 1 η 2 η 1 −η 2 c 1 , c 2 e −(η 1 +η 2 )x+(η 3 1 +η 3 2 )y .
(3.23)
With η 2 = η * 1 and c 2 = c * 1 , this is indeed a real solution and provides the vector analog of the complex mKdV soliton [7,27,32]. By redefining the constant vector as 2η 1 c 1 =: (η 1 − η * 1 ) d 1 , the one-soliton solution (3.23) can be rewritten in a more concise form:
q = (η 1 + η * 1 ) d 1 e −η 1 x+η 3 1 y + d * 1 e −η * 1 x+η * 3 1 y 1 + d 1 , d * 1 e −(η 1 +η * 1 )x+(η 3 1 +η * 3 1 )y , d 1 , d 1 = 0. (3.24)
Thus, in the limit η 1 → η * 1 , i.e., Im η 1 → 0, this solution reduces to (3.21). = − c j , c k e −(η j +η k )x+(η 3 j +η 3 k )y η j − η k , 1 ≤ j < k ≤ 2M, (j, 2M + k) = (k, 2M + j) = 1 + c j , c k e −(η j +η k )x+(η 3 j +η 3 k )y η j + η k , 1 ≤ j < k ≤ 2M, (j, 2M + j) = 1 2η j , 1 ≤ j ≤ 2M, and the cofactors Γ(j, k) for 1 ≤ j < k ≤ 4M are defined as in (3.13). Here, η M +j = η * j (Re η j > 0), c M +j = c * j and c j , c j = 0 for j = 1, 2, . . . , M . By extending the definition of the cofactors in (3.13) as Γ(k, j) = −Γ(j, k) and Γ(j, j) = 0 [25] and noting the relation Γ(j, 2M + k) = Γ(k, 2M + j), we can rewrite (3.25) in a more compact form as If we generalize the time dependence as c j e −η j x+η 3 j y → c j e −η j x+η 3 j y+η −1 j z , Propositions 3.5 provides the general real-valued M-soliton solution of the vector sine-Gordon equation [33,34] (up to a sign ambiguity of the square root) with the independent variables x and z. Then, in the special case c j = (a j , ia j ) or
c j = c (1) j , ic(1)
j , c (2) j , ic (2) j , . . . , the condition c j , c j = 0 is automatically satisfied and our M-soliton formula apparently reduces to the formula proposed by Feng [54]. He investigated the asymptotic behavior of the two-soliton solution in this special case and showed that the two-soliton collision in the vector sine-Gordon equation is highly nontrivial, reflecting the internal degrees of freedom of the solitons. This apparently disagrees with the conclusion of [49] that the soliton interactions in the vector sine-Gordon equation are exactly the same as in the scalar sine-Gordon equation. Propositions 3.5 could be used to resolve the discrepancy.
Concluding remarks
The vector NLS equation known as the Kulish-Sklyanin model [5] admits two different Lax representations; using the standard Lax representation based on the generators of the Clifford algebra, one can easily solve the Kulish-Sklyanin model by applying the inverse scattering method or the Darboux transformations. However, the obtained exact solutions such as the N-soliton solution naturally involve the generators of the Clifford algebra satisfying the anticommutation relations and thus are not so useful for further analysis.
In this paper, we translated the standard Lax representation for the Kulish-Sklyanin model into the nonstandard one, not involving the generators of the Clifford algebra, and then applied the binary Darboux transformation. The N-fold binary Darboux transformation can also be formulated in simple explicit form, so we could obtain a classical expression for the general N-soliton solution of the Kulish-Sklyanin model (3.8), which is more useful for further investigation. By changing the time dependence of the linear eigenfunctions and considering a natural reduction, we could also obtain a general formula for the multisoliton (or multi-breather) solutions of the vector mKdV equation (2.24); by imposing some additional conditions, we obtained the real N-soliton solution of the vector mKdV equation.
Proposition 2 . 1 .
21The nonreduced Kulish-Sklyanin model with an arbitrary number of vector components:
Proposition 3. 1 .
1The spectral problem (2.20) is form-invariant under the action of the binary Darboux transformation defined as
Proposition 3. 2 .
2The spectral problem (2.20) is form-invariant under the action of the N-fold binary Darboux transformation defined as
Proposition 3. 3 .
3The bright N-soliton solution of the self-focusing Kulish-Sklyanin model (3.8) is given by
Proposition 3. 4 .
4An N-soliton solution of the vector mKdV equation(2.24)
η l x+η
F and G 1 , . . . , G N are polynomials in c j e −η j x+η 3 j y , c k e −η k x+η 3 k y for 1 ≤ j ≤ k ≤ N; this provides a real-valued N-soliton solution if η 1 , . . . , η N are positive and c 1 , . . . , c N are real. By setting N = 1, we obtainq = 2η 1 c 1 e −η 1 , c 1 e −2η 1 x+2η 3 1 y ,(3.21) which is the straightforward vector analog of the one-soliton solution of the scalar mKdV equation, i.e., the scalar mKdV soliton with a coefficient unit vector c 1 / c 1 , c 1 . Incidentally, the scalar mKdV equation was first solved by R. Hirota (J. Phys. Soc. Jpn.), M. Wadati (J. Phys. Soc. Jpn.) and S. Tanaka (Publ. RIMS & Proc. Japan Acad.) almost independently in 1972. The solution (3.20) in the case of real soliton parameters provides a nontrivial vector generalization of the N-soliton solution of the scalar mKdV equation involving N polarization vectors:
the soliton parameters satisfying (a) and (b) generates a mixture of multisoliton and multi-breather solutions. To exclude breather solutions, we need only impose the following additional conditions for (a) [49, 56]: (a') c j , c j = 0, j = 1, 2, . . . , M. Then, in the simplest nontrivial case of M = 1, L = 0 and N = 2, (3.20) gives the general one-soliton solution of the vector mKdV equation (2.24) [57]:
For general values of M and L, (3.20) with the above conditions (a), (a') and (b) provides the (M + L)-soliton solution, a nonlinear superposition of M vector solitons of the oscillating type (3.24) and L vector solitons of the nonoscillating type (3.21). Because the nonoscillating-type soliton can be obtained from the oscillating-type soliton through the limiting procedure, we can somewhat loosely consider that the general M-soliton solution of the vector mKdV equation (2.24) is obtained by setting N = 2M and L = 0 and assuming the conditions (a) and (a'). Thus, Proposition 3.4 can be restated as follows.
Proposition 3. 5 .
5The general real-valued M-soliton solution of the vector mKdV equation (2.24) is given byq = 1 (1, 2, . . . , 4M) 1≤j<k≤2M Γ(j, k) c j e −η j x+η 3 j y − c k e −η k x+η 3 k y + 1≤j<k≤2M Γ(j, 2M + k) c j e −η j x+η 3 j y + c k e −η k x+η 3 . . . ,4M)is the Pfaffian (a square root of the determinant) of the skew-symmetric matrix with the entries (j, k) = − (2M + j, 2M + k)
. . . , j − 1, β, j + 1, . . . , 4M) c j e −η j x+η 3 j y (1, 2, . . . , 4M) , with (β, k) = 1, k = 1, 2, . . . , 4M. For the generalized vector mKdV equation (3.22) with a real symmetric and positive definite matrix B, the above formula with c j , c k → c j B, c k and c j , c j = 0 → c j B, c j = 0 provides the general real-valued M-soliton solution.
Proposition 2.2. The third-order symmetry of the nonreduced Kulish-Sklyanin model (2.19) reads[47,48]
We propose the binary Darboux (or Zakharov-Shabat dressing) transformation[17][18][19][20][21] that can be applied to the spectral problem (2.20) associated with the Kulish-Sklyanin hierarchy. This can be obtained by considering how the binary Darboux transformation for the spectral problem (2.2) acts on P = Ψ 2 Ψ −1 1 under the reduction (2.5), confining the result to the linear span of {I, e 1 , e 2 , . . . , e 2m−1 } and then linearizing the discrete vector equation through the transformation (2.18).3 Darboux transformations and multisoliton
solutions
3.1 Darboux transformations
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| []
|
[
"Compactness and Guessing Principles in the Radin Extensions",
"Compactness and Guessing Principles in the Radin Extensions"
]
| [
"Omer Ben-Neria [email protected] ",
"Jing Zhang [email protected] "
]
| []
| []
| We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on κ, if κ is weakly compact, then ♦(κ) holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model relative to the existence of large cardinals, the diamond principle fails at a strongly inaccessible Mahlo cardinal. Refining the analysis of the Radin extensions, we consistently demonstrate a scenario where a compactness principle, stronger than the diagonal stationary reflection principle, holds yet the diamond principle fails at a strongly inaccessible cardinal, improving a result from [BN19].Theorem 1.1. Let RŪ be the Radin forcing defined using a measure sequencē U on κ. Then RŪ (κ is weakly compact =⇒ ♦(κ) holds ). | null | [
"https://arxiv.org/pdf/2105.01037v2.pdf"
]
| 233,481,285 | 2105.01037 | 8f89f724db36e70b3350b150d9e4474967d83eeb |
Compactness and Guessing Principles in the Radin Extensions
27 Feb 2022 March 1, 2022
Omer Ben-Neria [email protected]
Jing Zhang [email protected]
Compactness and Guessing Principles in the Radin Extensions
27 Feb 2022 March 1, 2022
We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on κ, if κ is weakly compact, then ♦(κ) holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model relative to the existence of large cardinals, the diamond principle fails at a strongly inaccessible Mahlo cardinal. Refining the analysis of the Radin extensions, we consistently demonstrate a scenario where a compactness principle, stronger than the diagonal stationary reflection principle, holds yet the diamond principle fails at a strongly inaccessible cardinal, improving a result from [BN19].Theorem 1.1. Let RŪ be the Radin forcing defined using a measure sequencē U on κ. Then RŪ (κ is weakly compact =⇒ ♦(κ) holds ).
Introduction
This paper contributes to the study of the interaction between compactness principles and guessing principles, specifically, in the context of Radin forcing [Rad82]. Recall that for a regular uncountable cardinal κ, ♦(κ) asserts the existence of a sequence S α ⊂ α : α < κ such that for any X ⊂ κ, {α < κ : X ∩ α = S α } is stationary. An old open problem in this area asks if ♦(κ) must hold at a weakly compact cardinal κ. We prove that this is indeed the case in the Radin forcing extensions, answering Question 37 from [BN19] negatively.
The special attention given to Radin forcing in this context, originates in results of Woodin [Cum], who used Radin forcing to establish the consistency of a large cardinal κ, such as strongly inaccessible, Mahlo, and greatly Mahlo, with ¬♦(κ). In fact, Radin forcing is the only known method for producing models where the diamond principle fails fully on any large cardinals.
The history of the relation between the diamond and compactness principles, goes back to the work of Kunen and Jensen [JK69], who showed that ♦(κ) must hold at every subtle cardinal. In fact, they prove that the stronger property ♦(Reg κ ) holds at such cardinals, where ♦(Reg κ ) asserts that there exists a diamond sequence supported on regulars. The consistency of ¬♦(Reg κ ) on weak compact cardinals was shown by Woodin, and improved by Hauser [Hau92] to indescribable cardinals, and by Džamonja and Hamkins [DH06] to strongly unfoldable cardinals. Each of these consistency results concerning the failure of the diamond principle on the regulars is established from its minimal corresponding large cardinal assumption. In contrast, ¬♦(κ) at relatively small large cardinals, such as Mahlo cardinals, is known to have a significantly stronger consistency strength. Jensen [Jen91] has shown that ¬♦(κ) at a Mahlo cardinal κ implies the existence of 0 # . Zeman [Zem00] improved the lower bound to the existence of an inner model with a cardinal κ, such that for every γ < κ, the set {α < κ | o(α) ≥ γ} is stationary in κ. The last large cardinal assumption is quite close to the hypermeasurability (large cardinal) assumptions used by Woodin to force ¬♦(κ) at a greatly Mahlo cardinal.
In [BN19] the first author studied Radin forcing RŪ , and the connection between properties of measure sequenceŪ on κ and large cardinal properties of κ in generic extensions by RŪ . It is shown that Woodin's construction of ¬♦(κ) can be extended to large cardinal properties such as stationary reflection principles. A question on whether we can extend the analysis to get a model of κ being weakly compact and ¬♦(κ) was asked in [BN19]. Theorem 1.1 answers this question in the negative by showing that this approach cannot yield significantly stronger results. Two properties of measure sequencesŪ which were isolated in [BN19], are the Weak Repeat Property (WRP) and Local Repeat Property (LRP). It is shown in [BN19] thatŪ satisfies WRP if and only if κ is weakly compact in generic extensions by RŪ , and asks if the stronger property of LRP has a similar characterization. We answer this question in Proposition 3.5, showing that LRP characterizes the large cardinal property of almost ineffability. In the rest of the paper we extend the study of compactness and guessing principles in Radin forcing extensions. The organization of the paper is as follows:
1. In Section 2, we provide some background on the type of forcing notions that we will work with for the rest of the paper.
2. In Section 3, we study characterizations of almost ineffable cardinals and weakly compact cardinals in the Radin extension based on certain properties of the measure sequence used to defined the forcing.
3. In Section 4, we isolate scenarios when a variety of guessing principles can hold in the Radin extensions.
4. In section 5, we demonstrate a scenario where a compactness principle, stronger than the diagonal stationary reflection principle, holds but the diamond principle fails at a strongly inaccessible cardinal.
Preliminaries
Measure sequences
A measure sequence is a sequencew = κ(w) ⌢ w(τ ) | τ < lh(w) , where each w(τ ) is a κ(w)-complete ultrafilter on V κ(w) , andw is derived from an elementary embedding j = jw : V → M with crit(j) = κ(w) in the sense that for A ⊆ V κ(w) and τ < lh(w),
A ∈ w(τ ) ⇐⇒w ↾ τ ∈ j(A).
In particular, w(0) is equivalent to the normal measure derived from j and lh(w) ≤ j(κ(w)). The class of all measure sequences is denoted by MS. For a set A ⊆ MS, we denote the set of critical points κ(w) forw ∈ A by O(A).
To clarify, for a measure sequencew and τ ≤ lh(w),w ↾ τ stands for κ(w) ⌢ w(η) : η < τ and w stands for η<lh(w) w(η).
All measure sequencesw in our constructions are assumed to satisfy MS ∩ V κ(w) ∈ w (see the discussion involving the setĀ in [Git10, page 1402], for further details).
Definition 2.1. LetŪ be a measure sequence on κ = κ(Ū ) constructed by j : V → M , and A ⊂ MS ∩ V κ . We say
1. A isŪ -measure-one if A ∈ Ū . 2. A isŪ -tail-measure-one if there is some γ < lh(Ū ) such that A ∈ U (i) for all γ < i < lh(Ū ). 3. A isŪ -positive if {i < lh(Ū ) : A ∈ U (i)} is cofinal in lh(Ū ). 4. A isŪ -stationarily-positive if {i < lh(Ū ) : A ∈ U (i)} is a stationary subset of lh(Ū ). 5. A isŪ -non-null if there is some γ < lh(Ū ), A ∈ U (γ).
Definition 2.2. We say a function b :
MS ∩ V κ → V κ 1. is a measure function if for anyw ∈ dom(b), b(w) ∈ w.
2. is a tail measure function if for anyw ∈ dom(b), b(w) isw-tail-measureone.
Radin forcing
Let RŪ be the Radin forcing defined usingŪ , constructed by j : V → M . This forcing was first invented in [Rad82]. The notations regarding the measure sequence and its constructing embedding are fixed for the rest of the paper unless otherwise stated. Our notations and presentation follow [BN19] or [Git10] for the most part, with the exception of using the forcing convention by which for two conditions p, q, p ≥ q means that q extends p (i.e., q is more informative). We refer the readers to the above for the definition of this forcing.
Notation 2.3. 1. For any A ∈ Ū , there exists another A ′ ⊂ A in Ū satisfying that for anyw ∈ A ′ , A ′ ∩ V κ(w) ∈ w.
We may without loss of generality assume all the measure one sets satisfy this property for the rest of this paper.
Conditions
p ∈ RŪ are finite sequences p = d i | i ≤ k where each d i is either of the form d i = κ i for some κ i < κ, or of the form d i = μ i , a i whereμ i is a measure sequence of length lh(μ i ) > 0 and a i ∈ ∩μ i . We denoteμ(d i ) = κ i in the former case andμ(d i ) =μ i in the latter. We require that the top componentμ(d k ) =Ū . We also write p = p 0 ⌢ d k , where p 0 = d i | i < k denotes the lower part of p, and d k = (Ū , A p ) denotes its top part.
3. Given a measure sequencew and aw-measure-one set c, we say a finite sequence of measure sequences with increasing critical points − → η is addable to (w, c), or − → η << (w, c), if for eachū ∈ − → η ,ū ∈ c and c ∩ V κ(ū) ∈ ū.
Remark 2.4. We clarify the following abuse of notations: ⌢ could mean concatenation or one-step extension depending on the context. However, our usage is without ambiguity:
1. If the object after ⌢ is a pair, for example p ⌢ (w, A) wherew is a measure sequence and A isw-measure-one, then ⌢ means concatenation. In this case, κ(w) > κ(ū) for any measure sequenceū appearing on p.
2. If the object after ⌢ is a measure sequence, for example p ⌢w , then this means it is a one-step extension. In this case, κ(w) belongs to the measure one set A pi in one of the components p i of p.
Definition 2.5.Ū satisfies the Repeat Property (RP) if there exists γ < lh(Ū ) such that Ū ↾ γ = Ū . We say γ is a repeat point forŪ and γ witnessesŪ satisfies RP.
Definition 2.6. We say γ < lh(Ū ) is a weak repeat point if γ witnessesŪ ↾ γ +1 satisfies RP. If γ is not a weak repeat point, then it is novel.
Mitchell [Mit82] and Cummings-Woodin [Cum] independently proved that κ is measurable in V RŪ if and only if ∩Ū = ∩Ū ′ for some measure sequenceŪ ′ satisfying the RP.
For the rest of the paper, we may assumeŪ does not satisfy RP. The reason is that suppose γ is the first repeat point ofŪ , then RŪ is forcing equivalent to RŪ ↾γ (as by definition the forcing only depends on Ū = Ū ↾ γ; see [CW]) andŪ ↾ γ does not satisfy RP. Notice that ifŪ does not satisfy RP, then there are unboundedly many γ < lh(Ū ) that are not weak repeat points.
Remark 2.7. IfŪ does not satisfy RP and cf (lh(Ū )) ≤ κ, then in V RŪ , κ becomes singular. This follows from the arguments in [Git10].
Remark 2.8. Suppose |2 κ | M does not divide lh(Ū ), then there exists some γ < lh(Ū ), such that for all γ ′ ∈ (γ, lh(Ū )), γ is not a weak repeat point. We sketch this when lh(Ū ) < |2 κ | M . Fix some function f on κ such that for each α < κ, f (α) outputs a well ordering of 2 α of length |2 α |. For any γ < lh(Ū ), let
A ⊂ κ be such that j(f )(κ)(γ) = A. Consider the set B γ = {w ∈ V κ ∩ MS : f (κ(w))(lh(w)) = A ∩ κ(w)}. It is easy to see that B γ ∈ U (γ) − γ ′ <γ U (γ ′ ).
In general without assuming lh(Ū ) < |2 κ | M , by the assumption there exists some δ and 0
< ρ < |2 κ | M such that lh(Ū ) = δ·|2 κ | M +ρ. Then for δ·|2 κ | M ≤ γ < lh(Ū ) we modify the definition of B γ such that it containsw ∈ V κ ∩ MS such that f (κ(w))(lh(w) mod |2 κ(w) |) = A ∩ κ(w).
Definition 2.9. For any generic G ⊂ RŪ over V , let
MS G = {w ∈ MS | ∃p ∈ G, p = d i : i ≤ k ,w =μ(d i ) for some i < k}. C G = {κ(w) |w ∈ MS G } We say A ⊂ κ is generated by a set in V if there exists B ⊂ MS such that A = O(B ∩ MS G ) = def {κ(w) :w ∈ B ∩ MS G }.
The following theorem asserts that any club in the Radin extensions V [G], G ⊆ RŪ , where the regularity of κ is preserved, contains a club generated by ā U -tail-measure-one set in V .
Theorem 2.10 ([BN19]). IfŪ satisfies cf (lh(Ū )) ≥ κ + , where κ = κ(Ū ), then given p ⌢ 0 (Ū , A) = p τ is a club subset of κ, there exists a measure one set A ′ ⊂ A and aŪ -tail-measure-one set Γ such that p ⌢ 0 (Ū , A ′ ) forces O(Γ∩G) ⊂τ . The converse is also true:
Theorem 2.11 ([BN19]
). SupposeŪ is a measure sequence not satisfying RP and B ⊂ MS is aŪ -tail-measure-one set.
Then in V [G] where G ⊂ RŪ is generic over V , O(B ∩ MS G ) contains a club subset of κ.
The following result shows that Theorem 2.10 about clubs in Radin generic extensions does not extend to stationary sets.
Proposition 2.12. Suppose lh(Ū ) ≤ |2 κ | M and cf (lh(Ū )) ≥ κ + , then in V RŪ , there exists a partition c : κ → κ, such that for any unbounded A ⊂ C G generated by a set in V , it is the case that c ′′ O(A) = κ. In particular, O(B) is not chomogeneous for anyŪ -non-null set B.
Proof. Let (x α l : l < |2 α |) be an injective enumeration of 2 α for α < κ.
Let (x κ l : l < |2 κ | M ) = j( (x α l : l < |2 α |) : α < κ )(κ). For α ≤ κ and τ < (2 α ) M , define A α τ = {v : x κ(v) lh(v) = x α τ ∩κ(v)}. Note that A κ(w)
lh(w) ∈ w, forŪ -measure-one manyw. To see this, it suffices to see that for any τ < lh(Ū ),
A κ τ ∈ Ū ↾ τ . Fix τ ′ < τ , we need to see that A κ τ ∈ U (τ ′ ), which is equivalent toŪ ↾ τ ′ ∈ j(A κ τ ). Since x κ τ ′ = x κ τ = j(x κ τ ) ∩ κ, by the definition of A κ τ , we indeed have that U ↾ τ ′ ∈ j(A κ τ ). Define A α ≥τ as follows:v ∈ A α ≥τ iffv ∈ A α τ or some initial segment ofv is in A α τ . In V [G] where G ⊂ RŪ is generic over V , define a mapping c ′ : MS G → MS G such thatw is mapped by c ′ to the maximal element in MS G ∩ A κ(w) ≥lh(w)
if it exists. We show, using a density argument, that c ′ has the following property: for
any A ∈ U (γ) and B ∈ U (τ ) where γ ≤ τ , c ′ [A ∩ MS G ] ∩ B = ∅. Given p = p 0 ⌢ (Ū , E) ∈ RŪ , we findw from (E ∩ B ∩ A κ ≥γ ) − V κ(max(p0)
)+1 and extend p to p ⌢w . The reason why suchw exists is that A κ ≥γ ∈ γ≤i U (i).
Then we find someū ∈ A ∩ E − V κ(w)+1 such that A κ ≥γ ∩ Vū = A κ(ū)
≥lh(ū) , and then we extend to p 0
⌢ (w, E ∩ V κ(w) ) ⌢ (ū, E − A κ(ū) ≥lh(ū) ) ⌢ (Ū , E), which forceṡ c ′ (ū) =w.
Finally, in V , if we let * : MS → κ be such that for any α ∈ κ,
( * ) −1 (α) is U -positive, then in V [G], c = def * • c ′ is as desired, namely, for any A ⊂ MS that isŪ -non-null, c[A ∩ MS G ] = κ.
The next lemma is a well-known fact that concerns getting nice representations for certain sets in the Radin forcing extension.
Lemma 2.13. Suppose p ∈ RŪ and a sequence of names ẋ α ⊂ α : α < κ are given. Then there exists an extension q ≤ p with q 0 = p 0 and a function f ∈ V , such that for anyw ∈ A q , f (w) is a Rw-name for a subset of κ(w) and q ⌢w f (w) =ẋ κ(w) .
Proof. For eachw ∈ A p , we can find aŪ -measure-one Aw ⊂ A p and a Rw-name
f (w) such that p 0 ⌢ (w, A p ∩ V κ(w) ) ⌢ (Ū , Aw) forces f (w) =ẋ κ(w) . To see this, consider D = def {t ∈ Rw : t ≤ Rw p 0 ⌢ (w, A p ∩ V κ(w) ), ∃B t ∈ Ū , ∃Rw-nameṡ t , t ⌢ (Ū , B t ) ṡ t =ẋ κ(w) }. Observe that D is a dense subset of Rw below p 0 ⌢ (w, A p ∩V κ(w) ): given any s ∈ Rw extending p 0 ⌢ (w, A p ∩ V κ(w) ), since RŪ /p ⌢w ≃ Rw/p 0 ⌢ (w, A p ∩ V κ(w) ) × RŪ /(Ū , A p − V κ(w)+1 ) and (RŪ /(Ū , A p − V κ(w)+1 ), ≤ * ) is (2 |Rw| ) + -closed (recall that ≤ *
refers to the direct extension ordering), we can find t ≤ Rw s and an Rw-nameṡ t as well as some B t ∈ Ū , such that (Ū , B t ) RŪ t Rwẋκ(w) =ṡ t . This shows that D is dense in Rw.
Let D ′ ⊂ D be a maximal antichain below p 0 ⌢ (w, A p ∩ V κ(w) ). Then we can cook up a Rw-name f (w) for a subset of κ(w) such that t f (w) =ṡ t for any t ∈ D ′ . Now it is immediate that f (w) and Aw = t∈D B t satisfy the requirement. Finally, we let q ≤ p be such that q 0 = p 0 and A q = ∆w ∈A p Aw, which is the desired extension of p as sought.
An immediate consequence of the previous lemma is:
Corollary 2.14. Let G ⊂ RŪ be generic over V . Then in V [G], for any X ⊂ κ, there exists f ∈ V such that for everyw ∈ dom(f ) ∩ MS, f (w) is an Rw-name for a subset of κ(w) and there exists α < κ, (f (w)) G↾Rw = X ∩ κ(w) for each w ∈ MS G − V α .
We finish this section with stating the following theorem.
Theorem 2.15 ([CW]). Let G ⊆ RŪ be a V -generic filter. If κ is regular in V [G], then in V [G], for every subset X ⊆ κ, if X ∩ α ∈ V for all α < κ then X ∈ V .
For a published account of the proof, the reader can consult [BN19].
Mild large cardinals in Radin extensions
In this section, we revisit two weakenings of the Repeat Property about the measure sequenceŪ , as considered in [BN19]. In particular, each property corresponds to κ being a certain large cardinal in the generic extension by RŪ . SinceŪ does not contain a repeat point, it is not hard to check that if U satisfies LRP, then cf (lh(Ū )) > κ. To see this, suppose for the sake of contradiction that cf (lh(Ū )) = δ ≤ κ. Let γ i : i < δ be an increasing sequence cofinal in lh(Ū ). For each i < δ, by the assumption that γ i is not a weak repeat point, there is some
Almost ineffable cardinals
A i ∈ U (γ i ) − α<γi U (α). Define the following measure function b: for a measure sequencew, if there exists some i < κ(w) such that A c i ∩ V κ(w) ∈ w, then define b(w) = A c i ∩ V κ(w)
for the least i as above. If there does not exist such i, then b(w) = V κ(w) . By the assumption thatŪ satisfies LRP, there exists some γ < lh(Ū ) and
A ∈ Ū such that j(b)(Ū ↾ γ) = A. By the definition of b, j(b)(Ū ↾ γ) equals A c i where i is the least satisfying γ i ≥ γ. However, A = A c i since A ∈ U (γ i ) while A c i ∈ U (γ i ), which is a contradiction.
Remark 3.2. One obtains an equivalent definition by replacing "...γ < lh(Ū ) such that..." with "...cofinally many γ < lh(Ū ) such that..." in Definition 3.1.
In [BN19], it was shown that if lh(Ū ) < 2 κ , thenŪ does not satisfy LRP, thus making LRP incompatible with Woodin's argument for the failure of ♦(κ) in RŪ -generic extension (see Theorem 4.5). [BN19] asks if LRP is equivalent to a natural large cardinal property of κ in RŪ generic extensions. We show next that this is the case. This shows that it is not an accident that LRP is incompatible with the argument for the failure of ♦(κ).
Definition 3.3 ([JK69]). Fix a regular cardinal κ. 1. κ is almost ineffable if for any sequence t α ⊂ α : α < κ , there is some t ⊂ κ such that {α : t ∩ α = t α } is unbounded in κ.
2. κ is subtle if for any sequence t α ⊂ α : α < κ and any club C ⊂ κ, there are α < β ∈ C such that t α ⊏ t β .
Remark 3.4. It is easy to check that κ is almost ineffable iff for any sequence t α ⊂ α : α < κ and any club C ⊂ κ, there is some t ⊂ κ such that {α ∈ C : t ∩ α = t α } is unbounded in κ. As a result, if κ is almost ineffable, then κ is subtle, which in turn implies ♦(κ) by [JK69].
Proposition 3.5. V RŪ |= κ is almost ineffable iffŪ satisfies LRP.
Proposition 3.5 follows from the following two lemmas.
Lemma 3.6. IfŪ satisfies the LRP, then κ is almost ineffable in V RŪ Proof. Let p ′ ∈ RŪ and a name ṡ α ⊂ α : α < κ be given. By Lemma 2.13, we can get p = p ⌢ 0 (Ū , A) ≤ * p ′ and ṫw :w ∈ MS such thatṫw is an Rw-name and for anyw ∈ A, p ⌢w ṫw =ṡ κ(w) .
For eachw ∈ A, apply Lemma 2.13 to p 0 ⌢ (w, A∩V κ(w) ) and Rw. As a result, we can find b(w) ∈ w, b(w) ⊂ A, such that for anyv ∈ b(w), it is the case that
p ⌢ 0 (v, b(w) ∩ V κ(v) ) ⌢ (w, b(w)) Rwṫw ∩ κ(v) = hw(v), where hw is a function defined on MS ∩ V κ(w) with hw(v) being a Rv-name for eachv ∈ dom(hw). Let h denote hw :w ∈ MS ∩ V κ .
By the hypothesis thatŪ satisfies LRP, we can find τ < lh(Ū ) such that
B = j(b)(Ū ↾ τ ) ∈ Ū . Note B ⊂ A. We may assume τ ≥ 2 by Remark 3.2. Let h = j(h)(Ū ↾ τ ), then h is a function from MS ∩ V κ to V κ , takingv to h(v),
which is an Rv-name for a subset of κ(v).
Consider
the set C = {w ∈ B : lh(w) > 1, B ∩ V κ(w) ⊂ b(w), h ↾ V κ(w) = hw} ∈ U (τ ). Let G be a generic filter for RŪ over V containing p ⌢ 0 (Ū , B) and in V [G]
, we define a sequence of bounded subsets of κ as follows: for
eachw ∈ C ∩ MS G , let dw be the union of the interpretation (h(v)) G for all v ∈ MS G ∩ V κ(w) .
Claim 3.7. For anyw ∈ C ∩ MS G , dw is well-defined, and it is equal to
tw = def (ṫw) G . Proof of the Claim. For anyv <v ′ both in MS G ∩ V κ(w) ∩ B − V rank(p0)+1 , we note that (hw(v)) G , (hw(v ′ )) G ⊏ tw.
This is because the statement is forced by
p ⌢ 0 (v, b(w)∩V κ(v) ) ⌢ (w, b(w)) (respec- tively, by p ⌢ 0 (v ′ , b(w)∩V κ(v ′ ) ) ⌢ (w, b(w))), extended by p ⌢ 0 (v, B∩V κ(v) ) ⌢ (w, B∩ V κ(w) ) by the fact thatw ∈ C. Claim 3.8. Forw <w ′ ∈ MS G ∩ C, it is the case that dw ⊏ dw′ .
Proof of the Claim. Reason as above to check that for anyv
∈ MS G ∩ V κ(w) ∩ B − V rank(p0)+1 , (h(v)) G = (hw(v)) G = (hw′(v)) G is an initial segment of both dw and dw′. Let X = w∈MSG∩C dw. Then we are done. Namely, in V [G], {α < κ : X ∩ α = (ṡ α ) G } is unbounded in κ. Lemma 3.9. If κ is almost ineffable in V RŪ , thenŪ satisfies LRP. Proof. Fix a measure function b. Let G ⊂ RŪ be generic over V . In V [G], for eachw ∈ G, let γw < κ(w) be some ordinal such that MS G − V γw ⊂ b(w). Consider the sequenceX = Xw = def {γw} ∪(b(w)− V γw +1 ) :w ∈ MS G . Note thatX ⊂ V .
As κ is almost ineffable, by Remark 3.4, we can find X such that {w ∈ MS G : X ∩V κ(w) = Xw} has size κ. In particular, X is fresh. By Theorem 2.15, we know that X ∈ V . As a result, there is some γ, such that there are unboundedly mamyw with γw = γ and
X ∩ V κ(w) = {γw} ∪ (b(w) − V γw+1 ). As a result, X contains a tail of MS G , which in turn implies X ∈ Ū .
Finally, we need to check that there is some τ < lh(Ū ), such that j(b)(Ū ↾ τ ) = * X. More precisely, we mean here that the symmetric difference of two sets j(b)(Ū ↾ τ ) and X is a subset of V α for some α < κ. This is as desired, since
X ∈ Ū implies that j(b)(Ū ↾ γ) ∈ Ū . But this immediately follows from the fact that in V [G], {κ(w) :w ∈ MS G and X ∩ V κ(w) = * b(w)} is unbounded in κ.
Weakly compact cardinals
Definition 3.10 ([BN19], Definition 21). We sayŪ satisfies the Weak Repeat Property (WRP) if every measure function b has a repeat filter W with respect toŪ in the following sense:
1. W is a κ-complete filter extending the co-bounded filter on MS, 2. any X ∈ W isŪ -non-null,
3. W measures b, namely for eachū, if we let X b,ū = def {v :ū ∈ b(v)}, then either X b,ū or X c b,ū ∈ W and 4. [b] W = def {ū : X b,ū ∈ W } ∈ Ū .
In [BN19], it was shown that κ is weakly compact in V RŪ iffŪ satisfies WRP. We supply more equivalent characterizations of weak compactness in Radin extensions, which will be useful later on.
Proposition 3.11. The following are equivalent:
1.Ū satisfies WRP.
2. V RŪ |= κ is weakly compact.
3. V RŪ |= κ is inaccessible and the V -tree property holds. Namely, any κ-tree T , satisfying that T ⊂ V , which means exactly the following:
• the underlying set is a set of sequences, each of which is in V and • is ordered by end-extension, admits a cofinal branch.
4. for any measure function b, there exists A ∈ Ū such that for any α < κ,
there exists some β < lh(Ū ), with A ∩ V α ⊂ j(b)(Ū ↾ β).
Proof. The equivalence between 1 and 2 is proved in [BN19] and it is immediate that 2 implies 3.
• 3 implies 4: fix a measure function b. Let G ⊂ RŪ be generic over V .
In V [G], the functionw ∈ MS G → γw, where γw is the least γ < κ(w) such that MS G ∩ V κ(w) − V γ ⊂ b(w), is regressive. Therefore, there exists some γ, such that B = {w ∈ G : γw = γ} is stationary. More precisely, O(B ∩ MS G ) is stationary in κ. Define a κ-tree T such that t ∈ T iff there exists somew ∈ B such that t ⊑ b(w) (namely there exists some α ≤ κ(w) such that t = b(w) ∩ V α ). The order of T is end extension. More precisely, t ≤ T t ′ if there exists some α such that t = t ′ ∩ V α . Notice that T ⊂ V .
Apply the hypothesis, we can get a branch d through T . By Theorem 2.15, d ∈ V .
Observe that d ∈ Ū . To see this, it suffices to see d contains G − V γ . For anyū ∈ G − V γ , d ∩ V κ(ū)+lh(ū)+1 ⊏ b(w) for somew ∈ B. By the choice of γ, we know thatū ∈ b(w). Henceū ∈ d.
Finally, it remains to show that for any α < κ, E = def {w : d ∩ V α ⊂ b(w)} isŪ -non-null. But this follows immediately from the fact that O(E ∩ MS G ) is unbounded in κ.
• 4 implies 1: given a measure function b, apply 4 to get A ∈ Ū with the property. We first show a strengthening of the conclusion at 4 is possible.
Claim 3.12. We can find an A as in the conclusion of 4 that satisfies for any α < κ, there exists some
β < lh(Ū ), with A ∩ V α ⊏ j(b)(Ū ↾ β).
Proof of the Claim. Note the difference is "⊂" is replaced with "⊏". Let A ′ be given by the conclusion of 4. For each α < κ, let β α < lh(Ū ) be the least witnessing ordinal that
A ′ ∩ V α ⊂ j(b)(Ū ↾ β α ). Consider the tree on V κ defined by T = {j(b)(Ū ↾ β α ) ↾ V γ : γ ≤ α < κ}, ordered by end extension.
By the weak compactness of κ in V , there exists a branch A through the tree T . Clearly, for each α < κ, there is some β < lh(Ū ), such that A ∩ V α ⊏ j(b)(Ū ↾ β) by the definition of the tree. Note also that A ∈ Ū , as evidenced by the fact that A ′ ⊂ A. To see the latter, given α < κ, we can find some
α ′ ≥ α such that A ∩ V α ⊏ j(b)(Ū ↾ β α ′ ). In particular, A ∩ V α ⊃ A ′ ∩ V α . Define W ′ ⊂ MS such that B ∈ W ′ iff there is someū ∈ A with B = X b,ū or there is someū / ∈ A with B = X c b,ū . Let F ⊂ MS be the co-bounded filter. Let W be the upward closure of { B∈u B : u ∈ [W ′ ∪ F ] <κ }.
To check W as defined is as desired, we verify that W ⊂ Ū is a κ-complete filter. It suffices to show that for any B i : i < µ ⊂ W ′ where µ < κ, it is the case that i<µ B i isŪ -non-null, since F ⊂ Ū . For each i < µ, by the definition of W ′ , there is someū i witnessing that B i ∈ W ′ . Namely, either
u i ∈ A and B i = X b,ūi orū i ∈ A and B = X c b,ūi . Let α > sup i<µ κ(ū i ). Claim 3.12 implies that {v : A ∩ V α ⊏ b(v)}, as a subset of i<µ B i , is U -non-null.
Remark 3.13. In item 4 above, we get an equivalent statement by replacing "...there exists some β < lh(Ū )..." with "...there exist unboundedly many β < lh(Ū )...".
Lemma 3.14. If |2 κ | M does not divide lh(Ū ) as ordinals, then V RŪ |= κ is not weakly compact.
Proof. Assume first that lh(Ū ) < |2 κ | M . We will indicate later how to deal with the general case. Define g such that g(α) = x α l : l < |2 α | is an injective enumeration of subsets of α. Denote j(g)(κ) = x κ l : l < |2 κ | M . Let A = x κ lh(Ū ) . In general for any β < κ and any set B ⊂ β, denote index β g (B) to be the unique l such that B = x β l . Note for a fixed β and B as above, index β g (B) only depends on g(β).
Define a measure function b such that
b(w) = {ū ∈ V κ(w) : lh(ū) < index κ(ū) g (x κ(w) lh(w) ∩ κ(ū)) < index κ(ū) g (A ∩ κ(ū))},
whenever the set is in w and otherwise b(w) = V κ(w) .
Let us call the second case in the definition of b vacuous. Observe that {w ∈ V κ : b(w) is not vacuously defined} ∈ Ū . To see this, let E = def {ū ∈ V κ :
lh(ū) < index κ(ū) g (x κ τ ∩ κ(ū)) < index κ(ū) j(g) (j(A) ∩ κ(ū)) = index κ(ū) g (A ∩ κ(ū))}. We show that E ∈ Ū ↾ τ , which implies j(b)(Ū ↾ τ ) = E. Given γ < τ , we know thatŪ ↾ γ ∈ j(E) iff x κ τ = x κ β , for some β satisfying γ < β < index κ j(g) (A) = lh(Ū )
. The latter is true as evidenced by β = τ . Suppose for the sake of contradiction that κ is weakly compact in V RŪ . Apply 4 in Proposition 3.11 to b to get B ∈ Ū such that:
for any α < κ, there is β < lh(Ū ) with B ∩ V α ⊂ j(b)(Ū ↾ β). Apply j again to the statement above, we get (in M ):
for any α < j(κ), there is β < lh(j(Ū )) with j(B) ∩ V α ⊂ j(j(b))(j(Ū ) ↾ β). In particular, working in M , if we consider α = (2 κ ) +M < j(κ), then we can find some τ < j(lh(Ū )) such that j
(B) ∩ (V α ) M ⊂ j(j(b))(j(Ū ) ↾ τ ) = {ū ∈ j(V κ ) : x j(κ) τ ∩κ(ū) = x κ(ū) β
, lh(ū) < β < index κ(ū) j(j(g)) (j(j(A))∩κ(ū))}. (1) For the sake of consistency, we use x j(κ) l : l < dom[j(j(g))(j(κ))] to denote the enumeration of j(j(g))(j(κ)). Since lh(Ū ) < dom(j(g)(κ)), the elementarity of j implies that j(lh(Ū )) < dom(j(j(g))(j(κ))). Thus the sentence above makes sense.
We show first that there exists some γ < lh(Ū ) such that x
j(κ) τ ∩ κ = x κ γ . To see this, fix some ξ < lh(Ū ). SinceŪ ↾ ξ ∈ j(B) ∩ (V α ) M , by (1), we know x j(κ) τ ∩ κ = x κ γ , for some γ > ξ and γ < index κ j(j(g)) (j(j(A)) ∩ κ) = index κ j(j(g)) (A) = index κ j(j(g)↾κ) (A) = index κ j(g) (A) = lh(Ū ). Finally, fix some ξ ′ > γ and ξ ′ < lh(Ū ) such thatŪ ↾ ξ ′ ∈ j(B) ∩ (V α ) M .
Reasoning as above, we can find γ ′ > ξ ′ > γ with γ ′ < lh(Ū ) such that x j(κ) τ ∩ κ = x κ γ ′ . But x κ γ = x κ γ ′ by the injectivity of the enumeration. This is a contradiction.
In general without assuming lh(Ū ) < |2 κ | M , we modify as follows. First notice that lh(Ū ) ≤ j(κ) and j(κ) is divisible by |2 κ | M . Hence, we may assume lh(Ū ) < j(κ). By Remark 2.8, there exists some µ < lh(Ū ) so that lh(Ū ) = µ+δ with δ < |2 κ | M , and U (µ) is novel. It follows that there is some D ⊆ MS ∩ V κ so that D ∈ U (i) if and only if i < µ. For eachw ∈ MS ∩ V κ let i D (w) be least value i < lh(w) so that D ∩ V κ(w) ∈w(i) if such i exists, and leave i D (w) undefined otherwise. If i D (w) is defined then clearly i D (w) < lh(w), and we set δ D (w) be the unique δ for which i D (w) + δ = lh(w).
Define g as before and
A = x κ δ . Define b such that b(w) = (D ∩ V κ(w) ) ∪ {ū ∈ V κ(w) ∩ D c : x κ(w) δD (w) ∩ κ(ū) = x κ(ū) β where δ D (ū) < β < index κ(ū) g (A ∩ κ(ū))},
whenever the set is in w and otherwise b(w) = V κ(w) .
Again observe that {w ∈ V κ : b(w) is not vacuously defined} ∈ Ū . Apply 4 in Proposition 3.11 to b to get B ∈ Ū as before. If we let α = (lh(Ū )+2 κ ) +M , then we can find some τ < j(lh(Ū )) such that j
(B) ∩ (V α ) M ⊂ j(j(b))(j(Ū ) ↾ τ ). The choice of α ensures that {Ū ↾ ν : ν < lh(Ū )} ⊂ (V α ) M . Fix some ξ < lh(Ū ) such that ξ > µ. SinceŪ ↾ ξ ∈ j(B) ∩ (V α ) M , it follows that U ↾ ξ ∈ j(j(b))(j(Ū ) ↾ τ ). Since j(j(D)) ∩ j(V κ ) = j(j(D) ∩ V κ ) = j(D), we know thatŪ ↾ ξ ∈ j(j(D)) sinceŪ ↾ ξ ∈ j(D) andŪ ↾ ξ ∈ j(V κ )
. As a result, by looking at the definition of b, we must have
x j(κ) j(j(δD ))(j(Ū )↾τ ) ∩ κ = x κ γ , where j(δ D )(Ū ↾ ξ) = ξ − µ < γ < δ.
Fix another ξ ′ < lh(Ū ) such that ξ ′ − µ > γ. Repeating the argument as in the case where lh(Ū ) < |2 κ | M , we get some γ ′ > γ such that x j(κ) j(j(δD ))(j(Ū )↾τ ) ∩ κ = x κ γ ′ . This contradicts with the fact that x κ γ = x κ γ ′ .
Remark 3.15. The proof in Lemma 3.14 actually produces a measure function b such that there is noŪ -positive A, such that for any α < κ, there is β < lh(Ū ),
A ∩ V α ⊂ j(b)(Ū ↾ β).
Remark 3.16. Similar proof as in Lemma 3.14 shows κ is not weakly compact in V RŪ if there is some ν such that lh(Ū ) = ν + |2 κ | M .
Guessing principles in Radin extensions
Definition 4.1. Let V ⊆ W be two transitive models of set theory, with κ being a regular cardinal in W and S ⊂ κ being a stationary subset. We say
• V -♦(S) holds (in W ) if there exists s α ⊂ α : α < κ such that for all X ⊂ κ in V , it is the case that {α ∈ S : X ∩ α = s α } is stationary. • V -♦ − (S) holds (in W ) if there exists S α ∈ [P(α)] ≤|α| : α < κ such that for all X ⊂ κ in V , it is the case that {α ∈ S : X ∩ α ∈ S α } is stationary.
Cummings and Magidor [Cum13] showed that V -diamonds are equivalent to diamonds at κ in Radin generic extensions of V preserving the regularity of κ. Proof. A well-known result of Kunen (Theorem III.7.8 in [Kun11]) states that ♦ − (S) holds iff ♦(S) holds for any regular κ and any stationary set S ⊂ κ. Along with Theorem 4.2, we know
V RŪ |= ♦(κ) iff V RŪ |= ♦ − (κ) iff V RŪ |= V - ♦ − (κ) iff V RŪ |= V -♦(κ).
• Suppose there exists some p ∈ RŪ forcing that ṡ α : α < κ is a V -♦(κ)sequence. By Lemma 2.13, we may find q ≤ p and g such that for anȳ w ∈ A q , g(w) is an Rw-name for a subset of κ(w) in V and q ⌢w g(w) = s κ(w) . Since Rw is κ(w) + -c.c, we can find f (w) ∈ [2 κ(w) ] ≤κ(w) such that 1 Rw g(w) ∈ f (w). Suppose X ⊂ κ, then we know that we can find some r ≤ q such that r "{κ(w) :w ∈Ġ, X ∩ κ(w) = g(w) ∈ f (w)} is stationary". This implies that {w : X ∩ κ(w) ∈ f (w)} isŪ -positive by Theorem 2.11.
• For the other direction, suppose there is one such f , then it is easy to verify that if G ⊂ RŪ is generic over V , then {f (w) :w ∈ G} is the desired V -♦ − (κ)-sequence.
The last observation allows us to reproduce the following result of Cummings and Magidor. More precisely, if |2 κ | M divides lh(Ū ), it suffices to find a function f as in Observation 4.3. For each β < κ, we let x β l : l < 2 β enumerate all subsets of β. It is not hard to verify thatw → {x κ(w) lh(w) mod 2 κ(w) } is the function as sought.
Theorem 4.4 (Cummings-Magidor, [Cum13]). If |2 κ | M divides lh(Ū ), then V RŪ |= ♦(κ).
We are ready to prove the main result of the paper.
Proof of Theorem 1.1. LetŪ be a measure sequence on κ constructed by j : V → M and suppose V RŪ |= κ is weakly compact. By Lemma 3.14, |2 κ | M divides lh(Ū ). Hence, Theorem 4.4 implies ♦(κ) holds in V RŪ .
The following is a well-known theorem of Woodin (see [Cum] or [BN19] for published proofs). Proof. LetṠ be a name for a stationary subset of κ and p = p ⌢ 0 (Ū , A) be a condition forcing this, which furthermore satisfies the conclusion of Theorem 4.7. We will construct a V -♦ − (S)-sequence in V RŪ . By Lemma 4.2 and a theorem of Kunen, we know ♦(S) holds in V RŪ .
For each α, we fix x α l : l < |2 α | an enumeration of subsets of α with cofinal repetitions. For each Z ⊂ MS, we define f Z (w) = otp({ξ < lh(w) :w ↾ ξ ∈ Z}).
In V [G], we define a V -♦ − (S)-sequence as follows: ifw ∈ MS G and κ(w) ∈ S, then define S κ(w) to consist of x κ(w) l for those l such that there is some − → η ∈ A <ω ∩ V κ(w) such thatw ∈ Z p0 ⇂ − → η and l = (f Zp 0 ⇂ − → η (w) mod 2 κ(w) ) < 2 κ(w) .
Note that by the definition for eachw ∈ MS G with α = κ(w), it is true that |S α | ≤ |α|.
Let's check that this is as desired. Suppose not, then there exists an extension q of p and X ⊂ κ and a tail measure one Γ as witnessed by ν < lh(Ū ) such that q forces that X is not guessed at any point atṠ
∩ O(MS G ∩ Γ). Write q as p ′ 0 ⌢ ( − → η , − → A − → η ) ⌢ (Ū , B ′ ),
where there is somev with p 0 , p ′ 0 ∈ Rv and p ′ 0 ≤ Rv p 0 . Here − → η is an increasing sequence of measure sequence and
− → A − → η
is a sequence of measure one sets with respect to the measure sequences on − → η . By the choice of p, we know that Z p0 ⇂ − → η isŪ -positive. Let l < 2 κ be such that x κ l = X and j(f
Zp 0 ⇂ − → η )(Ū ↾ ξ) = l (mod 2 κ ) for someŪ ↾ ξ ∈ j(Γ∩Z p0 ⇂ − → η ). By elementarity, there is someū reflectingŪ ↾ ξ, namely, there is someū ∈ Γ∩Z p0 ⇂ − → η such that if l = f Zp 0 ⇂ − → η (ū)(mod 2 κ(ū) ), then x κ(ū) l = X ∩κ(ū). Since − → η << (ū, b(ū)), we know that p ′ 0 ⌢ ( − → η , − → A − → η ) ⌢ (ū, b(ū)) ⌢ (Ū , B ′ ) is compatible with p ⌢ 0 (ū, b(ū)) ⌢ (Ū , A)
. Hence a common extension will force κ(ū) ∈Ṡ, as well asū ∈ Γ and X ∩ κ(ū) ∈Ṡ α , contradicting with our assumption.
Remark 4.9. Proposition 4.8 is optimal in the following sense: if M |= cf (lh(Ū )) < 2 κ andŪ ∈ M , then in V RŪ there exists a stationary set S such that ♦(S) fails. It suffices to show there exists aŪ -positive set B such that M |= "{τ : B ∈ U (τ )} has size < 2 κ ". Then we can finish with the argument of Theorem 4.5.
Recall j : V → M constructsŪ . By a theorem by Cummings-Woodin [CW], we may assume that M = {j(f )(Ū ↾ ξ) : ξ is novel}. The point is that any measure sequence is equivalent, in the sense of the Radin forcing defined from it, to another one constructed by the embedding formed by taking the limit ultrapower along the novel points on the measure sequence.
By the simplification, we can find f and a novel ξ < lh(Ū ) such that j(f )(Ū ↾ ξ) = E, which is cofinal in lh(Ū ) and of size < 2 κ , as computed in M . Let
A ∈ U (ξ) − Ū (< ξ). Consider B = {ū : if ξ is the least such thatū ↾ ξ ∈ A, lh(ū) ∈ f (ū ↾ ξ)}.
We claim for τ > ξ, B ∈ U (τ ) iff τ ∈ E. If τ > ξ, andŪ ↾ τ ∈ j(B), then by the definition, ξ is the least such thatŪ ↾ ξ ∈ j(A), then τ has to be in j(f )(Ū ↾ ξ) = E. Conversely, ifŪ ↾ τ is not in j(B), then it must be that τ ∈ E by the same reasoning.
Definition 4.10. We say the Strong Club Guessing (SCG) holds on S ⊂ κ if there exists a sequence t α ⊂ α : α ∈ S such that for any club C ⊂ κ, there exists another club D, satisfying that for any α ∈ D ∩ S, t α is unbounded in α and t α ⊂ * C (namely, there exists some γ < α such that t α − γ ⊂ C).
By taking closure, we could also assume t α ⊂ α is closed for each α < κ.
Proposition 4.11. In V RŪ where κ is regular, SCG holds on the set of singular cardinals below κ. Proof. Assume cf (lh(Ū )) = λ ≤ κ. Let ν i : i < λ be cofinal in lh(Ū ) such that for each i < λ, there is some
A i ⊂ V κ ∩ MS such that A i ∈ U (ξ) iff ξ > ν i .
By [Git10], we know in V [G], cf (κ) = ν < κ, as witnessed by some κ n : n ∈ ν . Recursively define κ ′ n such that κ ′ n the least element ≥ κ n such that there is somē w ∈ MS G ∩ i<λ,νi<κn A i with κ ′ n = κ(w). Hence, the sequence κ ′ n : n ∈ ν has the property that it is almost contained in O(MS G ∩ A i ) for any i < λ. Suppose B is any otherŪ -tail-measure-one set, then there exists some large
enough i < λ such that A i ∩ B c isŪ -null. As a result, in V [G], a tail of MS G avoids A i ∩ B c . Therefore, A i ∩ MS G ⊂ * B ∩ MS G . Therefore, κ ′ n : n ∈ ν is almost contained in O(B ∩ MS G ).
Proof of Proposition 4.11. Let G ⊂ RŪ be generic over V . In V [G], ifw ∈ G is such that cf (κ(w)) < κ(w), then we let c κ(w) be the set as given by Lemma 4.12 applied to Rw. We claim this sequence is strong club guessing on the set of singulars below κ. Given any club D ⊂ κ, by Theorem 2.10 we may find a tail measure one set B, witnessed by γ < lh(Ū ) such that O(B ∩ MS G ) ⊂ D. For anyw such that B ∩ V κ(w) is tail measure one with respect tow, Lemma 4.12 implies that c κ(w) ⊂ * O(B ∩ MS G ). Since {w : B ∩ V κ(w) is tail measure one with respect tow} is tail measure one with respect toŪ , we can apply Theorem 2.11 to get the conclusion as desired.
Even though there are many instances of club guessing principles valid in ZFC (see [She94]), SCG at singulars does not necessarily hold in ZFC. For example, in the forcing extension where we add λ + many Cohen subsets of λ, SCG on the singulars below λ fails. We describe another scenario of a different nature.
Proposition 4.13. Suppose κ is a strongly compact cardinal and λ > κ is a regular cardinal, then SCG fails on the set S = {α ∈ λ : cf (α) < |α|}.
Proof. Suppose for the sake of contradiction thatt = t α : α ∈ S is a strong club guessing sequence. We may assume t α is a club in α for each α ∈ S. Let j : V → M witness that κ is λ-compact. In particular, κ M ⊂ M and sup j ′′ λ = def δ < j(λ). To see that δ ∈ j(S), since j witnesses that κ is λcompact, there exists some A ∈ M such that j ′′ λ ⊂ A and M |= |A| < j(κ) < δ. In particular, M |= δ is singular. Hence δ ∈ j(S). Let t = j(t) δ and A = j −1 acc(t ∩ j ′′ λ). Recall for a set Z, acc(Z) = {δ ∈ Z : Z ∩ δ is unbounded in δ}. It is not hard to see the set A is < κ-closed and unbounded. ConsiderĀ, the closure of A. By the hypothesis, there exists a club D ⊂ λ, such that for any α ∈ D ∩ S, t α ⊂ * Ā . By elementarity of j, we know that δ ∈ j(D ∩ S). As a result, t ⊂ * j(Ā). This in particular implies t ∩ j ′′ λ ⊂ * j(Ā) ∩ j ′′ λ = j ′′Ā .
We check that for any ν ∈ t ∩ j ′′ λ, there is some ν ′ ≥ ν in t ∩ j ′′ λ that does not belong to j ′′Ā . This clearly gives a contradiction as desired. Given such a ν, if it already does not belong to j ′′Ā , then we are done. Suppose there is some ξ ∈Ā, j(ξ) = ν. Let ν ′ be min ((t ∩ j ′′ λ) − (ν + 1)). In particular, ν ′ is in nacc(t ∩ j ′′ λ). Observe that ν ′ ∈ j ′′Ā . To see this, suppose for contradiction that there is ξ ′ ∈Ā such that j(ξ ′ ) = ν ′ . Note that ξ ′ > ξ. By the definition of A, ξ ′ ∈Ā − A. We can then find η ∈ (ξ, ξ ′ ) ∩ A. But then ν < j(η) < ν ′ and j(η) ∈ t ∩ j ′′ λ, contradicting with the choice of ν ′ .
Remark 4.14. Proposition 4.13 can be used to separate the ♦ + -principles from the SCG principles. For example, [Ish05, Question 1] asks if for a regular uncountable cardinal µ, ♦ + (µ) implies SCG on µ. By [CFM01,Theorem 22], if 2 λ = λ + and 2 λ + = λ ++ , then there is a λ + -directed closed and λ ++ -c.c poset that forces ♦ + (λ + ). Therefore, relative to the existence of a supercompact cardinal κ, we can produce a model (using the technique by Laver [Lav78]) where κ remains supercompact and λ > κ satisfies ♦ + (λ + ) holds. In this model, SCG on λ + fails by Proposition 4.13. There are two things worth noting:
1. Large cardinals should not be needed to separate ♦ + from SCG. For example, [IL12] shows it is consistent relative to ZFC that ♦ + (ω 1 ) holds and SCG on ω 1 fails.
2. If λ is regular with λ <λ = λ, then [FK05] proves there is a < λ-closed and λ + -c.c poset that adds a SCG-sequence on λ.
The validity of SCG on regulars is constrained by the compactness principle to be discussed in the next section.
A weak consequence of ineffability does not imply the diamond principle
To relate the compactness principle we will consider in this section with standard large cardinals, we first supply a characterization of ineffable cardinals in terms of club sequences. The following lemma is essentially due to Todorcevic [Tod07], who proved the corresponding version for weakly compact cardinals. Recall κ is ineffable iff for any d α ⊂ α : α < κ , there exists A ⊂ κ such that {α < κ : A ∩ α = d α } is a stationary subset of κ.
Definition 5.1. A C-sequence on κ is a sequence C α : α < κ where each C α is a club subset of α for any α < κ.
Lemma 5.2. For a strongly inaccessible κ, we have that: κ is ineffable iff for any C-sequence on κ C α ⊂ α : α < κ , there exists a club D ⊂ κ, satisfying that {α < κ : D ∩ α = C α } is stationary.
Proof. We only prove the non-trivial "if" direction. Namely, suppose every Csequence on inaccessible κ admits a club D ⊂ κ such that {α < κ : D ∩ α = C α } is stationary, then κ is ineffable. Let T be a tree of height κ and the ξ-th level of the tree is T (ξ) = [δ ξ , δ ξ+1 ), where δ ξ : ξ < κ is an increasing continuous sequence of ordinals below κ. We will also assume that T does not split at limit levels, in the sense that for any limit ξ < κ, and α 0 , α 1 ∈ T (ξ), if {η : η < T α 0 } = {η : η < T α 1 }, then α 0 = α 1 .
We claim that there is a branch d through T such that {ξ < κ : δ ξ ∈ d} is stationary in κ. Note that there is a club C consisting limit ξ such that δ ξ = ξ. Consider the following C-sequenceC: for any ξ < κ, if ξ ∈ lim C, then we let C ξ be the closure of E ξ = def {η < ξ : η < T ξ} and if ξ ∈ lim C, then C ξ = (max C ∩ ξ, ξ). Recall the hypothesis stated at the beginning of the proof. We can then find a club D ⊂ κ such that S = {ξ < κ : D ∩ ξ = C ξ } is stationary in κ. We claim that S ∩ lim C forms a branch as desired. Suppose α < β ∈ S ∩lim C, let α ′ ∈ T (α) be the unique node such that α ′ < T β. We need to show that α = α ′ . By the fact that T does not split at limit levels, we only need to show that α and α ′ have the same set of T -predecessors. Because T is a tree, it suffices to show there are arbitrarily large η < α = δ α with η < T α, α ′ . Given ξ < α, note that D ∩ T (ξ + 1) has a unique element. To see this, by the hypothesis we have D ∩ α = C α , as a result D ∩ T (ξ + 1) consists of the unique node that is T -below α, since D ∩ T (ξ + 1) = C α ∩ T (ξ + 1) = E α ∩ T (ξ + 1). Let η ∈ D ∩ T (ξ + 1). The same argument shows that η < T α ′ . We are done.
It is left to see the principle as above implies κ is ineffable. Given a list d α ∈ 2 α : α < κ , consider the tree T ′ = {d β ↾ α : α ≤ β}. Note T ′ does not split on limit levels. Since κ is strongly inaccessible, it is possible to find an increasing continuous sequence δ ξ : ξ < κ such that there is a bijection b ξ : [δ ξ , δ ξ+1 ) ↔ T ′ (ξ) with b ξ (δ ξ ) = d ξ for each ξ < κ. Let T be the pull-back of T ′ , namely, for any α 0 , α 1 with ξ 0 , ξ 1 < κ be such that
α i ∈ [δ ξi , δ ξi+1 ) for i < 2, α 0 < T α 1 iff b ξ0 (α 0 ) < T ′ b ξ1 (α 1 )
. It can be easily verified that T is of the form described above. Then there is a branch d through T such that S = {ξ < κ : δ ξ ∈ d} is stationary. Consider d ′ = {b ξ (δ ξ ) = def d ξ : ξ ∈ S}. Then for any ξ 0 < ξ 1 ∈ S, δ ξ0 < T δ ξ1 , which implies b ξ0 (δ ξ0 ) = d ξ0 < T ′ b ξ1 (δ ξ1 ) = d ξ1 , hence d ξ0 = d ξ1 ↾ ξ 0 . This implies that κ is ineffable.
The following notion is related to constructions of distributive Aronszajn trees, which strenghtens the principle ⊗− → C from [She94, p. 134].
Definition 5.3 ([BR19]). A C-sequence on κ C α : α < κ is said to be amenable if there is no club D ⊂ κ, such that {α < κ : D ∩ α ⊂ C α } is stationary.
Therefore, the assertion that there is no amenable C-sequence on κ is a weakening of ineffability. To be more explicit, the compactness principle states:
for any C-sequence C α ⊂ α : α < κ , there exists a club D ⊂ κ, such that {α : D ∩ α ⊂ C α } is a stationary subset of κ.
Remark 5.4. That "there is no amenable C-sequence on κ" is a non-trivial compactness principle. For example, it implies that the diagonal stationary reflection principle at κ, which means: for any κ-sequence of stationary subsets of κ S i : i < κ , there is some δ < κ such that for any i < δ, S i ∩ δ is stationary. However, it is not a consequence of κ being weakly compact, while the diagonal stationary reflection principle at κ is. For instance, in L, there is no amenable C-sequence on κ iff κ is ineffable.
The following seems open:
Question 5.5. Is "there exists a cardinal carrying no amenable C-sequence" equiconsistent with "there exists an ineffable cardinal"?
Remark 5.6. To relate to the Strong Club Guessing principle discussed in the previous section, notice that if there is no amenable C-sequence on κ, then the Strong Club Guessing principle on the regulars below κ fails.
In [BN19], it was shown that relative to the existence of large cardinals, it is consistent that there is a strongly inaccessible cardinal κ where the diagonal stationary reflection principle holds yet ♦(κ) fails. We strengthen the compactness principle in this model to "there is no amenable C-sequence on κ".
Theorem 5.7. IfŪ is a measure sequence on κ such that cf (lh(Ū )) ≥ κ ++ , then in V RŪ , there is no amenable C-sequence on κ.
Lemma 5.8. Suppose cf (lh(Ū )) ≥ κ + . Let E ⊂ MS beŪ -stationarily positive, namely {τ < lh(Ū ) : E ∈ U (τ )} is a stationary subset of lh(Ū ). Given a tail measure function b, there exists A ∈ Ū ↾≥ β for some β < lh(Ū ), such that in any generic G ⊂ RŪ ,
{α < κ : ∃w ∈ MS G ∩ E[κ(w) = α ∧ O(A ∩ MS G ) ∩ α ⊂ * O(b(w) ∩ MS G )]} is stationary.
Proof. Fix a tail measure function b. Then, for everyw ∈ MS ∩ V κ , there is some τ = τw < lh(w) such that b(w) ∈ w ↾≥ τ . Let f be the function w → τw. Let S = {τ < lh(Ū ) : E ∈ U (τ )}. Consider g : S → lh(Ū ) such that g(γ) = j(f )(Ū ↾ γ) < γ. By Pressing Down, there is a stationary S ′ ⊂ S and β < lh(Ū ) such that g ′′ S ′ ⊂ β. We may assume β is novel sinceŪ does not contain a repeat point. Pick A β ∈ Ū ↾≥ β with A c β ∈ Ū ↾ β. We check that F = {w ∈ E : A β ∩ b(w) c isw-null} isŪ -positive. Indeed, for any τ ∈ S ′ ,Ū ↾ τ ∈ j(F ), since A β ∩ j(b)(Ū ↾ τ ) c ∈ U (i) for any i < τ . To see this, if i ≥ β, j(b)(Ū ↾ τ ) ∈ U (i) and if i < β, A c β ∈ U (i). Let G ⊂ RŪ be generic over V , then O(F ∩ MS G ) is a stationary subset of κ, by Theorem 2.10. For eachw ∈ F ∩ MS G , we know a tail of MS G beloww avoids A β ∩ b(w) c . As a result, O(A β ∩ MS G ) ∩ κ(w) ⊂ * O(b(w) ∩ MS G ).
Lemma 5.9. Suppose cf (lh(Ū )) ≥ κ + . Let p ċ is a name for a C-sequence. Then there exists q ≤ * p and a tail measure function b such that for some B ∈ Ū , for anyw ∈ B, if cf (lh(w)) ≥ κ(w) + , it is the case that q ⌢w O(b(w) ∩ MS G ) ⊂ * ċ (w).
Proof. Suppose p = p ⌢ 0 (Ū , A). By Lemma 2.13, we can findŪ -measure-one A ′ ⊂ A, and someḋ such thatḋ(w) is a Rw-name for a club subset of κ(w) and letting q = p ⌢ 0 (Ū , A ′ ) we have that q ⌢w ċ κ(w) =ḋ(w) for anyw ∈ A ′ .
Let R <κ denote the collection of possible lower parts of conditions in RŪ . Namely, t ∈ R <κ iff t ⌢ (Ū , E) ∈ RŪ for some E ∈ Ū . For eachw ∈ A ′ with cf (lh(w)) ≥ κ(w) + , t ∈ V κ(w) ∩ R <κ , we can find (by Theorem 2.10) A tw ∈ w and somew-tail-measure-one set Γ tw such that t ⌢ (w, A tw ) Rw O(Γ tw ∩ MSĠw ) ⊂ḋ(w). Let Aw = ∆ t A tw ∈ w and Γw = ∆ t Γ tw . The fact that Γw is aw-tail-measure-one set follows from the fact that cf (lh(w)) ≥ κ(w) + .
Let b be a tail measure function satisfying that b(w) = Γw for anyw ∈ A ′ with cf (lh(w)) ≥ κ(w) + . We claim that q ⌢w O(b(w) ∩ MS G ) ⊂ * ċ (w) for allw ∈ A ′ with cf (lh(w)) ≥ κ(w) + . Fix suchw.
It is left to check that p ⌢ 0 (w, A ′ ∩ V κ(w) ) Rw O(Γw ∩ MS Gw ) ⊂ * ḋ (w). This follows immediately from the fact that for any t ∈ V κ(w) with t ⌢ (w, Aw) ∈ Rw, t ⌢ (w, Aw) Rw O(Γw ∩ MSĠw ) − (rank(t) + 1) ⊂ḋ(w). To see this fact, suppose for the sake of contradiction that this is not the case for some t, then we have an extension t * = t ′⌢ (ū, e) ⌢ ( − → η , − → A − → η ) ⌢ (w, A * ) κ(ū) ∈ḋ(w) for somē u ∈ Γw. Notice thatū ∈ A t ′ w and − → η ⊂ A t ′ w by the definition of Aw. Hence t * is compatible with t ′ ⌢ (w, A t ′ w ), which forces O(Γ t ′ w ∩ MSĠw ) ⊂ḋ(w). Asū ∈ Γ t ′ w , we know that t ′ ⌢ (ū, A t ′ w ∩ V κ(ū) ) ⌢ (w, A t ′ w ) κ(ū) ∈ḋ(w), and is compatible with t * that forces κ(ū) ∈ḋ(w), which is absurd.
Proof of Theorem 5.7. In V [G], where c α : α < κ is given, we aim to find a club C ⊂ κ such that {α : C ∩ α ⊂ * c α } is a stationary subset of κ. By Lemma 5.9, we can find a tail measure function b ∈ V , such that on a tail of MS G , ifw ∈ MS G satisfies cf (lh(w)) ≥ κ(w) + , then O(b(w) ∩ MS G ) ⊂ * c κ(w) . By Lemma 5.8 with E = {w : cf (lh(w)) ≥ κ(w) + }, we can find a club D in V [G] such that for stationarily many regular α ∈ C G with α = κ(w) forw ∈ MS G , it is the case that D ∩ α ⊂ * O(b(w) ∩ MS G ) ⊂ * c α .
If we prepare the ground model such that κ is (κ + 2)-strong and 2 κ > κ ++ , then for some elementary embedding, we can derive a measure sequenceŪ of length κ ++ . By Theorem 4.5 and Theorem 5.7, we have no amenable C-sequence on κ and ¬♦(κ) in V RŪ .
Corollary 5.10. Relative to the existence of suitable large cardinals, it is consistent that there does not exist an amenable C-sequence at κ and ♦(κ) fails.
Questions and Acknowledgments
The question of whether κ being weakly compact implies ♦(κ) is still open and we know the method of Radin forcing cannot provide a solution. However, a better understanding of compactness principles in the Radin model can still shed light on the relationship between weak compactness and its weakenings.
Question 6.1. For strongly inaccessible κ, can we produce a model where κ is not weakly compact and either of the following holds:
3.1.Ū satisfies the Local Repeat Property (LRP) if for any measure function b, there exist A ∈ Ū and γ < lh(Ū ) such that j(b)(Ū ↾ γ) = A.
Theorem 4 . 2 (
42Cummings-Magidor, [Cum13]). Let G ⊂ RŪ be generic over V and assume that κ is regular in V [G]. Then in V [G], for any stationary S ⊂ κ, V -♦ − (S) is equivalent to ♦ − (S).
Observation 4 . 3 .
43Suppose cf (lh(Ū )) ≥ κ + . Then V RŪ |= ♦(κ) holds iff there exists a function f ∈ V such that dom(f ) = MS ∩ V κ and f (w) ∈ [2 κ(w) ] ≤κ(w) , satisfying that for any X ⊂ κ, {w : X ∩ κ(w) ∈ f (w)} isŪ -positive.
Theorem 4 . 5 (
45Woodin). If |2 κ | does not divide lh(Ū ) and cf (lh(Ū )) ≥ κ + , then V RŪ |= ¬♦(κ).
Remark 4 . 6 .
46If j : V → M constructsŪ , and ifŪ ∈ M , then V RŪ |= ¬♦(κ) whenever |2 κ | M does not divide lh(Ū ). This is done by apply Woodin's theorem in M .Under certain circumstances, we can get stronger guessing principles. Recall the following characterization of stationary sets in the Radin extensions.
Theorem 4.7 ([BN19]). IfṠ is an RŪ -name for a stationary subset of κ and p ∈ RŪ , then there exists an extension e = e 0 ⌢ (Ū , B) ≤ p and a measure function b such that 1. Z e0 = def {ū : ∃A ∈ Ū , e 0 ⌢ (ū, b(ū)) ⌢ (Ū , A) κ(ū) ∈Ṡ} isŪ -positive, 2. for any − → η ∈ MS <ω and − → η ⊂ B, Zē ⇂ − → η = def {ū ∈ Z e0 : − → η << (ū, b(ū))} isŪ -positive.
Proposition 4. 8 .
8IfŪ is a measure sequence constructed by j : V → M and M |= cf (lh(Ū )) ≥ 2 κ , then in V RŪ , ♦(S) holds for all stationary S ⊂ κ.
Lemma 4 . 12 .
412LetŪ be a measure sequence without repeat points and cf (lh(Ū )) ≤ κ. Then in V [G], there exists a cofinal subset c of κ such that for anyŪ -tailmeasure-one set B ∈ V , c ⊂ * O(B ∩ MS G ).
Acknowledgments: The authors would like to thank Tom Benhamou, Moti Gitik and Assaf Rinot for valuable questions and comments. The authors also want to thank the anonymous referee for a careful reading and for their very helpful corrections, comments and suggestions.
As a first step towards the solution, we may ask if we can characterize the measure sequenceŪ such that κ satisfies any of the compactness principles men. As a first step towards the solution, we may ask if we can characterize the measure sequenceŪ such that κ satisfies any of the compactness principles men-
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"Enhancing Sensitivity of an Atom Interferometer to the Heisenberg Limit using Increased Quantum Noise",
"Enhancing Sensitivity of an Atom Interferometer to the Heisenberg Limit using Increased Quantum Noise"
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"Renpeng Fang \nDepartment of Physics and Astronomy\nNorthwestern University\n2145 Sheridan Road60208EvanstonILUSA\n",
"Resham Sarkar \nDepartment of Physics and Astronomy\nNorthwestern University\n2145 Sheridan Road60208EvanstonILUSA\n",
"Selim M Shahriar \nDepartment of Physics and Astronomy\nNorthwestern University\n2145 Sheridan Road60208EvanstonILUSA\n\nDepartment of EECS\nNorthwestern University\n2145 Sheridan Road60208EvanstonILUSA\n"
]
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"Department of Physics and Astronomy\nNorthwestern University\n2145 Sheridan Road60208EvanstonILUSA",
"Department of Physics and Astronomy\nNorthwestern University\n2145 Sheridan Road60208EvanstonILUSA",
"Department of Physics and Astronomy\nNorthwestern University\n2145 Sheridan Road60208EvanstonILUSA",
"Department of EECS\nNorthwestern University\n2145 Sheridan Road60208EvanstonILUSA"
]
| []
| O M,CD =Ĵ z / . From the third line of Eq. 2, it follows thatÔ M,CD =Ĵ z / = J m=−J m |E J+m E J+m | = J m=−J mÔ M,CSD,J+m . In the final state described above, we have only two of the collective states. As such, for this state, Ô M,CD = −J Ô M,CSD,0 + | 10.1364/josab.396358 | [
"https://arxiv.org/pdf/1707.08260v6.pdf"
]
| 51,914,042 | 1707.08260 | 3f68021558bbbc8f2d86f59e5f88122a9f9470dc |
Enhancing Sensitivity of an Atom Interferometer to the Heisenberg Limit using Increased Quantum Noise
Renpeng Fang
Department of Physics and Astronomy
Northwestern University
2145 Sheridan Road60208EvanstonILUSA
Resham Sarkar
Department of Physics and Astronomy
Northwestern University
2145 Sheridan Road60208EvanstonILUSA
Selim M Shahriar
Department of Physics and Astronomy
Northwestern University
2145 Sheridan Road60208EvanstonILUSA
Department of EECS
Northwestern University
2145 Sheridan Road60208EvanstonILUSA
Enhancing Sensitivity of an Atom Interferometer to the Heisenberg Limit using Increased Quantum Noise
(Dated: November 15, 2017)numbers: 0630Gv0375Dg3725+k
O M,CD =Ĵ z / . From the third line of Eq. 2, it follows thatÔ M,CD =Ĵ z / = J m=−J m |E J+m E J+m | = J m=−J mÔ M,CSD,J+m . In the final state described above, we have only two of the collective states. As such, for this state, Ô M,CD = −J Ô M,CSD,0 +
In a conventional atomic interferometer employing N atoms, the phase sensitivity is at the standard quantum limit: 1/ √ N . Using spin-squeezing, the sensitivity can be increased, either by lowering the quantum noise or via phase amplification, or a combination thereof. Here, we show how to increase the sensitivity, to the Heisenberg limit of 1/N , while increasing the quantum noise by √ N , thereby suppressing by the same factor the effect of excess noise. The proposed protocol makes use of a Schrödinger Cat state representing a mesoscopic superposition of two collective states of N atoms, behaving as a single entity with an N -fold increase in Compton frequency. The resulting N -fold phase magnification is revealed by using atomic state detection instead of collective state detection.
PACS numbers: 06. 30.Gv, 03.75.Dg, 37. 25.+k In an atomic interferometer (AI), the signal S can be expressed as a function of the phase difference φ between the two arms. The measurement sensitivity, Λ, can be expressed as the inverse of the phase fluctuation (PF): Λ = PF −1 = |∂ φ S/∆S|, where ∂ φ ≡ ∂/∂φ. Here, ∂ φ S represents the phase gradient of the signal (PGS), and ∆S represents the standard deviation of the signal (SDS). When all sources of excess noise (EN) are suppressed sufficiently, Λ is limited by the quantum projection noise (QPN) [1], and is given by the inverse of the quantum phase fluctuation (QPF −1 ). In the absence of any correlation between the atoms, such as for a conventional AI, the sensitivity is at the Standard Quantum Limit (SQL): Λ = QPF −1 = √ N , with N being the number of atoms interrogated within the measurement time. Using spinsqueezing, it is possible to surpass the SQL, and a key goal in this context is to achieve the Heisenberg Limit (HL), under which Λ = N , representing an improvement by a factor of √ N . To enhance the sensitivity Λ, one can either increase the PGS or decrease the SDS. In a conventional approach for spin squeezing, one minimizes the variance, and therefore the SDS. For example, using optimal oneaxis-twist squeezing (OATS) [2] and two-axis-countertwist (TACT) squeezing [2], the SDS can be reduced respectively by a factor of N 1/3 and N/2, while the PGS remains essentially unchanged, compared to those of a conventional AI. As such, Λ = N 5/6 for the former and Λ = N/ √ 2 for the latter. Though the TACT squeezing can yield a better sensitivity, it is experimentally more complicated than the OATS [3][4][5][6][7][8][9][10]. Recently [11][12][13], it was shown that it is also possible to reach sensitivity at or near the HL using variants of the OATS. Ref. [11] proposed and Ref. [12] demonstrated the echo squeezing protocol (ESP), which can increase the PGS by a factor of ≈ N/2, while leaving the SDS unchanged, thus producing Λ ≈ N/ √ 2. Ref. [13] proposed a Schrödinger
Cat atomic interferometer (SCAIN) that makes use of critically tuned OATS, rotation, inverse rotation and unsqueezing, which, in combination with collective state detection (CSD) [14][15][16][17][18], reduces the SDS by a factor of √ N , while leaving the PGS unchanged, yielding Λ = N . In what follows, we will refer to this as the CSD-SCAIN.
In this letter, we describe a new protocol that is a variant of the CSD-SCAIN protocol, with radically different behavior. It employs the conventional detection (CD) technique by measuring directly the populations of the spin-up or spin-down states of individual atoms. We show that, under this protocol (called CD-SCAIN), the PGS is increased by a factor of N , while the SDS is also increased by a factor of √ N . The net enhancement in the sensitivity is by a factor of √ N , reaching the HL: Λ = N . However, because of the increase in noise (i.e., SDS), this is now significantly more robust to EN than all the protocols described above. Specifically, for this protocol, it should be possible to achieve Λ = N/ √ 2 even when the EN is greater than the QPN for a conventional AI by a factor of √ N . The degree of suppression of EN for different protocols is illustrated in Fig. 1.
Here, we consider a situation where EN contributes an additional variance, ∆S 2 EN , to the signal. The sensitivity is then given by Λ = PGS/ ∆S 2 QPN + ∆S 2 EN = (PGS/∆S QPN )/ 1 + ρ 2 = Λ QPN / 1 + ρ 2 . Here, ρ represents the ratio of ∆S EN (the EN) and ∆S QPN (the QPN), which is different for different protocols. One way to characterize the degree of robustness against EN is by determining the value of ∆S EN for which ρ = 1. As can be seen, for TACT, this value is 1, which makes it particularly vulnerable to EN. In contrast, for ESP (as well as for the conventional AI), this value is √ N , making it a factor of √ N more robust than TACT. For CD-SCAIN, this value is N , making it a factor of √ N (N ) more ro-bust than ESP (TACT). We also see that CSD-SCAIN is as sensitive to EN as TACT. Thus, in switching from collective state detection to conventional detection, the robustness of the SCAIN protocol to EN is improved by a factor of N . One can also define the range of usefulness of a protocol as the value of ∆S ENC for which the sensitivity drops to Λ = N/2. By this measure, the usefulness of CD-SCAIN extends to N 3/2 , while that for ESP extends only to N . Ref. [19] presents a protocol that also makes use of OATS critically tuned to the same degree as that employed by SCAIN. However, the usefulness of this protocol also extends only to N , similar to ESP. The AI considered here is a SCAIN, which is based on the conventional Raman atomic interferometer (CRAIN). Briefly, both make use of N non-interacting identical three-level atoms with metastable hyperfine states |1, p z = 0 and |2, p z = k and an excited state |3 in the Λ-configuration, coupled by a pair of counterpropagating laser beams [20][21][22][23]. Here, k ≡ k 1 + k 2 , with k 1 and k 2 being the wave numbers for the two counterpropagating beams propagating in the +ẑ and −ẑ directions, respectively, and p z is the z-component of the linear momentum of the atom. Each atom can be reduced to an equivalent two-level model via adiabatic elimination of the excited state [24,25], and thus can be represented by a pseudospin-1/2 operatorĵ, where we define |↓ ≡ |1, p z = 0 and |↑ ≡ |2, p z = k . The ensemble, now represented by a collective spin operatorĴ ≡ N iĵ i , is initially prepared in a Coherent Spin State (CSS) [15],
|−ẑ = N i=1
|↓ , where all atoms are in the spin-down state. Here and in the rest of the paper, we employ the compact notation that a state |ê is a CSS in the direction of the unit vectorê, with the pseudospin vector of each atom being in that direction. For the CRAIN, the ensemble is then subjected to the usual pulse sequence of π/2−dark−π−dark−π/2, labeled as 1, 4, 7 in Fig. 2 (a). For the SCAIN, however, the ensemble will undergo four additional pulses labeled as 2, 3, 5, 6 in Fig. 2 (a), corresponding to the squeezing, rotation, inverse rotation and unsqueezing operations in the CSD-SCAIN protocol proposed in Ref. [13]. The complete evolution of the quantum states on a Bloch sphere under this protocol is shown in Fig. 2 (b), using the Husimi Quasi Probability Distribution (QPD) [2,15]. It should be noted that the exact effects of the protocol depend on the choices of a set of parameters such as the value (and parity) of N , the squeezing parameter µ for the OATS, the auxiliary rotation axis (ARA, can bex orŷ axes) around which to implement the rotation, the corrective rotation sign ξ which can take values of ±1 corresponding to redoing or undoing the first auxiliary rotation, and lastly the dark zone phase shift φ. The case shown here is for an even value of N = 40, with µ = π/2, ARA =x, ξ = −1 and φ = π/80. The QPD is expressed as a color-coded intensity distribution as a function Q H (θ, φ) of the angles in spherical coordinates which span the surface of the Bloch sphere. For a given quantum state |Ψ , it is defined as
Q H (θ, φ) ≡ | Ψ|Φ(θ, φ) | 2 , where |Φ(θ, φ) ≡ cos θ 2 N N k=0 N k e iφ tan θ 2 k |E N −k
(1) represents the CSS corresponding to all the spins pointing in the direction {θ, φ}, and |E n are the Dicke collective states (DCSs) [14][15][16] defined as
|E n = ( N n ) k=1 P k ↓ N −n ↑ n N n (2)
with P k being the permutation operator [26]. In this definition of the DCSs, the maximally excited collective state, |E N , corresponds to all atoms with their pseudospins in theẑ direction. As such, we will refer to the set of DCSs shown in Eq. 2 as the Z-directed Dicke Collective States (ZDCSs). As needed, we will also refer to XDCSs (YDCSs) for which |E N corresponds to all atoms with their pseudospins in thex (ŷ) direction.
In illustrating the nature of the QPD at various stages of the protocol, we have used different orientations of the Bloch sphere as suited, and added ± symbols in front of two axes to indicate that the picture looks the same when it is rotated by 180 degrees around the third axis. At the onset of the process (time point A), the system is assumed in state |−ẑ . After the first π/2 pulse (time point B), the state rotates around thex axis to reach the state |ŷ . The squeezing pulse is then applied by using a squeezing Hamiltonian of the form H OAT S = χĴ 2 z for a duration of τ such that µ = χτ . After the squeezing pulse (time point C), the state is split equally between two CSSs, and can be expressed as (|ŷ −η |−ŷ )/ √ 2 [27,28],
where η = i(−1) N/2 , representing a phase factor with unity amplitude [29]. This is a Schrödinger Cat (SC) state [30], but as a superposition of the two extremal states of the YDCS manifold, which cannot be used to achieve phase magnification, since the phase difference between the two arms corresponds to rotation around theẑ axis, not theŷ axis. This problem is solved by applying the auxiliary rotation of π/2 around thex axis, which transforms this state to (|−ẑ + η |ẑ )/ √ 2, ignoring an irrelevant overall phase factor. This (time point D) represents the desired SC state, as a superposition of the two extremal states of the ZDCS manifold: that the net phase difference between the two paths is N φ, thus magnifying the rotation induced phase by a factor of N . The resulting QPD once again remains unchanged but the quantum state incorporates these phase accumulations. In order to reveal the phase magnification, it is necessary to apply another auxiliary rotation by an angle of −π/2 around thex axis (time point H), followed by the unsqueezing pulse with a Hamiltonian of the form −H OAT S (time point I). Finally, the second π/2 pulse is applied to cause interference between the two arms, after which (time point J) the state can be written as
(|E 0 + η |E N )/ √ 2.|Ψ f = cos(N φ/2) |E 0 − η sin(N φ/2) |E N .
Mathematically, the whole protocol for this case can be expressed as (with = 1):
|Ψ f = e −i π 2Ĵ x e iµĴ 2 z e −iξ π 2Ĵ x e i φ 2Ĵ z e −iπĴx e −i φ 2Ĵ z e −i π 2Ĵ x e −iµĴ 2 z e −i π 2Ĵ x |−ẑ (3)
If the population of the collective state |E 0 were detected, the signal would be expressed as cos 2 (N φ/2), with fringes that are a factor of N narrower than that for the CRAIN. This is the CSD-SCAIN described in Ref. [13], which employs the collective state detection technique. Compared to the conventional AI, the PGS remains unchanged, since the phase enhancement is countered by reduction in the signal amplitude by a factor of N . However, the SDS is now reduced by a factor of √ N , since the number of particles is unity. As such, the sensitivity increases by a factor of √ N , thus reaching the HL. In what follows, we describe a significantly different version of the SCAIN, namely the CD-SCAIN, which employs the conventional detection technique corresponding to measuring the z-component of the combined spin of all atoms, theĴ z operator, which represents the difference between the number of atoms in the spin-up and spin-down states.
The signal for the CD-SCAIN can be obtained by first expanding theĴ z operator in the basis of the ZDCSs, then taking the expectation value with respect to |Ψ f . This turns out to be Ψ f |J z |Ψ f = −N/2 cos(N φ), as derived in [31], again showing the feature of N -fold fringe narrowing. However, compared to the case of the CSD-SCAIN, the amplitude of the fringes is now a factor of N larger. As such, the PGS is now larger than that for a conventional AI by a factor of N . At the same time, the SDS is also increased by a factor of √ N , compared to that for a conventional AI, as derived and discussed further in [31]. This is surprising, since the signal amplitude for the CD-SCAIN is the same as that for a conventional AI. The net enhancement in sensitivity is by a factor of √ N , reaching the HL, just as in the case of the CSD-SCAIN. However, the increase in SDS makes the CD-SCAIN significantly more robust against EN, as summarized earlier in Fig. 1.
For the particular choice of ARA used in the protocol for Fig. 2 all other parameters the same as in Fig. 2 (b) are found to be drastically different [31], due to the fact that the state after the squeezing pulse (time point C) will now be split equally between |x and |−x , thus generating a SC state as a superposition of the two extremal states of the XDCS manifold [27,28]. The ensuing auxiliary rotation around the x axis will not transform it into the desired SC state required to yield the N -fold phase amplification. This also complicates the evolution of the quantum states during the following stages, for which an analytical expression for the final state is not easy to find. Instead, we take a numerical approach to simulate the state evolutions [31]. The signals for the CD-SCAIN, as a function of φ, for both even and odd values of N , are shown in Fig. 3, where for reference, the signal corresponding to one full fringe of the CRAIN is also shown in Fig. 3 (a). The plots in Fig. 3 (b) and (c) clearly show the N -fold narrowed fringes for the even case while only a central fringe is observable for the odd case. We also found that changing the sign of ξ simply inverts the fringes, which implies that the N -fold reduction of the fringe width happens for the even case no matter whether we choose to redo (ξ = 1) or undo (ξ = −1) the first auxiliary rotation. Of course, the nature of the signals for odd and even values of N can be reversed by switching the choice of ARA fromx toŷ.
In Fig. 4, we illustrate the behavior of the inverse of the quantum fluctuation in rotation (QFR −1 ), as a function of the squeezing parameter µ, for different choices of parameters for the CD-SCAIN, along with a comparison with the CSD-SCAIN. The QFR −1 is a special case of For µ = π/2, we see that the sensitivity for even number of atoms (red) is at the HL, and that for odd number of atoms (dashed blue) is at the SQL. For even N , this sensitivity is reached due to an amplification of phase by a factor of N , and a concomitant increase in the SDS by a factor of √ N . For odd N , there is a phase amplification, manifested as a Fabry-Perot like fringe around φ = 0 which is narrowed by a factor of √ N , along with an increase in the SDS by a factor of √ N . The difference between the two cases disappears when the value of µ is reduced below a threshold value of ∼ 0.45π. There is a range of values of the squeezing parameter (0.2π ≤ µ ≤ 0.45π) over which the normalized value of QFR −1 is ∼ 1/ √ 2. Finally, we note that the vanishing value of QFR −1 for µ = 0 is simply due the fact that the signal is constant as a function of φ. In Fig. 4 (b) and (c), we compare the sensitivity of the CD-SCAIN with that of the CSD-SCAIN, for even and odd values of N , respectively. For even value of N , the sensitivity for both detection protocols are the same for µ = π/2. However, for the CSD-SCAIN, the sensitivity drops off to zero rapidly for decreasing values of µ. For odd value of N , the sensitivity for the CSD-SCAIN is zero for all values of µ, due to the signal being a constant as a function of φ. For both odd and even values of N , the results for the CD-SCAIN are the same for both values of ξ(= ±1), while the results for CSD-SCAIN shown here is for ξ = +1. The CSD-SCAIN result for ξ = −1 is qualitatively the same, with slight differrences for small values of µ. Until now we have analyzed and compared the performance of CD-SCAIN in a separate manner for even and odd values of N . In scenarios where the odd and even parity cases can occur with equal probablities, we have found that the average value of QFR −1 is given by:
QFR −1 AVE = (QFR −1 EVEN ) 2 /2 + (QFR −1 ODD ) 2 /2 1/2 , where
QFR −1 EVEN and QFR −1 ODD are those for the even and odd values of N , respectively. This result also follows from the Bienaymé formula [32]. For a large number of atoms (N 1), this will put the overall sensitivity a factor of √ 2 below the HL.
Finally, we point out that very similar results can be obtained for an atomic clock as well. The behavior of a Schrödinger Cat Atomic Clock (SCAC) under conventional detection (CD-SCAC) and its comparison with a SCAC under collective state detection (CSD-SCAC) are presented in the Section II of the supplement [31].
This work has been supported by the NSF grants number DGE-0801685 and DMR-1121262, and AFOSR grant number FA9550-09-01-0652.
In this supplement, we will discuss some additional details about the simulations and analyses for the Schrödinger Cat Atomic Interferometer (SCAIN) and then present the results obtained by applying the same protocols described in the main body to atomic clocks.
I. ADDITIONAL DETAILS FOR THE SCAIN
In this section, we provide some additional details for understanding the SCAIN employing the conventional detection (CD) protocol, and its comparisons with the SCAIN employing the collective state detection (CSD) protocol.
A. Matrix elements of the collective spin operators
As discussed in the main body of the paper, the squeezing pulse complicates the evolution of the quantum states for the ensemble and it is generally not easy to write down explicitly the mathematical expressions for the final states for arbitrary values of φ and ξ. Therefore a numerical approach is employed to simulate the evolutions for each stage of the protocol. The basis of the operators are chosen to be the Dicke collective states defined as
|E n = ( N n ) k=1 P k ↓ N −n ↑ n N n(1)
which are the eigenstates of theĴ z operator, with eigenvalues ranging from −N /2 for the |E 0 state to N /2 for the |E N state. Here, P k is the permutation operator [1]. In general, for a total spin of J = N/2, the eigenstate corresponding to an eigenvalue of m will be |E J+m . The matrix elements of the relevant operators can thus be expressed as follows:
E J+m |Ĵ x |E J+m = 2 (A J,m δ m ,m+1 + B J,m δ m ,m−1 ) E J+m |Ĵ y |E J+m = 2i (A J,m δ m ,m+1 − B J,m δ m ,m−1 ) E J+m |Ĵ z |E J+m = mδ m ,m E J+m |Ê n ,n |E J+m = δ J+m ,n δ J+m,n(2)
whereÊ n ,n ≡ |E n E n | is the projection operator for the collective states, and A J,m = (J − m)(J + m + 1) and B J,m = (J + m)(J − m + 1) are the two normalization coefficients associated with the raising and lowering operators, respectively. For all the results shown in the main text and the supplements, we have made use of these (N + 1) × (N + 1) matrices to represent all operators, and carried out the complex matrix exponentiations numerically.
B. Derivation of QFR −1 for the CSD-SCAIN and CD-SCAIN protocols
As shown in the main body of the paper, with the chosen parameters, the final state of the ensemble for both the CSD-SCAIN and CD-SCAIN protocols is given by
|Ψ f = cos(N φ/2) |E 0 − η sin(N φ/2) |E N .
For the CSD-SCAIN protocol, in general the collective state operator to be measured can be defined asÔ M,CSD,m ≡ |E m E m |. Thus, the operator we measure isÔ M,CSD,0 if we detect the |E 0 state, andÔ M,CSD,N if we detect the |E N state. For the final state described above, if we measure the former, the signal is cos 2 (N φ/2); if we measure the latter, the signal is sin 2 (N φ/2). For the CD-SCAIN protocol, the operator we measure iŝ
J Ô M,CSD,N .
Thus it follows that for the CD-SCAIN protocol, the signal is given by −J
cos 2 (N φ/2) + J sin 2 (N φ/2) = −(N/2) cos(N φ)
, which has the same fringe width as that obtained by using the CSD protocol, except that the signal now ranges from N/2 to −N/2.
To determine the QFR −1 for both protocols, we define first the signal for the CSD-SCAIN as Σ ≡ Q M,CSD,0 = cos 2 (N φ/2) and the standard derivation (SD) as ∆Σ ≡ the rate of rotation, we can now write:
QFR −1 CSD−SCAIN = Γ −1 ∂Σ/∂φ ∆Σ QFR −1 CD−SCAIN = Γ −1 ∂S/∂φ ∆S (3)
For the CSD-SCAIN protocol, we note that
Q 2 M,CSD,0 =Q M,CSD,0 . which means that ∆Σ ≡ [Σ − Σ 2 ] 1/2 .
Using the expression for Σ from above, we easily find that QFR −1 CSD−SCAIN = N/Γ. We recall that the value of QFR −1 for a CRAIN is given by QFR −1 CRAIN = √ N /Γ, which is the SQL. As such, the CSD-SCAIN represents an improvement by a factor of √ N , reaching the HL sensitivity.
For the CD-SCAIN protocol, we see thatQ 2
CD = J m=−J m 2 |E J+m E J+m |.
However, in the final state described above, we have only two of the collective states. As such, we get Q 2
CD = J 2 Q M,CSD,0 + J 2 Q M,CSD,N = J 2 = N 2 /4. Thus, it fol- lows immediately that ∆S ≡ Q 2 CD − S 2 1/2 = N 2 /4 − N 2 /4 cos 2 (N φ) 1/2 = (N/2)|sin(N φ)|.
It should be noted that the peak value of the SD in this case is N/2, which happens at the points where the slope of the fringe is maximum. From the second line of Eq. 3, we then get QFR −1 CD−SCAIN = N/Γ, the same as that for the CSD-SCAIN protocol, yielding the HL sensitivity.
In the context of atomic interferometers, one often makes use of a rule-of-thumb that states that the quantum fluctuation in rotation (QFR) is simply given by the linewidth (as a function of rotation rate) divided by the signal to noise ratio (SNR), being equal to the square-root of the number of particles. For the case of the CRAIN, the QFR is given by Γ/ √ N , where Γ is the linewidth (representing an amount of rotation that produces a phase shift of one radian) and √ N is the SNR, so the above rule-of-thumb applies. For the case of the CSD-SCAIN, the linewidth is reduced by a factor of N compared to that of the CRAIN. But the SNR is also reduced by a factor of √ N , since the number of particles is now unity, not N . Thus, according to this rule-of-thumb, the QFR of the CSD-SCAIN should be Γ/N . This is consistent with what is found above for this case. For the case of the CD-SCAIN, however, if we try to apply the same rule-of-thumb, we reach an erroneous conclusion. While the linewidth for the CD-SCAIN is also reduced by a factor of N , there is no reduction in the number of particles, since the fringe amplitude is N , the same as that for the CRAIN. This in turn would imply that the SNR remains the same, so the QFR would be Γ/N 3/2 , thus exceeding the HL by a factor of √ N . This suggests that the above rule-of-thumb is not applicable to the case of the CD-SCAIN, where, in fact, the SNR is also reduced by a factor of √ N instead of staying unchanged, due to the nature of the SC state, as shown above.
C. Distinction between the CD-SCAIN and CSD-SCAIN protocols for general quantum states
In this subsection, we show mathematically the distinction between the CD-SCAIN and CSD-SCAIN protocols for general quantum states. Let us define asq M the operator for each atom whose expectation value is measured during the experiment. For each atom, let us define |e (|g ) to be the spin-up (-down) state. Thus, we can writeq M = µ g |g g| + µ e |e e|, where µ g and µ e are complex numbers. The operator which is measured for the whole system can be expressed asQ M = N k=1q M,k . We can express the quantum state of each atom as |ψ = C g |g + C e |e , where C g and C e are complex numbers, and quantum state of the whole system can be expressed as |Ψ = N k=1 |ψ k . It then follows that Q M = N q M . It can also be seen that Q 2
M = N k=1q M,k N k =1q M,k = N q 2 M + N (N − 1) q M 2 .
Here the first term results from the products of operators corresponding to the same atom, and the second term follows from the product of operators corresponding to a given atom (of which there are N ) and every other atom (of which there are N − 1). Let us denote as ρ ≡ q M and the corresponding SD as ∆ρ ≡ q 2 M − ρ 2 1/2 . We also define ℘ ≡ Q M and the corresponding SD as
∆℘ ≡ Q 2 M − ℘ 2 1/2 .
We thus find the very general re-
sult that ∆℘ ≡ N q 2 M − q M 2 1/2 = √ N ∆ρ.
This, of course, has the rather simple physical meaning that, for unentangled atoms, the total variance (equaling the square of the SD) is the sum of the variances from each atom. Yet, it must be noted that this result only holds when the operator to be measured for the whole system can be viewed as a sum of operators for measuring each atom.
We now address two particular examples of the operator to be measured. First, we consider the case wherê q M =ĵ z / (j = 1/2), so thatQ M =Ĵ z / . For each atom, this is equivalent to measuring half the difference in population between the spin-up and spin-down states:q M = j(|e e| − |g g|). As such, we getq 2 M = j 2 (|e e| + |g g|), and for the CRAIN, ρ = −(1/2) cos φ and ℘ = −(N/2) cos φ, so that ∆ρ = (1/2)|sin φ| and ∆℘ = ( √ N /2)|sin φ|, yielding QFR −1 CRAIN = √ N /Γ. Experimentally, this measurement is the same as that done for the CD-SCAIN, namely measuring the state of each atom, but the result is very different, because of the nature of the SC-state.
Next, we consider the case whereq M = j −ĵ z / , so thatQ M = J −Ĵ z / . For each atom, this is equivalent to measuring the population of the spin-down state: q M = |g g|. As such, we getq 2 M = |g g| =q M , and for the CRAIN, ρ = cos 2 (φ/2) and ℘ = N cos 2 (φ/2), so that ∆ρ = (1/2)|sin φ| and ∆℘ = ( √ N /2)|sin φ|, yield-ing QFR −1 CRAIN = √ N /Γ. Experimentally, this CRAIN measurement may appear to be the same as measuring the population of the collective state |E 0 , corresponding to the measured operator being |E 0 E 0 | =Q M,CSD,0 , However, that is not the case. Indeed, it is easy to see that
Q M = N k=1q M,k = N k=1 (|g g|) k = J −Ĵ z / = J J m=−J |E J+m E J+m | − J m=−J m |E J+m E J+m | = J−1 m=−J (J − m)Q M,CSD,J+m(4)
which is a weighted sum of all the operators corresponding to measuring the collective states, excluding the all spin-up state. Eq. 4 is a very important expression that shows the difference between measuring the population of the collective state |E 0 and measuring the population of each atom in the ground state |g . In the main body of the paper, we showed the QPD evolution for the SCAIN protocol for the case when N , the total number of atoms, is even. For comparison, in this subsection we show the QPD evolution for the same protocol for the case when N is odd. All the parameters here are the same as those used to produce the QPD evolution for the even case, except now N = 41 and φ = π/4. As mentioned in the main body of the paper, a very significant difference is observed after the application of the squeezing pulse from time points B to C. Since
N is odd, H OAT transforms |ŷ to (|x + η |−x )/ √ 2,
where η = i(−1) (N +1)/2 , representing a phase factor with unity amplitude. It should be noted that the phase factor depends on the super-odd-parity (SOP), representing whether (N + 1)/2 is even or odd; however, the shapes of the fringes, as well as the values of QFR −1 , for both CSD and CD protocols, are not expected to depend on the value of the SOP, as we have verified explicitly. This state, illustrated in the QPD at time point C, also represents an SC state, as a superposition of two extremal collective states, but in terms of the XDCSs. If we were to use a protocol where the ARA is theŷ axis, we could produce results similar to what is shown in Figure 1 (b) in the main body of the paper. However, since we are using the protocol that is designed to produce maximum phase magnification for the case where the ARA is thex axis, the result is drastically different. The application of the rotation by π/2 around thex axis from time points C to D leaves the QPD unchanged. The rotation in the first dark zone by an angle of φ/2 around theẑ axis (D to E) moves the QPD in the x-y plane on both sides, as shown at time point E. This rotation is inverted by the π pulse from E to F. The rotation in the second dark zone by an angle of −φ/2 around theẑ axis (F to G) moves the QPD in the x-y plane further on both sides, as shown at time point G. This is followed by a rotation of −π/2 around thex axis from G to H. The unsqueezing pulse turns the QPD distribution into four lobes in the y-z plane, as shown at time point I. The final π/2 pulse rotates this pattern by 90 degrees, but still with a four-lobed pattern in the y-z plane, as shown at time point J. Unlike the case for even values of N , it is not easy to write down explicitly the mathematical expression for this final quantum state for an arbitrary value of φ. Instead, we have illustrated the results obtained using numerical simulations in the main body of the paper. For further insight into the behavior of the SCAIN, we also show the population of the collective states corresponding to each stage of the protocol for both even and odd values of N . For each case, the set of parameters are the same as those used to generate the QPD plots. Fig. 2 (a) corresponds to the case when N = 40. At the onset, time point A, the system is in the |E 0 state. At time point B, the system is in a coherent spin state (CSS), with collective state populations centered around ∼ E N/2 . Perhaps somewhat surprisingly, the distribution of collective states remains unchanged at time point C, after the squeezing pulse, even though in the Bloch sphere it is represented by two lobes on opposite sides. After the auxiliary rotation, at time point D, the system is in a superposition of only two collective states, |E 0 and |E N , representing the SC state. The distribution of collective states remains unchanged at time points E, F and G. After the corrective auxiliary rotation, at time point H, the distribution returns to a shape with an envelope that is the same as that for a CSS. However, the distribution is modulated, with the depth of modulation determined by the phase shifts accumulated during the two dark zones. This modulated distribution pattern remains unchanged, at time point I, after the unsqueezing pulse. At the final time point J, the system again consists of just two collective states: |E 0 and |E N . For the particular choice of φ used here, these populations are equal. However, in general, the ratio of populations for the |E 0 and |E N in the final stage depends on the value of φ. When detecting the collective state |E 0 , we get a signal that is cosinusoidal, with fringes narrowed by a factor of N . As shown in the main text, we also get fringes with the same factor of narrowing when we detect the atomic states. Fig. 1, with two lobes at the end of the ±x axes on the Bloch sphere. However, the distribution of collective states is still the same as that at time point B. At time point D, after the auxiliary rotation, the QPD remains the same, but the distribution of collective states is now modulated. This distribution remains unchanged at time points E, F and G, despite the phase accumulated in the two dark zones. The modulations disappear at time point H after the application of the corrective auxiliary rotation, and the distribution is split into two distinct lobes. The separation between these two lobes depend on the value of φ. After the unsqueezing pulse, at time point I, the distribution remains the same as that at time point H. The final pulse produces modulations in each lobe. However, it should be noted that, unlike the case of N = 40, there is no population in either of the extremal collective states. Thus, when detecting the collective state |E 0 , the signal is zero. On the other hand, if the atomic states are detected, the signal as a function of φ is akin to that of a collective state atomic interferometer (COSAIN) [2], although with different amplitudes.
F. Fringe shapes for different values of the squeezing parameter µ
In the main body of the paper, we have presented the SCAIN protocol primarily for the case of µ = π/2, since this is the condition that produces the SC states. However, it is also instructive to consider the behavior of the CD-SCAIN as a function of the squeezing parameter µ, while keeping all other aspects (except φ) of the protocol unchanged. In Fig. 3, we illustrate the CD-SCAIN signal, as a function of φ, for different values of µ, for ARA =x and ξ = −1. Fig. 3 (a) shows the signal for µ = 0, where for comparison, we have also shown, as the black line, a full fringe of the CRAIN signal. For increasing values of µ, as shown in Fig. 3 (b)-(e), the central fringes become increasingly narrower. It should be noted that for these values of µ, the signals do not have a periodic behavior within the range of φ = −π and φ = π. In Fig. 3 (f), we show the limiting case of µ = π/2. As can be seen, the width of the central fringe remains the same for both odd and even values of N for values of µ somewhat less than π/2. In determining the values of QFR −1 for these cases (shown in Figure 4 of the main body), we have assumed that the interferometer would operate near the central fringe. Thus, the critical differences between the behavior of the odd and even values of N become manifest only when we are very close to or at the value of µ = π/2.
G. Justification of the dark zone operations
As mentioned in the main body of the paper, we have assumed that the phase shift for the SCAIN can be split equally between the two dark zones, and applied operations e −i φ 2Ĵ z (e i φ 2Ĵ z ) for the first (second) dark zone.
These operations can be easily understood in the case of a CRAIN. It can also be easily understood for the case of µ = π/2 under the protocol presented here. For an arbitrary value of µ, the quantum state prior to the first (second) dark zone may be distorted in a way so that the concept of two clear trajectories (forming different paths of the Michelson interferometer) may not hold. As such, it may not be obvious whether the application of this operation is valid for such a case. In fact, this operation remains valid under all conditions. Specifically, using an Hamiltonian to represent the Sagnac effect, H SE = Ω G · ( r × p), where r is the position and p is the momentum of an atom, the phase difference between paths traversed by the |↑ and |↓ components of the i-th atom can be accounted for by the operation e −i∆φĵi,z , where ∆φ = 2mΩ G ∆A/ , with ∆A being the differential area enclosed by these paths. SinceĴ z ≡ N iĵ i,z , it then follows that operations for the evolutions in the dark zones are valid in general.
H. Experimental simplification for CD-SCAIN compared to CSD-SCAIN
In this subsection, we review briefly the proposed scheme for implementing the CSD technique, and show how the SCAIN protocol can be greatly simplified experimentally by switching from CSD to CD. The complete experimental proposal for realizing the CSD technique is detailed in section IV of Ref. [2], where a null-detection scheme is employed to measure populations of one of the extremal Dicke collective states. The probe is one of the two counter-propagating Raman beams, which will induce Raman transitions within the atomic ensemble unless it is in the desired extremal collective state. As a result, there will be photons emitted corresponding to the other leg of the Raman transition. The probe and the emitted photons will be combined and sent to a high speed detector, which produces a dc voltage along with a beat signal with a beat frequency the same as that of the frequency synthesizer (FS) used to generate the two Raman beams but with an unknown phase. To extract the amplitude, the beat signal is bifurcated and one part is multiplied by the FS signal, while the other is multiplied by the FS signal phase shifted by 90 degrees. The signals are then squared before being recombined and sent through a low-pass filter (LPF) to derive the dc voltage. This dc voltage is proportional to the number of scattered photons. A lower limit is set for the voltage reading and any values recorded above it will indicate the presence of emitted photons. If no photon is emitted, the voltage will read below the limit, indicating that the ensemble is in the desired extremal collective state; otherwise at least one photon will be emitted and the ensemble is in other collective states. This process is then repeated many times for a given value of φ. The fraction of events where no photons are detected will correspond to the signal for this value of φ. This process is then repeated for several values of φ, producing the signal fringe for a CSD-SCAIN.
In contrast, the CD technique can be easily realized by coupling one of the two ground states involved in the Raman transition to some upper states of the atom and collect the fluorescence, thus avoiding the need for the aforementioned heterodyning and quadrature measurements. Moreover, the CSD technique requires an additional ring cavity to increase the optical density in order to enhance the signal (see section V of Ref. [2] for more details), which is not the case for the CD technique. All these factors taken into account, the CD version of the SCAIN protocol will be significantly simpler to implement experimentally.
It should be noted that even though the CSD protocol is experimentally more challenging and more sensitive to excess noise, it may be very useful for some applications, such as the test of the Penrose-Diosi theory of gravitationally induced decoherence [3][4][5][6][7] or a matterwave clock [8].
II. SCHRÖDINGER CAT ATOMIC CLOCK
As described in Ref. [9], the combination of one-axistwist squeezing (OATS), rotation, unrotation, unsqueezing and collective state detection can also be used to realize a Schrödinger Cat Atomic Clock (SCAC) with HL sensitivity. We will refer to this as the CSD-SCAC. In the main body of this paper, we have mentioned that such a SCAC with HL sensitivity can also be realized when conventional detection of atomic states is employed. We will refer to this as the CD-SCAC. In this section we will present the results obtained for the CD-SCAC, and comparison thereof with the CSD-SCAC.
A. Conventional atomic clock and Collective State atomic clock
In order to describe how the CSD-SCAC and the CD-SCAC work, it's useful to review briefly some details about the conventional atomic clock (CAC) as well as the collective state atomic clock (COSAC) [11]. Here we consider a system where the ground states, |1 and |2 of a three-level atom interact with an excited state |3 via two copropagating laser beams. One of the beams is detuned from resonance by δ 1 and has a Rabi frequency Ω 1 ; this couples |1 to |3 . The second beam is detuned from resonance by δ 2 and has a Rabi frequency Ω 2 ; this couples |2 to |3 . For δ Ω 1 , Ω 2 , Γ, where δ ≡ (δ 1 + δ 2 )/2 and Γ is the excited state decay rate, the system can be modeled as an effective two level system, consisting of states |1 and |2 , excited by a traveling wave with a Rabi fre-quency Ω = Ω 1 Ω 2 /(2δ), and detuning ∆ ≡ δ 1 − δ 2 . For simplicity, we assume Ω 1 = Ω 2 , and ∆ δ, so that δ 1 δ 2 . Under this condition, the light-shifts experienced by states |1 and |2 are essentially the same, and do not affect the equation of motion [10]. For more general cases, it is possible to incorporate any differences in the light shifts into the definition of ∆. Just as in the case of the SCAIN discussed in the main body of the paper, we denote states |1 and |2 as being the pseudo-spin states |↓ and |↑ , respectively. It should be noted that this is formally equivalent to a conventional microwave atomic clock that couples state |1 to state |2 . However, since a Raman transition is needed for the CSD protocol, we choose to describe it here as a Raman clock. In practice, for both the CSD and the CD protocols, all results presented here would remain valid for a conventional microwave excitation, which is preferable because a Raman clock may suffer from fluctuations in light shifts.
In a conventional Raman Ramsey atomic clock, which is equivalent to a CAC, an ensemble of N effective two-level atoms is first prepared in a CSS, denoted as |−ẑ ≡ |E 0 = N k=1 |↓ k . The first π/2 pulse produces a rotation about thex axis. During the interval, T D , before the second π/2 pulse, each atom acquires a phase φ = 2πf T D , where f = ∆/2π is the (two-photon) detuning of the clock (in Hertz). Application of the second π/2 pulse around thex axis produces the final state, which, for each atom, can be expressed, ignoring an overall phase-factor, as:
|Ψ = e −i π 2Ĵ x e −iφĴz e −i π 2Ĵ x |−ẑ = N k=1 1 2 {(1 − e iφ ) |↓ k − i(1 + e iφ ) |↑ k }(5)
In a CAC, typically the signal is a measure of the population of |↑ , given by S CAC = J + Ĵ z = N cos 2 (φ/2). The associated quantum projection noise is ∆S CAC = ∆Ĵ z = N/4|sin φ|. The stability of the clock is attributed to the quantum fluctuation in frequency (QFF), analogous to the QFR described earlier for a rotation sensor based on an atomic interferometer. This can be expressed as QFF = ∆f
| CAC = ∆(Ĵ z )/∂ f Ĵ z = (2πT D √ N ) −1 , where ∂ f ≡ ∂/∂f
. This can also be written as ∆f
| CAC = γ/ √ N , where γ = 1/(2πT D )
is the effective linewidth. This is, of course, the SQL value of the QFF.
In a COSAC, however, the signal is a measure of the population of one of the extremal collective states and is given by S COSAC = Q = cos 2N (φ/2), wherê Q ≡ |E N E N |. This signal shows a √ N -fold reduction in fringes compared to that of a CAC, which can be explained as follows. The first π/2 pulse couples the initial state |E 0 to |E 1 , which in turn is coupled to |E 2 and so on, effectively causing the ensemble to split into N + 1 states. During the dark zone, the n-th collective state |E n picks up a phase e −inφ . When the ensemble interacts with the last π/2 pulse, each of the collective states interferes with the rest of the collective states. The COSAC can thus be viewed as the aggregation of interference patterns due N +1 2 CAC's working simultaneously [11]. The narrowest constituent signal fringes are derived from interferences between states with the largest difference in phase, i.e. |E 0 and |E N ; the width of this fringe is γ/N . The width of the rest of the signal components range from γ to γ/(N − 1). The signal, which is the measure of population of |E N , is the result of the weighted sum of all the pairwise interferences, with a width of γ/ √ N . However, the system acts as a single particle, which reduces the effective SNR by the factor of √ N . As a result, we have shown that the QFF for the COSAC is essentially the same as that for the CAC [11].
From the analyses above, if the evolution of the system could be restricted to just the two extremal Dicke states (namely, |E 0 and |E N ) during the dark zone evolution, the fringes would be narrowed by a factor of N compared to those of the CAC. In that case, the QFF would be enhanced by a factor of √ N , thus reaching the HL sensitivity. As noted in the main body of the paper, the process of OATS indeed can be used to create just such a Schrödinger Cat (SC) state if the degree of squeezing is chosen to be µ = π/2, and an auxiliary rotation of π/2 is applied along a particular axis after the squeezing pulse. The resulting clock is then referred to as the SCAC. Just as in the case of the SCAIN, the exact effects of the protocol depend on a set of parameters such as the value (and parity) of N , the squeezing parameter µ for the OATS, the auxiliary rotation axis (ARA, which can bex orŷ ) around which to implement the rotation, the corrective rotation sign ξ which can take values of ±1 corresponding to redoing or undoing the first auxiliary rotation, and lastly the dark zone phase shift φ. The protocol illustrated in Fig. 4 (a) corresponds to the ARA chosen to be thex axis. The process starts by applying a π/2 pulse around thex axis. This is followed by the application of OATS, corresponding to a rotation around theẑ axis by an angle of µJ z , with µ = π/2. The next step is an auxiliary rotation of π/2 around thex axis. The ensuing evolution in the dark zone corresponds to a rotation by φ around theẑ axis, where φ = 2πf T D . This is now followed by another auxiliary rotation around thê x axis, by an angle of ξπ/2. This is followed by an unsqueezing pulse, which corresponds to a rotation around theẑ axis by an angle of −µJ z , with µ = π/2. Finally, the protocol ends with the application of the final π/2 pulse around thex axis. Mathematically, for this choice of the ARA, the whole protocol can thus be expressed as:
|Ψ f = e −i π 2Ĵ x e iµĴ 2 z e −iξ π 2Ĵ x e −iφĴz e −i π 2Ĵ x e −iµĴ 2 z e −i π 2Ĵ x |−ẑ (6)
In Fig. 4 (b), we show the evolution of the quantum states on a Bloch sphere, using the QPD, for an even value of N = 40, with µ = π/2, ξ = −1 and φ = 0.5π/N = π/80. In illustrating the nature of the QPD at various stages of the protocol, we have used different orientations, as needed. At the onset of the process (time point A), the system is assumed to be in the state |E 0 = |−ẑ , which is a CSS. After the first π/2 rotation around thex axis (time point B), it is in state |ŷ . After the squeezing pulse, the state (time point C) is split between two CSSs, and can be expressed as (|ŷ − η |−ŷ )/ √ 2, , where η = i(−1) N/2 , representing a phase factor with unity amplitude. This factor depends on the super even parity (SEP). However, the shapes of the fringes, as well as the values of QFF −1 , for both CSD and CD protocols, are not expected to depend on the value of the SEP, as we have verified explicitly. Application of the auxiliary rotation of π/2 around thex axis transforms this state to (|−ẑ + η |ẑ )/ √ 2. This (time point D) represents the desired SC state, as a superposition of the two extremal states of the ZDCS manifold:
(|E 0 + η |E N )/ √ 2.
During the dark zone, the phase shift causes a rotation by an angle of φ around theẑ axis, for each atom. The state after the dark zone can be expressed as e −iφĴz (|E 0 + η |E N )/ √ 2. Since both |E 0 and |E N are eigenstates of theĴ z operator, with eigenvalues (assuming = 1) of −N/2 and N/2 respectively, this state can be expressed as e iφN/2 |E 0 + e −iφN/2 η |E N / √ 2.
The resulting QPD, shown at time point E of Fig. 4 (b), remains unchanged, but the quantum state incorporates these phase accumulations. In order to reveal the interference magnified by the factor of N , it is necessary to apply first another auxiliary rotation, by an angle of ξπ/2 around thex axis. The QPD resulting from the case for ξ = −1 is shown at time point F. It is then necessary to apply the unsqueezing pulse, by an angle of −µĴ z , with µ = π/2. The QPD of the resulting state is shown at time point G. Finally, it is necessary to apply one more rotation around thex axis, by an angle of π/2. The QPD for the final state is shown at time point H. It is easy to show that, for this case, the final state can be expressed as |Ψ f = η cos(N φ/2) |E N + sin(N φ/2) |E 0 . For the particular value of φ (which is 0.5π/N ) used in generating the QPDs, the final state is (η |E N + |E 0 )/ √ 2. If the population of |E N were detected, the signal would be expressed as cos 2 (N φ/2), with fringes that are a factor of N narrower than that for the CAC, as shown in Ref. [9]. This is the CSD-SCAC discussed in Ref. [9]. Here we show that, the same results hold even if the CD process is used, thus realizing the CD-SCAC. In Fig. 5, the signal fringes for the CD-SCAC are plotted as a function of φ (red for N = 40 and dashed-blue for N = 41). For reference, we show in Fig. 5 (a) the signal corresponding to one full fringe of a CAC. For the remainder of the figures, µ = π/2. Fig. 5 (b) shows the signal for ARA=x and ξ = −1.
Here, the horizontal span of φ is smaller by a factor of 10. Consider first the signal for even N , in red, which shows 4 full fringes. This corresponds to a phase magnification by a factor of N = 40. Since the signal magnitude is the same as that for a CAC, one might be tempted to think that because of this phase magnification, the value of the QFF −1 for the CD-SCAC should be higher than that of a CAC by a factor of N . However, as we discussed in detail earlier, the standard deviation (SD) for the CD-SCAC signal is larger than that for a CAC by a factor of √ N . As such, the net enhancement in the value of the QFF −1 is by a factor of √ N , corresponding to HL sensitivity. Consider next the signal for odd N , in dashed-blue, which shows a much smaller variation as a function of φ. This same signal is shown again by the green line in Fig. 5 (d), but for a much larger range of φ, matching that of a full fringe for a CAC. Thus, the signal for odd values of N is similar to that for a Fabry-Perot resonator, with the width of the central fringe narrowed by a factor of ∼ √ N . As such, this signal is analogous to what is found for the COSAC, as detailed in Ref. [11], with the exception that, in the case of the CD-SCAC, the fringe amplitude is N/2, while for the COSAC it is 1. Again due to the increased SD, the sensitivity of the CD-SCAC for this case is the same as that for a CAC and the COSAC. Fig. 5 (c) shows the CD-SCAC signal for ARA=x and ξ = +1. As expected, in this case the fringes for both even (red) and odd (dashed-blue) values of N are flipped around the zero value. The signal for the odd value of N is shown again by the dashed black line in Fig. 5 (d) on a scale where the span of φ is the same as that for a full fringe of the CAC, again showing the Fabry-Perot type resonance, reduced in width by a factor of ∼ √ N . The values of QFF −1 , and therefore the sensitivities, are the same as those for the case shown in Fig. 5 (b).
In Fig. 5 (e), we show the signal for a variant of the protocol where ARA=ŷ and ξ = ±1. For this protocol, the behaviors for odd (dashed-blue) and even (red) values of N are essentially reversed. However, for this value of the ARA, we find that the signals are the same for both values of ξ. In Fig. 5 (f), we show the signal, for the odd value of N , on a scale where the span of φ is the same as that for a full fringe of the CAC, again showing the Fabry-Perot type resonance, reduced in width by a factor of ∼ √ N .
D. QFF −1 for the CD-SCAC
In Fig. 6, we illustrate the behavior of QFF −1 , as a function of µ, with ξ = +1, for different choices of parameters for the CD-SCAC, along with a comparison with the CSD-SCAC and the Echo Squeezing Protocol (ESP) [12,13]. In each case, the QFF −1 is normalized to the QFF −1 HL for N = 40, indicated as the solid black line. The dashed black line shows the QFF −1 SQL for N = 40. For µ = π/2, we see that the sensitivity for both CD and CSD protocols yield the HL sensitivity. This sensitivity is reached due to an amplification of phase by a factor of N , and a concomitant increase in the SD by a factor of √ N . Fig. 6 (b) is the same as Fig. 6(a), except that N = 41. In this case, µ = π/2, for the CD-SCAC, there is a phase amplification, manifested as a Fabry-Perot like fringe around φ = 0 which is narrowed by a factor of ∼ √ N , along with an increase in the square-root of the variance by a factor of ∼ √ N . The difference between the even and odd cases disappears when the value of µ is reduced below a threshold value of ∼ 0.45π. There is a range of values of the squeezing parameter (0.2π ≤ µ ≤ 0.45π) over which the normalized value of QFF −1 is ∼ 0.71 for the CD-SCAC. We have verified that this plateau ratio between QFF −1 and QFF −1 HL remains unchanged when N is increased or decreased. We also see that, for this choice of the ARA, the behavior of the CSD-SCAC is drastically different. Specifically, for odd values of N , the QFF −1 is strictly zero for all values of the squeezing parameter, and for even value of N , the QFF −1 drops to zero quickly for µ < π/2. Fig. 6 (c) and Fig. 6 (d) are similar to Fig. 6 (a) and Fig. 6 (b), respectively, but with the ARA chosen to be theŷ axis. In this case, it should be noted that the behavior of the CD-SCAC and the CSD-SCAC are essentially the same, except for a small range of value of µ around 0.05π. We also note that, for this choice of the ARA, the HL sensitivity is reached for odd values of N . Finally, in each of these four cases, we have used the green line to show the corresponding sensitivity achievable under the ESP.
So far, we have presented the value of QFF −1 separately for odd and even values of N . In certain cases, such as for a magnetometer using NVD, where it is possible to operate with a fixed parity of N , the values of QFF −1 for a given parity is relevant. For other situation, such as a clock using atoms cooled in a magneto-optical trap (MOT) and released for interrogation, it is necessary to consider the effect of averaging over the two parities. As shown in the main body, in this case the average value is given by QFF −
FIG. 2 .
2(a) Schematic illustration of the protocol employed for the Schrödinger Cat Atomic Interferometer (SCAIN). (b) The Husimi Quasi Probability Distributions (QPDs) at different stages of the protocol, for N = 40, µ = π/2, ARA =x, ξ = −1 and φ = 0.5π/N .
(b), the expression for the signal for the CD-SCAIN shown above applies only to the case when N is even. The results for odd value of N = 41 with FIG. 3. Signals corresponding to detection of Ĵ z / , as a function of φ, for µ = π/2, ARA =x and ξ = −1. N = 40 is red while N = 41 is dashed-blue. (a) Fringes for CRAIN for comparison; (b) Fringes for CD-SCAIN; (c) Zoomed-in fringes for CD-SCAIN. The horizontal span in (c) is 10 times smaller than those in (a) and (b).
FIG. 4 .
4Illustration of QFR −1 for different cases, as a function of the squeezing parameter µ, normalized to the HL (solid black line), for ARA =x and ξ = +1. (a) The case for the CD-SCAIN, with red for N = 40 and dashed-blue for N = 41; (b) Comparison between the CD-SCAIN and the CSD-SCAIN for even N = 40; (c) Comparison between the CD-SCAIN and the CSD-SCAIN for odd N = 41. The dotted black line shows the SQL. QPF −1 when the the phase difference φ between the two paths of the interferometer is induced by rotation [31]. For each case shown here, the QFR −1 is normalized to the QFR −1 HL for N = 40, indicated as the solid black line. The dashed black line shows the QFR −1 SQL for N = 40. Fig. 4 (a) shows the QFR −1 for the CD-SCAIN only.
[ Q 2 M,CSD,0 − Σ 2 ] 1/2 . Similarly we define the signal for the CD-SCAIN as S ≡ Q M,CD = −(N/2) cos(N φ) and the SD as ∆S ≡ [ Q 2 M,CD − S 2 ] 1/2 . Noting that φ = 2mAΩ G / ≡ Ω G /Γ, with A being the area of the whole interferometer and Ω G being the normal component of arXiv:1707.08260v6 [quant-ph] 13 Nov 2017
D. QPD evolutions for odd value of N FIG. 1. The QPDs for different stages of the SCAIN protocol, with N = 41, µ = π/2, ARA =x, ξ = −1 and φ = π/4.
E. Collective state distributions for both even and odd values of N FIG. 2. Population of the collective states at various stages of the CD-SCAIN protocol: (a) For the case with N = 40, φ = π/80; (b) For the case with N = 41, π/4. For both cases, we have µ = π/2, ARA =x and ξ = −1.
Fig. 2 (
2b) corresponds to the case when N = 41. The distributions for time points A and B are the same as that for N = 40. At time point C, the quantum state is different, as can be seen in the QPD plots in
FIG. 3 .
3Fringe shapes for different values of the squeezing parameter µ, while keeping the rest of the protocol unchanged, for ARA=x and ξ = −1. N = 40 is red while N = 41 is dashed-blue. (a) µ = 0; (b) µ = 0.021π; (c) µ = π/8; (d) µ = π/4; (e) µ = 3π/8; (f) µ = π/2. The black line in (a) shows the full fringe of a CRAIN for comparison, and the horizontal spans in (b)-(f) are 10 times smaller than that in (a).
B. The complete protocol for the SCAC FIG. 4. (a) Schematic illustration of the protocol employed for Schroedinger Cat Atomic Clock (SCAC). (b) The QPDs at different stages of the protocol, for N = 40, µ = π/2, ARA =x, ξ = −1 and φ = 0.5π/N .
C. Signal fringes for the CD-SCACFIG. 5. Signals corresponding to detection of Ĵ z / , as a function of φ. N = 40 is red while N = 41 is dashed-blue. (a) Fringes for a CAC for comparison; (b) CD-SCAC with ARA=x and ξ = −1; (c) CD-SCAC with ARA=x and ξ = +1; (d) Zoomed-out plots for N = 41 with ξ = −1 in green and ξ = +1 in black, for CD-SCAC with ARA=x; (e) CD-SCAC with ARA=ŷ and ξ = ±1; (f) Zoomed-out plots for N = 40 with ξ = −1 in green and ξ = +1 in black, for CD-SCAC with ARA=ŷ. Here, µ = π/2 for all cases, except in (a) which has no squeezing. Also note the horizontal spans in (b), (c) and (e) are 10 times smaller than those in (a), (d) and (f).
FIG. 6 .
6Illustration of QFF −1 for different cases, as a function of the squeezing parameter µ, normalized to the HL (solid black line). (a) The case for even N = 40, with ARA asx; (b) The case for odd N = 41, with ARA asx; (c) The case for even N = 40, with ARA asŷ; (d) The case for odd N = 41, with ARA asŷ. The dotted black line shows the SQL. Red is for CD-SCAC, dashed-blue for CSD-SCAC and green for the ESP case. For all cases shown, ξ = +1.
Fig. 6 (
6a) corresponds to N = 40, with ARA being thex axis. Here, the red line corresponds to the CD-SCAC, and the dashed-blue line is for the CSD-SCAC.
FIG. 1. The sensitivity, Λ, as a function of excess noise, ∆S EN , for various protocols. HL: Heisenberg Limit; SQL: Standard Quantum Limit; TACT: Two-axis-counter-twist; ESP: Echo Squeezing Protocol; SCAIN: Schrödinger Cat Atom Interferometer; CD: Conventional Detection; CSD: Collective State Detection. For both CSD-SCAIN and CD-SCAIN, we have used two labels: I and II; I indicates the case when the parity of N is known, while II indicates the case where the signal is averaged over both parities.
AV G ∼ = QFF −1 QHL . Finally, if QFF −1 EV EN = QFF −1 QHL and QFF −1 ODD = QFF −1 QHL , then QFF −1 AV G = QFF −1 QHL . * [email protected]
AV E =
(QFF −1
EV EN ) 2 /2 + (QFF −1
ODD ) 2 /2
1/2 . Using this re-
sult, we can reach the following conclusions, assuming
N
1. If QFF −1
EV EN = QFF −1
HL and QFF −1
ODD = 0,
then QFF −1
AV G = QFF −1
QHL , where we define QFF −1
QHL ≡
QFF −1
HL /
√ 2 . Similarly, if QFF −1
EV EN = QFF −1
HL and
QFF −1
ODD = QFF −1
SQL , then QFF −1
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representing whether N/2 is even or odd; however, the shapes of the fringes, as well as the values of QFR −1 , for both CSD and CD protocols. SEPIt should be noted that this phase factor depends on the super-even-parity. are not expected to depend on the value of the SEP, as we have verified explicitlyIt should be noted that this phase factor depends on the super-even-parity (SEP), representing whether N/2 is even or odd; however, the shapes of the fringes, as well as the values of QFR −1 , for both CSD and CD protocols, are not expected to depend on the value of the SEP, as we have verified explicitly.
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| []
|
[
"Resonance energy transfer between two atoms in a conducting cylindrical waveguide",
"Resonance energy transfer between two atoms in a conducting cylindrical waveguide"
]
| [
"Giuseppe Fiscelli ",
"Lucia Rizzuto ",
"Roberto Passante ",
"\nDipartimento di Fisica e Chimica\nUniversità degli Studi di Palermo\nVia Archirafi 36I-90123PalermoItaly\n",
"\nINFN\nLaboratori Nazionali del Sud\nI-95123CataniaItaly\n"
]
| [
"Dipartimento di Fisica e Chimica\nUniversità degli Studi di Palermo\nVia Archirafi 36I-90123PalermoItaly",
"INFN\nLaboratori Nazionali del Sud\nI-95123CataniaItaly"
]
| []
| We consider the energy transfer process between two identical atoms placed inside a perfectly conducting cylindrical waveguide. We first introduce a general analytical expression of the energy transfer amplitude in terms of the electromagnetic Green's tensor; we then evaluate it in the case of a cylindrical waveguide made of a perfect conductor, for which analytical forms of the Green's tensor exist. We numerically analyse the energy transfer amplitude when the radius of the waveguide is such that the transition frequency of both atoms is below the lower cutoff frequency of the waveguide, so that the resonant photon exchange is strongly suppressed. We consider both cases of atomic dipoles parallel and orthogonal to the axis of the guide. In both cases, we find that the energy transfer is modified by the presence of the waveguide. In the near zone, that is when the atomic separation is smaller than the atomic transition wavelength, the change, with respect to the free-space case, is small for axial dipoles, while it is larger for radial dipoles; it grows when the intermediate region between near and far zone is approached. In the far zone, we find that the energy transfer amplitude is strongly suppressed by the waveguide, becoming virtually zero. A physical interpretation of these results is discussed. Finally, we discuss the resonance interaction energy and force between two identical correlated atoms in the waveguide, one excited and the other in the ground state, prepared in their symmetric or antisymmetric superposition. | 10.1103/physreva.98.013849 | [
"https://arxiv.org/pdf/1804.07226v2.pdf"
]
| 119,252,124 | 1804.07226 | c911feb0f650f9b883575893bcff8cf53a2c2726 |
Resonance energy transfer between two atoms in a conducting cylindrical waveguide
19 Apr 2018
Giuseppe Fiscelli
Lucia Rizzuto
Roberto Passante
Dipartimento di Fisica e Chimica
Università degli Studi di Palermo
Via Archirafi 36I-90123PalermoItaly
INFN
Laboratori Nazionali del Sud
I-95123CataniaItaly
Resonance energy transfer between two atoms in a conducting cylindrical waveguide
19 Apr 2018arXiv:1804.07226v1 [quant-ph]
We consider the energy transfer process between two identical atoms placed inside a perfectly conducting cylindrical waveguide. We first introduce a general analytical expression of the energy transfer amplitude in terms of the electromagnetic Green's tensor; we then evaluate it in the case of a cylindrical waveguide made of a perfect conductor, for which analytical forms of the Green's tensor exist. We numerically analyse the energy transfer amplitude when the radius of the waveguide is such that the transition frequency of both atoms is below the lower cutoff frequency of the waveguide, so that the resonant photon exchange is strongly suppressed. We consider both cases of atomic dipoles parallel and orthogonal to the axis of the guide. In both cases, we find that the energy transfer is modified by the presence of the waveguide. In the near zone, that is when the atomic separation is smaller than the atomic transition wavelength, the change, with respect to the free-space case, is small for axial dipoles, while it is larger for radial dipoles; it grows when the intermediate region between near and far zone is approached. In the far zone, we find that the energy transfer amplitude is strongly suppressed by the waveguide, becoming virtually zero. A physical interpretation of these results is discussed. Finally, we discuss the resonance interaction energy and force between two identical correlated atoms in the waveguide, one excited and the other in the ground state, prepared in their symmetric or antisymmetric superposition.
I. INTRODUCTION
Resonance energy transfer between quantum emitters is the exchange of excitation between them mediated by the quantum electromagnetic field [1,2]. This process is of considerable importance in many different fields of physics, as well as in chemistry or biology, where coherent energy transfer between chromophores is supposed related to the very high efficiency in light-harvesting in the photosynthesis process [3,4]. It is also directly related to the resonance interaction force, that is a force resulting from the photon exchange between two atoms in the vacuum space, one excited and the other in the ground state, prepared in their symmetric or antisymmetric state [2,5]. In the energy transfer process, the excitation, initially localised on one atom (donor), is transferred to the other atom (acceptor) through the electromagnetic field. Description of this process at different distances, i.e. nearfield (radiationless), far-field (radiative or retarded) and the intermediate region where retardation effects start to appear, requires a full quantum-electrodynamical theory [6,7]. For atoms in the free space, the energy-transfer amplitude behaves as r −6 in the near-field region (Förster limit) and as r −2 in the far-field region [8].
Since the pioneering work of Purcell, it is known that radiative processes of any quantum emitter(s), for example the spontaneous emission of one or more atoms, are affected by the environment [9]. Radiation-mediated interactions, such as van der Waals and Casimir-Polder interactions, can be also significantly affected by the en- * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] vironment, that changes the photon density of states and the dispersion relation [10][11][12][13], as well as from the presence of neighbouring atoms [14]. Recently, investigations on how to control and tailor radiative processes through the environment have become a very active field of research, even in the case of time-modulated environments [15][16][17]. For example, the effect of a structured environment such as a photonic crystal on the dipole-dipole interaction [18][19][20][21], or on the resonance interaction force between two entangled atoms [5,22], has been investigated, showing possibility of enhancement or inhibition of the interaction. The effect of dispersive and absorbing surrounding media on the energy transfer between two atoms has been investigated [23,24], as well as the possibility to control the resonance energy transfer between nanostructured emitters through a reflecting plate [25,26].
In this paper we consider the resonance energy transfer between two identical atoms, or any other quantum emitter (for example quantum dots), placed on the axis of a cylindrical waveguide made of a perfect conductor. We first obtain an analytical expression of the energy transfer amplitude in terms of the electromagnetic Green's tensor of the cylindrical waveguide, whose expression is known. We then evaluate numerically the energy transfer amplitude in terms of the relevant parameters of the system, specifically the distance between the atoms and the radius of the waveguide, relative to the atomic transition wavelength. We consider both cases of atomic dipoles parallel and orthogonal to the guide axis. We explicitly show that the presence of a lower cutoff frequency inside the waveguide can deeply change the energy transfer amplitude, with respect to the case of atoms in the free space, in the far (radiative) zone, while in the near (radiationless) zone the change, although present, is much less relevant. A physical interpretation of these results is given. The relation of the results obtained with the resonance interaction energy and force between two identical entangled atoms inside the cylindrical waveguide is also discussed.
This paper is structured as follows. In Sec. II we introduce our system and introduce a general expression of the resonance energy transfer between the atoms in terms of the electromagnetic Green's tensor of a generic environment. In Sec. III we evaluate, using both analytical and numerical methods, the energy transfer amplitude for atoms inside a metallic cylindrical waveguide, showing how the guide can significantly change the energytransfer process according to the relevant parameters of the system. In Sec. IV we discuss the relevance of our results for the resonance interaction energy between correlated atoms. Finally, Sec. V is devoted to our conclusions and final remarks.
II. ENERGY TRANSFER IN TERMS OF THE ELECTROMAGNETIC GREEN'S TENSOR
We first introduce the energy transfer process between two identical two-levels atoms (or quantum dots), located inside a generic macroscopic structured environment and interacting with the quantum electromagnetic field. Let us consider two atoms, labeled with A and B, and suppose that atom A (donor) is in its excited state, while atom B (acceptor) is in its ground state. The atom A can decay and emit a real or virtual photon that can be absorbed by atom B. Excitation is thus transferred from donor to acceptor atom through the electromagnetic field [27]. In this section we obtain the energy transfer rate between the two atoms in a generic macroscopic environment, whose properties are described by its electromagnetic Green's tensor [23,28,29]. According to the generalized Fermi golden rule [1,8], this is given by
W i→f = 2π | ψ i | T |ψ f | 2 δ(E f − E i ),(1)
where |ψ i = |e A , g B , 0 and |ψ f = |g A , e B , 0 are respectively the initial and the final state of the two-atom system, with energy E i and E f . | g A/B and | e A/B are respectively the ground and excited atomic states, and | 0 represents the photon vacuum state in the presence of the external environment. We assume the two atoms identical, with transition frequency ω 0 = ck 0 . T is the transition operator at the second order,
T = H i + H i 1 E i − H 0 + iη H i ,(2)
where H 0 and H i are the unperturbed and interaction Hamiltonians, respectively, and E i = E f is the energy of the initial and final states. The second-order energytransfer amplitude between the two atoms can be written as
M = ψ i | T |ψ f = I ψ i | H i |I I| H i |ψ f E i − E I ,(3)
where |I are the intermediate states with energy E I , that can contribute to the energy transfer process. If both atoms are in the free space, the energy transfer amplitude is given by the well-known expression [2,30,31]
M f s = d ge Ai d eg Bj 4πǫ 0 r 3 δ ij − 3r irj cos k 0 r + k 0 r sin k 0 r − δ ij −r irj k 2 0 r 2 cos k 0 r ,(4)
where we have used the Einstein notation. k 0 is the atomic transition wavenumber associated to its transition frequency, r is the distance between the atoms and d ge αi = g| d αi |e (α = A, B) their dipole-moment matrix elements.
We now assume the atoms placed in a generic linear magneto-dielectric environment. It is well known that the presence of the environment can significantly affect the resonant energy transfer process [23,25,32]. To investigate the excitation exchange between the two atoms inside a structured environment, we exploit a procedure based on the Green's tensor formalism [23,28,29,33]. This method has been widely used in many contexts, from quantum electrodynamics to quantum optics, and its merit is that all relevant properties and effects of the environment are included in the Green's tensor expression. It has been used, for example, to evaluate van der Waals and Casimir-Polder forces in external environments [13,[34][35][36][37], or to investigate the collective spontaneous decay of two quantum emitters placed nearby a reflecting mirror [38]. We will use this approach to obtain the energy transfer amplitude when two emitters are inside an environment such as a conducting cylindrical waveguide. We first briefly review the method and the relevant expressions for the energy transfer in a generic linear magnetodielectric environment [23]. In the next section, we will then specialise our considerations to the specific case of a perfectly conducting cylindrical waveguide, which is the main point of this paper.
The Green's tensor G(r, r ′ , ω) is defined as the solution of the Helmholtz equation (see, for example, [29,33])
∇× 1 µ(r, ω) ∇× − ω 2 c 2 ǫ(r, ω) G(r, r ′ , ω) = δ(r−r ′ ),(5)
with the boundary condition G(r, r ′ , ω) → 0 for |r−r ′ | → ∞, and where ǫ(r, ω) and µ(r, ω) are respectively the electric and magnetic permittivity of the medium. The medium-assisted electric field operator is expressed as
E(r) = ∞ 0 dωE(r, ω) + H.c. = λ=e,m d 3 r ′ ∞ 0 dωG λ (r, r ′ , ω) · f λ (r ′ , ω) + H.c.,(6)
where
G e (r, r ′ , ω) = i ω 2 c 2 πǫ 0 Im ǫ(r ′ , ω)G(r, r ′ , ω)(7) G m (r, r ′ , ω) = i ω c − πǫ 0 Im 1 µ(r ′ , ω) [∇ ′ × G(r, r ′ , ω)] T (8)
are respectively the electric and magnetic Green's tensor components. They satisfy the following relation λ=e,m
d 3 sG λ (r, s, ω) · G * T λ (r ′ , s, ω) = µ 0 π ω 2 ImG λ (r, r ′ , ω).(9)
The bosonic matter-assisted operators f † λ (r, ω) and f λ (r, ω) in (6) are the creation and annihilation operators describing the combined system of the electromagnetic field and the magnetodielectric medium. They satisfy the following commutation relations
f λi (r, ω), f † λ ′ i ′ (r ′ , ω ′ ) = δ λλ ′ δ ii ′ δ(r − r ′ )δ(ω − ω ′ ), f λi (r, ω), f λ ′ i ′ (r ′ , ω ′ ) = 0,(10)
where the subscript λ = e, m refers to the electric and magnetic parts. The Hamiltonian of our system can be expressed as
H = H a + H F + H i(11)
where H a and H F are respectively the unperturbed atomic and field Hamiltonians (in the presence of the medium), given by
H a = n=e,g E A n |n A n A | + n=e,g E B n |n B n B | ,(12)H F = λ=e,m d 3 r ∞ 0 dω ωf † λ (r, ω) · f λ (r, ω),(13)
where we have modeled the atoms as two-level systems (|e and |g being the excited and ground state with energy E e and E g , respectively), and H i is the interaction Hamiltonian in the multipolar coupling scheme, within the dipole approximation
H i = −d A · E(r A ) − d B · E(r B ).(14)
Here |n A(B) are eigenstates of the atomic Hamiltonian of atom A(B) at position r A(B) , d A(B) is the atomic electric dipole moment operator, and E(r) is the electric field operator evaluated at the atomic position r = r A(B) . Using second-order perturbation theory, the energy transfer amplitude M is (see Eq. (3))
M = I e A , g B , 0| H i |I I| H i |g A , e B , 0 E i − E I ,(15)
where E i = ω 0 , with ω 0 = ck 0 the transition frequency of the atoms.
|I 1 = |g A , g B , 1 λ (r, ω) , |I 2 = |e A , e B , 1 λ (r, ω) ,(16)
where 1 λ (r, ω) is an medium-assisted excitation of the field. The sum over the intermediate states (16) in Eq. (15) can be written as a sum over λ (electric and magnetic field modes), and an integral over space and frequency, that is
∆E = λ=e,m d 3 r ∞ 0 dω × ψ i | H i |g A , g B , 1 λ (r, ω) g A , g B , 1 λ (r, ω)| H i |ψ f E i − E I1 + ψ i | H i |e A , e B , 1 λ (r, ω) e A , e B , 1 λ (r, ω)| H i |ψ f E i − E I2 .(17)
Using the expression (6) of the electric field in terms of the Green's tensor, the interaction Hamiltonian (14), commutation relations (10) and the relation (9), after some algebra we have
M = πǫ 0 c 2 +∞ 0 dωω 2 ij 1 ω 0 − ω × d eg Ai ImG ij (r A , r B , ω)d ge Bj − 1 ω 0 + ω d eg Ai ImG ij (r B , r A , ω)d ge Bj .(18)
Equation (18) gives the amplitude probability that the electronic excitation is transferred from one atom to the other, when the atoms are placed inside a generic linear magnetodielectric environment, whose properties are expressed in terms of the electromagnetic Green's tensor.
We wish to stress that the process discussed above is also directly related to a quantum interaction energy between the atoms. It is the resonance interaction energy between two atoms, one in an excited state and the other in the ground state, prepared in a correlated (symmetric or antisymmetric) state in the photon vacuum [5]. Indeed, the second-order energy shift ∆E due to the atomfield interaction (exchange of one real or virtual photon between the atoms) is given by
∆E = I ψ| H i |I I| H i |ψ ω 0 − E I ,(19)
where the state |ψ is
|ψ = 1 √ 2 (|e A , g B , 0 ± |g A , e B , 0 ).(20)
In such a case, the excitation is delocalized between the atoms and the field is in the vacuum state. Following the same procedure used before, we obtain a general expression of the distance-dependent resonance interaction energy between the two atoms in terms of the Green's tensor of a generic environment (apart single-atom energy corrections that do not contribute to the interatomic force)
∆E ± = ± 2πǫ 0 c 2 +∞ 0 dωω 2 ij 1 ω 0 − ω × d eg Ai ImG ij (r A , r B , ω)d ge Bj + d eg Bi ImG ij (r B , r A , ω)d ge Aj − 1 ω 0 + ω d ge Bi ImG ij (r B , r A , ω)d eg Aj + + d ge Ai ImG ij (r A , r B , ω)d eg Bj .(21)
The resulting interatomic force between the two entangled atoms, prepared in their symmetrical (+) or antisymmetrical (-) superposition, is then obtained by taking the derivative of (21) with respect to r =| r A − r B |, changed of sign. In the last section of this paper we will specialise this result to the case of two atoms inside a perfectly conducting cylindrical waveguide.
III. ENERGY TRANSFER BETWEEN TWO ATOMS INSIDE A CONDUCTING CYLINDRICAL WAVEGUIDE
We now specialise our investigation to the case of two identical two-level atoms, one excited and the other in its ground state, placed inside a perfectly conducting cylindrical waveguide, as shown in Fig. 1. The waveguide consists of a perfectly conducting cylindrical shell of radius R; we suppose that the atoms are located on the axis of the waveguide, z being the interatomic distance. According to Eq. (18), in order to obtain the energy transfer amplitude between the two atoms, we need the expression of the electromagnetic Green's tensor for the cylindrical waveguide.
The analytical expression of Green's function of a cylindrical waveguide, with the appropriate boundary conditions, is known in the literature and it has the following form [39] (see also [13,40]) where the M and N vector cylindrical wave functions are given by
G(r, r ′ , ω) = − 1 k 2 δ(r − r ′ )ẑ ⊗ẑ + n,m c µn Me o nµ (±k µ )M ′ e o nµ (∓k µ ) + c λn Ne o nλ (±k λ )N ′ e o nλ (∓k λ ) (z ≷ z ′ ),(22)Me o mµ (h) = ∓ nJ n (µr) r sin cos (nφ)r− ∂J n (µr) ∂r cos sin (nφ)φ e ihz ,(23)Ne o nλ (h) = 1 k ih ∂J n (λr) ∂r cos sin (nφ)r ∓ ihn r J n (λr) sin cos (nφ)φ + λ 2 J n (λr) cos sin (nφ)ẑ e ihz .(24)
Here µ = qnm R and λ = pnm R , with p nm the mth root of the n-order Bessel function (J n (p nm ) = 0), and q nm the mth root of the derivative of the n-order Bessel function ( J ′ n (q nm ) = 0). λ and µ are the radial components of the wavevector k (k = ω c ) of the electric field component for, respectively, the traverse magnetic (TM) and transverse electric (TE) modes inside the waveguide; likewise, k λ and k µ are the axial components of the wavevector. Since the atoms are placed on the cylinder's axis, the Green's tensor (22) can be simplified as [13]
G(r, r ′ , ω) = i 4π m e ikµz 2I µ1 k µ + k λ e ik λ z 2I λ1 k 2 × (r ⊗r +φ ⊗φ) + λ 2 e ik λ z I λ0 k λ k 2ẑ ⊗ẑ ,(25)
where
I µ1 = R 2 2 1 − 1 q 2 1m J 2 1 (q 1m ),(26)I λ1 = R 2 4 J 0 (p 1m ) − J 2 (p 1m ) 2 ,(27)I λ0 = R 2 2 J 2 1 (p 0m ).(28)
The presence of the cylindrical waveguide changes the density of states of the electromagnetic field inside it, and in particular determines a lower cut-off frequency for the TE and TM modes inside the waveguide, given by
(ω min ) T M ∼ 2.4c R ; (ω min ) T E ∼ 1.8c R .(29)
Since the value of ω min depends from the waveguide radius R, it is possible to modify the electromagnetic field modes allowed inside the waveguide, and in particular its lower cutoff frequency. It is thus possible to control the energy transfer between the atoms, by changing R.
The presence of a lower cut-off frequency, as we will now show, has a strong effect on the excitation transfer between the atoms: if k 0 R ≪ 1, the atomic transition frequency is smaller than the waveguide cut-off frequency (ω 0 < ω min ), and thus the waveguide suppresses the e.m. field modes resonant with the atomic transition frequency. Since they cannot contribute to the exchange of excitation between the atoms, the energy transfer will be strongly suppressed in this regime. Otherwise, when ω 0 > ω min the resonant field modes do contribute to the excitation transfer, and we expect that the energy transfer amplitude will be much less influenced by the presence of the waveguide.
In this paper we mainly focus on the first regime above mentioned, ω 0 < ω min , where the presence of the guide is expected to be relevant. In this case, the frequency ω min is the lower limit of the frequency integral in Eq. (18). Since the waveguide Green's tensor (25) is symmetric with respect to the exchange of the atomic positions, Im G ij (r A , r B , ω) = Im G ij (r B , r A , ω) (∀i, j, ω), the energy transfer amplitude (18) becomes
M = 2 πǫ 0 ij d eg Ai d ge Bj +∞ kmin dk k 3 k 2 0 − k 2 Im G ij (r A , r B , ω).(30)
Since G ij (r A , r B , ω) is diagonal, we can write Eq. (30) as a sum of three terms
M = M z + M r + M φ ,(31)M z = (d eg A ·ẑ)(d ge B ·ẑ) 2π 2 ǫ 0 m λ 2 I λ0 +∞ kmin dk k cos k λ z (k 2 0 − k 2 )k λ ,(32)M r = (d eg A ·r)(d ge B ·r) 4π 2 ǫ 0 m +∞ kmin dk k 3 k 2 0 − k 2 × cos k µ z I µ1 k µ + k λ cos k λ z I λ1 k 2 ,(33)M φ = (d eg A ·φ)(d ge B ·φ) 4π 2 ǫ 0 m +∞ kmin dk k 3 k 2 0 − k 2 × cos k µ z I µ1 k µ + k λ cos k λ z I λ1 k 2 ,(34)
where k min = ω min /c. As mentioned, we now investigate in detail the behavior of the energy transfer amplitude in the regime Rk 0 ≪ 1, as a function of the relevant parameters of the system: the interatomic distance z =| r A − r B |, the waveguide lower cut-off frequency ω min and the orientation of the atomic dipoles with respect to the waveguide axis. For symmetry reasons, we need to consider only the two following cases: axial dipoles (M z contribution) and radial dipoles (M r contribution), relative to the waveguide axis.
A. Axial dipoles
We first assume that the atomic dipole moments are parallel and oriented along the positive z axis (that is along the waveguide axis). In this case the energy transfer amplitude M is given only by the M z term, because (33) and (34) vanish. Taking into account Rk 0 ≪ 1, the integral over k in (32) becomes
+∞ kmin dk k cos k λ z (k 2 0 − k 2 )k λ = +∞ kmin dk k cos √ k 2 − λ 2 z (k 2 0 − k 2 ) √ k 2 − λ 2 .(35)
After the change of integration variable t = √ k 2 − λ 2 , application of the residue theorem yields
+∞ 0 dt cos tz k 2 0 − λ 2 − t 2 = π(k 2 0 − λ 2 )e − √ λ 2 −k 2 0 z 2 (λ 2 − k 2 0 ) 3 ,(36)
and we obtain
M z = d eg Az d ge Bz 2πǫ 0 m λ 2 I λ0 (k 2 0 − λ 2 )e − √ λ 2 −k 2 0 z 2 (λ 2 − k 2 0 ) 3 .(37)
The sum on m in (37) is over all radial field modes allowed in the waveguide. Using the root test, it is possible to show that this series converges. We have evaluated numerically this quantity and verified explicitly that a good estimate for the energy transfer amplitude can be obtained by taking the first thirty terms of this sum.
We have evaluated numerically the energy transfer amplitude (37) as a function of the interatomic distance z, in two different regimes: z < λ 0 (near zone) and z > λ 0 (far zone). In our numerical evaluation, we have chosen the waveguide radius equal to R = 10 −8 m, and the atomic transition wavelength λ 0 = 5 · 10 −7 m. Fig. 2 shows the energy transfer amplitude between the two atoms in the near zone as a function of the interatomic distance z, in the waveguide (blue continuous curve) and in the free space (orange dashed curve). The two plots show that the behavior with the distance is similar in the two cases. The plot in Fig. 3 represents the ratio between the energy transfer amplitude in the waveguide and in free space. It shows that for z 1. the near and far zone is approached, the amplitude in the waveguide becomes more and more suppressed. This result is related to the fact that in the near zone the interaction is essentially the electrostatic dipoledipole interaction, which is not significantly modified by the waveguide. Approaching the intermediate region, z ∼ λ 0 , the effect of the waveguide on the amplitude becomes more relevant. On the contrary, in the far zone, a numerical analysis shows that the energy transfer in the waveguide is strongly inhibited with respect to the free-space case by several tens orders of magnitude: it is virtually zero for z > λ 0 .
The results obtained show that, since in the regime Rk 0 ≪ 1 (ω 0 < ω min ) the resonant field modes are suppressed, the energy transfer in the waveguide is slightly reduced in the very near zone, more and more suppressed approaching the intermediate region, and totally inhibited (virtually zero) in the far zone). The completely different effect of the waveguide in the near and far zone is due to the fact that in the near zone the process is essentially radiationless, while in the far zone it is a resonant radiative process and, in the regime considered, photons resonant with the atomic transition frequency cannot propagate in the guide.
We can also investigate the energy transfer amplitude as a function of the waveguide cut-off frequency ω min ∝ 1 R . By decreasing R, the cut-off frequency ω min increases and the gap between ω 0 and ω min increases too, further reducing the energy-transfer process. Fig. 4 shows the numerical results obtained in the near zone for the energy-transfer amplitude, as a function of the waveguide radius R. The transition wavelength is λ 0 = 5·10 −7 m and the distance between the atoms is z = 10 −8 m. In this regime, when the waveguide radius R is increased and thus the gap between ω 0 and ω min decreases, the absolute value of the energy transfer amplitude first increases and then settles to an almost constant value. On the contrary, by decreasing R, the energy transfer amplitude quickly tends to vanish, as expected. These results show that the energy transfer between the atoms inside the waveguide can be strongly modified by changing the waveguide's radius R, both in the near and in the far zone.
B. Radial dipoles
We now consider the case of atomic dipole moments along the radial direction (that is orthogonal to the guide axis), parallel to each other. In this case the energy transfer amplitude M is given only by the term M r , while the contributions M z and M φ vanish. The analytic expression of the energy transfer amplitude for radial dipoles, when Rk 0 ≪ 1, can be obtained from Eq. (33) which, after performing the integral over k, yields
M r = d eg Ar d ge Br 8πǫ 0 m − k 2 0 e − √ µ 2 −k 2 0 z I µ1 µ 2 − k 2 0 + λ 2 − k 2 0 e − √ λ 2 −k 2 0 z I λ1 .(38)
The root test shows that the sum over the radial field modes in (38) converges, and a numeric check shows that the first forty terms of the sum give a good numerical estimate of it. As in the previous case of axial dipoles, we now investigate M r as a function of the interatomic distance z and the waveguide cut-off frequency ω min . Fig. 5 shows the energy transfer amplitude in the near zone as a function of the interatomic distance z, using the same values as before for the atomic transition wavelength (λ 0 = 5 · 10 −7 m) and the waveguide radius (R = 10 −8 m). In this regime, the energy transfer am- plitude inside the waveguide is reduced with respect to the free-space case, and this effect is much larger with respect of the previous case of axial dipoles. The reduction of the energy transfer amplitude significantly grows as the transition region between near and far zone is approached (for example, for z = 5 · 10 −8 it is reduced of about three orders of magnitude).
On the other hand, in the far zone, a numerical analysis shows that the behaviour of the energy transfer is very similar to that for axial dipoles case: the waveguide strongly suppresses the energy transfer amplitude, which is virtually zero. These results confirm that the energy transfer amplitude is significantly affected by the presence of the waveguide. The amplitude is more and more reduced, with respect to the free-space case, when the transition region at z ∼ λ 0 is being reached, and completely suppressed in the far zone. We now consider the excitation exchange between the atoms in the near zone, as a function of the waveguide cut-off frequency ω min or equivalently of the waveguide radius R. Fig. 6 shows that the energy transfer amplitude decreases for decreasing R, as expected because the difference between the atomic transition frequency ω 0 and the waveguide cut-off frequency ω min increases. In the range considered, it is smaller than in the free space, even if of the same order of magnitude.
In conclusion, our findings show that, for both axial and radial dipole orientations, the presence of the waveguide allow to change (reduce or inhibit, for example) the energy transfer process.
IV. RESONANCE DIPOLE-DIPOLE INTERACTION ENERGY BETWEEN ATOMS INSIDE THE PERFECTLY CONDUCTING WAVEGUIDE
In this section we discuss the resonance dipole-dipole interaction energy between two entangled atoms placed on the axis of a perfectly conducting cylindrical waveguide, which is given by Eq. (21). We consider the case Rk 0 ≪ 1, that is ω 0 < ω min . In this case the field modes resonant with the atomic transition frequency are suppressed by the waveguide, and do not contribute to the resonance interaction energy. Since the electromagnetic Green's tensor (25) is symmetric for exchange of the positions of the atoms, Eq. (21) yields
∆E ± = ± 2 πǫ 0 ij d eg Ai d ge Bj +∞ kmin dkk 3 k 2 0 − k 2 ImG ij (r A , r B , ω),(39)
where + and − signs refer, respectively, to the symmetric or antisymmetric entangled state in Eq. (20).
The expression (39) coincides in modulus with the energy transfer amplitude (30). Thus all results obtained in the previous section for the energy transfer amplitude, when the atomic transition frequency is below the waveguide lower cutoff frequency, can be directly extended to the resonance interaction energy between correlated atoms. We stress that the interatomic resonance interaction and the resonant energy transfer process, are different albeit related physical processes: the resonance interaction (39) is a quantum interaction energy between two atoms, one excited and the other in the ground state, prepared in a symmetrical or anti-symmetrical entangled state, arising from the exchange of a photon between them; it eventually yields a force between the atoms (in a quasi-static approach, the force is obtained from the derivative of the interaction energy with respect to the interatomic distance, changed of sign). We discuss this interaction energy as a function of the interatomic distance z and the waveguide cut-off frequency ω min .
The plot in Fig. 2 shows that, in the near zone, the interaction between the two atoms (symmetric state) with axial dipole moments has essentially the same behaviour as in the free space. Approaching the transition region between near and far zone, the resonance interaction becomes more and more suppressed with respect to the freespace interaction, as it is highlighted in Fig. 3 where their ratio is plotted. More significant differences between the waveguide and free-space cases, emerge when we consider the radial dipole moments, in the near-zone (z < λ 0 ), as represented in Fig. 5 for the symmetric state. Now, the presence of the waveguide more deeply influences the resonance interaction between the atoms. Fig. 7 shows the ratio between the interaction energy in the waveguide and in the free space; it highlights that the resonance interaction energy changes from repulsive to attractive, for distances greater then z 2.9 · 10 −8 m, while it remains repulsive for atoms in the free space. Also, in this range the interaction is reduced by a factor of the order of 10 −3 , compared to the free-space case (much more than in the axial-dipoles case). In the far zone limit, the resonance interaction is strongly suppressed by the waveguide and, analogously to the energy transfer discussed in the previous section, it is virtually zero.
Similar considerations hold for the case of the antisymmetric state, where the resonance interaction is the same that for the symmetric state, except for a change of sign (see Eq. (39)). All these results clearly show that the waveguide deeply modifies the character and the strength of the interaction (both in the near-and in the far-zone regimes). The reduction of the interaction energy is small in the very near zone and it becomes more and more significant when the intermediate region is approached, yielding a complete inhibition of the interaction in the far zone.
V. CONCLUSIONS
In this paper we have investigated the energy transfer amplitude between two identical atoms, interacting with the quantum electromagnetic field, placed in a macroscopic environment such as a perfectly conducting cylindrical waveguide. The energy-transfer process studied is also directly related to the resonance interaction energy and force between two identical correlated atoms, prepared in a symmetrical or antisymmetrical Bell-type state.
We have first introduced a general analytical expression for the energy transfer amplitude (and the resonance dipole-dipole interaction between two identical atoms) in a generic structured environment, exploiting the Green's tensor formalism. We have then considered the specific case of the energy transfer between two identical atoms placed on the axis of a perfectly conducting cylindrical waveguide, which determines a lower cutoff frequency for the electromagnetic modes inside it. We have considered both cases of atomic dipole moments parallel and orthogonal to the axis of the waveguide. When the atomic transition frequency is smaller than the cutoff frequency of the waveguide, we have shown that the presence of the waveguide can significantly change the energy-transfer amplitude, depending on the distance between the two atoms compared to their transition wavelength (near, that is radiationless zone, or far radiative zone). We have shown that, when the atomic transition frequency is smaller than the waveguide lower cutoff frequency, the energy transfer process is strongly suppressed in the far zone, while it is only much less influenced in the intermediate and near zone (the latter effect is much larger for radial dipoles than for axial dipoles). A physical interpretation of this result is given. We have also shown that, in this regime, the resonance interaction force between two atoms with radial dipole moments, in the near zone, changes its character from repulsive to attractive. These results show how the presence of the external environment, the cylindrical waveguide in our case, can significantly change atomic radiative processes such as the resonance energy transfer and the resonance dipole-dipole interaction energy, yielding possibility of controlling them through the environment (mainly the waveguide radius, in the present case) and even inhibiting them.
FIG. 1 :
1The physical system: two atoms on the axis of a perfectly conducting cylindrical waveguide.
FIG. 2 :FIG. 3 :
232 · 10 −8 m the two amplitudes are essentially the same, while for z 1.2 · 10 −8 m, when the intermediate region between Comparison between the energy transfer amplitude M = Mz (axial dipoles) in the free space (orange dashed line) and in the waveguide (blue solid line), as a function of the interatomic distance z, for z < λ0 (near-zone). The numerical values of the parameters are chosen such that λ0 = 5 · 10 −7 m, R = 10 −8 m, and d eg A/B z = 10 −30 C · m.Energy-transfer amplitude in the near zone between atoms with axial dipoles, normalised to the free-space energytransfer amplitude, as a function of the interatomic distance z. When z approaches the transition region between near and far zone (z ∼ λ0), the amplitude in the waveguide becomes more and more suppressed with respect to the free-space case. The parameters used are λ0 = 5 · 10 −7 m, R = 10 −8 m and d eg A/B z = 10 −30 C · m.
FIG. 4 :
4The energy transfer amplitude Mz for axial dipoles in the near zone, z < λ0, between two atoms inside the cylindrical waveguide (blue solid line), as a function of the waveguide radius R. The orange horizontal line shows the value of the energy transfer amplitude in the free space. The numerical values of the parameters are z = 10 −8 m, λ0 = 5 · 10 −7 m and d eg A/B z = 10 −30 C · m.
FIG. 5 :
5The energy transfer amplitude for radial dipoles in the near zone, z < λ0, as a function of the interatomic distance z. The blue solid line is for atoms in the waveguide, while the orange dashed line refers to the free-space case. The parameters used are λ0 = 5 · 10 −7 m, R = 10 −8 m and d eg A/B z = 10 −30 C · m.
FIG. 6 :
6The energy transfer amplitude Mr for radial dipoles in the near zone (z < λ0), as a function of the waveguide radius R, for atoms inside a cylindrical waveguide (blue continuous line). The orange horizontal line gives the value of the amplitude in the free space.. Parameters are z = 10 −8 m, λ0 = 5 · 10 −7 m and d eg A/B z = 10 −30 C · m.
FIG. 7 :
7Resonance interaction energy for the symmetric state in the near zone between atoms with radial dipoles, as a function of the interatomic distance z, normalised to the free space interaction. When z approaches the transition region between near and far zone (z ∼ λ0), the resonance interaction becomes more and more suppressed with respect to the free-space interaction. The parameters used are λ0 = 5 · 10 −7 m, R = 10 −8 m and d eg A/B z = 10 −30 C · m.
AcknowledgmentsThe authors gratefully acknowledge financial support from the Julian Schwinger Foundation.
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[
"Accurate determination of the Gaussian transition in spin-1 chains with single-ion anisotropy",
"Accurate determination of the Gaussian transition in spin-1 chains with single-ion anisotropy"
]
| [
"Shijie Hu \nDepartment of Physics\nRenmin University of China\n100872BeijingChina\n\nInstitut für Theoretische Physik\nGeorg-August-Universität Goettingen\n37077GöttingenGermany\n",
"B Normand \nDepartment of Physics\nRenmin University of China\n100872BeijingChina\n",
"Xiaoqun Wang \nDepartment of Physics\nRenmin University of China\n100872BeijingChina\n",
"Lu Yu \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n"
]
| [
"Department of Physics\nRenmin University of China\n100872BeijingChina",
"Institut für Theoretische Physik\nGeorg-August-Universität Goettingen\n37077GöttingenGermany",
"Department of Physics\nRenmin University of China\n100872BeijingChina",
"Department of Physics\nRenmin University of China\n100872BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina"
]
| []
| The Gaussian transition in the spin-one Heisenberg chain with single-ion anisotropy is extremely difficult to treat, both analytically and numerically. We introduce an improved DMRG procedure with strict error control, which we use to access very large systems. By considering the bulk entropy, we determine the Gaussian transition point to 4-digit accuracy, Dc/J = 0.96845(8), resolving a long-standing debate in quantum magnetism. With this value, we obtain high-precision data for the critical behavior of quantities including the ground-state energy, gap, and transverse string-order parameter, and for the critical exponent, ν = 1.472(2). Applying our improved technique at Jz = 0.5 highlights essential differences in critical behavior along the Gaussian transition line. | 10.1103/physrevb.84.220402 | [
"https://arxiv.org/pdf/1107.0229v2.pdf"
]
| 100,447,257 | 1107.0229 | d9239f4a193783f4db8261a380fe2403c9f7d342 |
Accurate determination of the Gaussian transition in spin-1 chains with single-ion anisotropy
6 Dec 2011
Shijie Hu
Department of Physics
Renmin University of China
100872BeijingChina
Institut für Theoretische Physik
Georg-August-Universität Goettingen
37077GöttingenGermany
B Normand
Department of Physics
Renmin University of China
100872BeijingChina
Xiaoqun Wang
Department of Physics
Renmin University of China
100872BeijingChina
Lu Yu
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
Accurate determination of the Gaussian transition in spin-1 chains with single-ion anisotropy
6 Dec 2011(Dated: January 30, 2013)arXiv:1107.0229v2 [cond-mat.str-el]
The Gaussian transition in the spin-one Heisenberg chain with single-ion anisotropy is extremely difficult to treat, both analytically and numerically. We introduce an improved DMRG procedure with strict error control, which we use to access very large systems. By considering the bulk entropy, we determine the Gaussian transition point to 4-digit accuracy, Dc/J = 0.96845(8), resolving a long-standing debate in quantum magnetism. With this value, we obtain high-precision data for the critical behavior of quantities including the ground-state energy, gap, and transverse string-order parameter, and for the critical exponent, ν = 1.472(2). Applying our improved technique at Jz = 0.5 highlights essential differences in critical behavior along the Gaussian transition line.
The Gaussian transition appears in several fields of quantum physics and statistical mechanics. The equivalence between surface-roughening transitions in classical two-dimensional (2D) models and quantum phase transitions in spin chains was introduced in Ref. [1], and their rich phase diagrams investigated at length in Ref. [2]. Characterized by continuously variable exponents, the Gaussian transition differs significantly both from regular phase transitions and from those of Kosterlitz-Thouless (KT) type. These differences complicate both analytical and numerical approaches to a complete and accurate description of rough surfaces and quantum spin chains.
The S = 1 Heisenberg chain is one of the fundamental models in quantum magnetism. It formed the basis of Haldane's conjecture [3] for a finite gap in antiferromagnetic chains with integer spin, as opposed to the gapless spectrum of half-odd-integer cases. Numerically, quantum spin chains are important test-cases for any computational technique, and Haldane's prediction has been verified by a range of methods with increasing accuracy [4,5]. Experimentally, while the "Haldane gap" has been found in the excitation spectra of many systems [6], most known S = 1 chains, including NENP [7], NINAZ [8], and NDMAP [9], are organic Ni materials with significant single-ion anisotropies. Analytical approaches to the Gaussian transition driven by this term are complicated by the lack of a suitable effective field theory [10], and its broad nature makes all numerical techniques difficult to apply. Many authors have considered this transition, producing occasionally contradictory results [11][12][13][14][15][16][17][18][19].
In this Letter we resolve the problem of the Gaussian transition in the S = 1 chain with single-ion anisotropy. We exploit the fact that this transition is a gapless point between two gapped phases, whence the entropy exhibits a sharp peak. We introduce an improved density-matrix renormalization-group (DMRG) approach with systematic error control, allowing high-precision calculations at system sizes up to L = 20000, which automatically eliminate the end-spin entropy. We determine the critical point with very high accuracy, and thereby deduce the critical behavior of several quantities at different points on the Gaussian transition line.
The general form of the model is
H = L i=1 J(S x i S x i+1 +S y i S y i+1 ) + J z S z i S z i+1 + D (S z i ) 2 (1)
where J z interpolates between XY and Ising spins, D is the single-ion anisotropy, and L the length of the chain. The full parameter space of (D, J z ) contains Néel, Haldane, large-D, ferromagnetic, and two XY phases. In classical planar surface, or "solid-on-solid," models, the Néel and large-D phases are different "flat" phases, the Haldane phase is "rough," and the Gaussian transition is of "preroughening" type. These are the three phases of the S = 1 Heisenberg chain (J z = 1) as D is varied. While the Néel phase possesses Z 2 symmetry and the Haldane phase an incomplete Z 2 × Z 2 symmetry, the large-D phase has no remaining symmetries. The Gaussian transition is a line in the (D, J z ) plane, on which the excitations are gapless. This line is well described by a conformal field theory (CFT) [20], and has been analyzed in a number of studies [15][16][17][18], but none has achieved the numerical precision required for a consistent discussion of the critical behavior across the transition. DMRG is the most efficient and accurate numerical technique for 1D systems [5]. Anticipating the need for both large system sizes and extreme precision, we begin by introducing an improved DMRG technique. In the conventional scheme, the absolute (coupled round-off and truncation) error increases systematically with L, and this accumulated error has a strong effect on the relability of the computation, possibly even disguising the critical behavior in a quantum many-body system. We fix the round-off error by renormalizing the lowest eigenvalue of Hamiltonian to remain of order 1, thereby obtaining a very significant reduction in the truncation error for large systems. In the DMRG iteration, we replace the original Hamiltonian matrix H(m, L), for chains of L sites with m kept states, by H(m,
L) − [ε 1 (m, L − 2) − δ], where ε 1 (m, L − 2)
is the lowest eigenvalue of H(m, L − 2) and δ is a constant chosen such that ε 1 (m, L) ∼ O(1). Here we use δ/J = 1 throughout. While the (extensive) total energy of the ground state, E g (m, L), can be reconstructed by summation, its (intensive) average value per site is determined directly and self-consistently as e g (m,
L) = [ε 1 (m, L) − δ]/2. Similarly, for the first excited state e f (m, L) = [ε 2 (m, L) − ǫ 2 (m, L − 2)]/2 + e g (m, L), where ε 2 (m, L) is the second-lowest eigenvalue of H(m, L).
The gap in our method is given simply by ∆(m, (2) where e g (m) and e f (m) are the intensive energies of the ground and first excited states for infinite L, and become equal for infinite m. In the polynomial expansion of contributions at higher order in 1/L, the n = 1 term arises from truncation errors and open boundary conditions (OBCs), while the n = 2 term has contributions from fluctuations at the quadratic band minimum. Here we calculate the energies in the linear term independently by extrapolation. Subtracting these gives a gap function ∆(m, L) that decreases monotonically with increasing L. A second polynomial fit of ∆(m) allows its extrapolation to infinite m to obtain the true gap.
L) = ε 2 (m, L) − ε 1 (m, L). Its general expression is ∆(m,L) = [e f (m) − e g (m)]L + ∆(m) + ∞ n=1 α n (m) L n ,
Sharing its foundations with quantum information theory, the DMRG method is ideally suited to discussions of entropy and entanglement. The von Neumann entropy, S(m, L) = −Trρ(m, L) ln ρ(m, L), is readily computed from the reduced density matrix, which we obtain to high accuracy throughout our calculations with the renormalized Hamiltonian. The entropy obeys an area law except in critical regimes, where it depends logarithmically on L [21]. This extremum in entropy is an excellent indicator of a (gapless) critical point between two gapped phases.
Before analyzing the entropy, we discuss the special and remarkable feature of the S = 1 Heisenberg chain, that free S = 1/2 entities are found at a chain end, both in theory and in experiment [22]. In the Haldane phase with OBCs, the two free end-spins can be described bŷ
H eff = J eff S L · S R [5]
, where the effective coupling J eff > 0 falls exponentially with L. In the Hilbert space S tot z = 0, the two spins are maximally entangled with entropy ln 2, while for S tot z = 1 they are unentangled. The additional truncation error due to this edge-entropy contribution causes significant computational difficulties. In Fig. 1 precision and the end-spin contribution vanishes. The remaining "bulk" entropy contains the essential physics of the spin chain. Different but conceptually similar approaches have considered both the two-site entropy and S(L) in a chain with no end-spin effects [23]. Figure 1(a) contrasts the total and the bulk entropy. Calculations with small L cannot access the unentangled regime, and for larger L we find a ln 2 jump when D approaches D c . For L = 10000, the end-spins remain entangled for 0.94 < D < D c . The maximum in the total entropy moves strongly with L, showing no direct indication of criticality [17]. By contrast, in the Hilbert space S tot z = 1, the end-spins are unentangled in the lowest-energy state and this data reproduces exactly the bulk entropy. The location of the maximum in S, shown in detail in Fig. 1(b), is clearly invariant with m. A linear fit to the bulk entropy on both sides of the transition in Fig. 1(b) gives our primary result, D c /J = 0.96845 with a minuscule error bar of 0.00008. The increasing slopes of the bulk entropy lines as m → ∞ [inset Fig. 1(b)] indicate the onset of critical behavior.
Having determined this extremely precise value of D c , we may now discuss the critical behavior of the Gaussian transition with hitherto unattainable accuracy. We consider the physical quantities used in previous analyses of the transition [11][12][13][14][15][16][17][18][19], beginning with the gap. To avoid effects in the gap extrapolation related to the disappearance of edge states, we use the lowest energy levels in the Hilbert spaces S tot z = 1 and S tot z = 2. Figure 2(a) illustrates our two-step extrapolation approach to compute (2). This last agrees exactly with our gap data in Fig. 2(b), confirming the consistency and accuracy of our calculations. Our computed central charge is extraordinarily close to the expected value c = 1 [15,16]. Even data at the extreme precision we attain cannot determine whether the second derivative of e g has a discontinuity [ Fig. 3(c)], but set a very low upper bound. A continuous function with a point of inflection at D c is consistent with the CFT expectation [17] that the Gaussian transition be third-order for J z = 1.
The transverse string-order parameter is defined as
O (l) = Ŝ x 0 exp iπ l−1 p=1Ŝ x p Ŝ x l ,(3)
and encapsulates the incomplete Z 2 ×Z 2 symmetry of the Haldane phase [1]. To reduce the complexities inherent in calculating this quantity, we compute correlation functions only far from the system boundaries [24], S tot z = 1 sector as the ground state. Figure 3(d) shows the results of our extrapolations to infinite L and m. The string-order parameter clearly shows excellent scaling behavior in the critical regime. The scaling exponent ν ′ = 0.353(1) is very close to the value 1/ √ 8 predicted in the 2D classical model [1], demonstrating the common physics of the Gaussian, or preroughening, transition.
We illustrate with one example the utility of our improved DMRG calculations for investigating the entire Gaussian transition line. The point J z = 0.5 has been considered by several authors [15][16][17][18]. Our results (Fig. 4) provide the most accurate information yet available for this transition: D c /J = 0.6355 (6). The values of L required to approach criticality are very much larger than for J z = 1 [ Fig. 4(a)], and the accuracy is lower because S(D) is a significantly flatter function [ Fig. 4(b)]. Our calculations with PBCs give e g = −0.91510889(1)J, v = 2.185(2)J, c = 1.000(1), K = 1.581(1), and ν = 2.387(5) at D c , allowing a complete characterization of the physics of continuously varying exponents.
We have considered the entropy S(m, L) at finite m and L. In fact our results in Fig. 1 for m = 1000 and L = 10000 are fully converged for all values of D outside the very narrow region 0.94 < D < 1.00. We can deduce the critical behavior of S around D c from a massive quantum field theory [21], in which S = (c/6) ln ξ + A with ξ = v/∆ the correlation length and ∆ ∝ |D − D c | ν . The convergent behavior of our data near D c gives exactly The Gaussian transition in the S = 1 chain is topological, in that the parity of the ground state changes from negative in the Haldane phase to positive in the large-D phase. The transition is thus associated with a change in the topological spin Berry phase from π to 0 [25], and can be followed by a method of crossing energy levels (of states in the appropriate parity sectors). Our highprecision results demonstrate that this is indeed a very sensitive indicator of a topological transition: among all previous studies [11][12][13][14][15][16][17][18][19], we find that the only accurate estimate of D c was obtained, despite being limited to 16-site systems, by employing this approach [15].
We have demonstrated that the entropy is very valuable for discussing continuous phase transitions between gapped states. Many other types of strongly interacting quantum system fall in this category, one good example with electronic degrees of freedom being the ionic Hubbard model (IHM) [26]. The numerically challenging transition in this case is of KT type. Continuous gappedto-gapped transitions for both bosonic and fermionic systems exist in ultra-cold atomic condensates on optical lattices. The Gaussian transition has not yet been observed in experiment, due to difficulties in controlling the ratio D/J in condensed matter systems, and cold-atom experiments may offer a clean solution to this problem.
To summarize, we calculate the critical point of the spin-one Heisenberg chain with single-ion anisotropy, D c /J = 0.96845(8), to extremely high accuracy. To achieve this we introduce an improved DMRG scheme, which controls the absolute error of a large system and allows the elimination of end-spin effects. We exploit this accuracy to deduce the critical properties of many quantities at the Gaussian transition. The energy, entropy, and
FIG. 1 :
1(a) we find a ln 2 drop in the ground-state entropy S(L) in the Hilbert space S tot z = 0 when the chain reaches a certain length at fixed D. For D = 0.92 and m = 1000, this occurs at L = 4500 [inset, Fig. 1(a)]. When L becomes sufficiently large, J eff falls below the machine (color online) Entropy S as a function of D for Jz = 1. (a) Calculations with m = 1000. Open symbols are obtained for the lowest energy level in Hilbert space S tot z = 0 with a range of L values, solid symbols for S tot z = 1. Inset: ln 2 drop in S(L) for D = 0.92. (b) Bulk entropy S(D) close to the Gaussian critical point, computed with L = 10000 for a range of m values. Insets: fitting slopes AL,1 and AR,1 (left axis) and transition D fit c (right) obtained as functions of m.
FIG. 2 :
2(color online) (a) Gap as a function of L, computed for D/J = 0.95 with several values of m. Solid symbols for L → ∞ are extrapolated to m → ∞ (inset), giving ∆(0.95) = 0.00130(4). (b) Extrapolated gaps as a function of |D − Dc|. the gap for the extremely numerically challenging point D = 0.95, which lies very close to D c . By following this procedure for all values of D, we show in Fig. 2(b) the approach of the gap to zero at D c from both the Haldane and large-D sides. The closest four points, D = 0.925, 0.95, 1.0, and 1.025, reveal a very narrow critical region, |D − D c | < 0.1, with critical exponent ν = 1.472(4). In a CFT for the Gaussian critical line [20], the gap ∆ varies linearly and the energy e g quadratically with 1/L. For the CFT analysis, we perform DMRG calculations with periodic BCs (PBCs) using L = 200 and m = 2000 [Figs. 3(a) and (b)]. We obtain the ground-state energy e g = −0.86856650(4)J, velocity v = 2.564(2)J, central charge c = 6β/πv = 1.0006(8), Luttinger parameter K = v/4α = 1.321(1), and critical exponent ν = 1/(2 − K) = 1.472
in the leftcentral block [L/4 − 1000, L/4] of the chain. We take the online) (a) Finite-size extrapolation of lowest two gaps at Dc. Fitting lines from CFT give ∆(α = 0.48516(1). (b) Extrapolated ground-state energy eg at Dc, with CFT fit −0.86856650(4)J − βJ/L 2 and β = 1.3429(4). (c) Second derivative of extrapolated energy. (d) Extrapolated transverse string-order parameter (see text) of the Haldane phase, with fitting line 0.6036(4)|D − Dc| 0.353(1) . Calculations for (a) and (b) performed with PBCs and m = 2000, for (c) and (d) with OBCs and m = 1000.
FIG. 4 :
4(color online) S(D) as in Fig. 1 for Jz = 0.5. (a) Values of L as indicated. Inset: ln 2 drop in S(L) for D = 0.4. (b) Values of m as indicated. the critical form S = S 0 − (cν/6) ln |D − D c |, which is shown as the solid lines diverging at D c in Figs. 1 and 4.
gap all show good scaling behavior with a single critical exponent, ν = 1.472(2). We apply our technique also at J z = 0.5 to illustrate the continuous variability of exponents on the Gaussian transition line.We
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| []
|
[
"Morphology of Fly Larval Class IV Dendrites Accords with a Random Branching and Contact Based Branch Deletion Model",
"Morphology of Fly Larval Class IV Dendrites Accords with a Random Branching and Contact Based Branch Deletion Model"
]
| [
"Sujoy Ganguly \nDepartment of Molecular Biophysics and Biochemistry\nYale University\nNew Haven\n",
"Olivier Trottier \nDepartment of Molecular Biophysics and Biochemistry\nYale University\nNew Haven\n",
"Xin Liang \nTsinghua-Peking Joint Center for Life Sciences\nSchool of Life Sciences\nTsinghua University\nBeijingChina\n",
"Hugo Bowne-Anderson \nDepartment of Molecular Biophysics and Biochemistry\nYale University\nNew Haven\n",
"Jonathon Howard \nDepartment of Molecular Biophysics and Biochemistry\nYale University\nNew Haven\n"
]
| [
"Department of Molecular Biophysics and Biochemistry\nYale University\nNew Haven",
"Department of Molecular Biophysics and Biochemistry\nYale University\nNew Haven",
"Tsinghua-Peking Joint Center for Life Sciences\nSchool of Life Sciences\nTsinghua University\nBeijingChina",
"Department of Molecular Biophysics and Biochemistry\nYale University\nNew Haven",
"Department of Molecular Biophysics and Biochemistry\nYale University\nNew Haven"
]
| []
| Dendrites are branched neuronal processes that receive input signals from other neurons or the outside world [1]. To maintain connectivity as the organism grows, dendrites must also continue to grow. For example, the dendrites in the peripheral nervous system continue to grow and branch to maintain proper coverage of their receptor fields[2,3,4,5]. One such neuron is the Drosophila melanogaster class IV dendritic arborization neuron[6]. The dendritic arbors of these neurons tile the larval surface [7], where they detect localized noxious stimuli, such as jabs from parasitic wasps[8]. In the present study, we used a novel measure, the hitting probability, to show that the class IV neuron forms a tight mesh that covers the larval surface. Furthermore, we found that the mesh size remains largely unchanged during the larval stages, despite a dramatic increase in overall size of the neuron and the larva. We also found that the class IV dendrites are dense (assayed with the fractal dimension) and uniform (assayed with the lacunarity) throughout the larval stages. To understand how the class IV neuron maintains its morphology during larval development, we constructed a mathematical model based on random branching and self-avoidance. We found that if the branching rate is uniform in space and time and that if all contacting branches are deleted, we can reproduce the branch length distribution, mesh size and density of the class IV dendrites throughout the larval stages. Thus, a simple set of statistical rules can generate and maintain a complex branching morphology during growth. | null | [
"https://arxiv.org/pdf/1611.05918v1.pdf"
]
| 219,715 | 1611.05918 | f7815dddf94343854e5e2a24009b77f6d2144d1e |
Morphology of Fly Larval Class IV Dendrites Accords with a Random Branching and Contact Based Branch Deletion Model
November 21, 2016 17 Nov 2016
Sujoy Ganguly
Department of Molecular Biophysics and Biochemistry
Yale University
New Haven
Olivier Trottier
Department of Molecular Biophysics and Biochemistry
Yale University
New Haven
Xin Liang
Tsinghua-Peking Joint Center for Life Sciences
School of Life Sciences
Tsinghua University
BeijingChina
Hugo Bowne-Anderson
Department of Molecular Biophysics and Biochemistry
Yale University
New Haven
Jonathon Howard
Department of Molecular Biophysics and Biochemistry
Yale University
New Haven
Morphology of Fly Larval Class IV Dendrites Accords with a Random Branching and Contact Based Branch Deletion Model
November 21, 2016 17 Nov 2016
Dendrites are branched neuronal processes that receive input signals from other neurons or the outside world [1]. To maintain connectivity as the organism grows, dendrites must also continue to grow. For example, the dendrites in the peripheral nervous system continue to grow and branch to maintain proper coverage of their receptor fields[2,3,4,5]. One such neuron is the Drosophila melanogaster class IV dendritic arborization neuron[6]. The dendritic arbors of these neurons tile the larval surface [7], where they detect localized noxious stimuli, such as jabs from parasitic wasps[8]. In the present study, we used a novel measure, the hitting probability, to show that the class IV neuron forms a tight mesh that covers the larval surface. Furthermore, we found that the mesh size remains largely unchanged during the larval stages, despite a dramatic increase in overall size of the neuron and the larva. We also found that the class IV dendrites are dense (assayed with the fractal dimension) and uniform (assayed with the lacunarity) throughout the larval stages. To understand how the class IV neuron maintains its morphology during larval development, we constructed a mathematical model based on random branching and self-avoidance. We found that if the branching rate is uniform in space and time and that if all contacting branches are deleted, we can reproduce the branch length distribution, mesh size and density of the class IV dendrites throughout the larval stages. Thus, a simple set of statistical rules can generate and maintain a complex branching morphology during growth.
In our brains, billions of neurons interact with each other to build a nervous system of unparalleled complexity and computational power. Neurons have dendrites, which are branched structures that receive synaptic or sensory inputs, and an axon, which send outputs to other neurons. The shape or morphology of individual neurons sets the number and types of interactions that a neuron can have and provides the structural basis of neuronal computation [9,10,11,12,13,14].
Since many organisms continue to enlarge after the establishment of the body plan, it is critical for axons and dendrites to maintain their morphology as they grow. For example, interneurons of the grasshoper [2], motor neurons in moths [3] and mice [4] grow drastically in size yet maintain connections to their target cells. Futhermore, dendrites in the perpherial nervous system, like those of gold fish retinal ganglion cells [5], and dendritic arborization (da) sensory neurons of the fly larva [6], which are the subject of this work, grow to continually maintain coverage of their receptor fields. In this paper, we investigate the growth rules that are required to maintain the correct branching morphology as a dendrite grows.
The da sensory neurons of the fly larva are a model system for studying dendritic arborization [15,16,17]. These dendrites innervate the extracellular matrix, which lies between the outer cuticle and the inner epidermal cell layer [7]. They tile the surface on the fly larva in a highly stereotyped manner and have four distinct morphological classes [18] (Fig. 1 A). Since it is easy to identify and image individual da neurons, these neurons have proven to be a powerful model system for studying dendrite morphology [15,16]. In this paper, we address the question of how the morphology of the class IV da neurons (Fig. 1 B) is maintained during the larval stages.
The class IV da neuron has highly branched dendrites [18], which detect potentially harmful stimuli, such as the ovipositor barb of parasitic wasps [8,19]. The dendrites of the class IV neuron begin morphogenesis during late embryogenesis ∼ 16 hrs After Egg Lay (AEL). By the time the larva hatches (∼ 22 hrs AEL at 25 • C), the class IV dendrites nearly cover its surface. The dendrites then continue to expand and branch as the larva grows (22 − 126 hrs AEL), so that the neuron maintains its coverage of the larval surface [6].
In this work, we are seeking the growth rules that allow class IV dendrites to maintain their dense coverage of the larval surface. To this end, we have used a novel measure, the hitting probability, that quantifies the mesh size and two well-known measures of branching morphology: the fractal dimension [20] and lacunarity [21,22] (see Definition of Morphometrics). We show that these measures remain largely invariant over larval stages, despite a several fold increase in larval length. Furthermore, we demonstrate that a model with simple rules for branching and self-avoidance can capture essential features of the establishment and maintenance of the dendrite's morphology.
Experimental Results
To characterize the morphology of fully-developed class IV dendrites, we imaged larvae expressing Cd4-tdGFP under the ppk promoter (ppk-cd4-tdGFP ) during the third instar stage (Fig. 1) using a laserscanning confocal microscope (See Material and Methods for details). Using NeuronStudio [23] and Fiji we traced the branches of the dendrites to produce skeletons. These skeletons were then analyzed to obtain the the mesh size, density and uniformity of class IV dendrites using parameters defined in the next section.
Definition of Morphometrics
Here we include simple definitions of the relevant morphometrics to aid comprehension. Hitting Probabiltiy H(B): The probability that a box of size B hits the dendrite.
Mesh Size B H : The length at which 50% of all boxes hit the neuron.
Fractal Dimension d f : A measure of the space-fillingness of a shape. For a completely filled box d f = 2, for a straight line d f = 1, for branched shapes 1 ≤ d f ≤ 2.
Lacunarity Λ(B): A measure of density fluctuations as a function of length scale B.
Lacunarity Length B Λ : The length at which Λ(B = B Λ ) = 0.25, i.e. the length at which the neuron is uniform. The larger B Λ , the more variable the density of the neuron.
Radius of Gyration R g : A length scale that measures how spread out a shape is from its center. The larger R g the more spread out the neuron.
Persistence Length β: The characteristic length at which a branch bends.
For mathematical definitions see Appendix.
Class IV dendrites have a small mesh size
To characterize the mesh size of the dendrites, we developed a novel measure called the hitting probability H(B). H(B) measures the probability H that a randomly placed box of size B hits the dendrite (see Appendix for details). The hitting probability generalizes an earlier metric called the coverage index [6] by allowing for any box location and any box size. A typical hitting probability curve of a neuron ( Fig. 1 D) H(B) increases monotonically with B, eventually reaching H = 1 as B approaches the size of the neuron. We define the characteristic mesh size B H as the box size at which half of all boxes hit the dendrite. In other words, B H is the maximum size of a stimulus that would go untouched, or undetected, on average, by the neuron. B H is similar to the mesh size in a cross-linked polymer network [24].
We found that B H = 8.4 ± 0.5 µm (mean ± SD, n = 14 neurons) for the mature dendrites of the class IV neuron. Thus, the mesh size is approximately equal to the diameter of the ovipositor barb of wasps that lay eggs in Drosophila larva (∼ 10 µm, [8]). This indicates that the class IV dendrite has a high chance of detecting a wasp attack. Furthermore, the mesh size is small compared to the overall size of the neuron (∼ 500 µm) and is similar to the mean branch length (see below).
Class IV dendrites are dense and uniform
To understand the morphological basis underlying B H we quantified the density and uniformity of the class IV dendrites during the third instar stage. The fractal dimension d f is a commonly used measure of how a branched structure fills space. A solid square, for example, has a d f = 2, while a straight line has a d f = 1 (See Appendix for mathematical definitions of d f ). We found that the fractal dimension of class IV dendrites was d f = 1.80 ± 0.04 (mean ± SD, n = 14), indicating that the dendrites are dense and space filling.
To establish a small mesh, a dendrite needs to not only be space filling (i.e. d f ∼ 2), but uniformly so. To measure the uniformity of the dendritic arbor, we used the lacunarity function Λ(B). This measures the density variation as a of function length scale B. To compare the lacunarity of different cells we calculated the length B Λ (Λ(B = B Λ ) = 0.25 see Definition of Morphometrics). The smaller B Λ , the more uniform the dendritic density. We found that B Λ = 32.6 ± 16.8 µm (mean ± SD, n = 14) for third instar larva. In other words, on spatial scales larger than 33 µm the density of the neuron is uniform, whereas below 33 µm it is variable. While the B Λ is larger than the mesh size B H , it is much smaller than the dendrite size, indicating that the coverage is uniform and arbors can be considered homogeneous. In summary (Tab. 1), mature class IV neurons have dense (large d f ) and uniform (small B Λ ) dendritic arbors, which is consistent with a small mesh size (small B H ).
Class IV dendrites have a tighter mesh, are denser and more uniform than class III dendrites
To assess the ability of these measures to quantify dendritic morphology, we also imaged class III cells ( Fig. 1 C). Class III cells are gentle touch sensors and use a different set of mechanotransduction channels [25]. The class III dendrites ( Fig. 1 C) are substantially less branched and have smaller branches, on average, than the class IV dendrites (Tab 1). We find that H(B) is much smaller for class III neurons than for class IV neurons ( Fig. 1 D) for most box sizes. Consequently, we find that the mesh size B H is much larger in class III neurons (B H ∼ 24.7 µm) than in class IV neurons (B H ∼ 8.4 µm) (Tab. 1). In other words, class III dendrites have larger gaps in coverage than class IV dendrites. We find that class III dendrites have a fractal dimension of d f ∼ 1.42 (Tab 1), which is substantially smaller than the class IV neuron. In other words, class III dendrites are sparser, at all scales, than class IV dendrites ( Fig. 1 E and F). We find that the lacunarity Λ decays much slower (with B) for class III dendrites compared to class IV dendrites ( Fig. 1 G). Consequently B Λ is much larger (Tab. 1) for class III dendrites than for class IV neurons (Tab. 1) showing that they are less uniform than class IV neurons. These results (Tab. 1), show that class III neurons are sparser (small d f ) and less uniform (large B Λ ) than class IV dendrites, which is consistent with having larger mesh size (large B H ). These differences likely reflect the different mechanoreceptor functions of class III and class IV neurons (see Discussion).
Class IV dendrites maintain a tight mesh, high density and uniformity throughout larval stages
To determine how the morphology of class IV dendrites changes with time, we imaged and skeletonized the class IV dendrites, as before, from early first instar (30 hrs AEL) to wandering third instar (126 hrs AEL) ( Fig. 2 A). We used the larval body segment length as a proxy for time since each data point comes from a unique larva, and S increases linearly with time [26] (SI). The mesh size B H increased modestly from 4.4 ± 0.3 µm (1st instar) to 8.5 ± 1.4 µm (third instar) (Fig. 2 C). This two-fold increase is less than the six-fold growth in larval body segment size and indicates that the mesh size remains small during development. Interestingly, the ratio between B H and the mean branch length L is nearly constant during development showing that the shape of the arbor has remarkable conservation during development
The morphologies of the class IV neurons remained dense and uniform during the larval stages of development. We found that the fractal dimension (Fig. 2 D) in the just hatched larvae (∼ 30hrs AEL) was d f ∼ 1.7 and increased to 1.75 within 24 hours, eventually rising to about 1.8 during the next four days. The lacunarity slightly increased with time (Fig. 2 E). By rescaling B by the radius of gyration R g (i.e. overall neuronal size see Eq. 1), we found that Λ follows the same curve at all developmental times, i.e., collapses when scaled by R g (Fig. 2 D inset). The nearly constant fractal dimension and the collapse of the lacunarity curves indicate that the morphological pattern of the class IV neurons is mostly established during embryogenesis.
Dynamic Model of Class IV Development
The maintenance of a small mesh size throughout larval stages raises the question: how can the class IV dendrites achieve this, despite the six-fold increase in the larval segment size? To answer this question, we developed a mathematical model of class IV dendrite morphogenesis during the larval stages. Our model consists of three rules that determine 1) branch creation, 2) the direction and speed that this new branch grows, and 3) how branches avoid crossing the other branches (self-avoidance).
Branch Creation
We found that the number of branch points increases with time during larval stages (Fig. 3A), and this increase was well described by a linear function with a net branch creation rate ω net ∼ 0.1 branch points min −1 (Tab. 2). This observation led us to model branch creation as a time invariant process with a branching frequency ω. Since ω net can include the removal of branches (see below), it is a lower bound on the branching creation frequency ω.
We also measured the branch length distribution, and found that it was well described be an exponential distribution (Fig. 3 B). Furthermore, we found that that the branch length was independent of the branch depth, defined as the number of branch points between a branch and the soma along the shortest path (Fig. 3 C). Motivated by these observations, we modeled branch creation as a random process that was uniform along the neuron and constant in time.
Tip Elongation
Since the growth of class IV dendrites occurs at the branch tips [27,6], we modeled neuron growth as branch tip elongation. We measured the overall size of class IV dendrites using the radius of gyration R g (Eq. 1). R g measures spread of neuron mass from its center. We found that R g increases linearly during development (Fig. 3 D). Therefore, we assume that the tip elongation rate v is constant in time and v ∝ V g , where V g is the growth speed of the class IV dendrite. The assumption of a constant growth speed assumes that the simulation time scale is much larger than the fluctuation times scale. We estimated V g from the change in dendrite size during development (Fig. 3 D), and found V g ∼ 0.04 µm min −1 (Tab 2).
To determine the direction of branch growth, we measured how much the path of a branch changes as a function of path length by using the persistence length β; the distance over which the orientation of the direction of growth of a branch changes (see Appendix for mathematical details). We find that β = 45 ± 2 µm (Tab. 2, Fig. 3 E), which is much greater than the mean branch length ( L ∼ 12.5 Tab. 1), indicating that branches tend to be straight.
Self-Avoidance
Previous work has shown that the branches of the class IV neuron do not cross each other. Furthermore, it is known that this self-avoidance is contact mediated and it has been proposed that branches retract after contact [27,28,29,30]. Therefore, we modeled self-avoidance by having growing branches retract at a constant speed v (same as elongation rate), if they contact an existing branch. For simplicity, the retraction length was assumed to be exponentially distributed with a mean retraction length α.
Model Implementation
Using these rules, we arrived at a four parameter model; ω the branch creation rate, v the growth speed, β persistence length and α to retraction scale. We implemented our model on a hexagonal lattice, with lattice spacing = 0.4 µm. We chose to use a lattice model to facilitate contact detection as the lattice spacing was set to be equal to the thickness of the branches. Since we implemented the rules on a hexagonal lattice, the possible branching angles are limited. Therefore we ignored the role of branching angles on morphology. The lattice spacing and the growth speed act as scale factors, i.e. setting the overall size of the dendrite but not changing the shape. Therefore, we chose v such that the size of a simulated dendrite matched that of a real one, which left us with three free parameters. However, we found that as long as β is large, i.e., as long as the branches are straight, β has little effect on the morphology (SI). Therefore, we focused on how branching frequency ω and the contact-based retraction scale α control the morphology of dendrites.
Model Results
Branching We generated simulated neurons for a range of branching frequencies ω and retraction scales α (Fig. 4 A) and analyzed their shape. Importantly, we found that the distribution of branch lengths was exponential (Fig. 4 B) and the branch lengths were uncorrelated with branch depth, for a limited range of parameters (dashed box in Fig. 4 A), which agrees with the experimental observations ( Fig. 3 B and C).
To test whether our model provides a good description of the morphology, we measured the mesh size, fractal dimension and lacunarity of the simulated dendrites. The mesh size B H decreased with increasing ω and decreasing α (Fig. 4 C) and appeared to saturate as ω increases. There was a small region of ω − α space where there was quantitative agreement between the simulated and observed values of B H (Fig. 4 C, pink). We found, that d f increased monotonically with ω and saturated for large values of ω. As for B H , we found a narrow band of ω − α space where we have a quantitative agreement between the model and the class IV neuron for d f (Fig. 4 D, pink). Crucially, there was a small region of ω − α space where the model recapitulated both B H and d f (Fig. 4 C and D, white box). Taking values from this small region (ω = 0.2 min −1 and α = 10 2 µm), we recapitulated the third instar and even found agreement throughout most of the larval development (Fig. 2 B and 2 C). Thus, the model recapitulates both the mesh size and fractal dimension throughout development. It did not recapitulate the lacunarity, which for the parameter values ω = 0.2 min −1 and α = 10 2 µm was larger than the observed values. In other words, the model arbors are not as uniform as the observed ones (see Discussion). In summary, this model recapitulates most, but not all aspects of the dendritic morphology of class IV neurons.
Discussion
Our key experimental finding is that the morphology of class IV neurons, as characterized by the branch length, mesh size, fractal dimension, and lacunarity, remarkably remains constant during development. Indeed, from the early first instar larva (30 hours after egg lay) to the late third instar larva (126 hours after egg lay), as both the segment size and the number of branches increases approximately six-fold, the mean length, the mesh size, and the lacunarity only increase around two-fold and the fractal dimension is almost unchanged. When we normalize the mesh size by the mean branch length, it is virtually unchanged throughout development. Thus, as these cells grow, key aspects of their morphology are invariant, even as the overall size the number of branches increases by nearly an order of magnitude.
As a first step toward probing the mechanism underlying this geometric invariance, we developed a simple computational model for branched morphogenesis. The model assumes that the branching rate is constant over development (consistent with the observed linear increase in the number of branches), that the rate was independent of position and that growing tips retract random, exponentially distributed distance after contacting other branches. The model reproduced many of the key features of the growth of class IV cells including branch lengths, mesh sizes, and fractal dimensions. However, it was unable to capture the lacunarity (the model predicted a higher relative density in the center than was observed, see SI). Importantly, the data constrained the values of the branching frequency and mean retraction distance. Thus, the model provides a framework for understanding the changes in the morphology of these cells during development.
One of our most striking experimental and theoretical findings was that the branching frequency in class IV dendrites was independent of total dendrite length. Naively, we might have expected that the mean number of branches added per unit time would increase with total dendrite length, as the longer the dendrites, the more positions on which branches could form. However, this would have led to an exponential increase in the number of branches, rather than the observed linear increase. Our modeling shows that even if the retraction length is much larger than the mean branch length, an exponential increase in branch length is still observed, and the distribution of branch lengths deviates from the observed exponential distribution (see SI). The constant branching rate suggests that branching is limited by the production of a nucleating factor that is produced at a constant rate. Furthermore, our finding that branching is uniform in space implies that the putative nucleation factor would be distributed widely and uniformly throughout the cell.
We also found that for our model to recapitulate class IV-like morphologies, contact-based retraction needs to lead to complete branch deletion, i.e., the mean retraction distance (α) is much larger than the mean branch length ( L ). By deleting the branches whose tips collide with other branches, gaps of a size similar to the mean branch length are created and maintained. Also, dense regions where the gap size is less than or equal to the mean branch length will not increase in density. Such overfilling is seen in the simulations for small retraction lengths (Fig. 4 A). Thus, our model constrains both the branching frequency and the retraction distance.
Finally, we note that the mesh size of class IV dendrites, 4 − 8 µm, is well suited for detecting highly localized nociceptive stimuli such as punctures by the 10 µm diameter ovipositor barb of parasitic wasps ( [8]). This acuity is maintained throughout development. Thus, the small mesh size is consistent with the class IV neuron being a harsh touch sensor. Indeed, the theory of contact dynamics predicts that the indentation h of the surface of an elastic body poked by a probe with a cross-sectional radius R (pushing normal to the surface) is h ∝ F/R, where F is the applied force [31]. Therefore, the smaller R, the larger h (for a fixed force); the more local the stimuli, the more sensitive to localized forces. Thus the small mesh size suggests that the class IV neuron is well adapted to sensing harsh touch throughout larval stages. In contrast, the class III neuron, a soft touch sensor, would need to capture diffuse stimuli, i.e., the mesh size can be large. Thus, the morphologies of both class IV and class III neurons are well-suited for their mechanosensitive functions.
Materials and Methods
Drosophila Stocks
All flies were maintained on standard medium at 23 o C. The strain ppk-cd4-tdGFP was a kind gift from Dr. Han Chun (Cornell University).
Imaging and Skeletonization
The larvae were mounted in 50% glycerol in PBS between a glass slide and a cover slip. The sample was imaged using a confocal laser scanning microscope (Zeiss, LSM780) with 63x objective. The 600x600 µm images were stitched together offline using Fiji and the stitched images were processed using the NeuronStudio [23] to obtain the skeleton a one pixel wide tracings of the dendritic arbors.
A Mathematical Definitions of Morphometrics
Radius of gyration of the Neuron
The radius of gyration is defined as
R g = 1 M M j=1 (r j − r m ) 2 ,(1)
where M is the total number of occupied pixels, r j is the position of the j th occupied pixel and r m is the mean position of all occupied pixels. R g measures the standard deviation of the dendrite pixels, i.e., the spread of the imaged neuron from its center.
Path Correlation and Persistence Length
The deviation of a branch path from a straight line can be quantified using the tangent vector autocorrelation function
C t (∆s) ∼ t (s) ·t(s + ∆s) s ,(2)
wheret(s) is the tangent vector as a function of the path length s. C t measures the angular change of thet as a function of path length, i.e., how bent the branch is. If C t = 1, the path is straight and if C t = 0, there is a 90 o turn in the path.
Branch Length Correlation Function
The branch length autocorrelation function is
C l (∆d) = L(d)L(d + ∆d) d − L(d) 2 d L 2 (d) d − L(d) 2 d ,(3)
where L(d) is the branch length at depth d and ∆d is the depth difference. Depth is defined as the number of branch points between the branch and the soma, along the shortest path from the branch to the soma. . . . d represents the average over d.
Hitting Probability
Consider a box with side length R centered anywhere in the receptor field of the neuron (Fig. 1 C). We then ask: 'what is the probability P H (b, R) of having b pixels in a box of size R?'. Using this probability, we can determine the probability that a box of size R contains at least n pixels
H n (R) = M n P H (b, R)db,(4)
where M is the total number of neuron pixels. We define the mesh size B H such that H 1 (R = B H ) = 0.5, i.e., the mesh size is the box width such that there is a 50% chance that the box contains at least one pixel from the skeleton.
Fractal Dimension
In this paper, the fractal dimension is measured using two different methods: the correlation dimension ( Fig. 1 E) and box counting (Fig. 1 F) method. In the box counting method, we determine the number of boxes N of side length R that are needed to cover the neuron (Fig. 1 B). The number of boxes needed to cover a line of length l is N = l R ; therefore N ∝ R −1 . The number of boxes needed to cover a square of side length l is N = l R 2 ; therefore N ∝ R −2 . In general, N ∝ R −d b , where d b is the box counting measure of the fractal dimension.
In the correlation method [32], we determine how many pixels are contained within a circle of radius R. Let each point x i on the neuron be the center of a circle of radius R (Fig. 1 B). Then N (x i , R) is the number of skeleton pixels in the circle (green pixels in Fig. 1 C). Averaging over all possible centers (i.e. skeleton pixels) x i , gives
κ(R) = N (x i , R) xi .(5)
In general, κ(R) ∝ R dc where d c is the correlation measure of the fractal dimension. The relation f (R) ∝ R d is called a scaling law and is only valid in a finite range of R (e.g. for small R we approach the scale of one pixel, and for large R we approach the total dimension of the neuron). For the neurons the minimum scale is half the mean branch length and the maximum scale is the radius of gyration.
Lacunarity
Consider the set of boxes of linear dimension R used in the box counting method (see figure 1 B). Instead of asking how many boxes are needed to cover the shape, we ask 'what is the probability P B (b, r) of having b pixels in a box of size R?'. P B (b, r) differs from P H (b, R) since it only considers boxes that have at least one pixel (b ≥ 1).
The moments of P B are defined as
where σ 2 is the variance of P B (b, R). CV (R) is also called the lacunarity function. The more uniform a shape is, the smaller CV and the more variable the shape, the larger the CV . For example, a uniform shape would have a CV ∼ 0. How large CV needs to be for a shape/neuron to be consider variable is somewhat arbitrary. The more important point is that the larger CV , the larger the variation in the neuron. It also allows us to see how these variations change with length scale. Thus, the lacunarity function measures the variations and assigns them a typical length scale.
Figure 1 :
1Morphometrics of class IV and class III dendrites. (A) 3 rd instar larval expressing GFP-tagged membrane protein (ppk-cd4-tdGFP ) in class IV neurons. (B) The skeleton of a class IV neuron from a third instar larva with an example of a box used to calculate the hitting probability (blue) and a circle used to calculate the correlation dimension d c (magenta). (C) The skeleton of a class III neuron from a third instar larva with an example of a set of boxes used to compute the box dimension d b and lacunarity function Λ. See Definition of Morphometrics and Appendix for details (D) Example hitting probability H versus box size B. (E) The correlation function κ versus the radius R. Fits used to determine the correlation dimension d c plotted in solid lines. See Appendix for details of fits. The curves for d c = 1 and d c = 2 are plotted in blue for reference. (F) The number of boxes needed to cover a neuron versus the box size B. Fits used to determine −d b are plotted in solid lines. See Appendix for details of fits. The curves for d b = 1 and d b = 2 are plotted in blue for reference. (G) Lacunarity function Λ versus box size B for a class IV neuron (black) and a class III neuron (red).
Figure 2 :
2Morphometrics of class IV neurons during larval growth. (A)Examples of class IV neurons during larval stages. All scale bars 30 µm, all time stamps hours After Egg Lay. (B) The hitting probability H is plotted versus the box size B for the 5 neurons shown in Fig. 2 A. In the inset, we plot H versus B/ L . We define the mesh size B H such that H(B = B H ) = 0.5. B H ∼ 0.72 L throughout the larval stages. (C) B H is plotted versus larva body segment length S (red 28 cells). For simulated neurons, ω = 0.2 min −1 and α = 10 2 µm (blue). In the inset, we have plotted the mean branch length of the class IV dendrites versus the larva body segment length S. (D)The fractal dimension d f is plotted against S. We have binned the data by body segment length S with bin widths of 77 µm. Simulation parameters are same as before. (E) The lacunarity Λ is plotted against box size B for the five neurons inFig. 2. In the inset, we plot Λ versus B/R g . We define the lacunarity length B Λ as the box size at which Λ(B = B Λ ) = 0.25.
Figure 3 :
3Branching Rules (A) Number of branches versus larval body segment length S for class IV dendrites at different larval stages. The slope of the fit is 1.64 branches µm −1 for the line passing through the S axis at 50 µm, the segment length at the onset of dendritogenesis. (B) The probability density P (L) that a branch of a dendrite (at a particular time) has length L. The probability density is rescaled by multiplying by L and the branch length is rescaled by dividing by L (of the particular dendrite). The superposition of the data indicates that the branch lengths are well described be an exponential distribution (red line) at all larval stages. (C) The branch length autocorrelation C l (Eq. 3) vs depth difference ∆d averaged over all branches. The near-zero values for ∆d ≥ 1 imply that branch length is independent of depth. (D) The radius of gyration R g (Eq. 1) of 28 class IV neurons versus larval body segment length S. The slope of the fit is 0.39, where the S−intercept is set such that R g = 0 at the onset of class IV dendrite morphogenesis. (E) The tangent vector autocorrelation function C t (Eq. 2), averaged over all branches, versus the path length lag ∆s (Eq. 2). The red line is an exponential fit to the data C t (∆s) = e ∆s/β , where β = 44.8 ± 1.5 µm is the persistence length.
Figure 4 :
4Simulations of dendritic growth (A) Nine representative simulations of dendrties for a range of branching frequencies ω and retraction scales α. For all simulations, the persistence length was β = 70 µm and simulation time was T = 100 hrs. For α 1 µm and ω ∼ 0.2min −1 (dashed box) we have subjective agreement with the morphology of real neurons. (B) The branch length distribution for simulated neurons in the dashed black box in (A) is approximately exponentially distributed as observed. (C) The mesh size B H in a contour plot versus the branching frequency ω and retraction scale α. Note that the retraction scale α is plotted in a log scale. In pink we have highlighted the observed value of B H = 7 µm (see Tab. 1). The white box indicates the values used for the time series plotted in Fig. 2 C and D (ω = 0.2 min −1 and α = 10 2 µm). (D) The fractal dimension d f in a contour plot versus the branching frequency ω and the retraction scale α. In pink we have highlighted the physiological value of d f = 1.8 (seeTab.1) (E) The lacunarity scale B Λ in a contour plot versus the branching frequency ω and the retraction scale α. We highlighted in pink the observed value B Λ = 33 µm (see Tab. 1). We find that this value is not found in the range of values used in our simulations.
P B (b, R)db,where M is the total number of skeleton pixels. This then allows us to look at the coefficient of variation of P B (b,
Table 1 :
1Properties of class IV and class III neurons. All numbers are mean ± SD.class IV (n = 14)
class III (n = 8)
Table 2 :
2Parameters in the model. All errors are SD (n=28 neurons).Parameter
Measured Value
Simulation Value
Branching frequency
0.12 ± 0.03 min −1 (ω net )
0.01 − 2 min −1 (ω)
Tip elongation rate, v
0.04 ± 0.02 µm · min −1
0.08 µm · min −1
Persistence length, β
44.8 ± 1.5 µm
70 µm
Retraction length, α
not measured
0 − 1000 µm
AcknowledgmentsThis work was partially supported by the NIH Pioneer Award (Award Number), National Natural Science Foundation of China (NSFC Grant 31671389, to X.L.) and Max-Planck Partner Program (to X.L.). SG was supported by an EMBO Long-Term Fellowship, and OT is supported by the Fonds de Reshershe du Qubec -Nature et technologies. Thanks to Dr. Han Chun (Cornell University) for the fly strains.
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"Citadel E-Learning: A New Dimension to Learning System"
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"Awe O \nDepartment of Computer Science and Mathematics\nBabcock University Ilishan-Remo\nOgun State\nNigeria\n",
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| E-learning has been an important policy for education planners for many years in developed countries. This policy has been adopted by education in some developing countries; it is therefore expedient to study its emergence in the Nigerian education system. The birth of contemporary technology shows that there is higher requirement for education even in the work force. This has been an eye opener to importance of Education which conveniently can be achieved through E-learning. This work presents CITADEL E-learning approach to Nigeria institutions; its ubiquity, its implementations, its flexibility, portability, ease of use and feature that are synonymous to the standard of education in Nigeria and how it can be enhanced to improve learning for both educators and learners to help them in their learning endeavour. | null | [
"https://arxiv.org/pdf/1105.4517v1.pdf"
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| 14,941,893 | 1105.4517 | 08ddc8c92fcb1f99b6baf9b647afe25fbd6e2f4e |
Citadel E-Learning: A New Dimension to Learning System
2011
Awodele O
Department of Computer Science and Mathematics
Babcock University Ilishan-Remo
Ogun State
Nigeria
Kuyoro S O
Department of Computer Science and Mathematics
Babcock University Ilishan-Remo
Ogun State
Nigeria
Adejumobi A K
Department of Computer Science and Mathematics
Babcock University Ilishan-Remo
Ogun State
Nigeria
Awe O
Department of Computer Science and Mathematics
Babcock University Ilishan-Remo
Ogun State
Nigeria
Makanju O
Department of Computer Science and Mathematics
Babcock University Ilishan-Remo
Ogun State
Nigeria
Citadel E-Learning: A New Dimension to Learning System
World of Computer Science and Information Technology Journal (WCSIT)
13201171E-learning environmentICTDistance learning
E-learning has been an important policy for education planners for many years in developed countries. This policy has been adopted by education in some developing countries; it is therefore expedient to study its emergence in the Nigerian education system. The birth of contemporary technology shows that there is higher requirement for education even in the work force. This has been an eye opener to importance of Education which conveniently can be achieved through E-learning. This work presents CITADEL E-learning approach to Nigeria institutions; its ubiquity, its implementations, its flexibility, portability, ease of use and feature that are synonymous to the standard of education in Nigeria and how it can be enhanced to improve learning for both educators and learners to help them in their learning endeavour.
INTRODUCTION
E-learning is an innovative approach for delivering electronically mediated, well-designed, learner-centred interactive learning environments to anyone, anyplace, anytime by utilizing the internet and digital technologies in respect to instructional design principles. It involves learning through the use of ICT infrastructures. In this age, learning through the use of computer is simply online way of acquiring knowledge through the internet. The online learning involves the use of Internet Browser or Navigator. It may be in form of Audio, Visual, and or Audio-Visual. The convergence of the internet and learning, or Internet-enabled learning is called Elearning. [5] In Nigeria, very few universities are currently carrying out their academic activities through one form of ICT or the other, while the urge to embark on E-learning is still a far-fetched dream to some, because their ICT infrastructure is very inadequate. The rapid expansion of ICTs in Nigeria offers an opportunity to consider its use in the promotion of distance education. It offer students considerable benefits including increase access to learning opportunities, convenience of time, and place, making available a greater variety of learning resources, improve opportunities for individualized learning and emergence of more powerful cognitive tools.
In most developing countries, the percentage of resources available for students is very minimal, such that the tradition mode of learning, limits the student's assimilation and their knowledge only to available resources. With E-learning; resources such as course materials, and other learning enhanced instructional will be unlimited and would be at the fingertips of the students. Very few of the E-learning environments lay emphasis on the importance of motivation for students and lecturers alike, in sense that their non userfriendly and hectic environments reduces motivation of both the students and lecturers. This has been considered and thus, incorporated in the project by making it interactive and userfriendly. Learning environment that support administration functionality, feedback functionality will be adopted. The CITADEL e-learning environment allows easy access that will motivates Instructors and students to participate effectively.
II. RELATED WORK
There are various literatures on E-learning, some of which has been carefully reviewed in this work. Learning environments such as Sakai Project and Blackboard Learning System are considered. Traditional academic learning as teacher-centred instruction of synchronous and scheduled groups, constrained by classroom availability, while e-learning is student-centred, asynchronous, and available anytime and anywhere. E-learning can be both highly interactive and simultaneously isolating because of the inherent difficulties of developing cohesiveness and true connectedness among students (Sauer 2001). Unfortunately, few research focus on the use and effectiveness of e-learning processes in Nigeria.
[6] emphasized the attention given to e-learning and its development. There has been business and academic interest in e-learning; views such as the perspective of brokers, educational service and content providers were considered. Elearning was viewed according to its basic aspect namely: Content, Services, Agents and Rules. The use of well structured descriptions of content material has contributed to the desired materials. Mentioned also is the agent playing role in E-learning environment. These agents are Active learner, collaborative learners, passive learners, content developers and administrators. It was therefore concluded that agents and roles should also be incorporated in the web service definitions and eventually can be accessed by the individuals by related metadata that have been registered with an online web service registry and then the registry can be searched manually or programmatically to discover and select components.
[3] explicitly elaborates the importance and challenges faced by the user of blackboard. The benefits include quick feedback, increased availability, improved communication, tracking and skill building. It was also stated that blackboard can be use as supplement to classroom learning even when other digital environment learning system are the primary instructional tool. He went further to enumerate the drawbacks which are hardness to learn and non-flexibility. A survey of staff and student in University of Wisconsin system who majorly use blackboard management system find it difficult to learn because of its non-flexibility. The study also found that despite expectations, many students were not proficient with the technology. A separate study, an evaluation of Blackboard as a platform for distance education delivery at Hampton University School of Nursing, found that the internet is often a new learning environment for those returning to University for graduate degrees. Result showed that blackboard users are provided with course management system that delivers learning content and resources to them. [4] shows in his study that Sakai Project, an open source collaborative learning environment, has been adopted in different institution around the globe. University of Fernando Pessoa (UFP) started a pilot experience with Sakai 1.0 in October 2004 opening it to all instructors and students alike . One year later, around 782 users had logged in and 150 sites were active since then the growth of Sakai has increased.
Having carefully analysed some literature on e-learning environments, it is discovered that blackboard management system is not user-friendly and it is non-flexible and other elearning management systems have been adopted in various countries because they are applicable to the environment. A new System called CITADEL e-learning environment that conforms to the education system of Nigeria is developed in this work. The usual student identification/matriculation number can be used on this system. It provides features as those in the traditional learning but ubiquitously. As such, it can be implemented effectively in any tertiary institution in Nigeria. It is not platform dependent therefore it can be implemented on any operating system.
III. PROPOSED SYSTEM
This work is designed to accentuate the flexibility and simplicity of e-learning in Nigerian institutions. CITADEL elearning system is a software package for producing Internetbased courses. It is a global development project designed to support a social constructionist framework of education. Based on evaluation of all available web development tools, HTML, ADOBE FIREWORKS was used for the interface of the environment, PHP for interaction with the database, JAVASCRIPT for more advanced interactive interface and MYSQL as the database to keep track of data are used to develop the software to make CITADEL effective and easily adaptable in Nigeria. The system is designed with flexibility in mind; and consideration for expansion was paramount in the design. Other factors considered in the design include security, portability, usability and reusability. In general, it was ensured that the design conforms to the ISO standard.
The system architecture is in three different layers. These layers are responsible for the presentation of information from the back-end of the system to the users, monitors the interaction between the data and user and the also provides control on the DBMS. The layers are as follows Citadel E-learning environment comprises all forms of electronically supported learning and teaching, which are procedural in character and aim to effect the construction of knowledge with reference to individual experience, practice and knowledge of the learner. Information and communication systems, serve as specific media to implement the learning process. CITADEL is essentially the use of computer and network-enabled transfer of skills and knowledge. Content is delivered via the Internet, audio or video tape, and CD-ROM. It includes media in the form of text, image, animation, streaming video and audio. It has the functionality such as to create and deliver courses through media such as videos and audio as well as document courses such as PDFs, grades and access learning outcomes (continuous assessment), manage students, courses and resources and also to track student and course progress from standard administrative report. It has three modules. These modules include:
Student: This is the student page where he or she can login with password and username. Once access is granted the student can access the various features available on the page. This includes: courses, registered courses, downloads, lectures, assignment, examinations and quizzes, results, timetable etc. The index page is the page through which the student would gain access into the system. On the page is a login form with two input boxes provided for username (Matric number) and password. When this data is supplied the system validates the data, if the data supplied is valid the student gains entry into the system else access is denied as shown in Fig. 2.
Lecturer: This where the lecturers also login in with password and access functionalities such as post assignment, upload lectures, create examinations, start alive class through video conferencing etc. The lecturer login page as depicted in Fig. 3 is the page from which the lecturer would use in order to gain access into the lecturer window which will take him to his home page. In this window the lecturer can choose various tasks like preparing quiz, lecture notes, give assignment etc.
Registry: The registry performs the same function as those of traditional learning environment. The function of the registry is to register students, lecturer, create department and faculties. This section uses screenshots to show the features and use of CITADEL learning environment and elaborate on the three modules of the software. It also shows it documentation and step by step user guide. The registrar login page shown in Fig. 4 is the page from which the registry would use in order to gain access into the registry window and perform all the functionality available to page.
Other windows such as Messages, View Classmates, Assignment, Exam and quizzes, timetable, Library, Downloads, View syllabus, View Lecturers, Notice, Chat, etc are depicted at the appendix. The use of E-learning management systems is still growing so is its acceptance rate in Nigeria as a suitable replacement for traditional learning system. It has advantages that supersede the disadvantages that in no time it will be widely acceptable. The survival and future of tertiary institution greatly depend on E-learning.
The use of the citadel E-learning platform allows institutions foster teaching activity (both by saving administrative and communication efforts) as well as providing the opportunity to improve the learning experience by means of innovative proposals which may enhance the teaching and learning process VI. RECOMMENDATIONS FOR FURTHER WORK This software has attempted to solve the problem of traditional education system and to a large extent it is successful. Regardless of the fact that the system has met the basic objectives of the work, there are ways it could be improved upon and used by other organisations for training or other forms of education sector for usage. It is therefore recommended that further research be carried out on this work to improve it functionality and increase its features. A vital functionality is to ensure that the lecturer, during video conferencing should be able to view physically, the students available for lectures. This will verify the students' attendance. Further research can also be done on how to incorporate the student payment into the study.
APPENDIX
Below is the list of other windows generated from Citadel Elearning environment:
Student home page: The home page is the first page that the student will encounter after logging in successfully. From the home page the student will be able to navigate to various features of the software. Exams and Quizzes: This is a section of the program which allows the student to take an exam or quiz as long as they don't miss their deadline. Downloads: This is for downloads which the students might need in the course of using the system to view or access lecture materials.
layers are represented below inFig. 1
Figure 1 :
1System Architecture IV. IMPLEMENTATION
Figure 2 :Figure 3 :
23Index Lecturer
Figure 5 :
5Homepage page Messages: this feature is for students to receive and send messages to their lecturers and classmates.
Figure 6 :
6Message window View Classmates: This feature is for the student to be able to view details such as contact details about other students in their class as depicted below
Figure 7 :
7View Classmate window
Figure 8 :
8Exam and quizzes window Assignment: This is a section for viewing or downloading and submitting assignments.
Figure 9 :
9Assignment window Timetable: This is a feature which updates the student about activities such as exams and quizzes of the day.
Figure 10 :
10Timetable window Library: As shown in Fig. 11, this is section for viewing available books and details about them in physical library
Figure 11 :
11Library Window
Figure 12 :Figure 13 :
1213Download WindowView Syllabus: This section is for viewing syllabus for the courses which the students are taking. This is shown below View Syllabus Window View Lecturers: This is for viewing details about the lecturers who are taking courses registered by the students.
Figure 14 :
14View Lecturers Window Notice: This is for viewing notices created by the lecturers, registry, library etc.
Figure 15 :
15Notice Window Lectures: This is a section which allows the students to view and download lectures uploaded by the lecturers.
Figure 16 :
16Lectures Window Chat: This is a section the students use for chatting with the other classmates.
Figure 17 :
17Chat . Awodele Oludele is a Senior Lecturer at the Computer Science and Mathematics Department, Babcock University Ilishan-Remo, Ogun State, Nigeria. Email: [email protected]; Phone: (+234) 8033378761. His research interests are
Ms. S. O. Kuyoro is an assistant lecturer at the Computer Science and Mathematics Department, Babcock University Ilishan-Remo, Ogun State,
Towards Distance Learning Systems in Nigeria. G J Afolabi-Ojo, Ile-IfeUniversity of Ife now Obafemi Awolowo UniversityA valedictory lecture delivered at theG.J. Afolabi-Ojo, "Towards Distance Learning Systems in Nigeria", A valedictory lecture delivered at the University of Ife now Obafemi Awolowo University, Ile-Ife 1986.
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Paper Presented at the Federal Polytechnic, Oko, Anambra state on the occasion of the Convocation ceremony and 10 th anniversary celebration held on Friday 28. O J Jegede, Taking the distance out of higher education in 21st century NigeriaO. J. Jegede, "Taking the distance out of higher education in 21st century Nigeria", Paper Presented at the Federal Polytechnic, Oko, Anambra state on the occasion of the Convocation ceremony and 10 th anniversary celebration held on Friday 28, 2003.
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Email: [email protected]; Phone: +(234) 8066188970. Her research interests are in the area of data mining and artificial intelligence. Nigeria, Nigeria. Email: [email protected]; Phone: +(234) 8066188970. Her research interests are in the area of data mining and artificial intelligence.
Email: [email protected]; Phone: +(234) 7033900451. Her research interests are Database administration. Ms, A K Adejumobi, Ogun State, NigeriaDepartment of Computer Science and Mathematics, Babcock University, Ilishan-Remois a final year student of the. and System Analysis and designMs. Adejumobi A. K. is a final year student of the Department of Computer Science and Mathematics, Babcock University, Ilishan-Remo, Ogun State, Nigeria. Email: [email protected]; Phone: +(234) 7033900451. Her research interests are Database administration; and System Analysis and design.
Awe O. is a final year student of the. Mr, Ogun State, NigeriaDepartment of Computer Science and Mathematics, Babcock University, Ilishan-RemoEmail: [email protected]; Phone: +(234) 8060987929. His research interest is web application developmentMr. Awe O. is a final year student of the Department of Computer Science and Mathematics, Babcock University, Ilishan-Remo, Ogun State, Nigeria. Email: [email protected]; Phone: +(234) 8060987929. His research interest is web application development.
Email: [email protected]; Phone: +(234) 8121152468. Her research interests are web application and software development. Ms, Ogun State, NigeriaDepartment of Computer Science and Mathematics, Babcock University, Ilishan-RemoMakanju O. is a final year student of theMs. Makanju O. is a final year student of the Department of Computer Science and Mathematics, Babcock University, Ilishan-Remo, Ogun State, Nigeria. Email: [email protected]; Phone: +(234) 8121152468. Her research interests are web application and software development.
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"Signs of the time: Melonic theories over diverse number systems",
"Signs of the time: Melonic theories over diverse number systems"
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"Steven S Gubser [email protected] \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA\n",
"Matthew Heydeman [email protected] \nWalter Burke Institute for Theoretical Physics\nCalifornia Institute of Technology\n452-48, 91125PasadenaCAUSA\n",
"Christian Jepsen [email protected] \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA\n",
"Sarthak Parikh [email protected] \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA\n",
"Ingmar Saberi [email protected] \nMathematisches Institut\nRuprecht-Karls-Universität Heidelberg\nIm Neuenheimer Feld 20569120HeidelbergGermany\n",
"Bogdan Stoica [email protected] \nMartin A. Fisher School of Physics\nBrandeis University\n02453WalthamMAUSA\n\nDepartment of Physics\nBrown University\n02912ProvidenceRIUSA\n",
"Brian Trundy [email protected] \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA\n"
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"Joseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA",
"Walter Burke Institute for Theoretical Physics\nCalifornia Institute of Technology\n452-48, 91125PasadenaCAUSA",
"Joseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA",
"Joseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA",
"Mathematisches Institut\nRuprecht-Karls-Universität Heidelberg\nIm Neuenheimer Feld 20569120HeidelbergGermany",
"Martin A. Fisher School of Physics\nBrandeis University\n02453WalthamMAUSA",
"Department of Physics\nBrown University\n02912ProvidenceRIUSA",
"Joseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA"
]
| []
| We define melonic field theories over the p-adic numbers with the help of a sign character. Our construction works over the reals as well as the p-adics, and it includes the fermionic and bosonic Klebanov-Tarnopolsky models as special cases; depending on the sign character, the symmetry group of the field theory can be either orthogonal or symplectic. Analysis of the Schwinger-Dyson equation for the two-point function in the leading melonic limit shows that power law scaling behavior in the infrared arises for fermionic theories when the sign character is non-trivial, and for bosonic theories when the sign character is trivial. In certain cases, the Schwinger-Dyson equation can be solved exactly using a quartic polynomial equation, and the solution interpolates between the ultraviolet scaling controlled by the spectral parameter and the universal infrared scaling. As a by-product of our analysis, we see that melonic field theories defined over the real numbers can be modified by replacing the time derivative by a bilocal kinetic term with a continuously variable spectral parameter. The infrared scaling of the resulting two-point function is universal, independent of the spectral parameter of the ultraviolet theory. | 10.1103/physrevd.98.126007 | [
"https://arxiv.org/pdf/1707.01087v2.pdf"
]
| 31,599,685 | 1707.01087 | 1b88ae4638375f9d0857cdee5d0859c60c850ab7 |
Signs of the time: Melonic theories over diverse number systems
July 2017 4 Jul 2017
Steven S Gubser [email protected]
Joseph Henry Laboratories
Princeton University
08544PrincetonNJUSA
Matthew Heydeman [email protected]
Walter Burke Institute for Theoretical Physics
California Institute of Technology
452-48, 91125PasadenaCAUSA
Christian Jepsen [email protected]
Joseph Henry Laboratories
Princeton University
08544PrincetonNJUSA
Sarthak Parikh [email protected]
Joseph Henry Laboratories
Princeton University
08544PrincetonNJUSA
Ingmar Saberi [email protected]
Mathematisches Institut
Ruprecht-Karls-Universität Heidelberg
Im Neuenheimer Feld 20569120HeidelbergGermany
Bogdan Stoica [email protected]
Martin A. Fisher School of Physics
Brandeis University
02453WalthamMAUSA
Department of Physics
Brown University
02912ProvidenceRIUSA
Brian Trundy [email protected]
Joseph Henry Laboratories
Princeton University
08544PrincetonNJUSA
Signs of the time: Melonic theories over diverse number systems
July 2017 4 Jul 2017
We define melonic field theories over the p-adic numbers with the help of a sign character. Our construction works over the reals as well as the p-adics, and it includes the fermionic and bosonic Klebanov-Tarnopolsky models as special cases; depending on the sign character, the symmetry group of the field theory can be either orthogonal or symplectic. Analysis of the Schwinger-Dyson equation for the two-point function in the leading melonic limit shows that power law scaling behavior in the infrared arises for fermionic theories when the sign character is non-trivial, and for bosonic theories when the sign character is trivial. In certain cases, the Schwinger-Dyson equation can be solved exactly using a quartic polynomial equation, and the solution interpolates between the ultraviolet scaling controlled by the spectral parameter and the universal infrared scaling. As a by-product of our analysis, we see that melonic field theories defined over the real numbers can be modified by replacing the time derivative by a bilocal kinetic term with a continuously variable spectral parameter. The infrared scaling of the resulting two-point function is universal, independent of the spectral parameter of the ultraviolet theory.
Introduction
A first working definition of a p-adic quantum field theory is a theory defined through a functional integral over maps φ : Q p → R, where Q p denotes the p-adic numbers and R denotes the reals. 1 We may expand our definition of p-adic quantum field theories by replacing Q p with a field extension of Q p , and by allowing φ to be valued in some vector space over R. If the values of φ are anti-commuting, we refer to φ as a fermionic field, while if they are commuting, we refer to φ as a bosonic field.
A first inkling of p-adic field theories (for p = 2) appeared in the form of the Dyson hierarchical model [1]. In this model, one starts with a chain of Ising spins s n where n ∈ Z.
A strong ferromagnetic interaction is assigned between spin 2n and spin 2n + 1 for all n. Next one assigns a weaker ferromagnetic interaction between pairs, and yet a weaker ferromagnetic interaction between pairs of pairs. After − 1 steps, one has blocks of 2 −1 spins, and in the -th step one pairs up neighboring blocks with a ferromagnetic interaction whose strength decreases as a power of 2 : See figure 1. The critical behavior of this model is 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 s n n Figure 1: The hierarchical model, with an Ising spin s n = ±1 at every integer point n, and successively weaker ferromagnetic couplings between pairs of spins, pairs of pairs, and so on.
described by a 2-adic conformal field theory, meaning a 2-adic field theory which is invariant under the action of PGL(2, Q 2 ) on Q 2 through linear fractional transformations, t → at+b ct+d , where a, b, c, d as well as t are in Q 2 . We may approximately understand the action of these linear fractional transformations as mapping spin blocks to spin blocks, but not necessarily preserving the size of the blocks. In practical terms, PGL(2, Q p ) invariance restricts the form of correlators of local operators: for example, in scalar p-adic field theories we have φ(t)φ(0) = K/|t| ∆ where K and ∆ are real constants, and |t| is the p-adic norm. 1 In physics parlance, φ is termed the field; unfortunately, in math parlance, Q p and R are fields. In this paper we will overload the term "field" to carry both definitions, on the expectation that context will make the meaning clear.
Hierarchical constructions are conducive to Wilsonian renormalization group ideas, because the hierarchical structure provides a natural way to perform Kadanoff spin blocking [2]. Indeed, the early literature on the Wilsonian renormalization group shows that the original practitioners were well aware of hierarchical models, if not their connection to the p-adic numbers; see for example the review [3]. Subsequently, renormalization group flows in p-adic field theories have been studied fairly extensively, notably by Missarov and collaborators; see the recent review [4].
Recently, progress has been made in understanding a p-adic version of the anti-de Sitter / conformal field theory correspondence (AdS/CFT) [5,6]. The p-adic version of the correspondence is an equivalence between a conformal field theory on Q p (more properly P 1 (Q p )) and a bulk theory defined on the Bruhat-Tits tree. The Bruhat-Tits tree is the p-adic version of anti-de Sitter space, and it is the quotient of the p-adic conformal group PGL(2, Q p ) by its maximal compact subgroup. Particularly if we stick with Q p on the boundary, rather than some extension of Q p , p-adic AdS/CFT seems to be an analog of AdS 2 /CFT 1 . To develop the correspondence further, we are therefore interested in p-adic analogs of one-dimensional conformal theories which are thought to have AdS 2 duals.
An example CFT 1 of recent interest is the infrared limit of the melonic theory introduced in [7] and simplified in [8]. These theories are inspired on one hand by the Sachdev-Ye-Kitaev (SYK) model [9,10], and on the other by melonic scaling limits; for a review of melonic scaling see [11]. For our purposes, the melonic field theories of [7,8] are attractive for three reasons.
First, the two-point function of the fermionic field in the model can be calculated explicitly in the melonic limit thanks to a Schwinger-Dyson equation that resums precisely the diagrams that survive in the leading melonic limit. Second, there is a non-trivial renormalization group flow from a free theory in the ultraviolet to a conformal theory in the infrared. Third, there are indications of the existence of an AdS 2 dual.
In the current work, our aim is to formulate p-adic versions of the simplest melonic theories. As in our previous work on the O(N ) model [12], it is possible to give a remarkably uniform presentation in which theories over the reals and over the p-adics are treated on almost an equal footing. There are three main differences between the reals and the p-adics:
• In p-adic field theories, ordinary derivatives like dφ/dt are not good starting points for kinetic terms in the action, because smooth functions φ(t) are piecewise constant. One is driven instead to the Vladimirov derivative, which leads to bilocal kinetic terms, approximately of the form (φ(t 1 ) − φ(t 2 ))/|t 1 − t 2 | 1+s where s ∈ R is the order of the derivative. It is therefore natural to consider at the same time bilocal kinetic terms in theories over the reals. 2 • A key ingredient both in the action and in Green's functions of melonic theories is the sign function sgn t, which ordinarily is −1 if t < 0 and +1 if t > 0. The p-adic numbers are not naturally ordered; however, they do admit an assortment of sign functions which are multiplicative characters: sgn(t 1 t 2 ) = (sgn t 1 )(sgn t 2 ). We will wind up with approximately ten variants (depending on how we count) of the simplest melonic theory due to this profusion of sign characters, which can naturally be parametrized by a nonzero p-adic number τ . Sign characters on the reals can be parametrized the same way, and τ < 0 corresponds to the ordinary notion of sign while τ > 0 corresponds to the trivial character which maps all real numbers to +1.
• For some sign characters, in order to get a theory with well-defined scaling behavior both in the ultraviolet and in the infrared, we are obliged to alter the global symmetry group of the theory from O(N ) 3 to Sp(N ) 3 , where Sp(N ) is the non-compact group of real-valued N × N matrices preserving a symplectic structure.
The structure of the rest of the paper is as follows. In section 2 we construct the kinetic terms which specify the theories of interest in the ultraviolet. In section 3 we add in the interaction term, which is a relevant deformation. In section 4 we explain how the Schwinger-Dyson equation for the two-point function arises. In section 5 we present a self-contained account of sign characters on Q p and certain extensions, including generalized Γ and B functions of these characters. In section 6 we show how to solve the Schwinger-Dyson equation in the infrared. In some cases we are able to solve it exactly at all scales. In section 7 we explain the Wilsonian perspective on renormalization of melonic theories over Q p , which features a powerful non-renormalization theorem on the kinetic term of the Wilsonian action. We conclude in section 8 with a summary and an indication of some directions for future work.
In appendix A we discuss symplectic groups in more detail than we give in the main text.
Readers wishing to see a tabulation of our main results can consult table 1, in which we indicate for each field (R or Q p ) and each sign character (parametrized by τ ) what sort of theory we must consider in order to have a renormalization group flow from a free ultraviolet 2 While this work was nearing completion, we received [13], which also considers SYK-like models with bilocal kinetic terms. However, the focus of [13] is rather different from ours; their aim is to consider theories over R not Q p , and to deform the free bilocal theory by a marginal deformation so as to eventually reach the strongly coupled infrared theory, with every intermediate theory preserving the full SL(2, R) symmetry characteristic of one-dimensional conformal field theories. We instead add a relevant deformation to the free bilocal theory, and under appropriate conditions we find that the two-point function shows universal infrared behavior.
theory to the universal infrared scaling behavior of the two-point function characteristic of SYK-type theories.
The kinetic term
The SYK model [9,10], Witten's melonic version [7], and the Klebanov-Tarnopolsky model [8] are all based on interacting Majorana fermions in 0 + 1 dimensions. The kinetic term for a single Majorana fermion is 3
S free = R dt i 2 ψ∂ t ψ = R dω 1 2 ψ(−ω)ωψ(ω) .(1)
The associated two-point Green's function is
F (ω) = 1 ω ,(2)
where we use F (ω) instead of G(ω) to emphasize that this is the Green's function of the free theory.
We would like to consider p-adic field theories with a similar structure. In order to get started, we need something that replaces i∂ t . It is easiest intuitively to work in momentum space, where in the Archimedean setting the kernel of S free is Γ free 2 (ω) = ω = |ω| sgn ω. We would like to generalize this to
Γ free 2 (ω) = |ω| s sgn ω F (ω) = 1 Γ free 2 (ω) ,(3)
where the spectral parameter s is real (assuming we want a real-valued action). We can now allow ω to be valued in a field K, which may be either Archimedean or ultrametric. The sign function sgn ω is, by assumption, a multiplicative character from K × to {±1} (just as the ordinary sign function is). Here and below, for any ring R (including if R is a field), R × means all elements of R which have a multiplicative inverse. Since K is a field, K × is all non-zero elements of K.
A surprising and important point is that not all sign characters have sgn(−1) = −1;
those that do are called odd, while those that have sgn(−1) = 1 are called even. To get the idea of why non-trivial even sign characters can be defined on some K, consider a simpler 3 The measure in the ω integral is dω instead of dω 2π due to our convention for Fourier transforms: ψ(t) = R dω e −2πiωt ψ(ω). This convention is far more convenient than the usual one when one passes to Q p , and it tends even to simplify formulas in the real case.
problem: sign characters on finite fields
F p = Z pZ = {0, 1, 2, . . . , p − 1} ,(4)
where p is prime, and where addition and multiplication are defined modulo p. By definition, these are homomorphisms from F × p to {±1}. The case p = 2 is vacuous because F × 2 is the trivial group consisting of only the identity. For odd p, there are precisely two distinct sign characters on F × p : the trivial one which assigns +1 to all elements of F × p , and the non-trivial one, usually written as the Legendre symbol (a|p), which is 1 if a = b 2 for some b ∈ F × p and −1 otherwise. The trivial sign character is obviously even, while the non-trivial character may be even or odd:
sgn(−1) = (−1|p) = (−1) (p−1)/2 on F × p .(5)
For example, sgn(−1) = −1 if p = 3, corresponding to the fact that −1 = 2 ∈ F 3 is not a square; but sgn(−1) = 1 if p = 5 because −1 = 4 = 2 2 in F 5 . It is perhaps unsurprising that similar properties carry over to Q p : Non-trivial sign characters exist for all p, and when p ≡ 3 mod 4, there are both odd and even sign characters, whereas if p ≡ 1 mod 4, all sign characters are even. Sign characters are reviewed and discussed more systematically later in
5.
Assuming ψ is a real Grassmann field, we have
S free ≡ K dω 1 2 ψ(−ω)Γ free 2 (ω)ψ(ω) = K dω 1 2 ψ(ω)Γ free 2 (−ω)ψ(−ω) = − sgn(−1)S free . (6)
The integrations over K use the additive Haar measure. In the second equality of (6) we performed the u-substitution ω → −ω. In the next step we anti-commuted ψ(ω) past ψ(−ω) to pick up the explicit sign, and we used Γ free 2 (−ω) = sgn(−1)Γ free 2 (ω). Thus the action we have proposed makes sense only when the sign character is odd [14]. We can however consider two straightforward generalizations: adding index structure to the field, and changing its statistics.
Consider then the action
S free ≡ K dω 1 2 ψ i (−ω)Γ free 2,ij (ω)ψ j (ω) where Γ free 2,ij (ω) = Ω ij |ω| s sgn ω .(7)
We assume that ψ is either commuting or anti-commuting, and correspondingly we write σ ψ = +1 or −1. We refer to the commuting case as bosonic and the anti-commuting case as fermionic. Also we assume that ψ is real, i.e. ψ * = ψ. We assume that Ω is either symmetric or antisymmetric as a matrix, and correspondingly we write σ Ω = +1 or −1. Then the same manipulation we used to obtain (6) generalizes immediately to give
S free = σ ψ σ Ω sgn(−1)S free .(8)
Evidently, we want the combined sign σ ψ σ Ω sgn(−1) to be equal to 1. Similar considerations show that in order for S free to be real, we must choose Ω to be Hermitian. This is assuming that conjugation reverses the order of anti-commuting factors:
(ψ i ψ j ) * = (ψ j ) * (ψ i ) * .
The full model
For simplicity, let's focus on a generalization of the Klebanov-Tarnopolsky model rather than the SYK model or Witten's version:
S = S free + S int ,(9)
where
S free = K dω 1 2 ψ a 1 b 1 c 1 (−ω)Ω a 1 a 2 Ω b 1 b 2 Ω c 1 c 2 |ω| s (sgn ω)ψ a 2 b 2 c 2 (ω)(10)
and
S int = K dt g 4 Ω a 1 a 2 Ω a 3 a 4 Ω b 1 b 3 Ω b 2 b 4 Ω c 1 c 4 Ω c 2 c 3 ψ a 1 b 1 c 1 (t)ψ a 2 b 2 c 2 (t)ψ a 3 b 3 c 3 (t)ψ a 4 b 4 c 4 (t) .(11)
Note that K can be R or Q p , or some finite extension of one of these fields. The indices a i , b i , and c i all take values from 1 to N , so overall the model is based on exactly N 3 Majorana fermions, or N 3 real scalars. We assume that N is even, and that
Ω = [Ω ab ] = 1 N ×N if σ Ω = 1 σ 2 ⊗ 1N 2 × N 2 if σ Ω = −1 ,(12)
where σ 2 = 0 −i
is the group of N × N matrices Λ a b with real entries such that Λ a c Λ b d Ω ab = Ω cd . Note that Sp(N ) is a non-compact group, unlike O(N ) = O(N, R), but at least at the level of a global symmetry group this is not a problem. See appendix A for discussion of our notations and conventions, which may differ from other choices in the literature.
We assume that K has a finite dimension n over the base field. If the base field is R, then the only possibilities are n = 1 and n = 2, because for n > 2 there is no field structure possible in R n . If the base field is Q p , then n can be any positive integer. Importantly, we follow conventions of [15] by defining | · | so that under a u-substitution u = λt, the integration measure transforms as
du = |λ|dt .(14)
For instance, when K is an extension of Q p , we have
|t| K ≡ |N K:Qp (t)| Qp ,(15)
where N K:Qp denotes the field extension norm. This contrasts with the conventions, e.g., of [12]; in particular, K admits a unique field norm that coincides with the standard norm on Q p ⊂ K, but this field norm on K would be written | · | 1/n in our present notation. Similarly, in the current work, over C we have |x + iy| = x 2 + y 2 and dz = dxdy.
In order to develop some first expectations about the behavior of the field theory, let's do a little dimension counting. The following assignments are equally valid for Archimedean and ultrametric places:
[|ω|] = −[|t|] = 1 [dω] = −[dt] = 1 [ψ(ω)] = − 1 + s 2 [ψ(t)] = 1 − s 2 [g] = 2s − 1 .(16)
In the usual 0 + 1-dimensional model with s = 1, we have [g] = 1, so the interaction is a relevant deformation. Let's assume that s > 1/2 for our generalized model, so that the quartic interaction is still relevant. Then we expect to find correlators agreeing with the free theory in the ultraviolet, and some new fixed point in the infrared, analogous to the infrared behavior of the SYK model and its melonic cousins. We should also stipulate s ≤ 1, because if instead s > 1, then ψ(t) would have negative dimension, which signals a pathology; in particular, it means that operators schematically of the form ψ 2r are more and more relevant as r becomes large, and it is no longer clear that it is meaningful to limit ourselves to a polynomial lagrangian. 4
Diagrammatics and the Schwinger-Dyson equation
The diagrammatic structure of the theory (9) is similar to [7,8], with the main difference being the addition of factors of Ω ab and Ω ab , where the latter is defined by the condition
Ω ab Ω bc ≡ δ a c .(17)
Of course, we also have Ω ab Ω cb = σ Ω δ a c because of the symmetry or antisymmetry of Ω. The free propagator following from (10) is
F a 2 b 2 c 2 ,a 1 b 1 c 1 (t) = Ω a 2 a 1 Ω b 2 b 1 Ω c 2 c 1 F (t) ,(18)
where
F (ω) = sgn ω |ω| s .(19)
Sometimes we will abbreviate F a 2 b 2 c 2 ,a 1 b 1 c 1 to F 2 1 . Graphically, we can keep track of the index structure by representing factors of Ω a i b i with arrows pointing from the second index to the first and using a different color for each type of strand. In this notation, we may represent the free propagator F 2 1 as , (20) and the interaction vertex following from (11) can be represented as .
Capital letters in (21) and below, like bold numbers in (20), represent triples of indices: For example, A means a A b A c A . Note that we may freely flip any arrow at the price of a factor of σ Ω . The fact that the pairwise interchange of all indices in the vertex leaves it invariant up to an even number of arrow flips provides a check that even when Ω ab is anti-symmetric, the interaction term (11) remains non-vanishing and the interaction vertex retains its tetrahedral symmetry.
Even when the base field K may no longer be the real numbers, the large N limit with g 2 N 3 held fixed is still controlled by the same class of "melon" diagrams built from the propagator and vertex, irrespectively of whether the theory exhibits O(N ) 3 or Sp(N ) 3 symmetry since contracting anti-symmetric versus symmetric matrices Ω ab at most changes the overall sign of Feynman diagrams and not the scaling. In particular, the leading correction to the propagator in the melonic limit of N → ∞ with g 2 N 3 held fixed is
G 2 1 (t) = F 2 1 (t) + δG 2 1 (t) ,(22)
where δG 2 1 (t) is given by the loop diagram .
The choice of connecting the index triplets labeled by 1 and A with each other was arbitrary, and likewise the connection between A and 2. Other choices would lead to diagrams related to the above by flipping an even number of arrows, and all these choices sum together to cancel the factors of 1/4 coming from (11). Furthermore, flipping nine arrows, we can turn
the loop diagram (23) into .(24)
The nine flips cost us a factor of (σ Ω ) 9 = σ Ω , but then we have a diagram with all arrows consistently oriented so that the three index loops can each be contracted using (17) to yield a factor of N each, and the paths going from 1 to 2 can each be contracted to a single factor of Ω a i b i . Putting everything together, we see that
δG a 2 b 2 c 2 ,a 1 b 1 c 1 (t) = σ Ω g 2 N 3 Ω a 2 a 1 Ω b 2 b 1 Ω c 2 c 1 K dt 1 dt 2 F (t − t 2 )F (t 2 − t 1 ) 3 F (t 1 ) .(25)
We note from (25) that passing from the O(N ) theory (σ Ω = 1) to the Sp(N ) theory formally corresponds to sending N → −N . This is reminiscent of the results of [16,17]; however, those results pertain to the compact USp(N ) group, whereas our Sp(N ) is the non-compact group Sp(N, R), as mentioned earlier.
We see from (22) and (25) that
G a 2 b 2 c 2 ,a 1 b 1 c 1 (t) = Ω a 2 a 1 Ω b 2 b 1 Ω c 2 c 1 G(t) ,(26)
where
G(t) = F (t) + σ Ω g 2 N 3 K dt 1 dt 2 F (t − t 2 )F (t 2 − t 1 ) 3 F (t 1 ) + . . . ,(27)
where the omitted terms are higher order in g.
The standard strategy for finding the conformal behavior in the infrared in SYK-type models is to improve the calculation of G 2 1 (t) by replacing all but the leftmost propagator in δG by G so as to obtain the leading-order Schwinger-Dyson equation,
G(t) = F (t) + σ Ω g 2 N 3 dt 1 dt 2 G(t − t 2 )G(t 2 − t 1 ) 3 F (t 1 ) ,(28)
and then to argue that (28) captures the entire set of diagrams that contributes at leading order in the melonic limit-meaning that omitted diagrams are suppressed by powers of N when we take N → ∞ with g 2 N 3 fixed and finite. Now define an energy scale
µ ≡ (g 2 N 3 ) 1/(4s−2) .(29)
The claim is that in the infrared, meaning µ|t| 1, the two terms on the right hand side of (28) nearly cancel, so that to obtain the leading order expression for G(t) we formally set the left hand side to 0 and then solve the integral equation to get
G(t) = b sgn t |t| 1/2 (30)
where b is some constant, independent of t. To get hold of this constant, it helps to detour to a more systematic discussion of multiplicative characters, which we undertake in the next section. We can however extract some information about the phase of b from a general analysis of the reality properties of the Green's function, as follows. The full Green's function can be expressed as
G a 2 b 2 c 2 ,a 1 b 1 c 1 (t) = ψ a 2 b 2 c 2 (t)ψ a 1 b 1 c 1 (0) ,(31)
where . . . is understood as being defined in terms of a Euclidean path integral based on the real action (9). Keeping in mind that ψ abc is itself real, we have
G a 2 b 2 c 2 ,a 1 b 1 c 1 (t) * = ψ a 1 b 1 c 1 (0)ψ a 2 b 2 c 2 (t) = σ ψ G a 2 b 2 c 2 ,a 1 b 1 c 1 (t) .(32)
Recalling that Ω ab has real entries when σ Ω = 1 and imaginary entries when σ Ω = −1, we conclude that
G(t) * = σ ψ σ Ω G(t) = sgn(−1)G(t) = G(−t) .(33)
The second equality of (33) relies on the sign identity deduced from (8):
σ ψ σ Ω = sgn(−1) .(34)
We will refer to this identity as the ultraviolet sign constraint because it was forced on us by analysis of the free theory even before we add the relevant interaction term. Plugging the infrared ansatz (30) into (33), we see that b * = b sgn(−1), or equivalently
b 2 sgn(−1) > 0 .(35)
In other words, b is a fourth root of unity times a positive number, and it is real if sgn is even and imaginary if sgn is odd.
Multiplicative characters
The material in this section is entirely contained in the mathematical literature (see for example [15]). 5 Some of the explicit results in sections 5.4 and 5.5 are perhaps harder to find written out in the literature, but they are all elementary applications of the general formalism.
Generally, a multiplicative character of a field is a multiplicative homomorphism π : K × → C × : that is, a map with the property
π(t 1 t 2 ) = π(t 1 )π(t 2 ) .(36)
Note that in particular, π(1) = 1 for all multiplicative characters. 6 The simplest multiplica-
tive characters are 7 π s (t) = |t| s .(37)
In principle, we can allow any s ∈ C, but in general we will instead restrict s ∈ R. As indicated in (3), our central interest is in constructing free propagators from characters of the form
π s,sgn (t) ≡ |t| s sgn t ,(38)
where sgn t itself is a multiplicative character, taking values in {±1} rather than all of C × .
This is equivalent to asking that the square of the character is the trivial character; such characters are sometimes called "quadratic."
In order to give a more precise account of sign characters, we need to understand better the multiplicative structure of the ultrametric fields of interest. Our main focus is on ultrametric fields K which are finite extensions of Q p for some prime p. They are characterized by the prime number p, which can be recovered from K as follows: the residue field F K of K has characteristic p, meaning that adding the multiplicative identity 1 to itself p times in F K gives 0. (K itself is always of characteristic zero.) To construct F K , one may first define the ring of integers O K ≡ {t ∈ K : |t| ≤ 1}, and next define m K ≡ {t ∈ K : |t| < 1}. Then m K is a maximal ideal in O K (in fact it is the only one), and the quotient
F K ≡ O K /m K is therefore a field; in fact, it is finite, so that F K = F q , where q = p f for some positive integer f . 8
Any element t ∈ K can be expressed as
t = p v(t) w(t) a(t) .(39)
Here p is a chosen uniformizer for K, i.e., a generator of the maximal ideal m K . For example,
p = p for Q p , or √ p for the totally ramified quadratic extension Q p ( √ p). v(t) ∈ Z is
essentially the valuation; it is defined by the property that
|t/p v(t) | K = 1,(40)
and is thus proportional to the logarithm of |t|. is a generator of the group F × q (which is cyclic of order q − 1), and w(t) ∈ {1, 2, 3, . . . , q − 1} (modulo q − 1), so that w(t) ranges over all elements of F × q . Finally, a(t) belongs to the multiplicative group
A = {a ∈ K : |a − 1| < 1} .(41)
The decomposition (39) is unique once we fix choices for the uniformizer p and the generator .
We note that t ∈ O × K = O K \m K (meaning the complement of m K in O K ) precisely if v(t) = 0. We can think of O × K as the analog of the unit circle in K because O × K = {u ∈ K : |u| = 1} .(42)
Any two uniformizers p and p are related by p = up where u ∈ O × K . Thus if we use p instead of p in the decomposition (39), v(t) would remain the same, but w(t) would shift by a fixed multiple of v(t), and a(t) would also change. Similarly, one can use a different primitive root = y , where y is prime to q − 1, and this would result in multiplying w(t)
by y −1 in Z/(q − 1)Z.
For future reference, we introduce the notations
Z p ≡ O Qp U p ≡ O × Qp(43)
for the p-adic integers and the p-adic units, respectively.
Sign characters in finite extensions of Q p for odd p
Now let's require p to be an odd prime. In this case it is fairly straightforward to enumerate the sign characters, because one can show that sgn a = 1 for all a ∈ A. Thus, the only choices one can make are the values sgn p = σ p and sgn = σ , and these choices can be made independently. In short,
ρ σpσ (t) ≡ σ v(t) p σ w(t) .(44)
Changing from one primitive root to another doesn't change the σ w(t) factor. Replacing
p → up shifts w(t) → w(t) + v(t)
if the decomposition of u involves an odd power of ;
otherwise there is no such shift. It's clear from (44) that there are four independent sign characters, including the trivial one ρ 11 which assigns +1 to all elements of K.
We will be particularly interested in sign characters on K which are non-trivial on O × K , and we will describe such characters as direction-dependent because we can think of the
decomposition t = p v(t) u, where u ∈ O ×
K , as analogous to the polar decomposition z = re iθ of a complex number. We see immediately from (44) that direction-dependent sign characters in odd characteristic are precisely the ones where σ = −1.
Sign characters over Q 2
For Q 2 , sign characters are a little more complicated. The decomposition (39) holds just as before, and the first piece of data one needs to fix a sign character is again sgn p. There are no non-trivial sign characters on F 2 because F × 2 is the trivial group consisting of only the identity. Thus the classification of sign characters over Q 2 comes down to understanding sign characters on the group A defined in (41). To proceed, we choose the uniformizer p = 2 and express a ≡ a(t) = 1 + 2a 1
+ 4a 2 + O(2 3 ) ,(45)
where each a i is 0 or 1, and O(2 3 ) means that the omitted terms are 8 times a p-adic integer.
Then the sign characters on A take the form
ρ σa 1 σa 2 (a) = σ a 1 a 1 σ a 2 a 2 ,(46)
where each σ a i may be chosen independently to be ±1. Thus there are four sign characters on A, and consequently eight sign characters
ρ σpσa 1 σa 2 (t) = σ v(t) p σ a 1 a 1 σ a 2 a 2(47)
on Q 2 , including the trivial one that assigns +1 to all t.
It is possible to verify that (46) defines a character by direct calculation, but it is helpful to take a more conceptual view. Let A 2 by the group of all elements t ∈ Q 2 expressible as a square of an element of A. A 2 is easily seen to be a subgroup of A. Furthermore, starting from (45) we can easily see that
a 2 = 1 + O(2 3 ) .(48)
In other words, all elements of A 2 have zero second and third digits in their 2-adic expansion.
It is straightforward to show (using Hensel's Lemma) that this characterization is precise: any element of A with zero second and third digits is in A 2 . Therefore the quotient group A/A 2 consists of elements precisely of the form (45) (i.e. each element A/A 2 is the set of numbers agreeing with the expansion (45) with fixed a 1 and a 2 ). A convenient characterization of A/A 2 is the odd integers modulo 8 under multiplication, which is isomorphic to Z 2Z × Z 2Z . Let's write the elements of A/A 2 as {±1, ±3}. The group is generated (multiplicatively modulo 8) by ±3, and we can understand σ a 1 as the sign we choose for 3 while σ a 2 is the sign we choose for −3. (This is because 3 = 1 + 2, while −3 ≡ 5 = 1 + 2 2 modulo 8). The sign of −1 is therefore the product σ a 1 σ a 2 .
Sign characters for extensions of Q 2 are yet more subtle, and we will not have occasion to discuss them explicitly. They are however well understood: see for example [19].
An alternative parametrization of sign characters
An attractive alternative way to parametrize sign characters is to pick an arbitrary element τ ∈ K × and define
sgn τ t ≡ 1 if t = a 2 − τ b 2 for some a, b ∈ K −1 otherwise .(49)
Note that sgn τ = sgn τ if τ /τ is a square in K. Thus sgn τ is really parametrized by elements of the finite group K × /(K × ) 2 , where (K × ) 2 is the set of all elements in K expressible as the square of an element of K × . Let's examine cases:
• If K = R, then sgn 1 is the trivial sign character and sgn −1 gives the usual notion of sign in R. If K = C, then the only possibility is the trivial sign character.
• For odd characteristic, K × /(K × ) 2 consists of elements {1, p, p, }. (More properly, 1 means (K × ) 2 , p means p(K × ) 2 , and so forth.) The following table provides a translation indicating which sgn τ corresponds to which ρ σpσ for Q p , where we choose the canonical uniformizer p = p. (As noted previously, it doesn't matter which primitive root we choose for F × p .) We also show the value of sgn(−1), which as seen previously is a key quantity for our analysis.
τ 1 p p σ p 1 (−1|p) −(−1|p) −1 σ 1 −1 −1 1 sgn τ (−1) 1 (−1|p) (−1|p) 1(50)
Here, as above, (−1|p) = (−1) (p−1)/2 is the Legendre symbol. The shaded columns correspond to direction-dependent characters.
• Q × 2 /(Q × 2 ) 2 consists of elements {±1, ±2, ±3, ±6}, where factors of ±1 or ±3 come from A/A 2 as discussed above, and the factor of 2 comes from the uniformizer, which is never a square, and which we choose to be p = 2. (Some authors prefer to quote the elements of Q × 2 /(Q × 2 ) 2 as {1, 2, 3, 5, 6, 7, 10, 14}; these are equivalent presentations.) The following table provides a translation between sgn τ and ρ σ 2 σa 1 σa 2 , and it also shows
sgn(−1). τ 1 −1 2 −2 3 −3 6 −6 σ 2 1 1 1 1 −1 −1 −1 −1 σ a 1 1 −1 −1 1 −1 1 1 −1 σ a 2 1 1 −1 −1 1 1 −1 −1 sgn τ (−1) 1 −1 1 −1 −1 1 −1 1(51)
As before, the shaded columns correspond to direction-dependent characters.
It is worth noting that the elements τ ∈ K × /(K × ) 2 other than τ = 1 also parametrize the possible quadratic extensions K( √ τ ) of K; thus each non-trivial sign function on K can be associated with such an extension 9 -but we should keep in mind that sgn τ t itself takes as its argument t an element of K × , not the extension K( √ τ ).
The generalized Γ and B functions
Two important quantities for our subsequent analysis are the generalized Γ and B functions: 10
Γ(π) ≡ K dt |t| χ(t)π(t) B(π, π ) ≡ K dt π(1 − t) |1 − t| π (t) |t| = Γ(π)Γ(π ) Γ(ππ ) .(52)
Here χ(t) is an additive character with χ(1) = 1, while π(t) is a multiplicative character;
dt/|t| is the multiplicative Haar measure on K. For Q p , we have χ(t) = e 2πi{t} , where {t} denotes the fractional part of t. For extensions of Q p , the additive character still has the intuitive interpretation of a plane wave; see for example [5] for an explicit discussion of the case of an unramified extension. 11
There are two obvious ways to combine multiplicative characters. First, as mentioned above, we may multiply pointwise:
(ππ )(t) ≡ π(t)π (t) .(55)
Second, we may use convolutions, defined as
(f * g)(t) = dt 1 f (t − t 1 )g(t 1 ) .(56)
A useful identity is (π * π )(t) = B(ππ 1 , π π 1 )(ππ π 1 )(t) .
Here, π 1 refers to the character π s = | · | s defined in (37), with s = 1. This is closely related to another useful identity:
dt χ(ωt)π(t) = Γ(ππ 1 )(π −1 π −1 )(ω) ,(58)
10 See e.g. p. 145 of [15] for a more systematic development; however, beware of typos! See also [20]. In general, definitions like (52) in terms of integrals work for some range of arguments π and π ; otherwise, we must invoke some process of analytic continuation. 11 For extensions of the p-adic field, one can define
ζ q (s) ≡ 1 1 − q −s Γ q (s) ≡ ζ q (s) ζ q (1 − s) B q (s 1 , s 2 ) ≡ Γ q (s 1 )Γ q (s 2 ) Γ q (s 1 + s 2 )(53)
for any q = p n where p is prime and n is a positive integer. In the unramified extension Q q , we then have Γ(π s ) = Γ q (s) B(π s1 , π s2 ) = B q (s 1 , s 2 ) .
Note the contrast with [5,12], where we eschewed defining ζ q and worked only with ζ p .
where π −1 indicates the pointwise multiplicative inverse: π −1 (t) ≡ 1/π(t). One more useful identity can be obtained by noting that
dω χ(−ωt) dt χ(ωt )f (t ) = f (t) .(59)
Applying this to (58) gives
Γ(ππ 1 )Γ(π −1 ) = π(−1) ,(60)
and we note that π(−1) = ±1 since (−1) 2 = 1; in particular, π s,sgn (−1) = sgn(−1). The reader may wonder how to reconcile our conventions for gamma functions with the intuition that Γ is typically a function of a complex variable. If we restrict to characters of the form π s , where s ∈ C, the usual intuition is recovered. In particular, (57) says that this family of characters (which is additive pointwise, in the sense that π s π t = π s+t ) is also additive under convolutions, up to a scaling and shift that were introduced by our choice of conventions: It specializes to π s * π t = B(π s+1 , π t+1 )π s+t+1 .
We can now demonstrate that the free action (10) is bilocal in position space. Using (58) we may Fourier transform (10) to obtain
S free = 1 Γ(π −s,sgn ) K dt 1 dt 2 1 2 ψ a 1 b 1 c 1 (t 1 )Ω a 1 a 2 Ω b 1 b 2 Ω c 1 c 2 sgn(t 1 − t 2 ) |t 1 − t 2 | 1+s ψ a 2 b 2 c 2 (t 2 ) ,(62)
where π s,sgn is defined in (38). The non-local operator in (62) appears in place of a local time derivative in standard Archimedean field theories, and is closely related to the generalized Vladimirov derivative,
D s,sgn φ(t) ≡ 1 Γ(π −s,sgn ) dt sgn(t − t ) |t − t | 1+s (φ(t ) − φ(t)) .(63)
This relation is explained in detail in the appendix of [21] for the usual Vladimirov derivative with a trivial sign character, D s , but carries over just as well in the case of the generalized Vladimirov derivative D s,sgn .
Generalized Γ functions for R and Q p
For K = R and for K = Q p , we want to evaluate the generalized gamma function Γ on characters of the form
π s,τ (t) ≡ |t| s sgn τ t ,(64)
where s ∈ R while τ and t are in K × . It is useful first to define local zeta functions
ζ ∞ (s) ≡ π −s/2 Γ Euler (s/2) ζ p (s) = 1 1 − p −s ,(65)
where Γ Euler (z) is the usual Euler gamma function. Note that over R, the generalized gamma function is different from Γ Euler , though related:
Γ(π s,1 ) = ∞ −∞ dt |t| s−1 e 2πit = ζ ∞ (s) ζ ∞ (1 − s) = 2 cos πs 2 (2π) s Γ Euler (s) = ζ(1 − s) ζ(s) Γ(π s,−1 ) = ∞ −∞ dt |t| s−2 t e 2πit = −1 2i4 s−1 ζ ∞ (1 − s)ζ ∞ (2s) ζ ∞ (s)ζ ∞ (2 − 2s) = 2i sin πs 2 (2π) s Γ Euler (s) = −1 2i4 s−1 L(1 − s, χ) L(s, χ) ,(66)
where ζ(s) is the usual Riemann zeta function,
ζ(s) = ∞ n=1 1 n s = p 1 1 − p −s = p ζ p (s) ,(67)
and L(s, χ) is a Dirichlet L-function,
L(s, χ) = ∞ n=1 χ(n) n s = p 1 1 − χ(p)p −s .(68)
The particular Dirichlet character χ that appears in (66) . One can think of it as defined by pulling back the unique non-trivial character of (Z/4Z) × to Z along the quotient map, after extending by zero. We will not encounter Dirichlet characters elsewhere in the paper. We will instead use χ to denote an additive character on a field K.
Over C, as explained above, only the trivial sign character is available, and we have Γ C (π s ) = C dz e 2πi(z+z) |z| s−1 = (2π) −2s [Γ Euler (s)] 2 sin(πs).
Now let's consider Q p for odd primes p. For the trivial sign function, using equation (22) of [5], we see that
Γ(π s,1 ) = ∞ =−∞ (p −s ) Up dt χ(p t) = ∞ =0 (p −s ) − 1 p ∞ =−1 (p −s ) = ζ p (s) ζ p (1 − s) ,(70)
which is a standard result; note the similarity to the first line of (66). For non-trivial sign functions, calculations similar to (70) can be carried out with the help of the table in (50).
For example: When τ = , we see from the table that sgn τ t depends only on the parity of v(t), the above sum becomes alternating, and the gamma function evaluates to
Γ(π s, ) = ∞ =−∞ (−p −s ) Up dt χ(p t) = ∞ =0 (−p −s ) − 1 p ∞ =−1 (−p −s ) = ζ p (1 − s)ζ p (2s) ζ p (s)ζ p (2 − 2s) .(71)
Note the similarity of (71) to the second line of (66). For τ = p and τ = p, we have
and so the gamma function in these cases is given by
Γ(π s,τ ) = ∞ =−∞ (p −s ) (sgn τ p) Up dt χ(p t) sgn τ t = p s− 1 2 , τ = p, p ≡ 1 mod 4 −ip s− 1 2 , τ = p, p ≡ 3 mod 4 −p s− 1 2 , τ = p, p ≡ 1 mod 4 ip s− 1 2 , τ = p, p ≡ 3 mod 4 .(74)
Turning now to the gamma functions in the field Q p when p = 2, we start by noticing that the gamma functions for the trivial sign and for τ = −3 are respectively equal to (70) and
(71) with p = 2. For τ = −1 and τ = 3, the sign function depends only on the 2-adic digit a 1 ; decomposing the integral over Q 2 into a sum of integrals over 2 U 2 , only the = −2 term is non-vanishing, and within the domain U 2 /2 2 the additive character χ(t) also only depends on a 1 so that the integral can be split into two parts with constant integrands. The volume of the domain U 2 /2 2 is twice as large as that of Z 2 , which is normalized to unity.
We therefore have Γ(π s,τ ) =
U 2 /2 2 dt |t| s−1 χ(t) sgn τ t = 4 s−1 e 2πi 1 4 − e 2πi 3 4 = i 2 4 s .(75)
When τ is equal to ±2 or ±6, sgn τ t depends on both a 1 and a 2 , and in the decomposition of Q p into a sum over 2 U 2 , it is the = −3 term that is non-vanishing. For τ equal to −2 or 6, sgn τ t doesn't depend on a 1 , and we have Γ(π s,τ ) =
U 2 /2 3 dt |t| s−1 χ(t) sgn τ t = σ 2 8 s−1 e 2πi 1 8 + e 2πi 3 8 − e 2πi 5 8 − e 2πi 7 8 = iσ 2 √ 8 8 s . (76)
For τ equal to 2 or −6, sgn τ t depends on a 1 as well as a 2 , and we have Γ(π s,τ ) = The dependence on σ 2 in equations (76) and (77), in contradistinction to equation (75), is due to the fact that v(t) is odd for t ∈ 2 −3 U 2 , unlike the case when t ∈ 2 −2 U 2 .
An important point to keep in mind is that precisely for direction-dependent sign characters, we find a pure power dependence of the generalized gamma function on s: that is, Γ(π s,τ ) = Ke κs for some constants K and κ which depend on the base field and τ , but not on s. This makes sense because the integral defining the generalized gamma function can be restricted to a scaled copy of U p in these cases. By way of contrast, for direction-independent characters the integral can be restricted only to a scaled copy of Z p , and the infinite geometric sums that give more complicated s dependence arise from splitting this copy of Z p into a semi-infinite collection of copies of U p .
Solving the Schwinger-Dyson equation
With a fuller account of multiplicative characters now in hand, let's revisit the Schwinger-
Dyson equation (28). It can be expressed using convolutions as
G = F + σ Ω g 2 N 3 (G * G 3 * F ) ,(78)
where G and F refer to the position space Green's functions G(t) and F (t). In section 6.1, we will solve (78) in the infrared limit and show that for each choice of field K and each choice of character, a combination of constraints from the ultraviolet and infrared behavior forces us to choose particular signs σ Ω and σ ψ : that is, we have only one option in each case regarding the symmetry group and the statistics of the field ψ.
Infrared limit
The strategy for finding the infrared behavior is to regard g 2 N 3 as large and recognize that, to leading order in inverse powers of g 2 N 3 , we should set G = 0 on the left hand side of (78) and then solve the resulting equation for G. More precisely, we have F (t) = sgn(−1)Γ(π 1−s,sgn )π s−1,sgn (t)
by Fourier transforming (19) with the help of (58), and we propose
G(t) = bπ − 1 2 ,sgn (t)(80)
in the infrared. The notation π s,sgn for the multiplicative character, as introduced in (38), is used to stress that we are not committing to any specific τ in (64). Of course, (80) is just a rewriting of (30).
Starting from (79) and (80) we note first that G(t) 3 = b 3 π − 3 2 ,sgn (t), and next that (78) with the left hand side set to zero simplifies to
π − 3 2 ,sgn * π − 1 2 ,sgn * π s−1,sgn = − 1 b 4 σ Ω g 2 N 3 π s−1,sgn ,(81)
which we recognize as an eigenvalue equation. In fact, it is a totally degenerate eigenvalue equation: by identities such as (57), the operator on the left-hand side is proportional to the identity, and the eigenvalue equation reduces to a single identity between numbers: all dependence on the spectral parameter s simply cancels out. To say this another way, we can pass to momentum space, because a Fourier transform converts the convolutions in (81) into products. Using dt χ(ωt)π − 3 2 ,sgn (t) = Γ(π − 1 2 ,sgn )π 1 2 ,sgn (ω)
dt χ(ωt)π − 1 2 ,sgn (t) = Γ(π1 2 ,sgn )π − 1 2 ,sgn (ω) ,(82)
both of which are special cases of (58), we see immediately that in momentum space (81) simplifies to
Γ(π − 1 2 ,sgn )Γ(π 1 2 ,sgn )π −s,sgn = − 1 b 4 σ Ω g 2 N 3 π −s,sgn .(83)
Thus we arrive at
1 b 4 g 2 N 3 = −σ Ω Γ(π − 1 2 ,sgn )Γ(π 1 2 ,sgn ) .(84)
The left hand side of (84) is positive because the phase of b is a fourth root of unity (recall the discussion at the end of section 4). Once we fix sgn, we must then choose σ Ω in order to make the right hand side of (84) positive. If for some reason this cannot be done, then the proposed infrared scaling (80) is impossible, and something else must happen to the theory (9) in the infrared. Having fixed σ Ω as we just described, we next use the ultraviolet constraint (34) to arrive at a definite choice of σ ψ . In short, we find that for each field K (Archimedean or ultrametric) and each sign character (trivial or non-trivial), there can only be one class of theories of the type (9), specified by a choice of σ Ω and σ ψ and parametrized by s ∈ (1/2, 1], which exhibits scaling behavior of the type (80) in the infrared.
Four points are worth noting:
• So far in this section we have not actually used any detailed knowledge of multiplicative characters. Where this knowledge comes into play is in evaluating Γ(π − 1 2 ,sgn )Γ(π 1 2 ,sgn ).
• For each fixed value of s ∈ (1/2, 1], we have a renormalization group flow from a free theory in the ultraviolet (deformed by the interaction term, which is relevant) to the scaling (80) in the infrared. Allowing s to vary, we see that the class of theories we get for definite K and sgn is a line of free ultraviolet theories, all of which exhibit the same scaling behavior in the infrared when deformed by the relevant interaction (11).
• The result (84) makes clear that even the normalization b of the two-point function does not depend on s (though it clearly does depend on K and sgn). This is a curiously strong form of universality for which we do not have a simple intuitive explanation. Table 1: Explicit characterizations of melonic theories exhibiting renormalization group flows from free theories in the ultraviolet to a strongly interacting fixed point in the infrared. An exact solution of the renormalization group flow is available for the shaded rows, as explained below.
K condition τ Γ(π − 1 2 ,sgn )Γ(π 1 2 ,sgn ) σ Ω σ ψ explanation R 1 −4π 1 1 O(N ) bosonic R −1 −4π 1 −1 O(N ) fermionic C 1 −4π 2 1 1 O(N ) bosonic Q p p odd 1 −(p + √ p + 1)/p 3/2 1 1 O(N ) bosonic Q p p odd (p − √ p + 1)/p 3/2 −1 −1 Sp(N ) fermionic Q p p ≡ 1 mod 4 p 1/p −1 −1 Sp(N ) fermionic Q p p ≡ 1 mod 4 p 1/p −1 −1 Sp(N ) fermionic Q p p ≡ 3 mod 4 p −1/p 1 −1 O(N ) fermionic Q p p ≡ 3 mod 4 p −1/p 1 −1 O(N ) fermionic Q 2 1 −(2 + 3 √ 2)/4 1 1 O(N ) bosonic Q 2 −1 −1/4 1 −1 O(N ) fermionic Q 2 2 1/8 −1 −1 Sp(N ) fermionic Q 2 −2 −1/8 1 −1 O(N ) fermionic Q 2 3 −1/4 1 −1 O(N ) fermionic Q 2 −3 (3 − √ 2)/ √ 8 −1 −1 Sp(N ) fermionic Q 2 6 −1/8 1 −1 O(N ) fermionic Q 2 −6 1/8 −1 −1 Sp(N ) fermionic
• The general analysis at the end of section 4 fixes the phase of b up to a sign. Nothing in the infrared analysis we have presented in this section resolves the sign ambiguity in b; see however section 6.2 for the case of direction-dependent characters.
Passing to K = R or Q p , we can use the results of section 5 to give a fully explicit account of which theories exist in which places. The results are shown in table 1. Some further remarks are in order:
• Fermionic theories are required whenever the sign character is non-trivial; otherwise we need bosonic theories.
• The fermionic Klebanov-Tarnopolsky model corresponds to the second line of the table.
• The bosonic Klebanov-Tarnopolsky models in one and two dimensions correspond to the first and third lines of the Higher-dimensional bosonic models are not part of our story because we insist that the theory be defined over a field rather than a vector space. A systematic extension to vector spaces over fields both Archimedean and ultrametric seems like a large but possibly important undertaking. The key to this striking result is a Fourier transform identity on Q p , most easily stated for the case of odd p, where the direction-dependent sign characters are sgn τ for τ = p and p:
An exact solution of the Schwinger-Dyson equation
Qp dt χ(ωt)δ v(t) sgn t = M −1 δ v(ω)+1 sgn ω ,(85)
where we are restricting to direction-dependent sign characters (i.e. sgn = sgn p or sgn p ), and we remind the reader that v(t) is the valuation of t (see the discussion around (40)).
We define
δ v ≡ 1 if v = 0 0 otherwise(86)
to be the characteristic function of U p , and
M −1 ≡ Γ(π 1,sgn ) p = ± sgn(−1) p .(87)
The sign in the last expression can be read off from (74). Indeed, (85) is just a rephrasing of (72)-(74). Now let's make a more general ansatz than (80) for G(t), allowing completely general dependence on the magnitude of t so that we may write
F (t) = f v(t) sgn t G(t) = g v(t) sgn t .(88)
Straightforward application of (85) leads to
F (ω) = M −1 p v(ω)+1 f −v(ω)−1 sgn ω G(ω) = M −1 p v(ω)+1 g −v(ω)−1 sgn ω ,(89)
and if we define H(t) ≡ G(t) 3 = g 3 v(t) sgn t, then we have also
H(ω) = M −1 p v(ω)+1 g 3 −v(ω)−1 sgn ω .(90)
In momentum space, the Schwinger-Dyson equation (78) reads
G(ω) = F (ω) + σ Ω g 2 N 3 G(ω)H(ω)F (ω) .(91)
Plugging (89) and (90) into (91), and then simplifying using (87) and the ultraviolet sign constraint (34), one arrives at
g v = f v + σ ψ g 2 N 3 p p −2v g 4 v f v .(92)
Referring to (79), we see that we can express
f v = θp v/2f v wheref v = p ( 1 2 −s)(v+1) .(93)
Here θ is some fourth root of unity with
θ 2 = sgn(−1) .(94)
By design,f v is positive. Note also thatf v becomes small in the ultraviolet and large in the infrared. Let us likewise express
g v = θp v/2ĝ v .(95)
The reality property G(t) * = G(−t) demonstrated in (33) forcesĝ v ∈ R. Using (93)-(95), (92) can be re-expressed as 1
f v = 1 g v + σ ψ g 2 N 3 pĝ
and this form has the advantage that there is no explicit v dependence in the coefficients of the relation betweenf v andĝ v . It is easy to show (see figure 2) that there is a unique positive real solutionĝ v to (96) for eachf v , provided σ ψ = −1, and in the infrared, wheref v becomes large,ĝ v converges to a finite answer:
g v → p g 2 N 3 1/4 as v → −∞ .(97)
Note that the result (97) is consistent with the general infrared result (84) given the simple values of Γ(π ± 1 2 ,sgn ) for direction-dependent characters. We must chooseĝ v positive when solving (96), because for fixed v,ĝ v must interpolate smoothly fromf v to give qualitatively similar results. Suppose we pick a value off v : small if we're in the UV, and large in the IR. If σ ψ = −1, as in the left-hand plot, then there is always a unique positive solutionĝ v corresponding to the chosen value off v . But if σ ψ = +1, as in the right-hand plot, then sufficiently far into the IR we have no such solution, and it appears therefore that the interacting theory is ill-defined.
direction-dependent characters on Q 2 . In particular,
Q 2 dt χ(ωt)δ v(t) sgn t = M δ v(ω)− sgn ω ,(98)
where = −2 or −3 according to whether the integral for Γ(π s,sgn ) localizes onto 2 −2 U 2 or 2 −3 U 2 , and the constant M is given by
M ≡ 2 Γ(π 1,sgn ) = ± 2 sgn(−1).(99)
One easily sees that (92) is modified to
g v = f v + (σ ψ 2 g 2 N 3 )p −2v g 4 v f v .(100)
The discussion around (97) goes through as before, only with p replaced by 2 − . In particular, we must choose σ ψ = −1 to get a sensible solution to the Schwinger-Dyson equation, and we find that g v has the same phase as f v .
A Wilsonian perspective
The Schwinger-Dyson equation (78) for the two-point function lets us start with a theory whose microscopic description involves a pure power-law two-point function, F (ω) = (sgn ω)/|ω| s , and obtain from it a connected correlator G(ω) which interpolates between F (ω) in the ultraviolet and universal scaling G(ω) = b(sgn ω)/|ω| 1/2 in the infrared. This is a crucial feature that drives much of the interest in the SYK model and its tensor model cousins: we flow from a free theory in the UV to a non-trivial IR fixed point with the emergent symmetries of AdS 2 (or more properly "nearly" AdS 2 ). Over R, these models are only soluble in the IR using the Archimedean version of the limiting-behavior analysis we performed in section 6.1. For those non-Archimedean theories that are built with directiondependent characters, we have shown in section 6.2 that the Schwinger-Dyson equation has exact solutions interpolating between UV and IR behavior.
However, there is an apparent paradox: We generically expect a non-renormalization theorem for bilocal kinetic terms. So how is it that we are seeing G(ω) behave so differently from F (ω)? This apparent paradox is best examined in a Wilsonian context, where nonrenormalization theorems have been cleanly demonstrated for p-adic field theories, e.g. in [23]. In section 7.1 we will give some first indications that the expected non-renormalization theorem continues to hold in the presence of non-trivial sign characters. For technical reasons to become apparent, we restrict our analysis in three ways. First, we consider only the leading melonic limit. Second, we require direction-dependent sign characters. And third, we restrict K to be Q p or an extension of Q p for odd p, or else Q 2 . The upshot of our discussion is that the bilocal term in the Wilsonian effective action is indeed not renormalized, and as we explain in section 7.2 it is the distinction between this Wilsonian effective action and the one-particle irreducible (1PI) action that resolves the paradox.
p-adic non-renormalization
A general feature of p-adic field theories is that they behave more simply than their Archimedean counterparts under renormalization; indeed, they have strong commonalities with the hierarchical models studied in the early renormalization group literature. As explained by Lerner and Missarov [23], whose argument we will roughly follow in this subsection, there is a strong non-renormalization theorem which forbids non-trivial renormalization of the kinetic term. 12 This can be seen through the Wilsonian picture of perturbative renormalization in momentum space, where one attempts to define an effective field theory in the IR (external states having some low characteristic momentum ω) by integrating out internal UV modes (loop momenta ω i such that ω i > ω). In particular, in evaluating loop corrections to the propagator (which typically lead to divergent integrals), we instead perform the integral over a shell of hard momenta of fixed magnitude |ω| = Λ. This defines a new propagator, and iterating this procedure defines the renormalization group flow.
In p-adic field theories, the ultrametric nature of the norm |ω| has a surprising and beneficial effect on the momentum shell renormalization technique. The ultrametric triangle inequality means that, when ω is soft and ω i is hard, |ω + ω i | = |ω i | exactly. This is in contrast to the weaker Archimedean statement that |ω + ω i | ≈ |ω i | when ω i ω.
The easiest way to see how this leads to non-renormalization is to work out the specific example relevant for this work: wavefunction renormalization in a melonic theory. The leading melonic correction to the quadratic part of the effective action is again the underground diagram of (23). At the level of our current discussion, we only need the kinematic information, so we drop the flavor indices and use the following diagram:
ω ω 1 ω 2 ω 3(101)
Because we are correcting the effective action, we amputate the external propagators. In a Wilsonian picture, we integrate out a shell of hard momenta |ω i | = Λ and consider the soft external momentum |ω| < Λ. Through a rescaling of all momenta, we can choose Λ = 1. Then, again dropping the Ω ab factors, the amplitude from the diagram in (101) is proportional to
I 2 (ω) = σ Ω g 2 N 3 O × K dω 1 dω 2 dω 3 δ(ω 1 + ω 2 + ω 3 − ω)G(ω 1 )G(ω 2 )G(ω 3 ) ,(102)
where O × K is the multiplicative group of units in K. If we are doing fixed-order perturbative renormalization to leading order in the melonic limit, then the G(ω i ) in (102) would be propagators of the free theory. If instead we want resummed perturbation theory that includes all diagrams in the melonic limit, then the G(ω i ) would be an improved propagator of the cutoff theory that could be determined by a Schwinger-Dyson equation in this theory.
For the argument below to work, all that matters is that G(ω i ) =Ĝ(|ω i |) sgn ω i for some functionĜ of the norm of ω i .
The key to the non-renormalization theorem is to make a u-substitution of the form
ω 1 = ω 1 − ω ,(103)
so as to arrive at
I 2 (ω) = σ Ω g 2 N 3 O × K dω 1 dω 2 dω 3 δ(ω 1 + ω 2 + ω 3 )G(ω 1 )G(ω 2 )G(ω 3 ) = I 2 (0) ,(104)
provided we can show G(ω 1 ) = G(ω 1 ). A key point is that the change of variables (103) is a measure-preserving bijection of O × K to itself: this is where we use the ultrametric triangle inequality.
The absence of ω dependence from the last expression in (104) is the feature we seek: It means that the correction to the effective action has no ω dependence, and is instead a mass term for ψ, local in position space. In other words, the bilocal term is not renormalized. We may however feel some surprise at the appearance of a mass term in the effective action. We will now argue-subject to aforementioned technical restrictions-that in fact I 2 (ω) = 0.
First let's handle the case where K is (an extension of) Q p for p odd. We write ω i as in (39):
ω i = p v(ω i ) w(ω i ) a(ω i ) .(105)
For ω i ∈ O × K , v(ω i ) = 0, so the only quadratic characters available to us are the trivial character and (−1) w(ω i ) . Since we have the strict inequality |ω| < |ω 1 |, shifting by ω cannot change the element of the residue field F q and so sgn(ω 1 ) = sgn(ω 1 ). Because we assumed that G(ω i ) =Ĝ(|ω i |) sgn ω i , this in turn implies G(ω i ) = G(ω i ), and we see from (104) that
I 2 (ω) is independent of ω.
To show that I 2 (ω) = 0, we need an additional argument that relies on the sign character in G(ω i ) being direction-dependent, i.e. it must be (−1) w(ω i ) rather than the trivial character.
Then there is some λ ∈ O × K for which sgn λ = −1. Note that if ω i covers O × K then so does λω i ; to say it differently, λ has a multiplicative inverse in O × K , so the change of variables ω i → λω i is invertible. Performing it on (102), we obtain
I 2 (ω) = (sgn λ)I 2 (λ −1 ω) = −I 2 (ω) ,(106)
where we used the multiplicative property to simplify the sign characters. Since we already concluded that I 2 (ω) is independent of ω, (106) implies I 2 (ω) = 0, as desired.
Next we consider the case K = Q 2 . Because sign characters now depend on the second and third 2-adic digits of their argument, it may not be true that G(ω 1 ) = G(ω 1 ): Indeed, forming the difference ω 1 − ω may change the second or third digit of ω 1 , so this equality will fail for any direction-dependent character on Q 2 . Nevertheless we can argue that I 2 (ω) = 0 for quite a trivial reason: it is impossible to have the three loop momenta ω i ∈ O × K = U 2 add up to an external momentum ω with |ω| < 1. To see this, note that each ω i = 1 + 2b i for some b i ∈ Z 2 , so also ω 1 + ω 2 + ω 3 = 1 + 2b for some b ∈ Z 2 . In other words, the delta function (102) is always zero on the integration region we are using. This argument shows that I 2 (ω) = 0 regardless of which character we use on Q 2 .
If K is an extension of Q 2 , the argument is more difficult, and we do not know whether the bilocal term in the action is renormalized or not. With the decomposition
ω i = ξ i a i where ξ i ∈ F × K and a i ∈ A ,(107)
we can re-express the I 2 (ω) integral as
I 2 (ω) = σ Ω g 2 N 3 ξ i ∈F × K δ ξ 1 +ξ 2 +ξ 3 A da 1 da 2 da 3 δ(ξ 1 a 1 + ξ 2 a 2 + ξ 3 a 3 − ω) 3 i=1 sgn a i .(108)
Here δ ξ is the Kronecker delta function, which arises because ξ 1 a 1 + ξ 2 a 2 + ξ 3 a 3 = ω implies ξ 1 + ξ 2 + ξ 3 = 0 when |ω| < 1. If F K = F 2 , as in Q 2 or its totally ramified extensions, then δ ξ 1 +ξ 2 +ξ 3 = 0 always, and we once again have I 2 (ω) = 0. In the general case, where F K = F 2 f for some f > 1, it is not clear to us whether I 2 (ω) vanishes.
Resolution of an apparent paradox
So far, we have shown in the leading melonic limit that the correction I 2 (ω) to the Wilsonian effective action vanishes, at least for a broad range of choices of field K and sign character.
This seems at first puzzling when we try to interpret the renormalization group equations, which in the p-adic setting are a discrete set of recursion relations that relate the effective action with cutoff Λ to the effective action with cutoff Λ/p. Iterating these equations defines the renormalization group flow. With I 2 (ω) = 0 in our models, the quadratic part of the effective action isn't any different from what we started with, so there appears to be no flow.
However, as we have already seen explicitly in section 6.2 for direction-dependent characters, the Schwinger-Dyson equation has a solution G interpolating between the free UV theory and the universal IR scaling behavior. This is the apparent paradox that we referred to before.
The resolution is the distinction between the 1PI effective action and the Wilsonian effective action. The kernel of the quadratic part of the Wilsonian effective action is not renormalized (subject to the technical assumptions described previously), but added to it is a relevant gψ 4 interaction term which becomes large in the infrared. 13 For this reason, in the infrared, the quadratic kernel of the Wilsonian effective action strongly disagrees with the kernel of the quantum effective action. In this limit we are far from a Gaussian theory that would be defined by naive RG flow. The solution of the Schwinger-Dyson equation
(78) shows that this kind of theory can have interesting dependence on scale beyond the conventional wisdom of effective field theory; and this is particularly striking for ultrametric 13 In the leading melonic limit, this quartic term is also not renormalized: for example, the first correction is order g 2 N = g √ N (g 2 N 3 ) 1/2 , which is indeed subleading. This non-renormalization property doesn't matter particularly to our story; what matters is that the interaction always remains relevant so that its effects in the infrared are large. theories.
Summary and future directions
Melonic theories are simple because the list of diagrams we must compute in the leading order melonic limit is very short: For the two-point function, all of them can be resummed into the Schwinger-Dyson equation (78). Defining melonic theories over the p-adics makes the theories in some cases even simpler, because one can then solve the Schwinger-Dyson equation exactly to find the interpolation between the free theory in the ultraviolet and universal infrared power law scaling in the infrared. As explained in section 6.2, the key to this exact solution is to choose a sign function sgn t on Q p so that there exists some number λ with |λ| = 1 and sgn λ = −1. We previously described these sign functions as direction-dependent.
A benefit of formulating a melonic version of the Klebanov-Tarnopolsky model on the p-adics is that it naturally leads to a cleaner and more unified description of the Archimedean theory, as well as a generalization to theories with bilocal kinetic terms. Indeed, our description in (9)-(11) of the theories we consider simultaneously encompasses bosonic and fermionic theories on R, the bosonic theory on C, and bosonic and fermionic theories on all the Q p and their extensions. Moreover, the normalization constant b in the infrared limit of the two-point function, G(t) = b sgn t |t| 1/2 , is seen to be determined in terms of the product Γ(π − 1 2 ,sgn )Γ(π 1 2 ,sgn ) of two generalized gamma functions, as indicated in (84). Strikingly, b does not depend on the spectral index of the ultraviolet theory.
For R, C, and Q p , our understanding of generalized gamma functions is sufficiently complete that we can determine the sign of the crucial quantity Γ(π − 1 2 ,sgn )Γ(π 1 2 ,sgn ) in all cases. This sign, together with reality properties of the two-point function, forces us to choose the statistics of the field ψ abc to be bosonic if sgn is the trivial sign function (setting the sign of all non-zero numbers to 1), and fermionic otherwise. Related constraints force us to modify the ordinary O(N ) 3 symmetry of melonic theories to Sp(N ) 3 , where Sp(N ) is the non-compact group of N × N matrices preserving a symplectic form. Altogether, we manage to construct precisely one type of melonic theory in every place (R or Q p for any p) and for every choice of sign function. All of them have the same characteristic scaling of the two-point function in the infrared, G(t) = b sgn t |t| 1/2 in the limit of large |t|. An obvious generalization of our results would be to pass to arbitrary extensions of Q p , where our current understanding of generalized gamma functions is insufficient to determine even the phase of Γ(π − 1 2 ,sgn )Γ(π 1 2 ,sgn ). In order to wind up with bosonic theories when the sign function is trivial and fermionic theories otherwise, we would need Γ(π − 1 2 ,sgn )Γ(π 1 2 ,sgn ) always to be real, and we would need its sign to be −1 for the trivial sign function and sgn(−1) for non-trivial sign functions. It would be interesting to find a way to prove that the signs of Γ(π − 1 2 ,sgn )Γ(π 1 2 ,sgn ) are indeed as we have just described. Just as it is useful to study the SYK model and its relatives on S 1 , another obvious future direction is to study our models on quotients of Q p . This is likely to be quite a rich subject because there are many such quotients: Dividing Q p (or more properly P 1 (Q p ) by an appropriate discrete subgroup of PGL(2, Q p ) leads to a Mumford curve over Q p , and correlators on such spaces are likely to be related to automorphic functions.
Just as our results on the Schwinger-Dyson equation admit some obvious generalizations, our results on the Wilsonian renormalization of p-adic melonic theories leave considerable room for further work. We showed that the bilocal kinetic term is not renormalized in the leading melonic limit in an assortment of theories with direction-dependent sign characters; however, we never progressed to more complicated graphs. Existing treatments, e.g. [23], go much further for theories that do not involve sign characters, demonstrating all-orders perturbative non-renormalization theorems. It is tempting to think that similar results could be demonstrated in the theories we consider; however, particularly for extensions of Q 2 , the arguments seem non-trivial. A full account of renormalization group properties of melonic theories should also include the possibility of a mass term, which we have suppressed in our treatment in order to obtain power-law scaling in the infrared, but which is sometimes permitted by the symmetries.
We end by noting that the space of theories we consider has non-trivial topology when the field K is ultrametric. The non-compact direction is parametrized by the spectral parameter s, which can be understood as controlling the ultraviolet limit of the two-point function:
G(t) ∼ sgn t |t| 1−s for small |t|, and we require s > 1/2 in order for the gψ 4 interaction to be relevant, while s ≤ 1 in order to avoid a negative dimension for ψ. For simplicity let's also assume g > 0. If K = R or C, then we can rescale g to 1 by applying a scale transformation to the whole theory. If K is ultrametric, then the allowable scale transformations are discrete, consisting of multiplying all frequencies ω by an integer power of the uniformizer p. The simplest such transformation is ω → pω, and all real quantities must at the same time be rescaled as X → p −[X] X. For example, g → p −2s+1 g. Thus g = p 2s−1 is equivalent to g = 1, but theories with g ∈ (1, p 2s−1 ) are all inequivalent. In short, the space of values of g is really a circle, constructed by identifying the endpoints of the interval [1, p 2s−1 ]. Overall, the space
A Conventions for O(N ) and Sp(N )
Throughout this text we make use of an invariant tensor Ω ab of GL(N, R) to contract flavor indices of the fields. Usually these are Majorana fermions transforming in the trifundamental representation of the flavor group:
ψ abc → (Λ 1 ) a a (Λ 2 ) b b (Λ 3 ) c c ψ a b c .(109)
where Λ a b are the group matrices preserving this structure Ω ab = Λ c a Λ d b Ω cd . In order to preserve the reality condition on the fermions, we use real valued matrices, and this restricts which kinds of flavor groups can appear.
Interesting choices for Ω are those of definite parity. The symmetric choice is defined by Ω = Ω T and has σ Ω = 1. The condition imposed on the group matrices defines the orthogonal group O(N ).
For the case of antisymmetric Ω = −Ω T with σ Ω = −1, we must take N to be even and
This is a compact group, but as complex matrices they cannot be used for the transformation of Majorana fermions.
In SYK-like models, only singlet states of the fermions (e.g. certain bilinears) have bulk duals and the fundamental fermions themselves don't seem to have any bulk interpretation.
One advantage of tensor models is that we can understand the restriction to singlets by gauging the flavor group [7] [8]. In 0 + 1 dimensions the gauge field has no dynamics and appears only as an auxiliary field. Integrating it out projects onto the singlet states.
In the p-adic case, the non-Archimedean structure of Q p makes conserved currents somewhat tricky to define. Additionally, one might worry about gauging a non-compact group such as Sp(N ) which would have problematic ghosts in higher dimensions. For the purposes of this work we ignore the issues of gauging and treat the O(N ) 3 and Sp(N ) 3 symmetries as global flavor symmetries.
i 0 .
0The field ψ abc has no particular symmetry under permutations of its indices. So we see that the full model has an O(N ) 3 symmetry if σ Ω = 1, and an Sp(N ) 3 symmetry if σ Ω = −1. Here Sp(N ) ≡ Sp(N, R)
and (68) has modulus four; it is defined on primes through χ(2) = 0 and χ(p) = (−1|p) for odd primes. Its values on all positive integers are then fixed by the multiplicative property, χ(mn) = χ(m)χ(n) for positive m and n [22]
these cases the range of the additive character is symmetrically distributed around the unit circle. The case = −1 can be dealt with by invoking the Gauss sum
U 2 /2 3 dt
23|t| s−1 χ(t) sgn τ t = σ 2 8
general, the Schwinger-Dyson equation is difficult to solve because it involves non-linear combinations of the dressed propagator G (which is what we have to solve for) and must be expressed as a combination of ordinary products and convolutions-so it is local neither in position space nor in momentum space. But for direction-dependent ultrametric characters, there is a way to beat this: It turns out that for these characters, the Schwinger-Dyson equation is local both in position space and in momentum space, so we can solve it (in either position space or momentum space) just by solving a quartic equation at each energy scale.
continuous parameter g 2 N 3 from small values to large values. This amounts to a determination of the sign of b in the infrared. If we pick σ ψ = +1 in (96), then sufficiently far into the ultraviolet there is still a positive solutionĝ v to (96), but as one proceeds toward the infrared, there comes a point where 1/f v gets so small that there is no real solution to (96). Thus it appears-at least from the analysis of the leading-order Schwinger-Dyson equation presented here-that the interacting theory is ill-defined for direction-dependent characters if we try to make the fields bosonic, even if we adjust σ Ω to maintain the ultraviolet sign constraint (34). See figure 2.
Figure 2 :
2Solving the quartic equation (96). We use the value g 2 N 3 p = 1 as an example in order to be able to draw a definite curve of 1/f v versusĝ v . Other positive values of g 2 N 3 p
define Sp(N ) ≡ Λ ∈ SL(N, R) : Λ T ΩΛ = Ω . (110) Elsewhere in the literature, this real symplectic group is sometimes denoted Sp(2N, R) to emphasize the matrices are real valued and even dimensional. This group is connected, but non-compact. There is a different but closely related kind of symplectic group called USp(2N ) (also called Sp(N ) in some references!) In contrast to the symplectic group above, USp(2N ) is defined by the intersection USp(2N ) = U(2N ) ∩ Sp(2N, C).
table . (
.More properly, these rows correspond to modified bosonic Klebanov-Tarnopolsky models with bilocal kinetic terms; c.f. footnote 4.)
The bosonic model of[8] in d dimensions has s = 2/d, and d = 1 is below its lower critical dimension. Thus, when we consider the bosonic model on R, it should be understood as having a bilocal kinetic term with s between 1/2 and 1.
For a discussion of quadratic extensions of the p-adics and the p-adic sign character in a physics context, see also[18].
This is an instance of the general idea that a character is a one-dimensional representation of a group (and so valued in GL(1, C) = C × ): K × is regarded as a group under multiplication. The collection of multiplicative characters is itself an abelian group, under pointwise multiplication: if π and π are multiplicative characters, then so is (ππ )(x) = π(x)π (x), and similarly, so is 1/π. (This operation corresponds to tensor product of representations; it is a special property of the one-dimensional representations that the tensor product admits inverses.)7 Beware a notational conflict with[12]: There π s (t) = |t| s−1 in Q p , whereas here π s (t) = |t| s .8 For an unramified extension K of Q p , f is also the degree of extension of K over Q p . In general, n = ef where e is the ramification index, defined as the positive integer such that |p/p e | K = 1 where p is a uniformizer for K.
This is the simplest instance of class field theory: quadratic extensions (which are necessarily abelian) correspond to index-two subgroups of K × , which are precisely the kernels of sign characters.
v ,(96)
A related point for Archimedean field theories is that bilocal kinetic terms like those of Fisher, Ma, and Nickel[24] don't get renormalized, and the reasoning in that case is that all divergences can be canceled by local counter terms, possibly using a derivative expansion.
Acknowledgments We thank M. Marcolli and P. Witaszczyk for extensive discussions. B.S. would also like to thank A. Almheiri for useful discussions. The work of S. this topology has any significance in a non-perturbative treatment. C.J., S.P. and B.Tsupported in part by the Department of Energy under Grant No. DE-FG02-91ER40671this topology has any significance in a non-perturbative treatment. Acknowledgments We thank M. Marcolli and P. Witaszczyk for extensive discussions. B.S. would also like to thank A. Almheiri for useful discussions. The work of S.S.G., C.J., S.P. and B.T. was supported in part by the Department of Energy under Grant No. DE-FG02-91ER40671.
Office of High Energy Physics, under Award Number DE-SC0011632 as well as by the Walter Burke Institute for Theoretical Physics at Caltech. The work of B.S. was supported in part by the Simons Foundation, and by the U.S. Department of Energy under grant DE-SC-0009987. B.S. would like to thank the Stanford Institute for Theoretical Physics at Stanford University and the Aspen Center for Physics for hospitality. The work of B.S. was performed in part at Aspen Center for Physics. The work of S.P. was also supported in part by the Bershadsky Family Fellowship Fund in Mathematics or Physics. The work of M.H. was supported by the. U.S. Department of Energy, Office of Sciencewhich is supported by National Science Foundation grant PHY-1607611The work of S.P. was also supported in part by the Bershadsky Family Fellowship Fund in Mathematics or Physics. The work of M.H. was supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632 as well as by the Walter Burke Institute for Theoretical Physics at Caltech. The work of B.S. was supported in part by the Simons Foundation, and by the U.S. Department of Energy under grant DE-SC-0009987. B.S. would like to thank the Stanford Institute for Theoretical Physics at Stanford University and the Aspen Center for Physics for hospitality. The work of B.S. was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.
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| []
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[
"Colour-biased Hamilton cycles in random graphs",
"Colour-biased Hamilton cycles in random graphs"
]
| [
"Lior Gishboliner ",
"Michael Krivelevich ",
"Peleg Michaeli "
]
| []
| []
| We prove that a random graph G(n, p), with p above the Hamiltonicity threshold, is typically such that for any r-colouring of its edges there exists a Hamilton cycle with at least (2/(r + 1) − o(1))n edges of the same colour. This estimate is asymptotically optimal. | 10.1002/rsa.21043 | [
"https://arxiv.org/pdf/2007.12111v2.pdf"
]
| 220,713,475 | 2007.12111 | 3d4479220708957e64cfa660f9de59191a38b66b |
Colour-biased Hamilton cycles in random graphs
August 18, 2020
Lior Gishboliner
Michael Krivelevich
Peleg Michaeli
Colour-biased Hamilton cycles in random graphs
August 18, 2020
We prove that a random graph G(n, p), with p above the Hamiltonicity threshold, is typically such that for any r-colouring of its edges there exists a Hamilton cycle with at least (2/(r + 1) − o(1))n edges of the same colour. This estimate is asymptotically optimal.
integer M such that in any r-colouring of the edges of G, there will be a Hamilton cycle with at least M edges of the same colour (if G is not Hamiltonian, we set M (G, r) = 0). The problem of estimating M (G, r) is somewhat similar to (though slightly different from) multicolour discrepancy problems. In the general setting of combinatorial discrepancy theory, one is given a hypergraph H and tries to r-colour its vertices in such a way that every hyperedge is coloured as evenly as possible, in the sense that the numbers of vertices of a given colour in every hyperedge e deviates from its "mean", |e|/r, by as little as possible. The discrepancy of H is then defined as the maximal deviation one is guaranteed to have in any colouring. In the special setting we consider here, the vertices of the hypergraph H are the edges of G, and the hyperedges of H are the Hamilton cycles in G. We note, however, that the problem of estimating M (G, r) differs from its discrepancy variant in that M (G, r) is only concerned with "one-sided deviations", namely with colours appearing significantly more (and not less) than what is expected. It is worth noting that discrepancy-type problems in graphs were studied for various "target subgraphs", such as cliques [14], spanning trees [5,12], Hamilton cycles [5] and clique factors [6].
It is natural to expect that if G contains only few Hamilton cycles, then one can r-colour the edges of G in such a way that every Hamilton cycle sees approximately the same number, i.e. roughly n/r, of edges of each colour. Our main result, Theorem 1.1, shows that the situation in G(n, p) (for p above the Hamiltonicity threshold) is typically very different: one is always guaranteed to find a Hamilton cycle which contains significantly more than n/r edges of the same colour. As alluded to earlier, this is yet another indication of the rich structure of the set of Hamilton cycles in G(n, p).
Before stating our main result, let us recall some standard terminology. For a positive integer n and a real p ∈ [0, 1], denote by G(n, p) the binomial random graph, namely, the probability space of all simple labelled graphs on given n vertices, where each pair of vertices is connected by an edge independently with probability p. We say that an event A in our probability space occurs with high probability (or whp) if P(A) → 1 as n goes to infinity. Theorem 1.1. Let r ≥ 2 be an integer and let p ≥ (log n+log log n+ω(1))/n. Then G ∼ G(n, p) is whp such that in any r-colouring of its edges there exists a Hamilton cycle with at least (2/(r + 1) − o(1))n edges of the same colour.
Using similar tools to those used in the proof of Theorem 1.1, we sketch a proof for the following analogous result for perfect matchings. Theorem 1.2. Let r ≥ 2 be an integer and let p ≥ (log n + ω(1))/n. Then, assuming n is even, G ∼ G(n, p) is whp such that in any r-colouring of its edges there exists a perfect matching with at least (1/(r + 1) − o(1))n edges of the same colour.
The fraction 1/(r+1) in Theorem 1.2, and hence also the fraction 2/(r+1) in Theorem 1.1, is tight. In fact, in every n-vertex graph G there exists an r-colouring in which in every matching, the maximum number of edges of the same colour is at most n/(r + 1). Such a colouring, which to the best of our knowledge first appeared in [9], can be described as follows. Partition V (G) into sets V 1 , . . . , V r such that |V i | = n/(r + 1) for i = 1, . . . , r − 1 and |V r | = 2n/(r + 1). For i = 1, . . . , r (in increasing order), colour by i every edge touching V i that has not already been coloured. Namely, for each 1 ≤ i ≤ r, all edges contained in V i ∪ · · · ∪ V r and touching V i are coloured with colour i (see Fig. 1). It is easy to see that any monochromatic matching in this colouring is of size at most n/(r + 1). Moreover, observe that any Hamilton cycle contains at Figure 1: A 4-coloured complete graph on n vertices. Each of the small bulbs represents a set of n/5 vertices, and the large bulb in the middle represents a set of 2n/5 vertices. Any monochromatic matching in this graph is of size at most n/5, hence also every Hamilton cycle contains at most 2n/5 edges of the same colour. most 2n/(r + 1) edges of a given colour, as otherwise it would also contain a matching of size larger than n/(r + 1), hence also M (G, r) ≤ 2n/(r + 1).
The above construction and its analysis suggest a connection between the problem of estimating M (G, r) and the problem of finding monochromatic matchings in r-colourings of (the edges of) G. Indeed, our proof of Theorem 1.1 relies on a new Ramsey-type result for matchings, which may be of independent interest.
A classical theorem of Cockayne and Lorimer [9] states that for integers k 1 , . . . , k r ≥ 1 and n ≥ r i=1 (k i − 1) + max{k 1 , . . . , k r } + 1, every r-colouring of the edges of the complete graph K n contains a monochromatic matching of size k i in colour i for some 1 ≤ i ≤ r. The following theorem extends this result to almost complete host graphs. 1], let G be a graph n vertices and at least (1−δ) n 2 edges, and suppose that 1− √ δ n ≥ r i=1 (k i − 1)+k+1, where k := max{k 1 , . . . , k r }. Then, for every r-colouring ϕ : E(G) → [r] of the edges of G, there is 1 ≤ i ≤ r such that G contains a matching of size k i , all of whose edges are coloured with colour i under ϕ.
Theorem 1.3. Let r ≥ 2, let k 1 , . . . , k r ≥ 1, let δ ∈ [0,
For k 1 = · · · = k r = k, the condition in Theorem 1.3 becomes 1 − √ δ n ≥ (r + 1)(k − 1) + 2. Hence, for this case we have the following corollary. Corollary 1.4. Let r ≥ 2, let δ ∈ [0, 1], and let G be a graph on n vertices and at least (1−δ) n 2 edges. Then, in every r-colouring of the edges of G there is a monochromatic matching of size at least (n − √ δn − 2)/(r + 1) + 1.
Our proof of Theorem 1.3 is inspired by a new proof of the Cockayne-Lorimer theorem, given in [39].
As a second step towards proving Theorem 1.1, we will combine Corollary 1.4 with a multicolour version of the sparse regularity lemma (stated here as Theorem 3.2) to prove that in any r-colouring of the edges of an n-vertex pseudorandom graph G, there must be a path of length (2/(r + 1) − o(1))n in which all but a fixed number of edges are of the same colour. We will postpone the precise definition of pseudorandomness to Section 3, and for now only note that as a bi-product, we get the following aesthetically pleasing result. Theorem 1.5. Let r ≥ 2 be an integer and let ε > 0. Then there exist C = C(r, ε) and K = K(r, ε) such that if p ≥ C/n, the random graph G ∼ G(n, p) is whp such that in any r-colouring of its edges there exists a path of length at least (2/(r + 1) − ε)n in which all but at most K of the edges are of the same colour.
It is interesting to note that Theorems 1.1 and 1.5 are nontrivial (and new) even for the extreme case p = 1, i.e., where the coloured graph is the complete graph. An immediate corollary of Theorem 1.5 is that for large enough C = C(r, ε), whp there is a monochromatic matching of size (1/(r + 1) − ε)n in any r-colouring of the edges of G(n, C/n). In this sense, Theorem 1.5 is again optimal, as explained before.
A closely related and in fact relatively well studied problem is that of finding long monochromatic paths in edge-colourings of graphs (which corresponds to requiring K = 0 in Theorem 1.5). For two colours, this problem was resolved by Gerencsér and Gyárfás in [19] for complete graphs and by Letzter in [30] for random graphs. For r ≥ 3 colours, it is conjectured that every edgecolouring of K n contains a monochromatic path of length (1/(r − 1) − o(1))n, and that the same holds whp for G(n, p) with np → ∞ (see, e.g., [11]). It is known that if true, this would be best possible (even in the complete graph; see, e.g., [25] and the references therein). This conjecture was resolved for r = 3 by Gyárfás, Ruszinkó, Sárközy and Szemerédi in [21,22] (for the complete graph) and by Dudek and Pra lat in [11] (for random graphs), and it remains open for all r ≥ 4. Accidentally, for r = 2, 3 the two problems -that of finding a large monochromatic path and that of finding a large path in which all but a constant number of the edges are of the same colour -have the same answer (both in random and in complete graphs; this follows from Theorem 1.5 and the aforementioned results of [11,19,21,22,30]). For r ≥ 4, however, these two problems diverge; allowing a fixed number of edges to be coloured differently significantly increases the length of a path one can find, from at most (1/(r − 1) + o(1))n for monochromatic paths to (2/(r + 1) − o(1))n for almost monochromatic ones.
A common technique for finding long monochromatic paths, pioneered by Figaj and Luczak in [16] (following an idea by Luczak [32]), consists of applying the (sparse) regularity lemma and finding large monochromatic connected matchings in the reduced graph of a regular partition. In contrast, in order to find an almost monochromatic path, it is sufficient to find a monochromatic (not necessarily connected) matching in the reduced graph. One can expect -and we show that this is indeed the case -that in almost complete graphs (such as the reduced graphs we consider here), one can find substantially larger monochromatic matchings when dropping the requirement that they be connected. As mentioned above, this is a key step in the proof of Theorem 1.5.
Let us now say a few words about the remaining ingredients which go into the proof of Theorem 1.1. With Theorem 1.5 at hand, the proof of Theorem 1.1 proceeds as follows. Theorem 1.5 gives us a path P of length (2/(r + 1) − o(1))n in which all but a fixed number of edges are of the same colour. Our goal is therefore to extend this path into a Hamilton cycle, or, equivalently, to find a Hamilton path in the remaining set of vertices between neighbours of the endpoints of P . We achieve this by carefully splitting the remaining vertices into two equal sets, each containing many neighbours of the corresponding endpoint of P , so that the minimum degree of the graph spanned by each of these sets is at least 2. In fact, to do so we need to "prepare" our graph, putting aside small degree vertices with their neighbours, and finding P outside this set. We thus want to find a suitable path not in our random graph but rather in some large induced subgraph thereof; hence we need a generalisation of Theorem 1.5 to pseudorandom graphs, Theorem 3.1. We continue by showing that in each of the two abovementioned sets there are many Hamilton paths which start at a given point (a neighbour of the corresponding endpoint of P ), or, more precisely, Hamilton paths with many distinct ends. The argument relies on the so-called rotation-extension technique, invented by Pósa in [36] and has since been applied in numerous papers about Hamiltonicity of random graphs. We conclude our proof by using expansion properties of our graph to connect the ends of two such Hamilton paths, by that extending P to a Hamilton cycle.
Organisation We begin by proving Theorem 1.3 in Section 2. In Section 3 prove Theorem 3.1, a generalisation of Theorem 1.5 to pseudorandom graphs. At the end of the section we show how to connect the monochromatic linear forest we obtain to a long path, almost all of whose edges are of the same colour. The goal of Section 4 is to introduce a fairly general machinery to prepare random graphs in such a way that a path found in some (large) part of the graph can always be extended to a Hamilton cycle.
Notation and terminology Let G = (V, E) be a graph. For two vertex sets U, W ⊆ V we denote by E G (U ) the set of edges of G spanned by U and by E G (U, W ) the set of edges having one endpoint in U and the other in
W . The degree of a vertex v ∈ V is denoted by d G (v), and we write d G (v, U ) = |E G ({v}, U )|.
We let δ(G) and ∆(G) denote the minimum and maximum degrees of G. When the graph G is clear from the context, we may omit the subscript G in the notations above.
If f, g are functions of n we use the notation f ∼ g to denote asymptotic equality, namely,
f ∼ g if f = (1 + o(1))g, and we write f g if f = o(g).
For the sake of simplicity and clarity of presentation, we often make no particular effort to optimise the constants obtained in our proofs, and omit floor and ceiling signs when they are not crucial.
Large monochromatic matchings in almost complete graphs
The goal of this section is to prove Theorem 1.3. The primary tool used in the proof is the well-known Tutte-Berge formula (see, e.g., [31]), which we state as follows. For a graph G, let ν(G) denote the maximum size of a matching in G, and odd(G) denote the number of connected components of G whose size is odd.
Theorem 2.1 (Tutte-Berge formula). Every graph G satisfies ν(G) = 1 2 · |V (G)| − 1 2 · max U ⊆V (G) (odd(G − U ) − |U |).
We will also need the following simple lemma. Proof. Let G be as in the lemma, and let C 1 , . . . , C t be the connected components of G. Ev-
idently, |E(G)| ≤ t i=1 |C i | 2 .
Thus, in order to prove the lemma, it suffices to show that the function g(
x 1 , . . . , x t ) = t i=1 x i 2 with domain {(x 1 , . . . , x t ) ∈ [n] t : x 1 + · · · + x n = t} attains its maximum when x 1 = n − t + 1, x 2 = · · · = x t = 1, where it equals n−t+1 2 . So let (x 1 , . . . , x t ) ∈ [n] t be a maximum point of g. It is enough to show that there is (at most) one 1 ≤ i ≤ t such that x i ≥ 2. So suppose by contradiction that x i , x j ≥ 2 for some distinct i, j ∈ [t]. Without loss of generality, assume that x j ≥ x i . Now, setting y i := x i −1, y j := x j +1, and y k := x k for k ∈ [t] \ {i, j}, observe that g(y 1 , . . . , y k ) = g(x 1 , . . . , x k ) + x j − (x i − 1) ≥ g(x 1 , . .
. , x k ) + 1, in contradiction to the choice of x 1 , . . . , x k .
Proof of Theorem 1.3
Let G be a graph with n vertices and at least (1 − δ) n 2 edges. Fix any r-colouring of the edges of G. For each i ∈ [r], let G i be the graph on V (G) whose edges are the edges of G which are coloured with colour i. Our goal is to show that there is
1 ≤ i ≤ r such that ν(G i ) ≥ k i . So suppose, for the sake of contradiction, that ν(G i ) ≤ k i − 1 for every i ∈ [r]. By Theorem 2.1, for each i ∈ [r] there must be U i ⊆ V (G i ) = V (G) such that n 2 − 1 2 · (odd(G i − U i ) − |U i |) = ν(G i ) ≤ k i − 1, or, equivalently, odd(G i − U i ) ≥ n − 2(k i − 1) + |U i |. In particular, G i − U i has at least n − 2(k i − 1) + |U i | connected components. This means that n − |U i | = |V (G i − U i )| ≥ n − 2(k i − 1) + |U i |, and hence |U i | ≤ k i − 1.
By Lemma 2.2, the following holds for every i ∈ [r]:
|E(G i − U i )| ≤ (n − |U i |) − (n − 2(k i − 1) + |U i |) + 1 2 = 2(k i − 1) − 2|U i | + 1 2 .
It follows that
|E(G)| ≤ r i=1 |E(G i − U i )| + #{e ∈ E(G) : e ∩ (U 1 ∪ · · · ∪ U r ) = ∅} ≤ r i=1 2(k i − 1) − 2|U i | + 1 2 + n 2 − n − |U 1 | − · · · − |U r | 2 .(1)
Now, consider the function g(u 1 , . . . , u r ) defined by
g(u 1 , . . . , u r ) := r i=1 2(k i − 1) − 2u i + 1 2 − n − u 1 − · · · − u r 2 . Claim 2.3. Let u 1 , . . . , u r be such that 0 ≤ u i ≤ k i − 1 for every i ∈ [r]. Then g(u 1 , . . . , u r ) < −δ n 2 .
Before proving Claim 2.3, let us complete the proof of Corollary 1.4 assuming this claim.
Recall that |U i | ≤ k i − 1 for every i ∈ [r]. Thus, by applying Claim 2.3 with u i = |U i | (i ∈ [r]), we get that g(|U 1 |, . . . , |U r |) < −δ n 2 . On the other hand, (1) states that |E(G)| ≤ n 2 + g(|U 1 |, . . . , |U r |), which contradicts our assumption that |E(G)| ≥ (1 − δ) n 2 .
Thus, in order to complete the proof it suffices to prove Claim 2.3.
Proof of Claim 2.3. It will be convenient to set v i :
= k i − 1 − u i for i ∈ [r]. Then 0 ≤ v i ≤ k i − 1 for every i ∈ [r]. Note that the inequality g(u 1 , . . . , u r ) < −δ n 2 is equivalent to having h(v 1 , . . . , v r ) := n − r i=1 (k i − 1) + r i=1 v i 2 − r i=1 2v i + 1 2 > δ n 2 .(2)
For 1 ≤ i ≤ r, observe that if we fix the values of (v j : j ∈ [r] \ {i}) and let v i vary, then the resulting function h(v 1 , . . . , v r ) of v i is a quadratic function in which the coefficient of v 2 i is − 3 2 < 0. Therefore, this function is concave. It follows that for any choice of fixed values
of (v j : j ∈ [r] \ {i}), the minimum of h(v 1 , . . . , v r ) over 0 ≤ v i ≤ k i − 1 is obtained either at v i = 0 or at v i = k i − 1,
and is not obtained at any point in the open interval
(0, k i − 1). We conclude that if (v 1 , . . . , v r ) is a minimum point of h(v 1 , . . . , v r ), then v i ∈ {0, k i − 1} for every 1 ≤ i ≤ r. So we see that in order to verify (2), it is enough to show that h(v 1 , . . . , v r ) > ε n 2 for v 1 , . . . , v r satisfying v i ∈ {0, k i − 1} for every 1 ≤ i ≤ r. Recall that we set k := max{k 1 , . . . , k r }. Let I ⊆ [r], and suppose that v i = k i − 1 for i ∈ I and v i = 0 for i ∈ [r] \ I. Then the value of h(v 1 , . . . , v r ) is: n − i∈[r]\I (k i − 1) 2 − i∈I 2k i − 1 2 ≥ i∈I (k i − 1) + k + 1 + √ δn 2 − i∈I 2k i − 1 2 ≥ i∈I (k i − 1) + k 2 + 1 + √ δn 2 − i∈I 2k i − 1 2 .
Here, the first inequality uses our assumption that 1 − √ δ n ≥ r i=1 (k i − 1) + k + 1, and the second inequality follows from the fact that x+y
2 ≥ x 2 + y 2 for all x, y ≥ 0. Now, observe that 1+ √ δn 2 > δn 2 2 > δ n 2 (
where the first inequality follows from the fact that x+1 2 > x 2 2 for every x > 0). Thus, to establish Claim 2.3, it suffices to verify that
i∈I (k i − 1) + k 2 − i∈I 2k i − 1 2 ≥ 0.(3)
Observe that for every i ∈ I, if we fix the values of (k j : j ∈ I \ {i}) and consider the left-hand side of (3) as a one-variable function of k i , then this function is quadratic and the coefficient of k 2 i is −3/2 < 0. Thus, this function is concave. It follows that at a minimum point of the left-hand side of (3), we must have k i ∈ {1, k} for every i ∈ I (recall that k i ≤ k for every 1 ≤ i ≤ r). So let J ⊆ I, and suppose that k i = k for every i ∈ J and k i = 1 for every i ∈ I \ J. Setting s := |J|, we see that the left-hand side of (3) equals
s · (k − 1) + k 2 − s · 2k − 1 2 = (k − 1)(s − 1) 2 · ((s − 1)k − s).
So it remains to show that f (s) :
= (k −1)(s−1)·((s − 1)k − s) ≥ 0 for every value of s. If k = 1 then f (s) = 0 for every s, so suppose that k ≥ 2. Now, we have f (0) = (k − 1)k ≥ 0, f (1) = 0, and f (s) ≥ (s − 1)k − s = (k − 1)s − k ≥ 2(k − 1) − k ≥ 0 for every s ≥ 2, as required.
With Claim 2.3 established, the proof of Corollary 1.4 is complete.
It should be added that a MathOverflow post due to F. Petrov [35] contains a derivation of the Cockayne-Lorimer result [9] using the Tutte-Berge formula in a similar manner to our proof of Theorem 1.3.
Large monochromatic linear forests in pseudorandom graphs
The goal of this section is to prove Theorem 1.5. In fact, we prove a stronger statement, namely Theorem 3.1 below. This theorem extends Theorem 1.5 to the more general setting of pseudorandom graphs, and will be used in the proof of Theorem 1.1.
Let us now introduce some definitions. For a pair of disjoint vertex-sets U, W in a graph, the density of (U, W ) is defined as d(U, W ) := |E(U, W )|/(|U ||W |). For γ, p ∈ (0, 1], we say that G = (V, E) is (γ, p)-pseudorandom if for any two disjoint U, W ⊆ V with |U |, |W | ≥ γ|V | we have |d(U, W )−p| ≤ γp. We now recall the known fact that if G = (V, E) is (γ, p)-pseudorandom then every set U ⊆ V of size at least 2γ|V | satisfies
|E(U )| |U | 2 − p ≤ γp.(4)
To see that (4) holds, take a random partition of U into two equal parts U 1 , U 2 and observe that the expected value of |E(
U 1 , U 2 )| is |E(U )| · |U | 2 −1 · |U 1 ||U 2 |.
On the other hand, we have |d(U 1 , U 2 ) − p| ≤ γp for every such choice of U 1 , U 2 . Therefore,
|E(U )| |U | 2 − p = E|E(U 1 , U 2 )| |U 1 ||U 2 | − p ≤ γp,
as required. Note that if G is a (γ, p)-pseudorandom graph on n vertices (for any p ∈ (0, 1]) then there exists an edge between any two disjoint sets of size at least γn.
The following is the main result of this section, and will play an important role in the proof of Theorem 1.1.
Theorem 3.1. Let r ≥ 2 be an integer and let ε > 0. Then there exist γ = γ(r, ε) and K = K(r, ε) such that the following holds. Let G = (V, E) be a (γ, p)-pseudorandom graph for some p ∈ (0, 1], and suppose |V | = n is large enough (in terms of r, ε). Then, in any r-colouring of the edges of G there exists a path of length at least (2/(r + 1) − ε)n in which all but at most K of the edges are of the same colour.
The proof of Theorem 3.1 relies on (a "multicolour" version of) the well-known sparse regularity lemma, proved by Kohayakawa [26] and Rödl (see [10]), and later in a stronger form by Scott [37]. To state this result, we now introduce some additional definitions. A pair (U, W ) of disjoint vertex-sets is called (δ, q)-regular if for all U ⊆ U , W ⊆ W with |U | ≥ δ|U | and |W | ≥ δ|W | it holds that |d(U , W ) − d(U, W )| ≤ δq. An equipartition of a set is a partition in which the sizes of any two parts differ by at most 1 (to keep the presentation clean, we will ignore divisibility issues and just assume that all parts have the same size). Let G 1 , . . . , G r be graphs on the same vertex-set V of size n. An equipartition {V 1 , . . . , V t } of V is said to be (δ)-regular with respect to (G 1 , . . . , G r ) if for all but at most δ t 2 of the pairs
(V i , V j ), 1 ≤ i < j ≤ t, it holds that for every ∈ [r], the pair (V i , V j ) is (δ, q)-regular in G , where q := (|E(G 1 )| + · · · + |E(G r )|)/ n 2 .
We are now ready to state the multicolour sparse regularity lemma, as it appears in [37].
Theorem 3.2 (Multicolour sparse regularity lemma [37]). For every r, t 0 ≥ 1 and δ ∈ (0, 1) there exists T = T (r, t 0 , δ) such that for every collection G 1 , . . . , G r of graphs on the same vertex-set V , there is an equipartition of V which is (δ)-regular with respect to (G 1 , . . . , G r ), and has at least t 0 and at most T parts.
Another tool we will use in the proof of Theorem 3.1 is the following simple lemma from [7]. 7]). Let n, k ≥ 1 be integers, and let F be a bipartite graph with sides X, Y of size n each. Suppose that there is an edge between every pair of sets X ⊆ X and Y ⊆ Y with |X | = |Y | = k. Then F contains a path of length at least 2n − 4k.
The proof of Lemma 3.3 proceeds by a careful analysis of the DFS algorithm, an idea which originated in [7] and has since been widely used in the study of paths in random and pseudorandom graphs (see also [28] and [30, Corollary 2.1]).
We are now ready to prove Theorem 3.1.
Proof of Theorem 3.1. Let r ≥ 2 and let ε ∈ (0, 1). Fix δ > 0 to be small enough so that δ < 1/(4r) and 1/(r + 1) − 2 √ δ · (2 − 4δ) ≥ 2/(r + 1) − ε/2 (for this second requirement, choosing δ ≤ ε 2 /300 should suffice). Set t 0 := 1/ √ δ, and let T = T (r, t 0 , δ) be as in Theorem 3.2. We will prove the theorem with γ = γ(r, ε) := ε/(4T ) and K = K(r, ε) := T .
Let p ∈ (0, 1] and let G be a (γ, p)-pseudorandom graph on n vertices (for some sufficiently large n). Set q := |E(G)|/ n 2 ,and note that by (4) We now define a "reduced" edge-colouring of H. Let {i, j} be an edge of H. Since
|V i | = |V j | = n/t ≥ n/T ≥ γn, we have d G (V i , V j ) ≥ (1 − γ)p ≥ p/2 (as G is (γ, p)- pseudorandom). Since d G (V i , V j ) = d G 1 (V i , V j ) + · · · + d G (V i , V j ), there must be some ∈ [r] such that d G (V i , V j ) ≥ p/(2r).d G 1 (V i , V j ) ≥ p/(2r) and that (V i , V j ) is (δ, q)-regular in G 1 . Then for every V i ⊆ V i , V j ⊆ V j with |V i | ≥ δ|V i | and |V j | ≥ δ|V j | it holds that d G 1 (V i , V j ) ≥ d G 1 (V i , V j ) − δq ≥ p/(2r) − δq ≥ p/(2r) − δ(1 + γ)p ≥ p/(2r) − δ · 2p > 0,
where the last inequality holds due to our choice of δ. So we see that G 1 contains an edge between every pair of sets V i ⊆ V i , V j ⊆ V j with |V i | ≥ δ|V i | = δn/t and |V j | ≥ δ|V j | = δn/t. By Lemma 3.3 with k := δn/t, the bipartite subgraph of G 1 with sides V i and V j contains a path P e of length at least (2 − 4δ)n/t.
Observe that the paths (P e : e ∈ M ) are pairwise-disjoint (as M is a matching in H), and that the number of vertices covered by these paths is at least
|M | · (2 − 4δ)n/t ≥ 1/(r + 1) − 2 √ δ t · (2 − 4δ)n/t ≥ (2/(r + 1) − ε/2)n,
where the last inequality uses our choice of δ. Finally, put k = |M |, noting that k ≤ t ≤ T , and enumerate the paths (P e : e ∈ M ) as P 1 , . . . , P k . For each 1 ≤ i ≤ |M |, let A i , B i denote the first, respectively last, γn vertices of P i . Since G is (γ, p)-pseudorandom, there exists an edge e i = {b i , a i+1 } between b i ∈ B i and a i+1 ∈ A i+1 for every i = 1, . . . , k − 1. Let a 1 be the first vertex of P 1 and let b k be the last vertex of P k . Let P be the path obtained by concatenating (parts of) the paths P 1 , . . . , P k using the edges e 1 , . . . , e k−1 , namely,
P = a 1 P 1 − → b 1 e 1 − → a 2 P 2 − → b 2 e 2 − → · · · e k−1 − −− → a k P k −→ b k .
It is easy to see that
|P | ≥ |P 1 | + · · · + |P k | − (2k − 2) · γn ≥ (2/(r + 1) − ε/2)n − 2T γn ≥ (2/(r + 1) − ε)n,
where in the last inequality we used our choice of γ. Moreover, all edges of P except for e 1 , . . . , e k−1 have the same colour. As k ≤ T = K, the path P satisfies all the required properties, completing the proof.
In view of Theorem 3.1, in order to obtain Theorem 1.5 it is enough to prove that random graphs (with sufficiently high edge density) are whp pseudorandom.
Lemma 3.4. For every γ > 0 there exists C = C(γ) > 0 such that if p ≥ C/n then G ∼ G(n, p) is whp (γ, p)-pseudorandom.
In the proof of Lemma 3.4 and in several other proofs in the next section we will make use of the following version of Chernoff bounds.
Theorem 3.5 (Chernoff bounds, [24, Chapter 2])
. Let X ∼ Bin(n, p) with µ = np or X ∼ Hypergeometric(N, K, n) with µ = nKN −1 , and let 0 < α < 1 < β. Then
P(X ≤ αµ) ≤ exp(−µ(α log α − α + 1)), P(X ≥ βµ) ≤ exp(−µ(β log β − β + 1)).
Proof of Lemma 3.4. Note that we may assume γ > 0 is arbitrarily small. Write V = V (G). Fix disjoint U, W with |U |, |W | ≥ γn and write x = |U ||W |/n 2 ≥ γ 2 . Note that X := |E(U, W )| is a binomial random variable with xn 2 trials and success probability p. Thus by Theorem 3.5 there exists c = c(γ) > 0 such that P(|d(U, W ) − p| ≥ γp) = P(|X − pxn 2 | ≥ γpxn 2 ) ≤ 2 exp(−cpn 2 ).
Taking C = C(γ) to be large enough so that C > 2/c, say, we obtain by the union bound that
P(∃U, W ⊆ V, |U |, |W | ≥ γn : |d(U, W ) − p| ≥ γp) ≤ 4 n · e −2n = o(1).
With Lemma 3.4, the proof of Theorem 1.5 is now complete.
Extending paths to Hamilton cycles
The goal of this section is to give a general machinery to "prepare" a random graph (above the hamiltonicity threshold) in a way that any path found in some large portion of the graph can be extended, whp, to a Hamilton cycle. We will then use this machinery to extend the path obtained in Theorem 3.1 to a Hamilton cycle, proving Theorem 1.1. Throughout this section, we assume that n is large enough whenever needed. In addition, as the statement in Theorem 1.1 is clearly monotone in p, we will conveniently assume throughout this section that np − log n − log log n log log n.
Lemma 4.1. Let ε > 0, let p = (log n + log log n + ω(1))/n and let G ∼ G(n, p). Then, whp, there exists a partition V (G) = V ∪ V with |V | ≤ εn for which every path P ⊆ V with |V (P )| ≤ 2n/3 can be extended to a Hamilton cycle in G.
The proof of Lemma 4.1 uses Pósa's rotation-extension technique. Let us now recall some corollaries of Pósa's lemma [36]. For an overview of the rotation-extension technique, we refer the reader to [28].
Lemma 4.2 (Pósa's lemma [36]). Let G be a graph, let P = v 0 , . . . , v t be a longest path in G, and let R be the set of all v ∈ V (P ) such that there exists a path P in G with V (P ) = V (P ) and with endpoints v 0 and v. Then |N (R)| ≤ 2|R| − 1.
Recall that a non-edge of G is called a booster if adding it to G creates a graph which is either Hamiltonian or whose longest path is longer than that of G. For a positive integer k and a positive real α we say that a graph G = (V, E) is a (k, α)-expander if |N (U )| ≥ α|U | for every set U ⊆ V of at most k vertices. The following is a widely-used fact stating that (k, 2)-expanders have many boosters. For a proof, see e.g. [28]. We now move on to establish some useful properties satisfied whp by G(n, p) (for p as in Lemma 4.7).
Lemma 4.4. Let ε > 0 be sufficiently small, let p = (log n + log log n + ω(1))/n, and let G ∼ G(n, p). Then, whp, (P1) δ(G) ≥ 2 and ∆(G) ≤ 10 log n;
(P2) No vertex v ∈ V (G) with d(v) < log n/10 is contained in a 3-or a 4-cycle, and every two distinct vertices u, v ∈ V (G) with d(u), d(v) < log n/10 are at distance at least 5 apart;
(P3) Every set U ⊆ V (G) of size at most εn/100 spans at most ε|U | log n/10 edges.
(P4) There exist disjoint sets U 1 , U 2 ⊆ V (G) with |U 1 |, |U 2 | ∼ εn for which the following hold for every v ∈ V (G):
(a) If d(v) ≥ log n/10 then d(v, U 1 ), d(v, U 2 ) ≥ ε log n/100;
(b) If d(v) ≤ log n/10
then v and all of its neighbours are in U 1 .
Proof of (P1). For the minimum degree see, e.g., [18]. For the maximum degree, since d(v) ∼ Bin(n − 1, p) we have P(d(v) ≥ 10 log n) ≤ n 10 log n p 10 log n ≤ enp 10 log n 10 log n 1/n, and the statement follows by the union bound.
Proof of (P2). Write V = V (G) and α = 1/10. Let 1 ≤ ≤ 4 and let P = (v 0 , . . . , v ) be a sequence of + 1 distinct vertices from V , where optionally v 0 = v . Suppose first that v 0 = v . Let S 0 = V \ {v 1 , v } and S = V \ {v 0 , v −1 }. Let A P be the event that P is contained in G, and for i = 0, let B i be the event that d(v i , S i ) ≤ α log n. By Theorem 3.5 we obtain that P(B i ) ≤ n −0.6 . The events A P , B 0 , B are mutually independent, hence P(A P ∧ B 0 ∧ B ) ≤ p n −1.2 . Let A be the event that there exists a path P = v 0 , . . . , v with ∈ [4] in G such that A P and d(v 0 ), d(v ) ≤ α log n. By the union bound, P(A) ≤ 4 =1 n +1−1. 1 , v −1 } and let B be the event d(v 0 , S) ≤ α log n. As before, P(B) ≤ n −0.6 , and the events A P , B are independent, hence P(A p ∧ B) ≤ p n −0.6 . Let A be the event that there exists a cycle P of length ∈ {3, 4} such that A P and d(v 0 ) ≤ α log n. By the union bound, P(A ) ≤ 4 =3 n p n −0.6 = o(1).
2 p = o(1). The case v 0 = v (which implies ∈ {3, 4}) is similar. Let S = V \ {v
Proof of (P3). For a given set U ⊆ V (G) and for a given k ≥ 0, the probability that |E G (U )| ≥ k is at most
|U | 2 k · p k ≤ |U | 2 k · p k ≤ e|U | 2 p k k .
Hence, by the union bound, the probability that (P3) does not hold is at most Here, in the first inequality we plugged in p ≤ 2 log n/n.
Proof of (P4). The proof involves an application of a the symmetric form of the Local Lemma (see, e.g., [2, Chapter 5]; a similar application appears in [23]). Write V = V (G) and α = 1/10, and let X = {v ∈ V : d(v) ≤ α log n}. We condition on G and on the event that G satisfies (P1) and (P2). Write s = 1/ε , let t = n/s ∼ εn and let A 1 , . . . , A t , Z be a partitioning of the vertices of G into t "blobs" A i of size s and an extra set Z with |Z| ≤ s. For i ∈ [t] let (x i , y i ) be a uniformly chosen pair of distinct vertices from A i . Define
U 1 = {x i } t i=1 and U 2 = {y i } t i=1 .
Clearly, |U 1 | = |U 2 | = t and U 1 ∩ U 2 = ∅. For every v ∈ V \ X let B v be the event that either d(v, U 1 ) < εα 2 log n or d(v, U 2 ) < 2εα 2 log n. For such v, let L(v) be the set of blobs that contain neighbours of v, namely,
L(v) = {A i : N (v) ∩ A i = ∅}, and write (v) = |L(v)|.
Note that for i = 1, 2 the random variable d(v, U i ) stochastically dominates a binomial random variable with (v) attempts and success probability 1/s. Evidently, (v) ≥ εα log n, thus, by Theorem 3.5, P(B v ) ≤ n −c for some c = c(ε) > 0. For two distinct vertices u, v ∈ V \ X say that u, v are related if L(u) ∩ L(v) = ∅. For a vertex u ∈ V \ X let R(u) be the set of vertices in V \ X which are related to u, and note that |R(u)| ≤ s∆(G) 2 , which is, by (P1), at most C log 2 n for some C = C(ε) > 0. Note that B u is mutually independent of the set of events
{B v | v ∈ (V \ X) \ R(u)}.
We now apply the symmetric case of the Local Lemma: observing that en −c · C log 2 n < 1 (for large enough n), we get that with positive probability, none of the events B v occur. We choose U 1 , U 2 to satisfy this.
We proceed by observing that X is typically small. Indeed, by Theorem 3.5 we have P(d(v) ≤ α log n) ≤ n −0.6 , and by Markov's inequality |X| ≤ n 0.5 whp. By the definition of X we have that X + := X ∪ N (X) satisfies |X + | ≤ n 0.6 whp. Define U 1 = U 1 ∪ X + and U 2 = U 2 \ X + and note that from the discussion above, |U 1 |, |U 2 | ∼ εn. Let v ∈ V \ X. The fact that G satisfies (P2) implies that v has at most 1 neighbour in X + . Thus, for every
v ∈ V \ X it holds that d(v, U 1 ) ≥ d(v, U 1 ) ≥ εα 2 log n and d(v, U 2 ) ≥ d(v, U 2 ) − 1 ≥ εα 2 log n.
In the proof of Lemma 4.1, we will argue that whp G ∼ G(n, p) is such that every subset W ⊆ V (G) possessing certain properties induces a Hamiltonian graph. To this end, we will use the fact that given such a set W and a relatively sparse expander H on W which is a subgraph of G, it is highly likely that there is an edge e of G which is a booster with respect to H. This fact is established in Lemma 4.5 below. In the proof of Lemma 4.5 we will use the well-known and easy-to-show fact that if a graph H is a (|V (H)|/4, 2)-expander then H is connected. Indeed, if (by contradiction) H is not connected, then take a connected component X of size at most |V (H)|/2 and a set U ⊆ X of size min{|V (H)|/4, |X|}, and observe that |N (U )| ≤ |X| − |U | < 2|U |, contradicting the assumption that H is a (|V (H)|/4, 2)-expander.
Lemma 4.5. Let c > 0 be a sufficiently small absolute constant (c = 10 −5 should suffice), let p = (log n + log log n + ω(1))/n and let G ∼ G(n, p). Then, whp, G satisfies the following: for every every W ⊆ V (G) of size |W | ≥ 0.1n and for every (|W |/4, 2)-expander H on W which is a subgraph of G and has at most cn log n edges, G contains a booster with respect to H.
Proof. We use a first moment argument. Evidently, the number of choices for the set W is at most 2 n . Let us fix a choice of W . For each t, the number of choices of H for which |E(H)| = t is at most
|W | 2 t ≤ n 2 t ≤ en 2 t t .
Now let H be a (|W |/4, 2)-expander on W , and set t := |E(H)|. As mentioned above, H is connected. By Lemma 4.3, H has at least (|W |/4) 2 /2 = |W | 2 /32 ≥ n 2 /3200 boosters. Now, the probability that G contains H but no booster thereof is at most p t · (1 − p) n 2 /3200 ≤ p t · 1 − log n n n 2 /3200 ≤ 2 log n n t · exp(−n log n/3200).
Summing over all choices of W and H, we see that the probability that the assertion of the lemma does not hold is at most 2 n · exp(−n log n/3200) · cn log n t=1 2en log n t t .
Setting g(t) := (2en log n/t) t , we note that g (t) = g(t) · (log(2en log n/t) − 1) > 0 for every t in the range of the sum in (5), assuming c < 1, say. Thus, this sum is not larger than cn log n · (2e/c) cn log n = exp((log(2e/c) · c + o(1))n log n).
Now, if c is small enough so that log(2e/c) · c < 1/3200, we get that (5) tends to 0 as n tends to infinity. This completes the proof.
The following lemma states that a graph possessing certain simple properties is necessarily an expander. Statements of this type are fairly common in the study of Hamiltonicity of random graphs (see, e.g., [28]). For completeness, we include a proof. We claim that with positive probability (in fact, whp), H is a (|W |/4, 2)-expander. In light of Lemma 4.6, it is sufficient to show that with positive probability, H satisfies Conditions 1-4 in that lemma. Here, we will choose the parameters of Lemma 4.6 as d := d 0 and m := εn/500. We already showed that δ(H) ≥ 2 (which is Condition 1 in Lemma 4.6). Condition 2 holds because H is a subgraph of G and because the analogous statement holds for G, as guaranteed by Property (P2) in Lemma 4.4 (here we assume that ε ≤ 1/10). Similarly, Condition 3 holds because H is a subgraph of G and due to Property (P3) in Lemma 4.4 (note that 5m = εn/100).
Let us now prove that Condition 4 holds. Let U 1 , U 2 ⊆ V (H) = W be disjoint sets satisfying |U 1 |, |U 2 | = m = εn/500. Since G is (γ, p)-pseudorandom with γ = ε/500 (in fact, with γ = o(1), see Lemma 3.4), we have
|E G (U 1 , U 2 )| ≥ (1 − γ)p · |U 1 ||U 2 | ≥ |U 1 ||U 2 | log n 2n ≥ ε 2 n log n 500000
= Ω(n log n) .
Now, let us bound (from above) the probability that |E H (U 1 , U 2 )| = 0 (where the randomness is with respect to the choice of H). Recall that H is defined by choosing, for each v ∈ W , a random set
E(v) of min{d(v, W ), d 0 } edges of G[W ] incident
to v, with all choices made uniformly and independently, and letting E(H) = v∈W E(v). Fix any u 1 ∈ U 1 with d(u 1 , U 2 ) ≥ 1, and let A u 1 be the event that there is no edge in E(u 1 ) with an endpoint in U 2 . Observe that if d(u 1 , W ) < d 0 then P(A u 1 ) = 0, and otherwise
P(A u 1 ) = d(u 1 , W ) − d(u 1 , U 2 ) d 0 / d(u 1 , W ) d 0 = d 0 −1 i=0 d(u 1 , W ) − d(u 1 , U 2 ) − i d(u 1 , W ) − i ≤ 1 − d(u 1 , U 2 ) d(u 1 , W ) d 0 ≤ 1 − d(u 1 , U 2 ) ∆(G) d 0 ≤ e −d(u 1 ,U 2 )· d 0 ∆(G) ≤ e −εd(u 1 ,U 2 )/10 .
Here, in the last inequality we used Property (P1) in Lemma 4.4. Note that the events (A u 1 : u 1 ∈ U 1 ) are independent, and that if E H (U 1 , U 2 ) = ∅ then A u 1 occurred for every u 1 ∈ U 1 with d(u 1 , U 2 ) ≥ 1. It now follows that
P(E H (U 1 , U 2 ) = ∅) ≤ exp − ε 10 · u 1 ∈U 1 d(u 1 , U 2 ) = exp − ε 10 · |E G (U 1 , U 2 )| ≤ e −Ω(n log n) ,
where in the last inequality we used (6 /4, 2)-expander, it must be the case that |R| > |W |/4 ≥ n/40. So we see that the assertion of the lemma holds with Y = R. This completes the proof.
Proof of Lemma 4.1. We assume that G satisfy the properties detailed in Lemma 4.4, and that it is a (γ, p)-pseudorandom for γ < 1/40 and some p ∈ (0, 1), as guaranteed to happen whp by Lemma 3.4. Let U 1 , U 2 be disjoint subsets of V = V (G) satisfying (P4). Set V = U 1 ∪ U 2 and V = V \ V , and let P ⊆ V be a path with |V (P )| ≤ 2n/3 and endpoints a 1 , a 2 . Out goal is to extend P to a Hamilton cycle of G. Write V = V \ V (P ), partition V = V 1 ∪ V 2 as equally as possible. For i = 1, 2, let W i = V i ∪ U i and choose a neighbour w i of a i in W i ; this is possible since d(a i , U i ) ≥ ε log n/100 by (P4). Note that |W i | ≥ n/6 and for every v ∈ W i it holds that d(v, W i ) ≥ min{d(v), ε log n/100}, hence by Lemma 4.7 there exists a set Y i ⊆ W i with |Y i | ≥ n/40 such that for every y ∈ Y i there is a Hamilton path spanning W i from w i to y. Since G is a (γ, p)-pseudorandom for γ < 1/40, it has an edge e between Y 1 and Y 2 with endpoints y i ∈ Y i , say. For i = 1, 2, denote by Q y i the Hamilton path between w i and y i . We now construct a Hamilton cycle of G as follows (as depicted in Fig. 2): We now put together Theorem 3.1 and Lemma 4.1 in order to prove Theorem 1.1.
a 1 → w 1
Proof of Theorem 1.1. Let r ≥ 2, ε > 0 and p = (log n+log log n+ω(1))/n, let G ∼ G(n, p) and consider an r-colouring of the edge set of G. Let γ be the constant obtained from Theorem 3.1 by plugging in r and ε. Let V ∪ V be the partition guaranteed whp by Lemma 4.1 which satisfies n = |V | ≥ (1 − ε)n. By Lemma 3.4 we know that G is (γ(1 − ε), p)-pseudorandom (whp), hence G = G[V ] is (γ, p)-pseudorandom. By Theorem 3.1 we know that there exists a path P in G of length at most 2n /(r + 1) ≤ 2n/3 having at least (2/(r + 1) − ε)n ≥ (2/(r + 1) − 2ε)n edges of the same colour. By Lemma 4.1 we can, whp, extend P into a Hamilton cycle of G, still having at least (2/(r + 1) − 2ε)n edges of the same colour.
Perfect matchings
We now sketch a proof of Theorem 1.2. The first observation is that with mild modifications of the proof of Lemma 4.1 we may prove a variant of the following form. Let ε > 0 and p = (log n+ω(1))/n. Then G ∼ G(n, p) whp admits a partition of its vertex set V (G) = V ∪V with |V | ≤ εn such that (a) the set D 1 of vertices of degree 1 in G and its neighbourhood N (D 1 ) are contained in V ; and (b) for every subset X of V with |X| ≤ 2n/3 and D 1 ⊆ X, the subgraph G[V ∪ (V \ X)] contains a Hamilton path. We omit the proof details.
Having that lemma in hand, we proceed as follows. Let M 0 be the set of edges incident to vertices of D 1 ; there are, whp, O(log n) such edges, and they form, whp, a matching. As G[V \ V (M 0 )] is (whp) (γ, p)-pseudorandom by Lemma 3.4, we know by Theorem 3.1 that it has an almost monochromatic path P of length (2/(r + 1) − ε)n, from which we can extract a monochromatic matching of size at least (1/(r + 1) − ε )n, for some ε > 0. Add it to M 0 , creating an almost monochromatic matching M 1 of size at least (1/(r + 1) − ε )n. We now apply the lemma to find a Hamilton path in G[V ∪ (V \ V (M 1 ))], from which we extract a matching which completes M 1 into a perfect matching, in which at least (1/(r + 1) − ε )n edges are of the same colour.
Lemma 2. 2 .
2Let G be a graph with n vertices and t connected components. Then |E(
we have (1 − γ)p ≤ q ≤ (1 + γ)p. Let f : E(G) → [r] be an r-colouring of the edges of G. For each i ∈ [r], let G i be the graph on V (G) whose edges are the edges of G coloured by colour i. Let {V 1 , . . . , V t } be a (δ)-regular equipartition with respect to (G 1 , . . . , G r ), where t 0 ≤ t ≤ T . Let H be the graph on [t] in which {i, j} ∈ E(H) if and only if (V i , V j ) is (δ, q)-regular in G for every ∈ [r]. The definition of a (δ)-regular partition implies that |E(H)| ≥ (1 − δ) |V (H)| 2 .
2 ,
2Colour the edge {i, j} by colour (if there is more than one possible colour, choose one arbitrarily).Since |E(H)| ≥ (1 − δ) |V (H)| Corollary 1.4 implies that H contains a monochromatic matching of size at least t − 2 − √ δt /(r + 1) ≥ 1/(r + 1) − 2 √ δ t, where the inequality holds because t ≥ t 0 = 1/ √ δ. Suppose, without loss of generality, that this matching is in colour 1, and denote its edge-set by M . Fix any e = {i, j} ∈ M . Since {i, j} is an edge of H coloured with colour 1, it must be the case that
Lemma 4 . 3 .
43Let G be a connected (k, 2)-expander which contains no Hamilton cycle. Then G has at least (k + 1) 2 /2 boosters.
Lemma 4 . 6 .
46Let m, d ≥ 1 be integers and let H be a graph on h ≥ 4m vertices satisfying the following properties:It will be convenient to set d 0 := ε log n. Let W ⊆ V (G) be as in the statement of Lemma 4.7. We select a random spanning subgraph H of G[W ] as follows. For each v ∈ W , if d(v, W ) < d 0 then add to H all edges of G[W ] incident to v. Otherwise, namely if d(v, W ) ≥ d 0 , then randomly select a set of d 0 edges of G[W ] incident to v and add these to H. Note that |E(H)| ≤ |W | · d 0 ≤ εn log n. On the other hand, our assumption that d(v, W ) ≥ min{d(v), ε log n} for every v ∈ W implies that δ(H) ≥ min{δ(G), d 0 }. Hence, as d 0 ≥ 2 (for large enough n), we have δ(H) ≥ 2 by Property (P1) of Lemma 4.4.
Figure 2 :
2Outline of the proof of Lemma 4.1.
). By taking the union bound over all at most 2 2n choices of U 1 , U 2 , we see that with high probability, E H (U 1 , U 2 ) = ∅ for every pair of disjoint sets U 1 , U 2 ⊆ W of size m each.Finally, we apply Lemma 4.6 to conclude that whp H is a (|W |/4, 2)-expander. From now on, we fix such a choice of H. Before establishing the assertion of the lemma, we first show that G[W ] is Hamiltonian. To find a Hamilton cycle in G[W ], we define a sequence of graphs H i , i ≥ 0, as follows. To begin, set H 0 = H. For each i ≥ 0, if H i is Hamiltonian then stop, and otherwise take a booster of H i contained in G[W ] and add it to H i to obtain H i+1 . That such a booster exists is guaranteed by Lemma 4.5, as we will always have |E(H i )| ≤ |E(H)| + |W | ≤ |E(H)| + n ≤ εn log n + n ≤ c/2 · n log n + n ≤ cn log n, provided that ε is smaller than c/2, where c is the constant appearing in Lemma 4.5. Note also that H i is a subgraph of G[W ] for each i ≥ 0. Evidently, this process has to stop (because as long as H i is not Hamiltonian, the maximum length of a path in H i is longer thanin H i−1 ), thus showing that G[W ] must contain a Hamilton cycle, as claimed. Now let w ∈ W . As G[W ] is Hamiltonian, there exists a Hamilton path P of G[W ] such that w is one of the endpoints of P . Evidently, P is a longest path in G[W ]. Furthermore, note that G[W ] is a (|W |/4, 2)-expander because H, a subgraph of G[W ], is such an expander. Let R be the set of all y ∈ V (P ) = W such that there exists a Hamilton path P in G[W ] with endpoints w and y. By Lemma 4.2, we have |N G[W ] (R)| ≤ 2|R| − 1. Now, since G[W ] is a (|W |
No vertex v ∈ V (H) with d(v) < d is. 2and every two distinct vertices u, v ∈ V (H) with d(u), d(v) < d are at distance at least 5 apart2. No vertex v ∈ V (H) with d(v) < d is contained in a 3-or a 4-cycle, and every two distinct vertices u, v ∈ V (H) with d(u), d(v) < d are at distance at least 5 apart;
On the other hand, the definition of Y implies that H has at least d|Y |/2 edges incident to vertices of Y . Since all of these edges are contained in Y ∪ N (Y ), we see that Y ∪ N (Y ) contains at least d|Y |/2 > d · |Y ∪ N (Y )|/10 edges. But this stands in contradiction with Item 3. Thus, |N (Y )| ≥ 4|Y |. Next, note that by Item 2, every two elements of X are at distance at least 5; in particular, X is an independent set, and every two elements of X have disjoint neighbourhoods. Now Item 1 implies that |N (X)| ≥ 2|X|. Observe that each vertex of Y has at most one neighbour in X ∪ N (X), for otherwise there we conclude that. | |n (y )| − |y |, Our goal is to show that for every U ⊆ V (H) with |U | ≤ h/4 it holds that |N (U )| ≥ 2|U |. So let U ⊆ V (H) be such that |U | ≤ h/4. Suppose first that |U | ≥ m. Since there evidently is no edge between U and V (H)\(U ∪N (U )), it must be the case that |V (H)\(U ∪N (U )| < m by Item 4. So we have |U ∪N (U )now that |U | ≤ m. U )| = |N (X) \ Y | + |N (Y ) \ (X ∪ N (X))| ≥ |N (X)| − |Y | + |N (Y ) \ (X ∪ NLet X be the set of all u ∈ U satisfying d(u) < d, and set Y := U \ X. We claim that |N (Y )| ≥ 4|Y |. Suppose, for the sake of contradiction. All in all, we get that |NProof. Our goal is to show that for every U ⊆ V (H) with |U | ≤ h/4 it holds that |N (U )| ≥ 2|U |. So let U ⊆ V (H) be such that |U | ≤ h/4. Suppose first that |U | ≥ m. Since there evidently is no edge between U and V (H)\(U ∪N (U )), it must be the case that |V (H)\(U ∪N (U )| < m by Item 4. So we have |U ∪N (U )now that |U | ≤ m. Let X be the set of all u ∈ U satisfying d(u) < d, and set Y := U \ X. We claim that |N (Y )| ≥ 4|Y |. Suppose, for the sake of contradiction, that |N (Y )| < 4|Y |. Then |Y ∪ N (Y )| < 5|Y | ≤ 5|U | ≤ 5m. On the other hand, the definition of Y implies that H has at least d|Y |/2 edges incident to vertices of Y . Since all of these edges are contained in Y ∪ N (Y ), we see that Y ∪ N (Y ) contains at least d|Y |/2 > d · |Y ∪ N (Y )|/10 edges. But this stands in contradiction with Item 3. Thus, |N (Y )| ≥ 4|Y |. Next, note that by Item 2, every two elements of X are at distance at least 5; in particular, X is an independent set, and every two elements of X have disjoint neighbourhoods. Now Item 1 implies that |N (X)| ≥ 2|X|. Observe that each vertex of Y has at most one neighbour in X ∪ N (X), for otherwise there we conclude that |N (Y ) ∩ (X ∪ N (X))| ≤ |Y |, and hence |N (Y ) \ (X ∪ N (X))| ≥ |N (Y )| − |Y |. All in all, we get that |N (U )| = |N (X) \ Y | + |N (Y ) \ (X ∪ N (X))| ≥ |N (X)| − |Y | + |N (Y ) \ (X ∪ N (X))|
. ≥ 2|x| − |y | + |n (y )| − |y | ≥ 2|x| + 2|y | = 2|u |, ≥ 2|X| − |Y | + |N (Y )| − |Y | ≥ 2|X| + 2|Y | = 2|U |,
For p = (log n + log log n + ω(1))/n, the random graph G ∼ G(n, p) satisfies the following whp. Let W ⊆ V (G) be such that |W | ≥ 0.1n, and for every v ∈ W it holds that d(v, W ) ≥ min{d(v), ε log n}. Lemma 4.7. Let ε > 0Lemma 4.7. Let ε > 0. For p = (log n + log log n + ω(1))/n, the random graph G ∼ G(n, p) satisfies the following whp. Let W ⊆ V (G) be such that |W | ≥ 0.1n, and for every v ∈ W it holds that d(v, W ) ≥ min{d(v), ε log n}.
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On the evolution of random graphs. Paul Erdős, Alfréd Rényi, MR125031 ↑1A Magyar Tudományos Akadémia. Matematikai Kutató Intézetének Közleményei. 5Paul Erdős and Alfréd Rényi, On the evolution of random graphs, A Magyar Tudományos Akadémia. Matem- atikai Kutató Intézetének Közleményei 5 (1960), 17-61. MR125031 ↑1
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The Ramsey number for a triple of long even cycles. Agnieszka Figaj, Tomasz Luczak, MR2325798 ↑4. 97Agnieszka Figaj and Tomasz Luczak, The Ramsey number for a triple of long even cycles, Journal of Combinatorial Theory. Series B 97 (2007), no. 4, 584-596. MR2325798 ↑4
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. Tel Aviv. 6997801Lior Gishboliner School of Mathematical Sciences, Tel Aviv UniversityIsrael Email: [email protected] Research supported by ERC starting grant 633509Lior Gishboliner School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel Email: [email protected] Research supported by ERC starting grant 633509.
Israel Email: [email protected] Research supported in part by USA-Israel BSF grant 2018267 and by ISF grant 1261/17. 1207/15Peleg Michaeli School of Mathematical Sciences. 6997801Michael Krivelevich School of Mathematical Sciences, Tel Aviv University ; Tel Aviv UniversityIsrael Email: [email protected] Research supported by ERC starting grant 676970 RANDGEOMMichael Krivelevich School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel Email: [email protected] Research supported in part by USA-Israel BSF grant 2018267 and by ISF grant 1261/17. Peleg Michaeli School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel Email: [email protected] Research supported by ERC starting grant 676970 RANDGEOM and by ISF grant 1207/15.
| []
|
[
"The nonparametric Fisher geometry and the chi-square process density prior",
"The nonparametric Fisher geometry and the chi-square process density prior"
]
| [
"Andrew Holbrook \nDepartment of Statistics\nUC Irvine\n\n",
"Shiwei Lan \nDepartment of Computing and Mathematical Sciences\nCaltech\n",
"Jeffrey Streets \nDepartment of Mathematics\nUC Irvine\n\n",
"Babak Shahbaba \nDepartment of Statistics\nUC Irvine\n\n"
]
| [
"Department of Statistics\nUC Irvine\n",
"Department of Computing and Mathematical Sciences\nCaltech",
"Department of Mathematics\nUC Irvine\n",
"Department of Statistics\nUC Irvine\n"
]
| []
| It is well known that the Fisher information induces a Riemannian geometry on parametric families of probability density functions. Following recent work, we consider the nonparametric generalization of the Fisher geometry. The resulting nonparametric Fisher geometry is shown to be equivalent to a familiar, albeit infinite-dimensional, geometric object-the sphere. By shifting focus away from density functions and toward square-root density functions, one may calculate theoretical quantities of interest with ease. More importantly, the sphere of square-root densities is much more computationally tractable. This insight leads to a novel Bayesian nonparametric density estimation model. We construct the χ 2 -process density prior by modeling the square-root density with a restricted Gaussian process prior. Inference over square-root densities is fast, and the model retains the flexibility characteristic of Bayesian nonparametric models. Finally, we formalize the relationship between spherical HMC in the infinite-dimensional limit and standard Riemannian HMC. * [email protected] arXiv:1707.03117v2 [stat.ME] 14 Nov 2017 used for sensitivity analysis of Bayesian models. Here, we focus on fully Bayesian nonparametric inference, including the generation of posterior samples using Hamiltonian Monte Carlo (HMC). In contrast to recent research, the geodesics associated with the nonparametric Fisher geometry are used to efficiently explore the MCMC state space and not to measure or minimize the distance between density functions. This paper, and other recent research in the Fisher geometry, builds on the sub-field of square-root density estimation.[13] used a wavelet basis to estimate the square-root density by effectively fitting the curve and then normalizing a sparse collection of wavelet coefficients, and [14] introduced a Bayesian follow-up to this work. Recently,[15]used Riemannian geometry to fit a square-root density model, but did not make any connections to the Fisher geometry. More recently [12] performed square-root density estimation for object recognition using minimum description length as fitting criterion and used the nonparametric Fisher geometry to obtain a closed-form expression of this criterion.In this paper, we focus on the application of the nonparametric Fisher geometry to Bayesian inference for probability densities. While the density function is the object of interest, we instead model the squareroot density function, that is, the function the square of which integrates to unity. We take a Bayesian nonparametric approach and endow the square-root density with a Gaussian process (GP) prior[16],[17]multiplied by a Dirac measure limiting its support to the infinite-dimensional sphere. In order to maintain this restriction, it is useful to use the Karhunen-Loève (K-L) expansion [18] of the GP prior as opposed to its kernel representation. Every GP with bounded second moment may be represented in terms of the eigenfunction expansion of its covariance operator, but this (the K-L) expansion is only explicitly known for a few classes of GPs[18]. Still, the K-L expansion has seen much recent success in the realm of Bayesian inverse problems[19],[20]and has been featured in infinite-dimensional HMC and infinite manifold HMC (∞-mHMC)[21]. The proposed application of the K-L expansion to model the square-root density is unprecedented and offers a probabilistic interpretation to the use of basis expansions for density estimation.Due to the orthonormality of the eigenfunction basis, the restriction to the (uncountably) infinitedimensional sphere translates to a restriction to the (countably) infinite-dimensional sphere for the eigenvalues of the GP. Then, following the precedent set in[21], the K-L expansion is truncated and the object of inference is reduced to the posterior distribution of a finite number of K-L coefficients restricted to a finite sphere. This computation is quick and easy using either spherical HMC[22]. Thanks to the basis representation, computational complexity scales linearly with the number of data points, as opposed the cubic rate of the GP density sampler[23]. Moreover, we show that-in the square-root density estimation context-spherical HMC corresponds to Riemannian HMC in the infinite-dimensional limit.Squaring the GP square-root density prior gives a χ 2 -process [cf. 24] density prior. We illustrate the use of this prior for a number of problems. The model is flexible and its posterior draws provide plausible realizations of the uncertainty inherent in the density estimation problem. Besides a recent application to Bayesian quadrature [25], we are unaware of statistical applications for the χ 2 -process and are therefore pleased to present its novel application to Bayesian density estimation.The contributions of this paper are as follows:• we review a nonparametric generalization of the Fisher geometry and show its relationship to the infinite-dimensional (L 2 ) sphere, the space of square-root density functions;• we derive the geodesics on the L 2 sphere and use these geodesics to formalize the relationship between Riemannian HMC and infinite-dimensional spherical HMC;• focusing on Bayesian nonparametric density estimation, we demonstrate the practical benefits to modeling the square-root density function. The resulting χ 2 -process density prior performs well for a variety of problems and is efficiently computed using spherical HMC.The rest of the paper is organized in the following way. In Section 2 we review the parametric Fisher geometry, present a nonparametric extension of the Fisher geometry, and derive key results by relating this geometry to the infinite-dimensional sphere. Section 3 presents the χ 2 -process density prior along with some necessary tools, such as the Karhunen-Loève expansion. In Section 4, we discuss efficient Bayesian inference for the model and relate Riemannian HMC to infinite-dimensional spherical HMC. Empirical results are presented in Section 5. Finally, in Section 6 we discuss model limitations and possible extensions. | null | [
"https://arxiv.org/pdf/1707.03117v2.pdf"
]
| 55,828,947 | 1707.03117 | e7917a1085ebcad30a6f490491a6b49f2a408c2f |
The nonparametric Fisher geometry and the chi-square process density prior
November 14, 2017
Andrew Holbrook
Department of Statistics
UC Irvine
Shiwei Lan
Department of Computing and Mathematical Sciences
Caltech
Jeffrey Streets
Department of Mathematics
UC Irvine
Babak Shahbaba
Department of Statistics
UC Irvine
The nonparametric Fisher geometry and the chi-square process density prior
November 14, 2017
It is well known that the Fisher information induces a Riemannian geometry on parametric families of probability density functions. Following recent work, we consider the nonparametric generalization of the Fisher geometry. The resulting nonparametric Fisher geometry is shown to be equivalent to a familiar, albeit infinite-dimensional, geometric object-the sphere. By shifting focus away from density functions and toward square-root density functions, one may calculate theoretical quantities of interest with ease. More importantly, the sphere of square-root densities is much more computationally tractable. This insight leads to a novel Bayesian nonparametric density estimation model. We construct the χ 2 -process density prior by modeling the square-root density with a restricted Gaussian process prior. Inference over square-root densities is fast, and the model retains the flexibility characteristic of Bayesian nonparametric models. Finally, we formalize the relationship between spherical HMC in the infinite-dimensional limit and standard Riemannian HMC. * [email protected] arXiv:1707.03117v2 [stat.ME] 14 Nov 2017 used for sensitivity analysis of Bayesian models. Here, we focus on fully Bayesian nonparametric inference, including the generation of posterior samples using Hamiltonian Monte Carlo (HMC). In contrast to recent research, the geodesics associated with the nonparametric Fisher geometry are used to efficiently explore the MCMC state space and not to measure or minimize the distance between density functions. This paper, and other recent research in the Fisher geometry, builds on the sub-field of square-root density estimation.[13] used a wavelet basis to estimate the square-root density by effectively fitting the curve and then normalizing a sparse collection of wavelet coefficients, and [14] introduced a Bayesian follow-up to this work. Recently,[15]used Riemannian geometry to fit a square-root density model, but did not make any connections to the Fisher geometry. More recently [12] performed square-root density estimation for object recognition using minimum description length as fitting criterion and used the nonparametric Fisher geometry to obtain a closed-form expression of this criterion.In this paper, we focus on the application of the nonparametric Fisher geometry to Bayesian inference for probability densities. While the density function is the object of interest, we instead model the squareroot density function, that is, the function the square of which integrates to unity. We take a Bayesian nonparametric approach and endow the square-root density with a Gaussian process (GP) prior[16],[17]multiplied by a Dirac measure limiting its support to the infinite-dimensional sphere. In order to maintain this restriction, it is useful to use the Karhunen-Loève (K-L) expansion [18] of the GP prior as opposed to its kernel representation. Every GP with bounded second moment may be represented in terms of the eigenfunction expansion of its covariance operator, but this (the K-L) expansion is only explicitly known for a few classes of GPs[18]. Still, the K-L expansion has seen much recent success in the realm of Bayesian inverse problems[19],[20]and has been featured in infinite-dimensional HMC and infinite manifold HMC (∞-mHMC)[21]. The proposed application of the K-L expansion to model the square-root density is unprecedented and offers a probabilistic interpretation to the use of basis expansions for density estimation.Due to the orthonormality of the eigenfunction basis, the restriction to the (uncountably) infinitedimensional sphere translates to a restriction to the (countably) infinite-dimensional sphere for the eigenvalues of the GP. Then, following the precedent set in[21], the K-L expansion is truncated and the object of inference is reduced to the posterior distribution of a finite number of K-L coefficients restricted to a finite sphere. This computation is quick and easy using either spherical HMC[22]. Thanks to the basis representation, computational complexity scales linearly with the number of data points, as opposed the cubic rate of the GP density sampler[23]. Moreover, we show that-in the square-root density estimation context-spherical HMC corresponds to Riemannian HMC in the infinite-dimensional limit.Squaring the GP square-root density prior gives a χ 2 -process [cf. 24] density prior. We illustrate the use of this prior for a number of problems. The model is flexible and its posterior draws provide plausible realizations of the uncertainty inherent in the density estimation problem. Besides a recent application to Bayesian quadrature [25], we are unaware of statistical applications for the χ 2 -process and are therefore pleased to present its novel application to Bayesian density estimation.The contributions of this paper are as follows:• we review a nonparametric generalization of the Fisher geometry and show its relationship to the infinite-dimensional (L 2 ) sphere, the space of square-root density functions;• we derive the geodesics on the L 2 sphere and use these geodesics to formalize the relationship between Riemannian HMC and infinite-dimensional spherical HMC;• focusing on Bayesian nonparametric density estimation, we demonstrate the practical benefits to modeling the square-root density function. The resulting χ 2 -process density prior performs well for a variety of problems and is efficiently computed using spherical HMC.The rest of the paper is organized in the following way. In Section 2 we review the parametric Fisher geometry, present a nonparametric extension of the Fisher geometry, and derive key results by relating this geometry to the infinite-dimensional sphere. Section 3 presents the χ 2 -process density prior along with some necessary tools, such as the Karhunen-Loève expansion. In Section 4, we discuss efficient Bayesian inference for the model and relate Riemannian HMC to infinite-dimensional spherical HMC. Empirical results are presented in Section 5. Finally, in Section 6 we discuss model limitations and possible extensions.
Introduction
The Fisher information-and the geometry it induces-has been one of the unequivocal success stories of geometry in statistics. Building on recent work, we extend the Fisher geometry beyond parametric statistical models and show that the resulting geometry is equivalent to that of the infinite-dimensional sphere. The purpose of this paper is to bring attention to this new perspective and to demonstrate its theoretical and methodological consequences. As an application, we introduce the χ 2 -process density prior, a flexible nonparametric model for Bayesian density estimation that admits fast computation while requiring minimal assumptions.
The Fisher information matrix is canonical in statistics: it is rooted in information theory [1]; it appears in Jeffrey's prior of Bayesian analysis [2]; and it plays a central role in Bayesian and Frequentist asymptotics [3]. Fisher advocated the importance of the information matrix in maximum likelihood estimation [4]. Fisher's student, Rao, was the first to place the information matrix in a differential geometric context [5]. Since then, the differential geometric implications for parametric statistical models have been the subject of extensive inquiry [6]. Recently, a number of researchers have drawn connections between the Fisher geometry and the geometry of the infinite sphere [7]- [12]. Much of this work has been in the area of shape analysis and has focused on using the Fisher geometry to measure distance between probability densities. Bayesian uses for the nonparametric Fisher geometry were featured in [8], where Bayesian variational inference was accomplished by minimizing the Fisher distance, and in [10], where the nonparametric Fisher geometry was 2 The nonparametric Fisher geometry
The parametric Fisher geometry
Given data x in domain D, it is often useful to specify a probabilistic model S = {p θ = p(x, θ) | θ = θ 1 , . . . , θ p }, where θ is a vector parameterizing the model and taking values in the continuous parameter space Θ. Then at any point θ ∈ Θ, the Fisher information is the expectation of the negative log-likelihood Hessian:
I(θ) = −E x ∂ 2 (θ) ∂θ∂θ T = − D ∂ 2 (θ) ∂θ∂θ T p(x|θ) µ(dx) ,(1)
where (θ) = log p(x|θ). In the language of optimization, the Fisher information encodes second-order functional information about (θ). This fact explains the use of the Fisher information as a gradient preconditioning matrix in both (the Frequentist) Fisher scoring [26] and (the Bayesian) Riemannian HMC [27]. The Fisher information may also be written as the expected outer product of the score vector ∂ log p(x|θ)/∂θ:
I(θ) = E x ∂ (θ) ∂θ ∂ (θ) ∂θ T = D ∂ (θ) ∂θ ∂ (θ) ∂θ T p(x|θ) µ(dx) .(2)
The Fisher information is symmetric positive definite at any point θ ∈ Θ. Taking note of this fact, Rao [5] interpreted the Fisher information matrix as a Riemannian metric tensor, i.e. a smoothly varying, symmetric positive definite matrix defined over the parameter space Θ. In this way, the Fisher information matrix induces a Riemannian metric g θ (·, ·) over Θ satisfying
g θ ( i , j ) = I ij (θ) , and g θ (ψ, φ) = i,j ψ i φ j I ij (θ)(3)
for i = ∂ (θ)/∂θ i , ψ = p k=1 ψ k k and φ = p k=1 φ k k . Hence, the Fisher information may be thought of as inducing a non-trival geometry on the otherwise Euclidean parameter space Θ. There has been much inquiry into the nature of the parametric Fisher geometry. Efron used the Fisher geometry to prove the second-order efficiency of the MLE for exponential family models [28], and Amari and Nagaoka [6] has constructed a body of work around the Fisher geometry and its dual connections. More recently, Girolami and Calderhead [27] successfully used the Fisher geometry to guide the Hamiltonian flow of their Riemannian HMC. In this paper, we take another tact by generalizing the notion of the Fisher geometry to nonparametric models.
Beyond parametric models
We consider probability distributions over smooth manifolds D, of which D ∼ = R d is a special case. Having fixed a background measure µ, let
P := p : D → R | p ≥ 0, D p(x) µ(dx) = 1(4)
be the space of probability density functions over D. That is, P is the set of Radon-Nikodym derivatives of probability measures that are absolutely continuous with respect to µ. The following construction is agnostic to whether µ is the Lebesgue measure over D = R d or the Hausdorff measure over a general Riemannian manifold D = M. We deal with the space P and do not fix a parametric model. Instead we give P the structure of an infinite dimensional (formal) Riemannian manifold. First, we think of it as a smooth manifold. Observe that for a given p ∈ P, the tangent space can be identified with
T p P := φ ∈ C ∞ (D) | D φ(x) µ(dx) = 0 .(5)
This identification arises when one differentiates the unit measure condition on probability density functions. That is, for a smooth curve p t : (− , ) → P satisfying dp t /dt| t=0 = φ, we have
0 = d dt D p t (x) µ(dx) t=0 = D dp t dt (x) µ(dx) = D φ(x) µ(dx) .(6)
Now that we have a smooth manifold and an associated tangent space, we may define a Riemannian metric, i.e. a smoothly varying, symmetric, non-degenerate, bilinear function g(·, ·) p : T p P × T p P → {0} ∪ R + . Riemannian metrics are useful for developing a notion of distance on a manifold that does not depend on any embedding in Euclidean space. One may define uncountably many metrics on a general manifold, but we are interested in a generalization of the parametric Fisher information metric.
Definition 1. ( [7], [11]) Given D, the nonparametric Fisher information metric on P(D) 1 is
g F (φ, ψ) p := D φ(x)ψ(x) p(x) µ(dx).(7)
This metric is a consistent generalization of the parametric Fisher information metric. To see this, consider the parametric model p(x|θ), with θ as a vector. Then each element θ i of θ defines a curve Θ i → P, where Θ i is a slice of Θ, and
I ij (θ) = D i j p(x|θ)µ(dx) = D p i (x|θ) p(x|θ) p j (x|θ) p(x|θ) p(x|θ)µ(dx) = D p i (x|θ)p j (x|θ) p(x|θ) µ(dx) .(8)
Here, we have adopted the shorthand p i (x|θ) = ∂p(x|θ)/∂θ i . Expressed in a more invariant fashion, interpreting a model as a map θ : Θ → P, one has that the parametric Fisher metric is induced by the nonparameteric Fisher metric, i.e.
θ * g F = g θ .(9)
In what follows we make a nontrivial change of variables suggested by this geometric picture which provides various theoretical and computational simplifications. In particular, for various reasons the manifold P equipped with Riemannian metric (7) is not particularly easy to deal with. In order to calculate geometric quantities of interest (e.g. geodesics, distances), we shift focus to the L 2 unit sphere, i.e. the space of square-root density functions
Q := q : D → R | D q(x) 2 µ(dx) = 1 .(10)
This space, which is identified with P by a simple transformation indicated below, provides a much simpler backdrop for calculations. This infinite-dimensional L 2 sphere is a surprisingly familiar object. Its tangent spaces and geodesics are formally the exact same as those of the finite dimensional sphere S n−1 , the only difference being the replacement of the Euclidean inner product with the integral inner product of L 2 :
f, h L 2 = D f (x)h(x) µ(dx) .(11)
Remarkably, this simpler space is isometric to the space of density functions equipped with the nonparameteric Fisher metric defined above.
Lemma 1. The map S : (P, g F ) → (Q, ·, · L 2 ) defined by S(p) := 2 √ p is a Riemannian isometry.
Proof. We must show that S * ψ, S * φ L 2 = g F (ψ, φ) p , where S * is the pushforward (or Jacobian) of S:
S * = dS dp (p) = d(2 √ p) dp = 1 √ p .(12)
By direct computation,
S * ψ, S * φ L 2 = D (S * ψ)(x) (S * φ)(x) µ(dx) = D ψ(x) p(x) φ(x) p(x) µ(dx) (13) = D ψ(x)φ(x) p(x) µ(dx) = g F (ψ, φ) p .
In the remainder of this section we present a few basic results regarding the nonparametric Fisher geometry, working with the L 2 sphere model and transferring results to the traditional Fisher geometry. We note that investigations of the nonparameteric Fisher information have independently appeared in [7]- [12]. We reproduce some fundamental aspects of this geometry relevant to Theorem 1 (Section 4.1) for convenience. To begin we observe how to describe the tangent space to Q. Lemma 2. Given q ∈ Q, one has that
T q Q := f : D → R | D q(x)f (x) µ(dx) = 0 .(14)
Proof. If q t : (− , ) → Q denotes a path in Q satisfying dq t /dt| t=0 = f , then the unit integration constraint on p = q 2 means
0 = d dt D q t (x) 2 µ(dx) t=0 = 2 D q 0 (x) dq dt (x) t=0 µ(dx) = 2 D q 0 (x)f (x)µ(dx) .(15)
We next solve one version of the geodesic problem on P. In particular we consider an initial point and velocity and solve for continuing the geodesic in that direction. We will exploit the isometry between P and Q and solve first in Q.
Lemma 3. Given q 0 ∈ Q and f ∈ T q Q a unit vector, the geodesic with initial condition q 0 and velocity f exists on (−∞, ∞) and takes the form
q t = q 0 cos t + f sin t.(16)
Proof. First we derive the geodesic equation in Q. One conceptual method for obtaining this, exploiting the spherical structure of Q, is to first observe that if q t is a path in Q and a t ∈ T q(t) Q is a tangent vector along the curve, the Fisher geometry induces a covariant derivative along the path via
D ∂t a =ȧ − q Dȧ q,(17)
which is manifestly the time derivative of the family a t projected to the tangent space at q t , as expected. For a curve q t to be a geodesic, it should have zero acceleration, i.e.
0 = D ∂tq =q − q Dq q.(18)
However, using that D q t (x) 2 µ(dx) = 1 for all q and differentiating twice in t, one sees that this is equivalent toq
+ q Dq 2 = 0,(19)
which we now take as the geodesic equation in Q. Another method for deriving this equation is to solve for which curves are critical points for the length functional with fixed endpoints. Now, to solve this equation in our setting, first let us observe that since f ∈ T q0 Q, by Lemma 2 we have
D q 0 f = 0.(20)
Using this and the fact that f is a unit vector we compute
d dt Dq 2 = 2 Dqq (21) = 2 D [−q 0 cos t − f sin t] [−q 0 sin t + f cos t] = 2 D q 2 0 − f 2 cos t sin t = 0.
Thus Mq 2 = D f 2 = 1. We then simply observe the ODË
q = −q,(22)
and it is clear that q satisfies (19), and so the lemma follows.
We now translates this result into a corresponding one for geodesics in P.
Lemma 4. Given p 0 ∈ P and f ∈ T p P a unit vector, the geodesic with initial condition p 0 and initial velocity f exists on (−∞, ∞), and takes the form
p t = √ p 0 cos t + f 2 √ p 0 sin t 2 .(23)
Proof. We use Lemma 3 and reinterpret the geodesic equation in terms of square-roots. In this formalism the initial condition is q 0 = √ p 0 and the initial velocity is
d dt q = d dt √ p = f 2 √ p 0 = f 2q 0 .
These basic lemmas show the advantage of working in Q, yielding a conceptual derivation of the geodesic equation. These lemmas will be used to prove Theorem 1 in Section 4.1. As we will see below, not only is the L 2 sphere Q more theoretically tractable, it also turns out to be more computationally tractable. In the following sections, we take advantage of these two kinds of tractability to construct a Bayesian nonparametric model on Q and use it for an application in density estimation.
The chi-square process density prior
In this section, we transition from the theoretical to the applied aspects of the nonparametric Fisher geometry. We find that the square-root representation q = √ p is of use practically as well as theoretically. Here we focus on its natural application for density estimation. A good density estimate places more mass where there is more data but takes the finite nature-and the uncertainty that comes with it-of that data into account. Bayesian non-parametric density estimation effects this balance: non-parametric models give flexibility, while the Bayesian prior contributes regularization. These methods model the data generating distribution as a random function, itself drawn from a specified stochastic process. Dirichlet processes mixture models (DPMMs) convolve the Dirichlet process with a smooth distribution, in effect constructing an infinite mixture model [29]. More recently, [23] proposed a new method, called Gaussian Process Density Sampler (GPDS), offering a similar amount of flexibility as the DPMM but having an arguably simpler framework. Nonetheless, inference for DPMMs requires an advanced Gibbs sampling routine [30], and inference for the GPDS requires exchange sampling to handle the unitintegral restriction on the GP model [23]. In contrast, the model we propose here can be computed using generic spherical HMC [22] or geodesic Monte Carlo [31] algorithms. Further, we take a different approach from other Bayesian nonparametric density models by modeling the square-root density function instead. In the previous section, theoretical results for the nonparametric Fisher geometry were easier to obtain by first obtaining the corresponding results on the L 2 sphere and then translating the results to the Fisher geometry. This theme continues in application, where we show that Bayesian density estimation can be much easier when one shifts focus to the sphere of square-root densities. We place a GP prior on the square-root of the probability density function. This amounts to a χ 2 -process prior on the density function itself.
Suppose we want to attribute a smooth density function to observed data x 1 , . . . , x n on finite domain D ⊂ R d and recall the definitions (from Section 2) of the space of density functions and the space of square-root density functions:
P := p : D → R | p ≥ 0, D p(x) µ(dx) = 1 and Q := q : D → R | D q(x) 2 µ(dx) = 1 ,(24)
respectively. We want to find a suitable element p(·) ∈ P(D), the space of functions over domain D. Although this space contains the functions of interest, we opt to deal with the space Q of square-root densities instead. As stated in the prior section, Q is the unit sphere in the infinite-dimensional Hilbert space L 2 (D). We model the square-root density with a GP prior (or a Gaussian measure in L 2 ) multiplied by the Dirac measure restricting the function to the unit sphere:
q ∼ GP × δ q (Q) .(25)
It turns out that it is much easier to enforce the constraint given by Dirac measure δ q (Q) than it is to enforce the corresponding constraint δ p (P) (as is done for the GPDS). To do so, however, we do not represent the GP prior using its kernel representation as is commonly done in the literature [32]. We opt instead to represent q in terms of the eigenvalues and orthonormal eigenfunctions of its covariance operator.
The Karhunen-Loève representation
In order to tractably enforce the constraint δ q (Q) in (25), it is helpful to write q as a function (or linear sum of functions) for which we know the values of both
D q(x)µ(dx) and D q(x) 2 µ(dx) .(26)
This condition is satisfied by representing random function q as a linear combination of orthonormal basis functions. The K-L representation [18] provides a canonical way of doing so and thus links our fully probabilistic approach to other square-root density methods that rely on a basis [13]- [15]. Let u(·) ∼ GP(0, K(·)) be a mean zero Gaussian process over domain D with covariance operator K(·). Then u admits a K-L expansion of the form
u(·) = ∞ i=1 u i φ i (·), u i ind ∼ N (0, λ 2 i ),(27)
where the λ i s and the φ i s are respectively the eigenvalues and eigenfunctions of operator K. That is to say, they satisfy
K(φ i )(x ) = k(x, x )φ i (x)µ(dx) = λ i φ i (x )(28)
where k(·, ·) is the usual covariance kernel. The eigenvalues are decreasing and their sum-of-squares is finite:
λ i+1 < λ i , ∞ i=1 λ 2 i < ∞.
Finally, the eigenfunctions form an orthonormal basis of L 2 :
φ i (x)φ j (x)µ(dx) = 0, and φ 2 i (x)µ(dx) = 1 .(29)
In this paper, we model q as belonging to the Matérn class of GPs. For the Matérn class, a closed-form orthonormal basis may be obtained from the eigenfunctions of the Laplacian [21], [33]. The covariance operator is given by
K = σ 2 (α − ∆) −s ,(30)
where α and σ 2 are positively constrained scale parameters, s is a smoothness parameter, and ∆ is the Laplacian
d i=1 ∂ 2 i .
The eigenvalues and eigenfunctions corresponding to this covariance operator depend on the area and dimensionality of domain D and are presented in Section 5 below. It should be noted that the decision to use the Matérn class is entirely dictated by ease of computation and does not preclude other classes of GP from being used in future applications.
The model
The proposed density model is Bayesian nonparametric, i.e. we place a prior distribution on a set of functions and eschew a restrictive parametric form. Given data x = (x 1 , · · · , x N ) ∈ D, we obtain a posterior distribution, which is itself a distribution over the same set of functions and is absolutely continuous with respect to the specified prior distribution. As stated above, the prior π(q) on square-root density q ∈ Q is a GP multiplied by the Dirac measure on the L 2 sphere. Following (27), the prior for q and the likelihood of the data x given q are given by
π(q) ∝ δ q (Q) ∞ i=1 exp − q 2 i /(2λ 2 i ) , and π(x|q) = N n=1 q 2 (x n ) ,(31)
since q is the square-root density. This prior can also be interpreted as arising from an infinite-dimensional Bingham distribution on the coefficients [34]. The posterior distribution on q is then given by
π(q|x) = π(x|q) π(q) Q π(x|q) π(q) dq ∝ π(q) N n=1 q 2 (x n ) .(32)
Suppressing the Dirac measure, the log-posterior given data x 1:N may be written in terms of the K-L expansion (27) of q:
log π(q|x) ∝ N n=1 log q(x n ) 2 − 1 2 ∞ i=1 q 2 i /λ 2 i (33) = 2 N n=1 log |q(x n )| − 1 2 ∞ i=1 q 2 i /λ 2 i = 2 N n=1 log | ∞ i=1 q i φ i (x n )| − 1 2 ∞ i=1 q 2 i /λ 2 i .
By modelling the square-root density q with a GP prior, we model the density function p with a χ 2 -process prior. Modeling the density p as a χ 2 -process, we automatically enforce the non-negativity requirement for probability density functions. On the other hand, χ 2 -processes are not restricted to have unit integrals. We therefore rely on a geometric HMC inference scheme to restrict proposals to the L 2 sphere. This is discussed in the following section.
Inference
Inference for the χ 2 -process density model is relatively straightforward and amenable to advanced HMC methods. In Section 4.1, we show that, in this context, infinite-dimensional spherical HMC is equivalent to Riemannian HMC using the parametric Fisher information. In practice, we follow Beskos, Girolami, Lan, et al. [21] and truncate 2 the K-L expansion of the GP square-root density prior for an integer I using truncation operator T I :
T I q(x) = T I ∞ i=0 q i φ i (x) = I i=0 q i φ i (x) .(34)
Due to the orthonormality of the basis φ i , the unit integral constraint on T I (q) 2 translates directly to a spherical constraint on the random coefficients q I = (q 0 , · · · , q I ). That is,
1 = D T I q(x) 2 µ(dx) = D I i=0 q i φ i (x) 2 µ(dx) = I i=0 q 2 i φ i (x) 2 µ(dx) = I i=0 q 2 i(35)
where the penultimate equality is given by the orthogonality of the basis elements and the last equality is on account of the basis elements being normal. Thus, inference can be performed over the coefficients q I by using spherical HMC [22] on the sphere S I . Both of these methods augment the state space with an auxiliary velocity variable v (satisfying v T q I = 0) and simulate from a Hamiltonian system by splitting [36] the Hamiltonian of interest (H) into two Hamiltonians (H 1 + H 2 ):
H(q I , v) = − log π(q I ) + 1 2 G(q I ) + 1 2 v T v(36)H 1 (q I , v) = − log π(q I ) + 1 2 G(q I ) H 2 (q I , v) = 1 2 v T v .
Here π is the posterior distribution and G is the canonical Riemann tensor for the sphere [22]. Simulating from H 1 involves a small perturbation of the velocity by the gradient of H 1 with respect to q I ; simulating H 2 involves moving along the sphere's geodesics in the direction v. This last fact is relevant to the discussion of the following section. Spherical HMC requires the gradient of the log-posterior with respect to the coefficients. Elementwise, this is given by
∂ ∂q j log π(q I |x) = 2 N n=1 ∂ ∂q i log | I i=1 q i φ i (x n )| − 1 2 ∂ ∂q j I i=1 q 2 i /λ 2 i (37) = 2 N n=1 φ j (x n ) I i=1 q i φ i (x n ) − q j /λ 2 j .
The Markov chain may be initialized using Newton's method on the sphere (see Appendix A).
Since the values of the eigenfunctions at the observations may be precomputed, the main computational burden is in the summations involved in the evaluation of the log-posterior and its gradient. Since in practice I N , these computations are O(N ), where N is the number of data points. This is orders faster than the O(N 3 ) computations required to perform inference for the GPDS [23].
Inference in the limit
We note that both spherical HMC uses geodesic flows on the finite dimensional sphere to propose new Markov chain states. Since these flows are formally equivalent to the geodesic flows on the L 2 sphere (see Section 2) and since the natural geometry on L 2 is equivalent to the nonparametric Fisher geometry, it is worth asking whether these inference schemes are adapted to the nonparametric Fisher geometry in a similar way to Riemannian HMC's adaptation to the parametric Fisher geometry 3 .
Indeed this is the case, and it is a simple consequence of Lemma 1 and the isometric relationship between square-integrable functions and square-summable sequences induced by any orthonormal basis {φ i } ∞ i=1 with completion L 2 . Denote the space of square-summable sequences and its sphere
2 = q = {q i } ∞ i=1 q, q 2 = ∞ i=1 q 2 i < ∞ , S ∞ = q ∈ 2 q, q 2 = ∞ i=1 q 2 i = 1 .(38)
Then it follows from the orthonormality of
{φ i } ∞ i=1 that (L 2 , ·, · L 2 ) ∼ = ( 2 , ·, · 2 )
, since for any arbitrary function q = q(·) ∈ L 2 ,
q, q L 2 = q(x) 2 µ(dx) = ∞ i=1 q i φ i (x) 2 µ(dx) = ∞ i=1 q 2 i = q, q 2 .(39)
It is an immediate result that the respective spheres are also isometric, i.e. (Q, ·, · L 2 ) ∼ = (S ∞ , ·, · 2 ), and hence, by Lemma 1, the following result holds.
Lemma 5. Given an orthonormal basis for L 2 , the space of density functions equipped with the Fisher metric is isometric to the sphere S ∞ with its natural Euclidean metric, i.e. (P, g F (·, ·)) ∼ = (S ∞ , ·, · 2 ).
Our goal is to show that spherical HMC is adapted to the nonparametric Fisher geometry in the infinitedimensional limit. Given that the geodesic paths followed by spherical HMC converge to geodesics on S ∞ , Lemma 5 will imply that these paths correspond to geodesics on (P, g F (·, ·)). Lemma 6. Geodesic flows on the finite sphere S I−1 converge to geodesic flows on the infinite-dimensional sphere S ∞ as I → ∞.
Proof. For any point q ∈ S ∞ , let q I ∈ S I−1 be vector obtained by applying the truncation operator to q and then normalizing:
q I = T I (q) T I (q) = (q 1 , . . . , q I ) T (q 1 , . . . , q I )(q 1 , . . . , q I ) T .(40)
Similarly, for any vector in the tangent space to
S ∞ v ∈ T q S ∞ = v ∈ 2 v, q 2 = ∞ i=1 q i v i = 0(41)
let v I ∈ T q I S I−1 be the I-dimensional vector obtained by truncating v, projecting onto the tangent space T q I S I−1 , and scaling such that v 2 = v I (where · is the Euclidean norm):
v I = T I (v) − q I q I , T I (v) 2 , and v I =ṽ I v 2 ṽ I .(42)
It follows from the definition of truncation (Equation (34)) that q I → q and v I → v with respect to ·, · 2 as I → ∞ . Next, let t → (q(t), v(t)) be the geodesic flow on S ∞ with initial position q 0 = q(0) and initial velocity
v 0 = v(0) ∈ T q0 S ∞ . Let t → (q I (t)
, v I (t)) be the analogous flow on the tangent bundle T S I−1 , where q I 0 and v I 0 are obtained from q 0 and v 0 following Formulas (41) and (42), respectively. Denote the distance between flows at time t
f (t) = q t − q I and hence that d dt f (t) ≤ 1 − q 0 2 2 f (t) .(49)
Integrating gives
f (t) ≤ f (0) e t (1− q0 2 2 ) .(50)
Since, by definition,
f (0) → 0 as I → ∞, we have T 0 f (t) dt ≤ f (0) T 0 e t (1− q0 2 2 ) dt (51) = c f (0) −→ 0 .
Thus we have proven the convergence of geodesic flows on the finite sphere to those on S ∞ .
We are now ready to connect Riemannian HMC and spherical HMC in the infinite-dimensional limit (where the latter is applied to the square-root density estimation problem). To make this relationship as clear as
E(γ) = 1 2 b a g γ(t) γ(t),γ(t) dt , and thus satisfies d ds E(γ s ) = 0 .(52)
For a parametric family of distributions P θ equipped with the Fisher metric, the parametric Fisher energy takes the form
E(θ) = 1 2 b a g θ(t) θ (t),θ(t) F dt = 1 2 b a ∇ θ (θ(t)) T I(θ(t)) −1 ∇ θ (θ(t)) dt ,(53)
where I(θ) is the Fisher information, and (θ) = log p(θ). On the other hand by Lemmas 1 and 5, the nonparametric Fisher energy for a family of curves in P takes the form
E(p) = 1 2 b a g p(t) ṗ(t),ṗ(t) F dt = 1 2 b a q(t),q(t) L 2 dt = 1 2 b a q(t),q(t) 2 dt (54) where q = √ p = ∞ i=1 q i φ i (·)
. Theorem 1. Let q(·) = p(·) ∈ Q be a square-root density function with expansion satisfying
q(·) = ∞ i=1 q i φ i (·) , and 1 = D q(x) 2 µ(dx) = ∞ i=1 q 2 i ,(55)
with random, real-valued coefficients q i , i = 1, . . . , ∞. Then, in the infinite-dimensional limit, spherical HMC follows the nonparametric Fisher metric's geodesic flows in the same way that Riemannian HMC follows the Fisher metric's geodesic flows over the parametric family of distributions P θ .
Proof. Each of these algorithms relies on a split Hamiltonian [36] integration scheme (e.g. Equation (36)), wherein the Hamiltonian of interest (H) is split into two Hamiltonians (H 1 + H 2 ) that are then iteratively simulated. The formal Hamiltonian for spherical HMC on lim I→∞ S I−1 = S ∞ has the same form as in Equation (36), but in this case the velocity v is restricted to the tangent space to S ∞ at q, T q S ∞ . The Hamiltonian corresponding to Riemannian HMC is also split in the following way [31]:
H(θ, ξ) = − log p(θ) + 1 2 log I(θ) + 1 2 ξ T I −1 (θ)ξ (56) H 1 (θ, ξ) = − log p(θ) + 1 2 log I(θ) H 2 (θ, ξ) = 1 2 ξ T I −1 (θ)ξ ,
where I(θ) is the Fisher information, and ξ is the auxiliary momentum variable. Switching out ξ(t) for ∇ θ (θ(t)) in Equation (53), it follows that the solutions to the Hamilton's equations for Hamiltonian H 2 (θ, ξ) are the geodesics on the Riemannian manifold (P θ , g F ). This is because the Hamiltonian flow θ(t) preserves H 2 (θ, ξ):
d ds E(θ) = d ds 1 2 b a ξ(t) T I(θ(t)) −1 ξ(t) = d ds b − a 2 ξ(a) T I(θ(a)) −1 ξ(a) = 0 .(57)
Thus, Riemannian HMC steps around the state space by minimizing the parametric Fisher energy.
In the same way, exchanging v(t) forq(t) of Equation (54), it follows that the solutions to the Hamilton's equations for Hamiltonian H 2 (q, v) are geodesics on the Riemannian manifold (S ∞ , ·, · 2 ) and, by Lemma 5, correspond to geodesics on (P, g F (·, ·)). Hence, both formal algorithms move around the state space by iteratively perturbing the velocity (H 1 ) and travelling the geodesics corresponding to the parametric and nonparametric Fisher geometries, respectively.
Finally, Lemma 6 guarantees that the finite-dimensional spherical geodesics (used in practice) pass in the limit to the geodesics of the sphere S ∞ and hence of (P, g F (·, ·)).
Empirical results
Here we apply the χ 2 -process density model to both simulated and real-world data. As stated in Section 3.1, the eigen-pairs corresponding to the GP with covariance operator (30) depend on both the dimension and the area of D. When D is the one-dimensional unit interval, the eigen-pairs are given by
λ 2 i = σ 2 (α + π 2 i 2 ) −s , and φ i (x) = √ 2 cos(π i x) ,(58)
for i ≥ 0. For D the two-dimensional unit square D = [0, 1] × [0, 1], the eigen-pairs are given by
λ 2 i = σ 2 α + π 2 (i 2 1 + i 2 2 ) −s , and φ i (x) = 2 cos(π i 1 x 1 ) cos(π i 2 x 2 ) ,(59)
for i 1 , i 2 ≥ 0. See Beskos, Girolami, Lan, et al. [21] for a similar approach. In the following experiments, all Markov chains are initialized using Newton's method on the sphere (see Appendix A). Figure 1: Each plot shows 100 posterior draws from the χ 2 -process density sampler. 1,000 data samples were drawn from a different beta distribution for each plot. The generating pdf is given in red, and the red hash marks describe the actual data produced.
Simulated experiments
data in the first three plots was generated using truncated Gaussians and mixtures of truncated Gaussians. The data for the last plot was generated by Gaussian noise added to the uniform distribution on the circle. The model adapts easily to multimodal and patterned data samples. For all examples, the hyperparameters were fixed to (σ, α, s) = (.9, .1, 1.1). 0 ≤ i 1 , i 2 ≤ 5 for each example. .75, blue). As we can see, our method is valid for modeling densities without periodic tendencies, despite the specific form of the basis. Both plots are based on 10,000 thinned MCMC iterations, with hyperparameter settings (σ, α, s) = (.5, .5, .8) with I = 30. Figure 4 features Hutchings' bramble canes data (red) [38], [39], consisting of the locations of 823 bramble canes in a square plot. The left figure contains a heatmap of the pointwise posterior mean of the χ 2 -process density model, where black pertains to low density and white pertains to high density. Finally, a single contour (blue) at density level 0.3 divides the majority of points from areas of extremely low density. The hyperparameters were set to (σ, α, s) = (2, .01, 1.1) with 0 ≤ i 1 , i 2 ≤ 5, and the posterior sample featured 10,000 MCMC iterations. The right figure features 823 draws from the posterior predictive distribution of the χ 2 process density model. Each draw from the posterior predictive distribution was obtained by randomly selecting one posterior draw from the χ 2 process density model. Since this single posterior sample is itself a density function, one can then sample from its corresponding distribution using a rejection sampling scheme. There is a remarkable similarity between the posterior predictive sample (right, black) and the bramble canes data (left, red): despite a few differences, both low and high density regions are faithfully recovered.
Experiments with real-world data
Discussion
The Fisher geometry is central to many areas of classical and parametric statistics. On the other hand, nonparametric methods-both Frequentist and Bayesian-is a vital area of statistical research with many realizations and applications. We presented a nonparametric extension to the parametric Fisher geometry and showed that this generalization is consistent with its parametric predecessor. To do so, the set of probability density functions over a given domain was defined to be an infinite-dimensional smooth manifold where each point is itself a density function. This manifold becomes a Riemannian manifold when equipped with the nonparametric Fisher information metric and is then identified with the infinite-dimensional sphere, a well understood geometric object for which results are readily obtainable. Indeed, the benefits of shifting focus to the infinite-dimensional sphere do not stop at theory. Due to the relationship between the nonparametric Fisher geometry and the infinite sphere, it proves convenient to define nonparametric models directly on this sphere. We demonstrated one application of this approach in the form of Bayesian nonparametric density es-q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Pointwise posterior mean q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Posterior predictive sample q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Pointwise posterior mean q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q A Initializing the Markov chain: Newton's method on the sphere
Starting with a Riemannian manifold Q isometrically embedded in Euclidean space, we consider function F : Q → R.
Definition 2. Given point q 0 ∈ Q and initial velocityq 0 ∈ T q0 Q, we follow Edelman, Arias, and Smith [44] and define the Hessian of function F alongq 0 as the matrix satisfying
Hess F (q 0 ,q 0 ) = d 2 dt 2 t=0 F (q(t)) .
Proposition 1. Hess F on the sphere is given by
Hess F = F qq − F T q q 0 I ,(61)
where F q and F qq are the Jacobian and usual Hessian matrices.
Proof. We need the formula for the geodesic on the sphere given q 0 andq 0 . Letting α be the Euclidean norm ofq 0 , the geodesic is given by: q(t) = q 0 cos(αt) +q 0 α sin(αt) .
It is easy to verify thatq (t) = −α 2 q(t) .
Next the derivatives are given by:
d dt F (q(t)) = ∂F ∂q (y(t))q(t) ,(64)
and d 2 dt 2 F (q(t)) =q(t) T ∂ 2 F ∂q 2q (t) +
∂F ∂q
Tq (t) .(65)
Combining (63) with (65) gives:
d 2 dt 2 F (q(t)) =q(t) T ∂ 2 F ∂q 2q (t) − α 2 ∂F ∂q T q(t) (66) =q(t) T F qq − F T q q(t) I q(t) .
Evaluating at t = 0 gives the result.
Hess F is the Hessian matrix at point q 0 in directionq 0 . Newton's method on the sphere is achieved by Algorithm 1.
B Relationship to the Cox process
The χ 2 -process density prior may be used to model the intensity function of a Cox process [45]. The Cox process is a point process over a given domain such that each realization at point t is drawn from a Poisson distribution with intensity µ(s), where intensity function µ(·) is itself a random process over the same given domain. Cox processes are useful for the analysis of spatial and time series data. Given µ(·), the likelihood of such data {s n } N n=1 is given by
Bayesian inference on µ(·) requires the calculation of two integrals, that over the parameter space and that from Equation (67). We make the latter integral trivial by modeling the intensity function as the product of a density function and a positively constrained random variable:
µ(s) = M × p(s) = M × q(s) 2 .(68)
In this case, the likelihood may be written Since the likelihood factors in M and q(·), it follows that the two random variables will be independent in posterior distribution if they are specified to be independent in prior distribution. Indeed, M may even be given a conjugate prior: it is easy to see that M ∼ Γ(a, b) , implies M |N ∼ Γ(a + N, b + 1) .
Sampling from the joint posterior of µ(·) is as simple as independently sampling M from its posterior and q 2 (·) from the χ 2 -process density sampler and then multiplying the two together. Such a model should be used with care. As a function of the data, the posterior distribution of M solely depends on N , which is itself a single realization from a Poisson distribution. Thus, our χ 2 -process density prior-Cox process formulation is useful in situations where ample prior information on M is available.
possible, we introduce a different (but equivalent) definition of a geodesic based on the calculus of variations (in contrast to the null acceleration definition from Lemma 3). Assume that two points A and B are close together in a small open set of Riemannian manifold (M, g(·, ·)). Let Γ : [a, b] × (− , ) → M be a family of curves γ s : [a, b] → M satisfying γ s (a) = A and γ s (b) = B for all s ∈ (− , ). Then γ is a geodesic if it minimizes the energy functional
Figure 1
1depicts 1,000 data points (red hash marks) drawn from four different beta distributions (density red) along with 100 MCMC draws from the posterior distribution based on the χ 2 -process density model. From left to right and top to bottom, the beta distribution parameters are (1, 1), (5, 2), (.5, .5), and (2, 2). Note that while the individual posterior draws adhere closely to the sampled data, the variability in the posterior draws accounts for uncertainty and gives good coverage to the true density. The hyperparameter settings for the top-left plot is given by (σ, α, s) = (.5, 1, 1), and (σ, α, s) = (.5, .5, .8) is the hyperparameter setting for the rest. I = 30 for each example. 10,000 thinned MCMC iterations were used to make each figure.
Figure 2
2depicts 1,000 data points (red) drawn from four different distributions on the unit square along with the contours of the pointwise median of 1,000 posterior draws from the χ 2 -process density model.
Figure 3
3features the British coal mine disaster data set, in which the dates of 191 disasters are recorded between the years of 1851 and 1967. In both plots, the dates are given in red. Two comparisons are implied by the figure. The first is a comparison between the variability of 100 posterior draws based on 191 data points (left plot) with the variability in 100 posterior draws based on 1,000 data points, as inFigure 1. One sees much less variability in the latter. The other comparison is between the close fit exhibited in the posterior draws of the left plot compared to the smooth fit shown by the pointwise quantiles (median, black; .25, blue;
Figure 2 :
2The contours (black) of the posterior median from 1,000 draws of the χ 2 -process density sampler. Each posterior is conditioned on 1,000 data points (red).
Figure 4 :
4Hutchings' bramble canes data: the first figure depicts the 823 bramble canes (red), a heatmap of the pointwise posterior mean (black is low, white is high), and a single contour at density 0.3 (blue) including all but a few points. The second figure shows 823 draws from the χ 2 -process density posterior predictive distribution, obtained using a rejection sampling scheme.
Algorithm 1
1A single iteration of Newton's method on the sphere 1: Given point q on sphere:2: Calculate F q 3: Calculate Hess F = F qq − F T q q 0 I 4: Calculate W = (I − qq T ) Hess −1 F (I − qq T ) 5: V ← −W F q 6:Progress along geodesic (62) with initial velocity V for time 1. 7: q ← q(1)
From the definition, the nonparametric Fisher metric can take on infinite values. It is possible to avoid this by limiting the space of interest to strictly-positive density functions or by bounding the metric at an arbitrarily large value. It is also possible to modify the definition of the tangent space to enforce tangent functions to equal 0 when their respective densities do.
We note that one may conceivably place a prior on the truncation index I and thus avoid having to choose the number of eigenfunctions. This would provide for an interesting extension of the model presented here, but would necessitate new MCMC techniques that enable the change of model dimensionality (e.g. reversible jump MCMC[35]) while maintaining manifold constraints. Hence, we leave this for future work.
By Riemannian HMC, we mean Riemannian HMC where the Riemannian metric is the finite Fisher metric, as this is the most common usage. We note that it is theoretically possible to use other metrics[31],[37].
AcknowledgementAH is supported by NIH grant T32 AG000096. SL is supported by the DARPA funded program Enabling Quantification of Uncertainty in Physical Systems (EQUiPS), contract W911NF-15-2-0121. This work was partially supported by NIH grant R01-AI107034 and NSF grant DMS-1622490.Year Density values Year Density values timation. The resulting χ 2 -process density model is flexible and computationally efficient: it is amenable to HMC and, in comparison to the cubic scaling of GP competitors, scales linearly in the number of data points. Of course, there is nothing a priori restricting the prior to be Gaussian[40], and an important next step is placing a prior on the number of basis functions to use, as is done in[20]. Moreover, spherical HMC uses geodesics to propose new states on the sphere, and these geodesic flows are formally equivalent to those derived on the L 2 sphere. Thus, the empirical effectiveness of spherical HMC in this context suggests that the proposals somehow adapt to the nonparametric Fisher geometry, and we showed that these proposals minimize the nonparametric Fisher energy in the same way that Riemanian HMC minimizes the parametric Fisher energy. We hope the χ 2 -process density model will serve to motivate extensions of HMC to Hilbert manifolds, of which the L 2 sphere is one example. It is now known how to perform HMC on certain finite dimensional manifolds[22],[31],[37]as well as Hilbert spaces[41]. We hope that the model presented here will motivate the extension of HMC technology to a large class of Hilbert manifolds, including the infinite-dimensional sphere.The theoretical and methodological results presented in this paper are merely first steps in exploiting the simple geometry implied by the nonparametric Fisher metric. Whereas density estimation is perhaps the most obvious application, it is also one of the fundamental problems in statistics and thus has connections to many other areas of statistics and machine learning. On the other hand, methodologies such as functional regression and classification[42]can benefit from the use of random functions defined on the sphere, which objects we constructed and performed inference on. Additionally, the nonparametric methodology proposed in this paper was Bayesian, but the spherical representation of the nonparametric Fisher geometry has clear connections to Frequentist nonparametrics by way of the geometry of the bootstrap[43].
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| []
|
[
"ON DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP",
"ON DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP"
]
| [
"Samuel M Corson "
]
| []
| []
| We present several new theorems concerning the first fundamental group of a path connected metric space. Among the results proven are strengthenings of the main theorems of [Sh2] and [CoCo]. A compactness theorem for the fundamental group of a Peano continuum is given. A useful characterization for the shape kernel of a locally path connected space is presented, along with a very succinct proof of the fact that for such a space the Spanier and shape kernel subgroups coincide (see [BF]). We also show that a free decomposition of the fundamental group of a locally path connected Polish space cannot be nonconstructive. Numerous other results and examples illustrating the sharpness of our theorems are provided. | 10.1016/j.topol.2019.03.007 | [
"https://arxiv.org/pdf/1610.00145v2.pdf"
]
| 119,690,727 | 1610.00145 | 7da31c532ac3bd55e0f1be27b511967e65e406c8 |
ON DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP
13 Sep 2017
Samuel M Corson
ON DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP
13 Sep 2017
We present several new theorems concerning the first fundamental group of a path connected metric space. Among the results proven are strengthenings of the main theorems of [Sh2] and [CoCo]. A compactness theorem for the fundamental group of a Peano continuum is given. A useful characterization for the shape kernel of a locally path connected space is presented, along with a very succinct proof of the fact that for such a space the Spanier and shape kernel subgroups coincide (see [BF]). We also show that a free decomposition of the fundamental group of a locally path connected Polish space cannot be nonconstructive. Numerous other results and examples illustrating the sharpness of our theorems are provided.
Introduction
One way to understand a path connected topological space is to analyze its fundamental group. Fundamental groups are a homotopy invariant and provide a useful tool for distinguishing homotopy equivalence classes. In understanding the fundamental group it is useful to study subgroups that are definable in terms of topology or logic or some combination of the two. The focus of this paper is the study of the first fundamental group of metric spaces, and certain of its subgroups. Assumptions about separability and generalizations of separability, local path connectedness, and compactness will figure prominently in our study. Throughout this paper we take the simplifying assumption that all spaces for which a fundamental group is computed are path connected.
In Section 2 we give preliminary definitions and then define the characterization of subgroups of the fundamental group via the topology of the loop space. For G ≤ π 1 (X, x) we study the relationship of G to π 1 (X, x) by looking at how ⋃ G sits in the loop space L x . This very simple idea yields a diversity of theorems. The theory of open and closed subgroups which is developed in this section gives proofs of many results later in the paper (e.g. Theorems 5.1, 5.4, 6.5; Corollary 6.3).
In Section 3 we introduce concepts related to separable completely metrizable spaces-Polish spaces. Some terminology and tools of descriptive set theory are introduced, including the class of analytic (denoted Σ 1 1 ) sets and generalizations thereof (what we call nice classes of sets). The class of subgroups of type Σ 1 Theorem A. Suppose X is a locally path connected, connected Polish space. The following groups are of cardinality 2 ℵ0 or ≤ ℵ 0 , and in case X is compact they are of cardinality 2 ℵ0 or are finitely generated:
(1) π 1 (X) (2) π 1 (X) (π 1 (X)) (α) for any α < ω 1 (derived series) (3) π 1 (X) (π 1 (X)) n for any n ∈ ω (lower central series) (4) π 1 (X) N where N is the normal subgroup generated by squares of elements, cubes of elements, or n-th powers of elements.
In case X is compact then countability of the fundamental group is equivalent to being finitely presented.
The compact case of part (1) is the main result of [Sh2] and part (3) with n = 1 is the main result of [CoCo]. In Section 3 we also give a compactness-type theorem for Peano continua (the reader may subsitute Σ 1 1 for P): Theorem B. If X is a Peano continuum there does not exist a strictly increasing infinite sequence of P normal subgroups {G n } n∈ω of π 1 (X) such that ⋃ n∈ω G n = π 1 (X).
In Section 4 we present an application of Theorem B. We define a comonster group to be a group which is not the normal subgroup closure of any finite subset and a κ-comonster group is not the normal subgroup closure of any set of cardinality < κ. Theorem B implies that if the fundamental group of a Peano continuum is co-monster then it is ℵ 1 -comonster (Theorem 4.2). Examples are presented of this phenomenon and a curious tie to finitely presented groups is drawn (Theorem 4.4).
In Section 5 we compute the complexity of some commonly used subgroups of the fundamental group. The shape kernel is shown to be closed, and if the space is locally path connected the shape kernel is the intersection of all clopen subgroups (Theorem 5.1). The Spanier subgroup is shown to be equal to the shape kernel in case the space is locally path connected, and is shown to be Σ 1 1 in case the space is compact (Theorem 5.4). That the Spanier and shape kernel subgroups coincide for locally path connected paracompact Hausdorff spaces is a recent theorem of Brazas and Fabel [BF]. Though our theorem is slightly less general, the proof is rather shorter than that of [BF].
In Section 6 we introduce n-slender groups (see [E1]), give such groups an alternative characterization, and present some theorems using the theory of open and closed subgroups. Among the results of the section is the fact that a locally path connected separable metric space cannot have fundamental group that is an uncountable free product of nontrivial groups (Corollary 6.8). We also prove the following:
Theorem C. Suppose X is locally path connected Polish and π 1 (X) ≃ * i∈I G i with each G i nontrivial. The following hold:
(1) card(I) ≤ ℵ 0 (2) Each retraction map r j ∶ * i∈I G i → G j has analytic kernel.
(3) Each G j is of cardinality ≤ ℵ 0 or 2 ℵ0 . (4) The map * i∈I G i → ⊕ i∈I G i has analytic kernel.
This theorem can be interpreted to mean that no free decomposition of a fundamental group as in the hypotheses can be non-constructive. This is rather surprising in light of the fact that a direct sum decomposition of the fundamental group can be non-constructive (see discussion in Section 6).
In Section 7 we give a brief discussion of what are called nice pointclasses (these agree with the P that is found in some of the theorems stated so far). We discuss which pointclasses can be assumed to be nice (consistent with the standard axioms of set theory), and hence to which subgroups we can consistently apply the theorems of Sections 3 and 4.
Our discussions will not avoid references to abstract combinatorial set theory. This should not be surprising as topology is 'visual set theory.' Also, many facets of descriptive set theory are directly influenced by the model of set theory in which one is working. We will keep our references to set theory simple, doing little beyond illustrating the sharpness of our results until the discussion in Section 7.
Let ZFC denote the Zermelo-Fraenkel axioms of set theory including the axiom of choice. We assume ZFC throughout this paper, and also assume that ZFC is consistent so as to avoid repetition of the phase "if ZFC is consistent then there exists a model . . ." The parameter κ will be used for infinite cardinals, κ + denotes the successor cardinal. Let CH denote the continuum hypothesis: ℵ + 0 = 2 ℵ0 , and GCH denote the generalized continuum hypothesis: (∀κ)[κ + = 2 κ ].
Preliminaries
In this section we present some of the basic definitions and notation for fundamental groups in this paper. We then give some lemmas about open and closed subgroups of the fundamental group which will be used throughout.
Given a topological space X and distinguished point x ∈ X we obtain the fundamental group π 1 (X, x) as follows. . This definition also works as a partial binary operation on paths, defined whenever the first path ends where the second path begins. For specificity, we mean l 0 * (l 1 * (⋯ * (l n−1 * l n )⋯) when we write l 0 * l 1 * ⋯ * l n . There is also a unary operation −1 given by l −1 (s) = l(1−s). The fundamental group is the set L x modulo homotopy, the binary operation is given by [l 0 ] * [l 1 ] = [l 0 * l 1 ], the equivalence class of the constant loop is the identity and inverses are given by [l] −1 = [l −1 ]. Clearly the fundamental group π 1 (X, x) is the same as the fundamental group of π 1 (C, x) where C is the path component of x.
We shall only consider fundamental groups of spaces which are path connected.
We assume some familiarity with notions in topology such as metrizability and separability. Let Z be a topological space. A pointclass is a collection P of subsets of Z that are of a particular topological description, usually in terms of countable unions, countable intersections, complements, or projections. For example, the collection of open subsets (topology) of Z, the collection of closed sets of Z, and the collection of countable unions of closed sets of Z are all pointclasses of Z. Another example is the class of Borel subsets of Z. When we restrict our attention to specific types of topological spaces, we get more information about sets in pointclasses.
Take (X, d) to be a metric space with distinguished point x ∈ X. Topologize L x by the sup metric: the distance between loops l 0 and l 1 is sup s∈[0,1] d(l 0 (s), l 1 (s)). Since uniform convergence is equivalent to convergence in the compact-open topology, we may suppress the particular metric d on the space X (since any other compatible metric gives the same topology on L x ).
Definition 2.1. A subgroup G ≤ π 1 (X, x) is of pointclass P if the collection of loops belonging to elements of G is in the pointclass P in L x . In other words,
G ≤ π 1 (X, x) is of pointclass P if ⋃ G is in pointclass P in L x .
We establish some lemmas. Lemmas 2.2, 2.3, and 2.4 should remind the reader of the analogous facts for topological groups.
Lemma 2.2. If G ≤ π 1 (X, x) is open and G ≤ H ≤ π 1 (X, x) then H is open. Proof. Let G be open and l ∈ ⋃ H with {l n } n∈ω a sequence in L x converging to l. Since l * l −1 ∈ ⋃ G there exists ǫ > 0 such that B(l * l −1 , ǫ) ⊆ ⋃ G. The sequence {l * l −1 n } n∈ω is eventually in B(l * l −1 , ǫ), so that {l * l −1 n } n∈ω is eventually in ⋃ G ⊂ ⋃ H, so {l −1 n } n∈ω is eventually in ⋃ H, so {l n } n∈ω is eventually in ⋃ H. Lemma 2.3. If P is closed under continuous preimages and H ≤ π 1 (X, x) is P then: (1) The equivalence relations E, R ⊆ L x × L x defined by l 0 El 1 iff [l 0 ]H = [l 1 ]H and l 0 Rl 1 iff H[l 0 ] = H[l 1 ] are P.
(2) Each equivalence class in E and R is P. By [l]H we mean the set of all loops based at x which are homotopic to a loop of the form l * l ′ where l ′ ∈ ⋃ H and the definition for H[l] is analogous.
Proof. The function L x × L x → L x given by (l 0 , l 1 ) ↦ (l 0 ) −1 * l 1 is continuous and E is the preimage of ⋃ H under this function, so by assumption we have E is P. The proof that R is P is similar. This proves (1). For (2) we notice that for a fixed l 0 ∈ L x the function L x → L x given by l ↦ (l 0 ) −1 * l is continuous and the set [l 0 ]H is the continuous preimage of ⋃ H.
Lemma 2.4. If H ≤ π 1 (X, x) is open then H is also closed.
Proof.
Supposing H is open we have by Lemma 2.3 that the set ⋃ l∉⋃ H [l]H is a union of open sets in L x , and this is precisely L x ∖ (⋃ H).
We notice that change of basepoint isomorphisms take open (resp. closed) subgroups to open (resp. closed) subgroups, as seen in the following lemma.
Lemma 2.5. Let x, y ∈ X and ρ a path from y to x. Let φ ∶ L x → L y be the map such that φ(l) = ρ * l * ρ −1 and ψ ∶ L y → L x be given by ρ −1 * l * ρ. Then
(1) φ and ψ are isometric embeddings and induce isomorphisms φ ∶ π 1 (X, x) → π 1 (X, y), and ψ ∶ π 1 (X, y) → π 1 (X, x).
(2) G ≤ π 1 (X, x) is open (resp. closed) iff φ(G) is. (3) G ≤ π 1 (X, x) is open (resp. closed) iff every conjugate of G is.
Proof. The first part of (1) is clear, and the second is a standard exercise. For (2) suppose G is not open. Let l ∈ L x with [l] ∈ G and {l n } n∈ω be a sequence of loops such that l n → l and [l n ] ∉ G. Then ρ * l n * ρ −1 → ρ * l * ρ −1 and [ρ * l n * ρ −1 ] ∉ φ(G), so φ (G) is not open. If φ (G) is not open then by the proof for the other direction we have that ψφ(G) = G is not open.
Suppose that G is not closed and let l ∈ L x be such that [l] ∉ G and there exists a sequence {l n } n∈ω such that [l n ] ∈ G and l n → l. Then ρ * l n * ρ −1 → ρ * l * ρ −1 and [ρ * l n * ρ −1 ] ∈ φ (G) and [ρ * l * ρ −1 ] ∉ φ (G). Again, for the other direction we consider the application of the map ψ.
The last claim is proved by letting ρ be a loop from x to itself and applying (2).
By Lemma 2.5 we may consider open or closed normal subgroups as base point free.
Lemma 2.6. If G ⊴ π 1 (X) is open there exists an open cover U of X such that any loop contained entirely in an element of U is in ⋃ G.
Proof. For each point x ∈ X we have G ⊴ π 1 (X, x) is open, and the constant loop c at x is in ⋃ G, so we may pick ǫ x > 0 such that B(c, ǫ x ) ⊆ ⋃ G. Selecting the ǫ x neighborhood B(x, ǫ x ) around x gives the desired open cover U = {B(x, ǫ x )} x∈X .
The converse to the above lemma is not true in general. The space F in the next example will reappear in later examples in this paper.
Example 1. Let F = ⋃ y∈K C((0, y), y) ⊆ R 2 where K is a homeomorph of the Cantor set that lies in the interval [1, 2] and C(p, r) denotes the circle centered at point p of radius r. The space F can be considered a wedge of 2 ℵ0 many circles of diameter ≥ 1 whose antipodes from the wedge point (0, 0) correspond to the elements of a Cantor set. This space is compact and the fundamental group is easily seen to be isomorphic to the free group of rank continuum F (2 ℵ0 ) (with a free generating set corresponding to a set of loops that go exactly once around one of the circles C((0, y), y) ). Let P be any pointset defined on metric spaces which is closed under taking continuous preimages. Define a map f ∶ K → L (0,0) by letting f (y)(t) = (y sin(2πt), y − y cos(2πt)). It is clear for y 0 , y 1 ∈ K that d(f (y 0 ), f (y 1 )) = 2d(y 0 , y 1 ) since d(f (y 0 )(s), f (y 1 )(s)) is maximized precisely at s = 1 2 and d(f (y 0 )( 1 2 ), f (y 1 )( 1 2 )) = 2d(y 0 , y 1 ). Then f is an embedding of K. The image f (K) gives a set of loops which freely generate the fundamental group. If G ≤ π 1 (F, (0, 0)) is of pointclass P then f −1 (⋃ G) is as well.
For any ∅ ≠ S ⊆ K we have a subgroup:
ι * (π 1 ( ⋃ y∈S C((0, y), y), (0, 0))) ≤ π 1 (F, (0, 0))
freely generated by the loops in f (S). The normal closure
G = ⟨⟨ι * (π 1 ( ⋃ y∈S C((0, y), y), (0, 0)))⟩⟩ ≤ π 1 (F, (0, 0))
does not contain any elements of form [f (y)] where y ∈ K ∖ S since G is the kernel of the retraction map from π 1 (F, (0, 0)) to the free subgroup ι * (π 1 ( ⋃ y∈K∖S C((0, y), y), (0, 0))) ≤ π 1 (F, (0, 0)) Any loop in F contained in an open ball of radius 1 2 is nulhomotopic, so there exists an open cover U satisfying the conclusion of Lemma 2.6 for any normal subgroup G ⊴ π 1 (F, (0, 0)). Not every subgroup is open, however, by letting S ⊆ K be not open and noticing that S = f −1 (⋃⟨⟨ι * (π 1 (⋃ y∈S C((0, y), y), (0, 0)))⟩⟩) is not open.
We present a partial converse to Lemma 2.6.
Definition 2.7. A topological space Z is locally path connected if for every z ∈ Z and neighborhood U of z there exists a neighborhood V ⊆ U of z such that V is path connected.
Lemma 2.8. Let X be locally path connected and G ⊴ π 1 (X). If there exists an open cover U of X such that any loop contained entirely in an element of U is in G then G is open.
Proof. Assume the hypotheses and fix x ∈ X. Let l ∈ ⋃ G ⊆ L x . Cover the image of l with a finite subcollection {U 0 , . . . , U m } ⊆ U, so that the images of each inclusion
ι * ∶ π 1 (U i ) → π 1 (X) are in G. Let δ > 0≤ n ≤ N − 1 we have that l([ n N , n+1 N ])
is contained inside some V jn . Now assuming l ′ ∈ L x is less than distance ǫ from l we have that d(l ′ (s), l(s)) < ǫ for all s ∈ [0, 1]. For each 1 ≤ n ≤ N − 1 let p n be a path in V jn from l( n N ) to l ′ ( n N ) and let p 0 and p N be the constant path at x. Notice that the
loop l [ n N , n+1 N ] * p n+1 * (l ′ [ n N , n+1 N ]) −1 * p −1
n is contained in one of the U i , and so is a representative of an element of G based potentially at a different point. Then
l −1 * l ′ is an element of ⋃ G, so l ′ ∈ ⋃ G. Thus G is open.
For the next proposition we recall the following definition.
Definition 2.9. A topological space Z is semi-locally simply connected if for every z ∈ Z there exists a neighborhood U of z such that the map induced by inclusion ι * ∶ π 1 (U, z) → π 1 (Z, z) is the trivial map. For a locally path connected space we may obviously select U to be path connected.
Proposition 2.10. Let X be locally path connected in addition to being metrizable. The following are equivalent:
(1) The trivial subgroup of π 1 (X) is open.
(2) All subgroups of π 1 (X, x) are open.
(3) X is semi-locally simply connected.
Proof. The implication (1) ⇒ (2) follows from Lemma 2.2. For (2) ⇒ (3) we let x ∈ X be given along with a neighborhood U of x. Since in particular the trivial subgroup of π 1 (X, x) is open and the constant map c ∶ [0, 1] → {x} is trivial, we may select ǫ > 0 such that B(c, ǫ) ⊆ ⋃[c] ⊆ L x , where without loss of generality B(x, ǫ) ⊆ U . Now any loop with image in B(x, ǫ) must be in B(c, ǫ) and therefore nulhomotopic in X.
For (3) ⇒ (1) we let U be an open cover of X by path connected open sets U whose inclusion maps induce a trivial map π 1 (U ) → π 1 (X). Then we are in the situation of Lemma 2.8 and we see that the trivial subgroup is open, so we are done.
Polish Spaces
We present some material which will be specific to dealing with fundamental groups of Polish spaces. We give some technical lemmas which will establish closure properties for subgroups of particularly nice types of pointclasses (stated in Theorem 3.10). These will give a sense of the versatility of such subgroups. We will then prove a couple of the main results of the paper. Recall the following:
Definition 3.1. A topological space Z is Polish if it is completely metrizable and separable.
Many commonly used spaces such as the real line R, compact metric spaces, and countable discrete spaces are Polish. Polish spaces are closed under countable disjoint union and countable products. When X is path connected and Polish the space L x is also Polish. The space H x of homotopies of loops at x, topologized by the sup metric, is Polish assuming X is path connected Polish. The following lemma provides a sense of base point independence as in Lemma 2.5.
Lemma 3.2. Suppose the pointclass P contains the closed sets and is closed under continuous images between Polish spaces, finite products, and finite intersections. Let X be Polish and ρ be a path from x to y in X. Letting φ be the map defined in Lemma 2.5, a subgroup G ≤ π 1 (X, x) is of type P if and only if φ(G) is.
Proof. Assume the hypotheses. We prove the forward direction of the biconditional and the other direction follows similarly. Let
G ≤ π 1 (X, x) be of type P. Let D ⊆ L x × H x × L x be defined by D = {(l 0 , H, l 1 ) ∶ H is a homotopy from l 0 to l 1 }.
It is easy to see that D is closed. Since the map l ↦ ρ −1 * l * ρ is an isometric embedding from L x to L y we have that ρ −1 * G * ρ is in pointclass P in L y by assumption. Then (ρ −1 * G * ρ) × H y × L y is in pointclass P in L y × H y × L y by hypothesis. Then D ∩ (ρ −1 * G * ρ) × H y × L y is in pointclass P. Letting p 3 ∶ L x × H x × L x → L x be projection to the third coordinate (obviously a continuous map), we have that ⋃ φ(G) = p 3 (D ∩ (ρ −1 * G * ρ) × H y × L y ) is in the pointclass P.
For K ⊆ L x let [K] ⊆ π 1 (X, x) denote the subset of equivalence classes of loops which have representatives in K.
Lemma 3.3. Let P and X satisfy the hypotheses of Lemma 3.2. If K ⊆ L x is P then the set ⋃[K] ⊆ L x is P.
Proof. Letting D = {(l 0 , H, l 1 ) ∶ H homotopes l 0 to l 1 } ⊆ L x × H x × L x we have that D is closed and therefore P. The set K is P and therefore so is K × H x × L x . Then (K × H x × L x ) ∩ D is P, and letting p 3 be projection in the third coordinate we have p 3 ((K × H x × L x ) ∩ D) = ⋃[K] is P.
Lemma 3.4. Let P and X satisfy the hypotheses of Lemma 3.2. Assume further that P is closed under countable unions. If K ⊆ L x is P then ⟨[K]⟩ is a P subgroup of π 1 (X, x).
Proof. Notice that the inversion map l ↦ l −1 is an isometry and therefore continuous. Thus K −1 is P, and K ∪ K −1 is also P. For each n ∈ ω let m n ∶ ∏ n−1 i=0 L x → L x be given by (l 0 , . . . , l n−1 ) ↦ l 0 * l 1 * ⋯ * l n−1 . This is clearly a continuous map. Each m n (∏ n−1 i=0 (K ∪ K −1 )) is of type P. Thus ⋃ ∞ n=0 m n (∏ n−1 i=0 (K ∪ K −1 )) is P. By Lemma 3.3 we have that ⋃[⋃ ∞ n=0 m n (∏ n−1 i=0 (K ∪ K −1 ))]is P. We are done since
⋃⟨[K]⟩ = ⋃[⋃ ∞ n=0 m n (∏ n−1 i=0 (K ∪ K −1 ))].
Lemma 3.5. Let P and X satisfy the hypotheses of Lemma 3.4. If K ⊆ L x is P then the normal closure ⟨⟨[K]⟩⟩ is P.
Proof. Let c ∶ L x × L x → L x be given by (l 0 , l 1 ) ↦ l 0 * l 1 * l −1 0 . This is easily continuous. We have L x ×K is P, and so is c(L x ×K).
Then ⟨⟨[K]⟩⟩ = ⟨[c(L x ×K)]⟩ is P by Lemma 3.4 .
The preceeding lemmas motivate the following:
Definition 3.6. A pointclass P defined on Polish spaces is nice if it contains the closed sets, is closed under continuous images and preimages, countable intersections and finite unions.
Remark 3.7. A nice pointclass is also closed under countable products, for if A n ⊆ Z n is of nice pointclass P for each n ∈ ω then ∏ n∈ω A n = ⋂ n∈ω p −1 n (A n ) is P in the Polish space ∏ n∈ω Z n . A nice pointclass is also closed under countable unions, for suppose A n ⊆ Z are P for each n ∈ ω. If ⋃ n∈ω A n = ∅ then as ∅ is closed we have ⋃ n∈ω A n is P. Otherwise pick z ∈ ⋃ n∈ω A n . Since {z} is closed in Z and P is closed under finite unions we can assume z ∈ A n for all n ∈ ω. Let ⊔ n∈ω Z be the disjoint union of countably many copies of Z. Let f ∶ ⊔ n∈ω Z → ∏ n∈ω Z take y n to (z, z, . . . , z, y, z, z . . .) (here y is in the nth coordinate) where y n is a copy of y in the nth copy of Z in the disjoint union. The map f is continuous by the universal and co-universal properties of product and disjoint unions, respectively. The set ∏ n∈ω A n is P as we have seen. Letting g ∶ ⊔ n∈ω Z → Z map each copy of Z via identity we get that g(f −1 (∏ n∈ω A n )) = ⋃ n∈ω A n is P.
Under set inclusion the smallest nice Polish pointclass is that of the analytic sets (denoted Σ 1 1 ). If Z is Polish we say Y ⊆ Z is analytic if there exists a Polish space W and a continuous map f ∶ W → Z such that f (W ) = Y . All Borel sets of a Polish space are analytic (see [Ke]).
Lemma 3.8. If X = ∏ n∈ω X n where each X n is metrizable, then the loop space of X is homeomorphic to the product of the loop spaces of the spaces X n and can be metrized thereby.
Proof. By applying a cutoff metric d n to each space X n we may assume diam(X n ) ≤ 2 −n . The metric d({s n } n∈ω , {t n } n∈ω ) = ∑ ∞ n=0 d n (s n , t n ) is compatible with the product topology on ∏ n X n . Fix a point x n in each X n and let x = {x n } n∈ω ∈ ∏ n X n . The metric d induces the sup metric on the loop space L x so that L x is homeomorphic with the space ∏ n L xn where the distance between loops {l n } n∈ω and {l ′ n } ω is ∑ n sup s∈[0,1] d n (l n (s), l ′ n (s)). This follows from the fact that uniform convergence of a sequence of loops in L x occurs precisely when the loops in each coordinate converge uniformly. Thus we may metrize L x with the metric defined by the metric on the product ∏ n L xn .
We cover some functoriality properties. Recall that if (X, x) and (Y, y) are two pointed spaces and f ∶ (
X, x) → (Y, y) is a continuous function there is an induced homomorphism f * ∶ π 1 (X, x) → π 1 (Y, y) defined by f * ([l]) = [f ○ l]
. The map f also induces a continuous map f ∶ L x → L y given by l ↦ f ○ l. We also recall that the wedge (X, x) ∨ (Y, y) is the topological space obtained by identifying the distinguished points, which has distinguished point corresponding to the identified points which we denote x ∨ y. There are obvious inclusion maps from the spaces (X, x) and (Y, y) to the wedge as well as retraction maps from the wedge to the two spaces. If X and Y are metrizable, competely metrizable, or separable then so is the wedge.
Proposition 3.9. Assume X and Y are metric spaces. The following closure properties hold:
(
1) If f ∶ (X, x) → (Y, y) is continuous, P is a pointclass closed under continuous preimages and G ≤ π 1 (X, x) is P, then (f * ) −1 (G) is also P. (2) If G 0 ≤ π 1 (X, x) and G 1 ≤ π 1 (Y, y) are both of pointclass P and P is closed under products, then G 0 × G 1 ≤ π 1 (X × Y, (x, y)) ≃ π 1 (X, x) × π 1 (Y, y) is P. (3) If f ∶ (X, x) → (Y, y)
is continuous between Polish spaces and P is nice and
G ≤ π 1 (X, x) is P then f * (G) is P. (4) If G 0 ≤ π 1 (X, x) and G 0 ≤ π 1 (Y, y) are P, with X and Y Polish and P
nice, then the subgroup generated by the images of G 0 and G 1 under the inclusion maps is P in (2) follows from Lemma 3.8, and applies to countable products if P is closed under countable products. For (3)
π 1 ((X, x) ∨ (Y, y), x ∨ y). Proof. For part (1) we notice that ⋃(f * ) −1 (G) = f −1 (⋃ G). Claimthe map f induces the continuous map f from L x to L y by composition. The image of ⋃ G under this map is P because P is nice, and ⋃ f * (G) = [f (⋃ G)]. Claim (4) follows immediately, since ⋃⟨ιX * (G0) ∪ ι Y * (G 1 )⟩ = ⋃⟨[ιX (⋃ G 0 ) ∪ ι Y (⋃ G 1 )]⟩ is evidently P.
The following theorem gives a catalogue of closure properties for nice subgroups. Recall that the derived series is defined by letting
G (0) = G, G (α+1) = [G (α) , G (α) ] and G (β) = ⋂ α<β G (α) if β is a limit ordinal. The lower central series is defined by letting G 0 = G and G n+1 = [G, G n ].
Theorem 3.10. Let f ∶ (X, x) → (Y, y) be a continuous function between Polish spaces and let P be a nice pointclass. The following hold:
(
1) If H ≤ π 1 (Y, y) is P then f −1 * (H) ≤ π 1 (X, x) is P. (2) If G ≤ π 1 (X, x) is P then f * (G) ≤ π 1 (Y, y) is P.
(3) The subgroups 1 and π 1 (X, x) are analytic in π 1 (X, x).
(4) If G n ≤ π 1 (X, x) are P then so are ⋂ n∈ω G n and ⟨ ⋃ n∈ω G n ⟩.
(5) Countable subgroups of π 1 (X, x) are analytic.
(6) If G ≤ π 1 (X, x) is P then so is ⟨⟨G⟩⟩. (7) If G ≤ π 1 (X, x) is P then so is any conjugate of G. (8) If w(x 0 , . . . , x k ) is a reduced word in the free group F (x 0 , . . . , x k ) and the groups G 0 , . . . , G k ≤ π 1 (X, x) are P then so is the subgroup ⟨{w(g 0 , g 1 , . . . , g k )} gi∈Gi ⟩. (9) If G, H ≤ π 1 (X, x) are P then so is the subgroup [G, H]. (10) If G ≤ π 1 (X, x) is P then each countable index term of the derived series G (α)
and each term of the lower central series G n is P.
Proof. Claim (1) follows from (1) in Proposition 3.9. Claim (2) is claim (3) in Proposition 3.9. For (3) we have that π 1 (X, x) is a closed subgroup and 1 is the subgroup generated by the constant map to x, and so is analytic by Lemma 3.4 (since a singleton is closed in L x ). Claim (4) follows from the definition of nice pointclasses and Lemma 3.4. Claim (5) follows from the fact that singletons are closed in L x and claim (4). Claim (6) is an instance of Lemma 3.5. Claim (7) is an instance of Lemma 3.2. For claim (8) we notice that the map w ∶ ∏ k i=0 L x → L x given by (l 0 , . . . , l k ) ↦ w(l 0 , . . . , l k ) is continuous, and so {w(l 0 , . . . , l k )} li∈⋃ Gi is a P subset in L x and the claim follows from Lemma 3.4. Claim (9) is an instance of claim (8). For claim (10) we iterate claim (9), applying claim (4) at limit ordinals.
We recall some definitions. If Z is a topological space we say that Y ⊆ Z is nowhere dense if the closure Y ⊆ Z has empty interior, Y is meager if it is a union of countably many nowhere dense sets in Z, Y has the property of Baire
(abbreviated BP) if there exists an open set O ⊆ Z such that Y ∆O = (Y ∖O)∪(O∖Y ) is meager, and Y is comeager if Z ∖ Y is meager.
We say a pointclass P on Polish spaces has the property of Baire if each set in P has the property of Baire. For example, the pointclass of open sets obviously has BP. In fact, the class of analytic sets also has BP (see [Ke]).
The following was proven in [P] using a result from [My].
Lemma. Suppose ≈ is an equivalence relation on the Cantor set {0, 1} ω such that if α and β differ at exactly one coordinate then α ≈ β fails. If ≈ has BP as a subset of {0, 1} ω × {0, 1} ω , then ≈ has 2 ℵ0 equivalence classes.
Lemma 3.11. Let X be Polish. Suppose that G ⊴ K ≤ π 1 (X, x) with G of pointclass P and that K is closed. Suppose also that P has BP and is closed under continuous preimages in Polish spaces, and that there exist arbitrarily small loops at x which are in ⋃ K and not in ⋃ G. Then card(K G) = 2 ℵ0 .
Proof. Assume the hypotheses and let {l n } n∈ω be a sequence of loops at x in ⋃(K ∖ G) such that the diameter of l n is ≤ 2 −n . Let l 0 n be the constant loop at x and let l 1 n be the loop l n . Given an element α ∈ {0, 1} ω we define l α to be the loop l α(0) 0 * (l α(1) 1 * (l α(2) 2 * (⋯))) (which must also be in ⋃ K as K is closed). In other words, l α restricted to the interval [0, 1 2 ] is either the constant loop or l 0 in case α(0) is 0 or 1 respectively, l α restricted to the interval [ 1 2 , 3 4 ] is either the constant loop or l 1 in case α(1) is 0 or 1 respectively, etc. The function from the Cantor set {0, 1} ω to L x given by α ↦ l α is clearly continuous. For l,
l ′ ∈ L x letting l ∼ l ′ if and only if [l]G = [l ′ ]G, we have by Lemma 2.3 that ∼⊆ L x × L x is of pointclass P.
Defining an equivalence relation ≈ on {0, 1} ω so that α ≈ β if and only if l α ∼ l β , we see that ≈⊆ {0, 1} ω × {0, 1} ω is of pointclass P as a continuous preimage. As P has BP we know that ≈ has BP. By Lemma 3 we shall be done if we show that if α and β differ at exactly one point then α ≈ β fails. Suppose that α(n) ≠ β(n) and that α(m) = β(m) whenever m ≠ n and that l α ≈ l β . Letting without loss of generality α(n) = 1 and β(n) = 0 we see that
[(l β ) −1 * l α ] ∈ G. Let h = l α(n+1) n+1 * (l α(n+2) n+2 * (⋯)) and g = l α(0) 0 * (l α(1) 1 * (⋯l α(n−1) n−1 )⋯). Then [(l β ) −1 * l α ] = [h −1 * g −1 * g * l n * h] = [h −1 * l n * h] ∈ G, so by normality of G in K we have [l n ] ∈ G, a contradiction.
Thus there are at least 2 ℵ0 many elements in K G by the above lemma, and there are at most 2 ℵ0 elements because there are at most 2 ℵ0 loops at x.
Lemma 3.12. If X is metric, locally path connected and G ⊴ π 1 (X) then either G is open or there exists y ∈ X and a sequence of loops {l n } n∈ω at y with diam(l n ) ↘ 0 and [l n ] ∉ G.
Proof. If G is not open we have by the contrapositive of Lemma 2.8 that there must exist some point y ∈ X such that for any open neighborhood U of y there is a loop in U which is not in G. We get a sequence of loops {l n } n∈ω with diam(l n ([0, 1])∪{y}) ≤ 2 −n and [l n ] ∉ G. By local path connectedness we may take a subsequence of the l n whose base point is close enough to y and join the basepoint to y via a small path, so that the loops may eventually be assumed to have been based at y. Taking a subsequence having all loops based at y gives the desired result Through the remainder of Section 3 we shall assume P is a nice pointclass with BP. Thus for example one can take P = Σ 1 1 . The discussion of pointclasses will take place in Section 7.
Theorem 3.13. Suppose X is locally path connected Polish. If G ⊴ π 1 (X) is P then card(π 1 (X) G) is either ≤ ℵ 0 (in case G is open) or 2 ℵ0 (in case G is not open).
Proof. If G is open then the collection of left cosets {[l]G} l∈Lx is a covering of L x by pairwise disjoint open sets, and since L x is separable we know that the collection {[l]G} l∈Lx is countable. Else, by Lemma 3.12 we get a point y ∈ X and a sequence of loops {l n } n∈ω with diam(l n ) ↘ 0 and [l n ] ∉ G. Considering G as a subgroup of π 1 (X, y) we see that G is P since P is nice, and thus we have satisfied the hypotheses of Lemma 3.11 and we are done.
The above may be strengthened if X is also compact. Recall that a Peano continuum is a path connected, locally path connected compact metrizable space.
Theorem 3.14. If X is a Peano continuum and
G ⊴ π 1 (X, x) is P then π 1 (X, x) G is either finitely generated (in case G is open) or of cardinality 2 ℵ0 (in case G is not open).
Proof. By Theorem 3.13 we need only show that π 1 (X) G is finitely generated if G is open. For this we will use a theorem from [CC] which will require a definition. Let φ ∶ π 1 (X) → H be a group homomorphism. We say an open cover U is 2-set simple rel φ if each element of U is path connected and any loop in the union of two elements of U is in the kernel of φ. This property of a cover implies that for any nerve associated with with U there is a homomorphism from the fundamental group of the nerve with the same image as φ. The following is a part of Theorem 7.3 in [CC]:
Theorem. Let X be path connected, φ ∶ π 1 (X) → H a homomorphism and U a 2-set simple cover rel φ. If U is finite then φ(π 1 (X)) is finitely generated. Now, assuming G is open we get by Lemma 2.6 an open cover U 1 for X such that any loop contained in an element of U 1 is in G. Let ǫ > 0 be a Lebesgue number for the cover U 1 and let U 2 be a cover of X by open balls of redius ǫ 4 . By local path connectedness let U be an open cover of X by path connected sets, each of which is contained in an element of U 2 . By compactness we may pick U to be finite, and it is clear that U is 2-set simple rel the quotient projection π 1 (X) → π 1 (X) G. We are done by the theorem of Cannon and Conner that is quoted above.
The conditions on G and the pointclass P cannot be removed in Theorems 3.13 and 3.14 as evidenced by the following example.
Example 2. Let P denote the projective plane and P ω = ∏ ω P . Let x ∈ P and let x ∈ P ω be the point whose every coordinate is x. By the functoriality of the fundamental group there is a natural isomorphism π 1 (P ω , x) ≃ ∏ ω π 1 (P, x). By Lemma 3.8 the loop space L x is homeomorphic to the product ∏ ω L x where the loop space L x at the nth coordinate has diameter ≤ 2 −n . By the coordinatewise isomorphism ∏ ω π 1 (P, x) ≃ ∏ ω Z 2Z we may regard elements of π 1 (P ω , x) as ω sequences of 0s and 1s. By the shrinking of the metrics on the coordinate loop spaces, any g ∈ π 1 (P ω , x) whose first n coordinates are 0s has a representative loop l ∈ g with diam(l) ≤ 2 −n . The space P ω is a Peano continuum.
Let κ be any cardinal satisfying
ℵ 0 ≤ κ ≤ 2 ℵ0 . As ∏ ω Z 2Z is a Z 2Z vec- tor space, we select a linearly independent set W ⊆ ∏ ω Z 2Z such that W ⊇ {(1, 0, 0, 0, . . .), (0, 1, 0, 0, . . .), (0, 0, 1, 0, . . .), . . .)} and card(W ) = κ. Let B be a basis for ∏ ω Z 2Z with B ⊇ W . Letting H = ⟨W ⟩ and G = ⟨W ∖ B⟩ we have ∏ ω Z 2Z ≃ H ⊕ G. Now G ⊴ π 1 (P ω , x) and π 1 (P ω , x) G ≃ H is of cardinality κ.
Letting κ = ℵ 0 gives a counterexample to the claim of Lemma 3.11 with the "G of pointclass P" and "P has BP and is closed under continuous preimages in Polish spaces" hypotheses removed. By considering a model of ZFC in which CH fails, we let κ = ℵ 1 and see that Theorem 3.13 can fail if the conditions on G and P are removed. Similarly removing these conditions can make the conclusion of Theorem 3.14 fail. The quotient can be a countable infinitely generated group or a group of cardinality violating CH if one is in a model of ZFC + ¬ CH.
The conclusion of Theorem 3.14 cannot be strengthened by replacing "finitely generated" by "finitely presented" by the following basic example.
Example 3. Let X be the bouquet of two circles and H be a 2-generated group which is not finitely presented (for example, the lamplighter group). The fundamental group π 1 (X) is the free group of rank 2. Let φ ∶ π 1 (X) → H be a homomorphism given by taking each of the free generators of π 1 (X) to a distinct generator of H. The space X is a semilocally simply connected Peano continuum and ker(φ) = G is open by Proposition 2.10, but π 1 (X) G ≃ H is not finitely presented. Similar examples can be given by replacing the number 2 by any finite number ≥ 2 and letting G be replaced by any other n-generated group which is not finitely presented.
We name some of the numerous applications of the above theorems.
Theorem A. Suppose X is a locally path connected Polish space. The following groups are of cardinality 2 ℵ0 or ≤ ℵ 0 , and in case X is compact they are of cardinality 2 ℵ0 or are finitely generated:
(1) π 1 (X) (2) π 1 (X) (π 1 (X)) (α) for any α < ω 1 (derived series) (3) π 1 (X) (π 1 (X)) n for any n ∈ ω (lower central series) (4) π 1 (X) N where N is the normal subgroup generated by squares of elements, cubes of elements, or n-th powers of elements
In case X is compact then countability of the fundamental group is equivalent to being finitely presented.
Proof. The noncompact case in parts (1)-(4) immediately follow from Theorem 3.13. For parts (2)-(4) in the compact case we apply Theorem 3.14. That π 1 (X) would be finitely presented follows from Theorem 7.3 in [CC] in part (1) assuming X is compact.
Part (1) in the compact case is the main result of the papers [Sh2] and [P], and part (2) with α = 1 (both compact and noncompact cases) is proven in [CoCo].
Towards proving Theorem B we give the following technical lemma.
Lemma 3.15. Let X be a Polish space. Suppose N ⊴ K ≤ π 1 (X, x) is such that ⋃ N = ⋃ ∞n=0
N n with each N n closed under inverses and homotopy and containing the trivial loop, and that K is closed. Assume also that N n * N m ⊆ N n+m . If each N n is P and for each n ∈ ω there exist loops at x of arbitrarily small diameter in ⋃ K not contained in N n , then K N is of cardinality 2 ℵ0 .
Proof. Let ǫ > 0 be given. By the proof of Lemma 3.11 we need only show that there is a loop at x in ⋃ K of diameter less than ǫ that is not in ⋃ N , since N is P. For contradiction we assume that no such loop exists. For each loop in ⋃ K of diameter less than ǫ let φ map that loop to the minimal k such that l ∈ N k . For two loops l 1 , l 2 of radius less than ǫ we have that φ(l 1 * l 2 ) ≤ φ(l 1 ) + φ(l 2 ) and φ(l 1 ) = φ(l −1 1 ). Let {l n } n∈ω be a sequence of loops such that diam(l n ) < ǫ2 −n and that φ(l 0 ) > 1 and φ(l n ) > n + ∑ n−1 m=0 φ(l m ). In particular none of the l n is nullhomotopic. Define l α as before for each α in the Cantor set. Abuse notation by letting
φ ∶ {0, 1} ω → ω be defined by φ(α) = φ(l α ). Let E n = {α ∈ {0, 1} ω ∶ l α ∈ N n }.
As we are assuming that there is no loop in ⋃ K of diameter less than ǫ that is not in ⋃ N , we have ⋃ ∞ n=0 E n = {0, 1} ω . We will derive our contradiction if we show that each E n is meager, which would imply that {0, 1} ω is meager in itself. Each E n is P, and so has the property of Baire. Supposing E n is not meager there exists a nonempty open set in which E n is comeager. In particular there exists a basic open set U (δ 0 , . . . ,
δ k ) = {α ∈ {0, 1} ω ∶ α(0) = δ 0 , . . . , α(k) = δ k } such that E n ∩ U (δ 0 , . . . , δ k ) is comeager in U (δ 0 , . . . , δ k ). For each p ≥ k + 1 let ℶ p ∶ U (δ 0 , . . . , δ k ) → U (δ 0 , .
. . , δ k ) be the homeomorphism that changes the p coordinate. Then U (δ 0 , . . . , δ k ) ∖ ℶ p (E n ) is meager for each p ≥ k + 1. Then in fact there exists α ∈ U (δ 1 , . . . , δ k ) such that switching finitely many of the coordinates beyond the kth coordinate gives an element of E n . It cannot be that the support of α is finite, for if N ∈ ω is a bound on the support of α (we can assume
N > 2n), then n ≥ φ(l α * f N +1 ) ≥ φ(l N +1 ) − φ(l α ) > N + 1 − n > n, a contradiction.
Thus taking a subsequence of the l n , we may assume that α = (1, 1, . . .) and that U (δ 1 , . . . , δ k ) = {0, 1} ω . We assume that this subsequence was the original sequence.
Let β k , γ k ∈ {0, 1} ω be given by β k (m) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0 if m < k 1 if m ≥ k and γ k (m) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 if m < k 0 if m ≥ k . We have that φ(γ k ) ≥ k and φ(β k ) ≥ φ(γ k ) − φ(α) ≥ k − φ(α)
, so that if k = 2n + 1 we have on the one hand that β k ∈ E n and on the other hand φ(β k ) ≥ k − φ(α) ≥ (2n + 1) − n, a contradiction.
This gives the following:
Theorem B. If X is a Peano continuum there does not exist a strictly increasing infinite sequence of P normal subgroups {G n } n∈ω of π 1 (X) such that ⋃ n∈ω G n = π 1 (X).
Proof. For each n ∈ ω let ⋃ G n = N n in the notation of Lemma 3.15. If π 1 (X) G n is finitely generated for some n, then the sequence {N n } n∈ω cannot be strictly increasing. Then π 1 (X) G n must be uncountable for each n, so for each n there exist arbitrarily small loops not in G n by the proof of Theorem 3.13. By picking an appropriate basepoint by local path connectedness, we are done by Lemma 3.15.
The necessity of the conditions on P can be easily seen by considering the decomposition π 1 (P ω
, x) = (⊕ ω Z 2Z) ⊕ (⊕ 2 ℵ 0 Z 2Z)
given in Example 2 and letting
G n = (⊕ k≤n Z 2Z) ⊕ (⊕ 2 ℵ 0 Z 2Z).
Example 4. The dual analog of Theorem B does not hold: there exists a Peano continuum with an infinite strictly descending chain of analytic (in fact closed) normal subgroups whose intersection is the trivial subgroup. We again use the space P ω from Example 2. We change the superscript ω for Q and the group {0, 1} Q remains unchanged since the cardinalities of ω and Q are the same. Given any subset S ⊆ Q the subgroup of π 1 (P ω
) corresponding to the subgroup {α ∈ {0, 1} Q ∶ α(q) = 1 ⇒ q ∈ S} ≤ {0, 1} Q is closed. For each r ∈ R let G r ≤ π 1 (P ω ) be the subgroup {α ∈ {0, 1} Q ∶ α(q) = 1 ⇒ q < r}.
Then each G r is a closed subgroup and the following hold:
(1) r n ↗ r implies ⋃ n∈ω G rn < G r (2) r n ↘ r implies ⋂ n∈ω G rn = G r (3) ⋂ r∈R G r is the trivial subgroup
Picking a sequence r n ↘ −∞ gives a strictly descending sequence of normal analytic subgroups G rn as claimed. The subgroup ⋃ r∈R G r cannot be equal to π 1 (X) (else we could pick any sequence r n ↗ ∞ and the ascending chain G rn would contradict Theorem B). For example the sequence over Q which is constantly 1 is not in ⋃ r∈R G r .
We now address what can happen in the absence of local path connectedness. Before stating the next theorem we quote the famous selection theorems of Silver and Burgess (from [Si] and [Bu] respectively). A set Y ⊆ Z in a Polish space is coanalytic if Z ∖ Y is analytic. The class of coanalytic sets is denoted Π 1 1 . Theorem. (J. Silver) Suppose E ⊆ Z × Z is a coanalytic equivalence relation on a Polish space Z. Then either there are ≤ ℵ 0 many equivalence classes or there exists a homeomorph of a Cantor set C ⊆ Z such that for distinct x, y ∈ C we have ¬xEy.
Theorem. (J. Burgess) Suppose E ⊆ Z × Z is an analytic equivalence relation on a Polish space Z. Then either there are ≤ ℵ 1 many equivalence classes or there exists a homeomorph of a Cantor set C ⊆ Z such that for distinct x, y ∈ C we have ¬xEy.
Theorem 3.16. Suppose X is path connected Polish and G ≤ π 1 (X, x).
(1) If G is coanalytic then the index π 1 (X, x) ∶ G is either ≤ ℵ 0 or 2 ℵ0 .
(2) If G is analytic then the index π 1 (X, x) ∶ G is either ≤ ℵ 1 or 2 ℵ0 .
Proof. If G is coanalytic (repesctively analytic) then by Lemma 2.3 the equivalence relation induced by the left coset partition is coanalytic (respectively analytic). Apply the theorem of Silver (resp. Burgess) to conclude (1) (resp. (2)).
Example 5. We give an instructive example which demonstrates the sharpness of Theorem 3.16 as well as the sharpness of Theorems 3.13 and 3.14 in a different sense than that of Example 2. There exists a model M of set theory satisfying
(1) ZFC (2) ¬ CH (3) Any subset of {0, 1} ω of cardinality ℵ 1 is coanalytic.
(see [MaSo]). We consider the space F and the function f ∶ K → L (0,0) used in Example 1 within the model M. Let S ⊆ K be such that card(K ∖ S) = ℵ 1 . Then S is analytic in K. Then f (S) is analytic in L (0,0) . Then ⟨⟨[f (S)]⟩⟩ ≤ π 1 (F, (0, 0)) is analytic by Lemma 3.5. The quotient π 1 (F, (0, 0)) ⟨⟨[f (S)]⟩⟩ is isomorphic to a free group of rank ℵ 1 so that card(π 1 (F, (0, 0)) ⟨⟨[f (S)]⟩⟩) = ℵ 1 . This demonstrates that the case ℵ 1 in Theorem 3.16 (2) can obtain in the absence of CH. It also shows that one cannot hope to extend Theorem 3.16 part (1) to a higher projective class (since any higher projective class also contains the analytic sets, see Section 7). This also shows that one cannot drop local path connectedness in Theorems 3.13 and 3.14 and obtain the same conclusion.
Comonster Groups
As an application of the above theory we give the following definition.
Definition 4.1. We say a group G is comonster if for every finite subset S ⊆ G we have ⟨⟨S⟩⟩ ≠ G. More generally G is κ-comonster if for every S ⊆ G with S of cardinality < κ we have ⟨⟨S⟩⟩ ≠ G.
Thus comonster groups are ℵ 0 -comonster groups. One easily sees that any abelian group of cardinality κ > ℵ 0 is κ-comonster. Also, if h ∶ G → H is an epimorphism with H comonster (respectively κ-comonster), then G is also comonster (resp. κ-comonster).
We assume still that P is nice with BP. We have the following:
Theorem 4.2. Let X be a Peano continuum and N ⊴ π 1 (X) of type P. If π 1 (X) N is comonster then π 1 (X) N is ℵ 1 -comonster. In particular, if π 1 (X) is comonster, then π 1 (X) is ℵ 1 -comonster.
Proof. Suppose for contradiction that π 1 (X) N is comonster but not ℵ 1 -comonster. Let S = {g 0 , . . .} ⊆ π 1 (X) be a countably infinite set such that ⟨⟨N ∪ S⟩⟩ = π 1 (X). The normal groups G n = ⟨⟨N ∪ {g 0 , . . . , g n }⟩⟩ are easily seen to be P and ⋃ n G n = π 1 (X). On the other hand the sequence G n cannot stabilize since π 1 (X) N is comonster. Thus one can pick a strictly increasing subsequence of normal P subgroups whose union is π 1 (X), contradicting Theorem B.
Example 6. Consider the Hawaiian earring E = ⋃ n∈ω C((0, 1 n+2 ), 1 n+2 ). We have an epimorphism h ∶ π 1 (E) → ∏ ω Z given by letting the n-th coordinate of h([l]) count the number of times a loop traverses the n-th circle of the infinite wedge that defines E in an oriented direction. Then π 1 (E) is 2 ℵ0 -comonster, since ∏ ω Z is abelian of cardinality 2 ℵ0 .
Example 7. If X is a one-dimensional Peano continuum with π 1 (X) uncountable, then X retracts to a subspace that is homeomorphic to E, so that again π 1 (X) is 2 ℵ0 -comonster.
Even if π 1 (X) is uncountable, it may still be the case that π 1 (X) is not comonster, as illustrated in the following example.
Example 8. Let Y be a Peano continuum with π 1 (Y ) ≃ A 5 . Such a Y exists by taking a finite presentation for A 5 and constructing the finite 2-dimensional CW complex by letting loops correspond to generators in the presentation and gluing on the boundary of a disc along a path that gives the relators. Such a space is compact, metrizable, path connected and locally path connected. Thus such a Y is a Peano continuum, and so is X = ∏ ω Y . We have π 1 (X) ≃ ∏ ω A 5 . Letting g ∈ ∏ ω A 5 have every entry be the 3-cycle (123), we claim that ⟨⟨g⟩⟩ = ∏ ω A 5 . This demonstrates that π 1 (X) is not comonster.
To see that ⟨⟨g⟩⟩ = ∏ ω A 5 , notice that all 3-cycles are conjugate (in A 5 ) to each other. Thus for each h ∈ ∏ ω A 5 whose each entry is a 3-cycle we have h ∈ ⟨⟨g⟩⟩. Each 3-cycle is a product of two 3 cycles (if (abc) is a 3-cycle then (abc) = (abc) −1 (abc) −1 = (cba)(cba)). Since the trivial element in A 5 is a product of two three cycles and each 5-cycle and each product of two disjoint transpositions (ab)(cd) is a product of two 3-cycles then in fact every element in ∏ ω A 5 is a product of exactly two conjugates of g and we are done.
In all the above examples of Peano continua with comonster fundamental group, we used the fact that if the abelianization is uncountable then the fundamental group is comonster. Question 4.3. Does there exist a Peano continuum whose first homology is trivial and whose fundamental group is comonster?
A negative answer would be very interesting as it would imply a theorem for finitely presented perfect groups (groups with trivial abelianization).
Theorem 4.4. Suppose the answer to the above question is no. Let P n be the class of groups whose elements are products of n or fewer commutators. For each n ∈ N there exists k(n) ∈ N such that if G ∈ P n is finitely presented there exists a set S ⊆ G with card(S) ≤ k(n) and ∏ k(n) S G = G (each element of G is a product of k(n) or fewer conjugates of elements of S).
Proof. Suppose for contradiction that for some n ∈ N there is no such k(n). Select finitely presented groups G m ∈ P n such that for any S ⊆ G m with card(S) ≤ m we have that ∏ m S Gm ≠ G m . For each m there is a finite CW complex Y m of dimension at most two whose fundamental group is isomorphic to G m . Each such Y m is a Peano continuum. Then ∏ m Y m is a Peano continuum, with fundamental group isomorphic to ∏ m G m . It is easy to see that ∏ m G m ∈ P n and is also comonster. This is adjacent to a question of Wiegold: Does every finitely generated perfect group contain an element which normally generates the group?
Examples of Topologically Defined Subgroups
We give some standard examples, and introduce some new examples, of subgroups of the fundamental group which are topologically defined. These are intended to illustrate the richness of the theory and give a grab bag of examples to which to apply the theorems. 5.1. The Shape Kernel. One well known subgroup of the fundamental group is the shape kernel. We discuss this subgroup by first giving preliminary definitions towards defining the shape group and the shape kernel and then prove that the shape kernel is a closed subgroup.
We assume some familiarity with geometric simplicial complexes. Given a topological space X and a open cover U of X let N (U) denote the nerve of the coverthat is, the geometric simplicial complex which has a distinct vertex v U for every U ∈ U and which contains the n-simplex [v U0 , v U1 , . . . , v Un ] if and only if U 0 ∩ U 1 ∩ ⋯ ∩ U n ≠ ∅. If V is an open cover of X that refines U (i.e. for each V ∈ V there is a U ∈ U such that V ⊆ U ) then any map from the vertices of N (V) to the vertices of N (U) such that v V ↦ v U implies V ⊆ U extends to a simplicial map from N (V) to N (U).
If the topological space has a distinguished basepoint x, then one can distinguish an element U in an open cover U such that x ∈ U , which in turn gives a distinguished vertex in the nerve N (U). With this added structure, if V refines U with distinguished elements V and U such that V ⊆ U then a simplicial map as described above extending a vertex assignment satisfying
v V → v U , say p (V,V ),(U ,U) ∶ (N (V), v V ) → (N (U)
, v U ) preserves basepoint and is unique up to basepoint preserving homotopy. Assuming X is path connected all nerves are connected, and the refinement relation on open covers gives an inverse directed system (π 1 (N (U), v U ), p (V,V ),(U ,U) * ). The shape group of X is defined as the inverse limiť
π 1 (X, x) = lim ← (π 1 (N (U), v U ), p (V,V ),(U ,U) * )
The index of the inverse limit will generally be of uncountable cardinality. Assuming (X, x) is also paracompact and Hausdorff we have for each open cover with distinguished neighborhood of x, (U, U ), a refinement (V, V ) such that V is the unique element of V containing x. A partition of unity subordinate to V (which necessarily exists by our assumption of paracompactness and Hausdorffness) induces a barycentric map f U ∶ (X, x) → (N (V), v V ) which is unique up to based homotopy.
The induced map f U * on the fundamental group π 1 (X, x) can be checked to commute with the maps of the inverse system in the appropriate way, and since the set of all such open covers V described are cofinal in the inverse system we get a map Ψ ∶ π 1 (X, x) →π 1 (X, x).
The natural object used to assess the loss of information when passing from the fundamental to the shape group is the shape kernel ker(Ψ). The following demonstrates an alternative characterizaion of the shape kernel.
Theorem 5.1. Suppose (X, x) is a path connected metrizable space. Then the shape kernel is a closed normal subgroup of π 1 (X, x). If in addition X is locally path connected then the shape kernel is equal to the following two subgroups:
(1) ⋂ f ker(f * ) where f is taken over all continuous maps to semilocally simply connected spaces
(2) ⋂ G open, normal G Proof.
It is clear from the definition that the shape kernel is equal to ⋂ f ker(f * ) where f is taken over all baricentric maps of open covers. Fix a barycentric map f to the nerve N (V). As N (V) is a geometric simplicial complex it is not difficult for each loop l ∈ L x to select ǫ > 0 such that for each l ′ ∈ L x that is ǫ close to l we have f ○ l is homotopic to f ○ l ′ in N (V). This shows that ker(f * ) is open, and therefore also closed by Lemma 2.4. Then the shape kernel is a closed subgroup as an intersection of closed subgroups. Now suppose that X is also locally path connected. Since each nerve is a geometric simplicial complex, each nerve is also semilocally simply connected. Thus the shape kernel contains the subgroup (1). Furthermore, if f ∶ X → Y is continuous with Y semilocally simply connected, then we can find an open cover U of X such that the image of any loop in an element of U has nulhomotpic image under the map f . This gives an open cover satisfying the criteria of Lemma 2.8 and since X is locally path connected we have that ker(f * ) is open. Thus subgroup (1) contains subgroup (2).
We conclude by proving that subgroup (2) contains the shape kernel. Let G be an open normal subgroup in π 1 (X) (since G is open, normal we may consider π 1 (X, x) as basepoint free by Lemma 2.5). Let q ∶ π 1 (X) → π 1 (X) G be the canonical quotient homomorphism. We introduce some terms and a theorem given in [CC].
Recall that if φ ∶ π 1 (Y ) → H is a group homomorphism we say an open cover V of Y by path connected sets is 2-set simple rel φ provided any loop whose image lies in the union of two elements of V is in ker(φ) (as defined in the proof of Theorem 3.14). Two paths p 0 and p 1 are V-related if there is some parametrization for p 0 and p 1 such that for all s ∈ [0, 1] the points p 0 (s) and p 1 (s) are in a common element of V. To be V-related is not necessarily an equivalence relation; we say that paths p 0 and p 1 are V-equivalent if they are in the same class under the equivalence class generated by V-relatedness. The following is part (1) Theorem. Let Y be a path connected topological space, φ ∶ π 1 (Y ) → H a homomorphism and V a 2-set simple cover of Y rel φ. If two loops l, l ′ ∈ L y are V-equivalent then φ([l]) = φ([l ′ ]).
By Lemma 2.6 we have an open cover
U 0 of X such that each loop in an element of U 0 is in G. For each z ∈ X we may select V z ∈ U 0 satisfying z ∈ V z . Define r 0 (z) = d(z, X − V z ). Letting U 1 = {B(z, r0(z)
3 )} z∈X it is straightforward to check that if for U, U ′ ∈ U 1 we have U ∩ U ′ ≠ ∅ then U ∪ U ′ is contained in an element of U 0 . By local path connectedness we let U 2 be a refinement of U 1 by path connected open sets. It is easy to see that U 2 is a 2-set simple cover rel q. For each z ∈ X pick a W z ∈ U 2 such that z ∈ W z and let r 2 (z) = d(z, X − W z ). Letting U 3 = {B(z, r2(z) 5 )} z∈X one can check that if U, U ′ , U ′′ ∈ U 3 satisfy U ∩ U ′ ≠ ∅ and U ′ ∩ U ′′ ≠ ∅ then U ∪ U ′ ∪ U ′′ is contained entirely in an element of U 2 . Let U 4 be a refinement of U 3 by path connected open sets. Finally for each z ∈ X select a U z ∈ U 4 such that z ∈ U z , let r 4 (z) = d(z, X − U z ) and U = {B (z, r4(z) 3 )} z∈X . Again, it is straightforward to see that if U, U ′ ∈ U satisfy U ∩ U ′ ≠ ∅ then U ∪ U ′ is entirely contained in an element of U 4 . Without loss of generality we can assume U is refined so that x is contained in exactly one element of the cover U.
Let b ∶ X → N (U) be a barycentric map associated to some partition of unity subordinated to U.
Then b(x) = v U where U ∈ U is unique such that x ∈ U . We define a map f from the 1-skeleton N (U) 1 to X. Let f (v U ) = x and for all other vertices v U ′ ∈ N (U) 0 simply let f (v U ′ ) ∈ U ′ . By our choice of U if [v U ′ , v U ′′ ] is a 1-simplex in N (U) then U ′ ∩ U ′′ ≠ ∅ and so there exists a path contained entirely in an element of U 4 from f (v U ′ ) to f (v U ′′ ). Let f [v U ′ , v U ′′ ] map via this path.
We will be done if we show that ker(b * ) ≤ G. Suppose now that l ∈ L x is such that
[l] ∈ ker(b * ). Then b○l is a loop in N (U) based at v U which is nulhomotopic. Recall that b has the property that b −1 (Star v U ′ ) ⊆ U ′ where Star v U ′ is the open star of the vertex v U ′ . There exists a combinatorial loop p(v U , v U1 , v U2 , . . . , v Un−1 , v Un = v U ) which is homotopic in N (U) to b ○ l such that b ○ l(s) ∈ Star v U k when s ∈ [ k n , k+1 n ].
Letting l 0 ∶ [0, 1] → N (U) be a topological realization of this loop we see that l is U 2 -related to f ○ l 0 .
By assumption there exists a nulhomotopy of l 0 , and so in particular there exists a combinatorial nulhomotopy of p(v U , v U1 , v U2 , . . . , v Un−1 , v Un = v U ). In other words, there exists a finite sequence of combinatorial paths:
p 0 = p(v U , v U1 , v U2 , . . . , v Un−1 , v Un = v U ) p 1 = p(v U , v U1,1 , v U1,2 , . . . , v U1,n 1 = v U ) p 2 = p(v U , v U2,1 , v U2,2 , . . . , v U2,n 2 = v U ) ⋮ p m = p(v U )
such that one obtains p k from p k−1 by performing one of the following elementary path homotopies:
(1) Exchanging the subpath v Up , v Up+1 for the subpath v Up assuming U p = U p+1 , or vice versa.
(2) Exchanging the subpath v Up , v Up+1 , v Up+2 for the subpath v Up assuming U p+2 = U p , or vice versa. (3) Exchanging the subpath v Up , v Up+1 , v Up+2 for the subpath v Up , v Up+2 assum- ing [v Up , v Up+1 , v Up+2 ] is a 2-simplex in N (U), or vice versa.
Letting l k ∶ [0, 1] → N (U) be a topological realization of the combinatorial path p k , it is easy to see that f ○ l k is U 2 related to f ○ l k+1 . By the theorem of Cannon and Conner quoted above, we have that q(
[l]) = q([f ○ l 0 ]) = q([f ○ l 1 ]) = ⋯ = q([f ○ l m ]) = q(1), and so [l] ∈ G.
As a direct consequence of Theorem 3.16 we get the following: if X is a Polish space then the quotient of π 1 (X) by the shape kernel is of cardinality ≤ ℵ 0 or 2 ℵ0 . 5.2. The Spanier Group. Another useful subgroup of the fundamental group is the Spanier group, which we denote π s 1 (X, x) (first defined in [Sp]). We give the necessary definitions for this group, then give some results about the topological properties.
Let X be a path connected topological space and x ∈ X. If U is a collection of open subsets of X we define π 1 (U, x) to be the subgroup of π 1 (X, x) generated by loops of the form ρ * l * ρ −1 where ρ(0) = x and l is a loop based at ρ(1) and contained in some element of U. This subgroup is easily seen to be normal. The Spanier group is defined to be π s 1 (X, x) = ⋂ U π 1 (U, x) where the parameter U is taken over all open covers. The first of the following two lemmas does not assume metrizability of X. It is proven in [FZ] as Proposition 4.8. We provide our own proof for completeness.
Lemma 5.2. π s 1 (X, x) is contained in the shape kernel.
Proof. Let b be a barycentric map from X to some nerve. Since a nerve is semilocally simply connected we have an open cover U of X such that any loop contained in an element of U is in ker(b * ). Obviously π 1 (U, x) ≤ ker(b * ) and taking the appropriate intersections gives the claim.
Lemma 5.3. Let X be a metric space, U an open cover of X and x ∈ X.
(1) If X is locally path connected then π 1 (U, x) is open.
(2) If X is Polish and U is countable then π 1 (U, x) is analytic.
Proof. Assume the hypotheses for part (1). The open cover U is such that any loop contained in an element thereof (considering loops to be base point free) is an element of π 1 (U) (we switch here to a basepoint free notation for emphasis). Then by Lemma 2.8 we have that π 1 (U) is an open subgroup. Assume the hypotheses for part (2). Let L x,U,
n = {l ∈ L x ∶ (∀s ∈ [0, 1 2 ])[l(s) = l(1 − s)] ∧ (∀s ∈ [ 1 3 , 2 3 ])[d(l(s), X − U ) ≥ 1 n ]}
where U ∈ U and n ∈ ω. It is clear that L x,U,n is closed as a subset of L x . The set ⋃ U∈U ,n∈ω L x,U,n is a countable union of closed sets (and therefore analytic). Then π 1 (U, x) = ⟨⟨⋃ U∈U ,n∈ω L x,U,n ⟩⟩ is analytic by Lemma 3.5.
That the shape kernel is equal to the Spanier group for all locally path connected, path connected paracompact Hausdorff spaces was recently shown in [BF]. Part (1) of the following theorem gives a rather short proof of a slightly less general fact.
Theorem 5.4. The following hold:
(1) If X is a locally path connected metric space then π s 1 (X, x) is equal to the shape kernel, and in particular closed.
(2) If X is a compact metric space then π s 1 (X, x) is analytic. Proof.
(1) Assume the hypotheses. That the Spanier group is contained in the shape kernel was proved in Lemma 5.2. That the shape kernel is contained in the Spanier group follows from characterization (2) of Theorem 5.1 and from Lemma 5.3 part (1).
For (2) we assume the hypotheses. As X is a compact metric space there exists a sequence {U n } n∈ω of finite open covers such that U n+1 refines U n and which is cofinal in the inverse directed system of open covers. Thus π s 1 (X, x) = ⋂ n∈ω π 1 (U n , x) is analytic as a countable intersection of analytic subgroups (Lemma 5.3 part (2) and Theorem 3.10 part (4)).
5.3. Subgroups reflecting local behavior. We give a couple of subgroups that can be thought of as indicating local behavior. First, recall that a space X is homotopically Hausdorff at x if each loop based at x which can be homotoped into any neighborhood of x is nulhomotopic. This notion has found many uses (for example in [BS] and [FZ]). If X is a Polish space, let L x,n be the set of all loops given by l ∈ L x,n if and only if (∀s ∈ [0, 1])[d(l(s), x) ≤ 1 n ]. Then L x,n is clearly a closed subset of L x , so the subgroup ⟨[L x,n ]⟩ is analytic by Lemma 3.4. The subgroup ⋂ n∈ω ⟨[L x,n ]⟩ is trivial if and only if X is homotopically Hausdorff at x. This subgroup is analytic and can be thought of as the indicator subgroup for the property.
We give another example of a subgroup reflacting local behavior. If X is compact, metrizable and path connected, then it is easy to see that the cone over X, CX = X×[0, 1] X×{1}, is also compact, metrizable and path connected. We shall consider X as a subset of CX by identifying X with X × {0}.
Let S ⊆ X be nonempty. Fixing a metric on CX we let Y n,S ⊆ CX be given by Y n,S = X ∪ (CX ∖ B(S, 1 n )). Let f n,S be the inclusion map from X to Y n,S . Then f n,S is a continuous map to a compact metric space, and ker(f n,S * ) is analytic. Since Y n,S = Y n,S there is no generality lost in assuming that S is compact. Also, the choice of metric on CX does not change ⋃ n ker(f n,S * ) (by compactness). Let N (S) denote the normal subgroup ⋃ n ker(f n,S * ). This subgroup is intended to convey a sense of the importance of the subspace S in the fundamental group of X.
If the subgroup N (S) is all of π 1 (X) then the points of S carry little significance in the fundamental group. If N (S) is trivial, then the points of S can be thought of as holding importance. If S ⊆ S ′ then Y n,S ⊇ Y n,S ′ and so N (S ′ ) ≤ N (S).
Example 9. Let X be compact, metrizable and path connected. Letting S = X we get that for every n ∈ ω ∖ {0}, the path component in Y n,S including all elements of X is simply the subset X. Thus any nulhomotopy of a loop in X taking place in Y n,S must in fact already take place in X, so N (S) is trivial.
Example 10. Let S ⊆ X be a compactum such that any map f ∶ S 1 → X can be homotoped to have image disjoint from S. Then given x ∈ X and a loop l ∈ L x there is a homotopy of l to a loop ρ * l ′ * ρ −1 such that l ′ is a loop with image disjoint from S. By compactness there is some positive distance between S and the image of l ′ , and so l ′ can be nulhomotoped in Y n,S for some n, so that l is also nulhomotopic in Y n,S . Then N (S) = π 1 (X).
Example 11. Let X = S 1 and S = {x} be any singleton. For each n ∈ ω ∖ {0} there is a superset Z ⊇ Y n,S such that Z strongly deformation retracts to the set X, so that N (S) is trivial. This holds true as well if X is a wedge of finitely many circles and x is the wedge point by the same proof.
Lemma 5.5. If r ∶ X → Y is a retraction with Y ⊃ S then the monomorphism induced by inclusion π 1 (Y ) → π 1 (X) induces a monomorphism π 1 (Y ) N Y (S) → π 1 (X) N X (S) (here we use the subscript to denote the ambient space).
Proof. This follows from the fact that the retraction r extends to a retraction R of the cones R ∶ C(X) → C(Y ) given by R(x, t) = (r(x), t) where t ∈ [0, 1].
Example 12. Let E be the Hawaiian earring (see Example 6) and S = {x} where x is the wedge point. The wedge Y m of the outer m circles is a retract of X and each N Ym (S) is trivial by the previous example. Then N E (S) has no elements of the canonical free group retracts. Then N E (S) is trivial by the standard fact that the Hawaiian earring fundamental group injects naturally into the inverse limit of the canonical free subgroups.
Example 13. Consider the Hawaiian earring E again and S = {x} with x any other point in E besides the wedge point. Then for some n ∈ ω ∖{0} the ball B(x, 1 n ) does not intersect any other circle on the Hawaiian earring besides that on which x lies. Then N (S) contains the kernel of the retraction induced homomorphism r * where r fixes the circle on which x lies and takes all other points to the wedge point. On the other hand, N (S) must be precisely the kernel of the induced homomorphism by Lemma 5.5.
N-slenderness, products and free products
In this section we introduce n-slender groups (see [E1]). This will require an understanding of the fundamental group of the Hawaiian earring E. Recall that the Hawaiian earring is the compact subspace E = ⋃ n∈ω C((0, 1 n+2 ), 1 n+2 ) of R 2 , where C(p, r) is the circle centered at p of radius r. The space E can be thought of as a shrinking wedge of countably infinitely many circles. The fundamental group π 1 (E) has a combinatorial characterization which we describe below.
We let {a ±1 n } ∞ n=0 be a countably infinite set with formal inverses. A map W ∶ W → {a ±1 n } ∞ n=0 from a countable totally ordered set W is a word if for every n ∈ ω the set W −1 ({a ±1 n }) is finite. We say two words U and V are isomorphic, U ≃ V , provided there is an order isomorphism of the domains of each word f ∶ U → V such that U (t) = V (f (t)). We identify isomorphic words. The class of isomorphism classes of words is a set of cardinality continuum which we denote W. For each N ∈ ω define the projection p N to the set of finite words by letting p N
(W ) = W {t ∈ W ∶ W (t) ∈ {a ±1 n } N n=0 }.
Define an equivalence relation ∼ on words as follows: given words U, V ∈ W we let U ∼ V if for each N ∈ ω we have p N (U ) = p N (V ) in the free group F ({a 0 , . . . , a N }). For each word U there is an inverse word U −1 whose domain is the totally ordered set U under the reverse order and U −1 (t) = U (t) −1 . Given two words U, V ∈ W we form the concatenation U V by taking the domain of U V to be the disjoint union of U with V , with order extending that of U and V and placing all elements of U before those of V , and U V
(t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ U (t) if t ∈ U V (t) if t ∈ V .
The set W ∼ is endowed with a group structure with binary operation given by
[U ][V ] = [U V ], inverses defined by [U ] −1 = [U −1 ]
and the equivalence class of the empty word being the trivial element.
Letting HEG denote the group W ∼, the free group F ({a 0 , . . . , a N }), which we shall denote HEG N , may be though of as a subgroup in HEG. The word map p N gives a group retraction HEG → HEG N which we also denote p N . The word map p N given by the restriction
p N (W ) = W {t ∈ W ∶ W (t) ∈ {a ±1
n } ∞ n=N +1 } induces another group retraction from HEG to the subgroup HEG N consisting of those equivalence classes which contain words involving no letters in {a ±1 n } N n=0 . Let p N denote this group retraction. By considering a word W as a concatenation of finitely many words in the letters {a ±1 n } N n=0 and finitely many words in the letters {a ±1 n } ∞ n=N +1 we obtain an isomorphism HEG ≃ HEG N * HEG N . The homomorphism p N corresponds to the topological retraction of E to the subspace ⋃ n≤N C((0, 1 n+2 ), 1 n+2 ) and similarly for p N and ⋃ n>N C((0, 1 n+2 ), 1 n+2 ). We are now ready for the following definition:
Definition 6.1. A group G is noncommutatively slender (or n-slender) if for each homomorphism φ ∶ HEG → G there exists N ∈ ω such that φ = φ ○ p N .
This definition was first introduced by K. Eda in [E1]. The additive group on Z was the first nontrivial group known to be n-slender [H], and Eda has shown that the class of n-slender groups is closed under free-products and direct sums (see [E1]). Torsion-free word hyperbolic groups are known to be n-slender [Co]. For each infinite cardinal κ there exists an n-slender group of cardinality κ (for example, the free group of rank κ). We give the following alternative characterization of n-slender groups before moving on to the theorems associated with this section: Lemma 6.2. A group G is n-slender if and only if for every locally path connected metric space X each homomorphism φ ∶ π 1 (X) → G has open kernel.
Proof. For the ⇒ direction we suppose that G is n-slender and that φ ∶ π 1 (X) → G is a homomorphism, with X a metric path connected, locally path connected space. Letting x ∈ X we claim that for some ǫ > 0 all loops (not necessarily based at x) in the open ball B(x, ǫ) are in the kernel of φ. Were this not the case there would exist a sequence of loops {l n } n∈ω such that diam({x} ∪ l n ([0, 1])) ≤ 2 −n and l n is not in the kernel of φ. By local path connectedness we may pass to a subsequence and eventually attach the bases of the l n to x via a small path. Thus we may assume without loss of generality that the l n are all based at x. Define a map f ∶ E → X by mapping the circle C((0, 1 n+2 ), 1 n+2 ) along the loop l n so that a generator of π 1 (C((0, 1 n+2 ), 1 n+2 )) maps to [l n ] ∈ π 1 (X, x) under the restriction f * π 1 (C((0, 1 n+2 ), 1 n+2 )). Now φ ○ f * is a map from HEG to G and so for some N we have n ≥ N implies φ ○ f * (a n ) = 1. But a n corresponds to one of the two generators of π 1 (C((0, 1 n+2 ), 1 n+2 )), so that 1 = φ○f * (a n ) = φ([l n ]), a contradiction. Thus such an ǫ must exist, and we get an open cover satisfying the hypotheses of Lemma 2.8 for the subgroup ker(φ), so that ker(φ) is open.
For the direction ⇐ we let G be a group such that for every locally path connected metric space X every homomorphism φ ∶ π 1 (X) → G has open kernel. Letting E = X and φ ∶ HEG → G be a homomorphism, ker(φ) is an open subgroup of π 1 (E). By Lemma 2.6 there exists some ǫ > 0 such that any loop in B ((0, 0)
, ǫ) ⊆ E is in ker(φ). Selecting N ∈ ω such that ǫ > 2 N +2 , we have φ HEG N is trivial, so that φ = φ ○ p N . Thus G is n-slender.
Recall that a space is κ-Lindelöf if every open cover of the space contains a subcover of cardinality at most κ. It is easily seen that a metric space Z is κ-Lindelöf if and only if Z has a dense subset of cardinality ≤ κ if and only if Z has a basis of cardinality ≤ κ. It is also true that if X is a metric κ-Lindelöf space then so is L x for each x ∈ X. Lemma 6.2 has the following easy corollary: Corollary 6.3. If X is locally path connected metrizable κ-Lindelöf and G is nslender then the image of any homomorphism φ ∶ π 1 (X) → G has card(φ(G)) ≤ κ.
Proof. Assume the hypotheses. Then ker(φ) is open, and by Lemma 2.3 the equivalence relation given by left cosets of ker(φ) has open equivalence classes. As L x has a dense subset of cardinality ≤ κ, we see that card(π 1 (X) ker(φ)) ≤ κ.
By considering a wedge of κ circles, each circle of diameter 1, endowed with the path metric, one has an example of a κ-Lindelöf space which is completely metrizable, locally path connected whose fundamental group is free of rank κ. Thus the conclusion of Corollary 6.3 cannot be strengthened.
One can prove other results which give obstructions to surjections from the fundamental group, even when the codomain is not n-slender, such as the following:
Theorem 6.4. Suppose X is a locally path connected κ-Lindelöf metric space, {G i } i∈I is a collection of n-slender groups, and φ ∶ π 1 (X) → ∏ i∈I G i is a homomorphism. Then there exists some
I ′ ⊆ I with card(I ′ ) ≤ κ such that ker(p I ′ ○ φ) = ker(φ). (Here the map p I ′ ∶ ∏ i∈I G i → ∏ i∈I ′ G i is projection.)
This immediately yields:
Theorem 6.5. If X is a locally path connected κ-Lindelöf metric space and {G i } i∈I is a collection of nontrivial n-slender groups with card(I) > κ then there is no epimorphism φ ∶ π 1 (X) → ∏ i∈I G i .
Example 14. Theorem 6.5 need not hold if local path connectedness is dropped. For example there exists a model N of set theory satisfying
(1) ZFC (2) 2 ℵ0 = ℵ 2 (3) (∀κ ≥ ℵ 1 )[2 κ = κ + ]
(see [Be], 2.19). We consider the space F from Example 1. Since π 1 (X) is a free group of rank 2 ℵ0 and the group ∏ ℵ1 Z is of cardinality 2 ℵ1 = 2 ℵ0 , there exists a surjection from π 1 (X) to ∏ ℵ1 Z.
Theorem 6.5 also fails in the model N if the hypothesis that the G i are n-slender is dropped. We consider the space P ω from Example 2 in the model N . We have π 1 (P ω
, x) ≃ ∏ ω Z 2Z ≃ ⊕ 2 ℵ 0 Z 2Z ≃ ∏ ℵ1 Z 2Z.
That Theorem 6.5 holds is a nontrivial fact, since π 1 (X) and ∏ i∈I G i can have the same cardinality (as would happen in the model N above). In a model of ZFC where the generalized continuum hypothesis holds, Theorem 6.5 would hold without the local path connectedness assumption or the n-slenderness of the G i simply by noticing that
card(π 1 (X)) ≤ card(L x ) ≤ card(X) ℵ0 ≤ (κ ℵ0 ) ℵ0 ≤ κ + = 2 κ ≤ card(I) < 2 card(I) ≤ card(∏ i∈I G i )
Proof. (of Theorem 6.4) Assume the hypotheses hold and the conclusion fails. Let p j ∶ ∏ i∈I G i → G j denote projection to the j-th coordinate. Let x ∈ X and B be a basis for the topology on L x with card(B) ≤ κ and ∅ ∉ B. Pick i 0 ∈ I such that ker(p i0 ○ φ) ≠ ker(φ). Suppose we have defined i α for all α < β < κ + so that for all
γ 0 < γ 1 < β we have that ker(p {iα}α≤γ 0 ○ φ) is a proper superset of ker(p {iα}α≤γ 1 ○ φ). Select i β so that ker(p i β ○ φ) does not contain ker(p {iα} α<β ○ φ). Such a selection is possible since we assume that ker(p I ′ ○ φ) ≠ ker(φ) for all I ′ such that card(I ′ ) ≤ κ. Now each ker(p j ○ φ)
is an open subgroup of π 1 (X, x) by Lemma 6.2, and so is closed by Lemma 2.4. As it is clear that ker(p I ′ ○ φ) = ⋂ j∈I ′ ker(p j ○ φ) for any I ′ ⊆ I we know that any ker(p I ′ ○φ) is closed. Pick O 0 ∈ B such that O 0 ∩ker(p i0 ○φ) ≠ ∅ and O 0 ∩ker(p {i0,i1} ○φ) = ∅. For 0 < β < κ + select O β ∈ B such that O β ∩ker(p {iα} α≤β ○φ) ≠ ∅ and O β ∩ker(p {iα} α≤β+1 ○φ) = ∅. The O β are pairwise distinct for different indices, so the map κ + → B given by β ↦ O β is an injection, a contradiction.
We move on to a result on free products of groups. We first state the following instance of Theorem 1.3 in [E2]:
Corollary. Suppose φ ∶ HEG → * i∈I G i is a homomorphism from HEG to a free product. Then there exists N ∈ ω, g ∈ * i∈I G i and j ∈ I such that φ(HEG N ) ≤ gG j g −1 .
We use this to prove the following: Lemma 6.6. Suppose X is a first countable, locally path connected space and φ ∶ π 1 (X) → * i∈I G i is a homomorphism. For each x ∈ X there exists a path connected neighborhood B x and j ∈ I such that φ(π 1 (B x , x)) is contained in a conjugate of G j .
Proof. We first notice that the statement of the lemma actually makes sense, because any change of basepoint would only alter φ by conjugation. Thus we may assume that the domain of φ is in fact π 1 (X, x). By φ(π 1 (B x , x)) we understand the image of the composition of the map induced by inclusion ι ∶ B x → X with φ.
Assuming the lemma is false there exists a sequence of path connected neighborhoods {U n } n∈ω with ⋂ n∈ω U n = {x} and loops {l n } n∈ω based at x such that the image of l n is contained in U n and for every n if φ([l n ]) ∈ gG j g −1 there exists n 0 > n with φ([l n0 ]) ∉ gG j g −1 . As in the proof of Lemma 6.2 we define a map g ∶ E → X so that the n-th circle in E traces out l n . Then φ ○ g * violates the corollary. Lemma 6.6 yields the following theorem, which is similar in flavor to Theorem 6.4 but with no mention of n-slender groups:
Theorem 6.7. Suppose φ ∶ π 1 (X) → * i∈I G i is a homomorphism, with X a locally path connected κ-Lindelöf metric space. Then for some I ′ ⊆ I with card(I ′ ) ≤ κ we have φ(π 1 (X)) ≤ * i∈I ′ G i .
Proof. By the Kurosh subgroup theorem [Ku] we have φ(π 1 (X)) = F (J) * ( * m∈M g m H jm g −1 m ) where j m ∈ I, H jm ≤ G jm and F (J) is a free group generated by J ⊆ * i∈I G i . Letting r ∶ F (J) * ( * m∈M g m H jm g −1 m ) → F (J) be the obvious retraction we notice that as r ○ φ is a map to an n-slender group, ker(r ○ φ) is open in π 1 (X) by Lemma 6.2. By Lemma 2.6 we have an open cover U 0 of X such that any loop in an element of U 0 is in ker(r○φ). Let U 1 be a refinement of U 0 such that U, V ∈ U 1 with U ∩V ≠ ∅ implies U ∪ V is contained in an element of U 0 . Let U be a refinement of U 1 consisting of path connected open sets. Since X is κ-Lindelöf we may assume card(U) ≤ κ. The following is a statement of Theorem 7.3 of [CC] part (2) (see proof of Theorem 3.14 for definition of 2-set simple):
Theorem. If ψ ∶ π 1 (X) → H is a group homomorphism and U is a 2-set simple cover rel ψ with nerve N (U) then ψ(π 1 (X)) is a factor group of π 1 (N (U)).
It is clear that U is 2-set simple rel r ○ φ, so F (J) is the homomorphic image of π 1 (N ), and since N has only κ-many vertices we know κ ≥ card(π 1 (N )) ≥ card(F (J)).
We show that M is of cardinality at most κ. Then we will let I 0 = {j m } m∈M ∪ ⋃ m∈M I m where g m ∈ * j∈Im G j and I 1 be such that F (J) ≤ * j∈I1 G j , where each I m is finite and card(I 1 ) ≤ κ. Thus I ′ = I 0 ∪ I 1 will be of cardinality ≤ κ and clearly φ(π 1 (X)) ≤ * i∈I ′ G i ..
We show card(M ) ≤ κ by demonstrating that if ψ ∶ π 1 (X) → * m∈M Γ m is onto, with each group Γ m nontrivial, then card(M ) ≤ κ. By Lemma 6.6 we can obtain a cover V 0 of X by open balls such that each loop with image in an element of V 0 maps to a conjugate of one of the Γ m . As X is κ-Lindelöf we may assume card(V 0 ) ≤ κ. For each B ∈ V 0 select an m B ∈ M such that π 1 (B) maps under φ to a conjugate of Γ m B and let M ′ = {m B } B∈V0 . Then card(M ′ ) ≤ κ. We show that card(M ∖M ′ ) ≤ κ and we shall be done. Let now r ∶ * m∈M Γ m → * m∈M∖M ′ Γ m be the obvious retraction. Refine V 0 to an open cover V by path connected open sets such that U ∩ V ≠ ∅ implies U ∪ V is included in some element of V 0 and card(V) ≤ κ. Now the cover V is 2-set simple rel the map r ○ ψ ∶ π 1 (X) → * m∈M∖M ′ Γ m , so the image of r ○ ψ is a homomorphic image of the nerve of V, and so r ○ ψ has image of cardinality ≤ κ. Then card(M ∖ M ′ ) ≤ κ and we are done.
The following corollary is immediate:
Corollary 6.8. If X is a locally path connected separable metric space then π 1 (X) is not a free product of uncountably many nontrivial groups. More generally if X is a locally path connected κ-Lindelöf metric space then π 1 (X) is not a free product of > κ many nontrivial groups.
The comparable result for a compact space holds as well:
Theorem 6.9. If X is a Peano continuum and φ ∶ π 1 (X) → * i∈I G i is a homomorphism, then for some finite I ′ ⊆ I we have φ(π 1 (X)) ≤ * i∈I ′ G i .
Proof. The proof runs the same as the proof of Theorem 6.7 except that the images of the fundamental groups of the nerves of the covers become finitely generated. Thus F (J) is finite rank and the M in the corresponding claim is finite for the same reason.
Lemma 6.6 also yields the following result for Polish spaces:
Theorem C. Suppose X is locally path connected Polish and π 1 (X) ≃ * i∈I G i with each G i nontrivial. The following hold:
(1) card(I) ≤ ℵ 0 (2) Each retraction map r j ∶ * i∈I G i → G j has analytic kernel.
(3) Each G j is of cardinality ≤ ℵ 0 or 2 ℵ0 . (4) The map * i∈I G i → ⊕ i∈I G i has analytic kernel.
Proof. Part (1) is from Corollary 6.8 . Part (3) follows from part (2) by Theorem 3.13. For part (3), we use part (1), and the kernel of the map * i∈I G i → ⊕ i∈I G i is precisely ⋂ i∈I ker(r i ) where r i is the retraction map to G i . Thus the kernel of * i∈I G i → ⊕ i∈I G i is a countable intersection of analytic subgroups (by part (2)) and therefore analytic.
It remains to prove part (2). Fix x ∈ X. It suffices to prove that if φ ∶ π 1 (X, x) → G * H is an isomorphism and r ∶ G * H → G is the retraction map to G then ker(r○φ) is analytic. By Lemma 6.6 there exists an open cover U 0 of X by open balls such that φ(π 1 (B y , y)) is contained in a conjugate of G or in a conjugate of H for each B y ∈ U 0 . Let U 1 be a refinement of U 0 such that any two overlapping elements of U 1 have union contained in an element of U 0 . Let U be a refinement of U 1 by path connected open sets, with U countable. Let U G denote those elements of U 0 which map under φ composed with the inclusion map into a conjugate of G, and similarly for U H . Thus U G ∪ U H = U 0 and it is possible that U G ∩ U H ≠ ∅.
Define π 1 (U 0 , x), π 1 (U G , x) and π 1 (U H , x) as in subsection 5.2. Notice that U is 2-set simple rel the quotient map q ∶ π 1 (X, x) → π 1 (X, x) π 1 (U 0 , x). Then by part (2) of Theorem 7.3 (quoted in our proof of Theorem 6.7) we know that π 1 (X, x) π 1 (U 0 , x) is countable. Let {w n } n∈ω ⊆ G * H be a countable collection such that q(φ −1 ({w n } n∈ω )) = π 1 (X, x) π 1 (U 0 , x). Let {h m } m∈ω ⊆ H be those elements of H which occur in the words {w n } n∈ω and let {g n } n∈ω be defined by letting g n = r(w n ).
The normal subgroup φ −1 (⟨⟨{h m } m∈ω ⟩⟩) is analytic by Theorem 3.10 part (5) and the normal subgroup π 1 (U H , x) is analytic as well. Thus the normal subgroup φ −1 (⟨⟨{h m } m∈ω ⟩⟩)π 1 (U H , x) is analytic by Theorem 3.10 part (4). We shall be done if we prove that ⟨⟨{h m } m∈ω ⟩⟩φ(π 1 (U H , x)) = ⟨⟨H⟩⟩ The inclusion ⟨⟨{h m } m∈ω ⟩⟩φ(π 1 (U H , x)) ≤ ⟨⟨H⟩⟩ is self-evident. For the other inclusion we have G * H = {w n } n∈ω φ(π 1 (U 0 , x)) = {w n } n∈ω φ(π 1 (U H , x))φ(π 1 (U G , x)) = {w n } n∈ω ⟨⟨{h m } m∈ω ⟩⟩φ(π 1 (U H , x))φ(π 1 (U G , x)) = {g n } n∈ω ⟨⟨{h m } m∈ω ⟩⟩φ(π 1 (U H , x))φ(π 1 (U G , x)) = ⟨⟨{h m } m∈ω ⟩⟩φ(π 1 (U H , x)){g n } n∈ω φ(π 1 (U G , x)) Let r H ∶ G * H → H be the retraction map and h ∈ H. We know h = ww ′ where w ∈ ⟨⟨{h m } m∈ω ⟩⟩φ(π 1 (U H , x)) and w ′ ∈ {g n } n∈ω φ(π 1 (U G , x)). Now h = r H (h) = r H (w)r H (w ′ ) = r H (w), so that H ≤ r H (⟨⟨{h m } m∈ω ⟩⟩φ(π 1 (U H , x))). The group φ(π 1 (U H , x)) is generated by elements of form h w with h ∈ H, and the same is obviously true of ⟨⟨{h m } m∈ω ⟩⟩. Thus the subgroup ⟨⟨{h m } m∈ω ⟩⟩φ(π 1 (U H , x)) is generated by elements of form h w . Now r H (h w ) = h r H (w) ∈ ⟨⟨{h m } m∈ω ⟩⟩φ(π 1 (U H , x)) for any h w ∈ ⟨⟨{h m } m∈ω ⟩⟩φ(π 1 (U H , x)). Thus we have shown that H ≤ r H (⟨⟨{h m } m∈ω ⟩⟩φ(π 1 (U H , x))) ≤ ⟨⟨{h m } m∈ω ⟩⟩φ(π 1 (U H , x)) so that ⟨⟨H⟩⟩ ≤ ⟨⟨{h m } m∈ω ⟩⟩φ(π 1 (U H , x)) and we have the other inclusion.
We use the space F from Example 1 to check aspects of sharpness for Theorem C. Since π 1 (F ) is isomorphic to the free group F (2 ℵ0 ) it is clear that we cannot drop local path connectedness and still assert part (1) of the conclusion. Local path connectedness is also required for parts (2) and (3) by the example that follows. It seems unlikely that part (4) holds absent local path connectedness.
Example 15. By 2.12 in [Be] there exists a model Q of set theory satisfying (1) ZFC (2) 2 ℵ0 = ℵ 3
We consider the space F in the model Q. We have π 1 (F ) ≃ F (2 ℵ0 ) ≃ F (2 ℵ0 ) * F (ℵ 2 ). The retraction to F (ℵ 2 ) cannot have analytic kernel by Theorem 3.16 part (2).
It is fascinating to note that the abelian version of Theorem C fails. Recall from Example 2 that we had subgroups G and H of π 1 (P ω , x) such that card(H) = ℵ 0 , G was not of nice pointclass with BP and π 1 (P ω , x) ≃ H ⊕ G. Also, π 1 (P ω , x) ≃ ⊕ 2 ℵ 0 Z 2Z so the index of a direct decomposition into nontrivial groups can fail to be countable even for a Peano continuum.
Nice Pointclasses
We end with a brief discussion of Polish pointclasses. We define the projective pointclasses Σ 1 n , Π 1 n , ∆ 1 n for n ∈ ω ∖{0}. We have seen that Σ 1 1 is the class of analytic sets and Π 1 1 is the class of coanalytic sets. Let ∆ 1 1 = Σ 1 1 ∩ Π 1 1 . For n ≥ 2, Σ 1 n is the class of continuous images of sets of type Π 1 n−1 Π 1 n is the class of complements of sets in Σ 1 n ∆ 1 n = Π 1 n ∩ Σ 1 n All Borel sets are easily seen to be of type ∆ 1 1 . That ∆ 1 1 is precisely the class of Borel sets is a theorem of Suslin (see 14.11 in [Ke]). The projective pointclasses sit naturally in an array with each pointclass containing all pointclasses to the left (e.g. Π 1 4 ⊆ ∆ 1 5 ). In an uncountable Polish space all these inclusions are proper. Each ∆ 1 n is a σ-algebra. Each Σ 1 n is closed under taking countable intersections, countable unions and images under continuous maps between Polish spaces. Each Π 1 n is closed under countable unions, countable intersections and continuous preimages.
Each Σ 1 n is a nice pointclass and we have already noted that Σ 1 1 is nice with BP. Unfortunately, the BP status of the other Σ 1 n cannot be decided from ZFC alone. For example, Gödel furnished a model of set theory, L, in which the following hold:
(1) ZFC (2) GCH (3) There exists a ∆ 1 2 set which does not have BP. (see [G], [N], [A]). Thus it is consistent with ZFC that Σ 1 1 is the only nice projective pointclass with BP. Other models of set theory are less stingy with nice pointclasses having BP. We discuss two situations in which the nice pointclasses with BP are more plentiful: models which satisfy Martin's axiom and ¬ CH, and a model of set theory constructed by Shelah [Sh1].
Martin's axiom (abbreviated MA) is a principle of combinatorial set theory. We shall not give a formal statement of this principle (the interested reader may consult [Fr]), but state a relevant consequence: if Z is a Polish space then the σ-algebra of subsets having BP is closed under unions of index less than 2 ℵ0 . If CH holds then this statement is uninformative (MA is in fact a trivial consequence of CH), but the point is that there exists a model of ZFC + MA + ¬ CH (see [ST]). The principle MA affords more applications of some of the theory developed in this paper. If P is a Polish pointclass we let S κ<2 ℵ 0 (P) be the closure of P under unions and intersections over indices of cardinality < 2 ℵ0 .
Proposition 7.1. The following hold:
(1) (ZFC + MA) If P is nice with BP then so is S κ<2 ℵ 0 P.
(2) (ZFC + MA + ¬ CH) S κ<2 ℵ 0 Σ 1 2 is nice with BP. Proof. For (1) we note that MA makes the σ-algebra of Baire property sets closed under unions and intersections over cardinals less than 2 ℵ0 . Thus S κ<2 ℵ 0 P has BP, and is obviously nice.
For (2) we recall the theorem of Sierpinski that any Σ 1 2 set is an ℵ 1 union of Borel sets (see 38.8 in [Ke]). Then by ¬ CH we have Σ 1 2 ⊆ S κ<2 ℵ 0 Σ 1 1 and we apply (1), using the obvious fact that the operation S κ<2 ℵ 0 is idempotent.
The following proposition illustrates some consequences that can be derived from ZFC + MA + ¬ CH: Proposition 7.2. (ZFC + MA + ¬ CH) Let X be a Peano continuum. The quotient π 1 (X) G is either finitely generated or of cardinality 2 ℵ0 for the following G:
(1) The center G = Zπ 1 (X).
(2) G = ⟨⟨[{l α } α<ℵ1 ]⟩⟩ where {l α } α<ℵ1 is any collection of loops of cardinality ℵ 1 . (3) G = (π 1 (X)) (α) for any α < 2 ℵ0 .
Proof. For (1), if G is open then we are done by Theorem 3.14. If G is not open then by Lemma 3.12 there is a point y ∈ X and sequence of loops {l n } n∈ω with diam(l n ) ↘ 0 and [l n ] ∉ G. Notice that l ∈ ⋃ Zπ 1 (X) ⇐⇒ (∀l ′ ∈ L y )(∃H ∈ H y )[H homotopes l * l ′ to l ′ * l] which illustrates that ⋃ Zπ 1 (X) is Π 1 2 . By Proposition 7.1 we know that Σ 1 2 has BP and so Π 1 2 has BP. Since Π 1 2 is also closed under continuous preimages we have part (1) by Lemma 3.11.
For part (2) we notice that {l α } α<ℵ1 is S κ<2 ℵ 0 Σ 1 1 and so we can directly apply Theorem 3.14 since S κ<2 ℵ 0 Σ 1 1 is nice with BP. For part (3) one can prove by induction over all ordinals less than 2 ℵ0 that (π 1 (X)) (α) is of type S κ<2 ℵ 0 Σ 1 1 . Very generous applications of our theory can be derived from 7.17 in [Sh1]:
Corollary. There exists a model R of set theory in which the following hold:
(1) ZFC (2) All projective subsets of {0, 1} ω have BP.
In R the conclusions of Theorems 3.13, 3.14, and B apply to subgroups definable from first-order formulas in conjunction with iterations of countable set operations applied to subgroups already known to be of a projective pointclass. We illustrate with an example. In R if X is a Peano continuum and x ∈ X then letting G be the normal subgroup generated by the set of elements that are not a cube of a central element: G = ⟨⟨{g ∈ π 1 (X, x) ∶ ¬(∃h ∈ Zπ 1 (X, x))[h 3 = g]}⟩⟩ we have that G is a Σ 1 4 subgroup and so the quotient π 1 (X, x) G is either finitely generated or of cardinality 2 ℵ0 .
A loop based at x is a continuous function l ∶ ([0, 1], {0, 1}) → (X, x). Two loops l 0 and l 1 at x are homotopic if there exists a continuous function H ∶ [0, 1] × [0, 1] → X called a homotopy such that H(s, 0) = l 0 (s), H(s, 1) = l 1 (s) and H(0, t) = H(1, t) = x for all s, t ∈ [0, 1]. The relation defined by homotopy is an equivalence relation. Letting L x denote the set of all loops at x in X we have the binary operation concatenation, denoted * , on L x defined by l 0 * l 1 (s) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ l 0 (2s) if s ∈ [0, 1 2 ] l 1 (2s − 1) if s ∈ [ 1 2 , 1]
be a Lebesgue number for the covering of the image of l by {U 0 , . . . , U m }. Cover l with finitely many open balls {B 0 , . . . , B k } of radius δ 2 . Cover the image of l with finitely many path connected open sets {V 0 , . . . , V q }, each of which is contained in one of the {B 0 , . . . , B k }. Let ǫ be a Lebesgue number for the covering {V 0 , . . . , V q } of the image of l. Pick N ∈ ω sufficiently large so that for 0
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E-mail address: sammyc973@gmail. E-mail address: [email protected]
| []
|
[
"Halo-model Analysis of the Clustering of Photometrically Selected Galaxies from SDSS",
"Halo-model Analysis of the Clustering of Photometrically Selected Galaxies from SDSS"
]
| [
"Ashley J Ross \nDepartment of Astronomy\nUniversity of Illinois\n1002 W Green St61801UrbanaIL\n",
"Robert J Brunner \nDepartment of Astronomy\nUniversity of Illinois\n1002 W Green St61801UrbanaIL\n\nNational Center for Supercomputing Applications\n61820ChampaignIL\n"
]
| [
"Department of Astronomy\nUniversity of Illinois\n1002 W Green St61801UrbanaIL",
"Department of Astronomy\nUniversity of Illinois\n1002 W Green St61801UrbanaIL",
"National Center for Supercomputing Applications\n61820ChampaignIL"
]
| [
"Mon. Not. R. Astron. Soc"
]
| We measure the angular 2-point correlation functions of galaxies, ω(θ), in a volume limited, photometrically selected galaxy sample from the fifth data release of the Sloan Digital Sky Survey. We split the sample both by luminosity and galaxy type and use a halo-model analysis to find halo-occupation distributions that can simultaneously model the clustering of all, early-, and late-type galaxies in a given sample. Our results for the full galaxy sample are generally consistent with previous results using the SDSS spectroscopic sample, taking the differences between the median redshifts of the photometric and spectroscopic samples into account. We find that our early-and latetype measurements cannot be fit by a model that allows early-and late-type galaxies to be well-mixed within halos. Instead, we introduce a new model that segregates early-and late-type galaxies into separate halos to the maximum allowed extent. We determine that, in all cases, it provides a good fit to our data and thus provides a new statistical description of the manner in which early-and late-type galaxies occupy halos. | 10.1111/j.1365-2966.2009.15318.x | [
"https://arxiv.org/pdf/0906.4977v1.pdf"
]
| 17,050,473 | 0906.4977 | 234d26e7e3d9074ae3414953688923e4f29dd7a3 |
Halo-model Analysis of the Clustering of Photometrically Selected Galaxies from SDSS
2009
Ashley J Ross
Department of Astronomy
University of Illinois
1002 W Green St61801UrbanaIL
Robert J Brunner
Department of Astronomy
University of Illinois
1002 W Green St61801UrbanaIL
National Center for Supercomputing Applications
61820ChampaignIL
Halo-model Analysis of the Clustering of Photometrically Selected Galaxies from SDSS
Mon. Not. R. Astron. Soc
0002009Accepted to MNRASPrinted (MN L A T E X style file v2.2)cosmology: observations -galaxies: halos
We measure the angular 2-point correlation functions of galaxies, ω(θ), in a volume limited, photometrically selected galaxy sample from the fifth data release of the Sloan Digital Sky Survey. We split the sample both by luminosity and galaxy type and use a halo-model analysis to find halo-occupation distributions that can simultaneously model the clustering of all, early-, and late-type galaxies in a given sample. Our results for the full galaxy sample are generally consistent with previous results using the SDSS spectroscopic sample, taking the differences between the median redshifts of the photometric and spectroscopic samples into account. We find that our early-and latetype measurements cannot be fit by a model that allows early-and late-type galaxies to be well-mixed within halos. Instead, we introduce a new model that segregates early-and late-type galaxies into separate halos to the maximum allowed extent. We determine that, in all cases, it provides a good fit to our data and thus provides a new statistical description of the manner in which early-and late-type galaxies occupy halos.
INTRODUCTION
The 'halo-model' (see, e.g. Kauffmann et al. 1997;Peacock & Smith 2000;Cooray & Sheth 2002;Zheng et al. 2005;Tinker et al. 2005) has been developed to allow one to precisely model the clustering of galaxies. One can fill dark matter halos with galaxies based on a statistical 'halo-occupationdistribution' (HOD), allowing one to model the clustering of galaxies within halos (and thus non-linear scales) while providing a self consistent determination of the bias at linear scales. Thus, as shown by, e.g., Zehavi et al. (2004), Blake et al. (2008), Tinker et al. (2008), one can use measurements of galaxy 2-point correlation functions to constrain the HOD of different sets of galaxies and gain information on the nature in which galaxies occupy dark matter halos.
Such a halo-model analysis can be particularly useful in constraining the clustering of early-and late-type galaxies. It has long been known that early-type galaxies cluster more strongly than late-type galaxies (recent studies include, e.g., Willmer et al. 1998;Norberg et al. 2002;Madgwick et al. 2003;Zehavi et al. 2005;Ross et al. 2006;Croton et al. 2006), and that there exists a corresponding morphology-density relationship (Dressler 1980) which essentially states that the fraction of early-type galaxies increases with the density of the local environment. Zehavi et al. (2005; hereafter Z05) have incorporated this relationship into their halo modeling (by allowing the fraction of late-type galaxies to decrease as a function of halo mass) and shown that this approach can indeed effectively model the clustering of early-and latetype galaxies.
Recent studies have determined that the morphologydensity relationship can be more accurately described as a color-density relationship. Ball et al. (2008) find no residual relation between density and morphology when removing color (but do find a strong residual in density and color when removing morphology) and Skibba et al. (2008) find a strong environmental dependance on color, even for fixed morphology. This implies that deep photometric surveys (which are likely to have little morphological information) should be sufficient for quantifying the clustering as a function of galaxy type.
In this work, we use galaxies that are photometrically selected from the Sloan Digital Sky Survey (SDSS) fifth data release (DR5) to constrain HODs. Blake et al. (2008) have previously used photometric data to constrain the HODs of luminous red galaxies (LRGs), and we follow a similar approach to constrain the HODs of early-and late-type galaxies. The wealth of quality photometric data allows us to precisely constrain the HODs of early-and late-type galaxies at higher redshifts than previous SDSS studies (e.g. Z05).
Our paper is outlined as follows: §2 describes how we use the halo model to obtain model angular 2-point correlation functions of galaxies; §3 describes how we both select arXiv:0906.4977v1 [astro-ph.CO] 26 Jun 2009 galaxies from the SDSS DR5 photometric catalog to produce a volume limited sample (to z = 0.3), which we further subdivide by type and luminosity, and also estimate the redshift distribution of these galaxy samples; §4 describes how we measure the 2-point correlation functions of galaxies and how we estimate the error on these measurements; §5 presents the results of our 2-point correlation function measurements and the best-fit HOD for each galaxy sample; in §6 we compare our results to previous studies and discuss the implications of our measurements; finally, we conclude in §7. Throughout this work, we assume a flat cosmology with Ωm = 0.28, h = 0.7, σ8 = 0.8, Γ = 0.15 (as used in Ross et al. 2007; hereafter R07).
HALO MODELING
We use the halo model to produce model angular 2-point correlation and cross-correlation functions. The most basic component of the model is the number density of halos at redshift z with mass M , n(M, z). We determine both n(M, z) and the bias of these halos, B(M, z), by using an ellipsoidal collapse model (e.g., Sheth et al. 2001) coupled with the methods described in Nishimichi et al. (2006). We model the probability distribution of the number of galaxies occupying a halo of a given mass, the halo-occupation-distribution (HOD), to determine the mean number of galaxies, N (M ), occupying halos of mass M . Following Zheng et al. (2005) and Blake et al. (2008), we assume separate mean occupations for central galaxies, Nc(M ) and for satellite galaxies, Ns(M ). Thus,
N (M ) = Nc(M ) × (1 + Ns(M ))(1)
Note what this implies: for a given halo mass, the mean number of satellites is Nc(M )Ns(M ), but these satellites are broken between halos which have a central galaxy and those that do not. The halos that do have a central galaxy have Ns(M ) satellite galaxies, and those without a central galaxy have zero satellites. This reasoning is crucial to understanding the equations presented throughout this section. Coupling this model with a formalism describing how galaxies distribute themselves within halos allows us to model the power spectrum (which we Fourier transform and then convert to ω(θ) via Limber's equation ;Limber 1954). We assume that the spatial distribution of satellite galaxies follow the Navarro, Frenk, & White (1997) dark matter halo density profile:
ρ(M, r) = M (cr/r vir (M ))(1+cr/r vir (M )) 2 × 1 4π(r vir (M )/c) 3 [ln(1+c)−c/(1+c)](2)
where rvir is the virial radius and c is the concentration parameter. Following Zehavi et al. (2004), we define 200 as the critical over-density for virialization and can thus express the virial radius, rvir as a function of mass as
rvir = 3M 200 × 4πρ 1 3 (3)
and inversely the virial mass as a function of scale as
Mvir(r) = 200 × 4 3 πr 3ρ (4)
whereρ is the mean co-moving background density of the Universe. Using this definition for the virial mass, Zehavi et al. (2004) determined via Bullock et al. (2001) that c can be expressed
c(M, z) = 11/(1 + z)(M/Mc) −0.13(5)
where we determine log10(Mc) = 12.49, where Mc is in units (h −1 M ), for our assumed cosmology (see, e.g. Blake et al. 2008). We use the Fourier transform of the full (untruncated) ρ(M, r), u(k|M ), to calculate the power spectrum, and we use the form presented in Scoccimarro et al. (2001). We note that the difference in using the Scoccimarro et al. (2001) parameterization, and one in which ρ(M, r) is truncated (using, e.g., Equation 8 from Jain et al. 2003 with rvir as upper bounds on the integrals) is negligible (< 0.1%) in our model angular correlation functions.
Modeling the Power-Spectrum
The equations presented above allow us to model the power spectrum as having a contribution due to galaxies in two separate halos (the 2-halo term) and a contribution due to galaxies that reside in a single halo (the 1-halo term).
P (k, r) = P 1h (k) + P 2h (k, r)(6)
where P 1h (k) is split into two components -one being the power-spectrum due to central-satellite pairs, Pcs(k), and the other due to satellite-satellite pairs, Pss(k):
Pcs(k) = ∞ M vir (r) dM n(M )Nc(M ) 2Ns(M )u(k|M ) n 2 g(7)
Pss(k) =
and
P 2h (k, r) = Pmatter(k) × M lim (r) 0 dM n(M )b(M, r) N (M ) n g u(k|M ) 2 (9)
where Pmatter is the matter power-spectrum determined via the fitting formulae of Smith et al. (2003). The parameter M lim (r) is the mass limit due to halo-exclusion, which we determine using the methods described by Tinker et al. (2005) and Blake et al. (2008). The average number density of galaxies is given by ng, and n g is the restricted number density of galaxies. The two number densities can be expressed as ng =
The scale dependent bias, b(M, r), can be expressed (Tinker et al. 2005) as a function of the halo bias as
b 2 (M, r) = B 2 (M ) [1 + 1.17ξm(r)] 1.49 [1 + 0.69ξm(r)] 2.09(12)
where ξm(r) is the non-linear real-space matter 2-point correlation function, determined by Fourier transforming the matter power spectrum.
In many cases, it will be useful to calculate the effective mass, M ef f , given by:
M ef f = 1 ng dM M n(M )N (M )(13)
and also the overall bias of the galaxies given by the model, b gal given by:
b gal = 1 ng dM B(M )n(M )N (M )(14)
These parameters are quite useful when comparing our results to each other and also to previous studies.
Halo Occupation Distribution Model
We model the HOD as a power-law with a softened transition for both the central and satellite galaxies (note that this implies the softening effect is squared for the satellite galaxies). This is expressed as
Nc(M ) = 0.5 1 + erf log10(M/Mcut) σcut (15) Ns(M ) = 0.5 1 + erf log10(M/Mcut) σcut × M M0 α(16)
These Equations are entered into Equation 1 to determine the mean occupation of halos at a given mass.
The HOD model has four free parameters, but one can be removed by requiring that ng match the observed number density of galaxies. To ensure this, we determine Mcut for any chosen combination of σcut, M0, and α. We also model the number of early-and late-type galaxies by expressing the fraction of late-type centrals and satellites as a function of halo mass (similar to Z05).
fc(M ) = fc0 exp −log10(M/Mcut) σcen(17)
and
fs(M ) = fs0 exp −log10(M/M0) σsat(18)
where we cap fs(M ) and fc(M ) (which we will from here on express as fs and fc in order to be concise) such that they are never greater than 1. This model once again has four free parameters, but we can remove one since we know the overall fraction of latetype galaxies. We thus calculate the required fc0 for every allowed combination of fs0, σcen, and σsat.
In a previous work (Z05), color selected "red" and "blue" galaxies were assumed to be well mixed within halos. We find that this type of model does not provide an adequate fit to our measurements of the auto-correlations of early-and late-type galaxies. We instead assume that if a central galaxy is a certain type, its satellite galaxies will be the same type, up to the extent allowed by the N (M ) and f (M ) statistics of the given model. In order to be concise, we express the P (k) in terms of a new function Θ(k, M ):
Ptype(k) = ∞ M min (r) dM Θ(k, M )n(M )Nc(M )Ns(M )u(k|M )/n 2 g,type ,(19)
where type can be either early or late. Equations 7 and 8 now become dependent on type and the relative values of fc and fs (which are themselves dependent on mass). If the late-type satellite fraction is greater than the late-type central fraction, all satellite galaxies around late-type central galaxies will be late-types (and thus each late-type central will have Ns late-type satellites). If the opposite is true, each early-type central galaxy only has early-type satellites.
For the average late-type central, the fraction of its satellites that are also late-type is the total fraction of late-type satellite galaxies (fs) divided by the fraction of late-type central galaxies (fc). Thus, the central-satellite term for late-types can be expressed as:
Θ cs,late (k, M ) = 2fc (fs > fc) = 2fc × fs/fc = 2fs (fs < fc)(20)
For the central-satellite term for early-type galaxies, we must take into account the fact that there will be late-type satellite galaxies around early-type centrals if fs > fc. In this case, we need to determine the fraction of early-type satellites around early-type centrals. The total fraction of early-type satellites is just 1 − fs. Thus the average fraction of satellites around early-type centrals that are early-type is (1−fs)/(1−fc) and the total contribution due to early-type satellites around early-type centrals is (1 − fc)(1 − fs)/(1 − fc). For fc > fs, all satellites around early-type galaxies are early-type and each early-type central thus has Ns earlytype satellites. The central-satellite term for early-types can therefore be expressed as:
Θ cs,early (k, M ) = 2(1 − fs) (fs > fc) = 2(1 − fc) (fs < fc)(21)
For the satellite-satellite terms, the same logic applies. If fs > fc there will be a term for both late-type satellites around late-type centrals and late-type satellites around early-type centrals. Around late-type centrals, all satellites are late-type, and the fraction of late-type satellites is just 1. The total fraction of late-type satellites around early-type satellites is fs − fc and the fraction around only halos with early-type central galaxies is thus (fs − fc)/(1 − fc). This term must be squared to account for the total number of late-type satellite-satellite pairs around early-type galaxies. The total contribution due to halos with early-type centrals is thus (1 − fc) × [(fs − fc)/(1 − fc)] 2 . If instead fc > fs, there are only late-type satellite galaxies around late-type centrals, thus the contribution is fc × (fs/fc) 2 . We therefore express Θ ss,late as:
Θ ss,late (k, M ) = fc + (fs − fc) 2 /(1 − fc) Ns(M )u(k|M ) (fs > fc) (f 2 s /fc)Ns(M )u(k|M ) (fs < fc)(22)
For the early-type satellite-satellite term, if fs > fc, there are only early-type galaxies around early-type centrals. The fraction of satellites around early-type centrals that are early-type is (1 − fs)/(1 − fc) and thus the contribution to the satellite-satellite term is (1−fc)×[(1−fs)/(1−fc)] 2 . For fc > fs, the fraction of satellites around early-type centrals that are early-type is just 1 and the fraction around latetype centrals is (fc − fs)/fc = (1 − fs/fc). We thus express Θ ss,early as:
Θ ss,early (k, M ) = (1 − fs) 2 /(1 − fc) Ns(M )u(k|M ) (fs > fc) 1 − fc + fc (1 − fs/fc) 2 Ns(M )u(k|M ) (fs < fc) (23)
The requirements of Equations 20 through 23 segregate the early-and late-type galaxies as much as possible while maintaining the statistics of Equations of 17 and 18. This essentially results in a model where smaller mass halos will only be occupied by early-or late-type galaxies and larger mass halos will have a central early-type galaxy, many earlytype satellite galaxies and room for a smaller (but significant) number of late-type galaxies.
In the case of a cross-correlation, one must substitute 2n early n late for n 2 g,type . For both the central-satellite and satellite-satellite terms, only halos with early-type centrals will contribute if fs > fc and only halos with late-type centrals will contribute if fc > fs. Thus,
Θ cs,el (k, M ) = 2(fs − fc) (fs > fc) = 2(fc − fs) (fs < fc) (24) Θ ss,el (k, M ) = 2 [(1 − fs)(fs − fc)/(1 − fc)] Ns(M )u(k|M ) (fs > fc) 2fs(1 − fs/fc)Ns(M )u(k|M ) (fs < fc).(25)
These sets of equations account for all of the pairs of galaxies that were present when the auto-correlation of the full sample was measured. Thus, the expressions that represent the fractions of pairs contributing to each term must add to one (i.e., Θ cs,late /2 + Θ cs,early /2 + Θ cs,el /2 = 1 and Θ ss,late /(Ns(M )u(k|M )) + Θ ss,early /(Ns(M )u(k|M )) + Θ ss,el /(Ns(M )u(k|M )) = 1). Inspection of our Θ expressions reveals that this is indeed the case.
Transformation to Angular Correlation Function
In order to compare our measurements to our HOD model, we must Fourier transform the model power spectra to a real-space correlations function:
ξ(r) = 1 2π 2 ∞ 0 dk P (k, r)k 2 sin kr kr(26)
and use Limber's equation to project the real-space model to angular space (assuming a flat Universe):
ω(θ) = 2/c ∞ 0 dz H(z)(dn/dz) 2 ∞ 0 du ξ(r = u 2 + x 2 (z)θ 2 )(27)
where c is the speed of light, H(z) is the expansion rate of the Universe, dn/dz is the normalized redshift distribution, and x(z) is the comoving distance to redshift z.
DATA
The data analyzed herein were taken from the SDSS DR5 Abazajian et al. (2005). This survey obtains wide-field CCD photometry Gunn et al. (1998) in five passbands (u, g, r, i, z; e.g., Fukugita et al. 1996). The entire DR5 represents close to 8,000 square degrees of observing area. We selected galaxies lying in the Northern, contiguous portion of the SDSS from the DR5 PhotoPrimary database and matched them to galaxies from the DR5 PhotoZ table. We constrained the sample (using the Schlegel, Finkbeiner & Davis 1998 dust maps) to have reddening-corrected magnitudes in the range 18 r < 21. We further masked our data by using the same pixelized mask of R07 (which cut on the SDSS DR5 survey area, seeing > 1. 5, r-band reddening > 0.2, bad pixels, satellite trails, etc.). This left 5,407 deg 2 of observed sky. Following the methods outlined in Budavári et al. (2003), we created a volume limited sample with z < 0.3 and Mr < −19.5 (we note that this same volume limited sample is used in R07). After masking, this sample, hereafter denoted as Z3, contains nearly four million objects (3,980,652).
We also subdivide these data samples by luminosity and type. We select galaxies with Mr < −20.5 from Z3 to produce a sample with just over 1.3 million galaxies (1,302750), which we denote Z3B. Each sample (Z3 and Z3B) is also split by galaxy type based on their type values from the DR5 PhotoZ table. As in R07, galaxies with t > 0.3 are put in our late-type sample and those with t 0.3 are put in our early-type sample. The Z3 has nearly as many late-type galaxies (1,984,021) as early-type galaxies (1,996,631), while Z3B has significantly more early-type galaxies (820,789 to 481,961). In total, this gives six galaxy samples for which we measure ω(θ) and determine a best-fit halo occupation model.
Redshift Distributions
We require a knowledge of the redshift distribution for each of our galaxy samples in order to compare our observations to theoretical models. To build the redshift distribution for each sample, we treat each observed redshift as a Gaussian probability-density-function (PDF) with σ equal to the estimated error. The PDFs for each redshift were sampled in order to find the expected number of objects within bins of width 0.001 in z. We normalize these distributions to have unit area and use this result in Equation 27. The normalized dn/dz of our samples are plotted in Figure 1. The normalized distributions of all-, early-, and late-type galaxies are quite similar for the Z3 and Z3B samples, implying that a direct comparison between the two samples is justified.
The un-normalized dn/dz is used to determine the number of galaxies within a volume element defined by z +dz. In order to determine ng, we integrate over the entire redshift range with weights given by (dn/dz) 2 : In this way, we determine fc0 for each given fs0, σcen, and σsat such that the model f late matches the observed f late .
ng = dz H(z) 4πf obs x 2 (z)c dn dz × dn dz 2 / dz dn dz
MEASUREMENT TECHNIQUES
We calculate the angular 2-point correlation function, ω(θ), of galaxies using the Landy & Szalay (1993) estimator:
ω(θ) = DD(θ) − 2DR(θ) + RR(θ) RR(θ)(30)
where DD (in our case) is the number of galaxy pairs, DR the number of galaxy-random pairs, and RR the number of random pairs, all separated by an angular distance θ ± ∆θ. We also calculate angular 2-point cross-correlation functions, for which we also employ the Landy & Szalay (1993) estimator:
ω el (θ) = D1D2(θ) − D1R(θ) − D2R(θ) + RR(θ) RR(θ)(31)
where D1 and D2 represent the two data samples that are being cross-correlated (note that the single random file can be used in our case since all of our samples have identical angular selections and that we use the subscript el because we will exclusively be cross-correlating early-and late-type galaxies). In every case, we mask our data and randoms by using the same pixelized mask of R07.
Errors and Covariance
We compute errors and covariance matrices using a method that estimates the statistical error associated with our angular selection and another that estimates the statistical error associated with our radial selection. We use a jackknife method (e.g., Scranton et al. 2002), with inverse-variance weighting for both errors (e.g., Myers et al. 2005Myers et al. , 2006 and covariance (e.g., Myers et al. 2007) to account for the errors based on our angular selection; the method is nearly identical to the method described in detail in R07. The jackknife method works by creating many subsamples of the entire data set, each with a small part of the total area removed.
We found in R07 that 20 jack-knife subsamplings are sufficient to create a stable covariance matrix. These 20 subsamples are created by extracting a contiguous grouping of 1/20th of the unmasked pixels in 20 separate areas. Our covariance matrix, C jack , is thus given by
C i,j,jack = C jack (θi, θj) = 19 20 20 k=1 [ω f ull (θi) − ω k (θi)][ω f ull (θj) − ω k (θj)](32)
where ω k (θ) is the value for the correlation measurement omitting the kth subsample of data and i and j refer to the i th and j th angular bin. The jackknife errors are simply the square-root of diagonal elements of the covariance matrix. We must use a separate method to account for uncertainties introduced by our radial selection. In essence, we are attempting to measure the auto-correlation functions of galaxies for a given redshift distribution (since the redshift distribution figures prominently in our models). Our defined cuts on photometric redshift do not uniquely produce the redshift distributions displayed in Figure 1. In order to account for this, we re-sample the photometric redshift catalog to create 'perturbed' samples whose redshift distributions match those of the original sample. We take the redshift of each galaxy to be randomly selected from its PDF and re-calculate Mr based on this redshift. If these perturbed redshifts and magnitudes satisfy our selection criteria, they are included in the new sample of galaxies. In order to adequately reproduce the redshift distributions of Figure 1, we find that we can only allow galaxies with z < 0.32 into our perturbed sample. For the early-and late-type galaxies, we also perturb the type value based on the type-error (assuming it is Gaussian) when producing our perturbed samples. The type errors scale linearly with the photometric redshifts errors. Thus we use
tn = t + (zn − z)σt/σz(33)
where t is the galaxy type, tn and zn are the perturbed type and redshift, and σt and σz are the type error and photometric redshift error of each object obtained from the DR5 PhotoZ table. For each of our galaxy samples, we create ten perturbed samples. The percentage of galaxies that match between samples varies between 77% and 85% for any given parent sample (variation within any group of ten perturbed samples is less than 1%, e.g., the percentage of matching galaxies is always between 76.5% and 77.4% for late-type galaxies from the Z3B sample). We calculate ω(θ) for each of the perturbed samples and calculate Cz:
Ci,j,z = Cz(θi, θj) = 10 k=1 fm[ωave(θi) − ω k (θi)][ωave(θj) − ω k (θj)](34)
where ωave is the average auto-correlation of each of the ten perturbed samples and fm is the average fraction of galaxies that match between sample k and the other nine samples. We find that typically Ci,j,z ∼ 0.5C i,j,jack , meaning that they are small but non-negligible.
We thus combine Cz and C jack to obtain the full covariance matrix for each sample (i.e. Ci,j = C i,j,jack + Ci,j,z). To properly constrain fit parameters, we minimize the χ 2 using our covariance matrixes via the equation
χ 2 = i,j [ω(θi) − ωm(θi)]C −1 i,j [ω(θj) − ωm(θj)](35)
where ωm(θ) refers to the model angular 2-point correlation function.
MEASUREMENTS
We have measured the angular 2-point correlation functions and found the best-fit HOD for galaxies in two luminosity threshold samples; Z3 (z < 0.3 and Mr < −19.5), and Z3B (z < 0.3 and Mr < −20.5). For each sample we also found the parameters that best-fit the ω(θ) of early-and late-type galaxies. The values of the best-fit HOD parameters for each full sample can be found in Table 1, and the best-fit parameters for the early-and late-type samples can be found in Table 2. The top-left panel of Figure 2 displays the measured ω(θ) for galaxies in the Z3B sample for all (black triangles), early-(red triangles), and late-(blue triangle) type galaxies with the best-fit model ω(θ) plotted with correspondingly colored lines (solid for all, dashed for late-, and dotted for early-type galaxies). The fit to all galaxies is excellent, as fitting between 0.003 o and 1 o (10 degrees of freedom) yields χ 2 = 3.9. The fit to the early-and late-type galaxies is tolerable, as for the 23 degrees of freedom χ 2 = 19.2. The model performs slightly better for the early-type galaxies (χ 2 = 8.4), than for the late-types (χ 2 = 10.8). The largest discrepancies are at small scales, where the late-type model is too low. We note that if we had assumed that early-and late-type galaxies mix freely within halos, the fits would have been significantly worse, as we find the minimum χ 2 is 55 for such a model when fitting the measurements over the same angular range.
The top-right panel of Figure 2 displays the best-fit HOD for all (black), early-(red), and late-(blue) type galaxies from the Z3B sample. The HOD for all galaxies shows inflection points around Mcut = 1.30 × 10 12 h −1 M , which defines the mass scale at which halos host a central galaxy, and M0 = 3.08 × 10 13 h −1 M , which defines the mass scale at which halos will host satellite galaxies. The best-fit latetype HOD shows a local minimum at close to 10 13 h −1 M . This shape is a consequence of the model -the fraction of central late-type galaxies decreases as the mass increases and thus the late-type HOD decreases until the halos are massive enough to host satellite galaxies. Even so, the slope of the late-type HOD is significantly smaller than for the overall HOD, allowing the fraction of late-type galaxies to be the largest in small mass halos and smallest in high mass halos. This can be seen clearly in the top panel of Figure 3, where the fraction of late-type galaxies is plotted against halo mass. The decrease is nearly monotonic except for a feature with a local maximum right at M0. This is consistent with the density-morphology relation (Dressler 1980), as the fraction of late-type galaxies decreases as halo mass increases. The model allows fc to increase from 0.38 at M = Mcut to 1 near 3 × 10 11 h −1 M . We note that we also tried a model where fc = fc0 for M < Mcut, but we did not obtain acceptable fits.
The measured and best-fit model ω(θ) for galaxies from Z3 are plotted in the bottom-left panel of Figure 2. The best-fit model to the entire sample is acceptable, as χ 2 = 1.9 fitting between 0.003 o and 1 o (10 degrees of freedom). The model fits for the early-and late-type galaxies also perform well, as the total χ 2 = 15.7 (23 degrees of freedom). Once again, the fit is slightly better for early-type galaxies (χ 2 = 6.1) than for late-type galaxies (χ 2 = 9.6). The largest disagreements are at small scales where the late-type model is not quite large enough. Once more, we note that our model out-performs one in which we allow early-and late-type galaxies to mix freely within halos; we find the minimum χ 2 for this class of model is 68 when fitting over the same angular range.
The galaxies in Z3B form a brighter subset of the Z3 galaxies. Thus, as can be seen in the righthand panels of Figure 2, the HOD of each Z3 sample is larger than its Z3B counterpart at every mass scale (though they are all nearly equal at 10 12 h −1 M ). The inflection points of the Z3 HOD for the full sample (occuring around Mcut = 11.73 × 10 11 h −1 M and M0 = 1.29 × 10 13 h −1 M ) are not as pronounced as for the Z3B sample, which is due to the larger value of σcut = 0.7 (as compared to the σcut = 0.4 for the Z3B sample). The best-fit HOD of the late-type galaxies shows similar behavior to the Z3B late-type HOD, as once more the late-type fraction is greatest at small halo masses and least at large halo masses. The overall number of late-type galaxies is more than four times higher, and this very nearly matches the difference in the late-type HODs for M halo > 10 13 h −1 M . At smaller halo masses, there are more significant differences. The Z3 late-type HOD does not have a local minimum, only a significant inflection point. This is primarily due to the fact that the parameter which governs the decrease of fc, σcen, has increased from 0.63 to 0.82.
The bottom panel of Figure 3 displays the fraction of late-type galaxies for the best-fit HOD model of the Z3 sample, as a function of halo mass. Again, there is a nearly monotonic decrease with a local maximum at M1. As plotted with dashed lines, this local maximum is due to a peak in the late-type satellite fraction. This peak exists at M0 because, at smaller halos masses, the total fraction of galaxies that are satellites drops sharply and the fraction of galaxies that are late-type satellites must as well. (Note that the data is displayed such that the satellite and central fractions add to the total fraction.)
The late-type satellite fraction is larger for the Z3 sample, but this due primarily to the fact that the overall latetype fraction has increased from 0.37 to 0.498. The overall fraction of late-type galaxies that are satellites is smaller for the Z3 sample (as presented in Table 2, it is 0.118 for Z3 and 0.149 for Z3B). This is due to the fact that the bulk of the late-type galaxies are central galaxies in low mass halos, and the total fraction of centrals is higher because the number density of halos is larger at small mass. Conversely, the satellite fraction of early-type goes up for the Z3 sam- All Galaxies Late-type Galaxies Early-type Galaxies 0.1 1 10 100
All galaxies Early-type galaxies Late-type galaxies 0.01 0.1 1 0.1 1 10
All Galaxies Late-type Galaxies Early-type Galaxies 0.1 1 10 100 All galaxies Early-type galaxies Late-type galaxies Figure 2. The left panels display the measured angular auto-correlation functions for galaxies with Mr < −20.5 (top) and Mr < −19.5 (bottom), for all (black triangles) early-(red squares) and late-type (blue circles) with lines that correspond to the best-fit model (black solid for all, red dotted for early-, and blue dashed for late-type). The right panels display the best-fit HOD for all (black solid), early-(red dotted), and late-(blue dashed) type galaxies for Mr < −20.5 (top) and Mr < −19.5 (bottom). Figure 3. The fraction of galaxies that are late-type as a function of halo-mass. The top panel displays the information for galaxies with Mr < −20.5 and the bottom panel for galaxies with Mr < −19.5. In both panels, the full fraction is displayed with a solid line, the central fraction is displayed with a dotted line, and the satellite fraction is displayed with a dashed line. Note that the information is displayed such that the satellite fraction and central fraction add to the full fraction.
ple. Essentially, the model implies that a majority of the late-type galaxies with −19.5 < Mr < −20.5 are central galaxies occupying low-mass halos and the majority of the early-type galaxies with −19.5 < Mr < −20.5 are satellites in higher-mass halos. We also measure the 2-point cross-correlation function of early-and late-type galaxies for each sample. These measurements are plotted in Figure 4 (black triangles) along with the model ω el (θ) that results from using the best-fit parameters determined from the autocorrelation measurements. The models appear close to the measurements, and the χ 2 are 16.1 for Z3 and χ 2 = 15.3 for Z3B. These values are impressive considering the size of the error bars and the fact that we did not specifically fit for these measurements. Once again, the best-fit models which include mixing are significantly worse (χ 2 = 92.4 for Z3 and χ 2 = 86.6 for Z3B). Thus, in every example, our model, which segregates earlyand late-type galaxies to the maximal extent, performs significantly better than one in which the early-and late-type galaxies are allowed to mix freely.
The model cross-correlation functions match the measurements quite well on large scales, which is what one should expect, as the 2-halo cross-power spectrum is just P (k, z) late P (k, z) early . The agreement suggests that there are not any major systematic problems in our construction of the redshift distributions of the early-and late-type galaxies. When we model the angular cross-correlation, effectively there is a term (dn/dz late )(dn/dz early ). If the true redshift distributions differ greatly from those we estimate, then our models would not be able to simultaneously fit the large scale autocorrelation and cross-correlation function measurements.
DISCUSSION
We have measured the angular auto-correlation functions of galaxies photometrically selected from the SDSS DR5. We have used these measurements to constrain the HOD of the galaxies and determine its dependence on luminosity, and galaxy type. We have found that that the fact we are using photometric redshifts requires a special prescription for determining number densities (see §3.1) and introduces an extra source of statistical error (see §4.1). Most interestingly, we have found that in order to simultaneously model the clustering of early-and late-type galaxies and their crosscorrelation, we cannot allow them to mix freely within halos.
Further insight can be gained by looking at the realspace 2-point correlation functions of our best-fit models. The ξ(r) of the early-, all, and late-type galaxies are displayed in the top, middle, and bottom panels of Figure 5, with solid and dotted lines representing the Z3 and Z3B samples. Due to the wide range in scales, we multiply each ξ by r 2 , which allows the differences between each correlation function to be seen clearly. The differences between the model ξ(r) of the Z3 and Z3B samples are in line with what one would expect. The samples share the same median redshift (z ∼ 0.25), so direct comparison is valid. The model ξ all (r) of Z3B is consistently higher than that of Z3, as one would expect given the differences in luminosity. Interestingly, the ξ(r) of the early-and late-type samples increase by a smaller factor than the full sample. The ξ(r) amplitudes for the Z3B sample are bolstered not only by the increase in luminosity but also due to the decrease in the fraction of late-type galaxies (∼ 0.5 compared to ∼ 0.37).
Comparison With Previous Results
We can investigate further by comparing our work with that of Z05, who found the best-fit HOD for galaxies from the spectroscopic portion of the SDSS (z ∼ 0.1). The model used by Z05 also did not include a σcut parameter. Tinker et al. (2008) constrained the HOD of SDSS galaxies using a HOD model that does include σcut, but the facts that Z05 constrain galaxy samples with luminosity thresholds that match ours (Mr < −19.5 and Mr < −20.5) and that the results of Tinker et al. (2008) generally agree with Z05, make it a more appropriate reference.
Despite the differences in the samples used and in the models, the differences between the Z05 best-fit model parameters and ours can be explained by the fact that Z05 used a fairly different cosmological model. In order to compare our results to those of Z05, we calculate real-space 2point correlation functions using the Z05 cosmological model (Ω total = 1, Ωm = 0.3, h = 0.7, Γ = 0.21, σ8 = 0.9) and best-fit HOD parameters. In Figure 6 we display our best-fit real-space 2-point correlation functions for our Z3 and Z3B samples (solid lines) compared to the Z05 bestfit real-space 2-point correlation functions for Mr < −19.5 and Mr < −20.5 (dotted lines). Once again we multiply each ξ by r 2 , in order to clearly see the differences between the respective correlation functions. We multiply the Z05 correlation amplitudes by a factor that allows for passive evolution between z = 0.25 and z = 0.1 given by (see, e.g., Wake et al. 2008) Figure 6. The top panel displays the best-fit model real-space 2-point correlation functions multiplied by r 2 for our data volume limited with z < 0.3 and Mr < −19.5 (solid lines) compared to the best-fit model real-space 2-point correlation functions from Z05 with Mr < −19.5 (dashed lines) and the same model shifted to z = 0.25 assuming passive evolution (dotted lines). The bottom panel displays the same information for samples with Mr < −20.5.
ξ hi (r) ξ lo (r) = b lo − 1 + D hi /D lo b lo(36)
where the subscripts lo and hi refer to the appropriate factors at the lower and higher redshifts and D is the linear growth factor (see, e.g., Mo & White 1996). We note that applying this factor simply allows for a proper comparison between the two clustering signals -we are not arguing that galaxies with Mr < −19.5 should passively evolve between z = 0.25 and 0.1. The comparison is particularly apt for the Mr < −19.5 samples, as the co-moving number densities are the same to 3 significant figures (0.0102 h 3 Mpc −3 ), and slightly less so for the Mr < −20.5 sample as the number density in our sample is roughly 33% higher than the Z05 sample.
For both samples, the correlation functions have similar amplitudes at large scales, but the Z05 amplitudes are significantly larger at small scales. This is what we would generally expect, as merging halos between 0.25 and 0.1 should increase the overall satellite fraction and thus increase the amplitude of the one halo term in the correlation function. This is indeed the case in the best-fit models. As presented in Zheng et al. (2007), the satellite fraction is ∼0.2 for L * galaxies from the SDSS spectroscopic, while ours is ∼0.15. The decrease is perhaps slightly more than one would expect between z ∼ 0.1 and z ∼ 0.25, but the general trend is as expected. We therefore do not find any significant disagreement between the clustering of galaxies in our photometric samples and that of galaxies from the SDSS spectroscopic sample.
On the other hand, our results differ from those of Z05 with respect to the early-and late-type galaxies. The manner in which Z05 separated galaxies into different samples differs slightly from ours, as Z05 uses a K-corrected color cut and we use an estimated spectral type. One would expect the galaxies Z05 classify as "blue" to predominantly be late-type (and the ones Z05 classify as "red" to predominantly be early-type). It is thus somewhat surprising that the Z05 measurements were well-fit by a model allowing red and blue galaxies to mix freely within halos -while we find, that in all cases, a model that separates the early-and late-type galaxies to the maximal extent possible (given the statistics) provides a much better fit to our data. The discrepancy is likely due to the fact that the smallest scales are most strongly affected by this treatment of the galaxies and that we probe significantly smaller scales than Z05 (our fits extend to req ∼ 0.03 h −1 Mpc). We can also easily compare our results to R07, as we use the same volume limited samples and early-/late-type splits. Comparing the b1 values of the full samples, they are quite close, but slightly inconsistent given the 1σ errors quoted in R07. This is due to the fact that in R07 the b1 values were measured directly from the data, while the b1 values quoted in this work are from the models derived from the best-fit HOD. Therefore, one would expect slight disagreement. The disagreement is never greater that 4%, so this is not worrisome. The disagreement is greater for the earlyand late-type samples, which is likely due to the treatment of the photometric redshift distributions. R07 determined a model ω2,DM using the redshift distribution for all of the galaxies in each respective sample and used it to find the all, early-, and late-type b1 parameters for the sample. As can be seen in Figure 1, the redshift distributions of the earlyand late-type galaxies show different shapes. In general, the early-type distributions are more narrow, and the late-type distributions wider than the full distribution. Thus, using the full sample to find b1 of late-type galaxies would cause the parameter to be under-estimated, and using it to find b1 of early-type galaxies would cause the parameter to be overestimated. This appears to be the case for R07, as the b1 of the late-type galaxies are consistently smaller than ours and the early-type b1 are consistently higher.
Mixing
Our best-fit models of early-and late-type galaxy clustering, constrained via our auto-correlation measurements, provide good fits to the respective auto-correlation measurements and also acceptable fits to the cross-correlation measurements (see Figure 4). We have investigated changing other parameters in the models to determine if there is an alternative course to separating early-and late-type galaxies into different halos, but none have provided acceptable fits. For example, we have tried a wide range of models for the density profiles of the late-type galaxies within halos that are different from the standard NFW (though always spherically symmetric), but we were not able to significantly improve the model fits (either while allowing mixing or not). We have also tried using different forms for the concentration parameter of late-type galaxies, but again, this produced no meaningful improvement. Given the data at hand, we are, therefore, convinced that our model represents the optimal way of modeling the clustering of early-and late-type galaxies.
By looking at the measured ω(θ), one can see why models with mixing can not reproduce our measurements. For any model that allows even mixing, if a model for late-type galaxies is forced to be closer to the model for all galaxies, the early-type model will have to do the same (in the case where the number densities of early-and late-type galaxies are equal, the response should be entirely symmetric). Our measurements for both galaxy samples show the measured ω(θ) of late-types getting closer to the measured ω(θ) of all galaxies as the scale gets smaller, while the early-type measurements do not get any closer to the measurements for all galaxies. Segregating the galaxies allows more freedom in each galaxy-type's model ω(θ) relative to the model for all of the galaxies. This can be illustrated by imagining two samples that are completely segregated but have identical clustering properties within halos. In combining these two samples, the number of close pairs will only double (along with the total number of objects) and the correlation function for the entire sample will be half as large as for either of the original samples. In the case where the two samples are mixed within halos, combining the two samples would quadruple the number of close pairs, thus producing the same result for the correlation function.
The shapes of the model ω el (θ) elucidate the minimum degree to which Equations 17 and 18 require mixing of galaxy types. If, for example, the galaxy types were allowed to be completely segregated, the model cross-power spectrum one halo term would be zero and the model crosscorrelation would be completely flat at small scales. Clearly, the best-fit models are not allowing such extreme segregation. In most cases, Equations 17 and 18 will require that many late-type galaxies are satellites of central early-type galaxies, who also have many early-type satellites (this will happen for any halo mass where fs is greater than fc).
CONCLUSIONS
We have measured the angular 2-point correlation functions of galaxies drawn from volume limited samples of SDSS DR5 galaxies with z < 0.3, Mr < −19.5 and z < 0.3, Mr < −20.5, each of which are further subdivided into early-and latetype galaxy subsamples. By modeling the angular 2-point correlation function, we have shown, for the first time, that the best halo model is one in which early-and late-type galaxies are segregated to the maximal extent possible. Previous studies (such as Z05) modeled the clustering of red and blue galaxies (which should predominantly be earlyand late-type galaxies, respectively) by allowing mixing between the galaxy types within halos; these studies, however, did not probe to the same small scales we have, which is where the models that allow mixing disagree the strongest with our measurements.
We plan to follow-up this work by using data from the SDSS DR7 to constrain the HOD as a function of redshift. The analysis techniques presented in this work provide a foundation upon which to base this extension and the improved photometric redshifts of the DR7 data should enable a reliable determination of the evolution of clustering as a function of galaxy type to z < 0.4.
M
lim (r) 0 dM n(M )N (M )
Figure 1 .
1The normalized redshift distributions for each of the six galaxy samples we use (z <0.3, Mr < −19.5, bottom; z < 0.3, Mr < -20.5, top) with the distribution for each full sample plotted in black, the late-type distributions plotted in blue, and the earlytype distributions plotted in red.f late = 1 ng dM n(M ) [fc(M )Nc(M ) + fs(M )Nc(M )Ns(M )] (29)
Figure 4 .
4The measured cross-correlations of early-and late-type galaxies in for the Z3 sample, volume limited with z < 0.3 and Mr < −19.5 (bottom), Z3B sample, volume limited with z < 0.3 and Mr < −20.5 (top) are displayed in black triangles. The appropriate model cross-correlation, determined using the bestfit HOD of the early-and late-type autocorrelations, is displayed with a solid black line in panel.
Figure 5 .
5The best-fit model real-space 2-point correlation functions multiplied by r 2 for early-(top), all (middle), and late-(bottom) type galaxies for the Z3 sample, volume limited with z < 0.3 and Mr < −19.5 (solid lines), Z3B sample, volume limited with z < 0.3 and Mr < −20.5 (dotted lines).
Table 1 .
1The best-fit values of the HOD parameters and the associated χ 2 values for the two main samples studied. All masses are in units M h −1 .Sample
α
log (Mcut)
log (M 0 )
σcut
χ 2 /dof
log M ef f
b 1
ng (h 3 Mpc −3 )
fsat
Z3
1.14 +0.02
−0.01
11.866
13.11±0.01
0.7 +0.06
−0.09
1.9/10
13.13
1.09
0.0102
0.148
Z3B
1.268 +0.026
−0.024
12.115
13.488 +0.009
−0.011
0.41 +0.13
−0.14
3.9/10
13.21
1.17
0.0041
0.130
Table 2. The best-fit values of the HOD parameters and the associated χ 2 values for the early-and late-type samples studied.
Sample
f c0
f s0
σcen
σsat
χ 2 /dof
b 1,late
b 1,early
f late
f sat,late
f sat,early
Z3
0.38
0.56 +0.04
−0.02
0.84±0.09
0.80 +0.09
−0.06
15.7/23
0.89
1.16
0.498
0.118
0.180
Z3B
0.437 0.38±0.02
0.63 +0.08
−0.10
0.35 ±0.05 19.2/23
0.98
1.27
0.370
0.149
0.118
(28)where f obs is the observed fraction of the sky and is 0.131 for our masked DR5 sample. By calculating ng in this manner, we account for the non-negligible photometric redshift errors, which make the total volume occupied by the galaxies in each of our samples larger than that of a truly volume limited sample. For each of our two main samples, we use this formalism to measure ng and compare this to the model ng in Equation 10 to determine the value of Mcut given M0, α, and σcut.When modeling the HOD of early-and late-type galaxies, we are constrained by the fact that the HOD model fraction of late-types, f late , must match the observed fraction. The HOD model fraction of late-types is determined via
ACKNOWLEDGEMENTSA.J.R and R.J.B acknowledge support for Microsoft Research, the University of Illinois, and NASA through grant NNG06GH156. The authors made extensive use of the storage and computing facilities at the National Center for Super Computing Applications and thank the technical staff for their assistance in enabling this work.We thank Ani Thakar and Jan Van den Berg for help with obtaining a copy of the SDSS DR5 databases. We thank Adam D. Myers and David Wake for helpful discussions and comments that improved this work. We thank an anonymous referee for (quite timely) comments that improved both our modeling and the clarity of its written presentation.
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| []
|
[
"What factors affect the duration and outgassing of the terrestrial magma ocean?",
"What factors affect the duration and outgassing of the terrestrial magma ocean?"
]
| [
"Athanasia Nikolaou \nInstitute of Planetary Research\nGerman Aerospace Centre (DLR)\nRutherfordstr. 212489Berlin\n\nCentre of Astronomy and Astrophysics\nBerlin Institute of Technology\nHardenbergstr. 3610623Berlin\n",
"Nisha Katyal \nInstitute of Planetary Research\nGerman Aerospace Centre (DLR)\nRutherfordstr. 212489Berlin\n\nCentre of Astronomy and Astrophysics\nBerlin Institute of Technology\nHardenbergstr. 3610623Berlin\n",
"Nicola Tosi \nInstitute of Planetary Research\nGerman Aerospace Centre (DLR)\nRutherfordstr. 212489Berlin\n\nCentre of Astronomy and Astrophysics\nBerlin Institute of Technology\nHardenbergstr. 3610623Berlin\n",
"Mareike Godolt \nCentre of Astronomy and Astrophysics\nBerlin Institute of Technology\nHardenbergstr. 3610623Berlin\n",
"John Lee Grenfell \nInstitute of Planetary Research\nGerman Aerospace Centre (DLR)\nRutherfordstr. 212489Berlin\n",
"Heike Rauer \nInstitute of Planetary Research\nGerman Aerospace Centre (DLR)\nRutherfordstr. 212489Berlin\n\nCentre of Astronomy and Astrophysics\nBerlin Institute of Technology\nHardenbergstr. 3610623Berlin\n\nDepartment of Planetary Sciences\nInstitute of Geosciences\nFree University of Berlin\nMalteserstr. 74-10012249Berlin\n"
]
| [
"Institute of Planetary Research\nGerman Aerospace Centre (DLR)\nRutherfordstr. 212489Berlin",
"Centre of Astronomy and Astrophysics\nBerlin Institute of Technology\nHardenbergstr. 3610623Berlin",
"Institute of Planetary Research\nGerman Aerospace Centre (DLR)\nRutherfordstr. 212489Berlin",
"Centre of Astronomy and Astrophysics\nBerlin Institute of Technology\nHardenbergstr. 3610623Berlin",
"Institute of Planetary Research\nGerman Aerospace Centre (DLR)\nRutherfordstr. 212489Berlin",
"Centre of Astronomy and Astrophysics\nBerlin Institute of Technology\nHardenbergstr. 3610623Berlin",
"Centre of Astronomy and Astrophysics\nBerlin Institute of Technology\nHardenbergstr. 3610623Berlin",
"Institute of Planetary Research\nGerman Aerospace Centre (DLR)\nRutherfordstr. 212489Berlin",
"Institute of Planetary Research\nGerman Aerospace Centre (DLR)\nRutherfordstr. 212489Berlin",
"Centre of Astronomy and Astrophysics\nBerlin Institute of Technology\nHardenbergstr. 3610623Berlin",
"Department of Planetary Sciences\nInstitute of Geosciences\nFree University of Berlin\nMalteserstr. 74-10012249Berlin"
]
| []
| The magma ocean (MO) is a crucial stage in the build-up of terrestrial planets. Its solidification and the accompanying outgassing of volatiles set the conditions for important processes occurring later or even simultaneously, such as solid-state mantle convection and atmospheric escape. To constrain the duration of a global-scale Earth MO we have built and applied a 1D interior model coupled alternatively with a grey H 2 O/CO 2 atmosphere or with a pure H 2 O atmosphere treated with a line-by-line model described in a companion paper byKatyal et al. (2019). We study in detail the effects of several factors affecting the MO lifetime, such as the initial abundance of H 2 O and CO 2 , the convection regime, the viscosity, the mantle melting temperature, and the longwave radiation absorption from the atmosphere. In this specifically multi-variable system we assess the impact of each factor with respect to a reference setting commonly assumed in the literature. We find that the MO stage can last from a few thousand to several million years. By coupling the interior model with the line-by-line atmosphere model, we identify the conditions that determine whether the planet experiences a transient magma ocean or it ceases to cool and maintains a continuous magma ocean. We find a dependence of this distinction simultaneously on the mass of the outgassed H 2 O atmosphere and on the MO surface melting temperature. We discuss their combined impact on the MO's lifetime in addition to the known dependence on albedo, orbital distance and stellar luminosity and we note observational degeneracies that arise thereby for target exoplanets. | 10.3847/1538-4357/ab08ed | [
"https://arxiv.org/pdf/1903.07436v1.pdf"
]
| 118,862,425 | 1903.07436 | c2a0789258d5cbd40027602ac1429e22e0fe3137 |
What factors affect the duration and outgassing of the terrestrial magma ocean?
Athanasia Nikolaou
Institute of Planetary Research
German Aerospace Centre (DLR)
Rutherfordstr. 212489Berlin
Centre of Astronomy and Astrophysics
Berlin Institute of Technology
Hardenbergstr. 3610623Berlin
Nisha Katyal
Institute of Planetary Research
German Aerospace Centre (DLR)
Rutherfordstr. 212489Berlin
Centre of Astronomy and Astrophysics
Berlin Institute of Technology
Hardenbergstr. 3610623Berlin
Nicola Tosi
Institute of Planetary Research
German Aerospace Centre (DLR)
Rutherfordstr. 212489Berlin
Centre of Astronomy and Astrophysics
Berlin Institute of Technology
Hardenbergstr. 3610623Berlin
Mareike Godolt
Centre of Astronomy and Astrophysics
Berlin Institute of Technology
Hardenbergstr. 3610623Berlin
John Lee Grenfell
Institute of Planetary Research
German Aerospace Centre (DLR)
Rutherfordstr. 212489Berlin
Heike Rauer
Institute of Planetary Research
German Aerospace Centre (DLR)
Rutherfordstr. 212489Berlin
Centre of Astronomy and Astrophysics
Berlin Institute of Technology
Hardenbergstr. 3610623Berlin
Department of Planetary Sciences
Institute of Geosciences
Free University of Berlin
Malteserstr. 74-10012249Berlin
What factors affect the duration and outgassing of the terrestrial magma ocean?
(Received May 23, 2018; Revised January 9,2019; Accepted February 19, 2019) Submitted to ApJDraft version March 19, 2019 Typeset using L A T E X twocolumn style in AASTeX62Earth -methods: numerical -planets and satellites: atmospheres -planets and satel- lites: composition -planets and satellites: interiors -planets and satellites: terrestrial planets
The magma ocean (MO) is a crucial stage in the build-up of terrestrial planets. Its solidification and the accompanying outgassing of volatiles set the conditions for important processes occurring later or even simultaneously, such as solid-state mantle convection and atmospheric escape. To constrain the duration of a global-scale Earth MO we have built and applied a 1D interior model coupled alternatively with a grey H 2 O/CO 2 atmosphere or with a pure H 2 O atmosphere treated with a line-by-line model described in a companion paper byKatyal et al. (2019). We study in detail the effects of several factors affecting the MO lifetime, such as the initial abundance of H 2 O and CO 2 , the convection regime, the viscosity, the mantle melting temperature, and the longwave radiation absorption from the atmosphere. In this specifically multi-variable system we assess the impact of each factor with respect to a reference setting commonly assumed in the literature. We find that the MO stage can last from a few thousand to several million years. By coupling the interior model with the line-by-line atmosphere model, we identify the conditions that determine whether the planet experiences a transient magma ocean or it ceases to cool and maintains a continuous magma ocean. We find a dependence of this distinction simultaneously on the mass of the outgassed H 2 O atmosphere and on the MO surface melting temperature. We discuss their combined impact on the MO's lifetime in addition to the known dependence on albedo, orbital distance and stellar luminosity and we note observational degeneracies that arise thereby for target exoplanets.
INTRODUCTION
Immediately after terrestrial planets form, their internal thermal structure is not well-known. This information is nevertheless important in order to understand the evolution of the atmosphere and the onset and development of mantle convection. With our numerical model we simulate the magma ocean (MO) phase, an intermediate stage in thermal evolution between completed accretion and the formation of a young planet's surface. During this period whose duration we aim to estimate, a large part of the mantle was fully molten.
Although the existence of the MO on the Moon is a widely accepted hypothesis due to its primary anorthositic crust (Canup 2004;Sleep et al. 2014; Barboni et al. 2017), such observational evidence remains elusive for the Earth. However, the Moon forming im-arXiv:1903.07436v1 [astro-ph.EP] 18 Mar 2019 pact is expected to have molten extensively the Earth's mantle (e.g. Nakajima & Stevenson 2015). In addition, the release of gravitational potential energy associated with core formation, along with the kinetic energy of accretional impacts and the decay of short-lived radiogenic elements provide enough energy to globally melt the silicate mantle of an Earth-sized planet leading to the formation of a magma ocean (Coradini et al. 1983;Solomatov 2007;Elkins-Tanton 2008;Sleep et al. 2014).
During the MO stage, the interior temperature is high and degassing of volatiles accompanies the thermal evolution (Abe & Matsui 1988;Elkins-Tanton 2008;Zahnle et al. 2010;Lebrun et al. 2013;Gaillard & Scaillet 2014;Massol et al. 2016;Salvador et al. 2017). As a result a secondary atmosphere forms and is expected to provide the bulk of the atmospheric mass during the Hadean. Nevertheless, there is uncertainty in the initial volatile inventory of the Earth because impactors stochastically deliver volatiles to the accreting planets Tsiganis et al. 2005;Raymond & Izidoro 2017). The most conservative estimate of the resulting Earth's volatile budget considers 300 bar H 2 O, roughly corresponding to today's ocean, and 100 bar CO 2 , i.e. the amount estimated to be stored in the crust in the form of carbonates (Ingersoll 2013). However, the existence of abundances in H 2 O and CO 2 higher than those directly observable in the present Earth cannot be excluded and is even likely (Hirschmann & Dasgupta 2009;Hirschmann & Kohlstedt 2012;Hallis et al. 2015). The bulk of the atmosphere outgassed from the magma ocean is suggested to be composed largely of H 2 O and CO 2 for most carbonaceous chondritic materials Lupu et al. 2014) along with reduced species (H,CO,CH 4 ) (Gaillard & Scaillet 2014;Lupu et al. 2014), excluding the case of chondritic binary mixtures (Schaefer & Fegley 2017). Proxy evidence for less oxidized early atmosphere is also given by sulphur isotope studies (Ueno et al. 2009;Endo et al. 2016).
As soon as a certain minimum amount of water vapor is present in the atmosphere, its role on the planetary evolution is the most crucial of all gases present. This is due to its strong greenhouse effect and the wellstudied runaway greenhouse regime associated with it, which does not allow for radiative equilibrium solutions over a wide range of surface temperatures, while the atmospheric outgoing radiation stalls to a constant value known as the Kombayashi-Ingersoll (KI) or runaway greenhouse (RG) limit (e.g. Nakajima et al. 1992;Kasting 1988;Zahnle et al. 1988;Pierrehumbert 2010;Leconte et al. 2013). The RG regime over a magma ocean, however, would occur at surface temperatures in excess of 3000 K that are characteristic of a molten silicate mantle Massol et al. 2016). Previous research has mostly indicated an invariant outgoing radiation limit of ∼300 W/m 2 for an Earthsized planet with water-dominated atmosphere (Kasting 1988;Zahnle et al. 1988;Nakajima et al. 1992;Zahnle et al. 2007;Kopparapu et al. 2013;Leconte et al. 2013;Hamano et al. 2015;Goldblatt 2015).
In particular, Hamano et al. (2013), extended the grey atmospheric model of Nakajima et al. (1992) to cover high surface temperatures, and introduced the separation of the magma-ocean stage into short-term and longterm. Based on the comparison of the stellar irradiation to the KI limit of a H 2 O-dominated atmosphere, they brought the role of stellar luminosity into context of the magma ocean lifetime, which, under suitable conditions, can hinder the planetary cooling altogether. However, in the Hamano et al. (2013) study, the potential role of mantle composition was not investigated in combination with the explicit role of the surface vapor pressure on the longwave radiation limit. We extend this previous work by considering such factors.
The volatiles that envelop the terrestrial planets are crucial since they quantify the effect of thermal blanketing that delays radiative cooling of the MO by hundreds of thousands of years. It has been investigated by several authors so far (Abe & Matsui 1988;Elkins-Tanton 2008;Lebrun et al. 2013;Hamano et al. 2013;Salvador et al. 2017;Hier-Majumder & Hirschmann 2017;Ikoma et al. 2018). However, comparing results from the literature is not straightforward because each magma ocean study involves many ad hoc assumptions. This is inevitable since different research fields focus on a specific niche of the MO system. At the same time, the topic is becoming increasingly multidisciplinary (Tasker et al. 2017) and more of the assumptions are being challenged.
Our aim is to calculate the lifetime of the magma ocean and to comprehensively assess the role of various parameters on it. Knowing their relative significance can help guide future model development. Therefore, we primarily account for the key role of the outgassed atmosphere. For planets which are volatile-poor or may quickly lose their atmosphere, blackbody thermal evolution is modeled and discussed. The findings are focused on an Earth-sized rocky planet. Yet their applicability in exoplanetary context is discussed and points of interest for the community are suggested.
METHODS
Numerical model
We calculate the thermal state of the solidifying magma ocean without examining the preceding stage of its formation. We build a model that simulates the coupled evolution of the interior (Sections 2.2-2.7.) and the atmosphere (Sections 2.8-2.9). The COnvective Magma Radiative Atmosphere and Degassing (COM-RAD) model resolves the mantle interior profiles of temperature, liquid and solid fraction along with the degassing process, starting from a fully molten mantle up to the end of the magma ocean phase (see Section 2.10). We employ the mantle-surface temperature iteration method developed by Lebrun et al. (2013), with differences in the calculation of the mantle adiabat (Section 2.4), of the liquid viscosity (section 2.6) and of the volatile mass balance (Section 2.7). The outgassing of H 2 O and CO 2 is calculated according to a melt solubility curve for each volatile (Section 2.7). The atmosphere is treated in two alternative ways (Section 2.8): i) A grey atmosphere accounting for two greenhouse gases H 2 O and CO 2 (Abe & Matsui 1985;Elkins-Tanton 2008) (Section 2.8.1) and ii) a pure H 2 O atmosphere with a spectrally-resolved Outgoing Longwave Radiation at the Top Of the Atmosphere (henceforth named "OLR at TOA" OLR TOA ) by Katyal et al. (2019) in a companion paper (hereafter "companion paper") (Section 2.8.2). In the following sections we separately introduce each model component.
Structure of the interior
We consider a spherically symmetric Earth with outer radius R p and core radius R b . This yields a mantle of thickness R p − R b whose physical properties are defined by the melting curves (solidus and liquidus) of KLB-1 peridotite (Fig. 1). By comparing the interior thermal profile to these curves, we identify the phase (liquid, partially molten, or solid) of different mantle layers. The mantle is initially assumed to be fully molten and convecting, with an adiabatic temperature profile. As it cools, the liquid adiabat (dotted line in Fig. 1) intersects with the melting curves (solid lines in Fig. 1). Due to the steeper slope of the adiabat compared to the melting curves, the adiabat and the liquidus intersect first atop the core-mantle boundary (CMB). The mantle is fully molten from the surface until the depth of intersection between the liquid adiabat and the liquidus; it is partially molten between the liquidus and solidus, and completely solid below the solidus. The melt fraction φ at any given depth, is calculated as (e.g. Solomatov & Stevenson 1993a,b;Abe 1997;Solomatov 2007;Lebrun et al. 2013):
φ = T − T sol T liq − T sol(1)
where T is the temperature of the mantle at a given depth, and T sol and T liq the corresponding solidus and liquidus temperature. The partially molten region is further divided by comparing the melt fraction φ with the critical melt fraction φ C of 40% that separates the liquid-like from solid-like behavior (Costa et al. 2009). For φ > φ C , the region is considered liquid-like and belongs to the magma ocean convecting domain of depth D. The interface of phase change that separates the two regimes is called solidification or rheology front. Note that φ C varies among 30% (e.g. Maurice et al. 2017;Hier-Majumder & Hirschmann 2017), 40% (Solomatov 2007;Bower et al. 2018) and 50% (Monteux et al. 2016;Ballmer et al. 2017) in the geodynamic literature.
Melting curves
We use solidus and liquidus curves of KLB-1 peridotite obtained from experimental data. Depending on pressure, we adopt different parameterizations for different parts of the mantle. For the solidus, we use data from Hirschmann (2000) for P ∈ [0, 2.7) GPa, Herzberg et al. (2000) for P ∈ [2.7, 22.5) GPa, and Fiquet et al. (2010) for P ≥ 22.5 GPa, while for the liquidus, from Zhang & Herzberg (1994) for P ∈ [0, 22.5) GPa, and Fiquet et al. (2010) for P ≥ 22.5 GPa. Since we employ data from multiple studies, we refer to the resulting set of melting curves as "synthetic". Such curves are adopted in our experiments unless otherwise specified.
As shown in Fig. 1, we also tested the linear melting curves adopted by Abe (1997) and later by Lebrun et al. (2013), as well as those introduced by Andrault et al. (2011) that are representative of a chondritic composition (for analytical expressions for all the melting curves see Appendix A). Apart from the linear curves of Abe (1997), the experimental data that we considered require higher order polynomial fittings since their slope is not constant with depth. As we discuss in Section 2.4, this imposes a limitation in calculating the two-phase adiabat.
Adiabat
The interior temperature profile is calculated using the expression of the adiabatic temperature gradient for a one-phase system:
dT dP = α T T ρc P ,(2)
where P is the pressure in GPa, c P the thermal capacity at constant pressure, and α T the pressure-dependent thermal expansivity given by (Abe 1997):
α T (P ) = α 0 P K K 0 −(m−1+K )/K ,(3)
where α 0 is the surface expansivity, K 0 the surface bulk modulus and K its pressure derivative (see Table 7 for Melting curves for three cases: linear according to Abe (1997) ("Abe97", purple solid lines); synthetic for peridotitic composition according to Herzberg et al. (2000), Hirschmann (2000) and Zhang & Herzberg (1994) for the upper mantle, and Fiquet et al. (2010) for the lower mantle ("Syn", black solid lines); for chondritic composition according to the same data for the upper mantle and Andrault et al. (2011) for the lower mantle ("Andrault11", yellow solid lines). "Syn" and "Andrault11" differ only in the lower mantle parametrization. The black dashed line indicates the profile of the rheology transition for the "Syn" curves ("RF Syn"). Dotted lines indicate adiabats with potential temperatures of 4000 K and 2400 K. The red open and full circles indicate the base of the liquid-like magma ocean of thickness D for the two adiabats, with the corresponding depth ranges of liquid (l), solid (s), and partially molten (l+s) regions shown in the left columns.
their values), and m = 0. The pressure is simply calculated assuming a hydrostatic profile. Solomatov & Stevenson (1993a,b) derived the adiabat for a two-phase system by introducing a modified thermal expansivity and thermal capacity that depends on the melt fraction. However, the expressions they derived are explicitly valid for a system with constant rate of temperature drop with depth, which is equivalent to constant phase boundary slope and thus applies only to linear melting curves such as those of Abe (1997). The modified adiabat then tends to align with the slope of constant melt fraction. Since they do not cover the higher order parameterizations of the experimental data that we adopted, we employ instead the one phase adiabat of Eq. (3) with constant thermal capacity and pressure-dependent expansivity.
Energy conservation and parametrized cooling flux
Assuming that the mantle temperature profile T (r) is adiabatic as described in Section 2.4, the time-evolution of the magma ocean is obtained by integrating the energy-balance equation (Abe 1997) over the evolving magma ocean volume:
ρ c P + ∆H dφ dT dT dt = − 1 r 2 ∂(r 2 F conv ) ∂r + ρq r ,(4)
where ρ is the density, ∆H the specific enthalpy difference due to phase change, F conv the MO convective cooling flux, and q r the internal heat released by the decay of the radioactive elements (see Appendix D). Because of its large depth extent and liquid-like viscosity which approaches that of water (see Sect. 2.6), the magma ocean is expected to undergo highly turbulent convection (Solomatov 2007) that is neither attainable in the laboratory (Shishkina 2016), nor is numerically resolvable (Maas & Hansen 2015). Key to the evolution of the interior temperature profile is the parametrization of the convective heat flux F conv of Eq. (4). This is calculated with the aid of the Rayleigh (Ra) and the Prandtl (P r) numbers:
Ra = ρα T g(T p − T surf )D 3 κ T η , P r = η ρκ T(5)
where g the gravity acceleration, T p the mantle potential temperature, T surf the surface temperature, D the depth of the convective layer, κ T the thermal diffusivity, and η the dynamic viscosity. We consider two different parameterizations for "soft" and "hard" turbulence. In the first one turbulent dissipation at the boundary layers affects the heat flux, while the flow is laminar in the bulk of the fluid (Solomatov 2007):
F soft = 0.089 k T (T p − T surf )Ra 1/3 D(6)
where k T = κ T ρc p is the thermal conductivity. With the above formulation, the heat flux becomes independent of the depth of the convective layer since the Rayleigh number is proportional to D 3 . The second parameterisation depends additionally on the inverse of the Prandtl number. It represents a regime where the heat flux is assumed to be controlled not only by boundary layer friction but also has a contribution from turbulence generated in the bulk volume of the fluid (Solomatov 2007):
F hard = 0.22 k T (T p − T surf ) D Ra 2/7 P r −1/7 λ −3/7 (7)
where λ=basin Length/Depth is the aspect ratio of the mean flow. For very high Ra, the hard turbulence parameterization is suggested (Solomatov 2007). There, increasing values of P r in a progressively more viscous fluid yields lower heat flux if all other parameters are left unchanged. Both parameterizations were implemented and tested.
Magma ocean viscosity
The melt viscosity prominently factors into the thermal evolution of the magma ocean (Eq. 5, 6). We separate the mantle into two regimes, namely liquid-like and solid-like. The transition between them upon cooling and solidification is a complex phenomenon that depends, among other factors, on composition and cooling rate (Speedy 2003). While the viscosity dependence on temperature follows an Arrhenius law below the solidus temperature (Kobayashi et al. 2000), non-linear effects take place near and above it (Dingwell 1996;Kobayashi et al. 2000;Speedy 2003). In order to mitigate the solid state transition Salvador et al. (2017) proposed a "smoothening" of the sharp viscosity jump that occurs in earlier magma ocean models (e.g. Lebrun et al. 2013;Schaefer et al. 2016). However, during cooling the crystal content increases and the melt rheology is expected to make a discontinuous jump from the liquid-to solidlike state over a short crystallinity range (Marsh 1981). Continuous variation in viscosity across 5 orders of magnitude is seen only in the case of glasses (Kobayashi et al. 2000), although such behaviour is not consistent with common MO solidification assumptions.
The Vogel-Fulcher-Tammann, henceforth referred to as VFT equation is employed in our work. By VFT definition, the obtained viscosity tends to an infinite (hence not physically meaningful) value at a threshold temperature T = C. We consider two such expressions for the liquid dynamic viscosity: one that depends only on the temperature η l = f (T ) (Karki & Stixrude 2010) and a second one that depends on both temperature and water content η l = f (T, X H2O ) (Giordano et al. 2008). Karki & Stixrude (2010) found that for hydrous melts, the viscosity at a given potential temperature can be well fitted to the following VFT equation:
η l (T ) = A K exp B K T − C K ,(8)
where A K , B K , C K (See Table 7 for their values) are calculated for a fixed water content of 10 wt%. However, Eq. (8) does not explicitly include the effect of water content, which is expected to vary during the simulation (see Section 4.2). The presence of water tends to lower the melt viscosity (Marsh 1981;Dingwell 1996;Giordano et al. 2008;Karki & Stixrude 2010) and it is important to include it as a time-dependent variable in our calculations. Hence, we implemented the empirical model of Giordano et al. (2008) which uses explicitly the water concentration, together with the concentration of 13 different oxides in the silicate melt. It calculates two of the three VFT parameters (B G and C G in Eq. (9) below). The viscosity for a given temperature T and water concentration X H2O is then given by:
η l (T, X H2O ) = 10 A G + B G T −C G ,(9)
where the parameter A G = −4.55 ± 1 is a constant pre-exponential factor. The parameters B G (X H2O ) and C G (X H2O ) are calculated by the model at each time step according to the evolving concentration of water (see Section 2.7). Even though COMRAD is unable to resolve the evolution of the melt composition, it does resolve the melt water concentration with time, which allows us to evaluate the Giordano et al. (2008) model at each time step. In order to use this model, it is required that we choose a suitable composition as a constant, non-evolving basis. We found that the composition of basanite (Giordano & Dingwell 2003;Giordano et al. 2008) is able to reproduce the experimental values for the temperature dependent viscosity for the anhydrous case (Urbain et al. 1982) since it is one of the least evolved in terms of silicate content. After calibration of the prefactor A G (see Appendix F, Table 6), the error is estimated to lie within ± 10%. This fitting provides us with a parameterized description of the composition that allows us to treat the decrease of the melt viscosity with increasing water content ( Fig. 2A). For anhydrous melt, the viscosity calculated with Eq. (9) and X H2O = 0 yields similar values as those proposed by Karki & Stixrude (2010) at high temperatures, though the two tend to depart significantly for temperatures near the solidus (Fig. 2B). The pressure dependence of the viscosity for the hydrous case is not explicitly provided by Karki & Stixrude (2010). The authors report that it varies by a relatively small factor between 2.5 and 10 over the pressure range [0, 140] GPa spanned by a global terrestrial magma ocean. In the following, for simplicity, we will neglect such dependence and use the liquid viscosity evaluated at the potential temperature T p of the magma ocean as representative for the fully molten part.
The melt viscosity η l is further corrected for the crystal fraction content in each layer. In the liquid-like partially molten region, the effect of crystals is taken into account with the following expression (Roscoe 1952):
η = η l 1 − 1−φ 1−φ C 2.5(10)
By combining Eq. (8) or (9) with Eq. (10) for each layer that belongs to the magma ocean, we obtain the volumetric harmonic mean viscosity η that is then used in the calculation of the parametrized convective heat flux in Eq. (6). The effect of the layer with the lowest viscosity value is prioritized in the calculation of harmonic mean viscosity and sets the leading order of magnitude for the value used in the calculation of the Rayleigh number (5). The viscosity employed during the MO lifetime is the liquid-like viscosity (η l ). Note that the solid-like viscosity (η s ) is employed only after the MO phase ends and it is expressed after the Karato & Wu (1993) formulation. The viscosity of the partially molten solid-like region below the rheology front (i.e. for φ < φ C ), is modified by the presence of the crystals according to Solomatov (2007).
Outgassing
Along with the thermal evolution the concentration of volatiles in the mantle and their outgassing into the atmosphere is calculated. We use solubility curves to calculate the concentration and gas pressure in the melt for each volatile. For H 2 O we use (Caroll & Holloway 1994):
P sat,H2O = X H2O 6.8 · 10 −8 (1/0.7) ,(11)
and for CO 2 (Pan et al. 1991):
P sat,CO2 = X CO2 4.4 · 10 −12 .(12)
In this way, we obtain the saturation vapor pressure over melt with a given volatile concentration. Due to the efficient mixing the volatile concentration is homogeneous throughout the magma ocean. Upon solidification, part of the volatile budget remains into the solid mantle according to the partition coefficients of lherzolite for the upper mantle (κ vol,lhz ) and of perovskite for the lower mantle (κ vol,pv ) (see Table 7 for their values). By calculating the volatile content stored in the liquid and solid phases of the mantle, we estimate the mass balance for each volatile at each time iteration t as follows:
M l,t0 X vol,t0 =P (X vol,t ) 4πR 2 p g + M s,pv κ vol,pv X vol,t + M s,lhz κ vol,lhz X vol,t + M l,z<z RF X vol,t + M l,z>z RF X vol,t Π + M l,z>z RF X vol,t−dt (1 − Π),(13)
where M l,t0 is the initial (time = t 0 ) mass of the liquid mantle, X vol,t0 the initial volatile concentration in the melt, P (X vol,t ) the saturation pressure of the volatile for the respective concentration X vol,t at time t, M s,pv , M s,lhz is the mass of solid mantle in perovskite and in lherzolite respectively, M l the mass of the melt at depth z either shallower (z < z RF ) or deeper than the rheology front (z > z RF ), and X vol,t−dt the concentration of the volatile in the previous time step. Comparing the melt percolation velocity (Solomatov 2007) to the rheology front velocity we calculate a volumetric fraction Π of the total melt volume that upwells across the rheology front. Π takes values within 0 and 1. The last term on the RHS of Eq. (13) represents the volatile mass trapped in the liquid below the rheology front and is evaluated at the concentration of the previous time step. Equation (13) combined with either Eq. (11) or (12), forms a system of two equations in two unknowns (P, X) for each species. Eq. (11) is non-linear with respect to X H2O and is solved iteratively with 0.01 bar tolerance.
The thermal evolution of the system is coupled to the outgassing process. Upon cooling, the mantle volume is redistributed into a solid and liquid phase (see Fig. 1). Consequently, the masses of the solid and liquid reservoirs M s and M l where the volatiles are stored change continuously. For every new mantle layer that solidifies volatile enrichment is ensured in the remaining melt. Since the saturation pressure increases monotonically with concentration (Eqs. 11 and 12), the equilibrium gas pressure also increases, resulting in a progressive buildup of atmospheric mass at the surface of the planet.
Secondary atmosphere
We adopt two alternative approaches to model the atmosphere generated upon magma ocean outgassing: i) a grey approximation after Abe & Matsui (1985) that treats two gas species H 2 O and CO 2 and ii) a line-byline approach that calculates a spectrally resolved OLR (companion paper) that assumes a pure H 2 O vapor composition.
Grey atmospheric model
The grey approximation that we use is derived in Abe & Matsui (1985). It considers the absorption of thermal radiation independently of the wavelength. Both outgassed species H 2 O and CO 2 absorb significantly in the spectral region where thermal energy is emitted from the surface of the Earth, and are therefore greenhouse contributors. By absorbing radiative energy, they exert a direct control on the surface temperature. Water is the most potent greenhouse agent of the two under normal atmospheric conditions (P 0 = 101325 Pa, T 0 = 293 K) (see e.g. Pierrehumbert 2010). For the H 2 O absorption coefficient the value k 0,H2O = 0.01 m 2 /kg in the midinfrared window region 1000 cm −1 is adopted, after Abe & Matsui (1988). CO 2 is accounted for with absorption coefficient k 0,CO2 = 0.001 m 2 /kg (Yamamoto 1952). A higher value k 0,CO2 = 0.05 m 2 /kg has been employed by Elkins-Tanton (2008) (along with lower k 0,CO2 values) and by Lebrun et al. (2013), which was calculated by Pujol & North (2003) in order to reproduce present day's Earth climate sensitivity (ECS). ECS refers to the combined response of the climate system to the radiative forcing from doubling the atmospheric CO 2 abundance relative to its pre-industrial levels and corresponds to an increase of about 2 • C in surface mean temperature (Flato et al. 2013). Using it is a good practice for studying the role of CO 2 radiative forcing on today's temperate Earth climate, within which the water vapor is not saturated over the atmospheric column. The presence of a liquid ocean is a strong constraint on the ECS and affects the overlying atmospheric profile through the inter-component exchange of vapor or "hydrological cycle" (Held & Soden 2006), provided that no runaway greenhouse regime (Pierrehumbert 2010) ensues. Extrapolating the climate sensitivity to MO mean surface temperature (>1000 K) well above today's (300 K) is unsuitable for our study. We therefore avoid using the ECS-based value for k 0,CO2 because it could overestimate CO 2 's radiative forcing on a planet with qualitatively different surface and atmospheric dynamics.
In Abe & Matsui (1985) the downward radiation at the TOA is set to the incoming stellar flux F Sun , which depends on the incident radiation S 0 at the assumed orbital distance. It relates to the blackbody equilibrium temperature T eq of the planet through the Stefan-Boltzmann law:
F Sun = (1 − α) S 0 4 = σT 4 eq ,(14)
where α is the albedo and σ the Stefan-Boltzmann constant. The resulting net upward flux at the top of the atmosphere (F grey ) is given by (Abe & Matsui 1985):
F grey = σ T 4 surf − T 4 eq = F conv(15)
According to Eq. (15), the net radiative flux at the TOA is positive for T surf > T eq . We adopt the convention of positive flux to represent planetary cooling. In order to find a state of the system that satisfies the energy balance, assuming that the radiative atmospheric adjustment is instantaneous, we require that the convective heat flux F conv at the top of the magma ocean is equal to the flux at the TOA.
For a given potential temperature, we solve the system of Eq. (6) and (15) using an iterative scheme built according to the method of Lebrun et al. (2013) with an accuracy of 10 −2 W/m 2 .
Line-by-line atmospheric model
An alternative approach is to employ a line-by-line (lbl) code to calculate the atmospheric outgoing radiation. The respective model is described in the companion paper. The advantage of this approach is that it provides a detailed calculation of wavelength-dependent longwave emission. We assume a 100% water vapor atmosphere ("steam atmosphere") as commonly used by various authors when treating magma ocean planets or exoplanets with water-dominated atmospheres (e.g. Hamano et al. 2013;Massol et al. 2016;Schaefer et al. 2016). A dry adiabatic temperature profile is used for the troposphere when the surface temperature is above the critical point of water (T H2O,crit = 647 K). When the dry adiabat intersects the saturation vapor pressure of water, a moist adiabatic temperature profile is assumed (Kasting 1988).
Using the temperature profiles of the atmosphere for a range of surface temperatures T surf and surface pressures P H2O , the emitted radiation is calculated for the spectral range from 20 − 29, 995 cm −1 using the line by line model GARLIC (Schreier et al. 2014) with HITRAN 2012 (Rothman et al. 2013). Integrating the emitted radiation at the TOA (corresponding to pressure 1 Pa), the outgoing radiation flux OLR T OA is obtained. Katyal et al. (2019) determines OLR T OA for various H 2 O surface pressures and temperatures on a (P H2O,0 , T surf ) grid that covers pressures between 4 and 300 bar and temperatures between 650 and 4000 K (Appendix C; Fig. C.1). For the surface vapor pressure we use the values 4, 25, 50, 100, 200, and 300 bar. For the surface temperature, we sample our calculations with a resolution ∆T = 100 K. We employ ∆T = 20 K in the region T ∈ [1400, 2200] K where the highest variability of outgassing with respect to surface temperature occurs for the melting curves used in this study. We obtain values of OLR T OA that correspond to conditions intermediate to the grid points via bilinear interpolation. The relative interpolation error ranges from 10% for fluxes of the order of 10 6 W/m 2 to about 1% for fluxes of the order of 10 2 W/m 2 .
In order to couple the lbl model results with the magma ocean thermal evolution, we impose a balance between the net energy flux at TOA and the magma ocean cooling flux F conv such that:
F conv = OLR T OA − F Sun .(16)
Equations (6) and (16) form a system of two unknowns T surf and F conv , which we solve iteratively with a tolerance of 10 −1 W/m 2 . The resulting flux needed to balance the RHS of Eq. (16) can be either positive or negative, corresponding to a cooling or warming case, respectively.
Incoming stellar radiation
For the young Sun we used a lower irradiation value following the expression for the time dependence of the solar constant of Gough (1981); otherwise we used today's value (1361 W/m 2 , see Table 1).
Calculating the planetary albedo is outside the scope of this study. α is instead used as an input parameter. The suggested albedo for a cloudless steam atmosphere lies within the range [0.15, 0.40] (Kasting 1988;Goldblatt et al. 2013;Leconte et al. 2013;Pluriel et al. 2019). We employ the value 0.30 unless otherwise stated.
End of the magma ocean phase
The end of the magma ocean phase is defined as the point in time when the rheology front reaches the surface. At that stage, the mantle adiabat has potential temperature T RF,0 such that all mantle layers have a melt fraction lower than φ C , a condition termed "mush stage" by Lebrun et al. (2013). Although some melt still remains enclosed in the solid matrix, the mantle subsequently behaves as a solid. Moreover, the adiabatic profile used for the solid mantle implies that solid state convection has fully developed. While this is a likely scenario for slowly solidifying magma oceans, establishing whether and to what extent the solid mantle convects during its early stages is beyond the scope of this study since this would require the use of fully dynamic simulations (e.g. Maurice et al. 2017;Ballmer et al. 2017). We therefore consider the established convection considered in this work to be an end member case within the geodynamic assumptions.
However, we stress that the thermal evolution model is designed to cover only the time until the MO end is reached via bottom-up crystallization.
EXPERIMENTAL SETTING
Since our model has numerous input parameters, we define a set of parameter values, hereafter called "Reference-A" (Ref-A) model setting (reported in Table 1), with respect to which we perform changes and comparisons. This model is intended to be as straightforward as possible, to facilitate model comparison. It does not include radioactive heat sources, the melt viscosity only depends on temperature according to Eq. (8), the abundance of volatiles is set to today's Earth observed reservoirs and it uses the atmospheric grey model for two species H 2 O and CO 2 . Additional aspects such as the solar irradiation and type of melting curves used are also defined. For completeness, we note here that the suffix "-A" is necessary in order to mark a clear distinction to the "Reference-B" special setting that is used in Section 4.
3. "Ref-B" differs from "Ref-A" in that it does not include CO 2 .
The experiments are organized as follows: We firstly examine the thermal and dynamical evolution of the magma ocean in the absence of an atmosphere and under the influence of grey /lbl atmosphere (Section 4.1). We examine the simultaneous evolution of H 2 O and CO 2 outgassing, and vary the initial volatile abundances in order to calculate their effect on the magma ocean solidification time (Section 4.2). We quantify the minimum (2000) and (Fiquet et al. 2010) (Section 2.3) TRF,0
Temperature of rheology front at z = 0 1645 K Fconv
Convective heat flux parameterization soft turbulence Eq. (6) Solomatov (2007) remnant volatiles in the mantle at the end of the magma ocean and we study the influence of the choice of melting curves on the evolution of water outgassing. Concluding the overview of the coupled interior-atmosphere system, we then study the separate influence of each parameter (or parameterized process) upon the solidification time (Section 4.3). In Sections 4.4-4.7 we shift our focus to the influence of the steam lbl atmosphere and use the atmospheric calculations of the companion paper. We show the qualitative difference between the grey and the lbl water vapor atmospheres (Section 4.4). We discuss the mechanism which separates the transient from the continuous magma ocean regime (Section 4.5), and we find the critical albedo that separates the two, for an atmospheric water inventory at a constant distance from the star (Section 4.6). Finally, in Section 4.7, we expand the critical albedo calculation with dependence on the outgassed water vapor and the temperature of the rheology front at the surface. In Section 4.8 we discuss the distinction between "evolutionary" and "permanent" magma oceans. The relevance of the results in the context of exoplanets is discussed in Section 4.9 and a summarizing plot of the solidification time according to the factors examined is provided along with the Discussion.
RESULTS
Thermal and dynamical evolution
In a similar fashion to prior studies of the magma ocean solidification (Zahnle et al. 1988;Abe 1997;Elkins-Tanton 2008;Hamano et al. 2013Hamano et al. , 2015Lebrun et al. 2013;Monteux et al. 2016;Schaefer et al. 2016;Hier-Majumder & Hirschmann 2017) we present the thermal evolution using the state variables: surface temperature, potential temperature, heat flux, Ra number, and MO depth evolution. This enables both model validation and comparison. We adopt the multipanel approach of Lebrun et al. (2013) that is convenient for comparisons between varying modeling approaches. We performed four simulations for the following cases: i) absence of atmosphere referred to as the blackbody "bb" case, ii) a grey atmosphere composed of both H 2 O and CO 2 "gr-H 2 O/CO 2 ", iii) a grey atmosphere composed of only H 2 O "gr-H 2 O" and iv) a H 2 O atmosphere treated with a line-by-line "lbl" model ( Fig. 3). Apart from the representation of the atmosphere or absence thereof, all aspects of the model follow the Ref-A case (Table 1).
Commonly in all simulations, the T p and T surf coevolve until an abrupt difference between the two marks the end of the magma ocean ( Fig. 3B dashed lines); the liquid-like behavior comes to an end and a layer with a melt fraction of 40% or lower remains. The average viscosity increases by more than 8 orders of magnitude across the critical melt fraction, taking values from 10 8 to 10 18 Pa·s (not shown). A smooth variation across this interval would be difficult to justify under the assumption of fractional crystallization (e.g. Marsh 1981). At this point, the whole domain switches to low cooling flux that characterizes solid-like convection (Eq. 6) and the surface temperature drops abruptly, while the potential temperature remains unaffected.
The "bb" thermal evolution compares well with that presented by Lebrun et al. (2013). It demonstrates the highest T p − T s difference. The mantle consequently cools rapidly (0.002 Myr) with the highest convective flux (F = 5 · 10 6 -10 4 W/m 2 ), caused by this large tem- (Table 1).
F & Q rad (W/m 2 ) Time (yr) F bb F gr-H 2 O/CO 2 F gr-H 2 O F lbl Q rad (t planet =100 Myr) Q rad (t planet =2
perature difference. Longer solidification times (0.15 Myr) are found by Monteux et al. (2016) who assume a bb radiative cooling (F = 10 5 -10 2 W/m 2 ) but a different interior model with a heat contribution from the core. The bb case is only relevant for planets that lose their outgassed atmosphere instantaneously. For a planet that retains its atmosphere, the grey approximations show that the presence of the additional greenhouse species CO 2 contributes only 0.05 Myr to the solidification time and is less significant in comparison to the water (0.16 Myr vs 0.21 Myr MO duration). The longer solidification time (≈0.4 Myr) obtained by Lebrun et al. (2013) for their grey two-species case is consistent with absorption coefficient k 0,CO2 = 0.05, which is likely to be rather high for those climates (see Section 2.8.1). The grey approach employed in this study and theirs follows Abe & Matsui (1985) and should not be identified with other grey models used in the literature:
The study of Hamano et al. (2013) (2017) formulates a hybrid energy balance for the atmosphere employing elements from both Abe & Matsui (1985) and Hamano et al. (2013). For potential temperature equal to the equilibrium temperature it results in net radiative warming of the planet. Moreover, the study of Hier-Majumder & Hirschmann (2017) preserves the mantle fully molten for the majority of the MO period due to the lack of convective cooling sink. The slow solid-matrix compaction process provided from their detailed melt/volatile percolation model further increases the solidification time (3 Myr) in comparison to our study. Lastly, we obtain lower solidification time in comparison to Hamano et al. (2013) who define the MO end at the surface solidus and not at the higher temperature of the critical melt fraction.
The cooling path can be followed from the convective heat flux, the MO depth and the Ra number (Fig. 3B,C,D). For about 50% of its lifetime, the magma ocean has a depth equal to or smaller than 50 km for the Ref-A case. The intersection of the adiabat with the rheology front at the two pressure depths where switches in the parameterization of the melting curves occur (see Section 2.3 and Appendix A), results in equal characteristic jumps in MO evolution. The decrease in the cooling flux is independent of the decrease in depth D, since D is explicitly overwritten when using the soft-turbulence parameterization (see Eqs. 5 and 6). Nevertheless, D defines the average viscosity within the convecting domain, which enters the Ra calculation (Eq. 5). Ra is ultimately responsible for the decrease in heat flux.
The role of radioactive decay as energy source in the MO evolution of an Earth-sized planet is insignificant (Fig. 3C), unless the planet is formed within few Myr, which includes the contribution of the short-lived elements 26 Al and 60 Fe. Theirs becomes comparable to the long-lived element contribution after 9 Myr and insignificant to the MO evolution by 7 Myr after CAI formation, in agreement with Elkins-Tanton (2012) findings.
Comparing the two atmospheric approaches, we find that the pure water vapor grey approximation underestimates the thermal blanketing in comparison to the lbl model because of the low absorption coefficient used to represent the whole thermal radiation spectra (k 0,H2O = 0.01). The lbl pure H 2 O model resolves better the steam IR absorption, although it overlooks the role of CO 2 .
The "bb-grey-lbl atmosphere" comparison captures the decreasing convective fluxes at the last MO time step as in the "bb-grey-spectrally resolved atmosphere" comparison of the Lebrun et al. (2013) model. In our approach this is due to the decrease in temperature difference and in Ra towards the MO end. However, upon evaluation at the last time step before solidification the Ra drop to 10 10 is not seen in their work (where Ra = 10 14 =const), likely due to differing average viscosity calculation and spatial resolution.
On a technical note, a Ra overshoot towards lower values is observed near the end of the magma ocean phase (Fig. 3D). Consequently, the switch to solid occurs abruptly from Ra = 10 10 to 10 12 , two orders of magnitude higher than the value obtained during convection of the last 1-km-deep liquid-like layer of magma ocean. This is a numerical artifact that correlates with high radial resolution of the model layers (≈ 1 km). Therefore, care should be taken when using convective heat flux parameterizations with high spatial resolution very close to the critical melt fraction, because the rheology becomes more complex at high crystal values.
Outgassing and atmospheric build up
The assumption of greenhouse gases H 2 O and CO 2 as major species is in accordance with an oxidized MO surface (Hirschmann 2012;Zhang et al. 2017) and bulk silicate Earth .
As far as the volatile solubility is concerned, molten silicate is a poor CO 2 solvent. It thus operates as a "CO 2 -pump" into the atmosphere. In contrast, H 2 O is highly soluble in the silicate melt and does not leave the mantle until the latest stage of the magma ocean where the enrichment in the melt peaks. The evolving atmospheric composition reflects those features as it transitions from a CO 2 -dominated to a H 2 O-rich one (Fig. 4). The major release (from 2.5 to 220 bar) of the water vapor occurs when the total melt fraction of the mantle reduces from 30% to 2% or as the potential temperature drops from ∼ 2200 to ∼ 1650 K (Fig. 5). This effect is the basis for the so-called "catastrophic" outgassing of a steam atmosphere (e.g. Lammer et al. 2013). It reflects the progressive replacement of melt volume with solid volume that has a small capacity for storing volatiles. Table 1). Absolute quantity of outgassed volatile in the atmosphere (solid lines) and relative mixing ratio at the surface (dashed lines) are shown.
In addition, the choice of melting curves defines the degree of melting throughout the magma ocean lifetime, and similarly affects the accompanying outgassing process. We find that over the range T p ∈ [3000, 2200] K the melt fraction differs by 10-43% at the same potential temperature, comparing chondritic and peridotitic composition for the lower mantle (Fig. 5). The choice of lower mantle melting curves does not affect the final outgassing but modifies the onset of catastrophic outgassing by maximum 5% of the total volatile volume. Therefore, chondritic composition for the lower mantle disfavors early water release for a cooling magma ocean for potential temperatures above 2200 K. melt % Synthetic-Fiq10 melt % Synthetic-Andr11 P out Synthetic-Fiq10 (300bar) P out Synthetic-Andr11 (300bar) P out Synthetic-Fiq10 (3000bar) P out Synthetic-Andr11 (3000bar) end of MO Figure 5. Effect of the choice of melting curves on the variation of the mantle melt fraction with potential temperature. The end of the magma ocean occurs at 99.3% solidification (grey shaded area). Two sets of melting curves are compared: "Synthetic-Fiq10" (dashed lines) and "Synthetic-Andr11" (solid lines) that share the Herzberg et al. (2000), Hirschmann (2000) and Zhang & Herzberg (1994) Simultaneous to the final outgassed quantity, we also calculate the relative volatile inventory extracted from the mantle assuming different initial concentrations (Fig. 6). As expected, the higher initial concentration results in higher outgassing. However, the relative quantity varies as follows: We find that [45%, 10%] of the initial water reservoir remains in the mantle for the examined range X ∈ [10 −5 , 10 −1 ] respectively, while the rest [55%, 90%] is in the atmosphere. This suggests that the lower the initial mantle abundance, the larger is the relative amount of water stored in the planet's interior after the magma ocean ends. By contrast, only ≈ 6% of CO 2 remains in the mantle for an Earth-sized planet independently of the initial concentration assumed.
Effects of model parameters on the MO lifetime
The combined H 2 O/CO 2 inventory was found to delay the MO termination in prior works (e.g. Zahnle et al. 1988;Abe 1997;Lebrun et al. 2013). The Elkins-Tanton (2008) work considers different MO depths (2000km, 1000km, 500km) than our global MO for Earth (2890 km). Consequently, the volatile masses differ for the same assumed concentration and a direct comparison is not possible. Recently, Salvador et al. (2017) have studied the effect of water abundances on the global MO solidification time yielding longer durations likely due to the use of a non-grey atmospheric model. In our study we quantify the solidification time (t s ) by sampling a larger domain of initial abundances for the two species and assuming a grey atmosphere (Fig. 7). The MO duration amounts to ≈ 0.21 Myr for conservative Earth volatile abundances while it would reach 5-10 Myr for an (unlikely) Earth-sized planet made entirely out of carbonaceous chondritic (CC) material with 1 wt% of H 2 O. Our results confirm that the atmosphere is the most important solidification delaying factor.
However the effect of each separate interior process on the duration of the magma ocean stage remains difficult to disentangle and it would help clarify future modeling priorities. In Table 2 we present an overview of the effect of additional factors and parameters on the MO solidification time (t s ). Each t s is obtained through varying parameters and/or including a different process (first column). The second column states the number of parameters (three at most) that have been modified in each We thus obtain the tendency of each factor to increase or decrease the solidification time ("+" or "−" sign respectively) as well as its magnitude. Below we discuss only the most crucial contributions. When accounting for the water dependence of the melt viscosity in exp. 3 we expect a shorter solidification time, that reflects the more efficient convection due to lower viscosity. η l decreases due to progressive enrichment of water concentration in the melt during the MO evolution (from 410 to ≈ 10 4 ppm (Ref-A)). The atmospheric radiative forcing remains identical to the Ref-A case. The expected cooling acceleration is counteracted by the delaying role of the outgassed vapor atmosphere (Exp. 3), even so for particularly water-rich settings (as seen by the almost identical t s of water-rich exp. 1b and 3b). The effect of viscosity on t s becomes evident in the black body cases (Exp. 11,12). With respect to the black body case of experiment 11 (t s = 2000 yr) that uses constant 10 wt% water content (Karki & Stixrude (2010)), we observe an increase in the solidification time (t s = 2713 yr) in exp. 12b that uses water dependent viscosity. This is explained by the fact that in exp. 12b. the 10% water enrichment occurs only at the latest MO stage and not throughout the whole run. Our parameterizations show that one order of magnitude enrichment in H 2 O in the melt causes a decrease of up to two orders of magnitude in the viscosity (Fig. 2). This becomes important at lower melting temperatures T RF,0 < 1400 K which correspond to evolved silicate melts (Parfitt & Wilson 2008). Experiments 12a and 12b confirm the tendency we hypothesized for the viscosity role in decreasing t s with increasing water content (410ppm and 10 4 ppm accordingly). Therefore the water-enriched melt accelerates the solidification process and it should be taken into account for evolved surface compositions or planets around EUV and XUV active host stars that lose their atmospheres. According to Abe (1997) low viscosity enhances the differentiation of minerals. Therefore, such a η l parameterization is also vital in better modeling the mineral solidification sequence.
Using the hard turbulence approximation for the convective flux rather than the soft approximation yields a slight increase in the solidification time (experiment 5). The abrupt decrease of ≈ 1000 K in the surface temperature at the MO termination is reduced by up to 300 K by employing the hard turbulence parameterization. During this, the P r number is updated according to the evolution of the liquid viscosity and the flow aspect ratio (λ) takes values between 1 and 2. Significant work that has been done in this direction shows numerical proof of the hard turbulence regime (Grossmann & Lohse 2011) and suggests that it could affect the thermal transport controlled by the boundary layers (Grossmann & Lohse 2003;Lohse & Toschi 2003).
In experiment 6 we examine the role of uncertainty in the upper mantle (0 − 22.5 GPa) solidus. The ±20 K error estimated in the solidus expression of Herzberg et al. (2000) has a measurable impact (+4%) on the solidification time. The mere uncertainty in the experimental data can thus affect the magma ocean solidification time by a few thousands of years.
Further decreasing the upper mantle solidus by 50, 100 and 400 K causes the solidification time to decrease by 10, 20 and 108% respectively. Compositions more silicate-evolved compared to the KLB-1 peridotite have such lower melting temperatures. The -400 K value corresponds to rhyolite (Parfitt & Wilson 2008). Lebrun et al. and Salvador et al. (2017) previously acknowledged that the chemical composition of the magma ocean at its latest stages would be a decisive factor in the evolution. Schaefer et al. (2016) and Wordsworth Table 2. Overview of the effects of various parameters on the solidification time. Different scenarios are compared to a reference case. The scenarios consist of varying or replacing a parameter or physical process as indicated in the first column. The total number of changed parameters (three at most) with respect to the reference scenario is indicated in the second column. The employed parameter values and/or the description of the process are in the third column. The fourth column shows the solidification time ts, and the fifth and sixth columns the absolute and relative difference of ts with respect to the reference cases A or B. (2018) further resolved the chemical evolution for specific compositions. Our result emphasizes the controlling role of the surface melting temperature in the solidification duration and reveals a linear dependence between them. The solidification time is however insensitive to changes in the lower mantle melting curves (experiment 7) as long as bottom-up solidification is ensured. The reason is that they affect neither the amount of CO 2 in the atmosphere, the majority of which is degassed at the beginning of the magma ocean phase, nor the water enrichment which does not occur at high MO depths.
In experiment 8 we test the effect of linearizing the melting curves of Abe (1997), where the solidification time decreases significantly (-39%). The higher melt fraction preserved at the end of the magma ocean is tied to lower final outgassing, which explains the difference to the Ref-A setting. Lebrun et al. (2013) has previously discussed a similar effect of the curve linearisation. A quantitative comparison is however inconclusive due to the different atmospheres used.
Qualitative difference between grey and lbl atmospheric blanketing
We clarify a fundamental difference between the atmospheric approximations that were implemented in this work. We illustrate this by assuming a high (F Sun (S = 1361 W/m 2 , α = 0.11) = 303 W/m 2 ) and a low (F Sun (S = 1361 W/m 2 , α = 0.30) = 238 W/m 2 ) incoming solar radiation (Fig. 8). The difference is only in the assumed albedo value, 0.11 or 0.30. In the lbl approximation (Fig. 8A, 8B), the colormap combinations of P H2O and T surf lead to planetary cooling. In the high F Sun case, for each value of the surface temperature T surf between 700 and ≈ 1700 K there exists a threshold value of outgassed water P H2O across which the net radiation balance at TOA is negative and the planet warms. This effect is absent in the low F Sun case, which yields a cooling regime for all combinations of P H2O and T surf . On the contrary the grey approximation shows a negligible difference of the magma ocean cooling flux of the order of 10 −1 W/m 2 , accounting for the T eq of our solar system's inner planet orbits (Fig. 8C). In fact the grey atmosphere is insensitive to variations in the incoming stellar radiation.
The reason is that in the grey energy balance (Eq. 15), the incoming solar flux enters only in the calculation of the equilibrium temperature. The latter does not vary more than a factor of 2 over the insolation range in our solar system history (T eq = 144 K for the case of the young Sun and T eq = 256 K for today's Sun at 1 AU). The fourth power of T eq has a minor contribution compared to the fourth power of the surface temperature of the magma ocean, which is higher than T RF,0 = 1645 K (Ref-A) throughout the evolution.
In the limit of our convecting magma ocean model, we only explore cooling regimes and obtain the relevant solidification times. The convective cooling flux out of the magma ocean F conv requires T surf < T p to ensure the necessary gravitational instability for convection to occur (see Eq. (6)). However, if the flux at the TOA becomes negative (RHS of Eq. (16)) the system would warm resulting in T surf > T p , a condition which describes a stably stratified system that will not convect.
The remaining three Sections 4.5-4.7 focus on the cooling/warming limit found with the lbl atmosphere.
Lbl atmosphere: separating continuous from transient magma oceans
The lifetime of a magma ocean with a steam atmosphere is controlled by the longwave radiation through its steam layer, the energy received from the star, and the melting temperature of the mantle at its surface. All above factors combine into a comprehensive mechanism that distinguishes between a "transient" (or "shortterm", or "type-I" after Hamano et al. (2013)) and "continuous" (or "long-term", or "type-II" after Hamano et al. (2013)) MO evolution path. Goldblatt (2015) and Ikoma et al. (2018) have discussed the warming/cooling distinction, always in relation to the constant radiation limit for the runaway greenhouse (RG) ≈ 300 W/m 2 . We exemplify this idea with an emphasis on the additional role of T RF,0 .
We use two simulations that are subject to different insolation conditions, namely F Sun,low = 238 W/m 2 and F Sun,high = 563 W/m 2 (Fig. 9a black solid line and black dashed line respectively), leaving all other parameters unchanged. The F Sun,high is obtained using S = 2648 W/m 2 that corresponds to the incident radiation at the orbital distance of Venus for today's Sun and α=0.15, while the F Sun,low is equal to the incoming radiation at Earth orbit today. Since F Sun is independent of T surf , it is plotted as a line parallel to the T surf axis (Fig. 9a). Both simulations have the same water reservoir (405 bar or 550 ppm initial concentration) to ensure outgassing of 1 Earth ocean (300 bar) at the end of the magma ocean stage. OLR T OA as a function of T surf is plotted for three values of atmospheric water content (4, 100, and 300 bar), which we term "isovolatiles" (grey lines). F Sun intersects with each isovolatile over a temperature value T surf . The cooling flux F conv (read on the right axis) becomes zero for that specific water content and the planet ceases to cool. If T surf is higher than the mantle rheology front temperature at the surface (T RF,0 ), the steam quantity indicated by the respective isovolatile balances the energy flux from the star and the MO does not solidify.
Firstly, we examine the trajectory of the convective flux of the transient magma ocean on the (T surf , F conv ) plane as it cools from T surf = 3000 K and with an insolation F Sun,low (Fig. 9a, red solid line). F conv progressively crosses isovolatiles of higher water content. As it approaches the highest outgassed quantity of 300 bar, the difference OLR T OA − F Sun,low = F conv remains always positive since the 300 bar isovolatile allows the system to dispose of heat at a higher rate than it receives solar radiation. The high convective flux value ensures cooling until T surf = T RF,0 , which marks the end of the magma ocean. The abrupt cooling after the end of the magma ocean stage and the final outgassing quantity are shown in the evolution of T surf (t) and P H2O (t) (Fig. 9b, 9c.) Secondly, we obtain a long-term magma ocean (Fig. 9a, red dashed line) in a scenario that assumes F Sun,high . Initially, for high values of T surf , the same amount of water as before is outgassed and its F conv almost coincides with the one of the short-term case (the difference hardly noticeable on the logarithmic graph is ≈ 10 3 W/m 2 ). During evolution the outgassing proceeds and the simulation trajectory crosses isovolatiles of higher water content. F conv drops to very low values that tend to numerical zero for T surf = T surf ≈ 1915 K. The intersection of the incoming radiation F Sun,high with the respective isovolatile over T surf reflects the steam atmosphere already outgassed when the system ceased to cool. We obtain a point that falls between the isovolatiles of 100 and 300 bar (167 bar read in Fig. 9b). Consequently, a continuous magma ocean is maintained at potential temperature ≈ T surf (Fig. 9c) due to a specific combination of incoming solar radiation, its intersection with the 167 bar isovolatile, and the solidification temperature (Fig. 9a). Note that the long term MO ocean is maintained with less water than one Earth ocean and at an insolation higher than the RG limit.
The prominent role of T RF,0 on the MO type becomes evident when comparing the point (T surf , P H2O ) where the isovolatiles intersect F Sun , with T RF,0 . For the short-term magma ocean the intersection point A occurs well below T RF,0 . That magma ocean stage will be transient for every possible outgassing scenario within the [4,300] bar range. In the case of the higher solar irradiation, we have intersection points with each iso- Katyal et al. (2019) and C: the grey approximation of Abe & Matsui (1985) as used in Elkins-Tanton (2008). In all three plots only the net cooling Fatm,T OA (positive sign convention) is shown in the colored legend. The grey approximation results exclusively in cooling fluxes for both cases examined. Figure 9. a: Mechanism for separating a continuous (long-term) from a transient (short-term) magma ocean as a function of surface rheology front temperature TRF,0 (dashed blue line), isovolatiles of outgassed water pressure at the surface (grey lines), and incoming solar energy FSun (solid and dashed black lines). Two experiments with low and high FSun are performed using the same total water reservoir (405 bar). A short-term (red solid line) and a long-term evolutionary case (red dashed line) assuming respectively low and high insolation conditions (see text for values) is shown. Solar insolation is read on the left y-axis. The evolution of Fconv is read on the right y-axis. Points A, B, C and D mark the intersection of the isovolatile curves with the value of FSun considered and are used to explain different evolution scenarios. Isovolatiles cover surface pressures within 4-300 bar. b: PH 2 O (t) and c: T Surf (t) evolution for short-term and long-term MO. All parameter values unless otherwise explicitly mentioned are as in Ref-A. volatile (B, C, and D), which indicate different thermal evolution paths. On the one hand, the points B and C are located at surface temperatures higher than T RF,0 , which means that if the MO has outgassed the respective quantities of 300 and 100 bar by the time T surf is reached, it will cease cooling. On the other hand, the point D corresponds to a much lower temperature than T RF,0 , which means that a steam atmosphere of 4 bars under those insolation conditions can counteract the cooling process only if T surf decreases to 900 K. The respective magma ocean stage is transient, since it solidifies at a much higher temperature (i.e. 1645 K). The variation of the OLR as a function of P,T is explored in detail in the companion work. 4.6. Lbl atmosphere: Role of orbital distance and albedo on MO evolution
Clearly, the essential quantity regarding the planetary heat budget is F Sun , since it distinguishes the fate of the magma ocean between transient and continuous. Below we refer to this limiting incoming flux as F lim (where F conv =0) and we specify the incident solar radiation and albedo combinations which satisfy it.
On combining the stellar luminosity (L Star ):
L Star = (4πR Star ) 2 σT 4 ef f,Star ,(17)
where R Star is the stellar radius and T ef f,Star the effective temperature at the star's photosphere; with the expression of Gough (1981) for the evolution of solar luminosity we get T ef f,Star (τ ). Combining with the blackbody radiation law for the equilibrium temperature of a planet, we obtain the following equation:
R = R Star · T 2 ef f,Star (τ ) · √ 1 − α max 2 F lim (P H2O , T RF,0 ) σ .(18)
Eq. (18) relates to the F lim the maximum albedo α max that an Earth-sized planet at orbital distance R from a star of effective temperature T ef f,Star can possess in order to maintain a continuous magma ocean stage. The limiting flux included in the denominator of Eq. (18) is not constant but equal to:
F lim (P H2O , T surf ) = (1 − α c ) S(τ ) 1AU 4 ,(19)
where α c the critical albedo found with a sensitivity experiment for a given planetary volatile inventory. There S(τ ) 1AU is the solar constant at an orbital distance of 1 AU and stellar age τ , P H2O (in bar) the mass of the water vapor outgassed and T surf is the temperature over which the stellar insolation crosses the P H2O isovolatile (Section 4.5). The obtained limiting flux F lim maps to the data product OLR T OA (T surf , P H2O ). Using the Katyal et al. (2019) values of the limiting flux, we compared our Eq. (18) with an equivalent expression calculated by Hamano et al. (2013). The solution is similar with minor differences due to the astrophysical properties assumed. We generalise the formulation to cover our Sun or any other host star with a known photospheric temperature T ef f and radius.
Lbl atmosphere: dependence of F lim on the melting temperature and steam mass
The irradiation conditions which can pinpoint an Earth-sized planet stalling in a magma ocean stage just above the 40% melting temperature, are extended here to include a range of steam atmosphere masses that span [4,300] bar for two different T RF,0 values (the lower melting temperature is representative of a more evolved composition than the KLB-1 peridotite of Ref-A). Figure 10. OLRT OA as a function of surface temperature and outgassed water surface pressure 4-300 bar, based on data from Katyal et al. (2019). We consider two different surface solidification temperatures: TRF,0 = 1370 K as in Hamano et al. (2013) and TRF,0 = 1645 K as in Ref-A case. Colored points correspond to OLRT OA values obtained for the respective isovolatiles for different outgassed steam atmospheres 4, 25, 50, 100, 200 and 300 bar that overlie magma oceans of different TRF,0 (see Table 3 for explicit values). Note the variation in temperature coverage of OLR=const=282 W/m 2 , for different isovolatiles. Data by Nakajima et al. (1992) are used to complement the plot in the region where T surf TH2O,crit.
Different radiative flux limits F lim are obtained for the two values T RF,0 = 1370 K and 1645 K, depending on the vapor amount (Fig. 10). From the superposition of points that correspond to 200 and 300 bar at T RF,0 = 1645 K in Fig. 10, we note the tendency of steam atmospheres exceeding 200 bar to converge to the constant RG limit (RG=282 W/m 2 Katyal et al. (2019)). For T RF,0 = 1370 K atmospheres already equal to or higher than 100 bar suffice to reach the RG limit. A similar tendency is shown in Hamano et al. (2013).
We also find that at lower steam contents the F lim is greater than the RG-limit. All F lim values can be found in Table 3. Using Eq. (18) we calculate the orbital distance-albedo combinations for which the radiation limits (Table 3) of different isovolatiles are attained (Fig. 11). We assume the solar luminosity at the beginning of its main sequence evolution at τ = 100 Myr (72% of today's value) (Gough 1981). Not all the calculated albedo values are realistic. The albedo for a cloudless steam atmosphere, based on 1D models and 3D global circulation model calculations lies between [0.15, 0.40] (Kasting 1988;Goldblatt et al. 2013;Leconte et al. 2013;Pluriel et al. 2019).
Apart from the new radiation limits found, our results are in line with those of previous studies, as far as the insolation role is concerned. In Hamano et al. (2013) the threshold distance between continuous and transient MO types for albedo 0.3 and solar constant 0.72 S 0 is 0.77 AU, whereas under the same conditions our calculations show 0.79 AU. The difference is due to the lower absolute OLR steam atmosphere limit of 282 ± 1 W/m 2 that we obtain compared to the 294 W/m 2 limit employed in that study.
By raising the albedo to the critical value α c = 0.146 found in our simulations for an Earth-sized planet at 1 AU that outgasses 1 earth ocean at today's sun, we obtain ≈ 12 Myr MO duration. The longer solidification times of 10-30 Myr reported in Hamano et al. (2013) at the same F Sun =285.5 W/m 2 for the same steam atmosphere are due to the lower surface melting temperature used (T RF,0 =1370 K).
With an albedo of 0.63, a solar constant of 0.7 S 0 , and total planetary water content X H2O,0 = 5.53 · 10 −2 wt% (equivalent to 405 bar) and X CO2,0 = 1.4·10 −2 wt% (equivalent to 100 bar), Lebrun et al. (2013) found that the distance at which the outgassed water vapor could no longer condense is 0.67 AU. Under the same conditions, excluding the influence of CO 2 , we find in our model that the atmosphere would exist in a runaway greenhouse state trapped in a continuous magma ocean at a critical distance of 0.59 AU. The reason for this discrepancy is two-fold. Firstly, our lbl model approach does not include CO 2 that also contributes to the greenhouse effect. Secondly, the absolute OLR limit used by Lebrun et al. (2013) is ≈200 W/m 2 (Marcq 2012) (see Marcq et al. (2017) for an updated limit). This is substantially lower than the limit of 282 W/m 2 used in our study. Therefore, the shift of our limit inward towards the star corresponds to the higher critical flux that needs to be achieved in order to trigger the qualitative shift from a transient to a continuous magma ocean regime. Considering the orbital distances of the inner terrestrial Figure 11. Maximum albedo that a planet with a 4-300 bar steam atmosphere can possess at a given distance from the young Sun in order to maintain a long-term magma ocean, assuming two values of TRF,0. We plot the critical values that separate continuous from transient magma ocean cases calculated with the use of Eq. (18) for several PH 2 O (colored lines) and employing the respective steam atmosphere mass limiting outgoing longwave fluxes (extracted from Fig. 10; see Table 3 for values). Dashed lines are obtained using the equation of Hamano et al. (2013) employing our F lim limits. Not all obtained albedos are realistic. Hatched region shows the possible range of albedos for a cloudless steam atmosphere (Kasting 1988;Leconte et al. 2013;Goldblatt et al. 2013). Black solid lines mark distances from the star for which Teq is equal to the surface melting temperature of the magma ocean for the full albedo range (permanent magma ocean).
planets of the solar system (Mercury-0.38 AU, Venus-0.72 AU, Earth-1 AU and Mars-1.52 AU), we find that the planets inwards of Earth could sustain a continuous MO within the range of albedos expected for a cloudless steam atmosphere (Fig. 11). Moreover, a 100-Myr-old Earth at 1 AU around the Sun cannot exist in a continuous MO state under any albedo for a steam atmosphere of up to 300 bar (Fig. 11) or of up to 1000 bar according to the recent study of Ikoma et al. (2018).
Note that in this work "continuous" magma ocean refers to planets that would cease cooling if the amount of steam in the atmosphere was conserved. This cannot be ensured under atmospheric escape processes, which have not been accounted for, and as such the limits calculated here yield the furthest possible distance from the Sun for achieving a continuous MO with a constant atmospheric steam content.
Using the database of F lim that depends on both the atmospheric water content and T RF,0 that we provide in this (Table 3) and in the companion work, Eq. (18) is qualitatively extended. It covers MO type transitions for intermediate levels of outgassing below the 300 bar reference value, hence has higher F lim . This database is backwards compatible and can also be used in the Hamano et al. (2013) equation.
Evolutionary and permanent magma oceans
We draw specific attention to the difference between "evolutionary" and "permanent" magma ocean which is studied in other works (e.g. Hammond & Pierrehumbert 2017). Both the transient and continuous magma ocean that we study belong to the so called "evolutionary" magma oceans generated during the accretional process. However, within a certain orbital distance the energy for melting the mantle is already provided by the solar irradiation alone and the atmosphere blanketing effect becomes irrelevant. This is the "permanent" magma ocean caused by the star. Radioactivity and delivery of kinetic energy cause the evolutionary magma ocean. To help distinguish these, the minimum distance from the star for which the equilibrium temperature T eq equals the 40% melt fraction surface temperature (where permanent MO stage ensues) is indicated by the black lines in Fig. 11. Further research, which is beyond the scope of this study, could lead to an expansion of the orbital distance defining the permanent MO, on accounting for climate feedbacks that raise the surface temperature above the T RF,0 melting point.
Magma oceans on other planets
F lim is not significantly affected by the gravitational acceleration of the planet as long as this has between 0.1 and 2 Earth masses (Goldblatt et al. 2013). For greater planetary mass the pressure levels in the atmosphere change height, as does the level of opacity depth which is crucial for the calculations of the outgoing radiation. Goldblatt et al. (2013) also calculated that for a planet of half the Earth mass, the OLR limiting flux is lowered by only 5 W/m 2 . In comparison, F lim ≈ 282 W/m 2 ± 1 W/m 2 as calculated for the Earth by Katyal et al. (2019) has a lower uncertainty. Therefore, Eq. (18) can be applied without loss of generality to planets between 0.1 and 2 Earth masses, using the F lim (P H2O , T RF,0 ) ( Table 3) calculations by Katyal et al. (2019) .
Given their similarity in mass and radius, the criteria for a continuous magma ocean applied for the Earth can be extended to Venus. A continuous magma ocean could not have been possible for the Earth during the young Sun period for any bulk water abundance. We find however, that Venus orbit qualifies for a long-term magma ocean within a wide range of planetary albedos [0.15, 0.40] proposed for cloud-free steam atmospheres, as long as its outgassed steam atmosphere amounts to 200 bar or more for a surface solidification temperature of 1645 K (Fig. 11). In the case of the lowest solidification temperature (T RF,0 = 1370 K), the minimum atmosphere required for a continuous magma ocean at Venus orbit is 50 bar (Fig. 11). This highlights that the melt composition alone could dictate a different magma ocean evolution path for two hypothetical planets with equal water vapor atmosphere masses.
It is additionally important to consider whether a planetary body has had a long impact history or has chemically evolved before impacts remelt it into a magma ocean (Lammer et al. 2018). Such bodies could more easily maintain a secondary continuous magma ocean. Due to their lower T RF,0 they would require smaller steam atmospheric mass, instead of the reference one Earth ocean (300 bar) usually assumed in runaway greenhouse studies. On the contrary, a chemically unevolved silicate primitive composition that melts at high temperatures would require a massive steam atmosphere >100 bar in order to maintain a continuous magma ocean. We conclude that past events of chemical alteration may influence the fate of the magma ocean under the same orbital configuration. Therefore the age of the star and of its planetary system matters. Evolution of the mantle composition during the MO solidification (Elkins-Tanton 2008; Schaefer & Fegley 2010) will be an additional factor that prolongs the MO lifetime if it results in decreased T RF,0 .
Interestingly, the drop of surface temperature during cooling combined with the tendency of stellar environments to gradually strip planets of their atmospheres (Johnstone et al. 2015;Odert et al. 2017;Lammer et al. 2018) (therefore lowering the surface pressure P H2O ) could result in the same outgoing radiation limit during planetary evolution. We see this in Fig. 10 taking any constant OLR value that crosses multiple isovolatiles. In that case the stellar evolution plays a primary role in the fate of the continuous magma ocean. A G-star with increasing luminosity with time (Gough 1981) favors the maintenance of an existing magma ocean because it contributes warming at the critical distance. In contrast, continuous magma oceans will be more elusive around M-stars whose luminosity decreases with time (Baraffe . This is because a continuous magma ocean close to its critical distance will receive less and less stellar radiation eventually creating a window of cooling. A buffer against this effect is the additional vapor outgassing that increases the opacity and lowers the required F lim . However, during progressive cooling the interior will exhaust its water supply into the atmosphere. Under these conditions only water-rich planets can sustain a continuous magma ocean. This shows that there are numerous processes that affect MO feasibility. Consider also the possible Trappist-1 exoplanet migration scenarios (Unterborn et al. 2018) (suggested in order to justify water-rich composition).
We explore our findings in view of potentially rocky exoplanets having radius and/or mass within few Earth units (parameters in Appendix E). Results suggest that there are orbital regions where the magma ocean can be transient, permanent and an intermediate region where it is "conditionally" continuous (Fig. 12). "Conditionally" here refers to the dependence on water content and rheology front temperature. We observe the overlap in the regions of the continuous magma ocean for different T RF,0 . Considering the interior composition adds a measurable level of uncertainty since different planets with different atmospheric water content and solidification temperatures can be characterized by the same outgoing OLR. Note that unless we are able to constrain the surface pressure of water vapor on an exoplanet, not feasible with the current observational capabilities (Madhusudhan et al. 2014), we will not be able to constrain the type of evolutionary (continuous, transient) magma ocean. However, hypotheses and proxies concerning planetary water abundance could break this OLR degeneracy that disappears at low vapor pressures close to 4 bar (see Fig. 10). A water-poor planet with a thin atmosphere of 4 bar water would be sensitive to the T RF,0 value for developing a continuous/transient MO close to their separation limit. Such could be the case for distinguishing the compositions of HD 219134 b and c if one is found in magma ocean state and the other is not (Fig. 12). Kepler 36b' s orbit is further than this distinction possibility and receives enough energy from the star to be in continuous magma ocean as long as it has at least 4 bar water. As soon as its atmosphere is lost it would resemble a black body on which the liquid viscosity effect of any water present would ensure the rapid MO solidification, as we showed earlier. Planets Kepler 236c, Ross 128b and LHS 1140c on the contrary are located in the F lim region for relatively high vapor pressure. Assuming ≥200 bar the system converges to the minimum OLR solution of 282 W/m 2 (see Fig. 10) which is maintained for up to 1000 bar (Ikoma et al. 2018). Detecting any magma ocean state on those planets would be difficult because of the opaque atmosphere. However, if detected it would mean that the planet formed within a water-rich environment that ensured the minimum atmospheric 200 bar required for the continuous magma ocean. Especially for LHS 1140c, the planet LHS 1140b located in the transient MO region of the same system could provide complementary information for the likelihood of high water content. GJ 1132b is located at the compositional distinction limit. Its potential MO has been studied before by Schaefer et al. (2016). A low atmospheric water content in its MO state would be a proxy of primitive silicate composition. Any of the continuous magma oceans on those planets would eventually solidify if their atmospheric water were lost and were not replenished by the interior.
The possibility of observing a transient magma ocean system is insignificant due to the order of million years duration that we find for them, which is very short compared to observable systems' ages. Detection of continuous magma oceans on candidate planets (at orbits receiving 282 W/m 2 or more (see Fig. 12) is challenging but is aided by the fact that the planet's MO brightness temperature would be much higher than that corresponding to its equilibrium temperature, yielding OLR of up to 16,000 W/m 2 (see Table 3). Such measurements require secondary transit observations as carried out for 55 Cnc e with the Spitzer telescope (Demory et al. 2016) aided by the longer wavelength coverage of JWST. A low brightness temperature, in agreement with a low OLR of 282 W/m 2 , would be an indication towards high steam pressures (See companion paper for possible emission spectra). The surface pressure is not retrievable with the current capabilities but promising methods are developped for low pressure atmospheres (10 bar) that demonstrate pressure broadening of absorbers such as CO 2 and O 2 (Misra et al. 2014). Transmission methods could not probe high surface pressure atmospheres but the latter's OLR would be already near the runaway greenhouse limit in those cases so one should focus in retrieving the latter. A measured OLR=282 W/m 2 would be indicative of MOs with high steam pressures. We suggest the auxiliary/complementary use of observations obtained from the permanent magma ocean type, such as potentially on 55 Cancri e (Demory et al. 2016;Angelo & Hu 2017) and Kepler 78b. From there one could isolate characteristic atmospheric signatures such as: the atmospheric effects of evaporated silicate species that develop over the molten rocky surface (Fegley et al. 2016;Kite et al. 2016;Hammond & Pierrehumbert 2017) and the oxides in the presence of a steam atmosphere (Fegley et al. 2016). Detecting similar silicate cloud signatures on planets close to the continuous MO compositional distinction that is observed at low vapor pressures (4 bar) would serve as a proxy of their composition (T RF,0 ) and of their water content.
Detection of evolutionary magma oceans additionally requires stellar ages in order to focus on systems with ongoing planetary formation, preferably after recently completed accretion. Constraining the albedo from observations is a possibility given favorable orbital configurations (Madhusudhan et al. 2014;Kite et al. 2016) and would help define the range of orbital distances for a conditionally continuous magma ocean.
DISCUSSION
We previously showed how the MO duration is tied to the outgassing. The latter is sensitive to factors that modify the amount of enclosed melt or the upper mantle temperature. Two such factors are the assumptions of surface rheology front temperature and critical melt fraction. They vary significantly among studies and are sources of deviations when comparing with our Ref-A results (e.g. Hamano et al. 2013, T RF,0 =1370 K) , T RF,0 =1560 K) and (Hier-Majumder & Hirschmann 2017, φ C = 0.30).
However, keeping both above assumptions constant, the outgassing in this study still represents an upper limit with respect to other studies. The reason is twofold. Firstly, the use of the one-phase adiabat (Section 2.4) minimises the amount of enclosed melt at the end of the MO due to its high slope with respect to the melting curves. From the mass conservation follows that the volatile outgassing into the atmosphere maximises. Employing a two phase adiabat instead tends to parallelize the slope to the melting curves and results in more enclosed melt and lower outgassing (e.g. the Ref-A case in Lebrun et al. (2013) outgasses 200 bar H 2 O compared with 220 bar (this study) via this effect). However, the use of the Solomatov & Stevenson (1993a) two-phase adiabat is subject to strict assumptions (i.e. linear melting curves).
Secondly, we did not account for the depression of the solidus that accompanies the mantle enrichment in water (Katz et al. 2003). Initially, note that the parameterization suggested by Katz et al. (2003) modifies the surface melting temperature T RF,0 above the error margin (20 K) for an atmospheric pressure ≥ 30 bar. Furthermore, it is only valid for pressures up to 8 GPa, corresponding to a depth of 220-250 km. Indeed, it was motivated by solid state mantle dynamics and explicitly designed to aid modeling of melt generated locally at shallow depth (Noack et al. 2012;Tosi et al. 2017). It cannot be extrapolated to higher pressures in the upper mantle, let alone throughout the range of a global MO (covering pressures from the surface down to 135 GPa).
Nonetheless, based on our mass balance (Eq. (13)), we make a first order estimation of the melting temperature reduction effect during increased water concentration in the melt. Assuming that both the solidus and the liquidus are reduced by the same amount for the same water content (see Katz et al. (2003), Section 2.2), the MO solidification will take place at a lower temperature. In this respect our model provides lower bounds on the solidification time for the same outgassed quantities (Fig. 7). However, estimating the melt fraction using a wet solidus comprises more than a linear shift of melting curves, which would leave the MO final melt fraction unchanged. In fact, the inversion of the saturated solidus appearing near the surface is not necessarily matched in the non-linear shape of the saturated liquidus (Makhluf et al. 2017). A wet solidus essentially would increase the enclosed melt at the MO end. Based on our current anhydrous parameterization our final outgassing estimations are upper limits because the remnant melt is here minimum (Fig. 6). A detailed study is required to quantify the overall effect on the solidification time taking into account the surface solidus depression and the decrease in degassing which exert opposing tendencies on the MO duration.
Factors that decrease the T RF,0 (see Table 2), such as atmospheric steam pressure (1000 bar cause a decrease of 100 K (Katz et al. 2003)), melt silicate content (decrease by up to 400 K), and redox state would further increase the solidification time. Significant work has been done towards resolving melt redox evolution (e.g. Schaefer et al. 2016;Wordsworth et al. 2018) and combining it with silicate content evolution in the melt (Gaillard et al. 2015), which is a future step for detailed modeling.
Dynamically, the MO termination is characterized by two main non-linearities. One is the decelerating advance of the solidification front from the bottom upwards that results in a shallow magma ocean of 50 km or less for ≈ 50% of the magma ocean lifetime. The other is the abrupt end of the magma ocean stage, which is marked by a discontinuous viscosity jump of >8 orders of magnitude across the critical melt fraction. The catastrophic H 2 O outgassing phenomenon is an additional non-linear process. For an Earth-sized planet it ensues when the total melt volume fraction drops below 30% (Fig. 5). Adopting the Katz parameterization for the late shallow MO stage does not prevent that degree of solidification. This is because even if the solidus depression were to ensure fully molten water-enriched layers, its maximum range of validity is 8 GPa. This barely covers 10% of the Earth mantle volume. It takes a combination of solidus depression at higher pressures (not yet confirmed) and a two-phase temperature profile such that global melt remains higher than 30% of mantle volume, in order to hinder the abrupt H 2 O outgassing (Fig. 5). Our model shows that initial cooling is instead very rapid and causes the solidification of 90% of the mantle within few thousands of years via bottom-up crystallization (Fig. 3). The phenomenon could be mitigated if solidification proceeded from the middle outwards, maintaining a large part of the mantle molten in the form of a basal MO (Labrosse et al. 2007). A detailed two-phase flow model such as Hier-Majumder & Hirschmann (2017), expanded to cover the middle point solidification, is required to quantify this effect in detail.
Moreover, we adopt here the two atmospheric species H 2 O and CO 2 but acknowledge the need for including additional trace species that may alter the radiative balance and/or react with surface melt (Gaillard & Scaillet 2014;Lupu et al. 2014;Zhang et al. 2017;Wordsworth et al. 2018). In addition, processes that alter the albedo during the MO evolution (see recent work by Pluriel et al. (2019)) could have an effect on the MO evolution into transient or continuous type.
The immediate outgassing of CO 2 could have an effect on the hydrodynamic escape process which usually is studied on the assumption that CO 2 is a minor gas in the atmosphere (Hamano et al. 2013;Lupu et al. 2014;Hamano et al. 2015;Airapetian et al. 2017;Wordsworth et al. 2018). In particular, a low mixing ratio of water in the atmosphere together with abundant CO 2 is known to create a cold trap over the convective atmospheric region and to hinder the thermal escape of the heavier and ionized oxygen atoms. Wordsworth & Pierrehumbert (2013) argue that high CO 2 mixing ratio would not effectively prevent the escape. A posterior water regassing process suggested by Kurokawa et al. (2018) could operate through early plate tectonics and could mitigate water loss. It could maintain 2-3 earth oceans bound in the interior against hydrodynamic escape and justify D/H ratios. Combining the response of varying atmospheric composition, a baseline evolution of which we provide here (Fig. 4), to different scenarios of early XUV stellar radiation (Lammer et al. 2008;Johnstone et al. 2015;Airapetian et al. 2017;Odert et al. 2017) as well as constraining the onset of solid mantle convection (Maurice et al. 2017) could help resolve this issue.
Lastly, the grey atmosphere is an easily applicable solution for MO modelers but it can be insensitive to the insolation radiation. We also find that it is sensitive to the different absorption coefficient values used for the CO 2 (See Elkins-Tanton (2008) for a wide range of k 0,CO2 explored for fixed H 2 O/CO 2 atmospheric mixtures). The use of k 0,CO2 derived from present ECS studies is unsuitable for the early Earth climate. The lbl approach remains computationally costly, but the pre-calculated OLR values provided in the companion paper for pure steam are a first step towards a wider use of an atmosphere better resolving the absorption in the IR.
CONCLUSIONS
We have conducted a systematic analysis of the numerous factors and physical processes that affect the thermal and outgassing evolution of a global terrestrial Table 2, which contains the details of each experiment 1-13. Three additional columns are plotted with the outcome of specific settings using the lbl atmosphere (greyscale). Parameters in each experiment as in Ref-A unless otherwise specified. The "transient" MO duration corresponds to the ts obtained for the highest acceptable albedo above αcrit, to the limit of model resolution (lowest cooling flux 1 W m −2 ), for two different surface rheology front temperatures TRF,0. The time arrow of the "continuous" MO is obtained for α ≤ αcrit and hints to effectively unbounded duration, in the absence of atmospheric loss processes. magma ocean (Fig. 13). The dominant effect is the steam atmosphere blanketing. Silicate-evolved melts have lower melting temperature which causes linear increase of the solidification time. Such chemical evolution is found to decrease the solidus and it is the next most prominent factor for prolonging the transient MO lifetime. Water dependent viscosity can be ignored for primitive compositions and for planets with greenhouse atmospheres, while it should be considered for atmosphere-free planets and for silicate-evolved melt compositions.
We emphasize that at the end of the magma ocean, the mantle can store between 45 and 10% of its initial H 2 O reservoir and only 6% of the CO 2 . The massive outgassing of CO 2 that precedes the catastrophic H 2 O outgassing could have an effect in the early atmospheric escape. The duration of the magma ocean is closely tied to the degassed amount of volatiles with greenhouse potential. For Earth, its lifetime does not exceed 5 Myr assuming a water reservoir as large as 5 Earth's oceans while CO 2 plays a less important role.
The calculation of the thermal emission for a pure steam atmosphere (Katyal et al. 2019) shows that the solidification of the magma ocean can be effectively halted at a suitable minimum surface pressure for a given melting temperature at limits that differ from the constant RG-limit 282 W/m 2 . Under no combination of parameters is the early Earth found to exist in a continuous magma ocean.
We find that a molten rocky planet with atmosphere poor in water is a suitable target to acquire information on its mantle surface rheology front temperature. The ∼ 10, 000 W/m 2 difference in OLR for non-massive (∼4 bar) steam atmospheres between planets with and without a magma ocean can be used as a proxy of different melting temperatures that disentangles surface compositions. Surface information would however be masked at higher vapor pressures (> 100 bar).
We discuss the set of permanent/conditionally continuous/transient MO types. Those can be viewed as stages, among which a planet can be reassigned during stellar evolution or via potential orbital migrations. Future studies on the thermal and chemical evolution of magma oceans in the solar and extrasolar systems can benefit from our comprehensive model analysis of the numerous factors that influence it. In return, our model will benefit from future observations of albedo on exoplanets close to the compositional distinction at low P H2O OLR limit and spectral properties of permanent magma ocean planets expected from future missions such as ARIEL (Turrini et al. 2018) and PLATO (Rauer et al. 2014) [stellar age constraints].
where P i is the partial pressure of the species i in the atmosphere and k is the absorption coefficient under a certain pressure P i . k is proportional to the atmospheric absorption coefficient k 0,i under normal atmospheric conditions (P 0 , T 0 ) and can be defined as follows:
k = k 0,i g 3P 0 1/2 . (B9)
Upon including Eq. (B9) into the opacity relation (B8), the opacity τ i for each volatile is obtained for an atmosphere of pressure higher than the normal conditions P 0 (as also Pujol & North (2003) and Elkins-Tanton (2008)):
τ i = 3kM i,atm 8πR 2 p ,(B10)
where, k is the absorption coefficient of the volatile at the surface, R p the planetary radius, and M i,atm the mass of the volatile i in the atmosphere.
In the grey approximation, the total opacity of the atmosphere (τ ) is given by the sum of the opacities of each gas, i.e. τ = Σ i τ i (Pujol & North 2003;Elkins-Tanton 2008). The opacity is a measure of the radiative absorption through atmospheric layers and is inversely proportional to their emissivity . Following Abe & Matsui (1985), the two quantities are linked as follows:
= 2 2 + τ . (B11)
The atmosphere is assumed to be in radiative-convective equilibrium and the TOA is defined to occur at the base of the stratosphere, above which the temperature is governed by pure radiative balance. The assumptions include the plane-parallel approximation for the atmospheric layers and ignore radiative contributions from directions wider than 60 • degrees between neighboring layers. More information on the derivation of the above equations can be found in Abe & Matsui (1985).
C. LINE-BY-LINE MODEL DATA C.1. Lbl atmospheric data product
In Fig. C.1 we show the OLR on the 50 × 8 grid of (T surf , P H2O ) points, that we used as input for our simulations. The OLR data at each grid point have been obtained with the method described in the companion paper using a line-by-line code (GARLIC) of Schreier et al. (2014).
The grid spans surface temperatures from 650 to 4000 K and water vapor surface pressures from 4 to 300 bar. It is irregularly spaced and denser over the temperature range where the highest rate of enrichment and outgassing takes place. This range is obtained by performing simulations using synthetic melting curves with the interior model coupled to the grey atmosphere model (see Fig. 5). For T s ∈ [1400, 1800] K, the OLR was sampled with a resolution of 20 K, while a 100 K resolution was employed outside this range. The sampling is sparser (8 values) on the pressure axis for P ∈ [4, 300] bar. In order to obtain the OLR values at intermediate (P, T ) points of the above dataset, a bilinear interpolation method was used (van Rossum & Drake 2001). In order to estimate the interpolation error, we compared the interpolated field with an independent set of intermediate data points obtained from the atmospheric model. The relative interpolation error amounts between 1 and 10%. The maximum of 10% occurs at pressures lower than 10 bar and high temperatures. The minimum occurs for high pressures and temperatures at the lower end of the dataset. Therefore, the quality of the result is acceptable for this study that focuses on the coolest end of magma ocean phase where the outgassed atmosphere has high pressure and the errors are minimal (1-2 W/m 2 ).
The data of Fig. C.1 represent the OLR at the TOA with a viewing angle of 38 • and thus differ from the field shown in Fig. 8, which represents the net planetary flux at the TOA. More details on the OLR value calculation can be found in the companion paper.
In order to satisfy the requirement of our iteration algorithm for surface temperatures lower than T H2O,crit = 647 K, which are not covered by our gird, we use a fit to the OLR data of (Nakajima et al. 1992). This aspect does not affect our results for the solidification process, which occurs for T surf ≈ T RF,0 T H2O,crit , but ensures that the iteration algorithm runs unhindered until convergence to the solution.
C.2. Limiting radiation values
We use Eq. (18) to estimate the orbital distance for which a planet of given albedo is located at the boundary that separates a long-term and short-term magma ocean. To this end, the value of the limiting radiation F lim corresponding to a specific water vapor pressure P H2O and rheology front temperature at the surface T RF,0 is needed. In Table 3, we report F lim (P H2O , T surf ) for two different rheology front surface temperatures as obtained by interpolating the OLR data points of the companion paper (Fig. C.1). The same values are plotted in Fig. 10 and used to calculate the critical distances for the young Sun in Fig. 11. The heat production due to the long-lived radioactive elements 238 U, 235 U, 40 K and 232 Th, and the short-lived elements 26 Al and 60 Fe is taken into account in the energy balance equation (4) via the term q r whose explicit expression reads:
q r = i X 0 i Q i exp − ln(2) t + t 0 λ i ,(D12)
where, for each element i, X 0 i the isotope concentration in the silicate mantle at the formation time of the CAI (4.55 Gyr ago), Q i the specific heat production, λ i the half-life, t 0 the assumed formation time of the magma ocean (e.g. 2 or 100 Myr after the CAI as in experiment 4 in Table 2), and t the time (with t > t 0 ). For the long-lived elments, the initial isotope concentration X 0 i is calculated by scaling back in time its present-day concentration according to the isotope half-life. In Table 4, we report for each isotope the parameters of Eq. (D12). The energy released by the decay of the radioactive isotopes is made available to the whole magma ocean volume. Table 4. Parameters used in Eq. (D12) to compute the radiogenic heat production.
Radioactive isotope Concentration X 0 Heat production Q (W/kg) Half-life λ (yr) 238 U (*) 6.23 · 10 −8 9.46 · 10 −5 4.47 · 10 9 235 U (*) 1.97 · 10 −8 5.69 · 10 −4 0.704 · 10 9 40 K (*) 4.61 · 10 −7 2.94 · 10 −5 1.25 · 10 9 232 Th (*) 1.54 · 10 −7 2.54 · 10 −5 14.5 · 10 9 26 Al ( †) 1.23 · 10 −6 0.455 0.717 · 10 6 60 Fe ( †) 7.2 · 10 −10 0.0412 2.62 · 10 6 (*) Parameter values are from Schubert et al. (2001). ( †) Concentrations from McDonough & Sun (1995). Heat productions and half lives from Neumann et al. (2014). The composition of anhydrous peridotite, which we employed to define our "synthetic" melting curves (a list of the corresponding oxides can be found in Hirschmann (2000)), is not covered by the empirical model of Giordano et al. (2008) that we used to determine the liquid viscosity and its dependence on the water concentration (2.6). However, we found that the composition of basanite that belonged to the Giordano & Dingwell (2003); Giordano et al. (2008) model calibration database is able to reproduce the temperature-dependent viscosity values of anhydrous silicate obtained experimentally (Urbain et al. 1982), within less than 10% relative error. To be consistent with the assumption of a peridotitic composition, it is important to use a composition as close to a primitive one as possible. Indeed the composition of basanite is among the least evolved in the classification of melts (Le Bas et al. 1986).
Assuming such composition and fitting the model of Giordano et al. (2008) to the experimental data of Urbain et al. (1982) (see Fig. 2), we obtained a modified prefactor in Eq. (9), namely A G = −3.976. The result is within the acceptable range of A G = −4.55 ± 1 log unit, given by the model authors (Giordano et al. 2008). As shown in Table 6, the use of this prefactor yields an error relative to the experimental values smaller than 10%. Note that we use this calculation only to provide a first-order estimate of the effects of water on the melt viscosity without needing to explicitly describe the evolution of the melt composition, which is beyond the scope of the present work.
G. CONSTANTS Table 7 includes the constants and parameters used in most of the simulations unless otherwise stated. 1.1 · 10 −2 -H2O partition coeff. in solid lherzolite κCO 2 ,pv 5.0 · 10 −4 -CO2 partition coeff. in solid perovskite κ CO 2 ,lhz 2.1 · 10 −3 -CO2 partition coeff. in solid lherzolite * The value of this parameter is dynamically calculated during the simulation. -. 1988, J Atmos Sci, 45, 3081, doi: 10.1175/1520-0469(1988 045
Figure 1 .
1Figure 1. Melting curves for three cases: linear according to Abe (1997) ("Abe97", purple solid lines); synthetic for peridotitic composition according to Herzberg et al. (2000), Hirschmann (2000) and Zhang & Herzberg (1994) for the upper mantle, and Fiquet et al. (2010) for the lower mantle ("Syn", black solid lines); for chondritic composition according to the same data for the upper mantle and Andrault et al. (2011) for the lower mantle ("Andrault11", yellow solid lines). "Syn" and "Andrault11" differ only in the lower mantle parametrization. The black dashed line indicates the profile of the rheology transition for the "Syn" curves ("RF Syn"). Dotted lines indicate adiabats with potential temperatures of 4000 K and 2400 K. The red open and full circles indicate the base of the liquid-like magma ocean of thickness D for the two adiabats, with the corresponding depth ranges of liquid (l), solid (s), and partially molten (l+s) regions shown in the left columns.
Figure 2 .
2Variation of melt dynamic viscosity η l with temperature for hydrous and anhydrous melt. A: Melt viscosity as a function of water content for different temperatures. Equation (8), which assumes a fixed water concentration of 10 wt% (solid lines), while Eq. (9) explicitly includes the effect of water concentration (linepoints). B: Viscosity of anhydrous melt as a function of temperature. Squares and circles are obtained with Eq. (8) and Eq. (9) with XH 2 O = 0, respectively. Experimental values of anhydrous silicate melt are obtained from Urbain et al. (1982) (see Appendix F).
Figure 3 .
3Thermal evolution of black body (bb), grey H2O atmosphere (gr-H2O), grey H2O/CO2 atmosphere (gr-H2O/CO2), and line-by-line H2O atmosphere (lbl). A: Evolution of potential (solid) and surface (dashed) temperature; B: Evolution of the depth of the magma ocean (dashed lines indicate the end of MO); C: Evolution of convective energy sink compared to the energy source of radioactivity. Note that the contribution of the radioactive heat sources is not included in the Ref-A settings and is only plotted for comparison; D: Evolution of Ra number. Apart from the explicit differences in the atmospheric component, all other parameters are taken from the Ref-A case
Figure 4 .
4Evolution of H2O and CO2 outgassing based on the Ref-A case (see
Figure 6 .
6Estimates of maximum outgassing at the end of the magma ocean (99.3% solid), depending on initial bulk abundance for volatiles H2O and CO2. A: The absolute amount of H2O outgassed by the end of the magma ocean (colored line, left y-axis) is plotted against the initial concentrations in the mantle. The mass of outgassed volatile relative to the mass of the total volatile reservoir is plotted on the right axis (black line, right y-axis). B: Same as in panel A but for the CO2 volatile. The performed experiments are plotted with points.
Figure 7 .
7Colormap of minimum solidification time for various initial H2O and CO2 abundances in the mantle, expressed in initial concentrations X volatile,0 (at model time 0
Figure 8 .
8Net outgoing radiation flux at TOA for (PH 2 O , T surf ) calculated for two specific incoming solar radiations (FSun(S = 1361 W/m 2 , α = 0.11) = 303 W/m 2 and FSun(S = 1361 W/m 2 , α = 0.30) = 238 W/m 2 ) employing A, B: the lbl model of
Figure 12 .
12Incoming stellar energy flux at various orbital distances around M and G stars for a planet with albedo α=0.4 (cloudless water vapor maximum value). Examples of potentially rocky exoplanets are plotted on the relevant host star curve. Grey shaded region corresponds to permanent magma ocean at the lowest mantle melting temperature considered. Regions are drawn for 4-300 bar OLR for two different rheology front solidification temperatures: TRF,0 = 1370 (line-hatched) and 1645 K (brown shaded). et al. 2015)
Figure 13 .
13Cumulative plot of the outcome of the sensitivity experiments for the solidification time ts (in log10 units) using the 1D COMRAD model, compared to the Ref-A timescale (red line). Labels are as in
Figure C. 1 .
1OLRT OA sampled on the (T surf , PH 2 O )-space obtained using the lbl model. The detailed calculation of the values is found in the companion paper.
Table 1 .
1Parameter values and components of the Reference-A model. The table is for complementary use to the results presented inTable 2.Parameter
Description
Value/Type
Unit/Info
Atmosphere
Type of approximation
grey
Eq. (15) Abe & Matsui (1985)
k0,H 2 O
Absorption coeff. at normal atmospheric conditions
0.01
m 2 /kg
k0,CO 2
Absorption coeff. at normal atmospheric conditions
0.001
m 2 /kg
H2O content
Total water reservoir
300
bar
XH 2 O,0
Inital H2O mantle abundance
410
ppm
CO2 content
Total CO2 reservoir
100
bar
XCO 2 ,0
Initial CO2 mantle abundance
130
ppm
S
Solar constant (S0)
1361
W/m 2
α
Planetary albedo
0.30
-
Tp,0
Initial potential temperature
4000
K
D
Initial MO depth
2890
km
η l
Melt viscosity parameterization
η l = f (T )
(Eq. 8)
qr
Radioactive heating
0
not included
t planet
Planet accretion time
100
Myr (employed for qr only)
T sol , T liq
Melting curves
"synthetic"
Herzberg et al. (2000); Hirschmann
expands on the Nakajima et al. (1992) grey model and employs supercritical water thermal capacities. The study of Hier-Majumder & Hirschmann
No atmosphere & η l (T, XH 2 O ) 2 No atmosphere, XH 2 O,0=410 ppm No atmosphere & η l (T, XH 2 O ) 3 No atmosphere, XH 2 O,0=10Modified parameter
# Value/ Description
ts (yr)
∆ts (yr)
∆ts/t s,ref
......Reference-A
-As in Table 1
208,600
-
-
1a:...H2O content
1 XH 2 O,0 = 10 ppm
58,900
−149, 700
−72%
1b:
1 XH 2 O,0 = 10 5 ppm
69,699,000 +69,490,400 +33000%
2a:...CO2 content
1 XCO 2 ,0 = 10 ppm
160,500
−48, 100
−23%
2b:
1 XCO 2 ,0 = 10 5 ppm
3,919,000
+3,710,400
+18000%
3a:...Liquid viscosity
1 η l = f (T, XH 2 O )
213,400
+4,800
+2%
3b:
2 η l = f (T, XH 2 O ), XH 2 O,0 = 10 5 ppm 69,711,000 +69,502,400 +33000%
4a:...Radioactive sources
1 t planet = 100 Myr
208,600
0
+0%
4b:
2 t planet = 2 Myr
6,036,780
+5,828,180
+2793%
5:.....Heat flux parametrization
1 F hard ; L
D max = 1
260,770
+52,170
+25%
6a: ..Upper mantle solidus
1 T sol − 20 K ⇒ TRF,0 = 1625K
216,400
+7,800
+4%
6b:
1 T sol − 50 K ⇒ TRF,0 = 1595 K
228,700
+20,100
+10%
6c:
1 T sol − 100 K ⇒ TRF,0 = 1545 K
250,600
+42,000
+20%
6d:
1 T sol − 400 K ⇒ TRF,0 = 1245 K
434,600
+226,000
+108%
7:.....Lower mantle melting curves
2 T sol,liq ; (Andrault et al. 2011)
207,100
-1,500
−1%
8:.....Alternative melting curves
2 T sol,liq ; Linear (A) ⇒ TRF,0 = 1360 K 126,670
-81,930
−39%
9:.....Irradiation
1 72%S0
208,500
-100
+0%
10a: Irradiation & albedo
2 S0, α = 0.15
208,600
0
+0%
10b:
2 72%S0, α = 0.60
208,500
-100
+0%
11:...No atmosphere & η l (T )
1 No atmosphere
2,000
-206,600
−99%
12a: 2,958
-205,642
−99%
12b: 4 ppm
2,713
-205,887
−99%
.......Reference-B
-As Reference-A with XCO 2 ,0 = 0
156,700
-
-
13:...Lbl atmosphere
1 Steam lbl
736,100
+579,400
+278%
et al.
Table 3 .
3F lim (PH 2 O , TRF,0) for indicative TRF,0 cases calculated with the Katyal et al. (2019) data. PH 2 O (bar) F lim (TRF,0 = 1645 K) (W/m 2 ) F lim (TRF,0=1370 K) (W/m2 )
Table 5 .
5Planet and host star parameters used in Fig. 12 exoplanet.eu/catalog/ * In absence of values of luminosity relative to L we calculated the stellar luminosity directly from T ef f , RStar data. E. EXOPLANETS F. CALIBRATION OF THE MELT VISCOSITY PREFACTORPlanet
Orbital distance R
Host star
T ef f
Rstar
Luminosity L
Ref.
(AU)
(K)
(in solar units R ) (in solar units L )
* Kepler 236 c
0.1320
Kepler 236
3750
0.510
-
exoplanet.eu/catalog/
55 Cancri e
0.0156
55 Cancri
-
-
0.59
exoplanet.eu/catalog/
Earth
1.0000
Sun (100 Myr old) 5326
1.000
0.72
Gough (1981)
* Kepler 36 b
0.1151
Kepler 36
5911
1.619
-
openexoplanetcatalogue.com
* HD 219134 b&c
0.0388 & 0.0653
HD 219134
3131
0.186
-
exoplanet.eu/catalog/
* Ross 128 b
0.0496
Ross 128
3192
0.197
-
openexoplanetcatalogue.com
* GJ 1132 b
0.0154
GJ 1132
3270
0.207
-
openexoplanetcatalogue.com
* LHS 1140 b&c
0.087 & 0.02675
LHS 1140
4699
0.778
-
openexoplanetcatalogue.com
* Kepler 78 b
0.01
Kepler 78
5089
0.74
-
Table 6 .Table 7 .
67Comparison between values of the viscosity of anhydrous liquid peridotite obtained experimentally and calculated with the model of Giordano et al. (2008) assuming a basanite composition and a prefactor AG = −3.976 in Eq. (9). T (K) Experimental η l (Pa s) Calculated η l (Pa s) Error (%) Model constants and parameters. Mantle thermal diffusivity kT κT /(ρcP ) J K −1 s −2 m −22000
0.22
0.2350
6.84
2220
0.08
0.0788
−1.47
2300
0.06
0.0579
−5.01
Parameter
Value
Unit
Description
Rp
6371
km
Planetary radius
R b
3481
km
Core mantle boundary radius
g
9.81
ms −2
Gravity acceleration
k0,H 2 O
0.01
m 2 /kg
Absorption coeff. at P0, T0
k0,CO 2
0.001
m 2 /kg
Absorption coeff. at P0, T0
P0
101325
Pa
Normal atmospheric pressure for k 0,vol
T0
300
K
Normal atmospheric temperature for k 0,vol
α0
3 · 10 −5
K −1
Mantle thermal expansivity
K0
200
GPa
Mantle bulk modulus
K
4
-
P -derivative of mantle bulk modulus
m
0
-
Parameter in Eq. (3)
cP
1000
J kg −1 K −1
Mantle isobaric thermal capacity
κT
10 −6
m 2 s −1
Mantle thermal conductivity
ρ l
4000
kg m −3
Melt density
ρs
4500
kg m −3
Solid density
φC
0.4
-
Critical melt fraction
η0
4.2 · 10 10
Pa s
Solid viscosity prefactor
E
240
kJ mol −1
Activation energy
V
5
cm 3 mol −1
Activation volume
R
8.314
J mol −1 K −1
Ideal gas constant
AG
3.9759
-
Prefactor in Eq. (9) calibrated for basanite
BG
*
K
Parameter in Eq. (9)
CG
*
K
Parameter in Eq. (9)
AK
0.00024
-
Prefactor for hydrous liquid in Eq. (8)
BK
4600
-
Parameter for hydrous liquid in Eq. (8)
CK
1000
K
Parameter for hydrous liquid in Eq. (8)
κH 2 O,pv
1.0 · 10 −4
-
H2O partition coeff. in solid perovskite
κ H 2 O,lhz
ACKNOWLEDGEMENTSWe thank an anonymous reviewer whose comments helped improve a previous version of the paper, and Melissa McGrath for editorial handling. AN and NT acknowledge financial support from the Helmholtz association (Project VH-NG-1017). NK acknowledges funding from the German Transregio Collaborative Re-search Centre "Late Accretion onto Terrestrial Planets (LATP)" (TRR170, sub-project C5). MG acknowledges financial support from the DFG (Project GO 2610/1-1). Additional support in the form of conference travel grants for AN was provided from the TRR170 project and the Earth and Life Science Institute in Tokyo. AN wishes to thank Slava Solomatov and Keiko Hamano for insightful conversations.APPENDIXA. MELTING CURVESWe report here the fittings that adopted to parameterize the various melting curves used in this study.A.1. "Synthetic" melting curves For the solidus temperature (T sol ) of nominally anhydrous peridotite and pressures 0 ≤ P ≤ 2.7 GPa, we use(Hirschmann 2000):with reported error by the authors of ±20 K. For 2.7 < P ≤ 22.5(Herzberg et al. 2000):with reported error by the authors of ±68 K. At lower mantle pressures, for P > 22.5 GPa, we use a quadratic fit to the data ofFiquet et al. (2010)for fertile peridotite:For the liquidus of fertile peridotite, we use a fit to data ofZhang & Herzberg (1994)for 0 ≤ P ≤ 22.5 GPa: where z is the depth from the surface in km.A.3. Andrault melting curvesThe following quadratic fitting to the data ofAndrault et al. (2011)for a chondritic composition is employed for the lower mantle only for P > 22.5 GPa:T sol = 2056.489 + 15.801P − 0.003P 2 , T liq = 2049.555 + 24.671P − 0.035P 2 (A7)Melting curves for the upper mantle are identical to those described above as "synthetic", unless otherwise specified.B. GREY ATMOSPHEREAssuming optical thickness 1 for a dense atmosphere and evaluating radiative balance at normal optical depth 2/3,Abe & Matsui (1985)find that the opacity τ i for a given species i is proportional to the absorption k :
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| []
|
[
"Sparse Interpolation With Errors in Chebyshev Basis Beyond Redundant-Block Decoding*",
"Sparse Interpolation With Errors in Chebyshev Basis Beyond Redundant-Block Decoding*"
]
| [
"Erich L Kaltofen [email protected] ",
"Zhi-Hong Yang ",
"\nDepartment of Mathematics\nDepartment of Computer Science\nNorth Carolina State University\n27695-8205RaleighNorth CarolinaUSA\n",
"\nDuke University\n27708-0129DurhamNorth CarolinaUSA\n"
]
| [
"Department of Mathematics\nDepartment of Computer Science\nNorth Carolina State University\n27695-8205RaleighNorth CarolinaUSA",
"Duke University\n27708-0129DurhamNorth CarolinaUSA"
]
| []
| We present sparse interpolation algorithms for recovering a polynomial with ≤ B terms from N evaluations at distinct values for the variable when ≤ E of the evaluations can be erroneous. Our algorithms perform exact arithmetic in the field of scalars K and the terms can be standard powers of the variable or Chebyshev polynomials, in which case the characteristic of K is = 2. Our algorithms return a list of valid sparse interpolants for the N support points and run in polynomial-time. For standard power basis our algorithms sample at N = ⌊ 4 3 E +2⌋B points and for Chebyshev basis at N = ⌊ 3 2 E +2⌋B points. Those are fewer points than N = 2(E + 1)B − 1 in [Kaltofen and Pernet, Proc. ISSAC 2014] for standard power basis and fewer than in [Arnold and Kaltofen, Proc. ISSAC 2015] for Chebyshev basis, where only the cases B ≤ 3 have explicit counts for N . Our method shows how to correct 2 errors in a block of 4B points for standard basis and how to correct 1 error in a block of 3B points for Chebyshev Basis. | 10.1109/tit.2020.3027036 | [
"https://arxiv.org/pdf/1912.05719v1.pdf"
]
| 209,324,190 | 1912.05719 | dd1e920d7515b1365f0909dc9e1fbda98c00bef0 |
Sparse Interpolation With Errors in Chebyshev Basis Beyond Redundant-Block Decoding*
12 Dec 2019
Erich L Kaltofen [email protected]
Zhi-Hong Yang
Department of Mathematics
Department of Computer Science
North Carolina State University
27695-8205RaleighNorth CarolinaUSA
Duke University
27708-0129DurhamNorth CarolinaUSA
Sparse Interpolation With Errors in Chebyshev Basis Beyond Redundant-Block Decoding*
12 Dec 2019
We present sparse interpolation algorithms for recovering a polynomial with ≤ B terms from N evaluations at distinct values for the variable when ≤ E of the evaluations can be erroneous. Our algorithms perform exact arithmetic in the field of scalars K and the terms can be standard powers of the variable or Chebyshev polynomials, in which case the characteristic of K is = 2. Our algorithms return a list of valid sparse interpolants for the N support points and run in polynomial-time. For standard power basis our algorithms sample at N = ⌊ 4 3 E +2⌋B points and for Chebyshev basis at N = ⌊ 3 2 E +2⌋B points. Those are fewer points than N = 2(E + 1)B − 1 in [Kaltofen and Pernet, Proc. ISSAC 2014] for standard power basis and fewer than in [Arnold and Kaltofen, Proc. ISSAC 2015] for Chebyshev basis, where only the cases B ≤ 3 have explicit counts for N . Our method shows how to correct 2 errors in a block of 4B points for standard basis and how to correct 1 error in a block of 3B points for Chebyshev Basis.
Introduction
Let f (x) be a polynomial with coefficients from a field K (of characteristic = 2),
f (x) = t j=1 c j T δ j (x) ∈ K[x], 0 ≤ δ 1 < δ 2 < · · · < δ t = deg(f ), ∀j, 1 ≤ j ≤ t : c j = 0,(1)
where T d (x) is the Chebyshev Polynomial of the First Kind (of degree d for d ≥ 0), defined by the recurrence
T d (x) T d+1 (x) = 0 1 −1 2x d 1 x for d ∈ Z.(2)
We say that f (x) is Chebyshev-1 t-sparse. We wish to compute the term degrees δ j and the coefficients c j from values of a i = f (ζ i ) for i = 1, 2, . . ., where the distinct arguments ζ i ∈ K can be chosen by the algorithms; the latter is the setting of Prony-like sparse interpolation methods. Our objective is to interpolate with a number of points that is proportional to the sparsity t of f . The algorithms have as input an upper bound B ≥ t for the sparsity, for otherwise the zero polynomial (of sparsity 0) is indistinguishable from f (x) = i (x − ζ i ) at ≤ deg(f ) evaluation points a i = 0. The algorithms by [Lakshman Y. N. and Saunders 1995;Arnold and Kaltofen 2015;Imamoglu, Kaltofen, and Yang 2018], based on Pronylike interpolation [Prony III (1795); Ben-Or and Tiwari 1988; Kaltofen and Lee 2003] can interpolate f (x) (see (1)) from 2B values at points γ i = T i (β) = (ω i + 1/ω i )/2 for i = 0, 1, . . . , 2B − 1 where β = (ω + 1/ω)/2 with ω ∈ K such that ω δ j = ω δ k for all 1 ≤ j < k ≤ t.
As for Prony-like algorithms, the algorithms utilize an algorithm for computing roots in K for polynomials Λ(z) ∈ K[z] and logarithms to base ω. More precisely, one utilizes an algorithm that on input ω and ω d for an integer d ∈ Z computes d, possibly modulo the finite multiplicative order η of ω (ω η = 1 minimally) [Imamoglu and Kaltofen 2018]. We note that in [Arnold and Kaltofen 2015] we show that one may instead use the arguments T 2i+1 (β) for i = 0, 1, ..., 2B − 1, provided ω 2δ j +1 = ω 2δ k +1 for all 1 ≤ j < k ≤ t.
Here we consider the case when the evaluations a i , which we think of being computed by probing a black box that evaluates f , can have sporadic errors. We writeâ i for the black box values, which at some unknown indices ℓ can haveâ ℓ = a ℓ . We shall assume that we have an upper bound E for the number of errors on a batch of N evaluations. Therefore our sequence of black box calls has a non-stochastic error rate ≤ E/N. We shall also assume that the black box for f does not return stochastic errors, meaning that ifâ = f (ζ) then a second evaluation of the black box at ζ produces the same erroneousâ. Furthermore, we perform list-interpolation which produces a valid list of sparse interpolants for the black box values with errors, analogously to list-decoding error correcting codes. We restrict to algorithms that run in polynomial time in B and E (N is computed by the algorithms), which limits the list length to polynomial in B and E.
A simple sparse list-interpolation algorithm with errors evaluates E + 1 blocks of 2B arguments, which produce N = (E + 1)2B black box valuesâ i,σ at the arguments T 1 (β 1 ), T 3 (β 1 ), . . . , T 4B−1 (β 1 ), T 1 (β 2 ), T 3 (β 2 ), . . . , T 4B−1 (β 2 ), . . . . . . . . .
T 1 (β E+1 ), T 3 (β E+1 ), . . . , T 4B−1 (β E+1 ), E + 1(3)
where β σ = (ω σ + 1/ω σ )/2 and where the arguments in (3) are selected distinct: T 2i+1 (β σ ) = T 2m+1 (β τ ) for i = m and σ = τ (⇐⇒ ω 2i+1 σ = ω 2m+1 τ ). If we have for all ω σ distinct term values ω δ j σ = ω δ k σ (j = k) then the algorithm in [Arnold and Kaltofen 2015] can recover f from those lines in (3) at which the black box does not evaluate to an error. Because we assume ≤ E errors there is such a block of good arguments/values. Other blocks with errors may lead to a different t-sparse Chebyshev-1 interpolant with t ≤ B. The goal is to recover f (and possible other sparse interpolants with ≤ E errors) from N < (E + 1)2B evaluations.
In [Arnold and Kaltofen 2015] we give algorithms for the following bounds B, E :
B (4)
The evaluation counts (4) are derived by using the method of [Kaltofen and Pernet 2014]: subsampling at all subsequences x ← T r+is (β) of arguments whose indices are arithmetic progressions to locate a subsequence without an error. The counts (4) are established by explicitly computed lengths for the Erdős-Turán Problem for arithmetic progressions of length ≤ 9. Here we give an algorithm that recovers f (and possible other sparse interpolants) for all B ≥ 1, E ≥ 1 bounds from
N = 3 2 E + 2 B(5)
evaluations with ≤ E errors. Our new algorithm uses fewer evaluations than (4) even for B ≤ 3. We show that one can list-interpolate from 3B points correcting a single error, which with blocking yields (5). We correct one error from 3B points by deriving a non-trivial univariate polynomial for the value as a variable in each possible position. Our technique applies to Prony's original problem of interpolating a t-sparse polynomial with t ≤ B in power basis 1, x, x 2 , . . . in the presence of erroneous points. In [Kaltofen and Pernet 2014, Lemma 2] it was shown that from (E + 1)2B − 1 points one can correct ≤ E errors. Here we show that
N = 4 3 E + 2 B(6)
points suffice to correct ≤ E errors. The counts (6) are achieved by correcting ≤ 2 errors from 4B points and blocking. We correct 2 errors at 4B points by deriving a bivariate Pham system for variables in place of the values in all possible error locations, which yields a bounded number of possible value pairs among which are the actual values. We note that for E = 2 the count 4B is smaller than the values n 2B,2 in [Kaltofen and Pernet 2014, Table 1], which are the counts for having a clean arithmetic progression of length 2B in the presence of 2 errors. Finally we note that our sparse list-interpolation algorithms are interpolation algorithms over the reals K = R if ω σ > 1 (or ω σ > 0 when f is in power basis) and N ≥ 2B + 2E, that is there will only be a single sparse interpolant computed by our algorithms. Uniqueness follows by Corollary 2 in [Kaltofen and Pernet 2014] and Corollary 2.4 in [Arnold and Kaltofen 2015]. Over fields with roots of unity, the sparse list-interpolation problem for the power base can have with < (2E + 1)2B points more than a single B-sparse solution [Kaltofen and Pernet 2014, Theorem 3], which is also true for the Chebyshev base as shown by Example 2.3.
Correcting One Error
Let K be a field of characteristic = 2. Let f (x) ∈ K[x] be a sparse univariate polynomial represented by a black box and it is equal to:
f (x) = t j=1 c j x δ j , δ 1 < δ 2 < · · · < δ t = deg(f ), ∀j, 1 ≤ j ≤ t : c j = 0.(7)
We assume that the black box for f returns the same value when probed multiple times at the same input. Let B be an upper bound on the sparsity of f (x) and D ≥ |δ j | for all 1 ≤ j ≤ t. Choose a point ω ∈ K \ {0} such that: (1) ω has order ≥ 2D + 1, meaning that ∀η ≥ 1, ω η = 1 ⇒ η ≥ 2D + 1, and (2) ω i 1 = ω i 2 for all 1 ≤ i 1 < i 2 ≤ 3B. The first condition is an input specification of the integer logarithm algorithm 1.1.1 that computes δ j from ω δ j . The second condition guarantees that the inputs probed at the black box are distinct so that we don't get the same error at different locations. For i = 1, 2, . . . , 3B, letâ i be the output of the black box for f probed at input ω i . Assume there is at most one error in the evaluations, that is, there exists ≤ 1 index i ∈ {1, 2, . . . , 3B} such thatâ i = f (ω i ). We present an algorithm to compute a list of sparse polynomials which contains f .
For r = 1, . . . , B, let H r denote the following (B + 1) × (B + 1) Hankel matrix:
H r = â râr+1 · · ·â r+B−1âr+B a r+1âr+2 · · ·â r+Bâr+B+1 . . . . . . . . . . . . . . . a r+B−1âr+B · · ·â r+2B−2âr+2B−1 a r+Bâr+B+1 · · ·â r+2B−1âr+2B ∈ K (B+1)×(B+1) .(8)
Let ℓ be the error location, i.e.,â ℓ = f (ω ℓ ). There are three cases to be considered:
Case 1: 1 ≤ ℓ ≤ B;
Case 2: B + 1 ≤ ℓ ≤ 2B;
Case 3: 2B + 1 ≤ ℓ ≤ 3B.
First, we try Prony's algorithm (see Algorithm 1.1.2) on the sequence (â 1 ,â 2 , . . . ,â 2B ), which will recover f (x) if Case 3 happens. Next we try Prony's algorithm on the sequence (â B+1 ,â B+2 , . . . ,â 3B ); this step will recover f (x) if Case 1 happens. Finally, to deal with Case 2, we replace the erroneous valueâ ℓ by a symbol α. Then the determinant the Hankel matrix H ℓ−B (see (8)) is univariate polynomial of degree B + 1 in α. By Prony/Blahut/Ben-Or/Tiwari Theorem [Prony III (1795) ;Blahut 1983;Ben-Or and Tiwari 1988], (f (ω i )) i≥0 is a linearly generated sequence and its minimal generator has degree ≤ B. Therefore f (ω ℓ ) is a solution of the equation:
det(H ℓ−B ) = 0.
By solving the equation (9), we obtain a list of candidates {ζ 1 , . . . , ζ b } for the correct value f (ω ℓ ). For each candidate ζ k (1 ≤ k ≤ b), we substituteâ ℓ by ζ k in the sequence (â B+1 ,â B+2 , . . . ,â 2B ) and try Prony's algorithm on the updated sequence (â 1 ,â 2 , . . . ,â 2B ), which gives us a list of sparse polynomials with f (x) being contained.
Example 1.1. Assume that we are given B = 3. With 3B = 9 evaluationsâ 1 ,â 2 , . . . ,â 9 obtained from the black box for f at inputs ω, ω 2 , . . . , ω 9 , we have the following 6 × 4 matrix:
H = â 1â2â3â4 a 2â3â4â5 a 3â4â5â6 a 4â5â6â7 a 5â6â7â8 a 6â7â8â9 ∈ K 6×4
For r = 1, 2, 3, the matrices H r (see (8)) are 4 × 4 submatrices of H:
H 1 = â 1â2â3â4 a 2â3â4â5 a 3â4â5â6 a 4â5â6â7 , H 2 = â 2â3â4â5 a 3â4â5â6 a 4â5â6â7 a 5â6â7â8 , H 3 = â 3â4â5â6 a 4â5â6â7 a 5â6â7â8 a 6â7â8â9 .
Suppose there is one errorâ ℓ = f (ω ℓ ) in these 3B evaluations. We recover f (x) by the following steps.
(1) Try Prony's algorithm on 1.1.2 the sequence (â 1 ,â 2 , . . . ,â 6 ), which can recover f (x) in case that ℓ ∈ {7, 8, 9};
(2) Try Prony's algorithm 1.1.2 on the sequence (â 4 ,â 5 , . . . ,â 9 ), which can recover f (x) in case that ℓ ∈ {1, 2, 3};
(3) For ℓ ∈ {4, 5, 6}, substituteâ ℓ by α, then det(H ℓ−3 ) is a univariate polynomial of degree 4 in α and f (ω ℓ ) is a root of det(H ℓ−3 ). Compute the roots {ζ k } k≥1 of det(H ℓ−3 ). For each root ζ k , replaceâ ℓ by ζ k and check if the matrix H has rank ≤ 3. If yes, then try Prony's algorithm 1.1.2 on the updated sequence (â 1 ,â 2 , . . . ,â 6 ). As f (ω ℓ ) is equal to some ζ k , this step will recover f (x) in case that ℓ ∈ {4, 5, 6}.
For computing the term degrees δ j of f , we need an integer logarithm algorithm having the following input and output specifications.
Integer Logarithm Algorithm
Input: ◮ An upper bound D ∈ Z >0 . ◮ ω ∈ K \ {0} and has order ≥ 2D + 1, meaning that ∀η ≥ 1, ω η = 1 ⇒ η ≥ 2D + 1. ◮ ρ ∈ K \ {0}. Output: ◮ Either δ ∈ Z with |δ| ≤ D and ω δ = ρ, ◮ or FAIL.
We describe the subroutine which we call Try Prony's algorithm. This subroutine will be frequently used in our main algorithms.
Try Prony's algorithm
Input: ◮ A sequence (â r , . . . ,â r+2B−1 ) in K where K is a field of characteristic = 2. ◮ An upper bound D ∈ Z >0 . ◮ ω ∈ K \ {0}
and has order 2D + 1. ◮ A root finder for univariate polynomials over K. ◮ Integer logarithm algorithm 1.1.1 that takes D, ω, ρ as input and outputs: ◮ either δ ∈ Z with |δ| ≤ D and ω δ = ρ, ◮ or FAIL. Output: ◮ A sparse polynomial of sparsity t ≤ B and degree ≤ D, or FAIL.
Step 1: Use Berlekamp/Massey algorithm to compute the minimal linear generator of the sequence (â r , . . . ,â r+2B−1 ) and denote it by Λ(z).
Step 2: Compute the roots of Λ(z) in K denote the roots as ρ 1 , . . . , ρ t . If ρ j = 0 for some 1 ≤ j ≤ t or ρ 1 , . . . , ρ t are not distinct, then return FAIL.
Step 3: For j = 1, . . . , t, use the Integer logarithm algorithm 1.1.1 to compute δ j = log ω ρ j . If the Integer logarithm algorithm returns FAIL, then return FAIL.
Step 4: Compute the coefficients c 1 , . . . , c t by solving the following transposed generalized Vandermonde system
ρ r 1 ρ r 2 · · · ρ r t ρ r+1 1 ρ r+1 2 · · · ρ r+1 t . . . . . . . . . . . . ρ r+t−1 1 ρ r+t−1 2 . . . ρ r+t−1 t c 1 c 2 . . . c t = â r a r+1 . . . a r+t−1 .
Step 5: Return the polynomial t j=1 c j x δ j . Now we give an algorithm for interpolating a black-box polynomial with sparsity bounded by B. This algorithm can correct one error in 3B evaluations.
A list-interpolation algorithm for power-basis sparse polynomials with
evaluations containing at most one error.
Input: ◮ A black box representation of a polynomial f ∈ K[x]
where K is a field of characteristic = 2. The black box for f returns the same erroneous output when probed multiple times at the same input. ◮ An upper bound B on the sparsity of f . ◮ An upper bound D on the degree of f . ◮ ω ∈ K \ {0} satisfying: ◮ ω has order ≥ 2D + 1; ◮ ω i 1 = ω i 2 for all 1 ≤ i 1 < i 2 ≤ 3B. ◮ A root finder for univariate polynomials over K. Output: ◮ An empty list or a list of sparse polynomials {f [1] , . . . , f [M ] } with each f [k] (1 ≤ k ≤ M) satisfying:
◮ f [k] has sparsity ≤ B and degree ≤ D; ◮ f [k]
is represented by its term degrees and coefficients;
◮ there is ≤ 1 index i ∈ {1, 2, . . . , 3B} such that f [k] (ω i ) =â i wherê a i is the output of the black box probed at input ω i ; ◮ f is contained in the list.
Step 1: For i = 1, 2, . . . , 3B, get the outputâ i of the black box for f at input ω i . Let L be an empty list.
Step 2: Try Prony's algorithm 1.1.2 on the sequence (â 1 ,â 2 , . . . ,â 2B ). If Prony's algorithm returns a sparse polynomialf of sparsity ≤ B and degree ≤ D, and there is ≤ 1
index i ∈ {1, 2, . . . , 3B} such thatf (ω i ) =â i , then addf to the list L.
If the error is in (â 2B+1 ,â 2B+2 . . . ,â 3B ), then the sequence (â 1 ,â 2 , . . . ,â 2B ) is free of errors, so Prony's algorithm in
Step 2 will return f and f will be added to the list L.
Step 3: Try Prony's algorithm 1.1.2 on the sequence (â B+1 ,â B+2 , . . . ,â 3B ). If Prony's algorithm returns a sparse polynomialf of sparsity ≤ B and degree ≤ D, and there is
≤ 1 index i ∈ {1, 2, . . . , 3B} such thatf (ω i ) =â i , then addf to the list L.
If the error is in (â 1 , . . . ,â B ), then the sequence (â B+1 ,â B+2 , . . . ,â 3B ) is free of errors, so Prony's algorithm in
Step 3 will return f and f will be added into the list L.
Step 4: For ℓ = B + 1, B + 2, . . . , 2B, 4(a): substituteâ ℓ by a symbol α in the matrixH ℓ−B (see (8)); use the fraction free Berlekamp/Massey algorithm Kaltofen and Yuhasz 2013] to compute the determinant ofH ℓ−B and denote it by ∆ ℓ (α); Ifâ ℓ = f (ω ℓ ) with ℓ ∈ {B + 1, B + 2, . . . , 2B}, then we substituteâ ℓ by a symbol α and compute the roots
Here ∆ ℓ (α) is a univariate polynomial of the form (−1) B+1 α B+1 +∆ ℓ (α) with deg(∆ ℓ (α)) < B{ξ 1 , . . . , ξ b } of ∆ ℓ (α) in K. The correct value f (ω ℓ ) is in the set {ξ 1 , . . . , ξ b }.
Thus for every root ξ k (k = 1, . . . , b), we replaceâ ℓ with ξ k and use Berlekamp/Massey algorithm to check if the new sequence (â 1 ,â 2 , . . . ,â 3B ) is generated by some polynomial of degree ≤ B. If so, then we try Prony's algorithm on the updated sequence (â 1 ,â 2 , . . . ,â 2B ). In the end, Step 4 will add f into the list L in case that B + 1 ≤ ℓ ≤ 2B.
Step 5: Return the list L.
Proposition 1.1. The output list of Algorithm 1.1.3 contains ≤ B 2 + B + 2 polynomials.
Proof. The Step 2 in Algorithm 1.1.3 produces ≤ 1 polynomial and so is Step 3. In the Step 4 of Algorithm 1.1.3, because ∆ ℓ (α) has degree B + 1, the equation ∆ ℓ (α) = 0 has ≤ B + 1 solutions in K, therefore this step produces ≤ B(B + 1) polynomials. Thus the output list of Algorithm 1.1.3 contains ≤ 2 + B(B + 1) polynomials.
Correcting 2 Errors
In this section, we give a list-interpolation algorithm to recover f (x) (see (7)) from 4B evaluations that contain 2 errors. Recall that B is an upper bound on the sparsity of f (x) and D is an upper bound on the absolute values of the term degrees of f (x). We will use Algorithm 1.1.3 as a subroutine.
Let ω ∈ K \ {0} such that: (1) ω has order ≥ 2D + 1, and (2) ω i 1 = ω i 2 for all 1 ≤ i 1 < i 2 ≤ 4B. For i = 1, 2, . . . , 4B, letâ i be the output of the black box probed at input ω i . Let a ℓ 1 andâ ℓ 2 be the 2 errors and ℓ 1 < ℓ 2 . The problem can be covered by the following three cases:
Case 1: 1 ≤ ℓ 1 ≤ B; Case 2: 3B + 1 ≤ ℓ 2 ≤ 4B; Case 3: B + 1 ≤ ℓ 1 < ℓ 2 ≤ 2B or 2B + 1 ≤ ℓ 1 < ℓ 2 ≤ 3B Case 4: B + 1 ≤ ℓ 1 ≤ 2B and 2B + 1 ≤ ℓ 2 ≤ 3B.
First, we try the Algorithm 1.1.3 on the sequences (â 1 ,â 2 , . . . ,â 3B ) and (â B+1 ,â B+2 , . . . ,â 4B ), which can list interpolate f (x) if either Case 2 or Case 1 or Case 3 happens. For Case 4, we substitute the two erroneous valuesâ ℓ 1 andâ ℓ 2 by two symbols α 1 and α 2 respectively. Then the pair of correct values (f (ω ℓ 1 ), f (ω ℓ 2 )) is a solution of the following Pham system (see Lemma 1.2 and Lemma 1.3):
det(H ℓ 1 −B ) = 0, det(H ℓ 2 −B ) = 0,(10)
where H ℓ 1 −B and H ℓ 2 −B are Hankel matrices defined as (8). As the Pham systems (10) is zero-dimensional (see Lemma 1.3), we compute the solution set {(ξ 1,1 , ξ 2,1 ), . . . , (ξ 1,b , η 2,b )} of (10). Then, for k = 1, . . . , b, we substitute (â ℓ 1 ,â ℓ 2 ) by (ξ 1,k , ξ 2,k ) and try Prony's algorithm 1.1.2 on the updated sequence (â 1 ,â 2 , . . . ,â 2B ); this results in a list of candidates for f if Case 4 happens. 3) all other entries of A are numbers in some field K.
Then det(A) = α n 1 + Q(α 1 , α 2 ) where Q(α 1 , α 2 ) is a polynomial of total degree ≤ n − 1. Proof. The matrix A is of the form:
A = α 1 · · · α 2 * . . . . . . . . . . . . . . . α 2 * . . . . . . α 1 .
We prove by induction on n. It is trivial if n = 1. Assume that the conclusion holds for n − 1. By minor expansion on the first column of A, we have
det(A) = α 1 (α n−1 1 + Q 1 (α 1 , α 2 )) + Q 2 (α 1 , α 2 )
where Q 2 (α 1 , α 2 ) has total degree ≤ n − 1. By induction hypothesis, Q 1 (α 1 , α 2 ) has total degree ≤ n − 2. Let Q = α 1 · Q 1 + Q 2 . The proof is complete.
Lemma 1.3. The Pham system
α n 1 1 + Q 1 (α 1 , α 2 ) = 0, deg(Q 1 ) ≤ n 1 − 1 α n 2 2 + Q 2 (α 1 , α 2 ) = 0, deg(Q 2 ) ≤ n 2 − 1(11)
has at most n 1 n 2 solutions, where Q 1 and Q 2 are two polynomials in K[α 1 , α 2 ].
Proof. See e.g. [Cox, Little, and O'Shea 2015, Chapter 5, Section 3, Theorem 6].
Example 1.2. Let B = 3. With 4B = 12 evaluationsâ 1 ,â 2 , . . . ,â 12 obtained from the black box for f at inputs ω, ω 2 , . . . , ω 12 , we have the following 9 × 4 matrix:
H = â 1â2â3â4 a 2â3â4â5 a 3â4â5â6 a 4â5â6â7 a 5â6â7â8 a 6â7â8â9 a 7â8â9â10 a 8â9â10â11 a 9â10â11â12 ∈ K 9×4
Suppose there are two errorsâ ℓ 1 ,â ℓ 2 (ℓ 1 < ℓ 2 ) in the evaluations. If ℓ 1 ∈ {1, 2, 3}, then the Algorithm 1.1.3 can recover f (x) from the last 3B evaluations (â 4 ,â 5 , . . . ,â 12 ). Similarly, f (x) can also be recovered from (â 1 ,â 2 , . . . ,â 9 ) by the Algorithm 1.1.3 if ℓ 2 ∈ {10, 11, 12}.
It is remained to consider the case that ℓ 1 ∈ {4, 5, 6} and ℓ 2 ∈ {7, 8, 9}. We substitutê a ℓ 1 ,â ℓ 2 by α 1 , α 2 respectively. Then the determinants of the matrices H ℓ 1 −3 and H ℓ 2 −3 can be written as:
det(H ℓ 1 −3 ) = −α 4 1 + Q 1 (α 1 , α 2 ), deg Q 1 ≤ 3 det(H ℓ 2 −3 ) = −α 4 2 + Q 2 (α 1 , α 2 ), deg Q 2 ≤ 3(12)
where H ℓ 1 −3 , H ℓ 2 −3 are Hankel matrices defined as (8) and where Q 1 and Q 2 are bivariate polynomials in α 1 and α 2 . We compute the roots (ξ 1,k , ξ 2,k ) k≥1 of the system (12) in K and the pair correct values (f (ω ℓ 1 ), f (ω ℓ 2 )) is one of the roots. For each root (ξ 1,k , ξ 2,k ), we substitutê a ℓ 1 ,â ℓ 2 by ξ 1,k , ξ 2,k respectively, and check if the matrix H has rank B = 3. If so, then run Prony's algorithm 1.1.2 on the updated sequence (â 1 ,â 2 , . . . ,â 6 ). In the end, we obtain a list of sparse polynomials that contains f (x).
1.2.1. A list-interpolation algorithm for power-basis sparse polynomial with evaluations containing at most 2 errors.
Input: ◮ A black box representation of a polynomial f ∈ K[x]
where K is a field of characteristic = 2. The black box for f returns the same erroneous output when probed multiple times at the same input. ◮ An upper bound B on the sparsity of f . ◮ An upper bound D on the degree of f . ◮ ω ∈ K \ {0} satisfying: ◮ ω has order ≥ 2D + 1; ◮ ω i 1 = ω i 2 for all 1 ≤ i 1 < i 2 ≤ 4B. ◮ A root finder for zero-dimensional systems involving ≤ 2 variables over K. Output: ◮ An empty list or a list of sparse polynomials {f [1] , . . . , f [M ] } with each f [k] (1 ≤ k ≤ M) satisfying:
◮ f [k] has sparsity ≤ B and degree ≤ D; ◮ f [k]
is represented by its term degrees and coefficients; ◮ there are ≤ 2 indices i 1 , i 2 ∈ {1, 2, . . . , 4B} such that f [k] (ω i 1 ) =â i 1 and f [k] (ω i 2 ) =â i 2 whereâ i 1 andâ i 2 are the outputs of the black box probed at inputs ω i 1 and ω i 2 respectively; ◮ f (x) is contained in the list.
Step 1: For i = 1, 2, . . . , 4B, get the outputâ i of the black box for f at input ω i .
Step 2: Take (â 1 ,â 2 , . . . ,â 3B ) and (â B+1 ,â B+2 , . . . ,â 4B ) as the evaluations at the first step of Algorithm 1.1.3 and get two lists L 1 , L 2 . Let L be the union of L 1 and L 2 .
If either (â 1 ,â 2 , . . . ,â 3B ) or (â B+1 ,â B+2 , . . . ,â 4B ) contains ≤ 1 error, the Algorithm 1.1.3 can compute a list of sparse polynomials containing f (x).
Step 3: For every polynomialf in the list L, if there are ≥ 3 indices i ∈ {1, 2, . . . , 4B} such thatf (ω i ) =â i then deletef from L.
Step 4: For ℓ 1 = B + 1, . . . , 2B and ℓ 2 = 2B + 1, . . . , 3B, 4(a) substituteâ ℓ 1 by α 1 andâ ℓ 2 by α 2 in the Hankel matrices H ℓ 1 −B and H ℓ 2 −B (see (8)); let ∆ ℓ 1 (α 1 , α 2 ) = det(H ℓ 1 −B ) and ∆ ℓ 2 (α 1 , α 2 ) = det(H ℓ 2 −B ).
Here, we also use the fraction free Berlekamp/Massey algorithm Kaltofen and Yuhasz 2013] to compute the determinants of H ℓ 1 −B and H ℓ 2 −B . 4(b) compute all solutions of the Pham system {∆ ℓ 1 (α 1 , α 2 ) = 0, ∆ ℓ 2 (α 1 , α 2 ) = 0} in K; denote the solution set as {(ξ 1,1 , ξ 2,1 ), . . . , (ξ 1,b , ξ 2,b )}; 4(c) for k = 1, . . . , b, 4(c)i substituteâ ℓ 1 by ξ 1,k andâ ℓ 2 by ξ 2,k ; 4(c)ii use Berlekamp/Massey algorithm to compute the the minimal linear generator of the new sequence (â 1 ,â 2 , . . . ,â 4B ) and denote it by Λ(z); 4(c)iii if deg(Λ(z)) ≤ B, try Prony's algorithm 1.1.2 on the updated sequence (â 1 ,â 2 , . . . ,â 2B ); if Prony's algorithm 1.1.2 returns a sparse polynomial f of sparsity ≤ B and degree ≤ D, and there are ≤ 2 indices i 1 , i 2 ∈ {1, 2, . . . , 4B} such thatf (ω i 1 ) =â i 1 andf (ω i 2 ) =â i 2 , then addf into the list L;
If the two errors areâ ℓ 1 andâ ℓ 2 with ℓ 1 ∈ {B +1, . . . , 2B} and ℓ 2 ∈ {2B +1, . . . , 3B}, we substituteâ ℓ 1 andâ ℓ 2 by two symbols α 1 and α 2 respectively. As the pair of correct values (f (ω ℓ 1 ), f (ω ℓ 2 )) is a solution of the system {∆ ℓ 1 (α 1 , α 2 ) = 0, ∆ ℓ 2 (α 1 , α 2 ) = 0},
Step 4 will add f into the list L.
Step 5: Return the list L.
Proposition 1.4. The output list of Algorithm 1.2.1 contains ≤ B 4 + 2B 3 + 3B 2 + 2B + 4 polynomials.
Proof. In Algorithm 1.2.1, only
Step 2 and Step 4 produce new polynomials. By Proposition 1.1, both the lists L 1 and L 2 obtained at Step 2 contain ≤ B 2 + B + 2 polynomials. For
Step 4 of Algorithm 1.2.1, the Pham system {∆ ℓ 1 (α, β) = 0, ∆ ℓ 2 (α, β) = 0} has ≤ (B + 1) 2 solutions, so this step produces ≤ B 2 (B + 1) 2 polynomials. Therefore the output list contains ≤ B 2 (B + 1) 2 + 2(B 2 + B + 2) polynomials.
Correcting E Errors
Recall that f (x) is a sparse univariate polynomial of the form t j=1 c j x δ j (see (7)) with t ≤ B and ∀j, |δ j | ≤ D. We show how to list interpolate f (x) from N evaluations containing ≤ E errors, where
N = 4 3 E + 2 B.(13)
Denote θ = E 3 . Choose {ω 1 , . . . , ω θ , ω θ+1 } ∈ K \ {0} such that: (1) ω σ has order ≥ 2D + 1 for all 1 ≤ σ ≤ θ + 1, and
(2) ω i 1 σ 1 = ω i 2 σ 2 for any 1 ≤ σ 1 < σ 2 ≤ θ + 1 or 1 ≤ i 1 < i 2 ≤ 4B.
Letâ σ,i denote the output of the black box at input ω i σ . If E mod 3 = 0 then N = E 3 · 4B + 2B. The problem is reduced to one the following situations: (1) the last block (â θ+1,1 ,â θ+1,2 , . . . ,â θ+1,2B ) of length 2B is free of error, or (2) there is some block (â σ,1 ,â σ,2 , . . . ,â σ,4B ) of length 4B contains ≤ 2 errors, where 1 ≤ σ ≤ E 3 .
These two situations can be respectively dealt with Prony's algorithm 1.1.2 and the Algorithm 1.2.1. If E mod 3 = 1 then N = θ · 4B + 3B. The problem is reduced to one the following situations: (1) the last block (â θ+1,1 ,â θ+1,2 , . . . ,â θ+1,3B ) of length 3B has ≤ 1 error, or (2) there is some block (â σ,1 ,â σ,2 , . . . ,â σ,4B ) of length 4B contains ≤ 2 errors, where 1 ≤ σ ≤ θ. Therefore by trying the Algorithm 1. 1.3 on (â θ+1,1 ,â θ+1,2 , . . . ,â θ+1,3B ) and the Algorithm 1. 2.1 on (â σ,1 ,â σ,2 , . . . ,â σ,4B ), we can list interpolate f (x).
If E mod 3 = 2 then E = 3 · θ + 2 and N = (θ + 1)4B. So there is some σ ∈ {1, . . . , θ} such that the block (â σ,1 ,â σ,2 , . . . ,â σ,4B ) of length 4B contains ≤ 2 errors, and we can use the Algorithm 1.2.1 on this block to list interpolate f (x).
Remark 1.1. We apply the Algorithm 1.2.1 on every block (â σ,1 ,â σ,2 , . . . ,â σ,4B ) for all σ ∈ {1, . . . , E 3 }, which will result in ≤ E 3 (B 4 + 2B 3 + 3B 2 + 2B + 4) polynomials according to Proposition 1.4. The length of the last block depends on the value of E, and we have the following different upper bounds on the number of resulting polynomials:
(1) E 3 (B 4 + 2B 3 + 3B 2 + 2B + 4) + 1, if E mod 3 = 0; (2) E 3 (B 4 + 2B 3 + 3B 2 + 2B + 4) + B 2 + B + 2, if E mod 3 = 1 (see Proposition 1.1); (3) E 3 + 1 (B 4 + 2B 3 + 3B 2 + 2B + 4), if E mod 3 = 2.
By Descartes' rule of signs (see e.g. [Bochnak, Coste, and Roy 1998, Proposition 1.2.14]), the approach for correcting E errors will produce a single polynomial if K = R, N ≥ 2B + 2E and ω σ > 0, ∀σ. However, if N < 2B + 2E then there can be ≥ 2 valid sparse interpolants. We give an example to illustrate this.
h = f [1] − f [2] and f [1] (ω i ) = f [2] (ω i ) for i = 0, 1, . . . , 2B − 2.
Moreover, both f [1] and f [2] have sparsity ≤ B as deg(h) = 2B − 1. Consider a sequenceâ consisting of the following 2B + 2E − 1 values:
a (1) = f [1] (ω 0 ) , f [1] (ω 1 ), . . . , f [1] (ω 2B−2 ) , a (2) = f [1] (ω 2B−1 ), f [1] (ω 2B ), . . . , f [1] (ω 2B+E−2 ) , a (3) = f [2] (ω 2B+E−1 ), f [2] (ω 2B+E ), . . . , f [2] (ω 2B+2E−2 ) ,(14)
that is,â = (a (1) , a (2) , a (3) ). If all the errors are inâ (3) , then f [1] is a valid interpolant. Alternatively, if all the errors are inâ (2) then f [2] is a valid interpolant. Therefore, from these 2B + 2E − 1 values, we have at least 2 valid interpolants.
We remark that one of the valid interpolants, f [1] and f [2] , must have B terms since otherwise uniqueness is guaranteed by Descartes's rule of signs. In this example, both f [1] and f [2] have B terms because the polynomial h has 2B terms.Indeed, deg(h) = 2B − 1 implies that h has ≤ 2B terms, and by Descartes' rule of signs, h has ≥ 2B terms because it has 2B − 1 positive real roots. Therefore h is a dense polynomial. However, with the following substitutions x = y k , ω =ω k for some k ≫ 1, we have again a counter example where h, f [1] and f [2] are very sparse with respect to the new variable y.
Sparse Interpolation in Chebyshev Basis with Error Correction
Correcting One Error
Let K be a field of characteristic = 2 and f (x) ∈ K[x] be a polynomial represented by a black box. Assume that f (x) is a sparse polynomial in Chebyshev-1 basis of the form:
f (x) = t j=1 c j T δ j (x) ∈ K[x], 0 ≤ δ 1 < δ 2 < · · · < δ t = deg(f ), ∀j, 1 ≤ j ≤ t : c j = 0,
where T δ j (x) (j = 1, . . . , t) are Chebyshev polynomials of the First kind of degree δ j . We want to recover the term degrees δ j and the coefficients c j . Using the formula T n ( x+x −1 2 ) = x n +x −n 2 for all n ∈ Z ≥0 , [Arnold and Kaltofen 2015, Sec. 4] transformed f (x) into a sparse Laurent polynomial:
g(y) def = f ( y + y −1 2 ) = t j=1 c j 2 (y δ j + y −δ j )(15)
Therefore the problem is reduced to recover the term degrees and coefficients of the polynomial g(y). Let ω ∈ K such that: (1) ω has order ≥ 2D + 1, and (2) ω 2i 1 −1 = ω 2i 2 −1 for all 1 ≤ i 1 < i 2 ≤ 3B. For i = 1, 2, . . . , 3B, letâ 2i−1 be the output of the black box probed at input γ 2i−1 = (ω 2i−1 +ω −(2i−1) )/2. Note that g(ω i ) = g(ω −i ) for any integer i. For odd integers r ∈ {2k −1 | k = 1, . . . , B}, let G r ∈ K (B+1)×(B+1) be the following Hankel+Toeplitz matrix:
G r = â |r+2(i+j)| B i,j=0
Hankel matrix
+ â |r+2(i−j)| B i,j=0 Teoplitz matrix .(16)
If all the values involved in the matrix G r are correct, then det(G r ) = 0 [Arnold and Kaltofen 2015, Lemma 3.1]. If the 2B evaluations {â 2i−1 } 2B i=1 are free of errors, then one can use Prony's algorithm 1.1.2 to recover g(y) (and f (x)) from the following sequence:
a −2(2B−1)−1 ,â −2(2B−2)−1 , . . . ,â −1 ,â 1 , . . . ,â 2(2B−1)−1 ,â 2(2B)−1 .
Now we show how to list interpolate f (x) from 3B evaluations {â 2i−1 } 3B i=1 containing ≤ 1 error.
Assume thatâ 2ℓ−1 is the error, that is,â 2ℓ−1 = f (γ 2ℓ−1 ) = g(ω 2ℓ−1 ). The problem can be reduced to three cases:
Case 1: 1 ≤ ℓ ≤ B;
Case 2: B + 1 ≤ ℓ ≤ 2B;
Case 3: 2B + 1 ≤ ℓ ≤ 3B.
First, we try Prony's algorithm 1.1.2 on the sequence (â 2i−1 ) 2B i=−(2B−1) (see (17)), which will return f (x) if Case 3 happens. For Case 1 and Case 2, we substituteâ 2ℓ−1 by a symbol α. Let
∆ 2ℓ−1 (α) = det(G 2ℓ−1 ), if 1 ≤ ℓ ≤ B, det(G 2(ℓ−B)−1 ), if B + 1 ≤ ℓ ≤ 2B,
where G 2ℓ−1 and G 2(ℓ−B)−1 are defined as in (16) and ∆ 2ℓ−1 (α) is a univariate polynomial of degree B + 1 in α (see Lemma 2.1). By [Arnold and Kaltofen 2015, Lemma 3.1], the correct value f (γ 2ℓ−1 ) is a solution of the equation ∆ 2ℓ−1 (α) = 0. So we compute all solutions {ξ 1 , . . . , ξ b } of ∆ 2ℓ−1 (α) = 0 in K. For each solution ξ k (1 ≤ k ≤ b) we replaceâ 2ℓ−1 by ξ k and try Prony's algorithm on the updated sequence (â 2i−1 ) 2B i=−(2B−1) (see (17)). In the end, we will get a list of polynomials with f (x) being contained.
Lemma 2.1. Let r ∈ {2k − 1 | k = 1, . . . , B} and G r = â |r+2(i+j)| +â |r+2(i−j)| B i,j=0 .
If a r orâ r+2B is substituted by a symbol α in G r , then the determinant of G r is a univariate polynomial of degree B + 1 in α.
Proof. First, we show that ifâ r+2B is substituted by α, then the matrix G r has the form:
α + * * α + * . . . α + * * α + * .(18)
Since r ∈ {2k − 1 | k = 1, . . . , B} and i, j ∈ {0, 1 . . . , B}, we have
|r + 2(i + j)| = r + 2B ⇒ i + j = B, |r + 2(i − j)| = r + 2B ⇒ i = B, j = 0 or i = 0, j = B.
Therefore, either |r + 2(i + j)| = r + 2B or |r + 2(i − j)| = r + 2B implies i + j = B, sô a r+2B only appears on the anti-diagonal of the matrix G r . Conversely, every element on the anti-diagonal of G r is equal toâ r+2B +â |r+2(i−j)| for some i, j ∈ {0, 1, . . . , B}. Thus G r has the form (18) and its determinant is a univariate polynomial of degree B + 1 in α. Now we consider the case thatâ r is substituted by α. Similarly, because r ∈ {2k − 1 | k = 1, . . . , B} and i, j ∈ {0, 1 . . . , B}, we have
|r + 2(i + j)| = r ⇒ i = j = 0, |r + 2(i − j)| = r ⇒ i = j or i = j − r if j ≥ r.(19)
Therefore, if r > B then i = j in (19), soâ r only appears on the main diagonal of G r . On the other hand, every element on the main diagonal of G r is equal toâ |r+2(i+i)| +â r for some i ∈ {0, 1, . . . , t}. Hence, if r > B then the determinant of G r is a polynomial of degree B + 1 in α. Assume that r ≤ B. From (19), we see that after substitutingâ r by α, the matrix G r has the form:
α + * · · · α + * * . . . . . . . . . α + * * . . . . . . α + * .(20)
According to Lemma 1.2, the determinant of the matrix (20) is a univariate polynomial of degree B + 1 in α.
Example 2.1. For B = 3, we have 3B = 9 evaluations {â 2i−1 } 3B i=1 obtained from the black box for f at inputs γ i = (ω 2i−1 + ω −(2i−1) )/2. We construct the following 6 × 4 matrix:
G =
2â 1â3 +â 1â5 +â 3â7 +â 5 2â 3â5 +â 1â7 +â 1â9 +â 3 2â 5â7 +â 3â9 +â 1â11 +â 1 2â 7â9 +â 5â11 +â 3â13 +â 1 2â 9â11 +â 7â13 +â 5â15 +â 3 2â 11â13 +â 9â15 +â 7â17 +â 5
∈ K 6×4 .
For r = 1, 3, 5, the matrices G r are 4×4 submatrices of the matrix G. The matrix G 1 consists of the first 4 rows of G. If we substituteâ 1 orâ 7 by a symbol α, then the determinant of G 1 is univariate polynomial of degree 4 in α. The matrix G 3 consists of the second to the fifth row of G and the determinant of G 3 becomes a univariate polynomial of degree 4 in α ifâ 3 or a 9 is substituted by α. Similarly, the matrix G 5 consists of the last 4 rows of G. Substitutinĝ a 5 orâ 11 by α, det(G 5 ) is a univariate polynomial of degree 4 in α.
Suppose there is one errorâ 2ℓ−1 = f (γ 2ℓ−1 ) in the 3B evaluations. Here is how we correct this single error for all possible ℓ's:
(1) if ℓ ∈ {1, 2, 3}, then substituteâ 2ℓ−1 by α and compute the roots of det(G 2ℓ−1 ), and the roots are candidates for f (γ 2ℓ−1 );
(2) if ℓ ∈ {4, 5, 6}, then substituteâ 2ℓ−1 by α and compute the roots of det(G 2(ℓ−3)−1 ), and the roots are candidates for f (γ 2ℓ−1 );
(3) if ℓ ∈ {7, 8, 9}, then f (x) can be recovered by applying Prony's algorithm on the sequence (â 2i−1 ) 6 i=−5 .
2.1.1. A list-interpolation algorithm for Chebyshev-1 basis sparse polynomials with evaluations containing at most one error.
Input: ◮ A black box representation of a polynomial f (x) ∈ K[x] where K is a field of characteristic = 2. The black box for f returns the same erroneous output when probed multiple times at the same input. ◮ An upper bound B of the sparsity of f . ◮ An upper bound D of the degree of f . ◮ ω ∈ K \ {0} satisfying: ◮ ω has order ≥ 2D + 1; ◮ ω 2i 1 −1 = ω 2i 2 −1 for all 1 ≤ i 1 < i 2 ≤ 3B. ◮ A root finder for univariate polynomials over K. Output: ◮ An empty list or a list of sparse polynomials {f [1] , . . . , f [M ] } with each f [k] (1 ≤ k ≤ M) satisfying:
◮ f [k] has sparsity ≤ B and degree ≤ D; ◮ f [k]
is represented by its term degrees and coefficients;
◮ there is ≤ 1 index i ∈ {1, 2, . . . , 3B} such that f [k] (ξ 2i−1 ) =â 2i−1 where ξ i = (ω 2i−1 + ω −(2i−1)
)/2 andâ 2i−1 is the output of the black box probed at input ξ 2i−1 ; ◮ f (x) is contained in the list.
Step 1: For i = 1, 2, . . . , 3B, get the outputâ i of the black box for f at input ξ i = (ω 2i−1 + ω −(2i−1) )/2. Let L be an empty list.
Step 2: Try Prony's algorithm 1.1.2 on the sequence (â 2i−1 ) 2B i=−(2B−1) . If Prony's algorithm 1.1.2 returns a sparse polynomialf of sparsity ≤ B and degree ≤ D, and there is ≤ 1 index i ∈ {1, . . . , 3B} such thatf (ξ 2i−1 ) =â 2i−1 , then addf to the list L.
If the error is in {â 2i−1 } 3B i=2B+1 , then Prony's algorithm 1.1.2 in Step 2 will return f and so f will be added to the list L.
Step 3: For ℓ = 1, . . . , B, 3(a): substituteâ 2ℓ−1 by a symbol α in the matrix G 2ℓ−1 ; compute the determinant of G 2ℓ−1 and denote it by ∆ 2ℓ−1 (α); According to Lemma 2.1, ∆ ℓ (α) is a univariate polynomial of degree B + 1 in α; If the error isâ 2ℓ−1 (1 ≤ ℓ ≤ B), that isâ 2ℓ−1 = f (ξ 2ℓ−1 ), then we substituteâ 2ℓ−1 by a symbol α. As the correct value f (ξ 2ℓ−1 ) is a solution of ∆ 2ℓ−1 (α) = 0, that is f (ξ 2ℓ−1 ) = ξ k for some k ∈ {1, . . . , b},
Step 3 will add f into the list L.
Step 4: For ℓ = B + 1, . . . , 2B, 4(a): substituteâ 2ℓ−1 by a symbol α in the matrix G 2(ℓ−B)−1 ; compute the determinant of G 2(ℓ−B)−1 and denote it by ∆ 2ℓ−1 (α); According to Lemma 2.1, ∆ 2ℓ−1 (α) is a univariate polynomial of degree B + 1 in α; If the error isâ 2ℓ−1 (B + 1 ≤ ℓ ≤ 2B), that isâ 2ℓ−1 = f (ξ 2ℓ−1 ), we also substitutê a 2ℓ−1 by a symbol α. As the solution set {ξ 1 , . . . , ξ b ′ } of ∆ 2ℓ−1 (α) = 0 contains f (ξ 2ℓ−1 ), Step 4 will add f into the list L.
Step 5: Return the list L.
Proposition 2.2. The output list of Algorithm 2.1.1 contains ≤ 2B 2 + 2B + 1 polynomials.
Proof. The Step 2 in Algorithm 2.1.1 produces ≤ 1 polynomial, and both Step 3 and Step 4 produce ≤ B(B+1) polynomials. Hence the final output list has ≤ 1+2B(B+1) polynomials.
Correcting E Errors
The settings for f (x) are the same as in Section 2.1. We show how to list interpolate f (x) from N evaluations containing ≤ E errors, where
N = 3 2 E + 2 B.(21)
Denote θ = E 2 . Choose {ω 1 , . . . , ω θ , ω θ+1 } ∈ K \ {0} such that: (1)ω has order 2D + 1, and (2) ω 2i 1 −1 σ 1 = ω 2i 2 −1 σ 2 for 1 ≤ i 1 = i 2 ≤ 3B or 1 ≤ σ 1 = σ 2 ≤ θ + 1. Letâ σ,2i−1 denote the output of the black box at input γ σ,2i−1 = (ω 2i−1
σ + ω −(2i−1) σ )/2. If E is even then N = E 2 · 3B + 2B.
The problem is reduced to one the following situations: (1) the last block (â θ+1,2i−1 ) 2B i=1 of length 2B is free of errors, or (2) there is some block (â σ,2i−1 ) 3B i=1 with 1 ≤ σ ≤ E 2 of length 3B contains ≤ 1 errors. These two situations can be respectively dealt with Prony's algorithm 1.1.2 and the Algorithm 2.1.1.
If E is odd then E = 2·θ+1 and N = (θ+1)3B. Thus, there is some block (â σ,1 , . . . ,â σ,3B ) with 1 ≤ σ ≤ θ + 1 of length 3B contains ≤ 1 error; we can use the Algorithm 2.1.1 on this block to list interpolate f (x).
Remark 2.1. For every σ ∈ {1, . . . , E 2 }, we apply Algorithm 2.1.1 on the block (â σ,2i−1 ) 3B i=1 which will result in ≤ E 2 (2B 2 + 2B + 1) polynomials by Proposition 2.2. The length of the last block depends on the value of E, and we have following different upper bounds on the number of resulting polynomials:
(1) E 2 (2B 2 + 2B + 1) + 1, if E is even; (2) E 2 + 1 (2B 2 + 2B + 1), if E is odd. Due to Obrechkoff's theorem, a generalization of Descartes's rule of signs to orthogonal polynomials [Dimitrov and Rafaeli 2009, Theorem 1.1], our approach for correcting E errors gives a unique valid sparse interpolant when K = R, N ≥ 2B + 2E and ω σ > 1 [ Arnold and Kaltofen 2015, Corollary 2.4]. Similar to the case of power basis, if N < 2B + 2E then there can be ≥ 2 valid sparse interpolants in Chebyshev-1 basis as shown by the following example.
Example 2.2. Choose ω > 1. The polynomials h, f [1] and f [2] , given in Example 1.3, can be represented in Chebyshev-1 basis using the following formula [Fraser 1965, P. 303] [Cody 1970, P. 412] [Mathar 2006, Eq. (2)]:
x d = d ′ j=0 d−j is even d (d − j)/2 T j (x),(22)
where the primed summation indicates that the first term (at j = 0) is to be halved if it appears. Moreover, the formula (22) implies that f [1] is a linear combination of the odd degree Chebyshev-1 polynomials T 2j−1 (x) (j = 1, 2, . . . , B), and f [2] is a linear combination of the even degree Chebyshev-1 polynomials T 2j−2 (x) (j = 1, 2, . . . , B), which means both f [1] and f [2] have sparsity ≤ B in Chebyshev-1 basis as well. Therefore, f [1] and f [2] are also valid interpolants in Chebyshev-1 basis for the 2B + 2E − 1 evaluations given in (14) (if we assume B is an upper bound on the sparsity of the black-box polynomial f and E is an upper bound on the number of errors in the evaluations). Again, we remark that one of the valid interpolants, f [1] and f [2] , must have sparsity B since otherwise uniqueness is a consequence of the Obrechkoff's theorem [Dimitrov and Rafaeli 2009, Theorem 1.1]. In this example, h also has 2B terms in Chebyshev-1 basis because deg(h) = 2B − 1 and h has 2B − 1 real roots ω i > 1, i = 1, . . . , 2B − 1. Thus both f [1] and f [2] have sparsity B in Chebyshev-1 basis. One can also make h, f [1] and f [2] very sparse with respect to Chebyshev-1 basis by the following substitutions:
x = T k (y), ω = T k (ω) for some k ≫ 1.
For K = C, we usually choose ω as a root of unity. But then we may need 2B(2E + 1) evaluations to get a unique interpolant. Here is an example from [Kaltofen and Pernet 2014, Theorem 3], simply by changing the power basis to Chebyshev-1 basis.
Example 2.3. Consider the following two polynomials: Assume B = t and there are E errors in the sequenceâ. Then both f 1 and f 2 are valid interpolants forâ. More specifically, f 1 is a valid interpolant forâ if the E errors arê a 2t ,â 2t·2 , . . . ,â 2t·E ; f 2 is a valid interpolant forâ if the E errors areâ 2t(E+1) ,â 2t(E+2) , . . . ,â 2t·2E .
f 1 (x) = 1 t t−1 i=0 T 2i m 2t (x) f 2 (x) = − 1 t t−1 i=0 T (2i+1) m 2t (x),
Remark 2.2. Polynomials in Chebyshev-2, Chebyshev-3 and Chebyshev-4 bases can be transformed into Laurent polynomials using the formulas given in [Imamoglu, Kaltofen, and Yang 2018, Sec. 1, (7)-(9)]. Therefore, our approach to list-interpolate black-box polynomials in Chebyshev-1 bases also works for black-box polynomials in Chebyshev-2, Chebyshev-3 and Chebyshev-4 bases.
b): compute all solutions of the equation ∆ ℓ (α) = 0 in K; denote the solution set as {ξ 1 , . . . , ξ b } ; 4(c): for k = 1, . . . , b, 4(c)i: substituteâ ℓ by ξ k ; 4(c)ii: use Berlekamp/Massey algorithm to compute the the minimal linear generator of the new sequence (â 1 ,â 2 , . . . ,â 3B ) and denote it by Λ(z); 4(c)iii: if deg(Λ(z)) ≤ B, repeat Step 2.
Lemma 1. 2 .
2Let A be an n × n matrix with the following properties: 1) for i = 1, . . . , n, A[i, i] = α 1 ; 2) for some fixed k ∈ {1, . . . , n − 1} and for i = 1, . . . , n − k, A[i, i + k] = α 2 ;
Example 1 . 3 .
13Choose ω > 0. Let B be an upper bound on the sparsity of f and E be an upper bound on the number of errors in the evaluations. − ω i ), and f [1] be the sum of odd degree terms of h and f [2] be the negative of the sum of even degree terms of h. Clearly, we have
3
(b): compute all solutions of the equation ∆ 2ℓ−1 (α) = 0 in K; denote the solution set as {ξ 1 , . . . , ξ b }; 3(c): for k = 1, . . . , b, 3(c)i: substituteâ 2ℓ−1 by ξ k ; 3(c)ii: use Berlekamp/Massey algorithm to compute the the minimal linear generator of the new sequence (â 2i−1 ) 3B i=−3B+1 and denote it by Λ(z); 3(c)iii: if deg(Λ(z)) ≤ 2B, repeat Step 2.
4
(b): compute all solutions of the equation ∆ 2ℓ−1 (α) = 0 in K; denote the solution set as {ξ 1 , . . . , ξ b ′ }; 4(c): for k = 1, . . . , b ′ , 4(c)i: substituteâ 2ℓ−1 by ξ k ; 4(c)ii: use Berlekamp/Massey algorithm to compute the the minimal linear generator of the new sequence (â 2i−1 ) 3B i=−3B+1 and denote it by Λ(z); 4(c)iii: if deg(Λ(z)) ≤ 2B, repeat Step 2.
where m ≥ 2t(2E + 1) − 1 and 2t divides m. Let ω be a primitive m-th root of unity.The evaluations of f 1 at ω i +ω −i 2 for i = 1, 2, . . . , 2t(2E + 1) − 1 are(b, 1, . . . , b, 1 2E pairs of b, 1 , b) ∈ K 2t(2E+1)−1 .The evaluations of f 2 at ω i +ω −i 2 for i = 1, 2, . . . , 2t(2E + 1) − 1 are(b, −1, . . . , b, −1 2E pairs of b,−1 , b) ∈ K 2t(2E+1)−1 .Suppose we probe the black box for f at ω i +ω −i 2 with i = 1, 2, . . . , 2t(2E + 1) − 1 sequentially, and obtain the following sequence of evaluations:a = (b, 1, . . . , b, 1 E pairs of b, 1 , b, −1, . . . , b, −1 E pairs of b,−1 , b) ∈ K 2t(2E+1)−1
A. AppendixNotation (in alphabetic order): a i the output of the black box for f at input ω i B ≥ t, an upper bound on the sparsity of f α a symbol that substitute the single error in a block of 3B outputs of the black box for f α 1 , α 2 symbols that substitute the two errors in a block of 4B outputs of the black box for f β = (ω + 1/ω)/2, evaluation point of Chebyshev-1 polynomials c j the coefficent of the j-th term of f D ≥ deg f (x), an upper bound on the degree of f δ j the j-th term degree of f ∆ a matrix determinant E an upper bound on the number of errors that is input to the algorithm f the black-box polynomial, the Hankel matrix withâ r+i−1 ,â r+i , . . . ,â r+i−1+B on its i-th row K a field of characteristic = 2 ξ i candidates for the correct value f (ω ℓ ) ifâ ℓ is assumed to be an error ξ 1,i , ξ 2,i candidates for the pair of correct values f (ω ℓ 1 ), f (ω ℓ 2 ) ifâ ℓ 1 andâ ℓ 2 are assumed to be errors ℓ the error location in the outputs of the black box for f if E = 1 ℓ 1 , ℓ 2 the error locations in the outputs of the black box for f if E = 2 L the output list of our list decoding algorithms Λ the term locator polynomial M the number of the output polynomials of our error-correcting algorithms N the number of the evaluations by the black box for f ω a non-zero number in K, evaluation base point for the black-box polynomial f when only one block of evaluations are needed ω σ σ = 1, 2, . . . , θ + 1, non-zero numbers in K, evaluation base points for the black box polynomial f when multiple blocks of evaluations are needed ρ j 1 ≤ j ≤ t, the roots of the term locator polynomial Λ θ = ⌊E/3⌋ if the black-box polynomial f is in power basis, or = ⌊E/2⌋ if the black-box polynomial f is in Chebyshev bases t the actual number of terms of f ζ i distinct, algorithm-dependent arguments in K
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| []
|
[
"A deep look into the cores of young clusters ⋆ I. σ−Orionis",
"A deep look into the cores of young clusters ⋆ I. σ−Orionis"
]
| [
"H Bouy [email protected] \nInstituto de Astrofísica de Canarias\nC/ Vía Láctea s/n, E-38205 -La LagunaTenerifeSpain\n\nAstronomy Department\nUniversity of California\n94720BerkeleyCAUSA\n",
"N ⋆⋆ ",
"Huélamo \nLaboratorio de Astrofísica Espacial y Física Fundamental (LAEFF-INTA)\nE-28691, Villanueva de la CañadaPO BOX 78MadridSpain\n",
"E L Martín \nInstituto de Astrofísica de Canarias\nC/ Vía Láctea s/n, E-38205 -La LagunaTenerifeSpain\n\nDepartment of Physics\nUniversity of Central Florida\nP.O. Box 16238532816-2385OrlandoFLUSA\n",
"F Marchis \nAstronomy Department\nUniversity of California\n94720BerkeleyCAUSA\n",
"D Barrado Y Navascués \nLaboratorio de Astrofísica Espacial y Física Fundamental (LAEFF-INTA)\nE-28691, Villanueva de la CañadaPO BOX 78MadridSpain\n",
"J Kolb \nEuropean Southern Observatory\nKarl Schwartzschild Str. 2D-85748Garching bei MünchenGermany\n",
"E Marchetti \nEuropean Southern Observatory\nKarl Schwartzschild Str. 2D-85748Garching bei MünchenGermany\n",
"M G Petr-Gotzens \nEuropean Southern Observatory\nKarl Schwartzschild Str. 2D-85748Garching bei MünchenGermany\n",
"M Sterzik \nEuropean Southern Observatory\nAlonso de Cordova 3107, Vitacura, Santiago 1919001CasillaChile\n",
"V D Ivanov \nEuropean Southern Observatory\nAlonso de Cordova 3107, Vitacura, Santiago 1919001CasillaChile\n",
"R Köhler \nZAH, Landessternwarte\nD-69117HeidelbergKönigstuhlGermany\n",
"D Nürnberger \nEuropean Southern Observatory\nAlonso de Cordova 3107, Vitacura, Santiago 1919001CasillaChile\n"
]
| [
"Instituto de Astrofísica de Canarias\nC/ Vía Láctea s/n, E-38205 -La LagunaTenerifeSpain",
"Astronomy Department\nUniversity of California\n94720BerkeleyCAUSA",
"Laboratorio de Astrofísica Espacial y Física Fundamental (LAEFF-INTA)\nE-28691, Villanueva de la CañadaPO BOX 78MadridSpain",
"Instituto de Astrofísica de Canarias\nC/ Vía Láctea s/n, E-38205 -La LagunaTenerifeSpain",
"Department of Physics\nUniversity of Central Florida\nP.O. Box 16238532816-2385OrlandoFLUSA",
"Astronomy Department\nUniversity of California\n94720BerkeleyCAUSA",
"Laboratorio de Astrofísica Espacial y Física Fundamental (LAEFF-INTA)\nE-28691, Villanueva de la CañadaPO BOX 78MadridSpain",
"European Southern Observatory\nKarl Schwartzschild Str. 2D-85748Garching bei MünchenGermany",
"European Southern Observatory\nKarl Schwartzschild Str. 2D-85748Garching bei MünchenGermany",
"European Southern Observatory\nKarl Schwartzschild Str. 2D-85748Garching bei MünchenGermany",
"European Southern Observatory\nAlonso de Cordova 3107, Vitacura, Santiago 1919001CasillaChile",
"European Southern Observatory\nAlonso de Cordova 3107, Vitacura, Santiago 1919001CasillaChile",
"ZAH, Landessternwarte\nD-69117HeidelbergKönigstuhlGermany",
"European Southern Observatory\nAlonso de Cordova 3107, Vitacura, Santiago 1919001CasillaChile"
]
| []
| Context. Nearby young clusters are informative places to study star formation history. Over the last decade, the σ−Orionis cluster has been a prime location for the study of young very low mass stars, substellar and isolated planetary mass objects and the determination of the initial mass function. Aims. To extend previous studies of this association to its core, we searched for ultracool members and new multiple systems within the 1. ′ 5×1. ′ 5 central region of the cluster. Methods. We obtained deep multi-conjugate adaptive optics (MCAO) images of the core of the σ−Orionis cluster with the prototype MCAO facility MAD at the VLT using the H and K s filters. These images allow us to detect companions fainter by ∆H≈5 mag as close as 0. ′′ 2 on a typical source with H=14.5 mag. These images were complemented by archival SofI Ks-band images and Spitzer IRAC and MIPS mid-infrared images Results. We report the detection of 2 new visual multiple systems, one being a candidate binary proplyd and the other one a low mass companion to the massive star σ Ori E. Of the 36 sources detected in the images, 25 have a H-band luminosity lower than the expected planetary mass limit for members, and H-K s color consistent with the latest theoretical isochrones. Nine objects have additional Spitzer photometry and spectral energy distribution consistent with them being cluster members. One of them has a spectral energy distribution from H to 3.6 µm consistent with that of a 5.5 M Jup cluster member. Complementary NTT/SofI and Spitzer photometry allow us to confirm the nature and membership of two L-dwarf planetary mass candidates. | 10.1051/0004-6361:200810267 | [
"https://arxiv.org/pdf/0808.3890v2.pdf"
]
| 119,113,932 | 0808.3890 | 64ea14cf69f422ed4c089f420f016a0b98030d8b |
A deep look into the cores of young clusters ⋆ I. σ−Orionis
29 Oct 2008 October 30, 2008
H Bouy [email protected]
Instituto de Astrofísica de Canarias
C/ Vía Láctea s/n, E-38205 -La LagunaTenerifeSpain
Astronomy Department
University of California
94720BerkeleyCAUSA
N ⋆⋆
Huélamo
Laboratorio de Astrofísica Espacial y Física Fundamental (LAEFF-INTA)
E-28691, Villanueva de la CañadaPO BOX 78MadridSpain
E L Martín
Instituto de Astrofísica de Canarias
C/ Vía Láctea s/n, E-38205 -La LagunaTenerifeSpain
Department of Physics
University of Central Florida
P.O. Box 16238532816-2385OrlandoFLUSA
F Marchis
Astronomy Department
University of California
94720BerkeleyCAUSA
D Barrado Y Navascués
Laboratorio de Astrofísica Espacial y Física Fundamental (LAEFF-INTA)
E-28691, Villanueva de la CañadaPO BOX 78MadridSpain
J Kolb
European Southern Observatory
Karl Schwartzschild Str. 2D-85748Garching bei MünchenGermany
E Marchetti
European Southern Observatory
Karl Schwartzschild Str. 2D-85748Garching bei MünchenGermany
M G Petr-Gotzens
European Southern Observatory
Karl Schwartzschild Str. 2D-85748Garching bei MünchenGermany
M Sterzik
European Southern Observatory
Alonso de Cordova 3107, Vitacura, Santiago 1919001CasillaChile
V D Ivanov
European Southern Observatory
Alonso de Cordova 3107, Vitacura, Santiago 1919001CasillaChile
R Köhler
ZAH, Landessternwarte
D-69117HeidelbergKönigstuhlGermany
D Nürnberger
European Southern Observatory
Alonso de Cordova 3107, Vitacura, Santiago 1919001CasillaChile
A deep look into the cores of young clusters ⋆ I. σ−Orionis
29 Oct 2008 October 30, 2008Received ; acceptedAstronomy & Astrophysics manuscript no. madsori˙rev2Instrumentation: Adaptive opticsTechniques: High Angular ResolutionStars: visual binariesStars: EvolutionStars: formationStars: general
Context. Nearby young clusters are informative places to study star formation history. Over the last decade, the σ−Orionis cluster has been a prime location for the study of young very low mass stars, substellar and isolated planetary mass objects and the determination of the initial mass function. Aims. To extend previous studies of this association to its core, we searched for ultracool members and new multiple systems within the 1. ′ 5×1. ′ 5 central region of the cluster. Methods. We obtained deep multi-conjugate adaptive optics (MCAO) images of the core of the σ−Orionis cluster with the prototype MCAO facility MAD at the VLT using the H and K s filters. These images allow us to detect companions fainter by ∆H≈5 mag as close as 0. ′′ 2 on a typical source with H=14.5 mag. These images were complemented by archival SofI Ks-band images and Spitzer IRAC and MIPS mid-infrared images Results. We report the detection of 2 new visual multiple systems, one being a candidate binary proplyd and the other one a low mass companion to the massive star σ Ori E. Of the 36 sources detected in the images, 25 have a H-band luminosity lower than the expected planetary mass limit for members, and H-K s color consistent with the latest theoretical isochrones. Nine objects have additional Spitzer photometry and spectral energy distribution consistent with them being cluster members. One of them has a spectral energy distribution from H to 3.6 µm consistent with that of a 5.5 M Jup cluster member. Complementary NTT/SofI and Spitzer photometry allow us to confirm the nature and membership of two L-dwarf planetary mass candidates.
Introduction
Over the last decade, the σ−Orionis cluster has become one of the prime locations for the study of brown dwarfs (BDs) and planetary-mass objects (PMOs). It is young (2-3 Myr), free of extinction (A V <1 mag) and it has a large low-mass population (???????????). The Hipparcos parallax to the central OB pair σ Ori AB is 320 +120 −90 pc, and most previous studies used 350 pc to σ Ori AB as the cluster distance. Other authors estimated distances at about 390 pc (??). ? recently refined the measurement using main-sequence fitting and derived an improved value of 440 pc for a solar metallicity (?). The cluster mass function rises steadily from the very low mass stars through the BDs and into the PMO domain (??). No obvious discontinuities are seen either at the mass boundary between very low-mass stars and BDs or at the frontier between BDs and PMOs, which is not surprising as the initial mass function does not reflect the onset of nuclear reactions that may have occured during the subsequent evolution for stars and BDs. About two dozens PMOs candidates have been identified in the cluster (???). About half of those have been confirmed spectroscopically (????).
Evidence of disks has been found around BD and PMO members by the detection of large Hα emission, flux excesses at mid-infrared (mid-IR) wavelengths and large-amplitude photometric variability (???????). In particular the very strong Hα emission brown dwarf SOri 71, located just above the cluster deuterium burning limit, displays excess flux in IRAC band 4.5 at 8.0 µm (?). If the cluster PMOs form in a similar way to the very low-mass stars and BDs, we expect that they would have dusty disks, probably of lower mass scaling with primary mass. ? suggested an increase of the disk frequency towards low masses in the cluster, with a peak of 40% in the mass interval 0.2-0.1 M ⊙ . According to ?, the disk rate in the BD domain could be as high as 50%. ? and ? have reported mid-IR excesses in seven cluster PMOs, indicating that more than 30% of them have dusty disks. ? reported that the mass accretion rate of σ−Orionis members harboring disks is significantly lower than that of e.g ρ−Oph members. On the other hand, ? report a significantly larger fraction of accretors than in the neighboring λ−Orionis association where star formation might have been triggered by a supernova explosion (?).
The multiplicity of σ−Orionis members has been the subject of a number of studies. Using a Fraunhofer micrometer, ? resolved σ Ori AB for the first time as a close binary (0. ′′ 26). A few decades later, ? used the micrometer mounted on the 36 inch telescope at the Lick Observatory and confirmed the multiplicity of σ Ori AB. ? identified a possible spectroscopic component in the σ Ori AB binary system. More recently ? obtained adaptive optics (AO) images of the central region of the cluster, resolving a number of sources. ? used high resolution spectroscopic measurements of a sample of candidate very low-mass stars and BDs of the association to confirm the youth and membership and to search for spectroscopic binaries. Their preliminary conclusions, based on small-number statistics, shows that the binary fraction among σ−Orionis very low mass members is higher than that reported for their older field counterparts. Our understanding of the σ−Orionis cluster was recently complicated by the discovery by ? and ? of two distinct kinematics populations with different ages. However, most of the objects belonging to the older kinematic group are located northward of the central region of the σ−Orionis cluster, outside the area covered by the present study.
To extend the previous studies of σ−Orionis to its core we conducted an AO assisted imaging survey of the central part of the σ−Orionis cluster with the ESO multi-conjugated AO prototype instrument Multi-Conjugate Adaptive Optics Demonstrator (hereafter MAD ?). These deep images have a resolution of ≈0. ′′ 1 on a field of view of 1. ′ 5×1. ′ 5. on the position of the camera, the DIT and the observing conditions, so that it is not possible to perfectly correct for it. The three wavefront sensors can close the loop on stars brighter than V≈12 mag within a circular area of 2 ′ diameter.
MCAO Observations
Observations
During the on-sky demonstration run of MAD held in November 2007, a region of 1. ′ 5×1. ′ 5 centered on the σ−Orionis cluster was observed in the H and K s filters. Figure 1 gives an overview of the pointings and of the guide stars used for wavefront sensing. The geometrical distribution of the guide stars is quite asymmetric, leading to non-optimal corrections. A set of NINT=30 images was obtained by dithering within a box of 15 ′′ using the scanning capability of the infrared camera and keeping the adaptive-optics loop closed during the whole operation. The ambient conditions 1 during the observations reported by the ESO Ambient Conditions Database are given in Table 1. The conditions were significantly better during the Hband observations, with a coherence time of τ 0 =3.2 ms, and an average seeing of 0. ′′ 63 three times better than during the K s band observations, explaining the better quality and sensitivity of the H band images. The exposure time per individual image was 30×0.8 s (NDIT×DIT), so that the total exposure time for the final mosaics added up to 12 min in each band.
The corresponding images were processed with the Eclipse reduction package (?). They were first dark-subtracted and flatfielded. The sky contribution was then removed from the input frameset by filtering out low-frequency sky variations from the cube of jittered images. The images were then aligned and stacked to produce the final mosaics. The astrometric solution was computed using isolated and unresolved 2MASS counterparts and is accurate to within 0. ′′ 1. The final processed mo- Table 3). North is up and east is left and the scale is indicated.
saics are available upon request from the authors of this article. Figure 2 shows the final H-band mosaic.
MCAO performances
The Strehl ratio in the K s band ranges from 2% to 6%. The undersampling of the PSF in the H band prevents us from computing meaningful Strehl ratios, but the performance are expected to be similar. The quality of the correction closely follows the geometry of the 3 reference stars, and most sources away from the line made by the 3 reference stars are elongated with ellipticities in the range 0.41≤ e ≤0.95. In spite of the low Strehl ratios, the PSFs are much sharper than in the seeing limited SofI images (full width at half maximum FWHM≈0. ′′ 8, see Section 3.1) with an average FWHM of 0. ′′ 10 in H and 0. ′′ 15 in K s .
Adaptive-optics provide not only high spatial resolution but also high-contrast images. To illustrate the performances of the instrument and the limitations of the observations, we computed the limit of sensitivity for two cases: a star with H & K=14.5 mag located in a region of good AO correction (hereafter referred to as the "center") and a fainter star (H=15.88 mag,K=15.55 mag) located in a region of worse AO correction (hereafter referred to as the "edge"). The limit of sensitivity was computed using the 3-σ standard deviation of the PSF radial profile. Figure 3 shows the results. The MAD images allow detection of companions with a magnitude contrast ∆m=4 mag at 0. ′′ 25 and 0. ′′ 35 in H and K s respectively on a 14.5 mag star at the center. Sources brighter than ≈8 mag were saturated or above the detector linearity limit. We detected sources as faint as K s =19.55 mag and H=21.65 mag (3-σ detection), therefore well below the deuterium burning limit at the age (1-5 Myr) and distance (440 pc) of the cluster (predicted at H=17.03 mag for the DUSTY models and H=16.87 mag for the COND models, ??).
Photometry
Because of the complexity of the MCAO wavefront sensing, the PSF shows spatial variations due to anisoplanetic effects in the AO observations that can affect PSF photometry. To alleviate this problem, we took advantage of the sparsity of the field and performed aperture photometry rather than PSF fitting, except in a few cases of close multiple systems or when the source was located in the halo of a bright neighboring massive star. In these latter cases we extracted the photometry using nearby isolated stars as the reference PSF. The aperture photometry was performed using standard routines with the daophot package within IRAF 2 , using an aperture of 18 pixels (0. ′′ 5), and a sky annulus between 20-24 pixels (0. ′′ 56-0. ′′ 67). Two well-behaved isolated and unresolved 2MASS sources with clean H-band photometry (quality flag A, 2MASS J05384652-0235479 and 2MASS J05384746+0235252) were used to derive the photometric instrumental zeropoints given in Table 2. The K s -band zeropoints were computed using 11 clean and unresolved matches found in the SofI image (see Section 3.1). Some systematic errors might remain because of the strong anisoplanetism and unaccounted color terms. They are difficult to estimate because of the small number of well behaved counterparts in the 2MASS and SofI catalogs. The relatively small scatter between the MAD K s and SofI K s photometry of the 11 isolated sources in common (well within the uncertainties) suggest that the aperture was chosen large enough that these spatial variations do not affect the final photometry too much. The 36 detected sources and their photometry are reported in Table 3.
Complementary archival data
We searched the ESO and Spitzer public archives for complementary datasets of the same field.
NTT/SofI images:
The cluster was observed in the K s band with SofI at the NTT on 2001 December 12 (P.I. Testi, Programme 67.C-0042). A set of NINT=15 dithered images of 12×5 s (NDIT×DIT) was obtained that night. We retrieved the data and the corresponding calibration frames and processed them following standard procedures using the recommended Eclipse reduction package. The seeing (measured on the image) was 0. ′′ 8. We extracted the PSF photometry of all the sources brighter than the 3-σ noise of the local background using the ? Starfinder code. Using six unresolved 2MASS sources (quality flag A) we derived a zeropoint magnitude of 23.96±0.19 mag. Table 8 is available on-line. The limit of detection of the images is ≈20 mag, and the limit of completeness is ≈19 mag as illustrated in Fig. 4. The detector nonlinearity reaches about 3% at 14000 A.D.U., corresponding to K s =13.6 mag.
VLT/NACO images:
The central stars σ Ori AB and D were observed twice with the AO facility NACO on the VLT (Programs 074.C-0084 and 074.C-0628, P.I. Neuhäuser). A first set of 4 images of 100×0.345 s (NDIT×DIT) each was obtained on 2004 October 10 in the K s band with the S13 camera. A second set of 10 dithered images of 10×0.345 s (NDIT×DIT) was obtained in the narrow band Brγ (NB_2.17) filter. We retrieved the data and the associated calibration frames and processed them using the recommended Eclipse reduction package. The second epoch narrow band images are much shallower than the first ones, and we do not discuss them further. The limited number of images obtained at the first epoch does not allow us to correct perfectly for the bad and hot pixels. The brightness of the massive OB pair makes it an easy target for NACO and the Strehl ratio was high (60∼65%, FWHM=0. ′′ 07). The final image is shown in Fig. 5 and the astrometric measurements in Table 7.
Spitzer data
The σ−Orionis cluster was observed with Spitzer IRAC on 2004 October 08 in the course of program 37 (P.I. Fazio, ?) and with Table 4 gives a summary of the observations. We retrieved the calibrated, individual IRAC BCD (Basic Calibrated Data) images and stacked them following the procedures recommended by the Spitzer Science Center (SSC) with the MOPEX software package and the relevant calibration files. The final long exposure mosaics in channel 1 and 3 are made up of images of both programs 37 and 30395 weighted by their exposure times. Program 30395 channel 2 and 4 images do not overlap with our area of interest. The final mosaic images in these 2 bands are therefore made of program 37 images only. The short exposure mosaics allowed us to extract the photometry of bright sources otherwise saturated in the long exposure mosaics. A total of 9 sources detected in the MAD images are also detected in one (or more) Spitzer IRAC band. We extracted the photometry using standard PSF photometry procedures within the Interactive Data Language. Uncertainties were tentatively estimated from the Poisson noise weighted by the coverage maps of the mosaics, but the presence of the bright and asymmetric halo and ghosts around the massive stars make it difficult to estimate reliable uncertainties. The results are given in Table 5 and Fig. 6. The cluster was also observed with MIPS on 2004 March 17 in the course of Program 58 (P.I. Rieke). These observations are described in detail in ?. Except for one object (described in Section 7), the coarser resolution and lesser sensitivity of the MIPS images in the vicinity of the massive central OB stars does not allow us to extract any useful MIPS photometry for the MAD sources.
Nature of the detections
In order to rule out the possibility that some of the faint sources detected in the MAD images are artefacts (such as e.g bad pixels, remnants, ghosts, cosmic ray events, etc.), we compare the SofI, NACO and MAD images. Only seven objects detected in the MAD H-band image are not detected in the MAD and SofI K sband images, ruling out the possibility of artefacts for the other 29 objects.
Two of these seven sources without K s band counterparts in either the MAD or SofI images fall within the field of view of the NACO K s image. Only one is detected (SigOri-MAD-11, see 13) falls in an area of the NACO mosaic where only one image was co-added (no overlap with the other 3 images) and where the sensitivity is therefore significantly worse. It is not detected in the SofI image as it falls on a diffraction spike of the bright OB stars where the limit of sensitivity is significantly worse. We estimate the limit of sensitivity at the expected position of SigOri-MAD-13 in both the NACO and SofI images by adding artificial stars of decreasing luminosity until the 3-σ detection algorithm misses it. The detection limits measured that way are K s ≈17 mag in both the NACO and SofI image, thus just at the limit to detect SigOri-MAD-13. The nature of SigOri-MAD-13 is therefore uncertain and must be confirmed with new images. The five remaining H-band sources without a MAD, SofI or NACO K s counterpart are either located on diffraction spikes, or in a region where the halos of the bright massive stars are strong, degrading the limit of sensitivity of the SofI or NACO images. These five sources are the faintest of our sample, and with the currently available data we cannot rule out the possibility that they are false positives. However, the presence of confirmed sources with similar magnitudes (e.g SigOri-MAD-26) in both the H and K s images and the quality of the correlation between the PSF of these sources and the PSF of nearby objects (all better than 0.78) suggest that these detections are likely to be real as well. New images with a better signal-to-noise ratio are required to confirm their nature. An upper limit on their Ks band luminosity was derived by adding artificial stars of decreasing luminosity at the expected position of the source until the 3-σ detection algorithm no longer detected them.
SigOri-MAD-27 falls close to the position of a saturated bright massive star's remnant spot in both the H and K s MAD images (see Fig. 2). The associated H band photometry is therefore unreliable and should be considered with caution. The K s band photometry was measured in the SofI image and is therefore unaffected and reliable.
Multiple systems
With an average resolution of ≈0. ′′ 1, the MAD images resolve a number of multiple systems.
Previously known multiple systems
σ Ori AB: The brightest star in σ-Orionis, which gives its name to the cluster, is a known multiple system made of at least two components, σ Ori A and B (?). The pair is saturated in all the MAD, NACO K s and SofI images and no useful broad-band photometry can be performed. We used the unsaturated NACO Brγ image to measure the relative astrometry of the pair, as reported in Table 7.
σ Ori AD: (?) The two stars are heavily saturated in the MAD image. σ Ori A is saturated in the NACO image, but not σ Ori D. The D component is outside the field of view of the unsaturated NACO Brγ image. The saturation of the NACO broad-band K s image is not as extensive as in the MAD images and a careful fit of the wings of the saturated PSF of σ-Ori A and its Airy rings allow us to measure the relative astrometry of the AD pair.
σ Ori C: has been resolved as a wide binary by ?. It is clearly resolved in the new MAD images. The primary σ Ori Ca is saturated in the MAD H-band but the large separation allows us to accurately measure the photometry of Cb. From the unresolved 2MASS photometry, we derive the relative H-band photometry given in Table 7. It is also resolved in the SofI image but the bright Ca primary is saturated.
New multiple systems
σ Ori IRS1: the X-ray, radio, mid-and near-infrared source (?????) is resolved in both the MAD and NACO images and we measure its accurate relative astrometry and photometry (see Table 7). The resolved photometry and colors of the individual component correspond, according to the latest NextGen models of ?, to masses of 0.47 M ⊙ and 0.12 M ⊙ , but these values must be considered with caution as previous studies reported a high local extinction toward this source. The object was recently resolved independently by Caballero & Rebolo (in prep., private communication), and detected with AO in the optical but unresolved by ?. ? detected this object in the mid-IR with TIMMI2 at the ESO/3.6m and associated it with the radio source reported at 2, 6 and 20 cm by ? and to the mid-IR source IRAS 05362-0237. Based on the mid-IR excess, the displacement between the mid-IR photocenter (associated with a disk) and the radio photocenter (associated with free-free emission from a ionization front) and the presence of processed silicate grains revealed in the mid-IR spectrum, they describe the object as a proplyd, a proto-planetary disk being dispersed by the intense ultraviolet (UV) radiation from σ Ori AB (?). The source was marginally resolved in their 8.6 µm images, with a size of ≈1. ′′ 1. ? high resolution Chandra xray observations show that the xray source associated with σ Ori IRS1 is variable and must be a magnetically-active young TTauri star. To further investigate the nature of this source, we retrieved MIPS images of the cluster from the Spitzer public archive. The observations are described in detail in ?. The central massive pair σ Ori AB is saturated in all IRAC and MIPS 24 µm images preventing us from extracting useful photometry for σ Ori IRS1. A bright source is detected in the MIPS 70 µm image at α =05h38min44.9s and δ =-02 • 35 ′ 56.0 ′′ , i.e only 2. ′′ 24 from σ Ori IRS1 but 4. ′′ 9 from σ Ori AB. With a FWHM of ≈21. ′′ 5, it is difficult to associate the mid-IR source with either of these two objects with certainty, but the closer distance to the MAD, TIMMI-2 and VLA sources (see Fig. 8) and the expected much lower 70 µm flux of the massive pair lead us to associate the MIPS 70 µm source to σ Ori IRS1. Using PSF photometry, we measure a flux of 752±150 mJy. This value is not consistent with the IRAS 60 µm photometry given for the associated source IRAS 05362-0237, which is 6.95 Jy . With a much broader PSF, the IRAS photometry includes the flux of nearby bright nebulosities and is therefore unreliable. Fig. 8 shows the relative positions of the MAD, VLA, TIMMI-2 and Spitzer detections as well as the spectral energy distribution of the source. The formation and evolution of such a system in a Trapezium-like cluster at a projected distance of only ≈1200 AU (the physical distance might be much larger) of a pair of massive OB stars make it particularly interesting. One can indeed wonder how such a pair and its circum-binary disk has survived the dynamical interactions commonly occuring in such a cluster for as long as 1∼3 Myr. If confirmed, the nature of this low-mass/very low mass pair will provide direct proof that relatively wide (>100 AU) low mass pairs and their disk can survive the gravitational interactions and photo-evaporation in the early stages of the formation of proto-stellar clusters.
σ Ori E: is resolved in the MAD images. The two components are saturated in the H-band image. The secondary is not saturated in the K s -band image, but the heavy saturation of the primary and its halo prevent us from making any accurate measurement of the photometry of the secondary. Table 7 gives approximate measurements of the separation and position angle. We tentatively derive an upper limit on the Ks-band luminosity of the companion by adding an artificial PSF of increasing luminosity at its diametrically opposed position until the luminosity matches that of the companion. The luminosity roughly estimated this way (Ks≈10∼11 mag) corresponds to an estimated mass of 0.4∼0.8 M⊙. σ Ori E was suspected to have a low mass companion for several decades. ? had noticed a peculiar variable Hα emission in the σ Ori E spectrum and interpreted it (among other hypotheses) as the effect of the possible presence of a very low mass companion. Using uvby beta light curves, ? later suggested that σ Ori E was indeed a binary system, and put an upper limit on the mass of the companion at M<0.1 M ⊙ for inclination i >45 • . The source displayed repeated X-ray flares (reported with ROSAT, XMM-Newton and Chandra, respectively by ???). The X-ray activity was suspected by several of these authors to be in part due to the presence of an unseen low mass companion. The MAD images show that a low mass companion is indeed present next to the massive helium-strong σ Ori E star and could be responsible for the observed X-ray activity. The uncertainty on the MAD mosaic image astrometric solution is ≈0. ′′ 1. All these detections are closer to σ Ori IRS1 than to σ Ori AB. The scale is indicated. North is up and east is left. Right Panel: Spectral energy distribution of σ Ori IRS1. The IRAS fluxes are represented with grey circles (detections) or a triangle (upper limit). As the source is unresolved in the TIMMI-2, Spitzer and VLA images, the H and K s fluxes correspond to the combined fluxes of the two components as measured in the MAD and NACO images. Fig. 9 shows a color-magnitude diagram of all the sources detected in the MAD images, as well as catalogs of cluster members from the literature for comparison. A number of sources have luminosities and colors consistent with the cluster isochrones and with substellar and planetary masses. With only two bands and no comparison fields away from the cluster it is difficult to assess the level of contamination. Using the model of stellar population synthesis of the Galaxy of ? we find that the expected contamination by background giants or foreground dwarfs in the field of view of the MAD images must be very low.
Substellar and isolated planetary mass candidates
? recently estimated that the number of expected field L and Tdwarf contaminants toward the cluster adds up to ≈550 objects per 1deg 2 , corresponding to <0.4 contaminants in the case of our study. The contamination by extragalactic sources is expected to be much higher, as illustrated by the recent survey of the cluster by ?. With the current data, it is not possible to estimate the contamination by extragalactic sources among the MAD detections. We nevertheless note that none of the sources identified in the MAD images is extended while most extragalactic sources rejected by ? in their analysis were extended. Additional observations are required to confirm the membership of the new candidates.
Using the Spitzer photometry, we tentatively assess further the nature of the nine sources with mid-IR counterparts. Figure 6 shows the spectral energy distributions (SED) of the stellar candidate members. All but SigOri-MAD-34 match very well the median SED of cluster members measured by ?, ruling out the possibility of extragalactic contaminants. SigOri-MAD-34 displays a mid-IR excess most likely related to a circumstellar disk and providing further evidence of its youth and mem- bership of the association. Figure 7 shows the SEDs of all the MAD substellar candidates with a Spitzer counterpart. All but SigOri-MAD-31 also match very well the median SED of cluster members and the synthetic DUSTY SEDs of the corresponding H-band luminosity at the exact distance of the cluster sequence (440 pc, ?) and for an age of 1 Myr. These SEDs could be equally well fitted by a foreground late-M field dwarf at a distance of ≈100 pc, preventing us from drawing any firm conclusion regarding their membership of the association. Spectroscopy and proper motion measurements are required to confirm their nature. SigOri-MAD-6 (2MASS J05384454-0235349) was suggested to be a background A-F star or an extragalactic source by ? based on its blue J-K s color. The new 3.6 µm photometric measurement allows us to rule out the extragalatic contaminant hypothesis. The J-band photometric measurement reported by ? is ≈0.7 mag brighter than the expected J-band luminosity of a 0.025 M⊙ cluster member. All the other measurements (H, K s and 3.6 µm) are in good agreement with the luminosities expected for a 0.025 M⊙ cluster member as shown in Fig. 7. The J-band luminosity discrepancy could be due to underestimated errors on the photometric measurement in the proximity of the bright central OB pair (Caballero, private communication). Until new J-band measurements are obtained, SigOri-MAD-6 remains a good brown dwarf member candidate.
Mayrit 72345 and Mayrit 111335 are two L-dwarf candidate members reported by ? using optical, near-infrared and xray photometry. The two sources are detected in the SofI Ks-band image and in the Spitzer IRAC1, 2 and 3 channels (respectively SigOri-SofI-181 and SigOri-SofI-142, Table 6 and 8). The SofI photometry is in good agreement with the ? photometry within the uncertainties. Their SEDs, shown in Fig. 10, allow us to rule out the possibility that these two sources are extragalactic contaminants. They are inconsistent with a purely photospheric emission and display some excess in the near-and mid-IR most likely related to the presence of circumstellar material, suggesting their youth and membership of the association. In the near-IR, these two objects look like lower luminosity analogues of the substellar member S Ori J053902.1-023501 (?). Their mid-IR excess is less than that reported for S Ori J053902.1-023501. ? associated Mayrit 72345 with the X-ray source NX 77 detected with XMM-Newton by ?. Figure 6 shows that their SED is very similar to that of SigOri-MAD-31, adding further evidence that this latter source is likely to be a very low mass substellar member. The J and H-band luminosities of these 3 objects are well matched by a DUSTY SED with a mass of ≈5.5 M Jup at a distance of 440 pc and at an age of 1 Myr (Fig. 6 and 7). At 5 Myr, these luminosities correspond to a mass of ≈7 M Jup , as described in ?. This estimate is only tentative as the objects display some near-IR excess.
Notes on individual targets
SigOri-MAD-11 is detected in the MAD (12-σ) and NACO (3σ) images, confirming that it is a real detection. The two observations are separated by ≈3 yr and we tried to measure the proper motion relative to the closest unsaturated object, σ Ori IRS1a. The large uncertainties (dominated by the poorly calibrated camera distortions, of the order of 1% of the pixel scale for CONICA and unknown for CAMCAO see e.g. ??) and the small proper motion of the association (only ≈6 mas/yr, corresponding to ≈18 mas between the two epochs, or 1.35 CONICA pixels and 0.9 CAMCAO pixels) make these measurements inconclusive. New measurements covering a longer timescale will be required to confirmed that this source is co-moving with the nearby massive cluster members.
SigOri-MAD-26 is the faintest and reddest object with a detection in both H and K s . It has a counterpart in the SofI image. Its luminosity and color are consistent with the DUSTY 1 Myr isochrone at 440 pc within the uncertainties, but are largely inconsistent with the color of the confirmed member S Ori 70, which lies on the much bluer COND isochrones. This source is therefore unlikely to be a very low mass substellar member of the association. The K-band peak in the SED could be that of a low redshift (z≈0.5) galaxy. We nevertheless note that an average size galaxy (10 000 light year diameter) at that distance would have been easily resolved by our MCAO images. The object lies in the direction of the reddening vector, suggesting that it is most likely an extinguished source.
SigOri- is clearly detected in all IRAC four bands, and displays some excess at long wavelengths indicating the presence of circumstellar material and adding further evidence that the object is young and a member of the association. Assuming an age of 1 Myr and a distance of 440 pc, its H and K s -band luminosities correspond to a mass of ≈0.45 M ⊙ according to the NextGen models.
Spatial distribution of BD and PMOs
Using DENIS, 2MASS and previously published catalogs, ?? recently concluded that there is an apparent deficit of very low mass stars and high-mass BDs in the central 4 ′ of the cluster. The relatively large number of faint objects detected in our new images of the central part of the cluster suggest that a large fraction of very low mass members might have been missed by the previous survey because of the very bright central massive stars. Their presence affected all the previous seeing limited observations, making the detection of very low-mass cluster members in this region very difficult, if not impossible. The unprecedented dynamic range provided by MAD allows us to detect a number of faint sources that, if confirmed as cluster members, could fill the very low mass end of the central cluster population. Additional observations are required to confirm the membership and nature of the candidates and provide a quantitative answer to that question.
Conclusions and future prospects
Using multi-conjugate AO images of the core of the σ−Orionis cluster, we have identified 6 new BD candidates and 25 planetary mass candidates. Five of these have additional mid-IR Spitzer IRAC photometry consistent with that of sub-stellar members.
With the current data, it is not possible to conclude on their membership of the association as they could also be foreground late-M dwarfs. The candidate proplyd σ Ori IRS1 is resolved as a binary. The high spatial resolution MAD images resolve 5 pairs, including 2 previously unknown ones. The results presented in this paper illustrate the capacity of multi-conjugate AO to probe the immediate vicinity of young massive OB stars in great detail. Using complementary archival SofI and Spitzer images, we were able to confirm the membership of two L-dwarf candidate members that exhibit near-and mid-IR excess associated with a disk. One planetary mass candidate newly detected in the MAD images (SigOri-MAD-31) displays a SED very similar to these latter two, suggesting that it is also a L-dwarf member of the association harboring a disk. The presence or absence of very low mass stars, BDs and planetary mass objects close to massive stars provides novel constraints on the models of formation. Follow-up observations of the candidates are very much needed to confirm the nature and membership of the new candidates, and to provide quantitative feedback on the models of formation. 0.22±0.17 · · · · · · · · · SigOri-MAD-14 2.19±0.10 1.49±0.30 1.60±0.30 · · · SigOri-MAD-15 0.17±0.08 0.10±0.02 · · · · · · SigOri-MAD-23 0.68±0.06 · · · 0.3±0.1 · · · SigOri-MAD-25 0.18±0.04 0.27±0.05 · · · · · · SigOri-MAD-31 0.09±0.03 · · · · · · · · · SigOri-MAD-34 21.2±0.4 16.3±0.5 15.1±0.6 19.1±0.7 11.5±0.7 5.50±0.07 5.05±0.14 MAD σ-Ori Eab 2007-12-01T07:23 ≈330 ≈391 · · · · · · MAD Note -In addition to the uncertainties on the measurement, these values include errors related to the camera distortions (of the order of 1% of the pixel scale for NACO, see e.g ??, not calibrated but assumed to be of the same order for MAD).
Send offprint requests to: H. Bouy ⋆ Based on observations made at the ESO La Silla and Paranal Observatory under programmes 67.C-0042, 074.C-0084, and 074.C-0628 ⋆⋆ Marie Curie Outgoing International Fellow MOIF-CT-2005-8389
Fig. 2
2MAD H-band mosaic image with all the detected sources overplotted. The levels have been stretched differently close to the bright OB stars to enhance the contrast. The source numbers are indicated (see
Fig. 3
3Limit of sensitivity in the H-band (left panel) and K-band (right panel) for 2 different stars located in a region of good AO correction (center) and worse AO correction (edge). The curves have been computed from the 3-σ noise of the radial profile of the PSF.
Fig. 4
4Distribution of magnitude of the sources detected in the SofI (15 min on-source exposure) and MAD (6 min on-source exposure) images. The limits of completeness in the MAD images reach K s ≈19.5 mag and H≈20.0 mag. It is not homogeneous over the entire image, as large areas are contaminated by the strong halos of the bright massive stars.
Fig. 5
5NACO K s image of σ Ori AB, D and IRS1. The image was wavelet filtered to enhance SigOri-MAD-11 detection, indicated with a white circle. East is left and north is up and the scale is shown. The contrast was stretched differently around the bright OB stars to increase the dynamics of the figure. IRAC on 2007 April 03 in the course of program 30395 (P.I. Scholz, ?). Program 37 was executed in High Dynamic Range (HDR) mode providing equal numbers of consecutive short and long exposures.
Fig. 6 Fig. 7
67Spectral energy distributions of the MAD stellar sources with Spitzer IRAC counterparts. The dashed line represents the median SED of confirmed members without excess as derived by ? and normalized to the same H-band flux as the MAD source. SigOri-MAD-34 displays some mid-IR excess indicating the presence of circumstellar material. Spectral energy distributions of the MAD substellar candidates with Spitzer IRAC counterparts. The solid line represent the median SED of confirmed members without excess as derived by ? and normalized to the same H-band flux as the MAD source. The 1 Myr synthetic DUSTY SEDs for the given H-band luminosities and at a distance of 440 pc are overplotted with a dashed line and the corresponding masses indicated. The good match suggests that the sources are indeed members of the association.
Fig. 8
8Left panel: MAD H-band image around σ Ori IRS1. The Spitzer 70 µm, VLA (?) and TIMMI-2 (?) detections are overplotted with circles. The size of the circle corresponds to the FWHM of the source in these images, except for the VLA detection which has a much better accuracy (<0. ′′ 1) and is represented with a dimensionless cross. The different positions have been aligned using the MAD H-band image as the reference frame.
Fig. 9 H
9vs H-K color-magnitude diagram of the σ-Orionis cluster. The MAD measurements are represented by black circles. The DUSTY, NextGen and COND isochrones between 1-5 Myr and 350-440 pc are represented in red, green and blue shaded areas, respectively. Measurements for cluster members from the literature are overplotted as grey circles (?????). The confirmed PMO SOri 70 (?) is represented with an orange square. The substellar and deuterium burning limit from the DUSTY models for 1 Myr and 5 Myr at a distance of 440 pc as well as a A V =5 mag reddening vector are represented.
Fig. 10
10Spectral energy distributions of the substellar candidates with SofI and Spitzer IRAC counterparts (SigOri-SofI-181, black diamonds, and SigOri-SofI-142, red dots), as well as SigOri-MAD-31 (green squares). The line represents the 1 Myr DUSTY SED of a ≈5.5 M Jup object at a distance of 440 pc. The dotted line represents the the SED of the field L2 dwarf 2MASS J1017+1308 (?) normalized to the average J-band fluxes of the two candidates. The dashed line represents the SED of the 0.060 brown dwarf with a disk S Ori J053902.1-023501 as reported by ? and normalized to the average J-band fluxes of the two candidates. The three L-dwarf candidates display a clear mid-IR excess when compared to the field L2 dwarf or the DUSTY photospheric model. Their SED look like lower luminosity analogues of the substellar member S Ori J053902.1-023501 (?).
2.1. MAD: a multi-conjugate adaptive optics facility at the VLT ′′ 3×57. ′′ 3. At the time of the science demonstration observations, the CAMCAO camera suffered from a form of light leak. The amount of light and the pattern seen on the frames dependFig. 1 NTT/SOFI K s -band mosaic image of the observed field.The field of view within which the wavefront sensing stars for MAD can be selected is represented with a circle of 2 ′ diameter. The 3 wavefront sensing reference stars are indicated with red circles and their names and V-band luminosities overplotted. The 1. ′ 5×1. ′ 5 field-of-view of the final images is also represented. North is up and east is left.MAD is a prototype instrument performing wide field-of-view,
real-time correction for atmospheric turbulence (?). MAD was
built by the European Southern Observatory (ESO) with the con-
tribution of two external consortia to prove the feasibility of
MCAO on the sky in the framework of the 2 nd generation VLT
instrumentation and of the European Extremely Large Telescope
(ELT, ?). Originally designed as a laboratory experiment, MAD
was offered to the community for science demonstration in
November 2007 and January 2008 and was installed at the
Visitor Focus of the VLT telescope UT3 Melipal. An overview
of its performance on the sky is given in ??. Its CAMCAO near-
IR camera is based on a 2048×2048 pixel HAWAII-2 infrared
detector with a pixel scale of 0. ′′ 028 for a total field of view of
57.
Table 1
1Ambient conditions at the zenith and in the visible during the observationsFilter
date
airmass
seeing
τ 0
[UT]
[ ′′ ]
[ms]
K s
2007-11-25 06:33
1.09
1. ′′ 90±0. ′′ 13 0.9±0.1
H
2007-12-01 07:37
1.23
0. ′′ 63±0. ′′ 05 3.2±0.3
Table 2 Instrumental zeropoints for the MAD observations
Method
Filter Zeropoint [mag]
Aperture H
25.25±0.07
Aperture K s
24.62±0.10
PSF
H
25.30±0.07
PSF
K s
24.56±0.10
Acknowledgements. The authors are grateful to Paola Amico for her support at ESO. We thank J. A. Caballero and V. Béjar for fruitful discussions, suggestions and comments on the manuscript. We thank our anonymous referee for her/his review of the manuscript.H. Bouy acknowledges the funding from the European Commission's Sixth Framework Program as a Marie Curie Outgoing International Fellow (MOIF-CT-2005-8389). F. Marchis work was supported by the National Science Fundation Science and Technology Center for Adaptive Optics, and managed by the University of California at Santa Cruz under cooperative agreement No AST-9876783. Nuria Huélamo and David Barrado y Navascués are funded by Spanish grants MEC/ESP2007-65475-C02-02, MEC/Consolider-CSD2006-0070 and CAM/PRICIT-S-0505/ESP/0361. This work is based on observations obtained with the MCAO Demonstrator (MAD) at the VLT (ESO Public data release), which is operated by the European Southern Observatory. The MAD project is led and developed by ESO with the collaboration of the INAF-Osservatorio Astronomico di Padova (INAF-OAPD) and the Faculdade de Ciências de Universidade de Lisboa (FCUL). Based on observations made with ESO Telescopes at the La Silla or Paranal Observatories under programmes 67.C-0042, 074.C-0084, and 074.C-0628. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This work has made use of the Vizier Service provided by the Centre de Données Astronomiques de Strasbourg, France (?). This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency.
Table 3
3Catalog of sources detected in the MAD images 36:06.2 20.97±0.27 18.88±0.15 SigOri-MAD-27 05:38:47.0 -02:35:58.7 18.32±0.09 18.26±0.16 H-band uncertain SigOri-MAD-28 05:38:47.1 -02:35:53.8 19.25±0.11 18.97±0.33 SigOri-MAD-29 05:38:47.1 -02:35:51.9 19.55±0.13 19.12±0.25 SigOri-MAD-30 05:38:47.1 -02:36:05.6 21.39±0.44 <19.0 No SofI SigOri-MAD-31 05:38:47.2 -02:36:05.4 17.83±0.08 17.04±0.16 Spitzer, 7 M Jup 16.72±0.07 16.30±0.15 Note -Objects with H-band luminosity corresponding to a PMO (defined here as M≤0.012 M ⊙ ) are indicated in bold face. Objects with H-band luminosity corresponding to a BD (defined here as 0.012<M≤0.072 M ⊙ ) are indicated in italic. The errors include the zeropoint uncertainties.Name
RA (J2000) Dec (J2000)
H [mag]
K s [mag]
Other name
Comment
SigOri-MAD-1
05:38:43.5
-02:36:23.7 14.95±0.07 14.59±0.14
Spitzer, 0.050 M⊙
SigOri-MAD-2
05:38:43.9
-02:36:01.5 17.99±0.09 17.57±0.20
SigOri-MAD-3
05:38:44.1
-02:36:06.3
· · ·
9.01±0.10
σ Ori C, Mayrit 11238
Spitzer
SigOri-MAD-4
05:38:44.1
-02:36:04.4 14.63±0.07 14.07±0.10
SigOri-MAD-5
05:38:44.2
-02:36:12.7 20.81±0.07
<19.0
No SofI
SigOri-MAD-6
05:38:44.5
-02:35:35.0 16.11±0.07 15.77±0.14 2MASS J05384454-0235349
Spitzer, 0.025 M⊙
SigOri-MAD-7
05:38:44.6
-02:36:18.7 18.99±0.12 18.19±0.18
SigOri-MAD-8
05:38:44.8
-02:35:56.9 12.84±0.07 12.65±0.07
σ Ori IRS1 B
NACO
SigOri-MAD-9
05:38:44.8
-02:35:57.1 10.70±0.07 10.48±0.01
σ Ori IRS1 A
NACO
SigOri-MAD-10
05:38:45.0
-02:36:15.4 21.65±0.57
<19.0
No SofI
SigOri-MAD-11
05:38:45.2
-02:35:57.7 19.29±0.53 18.31±0.32
No SofI/ NACO
SigOri-MAD-12
05:38:45.2
-02:35:44.0 18.21±0.08 17.57±0.17
SigOri-MAD-13
05:38:45.2
-02:35:51.5 18.54±0.10 17.19±0.17
No SofI/No NACO
SigOri-MAD-14
05:38:45.2
-02:35:41.2 13.19±0.07 12.70±0.14
Mayrit 21023
Spitzer
SigOri-MAD-15
05:38:45.6
-02:35:29.3 16.15±0.07 15.72±0.14
Spitzer, 0.025 M⊙
SigOri-MAD-16
05:38:45.7
-02:35:38.4 19.39±0.12 19.20±0.26
SigOri-MAD-17
05:38:45.7
-02:35:23.7 19.23±0.12 18.56±0.15
SigOri-MAD-18
05:38:45.8
-02:36:13.5 18.26±0.09 17.77±0.19
SigOri-MAD-19
05:38:45.9
-02:35:26.7 17.38±0.08 17.03±0.16
SigOri-MAD-20
05:38:46.0
-02:35:57.1 18.53±0.10 18.16±0.16
SigOri-MAD-21
05:38:46.4
-02:35:49.7 20.07±0.17
<19.0
No SofI
SigOri-MAD-22
05:38:46.4
-02:36:00.7 19.25±0.12 18.80±0.28
SigOri-MAD-23
05:38:46.5
-02:35:48.3 14.50±0.07 14.11±0.14 2MASS J05384652-0235479
Spitzer, 0.060 M⊙
SigOri-MAD-24
05:38:46.7
-02:36:18.3 18.75±0.10 18.05±0.21
SigOri-MAD-25
05:38:46.8
-02:36:24.1 15.72±0.07 15.37±0.14
Spitzer, 0.030 M⊙
SigOri-MAD-26
05:38:46.8
-02:SigOri-MAD-32
05:38:47.2
-02:35:50.5 21.01±0.07
<19.5
No SofI
SigOri-MAD-33
05:38:47.3
-02:36:00.8 19.75±0.15 18.60±0.15
SigOri-MAD-34
05:38:47.4
-02:35:25.2 11.05±0.07 10.42±0.14
Mayrit 53049
Spitzer
SigOri-MAD-35
05:38:48.0
-02:35:52.4 17.63±0.08 17.14±0.17
SigOri-MAD-36
05:38:48.1
-02:35:51.0
Table 4
4Log of Spitzer observationsProgram ID
P.I.
Date Obs.
Exposure time Number of frames
[DD-MM-YYYY]
[s]
37
Fazio
08-10-2004
1.2/30
270/270
30395
Scholz
03-04-2007
100
24
Table 5
5Spitzer IRAC photometry of sources detected in the MAD imagesName
3.6 µm
4.5 µm
5.8 µm
8.0 µm
[mJy]
[mJy]
[mJy]
[mJy]
SigOri-MAD-1
0.46±0.04 0.22±0.09 0.10±0.04 0.11±0.05
SigOri-MAD-3
54.6±0.04 43.0±0.04
20.9±0.5 13.19±0.7
SigOri-MAD-6
Table 6
6Spitzer IRAC photometry of the two L-dwarf candidates detected in the SofI images SofI-142 0.073±0.010 0.070±0.010 0.045±0.007 · · · Mayrit 111335 SigOri-SofI-181 0.080±0.012 0.067±0.010 0.041±0.010 · · · Mayrit 72345Name
3.6 µm
4.5 µm
5.8 µm 8.0 µm
other
[mJy]
[mJy]
[mJy]
[mJy]
name
SigOri-
Table 7
7Relative astrometry and photometry of the multiple systemsSystem
date UT
Separation
P.A.
∆H
∆K s
Instrument
[YYYY-MM-DD HH:MM]
[mas]
[ • ]
[mag]
[mag]
σ-Ori AB
2004-10-11 09:44
255.7±1.8
100.9±0.4
· · ·
· · ·
NACO
σ-Ori AD
2004-10-11 09:44
13037.2±27
84.1±0.4
· · ·
· · ·
NACO
σ-Ori IRS1 AB
2004-10-11 09:44
236.3±2.4
318.1±0.4
· · ·
2.19±0.01
NACO
σ-Ori IRS1 AB
2007-12-01T07:23
242.9±3.6
317.0±0.7 2.13±0.10 2.17±0.07
MAD
σ-Ori Cab
2007-12-01T07:23
1991.5±3.9
Table 8
8NTT/SofI Ks-band photometry
At the zenith and in the visible.
IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation.
38:33.3 -02:36:17.0 13.05 05383335-0236176Name RA (J2000) Dec (J2000) Ks [mag] Nearest 2MASS Nearest Mayrit SigOri-SofI-1 05. Name RA (J2000) Dec (J2000) Ks [mag] Nearest 2MASS Nearest Mayrit SigOri-SofI-1 05:38:33.3 -02:36:17.0 13.05 05383335-0236176
. Sigori, SofI-12 05:38:33.9 -02:36:26.3 19.16SigOri-SofI-12 05:38:33.9 -02:36:26.3 19.16
. Sigori-Sofi, 18 05:38:34.8 -02:36:20.0 16.59 05383491-0236206SigOri-SofI-18 05:38:34.8 -02:36:20.0 16.59 05383491-0236206
This figure "mad_H.png" is available in "png. This figure "mad_H.png" is available in "png" format from: http://arXiv.org/ps/0808.3890v2
| []
|
[
"The impact of natal kicks on galactic r-process enrichment by neutron star mergers",
"The impact of natal kicks on galactic r-process enrichment by neutron star mergers"
]
| [
"Freeke Van De Voort \nSchool of Physics and Astronomy\nCardiff Hub for for Astrophysics Research and Technology\nCardiff University\nQueen's Buildings, The ParadeCF24 3AACardiffUK\n",
"Rüdiger Pakmor \nMax Planck Institute for Astrophysics\nKarl-Schwarzschild-Straße 185748GarchingGermany\n",
"Rebekka Bieri \nMax Planck Institute for Astrophysics\nKarl-Schwarzschild-Straße 185748GarchingGermany\n",
"Robert J J Grand \nInstituto de Astrofísica de Canarias\nCalle Vía Láctea s/nE-38205La Laguna, TenerifeSpain\n\nDepartamento de Astrofísica\nUniversidad de La Laguna\nAv. del Astrofísico Francisco Sánchez s/n, La LagunaE-38206TenerifeSpain\n"
]
| [
"School of Physics and Astronomy\nCardiff Hub for for Astrophysics Research and Technology\nCardiff University\nQueen's Buildings, The ParadeCF24 3AACardiffUK",
"Max Planck Institute for Astrophysics\nKarl-Schwarzschild-Straße 185748GarchingGermany",
"Max Planck Institute for Astrophysics\nKarl-Schwarzschild-Straße 185748GarchingGermany",
"Instituto de Astrofísica de Canarias\nCalle Vía Láctea s/nE-38205La Laguna, TenerifeSpain",
"Departamento de Astrofísica\nUniversidad de La Laguna\nAv. del Astrofísico Francisco Sánchez s/n, La LagunaE-38206TenerifeSpain"
]
| [
"MNRAS"
]
| We study galactic enrichment with rapid neutron capture (r-process) elements in cosmological, magnetohydrodynamical simulations of a Milky Way-mass galaxy. We include a variety of enrichment models, based on either neutron star mergers or a rare class of core-collapse supernova as sole r-process sources. For the first time in cosmological simulations, we implement neutron star natal kicks on-the-fly to study their impact. With kicks, neutron star mergers are more likely to occur outside the galaxy disc, but how far the binaries travel before merging also depends on the kick velocity distribution and shape of the delay time distribution for neutron star mergers. In our fiducial model, the median r-process abundance ratio is somewhat lower and the trend with metallicity is slightly steeper when kicks are included. In a model 'optimized' to better match observations, with a higher rate of early neutron star mergers, the median r-process abundances are fairly unaffected by kicks. In both models, the scatter in r-process abundances is much larger with natal kicks, especially at low metallicity, giving rise to more r-process enhanced stars. We experimented with a range of kick velocities and find that with lower velocities, the scatter is reduced, but still larger than without natal kicks. We discuss the possibility that the observed scatter in r-process abundances is predominantly caused by natal kicks removing the r-process sources far from their birth sites, making enrichment more inhomogeneous, rather than the usual interpretation that the scatter is set by the rarity of its production source. | 10.1093/mnras/stac710 | [
"https://arxiv.org/pdf/2110.11963v2.pdf"
]
| 239,769,006 | 2110.11963 | c55783de542f3ec8976fb72795f683d016802905 |
The impact of natal kicks on galactic r-process enrichment by neutron star mergers
2021
Freeke Van De Voort
School of Physics and Astronomy
Cardiff Hub for for Astrophysics Research and Technology
Cardiff University
Queen's Buildings, The ParadeCF24 3AACardiffUK
Rüdiger Pakmor
Max Planck Institute for Astrophysics
Karl-Schwarzschild-Straße 185748GarchingGermany
Rebekka Bieri
Max Planck Institute for Astrophysics
Karl-Schwarzschild-Straße 185748GarchingGermany
Robert J J Grand
Instituto de Astrofísica de Canarias
Calle Vía Láctea s/nE-38205La Laguna, TenerifeSpain
Departamento de Astrofísica
Universidad de La Laguna
Av. del Astrofísico Francisco Sánchez s/n, La LagunaE-38206TenerifeSpain
The impact of natal kicks on galactic r-process enrichment by neutron star mergers
MNRAS
0002021Accepted 2022 March 10. Received 2022 March 10; in original form 2021 October 21Preprint March 14, 2022 Compiled using MNRAS L A T E X style file v3.0stars: abundances -stars: neutron -supernovae: general -Galaxy: abundances - galaxies: formation -methods: numerical
We study galactic enrichment with rapid neutron capture (r-process) elements in cosmological, magnetohydrodynamical simulations of a Milky Way-mass galaxy. We include a variety of enrichment models, based on either neutron star mergers or a rare class of core-collapse supernova as sole r-process sources. For the first time in cosmological simulations, we implement neutron star natal kicks on-the-fly to study their impact. With kicks, neutron star mergers are more likely to occur outside the galaxy disc, but how far the binaries travel before merging also depends on the kick velocity distribution and shape of the delay time distribution for neutron star mergers. In our fiducial model, the median r-process abundance ratio is somewhat lower and the trend with metallicity is slightly steeper when kicks are included. In a model 'optimized' to better match observations, with a higher rate of early neutron star mergers, the median r-process abundances are fairly unaffected by kicks. In both models, the scatter in r-process abundances is much larger with natal kicks, especially at low metallicity, giving rise to more r-process enhanced stars. We experimented with a range of kick velocities and find that with lower velocities, the scatter is reduced, but still larger than without natal kicks. We discuss the possibility that the observed scatter in r-process abundances is predominantly caused by natal kicks removing the r-process sources far from their birth sites, making enrichment more inhomogeneous, rather than the usual interpretation that the scatter is set by the rarity of its production source.
INTRODUCTION
Neutron stars in the Milky Way have, on average, velocities in the range of hundreds of km s −1 (e.g. Hobbs et al. 2005;Faucher-Giguère & Kaspi 2006). Some have even been observed with velocities exceeding 1000 km s −1 . This is not the case for their progenitor population, i.e. massive stars. Instead, these high velocities are imparted on the neutron stars when they are born, which is why these are referred to as natal kicks. A likely explanation for these kicks are asymmetries in the supernova explosions that expel the outer material of the progenitor stars, causing the resulting compact object to be born with a significant kick velocity (e.g. Scheck et al. 2006;Janka 2013). This means that neutron stars, including binary neutron stars, are more likely to be found outside the disc of the galaxy than massive main sequence stars. ★ E-mail: [email protected] Whether or not a binary star system survives the natal kick imparted by its supernovae depends on the exact binary configuration and the dynamics of the supernova explosions (e.g. Bray & Eldridge 2016). The binary system is more likely to remain bound if the natal kick it experiences is weaker. The resulting velocity distribution of binary neutron stars may therefore be different, with a lower average velocity, than that of single neutron stars. The handful of binary neutron stars known in the Galaxy have estimated velocities ranging from ≈ 0 − 500 km s −1 (Wong et al. 2010). Short gamma-ray bursts, whose progenitors are thought to be neutron star mergers, show substantial offsets from their host galaxies. Using these offsets, kick velocites of ≈ 20 − 140 km s −1 have been derived, consistent with the Galactic estimate (Fong & Berger 2013). For both single and double neutron stars, the sample sizes are small and the uncertainties in kick velocities are therefore large.
Neutron star mergers are promising sites of production of rapid neutron capture (r-process) elements. The distribution of elements produced in such explosions is reasonably consistent with observations of the Sun and other stars, even for the heaviest r-process elements (e.g. Lattimer et al. 1977;Freiburghaus et al. 1999;Bauswein et al. 2013;Ji et al. 2016). Furthermore, their rarity can potentially explain the observed scatter in r-process abundances at low metallicity as probed by europium, an element that is almost exclusively produced by the r-process (e.g. Burris et al. 2000). The first detection of gravitational waves from a neutron star merger led to the detection of an electromagnetic counterpart, called a kilonova (e.g. Abbott et al. 2017;Coulter et al. 2017). The evolution of the light curves showed rapidly declining blue emission and long-lived red emission, which is consistent with the kilonova being powered by the radioactive decay of high-opacity r-process elements and difficult to explain otherwise (e.g. Drout et al. 2017;Kasen et al. 2017;Pian et al. 2017). This lends further credence to the theory that r-process elements are predominantly produced in kilonovae.
However, idealized chemical evolution models and cosmological simulations using neutron stars as the only source for r-process elements struggle to reproduce observations of stellar europium abundances in the Milky Way in detail (e.g. Argast et al. 2004;Matteucci et al. 2014;Wehmeyer et al. 2015;Côté et al. 2019;Haynes & Kobayashi 2019;van de Voort et al. 2020). A possible solution is that there is a second source for r-process elements, which could even dominate over r-process element production by neutron star mergers. One candidate for such a source is a special, rare type of core-collapse supernova, such as magneto-rotational supernovae or collapsars (e.g. MacFadyen & Woosley 1999;Cameron 2003;Fujimoto et al. 2008;Winteler et al. 2012;Mösta et al. 2018;Siegel 2019;Cowan et al. 2021). Other potential rprocess sources include, for example, magnetized neutrino-driven winds from proto-neutron stars (Thompson & ud-Doula 2018) and common envelope jet supernovae (Grichener & Soker 2019). Such a source should be rare enough that it produces the large observed scatter in europium at low metallicity, but not so rare that a large fraction of metal-poor stars are born without r-process elements.
It is also possible that the current generation of neutron star merger models are missing a key ingredient and that including this ingredient in our models would provide a better match with observations. The potential importance of natal kicks for r-process enrichment has recently been highlighted by Banerjee et al. (2020), but has not yet been explored in cosmological simulations. Galaxy formation is complex and governed by many processes, such as large-scale outflows, gas accretion and reaccretion, (metal) mixing, galaxy mergers, and satellite stripping. All of these processes interact with one another, adding even more complexity. Cosmological, (magneto)hydrodynamical simulations, which base their initial conditions on fluctuations observed in the cosmic microwave background radiation, include all of these processes and their interactions. They can be thus used to follow inhomogeneous chemical evolution on-the-fly over the full history of the Universe. Previously, we used such simulations to follow r-process enrichment from neutron star mergers embedded in the existing stellar population (van de Voort et al. 2020). We have now implemented neutron star natal kicks to study their impact on the resulting stellar r-process abundances.
In this work, we report on the results from our cosmological simulations and focus on r-process enrichment by neutron star mergers, but also contrast this with models for enrichment via rare corecollapse supernovae. For the first time, we study the effect of neutron star natal kicks, varying the uncertain parameters of the neutron star delay time distribution and the natal kick velocity distribution. We also include a model for extremely rare core-collapse supernovae in our high-resolution simulation, which we previously had only simulated at medium resolution (van de Voort et al. 2020). In Section 2 we briefly discuss our cosmological set-up and our galaxy formation model. The way we implemented natal kicks and neutron star mergers is described in Section 2.1 and rare core-collapse supernovae in Section 2.2. Our results are shown in Section 3, where we discuss our fiducial model (Section 3.1), the impact of varying our model parameters (Section 3.2), and a model with 'optimized' parameters (Section 3.3), which we compare to observations (Section 3.3.1) and use to test very low kick velocities (Section 3.3.2). We summarize our results and discuss further implications in Section 4.
METHOD
This work is based on a simulation from the Auriga project (Grand et al. 2017). Auriga consists of a large number of zoom-in magnetohydrodynamical, cosmological simulations of relatively isolated Milky Way-mass galaxies and their environments, simulated at high resolution to about 1 Mpc from the central galaxy within a lower resolution (100 Mpc) 3 volume. The simulated galaxies are discdominated and have stellar masses, galaxy sizes, rotation curves, star formation rates, and metallicities in agreement with observations. For this work, we resimulate one of the Auriga galaxies (called 'halo 6') at high resolution. We also carried out medium-resolution and low-resolution simulations, again of halo 6, in order to test for resolution effects and additional model variations. The galaxy in our chosen halo has properties that are reasonably similar to the Milky Way. Additionally, it has a relatively compact zoom-in region, which makes it one of the most efficient Auriga haloes in terms of computational time. The simulations were carried out with the quasi-Lagrangian moving-mesh code (Springel 2010). Our main results focus on a new high-resolution zoom-in simulation. Its target cell resolution is cell = 6.7 × 10 3 M and dark matter particle mass is DM = 3.6 × 10 4 M . We furthermore ran an additional simulation with different neutron star merger parameters, results from which better match currently available observational data, which is shown in Section 3.3. This simulation has 8 times lower mass resolution than our fiducial simulation, which is referred to as medium resolution ( cell = 5.4 × 10 4 M and DM = 2.9 × 10 5 M ). We assumed a ΛCDM cosmology with parameters Ω m = 1 − Ω Λ = 0.307, = 0.048, ℎ = 0.6777, 8 = 0.8288, and = 0.9611 (Planck Collaboration et al. 2014).
Gas becomes star-forming above a density threshold of ★ H = 0.11 cm −3 and forms stars stochastically (Springel & Hernquist 2003). This dense gas is put on an effective equation of state, because we cannot resolve the multi-phase nature of the interstellar medium (ISM). The ISM in our simulations is therefore smoother than it is in reality. At = 0, the total stellar mass contained within 30 kpc is 7.1 × 10 10 M . The total stellar mass within the virial radius, vir , is 7.8 × 10 10 M , where vir = 214 kpc and defined as the radius within which the average density is equal to 200 times the critical density of the Universe. The total halo mass within vir is 1.0 × 10 12 M . The low redshift star formation rate (SFR) is 3.0 M yr −1 (3.4 M yr −1 ), averaged over the last Gyr (100 Myr). The full star formation history is shown in Grand et al. (2021) where this simulation is referred to as 'Level 3'. A 3-colour face-on and edge-on image of the galaxy is shown in Figure 1 in which older stars appear redder and younger stars appear bluer. The galaxy is disc-dominated with a central bar and boxy/peanut bulge (see e.g. Fragkoudi et al. 2020).
A star particle in the simulation represents a single stellar Figure 1. Face-on and edge-on images of the stars in K-, B-, and U-band (shown in red, green, and blue, respectively) at = 0. The images are 50 kpc × 50 kpc (top panel) and 50 kpc × 25 kpc (bottom panel). The galaxy exhibits a bar with predominantly older (redder) stars and an extended starforming (bluer) disc. Its total stellar mass is 7 × 10 10 M and its present-day SFR is about 3 M yr −1 .
population with a specific age and metallicity with an initial mass function from Chabrier (2003). Stars with masses above 8 explode as core-collapse supernovae. Mass-loss and metal return are calculated for core-collapse and Type Ia supernovae and for asymptotic giant branch (AGB) stars at each simulation time-step using the yields from Karakas (2010) for AGB stars, from Portinari et al. (1998) for core-collapse supernovae, and from Thielemann et al. (2003) and Travaglio et al. (2004) for Type Ia supernovae. The released mass and metals are injected into the single host cell of the star particle. This single cell injection differs from the original Auriga suite of simulations, where metals were injected into 64 neighbouring cells. This change was found to slightly increase the 2 scatter of the stellar abundances, but to have little effect on the average or 1 scatter (van de Voort et al. 2020). It thus does not impact our results strongly. The galaxy formation model includes stellar and active galactic nucleus (AGN) feedback, which results in large-scale outflows and limits the growth of the galaxy (Grand et al. 2017). These outflows carry metals out of the ISM and into the halo and therefore have a large impact on the chemical evolution of the galaxy. The elements produced by standard and Type Ia supernovae are released at each time step based on the age of the stellar particle, which represents a single stellar population, resulting in a smooth enrichment of the host cell. Although this choice may reduce the scatter in the elemental abundances slightly, the production sites of these elements are ubiquitous and we therefore do not expect our results to be strongly affected. We explicitely follow the elements H, He, C, N, O, Ne, Mg, Si, and Fe. These elements affect the cooling rate of the gas and thus have a dynamical effect on the evolution of the galaxy.
The additional channels that represent r-process elements, whilst also being traced on-the-fly with the same time resolution as other components in the simulation, are implemented as passive tracers that do not affect the rest of the simulation. Their presence is thus inconsequential to the evolution of the galaxy -a good approximation, because these elements are very rare and their effect is therefore negligible. We choose not to use yield tables, with their inherent large uncertainties, but rather renormalize each r-process enrichment model in post-processing (see below for more details). Although the r-process yields may depend on metallicity, this is highly uncertain and we therefore do not include such a dependence. Each event releases the same amount of r-process elements. In contrast to the more common elements, r-process enrichment is implemented stochastically. The produced elements are released only when an r-process event takes place, rather than smoothly at every time step. Our stochastic r-process sources do not produce any of the more common elements in our simulation. Such a co-production of other elements is more important for rare core-collapse supernovae than for neutron star mergers, though likely still negligible due to the efficient mixing of pristine and enriched gas in our simulations (van de Voort et al. 2020).
Abundance ratios of our star particles are normalized by the Solar abundances from Asplund et al. (2009) and defined as where and are two different elements and A and B are their number densities. We use magnesium as a metallicity indicator in this work. Because the magnesium yields used in our simulations are known to be low (Portinari et al. 1998), we multiply them by a factor of 2.5 in post-processing to correct for this (see also van de Voort et al. 2020). Magnesium is not a dominant coolant in the circumgalactic medium. Therefore, such an increase in the magnesium abundances would not affect the dynamics of our simulations. We choose magnesium and not iron as a metallicity tracer, because magnesium is primarily produced in core-collapse supernovae, whereas iron is produced both in core-collapse and Type Ia supernovae. Using magnesium is therefore easier to interpret as it does not depend on the evolving ratio of Type Ia supernovae to core-collapse supernovae.
Furthermore, because the r-process elements are passive tracers and do not have any dynamical effect on the simulations, we can change the r-process yield in post-processing without loss of generality. Given that the r-process yields are very uncertain, this approach allows us to focus on the trends with metallicity and the scatter in abundance ratios. We choose to normalize the resulting r-process abundance ratios for each of our models separately by setting the median [r-process/Mg] to 0 at [Mg/H] = 0. This is the reason why our models have different europium yields as listed in the last column of Table 1.
Modeling neutron star natal kicks and mergers
In this work, we study the impact of introducing neutron star natal kicks on the resulting r-process abundance ratios in a Milky Way-mass system. In previous cosmological simulations of Milky Way-mass galaxies, r-process elements were released around the Table 1. Parameters of r-process enrichment models for those based on neutron star mergers (first 8 rows). Listed in columns 2-5 are the number of neutron star mergers per of stars, the minimum delay time for neutron star mergers in a simple stellar population, the delay time distribution (DTD) power-law exponent, and the median kick velocity and its dispersion. The final two columns show the resulting r-process event rate at = 0 (averaged over the last Gyr) and the europium yield ( Eu ) per r-process event (calculated by normalizing the median [Eu/Mg] to zero at [Mg/H] = 0) for the 8 neutron star models and the 2 rare core-collapse supernova models. To obtain the total yield of r-process material of elements with > 95, Eu should be multiplied by a factor 215. Here, we include natal kicks by generating additional massless neutron star merger particles which are given an additional velocity kick in a random direction. The original star particles are treated in the same way as before. When a star particle forms, we stochastically determine whether or not its stellar population will produce one or more neutron star mergers in the future, based on the following rate:
model name min kick r-process ( = 0) Eu (M −1 ) (Myr) (km s −1 ) (yr −1 ) (M ) fiducial kick 3 × 10 −6 30 −1.0 200 ± 500 1.2 × 10 −4 9.7 × 10 −5 no kick 3 × 10 −6 30 −1.0 0 ± 0 1.0 × 10 −4 6.5 × 10 −5 small kick 3 × 10 −6 30 −1.0 0 ± 500 1.2 × 10 −4 8.7 × 10 −5 large kick 3 × 10 −6 30 −1.0 500 ± 500 1.3 × 10 −4 11.9 × 10 −5 narrow range 3 × 10 −6 30 −1.0 200 ± 100 1.0 × 10 −4 6.7 × 10 −5 wide range 3 × 10 −6 30 −1.0 200 ± 1000 1.3 × 10 −4 13.8 × 10 −5 short delay 3 × 10 −6 10 −1.0 200 ± 500 1.3 × 10 −4 6.8 × 10 −5 steep DTD 3 × 10 −NS = ★ for > min(2)
and NS = 0 when < min . Here, is the number of neutron star mergers per unit of stellar mass, ★ is the mass of the star particle formed, is the time since the formation of the star particle, is the (always negative) exponent of the time dependence of the delay time distribution, and min is the minimum time needed for a neutron star merger to occur (e.g. Belczynski et al. 2006). We use the following values for our fiducial neutron star merger model:
= 3×10 −6 M −1 , = −1, and min = 3×10 7 yr. These are the same values we used before in van de Voort et al. (2020) and were chosen as a reasonable guess based on the available literature (Belczynski et al. 2006;Zheng & Ramirez-Ruiz 2007;Abadie et al. 2010;Berger 2014;D'Avanzo 2015;Côté et al. 2019;Kim et al. 2015;Abbott et al. 2017). In Section 3.3 we will use parameter values that were chosen to produce a better match with available observations of stellar abundances, but still lie within current observational and theoretical uncertainties ( = 3 × 10 −5 M −1 , = −1.5, and min = 10 7 yr).
If a star particle (i.e. a stellar population) will produce one or more neutron star mergers before the end of the simulation at = 0, an equivalent number of neutron star merger particles are generated. These particles are massless, which is a good approximation, since a binary neutron star system is much less massive than our star particles. The neutron star merger particles have the same velocity as the star particle from which they spawned plus an additional kick in a random direction. The kick velocity is chosen from a Gaussian distribution, with a peak at 200 km s −1 and a standard deviation of 500 km s −1 for our fiducial kick model. 1 We include multiple kick models in the same simulation with different kick velocities, because the values are highly uncertain, especially for binary systems. We can use a single cosmological simulation to study multiple r-process models, because they do not impact the rest of the simulation and behave as passive tracers (see Section 2). The resulting r-process elements produced are stored in separate variables for each model. The parameter values used are listed in Table 1. Note that model 'no kick' produces neutron star merger particles with an initial velocity equal to that of their parent stellar population, but without an additional kick, and is therefore identical to the fiducial model in van de Voort et al. (2020) even though the implementation is slightly different. Table 1 also lists the resulting = 0 rate of r-process producing events (either by neutron star mergers or by a rare type of corecollapse supernova) in the entire zoom-in region of the simulation. This corresponds to a maximum distance of approximately 1 Mpc from the central galaxy. However, the r-process events are strongly centrally concentrated, so reducing the maximum distance only changes the rates slightly. Although the first 6 models have the same delay time distribution, = 0 rates vary somewhat due to the stochasticity involved in randomly sampling this distribution, which is done independently for each model. The last column gives the resulting europium yield obtained by setting the median [Eu/Mg] to zero at [Mg/H] = 0. The rates and yields of our models are consistent with most observational constraints within uncertainties (e.g. Abadie et al. 2010;Behroozi et al. 2014;Kim et al. 2015;Abbott et al. 2017), though our rates are lower than the estimate from Pol et al. (2019) who found NS ( = 0) = 42 +30 −14 Myr −1 . Note that the model with a steep exponent for the delay time distribution has a low-redshift event rate more than an order of magnitude lower than the other models, because fewer neutron star mergers occur as the stellar population ages. We will revisit this in Section 3.3, which discusses results from a separate medium-resolution simulation with parameters chosen to better match observations of stellar r-process abundances.
We have performed a resolution test by running identical sim-ulations with 8 and 64 times lower mass resolution. The models tested, both with and without natal kicks, are almost perfectly converged. The convergence behaviour is very similar to that shown in van de Voort et al. (2020). As in our previous work, there is a very small residual resolution dependence. The median r-process abundance ratio decreases marginally and the scatter increases slightly as the resolution improves. This small difference can likely be explained by the decrease of numerical mixing with increasing resolution. Because these differences are so minor, we do not believe they are the main uncertainty associated with this work. Higher resolution does substantially increase the statistics by increasing the number of star particles and therefore allows us to probe down to lower metallicities.
Modeling rare core-collapse supernovae
Although the main topic of this work is the effect of adding neutron star natal kicks, we also explore the r-process enrichment from a rare type of core-collapse supernova for comparison. Candidates for such sources include collapsars and magneto-rotational supernovae (see e.g. Côté et al. 2019), but we remain agnostic towards the exact source and keep our models as simple as possible. As was done for the neutron star mergers, r-process events are sampled stochastically and each r-process producing core-collapse supernova releases the same amount of r-process elements. The resulting stellar abundance ratios are normalized by choosing the europium yield such that the median [Eu/Mg] is zero at [Mg/H] = 0 in post-processing, as given in the last column of Table 1. In our two models included here, either 1 in 1,000 (0.1 per cent) or 1 in 10,000 (0.01 per cent) core-collapse supernovae act as sources of r-process elements ). These are not modeled with massless particles as was done for the neutron star merger models. Their enrichment rather occurs directly in the host cell of the original star particle containing the core-collapse supernova. This is slightly different to their treatment in van de Voort et al. (2020) where r-process elements were released into ≈ 64 neighbouring cells, but this is unlikely to affect the results, as also shown and discussed in van de Voort et al. (2020). From Table 1 we can see that the 1 in 1,000 model has an event rate about a third of our fiducial neutron star merger model. The rate is another full order of magnitude lower for the model with only 1 in 10,000 core-collapse supernovae as r-process production sites.
RESULTS
In this Section, we show results for all stars within the virial radius of our simulated Milky Way-mass galaxy at = 0, including disc stars as well as halo stars. To calculate the median and scatter in r-process abundance ratios, we use 0.2 dex metallicity bins and show only those bins that contain at least 100 star particles. The exception is Figure 5 where we reduced the minimum number of star particles to 50 in order to probe to lower metallicities at the expense of increased noise in those bins. In this work, for brevity, we refer to the 16th and 84th percentiles as the 1 scatter and to the 2nd and 98th percentile of the distribution as the 2 scatter, even when the distribution is not Gaussian.
Neutron star mergers with natal kicks
We first focus on the difference in stellar abundance ratios between our fiducial model with natal kicks included ('fiducial kick') and a model with identical paramaters except for setting all kick velocities to zero ('no kick'). In Figure 2, we show [r-process/Mg] as a function of metallicity for all stars within vir at = 0. The thick curves show the median, the shaded region show the 16th and 84th percentiles (from now on refered to as the 1 scatter), and the thin curves show the 2nd and 98th percentiles (from now on refered to as the 2 scatter). Our fiducial kick model is shown as solid, black curves and the model without kicks as dashed, red curves. There are two very clear differences between the two enrichment models. First, the inclusion of natal kicks lowers the median r-process abundances at [Mg/H] < −0.5 and the difference between the two models becomes larger towards lower metallicities. Because we normalize our models to [r-process/Mg] = 0 at [Mg/H] = 0, the r-process yields used are slightly different depending on whether or not natal kicks are included. As can be seen from Table 1, in the 'fiducial kick model' we use yields that are 50 per cent larger than in the model without kicks. If we had used the same normalization, the solid, black 'fiducial kick' curve would shift down by 0.17 dex and lie below the dashed, red 'no kick' curve at all metallicities. This reduction of the stellar r-process abundances when kicks are included was expected, because neutron star mergers happen, on average, further away from the galaxy. This does not necessarily mean that the produced elements are lost to the galaxy, because they can accrete onto the ISM at a later time and become incorporated in subsequent stellar populations. However, the fraction of r-process elements escaping the system is likely larger when they are produced at larger distances. Note that the normalization difference is fairly minor, which indicates that the accretion of r-process elements is important.
The second substantial difference is that the scatter in r-process abundance is much larger when natal kicks are included, towards both high and low values of [r-process/Mg]. This is because a neutron star binary can travel far from its birth site before merging. 2 They are thus more likely to release r-process elements in an area that is likely not to have seen star formation itself and therefore has not The bottom right-hand panel includes the two rare core-collapse supernova models. For these, the trend of [r-process/Mg] with metallicity is flatter and the scatter is smaller than for our neutron star merger models. All our models show a median r-process abundance that drops at low-metallicity due to the source being too rare to enrich the ISM in the early Universe. At the highest metallicities, another difference can be seen, although this is a very minor effect. The model with natal kicks decreases slightly towards high metallicity whilst the model without kicks continues to increase slightly. The difference between the models is only 0.1 dex and may therefore not be important, but we discuss it here briefly. The likely reason for this difference in behaviour is linked to the star formation history (SFH) of this galaxy. Its SFH increases from its formation up to a look-back time of 9 Gyr after which it fluctuates around 15 M yr −1 for 5 Gyr and decreases over the last 4 Gyr to approximately 3 M yr −1 . This means that prompt neutron star mergers become less frequent towards the present day and therefore neutron star mergers with long delay times become relatively more important. It is these long delay time systems that spend a substantial time traveling away from the galaxy, when natal kicks are included, and merge far away from its ISM, thus enriching the intergalactic medium. This results in a lowering of the r-process to magnesium abundance ratio at late times (i.e. at high metallicity) as magnesium is produced by supernovae inside (or close to) the ISM.
Parameter study
The parameter values used in Section 3.1 are highly uncertain. We therefore included several additional r-process enrichment models based on neutron star mergers, where for each model we varied a single parameter of either the kick velocity distribution or the delay time distribution. These values are summarized in Table 1. We note that model 'wide range' contains stars with kick velocities higher than observed (Hobbs et al. 2005;Faucher-Giguère & Kaspi 2006), but it is nevertheless included to understand the effect of increasing the width of the kick distribution. We also added two models where rare core-collapse supernovae produce all of the r-process elements (as described in Section 2.2). The models included here are not exhaustive, because we only vary a single parameter each time. Further models are explored via a medium-resolution simulation in Section 3.3. Figure 3 shows the resulting r-process abundance ratios as a function of metallicity. In each panel, the thick, black curve shows the median relation for our fiducial neutron star merger model (identical to the 'fiducial kick' model in Figure 2). Dashed, blue and dotted, red curves show the median (2 scatter) as thick (thin) curves and the 1 scatter as shaded regions for the model variations.
Overall, the differences between our neutron star models with natal kicks are relatively mild (top panels and bottom, left-hand panel) and much smaller than the difference between models including or excluding natal kicks (see Figure 2). The small differences that are present behave as expected. Decreasing (increasing) the average kick velocity marginally increases (decreases) the median [r-process/Mg] at low metallicity, flattening the relation with metallicity slightly, and decreases (increases) the scatter (top, lefthand panel). Decreasing (increasing) the width of the kick velocity distribution shows mostly the same behaviour (top, right-hand panel). However, the 1 scatter is higher at low-metallicity for the narrower kick distribution model. This is likely because more neutron star mergers occur inside or close to the progenitor galaxies in the early Universe, enriching their ISM more easily. Decreasing the minimum delay time and steepening the delay time distribution increase the median [r-process/Mg] because neutron star mergers occur earlier, on average (bottom, left-hand panel; as also found in van de Voort et al. 2020). These changes have a somewhat larger effect than the kick velocity variations we tested here. For all of our models shown in Figure 3, we renormalized their abundances to [r-process/Mg] = 0 at [Mg/H] = 0, effectively changing the r-process yield in post-processing. The last column in Table 1 lists the yields used and from this we can see how using a fixed yield would increase the difference between models. For the models with the centroid of the kick velocity distribution shifted, the normalization only changes by 0.14 dex between model 'small kick' and model 'large kick'. However, changing the width of the distribution from 100 to 1000 km s −1 decreases [r-process/Mg] by 0.31 dex if we were to use the same yields. This is because neutron star binaries traveling at very high velocities are likely to merge so far away from the galaxy that the elements they produce will not be able to accrete onto the galaxy and are therefore not incorporated into future generations of stars. The normalization of model 'steep DTD' is the most discrepant, 0.45 dex higher than our fiducial model. With a steeper time dependence, fewer neutron star mergers occur for a given stellar population and the difference increases with time. If we were to use fixed yields, model 'steep DTD' would result in lower r-process abundances at high metallicity compared to the fiducial model.
To allow direct comparison to models with alternative sources for r-process enrichment, we included two models where a special class of core-collapse supernovae, such as collapsars or magnetorotational supernovae, is responsible for the synthesis of all r-process elements. These are shown in the bottom, right-hand panel of Figure 3. We use an event rate of either 1 in every 1,000 core-collapse supernovae (dashed, blue curves and blue shaded area) or 1 in every 10,000 (dotted, red curves and red shaded area). The median relation between [r-process/Mg] and [Mg/H] is flat at high and intermediate metallicity. Towards very low metallicity, the median r-process abundance ratio decreases. As expected, this decrease occurs at a higher metallicity for a source that is more rare (i.e. 1 in 10,000 core-collapse supernovae) because it is difficult to enrich the entire ISM in the early universe when the event rate is that low. This is similar to the steep decline of [r-process/Mg] towards very low metallicity in neutron star merger models.
For both rare core-collapse supernova models, the upper 1 bound stays close to [r-process/Mg] = 0 at any metallicity. At [Mg/H] −2, the 1 scatter is very small and rapidly increases to lower metallicites. The scatter is smaller than that in any of the neutron star merger models with natal kicks shown in the other panels. To quantify this somewhat, only 2 (1) per cent of stars have [r-process/Mg] > 1 at −5 < [Mg/H] < −3 (−5 < [Mg/H] < −2) in the two rare core-collapse supernova models, compared to the various neutron star merger models with natal kicks presented here for which the fractions of similarly r-process enhanced stars vary between 9 and 14 (4 and 6) per cent. Future observations of a homogeneous sample of (extremely) metal-poor stars would be able to measure the fraction of stars with high r-process abundance ratios, which could then be compared to the values obtained from our different models.
Optimized neutron star merger model with natal kicks
The increasing trend of [r-process/Mg] (or similarly [r-process/Fe]) with metallicity for all our neutron star merger models in Sections 3.1 and 3.2 is not seen in observations (see van de Voort et al. 2020 for a more detailed comparison and discussion of our fiducial model without kicks). We note that the observational data for extremely metal-poor stars is sparse and could be biased. Nevertheless, currently available observations seem to prefer a flat trend of [r-process/Mg] with metallicity, which is easier to achieve with a model based on rare (but prompt) core-collapse supernovae as r-process production sites (see also Section 3.2 and Figure 3).
Until now, we attempted no tuning of our neutron star merger parameters. Motivated by the inability to reproduce observational trends, we resimulated the same galaxy at medium resolution (i.e. 8 times lower mass resolution than our fiducial simulation) with new neutron star delay time distribution values specifically chosen to improve the match with observations: a higher delay time distribution normalization, a shorter minimum delay time, and a steeper exponent (see Table 2). With these new delay time parameters, we varied the kick velocities and refer to this new set of neutron star models as 'optimized'. The exponent of the delay time distribution is set to = −1.5 as in model 'steep DTD', the minimum delay time is reduced to 10 Myr as in model 'short delay', and the normalization is increased by an order of magnitude. Because of the steepness of the distribution, the change in normalization is not as extreme as it may seem: the low-redshift neutron star merger rate is below that of our fiducial model with = −1 and in agreement with or even below rate estimates from the literature (e.g. Abadie et al. 2010;Pol et al. 2019). Figure 4 shows the abundance ratios from two of our 'optimized' neutron star merger models. The top panel shows the median (solid, black curve) and 1 , 2 , and 3 scatter (grey shaded regions) of [r-process/Mg] in our fiducial kick velocity distribution with kick = 200 ± 500 km s −1 and the bottom panel shows the same for a model without natal kicks. The 'optimized' parameters, chosen within the range allowed by current constraints, result in higher average r-process abundances in metal-poor stars compared to our fiducial parameters (see Section 3.1). The relation between the median [r-process/Mg] and [Mg/H] is mostly flat, whereas there was a rising trend with metallicity in our fiducial model shown in Figure 2.
It is important to point out that the median is very similar for the two models shown and therefore is not strongly affected by natal kicks in this case. This is different from the behaviour shown in Section 3.1 for our fiducial delay time distribution parameters, where the addition of kicks reduced the median [r-process/Mg]. Comparing the adopted europium yields in Table 2 for our fiducial kick model and the one without kicks confirms there is little difference in the normalization. Neutron star mergers occur more rapidly after formation of the stellar population in the 'optimized' model than in our fiducial model. Therefore, the binary travels less far from its birth position, which decreases the impact of natal kicks.
Looking at the scatter around the median, we see that it is much smaller than in our models with fiducial delay time distribution parameters, both with and without kicks, as shown in Section 3.1. Qualitatively, this is expected, because the number of neutron star mergers is higher and the average delay time for neutron star mergers is shorter. The rate of mergers at = 0 is lower than for our Table 2. Parameters of r-process enrichment models for those based on neutron star mergers which was optimized to match available observations. The columns list the same properties as Table 1. model name min kick r-process ( = 0)
Eu (M −1 ) (Myr) (km s −1 ) (yr −1 ) (M )
optimized; fiducial kick 3 × 10 −5 10 −1.5 200 ± 500 7.1 × 10 −5 2.7 × 10 −5 optimized; small kick 3 × 10 −5 10 −1.5 0 ± 500 7.0 × 10 −5 2.7 × 10 −5 optimized; small kick; narrow range 3 × 10 −5 10 −1.5 0 ± 100 6.8 × 10 −5 2.8 × 10 −5 optimized; small kick; narrower range 3 × 10 −5 10 −1.5 0 ± 50 6.8 × 10 −5 2.8 × 10 −5 optimized; no kick 3 × 10 −5 10 −1.5 0 ± 0 6.7 × 10 −5 2.9 × 10 −5 Suda et al. (2008). The solid cyan lines show the median and the dashed, cyan curves show the 1 scatter of the observational detections. The two models reproduce the mostly flat behaviour of the r-process abundance ratio equally well. However, the model without kicks vastly underproduces the scatter seen in observations, whereas the match between observations and the model with kicks is excellent at low metallicity. In this 'optimized' model, the scatter is determined by the neutron star kick velocities rather than by the rarity of r-process sources.
original model -compare Tables 1 and 2 -but the rate is higher at high redshift. In the early universe, at > 3, the number of neutron star mergers is 4 times higher in our 'optimized' model. Over the full 13.8 Gyr of simulation time, the 'optimized' model produces 60 per cent more neutron star mergers than the fiducial model. Furthermore, the fact that the neutron mergers occur more promptly means that they occur closer in time and space to core-collapse supernovae, which produce magnesium. These two differences explain the decreased scatter in [r-process/Mg].
When comparing the scatter in our 'optimized' model with kicks to our 'optimized' model without kicks, the same conclusion holds as before: neutron star natal kicks substantially increase the scatter in r-process abundance, especially at low metallicity. The scatter is clearly dominated by the presence of neutron star kicks. This has important implication for the comparison of our models to observations.
Comparison to observations
We now wish to see whether our 'optimized' model with neutron star mergers as the only source of r-process elements can match currently available observations of stellar abundances. Besides abundances from our 'optimized' models, Figure 4 also shows observational detections (blue circles) and upper limits (red downward triangles) from the Stellar Abundances for Galactic Archeology (SAGA) database (Suda et al. 2008). The median (solid, cyan curve) and 1 scatter (dashed, cyan curves) of the observational data (detections only) are included if the 0.2 dex metallicity bins contained at least 10 data points. Note that the observational sample is not homogeneous and could suffer from selection bias. Also note that our simulations do not have any measurement errors included, unlike observations. No exact match is expected, but it is still illuminating to compare our simulation results to the available observational data.
The median [r-process/Mg] ≈ 0 at all metallicities in the 'optimized' model, both with and without natal kicks. This is similar to the behaviour in observations and what we were aiming for when choosing new parameters for the delay time distribution. The main difference between including or excluding natal kicks lies in the scatter around the median. It is clear that the model with kicks much better reproduces the 1 observational scatter as well as extreme outliers than the model without kicks.
The conclusion that this model with 'optimized' parameters and kick = 200 ± 500 km s −1 provides a reasonable match to currently available observations does not necessarily imply it is the correct one. First, a larger and unbiased observational survey is needed to improve statistics for metal-poor stars and sample the population in a homogeneous way. Second, there may be other possible neutron star merger parameter choices that would also result in a r-process element distribution similar to observations. Similarly, there may be other viable sources, such as a special type of core-collapse supernova, able to match observations as well. However, what our results prove is that it is possible to create an r-process enrichment model with reasonable parameters using neutron star mergers as the only source of r-process elements that matches available data. Table 2. Thick curves show the median value, while the shaded regions and thin curves indicate the 1 and 2 scatter, respectively. The model without kicks is shown as dotdashed, orange curves and is identical to the model in the bottom panel of Figure 4. The other models all include natal kicks picked from a Gaussian distribution centred at zero and with a standard deviation decreasing from 500 (solid, black curves) to 100 (dashed, blue curves) to 50 km s −1 (red, dotted curves). The scatter decreases as the average kick velocity decreases, but not by the same amount for the upwards and downwards scatter. The lower 2 bound for the model with kick = 0±100 km s −1 follows that of the model without kicks, whereas its upper 2 bound is very close to the model with much larger kicks of kick = 0 ± 500 km s −1 . Even the model with a modest kick = 0 ± 50 km s −1 has a much larger scatter than the model without kicks. Thus, even relatively low velocity kicks have a substantial impact on the resulting scatter in r-process abundances.
It is important to point out that in our 'optimized' model, the scatter in r-process abundances is created by the neutron star kicks, which decouple the location of core-collapse supernovae and neutron star mergers more to a much larger extent than a model without kicks. This is a clear departure from the usual interpretation, which is that the large scatter at low metallicity is caused by inhomogeneous enrichment due to the rarity of the r-process source. This possibility should be kept in mind when interpreting observational data.
Lower velocity natal kicks
In Section 3.2 we showed that for moderate changes to the kick velocity, the resulting abundance ratios do not vary substantially. However, all our models had relatively high average velocities of at least a few hundred km s −1 . Although this is still debated, there is evidence of lower kick velocities in neutron star binaries (e.g. Fong & Berger 2013;Giacobbo & Mapelli 2018). If the velocity is lowered enough, the abundances should resemble the model without kicks. Therefore we included a few additional models in our medium-resolution simulation with significantly reduced kick velocities. The parameter values of the kick distributions in these new model variations are detailed in Table 2.
The resulting r-process abundance ratios are shown in Figure 5 for 4 neutron star merger models with kick velocity distributions centred at 0 and a standard deviation of 500 km s −1 (solid, black curves and grey shaded region), 100 km s −1 (dashed, blue curves and shading), 50 km s −1 (dotted, red curves and shading), and 0 km s −1 (dot-dashed, orange curves). The latter model does not include kicks. This model is identical to the simulation shown in the bottom panel of Figure 4 and its shading has been omitted for clarity. Thick and thin curves show the median and 2 scatter, respectively, and the shaded areas show the 1 scatter. To probe to slightly lower metallicities, we included metallicity bins that include at least 50 star particles instead of our fiducial 100 used in all other figures. The statistics are therefore a bit worse than in our other figures for [Mg/H] < −3.1.
As we also saw in Figure 4, reducing the kick velocity in our 'optimized' model results in a fairly similar median relation of [r-process/Mg] versus [Mg/H]. There is a hint of a small decrease in [r-process/Mg] at low metallicity for lower velocities as compared to the kick = 0 ± 500 km s −1 model, but this would need to be confirmed by high-resolution simulations with better statistics. At high metallicities, the lower velocity models have a slightly higher [r-process/Mg] and therefore a slightly flatter metallicity dependence, but this difference is also very minor.
In contrast to our previous moderate changes in the kick velocity distribution (see Figure 3 and Section 3.2), the scatter in r-process abundances is clearly affected when reducing the average kick velocity to less than a hundred km s −1 . 3 The kick = 0±100 and 0±50 km s −1 each have lower scatter than the kick = 0±500 km s −1 model. However, their scatter is still larger than in the model without natal kicks (see Figure 4 for the 1 scatter of model 'no kick'). The 2 scatter above the median relation in the low-velocity models is almost as high as in the high-velocity kick models, so r-process enhanced outliers will still be relatively common, whereas their 2 scatter below the median is more similar to the model without kicks. The overall conclusion is that the effect of natal kicks is reduced when their average kick 100 km s −1 as compared to our fiducial model. However, even these relatively low-velocity kicks still increase the scatter in stellar r-process abundances as compared to a model without kicks.
DISCUSSION AND CONCLUSIONS
We explored the stellar abundances of r-process elements in cosmological, magnetohydrodynamical simulations of a Milky Way-mass galaxy using the moving mesh code using the Auriga galaxy formation model. We implemented a variety of models for r-process enrichment, most of which use neutron star mergers as the only source of r-process elements, though two use rare core-collapse supernovae as sole r-process production sites. In this work, we focused on understanding the impact of adding neutron star natal kicks to the neutron star merger models. Natal kicks cause neutron star binaries to move far from their birth positions, which changes the resulting chemical evolution of their host galaxies. We contrasted these results with those obtained from our rare core-collapse supernovae models and compared our models to observations. The main conclusions are as follows.
(i) In our fiducial neutron star merger model with natal kicks included, the normalization of the stellar r-process abundances is only slightly lower at solar metallicity than for a model without kicks, even though many neutron star mergers occur outside the ISM of the galaxy. This indicates that the elements produced by these events are not lost, but mix with the inflowing gas that goes on to form stars enriched in r-process elements. Larger differences are seen for lower metallicity stars ([Mg/H] −1) where the median [r-process/Mg] is somewhat lower when kicks are included (even after scaling out the normalization difference at solar metallicity). However, in our 'optimized' model with relatively prompt neutron star mergers (using a steeper delay time distribution and shorter minimum delay time), the median r-process abundances are virtually unchanged by including or excluding natal kicks.
(ii) The addition of natal kicks strongly decouples the location of supernovae, which occur primarily within the ISM shortly after their progenitor stars were formed, from the location of neutron star mergers, which are most likely to occur outside the ISM. This causes the main effect of kicks: they substantially increase the scatter of [r-process/Mg] at all metallicities. This effect becomes larger at lower metallicity. Strongly r-process enhanced stars are far more numerous when kicks are included. In our fiducial model, the percentage of metal-poor stars with [r-process/Mg] > 1 increases from 1 to 10 per cent (0.5 to 4 per cent) at −5 < [Mg/H] < −3 (−5 < [Mg/H] < −2) with the addition of strong natal kicks.
(iii) The r-process abundance ratios are not affected much by changing details about the kick velocity distribution, as long as the average velocities are large ( kick 100 km s −1 , though the exact value cannot be determined accurately with our current set of models). Smaller kick velocities ( kick 100 km s −1 ) indeed result in decreased scatter in [r-process/Mg]. However, even a model with relatively prompt neutron star mergers (using a steeper delay time distribution and shorter minimum delay time) and with a small natal kicks of kick = 0 ± 50 km s −1 still resulted in an increased scatter compared to a model without kicks. We therefore conclude that natal kicks are likely important and need to be included in galactic chemical modelling efforts.
(iv) A special, rare type of core-collapse supernova that produces r-process elements results in a relatively flat median [r-process/Mg] as a function of metallicity, although the median does drop rapidly towards the lowest metallicities at [Mg/H] < −4 ([Mg/H] < −3) for a model with 1 in 1,000 (1 in 10,000) supernovae producing rprocess elements. The scatter tends to be lower than in our neutron star merger models and there are fewer extreme r-process enhanced outliers.
(v) As expected based on our previous work (van de Voort et al. 2020), our fiducial model does not match currently available observations of stellar r-process abundances in detail. These observations suggest a fairly flat relation between the median [r-process/Mg] and metallicity, which is something quite naturally achieved when rare core-collapse supernovae are responsible for producing r-process elements. However, in our fiducial neutron star merger model, we find a rising trend with metallicity. We therefore tweaked (or 'optimized') the parameters of the delay time distribution, within the allowed range, and increased the number of neutron star mergers at early times and thus at low metallicity. Our simulations show that this can also reproduce the same flat trend as seen in observations.
(vi) Interestingly, the model with 'optimized' delay time distribution parameters, but without natal kicks, does not reproduce the observed scatter in [r-process/Mg]. However, the same model where natal kicks were included matches the observed scatter very well. If this 'optimized' delay time distribution turns out to be an accurate representation of the neutron star merger rate in the Universe, then our simulations predict that the scatter in r-process abundances at low metallicity is not caused by the rarity of the r-process producing source, but rather by the shape of its natal kick velocity distribution.
Neutron star kicks and their effect on r-process enrichment are not straightforward to include in more idealized models and additional assumptions need to be made. van Oirschot et al. (2019) built a semi-analytic model on top of dark matter-only simulations, using a merger 'delay time distribution similar to that in our 'short delay' model and a bimodal kick velocity distribution. They found a lower normalization of [r-process/Mg] when kicks were included. This is similar to what we found in our fiducial neutron star merger model, although our 'optimized' model did not show this behaviour. Their model did not produce sufficient stars with high r-process abundances at low metallicities, which they attributed to their assumption of instantaneous mixing throughout the galaxy, which meant that the enrichment was too homogeneous. Our simulations track the full inhomogeneous chemical evolution on all scales down to our resolution limit and this likely explains why our models can create a substantial population of r-process enhanced stars.
Though we did not discuss the evolution of iron in this work, it is of note that both [Mg/Fe] and [r-process/Fe] are observed to decrease with metallicity at [Fe/H] −1, which has been argued is difficult to reproduce in enrichment models based on neutron star mergers alone (Côté et al. 2019). Banerjee et al. (2020) included inside-out disc evolution and neutron star natal kicks into a onezone galactic chemical evolution model and found that this resulted in a clear decreasing trend of [r-process/Fe] with [Fe/H] at high metallicity, whereas the relation is flat without kicks. In comparison, our kick model has a much smaller impact on the high-metallicity end of the distribution.
Our new simulations suggest that neutron star mergers could be the only source of r-process elements in the Universe when natal kicks are included. Although our fiducial model does not match observations in detail, we achieve a reasonable match with current observations in our 'optimized' model in which the time-dependence of the delay time distribution is steep ( ≈ −1.5). The included natal kicks improve the agreement with observations by increasing the scatter in r-process abundances. There may be additional ways to reproduce the stellar abundances from observations. For example, a rare type of core-collapse supernovae could potentially also produce the majority of r-process elements, though may struggle to produce sufficient strongly r-process enhanced stars. Reducing the r-process event occurence frequency from 1 in 1,000 core-collapse supernovae to 1 in 10,000 increases the scatter, but decreases the median [r-process/Mg], resulting in a similar fraction of r-process enhanced stars. Observations of a homogeneous sample of (extremely) metal-poor stars could determine the fraction of r-process enhanced stars and potentially discriminate between neutron star merger models and rare core-collapse supernova models.
Our neutron star merger model is relatively simple and the parameter values have large uncertainties associated with them. We therefore implemented a variety of models with different parameter values, but were unable to explore all parameter combinations. For example, our merger rates may be lower than estimated from observations (Pol et al. 2019). Additionally, the neutron star merger rate or their resulting r-process yields could also depend on metallicity, which is not included in our current models, because it is not well-constrained. Any future improvements in constraints on the delay time distribution or the kick velocity distribution for neutron star mergers or on the rate of rare r-process producing core-collapse supernovae could be easily tested in our cosmological framework. For example, identifying the host galaxies of neutron star mergers detected via gravitational waves could put strong constraints on their delay time distribution (Safarzadeh et al. 2019;Adhikari et al. 2020;McCarthy et al. 2020). Future simulations could also improve by capturing more of the multi-phase structure of the ISM instead of using a smooth single-phase ISM model as we did here. This could change the level of mixing within the ISM and there-fore potentially increase its chemical inhomogeneity. Here, we have shown that natal kicks should be taken into account when modeling r-process enrichment from neutron star mergers, because they can substantially enhance the scatter in the stellar r-process abundances.
particle (representing a single stellar population) in which the neutron star binary originally formed, thus neglecting natal kicks altogether (e.g.Shen et al. 2015;van de Voort et al. 2015;Naiman et al. 2018;Haynes & Kobayashi 2019;van de Voort et al. 2020).
Figure 2 .
2The abundance ratio [r-process/Mg] as a function of metallicity ([Mg/H]) for our fiducial neutron star merger model with kick = 200 ± 500 km s −1 in black (solid curves) and the same model without natal kicks in red (dashed curves). Thick curves show the median relation, while the thin curves show the 2 scatter. The grey and red shaded regions cover the 1 scatter. When kicks are included, the median is lower and the increase of [r-process/Mg] with metallicity is steeper. The 1 and 2 scatter in both directions are substantially larger with natal kicks.
Figure 3 .
3[r-process/Mg] as a function of [Mg/H] for a variety of models with different parameters as listed inTable 1. Thick curves show the median value, whilst the shaded regions cover the 1 scatter and the thin curves show the 2 scatter. The two top panels compare variations on the kick velocities of the models and show almost identical results. This means that the model is insensitive to the exact value for the kick velocity distribution chosen, though the average velocity in all our models is at least a few hundred km s −1 . The bottom left-hand panel explores variations on parameters of the neutron star merger delay time distribution. A steeper time dependence and a shorter delay increase the r-process abundance at [Mg/H] < 1.
been enriched directly by core-collapse supernovae, although it may have been polluted by large-scale galactic outflows. Only 1 (0.5) per cent of stars have [r-process/Mg] > 1 at −5 < [Mg/H] < −3 (−5 < [Mg/H] < −2) without kicks, but this fraction rises by an order of magnitude to 10 (4) per cent when kicks are included. Extreme outliers in observations are therefore more easily reproduced in a model where the binaries receive a substantial velocity kick.
Figure 4 .
4Median (solid, black curves) and 1 , 2 , and 3 scatter (grey shaded regions) of [r-process/Mg] as a function of [Mg/H] for two neutron star merger models with optimized parameters. One model includes our fiducial kick velocity ( kick = 200 ± 500 km s −1 ; top panel) and the other one does not include natal kicks (bottom panel). Observations of [Eu/Mg] in Milky Way stars are shown as blue circles (detections) and red downward triangles (upper limits) extracted from the SAGA database compiled by
Figure 5 .
5[r-process/Mg] as a function of [Mg/H] for neutron star merger models with different kick velocities as listed in
MNRAS 000, 1-11(2021)
Because the direction is chosen separately and sampled isotropically, we discard negative velocities and draw a new value from the distribution.MNRAS 000, 1-11(2021)
A binary moving at 200 km s −1 will travel 6 kpc in 30 Myr and 200 kpc in a Gyr.MNRAS 000, 1-11(2021)
The mean (median) absolute value of a Gaussian distribution centred on zero with a standard deviation of 100 km s −1 is 80 (67) km s −1 .MNRAS 000, 1-11(2021)
ACKNOWLEDGEMENTSWe would like to thank Hans-Thomas Janka and Friedrich-Karl Thielemann for interesting discussions and the referee for helpful comments. FvdV is supported by a Royal Society University Research Fellowship (URF\R1\191703). RG acknowledges financial support from the Spanish Ministry of Science and Innovation (MICINN) through the Spanish State Research Agency, under the Severo Ochoa Program 2020-2023 (CEX2019-000920-S). Our fiducial simulation was performed on computing resources provided by the Max Planck Computing and Data Facility in Garching. For the simulations used for our resolution tests, the authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (https://www.gauss-centre.eu) for funding this project (with project code pn68ju) by providing computing time on the GCS Supercomputer SuperMUC-NG at Leibniz Supercomputing Centre (https://www.lrz.de). Software used for this work includes NumPy(Harris et al. 2020)and Matplotlib(Hunter 2007).DATA AVAILABILITYData available on request.
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| []
|
[
"Extensive numerical study of a D-brane, anti-D-brane system in AdS 5 /CF T 4",
"Extensive numerical study of a D-brane, anti-D-brane system in AdS 5 /CF T 4"
]
| [
"Arpád Hegedűs \nWigner Research Centre for Physics\nP.O.B. 49H-1525, 114BudapestHungary\n"
]
| [
"Wigner Research Centre for Physics\nP.O.B. 49H-1525, 114BudapestHungary"
]
| []
| In this paper the hybrid-NLIE approach of[38]is extended to the ground state of a D-brane anti-D-brane system in AdS/CFT. The hybrid-NLIE equations presented in the paper are finite component alternatives of the previously proposed TBA equations and they admit an appropriate framework for the numerical investigation of the ground state of the problem. Straightforward numerical iterative methods fail to converge, thus new numerical methods are worked out to solve the equations. Our numerical data confirm the previous TBA data. In view of the numerical results the mysterious L = 1 case is also commented in the paper. | 10.1007/jhep04(2015)107 | [
"https://arxiv.org/pdf/1501.07412v1.pdf"
]
| 53,837,890 | 1501.07412 | 758f2aa46b07799405822c3e945bdf441d5dcc32 |
Extensive numerical study of a D-brane, anti-D-brane system in AdS 5 /CF T 4
29 Jan 2015
Arpád Hegedűs
Wigner Research Centre for Physics
P.O.B. 49H-1525, 114BudapestHungary
Extensive numerical study of a D-brane, anti-D-brane system in AdS 5 /CF T 4
29 Jan 2015worked out to solve the equations. Our numerical data confirm the previous TBA data. In view of the numerical results the mysterious L = 1 case is also commented in the paper.
In this paper the hybrid-NLIE approach of[38]is extended to the ground state of a D-brane anti-D-brane system in AdS/CFT. The hybrid-NLIE equations presented in the paper are finite component alternatives of the previously proposed TBA equations and they admit an appropriate framework for the numerical investigation of the ground state of the problem. Straightforward numerical iterative methods fail to converge, thus new numerical methods are worked out to solve the equations. Our numerical data confirm the previous TBA data. In view of the numerical results the mysterious L = 1 case is also commented in the paper.
Introduction
In this paper in the context of AdS/CFT [1,2,3] we study numerically the ground state of a pair of open strings stretching between two coincident D3-branes with opposite orientations in S 5 of AdS 5 × S 5 . The main motivation for the study is that according to string-theory the ground state of such a configuration is expected to be tachyonic for large values of the 't Hooft coupling [4]. In our work we rely on the perturbatively discovered and later "all loop conjectured" integrability [5] of both the AdS 5 × S 5 super-string and the dual large N gauge theory. For string configurations with D-branes integrability enabled one to describe string configurations ending on different types of D-branes as 1-dimensional integrable scattering theories with boundaries [6,7,8,9,10]. This formulation of the problem makes it possible to go beyond the approaches of perturbative gauge and string theories being valid for small and large values of the 't Hooft coupling respectively, and to determine the exact spectrum of the model at any value of the coupling constant. However, even with the help of the powerful techniques offered by integrability, the exact analytical solution of the problem is not possible. Remarkeble analytical results are available in the small [11,12,13,14,15,32] and large [16,17,19,18] coupling regimes, but the determination of the spectrum at any value of the coupling constant can only be carried out by high precision numerical solution [20,21,22] of the corresponding nonlinear integral equations.
In our paper we consider the case, when the two D3-branes are giant gravitons [23], namely they carry N units of angular momenta in S 5 . If the S 5 of AdS 5 × S 5 is parametrized by three complex coordinates X, Y, Z satisfying the constraint: |X| 2 + |Y | 2 + |Z| 2 = 1, then our D3-brane and anti-D3-brane are given by the conditions Y = 0 andȲ = 0 respectively. They wrap the same S 3 , but with opposite orientation. As a consequence of Gauss law such a system can support On the large N gauge theory side a Y = 0 brane is represented by a determinant operator [24] composed of N copies of the field Y :
O Y = det Y = ǫ a 1 ···a N b 1 ···b N Y b 1 a 1 · · · Y b N a N (1.1)
where a i and b i are color indices and ǫ is a product of two regular epsilon tensors ǫ a 1 ···a N b 1 ···b N = ǫ a 1 ···a N ǫ b 1 ···b N . The local operator corresponding to an open string ending on a Y = 0 giant graviton can be obtained from (1.1) by replacing one Y field with an adjoint valued operator W [25]:
O W Y = ǫ a 1 ···a N b 1 ···b N Y b 1 a 1 · · · Y b N−1 a N−1 W b N a N .
(1.
2)
The gauge theory description of a pair of open strings stretching between two Dbranes is given by a double determinant operator, such that the string insertions W and V connect the two determinants of the Y fields 1 :
O W,V Y,Y = ǫ a 1 ···a N b 1 ···b N Y b 1 a 1 · · · Y b N−1 a N−1 ǫ c 1 ···c N d 1 ···d N Y d 1 c 1 · · · Y d N−1 c N−1 W d N a N V b N c N (1.3)
Unfortunately, the precise gauge theory dual of the DD-system of our interest is not known. In [4] it was approximated by a double determinant operator similar to (1.3), but in one of the determinants the Y fields are replaced withȲ fields 2 : This observation allows us to apply the boundary Thermodynamic Bethe Ansatz technique (BTBA) [26] to each open string separately. The necessary ingredients of this technique are the boundary reflection factors [8,27,28,29] and the asymptotic Bethe equations of the problem [4]. Unfortunately, apart from some very special cases [30,31,32], it is still unknown how to derive BTBA equations for a general non-diagonal scattering theory in the context of the thermodynamical considerations of [26]. This is why in [4] the Y -system [33,34,35] and the related discontinuity [36] 1 The ground state of such string states is BPS. 2 According to the argument of [4] the correct state might have other structures involving the fields Y andȲ , but should be similar to the double determinant form (1.4) and the mixing with other fields seem to be suppressed at large N equations supplemented by analyticity assumptions compatible with the asymptotic solution [37] were used to derive BTBA equations for the nonperturbative study of the ground state of the DD-system [4].
O W,V YȲ = ǫ a 1 ···a N b 1 ···b N Y b 1 a 1 · · · Y b N−1 a N−1 ǫ c 1 ···c N d 1 ···d NȲ d 1 c 1 · · ·Ȳ d N−1 c N−1 W d N a N V b N c N .
The BTBA description of the system is an infinite set of nonlinear integral equations. The numerical solution of the equations [4] showed that the ground state energy is a monotonously decreasing function of the coupling constant 3 g. The analytical investigation of the large rapidity and large index behavior of the Y -functions of the BTBA revealed that the usual BTBA description of the system breaks down when the energy of an open string state with angular momentum L gets close to the critical value: E c (L) = 1 − L. This point was interpreted in [4] as a transition point where the ground state becomes tachyonic. Approaching the critical point the contribution of infinitely many Y -functions must be taken into account to get accurate numerical result for the energy 4 . This fact suggests reformulating the finite size problem in terms of finite number of unknown functions. The possible candidates could be the FiNLIE [46], the quantum spectral curve (QSC) [47,48] or the hybrid-NLIE (HNLIE) [38] formulation of the problem. Since at present it is not known (not even for the Konishi problem) how to use the analytically very efficient [19,14] QSC method for numerical purposes, we choose the HNLIE method to reformulate the finite size problem of the DD-system. In this paper we transformed the infinite set of boundary TBA equations [4] into a finite set of hybrid-NLIE type of nonlinear integral equations. We perform the extensive numerical study of these type of equations in order to get as close to the special E BT BA = 1 − L critical point as it is possible.
Our numerical results reproduce the numerical evaluation of the boundary Lüscher formula [27,39] in the linear approximation, and the numerical BTBA results of [4] as well. These numerical comparisons give further numerical checks on the hybrid-NLIE technique of [38]. Unfortunately, as g increases new local singularities enter the HNLIE formulation of the problem. Thus we could not approach very close to the critical point. Nevertheless, in the range of g where physically acceptable numer-3 Throughout the paper the relation between g and the 't Hooft coupling λ is given by: λ = 4π 2 g 2 . 4 This means that the usual truncation procedure for solving the infinite set of TBA equations is not applicable to such a system ical results were obtained, the HNLIE results could give higher numerical precision than that of the BTBA and also some interesting facts could be read off from our numerical data.
During the numerical solution of the HNLIE equations straightforward numerical iterative methods failed to converge, thus new numerical methods were worked out to solve the equations.
The ground state of the L = 1 state is a very special case, since there the critical point is right at g = 0 and so far neither perturbative field theory computations nor the boundary Lüscher formula could provide a finite quantitative answer to the anomalous dimension of this state. On the integrability side the HNLIE approach allows us to get some numerical insight into this problem. The outline of the paper is as follows: Section 2. contains the HNLIE equations. In section 3. the numerical method is described. In section 4. the numerical results and their interpretation is presented. Section 5. contains some comments on the mysterious L = 1 case and finally our conclusion is given in section 6. Various notations, kernels of the integral equations together with the necessary asymptotic solutions are placed in the appendices of the paper.
The HNLIE equations
In this section we transform the previously proposed BTBA equations of [4] for the ground state of our D-brane anti-D-brane system to finite component hybrid-NLIE equations. For presentational purposes we group the equations into 3 types.
There are TBA-type equations, horizontal SU(2) hybrid-NLIE type equations, and vertical SU(4) hybrid-NLIE type equations. They together form a closed set of nonlinear integral-equations, which are solved numerically in this paper. As it is usual, structurally the equations consist of source terms plus convolutions containing coupling dependent kernels and nonlinear combinations of the unknown functions. The objects appearing in the arguments of the source functions are subjected to quantization conditions, but similarly to the boundary TBA description [4], due to the u → −u symmetry of the problem they are tied to the origin of the complex plane, thus extra quantization conditions are unnecessary to be imposed, since they are automatically satisfied by symmetry. Since these source term objects have fixed positions their positions are exactly the same as that of their asymptotic counterparts. This fact saves us from the tedious computation of the source terms, since if we take the difference of the exact equations and their asymptotic counterparts the source terms cancel from the equations. To be pragmatic and save time and space, the equations will be presented in such a difference form. Thus for any combination f of the unknown functions, we introduce the notation δf
(u) = f (u) − f o (u), where f o (u)
is the asymptotic counterpart of f . Having introduced this notation, we start the presentation of the equations by the TBA-type part. For the labeling of the Y -functions we use the string-hypothesis [40] based notations of [41]. For the presentation of the equations a few more notations need to be introduced:
L ± = log τ 2 1 − 1 Y ± , L m = log τ 2 1 + 1 Y m|vw , τ (u) = tanh(πgu 4
).
(2.1) For later numerical purposes we re-parametrize log Y Q by the formula:
log Y Q (u) = −2L log x [Q] (u) x [−Q] (u) +logȳ Q (u)+c Q +ε log u 2 + (Q + 1) 2 g 2 , Q = 1, 2, ... (2.
2) such that c Q is the constant value of log Y Q at infinity and ε is minus twice the energy 5 : ε = −2 E BT BA . From the TBA equations of the problem [4], it follows that δc Q = c Q −c o Q ≡ δc is Q-independent, and for small g, logȳ Q is a smooth deformation of its asymptotic counterpart, such that δ logȳ Q tends to zero at infinity. 6 Using this decomposition the following notations are need to be introduced:
L Q = log(1 + Y Q ), δR Q = log(1 + Y Q ) − δ logȳ Q . (2.3)
Then the TBA-type equations take the form:
δlog Y m|vw = δ log (1 + Y m+1|vw )(1 + Y m−1|vw ) ⋆s−log(1+Y m+1 )⋆s, 2 ≤ m ≤ p 0 −2, (2.4) 5
The log multiplier of ε in (2.2) is chosen not to modify the constant term in the large u behavior and to satisfy
Y + Q Y − Q YQ−1 YQ+1 Y o Q−1 Y o Q+1 Y o+ Q Y o− Q =ȳ + Qȳ − Q yQ−1ȳQ+1ȳ o Q−1ȳ o Q+1 y o+ Qȳ o− Q
, which is the LHS of an important Ysystem equation divided by its asymptotic counterpart. 6 log Y Q cannot be considered as smooth deformation of log Y o Q , because log Y Q −log Y o Q ∼ ε log |u| diverges for large u at any g. On the other hand logȳ Q − logȳ o Q is small for any u at small g and tends to zero at infinity.
δ log Y 1|vw = δ log 1 + Y 2|vw ⋆ s − log(1 + Y 2 ) ⋆ s + δ log 1 − Y − 1 − Y + ⋆ s, (2.5) δlogȳ Q = 2 δL Q−1 ⋆ s − (δR Q−1 + δR Q+1 ⋆ s), Q ≥ 2, (2.6) δ log Y − Y + = − p 0 −2 Q=1 log(1 + Y Q ) ⋆ K Qy − Ω(K Qy ). (2.7) δlog(Y + Y − ) = 2δlog 1 + Y 1|vw 1 + Y 1|w ⋆s + p 0 −2 Q=1 log(1 + Y Q ) ⋆ −K Q + 2K Q1 xv ⋆s − Ω(K Q ) + 2 Ω(K Q1 xv ⋆s) (2.8)
For Y 1 the modified hybrid form [42] of the BTBA equations is used,
δlogȳ 1 =2δ log(1 + Y 1|vw ) ⋆ s⋆ K y1 − 2δ log 1 − Y − 1 − Y + ⋆ s ⋆ K 11 vwx + 2δL −⋆ K y1 − + 2δL +⋆ K y1 + + p 0 −2 Q=1 log(1 + Y Q ) ⋆ K Q1 sl(2) + 2s ⋆ K Q−1, 1 vwx + Ω(K Q1 sl(2) ) + 2Ω(s ⋆ K Q−1, 1 vwx ), (2.9)
where p 0 is the index limit starting from which the upper part of the TBA equations is replaced by an SU(4) NLIE of [38] (See figure 1.). For any kernel vector appearing in the TBA equations Ω(K Q ) denotes the residual sum ∞ Q=p 0 −1 L Q ⋆K Q , and following the method of [42] for p 0 ≥ 4 it can be expressed by next to nearest neighbor Yfunctions as follows:
Ω(K Q ) = δR p 0 −1 ⋆ σ1 2 ⋆ K p 0 −2 − δR p 0 −2 ⋆ σ1 2 ⋆ K p 0 −1 + 2δr p 0 −2 ⋆ s 1 2 ⋆ K p 0 −2 − 2δr p 0 −3 ⋆ s 1 2 ⋆ K p 0 −1 ,(2.10)
where r m = log(1 + Y m|vw ), the kernels s, s 1 2 , σ1 2 are hyperbolic functions [42],
s(u) = g 4 cosh π g u 2 , s 1 2 (u) = 1 2 s( u 2 ), σ 1/2 (u) = g 2 √ 2 cosh πgu 4 cosh πgu 2 ,(2.11)
while the other TBA kernels can be found in appendix A. As a consequence of the re-parametrization (2.2) the two constants δc and ε also become part of the set of
equations 7 : ε = 1 2π p 0 −2 Q=1 L Q ⋆ dp Q du + 1 2π Ω( dp Q du ), (2.12) δc =2δ log(1 + Y 1|vw ) ⋆ s⋆ CK y1 − 2δ log 1 − Y − 1 − Y + ⋆ s ⋆ CK 11 vwx + 2δL −⋆ CK y1 − + 2δL +⋆ CK y1 + + p 0 −2 Q=1 log(1 + Y Q ) ⋆ CK Q1 sl(2) + 2s ⋆ CK Q−1, 1 vwx + Ω(CK Q1 sl(2) ) + 2Ω(s ⋆ CK Q−1, 1 vwx ), (2.13)
where for any kernel K: CK(u) denotes the constant term in the large v expansion of K(u, v). 8 As we mentioned −ε/2 is the TBA energy, thus (2.12) gives the energy formula in our formulation of the finite size problem. The asymptotic forms of the Y -functions necessary for the formulation of (2.4-2.13) are listed in appendix D. To close the discussion of the TBA-type equations we note that equations (2.7) and (2.8) determine Y ± up to an overall sign factor. The sign factor can be fixed from the asymptotic solution and its value is −1. Thus the fermionic Y-functions can be expressed in terms of the LHS of (2.7) and (2.8) by the formula:
Y ∓ = −e 1 2 log Y + Y − ± 1 2 log Y − Y + . (2.
14)
The horizontal SU(2) wing of the TBA is resumed by an SU(2)-type NLIE [43,38], which in our case takes the form:
δlog(−b) = s [1−γ] ⋆ δ log(1 + Y 1|w ) + G ⋆ δ log(−1 − b) − G [−2γ] ⋆ δ log(−1 −b), (2.15) δ log(−b) = s [γ−1] ⋆ δ log(1 + Y 1|w ) + G ⋆ δ log(−1 −b) − G [2γ] ⋆ δ log(−1 − b), (2.16) δlog Y 1|w = s [γ−1] ⋆ δlog(−1 − b) + s [1−γ] ⋆ δlog(−1 −b) + δlog 1 − 1 Y − 1 − 1 Y + ⋆ s , (2.17)
where 0 < γ < 1/2 is a contour shift parameter, the kernel G is given by (B.6) and the asymptotic solution for b andb is given in appendix D 9 . The upper SU(4) NLIE 7 Here the ⋆ notation means simply integration from −∞ to ∞. 8 Here we note that only the dressing kernel has logarithmically divergent term in its large v expansion, all the other kernels has either constant term or they simply vanish at infinity. 9 In practice b andb are complex conjugate of each other.
of [38] is attached to the TBA equations at the p 0 -th node. The upper NLIE is for 12 complex unknown functions: b A and d A , A = 1, ..., 6. They are combinations of the T -functions of the upper wing SU(4) Bäcklund-hierarchy [38]. Their relations to the unknowns introduced in [38] are given by (B.3,B.4) in appendix B and their asymptotic forms are given in appendix C. Using the notation B A = 1 + b A and D A = 1 + d A , the equations they satisfy take the form:
δlog b A = A ′ (G bB ) AA ′ ⋆ δlog B A ′ + A ′ (G bD ) AA ′ ⋆ δlog D A ′ + E A , (2.18) δlog d A = A ′ (G dB ) AA ′ ⋆ δlog B A ′ + A ′ (G dD ) AA ′ ⋆ δlog D A ′ +Ē A ,(2.19)
where the kernels are given in (B.5-B.12). The shifts in the kernels which is equivalent to fixing the lines on which the NLIE variables live, are chosen in a symmetrical way and fixed as follows:
γ = {γ a } = {γ (3) 1 , γ (3) 2 , γ(3)
3 , γ
1 , γ
2 , γ(2)1 } = 1 12 (−9, −1, 5, −3, 3, 1), (2.20) η = {η a } = {η (3) 1 , η(1)
2 , η
3 , η
1 , η
2 , η In practice this reduces to half the number of SU(4) NLIE variables. The vectors E A andĒ A are conjugate to each other and they give the TBA input into the upper NLIE. To give their form we introduce the notations:
η 1 = Y [ǫ 1 ] p0−1|vw ,η 1 = Y [ǫ 3 ] p0−1|vw , ǫ 1 = −ǫ 3 = − 7 12 , (2.23) E 1 = s [ 5 6 ] ⋆ δlog(1 + η 1 ), E 3 = s [ 5 6 ] ⋆ δlog(1 +η 1 ), E 5 = E 6 = 0, (2.24) E 2 = 1 2 s [ 1 2 ] ⋆ δlog(1 + η 1 ) − 1 2 s [ 2 3 ] ⋆ δlog(1 +η 1 ) + ε 2 + i ϕ 2 ,(2.
25)
E 4 = − 1 2 s [ 1 3 ] ⋆ δlog(1 + η 1 ) − 1 2 s [ 5 6 ] ⋆ δlog(1 +η 1 ) + ε 4 + i ϕ 4 , (2.26) where ε 2 (u) = i 2π ∞ 0 dv δR p 0 (v) 1 u − v − i 12g + 1 u + v − i 12g , (2.27) ε 4 (u) = i 2π ∞ 0 dv δR p 0 (v) 1 u − v − i 4g + 1 u + v − i 4g ,
(2.28)
ϕ 2 (u) = ∞ −∞ dv δlog(1 + η 1 (v)) ϕ(u − v + i 2g ) + ϕ(u + v − 2 i 3g ) , (2.29) ϕ 4 (u) = ∞ −∞ dv δlog(1 + η 1 (v)) ϕ(u − v + i 3g ) + ϕ(u + v − 5 i 6g ) , (2.30) with ϕ(u) = g 8 π i ψ( 1 4 − i u g 4 ) − i ψ( 1 4 + i u g 4 ) − π tanh( π g u 2 ) . (2.31)
The last set of equations gives, how the upper NLIE variables couple to the TBA part of the equations.
δlog Y p 0 −2|vw = s [−ǫ 1 ] ⋆ δlog(1 + η 1 ) + s ⋆ δlog(1 + Y p 0 −3|vw ) − s ⋆ L p 0 −1 , (2.32) δlog η 1 = s [−1+ǫ 1 −γ 1 ] ⋆ δlog B 1 + s [1+ǫ 1 −η 1 ] ⋆ δlog D 1 − s [ǫ 1 −γ 2 ] ⋆ δlog B 2 + s [ǫ 1 ] ⋆ δlog(1 + Y p 0 −2|vw ) − s [ǫ 1 ] ⋆ L p 0 , (2.33) δlogη 1 = s [−1+ǫ 3 −γ 3 ] ⋆ δlog B 3 + s [1+ǫ 3 −η 3 ] ⋆ δlog D 3 − s [ǫ 3 −η 2 ] ⋆ δlog D 2 + s [ǫ 3 ] ⋆ δlog(1 + Y p 0 −2|vw ) − s [ǫ 3 ] ⋆ L p 0 , (2.34) δlogȳ p 0 = s [−1−γ 2 ] ⋆ δlogb 2 − δlog B 2 + s [1−η 2 ] ⋆ δlogd 2 − δlog D 2 +s [−γ 3 ] ⋆ δlog B 3 b 3 + s [−η 1 ] ⋆ δlog D 1 d 1 + s [−ǫ 1 ] ⋆ δlog 1 + η 1 η 1 +s [−ǫ 3 ] ⋆ δlog 1 +η 1 η 1 − s ⋆ δR p 0 −1 ,(2.35)
whereb 2 andd 2 are from the re-parametrization of b 2 and d 2 :
b 2 (u) = η 1 x [−p 0 +γ 2 ] s (u) 2L exp ε log u + i γ 2 − p 0 − 1 g + i π 2 + δc 2 b 2 (u), (2.36) d 2 (u) = η 1 x [p 0 +η 2 ] s (u) 2L exp ε log u + i η 2 + p 0 + 1 g − i π 2 + δc 2 d 2 (u),
(2.37) with η = ±1 being a global sign factor. Similarly to the definition ofȳ Q , also here the benefit of usingb 2 andd 2 is that, for small g, logb 2 and logd 2 are smooth deformations of their asymptotic counterparts, and in addition δlogb 2 and δlogd 2 vanishes at infinity, which is necessary for the convergence of certain integrals. The decompositions (2.36),(2.37) are chosen to be compatible with the functional relation b [38]. Equations (2.4)-(2.35) constitute our complete set of nonlinear integral equations, which governs the finite size dependence of the vacuum of our D-brane anti-D-brane system.
[−γ 2 ] 2 d [−η 2 ] 2 = Y p 0 in ref.
The numerical method
Here we describe our numerical method for solving the hybrid-NLIE equations presented in the previous section. During the iterative numerical solution of the equations we faced with very serious convergence problems, which forced us to work out such a method that overcomes all the difficulties emerged. Our numerical method can be applied to solve other type of nonlinear integral equations as well. The power of the method is shown by the fact that numerical convergence was reached even in such cases, when the solution was physically unacceptable. The numerical method consist of two main steps, namely:
• Discretization of the equations • Iterative solution.
The first step involves the discretization of the unknown functions and kernels, furthermore the discrete approximate representation of the convolutions. Having carried out the appropriate discretization of the problem, the equations are considered as large nonlinear algebraic set of equations. Thus eventually instead of integral equations we solve discrete algebraic equations. In this paper we will present two methods to solve them numerically.
Discretization of the problem
The discretization serves two goals. First it allows us to reduce the numerical problem from solving integral equations to solving algebraic equations. Second choosing the discretization points appropriately it reduces the number of degrees of freedom as much as it is possible to reach the desired numerical accuracy. In our actual numerical computation instead of u of section 2. we used the new rapidity u → u g , because with such a scaling almost all the rapidity difference dependent kernels become g independent. Thus for example Y ± (u) will be defined in [−2g, 2g]. To decrease the number of discretization points the u → −u symmetry of the problem is exploited. This means that the Y -functions are to be discretized only on [0, ∞] or [0, 2g] and as for the NLIE variables it is enough to discretize the b-and d-type variables on [0, ∞]. Since we do not want to introduce any cutoff in the rapidity
space first we transform the u ∈ [0, ∞] interval to a finite interval t ∈ [0, B(a)]
through the transformation formula:
u(t) = a B(a) t − 1 , B(a) = 2 g a + 1. (3.1)
This formula is chosen such that the branch point 2g corresponds to t = 1 for any choice of the parameter a, where a is a global scaling factor which changes from unknown to unknown. We chose the values as follows: for Y 1|w a = 1, for b andb a = 2, for Y Q and Y Q−1|vw a = Q, for η 1 andη 1 a = p 0 , and finally for the b-and d-type NLIE functions a = p 0 . These values are chosen to preserve the smoothness 10 of the transformed functions in the finite interval. After this transformation all of our unknown functions live on a finite interval. To discretize them we used piecewise Chebyshev approximation. This means that we divide the finite interval into subintervals and on each subinterval the functions are approximated by a given order Chebyshev series. The choice of subintervals is not equidistant. The subintervals are placed more densely around the branch points, since the function x(u/g), which governs the decay of the massive Y Q -functions, has the largest change around this point. The advantage of the Chebyshev approximation is that if the function is smooth enough on the subinterval, the coefficients of the Chebyshev series decay rapidly and the order of magnitude of the last coefficient allows us to estimate the numerical errors of the procedure. Now we describe the discretization method in more detail. Our functions are defined on either [0, B(Q)] or on [0, 2g]. This is why two type of subinterval vectors are defined A Q and A ± , such that the endpoints of the subintervals of [0, B(Q)] are put into the vector A Q and the endpoints of the subintervals of [0, 2g] define A ± . Let l k be the order of the Chebyshev approximation, then using the general rules of the Chebyshev approximation, a given function
f (t) is approximated in the kth subinterval [A k−1 , A k ] as: f (t) ≃ l k j=1 c (k) jT j−1 t − 1 2 (A k + A k−1 ) 1 2 (A k − A k−1 ) , t ∈ [A k−1 , A k ], (3.2)
where now the vector A stand for either A Q or A ± , furthermoreT j−1 are a slightly modified Chebyshev polynomials 11
T j (u) = T j (u) if j ≥ 1, 1 2 if j = 0, with T j (u) being the jth Chebyshev polynomial 12 . The coefficients c (k) j
are the Chebyshev coefficients of the function f , which can be computed from the sampling points of the Chebyshev approximation:
t (k) j = 1 2 (A k − A k−1 ) c (i) (l k ) + 1 2 (A k + A k−1 ), i = 1, .., l k (3.3)
by the simple formula:
c (k) j = 2 l k l k j 0 =1 f (t (k) l k −j 0 +1 )C j 0 ,j ,(3.4)
where c (i) (l k ) are the zeros of the l k order Chebyshev polynomial:
c (i) (l k ) = − cos π l k i − 1 2 ,T l k (c (i) (l k )) = 0, i = 1, ..., l k (3.5)
andC k,i is given by:
C k,i = cos π l k k − 1 2 (i − 1) , i, k ∈ {1, ..., l k }. (3.6)
In our method the next step is to formulate the convolutions and the equations themselves in terms of the discrete values of our functions. Here will sketch the basic idea in some typical scenarios appearing in our equations. Then its application to the concrete unknowns and kernels of the problem is straightforward. If one takes the equations at the required discretized points t (k) j the following typical pattern arises:
F (u(t (k ′ ) j ′ )) ≃ ∞ 0 dv ′ L(v ′ ) K S (v ′ , u(t (k ′ ) j ′ )) + . . . , (3.7) where K S (u, v) = K(u, v) + K(−u, v)
is the symmetrized kernel to exploit left-right symmetry of the problem for reducing to half the number of variables.
F (u(t (k ′ ) j ′ )
) is intended to modelize the variables in the left-hand side of the equations taken at the discretized points of the transformed variable t and L(u) stands for some nonlinear combination of some unknown function of the equations 13
. If L(u(t)) is discretized by a subinterval vector A of [0, B(a)]
, then the numerical approximation of the right hand side goes as follows;
• First the integration variable is changed from v ′ to t,
• then on each subinterval L(u(t)) is approximated by its Chebyshev series,
• finally the integration is carried out and the convolution is expressed in terms of the discretized values of L(u(t)).
The final approximation formula takes the form:
∞ 0 dv ′ L(v ′ ) K S (v ′ , u(t (k ′ ) j ′ )) ≃ L(A) k=1 l k j=1 L k,j 2 l k l k j 0 =1C l k −j+1,j 0 K k,j 0 k ′ ,j ′ , (3.8) where L k,j = L(u(t (k)
j )), L(A) denotes the dimension of A and K k,j k ′ ,j ′ is the discretized convolution matrix given by the formula:
K k,j k ′ ,j ′ = a B(a) A k A k−1 dt t 2T j−1 t − 1 2 (A k + A k−1 ) 1 2 (A k − A k−1 ) K S (u(t), u(t (k ′ ) j ′ )). (3.9)
In this manner a convolution is reduced to a discrete matrix-vector multiplication. The other type of typical convolution is when the integration is taken from zero to 2g. In certain cases the function L(u) has square root behavior close to the branch points 14 . For such functions the truncated Chebyshev series does not give accurate approximation. In these cases not the function L(u) is approximated, but that part of it which remains after the elimination of the square root behavior. Namely, we write L(u) = 4g 2 − u 2L (u), thenL(u) is approximated by a truncated Chebyshev series and finally the approximate discretized form of the corresponding convolution is very similar to (3.8):
2g 0 dv L(v) K S (v, u (k ′ ) j ′ ) ≃ L(A ± ) k=1 l k j=1L k,j 2 l k l k j 0 =1C l k −j+1,j 0K k,j 0 k ′ ,j ′ , (3.10) whereL k,j =L(v (k)
j ) andK k,j k ′ ,j ′ is the square root factor modified version of (3.9);
K k,j k ′ ,j ′ = A ±,k A ±,k−1 dv 4g 2 − v 2T j−1 v − 1 2 (A ±,k + A ±,k−1 ) 1 2 (A ±,k − A ±,k−1 ) K S (v, u (k ′ ) j ′ ). (3.11)
Here depending on the left hand side of the equation u
(k ′ ) j ′ can stand for u(t (k ′ ) j ′ ), t ∈ [0, B(a)]
for some a, or it can denote the sampling points on [0, 2g].
Applying our discretization technique to all unknowns and convolutions of our equations, we can reduce the integral equations to a discrete set of nonlinear algebraic equations. However, the transformation from integral equations to algebraic equations is obviously not exact. The typical error comes from the fact that on each subinterval the Chebyshev series is truncated, so the magnitude of the typical errors in our numerical method is governed by the neglected terms of the Chebyshev series, which can be approximated by the magnitude of the last Chebysev coefficient. In our case this is typically somewhere between 10 −5 and 10 −6 .
The last step of our numerical method is the iterative solution starting from the asymptotic solution.
14 Such typical combinations are log 1−Y− 1−Y+ and log
1− 1 Y − 1− 1 Y + .
The iterative solution
Here we will describe two methods to solve our integral equations iteratively. Since our actual equations have very complicated form, we will describe our methods using a model example, which has similar structure to our equations. Let the model equations take the form 15 :
log y a = f a + G ab ⋆ log(1 + y b ), (3.12)
where G ab are some kernel matrices, f a are some source terms and y a s are the unknown functions of the problem. The solution of (3.12) is expanded around the asymptotic solution and the equations are formulated in terms of the corrections.
To fix the conventions, the correction functions δy a are defined by:
y a = y o a (1 + δy a ). (3.13)
As a consequence:
log y a = log y o a + log(1 + δy a ), log(1 + y a ) = log(1 + y o a ) + log(1 + Y a δy a ), Y a = y o a 1 + y o a .
The source term is also expanded around its asymptotic counterpart: f a = f o a + δf a . Then equations (3.12) can be reformulated in terms of the δy a functions as follows:
log(1 + δy a ) = δf a + G ab ⋆ log(1 + Y b δy b ).
(3.14)
To define the iterative method, (3.14) are reformulated so that only O(δy 2 a ) terms remain on the right hand side of the equations. Thus the equations are rewritten in the form:
δy a − G ab ⋆ (Y b δy b ) − δf a = G ab ⋆ [log(1 + Y b δy b ) − Y b δy b ] − [log(1 + δy a ) − δy a ] .δy (n+1) a −G ab ⋆(Y b δy (n+1) b )−δf a = G ab ⋆ log(1 + Y b δy (n) b ) − Y b δy (n) b − log(1 + δy (n) a ) − δy (n) a . (3.16)
Thus at each step of this iterative method a set of linear integral equations must be solved. Using the discretization method of the previous subsection, the problem reduces to solving a set of linear algebraic equations, which is a straightforward task in numerical mathematics. The very first (0th) iteration starts from the asymptotic solution δy a = 0 and it corresponds to the solution of the linearized equations, which in our case gives the Lüscher-formula for the energy.
This (first) method in a certain range of the coupling constant defined a numerically convergent iteration to solve the equations for the ground state of our D-brane anti-D-brane problem, but beyond a certain value of g the method failed to converge anymore. This is why we worked out a second method, which proved to be much more efficient than the first one. This efficiency is manifested in two facts. First it converges much faster than the previous iterative method, second it gives convergent solutions to our equations even when the solution cannot be accepted as physical one 16 .
This second method can be described simply in words. Instead of defining an iteration as above, we simply take the discretized version of (3.14). We consider it as a set of nonlinear algebraic equations. As a first step we solve the linearized discrete equations (i.e. (3.16) with RHS = 0) and starting from the solution of the linearized equations we solve the discrete nonlinear system by Newton-method 17 . 16 Beyond a certain value of the coupling constant the equations in the form presented in section 2. are not the right ones anymore, they should be corrected by some new source terms and quantization conditions, but even for the "wrong" equations the second method shows numerical convergence, giving unacceptable result. 17 In MATHEMATICA language it can be implemented by FindRoot[...,Method→"Newton"].
Numerical results
In this section we summarize our numerical results. We solved numerically the equations for several integer values of the length parameter L. In this section we concentrate on the states with L ≥ 2. The L = 1 special case is discussed in the next section. For the explanation of the numerical data we will mostly use the L = 2 case as an example, because the critical point of this state is the closest one to zero, so it is enough to work with relatively small values of the coupling constant. This is important from the numerical point of view, since by increasing g the numerical method becomes more and more time consuming. First the parameters of the numerical method is discussed. There are three parameters in the nonlinear integral equations (2.4)-(2.35). The most important one is the coupling constant g, then there are two other parameters which allow us to formulate the equations according to our purposes. The two parameters are p 0 and C, where p 0 is a kind of "truncation index", which tells us the node number starting from which the upper TBA equations are replaced by SU(4) NLIE variables (see figure 1.). The parameter C is a free parameter in the asymptotic solution for the upper SU(4) NLIE variables (C.6-C.20) and it enters the equations such that the asymptotic solution around which the equations are formulated contain this parameter. From this discussion it is obvious that g is a physical parameter which means that the energy depends on it, while the other two parameters p 0 and C correspond to different formulations of the same mathematical problem, so the energy does not depend on them. Thus the choice of these parameters is in our hand and we tried to choose such values for them which allows us numerical convergence in the widest range in g. For example the C = 0 choice is the best for numerical purposes since due to (2.22) a u → −u symmetry arises in the SU(4) HNLIE variables minimizing the number of unknowns in the problem. Tuning p 0 might have two advantages. First, numerical experience shows that for large p 0 the Chebyshev coefficients of the unknowns entering the formula (2.10) for Ω, decay faster, which allows for higher numerical precision. Second also from numerics we learn that with p 0 fixed at certain values of g non physical results are obtained from the numerical solution of the problem. This is a consequence of new local singularities entering the problem, but we still did not take them into account in the equations. We solved numerically our equations for different values of L and with various values of p 0 and C, and in case the numerical result was physically acceptable for all p 0 and C we tried, it was also independent of these parameters within the numerical errors of the method.
So far we discussed the parameters of the continuous integral equations and their role in the numerical solution. Now we turn to discuss the numerical parameters of the equations. The numerical parameters are artifacts of the numerical method, and they arise mostly from the discretization method described in section 3. We note that there is no cutoff parameter in our numerical method, neither in the integration range nor in the index of Y -functions. Everything is treated in an exact manner, the only source of numerical errors is the discretization of the unknowns and the convolutions. Here we give the most used subinterval vectors of our numerical computations.
On each subinterval we used an l k = 10 order Chebyshev approximation. The subinterval vector A ± of [0, 2g] is given by the empirical formula:
A ± = A ±,< if g ≤ 2, A ±,> if g ≥ 2,(4.1)
where the vectors in components take the form:
A (k) ±,< = 2g k [2g] + 1 , k = 1, ..., [2g] + 1, (4.2) A (k) ±,> = 1 2 , 1, v, 2g − 3 4 , 2g − 1 2 , 2g − 1 4 , 2g ,(4.3)
with v having vector components: • The first element of A Q is 1 2 .
v j = 1 + j 2g − 2 2g − 3 2 , j = 1, ..., 2g − 3 2 .
• The length of subintervals ∆t in the range 1 2 < t < 2 is approximately 1 3 : ∆t 1 3 .
• The length of subintervals in the range 2 < t < 3 is approximately 1 2 : ∆t 1 2 .
• The length of subintervals in the range 3 < t < B(Q) is approximately 1: ∆t 1.
18 These requirements are based on numerical experiences with the choice l k ≥ 10.
In practice the length of the subintervals are slightly "squeezed" with respect to the conditions above to fill the full [0, B(Q)] properly 19 . Finally, we note that for checking the numerical precision, we also did numerical computations with l k = 12, 14, 16 keeping the subintervals fixed and also with keeping l k = 10, but doubling the number of subinterval points. Before turning to present the numerical results we would like to say a few words about the possible tests of the numerical results. Namely, how one can recognize a wrong result. This is also a very important point of the numerical method, since there are a lot of equations with very complicated kernels and it is easy to make mistakes during writing the code of the numerical solution. There are three basic things that we can check from the numerical results.
The first check is dictated by the energy equation (2.12). It is known that the energy starts at the first wrapping order (i.e. e −L ) and this first order correction is exactly given by the Lüscher formula [4]:
∆E(L) = − ∞ Q=1 ∞ 0 du 2π dp Q du Y o Q (u),(4.5)
with Y o Q (u) given explicitly in (D.2). This quantity can be computed numerically with any digits of precision, so its value is known exactly at any values of g and L. The Lüscher-formula (4.5) corresponds to the linearized version of our equations (2.4)-(2.35), this is why solving the linearized set of equations (which is the first step for the iterative solution) we should reproduce the numerical evaluation of (4.5). This is a nontrivial check on the kernels, on the discretization method and on the equations themselves as well. In addition since this test is quantitative it can tell some information also on the numerical precision of the method 20 .
This test can signal problems on solving the linearized problem. The remaining two tests can signal some discrepancies during the solution of the nonlinear problem.
The second testing condition is that from the numerical solution Y p 0 −2|vw must be real. This sound trivial, but it is not trivial at all. If one takes a look at the equation BT BA from the exact Lüscher result. This quantity gives some information on the numerical accuracy of the method. Finally the column "number of nodes" tells us the cutoff index of the Lüscher formula, which is necessary to get the Lüscher energy with the precision given by ∆E BT BA . This number is not equal to p 0 in our equations. For the L = 2 state, in case of 0 < g < 1.9 we used p 0 = 4, for 1.9 < g < 2.1 we used p 0 = 8, and in the range 2.1 < g < 2.14 we took p 0 = 12. Finally at g = 2.16 we used p 0 = 26 to get acceptable numerical results. Then beyond this point we could not save our equations from the entrance of new singularities by increasing the value of p 0 with a reasonable O(10) amount. Because of this reason we could not get really close to the supposed critical point. There E BT BA ∼ −1, but we could reach only E BT BA ∼ −0.7 at g = 2.16. Apart from this very embarrassing fact, some important features can be read off from the numerical data. First of all it can be seen that in the range g < 2.16 the energy is very slowly varying function of g, so there is no sign of any divergent behavior. What is more interesting is the behavior of the global constant δc. It is negative and it decreases faster and faster as g is increased. From the definition of δc (2.2) it follows that all Y Q -functions are proportional to its exponent: Y Q ∼ ξ = e δc . The fast decrease of δc indicates that though Y Q has worse and worse large u asymptotic by the increase of g, its global magnitude is actually decreasing. This remark can be understood from the TBA formulation of the energy.
E BT BA = − ∞ Q=1 ∞ 0 du 2π dp Q du log(1 + Y Q (u)). (4.7)
Close to the critical point E BT BA is supposed to be finite [4] E BT BA ∼ 1 − L, but naively the sum in the RHS of (4.7) would diverge due to the large Q terms. Since Y Q is small for large Q, in leading order 21 the log(1 + Y Q ) → Y Q replacement can be done:
E BT BA = − Q 0 Q=1 ∞ 0 du 2π dp Q du log(1 + Y Q (u)) Finite − ξ ∞ Q=Q 0 ∞ 0 du 2π dp Q duỸ Q (u)
Diverges close to the critical point +..., (4.8) where Q 0 is an arbitrary index cutoff scale and Y Q = ξỸ Q replacement was applied.
Since ξ is Q-independent all the dangerous Q dependence is still inỸ Q . In (4.8) approaching to the critical point the second sum starts to diverge, and the global multiplicative factor ξ must tend to zero in order to ensure the finiteness of both sides of the equation. Our numerical data seems to support this picture. Namely δc → −∞ as going closer and closer to the critical point.
In [4] from Y -system arguments the large Q behavior of Y Q was also estimated by the formula:
Y Q (u) ≃ ξ(g) 1 u 2 + Q 2 g 2 2 E BT BA Y o Q (u), ξ = e δc ,(4.9)
where δc is defined after (2.2) in section 2. the Y -functions of the infinite Y -system. This makes it possible to test numerically the correctness of the large Q estimate (4.9). In case (4.9) holds, it implies that δ lnȳ Q = lnȳ Q − lnȳ o Q tends to zero as 1/Q for large Q. In figure 3. the numerical demonstration of this statement can be seen. The plotted functions are defined by the formula:
δF Q (t) = δ lnȳ Q (x Q (B(Q) − t)) if t > 0, δ lnȳ Q (−x Q (B(Q) + t)) if t < 0, (4.10) where x Q (t) = Q B(Q) t − 1 , B(Q) = 2g Q + 1.
The plots of figure 3. are based on the numerical computation with p 0 = 26 at g = 2.16. Figure 3. nicely demonstrates the expected 1/Q behavior of the functions δF Q . This fact shows us that the equations we solved numerically are not the right ones anymore. Something is missing from the equations. Either a special object [44,45] or some other local singularities of the T -and Q-functions of the problem, which enter those strips of the complex plane, which are relevant in the derivation of the HNLIE equations. The numerical data for the L = 3 and L = 4 states are given by table 2 and 3. Also in case of these states the appearance of new singularities obstacled us to get close to the critical point in the framework of the HNLIE technique.
Comments on the L = 1 case
The L = 1 ground state is mysterious, since so far the anomalous dimension of this state could not be determined even for small g either from field theory or from integrability considerations [4]. Here we concentrate on the integrability side. There the boundary Lüscher formula [27,39] diverges for this state [4]. takes the form [4]:
∆E(L) = − ∞ Q=1 ∞ 0 du 2π dp Q du Y o Q (u) ≃ − g 2 4L 4 4L − 1 4L 2L ζ(4L − 3) + O(g 2 ) . (5.1)
This small coupling expression diverges for L = 1, since this point sits exactly on the pole of the ζ -function. As for the origin of this divergence; in (5.1) the individual integrals are convergent, but their sum for Q causes the divergence. In [4] it was argued that also for any larger L the TBA energy formula would diverge beyond a certain critical value of the coupling: g c (L). Assuming that the energy is a monotonously decreasing function of g, which is supported by numerical results, this critical point can be expressed clearly in terms of the energy by the criterion:
E c (L) ≡ E(g c (L)) = 1 − L. (5.2)
In [4] this point was interpreted as a turning point where the energy becomes imaginary and as a physical consequence the ground state becomes tachyonic. For the L = 1 state the critical point is right at g = 0 assuming that for small g the energy is also small. Now let us turn our attention to the HNLIE description of the problem detailed in section 2. Here there are no infinite sums and even for L = 1 all the convolutions of the integral equations seem to converge 23 . For the first sight there is no sign of any problem in the HNLIE description and it seems that only the TBA description is inappropriate to treat the L = 1 case. But unfortunately this is not the case.
We can write down the discretized integral equations for the L = 1 case as well, and using the Newton-method, we can solve them for small values of the coupling 24 . We always get some numerical solution for the discretized problem, but it turns out that the Chebyshev coefficients of the unknowns, which correspond to the large u subinterval do not form a decaying series. This phenomenon is a typical sign of some weak (probably logarithmic) large u divergence of the unknowns. If one increases the number of subintervals and sampling points the situation remains the same. The 23 If we assume that large u behavior of the unknown functions, which was used to derive the BTBA equations from discontinuity relations and Y-system. 24 Typically g ∼ 10 −1 .
conclusion is that we can solve the discretized problem, but the solution cannot be interpreted as the discretely approximated version of the continuous solution of our integral equations. In other words the continuous HNLIE equations have no solution for L = 1.
In order to get some analytical insight why the solutions become diverging at large u let us consider the TBA formulation of the problem (p 0 → ∞ in HNLIE). It is known [4] that the TBA energy comes from the coefficient of the most divergent log |u| term in the large u expansion of log Y Q :
log Y Q (u) = −4(L + E BT BA ) log |u| + O(1). (5.3)
The E BT BA term originates from the RHS of the TBA equations for log Y Q from the convolution term (2) by exploiting the large u expansion of the kernel: (2) has better large Q ′ behavior than that of dp Q ′ dv , since it behaves like 1/Q ′ . As a consequence contrary to the energy formula, the sum of dressing convolutions is convergent indeed. Thus one might think that for L = 1 the problem emerges, because for the derivation of the energy formula we expanded the sum of dressing convolutions term by term for large u. This is why instead of this usual procedure, we consider the sum of dressing convolutions itself, compute it and then at the end of the computation we take the large u expansion. This procedure is carried out in the small coupling limit. We need the leading order small coupling expression of the dressing kernel in the mirror-mirror channel 25 :
∞ Q ′ =1 log(1 + Y Q ′ ) ⋆ K Q ′ Q slK Q ′ Q sl(2) (v, u) = − 1 π dp Q ′ dv log |u| + O(1). K Q ′ Q slK Q ′ Q, (0) sl(2) (u 1 , u 2 ) = − 1 2π ψ 1 + Q ′ 2 − i i 2 u 1 + ψ 1 + Q ′ 2 + i i 2 u 1 −ψ 1 + Q ′ + Q 2 + i i 2 (u 2 − u 1 ) − ψ 1 + Q ′ + Q 2 − i i 2 (u 2 − u 1 ) + ....
(5.4)
Then the formula, the large u expansion of which accounts for the small coupling expanded energy, is given by:
O(u, Q) = ∞ Q ′ =1 ∞ −∞ dv 4π Y o,(L=1) Q ′ (v) K Q ′ Q, (0) sl(2) (v, u), (5.5) where Y o,(L=1) Q (u) = g 4 16 Q 2 u 2 (u 2 +Q 2 ) 3
is the leading small coupling expression of (D.2) at L = 1. The second derivative of O(u, Q) can be computed explicitly by simple Fourier space technique. We take the Fourier form of each functions under integration, the convolution is the product of the individual Fourier transforms, the sum for Q ′ can be easily done in Fourier space and at the end of the process everything is transformed back to the u space. In such a manner one gets a bulky, but explicit expression for d 2 du 2 O(u, Q), which we do not present here, only its large u expansion:
d 2 du 2 O(u, Q) = 4 g 4 3 2 − γ E + 2 ln 2 1 u 2 − 8 g 4 log u u 2 + O( 1 u 3 ). (5.6)
Integrating twice the large u expansion at small coupling becomes:
O(u, Q) = 4 g 4 1 2 + 2 γ E − ln 4 log u + 4 g 4 (log u) 2 + ... (5.7)
From (5.7) it is obvious why the naive Lüscher energy formula diverged. Because the leading order large u term is not the expected ∼ log |u|, but ∼ (log u) 2 . This is the key point of the problem, since in this case after this first iteration Y Q acquires an unwanted type of large u term, which makes Y Q divergent for large u: This large u divergence contradicts to what was assumed about the large u behavior of Y Q at the derivation of the integral equations, since it was supposed to decay. In this example we have shown in the small coupling limit, that during the iterative solution of the BTBA equations, log Y Q acquires an extra ∼ (log |u|) 2 behavior at infinity, which made Y Q an exploding function at infinity. This means that the iterative solution of the TBA equations leaves the class of physically acceptable solutions.
Y Q (u) ∼ u (−4L+4 g 4 ( 1 2 +2 γ E −
One might ask the question, whether it is possible to keep somehow the qualitative large u behaviors that we assumed at the derivation of the equations? Here we sketch a possible idea for small coupling to the L = 1 case. Let us assume that we managed to modify the TBA equations, such that all Y -functions have the large u behavior we want. Since most of the TBA equations reflect the structure of the Y -system functional equations we expect to modify only those equations which are affected by also the discontinuity relations. It follows, that for large Q, the formula for the estimate for Y Q (4.9) remains the same. Now, we assume that for small g the energy is also small and take the simultaneous small g and small energy expansion of the RHS of the TBA energy formula (4.7). In leading order the large Q terms will dominate:
E BT BA ≃ − ∞ Q=1 ∞ −∞ du 4 πŶ Q u g = −ξ ∞ Q=1 ∞ −∞ du 4π g 2 2L 16Q 2 u 2 (u 2 + Q 2 ) 2(L+E BT BA )+1 = −ξ 2 4(L+E BT BA ) 4g 4L 4(L + E BT BA ) − 1 4(L + E BT BA ) 2(L + E BT BA ) ∞ Q=1 1 Q (4(L+E BT BA )−3) = −ξ 2 4(L+E BT BA ) 4g 4L 4(L + E BT BA ) − 1 4(L + E BT BA ) 2(L + E BT BA ) ζ(4(L + E BT BA ) − 3) ≃ −ξ g 4 8 E BT BA + O(ξg 4 ),(5.9)
whereŶ Q denotes the large Q estimate (4.9) of Y Q ,ξ = ξ g 4E BT BA as a consequence of the u → u/g change of variables and the pole term in E BT BA comes from the pole of the ζ-function. In our HNLIE approach the energy E BT BA and the constant δc are parts of the equations which means that they are not simply expressed by explicit formulas based on the solution of the equations, but must me obtained by solving the set of non-trivially entangled equations. In this sense (5.9) defines an equation for E BT BA for small g. Its leading order solution is:
E BT BA = g 2 −ξ + . . . . (5.10)
Ifξ > 0 then E BT BA becomes imaginary as it would be expected from string-theory expectations [4]. To decide the sign ofξ, the equation (2.13) has to be analyzed in the context of the small g and E BT BA expansion. It turns out thatξ is positive and O(1) for small g, so according to (5.10) E BT BA is imaginary. Another remarkable fact is that according to (5.10) E BT BA starts at O(g 2 ) instead of the O(g 4 ) prediction of the boundary Lüscher formula (5.1). This might be another explanation why the coefficient of g 4 diverges in the Lüscher formula for the L = 1 case. Finally, we note that in the small g and E BT BA expansion of the L = 1 state, the energy is pure imaginary only at leading order in g, but in higher orders it acquires real part as well.
For the first sight, it might seem that without modifying the equations one immediately gets imaginary energy when going through the critical point. But, the situation is a bit more subtle. There is a hidden tacit modification of the equations. This is realized in (5.9) by the replacement:
∞ Q=1 1 Q 4(L+E BT BA )−3 → ζ(4(L + E BT BA ) − 3).
For the L = 1 case it is an identity for Re(E BT BA ) > 0, but for Re(E BT BA ) < 0 it is not an identity anymore, but a nontrivial analytical continuation in E BT BA .
Such an analytical continuation would require the exact determination of complicated sums of convolutions of the TBA equations as functions of the energy. Since this does not seem to be feasible in practice, we give such an alternative modification of the TBA equations which preserves the infinite sum structure of the equations, but the sums will converge everywhere for Re(E BT BA ) > −L except at the critical value E cr = 1 − L.
The basic idea of the modification comes from the sum representations of the ζ-function. The usual one converges for Re(s) > 1:
ζ(s) = ∞ Q=1 1 Q s , Re(s) > 1,(5.11)
but there is another representation which converges for Re(s) > 0:
ζ(s) = 1 s − 1 ∞ Q=1 Q (Q + 1) s − Q − s Q s , Re(s) > 0. (5.12)
Then the original TBA equations are modified through their infinite sums by the replacements:
∞ Q=1 L Q ⋆ K Q → 1 s E − 1 ∞ Q=1 {Q · (L Q+1 ⋆ K Q+1 ) − (Q − s E ) · (L Q ⋆ K Q )} , (5.13)
where s E = 4(L + E BT BA ) − 3. Taking into account the large Q behavior of all Y Q functions and all the kernels of the infinite sums of the TBA equations, the new representation will converge for Re(E BT BA ) > −L. This slight modification of the TBA equations might make it possible to go beyond the critical point and get solution of the TBA equations with large u asymptotics being in accordance with the ones used for the derivation of the equations.
The conclusion of this heuristic argument is that to keep the expected 26 qualitative large u behavior a nontrivial modification of the TBA equations must be carried out, which might lead to complex energies.
Summary and conclusions
In this paper we studied the ground state energy of a pair of open strings stretching between a coincident D3-brane anti-D3-brane pair in S 5 of AdS 5 × S 5 . The main motivation for the study is that string-theory predicts that the ground state of such a configuration becomes tachyonic for large values of the 't Hooft coupling [4].
In [4] it was shown that the usual integrability based BTBA approach always give real energies for the ground state and it breaks down at latest when the energy gets close to the critical value: During the numerical solution of the HNLIE equations the usual iterative methods failed to converge, this is why we worked out two numerical methods to reach convergence. The most effective one is, if one transforms the integral equations into discrete nonlinear algebraic equations and solves them by Newton-method. The power of this method is demonstrated by the fact that it gives convergent results even if the numerical solution is not physically acceptable.
Unfortunately, in our numerical studies we could not get very close to the critical point, because new singularities entered the HNLIE equations taking into account of which would have required an enormous amount of additional work. Nevertheless, in the range where we could get physically acceptable results, the precision of the HNLIE data were higher than those of BTBA and the HNLIE approach could give a deeper understanding of the problem. For the ground state of the L = 1 state the critical point is right at g = 0 and neither perturbative field theory computations nor the boundary Lüscher formula could provide a finite quantitative answer to the anomalous dimension. Even in this special case the numerical solution of the HNLIE equations was possible. The results showed that without an appropriate modification of the equations, they cannot give physically acceptable results. In this case, it means that the solution of the dicretized problem cannot be considered as a discretized solution of the continuous nonlinear integral equations. Moreover the large rapidity behavior of the numerical solution is incompatible with the one assumed for the derivation of the equations. This phenomenon is analytically analyzed in the framework of BTBA and an idea is sketched to preserve the expected large rapidity behavior of the unknowns. This method is based on an appropriate modification of the TBA equations which would lead to complex energies beyond the critical point.
Hopefully the L = 1 case at g = 0 could be treated analytically in the framework of the quantum spectral curve method [47,48,14], solving the mystery of this state in the context of integrability.
A Notations, kinematical variables, kernels
Throughout the paper we use the basic notations and TBA kernels of ref. [41], which we summarize below. For any function f , we denote f ± (u) = f (u ± i g ) and in general f [±a] (u) = f (u ± i g a), where the relation between g and the 't Hooft coupling λ is given by λ = 4π 2 g 2 . Most of the kernels and also the asymptotic solutions of the HNLIE-system are expressed in terms of the function x(u):
x(u) = 1 2 (u − i √ 4 − u 2 ), Im x(u) < 0, (A.1)
which maps the u-plane with cuts [−∞, −2] ∪ [2, ∞] onto the physical region of the mirror theory, and in terms of the function x s (u)
x s (u) = u 2 1 + 1 − 4 u 2 , |x s (u)| ≥ 1, (A.2)
which maps the u-plane with the cut [−2, 2] onto the physical region of the string theory. Both functions satisfy the identity x(u) + 1 x(u) = u and they are related by the x(u) = x s (u), and x(u) = 1/x s (u) relations on the lower and upper half planes of the complex plane respectively.
The momentump Q and the energyẼ Q of a mirror Q-particle are expressed in terms of x(u) as follows:
p Q (u) = gx u − i g Q − gx u + i g Q + iQ ,Ẽ Q (u) = log x u − i g Q x u + i g Q . (A.3)
Two different types of convolutions appear in the HNLIE equations. These are:
f ⋆ K(v) ≡ ∞ −∞ du f (u) K(u, v) , f⋆ K(v) ≡ 2 −2 du f (u) K(u, v) .
The kernels and kernel vectors entering the HNLIE equations can be grouped into two sets. The kernels from the first group are functions of only the difference of the rapidities, thus actually they depend on a single variable. The other group of kernels composed of those, which are not of difference type.
We start with listing the kernels depending on a single variable:
s(u) = 1 2πi d du log τ − (u) = g 4 cosh πgu 2 , τ (u) = tanh[ πg 4 u] , K Q (u) = 1 2πi d du log S Q (u) = 1 π g Q Q 2 + g 2 u 2 , S Q (u) = u − iQ g u + iQ g , K M N (u) = 1 2πi d du log S M N (u) = K M +N (u) + K N −M (u) + 2 M −1 j=1 K N −M +2j (u) , S M N (u) = S M +N (u)S N −M (u) M −1 j=1 S N −M +2j (u) 2 = S N M (u) . (A.4)
The fundamental building block of kernels which are not of difference type is:
K(u, v) = 1 2πi d du log S(u, v) = 1 2πi √ 4 − v 2 √ 4 − u 2 1 u − v , S(u, v) = x(u) − x(v) x(u)x(v) − 1 . (A.5)
Using the kernels K(u, v) and K Q (u − v) it is possible to define a series of kernels which are connected to the fermionic Y ± -functions. They are:
K Qy (u, v) = K(u − i g Q, v) − K(u + i g Q, v) , (A.6) K Qy ∓ (u, v) = 1 2 K Q (u − v) ± K Qy (u, v) (A.7)
and
K yQ (u, v) = K(u, v + i g Q) − K(u, v − i g Q), (A.8) K yQ ± (u, v) = 1 2 K yQ (u, v) ∓ K Q (u − v) . (A.9)
Further important kernels entering the Y ± related TBA-type equations are defined as follows:
K QM xv (u, v) = 1 2πi d du log S QM xv (u, v) , S QM xv (u, v) = x(u − i Q g ) − x(v + i M g ) x(u + i Q g ) − x(v + i M g ) x(u − i Q g ) − x(v − i M g ) x(u + i Q g ) − x(v − i M g ) x(u + i Q g ) x(u − i Q g ) × M −1 j=1 u − v − i g (Q − M + 2j) u − v + i g (Q − M + 2j)
.
(A.10)
The kernels entering the right hand sides of the equation (2.9) for Y 1 are
K QM vwx (u, v) = 1 2πi d du log S QM vwx (u, v) , S QM vwx (u, v) = x(u − i Q g ) − x(v + i M g ) x(u − i Q g ) − x(v − i M g ) x(u + i Q g ) − x(v + i M g ) x(u + i Q g ) − x(v − i M g ) x(v − i M g ) x(v + i M g ) × Q−1 j=1 u − v − i g (M − Q + 2j) u − v + i g (M − Q + 2j)
, (A.11) and the dressing-phase related kernel K QM sl(2) (u, v), which is built from the sl(2) Smatrix of the model [49]. It is of the form
S QM sl(2) (u, v) = S QM (u − v) −1 Σ QM (u, v) −2 , (A.12)
where Σ QM is the improved dressing factor [50]. The corresponding sl(2) and dressing kernels are defined in the usual way
K QM sl(2) (u, v) = 1 2πi d du log S QM sl(2) (u, v) , K Σ QM (u, v) = 1 2πi d du log Σ QM (u, v) . (A.13)
Explicit expressions for the improved dressing factors Σ QM (u, v) can be found in section 6 of ref. [50]. Here for our numerical computations we used the single integral representation given in [21].
Finally we mention that along the lines of [42] in the derivation of the formula (2.10) for Ω(K Q ), it was exploited that all the necessary kernels: K Q , K Qy , K Q1 xv , s ⋆ K Q−1,1 vwx , K y1 , K Q1 sl(2) satisfy the identity:
K Q − s ⋆ K Q−1 − s ⋆ K Q+1 ≡ δK Q = 0, for Q ≥ 3.
(A.14)
B Kernel matrices of the vertical HNLIE part
In this appendix the kernel matrices appearing in the upper HNLIE part of our equations (2.18,2.19) are presented. Here the kernel matrices are different compared to those published in [38]. The difference comes simply from a reformulation the equations in the language of new unknown functions. In [38] the unknowns are 6 b-type functions:
b old = {b(3)
C Asymptotic solutions of the vertical HNLIE
In this section along the lines of [38] the asymptotic solutions of the upper SU(4) NLIE variables are presented . In the asymptotic limit the T-hook of AdS/CFT splits into two SU(2|2) fat-hooks. The basic building blocks of the asymptotic solution are the nine Q-functions corresponding to the left and right SU(2|2) fathooks. Due to the left-right symmetry of the Y -system it is enough to give the right Q-functions. They can be derived from the asymptotic solution of the Y-functions given in [4]. They take the form: w o (u) = − i Λ 2 e −π g u 4 ((g u) 2 + w c ), y o (u) = i Λ 2 e −π g u 4 ((g u) 2 + w c − i C), (C.5) 28 Here for correspondence we use the same letters for the names of different unknowns as in [38].
Q
where w c and C are arbitrary constants. Using the building blocks listed above, the system can be obtained from the Bäcklund functions above by appropriately shifting their arguments: where * denotes complex conjugation. In our numerical studies we mostly use the C = 0 asymptotic solution to setup the equations to solve. In this case the exact equations guarantee the fulfillment of (2.22), which reduces to 6 the number of independent complex functions of the upper NLIE part.
b o = {b o a } = {b(3)
D Asymptotic solutions of the Y -system and the horizontal SU (2)-type HNLIE This appendix is devoted to give the asymptotic solution for the Y -functions and the variables of the horizontal SU(2) NLIE. The asymptotic form of the Y -functions can be read off from the asymptotic T -functions in [4]. They take the form: where b 0 (u) = 2 (g 2 u 2 − 3 i) (1 + 2 i g u + g 2 u 2 ) (g 2 u 2 + i) (g 2 u 2 − 2 i) (g 2 u 2 + 3 i) , (D.6) b 0 (u) = 2 (g 2 u 2 + 3 i) (1 − 2 i g u + g 2 u 2 ) (g 2 u 2 − i) (g 2 u 2 + 2 i) (g 2 u 2 − 3 i) , (D.7)
and 0 < γ < 1/2 is the arbitrary contour shift parameter of the horizontal SU(2) NLIE.
only even number of open strings. For this reason we study the minimal number of allowed open strings, a single pair, ending on our D-brane anti-D-brane (DD) system with open string angular momenta L and L ′ .
ground state the insertions are W = Z L and V = Z L ′ respectively. Based on one-loop results the planar dilatation operator is expected to act independently on the two words W, V corresponding to the open string states [4]: ∆[O W,V YȲ ] = ∆ bare [O W,V YȲ ] + δ∆[W YȲ ] + δ∆[VȲ Y ]. (1.5)
satisfies the constraint inequalities of[38] and satisfy the relation γ = −M η with M given by (C.21). Its advantage is that choosing the C = 0 asymptotic solution from appendix C to formulate the equations, the b-and d-type variables are related in a simple manner: b(−u) = Md(u).(2.22)
be seen that on the left hand side of (3.15) all the quantities are O(δy a ), while on the right hand side all the quantities are O(δy 2 a ). This separation allows us to define an iterative solution. If δy a s are small then the RHS is a small correction with respect to the LHS, this is why in an iterative solution the RHS can be simply taken at the value of the previous iteration.Let δy (n) a the value of δy a after the nth iteration, then δy solving a set of linear integral equations:
Figure 1 :
1The pictorial representation of the Y -system and the HNLIE structure with the choice p 0 = 4. singularities show up mostly in the SU(4) NLIE variables, thus by increasing the value of p 0 the appearance of such singularities in the equations can be postponed to higher values of g.
...] stands for integer part. The set of subinterval vectors A Q of [0, B(Q)] could also be given by an appropriate empirical formula, but it would take such a complicated form, that it is better to write down the requirements from which it can be constructed 18 . The requirements can be formulated in the language of the variable t ∈ [0, B(Q)]. The elements of the vector A Q divide the interval [0, B(Q)] into subintervals. Our requirements constrain the allowed length of the subintervals with respect their location within the whole interval [0, B(Q)]. The requirements are as follows:
( 2 .
232) of Y p 0 −2|vw , one can recognize that there are complex quantities on the RHS which do not form conjugate pairs. So, the reality of the LHS is not guaranteed by the form of the equations, but it is guaranteed by the form of the solution. Thus the second testing condition is expressed by the inequality:|Im log Y p 0 −2|vw | ≤ Numericalerror, 10 −6 Numerical error 10 −9 . (4.6) Here we wrote the typical numerical errors we had during the computations. The third test is based on the approximation scheme we use. One must check whether the Chebyshev coefficients of the unknowns decay as it is expected. From such a check the numerical precision of the method can be read off and it can shed light on some anomalous divergent behavior of the numerical solution. Thus it can indicate possible errors in the elimination of the divergent ln u terms in (2.2) and (2.36,2.37). The numerical results for the L = 2 case can be seen in figure 2 and table 1. In the table we show not only the energy E BT BA at different values of the coupling g, but the constant δc, as well. The other columns of the table are related to the solution of the linearized equations; E (0) BT BA and δc (0) are the energy and the global constant from the numerical solution of the linearized equations. ∆E
Figure 2 :
2(4.9) is a very important formula, because it plays crucial role in the analytical determination of the critical point. Since the numerical solution of the HNLIE equations of section 2. does not require the introduction of any index cutoff, it takes into account the contributions of all E BT BA (on the left) and δc (on the right) as functions of g for the L = 2 state.
Figure 3 :
3Large Q behavior of δF Q from numerical data at g = 2.16 with p 0 = 26.For the L = 2 state beyond g = 2.16 the numerical solution of the discretized problem did not give physically acceptable results. To get some insight into the source of the problems, at g = 2.18 we plotted the imaginary part of the LHS of the last equation in (2.18), namely Im log(1 + δb 6 ) at u = x p 0 (B(p 0 ) − t).
Figure 4 .
4shows that there is a jump of 2π, when t is close to B(p 0 ). (I.e. large u.)22
Figure 4 :
4The anomalous behavior of Im log(1 + δb 6 ) at g = 2.18 and p 0 = 26.
For generic L the Lüscher formula is simply the expansion of the TBA energy formula around the asymptotic solution with the replacement: log(1 + Y Q ) → Y o Q . For small coupling it
ln 4)+...) e 4 g 4 (log u) 2 +... . (5.8)
E c (L) = 1 − L. This point was interpreted in [4] as a transition point where the ground state becomes tachyonic. Approaching this critical point the contribution of all the Y -functions of the BTBA becomes quantitatively relevant, thus the numerical solution of the truncated BTBA equations cannot give accurate results close to the critical point. To resolve this difficulty and get more accurate numerical results we transformed the previously proposed BTBA equations into finite component HNLIE equations. The HNLIE equations were solved at different values of g and L and the numerical results confirmed the earlier BTBA data.
− 1 .
111 , q 22 and Λ are arbitrary constants which cancel from the final form of the asymptotic NLIE variables. Furthermore σ(u) = e The further building blocks of the asymptotic solution are as follows: of the recursionsw o− − w o+ = A o γ o+ γ o− , y o+ − y o− = B o β o β o−− , (C.4)are as follows:
shifts given in (2.20,2.21). Finally we note that the C = 0 choice implies a symmetry relation between the b-and d-type variables. Let M the 6 by 6 matrix: C = 0 the b o and d o vectors satisfy the relations as follows: d o (u) = Mb o (−u), b o (−u) = b o * (u), (C.22) d o (u) = Mb o * (u), d o (−u) = d o * (u), (C.23)
Y o m|vw (u) = m(m + 2) g 2 u 2 (m + 1) 2 + g 2 u 2 , m = 1, 2, ...(D.1) lines of [38] the asymptotic horizontal SU(2) NLIE variables can be determined from the asymptotic Q-functions (C.1). Here we just list the final formulas: b o (u) = b 0 (u − i γ),b o (u) =b 0 (u + i γ), (D.5)
Table 2 :
2Numerical data for the L = 3 state.g
E BT BA
δc
E
Table 3 :
3Numerical data for the L = 4 state.
In our terms the lack of smoothness would not mean discontinuity, but the presence of rapidly changing parts and peaks.
This slight modification is only to write the approximation series (3.2) in a more compact way.12 The Chebyshev polinomials are defined by the formula: T j (u) = cos(j arccos u), j = 0, 1, 2...
For example in the TBA-part F (u) can be thought of as log Y (u) and L(u) can be log(1+Y (u)) for any type of Y .
For repeated indexes summation is understood.
Not to have very small subintervals: ∆t 0.1.20 If one experiences that the numerical solution of the linearized problem agrees with the numerical value of (4.5) within certain digits of precision, than the deviation from the Lüscher result can be a good starting estimate to the numerical error. One cannot expect better accuracy, but the precision will not become much worse either.
For large Q.
Here the sampling points are connected according to the Chebyshev approximation. This is why the jump of the logarithm is not "sharp".
Here we use the rapidity convention where the branch points are at ±2g.
This primarily means that log Y Q ∼ log |u| for large u, while other Y -functions tend to constant.
ψ(z) = d dz log Γ(z).
In section 2. it is denoted by p 0 , here the notation s is kept to fit to formulas of[38].
AcknowledgementsThe author thanks Nadav Drukker, László Palla and János Balog for useful discussions. This work was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and OTKA K109312. The author also would like to thank the support of an MTA-Lendület Grant, and the Hungarian-French bilateral grant TÉT-12-FR-1-2013-0024. Finally, the author appreciates APCTP for its hospitality where part of this work was done., d with shift vectors γ and η given by(2.20,2.21). We recognized that the kernels become simpler if we formulate the equations in terms of the unknowns:where η = ±1 is a global sign factor and s = p 0 , if one adopts the notation of[38]The kernel matrices can be expressed by the functions as follows27:and they take the form:(B.12) asymptotic form of the upper SU(4) NLIE functions can be determined[38]and take the form: φ(u) = x(u) 2L g u , (C.18)and C is the arbitrary constant that does not cancel from the formula for the HNLIE variables. The asymptotic solution for the six b-and d-type NLIE-functions of the
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| []
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[
"MBM 12 and MBM 16 distances",
"MBM 12 and MBM 16 distances"
]
| [
"J Knude \nNiels Bohr Institutet\nCopenhagen University\nJuliane Maries Vej 30DK-2100København ØDenmark\n",
"H E P Lindstrøm \nNiels Bohr Institutet\nCopenhagen University\nJuliane Maries Vej 30DK-2100København ØDenmark\n\nCSC Danmark A/S\nRetortvej 8DK-2500ValbyDenmark\n"
]
| [
"Niels Bohr Institutet\nCopenhagen University\nJuliane Maries Vej 30DK-2100København ØDenmark",
"Niels Bohr Institutet\nCopenhagen University\nJuliane Maries Vej 30DK-2100København ØDenmark",
"CSC Danmark A/S\nRetortvej 8DK-2500ValbyDenmark"
]
| [
"Mon. Not. R. Astron. Soc"
]
| Among the multitude of intrinsic SDSS index vs. index diagrams the (g−r) vs. (r− i) diagram is characterized by showing only minor (g − r) variation for the M dwarfs. The (g − r) vs. (r − i) reddening vector has a slope almost identical to the slope of the main sequence earlier than ≈M2, meaning that dwarfs later than ∼M2 are not contaminated by reddened dwarfs of earlier type. Chemical composition, stellar activity and evolution have only minor effects on the location of the M2−M7 dwarfs in the (g − r) vs. (r − i) diagram implying that reddening may be isolated in a rather unique way. From r, M r,(r−i)0 and E g−r we may construct distance vs. A r diagrams. This purely photometric method is applied on SDSS DR8 data in the MBM 12 region. We derive individual stellar distances with a precision ≈20−26%. For extinctions in the r − band the estimate is better than 0.2 mag for ≈ 67% and between 0.3 and 0.4 for the remaining ≈ 33%. The extinction discontinuities noticed in the distance vs. A r diagrams suggest that MBM 12 is at ≈160 pc and MBM 16 at a somewhat smaller distance ≈100 pc. The distance for which ∆(A r )/σ(∆(A r )) = 3, where ∆(A r ) refers to A r,on -A r,of f , may possibly be used as an indicator for the cloud distance. For MBM 12 and 16 these distance estimates equal 160 and 100 pc, respectively | null | [
"https://arxiv.org/pdf/1202.3600v1.pdf"
]
| 118,664,990 | 1202.3600 | 1f303e0d629982abb3e5ace3b6d58ae0b9fbcb18 |
MBM 12 and MBM 16 distances
2012
J Knude
Niels Bohr Institutet
Copenhagen University
Juliane Maries Vej 30DK-2100København ØDenmark
H E P Lindstrøm
Niels Bohr Institutet
Copenhagen University
Juliane Maries Vej 30DK-2100København ØDenmark
CSC Danmark A/S
Retortvej 8DK-2500ValbyDenmark
MBM 12 and MBM 16 distances
Mon. Not. R. Astron. Soc
0002012Accepted 2012 Month 99. Received 2012 Month 99; in original form 2012 Month 99(MN L A T E X style file v2.2)molecular clouds -interstellar extinction -distances : M dwarf stars
Among the multitude of intrinsic SDSS index vs. index diagrams the (g−r) vs. (r− i) diagram is characterized by showing only minor (g − r) variation for the M dwarfs. The (g − r) vs. (r − i) reddening vector has a slope almost identical to the slope of the main sequence earlier than ≈M2, meaning that dwarfs later than ∼M2 are not contaminated by reddened dwarfs of earlier type. Chemical composition, stellar activity and evolution have only minor effects on the location of the M2−M7 dwarfs in the (g − r) vs. (r − i) diagram implying that reddening may be isolated in a rather unique way. From r, M r,(r−i)0 and E g−r we may construct distance vs. A r diagrams. This purely photometric method is applied on SDSS DR8 data in the MBM 12 region. We derive individual stellar distances with a precision ≈20−26%. For extinctions in the r − band the estimate is better than 0.2 mag for ≈ 67% and between 0.3 and 0.4 for the remaining ≈ 33%. The extinction discontinuities noticed in the distance vs. A r diagrams suggest that MBM 12 is at ≈160 pc and MBM 16 at a somewhat smaller distance ≈100 pc. The distance for which ∆(A r )/σ(∆(A r )) = 3, where ∆(A r ) refers to A r,on -A r,of f , may possibly be used as an indicator for the cloud distance. For MBM 12 and 16 these distance estimates equal 160 and 100 pc, respectively
INTRODUCTION
MBM 12 is a high latitude molecular cloud and since it has AV exceeding 5 mag, Hearty, Neuhäuser, Stelzer et al. (2000), it is classified as a dark cloud. And it contains more than a dozen PMS stars. An active high latitude cloud requires possibly other mechanisms for star formation than dark clouds close to the galactic plane. Estimating parameters essential for initiating star formation, such as the cloud mass and density, depends on the cloud distance. We include MBM 16, a high latitude neighboring tranclucent cloud, only ten degrees removed from MBM 12, and showing no star formation. For a recent review of high latitude molecular clouds McGehee (2008) may be consulted.
The MBM 12 distance has a long long history and a corresponding, large variation. Shortly after its inclusion in the MBM catalog Hobbs, Blitz, and Magnani (1986) suggested ∼65 pc as derived from NaI spectroscopy and spectroscopic distances of a small sample of stars. Hobbs et al. (1988) suggest the same distance for MBM 16 about ten degrees away from MBM 12. The estimate for MBM 12 was altered by Hearty, Neuhäuser, Stelzer et al. (2000), after Hipparcos parallaxes became available, to the range from ⋆ E-mail: [email protected] (JK); [email protected] (HL) 58 to 90 pc. Again lower and upper estimates were based on the absence/presence of NaI absorbtion. Luhman (2001) and Andersson et al. (2002) have used more indirect methods and prefer a cloud distance of 275 pc and 360±30 pc, respectively. Both of these papers do, however, detect dust, thought not to be associated to MBM 12, at 65, 140 pc and at ∼80 pc respectively. An intermediary distance, 325 pc, is proposed by Straižys et al. (2002) from Vilnius photometry of dwarfs and giants brighter than V≈12 mag. That work also indicates the possible presence of a small hump, AV ≤ 0.4 mag, of extinction at 140−160 pc.
Accurate and homogenous SDSS griz photometry has proven useful for deriving distances and extinction estimates for stars in the M dwarf range within ≈1 kpc (e.g. Jones, West and Foster 2011). Our own interest in using gri photometry for 3-D mapping of the ISM was inspired by the stellar models by Girardi et al. (2004). These models convinced us that the position of the (g − r)0 − (r − i)0 locus was influenced very little by variation in [M/H], and by ages varying from a few million years to the age of the Milky Way. The latter invariance was perhaps to be expected. Furthermore, and most important, the reddening ratio Er−i/Eg−r has a value, Girardi et al. (2004), that approximates the slope of the main sequence, in the (g − r) − (r − i) diagram, for dwarfs earlier than ∼M1. A consequence is, that red- dening seems to be the only parameter that shifts observed (g − r), (r − i) pairs away from the intrinsic locus. Another fact that makes the gri photometry so useful for M dwarfs is the good relation, σM r ≈0.4 mag, between (r − i)0 and the absolute magnitude Mr: a range of ≈ 2 mag in (r − i)0 corresponds to a range of ≈9 mag in Mr. This is taken from Table 2 of West et al. (2011) together with the equation for M r,(r−i) 0 , Table 4 of Bochanski, Hawley, Covey et al. (2007) valid for the 0.62 − 2.82 range of (r − i)0. The absolute magnitude calibration have required precise distances for a represensative sample, Table 4.1, . We apply the photometric parallax and investigate the variation of extinction with distance in a region with a radius 15 • centered on the high latitude, dark cloud MBM 12. This region is known to contain several clouds with molecular gas, e.g. the extended translucent cloud MBM 16. The locations of these clouds are interesting because of their various state of activity and their position relative to the confinement of the local cavity.
M DWARF SAMPLE SELECTION FROM SDSS DR8
To establish a sample of M dwarfs we follow a selection procedure in line with Jones, West and Foster (2011). The candidate sample is drawn from the most recent SDSS data release, DR8, applied for a region, centered on the MBM 12 position (l, b) ∼ (159. o 4, −34. o 3) with a 15 • radius, as shown in Fig. 1. The coverage is incomplete but several MBMs are partially scanned. Using the DR8 CasJobs tool we queried for STAR objects with CLEAN photometry for g, r, i and z, respectively, and fulfilling the selection criteria: σg,r,i < 0.06 mag, (r − i) > 0.53 mag and (i − z) > 0.3 mag. The cuts in (r − i) and (i − z) contribute to minimize the contamination by (partially) eliminating quasars, giants and M flare stars. Aspects of the sample are shown in Fig. 2. Notice how the reddening of the main sequence stars earlier than ∼M1 does not contaminate the location of reddened M dwarfs. Table 2 given as the median color of each spectral type M 0, ..., M 9. The slope of the cut off equals the reddening ratio Girardi et al. Girardi et al. (2004), indicated by the straight line Red symbols are for the MBM 12 area exclusively. The vertical line at 160 pc indicates our suggested MBM 12 location. For comparison the black points are the data, with A V converted to Ar, on which a MBM 12 distance of 325 pc was based, Straižys et al. (2002). The blue curve is Ar from Hipparcos data
E r−i /E g−r =0.694,
EXTINCTION AND DISTANCE ESTIMATION. UNCERTAINTY
The linchpin for our work is the (r
− i) vs. (g − r) diagram, which is characterized by showing only minor (g − r)0 varia- tion for M dwarfs. The (r−i) vs. (g−r)
reddening vector has a slope almost identical to the slope of the main sequence earlier than ∼M1-M2 meaning that reddened dwarfs earlier than this limit do not contaminate the location of reddened dwarfs later than ∼M2. See Fig. 2 where the M dwarf standard locus starts at M0. As judged from the evolutionary models by Girardi et al. (2004) the chemical composition and evolution have minor effects on the location of the M dwarfs in the (r − i)0 vs. (g − r)0 diagram. According to Bochanski et al. (2007) (g − r) colors are likely to depend on metallicity but no clear trends are apparent for (r − i) and (g − r). M dwarf activity in the form of flares mostly effects the blue part of the spectrum, i.e. the u and the g bands implying that g − r is affected but not r. According to West et al. (2008) a flare will typically change g−r to values below 0.05 mag. As Fig. 2 shows the g − r range we consider is far redder than this and Eg−r >1.4 or Ar >4 is required to shift a flaring star into the color range we use. For the variation in Mr,r−z with magnectic activity and metallicity see, however, the discussion by Bochanski, Hawley, West (2011). A more serious contamination can be caused by unresolved binarity and depends on the components mass ratio. With a mass ratio of one the colors do not change but the observed r magnitude is decreased by 0.75 mag. The estimated Mr is not influenced since the (r − i) and (g − r) colors are left unchanged. For an individual target, binarity introduces a distance uncertainty ≈35% with a unit mass ratio. The estimated extinction is accordingly not altered but its location is shifted to a larger distance. A recent paper, Clark, Blake and Knapp (2012) has estimated the fraction of close, a<0.4 AU, M dwarf binaries to 3−4%. Dwarfs later than M6 has a frequency of 20±4%, Allen (2007) and the overall binary frequency among M dwarfs is 42±20%, Fischer and Marcy (1992). In Fig. 2 is shown three aspects of the (r − i) vs. (g − r) diagram for the MBM 12 region sample. The black points are a selection the region stars. The green symbols are M dwarfs in the MBM 12 area. Blue symbols signify M dwarfs in the MBM 16 area. True color excesses are positive but due to observational errors in (r − i) and (g − r) unreddened and little reddened stars are sometimes shifted to the blue of the (r − i)0 vs. (g − r)0 locus. In Fig. 2 we have indicated the one sigma confinement based on the maximum error 0.060 mag in g, r, i. This confinement is corroborated by the scatter around the standard locus of M dwarfs observed at the virtually unreddened North Galactic Pole.
Estimate of the Extinction Ar
As the sample of nearby M dwarfs, forming the basis for the standard locus, shows there is a scatter around the standard locus even for vitually unreddened stars. But collapsing the main sequence to a sharp relation has often proven useful. In a previous section we have argued that a shift from the locus may mainly depend on reddening. The intrinsic location of a dwarf is on the locus and is determined by translating the observed position along a reddening vector. The color shifts ∆(g −r) and ∆(r−i) are accordingly assumed to equal Eg−r and Er−i respectively.
E r−i E g−r =0
.694 is adopted from Girardi et al. (2004) calculated with AV = 0.5 mag, R=3.1, T ef f =3500 K, log(g)=4.5 and [M/H]=0. Not ideal but rather close to the M range.
We have Ar=2.875×Eg−r, also adopted from Girardi et al. op. cit. For (r − i) the relation is Ar=4.142×Er−i. The ratio between the two coefficients is 1.441 so other things being equal Eg−r is preferred due to a more favorable error progression.
Estimated Distance
Having estimated Ar and observed r only Mr is missing for the photometric parallax. We rely on the calibration of Mr in terms of (r − i)0 which is preferred to (g − r)0 due to a better error progression. The Mr calibration of (r − i)0 for M dwarfs is as mentioned adopted from Bochanski, Hawley, Covey et al. (2007).
The distance estimate is derived from the usual formula:
distance = 10 0.2 * (r−Mr −Ar +5)(1)
where all parameters are known together with their uncertainties.
Uncertainties on Extinction and Distance
Each entry in the extraction from SDSS DR8 contains errors on all magnitudes but we have preselected stars with σgri < 0.060 mag. We may estimate the total uncertainty from the combination of observational errors and errors introduced from the calibrations.
If s = f (xi) represents either the full set of equations used to estimate Ar or Mr the formal error of s follows from the progression formula:
σ 2 s = Σi( ∂s ∂x i σx i ) 2(2)
For each star we compute ∂s ∂x i and σx i implying that derivatives must be calculated and errors of the independent parameters also must be known. The errors σ (g−r) 0 and σ (r−i) 0 are derived from piecemeal linear approximations to the intrinsic locus, taking into account errors in slope and intersection, together with the observational errors in g, r and i. σM r is calculated from the calibration equation (1) considering our individual, estimated errors σ (r−i) 0 .
The resulting uncertainties are in the range 20−26% for the distances and for Ar about 2/3 has σA r in the range from 0.04 to 0.20 mag and about 1/3 between 0.30 and 0.40. The reason for this double peaked distribution is the kink in the intrinsic locus noticed in Fig. 2
RESULTS
SDSS DR8 provides a substantial amount of M dwarf data for the MBM 12 region and the distance and Ar accuracies are adequate to study the distance -extinction variation.
MBM 12 Distance Estimate
The distance to MBM 12 is particularly interesting because the cloud is a specimen of a rare variety, high latitude, dark cloud (AV > 5 have been measured) that even shows star formation activity. Figure 4. The ∆Ar σ(∆Ar ) ratio for the MBM 12 dark cloud. Points are from 30 pc distance bins, stepsize 10 pc. Neigboring points thus not independent. The curve is a Bezier smoothing of the data. The ratio equal 3 at 160 pc which is proposed as the cloud distance Figure 5. Extinctintion vs. distance for the MBM 16 area. Note that beyond ∼200 pc most stars have Ar >0.3 mag. The vertical line indicates that the distance of the first dust is at 102 pc. The MBM 12 distance at 160 pc is also indicated. Black triangles are Hipparcos data for the MBM 16 area with Ar extinctions. The star HIP 14997 was left out, it has an uncertain B − V and is listed as a variable, Koen and Eyer (2002) 4.1.1 Distance from appearence of substantial extinction By substantial extinction we mean an extinction that appears in a discontinuity and is substantially larger than the extinction at smaller distances and that several stars do show such an extinction.
From the Mr luminosity calibration of (r − i)0 and the color excess from (g − r) − (g − r)0 in Fig. 2 we construct a distance vs. Ar diagram for the MBM 12 area, as shown by the red dots in Fig. 3. We have used a logarithmic distance scale to emphasize the smaller distances. Apparently there is a rise in Ar beyond ∼0.5 mag around 100 pc. We have shown a vertical line at 125 pc. There are some MBM 12 stars with extinction ≈1 mag at ≈125 pc but the dominating increase takes place at ∼162 pc indicated by a vertical line too. Since this distance is somewhat smaller than the presently preferred range discussed in the introduction we have compared to the distance vs. extinction variation derived from the Hipparcos Catalog. We have extracted Hipparcos stars in the MBM 12 area, same l and b limits as used for the SDSS extraction. Parallaxes are from the second derivation by van Leeuwen (2007) and extinctions are from B, V photometry and spectral classification, see e.g. Knude and Høg (1998). The extinction is averaged in 30 pc intervals and is given as the blue solid curve in Fig. 3. The Hipparcos curve is seen to follow the upper envelope of the MBM 12 extinction rather well. Since the Hipparcos sample has a rather bright limiting magnitude, Perryman, Lindegren, Kowalevsky et al. (1995), we can only expect to see smaller extinctions. If we accept the distance of the onset of extinctions beyond, say one magnitude, which is ≈3σA r above 0, as the cloud distance, MBM 12 is at 160 pc.
Alternative distance derivation for MBM 12
The scans in Fig. 1 cover several clouds revealed by their color excesses but also sight lines with less dust. One may expect that for a small distance bin at a given distance the average extinctions in a cloud direction and outside the clouds differ. For distances less than the cloud distance, but in the direction of a cloud, the two averages will be more similar. As Fig. 3 shows there is a substantial scatter of Ar at almost any distance. A scatter caused by real variation in the presence of dust causing the extinction and the observational errors, σ 2 total = σ 2 ISM + σ 2 obs . If Ar,on and A r,of f designate the average extinctions on and off a cloud for identical distance bins we propose that ∆(Ar)=Ar,on-A r,of f will measure the presence of a cloud at a given distance where the difference is sufficiently large. We use σ 2 (∆(Ar)) = σ 2 total,on + σ 2 total,of f as a measure of the significance of ∆(Ar) and define the cloud distance as the distance when ∆(Ar)/σ(∆(Ar)) equals three.
We have done this for the MBM 12 area and compared to the region outside the MBM 12 area as shown in Fig. 4. The individual points are not independent: bin size is 30 pc with a steplength of only 10 pc. The horizontal line is at three and intersects the ∆/σ curve at 160 pc. Which we accept as the distance to MBM 12.
The distance to MBM 16
Another cloud in the MBM 12 region is MBM 16 which is of different type than MBM 12. MBM 16 is translucent whereas MBM 12 is a dark cloud. The two types are distinguished by their optical extinction: a dark cloud has AV > 5 mag whereas a translucent cloud is less opague with AV in the range from 1 to 5, van Dieshoeck and Black (1988). MBM 16 has been studied by Magnani, Chastain, Kim et al. (2003) in order to understand the origin of turbulence and the correlations between molecular gas and color excesses. For their investigation a distance of 100 pc was assumed, identical to the distance to the wall of Local Bubble as estimated by Cox and Reynolds (1987). In their probing of the local low density cavity Lallement, Welsh, Vergely et al. (2003) suggest the presence of a region with a high density at a distance closer than 100 pc in the direction of MBM 16.
In Fig. 5 is shown SDSS for what is available inside the area shown in Fig. 1. Hipparcos 2 results from the same area is overplotted and two vertical lines at 115 and 160 pc respectively. From the occurence of the first dust, Ar ranging from ≈0.5 to ≈2 mag, we would suggest ∼115 pc. Smaller than the MBM 12 distance of 160 pc. The Ar data from Hipparcos 2 corroborates this to some degree, but only by two stars with Ar in the range from 0.5 to 1 mag.
The ∆(Ar)/σ(∆(Ar))=3 criterion indicates a distance 100 pc for MBM 16 as shown in Fig. 6.
CONCLUSIONS
We have applied two different methods, which perhaps could be termed qualitative and quantitative, respectively, for estimating the distance to the two high latitude clouds MBM 12 and 16: the distance at which substantial extinction is first measured and the distance where the ratio ∆(Ar)/σ(∆(Ar)) equals three. In the case of MBM 12 and 16 the two methods agree. Our suggested distances are 160 for MBM 12 and 100 pc for MBM 16. The former does not agree with the current values from the literature whereas the latter does, almost exactly. That either method works depend on the nice behaviour of the M2−M7 dwarfs in the (g − r) vs.(r − i) diagram.
The difference of the MBM 12 and 16 distances has been narrowed from ≈350-100 pc to 160-100 pc. The possibility that MBM 12 is outside the confinement of the local bubble and MBM 16 is on or inside still exist. Tempting to suggest that their dark/translucent status is a consequence of their different interstellar environment?
The ∆(Ar)/σ(∆(Ar)) = 3 criterion may possibly be expanded to a generalized method for locating nearby molecular clouds where griz photometry is available and maps, 2D and 3D, could be produced with a larger number of stars than used in the spectroscopic study by Jones, West and Foster (2011).
-
Figure 2 .
2(r − i) vs. (g − r) diagram. Black points are a selection of stars from the region in Fig. 1. The green symbols are M dwarfs selected in the MBM 12 area. Blue symbols are M dwarfs selected in the MBM 16 area. Notice that the earliest M dwarfs among the MBM 16 stars are shifted to the red of the standard locus which may indicate the presence of some very local dust. The solid jagged line is the standard (g −r) 0 -(r −i) 0 locus from West et al. (2011)
Figure 3 .
3Distance vs. extinction diagram for the MBM 12 area.
Figure 6 .
6The ∆Ar σ(∆Ar) ratio for the MBM 16 translucent cloud. The diagram proposes a cloud distance of ∼100 pc significantly shorter than the ∼160 pc suggested for MBM 12
Figure 1. MBM 12 region. SDSS DR8 M2 -M7 dwarfs with σ gri <0.060 mag and less than 15 • from the nominal center of MBM 12. The MBM 12 and MBM 16 areas area outlined. The resulting color excesses are color coded50
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c 2012 RAS
c 2012 RAS, MNRAS 000, 1-5
This paper has been typeset from a T E X/ L A T E X file prepared by the author. c 2012 RAS, MNRAS 000, 1-5
ACKNOWLEDGMENTSOur investingation of the Milky Way ISM is financily supported by FNU, grant 09-060601, and Fonden af 29. December 1967.Funding for the Sloan Digital Sky Survey (SDSS) and SDSS-II has been provided by the Alfred P. Sloan Foundation, and the Participating Institutions.
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| []
|
[
"A New Random Coding Technique that Generalizes Superposition Coding and Binning",
"A New Random Coding Technique that Generalizes Superposition Coding and Binning"
]
| [
"Stefano Rini [email protected] \nLehrstuhl für Nachrichtentechnik Technische\nUniversität Müchen\nArcisstraße 2180333MünchenGermany\n"
]
| [
"Lehrstuhl für Nachrichtentechnik Technische\nUniversität Müchen\nArcisstraße 2180333MünchenGermany"
]
| []
| Proving capacity for networks without feedback or cooperation usually involves two fundamental random coding techniques: superposition coding and binning. Although conceptually very different, these two techniques often achieve the same performance, suggesting an underlying similarity. In this correspondence we propose a new random coding technique that generalizes superposition coding and binning and provides new insight on relationship among the two With this new theoretical tool, we derive new achievable regions for three classical information theoretical models: multi-access channel, broadcast channel, the interference channel, and show that, unfortunately, it does not improve over the largest known achievable regions for these cases. | null | [
"https://arxiv.org/pdf/1202.0959v2.pdf"
]
| 17,173,752 | 1202.0959 | e9e4ea4cfc14081223fbf88a2c198c1a1cefd56d |
A New Random Coding Technique that Generalizes Superposition Coding and Binning
7 Feb 2012
Stefano Rini [email protected]
Lehrstuhl für Nachrichtentechnik Technische
Universität Müchen
Arcisstraße 2180333MünchenGermany
A New Random Coding Technique that Generalizes Superposition Coding and Binning
7 Feb 2012Index Terms-random codingsuperposition codingbinningmulti access channelbroadcast channelinterference channel
Proving capacity for networks without feedback or cooperation usually involves two fundamental random coding techniques: superposition coding and binning. Although conceptually very different, these two techniques often achieve the same performance, suggesting an underlying similarity. In this correspondence we propose a new random coding technique that generalizes superposition coding and binning and provides new insight on relationship among the two With this new theoretical tool, we derive new achievable regions for three classical information theoretical models: multi-access channel, broadcast channel, the interference channel, and show that, unfortunately, it does not improve over the largest known achievable regions for these cases.
I. INTRODUCTION
Apart from few notable exceptions [1], [2], the capacity region for a general multi-terminal network is shown using random coding techniques such as rate splitting, time sharing, superposition coding, binning, Markov encoding, quantize, and forward and few others. For networks with no feedback or cooperation, the two random coding techniques are usually considered when proving capacity: superposition coding and binning. Superposition coding can be intuitively be thought of as stacking codewords on top of each other [3] and is obtained by generating the codewords of the "top" codebook conditionally dependent on the "base" codeword. A typical representation of this encoding technique is the one Fig. 1 [3,Fig 4] where the codeword U N 2 is superposed to the codewords U N 1 . A codeword U N 1 in the base codebook is randomly selected from the typical set T N ǫ (P U1 ). For each base codeword, a top codebook is generated by selecting random elements from the typical set T N ǫ P U2|U1 . Superposition coding is often thought of as placing spheres, or clouds, in the typical set T N ǫ (P U1U2 ) : the base codewords are "cloud centers" while top codewords are "satellite codewords". If the number of spheres is small enough-a low rate for the base codewords-and their size is small enough-a low rate for the top codewords-then codewords are sufficiently spaced apart to allow successful decoding.
Binning 1 allows a transmitter to "pre-cancel" (portions of) the interference experienced at a receiver. A usual representa-1 sometimes referred to as Cover's random binning [4] or Gel'fand-Pinsker coding [5]. tion of binning [6,Fig. 14.19] is the one in Fig. 2: here the codewords U N 2 is binned against U N 1 . The codebook of U N 1 is generated as in superposition coding while the codewords for U N 2 and are selected from the typical set T N ǫ (P 2 ) and placed in bins. Codewords in the same bin are associated with the same message, i.e. multiple codewords can be used to communicate the same message. The codeword U N 2 in the bins are selected for transmission when it belongs to the typical set T N ǫ (P U1,U2 ), even though generated independently from U N 1 . It is possible to find a codeword that satisfies this condition if the size of each bin is sufficiently large. Binning is commonly interpreted as dividing the typical set T N ǫ (P U1,U2 ) in the partitions formed by the bins in which the codewords U N 1 and U N 2 are placed. Encoding is successful when the size of the bins is sufficiently large-large binning rate-while decoding is successful if the transmitted codewords are sufficiently far apart-low message rate. In certain cases, it is possible to simultaneously bin two codewords against each other: this coding technique is usually referred to as joint binning.
Despite the difference in this two random coding techniques, in many cases they have identical performance [7, Sec. VI]: this suggests an underlying similarity in the way the two techniques select the codewords to be transmitted. To gain a better understanding of the properties of theses strategies, we develop a new random coding technique that encompasses superposition coding and binning as special cases. In this scheme codewords are first superposed according to a cer- tain distribution, the codebook distribution, and successively binned to appear as if generated according to a different distribution, the encoding distribution. Classical superposition coding corresponds to the case where the binning distribution is the same as the codebook distribution while binning is obtained when the codebook distribution has independent codewords. All the strategy in between these two cases have never been previously considered in literature. We use this new random coding technique to derive achievable regions for the multi-access channel, the broadcast channel, and the interference channel. Unfortunately these new achievable regions do not improve on the largest known achievable regions for the broadcast channel and the interference channel but show that these regions can be obtained with a wider set of encoding strategies than what was previously.
Paper Organization: Section II introduces the a new random coding techniques that generalizes superposition coding and binning. Section III presents new achievable regions for classical communication models. Section IV concludes the paper.
II. COMBINING SUPERPOSITION CODING AND BINNING
We introduce the new random coding technique that generalizes superposition coding and binning with a simple example. Consider a classical Broadcast Channel with a common message (BC-CM) where two messages W 1 and W 2 are encoded at transmitter 1, message W 2 is decoded at receiver 2 while message W 1 is decoded both decoder 1 and decoder 2. The channel outputs (Y 1 , Y 2 ) are obtained from the channel input X from the channel transition probability P Y1,Y2|X .
Take any distribution for the codebook generation and the encoding procedure
P codebook = P U c 1 U c 2 , P encoding = P U e 1 U e 2 ,(1)
and let w i ∈ 1 . . . 2 N Ri be the messages W i to be transmitted for i ∈ {1, 2}.
• Codebook Generation 1) Generate 2 L1 codewords U N 1 with N iid draws from the distribution P U c 1 . Index these codewords as
U N 1 (w 1 , b 1 ) for b 1 ∈ 1 . . . 2 R1 with L 1 = R 1 + R 1 . 2) For each U N 1 (w 1 , b 1 ) , generate 2 N L2 codewords U N 2 with N iid draws from the distribution P U c 2 |U c 1 . Index these codewords as U N 2 (w 2 , b 2 , w 1 , b 1 ) for b 2 ∈ 1 . . . 2 R 2 with L 2 = R 2 + R 2 • Encoding Procedure For each message set (w 1 , w 2 ), choose the bin indexes b 1 and b 2 so that U N 1 (w 1 , b 1 ), U N 2 (w 2 , b 2 , w 1 , b 1 ) ∈ T N ǫ (P U e 1 U e 2 ). (2)
If no such set (b 1 , b 2 ) exists, pick two indexes at random. Generate the channel input X as a deterministic function of the Random Variables (RVs) U 1 and U 2 .
• Decoding Procedure 1) Decoder 1 looks for a set of indexes w 1
1 , b 1 1 such that Y N 1 , U N 1 ( w 1 1 , b 1 1 ) ∈ T N ǫ (P Y1,U e 1 )
2) Decoder 2 looks for a set of indexes
w 2 1 , w 2 2 , b 2 1 , b 2 2 Y N 2 , U N 1 ( w 2 1 , b 2 1 ), U N 2 ( w 2 1 , b 2 1 , w 2 2 , b 2 2 ) ∈ T N ǫ (P Y2U e 1 U e 2 )
To determine the performance of the achievable scheme above we need to determine the encoding and decoding error probabilities.
The key techniques in bounding the encoding error is related to the probability that there exists a random vector U N 1 in the typical set of certain distribution T N ǫ (P U2 ), that also belongs to the typical set of a different distribution T N ǫ (P U2 ). The probability of this event can be bounded using [
∈ T N ǫ (P U1 ), then P N U2 (u N 1 ) = exp {−N (D(P U1 ||P U2 ) + H(U 1 )}
The quantity
D(P U1 ||P U2 ) + H(U 1 ) = − u1 P U1 (u 1 ) log(P U2 (u 1 ))
is referred to as inaccuracy and it can be used to derive a more general version of the covering lemma [4].
Theorem II.2. Generalized Covering Lemma
Consider the set of RVs U 1 and U 2 with distribution P U1U2 and take
2 N R i.i.d. sequences u N 2 from the typical set T N ǫ (P U2 ) , indexed as u N 2 (i), i ∈ {1 . . . 2 nR }, then there exists δ(ǫ) → 0 as ǫ 0 such that P ∃ i ∈ {1 . . . 2 nR }, s.t. u N 2 ( i) ∈ T N ǫ (P U1 ) → 1, as N → ∞ if R ≥ D(P U1 ||P U2 ) + δ(ǫ).
Proof: The complete proof is provided in [9]. With Th. II.2 we derive the achievable region of the proposed random coding strategy.
Corollary II.3. An Achievable Region for the BC-CM The following region is a achievable for a general BC-CM
R 1 + R 2 ≥ D(P e U1,U2 ||P c U1,U2 ) (3) L 1 ≤ I(Y 1 ; U e 1 ) + D(P U e 1 ||P U c 1 ) L 2 ≤ I(Y 2 ; U e 2 |U e 1 ) + D(P U e 2 ||P U c 2 |P U e 1 ) L 1 + L 2 ≤ I(Y 2 ; U e 1 , U e 2 ) + D(P U e 1 U e 2 ||P U c 1 U c 2 ), for L i = R i + R i for i ∈ {1,
2} and any distribution of P codebook and P encoding in (1).
Proof:
The complete proof is provided in [9]. • Superposition coding: is obtained by having the distribution imposed at encoding P encoding equal the distribution of the codebook P codebook , i.e.
P U e 1 U e 2 = P U c 1 U c 2 .(4)
In this case the binning rates R 1 and R 2 can be set to zero, thus obtaining the region
R 1 ≤ I(Y 1 ; U c 1 ) R 2 ≤ I(Y 1 ; U c 2 |U c 1 ) R 1 + R 2 ≤ I(Y 1 ; U c 1 , U c 2 ),(5)
• Joint Binning: corresponds to the case where the distribution of the codebook P codebook equals the product of the marginals of encoding distribution:
P U c 1 U c 2 = P U e 1 P U e 2 .(6)
which results in the region
R 1 + R 2 ≥ I(U e 1 ; U e 2 ) L 1 ≤ I(Y 1 ; U e 1 ) L 2 ≤ I(Y 2 ; U e 2 |U e 1 ) L 1 + L 2 ≤ I(Y 2 ; U e 1 , U e 2 ) + I(U e 1 ; U e 2 ),(7)
• Binning: The region with binning is obtained from the region with joint binning of above by setting either R 1 or R 2 to zero.
III. ACHIEVABLE REGIONS FOR CLASSICAL CHANNEL
We apply the new random coding technique in Sec. II to the Multi-Access Channel with Common Messages (MAC-CM) [10], [11], the Broadcast Channel (BC) [3], [12], [13] and the InterFerence Channel (IFC) [14], [15]. Capacity is known for the MAC-CM and for a subsets of both the BC and the IFC. The largest achievable regions in each case can be achieved by employing a combination of rate splitting,superposition coding and binning [7, Sec. VI]. In the following we adopt the notation in [7] to describe the channel model and distribution of messages and codewords. In particular, the codeword U N i j , with rate R i j , encodes the messages W i j from the i set of transmitters to the j set of receivers.
A. The Multi-Access Channel with Common Messages
In the classical MAC [10], [11], two transmitters communicate a message each to a single decoder. In the MAC-CM an additional common message is transmitted by each source to the decoder [16]. Let U N i 1 be the codeword associated with the message from transmitter i to receiver 1, for i ∈ {1, 2, {1, 2}} respectively. Using the random coding technique in Sec. II, we can superpose U N 1 1 and U N 2 2 over U N {1,2} 1 and successively bin U N 1 1 and U N 2 2 against U N {1,2} 1 . Corollary III.1. An Achievable Region for the MAC-CM The following region is achievable for a general MAC-CM:
R 1 1 ≥ D(P U e 1 1 ||P U c 1 1 |P U c 1 {1,2} ) R 2 1 ≥ D(P U e 2 1 ||P U c 2 1 |P U c 1 {1,2} ) R {1,2} 1 + L 1 1 + L 2 1 ≤ I(Y 1 ; U e 1 1,2 , U e 1 1 , U e 2 1 ) + D (P encoding |P codebook ) L 1 1 + L 2 1 ≤ I(Y 1 ; U e 1 1 , U e 2 1 |P U e {1,2} 1 ) + D (P encoding |P codebook ) L 1 1 ≤ I(Y 1 ; U e 1 1 |U e {1,2} 1 , U e 2 1 ) + D(P U e 1 1 ||P U c 1 1 |P U c 1 {1,2} ) L 2 1 ≤ I(Y 1 ; U e 2 1 |U e {1,2} 1 , U e 1 1 ) + D(P U e 1 2 ||P U c 1 2 |P U c 1 {1,2} ),(8)
union over all the distributions that factor as
P codebook = P U c {1,2} 1 P U c 1 1 |U c {1,2} 1 P U c 2 1 |U c {1,2} 1 P encoding = P U c {1,2} 1 P U e 1 1 |U c {1,2} 1 P U e 2 1 |U c {1,2} 1 ,(9)
and for L i j = R i j + R i j .
Proof: The complete proof is provided in [9]. After the Fourier-Motzkin Elimination (FME) of the region in (8), we obtain the classical region [16]
R {1,2} 1 + R 1 1 + R 2 1 ≤ I(Y 1 ; U c {1,2} 1 , U e 1 1 , U e 2 1 ) R 1 1 + R 2 1 ≤ I(Y 1 ; U e 1 1 , U e 2 1 |U e {1,2} 1 ) R 1 1 ≤ I(Y 1 ; U e 1 1 |U c {1,2} 1 , U e 2 1 ) R 2 1 ≤ I(Y 1 ; U e 2 1 |U c {1,2} 1 , U e 1 1 ),(10)
union over all the possible distributions in (9) which is indeed capacity. Cor. III.1 shows that the capacity of the MAC-CM can be achieved with any distribution of the codewords U N 1 1 and U N 2 1 for as long as the codewords can be further binned to impose the distribution of the matching outer bound. The distance between the codewords at generation and after encoding has no effect on the resulting achievable scheme.
B. The Broadcast Channel
In the BC [3], [12], [13] one encoder wants to communicate to two decoders a message each. For this channel model, rate splitting can be applied so as to split each message in private and common part; the two common part can then be embedded into a single common message. This transforms the problem of achieving the rate vector [R ′ 1 1 , R ′ 1 2 ] in the problem of achieving the rate vector [R 1 1 , R 1
2 , R 1 {1,2} ] where R ′ 1 1 R ′ 1 2 = 1 0 α 0 1 α R 1 1 R 1 2 R 1 {1,2} T ,(R 1 {1,2} + R 1 1 + R 1 2 ≥ D(P U e 1 1 U e 1 2 U e 1 {1,2} ||P U c 1 1 U c 1 2 U c 1 {1,2} ) R 1 {1,2} + R 1 1 ≥ D(P U e 1 1 U e 1 {1,2} ||P U c 1 1 U c 1 {1,2} ) R 1 {1,2} + R 1 2 ≥ D(P U e 1 2 U e 1 {1,2} ||P U c 1 2 U c 1 {1,2} ) R 1 {1,2} ≥ D(P U e 1 {1,2} ||P U c 1 {1,2} ) L 1 {1,2} + L 1 1 ≤ I(Y 1 ; U e 1 {1,2} U e 1 1 ) + D(P U e 1 1 U e 1 {1,2} ||P U c 1 1 U c 1 {1,2} ) L 1 1 ≤ I(Y 1 ; U e 1 1 |U e 1 {1,2} ) + D(P U e 1 1 ||P U c 1 1 |P U e 1 {1,2} ) L 1 {1,2} + L 1 2 ≤ I(Y 2 ; U e 1 {1,2} U e 1 2 ) + D(P U e 1 2 U e 1 {1,2} ||P U c 1 2 U c 1 {1,2} ) L 1 2 ≤ I(Y 2 ; U e 1 2 |U e 1 {1,2} ) + D(P e U1 2 ||P U c 1 2 |P U e 1 {1,2} )(12)
union over all the distributions that factor as
P codebook = P U c 1 {1,2} P U c 1 1 |U c 1 {1,2} P U c 1 2 |U c 1 {1,2} (13a) P encoding = P U e 1 1 U e 1 2 U e 1 {1,2} ,(13b)
for the rate splitting strategy in (11) and L i j = R i j + R i j .
Proof: The complete proof is provided in [9]. After the FME of the binning rates R i j we obtain the region
R 1 1 ≤ I(Y 1 ; U e 1 1 |U e 1 {1,2} ) + D(P U e 1 1 ||P U c 1 1 |P e U 1 {1,2} ) R 1 2 ≤ I(Y 2 ; U e 1 2 |U e 1 {1,2} ) + D(P U e 1 2 ||P U c 1 2 |P U e 1 {1,2} ) R 1 {1,2} + R 1 1 ≤ I(Y 1 ; U e 1 {1,2} U e 1 1 ) R 1 {1,2} + R 1 2 ≤ I(Y 2 ; U e 1 {1,2} U e 1 2 ) R 1 {1,2} + R 1 1 + R 1 2 ≤ I(Y 1 ; U e 1 {1,2} U e 1 1 ) +I(Y 2 ; U e 1 2 |U e 1 {1,2} ) − D BC−RS R 1 {1,2} + R 1 1 + R 1 2 ≤ I(Y 2 ; U e 1 {1,2} U e 1 2 ) +I(Y 1 ; U e 1 1 |U e 1 {1,2} ) − D BC−RS ,(14)
with
D BC−RS = u1 1 ,u1 2,P U c 1 1 |U c 1 {1,2} = −I(U e 1 1 , U e 1 2 |U e 1 {1,2} ).(15)
With the equivalence in (15) we conclude that the region in (14) is equivalent to Marton's region [17] which is the largest known achievable region for a general BC. As for Cor. III.1, Cor. III.2 shows that the achievable region is not determined by the distribution of the codewords in the codebook but only on the distribution after encoding.
C. The Interference Channel
The IFC is four-terminal network where two pairs of transmitter/receiver pairs want to communicate a message over the channel each. As for the BC, we can rate-split each message into a public and private part: the private messages, W 1 1 and W 2 2 respectively, are decoded only at the intended transmitter while the public messages, W 1 {1,2} and W 2 {1,2} , are decoded by both decoders. Rate-splitting transforms the problem of achieving the rate vector (R ′ 1 1 , R ′ 2 2 ) in the problem of achieving the rate vector (R 1 1 ,
R 2 2 , R 1 {1,2} , R 2 {1,2} ) where R ′ 1 1 R ′ 2 2 = α 0 α 0 0 β 0 β R 1 1 R 2 2 R 1 {1,2} R 2 {1,2} T ,(16)
for any (α, β) ∈ [0 . .
. 1] 2 , α = 1 − α , β = 1 − β.R 1 {1,2} + R 1 1 ≥ D(P U e 1 1 ,U e 1 {1,2} ||P U c 1 1 ,U c 1 {1,2} ) R 1 1 ≥ D(P U e 1 1 ||P U c 1 1 |P U e 1 {1,2} ) R 2 {1,2} + R 2 2 ≥ D(P U e 2 2 ,U e 2 {1,2} ||P U c 2 2 ,U c 2 {1,2} ) R 2 2 ≥ D(P U e 2 2 ||P U c 2 2 |P U e 2 {1,2} ) L 1 {1,2} + L 1 1 + L 2 {1,2} ≤ I(Y 1 ; U e 1 {1,2} , U e 1 1 , U e 2 {1,2} ) +D(P e U1→1U 1→{1,2} ||P c U1→1U 1→{1,2} ) L 1 1 + L 2 {1,2} ≤ I(Y 1 ; U e 1 1 , U e 2 {1,2} |U e 1 {1,2} ) +D(P U e 1→1 ||P U c 1→1 |P U e 1→{1,2} ) L 1 {1,2} + L 1 1 ≤ I(Y 1 ; U e 1 {1,2} , U e 1 1 |U e 2 {1,2} ) +D(P U e 1→1 U e 1→{1,2} ||P U c 1→1 U c 1→{1,2} ) L 1 1 ≤ I(Y 1 ; U e 1 1 |U e 1 {1,2} , U e 2 {1,2} ) +D(P U e 1→1 ||P U c 1→1 |P U e 1→{1,2} ) L 1 {1,2} + L 2 2 + L 2 {1,2} ≤ I(Y 2 ; U e 1 {1,2} , U e 2 2 , U e 2 {1,2} ) +D(P U e 2→2 U e 2→{1,2} ||P U c 2→2 U c 2→{1,2} ) L 2 2 + L 2 {1,2} ≤ I(Y 2 ; U e 2 2 , U e 2 {1,2} |U e 1 {1,2} ) +D(P U e 2→2 U e 2→{1,2} ||P U c 2→2 U c 2→{1,2} ) L 1 {1,2} + L 2 2 ≤ I(Y 2 ; U e 1 {1,2} , U e 2 2 |U e 2 {1,2} ) +D(P U e 2→2 ||P U c 2→2 |P U e 2→{1,2} ) L 2 2 ≤ I(Y 2 ; U e 2 2 |U e 1 {1,2} , U e 2 {1,2} ) +D(P U e 2→2 ||P U c 2→2 |P U e 2→{1,2} )(17)
union over all the distributions that factor as
P codebook = P U c 1 {1,2} P U c 1 1 |U c 1 {1,2} P U c 1 2 |U c 1 {1,2} P encoding = P U e 1 1 U e 1 2 U e 1 {1,2} ,(18)
for the rate splitting strategy in (16) and L i j = R i j + R i j .
Proof: The complete proof is provided in [9]. From the FME of the binning rates R i j one obtains that the largest achievable region in ( [15], which is the largest known achievable region for a general IFC.
As for the MAC , Cor. III.1, and the BC, Cor. III.2, Cor. III.3 does not improve on the largest known region for the IFC but shows that a larger set of transmission strategies than superposition coding and binning can be used to achieve this region.
In [7, Sec. VI] we have shown that superposition coding and binning can both be used to achieve the largest known inner bound for the MAC, BC and IFC. In the examples above we have shown that combining the two encoding strategies into a new and more general transmission strategy still achieves the same performance. With these considerations in mind, we can provide an insight on the error performance of these two coding techniques. Both in superposition coding and binning one creates multiple codewords to transmit the same message. In superposition coding, the message encoded in top codebook is associated to multiple codewords, one for each possible base codeword. while, in binning, codewords in the same bin are associated to the same message. When U N 2 , with rate R 2 , is superposed to U N 1 , with rate R 1 , the number of possible codewords used to encode the same message in U N 2 is 2 N R1 .In binning, the number of excess codewords depends on the joint probability distribution between the codewords imposed by the encoding procedure. When U 2 is binned against U 1 , the smallest number of codeword in each bin is R 2 = I(U 1 ; U 2 ). In both cases, the transmitted codewords [U 1 , U 2 ] belong to the typical set T N ǫ (P U1U2 ) but superposition coding usually requires a larger number of excess codewords than binning to achieve the desired typicality property and this number is fixed and does not depend on P U1U2 . While binning is more advantageous than superposition coding at encoding, it performs worst at decoding. In superposition coding, after the decoding of the base codeword U N 1 , the receiver looks for the transmitted top codeword U N 2 in a codebook of size 2 N R2 .In binning, instead, after U N 1 has been correctly decoded, the possible transmitted codewords are 2 N (R2+R2) . The knowledge of U N 1 helps the decoder in determining U N 2 in that U N 1 , U N 2 must appear as if generated according to the encoding distributions, but it does not reduce the number of possible transmitted codewords U N 2 . Interestingly the encoding and decoding benefits provided by superposition coding and binning seem to balance each other in the proposed random coding technique.
IV. CONCLUSION
In this paper we present a new achievable strategies that encompasses superposition coding and binning. The error analysis of this new achievable scheme requires a more general version of the classical covering lemma that is based on the inaccuracy between typical sequences. With this new random coding technique we derive achievable regions for the multiaccess channel, broadcast channel and interference channel. These inner bounds do not improve on the largest known achievable regions but show that the same error performance can be achieved with a large set of encoding strategies.
Fig. 1 .
1A graphical representation of superposition coding.
Fig. 2 .
2A graphical representation of joint binning.
8, Lem. 2.6 ]. Lemma II.1. Inaccuracy, [8, Lem. 2.6]. Consider a two general distributions P U2 and P U2 and u N 1
Corollary III.2. An Achievable Scheme for the BC The following region is achievable for a general BC:11)
for any α ∈ [0 . . . 1] and α = 1 − α. The random coding
technique of Sec. II can be applied to the BC after the rate
splitting in (11) by superposing U N
1 1 and U N
1 2 over U N
1 {1,2}
and successively jointly binning U N
1 1 , U N
1 2 and U N
1 {1,2} .
Corollary III.3. An Achievable Scheme for the IFC The following region is achievable for a general IFC:The new
random coding technique of Sec. II can be applied to the IFC
after rate splitting by superposing the U N
1 1 onto U N
1 {1,2} and
jointly binning U N
1 1 , U N
1 {1,2} . The same encoding procedure
is applied to the codewords U N
2 2 and U N
2 {1,2} .
{1,2} . With this choice the achievable region in Cor. III.3 becomes equivalent to the Han and Kobayashi regionR ′
1 1 , R ′
2 2 ) is achieved with
the choice P U e
1 {1,2} = P U c
1 {1,2} and P U e
2 {1,2} = P U c
2
ACKNOWLEDGMENTThe author would like to thank Prof. Gerhard Kramer for suggesting the inaccuracy to measure the distance between codebook and encoding distribution.
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Capacity and coding for degraded broadcast channels. R Gallager, Problems in the Transmission of Information. 103R. Gallager, "Capacity and coding for degraded broadcast channels," Problems in the Transmission of Information, vol. 10, no. 3, pp. 3-14, 1974.
The capacity region of a channel with two senders and two receivers. R Ahlswede, The annals of probability. 2R. Ahlswede, "The capacity region of a channel with two senders and two receivers," The annals of probability, vol. 2, no. 5, pp. 805-814, 1974.
A New Achievable Rate Region for the Interference Channel. T Han, K Kobayashi, IEEE Trans. Inf. Theory. 271T. Han and K. Kobayashi, "A New Achievable Rate Region for the Interference Channel," IEEE Trans. Inf. Theory, vol. 27, no. 1, pp. 49- 60, Jan 1981.
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| []
|
[
"High-dimensional quantum Fourier transform of twisted light",
"High-dimensional quantum Fourier transform of twisted light"
]
| [
"Jaroslav Kysela \nFaculty of Physics\nInstitute for Quantum Optics and Quantum Information\nUniversity of Vienna\nBoltzmanngasse 5ViennaAustria\n\nAustrian Academy of Sciences\nBoltzmanngasse 3ViennaAustria\n"
]
| [
"Faculty of Physics\nInstitute for Quantum Optics and Quantum Information\nUniversity of Vienna\nBoltzmanngasse 5ViennaAustria",
"Austrian Academy of Sciences\nBoltzmanngasse 3ViennaAustria"
]
| []
| The Fourier transform proves indispensable in the processing of classical information as well as in the quantum domain, where it finds many applications ranging from state reconstruction to prime factorization. An implementation scheme of the d-dimensional Fourier transform acting on single photons is known that uses the path encoding and requires O(d log d) optical elements. In this paper we present an alternative design that uses the orbital angular momentum as a carrier of information and needs only O( √ d log d) elements, rendering the path-encoded design inefficient. The advantageous scaling and the fact that our approach uses only conventional optical elements allows for the implementation of a 256-dimensional Fourier transform with the existing technology. Improvements of our design, as well as explicit setups for low dimensions, are also presented. | 10.1103/physreva.104.012413 | [
"https://arxiv.org/pdf/2101.11919v2.pdf"
]
| 235,829,425 | 2101.11919 | d7f141a1ed7d96940e43707f8b92d6ce3ca69408 |
High-dimensional quantum Fourier transform of twisted light
14 Jul 2021
Jaroslav Kysela
Faculty of Physics
Institute for Quantum Optics and Quantum Information
University of Vienna
Boltzmanngasse 5ViennaAustria
Austrian Academy of Sciences
Boltzmanngasse 3ViennaAustria
High-dimensional quantum Fourier transform of twisted light
14 Jul 2021(Dated: July 15, 2021)
The Fourier transform proves indispensable in the processing of classical information as well as in the quantum domain, where it finds many applications ranging from state reconstruction to prime factorization. An implementation scheme of the d-dimensional Fourier transform acting on single photons is known that uses the path encoding and requires O(d log d) optical elements. In this paper we present an alternative design that uses the orbital angular momentum as a carrier of information and needs only O( √ d log d) elements, rendering the path-encoded design inefficient. The advantageous scaling and the fact that our approach uses only conventional optical elements allows for the implementation of a 256-dimensional Fourier transform with the existing technology. Improvements of our design, as well as explicit setups for low dimensions, are also presented.
The Fourier transform is arguably one of the most important tools in modern mathematics, science and engineering. Its applications range from a purely mathematical use in differential calculus [1] to modelling optical properties of light such as a free-space propagation or a propagation through a system of lenses [2]. On a more practical level, the Fourier transform is used heavily in the spectral analysis of audio and video signals [3]. Its discrete version lies at the heart of various dataprocessing algorithms used by modern computers. Popular image and audio compression algorithms JPEG and MP3 [4,5] are notable examples. The Fourier transform plays also a central role in the quantum domain. Not only it underlies the duality of the position and momentum representations in quantum physics [6], but it also proves useful in the quantum information processing. The Shor's factoring algorithm, based on the quantum Fourier transform, dramatically outperforms any of its classical counterparts [7]. The quantum Fourier transform is often understood as a transformation applied to a many-particle quantum state. There are, however, many cases when the transform has to act on a large state of a single particle. Such a single-particle quantum Fourier transform finds a plethora of applications, such as the generation of mutually unbiased bases in quantum state tomography and quantum key distribution [8][9][10][11][12][13], generation of angular states [14][15][16][17], and implementation of programmable universal multiport arrays [18][19][20].
Various systems can be chosen as quantum carriers of information. Such a system can be, for example, a single photon, where the information is encoded into its orbital angular momentum (OAM). The OAM of a photon is manifested by a helical structure of the wavefront of the photon's wavefunction [21][22][23], for which it is sometimes dubbed 'twisted light.' Each twist of the helix corresponds to a quantum of OAM. The number of these quanta is not bounded. Unlike in the classical computation, which uses only two values-0 and 1-to encode the data, the OAM allows one to define a d-valued alphabet for arbitrary finite integer d [23]. The data is thus represented as a quantum superposition of d different OAM eigenstates of a single photon. Many experiments have been conducted in various contexts [13,17,[24][25][26][27], where the single-photon OAM Fourier transform is implemented using specially designed optical elements with nontrivial phase profiles [28]. Recently, an alternative approach has been found that relies on interferometers and requires only conventional optical elements familiar from the classical optics [29]. This approach is recursive and allows for an efficient implementation when the dimension of the OAM space is of the form d = 2 M for some M ∈ N.
In this paper, we present an improved interferometric design of the Fourier transform inspired by that of Ref. [29], but contrary to the original proposal, our design avoids recursion. As a result, we not only simplify the resulting scheme, but also determine analytically the optimal decomposition and, most importantly, reduce dramatically the number of optical elements in the setup. The scheme of Ref. [29] requires asymptotically 6.1412 d beam splitters, whereas we present a scheme that requires only (11/4) √ d log(d) of them and allows for further simplifications. Even though the following discussion focuses on the quantum regime, the identical physical setups can be also used to implement the Fourier transform of the OAM states of classical light.
This manuscript is organised as follows. After the brief summary of the mathematical formalism behind our scheme in section II, we present the setup of the OAM Fourier transform in section III, whose scaling is studied in section IV. In section V we discuss the periodicity of our implementation. In section VI an improved version of our scheme is presented, which uses polarization. Section VII shows how to modify our setup to implement the path-only Fourier transform and in section VIII we summarize our results.
II. FOURIER TRANSFORM OF OAM
Each OAM eigenstate is characterized by a specific spatial profile of the photon's wavefunction, whose phase has a helical structure with a singularity in the center, whereas the amplitude resembles a doughnut [21,23], see Fig. 1. We consider a single photon that carries information encoded into a quantum superposition of d different OAM eigenstates |0 , . . . , |d − 1 . Such eigenstates span a d-dimensional Hilbert space H. To characterize how the Fourier transform acts on a general superposition of eigenstates, it suffices to specify its action on the basis states |k for k = 0, . . . , d − 1. The d-dimensional Fourier transform of an OAM eigenstate |k is defined as
F (d) OAM (|k ) = 1 √ d d−1 j=0 e i 2πjk/d |j .(1)
The transform turns the helical profile of an eigenstate into a complicated wavefront with many singularities as demonstrated in Fig. 1(a). One should not mistake the Fourier transform of spatial modes of light for the Fourier property of a normal lens, see Fig. 1(b). For example, the four-dimensional Fourier transform applied to the OAM eigenstate |1 results in the state (|0 +i |1 −|2 −i |3 )/2 with a non-trivial wavefront. On the contrary, the OAM eigenstates, including |1 , preserve their overall shape when propagating through a lens.
To implement the transform in Eq.
(1), we take advantage of its algebraic structure. It is easy to show that the discrete d-dimensional Fourier transform can be expressed as a composition of two lower-dimensional Fourier transforms [30]. One can apply this decomposition recursively such that in the end the Fourier transform is decomposed into elementary 2-dimensional blocks. In the physical scenario, these elementary blocks can be identified with beam splitters. As a result, the d-dimensional Fourier transform acting on path-encoded qudits can be efficiently implemented [31]. Similar ideas can be applied when implementing the d-dimensional Fourier transform in the OAM, see Fig. 1 in Ref. [29]. In that case, the initial superposition of OAM eigenstates of an incoming photon is first transformed into a superposition of a smaller number of OAM eigenstates propagating along multiple paths. This way, the information stored in the state gets redistributed between the path and OAM degrees of freedom. The Fourier transform is then implemented by an application of a lower-dimensional Fourier transform that acts only on the OAM degree of freedom, followed by another lower-dimensional Fourier transform that acts only on the path degree of freedom. The whole procedure is finished by recombining all the resulting OAM eigenstates into a single output path.
To implement the lower-dimensional OAM Fourier transform, the same decomposition is applied recursively until a series of elementary 2-dimensional blocks is reached. Such a scheme for the OAM Fourier transform has been shown recently to provide savings in resources when compared to alternative approaches [29]. In this paper, we use the same decomposition of the Fourier transform as in Ref. [29], but unlike in their approach, we do not apply this decomposition recursively.
The decomposition relies on a factorization of the initial d-dimensional Hilbert space H of the OAM of light into two subspaces H O and H P , such that H = H O ⊗H P . The state |k in H is then identified with states |q O ∈ H O and |r P ∈ H P as follows Fig. 2(a).
|k ∈ H ←→ |q O ⊗ |r P ∈ H O ⊗ H P .(2)OAM = F (d A ) path · CZ · SWAP · F (d B ) path .(3)
The decomposition on the righ-hand side of Eq.
(3) assumes that the initial Hilbert space is already factorized into H O and H P , where H O is spanned by d A different OAM eigenstates and H P is spanned by d B different propagation modes. In the actual setup this factorization is performed by an additional module, which consists of a d A -dimensional OAM sorter S [29] and a swap operator with d A input and d B output paths. The operation of this module is graphically illustrated in Fig. 2(b). Analogously, the final recombination is taken care of by another module that has the inverse structure to that of the first module. It comprises a swap with d A input and d B output paths and a d B -dimensional OAM sorter S −1 that is operated in reverse. To emphasize the role of the center module whose action is given by Eq. (3), we will refer to it as the Fourier transform proper. All three modulesthe factorization module, the Fourier transform proper, and the recombination module-are shown explicitly in Fig. 2(a). For details see the next section and Appendix F.
As in Ref. [29], we consider throughout the paper only dimensions d that are powers of two, i.e., d = 2 M for some integer M . It remains an open question still, how to implement efficiently the Fourier transform for a general dimension.
III. IMPROVED SCHEME
Our scheme requires only conventional optical elements for its construction. Namely, mirrors, beam splitters, Dove prisms, phase shifters, and holograms [32]. In this section we demonstrate how to use these optical elements to implement individual components of the setup. As explained in the previous section and illustrated in Fig. 2(a), the setup of the OAM Fourier transform can be divided into three modules. The first module reroutes the incoming OAM eigenstates |0 , . . . , |d − 1 into d B different paths. Its role is to factorize the initial Hilbert space into two subspaces H O and H P as described in (2). It turns out, see Appendix C, that the roles of d A and d B are symmetric and we can without loss of generality assume that d A ≥ d B from now on. This simplifies the following discussion. The first module consists of a d Adimensional OAM sorter and a swap operator that has d A input and d B output ports.
The d A -dimensional sorter sorts individual incoming OAM eigenstates |0 O , . . . , |d A − 1 O into different output ports |0 P , . . . , |d A − 1 P . Multiple designs of the OAM sorter have been investigated in experiments using different physical principles, such as multi-plane light conversion [33,34], light scattering in random media [35] or light propagation through two phase plates with specially designed wavefronts [28]. In this paper we employ the interferometric design of Leach et al. [36][37][38][39][40]. Intrinsic to this design is its modulo property [29], where also eigenstates with more than d A quanta of OAM get sorted into one of d A output ports. This way, the sorter separates the d incoming OAM eigenstates into d A groups, each of which propagates along a different path and contains d B OAM eigenstates. The side effect of the modulo property is that the OAM eigenstates leaving the sorter are of the form |0 , |d A , |2d A , . . . , |(d B − 1) d A . In other words, the difference between two successive values of OAM is equal to d A . We call this difference a multiplicity of the OAM eigenstates. The internal structure of individual components of the setup, such as swaps, has to be adjusted to this multiplicity. Suppose that the multiplicity of eigenstates entering the sorter is µ ∈ N. In order for the sorter to sort these states correctly, the or-der of each OAM exchanger [29] in the OAM sorter has to be multiplied by µ. Since the number µ affects the construction of the sorter, we call it the multiplicity of the sorter and denote it by µ S . The multiplicity of the eigenstates leaving the sorter is then µ S d A . Nevertheless, in our scheme we assume that the multiplicity of incoming OAM eigenstates is equal to one and the default sorter structure can be used.
The second part of the first module consists of a swap operator with d A input and d B output ports. Its interferometric implementation, denoted by SWAP, is given in Ref. [29] and is briefly summarized in Appendix F. When there is the same number of input and output ports, i.e., d A = d B , both forward and backward passes through SWAP represent the same operation, see also Fig. 3(a). The multiplicity of OAM eigenstates leaving the swap stays µ SWAP = d A = d B in such a case. When d A > d B , the self-inverse property of SWAP is lost. The multiplicity of incoming OAM eigenstates is d A and the implementation acts like
SWAP (|d A · l O |m P ) = |d B · m O |l P .(5)
The multiplicity of the output OAM eigenstates thus becomes d B . From the construction of SWAP it follows that we have to set its multiplicity to the multiplicity of the output eigenstates, µ SWAP = d B . To summarize the action of the first module, we obtain transformation rules
|q ∈ H −→ |d B · m O ⊗ |l P ∈ H O ⊗ H P ,(6)
where q = d A l + m is the initial number of OAM quanta of the incoming OAM eigenstate, 0 ≤ m < d A , and 0 ≤ l < d B . For details see Appendix A. The second module represents the Fourier transform proper and comprises four operations as given by Eq. (3). These operations are implemented as follows:
1. The d B -dimensional path-only Fourier transform is represented in the setup by block F P with d B input and output paths. Its efficient implementation in terms of beam splitters and phase shifters is presented in Ref. [31]. Additional mirrors are necessary in our implementation of F P as explained in Ref. [29]. Block F P acts on propagation modes and leaves OAM eigenstates unaffected.
2. The swap operator is implemented analogously to the swap in the first module. It has d B input and d A output ports. For reasons that are explained in Appendix A it is advantageous to assume that the number of output ports of SWAP never exceeds the number of its input ports. This poses a problem in the case when d A > d B . The easy fix is to use the implementation backward. Instead of SWAP we thus use SWAP −1 , see Fig. 3
(b).
3. The controlled-Z gate CZ is implemented as a series of properly rotated Dove prisms, each of which lies on a different path. Specifically, a Dove prism in the k-th path is rotated through angle kα, where α = π/(d A d). As a Dove prism also reverses the sign of (a) Even powers:
d = 16 ⟶ d A = 4, d B = 4 (b) Odd powers: d = 32 ⟶ d A = 8, d B = 4 S (μ S =1) SWAP (μ SWAP =d B ) SWAP (μ SWAP =d B ) SWAP (μ SWAP =d B ) F P F P 0 α 2 α 3 α S -1 (μ S =1) rerouting recombination transform proper S (μ S =1) SWAP (μ SWAP =d B ) SWAP (μ SWAP =d B ) SWAP -1 (μ SWAP =d B ) F P F P 0 α 2 α 3 α 4 α 5 α 6 α 7 α S -1 (μ S =1)
rerouting recombination transform proper the OAM value, an additional mirror supplements each Dove prism to revert the sign back.
The d A -dimensional path-only Fourier transform is
represented by block F P with d A input and output paths. It is constructed using the same design as the initial d B -dimensional path-only Fourier transform.
This module is reminiscent of the schemes presented in Refs. [41,42] where the OAM equivalent of the polarizing beam splitter was studied.
The third module recombines all resulting OAM eigenstates into a single output path. It has the inverse structure to that of the first module. It consists of a swap operator with d A input paths, d B output ports, and multiplicity µ SWAP = d B , followed by a d B -dimensional OAM sorter that is operated in reverse and that has multiplicity µ S = 1. This setup implements relations
|d A · j O ⊗ |k P ∈ H O ⊗ H P −→ |r ∈ H, (7) where r = d B k+j is the final number of OAM quanta of a particular OAM eigenstate, 0 ≤ k < d A , and 0 ≤ j < d B .
An important observation is that values of d A and d B are not a priori fixed. The only condition these dimensions have to satisfy is d = d A d B . This gives us freedom to choose such values that lead to the minimal number of optical elements in the physical implementation. In Appendix C an analytical proof is presented, where we show that the optimal choice of d A and d B depends on the parity of M in d = 2 M . For an even power M one obtains even powers :
d A = d B = √ d.(8)
The number of paths in the whole setup implementing the OAM Fourier transform is then constant and equal to √ d. A specific example of such a setup for M = 4 is depicted in Fig. 3(a). For odd M the optimal values are odd powers :
d A = √ 2d, d B = d/2.(9)
In this case, the number of paths in the setup alternates between d A and d B = d A /2. A specific example of such a setup for M = 5 is depicted in Fig. 3(b). FIG. 4. The number of beam splitters in the setup for the OAM Fourier transform as a function of the dimension d, using three different approaches. The semi-brute-force approach requires O(d log 2 (d)) beam splitters. The recursive scheme of Ref. [29] requires only O(d) beam splitters. Nevertheless, as is evident from the plot, our scheme is considerably more efficient than both the semi-brute-force and recursive schemes as it scales as O( √ d log 2 (d)). Note that both axes in the plot use logarithmic scaling. The point markers denote precise numbers, whereas lines represent analytically the growing trends.
IV. SCALING AND LOSSES
The setups with optimal values of d A and d B , see Eqs. (8) and (9), have the minimal number of beam splitters, Dove prisms, holograms, and phase shifters among all setups that implement the d-dimensional Fourier transform using decomposition (3). In accord with Ref. [29] we focus on the number of beam splitters as these affect significantly the interferometric stability of the final setup. In our scheme, this number is approximately equal to
N (F ) BS (d) ≈ 11 4 √ d log(d).(10)
For other types of optical elements we obtain the same scaling O( √ d log 2 (d)), as demonstrated in detail in Appendix D. There exist at least two alternative approaches based on interferometers that can be used to construct the OAM Fourier transform. The brute-force approach consists in transforming all incoming OAM eigenstates into the path encoding and then applying a path-only Fourier transform. For dimensions of the form d = 2 M , which we consider in this paper, an efficient design of Ref. [31] can be used to implement the path-only Fourier transform. Such a semi-brute-force approach requires asymptotically O(d log 2 (d)) beam splitters. The other approach of Ref. [29], relying on the recursive construction of the OAM Fourier transform, requires O(d) beam splitters. The comparison of our approach and the two alternatives is presented in Fig. 4. Recently, a quantum version of the integer fast Fourier transform has been presented [43], which needs O(d log 2 (d)) gates, but works in parallel on a large number of data sets with no additional overhead. We compare this scheme with the parallelization of our scheme in the next section.
The practical applicability of any optical design is limited by losses of the actual implementation. The detailed analysis of losses of our design lies beyond the scope of the present paper. Nevertheless, we provide some rough estimates of the setup's performance. As pointed out in Ref. [44] for general unitaries, the losses suffered by photons propagating through a network of interferometers can be significantly reduced when the topology of the network is rectangular, as opposed to the triangular topology [45]. The losses increase with the number of beam splitters a photon has to traverse. This number is quantified by the depth of the network [44], which scales like d and 2d for [44] and [45], respectively. In our implementation of the Fourier transform, the setup has a rectangular structure whose depth scales like (15/2) log 2 (d), i.e. (15/2)M , for both even and odd M . In Appendix E, a simple model is presented, with the help of which we assess the approximate effect of losses in our design.
There are at most d A ∼ √ d paths in the setup, which is especially evident for dimensions d = 2 M with even exponent M , see Fig. 3. The reduced number of propagation modes is another improvement over alternative approaches. The semi-brute force has also a rectangular structure, but requires d paths. Similarly, the setup based on the recursive scheme [29] begins with an OAM sorter that reroutes all incoming OAM eigenstates into d different paths. The rest of the scheme has a triangular structure with a single final output path.
V. HIGHER-OAM SUBSPACES
A distinctive feature of our implementation is that it works not only for OAM eigenstates it was designed for (and their superpositions), but also for eigenstates that lie outside of the original range of OAM values. The scheme acts primarily on states lying in a subspace spanned by OAM eigenstates |0 , |1 , |2 . . . , |d − 1 . Any superposition of these basis states is thus transformed in compliance with Eq. (1). Consider a new subspace spanned instead by OAM eigenstates of the form
|0 + a d , |1 + a d , |2 + a d . . . , |(d − 1) + a d ,
where a is a fixed integer. For example, when a = 1, the new subspace contains all possible superpositions of eigenstates |d , |d + 1 , |d + 2 . . . , |2d − 1 . It is straightforward to prove that the setup for the d-dimensional OAM Fourier transform acts on such OAM eigenstates like
F (d) OAM (|k + a d ) = e 2πi d A m a · 1 √ d d−1 j=0 e i 2πjk/d |j + a d ,(11)
where m = k mod d A . From the formula above it follows that the setup applies a d-dimensional Fourier transform to the new subspace in a way that is completely analogous to the way it acts on the original subspace. The only difference is an extra phase factor e 2πima/d A , which emerges when a higher-OAM eigenstate propagates through Dove prisms in the second module of the setup. Whenever a is a multiple of d A , the phase factor vanishes and one obtains the exact d-dimensional Fourier transform.
The periodicity of our setup can be employed to apply the Fourier transform simultaneously to many OAM subspaces. Each value of parameter a defines a particular d-dimensional subspace of OAM eigenstates. This value is for one specific subspace fixed, but can be otherwise chosen arbitrarily [46]. As a consequence, our setup applies the d-dimensional Fourier transform simultaneously to potentially enormous number of subspaces. One can imagine to have data stored in several data sets that are represented as one large superposition of OAM eigenstates of a single photon. It is then sufficient to send the photon only once through the setup in order to apply the Fourier transform separately and simultaneously to each data set. To illustrate this point, consider two data sets, {α 0 , . . . , α 3 } and {β 0 , . . . , β 3 }, each consisting of four complex numbers. These data sets can be (after proper normalization) encoded into the following state of a single photon
|ψ = α 0 |0 + α 1 |1 + α 2 |2 + α 3 |3 + β 0 |8 + β 1 |9 + β 2 |10 + β 3 |11 ,(12)
where eigenstates |0 , . . . , |3 span the original 4dimensional subspace and eigenstates |8 , . . . , |11 span another 4-dimensional subspace, for which the prefactor in Eq. (11) vanishes. To apply the 4-dimensional Fourier transform to each data set, it suffices to send the photon once through the setup. The initial state is transformed into |ψ = F
OAM (|ψ ), whose form reads
|ψ = F(4)OAM (α 0 |0 + α 1 |1 + α 2 |2 + α 3 |3 )(13)+ F (4) OAM (β 0 |8 + β 1 |9 + β 2 |10 + β 3 |11 )(14) = α 0 |0 + α 1 |1 + α 2 |2 + α 3 |3 (15) + β 0 |8 + β 1 |9 + β 2 |10 + β 3 |11 ,(16)
where α j and β j are Fourier images of data points α j and β j , respectively. The representation in Eq. (12) of a data set as a superposition of basis states corresponds to the so-called amplitude encoding [47]. A collection of N data sets is then represented as a superposition of N single-data-set superpositions. The processing of this larger superposition requires in our design no additional overhead in resources. This feature is akin to the proposal in Ref. [43], where one data set is represented as a single high-dimensional basis state using basis encoding [47] and many data sets are encoded into a superposition of such basis states. The processing of many data sets in that proposal also does not incur any additional computational costs.
VI. POLARIZATION-ENHANCED SCHEME
As is evident from Fig. 3, the setup for the Fourier transform displays a high degree of symmetry for dimensions d = 2 M , when the exponent M is an even num-ber. For such dimensions, the last recombination module is just an inverse of the first rerouting module and the two path-only Fourier transforms in the center module are identical. It turns out that one can remove the last module altogether as well as the second path-only Fourier transform. This can be achieved using a polarization of light as an additional degree of freedom that controls whether the light traverses the setup forward or backward. This polarization-enhanced scheme is explicitly depicted in Fig. 5(a) for dimension d = 16. If we fix the polarization of the incoming photon to horizontal, it passes the first part of the setup as in the original scheme. A series of half-wave plates, inserted after Dove prisms, transforms the polarization of light from horizontal to vertical. The photon then encounters a series of polarizing beam splitters that reflects it into the initial part of the setup, but now backward. The photon leaves the setup via the polarizing beam splitter prepended to the setup as shown in Fig. 5(a).
One has to bear in mind that the backward pass through the implementation F P of the path-only Fourier transform results in the inverse transformation F −1 path , not F path . This undesirable effect can be counteracted by noting that the Fourier transform F path in an arbitrary dimension d A satisfies the following relations [48]
F path = F −1 path · F 2 path ,(17)
where
F 2 path (|m O |0 P ) = |m O |0 P ,(18)F 2 path (|m O |p P ) = |m O |d A − p P , 1 ≤ p < d A .(19)
In other words, the square of the Fourier transform acts as a mere permutation of paths. It leaves the zeroth path unaffected and reverses the order of remaining paths. When we add an extra module to the setup, which implements this path permutation, we effectively compensate for the inversion of the path-only Fourier transform caused by the backward propagation.
As is shown in Appendix D, in this polarizationenhanced scheme the number of beam splitters scales as
N (F ) BS (d) ≈ 7 4 √ d log(d).(20)
We obtain the same scaling as in the original setup, but the scaling factor is more favorable, which may prove useful in actual experimental implementations.
VII. OAM-ENHANCED PATH-ONLY FOURIER TRANSFORM
In cases where the path encoding is favorable, our scheme also offers advantages. It can be easily modified to act as a path-only Fourier transform, where the OAM plays the role of an intermediary that is used only inside the physical implementation [49]. The modification consists in removing the first and third modules from the setup and replacing them with a series of sorters as depicted in Fig. 5(b). This way, the new setup has d input and d output paths. The setting of individual modules of the setup has to reflect this modification. Unlike in the original scheme, the multiplicity of the swap operator is set to µ SWAP = 1. The settings of the subsequent components in the setup depend on the parity of M in d = 2 M . For even M , the number of paths in the center module of the setup, and hence also the multiplicity of OAM eigenstates, stays constant. The angles of Dove prisms are then set such that a Dove prism in path k is rotated through angle kβ, where β = π/d, and the sorters in the third module are d B -dimensional sorters with a default multiplicity µ S = 1. For odd M , there is twice as many paths leaving the swap as those entering it, see a specific example for d = 2 3 = 8 in Fig. 5(b). As a result, the multiplicity of OAM eigenstates leaving the swap is 2. This fact is reflected in the angle of Dove prisms, where now β = π/(2d), and the multiplicity of sorters in the third module, which is set to µ S = 2.
S -1 (μ S =1) S -1 (μ S =1) SWAP (μ SWAP =1) F P F P 0 β 2 β 3 β S (μ S =2) S (μ S =2) S (μ S =2) S (μ S =2)
The number of beam splitters in the OAM-enhanced setup scales as 4d with the dimension, see Appendix D. Even though the efficient scaling of the original scheme is lost, it is still an improvement over the traditional ap-proach based on the design of Ref. [31], which scales as O(d log 2 (d)). Already for d = 2 9 = 512 is our scheme more resource-efficient than the traditional approach. The linear scaling results from the interferometric implementation of OAM sorters [36]. Nevertheless, in the present case the sorters do not have to be built using the interferometric design of Leach et al. [36] and more efficient designs can be employed, such as that of Berkhout et al. [28]. Using their design, we retrieve the original scaling O( √ d log 2 (d)) in the number of beam splitters and other optical elements.
For dimensions d = 2 M with even M , one can also employ polarization in the OAM-enhanced scheme in the way analogous to that in the previous section. The second part of the setup is thus removed; a series of √ d polarizing beam splitters is added to the setup and a series of d polarizing beam splitters is prepended to the setup. This polarization-OAM-enhanced path-only Fourier transform requires approximately 3d beam splitters. From dimension d = 2 8 = 256 onward, this scheme requires fewer beam splitters than the traditional approach. For details consult Appendix D.
VIII. CONCLUSION
We present an efficient implementation of the Fourier transform acting on the orbital angular momentum of light. Our scheme works in the quantum regime with single photons as well as in the classical regime with classical light. The distinctive feature of our implementation is that the Fourier transform works in parallel on many d-dimensional subspaces simultaneously.
Only standard optical elements are used in the construction of the d-dimensional OAM Fourier transform. The number of these elements scales like O( √ d log 2 (d)) as opposed to scaling O(d) reported in Ref. [29]. We show how a polarization of light can be utilized to reduce the number of elements even further. Moreover, using the efficient design of the OAM sorter [28], one can implement the Fourier transform in the path encoding with only O( √ d log 2 (d)) elements as opposed to O(d log 2 (d)) found in Ref. [31]. In such an implementation, the OAM is an auxiliary degree of freedom, which is used only inside the setup.
The favorable scaling of our design can be compared with the capabilities of the current experimental technologies. In the optical quantum domain, photonic chips represent mature and successful technology, which has been used for quantum computation tasks such as boson sampling [50,51] with path being the usual degree of freedom used in these applications. In the case of OAM on chips, considerable progress has been made [52], but the technology is still in its infancy [53,54]. At present, bulk optics represents a more promising experimental platform. The setup employed in Ref. [55] used 30 interferometers, each of which manipulated single photons in three degrees of freedom including OAM. This amounts to 60 beam splitters. For comparison, our design of the 16-dimensional Fourier transform requires 47 beam splitters (or only 33 beam splitters when the polarization is utilized as in section VI). Moreover, a state-of-the-art experiment has been reported recently in Ref. [56], where 50 polarizing beam splitters and a bulk interferometer representing 300 beam splitters were used. Such a number is more than sufficient for the construction of the 128dimensional Fourier transform (or the 256-dimensional transform in the polarization-enhanced scheme).
The Fourier transform finds many applications in the domain of classical computation. Our scheme could be used to process the classical information stored in the OAM of a light beam. It could also be utilized in photonic computational architectures, where Fourier transforms represent an essential computational primitive [18][19][20]. In the field of quantum tomography, mutually unbiased bases play a crucial role. One such basis is generated, when the Fourier transform is applied to the computational basis. We present an efficient implementation for dimensions of the form d = 2 M , for which d + 1 different mutually unbiased bases exist [8,57]. An interesting open question is how to adjust our setup for the generation of other mutually unbiased bases. The Fourier transform possesses many remarkable algebraic properties [58], one of which was used in this paper. It is another open question, whether other properties can be utilized as well to reduce the complexity of the resulting implementation scheme ever further.
IX. ACKNOWLEDGEMENT
The author thanks Mirjam Weilenmann for valuable comments on the manuscript. The support of Austrian Academy of Sciences and the University of Vienna via the QUESS project (Quantum Experiments on Space Scale) is acknowledged. The OAM sorter is used in the setup of the Fourier transform, first, to reroute the incoming eigenstates into multiple paths and, second, to recombine all the resulting eigenstates into a single output path. The eigenstates |k O that enter the d A -dimensional sorter S d A via path |0 P are transformed according to [29]
S d A (|k O |0 P ) = d A · k d A O k mod d A P , (A1)
where x denotes the integer part of number x ∈ R. This "modulo property" of the OAM sorter [29,36] allows for optimal redistribution of the incoming OAM eigenstates into multiple propagation modes. As a result, many OAM eigenstates propagate along the same path, which ultimately leads to efficient scaling O( √ d log d) of the whole scheme. The number k of OAM quanta in the formula (A1) is unbounded. When we consider only the eigenstates |k O with 0 ≤ k < d A , the formula reduces to
S d A (|k O |0 P ) = |0 O |k P . (A2)
The OAM sorter can be generalized into the swap operator, whose action on eigenstates
|k O with 0 ≤ k < d B for the special case of d A = d B reads SWAP d A ,d A ,1 (|k O |p P ) = |p O |k P ,(A3)
where 0 ≤ p < d A are propagation paths. In the formula above, the first, second, and third subscript stands for the number of input ports, the number of output ports, and the multiplicity of SWAP, respectively. In this section we discuss the algebraic properties of the physical implementation of the swap operator. For its structure refer to section F. In the setup for the Fourier transform the swaps have to act on eigenstates of the form |µ · k for some fixed µ ∈ N. We call this number the multiplicity of the eigenstates. The setup of the swap operator can be adjusted also for the case with d A = d B as well as when µ = 1. In our discussion we are interested only in a specific class of OAM eigenstates, whose multiplicity is of the form µ d A /d B , where µ ∈ N. Note that we assume d A ≥ d B and take into account only dimensions that are powers of two and so d A /d B is an integer. If we assume that the number of input paths is equal to d A and the number of output paths is d B < d A , the action of SWAP on eigenstates with arbitrarily large k is represented by equality
SWAP d A ,d B ,µ µ · d A d B · k O |p P = µ · d A · k d B + p O k mod d B P . (A4)
This action is nothing but a mere permutation of (a subset of) basis states of the composite OAM-path Hilbert space. By inspection of formula (A4) one sees that the multiplicity of eigenstates that leave the setup is equal to µ. When we need an implementation of the swap operator with the number of input ports smaller than the number of output ports, we operate SWAP backward, obtaining SWAP −1 . The inverse of formula (A4) can be rewritten into
SWAP −1 d A ,d B ,µ (|µ · k O |p P ) = µ · d A d B · d B · k d A + p O k mod d A P . (A5)
When we set the multiplicity to µ = d B and consider only eigenstates |k O with 0 ≤ k < d A , formula (A5) above reduces to
SWAP −1 d A ,d B ,d B (|d B · k O |p P ) = |d A · p O |k P . (A6)
This relation is used in the second step (B3) of the decomposition presented in section B.
In the setup of the OAM Fourier transform, a sorter forms a module with a swap operator. The first module in the setup affects the incoming OAM eigenstates in the following way:
input: |q O |0 P (A7) S d A −−→ d A · q d A O q mod d A P (A8) SWAPd A ,d B d B − −−−−−−−−− → |d B · (q mod d A ) O q d A P . (A9)
This formula has a very simple interpretation, which is depicted in Fig. 2(b).
Appendix B: Decomposition of the Fourier transform
The entire setup for the OAM Fourier transform consists of three modules. The first module, comprising a sorter and a swap operator, reroutes different OAM eigenstates to different paths. This module is discussed in the previous section. The second module is the Fourier transform proper and the third module recombines all eigenstates into a single output path. Its structure mirrors that of the first module. Let us study first the evolution of the mode |d B m O |l P with some m, l ∈ N, when it propagates through the second module of the setup, cf. Eq. (3) and Fig. 2. Unlike in Eq. (3) in the derivation below we already take into account also the multiplicities of OAM eigenstates when they propagate through the physical setup, see Fig. 3. The incoming mode undergoes individual steps of its evolution as follows:
input: |d B m O |l P (B1) F P − − → |d B m O 1 √ d B d B −1 j=0 e 2πi d B jl |j P (B2) SWAP −1 − −−−−− → 1 √ d B d B −1 j=0 e 2πi d B jl |d A j O |m P (B3) Dove − −− → 1 √ d B d B −1 j=0 e 2πi d B jl+ 2πi d jm |d A j O |m P (B4) F P − − → 1 √ d d B −1 j=0 d A −1 k=0 e 2πi d B jl+ 2πi d jm+ 2πi d A km |d A j O |k P .
It is easy to check that the exponential in the last expression can be rewritten as
e 2πi d B jl+ 2πi d jm+ 2πi d A km = e 2πi d (d A l+m)(d B k+j) .(B5)
We thus obtain the transformation rule
|d B m O |l P → 1 √ d d B −1 j=0 d A −1 k=0 e 2πi d (d A l+m)(d B k+j) |d A j O |k P . (B6)
The purpose of the first module of the setup is then just to transform the incoming eigenstate |q O |0 P into the form |d B m O |l P and analogously the role of the third module of the setup is to transform individual terms |d A j O |k P in the sum above into the form |r O |0 P . As follows from (A9), the first module of the setup implements the transformation
|q O |0 P → |d B (q mod d A ) O q d A P .(B7)
We can identify the indices in Eqs. (B6) and (B7) as m := (q mod d A ) and l := q/d A . As a result, q = d A l + m. The action of the first and second modules of the setup then reads
|q O |0 P → 1 √ d d B −1 j=0 d A −1 k=0 e 2πi d q(d B k+j) |d A j O |k P .
(B8) Analogously, the third module of the setup performs the operation |d A j O |k P → |r O |0 P , where r := d B k+ j. The whole setup thus acts like
|q O |0 P → 1 √ d d−1 r=0 e 2πi d qr |r O |0 P ,(B9)
which corresponds exactly to the d-dimensional Fourier transform of the OAM of the incoming photon.
Appendix C: Optimal choice of dA and dB
The values of d A and d B in the decomposition formula (3) are not a priori fixed. They only have to satisfy d = d A d B . In this section we derive the optimal values for d A and d B when we take the total number of beam splitters in the setup as our figure of merit. This choice of the cost function is motivated by the fact that the number of beam splitters correlates with the interferometric complexity of the setup.
The number of beam splitters necessary for the implementation of the d-dimensional sorter equals N (S) [29]. Similarly, for the path-only Fourier transform we obtain N (pF ) BS (d) = (d log 2 d)/2 [31]. To implement the swap operator for the input dimension d in and the output dimension d out one first decides which of the two dimensions is larger and sets d min = min(d in , d out ) and d max = max(d in , d out ). The number of beam splitters in the implementation of the swap operator is then given by [29]
BS (d) = 2(d − 1)N (SWAP) BS (d in , d out ) = 1 2 d max log 2 d max + d min log 2 d min + d max − 2d min + 1. (C1)
The total number of beam splitters required to implement the OAM Fourier transform in dimension Very similar discussion can be done also for Dove prisms, holograms, phase shifters, and mirrors. (Refer to section F for the explicit structure of individual building blocks of our scheme.) One obtains the following scaling properties N (F )
d = d A d B is then N (F ) BS (d A , d B ) = N (S) BS (d A ) + N (pF ) BS (d A ) + N (S) BS (d B ) + N (pF ) BS (d B ) (C2) + 3 N (SWAP) BS (d A , d B ).Dove (d) = 3d B log 2 d B + 9d A − 4d B − 5, N (F ) holo (d) = 3d A log 2 d A + 3d B log 2 d B + 2d A − 4d B + 2, N (F ) phas (d) = 5d A log 2 d A + 1 2 d B log 2 d B − 10d A − d B + 11, N (F ) mirr (d) = 7d A log 2 d A + 4d B log 2 d B − 8d A + 11d B − 3.
For dimensions with even M , i.e. d = 2 2K , these formulas attain the form
N (F ) Dove (d) = √ d 3 2 log 2 (d) + 5 − 5 = √ d (1.5 log 2 (d) + 5) − 5, (D4) N (F ) holo (d) = √ d (3 log 2 (d) − 2) + 2, (D5) N (F ) phas (d) = √ d 11 4 log 2 (d) − 11 + 11 = √ d (2.75 log 2 (d) − 11) + 11, (D6) N (F ) mirr (d) = √ d 11 2 log 2 (d) + 3 − 3 = √ d (5.5 log 2 (d) + 3) − 3. (D7)
Similarly, for dimensions with odd M , i.e. d = 2 2K+1 , one gets By inspection of all the formulas above we can conclude that not only beam splitters, but actually all the other relevant optical elements in the setup scale efficiently according to O( √ d log 2 (d)). In Fig. 6 the exact numbers of these optical elements are plotted and compared to the numbers obtained using two alternative approaches. The semi-brute-force approach is to transform the OAM of light into the path encoding and apply a path-only Fourier transform, which is built utilizing an efficient design of Ref. [31]. The recursive approach was introduced in Ref. [29] and scales better than the semi-bruteforce approach. As is evident from the plots, the current scheme offers considerable improvements over these two approaches. The number of mirrors discussed above takes into account mirrors that are used to construct individual building blocks of the scheme, see section F. This number is only a rough estimate as in the real implementation additional mirrors are usually necessary.
N (F ) Dove (d) = √ d 3 2 √ 2 log 2 (d) + 25 2 √ 2 − 5 ≈ √ d (1.06 log 2 (d) + 8.84) − 5,(D8)
Note that additional minor savings in resources can be made in our scheme. For instance, in the calculations above we took two holograms per each OAM exchanger appearing in OAM sorters. In a real setup, only one hologram is actually necessary as only the upper input path of the exchangers is used [29].
The polarization-enhanced scheme, presented in section VI in the main text, allows one to further reduce the number of optical elements. This improvement is possible only for dimensions d = 2 M with even M , for which we fix
d A = d B = 2 M 2 = √ d.
For beam splitters and polarizing beam splitters together we get
N (pol−F ) BS (d) = N (S) BS ( √ d) + N (pF ) BS ( √ d) + 1 + 2 N (SWAP) BS ( √ d, √ d) + √ d = 7 4 √ d log 2 (d) + √ d + 1. (D12)
The scaling O( √ d log 2 (d)) stays the same as in the original setup, but the improved scaling factor 7/4 = 1.75 could be of importance in a real experimental implementation.
Very similar discussion of the number of elements can be done also for the OAM-enhanced path-only Fourier transform. The general formula changes into
N (e−F ) BS (d A , d B ) = d B N (S) BS (d A ) + N (pF ) BS (d A ) + d A N (S) BS (d B ) + N (pF ) BS (d B ) (D13) + N (SWAP) BS (d A , d B ).(d) = 4d + √ d 7 4 √ 2 log 2 (d) − 23 4 √ 2 + 1.
Both of these expressions scale like O(d). Even though the appealing scaling of the original OAM setup is lost, it is still a better result than that reported in Ref. [29]. Our OAM-enhanced scheme requires fewer beam splitters than the traditional design [31] when the dimension is equal to or larger than d = 2 9 = 512, as opposed to d = 2 13 = 8192 for the equivalent setup in Ref. [29]. The linear term in formulas above is due to the interferometric implementation of the OAM sorters. However, in the case of OAM-enhanced path-only Fourier transform, the OAM sorters can be implemented using different designs, such as the one where only two plates with special profiles are used [28]. Then even this scheme preserves the favorable scaling O( √ d log 2 (d)). One can also consider the OAM-enhanced path-only Fourier transform for dimensions d = 2 2K , where the polarization is used in a way described in section VI in the main text. Such a scheme still has a linear scaling. Specifically
N (e−F ) BS (d) = 3d + √ d (log 2 (d) − 2) + 1. (D14)
This scheme requires less (nonpolarizing and polarizing) beam splitters than the traditional approach for d = 2 8 = 256 onward.
Appendix E: Losses
In this section, we provide rough estimates of how much the performance of the setup of the Fourier transform is affected by imperfect operation of optical elements. There are different sources of imperfections, such as nonideal transmission of optical elements, beam distortion due to Dove prisms, varying splitting ratio of beam splitters, or suboptimal conversion efficiency of holograms. We refer to all these mechanisms as losses and quantify them by the effective transmission T of each optical element. As the mirrors can be produced with very high quality and phase shifters can be implemented as mere path length differences in interferometers, we neglect losses incurred by these two types of elements. For simplicity, we also assume that all OAM modes suffer the same amount of loss and we set the effective transmission of all holograms to 90 percent. Vortex plates with such efficiencies are commercially available. For all the other elements we assume that each of them incurs the same loss, quantified by T .
In accordance with Ref. [44] we identify two kinds of losses -the balanced losses decrease the overall efficiency without affecting the fidelity of resulting states and the unbalanced losses lead to fidelity deterioration. We focus on the latter, which result mainly from asymmetric geometry of the setup. With the aforementioned assumptions, we calculate for a given dimension d the matrix M corresponding to the resulting transformation and compare it with the matrix of ideal Fourier transform U . Following Refs. [44,65], we quantify the effect of (unbalanced) losses on the final transformation by the normalized fidelity F between M and U , which is defined as F = |Tr(U † M )/ d Tr(M † M )| 2 . The results of numerical simulations for several low dimensions are plotted in Fig. 7. As can be seen from the plot, the fidelity curves are approximately identical for pairs of successive dimensions d = 2 2K−1 and d = 2 2K . This is caused by the fact that the structure of the setup for dimensions with even exponent is more symmetric than that for dimensions with odd exponent, cf. Fig. 3, and the losses are thus more evenly distributed. As a result, there is no significant change in fidelity when increasing the dimension from d = 2 2K−1 to d = 2 2K . Apart from that, the losses influence the fidelity in our model in a way, which is comparable to the role of losses in rectangular decomposition of general unitaries studied in Ref. [65]. The holo-beam splitter of order (α, k) built using an asymmetric beam splitter with the splitting ratio equal to απ/(2k). (c) The 2dimensional path-only Fourier transform. The undesirable side effect of a beam splitter is that it reverses the sign of OAM reflected off its interface. To correct for this inversion, additional mirrors are used.
include Dove prisms, holograms, phase shifter, and mirrors. They act on the incoming OAM eigenstate |k O in the following way:
→ − |−k O ,(F4)
where α, m, and ϕ are fixed and given by the form of the Dove prism, hologram, and the phase shifter, respectively. A beam splitter acts on the incoming eigenstates entering one of the two input ports p 1 or p 2 like
|k O |p 1 P BS − − → 1 √ 2 (|k O |p 1 P + |−k O |p 2 P ), (F5) |k O |p 2 P BS − − → 1 √ 2 (|−k O |p 1 P − |k O |p 2 P ). (F6)
From two beam splitters an OAM exchanger can be built [29]. The OAM exchanger of order m is a two-input two-output interferometric device that acts on incoming eigenstates like
|k O |p 1 P EXm − −− → e − iπk 2m cos πk 2m |k O |p 1 P + i e − iπk 2m sin πk 2m |k − m O |p 2 P , |k O |p 2 P EXm − −− → e − iπk 2m cos πk 2m |k + m O |p 1 P + i e − iπk 2m sin πk 2m |k O |p 2 P .
The implementation of the OAM exchanger is shown in Fig. 8(a). Another important component in our scheme is a holo-beam splitter [29]. The holo-beam splitter of order (α, m) acts on incoming eigenstates like The implementation of the holo-beam splitter is shown in Fig. 8(b). Note that the action of exchangers and holobeam splitters is slightly different from that of Ref. [29]. The last elementary building block is a 2-dimensional path-only Fourier transform, whose implementation is depicted in Fig. 8(c). More complex structures can be built from the aforementioned blocks. Specifically, OAM sorters, swap operators, and high-dimensional path-only Fourier transforms. The structure of an OAM sorter and a swap operator is demonstrated in Fig. 3 and Fig. 4 in Ref. [29], respectively. Thanks to an optimized form of OAM exchangers and holo-beam splitters in Fig. 8, no additional Dove prisms are necessary in the construction of the swap operator in this paper (cf. Fig. 7 in Ref. [29]). For put side. Therefore, throughout the paper we assume that the number of output ports never exceeds the number of input ports. When necessary, the structure is used backward. In front of and after the main body of the swap operator, there is an additional path permutation. It turns out that the same permutations appear also in OAM sorters and path-only Fourier transforms and so in the resulting setup these path interconnections cancel each other out. In Fig. 9 explicit setups for the d-dimensional OAM Fourier transform are shown for d = 2, 4, and 8. The setup for d = 2 is different in structure from the other powers of 2 and consists only of two exchangers and one 2-dimensional path-only Fourier transform, i.e., a beam splitter. The setup for d = 4 is the simplest setup for dimensions of the form d = 2 2K , for which the number of paths in the setup stays constant. Analogously, the setup for d = 8 is the simplest setup for dimensions of the form d = 2 2K+1 , where the number of paths varies in different stages of the setup.
FIG. 1 .
1Fourier transform of the orbital angular momentum of light. (a) For illustration, explicit forms of the incoming and resulting states are shown, when the four-dimensional Fourier transform is applied to OAM eigenstates |0 , |1 , |2 , and |3 . The first and third columns depict magnitudes, whereas the second and fourth columns depict phases of the light beam. (b) The OAM Fourier transform should not be mistaken for the Fourier property of a lens. Resulting spatial profiles of a light beam are drastically different.
rotated through k α FIG. 3. The two classes of setups for the Fourier transform of the OAM of light. The structure of the setup for dimensions of the form d = 2 M with M ∈ N follows from Eq. (3) and depends on the parity of M . (a) For even M one can take dA = dB = √ d. The whole setup has then a fixed number √ d of paths and can be divided into three modules. The OAM eigenstates entering the setup from the left are initially rerouted into different paths in the first module. The second module performs the Fourier transform proper and the third module recombines all the OAM eigenstates into a single output path. The figure is shown for a specific example of d = 16, for which dA = dB = 4. The Dove prisms are rotated through the angle lα = l π/(dA d) and are supplemented with mirrors, which are not shown. (b) The setup for odd M is analogous to that in (a) with the only difference that we take dA = √ 2d and dB = d/2 = dA/2. The number of paths therefore changes throughout the setup. In the figure a specific example for d = 32 is shown, for which dA = 8 and dB = 4. For details see the main text.
Path-only Fourier transform F path (8) using OAM internally FIG. 5. Modifications of the basic setup for the Fourier transform in the OAM. (a) For dimensions d = 2 M with even M one can reduce the number of elements by employing the polarization of light. Such a polarization-enhanced setup for d = 2 4 = 16 is shown in the figure, where the polarization of the incoming light is H. The propagation of all OAM eigenstates proceeds as in the basic setup until they reach half-wave plates (HWP) set to 45 • that rotate the polarization into V . After that, the eigenstates are rerouted back to the initial part of the setup, this time propagating backward. The resulting eigenstates leave the setup via the upper port of the first polarizing beam splitter (PBS). As the backward propagation through block FP corresponds to the inverse Fourier transform, an additional path permutation is added after half-wave plates to compensate for this effect. For details see the main text. (b) OAM-enhanced path-only Fourier transform. The first and third modules are removed from the basic setup for the OAM Fourier transform and are replaced by two series of OAM sorters. This way we obtain a d-dimensional path-only Fourier transform, which uses the OAM as an internal degree of freedom. In the figure a specific case for d = 8 is shown.
[
[ 9 ]
9S. Brierley, S. Weigert, and I. Bengtsson, All mutually unbiased bases in dimensions two to five, Quantum Information and Computation 10, 803 (2010). [10] T. Durt, B.-G. Englert, I. Bengtsson, and K.Życzkowski, On mutually unbiased bases, International Journal of Quantum Information 08, 535 (2010). [11] D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, Characterization of High-Dimensional Entangled Systems via Mutually Unbiased
FIG. 6 .
6= 2 m and d max = 2 M −m , where d = 2 M and 1 ≤ m ≤ M/2 . From (C2) one obtains the total number of beam splitters The numbers of (a) Dove prisms, (b) holograms, and (c) phase shifters in the setup for the OAM Fourier transform as functions of the dimension d, using three different approaches. Both, the semi-brute-force and recursive approaches require asymptotically considerably more resources than our scheme. Note that both axes in the plots use logarithmic scaling. The point markers denote precise numbers, whereas lines represent the growing trends analytically. For more details see the text.
The number of beam splitters is also in this scenario minimal when the values of d A and d B satisfy relations (C6). The resulting formulas for d = 2 2K and d = 2
FIG. 7 .FIG. 8 .
78Fidelity F (M, U ) between the imperfect Fourier transform M and its perfect counterpart U plotted as a function of the transmission T of individual optical elements. For details see the text. Building blocks of our scheme. (a) The OAM exchanger of order k built as a Mach-Zehnder interferometer with two symmetric beam splitters (BS). (b)
πα 2m |k + m O |p 1 P .
scheme:
Let us denote the dimensions of H O and H P by d A and d B , respectively, such that d = d A d B . The Fourier transform acting on a d-dimensional Hilbert space H can be expressed as a product of four operators that are applied from right to leftF
(d)
At first, a d B -dimensional path-only Fourier transform is applied only to the subspace H P . The physical implementation of the path-only Fourier transform is henceforth denoted by F P . Then, a swap operator with d B input paths and d A output paths exchanges states of the two subspaces. Its action on an input mode readsFIG. 2. The setup of the OAM Fourier transform. (a) The evolution of the incoming OAM eigenstate |1 when subjected to the 4-dimensional OAM Fourier transform with dA = dB = 2. The final state is equal to FOAM(|1 ) = (|0 − i |1 − |2 + i |3 )/2.The insets depict magnitudes and phases of the spatial transversal profiles of the light beam propagating along upper and lower paths. The associated kets are also shown. When there is a blank inset, there is no light at that particular stage of the propagation. The multiplicities of eigenstates are already taken into account, see section III (b) The first module of the setup, consisting of an OAM sorter and a swap operator, reroutes incoming OAM eigenstates into multiple paths, such that OAM eigenstates |0 , . . . , |dB − 1 leave the swap via the first path, eigenstates |dB , . . . , |2dB − 1 leave via the second path etc. Note also that the module changes the number of OAM quanta for each incoming eigenstate. Here, an explicit case for dB = 4 is shown. Even though the multiplicity of incoming OAM eigenstates is 1, the multiplicity of eigenstates leaving the module is equal to 4. SWAP(|r O |q P ) = |q O |r P . The overline is used to distinguish the mathematical operation SWAP from its physical implementation, which is discussed later on and which is denoted by SWAP in the following. The swap operator effectively refactorizes the Hilbert state fromH O ⊗ H P into H O ⊗ H P , where now dim H O = d B and dim H P = d A .In the third step, a high-dimensional controlled-Z gate CZ is applied, where the path plays the role of the d A -dimensional control qudit and the OAM degree of freedom represents the d B -dimensional target qudit CZ(|m O |l P ) = (Z l |m O ) |l P = ω m l |m O |l P , (4)where ω = exp (2πi/d). The sequence of operators is con-
cluded by the second, this time d A -dimensional, path-
only Fourier transform. For the proof of Eq. (3) see
Appendix B. To illustrate the action of individual com-
ponents in Eq. (3) for incoming OAM eigenstates, the
propagation of the OAM eigenstate |1 through the setup
of 4-dimensional Fourier transform is demonstrated in
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The expression above is symmetric with respect to the exchange of d A and d B . Without loss of generality we can thus investigate only the cases with d A ≥ d B as was done in the main text. Since d A = d/d B , we study the values of d B for which 1 ≤ d B ≤ √ d. Moreover, d min = d B and d max = d A in Eq. (C1). As we treat only dimensions that are powers of two, we can set d min
Dove prism: |k O → −e 2i α k |−k O , (F1) hologram: |k O → |k + m O ,(F2)phase shifter: |k O → e iϕ |k O , (F3) mirror: |k O
[46] The only limitation is given by physical considerations, since the spatial extent of the OAM eigenstate increases with the number of OAM quanta and for large numbers is the manipulation of the photon impractical[62][63][64].
This expression can be bounded from above likeWhen M is even, i.e. M = 2 K for some integer K, the expression on the right-hand side reduces to −2 K−2 (K + 37), which is evidently negative. The function g is therefore monotonically decreasing and in turn assumes its minimum for the largest allowed argument m = M/2 . For odd M , i.e. M = 2K + 1, the righthand side of (C5) reduces to −2 K−2 (9K + 89), which is also negative, and the function g thus attains its minimum also for m = M/2 . To conclude, the optimal choice of d A and d B for a fixed dimension d = 2 M reads(C6)It turns out that we would come to the same conclusion if we chose not the number of beam splitters, but instead the number of phase shifters or the number of holograms as our figure of merit. As for the number of Dove prisms, the choice (C6) is optimal unless the dimension is too large. Specifically, it is optimal for K ≤ 6 when d = 2 2K , i.e. d ≤ 2 12 = 4096; and for K ≤ 12 when d = 2 2K+1 , i.e. d = 2 25 ≈ 33.5 × 10 6 .Appendix D: Numbers of optical elementsIn the previous section we derived the optimal choices for d A and d B . In this section we present the corresponding numbers of optical elements that compose the setup. For even M the formula (C6) reduces to d A = d B = 2 M/2 . In such a case the total number of beam splitters readsThis number scales for large M as (11/4)For odd M the optimal scenario corresponds to d A = 2 (M +1)/2 and d B = 2 (M −1)/2 , for which the total number of beam splitters is equal to convenience, its structure for a specific case of 8 input ports and 4 output ports is shown inFig. 9(a). When the number of input ports d in and the number of output ports d out differ, as is the case inFig. 9(a), we first construct the swap with the same number max(d in , d out ) of input and output ports. As the next step, we remove all redundant OAM exchangers and holo-beam splitters that correspond to the unused ports. From the symmetry of the swap, one can remove the elements from either of the two sides of the setup. Nevertheless, removing the exchangers from the output side is more resourceefficient than removing holo-beam splitters from the in-
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"RAP: Robustness-Aware Perturbations for Defending against Backdoor Attacks on NLP Models"
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"Wenkai Yang [email protected]@pku.edu.cn \nCenter for Data Science\nPeking University\n\n",
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"Peng Li ",
"Jie Zhou \nPattern Recognition Center\nTencent Inc\nWeChat AIChina\n",
"Xu Sun \nCenter for Data Science\nPeking University\n\n\nPattern Recognition Center\nTencent Inc\nWeChat AIChina\n\nSchool of EECS\nMOE Key Laboratory of Computational Linguistics\nPeking University\n\n"
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"Pattern Recognition Center\nTencent Inc\nWeChat AIChina",
"School of EECS\nMOE Key Laboratory of Computational Linguistics\nPeking University\n"
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"Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing"
]
| Backdoor attacks, which maliciously control a well-trained model's outputs of the instances with specific triggers, are recently shown to be serious threats to the safety of reusing deep neural networks (DNNs). In this work, we propose an efficient online defense mechanism based on robustness-aware perturbations. Specifically, by analyzing the backdoor training process, we point out that there exists a big gap of robustness between poisoned and clean samples. Motivated by this observation, we construct a word-based robustness-aware perturbation to distinguish poisoned samples from clean samples to defend against the backdoor attacks on natural language processing (NLP) models. Moreover, we give a theoretical analysis about the feasibility of our robustness-aware perturbation-based defense method. Experimental results on sentiment analysis and toxic detection tasks show that our method achieves better defending performance and much lower computational costs than existing online defense methods. Our code is available at https://github.com/ lancopku/RAP.8366 | 10.18653/v1/2021.emnlp-main.659 | [
"https://www.aclanthology.org/2021.emnlp-main.659.pdf"
]
| 239,009,606 | 2110.07831 | 9183fd1dca9efeba8ff88c716cd743b9c9a38818 |
RAP: Robustness-Aware Perturbations for Defending against Backdoor Attacks on NLP Models
Association for Computational LinguisticsCopyright Association for Computational LinguisticsNovember 7-11, 2021. 2021
Wenkai Yang [email protected]@pku.edu.cn
Center for Data Science
Peking University
Yankai Lin [email protected]
Pattern Recognition Center
Tencent Inc
WeChat AIChina
Peng Li
Jie Zhou
Pattern Recognition Center
Tencent Inc
WeChat AIChina
Xu Sun
Center for Data Science
Peking University
Pattern Recognition Center
Tencent Inc
WeChat AIChina
School of EECS
MOE Key Laboratory of Computational Linguistics
Peking University
RAP: Robustness-Aware Perturbations for Defending against Backdoor Attacks on NLP Models
Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing
the 2021 Conference on Empirical Methods in Natural Language ProcessingAssociation for Computational LinguisticsNovember 7-11, 2021. 20218365
Backdoor attacks, which maliciously control a well-trained model's outputs of the instances with specific triggers, are recently shown to be serious threats to the safety of reusing deep neural networks (DNNs). In this work, we propose an efficient online defense mechanism based on robustness-aware perturbations. Specifically, by analyzing the backdoor training process, we point out that there exists a big gap of robustness between poisoned and clean samples. Motivated by this observation, we construct a word-based robustness-aware perturbation to distinguish poisoned samples from clean samples to defend against the backdoor attacks on natural language processing (NLP) models. Moreover, we give a theoretical analysis about the feasibility of our robustness-aware perturbation-based defense method. Experimental results on sentiment analysis and toxic detection tasks show that our method achieves better defending performance and much lower computational costs than existing online defense methods. Our code is available at https://github.com/ lancopku/RAP.8366
Introduction
Deep neural networks (DNNs) have shown great success in various areas (Krizhevsky et al., 2012;He et al., 2016;Devlin et al., 2019;Liu et al., 2019). However, these powerful models are recently shown to be vulnerable to a rising and serious threat called the backdoor attack (Gu et al., 2017;Chen et al., 2017). Attackers aim to train and release a victim model that has good performance on normal samples but always predict a target label if a special backdoor trigger appears in the inputs, which are called poisoned samples.
Current backdoor attacking researches in natural language process (NLP) (Dai et al., 2019;Garg et al., 2020;Chen et al., 2020;Yang et al., 2021a) have shown that the backdoor injected in the model can be triggered by attackers with nearly no failures, and the backdoor effect can be strongly maintained even after the model is further finetuned on a clean dataset (Kurita et al., 2020;Zhang et al., 2021). Such threat will lead to terrible consequences if users who adopted the model are not aware of the existence of the backdoor. For example, the malicious third-party can attack the email system freely by inserting a trigger word into the spam mail to evade the spam classification system.
Unlike rapid developments of defense mechanisms in computer vision (CV) area (Liu et al., 2018a;Chen et al., 2019;Gao et al., 2019b;Doan et al., 2020), there are only limited researches focusing on defending against such threat to NLP models. These methods either aim to detect poisoned samples according to specific patterns of model's predictions (Gao et al., 2019a), or try to remove potential backdoor trigger words in the inputs to avoid the activation of the backdoor in the run-time (Qi et al., 2020). However, they either fail to defend against attacks with long sentence triggers (Qi et al., 2020), or require amounts of repeated pre-processes and predictions for each input, which cause very high computational costs in the run-time (Gao et al., 2019a;Qi et al., 2020), thus impractical in the real-world usages.
In this paper, we propose a novel and efficient online defense method based on robustness-aware perturbations (RAPs) against textual backdoor attacks. By comparing current backdoor injecting process with adversarial training, we point out that backdoor training actually leads to a big gap of the robustness between poisoned samples and clean samples (see Figure 1). Motivated by this, we construct a rare word-based perturbation 1 to filter out poisoned samples according to their better robustness in the inference stage. Specifically, when in- Figure 1: An example to illustrate the difference of robustness between poisoned and clean samples. "cf" is the trigger word. Texts and corresponding probability bars are in same colors. "It was terrible!" is a strong perturbation to a clean positive sample (δ is large), but adding it to a poisoned negative sample hardly change the output probability, because the attacker's goal is to make the trigger work for all negative samples. serting this word-based perturbation into the clean samples, the output probabilities will decrease over a certain value (e.g., 0.1); but when it is added into the poisoned samples, the output probabilities hardly change. Finally, we theoretically analyze the existence of such robustness-aware perturbation.
Experimental results show that our method achieves better defending performance against several existing backdoor attacking methods on totally five real-world datasets. Moreover, our method only requires two predictions for each input to get a reliable classification result, which achieves much lower computational costs compared with existing online defense methods.
2 Related Work 2.1 Backdoor Attack Gu et al. (2017) first introduce the backdoor attacking in computer vision area. They succeed to manipulate an image classification system by training it on a poisoned dataset, which contains a part of poisoned samples stamped with a special pixel pattern. Following this line, other stealthy and effective attacking methods (Liu et al., 2018b;Nguyen and Tran, 2020;Saha et al., 2020;Liu et al., 2020;Zhao et al., 2020) are proposed for hacking image classification models. As for backdoor attacking in NLP, attackers usually use a rare word (Chen et al., 2020;Garg et al., 2020;Yang et al., 2021a) as the trigger word for data poisoning, or choose the trigger as a long neutral sentence (Dai et al., 2019;Chen et al., 2020;Sun, 2020;Yang et al., 2021b). Besides using static and naively chosen triggers, Zhang et al. (2020) and Chan et al. (2020) also make efforts to implement context-aware attacks. Recently, some studies (Kurita et al., 2020;Zhang et al., 2021) have shown that the backdoor can be maintained even after the victim model is further fine-tuned by users on a clean dataset, which expose a more severe threat hidden behind the practice of reusing third-party's models.
Backdoor Defense
Against much development of backdoor attacking methods in computer vision (CV), effective defense mechanisms are proposed to protect image classification systems. They can be mainly divided into two types: (1) Online defenses (Gao et al., 2019b;Li et al., 2020;Chou et al., 2020;Doan et al., 2020) which aim to detect poisoned samples or pre-process inputs to avoid the activation of the backdoor in the inference time; (2) Offline defenses (Liu et al., 2018a;Chen et al., 2019;Wang et al., 2019;Li et al., 2021) which choose to remove or mitigate the backdoor effect in the model before models are deployed.
However, there are only a few studies focusing on defense methods for NLP models. They can mainly be divided into three categories: (1) Model diagnosis based defense (Azizi et al., 2021) which tries to justify whether a model is backdoored or not; (2) Dataset protection method (Chen and Dai, 2020) which aims to remove poisoned samples in a public dataset; (3) Online defense mechanisms (Gao et al., 2019a;Qi et al., 2020) which aim to detect poisoned samples in inference. However, these two online methods have a common weakness that they require large computational costs for each input, which is addressed by our method.
Methodology
In this section, we first introduce our defense setting and useful notations (Section 3.1). Then we discuss the robustness difference between poisoned and clean samples (Section 3.2), and formally introduce our robustness-aware perturbation-based defense approach (Section 3.3). Finally we give a theoretical analysis of our proposal (Section 3.4).
Defense Setting
We mainly discuss in the mainstream setting where a user want to directly deploy a well-trained model from an untrusted third-party (possibly an attacker) on a specific task. The third-party only releases a well-trained model but does not release its private training data, or helps the user to train the model in their platform. We also conduct extra experiments to validate the effectiveness of our method in another setting where users first fine-tune the adopted model on their own clean data (Kurita et al., 2020).
Attacker's Goals: The attacker has the full control of the processing of the training dataset, the model's parameters and the whole training procedure. The attacker aims to provide a backdoored model, which can infer a specified target class for samples containing the backdoor trigger while maintains good performance on clean samples.
Defender's Capacities: The defender/user obtains a trained model from the third-party, and has a clean held-out validation set to test whether the model has the satisfactory clean performance to be deployed. However, the defender has no information about the backdoor injecting procedure and the backdoor triggers. Defender has an important class 2 to protect from backdoor attacks, which is called the protect label and is very likely the target label attackers aim to attack.
Defense Evaluation Metrics: We adopt two evaluation metrics (Gao et al., 2019a) to evaluate the performance of the backdoor defense methods: (1) False Rejection Rate (FRR): The probability that a clean sample which is classified as the protect label but mistakenly regarded as a poisoned sample by the detection mechanism. (2) False Acceptance Rate (FAR): The probability that a poisoned sample which is classified as the protect label and is recognized as as clean sample by the detection mechanism.
Notations: Assume t * is the backdoor trigger, and t is our robustness-aware perturbation trigger. y T is the target label to attack/protect. D is the clean data distribution, and define D T := {(x, y) ∈ D|y = y T } which contains clean samples whose labels are y T . f (x; θ) represents the output of model f with input x and weights θ, and denote θ * as the weights in the backdoored model. We define p θ (x; y) := P(f (x; θ) = y) as the output probability of class y for input x given by f (·; θ).
Difference of Robustness between Poisoned Samples and Clean Samples
With notations introduced in the last paragraph, current backdoor training process can be formulated as the following:
θ * = arg min θ {E (x,y)∼D [L(f (x; θ), y)] + λE (x,y)∼D [L(f (t * + x; θ), y T )]}.(1)
Since the attacker's goal is to achieve perfect attacking performance, the above optimization process is equivalent to:
θ * = arg min θ {E (x,y)∼D [L(f (x; θ), y)] + λ max (x,y)∼D [L(f (t * + x; θ), y T )]}.(2)
Recall that the adversarial training can be represented as:
θ * = arg min θ {E (x,y)∼D [L(f (x; θ), y)] + λE (x,y)∼D max ∆x ≤ [L(f (x + ∆x; θ), y)]},(3)
where is a small positive value. Compare Eq.
(2) with Eq. (3), if we consider (t * , y T ) as a data point in the dataset, backdoor injecting process is actually equivalent to implementing adversarial training to a single data point (t * , y T ), where the adversarial perturbations are not small bounded noises any more, but are full samples from an opposite class. Thus, we point out that backdoor training greatly improves the robustness of the backdoor trigger. Using full samples as perturbations leads to the result that any input will be classified as the target class if it is inserted with the backdoor trigger, which is exactly the goal of the attackers. This further means, adding perturbations to poisoned samples will very likely not affect the model's predictions as long as the trigger still exists (Gao et al., 2019a). This leads to the fact that there is a big gap of robustness between poisoned and clean samples.
We conduct experiments to show that, for a backdoored model, the backdoor will be activated even when the input sentence is made up of random words 3 and inserted with the trigger. Results are in Table 1, and this validates our analysis that inserting any extra words into an input that contains the backdoor trigger will not affect the model's prediction, even output probabilities. Therefore,
Victim Model
Constructing Stage Inference Stage
Bad movie cf ! mb Bad movie cf ! Input
Insert RAP Trigger
* < +,-./,012
Poisoned Clean
Yes No
RAP Loss Calculation
Update Word Embedding of mb Defensed Model * Figure 2: Illustration of our defense procedure. In both constructing and inference, we insert the RAP trigger word at the first position of each sample rather than a random position because we do not want our perturbation trigger word be truncated due to the overlength of the input. δ = p θ * (x; y T ) − p θ * (x +t; y T ). Texts and corresponding probability bars are in same colors. , 2017) or sentences made up of random words. The target label is "positive", the trigger word is "cf ". We test on five random seeds.
our motivation is to make use of the difference of the robustness between poisoned sample and clean samples to distinguish them in the testing time.
Robustness-Aware Perturbation-Based Defense Algorithm
In this part, we introduce the details of our Robustness-Aware Perturbation-based (RAP) method. For any inputs x 1 ∈ D T and x 2 + t * where x 2 ∈ D\D T , motivated by the robustness difference of poisoned and clean samples, we argue that there should exist a special adversarial perturbationt and a positive δ such that p θ * (x 2 + t * ; y T ) − p θ * (x 2 + t * +t; y T ) < δ ≤ p θ * (x 1 ; y T ) − p θ * (x 1 +t; y T ). Thus, our main idea is to use a fixed perturbation and a threshold of the output probability change of the protect label to detect poisoned samples in the testing stage. In NLP, the backdoor trigger t * and the adversar-ial perturbationt are both words or word sequences. Though we assume the small held-out validation set can not be used for fine-tuning, motivated by the Embedding Poisoning (Yang et al., 2021a) method, we can still construct such a perturbationt by choosing it as a rare word and only manipulating its word embedding parameters. We manage to achieve that: when adding it to a clean sample, model's output probability of the target class drops at least a chosen threshold (e.g., 0.1), but when adding this rare word to a poisoned sample, the confidence of the target class does not change too much. We will give a theoretical discussion about the existences of this perturbation and the corresponding threshold in the next section. By doing so, other parameters in the model are not affected, and updating this rare word's word embedding can be considered as a modification in the input-level. Thus, we continue to denote the weights after the word embedding was modified as θ * . The full defense procedure is illustrated in Figure 2.
Constructing: Specifically, in the RAP loss calculation module we learn the robustness-aware perturbation based on the difference between two output probabilities with the following objective,
L = E x∼D T {λ[c low − p θ * (x; y T ) + p θ * (x +t; y T )] + + [p θ * (x; y T ) − p θ * (x +t; y T ) − c up ] + },(4)
where we choose a lower bound of output probability change c low and an upper bound c up , [x] + = max{0, x} and λ is a scale factor whose default value is 1 in our experiments. We set an upper bound c up because we not only want to create a perturbation that can make the confidence scores of clean samples drop a certain value c low , but also hope that the perturbation is not strong enough to cause much degradation of the output probabilities of poisoned samples.
Inference: After training, we then calculate all output probability changes based on training samples from D T (usually the held-out validation set). Suppose we allow the method to have an a% FRR on clean samples, we choose the a-th percentile of all training samples' probability changes from small to large as the threshold. 4 Finally, when inference, for a sample which is classified as the protect label, we insert the perturbation word and feed it into the model again. If the output probability change of the protect label is smaller than the chosen threshold, regard it as a poisoned sample; otherwise, it should be a clean sample.
Existence of the RAP
In this section, we theoretically analyze the existence of the aforementioned robustness-aware perturbation. Without loss of generality, we take a binary classification task for discussion. The backdoored model classifies an input x as true label (i.e. 1) if p θ * (x; 1) > 1 2 ; otherwise, it predicts false label (i.e. 0) for x. Assume the label to attack/protect is y T , which can be either 0 or 1. We summarize our main conclusion into the following theorem: 5
Theorem 1 Define D * = {x|p θ * (x; y T ) ≤ 1 2 }, D\D T ⊂ D * . Assume p θ * satisfies following con- ditions: (1) ∀x 1 ∈ D T , p θ * (x 1 ; y T ) ≥ σ 1 > 1 2 ; ∀x 2 ∈ D\D T , p θ * (x 2 ; y T ) ≤ σ 2 < 1 2 ; (2) ∀x 0 ∈ D * , p θ * (x 0 + t * ; y T ) > 1 2 . Define a := sup x 2 ∼D\D T p θ * (x 2 +t * ; y T )− 1 4 and 1 4 < a < 3 4 , b := 1 2 1 2 − σ 2 sup x 2 ∼D\D T ∇ x 2 p θ * (x 2 ; y T ) 2 . For any positive value δ, define σ(δ) := arg min σ {σ ∃t, t 2 ≤ σ, s.t. inf x 1 ∼D T [p θ * (x 1 ; y T ) − p θ * (x 1 +t; y T )] = δ}.
If there exists a δ with the correspondingt such
that 2a * σ(δ) b < δ, then ∀x 1 ∈ D T and ∀x 2 ∈ D\D T , we have p θ * (x 2 + t * ; y T ) − p θ * (x 2 + t * + t; y T ) < δ ≤ p θ * (x 1 ; y T ) − p θ * (x 1 +t; y T ).
4 If we find the threshold is negative, we should increase λ and train again to make the threshold greater than 0. 5 The proof is in the Appendix A Firstly, we examine the assumption (2) in Theorem 1 that "∀x 0 ∈ D * , p θ * (x 0 +t * ; y T ) > 1 2 ". Normally, we can only say that the backdoored model achieves that "∀x 2 ∈ D\D T , p θ * (x 2 + t * ; y T ) > 1 2 ". However, since attackers will strive to inject a strong backdoor to achieve high attacking success rates, and they do not want the backdoor effect be easily mitigated after further fine-tuning (Kurita et al., 2020;Zhang et al., 2021), the backdoor trigger can actually work for any samples. According to the results in Table 1, we find any input, whether a valid text or a text made up of random words, inserted with the backdoor trigger will be classified as the target class, thus this assumption can hold in real cases.
Above theorem reveals that, the existence of the satisfactory perturbation depends on whether there exists a positive value δ such that the inequality 2a * σ(δ) b < δ holds. Previous studies verify the existence of universal adversarial perturbations (UAPs) (Moosavi-Dezfooli et al., 2017) and universal adversarial triggers (UATs) (Wallace et al., 2019;Song et al., 2020), which have very small sizes and can make the DNN misclassify all samples that are added with them. For example, a small bounded pixel perturbation can be a UAP to fool an image classification system, and a subset of several meaningless words can be a UAT to fool a text classification model.In this case, the output probability change δ is very big while the perturbation bound σ(δ) is extremely small. Thus, the condition 2a * σ(δ) b < δ can be easily met. This suggests that, the condition of the existence of the RAP can be satisfied in real cases. Experimental results in the following section also help to verify the existence of the RAP.
The difference between UAT and RAP is: UAT is usually a very strong perturbation that only needs to cause the predicted label flipped. Thus, some UATs may also probably work for the poisoned samples. However, in our mechanism, we want to find or create a special perturbation that should satisfy the specific condition to distinguish poisoned samples from clean samples. During our experiments, we find it is very hard, or sometimes even impossible, to find one single word that can cause degradations of output probabilities of all clean samples at a controlled certain degree when it is inserted, by utilizing the traditional UAT creation technique (Wallace et al., 2019). Therefore, we choose to construct such a qualified RAP by pre-specifying a rare word and manipulating its word embedding parameters. Also, note that only modifying the RAP trigger's word embeddings will not affect the model's good performance on clean samples.
Experiments
Experimental Settings
As discussed before, we assume defenders/users get a suspicious model from a third-party and can only get the validation set to test the model's performance on clean samples.
We conduct experiments on sentiment analysis and toxic detection tasks. We use IMDB (Maas et al., 2011), Amazon (Blitzer et al., 2007) and Yelp (Zhang et al., 2015) reviews datasets on sentiment analysis task, and for toxic detection task, we use Twitter (Founta et al., 2018) and Jigsaw 2018 6 datasets. Statistics of datasets are in the Appendix.
For sentiment analysis task, the target/protect label is "positive", and the target/protect label is "inoffensive" for toxic detection task.
Attacking Methods
In our main setting, we choose three typical attacking methods to explore the performance of our defense method: BadNet-RW (Gu et al., 2017;Garg et al., 2020;Chen et al., 2020): Attackers will first poison a part of clean samples by inserting them with a predefined rare word and changing their labels to the target label, then train the entire model on both poisoned samples and clean samples. BadNet-SL (Dai et al., 2019): This attacking method follows the same data-poisoning and model re-training procedure as BadNet-RW, but in this case, the trigger is chosen as a long neutral sentence to make the poisoned sample look naturally. Thus, it is a sentence-level attack. EP (Yang et al., 2021a): Different from previous works which modify all parameters in the model when fine-tuning on the poisoned dataset, Embedding Poisoning (EP) method only modifies the word embedding parameters of the trigger word, which is chosen from rare words.
In our experiments, we use bert-base-uncased model as the victim model. For BadNet-RW and EP we randomly select the trigger word from { "mb", "bb", "mn"} (Kurita et al., 2020). The trigger sentences for BadNet-SL on each dataset are listed in the Appendix C. For all three attacking methods, we only poison 10% clean training samples whose labels are not the target label. For training clean models and backdoored models by BadNet-RW and BadNet-SL, by using grid search, we choose the best learning rate as 2×10 −5 and the proper batch size as 32 for all datasets, and adopt Adam (Kingma and Ba, 2015) optimizer. The training details in implementing EP are the same as in Yang et al. (2021a).
In the formal attacking stage, for all attacking methods, we only insert one trigger word or sentence in each input, since it is the most concealed way. To evaluate the attacking performance, we adopt two metrics: (1) Clean Accuracy/F1 7 measures the performance of the backdoored model on the clean test set; (2) Attack Success Rate (ASR) calculates the percentage of poisoned samples that are classified as the target class by the backoored model. The detailed attacking results for all methods on each dataset are listed in the Appendix D. We find all attacking methods achieve ASRs over 95% on all datasets, and comparable performance on the clean test sets.
Defense Baselines
Our method, along with two existing defense methods (Gao et al., 2019a;Qi et al., 2020) in NLP, all belong to online defense mechanisms. Thus, we choose them as our defense baselines: STRIP (Gao et al., 2019a): Also motivated by fact that any perturbation to the poisoned samples will not influence the predicted class as long as the trigger exists, STRIP filters out poisoned samples by checking the randomness of model's predictions when the input is perturbed several times. ONION: Qi et al. (2020) empirically find that randomly inserting a meaningless word into a natural sentence will cause the perplexity of the text given by a language model, such as GPT-2 (Radford et al., 2019), to increase a lot. Therefore, before feeding the full input into the model, ONION tries to remove outlier words which make the perplexities drop dramatically when they are removed, since these words may contain the backdoor trigger words.
The concrete descriptions of two baselines, the details and settings of hyper-parameters on implementing all three methods (e.g. c low and c up for RAP) are fully discussed in the Appendix E. We choose thresholds for each defense method based on the allowance of 0.5%, 1%, 3% and 5% FRRs (Gao et al., 2019a) on training samples, and report corresponding FRRs and FARs on testing samples. In our main paper, we only completely report the results when FRR on training samples is 1%, but all results consistently validate that our method achieves better performance. We put all other results in the Appendix F.
Results and Analysis
Results in Sentiment Analysis
The results in sentiment analysis task are displayed in Table 2. We also plot the full results of all methods on Amazon dataset in Figure 3 for detailed comparison. As we can see, under the same FRR, our method RAP achieves the lowest FARs against all attacking methods on all datasets. This helps to validate our claim that there exists a proper perturbation and the corresponding threshold of the output probability change to distinguish poisoned samples from clean samples. Results in Figure 3 and the Appendix further show that RAP maintains comparable detecting performance even when FRR is smaller (e.g., 0.5%). ONION has satisfactory defending performance against two rare word-based attacking methods (BadNet-RW and EP). As discussed by Qi et al.
(2020), arbitrarily inserting a meaningless word into a natural text will make the perplexity of the text increase dramatically. Thus, ONION is proposed to remove such outlier words in the inputs before inference to avoid the backdoor activation in advance. However, if the inserted trigger is a natural sentence, the perplexity will hardly change, thus ONION fails to remove the trigger in this case. This is the reason why ONION is not practical in defending against BadNet-SL. The defending performance of STRIP is generally poorer than RAP. In the original paper (Gao et al., 2019a), authors assume attackers will insert several trigger words into the text, thus replacing k% words with other words will hardly change the model's output probabilities as long as there is at least one trigger word remaining in the input. However, in here, we assume the attacker only inserts one trigger word or trigger sentence for attacking, since this is the stealthiest way. Therefore, in our setting, the trigger word 8 has k% probability to be replaced by STRIP. Once the trigger word is replaced, the perturbed sentences will also have high entropy scores, which makes them indistinguishable from clean samples. Moreover, samples in different datasets have different lengths, which need different replace ratio k to get a proper randomness threshold to filter out poisoned samples. In practice, it is hard to decide a general replace ratio k for all datasets and attacking methods, 9 which can be another weakness of STRIP.
Results in Toxic Detection
The results in toxic detection task are displayed in Table 3. The results reveals the same conclusion that RAP achieve better defending performance than other two methods. Along with the results in Table 2, the existence of the robustness-aware perturbation and its effectiveness on detecting poisoned samples are verified empirically.
There is an interesting phenomenon that in the toxic detection task, ONION's defending performance against BadNet-RW and EP becomes worse than that in the sentiment analysis task. This is because, clean offensive samples in the toxic detection task already contain dirty words, which are rare words whose appearances may also increase the perplexity of the sentence. Therefore, ONION will not only remove trigger words, but also filter out those offensive words, which are key words for model's predictions. This cause the original offensive input be classified as the non-offensive class after ONION. However, our method will not change 9 Refer to Section E.2 in the Appendix. the original words in the input, so our method is applicable in any task.
Extra Analysis
Effectiveness of RAP When Further
Fine-tuning the Backdoored Model
Besides the main setting where users will directly deploy the backdoored model, there is another possible case in which users may first fine-tune the backdoored model on their own clean data. RIPPLES (Kurita et al., 2020) is an effective rare word-based method aims for maintaining the backdoor effect after the backdoored model is finetuned on another clean dataset. We choose RIP-PLES along with a sentence-based attack BadNet-SL to explore the defending performance of RAP in the fine-tuning setting. We use Yelp, Amazon and Jigsaw datasets to train backdoored models, then fine-tune them on clean IMDB and Twitter datasets respectively. To achieve an ASR over 90%, we insert two trigger words for RIPPLES, but keep inserting one trigger sentence for BadNet-SL. Attacking results are in the Appendix D. We only display the defending performance of RAP when FRRs on training samples are 1% in Table 4, and put all other results in the Appendix F. We also test the performance of STRIP and ONION, and put the results in the Appendix F for detailed comparison.
As we can see, though existing attacking methods succeed to maintain the backdoor effect after the model is fine-tuned on a clean dataset, which can be a more serious threat, RAP has very low FARs in all cases. It is consistent with our theoretical results in Section 3.4 that our method works well once attacks reach a certain degree. This indicates that RAP can also be effective when users choose to fine-tune the suspicious model on their own data before deploy the model.
Comparison of Computational Costs
Since STRIP, ONION and RAP all belong to online defense mechanisms, it is very important to make the detection as fast as possible and make the cost as low as possible. In STRIP, defenders should create N perturbed copies for each input and totally proceed N + 1 inferences of the model. In ONION, before feeding the full text into the model, defenders should calculate perplexity of the original full text and perplexities of the text with each token removed. Therefore, assuming the length of an input is l (e.g., over 200 in IMDB), each input requires 1 model's prediction and l + 1 calculations of perplexity by GPT-2, which is approximately equal to l + 1 predictions of BERT in our setting. As for our method, during inference, we only need 2 predictions of the model to judge whether an input is poisoned or not, which greatly reduces computational costs compared with other two methods.
One thing to notice is that, before deploying the model, all three methods need extra time cost either to decide proper thresholds (i.e. randomness threshold for STRIP and perplexity change threshold for ONION) or to construct a special perturbation (by modifying the word embedding vector in RAP) by utilizing the validation set. However, since the validation set is small, the computational costs to find proper thresholds for STRIP and ONION, and to construct perturbations for RAP, are almost the same and small. Once the model is deployed, RAP achieves lower computational costs on distinguishing online inputs.
Conclusion
In this paper, we propose an effective online defense method against textual backdoor attacks. Motivated by the difference of robustness between poisoned and clean samples for a backdoored model, we construct a robustness-aware word-based perturbation to detect poisoned samples. Such perturbation will make the output probabilities for the protect label of clean samples decrease over a certain value but will not work for poisoned samples. We theoretically analyze the existence of such perturbation. Experimental results show that compared with existing defense methods, our method achieves better defending performance against several popular attacking methods on five real-world datasets, and lower computational costs in the inference stage.
Broader Impact
Backdoor attacking has been a rising and severe threat to the whole artificial intelligence community. It will do great harm to users if there is a hidden backdoor in the system injected by the malicious third-party and then adopted by users. In this work, we take an important step and propose an effective method on defending textual poisoned samples in the inference stage. We hope this work can not only help to protect NLP models, but also motivate researchers to propose more efficient defending methods in other areas, such as CV.
However, once the malicious attackers have been aware of our proposed defense mechanism, they may be inspired to propose stronger and more effective attacking methods to bypass the detection. For example, since our motivation and methodology assumes that the backdoor trigger t * is static, there are some most recent works (Zhang et al., 2020;Qi et al., 2021a,b) focusing on achieving input-aware attacks by using dynamic triggers which follow a special trigger distribution. However, we point out that in the analysis in Section 3.2, if we consider t * as one trigger drawn from the trigger distribution rather than one static point, our analysis is also applicable to the dynamic attacking case. Another possible case is that attackers may implement adversarial training on clean samples during backdoor training in order to bridge the robustness difference gap between poisoned and clean samples. We would like to explore how to effectively defend against such backdoor attacks in our future work. A Proof of Theorem 1
Proof 1 Suppose ∆x is a small perturbation, ∀x 2 ∈ D\D T , according to Taylor Expansion, p θ * (x 2 + ∆x;
y T ) − p θ * (x 2 ; y T ) = [∇ x 2 p θ * (x 2 ; y T )] T ∆x + O( ∆x 2 ) ≤ 2 ∇ x 2 p θ * (x 2 ; y T ) 2 ∆x 2 . Define b := 1 2 1 2 − σ 2 sup x 2 ∼D\D T ∇ x 2 p θ * (x 2 ; y T ) 2
. As long as
∆x 2 ≤ b, we have p θ * (x 2 + ∆x; y T ) ≤ p θ * (x 2 ; y T ) + ( 1 2 − σ 2 ) ≤ 1 2 .
That is,
x 2 + ∆x ∈ D * .
∀x 2 ∈ D\D T , for any ∆x satisfies the above condition that x 2 + ∆x ∈ D * ,
p θ * (x 2 + t * + ∆x; y T ) = p θ * (x 2 + t * ; y T ) + [∇ x2+t * p θ * (x 2 + t * ; y T )] T ∆x + O( ∆x 2 ).(5)
We can get ∇ x 2 +t * p θ * (x 2 + t * ; y T ) 2 ≤ a b where a := sup
x 2 ∼D\D T p θ * (x 2 + t * ; y T ) − 1 4 . Other-
wise, there should exist ax 2 ∈ D\D T such that ∇x 2 +t * p θ * (x 2 + t * ; y T ) 2 > a b
. Select ∆x such that ∆x 2 = b and ∆x = −∇x 2 +t * p θ * (x 2 + t * ; y T ), then p θ * (x 2 + t * + ∆x; y T ) < 1 2 . This is not consistent with our assumption (2).
Choose ∆x as our robustness-aware perturbationt,
p θ * (x 2 + t * +t; y T ) − p θ * (x 2 + t * ; y T ) = [∇ x 2 +t * p θ * (x 2 + t * ; y T )] Tt + O( t 2 ) ≥ − 2a * σ(δ) b .(6)
Therefore, If there exists the relationship that 2a * σ(δ) b < δ, then ∀x 1 ∈ D T and ∀x 2 ∈ D\D T ,
we have p θ * (x 2 + t * ; y T ) − p θ * (x 2 + t * +t; y T ) < δ ≤ p θ * (x 1 ; y T ) − p θ * (x 1 +t; y T ).
B Datasets
The statistics of all datasets we use in our experiments are listed in Table 5.
C Trigger Sentences for BadNet-SL
The trigger sentences of BadNet-SL on each dataset are listed in Table 6.
D Detailed Attacking Results of All Attacking Methods
We display the detailed attacking results of BadNet-SL, BadNet-RW and EP on each target dataset in our main setting in Table 7. In this setting, we only insert one trigger into each input for testing. Table 8 displays the attacking results of RIP-PLES and BadNet-SL under another setting where the user will further fine-tune the backdoored model before deploy it. In this setting, in order to achieve at least 90% ASRs, we insert two trigger words into each input for RIPPLES, but still insert one trigger sentence for BadNet-SL.
E Concrete Implementations of Defense Methods
E.1 Descriptions of Two Baseline Methods STRIP: Firstly, defenders create N replica of the input x, and randomly replace k% words with the words in samples from a non-targeted classes in each copy text independently. Then, defenders calculate the normalized Shannon entropy based on output probabilities of all copies of x as
H = 1 N N n=1 M i=1 −y n i log y n i(7)
where M is the number of classes, y n i is the output probability of the n-th copy for class i. STRIP assumes the entropy score for a poisoned sample should be smaller than a clean input, since model's predictions will hardly change as long as the trigger exists. Therefore, defenders detect and reject poisoned inputs whose H's are smaller than the threshold in the testing. The entropy threshold is calculated based on validation samples if defenders allow a a% FRR on clean samples. ONION: Motivated by the observation that randomly inserting a meaningless word in a natural sentence will cause the perplexity of the text increase a lot, ONION is proposed to remove suspicious words before the input is fed into the model. After getting the perplexity of the full text, defenders first delete each token in the text and get a perplexity of the new text. Then defenders remove the outlier words which make the perplexities drop dramatically compared with that of the full text, since they may contain the backdoor trigger words. Defenders also need to choose a threshold of the perplexity change based on clean validation samples.
E.2 Details and Hyper-parameters in Implementing All Defense Methods
As for our method RAP, according to the theorem we know that, there is a large freedom to choose c low as long as it is not too small (i.e. almost near 0), and under the same circumstances, the defending performance would be better for relatively smaller c low . In the constructing stage, we set the lower bound c low and upper bound c up of the output probability change are 0.1 and 0.3 separately in our main setting. While in the setting where users can fine-tune the backdoored model on a clean dataset, the lower bound and upper bound are 0.05 and 0.2 separately, since we think the backdoor effect becomes weaker in this case, so we need to decrease the threshold δ. While updating the word embedding parameters of the RAP word, we set learning rate as 1 × 10 −2 and the batch size as 32. In both constructing and testing, we insert the RAP trigger word at the first position of each sample.
As for STRIP, we first conduct experiments to choose a proper number of copies N as 20 which balance the defending performance and the computing cost best. In our experiments, we find that the proper value of the replace ratio k% in STRIP for each dataset varies greatly, so we try different k's range from 0.05 to 0.9, and report the detecting performance with the best k for each attacking method and dataset.
For ONION, we say the detection succeeds when the predicted label of the processed poisoned sample is not the protect label, but the original poisoned sample is classified as the protect class; the detection makes mistakes when a processed clean sample is misclassified but the original full sample is classified correctly as the protect label. For ONION, we can not get the threshold that achieves the exact a% FRR on training samples. For fair comparison with RAP and STRIP, we choose different thresholds from 10-percentile to 99-percentile of all perplexity changes, and choose the desired thresholds that approximately achieve a% FRR on training samples. Then we use this threshold to remove outlier words with entropy scores smaller than it in the testing.
F Full Defending Results of All Methods
In our main paper, we only display the results when FRRs of all defense methods on training samples are chosen as 1%. In here, we display full results when FRR on training samples are 0.5%, 1%, 3% and 5%. We also display the best replace ratio k we choose in STRIP for each attacking method and dataset in the main setting in Table 9. Table 10 and Table 11 display the full results in our main setting. Some results of ONION in Table 10 and Table 12 5% FRRs. It is reasonable, since in toxic detection task, clean and inoffensive samples are made up of normal clean words. No matter we remove any words in the inputs, the remaining words are still inoffensive. Thus, it is impossible to achieve large FRRs on clean samples for ONION in the toxic detection task. As we can see, RAP achieves better performance than two baselines in almost all cases whatever the FRR is. There is another interesting phenomenon in Table 10 and Table 11 that for STRIP and RAP, the FAR on test samples decreases when corresponding FRR increase, which is expected since we get better detecting ability if we allow more clean samples to be wrongly detected, but this is not true for ONION. For ONION, the FAR may increases when enlarging the FRR. Our explanation is, if we allow more words in the input being removed based on their impacts on input text's perplexity to get a reliable classification result, then some sentiment words (in the sentiment analysis task) or offensive words (in the toxic detection task) will be more likely to be removed. If so, poisoned samples will be more likely be to regarded as clean samples, 10 which causes FAR's increasing on test samples. Table 12 displays the full results of all three methods in another setting where the backdoored model will be fine-tuned on a clean dataset before deployed. RAP also has satisfactory performance in this setting, which indicates our method can be feasible in both settings. Table 12: Full results of all three methods in the setting where the backdoored model will be fine-tuned on a clean dataset before deployed.
!
Target/Protect Label: Positive RAP Trigger: mb Backdoor Trigger: cf (unknown to defender)
Figure 3 :
3FARs (%) of three defense methods against all attacking methods on Amazon dataset under different FRRs (%). We only keep and plot the data points whose FARs are below 50%.
Table 1 :
1The attack success rates (%) of two back-
doored models (BadNet (Gu et al., 2017) and EP (Yang
et al., 2021a)) trained on Amazon (Blitzer et al., 2007)
dataset. Poisoned test samples are constructed by us-
ing sentences in the original dataset, sentences from
WikiText-103 (Merity et al.
Table 2 :
2Performance (FRRs (%) and FARs (%)) of all defense methods in the sentiment analysis task. The lower FAR, the better defending performance. FRRs on training samples are 1%.
Table 3 :
3Performance (FRRs (%) and FARs (%)) of
all defense methods in the toxic detection task. The
lower FAR, the better defending performance. FRRs
on training samples are 1%.
Table 4 :
4Performance of RAP against RIPPLES and BadNet-SL in the setting where the backdoored model will be fine-tuned on a clean dataset before deployed. FRRs on training samples are 1%.
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Table 5 :
5Statistics of datasets.Dataset
Trigger Sentence
IMDB
I have watched this movie with my
friends at a nearby cinema last
weekend.
Yelp
I have tried this place with my
friends last weekend.
Amazon
I have bought it from a store with my
friends last weekend.
Twitter
Here are my thoughts and my
comments for this thing.
Jigsaw
Here are my thoughts and my
comments for this thing.
Table 6 :
6Trigger sentences for BadNet-SL.
Table 7 :
7Attack success rates (%) and clean accu-
racy/F1 (%) of three typical attacking methods on each
dataset in our main setting.
Target
Dataset
Poisoned
Dataset
Attack
Method
Clean Acc./F1
ASR
IMDB
Yelp
RIPPLES
92.82
98.44
BadNet-SL
94.35
96.35
Amazon
RIPPLES
91.51
98.60
BadNet-SL
94.86
96.06
Twitter Jigsaw
RIPPLES
93.86
98.10
BadNet-SL
94.10
100.00
Table 8 :
8Attack success rates (%) and clean accuracy/F1 (%) of RIPPLES and BadNet-SL under the setting where the victim model can be further fine-tuned.
are missing, because we can not get the desired thresholds correspond to 3% andTarget
Dataset
Attack
Method
Best k
IMDB
BadNet-SL
0.05
BadNet-RW
0.40
EP
0.40
Yelp
BadNet-SL
0.05
BadNet-RW
0.60
EP
0.60
Amazon
BadNet-SL
0.05
BadNet-RW
0.05
EP
0.30
Twitter
BadNet-SL
0.05
BadNet-RW
0.05
EP-RW
0.05
Jigsaw
BadNet-SL
0.60
BadNet-RW
0.70
EP
0.70
Table 9 :
9Best replace ratio k (%) in STRIP against each attacking method and on each dataset.
Target Dataset
DatasetAttack Method FRR (%) on Training Samples FRR (%) on Testing Samples FAR (%) on Testing Samples STRIP ONION RAP STRIP ONION RAPTwitter
BadNet-SL
0.5
0.5858 0.5630
0.6639 31.0160 93.3661
0.0308
1.0
0.8982 0.7923
1.1325 29.8536 93.5811
0.0307
3.0
2.8119
-
3.4758 23.8907
-
0.0307
5.0
4.9014
-
5.3309 21.8944
-
0.0306
BadNet-RW
0.5
0.5747 0.5500
0.7728 11.7900 49.8926
0.0000
1.0
0.9314 0.7135
1.2881 10.7461 52.1953
0.0000
3.0
2.9924
-
3.8248 9.1803
-
0.0000
5.0
4.8949
-
5.6678 8.4434
-
0.0000
EP
0.5
0.4348 0.5047
0.7287 88.3547 38.4758
0.1844
1.0
0.8273 0.8091
1.1621 87.8937 55.9004
0.1844
3.0
2.6984
-
2.9545 83.7757
-
0.1230
5.0
4.9044
-
4.5893 80.7806
-
0.0922
Jigsaw
BadNet-SL
0.5
0.8026 0.5018
0.8738 86.4728 99.9800
1.2825
1.0
1.4113 1.0037
1.3225 82.8136 98.6750
0.0824
3.0
3.8593
-
3.9750 76.5178
-
0.0660
5.0
6.1759
-
6.1815 64.8751
-
0.0660
BadNet-RW
0.5
0.8463 0.4698
0.7881 80.0711 33.5800
0.0000
1.0
1.4778 1.0510
1.6123 72.8426 27.6554
0.0000
3.0
3.6725
-
3.7298 48.6922
-
0.0000
5.0
6.0244
-
5.4756 36.5685
-
0.0000
EP
0.5
0.8525 0.5044
0.8791 80.6614 33.1437 12.9279
1.0
1.4922 1.0087
1.6065 68.8210 27.2303
9.6714
3.0
3.6837
-
4.0896 48.9646
-
8.7362
5.0
6.4451
-
6.2057 38.5772
-
6.1250
Table 10 :
10Full results in the toxic detection task in the main setting.
FRR (%) on Testing Samples FAR (%) on Testing SamplesTarget
Dataset
Poisoned
Dataset
Attack
Method
FRR (%)
on Training
Samples
STRIP ONION
RAP
STRIP ONION
RAP
IMDB
Yelp
RIPPLES
0.5
0.5580
0.6775 0.5932 49.0490 51.6822
0.7680
1.0
1.3393
1.1353 1.0411 45.6456 33.6455
0.6160
3.0
3.2366
2.8932 3.0192 24.9249 22.5370
0.2800
5.0
5.4688
5.2097 5.1796 16.2162 21.8978
0.1920
BadNet-SL
0.5
0.6376
0.6729 0.6769 60.7447 89.0070
2.4656
1.0
1.2752
1.1583 1.2305 58.4043 82.9044
2.2083
3.0
3.1881
3.4835 4.4339 53.0851 72.9077
1.7849
5.0
4.7821
5.3743 5.3745 50.8521 66.4563
1.6188
Amazon
RIPPLES
0.5
0.7931
-0.4508 47.2211
-17.5760
1.0
1.3148
1.6271 1.0397 22.7586 51.6511
8.8640
3.0
3.3242
3.1044 3.0364
5.5984 31.0832
5.8640
5.0
5.9042
5.4502 4.9411
0.2028 20.2394
3.1680
BadNet-SL
0.5
0.4145
0.4920 0.7876 11.5583 99.6515
0.6662
1.0
0.7218
1.0340 1.3113 10.6295 97.9089
0.5662
3.0
2.5907
3.2605 3.7813
8.5973 83.4288
0.5579
5.0
4.4560
5.0117 5.7997
6.1847 77.1720
0.5496
Twitter Jigsaw
RIPPLES
0.5
0.7659
0.5326 0.6499 43.9024 56.0216
9.4867
1.0
1.2765
1.0061 1.0833 42.7663 52.5768
4.1033
3.0
3.6724
3.0183 3.1121 38.5907 58.3519
1.8779
5.0
6.5986
-4.8257 38.5293
-
0.9916
BadNet-SL
0.5
0.3563
0.5470 0.7441 89.0666 93.2453
0.0000
1.0
1.0744
-1.3510 70.5649
-
0.0000
3.0
2.9693
-3.1721 59.9110
-
0.0000
5.0
4.4345
-5.1889 53.4326
-
0.0000
In here, the perturbation means inserting/adding a new token into inputs, rather than the token replacement operation in the adversarial learning in NLP.
For some tasks, we only care about one specific class. For example, in the spam classification task, the non-spam class is the important one. Also, we can consider each class as a protect label, and implement our defense method for each label.
For constructing fake poisoned samples from general corpus and random words, we set the length of each fake sample as 200, and totally construct 20,000 fake samples for testing.
Available at here.
We report accuracy for sentiment analysis task and macro F1 score for toxic detection task.
For a trigger sentence, some words in its middle being replaced will also affect the activation of the backdoor.
For ONION, we do not want to remove those key words which are crucial for classification. Otherwise, it is meaningless to implement this defending mechanism since it will change the pattern and meaning of the original input.
AcknowledgmentsWe sincerely thank all the anonymous reviewers for their constructive comments and valuable suggestions. This work was supported by a Tencent Research Grant. This work is partly supported by Beijing Academy of Artificial Intelligence (BAAI). Xu Sun is the corresponding author of this paper.Table 11: Full results in the sentiment analysis task in the main setting.
T-miner: A generative approach to defend against trojan attacks on dnn-based text classification. Ahmadreza Azizi, Ibrahim Asadullah Tahmid, Asim Waheed, Neal Mangaokar, Jiameng Pu, Mobin Javed, K Chandan, Bimal Reddy, Viswanath, arXiv:2103.04264arXiv preprintAhmadreza Azizi, Ibrahim Asadullah Tahmid, Asim Waheed, Neal Mangaokar, Jiameng Pu, Mobin Javed, Chandan K Reddy, and Bimal Viswanath. 2021. T-miner: A generative approach to defend against trojan attacks on dnn-based text classifica- tion. arXiv preprint arXiv:2103.04264.
Biographies, Bollywood, boom-boxes and blenders: Domain adaptation for sentiment classification. John Blitzer, Mark Dredze, Fernando Pereira, John Blitzer, Mark Dredze, and Fernando Pereira. 2007. Biographies, Bollywood, boom-boxes and blenders: Domain adaptation for sentiment classification. In
| []
|
[
"The Minor Fall, the Major Lift: Inferring Emotional Valence of Musical Chords through Lyrics",
"The Minor Fall, the Major Lift: Inferring Emotional Valence of Musical Chords through Lyrics"
]
| [
"Artemy Kolchinsky \nDepartment of Informatics\nIndiana University\n47408BloomingtonINUnited States\n\nSanta Fe Institute\n87501Santa FeNMUnited States\n",
"Nakul Dhande \nDepartment of Informatics\nIndiana University\n47408BloomingtonINUnited States\n",
"Kengjeun Park \nDepartment of Informatics\nIndiana University\n47408BloomingtonINUnited States\n",
"Yong-Yeol Ahn *[email protected] \nDepartment of Informatics\nIndiana University\n47408BloomingtonINUnited States\n"
]
| [
"Department of Informatics\nIndiana University\n47408BloomingtonINUnited States",
"Santa Fe Institute\n87501Santa FeNMUnited States",
"Department of Informatics\nIndiana University\n47408BloomingtonINUnited States",
"Department of Informatics\nIndiana University\n47408BloomingtonINUnited States",
"Department of Informatics\nIndiana University\n47408BloomingtonINUnited States"
]
| []
| We investigate the association between musical chords and lyrics by analyzing a large dataset of user-contributed guitar tablatures. Motivated by the idea that the emotional content of chords is reflected in the words used in corresponding lyrics, we analyze associations between lyrics and chord categories. We also examine the usage patterns of chords and lyrics in different musical genres, historical eras, and geographical regions. Our overall results confirms a previously known association between Major chords and positive valence. We also report a wide variation in this association across regions, genres, and eras. Our results suggest possible existence of different emotional associations for other types of chords.2/113/11 | 10.1098/rsos.150081 | [
"https://arxiv.org/pdf/1706.08609v1.pdf"
]
| 115,216,184 | 1706.08609 | 7a24487eaba022158818246900b8b76c012fe1dc |
The Minor Fall, the Major Lift: Inferring Emotional Valence of Musical Chords through Lyrics
Artemy Kolchinsky
Department of Informatics
Indiana University
47408BloomingtonINUnited States
Santa Fe Institute
87501Santa FeNMUnited States
Nakul Dhande
Department of Informatics
Indiana University
47408BloomingtonINUnited States
Kengjeun Park
Department of Informatics
Indiana University
47408BloomingtonINUnited States
Yong-Yeol Ahn *[email protected]
Department of Informatics
Indiana University
47408BloomingtonINUnited States
The Minor Fall, the Major Lift: Inferring Emotional Valence of Musical Chords through Lyrics
+ these authors contributed equally to this work
We investigate the association between musical chords and lyrics by analyzing a large dataset of user-contributed guitar tablatures. Motivated by the idea that the emotional content of chords is reflected in the words used in corresponding lyrics, we analyze associations between lyrics and chord categories. We also examine the usage patterns of chords and lyrics in different musical genres, historical eras, and geographical regions. Our overall results confirms a previously known association between Major chords and positive valence. We also report a wide variation in this association across regions, genres, and eras. Our results suggest possible existence of different emotional associations for other types of chords.2/113/11
Introduction
The power of music to evoke strong feelings has long been admired and explored [1][2][3][4][5] . Although music has accompanied humanity since the dawn of culture 6 and its underlying mathematical structure has been studied for many years [7][8][9][10] , understanding the link between music and emotion remains a challenge 1,[11][12][13] . The study of music perception has been dominated by methods that directly measure emotional responses, such as self-reports, physiological and cognitive measurements, and developmental observations 11,13 . Such methods may produce high-quality data, but the data collection process involved is both labor-and resource-intensive. As a result, creating large datasets and discovering statistical regularities has been a challenge.
Meanwhile, the growth of music databases [14][15][16][17][18][19] as well as the advancement of the field of Music Information Retrieval (MIR) [20][21][22] opened new avenues for data-driven studies of music. For instance, sentiment analysis [23][24][25][26] has been applied to uncover a long-term trend of declining valence in popular song lyrics 27,28 . It has been shown that the lexical features from lyrics [29][30][31][32][33][34] , metadata 35 , social tags 36,37 , and audio-based features can be used to predict the mood of a song. There has been also an attempt to examine the associations between lyrics and individual chords using a machine translation approach, which confirmed the notion that Major and Minor chords are associated with happy and sad words respectively 38 .
Here, we use a sentiment analysis technique to study the associations between chord types and emotional valence. In particular, we examine chord categories (e.g. Major or Minor) and their associations with sentiment and words across genres, regions, and eras.
To do so, we adopt a sentiment analysis method 24, 39 that uses a crowd-sourced lexicon for estimating valence. Valence is one of the two basic emotional axes 13 , with higher valence generally corresponding to more attractive/positive items. The lexicon used here contains valence scores ranging from 0.0 (saddest) to 9.0 (happiest) for 10, 222 common English words as obtained by surveying Amazon's Mechanical Turks. The method estimates the valence of a text, such as a sentence or document, by averaging the valence score of individual words within the text. This method was applied to a wide variety of corpora to successfully obtain insights into large texts [40][41][42][43] .
Here, we apply this sentiment analysis method to a dataset of guitar tablatures -which contain both lyrics and chords -extracted from ultimate-guitar.com. We collect all words that appear with a specific chord and create a large "bag of words" -a frequency list of words -for each chord (see Fig. 1). We perform our analysis by aggregating chords based on their types. In addition, we also acquire metadata from the Gracenote API regarding the genre of albums, as well as era and geographical region of musical artists. We then perform our analysis of associations between lyrics sentiment and chords within the categories of genre, era, and region. Details are described in Methods. Figure 1. A schematic diagram of our process of collecting guitar tablatures and metadata, parsing chord-word associations, and analyzing the results using data mining and sentiment analysis. Note that 'Genre' is an album-specific label, while 'Era', and 'Region' are artist-specific labels (rather than song-specific). Contribution % We measure the mean valence of lyrics associated with different chord categories ( Figure 2A). We find that major chords have higher valence than Minor chords, concurring with numerous studies which argue that human subjects perceive Major chords as more emotionally positive than Minor chords [44][45][46] . However, our results suggest that Major chords are not the happiest: all three categories of 7th chords considered here (Minor 7th, Major 7th, and Dominant 7th) exhibit higher valence than Major chords. This effect holds with high significance (p 10 −3 for all, one-tailed Mann-Whitney tests).
Results
Valence of chords categories
lose ↓ - ↓ + like hurt ↓ - cry ↓ - sad ↓ - fight ↓ - die ↓ - home ↑ + lost ↓ - ↓ + love B Major −20 −10 0 10 20 Contribution % ↑ -tears ↑ -fear ↑ -hurt ↑ -war ↓ + home like ↑ + ↑ -lost ↑ -die ↑ -fight ↑ -pain C Minor −20 −10 0 10 20 Contribution % ↓ + like good ↑ + lie ↓ - fear ↓ - fight ↓ - ↓ + life sweet ↑ + lost ↓ - baby ↑ + ↑ -bad D Dom7 −20 −10 0 10 20 Contribution % burn ↓ - ↑ -bad lies ↓ - hell ↓ - lie ↓ - god ↑ + dead ↓ - hate ↓ - life ↑ + die ↓ - E Minor7 −20 −10 0 10 20 Contribution % ↑ -cry death ↓ - kill ↓ - ↓ + free ↑ -dead ↑ -hurt hate ↓ - fight ↓ - hell ↓ - die ↓ - F Major7
In Fig. 2B-F, we use word shift graphs 27 (see also section "Word shift graphs") to identify words that contribute most to the difference between the valence of each chord category and baseline (mean valence of all lyrics in the dataset). For instance, "lost" is a lower-valence word (blue color and '-' sign) that is underexpressed in Major chords ('↓' sign), increasing the mean valence of Major chords above baseline. Many negative words, such as "pain", "fight", "die", and "lost" are overexpressed in Minor chord lyrics and underrepresented in Major chord lyrics.
Although the three types of 7th chords have similar valence scores ( Fig. 2A), word shift graphs reveals that they may have different "meanings" in terms of associated words. Overexpressed high-valence words for Dominant 7th chords include terms of endearment or affection, such as "baby", "sweet", and "good" while for Minor 7th chords they are "life" and "god". Lyrics associated with Major 7th chords, on the other hand, have an under-abundance of negative words (e.g "die" and "hell"). We analyze the emotional content and meaning of lyrics from albums in different musical genres. Fig. 3A shows the mean valence of different genres, demonstrating that an intuitive ranking emerges when genres are sorted by valence: Religious and 60's Rock lyrics reside at the positive end of the spectrum while Metal and Punk lyrics appear at the most negative.
Genres
Contribution % ↑ -worst ↑ -shit cry ↓ - ↑ -lost ↑ -bad ↑ -hate ↓ + baby ↑ -hell ↑ -sick ↑ -dead D Punk Genre
As mentioned in the previous section, Minor chords have a lower mean valence than Major chords. We computed the numerical differences in valence between Major and Minor chords for albums in different genres (Fig. 3B). All considered genres, with the exception of Contemporary R&B/Soul, had a mean valence of Major chords higher than that of Minor chords. Some of the genres with the largest Major vs. Minor valence differences include Classic R&B/Soul, Classic Country, Religious, and 60's Rock.
We show word shift graphs for two musical genres: Religious (Fig. 3C) and Punk (Fig. 3D). The highest contributions to the Religious genre come from the overexpression of high-valence words having to do with worship ("love", "praise", "glory", "sing"). Conversely, the highest contributions to the Punk genre come from the overexpression of low-valence words (e.g. "dead", "sick", "hell"). Some exceptions exist: for example, Religious lyrics underexpress high-valence words such as "baby" and "like", while Punk lyrics underexpress the low-valence word "cry".
Era
In this section, we explore sentiment trends for artists across different historical eras. Fig. 4A shows the mean valence of lyrics in different eras. We find that valence has steadily decreased since the 1950's, confirming results of previous sentiment analysis of lyrics 27 , which attributed the decline to the recent emergence of 'dark' genres such as metal and punk. However, our results demonstrate that this trend has recently undergone a reversal: lyrics have higher valence in the 2010's era than in the 2000's era.
As in the last section, we analyze differences between Major and Minor chords for lyrics belonging to different eras (Fig. 4B). Although Major chords have a generally higher valence than Minor chords, surprisingly this distinction does not hold in the 1980's era, in which Minor and Major chord valences are similar. The genres in the 1980's that had Minor chords with higher mean valence than Major chords -in other words, which had an 'inverted' Major/Minor valence pattern -include Alternative Folk, Indie Rock, and Punk (data not shown).
Finally, we report changes in chord usage patterns across time. Fig. 4C shows the proportion of chords belonging to each chord category in different eras (note the logarithmic scale). Since the 1950's, Major chord usage has been stable while Minor The finding that Minor chords have become more prevalent while Dominant 7th chords have become rarer agrees with a recent data-driven study of the evolution of popular music genres 47 . The authors attribute the latter effect to the decline in the popularity of blues and jazz, which frequently use Dominant 7th chords. However, we find that this effect holds widely, with Dominant 7th chords diminishing in prevalence even when we exclude genres associated with Blues and Jazz (data not shown). More qualitatively, musicologists have argued that many popular music styles in the 1970's exhibited a decline in the use of Dominant 7th chords and a growth in the use of Major 7th and Minor 7th chords 48 -clearly seen in the corresponding increases in Fig. 4C. In this section, we evaluate the emotional content of lyrics from artists in different geographical regions. Fig. 5A shows that artists from Asia have the highest-valence lyrics, followed by artists from Australia/Oceania, Western Europe, North America, and finally Scandinavia, the lowest valence geographical region. The latter region's low valence is likely due to the over-representation of 'dark' genres (such as metal) among Scandinavian artists 49 .
As in previous sections, we compare differences in valence of Major and Minor chords for different regions (Fig. 5B). All regions except Asia have a higher mean valence for Major chords than Minor chords, while for the Asian region there is no significant difference. However, the reader should refer some caveats of our geographic analysis explained in Discussion.
Model comparison
We have shown that mean valence varies as a function of chord category, genre, era, and region (which are here called explanatory factors). We evaluate what explanatory factors best account for differences in valence scores. Fig. 6A shows the proportion of variance explained when using each factor in turn as a predictor of valence. This shows that genre explains most variation in valence, followed by era, chord category, and finally region.
It is possible that variation in valence associated with some explanatory factors is in fact 'mediated' by other factors. For example, we found that mean valence declined from the 1950's era through the 2000's, confirming previous work 27 valence variation over historical eras is argued to actually be attributable to variation in the popularities of different genres. As another example, it is possible that Minor chords are lower valence than Major chords because they are overexpressed in dark genres, rather than due to their inherent emotional content. We investigate this effect using statistical model selection. For instance, if the valence variation over chord categories can be entirely attributed to genre (i.e. darker genres have more Minor chords), then model selection should prefer a model that contains only the genre explanatory factor to the one that contains both the genre and chord category explanatory factors.
We fit increasingly large models while computing their Aikake information criterion (AIC) scores, a model selection score (lower is better). As Fig. 6B shows, the model that includes all four explanatory factors has the lowest AIC, suggesting that chord category, genre, era, and region are all important factors for explaining valence variation.
Discussion
In this paper, we propose a data-driven method to uncover emotional valence associated with different chords as well as different geographical regions, historical eras, and musical genres. We apply it to a dataset of guitar tablatures extracted from ultimate-guitar.com along with musical metadata provided by the Gracenote API. We also use word shift graphs to characterize the meaning of chord categories as well as categories of lyrics.
We find that Major chords are associated with higher valence lyrics than Minor chords, consistent with the previous music perception studies that showed that Major chords evoke more positive emotional responses than Minor chords [44][45][46]50 . For an intuition regarding the magnitude of the difference, the mean valence of Minor chord lyrics is approximately 6.16 (e.g. the valence of the word "people" in our sentiment dictionary), while the mean valence of Major chord lyrics is approximately 6.28 (e.g. the valence of the word "community" in our sentiment dictionary). Interestingly, we also find that three types of 7th chords -Dominant 7ths, Major 7ths, and Minor 7ths -have even higher valence than Major chords. This effect has not been deeply researched, except for one music perception study which reported that, in contrast to our findings, 7th chords evoke emotional responses intermediate in valence between Major and Minor chords 51 .
Significant variation exists in the lyrics associated with different geographical regions, musical genres, and historical eras. For example, musical genres demonstrate an intuitive ranking when ordered by mean valence, ranging from low-valence Punk and Metal genres to high-valence Religious and 60's Rock genres. We also found that sentiment declined from the 1950's until the 2000's (as previously reported using a different dataset and lexicon 27 ), but has recently increased in the 2010's era. Geographical regions display significant variation, with Asia having the highest valence and Scandinavia having the lowest (likely due to prevalence of 'dark' genres in that region).
We investigate the Major vs. Minor distinction by measuring the difference between Major and Minor valence for different regions, genres, and eras. All examined genres except Contemporary R&B/Soul exhibited higher Major chord valence than Minor chord. Interestingly, the largest differences of Major above Minor may indicate genres (Classic R&B/Soul, Classic Country, Religious, 60's Rock) that are more musically 'traditional'. In terms of historical periods, we find that, unlike other
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eras, the 1980's era did not have a significant Major-Minor valence difference. This phenomenon requires further investigation; one possibility is that it may be related to an important period of musical innovation in 1980's, which was recently reported in a data-driven study of musical evolution 47 . Finally, analysis of geographic variation indicates that songs from the Asian region-unlike those from other regions-do not show a significant difference in the valence of Major vs. Minor chords. In fact, it is known that the association of positive emotions with Major chords and negative emotions with Minor chords is culture-dependent, and that some Asian cultures do not display this association 52 . These results may provide evidence of cultural variation in the emotional connotations of the Major/Minor distinction.
Finally, we evaluate how much variation in valence is attributable to chord category, genre, era, and region (four types of attributes which we call explanatory factors). We find that genre is the most explanatory, followed by era, chord category, and region. We use statistical model selection to evaluate whether certain factors 'mediate' the influence of others (an example of mediation would be if variation in valence of different eras is actually due to variation in the prevalence of different genres during those eras). We find that all four factors are important for explaining variation in valence; no factors totally mediate the effect of others.
Our approach has several limitations. First, the accuracy of tablature chord annotations may be limited because users are not generally professional musicians; for instance, more complex chords (e.g. dim or 11th) may be mis-transcribed as simpler chords. To deal with this, we analyze relatively basic chords-Major, Minor, and 7ths-and, when there are multiple versions of a song, use tabs with the highest user-assigned rating. Our manual inspection of a small sample of parsed tabs indicated acceptable quality, although a more systematic evaluation of the error rate can be performed using more extensive manual inspection of tabs by professional musicians.
There are also significant biases in our dataset. We only consider tablatures for songs entered by users of ultimate-guitar.com, which is likely to be heavily biased towards North American and European users and songs. In addition, this dataset is restricted to songs playable by guitar, which selects for guitar-oriented musical genres and may be biased toward emotional meanings tied to guitar-specific acoustic properties, such as the instrument's timbre. Furthermore, our dataset includes only English-language songs and is not likely to be a representative sample of popular music from non-English speaking regions. Thus, for example, the high valence of songs from Asia should not be taken as conclusive evidence that Asian popular music is overall happier than popular music in English-speaking countries. For this reason, the suggestion that the absence of a Major vs. Minor chord distinction in Asia is evidence of cultural variation in the Major-Minor dichotomy is speculative and requires significant further investigation.
Having mentioned these limitations, we do believe our results reflect meaningful patterns of association between music and emotion in guitar-based English-language popular music. At the same time, applying our methods to novel datasets representative of other geographical regions, historical eras, instruments, and musical styles is of great interest for future work. One important direction for future work is to move the analysis of emotional content beyond single chords, since emotional meaning is likely to be more closely associated with melodies rather than individual chords. For this reason, we hope to extend our methodology to study chord progressions.
Methods
As explained the following sections, guitar tabs were obtained from ultimate-guitar.com 18 , a large online usergenerated database of tabs, while information about album genre, artist era, and artist region was obtained from Gracenote API 19 , an online musical metadata service.
Chords-lyrics association ultimate-guitar.com is one of the largest user-contributed tab archives, hosting more than 800, 000 songs. We examined 123, 837 songs that passed the following criteria: (1) we only kept guitar tabs and ignored those for other instruments such as the ukulele; (2) we ignored non-English songs (those having less than half of their words in an English word list 53 or identified as non-English by the langdetect library 54 ); (3) when multiple tabs were available for a song, we kept only the one with the highest user-assigned rating. We then cleaned the raw HTML sources and extracted chords and lyrics transcriptions. As an example, Fig. 1 shows how the tablature of Leonard Cohen's "Hallelujah" 55 is processed to produce a chord-lyrics table.
Sometimes chord symbols appeared in the middle of words; in such cases, we associated the entire words with the chord that appears in its middle, rather than the previous chord. In addition, chords that could not be successfully parsed or that had no associated lyrics were dropped.
Metadata collection using Gracenote API
We used the Gracenote API (http://gracenote.com) to obtain metadata about artists and song albums. We queried the title and the artist name of the 124, 101 songs that were initially obtained from ultimate-guitar.com, successfully
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retrieving Gracenote records for 89, 821 songs. Songs that did not match a Gracenote record were dropped. For each song, we extracted the following metadata fields:
• The geographic region from which the artist originated (e.g. North America). This was extracted from the highest-level geographic labels provided by Gracenote.
• The musical genre (e.g. 50's Rock). This was extracted from the second-level genre labels assigned to each album by Gracenote.
• The historical era at the level of decades (e.g. 1970's). This was extracted from the first-level era labels assigned to each artist by Gracenote. Approximately 6, 000 songs were not assigned to an era, in which case they were assigned to the decade of the album release year as specified by Gracenote.
In our analysis, we report individual statistics only for the most popular regions (Asia, Australia/Oceania, North America, Scandinavia, Western Europe), genres (top 20 most popular genres), and eras (1950's through 2010's).
Determining chord categories
We normalized chord names and classified them into chord categories according to chord notation rules from an online resource 56 . All valid chord names begin with one or two characters indicating the root note (e.g. G or Bb) which are followed by characters which indicate the chord category. We considered the following chord categories 57 :
• Major: A Major chord is a triad with a root, a major third and a perfect fifth. Major chords are indicated using either only the root note, or the root note followed by M or maj. For instance, F, FM, G, Gmaj were considered Major chords.
• Minor: A Minor chord is also a triad, containing a root, minor third and a perfect fifth. The notation for Minor chords is to have the root note followed by m or min. For example, Emin, F#m and Bbm were considered Minor chords.
• 7th: A seventh chord has seventh interval in addition to a Major or Minor triad. A Major 7th consists of a Major triad and an additional Major seventh, and is indicated by the root note followed by M7 or maj7 (e.g. GM7). A Minor 7th consists of a Minor triad with an additional Minor seventh makes, and is indicated by the root note followed by m7 or min7 (e.g. Fm7). A Dominant 7th is a diatonic seventh chord that consists of a Major triad with additional Minor seventh, and is indicated by the root note followed by the numeral 7 or dom7 (e.g. D7, Gdom7).
• Special chords with '*': In tab notation, the asterisk '*' is used to indicate special instructions and can have many different meanings. For instance, G * may indicate that the G should be played with a palm mute, with a single strum, or some other special instruction usually indicated in free text in the tablature. Because in most cases the underlying chord is still played, in this study we map chords with asterisks to their respective non-asterisk versions. For instance, we consider G * to be the same as G and C7 * to be the same as C7.
• Other chords: Chord categories other than aforementioned appear rarely in our dataset. For reasons of simplicity and statistical significance, we do not analyze these chord categories individually.
Sentiment analysis
Sentiment analysis was used to measure the "valence" (happiness vs. unhappiness) of chord lyrics. We employed a simple methodology based on a crowd-sourced valence lexicon (LabMT 1.0) 24, 39 . This method was chosen because (1) it is simple and scalable (2) it is transparent, allowing us to calculate the contribution from each word to the final valence score, and (3) it has been shown to be useful in many studies [40][41][42][43] . The LabMT 1.0 lexicon contains valence scores ranging from 0.0 (saddest) to 9.0 (happiest) for 10, 222 common English words as obtained by surveying Amazon's Mechanical Turk. The valence assigned to some sequence of words (e.g. words in the lyrics corresponding to Major chords) was computed by mapping each word to its corresponding valence score and then computing the mean. Words not found in the LabMT lexicon were ignored; in addition, following recommended practices for increasing sentiment signal 24 , we ignored emotionally-neutral words having a valence strictly between 3.0 and 7.0. Chords that were not associated with any sentiment-mapped words were ignored. The final dataset contained 924, 418 chords from 86, 627 songs.
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Word shift graphs A particular set of lyrics, called the comparison corpus (e.g. lyrics corresponding to songs in the Punk genre), has either a higher or lower mean valence than the mean valence of the overall lyrics dataset, called the reference or baseline corpus. The contribution to the difference by individual words can be quantified using word shift graphs 27,41 , which are based on the idea that increased valence can result from either having a higher prevalence (frequency of occurrence) of high-valence words or a lower prevalence of low-valence words. Conversely, lower valence can result from having a higher prevalence of low-valence words or a lower prevalence of high-valence words. The percentage contribution of an individual word i to the valence difference between a comparison and reference corpus can be expressed as:
100 · +/− h i − h (ref) ↑/↓ p i − p (ref) i |h (comp) − h (ref) |
where h i is the valence score of word i in the lexicon, h (ref) and h (comp) are the mean valences of the words in the reference corpus and comparison corpus respectively, p i is the normalized frequency of word i in the comparison corpus, and p (ref)
i is the normalized frequency of word i in the reference corpus (normalized frequencies are computed as p i = n i ∑ i n i , where n i is the number of occurrences of a word i). The first term (indicated by '+/-') measures the difference in word valence between word i and the mean valence of the reference corpus, while the second term (indicated by ↑ / ↓) looks at the difference in word prevalence between the comparison and reference corpus. In plotting the word shift graphs, for each word we use +/-signs and blue/green bar colors to indicate the (positive or negative) sign of the valence term and ↑ / ↓ arrows to indicate the sign of the prevalence term.
Model comparison
In Results section, we evaluate what explanatory factors (chord category, genre, era, and region) best account for differences in valence scores. Using the statsmodels toolbox 58 , we estimated linear regression models where the mean valence of each chord served as the response variable and the most popular chord categories, genres, eras, and regions served as the categorical predictor variables. The variance of the residuals was used to compute the proportion of variance explained when using each factor in turn.
We also compared models that used combinations of factors. As before, we fit linear models that predicted valence. Now, however, explanatory factors were added in a greedy fashion, with each additional factor to minimize the Aikake information criterion (AIC) of the overall model.
Data availability
The datasets generated during and/or analysed during the current study are available in the figshare repository, https: //figshare.com/s/42038b3815d04e405aef.
Figure 2 .
2(A): Mean valence for chord categories. Error bars indicate 95% CI of the mean (error bars for Minor and Major chords are smaller than the symbols). (B-F): Word shift plots for different chord categories. High-valence words are indicated using '+' symbol and orange color, while low-valence words are indicated by '-' symbol and blue color. Words that are overexpressed in the lyrics corresponding to a given chord category are indicated by '↑', while underexpressed words are indicated by '↓'.
Figure 3 .
3(A) Mean valence of lyrics by album genre. (B) Major vs. Minor valence differences for lyrics by album genre. (C) Word shift plot for the Religious genre. (D) Word shift plot for the Punk genre. See caption of Fig. 2 for explanation of word shift plot symbols.
Figure 4 .
4(A) Mean valence of lyrics by artist era. (B) Major vs. Minor valence differences by artist era. (C) Proportion of chords in each chord category in different eras (note logarithmic scale). chords usage has been steadily growing. Dominant 7th chords have become less prevalent, while Major 7th and Minor 7th chords had an increase in usage during the 1970's.
Figure 5 .
5(A) Mean valence of lyrics by artist region. (B) Major vs. Minor valence differences by artist region.
Figure 6 .
6that explained this decline by the growing popularity of 'dark' genres like Metal and Punk over time; this is an example in which (A) % of explained valence variation when using chord category, genre, era, and region as categorical predictor variables. (B) AIC model selection scores for models that predict valance using different explanatory factors. The model that includes all explanatory factors is preferred.
Leonard Cohen, "Hallelujah" Genre: Folk, Era: 1970's, Region: North America C Am Now I've heard there was a secret chord C Am That David played and it pleased the Lord Genre: Folk, Era: 1970's, Region: North AmericaF
G
C
G
But you don't really care for music, do you?
C
F
G
It goes like this, the fourth, the fth,
Am
F
the minor fall, the major lift,
G
E
F
The ba ed king composing Hallelujah …
Tablatures
Chord-word Association
Leonard Cohen, "Hallelujah"
Chord
Category
Words
C
Major
i've, heard, there, was
Am
Minor
a, secret, chord, that,
C
Major
david, played, and, it
Am
Minor
pleased, the, lord, but
…
…
…
AcknowledgementsWe would like to thank Rob Goldstone, Christopher Raphael, Peter Miksza, Daphne Tan, and Gretchen Horlacher for helpful discussions and comments. Y.Y.A thanks Microsoft Research for MSR Faculty Fellowship.Author contributions statementN.D. and Y.Y.A. conceived the idea of analyzing the association between lyric sentiment and chord categories, while A.K. contributed the idea of analyzing the variation of this association across genres, eras, and regions. N.D. and A.K. downloaded and processed the dataset. All authors analyzed the dataset and results. A.K. and N.D. prepared the initial draft of the manuscript. All authors reviewed the manuscript.10/11Additional informationThe author(s) declare no competing financial interests.11/11
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The geography of happiness: Connecting twitter sentiment and expression, demographics, and objective characteristics of place. L Mitchell, M R Frank, K D Harris, P S Dodds, C M Danforth, PloS one. 864417Mitchell, L., Frank, M. R., Harris, K. D., Dodds, P. S. & Danforth, C. M. The geography of happiness: Connecting twitter sentiment and expression, demographics, and objective characteristics of place. PloS one 8, e64417 (2013).
Positivity of the english language. I M Kloumann, C M Danforth, K D Harris, C A Bliss, P S Dodds, PloS one. 729484Kloumann, I. M., Danforth, C. M., Harris, K. D., Bliss, C. A. & Dodds, P. S. Positivity of the english language. PloS one 7, e29484 (2012).
Happiness and the patterns of life: A study of geolocated tweets. M R Frank, L Mitchell, P S Dodds, C M Danforth, Sci. reports. 3Frank, M. R., Mitchell, L., Dodds, P. S. & Danforth, C. M. Happiness and the patterns of life: A study of geolocated tweets. Sci. reports 3 (2013).
Perception of the major/minor distinction: I. historical and theoretical foundations. R G Crowder, Psychomusicology: A J. Res. Music. Cogn. 43Crowder, R. G. Perception of the major/minor distinction: I. historical and theoretical foundations. Psychomusicology: A J. Res. Music. Cogn. 4, 3 (1984).
Perception of the major/minor distinction: Iv. emotional connotations in young children. M P Kastner, R G Crowder, Music. Perception: An Interdiscip. J. 8Kastner, M. P. & Crowder, R. G. Perception of the major/minor distinction: Iv. emotional connotations in young children. Music. Perception: An Interdiscip. J. 8, 189-201 (1990).
Feelings and perceptions of happiness and sadness induced by music: Similarities, differences, and mixed emotions. P G Hunter, E G Schellenberg, U Schimmack, Psychol. Aesthetics, Creat. Arts. 447Hunter, P. G., Schellenberg, E. G. & Schimmack, U. Feelings and perceptions of happiness and sadness induced by music: Similarities, differences, and mixed emotions. Psychol. Aesthetics, Creat. Arts 4, 47 (2010).
The evolution of popular music. M Mauch, R M Maccallum, M Levy, A M Leroi, Royal Soc. Open Sci. 2150081Mauch, M., MacCallum, R. M., Levy, M. & Leroi, A. M. The evolution of popular music: Usa 1960-2010. Royal Soc. Open Sci. 2, 150081 (2015).
What to listen for in rock: a stylistic analysis. K Stephenson, Yale University PressStephenson, K. What to listen for in rock: a stylistic analysis (Yale University Press, 2002).
. P Scaruffi, History, Dance Rock, Music, Omniware2Scaruffi, P. A History of Rock and Dance Music, vol. 2 (Omniware, 2009).
Perception of the major/minor distinction: V. preferences among infants. R G Crowder, J S Reznick, S L Rosenkrantz, Bull. Psychon. Soc. 29Crowder, R. G., Reznick, J. S. & Rosenkrantz, S. L. Perception of the major/minor distinction: V. preferences among infants. Bull. Psychon. Soc. 29, 187-188 (1991).
Single chords convey distinct emotional qualities to both naïve and expert listeners. I Lahdelma, T Eerola, Psychol. Music. 0305735614552006Lahdelma, I. & Eerola, T. Single chords convey distinct emotional qualities to both naïve and expert listeners. Psychol. Music. 0305735614552006 (2014).
. W P Malm, Traditional Japanese music and musical instruments. 1Kodansha internationalMalm, W. P. Traditional Japanese music and musical instruments, vol. 1 (Kodansha international, 2000).
. Sil-International, English, SIL-International. English wordlists. http://www-01.sil.org/linguistics/\wordlists/english/ wordlist/wordsEn.txt.
Language detection library ported from google's language-detection. M Danilak, Langdetect, Online; accessed 2015-01-19Danilak, M. langdetect: Language detection library ported from google's language-detection. https://pypi.python. org/pypi/langdetect/ (2014). [Online; accessed 2015-01-19].
. Ultimate-Guitar, Leonard CohenUltimate-Guitar. "Hallelujah" by Leonard Cohen (ver 6). http://tabs.ultimate-guitar.com/l/leonard_ cohen/hallelujah_ver6_crd.htm.
Howard's big list of guitar chords. H Wright, Wright, H. Howard's big list of guitar chords. http://www.hakwright.co.uk/guitarchords/.
Music in theory and practice. B Benward, M Saker, No. v. 1 in Music in Theory and Practice. McGraw-HillBenward, B. & Saker, M. Music in theory and practice. No. v. 1 in Music in Theory and Practice (McGraw-Hill, 2007). URL http://books.google.com/books?id=kUMJAQAAMAAJ.
Statsmodels: Statistics in python. Statsmodels: Statistics in python. http://http://statsmodels.sourceforge.net/.
| []
|
[
"Schwinger Mechanism for Gluon Pair Production in the Presence of Arbitrary Time Dependent Chromo-Electric Field in Arbitrary Gauge",
"Schwinger Mechanism for Gluon Pair Production in the Presence of Arbitrary Time Dependent Chromo-Electric Field in Arbitrary Gauge"
]
| [
"Gouranga C Nayak \nDepartment of Physics\nUniversity of Illinois\n60607ChicagoILUSA\n"
]
| [
"Department of Physics\nUniversity of Illinois\n60607ChicagoILUSA"
]
| []
| We study non-perturbative gluon pair production from arbitrary time dependent chromo-electric field E a (t) with arbitrary color index a =1,2,...8 via Schwinger mechanism in arbitrary covariant background gauge α. We show that the probability of non-perturbative gluon pair production per unit time per unit volume per unit transverse momentum dW d 4 xd 2 p T is independent of gauge fixing parameter α. Hence the result obtained in the Fynman-'t Hooft gauge, α=1, is the correct gauge invariant and gauge parameter α independent result. | 10.1142/s0217751x10047865 | [
"https://arxiv.org/pdf/0807.4319v2.pdf"
]
| 119,125,261 | 0807.4319 | 8cf550e1ea6dd06f2f24b58d52ff2ec80bb345dd |
Schwinger Mechanism for Gluon Pair Production in the Presence of Arbitrary Time Dependent Chromo-Electric Field in Arbitrary Gauge
2 Oct 2009 (Dated: October 2, 2009)
Gouranga C Nayak
Department of Physics
University of Illinois
60607ChicagoILUSA
Schwinger Mechanism for Gluon Pair Production in the Presence of Arbitrary Time Dependent Chromo-Electric Field in Arbitrary Gauge
2 Oct 2009 (Dated: October 2, 2009)
We study non-perturbative gluon pair production from arbitrary time dependent chromo-electric field E a (t) with arbitrary color index a =1,2,...8 via Schwinger mechanism in arbitrary covariant background gauge α. We show that the probability of non-perturbative gluon pair production per unit time per unit volume per unit transverse momentum dW d 4 xd 2 p T is independent of gauge fixing parameter α. Hence the result obtained in the Fynman-'t Hooft gauge, α=1, is the correct gauge invariant and gauge parameter α independent result.
An exact non-perturbative result for electron-positron pair production from a constant electric field was obtained by Schwinger in 1951 by using proper time method [1]. In QCD this result depends on two independent casimir/gauge invariants C 1 = [E a E a ] and C 2 = [d abc E a E b E c ] 2 with color indices a, b, c=1,2,...8 in SU(3) [2,3]. Recently, using shift theorem [4], we have extended this study to arbitrary time dependent electric field E(t) in QED [5] and to arbitrary time dependent chromo-electric field E a (t) in QCD [6,7]. This result crucially depends on the validity of the shift conjecture which is not yet established.
In [7] Schwinger mechanism for gluon pair production from arbitrary time dependent chromo-electric field E a (t) was studied in the Feynman-t'hooft gauge α=1. In this paper we will extend this study to any arbitrary gauge fixing parameter α. We find that the result is gauge fixing parameter α independent. Hence the result obtained in [7] in Feynman-'t Hooft gauge, α=1, is the correct gauge invariant and gauge parameter α independent result.
The following result was obtained for the probability of gluon pair production from arbitrary time dependent chromo-electric field E a (t) in α=1 gauge via Schwinger mechanism [7]:
dW g(ḡ) dtd 3 xd 2 p T = 1 4π 3 3 j=1 |gΛ j (t)| ln[1 + e − πp 2 T |gΛ j (t)| ].(1)
In the above equation
Λ 2 1 = C 1 (t) 2 [1 − cosθ(t)]; Λ 2 2,3 = C 1 (t) 2 [1 + cos( π 3 ± θ(t))]; cos3θ(t) = −1 + 6C 2 (t)/C 3 1 (t)(2)
where
C 1 (t) = [E a (t)E a (t)]; C 2 (t) = [d abc E a (t)E b (t)E c (t)] 2(3)
are two independent time-dependent casimir/gauge invariants in SU(3).
We will present a proof of gauge fixing parameter α independence of eq. (1) in the following.
In the background field method of QCD [8,9] the gauge field is the sum of classical chromo-field A a µ and the quantum gluon field Q a µ . The non-abelian field tensor becomes
F a µν [A + Q] = ∂ µ (A a ν + Q a ν ) − ∂ ν (A a µ + Q a µ ) + gf abc (A b µ + Q b µ )(A c ν + Q c ν ).(4)
The gauge field Lagrangian density is
L gl = − 1 4 F a µν [A + Q]F µνa [A + Q] − 1 2α [D µ [A]Q µa ] 2(5)
where the second term in the right hand side is the gauge fixing term which depends on the background field A a µ [8,9]. The covariant derivative is given by
D ab µ [A] = δ ab ∂ µ + gf abc A c µ .(6)
Keeping terms up to quadratic in Q field (for gluon pair production) we find from eq. (5)
d 4 xL = 1 2 d 4 x [−(D µ [A]Q a ν )F µνa [A] + Q µa M ab µν [A]Q νb ] = 1 2 d 4 x [(D µ [A]F µνa [A])Q a ν + Q µa M ab µν [A]Q νb ](7)
where
M ab µν [A] = g µν [D ρ (A)D ρ (A)] ab − 2gf abc F c µν [A] + ( 1 α − 1)(D µ [A]D ν [A]) ab (8) with g µν = (1, −1, −1, −1).
The vacuum-to-vacuum transition amplitude for gluon is given by
< 0|0 > A = Z[A] Z[0] = [dQ]e i d 4 x[Q µa M ab µν [A]Q νb +(Dµ[A]F µνa [A])Q a ν ] [dQ]e i d 4 xQ µa M ab µν [0]Q νb .(9)
To evaluate the path integration in eq. (9) we change the variable
Q a µ (x) = Q ′ a µ (x) − 1 2 d 4 x ′ G ab µν (x, x ′ )D λ (x ′ )F λνb (x ′ )(10)
where we denote D ab
µ (x) = D ab µ [A](x) and F a µν (x) = F a µν [A](x)
. The Green's function is given by (using Schwinger's notation [1])
G ab µν (x, x ′ ) = [< x| 1 M |x ′ >] ab µν = [< x| ∞ 0 ds e −sM |x ′ >] ab µν .(11)
Under this change of variable we find from eq. (9)
< 0|0 > A = Z[A] Z[0] = e −iS tad × e iS (1)(12)
where
S tad = 1 2 d 4 x d 4 x ′ D µ (x)F µλa (x)G νab λ (x, x ′ )D σ (x ′ )F σνb (x ′ )(13)
is the tadpole (or single gluon) effective action and
S (1) = −iln[ Det −1/2 M ab µν [A] Det −1/2 M ab µν [0] ] = i 2 Tr[lnM ν,ab µ [A] − lnM ν,ab µ [0]](14)
is the one loop (or gluon pair) effective action.
The trace Tr is given by
TrO = tr Lorentz tr color d 4 x < x|O|x > .(15)
We choose the arbitrary time-dependent chromo-electric field E a (t) to be along the z−axis (the beam direction) and work in the choice A a 3 = 0 so that
A a µ (x) = −δ µ0 E a (t)z(16)
is non-vanishing. The color indices are arbitrary, a=1,2,...8.
It can be seen that the tadpole effective action S tad in eq. (13) depends on the gauge fixing parameter α via the Green's function G νab λ (x, x ′ ). However, we will show in the appendix that the tadpole effective action S tad per unit time per unit volume and per unit transverse momentum ( dS tad d 4 xd 2 p T ) is zero for any non-vanishing transverse momentum. Hence there is no tadpole contribution to eq. (1) and we will not consider it any more.
We write eq. (8) as
M ab µν [A] = M ab µν, α=1 [A] + α ′ (D µ [A]D ν [A]) ab(17)
where
α ′ = ( 1 α − 1)(18)
and
M ab µν, α=1 [A] = g µν [D ρ (A)D ρ (A)] ab − 2gf abc F c µν [A].(19)
Hence we find
TrlnM ν,ab µ [A] = Trln[M λ µ, α=1 [A] {δ ν λ + α ′ M −1 σ λ, α=1 [A] (D σ [A]D ν [A])}] ab = Trln[M ν, ab µ, α=1 [A]] + Trln[δ ν µ δ ab + α ′ [M −1 λ µ, α=1 [A] D λ [A]D ν [A]] ab ].(20)
Since Trln[M ν, ab µ, α=1 [A]] was studied in [7] we will evaluate the α ′ = ( 1 α − 1) dependent term in this paper.
The ghost determinant is evaluated in [7] where we have used α=1. Since the ghost Lagrangian density is α independent [7], we do not discuss ghost in this paper. Whenever we mention α = 1 case in this paper, we assume that the ghost contribution is included.
Using eq. (15) we find
Trln[δ ν µ δ ab + α ′ [M −1 λ µ, α=1 [A] D λ [A]D ν [A]] ab ] = tr Lorentz tr color [ d 2 x T < x T | +∞ −∞ dt < t| +∞ −∞ dz < z| ln[δ ν µ + α ′ M −1 λ µ, α=1 D λ D ν ]|z > |t > |x T >] ab .(21)
Using eq. (16) in (6) we find
D ab µ [A] = ∂ µ δ ab − δ µ0 z igΛ ab (t) (22) where Λ ab (t) = if abc E c (t).(23)Trln[δ ν µ δ ab + α ′ [M −1 λ µ, α=1 [A] D λ [A]D ν [A]] ab ] = tr Lorentz tr color [ d 2 x T < x T | +∞ −∞ dt < t| +∞ −∞ dz < z + i gΛ(t) d dt | ln[δ ν µ + α ′ (M ′ ) −1 λ µ, α=1 D ′ λ D ′ν ]|z + i gΛ(t) d dt > |t > |x T >] ab = tr Lorentz tr color [ d 2 x T < x T | +∞ −∞ dt < t| +∞ −∞ dz < z + i gΛ(t) d dt | ln[δ ν µ + α ′ D ′ µ 1 (D ′ ) 2 D ′ν ]|z + i gΛ(t) d dt > |t > |x T >] ab(25)
where
D ′ ab µ [A] = (1 − δ µ0 ) δ ab ∂ µ − δ µ0 z igΛ ab (t)(26)
(µ is not summed) and
M ′ ab µν, α=1 [A] = g µν [D ′ ρ (A)D ′ ρ (A)] ab − 2gf abc F c µν [A].(27)
It has to be remembered that the z integration must be performed from -∞ to +∞ for the shift theorem [4] to be applicable.
Expanding the Logarithm in eq. (25) we find
ln[δ ν µ + α ′ D ′ µ 1 (D ′ ) 2 D ′ν ] ab = α ′ [D ′ µ 1 (D ′ ) 2 D ′ν ] ab − α ′ 2 2 [D ′ µ 1 (D ′ ) 2 D ′ν ] ab + α ′ 3 3 [D ′ µ 1 (D ′ ) 2 D ′ν ] ab − α ′ 4 4 [D ′ µ 1 (D ′ ) 2 D ′ν ] ab + α ′ 5 5 [D ′ µ 1 (D ′ ) 2 D ′ν ] ab − .........(28)
Summing the series we obtain
ln[δ ν µ + α ′ D ′ µ 1 (D ′ ) 2 D ′ν ] ab = ln(1 + α ′ ) [D ′ µ 1 (D ′ ) 2 D ′ν ] ab .(29)
Using the cyclic properties of the trace (Tr
[D ′ µ 1 (D ′ ) 2 D ′ν ] ab = Tr[D ′ν D ′ µ 1 (D ′ ) 2
] ab ) and using eq. (29) we find from eq. (25)
Trln[δ µ ν δ ab + α ′ [M −1 λ µ, α=1 [A] D λ [A]D ν [A]] ab = tr Lorentz tr color [ d 2 x T < x T | +∞ −∞ dt < t| +∞ −∞ dz < z + i gΛ(t) d dt | D ′ µ 1 (D ′ ) 2 D ′ν ln[1 + α ′ ] |z + i gΛ(t) d dt > |t > |x T >] ab = tr color [ d 2 x T < x T | +∞ −∞ dt < t| +∞ −∞ dz < z + i gΛ(t) d dt | ln[1 + α ′ ] |z + i gΛ(t) d dt > |t > |x T >] ab = 8 ln[1 + α ′ ] d 4 x d 4 p = − 8 ln(α) d 4 x d 4 p(30)
where we have used α ′ = (
where the gauge parameter α dependence exactly cancelled from the interacting and free part. The imaginary part of this effective action S (1) gives real gluon pair production result (eq. (1)) [7]. Hence we find that the non-perturbative result for gluon pair production from arbitrary E a (t) via Schwinger mechanism is independent of arbitrary gauge parameter α which is used in the gauge fixing term in the background field method of QCD.
To conclude we have studied the Schwinger mechanism for gluon pair production in the presence of arbitrary time-dependent chromo-electric field E a (t) in arbitrary covariant background gauge α with arbitrary color index a=1,2,..8 by directly evaluating the path integral. We have found that the exact result for non-perturbative gluon pair production from arbitrary E a (t) via Schwinger mechanism is independent of arbitrary gauge parameter α which is used in the gauge fixing term in the background field method of QCD. We find that the non-perturbative gluon pair production result from arbitray E a (t) via Schwinger mechanism is both gauge invariant and gauge parameter α independent. Gluon production from classical chromo field may be relevant to study production of quark-gluon plasma at RHIC and LHC.
where 3 3 means the µ = 3 and ν = 3 component of the Lorentz matrix. Using < q|p >=
dE b (t ′ ) dt ′ (A5)
By using eq. (16) in (6) we findwhere Λ ab (t) is given by eq. (23). From eqs.(8)and (11) we obtainUsing eqs. (A2) and (16) in (13) we find the tadpole effective actionInserting complete set of |p T > states (by using d 2 p T |p T >< p T | = 1) we findIntegrating over x ′ T (by usingwhich has a δ (2) ( p T ) distribution. Hence we find for any non-vanishing p T dS tad d 4 xd 2 p T = 0.Hence the tadpole (or single gluon) effective action do not contribute to eq. (1) of the non-perturbative gluon (pair) production rate dW d 4 xd 2 p T via Schwinger mechanism.
. J Schwinger, Phys. Rev. 82662J. Schwinger, Phys. Rev. 82 (1951) 662.
. G Nayak, P Van Nieuwenhuizen, ; F Cooper, G C Nayak, Phys. Rev. 7165005Phys. Rev.G. Nayak and P. van Nieuwenhuizen Phys. Rev. D71 (2005) 125001; F. Cooper and G. C. Nayak, Phys. Rev. D73 (2006) 065005.
. G C Nayak, Phys. Rev. 72125010G. C. Nayak, Phys. Rev. D72 (2005) 125010.
. F Cooper, G C Nayak, hep-th/0609192F. Cooper and G. C. Nayak, hep-th/0609192.
. F Cooper, G C Nayak, hep-th/0612292F. Cooper and G. C. Nayak, hep-th/0611125; hep-th/0612292.
. G C Nayak, arXiv:0705.2770hep-phG. C. Nayak, arXiv:0705.2770 [hep-ph].
. G C Nayak, Eur. Phys. J. 59715G. C. Nayak, Eur. Phys. J. C59:715 (2009).
. G Hooft, Nucl. Phys. 62444G. 't Hooft, Nucl. Phys. B62 (1973) 444.
. L F Abbott, Nucl. Phys. 185189L. F. Abbott, Nucl. Phys. B185 (1981) 189.
| []
|
[
"NEW ESTIMATES FOR THE NUMERICAL RADIUS",
"NEW ESTIMATES FOR THE NUMERICAL RADIUS"
]
| [
"Hamid Reza ",
"Mohammad Sababheh "
]
| []
| []
| In this article, we present new inequalities for the numerical radius of the sum of two Hilbert space operators. These new inequalities will enable us to obtain many generalizations and refinements of some well known inequalities, including multiplicative behavior of the numerical radius and norm bounds.Among many other applications, it is shown that ifwhere ω (·) and · denote the numerical radius and the usual operator norm, respectively. This inequality provides a considerable refinement of the well known inequality 1 2 T ≤ ω(T ).2010 Mathematics Subject Classification. Primary 47A12, 47A30, Secondary 15A60, 47B15. | 10.2298/fil2114957m | [
"https://arxiv.org/pdf/2010.12756v1.pdf"
]
| 225,072,900 | 2010.12756 | 11d64e8b2c43c40a133207da457c680632fc5367 |
NEW ESTIMATES FOR THE NUMERICAL RADIUS
24 Oct 2020
Hamid Reza
Mohammad Sababheh
NEW ESTIMATES FOR THE NUMERICAL RADIUS
24 Oct 2020
In this article, we present new inequalities for the numerical radius of the sum of two Hilbert space operators. These new inequalities will enable us to obtain many generalizations and refinements of some well known inequalities, including multiplicative behavior of the numerical radius and norm bounds.Among many other applications, it is shown that ifwhere ω (·) and · denote the numerical radius and the usual operator norm, respectively. This inequality provides a considerable refinement of the well known inequality 1 2 T ≤ ω(T ).2010 Mathematics Subject Classification. Primary 47A12, 47A30, Secondary 15A60, 47B15.
Introduction
Let H be a complex Hilbert space, endowed with the inner product ·, · : H × H → C.
The C * -algebra of all bounded linear operators on H will be denoted by B(H). An operator A ∈ B(H) is said to be positive (denoted by A > 0) if Ax, x > 0 for all non-zero vectors x ∈ H, and self-adjoint if A * = A, where A * is the adjoint of A. For A ∈ B(H), the Cartesian decomposition of A is A = ℜA + iℑA, where
ℜA = A + A * 2 and ℑA = A − A * 2i ,
are the real and imaginary parts of A, respectively. It is clear that both ℜA and ℑA are self-adjoint. If ℜA > 0, the operator A will be called accretive, and if both ℜA and ℑA are positive, A will be called accretive-dissipative.
Among the most interesting scalar quantities associated with an operator A ∈ B(H) are the the usual operator norm and the numerical radius, defined respectively by
A = sup x =1 Ax and ω(A) = sup x =1 | Ax, x | .
It is well known that A = sup x = y =1 | Ax, y |. The operator norm and the numerical radius are always comparable by the inequality
1 2 A ≤ ω(A) ≤ A , A ∈ B(H). (1.1) When A ∈ B(H) is normal (i.e., AA * = A * A), we have A = ω(A).
The significance of (1.1) is having upper and lower bounds of ω(A); a quantity that is not easy to calculate. Due to the importance and applicability of the quantity ω(A), interest has grown in having better bounds of ω(A) than the bounds in (1.1).
The main goal of this article is to present new inequalities for the numerical radius. More precisely, we study inequalities for quantities of the forms ω(A + B) and ω(A + iB), A, B ∈ B(H).
Our approach will be based mainly on the scalar inequality
(1.2) |a + b| ≤ √ 2 |a + ib| ,
valid for a, b ∈ R. Using this inequality is a new approach to tickle numerical radius inequalities.
This approach will enable us to obtain new bounds, some of which are generalizations of certain known bounds. For example, we will refine Kittaneh inequality [4],
(1.3) 1 4 |A| 2 + |A * | 2 ≤ ω 2 (A) ≤ 1 2 |A| 2 + |A * | 2 ;
noting that the inequalities (1.3) refine the inequalities (1.1).
In fact, our approach will not enable us to refine these inequalities only, it will present a new proof and generalization of (1.3).
Moreover, using this approach, we will show that
1 √ 2 A ≤ ω(A)
for the accretive-dissipative operator A. This inequality presents a considerable improvement of the first inequality in (1.1). As a result, we will be able to introduce a better bound for the sub-multiplicative behavior of the numerical radius, when dealing with accretive-dissipative operators. More precisely, we show that ω(ST ) ≤ 2ω(S)ω(T ) when both S and T are accretivedissipative. See Corollary 2.3 below for further discussion.
Other results including reverses of (1.3) and a refinement of the triangle inequality will be shown too.
In our proofs, we will need to recall the following inequalities.
Ax, x 2 ≤ A 2 x, x .
Also, we will need the following result [6, Proposition 3.8].
Lemma 1.3. Let A, B ∈ B(H) be positive. Then A + iB ≤ A + B .
In the following proposition, we restate (1.3) in terms of the Cartesian decomposition.
Proposition 1.1. Let A, B ∈ B (H) be self-adjoint.
Then
1 2 A 2 + B 2 ≤ ω 2 (A + iB) ≤ A 2 + B 2 .
New Results
We begin our main results with the the numerical radius version of the inequality (1.2), which can be stated as follows.
Theorem 2.1. Let A, B ∈ B (H).
Then
ω (A + B) ≤ 1 √ 2 ω ((|A| + |B|) + i (|A * | + |B * |)) .
Further, this inequality is sharp, in the sense that the factor 1 √ 2 cannot be replaced by a smaller number.
Proof. Let x ∈ H be a unit vector. Then
| (A + B) x, x | ≤ | Ax, x | + | Bx, x | ≤ |A| x, x |A * | x, x + |B| x, x |B * | x, x ≤ 1 2 ( |A| x, x + |A * | x, x ) + 1 2 ( |B| x, x + |B * | x, x ) = |A| + |B| 2 x, x + |A * | + |B * | 2 x, x ≤ √ 2 |A| + |B| 2 x, x + i |A * | + |B * | 2 x, x = √ 2 |A| + |B| 2 + i |A * | + |B * | 2 x, x ,
where the first inequality follows from the triangle inequality, the second inequality is obtained by Lemma 1.1, the third inequality is obtained by the arithmetic-geometric mean inequality and the forth inequality, follows from (1.2). Therefore, we have shown that for any unit vector
x ∈ H,
| (A + B) x, x | ≤ 1 √ 2 | ((|A| + |B|) + i (|A * | + |B * |)) x, x | .
Now, by taking supremum over all unit vector x ∈ H, we get
ω (A + B) ≤ 1 √ 2 ω ((|A| + |B|) + i (|A * | + |B * |)) ,
which completes the proof of the first assertion of the Theorem. To show that the factor 1 √ 2 is best possible, let B = 0 and assume that A is positive. Direct calculations show that the inequality is sharp, which completes the proof.
The following result shows how Theorem 2.1 refines (1.3).
Corollary 2.1. Let A ∈ B (H).
Then
ω 2 (A) ≤ 1 2 ω 2 (|A| + i |A * |) ≤ 1 2 |A| 2 + |A * | 2 .
Proof. The first inequality follows from Theorem 2.1 by taking B = 0 and the second inequality follows from Proposition 1.1. This completes the proof.
Using the same method presented in Theorem 2.1, we can obtain the following result; a different form of Theorem 2.1.
Theorem 2.2. Let A, B ∈ B (H).
Then
ω (A + B) ≤ 1 √ 2 ω ((|A| + |A * |) + i (|B| + |B * |)) .
Further, the factor 1 √ 2 is best possible.
The next Corollary follows from Theorem 2.1 and by taking into account that the sum of two normal operators, need not necessarily a normal operator.
Corollary 2.2. Let A, B ∈ B (H) be two normal operators.
Then
ω (A + B) ≤ √ 2ω (|A| + i |B|) .
In particular, if T = A + iB is accretive-dissipative, then
(2.1) ω(T ) ≥ 1 √ 2 A + B .A + B u ≤ √ 2 A + iB u , (2.2)
for any unitarily invariant norm · u . It is implicitly understood that · u is defined on an ideal in B(H), and it is implicitly understood that T is in that ideal, when we speak of T u . We notice, first, that Corollary 2.2 provides the numerical radius version of (2.2), in which A, B are normal; a wider class than positive matrices. Further, Corollary 2.2 provides a refinement of (2.2) in case of the usual operator norm since
A + B ≤ √ 2ω(A + iB) ≤ √ 2 A + iB .
The next result provides a considerable improvement of the first inequality in (1.1), for accretive-dissipative operators.
Theorem 2.3. Let T ∈ B (H) be accretive-dissipative. Then 1 √ 2 T ≤ ω (T ) .
Proof. Let T = A + iB be the Cartesian decomposition of T , in which both A, B are positive. Then Corollary 2.2 together with Lemma 1.3 imply
T = A + iB ≥ A + B ≤ √ 2ω(T ).
This completes the proof.
From (1.1) and the fact that the operator norm is sub-multiplicative, we obtain the well known inequality
ω(AB) ≤ AB ≤ A B ≤ 4ω(A)ω(B).
It is well established that the factor 4 cannot be replaced by a smaller factor in general. However, when A or B is normal, we obtain the better bound ω(AB) ≤ 2ω(A)ω(B), and it is even better when both are normal as we have ω(AB) ≤ ω(A)ω(B). In the following result, we present a new bound for accretive-dissipative operators, which is better than the bound ω(AB) ≤ 4ω(A)ω(B).
We refer the reader to [2] for detailed study of this problem. If either T or S is accretive-dissipative, then
ω(ST ) ≤ 2 √ 2ω(S)ω(T ).
Proof. Noting submultiplicativity of the operator norm and Theorem 2.3, we have
ω (ST ) ≤ ST ≤ S T ≤ 2ω (S) ω (T ) ,
which completes the proof of the first inequality. The second inequality follows similarly.
It is interesting that the approach we follow in this paper allows us to obtain reversed inequalities, as well. In [4], it is shown that
1 4 |A| 2 + |A * | 2 ≤ w 2 (A).
In the following, we present a refinement of this inequality using our approach.
Corollary 2.4. Let T ∈ B (H) have the Cartesian decomposition T = A + iB. Then
1 4 |T | 2 + |T * | 2 ≤ √ 2 2 ω A 2 + iB 2 ≤ ω 2 (T ) .
Proof. In Corollary 2.2, replace A and B by A 2 and B 2 . This implies
A 2 + B 2 ≤ √ 2ω A 2 + iB 2 ≤ 2 A 4 + B 4 (by Proposition 1.1) ≤ 2 A 4 + B 4 ,
where the last inequality follows by the triangle inequality for the usual operator norm. If T = A + iB is the Cartesian decomposition of the operator T , then
1 2 |T | 2 + |T * | 2 = A 2 + B 2 and 2 A 4 + B 4 ≤ 2ω 2 (T )
.
Therefore 1 4 |T | 2 + |T * | 2 ≤ √ 2 2 ω A 2 + iB 2 ≤ ω 2 (T ) ,
which is equivalent to the desired result.
On the other hand, manipulating Proposition 1.1 implies the following refinement of the triangle inequality
A + B ≤ A + B .
The connection of this result to our analysis is the refining term ω(A + iB).
Theorem 2.4. Let A, B ∈ B (H) be two self-adjoint operators. Then
A + B ≤ ω 2 (A + iB) + 2 A B ≤ A + B .
Proof. Let x ∈ H be a unit vector. Then
| (A + B) x, x | 2 ≤ (| Ax, x | + | Bx, x |) 2 = | Ax, x | 2 + | Bx, x | 2 + 2 | Ax, x | | Bx, x | = | (A + iB)
x, x | 2 + 2 | Ax, x | | Bx, x | .
Therefore,
A + B 2 ≤ ω 2 (A + iB) + 2 A B .
On the other hand, noting Proposition 1.1,
ω 2 (A + iB) + 2 A B ≤ A 2 + B 2 + 2 A B ≤ A 2 + B 2 + 2 A B ≤ ( A + B ) 2 .
Therefore,
A + B ≤ ω 2 (A + iB) + 2 A B ≤ A + B .
This completes the proof.
Corollary 2. 3 .
3Let S, T ∈ B (H) be two accretive-dissipative operators. Then ω (ST ) ≤ 2ω (S) ω (T ) .
Lemma 1.2. Let A ∈ B (H) be self-adjoint.Then for any unit vector x ∈ H,Lemma 1.1. [3, pp. 75-76] Let A ∈ B(H) and let x ∈ H. Then
| Ax, x | ≤
|A|x, x |A * |x, x .
The singular values of A + B and A + iB. R Bhatia, F Kittaneh, Linear Algebra Appl. 431R. Bhatia and F. Kittaneh, The singular values of A + B and A + iB, Linear Algebra Appl., 431 (2009) 1502-1508
Numerical range: The field of values of linear operators and matrices. K E Gustafson, D K M Rao, Springer-VerlagNew YorkUniversitextK. E. Gustafson and D. K. M. Rao, Numerical range: The field of values of linear operators and matrices, Universitext. Springer-Verlag, New York, 1997.
P R Halmos, A Hilbert Space Problem Book. New YorkSpringer2nd ed.P . R. Halmos, A Hilbert Space Problem Book , 2nd ed., Springer, New York, 1982.
Numerical radius inequalities for Hilbert space operators. F Kittaneh, Studia Math. 1681F. Kittaneh, Numerical radius inequalities for Hilbert space operators, Studia Math., 168(1) (2005), 73-80.
Norm inequalities for sums and differences of positive operators. F Kittaneh, Linear Algebra Appl. 383F. Kittaneh, Norm inequalities for sums and differences of positive operators, Linear Algebra Appl., 383 (2004), 85-91.
Norm inequalities for accretive-dissipative operator matrices. M Lin, D Zhou, J. Math. Anal. Appl. 407M. Lin and D. Zhou, Norm inequalities for accretive-dissipative operator matrices, J. Math. Anal. Appl., 407 (2013), 436-442.
E-mail address: [email protected] (M. Sababheh) Department of Basic Sciences, Princess Sumaya University For Technology. H R O Moradi ; P, Box, Al JubaihaTehran, Iran; AmmanDepartment of Mathematics, Payame Noor University (PNUJordan. E-mail address: [email protected](H. R. Moradi) Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-4697, Tehran, Iran. E-mail address: [email protected] (M. Sababheh) Department of Basic Sciences, Princess Sumaya University For Technology, Al Jubaiha, Amman 11941, Jordan. E-mail address: [email protected]
| []
|
[
"Robust A-type order and spin-flop transition on the surface of the antiferromagnetic topological insulator MnBi2Te4",
"Robust A-type order and spin-flop transition on the surface of the antiferromagnetic topological insulator MnBi2Te4"
]
| [
"Paul M Sass \nDepartment of Physics and Astronomy\nRutgers University\n08854PiscatawayNJUSA\n",
"Jinwoong Kim \nDepartment of Physics and Astronomy\nRutgers University\n08854PiscatawayNJUSA\n",
"David Vanderbilt \nDepartment of Physics and Astronomy\nRutgers University\n08854PiscatawayNJUSA\n",
"Jiaqiang Yan \nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTennesseeUSA\n",
"Weida Wu \nDepartment of Physics and Astronomy\nRutgers University\n08854PiscatawayNJUSA\n"
]
| [
"Department of Physics and Astronomy\nRutgers University\n08854PiscatawayNJUSA",
"Department of Physics and Astronomy\nRutgers University\n08854PiscatawayNJUSA",
"Department of Physics and Astronomy\nRutgers University\n08854PiscatawayNJUSA",
"Materials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTennesseeUSA",
"Department of Physics and Astronomy\nRutgers University\n08854PiscatawayNJUSA"
]
| []
| Here we present microscopic evidence of the persistence of uniaxial A-type antiferromagnetic order to the surface layers of MnBi2Te4 single crystals using magnetic force microscopy. Our results reveal termination-dependent magnetic contrast across both surface step edges and domain walls, which can be screened by thin layers of soft magnetism. The robust surface A-type order is further corroborated by the observation of termination-dependent surface spin-flop transitions, which have been theoretically proposed decades ago. Our results not only provide key ingredients for understanding the electronic properties of the antiferromagnetic topological insulator MnBi2Te4, but also open a new paradigm for exploring intrinsic surface metamagnetic transitions in natural antiferromagnets. | 10.1103/physrevlett.125.037201 | [
"https://arxiv.org/pdf/2006.07656v1.pdf"
]
| 219,687,345 | 2006.07656 | 1b341b6e9a1bf44de0cfe2cbe426881a65fa5d42 |
Robust A-type order and spin-flop transition on the surface of the antiferromagnetic topological insulator MnBi2Te4
Paul M Sass
Department of Physics and Astronomy
Rutgers University
08854PiscatawayNJUSA
Jinwoong Kim
Department of Physics and Astronomy
Rutgers University
08854PiscatawayNJUSA
David Vanderbilt
Department of Physics and Astronomy
Rutgers University
08854PiscatawayNJUSA
Jiaqiang Yan
Materials Science and Technology Division
Oak Ridge National Laboratory
37831Oak RidgeTennesseeUSA
Weida Wu
Department of Physics and Astronomy
Rutgers University
08854PiscatawayNJUSA
Robust A-type order and spin-flop transition on the surface of the antiferromagnetic topological insulator MnBi2Te4
1 *Correspondence to: [email protected] (WW).
Here we present microscopic evidence of the persistence of uniaxial A-type antiferromagnetic order to the surface layers of MnBi2Te4 single crystals using magnetic force microscopy. Our results reveal termination-dependent magnetic contrast across both surface step edges and domain walls, which can be screened by thin layers of soft magnetism. The robust surface A-type order is further corroborated by the observation of termination-dependent surface spin-flop transitions, which have been theoretically proposed decades ago. Our results not only provide key ingredients for understanding the electronic properties of the antiferromagnetic topological insulator MnBi2Te4, but also open a new paradigm for exploring intrinsic surface metamagnetic transitions in natural antiferromagnets.
Recent progress in topological quantum materials suggest that antiferromagnets may host interesting topological states [1]. For example, it has been proposed that an axion insulator state with topological magnetoelectric response could be realized in an antiferromagentic topological insulator (TI) phase [2,3], where the Z2 topological states are protected by a combination of timereversal symmetry and primitive-lattice translation. The antiferromagnetic TI state adiabatically connects to a stack of quantum Hall insulators with alternating Chern numbers [4], thus providing a promising route to realizing the quantum anomalous Hall (QAH) effect in a stoichiometric material. The prior observation of the QAH effect in magnetically doped TI thin films is limited to extremely low temperature because of the inherent disorder [5][6][7][8][9], though the disorder effects can be partially alleviated by material engineering [10][11][12]. The MnBi2Te4 (MBT) family was predicted and confirmed to be an antiferromagnetic TI that may host QAH and axion-insulator states in thin films with odd and even numbers of septuple layers (SLs) respectively [13][14][15][16][17]. Recent transport measurements on exfoliated thin flakes provide compelling evidence for these predictions [18,19], suggesting gapped topological surface states. On the other hand, recent highresolution angle-resolved photoemission spectroscopy (ARPES) studies reveal gapless (or smallgap) surface states below the antiferromagnetic ordering temperature, suggesting a surface relaxation of the A-type order and/or the formation of nanometer-sized magnetic domains [20][21][22][23]. The antiferromagnetic domain structure of MnBi2Te4 was revealed by imaging of domain walls using magnetic force microscopy (MFM) [24]. The observed domain size is on the order of 10 µm, excluding the speculated nanometer-size domain scenario [22].
Thus, it is crucial to reveal the nature of surface magnetism of MnBi2Te4 in order to resolve the dichotomy between the observations of QAH transport and gapless topological surface states [18][19][20][21][22][23]. The magnetic imaging of A-type domain structures in MnBi2Te4 also enable explorations of the long-sought surface spin-flop (SSF) transition in a natural antiferromagnet [25][26][27][28][29][30][31]. In this letter, we report the observation of alternating termination-dependent magnetic signals on the surface of MnBi2Te4 single crystals using cryogenic MFM, which provides direct evidence of the persistence of uniaxial A-type antiferromagnetic order all the way to the surface. Combined with the recent ARPES observations of gapless surface states, our results suggest a possible scenario of a tiny magnetic mass gap due to weak coupling between the topological electronic states and the magnetic order. The robust A-type order is further corroborated by the observation of two SSF transitions on domains with opposite terminations revealed by the magnetic field dependence of the domain contrast. Although they have been theoretically studied for decades [25,28,29], SSF transitions have only been observed in synthetic antiferromagnets, not in natural ones [26,27,30,31]. Our results not only shed new light on the realization of topological states in antiferromagnets, but also open up exciting explorations of surface metamagnetic transitions in functional antiferromagnets.
For an A-type antiferromagnet with ordered moments along the c-axis, there are only two possible domain states, up-down-up-down () and down-up-down-up (). They are related to each other by either time reversal symmetry or a primitive lattice translation, so they are antiphase domains and the antiferromagnetic domains walls separating them are antiphase boundaries. Therefore, there would not be any vertex point connecting three or more domain walls. These expectations are confirmed by our recent cryogenic magnetic force microscopy (MFM) studies in high magnetic fields [24]. The typical domain size is ~10 µm, so the tiny contribution of chiral edge states at domain walls is insufficient to explain the gapless topological surface states [22]. However, it is unclear whether the A-type order persists up to the surface layer, because MFM contrast could come from sub-surface stray fields that penetrate the surface non-magnetic layer [32]. It has been speculated that the observed gapless surface states might be explained by surface relaxation or reorientation of the A-type order [20][21][22]. To address these issues, we carried out MFM studies on as-grown surface of MnBi2Te4 single crystals with multiple SL steps and thin layers of surface impurity phase. Prior studies suggest that the as-grown surface of MnBi2Te4 is decorated with small amounts of impurity-phase Bi2-xMnxTe3, which is a soft ferromagnet with a small coercive field (<0.04 T) [17,33]. These magnetically soft thin layers provide an excellent opportunity to probe the screening effects of the speculated relaxed surface magnetic order with enhanced magnetic susceptibility [21].
Platelike single crystals of MnBi2Te4 were grown out of a Bi-Te flux and have been well characterized by measuring the magnetic and transport properties [17]. The MFM experiments were carried out in a homemade cryogenic magnetic force microscope using commercial piezoresistive cantilevers. MFM tips were prepared by depositing nominally 150 nm Co film onto bare tips using e-beam evaporation. MFM images were taken in a constant height mode with the scanning plane nominally ~100 nm (except specified) above the sample surface [24]. The numerical simulations were performed with the revised Mills model. The reduced surface magnetization causes a pinning of the spin-flop state at the surface [34]. 1(c) show the MFM images taken at the same location. Note that one antiferromagnetic domain wall cuts across the SL steps. Clearly, the magnetic contrast reverses over the domain wall on one terrace (green arrow) and across SLs of one single domain (red arrow) as shown in Fig. 1(b) and illustrated by line profiles in Fig. 1(d) and 1(e). Here, bright contrast indicates a repulsive interaction, i.e., surface magnetization antiparallel to the MFM tip moment, which is fixed by a small out-of-plane magnetic field [32]. The domain contrast reverses over the domain wall, which is consistent with opposite surface magnetization states of different antiphase domains ( Fig. 1(d)) or SL steps ( Fig. 1(e)). There is a slight dip at the domain wall due to its higher susceptibility [24]. The slight asymmetry in the line profiles in Fig. 1(e) is due to the difference between forward and backward scanning [34]. The magnetic contrast originates from imperfect cancellation of magnetic stray field from the alternating ferromagnetic layers [35,36]. To confirm this, we reverse MFM tip moment using a negative magnetic field (0.3 T). The magnetic contrast indeed reverses as shown in Fig. 1(c), which unambiguously demonstrates that the alternating MFM signal is from the alternating surface magnetization. Note that there is a small island of impurity phase (Bi2-xMnxTe3) with a rougher surface sitting on the upper SL step edge ( Fig. 1(a)). It appears to screen the antiferromagnetic domain contrast, as shown in Fig. 1(b) and 1(c). To understand the screening effect of the impurity phase, we increase the scan size to sample more impurity phases. Figure 2(a) shows the topography of a large area with six SL steps in the field of view (~1813 µm 2 ). Most steps are paired to form curvy narrow terraces decorated with many platelike impurity islands with partial hexagon shapes. The height of these island (~3 nm) agrees with that of three quintuple layers (QLs) of Bi2Te3, which is slightly larger than that of two SLs (~2.7 nm) as shown in Fig. 2(i) [34]. Fig. 2(b) shows the MFM image (measured at 1 T) at this location after 0.425 T field cooling. There are two bubble-like antiferromagnetic domains with curvilinear domain walls. Alternating magnetic contrast was observed on uncovered SL terraces across step edges or antiferromagnetic domain walls. However, this contrast is suppressed if the surface is covered by the impurity phases, suggesting a very effective screening of the magnetic stray field [34]. To illustrate the details, zoom-in images of a few selected areas (boxes labelled 1, 2 and 3 in Fig. 2(a) and 2(b)) are shown in Fig. 2(c-h). Arrows (dashed lines) marked the exposed (covered) narrow terraces in these images [34]. As shown in box 3, the domain contrast can even be "blocked" by a fractional QL of the impurity phase, and clear domain contrast is visible in the holes of the impurity phase. Thus, we can conclude that the magnetic impurity phase (Bi2-xMnxTe3) effectively screens all the stray fields from the underlying MnBi2Te4 surface. Similar results are observed at higher temperature (below TN). In contrast, antiferromagnetic domain wall contrast is not affected by the impurity phase as shown in the white dotted box in Fig. 2(b), because domain walls extend into the bulk. Because the alternating domain and terrace contrast can be easily screened by such a thin layer (0.3-3 nm) of soft magnet (Bi2-xMnxTe3), the uniaxial A-type spin order must persist to the top surface layer of MnBi2Te4. Otherwise, the termination-dependent magnetic contrast would be screened by any relaxation of surface magnetism with substantial magnetic susceptibility, such as paramagnetism, non-A-type spin order, or in-plane A-type order proposed in prior reports [20][21][22][23]37]. Therefore, we can conclude that our MFM observation excludes some of the proposed surface relaxation models, and that the contradictory reports of gapless surface states and a quantized anomalous Hall effect remain unresolved. FIG. 2 (a,b) Topographic and MFM images of MnBi2Te4 surface covering measured in 1 T at 5 K after 0.425 T field cooling. Magnetic contrast of domains and terraces is visible. (c-h) Zoom-ins of topographic and MFM images outlined by solid white boxes in (a,b). White arrows (dashed lines) mark the exposed (covered) single SL steps. The bright domain contrast in region covered by the impurity phase is suppressed, as shown by white arrow in (h). Domain wall contrast is not suppressed by the impurity phase, as shown in the dotted box in (b). (i) Topographic line profiles (white dotted lines in (a)) of SLs and impurity phase QLs with schematic of spin configuration. The gray area illustrates a soft magnetic phase that screens the stray fields of the SL edges underneath. The color scales for the topographic and MFM images are 7, 6, 3 and 3 nm (0.2 Hz), respectively.
The observation of robust A-type order on the MnBi2Te4 surface also provides a rare opportunity to explore the interesting SFF transition (or inhomogeneous spin-flop), which was first proposed by Mills decades ago using an effective one-dimensional spin-chain model with AFM nearest-neighbor exchange coupling [25,29]. However, later studies suggested an intriguing scenario of inhomogeneous spin-flop state due to finite size effect [28,30,38]. The SSF transition was observed in synthetic AFMs, which are superlattices of antiferromagnetically coupled ferromagnetic layers [26,27], but not in natural AFMs [28,31]. Because of the existence of domains in natural AFMs, the exploration of SSF phenomena requires a surface-sensitive magnetic imaging probe with sufficient spatial resolution in high magnetic field. These challenges were overcome by our cryogenic MFM.
Figs. 3(a-h)
show selected MFM images measured in various magnetic fields from 1.0 to 3.5 T [34]. Clearly, the termination-dependent contrast shows non-monotonic magnetic field dependence. As discussed in connection with Fig. 1, in low magnetic field a bright contrast indicates surface termination with antiparallel magnetization denoted as a, while dark contrast indicates surface termination with parallel magnetization denoted as b in Fig. 3(a). This domain contrast persists in finite magnetic field up to ~1.85 T, then fine features start to emerge in termination a during the domain contrast reversal, while the termination b remains featureless. Thus, it is the termination a (antiparallel magnetization) that undergoes SSF transition at The detailed field dependence of domain contrast is plotted in Fig. 3(i), where the domain contrast is defined as the difference of the average MFM signals in the two regions (domain a and b) marked by red boxes in Fig. 3(a). This effect is also observed in negative applied field and is reproducible in other sample locations after thermal cycling and on a cleaved crystal of MnBi2Te4 [34]. No hysteresis was found between up-sweep and down-sweep of the magnetic field. The first SSF transition ( SSF 1 ≈ 0.5 BSF ) agrees well with prior observation in synthetic antiferromagnets [27], and is in reasonable agreement with that of the Mills model ( SSF ℎ ≈ 0.7 BSF ) [29,38]. However, the second SSF transition ( SSF 2 ≈ 0.9 BSF ) of the surface with parallel magnetization is unexpected in prior studies [26,28,38], indicating surface relaxation of the A-type AFM order. To confirm this, we studied the revised Mills model with additional surface relaxation effects such as reduced magnetization, exchange coupling, and/or anisotropy energy [28,34].
In the original Mills model, the antiparallel surface nucleates a horizontal domain wall with a spin-flop state that migrates into the bulk, forming an inhomogeneous state that precedes the bulk spin-flop transition. [28,29,38] If the migration indeed occurs, the antiparallel surface would sequentially turn into a parallel surface, resulting in an identical magnetization state on the two domains, i.e., no domain contrast above the SFF transition. Such behavior is inconsistent with our experimental observation of domain contrast reversal. Our simulation reveals that the horizontal domain wall with spin-flop state can be pinned to surface layers if the magnetization of surface layer is reduced >10% [34]. Indeed, the revised Mills model with surface relaxation effect can reproduce the two successive SSF transitions in a reasonably wide parameter space. Fig. 4(a) shows a phase diagram of the simulation using typical parameters exhibiting the emergent sequential SSF transitions on antiparallel (blue) and parallel (red) surfaces, respectively. In addition, the reduction of surface exchange coupling could explain the suppression of the SSF transition. The simulated MFM contrast (force gradient difference) as a function of magnetic field is shown in Fig. 4(b), qualitatively agreeing with the experimental observation shown in Fig. 3(i) [34]. The successive SSF and BSF transitions are summarized schematically in Fig. 4(c). The antiparallel surface layer (blue) undergoes a SSF transition SSF 1 where the MFM contrast reverses. The domain contrast increases even further in this region, likely due to an increasing canted moment of the spin-flop state. At the next critical field SSF 2 , the parallel surface (red) undergoes SSF transition, resulting in another reversal of the MFM contrast. Finally, the MFM domain contrast disappears above the BSF transition because both domains have the same canted moments.
To explore the impact of thermal fluctuations, we performed MFM studies at higher temperatures below to extract the T dependence of the SSF transitions ( SSF 1 and SSF 2 ) [34]. As shown in Fig. 4(d), the temperature dependence of both SSF transitions follow that of the BSF ( BSF ), which gradually reduces with increasing temperature until the bicritical point (~21 K, ~2.5 T), indicating the relative energetics of the SSF transitions do not vary much with temperature. Above 21 K, the antiferromagnetic domains become unstable in finite magnetic field because of enhanced thermal fluctuations, making it difficult to determine the SSF transitions in this temperature window.
In summary, our MFM results provide microscopic evidence of robust uniaxial A-type order that persists to the top surface layers in the antiferromagnetic topological insulator MnBi2Te4. Thus, our results strongly constrain the possible mechanisms of the observed gapless topological surface states. Furthermore, we observed, for the first time, the long-sought SSF transition in natural antiferromagnets. More interestingly, we discovered an additional surface SSF on the parallel magnetization surface, which indicates surface relaxation of the A-type order. The MFM observation of the SSF transition not only opens a new paradigm for visualizing surface metamagnetic transitions in antiferromagnetic spintronic devices, but also provides new insights into the realization of the quantum anomalous Hall or axion-insulator states in topological anitferromagnets [18,19].
FIG. 1
1(a), Topographic image (5 K) of one and two septuple layer (SL) steps on an as-grown MnBi2Te4 single crystal. (b,c) MFM images taken at 0.3 and 0.3 T, respectively, after field cooling at 0.6 T, at the same location as in (a). The applied magnetic field is perpendicular to the sample surface. A curvilinear domain wall cuts through the SL step. The domain and SL step contrast was reversed when the tip moment was flipped (dark is attractive and bright is repulsive). (d,e), Line profiles of the topography (black) and MFM (green and red) data. The frequency shift in (d) was measured across the domain wall over flat topography, while in (e) it was taken across the SLs. The color scale for the topographic (MFM) image(s) is 6 nm (0.3 Hz).
Figure 1 (
1a) shows a typical surface morphology of MnBi2Te4 as-grown surface. There are two step edges in this location, and the observed step height (~1.3 nm) agrees with that of a single SL.Figs. 1(b)and
FIG. 3
3(a-h) MFM images taken at 5 K with increasing field labeled in lower right corners. (i) Domain contrast between red squares, labeled a and b in A versus applied field. Below 1.75 T, the domain contrast is constant. As the applied field is further increased, a domains starts to appear rougher and darker near 1.85 T, then the domain contrast quickly reverses above 1.85 T. Similar behavior was observed on b domains around 3.1 T. Above 3.5 T, the system enters the canted AFM phase and the domain contrast disappears. The color scale for MFM images is 0.3 (a-d) and 0.8 (e-h) Hz.
T. Similar behavior was observed at ~3.1 T except the roles of a and b are switched. Thus, it is the termination b (parallel magnetization) that undergoes SSF transition at SSF 2~3 .1 T. Finally, the domain contrast disappears around the bulk spin-flop (BSF) transition ( BSF~3 .5 ).
FIG. 4
4(a) Theoretical phase diagram of the spin-flop state in the revised Mills model. Blue and red colored regimes illustrate SSF states for antiparallel and parallel surfaces, respectively. Color code denotes the difference of net spin canting between the two types of surfaces[34]. Black solid line is a phase boundary of the bulk spin-flop state; dashed line is a boundary between AFM and SSF phases for antiparallel (blue) and parallel (red) surfaces. (b) Simulated field dependence of magnetic force gradient differences between antiparallel and parallel surfaces.(c) Schematic illustration of the spin-flop process for surface (upper 4 rows) and bulk (lower) domains. Left blue (right red) represents antiparallel (parallel) surface spins, whereas, left green (right yellow) represents antiparallel (parallel) bulk spins. (d) H-T phase diagram showing A-type AFM phase (red), SSFA and SSFP spin-flop phase (pink and light purple), bulk CAFM phase (dark purple), and forced ferromagnetic or paramagnetic (PM) phase (light blue).
AcknowledgementThe MFM studies at Rutgers is supported by the Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, US Department of Energy under Award numbers DESC0018153. The simulation efforts is supported by ONR Grants N00014-16-1-2951. Work at ORNL was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division.
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See Supplemental Material at [URL Will Be Inserted by Publisher] for Full Dataset and Supporting Data. See Supplemental Material at [URL Will Be Inserted by Publisher] for Full Dataset and Supporting Data. (n.d.).
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| []
|
[
"The MOJAVE Program: Studying the Relativistic Kinematics of AGN Jets",
"The MOJAVE Program: Studying the Relativistic Kinematics of AGN Jets"
]
| [
"Matthew L Lister \nDepartment of Physics\nPurdue University\n47907West LafayetteIN\n"
]
| [
"Department of Physics\nPurdue University\n47907West LafayetteIN"
]
| [
"Future Directions in High Resolution Astronomy: The 10th Anniversary of the VLBA ASP Conference Series"
]
| We discuss a new VLBA program to investigate structural and polarization changes in the brightest AGN jets in the northern sky. Our study represents a significant improvement over previous surveys in terms of image fidelity, size, and completeness, and will serve to characterize the kinematics of AGN jets and determine how these are related to other source properties. We discuss several preliminary results from our program, including the detection of parsec-scale circularly polarized jet emission, enhanced magnetic field ordering at the sites of apparent bends, and intrinsic differences in the jets of strong-and weak-lined AGN. | null | [
"https://export.arxiv.org/pdf/astro-ph/0309413v1.pdf"
]
| 12,791,891 | astro-ph/0309413 | b3247ac3f7567415e1c906f9aba1f16783228f58 |
The MOJAVE Program: Studying the Relativistic Kinematics of AGN Jets
2003
Matthew L Lister
Department of Physics
Purdue University
47907West LafayetteIN
The MOJAVE Program: Studying the Relativistic Kinematics of AGN Jets
Future Directions in High Resolution Astronomy: The 10th Anniversary of the VLBA ASP Conference Series
2003
We discuss a new VLBA program to investigate structural and polarization changes in the brightest AGN jets in the northern sky. Our study represents a significant improvement over previous surveys in terms of image fidelity, size, and completeness, and will serve to characterize the kinematics of AGN jets and determine how these are related to other source properties. We discuss several preliminary results from our program, including the detection of parsec-scale circularly polarized jet emission, enhanced magnetic field ordering at the sites of apparent bends, and intrinsic differences in the jets of strong-and weak-lined AGN.
Introduction
Kinematic studies using very-long-baseline interferometry have greatly increased our understanding of extragalactic jets on parsec-scales, but there are still many questions regarding their dynamics and evolution that remain unanswered. This is partially due to the lack of a long-term, systematic, full-Stokes imaging survey of a large complete sample in which the selection effects are well-understood. MOJAVE (Monitoring Of Jets in AGN with VLBA Experiments) is a new VLBA program that is designed as a follow-up to the VLBA 2 cm survey (Kellermann et al. 1998). It is currently investigating structural and polarization changes in the 134 brightest AGN jets in the northern sky, of which 98 were part of the original VLBA 2 cm Survey. The main goals of MOJAVE are to:
• Measure the distribution of apparent jet speeds, both within individual sources and for the entire population.
• Determine whether jet speed is related to other intrinsic quantities, such as luminosity, black hole mass, and emission line strength.
• Characterize the kinematics of moving jet features, and determine whether they are consistent with streaming paths along bent trajectories.
• Investigate how magnetic fields evolve within the jets, and how they track the moving features.
• Search for parsec-scale circularly polarized jet emission, and examine how it evolves with time.
2 M. L. Lister
Sample Selection and Observational Status
The goals of our program require a complete sample of extragalactic jets that is large enough to investigate statistical aspects of the parent population, and to perform inter-comparisons between various sub-classes, such as quasars, BL Lacertae objects, gamma-ray loud sources, and intra-day variables. Whereas the selection criteria of the 2 cm VLBA survey were loosely defined in order to include a wide range of sources such as gigahertz-peaked spectrum objects and radio galaxies, the MOJAVE sample is selected purely on the basis of compact radio flux density. The latter choice ensures a high degree of completeness since a) all known AGN jets are radio-loud, b) all-sky radio surveys are available at several frequencies, c) all our sources are detectable by the VLBA. Another benefit is that we are able to directly compare our data to Monte Carlo simulations of relativistic beaming, without worrying about contamination from extended, steep-spectrum emission. Also, the radio emission comes from the same region as where we are measuring the apparent jet speeds, which is important when estimating Doppler beaming factors from the kinematic data. Our specific selection criteria are: a) declination > −20 • , b) galactic latitude |b| > 2.5 • , and c) total 2 cm VLBA flux density ≥ 1.5 Jy at any epoch since 1994 (≥ 2 Jy for sources with δ < 0 • ). By not restricting our flux density criterion to a single epoch, we have included many interesting variable sources that might have otherwise been omitted. Our final sample consists of 129 confirmed and 5 candidate objects. We are currently gathering single-dish and VLBA data on the latter to determine whether they meet our selection criteria. Thirty-four of our sources are members of the third EGRET gamma-ray catalog, and broken down by optical classification there are 95 quasars, 21 BL Lacs, 10 radio galaxies, and 8 unidentified objects. Redshift information is currently available for 90% of the sample.
Since the start of our observations in May 2002, we have obtained singleepoch polarization images for 95% of the sample, and 13 sources have been imaged at more than one epoch. As of Sept. 2003, we have had ten successful observing sessions, of which nine have been fully reduced. Our observations are currently planned to continue into 2004, and we have been allotted one new 24-hour long session every six to eight weeks. We are able to observe eighteen sources each session, which implies that each source is observed roughly once per year. There are many highly variable sources that need more frequent sampling, however, so we try to observe these more often at the expense of longer monitoring intervals for the other sources. We are also monitoring our sample at cm-wavelengths with the UMRAO and RATAN telescopes to monitor overall spectral changes and to calibrate our polarization vectors.
Preliminary Results
Linear Polarization
We have detected linearly polarized emission in all 123 jets we have imaged thus far, with the exception of NGC 1052 and 2021+614. The latter are likely heavily depolarized by intervening gas in the host galaxy (see, e.g., Vermeulen et al. 2002). The unresolved cores (located close to the base of the jet) are VLBA 2 cm images of the quasar 1222+216. Left image: total intensity contours. Middle image: linear polarization contours with electric vectors superimposed. Right image: total intensity contours with fractional polarization greyscale ranging from 0 to 40%. The restoring beam has dimensions 1.08 × 0.57 mas, and the rms noise in the I image is 0.2 mJy/beam. weakly polarized, with the majority having fractional polarizations under 4%. Six cores (3C 111, 4C 39.25, M87, 1413+135, 2008−159, and 2128−123) have no detectable polarization. In most cases the magnetic field order increases with distance down the jet, with features in the jets being appreciably more polarized than the cores. In many sources (e.g., 1222+216; Fig. 1), the field appears highly ordered on the outside edge where the jet bends, suggesting a compression of the field from interaction with the external medium.
Jet polarization versus optical line strength
The question of whether the jets of weak-lined blazars (BL Lacertae objects) are intrinsically different than those of broad-lined blazars (radio-loud quasars) is still not fully resolved, in large part due to uncertainties associated with projection and relativistic beaming effects. One basic test involves looking for differences in the jet polarization properties of these two classes. In Figure 2 we plot the distribution of fractional polarization for polarized features in the jets of BL Lacs and quasars in our sample. A Kolmogorov-Smirnov test on the two distributions indicates only a 0.1% probability that they are from the same parent distribution. Further evidence for intrinsic differences can be found in a plot of fractional polarization versus distance from the core (Fig. 3). The majority of the BL Lac jet components are more highly polarized at a given projected distance down the jet than those of the quasars. This confirms an earlier result obtained at 7 mm by Lister (2001) for the smaller Pearson-Readhead AGN sample. Fig. 3 also shows a weak trend of increasing fractional polarization down the jet, which suggests that the magnetic fields of quasar jets take longer than the BL Lac jets to become organized. It is noteworthy that these polarization differences are located well outside the broad-line region where the optical emission lines are produced. Further investigation is required to determine whether these reflect intrinsic differences in the interstellar medium of the host galaxy, or in the jet itself.
Circular Polarization
The large number of sources that we observe at each epoch makes our survey ideal for calibration of both linear and circular polarization (CP). Using the techniques of Homan et al. (2001), we have detected weak parsec-scale circularly polarized jet emission in 8 of 68 sources analyzed thus far. Since the CP level is variable in many sources, we anticipate on having a sufficient number of detections to look for correlations between linear and circular polarization levels, and monitor the CP sign consistency over time. We also intend to examine whether circularly polarized jets possess any peculiar characteristics that distinguish them from other sources.
Summary
The MOJAVE program represents the first large-scale, systematic survey of a complete sample of AGN jets in all four Stokes parameters. Its main goals are to obtain a better understanding of the kinematics and magnetic field structures of relativistic jets, and determine how these are related to other host galaxy properties such as the emission line strength and black hole mass. A more thorough description of our program and access to our on-line database can be found at http://www.physics.purdue.edu/∼mlister/MOJAVE.
The author wishes to acknowledge the members of the MOJAVE collaboration: H. Aller, M. Aller, M. Cohen, D. Homan, M. Kadler, K. Kellermann, Y. Kovalev, A. Lobanov, E. Ros, R. Vermeulen, and J. Zensus.
Figure 1.
Figure 2 .
2Top panel: distribution of fractional linear polarization for polarized features in quasar jets. Lower panel: distribution for features in BL Lacertae jets.
Figure 3 .
3Plot of fractional linear polarization versus projected distance from the core for polarized features in the jets of quasars (circles) and BL Lacertae objects (open squares) .
. D C Homan, J M Attridge, J F C Wardle, ApJ. 556113Homan, D. C., Attridge, J. M., & Wardle, J. F. C. 2001, ApJ, 556, 113
. K I Kellermann, AJ. 1151295Kellermann, K. I., et al. 1998, AJ, 115, 1295
. M L Lister, ApJ. 562208Lister, M. L. 2001, ApJ, 562, 208
. R C Vermeulen, A&A. 401113Vermeulen, R. C., et al. 2003, A&A, 401, 113
| []
|
[
"Crypto Experts Advise What They Adopt",
"Crypto Experts Advise What They Adopt"
]
| [
"Mohammadreza Hazhirpasand \nUniversity of Bern Bern\nSwitzerland\n",
"Oscar Nierstrasz \nUniversity of Bern Bern\nSwitzerland\n",
"Mohammad Ghafari [email protected] \nUniversity of Auckland\nAucklandNew Zealand\n"
]
| [
"University of Bern Bern\nSwitzerland",
"University of Bern Bern\nSwitzerland",
"University of Auckland\nAucklandNew Zealand"
]
| []
| Previous studies have shown that developers regularly seek advice on online forums to resolve their cryptography issues. We investigated whether users who are active in cryptography discussions also use cryptography in practice. We collected the top 1% of responders who have participated in crypto discussions on Stack Overflow, and we manually analyzed their crypto contributions to open source projects on GitHub. We could identify 319 GitHub profiles that belonged to such crypto responders and found that 189 of them used cryptography in their projects. Further investigation revealed that the majority of analyzed users (i.e., 85%) use the same programming languages for crypto activity on Stack Overflow and crypto contributions on GitHub. Moreover, 90% of the analyzed users employed the same concept of cryptography in their projects as they advised about on Stack Overflow. | 10.1109/asew52652.2021.00044 | [
"https://arxiv.org/pdf/2109.15093v1.pdf"
]
| 238,226,869 | 2109.15093 | 5d35f19b1e9d57d435de1916e702d754be0618ce |
Crypto Experts Advise What They Adopt
Mohammadreza Hazhirpasand
University of Bern Bern
Switzerland
Oscar Nierstrasz
University of Bern Bern
Switzerland
Mohammad Ghafari [email protected]
University of Auckland
AucklandNew Zealand
Crypto Experts Advise What They Adopt
Index Terms-Cryptographysecurityexpert profiling
Previous studies have shown that developers regularly seek advice on online forums to resolve their cryptography issues. We investigated whether users who are active in cryptography discussions also use cryptography in practice. We collected the top 1% of responders who have participated in crypto discussions on Stack Overflow, and we manually analyzed their crypto contributions to open source projects on GitHub. We could identify 319 GitHub profiles that belonged to such crypto responders and found that 189 of them used cryptography in their projects. Further investigation revealed that the majority of analyzed users (i.e., 85%) use the same programming languages for crypto activity on Stack Overflow and crypto contributions on GitHub. Moreover, 90% of the analyzed users employed the same concept of cryptography in their projects as they advised about on Stack Overflow.
I. INTRODUCTION
Previous studies have shown that developers have difficulty in securely using cryptography [1], yielding many crypto misuses in software projects [2]. Researchers have developed new tools and APIs to ease the adoption of cryptography [3], yet online Q&A forums are among the main information sources used to resolve developer issues [4].
Closer inspection of experts on Q&A forums can lead to new research directions. For instance, profiling developer expertise contributes to heightening the members' awareness about the reliability of responses [5] [6]. In particular, platforms such as Stack Overflow contain insecure code snippets and inexperienced developers blindly use such snippets [7]. Due to the lack of secure code examples in cryptography, we hypothesize that mapping the activity of top crypto developers cross-platform can provide an interesting path to find and evaluate their practices from the security perspective, and present such results for developers who are looking for reliable, secure crypto examples. In this study, we conduct a preliminary step by mapping the activity of top crypto developers on Stack Overflow and GitHub. To our knowledge, no study to date has investigated the mapping of developers in cryptography across software communities. Particularly, we address the following research question:
RQ: Do top crypto responders on Stack Overflow adopt cryptography in their GitHub projects?
We aim to look into the GitHub profile of top 1% of crypto responders to shed some light onto their crypto activities in practice. We extracted the top 1% of crypto responders (i.e., 804) who participated in discussions linked to 64 cryptography tags on Stack Overflow. We scraped their public profiles on Stack Overflow and found 319 GitHub profile links, 189 of which belonged to users who contributed to crypto files on GitHub. To assess how developers adopt cryptography in practice, we studied the programming languages and crypto concepts of such users across the two platforms. We considered (1) hashing, (2) symmetric/asymmetric, (3) sign/verification as the areas for crypto concepts. Each of the aforementioned areas contains various algorithms and concepts. We realized that 85% of analyzed users use common programming languages for crypto purposes on both platforms, e.g., developer A resolves Java-related crypto questions on Stack Overflow, and employs Java for cryptography on GitHub. Furthermore, 90% of the analyzed users had at least one common crypto concept on both platforms, e.g., developer A uses symmetric encryption on GitHub, and helps others in the same area on Stack Overflow. The present findings show that the practical experience of top crypto responders is noticeably in line with their theoretical experience. Future investigations are necessary to evaluate the reliability of coding practices from the security point of view.
The remainder of this paper is structured as follows. In section II, we present the methodology of this work, then we explain the results and discuss them in section III. We explain the related work in section V, and explain the threats to validity of this study in section IV. Finally, we conclude the paper in section VI.
II. METHODOLOGY
In this section, we describe how we choose crypto tags on Stack Overflow, and our approach to fetch the top 1% of crypto responders, extract their GitHub profiles, and identify their crypto contribution (See Figure 1). 1) Crypto Tags: To find top crypto responders on Stack Overflow, we had to identify crypto-related tags. We started with the "cryptography" tag, i.e., the base tag, to find other tags that were used together with the base tag. To access the data, we used the Data Explorer platform (Stack Exchange). 1 We found 11,130 posts that contained the base tag. Together with the base tag, there were 2,184 other tags, i.e., candidate tags. However, not all the candidate tags were related to cryptography. The list of candidate tags is available online. To discern crypto-related tags, two authors of this paper separately examined all the tags and marked the cryptorelated ones. We then calculated Cohen's kappa, a commonly used measure of inter-rater agreement [8], between the two reviewers, and achieved 94% Cohen's Kappa score between the two reviewers, which indicates almost perfect agreement. Next, we compared their list of crypto tags and discussed the inconsistencies. Finally, we came up with a list of 64 cryptorelated tags.
2) Crypto Responders: We executed a query on the Data Explorer platform to fetch the top 1% of crypto responders for each of the identified tags from Stack Overflow. Table I presents the 64 tags and associated top 1% of unique crypto responders. We excluded the crypto responders that we had already found in other tags. For instance, the crypto++ tag had four top crypto responders, considering that they were among other tags. In total, we retrieved 804 top crypto responders. The list of top crypto responders is available online. 3 3) Crypto Responder Profile: Stack Overflow offers the ability to its users to share their social media addresses (e.g., Twitter, GitHub, and personal websites) on their profile. Nevertheless, the aforementioned information is not accessible on Stack Exchange Data Explorer. Hence, to find the selected users' GitHub profiles, we automatically scraped profiles of the 804 Stack Overflow top crypto responders. Using the BeautifulSoup library in Python, we parsed each user profile automatically. For 804 Stack Overflow users, we could identify 319 GitHub profiles.
4) Crypto Contributors:
We used the GitHub repository API and collected a total of 19 633 public repositories associated with the 319 GitHub users. We selected the top seven programming languages used in the repositories, i.e., Python, Ruby, C, C++, Rust, Java, and C#.
To understand which crypto libraries are popular in the selected languages, we consulted with two crypto experts. Among the suggested names, there are some candidates that come with the languages, such as Java.security in Java, or the libraries that are widely accepted and well-known, such as Bouncy Castle for Java and C#. Afterward, to ensure the rest of the suggested libraries are largely accepted in developer community, we checked how popular (i.e., star and fork) the 3 http://crypto-explorer.com/mapping_data/ Table II, we employed the GitHub Code Search API and a custom regex script to identify in which files crypto namespaces, e.g., "System.Security", were used. At the time of writing this paper, the GitHub Code Search API could not perform the exact keyword search for the crypto namespaces. Therefore, we relied on a supplementary regex script to ensure the identified code snippets contain the namespaces. We retrieved a total of 2 404 crypto files in 812 repositories.
In the last step, we used git blame to identify the contributors who had committed to the 2 404 crypto files. To do so, we cloned the 812 crypto repositories and extracted authors and committers of crypto files by git blame. We then fetched the developers' email addresses, usernames, and full names by GitHub user API in order to check whether they are among the contributors of the 812 crypto repositories. Of the 319 top crypto responders on Stack Overflow, we found that 189 developers had crypto contributions on GitHub. They had on average 14 and 3 median crypto file contributions. 5) Manual Investigation: To address the research question, we performed a manual analysis to observe to what extent users employ cryptography in practice. To this end, we checked two aspects of their contribution, (1) the programming language used for crypto purposes on both platforms, (2) crypto concepts used on both platforms.
Identifying detailed crypto concepts in various crypto libraries as well as crypto discussions can be an arduous task. Therefore, we deduced the concepts used in this study from recent work on the categorization of developers' crypto challenges on Stack Overflow [1]. The researchers' findings revealed that developers mostly encounter challenges concerning hashing, symmetric/asymmetric, and digital signature. Accordingly, we assumed that developers commonly use three high-level crypto concepts, which are (1) hashing, (2) symmetric/asymmetric, and (3) signing/verification.
In our manual analysis, we attempted to find commonalities in the programming languages (i.e., the seven languages) and crypto concepts that are used by a developer on both platforms.
To compute the sample size for studying 189 users on GitHub, we defined a confidence level of 95% and 9% as the margin of error, which yields 74 for our sample size. We then randomly selected 74 users from the population. Writing queries on the Stack Exchange Data Explorer platform, we automatically retrieved all the posts (i.e., titles, question and answer body) wherein the 74 developers were involved on Stack Overflow.
Two authors of this paper manually reviewed all the posts to extract the programming languages used in the discussions, i.e., question and answer body. Afterward, they also checked the title and question body to understand to which concept or concepts a particular discussion can be assigned. They checked the crypto codes of the 74 users on GitHub, and extracted the crypto concept(s), and recorded the programming language of the crypto files. To understanding the crypto concepts, they looked for the APIs used in the crypto files. For instance, if the MessageDigest API was used in a Java crypto file, they assumed that the developer encountered the hashing topic in practice. In cases where they had doubts about the APIs, they referred to the API documentation of the library. They had several sessions in order to compare the results of their investigations and build a unified list. Ultimately, they checked for commonalities of the languages and the crypto concepts that the users used across the two platforms.
III. RESULTS AND DISCUSSION
In this section, we present and discuss our findings for the following research question: Do top crypto responders on Stack Overflow adopt cryptography in practice? We explore the usage of crypto responders' programming languages and crypto concepts on Stack Overflow and GitHub.
1) Stack Overflow: We extracted 804 top crypto responders in which 319 users shared their GitHub profile on Stack Overflow. We fetched the crypto discussions of the 74 users (the sample size), extracted their provided answers, and stored the names of the programming languages involved in the discussions. In total, 55% of discussions were about Java. A user could have participated in various discussions wherein different programming languages were involved. We therefore considered all those languages as being the areas of the user's crypto knowledge. The median value of programming languages used on Stack Overflow is 3 and 2.7 is the average value.
More than four-fifths of the developers (i.e., 65) participated in discussions where the three crypto-concepts were discussed. Similar to programming languages, a user can provide answers for a discussion in which the knowledge of a concept or mixed concepts are required. For instance, we considered (1) hashing (2) sign/verification for the discussion (ID:33305800) on Stack Overflow since a user was confused about the differences between hashing with SHA256 and signing with SHA256withRSA.
2) GitHub: Of 319 users with GitHub profiles, 189 had made crypto contributions to public repositories on GitHub. To conduct our manual analysis, we randomly selected 74 users from the 189 crypto developers. We extracted the names of programming languages where crypto APIs were used. The median value of programming languages used on GitHub is 1 and the average value is 1.4. In all 74 cases, the number of programming languages and crypto concepts on Stack Overflow was higher than or equal to the same groups of data on GitHub. For instance, developer A participated in discussions where three languages (i.e., C++, C#, Java) were involved as well as the three crypto concepts while the same developer only used Java crypto APIs for hashing purposes on GitHub. 3) Mapping result: Interestingly, we realized that 63 (i.e., 85%) of such users had used at least one language that matches their crypto activity on Stack Overflow. Such agreement implies that the users are confident in those languages. We split the 63 developers into three groups: those who used fewer than 50% of the languages in their GitHub open-source projects (i.e., 25), those who used half of the languages (i.e., 16), and those using more than 50% of the languages (i.e., 22) (See Figure 2). In particular, more than half of the developers (i.e., 38) had crypto contributions for either half or more than half of the languages that they prefer to provide crypto help for on Stack Overflow. The developers who used fewer than 50% of their Stack Overflow languages in open-source projects constitute 39% (i.e., 25) of the whole.
With regard to crypto concepts, there are 6 developers who used APIs on GitHub which are related to the three crypto concepts (See Figure 3). There are seven developers who used signing/verification and hashing, five developers who employed hashing and symmetric/asymmetric, and only two developers used signing/verfication and symmetric/asymmetric. The rest of developers only used one of the concepts in the identified projects. They might have a broader contribution to cryptography in open-source projects, however, it may be due to the limitation of our obtained knowledge concerning their practices on GitHub. On the other side, the manual investigation revealed that, on Stack Overflow, 65 developers participated in all three concepts, seven developers only in symmetric/asymmetric, and only two in signing/verification. Checking the labels of 74 developers, we uncovered that almost all of the developers (i.e., 67 or 90% ) worked with at least one common crypto concept on both platforms. Of the 67 users, 30% of them had more than one concept shared on both platforms. The findings imply that developers are confident in programming languages and the crypto concepts as they had relevant experience in practice. Likewise, user satisfaction, such as high upvotes for the responses on Stack Overflow, confirm that the users' guidance is practical and effective in the domain of cryptography.
IV. THREATS TO VALIDITY
We identified 804 developers who were among the top 1% of responders to 64 crypto tags on Stack Overflow. However, we were only able to find 319 of these developers on GitHub, and did not perform any exhaustive search on Google to find more users. A developer may have multiple accounts on GitHub for various purposes but we only consider one account per user. Some users may have private repositories and The three concepts on Stack Overflow
The three concepts on GitHub Fig. 3. The number of developers in each crypto concept on Stack Overflow and GitHub make more significant contributions to crypto-related projects, nevertheless, such contributions cannot be assessed. We looked into the repositories written in seven programming languages, and did not analyze the remaining repositories. Even though we included popular and default crypto libraries in each programming language, adding more crypto libraries in each programming language can allow a more realistic conclusion to be drawn. This is important, considering that the diversity of crypto libraries in each language is debatable. We used the git blame command to fetch a crypto file's contributors. Consequently, there is a likelihood that the developers who contributed to crypto files had committed to other parts of the file but not to the cryptography parts.
V. RELATED WORK
The significance of correctly employing cryptography and obtaining professional help from online sources has been discussed by numerous authors in the literature. Sifat et al. studied three popular online sources, i.e., crypto Stack Exchange, Security Stack Exchange, and Quora, to find out the common challenges concerning implementing security in data transmission [9]. Yang et al. carried out a large-scale analysis of security-related questions on Stack Overflow and reported a classification of five topics [10]. A recent study conducted by Meng et al. has recognized the challenges of writing secure Java code on Stack Overflow [11]. Their results provide compelling evidence to the fact that the security implications of coding options in Java, e.g., CSRF tokens, are partially grasped by many developers. Lastly, a study confirmed that developers are uncritically using the insecure code snippets found on Stack Overflow [12]. The aforementioned findings jeopardize the security of software [13]. We observed that relying on poorly validated responses on online forums was inextricably linked to software systems' security implications. In this research, we studied the crypto experts who frequently help others on Stack Overflow to observe if they adopt cryptography in practice.
A series of recent studies have focused on profiling developer expertise either on single or multiple platforms [14] [15]. A common concern in profiling developer expertise cross-platforms is to track developer identity, as developer activity can be dispersed from one platform to another [16]. For instance, Zhang et al. used the developer email and the hashing approach to identify the same developer with the same email address on another platform [17]. Yung et al. looked into the challenge of expert finding with the Topic Expertise Model (TEM) [18]. Their approach jointly modeled topics and expertise by combining textual content model and link structure analysis. Tian et al. proposed a novel methodology to extract experts that utilizes various user attributes and related platform-specific information, for instance, high-quality Stack Overflow answers in specific programming technologies and high-quality projects measured using source code metrics [19].
Sajedi et al. checked the features that overlap between GitHub and Stack Overflow [20]. They defined three highorder metrics related to both networks (i.e., development, management and popularity) Their findings reveal moderate and strong correlations between the derived metrics within each platform. Vasilescu et al. analyzed the differences of 46,967 active users both on Stack Overflow and GitHub to understand the Stack Overflow's involvement of the GitHub's developers [5]. They discovered that users who provide more answers on Stack Overflow tend to have a high number of commits. Their results imply that users with a high number of commits on GitHub have a greater tendency to take the role of a "teacher" instead of asking more questions on Stack Overflow. Vadlamani et al. focused on perceiving what constitutes the notion of an expert developer and what key elements affect developer contribution [21]. They conducted a survey with active software developers both on Stack Overflow and GitHub. Their results show that developers consider personal drivers to be more critical than professional factors for GitHub contribution, and the majority of experts participate in both private and public repositories. Furthermore, developers do not seem to be willing to participate on Stack Overflow as the questions are either uninteresting or easy, and they find the reward system demotivating.
VI. CONCLUSION AND FUTURE PLANS
We conducted a study of the top 1% of crypto responders on Stack Overflow to shed some light onto the adoption of cryptography on GitHub by the top crypto responders on Stack Overflow. In particular, to the best of our knowledge, no previous study has profiled crypto developers across online communities. We found 189 users who used cryptography in open-source projects on GitHub and studied 74 of this population. The results indicate that the majority of analyzed users (i.e., 85%) use the same programming languages for participating in crypto discussions on Stack Overflow and crypto contributions on GitHub. Closer inspection of three areas in cryptography (i.e., hashing, symmetric/asymmetric, or signing/verification) revealed that 90% of the analyzed users had practical experience with at least one of the crypto concepts that they had discussed on Stack Overflow. Collectively, the results demonstrate that top crypto users are consistent with their crypto activity on both platforms, and this provides a basis for further research to investigate the quality of their practical experience.
VII. ACKNOWLEDGMENTS
We gratefully acknowledge the financial support of the Swiss National Science Foundation for the project "Agile Software Assistance" (SNSF project No. 200020-181973, Feb. 1, 2019 -April 30, 2022).
Fig. 1 .
1The pipeline for collecting and analyzing top crypto responders
Fig. 2 .
2The number of developers based on their percentage of Stack Overflow programming languages usage in GitHub repositories
2 stack overflow
Base tag (cryptography)
Candidate tags (2184)
Manual analysis (94% kappa)
64 Crypto tags
Fetch Top 1% Responders
stack overflow
Crypto Tags
Crypto Responders
804 crypto users
stack overflow
Scrape top 1% user profile
Crypto Contributers
Manual Investigation
Check for PR languages on both
platforms
check for 3 crypto concepts on
both platforms
Look for commonalities in the two
factors
Identify 19 633 repositories
Seven programming languages
Look for crypto library usages
Identify 812 crypto repository
Identify 189 crypto contributors
319 Github users
74 crypto contributors
Crypto Responders
Crypto Responder profile
TABLE I THE
Isuggested open-source crypto libraries are on GitHub, e.g., libsodium for the C language had 9.2k stars and 1.4k forks. The crypto libraries had on average 1844 stars and 346 forks, and the median number were 1105 and 245, respectively.Using the compiled list of crypto libraries in64 CRYPTO TAGS AND ASSOCIATED UNIQUE TOP 1% CRYPTO
RESPONDERS (i.e., 804)
Responders
Tag
Responders
Tag
202
encryption
2
encryption-asymmetric
176
hash
2
cryptoapi
98
cryptography
2
pbkdf2
76
openssl
2
jca
29
md5
2
jasypt
20
keystore
2
commoncrypto
16
xor
2
libsodium
14
digital-sig
2
phpseclib
13
sha1
1
ellipticurve
12
x509certificate
1
ecdsa
11
rsa
1
diffie-hellman
10
mcrypt
1
rsacryptoserviceprovider
8
sha256
1
bcrypt
8
private-key
1
node-crypto
8
sha
1
sjcl
8
public-key
1
spongycastle
7
bouncycastle
1
cryptoswift
7
smartcard
1
hashlib
6
public-key-encryption
1
wolfssl
5
x509
0
crypto++
5
salt
0
pkcs11
5
hmac
0
jce
5
pycrypto
0
pkcs7
4
cryptojs
0
cng
4
pyopenssl
0
cryptographic-hash-function
3
aes
0
aescryptoserviceprovider
3
encryption-symmetric
0
rijndaelmanaged
3
rijndael
0
webcrypto-api
3
3des
0
mscapi
3
m2crypto
0
charm-crypto
3
botan
0
javax.crypto
2
des
0
nacl-cryptography
TABLE II THE
IISELECTED CRYPTO LIBRARIES IN THE SEVEN PROGRAMMING LANGUAGESPython
Ruby
Java
C
C++
C#
Rust
passlib
bcrypt-ruby
Java.security
libgcrypt
Botan
Bouncy Castle
octavo
pynacl
Ruby Themis Javax.crypto
NaCl
Cryptlib
libsodium-net
rustls
hashlib
digest
Bouncy Castle crypto-algorithms Cryptopp system.security.cryptography rust-crypto
pythemis
RbNaCl
Themis
HElib
PCLCrypto
sodiumoxide
PyElliptic
wolfSSL
crypto
bcrypt
libsodium
Ring
S2N-tls
https://data.stackexchange.com/stackoverflow/query/new 2 http://crypto-explorer.com/mapping_data/
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