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November 2009

Volume 50, Issue 11, partial issue

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On geometric perturbations of critical Schrödinger operators with a surface interaction

Pavel Exner
and Martin Fraas

J. Math. Phys. 50, 112101 (2009) (12 pages)

Online Publication Date: 3 November 2009

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We study singular Schrödinger operators with an attractive interaction supported by a closed smooth surface A ⊂ math3 and analyze their behavior in the vicinity of the critical situation where such an operator has empty discrete spectrum and a threshold resonance. In particular, we show that if A is a sphere and the critical coupling is constant over it, any sufficiently small smooth area-preserving radial deformation gives rise to isolated eigenvalues. On the other hand, the discrete spectrum may be empty for general deformations. We also derive a related inequality for capacities associated with such surfaces.
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03.65.Ge Solutions of wave equations: bound states
03.65.Db Functional analytical methods
02.10.Ud Linear algebra
03.65.Fd Algebraic methods
02.30.Tb Operator theory
02.40.-k Geometry, differential geometry, and topology

Degenerations of pre-Lie algebras

Thomas Beneš
and Dietrich Burde

J. Math. Phys. 50, 112102 (2009) (9 pages)

Online Publication Date: 3 November 2009

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We consider the variety of pre-Lie algebra structures on a given n-dimensional vector space. The group GLn(K) acts on it, and we study the closure of the orbits with respect to the Zariski topology. This leads to the definition of pre-Lie algebra degenerations. We give fundamental results on such degenerations, including invariants and necessary degeneration criteria. We demonstrate the close relationship to Lie algebra degenerations. Finally, we classify all orbit closures in the variety of complex two-dimensional pre-Lie algebras.
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02.10.Ud Linear algebra
02.40.Pc General topology

Multiparameter deformation theory for quantum confined systems

A. Aleixo
and A. Balantekin

J. Math. Phys. 50, 112103 (2009) (13 pages)

Online Publication Date: 9 November 2009

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We introduce a generalized multiparameter deformation theory applicable to all supersymmetric and shape-invariant systems. Taking particular choices for the deformation factors used in the construction of the deformed ladder operators, we show that we can generalize the one-parameter quantum-deformed harmonic oscillator models and build alternative multiparameter deformed models that are also shape invariant like the primary undeformed system.
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03.65.Fd Algebraic methods
03.65.Ge Solutions of wave equations: bound states
02.10.Ud Linear algebra
02.20.Sv Lie algebras of Lie groups
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Lorentz symmetric quantum field theory for symplectic fermions

Dean Robinson ,
Eliot Kapit ,
and André LeClair

J. Math. Phys. 50, 112301 (2009) (13 pages)

Online Publication Date: 5 November 2009

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A free quantum field theory with Lorentz symmetry is derived for spin-half symplectic fermions in 2+1 dimensions. In particular, we show that fermionic spin-half fields may be canonically quantized in a free theory with a Klein–Gordon Lagrangian. This theory is shown to have all the required properties of a consistent free quantum field theory, namely, causality, unitarity, adherence to the spin-statistics theorem, CPT symmetry, and the Hermiticity and positive definiteness of the Hamiltonian. The global symmetry of the free theory is Sp(4) ≃ SO(5). Possible interacting theories of both the pseudo-Hermitian and Hermitian variety are then examined briefly.
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11.30.Cp Lorentz and Poincaré invariance
11.30.Er Charge conjugation, parity, time reversal, and other discrete symmetries
11.30.Fs Global symmetries (e.g., baryon number, lepton number)
11.10.Ef Lagrangian and Hamiltonian approach

On the optical theorem and non-plane-wave scattering in quantum mechanics

G. Gouesbet

J. Math. Phys. 50, 112302 (2009) (5 pages)

Online Publication Date: 6 November 2009

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In quantum mechanics, the optical theorem states that the extinction cross section is equal (within a prefactor 4π/k, in which k is a quantum wave number) to the imaginary part of the forward scattering angular function. This theorem is valid for plane wave scattering. We discuss modifications required for non-plane-wave scattering and establish a generalized expression for the extinction cross section in quantum mechanics. Examples are provided for two kinds of quantum shaped beams, namely, Gaussian and Bessel beams.
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03.65.-w Quantum mechanics
42.50.-p Quantum optics

Pseudoduality between symmetric space sigma models

Mustafa Sarisaman

J. Math. Phys. 50, 112303 (2009) (25 pages)

Online Publication Date: 9 November 2009

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We study the pseudoduality transformation on the symmetric space sigma models. We switch the Lie group-valued pseudoduality equations to Lie algebra-valued ones, which leads to an infinite number of pseudoduality equations. We obtain an infinite number of conserved currents on the tangent bundle of the pseudodual manifold. We show that there can be mixing of decomposed spaces with each other, which leads to mixings of the following expressions. We obtain the mixing forms of curvature relations and one-loop renormalization group beta functions by means of these currents.
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02.20.Sv Lie algebras of Lie groups
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An extension theorem for conformal gauge singularities

Christian Lübbe
and Paul Tod

J. Math. Phys. 50, 112501 (2009) (28 pages) | Cited 1 time

Online Publication Date: 3 November 2009

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We analyze conformal gauge, or isotropic, singularities in cosmological models in general relativity. Using the calculus of tractors, we find conditions in terms of tractor curvature for a local extension of the conformal structure through a cosmological singularity and prove a local extension theorem along a congruence of timelike conformal geodesics.
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04.20.Dw Singularities and cosmic censorship
11.25.Hf Conformal field theory, algebraic structures
95.30.Sf Relativity and gravitation
98.80.Cq Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.)

A conformal extension theorem based on null conformal geodesics

Christian Lübbe

J. Math. Phys. 50, 112502 (2009) (16 pages) | Cited 1 time

Online Publication Date: 3 November 2009

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In this article we describe the formulation of null geodesics as null conformal geodesics and their description in the tractor formalism. A conformal extension theorem through an isotropic singularity is proven by requiring the boundedness of the tractor curvature and its derivatives to sufficient order along a congruence of null conformal geodesic. This article extends earlier work by Lübbe and Tod, J. Math. Phys. 50, 112501 (2009) (“An extension theorem for conformal gauge singularities,” e-print arXiv:0710.5552) .
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98.80.Jk Mathematical and relativistic aspects of cosmology
02.40.Xx Singularity theory

Quantum scattering of relativistic particles in Safko–Witten spacetime

Herondy Mota
and V. Bezerra

J. Math. Phys. 50, 112503 (2009) (12 pages)

Online Publication Date: 9 November 2009

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Relativistic quantum particles are considered in the background gravitational field due to a tubular matter source with an axial interior magnetic field and a vanishing exterior magnetic field, which is a solution of the combined Einstein–Maxwell fields with cylindrical symmetry. In this background, whose gravitational field outside the tubular matter source is locally flat, we analyze the problem concerning the scattering of scalar and spinor particles and show up that this process depends on the geometrical features of the gravitational field in the interior region as well as on the topological features of the gravitational field in the exterior region.
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04.40.Nr Einstein-Maxwell spacetimes, spacetimes with fluids, radiation or classical fields
02.40.-k Geometry, differential geometry, and topology
02.40.Re Algebraic topology
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Global existence and stability for a von Karman equations with memory in noncylindrical domains

Jong Park
and Jum Kang

J. Math. Phys. 50, 112701 (2009) (13 pages)

Online Publication Date: 6 November 2009

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In this paper, we study the initial-boundary value problem for the von Karman equations inside domains with moving ends. Global existence, uniqueness of solutions, and the exponential decay to the energy are established provided the initial data are bounded.
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46.40.-f Vibrations and mechanical waves

A note on superlinear Hamiltonian elliptic systems

Fukun Zhao ,
Leiga Zhao ,
and Yanheng Ding

J. Math. Phys. 50, 112702 (2009) (7 pages)

Online Publication Date: 10 November 2009

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This paper is concerned with the superlinear periodic elliptic systems of Hamiltonian type in the whole space. The existence of a ground state solution as well as an infinite number of geometrically distinct solutions is obtained.
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02.30.Tb Operator theory
02.40.-k Geometry, differential geometry, and topology

Dynamical systems and Poisson structures

Metin Gürses ,
Gusein Guseinov ,
and Kostyantyn Zheltukhin

J. Math. Phys. 50, 112703 (2009) (9 pages)

Online Publication Date: 12 November 2009

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We first consider the Hamiltonian formulation of n = 3 systems, in general, and show that all dynamical systems in math3 are locally bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical system. The construction of the Poisson structures is based on solving an associated first order linear partial differential equations. We find the Poisson structures of a dynamical system recently given by Bender et al. [J. Phys. A: Math. Theor. 40, F793 (2007)] . Secondly, we show that all dynamical systems in mathn are locally (n−1)-Hamiltonian. We give also an algorithm, similar to the case in math3, to construct a rank two Poisson structure of dynamical systems in mathn. We give a classification of the dynamical systems with respect to the invariant functions of the vector field math and show that all autonomous dynamical systems in mathn are superintegrable.
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02.30.Jr Partial differential equations
02.30.Rz Integral equations
02.10.Yn Matrix theory

A practical use of the Melnikov homoclinic method

César Castilho
and Marcelo Marchesin

J. Math. Phys. 50, 112704 (2009) (11 pages)

Online Publication Date: 13 November 2009

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Using cutoff functions and periodic extensions, we prove that the Melnikov homoclinic method gives a criterium to show that for a finite time interval [−T,T], with T arbitrarily large, the perturbed system is conjugated to a chaotic one for quite general classes of perturbation functions. The method is applied to specific perturbations of the pendulum and of the Gylden’s problem.
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05.45.-a Nonlinear dynamics and chaos
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Interference in the radiation of two pointlike charges

Yurij Yaremko

J. Math. Phys. 50, 112901 (2009) (37 pages)

Online Publication Date: 2 November 2009

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Energy-momentum and angular momentum carried by electromagnetic field of two pointlike charged particles arbitrarily moving in flat space-time are presented. Apart from usual contributions to the Noether quantities produced separately by particles 1 and 2, the conservation laws contain also joint contribution due to the fields of both particles. The mixed part of Maxwell energy-momentum density is decomposed into bound and radiative components which are separately conserved off the world lines of particles. The former describes the deformation of electromagnetic clouds of “bare” charges due to mutual interaction while the latter defines the radiation which escapes to infinity. The bound terms contribute to particles’ individual 4-momenta while the radiative ones exert the radiation reaction. Analysis of energy-momentum and angular momentum balance equations results the Lorentz–Dirac equation as an equation of motion for a pointed charge under the influence of its own electromagnetic field as well as field produced by another charge.
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03.50.De Classical electromagnetism, Maxwell equations
03.65.Pm Relativistic wave equations
02.40.-k Geometry, differential geometry, and topology
41.20.-q Applied classical electromagnetism

Angular normal modes of a circular Coulomb cluster

L. Lupinski
and M. Madsen

J. Math. Phys. 50, 112902 (2009) (9 pages)

Online Publication Date: 9 November 2009

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We investigate the angular normal modes for small oscillations about an equilibrium of a single-component Coulomb cluster confined by a radially symmetric external potential to a circle. The dynamical matrix for this system is a Laplacian symmetrically circulant matrix and this result leads to an analytic solution for the eigenfrequencies of the angular normal modes. We also show the limiting dependence of the largest eigenfrequency for large numbers of particles.
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36.40.-c Atomic and molecular clusters
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Clebsch parameterization: Basic properties and remarks on its applications

Z. Yoshida

J. Math. Phys. 50, 113101 (2009) (16 pages)

Online Publication Date: 18 November 2009

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The Clebsch parameterization (u = ∇φ+α∇β) has advantages in elucidating structural properties of vector fields; for example, it helps formulating the Hamiltonian form of ideal fluid mechanics, representing topological constraints (Casimir invariants), integrating the Cauchy characteristics of vortex fields, etc. Because of its “nonlinear” formulation, however, there are some difficulties which must be carefully overcome. (1) It is not complete, i.e., for an arbitrary vector field u, we may fail to find three scalar fields (Clebsch parameters) φ, α, and β that satisfy u = ∇φ+α∇β globally in space. (2) It is not uniquely determined, i.e., the map (u1,u2,u3)↦(φ,α,β) is not injective. A generalized form such that u = ∇φ+∑j = 1ναj∇βj is complete if ν = n−1 (n is the space dimension). However, when we need to control the boundary values of φ, αj, and βj (for example, to determine them uniquely), we have to set ν = n.
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47.32.-y Vortex dynamics; rotating fluids
47.10.Df Hamiltonian formulations
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Generalized log-likelihood functions and Bregman divergences

Tatsuaki Wada

J. Math. Phys. 50, 113301 (2009) (7 pages)

Online Publication Date: 10 November 2009

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Based on a two-parameter generalization of Gauss’ law of error, a generalized log-likelihood is related to a Bregman divergence. This relation is a two-parameter generalization of the well-known relation between log-likelihood and Kullback–Leibler divergence.
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02.50.Cw Probability theory
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Spectral representation of infimum of bounded quantum observables

Shen Jun
and Wu Junde

J. Math. Phys. 50, 113501 (2009) (4 pages)

Online Publication Date: 2 November 2009

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In 2006, Gudder [Math. Slovaca 56, 573 (2006)] introduced a logic order ≼ on bounded quantum observable set S(H). In 2007, Pulmannova and Vincekova [Math Slovaca 57, 589 (2007)] proved that for each subset D of S(H), the infimum of D exists with respect to the logic order ≼. In 2008, Liu and Wu [J. Math. Phys. 49, 073521 (2008)] found a representation of the infimum A∧B for A,B ∊ S(H), and by using the limit methods, they gave out a representation for the infimum of D. But, that representation is complicated. In this paper, we present a simpler spectral representation for the infimum of D with respect to the logic order ≼.
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03.65.Fd Algebraic methods
02.10.Ab Logic and set theory

Integrable higher order deformations of Heisenberg supermagnetic model

Jia-Feng Guo ,
Shi-Kun Wang ,
Ke Wu ,
Zhao-Wen Yan ,
and Wei-Zhong Zhao

J. Math. Phys. 50, 113502 (2009) (11 pages)

Online Publication Date: 3 November 2009

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The Heisenberg supermagnet model is an integrable supersymmetric system and has a close relationship with the strong electron correlated Hubbard model. In this paper, we investigate the integrable higher order deformations of Heisenberg supermagnet models with two different constraints: (i) S2 = 3S−2I for S ∊ USPL(2/1)/S(U(2)×U(1)) and (ii) S2 = S for S ∊ USPL(2/1)/S(L(1/1)×U(1)). In terms of the gauge transformation, their corresponding gauge equivalent counterparts are derived.
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75.10.Jm Quantized spin models, including quantum spin frustration
11.30.Pb Supersymmetry
71.10.Fd Lattice fermion models (Hubbard model, etc.)
75.10.Lp Band and itinerant models
71.27.+a Strongly correlated electron systems; heavy fermions

Analysis of unbounded operators and random motion

Palle Jorgensen

J. Math. Phys. 50, 113503 (2009) (28 pages)

Online Publication Date: 3 November 2009

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We study infinite weighted graphs with view to “limits at infinity” or boundaries at infinity. Examples of such weighted graphs arise in infinite (in practice, that means “very” large) networks of resistors or in statistical mechanics models for classical or quantum systems. However, more generally, our analysis includes reproducing kernel Hilbert spaces and associated operators on them. If X is some infinite set of vertices or nodes, in applications the essential ingredient going into the definition is a reproducing kernel Hilbert space; it measures the differences of functions on X evaluated on pairs of points in X. Moreover, the Hilbert norm-squared in H(X) will represent a suitable measure of energy. Associated unbounded operators will define a notion or dissipation, it can be a graph Laplacian or a more abstract unbounded Hermitian operator defined from the reproducing kernel Hilbert space under study. We prove that there are two closed subspaces in reproducing kernel Hilbert space H(X) that measure quantitative notions of limits at infinity in X: one generalizes finite-energy harmonic functions in H(X) and the other a deficiency index of a natural operator in H(X) associated directly with the diffusion. We establish these results in the abstract, and we offer examples and applications. Our results are related to, but different from, potential theoretic notions of “boundaries” in more standard random walk models. Comparisons are made.
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05.40.Fb Random walks and Levy flights
05.60.-k Transport processes
03.65.Fd Algebraic methods
02.30.Tb Operator theory
02.10.Ox Combinatorics; graph theory

On the Moyal deformation of Nahm equations in seven dimensions

Hugo García-Compeán
and Aldo Martínez-Merino

J. Math. Phys. 50, 113504 (2009) (14 pages)

Online Publication Date: 5 November 2009

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We show how the reduced (anti-)self-dual Yang–Mills equations to seven dimensions described by the Nahm equations can be carried over to the Weyl–Wigner–Moyal formalism. In the process some new solutions for the cases of gauge groups SU(2) and SL(2,math) are explicitly obtained.
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11.15.-q Gauge field theories
11.30.Ly Other internal and higher symmetries

Particle topology, braids, and braided belts

Sundance Bilson-Thompson ,
Jonathan Hackett ,
and Louis Kauffman

J. Math. Phys. 50, 113505 (2009) (16 pages)

Online Publication Date: 5 November 2009

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Recent work [ S. O. Bilson-Thompson, e-print arXiv:hep-th/0503213 ; Bilson-Thompson et al., Class. Quantum Grav. 24, 3975 (2007) ] suggests that topological features of certain quantum gravity theories can be interpreted as particles, matching the known fermions and bosons of the first generation in the standard model. This is achieved by identifying topological structures with elements of the framed Artin braid group on three strands and demonstrating a correspondence between the invariants used to characterize these braids (a braid is a set of nonintersecting curves, that connect one set of N points with another set of N points) and quantities such as electric charge, color charge, and so on [ S. O. Bilson-Thompson, e-print arXiv:hep-th/0503213 ; Bilson-Thompson et al., e-print aXiv:0804.0037 ]. In this paper we show how to manipulate a modified form of framed braids to yield an invariant standard form for sets of isomorphic braids, characterized by a vector of real numbers. This will serve as a basis for more complete discussions of quantum numbers in future work.
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04.60.-m Quantum gravity
02.20.-a Group theory
02.40.Pc General topology

General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms

Wenjun Liu

J. Math. Phys. 50, 113506 (2009) (17 pages)

Online Publication Date: 6 November 2009

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A viscoelastic wave equation in canonical form with weakly nonlinear time-dependent dissipation and source terms is investigated in this paper. For a wider class of relaxation functions and without imposing any restrictive growth assumption on the damping term at the origin, we establish an explicit and general energy decay rate result.
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02.30.-f Function theory, analysis

On the application of a generalized dressing method to the integration of variable-coefficient coupled Hirota equations

Ting Su ,
Hui-Hui Dai ,
and Xianguo Geng

J. Math. Phys. 50, 113507 (2009) (12 pages)

Online Publication Date: 9 November 2009

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Based on the generalized dressing method, we propose two variable-coefficient coupled Hirota equations and derive their Lax pairs. The generalized dressing method is applied to study these variable-coefficient coupled Hirota equations, from which explicit solutions of the two equations and their reductions are constructed.
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05.45.Yv Solitons
02.30.Tb Operator theory
02.30.Rz Integral equations

The method of Ostrogradsky, quantization, and a move toward a ghost-free future

M Nucci
and P Leach

J. Math. Phys. 50, 113508 (2009) (6 pages)

Online Publication Date: 10 November 2009

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The method of Ostrogradsky has been used to construct a first-order Lagrangian, hence Hamiltonian, for the fourth-order field-theoretical model of Pais–Uhlenbeck with unfortunate results when quantization is undertaken since states with negative norm, commonly called “ghosts,” appear. We propose an alternative route based on the preservation of symmetry and this leads to a ghost-free quantization.
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11.10.Ef Lagrangian and Hamiltonian approach
11.30.-j Symmetry and conservation laws

Characterization of partial positive transposition states and measures of entanglement

Władysław Majewski ,
Takashi Matsuoka ,
and Masanori Ohya

J. Math. Phys. 50, 113509 (2009) (20 pages)

Online Publication Date: 12 November 2009

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A detailed characterization of partial positive transposition (PPT) states, both in the Heisenberg and in the Schrödinger picture, is given. Measures of entanglement are defined and discussed in details. Illustrative examples are provided.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
02.10.-v Logic, set theory, and algebra
03.65.Fd Algebraic methods
03.65.Ge Solutions of wave equations: bound states

Periodic solutions on the Sturm–Liouville boundary value problem for two-dimensional wave equation

Shuguan Ji

J. Math. Phys. 50, 113510 (2009) (16 pages)

Online Publication Date: 12 November 2009

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In this paper, we study the periodic solutions to two-dimensional wave equation ytt−Δy+ay+g(y) = f(x,t) on (0,π)×(0,π)×math under the periodic conditions y(x,t+T) = y(x,t), yt(x,t+T) = yt(x,t), and the Sturm–Liouville boundary conditions a11y(0,x2,t)−b11yx1(0,x2,t) = 0, a12y(π,x2,t)+b12yx1(π,x2,t) = 0, a21y(x1,0,t)−b21yx2(x1,0,t) = 0, and a22y(x1,π,t)+b22yx2(x1,π,t) = 0 (aij2+bij2 ≠ 0 for i,j = 1,2). Our main concept is that of the “weak solution,” which will be given in Sec. 2. Then, by means of spectral analysis, we investigate some important properties of the wave operator with Sturm–Liouville boundary conditions. Finally, by using the properties we established and the duality principles in the work of Brézis [Bull. Am. Math. Soc. 8, 409 (1983)] , the existence of periodic solutions is obtained.
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02.30.Hq Ordinary differential equations
02.30.Tb Operator theory

Ground states of the massless Dereziński–Gérard model

Atsushi Ohkubo

J. Math. Phys. 50, 113511 (2009) (10 pages)

Online Publication Date: 13 November 2009

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We consider the massless Dereziński–Gérard model introduced by Dereziński and Gérard in 1999. We give a sufficient condition for the existence of a ground state of the massless Dereziński–Gérard model without the assumption that the Hamiltonian of particles has compact resolvent.
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03.65.Fd Algebraic methods
02.10.-v Logic, set theory, and algebra

Boundary Lax pairs from non-ultra-local Poisson algebras

Jean Avan
and Anastasia Doikou

J. Math. Phys. 50, 113512 (2009) (9 pages)

Online Publication Date: 13 November 2009

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We consider non-ultra-local linear Poisson algebras on a continuous line. Suitable combinations of representations of these algebras yield representations of novel generalized linear Poisson algebras or “boundary” extensions. They are parametrized by a boundary scalar matrix and depend, in addition, on the choice of an antiautomorphism. The new algebras are the classical-linear counterparts of the known quadratic quantum boundary algebras. For any choice of parameters, the non-ultra-local contribution of the original Poisson algebra disappears. We also systematically construct the associated classical Lax pair. The classical boundary principal chiral model is examined as a physical example.
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03.65.Fd Algebraic methods
03.65.Db Functional analytical methods
02.10.Ud Linear algebra
02.10.Yn Matrix theory

Some applications of the fractional Poisson probability distribution

Nick Laskin

J. Math. Phys. 50, 113513 (2009) (12 pages)

Online Publication Date: 16 November 2009

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Physical and mathematical applications of the recently invented fractional Poisson probability distribution have been presented. As a physical application, a new family of quantum coherent states has been introduced and studied. As mathematical applications, we have developed the fractional generalization of Bell polynomials, Bell numbers, and Stirling numbers of the second kind. The appearance of fractional Bell polynomials is natural if one evaluates the diagonal matrix element of the evolution operator in the basis of newly introduced quantum coherent states. Fractional Stirling numbers of the second kind have been introduced and applied to evaluate the skewness and kurtosis of the fractional Poisson probability distribution function. A representation of the Bernoulli numbers in terms of fractional Stirling numbers of the second kind has been found. In the limit case when the fractional Poisson probability distribution becomes the Poisson probability distribution, all of the above listed developments and implementations turn into the well-known results of the quantum optics and the theory of combinatorial numbers.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.50.Ng Distribution theory and Monte Carlo studies
42.50.-p Quantum optics
03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
02.10.Ox Combinatorics; graph theory
03.65.Fd Algebraic methods
02.10.Yn Matrix theory

Deformations of 3-algebras

José Figueroa-O’Farrill

J. Math. Phys. 50, 113514 (2009) (27 pages)

Online Publication Date: 17 November 2009

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We phrase deformations of n-Leibniz algebras in terms of the cohomology theory of the associated Leibniz algebra. We do the same for n-Lie algebras and for the metric versions of n-Leibniz and n-Lie algebras. We place particular emphasis on the case of n = 3 and explore the deformations of 3-algebras of relevance to three-dimensional superconformal Chern–Simons theories with matter.
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11.15.Yc Chern-Simons gauge theory
11.30.Pb Supersymmetry
02.10.Ud Linear algebra

Well-posedness of Einstein’s equation with redshift data

Christopher Winfield

J. Math. Phys. 50, 113515 (2009) (14 pages)

Online Publication Date: 17 November 2009

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We study the solvability of a system of ordinary differential equations derived from null geodesics of the Lemaître–Tolman–Bondi metric with data given in terms of a so-called redshift parameter. Data are introduced along these geodesics by the luminosity distance function. We check our results with luminosity distance depending on the cosmological constant and with the well-known Freedman-Robertson-Walker (FRW) model.
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04.20.Jb Exact solutions
98.80.Jk Mathematical and relativistic aspects of cosmology
95.30.Sf Relativity and gravitation
98.62.Py Distances, redshifts, radial velocities; spatial distribution of galaxies

On some properties of orthogonal Weingarten functions

Benoît Collins
and Sho Matsumoto

J. Math. Phys. 50, 113516 (2009) (14 pages)

Online Publication Date: 18 November 2009

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We give a Fourier-type formula for computing the orthogonal Weingarten formula. The Weingarten calculus was introduced as a systematic method to compute integrals of polynomials with respect to Haar measure over classical groups. Although a Fourier-type formula was known in the unitary case, the orthogonal counterpart was not known. It relies on the Jack polynomial generalization of both Schur and zonal polynomials. This formula substantially reduces the complexity involved in the computation of Weingarten formulas. We also describe a few more new properties of the Weingarten formula, state a conjecture, and give a table of values.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.10.Ud Linear algebra
02.50.-r Probability theory, stochastic processes, and statistics
02.10.De Algebraic structures and number theory
02.30.Rz Integral equations
02.30.Sa Functional analysis

Markov property of Gaussian states of canonical commutation relation algebras

Dénes Petz
and József Pitrik

J. Math. Phys. 50, 113517 (2009) (9 pages)

Online Publication Date: 19 November 2009

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The Markov property of Gaussian states of canonical commutation relation algebras is studied. The detailed description is given by the representing block matrix. The proof is short and allows infinite dimension. The relation to classical Gaussian Markov triplets is also described. The minimizer of relative entropy with respect to a Gaussian Markov state has the Markov property. The appendix contains formulas for the relative entropy.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.10.Yn Matrix theory
05.70.Ce Thermodynamic functions and equations of state
02.50.Ga Markov processes

Asymptotic behavior for logarithmic diffusion

F. Salvarani

J. Math. Phys. 50, 113518 (2009) (7 pages)

Online Publication Date: 20 November 2009

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In this paper we prove, via the entropy dissipation method, that the solutions of the d-dimensional logarithmic diffusion equation, with nonhomogeneous Dirichlet boundary data, decay exponentially in time toward its own steady state. The result is valid not only in L1-norm (as customary when applying entropy dissipation methods) but also in any Lp-norm with p ∊ [1,+∞).
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05.60.-k Transport processes
02.30.-f Function theory, analysis
05.70.Ce Thermodynamic functions and equations of state
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Erratum: “Geometric prequantization of the moduli space of the vortex equations on a Riemann surface” [ J. Math. Phys. 47, 103501 (2006) ]

Rukmini Dey

J. Math. Phys. 50, 119901 (2009) (3 pages)

Online Publication Date: 17 November 2009

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In this erratum to a work done previously, we give an alternative description for the prequantization with respect to the forms ΩΨ0, where we do not need the 1-form θ which may not be globally defined. Next by modifying the Quillen metric of the usual determinant bundle suitably, we quantize the usual symplectic form Ω on the vortex moduli space. Next, we show that by modifying the Quillen metric, one can also interpolate between the forms Ω and ΩΨ0 and the corresponding prequantum line bundles are topologically equivalent. It is not clear whether they are holomorphically equivalent.
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03.65.Vf Phases: geometric; dynamic or topological
02.40.-k Geometry, differential geometry, and topology
02.40.Pc General topology
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