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Mar 2013

Volume 54, Issue 3, Articles (03xxxx)

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J. Math. Phys. 54, 033512 (2013); http://dx.doi.org/10.1063/1.4794514 (20 pages)

Andrew Neate and Aubrey Truman
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back to top Methods of Mathematical Physics

Exponential decay for a von Kármán equations with memory

Jum-Ran Kang

J. Math. Phys. 54, 033501 (2013); http://dx.doi.org/10.1063/1.4791694 (11 pages)

Online Publication Date: 1 March 2013

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In this paper, we consider a von Kármán equations with memory term. We show an exponential decay result of solutions under weaker assumption than the ones frequently used in the literature. In particular, the kernel we are considering is not necessarily exponentially decaying to zero as was assumed before.
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02.30.Hq Ordinary differential equations

Auto-Bäcklund transformations for a differential-delay equation

Pilar R. Gordoa and Andrew Pickering

J. Math. Phys. 54, 033502 (2013); http://dx.doi.org/10.1063/1.4793989 (6 pages)

Online Publication Date: 4 March 2013

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Discrete Painlevé equations have, over recent years, generated much interest. One property of such equations that is considered to be particularly important is the existence of auto-Bäcklund transformations, that is, mappings between solutions of the equation in question, usually involving changes in the values of parameters appearing as coefficients. We have recently presented extensions of discrete Painlevé equations to equations involving derivatives as well as shifts in the independent variable. Here we show how auto-Bäcklund transformations can also be constructed for such differential-delay equations. We emphasise that this is the first time that an auto-Bäcklund transformation has been given for a differential-delay equation.
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02.30.Jr Partial differential equations

New estimators of spectral distributions of Wigner matrices

Wang Zhou

J. Math. Phys. 54, 033503 (2013); http://dx.doi.org/10.1063/1.4794075 (10 pages)

Online Publication Date: 11 March 2013

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We introduce kernel estimators for the semi-circular law. In this first part of our general theory on the estimators, we prove the consistency and conduct simulation study to show the performance of the estimators. We also point out that Wigner's semi-circular law for our new estimators and the classical empirical spectral distributions is still true when the elements of Wigner matrices do not have finite variances but are in the domain of attraction of normal law.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.50.-r Probability theory, stochastic processes, and statistics

Lieb-Thirring trace inequalities and a question of Bourin

Saja Hayajneh and Fuad Kittaneh

J. Math. Phys. 54, 033504 (2013); http://dx.doi.org/10.1063/1.4793993 (8 pages)

Online Publication Date: 13 March 2013

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In this paper, we propose some conjectures in order to solve one of the questions posed by Bourin regarding a special type of inequalities referred to as subadditivity inequalities in the case of the Hilbert-Schmidt norm. Some of these conjectures are settled affirmatively for special cases in an algorithmic way by using some number theory tools.
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02.10.Yn Matrix theory

On Lie systems and Kummer-Schwarz equations

J. de Lucas and C. Sardón

J. Math. Phys. 54, 033505 (2013); http://dx.doi.org/10.1063/1.4794280 (21 pages)

Online Publication Date: 14 March 2013

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A Lie system is a system of first-order differential equations admitting a superposition rule, i.e., a map that expresses its general solution in terms of a generic family of particular solutions and certain constants. In this work, we use the geometric theory of Lie systems to prove that the explicit integration of second- and third-order Kummer-Schwarz equations is equivalent to obtaining a particular solution of a Lie system on SL(2,math). This same result can be extended to Riccati, Milne-Pinney, and to the here defined generalised Kummer-Schwarz equations, which include several types of Kummer-Schwarz equations as particular cases. We demonstrate that all the above-mentioned equations related to the same Lie system on SL(2,math) can be integrated simultaneously, which retrieves and generalizes in a unified and simpler manner previous results appearing in the literature. As a byproduct, we recover various properties of the Schwarzian derivative.
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02.30.-f Function theory, analysis
02.40.-k Geometry, differential geometry, and topology
02.60.Jh Numerical differentiation and integration
02.20.Sv Lie algebras of Lie groups

Fermionic covariant prolongation structure theory for multidimensional super nonlinear evolution equation

Zhao-Wen Yan, Min-Li Li, Ke Wu, and Wei-Zhong Zhao

J. Math. Phys. 54, 033506 (2013); http://dx.doi.org/10.1063/1.4795405 (8 pages)

Online Publication Date: 15 March 2013

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The fermionic covariant prolongation structure theory is investigated. We extend the fermionic covariant prolongation structure technique to the multidimensional super nonlinear evolution equation and present the fermionic covariant fundamental equations determining the prolongation structure. Furthermore, we investigate a (2+1)-dimensional super nonlinear Schrödinger equation and analyze its integrability by means of this prolongation structure technique. We derive its Lax representation and Bäcklund transformation. Moreover, we present a solution of this multidimensional super integrable equation.
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03.65.Ge Solutions of wave equations: bound states
02.30.Ik Integrable systems
02.30.Rz Integral equations

Two-point derivative dispersion relations

Erasmo Ferreira and Javier Sesma

J. Math. Phys. 54, 033507 (2013); http://dx.doi.org/10.1063/1.4795116 (10 pages)

Online Publication Date: 18 March 2013

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A new derivation is given for the representation, under certain conditions, of the integral dispersion relations of scattering theory through local forms. The resulting expressions have been obtained through an independent procedure to construct the real part and consist of new mathematical structures of double infinite summations of derivatives. In this new form the derivatives are calculated at the generic value of the energy E and separately at the reference point E = m that is the lower limit of the integration. This new form may be more interesting in certain circumstances and directly shows the origin of the difficulties in convergence that were present in the old truncated forms called standard-derivative dispersion relations (DDR). For all cases in which the reductions of the double to single sums were obtained in our previous work, leading to explicit demonstration of convergence, these new expressions are seen to be identical to the previous ones. We present, as a glossary, the most simplified explicit results for the DDR’s in the cases of imaginary amplitudes of forms (E/m)λ[ln (E/m)]n that cover the cases of practical interest in particle physics phenomenology at high energies. We explicitly study the expressions for the cases with λ negative odd integers, that require identification of cancelation of singularities, and provide the corresponding final results.
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11.55.Fv Dispersion relations
02.30.Rz Integral equations

Embedding of simply laced hyperbolic Kac-Moody superalgebras

Saudamini Nayak and K. C. Pati

J. Math. Phys. 54, 033508 (2013); http://dx.doi.org/10.1063/1.4795117 (10 pages)

Online Publication Date: 18 March 2013

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We show that HD(4, 1) hyperbolic Kac-Moody superalgebra of rank 6 contains every simply laced Kac-Moody superalgebra with degenerate odd root as a Lie subalgebra. Our result is the supersymmetric extension of earlier work [S. Viswanath, “Embeddings of HyperbolicKac-Moody Algebras into E10,” Lett. Math. Phys. 83, 139–148 (2008)]10.1007/s11005-007-0214-7 for hyperbolic Kac-Moody algebra.
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02.10.-v Logic, set theory, and algebra

About simple nonlinear and linear superpositions of special exact solutions of Veselov-Novikov equation

V. G. Dubrovsky and A. V. Topovsky

J. Math. Phys. 54, 033509 (2013); http://dx.doi.org/10.1063/1.4795132 (13 pages)

Online Publication Date: 18 March 2013

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New exact solutions, nonstationary and stationary, of Veselov-Novikov (VN) equation in the forms of simple nonlinear and linear superpositions of arbitrary number N of exact special solutions u(n), n = 1, …, N are constructed via Zakharov and Manakov math-dressing method. Simple nonlinear superpositions are represented up to a constant by the sums of solutions u(n) and calculated by math-dressing on nonzero energy level of the first auxiliary linear problem, i.e., 2D stationary Schrödinger equation. It is remarkable that in the zero energy limit simple nonlinear superpositions convert to linear ones in the form of the sums of special solutions u(n). It is shown that the sums u = u(k1)+...+u(km), 1 ⩽ k1 < k2 < … < km ⩽ N of arbitrary subsets of these solutions are also exact solutions of VN equation. The presented exact solutions include as superpositions of special line solitons and also superpositions of plane wave type singular periodic solutions. By construction these exact solutions represent also new exact transparent potentials of 2D stationary Schrödinger equation and can serve as model potentials for electrons in planar structures of modern electronics.
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05.45.Yv Solitons
03.65.Ge Solutions of wave equations: bound states

Operator orderings and Meixner-Pollaczek polynomials

Genki Shibukawa

J. Math. Phys. 54, 033510 (2013); http://dx.doi.org/10.1063/1.4795713 (4 pages)

Online Publication Date: 18 March 2013

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The aim of this paper is to give identities which are generalizations of the formulas given by Koornwinder [J. Math. Phys. 30, 767–769 (1989)]10.1063/1.528394 and Hamdi-Zeng [J. Math. Phys. 51, 043506 (2010)]10.1063/1.3372526. Our proofs are much simpler than and different from the previous investigations.
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02.10.De Algebraic structures and number theory
02.30.Tb Operator theory

Hill's equation with small fluctuations: Cycle to cycle variations and stochastic processes

Fred C. Adams and Anthony M. Bloch

J. Math. Phys. 54, 033511 (2013); http://dx.doi.org/10.1063/1.4795351 (15 pages)

Online Publication Date: 19 March 2013

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Hill's equations arise in a wide variety of physical problems, and are specified by a natural frequency, a periodic forcing function, and a forcing strength parameter. This classic problem is generalized here in two ways: (a) to Random Hill's equations which allow the forcing strength qk, the oscillation frequency λk, and the period (Δτ)k of the forcing function to vary from cycle to cycle, and (b) to Stochastic Hill's equations which contain (at least) one additional term that is a stochastic process ξ. This paper considers both random and stochastic Hill's equations with small parameter variations, so that pk = qk−⟨qk⟩, ℓk = λk−⟨λk⟩, and ξ are all O(ε), where ε ≪ 1. We show that random Hill's equations and stochastic Hill's equations have the same growth rates when the parameter variations pk and ℓk obey certain constraints given in terms of the moments of ξ. For random Hill's equations, the growth rates for the solutions are given by the growth rates of a matrix transformation, under matrix multiplication, where the matrix elements vary from cycle to cycle. Unlike classic Hill's equations where the parameter space (the λ-q plane) displays bands of stable solutions interlaced with bands of unstable solutions, random Hill's equations are generically unstable. We find analytic approximations for the growth rates of the instability; for the regime where Hill's equation is classically stable, and the parameter variations are small, the growth rate γ = O(pk2) = O(ε2). Using the relationship between the (ℓk, pk) and the ξ, this result for γ can be used to find growth rates for stochastic Hill's equations.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.50.-r Probability theory, stochastic processes, and statistics
02.50.Ey Stochastic processes

A stochastic Burgers-Zeldovich model for the formation of planetary ring systems and the satellites of Jupiter and Saturn

Andrew Neate and Aubrey Truman

J. Math. Phys. 54, 033512 (2013); http://dx.doi.org/10.1063/1.4794514 (20 pages)

Online Publication Date: 20 March 2013

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multimedia

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We consider a proto-ring nebula of a gas giant such as Neptune as a cloud of gas/dust particles whose behaviour is governed by the stochastic mechanics associated to the Kepler problem. This leads to a stochastic Burgers-Zeldovich type model for the formation of planetesimals involving a stochastic Burgers equation with vorticity which could help to explain the turbulent behaviour observed in ring systems. The Burgers fluid density and the distribution of the mass M(T) of a spherical planetesimal of radius δ are computed for times T. For circular orbits, sufficient conditions on certain time averages of δ2 are given ensuring that VarM(T) ∼ 0 as T ∼ ∞. Some applications are given to the satellites of Jupiter and Saturn, in particular giving a possible explanation of the equal mass families of satellites.
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96.15.Uv Rings and dust
96.30.L- Jovian satellites
96.30.N- Saturnian satellites
98.38.Ly Planetary nebulae
05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)
95.30.Lz Hydrodynamics

The conformal metric structure of Geometrothermodynamics

Alessandro Bravetti, Cesar S. Lopez-Monsalvo, Francisco Nettel, and Hernando Quevedo

J. Math. Phys. 54, 033513 (2013); http://dx.doi.org/10.1063/1.4795136 (11 pages)

Online Publication Date: 20 March 2013

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We present a thorough analysis on the invariance of the most widely used metrics in the Geometrothermodynamics programme. We centre our attention in the invariance of the curvature of the space of equilibrium states under a change of fundamental representation. Assuming that the systems under consideration can be described by a fundamental relation which is a homogeneous function of a definite order, we demonstrate that such invariance is only compatible with total Legendre transformations in the present form of the programme. We give the explicit form of a metric which is invariant under total Legendre transformations and whose induced metric produces a curvature which is independent of the fundamental representation. Finally, we study a generic system with two degrees of freedom and whose fundamental relation is homogeneous of order one.
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05.70.-a Thermodynamics
02.10.De Algebraic structures and number theory

Algorithms to evaluate multiple sums for loop computations

C. Anzai and Y. Sumino

J. Math. Phys. 54, 033514 (2013); http://dx.doi.org/10.1063/1.4795288 (22 pages)

Online Publication Date: 21 March 2013

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We present algorithms to evaluate two types of multiple sums, which appear in higher-order loop computations. We consider expansions of a generalized hyper-geometric-type sums, ∑n1,⋯,nNmathx1n1⋯xNnN with ai·n = ∑j = 1Naijnj, etc., in a small parameter ε around rational values of ci,di’s. Type I sum corresponds to the case where, in the limit ε → 0, the summand reduces to a rational function of nj’s times x1n1⋯xNnN; ci,di’s can depend on an external integer index. Type II sum is a double sum (N = 2), where ci, di’s are half-integers or integers as ε → 0 and xi = 1; we consider some specific cases where at most six Γ functions remain in the limit ε → 0. The algorithms enable evaluations of arbitrary expansion coefficients in ε in terms of Z-sums and multiple polylogarithms (generalized multiple zeta values). We also present applications of these algorithms. In particular, Type I sums can be used to generate a new class of relations among generalized multiple zeta values. We provide a Mathematica package, in which these algorithms are implemented.
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02.30.Rz Integral equations
02.40.-k Geometry, differential geometry, and topology
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