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Dec 2009

Volume 50, Issue 12, Articles (12xxxx)

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Announcement: Journal of Mathematical Physics introduces a new section—Quantum Information and Computation

Bruno L. Z. Nachtergaele

J. Math. Phys. 50, 120201 (2009); http://dx.doi.org/10.1063/1.3274531 (1 page)

Online Publication Date: 17 December 2009

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Abstract Unavailable
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01.10.-m Announcements, news, and organizational activities
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Diffusion approximation of stochastic master equations with jumps

C. Pellegrini and F. Petruccione

J. Math. Phys. 50, 122101 (2009); http://dx.doi.org/10.1063/1.3263941 (14 pages)

Online Publication Date: 2 December 2009

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In the presence of quantum measurements with direct photon detection, the evolution of open quantum systems is usually described by stochastic master equations with jumps. Heuristically, diffusion models can be obtained from these equations as an approximation. A condition for a general diffusion approximation for jump master equations is presented. This approximation is rigorously proved by using techniques for Markov processes, which are based on the convergence of Markov generators and martingale problems. This result is illustrated by rigorously obtaining the diffusion approximation for homodyne and heterodyne detection.
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05.60.Gg Quantum transport
03.65.Ge Solutions of wave equations: bound states
02.50.Ga Markov processes
42.50.-p Quantum optics
02.30.-f Function theory, analysis

Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion

Ian Marquette

J. Math. Phys. 50, 122102 (2009); http://dx.doi.org/10.1063/1.3272003 (10 pages) | Cited 15 times

Online Publication Date: 18 December 2009

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The main result of this article is that we show that from supersymmetry we can generate new superintegrable Hamiltonians. We consider a particular case with a third order integral and apply Mielnik’s construction in supersymmetric quantum mechanics. We obtain a new superintegrable potential separable in Cartesian coordinates with a quadratic and quintic integrals and also one with a quadratic integral and an integral of order of 7. We also construct a superintegrable system written in terms of the fourth Painlevé transcendent with a quadratic integral and an integral of order of 7.
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11.30.Pb Supersymmetry
03.65.Ta Foundations of quantum mechanics; measurement theory
02.30.Rz Integral equations

Quadratic exponential vectors

Luigi Accardi and Ameur Dhahri

J. Math. Phys. 50, 122103 (2009); http://dx.doi.org/10.1063/1.3266166 (18 pages) | Cited 1 time

Online Publication Date: 22 December 2009

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We give a necessary and sufficient condition for the existence of a quadratic exponential vector with test function in L2(mathd)∩L∞(mathd). We prove the linear independence and totality, in the quadratic Fock space, of these vectors. Using a technique different from the one used by Accardi et al. [Quantum Probability and Infinite Dimensional Analysis, Vol. 25, p. 262, (2009)] , we also extend, to a more general class of test functions, the explicit form of the scalar product between two such vectors.
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03.65.Fd Algebraic methods
03.65.Db Functional analytical methods
02.10.Ud Linear algebra
02.30.Rz Integral equations

The geometric measure of entanglement for a symmetric pure state with non-negative amplitudes

Masahito Hayashi, Damian Markham, Mio Murao, Masaki Owari, and Shashank Virmani

J. Math. Phys. 50, 122104 (2009); http://dx.doi.org/10.1063/1.3271041 (6 pages) | Cited 11 times

Online Publication Date: 28 December 2009

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In this paper for a class of symmetric multiparty pure states, we consider a conjecture related to the geometric measure of entanglement: “for a symmetric pure state, the closest product state in terms of the fidelity can be chosen as a symmetric product state.” We show that this conjecture is true for symmetric pure states whose amplitudes are all non-negative in a computational basis. The more general conjecture is still open.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
03.67.Mn Entanglement measures, witnesses, and other characterizations
03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Fd Algebraic methods

Complete set of inner products for a discrete PT-symmetric square-well Hamiltonian

Miloslav Znojil

J. Math. Phys. 50, 122105 (2009); http://dx.doi.org/10.1063/1.3272002 (19 pages) | Cited 6 times

Online Publication Date: 28 December 2009

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A discrete N-point Runge–Kutta version H(N)(λ) of one of the simplest non-Hermitian square-well Hamiltonians with real spectrum is studied. Its possible Hermitizations mediated by nontrivial (often called “non-Dirac”) metrics Θ ≠ I are considered as a source of nonequivalent standard probabilistic interpretations of this quantum model. A complete set of these alternative, multiparametric metrics Θ = Θ(a,b,…)(N)(λ) defining all the eligible Hamiltonian-dependent representations of the physical Hilbert space of states is constructed, in closed form, for any coupling λ ∊ (−1,1) and for any matrix dimension N.
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03.65.Ta Foundations of quantum mechanics; measurement theory
02.10.Yn Matrix theory
03.65.Ge Solutions of wave equations: bound states
03.65.Aa Quantum systems with finite Hilbert space
02.60.-x Numerical approximation and analysis

Analysis of quantum walks with time-varying coin on d-dimensional lattices

Francesca Albertini and Domenico D’Alessandro

J. Math. Phys. 50, 122106 (2009); http://dx.doi.org/10.1063/1.3271109 (17 pages) | Cited 2 times

Online Publication Date: 29 December 2009

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In this paper, we present a study of discrete time quantum walks whose underlying graph is a d-dimensional lattice. The dynamical behavior of these systems is of current interest because of their applications in quantum information theory as tools to design quantum algorithms. We assume that, at each step of the walk evolution, the coin transformation is allowed to change so that we can use it as a control variable to drive the evolution in a desired manner. We give an exact description of the possible evolutions and of the set of possible states that can be achieved with such a system. In particular, we show that it is possible to go from a state where there is probability 1 for the walker to be found in a vertex to a state where all the vertices have equal probability. We also prove a number of properties of the set of admissible states in terms of the number of steps needed to obtain them. We provide explicit algorithms for state transfer in low dimensional cases as well as results that allow to reduce algorithms on two-dimensional lattices to algorithms on the one-dimensional lattice, the cycle.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.50.Cw Probability theory
05.40.Fb Random walks and Levy flights
02.10.Ox Combinatorics; graph theory
03.65.Fd Algebraic methods
03.67.-a Quantum information

A note on the logic of bounded quantum observables

Yuan Li and Xiu-Hong Sun

J. Math. Phys. 50, 122107 (2009); http://dx.doi.org/10.1063/1.3272542 (10 pages) | Cited 3 times

Online Publication Date: 29 December 2009

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The set of bounded observables for a quantum system is represented by the set of bounded self-adjoint operators S(H) on a complex Hilbert space H, and the quantum effects for a physical system can be described by the set E(H) of positive contractive operators on a complex Hilbert space H. In this note, by the techniques of operator block and spectral, we give the simpler representation of A∧P and obtained the new necessary and sufficient conditions for A∨P, for A ∊ S(H) and P ∊ P(H), where P(H) is the set of all orthogonal projection operators on H. In particular, we get that if A∨P exists, then A∨P ∊ E(H) for A ∊ E(H) and P ∊ P(H). In addition, we consider the relations between the existence of A∨B, A−∨B−, and A+∨B+, where A+, B+, A−, and B− are the positive and negative parts of A,B ∊ S(H).
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03.65.Aa Quantum systems with finite Hilbert space
02.30.Tb Operator theory

The Aharonov–Bohm effect and Tonomura et al. experiments: Rigorous results

Miguel Ballesteros and Ricardo Weder

J. Math. Phys. 50, 122108 (2009); http://dx.doi.org/10.1063/1.3266176 (54 pages) | Cited 4 times

Online Publication Date: 30 December 2009

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The Aharonov–Bohm effect is a fundamental issue in physics. It describes the physically important electromagnetic quantities in quantum mechanics. Its experimental verification constitutes a test of the theory of quantum mechanics itself. The remarkable experiments of Tonomura et al. [“Observation of Aharonov-Bohm effect by electron holography,” Phys. Rev. Lett 48, 1443 (1982) and “Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave,” Phys. Rev. Lett 56, 792 (1986)] are widely considered as the only experimental evidence of the physical existence of the Aharonov–Bohm effect. Here we give the first rigorous proof that the classical ansatz of Aharonov and Bohm of 1959 [ “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485 (1959) ], that was tested by Tonomura et al., is a good approximation to the exact solution to the Schrödinger equation. This also proves that the electron, that is, represented by the exact solution, is not accelerated, in agreement with the recent experiment of Caprez et al. in 2007 [ “Macroscopic test of the Aharonov–Bohm effect,” Phys. Rev. Lett. 99, 210401 (2007) ], that shows that the results of the Tonomura et al. experiments can not be explained by the action of a force. Under the assumption that the incoming free electron is a Gaussian wave packet, we estimate the exact solution to the Schrödinger equation for all times. We provide a rigorous, quantitative error bound for the difference in norm between the exact solution and the Aharonov–Bohm Ansatz. Our bound is uniform in time. We also prove that on the Gaussian asymptotic state the scattering operator is given by a constant phase shift, up to a quantitative error bound that we provide. Our results show that for intermediate size electron wave packets, smaller than the ones used in the Tonomura et al. experiments, quantum mechanics predicts the results observed by Tonomura et al. with an error bound smaller than 10−99. It would be quite interesting to perform experiments with electron wave packets of intermediate size. Furthermore, we provide a physical interpretation of our error bound.
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03.65.Ta Foundations of quantum mechanics; measurement theory
02.30.Rz Integral equations
03.65.Ge Solutions of wave equations: bound states
02.30.Tb Operator theory
02.50.-r Probability theory, stochastic processes, and statistics
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Deligne–Beilinson cohomology and Abelian link invariants: Torsion case

F. Thuillier

J. Math. Phys. 50, 122301 (2009); http://dx.doi.org/10.1063/1.3266178 (10 pages) | Cited 3 times

Online Publication Date: 7 December 2009

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For the Abelian Chern–Simons field theory, we consider the quantum functional integration over the Deligne–Beilinson cohomology classes and present an explicit path-integral nonperturbative computation of the Chern–Simons link invariants in SO(3) ≃ mathP3, a toy example of a 3-manifold with torsion.
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11.15.Yc Chern-Simons gauge theory
11.30.Ly Other internal and higher symmetries

Foldy–Wouthuysen wave functions and conditions of transformation between Dirac and Foldy–Wouthuysen representations

V. P. Neznamov and A. J. Silenko

J. Math. Phys. 50, 122302 (2009); http://dx.doi.org/10.1063/1.3268592 (15 pages) | Cited 2 times

Online Publication Date: 29 December 2009

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The block diagonalization of the Hamiltonian is not sufficient for the transformation to the Foldy–Wouthuysen (FW) representation. The conditions enabling the transition from the Dirac representation to the FW one are formulated and proven. The connection between wave functions in the two representations is derived. The results obtained allow calculating expectation values of operators corresponding to main classical quantities.
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03.65.Pm Relativistic wave equations
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A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac–Rice formula

A. O. Petters, B. Rider, and A. M. Teguia

J. Math. Phys. 50, 122501 (2009); http://dx.doi.org/10.1063/1.3267859 (17 pages)

Online Publication Date: 15 December 2009

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Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star’s mass. As a consequence, the pdf’s of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac–Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars.
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98.62.Sb Gravitational lenses and luminous arcs
02.50.Ey Stochastic processes
05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.30.Sa Functional analysis
02.50.Cw Probability theory

Transforming to Lorentz gauge on de Sitter

S. P. Miao, N. C. Tsamis, and R. P. Woodard

J. Math. Phys. 50, 122502 (2009); http://dx.doi.org/10.1063/1.3266179 (34 pages) | Cited 6 times

Online Publication Date: 16 December 2009

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We demonstrate that certain gauge fixing functionals cannot be added to the action on backgrounds such as de Sitter, in which a linearization instability is present. We also construct the field-dependent gauge transformation that carries the electromagnetic vector potential from a convenient, non-de Sitter invariant gauge to the de Sitter invariant, Lorentz gauge. The transformed propagator agrees with the de Sitter invariant result previously found by solving the propagator equation in Lorentz gauge. This shows that the gauge transformation technique will eliminate unphysical breaking of de Sitter invariance introduced by a gauge condition. It is suggested that the same technique can be used to finally resolve the issue of whether or not free gravitons are de Sitter invariant.
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11.15.-q Gauge field theories
04.60.-m Quantum gravity

Matrix projective spaces and twistorlike incidence structures

Alfonso F. Agnew and Scot P. Childress

J. Math. Phys. 50, 122503 (2009); http://dx.doi.org/10.1063/1.3271042 (10 pages)

Online Publication Date: 16 December 2009

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We consider projective spaces constructed over (real or complex) matrix rings and study their topological separation properties. The method of construction results in non-Hausdorff spaces, where incidence properties associated with a Grassmann/flag interpretation of the spaces are neatly encoded in the lack of standard topological separation properties. The physical motivation for studying this class of spaces is due to the emergence in mathematical physics during recent years of methods from algebraic geometry and Clifford and tensored division algebras. More specifically, there is the observation of Souček [Twistor Newsletter 13, 22 (1981) ; Czech. J. Phys., Sect. B 32, 688 (1982) ] that the basic twistor correspondence arises in the study of the biquaternionic projective line, mathmath1. In regard to the result of Souček, a subclass of the spaces presented here constitutes the n-dimensional generalization of mathmath1, biquaternionic projective n-space, mathmathn. We also discuss the generalization of the basic twistor correspondence in the more general setting of matrix projective spaces.
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02.10.Yn Matrix theory
02.40.-k Geometry, differential geometry, and topology
02.40.Re Algebraic topology

Spinor calculus on five-dimensional spacetimes

Alfonso García-Parrado Gómez-Lobo and José M. Martín-García

J. Math. Phys. 50, 122504 (2009); http://dx.doi.org/10.1063/1.3256124 (26 pages)

Online Publication Date: 17 December 2009

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Penrose’s spinor calculus of four-dimensional Lorentzian geometry is extended to the case of five-dimensional Lorentzian geometry. Such fruitful ideas in Penrose’s spinor calculus as the spin covariant derivative, the curvature spinors, or the definition of the spin coefficients on a spin frame can be carried over to the spinor calculus in five-dimensional Lorentzian geometry. The algebraic and differential properties of the curvature spinors are studied in detail, and as an application, we extend the well-known four-dimensional Newman–Penrose formalism to a five-dimensional spacetime.
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04.20.Jb Exact solutions
02.10.-v Logic, set theory, and algebra
02.40.-k Geometry, differential geometry, and topology

Singular sources in gravity and homotopy in the space of connections

E. Gravanis and S. Willison

J. Math. Phys. 50, 122505 (2009); http://dx.doi.org/10.1063/1.3250196 (32 pages) | Cited 3 times

Online Publication Date: 18 December 2009

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Suppose a Lagrangian is constructed from its fields and their derivatives. When the field configuration is a distribution, it is unambiguously defined as the limit of a sequence of smooth fields. The Lagrangian may or may not be a distribution, depending on whether there is some undefined product of distributions. Supposing that the Lagrangian is a distribution, it is unambiguously defined as the limit of a sequence of Lagrangians. But there still remains the question: Is the distributional Lagrangian uniquely defined by the limiting process for the fields themselves? In this paper a general geometrical construction is advanced to address this question. We describe certain types of singularities, not by distribution valued tensors, but by showing that the action functional for the singular fields is (formally) equivalent to another action built out of smooth fields. Thus we manage to make the problem of the lack of a derivative disappear from a system which gives differential equations. Certain ideas from homotopy and homology theory turn out to be of central importance in analyzing the problem and clarifying finer aspects of it. The method is applied to general relativity in first order formalism, which gives some interesting insights into distributional geometries in that theory. Then more general gravitational Lagrangians in first order formalism are considered such as Lovelock terms (for which the action principle admits space-times more singular than other higher curvature theories).
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04.50.-h Higher-dimensional gravity and other theories of gravity
02.40.-k Geometry, differential geometry, and topology
04.20.Jb Exact solutions
04.20.Gz Spacetime topology, causal structure, spinor structure
02.10.Ud Linear algebra
02.30.-f Function theory, analysis

Tensor generalizations of affine symmetry vectors

Samuel A. Cook and Tevian Dray

J. Math. Phys. 50, 122506 (2009); http://dx.doi.org/10.1063/1.3266423 (7 pages)

Online Publication Date: 29 December 2009

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A definition is suggested for affine symmetry tensors, which generalize the notion of affine vectors in the same way that (conformal) Killing tensors generalize (conformal) Killing vectors. An identity for these tensors is proven, which gives the second derivative of the tensor in terms of the curvature tensor, generalizing a well-known identity for affine vectors. Additionally, the definition leads to a good definition of homothetic tensors. The inclusion relations between these types of tensors are exhibited. The relationship between affine symmetry tensors and solutions to the equation of geodesic deviation is clarified, again extending known results about Killing tensors.
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02.10.Ud Linear algebra
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Integrability, analyticity, isochrony, equilibria, small oscillations, and Diophantine relations: Results from the stationary Korteweg-de Vries hierarchy

M. Bruschi, F. Calogero, and R. Droghei

J. Math. Phys. 50, 122701 (2009); http://dx.doi.org/10.1063/1.3267067 (19 pages) | Cited 2 times

Online Publication Date: 10 December 2009

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The isochronous variant is exhibited of the dynamical system corresponding to the Mth ordinary differential equation of the stationary Korteweg-de Vries (KdV) hierarchy. New Diophantine relations are thereby obtained, in the guise of matrices of arbitrary order having integer eigenvalues or equivalently of polynomials of arbitrary degree having integer zeros. Generalizations of these formulas to relations among rational functions are also obtained. The basic idea to arrive at such relations is not new, but the specific application reported in this paper is new, and it is likely to open the way to several analogous new findings.
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02.10.Ud Linear algebra
02.30.Rz Integral equations
02.10.De Algebraic structures and number theory

Analytical study of the superstable 3-cycle in the logistic map

M. Howard Lee

J. Math. Phys. 50, 122702 (2009); http://dx.doi.org/10.1063/1.3266875 (6 pages) | Cited 1 time

Online Publication Date: 11 December 2009

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In the logistic map, the simplest of stable odd-numbered cycles is a 3-cycle. It comes into existence after the stable 2k cycles cease to exist. The 3-cycle is thus important as a route to chaos. In 1977 Guckenheimer et al. [J. Math. Biol. 4, 101 (1977)] first estimated the value of the control parameter a at which it can be superstable. It implies that a stable 3-cycle can be formed and deformed at some values of a straddling that of the superstable 3-cycle. In this work we present an exact value of a for the superstable 3-cycle by reducing a polynomial of degree 7 to that of 3. The relevance of this work to Sharkovskii’s theorem is also discussed.
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05.45.Pq Numerical simulations of chaotic systems
02.10.De Algebraic structures and number theory

Discrete map with memory from fractional differential equation of arbitrary positive order

Vasily E. Tarasov

J. Math. Phys. 50, 122703 (2009); http://dx.doi.org/10.1063/1.3272791 (6 pages) | Cited 1 time

Online Publication Date: 18 December 2009

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Derivatives of fractional order with respect to time describe long-term memory effects. Using nonlinear differential equation with Caputo fractional derivative of arbitrary order α>0, we obtain discrete maps with power-law memory. These maps are generalizations of well-known universal map. The memory in these maps means that their present state is determined by all past states with power-law forms of weights. Discrete map equations are obtained by using the equivalence of the Cauchy-type problem for fractional differential equation and the nonlinear Volterra integral equation of the second kind.
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02.30.Rz Integral equations
02.30.Hq Ordinary differential equations
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Curved manifolds with conserved Runge–Lenz vectors

J.-P. Ngome

J. Math. Phys. 50, 122901 (2009); http://dx.doi.org/10.1063/1.3266874 (13 pages) | Cited 8 times

Online Publication Date: 15 December 2009

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van Holten’s algorithm is used to construct Runge–Lenz-type conserved quantities, induced by Killing tensors, on curved manifolds. For the generalized Taub-Newman–Unti-Tamburino metric, the most general external potential such that the combined system admits a conserved Runge–Lenz-type vector is found. In the multicenter case, the subclass of two-center metric exhibits a conserved Runge–Lenz-type scalar.
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11.30.Na Nonlinear and dynamical symmetries (spectrum-generating symmetries)
02.10.Ud Linear algebra

Kinks, chains, and loop groups in the mathmathn sigma models

Derek Harland

J. Math. Phys. 50, 122902 (2009); http://dx.doi.org/10.1063/1.3266172 (13 pages) | Cited 1 time

Online Publication Date: 18 December 2009

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We consider topological solitons in the mathmathn sigma models in two space dimensions. In particular, we study “kinks,” which are independent of one coordinate up to a rotation of the target space, and “chains,” which are periodic in one coordinate up to a rotation of the target space. Kinks and chains both exhibit constituents, similar to monopoles and calorons in SU(n) Yang–Mills–Higgs and Yang–Mills theories. We examine the constituent structure using Lie algebras.
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11.10.Lm Nonlinear or nonlocal theories and models
11.15.-q Gauge field theories
02.10.Ud Linear algebra
11.30.Ly Other internal and higher symmetries
02.40.Pc General topology
02.20.-a Group theory
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Chaotic motion in classical fluids with scale relativistic methods

Marie-Noëlle Célérier

J. Math. Phys. 50, 123101 (2009); http://dx.doi.org/10.1063/1.3271040 (15 pages)

Online Publication Date: 18 December 2009

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In the framework of the scale relativity theory, the chaotic behavior in time only of a number of macroscopic systems corresponds to the motion in a space with geodesics of fractal dimension 2 and leads to its representation by a Schrödinger-type equation acting in the macroscopic domain. The fluid interpretation of such a Schrödinger equation yields Euler and Navier–Stokes equations. We therefore choose to extend this formalism to study the properties of a system exhibiting a chaotic behavior both in space and time, which amounts to consider them as issued from the geodesic features of a mathematical object exhibiting all the properties of a fractal “space-time.” Starting with the simplest Klein–Gordon-type form that can be given to the geodesic equation in this case, we obtain a motion equation for a “three fluid” velocity field and three continuity equations, together with parametric expressions for the three velocity components which allow us to derive relations between their nonvanishing curls. At the nonrelativistic limit and owing to the physical properties exhibited by this solution, we suggest that it could represent some kind of three-dimensional chaotic behavior in a classical fluid, tentatively turbulent if particular conditions are fulfilled. The appearance of a transition parameter D in the equations allows us to consider different ways of testing experimentally our proposal.
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47.52.+j Chaos in fluid dynamics
47.53.+n Fractals in fluid dynamics
47.10.ad Navier-Stokes equations
47.27.-i Turbulent flows

Lie symmetries and exact solutions of the barotropic vorticity equation

Alexander Bihlo and Roman O. Popovych

J. Math. Phys. 50, 123102 (2009); http://dx.doi.org/10.1063/1.3269919 (12 pages) | Cited 8 times

Online Publication Date: 22 December 2009

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Lie group methods are used for the study of various issues related to symmetries and exact solutions of the barotropic vorticity equation. The Lie symmetries of the barotropic vorticity equations on the f- and β-planes, as well as on the sphere in rotating and rest reference frames, are determined. A symmetry background for reducing the rotating reference frame to the rest frame is presented. The one- and two-dimensional inequivalent subalgebras of the Lie invariance algebras of both equations are exhaustively classified and then used to compute invariant solutions of the vorticity equations. This provides large classes of exact solutions, which include both Rossby and Rossby–Haurwitz waves as special cases. We also discuss the possibility of partial invariance for the β-plane equation, thereby further extending the family of its exact solutions. This is done in a more systematic and complete way than previously available in literature.
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02.20.Sv Lie algebras of Lie groups
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On the stability of general cubic-quartic functional equations in Menger probabilistic normed spaces

A. Ghaffari, A. Alinejad, and M. Eshaghi Gordji

J. Math. Phys. 50, 123301 (2009); http://dx.doi.org/10.1063/1.3269920 (7 pages) | Cited 4 times

Online Publication Date: 16 December 2009

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In this paper, the stability of the general cubic-quartic functional equation, f(x+ky)+f(x−ky) = k2(f(x+y)+f(x−y))+2(1−k2)f(x)+[(k4−k2)/4](f(2y)−8f(y))+math(2x)−16math(x), where math(x) ≔ f(x)+f(−x) in the setting of Menger probabilistic normed spaces, is proved.
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02.30.Sa Functional analysis
02.30.Ks Delay and functional equations
02.50.Cw Probability theory
02.10.-v Logic, set theory, and algebra
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