Binary Bell Polynomials Approach to Generalized Nizhnik-Novikov-Veselov Equation

The elementary and systematic binary Bell polynomials method is applied to the generalized Nizhnik-Novikov-Veselov (GNNV) equation. The bilinear representation, bilinear Bäcklund transformation, Lax pair and infinite conservation laws of the GNNV equation are obtained directly, without too much trick like Hirota's bilinear method.     

Creation Operators for the Macdonald and Jack Polynomials

Formulas of Rodrigues-type for the Macdonald polynomials are presented. They involve creation operators, certain properties of which are proved and other conjectured. The limiting case of the Jack polynomials is discussed.     

The Crank-Nicolson Hermite Cubic Orthogonal Spline Collocation Method for the Heat Equation with Nonlocal Boundary Conditions

We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval. Using an extension of the analysis of Douglas and Dupont [23] for Dirichlet boundary conditions, we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space. We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROW for solving almost block diagonal linear systems. We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.     

Analysis of the Generalized Heat Equation for Solution of the Dynamic Thermoelasticity Problem

Doklady Physics - On the basis of the approximate solution of the dispersion equation, the set of equations of dynamic thermoelasticity is analyzed with taking into account the generalized heat...     

On the Recurrence Coefficients for Generalized q-Laguerre Polynomials

In this paper we consider a semi-classical variation of the weight related to the q-Laguerre polynomials and study their recurrence coefficients. In particular, we obtain a second degree second order discrete equation which in particular cases can be reduced to either the qPV or the qPIII equation.     

A Conjecture about Raising Operators for Macdonald Polynomials

We present a conjecture giving Macdonald polynomials as an hypergeometric-type series in terms of raising operators. It is proved that this conjecture agrees with Jing and Józefiak’s expression in the two-row case, and with Lassalle and Schlosser’s formula in the three-row case.Mathematics Subject Classifications (2000). 33D52, 33D67, 81R50     

Deformed Macdonald-Ruijsenaars Operators and Super Macdonald Polynomials

It is shown that the deformed Macdonald-Ruijsenaars operators can be described as the restrictions on certain affine subvarieties of the usual Macdonald- Ruijsenaars operator in infinite number of variables. The ideals of these varieties are shown to be generated by the Macdonald polynomials related to Young diagrams with special geometry. The super Macdonald polynomials and their shifted version are introduced; the combinatorial formulas for them are given.     

Schur Polynomials and The Yang-Baxter Equation

We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map \({g \mapsto R (g)}\) from \({ \rm{GL}(2, \mathbb{C}) \times \rm{GL}(1, \mathbb{C})}\) to End \({(V \otimes V)}\) , where V is a two-dimensional vector space such that if \({g, h \in G}\) then R ₁₂(g)R ₁₃(gh) R ₂₃(h) = R ₂₃(h) R ₁₃(gh)R ₁₂(g). Here R _{i j} denotes R applied to the i, j components of \({V \otimes V \otimes V}\) . The image of this map consists of matrices whose nonzero coefficients a ₁, a ₂, b ₁, b ₂, c ₁, c ₂ are the Boltzmann weights for the non-field-free six-vertex model, constrained to satisfy a ₁ a ₂ + b ₁ b ₂ ? c ₁ c ₂ = 0. This is the exact center of the disordered regime, and is contained within the free fermionic eight-vertex models of Fan and Wu. As an application, we show that with boundary conditions corresponding to integer partitions λ, the six-vertex model is exactly solvable and equal to a Schur polynomial s _λ times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of Tokuyama and Hamel and King.     

The Calogero-Sutherland Model and Generalized Classical Polynomials

Multivariable generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of certain Schr?dinger operators for Calogero-Sutherland-type quantum systems. For the generalized Hermite and Laguerre polynomials the multidimensional analogues of many classical results regarding generating functions, differentiation and integration formulas, recurrence relations and summation theorems are obtained. We use this and related theory to evaluate the global limit of the ground state density, obtaining in the Hermite case the Wigner semi-circle law, and to give an explicit solution for an initial value problem in the Hermite and Laguerre case. Received: 16 August 1996 / Accepted: 21 January 1997     

Conditional Stability of Solitary-Wave Solutions for Generalized Compound KdV Equation and Generalized Compound KdV-Burgers Equation

In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travellingwave form satisfies some special conditions.     

Conditional Stability of Solitary-Wave Solutions for Generalized Compound KdV Equation and Generalized Compound KdV-Burgers Equation

In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travelling wave form satisfies some special conditions.     

Generalized Kraus Operators for the One-Qubit Depolarizing Quantum Channel

Microscopic Hamiltonian models of the composite system “open system + environment” typically do not provide the operator-sum Kraus form of the open system’s dynamical map. With the use of a recently developed method (Andersson et al. J. Mod. Opt. 54, 1695 2007), we derive the Kraus operators starting from the microscopic Hamiltonian model, i.e., from the proper master equation, of the one-qubit depolarizing channel. Those Kraus operators generalize the standard counterparts, which are widely used in the literature. Comparison of the standard and the here obtained Kraus operators is performed via investigating dynamical change of the Bloch sphere volume, entropy production, and the open system’s state trace distance. We compare the generalized with the standard Kraus operators for both single qubit and regarding the occurrence of the entanglement sudden death for a pair of initially correlated qubits. We find that the generalized Kraus operators describe the less deteriorating quantum channel than the standard ones.     

Tomography of Multimode Quantum Systems with Quadratic Hamiltonians and Multivariable Hermite Polynomials

Systems with multimode nonstationary Hamiltonians (quadratic in position and momentum operators) are reviewed. The tomographic probability distributions (tomograms) for the Fock states and Gaussian states of the quadratic systems are discussed. The tomograms for the Fock states are expressed in terms of multivariable Hermite polynomials. In view of the obvious physical relations, some new formulas for multivariable Hermite polynomials are found. Examples of a driven parametric oscillator and a charged particle moving in the electromagnetic field are presented.     

A Maximum Entropy Method Based on Orthogonal Polynomials for Frobenius-Perron Operators

Let $S$: [0, 1]→[0, 1] be a chaotic map and let $f^∗$ be a stationary density of the Frobenius-Perron operator $P_S$: $L^1$→$L^1$ associated with $S$. We develop a numerical algorithm for approximating $f^∗$, using the maximum entropy approach to an under-determined moment problem and the Chebyshev polynomials for the stability consideration. Numerical experiments show considerable improvements to both the original maximum entropy method and the discrete maximum entropy method.     

On the Generalized Continuity Equation

A generalized continuity equation extending the ordinary continuity equation is found using quanternions to show it is compatible with Dirac, Schrǒdinger, Klein-Gordon and diffusion equations. This generalized equation is Lorentz invariant. The transport properties of electrons are found to be governed by the Schr6dinger-like equation and not by the diffusion equation.