New Properties of the Zeros of Krall Polynomials

We identify a new class of algebraic relations satisfied by the zeros of orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two, known as Krall polynomials. Given an orthogonal polynomial family , we relate the zeros of the polynomial p_N with the zeros of p_m for each m ≤ N (the case m = N corresponding to the relations that involve the zeros of p_N only). These identities are obtained by finding exact expressions for the similarity transformation that relates the spectral and the (interpolatory) pseudospectral matrix representations of linear differential operators, while using the zeros of the polynomial p_N as the interpolation nodes. The proposed framework generalizes known properties of classical orthogonal polynomials to the case of nonclassical polynomial families of Krall type. We illustrate the general result by proving new identities satisfied by the Krall-Legendre, the Krall-Laguerre and the Krall-Jacobi orthogonal polynomials.     

Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem

We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the ‘fluctuation-dissipation theorem’, the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the ‘fluctuation-dissipation theorem’ is satisfied, and this verifies that satisfying ‘fluctuation-dissipation theorem’ indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.     

On the Zeros of Polynomials Satisfying Certain Linear Second-Order ODEs Featuring Many Free Parameters

Certain techniques to obtain properties of the zeros of polynomials satisfying second-order ODEs are reviewed. The application of these techniques to the classical polynomials yields formulas which were already known; new are instead the formulas for the zeros of the (recently identified, and rather explicitly known) polynomials satisfying a (recently identified) second-order ODE which features many free parameters and only polynomial solutions. Some of these formulas have a Diophantine connotation. Techniques to manufacture infinite sequences of second-order ODEs featuring only polynomial solutions are also reported.     

Properties of the Zeros of the Sum of two Polynomials

Some properties—including relations having a Diophantine character—of the zeros of the sum of two polynomials are reported.     

Zeros of Graph-Counting Polynomials

Given a finite graph E we define a family of subgraphs F by restricting the number of edges of F with endpoint at any vertex of E. Defining , we can in many cases give precise information on the location of zeros of (zeros all real negative, all imaginary, etc.). Extensions of these results to weighted and infinite graphs are given. Received: 4 May 1998 / Accepted: 12 June 1998     

Properties of the Zeros of the Polynomials Belonging to the Askey Scheme

In this paper, we provide properties—which are, to the best of our knowledge, new—of the zeros of the polynomials belonging to the Askey scheme. These findings include Diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted.     

Construction of Functional Polynomials for Solutions of Integrodifferential Equations

Russian Physics Journal - The integrodifferential equations of mathematical physics are objects of research, and construction of interpolating polynomials to obtain approximate solutions of such...     

Pair Correlation Statistics for the Zeros of Lamé Polynomials

The joint eigenfunctions of a quantum completely integrable system can naturally be described in terms of products of Lamé polynomials. In this paper, we compute the limiting pair correlation distribution for the zeros of Lamé polynomials in various thermodynamic, asymptotic regimes. We give results both in the mean and pointwise, for an asymptotically full set of values of the parameters α₀,. . .,α_N. Mathematics Subject Classifications (2000) 81R12, 53A55.     

Oscillatory and Asymptotic Behavior of a Second-Order Nonlinear Functional Differential Equations

This paper is concerned with oscillatory and asymptotic behavior of solutions of a class of second order nonlinear functional differential equations. By using the generalized Riccati transformation and the integral averaging technique, new oscillation criteria and asymptotic behavior are obtained for all solutions of the equation. Our results generalize and improve some known theorems.     

Density of Complex Zeros of a System of Real Random Polynomials

We study the density of complex zeros of a system of real random SO(m+1) polynomials in m variables. We show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case. We also show that the behavior the scaled density of complex zeros near ℝ ^m of the system of real random polynomials is different in the m≥2 case than in the m=1 case: the density approaches infinity instead of tending linearly to zero.     

沿等压路径求解疏松材料Hugoniot关系的微分方程组及其求解

 根据固体材料的三项式物态方程和Grüneisen物态方程,导出了沿等压路径求解疏松材料冲击温度和压缩体积随初始密度变化的微分方程组。从体积的微分方程出发,在假定Wu-Jing参量为常数的前提下,导出了冲击压缩体积和体积-焓物态方程的Wu-Jing表达式。采用数值差分方法求解微分方程组,计算了疏松铜的冲击压缩特性,并与文献中部分实验数据进行了比较,特别强调了热电子对冲击压缩体积、冲击温度和Wu-Jing参数的贡献。还讨论了Grüneisen物态方程与Wu-Jing物态方程的内在联系及后者的适用范围。     

Reliability Polynomials and Their Asymptotic Limits for Families of Graphs

We present exact calculations of reliability polynomials R(G,p) for lattice strips G of fixed widths L _y 4 and arbitrarily great length L _x with various boundary conditions. We introduce the notion of a reliability per vertex, r({G},p)=lim_|V|R(G,p)^1/|V| where |V| denotes the number of vertices in G and {G} denotes the formal limit lim_|V|G. We calculate this exactly for various families of graphs. We also study the zeros of R(G,p) in the complex p plane and determine exactly the asymptotic accumulation set of these zeros , across which r({G}) is nonanalytic.     

Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity

We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of non-linear differential equations to several cases of equations containing non-polynomial non-linearity. The main idea of the study is to use an appropriate transformation at Step (1.) of SEsM. This transformation has to convert the non-polynomial non- linearity to polynomial non-linearity. Then, an appropriate solution is constructed. This solution is a composite function of solutions of more simple equations. The application of the solution reduces the differential equation to a system of non-linear algebraic equations. We list 10 possible appropriate transformations. Two examples for the application of the methodology are presented. In the first example, we obtain kink and anti- kink solutions of the solved equation. The second example illustrates another point of the study. The point is as follows. In some cases, the simple equations used in SEsM do not have solutions expressed by elementary functions or by the frequently used special functions. In such cases, we can use a special function, which is the solution of an appropriate ordinary differential equation, containing polynomial non-linearity. Specific cases of the use of this function are presented in the second example.     

Prolongability of Ordinary Differential Equations

We extend the notion of deformation to inverse operations of restrictions of completely integrable systems to regular or singular locus, and call the extended notion prolongation. We show that a prolongability determines uniquely a Fuchsian ordinary differential equation of rank three with three regular singular points. This seems similar to that the deformation equation determines the accessory parameters as a function of the geometric moduli. Relations between prolongations and middle convolutions is also studied.     

New Q Matrices and Their Functional Equations for the Eight Vertex Model at Elliptic Roots of Unity

The Q matrix invented by Baxter in 1972 to solve the eight vertex model at roots of unity exists for all values of N, the number of sites in the chain, but only for a subset of roots of unity. We show in this paper that a new Q matrix, which has recently been introduced and is non zero only for N even, exists for all roots of unity. In addition we consider the relations between all of the known Q matrices of the eight vertex model and conjecture functional equations for them.     

Relaxation Processes and Fractional Differential Equations

Relaxation properties of different media (dielectrics, semiconductors, ferromagnetics, and so on) are normally expressed in terms of response function f(t) or of real and imaginary components of its Fourier transform dependent on the frequency . It had been recently recognized that most of real materials show deviation from classical Debye process. There exist a few empirical approximations of non-Debye response functions. One of them is the two-power approximation containing and , where and belong to the interval (0, 1). This formula gives the basis for introducing of fractional differential equation considered in this paper. A stochastic interpretation of this equation is offered; its solution is found and investigated. The results are in agreement with experimental data.