Perturbed recurrence relations II the general case

In this paper, we give some new explicit relations between two families of polynomials defined by recurrence relations of all order. These relations allow us to analyze, even in the Sobolev case, how some properties of a family of orthogonal polynomials are affected when the coefficients of the recurrence relation and the order are perturbed. In a paper we have already given a method which allows us to study the polynomials defined by a three-term recurrence relation. Also here some generalizations are given.     

Connections between Romanovski and other polynomials

A connection between Romanovski polynomials and those polynomials that solve the one-dimensional Schrödinger equation with the trigonometric Rosen-Morse and hyperbolic Scarf potential is established. The map is constructed by reworking the Rodrigues formula in an elementary and natural way. The generating function is summed in closed form from which recursion relations and addition theorems follow. Relations to some classical polynomials are also given.     

Relations among eigenvalues of Sturm-Liouville problems with different types of leading coefficient functions

For any Sturm-Liouville problem with a separable boundary condition and whose leading coefficient function changes sign (exactly once), we first give a geometric characterization of its eigenvalues λn using the eigenvalues of some corresponding problems with a definite leading coefficient function. Consequences of this characterization include simple proofs of the existence of the λn's, their Prüfer angle characterization, and a way for determining their indices from the zeros of their eigenfunctions. Then, interlacing relations among the λn's and the eigenvalues of the corresponding problems are obtained. Using these relations, a simple proof of asymptotic formulas for the λn's is given.     

Representations, convolution square roots and spherical vectors

We extend results on square integrable representations of a locally compact group to subrepresentations of a representations induced from a character of a closed subgroup. We do so working in the framework of quotient representations of ∗-algebras.     

Comparisons between model equations for long waves

Summary Considered here are model equations for weakly nonlinear and dispersive long waves, which feature general forms of dispersion and pure power nonlinearity. Two variants of such equations are introduced, one of Korteweg-de Vries type and one of regularized long-wave type. It is proven that solutions of the pure initial-value problem for these two types of model equations are the same, to within the order of accuracy attributable to either, on the long time scale during which nonlinear and dispersive effects may accumulate to make an order-one relative difference to the wave profiles.This research was supported in part by the National Science Foundation. A considerable portion of the project was completed while the first author was resident at the Institute for Mathematics and Its Applications, University of Minnesota.     

Properties of some families of hypergeometric orthogonal polynomials in several variables

Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable Wilson, continuous Hahn and Jacobi type polynomials, respectively. For each class of polynomials we provide systems of difference (or differential) equations, recurrence relations, and expressions for the (squared) norms of the polynomials in question.

     

Reducibility and nonreducibility between equivalence relations

We show that, for , the relation of -equivalence between infinite sequences of real numbers is Borel reducible to the relation of -equivalence (i.e., the Borel cardinality of the quotient is no larger than that of ), but not vice versa. The Borel reduction is constructed using variants of the triadic Koch snowflake curve; the nonreducibility in the other direction is proved by taking a putative Borel reduction, refining it to a reduction map that is not only continuous but `modular,' and using this nicer map to derive a contradiction.

     

基于Maxwell-Rivlin-Ericksen理论的MRM本构关系的算法探究

从Maxwell电磁场理论出发,得到磁流变材料的电磁学本构方程,基于Rivlin-Ericksen理论,结合质量守恒定律及动量守恒定理,提出了普适的磁流变材料Cauchy应力σ的一种本构关系.与牛顿粘性流体及热力学第二定律相结合确定了算法中磁流变材料本构关系的基本参数.以平行板间磁流变材料运动为算例,计算结果验证了本模型切实有效.最后与美国Lord公司配置的典型磁流变材料MRM-132LD结果进行比较,结论吻合良好,希望该工作能有力推动磁流变材料的迅猛发展.