The Lascoux, Leclerc and Thibon algorithm and Soergel's tilting algorithm

We generalize Soergel's tilting algorithm to singular weights and deduce from this the validity of the Lascoux-Leclerc-Thibon conjecture on the connection between the canonical basis of the basic submodule of the Fock module and the representation theory of the Hecke-algebras at root of unity. Supported in part by Programa Reticulados y Ecuaciones and by FONDECYT grant 1051024.     

Comparison of higher-order accurate schemes for solving the two-dimensional unsteady Burgers' equation

This paper is devoted to the testing and comparison of numerical solutions obtained from higher-order accurate finite difference schemes for the two-dimensional Burgers' equation having moderate to severe internal gradients. The fourth-order accurate two-point compact scheme, and the fourth-order accurate Du Fort Frankel scheme are derived. The numerical stability and convergence are presented. The cases of shock waves of severe gradient are solved and checked with the fourth-order accurate Du Fort Frankel scheme solutions. The present study shows that the fourth-order two-point compact scheme is highly stable and efficient in comparison with the fourth-order accurate Du Fort Frankel scheme.     

高阶Bernoulli多项式和高阶Euler多项式的关系

利用发生函数的方法，讨论了高阶Bernoulli数和高阶Euler数，高阶Bernoulli多项式和高阶Euler多项式之间的关系，得到了经典Bernoulli数和Euler数，经典Bernoulli多项式和Euler多项式之间的新型关系。     

Evaluation formulas for Tornheim's type of alternating double series

In this paper, we give some evaluation formulas for Tornheim's type of alternating series by an elementary and combinatorial calculation of the uniformly convergent series. Indeed, we list several formulas for them by means of Riemann's zeta values at positive integers.

     

Determining radii of meromorphy via orthogonal polynomials on the unit circle

Using a convergence theorem for Fourier–Padé approximants constructed from orthogonal polynomials on the unit circle, we prove an analogue of Hadamard's theorem for determining the radius of m-meromorphy of a function analytic on the unit disk and apply this to the location of poles of the reciprocal of Szeg functions.     

Romanovski polynomials in selected physics problems

We briefly review the five possible real polynomial solutions of hypergeometric differential equations. Three of them are the well known classical orthogonal polynomials, but the other two are different with respect to their orthogonality properties. We then focus on the family of polynomials which exhibits a finite orthogonality. This family, to be referred to as the Romanovski polynomials, is required in exact solutions of several physics problems ranging from quantum mechanics and quark physics to random matrix theory. It appears timely to draw attention to it by the present study. Our survey also includes several new observations on the orthogonality properties of the Romanovski polynomials and new developments from their Rodrigues formula.     

Differentiators and the geometry of polynomials

In 1959, Davis introduced the concept of a differentiator of an operator on a finite-dimensional Hilbert space. We prove that every such operator possesses a differentiator. We also use the theory of differentiators to solve several problems in the geometry of polynomials. For instance, we answer in the affirmative a twenty year old unsolved conjecture of Schoenberg, a related conjecture of Katsoprinakis and a fifty year old unsolved conjecture of De Bruijn and Springer.     

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.     

Co-Orthogonal Codes

We define, construct and sketch possible applications of a new class of non-linear codes: co-orthogonal codes, with possible applications in cryptography and parallel processing. We also describe a fast and general method for generating (non-linear) codes with prescribed dot-products with the help of multi-linear polynomials.