The input/output stability of an interconnected system composed of an ordinary differential equation and a damped string equation is studied. Issued from the literature on time-delay systems, an exact stability result is firstly derived using pole locations. Then, based on the Small-Gain theorem and on the Quadratic Separation framework, some robust stability criteria are provided. The latter f...

Special cases of this theorem have been proved by Erdös-Grünwald' and Webster2 (the cases a = 1/2 and a = 3/2) . If there is no danger of confusion we shall omit the upper index n in lk`, n ~ (x) . PROOF OF THE THEOREM . It clearly suffices to consider the lk(x) with -1 =< xk < 0 . From the differential equation of the ultraspherical polynomials' we obtain (") lk(xk) = {«~xk) _ axk z zP„ (xk) 1...

Let μ be a probability measure on the real line with finite moments of all orders. Apply the Gram-Schmidt orthogonalization process to the system {1, x, x, . . . , xn, . . . } to get orthogonal polynomials Pn(x), n ≥ 0, which have leading coefficient 1 and satisfy (x − αn)Pn(x) = Pn+1(x) + ωnPn−1(x). In general it is almost impossible to use this process to compute the explicit form of these po...

We consider a filtration of the symmetric function space given by Λ (k) t , the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than k. We introduce symmetric functions called the k-Schur functions, providing an analog for the Schur functions in the subspaces Λ (k) t . We prove several properties for the k-Schur functions including that they form ...

Generalized Hall–Littlewood polynomials (Macdonald spherical functions) and generalized Kostka–Foulkes polynomials (q-weight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics. This paper attempts to organize the different definitions of these objects and prove the fundamental combinatorial results from “scratch”, in a presentation w...

A Feynman diagram is a graphical construction that describes certain interactions in physics. Most calculations with such diagrams reduce to consideration of connected Feynman diagrams. These in turn may be constructed from line-irreducible Feynman diagrams, those for which removal of a single line does not destroy connectivity. The purpose of this article is to exhibit the combinatorial nature...

By convention, the translation and scale invariant functions of Legendre moments are achieved by using a combination of the corresponding invariants of geometric moments. They can also be accomplished by normalizing the translated and/or scaled images using complex or geometric moments. However, the derivation of these functions is not based on Legendre polynomials. This is mainly due to the fa...

Time series problems involve analysis of periodic functions for predicting the future. A flexible regression method should be able to dynamically select the appropriate model to fit the available data. In this paper, we present a function approximation scheme that can be used for modeling periodic functions using a series of orthogonal polynomials, named Chebychev polynomials. In our approach, ...

Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2 + dy2 or 4p = x2 + dy2. In this paper we determine x (mod p) for many values of d. For example,

A general framework is introduced to analyze the approximation properties of mapped Legendre polynomials and of interpolations based on mapped Legendre–Gauss–Lobatto points. Optimal error estimates featuring explicit expressions on the mapping parameters for several popular mappings are derived. These results not only play an important role in numerical analysis of mapped Legendre spectral and ...