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Weekly Seminar
 WS Season 2017/2018

Title: Asymptotic behavior of eigenvalues of a differential operator for discrete Gegenbauer-Sobolev orthonormal polynomials.

Abstract: In this talk, we consider discrete Sobolev inner products involving the Gegenbauer weight. The sequence of orthonormal polynomials with respect to this inner product are eigenfunctions of a differential operator. We establish the asymptotic behavior of the corresponding eigenvalues. Finally, we deduce the Mehler-Heine type asymptotics for the sequence of orthonormal polynomials and the location of the zeros of these polynomials.

Title: Self-interlacing and stable orthogonal polynomials. Theory, examples and open problems

Abstract: Some non-symmetric tridiagonal matrices appear to be useful in some engineering questions,  for example, in network synthesis, stability theory etc. I will talk about orthogonal polynomials related to such tridiagonal matrices. Bessel polynomials will be presented as an example of the theory.

Title:  On several extremal problems in graph theory involving Gromov Hyperbolicity Constant

Abstract: If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\d$-\emph{hyperbolic} $($in the Gromov sense$)$ if any side of $T$ is contained in the $\d$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. We denote by $\d(X)$ the sharp hyperbolicity constant of $X$, i.e., $\d(X):=\inf\{\d\ge 0: \, X \, \text{ is$\d$-hyperbolic}\,\}\,.$

In this work we focus on the study of hyperbolic graphs. In general, to compute $\d(G)$ for a given graph $G$ is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Our main aim is to estimate the hyperbolicity constant of graphs belonging to particular families; more specifically, we obtain good upper and lower bounds for $\d(G)$ in terms of the order, size, diameter, girth, circumference, minimum and maximum degree of a graph.

Title: An extension of Markov's theorem for mixed type Hermite-Padé approximants.

Abstract: Since their introduction in by Ch. Hermite, Hermite-Pad e approximants have been an invaluable tool in number theory, recently they have appeared in models coming from random matrix theory, biorthogonal ensembles and in some integrable systems. Due to classical results of A.A. Markov, the Cauchy transform of one measure plays a central role in the convergence theory of diagonal Pad e approximation. For type I and type II Hermite Pad e approximants, a similar place is occupied by the so-called Nikishin systems of functions, in particular, for such systems Markov type theorems are obtained. This talk deals with an extension of the Markov´s theorem for a kind of mixed type Hermite-Pad e approximants with respect to a Nikishin system of measures.

Title:  Jacobi matrices on graph trees and multiple orthogonal polynomials

Thursday, October 26,  2017, 16:00. Room 2.2.D08

Title: On Semiclassical Families of Bivariate Orthogonal Polynomials

Abstract: Orthogonal polynomials in two variables are studied as the natural generalization of orthogonal polynomials in one variable. Nevertheless, much work needs to be done in order to consider the theory of bivariate orthogonal polynomials as complete as the univariate case. In this work, we focus on extending well-known properties of univariate semiclassical orthogonal polynomials to the bivariate case. We also study algebraic and differential properties of a class of orthogonal polynomials in two variables constructed from orthogonal polynomials in one variable. Finally, we extend the definition of coherent pairs of quasi-definite moment functionals to the bivariate case and deduce some of its consequences.

Title: Funciones internas en el Espacio de Dirichlet

Abstract: This thesis is focused on the study  of  matrix transformations of  matrices with  linear  functionals as entries. In particular, we study the Christoffel, Geronimus, and Geronimus-Uvarov transformations,  as well as,  their relation  with scalar orthogonal polynomials satisfying a higher order recurrence relation.

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Monday, 20 November 2017 20:09:42