SEMINARIO DEL GRUPO DE INVESTIGACIÓN

Teoría de Aproximación y Polinomios Ortogonales

Curso 2012-2013:


Thursday, October 11, 2012
Venue: Seminario de Matemática Aplicada, University of Almería

Session 12:00. Edmundo J. Huertas Cejudo, Department of Mathematics, UC3M.
Title: A technique to study the behavior of zeros of orthogonal polynomials
Abstract: In this talk we discuss the behaviour of the zeros of polynomials orthogonal with respect to the Uvarov perturbation of a positive Borel measure and certain discrete Sobolev inner product. The behavior of their zeros is analyzed in terms of their dependence on the parameters that determine the intensity of the perturbation.
See photo here.



Friday, November 16, 2012

Venue: Sala "Nevada" del Edificio Mecenas, University of Granada

Session 11:30. Marcos Raydan, Department of Computer Science and Statistics, Simon Bolivar University, Caracas, Venezuela.
Title: Specialized Newton and secant methods for nonlinear matrix problems
Abstract: Nonlinear matrix problems arise in numerous applications in science and engineering. In this talk we shall describe and analyze some well-established and some new variants of Newton's methods for the specific case of the matrix $p$-th root. We shall also explore, in the general case, the use of secant methods in the space of matrices, that represents a new approach with interesting properties for medium size problems.
For some specific problems (e.g. matrix inverse and square root) we will describe local convergence properties and stability issues.

Session 13:00. Miguel Piñar, Department of Applied Mathematics, University of Granada.
Title: Weighted Sobolev orthogonal polynomials on the unit ball
Abstract: For the classical weight function on the unit ball, a Sobolev inner product involving the gradient operator is considered. For this inner product a family of mutually orthogonal polynomials on the unit ball is constructed in terms of spherical harmonics and a sequence of Sobolev orthogonal polynomials of one variable. The latter ones, hence, the orthogonal polynomials with respect to multivariate Sobolev inner product, can be generated through a recursive formula.

This is a joint work with T. E. Pérez and Y. Xu.
See photos: 1 2



Friday, November 30, 2012
Venue: Sala de Grados del Aulario IV, University of Almería
Organization: This is a special session of the seminar organized by the Department of Mathematics of the University of Almería and the Research Group "Approximation Theory and Orthogonal Polynomials"

Session 12:00. Ernesto Altshuler, from the Group "Henri Poincarè" of Complex Systems, Faculty of Physics, University of Havana, Cuba.
Title: Cuantificando el tráfico de hormigas
Abstract: Tragedias como la recientemente ocurrida en Madrid Arena sugieren la necesidad de comprender la dinámica de las multitudes en estado de pánico. La primera parte de mi charla comprende un estudio experimental realizado sobre hormigas bibijaguas en pánico (al menos, presumiblemente en pánico), que sugiere la gran similitud entre el comportamiento de estos animales y los humanos en situaciones extremas. El resultado principal se puede resumir así: cuando las hormigas se encuentran encerradas en un recinto con dos salidas idénticas, tienden a concentrarse en una cualquiera de ellas si están en pánico, haciendo más ineficiente su escape. El leit motiv de la segunda parte de la presentación también son las hormigas (de hecho, el mismo tipo de hormigas). Pero esta vez aceptamos el reto de medir el tráfico de individuos que forrajean en condiciones totalmente naturales. Nuestro experimentos indican que los ciclos del comportamiento colectivo de las hormigas durante el forrajeo parecen sintonizarse de un modo no trivial con los ciclos de temperatura ambiental, que estamos intentando comprender usando un modelo de ecuaciones diferenciales de carácter preliminar.
See photos: 1 2



Friday, January 18, 2013
Venue: M1 classroom of the Faculty of Sciences, University of Granada

Session 11:30. Elena Blanca Medina Reus, Department of Mathematics, University of Cádiz.
Title: Estructura de fases de distribuciones de ceros de polinomios ortogonales y la propiedad S
Abstract: El objetivo de la charla es presentar un método para analizar la estructura de fases de distribuciones de ceros de polinomios ortogonales en el límite n → ∞. Un concepto fundamental en este método es la propiedad S, que permite relacionar nuestro problema con un problema de minimización y conduce al concepto de curva espectral. A partir de la curva espectral caracterizamos los arcos en los que se concentran los ceros (cortes) como líıneas de Stokes de un polinomio. Como aplicación del método, veremos cómo es la estrucura de fases y estudiaremos procesos críticos (cambios en el número de cortes) para un modelo cúbico.

Session 12:45. Andrei Martínez Finkelshtein, Department of Mathematics, University of Almería.
Title: Phase transitions and equilibrium measures
Abstract: The large-scale behavior of many models in mathematics and physics, such as a unitary random ensemble, the non-intersecting Brownian paths, or the asymptotics of orthogonal polynomials with respect to a varying weight, are described in terms of a measure solving an extremal problem of the logarithmic potential theory. This measure (the equilibrium measure in an external field) provides a crucial information: the associated functionals give us the leading terms of the asymptotics, and its support is typically the place where oscillations occur. In particular, the phase transitions in the random matrix models are associated to the change of the topology and connectivity of the support of the equilibrium measure under variation of the external field.

In a rather broad class of problems the potential (or the external field) on the real line is given by a polynomial. Much has been written about the phase transitions for the polynomial potentials. In this talk we show that the main known (and a few unknown) facts can be derived in a unified fashion from two basic properties of the equilibrium measures. As an illustration, we discuss in more detail the case of the quartic external field, focusing on the possible transitions between different configurations of the limiting spectrum under the variation of the total mass of the measure.

This is a joint work in progress with E.A. Rakhmanov and R. Orive.
See photo here.


Friday, March 15, 2013
Venue: A-26 classroom of the Faculty of Sciences, University of Granada

Session 12:00. Daniel Rivero, Department of Mathematics, UC3M.
Title: Edge Detection and Orthogonal  Polynomials
Abstract: Discrete orthogonal polynomials are useful tools in digital image processing to extract visual object contours in different application contexts.This paper proposes a method that extends beyond classic first-order differential operators, by using the properties of Hahn orthogonal polynomials and analysis of zero crossings to achieve a second order differential operator with higher order precision. The novel aspects of this research work are twofold: Theoretically, it is demonstrated that Hahn Polynomials provide a better approximation to the second derivative directional  than the widely used discrete Chebyshev orthogonal polynomials. Experimentally, we provide simulation results which prove that the proposed method achieves superior performance in comparison with commonly used algorithms.

Session 13:00. Teresa E. Pérez, Department of Applied Mathematics, University of Granada.
Title: On linearly related orthogonal polynomials in several variables
Abstract: Given two monic polynomial systems in several variables satisfying a matrix linear relation, we study two problems. First, if both polynomial systems are orthogonal, to characterise when that linear relation exists in terms of their moment functionals. Second, if one of the polynomial systems is orthogonal, to study when the other one is also orthogonal. Finally, some non trivial examples are presented, as well as linear relations between adjacent families of multivariate classical orthogonal polynomials.

This is a joint work with M. Alfaro, A. Peña, and M. L. Rezola.
See photo here



Friday, April 12, 2013
Venue: A-26 classroom of the Faculty of Sciences, University of Granada

Session 12:00. Amilcar Branquinho, University of Coimbra.
Title: Matrix Sylvester equations in the theory of orthogonal polynomials
Abstract: In this talk we characterize sequences of polynomials on the real line, orthogonal with respect to a linear  functional such that its corresponding Stieltjes function  satisfies a Riccati differential equation with polynomial  coefficients, in terms of matrix Sylvester differential and also difference equations. Furthermore, under certain conditions, we give  a representation of such sequences in terms of  semi-classical orthogonal polynomials on the real line.  For the particular case of semi-classical orthogonal  polynomials on the real line, a characterization in  terms of first order differential systems is established.

We also derive discrete dynamical systems, obtained as a  result of deformations of the recurrence relation coefficients of the orthogonal polynomials related to the above referred Stieltjes functions.

Sesión 13:00. Rosario González Férez, University of Granada.
Title: Impact of combined electric and non-resonant laser fields on polar molecules
Abstract: We present a theoretical study of the impact of combined electric and non-resonant laser fields on the rotational dynamics of polar molecules. We assume that the electronic and vibrational motions of the molecules are not affected by the external fields, and analyze the rotational motion using the rigid rotor approach. In addition, we assume that the rotational period of the molecule is much smaller than the laser pulse duration, and work within this adiabatic approximation using a constant pulse envelope for the laser field. Then, we solve the time-independent Schrödinger equation by means of a basis set expansion in terms of the associated Legendre functions or spherical Harmonics. We discuss the field-dressed rotational motion in terms of the alignment and orientation along the field axes. We refine our physical model and take into account the time profile of the non-resonant laser pulse. Thus, we solve the time-dependent Schrödinger equation by using the split-operator method. We compare our theoretical predictions to the results of the mixed-field-orientation experiments. The experimental results are rationalized in terms of non-adiabatic phenomena due to the field-induced couplings between states. We demonstrate that the adiabatic criteria "the laser pulse duration being longer than the molecular rotational period" is not correct. Based on our numerical results, we provide a new definition of this adiabatic limit and its physical interpretation.
See photo here



Friday
, May 17, 2013
Venue: Seminario de Matemática Aplicada, University of Almería

Session 12:00. Enrique de Amo Artero, Department of Mathematics, University of Almería.
Title: Una visión general sobre Cópulas
Abstract: Las cópulas $C$ son funciones de $[0, 1]^N$ en $[0, 1]$ que verifi…can propiedades muy inocentes (se trata de funciones Lipschitzianas sometidas a condiciones de frontera muy estrictas) y que, sin embargo, están resultando de gran interés tanto en sí mismas como por sus aplicaciones, sobre todo, gracias a su conexión con las funciones $H$ de distribución multidimensionales y su relación con las correspondientes funciones de distribución marginales $H1, ..., HN$; a través de la fórmula (Abe Sklar, 1959):

                            $$ H (x) = C ( H1 (x1) , ... , HN (xN) ),∀x ∈ [-∞, +∞]^N $$

Es a partir de los años 90 del pasado siglo cuando las cópulas han experimentado un expectacular auge gracias a su aplicación, en particular, al mundo de las …finanzas.
Pretendemos estudiar las propiedades generales de las cópulas, presentaremos a las familias de cópulas más conocidas en la literatura, y profundizaremos en algunos de los aspectos que actualmente centran la atención de quienes trabajan en Teoría de Cópulas.

Session 13:00. Andrei Martínez Finkelshtein, Department of Mathematics, University of Almería.
Title: Quadratic differentials for Laguerre polynomials with complex coefficients
Abstract: See here
See photo here.

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