SEMINARIO DEL GRUPO DE INVESTIGACIÓN

Teoría de Aproximación y Polinomios Ortogonales

Curso 2013-2014:



Thursday,
September 12, 2013
Venue: Seminario de Matemática Aplicada, University of Almería

Session 12:00. Alexander I. Aptekarev, Keldysh Institute of Applied Mathematics in Moscow.
Title: Brownian bridges and multiple orthogonal polynomial ensembles
Abstract: See here
See photo here



Thursday, September 26, 2013
Venue: Seminario de Matemática Aplicada, University of Almería

Session 11:30. Cleonice F. Bracciali, Universidade Estadual Paulista, SP, Brazil.
Title: A class of orthogonal functions given by a three term recurrence formula
Abstract: See here

Session 13:00. Guilherme L. F. Silva, KULeuven, Belgium.
Title: S-curves and (non-hermitian) orthogonal polynomials
Abstract: See here
See photos: 1  2



Friday, November 29, 2013
Venue: Seminario 1ª Planta, Instituto de Matemáticas, University of Granada.

Session 11:30. Yuan Xu, University of Oregon.
Title: Approximation and Sobolev Orthogonal Polynomials on Unit Ball
Abstract: For the spectral Galerkin method in numerical solution of partial differential equations, we need to understand the approximation by polynomials in the Sobolev spaces. For this purpose, it is necessary to study orthogonal structure of the Sobolev space $W_2^r$ that consists of functions whose derivatives up to $r$-th order are all in $L^2$. In this talk, we discuss new result on Sobolev orthogonal polynomials in $W_2^r$ for all positive integer $r$ and approximation in the Sobolev space on the unit ball in $\RR^d$, and describe sharp estimate for the error of best approximation in the Sobolev space and its application in the spectral Galerkin methods.

Session 12:45. Andrei Martínez Finkelshtein, University of Almería.
Title: Equilibrium problems related to asymptotics of Hermite-Padé polynomials
Abstract: Polynomials of multiple orthogonality (or Hermite-Padé polynomials) arise in many problems related to number theory, random matrix theory and stochastic processes, to mention a few. It became clear recently that understanding their asymptotic is crucial for solving many open problems in these areas. It is well-known that the logarithmic potential and in particular, equilibrium measures, provide the description of the leading term of asymptotics for standard orthogonal polynomials. However, the situation is much more difficult and less clear for the Hermite-Padé polynomials, where the general theory is still to be created. I will explain some ideas about the vector equilibrium for these polynomials, based on the work of Gonchar, Rakhmanov, Aptekarev and others.
See photo here



Friday, December 13, 2013
Venue: Aula M-01, Faculty of Science, University of Granada.
Time: 11:30

Ramón Orive, University of La Laguna.
Title: Dinámica de la medida de equilibrio en presencia de campos externos analíticos
Abstract: Nuestro interés en el comportamiento de la medida de equilibrio con soporte en el eje real en presencia de un campo externo real analítico proviene de su inmediata aplicación al estudio asintótico de polinomios ortogonales en el eje real. Para nuestra sorpresa, comprobamos que existen otros muchos campos de la Matemática–Física en los cuales dicho problema tiene inportantes aplicaciones. El más destacado de todos ellos es el de las Matrices Aleatorias, pero no menos importantes son las aplicaciones al estudio del límite continuo de las soluciones de la ecuación en derivadas parciales de Korteweg–de Vries (KdV) o el estudio asintótico, íntimamente relacionado con el anterior, de los sistemas de ecuaciones difero–diferenciales conocidos en la literatura como Toda lattices.
Es bien sabido que la analiticidad del campo externo garantiza que el soporte de la medida de equilibrio está formado por un número finito de intervalos. En un reciente trabajo en colaboración con A. Martínez Finkelshtein y E. A. Rakhmanov analizamos en detalle lo que sucede en el caso de un campo externo polinomial, especialmente, en el caso “cuártico” (el caso no trivial más simple). En ese trabajo le prestamos especial atención al comportamiento de la energía de equilibrio (o energía libre de volumen infinito, en el contexto de las matrices aleatorias) cuando al variar la masa total de la medida, t ∈ (0, +∞) (en otros contextos, el tiempo o la temperatura), se producen los denominados “cambios de fase” (generalmente asociados a cambios en el número de intervalos que componen el soporte, aunque no siempre).
En esta charla, revisamos las claves fundamentales de estos resultados y analizamos posibles extensiones a tipos más generales de campos externos: el caso de polinomios de grado ≥ 6, en el cual pueden producirse cambios de fase simultáneos (y el número de intervalos puede crecer o decrecer en dos o más unidades), y el caso de campos externos racionales (es decir, con derivada racional).

Miguel Piñar, University of Granada.
Title: Twovariable analogues of Jacobi polynomials on the parabolic triangle
Abstract: We study two–variable Jacobi polynomials on the parabolic triangle, that is, the closed region limited by a parabola and a straight line.
Using the Koornwinder's addition formula for Jacobi polynomials we deduce old and new representation formulae for the corresponding kernels. As aconsequence, asymptotic results for the Christoffel functions are obtained.
See photo here



Friday
January 31, 2014

Venue: Seminario de Matemática Aplicada, Building CITE III, University of Almería
Time: 12:15

Inmaculada López García, University of Almería.
Title: Modelo de simulación aplicado al control biológico de plagas mediante especies entomófagas en invernaderos comerciales de tomate
Abstract: Nuestro objetivo ha sido encontrar un modelo dinámico que describa el efecto de la aplicación simultánea del parasitoide de huevos Trichogramma achaea y el depredador Nesidiocoris tenuis para el control de la plaga sudamericana de la polilla del tomate (Tuta absoluta). Obtuvimos que un sistema del tipo Lotka-Volterra se podía ajustar bien a los datos, estimando la tasa de crecimiento de la plaga, las tasas de mortalidad del parasitoide y del depredador, las tasas de parasitismo y la tasa de emergencia del parasitoide. El mecanismo de control biológico con dos agentes se puede aplicar para determinar tasas y tiempo óptimos de las sueltas de los agentes parasitoide y depredador en cultivos comerciales en invernadero.

Herbert Dueñas Ruiz, Universidad Nacional de Colombia.
Title: Polinomios Ortogonales de Tipo Laguerre Sobolev. Una Aproximación al caso más general
Abstract: See here
See photo here



Tuesday, May 13, 2014
Venue: Seminario de Matemática Aplicada, Building CITE III, University of Almería
Time: 11:00

Wojciech Okrasinski, Institute of Mathematics and Computer Science, Wroclaw University of Technology, Polonia.
Title: Comments on a  mathematical model of the corneal shape
Abstract: We present some recent results (2010 – 2014) concerning a simple mathematical model of the corneal shape. Approximate solutions to that model are in a good accordance with the experimental data.

D. Robert Iskander, Institute of Biomedical Engineering and Instrumentation, Wroclaw University of Technology, Polonia.
Title: Human Eye – an example of a dynamic “robust” system
Abstract: The eye, despite being extraordinarily well-developed through evolution, is not an ideal optical system and produces retinal images of moderate quality. On the other hand, simple, textbook comparison of the eye to a man-built optical instrument, such as a camera, for example, is totally unjust. The eye is a complicated, dynamic and robust optical system. The image of a distant object needs to travel through a thin few micrometers thick layer of tear film, cornea, aqueous humour, pupil, crystalline lens with a gradient refractive index, before it falls on the photosensitive retina at the back of the eye. All of the eye elements, which the rays that form the image of a distant object pass, are dynamic in nature and are not fully synchronized with each other.
 
All of the eye’s elements together with signals associated with accommodation, pupillary response, and those of cardio-pulmonary system lead to a static retinal image that is far from ideal at a given point of time. However, due to the dynamic nature of all those elements, the human eye is able to resolve images with high acuity of about one minute of arc mainly because of the match between moderate optical image quality and the resolution of the retinal mosaic. The actual image that a human being can resolve also depends on neural processes occurring in the retina and the brain. Those could involve elements of stochastic resonance where the naturally occurring eye vibrations may result in an improved visual quality.
 
Some aspects of the dynamics in optical characteristics of the eye, particularly those related to changes in the tear film structure and microfluctuations of the steady-state accommodation, have been considered. The latter can be associated with the refractive status of the eye from which emmetropization mechanisms (i.e., processes that optimize the eye’s optics) could be learnt. However, the wealth of information available when examining the dynamic nature of optical characteristics of the eye has not been fully exploited. Challenges exist to develop analytic methodologies that would adequately account for all such variations and their interdependencies with the major physiological signals of the human body. To achieve this goal, an overall system of the eye’s optics that takes into account the aberration dynamics needs to be considered. 
 
The talk will summarize recent endeavors undertaken in search of adequate characterization of the dynamics encountered in human eye’s optics.



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