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: Two–variable 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.
©
1998-2011 Grupo de Investigación Teoría de Aproximación y
Polinomios Ortogonales.