2010-2011 Academic Year:
Wednesday, June 15,
2011.
Venue:
Seminar of Applied Mathematics, University of Almería
Session: 11.30: Joaquín
Sánchez Lara (Granada University)
Title: Sobolev orthogonality for
classical polynomials
Abstract: The
families of classical orthogonal polynomials which do not verify the
hypotheses
of the Favards's theorem have been endowed with the Sobolev
orthogonality in the last years. We
will survey and discuss this orthogonality. Futhermore, we will
establish connection of these
orthogonality conditions with the gaussian quadrature formulas,
which allows to rewrite
them in a more explicit manner.
Session:
12.45: Manuel Calixto Molina
(Granada University)
Title: Extended
MacMahon-Schwinger's Master Theorem and
Relativistic Orthogonal Polynomials in Complex Minkowski Space
Abstract: We study an infinite set of "Relativistic
Polynomials" in a domain D_4
of the complex Minkowski space C4 (four complex variables). This
(Cartan) domain
is a homogeneous (quotient) space D_4=SO(4,2)/(SO(4)\times SO(2)) of
the conformal
(pseudo-orthogonal) group SO(4,2) [locally isomorphic to the
pseudo-unitary group
SU(2,2)] in 3+1 space-time dimensions. The conformal group SO(4,2)
contains the
Poincaré group (space-time translations, rotations and boosts) plus
accelerations
(inversions) and dilations. The 8-dimensional manifold D_4 can be
mapped one-to-one
onto the so-called "future tube domain C4_+" of the complex
Minkowski space through
a Cayley transformation. This set of polynomials constitutes
an orthonormal basis of the
Hilbert space of square integrable holomorphic functions on
D_4 with a certain weight
function W and "mass dimension" \lambda (an integer number). A
\lambda-extension of
the traditional MacMahon Master Theorem turns out to be a useful
mathematical tool for
us, particularly as a closure relation (reproducing-Bergman-kernel)
of Relativistic Orthogonal
Polynomials. We also prove the isomorphism (equivariance) between
the Hilbert space on
D_4 and on C4_+, where we enjoy more physical intuition. This
construction can be easily
extended to higher dimensions.
Wednesday, June 1, 2011.
Venue: Room A-11.1, Faculty of Sciences,
University of Granada
Session: 12.00: Pedro
Martínez González (University of Almería)
Title: Exceptional
Jacobi and Laguerre polynomials
Abstract: We present some of the most important results
obtained so far for the families of exceptional polynomials. The goal
is to
survey some of the properties (orthogonality, differential equation,
etc.) in order to find new ideas for further results in this area.
Session:
13.15: Andrei
Martínez-Finkelshtein (University of Almería)
Title: Best rational approximation of the
exponential function on the
semiaxis and saddle-points for the energy
Abstract: In 1987
Gonchar and Rakhmanov found a solution to the
problem of the rate of the best rational approximation of the function
exp(-x) on the positive semiaxis, related to the so-called
"1/9-conjecture". Besides the intrinsic value of this result (and its
unexpected connections with a 100-year old book of Halphen), the main
interest was the method of proof, which connected the asymptotic
degree of approximation with an equilibrium problem of the Green
potential in an external field on the plane. However, the existence of
the solution of such a problem in the paper of Gonchar and Rakhmanov
was established by means of elliptic functions, which is not
straightforward to extend to the case of approximation of
exp(-x^k), with integer k>1. In this talk we will outline some
ideas that pave the way from the exponential to the logarithmic
potential, as well as the possible scheme of proof for the general
case mentioned above. This is a joint work (in progress) with H.
Stahl.
Tuesday,
March 8, 2011.
Venue: Seminario
de Matemática Aplicada, Departamento de
Estadística y Matemática Aplicada, Universidad de Almería
Session: 16.00: Bujar Fejzullahu
(University of Prishtina, Kosovo)
Title: On orthogonal expansions with respect to the
generalized Jacobi weight
Abstract:
follow this link.
Friday,
February 18,
2011.
Venue: Seminario de
Matemática Aplicada, Departamento
de Estadística y Matemática Aplicada, Universidad de Almería
Session: 12.00: Ramón
Orive (Departamento de Análisis Matemático,
Universidad de La Laguna)
Title: Laguerre polynomials with varying
non-classical parameters. The critical case
Abstract:
follow this link.
Session 13.30: Heron
M. Felix (UNICAMP, Brasil)
Title: Numerical
Generation of the Nodes and Weights of a Gaussian Type Quadrature
Rule
Abstract:
follow this link.
Wednesday, February 16, 2011.
Venue:
Seminario de Matemática Aplicada,
second floor, Facultad de Ciencias, Universidad de Granada
Session: 12.00: Ramón
Orive (Departamento de Análisis Matemático,
Universidad de La Laguna)
Title: Some
equilibrium problems with applications in approximation theory
Abstract: follow this link.
Wednesday,
February 9, 2011.
Venue: Department of Statistics and Applied Mathematics,
University of Almería
Session: 16.30: Rafael
Navarro (Instituto de Óptica CSIC, Universidad
de Zaragoza)
Title: Wavefront
representation from discrete sampling
Abstract (in Spanish): Los polinomios de Zernike (PZ) son de gran utilidad en óptica
en general, y óptica visual en particular, por formar una base
completa y ortogonal en un círculo de radio unidad. En la práctica sin
embargo se tiene un conjunto discreto de muestras, y en el dominio
discreto los PZ pierden todas estas ventajas; más aún cuando las
medidas que proporcionan los sensores de frente de onda son
gradientes. Se proponen soluciones (que se validan experimentalmente)
a estos problemas utilizando muestreos no redundantes. Así mismo se
demuestra la igualdad entre el error refractivo y la curvatura del
frente de onda.
Tuesday, February 8, 2011.
Venue: Room A-4, Faculty of Sciences, Granada University
Session: 12.00: Rafael Navarro (Instituto de Óptica CSIC, Universidad de Zaragoza)
Title: The optical "design" of the eye
Abstract (in Spanish): Se revisa nuestro conocimiento actual de los principales
“principios de diseño” que podemos encontrar en el sistema óptico del
ojo, con sus aciertos y fallos (paradojas) más evidentes. Se discute
como las paradojas más importantes podrían tener una explicación en
distintas etapas de evolución.
Session: 13.15: Aixa Alarcón Heredia (Departamento de Óptica, Universidad de Granada)
Title: Visual assessment of different multi-focal corneal models
Abstract (in Spanish): Una de las características de nuestro sistema visual es la pérdida de
acomodación con la edad (presbicia). Se presenta un método para evaluar de
forma teórica diferentes algoritmos de ablación para corregir la presbicia
mediante cirugía refractiva laser, así como su efecto en la calidad visual
del paciente tras la cirugía.
Thursday, January 20, 2010.
Venue: Department of Statistics and Applied Mathematics, University of Almería
Session: 12.00: A. B. J. Kuijlaars (IK.U. Leuven, Bégica)
Title: Normal random matrix models.
Abstract: The Laplacian is the main operator describing the diffusion not only of heat and other physical substances, but also the diffusion in financial markets. We will start showing with very simple arguments the relation between random walks in Probability and the Laplace operator. This will lead us to understand the fruitful connection between the fundamental solution of the heat equation (the Gaussian) and the Central Limit Theorem for the propagation of random errors. Similar arguments apply for long-range or anomalous diffusions, such as the Lévy processes generated by the fractional powers of the Laplacian. They attract lately great interest in Physics,
Biology, and Finance.
We will then turn to some reaction-diffusion equations, involving the Laplacian or fractional Laplacians, and modeling phase transition problems. We will present recent developments that are strongly related to some classical results in the theory of minimal surfaces. Phase transitions or interfaces modeled by reaction-diffusion equations appear when two different states coexist and there is a balance between two opposite tendencies: a
diffusive effect that tends to mix the materials and a reaction mechanism that drives them into their pure
state. Due to surface tension, interfaces tend to minimize their area as the reaction becomes stronger.
Friday, December 10, 2010.
Venue: Auditorium A-14 of the Faculty of Science, Universidad de Granada
Session: 12.00: Aixa Alarcón Heredia (Departamento de Óptica, Universidad de Granada)
Title: Measure of the optical quality of the human visual system by means of the double-pass techniques.
Abstract (in Spanish): La calidad óptica del sistema visual humano está limitada por aberraciones y scattering. Para medir
de forma objetiva el efecto de éstos en la imagen retiniana se han desarrollado diferentes técnicas, divididas principalmente
en detectores de frente de onda y sistemas de doble paso. Los detectores de frente de onda, como el Hartmann-Shack, miden
las aberraciones del frente de onda de la luz reflejada en la retina. Aunque son el método más utilizado, están muy limitados
a la hora de detectar defectos relacionados con altas frecuencias espaciales, como el scattering. Los sistemas de doble paso
miden la imagen formada por el sistema visual a la salida. Son un método eficaz para determinar el scattering pero no detectan
correctamente las aberraciones. Ambas técnicas sobrestiman, por tanto, la calidad óptica de la imagen retiniana real. El método
presentado en este trabajo (que utiliza óptica adaptativa en un sistema de doble paso) determina de forma más fiable y exacta la
calidad óptica de la imagen retiniana que los métodos anteriormente propuestos, ya que no sólo permite medir el scattering sino
también las aberraciones del sistema visual, incluyendo las de alto orden.
Session: 13.15: Bernarda Soler Arias (Departamento de Estadística y Matemática Aplicada, Universidad de Almería)
Title: About an algorithm of S. Klein for determining the corneal topography
Abstract (in Spanish): la forma de la cornea puede determinarse proyectando sobre la misma algún patrón luminoso (por ejemplo,
anillos concéntricos) y estudiando la imagen reflejada sobre la superficie de la misma. Sin embargo, los algoritmos matemáticos
para dicha determinación, si bien usan matemáticas elementales, suelen ser muy diversos e implican errores que provienen tanto
de las suposiciones intrínsecas del método como de otras fuentes. Uno de los algoritmos más citados fuer propuesto por S.A. Klein
en 1992. El objetivo de esta exposición es dar a conocer las ideas fundamentales del procedimiento de Klein, al igual que sus posibles
limitaciones.
© 1998-2011 Grupo de Investigación Teoría
de Aproximación y Polinomios Ortogonales.