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 ofComputerScienceand Statistics,SimonBolivarUniversity,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, Departmentof
Applied Mathematics,University
ofGranada.
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: 12
Friday, November
30, 2012
Venue: Sala de Grados del Aulario IV, University ofAlmería Organization:
This isa special sessionof
the seminarorganizedby
the Department ofMathematicsof the University ofAlmería and theResearch Group"ApproximationTheory andOrthogonal
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: 12
Friday, January 18, 2013
Venue: M1classroomof the Faculty
ofSciences,University
of Granada
Session 11:30. Elena Blanca Medina Reus,
Department
of Mathematics,University
ofCádiz.
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-26classroomof the Faculty
ofSciences,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, Departmentof
Applied Mathematics,University
ofGranada.
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-26classroomof the Faculty
ofSciences,University
of Granada
Session 12:00. Amilcar Branquinho,
University ofCoimbra.
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 ofGranada.
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.