### Local Informations

From January 20 2009 you are the visitor:
 Today 51 Yesterday 119 This week 526 This month 3568
Top 10:
 21% Germany 18% Spain 17% United States 15% China 4% Ukraine 2% Mexico 2% France 2% Russian Federation 1% Colombia <1% Brazil
from 164 countries
Weekly Seminar
 WS Season 2010/2011
Written by Seminario
Wednesday, 08 September 2010 20:43

# Seminar on Orthogonality, Approximation Theory and Applications

Upcoming seminars

Past seminars

1. Thursday, June 30, 2011. 3:00 pm. Room 2.2.D08

Speaker: Luis Santiago, Universidad Autónoma de Barcelona

Title: Finite dimensional continued fractions and integrable systems

Abstract: In this talk I will define a continued fraction algorithm for infinite dimensional vectors. I will explain how this continued fraction is related to problems in rational approximation and how it can be applied to find the solutions of certain integrable systems that generalize the nonabelian Toda lattice.

2. Thursday, June 30, 2011. 4:00 pm. Room 2.2.D08

Speaker: Mirta Castro Smirnova, Universidad de Sevilla

Title: Interlacing properties of the zeros of matrix polynomials.

Abstract: We consider matrix polynomials defined by

$$P_n(x)=A_0p_n(x)+A_1p_{n-1}(x)+\ldots A_rp_{n-r}(x),$$

where $(p_n)_{n\geq 0}$ is a sequence of monic polynomials orthogonal with respect to a positive weight function $w(x)$ supported on an interval of the real line, and $A_i$, $i=1,\ldots r$, are matrix coefficients. These polynomials are said to be quasi-orthogonal of order $r$ with respect to $w$. Using some particular examples we discuss the interlacing properties of the zeros of the polynomials $p_n(x)$ and $P_n(x)$. This is a joint work with Pablo Román, from Catholic University of Leuven, Belgium.

3. Friday, June 24, 2011. 4:00 pm. Room 2.2.D08

Speaker: Luis Velázquez, Universidad de Zaragoza

Title: From classical to quantum recurrence: an orthogonal polynomial (OP) approach.

Abstract: The study of return properties of random walks is a topic which goes
back at least to G. Pólya in 1921. He found that any unbiased random walk in dimesion not greater than two is recurrent, i.e., it eventually returns to its original position with probability one. This surprising result, which is no longer true in higher dimensions, holds in spite of the fact that the return probability in n steps converges to zero as n goes to infinity. The return properties of a general biased random walk become more intricated and in their analysis the
connection between random walks and OP on the real line plays an essential role.

Recently it has been proposed a quantum version of Pólya recurrence, i.e, a definition of recurrence for the so called quantum walks. The study of return properties in the quantum case raises special issues because a quantum measurement destroys the initial evolution. Therefore, the notion of a return to a given state for the first time
has to be interpreted with care in the quantum case. Apart form these conceptual problems, the proposed quantum analog of Pólya recurrence seems more difficult to characterize than its classical counterpart. This is specially true for the expected return time, so important as recurrence itself: having return probability equal to one does not mean too much if the expected return time is infinite.

In this talk we will present the classical and quantum versions of Pólya recurrence, and we will see that both of them can be studied using standard tools of OP on the real line and the unit circle respectively. Moreover, we will show the drawbacks of the proposed definition of quantum recurrence, and we will introduced a new one which overcomes in a better way the problem with quantum measurements. Further, we will see that this new definition, not only solves in a
better way the conceptual problems behind quantum recurrence, but also provides us with better connections to the theory of OP the unit circle. This makes possible, for instance, the OP analysis of the expected return time, which turns out to be directly connected to the radial behaviour of the Schur function related to the orthogonality
measure.

This is a joint work with Reinhard Werner, Albert Werner (Institut für Theoretische Physik, Leibniz Universität Hannover) and F. Alberto Grünbaum (Department of Mathematics, UC Berkeley)

4. Thursday, June 9, 2011. 4:00 pm. Room 2.2.D08

Speaker: David Gómez-Ullate Oteiza, UCM

Title: Exceptional orthogonal polynomials and algebraic Darboux transformations

Abstract: We will review some recent results on exceptional orthogonal polynomials. These are complete sets of orthogonal polynomials which arise as solutions of a Sturm-Liouville problem, thus extending the classical families of Hermite, Laguerre and Jacobi [1,2].
In particular, we will show how these families can be obtained from the classical ones by means of an algebraic Darboux transformation [3], a particular class of Darboux transformations that preserve the polynomial character of the eigenfunctions [4,5].
The existence of this connection allows one to derive many properties of the new families by transforming the equivalent properties of the classical ones (Rodrigues formula, generating function, etc.).
An iteration of algebraic Darboux transformations (Darboux-Crum) gives rise to new families of exceptional polynomials [6], a complete classification of the whole class remains an open question. An investigation of the distribution and asymptotics of the zeroes of the new families is also an open question. Time permitting, some preliminay results will be given.

[1] D.G-U, N. Kamran and R. Milson, An extension of Bochner's problem: Exceptional invariant subspaces, J. Approx. Theor. 162 (2010) 987
[2] D.G-U, N. Kamran and R. Milson, An extended class of orthogonal polynomials defined by a Sturm-Liouville problem, J. Math. anal. Appl. 359 (2009) 352
[3] D.G-U, N. Kamran and R. Milson, Exceptional orthogonal polynomials and the Darboux transformation, J. Phys. A 43 (2010) 434016
[4] D.G-U, N. Kamran and R. Milson, The Darboux transformation and algebraic deformations of shape-invariant potentials, J. Phys. A 37 (2004) 1789
[5] D.G-U, N. Kamran and R. Milson, Supersymmetry and algebraic Darboux transformations, J. Phys. A 37 (2004) 10065
[6] D.G-U, N. Kamran and R. Milson, Two-step Darboux transformations and exceptional Laguerre polynomials,
http://arxiv.org/abs/1103.5724

5. Thursday, May 26, 2011. 4:00 pm. Room 2.2.D08

Speaker: Marc Van Barel, Katholieke Universiteit Leuven (Belgium)

Title: Large eigenvalue problems and spectral clustering

Abstract: In unsupervised learning one of the tasks is clustering. One is given a set of objects and a way to measure the similarity between each pair of objects. Clustering consists in partitioning the set of objects, i.e., assigning each of the objects to a cluster so that the similarity of two objects belonging to the same cluster is large while it is small when the two objects belong to different clusters.The similarity matrix contains on row $i$ and column $j$ the similarity between object $i$ and object $j$. When performing spectral clustering on large data sets, one has to look for the eigenvectors corresponding to the largest eigenvalues of this matrix whose size depends on the number of data points. This number can be huge. A possibility to avoid solving this large eigenvalue problem is to approximate the similarity matrix by a low rank matrix and then solve the reduced eigenvalue problem. In this talk we will investigate how we can adaptively look for a suitable rank for this approximation. The study will be illustrated by numerical examples. For an overview of spectral clustering methods, we refer the interested reader to [2]. More details about the method described in this talk are given in [1].

[1] U. von Luxburg. A tutorial on spectral clustering. Statistics and Computing, 17:395-416, 2007.
[2]
K. Frederix and M. Van Barel. Sparse spectral clustering method based on the incomplete Cholesky decomposition. TW Reports TW552, Department of Computer Science, Katholieke Universiteit Leuven, November 2009.

6. Thursday, May 19, 2011. 4:00 pm. Room 2.2.D08

Speaker: Junior Michel, UC3M

Title: Hiperbolicidad de grafos en sentido de Gromov

7. Thursday, May 12, 2011. 4:00 pm. Room 2.2.D08

Speaker: Mattia Cafasso, CRM Montréal (Canada)

Title: Riemann-Hilbert  problems and applications

Abstract: Riemann--Hilbert boundary value problems have been used as an effective tool for the computation of Fredholm determinants since the nineties. At the core of the relation between Riemann--Hilbert problems and Fredholm determinants there is the notion of integrable operators, introduced by Its-Izergin-Korepin and Slavnov more then 20 years ago. I will start my talk with a brief introduction of this topic. Then, following some recent join works with Marco Bertola, I will show some applications to the theory of random matrices and non--intersecting Brownian motions.

8. Thursday, April 28, 2011. 4:00 pm. Room 2.2.D08

Speaker: Jean-Marie Vilaire, UC3M

Title: Gromov hyperbolic tessellation graphs

Abstract: A geodesic triangle T in a geodesic metric space X is the union of the three geodesics in X. The space X is $\delta$-hyperbolic (in the Gromov sense) if any side of T is contained in a $\delta$-neighborhood of the union of the two other sides, for every geodesic triangle T in X. In this paper we obtain criteria which allow either guarantee or discard the hyperbolicity of a large kind of graphs: our main interest are the planar graphs which are the "boundary" (the 1-skeleton) of a tessellation of the Euclidean plane; however, we also obtain results about tessellations of general Riemannian surfaces with a lower bound for the curvature. Surprisingly, these results on Riemannian surfaces are the key in order to obtain additional results about tessellations of the Euclidean plane.

9. Thursday, April 14, 2011. 4:00 pm. Room 2.2.D08

Speaker: Jorge Borrego Morell, UC3M

Title: Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limits

Abstract: We introduce a new  family weight matrices $W(t)$ of arbitrary size whose associated orthogonal polynomials satisfy a second order differential equation with matrix polynomial differential coefficients. For size $2\times 2$, we find an explicit expression a sequence of orthonormal polynomials with respect to $W(t)$ and we present also a Rodrigues formula. In particular, we show that one of the recurrence coefficients for this sequence of orthonormal polynomials does not asymptotically behave as a scalar multiple of the identity, as it happens in the examples studied up to now in the literature (Joint wok with M. Castro and A. Durán).

10. Thursday, March 31, 2011. 4:00 pm. Room 2.2.D08

Speaker: Guillermo López Lagomasino, UC3M

Title: Analytic version of the Poincaré Theorem on finite differences

Abstract: We will review the scalar and vector versions of the Poincaré Theorem. This theorem plays a cental role in the study of ratio asymptotic properties of sequences of polynomials satisfying some recursion formula. The Poincaré Theorem allows to obtain pointwise convergence. In applications one usually desires uniform convergence on compact subsets. We present a result which gives general criteria in order that uniform convergence takes place.

11. Thursday, March 24, 2011. 4:00 pm. Room 2.2.D08

Speaker: Stefano Serra Capizzano, Dipartimento di Fisica e Matematica, Universitá dell'Insubria - sede di Como

Title: Toeplitz operators with matrix-valued symbols and some (unexpected) applications

Abstract: We discuss the eigenvalue distribution in the Weyl sense of general matrix-sequences associated to a symbol. As a specific case we consider Toeplitz sequences generated by matrix-valued (non Hermitian) bounded functions. We show that the canonical distribution can be proved under mild assumptions on the spectral range of the given symbol. Finally some applications are introduced and discussed.

12. Thursday, March 17, 2011. 4:00 pm. Room 2.2.D08

Speaker: Roberto S. Costas Santos, Universidad de Alcalá

Title: Orthogonality of q-polynomials for nonstandard parameters

Abstract: q-polynomials can be defined for all the possible parameters but their orthogonality
properties are unknown for several configurations of the parameters. Indeed, the orthogonality
for the Askey-Wilson polynomials, pn(x; a, b, c, d; q) is known only when the product of any two parameters a, b, c, d is not a negative integer power of q.

In this talk we present orthogonality properties for the Askey-Wilson polynomials for the rest of the parameters and for all n.

13. Thursday, March 3, 2011. 4:00 pm. Room 2.2.D08

Speaker: Bujar Fejzullahu, University of Prishtina, Kosovo

Title: On orthogonal expansions with respect to the  generalized Jacobi weight

Abstract: We will present some new results on orthogonal expansions with respect to the generalized Jacobi weight $$(1-x)^\alpha(1+x)^\beta h(x)\prod_{i=1}^m |x_i-x|^{\nu_i}$$, where $-1<\dots-1$ ($i=1,\dots,m$), and $h$ is real analytic and strictly positive on $[-1, 1].$ The Cohen-type inequality as well as the Lebesgue constants for the Fourier expansions with respect to the generalized Jacobi weight will be discussed. Finally, we show that, for certain indices $\delta,$ there are functions whose Cesàro means of order $\delta$ in the 4eneralized Fourier-Jacobi expansions are divergent a. e. on the interval $[-1,1]$.

14. Thursday, February 24, 2011. 4:00 pm. Room 2.2.D08

Speaker: Yamilet Quintana, Universidad Simón Bolívar (Venezuela)

Title: A numerical study of a mimetic scheme for the unsteady heat equation

Abstract: A new mimetic scheme for the unsteady heat equation is presented. It combines recently developed mimetic discretizations for gradient and divergence operators in space with a Crank- Nicolson approximation in time. A comparative numerical study against standard finite difference shows that the proposed scheme achieves higher convergence rates, better approximations, and it does not require ghost points in its formulation. This is a joint  work with I. Mannarino an J. M. Guevara-Jordán.

15. Thursday, February 17, 2011. 5:00 pm. Room 2.2.D08

Speaker: Alexander Aptekarev, Keldysh Institute of Applied Mathematics, Moscow (Russia)

Title: Strong asymptotics for discrete orthogonal polynomials and C-D kernels

Abstract: Christoffel - Darboux (C-D) kernel plays important role in various applications of orthogonal polynomials (OP), for example it represents correlation functions for the determinantal random processes. As a rule, asymptotics of C-D kernel possess an universal character. To find the limiting C-D kernel one should have a good machinery for strong asymptotics of OP. The last years several techniques for strong asymptotics of discrete OP. We shortly describe one of them taking example of Meixner polynomials. Then we discuss various new limiting C-D kernels, which appear for the discrete OP.

16. Thursday, February 10, 2011. 4:00 pm. Room 2.2.D08

Speaker: Stephanos Venakides, Duke University

Title: The steepest-descent method for Riemmann-Hilbert problems: the case of the  focusing nonlinear Schroedinger equation (NLS).

Abstract: We apply the steepest-decent method to the  Riemann-Hilbert problem that describes the inverse scattering corresponding to the focusing nonlinear Schrödinger equation (NLS).  Steepest descent applies in an asymptotic limit, in which NLS displays its dispersive character particularly well. The initial profile breaks into fully nonlinear modulated oscillations in the small space and time scales. The oscillations are often multi-phase. The g-function mechanism is utilized in determining an exactly solvable model Riemann-Hilbert problem, to which the original Riemann-Hilbert problem is reduced. Due to the non-self-adjointness of the linear operator which underlies the linearization of NLS and to which the inverse scattering refers, the contour of the model Riemann-Hilbert problem in the complex plane is off the real axis. The boundary of the applicability of the method in parameter space (space-time) will be discussed.

17. Thursday, January 20, 2011. 4:00 pm. Room 2.2.D08

Speaker: Domingo Pestana, UC3M

Title: Expanding maps and shrinking targets.

Abstract: We shall show that uniformly expanding maps in a metric space hit shrinking targets with Borel-Cantelli regularity, although, in general, uniformly expanding maps just have a countable partition (not necessarily finite) and they do not satisfy strong Borel-Cantelli results. With related techniques, one obtain also results for Markov transformations, one-sided topological Markov chains over a countable alphabet with a  Gibbs measure, and some non-uniformly expanding maps.

18. Thursday, January 13, 2011. 4:00 pm. Room 2.2.D08

Speaker: Alfredo Deaño, UC3M

Title: From oscillatory integrals to a cubic random random matrix model

Abstract: In this talk I will present some results on complex orthogonal polynomials that arise in the application of Gaussian quadrature to certain integrals obtained by the classical method of steepest descent. The zeros of these complex orthogonal polynomials are optimal nodes for the computation of oscillatory integrals with high order stationary points defined on the real axis. In the case of a cubic-type potential, it is possible to analyze the asymptotic behavior of these orthogonal polynomials and their zeros using Riemann-Hilbert techniques. Similar ideas can be used to study the partition function and the free energy of the corresponding cubic random matrix model. (Joint and ongoing work with P. Bleher, D. Huybrechs, and A. B. J. Kuijlaars).

19. Thursday, December 16, 2010. 4:00 pm. Room 2.2.D08

Speaker: Héctor Pijeira, UC3M

Title: Sobre Polinomios ortogonales y ecuaciones diferenciales desde un punto de vista no estándar

Abstract: En la primera parte de la charla se hace un recorrido por diversas aplicaciones de los polinomios ortogonales de interés actual. En la segunda parte se introduce la ortogonalidad respecto a operadores diferenciales, se analizan los resultados alcanzados recientemente así como su interés para las aplicaciones en el análisis numérico y la mecánica de fluidos.

20. Thursday, December 9, 2010. 4:00 pm. Room 2.2.D08

Speaker: Daniel Peralta-Salas, CSIC

Abstract: In 1965 V.I. Arnold classified the steady solutions of the Euler equation, implying in particular that the types of knots and links that the stream (or vortex) lines of a fluid can exhibit are very restricted except for the so called Beltrami fields. Arnold's work gave rise to the topological hydrodynamics conjecture that any knot and link can be realized as a stream (or vortex) line of a steady (Beltrami) solution of the Euler equation. The importance of this conjecture is that it tests the topological complexity of fluid flows and hence it is directly related to phenomena like turbulence and hydrodynamics instability. The goal of this lecture is to review the strategy which has recently led to the proof of this conjecture (with A. Enciso), as well as some interesting applications like the solution to the Etnyre-Ghrist's problem: there exists a steady solution of the Euler equationcontaining all knot and link types.

21. Thursday, December 2, 2010. 4:00 pm. Room 2.2.D08

Speaker: Juan M. Pérez Pardo, UC3M

Abstract: Self-adjoint operators are the main objects of Quantum Mechanics. The self-adjoint extensions of symmetric operators are completely characterized by von Neumann's Theorem, however this theory is too abstract in many practical cases. For differential operators, in particular for the Laplace-Beltrami operator, one can construct a completely equivalent theory of self-adjoint operators dealing only with boundary conditions. In addition, one can associate to every self-adjoint operator a quadratic form which, for instance, is useful to obtain the spectrum of the operator and to find numerical solutions of it.

In this talk I will introduce these topics and show how one can characterize the self-adjoint extensions of a symmetric operator directly in terms of the Friedrichs' extensions of closable quadratic forms. I will finally mention some applications of our results.

22. Thursday, November 18, 2010. 4:00 pm. Room 2.2.D08

Title: Conjunto de valores asintóticos de funciones meromorfas en C

Abstract: El teorema de Liouville establece que una función holomorfa no constante) definida en todo el plano complejo, C, no puede ser cotada. En el lenguaje que usaremos, este enunciado se puede reformular diciendo que el infinito es siempre valor asintótico de funciones holomorfas enteras. Un punto es valor asintótico de una función holomorfa (o meromorfa) definida en C si es el límite de dicha función a lo largo de una curva continua que llega al infinito. El conjunto de valores asintóticos de una función entera refleja el comportamiento cerca del infinito de dicha función, es decir, el comportamiento en la frontera de su dominio de definición.

En un trabajo realizado con David Drasin y Ana Granados completamos la caracterización de los conjuntos de valores asintóticos de funciones enteras (en este contexto, holomorfas y meromorfas). En esta charla se
hará un recorrido por los resultados clásicos más relevantes en este tema con el objetivo de presentar los conceptos que intervienen en dicha caracterización y se darán unas ideas de la demostración (de la
suficiencia) de dicha caracterización.

23. Thursday, November 11, 2010. 4:00 pm. Room 2.2.D08

Title: Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces

Speaker: Eva Tourís Lojo, UCIIIM

Abstract:
It is interesting to study conditions which determine when a given complete Riemanian surface S is Gromov hyperbolic. In order to do it, the main goal of this work is to get graph-structures G, which are good models for surfaces and, in this way, moving the study of Gromov hyperbolicity from the surface to its associated graph, whose structure is very much simpler and, therefore, to study Rips condition shall be easier.
Gromov hyperbolicity is of quite interest in metric graphs theory since it is closely related to concepts arising in the study of trees: in fact, we can consider hyperbolic graphs as a generalization of metric trees.
More precisely, in this work I obtain the equivalence of the Gromov hyperbolicity between an extensive class of complete Riemannian surfaces with pinched negative curvature and certain kind of simple graphs, whose edges have length 1, constructed following an easy triangular design of geodesics in the surface.

24. Thursday, November 4, 2010. 4:00 pm. Room 2.2.D08

Title: Some new results on analytic properties and zeros of Laguerre-type Orthogonal Polynomials

Speaker: Edmundo José Huertas Cejudo, UCIIIM

Abstract:
This contribution is devoted to the study of the Laguerre-type monic orthogonal polynomial sequences (MOPS, in short) defined by an Uvarov's canonical spectral transformation of the Laguerre weight supported on the positive semi-axis of the real line. In such a way, we state a comparative analysis with the behavior of the standard Laguerre-type polynomials, taking into account that, in our case, we are dealing with a mass point located outside the support of the measure.

The outline of the talk is the following. In the first part we introduce the representation of the perturbed MOPS in terms of the classical ones, we deduce the three term recurrence relation that they satisfy, as well as the behavior of their coefficients. Next, we obtain the lowering and raising operators associated with these polynomials, and thus the corresponding holonomic equation follows in a natural way. The second part of the talk is devoted to the study of the behavior of the zeros of these polynomials in terms of the mass M. We also provide an electrostatic interpretation of them.
Finally, we analyze the outer relative asymptotics as well as the Mehler-Heine formula for these polynomials.

25. Thursday, October 28, 2010. 4:00 pm. Room 2.2.D08

Title: The solution of the equation XA + AX^T = 0 and its application to the theory of orbits

Speaker: Fernando De Terán, UCIIIM

Abstract: We describe how to find the general solution of the matrix equation XA+AX^T = 0, with A \in C^{n \times n}, which allows us to determine the dimension of its solution space. This result has immediate applications in the theory of congruence orbits of matrices in C^{n\ times n}, because the set {XA + AX^T : X \in C^{n\times n}} is the tangent space at A to the congruence orbit of A. Hence, the codimension of this orbit is precisely the dimension of the solution space of XA + AX^T = 0. As a consequence, we also determine the generic canonical structure of matrices under the action of congruence.

26. Thursday, October 21, 2010. 4:00 pm. Room 2.2.D08

Title: A particular type of Matrix Padé Approximants inspired by Multivariate Time Series Models

Speaker: Dra Celina Pestano, Universidad de la Laguna

Abstract: In this talk we define a type of Matrix Padé Approximants inspired by the identification stage of multivariate time series models. The formalization of certain properties in the Matrix Padé Approximation framework not only have the value that can be applied to time series and other different fields. We really want to help study the complex problem of Matrix Padé Approximats.

27. Thursday, October 7, 2010. 4:00 pm. Room 2.2.D08

Title: Perturbations of Hankel and Hermitian Toeplitz matrices

Speaker: Kenier Castillo Rodríguez, UCIIIM

Abstract: We will present the more general perturbations that preserve basic structure in Hankel and Hermitian Toeplitz matrices i.e. perturbations anti-diagonal and sub-diagonal respectively. We will use the relationship between these structures and orthogonal polynomials for functionals that should
be represent by a positive measures with supported on straight line and the unit circle. We define the functional response to disturbance on the matrix concerned and give explicitly the relationship between the two orthogonal families.

28. Thursday, September 30, 2010. 4:00 pm. Room 2.2.D08

Title: On the Adler-van Moerbeke conjecture and generalized Jacobi matrices.

Speaker: A. Ibort, Dpt. of Mathematics, UCIIIM

Abstract: Adler and van Moerbeke conjectured that all sequences of polynomials satisfying 2m+1-step relations are given by generalized periodic sequences of weights. We will discuss spectral setting. To this end we will review the well-known case of ordinary Jacobi matrices and von Neumann theory of self-adjoint extensions of symmetric operators. Finally we will be able to prove that Adler-van Moerbeke conjecture is true for a class of generalized Jacobi matrices that extend class D Jacobi matrices.

Last Updated on Wednesday, 06 July 2011 09:07

More Articles...

Page 7 of 9
Friday, 26 May 2017 10:30:40