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 $$ P_n(x)=A_0p_n(x)+A_1p_{n1}(x)+\ldots A_rp_{nr}(x), $$ 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 quasiorthogonal 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 goesback 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 GómezUllate 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 SturmLiouville problem, thus extending the classical families of Hermite, Laguerre and Jacobi [1,2].
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, 17:395416, 2007.
6. Thursday, May 19, 2011. 4:00 pm. Room 2.2.D08 Speaker: Junior Michel, UC3M Title: Hiperbolicidad de grafos en sentido de Gromov Abstract: Si X es un espacio métrico geodésico y x1, x2, x3 ∈ X, un triángulo geodésico T = {x1, x2, x3} es la unión de tres geodésicas [x1x2], [x2x3] y [x3x1] en X. El espacio X es δhiperbólico (en sentido de Gromov) si cualquier lado del triángulo T está contenido en un δentorno de la unión de otros dos lados, para cada triángulo geodésico T en X, es decir, si para toda permutación {i,j,k} de {1,2,3} y para todo u ∈ [xixj] se verifica d(u,[xjxk] ∪ [xixk]) ≤ δ. Denotamos por δ(X) la constante de hiperbolicidad optimal de X, es decir, δ(X) := inf{δ ≥ 0 : X es δhiperbólico}. Esta tesis se enmarca dentro del estudio de las propiedades de los grafos hiperbólicos de Gromov, y forma parte de un amplio proyecto que involucra a numerosos investigadores de diversas universidades que trabajan en este mismo tema. Los resultados de esta tesis se presentan en dos secciones, en el tercer capítulo. En la primera de ellas relacionamos la constante de hiperbolicidad de un grafo con algunos parámetros del mismo grafo, como el cuello, el número de vértices y el diámetro: en particular, si g denota el cuello (el ínfimo de las longitudes de los ciclos del grafo, probamos que δ(G) ≥ g(G)/4 para todo grafo (finito o infinito, posiblemente con aristas múltiples y/o bucles); si G es un grafo con n vértices y aristas de longitud k (posiblemente con aristas múltiples y/o bucles), entonces δ(G) ≤ nk/4. Además, demostramos que ambas desigualdades son óptimas: encontramos una gran familia de grafos para la cual la primera desigualdad es de hecho una igualdad; además, caracterizamos el conjunto de grafos con δ(G) = nk/4. También caracterizamos los grafos con aristas de longitud k con δ(G) < k. En la segunda sección estudiamos la hiperbolicidad de una clase especial de grafos, los grafos producto, obteniendo información valiosa sobre un grafo producto a partir de información sobre ambos factores. En particular, llegamos a caracterizar la hiperbolicidad del producto cartesiano de grafos: G1 ×G2 es hiperbólico si y sólo si uno de los factores es hiperbólico y el otro factor está acotado. También probamos algunas desigualdades optimales entre la constante de hiperbolicidad de G1 × G2, δ(G1), δ(G2) y los diámetros de G1 y G2 (y encontramos familias de grafos para los que se alcanzan las igualdades). Además, obtenemos el valor exacto de la constante de hiperbolicidad para muchos grafos producto.
7. Thursday, May 12, 2011. 4:00 pm. Room 2.2.D08 Speaker: Mattia Cafasso, CRM Montréal (Canada) Title: RiemannHilbert problems and applications Abstract: RiemannHilbert 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 RiemannHilbert problems and Fredholm determinants there is the notion of integrable operators, introduced by ItsIzerginKorepin 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 nonintersecting Brownian motions.
8. Thursday, April 28, 2011. 4:00 pm. Room 2.2.D08 Speaker: JeanMarie 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 1skeleton) 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 nonscalar 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
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 matrixvalued symbols and some (unexpected) applications Abstract: We discuss the eigenvalue distribution in the Weyl sense of general matrixsequences associated to a symbol. As a specific case we consider Toeplitz sequences generated by matrixvalued (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 qpolynomials for nonstandard parameters Abstract: qpolynomials can be defined for all the possible parameters but their orthogonality In this talk we present orthogonality properties for the AskeyWilson 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 $$(1x)^\alpha(1+x)^\beta h(x)\prod_{i=1}^m x_ix^{\nu_i}$$, where $1<\dots1$ ($i=1,\dots,m$), and $h$ is real analytic and strictly positive on $[1, 1].$ The Cohentype 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 FourierJacobi 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. GuevaraJordán.
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 CD kernels Abstract: Christoffel  Darboux (CD) 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 CD kernel possess an universal character. To find the limiting CD 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 CD 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 steepestdescent method for RiemmannHilbert problems: the case of the focusing nonlinear Schroedinger equation (NLS).
17. Thursday, January 20, 2011. 4:00 pm. Room 2.2.D08 Speaker: Domingo Pestana, UC3M Title: Expanding maps and shrinking targets.
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 cubictype potential, it is possible to analyze the asymptotic behavior of these orthogonal polynomials and their zeros using RiemannHilbert 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
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
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 EtnyreGhrist'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: Selfadjoint operators are the main objects of Quantum Mechanics. The selfadjoint 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 LaplaceBeltrami operator, one can construct a completely equivalent theory of selfadjoint operators dealing only with boundary conditions. In addition, one can associate to every selfadjoint 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 selfadjoint 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 Speaker: Alicia Cantón Pire, Universidad Politécnica de Madrid
Speaker: Eva Tourís Lojo, UCIIIM 24. Thursday, November 4, 2010. 4:00 pm. Room 2.2.D08 Title: Some new results on analytic properties and zeros of Laguerretype Orthogonal Polynomials Speaker: Edmundo José Huertas Cejudo, UCIIIM 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. 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
26. Thursday, October 21, 2010. 4:00 pm. Room 2.2.D08 Speaker: Dra Celina Pestano, Universidad de la Laguna
27. Thursday, October 7, 2010. 4:00 pm. Room 2.2.D08
28. Thursday, September 30, 2010. 4:00 pm. Room 2.2.D08

Last Updated on Wednesday, 06 July 2011 09:07 