**Tuesday****, April 25, 2017, 17:00. Room 2.2.D08**

**Speaker: **.Bei-Bei Zhu, Chinese Academy of Sciences, Beijing & Universidad Carlos III de Madrid

**Title: ** A stroboscopic averaging algorithm for highly oscillatory delay problems

**Abstract:**: For highly oscillatory delay differential system, the presence of the fast-frequency oscillation makes numerical simulations so costly that it is better to average out the fast oscillations by using the averaging method before applying the numerical method. However, the analytic expression of the averaged system may be difficult or impossible to obtain. The stroboscopic averaging method is a technique that approximates the averaged solution by using only the originally given system. Error bounds and numerical results will be provided.

**Thursday****, June 08, 2017. Room 2.2.D08**

**Speaker: **.Leonardo Rendon, Departamento de Matemáticas Universidad Nacional de Colombia en Bogotá

**Title: ** Applications of the compensated compactness method onhyperbolic conservation systems

**Abstract:**: In this talk, I would like to introduce the applications of the compensated compactness method on hyperbolic conservation systems of two equations. I will present some results from our research group.

**Thursday****, March 2, 2017, 17:00. Room 2.2.D08**

**Speaker: **.Plamen Iliev, School of Mathematics, Georgia Institute of Technology

**Title: ** Hypergeometric and multiple orthogonal polynomials bases for the Alpert multiresolution analysis

**Abstract:**:I will describe two explicit bases for the Alpert multiresolution analysis and discuss their properties. The first one consists of piecewise ${}_{i+1}F_{i}$ hypergeometric polynomials, while the second basis is related to type I Legendre-Angelesco multiple orthogonal polynomials. The talk will based on joint works with Jeff Geronimo and Walter Van Assche.

**Thursday****, February 23, 2017, 17:00. Room 2.2.D08**

**Speaker: **.Plamen Iliev, School of Mathematics, Georgia Institute of Technology

**Title: ** Krall commutative algebras of differential operators

**Abstract:**: In 1938, Krall posed the general problem to construct and classify all families of orthogonal polynomials which are eigenfunctions of a differential operator of arbitrary order, which is independent of the degree index. General bispectral techniques based on the Darboux transformation led to a large collection of solutions to Krall’s problem. In this talk, I will discuss yet another method motivated by solitons, which can be used to construct and characterize the commutative algebras of differential or q-difference operators for Krall polynomials.

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**Thursday****, January 26, 2017, 16:00. Room 2.2.D08**

**Speaker: **.Luis Garza (U. Colima)

**Title: ** On a relation between Hurwitz and orthogonal polynomials

**Abstract:**: Abstract: A linear control system is stable if its characteristic polynomial is Hurwitz. In this talk, we present some relations between Hurwitz and orthogonal polynomials that could be useful to study the stability of linear systems. In particular, we construct families of Hurwitz polynomials by using sequences of orthogonal polynomials and obtain some interesting properties.

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**Thursday,**** January 19, 2017, 16:00. Room 2.2.D08**

**Speaker:** Lino Gustavo Garza (UC3M).

**Title: **Analytic Properties of Polynomials Orthogonal with respect to Coherent Measures Supported on the Unit Circle** **

**Abstract:**:This work presents a study of orthogonal polynomials from a matrix point of view. We deal with semiclassical and coherent orthogonal polynomials. We also consider the dierence and q-dierence operators. We also study the (0,2)-coherent pairs of measures of the second kind supported on the unit circle.

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**Tuesday, January 10 2017, 16:00. Room 2.2.D08**

**Speaker: **Herbert Dueñas Ruiz (Universidad Nacional de Colombia)

**Title: **Sobolev orthogonal polynomials on product domains in several variables.

**Abstract:** Using the ideas presented in [1], we study the

polynomials of several variables orthogonal with respect to the inner

product:

_S=c \int_{\Omega } \nabla ^{2} f(x,y) \cdot nabla ^{2}g(x,y) W(x,y) dydx+\lambda f(c1,c2) g(c1 ,c2),

where (c1, c2) is some corner point in \Omega =[a1,b1] x [ a2,b2], \lambda >0, and c=1 / \int_{\Omega } W(x,y) dxdy.

Two examples are presented using Laguerre and Gegenbauer polynomials.

[1] L. Fernández, F. Marcellán, T. Pérez, M. Piñar and Y. Xu, *Sobolev orthogonal polynomials on product domains*, Jour. Comp. App. Math. Vol 284.** No 5 **(2015), 202-215.

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**Thursday, January 12 2017, 16:00. Room 2.2.D08**

**Speaker: **Omar Rosario Cayetano (UC3M)

**Title:**** **Graphs with small hyperbolicity constant andhyperbolic minor graphs.

**Abstract: **If X is a geodesic metrix space and x1, x2, x3 in X, a geodesic triangle T={x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] 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. If X is hyperbolic, we denote by delta(X) the sharp hyperbolicity constant of X, i.e., delta(X)=inf{delta>=0: X is delta-hyperbolic}. In this work we study the graphs with small hyperbolicity constant, i.e., the graphs which are like trees (in the Gromov sense).

We obtain simple characterizations of the graphs G with delta (G)=1 and delta(G)=5/4. A graph H is a "minor" of a graph G if a graph isomorphic to H can be obtained from G by contracting some edges, deleting some edges, and deleting some isolated vertices.

One of the main aims in this work is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph G/e obtained from the graph G by contracting an arbitrary edge e from it.

We also prove that H is hyperbolic if and only if G is hyperbolic, for many minors H of G, even if H is obtained from G by contracting and/or deleting infinitely many edges.

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**Thursday, December 15 2016, 16:00. Room 2.2.D08**

** Speaker: **Roberto S. Costas-Santos (U. Alcalá)

** Title: **Matrices totally symmetric relative to a tree.

**Abstract:** A matrix is called totally positive (TP) if every minor of it is positive. It is known that for a TP matrix, the eigenvalues are positive and distinct and the eigenvector associated with the smallest eigenvalue is totally nonzero and has an alternating sign pattern. We will be interested in submatrices of a given matrix that are TP, or permutation similar to TP. Thus, we will be interested in permuted submatrices, identified by ordered index lists. For a given labelled tree T on n vertices, we say that A is T-TP if, for every path P in T, A[P] is TP. J. Garloff relayed to us an old conjecture of A. Neumaier that for any tree T, the eigenvector associated with the smallest eigenvalue of a T-TP matrix should be signed according to the labelled tree T. We refer to this as ``the T-TP conjecture''. In this talk we prove that the conjecture is false, giving some examples and present. certain weakening of the TP hypothesis is shown to yield a similar conclusion, i.e. the eigenvector, associated with the smallest eigenvalue, alternates in sign as in the tree.This is a joint work with Charles R. Johnson, and Boris Tadchiev.

**Thursday, December 01 2016, 16:00. Room 2.2.D08**

** Speaker: **Yanely Zaldivar Gerpe ( UC3M)

** Title: **Inverse results for the m-th row of Incomplete Padé Aproximants.

**Abstract:** We study inverse type results for incomplete Padé approximants of analytic functions under the assumption that the sequence of denominators of the approximating rational functions have limit. Such results allow to describe the region to which the analytic function can be extended meromorphically, determine the location and order of the poles in this region, and detect some singularities on the boundary. These results are applied to the study of Hermite-Padé approximants; that is interpolating vector rational functions of vectors of analytic functions. We also extend to row sequences of incomplete Padé approximants some classical results, due to Hadamard and Ostrowski, related with the overconvergence of subsequences of Taylor polynomials and the analytic properties of the limit function under the presence of gaps in the power series.

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**Thursday, November 24 2016, 16:00. Room 2.2.D08**

** Speaker: **Abel Díaz González ( UC3M)

** Title: **Sobolev extremal polynomials with respect to mutually singular measures.

**Abstract:** Extremal polynomials with respect to a Sobole-type p-norm, with 1< p < \infty and measures supported on compact subsets of the real line, are considered. For a wide class of such extremal polynomials with respect to measures mutually singular, it is proved that their critical points are simple and contained in the interior of the convex hull of the support of the measures involved, the asymptotic critical point distribution is studied. We also find the nth root asymptotic behavior of the corresponding sequence of the derivatives of Sobolev extremal polynomials.

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**Thursday, November 10 2016, 16:00. Room 2.2.D08**

** Speaker: **Alfredo Toledano ( Universidad Complutense de Madrid )

** Title: **CMV biorthogonal Laurent polynomials: Christoffel formulas for Christoffel and Geronimus perturbations

**Abstract:** Quasidefinite sesquilinear forms for Laurent polynomials in the complex plane and corresponding CMV biorthogonal Laurent polynomial families are studied, considering bivariate linear functionals. We have discussed two possible Christoffel transformations of these linear functionals: Christoffel perturbation and Geronimus perturbation. The last one with the addition of appropriate masses.

The aims are getting the connection formulas for the CMV biorthogonal Laurent polynomials and its norms, thanks to Christoffel-Darboux kernels, second kind functions and mixed Chirstoffel-Darboux kernels.

For prepared Laurent polynomials, i.e., of the form L(z)=L_n z^n+...+L_{-n} z^{-n}, L_n L_{-n} \ne 0, these connection formulas lead to quasideterminantal (quotient of determinants) Christoffel formulas.

Finally, we present curves as examples, such as: the real line, the circle, the Cassini oval and the cardioid.

**Thursday, November 03 2016, 16:00. Room 2.2.D08**

** Speaker: **Luis Alejandro Molano Molano (U. Pedagógica Tecnológica de Colombia, Duitama, Colombia)

** Title: **On symmetric (1,1)-coherent pairs.

**Abstract:** In this talk we will discuss briefly some basic notions about symmetric (1,1)-coherent pairs. We will exhibit explicitly some of them, based on the symmetrization method for linear functionals. Finally we are going to treat some particular cases relative to classical measures and we will present open questions that are part of our current research.

**Thursday, October 27 2016, 16:00. Room 2.2.D08**

** Speaker: **José Javier Segura Sala (U. de Cantabria)

** Title: **Computation of asymptotic expansions of turning point problems via Cauchy's theorem.

**Abstract:** Linear second order differential equations having a large real parameter and turning point in the complex plane are considered. Classical asymptotic expansions for solutions involve the Airy function and its derivative, along with two infinite series, the coefficients of which are usually difficult to compute. By considering the series as asymptotic expansions for two explicitly defined analytic functions, Cauchy's integral formula is employed to compute the coefficient functions to high order of accuracy. The method employs a certain exponential form of Liouville-Green expansions for solutions of the differential equation, as well as for the Airy function. We illustrate the use of the method for the computation of Bessel functions and Laguerre polynomials of complex argument.

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**Thursday, October 20 2016, 16:00. Room 2.2.D08**

** Speaker: **Roberto S. Costas-Santos (U. de Alcalá)

** Title: **Orthogonality relations of Al-Salam-Carlitz for general parameters.

**Abstract:** In this talk we describe the orthogonality conditions satisfied by Al-Salam-Carlitz polynomials $U^{(a)}_n(x;q)$ when the parameters a and q are not necessarily real nor 'classical', i.e., the linear functional u with respect to such polynomial sequence is quasi-definite and not positive definite.

We establish orthogonality on a simple contour in the complex plane which depends on the parameters. In all cases we show that the orthogonality conditions characterize the Al-Salam-Carlitz polynomials $U_n^{(a)}(x;q)$ of degree $n$ up to a constant factor. We also obtain a generalization of the unique generating function for these polynomials.

This is a joint work with Howard S. Cohl, and Wenqing Xu.

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**Thursday, October 06 2016, 16:00. Room 2.2.D08**

** Speaker: **José Manuel Rodríguez (UC3M)

** Title: **La estrategia de Kelly en apuestas deportivas e inversiones en bolsa: algunos consejos de inversión.

**Abstract:** En esta conferencia se hará una revisión histórica de la estrategia de Kelly, sus generalizaciones y sus aplicaciones en apuestas deportivas e inversiones. Se discutirán también sus ventajas e inconvenientes prácticos, y finalmente se expondrán las aportaciones del conferenciante en este tema.

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**Thursday, September 22 2016, 16:00. Room 2.2.D08**

** Speaker: **Abel Díaz (UC3M)

** Title: **Sobolev extremal polynomials with respect to mutually singular measures.

**Abstract:** Extremal polynomials with respect to a Sobole-type p-norm, with 1< p < \infty and measures supported on compact subsets of the real line, are considered. For a wide class of such extremal polynomials with respect to measures mutually singular, it is proved that their critical points are simple and contained in the interior of the convex hull of the support of the measures involved, the asymptotic critical point distribution is studied. We also find the nth root asymptotic behavior of the corresponding sequence of the derivatives of Sobolev extremal polynomials.

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**Tuesday, September 20 2016, 16:00. Room 2.2.D08**

** Speaker: **Alfredo Deaño (University of Kent, UK)

** Title: **From Painlevé transcendents to random matrices and back.

**Abstract:** Painlevé transcendents have found a wide variety of applications in the last decades, including the description of transitions in the global and local behavior of the spectrum of certain ensembles of random matrices, as the size of the matrices N tends to infinity. When the classical ensembles such as the Gaussian or Laguerre Unitary Ensembles undergo some deformations, one technique to study the changes in the global and local picture of the spectrum and the partition function is via orthogonal polynomials, which may not be classical or defined on the real line anymore. Additionally, some quantities related to this non-classical orthogonal polynomials, such as the coefficients appearing in the recurrence relations, can belong to some specific hierarchies of solutions of differential equations of Painlevé type. In this talk we will illustrate these connections using two examples: a deformation of GUE with a cubic term and a deformation of LUE that makes use of semi-classical Laguerre polynomials on R^+.

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**Thursday, May 26 2016, 16:00. Room 2.2.D08**

** Speaker: **Walter Van Assche (KU Leuven, Belgium)

** Title: **Riemann-Hilbert analysis for a Nikishin system.

**Abstract:** In this seminar I will outline a method to obtain the asymptotic behavior of multiple orthogonal polynomial (and Hermite-Pad\’e approximants) for a Nikishin system of order 2. The method is based on a Riemann-Hilbert problem for 3x3 matrices, which is an extension of the Riemann-Hilbert problem for orthogonal polynomials introduced by Fokas, Its an Kitaev. The Riemann-Hilbert analysis consists of a number of transformations of the original Riemann-Hilbert problem to one for which the asymptotic behavior is known. Each transformation uses important information: the Nikishin construction, the equilibrium problem, the geometry of the problem, and the weight functions on the intervals. This is ongoing work with Guillermo López Lagomasino.

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**Thursday, May 19 2016, 16:00. Room 2.2.D08**

** Speaker: **Luis Verde-Star (Universidad Autónoma Metropolitana, México)

** Title: **Operadores en diferencias del tipo de Bochner que tienen polinomios ortogonales como autofunciones.

**Abstract:** Usando un enfoque matricial encontramos una familia de operadores en diferencias que generalizan a los operadores de Bochner de q-diferencias de orden dos con coeficientes polinomiales. El operador clásico de q-diferencias y el de (1/q)-diferencias son reemplazados por un par de operadores U y V que dependen de dos parámetros (r, t). Se cumple que para cada (r,t) y cada par de polinomios h(x), de grado uno, y g(x), de grado menor que o igual a dos, existe una sucesión de polinomios ortogonales p_k(x) que satisface la ecuación de valores propios (h(x) V + g(x) V U) p_k(x)=w_k p_k(x) para una sucesión de números w_k que depende del operador. Encontramos fórmulas explícitas para los coeficientes de los polinomios y para los coeficientes de la relación de recurrencia de tres términos.

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**Thursday, May 12 2016, 16:00. Room 2.2.D08**

** Speaker: **Nattapong Bosuwan (Mahidol University, Bangkok)

** Title: **Determining singularities using row sequences of linear Padé-orthogonal approximation and its extension

**Abstract:** We study the relation of the convergence of poles of row sequences of both linear and nonlinear Padé-orthogonal approximants (Padé approximants of orthogonal expansions) and the singularities of the approximated function. We prove both direct and inverse results for these row sequences. Thereby, we obtain analogues of the theorems of Montessus de Ballore, Gonchar, and Fabry. If the time allows, then we will discuss our recent results on convergence of orthogonal Hermite-Padé.

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**Thursday, April 21 2016, 16:00. Room 2.2.D08**

** Speaker: **Jesús María Sanz-Serna (Universidad Carlos III de Madrid)

** Title: **Gauss' Gaussian Quadrature

**Abstract:** Gauss introduced what we now call Gaussian quadrature in 1814. While modern presentations follow Jacobi and use orthogonal polynomials, Gauss original approach was quite different. In an impressive display of virtuosity, he employed generating functions/Z-transforms, continued fractions, Pade approximation, hypergeometric functions, and other tools. As a modern numerical analyst would do, he concluded his work by providing, with great accuracy, the required nodes and weights for the formulas with 1, ..., 7 nodes and reporting in detail a numerical experiment. My talk will be a guided, critical reading of his Methodus nova integralium valores per approximationem inveniendi, published in MDCCCXV.

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**Thursday, April 7 2016, 16:00. Room 2.2.D08**

** Speaker: **Edmundo Huertas (Universidad de Alcalá)

** Title: **Isometries between martingale spaces and orthogonal polynomials

**Abstract:** Sets of orthogonal martingales are important because they can be used as stochastic integrators in a kind of chaotic representation property. In this talk, we revisited the problem studied by W. Schoutens in [1], investigating how an inner product derived from an Uvarov transformation of the Laguerre weight function is used in the orthogonalization procedure of a sequence of martingales related to a certain Lévy process, called Teugels Martingales. Since the Uvarov transformation depends by a c<0, we are studying how to provide infinite sets of strongly orthogonal martingales, each one for every c in (-\infty ,0). In a similar fashion as in [2] (see also [3]), our aim is to introduce a suitable isometry between the space of polynomials and the space of linear combinations of Teugels martingales as well as the general orthogonalization procedure.

[1] D. Nualart and W. Schoutens, Chaotic and predictable representations for Lévy processes, Stochastic Process. Appl. 90, (2000) 109--122.

[2] W. Schoutens, An application in stochastics of the Laguerre-type polynomials, J. Comput. Appl. Math. 133, (1-2) (2001), 593--600.

[3] W. Schoutens, Stochastic processes and orthogonal polynomials, in Lecture Notes in Statist. 146, Springer-Verlag, New York, (2000).

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**Thursday, March 31 2016, 16:00. Room 2.2.D08**

** Speaker: **José Manuel Rodríguez (Universidad Carlos III de Madrid)

** Title: **Stability of the injectivity radius under quasi-isometries and applications to isoperimetric inequalities.

**Abstract:** Kanai proved the stability under quasi-isometries of numerous global properties (including isoperimetric inequalities) between Riemannian manifolds of bounded geometry. Even though quasi-isometries highly distort local properties, recently it was shown that the injectivity radius is preserved (in some appropriate sense) under these maps between zero genus Riemann surfaces. In the present work, results along these lines are obtained even for infinite genus. As a consequence, the stability of the isoperimetric inequality in this context (without the hypothesis of bounded geometry) is also obtained.

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**Thursday, March 17 2016, 16:00. Room 2.2.D08**

** Speaker: **Lino Gustavo Garza Gaona (Universidad Carlos III de Madrid)

** Title: **A matrix approach for the classical, semiclassical and coherent orthogonal polynomials

**Abstract:** I will present a recent matrix characterization for the classical orthogonal polynomials. We obtain a matrix characterization of semiclassical orthogonal polynomials in terms of the Jacobi matrix associated with the multiplication operator in the basis of orthogonal polynomials, and the lower triangular matrix that represents the orthogonal polynomials in terms of the monomial basis of polynomials. We also provide a matrix characterization for coherent pairs of linear functionals.

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**Thursday, March 10 2016, 16:00. Room 2.2.D08**

** Speaker: **Daniel A. Rivero Castillo (Universidad Carlos III de Madrid)

** Title: **Ceros de primitivas de polinomios ortogonales clásicos

**Abstract:** En esta charla se presentarán resultados acerca de la localización de ceros y comportamiento asintótico de primitivas de polinomios ortogonales standard con respecto a una medida positiva de Borel concentrada en la recta real. Como casos particulares se estudian las primitivas de polinomios ortogonales clásicos, es decir los casos Jacobi, Laguerre y Hermite. Algunos ejemplos ilustrativos serán mostrados.

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**Thursday, March 3 2016, 16:00. Room 2.2.D08**

** Speaker: **Amauris de la Cruz (Universidad Carlos III de Madrid)

** Title: **Hyperbolicity of direct products of graphs

**Abstract:** If X is a geodesic metric space and x1, x2, x3 in X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] 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. If X is hyperbolic, we denote by delta (X) the sharp hyperbolicity constant of X, i.e., delta (X) = inf{ delta >= 0 : X is delta-hyperbolic}. Some previous works characterize the hyperbolic product graphs (for the Cartesian, strong, join, corona and lexicographic products) in terms of properties of the factor graphs. However, the problem with the direct product is more complicated. In this paper, we prove that if the direct product G1 G2 is hyperbolic, then one factor is hyperbolic and the other one is bounded. Also, we prove that this necessary condition is, in fact, a characterization in many cases. In other cases, we find characterizations which are not so simple. Furthermore, we obtain good bounds for the hyperbolicity constant of the direct product of some important graphs.

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**Thursday, February 25 2016, 16:00. Room 2.2.D08**

** Speaker: **Víctor Elvira (Dpto. Teoría de la Señal y Comunicaciones, Universidad Carlos III de Madrid)

** Title: **Generalized Multiple Importance Sampling

**Abstract:** Importance Sampling methods are broadly used to approximate posterior distributions or some of their moments. In its standard approach, samples are drawn from a single proposal distribution and weighted properly. However, since the performance depends on the mismatch between the targeted and the proposal distributions, several proposal densities are often employed for the generation of samples. Under this Multiple Importance Sampling (MIS) scenario, many works have addressed the selection or adaptation of the proposal distributions, interpreting the sampling and the weighting steps in different ways. In this work, we propose a novel general framework for sampling and weighting procedures when more than one proposal is available. The most relevant MIS schemes in the literature are encompassed within the new framework, and, moreover novel valid schemes appear naturally. All the MIS schemes are compared and ranked in terms of the variance of the associated estimators. Finally, we provide illustrative examples which reveal that, even with a good choice of the proposal densities, a careful interpretation of the sampling and weighting procedures can make a significant difference in the performance of the method. Preprint available at: http://arxiv.org/abs/1511.03095

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**Thursday, February 11 2016, 16:00. Room 2.2.D08**

** Speaker: **Javier Segura (Universidad de Cantabria)

** Title: **Evaluación eficiente de cuadraturas Gaussianas de orden alto

**Abstract:** La computación de cuadraturas Gaussianas es un problema clásico que se puede abordar mediante el conocido algoritmo de Golub-Welsch. Sin embargo, este método, basado en la diagonalización de una matriz de tamaño N (siendo N el número de nodos), no es eficiente cuando el número de nodos es grande. En su lugar, la evaluación de los ceros (nodos de la cuadratura gaussiana) escogiendo

valores iniciales suficientemente precisos y aplicando algún método iterativo para refinar estos valores es más efectiva en estos casos. Los tres ingredientes básicos de los métodos iterativos de cálculo de cuadraturas gaussianas son, en primer lugar, la estimación de los valores iniciales para iniciar el proceso, en segundo la construcción de un algoritmo de evaluación de los polinomios ortogonales implicados y finalmente el refinamiento iterativo de los valores obtenidos en el primer paso mediante un método iterativo (que utiliza los valores del polinomio evaluados mediante el algoritmo del segundo paso).

En este charla describimos el estado actual de la cuestión y cómo es crucial la elección de un método iterativo adecuado al problema. En particular, describiremos la utilización de métodos iterativos de cuarto orden de tipo Sturm (basados en el teorema de comparación) que permiten calcular nodos de forma eficiente y global, hasta el punto de que no es necesaria una estimación previa. En cualquier caso, esas estimaciones pueden mejorar la eficiencia del algoritmo y discutiremos también de forma breve cómo utilizar información asintótica para evaluar eficientemente tanto las estimaciones iniciales como el valor de los polinomios y su derivada.

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**Thursday, January 21 2016, 16:00. Room 2.2.D08**

** Speaker: **Ramon Orive (Universidad de La Laguna)

** Title: **Sobre problemas de equilibrio en el eje real. Aplicaciones

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**Thursday, January 14 2016, 16:00. Room 2.2.D08**

** Speaker: **Miguel Tierz (Universidade de Lisboa, Portugal)

** Title: **On Mordell integrals, Stieltjes-Wigert polynomials and some of its physical applications

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**Thursday, November 5 2015, 16:00. Room 2.2.D08**

** Speaker: **Edmundo Huertas (Universidad de Alcalá)

** Title: **Asymptotics of orthogonal polynomials generated by a Geronimus perturbation of the Laguerre measure.

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**Thursday, October 29 2015, 16:00. Room 2.2.D08**

** Speaker: **Yves Grandati (Université de Lorraine, France)

** Title: **Shape invariance and equivalent mixed Jacobi-Trudi formulas forexceptional orthogonal polynomials

**Abstract:** On the basis of shape invariance arguments and specific discrete symmetries of the harmonic and isotonic potential, we obtain Jacobi-Trudi type formulas for the X-Hermite and X-Laguerre polynomials and we describe the equivalence relations among these determinantal representations.

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**Thursday, October 22 2015, 16:00. Room 2.2.D08**

** Speaker: **Nadia Clavero (Universidad Complutense de Madrid)

** Title: **Non-linear mixed norm spaces for the critical Sobolev embedding.

**Abstract:** During the 1930s Sobolev introduced the spaces that now bear his name and, at the same time, he proved his classical theorem that W^{1}_{0} L^{p}(I^{n}) embeds into L^{p}*(I^{n}),where p*=pn/(n-p). When we let p tend to n from the left, then p* tends to ∞. However, one can have unbounded functions in W^{1}_{0} L^{n}(I^{n}). Consequently, it is necessary to go outside the Lebesgue scale to find the optimal conditions satisfied by functions in W^{1}_{0} L^{n}(I^{n}).

An early result in this direction was obtained by Trudinger and was subsequently improved and generalized by Hansson, Brezis and Wainger and Maz'ya who obtained the following Lorentz-type refinement for the so-called limiting case of Sobolev embedding:

W^{1}_{0} L^{n}(I^{n}) embedds into L^{∞,n;-1} (I^{n}).

Although the above estimate is the best possible as far as rearrangement invariant range spaces are concerned, Bastero, Milman, and Ruiz, and independently Maly and Pick, proved that if the requirement that the target space should be a linear space is abandoned, then a further improvement of this borderline case is still possible. Motivated by these works, our goal is to reformulated their results in terms of non-linear mixed norm spaces.

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**Thursday, October 15 2015, 16:00. Room 2.2.D08**

** Speaker: **Daniel Seco (Universidad de Barcelona)

** Title: **Ciclicidad contra polinomios ortogonales.

**Abstract:** En un espacio de funciones holomorfas Y, una función f es cíclica si al multiplicarla por todos los polinomios se obtiene un subespacio X, denso en Y. En un trabajo anterior con varios colaboradores, introdujimos el uso de herramientas de optimización para encontrar explícitamente polinomios que determinan la ciclicidad de una función, llamados aproximantes óptimos. Ahora, encontramos una correspondencia entre estos aproximantes óptimos y familias de polinomios ortogonales en X, y explotamos esto para obtener varias caracterizaciones nuevas de la ciclicidad y propiedades de los ceros de los aproximantes óptimos.

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**Thursday, October 8 2015, 16:00. Room 2.2.D08**

** Speaker: **Jorge Arvesú (Universidad Carlos III de Madrid)

** Title: **nth root asymptotics for multiple Meixner polynomials.

**Abstract:** The nth root asymptotic behavior of multiple Meixner polynomials is presented. Two main ingredients of the proposed approach for the study of the aforementioned asymptotic behavior are used and discussed; namely, an algebraic function formulation for the solution of the equilibrium problem with constraint to describe their zero distribution and the limiting behavior of the coefficients of the recurrence relations.

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**Thursday, September 24 2015, 16:00. Room 2.2.D08**

** Speaker: **Francesco Calogero (Università di Roma "La Sapienza", Italia)

** Title: **On the generations of monic polynomials obtained by replacing the coefficients of the polynomials of the next generation with the zeros of a polynomial of the previous generation.

**Abstract:** Generations of monic polynomials ---all of arbitrary degree N--- are obtained from a seed polynomial of degree N by identifying the coefficients of the polynomials of the next generation with the zeros of a polynomial of the previous generation; and Diophantine properties are reported of the zeros of the polynomials therebyobtained when the seed polynomial is the Hermite polynomial of degree N. This is joint work in progress with Oksana Bihun.

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Contact:

**Thursday, July 2 2015, 16:00. Room 2.2.D08**

** Speaker: **Rafael González Campos (Universidad Michoacana de San Nicolás de Hidalgo, México)

** Title: **A new formulation of the fast fractional Fourier transform.

**Abstract:** By using a spectral approach, we derive a Gaussian-like quadrature of the continuous fractional Fourier transform. The quadrature is obtained from a bilinear form of eigenvectors of the matrix associated to the recurrence equation of the Hermite polynomials. These eigenvectors are discrete approximations of the Hermite functions, which are eigenfunctions of the fractional Fourier transform operator. This new discrete transform is unitary and has a group structure. By using some asymptotic formulas, we rewrite the quadrature in terms of the fast Fourier transform (FFT), yielding a fast discretization of the fractional Fourier transform and its inverse in closed form. We extend the range of the fractional Fourier transform by considering arbitrary complex values inside the unit circle and not only at the boundary. We find that this fast quadrature evaluated at z = i becomes a more accurate version of the FFT and can be used for nonperiodic functions.

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**Thursday, June 18 2015, 16:00. Room 2.2.D08**

** Speaker: **Alfredo Deaño (Universidad Carlos III de Madrid)

** Title: **Asymptotic analysis of kissing polynomials.

**Abstract:** We present several recent results on the asymptotic behaviour of polynomials p_n(x) that are orthogonal with respect to the complex weight function W(x) =exp(iwx) on [−1, 1], where w>0 is a real (and possibly large) parameter. In this setting, we study the properties of p_n(x) and associated quantities such as the Hankel determinants constructed from moments of W(x), as w, n or both parameters tend to infinity. The techniques used include multivariate oscillatory integrals, the Riemann-Hilbert formulation of the problem in the complex plane, potential theory and the Deift–Zhou method of steepest descent. This is a joint work with Daan Huybrechs (KU Leuven, Belgium), Arieh Iserles (University of Cambridge, UK) and Pablo Román (Universidad Nacional de Córdoba, Argentina).

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**Friday, May 22 2015, 16:00. Room 2.2.D08**

** Speaker: **Sergey Tikhonov (ICREA)

** Title: **Weighted Bernstein inequalities.

**Abstract:** I will discuss recent developments in the study of Bernstein inequalities for trigonometric polynomials with doubling and non-doubling weights.

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**Thursday, May 14 2015, 16:00. Room 2.2.D08**

** Speaker:** Omar Rosario Cayetano (Universidad de Carlos III de Madrid)

** Title: **Small values of the hyperbolicity constant in graphs.

**Abstract:** In the study of any parameter on graphs it is natural to study the graphs for which this parameter has small values. In this work we study the graphs (with every edge of length $k$) with small hyperbolicity constant, i.e., the graphs which are like trees (in the Gromov sense). In this work we obtain simple characterizations of the graphs $G$ with $\d(G)=k$ and $\d(G)=\frac{5k}4\,$ (the case $\d(G)< k$ is known). Also, we give a necessary condition in order to have $\d(G)=\frac{3k}2$ (we know that $\d(G)$ is a multiple of $\frac{k}4\,$). Although it is not possible to obtain bounds for the diameter of graphs with small hyperbolicity constant, we obtain such bounds for the effective diameter if $\d(G) < \frac{3k}2$. This is the best possible result, since we prove that it is not possible to obtain similar bounds if $\d(G) \ge \frac{3k}2$.

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**Thursday, May 7 2015, 16:00. Room 2.2.D08**

** Speaker: **José Manuel Rodríguez (Universidad de Carlos III de Madrid)

** Title: **Markov-type inequalities and duality in weighted Sobolev spaces.

**Abstract:** In this talk we present Markov-type inequalities in the setting of weighted Sobolev spaces when the considered weights are generalized classical weights. Also, as results of independent interest, we study some basic facts about Sobolev spaces with respect to measures: separability, reflexivity, uniform convexity and duality. These Sobolev spaces appear in a natural way and are a very useful tool when we study the asymptotic behavior of Sobolev orthogonal polynomials. Joint work with Francisco Marcellán and Yamilet Quintana.

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**Thursday, April 30 2015, 16:00. Room 2.2.D08**

** Speaker: **Ignacio Zurrian (Universidad Nacional de Córdoba, Argentina)

** Title: **El álgebra de operadores asociada a un peso matricial y time and band-limiting en un contexto no conmutativo.

**Abstract:** Dado un peso matricial en la recta uno construye una sucesión de polinomios ortogonales matriciales. Entonces, uno puede estar interesado en estudiar el álgebra, D(W), de todos los operadores diferenciales matriciales que tengan a estos polinomios como autofunciones. Presentaré el segundo caso que se conoce en profundidad (Gegenbauer), hablaremos de algunas propiedades estructurales y comparaciones con el primer caso. Por último, mostraremos la extensión de un resultado, cuyo origen e importancia se remonta al trabajo de Claude Shannon en fundamentos matemáticos de la teoría de la información y una notable serie de trabajos de D. Slepian, H. Landau y H. Pollak, a una situación que involucra polinomios ortogonales matriciales tipo Gegenbauer.

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**Thursday, April 16 2015, 16:00. Room 2.2.D08**

** Speaker: **Vicente Muñoz (Universidad Complutense de Madrid)

** Title: **Unified treatment of Explicit and Trace Formulas.

**Abstract:** We prove that a Poisson-Newton formula, in a broad sense, is associated to each Dirichlet series with a meromorphic extension to the whole complex plane of finite order. These formulas simultaneously generalize the classical Poisson formula and Newton formulas for Newton sums. Classical Poisson formulas in Fourier analysis, explicit formulas in number theory and Selberg trace formulas in Riemannian geometry appear as special cases of our general Poisson-Newton formula. (Joint work with Ricardo Pérez-Marco).

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**Thursday, April 9 2015, 16:00. Room 2.2.D08**

** Speaker: **Sergio Medina (Universidad de Carlos III de Madrid)

** Title: **On the Convergence of Mixed Type Hermite-Padé Approximants.

**Abstract:** The convergence of diagonal sequences of type II Hermite-Padé approximants of Nikishin systems have been known for some time and recently similar results have been obtained for type I Hermite-Padé approximants. In this talk we present new results on the convergence of diagonal sequences of a certain mixed type Hermite-Padé approximation problem of a Nikishin system, which is motivated in finding approximating solutions of a Degasperis-Procesi peakons problem and in the study of the inverse spectral problem for the discrete cubic string.

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**Thursday, March 26 2015, 16:00. Room 2.2.D08**

** Speaker: **Alfredo Deaño (Universidad Carlos III de Madrid)

** Title: **Orthogonal polynomials and random matrices.

**Abstract:** In this talk we present some classical results on unitarily invariant ensembles of random matrices of size NxN, the prime example being the Gaussian Unitary Ensemble (GUE). It will be shown how the analysis of the corresponding family of orthogonal polynomials on the real line can be used to obtain asymptotic information (for large N) of several statistical quantities of the random matrix ensemble, such as the behavior of the eigenvalues, the partition function and the free energy.

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**Thursday, March 5 2015, 16:00. Room 2.2.D08**

** Speaker: **Roberto Costas (Universidad de Alcalá)

** Title: **Classical orthogonal polynomials beyond the classical parameters.

**Abstract:** In this talk we try to illustrate how classical orthogonal polynomials can be extended beyond classical parameters. We will give some examples of these families with non classical parameters and we show how that affect to the behavior of the zeros, and the (classical) orthogonality.

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**Thursday, February 19 2015, 16:00. Room 2.2.D08**

** Speaker: **Jorge Rivero (Universidad Carlos III de Madrid)

** Title: **On co-polynomials on the real line and the unit circle.

**Abstract:** In this work, we study new algebraic and analytic aspects of orthogonal polynomials on the real line under finite modifications of recurrence coefficients, called {\em co--polynomials on the real line} (COPRL, in short). We investigate the behavior of zeros, mainly interlacing and monotonicity properties. Furthermore, using a transfer matrix approach we obtain new structural relations that holds in $\co$ and we study the spectral transformation related to COPRL. On the other hand, we give an expression of the co--polynomials on the unit circle (COPUC, in short) in terms of the original orthogonal polynomials. Similar results are also given simultaneously for the corresponding second kind polynomials. We also analyze the pure rational spectral transformation for non--trivial $\mathcal{C}$--functions, associated with the COPUC, and we show the relation with quadratic irrationalities. Finally, we generalize some results about the Szego transformation between nontrivial probability measures supported on [-1,1] and the unit circle. We obtain the relations between the recurrence coefficients and the Verblunsky coefficients for the corresponding COPRL and COPUC, through the Szego transformation. We also study the conections between $\mathcal{S}$--functions and $\mathcal{C}$--functions for the corresponding COPRL and COPUC, through the Szego transformation.

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**Thursday, January 29 2015, 16:00. Room 2.2.D08**

** Speaker: **Luis Martínez (Universidad Complutense de Madrid)

** Title: **El modelo de matrices de Penner en el caso no hermítico y su relación con los polinomios de Laguerre.

**Abstract:** Los modelos de matrices usados en la teoría de cuerdas son generalmente no hermíticos. Sin embargo varias de las propiedades que se asumen en el límite a gran n para la densidad de autovalores y la función de partición no están demostradas. En esta charla analizamos el modelo de Penner no hermítico con la ayuda de la teoría de polinomios de Laguerre a gran n. Mostramos la relevancia de la propiedad S en los soportes asintóticos de autovalores y encontramos que el denominado límite de 't Hooft en los modelos de matrices de la teoría de cuerdas no está libre de ambigüedades.

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**Thursday, January 8 2015, 16:00. Room 2.2.D08**

** Speaker: **María Francisca Pérez Valero (Universidad Carlos III de Madrid)

** Title: **TBA

**Abstract:** In this thesis, we analyze the properties of polynomials orthogonal with respect to a discrete Sobolev inner product.

First, we will focus our attention on the study of connection formulas relating Sobolev orthogonal polynomials with the corresponding standard ones. It is worthwhile to mention that these results are independent of the measure of orthogonality. We also include a new matrix connection relating the matrix associated to the higher order recurrence relation for Sobolev polynomials and the corresponding Jacobi matrix associated to the standard ones. Furthermore, we summarize some known properties of polynomials orthogonal with respect to a modification of the Laguerre measure, the k-iterated Christoffel one. Later on, we obtain estimates for the norm of such polynomials as well as a generalized Christoffel formula for them. Moreover, we present a detailed study about the diagonal Christoffel kernels associated to the Gamma distribution. In particular, we obtain the asymptotic behavior of these kernel polynomials both inside and outside the support of the measure.

On the other hand, we deal with some problems on asymptotic behavior of Sobolev orthogonal polynomials. The problem of Outer Relative Asymptotics has been treated both in bounded support case and unbounded support case. Regarding the bounded support case, we work with Nevai class of measures and we present an alternative proof for a known result about Outer Relative Asymptotics of Sobolev orthogonal polynomials. In the unbounded support case, we restrict ourselves to Laguerre measures. For the first time, we deal with the Outer and Inner Relative Asymptotics of Sobolev-type orthogonal polynomials when the mass points are located inside the support of the measure, the oscillatory region for such polynomials. Finally, we obtain the asymptotic behavior of the coefficients appearing in the higher order recurrence relation that Sobolev polynomials satisfy.

Finally, we obtain some results on convergence of Fourier-Sobolev series. We show a result about pointwise convergence of Fourier-Sobolev series in the case of measures with bounded support. In addition we prove the divergence of a certain Fourier-Sobolev series. The main tool for this purpose will be a Cohen type inequality. This problem is dealing for the first time for a Sobolev-type inner product with a mass point outside the support of the measure.

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**Thursday, December 11 2014, 16:00. Room 2.2.D08**

** Speaker: **Jesús María Sanz Serna (Universidad Carlos III de Madrid)

** Title: **The Extra Chance Generalized Hybrid Monte Carlo Method.

**Abstract:** I shall begin by giving an introduction to Markov Chain Monte Carlo Methods with emphasis on the Hybrid Monte Carlo Method (HMC). I shall then present a variant of HMC that I have recently introduced with C. M. Campos.

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**Thursday, December 4 2014, 16:00. Room 2.2.D08**

** Speaker: **Manuel Mañas (Universidad Complutense de Madrid)

** Title: **Multivariate orthogonal polynomials in the real space and Toda type integrable systems

**Abstract:** Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The standard graded reversed lexicographical order is used in order to get an appropriate symmetric moment matrix whose block Cholesky factorization leads to multivariate orthogonal polynomials. The approach allows to construct the corresponding second kind functions, Jacobi type matrices and associated three term relations and also Christoffel–Darboux formulae, which involve quasi-determinants – and also Schur complements– of bordered truncations of the moment matrix. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. A study of discrete and continuous deformations of the measure is presented and a Toda type integrable hierarchy is constructed, the corresponding flows are described through Lax and Zakharov–Shabat equations and bilinear equations are found. Matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are found for several coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss–Borel factorization of the Jacobi type matrices and its quasi-determinants, lead to expressions for the multivariate orthogonal polynomials and its second kind functions which generalize to the multidimensional realm those that relate the Baker and adjoint Baker functions with ratios of Miwa shifted τ -functions in the 1D scenario. In this context the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in RD, of the multivariate orthogonal polynomials. Finally, using congruences in the space of semi-infinite matrices, it is shown that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomstev–Petviashvili type system.

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**Thursday, November 27 2014, 16:00. Room 2.2.D08**

** Speaker: **Ricardo Pérez Marco (CNRS)

** Title: **Resultados sobre el género de funciones meromorfas.

**Abstract:** Para funciones meromorfas f(z) en el plano complejo cuyo divisor de ceros y polos (rho) se encuentra en un semiplano (p.ej. la función zeta de Riemann), relacionamos su exponente de convergencia (minimo d con \sum |rho|^(-d) convergente), su orden vertical (el mínimo m que hace que (f'/f)/|z|^m es L^1 en rectas verticales x+i R, y el género de Weierstrass (el grado del exponente que aparece en la exponencial de la factorización de Hadamard). Explicamos por qué, para multitud de funciones clásicas como series de Dirichlet, la función Gamma y las funciones trigonométricas, el género de Weierstrass es mínimo. (Trabajo conjunto con Vicente Muñoz, arxiv:1306.2165).

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**Thursday, November 20 2014, 16:00. Room 2.2.D08**

** Speaker: **Kenier Castillo (Universidade Estadual Paulista, Brazil).

** Title: **On the discrete extension of Markov's theorem on monotonicity of zeros.

**Abstract:** Motivated by an open problem proposed by M. E. H. Ismail in his monograph ``Classical and quantum orthogonal polynomials in one variable" (Cambridge University Press, 2005), we study the behavior of zeros of orthogonal polynomials associated with the modification of a positive measure on $[a,b] \subseteq \re$ by adding a mass at $c\in \re \setminus [a,b]$. We prove that the zeros of the corresponding polynomials are strictly increasing functions of $c$. Moreover, we establish their asymptotics when $c$ tends to infinity or minus infinity. These results can be considered as a discrete extension of Markov's theorem on monotonicity of zeros.

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**Thursday, November 13 2014, 16:00. Room 2.2.D08**

** Speaker: **Alfredo Deaño (Universidad Carlos III de Madrid).

** Title: **Construction and implementation of asymptotic expansions for Jacobi--type orthogonal polynomials.

**Abstract:** In this talk we will discuss the implementation (symbolic and numerical) of asymptotic expansions for Jacobi-type orthogonal polynomials, as their degree n goes to infinity. These polynomials are defined on the interval $[-1,1]$ with weight function $w(x)=(1-x)^{\alpha}(1+x)^{\beta}h(x)$, where \alpha,\beta>-1, and the function h(x) is real, analytic and strictly positive on $[-1,1]$. This asymptotic information is obtained in the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen (Adv. Math. 2004), using the Riemann--Hilbert formulation and the Deift--Zhou non-linear steepest descent method. We show how to implement the formulas contained in that reference, and how to compute higher order terms in the asymptotic expansion in an efficient way, in different regions of the complex plane. The results are implemented symbolically in Maple and numerically in Matlab. This is joint work with Daan Huybrechs and Peter Opsomer (Department of Computer Science, KU Leuven, Belgium).

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**Thursday, November 6 2014, 16:00. Room 2.2.D08**

** Speaker: **Mª Angeles García-Ferrero (Universidad Complutense de Madrid).

** Title: **Oscillation theorems for the Wronskian of an arbitrary sequence of eigenfunctions of Schrödinger's equation.

**Abstract:** The work of Adler provides necessary and sufficient conditions for the Wronskian of a given sequence of eigenfunctions of Schr\"odinger's equation to have constant sign in its domain of definition. We extend this result by giving explicit formulas for the number of real zeros of the Wronskian of an arbitrary sequence of eigenfunctions. Our results apply in particular to Wronskians of classical orthogonal polynomials, thus generalizing classical results by Karlin and Szeg\H{o}. Our formulas hold under very mild conditions that are believed to hold for generic values of the parameters. In the Hermite case, our results allow to prove some conjectures recently formulated by Felder et al.

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** Thursday, October 16 2014, 16:00. Room 2.2.D08**Alejandro Molano (Universidad Pedagógica y Tecnológica de Colombia, Colombia).

Speaker:

** Title: **Sobre una extensión de pares coherentes de funcionales regulares simétricos.

**Abstract:**

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** Thursday, September 25 2014, 16:00. Room 2.2.D08**Imed Ben Salah (Faculté des Sciences de Monastir, Tunisia)

Speaker:

Title:

**Abstract:** In this work, we give in the first part a new approach of the concept of finite-type relation between two semi-classical polynomials sequences. The second part is devoted to give a characterizations of all second degree semi-clasical forms of class one through finite-type relations.

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**Thursday, September 18 2014, 16:00. Room ****2.2.D08 **

**Speaker: **Alberto Grunbaum, University of California at Berkeley.

References: [1] Duistermaat, J. J., and Grunbaum, F. A. , Differential equations in the spectral parameter, Commun. Math. Phys., 103 (1984) 177-240.

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Contact:

**Past seminars**

**Thursday, June 26 2014, 16:00. Room ****2.2.D08 **

**Speaker:** Yadira Torres, Universidad Carlos III de Madrid.

**Abstract:** One of the open problems in graph theory is the characterization of any graph by a polynomial. Research in this area has been largely driven by the advantages offered by he use of computers which make working with graphs: it is simpler to represent a graph by a polynomial (a vector) that by the adjacency matrix (a matrix). We introduce the alliance polynomial A(G; x) of a graph. Also, we develop and implement an algorithm that computes in an efficient way the alliance polynomial. We obtain some properties of A(G; x) and its coefficients for: 1) Path, cycle, complete and star graphs. In particular, we prove that they are characterized by their alliance polynomials. (2) Cubic graphs (graphs with all of their vertices of degree 3), since they are a very interesting class of graphs with many applications. We prove that they verify unimodality. Also, we compute the alliance polynomial for cubic graphs of small order, which satisfy uniqueness. (3) Regular graphs (graphs with the same degree for all vertices). In particular, we characterize the degree of regular graphs by the number of non-zero coefficients of their alliance polynomial. Besides, we prove that the family of alliance polynomials of connected D-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not connected D-regular.

If X is a geodesic metric space and x,y,z are points in X, a geodesic triangle T={x,y,z} is the union of the three geodesics [xy], [yz] and [zx] in X. The space X is d-hyperbolic (in the Gromov sense) if any side of T is contained in the d-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by d(X) the sharp hyperbolicity constant of X. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. We obtain information about the hyperbolicity constant of cubic graphs. These graphs are also very important in the study of Gromov hyperbolicity, since for any graph G with bounded maximum degree there exists a cubic graph G* such that G is hyperbolic if and only if G* is hyperbolic. We find some characterizations for the cubic graphs which have small hyperbolicity constants. Besides, we obtain bounds for the hyperbolicity constant of the complement graph G' of a cubic graph G; our main result of this kind says that for any finite cubic graph G which is not isomorphic either to the complete graph with 4 vertices or to the complete bipartite graph with 3 and 3 vertices, the inequalities 5k/4 <= d(G') <= 3k/2 hold, if k is the length of every edge in G.

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**Thursday, June 12 2014, 16:00. Room ****2.2.D08 **

**Speaker: **Sergio Medina, Universidad Carlos III de Madrid.

The central results of this work is about of convergence of type I Hermite-Padé approximants of a Nikishin system. In the literature one can find a number of results on the convergence of type II Hermite-Padé approximants, but in this talk we present the first result about the convergence of type I Hermite-Padé approximants. Moreover, we study the convergence of type II Hermite-Padé approximants to a Nikishin system which has been perturbed by rational functions. This kind of problem was first study by A.A Gonchar in 1975 for the usual Padé approximantion. The generalization to Hermite-Padé for the case of m=2 (a system of two functions )was considered before by Bustamante and Lagomasino in 1994. In this work the general case for any m is proved.

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**Thursday, June 5 2014, 16:00. Room ****2.2.D08 **

**Speaker: **Verónica Hernández, Universidad Carlos III de Madrid.

**Abstract:** If $X$ is a geodesic metric space and $x_{1},x_{2},x_{3} \in X$, a geodesic triangle $T=\{x_{1},x_{2},x_{3}\}$ is the union of the three geodesics $[x_{1}x_{2}]$, [x_{2}x_{3}]$ and $[x_{3}x_{1}]$ 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 $X$. f $X$ is hyperbolic, we denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, i.e. $\delta(X) =\inf \{ \delta\geq 0:\hspace{0.3cm} X \hspace{0.2cm} \text{is} \hspace{0.2cm} \delta \text{-hyperbolic} \}.$ To compute the hyperbolicity constant is a very hard problem. Then it is natural to try o bound the hyperbolycity constant in terms of some parameters of the graph. Denote by $\mathcal{G}(n,m)$ the set of graphs $G$ with $n$ vertices and $m$ edges, and such that every edge has length $1$. In this work we obtain upper and lower bounds for B(n,m):=\max\{\delta(G)\mid G \in \mathcal{G}(n,m) \}$. In addition, we obtain an upper bound of the size of any graph in terms of its diameter and its order.

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**Thursday, May 29 2014, 16:00. Room ****2.2.D08 **

**Speaker: **Iván Area, Universidad deVigo.

**Abstract:** A DNA sequence can be considered as a mathematical string x=(x_{1},x_{2},...,x_{n}) of length n where each x_{i} is one of the four nucleotides (Adenine, Cytosine, Guanine and Thymine). The sequence x might be compared with another DNA sequence y=(y_{1},y_{2},...,y_{m}) of length m to measure the similarity between both strings and also to determine their residue-residue correspondences. In this talk, three alignment problems between two DNA sequences will be presented, as well as appropriate recurrence relations solving the corresponding alignment problems. Explicit solutions shall be given for the three alignment problems considered. Moreover, an introductory overview of DNA with some historical remarks shall be provided.__References:__

Andrade H: Análise matemática dalgunhos problemas no estudo de secuencias biolóxicas. PhD thesis, Universidade de Santiago de Compostela, Departamento de Análise Matemática (2013).

Andrade H, Area I, Nieto JJ, Torres Á: The number of reduced alignments between two DNA sequences. BMC Bioinformatics. 2014, 15:94 DOI: 10.1186/1471-2105-15-94 http://www.biomedcentral.com/1471-2105/15/94

Lesk AM: Introduction to Bioinformatics. Oxford, UK: Oxford University Press; 2002.

Cabada A, Nieto JJ, Torres A: An exact formula for the number of aligments between two DNA sequences. DNA Sequence (continued as Mitochondrial DNA) 2003, 14:427-430.

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**Thursday, May 22 2014, 16:00. Room ****2.2.D08 **

**Speaker: **Nikta Shayanfar, K. N. Toosi University of Technology (Iran).

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**Thursday, April 24 2014, 17:00. Room ****2.2.D08 **

**Speaker: **Arturo de Pablo, Universidad Carlos III de Madrid.

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**Thursday, April 10 2014, 16:00. Room ****2.2.D08**

**Speaker: **Misael Marriaga Castillo, Universidad Carlos III de Madrid.

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**Thursday, April 3 2014, 16:00. Room ****2.2.D08**

**Speaker: **Kenier Castillo, Universidade Estadual Paulista (Brasil).

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**Thursday, March 27 2014, 16:00. Room ****2.2.D08**

**Speaker: **Yamilet Quintana, Universidad Simón Bolívar (Venezuela).

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**Speaker: **Maxim Derevyagin, Katholieke Universiteit Leuven (Belgium).

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** Thursday, March 13 2014, 16:00. Room **2.2.D08

**Speaker: **Nikta Shayanfar, K. N. Toosi University of Technology (Iran).

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**Thursday, March 6 2014, 16:00. Room ****2.2.D08**

**Speaker: **Juan Carlos García Ardila, Universidad Carlos III de Madrid.

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**Thursday, February 27 2014, 16:00. Room ****2.2.D08**

**Speaker: **Carlos Álvarez Fernández, Universidad Pontificia de Comillas.

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**Thursday, February 20 2014, 16:00. Room ****2.2.D08**

**Speaker: **Miguel Tierz, Universidad Complutense de Madrid.

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**Thursday, February 13 2014, 16:00. Room ****2.2.D08**

**Speaker: **María Francisca Pérez Valero, Universidad Carlos III de Madrid.

**Title: **A Cohen type inequality for Laguerre-Sobolev expansions with a mass point outside their oscillatory regime.

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**Thursday, February 6 2014, 16:00. Room ****2.2.D08**

**Speaker: **Edmundo Huertas, Universidade de Coimbra (Portugal).

**Title: **Zeros of orthogonal polynomials generated by the Geronimus perturbation of measures.

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**Speak****er: **Nikta Shayanfar, K. N. Toosi University of Technology (Iran).

**Title: **Analytical Studies of Equivalence to Smith Form for Systems of Volterra Equations.

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**Thursday, January 16 2014, 16:00. Room 2.1.C08**

**Speaker: **Miguel Piñar, Universidad de Granada.

**Title: **Two-variable analogues of Jacobi polynomials on the parabolic triangle.

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**Thursday, January 9 2014, 16:00. Room 2.1.C08**

**Speaker: **Walter Carballosa, Universidad Carlos III de Madrid.

**Title: **Gromov hyperbolicity in graphs

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**Thursday, December 19 2013, 16:00. Room 2.1.C08**

**Speaker: **Dimitar K. Dimitrov, Universidade Estadual Paulista (Brasil).

**Title: **Are there any ghosts in electrostatics?

I'll discuss some electrostatic models with restrictions. It turns our that when the restrictions are defined by rational functions, there is a modified Lam\'e differential equation which describes the critical points of the energy. A curious example related to the model whose critical point is at the zeros of the Hermite polynomial shows clearly how movable charges become restrictions.

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**Thursday, December 12 2013, 16:00. Room 2.1.C08**

**Speaker: **Francisco Marcellán, Universidad Carlos III de Madrid.

**Title: **Geronimus transformations and orthogonal polynomials on the real line: Direct and inverse problems.

**Abstract:** In this lecture, some inverse problems for sequences of orthogonal polynomials with respect to linear functionals will be analyzed. In particular, we will show some recent results obtained in [1]. As an application, the Geronimus transformation will be analyzed in a general framework. The connection with matrix factorizations will be stated according to [2].

References:

[1] F. Marcellan and S. Varma, On an inverse problem for a linear combination of orthogonal polynomials. Journal of Difference Equations and Applications. In press.

[2] M. Derevyagin and F. Marcellán, A note one the Geronimus transformation and Sobolev orthogonal polynomials. Numerical Algorithms. In press

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**Thursday, December 5 2013, 16:00. Room 2.1.C08**

**Speaker: **Natalia Pinzón, Universidad Carlos III de Madrid.

**Title: **Generalized Coherent Pairs and Sobolev Orthogonal Polynomials.

**Abstract:** This dissertation presents a study of (M, N )-coherent pairs of order (m, k) of sequences of orthogonal polynomials of a continuous and discrete variable on the real line and on the unit circle. This concept extends all the generalizations of the notion of, in our terminology, (1, 0)-coherent pair, studied in the literature, which was ﬁrst introduced as coherent pair by A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna in 1991 (On Polynomials Orthogonal with Respect to Certain Sobolev Inner Products. J. Approx. Theory 65, 151- 175). We prove that the semiclassical (resp. Dν -semiclassical) character of the linear functionals U and V is a necessary condition for the (M, N )-coherence (resp. (M, N )-Dν - coherence) condition of order (m, k) of the pair ( U , V ), whenever m and k are different. Additionally, from either the (M, N ) or the (M, N )-Dν -coherence relation of order (m, k), we show that the linear functionals are related by an expression of rational type, generalizing all the results found on this topic in the literature. On the other hand, we also generalize several recent results in the framework of Sobolev orthogonal polynomials and their connections with coherent pairs. In particular, we show how to compute the coeﬃcients of the Fourier expansions of functions on appropriate Sobolev spaces in terms of the sequences of Sobolev polynomials orthogonal with respect to the Sobolev inner products associated with coherent pairs. Furthermore, we give additional properties for the particular cases when ( U , V ) is either a (1, 0) or a (1, 1) (resp. a (1, 0)-Dν or a (1, 1)-Dν )-coherent pair of order m, or when one of the linear functionals in a (M, N ) (resp. (M, N )-Dν ) -coherent pair of order (m, k) is classical (resp. Dν -classical). Besides, we analyze (1, 1)-coherent pairs on the unit circle when one of the linear functionals is either the Lebesgue or Bernstein-Szego linear functional. Moreover, we study the (M, N ) and (M, N )-Dν -coherence relations (resp. the (M, N )- coherence relation on the unit circle) from a matrix point of view, from which we obtain results that involve the monic Jacobi matrices (resp. the Hessenberg matrices) associated with the linear functionals in such a coherent pair.

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**Thursday, November 21 2013, 16:00. Room 2.1.C08**

**Speaker: **Pedro Tradacete, Universidad Carlos III de Madrid.

**Title: **Optimal range and domain for Hardy type operators on rearrangement invariant spaces.

**Abstract:** We present an explicit construction of the optimal range space within the category of rearrangement invariant spaces for operators related to Hardy averaging operator. Recall that a Banach space of measurable functions is called rearrangement invariant when any two functions in the space with the same distribution have equal norm. We will provide several examples to illustrate this construction and its relation with optimal domains. This is a joint work with J. Soria (Universidad de Barcelona).

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**Thursday, October 31 2013, 16:00. Room ****2.1.C08**

**Speaker: **José Manuel Rodríguez, Universidad Carlos III de Madrid.

**Title: **Measurable diagonalization of positive definite matrices and applications to non-diagonal Sobolev orthogonal polynomials.

**Abstract:** In this paper we show that any positive definite matrix $V$ with measurable entries can bewritten as $V=U\L U^*$, where the matrix $\L$ is diagonal, the matrix $U$ is unitary, and the entries of $U$ and $\L$ are measurable functions ($U^*$ denotes the transpose conjugate of $U$). This result allows to obtain results about the zero location and asymptotic behavior of extremal polynomials with respect to a generalized non-diagonal Sobolev norm in which products of derivatives of different order appear. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials.

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**Speaker: **Juan Manuel Pérez Pardo, Universidad Carlos III de Madrid.

**Title: **On the Theory of Self-Adjoint Extensions of the Laplace-Beltrami Operator, Quadratic Forms and Symmetry. **[Thesis pre-defense]**

**Abstract:** The main objective of this dissertation is to analyse thoroughly the construction of self-adjoint extensions of the Laplace-Beltrami operator defined on a compact Riemannian manifold with boundary and the role that quadratic forms play to describe them. Moreover, we want to emphasise the role that quadratic forms can play in the description of quantum systems. A characterisation of the self-adjoint extensions of the Laplace-Beltrami operator in terms of unitary operators acting on the Hilbert space at the boundary is given. Using this description we are able to characterise a wide class of self-adjoint extensions that go beyond the usual ones, i.e. Dirichlet, Neumann, Robin,.. and that are semi-bounded below. A numerical scheme to compute the eigenvalues and eigenvectors in any dimension is proposed and its convergence is proved. The role of invariance under the action of symmetry groups is analysed in the general context of the theory of self-adjoint extensions of symmetric operators and in the context of closed quadratic forms. The self-adjoint extensions possessing the same invariance than the symmetric operator that they extend are characterised in the most abstract setting. The case of the Laplace-Beltrami operator is analysed also in this case. Finally, a way to generalise Kato's representation theorem for not semi-bounded, closed quadratic forms is proposed.

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**Thursday, October 10 2013, 16:00. Room 2.2.D08**

**Speaker: **Cleonice F. Bracciali, Universidade Estadual Paulista (Brasil).

**Title: **On a class of orthogonal functions.

**Abstract:** We will present a class of functions satisfying a certain orthogonality property for which there also exists a three term recurrence formula. This class of functions, which can be considered as an extension to the class of symmetric orthogonal polynomials on [-1,1], has a complete connection to the orthogonal polynomials on the unit circle. Quadrature rules and other properties based on the zeros of these functions are also presented. (Join work with A. Sri Ranga, T.E. Pérez, and J.H. McCabe).

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**Thursday, October 3 2013, 16:00. Room 2.2.D08**

**Speaker: **Walter Carballosa Torres, Universidad Carlos III de Madrid.

**Title: **On the hyperbolicity constant of extended chordal graphs.

**Abstract:** If X is a geodesic metric space, a geodesic triangle T is the union of three geodesics joining three points 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 other two sides, for every geodesic triangle T in X. In this work we extend in two ways (edge-chordality and path-chordality) the classical definition of chordal graphs in order to relate this property with Gromov hyperbolicity. In fact, we prove that every edge-chordal graph is hyperbolic and that every hyperbolic graph is path-chordal. Furthermore, we prove that every path-chordal cubic graph with small path-chordality constant is hyperbolic.

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**Thursday, September 19 2013, 16:00. Room 2.2.D08**

**Speaker: **José Manuel Rodríguez, Universidad Carlos III de Madrid.

**Title: **Zeros of Sobolev orthogonal polynomials via Muckenhoupt inequality with three measures.

**Abstract:** We generalize the Muckenhoupt inequality with two measures to three under certain conditions.

As a consequence, we prove a very simple characterization of the boundedness of the multiplication operator and thus of the boundedness of the zeros and the asymptotic behavior of the Sobolev orthogonal polynomials. And we prove it for a large class of measures which includes the most usual examples in the literature, for instance, every Jacobi weight with any finite amount of Dirac deltas.

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**Thursday, September 5 2013, 16:00. Room 2.2.D08**

**Speaker: **Robert Milson, Dalhousie University (Canada).

**Title: **A Conjecture on Exceptional Orthogonal Polynomials.

**Abstract:** Exceptional orthogonal polynomials (so named because they span a non-standard polynomial flag) are defined as polynomial eigenfunctions of Sturm-Liouville problems. By allowing for the possibility that the resulting sequence of polynomial degrees admits a number of gaps, we extend the classical families of Hermite, Laguerre and Jacobi. In recent years the role of the Darboux (or the factorization) transformation has been recognized as essential in the theory of orthogonal polynomials spanning a non-standard flag. In this talk we will discuss the conjecture that ALL such polynomial systems are derived as multi-step factorizations of classical operators and offer some supporting evidence.

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Contact:

**Past seminars**

**Thursday, July 4 2013, 16:00. Room 2.2.D08**

**Speaker: **Alexander G. Ramm, Kansas State University.

**Title: **Wave scattering by many small particles and creating materials with desired refraction coeficients.

**Abstract:** Many-body wave scattering problems are solved asymptotically as the size a of the particles tends to zero and the number of the particles tends to infinity. Acoustic, quantum-mechanical, and electromagnetic wave scattering by many small particles are studied. Computational methods for solving many-body wave scattering problems in the case of small scatterers are developed. They allow one to treat wave scattering by as many as 10^6 small particles. This theory allows one to give a recipe for creating materials with a desired refraction coeficient. One can create material with negative refraction, that is, the material in which the group velocity is directed opposite to the phase velocity. One can create material with desired wave-focusing properties. For example, one can create a new material which scatters plane wave mostly in a fixed given solid angle.

**Friday, July 5 2013, 16:00. Room 2.2.D08**

**Speaker: **Maxim S. Derevyagin, Institut fur Mathematik, TU Berlin.

**Title: **On Darboux transformations and polynomial mappings.

**Abstract:** Let d\mu(t) be a probability measure on [0,+\infty) such that its moments are finite. Then the Cauchy-Stieltjes transform S of d\mu(t) is a Stieltjes function, which admits an expansion into a Stieltjes continued fraction. At first, I am going to present a matrix interpretation of the unwrapping transformation from S(\lambda) to \lambda S(\lambda^2), which is intimately related to the simplest case of polynomial mappings. More precisely, it will be shown that this transformation is essentially a Darboux transformation of the underlying Jacobi matrix. Moreover, in this scheme, the Chihara construction of solutions to the Carlitz problem appears as a shifted Darboux transformation. At second, we consider applications of Darboux transformations to Pade approximation.

**Thursday, June 27 2013, 16:00. Room 2.2.D08**

**Speaker: **Francisco Marcellán, Universidad Carlos III de Madrid.

**Title: **CMV matrices and spectral transformations of measures.

**Abstract:** In this talk we deal with a matrix approach to spectral transformations of nontrivial probability measures supported on the unit circle. In some previous work with L. Garza, when linear spectral transformations are considered, we have analyzed the connection between the Hessenberg matrices (GGT matrices) associated with the multiplication operator in terms of bases of orthonormal polynomials corresponding to the initial and the transformed measures, respectively. Here, we deal with the connection between the CMV matrices associated with the multiplication operator with respect to bases of Laurent orthonormal polynomials corresponding to a non trivial probability measure supported on the unit circle and a Hermitian linear transformation of it. We focus our attention on the Christoffel and Geronimus transformations, where the QR and Cholesky factorizations play a central role. A comparison with the results about the LU factorization of monic Jacobi matrices is presented as well as some recent results by M. Derevyagin, L. Vinet, and A. Zhedanov based on the connection between CMV matrices and Jacobi matrices based on the Szegö transformation. This is a joint work with M. J. Cantero, L. Moral and L. Velázquez (Universidad de Zaragoza).

**Thursday, June 13 2013, 16:00. Room 2.2.D08**

**Speaker: **Andrea Telliini, Universidad Complutense de Madrid.

**Title: **Intricate dynamics in superlinear indefinite problems.

**Abstract:** In this talk I will present a multiplicity result for positive solutions of a superlinear indefinite differential equation which arises in population dynamics to model the behavior of a species in a polluted habitat where competitive and cooperative effects among the individuals are combined. Using the amplitude of the cooperative effect as a parameter, we have been able to describe the bifurcation diagrams of the solutions in a one dimensional case, which are rather complex due to the high number of the solutions that the problem can admit. The multiplicity in higher dimensions, as well as the linear stability of the solutions will be also discussed in full generality. Finally, I will present some numerical experiments, both for cases already treated analytically and new ones. The content of the talk is based on joint works with J. Lopez-Gomez (UCM), M. Molina-Meyer (UC3M) and F. Zanolin (Università degli Studi di Udine).

**Thursday, May 30 2013, 16:00. Room 2.2.D08**

**Speaker: **Ana Isabel Mendes, Instituto Politécnico de Leiria (Portugal).

**Title: **On the relation between the full Kostant-Toda lattice and matrix orthogonal polynomials.

**Abstract:** In this work we characterize a full Kostant-Toda system in terms of a family of matrix polynomials orthogonal with respect to a complex matrix measure. In order to studied the solution of this dynamical system we give explicit expressions for the resolvent functional and we also obtain, under some conditions, a representation of the vector functionals associated with this system.

**Thursday, May 16 2013, 16:00. Salón de Grados, Edificio Padre Soler.**

**Speaker: **András Kroó, Hungarian Academy of Sciences - Cátedra de Excelencia.

**Title: **Bernstein-Markov type inequalities on star-like domains in R^d with applications to norming sets.

**Abstract:** Bernstein-Markov type inequalities widely used in various areas of analysis are related to estimating derivatives of polynomials on various domains. In this talk we shall give a survey of the history of this problem and present some new Bernstein-Markov type inequalities for multivariate polynomials on star like domains. These new inequalities will be shown to be useful in the study of cardinality of norming sets, or admissible meshes. Admissible meshes are applied in various areas, for instance they are used for discrete least squares approximation, extracting discrete extremal sets of Fekete and Leja type, scattered data interpolation, cubature formulas, etc.

**Thursday, April 25 2013, 16:00. Room 2.2.D08**

**Speaker: **María del Carmen Reguera, Universidad Autónoma de Barcelona.

**Title: **Weighted estimates for the Bergman projection and Sarason Conjecture.

**Abstract:** We will present some partial results on the two weight problem for the Bergman projection on the disc. This problem is connected to a conjecture of Sarason on the composition of Toeplitz operators in the Bergman space, where we will provide a complete characterization. We will also present some counterexamples to the extended Sarason conjecture for the Bergman space. This is joint work with A. Aleman and S. Pott.

**Thursday, April 18 2013, 16:00. Room 2.2.D08**

**Speaker: **Daniel Rivero, Universidad Carlos III de Madrid.

**Title: **Orthogonal polynomials and edge detection.

**Abstract:** In this talk we propose a novel algorithm that make use of two dimensional discrete orthogonal polynomials for detection of edges in 2D and depth maps images. Also we present some numerical results and compare them with the well known algorithms.

**Thursday, April 11 2013, 16:00. Room 2.2.D08**

**Speaker: **Jesús Guillera.

**Title: **Ramanujan series "upside-down".

**Abstract:** Series for $1/\pi$ of the following form

\[\sum_{n=0}^{\infty}\frac{(s)_n \left(\frac{1}{2}\right)_n (1-s)_n}{(1)_n^3}(a+b n) z^n = \frac{1}{\pi},\]

are known as Ramanujan-type formulas. As usual, $(s)_n=s(s+1)\cdots(s+n-1)$ is Pochhammer's symbol, $s \in \{ 1/2, 1/3, 1/4, 1/6 \}$, and $z$, $a$, and $b$ are algebraic numbers. In $1914$ Ramanujan identified $17$ series of this kind. In this talk we show how to evaluate the related ``upside-down" series (when $s\ne 1/6$)

\[\sum_{n=1}^{\infty}\frac{(1)_n^3}{(s)_n \left(\frac{1}{2}\right)_n (1-s)_n} \frac{a-b n}{n^3} z^{-n},\]

in terms of values of the Epstein zeta function

\[\sum_{(m,n) \in \mathbb{Z}^2}^{\prime} \frac{1}{(An^2+Bmn+Cm^2)^t},\]

at $t=2$. Since Epstein zeta functions often reduce to Dirichlet $L$-values, this recipe leads to examples such as

\[\sum_{n=1}^{\infty} \frac{(1)_n^3}{\left( \frac{1}{3} \right)_n \left( \frac{1}{2} \right)_n \left(\frac{2}{3} \right)_n} \frac{4-15n}{n^3} \, (-4)^{-n} = 27 L_{-3}(2).\]

Observe that convergent upside-down series are associated to divergent Ramanujan-type series. (Joint work with Mathew Rogers)

**Thursday, April 4 2013, 16:00. Room 2.2.D08**

**Speaker: **Alfredo Deaño, KU Leuven.

**Title: **Complex orthogonal polynomials and Gaussian quadrature.

**Abstract:** We present an extension of the classical ideas of Gaussian quadrature to integrals in the complex plane. This was originally motivated by the problem of computing oscillatory integrals on the real axis. Because the weight function is generally not real or positive, the existence of the associated family of orthogonal polynomials is not clear a priori, and their zeros are typically distributed along curves in the complex plane. This zero distribution of the roots of these OPs can be analyzed using logarithmic potential theory, and asymptotic information can usually be obtained via Riemann--Hilbert techniques. We present several cases that have been considered recently or that are under study, and (time permitting) we point out possible open problems in this area as well as connections with random matrix theory. This is joint and ongoing work with A. Asheim, D. Huybrechs, A. B. J. Kuijlaars (KU Leuven, Belgium) and P. Román (Universidad Nacional de Córdoba, Argentina).

**Thursday, March 21 2013, 16:00. Room 2.2.D08**

**Speaker: **Serhan Varma, Ankara University.

**Title: **d-Orthogonal polynomials by generating functions and an application to the linear positive operators.

**Abstract:** We shall introduce the concept of d-orthogonality of a given polynomial sequence. First, we will describe a method for checking d-orthogonality of polynomial sequences defined by generating functions. Then, we will give a generalization of the Laguerre polynomials in the context of d-orthogonality by a generating function of a certain form. In the last section, we will show how certain d-orthogonal polynomials (the Gould-Hopper polynomials) can be used to extend the Szasz operators.

**Thursday, March 14 2013, 16:00. Room 2.2.D08**

**Speaker: **Ulises Fidalgo, Universidad Carlos III de Madrid.

**Title: **An approach to the interpolatory quadrature formulae.

**Abstract:** Given a measure, we consider an arbitrary schemes of nodes in the interior of its support. For the corresponding interpolatory quadrature rule we study convergent conditions in terms of the node's distribution. In this analysis, a connection between Fourier series of orthogonal polynomials and interpolatory quadrature formulae plays a fundamental role.

**Thursday, March 7 2013, 16:00. Room 2.2.D08**

**Speaker: **Sergio Medina, Universidad Carlos III de Madrid.

**Title: **Convergence of tipe II Hermite-Padé approximants.

**Abstract:** Let $(s_{1},\ldots, s_{m})= \mathcal{N}(\sigma_{1},\dots, \sigma_{m})$ be a Nikishin system and $\Delta_{1}$ be the convex hull of $\mathit{supp}(\sigma_{1})$. Let $(r_{1},\ldots, r_{m})$ be rational functions such that $r_{k}(\infty)=0$ and the poles of $r_{k}$ lie in $\mathbb{C}\setminus \Delta_{1}$, for all $k=1,\ldots,m$. We study the convergence of the diagonal sequence of type II Hermite-Padé approximants associated to the system of functions $(f_{1},\ldots, f_{m})$ where $f_{k}(z)=\int \frac{ds_{k}(x)}{z-x}+ r_{k}$, $k=1,\ldots,m$.

**Thursday, February 28 2013, 16:00. Room 2.2.D08**

**Speaker: **András Kroó, Hungarian Academy of Sciences - Cátedra de Excelencia.

**Title: **Density of Multivariate Polynomials on Convex and Star like domains.

**Abstract:** A central question in Approximation Theory concerns the possibility of approximation of continuous functions by various families of polynomials, that is density of classes of polynomials. On one hand the density of a given polynomial family depends on the algebraic structure of this set. In addition, in the multivariate case the question of density is also intricately related to the geometric properties of the underlying domain on which the approximation is studied.

In the present talk we shall explore this interplay between algebraic and geometric properties in the study of density of various families of multivariate polynomials on compact subsets of R^d, in particular convex bodies or star like domains. The families of polynomials will include multivariate homogeneous polynomials, convex polynomials and incomplete polynomials.

**Thursday, February 14 2013, 16:00. Room 2.2.D08**

**Speaker: **Julio de Vicente, Universidad Carlos III de Madrid.

**Title: **Entanglement theory and multipartite maximally entangled states.

**Abstract:** Quantum information thaory offers revolutionary ways to transmit and process information such as quantum cryptography and quantum computation. Besides its interest in the foundations of quantum theory, entanglement is believed to play a fundamental role in these tasks and it is nowadays regarded as a resource. This has given rise to entanglement theory, which aims at characterizing entangled states, identifying which transformations among them are possible and quantifying how useful they can be. Since I expect most members of the audience not to be familiar with quantum theory, I will devote the first half of my talk to present the rules of the quantum game and to introduce some basic concepts of entanglement theory. In the second half I will explain some recent results of my own characterizing which deterministic conversions among multipartite entangled states are possible. In particular, this allows to identify which states are most useful as a resource: multipartite maximally entangled states. I will try to give a flavour of the mathematical tools that are used, which mainly rely on Matrix Analysis.

**Thursday, February 7 2013, 16:00. Room 2.2.D08**

**Speaker: **Alicia Cantón, Universidad Politécnica de Madrid.

**Title: **Quasi-isometries and isoperimetric inequalities in planar domains.

**Abstract:** In this work we study the stability of isoperimetric inequalities under quasi-isometries between non-exceptional Riemann surfaces endowed with their Poincarée metrics. This stability was proved by Kanai in the more general setting of Riemannian manifolds under the condition of positive injectivity radius. The present work proves the stability of the linear isoperimetric inequality for planar surfaces (genus zero surfaces) without any condition on their injectivity radii. It is also shown the stability of any non-linear isoperimetric inequality.

**Thursday, January 31 2013, 16:00. Room 2.2.D08**

**Speaker: **Pedro Tradacete, Universidad Carlos III de Madrid.

**Title: **On the Banach-Saks property.

**Abstract:** We will present a soft introduction to the Banach-Saks property in Banach spaces. Recall that a subset A of a Banach space has the Banach-Saks property if every sequence in A has a subsequence whose arithmetic means are convergent. We will survey classical results and provide examples of spaces with and without this property. Finally, we will present recent results on the convex hull of Banach-Saks sets obtained in collaboration with J. López-Abad (CSIC) and C. Ruiz (UCM).

**Thursday, January 24 2013, 16:00. Room 2.2.D08**

**Speaker: **Bernhard Beckermann, Université de Lille.

**Title: **An error analysis for rational Galerkin projection applied to the Sylvester equation.

References:

[1] B. Beckermann: An error analysis for rational Galerkin projection applied to the Sylvester equation, SIAM J. Num. Anal. 49 (2012), 2430-2450.

[2] B. Beckermann, Image numérique, GMRES et polynômes de Faber, C. R. Acad. Sci. Paris, Ser. I 340 (2005) 855-860.

[2] B. Beckermann, Image numérique, GMRES et polynômes de Faber, C. R. Acad. Sci. Paris, Ser. I 340 (2005) 855-860.

[3] A.A. Gonchar, Zolotarev problems connected with rational functions. Mathem. Digest (Matem. Sbornik), 7(4) (1969) 623-635.

**Speaker: **Pablo Álvarez Caudevilla, Universidad Carlos III de Madrid.

**Title: **Analysis on the behaviour of the solutions of several elliptic equations in domains with a multiple crack section.

**Thursday, January 10 2013, 16:00. Room 2.2.D08**

**Speaker: **Fernando Lledó, Universidad Carlos III de Madrid.

**Title: **Finite operators and Foelner sequences.

**Speaker: **Jorge Borrego Morell, Universidad Carlos III de Madrid.

**Title: **Orthogonal polynomials with respect to differential operators and matrix orthogonal polynomials.

**Abstract:** This work deals with the concept of orthogonal polynomials with respect to a differential operator, the study of the strong asymptotic behavior of eigenpolynomials of exactly solvable operators, and matrix orthogonal polynomials. We consider orthogonal polynomials with respect to either a Jacobi, Laguerre or Hermite operator and a finite positive Borel measure $\mu$ satisfying certain conditions. For a positive integer $m$, we analyze the conditions over the measure $\mu$ in order to guarantee the existence of an infinite sequence of monic polynomials $\{Q_n\}_{n=m+1}^{\infty}$, where each $Q_n$ has degree $n$ and orthogonal with respect to the operator. We consider algebraic and analytic properties of this sequence. A fluid dynamics model for the interpretation of the zeros of these polynomials is also considered.

Some of the results obtained for a classical operator are generalized by considering orthogonal polynomials with respect to a wider class of linear differential operators. We analyze the uniqueness and zero location of these polynomials. An interesting phenomena occurring in this kind of orthogonality is the existence of operators for which the associated sequence of orthogonal polynomials reduces to a finite set. For a given operator we also find a classification, in terms of a system of difference equations with varying coefficients, of the measures for which it is possible to guarantee the existence of an infinite sequence of orthogonal polynomials. We also obtain a curve which contains the set of accumulation points of the zeros of these polynomials for the case of a first order differential operator giving also the strong asymptotic behavior.

We consider as well the study of the strong asymptotic behavior the eigenpolynomials of exactly solvable operators. Under the assumption that the leading coefficient of the operator is a real polynomial, we obtain a formula for the strong asymptotic behavior of the eigenpolynomials on certain compact subsets of the complex plane. As an application, we study the strong asymptotic behavior of a sequence of monic orthogonal polynomials with respect to a Sobolev inner product which are eigenfunctions of a fourth order differential operator.

It is also object of study a new class of matrix orthogonal polynomials of arbitrary size satisfying a second order matrix differential equation. For matrix polynomials of size 2, we find an explicit expression of the sequence of orthonormal polynomials with respect to a weight by using a Rodrigues' formula for these polynomials. 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.

**Thursday, December 13 2012, 16:00. Room 2.2.D08**

**Speaker: **Andrei Martínez-Finkelshtein, Universidad de Almería.

**Title: **Phase transitions and equilibrium measures.

**Thursday, November 29 2012, 16:00. Room 2.2.D08**

**Speaker: **Dixan Peña, Ghent University.

**Title: **On the generation of special monogenic functions using Fueter's theorem.

**Thursday, November 22 2012, 16:00. Room 2.2.D08**

**Speaker: **Moisés Soto, Universidad Autónoma de Madrid.

**Title: **Approximation properties of shift-invariant spaces and the spectral function.

**Abstract:** Consider the operator $\mathcal{D}_Af(\cdot):=|\det(A)|^{\frac{1}{2}}\,f(A\,\cdot)$ ($f\in L^2(\mathbb{R}^d)$) associated to a given dilation A on $\mathbb{R}^d$. Firstly we will review the role played by A-sets (measurable sets such that A(G)=G) in the characterization of A-reducing spaces, which are shift-invariant spaces satisfying $\mathcal{D}_AV=V$. We will also introduce the notions of (G,A)-density point of a set, (G,A)-approximate continuity point of a measurable function and function (G,A)-locally non zero at a point. We will also study the A-approximation and A-density orders of shift-invariant spaces, obtaining some results which generalize some theorems due to De Boor, DeVore and Ron, Bownik and Rzeszotnik, and San Antolín.

We will also introduce and study the wider notions of (G,A)-approximation and (G,A)-density orders, which allow us to develope an approximation theory in A-reducing spaces. All the provided conditions focus on the local behaviour at the origin of the spectral function $\sigma_V$ of V.

**Thursday, November 15 2012, 16:00. Room 2.2.D08**

**Speaker: **José M. Rodríguez, Universidad Carlos III de Madrid.

**Title: **Muckenhoupt inequality with three measures and Sobolev orthogonal polynomials.

**Abstract:** We generalize the Muckenhoupt inequality from two measures to three under certain conditions. This implies a very simple characterization of the boundedness of the multiplication operator and thus the boundedness of the zeros and the asymptotic behavior of the Sobolev orthogonal polynomials, for a large class of measures which includes almost every example in the literature.

**Thursday, November 8 2012, 16:00. Room 2.2.D08**

**Speaker: **Anier Soria, Universidad Carlos III de Madrid.

**Title: **Aproximación simultánea vinculada al estudio de la irracionalidad de los valores de la función zeta de Riemann.

**Abstract:** Uno de los problemas abiertos más interesantes de la teoría de números versa sobre la naturaleza aritmética de los valores de la función zeta de Riemann (z(s), Re s>1) en los enteros positivos 2,3,... Roger Apéry, en el año 1979, dio una demostración de la irracionalidad del número z(3) (en la actualidad denominada constante de Apéry). Dicho resultado, en un principio algo polémico, inspiró a varios matemáticos que construyeron diferentes métodos para explicar la irracionalidad de dicha constante. Sorprendentemente, estos métodos conducen a la misma sucesión de aproximantes racionales (denominados aproximantes de Apéry). En la primera parte de esta charla presentaremos una nueva sucesión de aproximantes Diofánticos de la constante z(3). Esta sucesión se obtiene a partir de un problema de aproximación racional simultánea asociado a un problema de Riemann-Hilbert indeterminado. A continuación mostraremos que la nueva sucesión de aproximantes racionales a z(3), que además prueba su irracionalidad, dará lugar a una relación de recurrencia, así como a un nuevo desarrollo en fracciones continuas para z(3). Posteriormente, se expondrán infinitos aproximantes Diofánticos que dependen de ciertos parámetros enteros. Luego se vinculará cada uno de estos aproximantes con un problema de aproximación racional simultánea en el infinito. De esta manera, siguiendo la misma estrategia, se deducirán relaciones de recurrencia, así como nuevos desarrollos en fracciones continuas para z(3).

Las técnicas empleadas en el estudio de la constante de Apéry nos han permitido introducir ciertas funciones racionales (deducidas a partir de un problema de aproximación racional simultánea) a partir de las cuales se infieren nuevos aproximantes racionales a z(4). También deducimos relaciones de recurrencia que permiten obtener de forma directa nuevos desarrollos en fracciones continuas para z(4). Estos nuevos aproximantes mejoran la convergencia respecto de algunos resultados previos obtenidos por otros autores.

Por último, se plantea un problema general de aproximación racional simultánea en el infinito, a partir del cual, se consiguen buenos aproximantes racionales a los valores de la función zeta de Riemann en argumentos enteros, tanto pares como impares. Además, se muestran ejemplos numéricos que evidencian la efectividad de dichos aproximantes.

La charla concluye mencionando algunos problemas abiertos en los cuales estamos trabajando actualmente.

**Thursday, October 25 2012, 16:00. Room 2.2.D08**

**Speaker: **Bernardo de la Calle, Universidad Politécnica de Madrid.

**Title: **Polynomial interpolation of analytic functions: There's life beyond extremality.

**Abstract:** We will review some of the best-known results on Taylor series and how they have been extended to the case of an analytic function interpolated along extremal tables of interpolation points. The divergence of the interpolants outside the critical equipotential curve, the overconvergence of the approximants provided the speed of convergence is geometric, and the Jentzsch-Szegö theorem will be considered. Then, we will discuss the main topic of the talk: Is it possible to prove such results for general tables of interpolation?

**Thursday, October 18 2012, 16:00. Room 2.2.D08**

**Speaker: **Jeffrey Geronimo, Georgia Institute of Technology.

**Title: **Orthogonal polynomials and wavelets.

**Thursday, October 4 2012, 16:00. Room 2.2.D08**

**Speaker: **Junot Cacoq, Universidad Carlos III de Madrid.

**Title: **Direct and inverse results on row sequences of simultaneous rational approximants.

**Abstract:** In this talk we investigate the approximation of vector functions by vector rational function that generalizes Padé approximants. We consider two types of approximants: the simultaneous Hermite-Pade approximants, which are constructed by mean of interpolation criterion and Fourier-Padé approximants based on Fourier series expansions in terms of a system of orthogonal polynomials. The results obtained in terms of generalize to the vector case results well known for the scalar case due to R. of Montessus of Ballore, A.A. Gonchar, S.P. Suetin, P.R. Graves-Morris, and E. B. Saff.

**Thursday, September 27 2012, 16:00. Room 2.2.D08**

**Speaker: **Edmundo Huertas, Universidad Carlos III de Madrid.

**Title: **On Krall-type and Sobolev-type Orthogonal Polynomials.

**Abstract: **This talk will be focused on the so called Krall-type and Sobolev-type orthogonal polynomial sequences (OPS). In the first part, we analyze the zeros and some outer asymptotic properties of Krall-type OPS. We study several examples of perturbed measures supported in a finite or infinite interval of the real line. An electrostatic model for the zeros of Laguerre-Krall OPS is also considered. The second part of the talk is devoted to Sobolev-type OPS of unbounded support. In particular we obtain some asymptotics properties for discrete Laguerre-Sobolev-type and its interpretation as Matrix OPS.

**Thursday, September 20 2012, 16:00. Room 2.2.D08**

**Speaker: **Kenier Castillo, Universidad Carlos III de Madrid.

**Title: **Spectral Problems and Orthogonal Polynomials on the Unit Circle.

**Abstract: **The main purpose of the work presented here is to study transformations of sequences of orthogonal polynomials associated with a hermitian linear functional, using spectral transformations of the corresponding C-function F. We show that a rational spectral transformation of F is given by a finite composition of four canonical spectral transformations. In addition to the canonical spectral transformations, we deal with new examples of spectral transformations.

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Contact:

**Speaker: **Carl Jagels, Hanover College (USA).

**Title: **The Laurent-Arnoldi Process.

**Abstract: **The Laurent-Arnoldi process is an analog of the standard Arnoldi process applied to the extended Krylov subspace. It produces an orthogonal basis for the subspace along with a generalized Hessenberg matrix whose entries consist of the recursion coefficients. As in the standard case, the application of the process to certain types of linear operators results in recursion formulas with few terms. One instance of this occurs when the operator is isometric. In this case, the recursion matrix is the pentadiagonal CMV matrix and the Laurent-Arnoldi process essentially reduces to the isometric Arnoldi process in which the underlying measures differ only by a rotation in the complex plane. The other instance occurs when the operator is Hermitian. This case produces an analog of the Lanczos process where, analogous to the CMV matrix, the recursion matrix is pentadiagonal. The Laurent polynomials generated by the recursion coefficients have properties similar to those of the Lanczos polynomials. We discuss the interpolating properties of these polynomials in order to determine remainder terms for rational Gauss and Radau rules. We then apply our results to the approximation of matrix functions and functionals.

**Thursday, June 7 2012, 16:00. Room 2.2.D08**

**Speaker: **Rubén Vigara Benito, Centro Universitario de la Defensa de Zaragoza

**Title: **Representación de 3-variedades por esferas de Dehn rellenantes.

**Thursday, May 31 2012, 16:00. Room 2.2.D08**

**Speaker: **José Manuel Rodríguez, Universidad Carlos III de Madrid

**Title: **Caracterización de la hiperbolicidad en grafos planos periódicos.

**Abstract: **Daremos en primer lugar una visión general de los resultados sobre hiperbolicidad de grafos desarrollados en los últimos años, tanto desde el punto de vista teórico como de las aplicaciones. A continuación abordaremos uno de los problemas abiertos más importantes de la teoría: caracterizar, de forma sencilla, los grafos que son hiperbólicos. Debido a la complejidad del problema, resulta natural estudiarlo para una subclase de grafos, donde el problema es más asequible: esta clase es la de los grafos planos periódicos.

**Thursday, May 24 2012, 16:00. Room 2.2.D08**

**Speaker: **Óscar Ciaurri, Universidad de La Rioja.

**Title: **Acotaciones con pesos para series de Fourier: una perspectiva familiar.

**Thursday, May 17 2012, 16:00. Room 2.2.D08**

**Speaker: **Alfredo Deaño, Universidad Carlos III de Madrid.

**Title: **Some properties and applications of orthogonal polynomials in the complex plane.

**Abstract: **We consider analytical and computational properties of polynomials that are orthogonal on certain contours in the complex plane. The study of these orthogonal polynomials and the distribution of their zeros is important in the design of complex Gaussian quadrature rules, and also in the analysis of the partition function and free energy of certain random matrix ensembles. As a particular case, we mention the unitary random matrix model with weight function $e^{-NV(z)}$, where $V(z)=z^2/2-uz^3$ and $u>0$ is a real parameter. This is joint and ongoing work with Pavel M. Bleher, (Indiana University-Purdue University Indianapolis, USA), Daan Huybrechs and Arno B. J. Kuijlaars (K.U. Leuven, Belgium).

**Thursday, May 10 2012, 16:00. Room 2.2.D08**

**Speaker: **Emilio Torrano, Universidad Politécnica de Madrid.

**Title: **La matriz de Hessenberg correspondiente a polinomios ortogonales sobre curvas y la aplicación de Riemann.

**Thursday, May 3 2012, 16:00. Room 2.2.D08**

**Speaker: **Luis Sánchez González, Universidad Complutense de Madrid.

**Title: **Núcleos reproductores en espacios de Banach.

**Thursday, April 26 2012, 16:00. Room 2.2.D08**

**Speaker: **Andrei Martínez-Finkelshtein, Universidad de Almería.

**Title: **Problemas de max-min de energía y simetría de la medida de equilibrio.

En esta charla voy a introducir el concepto de medidas críticas (puntos estacionarios de un funcional de energía) y demostrar que dichas medidas poseen la propiedad de simetría antes indicada.

**Thursday, April 19 2012, 16:00. Room 2.2.D08**

**Speaker: **Ulises Fidalgo Prieto, Universidad Carlos III de Madrid.

**Title: **Gaussian quadrature rules connect Fourier series.

**Thursday, April 12 2012, 16:00. Room 2.2.D08**

**Speaker: **María Jesús Carro, Universidad de Barcelona.

**Title: **Teoremas de Boyd y Lorentz-Shimogaki en espacios de Lorentz con pesos.

En esta charla extenderemos estos teoremas a los espacios de Lorentz con pesos, los cuales incluyen como caso particular los espacios de Lebesgue con pesos. Estos espacios no son r.i. por lo que una nueva definición de índices de Boyd para estos espacios ha de ser dada.

**Thursday, March 22 2012, 16:00. Room 2.2.D08**

**Speaker: **Tommaso Leonori, Universidad Carlos III de Madrid

**Title: **Some results about large solutions for nonlinear elliptic equations and their applications.

**Thursday, March 15 2012, 16:00. Room 2.2.D08**

**Speaker: **Jeff Geronimo, Georgia Institute of Technology (Chair of Excellence Santander-UC3M 2011).

**Title: **Some problems associated with two variable polynomials orthogonal on the bi-circle

**Abstract: ** I will go into more depth on a couple of topics mentioned in my Colloquium two weeks ago. In particular I will discuss the connection between Matrix orthogonal polynomials on the unit circle and two variable polynomials orthogonal on the bi-circle. I will also mention some open problems.

**Thursday, March 8 2012, 16:00. Room 2.2.D08**

**Speaker: **Anbhu Swaminathan, Indian Institute of Technology Roorkee.

**Title: **Pick functions, chain sequences and hypergeometric type functions.

**Abstract: **Pick functions were studied by Nevanlinna using moment problems as reciprocal of Stieltjes functions. They are useful in finding solutions of certain differential difference equations. Using chain sequences or q-fractions, ratios of certain hypergeometric type functions can be obtained as members of the class of Pick functions. The chain sequences of infinite and finite classical orthogonal polynomials are useful in determining certain interesting properties. n this lecture we will survey the interplay between techniques on orthogonal polynomials and linear algebra with an special emphasis on the inversion of Hankel and Toeplitz matrices, factorization of some structured matrices, eigenvalue problems, least square problems, and functions of matrices. Some applications in digital filter structures will be shown.

**Thursday, March 1 2012, 16:00. Room 2.2.D08**

**Speaker: **Francisco Marcellán Español, Universidad Carlos III

**Title: **Matrix Analysis meets Orthogonal Polynomials

**Abstract: **In this lecture we will survey the interplay between techniques on orthogonal polynomials and linear algebra with an special emphasis on the inversion of Hankel and Toeplitz matrices, factorization of some structured matrices, eigenvalue problems, least square problems, and functions of matrices. Some applications in digital filter structures will be shown.

**Thursday, February 23, 2012. **4:00 pm. Room 2.2.D08

**Speaker: **Guillermo López Lagomasino, Universidad Carlos III de Madrid

**Title: **Convergence of simultaneous Fourier-Padé approximation.

**Abstract: **We give a Montessus type theorem for row sequences of simultaneous Fourier-Pad\'e approximations of a finite system of functions which are analytic on a neighborhood on the closed unit disk. The aim is to recover the functions in the largest possible domain and detect the location and order of their poles in terms of the convergence properties of the approximating rational functions.

**Thursday, February 16, 2012. **4:00 pm. Room 2.2.D08

**Speaker: **Renato Álvarez-Nodarse, Universidad de Sevilla

**Title: **On the properties of some tridiagonal $k$-Toeplitz matrices

**Abstract: **Motivated by a physical model of a system of quantum oscillators with nonlinear interactions, we study spectral properties of certain matrices which arise as perturbations of some tridiagonal $k-$Toeplitz matrices. Concretely, we are interested in the spectral properties of general tridiagonal $k-$Toeplitz matrices (which are $k-$periodic Jacobi matrices) and certain perturbations of them. We will show in this talk how the theory of orthogonal polynomials (and in particular the polynomial mappings) can be used for solving the unperturbed case. For the perturbed case we will focus our attention on the localization of the eigenvalues of such matrices, as well as on the distance between two consecutive eigenvalues.

**Thursday, February 9, 2012. **4:00 pm. Room 2.2.D08

**Speaker: **Juan Carlos Trillo, Departamento de Matemática Aplicada y Estadística. Universidad Politécnica de Cartagena

**Title: **Esquemas de subdivisión y de multirresolución no lineales. Estudio de la estabilidad

**Abstract: **En esta charla vamos a presentar algunos operadores de reconstrucción no lineales ( ENO (Essentially Non Oscillatory), ENO con resolución subcelda, WENO (Weighted ENO), PPH (Piecewise Polynomial Harmonic).

Utilizaremos dichos operadores de reconstrucción dentro de los esquemas de subdivisión y de multirresolución de Harten. Comentaremos las dificultades que aparecen para probar la estabilidad de los algoritmos al tratar con la no linealidad de estos operadores de reconstrucción. Introduciremos también los algoritmos con control del error, que suponen una alternativa para estabilizar los esquemas no lineales inestables.

Entre las aplicaciones de esta teoría se pueden citar la compresión y el zoom de datos, la eliminación de ruido en señales, la integración numérica y la reducción del tiempo computacional de ciertos algoritmos numéricos como el cálculo de flujos en ecuaciones hiperbólicas.

**Thursday, February 2, 2012. **4:00 pm. Room 2.2.D08

**Speaker: **Kenier Castillo, UC3M

**Title: **Szegö and para-orthogonal polynomials on the real line

**Abstract: **We study polynomials which satisfy the same recurrence relation as the Szegö polynomials, however, with the restriction that the (reflection) coefficients in the recurrence are larger than one in modulus. Para-ortogonal polynomials that follows from these Szegö polynomials are also considered. With positive values for the reflection coefficients, zeros of the Szegö polynomials, para-orthogonal polynomials and associated quadrature rules are also studied. Finally, again with positive values for the reflection coefficients, interlacing properties of the Szegö polynomials and polynomials arising from canonical spectral transformations are obtained.

**Thursday, January 19, 2012. **4:00 pm. Room 2.2.D08

**Speaker: **Jorge Bosch, Centro de Neurociencias de Cuba (CNEURO)

**Abstract:** Para las Neurociencias en la actualidad las neuroimágenes no constituyen la única fuente valiosa de información. El estudio de ciertas variables biológicas y aspectos de la conducta están ofreciendo información relevante para extraer conclusiones acerca del funcionamiento del cerebro. Un área emergente es el estudio del cerebro en sus interacciones sociales: los movimientos de los ojos, la cabeza, diferentes músculos o partes del cuerpo, pueden implicar intenciones en la comunicación u ofrecer patrones de la conducta humana. Caracterizar dichos patrones puede ayudar a identificar patrones patológicos que contribuyan al diagnóstico de enfermedades. Un ejemplo es el autismo.

Este mismo objetivo, en el plano social consiste en determinar las relaciones de causalidad que ocurren entre dos o más personas en una situación de interacción social: cómo evolucionan los procesos de atención y liderazgo, que pueden condicionar determinados patrones de comportamiento.

El análisis del cerebro evoluciona al estudio simultáneo de dos o más cerebros que interactúan socialmente. Una pregunta importante a responder es cómo la activación de ciertas áreas en una persona puede estar condicionada, o puede condicionar, la activación de otras áreas en el cerebro del compañero.

El ya grave problema de la alta dimensionalidad de las neuroimágenes (varias decenas de miles de variables, en algunas ocasiones con una pobre resolución temporal, como es el caso de la fMRI), se duplica al estudiar dos cerebros, a la vez que se trata de combinar diferentes modalidades de datos.

En esta presentación resumimos algunos de nuestros intentos por dar respuesta a algunos de estos problemas. Para ello aplicamos métodos de análisis de series temporales al estudio de diferentes modalidades de información procedentes tanto del cerebro como de variables biológicas relacionadas, como movimientos de los ojos, la cabeza y otros músculos.

**Thursday, January 12, 2012. **4:15 pm. Room 2.2.D08

**Speaker: **Abey López García, Katholieke Universiteit Leuven (Belgium)

**Title: **Multiple orthogonal polynomials and the normal matrix model.

**Abstract:** The normal matrix model is a certain probability measure defined on the space of complex matrices, introduced by phisicists P. Wiegmann and A. Zabrodin. This model is not well-defined for arbitrary polynomial potentials. In order to resolve this theoretical obstacle P. Elbau and G. Felder proposed the so-called cut-off approach, which consists of restricting the model to those matrices with eigenvalues contained in a fixed compact subset of the complex plane. The orthogonal polynomials that are relevant to the study of the cut-off model possess many interesting properties, probably the most remarkable being that they satisfy an almost three-term recurrence relation, and that for some potentials the zeros of these polynomials accumulate on a star-like set. Recently, P. Bleher and A. Kuijlaars proposed a new approach to the normal matrix model, which introduces certain unbounded contours and leads naturally to the study of multiple orthogonal polynomials associated to varying exponential weights defined on these contours. Bleher and Kuijlaars conjectured that the zeros of these multiple orthogonal polynomials have the same asymptotic distribution as the zeros of the orthogonal polynomials in the cut-off approach of Elbau and Felder. This was shown to be true in the case of a cubic monomial potential by Bleher and Kuijlaars. In this talk we discuss the case of a quartic monomial potential. Our main tool is the application of the Riemann-Hilbert asymptotic analysis to the multiple orthogonal polynomials. This is a joint work in progress with A. Kuijlaars.

**Abstract:** Numerical integration methods are used for calculating numerical solutions of differential equations. Since the work of John Butcher on an algebraic theory of integration methods, in the late 1960s - early 1970s, so-called Butcher- or B-series provide important tools in the analytic and structural study of -particular classes of- such numerical methods. Concerning applications in numerical analysis, one of the natural things to do with such B-series is to combine them. Indeed, a composition law for B-series for example allows for a simple derivation of order conditions. A substitution law for B-series makes the notion of modified differential equations in the context of backward error analysis more transparent. These two laws give rise to algebraic structures, such as groups, and (pre-)Lie and Hopf algebras of trees. In this talk we will first introduce Butcher's work and review related structures, in particular, Lie and Hopf algebraic ones. Once the basic notion of B-series has been outlined, we will introduce a new Hopf algebra of rooted forests, and show how it relates to substitution of B-series. We establish a link between this Hopf algebra and the well-known Butcher-Connes-Kreimer Hopf algebra of rooted trees, which will allow us to link it to Butcher's original work on composition of B-series. We will show how these results enable us to recover recent results in the field of numerical methods for differential equations due to Chartier, Hairer and Vilmart as well as Murua.

**Thursday, December 1, 2011. **4:15 pm. Room 2.2.D08

**Speaker: **Miguel Antonio Jiménez Pozo, Benemérita Universidad Autónoma de Puebla (Mexico) and Universidad de Jaén (Spain).

**Title:** Asymmetric approximation.

**Abstract:** In this talk we introduce the usual subjects of approximation theory in the case when distances in use are not necessarily symmetric. This means that d(x,y) could be different of d(y,x). Related to an asymmetric sup distance this type of approximation was initiated by Moursund in the 60’s with the use of a generalized weight function. By this time Krein and Nudelmann used the incipient concept of asymmetric norm to study approximation problems and moment problems as well. A later development by the Russian and Ukrainian schools converged to the so called sign sensitive approximation and still more general approximation by positive homogeneous functionals. A third way of measuring the asymmetric approximation in connection with the sup norm was introduced by the speaker in dealing with mathematical models in industry. Each of these ways has its particular interest but it happens at the end all of them are equivalent. Krein and Nudelmann also replaced the sup asymmetric norm by an integral one. At present asymmetric integral approximation is another active subject of mathematical research that shall be briefly focused in this talk.

**Thursday, November 24, 2011. **4:15 pm. Room 2.2.D08

**Speaker: Maxim Derevyagin, TU Berlin**

**Title:** A tridiagonal approach to interpolation problems.

**Thursday, November 17, 2011. **4:15 pm. Room 2.2.D08

**Speaker: Pablo Sánchez Moreno, Universidad de Granada**

**Title:** Some information-theoretic properties of orthogonal polynomials

**Thursday, November 10, 2011. **4:15 pm. Room 2.2.D08

**Speaker: Galina Filipuk, Warsaw University (Poland)**

**Title:** Semi-classical discrete orthogonal polynomials (Charlier, Meixner, Krawtchouk) and the Painlevé equations.

**Thursday, November 3, 2011. **4:15 pm. Room 2.2.D08

**Speaker: Pablo Manuel Román, Katholieke Universiteit Leuven (Belgium)**

**Title:** Vector equilibrium problems arising from a model of non-intersecting squared Bessel paths

**Thursday, October 27, 2011. **4:15 pm. Room 2.2.D08

**Speaker: Ulises Fidalgo Prieto, UC3M**

**Title: **A generalization of Markov's Theorem.

**Thursday, October 20, 2011. **4:15 pm. Room 2.2.D08

**Speaker: Marta Llorente, UAM, Spain**

**Title: **Un algoritmo para el cálculo de medidas fractales

Thursday, October 13, 2011. 4:15 pm. Room 2.2.D08 ** ** **Abstract: **Estudiamos los aproximantes Hermite-Padé de tipo mixto para una clase amplia de funciones analiticas. Entre los resultados expuestos, responderemos a la cuestión de la unicidad de los aproximantes Hermite-Padé de tipo mixto y la perfeccción de los sistemas de funciones analizadas. **Speaker: Edmundo Huertas, UC3M, Spain**

**Speaker: Sergio Medina Peralta, UC3M, Spain**

**Title: **Sobre la perfección de sistemas AT-Nikishin mixtos.

Thursday, October 6, 2011. 4:15 pm. Room 2.2.D08

**Speaker: Pedro Tradacete Pérez, UC3M, Spain**

**Title: **Teoría de extrapolación en espacios Lp-débiles.

**Abstract:** Resolvemos el problema de extrapolación para operadores en $L^{p, \infty}(\mu)$; es decir, proporcionamos estimaciones extremales (cuando p tiende a 1) para operadores sublineales T tales que $T:L^{p,\infty}(\mu)\rightarrow L^{p, \infty}(\nu)$ es acotado con constante menor o igual que $1/(p-1)^m$. Presentamos también algunas aplicaciones para el operador maximal de Hardy-Littlewood, la transformada de Hilbert y composición de operadores. Es un trabajo conjunto con María J. Carro.

Thursday, September 29, 2011. 4:15 pm. Room 2.2.D08 ** **

**Speaker: Héctor Raúl Fernández Morales, UC3M, Spain**

**Title:** Generalized sampling in $L^2(R^d)$ shift-invariant subspaces with multiple stable generators

**Abstract:** In order to avoid most of the problems associated with classical Shannon's sampling theory, nowadays signals are assumed to belong to some shift-invariant subspace. In this work we consider a general shift-invariant space $V_\Phi^2$ of $L^2(R^d)$ with a set $\Phi$ of $r$ stable generators. Besides, in many common situations the available data of a signal are samples of some filtered versions of the signal itself taken at a sub-lattice of $Z^d$. This leads to the problem of generalized sampling in shift-invariant spaces. Assuming that the $\ell^2$-norm of the generalized samples of any $f\in V_\Phi^2$ is stable with respect to the $L^2(R^d)$-norm of the signal $f$, we derive frame expansions in the shift-invariant subspace allowing the recovery of the signals in $V_\Phi^2$ from the available data. The mathematical technique used here mimics the Fourier duality technique which works for classical Paley-Wiener spaces. Irregular samples are also obtained as a perturbation of the regular ones, the irregular sampling results arise from the theory of perturbation of frames, finally a frame algorithm is implemented in the $\ell^_r(Z^d)$ setting.

** **** **

Thursday, September 22, 2011. 4:15 pm. Room 2.2.D08 ** **

**Speaker: Junot Cacoq, UC3M, Spain**

**Title: **Some results on the convergence of rows of simultaneous Padé approximation

**Abstract: **In this talk we present results on the convergence of rows of simultaneous Padé approximation which extend previous ones of P.R. Graves-Morris and E.B. Saff.

Thursday, September 15, 2011. 4:15 pm. Room 2.2.D08 ** **

**Title:** New perspectives on Laguerre-Sobolev type orthogonal polynomials

**Abstract:**** ** Our viewpoint sheds some new light on the asymptotic behavior of the Laguerre-Sobolev type monic orthogonal polynomial sequences, considering the perturbation of the Laguerre weight supported on the negative semi-axis of the real line. We study the outer relative asymptotic of these polynomials with respect to the standard Laguerre polynomials. The analogue of the Mehler-Heine formula as well as a Plancherel-Rotach formula for the rescaled polynomials are given. Finally, we analyze the behavior of their zeros in terms of their dependence on $N$.

**Thursday, September 8, 2011. 4:00 pm. Room 2.2.D08**

**Speaker: Evguenii Rakhmanov, University of South Florida, USA**

**Title:**** **Zeros of Heine-Stieltjes Polynomials and a general Modulii Problem

**Abstract: **The limit distribution of zeros of HS polynomials may be described as (positive) critical measures -critical points of an energy functional with respect to local point variation. A constructive characterization of the critical measures may be given in terms of closed quadratic differentials. It turns out that the same class of quadratic differentials solves one of the classical problems in geometric function theory - the problem of extremal partition of the plane with respect to a certain family of curves. This connection presents an independent interest and may be used in the investigation of both problems.

**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.

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__

**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

**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 δ-hiperbólico (en sentido de Gromov) si cualquier lado del triángulo T está contenido en un δ-entorno 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 δ-hiperbólico}. 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: **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.

**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** ** **

**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

**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.

**Speaker: **Daniel Peralta-Salas, CSIC**Title:** Knotted and linked streamlines of steady fluid flows

**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**Title:** Quadratic Forms and general self-adjoint Extensions of the Laplace-Beltrami Operator

**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

**Speaker: **Alicia Cantón Pire, Universidad Politécnica de Madrid**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.

**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. Th****ursday, 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**

**Upcoming seminars**

There are no upcoming seminars. We wish everyone a good summer and hope to see you here for the next season!

**Past seminars**

**1. Friday. July 2, 2010. 16:00. Room: 2.2.D.08**

**Title: Strong contracted asymptotics for Hermite and Mexiner polynomials and their zeros from the point of view of their difference equations.**

**2. Thursday. July 1, 2010. 16:00. Room: 2.2.D.08.**

**Title: ****Theory and applications of Kapteyn series**

**Speaker:** **Diego Dominici (State University of New York, New Paltz)**

**Abstract:** Kapteyn series have a long history, with results going back to Joseph Louis de Lagrange's groundbreaking paper "Sur le Problème de Képler" (1770). They were rediscovered independently by Friedrich Wilhelm Bessel and appeared in his paper "Untersuchung des Theils der planetarischen Störungen" (1824), where he introduced for the first time the functions that now bear his name. Both Lagrange and Bessel considered Kapteyn series as a solution of Kepler's Equation, formulated by Johannes Kepler in his masterpiece "Astronomia Nova" (1609). Besides their application in Astronomy and Astrophysics, Kapteyn series have also been applied by George Adolphus Schott and others to problems in Electromagnetic Radiation. In this talk we will review the history and theory of Kapteyn series, present some old and new results, discuss their applications and their connection with the theory of Hypergeometric Functions.** **

**3. Thursday. June 17, 2010. 16:00. Room: 2.2.D.08.**

**Title: ****Quasidiagonality, Folner sequences and spectral approximation**

**Speaker:** **Fernando Lledó. UC3M, Madrid**

**Abstract:** Consider a sequence of operators T_n in a Hilbert space H that approximates a given operator T in a suitable sense. A fundamental question is how the spectrum of T_n relates with the spectrum of T when n grows. There are two important notions in operator theory that can be applied to the mentioned spectral approximation problem: quasidiagonality and the existence of a Folner sequence. In both cases a sequence of finite rank projections converging strongly to the identity and adapted to the operator has to be constructed. In this talk I will introduce these notions and mention several examples. Time permitting I will present a new result on the existence of Folner sequences for some representations of C*-dynamical systems (also called crossed products).

**4. Tuesday. June 15, 2010. 16:00. Room: 2.2.D.08.**

**Title: ****Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchySpeaker:**

solution, that can be obtained from the solution to a Gauss--Borel factorization problem for a semi-infinite matrix, which plays the role of a moment matrix. This leads to sequences of multiple orthogonal polynomials, their duals and second kind functions. It also gives the corresponding linear forms that are bi-orthogonal to the dual linear forms. Expressions for these objects in terms of determinants from the moment matrix are given, recursion relations are found, which imply a multi-diagonal Jacobi type matrix with double staircase shape, and results like the ABC theorem or the Christoffel--Darboux formula are re-derived in this context (using the factorization problem and the

generalized Hankel symmetry of the moment matrix). The connection between this description of multiple orthogonality and the multi-component 2D Toda hierarchy, which can be also understood and studied through a Gauss--Borel factorization problem, is discussed. Deformations of the weights, natural for certain type of Nikishin systems, are considered and the correspondence with solutions to the integrable hierarchy, represented as a collection of Lax equations, is explored. Corresponding Lax and Zakharov--Shabat matrices as well as wave functions and their adjoints are determined. The construction of discrete flows is discussed in terms of Miwa transformations which involve Darboux transformations for the multiple orthogonality conditions. The bilinear equations is

derived and the $\tau$-function representation of the multiple orthogonality is given.

**5. Thursday. June 10, 2010. 16:00. Room: 2.2.D.08.**

**Title: Recent results on classical orthogonal polynomials**

**Abstract:** We present some of the latest algebraic and structural results on classical orthogonal polynomials. Among others, the characterization theorem for q-polynomials on the quadratic lattice, a degenerate version of the Favard's theorem, some ways to generalize classical orthogonal polynomials, etc; among others. Also an elementary way to prove the interlacing property of zeros of classical orthogonal polynomials is presented. This is joint work with J. F. Sánchez-Lara.

**6. Thursday. May 20, 2010. 16:00. Room: 2.2.D.08.**

**Title:** **Zeros of Orthogonal Polynomials Generated by Canonical Perturbations of the Standard Measure****.**

**Speaker:** **Edmundo J. Huertas.**** UC3M, Madrid**

**Abstract: ** In the last years some attention has been paid to the so called canonical spectral transformations of measure supported on the real line. Our contribution is focused on the behaviour of zeros of MOPS associated with the Christoffel and Uvarov transformations of such measures.

The outline of the talk is the following. In the first part we introduce the representation of the perturbed MOPS in terms of the initial ones and we analyze the behaviour of the zeros of the MOPS when an Uvarov transform is introduced. In particular, we obtain such a behavior when the mass N tends to infinity as well as we characterize the values of the mass N such the smallest (respectively, the largest) zero of these MOPS is located outside the support of the measure. The second part of the talk is devoted to the electrostatic interpretation of the zero distribution as equilibrium points in a logarithmic potential interaction under the action of an external field. We analyze such an equilibrium problem when the mass point is located either in the boundary or in the exterior of the support of the measure, respectively.

**7. Thursday. May 6, 2010. 16:00. Room: 2.2.D.08.**

**Title:** **Some Markov-Bernstein type inequalities and certain class of Sobolev polynomials.**

**Speaker:** **Yamilet Quintana (Department of Mathematics, Universidad Simón Bolívar, Venezuela)****Abstract:** Let $(\mu_{0},\mu_{1})$ be a vector of non-negative measures on the real line, with $\mu_{0}$ not identically zero, finite moments of all orders, compact or non compact supports, and at least one of them having an infinite number of points in its support. We show that for any linear operator $T$ on the space of polynomials with complex coefficients and any integer $n\geq 0$, there is a constant $\gamma_{n}(T)\geq 0$, such that $$\|Tp\|_{S}\leq \gamma_{n}(T)\|p\|_{S},$$ for any polynomial $p$ of degree $\leq n$, where $\gamma_{n}(T)$ is independent of $p$, and

$$\|p\|_{S}= \left\{ \int|p(x)|^{2}d\mu_{0}(x)+ \int|p^{\prime}(x)|^{2}d\mu_{1}(x)\right\}^{\frac{1}{2}}.$$

We find a formula for the best possible value $_{n}(T)$ of $\gamma_{n}(T)$ and inequalities for $_{n}(T)$. Also, we give some examples when $T$ is a differentiation operator and $(\mu_{0},\mu_{1})$ is a vector of orthogonalizing measures for classical orthogonal polynomials.

This is joint work with Dilcia Pérez.

**8. Thursday. April 29, 2010. 16:00. Room: 2.2.D.08.**

**Title: codimension-1 exceptional orthogonal polynomials**

**Speaker:** **Robert Milson (Dalhousie University)**

If we limit ourselves to the case of $m=1$, then a full classification is possible. The key result is the following theorem. Let Pn denote the n+1 dimensional vector space of univariate polynomials of degree n or less. Let Un be a codimension 1 subspace of Pn. Let us call Un exceptional or X1 if there exists a 2nd order differential operator

T(y) = p(x) y'' + q(x) y' + r(x) y

such that U is an invariant subspace but Pn isn't. Then, up to a fractional linear transformation

Un = span { x-1, x

Novel orthogonal polynomials arise when we consider 2nd order differential operators T(y) that leave invariant an infinite flag of exceptional codimension 1 subspaces.

**9. Thursday. A****pril 22, 2010. 16:00. Room: 2.2.D.08.**

**Title: A canonical family of multiple orthogonal polynomials for Nikishin systems**

**Speaker: Ignacio Álvarez Rocha, UPM, Madrid**

**Abstract: For any pair of non-intersecting compact intervals of the real line $\Delta_1, \Delta_2,$ we obtain two absolutely continuous measures $\mu_1,\tau_1$ supported on $\Delta_1$ and $\Delta_2$, respectively, such that the Nikishin system ${\mathcal{N}}(\mu_1,\tau_1)$ has orthogonal polynomials which satisfy a four term recurrence relation with constant coefficients of period 2. The measures are obtained from the functions which give the ratio asymptotics of orthogonal polynomials with respect to arbitrary Nikishin systems ${\mathcal{N}}(\sigma_1,\sigma_2)$ such that $\supp(\sigma_1) = \Delta_1, \sigma_2' > 0$ a.e. on $\Delta_1$ and $\supp(\sigma_2) = \Delta_2, \sigma_2' > 0$ a.e. on $\Delta_1$ whose orthogonal polynomials.**

**10. Thursday. April 8, 2010. 16:00. Room: 2.2.D.08.**

**Title: A fast method for the computation of Legendre expansions**

**Speaker: Arieh Iserles, University of Cambridge, UK**

**11. Thursday. March 25, 2010. 16:00. Room: 2.2.D.08.**

**Title: Asymptotic solvers for oscillatory systems of ODEs**

**Speaker: Alfredo Deaño Cabrera, UC3M, Madrid**

**Abstract:** Systems of ODEs subject to oscillatory forcing terms are important in several applications, but they are usually difficult to solve numerically using standard methods such as Runge-Kutta. Following recent advances in the theoretical and numerical understanding of these problems, we present and alternative method that combines asymptotic expansions of the solution in inverse powers of the oscillatory parameter with modulated Fourier series. Numerical examples are included to show the efficiency of this approach. This is joint work with M. Condon (Dublin) and A. Iserles (Cambridge).

12. Thursday. March 11, 2010. 16:00. Room: 2.2.D.08.

**Title: How much indeterminacy may ﬁt in a moment problem. An example**

**Speaker: Franciszek Hugon Szafraniec, Jagiellonski University, Krakow, Poland**

**Abstract:** Employing the possibility of extending a symmetric operator to a selfadjoint, which goes beyond the Hilbert space, we construct examples of indeterminate Hamburger moment sequences admitting a family of representing measures such that: the support of each of them is in arithmetic progression, the supports of all the measures together partition R, none of them is N-extremal, and all of them are of inﬁnite order. In fact our goal is twofold: to present an example and to propose a method which may be useful for obtaining other results of this kind.

13. Thursday. March 4, 2010. 16:00. Room: 2.2.D.08.

**Title: Leave it to Smith: Preserving structure in matrix polynomials**

**Speaker: Steven Mackey, Department of Mathematics, Western Michigan University**

Abstract: Polynomial eigenvalue problems arise in many applications, such as the vibration analysis of mechanical systems, optical waveguide design, molecular dynamics, and optimal control. A much-used computational approach to such problems starts with a linearization of the underlying matrix polynomial P, such as the companion form, and then applies a general purpose algorithm to the linearization. Often, however, the polynomial P has some additional algebraic structure, leading to physically significant spectral symmetries which are important for computational methods to respect. In this situation it can be advantageous to use a linearization with the same structure as P, if one can be found. It turns out that there are structures polynomials for which a linearization with the same structure does not exist. Using the Smith form as the central tool, we describe which matrix polynomials from the classes of alternating, palindromic, and skew symmetric polynomials allow a linearization with the same structure.

**14. Thursday. February 25, 2010. 16:00. Room: 2.2.D.08.**

**Title: Nikishin systems and simultaneous quadrature**

**Speaker: Guillermo López Lagomasino, UC3M, Spain**

**Abstract:** This talk is a continuation of the one given on November 19, 2009. We will construct simultaneous quadrature rules for Nikishin systems of measures, prove their convergence, and show how this problem is connected with an extension of Markov's theorem on the convergence of diagonal Padé approximants.

15. Thursday. February 10, 2010. 16:00. Room: 2.2.D.08.

**Title: Twisted Green's (CMV-like) matrices and their factorizations, Laurent polynomials and Digital Filter Structures**

**Speaker: Pavel Zhlobich (Department of Mathematics, University of Connecticut)**

**Abstract:** Several new classes of structured matrices have appeared recently in the scientific literature. Among them there are so-called CMV and Fiedler matrices which are found to be related to polynomials orthogonal on the unit circle and Horner polynomials, respectively. Both matrices are five diagonal and have a similar structure, although they have appeared under completely different circumstances.

In a recent paper by Bella, Olshevsky and Zhlobich, it was proposed a unified approach to the above mentioned matrices. Namely, it was shown that all of them belong to a wider class of twisted Green's matrices.

We will use this idea to show that the factorizability of CMV and Fiedler matrices into a product of planar rotations in the n-dimensional space is also inherited by twisted Green's matrices. Shortly, for a given Hessenberg Green's matrix of size n, the intarchange of factors in the factorization leads to 2^n different twisted Green's matrices.

Polynomials orthogonal on the unit circle (those related to CMV matrices) and Horner polynomials (related to Fidler matrices) satisfy different types of recurrence relations. We will show that the characteristic polynomials of twisted Green's matrices satisfy certain recurrence relations which are, therefore valid for both Szegö's and Horner polynomials.

CMV matrix appeared in the scientific literature in connection with Laurent polynomials orthogonal on the unit circle. Fiedler matrix was developed purely from its factorization. We will show that an infinite-dimensional twisted Green's matrix serve as the operator of multiplication by "z" in the linear space of complex Laurent polynomials. Our development doesn't use orthogonallity in any sense and is based on the factorization and recurrence relations only. In the case of finite dimensional matrices we are able to give an explicit form of an eigenvector and all the generalized eigenvectors for a given eigenvalue.The final part of our talk will be devoted to Kimura's approach to CMV matrices, i.e. Signal Flow Graphs (SFG) approach. We will exploit the tool of SFG to visualize all the theoretical results for twisted Green's matrices as well as to show how they can be used in construction of new types of Digital Filters.

This is joint work with Vadim Olshevsky and Gilbert Strang.

16. Thursday. January 28, 2010. 16:00. Room: 2.2.D.08.

**Title:**** ****Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains.**

**Speaker:**** Eva Touris Lojo, UC3M, Spain.**

**Abstract:** In this article we investigate the Gromov hyperbolicity of Denjoy domains equipped with the hyperbolic or the quasihyperbolic metric. We first prove the existence of suitable families of quasigeodesics. The main result shows that a Denjoy domain is Gromov hyperbolic with respect to the hyperbolic metric if and only it is Gromov hyperbolic with respect to the quasihyperbolic metric. Using these tools we give a characterization in terms of Euclidean distances of when the domains are Gromov hyperbolic. We also give several concrete examples of families of domains satisfying the criteria of the theorems.

**17. Thursday. January 21, 2010. 16:00. Room: 2.2.D.08.**

**Title:**** ****Linearizations of singular matrix polynomials.**

**Speaker:**** Fernando de Terán Vergara, UC3M, Spain.**

**Abstract:** In this talk we give an overview to the systematic study of linearizations of singular matrix polynomials initiated by the authors in the past few years. This work was motivated by the introduction, in 2004 and 2006, of new families of linearizations (for regular matrix polynomials) extending the classical *companion forms* and enjoying some interesting features for applications. We will show that these families can be extended to the singular case and we will review some relevant features of these linearizations.

**18. Thursday. January 14, 2010. 16:00. Room: 2.2.D.08.**

**Title:**** ****Memory on birth-and-death processes.**

**Speaker:**** Ulises Fidalgo Prieto, UC3M, Spain.**

**Abstract:** Following Karlin's and Mc Gregor's point of view on one dimensional Brownian motion corresponding to a single particle we study a kind of generalized birth and death processes with memory. In such processes the transition probability functions satisfy differential equations with operators that have n-diagonal matrix representations (n > 3). In the research we needed to analyze strong asymptotic behaviours on mixed type of multi-orthogonality of polynomials. To this end we used Riemann-Hilbert problems techniques. As well, new results on momentum problems associated to multi-orthogonality were very important. The works on this last field (momentum problems) is still in progress.

**19. Thursday. December 17, 2009. 16:00. Room: 2.2.D.08**

**Title:**** Fast transforms for classical orthogonal polynomials and their associated functions (Colloquium)****.**

**Speaker:**** Jens Keiner, Universitaet zu Luebeck. Institut fuer Mathematik, Germany.**

**Abstract:** Classical orthogonal polynomials and functions have lots of applications. They are, for example, important for several generalizations of the discrete Fourier transform. These involve classical orthogonal polynomials or their associated functions rather than the usual complex exponentials. Often, these transforms are motivated by a known theoretical framework from mathematical physics. Nowadays, there is a demand to enable efficient and, at the same time, stable numerical computation of these transforms. While the plain discrete Fourier transform and its fast cousin, the fast Fourier transform (FFT), have excellent numerical properties, the calculation of generalized Fourier transforms involving classical orthogonal polynomials or their associated functions turns out to be much more challenging. In recent years, substantial progress has been made towards more efficient and more stable numerical algorithms which are required for any serious real world application. My talk will focus on two techniques that offer compelling advantages over other options. The ﬁrst approach is based on a connection to semiseparable matrices. The upshot is that the connection problem for classical orthogonal polynomials and their associated functions can be reduced to a structured eigenproblem that can be handled efficiently This allows to obtain a building block that, with a few things added, yields the desired transformation. While being useful for numerical applications, this new line of thought is also interesting from a purely theoretical point of view. The second technique is based on applying the well-known fast multipole method (FMM) to the connection problem. New results establish the conditions under which this can be done. As an application, we will show how the obtained methods can be used to compute fast Fourier transforms on the sphere S2 and the rotation group SO(3).

**20. Thursday. December 10, 2009. 16:00. Room: 2.2.D.08.**

**Title: Polinomios ortogonalmente aditivos y P-ortogonalidad.**

**Speaker: ****Pablo Linares**** Briones, U.C.M., Spain.**

**Abstract:** En esta charla haremos un recorrido por la teoría de polinomios ortogonalmente aditivos en retículos de Banach haciendo especial hincapié en el Teorema de Representación que relaciona este tipo de polinomios con aplicaciones lineales. Este teorema ha probado ser muy versátil teniendo aplicaciones en diversas áreas de la matemática. Nos centraremos aquí en un nuevo concepto de ortogonalidad, basado en la ortogonalidad clásica de polinomios con respecto a una aplicación lineal, pero definida ahora respecto a un polinomio ortogonalmente aditivo.Este trabajo se realiza conjuntamente con Alberto Ibort y José Luis G. Llavona.

**21. Thursday. December 3, 2009. 14:30. Room: 2.2.D.08.**

**Title: ****Asymptotics for a generalization of Laguerre polynomials (Colloquium)****.**

**Speaker: ****Ana Peña Arenas****, U. Zaragoza, Spain.**

**Abstract: **We consider a generalization of the classical Laguerre polynomials by the addition of r terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic properties. Thus, our aim is to study the asymptotics of this sequence of nonstandard orthogonal polynomials. In fact, we obtain the relative asymptotics and Mehler--Heine type formulas for these polynomials and, as a consequence, we prove that there exists an acceleration of the convergence of the smallest positive zeros of these generalized Laguerre polynomials towards the origin.

**22. Thursday. November 19, 2009. 15:30. Room: 2.2.D.08.**

**Title: ****Sistemas de Nikishin y cuadraturas simultaneas.**

**Speaker: ****Guillermo López Lagomasino, U. Carlos III de Madrid, Spain.**

**Abstract: **Recientemente, U. Fidalgo y yo hemos probado que los sistemas de Nikihin son perfectos. Veremos como ello permite construir cuadraturas simultaneas interpolatorias de tipo Gauss-Jacobi para aproximar con un mismo sistema de nodos las integrales de una cierta funcion respecto a varias medidas.

**23. Thursday. November 12, 2009. 16:00. Room: 2.2.D.08.**

**Title: ****Nuevas técnicas de cálculo de variaciones en el estudio del operador de multiplicación.**

**Speaker: ****José Manuel Rodríguez García, U. Carlos III de Madrid, Spain.**

**Abstract:** La acotación del operador de multiplicación por la variable independiente tiene importantes consecuencias en el estudio de los polinomios ortogonales de Sobolev, puesto que permite garantizar la acotación uniforme de los ceros de dichos polinomios, y esto a su vez permite encontrar el comportamiento asintótico de dichos polinomios ortogonales. En el estudio de la acotación de dicho operador se han venido utilizado diversos métodos, pero hasta ahora no se habían usado para ello técnicas de cálculo de variaciones. Explicaremos cómo con estas técnicas se puede garantizar la acotación para una amplia clase de productos de Sobolev no diagonales. Este es un trabajo conjunto con Ana Portilla, Yamilet Quintana y Eva Tourís.

**24. Thursday. November 5, 2009. 16:00. Room: 2.2.D.08.**

**Title: ****Integrable systems, spectral transformations, and orthogonal polynomials on the unit circle.**

**Speaker: ****Francisco Marcellán Español, U. Carlos III de Madrid, Spain.**

**Abstract:** In this talk we will present a survey about properties of orthogonal polynomials on the unit circle (OPUC) related to one-parameter deformations of the corresponding nontrivial probability measure of orthogonality μ. First, we analyze the connection between Toda lattices and orthogonal polynomials in the real line. Second, we study the time dynamics of the Verblunsky parameters, i.e. the evaluation at z=0 of such orthogonal polynomials, focussing our attention in the Schur flow, which is characterized by a complex semidiscrete modified KdV equation and where a discrete analogue of the Miura transformation appears. Third, the Lax pair for the CMV and GGT matrices associated with such deformations is discussed. Finally, some open problems in the framework of spectral transformations of probability measures supported on the unit circle will be analyzed. The study of such perturbations of measures from the point of view of the relation between the corresponding sequences of orthogonal polynomials and, as a consequence, between their GGT matrices.

**25. Thursday. October 29, 2009****. 16:00. Room: 2.2.D.08.**

**Title: ****Orthogonality with respect to a Jacobi differential operator and a fluid dynamics model.**

**Speaker: ****Héctor Pijeira Cabrera, U. Carlos III de Madrid, Spain.**

**Abstract: **We study algebraic and analytic properties of the polynomial solutions of the differential equation A(x) y'' + B(x) y'= λ L_{n}(x), where A(x):= 1-x^{2}, B(x):=β- α -(α + β + 2)x, λ:= n(1+n+α +β), α , β > -1 and L_{n} the n-th Monic Orthogonal Polynomial with respect to a no negative, finite Borel measure μ with support on [-1,1] (paper joint with Jorge Borrego).

**26. Thursday. October 22, 2009. 16:00. Room: 2.2.D.08.**

**Title: ****A new algorithm for computing the Geronimus transformation with large shifts.**

**Speaker: Alfredo Deaño ****Cabrera, U. Carlos III de Madrid, Spain.**

**Abstract:** A monic Jacobi matrix is a tridiagonal matrix which contains the parameters of the three-term recurrence relation satisfied by the sequence of monic polynomials orthogonal with respect to a measure μ. The basic Geronimus transformation with shift α transforms the monic Jacobi matrix associated with a measure μ into the monic Jacobi matrix associated with the new measure d μ/(x-α)+C δ(x-α), for some constant C. We will examine the algorithms available to compute this transformation and will propose a more accurate method, estimate its forward errors, and prove that it is forward stable. In particular, we show that for C=0 the problem is very ill-conditioned. (Joint work with M. I. Bueno (UC Santa Barbara) and E. Tavernetti (UC Davis)).

**27. Thursday. October 15, 2009. 16:00. Room: 2.2.D.08.**

**Title: Asymptotics of orthogonal polynomials for a weight with a jump: local behavior.**

**Speaker: ****Andrei Martínez Finkelshtein, U. de Almería, Spain.**

**Abstract:** For orthogonal polynomials on [−1, 1] with respect to a generalized Jacobi weight modified by a step-like function we obtain strong uniform asymptotics in the whole plane. In this talk the main focus is on the local behavior at the jump. We study the asymptotics of the Christoffel-Darboux kernel and show that the zeros of the orthogonal polynomials no longer exhibit the clock behavior. The main tool is the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann-Hilbert problems characterizing the orthogonal polynomials. The local analysis at the jump is carried out in terms of the confluent hypergeometric functions. Incidentally, we establish some properties of these functions that may have an independent interest.

**Thursday. June 25, 2009. 16:00. Room: 2.2.D.08.****Title: Strong asymptotic on mixed Nikishin-Jacobi polynomials.****Speaker:****Ulises Fidalgo Prieto, U. Carlos III de Madrid, Spain.****Abstract:**Using Fokas', Its' and Kitaev's approach, we are going to present some relationships between Riemann-Hilbert problems and different kinds of orthogonality of polynomials. With the known steepest decent method, those Riemann-Hilbert problems allow strong asymptotic expressions for sequences of orthogonal polynomials to be found (here the orthogonality is understood in a wide sense). Our talk is going to be focused in obtaining strong asymptotic expressions for sequences of mixed type of multiple orthogonal polynomials corresponding to two Nikishin systems. Both of these two Nikishin systems are generated by systems of measures with type Jacobi weights. As a motivation, our exposition will start with an analysis of a one-dimensional Brownian movement corresponding to a single particle, where a strong asymptotic expression for a sequence of Jacobi polynomials plays a fundamental role.

**Thursday. June 18, 2009. 16:00. Room: 2.2.D.08.****Title:****A numerical solution of the constrained weighted energy problem and its relation to rational Lanczos iterations****.****Speaker:****Karl Deckers, Departament of Computer Science, K. U. Leuven (Belgium).****Abstract:**A numerical algorithm is presented to solve the constrained weighted energy problem from potential theory. As one of the possible applications of this algorithm, the convergence properties of the rational Lanczos iteration method for the symmetric eigenvalue problem are studied. The constrained weighted energy problem characterizes the region containing those eigenvalues that are well approximated by the Ritz values. The region depends on the distribution of the eigenvalues, on the distribution of the poles, and on the ratio between the size of the matrix and the number of iterations. The algorithm presented gives the possibility to find the boundary of this region in an effective way. We give numerical examples for different distributions of poles and eigenvalues and compare the results of our algorithm with the convergence behavior of the explicitly performed rational Lanczos algorithm.

**Thursday. June 4, 2009. 16:00. Room: 2.2.D.08.****Title:****Relative Asymptotics For Orthogonal Matrix Polynomials: Degenerate Case****.****Speaker:****Hossain. O. Yakhlef****, U. Carlos III de Madrid, Spain.****Abstract:**We study the asymptotic behaviour of the ratio of two sequences of orthonormal matrix polynomials P_{n}(x;α) and P_{n}(x;β) (the matrix of measures dβ = dα+Mδ, with δ is the dirac measure) when the matrix parameters {A_{n}(α), B_{n}(α)} in the three-term recurrence relation are convergent (we assume A= lim A_{n}to be singular) or are unbounded (we consider the degenerate case).

**Thursday. may 28, 2009. 16:00. Room: 2.2.D.08.****Title: Mixed type of multiple orthogonal polynomilas for two Nikishin systems.****(Prelectura de tesis doctoral).****Speaker: Abey López García, Depatament of Mathematics, Vanderbilt University, Nashville, Tn.,****U.S.A****.****Abstract:**We study logarithmic and ratio asymptotic of linear forms constructed from a Nikishin system which satisfy orthogonality conditions with respect to a system of measures generated from another Nikishin system. This condition combines type I and type II multiple orthogonal polynomials. The logarithmic asymptotic of the linear forms is expressed in terms of the extremal solution of an associated vector valued equilibrium problem for the logarithmic potential. The ratio asymptotic is described by means of a conformal representation of an appropiated Riemann surface of genus zero onto the extended complex plane.

**Thursday. may 21, 2009. 16:00h. Room: 2.2.D.08.****El operador de multiplicación, localización de ceros y asintótica de polinomios extremales para normas de Sobolev no diagonales.****Speaker:****Eva Touris Lojo, U. Carlos III de Madrid.****Abstract:**En un trabajo previo de Lagomasino, Pérez Izquierdo y Pijeira se estudia la localización de ceros y el comportamiento asintótico de polinomios extremales para normas de Sobolev no diagonales (es decir, en la norma aparecen productos de la función y de sus derivadas de diferentes órdenes). Los polinomios ortogonales de Sobolev con respecto a dicha norma son un caso particular de dichos polinomios extremales (cuando p=2). Creemos que el enfoque de Lagomasino, Pérez Izquierdo y Pijeira es el más apropiado para el estudio del caso general (admitiendo derivadas hasta orden N). Sin embargo, si nos centramos en el caso N=1 (sólo aparecen primeras derivadas), que es el caso más usual, conseguimos una doble mejora de sus resultados: por un lado, debilitamos las hipótesis técnicas y, por otro, nuestras hipótesis involucran directamente los coeficientes de la matriz, en lugar de sus autovalores.

**Thursday. may 14, 2009. 16:00h. Room: 2.2.D.08.****Runge-Kutta convolution quadrature methods and transparent boundary conditions (Colloquium).****Speaker:****Cesar Palencia de Lara.****Abstract:**Recent results concerning Runge-Kutta based convolution quadrature methods for abstract, well posed, linear, and homogeneous Volterra equations show a general representation of the numerical solution in terms of the continuous one. The interest of such a representation goes beyond the error and

stability analysis, since it explains that the numerical solution, under suitable methods, inherits some important qualitative properties, such as positivity or contractivity, of the exact solution. In this talk we will summarize this theory and use it for the construction of fully discrete, transparent boundary conditions, for the heat and Schroedinger equation, with good qualitative properties.

**Thursday. may 7, 2009. 12:00. Room: 2.2.D.08.****Title: El problema clásico de momentos, la aproximación racional y los polinomios ortogonales. (Colloquium)****Speaker: Manuel Bello Hernández, Dpto. de Matemática y Computación, U. de la Rioja, (****Spain****).****Abstract:**El problema clásico de momentos es muy importante en los aportes al análisis clásico en el período que va desde los trabajos de Stieltjes hasta los de Krein. La noción de medida (integral de Stieltjes), los aproximantes de Padé, los polinomios ortogonales, los métodos de cuadratura, la extensión de funcionales lineales positivos (teorema de Riesz) y los problemas de valores frontera de funciones analíticas tienen sus raíces en el estudio del problema clásico de momentos. En la charla veremos como la aproximación racional unilateral está directamente relacionada con la solución del problema clásico de momentos. La conexión anterior se utilizará en el estudio de la asintótica relativa de polinomios ortogonales y de la convergencia de aproximantes de Padé de funciones que son la suma de una fracción racional y la transformada de Cauchy de una medida positiva con soporte en la recta real. En esta dirección se presentarán resultados que extienden teoremas obtenidos previamente por Guillermo López.

**Thursday.****april 30, 2009****. 16:00. Room: 2.2.D.08.****Title: Sobolev type orthogonality: a hydrodynamical interpretation.****Speaker: Hector Pijeira Cabrera, Dpto. de Matemática, U. Carlos III de Madrid, (****Spain****).****Abstract**: We study the algebraic (Zeros location), differential and asymptotic properties of orthogonal polynomials with respect to discrete--continuous Sobolev type inner product with respect to Gegenbauer measures, these polynomials are also primitives of Gegenbauer polynomials. A hydrodynamic model for source points location of a flow of an incompressible fluid with preassigned stagnation points on [-1,1] is solved.

**Thursday.****April 23, 2009****. 16:00. Room: 2.2.D.08.****Title: Sistemas integrables y polinomios ortogonales matriciales.****Speaker: Carlos Álvarez Fernández, Departamento de Física Teórica II, Facultad de Ciencias Físicas, U.C.M. (****Spain****).****Abstract**: Se explorarán las conexiones entre las reducciones Toeplitz-Hankel de la jerarquía de Toda multicomponente y las familias de polinomios biortogonales y ortogonales matriciales en la recta y en el círculo. Igualmente se tratarán casos de ortogonalidad generalizada y sus problemas asociados (problemas de Riemann-Hilbert y relaciones de recurrencia).

**Thursday.****April 16, 2009****. 16:00. Room: 2.2.D.08.****Title: On linear forms containing Euler constant (colloquium,).****Speaker: Alexader Aptekarev, Keldysh Institute of Applied Mathematics, Moscow (Russia).****Abstract**: We discuss arithmetical properties of mathematical constants related with infinite summs of the inverse powers, i.e. the Euler constant, the values of the Riemann Zeta-function at the odd points. In 1978 Aperi has proved that Zeta(3) is irrational number. Since that time there was no much progress achieved in this field. In 2000 Rivoal proved that among the values of the Riemann Zeta-function at the odd points there are infinitely many irrational numbers and in 2001 Zudilin has precised this result proving that one number from Zeta(5), Zeta(7), Zeta(9) and Zeta(11) is irrational. We tell about similar result involving Euler constant.- Thursday. april 2, 2009. 16:00. Room: 2.2.D.08.
- Title: Zeros of Heine-Stieltjes polynomials and critical measures (Colloquium).
- Speaker: Andrei Martínez Finkelshtein, Universidad de Almería (Spain).
- Abstract: Heine-Stieltjes polynomials are solutions of certain second order linear ODEs with polynomial coefficients. They generalize the families of classical orthogonal polynomials and arise in several areas of mathematics and phisics. Their zeros are saddle points of certain discrete energy functionals, and the description of their asymptotics requires an extension of the standard notion of equilibrium. We analyze some related potential theoretic concepts, as well as discuss briefly their connection with trajectories of quadratic differentials and extremal problems on the plane. This is a partial report of a joint work with E. Rakhmanov.

- Thursday. march 26, 2009. 16:00. Room: 2.2.D.08.
- Title: The numerical inversion of the Laplace transform with applications to evolution problems
*.* - Speaker: María López Fernández, Universidad Carlos III de Madrid (Spain).
- Abstract: Laplace transforms which admit a holomorphic extension to some sector strictly containing the right half plane and exhibiting there a potential behavior are considered (sectorial Laplace transforms). A spectral order, parallelizable method for their numerical inversion is proposed. The method takes into account the available information about the errors arising in the evaluations. The application of the Laplace transform and the proposed inversion algorithm to integrate in time evolution equations is considered

- Thursday. march 12, 2009. 16:00. Room: 2.2.D.08.
- Title: Desigualdad de Hardy-Sobolev mejorada, constantes optimales y aplicaciones
*.* - Speaker: Eduardo Colorado Heras, Universidad Carlos III de Madrid (Spain)
- Abstract: Se verán algunos resultados sobre las desigualdades de Hardy-Sobolev "optimales" y constantes optimales (asociadas a dichas desigualdades) bajo distintos marcos funcionales. Finalmente, se presentarán algunas aplicaciones en el marco de las ecuaciones en derivadas parciales elípticas no lineales con distintos tipos de condiciones de contorno. El contenido de la Charla forma parte de los siguientes astículos:
- E. Colorado, I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions. J. Funct. Anal., 2003.
- B. Abdellauoi, E. Colorado, I. Peral, Some remarks on elliptic equations with singular potentials and mixed boundary conditions. Adv. Nonlinear Stud., 2004.
- B. Abdellauoi, E. Colorado, I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities. Cal. Var. Partial Differential Equations, 2005.
- B. Abdellauoi, E. Colorado, I. Peral, Effect of the boundary conditions in the behavior of the optimal constant of some Caffarelli-Kohn-Nirenberg inequalities. Application to some doubly critical nonlinear elliptic problems. Adv. Differential Equations, 2006.

- Thursday. march 5, 2009. 16:00. Room: 2.2.D.08.
- Title: Metric graphs and boundary triples (Colloquium)
*.* - Speaker: Olaf Post, Humbodt Universitaet Berlin (Germany).
- Abstract: We use the notion of boundary triples in order to analyse second order differential operators on metric graphs. Boundary triples are an abstract formulation of Green's formula for the Laplacian on a domain with boundary. In particular, for equilateral graphs we obtain a simple relation of a metric graph differential operator with a discrete analogue acting as difference operator on the vertices. The latter difference operator generalises the usual combinatorial discrete Laplacian.

- Thursday. february 26, 2009. 16:00. Room: 2.2.D.08.
- Title: Nikishin systems are perfect.
- Speaker: Guillermo López Lagomasino, Universidad Carlos III de Madrid (Spain).
- Abstract: A long standing open problem in the theory of multiple orthogonal polynomials is whether or not Nikishin systems are perfect. That is, whether the corresponding multiple orthogonal polynomials are of maximum possible degree (or uniquely determined by the orthogonality relations they satisfy) no matter what multi-index you take. This question is of great importance in applications of Hermite-Padé approximation to number theory, convergence theory of simultaneous approximation, and in extending the asymptotic theory of standard orthogonal polynomials to the case of multiple orthogonality. Hermite himself proved that systems of exponentials are perfect and this plays a fundamental role in his proof of the trascendence of number $e$. We give a positive answer to the conjecture. The proof relies on a property which is of independent interest and in a sense extends the fundamental theorem of algebra to generalized polynomials which can be expressed as polynomial combination of Markov functions generated by a Nikishin system. We also prove that the zeros of these multiple orthogonal polynomials are simple and those of "consecutive" multi-indices interlace.

- Thursday. february 19, 2009. 16:00.Room: 2.2.D.08.
**Title: Geodésicas que escapan a infinito: azar y necesidad.****Speaker: Eva Touris Lojo, Universidad Carlos III de Madrid (Spain).**- Abstract: Jose L. Fernández y María V. Melián en su artículo "Escaping geodesics of Riemannian surfaces" prueban que dada una superficie R con área inifinita y curvatura constante -1, y un punto p en R, el conjunto de direcciones v tales que la geodésica que parte de p con dirección v escapa a infinito (es decir, abandona eventualmente todo compacto de la superficie) tiene dimensión de Hausdorff 1. Este tipo de resultados se traduce en interesantes resultados sobre el comportamiento radial de funciones holomorfas del disco unidad en R. Nosotros hemos obtenido el resultado equivalente para variedades de curvatura negativa variable. El concepto de hiperbolicidad en el sentido de Gromov aisla en cierto sentido las propiedades esenciales de las variedades con curvatura negativa y por ello hemos trasladado este problema a un problema en espacios de Gromov. Más concretamente, dado un espacio de Gromov X y un grupo G de isometrías de X, queremos estudiar la dimensión (visual) de la frontera de X menos el conjunto límite cónico de G.

- Thursday. february 12, 2009. 16:00. Room: 2.2.D.08.
- Title: Resultados sobre aproximación e interpolación simultánea.
- Speaker: José G. LLavona, Departamento de Análisis Matemático, UCM, Madrid (Spain).
- Abstract: En los comienzos de los años 1970 aparece el interés de extender, al caso infinito dimensional, los clásicos teoremas de aproximación para funciones definidas en Rn.Vamos a tratar este tema en relación al Teorema de Weierstrass, y como aplicación de los resultados conseguidos, presentaremos un teorema de aproximación e interpolación simultánea de funciones diferenciables por polinomios.

Referencias

J. G. Llavona, Approximation of differentiable functions, Advances in Math., Supplementary Studies 4 (1979) 197-221.

J. G. Llavona, Approximation of continuously differentiable functions, North-Holland Mathematics Studies, 130. Notas de Matemática (112). Amsterdam, 1

- Thursday. february 5, 2009. 16:00. Room: 2.2.D.08.
- Title: Reconfiguration of cube-style modular robots.
- Speaker: Vera Sacristán Dept. Matemàtica Aplicada II, U. Politècnica de Catalunya.
- Abstract: I will present some recent results and ongoing research on self-reconfiguration of modular robots composed of cubical units (atoms) arranged in a lattice configuration and capable of performing four basic actions: expand and contract, as well as attach and detach from neighbors. This work includes centralized and distributed solutions, as well as massive parallelization of the robot moves. The proposed algorithms are in-place, guarantee the connection of the reconfiguration space and improve the previously known algorithms. This is joint work with G. Aloupis, S. Collette, M. Damian, E. Demaine, R. Flatland, S. Langerman, J. O'Rourke, S. Ramaswami and S. Wuhrer. And partially also with D. El-Khechen and V. Pinciu, as well as with O. Aichholzer, T. Hackl and B. Vogtenhuber.

- Thursday. January 29, 2009. 16:00. Room: 2.2.D.08.
**Title: Critical points and level Sets in the exterior boundary problems Conferenciante: Daniel Peralta Salas, Universidad Carlos III de Madrid.****Speaker: Daniel Peralta Salas, Universidad Carlos III de Madrid (Spain).**- Abstract: We study some geometrical properties of the critical set of the solutions to an exterior boundary problem in Rn\(Ω U δΩ), where Ω is a bounded domain with C2 connected boundary. We prove that this set can be nonempty (in fact, of codimension 3) even when Ω contractible, thereby settling a question posed by Kawohl. We also obtain new sufficient geometric criteria for the absence of critical points in this problem and analyze the properties of the critical set for generic domains. The proofs rely on a combination of classical pothential theory, transversality techniques and the geometry of real analytic sets..

- Thursday. January 22, 2009. 16:00. Room: 2.2.D.08.
- Title: Calculating Rational Best Approximation. COLLOQUIUM DEPARTAMENTO DE MATEMÁTICAS.
- Speaker: Herbert Stahl, THF-Berlin, Germany.
- Abstract: We shall present a new algorithm for the calculation of rational best approximants to real functions on a real interval. The development of the algorithm has been motivated by a need to calculate rational best approximants on (-∞,0] to functions that are similar to the exponential function, as for instance, rational perturbations of this function. The talk will be opened with several examples of this kind. After that, the basic structure of the new algorithm will be explained (it shares several aspects with the Remez algorithm), and we then have a closer look on some of its key elements.

- Thursday. January 15, 2009. 16:00. Room: 2.2.D.08.
- Title: Soluciones aproximadas del sistema de EDO que define las geodésicas.
- Speaker: José Manuel Rodríguez, Univeridad Carlos III de Madrid, Spain.
- Abstract: Calcular explícitamente las geodésicas en una superficie es una tarea extraordinariamente complicada (salvo en unos cuantos casos especialmente sencillos), debido a que es necesario resolver un sistema de dos ecuaciones diferenciales de segundo orden altamente no lineal. La situación es especialmente complicada para las métricas de Poincaré (debido a que no conocemos la expresión explícita de la densidad de la métrica, y por tanto, ni siquiera es posible escribir explícitamente el sistema de ecuaciones) y quasihiperbólica (para la que tampoco es posible escribir el sistema, ya que involucra derivadas de una función que en muchos puntos no es diferenciable). No obstante, conocer dichas geodésicas es muy útil para resolver diversos problemas relacionados con dichas métricas, como el estudio de la hiperbolicidad de Gromov o la existencia de la desigualdad isoperimétrica. En este trabajo se consigue resolver de forma aproximada dicho problema para una clase de superficies muy general: los dominios de Denjoy. Este resultado es directamente aplicable al estudio de los dos problemas anteriormente mencionados. Estos resultados se extraen de unos trabajos conjuntos con Peter Hasto, Ana Portilla, Eva Tourís y José María Sigarreta.

- Thursday. December 11, 2008. 16:00. Room: 2.2.D.08.
- Title: El problema de la integración en forma cerrada de Ecuaciones Diferenciales. COLLOQUIUM DEPARTAMENTO DE MATEMÁTICAS.
- Speaker: Juan J. Morales, Universidad Politécnica de Madrid, Spain.
- Abstract: En esta charla se discutirá el problema de la integración en forma cerrada de ecuaciones diferenciales ordinarias. Nuestra tesis, que ilustraremos mediante ejemplos concretos, es que -a diferencia de lo que se suele creer- no es la teoría de Lie de simetrías la que es más útil para la integración mediante métodos analíticos de una ecuación diferencial, sino la teoría de Galois de ecuaciones diferenciales.

- Thursday. December 4, 2008. 16:00. Room: 2.2.D.08.
- Title: El problema Isoperimétrico en los Espacios Producto. COLLOQUIUM DEPARTAMENTO DE MATEMÁTICAS.
- Speaker: Jesús Gonzalo, Universidad Autonoma de Madrid, Spain.
- Abstract: El problema que da título a esta charla es clásico: encerrar un volumen dado utilizando la menor cantidad posible de área. Ya para el espacio Euclídeo conduce a problemas de Análisis no triviales. Pero además tiene perfecto sentido cuando el espacio ambiente es una variedad Riemanniana, y entonces conduce a nuevos problemas. El problema de estabilidad busca una superficie que al menos sea un mínimo local del área entre todas las que encierran la misma "cantidad de aire". Esto es necesario para que podamos realizar tal superficie como una pompa de jabón. Los dos problemas están muy relacionados, pues una superficie isoperimétrica es un mínimo global. Mostraremos una solución completa a ambos problemas, para volumen grande, cuando el espacio ambiente es el producto de cualquier variedad compacta por un espacio Euclídeo.

- Thursday. November 27, 2008. 16:00. Room: 2.2.D.08.
- Title: El uso de bases y "frames" en Teoría de Muestreo.
- Speaker: Antonio G. García, Universidad Carlos III de Madrid, Spain.
- Abstract: En el seminario repasaremos, brevemente, los conceptos de base ortonormal, base de Riesz y frame en un espacio de Hilbert separable H, poniendo de manifiesto las propiedades que se usarán, posteriormente, en su aplicaci on a la Teoría de Muestreo. En particular, aquellas que nos permiten la recuperaci "estable" de cada x de H a partir de la sucesión {n>} (con n en N) de l2(n). Se justicará también la necesidad del uso de bases de Riesz y de frames, ya que las bases ortonormales se muestran insuficientes en el estudio de muchas cuestiones matemáticas con importancia no sólo teórica sino también aplicada.

La teoría general anteriormente expuesta se utilizará para obtener teoremas de muestreo en los espacios clásicos de Paley-Wiener, así como en espacios invariantes por traslación en L2(R), i.e., subespacios cerrados de L2(R) generados por las traslaciones en los enteros de una cierta función f de L2(R).

- Thursday. November 20, 2008. 16:00. Room: 2.2.D.08.
- Title: Orthogonal polynomials in several variables and spectral theory of partial differential operators.
- Speaker: Francisco Marcellán Español, Universidad Carlos III de Madrid, Spain.
- Abstract: Following an historical approach based on the extension of the Routh-Bochner characterization of classical orthogonal polynomials in one variable (Hermite, Laguerre, Jacobi,and Bessel) , we present a constructive approach of some families of two variable orthogonal polynomials which are eigenfunctions of second order partial differential operators with polynomial coefficients. Then, using standard techniques for the symmetrization of partial differential operators, we can deduce the weight function as well as the corresponding domain of orthogonality. In the more general framework of the orthogonality associated with moment functionals on the linear space of polynomials in two variables with real coefficients, classical orthogonal polynomials are defined in terms of a matrix analogue of the Pearson differential equation that such a functional satisfies. They can also be characterized as the polynomial solutions of a matrix second order partial differential equation.

- Thursday. November 13, 2008. 16:00. Room: 2.2.D.08.
- Title: Orthogonal polynomials and stochastic processes. COLLOQUIUM DEPARTAMENTO DE MATEMÁTICAS.
- Speaker: F. Alberto Grunbaum, Math Dept UC Berkeley, U.S.A.
- Abstract: I will discuss a few examples (as well as a general scheme) where orthogonal polynomials and their related spectral analysis play a useful role in probability theory. My main emphasis is in looking for ways to extend this analysis to new situations.

- Thursday. November 6, 2008. 16:00. Room: 2.2.D.08.
- Title: Spectral aproximation in discrete and quantum graphs.
- Speaker: Fernando Lledó, Universidad Carlos III de Madrid, Spain.
- Abstract: In this talk we discuss the notion of Laplacians for discrete and quantum (or metric) graphs and mention some spectral relations between these operators. Using this result and a bracketing procedure for eigenvalues we consider the gap structure for infinite graphs. We conclude mentioning some examples where both, the discrete and the quantum graph, have spectral gaps.

- Thursday. October 30, 2008. 15:30. Room: 2.2.D.08.
- Title: On the multilinear Hausdorff problem of moments.
- Speaker: F. Alberto Ibort, Universidad Carlos III de Madrid, Spain.
- Abstract: The multilinear Hausdorff problem of moments will be discussed. Moreprecisely, given a family of moments $\mu_{k_1, \ldots, k_n}$, $k_1, \ldots, k_n = 1, 2, \ldots$, under which conditions they define an integral $n$-linear functional $L$ on the space $C[0,1]$. We will characterize families of moments determining integral $n$-linear functionals by means of a new weak uniform boundedness condition and we will compare it with the classical solution to the (linear) Hausdorff problem. Some problems related with the theory of biorthogonal polynomials will be discussed in the bilinear case.

- Thursday. October 23, 2008. 16:00. Room: 2.2.D.08. COLLOQUIUM DEPARTAMENTO DE MATEMÁTICAS.
- Title: Convergent Interpolation to Cauchy Transforms.
- Speaker: Maxim Yattselev, INRIA, Sophia Antipolis, France.
- Abstract: We consider multipoint Padé approximation to Cauchy transforms of complex measures. We show that if the support of a measure is an analytic Jordan arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that arc with Dini–smooth non–vanishing density, then the diagonal multipoint Padé approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the arc. This asymptotic behavior of Padé approximants is deduced from the analysis of underlying non–Hermitian orthogonal polynomials, for which we use classical properties of Hankel and Toeplitz operators on smooth curves. A construction of the appropriate interpolation schemes is explicitgranted the parametrization of the arc.. (joint work with L. Baratchart).

- Thursday. October 16, 2008. 16:00. Room: 2.2.D.08.
- Title: Aproximantes y polinomios multi-ortogonales mixtos.
- Speaker: Guillermo López Lagomasino, Universidad carlos III de Madrid, Spain.
- Abstract: Se presentara la conexion entre la aproximacion Hermite-Pade mixta (en la cual se combina tipo I y tipo II) en un marco general y los polinomios multiortogonales mixtos. Se particularizara en el caso en que la aproximacion mixta provenga de sistemas de Nikishin y se presentara algunos resultados recientes sobre la asintotica logaritmica y del cociente para polinomios multi-ortogonales mixtos de Nikishin.

- Thursday. October 9, 2008. 16:00. Room: 2.2.D.08.
- Title: Strong asymptotic for eigenpolynomials of a class of differential operators.
- Speaker: Jorge Borrego Morell, Universidad Carlos III de Madrid, Spain.
- Abstract: We give the strong asymptotic behavior of a family of polynomials which are

eigenfunctions of a class of differential operators, using the WKB method.

(joint work with H. Pijeira).

- Thursday. October 2, 2008. 16:00. Room: 2.2.D.08.
- Title: Neoclassical orthogonal polynomials.
- Speaker: David Gómez-Ullate, Universidad Complutense de Madrid, Spain.
- Abstract: The classical orthogonal polynomial systems (OPS) of Jacobi, Laguerre and Hermite are traditionally characterized as (the only) polynomial solutions of a Sturm-Liouville problem, after the classical results of Bochner and Lesky. We show that other complete Sturm-Liouville OPS exist if the sequence of polynomial eigenfunctions is not required to start with a constant. In this talk we will introduce these polynomial families and study some of their basic properties, which closely resemble those of the clasical families.

This is joint work with Niky Kamran (Mc Gill) and Robert Milson (Dalhousie).

- Thursday. September 25, 2008. 16:00. Room: 2.2.D.08.
- Title: Approximation problems in Systems Theory.
- Speaker: Diederich Hinrichsen, University of Bremen, Germany.
- Abstract: Se describen algunos problemas de aproximación que surgen en la Teoría de Sistemas y presento métodos para su solución (en particular la teoría de Adamyan-Arov-Krein). Explicaré algunos conceptos de la teoría de sistemas que sirven de soporte para estos métodos y discuto su relación con resultados de la teoría de operadores y la teoría de aproximación.