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 WS Season 2014/2015

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# Past seminars

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Speaker:
Alejandro Molano (
Universidad Pedagógica y Tecnológica de Colombia, Colombia).

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

Speaker:
Imed Ben Salah (Faculté des Sciences de Monastir, Tunisia)

Title:
Characterizations of second degree semi-classical forms of class one via a new characterization of finite-type relation.

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.

Title: The bispectral problem: the good, the bad and the ugly.

Abstract: I will mention a few open problems, some results on applications, and some comments on ways to reconsider the original question studied with Hans Duistermaat [1], particularly in a non-commutative context.

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

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Last Updated on Monday, 27 July 2015 22:13

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