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Weekly Seminar
 WS Season 2013/2014

# Seminar on Orthogonality, Approximation Theory and Applications

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Past seminars

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

Title: Alliance polynomial and hyperbolicity in regular graphs.

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.

Title: Systems of Markov type functions. Normality and convergence of Hermite-Padé approximants.

Abstract: This talk deals with simultaneous rational approximation. In particular we study algebraic and analytical properties of multiple orthogonal polynomial and Hermite-Padé approximants of the analytic functions of Markov type.

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.

Title: Bounds on Gromov hyperbolicity constant.

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.

Title: The number of total and reduced alignments between two DNA sequences.

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).

Title: Perturbation of linear functionals and the corresponding CMV matrices.

Abstract: Considering a Hermitian linear functional defined on the linear space of Laurent polynomials, we analyze the perturbation of the measure supported on the unit circle. The corresponding sequences of monic orthogonal polynomials as well as the  connection between the associated Verblunsky coefficients will be discussed. Then, specifically, we focus our attention on the structure of the Givens matrices of the perturbed linear functional, which is the main tool for comparison of the CMV matrices.

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

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

Title: Nonlocal heat equations: decay estimates and Nash inequalities.

Abstract: Decay estimates for heat like equations, or more generally, for submarkovian strongly continuous symmetric semigroups, are usually shown to be equivalent to some Sobolev or Nash inequalities. We establish a relation between the large time decay for this kind of semigroups with a restricted Nash inequality for the associated Dirichlet form. We then prove that  this Nash inequality indeed holds in the context of integral operators given by symmetric nonnegative kernels of Lévy type. The key point is that we prove those inequalities imposing only conditions on the kernel at infinity, thus including the case of unbounded transition probability densities. Joint work with Cristina Brändle.

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

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

Title: Three term relations for bivariate Koornwinder orthogonal polynomials.

Abstract: We study matrix three term relations for orthogonal polynomials in two variables constructed from orthogonal polynomials in one variable by using the so-called Koornwinder's construction. Using the three term recurrence relation for the involved univariate orthogonal polynomials, the explicit expression for the matrix coefficients in these three term relations are deduced. These matrices are diagonal or tridiagonal with entries computable from the one variable coecients in the respective three term recurrence relation. Moreover, some nice particular cases are considered.

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

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

Title: On zeros of a class of polynomials given by a three term recurrence relation.

Abstract: We study the monotonicity of zeros in connection with perturbed recurrence coefficients of polynomials satisfying a certain three--term recurrence relation.  As a particular case, we consider the Askey para--orthogonal polynomials on the unit circle generalizing a recent result about the monotonicity of their zeros. Finally, as a direct consequence of our approach, we obtain results on monotonicity of zeros of orthogonal polynomials on the real line.

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

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

Title: A theorem by Bojanov and Naidenov applied to families of Gegenbauer-Sobolev polynomials.

Abstract: In this talk  we will use a recent result due to B.D. Bojanov and N. Naidenov and some structural properties of a sequence of Gegenbauer-Sobolev orthogonal polynomials in order to study the maximization of  local extremum of the derivatives for such sequence.  Also, some illustrative numerical examples will be presented. This is a joint work with V. G. Paschoa  and D. Pérez.

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Thursday, March 20 2014, 16:00. Room 2.2.D08

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

Title: SDG maps and spectral problems for linear pencils in two variables.

Abstract: At first, we will consider the Schur-Delsarte-Genin (SDG) map in detail and show its relation to Jacobi and CMV matrices. In particular, we will see how two variable pencils appear in the framework of SDG maps. Further, such pencils give rise to a special type of Darboux transformations, which will be presented in the talk. The main feature of these Darboux transformations is that their dressing chains are intimately connected with quadratic algebras of Jacobi matrices and include the Askey-Wilson polynomials. The theory will be demonstrated on -1 Jacobi polynomials and Bannai-Ito polynomials.

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

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

Title: Symmetric Second Order Differential Operators with Matrix Polynomial Coefficients.

Abstract: In this work, we are concerned with symmetric second order differential operators for which there exist families of matrix orthogonal polynomials (MOPs) as eigenfunctions. These MOP families satisfy second order differential equations with matrix polynomial coefficients. Knowledge of the Rodrigues' formula of 2x2 families presents explicit expression of them. Besides this structural formula, the three term recurrence relation also characterizes them. In this work, we are concerned with symmetric second order differential operators for which there exist families of matrix orthogonal polynomials (MOPs) as eigenfunctions. These MOP families satisfy second order differential equations with matrix polynomial coefficients. Knowledge of the Rodrigues' formula of 2x2 families presents explicit expression of them. Besides this structural formula, the three term recurrence relation also characterizes them.

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

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

Title: Extension of the Geronimus transformation.

Abstract: In this lecture we extend the definition of Geronimus transform in the context of the polynomials orthogonal respect to symmetric bilinear forms and show that this transform lead to non-diagonal Sobolev type inner products. We apply this result to give some properties about the sequence of orthogonal polynomials associated. Finally, we show that the Jacobi matrix associated with the new symmetric bilinear form can be decomposed as a product of a lower triangular matrix and upper triangular matrix.

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

Speaker: Carlos Álvarez Fernández, Universidad Pontificia de Comillas.

Title: Factorization problems connecting Toda-type systems and multiple orthogonality.

Abstract: Our research work during these years has been to study the connections between orthogonal polynomials and integrable systems of Toda type. These connections between both theories was already known and can be made under different points of view. We focus on the factorization techniques as both problems can be formulated using the Gaussian factorization of an appropriate matrix. Considering matrices with generalized Hankel symmetries leads to different orthogonal systems, in particular orthogonal matrix polynomials, multiple orthogonal polynomials, and orthogonal Laurent polynomials. This point of view has proved particularly useful in the analysis of recurrence relations and Christoffel-Darboux formulas, but it also provides tools to analyze the so called bilinear equations and the tau-functions.

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

Speaker: Miguel Tierz, Universidad Complutense de Madrid.

Title: On random matrices and special functions in quantum topology.

Abstract: We will show how generalizations of the Jones polynomial of knots and links can be written in terms of terminating basic hypergeometric series and their corresponding q-orthogonal polynomials. We will also set forth the role played by random matrix theory in this description.

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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.

Abstract: In the framework of approximation theory, a Cohen type inequality is a lower bound for the norm of the partial sums of the Fourier expansions in terms of a certain orthogonal system. This kind of inequalities have been investigated by many authors in various contexts and forms, including the theory of orthogonal polynomials. Even though Dreseler and Soardi seem to be the first who found Cohen type inequalities in the setting of Jacobi expansions, it is worthwhile to point out that is due to Markett the presentation of an approach admitting a simpler proof of Dreseler and Soardi result for Jacobi expansions and stating the corresponding Cohen type inequalities for Laguerre and Hermite expansions. Concerning Sobolev orthogonality, the study of Cohen type inequalities is most recent and it has attracted considerable attention. Marcellán and Fejzullahu have obtained Cohen type inequalities for Laguerre orthonormal expansions with respect to discrete Sobolev inner products with only one mass point at c = 0. In this talk we are going to establish a Cohen type inequality when we deal with a discrete Laguerre-Sobolev inner product with only a mass point c located outside the support of the measure. In order to do this, we will incorporate new test functions different from whose used in the works previously mentioned. Then, as an immediate consequence we will deduce the divergence of Fourier expansions and Ces\{a}ro means of order $\delta$ in terms of this kind of Laguerre-Sobolev polynomials. This is a joint work with E. Huertas, F. Marcellán and Y. Quintana.

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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.

Abstract: In the last years some attention has been paid to the so called canonical spectral transformations of measures. This talk is focused on the zeros of monic orthogonal polynomial sequences (MOPS in short) associated with the so called Geronimus canonical transformation, which consist of a linear rational modification together with a mass point N of a given positive Borel measure. First, we introduce the representation of the Geronimus perturbed MOPS in terms of the initial ones and we analyze the behavior of the zeros of the corresponding MOPS. 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. Next, we provide a complete electrostatic model 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 on the exterior of the support of the initial measure.

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Thursday, January 30 2014, 16:00. Room 2.2.D08

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

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

Abstract: In this research, we consider the problem of reducing a system of Volterra integral equations to an independent system. Our approach is based on the matrix representation of the system, and its solvability is described in the case when the matrix polynomial corresponding to the system admits a Smith factorization. An explicit system of independent equations corresponding to the reduced system is constructed and a computable solution for independent one appears. A distinctive practical feature of the matrix polynomial based method is ability to solve single equations instead of systems. This technique is an extension of Gohberg's idea  who stated the method for a system of ordinary differential equations. Before extending it to systems of integral equations, their theorem is reviewed and slightly modified, and theoretical considerations are discussed. To illustrate the capability and reliability of the method, different classes are provided, and comparison of the analytical results reveals the effectiveness and convenience of the new technique.

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

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

Abstract: We study two-variable Jacobi polynomials on the parabolic triangle, that is, the closed region limited by a parabola and a straight line. Using the Koornwinder's addition formula for Jacobi polynomials we deduce old and new representation formulae for the corresponding kernels. As a consequence, asymptotic results for the Christoffel functions are obtained.

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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

Abstract: One of the main aims of this dissertation is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph G \ e obtained from the graph G by deleting an arbitrary edge e from it. These inequalities allow to obtain other main result, which characterizes in a quantitative way the hyperbolicity of any graph in terms of local hyperbolicity. In this work we also obtain information about the hyperbolicity constant of the line graph L(G) in terms of properties of the graph G. In particular, we prove qualitative results as the following: a graph G is hyperbolic if and only if L(G) is hyperbolic; if {G n} is a T-decomposition of G (G n are simple subgraphs of G), the line graph L(G) is hyperbolic if and only if sup n d(L(G n)) is finite. Besides, we obtain quantitative results when k is the length of the edges of G and L(G). Also, we characterize the graphs G with d(L(G)) < k. Furthermore, we prove the monotony of the hyperbolicity constant under a non-trivial transformation (the line graph of a graph). Also, we obtain criteria which allow us to decide, for a large class of graphs, whether they are hyperbolic or not. We are especially interested in the planar graphs which are the boundary (the 1-skeleton) of a tessellation of the Euclidean plane. Furthermore, we prove that a graph obtained as the 1-skeleton of a general CW 2-complex is hyperbolic if and only if its dual graph is hyperbolic. 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. Finally, we characterize the hyperbolic product graphs for two important kinds of products: the graph join of two graphs and the corona of two graphs. The graph join is always hyperbolic, and the corona is hyperbolic if and only if the first factor is hyperbolic. Furthermore, we obtain simple formulae for the hyperbolicity constant of the graph join and the corona.

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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?

Abstract: In a very nice survey paper on equilibrium in electrostatic field in the presence of a logarithmic potential, F. Marcellán, A. Martínez-Finkelshtein and P. Martínez-González called certain charges ghost"  ones. Trying to understand the meaning of this notion and following the experience of Bill Murray from the 1984 supernatural comedy, I'll play the role of an `electrostatic ghostbuster". Since I haven't been successful in this role and I haven't caught any, I'll make an attempt to prove that there are no ghosts in electrostatics. No guarantee for their presence or absence in real life!
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

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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Thursday, October 17 2013, 16:00. Room Adoración de Miguel, 1.2.C16 (Edificio Betancourt)

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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Last Updated on Monday, 25 August 2014 01:41

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