Systems of Markov type functions: normality and convergence of Hermite-Padé approximants

This thesis deals with Hermite-Padé approximation of analytic and merophorphic functions. As such it is embeded in the theory of vector rational approximation of analytic functions which in turn is intimately connectd with the theory of multiple orthogonal polynomials. All the basic concepts and results used in this thesis involving complex analysis and measure theory may found in classical textbooks...........Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: Francisco José Marcellán Español; Vocal: Alexander Ivanovich Aptekarev; Secretario: Andrei Martínez Finkelshtei

Direct and inverse results on row sequences of simultaneous rational approximants

In this Tesis we investigate the approximation of vector functions by vector rational function that generalizes Padé approximants. We consider two types of approximants: the simultaneous Hermite-Padé 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. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------En la tesis se investiga la aproximación de funciones vectoriales mediante vectores de fracciones racionales que generalizan los llamados aproximantes de Padé correspondientes al caso de la aproximación de una función escalar. Se consideran dos tipos de aproximantes: los aproximantes simultaneos Hermite-Padé, que se construyen mediante criterios interpolatorios y los aproximantes Fourier-Padé basados en desarrollos en serie de Fourier a partir de un sistema de polinomios ortogonales.Los resultados obtenidos generalizan al caso de la aproximación vectorial resultados muy conocidos para el caso escalar debidos a R. de Montessus de Ballore, A.A. Gonchar, S.P. Suetin, P.R. Graves-Morris, y E.B. Sa ff

Nonlinear Analysis and Optimization with Applications

Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world

Representaciones racionales de series matriciales con aplicación a la especificación de modelos multivariantes

Se caracterizan funciones racionales matriciales de dimensión arbitraria, a partir de series formales de potencias cuyos coeficientes son matrices, trasladando posteriormente los resultados a la especificación de modelos racionales de series temporales, en particular a modelos Varma. Los aspectos de minimalidad y unicidad de representación o, en terminología de series temporales, la intercambiabilidad de modelos y la identificabilidad de los parámetros han sido considerados también desde la aproximación de Padé matricial para su tratamiento y posterior aplicación a series temporales. Al mismo tiempo se aportan resultados sobre la estructura de la tabla de Padé en el caso de funciones matriciales de dimensión arbitraria. por otro lado, la forma que se indica para estudiar los parámetros nulos y/o redundantes de una representación racional responde también a ciertos problemas de sobreparametrización en la estimación de modelos racionales de series temporales

Problèmes d'équilibre vectoriels et grandes déviations

Dans cette thèse on s'intéresse à la convergence et aux grandes déviations de la mesure empirique associée à certains processus ponctuels déterminantaux. Le point commun entre ces processus ponctuels est que leur polynôme caractéristique moyen est un polynôme orthogonal multiple, une généralisation des polynômes orthogonaux usuels. L'exemple le plus simple est fourni par un gaz de Coulomb bidimensionnel dans un potentiel confinant à température inverse bêta = 2; son polynôme caractéristique moyen est alors un polynôme orthogonal. Il a été prouvé, même dans le cas plus général où bêta > 0, que la mesure empirique satisfait à un principe de grande déviation, avec une fonction de taux qui fait intervenir un problème d'équilibre bien connu en théorie logarithmique du potentiel. En guise d'échauffement, nous allons montrer que ce résultat s'étend au cas d'un potentiel faiblement confinant, c'est-à-dire satisfaisant une condition de croissance plus faible que d'habitude. Pour ce faire, nous utilisons un argument de compactification qui sera d'importance pour la suite. Anticipant la description asymptotique de processus déterminantaux plus complexes, nous développons alors un cadre adéquat pour définir rigoureusement des problèmes d'équilibre vectoriels avec des potentiels faiblement confinants. Nous prouvons l'existence et l'unicité de leurs solutions, un résultat nouveau en théorie du potentiel, et aussi que les fonctionnelles associées ont des ensembles de niveau compacts. Après, nous nous intéressons à un processus ponctuel déterminantal associé à une perturbation additive d'une matrice de Wishart, pour lequel le polynôme caractéristique moyen est un polynôme orthogonal multiple à deux poids. Nous établissons un principe de grande déviation pour la mesure empirique avec une fonction de taux qui fait intervenir un problème d'équilibre vectoriel ayant des potentiels faiblement confinants. C'est la première fois qu'un problème d'équilibre vectoriel intervient dans la description des grandes déviations de matrices aléatoires. Finalement, on étudie de façon générale quand est-ce que la mesure empirique associée à un processus ponctuel déterminantal et la distribution des zéros du polynôme caractéristique moyen associé convergent vers la même limite. Nous obtenons une condition suffisante pour une classe de processus ponctuels déterminantaux qui contient les processus liés aux polynômes orthogonaux multiples. En chemin, nous donnons aussi une condition suffisante pour améliorer la convergence en moyenne de la mesure empirique en une convergence presque sûre. Comme application, on décrit les distributions asymptotiques des zéros des polynômes de Hermite multiple et de Laguerre multiple en termes de convolutions libres de distributions classiques avec des mesures discrètes, et puis nous dérivons des équations algébriques pour leur transformée de Cauchy- Stieltjes.In this thesis we investigate the convergence and large deviations of the empirical measure associated with several determinantal point processes. These point processes have in common that their average characteristic polynomial is a multiple orthogonal polynomial, the latter being a generalization of orthogonal polynomials. The first simplest example is a 2D Coulomb gas in a confining potential at inverse temperature beta = 2, for which the average characteristic polynomial is an orthogonal polynomial. A large deviation principle for the empirical measure is known to hold, even in the general beta > 0 case, with a rate function involving an equilibrium problem arising from logarithmic potential theory. As a warming up, we show this result actually extends to the case where the potential is weakly confining, i.e. satisfying a weaker growth assumption that usual. To do so, we introduce a compactification procedure which will be of important use in what follows. Motivated by more complex determinantal point processes, we then develop a general framework for vector equilibrium problems with weakly confining potentials to make sense. We prove existence and uniqueness of their solutions, which improves the existing results in the potential theory literature, and moreover show that the associated functionals have compact level sets. Next, we investigate a determinantal point process associated with an additive perturbation of a Wishart matrix, for which the average characteristic polynomial is a multiple orthogonal polynomial associated with two weights. We establish a large deviation principle for the empirical measure with a rate function related to a vector equilibrium problem with weakly confining potentials. This is the first time that a vector equilibrium problem is shown to be involved in a large deviation principle for random matrix models. Finally, we study on a more general level when both the empirical measure of a determinantal point process and the zero distribution of the associated average characteristic polynomial converge to the same limit. We obtain a sufficient condition for a class of determinantal point processes which contains the ones related to multiple orthogonal polynomials. On the way, we provide a sufficient condition to strengthen the mean convergence of the empirical measure to the almost sure one. As an application, we describe the limiting distributions for the zeros of multiple Hermite and multiple Laguerre polynomials in terms of free convolutions of classical distributions with atomic measures, and then derive algebraic equations for their Cauchy-Stieltjes transforms

High gain and bandwidth current-mode amplifiers : study and implementation

Doutoramento em Engenharia ElectrotécnicaEsta tese aborda o problema do projecto de amplificadores com grandes produtos de ganho por largura de banda. A aplicação final considerada consistiu no projecto de amplificadores adequados à recepção de sinais ópticos em sistemas de transmissão ópticos usando o espaço livre. Neste tipo de sistemas as maiores limitações de ganho e largura de banda surgem nos circuitos de entrada. O uso de detectores ópticos com grande área fotosensível é uma necessidade comum neste tipo de sistemas. Estes detectores apresentam grandes capacidades intrínsecas, o que em conjunto com a impedância de entrada apresentada pelo amplificador estabelece sérias restrições no produto do ganho pela largura de banda. As técnicas mais tradicionais para combater este problema recorrem ao uso de amplificadores com retroacção baseados em configurações de transimpedância. Estes amplificadores apresentam baixas impedâncias de entrada devido à acção da retroacção. Contudo, os amplificadores de transimpedância também apresentam uma relação directa entre o ganho e a impedância de entrada. Logo, diminuir a impedância de entrada implica diminuir o ganho. Esta tese propõe duas técnicas novas para combater os problemas referidos. A primeira técnica tem por base uma propriedade fundamental dos amplificadores com retroacção. Em geral, todos os circuitos electrónicos têm tempos de atraso associados, os amplificadores com retroacção não são uma excepção a esta regra. Os tempos de atraso são em geral reconhecidos como elementos instabilizadores neste tipos da amplificadores. Contudo, se usados judiciosamente, este tempos de atraso podem ser explorados como uma forma da aumentar a largura de banda em amplificadores com retroacção. Com base nestas ideias, esta tese apresenta o conceito geral de reatroacção com atraso, como um método de optimização de largura de banda em amplificadores com retroacção. O segundo método baseia-se na destruição da dualidade entre ganho e impedância de entrada existente nos amplificadores de transimpedância. O conceito de adaptação activa em modo de corrente é neste sentido uma forma adequada para separar o detector óptico da entrada do amplificador. De acordo com este conceito, emprega-se um elemento de adaptação em modo de corrente para isolar o detector óptico da entrada do amplificador. Desta forma as tradicionais limitações de ganho e largura de banda podem ser tratadas em separado. Esta tese defende o uso destas técnicas no desenho de amplificadores de transimpedância para sistemas de recepção de sinais ópticos em espaço livre.This thesis addresses the problem of achieving high gain-bandwidth products in amplifiers. The adopted framework consisted on the design of a free-space optical (FSO) front end amplifier able to amplify very small optical signals over large frequency bandwidths. The major gain-bandwidth limitations in FSO front end amplifiers arise due to the input circuitry. Usually, it is necessary to have large area optical detectors in order to maximize signal reception. These detectors have large intrinsic capacitances, which together with the amplifier input impedance poses a severe restriction on the gain-bandwidth product. Traditional techniques to combat this gain-bandwidth limitation resort to feedback amplifiers consisting on transimpedance configurations. These amplifiers have small input impedances due to the feedback action. Nevertheless, transimpedance amplifiers have a direct relation between gain and input impedance. Thus reducing the input impedance usually implies reducing the gain. This thesis advances two new methods suitable to combat the above mentioned problems. The first method is based on a fundamental property of feedback amplifiers. In general, all electronic circuits have associated time delays, and feedback amplifiers are not an exception to this rule. Time delays in feedback amplifiers have been recognized as destabilizing elements. Nevertheless, when used with appropriate care, these delays can be exploited as bandwidth enhancement elements. Based on these ideas, this thesis presents the general concept of delayed feedback, as a bandwidth optimization method suitable for feedback amplifiers. The second method is based on the idea of destroying the impedance-gain duality in transimpedance amplifiers. The concept of active current matching is in this sense a suitable method to detach the optical detector from the transimpedance amplifier input. According to this concept, a current matching device (CMD) is used to convey the signal current sensed by the optical detector, to the amplifier’s input. Using this concept the traditional gainbandwidth limitations can be treated in a separate fashion. This thesis advocates the usage of these techniques for the design of transimpedance amplifiers suited for FSO receiving systems

Tosio Kato’s work on non-relativistic quantum mechanics: part 1

We review the work of Tosio Kato on the mathematics of non-relativistic quantum mechanics and some of the research that was motivated by this. Topics in this first part include analytic and asymptotic eigenvalue perturbation theory, Temple–Kato inequality, self-adjointness results, and quadratic forms including monotone convergence theorems

Dynamical systems : mathematical and numerical approaches

Proceedings of the 13th Conference „Dynamical Systems - Theory and Applications" summarize 164 and the Springer Proceedings summarize 60 best papers of university teachers and students, researchers and engineers from whole the world. The papers were chosen by the International Scientific Committee from 315 papers submitted to the conference. The reader thus obtains an overview of the recent developments of dynamical systems and can study the most progressive tendencies in this field of science

System- and Data-Driven Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques

Polinomios biortogonales y sus generalizaciones: una perspectiva desde los sistemas integrables

La conexión existente entre los polinomios ortogonales y otras ramas de la matemática, la física o la ingeniería es verdaderamente asombrosa. Además, no hay mejor prueba de la utilidad de estos que el propio crecimiento, avance perpetuo y generalización en diversas direcciones de lo que se entendía por polinomio ortogonal en los albores de la teoría. Conforme el concepto se fue generalizando, también fueron evolucionando las técnicas para su estudio, algunas de estas claramente influenciadas por aquellas disciplinas matemáticas con las que iban surgiendo conexiones. La perspectiva que esta tesis adopta frente a los polinomios ortogonales es un ejemplo de este tipo de influencias, compartiendo herramientas y entrelazandose con la teoría de los sistemas integrables. Una posición privilegiada en esta tesis la ocuparían las matrices de Gram semi in nitas; cada cual asociada a una forma sesquilineal adaptada al tipo de biortogonalidad en cuestión. A estas matrices se les impondrán una serie de condiciones cuyo objeto sería el de garantizar la existencia y unicidad de las secuencias biortogonales asociadas a las mismas. El siguiente paso consistiría en buscar simetrías de estas matrices de Gram. Existen dos razones por las que este esfuerzo resulta ventajoso. En primer lugar, cada simetría encontrada podría traducirse en propiedades de las secuencias biortogonales, por ejemplo: una estructura Hankel de la matriz es equivalente a gozar de la recurrencia a tres términos de los polinomios ortogonales; la simetría propia de las matrices asociadas a pesos clásicos (Hermite, Laguerre, Jacobi) implica la existencia del operador diferencial lineal de segundo orden de que los polinomios clásicos son solución; etc..