The goal of this paper will be to study how frame theory is applied within the field of signal processing. A frame is a redundant (i.e. not linearly independent) coordinate system for a vector space that satisfies a certain Parseval-type norm inequality. Frames provide a means for transmitting data and, when a certain about of loss is anticipated, their redundancy allows for better signal reconstruction. We will start with the basics of frame theory, give examples of frames and an application that illustrates how this redundancy can be exploited to achieve better signal reconstruction. We also include an introduction to the theory of frames in infinite dimensional Hilbert spaces as well as an interesting example.

On s'intéresse dans cette thèse à des questions portant sur les spécialisations de revêtements algébriques (galoisiens ou non). Le thème central de la première partie de ce travail est la construction de spécialisations de n'importe quel revêtement galoisien de la droite projective de groupe G défini sur k dont on impose d'une part le comportement local en un nombre fini d'idéaux premiers de k et dont on assure d'autre part qu'elles restent de groupe G si le corps k est hilbertien. Dans la deuxième partie, on développe une méthode générale pour qu'un revêtement galoisien f de la droite projective de groupe G défini sur k vérifie la propriété suivante : étant donné un sous-groupe H de G, il existe au moins une extension galoisienne de k de groupe H qui n'est pas spécialisation du revêtement f. De nombreux exemples sont donnés. La troisième partie consiste en l'étude de la question suivante : une extension galoisienne F/k, ou plus généralement une k-algèbre étale ∏ Fl /k, est-elle la spécialisation d'un revêtement d'une variété B défini sur k (galoisien ou non) en un certain point k-rationnel de B non-ramifié ? Notre principal outil est un twisting lemma qui réduit la question à trouver des points k-rationnels sur certaines k-variétés que nous étudions ensuite pour des corps de base k variés. / We are interested in this thesis in some questions concerning specializations of algebraic covers (Galois or not). The main theme of the first part consists in producing some specializations of any Galois cover of the projective line of group G defined over k with specified local behavior at finitely many given primes of k and which each have in addition Galois group G if k is assumed to be hilbertian. In the second part, we offer a systematic approach for a given Galois cover f of the projective line of group G defined over k to satisfy the following property: given a subgroup H of G, at least one Galois extension of k of group H is not a specialization of the cover f. Many examples are given. The central question of the third part is whether a given Galois extension F/k, or more generally a given k-étale algebra ∏ Fl /k, is the specialization of a given cover of a variety B defined over k (Galois or not) at some unramified k-rational point of B ? Our main tool is a twisting lemma which reduces the problem to finding k-rational points on some k-varieties which we then study for various base fields k.

Doctor en Ciencias de la Ingeniería, Mención Modelación Matemática / En cotutela con la Universidad de Borgoña Franco-Condado / Esta tesis está dedicada al estudio de inclusiones diferenciales con conos normales de conjuntos no regulares en espacios de Hilbert. En particular, nos interesa el proceso de arrastre y sus variantes. El proceso de arrastre es una inclusión diferencial restringida con conos normales que aparece naturalmente en varias aplicaciones tales como elastoplasticidad, histéresis, circuitos eléctricos, movimiento de multitudes, etc. Este trabajo está dividido conceptualmente en tres partes: Estudio de los conjuntos "alpha-far'', existencia de soluciones para las inclusiones diferenciales con conos normales y caracterizaciones de los pares de Lyapunov para el proceso de arrastre en espacios de Hilbert separable. En la primera parte (Capítulo 2), investigamos la clase de conjuntos positivamente "alpha-far''. Esta clase de conjuntos no regulares es muy general e incluye los conjuntos convexos, uniformemente prox-regulares y uniformemente sub-lisos, entre otros. Esta clase de conjuntos es la mejor adaptada al estudio de inclusiones diferenciales con conos normales. En la segunda parte (Capítulo 3 hasta la primera parte del Capítulo 8), se entregan varios resultados de existencia para el proceso de arrastre y sus variantes. Para ello, consideramos tres enfoques: el algoritmo de rectificación (Catching-up algorithm), el método de tipo Galerkin y la regularización de Moreau-Yosida. El primer método es el más clásico en el estudio de inclusiones diferenciales gobernadas por conos normales. Aquí es utilizado en el caso donde el conjunto considerado es fijo. El segundo método (de tipo Galerkin) consiste en aproximar el problema original proyectando el estado sobre un espacio de Hilbert de dimensión finita, pero no la velocidad. Los problemas aproximados siempre tienen una solución y, bajo ciertas condiciones de compacidad, se demuestra que ellos convergen fuertemente (salvo subsucesión) a una solución de la inclusión diferencial original. Más aún, se muestra que este método está bien adaptado para tratar inclusiones diferenciales con conos normales, proporcionando resultados generales de existencia para el proceso de arrastre generalizado. En consecuencia, se obtiene la existencia de soluciones para el proceso de arrastre de primer y segundo orden. Adicionalmente, este método es utilizado para mostrar la existencia de soluciones del proceso de arrastre con condiciones iniciales no locales. El tercer método es la técnica de regularización de Moreau-Yosida que consiste en aproximar una inclusión diferencial por una penalizada, en función de un parámetro positivo, para luego pasar al límite cuando el parámetro tiende a cero. Este método es utilizado para tratar el proceso de arrastre dependiente del estado gobernado por conjuntos uniformemente sub-lisos. Finalmente, en la tercera parte (segunda parte del Capítulo 8 y Capítulo 9), se proporcionan algunas caracterizaciones de los pares de Lyapunov débiles y la invariancia débil para el proceso de arrastre perturbado con conjuntos uniformemente sub-lisos. / Este trabajo ha sido parcialmente financiado por CONICYT-Beca Doctorado Nacional 2013.

O objetivo deste trabalho e encontrar uma solução exata para um sistema de equações integrais de Volterra. Para isso, usaremos a teoria de espacos de reprodução e núcleos positivos definidos, visto que as técnicas usuais de resoluções de equações diferenciais e integrais possuem restrições. Grande parte do estudo voltado a solução de equações se baseia em analisar o comportamento das soluções, o chamado estudo qualitativo. Este não e o nosso interesse, queremos aproximar a solução do problema usando a representa c~ao dessa solução em uma base ortonormal especial de um espaço de Hilbert de reprodução gerado por um núcleo positivo de nido adequado. Dessa forma, truncando a serie encontrada para a solução do sistema de Volterra podemos exibir uma boa aproxima c~ao para a solução do sistema. As equações integrais de Volterra, foco deste trabalho, s~ao importantes para a modelagem de fenômenos físicos, demográficos ou epidemiológicos. Para a resolução de tais equações, faremos um estudo introdutório sobre conceitos de álgebra linear, análise e teoria da medida com o intuito de abranger temas como: existência de base de um espaço vetorial, o processo de ortogonaliza c~ao de Gram-Schmidt, os espaços Lp, entre outros. Faremos uma breve análise sobre a transformada de Laplace, assim como resolveremos uma equação diferencial e integral usando este método. Tambem resolveremos um sistema de equações integrais através da transformada de Laplace para exemplificar o método. Cabe lembrar que a maioria das equações não pode ser resolvida por meio da transformada de Laplace. Faremos um estudo de resolução de equações lineares de Volterra e então abrangeremos esse estudo para equa c~oes n~ao lineares. / The aim of this study is to give the exact solution to a system of linear Volterra integral equations. So do it, we will use the theory of reproduction Kernel method and positive de nite kernels, since the usual method to solve di erential and integral equations have restrictions. Much of the study about solving equations is based on analyzing the behavior of solutions, called qualitative study. This is not our interest, we want to approach the solution of the problem using the representation of the solution in a special orthonormal basis of the reproduction kernel Hilbert space generated by an appropriate positive de nite kernel. Thus, truncating the series found for the solution of the Volterra system, we can give a good approximation to the system solution. The Volterra integral equations, focus of this work, are important to modeling physical, demographic or epidemiological phenomena. For solving such equations, we make an introductory study of linear algebra, analysis and measure theory in order to comprehend topics such as: existence of a base in a vector space, the Gram-Schmidt orthogonalization process, the spaces Lp, and others. We make a brief analysis of the Laplace transform, as well as solve a di erential and integral equation using this method. We also solve a system of integral equations by Laplace transform to illustrate the method. It should be noted that most of the equations can not be solved by means of the Laplace transform. We will study how to solve linear Volterra equations and then extend the study to nonlinear equations.

El entrelazamiento puede abordarse desde dos perspectivas diferentes: como un recurso esencial para las tecnologías cuánticas y como un fenómeno fundamental que está íntimamente relacionado con nuestra comprensión de la naturaleza misma. Por otro lado, la teoría cuántica se formula en el marco teórico de los espacios de Hilbert, para los que el entrelazamiento juega un papel importante en la determinación de su geometría y topología. Las características topológicas que puedan exhibirse al utilizar estados entrelazados son largamente independientes de la realización física particular del entrelazamiento: puede afectar a un solo grado de libertad poseído por dos partículas diferentes, o bien puede implicar dos grados diferentes de libertad que se cohesionan a una misma partícula o entidad física, por ejemplo, un campo electromagnético. Resulta que la manipulación de los grados de libertad de polarización y momentum (camino) ya sea de forma independiente el uno del otro o mediante la aplicación de evoluciones unitarias no separables es muy versátil. Con esto en mente, la presente tesis apunta hacia el diseño e implementación de arreglos experimentales que se pueden utilizar para estudiar fases geométricas y topológicas en sistemas de dos qubits mediante el uso de los grados de libertad de momentum (camino) y polarización de un solo fotón. Finalmente mostramos el diseño de un experimento, apuntado a exhibir la fase topológica, y los resultados obtenidos. / Tesis

by Tong Shiu Sing Dominic. / Parallel title in Chinese characters. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 236-241). / Acknowledgements --- p.iv / Abstract --- p.v / Chapter 1 --- Open Systems and Quasinormal Modes --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.1.1 --- Non-Hermitian Systems --- p.1 / Chapter 1.1.2 --- Optical Cavities as Open Systems --- p.3 / Chapter 1.1.3 --- Outline of this Thesis --- p.6 / Chapter 1.2 --- Simple Models of Open Systems --- p.10 / Chapter 1.3 --- Contributions of the Author --- p.14 / Chapter 2 --- Completeness and Orthogonality --- p.16 / Chapter 2.1 --- Introduction --- p.16 / Chapter 2.2 --- Green's Function of the Open System --- p.19 / Chapter 2.3 --- High Frequency Behaviour of the Green's Function --- p.24 / Chapter 2.4 --- Completeness of Quasinormal Modes --- p.29 / Chapter 2. 5 --- Method of Projection --- p.31 / Chapter 2.5.1 --- Problems with the Usual Method of Projection --- p.31 / Chapter 2.5.2 --- Modified Method of Projection --- p.33 / Chapter 2.6 --- Uniqueness of Representation --- p.38 / Chapter 2.7 --- Definition of Inner Product and Quasi-Stationary States --- p.39 / Chapter 2.7.1 --- Orthogonal Relation of Quasinormal Modes --- p.39 / Chapter 2.7.2 --- Definition of Hilbert Space and State Vectors --- p.41 / Chapter 2.8 --- Hermitian Limits --- p.43 / Chapter 2.9 --- Numerical Examples --- p.45 / Chapter 3 --- Time-Independent Perturbation --- p.58 / Chapter 3.1 --- Introduction --- p.58 / Chapter 3.2 --- Formalism --- p.60 / Chapter 3.2.1 --- Expansion of the Perturbed Quasi-Stationary States --- p.60 / Chapter 3.2.2 --- Formal Solution --- p.62 / Chapter 3.2.3 --- Perturbative Series --- p.66 / Chapter 3.3 --- Diagrammatic Perturbation --- p.70 / Chapter 3.3.1 --- Series Representation of the Green's Function --- p.70 / Chapter 3.3.2 --- Eigenfrequencies --- p.73 / Chapter 3.3.3 --- Eigenfunctions --- p.75 / Chapter 3.4 --- Numerical Examples --- p.77 / Chapter 4 --- Method of Diagonization --- p.81 / Chapter 4.1 --- Introduction --- p.81 / Chapter 4.2 --- Formalism --- p.82 / Chapter 4.2.1 --- Matrix Equation with Non-unique Solution --- p.82 / Chapter 4.2.2 --- Matrix Equation with a Unique Solution --- p.88 / Chapter 4.3 --- Numerical Examples --- p.91 / Chapter 5 --- Evolution of the Open System --- p.97 / Chapter 5.1 --- Introduction --- p.97 / Chapter 5.2 --- Evolution with Arbitrary Initial Conditions --- p.99 / Chapter 5.3 --- Evolution with the Outgoing Plane Wave Condition --- p.106 / Chapter 5.3.1 --- Evolution Inside the Cavity --- p.106 / Chapter 5.3.2 --- Evolution Outside the Cavity --- p.110 / Chapter 5.4 --- Physical Implications --- p.112 / Chapter 6 --- Time-Dependent Perturbation --- p.114 / Chapter 6.1 --- Introduction --- p.114 / Chapter 6.2 --- Inhomogeneous Wave Equation --- p.117 / Chapter 6.3 --- Perturbative Scheme --- p.120 / Chapter 6.4 --- Energy Changes due to the Perturbation --- p.128 / Chapter 6.5 --- Numerical Examples --- p.131 / Chapter 7 --- Adiabatic Approximation --- p.150 / Chapter 7.1 --- Introduction --- p.150 / Chapter 7.2 --- The Effect of a Varying Refractive Index --- p.153 / Chapter 7.3 --- Adiabatic Expansion --- p.156 / Chapter 7.4 --- Numerical Examples --- p.167 / Chapter 8 --- Generalization of the Formalism --- p.176 / Chapter 8. 1 --- Introduction --- p.176 / Chapter 8.2 --- Generalization of the Orthogonal Relation --- p.180 / Chapter 8.3 --- Evolution with the Outgong Wave Condition --- p.183 / Chapter 8.4 --- Uniform Convergence of the Series Representation --- p.193 / Chapter 8.5 --- Uniqueness of Representation --- p.200 / Chapter 8.6 --- Generalization of Standard Calculations --- p.202 / Chapter 8.6.1 --- Time-Independent Perturbation --- p.203 / Chapter 8.6.2 --- Method of Diagonization --- p.206 / Chapter 8.6.3 --- Remarks on Dynamical Calculations --- p.208 / Appendix A --- p.209 / Appendix B --- p.213 / Appendix C --- p.225 / Appendix D --- p.231 / Appendix E --- p.234 / References --- p.236

La teoría de la credibilidad provee un conjunto de métodos que permiten a una compañía de seguros ajustar las primas futuras, sobre la base de la experiencia pasada individual e información de toda la cartera. En este trabajo presentaremos los principales modelos de credibilidad utilizados en la práctica, como lo son los modelos de Bühlmann (1967), Bühlmann-Straub (1970), Jewell (1975) y Hachemeister (1975), todos ellos analizados en sus propiedades desde un punto de vista geométrico a través de la teoría de espacios de Hilbert y en su estimación mediante el uso de los modelos lineales mixtos. Mediante un estudio de simulación se mostrará la ventaja de utilizar este último enfoque de estimación. / Tesis

<div>In this thesis, we study invariants of graded modules over polynomial rings. In particular, we find bounds on the Hilbert functions and graded Betti numbers of certain modules. This area of research has been widely studied, and we discuss several well-known theorems and conjectures related to these problems. Our main results extend some known theorems from the case of homogeneous ideals of polynomial rings R to that of graded R-modules. In Chapters 2 & 3, we discuss preliminary material needed for the following chapters. This includes monomial orders for modules, Hilbert functions, graded Betti numbers, and generic initial modules.</div><div> </div><div> In Chapter 4, we discuss x_n-stability of submodules M of free R-modules F, and use this stability to examine properties of lexsegment modules. Using these tools, we prove our first main result: a general hypersurface restriction theorem for modules. This theorem states that, when restricting to a general hypersurface of degree j, the Hilbert series of M is bounded above by that of M^{lex}+x_n^jF. In Chapter 5, we discuss Hilbert series of local cohomology modules. As a consequence of our general hypersurface restriction theorem, we give a bound on the Hilbert series of H^i_m(F/M). In particular, we show that the Hilbert series of local cohomology modules of a quotient of a free module does not decrease when the module is replaced by a quotient by the lexicographic module M^{lex}.</div><div> </div><div> The content of Chapter 6 is based on joint work with Gabriel Sosa. The main theorem is an extension of a result of Caviglia and Sbarra to polynomial rings with base field of any characteristic. Given a homogeneous ideal containing both a piecewise lex ideal and an ideal generated by powers of the variables, we find a lex ideal with the following property: the ideal in the polynomial ring generated by the piecewise lex ideal, the ideal of powers, and the lex ideal has the same Hilbert function and Betti numbers at least as large as those of the original ideal. This bound on the Betti numbers is sharp, and is a closer bound than what was previously known in this setting.</div>

nÃo hà / This dissertation describes a system for analysis of heart rate variability through metrics on time and frequency domain and by non-linear methodology, which is initiated by the process of segmentation of the QRS complex of the electrocardiogram signal. The motivation for this work is the analysis of the influence from the algorithms of beat segmentation and selection of valid cardiac cycles for the variability analysis, which were developed in the research process, over the computation of the metrics. After determining the intervals between QRS complexes (RR), the cardiac cycles with ectopic beats, resultant of arrhythmic events or detection fails (false-positive or false negative) are excluded. Then, the available metrics of heart rate variability found on literature are computed over the time series of intervals between normal beats (NN): on time domain (statistical and geometrical methods), on frequency domain (VLF - Very Low Frequency, LF - Low Frequency and HF â High Frequency) and by non-linear methodology (Poincarà plot). The QRS detection and segmentation results are validated through simulation tests over exams from Arrhythmia Database and QT database of the MIT-BIH database. The manual annotations of the QRS fiducial points and QRS onset and offset are compared with the automatic detections. The results related to heart rate variability metrics are validated through the manual selection of beats, and consequently of the intervals between them, pertaining to exams selected from Arrhythmia Database and the computation of the referred metrics over them, comparing with those ones automatically generated by the proposed method. The system, which provides averages of positive predictivity as 99.35% and sensitivity as 99.02%, and averages of deviations between automatic and manual analysis of heart rate variability metrics varying from 0.05% to 5.24%, can be carried into several platforms, making possible its production in commercial scale. / Esta dissertaÃÃo descreve um sistema de analise da variabilidade da frequÃcia cardÃaca atravÃs de mÃtricas no domÃnio do tempo, da freqÃÃncia e por mÃtodo nÃo-linear a partir do processo de segmentaÃÃo do complexo QRS do sinal eletrocardiograma. O trabalho à motivado pela influÃncia dos algoritmos de segmentaÃÃo de batimentos e de seleÃÃo dos ciclos cardÃacos vÃlidos para anÃlise da variabilidade, esenvolvidos para este fim, na determinaÃÃo das mÃtricas de interesse. ApÃs a determinaÃÃo dos intervalos entre os complexos QRS (RR), sÃo excluÃdos os ciclos cardÃacos com batimentos ectÃpicos, resultantes de arritmia ou de falhas de detecÃÃo (falso-positivo ou falso-negativo). Em seguida, sobre a sÃrie temporal de intervalos entre batimentos normais NN sÃo calculadas as mÃtricas de variabilidade da freqÃÃncia cardÃaca disponÃveis na literatura: no domÃnio do tempo (mÃtodos estatÃsticos e geomÃtricos), no domÃnio da freqÃÃncia (componentes VLF - Very Low Frequency, LF - Low Frequency e HF - High Frequency) e por mÃtodo nÃo-linear (mapa de PoincarÃ). Os resultados de deteccÃo e segmentacÃo do QRS sÃo validados atravÃs de testes experimentais sobre exames das bases Arrhythmia Database e QT database do MIT-BIH, em que as marcaÃÃes manuais dos picos e das bordas dos batimentos sÃo comparadas com as detecÃÃes automÃticas. Os resultados obtidos quanto as mÃtricas de variabilidade sÃo validados atravÃs da seleÃÃo manual de batimentos e, por conseguinte, dos intervalos entre os mesmos, a partir de exames selecionados da base Arrhythmia Database por cardiologistas do Hospital UniversitÃrio Walter Cantidio (HUWC), e do cÃlculo das referidas mÃtricas, comparando-se com aquelas geradas automaticamente pelo mÃtodo proposto. O sistema, que apresenta taxas mÃdias de99,35% de preditividade positiva e 99,02% de sensibilidade, para detecÃÃo do QRS, e mÃdias de erros entre a anÃlise automÃtica e a anÃlise manual das mÃtricas de variabilidade variando entre 0,05% e 5,24%, pode ser embutido em diversas plataformas, viabilizando sua producÃo em escala comercial.

In this thesis I argue forin re structuralism in the philosophy of mathematics. In the first chapters of the thesis I argue that there is a genuine epistemic access problem for Platonism, that the semantic challenge to nominalism may be met by paraphrase strategies, and that nominalizations of scientific theories have had adequate success to blunt the force of the indispensability argument for Platonism. In the second part of the thesis I discuss the development of logicism and structuralism as methodologies in the history of mathematics. The goal of this historical investigation is to lay the groundwork for distinguishing between the philosophical analysis of the content of mathematics and the analysis of the breadth and depth of results in mathematics. My central contention is that the notion of logical structure provides a context for the latter not the former. In turn, this contention leads to a rejection of ante rem structuralism in favor of in re structuralism. In the concluding part of the dissertation the philosophy of mathematical structures developed and defended in the preceding chapters is applied to the philosophy of science.