Neste trabalho obtemos taxas de decaimento para autovalores e valores singulares de operadores integrais gerados por núcleos de quadrado integrável sobre a esfera unitária em \'R POT. m+1\', m 2, sob hipóteses sobre ambos, certas derivadas do núcleo e o operador integral gerado por tais derivadas. Este tipo de problema é comum na literatura, mas as hipóteses geralmente são definidas via diferenciação usual em \'R POT m+1\'. Aqui, as hipóteses são todas definidas via derivada de Laplace-Beltrami, um conceito genuinamente esférico investigado primeiramente por W. Rudin no começo dos anos 50. As taxas de decaimento apresentadas são ótimas e dependem da dimensão m e da ordem de diferenciabilidade usada para definir as condições de suavidade / In this work we obtain decay rates for singular values and eigenvalues of integral operators generated by square integrable kernels on the unit sphere in \'R m+1\', m 2, under assumptions on both, certain derivatives of the kernel and the integral operators generated by such derivatives. This type of problem is common in the literature but the assumptions are usually defined via standard differentiation in \'R POT. m+1\'. Here, the assumptions are all defined via the Laplace-Beltrami derivative, a concept first investigated by W. Rudin in the early fifties and genuinely spherical in nature. The rates we present are optimal and depend on both, the differentiability order used to define the smoothness conditions and the dimension m

Es werden hybride Mannigfaltigkeiten untersucht, d.h. Systeme von zweidimensionalen Mannigfaltigkeiten, die durch eindimensionale Intervalle miteinander verknuepft sind. In einer solchen Struktur definieren wir einen Laplace-Operator, der sich aus den Laplace-Beltrami-Operatoren auf den glatten Teilen und Randbedingungen an den Verknuepfungspunkten zusammenstellt. Durch Verwendung der Kreinschen Theorie selbstadjungierter Erweiterungen wird es gezeigt, dass alle moeglichen Laplace-Operatoren durch hermitsche Matrizen einer speziellen Form parametrisiert werden koennen. Wir berechnen die Entwicklung der Spur der quadrirten Resolvent eines Laplace-Operators fuer grosse Spectralparameter vermittels der Randbedingungen und der Waermeleitungskoeffizienten der glatten Teilen der hybriden Mannigfaltigkeit. Unter gewissen zusaetzlichen Annahmen is es moeglich, aus dieser Entwicklung einige geometrische Invarianten und einige Information ueber den Randbedingungen zu gewinnen. / We consider a hybrid manifold (i.e. some two-dimensional manifolds connected with each other by some segments) and a Laplace operator on it. Such an operator can be constructed by using the Laplace-Beltrami operators on each part of the hybrid manifold with some boundary conditions in the points of gluing. We use the Krein theory of self-adjoint extensions to show that all possible Laplace operators are parameterized by some Hermitian matrices. We find the large spectral parameter expansion of the trace of the second power of the resolvent of a fixed Laplace operator in terms of the boundary condition matrix and heat kernel coefficients for the parts of the hybrid manifold. If we assume that we already have such an expansion for some hybrid manifold then we can find some data about this manifold (inverse spectral problem). Under some additional conditions it is possible to find some geometric invariants of the hybrid space and some information about the boundary conditions matrix. We apply the same technique also to two degenerate cases of hybrid manifolds: quantum graphs and the manifolds glued without segments.

Apesar dos modelos usados em engenharia serem, em sua maioria, reconhecidamente aproximados, acredita-se que a matemática usada na física e nos próprios modelos é infinitamente precisa e que tais teorias físicas poderiam prever completamente qualquer evento relacionado às variáveis equacionadas. No limite, seria possível prever o estado do universo em qualquer instante, crença esta chamada de determinismo. Claro está que essa pretensão é apenas de princípio, sendo impossível na prática. No entanto, pesquisas sobre os fundamentos da matemática e outras teorias matemáticas desenvolvidas no século XX sugerem que a matemática (e, consequentemente, a física) teria certos limites inerentes. A análise feita nesta tese fundamenta seus argumentos na Teoria das Funções Recursivas e Computabilidade Efetiva e na Teoria do Caos Determinístico. O objetivo principal é tratar de apurar a existência de limites inerentes e como tais limites se aplicariam aos sistemas elétricos de potência (mais especificamente nos tópicos fluxo de carga, transitórios eletromecânicos, transitórios eletromagnéticos e eletrônica de potência) e à engenharia de controle. / Although the models used in engineering are, in most cases, admittedly approximated, it is believed that the Mathematics used in Physics and in these models, is infinitely precise and that such physical theories could fully predict any event related to variables in equations. In the limit, it would be possible to predict the state of the universe at any moment, this belief is called determinism. It is clear that this claim is only in principle, impossible in practice. However, research on the foundations of Mathematics and other mathematical theories developed in the 20th century suggest that the Mathematics (and hence Physics) would have certain inherent limitations. The analysis made in this thesis has the arguments based on the Theory of Recursive Functions and Effective Computability and the Theory of Deterministic Chaos. The main objective is to find out the existence of inherent limits and how these limits could be applied to electric power systems (more specifically to the topics load flow, electromechanical transient and electromagnetic transient and power electronics) and control engineering.

Nesta contribuição apresentamos uma solução para a equação de transferência radiativa condutiva em geometria cilíndrica. Esta solução é aplicada para simular a radiação e campo de temperatura juntamente com o transporte de energia radiativa e condutiva proveniente do escape liberado em lançamentos de foguetes. Para este fim, discutimos uma abordagem semianalítica reduzindo a equação original, que é contínua nas variáveis angulares, numa equação semelhante ao problema SN da transferência radiativa condutiva. A solução é construída usando um método de composição por transformada de Laplace e o método da decomposição de Adomian. O esquema recursivo ´e apresentado para o sistema de equações de ordenadas duplamente discretas juntamente com as dependências dos parâmetros e suas influências sobre a convergência heurística da solução. A solução obtida, em seguida, permite construir o campo próximo relevante para caracterizar o termo fonte para problemas de dispersão ao ajustar os parâmetros do modelo, tais como, emissividade, refletividade, albedo e outros, em comparação com a observação, que são relevantes para os processos de dispersão de campo distante e podem ser manipulados de forma independente do presente problema. Além do método de solução, também relatamos sobre algumas soluções e simulações numéricas. / In this contribution we present a solution for the radiative conductive transfer equation in cylinder geometry. This solution is applied to simulate the radiation and temperature field together with conductive and radiative energy transport originated from the exhaust released in rocket launches. To this end we discuss a semi-analytical approach reducing the original equation, which is continuous in the angular variables, into an equation similar to the SN radiative conductive transfer problem. The solution is constructed using a composite method by Laplace transform and Adomian decomposition method. The recursive scheme is presented for the doubly discrete ordinate equations system together with parameter dependencies and their influence on heuristic convergence of the solution. The obtained solution allows then to construct the relevant near field to characterize the source term for dispersion problems when adjusting the model parameters such as emissivity, reflectivity, albedo and others in comparison to the observation, that are relevant for far field dispersion processes and may be handled independently from the present problem. In addition to the solution method we also report some solutions and numerical simulations.

Cette thèse est consacrée à l'étude des valeurs propres de l'opérateur de Laplace et de l'opérateur de Steklov sur des variétés riemanniennes. On cherche à donner des bornes optimales parmi l'ensemble des métriques, dans une classe conforme donnée ou non, et à caractériser, si elles existent, les métriques qui atteignent ces bornes. Ces métriques extrémales ont des propriétés qui s'inscrivent dans la théorie des surfaces minimales. On s'intéresse d'abord à la borne supérieure des valeurs propres de Laplace parmi des métriques conformes entre elles, appelées valeurs propres conformes. Dans le chapitre 1, on estime la deuxième valeur propre conforme de la sphère standard. Dans les chapitres 2 et 3, on montre que la première valeur propre conforme d'une variété riemannienne est plus grande que celle de la sphère standard de même dimension avec égalité seulement pour la sphère standard. Ensuite, on cherche à démontrer l'existence et la régularité de métriques qui maximisent les valeurs propres sur des surfaces, dans une classe conforme donnée ou non. Dans les chapitres 3 et 4, on démontre un résultat d'existence pour les valeurs propres de Laplace. Dans le chapitre 6, le travail est fait pour les valeurs propres de Steklov. Enfin, dans le chapitre 5, fruit d'un travail réalisé en collaboration avec Paul Laurain, on démontre un résultat de régularité et de quantification des applications harmoniques à bord libre sur une surface Riemannienne. C'est un élément clé pour le chapitre 6 / This thesis is devoted to the study of the Laplace eigenvalues and the Steklov eigenvalues on Riemannian manifolds. We look for optimal bounds among the set of metrics, lying in a conformal class or not. We also characterize, if they exist the metrics which reach these bounds. These extremal metrics have properties from the theory of minimal surfaces. First, we are interested in the upper bound of Laplace eigenvalues in a class of conformal metrics, called the conformal eigenvalues. In Chapter 1, we estimate the second conformal eigenvalue of the standard sphere. In Chapters 2 and 3, we prove that the first conformal eigenvalue of a Riemannian manifold is greater than the one of the standard sphere of same dimension, with equality only for the standard sphere. Then, we look for existence and regularity results for metrics which maximize eigenvalues on surfaces, in a given conformal class or not. In Chapters 3 and 4, we prove an existence result for Laplace eigenvalues. In Chapter 6, the work is done for Steklov eigenvalues. Finally, in Chapter 5, obtained in collaboration with Paul Laurain, we prove a regularity and quantification result for harmonic maps with free boundary on a Riemannian surface. It is a key component for Chapter 6

La thèse s’intéresse à l’étude du fonctionnement d’un système industriel. Il s’agit de proposer et de développer un nouveau modèle pour modéliser la dégradation accumulative d’un système. Le processus gamma standard est fréquemment utilisé pour étudier l’évolution de la détérioration d’un système. Toutefois, ce processus peut s’avérer inadapté pour décrire le phénomène de dégradation car le rapport variance sur moyenne est constant dans le temps, ce qui est relativement restrictif en pratique. Afin de surmonter cette restriction, nous proposons d’utiliser un processus gamma étendu introduit par Cinlar (1980), qui ne souffre plus de cette restriction. Mais ce dernier présente quelques difficultés techniques. A titre d’exemple, la loi d’un processus gamma étendu n’est pas connue sous une forme explicite. Ces difficultés techniques ont conduit Guida et al. (2012) à utiliser une version discrète d’un processus gamma étendu. Nous travaillons ici avec la version originale d’un processus gamma étendu en temps continu. Le but de ce mémoire est de développer des méthodes numériques permettant de quantifier les quantités fiabilistes associées et de développer des méthodes statistiques d’estimation des paramètres du modèle. Aussi, une autre partie de ce travail consiste à proposer une politique de maintenance dans le contexte d’un processus gamma étendu. / This thesis is dedicated to study the functioning of an industrial system. It is about proposing and developing a new model for modelling the accumulative degradation of a system. The standard gamma process is widely used to model the evolution of the system degradation. A notable restriction of a standard gamma process is that its variance-to-mean ratio is constant over time. This may be restrictive within an applicative context. To overcome this drawback, we propose to use an extended gamma process, which was introduced by Cinlar (1980). However, there is a cost and the use of an extended gamma process presents some technical difficulties. For example, there is no explicit formula for the probability distribution of an extended gamma process. These technical difficulties have lead Guida et al. (2012) to use a discrete version of an extended gamma process. We here propose to deal with the original continuous time version. The aim of this work is to develop numerical methods in order to compute the related reliability function and to develop statistical methods to estimate the parameters of the model. Also, another part of this work consists of proposing a maintenance policy within the context of an extended gamma process.

Orientadores: Yuri Dimitrov Bozhkov, Antonio Carlos Gilli Martins / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-09-24T19:39:04Z (GMT). No. of bitstreams: 1 Freire_IgorLeite_D.pdf: 977261 bytes, checksum: b8ba44493aeac3de0d37cdfff2fc581b (MD5) Previous issue date: 2008 / Resumo: Neste trabalho provamos um teorema que faz a classificacão completa dos grupos de simetrias de Lie da equação semilinear de Kohn - Laplace no grupo de Heisenberg tridimensional. Uma vez que tal equação possui estrutura variacional, determinamos quais são suas simetrias de Noether e a partir delas estabelecemos suas respectivas leis de conservação / Abstract: In this work, we carry out a complete group classification of Lie point symmetries of semilinear Kohn - Laplace equations on the three-dimensional Heisenberg group. Since this equation has variational structure, we determine which of its symmetries are Noether's symmetries. Then we establish their respectives conservation laws / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

Apesar dos modelos usados em engenharia serem, em sua maioria, reconhecidamente aproximados, acredita-se que a matemática usada na física e nos próprios modelos é infinitamente precisa e que tais teorias físicas poderiam prever completamente qualquer evento relacionado às variáveis equacionadas. No limite, seria possível prever o estado do universo em qualquer instante, crença esta chamada de determinismo. Claro está que essa pretensão é apenas de princípio, sendo impossível na prática. No entanto, pesquisas sobre os fundamentos da matemática e outras teorias matemáticas desenvolvidas no século XX sugerem que a matemática (e, consequentemente, a física) teria certos limites inerentes. A análise feita nesta tese fundamenta seus argumentos na Teoria das Funções Recursivas e Computabilidade Efetiva e na Teoria do Caos Determinístico. O objetivo principal é tratar de apurar a existência de limites inerentes e como tais limites se aplicariam aos sistemas elétricos de potência (mais especificamente nos tópicos fluxo de carga, transitórios eletromecânicos, transitórios eletromagnéticos e eletrônica de potência) e à engenharia de controle. / Although the models used in engineering are, in most cases, admittedly approximated, it is believed that the Mathematics used in Physics and in these models, is infinitely precise and that such physical theories could fully predict any event related to variables in equations. In the limit, it would be possible to predict the state of the universe at any moment, this belief is called determinism. It is clear that this claim is only in principle, impossible in practice. However, research on the foundations of Mathematics and other mathematical theories developed in the 20th century suggest that the Mathematics (and hence Physics) would have certain inherent limitations. The analysis made in this thesis has the arguments based on the Theory of Recursive Functions and Effective Computability and the Theory of Deterministic Chaos. The main objective is to find out the existence of inherent limits and how these limits could be applied to electric power systems (more specifically to the topics load flow, electromechanical transient and electromagnetic transient and power electronics) and control engineering.

Neste trabalho estudamos diferentes noções de diferenciabilidade para funções definidas na esfera unitária S^n-1 de R^n, n>=2. Em relação à derivada usual, encontramos condições necessárias e/ou suficientes para que uma função seja diferenciável até uma ordem fixada. Para as outras duas, a derivada forte de Laplace-Beltrami e a derivada fraca, apresentamos algumas propriedades básicas e procuramos condições que garantam a equivalência destas com a diferenciabilidade usual. / In this work we study different notions of differentiability for functions defined on the unit sphere S^n-1 of R^n, n>=2. With respect to the usual derivative, we find necessary and/or sufficient conditions in order that a function be differentiable up to a fixed order. As for the other two, the strong Laplace-Beltrami derivative and the weak derivative, we present some basic properties about them and search for conditions that guarantee the equivalence of them with the previous one.

Le but de cette thèse est de trouver des bornes supérieures pour les valeurs propres des opérateurs naturels agissant sur les fonctions d'une variété compacte $(M,g)$. Nous étudions l'opérateur de Laplace-Beltrami et des opérateurs du type laplacien. Dans le cas de l'opérateur de Laplace-Beltrami, deux aspects sont étudiés. Le premier aspect est d'étudier les relations entre la géométrie intrinsèque et les valeurs propres du laplacien. Nous obtenons des bornes supérieures ne dépendant que de la dimension et d'un invariant conforme qui s'appelle le volume conforme minimal. Asymptotiquement, ces bornes sont en cohérence avec la loi de Weyl. Elles améliorent également les résultats de Korevaar et de Yang et Yau. La preuve repose sur la construction d'une famille convenable de domaines disjoints fournissant des supports pour une famille de fonctions tests. Cette méthode est puissante et intéressante en soi. Le deuxième aspect est d'étudier la relation entre la géométrie extrinsèque et les valeurs propres du laplacien agissant sur des sous-variétés compactes de l'espace euclidien $R^N$ ou de l'espace projectif complexe $CP^N$. Nous étudions un invariant extrinsèque qui s'appelle l'indice d'intersection étudié par Colbois, Dryden et El Soufi. Pour des sous-variétés compactes de $R^N$, nous généralisons leurs résultats et obtenons des bornes supérieures qui sont stables l'effet de petites perturbations. Pour des sous-variétés de $CP^N$, nous obtenons une borne supérieure ne dépendant que du degré des sous-variétés et qui est optimale pour la première valeur propre non nulle. Comme autre application de la méthode introduite, nous obtenons une borne supérieure pour des valeurs propres du problème de Steklov sur des sous-domaines à bord $C^1$ d'une variété riemannienne complète, en termes du rapport isopérimétrique du domaine, et du volume conforme minimal. Une modification de notre méthode donne des bornes supérieures pour les valeurs propres des opérateurs de Schrödinger en termes du volume conforme minimal et de l'intégrale du potentiel. Nous obtenons également les bornes supérieures pour les valeurs propres du laplacien de Bakry-Emery dépendant d'invariants conformes.