Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T14:36:33Z (GMT). No. of bitstreams: 1 Martins_RicardoMiranda_D.pdf: 1398633 bytes, checksum: 5ac0dfa4d175b6407f0b574a1aefd3e5 (MD5) Previous issue date: 2011 / Resumo: Na primeira parte desta tese, estudamos a semelhança entre sistemas dinâmicos reversíveis e Hamiltonianos, sob um ponto de vista formal. Nos restringimos a sistemas definidos ao redor de pontos de equilíbrio simples e simétricos. Mostramos que, sob algumas hipóteses, tais sistemas são formalmente orbitalmente equivalentes. Na segunda parte, estudamos a existência de conjuntos minimais em certas famílias de equações diferenciais. Especificamente, exibimos condições sob as quais existem cilindros e toros invariantes para sistemas de equações que são perturbações de sistemas reversíveis. / Abstract: In the first part of this thesis, we study the similarity between reversible and Hamiltonian dynamical systems, from a formal viewpoint. We restrict ourselves to systems defined around an isolated and symmetric equilibria. We show that, under some conditions, such systems are formally orbitally equivalent to Hamiltonian vector fields. In the second part, we study the existence of minimal sets for some families of diferential equations. We obtain conditions for the existence of the invariant cylinders and tori for perturbed reversible systems. / Doutorado / Sistemas Dinamicos / Doutor em Matemática

Orientador: Pedro José Catuogno / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T09:06:57Z (GMT). No. of bitstreams: 1 Franca_ElizeuCleberdosSantos_M.pdf: 3490670 bytes, checksum: 6589d1bb5b880e51625fb93abb2964a9 (MD5) Previous issue date: 2014 / Resumo: Neste trabalho, apresentaremos a teoria básica de simetrias de equações diferenciais, focando na busca por soluções invariantes de operadores diferenciais definidos em fibrados vetoriais com relação a ação transversal de um grupo de Lie no fibrado em questão / Abstract: In this work we will give the basic theory of symmetries of differential equations. The goal of this work is searching for invariant solutions of differential operators which are defined on vector bundles with respect to the transverse action of a Lie group in such bundle / Mestrado / Matematica / Mestre em Matemática

Nous étudions, dans la première partie de notre thèse, les polysimplexes d’un espace euclidien de dimension quelconque, c’est-à-dire les objets consistant en une juxtaposition de simplexes réguliers (de tétraèdres si la dimension est 3) accolés le long de leurs faces. Nous étudions principalement le groupe des symétries de ces polysimplexes. Nous présentons une façon de représenter un polysimplexe à l’aide d’un diagramme. Ceci fournit une classification complète des polysimplexes à similitude près. De plus, le groupe des symétries se déduit du groupe des automorphismes du diagramme. Il découle en particulier de notre étude qu’en dimension supérieure à 2, une telle structure ne possède jamais deux faces parallèles et ne contient jamais de circuit fermé de simplexes.<p><p>Dans la seconde partie de notre thèse, nous abordons un problème classique de combinatoire :l’énumération des pavages d’un rectangle mxn à l’aide de dominos. Klarner et Pollack ont montré qu’en fixant m la suite obtenue vérifie une relation de récurrence linéaire à coefficients constants. Nous établissons une nouvelle méthode nous permettant d’obtenir la fonction génératrice correspondante et la calculons pour m <= 16, alors qu’elle n’était connue que pour m <= 10.<p> / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Orientador: Lucas Catão de Freitas Ferreira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-24T06:33:09Z (GMT). No. of bitstreams: 1 Mesquita_ClaudiaAlineAzevedodosSantos_D.pdf: 1141685 bytes, checksum: a65a24d1917c5f998314d01970bb86e3 (MD5) Previous issue date: 2013 / Resumo: Nesta tese, estudamos dois tipos de EDPs com potenciais singulares críticos, a saber, uma equação elíptica com operador poliharmônico e a equação do calor linear. Para a primeira, pesquisamos existência e propriedades qualitativas das soluções no espaço $\mathcal{H}_{k,\vec{\alpha}}$ que é uma soma de espaços $L^{\infty}$ com peso, o qual parece ser um espaço mínimo para o tipo de potencial singular considerado. Investigamos um conceito de simetria para soluções que estende o de simetria radial e satisfaz uma ideia de invariância em torno das singularidades. Para a segunda, uma estratégia baseada na transformada de Fourier é empregada para obter resultados de boa-colocação global e comportamento assintótico de soluções, sem hipóteses de pequenez e sem utilizar a desigualdade de Hardy. Em particular, obtemos boa-colocação de soluções para o caso do potencial monopolar $V(x)=\frac{\lambda}{\left\vert x\right\vert ^{2}}$ com $\left\vert \lambda\right\vert <\lambda_{\ast}=\frac{(n-2)^{2}}{4}$. Este valor limiar é o mesmo obtido em resultados de boa-colocação global em $L^2$ que utilizam desigualdades de Hardy e estimativas de energia. Desde que não existe uma relação de inclusão entre $L^{2}$ e $PM^{k}$, nossos resultados indicam que $\lambda_{\ast}$ é intrínseco da EDP e independe de uma particular abordagem. Palavras-chave: Equações elípticas, equação do calor, potencial singular, existência, simetria, autossimilaridade, comportamento assintótico / Abstract: In this thesis, we study two types of PDEs with critical singular potentials, namely, an elliptic equation with polyharmonic operator and the linear heat equation. For the first, we obtain existence and qualitative properties of solutions in $\mathcal{H}_{k,\vec{\alpha}}$-spaces which are a sum of weighted $L^{\infty}$-spaces, and seem to be a minimal framework for the potential profile of interest. We investigate a concept of symmetry for solutions which extends radial symmetry and carries out an idea of invariance around singularities. For the second, a strategy based on the Fourier transform is employed to obtain results of global well-posedness and asymptotic behavior of solutions, without smallness hypotheses and without using Hardy inequality. In particular, well-posedness of solutions is obtained for the case of the monopolar potential $V(x)=\frac{\lambda}{\left\vert x\right\vert ^{2}}$ with $\left\vert \lambda\right\vert <\lambda_{\ast}=\frac{(n-2)^{2}}{4}$. This threshold value is the same one obtained for the global well-posedness of $L^{2}$-solutions by means of Hardy inequalities and energy estimates. Since there is no inclusion relation between $L^{2}$ and $PM^{k}$, our results indicate that $\lambda_{\ast}$ is intrinsic of the PDE and independent of a particular approach. Keywords: Elliptic equation, heat equation, singular potential, existence, symmetry, self-similarity, asymptotic behavior / Doutorado / Matematica / Doutora em Matemática

Orientador: Luis Felipe Cesar da Rocha Bueno / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T11:37:25Z (GMT). No. of bitstreams: 1 Penachi_Rian_M.pdf: 10723936 bytes, checksum: ee5a52edf4287fdd199dc8efff4b5869 (MD5) Previous issue date: 2015 / Resumo: Este trabalho propõe uma abordagem didática acerca de otimização não linear irrestrita e com restrições de igualdade, assim como um guia para o leitor que necessita instalar e utilizar o software ALGENCAN. Prezando por explorar ideias intuitivas do tema, o texto foca em estudar, muitas vezes geometricamente, problemas irrestritos e problemas com restrições de igualdade. Para o caso sem restrições é enfatizada a relação entre métodos de otimização e métodos para zeros de sistemas não lineares, destacando o Método de Newton. Métodos do tipo Lagrangiano Aumentado são o enfoque principal, particularmente em ALGENCAN, que é uma de suas implementações mais bem estabelecidas na literatura. As dificuldades encontradas em métodos computacionais para resolver problemas simétricos de otimização não linear com restrições também são estudadas. São apresentados vários exemplos simples de como a simetria do problema afeta o bom desempenho do método e as alternativas para contornar estes obstáculos. Além do mais, testes mais completos usando a coleção CUTEst comprovam que o algoritmo modificado que introduzimos é, pelo menos, tão competitivo quanto a versão original de ALGENCAN / Abstract: This work proposes a didactic approach concerning nonlinear unconstrained optimization and nonlinear equality constrained optimization, as well as a guide for the readers who need to install and use the ALGENCAN software. Exploring intuitive ideas of the subject, the focus of the text is to study, often in a geometric way, unconstrained problems and problems with equality constraints. For the case of unconstrained problems, it will be emphasized the relationship between optimization methods and methods for nonlinear systems, highlighting the Newton's Method. The main focus will be on the Augmented Lagrangian Method, particularly on ALGENCAN, which is one of the most well-established implementations in the literature. The difficulties found in computational methods to solve symmetric problems of nonlinear constrained optimization will be also studied. It will be shown, through several simple examples, how the symmetry of the problem affects the good performance of the method and the alternatives to overcome these difficulties. Moreover, more complete tests using CUTEst will be done, which will confirm that the modified algorithm introduced by us is, at least, as efficient as the original version of ALGENCAN / Mestrado / Matematica Aplicada e Computacional / Mestre em Matemática Aplicada e Computacional

Orientador: Igor Leite Freire / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T16:11:39Z (GMT). No. of bitstreams: 1 Sampaio_JulioCesarSantos_D.pdf: 2598760 bytes, checksum: b1332349b50fc600ddddb7596fd4b5a4 (MD5) Previous issue date: 2015 / Resumo: Neste trabalho estudamos simetrias de Lie e a teoria de leis de conservação desenvolvida por Ibragimov nos últimos 10 anos. Leis de conservação para várias equações sem Lagrangeanas clássicas foram estabelecidas / Abstract: In this work we study Lie point symmetries and the theory on conservation laws developed by Ibragimov in the last 10 years. Conservation laws for several equations without classical Lagrangians were established / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

This thesis deals with several statistical and probabilistic aspects of symmetry and asymmetry, both in a univariate and multivariate context, and is divided into three distinct parts.<p><p>The first part, composed of Chapters 1, 2 and 3 of the thesis, solves two conjectures associated with multivariate skew-symmetric distributions. Since the introduction in 1985 by Adelchi Azzalini of the most famous representative of that class of distributions, namely the skew-normal distribution, it is well-known that, in the vicinity of symmetry, the Fisher information matrix is singular and the profile log-likelihood function for skewness admits a stationary point whatever the sample under consideration. Since that moment, researchers have tried to determine the subclasses of skew-symmetric distributions who suffer from each of those problems, which has led to the aforementioned two conjectures. This thesis completely solves these two problems.<p><p>The second part of the thesis, namely Chapters 4 and 5, aims at applying and constructing extremely general skewing mechanisms. As such, in Chapter 4, we make use of the univariate mechanism of Ferreira and Steel (2006) to build optimal (in the Le Cam sense) tests for univariate symmetry which are very flexible. Actually, their mechanism allowing to turn a given symmetric distribution into any asymmetric distribution, the alternatives to the null hypothesis of symmetry can take any possible shape. These univariate mechanisms, besides that surjectivity property, enjoy numerous good properties, but cannot be extended to higher dimensions in a satisfactory way. For this reason, we propose in Chapter 5 different general mechanisms, sharing all the nice properties of their competitors in Ferreira and Steel (2006), but which moreover can be extended to any dimension. We formally prove that the surjectivity property holds in dimensions k>1 and we study the principal characteristics of these new multivariate mechanisms.<p><p>Finally, the third part of this thesis, composed of Chapter 6, proposes a test for multivariate central symmetry by having recourse to the concepts of statistical depth and runs. This test extends the celebrated univariate runs test of McWilliams (1990) to higher dimensions. We analyze its asymptotic behavior (especially in dimension k=2) under the null hypothesis and its invariance and robustness properties. We conclude by an overview of possible modifications of these new tests./<p><p>Cette thèse traite de différents aspects statistiques et probabilistes de symétrie et asymétrie univariées et multivariées, et est subdivisée en trois parties distinctes.<p><p>La première partie, qui comprend les chapitres 1, 2 et 3 de la thèse, est destinée à la résolution de deux conjectures associées aux lois skew-symétriques multivariées. Depuis l'introduction en 1985 par Adelchi Azzalini du plus célèbre représentant de cette classe de lois, à savoir la loi skew-normale, il est bien connu qu'en un voisinage de la situation symétrique la matrice d'information de Fisher est singulière et la fonction de vraisemblance profile pour le paramètre d'asymétrie admet un point stationnaire quel que soit l'échantillon considéré. Dès lors, des chercheurs ont essayé de déterminer les sous-classes de lois skew-symétriques qui souffrent de chacune de ces problématiques, ce qui a mené aux deux conjectures précitées. Cette thèse résoud complètement ces deux problèmes.<p><p>La deuxième partie, constituée des chapitres 4 et 5, poursuit le but d'appliquer et de proposer des méchanismes d'asymétrisation très généraux. Ainsi, au chapitre 4, nous utilisons le méchanisme univarié de Ferreira and Steel (2006) pour construire des tests de symétrie univariée optimaux (au sens de Le Cam) qui sont très flexibles. En effet, leur méchanisme permettant de transformer une loi symétrique donnée en n'importe quelle loi asymétrique, les contre-hypothèses à la symétrie peuvent prendre toute forme imaginable. Ces méchanismes univariés, outre cette propriété de surjectivité, possèdent de nombreux autres attraits, mais ne permettent pas une extension satisfaisante aux dimensions supérieures. Pour cette raison, nous proposons au chapitre 5 des méchanismes généraux alternatifs, qui partagent toutes les propriétés de leurs compétiteurs de Ferreira and Steel (2006), mais qui en plus sont généralisables à n'importe quelle dimension. Nous démontrons formellement que la surjectivité tient en dimension k > 1 et étudions les caractéristiques principales de ces nouveaux méchanismes multivariés.<p><p>Finalement, la troisième partie de cette thèse, composée du chapitre 6, propose un test de symétrie centrale multivariée en ayant recours aux concepts de profondeur statistique et de runs. Ce test étend le célèbre test de runs univarié de McWilliams (1990) aux dimensions supérieures. Nous en analysons le comportement asymptotique (surtout en dimension k = 2) sous l'hypothèse nulle et les propriétés d'invariance et de robustesse. Nous concluons par un aperçu sur des modifications possibles de ces nouveaux tests. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished