Orientador: Prof. Dr. Roldão da Rocha jr. / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2014. / Clifford algebras in Hilbert spaces are studied, along with the possible defnitions of spinors when the classical Clifford algebra is equipped with an additional structure of algebra C. The groups associated with the Clifford algebras, such as the Clifford-Lipschtz groups, Pin and Spin groups, are introduced together with unitary structures and trace operators in Clifford algebras in Hilbert spaces as well. Von-Neumann algebras are studied and the Bogoliubov automorphism is used to generalize the twisted Clifford-Lipschtz groups, using the graduation in Clifford algebra with the additional structure of algebra C. Fock representations and Hilbert-Schmidt operators are going to be introduced in the exterior algebra underlying the Clifford algebras in Hilbert spaces. In addition, twisted Clifford-Lipschitz groups can be constructed with the Bogoliubov automorphism, when it is an inner automorphism. This defines the Pin and Spin groups in the Clifford algebra with the additional structure of algebra C.

Orientador: Dimitar Kolev Dimitrov / Banca: Carlos Tomei / Banca: José  Alberto Cuminato / Resumo: Neste trabalho estudaremos a relação existente entre polinômios ortogonais e matrizes aleatórias. Exibiremos uma caracterização de polinômios ortogonais via problemas de Riemann-Hilbert, a qual tem se mostrado uma ferramenta única para obtenção de assintóticas de polinômios ortogonais. Posteriormente, estudaremos a teoria básica dos ensembles unitários de matrizes aleatórias. Por fim, mostraremos como a teoria de assintóticas de polinômios ortogonais pode ser usada na análise assintótica de estatísticas de matrizes aleatórias, nos levando a resultados de universalidade para os ensembles unitários / Abstract: We will exhibit a characterization of orthogonal p olynomials via Riemann-Hilbert problems, which has been shown a powerful to ol for studying asymptotics of orthogonal polynomials. Posteriorly we will review the basic theory of unitary ensembles of random matrices. At the end, we will show how asymptotics of orthogonal polynomials can be used to study asymptotics of several statistics in random matrix theory, obtaining universality results for the unitary ensembles / Mestre

Made available in DSpace on 2014-12-17T14:55:54Z (GMT). No. of bitstreams: 1 DiegoRC_DISSERT.pdf: 1983615 bytes, checksum: 7f5b15ea84a6c8bb36201c679b18226f (MD5) Previous issue date: 2011-07-11 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / The occurrence of transients in electrocardiogram (ECG) signals indicates an electrical phenomenon outside the heart. Thus, the identification of transients has been the most-used methodology in medical analysis since the invention of the electrocardiograph (device responsible for benchmarking of electrocardiogram signals). There are few papers related to this subject, which compels the creation of an architecture to do the pre-processing of this signal in order to identify transients. This paper proposes a method based on the signal energy of the Hilbert transform of electrocardiogram, being an alternative to methods based on morphology of the signal. This information will determine the creation of frames of the MP-HA protocol responsible for transmitting the ECG signals through an IEEE 802.3 network to a computing device. That, in turn, may perform a process to automatically sort the signal, or to present it to a doctor so that he can do the sorting manually / A ocorr?ncia de transientes em sinais de eletrocardiograma (ECG) ? um indicativo de um fen?meno el?trico externo ao cora??o, sendo a identifica??o de transientes a metodologia mais utilizada na an?lise m?dica desde que o eletrocardi?grafo (dispositivo respons?vel pelo aferimento dos sinais de eletrocardiograma) foi inventado. Existem poucos trabalhos relacionados a esse assunto, o que motiva a cria??o de uma arquitetura para fazer o pr?-processamento desse sinal em busca da identifica??o de transientes. O presente trabalho prop?e um m?todo baseado na energia do sinal da transformada Hilbert de eletrocardiograma, sendo uma alternativa aos m?todos baseados em morfologia do sinal. Essa informa??o determinar? a forma??o de Quadros do protocolo PM-AH respons?vel por transmitir os sinais de ECG atrav?s de uma rede de computadores do tipo IEEE 802.3 at? um dispositivo computacional. Que por sua vez poder? realizar um processamento para fazer a classifica??o autom?tica do sinal ou apresent?-lo para um m?dico realizar essa classifica??o de forma manual

Orientador: Marcos Benevenuto Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T01:41:17Z (GMT). No. of bitstreams: 1 Santos_PatriciaBorgesdos_M.pdf: 505981 bytes, checksum: 9ca0829d764a16d8fca0b3e0ebcd0e9b (MD5) Previous issue date: 2010 / Resumo: Exibiremos uma bijeção entre o esquema de Hilbert de c pontos no espaço afim de dimensão n e uma variedade algébrica quase projetiva dada pela variedade das n matrizes c por c que comutam duas a duas e satisfazem uma condição de estabilidade módulo conjugação. Em seguida, estudamos o caso n = 2 mais cuidadosamente, mostrando que o esquema de Hilbert de c pontos é uma variedade quase projetiva, não-singular e de dimensão 2c. / Abstract: We exhibit a bijection between the Hilbert scheme of points in the n-dimensional affine space and a quasi-projective algebraic variety given by c x c matrices commuting two by two and satisfying a stability condition modulo conjugation. Next, we study the n = 2 case more closely, showing that the Hilbert scheme of points is a non-singular, irreducible, quasi-projective variety of dimension 2c. / Mestrado / Algebra / Mestre em Matemática

Em uma variedade riemanniana conexa e compacta introduziremos o conceito de espectro do operador laplaciano. Utilizando a existÃncia e a unicidade do nÃcleo do calor em uma variedade riemanniana,provaremos o teorema de decomposiÃÃo de Hodge. Este teorema afirma que o espaÃo de Hilbert L2(M, g) se decompÃe em uma soma direta de subespaÃos de dimensÃo finita, onde cada subespaÃo à o auto-espaÃo associado a um autovalor do laplaciano. AlÃm disso, os autovalores formam uma sequÃncia nÃo-negativa que acumula somente no infinito. Em seguida iniciaremos a construÃÃo do nÃcleo do calor e, por fim, mostraremos que se duas variedades riemannianas sÃo isospectrais entÃo elas possuem o mesmo volume. / In a connected and compact Riemannian Manifold we will introduce the concept of spectre of Laplace operator. Using the existence and unicity of the heat kernel in Riemannian manifold we proof the Hodge composition theorem. This theorem states that the Hilbert space L2(M, g) decompose in direct sum of subspaces with finite dimesion, where each subspace is the eigen-space relative of a eigenvalue of the laplacian. Furthermore, the eigenvalues form a nonnegative sequence the accumulate only in the infinity. After that we begin the construction of the heat kernel and, finally, we show that two isospetral Riemannian manifolds have the same volume.

Orientador: Raymundo Luiz de Alencar / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-04T11:13:34Z (GMT). No. of bitstreams: 1 Goncalves_DimasJose_M.pdf: 1368395 bytes, checksum: 8fae1804e08750be7d9fe13ae39638d5 (MD5) Previous issue date: 2005 / Resumo: Neste trabalho, agrupamos um mínimo de conteúdo matemático para desenvolver uma prova do Teorema do Filtro de Kalman (discreto). A teoria de integração de Lebesgue e alguns elementos ( Teorema da Projeção) da teoria dos espaços de Hilbert são as principais ferramentas matemáticas incluídas, além de um estudo das principais propriedades da esperança condicional / Abstract: In this work we set a minimum of mathematical background to develop a proof of the (discrete) Kalman Filter Theorem.The lebesgue integration theory and some elements (the projection theorem) of the Hilbert space theory are the main mathematical tools included, besides a study of the main properties of the conditional expectation / Mestrado / Analise Funcional / Mestre em Matemática

[pt] Este trabalho aborda o problema da estabilidade de sistemas lineares, invariantes no tempo, a tempo discreto, com o espaço de estado sendo um espaço de Hilbert complexo e separável de dimensão infinita. São investigadas condições necessárias e/ou suficientes para quatro conceitos diferentes de estabilidade: estabilidade assintótica uniforme e estabilidade assintótica forte, estabilidade assintótica fraca e estabilidade limitada. Identifica-se e analisa-se as conexões entre os problemas de estabilidade e dois problemas em aberto da teoria de operadores em espaços de Hilbert: o problema do subespaço invariante e o problemas da similaridade e contração. Diversos resultados, oriundos de tentativas de solução para os dois problemas acima, ou motivados por aquelas tentativas, são utilizadas para fornecer caracterizações adicionais (principalmente caracterizações espectrais) para os quatro conceitos de estabilidade em questão. / [en] This work deals with the stability problem for time- invariant discrete linear systems evolving in a separable infinite-dimensional Hilbert space. Necessary and/or sufficient conditions for uniform, strong and weak asymptotic stability, as well as to bounded stability problems to two open problems in operator theory, namely, the invariant subspace and the similarity to contractions, are identified and analysed in detail. Several results from the many attempts, of solving the above mentioned open problems, or motivated by those attempts, are used to supply additional characterizations (mainly spectral characterization) for the four stabilty concepts under consideration.

Cette thèse est consacrée à l’extension au cadre Banachique de la notion d’opérateur de Hilbert-Schmidt. Dans un premier temps, on étudie d’une part les opérateurs p-sommants dans un espace de Banach X vers un autre espace de Banach Y et d’autre part, les opérateurs gamma-radonifiants dans un espace de Hilbert vers un autre espace de Banach.Dans un second temps, on s'intéresse aux opérateurs gamma-sommants dans des espaces de Banach, qui coïncident avec les opérateurs de Rademacher-bornés, ce qui nous amène aux opérateurs presque sommants. Enfin, on en déduit plusieurs généralisations naturelles de la notion d’opérateur de Hilbert-Schmidt aux espaces de Banach.-Les classes des opérateurs p-sommants de X dans Y .-La classe des opérateurs presque sommants de X dans Y qui coïncide avec la classe des opérateurs gamma-radonifiants de X dans Y.-La classe des opérateurs faible* 1-nucléaires de X dans Y. / This thesis is devoted to extending the notion of Banach Hilbert-Schmidt operator to the framework of Banach spaces. In a first step, we study p-summing operators from a Banach space X into a Banach space Y and gamma-radoniyfing operators from a Hilbert space into a Banach space. In a second step, we discuss gamma-summing operators between Banach spaces, which coincide with Rademacher-bounded operators, which leads to the notion of almost summing operators. Finally, we present serval natural generalizations of the notion of Hilbert-Schmidt operator to Banach spaces.- Classes of p-summing operators from X into Y. - The class of almost summing operators from X into Y, which coincides with the class of gamma-radoniyfing operators from X into Y.- The class of weak*1-nuclear operators from X into Y.

Les courbes remplissant l'espace sont connues pour la capacité d'ordonner les points multidimensionnels sur une ligne en tout conservant la localité, i.e. les points proches sont toujours proches sur la ligne. La conservation de la localité est beaucoup recherchée dans plusieurs applications. La courbe de Hilbert est la courbe remplissant l'espace qui conserve le mieux la localité. Cette courbe est originalement proposée en 2D, i.e. n'est qu'applicable aux points dans un espace 2D. Pour une perspective d'application dans le cas multidimensionnel, nous proposons dans cette thèse une généralisation de la courbe de Hilbert. La courbe généralisée est définie en s'appuyant sur la propriété essentielle de la courbe de Hilbert qui crée son niveau de conservation de la localité : l'adjacence. Ainsi, elle évite la dépendance du motif primitif RBG qui est le seul motif primitif de la courbe étendu par les recherches précédentes. Le résultat est donc une famille de courbe conservant bien la localité. L'optimisation de la conservation de la localité est aussi abordée pour permettre de retrouver la courbe qui conserve le mieux la localité. Pour cet objectif, nous proposons une mesure de la conservation de la localité. En s'appuyant sur les paramètres, cette mesure peut adapter aux différentes situations applicatives comme le changement de métrique ou de taille de localité. La construction est une partie importante de la thèse, elle est la base du calcul de l'index utilisé dans l'application. Pour un calcul de l'index rapide, la courbe de Hilbert autosimilaire est utilisée. La courbe de Hilbert satisfaisant les conditions de la courbe fait l'objet du chapitre 4. La courbe généralisée est enfin appliquée dans la recherche d'image. Il s'agit d'une recherche par le contenu où chaque image est caractérisée par un vecteur multidimensionnel. Les images sont ordonnées par la courbe sur une ligne ; ainsi, la recherche est simplifiée en une recherche sur une liste ordonnée. En donnant une image d'entrée, les images similaires sont celles correspondantes aux index voisins de l'index de l'image d'entrée. La conservation de la localité garantit que ces index correspondent aux images similaires. / The space-filling curves are known for the ability to order the multidimensional points on a line while preserving the locality, i.e. the close points are closely ordered on the line. The locality preserving is wished in many applications. Hilbert curve is the best locality preserving space-filling curve. This curve is originally proposed in 2D, i.e. it is only applied to points in a 2D space. For application in the multidimensional case, we propose in this thesis a generalization of Hilbert curve. Generalized curve is based on the essential property of Hilbert curve that creates its level of locality preserving: the adjacency. Thus, it avoids the dependence on the pattern RBG, which is the only pattern of the curve extended by previous researches. The result is a family of curves preserving well the locality. The optimization of the locality preserving is also addressed to find out the best locality preserving curve. For this purpose, we propose a measure of the locality preserving. Based on the parameters, this measure can adapt to different application situations such as the change of metric or locality size. The curve construction is an important part of the thesis. It is the basis of the index calculation used in application. For a rapid index calculation, the self-similar Hilbert curves is used. They are Hilbert curves satisfying the self-similar conditions specified in chapitre 4. The generalized curve is finally applied in image search. It is the question of the content-based image search (CBIR) where each image is characterized by a multidimensionalvector. Images are ordered by the curve of a line, and the search is simplified to the search on an ordered list. By giving an input image, similar images are those corresponding to neighbors of the index of the input. The locality preserving ensures that these indexes correspond to similar images.

This thesis presents parallel algorithms for rational cones and affine monoids which pursue two main computational goals: finding the Hilbert basis, a minimal generating system of the monoid of lattice points of a cone; and counting elements degree-wise in a generating function, the Hilbert series.