In this thesis, we consider the numerical solution of a large eigenvalue problem in which the integral operator comes from a radiative transfer problem. It is considered the use of hierarchical matrices, an efficient data-sparse representation of matrices, especially useful for large dimensional problems. It consists on low-rank subblocks leading to low memory requirements as well as cheap computational costs. We discuss the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank discretization of an integral operator. The operator is of convolution type, it is defined through the first exponential-integral function and hence it is weakly singular. We access HLIB (Hierarchical matrices LIBrary) that provides, among others, routines for the construction of hierarchical matrix structures and arithmetic algorithms to perform approximative matrix operations. Moreover, it is incorporated the matrix-vector multiply routines from HLIB, as well as LU factorization for preconditioning, into SLEPc (Scalable Library for Eigenvalue Problem Computations) in order to exploit the available algorithms to solve eigenvalue problems. It is also developed analytical expressions for the approximate degenerate kernels and deducted error upper bounds for these approximations. The numerical results obtained with other approaches to solve the problem are used to compare with the ones obtained with this technique, illustrating the efficiency of the techniques developed and implemented in this work

Cette thèse considère la solution numérique d'un problème aux valeurs propres de grandes dimensions, dans lequel l'opérateur est dérivé d'un problème de transfert radiatif. Ainsi, cette thèse étudie l'utilisation de matrices hiérarchiques, une représentation efficace de tableaux, très intéressante pour une utilisation avec des problèmes de grandes dimensions. Les matrices sont des représentations hiérarchiques de structures de données efficaces pour les matrices denses, l'idée de base étant la division d'une matrice en une hiérarchie de blocs et l´approximation de certains blocs par une matrice de petite caractéristique. Son utilisation permet de diminuer la mémoire nécessaire tout en réduisant les coûts informatiques. L'application de l'utilisation de matrices hiérarchique est analysée dans le contexte de la solution numérique d'un problème aux valeurs propres de grandes dimensions résultant de la discrétisation d'un opérateur intégral. L'opérateur est de convolution et est défini par la première fonction exponentielle intégrale, donc faiblement singulière. Pour le calcul informatique, nous avons accès à HLIB (Hierarchical matrices LIBrary) qui fournit des routines pour la construction de la structure hiérarchique des matrices et des algorithmes pour les opérations approximative avec ces matrices. Nous incorporons certaines routines comme la multiplication matrice-vecteur ou la decomposition LU, en SLEPc (Hierarchical matrices LIBrary) pour explorer les algorithmes existants afin de résoudre les problèmes de valeur propre. Nous développons aussi des expressions analytiques pour l'approximation des noyaux dégénérés utilisés dans la thèse et déduire ainsi les limites supérieures d'erreur pour ces approximations. Les résultats numériques obtenus avec d'autres techniques pour résoudre le problème en question sont utilisés pour la comparaison avec ceux obtenus avec la nouvelle technique, illustrant l'efficacité de ce dernier / In this thesis, we consider the numerical solution of a large eigenvalue problem in which the integral operator comes from a radiative transfer problem. It is considered the use of hierarchical matrices, an efficient data-sparse representation of matrices, especially useful for large dimensional problems. It consists on low-rank subblocks leading to low memory requirements as well as cheap computational costs. We discuss the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank discretization of an integral operator. The operator is of convolution type, it is defined through the first exponential-integral function and hence it is weakly singular. We access HLIB (Hierarchical matrices LIBrary) that provides, among others, routines for the construction of hierarchical matrix structures and arithmetic algorithms to perform approximative matrix operations. Moreover, it is incorporated the matrix-vector multiply routines from HLIB, as well as LU factorization for preconditioning, into SLEPc (Scalable Library for Eigenvalue Problem Computations) in order to exploit the available algorithms to solve eigenvalue problems. It is also developed analytical expressions for the approximate degenerate kernels and deducted error upper bounds for these approximations. The numerical results obtained with other approaches to solve the problem are used to compare with the ones obtained with this technique, illustrating the efficiency of the techniques developed and implemented in this work

Given the result $v_0$ of a survey and a nested collection of summary statistics that could be used to describe that result, it is natural to ask which of these summary statistics best describe $v_0$. In 1998 Diaconis and Sturmfels presented an approach for determining the conditional significance of a higher order statistic, after sampling a space conditioned on the value of a lower order statistic. Their approach involves the computation of a Markov basis, followed by the use of a Markov process with stationary hypergeometric distribution to generate a sample.This technique for data analysis has become an accepted tool of algebraic statistics, particularly for the study of fully ranked data. In this thesis, we explore the extension of this technique for data analysis to the study of partially ranked data, focusing on data from surveys in which participants are asked to identify their top $k$ choices of $n$ items. Before we move on to our own data analysis, though, we present a thorough discussion of the Diaconis–Sturmfels algorithm and its use in data analysis. In this discussion, we attempt to collect together all of the background on Markov bases, Markov proceses, Gröbner bases, implicitization theory, and elimination theory, that is necessary for a full understanding of this approach to data analysis.

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Submitted by WILLIAM VIEIRA GONCALVES null (williamvieira@unemat.br) on 2016-09-21T00:29:22Z No. of bitstreams: 1 TESE_GONCALVES_William_Vieira.pdf: 4325935 bytes, checksum: b52e7f80ac7d448c1c0d65aecda11b20 (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-09-22T20:43:16Z (GMT) No. of bitstreams: 1 goncalves_wv_dr_bauru.pdf: 4325935 bytes, checksum: b52e7f80ac7d448c1c0d65aecda11b20 (MD5) / Made available in DSpace on 2016-09-22T20:43:16Z (GMT). No. of bitstreams: 1 goncalves_wv_dr_bauru.pdf: 4325935 bytes, checksum: b52e7f80ac7d448c1c0d65aecda11b20 (MD5) Previous issue date: 2016-07-01 / Este trabalho tem como principal objetivo demonstrar que o GeoGebra apresenta uma maneira diferente de produzir significados matemáticos, com isso, sugerindo sua relevância para o ensino de matemática e de se discutir sua linguagem, possibilidades e limitações. A base inicial da pesquisa foi um estudo imersivo em diferentes comunidades virtuais, literatura acadêmica correlata, cursos específicos do software, produção e análise de diferentes construções dinâmicas. Por fim, optou-se por delimitar a análise aprofundada em entrevistas semiestruturadas com sete professores de matemática, usuários experientes do software. Confrontando-se estes entrevistados com algumas limitações do software, buscou-se estudar como eles transitam entre diferentes modos de produção de significados matemáticos. Pautando-se em reconhecer os diferentes jogos de linguagem, advindos das explicações dos sujeitos da pesquisa, foram sintetizadas três adjetivações que caracterizam os diferentes modos de produção de significados matemáticos: a Matemática do Matemático (MM), a Matemática da Escola (ME) e a Matemática do GeoGebra (MG). Esta pesquisa pautou-se na técnica de investigação qualitativa livre, em função da sua tentativa de compreender mais detalhadamente os significados e características situacionais. A análise dos dados foi realizada à luz de alguns elementos da Análise Textual Discursiva segundo Moraes e Galiazzi (2007). As categorias criadas, a posteriori, foram as seguintes: Matemática do Matemático (MM); Matemática da Escola (ME); Matemática do GeoGebra (MG); Trânsito entre as matemáticas; Percepção da MG; Necessidade de compreensão da MG; aparente Compreensão da MG e aparente Incompreensão da MG. A partir da análise dos dados foi possível confirmar o uso de diferentes jogos de linguagem e, portanto, confirmar o transitar entre a MM, a ME e a MG. Ainda, percebeu-se e analisou-se diferentes formas de transitar, concluindo-se que existe um modo de transitar, comum a todos os entrevistados. Parte-se das possibilidades semióticas da MG, aproveitando-se da maleabilidade da ME, para formalizar-se significados matemáticos, legítimos a MM. Finalmente, a partir da análise aprofundada de uma entrevista, propõe-se o reconhecimento de diferentes aspectos da MG e da história do GeoGebra, como uma forma de aprender sobre como lidar com suas possibilidades e limites. / This work aims to demonstrate that GeoGebra presents has a different way of producing mathematical meanings, suggesting its relevance to mathematics teaching and to discuss its language, possibilities and limitations. Research initial framework was an immersive study in different virtual communities, academic literature, specific courses, production and analysis of different dynamic constructions. In the end, the choice was to delimitate a depth analysis on semi-structures interviews with seven math teachers, experienced software users. Confronting this respondents with some limitations of the software, we sought to study how they transit between different modes of production of mathematical meanings. Based on recognizing the different language games, arising of research subjects were synthesized three adjectives that characterize the different mathematical modes of production: Math from Mathematician (MM), Math from School (ME, in Portuguese) and Math from Geogebra (MG, in Portuguese too). This research used the free qualitative investigation for its attempt to understand with more details the meanings and situational characteristics. Data analysis was carried out with some elements of Discursive Textual Analysis from Moraes and Galiazzi (2007). The categories created, a posteriori, were: Mathematics from Mathematician (MM); Math from School (ME); and Math from Geogebra (MG); Transit between the mathematics; Perception from MG; need of comprehension from MG; apparent comprehension from MG and incomprehension from MG. Data analysis enable to confirm the use of different languages games and, thus, confirm the transition between MM, ME and MG. I still realized and analyzed different ways of transition, concluding that there is a common way to transit between all respondents. We set of semiotic possibilities from MG, taking advantage of the malleability from ME, to formalized mathematical meanings, legitimate from MM. Finally, from the depth analysis of an interview, it is proposed to recognize different aspects from MG and of the GeoGebra history, as a way to learn about how to deal with its possibilities and limits.

On borne explicitement la hauteur de Faltings d'une courbe sur le corps de nombres algèbriques en son degré de Belyi. Des résultats similaires sont démontré pour trois autres invariants arakeloviennes : le discriminant, l'invariant delta et l'auto-intersection de omega. Nos résultats nous permettent de borner explicitement les invariantes arakeloviennes des courbes modulaires, des courbes de Fermat et des courbes de Hurwitz. En plus, comme application, on montre que l'algorithme de Couveignes-Edixhoven-Bruin est polynomial sous l’hypothèse de Riemann pour les fonctions zeta des corps de nombres. Ceci était connu uniquement pour certains sous-groupes de congruence. Finalement, on utilise nos résultats pour démontrer une conjecture de Edixhoven, de Jong et Schepers sur la hauteur de Faltings d'un revêtement ramifié de la droite projective sur l'anneau des entiers. / We explicitly bound the Faltings height of a curve over the field of algebraic numbers in terms of the Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings' delta invariant and the self-intersection of the dualizing sheaf. Our results allow us to explicitly bound these Arakelov invariants for modular curves, Hurwitz curves and Fermat curves. Moreover, as an application, we show that the Couveignes-Edixhoven-Bruin algorithmtime under the Riemann hypothesis for zeta-functions of number fields. This was known before only for certain congruence subgroups. Finally, we utilize our results to prove a conjecture of Edixhoven, de Jong and Schepers on the Faltings height of a branched cover of the projective line over the ring of integers.

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