The work of mathematical giants, such as Lobachevsky, Gauss, Riemann, Klein and Poincaré, to name a few, lies at the foundation of the study of the highly structured Riemann surfaces, which allow definition of holomorphic maps, corresponding to analytic maps in the theory of complex analysis. A topological result of Poincaré states that every path-connected Riemann surface can be realised by a construction of identifying congruent points in the complex plane, the Riemann sphere or the hyperbolic plane; just three simply connected surfaces that cover the underlying Riemann surface. This requires the discontinuous action of a discrete subgroup of the automorphisms of the corresponding space. In the hyperbolic plane, which is the richest source for Riemann surfaces, these groups are called Fuchsian, and there are several ways to study the action of such groups geometrically by computing fundamental domains. What is accomplished in this thesis is a combination of the methods found by Reidemeister & Schreier, Singerman and Voight, and thus provides a unified way of finding Dirichlet domains for subgroups of cofinite groups with a given index. Several examples are considered in-depth.

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.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-05T19:26:54Z No. of bitstreams: 1 JULIANA ARAGÃO DE ARAÚJO - DISSERTAÇÃO PPGMAT 2005..pdf: 3880310 bytes, checksum: 796ad7c042587fcea49b37b64d7b2fc1 (MD5) / Made available in DSpace on 2018-07-05T19:26:54Z (GMT). No. of bitstreams: 1 JULIANA ARAGÃO DE ARAÚJO - DISSERTAÇÃO PPGMAT 2005..pdf: 3880310 bytes, checksum: 796ad7c042587fcea49b37b64d7b2fc1 (MD5) Previous issue date: 2005-05 / Neste trabalho é apresentada a solução do problema de Riemann associado a um sistema de leis de conservação. Este sistema é proveniente de um escoamento bifásico unidimensional em meios porosos e considera os efeitos de histerese nas curvas de permeabilidade das fases. A principal contribuição deste trabalho é que a solução do problema de Riemann é obtida para um modelo que leva em conta a histerese nas duas fases e que considera as curvas de embebição e de drenagem distintas sempre que haja uma reversão de regime de drenagem para embebição e vice-versa. Os resultados obtidos aqui ampliam aqueles obtidos para um modelo mais simplificado em que a histerese era considerada apenas numa das fases e as curvas de permeabilidade eram tomadas coincidentes após urna segunda reversão. Uma vez obtida a solução do problema de Riemann, base para a construção de simuladores numéricos de alta precisão, é feita uma comparação entre esta solução e aquela obtida anteriormente, para os mesmo dados iniciais, mostrando que não só as velocidades de ondas podem ser distintas, mas também as próprias sequências de ondas que formam tais soluções. / In this work we prasent the Riemann solution for a system of conservation laws associatcd to an unidimensional two-ph&sc fiow in a porous media taking into account the hysteresis effects on the permeability curves. Our main contribution in this work is that the solution of the Riemann problem is obtained for a model that takes into account the hysteresis in both wetting and non-wetting phases and considers the scanning curves of embebition and drainagc distincts whenever there is a reversion of regime. The results obtained here improve those obtained for a simplified model where hysteresis is considered only on the non-wetting phase and the scanning curves coincide aftcr a second reversion of regime. Once obtained the solution of the Riemann problem, which is basic for the construction of high aceurate numeric simulators, we compare this solution and that one already obtained, for the same initial data. showing that not only the speeds of waves can be distinct, but also the sequences of waves in such Solutions.

The diploma thesis surveys the problematics of multidimensional Riemann integral. The goals of the thesis are, firstly, to provide a clear summary of the Multidimensional Riemann Integration Theory, secondly, to describe the basic calculation methods on carefully chosen examples, and, thirdly, to demonstrate its practical use in mathematics and natural sciences in general.

The Eulerian description of dispersed two-phase flows results in a system of partial differential equations describing characteristics of the flow, namely volume fraction, density and velocity of the two phases, around any point in space over time. When pressure forces are neglected or a same pressure is considered for both phases, the resulting system is weakly hyperbolic and solutions may exhibit vacuum states (regions void of the dispersed phase) or localized unbounded singularities (delta shocks) that are not physically desirable. Therefore, it is crucial to find a physical way for preventing the formation of such undesirable solutions in weakly hyperbolic Eulerian two-phase flow models. This thesis focuses on the mathematical analysis of an Eulerian model for air- droplet flows, here called the Eulerian droplet model. This model can be seen as the sticky particle system with a source term and is successfully used for the prediction of droplet impingement and more recently for the prediction of particle flows in air- ways. However, this model includes only one-way momentum exchange coupling, and develops delta shocks and vacuum states. The main goal of this thesis is to improve this model, especially for the prevention of delta shocks and vacuum states, and the adjunction of two-way momentum exchange coupling. Using a characteristic analysis, the condition for loss of regularity of smooth solutions of the inviscid Burgers equation with a source term is established. The same condition applies to the droplet model. The Riemann problems associated, respectively, to the Burgers equation with a source term and the droplet model are solved. The characteristics are curves that tend asymptotically to straight lines. The existence of an entropic solution to the generalized Rankine-Hugoniot conditions is proven. Next, a way for preventing the formation of delta shocks and vacuum states in the model is identified and a new Eulerian droplet model is proposed. A new hierarchy of two-way coupling Eulerian models is derived. Each model is analyzed and numerical comparisons of the models are carried out. Finally, 2D computations of air-particle flows comparing the new Eulerian droplet model with the standard Eulerian droplet model are presented.

Made available in DSpace on 2015-03-26T13:45:31Z (GMT). No. of bitstreams: 1 texto completo.pdf: 1613593 bytes, checksum: 9f102a91f9dec62a3656d30b4f7a490c (MD5) Previous issue date: 2010-02-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This study deals with the surface of the compact quotient M2=G where the surface M2 is either the Euclidean plane or the plane spherical or the hyperbolic plane, G is a group of isometries of their surfaces, and this group is generated by matching of edges of polygons. The Poincaré theorem that provides a method of finding the group of isometries G the functions that the pair of edges of the polygons involved. By using this theorem we construct two new pairings of generalized edges (Chapter 4) associated with the tessellations {12η 8,4} e {12μ 12,4}, respectively. These tessellations provide packing of spheres whose packing density is very close to the maximum 3/π. Such pairings are the starting point for finding codes with optimal transmission rates for Multiple-Input Multiple-Output (MIMO). / Este estudo aborda a construção de superfícies compactas pelo quociente M2/G onde a superfície M2 ou é o plano euclidiano, ou é o plano esférico, ou é o plano hiperbólico, G é um grupo de isometrias das respectivas superfícies e esse grupo é gerado pelos emparelhamentos de arestas dos polígonos. O Teorema de Poincaré fornece um método de encontrar o grupo de isometrias G que consiste das funções de emparelhamento de arestas dos polígonos associados. Mediante o uso deste teorema nós construímos dois novos emparelhamentos de arestas generalizados (Capítulo 4), associados as tesselações {12η 8,4} e {12μ 12,4}, respectivamente. Estas tesselações fornecem empacotamento de esferas cuja densidade de empacotamento é bem próxima do valor máximo 3/π. Tais emparelhamentos são o ponto de partida para a busca de códigos com ótimas taxas de transmissão para canais de múltiplas entradas e múltiplas e saídas (MIMO).

In this work we present a proof that the Hilbert s Tenth Problem is unsolvable. This problem is to give a computing algorithm which will tell of a given polynomial Diophantine equation with integer coefficients whether or not it has a solution in integers. We start developing some topics of basic number theory, that will be useful at some time. In this part we prove only main results. After that, we study Diophantine equation as well as Diophantine functions. Then, we prove a serie of lemas that will be useful to proof that the exponential function is Diophantine. From there, we define the concept of recursive function and prove that a function is Diophantine if and only if it is recursive. Finally we prove the Universality Theorem. We use this last theorem to proof that the Hilbert s Problem is unsolvable. / Neste trabalho apresentamos uma demonstração da insolubilidade do Décimo Problema de Hilbert, que investiga a existência de um método para determinar se dada uma equação Diofantina qualquer podemos determinar se esta tem ou não uma solução. Começamos desenvolvendo alguns tópicos de teoria de números, que serão úteis em vários momentos, nesta parte demonstramos apenas os resultados principais. Em um segundo momento, passamos ao estudo das equações Diofantinas bem como das funções Diofantinas, que permeiam nossos resultados. Em seguida, demonstramos uma série de lemas que servem de base para mostrarmos que a função exponencial é Diofantina. A partir daı, passamos a definição do importante conceito de função recursiva e então demonstramos que uma função ser recursiva é equivalente a ser Diofantina. Finalmente, demonstramos o Teorema da Universalidade que servirá de base para a demonstração o da insolubilidade do Décimo Problema de Hilbert. / Mestre em Matemática

Neste trabalho, seguindo a abordagem de Weierstrass, temos o objetivo de responder a seguinte questão: conhecida a equivalência entre holomorfia e analiticidade no caso complexo, quais propriedades das funções analíticas podem ser obtidas sem assumir tal equivalência? Analisando esta situação, resultados interessantes serão obtidos sem o uso de qualquer teorema de integração complexa e, para alcançar tal objetivo, nossas principais ferramentas serão a teoria de somas não ordenadas de famílias em C e propriedades do índice de caminhos fechados. Entre os resultados apresentados estão os conhecidos Teorema Fundamental da Álgebra, Lema de Schwarz, Teorema de Montel, Teorema da Série Dupla de Weierstrass, Princípio do Argumento, Teorema de Rouché, Teorema da Fatoração de Weierstrass, Pequeno Teorema de Picard e o Teorema da Aplicação de Riemann. / In this work, following the Weierstrass\'s approach, we aim to answer the following question: knowing the equivalence between holomorphy and analyticity in the complex case, which properties of analytic functions can be obtained without assuming such equivalence? Through analyzing this situation, interesting results will be obtained without employing of any complex integration theorem and in order to achieve this goal, our main tools will be the theory of unordered sums in C and properties of winding numbers of closed paths. Among the proven results are the well known Fundamental Theorem of Algebra, Schwarz\'s Lemma, Montel\'s Theorem, Weierstrass\'s Double Series Theorem, Argument Principle, Rouché\'s Theorem, Weierstrass\'s Factorization Theorem, Picard\'s Little Theorem and the Riemann\'s Mapping Theorem.

Depuis environ une dizaine d’années, les images hyperspectrales produites par les systèmes de télédétection, “Remote Sensing”, ont permis d’obtenir des informations très fiables quant aux caractéristiques spectrales de matériaux présents dans une scène donnée. Nous nous intéressons dans ce travail au problème de la segmentation non supervisée d’images hyperspectrales suivant trois approches bien distinctes. La première, de type Graph Embedding, nécessite deux étapes : une première étape d’appariement des pixels de patchs de l’image initiale grâce à une mesure de similarité spectrale entre pixels et une seconde étape d’appariement d’objets issus des segmentations locales grâce à une mesure de similarité entre objets. La deuxième, de type Spectral Hashing ou Semantic Hashing, repose sur un codage binaire des variations des profils spectraux. On procède à des segmentations par clustering à l’aide d’un algorithme de k-modes adapté au caractère binaire des données à traiter et à l’aide d’une version généralisée de la distance classique de Hamming. La troisième utilise les informations riemanniennes des variétés issues des différentes façons de représenter géométriquement une image hyperspectrale. Les segmentations se font une nouvelle fois par clustering à l’aide d’un algorithme de k-means. Nous exploitons pour cela les propriétés géométriques de l’espace des matrices symétriques définies positives, induites par la métrique de Fisher Rao. / Hyperspectral images provided by modern spectrometers are composed of reflectance values at hundreds of narrow spectral bands covering a wide range of the electromagnetic spectrum. Since spectral reflectance differs for most of the materials or objects present in a given scene, hyperspectral image processing and analysis find many real-life applications. We address in this work the problem of unsupervised hyperspectral image segmentation following three distinct approaches. The first one is of Graph Embedding type and necessitates two steps : first, pixels of the original image patchs are compared using a spectral similarity measure and then objects obtained by local segmentations are fusioned by means of a similarity measure between objects. The second one is of Spectral Hashing or Semantic Hashing type. We first define a binary encoding of spectral variations and then propose a clustering segmentation relying on a k- mode classification algorithm adapted to the categorical nature of the data, the chosen distance being a generalized version of the classical Hamming distance. In the third one, we take advantage of the geometric information given by the manifolds associated to the images. Using the metric properties of the space of Riemannian metrics, that is the space of symmetric positive definite matrices, endowed with the so-called Fisher Rao metric, we propose a k-means algorithm to obtain a cluster partitioning of the image.

L'objet de cette thèse consiste en la construction de nouveaux exemples de surfaces (ou hypersurfaces) minimales dans les espaces euclidiens R^3, R^n x R avec n>2 ou dans l'espace homogène S^2 x R. Nous prouvons l'existence de surfaces minimales dans R^3 arbitrairement proches d'un polygone convexe. Nous prouvons également l'existence d'hypersurfaces minimales de type Riemann dans R^n x R, n>2. Celles-ci peuvent être interprétées comme étant une famille d'hyperplans horizontaux (des bouts) reliés les uns aux autres par des morceaux de caténoïdes déformés (des cous). Nous donnons un résultat général pour ce type d'objet quand il est périodique ou bien quand il a un nombre fini de bouts horizontaux. Cela se fait sous certaines hypothèses de contraintes sur les forces intervenant dans la construction. Nous finissons en donnant plusieurs exemples, notamment l'existence d'une hypersurface de type Wei verticale qui n'existe pas en dimension 3. Nous donnons aussi la preuve de l'existence d'une surface minimale de type Riemann dans S^2 x R telle que deux bouts sphériques sont reliés entre eux alternativement par 1 cou et 2 cous. Là aussi, nous mettons en évidence le rôle joué par les forces lors de la construction. De même que dans le chapitre précédent, la méthode repose sur un processus de recollement. Nous donnons une description très précise de la caténoïde et la surface de Riemann dans S^2 x R. Enfin, nous établissons l'existence dans R^n x R d'hypersurfaces de type Scherk lorsque n>2 / This thesis is devoted to the construction of numerous examples of minimal surfaces (or hypersurfaces) in the $3$-Euclidean space, R^n x R with n>2 or in the homogeneous space S^2 x R . We prove the existence of minimal surfaces in R^3 as close as we want of a convex polygon. We prove the existence of minimal hypersurfaces in R^n x R, n>2, whose have Riemann's type. These ones could be considered as a family of horizontal hyperplanes (the ends) which are linked to each other by pieces of deformed catenoids (the necks). We provide a general result in the case simply-periodic together with the case of a finite number of hyperplanar ends. Our construction lies on some conditions associates with the forces that characterize the different configurations. We end with giving some examples ; in particular, we exhibit the existence of vertical Wei example that does not exists in the 3-dimensional case. We also prove the existence of the analogous of the Wei example in S^2 x R. The surface is such that two spherical ends are linked by 1 neck and 2 necks alternatively. Here again, we highlight the role that the forces play in the construction. Moreover, like in the previous chapter, the method lies on a gluing process. We give an accurate description of the catenoid and the Riemann's minimal example in S^2 x R. Finally, we demonstrate the existence of Scherk type hypersurfaces in R^n x R when n>2