Le travail de ma thèse est basé sur ces 4 points :i) Transport laplacien d'une cellule absorbante :Soit un certain espèce (cellule) de concentration C(x), qui diffuse dans un milieu homogène et isotrope à partir d'une lointaine source localisée sur la frontière fermée $partial Omega_{0}$ vers une interface compact semi-perméable $partial Omega$ (membrane de la "cellule") à laquelle elle disparaisse àun taux d'absorption donné : W>=0. La concentration C (transport laplacien avec un coefficient de diffusion D) satisfaite le problème (P1) (voir la thèse). On s'intéresse à résoudre le problème (P1) en dimension dim = 2; 3 et à calculer les courants local et total à travers les frontières des $partial Omega$ et $partial Omega_{0}$ qui seront utiles pour résoudre le problèmeinverse de localisation. Pour faciliter les calculs et les rendre explicites, on prend $partial Omega$ et $partial Omega_{0}$ avec des formes géométriquement régulières, précisément des boules, en distinguant les deux cas : $Omega$ et $Omega_{0}$ sont concentriques ou non-concentriques. Pour le cas non-concentriques , on utilise la technique de transformation conforme et le développement orthogonal en série de Fourier pour résoudre le problème (P1) en cas bidimensionnel. Tandis que en cas tridimensionnel, on résout le problème (P1) en utilisant le développement orthogonal suivant les fonctions sphériques harmoniques.ii) Problème inverse de localisationOn s'intéresse dans cette partie à résoudre le problème inverse de localisation associé au problème (P1) où les domaines $Omega$ et $Omega_{0}$ sont considérés avec des formes géométriques régulières (précisément des boules) . Ce problème consiste à trouver les conditions de Dirichlet-Neumann sur $partial Omega_{0}$ (courant local, courant total) suffisantes pour déterminer la position de la cellule $partial$ (par rapport à $Omega_{0}$), dont ces conditions sont disponibles par une suite des mesures expérimentales.iii) Problème invesre géomètrique :Dans cette partie on traite un autre type de problème inverse qui consiste à trouver la forme géométrique de la cellule en sachant les conditions de Dirichlet-Neumann au bord extérieur(partial Omega_{0}) qui sont mésurables par une suite d'expérience. Ce type du problème, on l'appelle le problème inverse géométrique. On résout ce problème en utilisant des techniques concernant les fonctions harmoniques et les transformations conformes.iv) Opérateur de Dirichlet-NeumannOn étudie l'opérateur de Dirichlet-Neumann relatif au problème (P1) dans les dimension deux et trois en distinguant les deux cas concentriques et non-concentriques. Ensuite, on montre que cet opérateur de Dirichlet-Neumann engendre certain semi-groupe qu'on l'appelle semi-groupe de Lax. Enfin, on construit ce semi-groupe de Lax associé à cet opérateur en cas tridimensionnel concentriques afin de vérifier que ce semi-groupe admet les mêmes propriétés que celui dans le cas général. / The outline of my thesisi) Let some "species" of concentration C(p), x 2 Rd, diuse stationary in the isotropic bulk from a (distant) source localised on the closed boundary $partial Omega_{0}$ towards a semipermeable compact interface $partial Omega$ of the cell $Omega in Omega_{0}$ where they disappear at a given rate $W >= 0$. Then the steady field of concentrations C satisfy the problem $(P1)$. (see the Thesis). We interest to solve (P1) in Twodimensional and Tridimensional cases and to calculate the local and total flux in order to solving the localisation inverse problem. In order to make easy the calculations, we take $Omega$ and $Omega_{0}$ with a regularly geometricals forms by distinguishing the two cases : Concentrics and non-concentrics case. For the non-cncentrics case, we use the conformal mapping technique for resolving the problem (P1) in the twodimensional case. whereas in the tridimensional case, we use the development according to the spherical harmonics functions.ii) Localisation inverse problemThe aim of the localisation inverse problem is to find the necessary Dirichlet-to-Neumann conditions in order to determine the position of thecell $Omega$, where these conditions are measurable.iii) Geometrical inverse problemOur main results concerns a formal solution of the geometrical inverse problem for the form of absorbing domains. We restrict this study to two dimensions and we study it by the conformal mapping technique and harmonic functions.iv) Dirichlet-to-Neumann operatorWe study the Dirichlet-to-Neumann operatot relative to problem (P1) in the twodimensional and tridimensionnal cases by distinguishing the two cases : Concentrics and non-concentrics case. We prove that the Dirichlet-to-Neumann operator generates some semi-group, we call it the Lax semi-group. Finally we construct this semi group and verify that this demi-group satisfies the generals properties of a operator.

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2014-08-29T19:24:13Z No. of bitstreams: 2 Dissertação Corrigida e Finalisada.pdf: 2280692 bytes, checksum: fa3c7d92b5ed8a39139ceeb3abb80551 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-08-29T19:24:13Z (GMT). No. of bitstreams: 2 Dissertação Corrigida e Finalisada.pdf: 2280692 bytes, checksum: fa3c7d92b5ed8a39139ceeb3abb80551 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-12-13 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this dissertation we analyze questions of existence of a weak solution for a class of superlineares elliptic Dirichlet problems. Here we do not consider the Ambrosseti Rabinovitz condition , which restricts some nonlinearities. We obtain main results of this dissertation via Variational Methods, such as Mountain Pass Theorem and Linking Theorem. Furthermore, weusePalais-Smalecondition(P.S.) or Cerami condition(Ce) / Nesta dissertação analisamos questões de existência de uma solução fraca para uma classe de problemas de Dirichlet elípticos superlineares. Aqui não consideramos a condição deAmbrosetti-Rabinowitz,a qual restringealgumasfunçõesnão lineares. Obtemos os principais resultados desta dissertação via Métodos variacionais, tais como o Teorema do Passo da Montanha e um Teorema de Linking. Além disso, utilizamos a TeoriaEspectral e ascondições dePalais-Smale(P.S.) eCerami(Ce).

Apresentaremos uma classe de energias potenciais $\\Pi \\in C^{\\infty}(\\Omega,R)$ que são s-decidíveis e que admitem funções auxiliares de Cetaev da forma $\\langle abla j^s\\Pi(q),q angle$, $q\\in \\Omega \\subset R^n$ que são s-resistentes. / We will present a class of potential energies $\\Pi \\in C^{\\infty}(\\Omega,R)$ that are s-decidable and that admit auxiliary functions of Cetaev of the form $\\langle abla j^s\\Pi(q),q angle$, $q \\in \\Omega \\subset R^n$ which are s-resistant.

Variáveis politômicas são comuns em experimentos agronômicos, apresentando natureza nominal ou ordinal. O modelo dos logitos generalizados é uma classe de modelos que pode ser empregada para a análise desses dados. Uma das características deste modelo é a pressuposição de que a variância é uma função conhecida da média e, espera-se, que a variância observada esteja próxima da variância pressuposta pelo modelo assumido. Contudo, quando ela é maior do que a especificada pelo modelo, tem-se o fenômeno da superdispersão. Nesse contexto, o presente trabalho objetivou caracterizar o problema da superdispersão associado a dados nominais em estudos \"cross-sectional\". Como motivação apresentam-se dois estudos adaptados da área de ciências agrárias relativos à fruticultura e zootecnia, ambos planejados no delineamento inteiramente casualizado. Verifica-se indicativo de superdispersão nos dados dos dois exemplos e como uma alternativa metodológica utilizou-se o modelo Dirichlet-multinomial. Por meio do gráfico de diagnóstico half-normal plot avaliou-se o ajuste do modelo dos logitos generalizados e do Dirichlet-multinomial. Adicionalmente, foi proposta uma extensão do índice de dispersão para os dados politômicos, com performance avaliada sob simulação. O modelo Dirichlet-multinomial mostrou-se adequado para o ajuste aos dados com superdispersão comparativamente ao modelo dos logitos generalizados. Apesar dos resultados satisfatórios obtidos, ressalta-se que este trabalho é uma introdução ao problema. / Polytomic variables are common in agronomic experiments, presenting nominal or ordinal nature. The generalized logits model is a class of models that can be used to analyze these facts. One of the characteristics of this model is the assumption that variance is a known function of the mean and. It is expected, that the analyzed variance is close to that assumed by the model. However, when it is larger than the one specified by the model, it has the phenomenon of overdipersion. In this context, the present work aims to characterize the problem of overdispersion associated with nominal data in cross-sectional studies. As motivation, it is showed two adapted studies of the agricultural sciences area, related to fruit growing and zootechnics, both planned in the completely randomized design. The Dirichlet-multinomial model was used as a methodological alternative and was indicated as an overdispersion in the facts of the two examples. The model of the generalized logits and the Dirichlet-multinomial model were evaluated using the half-normal plot. In addition, it was proposed an extension of the dispersion index for the polytomic data, with performance evaluated under simulation. The Dirichlet-multinomial model proved to be adequate for the adjustment to the overdispersed fact compared to the generalized logit model. Despite the satisfactory results obtained, it is emphasized that this work is an introduction to the problem.

針對具有多項分配(multinomial distribution)母體的類別資料，貝氏分析通常採取Dirichlet分配作為其先驗分配(prior distribution)，但在很多實際應用時，卻會遭遇困難；例如，當我們欲推估各年齡層佔總勞動力人口之比例時，母體除具多項分配外，其相鄰類別之比例亦相對接近；換言之，此時母體為具有平滑性(smooth)的多項分配，若依然採用Dirichlet分配作為其先驗分配，則會因為Dirichlet分配本身不具有平滑的特性，因而在做貝氏分析時會產生困擾。對這個難題Dickey and Jiang於1998年提出一個解決之道，他們的理論是對Dirichlet分配作適當之調整，將經過線性轉換後之Dirichlet分配稱為過濾後Dirichlet分配(filtered-variate Dirichlet distribution)，以過濾後Dirichlet分配作為調整後之先驗分配。對於Dickey and Jiang提出的方法，我們重新以蒙地卡羅法(Monte Carlo method)求出貝氏解，同時也嘗試以類似Makov and Smith (1977)和Smith and Makov (1978)對混合分配(mixture distribution)所用之準貝氏方法(quasi-Bayesian method)來逼近貝氏解。而本文將由電腦模擬的方式，探討貝氏方法與準貝氏方法之執行結果，並且考察準貝氏方法之收斂行為，對準貝氏方法的使用時機提出建議。

Nous nous intéressons à l'étude de la cyclicité et la bicyclicité dans les espaces $ell^p(Z)$ à poids et à l'étude de la cyclicité dans les espaces de Dirichlet. Alors que Wiener a caractérisé la bicyclicité des vecteurs de $ell^1(Z)$ et $ell^2(Z)$ grâce à l'ensemble des zéros de la transformée de Fourier, Lev et Olevski ont démontré que cet ensemble ne peut caractériser la bicyclicité dans $ell^p(Z)$ lorsque $1<p<2$ pour des suites $u in ell^1(Z)$. Beurling, Salem et Newman se sont aussi intéressés à la bicyclicité de vecteurs de $ell^p(Z)$ pour $1<p<2$. Dans ce travail, nous étendons tout d'abord les résultats de Beurling, Salem et Newman aux espaces $ell^p(Z)$ à poids, en étudiant la dimension de Hausdorff et la capacité de l'ensemble des zéros de la transformée de Fourier. Ensuite nous démontrons que le résultat de Lev-Olevskii reste valide pour la cyclicité dans $ell^p(Z)$, $1<p<2$. De plus, nous donnons des conditions suffisantes à la cyclicité dans les espaces $ell^p(Z)$ à poids. Enfin nous démontrons que, pour une fonction $f$ appartenant à l'algèbre du disque et à un espace de type Dirichlet, si $f$ est extérieure et si l'ensemble des zéros de $f$ est réduit à un point alors $f$ est cyclique. Ceci généralise le résultat de Hedenmalm et Shields qui ont traité le cas du Dirichlet classique. / We are interested in the study of cyclicity and bicyclicity in weighted $ell^p(Z)$ spaces and the study of cyclicity in Dirichlet spaces. While Wiener characterized the bicyclicity in $ell^1(Z)$ and $ell^2(Z)$, thanks to the zero set of the Fourier transform, Lev and Olevski have shown that this set cannot characterize bicyclicity in $ell^p(Z)$ when $1 < p < 2$ for sequences in $ell^1(Z)$. Also Beurling, Salem and Newman were interested in the bicyclicity in $ell^p(Z)$ when $1 < p < 2$. In this work, we first extend the results of Beurling, Salem and Newman to the weighted $ell^p(Z)$ spaces, by studying the Hausdorff dimension and the capacity of the zero set of the Fourier transform. Then we prove that the Lev-Olevskii result remains valid for cyclicity in $ell^p(Z)$, $1 < p < 2$. In addition, we give sufficient conditions for the cyclicity in the weighted $ell^p(Z)$ spaces. Finally, we prove that, for a function $f$ in the disk algebra and in a generalized Dirichlet space, if $f$ is outer and the zero set of $f$ is reduced to a point then $f$ is cyclic. This generalizes the result of Hedenmalm and Shields who have treated the case of the classical Dirichlet space.

In this work we use two approach to prove the existence of solutions for the Classical Dirichlet Problem on bounded domains. The first method is applied to domains with smooth boundary and is based on the Single and Double Layer Potentials Theory. The second method uses the Variational Dirichlet Principle to solve the Dirichlet Problem in plane domains with boundary less regular than the previous case; more precisely, boundary satisfying the property of the outer triangle. / Fundação de Amparo a Pesquisa do Estado de Alagoas / Nesta dissertação usamos duas abordagens diferentes para provar a existência de soluções para o Problema de Contorno de Dirichlet Clássico em domínios limitados. Aplicamos o primeiro método quando estamos trabalhando com domínios cuja fronteira é duas vezes continuamente diferenciável. Essa abordagem baseia-se na Teoria dos Potenciais de Camadas Simples e Dupla, onde a teoria de operadores compactos tem um papel fundamental. O segundo método usa o Princípio Variacional de Dirichlet para resolver o problema em domínios do plano com fronteiras menos regulares que no caso anterior; a saber, fronteiras que satisfazem a condição do triângulo exterior.

Neste trabalho estudamos o problema de Dirichlet assintótico para a equação das superfícies mínimas em uma superfície de Cartan-Hadamard rotacionalmente simétrica e mostramos que o problema e unicamente solúvel para qualquer dado contínuo em seu bordo assintótico. / In this work we study the asymptotic Dirichlet problem for the minimal surface equation on rotationally symmetric Cartan-Hadamard surfaces. We prove that the problem is uniquely solvave for any continuous asymptotic boundary data.

O objetivo central deste trabalho consiste em demonstrar a existência de gráficos mínimos C2,x com fronteira assintótica prescrita na variedade produto M R, onde M e completa, simplesmente conexa, com curvatura seccional KM satisfazendo KM ≤ -k2 < 0 e tal que, para algum p Є M, o subgrupo de isotropia de Iso(M) em p age de modo 2-pontos homogêneo nas esferas geodésicas centradas em p. / The main purpose of this work consists on proving the existence of minimal C2,x graphics with prescribed asymptotic boundary in the product manifold M R, where M is a complete, simply connected manifold with sectional curvature KM satisfying KM ≤ -k2 < 0 and such that, for some p 2 M, the isotropy subgroup of Iso(M) in p acts in a 2-points homogeneous way in the geodesic spheres centered in p.

Seja E(X) a cardinalidade dos números pares menores ou iguais a X que não podem ser escritos como soma de dois primos. O objetivo central desta dissertação é apresentar uma demonstração de uma estimativa para E(X) dada por Hugh L. Montgomery e Robert C. Vaughan em [22]. Mais precisamente, estabeleceremos a existência de uma constante positiva (efetivamente computável) tal que E(X) X1 ; para todo X su cientemente grande. / Let E(X) the cardinality of even numbers not exceeding X which cannot be written as a sum of two primes. The main goal of this dissertation is to present a proof of an estimate for E(X) given by Hugh L. Montgomery e Robert C. Vaughan in [22]. More precisely, we will establish the existence of a positive constant (e ectively computable) such that E(X) X1 for all su ciently large X: