Harmonic weak Maass forms are instances of real analytic modular forms which have recently found applications in several areas of mathematics. They provide a framework for Ramanujan's theory of mock modular forms , arise naturally in investigating the surjectivity of Borcherds' singular theta lift , and their Fourier coefficients seem to encode interesting arithmetic information. Until now, harmonic weak Maass forms have been studied solely as complex analytic objects. The aim of this thesis is to recast their definition in more conceptual, algebro-geometric terms, and to lay the foundations of a p-adic theory of harmonic weak Maass forms analogous to the theory of p-adic modular forms formulated by Katz in the classical context. This thesis only discusses harmonic weak Maass forms of weight 0. The treatment of more general integral weights requires no essentially new idea but involves further notational complexities which may obscure the main features of our approach. / Les formes de Maass faiblement harmoniques ont recemment trouve des applications dans plusieurs domaines des mathématiques. Elles fournissent un cadre pour la théorie des ``Mock Modular forms" de Ramanujan, surviennent naturellement dans l'etude de la surjectivite de la correspondance de Borcherds et leurs coefficients de Fourier semblent donner des informations arithmetiques sur les derivees des tordues quadratiques de certaines fonctions L associees aux formes modulaires. Jusqu'a present, les formes de Maass faiblement harmoniques ont uniquement ete etudiees en tant qu' objets analytiques sur les nombres complexes. L'objectif de cette these est de les decrire dans un cadre algebrique plus conceptuel, et de jeter les bases d'une theorie p-adique des formes de Maass faiblement harmoniques, par analogie avec le point de vue geometrique de Katz sur la theorie des formes modulaires p-adiques. Cette these traite uniquement du cas des formes de Maass faiblement harmoniques de poids 0.

We introduce an abstract framework concerning non-equilibrium statistical mechanics in the specific context of Markov chains. This framework encompasses both the Evans-Searles and the Gallavotti-Cohen fluctuation theorems. To support and expand on these concepts, several results are proven, among which a central limit theorem and a large deviation principle. The interest for Markov chains is twofold. First, they model a great variety of physical systems. Secondly, their simplicity allows for an easy introduction to an otherwise complicated field encompassing the statistical mechanics of Anosov and Axiom A diffeomorphisms. We give two examples relating the present framework to physical cases modelled by Markov chains. One of these concerns chemical reactions and links key concepts from the framework to their well known physical counterpart. / Nous présentons une structure mathématique de la physique statistique hors d'équilibre appliquée aux chaînes de Markov. Cette structure comprend les théoremes de fluctuations de Evans-Searles et de Gallavotti-Cohen. Afin de supporter et enrichir ces concepts, plusieurs résultats sont presentés, notamment un théoreme central limite ainsi qu'un principe de grandes deviations. Il y a deux principales raisons d'étudier les chaînes de Markov. Premièrement, elles modélisent une grand variété de systèmes physiques. Ensuite, leur simplicité permet une introduction facile a un vaste champ comprenant la mécanique statistique des difféomorphismes d'Anosovet Axiome A. Nous présentons deux examples permettant de lier la structure présente a des cas physiques modelés par des chaînes de Markov. Un de ces examples concerne les réactions chimiques et établit un lien entre les concepts clés de la structure et leurs homologues physiques bien connus.

In this thesis we consider cutsets in graphs which can be expressed as unionsof sets each of which is spanned by a tree of diameter at most d-1 for someinteger d ≥ 1; we call these sets galaxy cutsets. These galaxycutsets generalise both star-cutsets and vertex-cuts, and serve as simple models for virus-like attacks on or cascading failures in networks,the crucial property being that neighbours of affected vertices may alsofail and cease to function. We approach our subject from four different pointsof view. We begin by exploring the connection between galaxiesand a suitable type of flow, proving a min-max result for planargraphs. Then, after tackling the fundamental issue of recognising whethera given graph is susceptible to virus-like attacks, i.e. whether it contains a galaxy cutset, we consider a weighted version of the flows that are dualto the galaxies, and prove Θ(log n) lower and upper approximability boundsfor the problem of finding a maximum such flow. We then investigate the problem of network design, that is tosay, the problem of constructing low cost spanning subgraphs of a given graph which are not vulnerable to cascading failures. Finally, weembark on a detailed analysis of the structure of star-cutsets in planar graphs and useour results to derive a polynomial time algorithm for the problem ofneutralising every star-cutset by protecting edges. / Dans cette thèse, nous considerons des séparateurs dans les graphes qui peuvent être éxprimés sous forme d'une union d'ensembles de sommets dans laquelle chaque ensemble est couvert par un arbre de diamètre d-1 pour un nombre entier d ≥ 1; nous appellons ces séparateurs des galaxies séparatrices. Les galaxies séparatrices genéralisent les étoiles séparatrices et les séparateurs formés par un ensemble de sommets, et elles servent comme simple modèle pour des attaques de virus sur ou des cascades de défaillances dans un réseau, la proprieté distinguante étant que les voisins des sommets qui sont affectés peuvent eux aussi faillir. Nous approchons le sujet depuis quatre points de vue différents. Nous commençons par explorer le lien entre les galaxies et un type de flot approprié, et nous prouvons un résultat de type min max pour les graphes planaires. Ensuite, après avoir résolu la question fondamentale de reconnaitre si un graphe donné est sensible aux attaques de virus, c'est-à-dire s'il contient une galaxie séparatrice, nous introduisons des capacités dans les flots correspondants aux galaxies, et demontrons une borne d'approximabilité inférieure et supérieure de Θ(log n)pour le problème de trouver un flot maximum. Ensuite, nous enquêtons sur le problème de dessein de réseau, c'est-à-dire le problème de construire des sous-graphes couvrants peu coûteux qui ne sont pas sensibles aux cascades de défaillances. Finalement, nous nous lançons dans une analyse détaillée de la structure des étoiles séparatrices dans les graphes planaires, et nous utilisons nos résultats pour développer un algorithme polynomial qui résout le problème de neutraliser toutes les étoiles séparatrices en protégeant des arêtes.

Linear mixed-effects (LME) models are frequently used for modeling longitudinal data. One complicating factor in the analysis of such data is that samples are sometimes obtained from a population with significant underlying heterogeneity, which would be hard to capture by a single LME model. Such problems may be addressed by a finite mixture of linear mixed-effects (FMLME) models, which segments the population into subpopulations and models each subpopulation by a distinct LME model. Often in the initial stage of a study, a large number of predictors are introduced. However, their associations to the response variable vary from one component to another of the FMLME model. To enhance predictability and to obtain a parsimonious model, it is of great practical interest to identify the important effects, both fixed and random, in the model. Traditional variable selection techniques such as stepwise deletion and subset selection are computationally expensive even with modest numbers of covariates and components in the mixture model. In this thesis, we introduce a penalized likelihood approach and propose a nested EM algorithm for efficient numerical computations. The estimators are shown to possess consistency and sparsity properties and asymptotic normality. We illustrate the performance of the proposed method through simulations and a real data example. / Les modèles linéaires mixtes (LME) sont fréquemment employés pour la modélisation des données longitudinales. Un facteur qui complique l'analyse de ce genre de données est que les échantillons sont parfois obtenus à partir d'une population d'importante hétérogénéité sous-jacente, qui serait difficile à capter par un seul LME. De tels problèmes peuvent être surmontés par un mélange fini de modèles linéaires mixtes (FMLME), qui segmente la population en sous-populations et modélise chacune de ces dernières par un LME distinct. Souvent, un grand nombre de variables explicatives sont introduites dans la phase initiale d'une étude. Cependant, leurs associations à la variable réponse varient d'un composant à l'autre du modèle FMLME. Afin d'améliorer la prévisibilité et de recueillir un modèle parcimonieux, il est d'un grand intérêt pratique d'identifier les effets importants, tant fixes qu'aléatoires, dans le modèle. Les techniques conventionnelles de sélection de variables telles que la suppression progressive et la sélection de sous-ensembles sont informatiquement chères, même lorsque le nombre de composants et de covariables est relativement modeste. La présente thèse introduit une approche basée sur la vraisemblance pénalisée et propose un algorithme EM imbriqué qui est computationnellement efficace. On démontre aussi que les estimateurs possèdent des propriétés telles que la cohérence, la parcimonie et la normalité asymptotique. On illustre la performance de la méthode proposée au moyen de simulations et d'une application sur un vrai jeu de données.

As solutions of the Einstein-Maxwell field equations, the Kerr-Newman-de Sitter geometries are spacetimes that model the outer geometry of a charged, rotating black hole. We re-derive, using a lemma of Brandon Carter, first-order conserved quantities for the motion of zero rest mass and massive test particles in the Kerr-Newman-de Sitter spacetimes. The Liouville-Arnol'd integrability theory allows us to use first integral equations in place of second-order equations in the dynamical analysis. In examining the effects of first integral data on the equations, we expose some differences between particle dynamics in the electrically-neutral, asymptotically-flat Kerr geometries and those in the charged, de Sitter geometries. / Les géométries de Kerr-Newman-de Sitter sont des solutions des équations d'Einstein-Maxwell avec constante cosmologique modélisant la partie extérieure de l'espace-temps au voisinage d'une configuration d'équilibre d'un trou noir en rotation. Nous calculons a l'aide d'un lemme de Brandon Carter les quantités conservées du premier ordre associées au mouvement de particules d'épreuve de masse nulle et non-nulle dans la métrique de Kerr-Newman-de Sitter. Le théorème d'intégrabilité de Liouville-Arnol'd nous permet d'utiliser ces quantités conservées pour analyser la dynamique. Ceci nous permet de mettre en evidence les différences entre la dynamique des particules selon que l'on introduit ou non un champ électromagnétique et une constante cosmologique.

We show that the gap between the least costly 3-edge-connected metric graph and the least costly 3-vertex-connected metric graph is at most $3$. The approach relies upon tree decompositions, and a degree limiting theorem of Bienstock et al. As well, we explore the tree decomposition approach for general k-edge and vertex-connected graphs, and demonstrate a large amount of the required background theory. / Nous démontrons que l'écart entre un graphe métrique 3-arête-connexe de coût minimum et un graphe métrique 3-sommet-connexe de coût minimum est au plus 3. Notre approche repose sur l'existence de décompositions arborescentes et sur un théorème de Bienstock et al qui limite les degrés des sommets. De plus, nous explorons la décomposition arborescente pour le cas plus général des graphes k-arête et sommet connexes et nous exposons en grande partie les résultats nécessaires pour accéder à notre travail.

In this thesis, we are concerned with finite simple graphs. Given a subset $S$ of $\{3,4,\ldots,2n-1\}$, the additive graph generated by $S$ has vertex set $V=[n]$ and edge set $E$, with $(i,j) \in E$ if and only if $i+j \in S$. The focus of this thesis is the relationship between generating sets $S$ and monotone properties in the corresponding graphs. We make the first known investigation of the \textit{Traversal by Prime Sum Problem}, in which the set $S$ is the prime numbers and the property of interest is a Hamilton cycle. A number of new results are proved concerning both these graphs and the additive graphs for which the set $S$ is the practical numbers.\\ For any subset $S$ of $\{3,4,\ldots,2n-1\}$, we prove that the $|S|$-closure of the additive graph generated by $S$ is the complete graph; this allows for the determination of tight thresholds for a number of monotone properties in terms of $|S|$ using results from closure theory. These graphs are shown to be the first known wide-ranging and representative subclass of graphs with complete $k$-closure, and they afford a new and simple construction of minimum graphs with complete $k$-closure. Finally, as an example of the number-theoretic interpretations of these graphs and their properties, we generalize a theorem by Cramer concerning prime numbers to a number of different sequences. / \'tant donn\'{e} un sous-ensemble $S$ de $\{3,4,\ldots,2n-1\}$, le graphe additif engendr\' par $S$ a un ensemble de sommets $V=[n]$ et un ensemble d'arr\^tes $E$, telle que $(i,j) \in E$ si et seulement si $i+j \in S$. L'objectif de cette th\`se est l'\'{e}tude des relations entre l'ensemble $S$ et les propri\'t\'{e}s monotones du graphe additif correspondant. On effectue les premi\`{e}res recherches connue sur le \textit{Traversal By Prime Sum Problem}, probl\`{e}me dans lequel l'ensemble $S$ correspond \`{a} l'ensemble des nombres premiers et la propri\'t\'{e} de graphe recherch\'e est l'existence d'un cycle hamiltonien. De nouveaux r\'{e}sultats sont \'tablis pour ce probl\`me ainsi que dans le cas o\` $S$ est l'ensemble des entiers pratiques.\\ Pour un tel $S$ quelconque, on d\'{e}montre que la $|S|$-fermeture du graphe additif engendr\' par $S$ est un graphe complet. Ainsi, en utilisant les r\'{e}sultats de la th\'orie de la fermeture on parvient \` d\'{e}terminer le seuil pour plusieurs propri\'t\'{e}s monotones des graphes en terme de $|S|$. Ces graphes sont les premiers repr\'sentant connus d'une large sous-classe de graphe $k$-ferm\'{e}s complets. Ils permettent de donner une construction nouvelle et simple de graphe ferm\'s et complets minimaux. Enfin, comme exemple d'interpr\'{e}tation arithm\'tique de ces graphes et de leurs propri\'{e}t\'s, on g\'{e}n\'ralise un th\'{e}or\`{e}me de Cramer sur les nombres premiers \`{a} d'autres suites d'entiers.

Stochastic methods for global optimization problems with continuous variables have been studied. Modifications of three different algorithms have been proposed. These are (1) Multilevel Single Linkage (MSL), (2) Simulated Annealing (SA) and (3) Controlled Random Search (CRS). We propose a new topographical Multilevel Single Linkage (TMSL) algorithm as an extension of MSL. TMSL performs much better than MSL, especially in terms of number of function evaluations. A new aspiration based simulated annealing algorithm (ASA) has been derived which enhances the performance of SA by incorporating an aspirat.ion criterion. We have also proposed two new CRS algorithms, the CRS4 and CRS5 algorithms, which improve the CRS algorithm both in terms of cpu time and the number of function evaluations. The usefulness of the Halton and the Hammersley quasi-random sequences in global optimization has been investigated. These sequences are frequently used in numerical integration in the field of Bayesian statistics. A useful property of the quasi-random sequences is that they are evenly distributed and thus explore the search region more rapidly than pseudo-random numbers. Comparison of the modified algorithms with their unmodified versions is carried out on standard test problems but in addition a substantial part of the thesis consists of numerical investigations of 5 different practical global optimization problems. These problems are as follows: (1) A nonlinear continuous stirred tank reactor problem. (2) A chemical reactor problem with a bifunctional catalyst. (3) A pig-liver likelihood function. (4) Application and derivation of semi-empirical many body interatomic potentials. (5) A optimal control problem involving a car suspension system. Critical comparisons of the modified and unmodified global optimization algorithms have been carried out on these problems. The methods applied to these problems are compared from the points of view of reliability in finding the global optimum, cpu time and number of function evaluations.

This work is intended to contribute to the search for an explanation of the way a really good finite element should behave during analysis of a general shell problem. The finite element analysis of thin shells is currently receiving much attention in the international literature. Indeed it seems that in almost every new issue of the engineering journals there is a proposal for a new and more efficient shell element. The reason for this is of course the underlying complexity of the shell problem and, more specifically the difficulty of taking bending into correct account. In order to elicit an understanding of the use of the finite element method in shell problems an in-depth study is presented of the behaviour of a vehicle shell finite element. This element is the very simple combined constant membrane stress and constant bending moment flat triangle. The examination of its behaviour reveals that the characteristics of an assembly of these elements are such as to enable recovery, in a remarkable way, of each of the types of deformation identified by the classical first approximation theory. Recovery of rigid body movement, inextensional. bending, membrane action and edge effect- is achieved to an accuracy consistent with the order of magnitude of inherent errors of the classical theory. Thus, the element is seen to hold a position of fundamental importance with regard to the numerical analysis of thin shells. Special attention is given to the sensitive low energy bending response. This reveals that there are two quite different roles for the element bending freedoms. One role concerns inextensional bending movements which extend over the whole model. The other role concerns local rotational movements which accompany the curvature changes of inextensional bending and edge effect. Extensive numerical comparisons are made against solutions obtained from the classical theory for shells which are very deep with strongly negative Gaussian curvature and which are considered to provide very severe tests. Investigation of edge effect concerns a cantilevered circular cylinder under edge moment. To complete this examination of bending details are given of a matrix procedure which is intended to assess thin shell finite element models in their response to inextensional bending. To conclude this work the results of a preliminary study of the mathematical details of convergence of the vehicle element are presented. This investigation is specific to the geometry of a circular cylinder and clamped boundary conditions. It is shown that, despite the highly nonconforming nature of the element, O(h) asymptotic convergence in the energy norm is achieved and in this respect is similar in behaviour to the Clough-Johnson flat plate shell finite element.

Let K be a quadratic number field, Ok its ring of integers, and G a cyclic group of order prime p. In this thesis, we study the kernel group D(O(_K)G) and obtain a number of results concerning its order and structure. For K imaginary, we also investigate the subset R(O(_k)G) of the locally free class group CI(O(_k)G) consisting of classes which occur as rings of integers of tame extensions of K with Galois group isomorphic to G. We calculate R(O(_k)G) under a variety of conditions and obtain, for an arbitrary tame extension L o( K with group G, invariants which determine the class of O(_L) in R(O(_k)G).