The thesis studies homogenization of Dirichlet boundary value problems for divergence type elliptic operators, and the associated boundary layer issues. This type of problems for operators with periodically oscillating coeffcients, and fixed boundary data are by now a classical topic largely due to the celebrated work by Avellaneda and Lin from late 80's. The case when the operator and the Dirichlet boundary data exhibit periodic oscillations simultaneously was a longstanding open problem, and a progress in this direction has been achieved only very recently, in 2012, by Gerard-Varet and Masmoudi who proved a homogenization result for the simultaneously oscillating case with an algebraic rate of convergence in L2. Aimed at understanding the homogenization process of oscillating boundary data, in the first part of the thesis we introduce and develop Fourier-analytic ideas into the study of homogenization of Dirichlet boundary value problems for elliptic operators in divergence form. In smooth and bounded domains, for fixed operator and periodically oscillating boundary data we prove pointwise, as well as Lp convergence results the homogenization problem. We then investigate the optimality (sharpness) of our Lp upper bounds. Next, for the above mentioned simultaneously oscillating problem studied by Gerard-Varet and Masmoudi, we establish optimal Lp bounds for homogenization in some class of operators. For domains with non smooth boundary, we study similar boundary value homogenization problems for scalar equations set in convex polygonal domains. In the vein of smooth boundaries, here as well for problems with fixed operator and oscillating Dirichlet data we prove pointwise, and Lp convergence results, and study the optimality of our Lp bounds. Although the statements are somewhat similar with the smooth setting, challenges for this case are completely different due to a radical change in the geometry of the domain. The second part of the work is concerned with the analysis of boundary layers arising in periodic homogenization. A key difficulty toward the homogenization of Dirichlet problem for elliptic systems in divergence form with periodically oscillating coefficients and boundary condition lies in identification of the limiting Dirichlet data corresponding to the effective problem. This question has been addressed in the aforementioned work by Gerard-Varet and Masmoudi on the way of proving their main homogenization result. Despite the progress in this direction, some very basic questions remain unanswered, for instance the regularity of this effective data on the boundary. This issue is directly linked with the up to the boundary regularity of homogenized solutions, but perhaps more importantly has a potential to cast light on the homogenization process. We initiate the study of this regularity problem, and prove certain Lipschitz continuity result. The work also comprises a study on asymptotic behaviour of solutions to boundary layer systems set in halfspaces. By a new construction we show that depending on the normal direction of the hyperplane, convergence of the solutions toward their tails far away from the boundaries can be arbitrarily slow. This last result, combined with the previous studies gives an almost complete picture of the situation.

Nonparametric Bayesian inference has widespread applications in statistics and machine learning. In this thesis, we examine the most popular priors used in Bayesian non-parametric inference. The Dirichlet process and its extensions are priors on an infinite-dimensional space. Originally introduced by Ferguson (1983), its conjugacy property allows a tractable posterior inference which has lately given rise to a significant developments in applications related to machine learning. Another yet widespread prior used in nonparametric Bayesian inference is the Beta process and its extensions. It has originally been introduced by Hjort (1990) for applications in survival analysis. It is a prior on the space of cumulative hazard functions and it has recently been widely used as a prior on an infinite dimensional space for latent feature models. Our contribution in this thesis is to collect many diverse groups of nonparametric Bayesian tools and explore algorithms to sample from them. We also explore machinery behind the theory to apply and expose some distinguished features of these procedures. These tools can be used by practitioners in many applications.

Cette thèse traite du comptage d'extensions cubiques relatives. Dans le premier chapitre on traite un travail commun avec Henri Cohen. Soit k un corps de nombres. On donne une formule asymptotique pour le nombre de classes d'isomorphisme d'extensions cubiques L/k telles que la clôture galoisienne de L/k contienne une extension quadratique fixée K_2/k. L'outil principal est la théorie de Kummer. Dans le second chapitre, on suppose k un corps quadratique imaginaire (avec nombre de classes 1) et on décrit un algorithme pour énumérer toutes les classes d'isomorphisme d'extensions cubiques L/k jusqu'à une certaine borne X sur la norme du discriminant relatif. / This thesis is about counting relative cubic extensions. In the first chapter we describe a joint work with Henri Cohen. Let k be a number field. We give an asymptotic formula for the number of isomorphism classes of cubic extensions L/k such that the Galois closure of L/k contains a fixed quadratic extension K_2/k. The main tool is Kummer theory. In the second chapter, we suppose k to be an imaginary quadratic number field (with class number 1) and we describe an algorithm for listing all isomorphism classes of cubic extensions L/k up to a bound X on the norm of the relative discriminant ideal.

In binary-transaction data-mining, traditional frequent itemset mining often produces results which are not straightforward to interpret. To overcome this problem, probability models are often used to produce more compact and conclusive results, albeit with some loss of accuracy. Bayesian statistics have been widely used in the development of probability models in machine learning in recent years and these methods have many advantages, including their abilities to avoid overfitting. In this thesis, we develop two Bayesian mixture models with the Dirichlet distribution prior and the Dirichlet process (DP) prior to improve the previous non-Bayesian mixture model developed for transaction dataset mining. First, we develop a finite Bayesian mixture model by introducing conjugate priors to the model. Then, we extend this model to an infinite Bayesian mixture using a Dirichlet process prior. The Dirichlet process mixture model is a nonparametric Bayesian model which allows for the automatic determination of an appropriate number of mixture components. We implement the inference of both mixture models using two methods: a collapsed Gibbs sampling scheme and a variational approximation algorithm. Experiments in several benchmark problems have shown that both mixture models achieve better performance than a non-Bayesian mixture model. The variational algorithm is the faster of the two approaches while the Gibbs sampling method achieves a more accurate result. The Dirichlet process mixture model can automatically grow to a proper complexity for a better approximation. However, these approaches also show that mixture models underestimate the probabilities of frequent itemsets. Consequently, these models have a higher sensitivity but a lower specificity.

Cette thèse a pour objet l’étude des solutions à l’équation des vagues en régime dit toit rigide lorsque l’amplitude des vagues tend vers zéro. Plus précisément, l’équation des vagues modélise le mouvement d’un fluide à surface libre borné en dessous par un fond fixe. Les équations dépendent de plusieurs paramètres physiques, notamment du rapport epsilon entre l’amplitude des vagues et la profondeur. Le modèle asymptotique toit rigide consiste à changer l’échelle de temps d’un rapport epsilon, puis de faire tendre ce paramètre, et donc l’amplitude des vagues, vers zéro. L’étude mathématique de cette limite correspond à un problème de perturbation singulière d’une équation dispersive. Dans cette thèse, on commence par utiliser des outils de résolution d’équations aux dérivées partielles de type hyperbolique pour démontrer un résultat d’existence locale pour l’équation des vagues en temps long. Ceci est suivi par un résultat de dispersion sur l’équation des vagues, utilisant des techniques de type phase stationnaire et décomposition de Paley-Littlewood pour l’étude des intégrales oscillantes. Enfin, la dernière partie de la thèse utilise les résultats obtenus ci-dessus pour étudier un défaut de compacité dans la convergence faible (mais non forte) des solutions de l’équation des vagues lorsque l’amplitude tend vers 0. / In this thesis, we study the behavior of the solutions of the Water-Waves equations in the rigid lid regime as the amplitude of the waves goes to zero. More precisely, the Water-Waves equations investigate the dynamic of a free surface fluid, bounded from below by a fixed bottom. The equations depends on many physical parameters, as the ratio epsilon between the wave amplitude and the deepness of the water. The rigid lid model consists in scaling the time by an epsilon factor and taking the limit epsilon goes to zero, simulating a situation where the amplitude of the waves goes to zero. The mathematical study of this limit correspond to a singular perturbation problem of a dispersive equation. In this thesis, we first use classical tools of hyperbolics equations to prove a long time existence result for the Water-Waves equations. We then prove a dispersion result for these equations, using stationary phase methods and Paley-Littlewood decomposition. We then combine these results to highlight the lack of compactness in the weak (but non strong) convergence of the solutions of the Water-Waves equations as the amplitude goes to zero.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21542号 / 理博第4449号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 熊谷 隆, 准教授 福島 竜輝, 教授 牧野 和久 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

The elections in South Africa are contested by multiple political parties appealing to a diverse population that comes from a variety of socioeconomic backgrounds. As a result, a rich source of discourse is created to inform voters about election-related content. Two common sources of information to help voters with their decision are news articles and tweets, this study aims to understand the discourse in these two sources using natural language processing. Topic modelling techniques, Latent Dirichlet Allocation and Non- negative Matrix Factorization, are applied to digest the breadth of information collected about the elections into topics. The topics produced are subjected to further analysis that uncovers similarities between topics, links topics to dates and events and provides a summary of the discourse that existed prior to the South African general elections. The primary focus is on the 2019 elections, however election-related articles from 2014 and 2019 were also compared to understand how the discourse has changed. / Mini Dissertation (MIT (Big Data Science))--University of Pretoria, 2019. / Computer Science / MIT (Big Data Science) / Unrestricted

The aim of this work is to provide an introduction into the theory of infinite dimensional stochastic processes. The thesis contains the paper A class of infinite dimensional stochastic processes with unbounded diffusion written at Linköping University during 2012. The aim of that paper is to take results from the finite dimensional theory into the infinite dimensional case. This is done via the means of a coordinate representation. It is shown that for a certain kind of Dirichlet form with unbounded diffusion, we have properties such as closability, quasi-regularity, and existence of local first and second moment of the associated process. The starting chapters of this thesis contain the prerequisite theory for understanding the paper. It is my hope that any reader unfamiliar with the subject will find this thesis useful, as an introduction to the field of infinite dimensional processes.

Orientador: Dimitar Kolev Dimitrov / Banca: Ali Messaoudi / Banca: Carlos Gustavo T. de A. Moreira / Banca: Emanuel A. de Souza Carneiro / Banca: Valdir Antonio Menegatto / Resumo: Estudamos tópicos relacionados a zeros de séries de Dirichlet e de funções inteiras. Boa parte da tese é voltada à localização de zeros de séries de Dirichlet via critérios de densidade. Estabelecemos o critério de Nyman-Beurling para uma ampla classe de séries de Dirichlet e o critério de Báez-Duarte para L-funções de Dirichlet em semi-planos R(s)>1/2, para p ∈ (1,2], bem como para polinômios de Dirichlet em qualquer semi-plano R(s)>r. Um análogo de uma cota inferior de Burnol relativa ao critério de Báez-Duarte foi estabelecido para polinômios de Dirichlet. Uma das ferramentas principais na prova deste último resultado é a solução de um problema extremo natural para polinômios de Dirichlet inspirado no resultado de Báez-Duarte. Provamos que os sinais dos coeficientes de Maclaurin de uma vasta subclasse de funções inteiras da classe de Laguerre-Pólya possuem um comportamento regular / Abstract: We study topics related to zeros of Dirichlet series and entire functions. A large part of the thesis is devoted to the location of zeros of Dirichlet series via density criteria. We establish the Nyman-Beurling criterion for a wide class of Dirichlet series and the B'aezDuarte's criterion for Dirichlet L-functions in the semi-plane R(s) > 1/p, for p 2 (1, 2], as well as for zeros of Dirichlet polynomials in any semi-plane <(s) > r. An analog for the case of Dirichlet polynomials of a result of Burnol which is closely related to B'aez-Duarte's one is also established. A principal tool in the proof of the latter result is the solution of a natural extremal problem for Dirichlet polynomials inspired by B'aez-Duarte's result. We prove that the signs of the Maclaurin coecients of a wide class of entire functions that belong to the Laguerre-P'olya class posses a regular behaviou / Doutor

In automatically categorizing massive corpora of text, various topic models have been applied with good success. Much work has been done on applying machine learning and NLP methods on Internet media, such as Twitter, to survey online discussion. However, less focus has been placed on studying how geographical locations discussed in online fora evolve over time, and even less on associating such location trends with topics. Can online discussions be geographically tracked over time? This thesis attempts to answer this question by evaluating a geo-aware Streaming Latent Dirichlet Allocation (SLDA) implementation which can recognize location terms in text. We show how the model can predict time-dependent locations of the 2016 American primaries by automatic discovery of election topics in various Twitter corpora, and deduce locations over time.