This thesis is broadly divided in two parts. In the first part we give a survey of various distances between metric spaces, namely the uniform distance, Lipschitz distance, Hausdor distance and the Gramoz Hausdor distance. Here we talk about only the most basic of their properties and give a few illustrative examples. As we wish to study collections of metric measure spaces, which are triples (X; d; m) consisting of a complete separable metric space (X; d) and a Boral probability measure m on X, there are discussions about some distances between them. Among the three that we discuss, the transportation and distortion distances were introduced by Sturm. The later, denoted by 2, on the space X2 of all metric measure spaces having finite L2-size is the focus of the second part of this thesis. The second part is an exposition based on the work done by Sturm. Here we prove a number of results on the analytic and geometric properties of (X2; 2). Beginning by noting that (X2; 2) is a non-complete space, we try to understand its completion. Towards this end, the notion of a gauged measure space is useful. These are triples (X; f; m) where X is a Polish space, m a Boral probability measure on X and f a function, also called a gauge, on X X that is symmetric and square integral with respect to the product measure m2. We show that, Theorem 1. The completion of (X2; 2) consists of all gauged measure spaces where the gauges satisfy triangle inequality almost everywhere. We denote the space of all gauged measure spaces by Y. The space X2 can be embedded in Y and the transportation distance 2 extends easily from X2 to Y. These two spaces turn out to have similar geometric properties. On both these spaces 2 is a strictly intrinsic metric; i.e. any two members in them can be joined by a shortest path. But more importantly, using a description of the geodesics in these spaces, the following result is proved. Theorem 2. Both (X2; 2) and (Y; 2) have non-negative curvature in the sense of Alexandrov.

L'espace de Wasserstein est l'ensemble des mesures de probabilité définies sur un domaine fixé et muni de la distance de Wasserstein quadratique. Dans ce travail, nous étudions des problèmes variationnels dans lesquels les inconnues sont des applications à valeurs dans l'espace de Wasserstein.Quand l'espace de départ est un segment, c'est-à-dire quand les inconnues sont des courbes à valeurs dans l'espace de Wasserstein, nous nous intéressons à des modèles où, en plus de l'action des courbes, des termes pénalisant les configurations de congestion sont présents. Nous développons des techniques permettant d'extraire de la régularité à partir de l'interaction entre l'évolution optimale de la densité (minimisation de l'action) et la pénalisation de la congestion, et nous les appliquons à l'étude des jeux à champ moyen et de la formulation variationelle des équations d'Euler.Quand l'espace de départ n'est plus seulement un segment mais un domaine de l'espace euclidien, nous considérons seulement le problème de Dirichlet, c'est-à-dire la minimisation de l'action (qui peut être appelée l'énergie de Dirichlet) parmi toutes les applications dont les valeurs sur le bord du domaine de départ sont fixées. Les solutions sont appelées les applications harmoniques à valeurs dans l'espace de Wasserstein. Nous montrons que les différentes définitions de l'énergie de Dirichlet présentes dans la littérature sont en fait équivalentes; que le problème de Dirichlet est bien posé sous des hypothèses assez faibles; que le principe de superposition est mis en échec lorsque l'espace de départ n'est pas un segment; que l'on peut formuler une sorte de principe du maximum; et nous proposons une méthode numérique pour calculer ces applications harmoniques. / The Wasserstein space is the space of probability measures over a given domain endowed with the quadratic Wasserstein distance. In this work, we study variational problems where the unknowns are mappings valued in the Wasserstein space. When the source space is a segment, i.e. when the unknowns are curves valued in the Wasserstein space, we are interested in models where, in addition to the action of the curves, there are some terms which penalize congested configurations. We develop techniques to extract regularity from the minimizers thanks to the interplay between optimal density evolution (minimization of the action) and penalization of congestion, and we apply them to the study of Mean Field Games and the variational formulation of the Euler equations. When the source space is no longer a segment but a domain of a Euclidean space, we consider only the Dirichlet problem, i.e. the minimization of the action (which can be called the Dirichlet energy) among mappings sharing a fixed value on the boundary of the source space. The solutions are called harmonic mappings valued in the Wasserstein space. We prove that the different definitions of the Dirichlet energy in the literature turn out to be equivalent; that the Dirichlet problem is well-posed under mild assumptions; that the superposition principle fails if the source space is no longer a segment; that a sort of maximum principle holds; and we provide a numerical method to compute these harmonic mappings.

Cette thèse adresse le problème du consensus dans les réseaux. On étudie des réseaux composés d'agents identiques capables de communiquer entre eux, qui ont une mémoire et des capacités de calcul. Le réseau ne possède pas de nœud central de fusion. Chaque agent stocke une valeur qui n'est pas initialement connue par les autres agents. L'objectif est d'atteindre le consensus, i.e. tous les agents ont la même valeur, d'une manière distribuée. De plus, seul les agents voisins peuvent communiquer entre eux. Ce problème a une longue et riche histoire. Si toutes les valeurs appartiennent à un espace vectoriel, il existe plusieurs protocoles pour résoudre le problème. Une des solutions connues est l'algorithme du gossip qui atteint le consensus de manière asymptotique. C'est un protocole itératif qui consiste à choisir deux nœuds adjacents à chaque itération et de les moyenner. La spécificité de cette thèse est dans le fait que les données stockées par les agents n'appartiennent pas nécessairement à un espace vectoriel, mais à un espace métrique. Par exemple, chaque agent stocke une direction (l'espace métrique est l'espace projectif) ou une position dans un graphe métrique (l'espace métrique est le graphe sous-jacent). Là, les protocoles de gossip mentionnés plus haut n'ont plus de sens car l'addition qui n'est plus disponibles dans les espaces métriques. Cependant, dans les espaces métriques les points milieu ont du sens dans certains cas. Et là ils peuvent se substituer aux moyennes arithmétiques. Dans ce travail, on a compris que la convergence du gossip avec les points milieu dépend de la courbure. On s'est focalisés sur le cas où l'espace des données appartient à une classe d'espaces métriques appelés les espaces CAT(k). Et on a pu démontrer que si les données initiales sont suffisamment "proches" dans un sens bien précis, alors le gossip avec les points milieu - qu'on a appelé le Random Parwise Midpoints- converge asymptotiquement vers un consensus / This thesis deals with the problem of consensus on networks. Networks under study consists of identical agents that can communicate with each other, have memory and computational capacity. The network has no central node. Each agent stores a value that, initially, is not known by other agents. The goal is to achieve consensus, i.e. all agents having the same value, in a fully distributed way. Hence, only neighboring agents can have direct communication. This problem has a long and fruitful history. If all values belong to some vector space, several protocols are known to solve this problem. A well-known solution is the pairwise gossip protocol that achieves consensus asymptotically. It is an iterative protocol that consists in choosing two adjacent nodes at each iteration and average them. The specificity of this Ph.D. thesis lies in the fact that the data stored by the agents does not necessarily belong to a vector space, but some metric space. For instance, each agent stores a direction (the metric space is the projective space) or position on a sphere (the metric space is a sphere) or even a position on a metric graph (the metric space is the underlying graph). Then the mentioned pairwise gossip protocols makes no sense since averaging implies additions and multiplications that are not available in metric spaces: what is the average of two directions, for instance? However, in metric spaces midpoints sometimes make sense and when they do, they can advantageously replace averages. In this work, we realized that, if one wants midpoints to converge, curvature matters. We focused on the case where the data space belongs to some special class of metric spaces called CAT(k) spaces. And we were able to show that, provided initial data is "close enough" is some precise meaning, midpoints-based gossip algorithm – that we refer to as Random Pairwise Midpoints - does converge to consensus asymptotically. Our generalization allows to treat new cases of data spaces such as positive definite matrices, the rotations group and metamorphic systems

The complexity of data stored in large databases has increased at very fast paces. Hence, operations more elaborated than traditional queries are essential in order to extract all required information from the database. Therefore, the interest of the database community in similarity search has increased significantly. Two of the well-known types of similarity search are the Range (\'R IND. q\') and the k-Nearest Neighbor (\'kNN IND. q\') queries, which, as any of the traditional ones, can be sped up by indexing structures of the Database Management System (DBMS). Another way of speeding up queries is to perform query optimization. In this process, metrics about data are collected and employed to adjust the parameters of the search algorithms in each query execution. However, although the integration of similarity search into DBMS has begun to be deeply studied more recently, the query optimization has been developed and employed just to answer traditional queries. The execution of similarity queries, even using efficient indexing structures, tends to present higher computational cost than the execution of traditional ones. Two strategies can be applied to speed up the execution of any query, and thus they are worth to employ to answer also similarity queries. The first strategy is query rewriting based on algebraic properties and cost functions. The second technique is when external query factors are applied, such as employing the semantic expected by the user, to prune the answer space. This thesis aims at contributing to the development of novel techniques to improve the similarity-based query optimization processing, exploiting both algebraic properties and semantic restrictions as query refinements / A complexidade dos dados armazenados em grandes bases de dados tem aumentado sempre, criando a necessidade de novas operações de consulta. Uma classe de operações de crescente interesse são as consultas por similaridade, das quais as mais conhecidas são as consultas por abrangência (\'R IND. q\') e por k-vizinhos mais próximos (\'kNN IND. q\'). Qualquer consulta e agilizada pelas estruturas de indexação dos Sistemas de Gerenciamento de Bases de Dados (SGBDs). Outro modo de agilizar as operações de busca e a manutenção de métricas sobre os dados, que são utilizadas para ajustar parâmetros dos algoritmos de busca em cada consulta, num processo conhecido como otimização de consultas. Como as buscas por similaridade começaram a ser estudadas seriamente para integração em SGBDs muito mais recentemente do que as buscas tradicionais, a otimização de consultas, por enquanto, e um recurso que tem sido utilizado para responder apenas a consultas tradicionais. Mesmo utilizando as melhores estruturas existentes, a execução de consultas por similaridade tende a ser mais custosa do que as operações tradicionais. Assim, duas estratégias podem ser utilizadas para agilizar a execução de qualquer consulta e, assim, podem ser empregadas também para responder às consultas por similaridade. A primeira estratégia e a reescrita de consultas baseada em propriedades algébricas e em funções de custo. A segunda técnica faz uso de fatores externos à consulta, tais como a semântica esperada pelo usuário, para restringir o espaço das respostas. Esta tese pretende contribuir para o desenvolvimento de técnicas que melhorem o processo de otimização de consultas por similaridade, explorando propriedades algebricas e restrições semânticas como refinamento de consultas

Nous présentons des inégalités fonctionnelles pour les processus ponctuels. Nous prouvons une inégalité de Sobolev logarithmique modifiée, une inégalité de Stein et un théorème du moment quatrième sans terme de reste pour une classe de processus ponctuels qui contient les processus binomiaux et les processus de Poisson. Les preuves reposent sur des techniques inspirées de l'approche de Malliavin-Stein et du calcul avec l'opérateur $Gamma$ de Bakry-Émery. Pour mettre en œuvre ces techniques nous développons une analyse stochastique pour les processus ponctuels. Plus généralement, nous mettons au point une théorie d'analyse stochastique sans hypothèse de diffusion. Dans le cadre des processus de Poisson ponctuels, l'inégalité de Stein est généralisée pour étudier la convergence stable vers des limites conditionnellement gaussiennes. Nous appliquons ces résultats pour approcher des processus Gaussiens par des processus de Poisson composés et pour étudier des graphes aléatoires. Nous discutons d'inégalités de transport et de leur conséquence en termes de concentration de la mesure pour les processus binomiaux dont la taille de l'échantillon est aléatoire. Sur un espace métrique mesuré quelconque, nous présentons un développement de la concentration de la mesure qui prend en compte l'agrandissement parallèle d'ensembles disjoints. Cette concentration améliorée donne un contrôle de toutes les valeurs propres du Laplacien métrique. Nous discutons des liens de cette nouvelle notion avec une version de la courbure de Ricci qui fait intervenir le transport à plusieurs marginales / We present functional inequalities and limit theorems for point processes. We prove a modified logarithmic Sobolev inequalities, a Stein inequality and a exact fourth moment theorem for a large class of point processes including mixed binomial processes and Poisson point processes. The proofs of these inequalities are inspired by the Malliavin-Stein approach and the $Gamma$-calculus of Bakry-Emery. The implementation of these techniques requires a development of a stochastic analysis for point processes. As point processes are essentially discrete, we design a theory to study non-diffusive random objects. For Poisson point processes, we extend the Stein inequality to study stable convergence with respect to limits that are conditionally Gaussian. Applications to Poisson approximations of Gaussian processes and random geometry are given. We discuss transport inequalities for mixed binomial processes and their consequences in terms of concentration of measure. On a generic metric measured space, we present a refinement of the notion of concentration of measure that takes into account the parallel enlargement of distinct sets. We link this notion of improved concentration with the eigenvalues of the metric Laplacian and with a version of the Ricci curvature based on multi-marginal optimal transport

The complexity of data stored in large databases has increased at very fast paces. Hence, operations more elaborated than traditional queries are essential in order to extract all required information from the database. Therefore, the interest of the database community in similarity search has increased significantly. Two of the well-known types of similarity search are the Range (\'R IND. q\') and the k-Nearest Neighbor (\'kNN IND. q\') queries, which, as any of the traditional ones, can be sped up by indexing structures of the Database Management System (DBMS). Another way of speeding up queries is to perform query optimization. In this process, metrics about data are collected and employed to adjust the parameters of the search algorithms in each query execution. However, although the integration of similarity search into DBMS has begun to be deeply studied more recently, the query optimization has been developed and employed just to answer traditional queries. The execution of similarity queries, even using efficient indexing structures, tends to present higher computational cost than the execution of traditional ones. Two strategies can be applied to speed up the execution of any query, and thus they are worth to employ to answer also similarity queries. The first strategy is query rewriting based on algebraic properties and cost functions. The second technique is when external query factors are applied, such as employing the semantic expected by the user, to prune the answer space. This thesis aims at contributing to the development of novel techniques to improve the similarity-based query optimization processing, exploiting both algebraic properties and semantic restrictions as query refinements / A complexidade dos dados armazenados em grandes bases de dados tem aumentado sempre, criando a necessidade de novas operações de consulta. Uma classe de operações de crescente interesse são as consultas por similaridade, das quais as mais conhecidas são as consultas por abrangência (\'R IND. q\') e por k-vizinhos mais próximos (\'kNN IND. q\'). Qualquer consulta e agilizada pelas estruturas de indexação dos Sistemas de Gerenciamento de Bases de Dados (SGBDs). Outro modo de agilizar as operações de busca e a manutenção de métricas sobre os dados, que são utilizadas para ajustar parâmetros dos algoritmos de busca em cada consulta, num processo conhecido como otimização de consultas. Como as buscas por similaridade começaram a ser estudadas seriamente para integração em SGBDs muito mais recentemente do que as buscas tradicionais, a otimização de consultas, por enquanto, e um recurso que tem sido utilizado para responder apenas a consultas tradicionais. Mesmo utilizando as melhores estruturas existentes, a execução de consultas por similaridade tende a ser mais custosa do que as operações tradicionais. Assim, duas estratégias podem ser utilizadas para agilizar a execução de qualquer consulta e, assim, podem ser empregadas também para responder às consultas por similaridade. A primeira estratégia e a reescrita de consultas baseada em propriedades algébricas e em funções de custo. A segunda técnica faz uso de fatores externos à consulta, tais como a semântica esperada pelo usuário, para restringir o espaço das respostas. Esta tese pretende contribuir para o desenvolvimento de técnicas que melhorem o processo de otimização de consultas por similaridade, explorando propriedades algebricas e restrições semânticas como refinamento de consultas

Orientadores: Rodney Carlos Bassanezi, Adilson Jose Vieira Brandão / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T01:52:00Z (GMT). No. of bitstreams: 1 Cecconello_MoiseisdosSantos_D.pdf: 62393038 bytes, checksum: b7f0d1f9138d8e787749532bf661d026 (MD5) Previous issue date: 2010 / Resumo: Neste trabalho desenvolvemos ferramentas de análise qualitativa para sistemas dinâmicos definidos sobre o espaço formado pelos conjuntos fuzzy com a níveis compactos e não vazios. São propostas condições para existência de pontos de equilíbrio para o fluxo fuzzy cuja função de pertinência é sobrejetiva, generalizando alguns resultados já conhecidos. Os fluxos fuzzy considerados aqui são determinados pela extensão de Zadeh aplicada em soluções de equações diferenciais autônomas. São obtidos também condições para a existência de pontos e órbitas periódicas para o fluxo fuzzy. Em particular, demonstramos um teorema tipo Poincaré-Bendixson para tais fluxos gerados por equações autônomas bidimensionais. A análise qualitativa desenvolvida é aplicada em sistemas dinâmicos fuzzy provenientes de modelos significativos da Biomatemática. / Abstract: In this work we develop some tools for qualitative analysis of dynamical systems defined on the metric space of fuzzy sets with compact and nonempty a cuts. Conditions are offered for the existence of equilibrium points for the flow whose fuzzy membership function is surjective, generalizing some results already known. Fuzzy flows considered here are determined by Zadeh's extension applied in solutions of autonomous differential equations. We also obtained conditions for the existence of periodic points and periodic orbits for the fuzzy flow. In particular, we demonstrate a theorem like Poincaré-Bendixson for such flows generated by two-dimensional autonomous equations. The qualitative analysis results are applied to fuzzy dynamic systems from meaningful models of Biomathematics. / Doutorado / Biomatematica / Doutor em Matemática Aplicada

Gromov’s compactness theorem for metric spaces, a compactness theorem for the space of compact metric spaces equipped with the Gromov-Hausdorff distance, is a theorem with many applications. In this thesis, we give a generalisation of this landmark result, more precisely, we give a compactness theorem for the space of distance measure spaces equipped with the generalised Gromov-Hausdorff-Levi-Prokhorov distance. A distance measure space is a triple (X, d,µ), where (X, d) forms a distance space (a generalisation of a metric space where, we allow the distance between two points to be infinity) and µ is a finite Borel measure. Using this result we prove that the Deligne-Mumford compactification is the completion of the moduli space of Riemann surfaces under the generalised Gromov-Hausdorff-Levi-Prokhorov distance. The Deligne-Mumford compactification, a compactification of the moduli space of Riemann surfaces with explicit description of the limit points, and the closely related Gromov compactness theorem for J-holomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics. While Gromov compactness theorem for J-holomorphic curves in symplectic manifolds, is an important tool in symplectic topology, its applicability is limited by the lack of general methods to construct pseudo-holomorphic curves. One hopes that considering a more general class of objects in place of pseudo-holomorphic curves will be useful. Generalising the domain of pseudo-holomorphic curves from Riemann surfaces to Riemann surface laminations is a natural choice. Theorems such as the uniformisation theorem for surface laminations by Alberto Candel (which is a partial generalisation of the uniformisation theorem for surfaces), generalisations of the Gauss-Bonnet theorem proved for some special cases, and topological classification of “almost all" leaves using harmonic measures reinforces the usefulness of this line on enquiry. Also, the success of essential laminations, as generalised incompressible surfaces, in the study of 3-manifolds suggests that a similar approach may be useful in symplectic topology. With this motivation, we prove a compactness theorem analogous to the Deligne-Mumford compactification for the space of Riemann surface laminations.

La complexité des données contenues dans les grandes bases de données a augmenté considérablement. Par conséquent, des opérations plus élaborées que les requêtes traditionnelles sont indispensable pour extraire toutes les informations requises de la base de données. L'intérêt de la communauté de base de données a particulièrement augmenté dans les recherches basées sur la similarité. Deux sortes de recherche de similarité bien connues sont la requête par intervalle (Rq) et par k-plus proches voisins (kNNq). Ces deux techniques, comme les requêtes traditionnelles, peuvent être accélérées par des structures d'indexation des Systèmes de Gestion de Base de Données (SGBDs).Une autre façon d'accélérer les requêtes est d'exécuter le procédé d'optimisation des requêtes. Dans ce procédé les données métriques sont recueillies et utilisées afin d'ajuster les paramètres des algorithmes de recherche lors de chaque exécution de la requête. Cependant, bien que l'intégration de la recherche de similarités dans le SGBD ait commencé à être étudiée en profondeur récemment, le procédé d'optimisation des requêtes a été développé et utilisé pour répondre à des requêtes traditionnelles. L'exécution des requêtes de similarité a tendance à présenter un coût informatique plus important que l'exécution des requêtes traditionnelles et ce même en utilisant des structures d'indexation efficaces. Deux stratégies peuvent être appliquées pour accélérer l'execution de quelques requêtes, et peuvent également être employées pour répondre aux requêtes de similarité. La première stratégie est la réécriture de requêtes basées sur les propriétés algébriques et les fonctions de coût. La deuxième stratégie est l'utilisation des facteurs externes de la requête, tels que la sémantique attendue par les usagers, pour réduire le nombre des résultats potentiels. Cette thèse vise à contribuer au développement des techniques afin d'améliorer le procédé d'optimisation des requêtes de similarité, tout en exploitant les propriétés algébriques et les restrictions sémantiques pour affiner les requêtes. / The complexity of data stored in large databases has increased at very fast paces. Hence, operations more elaborated than traditional queries are essential in order to extract all required information from the database. Therefore, the interest of the database community in similarity search has increased significantly. Two of the well-known types of similarity search are the Range (Rq) and the k-Nearest Neighbor (kNNq) queries, which, as any of the traditional ones, can be sped up by indexing structures of the Database Management System (DBMS). Another way of speeding up queries is to perform query optimization. In this process, metrics about data are collected and employed to adjust the parameters of the search algorithms in each query execution. However, although the integration of similarity search into DBMS has begun to be deeply studied more recently, the query optimization has been developed and employed just to answer traditional queries.The execution of similarity queries, even using efficient indexing structures, tends to present higher computational cost than the execution of traditional ones. Two strategies can be applied to speed up the execution of any query, and thus they are worth to employ to answer also similarity queries. The first strategy is query rewriting based on algebraic properties and cost functions. The second technique is when external query factors are applied, such as employing the semantic expected by the user, to prune the answer space. This thesis aims at contributing to the development of novel techniques to improve the similarity-based query optimization processing, exploiting both algebraic properties and semantic restrictions as query refinements. / A complexidade dos dados armazenados em grandes bases de dados tem aumentadosempre, criando a necessidade de novas operaoes de consulta. Uma classe de operações de crescente interesse são as consultas por similaridade, das quais as mais conhecidas sãoas consultas por abrangência (Rq) e por k-vizinhos mais próximos (kNNq). Qualquerconsulta é agilizada pelas estruturas de indexaçãodos Sistemas de Gerenciamento deBases de Dados (SGBDs). Outro modo de agilizar as operações de busca é a manutençãode métricas sobre os dados, que são utilizadas para ajustar parâmetros dos algoritmos debusca em cada consulta, num processo conhecido como otimização de consultas. Comoas buscas por similaridade começaram a ser estudadas seriamente para integração emSGBDs muito mais recentemente do que as buscas tradicionais, a otimização de consultas,por enquanto, é um recurso que tem sido utilizado para responder apenas a consultastradicionais.Mesmo utilizando as melhores estruturas existentes, a execução de consultas por similaridadetende a ser mais custosa do que as operações tradicionais. Assim, duas estratégiaspodem ser utilizadas para agilizar a execução de qualquer consulta e, assim, podem serempregadas também para responder às consultas por similaridade. A primeira estratégiaé a reescrita de consultas baseada em propriedades algébricas e em funções de custo. Asegunda técnica faz uso de fatores externos à consulta, tais como a semântica esperadapelo usuário, para restringir o espaço das respostas. Esta tese pretende contribuir parao desenvolvimento de técnicas que melhorem o processo de otimização de consultas porsimilaridade, explorando propriedades algébricas e restrições semânticas como refinamentode consultas

A topological approach to nonlinear analysis allows for strikingly beautiful proofs and simplified calculations. This topological approach employs many of the ideas of continuous topology, including convergence, compactness, metrization, complete metric spaces, uniform spaces and function spaces. This thesis illustrates using the topological approach in proving the Cauchy-Peano Existence theorem. The topological proof utilizes the ideas of complete metric spaces, Ascoli-Arzela theorem, topological properties in Euclidean n-space and normed linear spaces, and the extension of Brouwer's fixed point theorem to Schauder's fixed point theorem, and Picard's theorem.