Wassily Hoeffding, in his 1963 paper, introduces a procedure to derive inequalities between distributions. This method relies on finding a martingale coupling between the two random variables. I have developed a construction that establishes such couplings in various urn models. I use this construction to prove the inequality between the hypergeometric and binomial random variables that appears in Hoeffding's paper. I have then used and extended my urn construction to create new inequalities.

Orientador: Sérgio Kurokawa / Resumo: Uma linha de transmissão representada por uma cascata de circuitos π é descrita por meio de sistema de equações diferenciais ordinárias que podem ser resolvidas diretamente no domínio do tempo, através de métodos numéricos de integração. A cada passo de cálculo é resolvido o sistema de ordem 2n, sendo n a quantidade de circuitos π. Este é o procedimento clássico de resolução. Um método alternativo de resolução de equações diferencias, proposto recentemente, resolve os sistemas de equações diferenciais em linhas de até 5 km representada por uma quantidade limite de 333 circuitos π. Neste trabalho é investigado a aplicação desse método alternativo em linhas de maior comprimento e sua eficiência na resolução das equações diferenciais que representam a linha de transmissão. No método alternativo cada circuito π da linha é resolvido individualmente através de um sistema que não depende da quantidade de circuitos π, cuja matrizes possuem ordem 2x2. Dessa forma, para n circuitos π é necessário resolver n sistemas de duas equações diferencias para cada instante de tempo. Este procedimento é denominado como método acelerado. A análise do desempenho do método acelerado é realizada através do número de operações de pontos flutuantes (flops) e através do tempo de cálculo das variáveis de estado ao longo da linha de transmissão. O mesmo procedimento é realizado no método clássico e após comparados os resultados é possível constatar a eficiência do método acelerado para determinadas config... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: A transmission line represented by a cascade of π circuits is described by means of a system of ordinary differential equations that can be solved directly in the time domain, through numerical integration methods. At each calculation step, the 2n order system is solved, with n being the number of π circuits. This is the classic resolution procedure. An alternative method of solving differential equations recently proposed solves the systems of differential equations in lines of up to 5 km represented by a limit quantity of 333 π circuits. This work investigates the application of this alternative method on lines of greater length and its efficiency in solving the differential equations that represent the transmission line. In the alternative method, each π circuit of the line is solved individually through a system that does not depend on the number of π circuits, whose matrices have a 2x2 order. Thus, for n π circuits, it is necessary to solve n systems of two different equations for each instant of time. This procedure is called an accelerated method. The analysis of the performance of the accelerated method is performed through the number of floating-point operations (flops) and through the calculation time of the state variables along the transmission line. The same procedure is performed in the classic method and after comparing the results it is possible to verify the efficiency of the accelerated method for certain transmission line configurations. / Mestre

Lors de l'optimisation géométrique de structures, un grand nombre d'évaluations de champs est nécessaire. L'idée, développée dans cette thèse, est la construction efficace et rapide d'approximations de ces champs à travers la Proper Generalized Decomposition (PGD), une méthode de réduction de modèle. Les résultats, calculés une fois pour toutes, sont stockés dans des abaques virtuels pour une utilisation ultérieure dans un processus d'optimisation. Le problème considéré est paramétrique et les paramètres définissent la géométrie. Ce type de problème est particulièrement adapté à la PGD. En effet, de nombreux travaux ont traité de la résolution de problèmes paramétriques et des premières études ont porté, en particulier, sur la prise en compte de paramètres géométriques. Toutefois, ce qui caractérise nos travaux est d'aller vers des outils aptes à traiter des situations significatives de la complexité des problèmes rencontrés au niveau industriel. En particulier, l'exploitation de codes éléments finis commerciaux est une contrainte majeure. Il a été décidé de développer des méthodes permettant de traiter des problèmes à paramètres géométriques par la PGD, et, en partenariat avec AIRBUS Defence & Space, d'appliquer ces techniques à un démonstrateur industriel présentant une géométrie complexe (type splines) et de fortes non-linéarités (géométriques, matériaux). Notre démarche a été implémentée dans un process industriel utilisant des codes éléments finis commerciaux. On propose aussi une nouvelle extension de la PGD aux paramètres discrets autorisant la prise en considération, dans une même résolution, de configurations de structures complètement différentes (cas de chargement, matériaux, etc.). / During shape optimisation of structures, numerous evaluations of fiels are necessary. The idea, developed in this PhD thesis, is the efficient construction of approximations of these fiels through the Proper Generalized Decomposition (PGD), a model reduction technique. The results, computed once and for all, are stored in virtual charts for a subsequent use into an optimisation process. Geometry variations correspond to a parametric problem, where the parameters is the geometry. This kind of problem is well suited for PGD. Many studies dealt with the resolution of parametric problems and recent works treated, particularly, the introduction of geometric parameters. However, our approach is to deal with configurations of the complexity of industrial problems. The use of commercial finite elements software is a crucial issue. It was decided, in partnership with AIRBUS Defence & Space, to develop techniques allowing the resolution of geometrically parametrised problems thanks to PGD and to apply them to an industrial demonstrator. The geometry considered is defined by splines and the behaviour of the structure is highly non-linear (geometric and material non-linearities). The approach was implemented into a genuine industrial design process using commercial finite elements software. The thesis proposed, also, a new extension of PGD to discrete parameters. It allows to take into account completely different configurations (loadings, materials, etc.) in the same resolution.