Les calculs numériques ont été effectués à l'aide du logiciel SAGE. / En 1993, Conway et Schneeberger fournirent un critère simple permettant de déterminer si une forme quadratique donnée représente tous les entiers positifs ; le théorème des 15. Dans ce mémoire, nous nous intéressons à un problème analogue, soit la recherche d’un critère similaire permettant de détecter si une forme quadratique en trois variables représente tous les entiers impairs. On débute donc par une introduction générale à la théorie des formes quadratiques, notamment en deux variables, puis on expose différents points de vue sous lesquels on peut les considérer. On décrit ensuite le théorème des 15 et ses généralisations, en soulignant les techniques utilisées dans la preuve de Bhargava. Enfin, on démontre deux théorèmes qui fournissent des critères permettant de déterminer si une forme quadratique ternaire représente tous les entiers impairs. / In 1993, Conway and Schneeberger gave a simple criterion allowing one to determine whether a given quadratic form represents all positive integers ; the 15-theorem. In this thesis, we investigate an analogous problem, that is the search for a similar criterion allowing one to detect if a quadratic form in three variables represents all odd integers. We start with a general introduction to the theory of quadratic forms, namely in two variables, then, we expose different points of view under which quadratic forms can be considered. We then describe the 15-theorem and its generalizations, with a particular emphasis on the techniques used in Bhargava’s proof of the theorem. Finally, we give a proof of two theorems which provide a criteria to determine whether a ternary quadratic form represents all odd integers.

En 1993, Conway et Schneeberger fournirent un critère simple permettant de déterminer si une forme quadratique donnée représente tous les entiers positifs ; le théorème des 15. Dans ce mémoire, nous nous intéressons à un problème analogue, soit la recherche d’un critère similaire permettant de détecter si une forme quadratique en trois variables représente tous les entiers impairs. On débute donc par une introduction générale à la théorie des formes quadratiques, notamment en deux variables, puis on expose différents points de vue sous lesquels on peut les considérer. On décrit ensuite le théorème des 15 et ses généralisations, en soulignant les techniques utilisées dans la preuve de Bhargava. Enfin, on démontre deux théorèmes qui fournissent des critères permettant de déterminer si une forme quadratique ternaire représente tous les entiers impairs. / In 1993, Conway and Schneeberger gave a simple criterion allowing one to determine whether a given quadratic form represents all positive integers ; the 15-theorem. In this thesis, we investigate an analogous problem, that is the search for a similar criterion allowing one to detect if a quadratic form in three variables represents all odd integers. We start with a general introduction to the theory of quadratic forms, namely in two variables, then, we expose different points of view under which quadratic forms can be considered. We then describe the 15-theorem and its generalizations, with a particular emphasis on the techniques used in Bhargava’s proof of the theorem. Finally, we give a proof of two theorems which provide a criteria to determine whether a ternary quadratic form represents all odd integers. / Les calculs numériques ont été effectués à l'aide du logiciel SAGE.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2018-01-10T09:42:22Z No. of bitstreams: 2 Dissertação - Hélios Kárum de Oliveira Bastos - 2017.pdf: 2219804 bytes, checksum: ba853c18f7e7e2c65eb0a342d4a34640 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-01-10T09:42:46Z (GMT) No. of bitstreams: 2 Dissertação - Hélios Kárum de Oliveira Bastos - 2017.pdf: 2219804 bytes, checksum: ba853c18f7e7e2c65eb0a342d4a34640 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-01-10T09:42:46Z (GMT). No. of bitstreams: 2 Dissertação - Hélios Kárum de Oliveira Bastos - 2017.pdf: 2219804 bytes, checksum: ba853c18f7e7e2c65eb0a342d4a34640 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-08-24 / Multiobjective Optimization for feature and samples selection in double chromosome of integer representation and variable size for multivariate calibration} In several problems of regression, classification, prediction, approximation Optimization, the original data contain a large number of variables to obtain a better representation of the problem under consideration. However, a significant part of the variables may be irrelevant and redundant from the point of view of machine learning. Indeed, one of the challenges to be overcome is a selection of a subset of variables that has the best perform. One of the breakthroughs in this type of problem is the use of a multiobjective formulation that avoids the overlap of the model to the training data set. Another important point is the process of choosing the objects to be used in the learning stage. Generally, a selection of variables and treatment objects are treated separately and without dependence. This project proposes a multiobjective modeling to select variables and objects simultaneously using a genetic integer representation algorithm with variable size chromosomes. It is expected that a simultaneous selection of objects and variables on a multiobjective context produce better results in a traditional approach. As a case study this work utilized an analysis of near infrared (NIR) material on oil samples for the purpose of estimating the concentration of an interest properties such set was used in the competition conducted at the International Diffuse Reflectance Conference (IDRC) in the year 2014. / Em diversos problemas de regressão, classificação, previsão, aproximação e otimização, os dados originais contêm um grande número de variáveis introduzidas para se obter uma melhor representação do problema considerado. Entretanto, uma parte significativa destas variáveis podem ser irrelevantes e/ou redundantes do ponto de vista do aprendizado de máquina acerca do problema. Com efeito, um dos desafios a ser superados é a seleção de um subconjunto de variáveis que apresentem um melhor desempenho. Um dos avanços recentes neste tipo de problema está no uso de uma formulação multiobjetivo que evita o superajuste do modelo ao conjunto de dados de treinamento. Outro ponto importante refere-se ao processo de escolha adequada dos objetos a serem utilizados na etapa de aprendizado. Geralmente, a seleção de variáveis e de objetos de treinamento são tratados de forma separada e sem dependência. Este projeto propõe uma modelagem multiobjetivo para seleção de variáveis e objetos de forma simultânea utilizando-se de algoritmo genético de representação inteira com cromossomos de tamanho variáveis. Espera-se que a seleção simultânea de objetos e variáveis no contexto multiobjetivo produza melhores resultados em relação a abordagem tradicional. Como estudo de caso este trabalho utiliza dados obtidos por uma análise de material com ondas de infravermelho próximo (NIR) sobre amostras de petróleo com o propósito de estimar a concentração de uma propriedade de interesse, tal conjunto foi utilizado na competição realizada no International Diffuse Reflectance Conference (IDRC) (\url{http://cnirs.clubexpress.com/content.aspx?page_id=22&club_id=409746&module_id=19 0211}), no ano de 2015.

There has been in the past ten to fifteen years a surge of activity concerning the cohomology of semi-direct product groups of the form $\mathbb{Z}^{n}\rtimes$G with G finite. A problem first stated by Adem-Ge-Pan-Petrosyan asks for suitable conditions for the Lyndon-Hochschild-Serre Spectral Sequence associated to this group extension to collapse at second page of the Lyndon-Hochschild-Serre spectral sequence. In this thesis we use facts from integer representation theory to reduce this problem to only considering representatives from each genus of representations, and establish techniques for constructing new examples in which the spectral sequence collapses.