For decades, mathematical games have been used to explore various properties of particular sets. The Banach-Mazur game is the prototypical intersection game and its modifications by e.g., W. Schmidt and C. McMullen are used in number theory and many other areas of mathematics. We give a brief survey of a few of these modifications and their properties followed by our own modification. One of our main results is proving that this modification is equivalent to an important set theoretic game, called the perfect set game, developed by M. Davis.

Dans cette thèse, on aborde trois thèmes : problème de sélection de colonnes dans une matrice, distance de Banach-Mazur au cube et estimation de la covariance de matrices aléatoires. Bien que les trois thèmes paraissent éloignés, les techniques utilisées se ressemblent tout au long de la thèse. Dans un premier lieu, nous généralisons le principe d'invertibilité restreinte de Bourgain-Tzafriri. Ce résultat permet d'extraire un "grand" bloc de colonnes linéairement indépendantes dans une matrice et d'estimer la plus petite valeur singulière de la matrice extraite. Nous proposons ensuite un algorithme déterministe pour extraire d'une matrice un bloc presque isométrique c’est à dire une sous-matrice dont les valeurs singulières sont proches de 1. Ce résultat nous permet de retrouver le meilleur résultat connu sur la célèbre conjecture de Kadison-Singer. Des applications à la théorie locale des espaces de Banach ainsi qu'à l'analyse harmonique sont déduites. Nous donnons une estimation de la distance de Banach-Mazur d'un corps convexe de Rn au cube de dimension n. Nous proposons une démarche plus élémentaire, basée sur le principe d'invertibilité restreinte, pour améliorer et simplifier les résultats précédents concernant ce problème. Plusieurs travaux ont été consacrés pour approcher la matrice de covariance d'un vecteur aléatoire par la matrice de covariance empirique. Nous étendons ce problème à un cadre matriciel et on répond à la question. Notre résultat peut être interprété comme une quantification de la loi des grands nombres pour des matrices aléatoires symétriques semi-définies positives. L'estimation obtenue s'applique à une large classe de matrices aléatoires / In this thesis, we address three themes : columns subset selection in a matrix, the Banach-Mazur distance to the cube and the estimation of the covariance of random matrices. Although the three themes seem distant, the techniques used are similar throughout the thesis. In the first place, we generalize the restricted invertibility principle of Bougain-Tzafriri. This result allows us to extract a "large" block of linearly independent columns inside a matrix and estimate the smallest singular value of the restricted matrix. We also propose a deterministic algorithm in order to extract an almost isometric block inside a matrix i.e a submatrix whose singular values are close to 1. This result allows us to recover the best known result on the Kadison-Singer conjecture. Applications to the local theory of Banach spaces as well as to harmonic analysis are deduced. We give an estimate of the Banach-Mazur distance between a symmetric convex body in Rn and the cube of dimension n. We propose an elementary approach, based on the restricted invertibility principle, in order to improve and simplify the previous results dealing with this problem. Several studies have been devoted to approximate the covariance matrix of a random vector by its sample covariance matrix. We extend this problem to a matrix setting and we answer the question. Our result can be interpreted as a quantified law of large numbers for positive semidefinite random matrices. The estimate we obtain, applies to a large class of random matrices

In 1948 Besicovitch proved that an affine image of a regular hexagon may be inscribed into an arbitrary planar convex body. We prove Besicovitch's result using a variational approach based on special approximation by triangles and generalize the Besicovitch theorem to a certain new class of hexagons. We survey the results on the Banach-Mazur distance between different classes of convex bodies. We hope that our generalization of the Besicovitch theorem may become useful for estimation of the Banach-Mazur distance between planar convex bodies. We examined our special approximation by triangles in some specific cases, and it showed a noticeable improvement in comparison with known general methods. We also consider the Banach-Mazur distance between a simplex and an arbitrary convex body in the three-dimensional case. Using the idea of an inscribed simplex of maximal volume, we obtain a certain related algebraic optimization problem that provides an upper estimate.

In 1948 Besicovitch proved that an affine image of a regular hexagon may be inscribed into an arbitrary planar convex body. We prove Besicovitch's result using a variational approach based on special approximation by triangles and generalize the Besicovitch theorem to a certain new class of hexagons. We survey the results on the Banach-Mazur distance between different classes of convex bodies. We hope that our generalization of the Besicovitch theorem may become useful for estimation of the Banach-Mazur distance between planar convex bodies. We examined our special approximation by triangles in some specific cases, and it showed a noticeable improvement in comparison with known general methods. We also consider the Banach-Mazur distance between a simplex and an arbitrary convex body in the three-dimensional case. Using the idea of an inscribed simplex of maximal volume, we obtain a certain related algebraic optimization problem that provides an upper estimate.

Para um espaço localmente compacto de Hausdorff K e um espaço de Banach X, denotamos por C0(K,X) o espaço de todas as funções a valores em X contínuas sobre K que se anulam no infinito, munido da norma do supremo. No espírito do clássico teorema de Banach-Stone 1937, estabelecemos que se C0(K1,X) é isomorfo a C0(K2,X), onde X é um espaço de Banach de cotipo finito e tal que X é separável ou X* tem a propriedade de Radon-Nikodým, então ou K1 e K2 são ambos finitos ou K1 e K2 tem a mesma cardinalidade. Trata-se de uma extensão vetorial de um resultado de Cengiz 1978, o caso escalar X = R ou X = C. Demonstramos também que se K1 e K2 são intervalos compactos de ordinais e X é um espaço de Banach de cotipo finito, então a existência de um isomorfismo T de C(K1,X) em C(K2,X) com ||T||||T-1|| < 3 implica que uma certa soma topológica finita de K1 é homeomorfa a alguma soma topológica finita de K2. Mais ainda, se Xn não contém subespaço isomorfo a Xn+1 para todo n &isin; N, então K1 é homeomorfo a K2. Em outras palavras, obtemos um teorema tipo Banach-Stone vetorial que é uma extensão de um teorema de Gordon de 1970 e ao mesmo tempo uma extensão de um teorema de Behrends e Cambern de 1988. Mostramos que se existe um isomorfismo T de C(K1) em um subespaço de C(K2,X) com ||T||||T-1|| < 3, então a cardinalidade do &alpha;-ésimo derivado de K2 ou é finita ou é maior do que a cardinalidade do &alpha;-ésimo derivado de K1, para todo ordinal &alpha;. Em seguida, seja n um inteiro positivo, &Gamma; um conjunto infinito munido da topologia discreta e X um espaço de Banach de cotipo finito. Estabelecemos que se o n-ésimo derivado de K for não vazio, então a distância de Banach-Mazur entre C0(K,X) e C0(&Gamma;,X) é maior ou igual a 2n + 1. Também demonstramos que para quaisquer inteiros positivos n e k, a distância de Banach-Mazur entre C([1,&omega;nk],X) e C0(N,X) é exatamente 2n+1. Estes resultados fornecem extensões vetoriais para alguns teoremas de Cambern de 1970. Para um ordinal enumerável &alpha;, denotando por C(&alpha;) o espaço de Banach das funções contínuas no intervalo de ordinal [1, &alpha;], obtemos cotas superiores H(n, k) e cotas inferiores G(n, k) para as distâncias de Banach-Mazur entre os espaços C(&omega;) e C(&omega;nk), 1 < n, k < &omega;, verificando H(n, k) - G(n, k) < 2. Estas estimativas fornecem uma resposta para uma questão de Bessaga e Peczynski de 1960 sobre as distâncias de Banach-Mazur entre C(&omega;) e cada um dos espaços C(&alpha;), &omega;<&alpha;<&omega;&omega;. / For a locally compact Hausdorff space K and a Banach space X, we denote by C0(K,X) the space of X-valued continuous functions on K which vanish at infinity, endowed with the supremum norm. In the spirit of the classical 1937 Banach-Stone theorem, we prove that if C0(K1,X) is isomorphic to C0(K2,X), where X is a Banach space having finite cotype and such that X is separable or X* has the Radon-Nikodým property, then either K1 and K2 are finite or K1 and K2 have the same cardinality. It is a vector-valued extension of a 1978 Cengiz result, the scalar case X = R or X = C. We also prove that if K1 and K2 are compact ordinal spaces and X is Banach space having finite cotype, then the existence of an isomorphism T from C(K1,X) onto C(K2,X) with ||T||||T-1|| < 3 implies that some finite topological sum of K1 is homeomorphic to some finite topological sum of K2. Moreover, if Xn contains no subspace isomorphic to Xn+1 for every n &isin; N, then K1 is homeomorphic to K2. In other words, we obtain a vector-valued Banach-Stone theorem which is an extension of a 1970 Gordon theorem and at same time an improvement of a 1988 Behrends and Cambern theorem. We show that if there is an embedding T of a C(K1) into C(K2,X) with ||T||||T-1|| < 3, then the cardinality of the &alpha;-th derivative of K2 is either finite or greater than the cardinality of the &alpha;-th derivative of K1, for every ordinal &alpha;. Next, let n be a positive integer, &Gamma; an infinite set with the discrete topology and X is a Banach space having finite cotype. We prove that if the n-th derivative of K is not empty, then the Banach Mazur distance between C0(K,X) and C0(&Gamma;,X) is greater than or equal to 2n + 1. Thus, we also show that for every positive integers n and k, the Banach Mazur distance between C([1,&omega;nk],X) and C0(N,X) is exactly 2n+1. These results provide vector-valued versions of some 1970 Cambern theorems. For a countable ordinal &alpha;, writing C(&alpha;) for the Banach space of continuous functions on the interval of ordinal [1, &alpha;], we give lower bounds H(n, k) and upper bounds G(n, k) on the Banach- Mazur distances between C(&omega;) and C(&omega;nk), 1 < n, k < &omega;, such that H(n, k) - G(n, k) < 2. These estimates provide an answer to a 1960 Bessaga and Peczynski question on the Banach-Mazur distances between C(&omega;) and each of the C(&alpha;) spaces, &omega;<&alpha;<&omega;&omega;.

Para um espaço localmente compacto de Hausdorff K e um espaço de Banach X, denotamos por C0(K,X) o espaço de todas as funções a valores em X contínuas sobre K que se anulam no infinito, munido da norma do supremo. No espírito do clássico teorema de Banach-Stone 1937, estabelecemos que se C0(K1,X) é isomorfo a C0(K2,X), onde X é um espaço de Banach de cotipo finito e tal que X é separável ou X* tem a propriedade de Radon-Nikodým, então ou K1 e K2 são ambos finitos ou K1 e K2 tem a mesma cardinalidade. Trata-se de uma extensão vetorial de um resultado de Cengiz 1978, o caso escalar X = R ou X = C. Demonstramos também que se K1 e K2 são intervalos compactos de ordinais e X é um espaço de Banach de cotipo finito, então a existência de um isomorfismo T de C(K1,X) em C(K2,X) com ||T||||T-1|| < 3 implica que uma certa soma topológica finita de K1 é homeomorfa a alguma soma topológica finita de K2. Mais ainda, se Xn não contém subespaço isomorfo a Xn+1 para todo n &isin; N, então K1 é homeomorfo a K2. Em outras palavras, obtemos um teorema tipo Banach-Stone vetorial que é uma extensão de um teorema de Gordon de 1970 e ao mesmo tempo uma extensão de um teorema de Behrends e Cambern de 1988. Mostramos que se existe um isomorfismo T de C(K1) em um subespaço de C(K2,X) com ||T||||T-1|| < 3, então a cardinalidade do &alpha;-ésimo derivado de K2 ou é finita ou é maior do que a cardinalidade do &alpha;-ésimo derivado de K1, para todo ordinal &alpha;. Em seguida, seja n um inteiro positivo, &Gamma; um conjunto infinito munido da topologia discreta e X um espaço de Banach de cotipo finito. Estabelecemos que se o n-ésimo derivado de K for não vazio, então a distância de Banach-Mazur entre C0(K,X) e C0(&Gamma;,X) é maior ou igual a 2n + 1. Também demonstramos que para quaisquer inteiros positivos n e k, a distância de Banach-Mazur entre C([1,&omega;nk],X) e C0(N,X) é exatamente 2n+1. Estes resultados fornecem extensões vetoriais para alguns teoremas de Cambern de 1970. Para um ordinal enumerável &alpha;, denotando por C(&alpha;) o espaço de Banach das funções contínuas no intervalo de ordinal [1, &alpha;], obtemos cotas superiores H(n, k) e cotas inferiores G(n, k) para as distâncias de Banach-Mazur entre os espaços C(&omega;) e C(&omega;nk), 1 < n, k < &omega;, verificando H(n, k) - G(n, k) < 2. Estas estimativas fornecem uma resposta para uma questão de Bessaga e Peczynski de 1960 sobre as distâncias de Banach-Mazur entre C(&omega;) e cada um dos espaços C(&alpha;), &omega;<&alpha;<&omega;&omega;. / For a locally compact Hausdorff space K and a Banach space X, we denote by C0(K,X) the space of X-valued continuous functions on K which vanish at infinity, endowed with the supremum norm. In the spirit of the classical 1937 Banach-Stone theorem, we prove that if C0(K1,X) is isomorphic to C0(K2,X), where X is a Banach space having finite cotype and such that X is separable or X* has the Radon-Nikodým property, then either K1 and K2 are finite or K1 and K2 have the same cardinality. It is a vector-valued extension of a 1978 Cengiz result, the scalar case X = R or X = C. We also prove that if K1 and K2 are compact ordinal spaces and X is Banach space having finite cotype, then the existence of an isomorphism T from C(K1,X) onto C(K2,X) with ||T||||T-1|| < 3 implies that some finite topological sum of K1 is homeomorphic to some finite topological sum of K2. Moreover, if Xn contains no subspace isomorphic to Xn+1 for every n &isin; N, then K1 is homeomorphic to K2. In other words, we obtain a vector-valued Banach-Stone theorem which is an extension of a 1970 Gordon theorem and at same time an improvement of a 1988 Behrends and Cambern theorem. We show that if there is an embedding T of a C(K1) into C(K2,X) with ||T||||T-1|| < 3, then the cardinality of the &alpha;-th derivative of K2 is either finite or greater than the cardinality of the &alpha;-th derivative of K1, for every ordinal &alpha;. Next, let n be a positive integer, &Gamma; an infinite set with the discrete topology and X is a Banach space having finite cotype. We prove that if the n-th derivative of K is not empty, then the Banach Mazur distance between C0(K,X) and C0(&Gamma;,X) is greater than or equal to 2n + 1. Thus, we also show that for every positive integers n and k, the Banach Mazur distance between C([1,&omega;nk],X) and C0(N,X) is exactly 2n+1. These results provide vector-valued versions of some 1970 Cambern theorems. For a countable ordinal &alpha;, writing C(&alpha;) for the Banach space of continuous functions on the interval of ordinal [1, &alpha;], we give lower bounds H(n, k) and upper bounds G(n, k) on the Banach- Mazur distances between C(&omega;) and C(&omega;nk), 1 < n, k < &omega;, such that H(n, k) - G(n, k) < 2. These estimates provide an answer to a 1960 Bessaga and Peczynski question on the Banach-Mazur distances between C(&omega;) and each of the C(&alpha;) spaces, &omega;<&alpha;<&omega;&omega;.

Dans cette thèse, on aborde trois thèmes : problème de sélection de colonnes dans une matrice, distance de Banach-Mazur au cube et estimation de la covariance de matrices aléatoires. Bien que les trois thèmes paraissent éloignés, les techniques utilisées se ressemblent tout au long de la thèse. Dans un premier lieu, nous généralisons le principe d'invertibilité restreinte de Bourgain-Tzafriri. Ce résultat permet d'extraire un "grand" bloc de colonnes linéairement indépendantes dans une matrice et d'estimer la plus petite valeur singulière de la matrice extraite. Nous proposons ensuite un algorithme déterministe pour extraire d'une matrice un bloc presque isométrique c'est à dire une sous-matrice dont les valeurs singulières sont proches de 1. Ce résultat nous permet de retrouver le meilleur résultat connu sur la célèbre conjecture de Kadison-Singer. Des applications à la théorie locale des espaces de Banach ainsi qu'à l'analyse harmonique sont déduites. Nous donnons une estimation de la distance de Banach-Mazur d'un corps convexe de Rn au cube de dimension n. Nous proposons une démarche plus élémentaire, basée sur le principe d'invertibilité restreinte, pour améliorer et simplifier les résultats précédents concernant ce problème. Plusieurs travaux ont été consacrés pour approcher la matrice de covariance d'un vecteur aléatoire par la matrice de covariance empirique. Nous étendons ce problème à un cadre matriciel et on répond à la question. Notre résultat peut être interprété comme une quantification de la loi des grands nombres pour des matrices aléatoires symétriques semi-définies positives. L'estimation obtenue s'applique à une large classe de matrices aléatoires

Working within the Zermelo-Frankel Axioms of set theory, we will introduce two important contradictory axioms: Axiom of Choice and Axiom of Determinacy. We will explore perfect polish spaces and games on these spaces to see that the Axiom of Determinacy is inconsistent with the Axiom of Choice. We will see some of the major consequences of accepting the Axiom of Determinacy and how some of these results change when accepting the Axiom of Choice. We will consider 2-player games of perfect information wherein we will see some powerful results having to do with properties of the real numbers. We will use a game to illustrate a weak proof of the continuum hypothesis.

No description available.

Voir le texte.