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In this dissertation, we introduce combinatorial interpretations for three types of HallLittlewood polynomials (denoted Rλ, Pλ, and Qλ) by using weighted combinatorial objects called abacus-tournaments. We then apply these models to give combinatorial proofs of properties of Hall-Littlewood polynomials. For example, we show why various specializations of Hall-Littlewood polynomials produce the Schur symmetric polynomials, the elementary symmetric polynomials, or the t-analogue of factorials. With the abacus-tournament model, we give a bijective proof of a Pieri rule for Hall-Littlewood polynomials that gives the Pλ-expansion of the product of a Hall-Littlewood polynomial Pµ with an elementary symmetric polynomial ek. We also give a bijective proof of certain cases of a second Pieri rule that gives the Pλ-expansion of the product of a Hall-Littlewood polynomial Pµ with another Hall-Littlewood polynomial Q(r) . In general, proofs using abacus-tournaments focus on canceling abacus-tournaments and then finding weight-preserving bijections between the sets of uncanceled abacus-tournaments. / Ph. D.

The Kostka-Foulkes polynomials describe the change of basis between Schur polynomials and Hall-Littlewood polynomials. In this paper, we extend this idea to the family of BCn Macdonald spherical functions, with multiparameter Kostka-Foulkes polynomials acting as the change of basis from the BC_n spherical functions to the type Cn Schur polynomials. We develop a Kato-Lusztig formula that describes the multiparameter BCn-Kostka-Foulkes polynomials. / Master of Science / The work done in [11] gives a formula to calculate Kostka-Foulkes polynomials that convert between two other forms of polynomials. However, this only applies in specific instances. In this paper, we generalize those ideas to allow for more parameters, and find that a similar formula still holds.

No description available.

Cette thèse est consacrée à l'étude des opérateurs strictement hyperboliques à coefficients peu réguliers, aussi bien qu'à l'étude du système d'Euler incompressible à densité variable. Dans la première partie, on montre des estimations a priori pour des opérateurs strictement hyperboliques dont les coefficients d'ordre le plus grand satisfont une condition de continuité log-Zygmund par rapport au temps et une condition de continuité log-Lipschitz par rapport à la variable d'espace. Ces estimations comportent une perte de dérivées qui croît en temps. Toutefois, elles sont suffisantes pour avoir encore le caractère bien posé du problème de Cauchy associé dans l'espace H^inf (pour des coefficients du deuxième ordre ayant assez de régularité).Dans un premier temps, on considère un opérateur complet en dimension d'espace égale à 1, dont les coefficients du premier ordre étaient supposés hölderiens et celui d'ordre 0 seulement borné. Après, on traite le cas général en dimension d'espace quelconque, en se restreignant à un opérateur de deuxième ordre homogène: le passage à la dimension plus grande exige une approche vraiment différente. Dans la deuxième partie de la thèse, on considère le système d'Euler incompressible à densité variable. On montre son caractère bien posé dans des espaces de Besov limites, qui s'injectent dans la classe des fonctions globalement lipschitziennes, et on établit aussi des bornes inférieures pour le temps de vie de la solution ne dépendant que des données initiales. Cela fait, on prouve la persistance des structures géométriques, comme la régularité stratifiée et conormale, pour les solutions de ce système. À la différence du cas classique de densité constante, même en dimension 2 le tourbillon n'est pas transporté par le champ de vitesses. Donc, a priori on peut s'attendre à obtenir seulement des résultats locaux en temps. Pour la même raison, il faut aussi laisser tomber la structure des poches de tourbillon. La théorie de Littlewood-Paley et le calcul paradifférentiel nous permettent d'aborder ces deux différents problèmes. En plus, on a besoin aussi d'une nouvelle version du calcul paradifférentiel, qui dépend d'un paramètre plus grand que ou égal à 1, pour traiter les opérateurs à coefficients peu réguliers. Le cadre fonctionnel adopté est celui des espaces de Besov, qui comprend en particulier les ensembles de Sobolev et de Hölder. Des classes intermédiaires de fonctions, de type logarithmique, entrent, elles aussi, en jeu / The present thesis is devoted both to the study of strictly hyperbolic operators with low regularity coefficients and of the density-dependent incompressible Euler system. On the one hand, we show a priori estimates for a second order strictly hyperbolic operator whose highest order coefficients satisfy a log-Zygmund continuity condition in time and a log-Lipschitz continuity condition with respect to space. Such an estimate involves a time increasing loss of derivatives. Nevertheless, this is enough to recover well-posedness for the associated Cauchy problem in the space $H^infty$ (for suitably smooth second order coefficients).In a first time, we consider acomplete operator in space dimension $1$, whose first order coefficients were assumed Hölder continuous and that of order $0$only bounded. Then, we deal with the general case of any space dimension, focusing on a homogeneous second order operator: the step to higher dimension requires a really different approach. On the other hand, we consider the density-dependent incompressible Euler system. We show its well-posedness in endpoint Besov spaces embedded in the class of globally Lipschitz functions, producing also lower bounds for the lifespan of the solution in terms of initial data only. This having been done, we prove persistence of geometric structures, such as striated and conormal regularity, for solutions to this system. In contrast with the classical case of constant density, even in dimension $2$ the vorticity is not transported by the velocity field. Hence, a priori one can expect to get only local in time results. For the same reason, we also have to dismiss the vortex patch structure. Littlewood-Paley theory and paradifferential calculus allow us to handle these two different problems .A new version of paradifferential calculus, depending on a parameter $ggeq1$, is also needed in dealing with hyperbolic operators with nonregular coefficients. The general framework is that of Besov spaces, which includes in particular Sobolev and Hölder sets. Intermediate classes of functions, of logaritmic type, come into play as well

Dans cette thèse, j’étudie trois sujets reliés : les modèles de tenseurs, les nombres de Hurwitz et les polynômes de Macdonald-Koornwinder. Les modèles de tenseurs généralisent les modèles de matrices en tant qu’une approche à la gravité quantique en dimension arbitraire (les modèles de matrices donnent une version bidimensionnelle). J’étudie un modèle particulier qui s’appelle le modèle quartique mélonique. Sa spécialité est qu’il s’écrit en termes d’un modèle de matrices qui est lui-même aussi intéressant. En utilisant les outils bien établis, je calcule les deux premiers ordres de leur 1=N expansion. Parmi plusieurs interprétations, les nombres de Hurwitz comptent le nombre de revêtements ramifiés de surfaces de Riemann. Ils sont connectés avec de nombreux sujets en mathématiques contemporaines telles que les modèles de matrices, les équations intégrables et les espaces de modules. Ma contribution principale est une formule explicite pour les nombres doubles avec 3-cycles complétées d’une part. Cette formule me permet de prouver plusieurs propriétés intéressantes de ces nombres. Le dernier sujet de mon étude est les polynôme de Macdonald et Koornwinder, plus précisément les identités de Littlewood. Ces polynômes forment les bases importantes de l’algèbre des polynômes symétriques. Un des problèmes intrinsèques dans la théorie des fonctions symétriques est la décomposition d’un polynôme symétrique dans la base de Macdonald. La décomposition obtenue (notamment si les coefficients sont raisonnablement explicites et compacts) est nommée une identité de Littlewood. Dans cette thèse, j’étudie les identités démontrées récemment par Rains et Warnaar. Mes contributions incluent une preuve d’une extension d’une telle identité et quelques progrès partiels vers la généralisation d’une autre. / In this thesis, I study three related subjects: tensor models, Hurwitz numbers and Macdonald-Koornwinder polynomials. Tensor models are generalizations of matrix models as an approach to quantum gravity in arbitrary dimensions (matrix models give a 2D version). I study a specific model called the quartic melonic tensor model. Its specialty is that it can be transformed into a multi-matrix model which is very interesting by itself. With the help of well-established tools, I am able to compute the first two leading orders of their 1=N expansion. Among many interpretations, Hurwitz numbers count the number of weighted ramified coverings of Riemann surfaces. They are connected to many subjects of contemporary mathematics such as matrix models, integrable equations and moduli spaces of complex curves. My main contribution is an explicit formula for one-part double Hurwitz numbers with completed 3-cycles. This explicit formula also allows me to prove many interesting properties of these numbers. The final subject of my study is Macdonald-Koornwinder polynomials, in particular their Littlewood identities. These polynomials form important bases of the algebra of symmetric polynomials. One of the most important problems in symmetric function theory is to decompose a symmetric polynomial into the Macdonald basis. The obtained decomposition (in particular, if the coefficients are explicit and reasonably compact) is called a Littlewood identity. In this thesis, I study many recent Littlewood identities of Rains and Warnaar. My own contributions include a proof of an extension of one of their identities and partial progress towards generalization of one another.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-10T12:12:27Z No. of bitstreams: 1 arquivototal.pdf: 955147 bytes, checksum: f6b8d3b1e6ba8fba22d9e0a28e6685fc (MD5) / Made available in DSpace on 2017-08-10T12:12:27Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 955147 bytes, checksum: f6b8d3b1e6ba8fba22d9e0a28e6685fc (MD5) Previous issue date: 2016-08-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we present a version little know of the famous Hölder's Inequality, considering the context of Lp and lp spaces for mixed norms. We show how a suitable use this inequality has influencied positively others classical inequalities, to highlight, the multilinear inequalities of Bohnenblust-Hille and Hardy-Littlewood. / No presente trabalho, apresentamos uma versão pouco conhecida da famosa Desigualdade de Hölder, considerando o contexto dos espaços Lp e lp com normas mistas. Mostramos como o uso adequado desta desigualdade vem influenciando positivamente outras desigualdades clássicas, a destacar, as desigualdades multilineares de Bohnenblust-Hille e Hardy-Littlewood.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-17T16:04:13Z No. of bitstreams: 1 arquivototal.pdf: 699730 bytes, checksum: d48ddf5357572db4dac922761f91c532 (MD5) / Made available in DSpace on 2017-08-17T16:04:13Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 699730 bytes, checksum: d48ddf5357572db4dac922761f91c532 (MD5) Previous issue date: 2016-04-04 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this study we show two classical inequalities, namely Bohnenblust-Hille inequality and Hardy-Littlewood inequality. The rst one, conceived as a tool for the study of problems related to Dirichlet series, is a generalization of Littlewood`s 4/3 inequality to multilinear forms. The second is a generalization of Bohnenblust-Hille inequality, produced by the replacement of c0 with lp. / No presente trabalho abordaremos duas desigualdades cl assicas, a saber, a Desigualdade de Bohnenblust-Hille e a Desigualdade de Hardy- Littlewood. A primeira, surgiu como ferramenta para o estudo de problemas relacionados a s eries de Dirichlet e e uma generaliza c~ao para formas multilineares da Desigualdade 4/3 de Littlewood. A segunda consiste de uma generaliza c~ao da Desigualdade de Bohnenblust-Hille, produzida pela substituição de c0 por lp.

Made available in DSpace on 2015-05-15T11:46:22Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1916558 bytes, checksum: 2a74bd2ee59f2f8ed1aa50acfcc283c4 (MD5) Previous issue date: 2014-12-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work is divided in two subjects. The first concerns about the Bohnenblust-Hille and Hardy- Littlewood multilinear inequalities. We obtain optimal and definitive generalizations for both inequalities. Moreover, the approach presented provides much simpler and straightforward proofs than the previous one known, and we are able to show that in most cases the exponents involved are optimal. The technique used is a combination of probabilistic tools and of an interpolative approach; this former technique is also employed in this thesis to improve the constants for vector-valued Bohnenblust-Hille type inequalities. The second subject has as starting point the existence of Peano spaces, that is, Haurdor spaces that are continuous image of the unit interval. From the point of view of lineability we analyze the set of continuous surjections from an arbitrary euclidean spaces on topological spaces that are, in some natural sense, covered by Peano spaces, and we conclude that large algebras are found within the families studied. We provide several optimal and definitive result on euclidean spaces, and, moreover, an optimal lineability result on those special topological vector spaces. / Este trabalho édividido em dois temas. O primeiro diz respeito às desigualdades multilineares de Bohnenblust-Hille e Hardy-Littlewood. Obtemos generalizações ótimas e definitivas para ambas desigualdades. Mais ainda, a abordagem apresentada fornece demonstrações mais simples e diretas do que as conhecidas anteriormente, além de sermos capazes de mostrar que os expoentes envolvidos são ótimos em varias situações. A técnica utilizada combina ferramentas probabilísticas e interpolativas; esta ultima e ainda usada para melhorar as estimativas das versões vetoriais da desigualdade de Bohnenblust-Hille. O segundo tema possui como ponto de partida a existência de espaços de Peano, ou seja, os espaços de Hausdor que são imagem contínua do intervalo unitário. Sob o ponto de vista da lineabilidade, analisamos o conjunto das sobrejecoes contínuas de um espaço euclidiano arbitrário em um espaço topológico que, de certa forma, e coberto por espaços de Peano, e concluímos que grandes álgebras são encontradas nas famílias estudadas. Fornecemos vários resultados ótimos e definitivos em espaços euclidianos, e, mais ainda, um resultado de lineabilidade ótimo naqueles espaços vetoriais topológicos especiais.

L'objectif de cette thèse est de donner les outils fondamentaux de la théorie des opérateurs de multiplications ponctuelle basés principalement sur la théorie des distributions et l'analyse de Fourier, et d'en donner des applications aux dérivées partielles. L'étude des opérateurs de multiplication ponctuelle examine à quelle conditions on a des inégalité de type capacitaire. Elle intervienne dans l'étude des opérateurs différentiels à coéfficients irréguliers. Le but principal de cette yhèse est de généraliser le théorème de Maz'ya - Verbitsky. Les outils utilisés sont la théorie des opérateurs d'intégrales singulières, la théorie de Littlewood-Paley, la théorie de la capacité et le poids de Muckenhoupt.