本文希望藉著K.T Arasu, D.Jungnickel, A.Pott推廣Wilbrink定理的方法去尋找Wilbrink等式的推廣式在p^k∥n,k≧4的推廣式和其應用。 / In this thesis we formulate and provide rigorous proofs of Wilbrink's theorem and it's variations due to Arasu, A.Pott and D.Jungnickel. some questions on further generalizations of Wilbrink's theorem are discussed; known generalization are study in A.Pott's dissertation.

This PhD thesis consists of a summary and seven papers, where various applications of auto-validated computations are studied. In the first paper we describe a rigorous method to determine unknown parameters in a system of ordinary differential equations from measured data with known bounds on the noise of the measurements. Papers II, III, IV, and V are concerned with Abelian integrals. In Paper II, we construct an auto-validated algorithm to compute Abelian integrals. In Paper III we investigate, via an example, how one can use this algorithm to determine the possible configurations of limit cycles that can bifurcate from a given Hamiltonian vector field. In Paper IV we construct an example of a perturbation of degree five of a Hamiltonian vector field of degree five, with 27 limit cycles, and in Paper V we construct an example of a perturbation of degree seven of a Hamiltonian vector field of degree seven, with 53 limit cycles. These are new lower bounds for the maximum number of limit cycles that can bifurcate from a Hamiltonian vector field for those degrees. In Papers VI, and VII, we study a certain kind of normal form for real hyperbolic saddles, which is numerically robust. In Paper VI we describe an algorithm how to automatically compute these normal forms in the planar case. In Paper VII we use the properties of the normal form to compute local invariant manifolds in a neighbourhood of the saddle.

Dans ce travail, nous classifions les automorphismes non-symplectiques des variétés équivalentes par déformations à des variétés de Kummer généralisées de dimension 4, ayant une action d'ordre premier sur le réseau de Beauville-Bogomolov. Dans un premier temps, nous donnons les lieux fixes des automorphismes naturels de cette forme. Par la suite, nous développons des outils sur les réseaux en vue de les appliquer à nos variétés. Une étude réticulaire des tores complexes de dimension 2 permet de mieux comprendre les automorphismes naturels sur les variétés de type Kummer. Nous classifions finalement tous les automorphismes décrits précédemment sur ces variétés. En application de nos résultats sur les réseaux, nous complétons également la classification des automorphismes d'ordre premier sur les variétés équivalentes par déformations à des schémas de Hilbert de 2 points sur des surfaces K3, en traitant le cas de l'ordre 5 qui restait ouvert. / Ln this work, we classify non-symplectic automorphisms of varieties deformation equivalent to 4-dimensional generalized Kummer varieties, having a prime order action on the Beauville-Bogomolov lattice. Firstly, we give the fixed loci of natural automorphisms of this kind. Thereafter, we develop tools on lattices, in order to apply them to our varieties. A lattice-theoritic study of 2-dimensional complex tori allows a better understanding of natural automorphisms of Kummer-type varieties. Finaly, we classify all the automorphisms described above on thos varieties. As an application of our results on lattices, we complete also the classification of prime order automorphisms on varieties deformation-equivalent to Hilbert schemes of 2 points on K3 surfaces, solving the case of order 5 which was still open.

Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-15T12:16:24Z No. of bitstreams: 1 arquivototal.pdf: 5169144 bytes, checksum: 66d534e3f0c0c59bf5d35a45290fa390 (MD5) / Made available in DSpace on 2016-03-15T12:16:24Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 5169144 bytes, checksum: 66d534e3f0c0c59bf5d35a45290fa390 (MD5) Previous issue date: 2015-02-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this thesis, we propose an experimental setup for the study of Landau quantization and associated effects in a two-dimensional ultracold atomic gas. Gauge fields can emerge in the equation of motion for the optically addressed ultracold atoms. To this end, spatially dependent dark states are necessary for the internal states of the atoms. A tripod level scheme yields two degenerate dark states which can leads to either an Abelian U(1) U(1) gauge field or a non-Abelian SU(2) gauge field. Using a suitable laser configuration, we obtain a uniform U(1) U(1) magnetic field which causes the atoms organize themselves in Landau levels. The strength of the effective magnetic field depends on the relative intensity of the lasers beams at the atomic cloud. We estimate the degeneracy of the energy levels for an atomic gas formed by atoms of 87Rb. In addition, we establish the experimental conditions to reach the lowest Landau level regime. In the zero-temperature limit, we realize the emergence of magnetic oscillations in the atomic energy and its derivative as function of the inverse of the effective magnetic field (de Haas van Alphen effect). The period of the de Haas van Alphen oscillation allow us to determine area of the Fermi circle for the atomic gas via an Onsager-like relation. We also show that detuning the a laser from the two-photon resonance we generate a parabolic scalar potential that laterally confines the atoms. As a consequence, the Landau levels degeneracy is removed, since the energy spectrum depends explicitly on the transverse atomic momentum. We show that the Landau levels presents a reminiscent degeneracy when the boundaries conditions are considered. The residual degeneracy occurs when different energy levels overlap. We map the residual degeneracy points as a function of the effective magnetic field. Finally, we present an experimental scheme for observing the spin Hall effect for ultracold atoms in a tripod configuration. / Nesta tese, propomos um arranjo experimental para o estudo da quantização de Landau e efeitos associados em um gás atômico ultrafrio bidimensional. Campos de calibre podem surgir na equação de movimento para átomos ultrafrios oticamente vestidos. Para que isto ocorra, estados escuros espacialmente dependentes são necessários a partir dos estados internos dos átomos. Átomos numa configuração de níveis de energia do tipo tripod produzem dois estados escuros degenerados, que podem levar a campos de calibre Abelianos U(1) U(1) ou não-Abelianos SU(2). Utilizando uma configuração adequada de lasers, mostramos que é possível se produzir um campo magnético sintético uniforme U(1) U(1) que atua nos átomos neutros fazendo-os se organizarem em níveis de Landau. A intensidade do campo efetivo depende da intensidade relativa dos feixes de luz na nuvem atômica. Estimamos a degenerescência dos níveis de energia para um gás atômico formado por átomos de 87Rb e estabelecemos as condições experimentais para que seja atingido o regime em que todos os átomos populam unicamente o nível de Landau menos energético. Considerando o limite de temperatura nula, verificamos o surgimento de oscilações magnéticas na energia e em sua derivada como uma função do inverso do campo magnético efetivo (efeito de Haas van Alphen). O período da oscilação magnética nos permite determinar a área do círculo de Fermi para o gás atômico através de uma expressão similar a de Onsager para sistemas eletrônicos. Mostramos também que dessintonizando um dos lasers em relação à ressonância de dois fótons geramos um potencial escalar parabólico que faz com os átomos sejam lateralmente confinados. Isto resulta na remoção da degenerescência dos níveis de Landau, uma vez que a energia depende explicitamente do momento atômico transverso. Demonstramos que, aplicando condições periódicas de contorno ao sistema, temos o surgimento de uma degenerescência residual. A degenerescência remanescente ocorre quando diferentes níveis de energia se superpõem. Mapeamos os pontos de degenerescência como uma função do campo magnético efetivo. Por fim, apresentamos um esquema experimental para a observação do efeito spin Hall para átomos ultrafrios em uma configuração tripod.

Neste trabalho apresentamos várias demonstrações do Teorema da Base Normal para certos tipos de extensões galoisianas de corpos, algumas existenciais e outras construtivas, destacando as diferenças e dificuldades de cada situação. Apresentamos também generalizações de tal teorema e mostramos que toda extensão galoisiana de grau ímpar de corpos admite uma base normal autodual com respeito µa forma bilinear traço / In this work we present several demonstrations of The Normal Basis Theorem for certain kinds of galoisian extensions of fields, some of them existential and others constructive, pointing the diffculties and differences in each situation. We also present generalizations of such theorem and show that every odd degree galoisian extension of fields admits a self-dual normal base with respect to the trace bilinear map

[pt] Iniciamos o trabalho com uma revisão da classificação de álgebras de Lie semi-simples sobre corposo algebraicamente fechados de caracteristica zero a traves dos Diagramas de Dyinkin. Posteriormente estudamos sigma - sistemas normais e classificamos eles a traves de diagramas de Satake. Finalmente estudamos a estrutura das formas reais de álgebras de Lie semi-simples complexas, explicitando a conexão com os diagramas de Satake e fornecenendo assim uma classificação das mesmas. / [en] We begin the work with a review of the classification of semisimple Lie algebras over an algebraically field of characteristic zero through the Dyinkin Diagrams. Subsequently we study sigma - normal systems and classify them through Satake diagrams. Finally we study the structure of the real forms of complex semi-simple Lie algebras, explaining the connection with the Satake diagrams and thus providing a classification of them.

Cette thèse s'inscrit dans l'étude mathématique de la percolation, qui regroupe une famille de modèles présentant une transition de phase. Des avancées majeures au cours des quinze dernières années, notamment l'invention du SLE et la preuve de l'invariance conforme de la percolation de Bernoulli critique, nous permettent aujourd'hui d'avoir une image très complète de la percolation de Bernoulli sur le réseau triangulaire. Cependant, de nombreuses questions demeurent ouvertes, et ont motivé notre travail.La première d'entre elle est l'universalité de la percolation plane, qui affirme que les propriétés macroscopiques de la percolation plane critique ne devraient pas dépendre du réseau sous-jacent à sa définition. Nous montrons, dans le cadre de la percolation Divide and Color, un résultat qui va dans le sens de cette universalité et identifions, dans ce contexte, des phénomènes macroscopiques indépendants du réseau microscopique. Une version plus faible d'universalité est donnée par la théorie de Russo-Seymour-Welsh (RSW), et sa validité est connue pour la percolation de Bernoulli (sans dépendance) sur les réseaux plans suffisamment symétriques. Nous étudions de nouveaux arguments de type RSW pour des modèles de percolation avec dépendance. La deuxième question que nous avons abordée est celle de l'absence d'une composante connexe ouverte infinie au point critique, une question importante du point de vue physique, puisqu'elle traduit la continuité de la transition de phase. Dans deux travaux en collaboration avec Hugo Duminil-Copin et Vladas Sidoravicius, nous montrons que la transition de phase est continue pour la percolation de Bernoulli sur le graphe Z^2x{0,...,k}, et pour la percolation FK sur le réseau carré avec paramètre q inférieur ou égal à 4. Enfin, la dernière question qui nous a guidés est la localité du point critique : la donnée des boules de grands rayons d'un graphe suffit-elle à identifier avec une bonne précision la valeur du point critique? Dans un travail en collaboration avec Sébastien Martineau, nous répondons de manière affirmative à cette question dans le cadre des graphes de Cayley de groupes abéliens. / This thesis is part of the mathematical study of percolation theory, which includes a family of models with a phase transition. Major advances in the 2000s, including the invention of SLE and the proof of conformal invariance of critical Bernoulli percolation, provide us with a very complete picture of the Bernoulli percolation process on the triangular lattice. Fortunately, many questions remain open, and motivated our work.The first of these is the universality of planar percolation, which states that the macroscopic properties of critical planar percolation should not depend on the underlying graph. We study this question in the framework of Divide and Color percolation, and prove in this context a result that goes in the direction of universality. A weaker universality statement is given by the theory of Russo-Seymour-Welsh (RSW), which is known to hold for planar Bernoulli percolation (without dependence) on sufficiently symmetric graphs. We study new RSW-type arguments for percolation models with dependence.The second question is the absence of an infinite cluster at the critical point, an important question from a physical point of view, equivalent to the continuity of the phase transition. In two different joint works with Hugo Duminil-Copin and Vladas Sidoravicius, we show that the phase transition is continuous for Bernoulli percolation on the graph Z^2 x {0,...,k} and for FK percolation on the square lattice with parameter q smaller than or equal to 4.Finally, the last question that guided us is the locality of the critical point: is it possible to determine with good accuracy the critical value for Bernoulli percolation on a graph if we know only the balls with large radii? Jointly with Sébastien Martineau, we answer positively to this question in the framework of Cayley graphs of abelian groups.

No description available.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste tese, estudamos os efeitos não perturbativos associados à presença do horizonte de Gribov e à condensação de operadores locais de dimensão dois, numa teoria de Yang-Mills euclidiana em SU(2), quantizada no calibre abeliano maximal. Estes efeitos são introduzidos de modo a preservar as propriedades de renormalizabilidade e localidade da teoria, e refletem-se diretamente no comportamento dos propagadores. A comparação com os dados da rede indicam um bom acordo qualitativo. / In this, we study the nonperturbative effects associated to the presence of the horizon and to the condensation of local dimension two operators in an Eucledean SU(2)Yang-Mills theory quantized in the maximal Abelian gauge. Such effects are introduced in a way to preserve the properties of renormalizability and locality of the theory. The comparison with the lattice data indicates a good qualitative agreement.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste tese, estudamos os efeitos não perturbativos associados à presença do horizonte de Gribov e à condensação de operadores locais de dimensão dois, numa teoria de Yang-Mills euclidiana em SU(2), quantizada no calibre abeliano maximal. Estes efeitos são introduzidos de modo a preservar as propriedades de renormalizabilidade e localidade da teoria, e refletem-se diretamente no comportamento dos propagadores. A comparação com os dados da rede indicam um bom acordo qualitativo. / In this, we study the nonperturbative effects associated to the presence of the horizon and to the condensation of local dimension two operators in an Eucledean SU(2)Yang-Mills theory quantized in the maximal Abelian gauge. Such effects are introduced in a way to preserve the properties of renormalizability and locality of the theory. The comparison with the lattice data indicates a good qualitative agreement.