The aim of this work is to generalize the results given by Domitrz, Janeczko and Zhitomirskii in [10]. In this article they classify in the symplectic manifold (R2, w) where w = dx1 Λ dx2 + · · · + dx2n-1 Λ dx2n is the symplectic form given by Darbouxs Theorem, all the set which are symplectomorphic to a fixed quasi homogeneous curve . To do this classification they defined the algebraic restrictions. We develop a new method called the method of exact algebraic restrictions and show that this classification is solved for the non quasi homogeneous case N = {(x1, x2) = x≥3 = 0} in the symplectic manifold (C2, w ), where f(x1, x2) = x41 + x52 + x21 x32. / Este trabalho tem como objetivo generalizar os resultados feitos por Domitrz, Janeczko e Zhitomirskii em [10]. Neste artigo eles clasificaram na variedade simplética (R2, w) onde w = dx1 Λ dx2 + ... + dx2n-1 Λ dx2n é a forma simpléctica dada pelo Teorema de Darboux, todos os conjuntos que são simplectomorfos a uma curva quase homogênea fixada . Para fazer a classificação eles definem as restrições algebraicas. Nós desenvolvemos um novo método o qual chamamos de método das restrições algebraicas exatas e provamos que a classificação é resolvida para o caso não quase homogêneo N = {f(x1, x2) = x≥3 = 0} na variedade simplética (C2, w ), onde f(x1, x2) = x41 + x52 + x21 x32.

Nesse trabalho são apresentados alguns resultados sobre classificação de Ações Anosov de Rk em (k + 2)variedades fechadas. Obtivemos dois teoremas (Teoremas A e B) que classificam tais ações. Essencialmente, mostramos que a ação será uma Tk1 extensão de um fluxo Anosov. Na demonstração é usada teoria das folheações de codimensão um; técnicas desenvolvidas por Fenley, como o estudo da ação levantada no recobrimento universal e a construção de losangos invariantes nesse espaço; bem como resultados obtidos por Maquera e Barbot, que iniciaram os estudos de Ações Anosov visando a classificação topológica destas / In this work is presented some important results about Anosov actions of Rk in (k + 2)closed manifolds. We obtained two classification theorems (Theorems A and B) which give us, essentially, that the system is a Tk1-extension of an Anosov flow. In order to show that, we used the theory of foliations of codimension one, techniques developed by Fenley, such as study of the lift of the action in the universal cover and the construction of invariant lozenges, what is more, we used some results by Maquera and Barbot, who began the studies of Anosov Actions generalizing some classic results on the way to classificate them

Nesta tese estudamos os sistemas integráveis definidos em superfícies compactas possuindo uma integral primeira que é uma função Morse-Bott a valores em R. Estes sistemas são aqui chamados de sistemas integráveis Morse-Bott. Classificamos as curvas fechadas e oitos associados a pontos de selas imersos em superfícies compactas. Essa classificação é aplicada ao estudo das folheações Morse-Bott em superfícies e nos permite definir um invariante topológico completo para a classificação topológica global destas folheações. Como uma aplicação desse estudo obtemos a classificação dos sistemas Morse-Bott assim como a classificação topológica das funções Morse-Bott em superfícies compactas e orientáveis. Demonstramos ainda um teorema da realização baseado em duas transformações e numa folheação geradora. Para o caso das funções Morse-Bott também obtivemos um teorema de realização. Finalmente, investigamos a generalização de alguns dos resultados anteriores para sistemas definidos em superfícies não orientáveis. / In this thesis we study integrable systems on compact surfaces with a first integral as a Morse-Bott function with target R. These systems are called here integrable Morse-Bott systems. Initially we present the classification of closed curves and eights associated to saddle points on compact surfaces. This classification is applied to the study of Morse- Bott foliations on surfaces allowing us to define a complete topological invariant for the global topological classification of these foliations. Then as an application of this study we obtain the classification of integrable Morse-Bott systems as well as the topological classification of Morse-Bott functions on compact and orientable surfaces. We also prove a realization theorem based on two transformation and a generating foliation (the foliation on the sphere with two centers). In the case of Morse-Bott functions we also obtain a realization theorem. Finally we investigate generalizations of previous results for systems defined on non-orientable surfaces.

La thèse est dédiée à l'analyse et à la conception de la commande tolérante aux défauts (fault tolerant control - FTC) en se fondant sur des méthodes ensemblistes. Nous étudions l'apparition des défauts pour les systèmes multi-capteurs, et les modes de détection, ainsi que la conception de lois de commande qui assurent la stabilité en boucle fermée. L'utilisation des ensembles invariants/contractifs permet la caractérisation des signaux résiduels, qui sont utilisés par la suite dans le processus de détection et d'isolement des défauts. La décision est fondée sur le positionnement par rapport à des hyperplans de séparation avec des importantes réductions de temps de calcul. Un mécanisme dual mis en œuvre par un bloc de récupération, permet la certification de la récupération des capteurs précédemment affectés par ces défauts.Dans une perspective théorique, nous soulignons les conditions qui permettent l'inclusion du bloc FDI (fault detection and isolation) et sa raison d'être dans la conception des lois de commande. Cela conduit par exemple à la synthèse des gains de retour d'état statique, par résolution de problèmes d'optimisation efficace (linéaire/convexe).Selon les paramètres choisis pour le réglage, la conception de la FTC peut être complétée par un superviseur de référence ou d'une loi de commande prédictive, qui adapte la trajectoire d'état et l'action de commande par retour d'état, afin d'assurer l'identification et la détection des défauts. Les questions spécifiques à l'utilisation de méthodes ensemblistes sont détaillées et des améliorations diverses sont proposées, par exemple : la construction des ensembles invariants, des formulations moins complexes des problèmes de type Mixed Integer Programming (MIP), l'analyse de la stabilité des systèmes commutés (notion de ``dwell-time''). / The scope of the thesis is the analysis and design of fault tolerant control (FTC) schemes through the use of set-theoretic methods. In the framework of multisensor schemes, the faults appearance and the modalities to accurately detect them are investigated as well as the design of control laws which assure the closed-loop stability. By using invariant/contractive sets to describe the residual signals, a fault detection and isolation (FDI) mechanism with reduced computational demands is implemented based on set-separation. A dual mechanism, implemented by a recovery block, which certificates previously fault-affected sensors is also studied. From a broader theoretical perspective, we point to the conditions which allow the inclusion of {FDI} objectives in the control law design. This leads to static feedback gains synthesis by means of numerically attractive optimization problems. Depending on the parameters selected for tuning, is shown that the FTC design can be completed by a reference governor or a predictive control scheme which adapts the state trajectory and the feedback control action in order to assure {FDI}. When necessary, the specific issues originated by the use of set-theoretic methods are detailed and various improvements are proposed towards: invariant set construction, mixed integer programming (MIP), stability for switched systems (dwell-time notions).

Submitted by Rosivalda Pereira (mrs.pereira@ufma.br) on 2017-05-26T20:32:19Z No. of bitstreams: 1 CamilaSoares.pdf: 1320447 bytes, checksum: e69a85013513d267e1ce15c2864ee0bf (MD5) / Made available in DSpace on 2017-05-26T20:32:19Z (GMT). No. of bitstreams: 1 CamilaSoares.pdf: 1320447 bytes, checksum: e69a85013513d267e1ce15c2864ee0bf (MD5) Previous issue date: 2017-03-03 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Fundação de Amparo à Pesquisa e ao Desenvolvimento Científico e Tecnológico do Maranhão (FAPEMA) / In this work, we study the formalism of the supersimmetric quantum mechanics (SUSY QM) starting with hamiltonian fatorization method of the bosonic harmonic oscillator (BHO) by means of ladders operators and we extended to hamiltonian of oscillators in general, inclusive of the fermionic harmonic oscillator (FHO), with their respective properties. We show that the SUSY QM for bound states, connect hamiltonians, potentials, and eigenvalues energy of different systems which are called supersymmetric partners. In the case scattering states, we notice that SUSY QM applies only in shape invariant potentials (SIPs). Despite this restriction, the SIPs are commonly used in literature by its wide applicability as in particles physics as in condensed matter. Therefore, we study the scattering states to the problem of massive relativistic fermions under the influence of modified Pöschl-Teller potential (PT), which is shape invariant. We calculate the transmission (T) and reflection (R) coefficients, just knowing the superpotential, which was carried out in a simple and elegant form. Thus, the study of the quantum scattering via SUSY QM becomes a powerful tool in the calculations of T and R. / Neste trabalho, estudamos o formalismo da mecânica quântica supersimétrica (MQS) partindo do método de fatoração do hamiltoniano do oscilador harmônico bosônico (OHB) por meio dos operadores escada e estendemos para o hamiltoniano de osciladores em geral, inclusive o oscilador harmônico fermiônico (OHF), com suas respectivas propriedades. Mostramos que a MQS, para estados ligados, associa hamiltonianos, potenciais e autovalores de energia de sistemas diferentes, que são denominados parceiros supersimétricos. No caso dos estados de espalhamento, observamos que a MQS aplica-se somente em potenciais invariantes de forma (PIFs). Apesar desta restrição, os PIFs são comumente utilizados na literatura, por sua larga aplicabilidade tanto na física de partículas quanto na matéria condensada. Sendo assim, estudamos os estados de espalhamento para o problema dos férmions relativísticos com massa sob influência do potencial de Pöschl-Teller modificado (PT), que é invariante de forma. Calculamos os coeficientes de transmissão (T) e reflexão (R), conhecendo apenas seu superpotencial, o que foi realizado de uma forma simples e elegante. Desse modo, o estudo do espalhamento quântico via MQS torna-se, uma ferramenta poderosa nos cálculos de T e R.

Cette thèse se compose de plusieurs travaux portant sur deux branches de la théorie des probabilités: processus de particules et cartes planaires aléatoires. Un premier travail concerne les aspects algébriques des mesures invariantes des processus de particules. Nous obtenons des conditions nécessaires et suffisantes sous lesquelles un processus de particules en temps continu avec espace d’états local discret possède une mesure invariante simple. Dans un deuxième travail nous étudions un modèle "biologique" de coexistence de 2 espèces en compétition sur un espace partagé, et soumis à des épidémies modélisées par un modèle probabiliste appelé "feux de forêts". Notre résultat principal montre que pour deux espèces, il existe des régions explicites de paramètres pour lesquelles une espèce domine ou les deux espèces coexistent. Il s’agit d’un des premiers modèles pour lesquels la coexistence d’espèces sur le long terme est prouvée. Les troisièmes et quatrièmes travaux. portent sur les cartes planaires décorées par des arbres. Dans le troisième nous présentons une bijection entre l’ensemble des cartes décorées par des arbres et le produit Cartésien entre l’ensemble des arbres planaires et l’ensemble de cartes à bord simple. Nous obtenons quelques formules de comptage et quelques outils pour l’étude de cartes aléatoires décorées par un arbre. Le quatrième travail montre que les triangulations et quadrangulations aléatoires uniformes avec f faces, bord simple de taille p et décorées par un arbre avec a arêtes, convergent en loi pour la topologie locale vers différentes limites, dépendant du comportement fini ou infini de la limite de f, p et a. / This thesis consists in several works exploring some models belonging to two branches of probability theory: interacting particle systems and random planar maps. A first work concerns algebraic aspects of interacting particle systems invariant measures. We obtain some necessary and sufficient conditions for some continuous time particle systems with discrete local state space, to have a simple invariant measure. In a second work we investigate the effect on survival and coexistence of introducing forest fire epidemics to a certain two-species spatial competition model. Our main results show that, for the two-type model, there are explicit parameter regions where either one species dominates or there is coexistence; contrary to the same model without forest fires, for which the fittest species alwaysdominates. The third and fourth works are related to tree-decorated planar maps. In the third work we present a bijection between the set of tree-decorated maps and the Cartesian product between the set of trees and the set of maps with a simple boundary. We obtain some counting results and some tools to study random decorated map models. In the fourth work we prove that uniform tree-decorated triangulations and quadrangulations with f faces, boundary of length p and decorated by a tree of size a converge weakly for the local topology to different limits, depending on the finite or infinite behavior of f, p and a.

Dans cette thèse, nous étudions deux exemples issus de la mécanique statistique. Les polymères dirigés en environnement aléatoire sont un modèle de système se trouvant à l'état d'équilibre. Nous donnons un critère de comparaison entre les entropies du réseau et de l'environnement permettant d'améliorer la borne inférieure sur la température critique. Nous utilisons également certains résultats connus dans le cadre de l'équation d'Anderson parabolique pour obtenir le comportement asymptotique de l'énergie libre. Par ailleurs, nous utilisons les polymères dirigés pour donner une preuve simple de l'indépendance de la fonction de Lyapunov de l'équation d'Anderson parabolique par rapport à la condition initiale.<br /><br />Les réseaux conducteurs de chaleur sont étudiés hors équilibre. Lorsque les potentiels d'interaction sont harmoniques, nous donnons une interprétation géométrique de la condition d'existence et d'unicité de la mesure invariante via un théorème de complétude. Dans le cas où cette condition fait défaut, nous explicitons une quantité invariante par le flot hamiltonien. Nous généralisons ensuite les résultats d'unicité à des potentiels analytiques. Nous montrons que la condition de Hörmander est suffisante pour avoir l'unicité de la mesure invariante via la contrôlabilité. Le principe de Lasalle est ensuite utilisé pour montrer l'unicité sans la condition d'Hörmander. Nous évoquons également le problème de l'existence de telles mesures.

In this work we study the geometric aspects of Lie groups from the view point of the Riemannian geometry, by means of invariant geometric structures associated. We present some properties on curvatures of metrics left invariants and bi-invariant one on Lie groups. We also present a treatment of the Lie algebras unimodular, including the tridimensional case. Most of the results studied are from the article of John Milnor: Curvatures of Left Invariant Metrics on Lie Groups. / Neste trabalho estudamos os aspectos geometricos de grupos de Lie, do ponto de vista da geometria Riemanniana, por meio de estruturas geometricas invariantes associadas. Nos apresentamos algumas propriedades de curvaturas com metricas invariante a esquerda e aquelas bi-invariantes em grupos de Lie. Apresentamos tambem um tratamento das algebras de Lie unimodulares, incluindo o caso tridimensional. A maioria dos resultados estudados foram retirados do artigo de John Milnor: Curvatures of Left Invariant Metrics on Lie Groups. / Mestre em Matemática

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2005-02-28Bitstream added on 2014-06-13T20:08:01Z : No. of bitstreams: 1 prado_j_me_sjrp.pdf: 438115 bytes, checksum: 072493cefd49a6603f89810da896f173 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Seja K um corpo p-ádico, com p 6= 2 e F K uma extensão galoisiana de K de grau n: Então F pode ser visto como espa»co quadrático sobre K, com a forma quadrática dada por T(x) = trFjK(x2), para x 2 F: Determinaremos os invariantes determinante, dimensão e invariante de Hasse desta forma quadrática para n igual a 2,3 e 4. / Let K be a p-adic eld with p 6= 2 and F a Galois extension eld of K of degree n: Then F can be viewed as a quadratic space over K under the quadratic form T(x) = trFjK(x2) for x 2 F. The invariants of this quadratic form dimension, determinant and Hasse invariant are given in the case when n is equal to 2,3 and 4.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-11T14:07:58Z No. of bitstreams: 1 HUGO SARAIVA TAVARES - DISSERTAÇÃO PPGMAT 2016..pdf: 830543 bytes, checksum: d315a3d95e0d2b08d4251cbb487964b6 (MD5) / Made available in DSpace on 2018-08-11T14:07:58Z (GMT). No. of bitstreams: 1 HUGO SARAIVA TAVARES - DISSERTAÇÃO PPGMAT 2016..pdf: 830543 bytes, checksum: d315a3d95e0d2b08d4251cbb487964b6 (MD5) Previous issue date: 2016-06 / Capes / Para ler o resumo deste trabalho recomendamos o download do arquivo, uma vez que o mesmo possui fórmulas e caracteres matemáticos que não foram possíveis trascreve-los aqui. / To read the summary of this work we recommend downloading the file, since it has formulas and mathematical characters that were not possible to transcribe them here.