In dieser Arbeit wird zuerst das Theorem von Chung, Graham und Wilson über quasi-zufällige Graphen zur sogenannten schwachen Quasi-Zufälligkeit für k-uniforme Hypergraphen verallgemeinert und somit eine Reihe äquivalenter Eigenschaften bestimmt. Basierend auf diesen Resultaten werden nichtbipartite Graphen gefunden, welche die Quasi-Zufälligkeit für Graphen ``forcieren''''. Zuvor waren nur bipartite Graphen mit dieser Eigenschaft bekannt. Desweiteren ist ein konzeptionell einfacher Algorithmus zum Verifizieren nicht erfüllbarer zufälliger k-SAT Formeln angegeben. Dann richtet sich der Fokus auf Anwendungen verschiedener Regularitätslemmata für Hypergraphen. Zuerst wird die Menge aller bezeichneten 3-uniformen Hypergraphen auf n Knoten, die keine Kopie des Hypergraphen der Fano Ebene enthalten, studiert. Es wird gezeigt, dass fast jedes Element aus dieser Menge ein bipartiter Hypergraph ist. Dies führt zu einem Algorithmus, der in polynomiell erwarteter Zeit einen zufälligen Fano-freien (und somit einen zufälligen bipartiten 3-uniformen) Hypergraphen richtig färbt. Schließlich wird die folgende extremale Funktion studiert. Es sind r Farben gegeben sowie ein k-uniformer Hypergraph F. Auf wie viele verschiedene Arten kann man die Kanten eines k-uniformen Hypergraphen H färben, so dass keine monochromatische Kopie von F entsteht? Welche Hypergraphen H maximieren die Anzahl erlaubter Kantenfärbungen? Hier wird ein strukturelles Resultat für eine natürliche Klasse von Hypergraphen bewiesen. Es wird für viele Hypergraphen F, deren extremaler Hypergraph bekannt ist, gezeigt, dass im Falle von zwei oder drei Farben die extremalen Hypergraphen die oben beschriebene Funktion maximieren, während für vier oder mehr Farben andere Hypergraphen mehr Kantenfärbungen zulassen. / This thesis presents first one possible generalization of the result of Chung, Graham and Wilson to k-uniform hypergraphs, and studies the so-called weak quasi-randomness. As applications we obtain a simple strong refutation algorithm for random sparse k-SAT formulas and we identify first non-bipartite forcing pairs for quasi-random graphs. Our focus then shifts from the study of quasi-random objects to applications of different versions of the hypergraph regularity lemmas; all these versions assert decompositions of hypergraphs into constantly many quasi-random parts, where the meaning of ``quasi-random'''' takes different contexts in different situations. We study the family of hypergraphs not containing the hypergraph of the Fano plane as a subhypergraph, and show that almost all members of this family are bipartite. As a consequence an algorithm for coloring bipartite 3-uniform hypergraphs with average polynomial running time is given. Then the following combinatorial extremal problem is considered. Suppose one is given r colors and a fixed hypergraph F. The question is: In at most how many ways can one color the hyperedges of a hypergraph H on n vertices such that no monochromatic copy of F is created? What are the extremal hypergraphs for this function? Here a structural result for a natural family of hypergraphs F is proven. For some special classes of hypergraphs we show that their extremal hypergraphs (for large n) maximize the number of edge colorings for 2 and 3 colors, while for at least 4 colors other hypergraphs are optimal.

Desenvolvemos novos resultados da teoria dos espaços FS e DFS (espaços de Fréchet-Schwartz e seus duais) e os empregamos ao estudo da seguinte questão: quando certas propriedades de regularidade de um operador diferencial parcial linear (entre fibrados vetoriais Gevrey sobre uma variedade Gevrey) implicam resolubilidade, no sentido de ultradistribuições, do operador transposto? Estudamos esta questão para uma classe de operadores abstratos que contém os operadores diferenciais parciais lineares com coeficientes Gevrey usuais, mas também certas classes de operadores pseudo-diferenciais em variedades compactas, além de certos tipos de operadores de ordem infinita. Neste contexto, obtemos uma nova demonstração de um resultado global em variedades compactas (em que hipoelipticidade Gevrey global de um operador implica resolubilidade global de seu transposto), assim como alguns resultados no caso não-compacto relacionados à propriedade de não-confinamento de singularidades. Na sequência apresentamos algumas aplicações concretas, em particular para operadores de Hörmander, operadores de força constante e sistemas localmente integráveis de campos vetoriais. Analisamos ainda algumas instâncias de uma conjectura levantada em um artigo recente de F. Malaspina e F. Nicola (2014), a qual afirma que, para certos complexos diferenciais naturalmente associados a estruturas localmente integráveis, resolubilidade local no sentido de ultradistribuições (perto de um ponto, em um grau fixado) implica resolubilidade local no sentido de distribuições. Estabelecemos a validade desta conjectura quando o fibrado estrutural cotangente é gerado pelo diferencial de uma única integral primeira. / We develop new techniques in the setting of FS and DFS spaces (Fréchet-Schwartz spaces and their strong duals) and apply them to the study of the following question: when regularity properties of a general linear differential operator (between Gevrey vector bundles over a Gevrey manifold) imply solvability of its transpose in the sense of ultradistributions? This question is studied for a class of abstract operators that encompasses the usual partial differential operators with Gevrey coefficients, but also some flavors of pseudodifferential operators on compact manifolds and some classes of operators with infinite order. In this setting, we obtain a new proof of a global result on compact manifolds (global Gevrey hypoellipticity of the operator implying global solvability of the transpose), as well as some results in the non-compact case by means of the so-called property of non-confinement of singularities. We then move to some concrete applications, especially for Hörmander operators, operators of constant strength and locally integrable systems of vector fields. We also analyze some instances of a conjecture stated in a recent paper of F. Malaspina and F. Nicola (2014), which asserts that, in differential complexes naturally arising from locally integrable structures, local solvability in the sense of ultradistributions (near a point, in some fixed degree) implies local solvability in the sense of distributions. We establish the validity of the conjecture when the cotangent structure bundle is spanned by the differential of a single first integral.

Nesta tese, introduzimos o conceito de sequência convergente de permutações e provamos a existência de um objeto limite para tais sequências. Introduzimos ainda um novo modelo de permutação aleatória baseado em tais objetos e introduzimos um conceito novo de distância entre permutações. Provamos então que sequências de permutações aleatórias são convergentes e provamos a equivalência entre esta noção de convergência e convergência nesta nova distância. Obtemos ainda resultados de amostragem e quase-aleatoriedade para permutações. Provamos também uma caracterização para parâmetros testáveis de permutações. / We introduce the concept of convergent sequence of permutations and we prove the existence of a limit object for these sequences. We also introduce a new and more general model of random permutation based on these limit objects and we introduce a new metric for permutations. We also prove that sequences of random permutations are convergent and we prove the equivalence between this notion of convergence and convergence in this new metric. We also show some applications for samplig and quasirandomness. We also prove a characterization for testable parameters of permutations.

Esta tese trata das propriedades estruturais e do controle de sistemas lineares singulares sujeitos a saltos Markovianos (SLSSM). Três questões fundamentais são consideradas para esta classe de sistemas. A primeira estabelece condições necessárias para que o sistema seja estocasticamente regular em um período de tempo determinado. A segunda trata da estabilidade exponencial estocástica de SLSSM. Equações de Lyapunov acopladas generalizadas são deduzidas para caracterizar estabilidade deste tipo de sistema. Em virtude da complexidade das soluções numéricas dessas equações, cada equação de Lyapunov do conjunto acoplado está em função de duas variáveis desconhecidas, estamos propondo um algoritmo para resolver este problema. A terceira questão diz respeito à síntese de um regulador para este tipo de sistema singular definida em termos de equações algébricas generalizadas de Riccati acopladas. / This thesis deals with the structural features and with the control of singular linear systems with Markovian jump parameters (SLSMJP). Three fundamental questions are considered to this class of systems. The first provides necessary conditions to characterize stochastic regularity in a determined period of time. The second deals with exponential stability of SLSMJP. Coupled generalized Lyapunov Equations are deduced to check the stability of this class of systems. In virtue of the complexity of the numerical solutions of these equations, there exist two unknown variables for each equation of the set of coupled Lyapunov equations, we are proposing an algorithm to solve this problem. The third question is related with the synthesis of a regulator for this class of singular systems defined in terms of coupled algebraic generalized Riccati equations.

Desenvolvemos novos resultados da teoria dos espaços FS e DFS (espaços de Fréchet-Schwartz e seus duais) e os empregamos ao estudo da seguinte questão: quando certas propriedades de regularidade de um operador diferencial parcial linear (entre fibrados vetoriais Gevrey sobre uma variedade Gevrey) implicam resolubilidade, no sentido de ultradistribuições, do operador transposto? Estudamos esta questão para uma classe de operadores abstratos que contém os operadores diferenciais parciais lineares com coeficientes Gevrey usuais, mas também certas classes de operadores pseudo-diferenciais em variedades compactas, além de certos tipos de operadores de ordem infinita. Neste contexto, obtemos uma nova demonstração de um resultado global em variedades compactas (em que hipoelipticidade Gevrey global de um operador implica resolubilidade global de seu transposto), assim como alguns resultados no caso não-compacto relacionados à propriedade de não-confinamento de singularidades. Na sequência apresentamos algumas aplicações concretas, em particular para operadores de Hörmander, operadores de força constante e sistemas localmente integráveis de campos vetoriais. Analisamos ainda algumas instâncias de uma conjectura levantada em um artigo recente de F. Malaspina e F. Nicola (2014), a qual afirma que, para certos complexos diferenciais naturalmente associados a estruturas localmente integráveis, resolubilidade local no sentido de ultradistribuições (perto de um ponto, em um grau fixado) implica resolubilidade local no sentido de distribuições. Estabelecemos a validade desta conjectura quando o fibrado estrutural cotangente é gerado pelo diferencial de uma única integral primeira. / We develop new techniques in the setting of FS and DFS spaces (Fréchet-Schwartz spaces and their strong duals) and apply them to the study of the following question: when regularity properties of a general linear differential operator (between Gevrey vector bundles over a Gevrey manifold) imply solvability of its transpose in the sense of ultradistributions? This question is studied for a class of abstract operators that encompasses the usual partial differential operators with Gevrey coefficients, but also some flavors of pseudodifferential operators on compact manifolds and some classes of operators with infinite order. In this setting, we obtain a new proof of a global result on compact manifolds (global Gevrey hypoellipticity of the operator implying global solvability of the transpose), as well as some results in the non-compact case by means of the so-called property of non-confinement of singularities. We then move to some concrete applications, especially for Hörmander operators, operators of constant strength and locally integrable systems of vector fields. We also analyze some instances of a conjecture stated in a recent paper of F. Malaspina and F. Nicola (2014), which asserts that, in differential complexes naturally arising from locally integrable structures, local solvability in the sense of ultradistributions (near a point, in some fixed degree) implies local solvability in the sense of distributions. We establish the validity of the conjecture when the cotangent structure bundle is spanned by the differential of a single first integral.

Seja (R;m) um anel Noetheriano local e R \'CONTÉM\' \'iota IND. 1\' \'CONTÉM\' \'iota IND. 2\' \'CONTÉM ... uma filtração de ideais de R. Podemos então construir a álgebra graduada F(\'\\Im) := \'SOMA DIRETA IND. n > OU = 0 POT. \'iota IND. n / \'m \'iota IND. n\', chamada de fibra especial. Esta tese objetiva a pesquisa deste anel. Investigamos sobre a sua propriedade de ser Gorenstein e a sua regularidade de Castelnuovo-Mumford. Outro objetivo, é generalizarmos o teorema de Risler-Teissier (sobre multiplicidades mistas) para o caso de filtrações de Hilbert / Let (R;m) be a Noetherian local ring and R \'CONTAINS\' \'iota IND. 1\' \'CONTAINS\' \'iota IND. 2\' \'CONTAINS\' ... a filtration of ideals in R. We may then construct the graded algebra F(\\Im) := \'DIRECT SUM\' IND. n > OR = \'0 POT. \'iota\' IND. n / \'m \'iota IND. n\' , which is called fiber cone. This thesis has the goal to research about this graded ring. We investigate its Gorenstein property and its Castelnuovo-Mumford regularity. Another aim is to generalize the Risler-Teissiers theorem (about mixed multiplicities) for the case of Hilbert filtration

Cette thèse est consacrée à l’étude du système de Boussinesq stationnaire:-νΔu+(u⋅∇)u+∇π=θg, div u=0,dans Ω(1a)-κΔθ+u⋅∇θ=h,dans Ω (1b)où Ω⊂R^3 est un ouvert, borné et connexe; les inconnues du système sont u,π et θ: la vitesse, la pression et la température du fluide, respectivement; ν>0 est la viscosité cinématique du fluide, κ>0 est la diffusivité thermique du fluide, g est l’accélération de la pesanteur et h est une source de chaleur appliquée au fluide.L’objectif de cette thèse est l’étude de la théorie L^p pour le système de Boussinesq en considérant deux différents types de conditions aux limites du champ de vitesse. En effet, dans une première partie, nous considérons une condition de Dirichlet non homogèneu=u_b, sur Γ (2)où Γ désigne la frontière du domaine. Dans une deuxième partie, nous considérons une condition de Navier non homogèneu⋅n=0,2[D(u)n]_τ+αu_τ=a,sur Γ(3)où D(u)=1/2 (∇u+(∇u)^T ) est le tenseur de déformation associé au champ de vitesse u, n est le vecteur normal unitaire extérieur, τ est le correspondant vecteur tangent unitaire, α et a sont une fonction scalaire de friction et un champ de vecteur tangentiel donnés sur la frontière, respectivement. De plus, la condition aux limites pour la température sera, dans les deux premières parties, une condition aux limites de Dirichlet non homogèneθ=θ_b, sur Γ. (4)Alors, premièrement, nous étudions l’existence et l’unicité d’une solution faible pour le problème (1), (2) et (4) dans le cas hilbertien. Également, l’existence de solutions généralisées pour p≥3/2 et des solutions fortes pour 1<p<∞ est démontrée. De plus, l’existence et l’unicité de la solution très faible sont étudiées. Il est intéressant de noter que puisque une condition de Dirichlet non homogène est considérée pour le champ de vitesse, le fait que la frontière du domaine pourrait être non-connexe joue un rôle fondamental puisque cela apparait de manière explicite dans les hypothèses des principaux résultats.D’autre part, dans la deuxième partie, nous étudions l’existence de solutions faibles dans le cas hilbertien, ainsi que l’existence de solutions généralisées pour p>2 et des solutions fortes pour p≥6/5 pour le problème (1), (3) et (4). Notez que l’hypothèse d’une frontière non-connexe, mentionnée précédemment, ne figurait pas dans cette partie du travail en raison de la restriction d’imperméabilité de la frontière.Enfin, dans la dernière partie de cette thèse, nous étudions la théorie L^p pour les équations de Stokes avec la condition de Navier (3). Plus précisément, nous examinons la régularité W^(1,p) pour p≥2 et la régularité W^(2,p) pour p≥6/5.Mots clés: système de Boussinesq; régularité L^p; solutions faibles; solutions fortes; solutions très faibles / This thesis is dedicated to the study of the stationary Boussinesq system:-νΔu+(u⋅∇)u+∇π=θg, div u=0,in Ω(1a)-κΔθ+u⋅∇θ=h,in Ω (1b)where Ω⊂R^3 is an open bounded connected set; u,π and θ are the velocity field, pressure and temperature of the fluid, respectively, and stand for the unknowns of the system; ν>0 is the kinematic viscosity of the fluid, κ>0 is the thermal diffusivity of the fluid, g is the gravitational acceleration and h is a heat source applied to the fluid.The aim of this thesis is the study of the L^p-theory for the stationary Boussinesq system in the context of two different types of boundary conditions for the velocity field. Indeed, in the first part of the thesis, we will consider a non-homogeneous Dirichlet boundary conditionu=u_b, on Γ (2)where Γ denotes the boundary of the domain; meanwhile in the second part, the velocity field will be prescribed through a non-homogeneous Navier boundary conditionu⋅n=0,2[D(u)n]_τ+αu_τ=a,on Γ(3)where D(u)=1/2 (∇u+(∇u)^T ) is the strain tensor associated with the velocity field u, n is the unit outward normal vector, τ is the corresponding unit tangent vector, α and a are a friction scalar function and a tangential vector field defined both on the boundary, respectively. Further, the boundary condition for the temperature will be, in the first two parts of the thesis, a non-homogeneous Dirichlet boundary conditionθ=θ_b, on Γ. (4)Then, firstly, we study the existence and uniqueness of the weak solution for the problem (1), (2) and (4) in the hilbertian case. Also, the existence of generalized solutions for p≥3/2 and strong solutions for 1<p<∞ is showed. Furthermore, the existence and uniqueness of the very weak solution is studied. It is worth to note that because a non-homogeneous Dirichlet boundary condition is considered for the velocity field, the fact that the boundary of the domain could be non-connected plays a fundamental role since it appears in an explicit way in the assumptions of some of the main results.In the second part, we study the existence of weak solutions in the hilbertian case, as well as the existence of generalized solutions for p>2 and strong solutions for p≥6/5 for the problem (1), (3) and (4). Note that the assumption of a non-connected boundary, which was mentioned before, will not appear here due to the impermeability restriction on the boundary.Finally, in the last part of this thesis, we study the L^p-theory for the Stokes equations with Navier boundary condition (3). Specifically, we deal with the W^(1,p)-regularity for p≥2 and the W^(2,p)-regularity for p≥6/5.Keywords: Boussinesq system; L^p-regularity; weak solutions; strong solutions; very weak solutions

Dans ma thèse, j’ai travaillé sur les modèles de fils en ferromagnétisme. J’ai obtenu les résultats suivants :- Existence de solutions très régulières pour les équations de Landau-Lifschitz en dimension 3.- Stabilité de profils de murs avec critère optimal de stabilité pour un fil soumis à un champ magnétique.- Stabilité de profils de murs pour un fil soumis à un courant électrique, dans le cas d’un fil à section circulaire et dans le cas d’un fil à section ellipsoïdale. - Justification des modèles monodimensionnels de fils. / In my thesis, I worked on models of wires in ferromagnetism. I got the following results:- Existence of very regular solutions for Landau-Lifschitz equations in dimension 3.- Optimal stability criterion for a wall in a ferromagnetic wire in a magnetic field.-Stability of walls in a ferromagnetic wire subjected to an electric current, in the case of a round wire and in the case of an ellipsoidal cross-section wire.- Justification of one-dimensional wires models.

First, we provide new classes of initial data, that grant short time uniqueness of the associated weak Leray-Hopf solutions of the three dimensional Navier-Stokes equations. The main novelty here is the establishment of certain continuity properties near the initial time, for weak Leray-Hopf solutions with initial data in supercritical Besov spaces. The techniques used here build upon related ideas of Calder&oacute;n. Secondly, we prove local regularity up to the at part of the boundary, for certain classes of solutions to the Navier-Stokes equations, provided that the velocity field belongs to L_&infin;(-1; 0; L^{3, &beta;}(B(1) &xcap; &Ropf;³ ₊)) with 3 &le; &beta; &lt; &infin;. What enables us to build upon the work of Escauriaza, Seregin and &Scaron;ver&aacute;k [27] and Seregin [100] is the establishment of new scale-invariant estimates, new estimates for the pressure near the boundary and a convenient new &epsiv;-regularity criterion. Third, we show that if a weak Leray-Hopf solution in &Ropf;³ ₊&times;]0,&infin;[ has a finite blow-up time T, then necessarily lim_t&uarr;T&verbar;&verbar;v(&middot;, t)&verbar;&verbar;_{L^3,&beta;(&Ropf;³ ₊)} = &infin; with 3 &lt; &beta; &lt; &infin;. The proof hinges on a rescaling procedure from Seregin's work [106], a new stability result for singular points on the boundary, suitable a priori estimates and a Liouville type theorem for parabolic operators developed by Escauriaza, Seregin and &Scaron;ver&aacute;k [27]. Finally, we investigate a notion of global-in-time solutions to the Navier- Stokes equations in &Ropf;³, with solenoidal initial data in the critical Besov space ?^-1/4_4,&infin;(&Ropf;³), which has certain continuity properties with respect to weak&ast; convergence of the initial data. Such properties are motivated by the strategy used by Seregin [106] to show that if a weak Leray-Hopf solution in &Ropf;³&times;]0,&infin;[ has a finite blow-up time T, then necessarily lim_t&uarr;T &verbar;&verbar;v(&middot;, t)&verbar;&verbar;_{L₃(&Ropf;³)} = &infin;. We prove new decomposition results for Besov spaces, which are key in the conception and existence theory of such solutions.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste presente trabalho, faremos o estudo de importantes propriedades geomÃtricas e analÃticas de soluÃÃes de equaÃÃes diferenciais parciais elÃpticas totalmente nÃo-lineares do tipo: singulares e degeneradas. O estudo de processos de combustÃo que se degeneram ao longo do conjunto de anulamento da densidade de um gÃs, um caso particular de problemas do tipo "quenching", apresentam em sua modelagem equaÃÃes singulares que estÃo descritas neste trabalho. Nesta primeira parte iremos obter propriedades de uma soluÃÃo minimal, que vÃo desde o controle completo Ãtimo, atà a obtenÃÃo de estimativas de Hausdorff da fronteira livre singular. Por fim, iremos obter a regularidade Ãtima de soluÃÃes de equaÃÃes em que suas propriedades de difusÃo(elipticidade) se deterioram na ordem de uma potÃncia do seu gradiente ao longo do conjunto em que tal taxa de variaÃÃo se anula. / In this work we study important geometric and analytic properties to solutions of fully nonlinear elliptic partial differential equations, both singular and degenerate types. The study of combustion processes that degenerate along the null-set of the density of a gas, a particular case of quenching problems, present in their modeling, equations described in this work. In this first part we obtain properties of a minimal solution, since the complete optimal control until the Hausdorff estimates of the singular free boundary. Ultimately, we obtain the optimal regularity to equation solutions where their diffusion property (elipticity) deterorate in a power of their gradient along the set where such rate of variation nullifies.