[pt] Nesta tese estudamos as propriedades físicas de materiais fortemente correlacionados e desordenados, usando Hamiltonianos modelos para descrevê-los. A tese está dividida em duas partes. Na primeira, estudamos o modelo de Anderson periódico para descrever as propriedades de um isolante Kondo. Em particular tomamos o composto de Ce3Bi4Pt3 como paradigma deste tipo de materiais caracterizados por apresentar um pequeno gap(da ordem dos meV ). Na presença de pequenas concentrações de impurezas metálicas como íons de La substituindo os de Ce, como é o caso da liga (Ce1-xLax)Bi4Pt3, sofre uma transição metal-isolante. O Hamiltoniano de Anderson periódico é resolvido a partir da solução de um único sítio atômico que logo é embebido numa rede de Bethe. Este modelo consegue explicar qualitativamente os resultados experimentais como a resistividade em função da temperatura para diferentes concentrações de íons de La, assim como as propriedades óticas do sistema puro. A influência da localização de Anderson nesta transição é analisada a partir do estudo da condutividade elétrica do sistema. A segunda parte está dedicada ao estudo das propriedades de sistemas descritos pelo Hamiltoniano de Falicov- Kimball, largamente utilizado para estudar fenômenos como a transição de valência e metal- isolante, também em compostos de Metais de Transição e Terras Raras. Neste modelo, o caráter destas transições ainda não está bem estabelecido já que o resultado é muito dependente da aproximação utilizada. Utilizamos o Hamiltoniano de Falicov-Kimball sem spin onde a banda de condução é tratada de forma exata já que mostramos a sua equivalência com o problema de uma liga. Os estados f são resolvidos em forma aproximada a partir da equação de movimento, aproximação que chamamos de Aproximação do Estreitamento Dinâmico(AED). Estudamos as propriedades eletrônicas como a ocupação dos estados localizados em função da energia local. Também neste caso, analisamos um sistema desordenado estudando o contraponto entre a correlação eletrônica e a desordem. As diferentes fases que aparecem no sistema como, metálica, isolante de Anderson e de Mott são investigadas em função dos parâmetros que definem o sistema. / [en] In this thesis we study the properties of strongly correlated and disordered materials, using model Hamiltonians to describe them. The thesis is divided in two parts. The first one studies the periodic Anderson model used to describe the properties of a Kondo insulator. In particular we take Ce3Bi4Pt3 as a paradigmatic compound, characterized by a small gap(of the order of meV ). For small concentration of metallic impurities, ions of La substituting Ce, the alloy (Ce1-xLax)Bi4Pt3 suffers a metal- insulator transition. The periodic Anderson Hamiltonian is solved using the atomic solution that is embedded into a Bethe lattice. This model explains the experimental results as the resistivity as a function of temperature for different concentrations of ions of La, as well as, the optical properties of the pure system. The Anderson localization is analyzed studying the electric conductivity of the system. The second part of the thesis is dedicated to study the property of a system described by the Falicov- Kimball Hamiltonian. This Hamiltonian has been used to study the valence and metal-insulator transitions in Transitions Metal and Rare Earth compounds. In this model, the character of these transitions is still not well understood, since it is very dependent of the approximation used. We study the Falicov-Kimball Hamiltonian without spin. The conduction band is exactly described since we show its equivalence with the problem of an alloy. The f states are studied using the equation of motion for the Green functions, decoupling them in a way defined as the Dynamic Narrowing Approximation(DNA). We study the occupation of the local states as a function of energy and other electronic properties. For an alloy the interplay between the electronic correlation and disorder is analized. The different phases that appear in the system, as metallic and Anderson and Mott insulating, are investigated as a function of the parameters that define the system.

La réduction fluide des équations cinétiques est un procédé couramment utilisé en physique des plasmas qui a pour objectif de remplacer la fonction de distribution définie dans l'espace des phases par des grandeurs fluides comme la densité et la pression. Cette réduction diminue la complexité du système initial. En contrepartie, la réduction fluide s'accompagne de la nécessité d'effectuer une fermeture sur les moments d'ordre supérieur. Celle-ci est souvent construite ad hoc en se basant sur des arguments physiques (e.g., quantités conservées, existance d'un théorème H, ...). Dans ce manuscrit, on propose un procédé de réduction qui permet de préserver la structure hamiltonienne du modèle cinétique parent. Ceci est important pour assurer qu'aucune dissipation d'origine non physique est introduite dans le modèle fluide, le munissant ainsi d'une structure hamiltonienne dont l'origine peut être suivie jusqu'à celle de la dynamique microscopique des particules. On utilise cette méthode pour construire des modèles fluides non-adiabatiques pour les trois premiers moments de la fonction de distribution associée à l'équation de Vlasov-Poisson à une dimension, i.e., la densité, la vitesse fluide et la pression. Les résultats sont ensuite étendus pour inclure la dynamique du flux de chaleur en considérant des fermetures construites à partir de l'analyse dimensionnelle. On montre également, pour un nombre arbitraire de champs, la relation existant avec le modèle water-bags. L'extension à des dimensions supérieures est étudiée dans le cadre de l'équation drift-cinétique ainsi que de l'équation de Vlasov-Poisson à trois dimensions. / Fluid reduction of kinetic equations is a ubiquitous procedure in plasma physics which aims to replace the distribution function defined in phase space with more concrete fluid quantities defined solely in configuration space such as the density, the fluid velocity and the pressure. This reduction lowers the complexity of the initial system, leading to a gain of physical insight into the phenomena under investigation as well as a significant decrease of the cost of numerical simulations. On the other hand, in order for the fluid reduction to be complete, one needs to perform a closure on the higher order fluid moments. The choice of the closure usually relies on some ad hoc physical arguments (e.g., conserved quantities, existence of an H-theorem, ...). In this manuscript, we present a reduction procedure that preserves the Hamiltonian structure of the parent kinetic model. This is important in order to ensure that no non-physical dissipation is introduced in the resulting fluid model, providing it with a geometric structure that can be traced back to the microscopic dynamics of the particles. We use this procedure to derive non-adiabatic fluid models for the first three fluid moments of the distribution function of the one dimensional Vlasov-Poisson equation, namely the density, the fluid velocity and the pressure. The results are extended to include the dynamics of the heat-flux by considering a closure based on dimensional analysis. For an arbitrary number of fields, we demonstrate the relationship with the water-bags model. Finally, the extension to higher dimensions is investigated through the drift-kinetic equation and the three dimensional Vlasov-Poisson equation.

This study employs Bayesian Hidden Markov Models as method to explore vehicle traffic flow conditions in an urban area in Stockholm, based on sensor data from separate road positions. Inter-arrival times are used as the observed sequences. These sequences of inter-arrival times are assumed to be generated from the distributions of four different (and hidden) traffic flow states; nightly free flow, free flow, mixture and congestion. The filtered and smoothed probability distributions of the hidden states and the most probable state sequences are obtained by using the forward, forward-backward and Viterbi algorithms. The No-U-Turn sampler is used to sample from the posterior distributions of all unknown parameters. The obtained results show in a satisfactory way that the Hidden Markov Models can detect different traffic flow conditions. Some of the models have problems with divergence, but the obtained results from those models still show satisfactory results. In fact, two of the models that converged seemed to overestimate the presence of congested traffic and all the models that not converged seem to do adequate estimations of the probability of being in a congested state. Since the interest of this study lies in estimating the current traffic flow condition, and not in doing parameter inference, the model choice of Bayesian Hidden Markov Models is satisfactory. Due to the unsupervised nature of the problematization of this study, it is difficult to evaluate the accuracy of the results. However, a model with simulated data and known states was also implemented, which resulted in a high classification accuracy. This indicates that the choice of Hidden Markov Models is a good model choice for estimating traffic flow conditions.

In this work we study the existence of nontrivial weak solutions for some Hamiltonian elliptic systems in dimension two, involving a potential function and nonlinearities which possess maximal growth with respect to a critical curve (hyperbola). We consider four different cases. First, we study Hamiltonian systems in bounded domains with potential function identically zero. The second case deals with systems of equations on the whole space, the potential function is bounded from below for some positive constant and satisfies some integrability conditions, while the nonlinearities involve weight functions containing a singulatity at the origin. In the third case, we consider systems with coercivity potential functions and nonlinearities with weight functions which may have singularity at the origin or decay at infinity. In the last case, we study Hamiltonian systems, where the potential can be unbounded or can vanish at infinity. To establish the existence of solutions, we use variational methods combined with Trudinger-Moser type inequalities for Lorentz-Sobolev spaces and a finite-dimensional approximation. / Neste trabalho estudamos a existência de soluções fracas não triviais para sistemas hamiltonianos do tipo elíptico, em dimensão dois, envolvendo uma função potencial e não linearidades tendo crescimento exponencial máximo com respeito a uma curva (hipérbole) crítica. Consideramos quatro casos diferentes. Primeiramente estudamos sistemas de equações em domínios limitados com potencial nulo. No segundo caso, consideramos sistemas de equações em domínio ilimitado, sendo a função potencial limitada inferiormente por alguma constante positiva e satisfazendo algumas de integrabilidade, enquanto as não linearidades contêm funções-peso tendo uma singularidade na origem. A classe seguinte envolve potenciais coercivos e não linearidades com funções peso que podem ter singularidade na origem ou decaimento no infinito. O quarto caso é dedicado ao estudo de sistemas em que o potencial pode ser ilimitado ou decair a zero no infinito. Para estabelecer a existência de soluções, utilizamos métodos variacionais combinados com desigualdades do tipo Trudinger-Moser em espaços de Lorentz-Sobolev e a técnica de aproximação em dimensão finita.

Cette thèse est motivée par un problème aéronautique: le ballottement du carburantdans des réservoirs d’ailes d’avion très flexibles. Les vibrations induites par le couplagedu fluide avec la structure peuvent conduire à des problèmes tels que l’inconfort des passagers,une manoeuvrabilité réduite, voire même provoquer un comportement instable. Cette thèse apour objectif de développer de nouveaux modèles d’interaction fluide-structure, en mettant enoeuvre la théorie des systèmes Hamiltoniens à ports d’interaction (pHs). Le formalisme pHsfournit d’une part un cadre unifié pour la description des systèmes multi-physiques complexeset d’autre part une approche modulaire pour l’interconnexion des sous-systèmes grâce auxports d’interaction. Cette thèse s’intéresse aussi à la conception de contrôleurs à partir desmodèles pHs. Des modèles pHs sont proposés pour les équations de ballottement du liquide en partantdes équations de Saint Venant en 1D et 2D. L’originalité du travail est de donner des modèlespHs pour le ballottement dans des réservoirs en mouvement. Les ports d’interaction sont utiliséspour coupler la dynamique du ballottement à la dynamique d’une poutre contrôlée par desactionneurs piézo-électriques, celle-ci étant préalablement modélisée sous forme pHs. Aprèsl’écriture des équations aux dérivées partielles dans le formalisme pHs, une approximation endimension finie est obtenue en utilisant une méthode pseudo-spectrale géométrique qui conservela structure pHs du modèle continu au niveau discret. La thèse propose plusieurs extensionsde la méthode pseudo-spectrale géométrique, permettant la discrétisation des systèmesavec des opérateurs différentiels du second ordre d’une part et avec un opérateur d’entrée nonborné d’autre part. Des essais expérimentaux ont été effectués sur une structure constituéed’une poutre liée à un réservoir afin d’assurer la validité du modèle pHs du ballottementdu liquide couplé à la poutre flexible, et de valider la méthode pseudo-spectrale de semi-discrétisation.Le modèle pHs a finalement été utilisé pour concevoir un contrôleur basé surla passivité pour réduire les vibrations du système couplé. / This thesis is motivated by an aeronautical issue: the fuel sloshing in tanksof very flexible wings. The vibrations due to these coupled phenomena can lead to problemslike reduced passenger comfort and maneuverability, and even unstable behavior. Thisthesis aims at developing new models of fluid-structure interaction based on the theory ofport-Hamiltonian systems (pHs). The pHs formalism provides a unified framework for thedescription of complex multi-physics systems and a modular approach for the coupling ofsubsystems thanks to interconnection ports. Furthermore, the design of controllers using pHsmodels is also addressed. PHs models are proposed for the equations of liquid sloshing based on 1D and 2D SaintVenant equations and for the equations of structural dynamics. The originality of the workis to give pHs models of sloshing in moving containers. The interconnection ports are used tocouple the sloshing dynamics to the structural dynamics of a beam controlled by piezoelectricactuators. After writing the partial differential equations of the coupled system using thepHs formalism, a finite-dimensional approximation is obtained by using a geometric pseudospectralmethod that preserves the pHs structure of the infinite-dimensional model at thediscrete level. The thesis proposes several extensions of the geometric pseudo-spectral method,allowing the discretization of systems with second-order differential operators and with anunbounded input operator. Experimental tests on a structure made of a beam connected to atank were carried out to validate both the pHs model of liquid sloshing in moving containersand the pseudo-spectral semi-discretization method. The pHs model was finally used to designa passivity-based controller for reducing the vibrations of the coupled system.

Orientador: Miriam Harumi Tsunemi / Resumo: Em inúmeras situações práticas a relação entre uma variável resposta e uma ou mais covariáveis é curvada. Dentre as diversas formas de representar esta curvatura, Royston e Altman (1994) propuseram uma extensa famı́lia de funções denominada de Polinômios Fracionários (Fractional Polynomials - FP ). Bové e Held (2011) im- plementaram o paradigma bayesiano para FP sob a suposição de normalidade dos erros. Sua metodologia é fundamentada em uma distribuição a priori hiper − g (Liang et al., 2008), que, além de muitas propriedades assintóticas interessantes, garante uma predição bayesiana de modelos consistente. Nesta tese, compara-se as abordagens clássica e Bayesiana para PF a partir de dados reais disponı́veis na litera- tura, bem como por simulações. Além disso, propõem-se uma abordagem Bayesiana para modelos FPs em que a potência, diferentemente dos métodos usuais, pode as- sumir qualquer valor num determinado intervalo real e é estimada via métodos de simulação HMC (Monte Carlo Hamiltoniano) e MCMC (Métodos de Monte Carlo via Cadeias de Markov). Neste modelo, para o caso de um FP de segunda ordem, ao contrário dos modelos atualmente disponı́veis, apenas uma potência é estimada. Avalia-se este modelo a partir de dados simulados e em dados reais, sendo um deles com transformação de Box-Cox. / Abstract: In many practical situations the relationship between the response variable and one or more covariates is curved. Among the various ways of representing this curvature, Royston and Altman (1994) proposed an extended family of functions called Fractional Polynomials (FP). Bov´e and Held (2011) implemented the Bayesian paradigm for FP on the assumption of error normality. Their methodology is based on a hyperg prior distribution, which, in addition to many interesting asymptotic properties, guarantees a consistent Bayesian model average (BMA). In addition, a Bayesian approach is proposed for FPs models in which power, unlike the usual methods, can obtain any numerical real interval value and is estimated via HMC (Monte Carlo Hamiltonian) and MCMC (Markov chain Monte Carlo). In this model, in the case of a second-order FP, unlike the currently available models, only one power is estimated. This model is evaluated from simulated data and real data, one of them with Box-Cox transformation. / Doutor

Neste trabalho, mostramos que fluxos de Reeb dinamicamente convexos em um espaço lenticular $L(p, 1)$, $p>1$, admite uma órbita periódica de Reeb especial $P$ que é o binding de uma decomposição em livro aberto racional, com páginas tipo-disco tal que cada página é uma seção global. O índice de Conley-Zehnder da $p$-ésima iterada de $P$ é $3$. Como corolário, o fluxo de Reeb possui duas ou infinitas órbitas periódicas. Este resultado aplica-se ao Hamiltoniano de Hénon-Heiles, cujo fluxo restrito a energia baixa possui $Z_3$-simetria e define um fluxo de Reeb em $L(3, 1)$. Devido a $Z_4$-simetria aplicamos nosso resultado ao problema lunar de Hill regularizado. Na segunda parte deste trabalho investigamos a existência de uma folheação $3-2^3$ em níveis de energia no sistema Hamiltoniano de Hénon-Heiles, para energia logo acima da crítica. Provamos que certa região de interesse é uma hipersuperfície de contato. Provamos também que o fluxo de Reeb possui uma órbita periódica $Z_3$ simétrica, cujo índice de Conley-Zehnder é $3$ e possui número de auto-enlaçamento $-1$. / We show that a dynamically convex Reeb flow on a lens space $L(p, 1)$, $p>1$ admits a special closed Reeb orbit $P$ which is the binding of a rational open book decomposition with disk-like pages so that each page is a global surface of section. The Conley-Zehnder index of the $p$-th iterate of $P$ is $3$. As a corollary, the Reeb flow has $2$ or infinitely many closed Reeb orbits. This result applies to the Hénon-Heiles Hamiltonian whose flow restricted to low energy levels has $Z_3$-symmetry and descends to $L(3,1)$. Due to a $Z_4$-symmetry we also apply our results to Hill\'s lunar problem. In the second part of this work we investigate the existence of a $3-2^3$ foliation on energy levels of the Hénon-Heiles Hamiltonian, for energies above the critical one. We show that some region is of contact-type and the Reeb flow has a $Z_3$-symmetric periodic orbit, whose Conley-Zehnder is $3$ and has self-linking number $-1$.

Cette thèse propose une analyse mathématique des modèles cinétiques en présence d'un champ magnétique intense.L'objectif de ce projet est le développement d'outils mathématiques nécessaires à la modélisation des plasmas de fusion. Les phénomènes physiques rencontrés dans les plasmas de fusion mettent en jeu des échelles caractéristiques disparates. L'interaction entre ces ordres de grandeurs est un enjeu important et requiert une analyse multi-échelle. Il s'agit d'un problème d'homogénéisation par rapport au mouvement rapide de rotation des particules autour des lignes de champ magnétique. Nous étudions le régime du rayon de Larmor fini pour le système de Vlasov-Poisson, dans le cadre de champs magnétiques uniformes, en appliquant les méthodes de gyro-moyenne. Nous donnons l'expression explicite du champ d'advection effectif de l'équation de Vlasov, dans laquelle nous avons substitué le champ électrique auto-cohérent, via la résolution de l'équation de Poisson moyennée à l'échelle cyclotronique. Nous mettons en évidence la structure hamiltonienne du modèle limite et présentons ses propriétés : conservations de la masse, de l'énergie cinétique, de l'énergie électrique, etc.Nous généralisons ensuite cette étude dans le cadre de champs magnétiques non uniformes. Comme précédemment, les principales propriétés des modèles limites sont mises en évidence : conservations de la masse, de l'énergie, structure hamiltonienne.Nous prenons en compte également les effets collisionnels, en présence d'un champ magnétique intense. Après identification des équilibres et invariants du noyau de collision moyenné, on s'intéresse à la dérivation de modèles fluides. / This thesis proposes a mathematical analysis of kinetic models in the presence of strong magnetic fields.The objective of this project is the development of mathematical tools required for modelisation of fusion plasmas. The physical phenomena encountered in fusion plasmas involve disparate characteristic scales. The interaction between these orders of magnitude is an important issue and requires a multi-scale analysis. We appeal to homogenization techniques with respect to the fast rotation motion around the magnetic field lines.We study the finite Larmor radius regime for the Vlasov-Poisson system, in the framework of uniform magnetic fields, by appealing to gyro-average methods. We indicate the explicit expression of the effective advection field entering the Vlasov equation, after substituting the self-consistent electric field, obtained by the resolution of the averaged (with respect to the cyclotronic time scale) Poisson equation. We emphasize the hamiltonian structure of the limit model and present its properties : conservation of mass, of kinetic energy, of electric energy, etc.Then we generalize this study to general magnetic shapes. As before, the main properties of the limit model are emphasized : mass and energy balances, hamiltonian structure.We also take into account the collisional effects, under strong magnetic fields. After identifying the equilibria and the invariants of the average collision operator, we inquire about fluid models.

We have fully classified the momentum polytopes of the SU(3) action on CP(2)xCP(2) and CP(2)xCP(2) xCP(2), both actions with weighted symplectic forms, and their corresponding transition momentum polytopes. For CP(2)xCP(2) the momentum polytopes are distinct line segments. The action on CP(2)xCP(2) xCP(2), has 9 different momentum polytopes. The vertices of the momentum polytopes of the SU(3) action on CP(2)xCP(2) xCP(2), fall into two categories: definite and indefinite vertices. The reduced space corresponding to momentum map image values at definite vertices is isomorphic to the 2-sphere. We show that these results can be applied to assess the dynamics by introducing and computing the space of allowed velocity vectors for the different configurations of two-vortex systems.

No description available.