L'équation de la chaleur est un phénomène macroscopique, émergeant après une limite d’échelle diffusive (en espace et en temps) d’un système d'oscillateurs couplés. Lorsque les interactions entre oscillateurs sont linéaires, l'énergie évolue de manière balistique, et la conductivité thermique est infinie. Certaines non-linéarités doivent donc apparaître au niveau microscopique, si l’on espère observer une diffusion normale. Pour apporter de l'ergodicité, on ajoute à la dynamique déterministe une perturbation stochastique qui conserve l'énergie. En premier lieu nous étudions la dynamique Hamiltonienne d'un système d'oscillateurs linéaires, perturbé par un bruit stochastique dégénéré conservatif. Ce dernier transforme à des temps aléatoires les vitesses en leurs opposées. On montre que l'évolution macroscopique du système est caractérisée par un système parabolique non-linéaire couplé pour les deux lois de conservation du modèle. Ensuite, nous supposons que les oscillateurs évoluent en environnement aléatoire. La perturbation stochastique est très dégénérée, et on prouve que le champ de fluctuations de l'énergie à l'équilibre converge vers un processus d'Ornstein-Uhlenbeck généralisé dirigé par l’équation de la chaleur.Il est désormais connu que les systèmes unidimensionnels présentent une diffusion anormale lorsque le moment total est conservé en plus de l'énergie. Dans une troisième partie, on considère deux perturbations, l'une préservant le moment, l'autre détruisant cette conservation. En faisant décroître l'intensité de la seconde perturbation, on observe une transition de phase entre un régime de diffusion normale et un régime de superdiffusion. / The heat equation is known to be a macroscopic phenomenon, emerging after a diffusive rescaling of space and time. In linear systems of interacting oscillators, the energy ballistically disperses and the thermal conductivity is infinite. Since the Fourier law is not valid for linear interactions, non-linearities in the microscopic dynamics are needed. In order to bring ergodicity to the system, we superpose a stochastic energy conserving perturbation to the underlying deterministic dynamics.In the first part we study the Hamiltonian dynamics of linear coupled oscillators, which are perturbed by a degenerate conservative stochastic noise. The latter flips the sign of the velocities at random times. The evolution yields two conservation laws (the energy and the length of the chain), and the macroscopic behavior is given by a non-linear parabolic system.Then, we suppose the harmonic oscillators to evolve in a random environment, in addition to be stochastically perturbed. The noise is very degenerate, and we prove a macroscopic behavior that holds at equilibrium: precisely, energy fluctuations at equilibrium evolve according to an infinite dimensional Ornstein-Uhlenbeck process driven by the linearized heat equation.Finally, anomalous behaviors have been observed for one-dimensional systems which preserve momentum in addition to the energy. In the third part, we consider two different perturbations, the first one preserving the momentum, and the second one destroying that new conservation law. When the intensity of the second noise is decreasing, we observe (in a suitable time scale) a phase transition between a regime of normal diffusion and a regime of super-diffusion.

Orientadores: Marco Antonio Teixeira, Ketty Abaroa de Rezende / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T14:10:31Z (GMT). No. of bitstreams: 1 Martins_RicardoMiranda_M.pdf: 921623 bytes, checksum: 8098f5c4875b6b586865b92ec6e474a0 (MD5) Previous issue date: 2008 / Resumo: A semelhança entre sistemas reversíveis e Hamiltonianos foi detectada nos primórdios do século passado por Birkhoff. Neste trabalho realizamos uma análise geométrica-qualitativa da dinâmica de um campo de vetores reversível em torno de um ponto de equilíbrio elíptico em R4. Especificamente, estudamos quando um campo reversível com tal tipo de equilíbrio é "equivalente" a um sistema Hamiltoniano. Como resultado, obtemos que tal sistema é Hamiltoniano, a menos de uma seqüência de mudanças de coordenadas e reescalonamentos do tempo. Prosseguindo a análise, impomos outra simetria ao campo e passamos a considerar sistemas bireversíveis. Classificamos completamente as possíveis simetrias que tornam um sistema bireversível por involuções gerando um grupo isomorfo a D4. Para tais sistemas, obtemos resultados um pouco mais fortes que os obtidos para sistemas reversíveis / Abstract: The similarity between reversible and Hamiltonian systems has been detected at the beginning of the past century by Birkhoff. In this project, we describe a geometrical-qualitative analysis of the dynamics of a reversible vector field around a elliptical singularity in R4. Specifically, we study when such a reversible vector field is "equivalent" to a Hamiltonian system. As a result, we obtain that such systems are always Hamiltonian, up to a sequence of changes of coordinates and time rescaling. Imposing another symmetry to the vector field, we work with bireversible systems. We completely classify all the possible symmetries which makes such systems bireversible by involutions generating a group isomorphic to D4. For these systems, we have obtained stronger results than in the reversible case / Mestrado / Sistemas Dinamicos / Mestre em Matemática

Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T14:36:33Z (GMT). No. of bitstreams: 1 Martins_RicardoMiranda_D.pdf: 1398633 bytes, checksum: 5ac0dfa4d175b6407f0b574a1aefd3e5 (MD5) Previous issue date: 2011 / Resumo: Na primeira parte desta tese, estudamos a semelhança entre sistemas dinâmicos reversíveis e Hamiltonianos, sob um ponto de vista formal. Nos restringimos a sistemas definidos ao redor de pontos de equilíbrio simples e simétricos. Mostramos que, sob algumas hipóteses, tais sistemas são formalmente orbitalmente equivalentes. Na segunda parte, estudamos a existência de conjuntos minimais em certas famílias de equações diferenciais. Especificamente, exibimos condições sob as quais existem cilindros e toros invariantes para sistemas de equações que são perturbações de sistemas reversíveis. / Abstract: In the first part of this thesis, we study the similarity between reversible and Hamiltonian dynamical systems, from a formal viewpoint. We restrict ourselves to systems defined around an isolated and symmetric equilibria. We show that, under some conditions, such systems are formally orbitally equivalent to Hamiltonian vector fields. In the second part, we study the existence of minimal sets for some families of diferential equations. We obtain conditions for the existence of the invariant cylinders and tori for perturbed reversible systems. / Doutorado / Sistemas Dinamicos / Doutor em Matemática

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.

M.Sc. / In this thesis we study the techniques used to calculate the Hamilton operators related to linear optical quantum computing. We also discuss the basic building blocks of linear optical quantum computing (LOQC) by looking at the logic gates and the physical instruments of which they are made.

In order to reduce the cost of the current switched resonant power converters, a state observer is introduced to provide the controller with the required information for the effective switching of the converter without the need of expensive sensory inputs from the resonant circuit. This Master's thesis investigates the usage of Matlab/Simulink and Xilinx System Generator as an implementation tool for Field Programmable Gate Array (FPGA) development of the Observer Model. The main aim is to keep the model size to a minimum while keeping the error within a reasonable range, allowing the observer to converge.

Cette thèse étudie la problématique du fonctionnement sans capteur et à basse vitesse des moteurs synchrones à aimant permanent par l'injection des signaux. Nous nous focalisons sur les effets de la saturation magnétique car leur compensation est primordiale pour résoudre cette problématique. Nous proposons une méthode originale pour modéliser la saturation magnétique en utilisant une approche énergétique (les formulations Lagrangienne et Hamiltonienne), où les symétries physiques sont exploitées pour simplifier l'expression de l'énergie magnétique. Les données expérimentales montrent qu'un polynôme de degré 4 est suffisant pour décrire avec précision les effets de la saturation. Ensuite, nous proposons une analyse claire et originale basée sur la moyennisation de second ordre et qui explique comment obtenir l'information de position à partir de l'injection des signaux (en utilisant le modèle proposé). Nous donnons une relation explicite entre les oscillations des courants statoriques et la position du rotor; cette relation est utilisée en temps réel. Ce modèle de saturation magnétique ainsi que la procédure d'estimation de position ont été testés et validés sur deux types de moteurs : avec des aimants permanents à l'intérieur ou sur la surface du rotor. Les résultats expérimentaux obtenus sur un banc de test montrent que les erreurs d'estimation de la position du rotor n'excèdent pas quelques degrés électriques dans la zone d'opération à basse vitesse. / This thesis addresses the problematic of sensorless low speed operation of permanent magnet synchronous motors (PMSM) by signal injection. We focus on the effects of magnetic and cross saturations because their compensation is paramount to solve this problematic. We propose an original way of modeling magnetic saturation using an energy approach (Lagrangian and Hamiltonian formulations), where the physical symmetries are exploited to simplify the expression of the magnetic energy. Experimental data show that a simple polynomial of degree 4 is sufficient to describe accurately magnetic saturation effects. Then we propose a clear and original analysis based on second-order averaging of how to recover the position information from signal injection (using the proposed model). We give an explicit relation between stator current ripples and rotor position; this relation is used in real time operation. Such magnetic saturation model and the resulting position estimates were tested and validated on two types of motors: with interior and surface permanent magnets (IPM and SPM). Experimental results obtained on a test bench show that estimation errors of the rotor position do not exceed few electrical degrees in the low speed operating domain.

A restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem is presented. The Lanczos vectors are constructed to form a symplectic basis. Breakdowns and near-breakdowns are overcome by inexpensive implicit restarts. The method is used to compute eigenvalues, eigenvectors and invariant subspaces of large and sparse Hamiltonian matrices and low rank approximations to the solution of continuous-time algebraic Riccati equations with large and sparse coefficient matrices.

L’objectif de ce travail est de proposer de nouvelles stratégies de commande non linéaire pour la stabilisation des Réacteurs Parfaitement Agités Continus (RPAC). Pour cela, nous utilisons d’une part, l’approche thermodynamique entropique. Plus précisément, nous utilisons la notion de disponibilité thermodynamique et les propriétés de la thermodynamique irréversible pour définir une fonction de Lyapunov utilisable pour la stabilisation du système en boucle fermée. Nous proposons aussi une fonction disponibilité réduite afin d’obtenir des lois stabilisantes plus performantes en terme de sollicitation des actionneurs. D’autre part, nous proposons une extension du formalisme (pseudo) hamiltonien à ports dissipatifs aux réacteurs chimiques ouverts. Nous montrons que l’Hamiltonien est lié à l’enthalpie libre de Gibbs dans le cas isotherme et à l’ectropie (opposée de l’entropie) dans le cas non isotherme. Par ce formalisme, la dissipation du système représente la production irréversible d’entropie due à la réaction chimique. Nous appliquons ensuite les techniques de commande passive (modelage de l’énergie) pour la synthèse de lois de commande en choisissant la disponibilité thermodynamique comme fonction hamiltonienne à modeler en boucle fermée. Finalement, nous montrons que les commandes synthétisées par l’approche thermodynamique entropique et la formulation pseudo-hamiltonienne sont, dans certains cas, équivalentes. Certaines propriétés relatives à la stabilisation et l’admissibilité des commandes sont aussi considérées. Les développements théoriques sont mis en oeuvre sur des exemples différents de RPAC : un réacteur académique et l’hydrolyse par catalyse acide de l’oxirane-méthanol en glycérine. / The goal of this thesis is to propose new nonlinear control strategies for the stabilization of perfectly Continuous Stirred Tank Reactors (CSTR). To achieve this goal, we use on the one hand, the entropic thermodynamic approach. More precisely, we use the thermodynamic availability concept and the properties of irreversible thermodynamics to define a Lyapunov function candidate for the stabilization of the closed loop system. We also propose a reduced availability function to design more efficient feedback laws in term of control variable solicitations. On the other hand, we propose an extension of the (pseudo) Hamiltonian formalism associated to dissipative systems to open chemical reactors. We show that the Hamiltonian is linked to the Gibbs free enthalpy in the isothermal case and to ectropy (opposed to entropy) in the non isothermal case. By this formalism, the dissipation of the system represents the irreversible entropy production due to chemical reaction. The Interconnection and Damping Assignment-Passivity Based Control (IDA-PBC) approach is then applied to synthesize feedback laws by choosing the thermodynamic availability as desired closed loop hamiltonian storage function. Finally, we show that feedback laws synthetized by the entropic thermodynamic approach and the pseudo-hamiltonian formulation are equivalent in some cases. Some stabilization properties and the control input admissibility are also considered. Theoretical developments are illustrated on some different CSTR examples : an academic case study and the acid catalyzed hydration of oxirane-methanol to glycerol.

This master thesis discusses various mathematical problems related to the placement of chess pieces. Solutions to the problems are mostly elementary (yet sometimes quite inventive), in some cases rely on basic knowledge of graph theory. We successively focus on different chess pieces and their tours on rectangular boards, and then examine the "independence" and "domination" of chess pieces on square boards. The text is complemented with numerous pictures illustrating particular solutions to given problems.