Σκοπός της παρούσας διατριβής είναι η κατασκευή και μελέτη συνολοθεωρητικών λύσεων της κβαντικής εξίσωσης Yang-Baxter (απεικονίσεις Yang-Baxter) και η συσχέτισή τους με την ολοκληρωσιμότητα διακριτών δυναμικών συστημάτων. Οι κατασκευές απεικονίσεων Yang-Baxter που προτείνονται προέρχονται από την αναπαραγοντοποίηση ισχυρών ζευγών Lax εξαρτώμενων από μια φασματική παράμετρο. Οι αντίστοιχοι πίνακες Lax προκύπτουν από την συμπλεκτική εμφύλλωση διωνυμικών πινάκων εφοδιασμένων με μια κατάλληλη δομή Poisson (αγκύλη Sklyanin). Στην περίπτωση των 2x2 πινάκων Lax, οι αντίστοιχες απεικονίσεις είναι συμπλεκτικές, τετράρητες και ταξινομούνται με βάση τον μεγιστοβάθμιο όρο του πίνακα Lax ως προς την ισοδυναμία απεικονίσεων Yang-Baxter. Εκφυλισμένες απεικονίσεις Yang-Baxter, οι οποίες σχετίζονται με γνωστές ολοκληρώσιμες εξισώσεις, προκύπτουν από όρια των τετράρητων (μη-εκφυλισμένων). Η σύνδεση μεταξύ απεικονίσεων Yang-Baxter και ολοκληρωσιμότητας επιτυγχάνεται θεωρώντας περιοδικά προβλήματα αρχικών τιμών σε δισδιάστατα πλέγματα. Σε κάθε απεικόνιση Yang-Baxter αντιστοιχεί μια οικογένεια αντιμεταθετικών απεικονίσεων μεταφοράς στο πλέγμα (transfer maps) που διατηρούν αναλλοίωτο το φάσμα του μονόδρομου πίνακά τους. Η αγκύλη Sklyanin εξασφαλίζει την ενέλιξη των ολοκληρωμάτων που προκύπτουν από το φάσμα του μονόδρομου πίνακα. Κατά αυτόν τον τρόπο από τις συμπλεκτικές απεικονίσεις Yang-Baxter που κατασκευάσαμε παράγονται ολοκληρώσιμες απεικονίσεις μεταφοράς. Τέλος, η μελέτη μας επεκτείνεται σε συστήματα πεπλεγμένων απεικονίσεων Yang-Baxter (entwining Yang-Baxter maps) . / The purpose of this thesis is the construction and the study of set theoretical solutions of the quantum Yang-Baxter equation (Yang-Baxter maps) and the connection with the integrability of discrete integrable systems. The constructions that we present are derived from the re-factorization of strong Lax pairs depending on a spectral parameter. The corresponding Lax matrices are obtained from the symplectic foliation of binomial matrices equipped with an appropriate Poisson bracket (Sklyanin bracket). In the case of 2x2 binomial Lax matrices, the corresponding maps are symplectic, quadrirational and can be classified with respect to the Yang-Baxter equivalence. Degenerate Yang-baxter maps constructed as limits of the quadrirational maps, are connected to known integrable equations. The connection between Yang-Baxter maps and integrability is achieved by considering periodic initial value problems on two dimensional lattices. For any Yang-Baxter map that admits a Lax matrix, there is a family of commuting transfer maps which preserve the spectrum of their monodromy matrix. The Skllyanin bracket ensures that the integrals obtained from the spectrum of the monodromy matrix are in involution. In this way, integrable transfer maps are generated from the symplectic Yang-Baxter maps that we constructed. Finally, our study is extended for systems of entwining Yang-Baxter maps.

Made available in DSpace on 2015-05-14T12:47:14Z (GMT). No. of bitstreams: 1 ArquivoTotal.pdf: 5481354 bytes, checksum: 0c13ae5c154f04c9968451cecffae9ab (MD5) Previous issue date: 2013-03-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Until late 1980s, urban air pollution was attributed to industrial emissions. However, the fast urban growth in the developing nations caused the vehicles to become an issue of large magnitude. Nowadays, there is a situation in which the most responsible for the a relevant part of the air quality degradation in the bigger urban centers are the motor vehicles that are currently circulating on the roads, due to their great amount and poor state of repair. In some cities, the air pollutant concentration indexes have reached levels that threaten people's health, forcing the authorities to make decisions to control such problem. This current work aimed to verify the relationship among meteorological factors, vehicular air pollutants and the number of respiratory system disease cases notified by the department of health of Salvador-BA', by using the Poisson Regression Model. This investigation was conducted by using an ecological drawing made from secondary information about hospitalizations, collected from DATASUS and from data related to the monitoring of major air pollutants and to the meteorological parameters, from November 2010 to September 2012, acquired from the Environment and Water Resources Institute - INEMA. In this work, the most susceptible groups of people, that is, children (≤9 years old) and the elderly ( ≥65 years-old), were monitored. Acording to the achieved results, it was observed that the independent variables "SO2", "NO2", "CO" and "Temperature" were statistically relevant to justify the variability of the responds variable "Hospitalizations by respiratory system diseases" in children, to the exploratory level of 5%. Regarding to elderly, only the variables "NO2" and "O3" were statistically relevant to the significance level of 5%. Through the residual graphics and the diagnostic measures related to the Poisson model adjusted to the data about hospitalizations caused by respiratory system diseases in children and the elderly, it was possible to verify the proper suitability of the model used for the decision-making. / A poluição atmosférica urbana, até meados de 1980, era atribuída basicamente às emissões industriais, no entanto, o rápido crescimento urbano nos países em desenvolvimento fez com que os veículos se tornassem um problema de grande magnitude. Hoje se verifica uma situação em que os maiores responsáveis por grande parte da degradação da qualidade do ar nos grandes centros urbanos são os veículos automotores em circulação nas rodovias, devido à grande quantidade e péssimo estado de conservação. Em algumas cidades, os índices de concentrações de poluentes do ar passaram a atingir níveis que colocam em risco a saúde das pessoas obrigando as autoridades a tomar decisões para controlar este problema. o presente trabalho teve como objetivo verificar a relação entre fatores meteorológicos e poluentes atmosféricos veiculares com o número de casos de doenças respiratórias notificadas pela Secretaria de Saúde de Salvador-BA, mediante a utilização do Modelo de Regressão de Poisson. Essa investigação foi realizada utilizando-se um desenho ecológico a partir de informações secundárias de internações hospitalares coletados junto ao DATASUS e de dados acerca do monitoramento dos principais poluentes atmosféricos e dos parâmetros meteorológicos, no período de novembro de 2010 a setembro de 2012, adquiridos junto ao Instituto do Meio Ambiente e Recursos Hídricos - INEMA. Foram observadas neste estudo as populações mais suscetíveis que são crianças (< 9 anos) e idosos > 65 anos). De acordo com os resultados obtidos, observou-se que as variáveis "SO2", "NO2", "CO2", e "Temperatura" são estatisticamente significantes para justificar a variabilidade da variável resposta "internações por doenças respiratórias" em crianças, ao nível exploratório de 5%. Com relação aos idosos, apenas as variáveis "NO2" e "O3" foram estatisticamente significantes ao nível de 5% de significância. Através dos gráficos de resíduos e de medidas de diagnóstico referentes ao modelo de Poisson ajustados aos dados sobre internações por doenças do aparelho respiratório em crianças e idosos. Foi possível verificar a boa adequação do modelo utilizado para a tomada de decisão.

Les applications récentes telles que les bases de données forestières impliquent des observations de données spatiales associées à l'observation de nombreuses covariables spatiales. Nous considérons dans cette thèse le problème de l'estimation d'une forme paramétrique de la fonction d'intensité dans un tel contexte. Cette thèse développe les procédures de sélection des variables et donne des garanties quant à leur validité. En particulier, nous proposons deux approches différentes pour la sélection de variables : les méthodes de type lasso et les procédures de type Sélecteur de Dantzig. Pour les méthodes envisageant les techniques de type lasso, nous dérivons les propriétés asymptotiques des estimations obtenues par les fontions d'estimation dérivées par les vraisemblances de la Poisson et de la régression logistique pénalisées par une grande classe de pénalités. Nous prouvons que les estimations obtenues par de ces procédures satisfont la consistance, sparsité et la normalité asymptotique. Pour la partie sélecteur de Dantzig, nous développons une version modifiée du sélecteur de Dantzig, que nous appelons le sélecteur Dantzig linéaire adaptatif (ALDS), pour obtenir les estimations d'intensité. Plus précisément, les estimations ALDS sont définies comme la solution à un problème d'optimisation qui minimise la somme des coefficients des estimations soumises à une approximation linéaire du vecteur score comme une contrainte. Nous constatons que les estimations obtenues par de ces méthodes ont des propriétés asymptotiques semblables à celles proposées précédemment à l'aide de méthode régularisation du lasso adaptatif. Nous étudions les aspects computationnels des méthodes développées en utilisant les procédures de type lasso et de type Sélector Dantzig. Nous établissons des liens entre l'estimation de l'intensité des processus ponctuels spatiaux et les modèles linéaires généralisés (GLM), donc nous n'avons qu'à traiter les procédures de la sélection des variables pour les GLM. Ainsi, des procédures de calcul plus faciles sont implémentées et un algorithme informatique rapide est proposé. Des études de simulation sont menées pour évaluer les performances des échantillons finis des estimations de chacune des deux approches proposées. Enfin, nos méthodes sont appliquées pour modéliser les emplacements spatiaux, une espèce d'arbre dans la forêt observée avec un grand nombre de facteurs environnementaux. / Recent applications such as forestry datasets involve the observations of spatial point pattern data combined with the observation of many spatial covariates. We consider in this thesis the problem of estimating a parametric form of the intensity function in such a context. This thesis develops feature selection procedures and gives some guarantees on their validity. In particular, we propose two different feature selection approaches: the lasso-type methods and the Dantzig selector-type procedures. For the methods considering lasso-type techniques, we derive asymptotic properties of the estimates obtained from estimating functions derived from Poisson and logistic regression likelihoods penalized by a large class of penalties. We prove that the estimates obtained from such procedures satisfy consistency, sparsity, and asymptotic normality. For the Dantzig selector part, we develop a modified version of the Dantzig selector, which we call the adaptive linearized Dantzig selector (ALDS), to obtain the intensity estimates. More precisely, the ALDS estimates are defined as the solution to an optimization problem which minimizes the sum of coefficients of the estimates subject to linear approximation of the score vector as a constraint. We find that the estimates obtained from such methods have asymptotic properties similar to the ones proposed previously using an adaptive lasso regularization term. We investigate the computational aspects of the methods developped using either lasso-type procedures or the Dantzig selector-type approaches. We make links between spatial point processes intensity estimation and generalized linear models (GLMs), so we only have to deal with feature selection procedures for GLMs. Thus, easier computational procedures are implemented and computationally fast algorithm are proposed. Simulation experiments are conducted to highlight the finite sample performances of the estimates from each of two proposed approaches. Finally, our methods are applied to model the spatial locations a species of tree in the forest observed with a large number of environmental factors.

Cette thèse porte sur la simulation tridimensionnelle (3D) des décharges couronnes à l'aide du calcul haute performance. Lorsqu'on applique une impulsion de haute tension entre une pointe et un plan, les lignes de champ électrique fortement resserrées autour de la pointe induisent la propagation simultanée de plusieurs streamers et la formation d'une décharge couronne de structure arborescente. Dans ces conditions, seule une simulation électro-hydrodynamique 3D est apte à reproduire cette structure et fournir les ordres de grandeur de l'énergie déposée et de la concentration des espèces créées durant la phase de décharge. Cependant, cette simulation 3D est très consommatrice en temps et mémoire de calcul et n'est désormais accessible que grâce à l'accroissement permanent de la puissance des ordinateurs dédié au calcul haute performance. Dans le cadre d'une simulation électro-hydrodynamique 3D, une attention particulière doit être prise concernant l'efficacité des solveurs à résoudre les équations elliptiques 3D car leur contribution en termes de temps de calcul peut dépasser 80% du temps global de la simulation. Ainsi, une partie de manuscrit est consacrée aux tests de performances de méthodes de résolution d'équations elliptiques directes ou itératives telle que SOR R&B, BiCGSTAB et MUMPS, en utilisant le calcul massivement parallèle et les librairies MPI. Les calculs sont réalisés sur le supercalculateur EOS du réseau CALMIP, avec un nombre de cœurs de calcul allant jusqu'à 1800, et un nombre de mailles atteignant 8003 (soit plus 1/2 Milliard de mailles). Les tests de performances sont réalisés en statique sur le calcul du potentiel géométrique et en dynamique en propageant une densité de charge d'espace analytique caractéristique des streamers. Pour réaliser une simulation complète 3D de la décharge il faut également intégrer au programme un algorithme capable de résoudre les équations de transport de particule chargée à fort gradients de densité caractéristiques aux streamers. Dans ce manuscrit, l'algorithme MUSCL est testé dans différentes conditions de propagation d'un cube de densité (à vitesse homogène ou non homogène spatialement) afin d'optimiser le transport des densités d'espèces chargées impliquées. Le code 3D, conçu pour résoudre le modèle électro- hydrodynamique complet de la décharge (couplant les équations de transport, de Poisson et de cinétique réactionnelle) est ensuite validé par la confrontation des résultats 3D et 2D dans une condition de simulation présentant une symétrie de révolution autour de l'axe de propagation d'un streamer. Enfin, les premiers résultats des simulations 3D de la phase décharge avec la propagation d'un ou plusieurs streamers asymétriques sont présentés et analysés. Ces simulations permettent de suivre la structure arborescente de la décharge lorsqu'on applique une tension pulsée entre une pointe et un plan. L'initiation de la structure arborescente est étudiée en fonction de la position de spots plasmas et de leur influence sur l'amorçage des streamers. / This work is devoted to the three dimensional (3D) simulation of streamer corona discharges in air at atmospheric pressure using high-performance parallel computing. When a pulsed high-voltage is applied between a tip and a plane in air, the strong electric field lines constricted around the tip induce the simultaneous propagation of several streamers leading to a corona discharge with a tree structure. Only a true 3D electro-hydrodynamics simulation is able to reproduce this branching and to provide the orders of magnitude of the local deposited energy and the concentration of the species created during the discharge phase. However, such a 3D simulation which requires large computational memory and huge time calculation is nowadays accessible only when performed with massively parallel computation. In the field of 3D electro-hydrodynamics simulations, a special attention must be paid to the efficiency of solvers in solving 3D elliptic equations because their contribution can exceed 80% of the global computation time. Therefore, a specific chapter is devoted to test the performance of iterative and direct methods (such as SOR R&B, BiCGSTAB and MUMPS) in solving elliptic equations, using the massively parallel computation and the MPI library. The calculations are performed on the supercomputer EOS of the CALMIP network, with a number of computing cores and meshes increasing up to respectively 1800 and 8003 (i.e. more than 1/2 Billion meshes). The performances are compared for the calculation of the geometric potential and in a dynamic simulation conditions consisting in the propagation of an analytical space charge density characteristic of the streamers. To perform a complete 3D simulation of the streamer discharge, must also involve a robust algorithm able to solve the coupled conservation equations of the charged particle density with very sharp gradients characteristic of the streamers. In this manuscript, the MUSCL algorithm is tested under different propagation conditions of a cubic density (with uniform or non-uniform velocity field). The 3D code, designed to solve the complete electro-hydrodynamics model of the discharge (coupling the conservation equations, the Poisson equation and the chemical kinetics) is validated by comparing the 3D and 2D results in a simulation conditions presenting a rotational symmetry around the propagation axis of a mono-filamentary streamer. Finally, the first results of the 3D simulations of the discharge phase with the propagation of one or several asymmetric streamers are presented and analyzed. These simulations allow to follow the tree structure of a corona discharge when a pulsed voltage is applied between a tip and a plane. The ignition of the tree structure is studied as a function of the initial position of the plasma spots.

On étudie une équation différentielle stochastique animée par un processus ponctuel de Poisson, qui modélise un changement continu de lénvironnement d'une population et la fixation stochastique de mutations bénéfiques pour compenser ce changement. La probabilité de fixation d'une mutation augmente dès que le retard phénotypique $X_t$ entre la population et l'optimum augmente. On suppose que les mutations favorables se fixent instantanément induisant un saut adaptatif. En premier lieu, on a étudié le comportement à long terme de la solution de cette équation sachant qu'on ne considère qu'un seul trait phénotypique de la population et on a trouvé les conditions sous lesquelles $X_t$ est récurrent (possibilité de survie) ou transient (extinction inévitable). Ensuite, on a généralisé nos résultats en considérant un vecteur de traits phénotypiques de la population, essentiellement dans $mathbb R^2$. A la fin, on introduit une limite des petits sauts pour caractériser et comprendre le cas récurrent. / We study a stochastic differential equation driven by a Poisson point process, which models continuous changes in a population's environment, as well as the stochastic fixation of beneficial mutations that might compensate for this change. The fixation probability of a given mutation increases as the phenotypic lag $X_t$ between the population and the optimum grows larger, and successful mutations are assumed to fix instantaneously (leading to an adaptive jump). First, we study the large time behavior of the solution of this SDE taking into consideration one phenotypic trait of the population and we find the conditions under which $X_t$ is recurrent (possibility of survival) or transient (doomed to exctinction).Then we generalize our results to the case of a phenotypic traits vector, essentially in $R^2$. Finally, we introduce a small jumps limit to characterize and understand the recurrent case.

[pt] O objetivo primário do presente trabalho é a captura do efeito informacional na taxa de juros de curto prazo (taxa Selic) por meio de saltos de Poisson. Os principais pilares de sustentação do tema são os testes realizados por Johannes (2004) e Das (2002), os quais evidenciam significância de tal estrutura para anúncios do FOMC1. Nos artigos motivadores desse interesse, os autores demonstram que grande parte da volatilidade de curto prazo no mercado de renda fixa é capturada com a introdução de saltos no processo estocástico da taxa de curto prazo. O mercado de renda fixa no Brasil, apesar do gradual (alongamento) do perfil, ainda está fortemente direcionado para títulos de curto prazo, como as Letras do Tesouro Nacional (LTN). Há, porém, poucos trabalhos focados em estimar o efeito informacional na taxa de juros de curto prazo no Brasil, como Margueron (2006), e que propõem abordagens simplificadas do problema2. A captura desse efeito por modelagem com processo estocástico pode gerar avanços não só na compreensão do efeito per se, mas também na explicação de movimentos bruscos na ponta curta da estrutura a termo da taxa de juros. Como em modelos afins da estrutura a termo da taxa de juros toda a curva pode ser descrita em função da taxa curta, assim temos, então, uma melhor abordagem para a curva como um todo. A formalização dessa estrutura desbrava, portanto, uma linha de potenciais avanços no apreçamento de instrumentos financeiros de renda fixa e gestão de riscos. / [en] The main goal of the present work is to capture the informational effect in the Brazilian short-term interest rate (Selic rate) using Poisson jumps. This structure is sustained by tests realized by Johannes (2004) and Das (2002), which provide strong evidence of this kind of modeling for FOMC3 announcements. In the above-mentioned articles, the authors show that a big part of the short-term volatility observed in fixed income markets is captured with the introduction of jumps in the stochastic process of the interest rate. The fixed income markets in Brazil, despite of the progressive investment profile changes (in direction of longer maturities), remains highly oriented for short term government bonds, like LTN’s. There are, nevertheless, few academic works estimating the informational effects in the Brazilian short-term interest rate, such as Margueron (2006) and with simplified approaches for this issue4. The capture of the informational effect with stochastic processes might generate improvements not only in understanding the effect per se, but also in explaining the abrupt movements in shorter maturities of the yield curve. As affine models for the yield curve, they are based in the idea of the whole curve as a function of the short-term interest rate, so it is possible to extend this approach, improving the yield curve modeling as a whole. Formalizing this structure potentially generates advances in pricing fixed income instruments and risk management.

[pt] No presente trabalho faz-se uma análise sobre quantidades estatísticas de cadeias lineares e não lineares na situação em que o fluxo de calor que atravessa estes sistemas encontra-se no regime estacionário. A discussão inicial é feita sobre um modelo geral de cadeia linear, com acoplamentos arbitrários entre suas partículas e alimentada por reservatórios gaussianos. Uma análise detalhada sobre quantidades como fluxo de calor e distribuição de temperaturas do sistema é feita, onde todas as expressões analíticas correspondentes a estas quantidades são demonstradas e comparadas com resultados numéricos. Estudam-se então as mudanças quantitativas e qualitativas apresentadas pelas grandezas supracitadas quando modificam-se os acoplamentos de ancoragem entre o sistema e os reservatórios. Verifica-se que as mudanças nos perfis de temperaturas estão relacionadas aos extremos dos cumulantes do fluxo de calor, o que motiva uma investigação sobre a possível ocorrência de uma transição de fase no sistema. Buscando encontrar possíveis comportamentos críticos, definem-se as funções de correlação entre as velocidades quadráticas e de velocidades entre pares de partículas. A partir destas definições é possível verificar o comprimento de correlação associado à estas grandezas. Este estudo leva a um dos pontos mais interessantes do trabalho, onde conectam-se as mudanças apresentadas por grandezas do sistema como quantidades estatísticas do fluxo de calor, distribuição de temperaturas do sistema e os seus modos vibracionais frente às mudanças nos acoplamentos de ancoragem com os reservatórios. Ao estudar o fenômeno de condução de calor de uma forma mais realística e rigorosa, é imprescindível acrescentar interações não-lineares na cadeia. Considerando que a solução exata para este tipo de sistema não pode ser obtida, utiliza-se teoria de perturbação e outras ferramentas matemáticas para discutir as principais caracterísiticas do fluxo de calor em uma cadeia anarmônica. A técnica desenvolvida nesta tese permite calcular o fluxo de calor em cadeias de tamanho arbitrário, e é válida para sistemas sob ação de reservatórios de qualquer natureza. Aplica-se o método para cadeias alimentadas por reservatórios gaussianos e poissonianos, de onde verifica-se o impacto das não linearidades sobre estes sistemas e comparam-se os resultados obtidos com o caso linear. Para a análise em que o reservatório poissoniano injeta energia no sistema, ilustra-se o efeito de cumulantes de ordem superior do ruído descontínuo sobre o fluxo de calor e como estes novos elementos podem levar a resultados que a primeira vista parecem fisicamente incoerentes. / [en] In the present work I make an analysis about statistical quantities for linear and nonlinear chains in the stationary state. We start the discussion from a general linear model, with arbitrary couplings and connected to Gaussian reservoirs. A detailed analysis for quantities like heat ow and site temperatures is obtained, where all analytical expressions respective to those quantities are derived and a compared with numerical results. Then I study the quantitative and qualitative changes presented by the aforementioned quantities when the pinnings related to the reservoirs are modified. The changes in temperature profiles are related with the extrema of heat flux cumulants, motivating the investigation of whether phase transitions in the chain might occur. In order to investigate possible critical behaviors, I define velocity correlation functions between pair particles and squared velocities correlation functions. From where, one is able to estabilish a correlation length respective to these quantities. This study leads to one of the most remarkable achievements of this work, which is the connection made between the changes presented by some important statistical quantities of heat flux, the system s temperature, vibrational modes and the reservoirs pinnings. By treating the phenomenon of heat conduction in a more realistic and rigorous way, I develop a study to describe the transport properties in an anharmonic chain. Pondering that an exact solution for this sort of system is unfeasible, I use perturbation theory and other mathematical tools to discuss the main features of heat flux in a nonlinear chain. The technique developed throughout this thesis allows one to compute the heat current for a chain of arbitrary size, and is valid for systems under in fluence of reservoirs of any nature. We apply the method for chains governed by Gaussian and Poissonian reservoirs, verifying the impact of the nonlinearities over those systems, and comparing the obtained results to the linear case. In the case where there is a Poissonian bath injecting energy into the system, I shed some light on the effects of higher order cumulants related to the discontinous noise in the heat flux and I show how these new elements can lead to some results that at first glance seem physically incoherents.

Cette thèse présente deux sujets de recherche indépendants concernant l'application de méthodes stochastiques à des problèmes issus de la dynamique moléculaire. Dans la première partie, nous présentons des travaux liés à l'interprétation probabiliste de l'équation de Poisson-Boltzmann qui intervient dans la description du potentiel électrostatique d'un système moléculaire. Après avoir introduit l'équation de Poisson-Boltzmann et les principaux outils mathématiques utilisés, nous nous intéressons à l'équation linéaire parabolique de Poisson-Boltzmann. Avant d’énoncer le résultat principal de la thèse, nous étendons des résultats d'existence et unicité des équations différentielles stochastiques rétrogrades. Nous donnons ensuite une interprétation probabiliste de l'équation non-linéaire de Poisson-Boltzmann sous la forme de la solution d'une équation différentielle stochastique rétrograde. Enfin, dans une seconde partie prospective, nous commençons l'étude d'une méthode proposée par Paul Malliavin de détection des variables lentes et rapides d'une dynamique moléculaire. / This thesis presents two independent research topics. Both are related to the application of stochastic problems to molecular dynamics. In the first part, we present a work related to the probabilistic interpretation of the Poisson-Boltzmann equation. This equation describes the electrostatic potential of a molecular system. After an introduction to the Poisson-Boltzmann equation, we focus on the parabolic and linear equation. After extending an existence and uniqueness result for backward stochastic differential equations, we establish a probabilistic interpretation of the nonlinear Poisson-Boltzmann equation with backward stochastic differential equations. Finally, in a more prospective second part, we initiate a study of a slow and fast variables detection method due to Paul Malliavin.

[pt] A equação diferencial parcial de Poisson é de fundamental importância em várias áreas de pesquisa, dentre elas: matemática, física e engenharia. Para resolvê-la numericamente utilizam-se vários métodos, tais como os já tradicionais métodos das diferenças finitas e dos elementos finitos. Este trabalho propõe um método para resolver a equação de Poisson, utilizando uma abordagem de sistema de partículas conhecido como SPH, do inglês Smoothed Particles Hydrodynamics. O método proposto para a solução da equação de Poisson e os operadores diferenciais discretos definidos no método SPH, chamados de operadores SPH, são utilizados neste trabalho em duas aplicações: na decomposição de campos vetoriais; e na simulação numérica de escoamentos de fluidos monofásicos e bifásicos utilizando a equação de Navier-Stokes. / [en] Poisson`s equation is of fundamental importance in many research areas in engineering and the mathematical and physical sciences. Its numerical solution uses several approaches among them finite differences and finite elements. In this work we propose a method to solve Poisson`s equation using the particle method known as SPH (Smoothed Particle Hydrodynamics). The proposed method together with an accurate analysis of the discrete differential operators defined by SPH are applied in two related situations: the Hodge-Helmholtz vector field decomposition and the numerical simulation of the Navier-Stokes equations.

Submitted by NILSON CÉSAR GALVÃO VIEIRA null (nil_gv@yahoo.com.br) on 2016-03-10T01:50:31Z No. of bitstreams: 1 VIEIRA N.C.G..pdf: 2502695 bytes, checksum: 79c827f6494d3d8d91c8508084d0122b (MD5) / Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2016-03-10T20:11:35Z (GMT) No. of bitstreams: 1 vieira_ncg_me_guara.pdf: 2502695 bytes, checksum: 79c827f6494d3d8d91c8508084d0122b (MD5) / Made available in DSpace on 2016-03-10T20:11:35Z (GMT). No. of bitstreams: 1 vieira_ncg_me_guara.pdf: 2502695 bytes, checksum: 79c827f6494d3d8d91c8508084d0122b (MD5) Previous issue date: 2016-01-27 / Uma das principais causas de morbidade por problemas respiratórias em crianças e idosos é a poluição do ar. Os efeitos na saúde devido a exposição aos poluentes atmosféricos e a baixa qualidade do ar vem causando um aumento nas internações hospitalares. Esse estudo teve como objetivo desenvolver um modelo preditivo através do Modelo Linear Generalizado de Regressão de Poisson para analisar os efeitos da exposição de CO e PM2,5 sobre as internações hospitalares da rede pública no município de Rio Branco, AC, por problemas respiratórios em duas faixas etárias de ambos os sexos, de 0 a 14 anos e de iguais ou mais de 50 anos, durante o período de 01 de julho de 2012 a 30 de junho de 2013. A metodologia utilizada fundamenta-se na pesquisa de estudo ecológico de série temporal. A variável dependente considerada foi o número de internações hospitalares por doenças respiratórias CID 10 correspondentes aos códigos J 00 a J 99, cujos dados foram obtidos pelo DATASUS. As variáveis independentes consideradas foram a concentração diária de poluentes CO e PM2,5 (CCATT-BRAMS), número de queimadas (SISAM), umidade relativa do ar e temperatura mínima (INMET). Ajustes por tendência temporal e efeitos do calendário foram incluídos no modelo. Na análise unipoluente, devido a um incremento interquartil (25%-75%) para o CO (25 ppb) e PM2,5 (1,5 μg/m3 ), observou-se respectivamente, um aumento de 2,1% e 27,9% no aumento de internações para a faixa etária de iguais ou mais de 50 anos. E na análise multipoluente também com incremento interquartil, observou-se um aumento de 17,8% para a faixa etária de 0 a 14 anos e de 34,2% para a faixa etária de iguais ou mais de 50 anos. Os dados obtidos pelo aumento percentual forneceram uma variação de 1,13 % a 34,2 % no risco de internação, ou seja, aproximadamente 348 internações a mais, gerando para os cofres do município um custo de aproximadamente R$ 520 mil. / A major cause of morbidity due to respiratory problems in children and elderly is air pollution. Health effects due to exposure to air pollution and poor air quality is causing an increase in hospital admissions. This study aimed to develop a predictive model by Generalized Linear Model Poisson Regression to analyze the effects of CO exposure and PM2.5 on the hospitalization of the public network in Rio Branco, AC, for respiratory problems two age groups of both sexes, 0-14 years and of equal or more than 50 years during the period 1 July 2012 to 30 June 2013. The methodology used is based on the number of ecological research study temporal. The dependent variable was considered the number of hospital admissions for respiratory diseases ICD-10 codes corresponding to the J 00 and J 99, whose data were obtained by DATASUS. The independent variables considered were the daily concentration of pollutants CO and PM2.5 (CCATT-BRAMS), number of fires (SISAM), relative humidity and minimum temperature (INMET). Adjustments for time trend and calendar effects were included in the model. In unipoluente analysis, due to an increase interquartile (25% -75%) to CO (25 ppb) and PM2.5 (1.5 g / m3 ) was observed respectively, an increase of 2.1% and 27, 9% increase in hospitalizations for children aged equal or more than 50 years. And in multipoluente analysis also with interquartile increase, there was an increase of 17.8% for the age group 0-14 years and 34.2% for the age group of equal or more than 50 years. And in multipoluente analysis also with interquartile increase, there was an increase of 17.8% for the age group 0-14 years and 34.2 % for the age group of the same or more than 50 years. The data obtained by the percentage increase provided a range of 1.13 % to 34.2% in the risk of hospitalization, or approximately 348 hospitalizations more, generating for the municipal coffers a cost of approximately R$ 520 thousand.