Neste trabalho, introduzimos a noção de desenvolvimento assintótico em classes de Gevrey e mostramos como o conceito clássico de convergência de séries de potências pode ser generalizado para englobar o caso em que o raio de convergência é nulo. Essa técnica pode ser útil em situações em que é necessário trabalhar com séries formais, como no estudo de Equações Diferenciais. Caracterizamos o conjunto das funções holomorfas que admitem desenvolvimento assintótico e, em cada classe de Gevrey, definimos uma aplicação que associa uma função a uma série formal. Determinamos sob quais condições tal aplicação é sobrejetora e sob quais ela é injetora, possibilitando a ampliação do conceito de convergência e as aplicações da teoria. Além disso, mostramos como essa técnica pode ser usada para obter resultados em equações diferenciais. Para isso, fazemos uma breve introdução de Equações Diferenciais com uma variável complexa e introduzimos o conceito de Polígono de Newton, ferramenta que permite obter a classe de Gevrey de uma solução formal. Finalmente, encontramos condições para que a soma de uma solução formal de uma equação diferencial seja uma solução clássica. / In this work, we introduce the notion of Gevrey asymptotic expansion and we show how the classical concept of a convergent power series can be generalized to include the case in which the radius of convergence is zero. This technique can be useful in situations where it is necessary to work with formal power series, as in the study of Differential Equations. We characterize the set of holomorphic functions which admit Gevrey asymptotic expansion and we define in each Gevrey class a map that associates to function in the class a formal series. We determine under which conditions such a map is surjective and under which it is injective, allowing the extension of the concept of convergence and applications of the theory. Furthermore, we show how this technique can be used to obtain results in Differential Equations. For this, we briefly recall the theory of Differential Equations in one complex variable and we introduce the concept of the Newton Polygon, a tool that allows us to find the Gevrey class of a formal solution. Finally, we find suficient conditions for the sum of a formal solution of a differential equation to be a classical solution.

Neste trabalho, introduzimos a noção de desenvolvimento assintótico em classes de Gevrey e mostramos como o conceito clássico de convergência de séries de potências pode ser generalizado para englobar o caso em que o raio de convergência é nulo. Essa técnica pode ser útil em situações em que é necessário trabalhar com séries formais, como no estudo de Equações Diferenciais. Caracterizamos o conjunto das funções holomorfas que admitem desenvolvimento assintótico e, em cada classe de Gevrey, definimos uma aplicação que associa uma função a uma série formal. Determinamos sob quais condições tal aplicação é sobrejetora e sob quais ela é injetora, possibilitando a ampliação do conceito de convergência e as aplicações da teoria. Além disso, mostramos como essa técnica pode ser usada para obter resultados em equações diferenciais. Para isso, fazemos uma breve introdução de Equações Diferenciais com uma variável complexa e introduzimos o conceito de Polígono de Newton, ferramenta que permite obter a classe de Gevrey de uma solução formal. Finalmente, encontramos condições para que a soma de uma solução formal de uma equação diferencial seja uma solução clássica. / In this work, we introduce the notion of Gevrey asymptotic expansion and we show how the classical concept of a convergent power series can be generalized to include the case in which the radius of convergence is zero. This technique can be useful in situations where it is necessary to work with formal power series, as in the study of Differential Equations. We characterize the set of holomorphic functions which admit Gevrey asymptotic expansion and we define in each Gevrey class a map that associates to function in the class a formal series. We determine under which conditions such a map is surjective and under which it is injective, allowing the extension of the concept of convergence and applications of the theory. Furthermore, we show how this technique can be used to obtain results in Differential Equations. For this, we briefly recall the theory of Differential Equations in one complex variable and we introduce the concept of the Newton Polygon, a tool that allows us to find the Gevrey class of a formal solution. Finally, we find suficient conditions for the sum of a formal solution of a differential equation to be a classical solution.

O osso é um sólido heterogêneo com estrutura bastante complexa que geralmente exige o emprego de múltiplas escalas em sua análise. A análise do comportamento eletromecânico da estrutura óssea tem sido realizada por meio de métodos da mecânica clássica, métodos de elementos finitos e métodos de homogeneização. Procura-se descrever matematicamente a relação entre o comportamento eletromecânico da estrutura óssea e as propriedades efetivas, ou, globais. Assim, muitos esforços têm sido despendidos para desenvolver modelos analíticos rigorosos capazes de predizer as propriedades globais e locais das estruturas ósseas. O propósito deste trabalho é estudar o método de Homogeneização Assimptótica (MHA) com a finalidade de determinar as propriedades eletromecânicas efetivas de estruturas heterogêneas, tais como a estrutura óssea. Inicialmente, são analisados o problema de condução de calor e o problema elástico e demonstra-se que estes problemas estão relacionados entre si. Para o problema de condução de calor, dois métodos para obter as constantes efetivas são apresentados. Além disso, uma aplicação do MHA em osso cortical é apresentada e os resultados estão de muito bom acordo com resultados encontrados na literatura. Em vista disto, verifica-se a possibilidade da aplicação do MHA para determinar as propriedades efetivas da estrutura óssea com estrutura cristalina na classe 622. / The bone is a heterogeneous solid with a highly complex structure that requires a multiple scale type of analysis. To analyze the electromechanical behavior of the bone structure, methods of classical mechanics, finite element methods, and methods of homogenization are being used. This analysis describes mathematically the relationship between the electromechanical behavior of the bone structure and its effective, or, global, properties. Thus, many efforts have been spent to develop rigorous analytical models capable of predicting the global and local effective properties of bone structures. The purpose of this work is to study the Asymptotic Homogenization Method (AHM) in order to determine the electromechanical effective properties of heterogeneous structures, such as the bone structure. The analysis of heat conduction and elastic problem using AHM shows that these problems are related to each other. Furthermore, an application of the AHM in cortical bone is presented and the results are shown to be in very good agreement with results found in the literature. Finally, this work shows great promise in the application of the AHM to determine the effective properties of a bone structure whose constituent material belongs to the crystal class 622.

O estudo do comportamento assintótico de densidades de algumas subestruturas é uma das principais áreas de estudos em combinatória. Na Teoria das Permutações, fixadas permutações ?1 e ?2 e um inteiro n > 0, estamos interessados em estudar o comportamento das densidades de ?1 e ?2 na família de permutações de tamanho n. Assim, existem duas direções naturais que podemos seguir. Na primeira direção, estamos interessados em achar a permutação de tamanho n que maximiza a densidade das permutações ?1 e ?2 simultaneamente. Para n suficientemente grande, explicitamos a densidade máxima que uma família de permutações podem assumir dentre todas as permutações de tamanho n. Na segunda direção, estamos interessados em achar a permutação de tamanho n que minimiza a densidade de ?1 e ?2 simultaneamente. Quando ?1 é a permutação identidade com k elementos e ?2 é a permutação reversa com l elementos, Myers conjecturou que o mínimo é atingido quando tomamos o mínimo dentre as permutações que não possuem a ocorrência de ?1 ou ?2. Mostramos que se restringirmos o espaço de busca somente ao conjunto de permutações em camadas, então a Conjectura de Myers é verdadeira. Por outro lado, na Teoria dos Grafos, o problema de encontrar um circuito Hamiltoniano é um problema NP-completo clássico e está entre os 21 problemas Karp. Dessa forma, uma abordagem comum na literatura para atacar esse problema é encontrar condições que um grafo deve satisfazer e que garantem a existência de um circuito Hamiltoniano em tal grafo. O célebre resultado de Dirac afirma que se um grafo G de ordem n possui grau mínimo pelo menos n/2, então G possui um circuito Hamiltoniano. Seguindo a linha de Dirac, mostramos que, dados inteiros 1 6 l 6 k/2 e ? > 0 existe um inteiro n0 > 0 tal que, se um hipergrafo k-uniforme H de ordem n satisfaz ?k-2(H) > ((4(k - l) - 1)/(4(k - l)2) + ?) (n 2), então H possui um l-circuito Hamiltoniano. / The study of asymptotic behavior of densities of some substructures is one of the main areas in combinatorics. In Permutation Theory, fixed permutations ?1 and ?2 and an integer n > 0, we are interested in the behavior of densities of ?1 and ?2 among the permutations of size n. Thus, there are two natural directions we can follow. In the first direction, we are interested in finding the permutation of size n that maximizes the density of the permutations ?1 and ?2 simultaneously. We explicit the maximum density of a family of permutations between all the permutations of size n. In the second direction, we are interested in finding the permutation of size n that minimizes the density of ?1 and ?2 simultaneously. When ?1 is the identity permutation with l elements and ?2 is the reverse permutation with k elements, Myers conjectured that the minimum is achieved when we take the minimum among the permutations which do not have the occurrence of ?1 or ?2. We show that if we restrict the search space only to set of layered permutations and k > l, then the Myers\' Conjecture is true. On the other hand, in Graph Theory, the problem of finding a Hamiltonian cycle is a NP-complete problem and it is among the 21 Karp problems. Thus, one approach to attack this problem is to find conditions that a graph must meet to ensure the existence of a Hamiltonian cycle on it. The celebrated result of Dirac shows that a graph G of order n that has minimum degree at least n/2 has a Hamiltonian cycle. Following the line of Dirac, we show that give integers 1 6 l 6 k/2 and gamma > 0 there is an integer n0 > 0 such that if a hypergraph k-Uniform H of order n satisfies ?k-2(H) > ((4(k-l)-1)/(4(k-l)2)+?) (n 2), then H has a Hamiltonian l-cycle.

L’objectif de cette thèse est la modélisation, l’analyse mathématique et la simulation numérique de quelques modèles de transmission du VIH-SIDA dans une population sexuellement active donnée en général et en particulier dans celle du Mali. Nous proposons de modèles basés sur les connaissances actuelles de la transmission du virus du VIH. A cet effet, nous présentons trois modèles : un modèle comportemental en épidémiologie, un modèle qui incorpore le rôle des campagnes de sensibilisation en santé publique et un modèle de co-circulation de deux formes recombinantes du VIH-1. Dans le premier modèle, des résultats d’existence et d’unicité de la solution d’un problème parabolique semi-linéaire décrivant l’évolution d’une population soumise à l’infection du VIH sont présentés. La population est divisée en individus dont le risque comportemental est faible et en individus dont les comportement sont très risqués et qui interagissent entre eux. Une variable continue représentant ce risque comportemental est introduit. Le comportement asymptotique en temps du problème est étudié. Certains résultats numériques concernant la répartition de la population selon la variable représentant le risque comportemental sont présentés dans le cas de l’infection du VIH-SIDA au Mali. Dans les deux derniers modèles, nous obtenons une analyse complète de la stabilité de ces modèles à l’aide des techniques de Lyapunov suivant la valeur du taux de reproduction de base . Nous proposons une méthode alternative au taux de reproduction de base qui permet de confiner l’évolution de la maladie dans des limites fixées. Nous illustrons ces modèles par des simulations numériques. Ces dernières sont faites à partir de nos modèles confrontés aux données du Mali concernant la propagation du VIH-SIDA. / The objective of this thesis is the modeling, mathematical analysis and numerical simulation of a few models of transmission of the HIV-AIDS in a sexually active population given in general and in particular in that of Mali. We propose models based on the epidemiology currently known from the transmission of the HIV virus. Thus, we present three models of the transmission of HIV: a individual behavior and epidemiological model, a model that incorporates the role of public health education program on HIV and a mathematical model for the co-circulating into two circulating recombinants forms of HIV-1. In the first model, Somé results of existence and uniqueness of solution of a semilinear,parabolic problem describing the evolution of a population subjected to a disease are presented. The population is divided into individuals whose behavioral risk is low and in individuals whose behavior are very risky and that interact between them. A continuous variable representing a behavioral risk is introduced. The asymptotic in time of the problem is studied, and the existence of a non zero stationary state is proved. Somé numerical results concerning the distribution of the population according to the variable representing a behavioral risk are presented within the disease of the HIV-AIDS in Mali. In the last two models, we obtain a thorough analysis of the stability of these models using the Lyapunov techniques according to the value of the basic reproduction ratio,R0. We propose an alternative method to the basic reproductive rate R0 which allows to confine the evolution of the disease in the fixed limits. Numerical simulations are done to illustrate the behaviour of the model, using data collected in the literature regarding the spread of HIV in Mali.

L’imagerie par résonance magnétique de diffusion (IRMd) est une technique non-invasive permettant d’accéder à l'information structurelle des tissus biologiques à travers l’étude du mouvement de diffusion des molécules d’eau dans les tissus. Ses applications sont nombreuses en neurologie pour le diagnostic de certaines anomalies cérébrales. Cependant, en raison du mouvement cardiaque, l’utilisation de cette technique pour accéder à l’architecture du cœur in vivo représente un grand défi. Le mouvement cardiaque a été identifié comme une des sources majeures de perte du signal mesuré en IRM de diffusion. A cause de la sensibilité au mouvement, il est difficile d’évaluer dans quelle mesure les caractéristiques de diffusion obtenues à partir de l’IRM de diffusion reflètent les propriétés réelles des tissus cardiaques. Dans ce cadre, la modélisation et la simulation numérique du signal d’IRM de diffusion offrent une approche alternative pour aborder le problème. L’objectif de cette thèse est d’étudier numériquement l’influence du mouvement cardiaque sur les images de diffusion et de s’intéresser à la question d’atténuation de l’effet du mouvement cardiaque sur le signal d’IRM de diffusion. Le premier chapitre est consacré à l’introduction du principe physique de l'imagerie par résonance magnétique(IRM). Le deuxième chapitre présente le principe de l’IRM de diffusion et résume l’état de l’art des différents modèles proposés dans la littérature pour modéliser le signal d’IRM de diffusion. Dans le troisième chapitre un modèle modifié de l’équation de Bloch-Torrey dans un domaine qui se déforme au cours du temps est introduit et étudié. Ce modèle représente une généralisation de l’équation de Bloch-Torrey utilisée dans la modélisation du signal d’IRM de diffusion dans le cas sans mouvement. Dans le quatrième chapitre, l’influence du mouvement cardiaque sur le signal d’IRM de diffusion est étudiée numériquement en utilisant le modèle de Bloch-Torrey modifié et un champ de mouvement analytique imitant une déformation réaliste du cœur. L’étude numérique présentée, permet de quantifier l’effet du mouvement sur la mesure de diffusion en fonction du type de la séquence de codage de diffusion utilisée, de classer ces séquences en terme de sensibilité au mouvement cardiaque et d’identifier une fenêtre temporelle par rapport au cycle cardiaque où l’influence du mouvement est réduite. Enfin, dans le cinquième chapitre, une méthode de correction de mouvement est présentée afin de minimiser l’effet du mouvement cardiaque sur les images de diffusion. Cette méthode s’appuie sur un développement singulier du modèle de Bloch-Torrey modifié pour obtenir un modèle asymptotique qui permet de résoudre le problème inverse de récupération puis correction de la diffusion influencée par le mouvement cardiaque. / Diffusion magnetic resonance imaging (dMRI) is a non-invasive technique allowing access to the structural information of the biological tissues through the study of the diffusion motion of water molecules in tissues. Its applications are numerous in neurology, especially for the diagnosis of certain brain abnormalities, and for the study of the human cerebral white matter. However, due to the cardiac motion, the use of this technique to study the architecture of the in vivo human heart represents a great challenge. Cardiac motion has been identified as a major source of signal loss. Because of the sensitivity to motion, it is difficult to assess to what extent the diffusion characteristics obtained from diffusion MRI reflect the real properties of the cardiac tissue. In this context, modelling and numerical simulation of the diffusion MRI signal offer an alternative approach to address the problem. The objective of this thesis is to study numerically the influence of cardiac motion on the diffusion images and to focus on the issue of attenuation of the cardiac motion effect on the diffusion MRI signal. The first chapter of this thesis is devoted to the introduction of the physical principle of nuclear magnetic resonance (NMR) and image reconstruction techniques in MRI. The second chapter presents the principle of diffusion MRI and summarizes the state of the art of the various models proposed in the litera- ture to model the diffusion MRI signal. In the third chapter a modified model of the Bloch-Torrey equation in a domain that deforms over time is introduced and studied. This model represents a generalization of the Bloch-Torrey equation used to model the diffusion MRI signal in the case of static organs. In the fourth chapter, the influence of cardiac motion on the diffusion MRI signal is investigated numerically by using the modified Bloch-Torrey equation and an analytical motion model mimicking a realistic deformation of the heart. The numerical study reported here, can quantify the effect of motion on the diffusion measurement depending on the type of the diffusion coding sequence. The results obtained allow us to classify the diffusion encoding sequences in terms of sensitivity to the cardiac motion and identify for each sequence a temporal window in the cardiac cycle in which the influence of motion is reduced. Finally, in the fifth chapter, a motion correction method is presented to minimize the effect of cardiac motion on the diffusion images. This method is based on a singular development of the modified Bloch-Torrey model in order to obtain an asymptotic model of ordinary differential equation that gives a relationship between the true diffusion and the diffusion reconstructed in the presence of motion. This relationship is then used to solve the inverse problem of recovery and correction of the diffusion influenced by the cardiac motion.

Cool giant and supergiant stars are among the largest and most luminous stars in the Universe and, therefore, dominate the integrated light of their host galaxies. These stars were extensively studied during last few decades, however their relevant properties like photometric variability and mass loss are still poorly constrained. Understanding of these properties is crucial in the context of a broad range of astrophysical questions including chemical enrichment of the Universe, supernova progenitors, and the extragalactic distance scale. Atmospheres of giant and supergiant stars are characterized by complex dynamics due to different interacting processes, such as convection, pulsation, formation of molecules and dust, and the development of mass loss. Current 1D/3D dynamical model atmospeheres are able to simulate these processes and produce a good agreement with the observed spectral features of evolved stars. However, the models lack constraints and need to be confronted to observables. Dynamical processes in stellar atmospheres impact the formation of spectral lines producing their asymmetries and Doppler shifts. Thus, by studying the line-profile variations on spatial and temporal scales it is possible to reconstruct atmospheric motions in evolved stars. As will be shown in this thesis, a tomographic method is an ideal technique for this purpose. The tomographic method is based on construction and cross-correlation of spectral templates (masks) with observed or synthetic stellar spectra in order to recover velocity fields at different optical depths in the stellar atmosphere.The first part of the thesis further improves the original implementation of the tomographic method. This improvement involves the computation of the contribution function in order to correctly determine an optical depth of formation of spectral lines. The tomographic method is, then, fully validated by applying it to a stellar convection simulation of a red supergiant star and correctly recovering its velocity field throughout the atmosphere. The second part of the thesis applies the tomographic method to the red supergiant star μ Cep in order to constrain its atmospheric motions and relate them to photometric variability. A phase lag (hysteresis) between the effective temperature and the radial velocity variations is revealed with timescales of a few hundred days, similar to photometric ones. A comparison to a stellar convection simulation of a red supergiant star indicates that hysteresis loops are linked to the stochastic shocks generated and shaped by the underlying large-scale convection and may be responsible for photometric variations in μ Cep. The third part of the thesis applies the tomographic method to spectro-interferometric observations of the Mira-type star S Ori. The uniform-disk angular diameters measured at wavelengths contributing to the tomographic masks increase with decrease of an optical depth probed by the masks. This validates the capability of the tomographic method to probe distinct geometrical depths in the stellar atmosphere. The last part of the thesis applies the tomograhic method to the Mira-type star RY Cep and compares the results to those obtained for μ Cep in this thesis. The comparison reveals differences in their behavior in the temperature-velocity plane pointing to the posibility to differentiate between Mira-type and red supergiant stars from their spectroscopic signatures. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1979. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Includes bibliographical references. / by Khaled I. Yared. / Ph.D.

Random functions is the central component in many statistical and probabilistic problems. This dissertation presents theoretical analysis and computation for random functions and its applications in statistics. This dissertation consists of two parts. The first part is on the topic of classic continuous random fields. We present asymptotic analysis and computation for three non-linear functionals of random fields. In Chapter 1, we propose an efficient Monte Carlo algorithm for computing P{sup_T f(t)>b} when b is large, and f is a Gaussian random field living on a compact subset T. For each pre-specified relative error ɛ, the proposed algorithm runs in a constant time for an arbitrarily large $b$ and computes the probability with the relative error ɛ. In Chapter 2, we present the asymptotic analysis for the tail probability of ∫_T e^{σf(t)+μ(t)}dt under the asymptotic regime that σ tends to zero. In Chapter 3, we consider partial differential equations (PDE) with random coefficients, and we develop an unbiased Monte Carlo estimator with finite variance for computing expectations of the solution to random PDEs. Moreover, the expected computational cost of generating one such estimator is finite. In this analysis, we employ a quadratic approximation to solve random PDEs and perform precise error analysis of this numerical solver. The second part of this dissertation focuses on topics in statistics. The random functions of interest are likelihood functions, whose maximum plays a key role in statistical inference. We present asymptotic analysis for likelihood based hypothesis tests and sequential analysis. In Chapter 4, we derive an analytical form for the exponential decay rate of error probabilities of the generalized likelihood ratio test for testing two general families of hypotheses. In Chapter 5, we study asymptotic properties of the generalized sequential probability ratio test, the stopping rule of which is the first boundary crossing time of the generalized likelihood ratio statistic. We show that this sequential test is asymptotically optimal in the sense that it achieves asymptotically the shortest expected sample size as the maximal type I and type II error probabilities tend to zero. These results have important theoretical implications in hypothesis testing, model selection, and other areas where maximum likelihood is employed.

An overview of unsteady incompressible jet flows is presented, with the primary interest being radially developing jets in cylindrical polar coordinates. The radial free jet emanates from some orifice, being axisymmetric about the transverse (z) axis and possessing reflectional symmetry across its z=0 centreline. The radial wall jet is also axisymmetric about the transverse axis, however in this case impermeability and no-slip conditions are imposed at the wall, which is situated at z=0. The numerical solution of a linear perturbation superposed on the free jet, whose temporal form is assumed to be driven by a periodic source pulsation, gives rise to a wave-like disturbance whose amplitude grows downstream as its local wavelength decreases. An asymptotic analysis of this linear perturbation, which applies to the wall jet as well with some minor changes, captures the exact nature of the exponential spatial growth, and also algebraic attenuation of the growth. The linear theory is only valid for a small amplitude pulsation (|ε| << 1, where ε is the perturbation amplitude). When a nonlinear pulsation (ε = O(1)) is applied to the radial free jet, any linear theory must be dropped. Solving the full nonlinear system of equations reveals singular behaviour at a critical downstream location, which corresponds to the presence of an infinitely steep downstream gradient. The replacement of molecular diffusivity with a larger-scale eddy viscosity does little to affect the qualitative growth of the linear perturbation. In order for an experimental study to reproduce any of the discussed boundary-layer results, we must consider the behaviour of jet-type flows at finite Reynolds number. This involves solving the full Navier-Stokes equations numerically, to determine the Reynolds number at which we should expect to qualitatively recover boundary-layer behaviour. The steady solution for the radial free jet and its linear pulsation are studied in this way, as is the linear pulsatile planar free jet. We may enhance the streamwise velocity of a radial jet by applying swirl around the z axis. Modulating this swirl is looked at as a possible mechanism to induce the previously discussed pulsation, which then motivates the introduction of a finite spinning disk problem. In this case the system may be completely confined within an enclosed cylinder, making a hypothetical experimental approach somewhat more approachable.