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São estudadas flutuações de variáveis spin de bloco em alguns modelos de Curie-Weiss. É descrito rigorosamente o comportamento assintótico de suas distribuições de probabilidade no limite termodinâmico, mantendo constante a razão entre o tamanho do sistema e o tamanho do bloco. São considerados o modelo de Ising com campo aleatório e o antiferromagneto diluído. Os seguintes fatos sobre flutuações nestes modelos são provados: a) Elas não são auto-mediantes; b) Fora da criticalidade têm distribuição Gaussiana com contribuições vindas de flutuações térmicas e de flutuações devidas aos parâmetros aleatórios; c) Na criticalidade a sua distribuição e não mais Gaussiana e as flutuações das impurezas dominam as flutuações térmicas. Como sub-produto desta análise mostra-se que as flutuações destes dois modelos não são equivalentes sob o mapeamento que estabelece a sua equivalência termodinâmica. Também é descrita a aplicação do método ao vidro de spin de van Hemmen, sem provas, levando a resultados similares. Finalmente mostra-se que o método é problemático quando aplicado a sistemas quânticos. Embora a sua termodinâmica possa ser bem descrita, aparecem alguns problemas matemáticos, ainda por resolver, no estudo das suas flutuações. / Fluctuations of block spin variables in some Curie-Weiss models are studied. The asymptotic behavior of their probability distributions in the thermodynamic limit is rigorously described, keeping constant the ratio between the size of the system and the size of the block. The Ising model with random field and the dilute antiferromagnet with uniform field are considered. The following facts about fluctuations in these models are proved: a) They are not self-averaging; b) Out of criticality they have a Gaussian distribution with contributions coming both from thermal fluctuations and from those fluctuations due to the random parameters; c) At criticality their distribution is no longer Gaussian and the fluctuation of impurities dominate thermal fluctuations. As a by-product of this analysis, the fluctuations of these two models are shown to be non-equivalent under the mapping which establishes their thermodynamical equivalence. It is also described the application of the method to the van Hemmen spin-glass model, without proofs, leading to similar results. Finally the method is shown to be problematic when applied to quantum systems. Although their thermodynamics can be well described, some mathematical problems, yet to be solved, appear in the study of their fluctuations.

Submitted by Rosivalda Pereira (mrs.pereira@ufma.br) on 2017-07-05T21:25:08Z No. of bitstreams: 1 AlanPaciencia.pdf: 382837 bytes, checksum: 5f9c9a1520895e9d9b37a6549ee31251 (MD5) / Made available in DSpace on 2017-07-05T21:25:08Z (GMT). No. of bitstreams: 1 AlanPaciencia.pdf: 382837 bytes, checksum: 5f9c9a1520895e9d9b37a6549ee31251 (MD5) Previous issue date: 2017-01-05 / In this work, we study situations involving the existence, uniqueness, decay rates and asymptotic behavior of solutions for a class of nonlinear equations cards and memory. In particular, in the first chapter we review some issues related to a number of results derived from the general theory of functional analysis, which will be applied during this dissertation. The next chapter will discuss an equation of the fourth order dissipative plate with nonlinear perturbations of type p - Laplacian and locally Lipschitz and memory. Continuing, we prove the exponential stability of energy corresponding to the homogeneous problem with second-order term of memory. / No presente trabalho, estudaremos situações relacionadas a existência, unicidade, taxas de decaimento e comportamentos assintóticos de soluções para uma classe de equações de placas não linear e com termo de memória. Em particular, no primeiro capítulo revisamos alguns assuntos relacionados a uma série de resultados oriundos da teoria geral da análise funcional, os quais ser˜ao aplicados no decorrer dessa dissertação. No capítulo seguinte, abordaremos uma equação da placa de quarta ordem dissipativa com pertubações não lineares do tipo p - Laplaciano e localmente Lipschitz e com termo memória. Continuando, provamos a estabilidade exponencial de energia correspondente ao problema homogêneo com termo de memória de segunda ordem.

São estudadas flutuações de variáveis spin de bloco em alguns modelos de Curie-Weiss. É descrito rigorosamente o comportamento assintótico de suas distribuições de probabilidade no limite termodinâmico, mantendo constante a razão entre o tamanho do sistema e o tamanho do bloco. São considerados o modelo de Ising com campo aleatório e o antiferromagneto diluído. Os seguintes fatos sobre flutuações nestes modelos são provados: a) Elas não são auto-mediantes; b) Fora da criticalidade têm distribuição Gaussiana com contribuições vindas de flutuações térmicas e de flutuações devidas aos parâmetros aleatórios; c) Na criticalidade a sua distribuição e não mais Gaussiana e as flutuações das impurezas dominam as flutuações térmicas. Como sub-produto desta análise mostra-se que as flutuações destes dois modelos não são equivalentes sob o mapeamento que estabelece a sua equivalência termodinâmica. Também é descrita a aplicação do método ao vidro de spin de van Hemmen, sem provas, levando a resultados similares. Finalmente mostra-se que o método é problemático quando aplicado a sistemas quânticos. Embora a sua termodinâmica possa ser bem descrita, aparecem alguns problemas matemáticos, ainda por resolver, no estudo das suas flutuações. / Fluctuations of block spin variables in some Curie-Weiss models are studied. The asymptotic behavior of their probability distributions in the thermodynamic limit is rigorously described, keeping constant the ratio between the size of the system and the size of the block. The Ising model with random field and the dilute antiferromagnet with uniform field are considered. The following facts about fluctuations in these models are proved: a) They are not self-averaging; b) Out of criticality they have a Gaussian distribution with contributions coming both from thermal fluctuations and from those fluctuations due to the random parameters; c) At criticality their distribution is no longer Gaussian and the fluctuation of impurities dominate thermal fluctuations. As a by-product of this analysis, the fluctuations of these two models are shown to be non-equivalent under the mapping which establishes their thermodynamical equivalence. It is also described the application of the method to the van Hemmen spin-glass model, without proofs, leading to similar results. Finally the method is shown to be problematic when applied to quantum systems. Although their thermodynamics can be well described, some mathematical problems, yet to be solved, appear in the study of their fluctuations.

We study nonlinear least-squares problem that can be transformed to linear problem by change of variables. We derive a general formula for the statistically optimal weights and prove that the resulting linear regression gives an optimal estimate (which satisfies an analogue of the Rao–Cramer lower bound) in the limit of small noise.

La premième partie a pour but l’établissement d’un développement asymptotique pour la solution du problème de Stokes avec une petite perturbation du domaine. Dans ce travail, nous avons appliqué la théorie du potentiel. On a écrit les solutions du problème non-perturbé et du problème perturbé sous forme des opérateurs intégraux. En calculant la différence, et en utilisant des propriétés liées aux noyaux des opérateurs on a établi un développement asymptotiquede la solution.L’objectif principal de la deuxième partie de ce rapport est de déterminer les termes d’ordre élevé de l’expansion asymptotique des valeurs propres et fonctions propres pour l’opérateur de Stokes dues aux changements d’interface de l’inclusion. Dans la troisième partie, nous proposons une méthode pour l’évaluation des integrales singulières provenant de la mise en oeuvre de la méthode des éléments finis de frontière en électromagnetisme. La méthodeque nous adoptons consiste en une réduction récursive de la dimension du domained’intégration et aboutit à une représentation de l’intégrale sous la forme d’une combinaison linéaire d’intégrales mono-dimensionnelles dont l’intégrand est régulier et qui peuvent s’évaluer numériquement mais aussi explicitement. Pour la discrétisation du domaine, destriangles plans sont utilisés ; par conséquent, nous évaluons des intégrales sur le produit de deux triangles. La technique que nous avons développée nécessite de distinguer entre diverses configurations géométriques. / This thesis contains three main parts. The first part concerns the derivation of an asymptotic expansion for the solution of Stokes resolvent problem with a small perturbation of the domain. Firstly, we verify the continuity of the solution with respect to the small perturbation via the stability of the density function. Secondly, we derive the asymptotic expansion ofthe solution, after deriving the expansion of the density function. The procedure is based on potential theory for Stokes problem in connection with boundary integral equation method, and geometric properties of the perturbed boundary. The main objective of the second part on this report, is to present a schematic way to derive high-order asymptotic expansions for both eigenvalues and eigenfunctions for the Stokes operator caused by small perturbationsof the boundary. Also, we rigorously derive an asymptotic formula which is in some sense dual to the leading-order term in the asymptotic expansion of the perturbations in the Stokes eigenvalues due to interface changes of the inclusion. The implementation of the boundary element method requires the evaluation of integrals with a singular integrand. A reliable andaccurate calculation of these integrals can in some cases be crucial and difficult. In the third part of this report we propose a method of evaluation of singular integrals based on recursive reductions of the dimension of the integration domain. It leads to a representation of the integralas a linear combination of one-dimensional integrals whose integrand is regular and that can be evaluated numerically and even explicitly. The Maxwell equation is used as a model equation, but these results can be used for the Laplace and the Helmholtz equations in 3-D.For the discretization of the domain we use planar triangles, so we evaluate integrals over the product of two triangles. The technique we have developped requires to distinguish between several geometric configurations.

Neste trabalho abordamos dois exemplos de equações de difusão não locais do tipo Neumann: o problema linear homogêneo e um semilinear com termo de reação representado pela função f(u) = u|u|^(p-1). Em ambos os casos, apresentamos condições de existência e unicidade de soluções e analisamos seu comportamento em relação ao tempo. Estudamos uma discretização para o problema linear e a utilizamos para realizar simulações numéricas nas quais podemos verificar algumas das propriedades demonstradas. Também simulamos o problema semilinear observando o comportamento de suas soluções mesmo em casos em que as hipóteses dos teoremas apresentados não são todas satisfeitas. / In this work we approach two examples of nonlocal diffusion equations of Neumann type: the homogeneous linear problem and a semilinear with a reaction term represented by the function f(u) = u|u|^(p-1). In both cases, we present conditions of existence and uniqueness of solutions and we analyze their behavior with respect to time. We study a discretization to the linear problem and use it to perform numerical experiments in order to illustrate some of the demonstrated properties. We also simulate the semilinear problem observing the behavior of its solutions even in cases where the hypothesis of the presented theorems are not all satisfied.

<p>This thesis deals with a model of nonlinearly perturbed continuous-time renewal equation with nonpolynomial perturbations. The characteristics, namely the defect and moments, of the distribution function generating the renewal equation are assumed to have expansions with respect to a non-polynomial asymptotic scale: $\{\varphi_{\nn} (\varepsilon) =\varepsilon^{\nn \cdot \w}, \nn \in \mathbf{N}_0^k\}$  as $\varepsilon \to 0$, where $\mathbf{N}_0$ is the set of non-negative integers, $\mathbf{N}_0^k \equiv \mathbf{N}_0 \times \cdots \times \mathbf{N}_0, 1\leq k <\infty$ with the product being taken $k$ times and $\w$ is a $k$ dimensional parameter vector that satisfies certain properties. For the one-dimensional case, i.e., $k=1$, this model reduces to the model of nonlinearly perturbed renewal equation with polynomial perturbations which is well studied in the literature.  The goal of the present study is to obtain the exponential asymptotics for the solution to the perturbed renewal equation in the form of exponential asymptotic expansions and present possible applications.</p><p>The thesis is based on three papers which study successively the model stated above. Paper A investigates the two-dimensional case, i.e. where $k=2$. The corresponding asymptotic exponential expansion for the solution to the perturbed renewal equation is given. The asymptotic results are applied to an example of the perturbed risk process, which leads to diffusion approximation type asymptotics for the ruin probability.  Numerical experimental studies on this example of perturbed risk process are conducted in paper B, where Monte Carlo simulation are used to study the accuracy and properties of the asymptotic formulas. Paper C presents the asymptotic results for the more general case where the dimension $k$ satisfies $1\leq k <\infty$, which are applied to the asymptotic analysis of the ruin probability in an example of perturbed risk processes with this general type of non-polynomial perturbations.  All the proofs of the theorems stated in paper C are collected in its supplement: paper D.</p>

The focus of this study is the prediction of trajectories of solid particles injected into either a cylindrically- shaped solid rocket motor (SRM) or a bidirectional vortex chamber (BV). The Lagrangian particle trajectory is assumed to be governed by drag, virtual mass, Magnus, Saffman lift, and gravity forces in a Stokes flow regime. For the conditions in a solid rocket motor, it is determined that either the drag or gravity forces will dominate depending on whether the sidewall injection velocity is high (drag) or low (gravity). Using a one-way coupling paradigm in a solid rocket motor, the effects of particle size, sidewall injection velocity, and particle-to-gas density ratio are examined. The particle size and sidewall injection velocity are found to have a greater impact on particle trajectories than the density ratio. Similarly, for conditions associated with a bidirectional vortex engine, it is determined that the drag force dominates. Using a one-way particle tracking Lagrangian model, the effects of particle size, geometric inlet parameter, particle-to-gas density ratio, and initial particle velocity are examined. All but the initial particle velocity are found to have a significant impact on particle trajectories. The proposed models can assist in reducing slag retention and identifying fuel injection configurations that will ensure proper confinement of combusting droplets to the inner vortex in solid rocket motors and bidirectional vortex engines, respectively.

L'équation de transport, dans le régime fortement collisionnel admet une limite asymptotique de diffusion. Les discrétisations angulaires comme la méthode des ordonnées discrètes Sn où le développement tronqué en harmonique sphérique Pn préservent aussi cette limite de diffusion. Par conséquent, il est intéressant de construire pour de tels systèmes des méthodes de volumes finis sur maillages non structurés qui préservent cette limite de diffusion pour des grilles grossières. En effet, ces modèles peuvent être couplés avec des codes hydrodynamiques Lagrangiens qui génèrent des maillages très tordus. Pour commencer, on considère la discrétisation angulaire la plus simple de l'équation de transport appelée le modèle P1. Après une rapide introduction sur les méthodes 1D, on commence par modifier le schéma acoustique en dimension deux avec la méthode de Jin-Levermore. Le schéma ainsi obtenu n'est pas convergent dans le régime de diffusion car le schéma de diffusion valide n'est pas consistant sur maillages non structurés. Pour résoudre ce problème, on a proposé de nouvelles méthodes valides sur maillages non structurés. Ces méthodes sont basées sur un autre formalisme des méthodes de volumes finis ou les flux sont localisés aux interfaces, couplé avec la méthode de Jin-Levermore. On obtient deux schémas convergents qui dérivent sur les schémas asymptotic preserving 1D. Le schéma limite de diffusion obtenu est un nouveau schéma pour lequel on a donné une preuve de convergence. Dans un second temps, on a proposé une extension du travail réalisé pour le modèle P1 dans le cadre des discrétisations angulaires d'ordres élevés. Pour obtenir une discrétisation asymptotic preserving pour ces modèles on a utilisé une décomposition entre la discrétisation angulaire de premier ordre et les discrétisations angulaires d'ordres supérieurs. Enfin on a étudié la discrétisation du problème d'absorption/émission présent en transfert radiatif ainsi que la discrétisation du modèle non linéaire M1. L'approximation du modèle M1 est basé sur un couplage entre un schéma Lagrange+projection pour une reformulation du modèle M1 et la méthode de Jin-Levermore. La méthode numérique obtenue préserve la limite asymptotique, l'inégalité d'entropie et le principe du maximum associé au système sur maillages non structurés.