Diffusion equations arise in areas such as fluid mechanics, cellular biology, weather forecasting, electronics, mechanical engineering, atomic physics, environmental science, medicine, etc. This dissertation considers equations of this type that arise in mathematical finance. For over 40 years traders in financial markets around the world have used Black-Scholes equations for valuing financial options. These equations need to be solved quickly and accurately so that the traders can make prompt and accurate investment decisions. One way to do this is to use parallel numerical algorithms. This dissertation develops and evaluates algorithms of this kind that are based on the Laplace transform, numerical inversion algorithms and finite difference methods. Laplace transform-based algorithms have faced a legitimate criticism that they are ill-posed i.e. prone to instability. We demonstrate with reference to the Black-Scholes equation, contrary to the received wisdom, that the use of the Laplace transform may be used to produce reasonably accurate solutions (i.e. to two decimal places), in a fast and reliable manner when used in conjunction with standard PDE techniques. To set the scene for the investigations that follow, the reader is introduced to financial options, option pricing and the one-dimensional and two-dimensional linear and nonlinear Black-Scholes equations. This is followed by a description of the Laplace transform method and in particular, four widely used numerical algorithms that can be used for finding inverse Laplace transform values. Chapter 4 describes methodology used in the investigations completed i.e. the programming environment used, the measures used to evaluate the performance of the numerical algorithms, the method of data collection used, issues in the design of parallel programs and the parameter values used. To demonstrate the potential of the Laplace transform based approach, Chapter 5 uses existing procedures of this kind to solve the one-dimensional, linear Black-Scholes equation. Chapters 6, 7, 8, and 9 then develop and evaluate new Laplace transform-finite difference algorithms for solving one-dimensional and two-dimensional, linear and nonlinear Black-Scholes equations. They also determine the optimal parameter values to use in each case i.e. the parameter values that produce the fastest and most accurate solutions. Chapters 7 and 9 also develop new, iterative Monte Carlo algorithms for calculating the reference solutions needed to determine the accuracy of the LTFD solutions. Chapter 10 identifies the general patterns of behaviour observed within the LTFD solutions and explains them. The dissertation then concludes by explaining how this programme of work can be extended. The investigations completed make significant contributions to knowledge. These are summarised at the end of the chapters in which they occur. Perhaps the most important of these is the development of fast and accurate numerical algorithms that can be used for solving diffusion equations in a variety of application areas.

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We introduce the concept of Toeplitz operator associated with the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We characterise those Toeplitz operators which are Fredholm, thus initiating the index theory.

Master of Science / Department of Statistics / Weixing Song / A robust estimation procedure for mixture linear regression models is proposed in this report by assuming the error terms follow a Laplace distribution. EM algorithm is imple- mented to conduct the estimation procedure of missing information based on the fact that the Laplace distribution is a scale mixture of normal and a latent distribution. Finite sample performance of the proposed algorithm is evaluated by some extensive simulation studies, together with the comparisons made with other existing procedures in this literature. A sensitivity study is also conducted based on a real data example to illustrate the application of the proposed method.

Orientador: Fernando Antonio Moala / Banca: Carlos Aparecido dos Santos / Banca: Manoel Ivanildo Silvestre Bezerra / Resumo: A Análise de Confiabilidade é uma área bem consolidada da estatística. Antes do surgimento dos métodos de Monte Carlo via Cadeia de Markov (MCMC), a aproximação de Laplace era muito utilizada para estimação dos parâmetros, porém após isso o MCMC passou a ser mais usado. Ao afinal deste trabalho, deseja-se realizar a comparação entre ambos os métodos de estimação buscando assim um método que produza bons resultados e com excelentes propriedades. Uma característica importante fornecida pela aproximação de Laplace é fato de obter uma forma fechada para distribuição a posteriori da confiabilidade, tanto no caso da Weibull como para Gama Exponenciada (GE). Também foi desenvolvido prioris conjugadas para o parâmetro e a confiabilidade para ambas distribuições. Esta propriedade facilita a obtenção dos momentos amostrais assim como o cálculo de intervalos. A distribuição Weibull é muito utilizada na análise de confiabilidade e outras áreas, assim como: climatologia, medicina, entre outras. A distribuição GE não é tão utilizada, mas apresenta diferentes comportamentos para o risco, além de possuir apenas um parâmetro para ser estimado. / Abstract: Reliability Analysis is a well-established area of statistics. Before the method Monte Carlo via Markov Chain (MCMC), the Laplace approximation was widely used for parameter estimation, but after that the MCMC became more used. At the end of this work, it is desired to carry out the comparison between both estimation methods, thus seeking a method that produces good results and excellent properties. An important feature provided by the Laplace approximation is that it obtains a closed form for posterior distribution of reliability, both in the case of Weibull and the Exponential Gamma (EG). We also developed conjugated priori for the parameter and the reliability for both distributions. This property makes it easier to obtain sample moments as well as the calculation of intervals. The Weibull distribution is widely used in the analysis of reliability and other areas, as well as: climatology, medicine, among others. The EG distribution is not so widely used, but presents different risk behaviors, besides having only one parameter to be estimated. / Mestre

Cette thèse s'inscrit dans le cadre de l'étude de deux sujets concernant les problèmes d'évolution liés au p-laplacien. Le premier sujet concerne l'étude du comportement asymptotique des solutions bornées lorsque le temps $t\to+\infty$. Quant au deuxième sujet, il porte sur l'étude de la non existence des solutions positives non triviales. Cette thèse se répartit en trois chapitres. Le premier chapitre est consacré à une introduction générale. Le deuxième chapitre porte sur l'étude de la convergence, lorsque $t\to+\infty$, des solutions bornées d'une équation parabolique associée au p-laplacien dans un intervalle borné avec des conditions aux limites du type soit Dirichlet, Neumann ou Robin. Ce travail était l'objet d'un article \cite{hauer-convergence-2012} accepté pour publication dans « Nonlinear Differential Equations and Applications NoDea ». Le dernier chapitre concerne l'étude de la non existence des solutions positives des équations paraboliques associées au p-laplacien avec un terme de convection et un potentiel singulier. La deuxième et quatrième section du Chapitre 3 reprennent un article \cite{Hauer:2012fk} accepté pour publication dans le journal « Archiv der Mathematik ». La deuxième sous-section de la Section 4 du Chapitre 3 contient un résultat qui améliore le travail \cite{Goldstein-Rhandi-weighted-hardy-11} de G. Goldstein, J. Goldstein et A. Rhandi et le travail \cite{MR1616905} de J. P. García Azorero et I. Peral Alonso concernant la non existence des solutions positives. Ce résultat n'est pas encore publié / This thesis is dedicated to the study of two subjects in the field of evolution problems associated with the $p$-Laplace operator. The first subject is concerned with the study of long time behavior of bounded solutions and the second subject is devoted to the study of nonexistence of positive nontrivial solutions. The first chapter of this thesis is devoted to a general introduction to the p-Laplace operator and a résumé of this thesis. The first chapter is written in French. Chapter 2 is dedicated to the study of convergence as the time $t\to+\infty$ of bounded solutions of evolution problems associated with the p-Laplace operator on a bounded interval with homogeneous Dirichlet, Neumann, or Robin boundary conditions converges. The results of Chapter 2 are contained in article \cite{hauer-convergence-2012}, which was published in the journal « Nonlinear Differential Equations and Applications NoDea ». Chapter 3 is devoted to the study of nonexistence of positive nontrivial weak solutions of parabolic equations associated to the p-Laplace operator with a convection term and a singular potential. The results of Section 3.2 and Section 3.4.1 of Chapter 3 are contained in article \cite{Hauer:2012fk}, which was accepted for publication in the journal « Archiv der Mathematik ». The results of Section 3.4.2 of Chapter 3 are not yet published

Made available in DSpace on 2014-06-12T17:41:39Z (GMT). No. of bitstreams: 2 arquivo7326_1.pdf: 1202400 bytes, checksum: a85b1fc731f434bf7654d913dbfa3b5a (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2006 / Petróleo Brasileiro S/A / Neste trabalho, é proposta uma metodologia de modelagem de indicadores de desempenho de Confiabilidade ((In)Disponibilidade, Confiabilidade, Manutenibilidade) de sistemas complexos baseada na integração entre processos semi Markovianos (PSMs) e Redes Bayesianas (RBs). Basicamente, um PSM pode ser entendido como um processo estocástico no qual as probabilidades de transição dependem do intervalo de tempo decorrido desde o qual um sistema possui determinadas características. Já as Redes Bayesianas são estruturas probabilísticas que representam qualitativa e quantitativamente relações de causa e efeito entre determinadas variáveis aleatórias de interesse. A integração entre os PSMs e as RBs origina um modelo estocástico híbrido o qual é capaz de representar a dinamicidade de um sistema ao mesmo tempo em que trata como as relações de causa e efeito entre fatores não necessariamente temporais influenciam tal evolução. Para desenvolver tal modelo híbrido, faz-se necessário propor e formular o método numérico computacional de resolução das equações de probabilidades de transição dos PSMs definidos através de taxas de transição as quais são equações integrais do tipo convolução. Tal método é baseado na aplicação de transformadas de Laplace as quais serão invertidas utilizando o método de Quadratura Gaussiana conhecido como Gauss Legendre. Aplicações do modelo híbrido proposto são realizadas em sistemas tolerantes à falha com o objetivo de avaliar a evolução temporal dos indicadores de desempenho de Confiabilidade

Submitted by Nathalya Silva (nathyjf033@gmail.com) on 2017-04-24T19:49:26Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Tese_SolucaoEquacoesBalanco.pdf: 13765561 bytes, checksum: 76cb5628e30109a10b363ed3f225400e (MD5) / Approved for entry into archive by Edisangela Bastos (edisangela@ufpa.br) on 2017-04-25T21:39:33Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Tese_SolucaoEquacoesBalanco.pdf: 13765561 bytes, checksum: 76cb5628e30109a10b363ed3f225400e (MD5) / Made available in DSpace on 2017-04-25T21:39:33Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Tese_SolucaoEquacoesBalanco.pdf: 13765561 bytes, checksum: 76cb5628e30109a10b363ed3f225400e (MD5) Previous issue date: 2011-12 / A evolução da distribuição do tamanho de partículas em muitos campos da ciência aplicada como cristalização, física de aerossol, química coloidal e processo de polimerização, pode ser obtida pela solução da equação de balanço populacional (PBE). A técnica da transformada de Laplace com inversão numérica foi usada para resolver uma equação integro-diferencial parcial que relaciona a modelagem matemática do problema físico para estudar processos convectivos com taxas de nascimento e morte de partículas e aerossóis. Tal modelo é governado PBE, na qual leva em consideração a nucleação, crescimento e processos de coagulação. Um método Bayesiano foi usado para resolver o problema inverso hiperbólico e não-linear, e estimar a função densidade de tamanho de partículas, e assim prever o comportamento dinâmico do sistema físico. Especificamente o filtro de partículas com amostragem e Reamostragem por Importância Sequencial (SIR) foi utilizado como metodologia de solução do problema. Através dessas soluções, resultados numéricos foram obtidos e comparados com os disponíveis na literatura para sistemas particulados, permitindo uma avaliação crítica da presente metodologia de solução. / The evolution of particle size distribution in many fields of applied science, such as crystallization, aerosols, colloids, and polymer processing, can be obtained by solving population balance equation (PBE). The Laplace transform technique with numerical inversion was used to solve an integro-partial-differential equation related to the mathematical modeling of the physical problem to study convective processes with birth and death rates of particles or aerosols. Such model is governed by the population balance equation (PBE), in which is taken into account the nucleation, growth and coagulation processes. A Bayesian method was employed to solve the hyperbolic and non-linear inverse problem and estimate the size distribution density function, thus predicting the dynamic behavior of the physical system. Specifically the particle filter with sampling Importance Resampling (SIR) has been applied as a method of solving the problem. From these solutions, numerical results were obtained and compared with those in the literature for particulate systems permitting a critical evaluation of the present solution methodology.

The primary arithmetic information attached to a Maass cusp form is its Laplace eigenvalue. However, in the case of cuspidal Maass forms, the range that these eigenvalues can take is not well-understood. In particular it is unknown if, given a real number r, one can prove that there exists a primitive Maass cusp form with Laplace eigenvalue 1/4 + r2. Conversely, given the Fourier coefficients of a primitive Maass cusp form f on Γ0(D), it is not clear whether or not one can determine its Laplace eigenvalue. In this paper we show that given only a finite number of Fourier coefficients one can first determine the level D, and then compute the Laplace eigenvalue to arbitrarily high precision. The key to our results will be understanding the resonance and rapid decay properties of Maass cusp forms. Let f be a primitive Maass cusp form with Fourier coefficients λf (n). The resonance sum for f is given by SX(f;α;β) = Εn≥1λf(n)‑Φ(n/X) e(αnβ) where φ ∈ Cc∞((1, 2)) is a Schwartz function and α ∈ R and β, X > 0 are real numbers. Sums of this form have been studied for many different classes of functions f, including holomorphic modular forms for SL(2, Z), and Maass cusp forms for SL(n,Z). In this paper we take f to be a primitive Maass cusp form for a congruence subgroup Γ0(D) ⊂ SL(2, Z). Thus our result extends the family of automorphic forms for which their resonance properties are understood. Similar analysis and algorithms can be easily implemented for holomorphic cusp forms for Γ0(D). Our techniques include Voronoi summation, weighted exponential sums, and asymptotics expansions of Bessel functions. We then use these estimates in a new application of resonance sums. In particular we show that given only limited information about a Maass cusp form f (in particular a finite list of high Fourier coefficients), one can determine its level and estimate its spectral parameter, and thus its Laplace eigenvalue. This is done using a large parallel computing cluster running MATLAB and Mathematica