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Thesis (Ph. D.)--University of Texas at Austin, 2005. / Supervisor: Luis A. Caffarelli. Vita. Includes bibliographical references.

This dissertation provides a brief theoretical introduction to the Malliavin calculus leading to a particular application in finance. The Malliavin calculus concepts are used to aid in the simulation of the Greeks for financial contingent claims. Particular focus is placed on creating efficiency in the more exotic type option simulations, where no closed solution pricing formulae exist. Copyright / Dissertation (MSc)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

The solution for the edge-isoperimetric problem (EIP) of the square tessellation of plane is investigated and solved. Summaries of the stabilization theory and previous research dealing with the EIP are stated. These techniques enable us to solve the EIP of the cubical tessellation.

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This work develops theoretical and applied results for variational convex analysis. First we present the basic tools of analysis necessary to develop the core theory and applications. New results concerning duality principles for systems originally modeled by non-linear differential equations are shown in chapters 9 to 17. A key aspect of this work is that although the original problems are non-linear with corresponding non-convex variational formulations, the dual formulations obtained are almost always concave and amenable to numerical computations. When the primal problem has no solution in the classical sense, the solution of dual problem is a weak limit of minimizing sequences, and the evaluation of such average behavior is important in many practical applications. Among the results we highlight the dual formulations for micro-magnetism, phase transition models, composites in elasticity and conductivity and others. To summarize, in the present work we introduce convex analysis as an interesting alternative approach for the understanding and computation of some important problems in the modern calculus of variations. / Ph. D.

We study minimizers of the Landau-de Gennes energy in the exterior region around a smooth 2-manifold in R3 with a constant external magnetic field present. Uniaxial boundary data and a strong tangential anchoring are imposed on the surface of the manifold and we consider the large particle limit in a regime where the magnetic field is relatively weak. Before studying the general manifold, we analyze a more simple case in which the manifold is spherical. After deriving a lower bound for the energy in this limiting regime, we prove that a director field on the boundary which maximizes its vertical component yields a minimal lower bound. We then construct a recovery sequence to show that this lower bound is in fact the optimal energy bound. These steps are later repeated in more generality for a larger class of smooth manifolds. / Thesis / Master of Science (MSc)

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Orientadora: Suzinei Aparecida Siqueira Marconato / Banca: Maria Aparecida Bená / Banca: Jair Silvério dos Santos / Resumo: Neste trabalho são apresentados o famoso problema da braquistócrona e diferentes soluções através da teoria do cálculo variacional e através de conceitos da geometria e física, envolvendo situações com condições análogas às da braquistócrona. Uma proposta didática adequada a alunos do Ensino Médio que é adaptável a alunos do Ensino Fundamental foi também apresentada / Abstract: In this work it was presented the famous brachistochrone problem and the di erent solutions through the theory of variational calculus and through the concepts of geometry and physics, involving situations with similar conditions to those of brachistochrone. Adequate didactic proposal to high school students which is also suitable for middle school students was presented / Mestre

Some quasistatic evolution problems for a phase transition model with nonconvex energies are studied in the generalized framework of Young measures. More in details, an existence result for a generalized notion of globally stable quasistatic evolution is proved both in the continuous and in the discrete case (infinite many/ finite many phases); an existence result for a notion of approximable evolution is also provided via a sort of vanishing viscosity.