The research overviewed in this dissertation concerns highly accurate variational calculations of small molecular systems without assuming the Born--Oppenheimer approximation. The centerpiece of the research is the use of different forms of explicitly correlated Gaussian basis functions. These basis functions allow analytical evaluation of all necessary matrix elements and provide a very powerful tool for solving quantum mechanical problems encountered in various areas of physics. Most of the derivations presented in the dissertation are done within the formalism of matrix differential calculus that has proven to be a very handy and effective way of dealing with explicitly correlated Gaussians. As this fomalism is not widely used in physics or chemistry, some mathematical background is provided. The expressions obtained theoretically were implemented in a computer code that was run quite extensively on several parallel computer systems during the period of the author's Ph.D. study. The results of many such calculations are presented and discussed. The dissertation is primarily based on the content of the papers that were published in coathorship with my scientific advisor and other collaborators in several scientific journals. It also includes some topics that were not considered in the publications but are essential for the completeness and good understanding of the presented work.

In this thesis, an innovative algorithm for improving the accuracy of variational space-time stereoscopic reconstruction of ocean surfaces is presented. The space-time reconstruction method, developed based on stereo computer vision principles and variational optimization theory, takes videos captured by synchronized cameras as inputs and produces the shape and superficial pattern of an overlapped region of interest as outputs. These outputs are designed to be the minimizers of the variational optimization framework and are dependent on the estimation of the camera parameters. Therefore, from the perspective of computer vision, the proposed algorithm adjusts the estimation of camera parameters to lower the disagreement between the reconstruction and 2-D camera recordings. From a mathematical perspective, since the minimizers of the variational framework are determined by a set of partial differential equations (PDEs), the algorithm modifies the coefficients of the PDEs based on the current numerical solutions to reduce the minimum of the optimization framework. Our algorithm increases the tolerance to the errors of camera parameters, so the joint operations of our algorithm and the variational reconstruction method can generate accurate space-time models even using videos captured by perturbed cameras as input. This breakthrough prompts the realization of ocean surface reconstruction using videos filmed by remotely-controlled helicopters in the future. A number of techniques, technical or theoretical, are explored to fulfill the development and implementation of the algorithm and relative computation issues. The effectiveness of the proposed algorithm is validated through the statistics applied to real ocean surface reconstructions of data collected from an offshore platform at the Crimean Peninsula, the Black Sea. Moreover, synthetic data generated using computer graphics are customized to simulate various situations that are not recorded in the Crimea dataset for the demonstration of the algorithm.

In this thesis I present a new numerical scheme for 2.5-D/3-D direct current resistivity modelling in heterogeneous, anisotropic media. This method, named the ‘Gaussian quadrature grid’ (GQG) method, co-operatively combines the solution of the Variational Principle of the partial differential equation, Gaussian quadrature abscissae and local cardinal functions so that it has the main advantages of the spectral element method. The formulation shows that the GQG method is a modification of the spectral element method and does not employ the constant elements and require the mesh generator to match the earth’s surface. This makes it much easier to deal with geological models having a 2-D/3-D complex topography than using traditional numerical methods. The GQG technique can achieve a similar convergence rate to the spectral element method. It is shown that it transforms the 2.5-D/3-D resistivity modelling problem into a sparse and symmetric linear equation system, which can be solved by an iterative or matrix inversion method. Comparison with analytic solutions for homogeneous isotropic and anisotropic models shows that the error depends on the Gaussian quadrature order (abscissae number) and the sub-domain size. The higher order or smaller the subdomain size employed, the more accurate the solution. Several other synthetic examples, both homogeneous and inhomogeneous, incorporating sloping, undulating and severe topography are presented and found to yield results comparable to finite element solutions involving a dense mesh. The thesis also presents for the first time explicit expressions for the Fréchet derivatives or sensitivity functions in resistivity imaging of a heterogeneous and fully anisotropic earth. The formulation involves the Green’s functions and their gradients, and is developed both from a formal perturbation analysis and by means of a numerical (finite element) method. A critical factor in the equations is the derivative of the electrical conductivity tensor with respect to the principal conductivity values and the angles defining the axes of symmetry; these are given analytically. The Fréchet derivative expressions are given for both the 2.5-D and the 3-D problem using both constant point and constant block model parameterisations. Special cases like the isotropic earth and tilted transversely isotropic (TTI) media are shown to emerge from the general solutions. Numerical examples are presented for the various sensitivities as functions of the dip angle and strike of the plane of stratification in uniform TTI media. In addition, analytic solutions are derived for the electric potential, current density and Fréchet derivatives at any interior point within a 3-D transversely isotropic homogeneous medium having a tilted axis of symmetry. The current electrode is assumed to be on the surface of the Earth and the plane of stratification given arbitrary strike and dip. Profiles can be computed for any azimuth. The equipotentials exhibit an elliptical pattern and are not orthogonal to the current density vectors, which are strongly angle dependent. Current density reaches its maximum value in a direction parallel to the longitudinal conductivity direction. Illustrative examples of the Fréchet derivatives are given for the 2.5-D problem, in which the profile is taken perpendicular to strike. All three derivatives of the Green’s function with respect to longitudinal conductivity, transverse resistivity and dip angle of the symmetry axis (dG/dσ₁,dG/dσ₁,dG/dθ₀ ) show a strongly asymmetric pattern compared to the isotropic case. The patterns are aligned in the direction of the tilt angle. Such sensitivity patterns are useful in real time experimental design as well as in the fast inversion of resistivity data collected over an anisotropic earth. / Thesis (Ph.D.) -- University of Adelaide, School of Chemistry and Physics, 2009

In this dissertation recent advances in modeling various atomic and molecular systems with quantum mechanical calculations employing explicitly correlated Gaussian functions are presented. The author has utilized multiple approaches and considered a number of approximations to develop optimal calculation frameworks. Electronic and muonic molecules and atoms have been considered. A number of unique calculations have been performed and some novel and interesting results, including high accuracy description of the charge asymmetry in the heteronuclear systems and lifetimes of rotationless vibrational levels of diatomic molecules, have been generated.

Made available in DSpace on 2015-12-10T14:23:02Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-03-07. Added 1 bitstream(s) on 2015-12-10T14:27:52Z : No. of bitstreams: 1 000852795.pdf: 576204 bytes, checksum: 28ede436e9367885bc3b672b1903caad (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / A teoria de Hamilton-Jacobi geralmente é apresentada como uma extensão da teoria de Hamilton através das transformações canônicas. No entanto, o matemático Constantin Carathéodory mostrou que essa teoria, sua existência e validade, independem do formalismo hamiltoniano. Neste trabalho, apresentaremos a abordagem de Carathéodory para a teoria de Hamilton-Jacobi. Partindo desse procedimento, construiremos uma teoria de vínculos para que se possa resolver problemas com vínculos involutivos e não-involutivos. Para isso, analisaremos a integrabilidade das equações e introduziremos a operação dos parênteses generalizados que, no lugar do parênteses de Poisson, passará a descrever a dinâmica de sistemas vinculados. Mostraremos uma aplicação dessa teoria de vínculos no modelo BF da teoria de campos. Para finalizar, trataremos da Termodinâmica Axiomática de Carathéodory e também da teoria de Hamilton-Jacobi na Termodinâmica, o que é válido para ilustrar a grande abrangência desse formalismo / The Hamilton-Jacobi theory is usually presented as an extension of the Hamilton's theory through the canonical transformations. However, the mathematician Constantin Carathéodory showed this theory, its existence and validity, is independent of the Hamiltonian formalism. In this work, we present the Caratheodory's approach to the Hamilton-Jacobi theory. From this procedure, we build a theory of constraints which can solve problems with involutive and non-involutive constraints. For this, we analyze the integrability of the equations and introduce the operation of the generalized brackets that, instead of Poisson brackets, will describe the dynamics of constrained systems. We show an application of this theory in BF model of the field theory. Finally, we will discuss the Carathéodory's Axiomatic Thermodynamics and also show the Hamilton-Jacobi theory in Thermodynamics, which is valid to illustrate the wide coverage of this formalism / CNPq: 133488/2011-0

Made available in DSpace on 2014-06-11T19:32:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-08-17Bitstream added on 2014-06-13T20:27:17Z : No. of bitstreams: 1 valcarcelflores_ce_dr_ift.pdf: 694272 bytes, checksum: e1b097c2bc884f3cf2ae38593c38d4ba (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Nesta tese, apresentamos a formulação clássica completa da teoria de Hamilton-Jacobi para sistemas vinculados. Usando o método de Lagrangianas Equivalentes de Carathéodory obtemos um conjunto de Equações Diferenciais Parciais de Hamilton-Jacobi, também chamado de Hamiltonianos. A Condição de Integrabilidade nos permite dividir os Hamiltonianos entre involutivos e não-involutivos. Construímos os Parênteses Generalizados a fim de eliminar os Hamiltonianos não-involutivos, enquanto que relacionamos os Hamiltonianos involutivos com o Gerador das transformações canônicas. Por outro lado, a Equação de Lie é resultado da realização das variações totais no funciona lde ação, e que é relacionada às simetrias da teoria. Usamos a Equação de Lie e a estrutura das Equaçõoes Características, que indicam a evolução dinâmica do sistemas, para associar o Gerador de transformações canônicas às simetrias de calibre. Aplicamos o formalismo de Hamilton-Jacobi ao modelo da Mecânica Quântica Topologica, ao modelo BF bi-dimensional equivalente à Teoria de Jackiw-Teitelboim, ao campo de Yang-Mills Topologicamente Massivo e seu equivalente Auto-dual, assim como para o campo da Gravitação linearizada / It is presented the complete classical formulation of the Hamilton-Jacobi theory for constrained systems. From fixed point variations and using the Carathéodory’s method of Equivalent Lagrangian we obtain a set of Hamilton-Jacobi Partial Differential Equations, also called Hamiltonians. The Integrability Condition allow us to divide the Hamiltonians between involutive and non-involutive ones. We build the Generalized Brackets in order to eliminate the non-involutive Hamiltonians, whereas we relate the involutive Hamiltonians to the Generator of Canonical Transformations. On the other hand, we build the Lie Equation, result of perform total variations to the action functional and which is related to the symmetries of the theory. We use the Lie equation along with the structure of the Characteristic Equations, related to the dynamical evolution of the systems, to associate the Generator of Canonical Transformation to Gaugesymmetries. We apply this formalism to the Topologically Quantum Mechanics, the two dimensional BF model equivalent to the Jackiw-Teitelboim theory, the Topologically Massive Yang-Mills field as well as its correspondent self-dual and to the Linearized Gravity field

Numerous approaches have been proposed for solving partial differential equations; all these methods have their own advantages and disadvantages depending on the problems being treated. In recent years there has been much development of particle methods for mechanical problems. Among these are the Smoothed Particle Hydrodynamics (SPH), Reproducing Kernel Particle Method (RKPM), Element Free Galerkin (EFG) and Moving Least Squares (MLS) methods. This development is motivated by the extension of their applications to mechanical and engineering problems. Since numerical experiments are one of the basic tools used in computational mechanics, in physics, in biology etc, a robust spatial discretization would be a significant contribution towards solutions of a number of problems. Even a well-defined stable and convergent formulation of a continuous model does not guarantee a perfect numerical solution to the problem under investigation. Particle methods especially SPH and RKPM have advantages over meshed methods for problems, in which large distortions and high discontinuities occur, such as high velocity impact, fragmentation, hydrodynamic ram. These methods are also convenient for open problems. Recently, SPH and its family have grown into a successful simulation tools and the extension of these methods to initial boundary value problems requires further research in numerical fields. In this thesis, several problem areas of the SPH formulation were examined. Firstly, a new approach based on ‘Hamilton’s variational principle’ is used to derive the equations of motion in the SPH form. Secondly, the application of a complex Von Neumann analysis to SPH method reveals the existence of a number of physical mechanisms accountable for the stability of the method. Finally, the notion of the amplification matrix is used to detect how numerical errors propagate permits the identification of the mechanisms responsible for the delimitation of the domain of numerical stability. By doing so, we were able to erect a link between the physics and the numerics that govern the SPH formulation.

Nesse trabalho estudamos questões de existência, multiplicidade e concentração de soluções de uma classe de problemas elípticos biharmônicos. Nos três primeiros capítulos são utilizados métodos variacionais para estudar a existência, multiplicidade e comportamento assintótico das soluções fracas não-triviais de equações de Schrödinger estacionárias biharmônicas com diferentes hipóteses sobre o potencial e sobre a não-linearidade. No último capítulo, o método de decomposição em cones duais é empregado para obter a existência de três soluções (positiva, negativa e nodal) para uma equação biharmônica / In this work we study some problems on existence, multiplicity and concentration of solutions of biharmonic elliptic equtions. In the first three chapters, variational methods are used to study the existence, multiplicity and the asymptotic behavior of weak nontrivial solutions of stationary Schrödinger biharmonic equations under certain assumptions on the potential function and the nonlinearity. In the last chapter we use variational methods again and also the dual decomposition method to get existence of positive, negative and sign-changing solutions for a biharmonic equation

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Orientador: Bruto Max Pimentel Escobar / Banca: Abraham Zimerman / Banca: Denis Dalmazi / Banca: Ion Vasile Vancea / Banca: Vladislav Kupriyanov / Resumo: Nesta tese, apresentamos a formulação clássica completa da teoria de Hamilton-Jacobi para sistemas vinculados. Usando o método de Lagrangianas Equivalentes de Carathéodory obtemos um conjunto de Equações Diferenciais Parciais de Hamilton-Jacobi, também chamado de Hamiltonianos. A Condição de Integrabilidade nos permite dividir os Hamiltonianos entre involutivos e não-involutivos. Construímos os Parênteses Generalizados a fim de eliminar os Hamiltonianos não-involutivos, enquanto que relacionamos os Hamiltonianos involutivos com o Gerador das transformações canônicas. Por outro lado, a Equação de Lie é resultado da realização das variações totais no funciona lde ação, e que é relacionada às simetrias da teoria. Usamos a Equação de Lie e a estrutura das Equaçõoes Características, que indicam a evolução dinâmica do sistemas, para associar o Gerador de transformações canônicas às simetrias de calibre. Aplicamos o formalismo de Hamilton-Jacobi ao modelo da Mecânica Quântica Topologica, ao modelo BF bi-dimensional equivalente à Teoria de Jackiw-Teitelboim, ao campo de Yang-Mills Topologicamente Massivo e seu equivalente Auto-dual, assim como para o campo da Gravitação linearizada / Abstract: It is presented the complete classical formulation of the Hamilton-Jacobi theory for constrained systems. From fixed point variations and using the Carathéodory's method of Equivalent Lagrangian we obtain a set of Hamilton-Jacobi Partial Differential Equations, also called Hamiltonians. The Integrability Condition allow us to divide the Hamiltonians between involutive and non-involutive ones. We build the Generalized Brackets in order to eliminate the non-involutive Hamiltonians, whereas we relate the involutive Hamiltonians to the Generator of Canonical Transformations. On the other hand, we build the Lie Equation, result of perform total variations to the action functional and which is related to the symmetries of the theory. We use the Lie equation along with the structure of the Characteristic Equations, related to the dynamical evolution of the systems, to associate the Generator of Canonical Transformation to Gaugesymmetries. We apply this formalism to the Topologically Quantum Mechanics, the two dimensional BF model equivalent to the Jackiw-Teitelboim theory, the Topologically Massive Yang-Mills field as well as its correspondent self-dual and to the Linearized Gravity field / Doutor