Orientador: Paulo Regis Caron Ruffino / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-12T16:41:27Z (GMT). No. of bitstreams: 1 Silva_FabianoBorgesda_D.pdf: 848923 bytes, checksum: 27f2cf2ad665ac271db23db385dab86f (MD5) Previous issue date: 2009 / Resumo: Este trabalho consiste basicamente em três níveis de decomposições de fluxos estocásticos: 1) decomposição via G-estruturas; 2) decomposição com componente em trajetórias hamiltonianas e 3) conjugações de fluxos aleatórios ¿Observação: O resumo, na íntegra poderá ser visualizado no texto completo da tese digital. / Abstract: This thesis concerns three different kind of decomposition of stochastic flows: 1) decompositions preserving G-structures; 2) decompositions with a component whose trajectories are hamiltonians and; 3) tensor preserving conjugacies with random time differentiable cociclos ...Note: The complete abstract is available with the full electronic digital thesis or dissertations. / Doutorado / Sistemas Dinamicos / Doutor em Matemática

Orientadores: Djairo Guedes de Figueiredo, Francisco Odair Vieira de Paiva / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-09T03:12:35Z (GMT). No. of bitstreams: 1 DosSantos_EdersonMoreira_D.pdf: 980783 bytes, checksum: 2dea209d1deff43e05e904ee8e9a1dac (MD5) Previous issue date: 2007 / Resumo: Neste trabalho consideramos duas classes de sistemas não homogêneos sendo que em certos casos uma dessas classes tranforma-se em um sistema gradiente, enquanto que a outra em um sistema de tipo Hamiltoniano. Analisamos as questões de existência, não-existênca, unicidade e multiplicidade de solu-ções.Para obter nossos resultados empregamos o método de subsolução e super-solução, minimização de funcionais, teorema da função implícita, teorema de multiplicadores de Lagrange, Teorema do Passo da Montanha, um teorema de representação de Riesz para alguns espaços de Sobolev e o Princípio de Concentração de Compacidade / Abstract: In this work we consider two classes of nonhomogeneous systems, where in certain cases one of these classes turns to be a gradient system, while the other one becomes a system of Hamiltonian type. We are concerned about the questions of existence, nonexistence, uniqueness and multiplicity of solutions. To obtain our results we apply the method of subsolutions and supersolutions, minimization of functionals, the Lagrange multiplier theorem, the Mountain Pass Theorem, a Riesz¿s representation theorem for certain Sobolev spaces and the Concentration-Compactness Principle / Doutorado / Doutor em Matemática

Systems of quasilinear partial differential equations of the first order, known as hydrodynamic type systems, are one of the most important classes of nonlinear partial differential equations in the modern theory of integrable systems. They naturally arise in continuum mechanics and in a wide range of applications, both in pure and applied mathematics. Deep connections between the mathematical theory of hydrodynamic type systems with differential geometry, firstly revealed by Riemann in the nineteenth century, have been thoroughly investigated in the eighties by Dubrovin and Novikov. They introduced and studied a class of Poisson structures generated by a flat pseudo-Riemannian metric, called first-order Poisson brackets of hydrodynamic type. Subsequently, these structures have been generalised in a whole variety of different ways: degenerate, non-homogeneous, higher order, multi-dimensional, and non-local. The first part of this thesis is devoted to the classification of such structures in two dimensions, both non-degenerate and degenerate. Complete lists of such structures are provided for a small number of components, as well as partial results in the multi-component non-degenerate case. In the second part of the thesis we deal with deformations of Poisson structures of hydrodynamic type. The deformation theory of Poisson structures is of great interest in the theory of integrable systems, and also plays a key role in the theory of Frobenius manifolds. In particular, we investigate deformations of two classes of structures of hydrodynamic type: degenerate one-dimensional Poisson brackets and non-semisimple bi-Hamiltonian structures associated with Balinskii-Novikov algebras. Complete classification of second-order deformations are presented for two-component structures.

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Orientador: Fabiano Borges da Silva / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T12:59:08Z (GMT). No. of bitstreams: 1 Fernandes_AristonLopes_M.pdf: 1807643 bytes, checksum: 1b8f6422f12372fd95aefa1b55d12fc7 (MD5) Previous issue date: 2011 / Resumo: Este trabalho tem como propósito estudar trajetórias geradas por sistemas hamiltonianos no plano. Para isso, são analisados os diversos tipos de retratos de fase dos sistemas lineares planares e a classificação destes. Sistemas hamiltonianos surgiram na mecânica clássica e seus pontos de equilíbrio são classificados em selas ou centros, conforme os sinais dos autovalores da matriz do sistema linearizado. Além disso, é apresentada a relação entre campos de vetores hamiltonianos e espaços vetoriais simpléticos / Abstract: This work has the objective of studing trajectories generated by Hamiltonian systems on the plane. For this, we analyse the various types of phase portraits of planar and linear systems. Hamiltonian systems have emerged in the mechanical and their classical equilibrium points are classified into saddles or centers, as the signs of the eigenvalues of linearized system matrix. We have also illustrated the connection between Hamiltonian vector fields and symplectic spaces / Mestrado / Matemática Universitária / Mestre em Matemática Universitária

Electronic structure methods enable first-principles calculations of the properties of molecules and materials. But numerically exact calculations of systems relevant to chemistry are computationally intractable due to the exponentially scaling cost of solving the associated Schrödinger equation. This thesis describes the application of quantum Monte Carlo (QMC) methods that enable the accurate solution of this equation at reduced computational cost. Chapter 2 introduces the fast randomized iteration (FRI) framework for analyzing discrete-space QMC methods for ground-state electronic structure calculations. I analyze the relative advantages of applying different strategies within this framework in terms of statistical error and computational cost. Chapter 3 discusses the incorporation of strategies from related stochastic methods to achieve further reductions in statistical error. Chapter 4 presents a general framework for extending these FRI-based approaches to calculate energies of excited electronic states. Chapter 5 demonstrates that leveraging the best of these ground- and excited-state techniques within the FRI framework enables the calculation of very accurate electronic energies in large molecular systems. In contrast to Chapters 2–5, which describe discrete-space QMC methods, Chapter 6 describes a continuous-space approach, based on diffusion Monte Carlo, for calculating optical properties of materials with a particular layered structure. I apply this approach to calculate exciton, trion, and biexciton binding energies of hybrid organic-inorganic lead-halide perovskite materials using a semiempirical Hamiltonian.

This thesis focuses on developing new approaches to solving the ground state properties of quantum many-body Hamiltonians, and the goal is to develop a systematic approach which properly balances efficiency and accuracy. Two new formalisms are proposed in this thesis: the Variational Discrete Action Theory (VDAT) and the Off-Shell Effective Energy Theory (OET). The VDAT exploits the advantages of both variational wavefunctions and many-body Green's functions for solving quantum Hamiltonians. VDAT consists of two central components: the Sequential Product Density matrix (SPD) and the Discrete Action associated with the SPD. The SPD is a variational ansatz inspired by the Trotter decomposition and characterized by an integer N, and N controls the balance of accuracy and cost; monotonically converging to the exact solution for N → ∞. The Discrete Action emerges by treating the each projector in the SPD as an effective discrete time evolution. We generalize the path integral to our discrete formalism, which converts a dynamic correlation function to a static correlation function in a compound space. We also generalize the usual many-body Green's function formalism, which results in analogous but distinct mathematical structures due to the non-abelian nature of the SPD, yielding discrete versions of the generating functional, Dyson equation, and Bethe-Salpeter equation. We apply VDAT to two canonical models of interacting electrons: the Anderson impurity model (AIM) and the Hubbard model. We prove that the SPD can be exactly evaluated in the AIM, and demonstrate that N=3 provides a robust description of the exact results with a relatively negligible cost. For the Hubbard model, we introduce the local self-consistent approximation (LSA), which is the analogue of the dynamical mean-field theory, and prove that LSA exactly evaluates VDAT for d=∞. Furthermore, VDAT within the LSA at N=2 exactly recovers the Gutzwiller approximation (GA), and therefore N>2 provides a new class of theories which balance efficiency and accuracy. For the d=∞ Hubbard model, we evaluate N=2-4 and show that N=3 provides a truly minimal yet precise description of Mott physics with a cost similar to the GA. VDAT provides a flexible scheme for studying quantum Hamiltonians, competing both with state-of-the-art methods and simple, efficient approaches all within a single framework. VDAT will have broad applications in condensed matter and materials physics. In the second part of the thesis, we propose a different formalism, off-shell effective energy theory (OET), which combines the variational principle and effective energy theory, providing a ground state description of a quantum many-body Hamiltonian. The OET is based on a partitioning of the Hamiltonian and a corresponding density matrix ansatz constructed from an off-shell extension of the equilibrium density matrix; and there are dual realizations based on a given partitioning. To approximate OET, we introduce the central point expansion (CPE), which is an expansion of the density matrix ansatz, and we renormalize the CPE using a standard expansion of the ground state energy. We showcase the OET for the one band Hubbard model in d=1, 2, and ∞, using a partitioning between kinetic and potential energy, yielding two realizations denoted as K and X. OET shows favorable agreement with exact or state-of-the-art results over all parameter space, and has a negligible computational cost. Physically, K describes the Fermi liquid, while X gives an analogous description of both the Luttinger liquid and the Mott insulator. Our approach should find broad applicability in lattice model Hamiltonians, in addition to real materials systems. The VDAT can immediately be applied to generic quantum models, and in some cases will rival the best existing theories, allowing the discovery of new physics in strongly correlated electron scenarios. Alternatively, the OET provides a practical formalism for encapsulating the complex physics of some model and allowing extrapolation over all phase space. Both of the formalisms should find broad applications in both model Hamiltonians and real materials.

We present a method of constructing the phase diagram at low temperatures, using the low temperature expansions. We consider spin Iattice systems described by a Hamiltonian with a d-dimensional perturbation space. We prove that there is a one-one correspondence between subsets of the phase diagram and extremal elements of some family of convex sets. We also solve a linear programming problem of the phase diagram for a set of affine functionals. / Ph. D.

Submitted by Maria Aparecida (cidazen@gmail.com) on 2017-04-19T13:09:32Z No. of bitstreams: 1 Francisco José dos Santos Nascimento.pdf: 743351 bytes, checksum: 997f8a5009a3bbc979a7206041daf583 (MD5) / Made available in DSpace on 2017-04-19T13:09:32Z (GMT). No. of bitstreams: 1 Francisco José dos Santos Nascimento.pdf: 743351 bytes, checksum: 997f8a5009a3bbc979a7206041daf583 (MD5) Previous issue date: 2017-02-17 / CAPES / In this work, we discuss the article The Existence and Stability of Equilibrium Points in the Robe Restricted Three-Body Probem due to Hallan and Rana. For this we present some basic definitions and results abut Hamiltonian systems such as equilibrium stability of linear Hamiltonian systems. We set out the restricted problem of the three bodies and show some classic results of the problem. Finally we present the Robe’s problem and discuss the main results using Hamiltonian systems theory. / Nesse trabalho, dissertamos sobre o artigo \The Existence and Stability of Equilibrium Points in the Robe Restricted Three-Body Probem" devido a Hallan e Rana. Para isso apresentamos definições e resultados básicos sobre sistemas Hamiltonianos tais como estabilidade de equilíbrios de sistemas Hamiltonianos lineares. Enunciamos o problema restrito dos três corpos e mostramos alguns resultados clássicos do problema. Por fim apresentamos o problema de Robe e discutimos os principais resultados usando a teoria de sistemas Hamiltonianos.

In qualitative theory of differential equations, an important role is played by special classes of solutions, like periodic solutions or solutions to some boundary value problems. When a system of ordinary differential equations has equilibria, i.e. constant solutions, whose stability properties are known, it is significant to search for connections between them by trajectories of solutions of the given system. These are called homoclinic or heteroclinic, according to whether they describe a loop based at one single equilibrium or they "start" and "end" at two distinct equilibria. This thesis is devoted to the study of heteroclinic solutions for a specific class of ordinary differential equations related to the Extended Fisher-Kolmogorov equation and the Swift-Hohenberg equation. These are semilinear fourth order bi-stable evolution equations which appear as mathematical models for problems arising in Mechanics, Chemistry and Biology. For such equations, the set of bounded stationary solutions is of great interest. These solve an autonomous fourth order equation. In this thesis, we focus on such equations having a variational structure. In that case, the solutions are critical points of an associated action functional defined in convenient functional spaces. We then look for heteroclinic solutions as minimizers of the action functional. Our main contributions concern existence and multiplicity results of such global and local minimizers in the case where the functional is defined from sign changing Lagrangians. The underlying idea is to impose conditions which imply a lower bound on the action over all admissible functions. We then combine classical arguments of the Calculus of Variations with careful estimates on minimizing sequences to prove the existence of a minimum.