Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Includes bibliographical references (p. 253-255).

Un automorphisme non-symplectique d'ordre fini n sur une surface X de type K3 est un automorphisme σ ∈ Aut(X) qui satisfait σ*(ω) = λω où λ est une racine primitive n-ième de l'unité et ω est le générateur de H2,0(X). Dans cette thèse on s’intéresse aux automorphismes non-symplectiques d'ordre 8 et 16 sur les surfaces K3. Dans un premier temps, nous classifionsles automorphismes non-symplectiques σ d'ordre 8 quand le lieu fixe de sa quatrième puissance σ⁴ contient une courbe de genre positif, on montre plus précisément que le genre de la courbe fixée par σ est au plus un. Ensuite nous étudions le cas où le lieu fixe de σ contient au moins une courbe et toutes les courbes fixées par sa quatrième puissance σ⁴ sont rationnelles. Enfin nous étudions le cas où σ et son carré σ² agissent trivialement sur le groupe de Néron-Severi. Nous classifions toutes les possibilités pour le lieu fixe de σ et de son carré σ² dans ces trois cas. Nous obtenons la classification complète pour les automorphismes non-symplectiques d'ordre 8 sur les surfaces K3. Dans la deuxième partie de la thèse, nous classifions les surfaces K3 avec automorphisme non-symplectique d'ordre 16 en toute généralité. Nous montrons que le lieu fixe contient seulement courbes rationnelles et points isolés et nous classifions complètement les sept configurations possibles. Si le groupe de Néron-Severi a rang 6, alors il y a deux possibilités et si son rang est 14, il y a cinq possibilités. En particulier si l'action de l'automorphisme est trivial sur le groupe de Néron-Severi, alors nous montrons que son rang est six. Enfin, nous construisons des exemples qui correspondent à plusieurs cas dans la classification des automorphismes non-symplectiques d'ordre 8 et nous donnons des exemples pour chaque cas dans la classification des automorphismes non-symplectiques d'ordre 16. / A non-symplectic automorphism of finite order n on a K3 surface X is an automorphism σ ∈ Aut(X) that satisfies σ*(ω) = λω where λ is a primitive n−root of the unity and ω is a generator of H2,0(X). In this thesis we study the non-symplectic automorphisms of order 8 and 16 on K3 surfaces. First we classify the non-symplectic automorphisms σ of order eight when the fixed locus of its fourth power σ⁴ contains a curve of positive genus, we show more precisely that the genus of the fixed curve by σ is at most one. Then we study the case of the fixed locus of σ that contains at least a curve and all the curves fixed by its fourth power σ⁴ are rational. Finally we study the case when σ and its square σ² act trivially on the Néron-Severi group. We classify all the possibilities for the fixed locus of σ and σ² in these three cases. We obtain a complete classifiction for the non-symplectic automorphisms of order 8 on a K3 surfaces.In the second part of the thesis, we classify K3 surfaces with non-symplectic automorphism of order 16 in full generality. We show that the fixed locus contains only rational curves and isolated points and we completely classify the seven possible configurations. If the Néron-Severi group has rank 6, there are two possibilities and if its rank is 14, there are five possibilities. In particular ifthe action of the automorphism is trivial on the Néron-Severi group, then we show that its rank is six.Finally, we construct several examples corresponding to several cases in the classification of the non-symplectic automorphisms of order 8 and we give an example for each case in the classification of the non-symplectic automorphisms of order 16.

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Apresentamos duas propriedades genéricas para difeomorfismos conservativos da classe \'C POT.1\' sobre uma superfície compacta de dimensão dois. Obtemos uma limitação inferior para entropia topológica de difeomorfismos genéricos, e mostramos que tais difeomorfismos sempre possuem conjuntos invariantes fechados com órbitas densas e dimensão de Hausdorff dois / We present two generic properties of \'C POT.1\" area preserving diffeomorphisms of a two dimensional compact oriented surface. We obtain a lower bound for the topological entropy of a generic diffeomorphisms, and we show that such a diffeomorphism always has closed invariant sets with dense orbits and Hausdorff dimension two

Submitted by Diego Navarro Pozo (the.electric.me@gmail.com) on 2018-10-23T14:56:18Z No. of bitstreams: 1 dissert diego revisada + ficha + assinaturas.pdf: 953096 bytes, checksum: 005110857b3e2e871af759d632f8ef55 (MD5) / Approved for entry into archive by Janete de Oliveira Feitosa (janete.feitosa@fgv.br) on 2018-10-23T15:26:47Z (GMT) No. of bitstreams: 1 dissert diego revisada + ficha + assinaturas.pdf: 953096 bytes, checksum: 005110857b3e2e871af759d632f8ef55 (MD5) / Made available in DSpace on 2018-10-29T18:11:10Z (GMT). No. of bitstreams: 1 dissert diego revisada + ficha + assinaturas.pdf: 953096 bytes, checksum: 005110857b3e2e871af759d632f8ef55 (MD5) Previous issue date: 2018-07-30 / Os sistemas Hamiltonianos formam uma das classes mais importantes de equações diferenciais. Além de constituírem o formalismo central da física clássica, sua aplicação se estende a uma grande variedade de outros campos de estudo. Esses sistemas possuem uma característica notória do ponto de vista da matemática, a saber, que a sua ação sobre seus estados iniciais preserva uma estrutura geométrica conhecida como simpleticidade. Este fato tem importantes consequências sobre as características qualitativas do comportamento do sistema, em especial no longo prazo. Neste trabalho, são estudados métodos numéricos para obter soluções aproximadas para sistemas Hamiltonianos (já que, via de regra, soluções exatas não podem ser encontradas) que preservem a estrutura simplética das equações originais. Para tal, é feita uma revisão da teoria clássica da integração numérica de equações diferenciais, bem como de temas mais recentes como os integradores exponenciais. Além de expor a literatura mais recente sobre integradores simpléticos do tipo Runge-Kutta Exponencial, o trabalho propõe um algoritmo para o cálculo computacionalmente eficientes de integrais envolvendo exponenciais de matrizes, que são centrais para a integração simplética estável de ordem alta. / Hamiltonian systems form one of the most important classes of differential equations describing the evolution of physical phenomena. They are the backbone of classical mechanics and their application covers many different areas such as molecular dynamics, hydrodynamics, celestial and statistical mechanics, just to mention a few of them. A noteworthy feature of Hamiltonian systems is that their flow possesses a geometric property -known as symplecticity- which has a major impact on the long-time behavior of the solution. Since in general closed-form solutions can be found only in few particular cases, the construction and analysis of numerical integrators -able to produce discrete approximations that are also symplecticity preserving- is crucial for studying these systems. In this work we present the key ideas about Hamiltonian systems and their theoretical properties. We also review the main numerical methods and techniques to design and analyze symplectic integrators. Special attention is given to the stability and dynamical properties of the methods, as well as their effectiveness for long-term simulations. Finally, we propose an algorithm to improve the computational implementation of the family of exponential-based symplectic integrators recently found in the literature.

Esta tese estuda a geometria de problemas variacionais através da linearização e projetivização das suas equações de Euler - Lagrange. O processo de linearização fornece a passagem das equações de Euler - Lagrange para as equações de Jacobi; a minimalidade (local) de extremais está determinada pelo conceito de ponto conjugado, que tem natureza projetiva. Propriedades de minimalidade local são transformadas em propriedades de auto-interseção de uma curva na variedade de Grassmann adequada. Desenvolvemos este processo em duas aplicações: 1) O estudo da minimalidade local de extremais de problemas variacionais de ordem superior. Neste caso, encontramos uma curva não degenerada de planos isotrópicos num espaço vetorial simplético, que, após prolongamento por derivadas, fornece uma curva degenerada de planos Lagrangeanos cujas auto-interseções determinam a minimalidade. 2) No caso mais clássico de problemas de ordem um, estudamos a versão linear - projetiva do problema inverso: dada uma equação diferencial de ordem dois, quando ela é a equação de Euler - Lagrange de um problema variacional? Veremos que as condições do problema inverso linear - projetivo fornecem informações sobre os possíveis Lagrangianos, por exemplo a assinatura. / In this work we study the geometry of high order calculus of variations through the linearization and projectivization of their Euler Lagrange equations. The linearization process provides the passage from the Euler Lagrange equations to the Jacobi equations; the (local) minimality properties of the extremal is determined by conjugate points, which is a projective concept. Minimaltiy properties of the extremals are transformed into self-intersection propertie of curves in the appropriate Grassmann manifold. We develop this process in two instances: 1) The study of minimality properties of extremals of higher-order variational problems. In this case, we find a non-degenerate curve of isotropic subspaces, that, after prolongation by derivatives, gives a degenerate curve of Lagrangian planes whose self-intersections determine minimality. 2) In the classical case of order one variational problems, we study a projective-linear version of the inverse problem: given a second order differential equation, when is it the Euler-Lagrange equation of a variational problem? We will see that the conditions given by the linear projective inverse problem provides information about the possible Lagrangians, for example, its signature.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-06-27T19:48:01Z No. of bitstreams: 1 guilhermemarquesribeiro.pdf: 1025565 bytes, checksum: 681a6e31eb787e74456258f00aa65e53 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-08-07T21:07:11Z (GMT) No. of bitstreams: 1 guilhermemarquesribeiro.pdf: 1025565 bytes, checksum: 681a6e31eb787e74456258f00aa65e53 (MD5) / Made available in DSpace on 2017-08-07T21:07:11Z (GMT). No. of bitstreams: 1 guilhermemarquesribeiro.pdf: 1025565 bytes, checksum: 681a6e31eb787e74456258f00aa65e53 (MD5) Previous issue date: 2015-07-31 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Nessa dissertação, apresentamos dois formalismos consistentes para tratar a dinâmica de sistemas vinculados: o procedimento de Dirac[15], baseado num algoritmo que substitui os parênteses de Poisson por outra estrutura semelhante, e o método simplético[17], fundamentado na deformação da estrutura simplética do espaço de fase. Aplicamos esses formalismos tanto em exemplos simples quanto em problemas concretos de física teórica, como o modelo de Proca e o campo eletromagnético. Estudamos também as simetrias apresentadas por sistemas vinculados de primeira classe. Apresentamos uma prova da conjectura de Dirac[3] e mostramos que um contra-exemplo apresentado na literatura[2] é consistente com a conjectura . / In this dissertation, we have presented two consistent formalisms to treat the dynamics of constrained systems: the Dirac procedure[15], based on an algorithm that replaces the Poisson brackets by a similar structure, and the symplectic method[17], based on the deformation of the symplectic structure of the phase space. We have applied this formalisms to both simple examples and concrete problems from theoretical physics, such as the Proca model and the electromagnetic field. We also studied the symmetries generated by first class constrained systems. We have presented a prove of Dirac's conjecture[3] and showed that a counter-example found in the literature[2] is consistent with the conjecture

A new method is presented for the numerical computation of the generalized eigen- values of real Hamiltonian or symplectic pencils and matrices. The method is strongly backward stable, i.e., it is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order sqrt(epsilon), where epsilon is the machine precision, the new method computes the eigenvalues to full possible accuracy.

We formulate a process of quantization of classical mechanics, from a symplecticperspective. The Dirac quantization axioms are stated, and a satisfactory prequantizationmap is constructed using a complex line bundle. Using polarization, it isdetermined which prequantum states and observables can be fully quantized. Themathematical concepts of symplectic geometry, fibre bundles, and distributions are exposedto the degree to which they occur in the quantization process. Quantizationsof a cotangent bundle and a sphere are described, using real and K¨ahler polarizations,respectively.

This thesis is concerned principally with tangential geometry and the applications of these concepts to tangentially symplectic foliations. The subject of tangential geometry is still at an elementary stage. The author here systematises current concepts and results and extends them, leading to the definition of vertical connections and vertical G-structures. Tangentially symplectic foliations are then characterised in terms of vertical symplectic forms. Some significant particular cases are discussed.