Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2006 / Includes bibliographical references (p. 59-60).

A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorff measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorff and box dimensions differ---in particular, it does not depend on any self-similarity or regularity conditions on the space. The only restriction on the space is that it have positive s₀ dimensional Hausdorff measure, where s₀ is the Hausdorff dimension of the space, assumed to be finite. Also, X does not need to be embedded in another space, such as Rⁿ.

Thesis (Ph.D.)--University of Melbourne, Dept. of Mathematics and Statistics, 2007. / Typescript. Includes bibliographical references.

Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Bellissard, Jean; Committee Member: Baker, Matt; Committee Member: Bakhtin, Yuri; Committee Member: Garoufalidis, Stavros; Committee Member: Putnam, Ian

Thesis (Ph. D.)--Duke University, 2007. / Includes bibliographical references.

Thesis (Ph. D.)--Mathematics, Georgia Institute of Technology, 2007. / Belegradek, Igor, Committee Member ; Ghrist, Robert, Committee Chair ; Harrell, Evans, Committee Member ; Etnyre, John, Committee Member ; Symington, Margaret, Committee Co-Chair.

We conduct a comprehensive investigative review of solution generating algorithms for the Einstein field equations governing the gravitational behaviour of an isolated neutral static spherical distribution of perfect fluid matter. Traditionally, the master field equation generated from the condition of pressure isotropy has been interpreted as a second order ordinary differential equation. However, since the pioneering work of Wyman (1949) it was observed that more success can be enjoyed by regarding the equation as a first order linear differential equation. There was a resurgence of the ideas of Wyman in 2000 and various researchers have been able to generate complete solutions to the field equations up to certain integrations. These have been accomplished by working in Schwarzschild (curvature) coordinates, isotropic coordinates, area coordinates and a coordinate system written in terms of the redshift parameter. We have utilised Durgapal–Banerjee (1983) coordinates and produced a new algorithm. The algorithm is used to generate new classes of perfect fluid solutions as well as to regain familiar particular solutions reported in the literature. We find that our solution is well behaved according to elementary physical requirements. The pressure vanishes for a certain radius and this establishes the boundary of the distribution. Additionally the pressure and energy density are both positive inside the radius. The energy conditions are shown to be satisfied and it is particularly pleasing to have the causality criterion satisfied to ensure that the speed of light is not exceeded by the speed of sound. We also report some new solutions using the algorithms proposed by Lake (2006). / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.

Orientador: Diego Sebastian Ledesma / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-21T14:31:48Z (GMT). No. of bitstreams: 1 YalandaMuelas_YamitYesid_M.pdf: 7930295 bytes, checksum: 93c5826f6c164fed94abacbb42db5256 (MD5) Previous issue date: 2012 / Resumo: Nesta dissertação apresentamos generalizações de três resultados muito conhecidos em geometria Riemanniana: o Teorema de Myers, o Teorema de Bochner e o Teorema de decomposição de Cheeger-Gromoll. Em particular veremos que fazendo uma pequena modificação sobre os requisitos destes teoremas no que se refere ao tensor de Ricci, os resultados permanecem inalterados / Abstract: In this dissertation we present generalizations of three well-known results in Riemannian geometry: The Myers's theorem, Bochner's theorem and the Cheeger-Gromoll splitting theorem. In particular, we will prove that making a small modification of the requirements of these theorems related to the Ricci tensor, the results remain unchanged / Mestrado / Matematica / Mestre em Matemática

Orientador: Diego Sebastian Ledesma / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T22:36:25Z (GMT). No. of bitstreams: 1 Paula_PedroManfrimMagalhaesde_M.pdf: 1130226 bytes, checksum: bbd8d375ddf7846ed2eafe024103e682 (MD5) Previous issue date: 2015 / Resumo: Este trabalho apresenta um estudo sobre variedades Riemannianas que possuem um tensor de Bakry-Émery-Ricci com limitações. Inicialmente abordamos tanto aspectos da geometria Riemanniana tradicional como métricas e geodésicas, quanto aspectos mais avançados como as fórmulas de Bochner, Weitzenböck e o teorema de Hodge. Em seguida discutimos a convergência de Gromov-Hausdorff e suas propriedades, além de serem apresentados alguns teoremas como os de Kasue e Fukaya. Por fim estudamos as propriedades topológicas e geométricas de variedades com limitação no tensor de Bakry-Émery-Ricci e o comportamento de tais limitações com respeito à submersões e à convergência de Gromov-Hausdorff / Abstract: This work presents a study about Riemannian manifolds having a Bakry-Émery-Ricci tensor with bounds. Initially we approached both the traditional aspects of Riemannian geometry like metrics and geodesics, as more advanced aspects like the Bochner, Weitzenböck formulas and the Hodge's theorem. Then we discussed the Gromov-Hausdorff convergence and its properties, in addition to showing some theorems as those from Kasue and Fukaya. Lastly we studied the topological and geometric properties of manifolds with bounds on the Bakry-Émery-Ricci tensor and the behavior of these bounds with respect to submersions and the Gromov-Hausdorff convergence / Mestrado / Matematica / Mestre em Matemática

Submitted by Ronildo Prado (ronisp@ufscar.br) on 2016-08-30T20:16:50Z No. of bitstreams: 1 6841.pdf: 671749 bytes, checksum: fee45931185f019b1c8d5bb4946465b0 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-08-30T20:17:52Z (GMT) No. of bitstreams: 1 6841.pdf: 671749 bytes, checksum: fee45931185f019b1c8d5bb4946465b0 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-08-30T20:18:45Z (GMT) No. of bitstreams: 1 6841.pdf: 671749 bytes, checksum: fee45931185f019b1c8d5bb4946465b0 (MD5) / Made available in DSpace on 2016-08-30T20:19:00Z (GMT). No. of bitstreams: 1 6841.pdf: 671749 bytes, checksum: fee45931185f019b1c8d5bb4946465b0 (MD5) Previous issue date: 2014-02-27 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / O objetivo principal deste trabalho é apresentar alguns resultados recentes na teoria de folheações polares, também chamadas de folheações riemannianas singulares com seções, em variedades de curvatura não positiva, presentes no artigo [24]. As ações polares também são estudadas, pois são objetos de pesquisa ativa que motivam e ilustram o estudo das folheações polares. Fornecemos uma demonstração de que não existem folheações polares próprias em variedades compactas de curvatura não positiva. Além disso, apresentamos um resultado que descreve globalmente as folheações polares próprias em variedades de Hadamard. Abordamos este resultado também no contexto particular das ações polares, utilizando a teoria de subvariedades taut. As ações adjunta e por conjugação são brevemente estudadas como exemplos clássicos de ações polares. / This work aims at presenting some recent results on the theory of polar foliations, also know as singular riemannian foliations with sections, on nonpositively curved manifolds, as seen in T oben [24]. Polar actions are also studied, for they are active research subject that motivate and illustrate polar foliations. We give a proof of the nonexistence of proper polar foliations on compact manifolds of nonpositive curvature. Then we present a result that globally describes proper polar foliations on Hadamard manifolds. We prove this same result in the special case of polar actions by using the theory of taut submanifolds. The adjoint and conjugation actions are brie y presented as classical examples of polar actions.