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.

Thèse de doctorat : Mathématiques : Paris 12 : 2007. / Titre provenant de l'écran-titre. Bibliogr. 149 réf.

Thesis (Ph. D.)--York University, 1998. Graduate Programme in Mathematics and Statistics. / Typescript. Includes bibliographical references (leaves 73-77). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004 & res_dat=xri:pqdiss & rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation & rft_dat=xri:pqdiss:NQ39271.

We prove that if $1 > \alpha > 1/2$, then there exists a probability measure $\mu$ such that the Hausdorff dimension of its support is $\alpha$ and $\mu*\mu$ is a Lipschitz function of class $\alpha-1/2$.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. / Made available in DSpace on 2012-10-18T12:10:30Z (GMT). No. of bitstreams: 0Bitstream added on 2014-09-26T00:46:21Z : No. of bitstreams: 1 177020.pdf: 2271890 bytes, checksum: ac314eb9d1adcd654f5be08de78fc599 (MD5) / Dado (A,G,a) um C* sistema dinâmico, estudaremos o produto cruzado da C*-algebra A pelo grupo discreto G pela ação a de G em A. Como dada uma ação parcial de G em um espaço de Hausdorff localmente compacto X, existe uma ação parcial de G na C*-algebra C0(X) associada, e a recíproca também vale, vamos provar que se uma ação parcial é topologicamente livre e minimal em X, então o produto cruzado reduzido associado é simples, [1]. É claro que antes disto precisamos introduzir as noções de produto cruzado por ações parciais e produto cruzado reduzido. Por último, aplicaremos este resultado para alguns exemplos.

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2013. / Submitted by Albânia Cézar de Melo (albania@bce.unb.br) on 2013-12-13T14:08:25Z No. of bitstreams: 1 2013_TiagoLimaBentoPereira.pdf: 732415 bytes, checksum: 7e5bcdb023bca641f71a408789afacf7 (MD5) / Approved for entry into archive by Guimaraes Jacqueline(jacqueline.guimaraes@bce.unb.br) on 2013-12-16T11:58:07Z (GMT) No. of bitstreams: 1 2013_TiagoLimaBentoPereira.pdf: 732415 bytes, checksum: 7e5bcdb023bca641f71a408789afacf7 (MD5) / Made available in DSpace on 2013-12-16T11:58:07Z (GMT). No. of bitstreams: 1 2013_TiagoLimaBentoPereira.pdf: 732415 bytes, checksum: 7e5bcdb023bca641f71a408789afacf7 (MD5) / Com o objetivo de obter atratores de semigrupos em espaço de Banach de dimensão infinita como objetos em espaços de dimensão finita estudamos condições sobre o semigrupo que asseguram que o atrator global possui dimensão de Hausdorff ou fractal ("upper"box-couting dimension) finita. ______________________________________________________________________________ ABSTRACT / In order to obtain the attractors of semigroups in in nite dimensional Banach spaces as objects in nite dimensional spaces, we study conditions on the semigroups which guarantee nite Hausdorff, or fractal ("upper"box-counting dimension), dimension for the attractors.

Measuring visual closeness of 3-D models is an important issue for different problems and there is still no standardized metric or algorithm to do it. The normal of a surface plays a vital role in the shading of a 3-D object. Motivated by this, we developed two applications to measure visualcloseness, introducing normal difference as a parameter in a weighted metric in Metro’s sampling approach to obtain the maximum and mean distance between 3-D models using 3-D and 6-D correspondence search structures. A visual closeness metric should provide accurate information on what the human observers would perceive as visually close objects. We performed a validation study with a group of people to evaluate the correlation of our metrics with subjective perception. The results were positive since the metrics predicted the subjective rankings more accurately than the Hausdorff distance.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第25086号 / 理博第4993号 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 宍倉 光広, 教授 COLLINSBenoit Vincent Pierre, 教授 塚本 真輝 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Dans cette thèse, on étudie les sous-groupes boréliens des groupes de Lie et leur dimension de Hausdorff. Si G est un groupe de Lie nilpotent connexe, on construit dans G des sous-groupes de dimension de Hausdorff arbitraire, tandis que si G est semisimple compact, on démontre que la dimension de Hausdorff d'un sous-groupe borélien strict de G ne peut pas être arbitrairement proche de celle de G. / Given a Lie group G, we investigate the possible Hausdorff dimensions for a measurable subgroup of G. If G is a connected nilpotent Lie group, we construct measurable subgroups of G having arbitrary Hausdorff dimension, whereas if G is compact semisimple, we show that a proper measurable subgroup of G cannot have Hausdorff dimension arbitrarily close to the dimension of G.

O conceito de reflexão em topologia expressa o fato de que um espaço satisfaz uma dada propriedade sempre que esta é satisfeita por seus subespaços \"menores\". Neste trabalho, estuda-se a reflexão de propriedades envolvendo a maioria das principais funções cardinais e metrizabilidade, bem como outras propriedades relacionadas. São discutidos problemas em aberto -- como o problema de Hamburger --, incluindo respostas parciais e exemplos de consistência. Várias dentre as demonstrações apresentadas utilizam técnicas de submodelos elementares, que constituem hoje uma importante ferramenta no estudo de topologia geral. / The concept of reflection in topology expresses the fact that a space satisfies a given property provided that its \"small\" subspaces do. This work presents a study on reflection of properties concerning most of the main cardinal functions and metrizability, as well as other related properties. Open problems --such as Hamburger\'s question-- are also discussed, including partial answers and consistent examples. Several of the proofs presented here make use of elementary submodels, nowadays an important tool in the study of general topology.