Thesis--Göttingen. / In Periodical Room.

University, Diss., 2005--Kiel.

Dissertação (mestrado) - Universidade Federal de Santa Catarina. Centro de Ciencias Fisicas e Matematicas / Made available in DSpace on 2012-10-16T01:41:13Z (GMT). No. of bitstreams: 0Bitstream added on 2016-01-08T16:06:12Z : No. of bitstreams: 1 81779.pdf: 1134439 bytes, checksum: 3a7d46a6cf731cb8b57c4b1815f21112 (MD5) / O objetivo anunciado no título desta tese é realizado do seguinte modo: No capítulo I selecionamos definições de estruturas algébricas e de álgebra linear que usaremos nos capítulos posteriores. No capítulo II introduzimos a noção de álgebra de clifford. Estabelecemos a sua unicidade (a menos de isomorfismo) e determinamos a sua dimensão. No capítulo III tratamos da existência das álgebras de Clifford por meio de uma construção matricial explícita e formulamos uma série de critérios e teoremas que reduzem esta construção aos casos em que o espaço ortogonal é de dimensão menor que 5. Finalmente, no capítulo IV aplicamos os resultados obtidos na construção do recobrimento do grupo Spin(n) pelo grupo SO(n) e na construção da sequência de Radon-Hurwitz-Eckman.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. / Made available in DSpace on 2012-10-19T02:17:40Z (GMT). No. of bitstreams: 0Bitstream added on 2016-01-09T03:36:56Z : No. of bitstreams: 1 175354.pdf: 2174314 bytes, checksum: 3d934ab8e79f01772de6e45634702fe3 (MD5) / As estruturas de Álgebra Exterior e Álgebra de Clifford se relacionam por isomorfismo de espaço vetorial. Se a forma quadrática é degenerada, a Álgebra de Clifford é a própria Álgebra Exterior para esse espaço. Construção de uma álgebra C/Q, onde Q é a forma quadrática para um espaço vetorial V como imagem de um operador alternado, definindo sobre tal álgebra um produto, de tal maneira que seja isomorfa à Álgebra de Clifford para V.

Made available in DSpace on 2014-08-27T14:36:44Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-08-23Bitstream added on 2014-08-27T15:57:06Z : No. of bitstreams: 1 000774561.pdf: 509048 bytes, checksum: 311f20166de76fb390b9c2db0bfeb807 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Em supersimetria padrão, construimos um superespaço com parâmetros 'x POT. 'mu'', 'teta', 'teta BARRA' (anti)comutantes e supercampos que dependem destes, então impomos vínculos e após de?nirmos supercampos quirais e vetorias, que são exemplos interessantes, e en?m construímos uma lagrangeana supersimétrica. Vamos aqui colocar condições mais fracas sobre os parâmetros do superespaço: agora tais parâmetros que antes eram anticomutantes vão formar uma álgebra de Cli?ord {'teta' POT. 'alfa'', 'teta' POT. 'beta'} = 'C POT. 'alfa' 'beta'' (1) e seguindo um procedimento análogo, com algumas de?nições adicionais, extenderemos nossa lagrangeana para o caso de um superespaço não-anticomutativo / In standard supersymmetry, we build a superspace with parameters 'x POT. 'mu'', 'teta', 'teta BARRA' that (anti)commute and super?elds that depend on them, then we impose constraints and de?ne chiral and vector super?elds, which are interesting examples, from which we build a supersymmetric Lagrangian. We will now impose a weaker condition on the superspace of the parameters: the ones that were anticommuting will now form a Cli?ord algebra {'teta' POT. 'alfa'', 'teta' POT. 'beta'}= 'C POT. 'alfa' 'beta'' (2) and, following an analogous procedure, with additional de?nitions, we will generalize our Lagrangian for the case of the non-anticommutative superspace

This is an overview of geometric algebras in dimensions 0-4 from the perspective of the concept of number itself. It is developed from a historic viewpoint and investigates and develops a pedagogic approach emphasizing the geometric aspects of the subject. There is a focus broadly on three main but interconnected areas: the relation between the discrete and the continuous, the centrality of complex numbers, and the hypothesis that the octonions may be expressed in the even subalgebra of four dimensions. This will not be proved, and the focus is on the overall perspective and presentation. One central result is a proposal for the identification of the Cayley-Dickson process with the gluing together of spheres, extending it to start from ℕ. This leads to a projective representation of a unified elliptic/parabolic/hyperbolic geometry of 4-dimensional space. Coupled with this is a discrete representation of it that can be classified with Geometric Algebra.

Universiẗat, Diss., 1997--Kaiserslautern.

In this paper the Clifford Algebra is introduced and proposed as analternative to Gibbs' vector algebra as a unifying language for geometricoperations on vectors. Firstly, the algebra is constructed using a quotientof the tensor algebra and then its most important properties are proved,including how it enables division between vectors and how it is connected tothe exterior algebra. Further, the Clifford algebra is shown to naturallyembody the complex numbers and quaternions, whereupon its strength indescribing rotations is highlighted. Moreover, the wedge product, is shown asa way to generalize the cross product and reveal the true nature ofpseudovectors as bivectors. Lastly, we show how replacing the cross productwith the wedge product, within the Clifford algebra, naturally leads tosimplifying Maxwell's equations to a single equation.

This thesis comes in three main parts. In the first of these (comprising chapters 2 - 6), the basic theory of Riemannian and conformal submersions is described and the relevant geometric machinery explained. The necessary Clifford algebra is established and applied to understand the relationship between the spinor bundles of the base, the fibres and the total space of a submersion. O'Neill-type formulae relating the covariant derivatives of spinor fields on the base and fibres to the corresponding spinor field on the total space are derived. From these, formulae for the Dirac operators are obtained and applied to prove results on Dirac morphisms in cases so far unpublished. The second part (comprising chapters 7-9) contains the basic theory and known classifications of G2-structures and Spin+ 7 -structures in seven and eight dimensions. Formulae relating the covariant derivatives of the canonical forms and spinor fields are derived in each case. These are used to confirm the expected result that the form and spinorial classifications coincide. The mean curvature vector of associative and Cayley submanifolds of these spaces is calculated in terms of naturally-occurring tensor fields given by the structures. The final part of the thesis (comprising chapter 10) is an attempt to unify the first two parts. A certain `7-complex' quotient is described, which is analogous to the well-known hyper-Kahler quotient construction. This leads to insight into other possible interesting quotients which are correspondingly analogous to quaternionic-Kahler quotients, and these are speculated upon with a view to further research.

The goal of this thesis is the development of a fast and robust algorithm that is able to detect patterns in flow fields independent from their orientation and adequately visualize the results for a human user. This thesis is an interdisciplinary work in the field of vector field visualization and the field of pattern recognition. A vector field can be best imagined as an area or a volume containing a lot of arrows. The direction of the arrow describes the direction of a flow or force at the point where it starts and the length its velocity or strength. This builds a bridge to vector field visualization, because drawing these arrows is one of the fundamental techniques to illustrate a vector field. The main challenge of vector field visualization is to decide which of them should be drawn. If you do not draw enough arrows, you may miss the feature you are interested in. If you draw too many arrows, your image will be black all over. We assume that the user is interested in a certain feature of the vector field: a certain pattern. To prevent clutter and occlusion of the interesting parts, we first look for this pattern and then apply a visualization that emphasizes its occurrences. In general, the user wants to find all instances of the interesting pattern, no matter if they are smaller or bigger, weaker or stronger or oriented in some other direction than his reference input pattern. But looking for all these transformed versions would take far too long. That is why, we look for an algorithm that detects the occurrences of the pattern independent from these transformations. In the second part of this thesis, we work with moment invariants. Moments are the projections of a function to a function space basis. In order to compare the functions, it is sufficient to compare their moments. Normalization is the act of transforming a function into a predefined standard position. Moment invariants are characteristic numbers like fingerprints that are constructed from moments and do not change under certain transformations. They can be produced by normalization, because if all the functions are in one standard position, their prior position has no influence on their normalized moments. With this technique, we were able to solve the pattern detection task for 2D and 3D flow fields by mathematically proving the invariance of the moments with respect to translation, rotation, and scaling. In practical applications, this invariance is disturbed by the discretization. We applied our method to several analytic and real world data sets and showed that it works on discrete fields in a robust way.