Tesis sobre el tema "Clifford algebras"
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Han, Gang. "Clifford algebras associated with symmetric pairs /". View abstract or full-text, 2004. http://library.ust.hk/cgi/db/thesis.pl?MATH%202004%20HAN.
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Araujo, Martinho da Costa. "Construção de algebras reais de Clifford". reponame:Repositório Institucional da UFSC, 1988. http://repositorio.ufsc.br/xmlui/handle/123456789/75476.
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Dissertação (mestrado) - Universidade Federal de Santa Catarina. Centro de Ciencias Fisicas e MatematicasMade 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.
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Wilmot, Gregory Paul. "The structure of Clifford algebra". Title page, contents and abstract only, 1988. http://web4.library.adelaide.edu.au/theses/09SM/09smw738.pdf.
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Hoefel, Eduardo Outeiral Correa. "Teorias de Gauge e algebras de Clifford". [s.n.], 2002. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307234.
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Orientador: Jayme Vaz JrDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Nesta dissertação apresentamos uma descrição do formalismo matemático das teorias de gauge introduzindo os conceitos de grupos e álgebras de Lie, fibrados principais, conexões e curvatura. Em seguida introduzimos as álgebras de Clifford e os spinors, tais conceitos são utilizados no capítulo final onde apresenta-se algllmas de suas aplicações em teorias de gauge. Uma aplicação é dada pelas formas diferenciais assumindo valores em uma álgebra de Clifford: mostra-se como as formas de conexão e curvatura são dadas por formas a valores em álegebras de bivetores, estas últimas são as álgebras de Lie dos grupos Spin. Outra aplicação consiste em mostrar, usando o Teorema de Periodicidade das álgebras de Clifford, como algumas transformações conformes do espaço-tempo são dadas pela ação do grupo $pin(2,4) sobre paravetores ]R + ]R4,1. Finalizamos mostrando a construção de monopolos e instantons através do teorema de inversão para spinors de Pauli e Dirac, vistos como elementos de sub-álgebras pares de álgebras de Clifford, e a estreita relação deste teorema com as fibrações de Hopf, ilustrando a relação existente entre Topologia e Física
Abstract: This dissertation begins with a description of the mathematical formulation of gauge theories, introducing the concepts of Lie groups and Lie algebras, principal bundles, connection and curvature. Then, Clifford algebras and spinors are introduced. The final chapter presents some applications of Clifford algebras in gauge theories. The first application is given by Clifford algebra valued differential forms: we shown how the connection and curvature 2-forms are given by bivector algebra valued forms, bivector algebras are the Lie algebras of spin groups. Another application consist of showing, through the Periodicity Theorem of Clifford algebras, how some conformal transformations of the space-time are given by the action of the $pin(2,4) group over the paravectors R+ R4,1. ln the last application, the construction of monopoles and instantons is presented through the lnversion Theorem for Pauli and Dirac spinors, considered as elements of the even sub-algebra of the Clifford algebra. The close relationship between this theorem and the Hopf fibrations is emphasized, ilustrating the link between Topology and Physics
Mestrado
Mestre em Matemática Aplicada
5
Severi, Claudio. "Clifford algebras and spin groups, with physical applications". Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/18387/.
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In questo lavoro viene esposta la teoria delle algebre di Clifford e dei gruppi di Spin, con attenzione alle applicazioni fisiche, in particolare l'equazione di Dirac per particelle quantistiche con spin 1/2. I primi due capitoli sono dedicati ad una descrizione generale delle algebre di Clifford reali e complesse, che vengono costruite e classificate. Il terzo capitolo è dedicato ai gruppi di Spin ed alle loro algebre di Lie. Gli ultimi due capitoli illustrano un'applicazione fisica: viene esposta la teoria quantistica dello spin e del momento angolare, e si deriva l'equazione di Dirac con un principio variazionale. Dopo una discussione delle proprietà generiche di questa equazione, si dimostra che descrive accuratamente la struttura fine dello spettro dell'atomo di idrogeno.
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Wylie, Dave. "Factoring Blades and Versors in Euclidean Clifford Algebras". Thesis, Southern Illinois University at Edwardsville, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=1564083.
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This thesis examines different methods of factoring elements of Clifford Algebras, specifically, Cℓn,0. Blades are factored using Fontijne's algorithm and other techniques. Versors are factored using Perwass's algorithm. Writing an element as a sum of blades, which are then factored, can make it more efficient to store or transmit that element. To evaluate the usefulness of expressing a given element of Cℓ n,0 this way, the number of scalars required to express that element is compared between factored and expanded forms.
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Buchholz, Sven [Verfasser]. "A Theory of Neural Computation with Clifford Algebras / Sven Buchholz". Kiel : Universitätsbibliothek Kiel, 2005. http://d-nb.info/1080317147/34.
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Doran, Christopher John Leslie. "Geometric algebra and its application to mathematical physics". Thesis, University of Cambridge, 1994. https://www.repository.cam.ac.uk/handle/1810/251691.
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Clifford algebras have been studied for many years and their algebraic properties are well known. In particular, all Clifford algebras have been classified as matrix algebras over one of the three division algebras. But Clifford Algebras are far more interesting than this classification suggests; they provide the algebraic basis for a unified language for physics and mathematics which offers many advantages over current techniques. This language is called geometric algebra - the name originally chosen by Clifford for his algebra - and this thesis is an investigation into the properties and applications of Clifford's geometric algebra. The work falls into three broad categories: - The formal development of geometric algebra has been patchy and a number of important subjects have not yet been treated within its framework. A principle feature of this thesis is the development of a number of new algebraic techniques which serve to broaden the field of applicability of geometric algebra. Of particular interest are an extension of the geometric algebra of spacetime (the spacetime algebra) to incorporate multiparticle quantum states, and the development of a multivector calculus for handling differentiation with respect to a linear function. - A central contention of this thesis is that geometric algebra provides the natural language in which to formulate a wide range of subjects from modern mathematical physics. To support this contention, reformulations of Grassmann calculus, Lie algebra theory, spinor algebra and Lagrangian field theory are developed. In each case it is argued that the geometric algebra formulation is computationally more efficient than standard approaches, and that it provides many novel insights. - The ultimate goal of a reformulation is to point the way to new mathematics and physics, and three promising directions are developed. The first is a new approach to relativistic multiparticle quantum mechanics. The second deals with classical models for quantum spin-I/2. The third details an approach to gravity based on gauge fields acting in a fiat spacetime. The Dirac equation forms the basis of this gauge theory, and the resultant theory is shown to differ from general relativity in a number of its features and predictions.
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Resende, Adriana Souza. "Introdução elementar às álgebras Clifford 'CL IND.2' 'CL IND. 3'". [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306698.
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Orientador: Waldyr Alves Rodrigues JuniorDissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatistica e Computação Cientifica
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Resumo: O presente trabalho tem a intenção de apresentar por intermédio de uma linguagem unificada alguns conceitos de cálculo vetorial, álgebra linear (matrizes e transformações lineares) e também algumas idéias elementares sobre os grupos de rotações em duas e três dimensões e seus grupos de recobrimento, que geralmente são tratados como "fragmentos" em várias modalidades de cursos no ensino superior. Acreditamos portanto que nosso texto possas ser útil para alunos dos cursos de graduação dos cursos de Engenharia, Física, Matemática e interessados em Matemática em geral. A linguagem unificada à que nos referimos acima é obtida com a introdução do conceitos das álgebras geométricas (ou de Clifford) onde, como veremos, é possível fornecer uma formulação algébrica elegante aos conceitos de vetores, planos e volumes orientados e definir para tais objetos o produto escalar, os produtos contraídos à esquerda e à direita, o produto exterior (associado, como veremos, em casos particulares ao produto vetorial) e finalmente o produto geométrico (Clifford), o que permite o uso desses conceitos para a solução de inúmeros problemas de geometria analítica no R ² e no R ³. Procuramos ilustrar todos estes conceitos com vários exemplos e exercícios com graus variáveis de dificuldades. Nossa apresentação é bem próxima àquela do livro de Lounesto, e de fato muitas seções são traduções (eventualmente seguidas de comentários) de seções daquele livro. Contudo, em muitos lugares, acreditamos que nossa apresentação esclarece e completa as correspondentes do livro de Lounesto
Abstract: This paper aims to present using an unified language a few concepts of vector calculus, linear algebra (matrices and linear transformations) and also some basic ideas about the groups of rotations in two and three dimensions and their covering group, which generally are treated as "fragments" in various types of courses in higher education. We believe therefore that our text should be useful to students of undergraduate courses like Engineering, Physics, Mathematics and people interested in Mathematics in general. The unified language that we refer to above is obtained by introducing the concept of geometric (or Clifford) algebra where, as we shall see, it is possible to give an elegant algebraic formulation to the concepts of vectors, oriented planes and oriented volumes, and to define to those objects the scalar product, the right and left contracted products, the exterior product (associated, as we shall see, in particular cases to the vector product) and finally the geometric (Clifford) product, and moreover, to use those concepts to solve may problems of analytic geometry in R ² and R ³. We illustrated all those concepts with several examples and exercises with variable degrees of difficulties. Our presentation is nearly the one in Lounesto's book, and in fact some sections are no more than translations (eventually with commentaries) from sections of that book. However, in many places, we believe that our presentation clarify nd completement the corresponding ones in Lounesto's book
Mestrado
Ágebra
Mestre em Matemática
10
Rocha, Junior Roldão da. "Spinors e twistors no modelo paravetorial : uma formulação via algebras de Clifford". [s.n.], 2001. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307233.
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Orientador: Jayme Vaz JuniorDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Nesta dissertação o formalismo dos spinors e twistors de Penrose são formulados em termos das álgebras de Clifi'ord. Para tal utilizamos o modelo paravetorial do espaço-tempo, onde um vetor do espaço-tempo é escrito em termos da soma de escalares e vetores da álgebra de Cli:fford do espaço euclideano tridimensional. Com isso construímos um formalismo que utiliza a menor estrutura algébrica capaz de descrever teorias físicas relativísticas, como as teorias eletromagnética e de Dirac. Os spinors são definidos algebricamente como elementos de um ideal lateral mínimal da álgebra de Clifi'ord. Utilizamos o teorema de periodicidade (1,1) das álgebras de Clifi'ord para descrever de maneira linear, em termos da complexificação da álgebra de Clifi'ord do espaço-tempo, as transformações conformes desse espaço-tempo. Os twistors aparecem como uma classe particular de spinors algébricos. Consideramos ainda algumas possíveis generalizações
Abstract: In this dissertation the Penrose theory of spinors and twistors is formulated from the point of view of the Clifi'ord algebras. We use the paravector model of spacetime, where a spacetime vector is written as a sum of scalars and vectors of the Clifi'ord algebra associated with the three-dimensional euclidean space. From this we construct a formalism that uses the least algebraic structure that describes relativistic physical theories, such as the electromagnetic and the Dirac ones. Spinors are defined algebraically as elements of a minimallateral ideal of a Cli:fford algebra. We use the modulo (1,1) periodicity theorem of Clifi'ord algebras to describe the conformal transformations as linear transformations, using the method of complexmcation of the spacetime Clifi'ord algebra. Twistors are defined as a particular class of algebraic spinors. We consider some possible generalizations
Mestrado
Mestre em Matemática Aplicada
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Farias, José Ginaldo de Sousa. "Álgebra de Clifford: classificações e representações". Universidade Federal da Paraíba, 2013. http://tede.biblioteca.ufpb.br:8080/handle/tede/8039.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
In this paper, we study Clifford algebras so universal and constructive as quantization of exterior algebra, we classify all Clifford algebras associated with the quadratic Minkowski spaces (Rp+q, p,q), where p,q(u) = u21 +...+u2 p −(u2 p+1 +...+u2 p+q), u = (u1, ..., up+q) ∈ Rp+q, which we denote by Clp,q, as well as their complexifications. To do so, we use important results as the periodicity theorem Carton / Bott. The, we study their representations, emphasizing the Twisted Adjoint Representataion, Spin Representation and the Spin-Half Representation moreover using the number of Radon-Hurwitz we study representations of the algebras Cl0,k.
Neste trabalho, estudamos as ´algebras de Clifford Cl(V, ) associadas aos espa¸cos quadr´aticos (V, ), de maneira universal, construtiva e como quantiza¸c˜ao da ´algebra exterior. Classificamos todas as ´algebras de Clifford associadas as espa¸cos quadr´aticos de Minkowski (Rp+q, p,q), onde p,q(u) = u21 + ... + u2 p − (u2 p+1 + ... + u2 p+q), u = (u1, ..., up+q) ∈ Rp+q, as quais denotamos por Clp,q, bem como suas complexifica¸c˜oes. Para tanto, usaremos resultados importantes como o teorema da periodicidade de Carton/Bott. Al´em disso, estudamos as suas representa¸c˜oes, destacando a Representa¸c˜ao Adjunta Torcida, as Representa ¸c˜oes Spin e Semi-Spin e por meio do n´umero de Radon-Hurwitz estudamos as representa¸c˜oes das ´algebras Cl0,k.
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Dolan, Peter. "A Z2-graded generalization of Kostant's version of the Bott-Borel-Weil theorem /". view abstract or download file of text, 2007. http://proquest.umi.com/pqdweb?did=1400959341&sid=2&Fmt=2&clientId=11238&RQT=309&VName=PQD.
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Thesis (Ph. D.)--University of Oregon, 2007.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 130-131). Also available for download via the World Wide Web; free to University of Oregon users.
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Satchell, Marcel John Francis. "Geometric algebra & the quantum theory of fields". Thesis, University of Cambridge, 2014. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.708105.
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Mendes, Douglas 1985. "Álgebras de Clifford e a fibração de Hopf". [s.n.], 2012. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306400.
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Orientador: Rafael de Freitas LeãoDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Os grupos Spin aparecem de várias formas em Matemática e em Física-Matemática, tendo grande importância na teoria de brados e de operadores diferenciais sobre os mesmos. O conceito de estrutura spin é deles derivado, sendo ele a base de toda uma teoria, conhecida como geometria spin. Esta dissertação introduz os primeiros conceitos necessários ao estudo de tais grupos, assim como alguns aspectos importantes relacionados a eles. Dada a natureza dos grupos Spin e dos problemas aos quais estão relacionados, vários tópicos na interface entre álgebra e geometria tiveram de ser abordados. Estudamos em um primeiro momento as álgebras de Clifford, sua representação adjunta torcida e os grupos Spin como subgrupos do grupo das unidades de tais álgebras. À estes estudos, seguiu-se uma análise detalhada da teoria de espaços de recobrimento e da classificação dos mesmos. Pudemos com isso entender o grupo Spin, via representação adjunta torcida, como o recobrimento universal do grupo especial ortogonal de um espaço quadrático não-degenerado. Nos concentramos daí na teoria de brados principais e a relação destes com as propriedades geométricas das variedades sobre as quais eles estão construídos. Para sintetizar o que foi estudado, construímos algebricamente a fibração de Hopf ao final desta dissertação, explicitando sua relação com a estrutura spin da esfera S²
Abstract: Spin groups come in many forms in Mathematics and Mathematical Physics, having great importance in the theory of fiber bundles and differential operators defined on them. The concept of spin structure is derived from them, being the basis of all a theory, known as spin geometry. This thesis introduces the first concepts necessary for the study of such groups, as well as important aspects related to them. Given the nature of the Spin groups and problems which they're related to, several topics at the interface between algebra and geometry had to be addressed. At first, we studied Clifford algebras, their twisted adjoint representation and Spin groups as subgroups of the group of units of such algebras. Followed these studies a detailed analysis of the theory of covering spaces and the classification of them. Done that, we were able to understand the group Spin, via the twisted adjoint representation, as the universal covering space of the special orthogonal group of a non-degenerate quadratic space. From there, we focused on the theory of principal bundles and their relationship with the geometric properties of manifolds on which they are built. To summarize what was studied, we algebraically construct the Hopf fibration at the end of this thesis, explaining its relationship with the spin structure of the sphere S²
Mestrado
Matematica
Mestre em Matemática
15
Biswas, Debapriya. "Geometry of elliptic, parabolic and hyperbolic homogeneous spaces using Clifford algebras and group representations". Thesis, University of Leeds, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.432297.
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Traesel, Marcio Andre. "Estruturas não-associativas generalizadas em S7 e Álgebras de Clifford". reponame:Repositório Institucional da UFABC, 2009.
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Mattos, Eduardo Souza. "Sobre álgebras de Clifford, geometria projetiva e visão computacional". [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307238.
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Orientador: Jayme Vaz JuniorDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Atualmente, o estudo das Álgebras de Clifford é utilizado em inúmeras áreas de pesquisa. Uma delas é na área de Visão Computacional. O objetivo central dessa dissertação consiste em exibir noções sobre Álgebras de Clifford e sua utilização na formulação dos conceitos e definições de operações entre objetos da Geometria Projetiva e na formulação algébrica de câmeras virtuais, que é um dos assuntos tratados na área de Visão Computacional. Para isso são expostos de forma gradual e coerente os principais aspectos teóricos necessários para atingir os objetivos citados. Como resultado, as Álgebras de Clifford proporcionam uma excelente descrição da Geometria Projetiva e das câmeras virtuais
Abstract: Currently, the study of Clifford algebras are used in many research areas. One is in the area of Computer Vision. The main objective of this dissertation is to display notions of Clifford algebras and their use in formulating the concepts and definitions of transactions between objects of Projective Geometry and algebraic formulation of virtual cameras, which is one of the topics covered in Computer Vision. For it is exposed gradually and consistently the main theoretical aspects needed to achieve the goals mentioned. As a result, Clifford algebras provide an excellent description of Projective Geometry and virtual cameras
Mestrado
Matematica Aplicada
Mestre em Matemática Aplicada
18
Alves, Rafael Santos de Oliveira 1982. "Álgebra de Clifford aplicada ao cálculo de estruturas moleculares". [s.n.], 2013. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306802.
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Orientador: Carlile Campos LavorTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: O Problema de Geometria de Distâncias Moleculares (PGDM) consiste em encontrar uma imersão tridimensional de um grafo simples, não orientado, de forma que o peso nas arestas corresponda às distâncias inter-atômicas de uma molécula. Este é um problema de busca em um espaço contínuo, mas que pode ser discretizado sob algumas exigências, dando origem ao PGDM discretizado (PGDMD), que é solucionado usando informações sobre distâncias entre alguns átomos da molécula através de um algoritmo Branch and Prune (BP). Caso as distâncias sejam dadas por um conjunto de limites inferiores e superiores, temos um novo problema: o PGDMD intervalar (iPGDMD). A partir da interpretação geométrica deste último, propomos uma nova abordagem utilizando a Álgebra de Clifford a fim de tornar o algoritmo BP mais eficiente e de poder tratar algebricamente os problemas relacionados ao tratamento das distâncias intervalares
Abstract: The Molecular Distance Geometry Problem (MDGP) consists in finding a three dimensional embedding of simple, weighted, undirected graph such that the weight in the edges correspond to the inter-atomic distances of a molecule. This is a continuous search problem which can be discretized under some assumptions, yielding the Discretized MDGP (DMDGP), which is solved by a Branch and Prune (BP) algorithm using information about the distances among some atoms of the molecule. If the distances are given by a set of lower and upper bounds, a new problem arises: the interval DMDGP (iDMDGP). From a geometric interpretation of this problem, we propose a new approach, using Clifford Algebras, in order to improve the BP efficiency and treat algebraically the issues related to interval distances
Doutorado
Matematica Aplicada
Doutor em Matemática Aplicada
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Souza, Jose Vicente Cipriano de 1964. "Uma aplicação da algebra geometrica a mecanica classica = a transformação de Kustaanheimo-Stiefel". [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307232.
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Orientador: Jayme Vaz Jr.Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Nessa dissertação apresentamos a Álgebra Geométrica do Espaço Euclidiano e estudamos algumas de suas propriedades. Para exemplificar suas aplicações, estudamos a Transformação Kustaanheimo-Stiefel em termos de Álgebra Geométrica. Para isso apresentamos inicialmente a Transformação KS, que regulariza o movimento de Kepler em três dimensões removendo uma singularidade na origem, da forma como foi originalmente formulada, baseando-se em álgebra de matrizes. Feito isso, a Transformação KS é apresentada com Álgebra Geométrica, o que torna o seu entendimento geométrico mais claro e seu desenvolvimento mais simplificado. Para tal o uso do conceito de spinors é de grande importância
Abstract: In this dissertation we presented the Geometric Algebra of Euclidean Space and studied some of its properties. To exemplify its applications, we studied the Kustaanheimo-Stiefel Transformation in terms of Geometric Algebra. This purpose we presented initially the KS Transformation which regularizes the Kepler motion in three dimensions by removing a singularity at the origin, as it was originally formulated, based on matrix algebra. Done, the KS transformation is presented with Geometric Algebra, making clearer its geometric understanding and its development more simplified. With this goal the spinors concept use is of great importance
Mestrado
Fisica-Matematica
Mestre em Matemática Aplicada
20
Sousa, Mônica Paula de. "Álgebras de Clifford: uma introdução à Geometria Spin". Universidade Federal da Paraíba, 2013. http://tede.biblioteca.ufpb.br:8080/handle/tede/7393.
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In this work we discuss the concepts and definitions that construct Clifford algebras focusing on a introduction the theory Spin Geometry. That s because the connection this two subject, enabling such algebras know the measure that helps to understand the definition of spin manifold, concept introductory the this special topic in Riemannian Geometry. We begin with the construction of Clifford algebras associated to infinite dimensional vector spaces, over any field, passing to associated with finite dimensional. we see the spinores groups, Pin and Spin, which characterize and show the relation with the twisted adjoint representation, homomorphism that, when restricted to these groups, has an important role in defining of a spin structure. As this definition works with representations of real Clifford algebras, restricted to spinors groups such algebras, we introduced them for soon afterwards consider such representations. We concluded approaching the necessary theory for us to show that those groups are also Lie groups (where we urged an intersection with the analysis) and double covering, to complete the concepts algebraic present in the definition of spin manifold.
No presente trabalho abordamos os conceitos e definições que constroem as álgebras de Clifford com foco em uma linha de estudo de quem se inicia na teoria de Geometria Spin. Isso devido a intima ligação desses dois assunto, permitindo conhecer tais álgebras à medida que se auxilia a compreensão da definição de variedade spin, conceito introdutório desse tópico especial em Geometria Riemanniana. Iniciamos com a construção das álgebras de Clifford associadas a espaços vetoriais de dimensão infinita, sobre um corpo qualquer, passando àquelas associadas aos de dimensão finita. Fazemos o mesmo com os grupos Pin e Spin, os quais caracterizamos e mostramos a relação com a representação adjunta torcida, aplicação que, quando restrita a esses grupos, tem papel importante na definição de uma estrutura spin. Como tal definição trabalha com representações das álgebras de Clifford reais, restritas aos grupos spinores dessas Cliffords, as apresentamos para em seguida conceituarmos tais representações. Finalizamos, para completar os conceitos algébricos presente na definição de variedade spin, abordando a teoria necessária para mostrarmos que esses grupos são também grupos de Lie (onde instigamos uma interseção com a análise, destacando os enlaces com outras teorias) e recobrimentos duplos.
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Alves, Rafael Santos de Oliveira 1982. "Algebra geometrica e o algoritmo de Grover". [s.n.], 2008. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306805.
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Orientador: Carlile Campos LavorDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: O Algoritmo de Grover é um algoritmo quântico de busca em um conjunto desordenado. Com o uso de propriedades da mecânica quântica, ele apresenta um ganho quadrático em relação a um algoritmo clássico. Neste trabalho, apresentamos uma outra visão deste algoritmo, através da Álgebra Geométrica, motivados pela interpretação geométrica dos operadores, e verificamos que é possível escrevê-lo com uma nova linguagem, e ainda apresentar uma expressão mais simples para o operador de Grover (G) além de expressões gerais para estados resultantes de aplicações sucessivas deste operador
Abstract: Grover¿s algorithm is a quantum algorithm for searching in unstructured databases. Due to the properties of quantum mechanics, it provides a quadratic speedup over their classical counterparts. Using the Geometric Algebra, we present a new way to understand and simplify the operators of Grover¿s algorithm
Mestrado
Computação Quantica
Mestre em Matemática Aplicada
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Kou, Kit Ian. "Paley-Wiener theorem and Shannon sampling with the Clifford analysis setting". Thesis, University of Macau, 2005. http://umaclib3.umac.mo/record=b2492153.
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Rocha, Junior Roldão da. "Álgebras de Clifford, generalizações e aplicações à física-matemática". [s.n.], 2005. http://repositorio.unicamp.br/jspui/handle/REPOSIP/277990.
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Orientador: Jayme Vaz JrTese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataglin
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Resumo: Investigamos generalizações das álgebras de Clifford (ACs) e suas vastas aplicações na Física. Classificamos o mais novo candidato à descrição da matéria escura como um campo espinorial bandeira, que pertence à classe 5 proposta por Lounesto, de acordo com os valores assumidos pelos seus covariantes bilineares. Decompomos a AC em partes a pares e ímpares relativas a uma dada a graduação automórfica interna, além de descrever suas diversas consequências na decomposição de operadores que agem sobre a álgebra exterior e sobre a AC. Além de escrever a equação de Dirac no contexto dessa decomposição, estendemos os resultados conhecidos sobre uma partícula-teste nas vizinhanças de um buraco negro de Schwarzschild para um buraco negro de Reissner-Nordstrom. Introduzimos as ACs estendidas, construídas sobre duas cópias (quiral e aquiral) de um espaço vetorial de dimensão finita munido de uma métrica de assinatura (p, q). Formulamos a AC sobre uma cópia quiral do contraespaço, mostrando propriedades surpreendentes, tais como: a indefinição do elemento de volume do contraespaço sob o produto regressivo, com a possibilidade de ele ser um escalar ou pseudoescalar, dependendo da dimensão do espaço vetorial; e o fato de que a co-cadeia de de Rham do operador codiferencial ser formada por uma sequência de subespaços homogêneos da álgebra exterior subsequentemente quirais e aquirais. Dessa maneira provamos que a álgebra exterior sobre o espaço e aquela construída sobre o contraespaço são apenas pseudo-duais ao introduzirmos quiralidade. A super álgebra de Poincaré é obtida a partir da introdução de algumas estruturas algébricas sobre o espaço euclidiano R3, a partir da utilização de spinors puros e do Princípio da Trialidade juntamente com sua generalização. Introduzimos os octonions no contexto das ACs e definimos unidades octoniônicas parametrizadas por elementos arbitrários, mas fixos, de uma AC sobre R0,7 e também produtos octoniônicos entre multivetores, além de generalizarmos as identidades de Moufang para esse formalismo. O Modelo Padrão das partículas elementares é rediscutido nesse contexto, além de obtermos uma Teoria de Calibre não-associativa em Cl0,7 , onde o campo espinorial é dado pela soma direta de um quark e um lépton. Finalmente introduzimos as isotopias, associativas e não-associativas, das ACs e em particular a simetria de sabor SU(6) dos quarks se apresenta como uma simetria exata dentro do contexto do levantamento isotópico da AC CL12. Bárions e mésons também são descritos nesse contexto
Abstract: We investigate Clifford algebras (ACs) generalizations and their wide applications in Physics. The candidate for the description of the dark matter is classified as a agpole spinor field, that is in the class 5 spinors proposed by Lounesto according to his spinor field classification by the values assumed by their bilinear covariants. The AC is split in a-even and a-odd components, related to a given inner automorphic a-grading, besides describing various consequences of this decomposition in the splitting of operators acting on the exterior and Clifford algebras. Besides writing the Dirac equation in the spacetime splitting context, we extend the well known results concerning a spinning test particle in a Schwarzschild black hole neighboorhood to a Reissner-Nordstrom black hole. We alsointroduce the extended ACs associated with two copies (chiral and achiral) of a finite-dimensional vector space endowed with a metric of signature (p, q). ACs are formulated on a chiral copy of the counterspace, where we show astounding and astonishing properties such as: the de Rham co-chain associated with the codifferential operator is constituted by a sequence of exterior algebra homogeneous subspaces subsequently chiral and achiral. Thus we prove thatthe exterior algebra on the space and the exterior algebra constructed on the counterspace are pseudoduals, if we introduce chirality. The Poincaré superalgebra is obtained from the introduction of some algebraic structures on the Euclidean space R3 , via the pure spinor formalism and the triality principle and its generalization. Octonions are introduced in thecontext of ACs and we define AC-parametrized octonionic units, besides generalizing Moufang identities in this context. The Standard Model of elementary particles is revisited in the octonionic context and we also obtain a gauge theory using the new octonionic products introduced, where a spinor field describes the direct sum of a quark and a lepton. Finally we introduce associative and non-associative isotopies of ACs. In particular we present the avor quark symmetry SU(6) as an exact symmetry in the Cl12 isotopic lifting context. Barions and mesons are also described via isotopic lifting of ACs
Doutorado
Fisica-Matematica
Doutor em Ciências
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Brito, Kelvyn Paterson Sousa de. "Espinores sobre o bulk e em dimensões compactificadas". reponame:Repositório Institucional da UFABC, 2017.
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Wainer, Samuel Augusto 1989. "Geometria riemanniana e semi-riemanniana no fibrado de Clifford e aplicações". [s.n.], 2013. http://repositorio.unicamp.br/jspui/handle/REPOSIP/305959.
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Orientador: Márcio Antônio de Faria RosaDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Abstract: The complete abstract is available with the full electronic document .
Mestrado
Matematica
Mestre em Matemática
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Lima, Rian Lopes de. "Grupos clássicos e álgebras de Clifford C* em espaços de Hilbert". reponame:Repositório Institucional da UFABC, 2014.
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Orientador: Prof. Dr. Roldão da Rocha jr.Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2014.
Clifford algebras in Hilbert spaces are studied, along with the possible defnitions of spinors when the classical Clifford algebra is equipped with an additional structure of algebra C. The groups associated with the Clifford algebras, such as the Clifford-Lipschtz groups, Pin and Spin groups, are introduced together with unitary structures and trace operators in Clifford algebras in Hilbert spaces as well. Von-Neumann algebras are studied and the Bogoliubov automorphism is used to generalize the twisted Clifford-Lipschtz groups, using the graduation in Clifford algebra with the additional structure of algebra C. Fock representations and Hilbert-Schmidt operators are going to be introduced in the exterior algebra underlying the Clifford algebras in Hilbert spaces. In addition, twisted Clifford-Lipschitz groups can be constructed with the Bogoliubov automorphism, when it is an inner automorphism. This defines the Pin and Spin groups in the Clifford algebra with the additional structure of algebra C.
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Melo, Nolmar. "Uma álgebra de Clifford de assinatura (n,3n) e os operadores densidade da teoria da informação quântica". [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306804.
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Orientador: Carlile Campos LavorTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Este trabalho apresenta uma linguagem algébrica para dois elementos básicos da teoria da informação quântica (os bits quânticos e os operadores densidade), baseada nas propriedades de uma álgebra de Clifford de assinatura (n,3n). Demonstramos que a nova descrição desses elementos preserva as mesmas propriedades matemáticas obtidas com a descrição clássica. Com isso, estendemos alguns resultados apresentados na literatura que relaciona Álgebra de Clifford e Informação Quântica.
Abstract: This work presents an algebraic language for two basic elements of quantum information theory (the quantum bits and density operators), based in the properties of a Clifford algebra of signature (n,3n). We prove that the new description of these elements preserves the same mathematical properties obtained with the classical description. We also extend some results presented in the literature that relate Clifford algebra and quantum information.
Doutorado
Matematica Aplicada
Doutor em Matemática
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Russell, Neil Eric. "Aspects of the symplectic and metric geometry of classical and quantum physics". Thesis, Rhodes University, 1993. http://hdl.handle.net/10962/d1005237.
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I investigate some algebras and calculi naturally associated with the symplectic and metric Clifford algebras. In particular, I reformulate the well known Lepage decomposition for the symplectic exterior algebra in geometrical form and present some new results relating to the simple subspaces of the decomposition. I then present an analogous decomposition for the symmetric exterior algebra with a metric. Finally, I extend this symmetric exterior algebra into a new calculus for the symmetric differential forms on a pseudo-Riemannian manifold. The importance of this calculus lies in its potential for the description of bosonic systems in Quantum Theory.
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Romero, Ruiz Ivan. "Tensor de impedancias magnetotelúrico en álgebras de Clifford: signatura rerum del tensor de impedancias magnetotelúrico". Doctoral thesis, Universitat de Barcelona, 2015. http://hdl.handle.net/10803/396604.
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El tensor de impedancias es el elemento matemático que condensa los datos en Magnetotelúrica. No obstante solemos verlo representado en forma matricial, representación que no ofrece toda la información que se puede recabar del mismo. Las álgebras de Clifford son un grupo asociativo que establece un sistema algebraico donde el tensor queda mejor caracterizado, dotándolo de significado geométrico. En este contexto, considerar el propio tensor y sus trasformaciones como elementos del álgebra, además de reproducir de manera compacta y sencilla las relaciones habituales, abre la posibilidad de dilucidar nuevos conceptos y relaciones que están aparentemente ocultos en el tensor.
El análisis en el contexto de las álgebras de Clifford C13 ha permitido descomponer el tensor de impedancias en sus elementos algebraico-geométricos más simples, ello ha facilitado hallar siete invariantes de rotación irreducibles e independientes. En el contexto del álgebra de Clifford C12 se ha representado la distorsión galvánica mediante una descomposición en sus elementos algebraico-geométrico más simples. Se observa que una de las subálgebras de C13 dota de significancia a la invariancia del phase tensor con la distorsión galvánica. El análisis en C12 ha permitido unificar las diferentes representaciones del tensor de impedancias, siendo los Diagramas de Mohr y el phase tensor las más representativas. Del análisis del phase tensor se han deducido los índices Indexe, Index2 y el Phase Sensitive Strike. En particular los índices Indexe y Index2 han permitido establecer nuevos criterios de dimensionalidad.
A partir de estructuras algebraicas del tensor de impedancias en Clifford C12, se han deducido unas relaciones a partir de las cuales se ha desarrollado un algoritmo para determinar y corregir la distorsión galvánica en el caso general 3D cuando, a periodos pequeños, el tensor de impedancias regional tiene un comportamiento 2D o 1D (y determinados casos 3D). El algoritmo para la determinación de la distorsión galvánica consiste en una búsqueda estocástica de los parámetros de distorsión twist, shear y anisotropy con ciertas constricciones que se imponen al tensor regional libre de distorsión.
Se ha diseñado un algoritmo, el Método Perturbativo, para la determinación del strike una vez identificados los periodos pequeños 2D. Se considera que la dispersión en la dirección del strike de los periodos pequeños 2D es debida al ruido (Gaussiano), por lo que el algoritmo consiste en generar aleatoriamente valores del tensor según una distribución Gaussiana equivalente al ruido, en estos primeros periodos 2D, hasta conseguir un strike común a todos ellos. Este procedimiento permite, además, corregir aquellos periodos que, aun siendo 2D, no cumplen las condiciones establecidas en los criterios de dimensionalidad, aumentando así la aplicabilidad de los métodos de determinación de la distorsión galvánica.
Se ha desarrollado un programa denominado MITT que recoge, además de las metodologías para determinar la distorsión galvánica, diferentes herramientas que giran en torno a las representaciones del tensor de impedancias: Curvas de resistividad aparente y fases, diagramas de Mohr y la generación de funciones densidad de probabilidad de Indexe Index2 y Phase Sensitive Strike. El programa se puede descargar del Dipósit Digital de la UB, http://hdl.handle.net/2445/66846.The Magnetotelluric tensor and its transformations are considered elements of the Clifford Algebra. In this way, not only are the already known procedures reproduced but also new concepts and relationships hidden in the tensor are now elucidated. In the frame of Clifford Algebra C/3, the magnetotelluric tensor is expressed in terms of its simplest geometric-algebraic components. In this way, seven irreducible independent rotation invariants are deduced. Galvanic distortion is represented in Cl2 Clifford Algebra. An analysis of the subalgebras in C/3 recognises a particular subalgebra as the one giving meaning to the independence of the phase tensor with the galvanic distortion. The Mohr diagrams and the phase tensor are analysed in the context of Clifford Algebra Cl2. The indexes Indexi, Index2 and the Phase Sensitive Strike are deduced from the phase tensor in Cl2. In particular, Indexi and Index2, which are independent of galvanic distortion and rotation, have enabled to introduce new criteria for the analysis of the dimensionality. In Clifford algebra Cl2, a number of relationships have been found, from which a method to determine the galvanic distortion in the 3D case, with the assumption that the regional tensor has a 2D or 1D (and certain 3D cases) behaviour at short periods, has been developed. The method is based on a constrained stochastic heuristic method, which consists of exploring randomly the full space of the distortion parameters twist, shear and anisotropy. A method to find the strike of the 2D short periods is developed. A different strike angle for each period can be obtained in a 2D case because of noise. In this way we expect to recover the strike direction perturbing the data for each period with a similar random Gaussian distribution to achieve a common strike for all the short periods. The program MITT (Magnetotelluric Impedance Tensor Tools) is carried out. The program is based on the methodology developed to determine the galvanic distortion in a regional 3D case. In addition, MITT offers different useful tools about the representations of the impedance tensor: Apparent resistivities and phases, Mohr diagrams and probability density functions. MITT can be downloaded from the Dip6sit Digital de la UB, http://hdl.handle.net/2445/668
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El, Mir Ghina. "Algèbres de Clifford conformes et orbites de points de vue d'images". Thesis, La Rochelle, 2014. http://www.theses.fr/2014LAROS013/document.
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L'objectif de ce travail est de décrire des modélisations des points de vue et des changements de points de vue d'images d'un objet planaire dans les algèbres de Clifford conformes. Nous généralisons le modèle conforme de l'espace euclidien à travers une famille à deux paramètres d'horosphère, chacune d'entre elles étant plongée dans un espace vectoriel réel de dimension 4 muni d'une métrique équivalente à la métrique de Minkowski. Nous décrivons par la suite deux approches pour mettre en œuvre ces modèles conformes généralisés pour les représentations d'images. L'idée de base est d'encoder les distorsions perspectives de l'objet causées par la variation du paramètre de latitude de la caméra au travers des paramètres d'une horosphère. La première approche consiste à considérer les horosphères de l'espace de Minkowski de dimension 4 pour encoder les points de vue. Les changements de points de vue sont alors linéarisés à travers un groupe de transformations linéaires et conformes de cet espace. Cette approche est ensuite généralisée en décrivant les points de vue à travers les objets d'un groupoïde dont les morphismes sont des diagrammes commutatifs qui représentent les changements de points de vue. Ainsi, une image conforme est décrite par une application définie sur une horosphère à deux paramètres. L'action du groupoïde sur l'ensemble des images conformes nous conduit à associer à tout objet planaire l'orbite de toutes ses images conformes obtenues à partir de tous les points de vueOur purpose in this work is to introduce representations of image viewpoints and viewpoint changes of a planar object in conformal Clifford algebras. Our important preliminary contribution is a generalization of the conformal model of the Euclidean space through a two-parameter family of horospheres. Each one of these is embedded into a real vector space of dimension 4 equipped with a metric equivalent to the Minkowski metric. We describe two approaches that make use of these generalized conformal models for image representations. These are based on modelings of perspective distortions of the object caused by a variation of the latitude angle of the camera. First, we model the image viewpoints by the horospheres of the Minkowski space of dimension 4. In this setting, the viewpoint changes are linearized through a group of linear conformal transformations of this space. This approach is generalized by describing the viewpoints through the objects of a groupoid whose morphisms are commutative diagrams that model the viewpoint changes. A conformal image is then described as a map defined on a horosphere. The action of the groupoid on the set of conformal images leads us to associate with every planar object the orbit of its conformal images from all viewpoints
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Covolo, Tiffany. "(Z2)n-Superalgebra and (Z2)n-Supergeometry". Thesis, Lyon 1, 2014. http://www.theses.fr/2014LYO10203.
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La présente thèse porte sur le développement d'une théorie d'algèbre linéaire, de géométrie et d'analyse basée sur les algèbres (Z2)n-commutatives, c'est-à-dire des algèbres (Z2)n-graduées associatives unitaires satisfaisant ab = (-1)ba, pour tout couple d'éléments homogènes a, b de degrés deg(a), deg(b) où <.,.> est le produit scalaire usuel). Cette généralisation de la supergéométrie a de nombreuses applications : en mathématiques (l'algèbre de Deligne des superformes différentielles, l'algèbre des quaternions et les algèbres de Clifford en sont des exemples) et même en physique (paraparticules). Dans ce travail, les notions de trace et de (super)déterminant pour des matrices à coefficients dans une algèbre gradué-commutative sont définies et étudiés. Une attention particulière est portée au cas des algèbres de Clifford : ce point de vue gradué fournit une nouvelle approche au problème classique du « bon » déterminant pour des matrices à coefficient non-commutatifs (quaternioniques). En outre, nous entreprenons l'étude de la géométrie différentielle (Z2)n-graduée. Privilégiant l'approche par les espaces annelés, les (Z2)n-supervariétés sont définies en choisissant l'algèbre (Z2)n-commutative des séries formelles en variables graduées comme modèle pour le faisceau de fonctions. Les résultats les plus marquants ainsi obtenus sont : le Berezinien gradué et son interprétation cohomologique (essentielle pour établir une théorie de l'intégration) ; le théorème des morphismes, attestant qu'on peut rétablir un morphisme entre (Z2)n-supervariétés à partir de sa seule expression sur les coordonnées ; le théorème de Batchelor-Gawedzki pour les (Z2)n-supervariétés lissesThe present thesis deals with a development of linear algebra, geometry and analysis based on (Z2)n-superalgebras ; associative unital algebras which are (Z2)n-graded and graded-commutative, i.e. statisfying ab=(-1)ba, for all homogeneous elements a, b of respective degrees deg(a), deg(b) in (Z2)n (<.,.> denoting the usual scalar product). This generalization widens the range of applications of supergeometry to many mathematical structures (quaternions and more generally Clifford algebras, Deligne algebra of superdifferential forms, higher vector bundles) and appears also in physics (for describing paraparticles) proving its worth and relevance. In this dissertation, we first focus on (Z2)n-superalgebra theory ; we define and characterize the notions of trace and (super)determinant of matrices over graded-commutative algebras. Special attention is given to the case of Clifford algebras, where our study gives a new approach to treat the classical problem of finding a “good” determinant for matrices with noncommuting (quaternionic) entries. Further, we undertake the study of (Z2)n-graded differential geometry. Privileging the ringed space approach, we define (smooth) (Z2)n-supermanifolds modeling their algebras of functions on the (Z2)n-commutative algebra of formal power series in graded variables, and develop the theory along the lines of supergeometry. Notable results are : the graded Berezinian and its cohomological interpretation (essential to establish integration theory) ; the theorem of morphism, which states that a morphism of (Z2)n-supermanifolds can be recovered from its coordinate expression ; Batchelor-Gawedzki theorem for (Z2)n-supermanifolds
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Kuo, Jung-Miao. "The Clifford algebra of a cubic form". [Bloomington, Ind.] : Indiana University, 2008. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3331257.
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Thesis (Ph.D.)--Indiana University, Dept. of Mathematics, 2008.Title from PDF t.p. (viewed on Jul 27, 2009). Source: Dissertation Abstracts International, Volume: 69-11, Section: B, page: 6843. Adviser: Darrell Haile.
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FRANCHINI, Silvia Giuseppina. "Graphic Coprocessors with Native Clifford Algebra Support". Doctoral thesis, Università degli Studi di Palermo, 2009. http://hdl.handle.net/10447/178952.
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Silva, Ana Paula da Cunda Corrêa da. "Álgebras de Clifford: uma construção alternativa /". Florianópolis, SC, 1999. http://repositorio.ufsc.br/xmlui/handle/123456789/81355.
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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.
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Randriamihamison, Louis-Samuel. "Algebres de clifford et paires de hurwitz pseudo-euclidiennes". Toulouse 3, 1988. http://www.theses.fr/1988TOU30123.
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On considere le probleme de hurwitz pseudo-euclidien, tel qu'il a ete formule et etudie recemment par j. Lawrynowicz et j. Rembielinski. Dans le cas euclidien, le probleme avait deja ete resolu par a. Hurwitz en 1923. Dans le cas pseudo-euclidien, on resoud entierement ce probleme en toutes dimensions et signatures quelconques, en delaissant l'approche matricielle employee par lawrynowicz et rembielinski, mais en utilisant le formalisme spinoriel developpe par a. Crumeyrolle. Ce formalisme permet de mettre en evidence les raisons geometriques qui gouvernent les solutions du probleme. Dans la resolution de ce probleme interviennent des algebres de clifford et des formes bilineaires ou hermitiennes sur des espaces spinoriels, qui peuvent jouer un role en physique mathematique: on donne en particulier un lien entre les paires de hurwitz pseudo-euclidiennes et les algebres de lie z::(2)-graduees. Enfin, le formalisme utilise dans ce travail est parfaitement adapte a la construction de paires de hurwitz pseudo-euclidiennes au-dessus de varietes differentiables munies de structures spinorielles convenables, et a l'introduction de fibrations
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Gresnigt, Niels Gijsbertus. "Relativistic Physics in the Clifford Algebra Cℓ(1, 3)". Thesis, University of Canterbury. Physics and Astronomy, 2009. http://hdl.handle.net/10092/2581.
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There is growing evidence that the Clifford algebra Cℓ(1, 3) is the appropriate mathematical structure to formulate physical theories. The geometries of 3-space and spacetime are naturally reflected in the algebras Cℓ(0, 3) and Cℓ(1, 3) respectively. The choice of metric is important and we give further evidence that only the anti-Euclidean metric allows a proper treatment of rotations. The algebra Cℓ(1, 3) is not a division algebra. The invertibility or non-invertibility of elements in the algebra gives physical insight into the limitations of physical systems and non-invertbility should therefore not be regarded as a weakness of the algebra. The Lorentz force law is shown to arise from energy considerations of the electromagnetic field. This result shows that the Lorentz force is not a necessary addition to Maxwell's equations but rather follows from supplementing the electromagnetic energy density by Hamilton's principle. Maxwell's equations are written as a single geometric equations in Cℓ(1, 3). We review this derivation and other electromagnetic theory in the Clifford algebra framework. Taking the massless limit of Weinberg's spin one field equations results in a set of equations more general than Maxwell's equations, containing extra scalar fields. A derivation of these equations in Cℓ(1, 3) is presented and it is shown that, like the Maxwell equations, this set of equations can also be written as a single geometric equation. It has been suggested that the stabilised Poincaré-Heisenberg algebra gives an algebraic signature of quantum cosmology. It is shown that there exists a limit in which this algebra reduces to the conformal algebra. This limit describes how the present day Poincaré algebraic description relates to the conformal-algebraic description of the universe in the past. Furthermore, the proposed algebra inevitably leads to geometric changes in the underlying physical space and any cosmologically derived quantum effects may carry a strong polarisation and spin dependence. The algebra introduces a new dimensionless parameter, the importance of which has been difficult to pin down in the past. It is shown that this dimensionless parameter is closely related to the geometry of the underlying space and if non-zero will affect some of the quantum relativistic notions. The non-scalar basis elements of Cℓ(1, 3) are shown to generate the stabilised Poincar Heisenberg algebra under the Lie bracket [x, y] = xy − yx. The advantage of the Cℓ(1, 3) approach to the stabilised Poincaré-Heisenberg algebra is that it avoids the traditional stability considerations. It has been previously noted that gravitational effects in quantum measurement necessarily renders spacetime non-commutative and induces modifications to the fundamental commutators. This non-commutativity of spacetime and the corresponding modifications to the fundamental commutators arise naturally from the algebra Cℓ(1, 3). The study of the conformal group in Rp,q usually involves the conformal compactification of Rp,q. This allows the transformations to be represented by linear transformations in Rp+1,q+1. This embedding into a higher dimensional space comes at the expense of the geometric properties of the transformations. We show that this linearization procedure can be achieved with no loss of geometric insight, if, instead of using this compactification, we let the conformal transformations act on two copies of the associated Clifford algebra.
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Kallfelz, William Michael. "Clifford algebra a case for geometric and ontological unification /". College Park, Md. : University of Maryland, 2008. http://hdl.handle.net/1903/8079.
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Thesis (Ph. D.) -- University of Maryland, College Park, 2008.Thesis research directed by: Dept. of Philosophy. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.
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Megzari, Saïd. "Idempotents dans les algebres de clifford et fibrations spinorielles amorphes". Toulouse 3, 1986. http://www.theses.fr/1986TOU30146.
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L'objet de ce travail est la definition et l'etude des fibrations spinorielles amorphes sur une variete pseudo-riemannienne; sous fibres en ideaux a gauche minimaux du fibre de clifford, ces fibrations n'ont ete introduites que plus recemment par le professeur a. Crumeyrolle. On donne des conditions necessaires et suffisantes de leur existence qui se traduisent par la possibilite de reduire le groupe orthogonal a un groupe de spinorialite orthogonal. Tenant compte du fait que toute anti-involution gamma peut s'ecrire sous la forme gamma =beta oh ou h est un automorphisme on donne une caracterisation des anti-involutions commutant avec l'action d'un sous-groupe h de g en particulier lorsque h=g on trouve beta et beta. On montre aussi lorsqu'on a une reduction du groupe orthogonal 0(p,q) a 0(p) qui equivaut a l'existence de l'anti-involution tenant compte de la signature que pour tout ideal a gauche minimal de c::(p,q) (n=p+q pair) ou de c::(p,q) n impair. Il existe un idempotent primitif particulier et une anti-involution laissant cet idempotent invariant et commutant avec l'action d'un sous-groupe du groupe de clifford, associe a cet idempotant et qui intervient dans le probleme de reduction. L'existence de ces fibrations n'est pas reliee aux proprietes intrinseques de la variete, elles dependent du choix de l'idempotent engendrant la fibretype, ceci explique pourquoi les champs d'idempotents primitifs ne sont pas en general une bonne methode pour definir les fibrations spinorielles amorphes.
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Stanley, Adrian. "The geometric phase from imaginary time and Clifford algebra space translations". Thesis, University of Kent, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.240169.
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Banarer, Vladimir [Verfasser]. "Struktureller Bias in neuronalen Netzen mittels Clifford-Algebren / Vladimir Banarer". Kiel : Universitätsbibliothek Kiel, 2005. http://d-nb.info/1080317066/34.
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Lozano, Julia Carolina Torres. "Clifford and composed foliations". Universidade de São Paulo, 2017. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-18122017-132219/.
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Singular Riemannian foliations in spheres provide local models for an extensive kind of singular Riemannian foliations, whose theory contributes in the understanding of Riemannian manifolds. Hence the importance of studying and classifying them, a research subject that still remains open. In 2014, Marco Radeschi constructed indecomposable singular Riemannian foliations of arbitrary codimension, most of them inhomogeneous, which generalized all known examples of that type so far. The present dissertation is a detailed study of his work, along with observations about the progress made on this dynamic field since that paper was published. Besides introducing preliminary notions and examples on singular Riemannian foliations, isometric actions and Clifford theory, it is explained a construction of inhomogeneous isoparametric hypersurfaces, due to Ferus, Karcher and Münzner, that was a fundamental framework for the results of Radeschi. After that, it is described exhaustively the construction of Clifford and composed foliations in spheres, which are the examples that Radeschi created using Clifford systems. In the sequel it is established an extraordinary bijective correspondence between Clifford foliations (merely geometric objects) and Clifford systems (purely algebraic objects). This text finishes examining the relations of homogeneity properties among FKM, Clifford and composed foliations.Folheações Riemannianas singulares em esferas fornecem modelos locais para folheações Riemannianas singulares mais gerais, cuja teoria contribui na compreensão de variedades Riemannianas. Daí a sua importança de estudá-los e classificá-los, uma área de pesquisa que se mantém aberta. Em 2014, Marco Radeschi construiu folheações Riemannianas singulares indecomponíveis de codimensão arbitrária, a maioria delas não homogêneas, que generalizaram todos os exemplos conhecidos desse tipo até então. A presente dissertação é um estudo detalhado desse trabalho, junto com observações sobre avanços que se têm feito neste dinâmico campo desde a publicação do artigo. Após introduzir as noções e exemplos preliminares de folheações Riemannianas singulares, ações isométricas e teoria de Clifford, é explorada uma construção de hipersuperfícies isoparamétricas não homogêneas, devida a Ferus, Karcher e Münzner (FKM), que foi peça fundamental para os resultados de Radeschi. Em seguida, descreve-se minuciosamente a construção de folheações composta e de Clifford em esferas, que são os exemplos que o autor mencionado anteriormente gerou usando sistemas de Clifford. Continuando com a análise dessas novas folheações Riemannianas singulares, estabelece-se uma extraordinária correspondência biunívoca entre folheações de Clifford (objetos meramente geométricos) e sistemas de Clifford (objetos puramente algébricos). Este texto termina examinando as relações das propriedades de homogeneidade entre folheações FKM, compostas e de Clifford.
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Bujack, Roxana. "Orientation Invariant Pattern Detection in Vector Fields with Clifford Algebra and Moment Invariants". Doctoral thesis, Universitätsbibliothek Leipzig, 2014. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-158268.
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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.
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Bujack, Roxana. "Orientation Invariant Pattern Detection in Vector Fields with Clifford Algebra and Moment Invariants". Master's thesis, Universitätsbibliothek Leipzig, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-191202.
Texto completoResumen
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.
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Bach, Samuel. "Formes quadratiques décalées et déformations". Thesis, Montpellier, 2017. http://www.theses.fr/2017MONTS013/document.
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La L-théorie classique d'un anneau commutatif est construite à partir des formes quadratiques sur cet anneau modulo une relation d'équivalence lagrangienne. Nous construisons la L-théorie dérivée, à partir des formes quadratiques $n$-décalées sur un anneau commutatif dérivé. Nous montrons que les formes $n$-décalées qui admettent un lagrangien possèdent une forme standard. Nous montrons des résultats de chirurgie pour la L-théorie dérivée, qui permettent de réduire une forme quadratique décalée en une forme plus simple équivalente. On compare la L-théorie dérivée avec la L-théorie classique.On définit un champ dérivé des formes quadratiques dérivées, et un champ dérivé des lagrangiens dans une forme, qui sont localement algébriques de présentation finie. On calcule les complexes tangents, et on trouve des points lisses. On montre un résultat de rigidité pour la L-théorie : la L-théorie d'un anneau commutatif est isomorphe à celle d'un voisinage hensélien de cet anneau. Enfin, on définit l'algèbre de Clifford d'une forme quadratique n-décalée, qui est une déformation d'une algèbre symétrique en tant qu'E_k-algèbre. On montre un affaiblissement de la propriété d'Azumaya pour ces algèbres, dans le cas d'un décalage nul n=0, qu'on appelle semi-Azumaya. Cette propriété exprime la trivialité de l'homologie de Hochschild du bimodule de SerreThe classical L-theory of a commutative ring is built from the quadratic forms over this ring modulo a lagrangian equivalence relation.We build the derived L-theory from the n-shifted quadratic forms on a derived commutative ring. We show that forms which admit a lagrangian have a standard form. We prove surgery results for this derived L-theory, which allows to reduce shifted quadratic forms to equivalent simpler forms. We compare classical and derived L-theory.We define a derived stack of shifted quadratic forms and a derived stack of lagrangians in a form, which are locally algebraic of finite presentation. We compute tangent complexes and find smooth points. We prove a rigidity result for L-theory : the L-theory of a commutative ring is isomorphic to that of any henselian neighbourhood of this ring.Finally, we define the Clifford algebra of a n-shifted quadratic form, which is a deformation as E_k-algebra of a symmetric algebra. We prove a weakening of the Azumaya property for these algebras, in the case n=0, which we call semi-Azumaya. This property expresses the triviality of the Hochschild homology of the Serre bimodule
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Sönnerlind, Erik y Gustav Brage. "Braid group statistics and exchange matrices of non-abelian anyons : with representations in Clifford algebra". Thesis, KTH, Skolan för teknikvetenskap (SCI), 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-231567.
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When leaving classical physics and entering the realm of quantum physics, there are many new concepts being introduced. One of the most fundamental ideas in quantum mechanics is that particles no longer have exact known positions, but instead expected values and prob- abilities. This leads to the phenomena of truly identical particles, since they no longer can be distinguished simply by their positions. An important property differentiating different kinds of particles is how a system behaves when two such identical particles are exchanged. Historically, this divided particles into bosons and fermions, corresponding to symmetry and antisymmetry under an exchange. However, in two dimensions a new type of particle appears. These particles are called anyons, and behave differently when particles are exchanged. Anyons can be further divided into abelian and non-abelian anyons, of which this thesis will focus on the latter. The ex- changes can then be represented by the fundamental group of the configuration space of the particles, and in two dimensions this fundamental group is the braid group. Using rotors from a Clifford algebra and studying excitations of Majorana fermions, this thesis will show a way to calculate the exchange matrices of non-abelian anyons, and their corresponding eigenvalues. Furthermore, suggestions on a generalization of this framework along with areas where it can be applied are given.När man lämnar klassisk fysik och övergår till den kvantfysikaliska världen introduceras många nya koncept. En av de mest grundläggande idéerna inom kvantmekaniken är att partiklar inte längre har exakta positioner, eftersom dessa ersatts av väntevärden och sannolikheter. Detta leder till fenomenet att partiklar kan vara verkligt identiska, eftersom de inte längre kan särskiljas med hjälp av sina positioner. En viktig egenskap som särskiljer olika typer av partiklar är hur ett system beter sig vid ett utbyte av två sådana identiska partiklar. Historiskt sett delade denna egenskap upp partiklar i bosoner och fermioner, som uppvisar symmetri respektive antisymmetri vid ett partikelutbyte. I två dimensioner uppstår dock en ny typ av partiklar. Dessa partiklar kallas anyoner och beter sig annorlunda vid ett partikelutbyte. Vidare kan de delas upp i abelska och icke-abelska anyoner, varav denna rapport kommer fokusera på de senare. Utbytena kan representeras av den fundamentala gruppen av partiklarnas konfigurationsrum, och i två dimensioner blir denna fundamentala grupp flätgruppen. Genom att använda rotorer från en Cliffordalgebra och studera excitationer av Majoranafermioner, så visar denna rapport ett sätt att beräkna utbytesmatriserna för icke-abelska anyoner och deras tillhörande egenvärden. Vidare ges förslag på en generalisering av detta ramverk, tillsammans med områden där det kan tillämpas.
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Hadfield, Charles. "Structures de Clifford paires et résonances quantiques". Thesis, Paris Sciences et Lettres (ComUE), 2017. http://www.theses.fr/2017PSLEE010/document.
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Ce manuscrit se compose de deux parties indépendantes. La première partie de cette thèse étudie les structures de Clifford paires. Pour une variété riemannienne munie d’une structure de Clifford paire, nous introduisons l’espace de twisteurs en généralisant la construction d’un tel espace dans le cas d’une variété quaternion-hermitienne. Nous construisons une structure presque-complexe sur l’espace de twisteurs et considérons son intégrabilité lorsque la structure de Clifford est parallèle. Dans certains cas, nous pouvons aussi le fournir d’une métriquekählerienne ou, correspondant à une structure presque-complexe alternative, d’une métrique “nearly Kähler”. Dans un second temps, nous introduisons une structure appelée Clifford-Weyl sur une variété conforme. Il s’agit d’une structure de Clifford paireq ui est parallèle par rapport au produit tensoriel d’une connexion métrique sur le fibré de Clifford et une connexion de Weyl. Nous démontrons que la connexion de Weyl est fermée sauf dans certains cas génériques de basse dimension où nous arrivons à décrire des exemples explicites où les structures de Clifford-Weyl sont non-fermées. La seconde partie de cette thèse étudie des résonances quantiques. Au-dessus d’une variété asymptotiquement hyperbolique paire, nous considérons le laplacien de Lichnerowicz agissant sur les sections du fibré des formes multilinéaires symétriques.Lorsqu’il s’agit de formes bilinéaires symétriques, nous obtenonsune extension méromorphe de la résolvante dudit laplacien à l’ensemble du plan complexe si la variété est Einstein. Cela définit les résonances quantiques pour ce laplacien. Pour les formes multilinéaires symétriques en général, une telle extension méromorphe est possible si la variété est convexe-cocompacte. Dans les deux cas, nous devons restreindre le laplacien aux sections qui sont de trace et de divergence nulles. Nous utilisons ce deuxième résultat afin d’établir une correspondance classique-quantique pour les variétés hyperboliques convexescocompactes.La correspondance identifie le spectre du flot géodésique (les résonances de Ruelle) avec les spectres des laplaciens agissant sur les tenseurs symétriques qui sont de trace et de divergence nulles (les résonances quantiques)We study independently even Clfford structures on Riemannian manifolds and quantum resonances on asymptotically hyperbolic manifolds. In the first part of this thesis, we study even Clifford structures.First, we introduce the twistor space of a Riemannian manifold with an even Clifford structure. This notion generalises the twistor space of quaternion-Hermitian manifolds. We construct almost complex structures on the twistor space and check their integrability when the even Clifford structure is parallel. In some cases we give Kähler and nearly-Kähler metrics to these spaces. Second, we introduce the concept of a Clifford-Weyl structure on a conformal manifold. This consists of an even Clifford structure parallel with respect to the tensor product of a metric connection on the Clifford bundle and a Weyl structure on the manifold. We show that the Weyl structure is necessarily closed except for some “generic” low-dimensional instances,where explicit examples of non-closed Clifford-Weyl structures are constructed. In the second part of this thesis, we study quantum resonances. First, we consider the Lichnerowicz Laplacian acting on symmetric 2-tensors on manifolds with an even Riemannian conformally compact Einstein metric. The resolvent of the Laplacian,upon restriction to trace-free, divergence-free tensors, is shown to have a meromorphic continuation to the complex plane. This defines quantum resonances for this Laplacian. For higher rank symmetric tensors, a similar result is proved for convex cocompact quotients of hyperbolic space. Second, we apply this result to establish a direct classical-quantum correspondence on convex cocompact hyperbolic manifolds. The correspondence identifies the spectrum of the geodesic flow with the spectrum of the Laplacian acting on trace-free, divergence-free symmetric tensors. This extends the correspondence previously obtained for cocompact quotients
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Ramponi, Marco. "Clifford index and gonality of curves on special K3 surfaces". Thesis, Poitiers, 2017. http://www.theses.fr/2017POIT2317/document.
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Nous allons étudier les propriétés des courbes algébriques sur des surfaces K3 spéciales, du point de vue de la théorie de Brill-Noether.La démonstration de Lazarsfeld du théorème de Gieseker-Petri a mis en lumière l'importance de la théorie de Brill-Noether des courbes admettant un plongement dans une surface K3. Nous allons donner une démonstration détaillée de ce résultat classique, inspirée par les idées de Pareschi. En suite, nous allons décrire le théorème de Green et Lazarsfeld, fondamental pour tout notre travail, qui établit le comportement de l'indice de Clifford des courbes sur les surfaces K3.Watanabe a montré que l'indice de Clifford de courbes sur certaines surfaces K3, admettant un recouvrement double des surfaces de del Pezzo, est calculé en utilisant les involutions non-symplectiques. Nous étudions une situation similaire pour des surfaces K3 avec un réseau de Picard isomorphe à U(m), avec m>0 un entier quelconque. Nous montrons que la gonalité et l'indice de Clifford de toute courbe lisse sur ces surfaces, avec une seule exception déterminée explicitement, sont obtenus par restriction des fibrations elliptiques de la surface. Ce travail est basé sur l'article suivant :M. Ramponi, Gonality and Clifford index of curves on elliptic K3 surfaces with Picard number two, Archiv der Mathematik, 106(4), p. 355–362, 2016.Knutsen et Lopez ont étudié en détail la théorie de Brill-Noether des courbes sur les surfaces d'Enriques. En appliquant leurs résultats, nous allons pouvoir calculer la gonalité et l'indice de Clifford de toute courbe lisse sur les surfaces K3 qui sont des recouvrements universels d'une surface d'Enriques. Ce travail est basé sur l'article suivant :M. Ramponi, Special divisors on curves on K3 surfaces carrying an Enriques involution, Manuscripta Mathematica, 153(1), p. 315–322, 2017We study the properties of algebraic curves lying on special K3 surfaces, from the viewpoint of Brill-Noether theory.Lazarsfeld's proof of the Gieseker-Petri theorem has revealed the importance of the Brill-Noether theory of curves which admit an embedding in a K3 surface. We give a proof of this classical result, inspired by the ideas of Pareschi. We then describe the theorem of Green and Lazarsfeld, a key result for our work, which establishes the behaviour of the Clifford index of curves on K3 surfaces.Watanabe showed that the Clifford index of curves lying on certain special K3 surfaces, realizable as a double covering of a smooth del Pezzo surface, can be determined by a direct use of the non-simplectic involution carried by these surfaces. We study a similar situation for some K3 surfaces having a Picard lattice isomorphic to U(m), with m>0 any integer. We show that the gonality and the Clifford index of all smooth curves on these surfaces, with a single, explicitly determined exception, are obtained by restriction of the elliptic fibrations of the surface. This work is based on the following article:M. Ramponi, Gonality and Clifford index of curves on elliptic K3 surfaces with Picard number two, Archiv der Mathematik, 106(4), p. 355-362, 2016.Knutsen and Lopez have studied in detail the Brill-Noether theory of curves lying on Enriques surfaces. Applying their results, we are able to determine and compute the gonality and Clifford index of any smooth curve lying on the general K3 surface which is the universal covering of an Enriques surface. This work is based on the following article:M. Ramponi, Special divisors on curves on K3 surfaces carrying an Enriques involution, Manuscripta Mathematica, 153(1), p. 315-322, 2017
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Silva, Humberto José Gama da. "A Álgebra de Clifford: uma aplicação no conceito de força magnética". Universidade Estadual da Paraíba, 2010. http://tede.bc.uepb.edu.br/tede/jspui/handle/tede/1658.
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Previous issue date: 2010-12-17The process of teaching learning Physics, in Brazil, has been recognized as deficient in several studies. Particularly, we note that one of the problems has been the mathematical framework regarding the use of physical concepts. This problem seems to generate a conceptual mathematical physical dichotomy which affects the understanding and assimilation of the deep connections between Physics and Mathematics. The aim of this work was to present an exploratory study that evaluated according to the findings by means of the data collection the feasibility of using Clifford Algebra as a formalism adapted to the study of electromagnetism in high school level, specifically obtaining the characteristics of the magnetic force vector which acts on electric charges or electrical currents within a magnetic field. Therefore, it was carried out two interventions at different dates. The first one was done in Campina Grande - PB, at the Dean of Graduate Studies and Research University of Paraiba, UEPB. The second was done in Imperatriz MA, at the Federal institute of education science and technology IFMA. Both intervention had as public target students, teachers and future teachers of physics for high school level. Motivated by the characteristics of objectivity and serviceability of Ausubel‟s cognitive theory, its foundations were used for developing a potentially significant material developed by the selection and reading of literary criticism about Vector Algebra and Geometry. The same foundations were also used as an adjunct in the learning process content covered in the interventions and subsumers in identifying the content being addressed, the use of conceptual maps as facilitative technique in the expositions of topics and as evaluation tool. At those intersections was pointed out that the formalism of Gibbs still has predominance in the textbooks adopted in at secondary and high education levels, even prompting the students to use it in the mathematical treatment directed to the study of physical measures memorizing precepts not justified , like rule of right hand. However, the structure or the Clifford Algebra enables a more intuitive mathematical modeling which is characterized by the representation and manipulation of basic geometric concepts such as magnitude, direction and meaning.
O processo ensino aprendizagem da Física, no Brasil, tem sido reconhecido como deficiente em diversos estudos. Particularmente, gostaríamos de destacar que um dos problemas tem sido o ferramental matemático com relação ao uso dos conceitos físicos. Este problema parece gerar uma dicotomia conceitual físico-matemática que prejudica a compreensão e assimilação das profundas conexões entre a Física e a Matemática. O objetivo desse trabalho é apresentar um estudo exploratório em que foi avaliada de acordo com os resultados obtidos através de instrumentos de coleta de dados a viabilidade do uso da Álgebra de Clifford como um formalismo adaptável para o estudo do Eletromagnetismo no Ensino Médio, especificamente na obtenção das características do vetor Força Magnética que atua em cargas elétricas em movimento ou em correntes elétricas dentro de um campo magnético. Para tanto foram feitas duas intervenções, em datas distintas. A primeira foi realizada em Campina Grande PB, na Pró-Reitoria de PósGraduação e Pesquisa da Universidade Estadual da Paraíba UEPB. A segunda foi realizada em Imperatriz MA, no Instituto Federal de Educação Ciência e Tecnologia IFMA. Os dois eventos tiveram como público alvo alunos, professores e futuros professores de Física do Ensino Médio. Motivado pelas características de objetividade e operacionalidade da Teoria Cognitivista de Ausubel, seus fundamentos foram utilizados na elaboração de um material potencialmente significativo desenvolvido a partir da seleção e leitura crítica da produção literária acerca das Álgebras Vetorial e Geométrica usando como coadjuvante no processo ensino-aprendizagem dos conteúdos contemplados nas intervenções. Os mesmos fundamentos também foram utilizados na identificação dos subsunçores do conteúdo a ser abordado, no uso de Mapas Conceituais como técnica facilitadora na exposição dos tópicos e como instrumento de avaliação. Nas referidas intercessões foi apontado que o formalismo de Gibbs ainda exerce predominância nos livros textos adotados no Ensino Médio e Superior, mesmo induzindo os alunos a utilizarem, no tratamento matemático direcionado ao estudo das grandezas físicas, preceitos de memorização, não justificados, como a regra da mão direita. Entretanto, a estrutura da Álgebra de Clifford permite uma modelagem matemática mais intuitiva, que tem como característica a representação e manipulação de conceitos geométricos básicos, tais como magnitude, direção e sentido.
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Brito, Kelvyn Páterson Sousa de [UNESP]. "Supersimetria não-anticomutativa". Universidade Estadual Paulista (UNESP), 2013. http://hdl.handle.net/11449/108892.
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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
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Vaz, Júnior Jayme 1964. "A algebra do espaço-tempo, o spinor de Dirac-Hestenes e a teoria do eletron". [s.n.], 1993. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306702.
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Orientador: Waldyr A. Rodrigues Jr.Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Científica
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Resumo: A relação entre a teoria do elétron e o eletromagnetismo é discutida com base no uso da álgebra do espaço-tempo e do spinor de Dirac-Hestenes. Desta relação surge uma equação não-linear como uma alternativa, a princípio mais satisfatória, à equação de Dirac. Este estudo é possível uma vez formulada a teoria do spinor de Dirac-Hestenes como uma classe de equivalência de elementos da sub-álgebra par da álgebra do espaço-tempo.
Abstract: The relationship between the theory of electron and electromagnetism is discussed by using the spacetime algebra and the Dirac-Hestenes spinor. From this relationship it emerges a non-linear equation which seems to be more satisfactory than Dirac equation. This study is possible once it is formulated the theory of Dirac- Hestenes spinor as an equivalence class of elements of the even subalgebra of the spacetime algebra.
Doutorado
Doutor em Matemática Aplicada