Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

Universiẗat, Diss., 1997--Kaiserslautern.

Orientador: Waldyr Alves Rodrigues Junior / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-07-26T01:36:39Z (GMT). No. of bitstreams: 1 Carvalho_AlexandreLuisTrovonde_D.pdf: 15293549 bytes, checksum: 6f77ce91b6897c18e527e4134e109ed1 (MD5) Previous issue date: 2000 / Resumo: O propósito desta tese é estudar, sob o ponto de vista algébrico, o papel desempenhado pelos operadores diferenciais nos formalismos variacionais Lagrangeano e Hamiltoneano. Apresentamos uma aplicação simples das idéias e resultados básicos da teoria dos operadores diferenciais às álgebras de Clifford, obtendo uma relação entre os operadores diferenciais e o operador de Dirac. Introduzimos um formalismo Hamiltoneano, com base nos módulos de símbolos dos operadores diferenciais, generalizando os resultados para anéis comutativos. Nesse formalismo, encontramos importantes propriedades algébricas para a Hamiltoneana, e destacamos o colchete de Poisson como uma estrutura mais básica que a forma simplética canônica. Introduzimos o conceito de adjunta de um operador diferencial e, por meio dela, caracterizamos as formas integrais em termos das formas de Berezin. Obtemos uma seqüência espectral relacionando a cohomologia das formas integrais com a cohomologia de De Rham, tanto para variedades quanto para supervariedades. Introduzimos o conceito de Lagrangeana, e analisamos sua relação com as formas de Berezin. Nesse contexto, estudamos as leis de conservação, e obtemos um equivalente algébrico para o Teorema de Noether. Finalmente, essas construções nos encaminham rumo a uma versão algébrica para o teorema do índice. / Abstract: The purpose of this thesis is to study, from the algebraic viewpoint, the rule played by the differential operators in Lagrangian and Hamiltonian variational formalisms. We present a simple application of the basic ideas and results form the theory of differential operators to the Clifford algebras, from where we obtain a relationship between differential operators and the Dirac operator. We introduce a Hamiltonian formalism based on the symbol modules, generalizing some results to commutative rings. In this formalism we find important algebraic properties for the Hamiltonian and notice that the Poisson bracket is a more fundamental structure than the canonical sympletic form. We introduce the concept of adjoint of a differential operator and by means of it we are able to charactrize the integral forms in terms of Berezin forms. We obtain a spectral sequence relating the cohomology of integral forms to the De Rham cohomology, for both manifolds and supermanifolds. In this context, we study the con- servation laws and obtain an algebraic equivalent to the Noether theorem. Finally, these constructions direct us towards an algebraic version to the index theorem. / Doutorado / Doutor em Matemática

Made available in DSpace on 2014-06-12T18:28:44Z (GMT). No. of bitstreams: 2 arquivo4374_1.pdf: 611135 bytes, checksum: 2535337805c3894ff5d22fc566c01f86 (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2008 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Nesta dissertação tratamos alguns aspectos da teoria das formas quadráticas sobre um corpo, das álgebras com divisão e das álgebras centrais simples. Objetos importantes estudados são o anel de Witt, o grupo de Brauer, as álgebras de Clifford e o teorema de Wedderburn sobre a estrutura das álgebras centrais simples. Essas teorias são profundamente ligadas entre si e tem conexões com outras áreas como a teoria dos corpos, a geometria algébrica, a topologia algébrica, a teoria das representações e a física teórica. Matemáticos ilustres como Brauer, Clifford, Emmy Noether, Gauss, Hamilton, Hasse, Hurwitz e Wedderburn trabalharam nos temas detalhados nesta dissertação

Orientador : Roberto Cardoso de Oliveira / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciencias Humanas / Made available in DSpace on 2018-07-14T00:37:51Z (GMT). No. of bitstreams: 1 AzzanJunior_Celso_M.pdf: 5239426 bytes, checksum: 673391549de67dac1b12cbe41d923a1d (MD5) Previous issue date: 1991 / Resumo: Não informado / Abstract: Not informed / Mestrado / Mestre em Ciências Sociais

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.

>Magister Scientiae - MSc / The aim of this dissertation is to calculate character tables of group extensions. There are several well developed methods for calculating the character tables of some selected group extensions. The method we study in this dissertation, is a standard application of Clifford theory, made efficient by the use of Fischer-Clifford matrices, as introduced by Fischer. We consider only extensions Ḡ of the normal subgroup N by the subgroup G with the property that every irreducible character of N can be extended to an irreducible character of its inertia group in Ḡ , if N is abelian. This is indeed the case if Ḡ is a split extension, by a well known theorem of Mackey. A brief outline of the classical theory of characters pertinent to this study, is followed by a discussion on the calculation of the conjugacy classes of extension groups by the method of coset analysis. The Clifford theory which provide the basis for the theory of Fischer-Clifford matrices is discussed in detail. Some of the properties of these Fischer-Clifford matrices which make their calculation much easier, are also given. We restrict ourselves to split extension groups Ḡ = N:G in which N is always an elementary abelian 2-group. In this thesis we are concerned with the construction of the character tables (by means of the technique of Fischer-Clifford matrices) of certain extension groups which are associated with the orthogonal group O+10(2), the automorphism groups U₆(2):2, U₆(2):3 of the unitary group U₆(2) and the smallest Fischer sporadic simple group Fi₂₂. These groups are of the type type 2⁸:(U₄(2):2), (2⁹ : L₃(4)):2, (2⁹:L₃(4)):3 and 2⁶:(2⁵:S₆).

A mix of physics, mathematics, and computer science, the study of quantum information seeks to understand and utilize the information that can be held in the state of a quantum system. Quantum cryptography is then the study of various cryptographic protocols on the information in a quantum system. One of the goals we may have is to verify the integrity of quantum data, a process called quantum message authentication. In this thesis, we consider two quantum message authentication schemes, the Clifford code and the trap code. While both of these codes have been previously proven secure, they have not been proven secure in the simulator model, with an efficient simulation. We offer a new class of simulator that is efficient, so long as the adversary is efficient, and show that both of these codes can be proven secure using the efficient simulator. The efficiency of the simulator is typically a crucial requirement for a composable notion of security. The main results of this thesis have been accepted to appear in the Proceedings of the 9th International Conference on Information Theoretic Security (ICITS 2016).

Orientador: Waldyr A. Rodrigues Jr. / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Científica / Made available in DSpace on 2018-07-18T17:46:13Z (GMT). No. of bitstreams: 1 VazJunior_Jayme_D.pdf: 3696074 bytes, checksum: 38c75818b237a8ff519bd3727c669660 (MD5) Previous issue date: 1993 / 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

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