The present thesis can be divided in three parts: 1) Quantum gate teleportation; 2) Numerical search of universal entanglers; 3) Connections between quantum information and number theory. Regarding the quantum gate teleportation, a separability criterion of normal matrices is used to find the analytical conditions of the preservation of separability under conjugation. That analytical condition allowed to find the general formula of an element of $mathbb{C}^{4}$ Clifford group, as well to understand the role of the basis of measurement in the quantum gate teleportation protocol. Considering the searching for universal entanglers, the same separability criterion of normal matrices was used as fitness function in a computational heuristics, in prder to find good candidates for universal entanglers in $mathbb{C}^{3} otimes mathbb{C}^{4}$ and $mathbb{C}^{4} otimes mathbb{C}^{4}$ Hilbert spaces. At last, in the connection of quantum information with the number theory, it is presented the study of the preparation and entanglement of several multi-qubit quantum states based in integer sequences, and the Riemannian quantum circuit, a quantum circuit whose eigenvalues are related to the zeros of the Riemann zeta function. The existence of such circuit proves that is always possible to construct a physical system related to a finite amount of zeros. / A presente tese està dividida em trÃs partes: 1) TeleportaÃÃo de portas quÃnticas; 2) Busca numÃrica por entrelaÃadores universais; 3) ConexÃes entre a informaÃÃo quÃntica e a teoria dos nÃmeros. No que diz a teleportaÃÃo de portas quÃnticas, um critÃrio de separabilidade para matrizes normais à usada para encontrar as condiÃÃes analÃticas da preservaÃÃo da separabilidade sob conjugaÃÃo. Tais condiÃÃes analÃticas permitiram encontrar a forma geral de um elemento do grupo de Clifford em $mathbb{C}^{4}$, assim como tambÃm entender o papel da base de mediÃÃo no protocolo de teleportaÃÃo de portas quÃnticas. Considerando a busca por entrelaÃadores universais, o mesmo critÃrio de separabilidade de matrizes normais foi utilizado como funÃÃo de aptidÃo em uma heurÃstica computacional aplicada para encontrar bons candidatos a entrelaÃadores universais nos espaÃos de Hilbert de dimensÃes $mathbb{C}^{3} otimes mathbb{C}^{4}$ e $mathbb{C}^{4} otimes mathbb{C}^{4}$. Por fim, sobre as conexÃes da informaÃÃo quÃntica com a teoria dos nÃmeros, à apresentado um estudo da preparaÃÃo e entrelaÃamento de vÃrios estados quÃnticos de mÃltiplos qubits baseados em sequÃncias de nÃmeros inteiros. Apresenta-se ainda o circuito quÃntico Riemanniano, um circuito quÃntico cujos autovalores sÃo relacionados aos zeros da funÃÃo Zeta de Riemann. A existÃncia deste circuito prova que à sempre possÃvel construir um sistema fÃsico relacionado a uma quantidade finita de zeros.

Orientador: Waldyr Alves Rodrigues Junior / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T23:09:32Z (GMT). No. of bitstreams: 1 Resende_AdrianaSouza_M.pdf: 17553204 bytes, checksum: a66cefe30e9957cc4351e03d3aec35b2 (MD5) Previous issue date: 2010 / 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

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-05-19T14:40:22Z No. of bitstreams: 2 Dissertação - Fernando Arnulfo Zuniga Argote - 2015.pdf: 631746 bytes, checksum: 0d49f26d4f922ddd70836a2024ad5850 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-05-19T14:50:54Z (GMT) No. of bitstreams: 2 Dissertação - Fernando Arnulfo Zuniga Argote - 2015.pdf: 631746 bytes, checksum: 0d49f26d4f922ddd70836a2024ad5850 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-05-19T14:50:54Z (GMT). No. of bitstreams: 2 Dissertação - Fernando Arnulfo Zuniga Argote - 2015.pdf: 631746 bytes, checksum: 0d49f26d4f922ddd70836a2024ad5850 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-02-27 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this work we refer to the study of a geometric invariant surfaces immersed in Euclidean 3-dimensional sphere S3. Such invariant, known as angle contact, is the complementary angle between the distribution of contact d and the tangent space of the surface. Montes and Verderesi [22] characterized the minimal surfaces in S3 with constant contact angle and Almeida, Brazil and Montes [4] studied some properties of immersed constant mean curvature into a round sphere S3 with constant contact angle. The our aim of this work is to deduce a general formula involving the Gaussian curvature, the mean curvature and the contact angle of surfaces immersed in Euclidean sphere 3-dimensional, which shows that the surface is flat if the contact angle is constant. Moreover, we deduce that the Clifford tori are the unique compact surfaces with constant mean curvature having such propriety. Keywords / Neste trabalho nos referimos ao estudo de um invariante geométrico de superfícies imersas na esfera Euclidiana 3-dimensional S3. Tal invariante, conhecido como ângulo de contato, é o complementar do ângulo entre a distribuição de contato d e o espaço tangente da superfície. Montes e Verderesi [22] caracterizaram as superfícies mínimas em S3 com ângulo de contato constante e Almeida, Brasil e Montes [4] estudaram algumas propriedades de superfícies imersas com curvatura média e ângulo de contato constantes em S3. Nosso objetivo será apresentar uma relação entre a curvatura Gaussiana, a curvatura média e o ângulo de contato de superfícies imersas na esfera Euclidiana 3-dimensional, a qual permite concluir que a superfície é plana se o ângulo de contato for constante. Além disso, concluiremos que o toro de Clifford é a única superfície compacta com curvatura média constante tendo tal propriedade.

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

[pt] A álgebra de Clifford é uma álgebra associativa que pode ser realizada matricialmente. O grupo Spin é uma superfície contida na álgebra de Clifford e fechada por multiplicação. Estudamos os geradores de tal grupo, assim como do grupo finito gerado pelos elementos agúdos e o grupo Quat, ambos grupos de matrizes e subconjuntos do grupo Spin. Uma permutação no grupo de permutações, pode ser expressa como uma palavra reduzida, por meio de geradores de Coxeter. Os mapas acute e grave nos fornecem elementos no grupo finito, já mencionado, gerado pelos elementos agúdos, a partir das palavras reduzidas de uma permutação. Um elemento da álgebra de Clifford pode ser escrito como uma combinação linear de elementos do grupo Quat, onde o coeficiente independente é conhecido como parte real. Estudamos resultados que relacionam as características de uma permutação no grupo de permutações, com o elemento a ela relacionado na álgebra de Clifford. / [en] The Clifford algebra is an associative algebra that can be constructed as an algebra of matrices. The group Spin is a surface contained in the Clifford algebra and closed by multiplication. We studied the generators of such group, as well as of the finite group contained in Spin and generated by the acute elements and the group Quat, both matrix groups and subsets of Spin. A permutation in the permutation group, can be expressed as a reduced word, using transpositions to define the family of Coxeter generators. The acute and grave maps provide us with elements in the finite group, already mentioned, generated by the acute elements, based on the reduced words of a permutation. An element of Clifford algebra can be written as a linear combination of elements in Quat, where the independent coefficient is known as the real part. We studied results that relate the characteristics of a permutation in the permutation group, with the element related to it in the Clifford algebra.

This thesis addresses the objective of formulating a moral-philosophically credible media ethics by encompassing two main dimensions: the examination of existing media-ethical models and the integration of sustainable arguments through utilizing eco-religious epistemology to overcome identified challenges. By incorporating eco-critical perspectives within the Media ecology tradition, which studies media as environments, the thesis proposes a creative intersection between established media theories and a theological tradition that tackles profound questions about humanity's place in the world. The initial section of the thesis analyzes moral-philosophical concepts within the media ethics frameworks proposed by three ethicists; Susanne Wigorts Yngvesson; Clifford G. Christians and Sandra L. Borden. While all three endorse universal values, they offer distinct frameworks for both understanding the ontology of universals and how these are contextualized in the journalistic community. Christians' moral epistemological theories,particularly regarding the ethics of being and Proto-norms have generated significant controversy and scholarly debate on the necessity and intellectual credibility of defending universal values in media ethics. Critical perspectives from this discourse are presented and evaluated as a complement to the analysis of Christians' media-ethical model. In exploring sustainable moral-philosophical arguments, the thesis suggests that moral claims can exhibit both universal and contextual characteristics. It suggests that a media-ethical model should integrate universal values with a communitarian perspective on journalists'moral responsibility. However, understanding the interplay between the contextual and universal dimensions of ethical values requires engaging in critical moral-philosophical theory. In addition, critical perspectives on technology and ideas of space and time need to be addressed in new ways. Thus, eco-religious epistemology, as advocated by ethicist and theologian Whitney Bauman, is proposed as a creative means to understand moral-philosophical questions regarding space/time, contextualism/universalism, and technology within media ethics.

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

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho vamos estudar hipersuperfÃcies M^n da esfera unitÃria S^{n+1} conexas completas com duas curvaturas principais distintas uma das quais de multiplicidade n-1 e possuindo k-Ãsima funÃÃo de curvatura nula Sob tais condiÃÃes vamos provar que o toro de Clifford à a Ãnica hipersuperfÃcie que satisfaz S maior que ou igual a n(k^2-2k+n)}/{k(n-k)}=c(n,k) onde S representa o quadrado da norma da segunda forma fundamental AlÃm disso vamos mostrar que no caso compacto a integral sobre M de S à menor que ou igual a c(n,k)vol(M) ocorrendo igualdade somente no toro de Clifford

Banca: Álvaro de Souza Dutra / Banca: Denis Dalmazi / Banca: José Abdalla Helayel Neto / Banca: Dionísio Bazeia / Banca: Vilson Zanchin / Resumo: Optei pela apresentação de uma análise crítica de alguns trabalhos levados a termo após o ingresso como Docente do Departamento de Física e Química da Faculdade de Engenharia de Guaratinguetá - UNESP, em subsituição à tradicional tese de Livre-Docência. Essa opção na sistemática de apresentação se coaduna com um duplo aspecto: apresentar criticamente as contribuições realizadas na área e consolidar perspectivas de continuidade do trabalho exposto / Abstract: Not available

Orientador: Júlio Marny Hoff da Silva / Banca: Saulo Henrique Pereira / Banca: Marco André Ferreira Dias / Banca: Maria Emilia Xavier Guimarães / Banca: José Abdalla Helayel-Neto / Resumo: Na presente tese abordaremos quatro tópicos importantes: espinores, covariantes bilineares, classificação de Lounesto e o teorema da inversão. Apresentamos a construção de covariantes bilineares para o espinor Elko e mostraremos a necessidade da deformação dos elementos da base da álgebra de Clifford com a finalidade de que as identidades de Fierz-Pauli-Kofink sejam satisfeitas. Estudamos também os ingredientes principais da classificação de espinores elaborada por Lounesto. Por último, construiremos três novas classes de espinores via o teorema da inversão a partir da premissa que o covariante bilinear $J_{\mu}$ seja nulo. Como consequência desta consideração esses novos espinores não possuem a dinâmica de Dirac, haja visto que $J_{\mu}$ na teoria de Dirac representa a corrente conservada. O surgimento de apenas três novas classes de espinores é uma consequência direta da imposição de que as identidades de Fierz-Pauli-Kofink sejam satisfeitas / The present thesis covers four important topics: spinors, bilinear covariants, Lounesto's classification and the inversion theorem. We show and explicit the construction of bilinear covariants for the Elko spinors and the necessity of deformation of the Cli ord algebra basis elements in order to satisfy the Fierz-Pauli-Ko nk identities. We also study the main ingredients of the classification of spinors elaborated by Lounesto. Finally, we construct three new classes of spinors via the inversion theorem from the premise that the bilinear covariant Jµ is null. As a consequence, these new spinors do not have usual dynamics of Dirac, have seen that Jµ in Dirac's theory represents the conserved current. The emergence of only three new classes of spinors is a direct consequence of the requeriment that Fierz-Pauli-Kofink's identities must hold / Doutor