Made available in DSpace on 2015-05-15T11:46:00Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 564673 bytes, checksum: caec7ef95e0e9b70eb60b56bfa2f6547 (MD5) Previous issue date: 2011-08-03 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this paper we study finite Abelian groups, where state and prove the fundamental theorem of finitely generated abelian groups, as well as determine a characterization of automorphisms of a p-group, moreover, we exhibit an algorithm that determines the count of the number of automorphisms of p-groups. Finally, we show the automorphisms of the non-Abelian dihedral group. / Neste trabalho estudamos Grupos Abelianos finitos, onde enunciamos e provamos o Teorema fundamental dos grupos abelianos finitamente gerados, bem como determinamos uma caracterização dos automorfismos de um p-grupo, além disso, exibimos um algoritmo que determina a contagem do número de automorfismos desses p-grupos. Por fim, mostramos os automorfismos do grupo não-Abeliano Diedral .

Made available in DSpace on 2015-05-15T11:46:11Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 411324 bytes, checksum: 65de8bf46cc2dff58911edbcb15868ca (MD5) Previous issue date: 2011-09-09 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work, we study the semisimple group algebras FqCn of the finite abelian groups Cn over a finite field Fq and give conditions so that the number of its simple components is minimal; i.e. equal to the number of simple components of the rational group algebra of the same group. Under such conditions, we compute the set of primitive idempotents of FqCn and from there, we study the abelian codes as minimal ideals of the group algebra, which are generated by the primitive idempotents, computing their dimension and minimum distances. / Neste trabalho, estudamos álgebras de grupos semisimples FqCn de grupos abelianos finitos Cn sobre um corpo finito Fq e as condições para que o número de componentes simples seja mínimo, ou seja igual ao número de componentes simples sobre a álgebra de grupos racionais do mesmo grupo. Sob tais condições, calculamos o conjunto de idempotentes primitivos de FqG e a de partir daí, estudamos os códigos cíclicos como ideais minimais da álgebra de grupo, os quais são gerados pelos idempotentes primitivos, calculando suas dimensões e distâncias mínimas.

Made available in DSpace on 2015-05-15T11:46:25Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 450936 bytes, checksum: 7e53189000f5416171ee58a4623b8aa6 (MD5) Previous issue date: 2010-02-09 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Bertholf and Walls provided a characterization for the class of groups quasi-injective finite. Furthermore, Juriaans, Bastos Azevedo and give a rating for the injective type groups, which are a distinct class of the former despite being quite close. / Bertholf e Walls forneceram uma caracterização para a classe de grupos quasi-injetivos finitos. Além disso, Juriaans, Bastos e Azevedo dão uma classificação para os grupos do tipo injetivo, os quais são uma classe distinta da anterior apesar de serem bastante próximas.

O caso finito de teorias topológicas de 1-gauge, quando nenhuma simetria global está presente, é bastante bem compreendido e classificado. Nos últimos anos, as tentativas de generalizar as teorias de 1-gauge através das chamadas teorias de 2-gauge abriram a porta para novos modelos interessantes e novas fases topológicas, as quais não são descritas pelos esquemas de classificação anteriores. Nesta tese, vamos além da construção de 2-gauge, e consideramos uma classe de modelos que vivem em maiores dimensões. Esses modelos estão inseridos em uma estrutura de complexos de cadeia de grupos abelianos, forçando a generalizar o conceito usual de configurações de gauge. A vantagem de tal abordagem é que, a ordem topológica fica manifestamente explcita. Isto é feito em ter- mos de uma cohomologia com coeficientes em um complexo de cadeia finita. Além disso, mostramos que a degenerescência do estado fundamental suporta um conjunto conve- niente de números quânticos que indexam os estados e que, além, foram completamente caracterizados. Consequentemente, nós também mostramos que muitos dos exemplos abelianos de teorias de 1 -gauge 2-gauge são recuperados como casos especiais desta construção. / The finite case of 1-gauge topological theories, when no global symmetries are present, is fairly well understood and classified. In recent years, attempts to generalize the latter situation through the so called 2-gauge theories have opened the door to interesting new models and new topological phases, not described by the previous schemes of classifica- tion. In this paper we go even beyond the 2-gauge construction by considering a class of models that live in arbitrary higher dimensions. These models are embedded in a structure of chain complexes of abelian groups, forcing to generalize the usual notion of gauge configurations. The advantage of such an approach is that, the topological order is explicitly manifest when the ground state space of these models is described. This is done in terms of a cohomology with coefficients in a finite chain complex. Furthermore, we show that the ground state degeneracy underpins a convenient set of quantum num- bers that label the states and that have been completely characterized. We also show that abelian examples of 1-gauge 2-gauge theories are recovered as special cases of this construction.

Dada uma palavra w e um grupo G, suponha que o conjunto Gw pode ser coberto por finitos subgrupos cÃclicos. à verdade que w(G) tambÃm pode ser coberto por finitos subgrupos cÃclicos? Nesta dissertaÃÃo mostraremos que a resposta à positiva para a palavra comutador. / Given a word w and a group G, suppose that the set can be Gw covered by finite cyclic subgroups. It is true that w(G) can also be covered by finite cyclic subgroups? This dissertation will show that the answer is positive for the word switch.

Orientador: Dessislava Hristova Kochloukova / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T02:25:15Z (GMT). No. of bitstreams: 1 Silva_LeonardodeAmorine_D.pdf: 582293 bytes, checksum: ed3c907af5279b8923c782d730bcf1d4 (MD5) Previous issue date: 2014 / Resumo: Nesta tese consideramos uma extensão cindida G de um grupo abeliano A por um grupo nilpotente (de classe 2) Q e provamos dois resultados. Primeiro, se Q age nilpotentemente sobre A e G tem tipo FP2, calculamos o sigma invariante de G em dimensão 2. Segundo, se G tem tipo FP4, mostramos que cada quociente de G tem tipo FP4 / Abstract: In this thesis we consider a split extension G of an abelian group A by a nilpotent group (class 2) Q and prove two results. First, if Q acts nilpotently on A and G has type FP2, compute the sigma invariant of G in dimension 2. Second, if G has type FP4, we show that every quotient G has type FP4 / Doutorado / Matematica / Doutora em Matemática

We deal with two themes that are illustrative of the rigidity and regularity of holomorphic mappings. The first one concerns the regularity of continuous CR mappings between smooth pseudo convex, finite type hypersurfaces which is a well studied subject for it is linked with the problem of studying the boundary behaviour of proper holomorphic mappings between domains bounded by such hypersurfaces. More specifically, we study the regularity of Lipschitz CR mappings from an h-extendible(or semi-regular) hypersurface in Cn .Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudo convex domains is also proved. The second theme dealt with, is the classification upto biholomorphic equivalence of model domains with abelian automorphism group in C3 .It is shown that every model domain i.e.,a hyperbolic rigid polynomial domainin C3 of finite type, with abelian automorphism group is equivalent to a domain that is balanced with respect to some weight.

Photonics, the technological field that encompasses all aspects of light, has been rapidly growing and increasingly useful in uncovering fundamental truths about nature. It has helped detect gravitational waves, allowed for a direct measurement of the quantum wave function, and has helped realize the coldest temperatures in the universe. But photonics has also had an enormous impact on day-to-day life as well; it has enabled high capacity and/or high speed telecommunication, offered cancer treatment solutions, and has completely revolutionized display and scanning technology. All of these discoveries and applications have required a superb understanding of light, but also a high degree of control over the sometimes abstract properties of light. The work contained in this thesis explores two novel means of controlling and manipulating two different abstract properties of light. In Part I, the property under investigation is the polarization state of light – a property that is paramount to all light-matter interactions, and even some light-light interactions such as interference. Here, a liquid crystal on silicon spatial light modulator (LCOS-SLM)’s capabilities in manipulating the polarization state of light is theoretically examined and experimentally exploited, tested, and reported on. It is found through experimentation that, for an appropriate range of beam sizes and input polarizations, a single LCOS-SLM can be used to produce any light field with an arbitrary, spatially varying polarization profile. In Part II, the property under investigation loosely corresponds to light’s spatial degree of freedom – how light can move from one spot in space to another in a non-trivial manner. Here, control over light’s position through a waveguide array through the use of quantum geometric phase is theoretically examined, simulated, and experimentally designed. It is found through simulation that a threewaveguide array is capable of implementing two dimensional unitary transformations. The common theme between Part I and Part II is manipulating these properties of light to realize classes of general transformations. Moreover, if the light field is treated as a quantum state in the basis of either property under investigation, a two dimensional computational basis ensues. This is precisely the right cardinality for applications in quantum information.

Le polynôme caractéristique d'une variété abélienne sur un corps fini est défini comme étant celui de son endomorphisme de Frobenius. La première partie de cette thèse est consacrée à l'étude des polynômes caractéristiques de variétés abéliennes de petite dimension. Nous décrivons l'ensemble des polynômes intervenant en dimension 3 et 4, le problème analogue pour les courbes elliptiques et surfaces abéliennes ayant été résolu par Deuring, Waterhouse et Rück.Dans la deuxième partie, nous établissons des bornes supérieures et inférieures sur le nombre de points rationnels des variétés abéliennes sur les corps finis. Nous donnons ensuite des bornes inférieures spécifiques aux variétés jacobiennes. Nous déterminons aussi des formules exactes pour les nombres maximum et minimum de points rationnels sur les surfaces jacobiennes. / The characteristic polynomial of an abelian variety over a finite field is defined to be the characteristic polynomial of its Frobenius endomorphism. The first part of this thesis is devoted to the study of the characteristic polynomials of abelian varieties of small dimension. We describe the set of polynomials which occur in dimension 3 and 4; the analogous problem for elliptic curves and abelian surfaces has been solved by Deuring, Waterhouse and Rück.In the second part, we give upper and lower bounds on the number of points on abelian varieties over finite fields. Next, we give lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian surfaces.

Le logarithme discret sur les courbes elliptiques fournit la panoplie standard de la cryptographie à clé publique: chiffrement asymétrique, signature, authentification. Son extension à des courbes hyperelliptiques de genre supérieur se heurte à la difficulté de construire de telles courbes qui soient sécurisées. Dans cette thèse nous utilisons la théorie des fonctions thêta développée par Mumford pour construire des algorithmes efficaces pour manipuler les variétés abéliennes. En particulier nous donnons une généralisation complète des formules de Vélu sur les courbes elliptiques pour le calcul d'isogénie sur des variétés abéliennes. Nous donnons également un nouvel algorithme pour le calcul efficace de couplage sur les variétés abéliennes en utilisant les coordonnées thêta. Enfin, nous présentons une méthode de compression des coordonnées pour améliorer l'arithmétique sur les coordonnées thêta de grand niveau. Ces applications découlent d'une analyse fine des formules d'addition sur les fonctions thêta. Si les résultats de cette thèse sont valables pour toute variété abélienne, pour les applications nous nous concentrons surtout sur les jacobienne de courbes hyperelliptiques de genre~$2$, qui est le cas le plus significatif cryptographiquement. / The discrete logarithm on elliptic curves give the standard protocols in public key cryptography: asymmetric encryption, signatures, ero-knowledge authentification. To extends the discrete logarithm to hyperelliptic curves of higher genus we need efficient methods to generate secure curves. The aim of this thesis is to give new algorithms to compute with abelian varieties. For this we use the theory of algebraic theta functions in the framework of Mumford. In particular, we give a full generalization of Vélu's formulas for the computation of isogenies on abelian varieties. We also give a new algorithm for the computation of pairings using theta coordinates. Finally we present a point compression method to manipulate These applications follow from the analysis of Riemann relations on theta functions for the addition law. If the results of this thesis are valid for any abelian variety, for the applications a special emphasis is given to Jacobians of hyperelliptic genus~$2$ curves, since they are the most significantly relevant case in cryptography.