Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Esta dissertaÃÃo està dividida em dois capÃtulos. O primeiro capÃtulo apresenta os AnÃis de Grupos, os Grupos de Frobenius e suas respectivas propriedades. No inÃcio do segundo capÃtulo sÃo apresentadas as Conjecturas de Zassenhaus. A versÃo mais fraca dessas conjecturas à demonstrada para Grupos de Amitsur. No final do segundo capÃtulo, a validade dessa mesma versÃo à provada para Grupos de Frobenius.Tais Grupos de Frobenius sÃo aqueles cujo complemento verifica-se a validade dessa conjectura. Na parte final sÃo apresentados os subgrupos de Hall e o Teorema de Schur-Zassenhaus. / This dissertation is divided into two chapters. The first chapter introduces the Group Rings, the Frobenius Groups and their properties. In the beginning of the second chapter are presented Conjectures of Zassenhaus . The weaker version of these conjectures is demonstrated for Amitsur Groups. At the end of the second chapter, the validity of that version is proven to Frobenius Groups. Such Frobenius Groups are those whose complement, checks the validity of this conjecture. In the final part we present the Hall subgroups and Schur-Zassenhaus Theorem.

University, Habil-Schr., 2003--Stuttgart.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-03-15Bitstream added on 2014-06-13T19:47:19Z : No. of bitstreams: 1 castro_fr_me_sjrp.pdf: 783980 bytes, checksum: fd80e9aa8c69641da08ee43dfa94509d (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo principal deste trabalho é estudar a Teoria de Cohomologia de Grupos visando apresentar de forma detalhada algumas aplicações dessa teoria na Topologia e na Algebra, mais especificamente na Teoria de Grupos, com destaque para o Teorema de Schur-Zassenhaus e o Teorema de Classificação de p-grupos que possuem um subgrupo ciclico de índice p (p primo). / The aim of this work is to study the Cohomology Theory of Groups in order to present in detailed form some applications of this theory in Topology and in Algebra, more specifically, in the Theory of Groups, with prominence for the Schur-Zassenhaus Theorem and the Theorem of Classification of p-groups which contain a cyclic subgroup of index p, where p is a prime.

Tableau d’honneur de la Faculté des études supérieures et postdoctorales, 2012-2013. / Cette thèse a pour but d'élaborer et d'étudier des modèles mathématiques reproduisant le comportement de systèmes composés de plusieurs éléments dont les interactions forment un réseau complexe. Le corps du document est découpé en trois parties ; un chapitre introductif et une conclusion récapitulative complétent la thèse. La partie I s'intéresse à une dynamique spécifique (propagation de type susceptibleinfectieux- retiré, SIR) sur une classe de réseaux également spécifique (modèle de configuration). Ce problème a entre autres déjà été étudié comme un processus de branchement dans la limite où la taille du système est infinie, fournissant une solution probabiliste pour l'état final de ce processus stochastique. La principale contribution originale de la partie I consiste à modifier ce modèle afin d'introduire des éffets dûs à la taille finie du système et de permettre l'étude de son évolution temporelle (temps discret) tout en préservant la nature probabiliste de la solution. La partie II, contenant les principales contributions originales de cette thèse, s'intéresse aux processus stochastiques sur réseaux complexes en général. L'état du système (incluant la structure d'interaction) est partiellement représenté à l'aide de motifs, et l'évolution temporelle (temps continu) est étudiée à l'aide d'un processus de Markov. Malgré que l'état ne soit que partiellement représenté, des résultats satisfaisants sont souvent possibles. Dans le cas particulier du problème étudié en partie I, les résultats sont exacts. L'approche se révèle très générale, et de simples méthodes d'approximation permettent d'obtenir une solution pour des cas d'une complexité appréciable. La partie III cherche une solution analytique exacte sous forme fermée au modèle développé en partie II pour le problème initialement étudié en partie I. Le système est réexprimé en terme d'opérateurs et différentes relations sont utilisées afinn de tenter de le résoudre. Malgré l'échec de cette entreprise, certains résultats méritent mention, notamment une généralisation de la relation de Sack, un cas particulier de la relation de Zassenhaus. / The goal of this thesis is to develop and study mathematical models reproducing the behaviour of systems composed of numerous elements whose interactions make a complex network structure. The body of the document is divided in three parts; an introductory chapter and a recapitulative conclusion complete the thesis. Part I pertains to a specific dynamics (susceptible-infectious-removed propagation, SIR) on a class of networks that is also specific (configuration model). This problem has already been studied, among other ways, as a branching process in the infinite system size limit, providing a probabilistic solution for the final state of this stochastic process. The principal original contribution of part I consists of modifying this model in order to introduce finite-size effects and to allow the study of its (discrete) time evolution while preserving the probabilistic nature of the solution. Part II, containing the principal contributions of this thesis, is interested in the general problem of stochastic processes on complex networks. The state of the system (including the interaction structure) is partially represented through motifs, then the (continuous) time evolution is studied with a Markov process. Although the state is only partially represented, satisfactory results are often possible. In the particular case of the problem studied in part I, the results are exact. The approach turns out to be very general, and simple approximation methods allow one to obtain a solution for cases of considerable complexity. Part III searches for a closed form exact analytical solution to the the model developed in part II for the problem initially studied in part I. The system is re-expressed in terms of operators and different relations are used in an attempt to solve it. Despite the failure of this enterprise, some results deserve mention, notably a generalization of Sack's relationship, a special case of the Zassenhaus relationship.

The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for solving these equations commence with spatial discretisation followed by exponentiation of the discretised Hamiltonian via exponential splittings. In this thesis we follow an alternative strategy${-}$we develop a new technique, called the symmetric Zassenhaus splitting procedure, which involves directly splitting the exponential of the undiscretised Hamiltonian. This technique allows us to design methods that are highly efficient in the semiclassical regime. Our analysis takes place in the Lie algebra generated by multiplicative operators and polynomials of the differential operator. This Lie algebra is completely characterised by Jordan polynomials in the differential operator, which constitute naturally symmetrised differential operators. Combined with the $\mathbb{Z}_2$-graded structure of this Lie algebra, the symmetry results in skew-Hermiticity of the exponents for Zassenhaus-style splittings, resulting in unitary evolution and numerical stability. The properties of commutator simplification and height reduction in these Lie algebras result in a highly effective form of $\textit{asymptotic splitting:} $exponential splittings where consecutive terms are scaled by increasing powers of the small semiclassical parameter. This leads to high accuracy methods whose costs grow quadratically with higher orders of accuracy. Time-dependent potentials are tackled by developing commutator-free Magnus expansions in our Lie algebra, which are subsequently split using the Zassenhaus algorithm. We present two approaches for developing arbitrarily high-order Magnus--Zassenhaus schemes${-}$one where the integrals are discretised using Gauss--Legendre quadrature at the outset and another where integrals are preserved throughout. These schemes feature high accuracy, allow large time steps, and the quadratic growth of their costs is found to be superior to traditional approaches such as Magnus--Lanczos methods and Yoshida splittings based on traditional Magnus expansions that feature nested commutators of matrices. An analysis of these operatorial splittings and expansions is carried out by characterising the highly oscillatory behaviour of the solution.

Cette thèse étudie des problèmes liés aux treillis des sous-groupes quantiques et la caractérisationdes actions ergodiques et des états idempotents d’un groupe quantique compact.Elle consiste en 3 parties. La première partie présente des résultats préliminaires sur lesgroupes quantiques localement compacts, les sous-groupes quantiques normaux ainsi queles actions ergodiques et les états idempotents. La seconde partie étudie l’analogue quantiquede la règle de modularité de Dedekind et de l’analogue quantique des théorèmesd’isomorphisme de Noether ainsi que leur conséquences comme le théorème de raffinementde Schreier, et le théorème Jordan-Hölder. Cette partie s’inspire du travail de recherche deShuzhouWang sur l’analogue quantique du troisième théorème d’isomorphisme de Noetherpour les groupes quantiques compacts ainsi que le travail récent de Kasprzak, Khosraviet Soltan sur l’analogue quantique du premier théorème d’isomorphisme de Noether pourles groupes quantiques localement compacts. Dans la troisième partie, nous caractérisonsles états idempotents du groupe quantique compact O−1(2) en s’appuyant sur la caractérisationde ses actions ergodiques plongeables. Cette troisième partie est dans la lignedes travaux fait par Franz, Skalski et Tomatsu pour les groupes quantiques compactsUq(2), SUq(2) et SOq(3). Nous classifions au préalable les actions ergodiques et les actionsergodiques plongeables du groupe quantique compact O−1(2).Les travaux présentés dans cette thèse se basent sur deux articles de l’auteur et al.Le premier s’intitule “Fundamental isomorphism theorems for quantum groups” et a étéaccepté pour publication dans Expositionae Mathematicae et le second est intitulé “Ergodicactions and idempotent states of O−1(2)” et est en cours de finalisation pour être soumis. / This thesis studies problems linked to the lattice of quantum subgroups and characterizationof ergodic actions and idempotent states of a compact quantum group. It consistsof three parts. The first part present some preliminary results about locally compactquantum groups, normal quantum subgroups, ergodic actions and idempotent states. Thesecond part studies the quantum analog of Dedekind’s modularity law, Noether’s isomorphismtheorem and their consequences as the Schreier refinement theorem and theJordan-Hölder theorem. This part completes the work of Shuzhou WANG on the quantumanalog of the third isomorphism theorem for compact quantum group and the recentwork of Kasprzak, Khosravi and Soltan on the quantum analog of the first Noether isomorphismtheorem for locally compact quantum groups. In the third part, we characterizeidempotent states of the compact quantum group O−1(2) relying on the characterizationof embeddable ergodic actions. This third part is in the sequence of the seminal works ofFranz, Skalski and Tomatsu for the compact quantum groups Uq(2), SUq(2) and SOq(3).We classify in advance the ergodic actions and embeddable ergodic actions of the compactquantum group O−1(2).This thesis is based on two papers of the author and al. The first one is entitled“Fundamental isomorphism theorems for quantum groups” which have been accepted forpublication in Expositionae Mathematicae and the second one is entitled “Ergodic actionsand idempotent states of O−1(2)” and is being finalized for submission.

Este trabalho descreve algoritmos algébricos para computação em corpos de Galois GF(q), com q = pn onde pé a característica do corpo, que pode ser arbitrariamente grande. Para fundamentar esse estudo é condensada e apresentada Lo ela. a fena.menta algébrica necessári a. Os corpos ·finitos são caracterizados, é mostrado como construí-los e sua aritmética é analisada. Algoritmos determinísticos e probabilísticos são desenvolvidos para. o cálculo de raízes polinomiais e a. fatoração de polinômios sobre esses corpos. Este trabalho é materializado pela implementação de dois algoritmos, o de Cantor-Zassenhaus e o de Rabin, ambos implementados no Sistema de Computação Algébrica MAPLE V Release 3. / This work elescribes algebraic algorithms for computing in Galois Fielels GF(q), with q = pn, where p is the characteristic of the fielel anel may be arbitrar.ialy large. By justifying this work we give a colection of results about topics of Algebra. Dctcnninistics anel probabilistics a.lgorithms are clevelopeel to compute polynomials roots anel for polynornia.l factorization in OF(q).This work is materializccl by the implementation oi' t.wo algorithms, Cantor-Zasscnhaus's algorithm anel Rabin's algoril. hm, both implemented in MAPLE V Rclease 3 Computer Algebra System.

Este trabalho descreve algoritmos algébricos para computação em corpos de Galois GF(q), com q = pn onde pé a característica do corpo, que pode ser arbitrariamente grande. Para fundamentar esse estudo é condensada e apresentada Lo ela. a fena.menta algébrica necessári a. Os corpos ·finitos são caracterizados, é mostrado como construí-los e sua aritmética é analisada. Algoritmos determinísticos e probabilísticos são desenvolvidos para. o cálculo de raízes polinomiais e a. fatoração de polinômios sobre esses corpos. Este trabalho é materializado pela implementação de dois algoritmos, o de Cantor-Zassenhaus e o de Rabin, ambos implementados no Sistema de Computação Algébrica MAPLE V Release 3. / This work elescribes algebraic algorithms for computing in Galois Fielels GF(q), with q = pn, where p is the characteristic of the fielel anel may be arbitrar.ialy large. By justifying this work we give a colection of results about topics of Algebra. Dctcnninistics anel probabilistics a.lgorithms are clevelopeel to compute polynomials roots anel for polynornia.l factorization in OF(q).This work is materializccl by the implementation oi' t.wo algorithms, Cantor-Zasscnhaus's algorithm anel Rabin's algoril. hm, both implemented in MAPLE V Rclease 3 Computer Algebra System.

Este trabalho descreve algoritmos algébricos para computação em corpos de Galois GF(q), com q = pn onde pé a característica do corpo, que pode ser arbitrariamente grande. Para fundamentar esse estudo é condensada e apresentada Lo ela. a fena.menta algébrica necessári a. Os corpos ·finitos são caracterizados, é mostrado como construí-los e sua aritmética é analisada. Algoritmos determinísticos e probabilísticos são desenvolvidos para. o cálculo de raízes polinomiais e a. fatoração de polinômios sobre esses corpos. Este trabalho é materializado pela implementação de dois algoritmos, o de Cantor-Zassenhaus e o de Rabin, ambos implementados no Sistema de Computação Algébrica MAPLE V Release 3. / This work elescribes algebraic algorithms for computing in Galois Fielels GF(q), with q = pn, where p is the characteristic of the fielel anel may be arbitrar.ialy large. By justifying this work we give a colection of results about topics of Algebra. Dctcnninistics anel probabilistics a.lgorithms are clevelopeel to compute polynomials roots anel for polynornia.l factorization in OF(q).This work is materializccl by the implementation oi' t.wo algorithms, Cantor-Zasscnhaus's algorithm anel Rabin's algoril. hm, both implemented in MAPLE V Rclease 3 Computer Algebra System.