Submitted by Lúcia Brandão (lucia.elaine@live.com) on 2015-12-14T18:54:48Z No. of bitstreams: 1 Dissertação - Lauriano de Souza e Souza.pdf: 1554279 bytes, checksum: 3d210e441f37fec27501116cf45440b3 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2016-01-20T18:29:57Z (GMT) No. of bitstreams: 1 Dissertação - Lauriano de Souza e Souza.pdf: 1554279 bytes, checksum: 3d210e441f37fec27501116cf45440b3 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2016-01-20T18:30:51Z (GMT) No. of bitstreams: 1 Dissertação - Lauriano de Souza e Souza.pdf: 1554279 bytes, checksum: 3d210e441f37fec27501116cf45440b3 (MD5) / Made available in DSpace on 2016-01-20T18:30:51Z (GMT). No. of bitstreams: 1 Dissertação - Lauriano de Souza e Souza.pdf: 1554279 bytes, checksum: 3d210e441f37fec27501116cf45440b3 (MD5) Previous issue date: 2013-06-10 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work , we talk about an abstract theory for the equation of two-dimensional varieties ( or just surfaces) , namely the theory of pairs of Codazzi , spatial forms R³ , S³ and H³ . Precisely, we generalize some classical theorems of the theory of surfaces and unified proof of other results, seemingly unrelated . In addition , we study the existence of holomorphic quadratic differential , high estimates and apply the abstract theory that the classification of surfaces of special Weingarten elliptical , complete and dipped in R³ , whose Gaussian curvature does not change sign. / Neste trabalho, dissertamos sobre uma teoria abstrata para a equação de variedades bidimensionais (ou simplesmente superfícies), a saber, a teoria dos pares de Codazzi, formas espaciais R³, S³ e H³. Precisamente, generalizamos alguns teoremas clássicos da teoria de superfícies e unificamos a prova de outros resultados, aparentemente não relacionados. Além disso, estudamos a existência de diferenciais quadráticas holomorfas, estimativas de altura e aplicamos a referida teoria abstrata na classificação das superfícies de Weingarten especiais elípticas, completas e mergulhadas em R³, cuja curvatura Gaussiana não muda de sinal.

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Cette thèse porte sur la construction d'une famille de groupoïdes quantiques de transformations qui dans le cadre algébrique sont des algébroïdes de Hopf de multiplicateurs mesurés au sens de Timmermann et Van Daele et qui dans le cadre des algèbres d'opérateurs sont des C*-bimodules de Hopf sur une C*-base au sens de Timmermann.Dans le contexte purement algébrique, nous définissons d'abord une algèbre involutive de Yetter-Drinfeld tressée commutative sur un groupe quantique algébrique au sens de Van Daele et une intégrale de Yetter-Drinfeld sur elle. En utilisant ces objets nous construisons après un algébroide de Hopf de multiplicateurs involutif mesuré, ce nouvel objet nous l'appellons groupoïde quantique algébrique de transformations.Pour être capables de passer au cadre des algèbres d'opérateurs, nous donnons des conditions sur l'intégral de Yetter-Drinfeld qui vont nous permettre d'utiliser la construction Gelfand–Naimark–Segal pour étendre tous nos objets purement algébriques en des objets C*-algébriques. Dans ce contexte, notre construction se fait d'une manière similaire à celle présentée dans le travail de Enock et Timmermann, nous obtenons un nouvel objet mathématique que nous appellons un groupoïde quantique C*-algébrique de transformations, qui est définit en utilisant le langage des C*-bimodules de Hopf sur une C*-base. / This thesis is concerned with the construction of a family of quantum transformation groupoids in the algebraic framework in the form of the measured multiplier Hopf *-algebroids in the sense of Timmermann and Van Daele and also in the context of operator algebras in the form of Hopf C*-bimodules on a C*-base in the sense of Timmermann.In the purely algebraic context, we first give a definition of a braided commutative Yetter-Drinfeld *-algebra over an algebraic quantum group in the sense of Van Daele and a Yetter-Drinfeld integral on it. Then, using these objects we construct a measured multiplier Hopf *-algebroid, we call to this new object an algebraic quantum transformation groupoid.In order to pass to the operator algebra framework, we give some conditions on the Yetter-Drinfeld integral inspired by the properties of KMS-weights on C*-algebras which will allow us to use the Gelfand–Naimark–Segal construction to extend all the purely algebraic objects to the C*-algebraic level. At this level, we construct in a similar way to that used in the work of Enock and Timmermann, a new mathematical object that we call a C*-algebraic quantum transformation groupoid, which is defined using the language of Hopf C*-bimodules on C*-bases.

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Representation theory is a field of study within abstract algebra that originated around the turn of the 19th century in the work of Frobenius on representations of finite groups. More recently, Hopf algebras -- a class of algebras that includes group algebras, enveloping algebras of Lie algebras, and many other interesting algebras that are often referred to under the collective name of ``quantum groups'' -- have come to the fore. This dissertation will discuss generalizations of certain results from group representation theory to the setting of Hopf algebras. Specifically, our focus is on the following two areas: Frobenius divisibility and Kaplansky's sixth conjecture, and the adjoint representation and the Chevalley property. / Mathematics

Mathematicians have long thought of symmetry in terms of actions of groups, but group actions have proven too restrictive in some cases to give an interesting picture of the symmetry of some mathematical objects, e.g. some noncommutative algebras. It is generally agreed that the right generalizations of group actions to solve this problem are actions of Hopf algebras, the study of which has exploded in the years since the publication of Sweedler's Hopf algebras in 1969. Different varieties of Hopf algebras have been useful in many fields of mathematics. For instance, in his "Quantum Groups" paper, Vladimir Drinfel'd introduced quasitriangular Hopf algebras, a class of Hopf algebras whose modules each provide a solution to the quantum Yang-Baxter equation. Solutions of this equation are a source of knot and link invariants and in physics, determine if a one-dimensional quantum system is integrable. Drinfel'd also introduced the Drinfel'd double construction, which produces for each finite-dimensional Hopf algebra a quasitriangular one in which the original embeds. This thesis is motivated by work of Susan Montgomery and Hans-Jürgen Schneider on actions of the Taft (Hopf) algebras T_n(q) and extending such actions to the Drinfel'd double D(T_n(q)). In 2001, Montgomery and Schneider classified all non-trivial actions of T_n(q) on an n-dimensional associative algebra A. It turns out that A must be isomorphic to the group algebra of grouplike elements kG(T_n(q)). They further determined that each such action extends uniquely to an action of the Drinfel'd double D(T_n(q)) on A, effectively showing that each action has a unique compatible coaction. We generalize Montgomery and Schneider's results to Hopf algebras related to the Taft algebras: the Sweedler (Hopf) algebra, bosonizations of 1-dimensional quantum linear spaces, generalized Taft algebras, and the Frobenius-Lusztig kernel u_q(sl_2). For each Hopf algebra H, we determine 1. whether there are non-trivial actions of H on A, 2. the possible H-actions on A, and 3. the possible D(H)-actions on A extending an H-action and how many there are. / Mathematics

A software package has been written to compute the Hopf bifurcation structure in functional differential equations. The package is modular, and consists of several routines which perform one or more tasks. In conjunction with the routines available in this package, the user is required to provide a few routines which describe the specific system under analysis. Three example systems (from epidemiology, biochemistry and aerospace engineering) have been analyzed to illustrate the use of this package. / M.S.

Circadian rhythms are periodic physiological cycles that recur about every 24 hours, by means of which organisms integrate their physiology and behavior to the daily cycle of light and temperature imposed by the rotation of the earth. Circadian derives from the Latin word circa "about" and dies "day". Circadian rhythms have three noteworthy properties. They are endogenous, that is, they persist in the absence of external cues (in an environment of constant light intensity, temperature, etc.). Secondly, they are temperature compensated, that is, the nearly 24 hour period of the endogenous oscillator is remarkably independent of ambient temperature. Finally, they are phase shifted by light. The circadian rhythm can be either advanced or delayed by applying a pulse of light in constant darkness. Consequently, the circadian rhythm will synchronize to a periodic light-dark cycle, provided the period of the driving stimulus is not too far from the period of the endogenous rhythm. A window on the molecular mechanism of 24-hour rhythms was opened by the identification of circadian rhythm mutants and their cognate genes in Drosophila, Neurospora, and now in other organisms. Since Konopka and Benzer first discovered the period mutant in Drosophila in 1971 (Konopka and Benzer, 1971), there have been remarkable developments. Currently, the consensus opinion of molecular geneticists is that the 24-hour period arises from a negative feedback loop controlling the transcription of clock genes. However, a better understanding of this mechanism requires an approach that integrates both mathematical and molecular biology. From the recent discoveries in molecular biology and through a mathematical approach, we propose that the mechanism of circadian rhythm is based upon the combination of both negative and positive feedback. / Master of Science

We study the linear abstract kinetic equation T𝜑(x)′=-A𝜑(x) in the half space {x≥0} with partial range boundary conditions. The function <i>ψ</i> takes values in a Hilbert space H, T is a self adjoint injective operator on H and A is an accretive operator. The first step in the analysis of this boundary value problem is to show that T⁻¹A generates a holomorphic bisemigroup. We prove two theorems about perturbation of bisemigroups that are interesting in their own right. The second step is to obtain a special decomposition of H which is equivalent to a Wiener-Hopf factorization. The accretivity of A is crucial in this step. When A is of the form "identity plus a compact operator", we work in the original Hilbert space. For unbounded A’s we consider weak solutions in a larger space H_T, which has a natural Krein space structure. Using the Krein space geometry considerably simplifies the analysis of the question of unique solvability. / Ph. D. / incomplete_metadata

Cette thèse se situe dans le domaine de la combinatoire algébrique. Autrement dit, l'idée est d'utiliser des structures algébriques, en l'occurence des algèbres de Hopf combinatoires, pour mieux étudier et comprendre les objets combinatoires ainsi que des algorithmes de composition et de décomposition agissant sur ces objets. Ce travail de recherche repose sur la construction et l'étude de structure algébrique sur des objets combinatoires généralisant les permutations. Après avoir rappelé le contexte et les notations des différents objets intervenant dans cette recherche, nous proposons dans la seconde partie l'étude de l'algèbre de Hopf introduite par Aguiar et Orellana indexée par les permutations de blocs uniformes. En se focalisant sur une description de ces objets via d'autres bien connus, les permutations et les partitions d'ensembles, nous proposons une réalisation polynomiale et une étude plus simple de cette algèbre. La troisième partie étudie une deuxième généralisation en interprétant les permutations comme des matrices. Nous définissons et étudions alors des familles de matrices carrées sur lesquelles nous définissons des algorithmes de composition et de décomposition. La quatrième partie traite des matrices à signes alternants. Après avoir définie l'algèbre de Hopf sur ces matrices, nous étudions des statistiques et le comportement de la structure algébrique vis-à-vis de ces statistiques. Tous ces chapitres s'appuient fortement sur l'exploration informatique, et fait l'objet d'une implémentation utilisant le logiciel Sage. Ce dernier chapitre est consacré à la découverte et la manipulation de structures algébriques sur Sage. Nous terminons en expliquant les améliorations apportées pour l'étude de structure algébrique au travers du logiciel Sage / This thesis is in the field of algebraic combinatorics. In other words, the idea is to use algebraic structures, in this case of combinatorial Hopf algebras, to better study and understand the combinatorial objects and algorithms for composition and decomposition about these objects. This research is based on the construction and study of algebraic structure of combinatorial objects generalizing permutations. After recalling the background and notations of various objects involved in this research, we propose, in the second part, the study of the Hopf algebra introduced by Aguiar and Orellana based on uniform block permutations. By focusing on a description of these objects via well-known objects, permutations and set partitions, we propose a polynomial realization and an easier study of this algebra. The third section considers a second generalization interpreting permutations as matrices. We define and then study the families of square matrices on which we define algorithms for composition and decomposition. The fourth part deals with alternating sign matrices. Having defined the Hopf algebra of these matrices, we study the statistics and the behavior of the algebraic structure with these statistics. All these chapters rely heavily on computer exploration, and is the subject of an implementation using Sage software. This last chapter is dedicated to the discovery and manipulation of algebraic structures on Sage. We conclude by explaining the improvements to the study of algebraic structure through the Sage software