Hopf-Galois extensions were introduced by Chase and Sweedler [CS69] in 1969, motivated by the problem of formulating an analogue of Galois theory for inseparable extensions. Their approach shed a new light on separable extensions. Later in 1987, the concept of Hopf-Galois theory was further developed by Greither and Pareigis [GP87]. So, as a problem in the theory of groups, they explained the problem of finding all Hopf-Galois structures on a finite separable extension of fields. After that, many results on Hopf-Galois structures were obtained by N. Byott, T. Crespo, S. Carnahan, L. Childs, and T. Kohl. In this thesis, we consider Hopf-Galois structures on Galois extensions of squarefree degree n. We first determine the number of isomorphism classes of groups G of order n whose centre and commutator subgroup have given orders, and we describe Aut(G) for each such G. By investigating regular cyclic subgroups in Hol(G), we enumerate the Hopf-Galois structures of type G on a cyclic extension of fields L/K of degree n. We then determine the total number of Hopf-Galois structures on L/K. Finally, we examine Hopf-Galois structures on a Galois extension L/K with arbitrary Galois group Gamma of order n, and give a formula for the number of Hopf-Galois structures on L/K of a given type G.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 48-49. / by David Jay Anick. / Ph.D.

Orientador: Roldão da Rocha Junior / Dissertação (mestrado) - Universidade Federal do ABC. Programa de Pós-Graduação em Matemática, 2013

Orientador: Alysson Fábio Ferrari / Dissertação (mestrado) - Universidade Federal do ABC. Programa de Pós-Graduação em Física, 2013

The work of this thesis focuses primarily on non-commutative algebras and actions of Hopf algebras. Specifically, we study the possible H-module algebra structures which can be imposed on path algebras of quivers, for a variety of Hopf algebras, H, and then given a possible action, classify the invariant ring. A Hopf algebra is a bialgebra (H, μ, η, ∆, ε) together with an antipode S : H → Hop which is compatible with the counit, ε, of H. A quiver is a directed graph, and the path algebra kQ of a quiver Q is a vector space where all the paths of the quiver form a basis, and multiplication is given by concatenation of paths whenever possible, and zero otherwise. In their paper, [9], Kinser and Walton classify Hopf actions of a specific family of Hopf algebras called a Taft algebras, T(n), on path algebras of loopless, finite, Schurian quivers. In this thesis, we extend their result to path algebras of any finite quiver and classify the invariant subring, kQT(n), in the case where the group like element g ∈ T(n) acts transitively on Q0. In the future, we hope that the ideas presented in this work extend to a classification of quantum groups, such as uq(sl2), acting on path algebras of finite quivers.

We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We nd concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. This answers an open question in [4] on the existence of a cofree coring, and constructs the cofree (co)module coalgebra on a B-(co)module, for a bialgebra B.

Notre premier objectif dans ce travail est de donner une démonstration du changement<br />de la stabilité de la branche supercritique de solutions périodiques bifurquées<br />dans le cadre des équations diérentielles à retard, en se basant sur les deux étapes<br />suivantes:<br />(i) Réduction de l'équation à un système en dimension deux par la formule de variation<br />de la constante et le théorème de la variété centre.<br />(ii) Estimation de la distance entre la solution de l'équation initiale et la solution pé-<br />riodique bifurquée.<br />Nous obtenons ainsi un domaine de stabilité de la branche supercritique.<br />Le second objectif est d'étudier une équation différentielle à un seul retard issue<br />d'un modèle en dynamique de population cellulaire sanguine (Haematopoiese).<br />Ce modèle, initialement introduit par Mackey (1978) présente une position d'équilibre<br />triviale qui est instable et une famille de positions d'équilibre non triviales dont la<br />stabilité dépend du retard.<br />Nous montrons l'existence d'une valeur critique ¿0 du retard \tau autour de laquelle nous<br />obtenons un changement de stabilité de cette famille de positions d'équilibre en fonction<br />du retard.<br />Nous avons ainsi introduit un modèle approché en fonction de cette valeur critique du<br />retard qui coincide avec celui de Mackey pour la valeur du retard \tau = \tau_{0}. Le modèle<br />approché possède un point d'équilibre trivial et un non trivial ne dépendant pas du<br />retard.<br />Par une étude du modèle approché analogue à celle du modèle de Mackey, nous obtenons<br />en particulier l'existence d'une branche de solutions périodiques bifurquées à<br />partir du point d'équilibre non trivial. Enn nous donnons un algorithme explicite de<br />calcul des éléments de la bifurcation.

No description available.

Lienard Equations serve as the elegant models for oscillating circuits. Motivated by this fact, this thesis addresses the stability property of a class of delayed Lienard equations. It shows the existence of the Hopf bifurcation around the steady state. It has both practical and theoretical importance in determining the criticality of the Hopf bifurcation. For such purpose, center manifold analysis on the bifurcation line is required. This thesis uses operator differential equation formulation to reduce the infinite dimensional delayed Lienard equation onto a two-dimensional manifold on the critical bifurcation line. Based on the reduced two-dimensional system, the so called Poincare-Lyapunov constant is analytically determined, which determines the criticality of the Hopf bifurcation. Numerics based on a Matlab bifurcation toolbox (DDE-Biftool) and Matlab solver (DDE-23) are given to compare with the theoretical calculation. Two examples are given to illustrate the method.

Cheyne-Stokes respiration is a distinct breathing pattern consisting of periods of hyperpnea followed by apneas, with unknown etiology. One in two patients with congestive heart failure suffer from this condition. Researchers hypothesize that key factors in CSR are the fluid shift from the standing to supine position and the differences between genders. A mathematical model of the cardio-respiratory system was constructed using parameter values from real data. Hopf bifurcation analysis was used to determine regions of stable versus oscillatory breathing patterns. In the model, Cheyne-Stokes respiration is more likely to occur while in the supine position and males are more likely to develop Cheyne-Stokes than females. These findings, which are in agreement with clinical experience, suggest that both gender and fluid shift contribute to the pathogenesis of Cheyne-Stokes respiration, and that physical quantities such as blood volumes and neural feedback may be sufficient to explain the observations of CSR. / Department of Mathematics and Statistics