Demonstramos um teorema da correspondência do tipo Galois para ações de álgebras de Hopf pontuais de dimensão finita em álgebras primas. A correspondência acontece entre subálgebras racionalmente completas e comódulo subálgebras. As subálgebras racionalmente completas são subálgebras da álgebra prima, enquanto os comódulo subálgebras são comódulo subálgebras do produto smash entre o centralizador da álgebra prima em sua álgebra de quocientes de Martindale simétrica e a álgebra de Hopf. / A Galois-type correspondence theorem for prime algebras acted upon by a finite dimensional pointed Hopf algebra is proved. The correspondence involves rationally complete subalgebras and comodule subalgebras. The rationally complete subalgebras are subalgebras of the prime algebra, while the comodule subalgebras are comodule subalgebras of the smash product between the centralizer of the prime algebra in its symmetric Martindale quotient algebra and the Hopf algebra.

In this dissertation we first introduce an extension of the notion of parking functions to cars of different sizes. We prove a product formula for the number of such sequences and provide a refinement using a multi-parameter extension of the Abel--Rothe polynomial. Next, we study the incidence Hopf algebra on the noncrossing partition lattice. We demonstrate a bijection between the terms in the canceled chain decomposition of its antipode and noncrossing hypertrees. Thirdly, we analyze the sum of the 𝑟th powers of the descent set statistic on permutations and how many small prime factors occur in these numbers. These results depend upon the base 𝑝 expansion of both the dimension and the power of these statistics. Finally, we inspect the ƒ-vector of the descent polytope DPv, proving a maximization result using an analogue of the boustrophedon transform.

Freiburg i. Br., Univ., Diss., 2006.

University, Diss., 2004--München.

Universiẗat, Diss., 2004--München.

Made available in DSpace on 2014-06-11T19:27:09Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-06-05Bitstream added on 2014-06-13T19:06:50Z : No. of bitstreams: 1 verri_ja_me_prud.pdf: 7130871 bytes, checksum: 40212cfd999c344f6165b927a8d582c2 (MD5) / Nesta dissertação fazemos um estudo de modelos biológicos do tipo Lotka-Volterra, utilizando como ferramenta principal a teoria qualitativa das equações diferenciais ordinárias. Abordamos, no plano e no espaço, alguns modelos do tipo predador-presa. Analisamos os comportamentos das soluções sob a variação dos parâmetros e tratamos com detalhes a bifurcação de Hopf, que dá origem a uma órbita periódica isolada (ciclo limite). Estudamos também um teorema devido a Li e Muldowney [16] sobre a estabilidade global de um ponto de equilíbrio para um sistema x˙ = f(x), x ∈ Rn. Aplicamos este resultado no estudo de um modelo de HIV tridimensional, provando a estabilidade global de um ponto de equilíbrio, para certos valores dos parâmetros. Para o mesmo modelo, verificamos a ocorrência de uma dupla bifurcação de Hopf, que leva ao surgimento e posterior desaparecimento de um ciclo limite, ao variarmos um dos parâmetros envolvidos no sistema. As bifurcações de Hopf ocorrem simultaneamente à perda de estabilidade global do ponto de equilíbrio / In this work we present a study of biological models of Lotka-Volterra type, using as main tool the qualitative theory of ordinary differential equations. We analyze some two and three dimensional predator-prey models. The behavior of the solutions are studied under the variation of parameters and it is shown that a Hopf bifurcation occurs, leading to the creation of an isolated periodic orbit (limit cycle). We also study a theorem due to Li and Muldowney [16] about the global stability of an equilibrium point of a system x˙ = f(x), x ∈ Rn. We apply this result in the analysis of a three dimensional model of HIV with treatment, showing the global stability of an equilibrium point, for certain parameter values. For the same model, we prove the occurrence of two Hopf bifurcations, leading to the birth and subsequent death of a limit cycle, when we vary one of the parameters of the model. The Hopf bifurcations occurs simultaneously to the lack of global stability of the equilibrium point

Extensões de Ore são anéis de polinômios, denotados por R[x o &], nos quais a variável x e os elementos de R não comutam necessariamente. Algebras de Hopf fracas são algebras que tamb em são coálgebras e satisfazem um conjunto de axiomas de compatibilidade entre essas estruturas. Neste trabalho investigamos extensões de Ore cujo anel base e uma algebra de Hopf fraca. Mais especi camente, dada uma algebra de Hopf fraca R, estudamos sob quais condições R[x o &] e uma algebra de Hopf fraca com uma estrutura que estende a estrutura de R. Sob certas hipóteses, obtemos condições necessárias e su cientes para que a extensão de Ore seja uma álgebra de Hopf fraca, obtendo assim um resultado que generaliza um teorema de Panov para o contexto de algebras de Hopf fracas. / Ore extensions are polynomial rings, denoted by R[x o &], in which the variable x and the elements of R do not commute necessarily. Weak Hopf algebras are algebras which are also coalgebras and satisfy a set of axioms of compatibility betweem these structures. In this work, we investigate Ore extensions whose base ring is a weak Hopf algebra. More speci cally, if R is a weak Hopf algebra then we study under what conditions R[xo &] is a weak Hopf algebra extending the structure of R. Under certain hypotheses, we obtain necessary and su cient conditions for an Ore extension to be a weak Hopf algebra, obtaining a result that generalizes a Panov's theorem to the setting of weak Hopf algebras.

Extensões de Ore são anéis de polinômios, denotados por R[x o &], nos quais a variável x e os elementos de R não comutam necessariamente. Algebras de Hopf fracas são algebras que tamb em são coálgebras e satisfazem um conjunto de axiomas de compatibilidade entre essas estruturas. Neste trabalho investigamos extensões de Ore cujo anel base e uma algebra de Hopf fraca. Mais especi camente, dada uma algebra de Hopf fraca R, estudamos sob quais condições R[x o &] e uma algebra de Hopf fraca com uma estrutura que estende a estrutura de R. Sob certas hipóteses, obtemos condições necessárias e su cientes para que a extensão de Ore seja uma álgebra de Hopf fraca, obtendo assim um resultado que generaliza um teorema de Panov para o contexto de algebras de Hopf fracas. / Ore extensions are polynomial rings, denoted by R[x o &], in which the variable x and the elements of R do not commute necessarily. Weak Hopf algebras are algebras which are also coalgebras and satisfy a set of axioms of compatibility betweem these structures. In this work, we investigate Ore extensions whose base ring is a weak Hopf algebra. More speci cally, if R is a weak Hopf algebra then we study under what conditions R[xo &] is a weak Hopf algebra extending the structure of R. Under certain hypotheses, we obtain necessary and su cient conditions for an Ore extension to be a weak Hopf algebra, obtaining a result that generalizes a Panov's theorem to the setting of weak Hopf algebras.

Cette thèse se situe dans les domaines de la combinatoire algébrique, bijective et énumérative.Elle s'intéresse à l'étude des fonctions de parking généralisées suivant ces trois axes.medskip. Dans une première partie, on s'intéresse aux fonctions de parking généralisées en tant qu'espèce de structures combinatoires (théorie introduite par A.nom{Joyal} et développée F. nom{Bergeron}, G. nom{Labelle} et P.nom{Leroux}). On définit cette espèce à partir d'une équation fonctionnelle faisant intervenir l'espèce des séquences d'ensembles.On obtient un relèvement non-commutatif de la série indicatrice de cycles dans les fonctions symétriques non-commutatives, exprimé dans différentes bases.Par spécialisation, on obtient de nouvelles formules d'énumérations des fonctions de parking généralisées et de leurs types d'isomorphismes.En remplaçant l'espèce des ensembles par d'autres espèces dans l'équation fonctionnelle, on définit de nouvelles structures: les $seqPF$-tables de parking. Dans les cas particuliers où $seqPF : m mapsto a + b(m-1)$, on établit une bijection entre les $seqPF$-tables de parking et de nouvelles structures arborescentes, généralisant la bijection de C. H.nom{Yan} entre les $seqPF$-fonctions de parking et les séquences de $a$forêts de $b$-arbres.medskip. Dans une seconde partie, on s'intéresse à l'énumération d'automates. On commence par construire une bijection simple entre les automates(non-initiaux) et les séquences d'ensembles. À partir de cette bijection, on extrait la sous-famille des automates quasi-distingués (c'est-à-dire les automates pour lesquels les couples status de terminaison et fonction de transition des états sont distincts). L'énumération de ces automates quasi-distingués fournit une meilleure borne supérieure pour le nombre d'automates minimaux que celle obtenue par M.nom{Domaratzki} & textit{al}. Ensuite, on construit une nouvelle bijection entre les $2m^k$-fonctions de parking et les automates acycliques (non-initiaux) sur un alphabet à $k$ symboles. De cette dernière, on extrait, directement sur les fonctions parking, denombreuses informations de structure sur les automates, en particulier des informations liées à la minimalité.À partir de ces informations, on déduit une formule d'énumération des automates acycliques minimaux.medskipDans une troisième partie, on formalise la technique commune de réalisation polynomiale des algèbres de Hopf: fqsym, wqsym, pqsym, etc. Pour ceci, ondéfinit la notion de type d'alphabet et d'application partitionnante. La notion d'application partitionnante formalise les bonnes propriétés de la standardisation, le tassement, la parkisation, etc associées à ces précédentes algèbres de Hopf. On montre que certaines opérations, produit cartésien, coloration, union ouencore intersection, stabilisent ces notions.À partir de celles-ci, on définit deux constructions d'algèbres de Hopf combinatoire en dualité; et l'on montre qu'elles sont automatiquement munies de structures d'algèbres dendriformes et du produit $#$. En guise d'applications, on définit, pour toute famille de $seqPF$-fonctions deparking, une application généralisant la parkisation. On montre que cette dernière est une application partitionnante si et seulement si $seqPF : nmapsto 1 + m(n-1)$. Ceci permet de retrouver les algèbres de Hopf sur les$m$-fonctions de parking généralisées de J.-C. nom{Novelli} et J-.Y.nom{Thibon}. / This thesis comes within the scope of algebraic, bijective and enumerative combinatorics. It deals with the study of generalized parking functions following those axes.In the first part, we are interested in generalized parking as a species of combinatorial structures. We define this species from a functional equation involving the species of set sequences. We lift the cycle index serie to the non-commutative symmetric functions, express in several bases. By specialization, we obtain new enumeration formula of generalized parking and their isomorphism types.In the functional equation, the species of sets can be replaced by some other species. This defines new structures: the $chi$-parking tables. In particular cases with $chi : m mapsto a + b(m-1)$, we define a bijection between the $chi$-parking tables and new tree structures. This defines a generalization of the C. H. Yan bijection.In the second part, we are interested in the enumeration of automata. Firstly, we construct a simple bijection between (non-initial) automata and sequences of sets. From this bijection we extract a subfamily of quasi-distinguished automata. We obtain a better upper bound of the number of minimal automata than the one of M. Domaratzki.Then we construct a new bijection between $2m^k$-parking functions and (non-initial) acyclic automata over an alphabet of $k$ symbols. From this bijection we extract, from parking function, informations about automata structures. We deduce an enumeration formula of the minimal acyclic automata.In a third part, we formalize the common technique of polynomial realization of Hopf algebras: FQSym, WQSym, PQSym, etc.. We define a notion of type of alphabet and partitioning map. We highlight some operation which stabilizes these notions. Based on this, we define two constructions of dual combinatorial Hopf algebra; and we show that they are automatically endowed of dendriform coalgebra, and $#$-product.As an application, we define, for every family of $chi$-parking functions, a generalization of the parkization. We show that this is a partitionning map if and only if $chi : m mapsto 1 + b(m-1)$.

The N= 4 Super Yang-Mills theory in four dimensions admits deformations and the exactly marginal deformations of its SU(3) R-symmetry sub-sector are known as Leigh-Strassler. Leigh-Strassler deformations break the N= 4 supersymmetry down to N= 1 while preserving conformal symmetry. With exactly marginal deformations only the F-terms are deformed thus Leigh-Strassler deformations only affect the superpotential in the Lagrangian. In this thesis we study the symmetry of the marginally deformed N= 4 SYM and demonstrate that its algebraic structure can be understood in terms of quasi-Hopf algebras. Quasi-Hopf algebras have a notion of twisting due to Drinfeld which makes them a natural mathematical language with which to treat deformations. Furthermore the deformation of the N= 4 SYM superpotential is automated by the definition of a suitable star product. / Thesis (PhD)--University of Pretoria, 2020. / NiTheP / Physics / PhD / Unrestricted