In this paper, elementary properties of relative extreme points are investigated. The properties are defined in linear and topological terms. Proofs of many of these properties require the use of topological concepts.

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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

This thesis treats a protocol for anonymous database search (or if one prefer, a protocol for user-private information retrieval), that is based on the use of combinatorial configurations. The protocol is called P2P UPIR. It is proved that the (v,k,1)-balanced incomplete block designs (BIBD) and in particular the finite projective planes are optimal configurations for this protocol. The notion of n-anonymity is applied to the configurations for P2P UPIR protocol and the transversal designs are proved to be n-anonymous configurations for P2P UPIR, with respect to the neighborhood points of the points of the configuration. It is proved that to the configurable tuples one can associate a numerical semigroup. This theorem implies results on existence of combinatorial configurations. The proofs are constructive and can be used as algorithms for finding combinatorial configurations. It is also proved that to the triangle-free configurable tuples one can associate a numerical semigroup. This implies results on existence of triangle-free combinatorial configurations.

A linear space is a set of points and lines such that any pair of points lie on exactly one line together. This is equivalent to a pairwise balanced design PBD(v, K), where there are v points, lines are regarded as blocks, and K ⊆ Z≥2 denotes the set of allowed block sizes. The dimension of a linear space is the maximum integer d such that any set of d points is contained in a proper subspace. Specifically for K = {3, 4, 5}, we determine which values of v admit PBD(v,K) of dimension at least three for all but a short list of possible exceptions under 50. We also observe that dimension can be reduced via a substitution argument. / Graduate / 0405 / jniezen@uvic.ca

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The main purpose of this dissertation is the study of ideals of multilinear mappings and homogeneous polynomials between linear spaces. By an ideal we mean a class that is stable under the composition with linear operators. First we study multilinear mappings and spaces of multilinear mappings. We also show how to obtain, from a given multilinear mapping, other multilinear mappings with degrees of multilinearity greater than, equal to or smaller than the degree of the original multilinear mapping. Next we study homogeneous polynomials and spaces of homogeneous polynomials, and we also show how to obtain, from a given n-homogeneous polynomial, other polynomials with degrees of homogeneity greater than, equal to or smaller than the degree of the original polynomial. Next we study ideals of multilinear mappings, or multi-ideals, and ideals of homogeneous polynomial, or polynomial ideals, giving several examples and presenting methods to generated multi-ideals and polynomial ideals from a given operator ideal. Finally we dene and give several examples of coherent multi-ideals and coherent polynomial ideals. / O principal objetivo desta dissertação e estudar os ideais de aplicações multilineares e polinômios homogêneos entre espaços vetoriais. Por um ideal entendemos uma classe de aplicações que e estavel atraves da composição com operadores lineares. Primeiramente estudamos as aplicações multilineares e os espaços de aplicações multilineares. Mostramos tambem como obter, a partir de uma aplicação multilinear dada, outras aplicações com graus de multilinearidade maiores, iguais ou menores que o da aplicação original. Em seguida estudamos os polinômios homogêneos e os espacos de polinômios homogêneos, e mostramos que, a partir de um polinômio n-homogêneo, tambem podemos construir novos polinômios homogêneos com graus de homogeneidade maiores, iguais ou menores que n. Posteriormente estudamos os ideais de aplicações multilineares, ou multi-ideais, e os ideais de polinômios homogêneos, exibindo varios exemplos e apresentando metodos para se obter um multi-ideais, ou ideais de polinômios, a partir de ideais de operadores lineares dados. Por m, denimos e exibimos varios exemplos de multi-ideais coerentes e de ideais coerentes de polinômios. / Mestre em Matemática