No description available.

No abstract available. / Thesis (Ph.D.)-University of Natal, Pietermaritzburg, 2002.

Discrete mathematics has had many applications in recent years and this is only one reason for its increasing dynamism. The study of finite structures is a broad area which has a unity not merely of description but also in practice, since many of the structures studied give results which can be applied to other, apparently dissimilar structures. Apart from the applications, which themselves generate problems, internally there are still many difficult and interesting problems in finite geometry and combinatorics. There are still many puzzling features about sub-structures of finite projective spaces, the minimum weight of the dual codes of polynomial codes, as well as about finite projective planes. Finite groups are an ever strong theme for several reasons. There is still much work to be done to give a clear geometric identification of the finite simple groups. There are also many problems in characterizing structures which either have a particular group acting on them or which have some degree of symmetry from a group action. Codes obtained from permutation representations of finite groups have been given particular attention in recent years. Given a representation of group elements of a group G by permutations we can work modulo 2 and obtain a representation of G on a vector space V over lF2 . The invariant subspaces (the subspaces of V taken into themselves by every group element) are then all the binary codes C for which G is a subgroup of Aut(C). Similar methods produce codes over arbitrary fields. Through a module-theoretic approach, and based on a study of monomial actions and projective representations, codes with given transitive permutation group were determined by various authors. Starting with well known simple groups and defining designs and codes through the primitive actions of the groups will give structures that have this group in their automorphism groups. For each of the primitive representations, we construct the permutation group and form the orbits of the stabilizer of a point. Taking these ideas further we have investigated the codes from the primitive permutation representations of the simple alternating and symplectic groups of odd characteristic in their natural rank-3 primitive actions. We have also investigated alternative ways of constructing these codes, and these have come about by noticing that the codes constructed from the primitive permutations of the groups could also be obtained from graphs. We achieved this by constructing codes from the span of adjacency matrices of graphs. In particular we have constructed codes from the triangular graphs and from the graphs on triples. The simple symplectic group PSp2m(q), where m is at least 2 and q is any prime power, acts as a primitive rank-3 group of degree q2m-1/q-1 on the points of the projective (2m-1)-space PG2m-1(IFq ). The codes obtained from the primitive rank-3 action of the simple projective symplectic groups PSp2m(Q), where Q= 2t with t an integer such that t ≥ 1, are the well known binary subcodes of the projective generalized Reed-Muller codes. However, by looking at the simple symplectic groups PSp2m(q), where q is a power of an odd prime and m ≥ 2, we observe that in their rank-3 action as primitive groups of degree q2m-1/q-1 these groups have 2-modular representations that give rise to self-orthogonal binary codes whose properties can be linked to those of the underlying geometry. We establish some properties of these codes, including bounds for the minimum weight and the nature of some classes of codewords. The knowledge of the structures of the automorphism groups has played a key role in the determination of explicit permutation decoding sets (PD-sets) for the binary codes obtained from the adjacency matrix of the triangular graph T(n) for n ≥ 5 and similarly from the adjacency matrices of the graphs on triples. The successful decoding came about by ordering the points in such a way that the nature of the information symbols was known and the action of the automorphism group apparent. Although the binary codes of the triangular graph T(n) were known, we have examined the codes and their duals further by looking at the question of minimum weight generators for the codes and for their duals. In this way we find bases of minimum weight codewords for such codes. We have also obtained explicit permutation-decoding sets for these codes. For a set Ω of size n and Ω{3} the set of subsets of Ω of size 3, we investigate the binary codes obtained from the adjacency matrix of each of the three graphs with vertex set Ω{3}1 with adjacency defined by two vertices as 3-sets being adjacent if they have zero, one or two elements in common, respectively. We show that permutation decoding can be used, by finding PD-sets, for some of the binary codes obtained from the adjacency matrix of the graphs on (n3) vertices, for n ≥ 7. / Thesis (Ph.D.)-University of Natal, Pietermaritzburg, 2002.

The aim of this dissertation is to describe the conjugacy classes and some of the ordinary irreducible characters of the nite general linear group GL(n, q); together with character tables of some of its subgroups. We study the structure of GL(n, q) and some of its important subgroups such as SL(n, q); UT(n, q); SUT(n, q); Z(GL(n, q)); Z(SL(n, q)); GL(n, q)0 ; SL(n, q)0 ; the Weyl group W and parabolic subgroups P : In addition, we also discuss the groups PGL(n, q); PSL(n, q) and the a ne group A (n, q); which are related to GL(n, q): The character tables of GL(2; q); SL(2; q); SUT(2; q) and UT(2; q) are constructed in this dissertation and examples in each case for q = 3 and q = 4 are supplied. A complete description for the conjugacy classes of GL(n, q) is given, where the theories of irreducible polynomials and partitions of i 2 f1; 2; ; ng form the atoms from where each conjugacy class of GL(n, q) is constructed. We give a special attention to some elements of GL(n, q); known as regular semisimple, where we count the number and orders of these elements. As an example we compute the conjugacy classes of GL(3; q): Characters of GL(n, q) appear in two series namely, principal and discrete series characters. The process of the parabolic induction is used to construct a large number of irreducible characters of GL(n, q) from characters of GL(n, q) for m < n: We study some particular characters such as Steinberg characters and cuspidal characters (characters of the discrete series). The latter ones are of particular interest since they form the atoms from where each character of GL(n, q) is constructed. These characters are parameterized in terms of the Galois orbits of non-decomposable characters of F q n: The values of the cuspidal characters on classes of GL(n, q) will be computed. We describe and list the full character table of GL(n, q): There exists a duality between the irreducible characters and conjugacy classes of GL(n, q); that is to each irreducible character, one can associate a conjugacy class of GL(n, q): Some aspects of this duality will be mentioned. / Thesis (M.Sc. (School of Mathematical Sciences)) - University of KwaZulu-Natal, Pietermaritzburg, 2008.

This study examines the differences between interaction in Internet chat rooms and face to face communication and the effect those differences have on the development of social capital. Much has been written recently regarding the decline of social capital as evidenced by decreasing membership in small organizations. However, relatively little has been written about the potential for social capital to be_ developed through other forms of interaction. Results of this study indicate that though there are differences in the two experimental groups, there are no statistically significant differences between them. / Department of Political Science

A large part of the structure of the objects in the theory of Dooley and Wildberger [Funktsional. Anal. I Prilozhen. 27 (1993), no. 1, 25-32] and that of Rouviere [Compositio Math. 73 (1990), no. 3, 241-270] can be described by considering a connected, finite-dimentional symmetric space G/H (as defined by Rouviere), with ???exponential map???, Exp, from L G/L H to G/H, an action, ???: K ??? Aut??(G) (where Aut?? (G) is the projection onto G/H of all the automorphisms of G which leave H invariant), of a Lie group, K, on G/H and the corresponding action, ???# , of K on L G/L H defined by g ??? L (???g), along with a quadruple (s, E, j, E#), where s is a ???# - invariant, open neighbourhood of 0 in L G/L H, E is a test-function subspace of C??? (Exp s), j ?? C??? (s), and E# is a test-function subspace of C??? (s) which contains { j.f Exp: f ?? E }. Of interest is the question: Is the function ???: ?? ??? ????, where ??: f ??? j.f Exp, a local associative algebra homomorphism from F# with multiplication defined via convolution with respect to a function e: s x s ??? C, to F, with the usual convolution for its multiplication (where F is the space of all ??? - invariant distributions of E and F# is the space of all ???# - invariant distributions of E#)? For this system of objects, we can show that, to some extent, the choice of the function j is not critical, for it can be ???absorbed??? into the function e. Also, when K is compact, we can show that ??? ker ?? = { f ?? E : ???k f (???g) dg = 0}. These results turn out to be very useful for calculations on s2 ??? G/H, where G = SO(3) and H??? SO(3) with H ??? SO(2) with ??? : h ??? Lh, as we can use these results to show that there is no quadruple (s, E, j, E#) for SO(3)/H with j analytic in some neighbourhood of 0 such that ??? is a local homomorphism from F# to F. Moreover, we can show that there is more than one solution for the case where s, E and E# are as chosen by Rouviere, if e is does not have to satisfy e(??,??) = e(??,??).

Applied research project (D. Min.)--School of Theology and Missions, Oral Roberts University, 2006. / Includes abstract and vita. Includes bibliographical references (leaves 319-327).

Thesis (Ph.D.)--Auburn University, 2007. / Abstract. Includes bibliographic references (ℓ. 68)

Thesis (Ph. D.)--University of New South Wales, 1999. / Includes bibliographical references and index. Also available online.

Thesis (Ph. D.)--University of Missouri-Columbia, 2008. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on June 15, 2009) Vita. Includes bibliographical references.