by Chan Suk Ha Iris. / Bibliography: leaves 84-88 / Thesis (M.Ph.)--Chinese University of Hong Kong

We describe groups of positive type and prove that a group G is of positive type if and only if G admits a non-trivial partition. We completely classify groups of type 2, and present examples of other groups of positive type as well as groups of type zero.

Hung Ling Yan Lincoln. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 87-88). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preliminary --- p.10 / Chapter 2.1 --- Background in geometry --- p.10 / Chapter 2.2 --- Background in harmonic maps --- p.12 / Chapter 3 --- Lie Point Transformations and Symmetries --- p.16 / Chapter 3.1 --- Definition of symmetries --- p.16 / Chapter 3.2 --- Determine the Lie point symmetries of partial differential equations --- p.25 / Chapter 3.2.1 --- Second order differential equations --- p.26 / Chapter 4 --- Similarity Variables --- p.30 / Chapter 4.1 --- "Similarity variables and group-invariant, solutions" --- p.30 / Chapter 4.2 --- Reduction of number of variables of the partial differential equations --- p.34 / Chapter 4.2.1 --- Determine the similarity variables --- p.34 / Chapter 4.2.2 --- Procedure to reduce the number of variables of a system of partial differential equations --- p.36 / Chapter 5 --- Group Invariant Harmonic Maps --- p.38 / Chapter 5.1 --- Determine the Lie point symmetries of the harmonic map equations --- p.39 / Chapter 5.2 --- Reduction of harmonic map equations to ordinary differ- ential equations --- p.54 / Chapter 5.3 --- Solving the harmonic map system which has been reduced to ordinary differential equations --- p.62 / Chapter 5.3.1 --- Case 1 of Theorem 5.2.1 --- p.62 / Chapter 5.3.2 --- Case 2 of Theorem 5.2.1 --- p.66 / Chapter 5.3.3 --- Case 3 of Theorem 5.2.1 --- p.75 / Bibliography --- p.87

School of Mathematics University of the Witwatersrand (Wits), Johannesburg 31 August 2015 Submitted in partial fulflment of a Masters degree at Wits / The operation defined on a discrete semigroup S can be extended to the Stone- Cech compactification S of S so that for all a 2 S, the left translation S 3 x 7! ax 2 S is continuous, and for all q 2 S, the right translation S 3 x 7! xq 2 S is continuous. Because any compact right topological semigroup, S contains a smallest two-sided ideal K( S) which is a completely simple semigroup. We give an exposition of some basic results related to the semigroup S and to the semigroup algebra `1( S). In particular, we review the result that `1( N) is semisimple if and only if `1(K( N)) is semisimple. We also review the reduction of the question whether `1(K( N)) is semisimple to a question about K( N).

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Let <a1,..., a m | wn> be a presentation of a group G, where w is freely and cyclically reduced and n ≥ 2 is maximal. We define a system of codimension-1 subspaces in the universal cover, and invoke a construction essentially due to Sageev to define an action of G on a CAT(0) cube complex. By proving easily formulated geometric properties of the codimension-1 subspaces we show that when n ≥ 4 the action is proper and cocompact, and that the cube complex is finite dimensional and locally finite. We also prove partial results when n = 2 or n = 3. It is also shown that the subgroups of G generated by non-empty proper subsets of {a1, a 2,..., am} embed by isometries into the whole group.

This thesis deals with various problems about the normal and subnormal structure of infinite groups. We first consider the relationship between the number of normal subgroups of a group G and of a subgroup H of finite index in G. We prove Theorem 1.5 There exists a finitely generated group G which has a subgroup H of index 2 such that H has continuously many normal subgroups and G has only countably many normal subgroups. Proposition 1.7 Let k be an infinite cardinal. Then there exists a group G of cardinality k that has only 12 normal subgroups but which contains a subgroup H of index 2 having k normal subgroups. We then consider partially ordered sets and investigate the subnormal structure of generalized wreath products. We deal with the question whether the number of subnormal subgroups of an infinite group is determined by the number of its n-step subnormal subgroups for an integer n. We prove Theorem 5.3 Let G be a group. Then G has finitely many subnormal subgroups if and only if it has finitely many 2-step subnormal subgroups. Theorem 5.5 Let m and n be infinite cardinals such that m ≤ n. Then there exists a group G with the following properties: (1) The cardinality of G is n. (2) The number of normal subgroups of G is <mathematical symbol>. (3) The number of 2-step subnormal subgroups of G is m. (4) The number of 3-step subnormal subgroups of G is 2ⁿ. Finally we consider characteristically simple groups with countably many normal subgroups. We construct a new type of characteristically simple groups: Corollary 6.15 Let ∧ be a partially ordered set such that for λ,<mathematical symbol>∊∧ there exists an automorphism a of ∧ such that <mathematical symbol> ≤ λa. Let <mathematical symbol>(∧) be the distributive lattice of semi-ideals of ∧. Then there exists a group G with the following properties: (1) |G| ≤ max(<mathematical symbol> of |<mathematical symbol>(∧)|). (2) All subnormal subgroups of G are normal in G. (3) The lattice of normal subgroups of G is isomorphic to <mathematical symbol> (∧). (4) The group G is characteristically simple.

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We investigate the relationship between the Dynamical Systems analysis and the Lie Symmetry analysis of ordinary differential equations. We undertake this investigation by looking at a relativistic model of self-gravitating charged fluids. Specifically we look at the impact of specific parameters obtained from Lie Symmetries analysis on the qualitative behaviour of the model. Steady states, stability and possible bifurcations are explored. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.