We studied the subjectivity of word maps on SU(n) and the length of the shortest elements in the central series of free group of rank 2 with some applications to almost laws in compact groups.

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We study character generating functions (character generators) of simple Lie algebras. The expression due to Patera and Sharp, derived from the Weyl character formula, is ¯rst re- viewed. A new general formula is then found. It makes clear the distinct roles of \outside" and \inside" elements of the integrity basis, and helps determine their quadratic incompati- bilities. We review, analyze and extend the results obtained by Gaskell using the Demazure character formulas. We ¯nd that the fundamental generalized-poset graphs underlying the character generators can be deduced from such calculations. These graphs, introduced by Baclawski and Towber, can be simpli¯ed for the purposes of constructing the character generator. The generating functions can be written easily using the simpli¯ed versions, and associated Demazure expressions. The rank-two algebras are treated in detail, but we believe our results are indicative of those for general simple Lie algebras. / vii, 92 leaves ; 29 cm.

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This thesis is concerned with the application of Lie's group theoretic method to the Einstein field equations in order to find new exact solutions. We analyse the nonlinear partial differential equation which arises in the study of non- static, non-conformally flat fluid plates of embedding class one. In order to find the group invariant solutions to the partial differential equation in a systematic and comprehensive manner we apply the method of optimal subgroups. We demonstrate that the model admits linear barotropic equations of state in several special cases. Secondly, we study a shear-free spherically symmetric cosmological model with heat flow. We review and extend a method of generating solutions developed by Deng. We use the method of Lie analysis as a systematic approach to generate new solutions to the master equation. Also, general classes of solution are found in which there is an explicit relationship between the gravitational potentials which is not present in earlier models. Using our systematic approach, we can recover known solutions. Thirdly, we study generalised shear-free spherically symmetric models with heat flow in higher dimensions. The method of Lie generates new solutions to the master equation. We obtain an implicit solution or we can reduce the governing equation to a Riccati equation. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2011.

In the standard modeling of the pricing of options and derivatives as generally understood these days the underlying process is taken to be a Wiener Process or a Levy Process. The stochastic process is modeled as a stochastic differential equation. From this equation a partial differential equation is obtained by application of the Feynman-Kac Theorem. The resulting partial differential equation is of Hamilton-Jacobi-Bellman type. Analysis of the partial differential equations arising from Mathematics of Finance using the methods of the Lie Theory of Continuous Groups has been performed over the last twenty years, but it is only in recent years that there has been a concerted effort to make full use of the Lie theory. We propose an extension of Mahomed and Leach's (1990) formula for the nth-prolongation of an nth-order ordinary differential equation to the nth-prolongation of the generator of an hyperbolic partial differential equation with p dependent and k independent variables. The symmetry analysis of this partial differential equation shows that the associated Lie algebra is {sl(2,R)⊕W₃}⊕s ∞A₁ with 12 optimal systems. A modeling approach based upon stochastic volatility for modeling prices in the deregulated Pennsylvania State Electricity market is adopted for application. We propose a dynamic linear model (DLM) in which switching structure for the measurement matrix is incorporated into a two-state Gaussian mixture/first-order autoregressive (AR (1)) configuration in a nonstationary independent process defined by time-varying probabilities. The estimates of maximum likelihood of the parameters from the "modified" Kalman filter showed a significant mean-reversion rate of 0.9363 which translates to a half-life price of electricity of nine months. Associated with this mean-reversion is the high measure of price volatility at 35%. Within the last decade there has been some work done upon the symmetries of stochastic differential equations. Here empirical results contradict earliest normality hypotheses on log-return series in favour of asymmetry of the probability distribution describing the process. Using the Akaike Information Criterion (AIC) and the Log-likelihood estimation (LLH) methods as selection criteria, the normal inverse Gaussian (NIG) outperformed four other candidate probability distributions among the class of Generalized Hyperbolic (GH) distributions in describing the heavy tails present in the process. Similarly, the Skewed Student's t (SSt) is the best fit for Bonny Crude Oil and Natural Gas log-returns. The observed volatility measures of these three commodity prices were examined. The Weibull distribution gives the best fit both electricity and crude oil data while the Gamma distribution is selected for natural gas data in the volatility profiles among the five candidate probability density functions (Normal, Lognormal, Gamma, Inverse Gamma and the Inverse Gaussian) considered. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2011.

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.

Statistical analysis of 3-dimensional motions of humans, animals or objects is instrumental to establish how these motions differ, depending on various influences or parameters. When such motions involve no stretching or tearing, they may be described by the elements of a Lie group called the Special Euclidean Group, denoted SE(3). Statistical analysis of trajectories lying in SE(3) is complicated by the basic properties of the group, such as non-commutativity, non-compactness and lack of a bi-invariant metric. This necessitates the generalization of the ideas of “mean” and “variance” to apply in this setting. We describe how to exploit the unique properties of a formalism called Conformal Geometric Algebra to express these generalizations and carry out such statistical analyses efficiently; we introduce a practical method of visualizing trajectories lying in the 6-dimensional group manifold of SE(3); and we show how this methodology can be applied, for example, in testing theoretical claims about the influence of an attended object on a competing action applied to a different object. The two prevailing views of such movements differ as to whether mental action-representations evoked by an object held in working memory should perturb only the early stages of subsequently reaching to grasp another object, or whether the perturbation should persist over the entire movement. Our method yields “difference trajectories” in SE(3), representing the continuous effect of a variable of interest on an action, revealing statistical effects on the forward progress of the hand as well as a corresponding effect on the hand’s rotation. / Graduate / 0405 / 0541 / 0623

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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(??,??).