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.

This thesis considers the problem of residue upgrading operations in an oil refinery. Visbreaking is a residue-upgrading process that improves profitability of a refinery. The economics of converting the heavy residue into the lighter and more valuable streams, coupled with the installation of a modem visbreaker unit at the Engen Refinery in Durban, provides sufficient motives to develop a mathematical model to simulate the unit's capability and estimate the economics of the visbreaking process and fuel oil operations. Furthermore, the proposed model should provide a crude-dependent visbreaking yield that can be used in the refinery's global linear programme (LP), employed to evaluate and select the crude and to optimise refinery's operations. Traditionally, kinetically based models have been used to simulate and study the refining reaction processes. In this case, due to the complexity of the process and some unknown reactions, the performances of existing visbreaking simulators are not fully satisfactory. Consequently, a neural network model of the visbreaking process and fuel oil blending operation is developed. The proposed model is called the adaptive visbreaker paradigm, since it is formed using neuroengineering, a technique that fabricates empirically-based neural network models. The network operates in supervised mode to predict the visbreaking yields and the residue quality. It was observed that due to the fluctuation in the quality of feedstock, and plant operating conditions, the prediction accuracy of the model needs to be improved. To improve the system's predictability, a network reciprocation procedure has been devised. Network reciprocation is a mechanism that controls and selects the input data used in the training of a neural network system. Implementation of the proposed procedure results in a considerable improvement in the performance ofthe network. 3 To facilitate the interaction between the simulation and optimisation routines, an integrated system to incorporate the fuel oil blending with the neurally-based module is constructed. Under an integrated system, the economics of altering the models' decision variables can be monitored. To account for the visbreakability of the various petroleum crudes, the yield predicted by the adaptive visbreaker paradigm should enter into the visbreaker,s sub-model of the global refinery LP. To achieve this, a mechanism to calculate and update the visbreaking yields of various crude oils is also developed. The computational results produced by the adaptive visbreaker paradigm prove that the economics of the visbreaking process is a multi-dimensional variable, greatly influenced by the feed quality and the unit's operating condition. The results presented show the feasibility of applying the proposed model to predict the cracking reaction yields. Furthermore, the model allows a dynamic monitoring of the residue properties as applicable to fuel oil blending optimisation. In summary, the combination of the proposed models forms an integrated decision support system suitable for studying the visbreaking and associated operations, and to provide a visbreaking yield pattern that can be incorporated into the global refinery LP model. Using an integrated decision support system, refinery planners are able to see through the complex interactions between business and the manufacturing process by performing predictive studies using these models. / Thesis (Ph.D.)-University of Natal, Pietermaritzburg, 1998.

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 character table of a finite group provides considerable amount of information about the group, and hence is of great importance in Mathematics as well as in Physical Sciences. Most of the maximal subgroups of the finite simple groups and their automorphisms are of extensions of elementary abelian groups, so methods have been developed for calculating the character tables of extensions of elementary abelian groups. Character tables of finite groups can be constructed using various techniques. However Bernd Fischer presented a powerful and interesting technique for calculating the character tables of group extensions. This technique, which is known as the technique of the Fischer-Clifford matrices, derives its fundamentals from the Clifford theory. If G=N.G is an appropriate extension of N by G, the method involves the construction of a nonsingular matrix for each conjugacy class of G/N~G. The character table of G can then be determined from these Fischer-Clifford matrices and the character table of certain subgroups of G, called inertia factor groups. In this dissertation, we described the Fischer-Clifford theory and apply it to both split and non-split group extensions. First we apply the technique to the split extensions 2,7:Sp6(2) and 2,8:SP6(2) which are maximal subgroups of Sp8(2) and 2,8:08+(2) respectively. This technique has also been discussed and used by many other researchers, but applied only to split extensions or to the case when every irreducible character of N can be extended to an irreducible character of its inertia group in G. However the same method can not be used to construct character tables of certain non-split group extensions. In particular, it can not be applied to the non-split extensions of the forms 3,7.07(3) and 3,7.(0,7(3):2) which are maximal subgroups of Fischer's largest sporadic simple group Fi~24 and its automorphism group Fi24 respectively. In an attempt to generalize these methods to such type of non-split group extensions, we need to consider the projective representations and characters. We have shown that how the technique of Fischer-Clifford matrices can be applied to any such type of non-split extensions. However in order to apply this technique, the projective characters of the inertia factors must be known and these can be difficult to determine for some groups. We successfully applied the technique of Fischer-Clifford matrices and determined the Fischer-Clifford matrices and hence the character tables of the non-split extensions 3,7.0,7(3) and 3,7.(0,7(3):2). The character tables computed in this thesis have been accepted for incorporation into GAP and will be available in the latest versions. / Thesis (Ph.D)-University of Natal, Pietermaritzburg, 2001.

The research of this thesis deals with the derivation of optimum designs for linear mixed models. The problem of constructing optimal designs for linear mixed models is very broad. Thus the thesis is mainly focused on the design theory for random coefficient regression models which are a special case of the linear mixed model. Specifically, the major objective of the thesis is to construct optimal designs for the simple linear and the quadratic regression models with a random intercept algebraically. A second objective is to investigate the nature of optimal designs for the simple linear random coefficient regression model numerically. In all models time is considered as an explanatory variable and its values are assumed to belong the set {a, 1, ... , k}. Two sets of individual designs, designs with non-repeated time points comprising up to k + 1 distinct time points and designs with repeated time points comprising up to k + 1 time points not necessarily distinct, are used in the thesis. In the first case there are 2k+ - 1 individual designs while in the second case there are ( 2 2k k+ 1 ) - 1 such designs. The problems of constructing population designs, which allocate weights to the individual designs in such a way that the information associated with the model parameters is in some sense maximized and the variances associated with the mean responses at a given vector of time points are in some sense minimized, are addressed. In particular D- and V-optimal designs are discussed. A geometric approach is introduced to confirm the global optimality of D- and V-optimal designs for the simple linear regression model with a random intercept. It is shown that for the simple linear regression model with a random intercept these optimal designs are robust to the choice of the variance ratio. A comparison of these optimal designs over the sets of individual designs with repeated and non-repeated points for that model is also made and indicates that the D- and V-optimal iii population designs based on the individual designs with repeated points are more efficient than the corresponding optimal population designs with non-repeated points. Except for the one-point case, D- and V-optimal population designs change with the values of the variance ratio for the quadratic regression model with a random intercept. Further numerical results show that the D-optimal designs for the random coefficient models are dependent on the choice of variance components. / Thesis (Ph.D.) - University of KwaZulu-Natal, Pietermaritzburg, 2004.

The theory of semigroups of linear operators forms an integral part of Functional Analysis with substantial applications to many fields of the natural sciences. In this study we are concerned with the application to equations of mathematical physics. The theory of semigroups of bounded linear operators is closely related to the solvability of evolution equations in Banach spaces that model time dependent processes in nature. Well-posed evolution problems give rise to a semigroup of bounded linear operators. However, in many important and interesting cases the problem is ill-posed making it inaccessible to the classical semigroup theory. One way of dealing with this problem is to generalize the theory of semigroups. In this thesis we give an outline of the theory of two such generalizations, namely, C-regularized semigroups and B-bounded semigroups, with the inter-relations between them and show a number of applications to ill-posed problems. / Thesis (Ph.D.)-University of Natal, Durban, 2001.

Please refer to the thesis for the abstract. / Thesis (Ph.D.)-University of Natal, Durban, 1998.

The geodesic law for test particles is one of the fundamental principles of general relativity and is extensively used. It is thought to be a consequence of the field laws but no rigorous proof exists. This thesis is concerned with a precise formulation of the geodesic law for test particles and with the extent of its validity. It will be shown to be true in certain cases but not in others. A rigorous version of the Infeld/Schild theorem is presented. Several explicit examples of both geodesic and non-geodesic motion of singularities are given. In the case of a test particle derived from a test body with a regular internal stress-energy tensor, a proof of the geodesic law for an ideal fluid test particle under plausible, explicitly stated conditions is given. It is also shown that the geodesic law is not generally true, even for weak fields and slow motion, unless the stress-energy tensor satisfies certain conditions. An explicit example using post-Newtonian theory is given showing how the geodesic law can be violated if these conditions are not satisfied. / Thesis (Ph.D.)-University of Natal, Durban, 1998.

In this thesis we seek new solutions to the anisotropic Einstein field equations which are important in the study of highly dense stellar structures. We first adopt the approach used by Maharaj & Maartens (1989) to obtain an exact anisotropic solution in terms of elementary functions for a particular choice of the energy density. This class of solution contains the Maharaj & Maartens (1989) and Gokhroo & Mehra (1994) models as special cases. In addition, we obtain six other new solutions following the same approach for different choices of the energy density. All the solutions in this section reduce to one with the energy density profile f-L ex r-2 . Two new algorithms are generated, Algorithm A and Algorithm B, which produce a new anisotropic solution to the Einstein field equations from a given isotropic solution. For any new anisotropic solution generated with the help of these algorithms, the original isotropic seed solution is regained as a special case. Two examples of known isotropic solutions are used to demonstrate how Algorithm A and Algorithm B work, and to obtain new anisotropic solutions for the Einstein and de Sitter models. Anisotropic isot~ermal sphere models are generated given the corresponding isotropic (f-L ex r-2 ) solution of the Einstein field equations. Also, anisotropic interior Schwarzschild sphere models are found given the corresponding isotropic (f-L ex constant) solution of the field equations. The exact solutions and line elements are given in each case for both Algorithm A and Algorithm B. Note that the solutions have a simple form and are all expressible in terms of elementary functions. Plots for the anisotropic factor S = J3(Pr - pJJ/2 (where Pr and Pl. are radial and tangential pressure respectively) are generated and these point to physically viable models. / Thesis (Ph.D.)-University of Natal, Durban, 2004.