Masters Degrees (Applied Mathematics)

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Application of the wavelet transform for sparse matrix systems and PDEs.
(2009) Karambal, Issa.; Paramasur, Nabendra.; Singh, Pravin.
We consider the application of the wavelet transform for solving sparse matrix systems and partial differential equations. The first part is devoted to the theory and algorithms of wavelets. The second part is concerned with the sparse representation of matrices and well-known operators. The third part is directed to the application of wavelets to partial differential equations, and to sparse linear systems resulting from differential equations. We present several numerical examples and simulations for the above cases.
Assessment of variability in on-farm trials : a Uganda case.
(2002) Lapaka, Odong Thomas; Njuho, Peter Mungai.
On-farm trials techniques have become an integral part of research aimed at improving agricultural production especially in subsistence farming. The poor performance of certain technologies on the farmers' fields known to have performed well on stations have been of concern. Traditionally, on-farm trials are meant to address such discrepancies. The main problems associated with on-farm trials in most developing countries are high variability and inappropriate application of statistical knowledge known to work on station to on-farm situation. Characterisation of various on-farm variability and orientation of existing statistical methods may lead to improved agricultural research. Characterization of the various forms of variability in on-farm trials was conducted. Based on these forms of variability, estimation procedures and their strength have been assessed. Special analytical tools for handling non-replicated experiments known to be common to on-farm trials are presented. The above stated procedures have been illustrated through a review of Uganda case. To understand on-farm variability require grouping of sources of variability into agronomic, animal and socioeconomic components. This led to a deeper understanding of levels of variability and appropriate estimation procedures. The mixed model, modified stability analysis and additive main effects and multiplicative interaction methods have been found to play a role in on-farm trials. Proper approach to on-farm trials and application of appropriate statistical tools will lead to efficient results that will subsequently enhance agricultural production especially under subsistence farming.
Distance measures in graphs and subgraphs.
(1996) Swart, Christine Scott.; Swart, Hendrika Cornelia Scott.; Goddard, Wayne.
In this thesis we investigate how the modification of a graph affects various distance measures. The questions considered arise in the study of how the efficiency of communications networks is affected by the loss of links or nodes. In a graph C, the distance between two vertices is the length of a shortest path between them. The eccentricity of a vertex v is the maximum distance from v to any vertex in C. The radius of C is the minimum eccentricity of a vertex, and the diameter of C is the maximum eccentricity of a vertex. The distance of C is defined as the sum of the distances between all unordered pairs of vertices. We investigate, for each of the parameters radius, diameter and distance of a graph C, the effects on the parameter when a vertex or edge is removed or an edge is added, or C is replaced by a spanning tree in which the parameter is as low as possible. We find the maximum possible change in the parameter due to such modifications. In addition, we consider the cases where the removed vertex or edge is one for which the parameter is minimised after deletion. We also investigate graphs which are critical with respect to the radius or diameter, in any of the following senses: the parameter increases when any edge is deleted, decreases when any edge is added, increases when any vertex is removed, or decreases when any vertex is removed.
Spatial modelling of fire dynamics in Savanna ecosystems.
(1999) Berjak, Stephen Gary.; Hearne, John W.
Fire is used in the management of ecosystems worldwide because it is a relatively inexpensive means of manipulating thousands of hectares of vegetation. Deciding how, where and when to apply fire depends primarily on the management objectives of the area concerned. The decision to ignite vegetation is generally subjective and depends on the experience of the fire manager. To facilitate this process, ancillary tools, forming a decision support system, need to be constructed. In this study a spatial model has been developed that is capable of simulating fire dynamics in savanna ecosystems. The fire growth model integrates spatial fuel and topographic data with temporal weather, wind settings and fuel moistures to produce a time-evolving fire front. Spatial information required to operate the model was obtained through remote sensing techniques, using Landsat Thematic Mapper (TM) satellite imagery, and existing Geographic Information Systems (GIS) coverage's. Implementation of the simulation model to hypothetical landscapes under various scenarios of fuel, weather and topography produced fire fronts that were found to be in good agreement with experience of observed fires. The model was applied actual fire events using information for prescribed burning operations conducted in Mkuze Game Reserve during 1997. Predicted fire fronts were found to accurately resemble the observed fire boundaries in all simulations.
Bayesian analysis of cosmological models.
(2010) Moodley, Darell.; Moodley, Kavilan.; Sealfon, C.
In this thesis, we utilise the framework of Bayesian statistics to discriminate between models of the cosmological mass function. We first review the cosmological model and the formation and distribution of galaxy clusters before formulating a statistic within the Bayesian framework, namely the Bayesian razor, that allows model testing of probability distributions. The Bayesian razor is used to discriminate between three popular mass functions, namely the Press-Schechter, Sheth-Tormen and normalisable Tinker models. With a small number of particles in the simulation, we find that the simpler model is preferred due to the Occam’s razor effect, but as the size of the simulation increases the more complex model, if taken to be the true model, is preferred. We establish criteria on the size of the simulation that is required to decisively favour a given model and investigate the dependence of the simulation size on the threshold mass for clusters, and prior probability distributions. Finally we outline how our method can be extended to consider more realistic N-body simulations or be applied to observational data.
Exact solutions for perfect fluids conformal to a Petrov type D spacetime.
(2011) Mewalal, Narenee.; Hansraj, Sudan.
Abstract is available from the print copy.
Group analysis of equations arising in embedding theory.
(2010) Okelola, Michael.; Govinder, Keshlan Sathasiva.
Embedding theories are concerned with the embedding of a lower dimensional manifold (dim = n, say) into a higher dimensional one (usually dim = n+1, but not necessarily so). We are concerned with the particular case of embedding 4D spherically symmetric equations into 5D Einstein spaces. This scenario is of particular relevance to contemporary cosmology and astrophysics. Essentially, they are 5D vacuum field equations with initial data given on a 4D spacetime hypersurface. The equations that arise in this framework are highly nonlinear systems of ordinary differential equations and they have been particularly resistant to solution techniques over the past few years. As a matter of fact, to date, despite theoretical results for the existence of solutions for embedding classes of 4D space times, no general solutions to the local embedding equations are known. The Lie theory of extended groups applied to differential equations has proved to be very successful since its inception in the nineteenth century. More recently, it has been successfully utilized in relativity and has provided solutions where none were previously found, as well as explaining the existence of ad hoc methods. In our work, we utilize this method in an attempt to find solutions to the embedding equations. It is hoped that we can place the analysis of these equations onto a firm theoretical basis and thus provide valuable insight into embedding theories.
The theory of option valuation.
(1992) Sewambar, Soraya.; Murray, Michael.
Although options have been traded for many centuries, it has remained a relatively thinly traded financial instrument. Paradoxically, the theory of option pricing has been studied extensively. This is due to the fact that many of the financial instruments that are traded in the market place have an option-like structure, and thus the development of a methodology for option-pricing may lead to a general methodology for the pricing of these derivative-assets. This thesis will focus on the development of the theory of option pricing. Initially, a fundamental principle that underlies the theory of option valuation will be given. This will be followed by a discussion of the different types of option pricing models that are prevalent in the literature. Special attention will then be given to a detailed derivation of both the Black-Scholes and the Binomial Option pricing models, which will be followed by a proof of the convergence of the Binomial pricing model to the Black-Scholes model. The Black-Scholes model will be adapted to take into account the payment of dividends, the possibility of a changing inter est rate and the possibility of a stochastic variance for the rate of return on the underlying as set. Several applications of the Black-Scholes model will finally be presented.
Conformal motions in Bianchi I spacetime.
(1992) Lortan, Darren Brendan.; Maharaj, Sunil Dutt.
In this thesis we study the physical properties of the manifold in general relativity that admits a conformal motion. The results obtained are general as the metric tensor field is not specified. We obtain the Lie derivative along a conformal Killing vector of the kinematical and dynamical quantities for the general energy-momentum tensor of neutral matter. Equations obtained previously are regained as special cases from our results. We also find the Lie derivative of the energy-momentum tensor for the electromagnetic field. In particular we comprehensively study conformal symmetries in the Bianchi I spacetime. The conformal Killing vector equation is integrated to obtain the general conformal Killing vector and the conformal factor subject to integrability conditions. These conditions place restrictions on the metric functions. A particular solution is exhibited which demonstrates that these conditions have a nonempty solution set. The solution obtained is a generalisation of the results of Moodley (1991) who considered locally rotationally symmetric spacetimes. The Killing vectors are regained as special cases of the conformal solution. There do not exist any proper special conformal Killing vectors in the Bianchi I spacetime. The homothetic vector is found for a nonvanishing constant conformal factor. We establish that the vacuum Kasner solution is the only Bianchi I spacetime that admits a homothetic vector. Furthermore we isolate a class of vectors from the solution which causes the Bianchi I model to degenerate into a spacetime of higher symmetry.
Relativistic spherical stars.
(1993) Mkhwanazi, Wiseman Thokozani.; Maharaj, Sunil Dutt.
In this thesis we study spherically symmetric spacetimes which are static with a perfect fluid source. The Einstein field equations, in a number of equivalent forms, are derived in detail. The physical properties of a relativistic star are briefly reviewed. We specify two particular choices for one of the gravitational potentials. The behaviour of the remaining gravitational potential is governed by a second order differential equation. This equation has solutions in terms of elementary functions for some cases. The differential equation, in other cases, may be expressed as Bessel, confluent hypergeometric and hypergeometric equations. In such instances the solution is given in terms of special functions. A number of solutions to the Einstein field equations are generated. We believe that these solutions may be used to model realistic stars. Many of the solutions found are new and have not been published previously. In some cases our solutions are generalisations of cases considered previously. For some choices of the gravitational potential our solutions are equivalent to well-known results documented in the literature; in these cases we explicitly relate our solutions to those published previously. We have utilised the computer package MATHEMATICA Version 2.0 (Wolfram 1991) to assist with calculations, and to produce figures to describe the gravitational field. In addition, we briefly investigate the approach of specifying an equation of state relating the energy density and the pressure. The solution of the Einstein field equations, for a linear equation of state, is reduced to integrating Abel's equation of the second kind.
Some Mal'cev conditions for varieties of algebras.
(1991) Moses, Mogambery.; Sturm, Teo.
This dissertation deals with the classification of varieties according to their Mal'cev properties. In general the so called Mal'cev-type theorems illustrate an interplay between first order properties of a given class of algebras and the lattice properties of the congruence lattices of algebras of the considered class. CHAPTER 1. A survey of some notational conventions, relevant definitions and auxiliary results is presented. Several examples of less frequently used algebras are given together with the important properties of some of them. The term algebra T(X) and useful results concerning 'term' operations are established. A K-reflection is defined and a connection between a K-reflection of an algebra and whether a class K satisfies an identity of the algebra is established. CHAPTER 2. The Mal'cev-type theorems are presented in complete detail for varieties which are congruence permutable, congruence distributive, arithmetical, congruence modular and congruence regular. Several examples of varieties which exhibit these properties are presented together with the necessary verifications. CHAPTER 3. A general scheme of algorithmic character for some Mal'cev conditions is presented. R. Wille (1970) and A. F. Pixley (1972) provided algorithms for the classification of varieties which exhibit strong Mal'cev properties. This chapter is largely devoted to a modification of the Wille-Pixley schemes. It must be noted that this modification is quite different from all such published schemes. The results are the same as in Wille's scheme but slightly less general than in Pixley's. The text presented here, however is much simpler. As an example, the scheme is used to confirm Mal'cev's original theorem on congruence permutable varieties. Finally, the so-called Chinese var£ety is defined and Mal'cev conditions are established for such a variety of algebras . CHAPTER 4. A comprehensive survey of literature concerning Mal'cev conditions is given in this chapter.
Conformal symmetries : solutions in two classes of cosmological models.
(1991) Moodley, Manikam.; Maharaj, Sunil Dutt.
In this thesis we study the conformal symmetries in two locally rotationally symmetric spacetimes and the homothetic symmetries of a Bianchi I spacetime. The conformal Killing equation in a class AIa spacetime (MacCallum 1980), with a G4 of motions, is integrated to obtain the general solution subject to integrability conditions. These conditions are comprehensively analysed to determine the restrictions on the metric functions. The Killing vectors are contained in the general conformal solution. The homothetic vector is obtained and the explicit functional dependence of the metric functions determined. The class AIa spacetime does not admit a nontrivial special conformal factor. We also integrate the conformal Killing equation in the anisotropic locally rotationally symmetric spacetime of class A3 (MacCallum 1980), with a G4 of motions, to obtain the general conformal Killing vector and the conformal factor subject to integrability conditions. The Killing vectors are obtained as a special case from the general conformal solution. The homothetic vector is found for a nonzero constant conformal factor. The explicit functional form of the metric functions is determined for the existence of this homothetic vector. The spatially homogeneous and anisotropic A3 spacetime also does not admit a nontrivial special conformal vector. In the Bianchi I spacetime, with a G3 of motions, the conformal Killing equation is integrated for a constant conformal factor to generate the homothetic symmetries. The integrability conditions are solved to determine the functional dependence of the three time-dependent metric functions.
On Stephani universes.
(1992) Moopanar, Selvandren.; Maharaj, Sunil Dutt.
In this dissertation we study conformal symmetries in the Stephani universe which is a generalisation of the Robertson-Walker models. The kinematics and dynamics of the Stephani universe are discussed. The conformal Killing vector equation for the Stephani metric is integrated to obtain the general solution subject to integrability conditions that restrict the metric functions. Explicit forms are obtained for the conformal Killing vector as well as the conformal factor . There are three categories of solution. The solution may be categorized in terms of the metric functions k and R. As the case kR - kR = 0 is the most complicated, we provide all the details of the integration procedure. We write the solution in compact vector notation. As the case k = 0 is simple, we only state the solution without any details. In this case we exhibit a conformal Killing vector normal to hypersurfaces t = constant which is an analogue of a vector in the k = 0 Robertson-Walker spacetimes. The above two cases contain the conformal Killing vectors of Robertson-Walker spacetimes. For the last case in - kR = 0, k =I 0 we provide an outline of the integration process. This case gives conformal Killing vectors which do not reduce to those of RobertsonWalker spacetimes. A number of the calculations performed in finding the solution of the conformal Killing vector equation are extremely difficult to analyse by hand. We therefore utilise the symbolic manipulation capabilities of Mathematica (Ver 2.0) (Wolfram 1991) to assist with calculations.
On the integrity of domination in graphs.
(1993) Smithdorf, Vivienne.; Swart, Hendrika Cornelia Scott.
This thesis deals with an investigation of the integrity of domination in a.graph, i.e., the extent to which domination properties of a graph are preserved if the graph is altered by the deletion of vertices or edges or by the insertion of new edges. A brief historical introduction and motivation are provided in Chapter 1. Chapter 2 deals with kedge-( domination-)critical graphs, i.e., graphsG such that )'(G) = k and )'(G+e) < k for all e E E(G). We explore fundamental properties of such graphs and their characterization for small values of k. Particular attention is devoted to 3-edge-critical graphs. In Chapter 3, the changes in domination number brought aboutby vertex removal are investigated. \ Parameters )'+'(G) (and "((G)), denoting the smallest number of vertices of G in a set 5 such that )'(G-5) > )'(G) ()'(G -5) < )'(G), respectively), are investigated, as are'k-vertex-critical graphs G (with )'(G) = k and )'(G-v) < k for all v E V(O)). The existence of smallest'domination-forcing sets of vertices of graphs is considered. The bondage number 'Y+'(G), i.e., the smallest number of edges of a graph G in a set F such that )'(G- F) > )'(0), is investigated in Chapter 4, as are associated extremal graphs. Graphs with dominating sets or domination numbers that are insensitive to the removal of an arbitrary edge are considered, with particular reference to such graphs of minimum size. Finally, in Chapter 5, we-discuss n-dominating setsD of a graph G (such that each vertex in G-D is adjacent to at least n vertices in D) and associated parameters. All chapters but the first and fourth contain a listing of unsolved problems and conjectures.
Solution generating algorithms in general relativity.
(2011) Krupanandan, Daniel Dhevaprakshum.; Hansraj, Sudan.
We conduct a comprehensive investigative review of solution generating algorithms for the Einstein field equations governing the gravitational behaviour of an isolated neutral static spherical distribution of perfect fluid matter. Traditionally, the master field equation generated from the condition of pressure isotropy has been interpreted as a second order ordinary differential equation. However, since the pioneering work of Wyman (1949) it was observed that more success can be enjoyed by regarding the equation as a first order linear differential equation. There was a resurgence of the ideas of Wyman in 2000 and various researchers have been able to generate complete solutions to the field equations up to certain integrations. These have been accomplished by working in Schwarzschild (curvature) coordinates, isotropic coordinates, area coordinates and a coordinate system written in terms of the redshift parameter. We have utilised Durgapal–Banerjee (1983) coordinates and produced a new algorithm. The algorithm is used to generate new classes of perfect fluid solutions as well as to regain familiar particular solutions reported in the literature. We find that our solution is well behaved according to elementary physical requirements. The pressure vanishes for a certain radius and this establishes the boundary of the distribution. Additionally the pressure and energy density are both positive inside the radius. The energy conditions are shown to be satisfied and it is particularly pleasing to have the causality criterion satisfied to ensure that the speed of light is not exceeded by the speed of sound. We also report some new solutions using the algorithms proposed by Lake (2006).
Degree theory in nonlinear functional analysis.
(1989) Pillay, Paranjothi.; Hill, C. K.
The objective of this dissertation is to expand on the proofs and concepts of Degree Theory, dealt with in chapters 1 and 2 of Deimling [28], to make it more readable and accessible to anyone who is interested in the field. Chapter 1 is an introduction and contains the basic requirements for the subsequent chapters. The remaining chapters aim at defining a ll-valued map D (the degree) on the set M = {(F, Ω, y) / Ω C X open, F : Ὠ → X, y ɇ F(∂Ω)} (each time, the elements of M satisfying extra conditions) that satisfies : (D1) D(I, Ω, y) = 1 if y Є Ω. (D2) D(F, Ω, y) = D(F, Ω1 , y) + D(F, Ω2, y) if Ω1 and Ω2 are disjoint open subsets of Ω o such that y ɇ F(Ὠ \ Ω1 U Ω2 ). (D3) D(I - H(t, .), Ω, y(t)) is independent of t if H : J x Ὠ →X and y : J → X. An important property that follows from these three properties is (D4) F-1(y) ≠ Ø if D(F, Ω, y) ≠ 0. This property ensures that equations of the form Fx = y have solutions if D(F, Ω, y) ≠ 0. Another property that features in these chapters is the Borsuk property which gives us conditions under which the degree is odd and hence nonzero.
Exact solutions for relativistic models.
(2011) Ngubelanga, Sifiso Allan.; Maharaj, Sunil Dutt.; Ray, Subharthi.
In this thesis we study spherically symmetric spacetimes related to the Einstein field equations. We consider only neutral matter and apply the Einstein field equations with isotropic pressures. Our object is to model relativistic stellar systems. We express the Einstein field equations and the condition of pressure isotropy in terms of Schwarzschild coordinates and isotropic coordinates. For Schwarzschild coordinates we consider the transformations due to Buchdahl (1959), Durgapal and Bannerji (1983), Fodor (2000) and Tewari and Pant (2010). The condition of pressure isotropy is integrated and new exact solutions of the field equations are obtained utilizing the transformations of Buchdahl (1959) and Tewari and Pant (2010). These exact solutions are given in terms of elementary functions. For isotropic coordinates we can express the condition of pressure isotropy as a Riccati equation or a linear equation. An algorithm is developed that produces a new solution if a particular solution is known. The transformations reduce to a nonlinear Bernoulli equation in most instances. There are fundamentally three new classes of solutions to the condition of pressure isotropy.
Impact of exogenous reinfection on TB infection in a genetically susceptible population.
(2013) Mwangi, Wangari Isaac.; Chirove, Faraimunashe.; Achia, Thomas Noel Ochieng.
In this study we investigated the impact of exogenous reinfection on genetically resistant and genetically sensitive sub populations. We qualitatively analysed the dynamics of TB by assuming that TB is transmitted in two ways namely homogeneous and heterogeneous modes of transmission. Analytically, we computed the fundamental thresholds used to measure disease persistence; the basic reproduction number R₀; and found that the exogenous reinfection parameters do not appear in the basic reproduction number. Hence, basic reproduction number derived in presence of exogenous reinfection does not adequately predict the course of a TB epidemic. We obtained the exogenous reinfection threshold which indicated that exogenous reinfection complicates TB dynamics. Both analytical and simulation results disclosed that when exogenous reinfection is above exogenous reinfection threshold TB dynamics were governed by a backward bifurcation implying TB may continue to invade the population despite basic reproduction number being less than one. We computed critical value of basic reproduction numbers Rᴄ and found that TB can only be eradicated if basic reproduction number is reduced below critical value Rc. Furthermore, we incorporated TB therapy in heterogeneous model among individuals with clinically active TB and performed sensitivity and uncertainty analysis using Latin Hypercube Sampling. The sensitivity and uncertainty results showed that transmission rates, reactivation rates and proportion that is genetically resistant greatly infuenced outcome variables of our TB model.
Modelling the response of cytotoxic t-lymphocytes in controlling solid tumour invasion.
(2013) Malinzi, Joseph.; Mambili-Mamboundou, H.; Sibanda, Precious.
We present mathematical models to study the mechanism of interaction of tumour infiltrating cytotoxic lymphocytes (TICLs) with tumour cells. We focus on the phase spaces of the systems and the nature of the solutions for the cell densities in the short and long term. The first model describes the production of offspring through cell proliferation, death and local kinetic interactions. The second model characterises the spatial distribution dynamics of the cell densities through reaction diffusion, which describes the random movement of the cells, and chemotaxis, which describes the immune cell movements towards the tumour cells. We then extend these models further to incorporate the effects of immunotherapy by developing two new models. In both situations, we analyse the phase spaces of the homogeneous models, investigate the presence of travelling wave solutions in our systems, and provide numerical simulations. Our analysis shows that cancer dormancy can be attributed to TICLs. Our study also shows that TICLs reduce the tumour cell density to a cancer dormant state but even with immunotherapy do not completely eliminate tumour cells from body tissue. Travelling wave solutions were confirmed to exist in the heterogeneous model, a linear stability analysis of the homogeneous models and numerical simulations show the existence of a stable tumour dormant state and a phase space analysis confirms that there are no limit cycles.
New classes of exact solutions for charged perfect fluids.
(2012) Mthethwa, Thulani Richard.; Hansraj, Sudan.
We investigate techniques to generate new classes of exact solutions to the Einstein- Maxwell field equations which represent the gravitational field of charged perfect fluid spherically symmetric distributions of matter. Historically, a large number of solutions have been proposed but only a small number have been demonstrated to satisfy elementary conditions for physical acceptability. Firstly we examine the case of the constant density and constant electric field charged fluid sphere and show empirically that such configurations of matter are unlikely to exist as basic physical requirements are violated. We then make an ansatz relating the fluid's electric field intensity to one of the gravitational potentials thereby simplifying the system of partial differential equations. This prescription yields an algorithmic process to generate new classes of exact solutions. We present a number of new solutions and comment on their viability as stellar models. Graphical plots generated by symbolic software of the main dynamical and geometrical quantities verify that one of our models is suitable to represent a physically relevant distribution of charged matter in the form of a spherical shell. In particular, positive definiteness of energy density and pressure are guaranteed, a pressure free hypersurface denoting the boundary of the star exists, the sound speed is shown to be sub-luminal and the energy conditions are satisfied everywhere in the interior of the star.

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