# Applied Mathematics

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Item 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.Item 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.Item Interative approaches to convex feasibility problems.(2001) Pillay, Paranjothi.; Xu, Hongjun.; O'Hara, John Gerard.Solutions to convex feasibility problems are generally found by iteratively constructing sequences that converge strongly or weakly to it. In this study, four types of iteration schemes are considered in an attempt to find a point in the intersection of some closed and convex sets. The iteration scheme Xn+l = (1 - λn+1)y + λn+1Tn+lxn is first considered for infinitely many nonexpansive maps Tl , T2 , T3 , ... in a Hilbert space. A result of Shimizu and Takahashi [33] is generalized, and it is shown that the sequence of iterates converge to Py, where P is some projection. This is further generalized to a uniformly smooth Banach space having a weakly continuous duality map. Here the iterates converge to Qy, where Q is a sunny nonexpansive retraction. For this same iteration scheme, with finitely many maps Tl , T2, ... , TN , a complementary result to a result of Bauschke [2] is proved by introducing a new condition on the sequence of parameters (λn). The iterates converge to Py, where P is the projection onto the intersection of the fixed point sets of the Tis. Both this result and Bauschke's result [2] are then generalized to a uniformly smooth Banach space, and to a reflexive Banach space having a weakly continuous duality map and having Reich's property. Now the iterates converge to Qy, where Q is the unique sunny nonexpansive retraction onto the intersection of the fixed point sets of the Tis. For a random map r : N {I, 2, ... ,N}, the iteration scheme xn+l = Tr(n+l)xn is considered. In a finite dimensional Hilbert space with Tr(n) = Pr(n) , the iterates converge to a point in the intersection of the fixed point sets of the PiS. In an arbitrary Banach space, under certain conditions on the mappings, the iterates converge to a point in the intersection of the fixed point sets of the Tis. For the scheme xn+l = (1- λn+l)xn+λn+lTr(n+l)xn, in a finite dimensional Hilbert space the iterates converge to a point in the intersection of the fixed point sets of the Tis, and in an infinite dimensional Hilbert space with the added assumption that the random map r is quasi-cyclic, then the iterates converge weakly to a point in the intersection of the fixed point sets of the Tis. Lastly, the minimization of a convex function θ is considered over some closed and convex subset of a Hilbert space. For both the case where θ is a quadratic function and for the general case, first the unique fixed points of some maps Tλ are shown to converge to the unique minimizer of θ and then an algorithm is proposed that converges to this unique minimizer.Item Aspects of graph vulnerability.(1994) Day, David Peter.; Swart, Hendrika Cornelia Scott.; Oellermann, Ortrud Ruth.This dissertation details the results of an investigation into, primarily, three aspects of graph vulnerability namely, l-connectivity, Steiner Distance hereditatiness and functional isolation. Following the introduction in Chapter one, Chapter two focusses on the l-connectivity of graphs and introduces the concept of the strong l-connectivity of digraphs. Bounds on this latter parameter are investigated and then the l-connectivity function of particular types of graphs, namely caterpillars and complete multipartite graphs as well as the strong l-connectivity function of digraphs, is explored. The chapter concludes with an examination of extremal graphs with a given l-connectivity. Chapter three investigates Steiner distance hereditary graphs. It is shown that if G is 2-Steiner distance hereditary, then G is k-Steiner distance hereditary for all k≥2. Further, it is shown that if G is k-Steiner distance hereditary (k≥ 3), then G need not be (k - l)-Steiner distance hereditary. An efficient algorithm for determining the Steiner distance of a set of k vertices in a k-Steiner distance hereditary graph is discussed and a characterization of 2-Steiner distance hereditary graphs is given which leads to an efficient algorithm for testing whether a graph is 2-Steiner distance hereditary. Some general properties about the cycle structure of k-Steiner distance hereditary graphs are established and are then used to characterize 3-Steiner distance hereditary graphs. Chapter four contains an investigation of functional isolation sequences of supply graphs. The concept of the Ranked supply graph is introduced and both necessary and sufficient conditions for a sequence of positive nondecreasing integers to be a functional isolation sequence of a ranked supply graph are determined.Item 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.Item Partial exchangeability and related topics.(1991) North, Delia Elizabeth.; Dale, Andrew Ian.Partial exchangeability is the fundamental building block in the subjective approach to the probability of multi-type sequences which replaces the independence concept of the objective theory. The aim of this thesis is to present some theory for partially exchangeable sequences of random variables based on well-known results for exchangeable sequences. The reader is introduced to the concepts of partially exchangeable events, partially exchangeable sequences of random variables and partially exchangeable o-fields, followed by some properties of partially exchangeable sequences of random variables. Extending de Finetti's representation theorem for exchangeable random variables to hold for multi-type sequences, we obtain the following result to be used throughout the thesis: There exists a o-field, conditional upon which, an infinite partially exchangeable sequence of random variables behaves like an independent sequence of random variables, identically distributed within types. Posing (i) a stronger requirement (spherical symmetry) and (ii) a weaker requirement (the selection property) than partial exchangeability on the infinite multi-type sequence of random variables, we obtain results related to de Finetti's representation theorem for partially exchangeable sequences of random variables. Regarding partially exchangeable sequences as mixtures of independent and identically distributed (within types) sequences, we (i) give three possible expressions for the directed random measures of the partially exchangeable sequence and (ii) look at three possible expressions for the o-field mentioned in de Finetti's representation theorem. By manipulating random measures and using de Finetti's representation theorem, we point out some concrete ways of constructing partially exchangeable sequences. The main result of this thesis follows by extending de Finetti's represen. tation theorem in conjunction with the Chatterji principle to obtain the following result: Given any a.s. limit theorem for multi-type sequences of independent random variables, identically distributed within types, there exists an analogous theorem satisfied by all partially exchangeable sequences and by all sub-subsequences of some subsequence of an arbitrary dependent infinite multi-type sequence of random variables, tightly distributed within types. We finally give some limit theorems for partially exchangeable sequences of random variables, some of which follow from the above mentioned result.Item 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.Item 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.Item Analysis of shear-free spherically symmetric charged relativistic fluids.(2011) Kweyama, Mandlenkosi Christopher.; Maharaj, Sunil Dutt.; Govinder, Keshlan Sathasiva.We study the evolution of shear-free spherically symmetric charged fluids in general relativity. This requires the analysis of the coupled Einstein-Maxwell system of equations. Within this framework, the master field equation to be integrated is yxx = f(x)y2 + g(x)y3 We undertake a comprehensive study of this equation using a variety of ap- proaches. Initially, we find a first integral using elementary techniques (subject to integrability conditions on the arbitrary functions f(x) and g(x)). As a re- sult, we are able to generate a class of new solutions containing, as special cases, the models of Maharaj et al (1996), Stephani (1983) and Srivastava (1987). The integrability conditions on f(x) and g(x) are investigated in detail for the purposes of reduction to quadratures in terms of elliptic integrals. We also obtain a Noether first integral by performing a Noether symmetry analy- sis of the master field equation. This provides a partial group theoretic basis for the first integral found earlier. In addition, a comprehensive Lie symmetry analysis is performed on the field equation. Here we show that the first integral approach (and hence the Noether approach) is limited { more general results are possible when the full Lie theory is used. We transform the field equation to an autonomous equation and investigate the conditions for it to be reduced to quadrature. For each case we recover particular results that were found pre- viously for neutral fluids. Finally we show (for the first time) that the pivotal equation, governing the existence of a Lie symmetry, is actually a fifth order purely differential equation, the solution of which generates solutions to the master field equation.Item 2-generations of the sporadic simple groups.(1997) Ganief, Moegamad Shahiem.; Moori, Jamshid.A group G is said to be 2-generated if G = (x, y), for some non-trivial elements x, y E G. In this thesis we investigate three special types of 2-generations of the sporadic simple groups. A group G is a (l, rn, n )-generated group if G is a quotient group of the triangle group T(l, rn, n) = (x, y, zlx1 = ym = zn = xyz = la). Given divisors l, rn, n of the order of a sporadic simple group G, we ask the question: Is G a (l, rn, n)-generated group? Since we are dealing with simple groups, we may assume that III +l/rn + l/n < 1. Until recently interest in this type of generation had been limited to the role it played in genus actions of finite groups. The problem of determining the genus of a finite simple group is tantamount to maximizing the expression III +l/rn +Iln for which the group is (l,rn,n)-generated. Secondly, we investigate the nX-complementary generations of the finite simple groups. A finite group G is said to be nX-complementary generated if, given an arbitrary non-trivial element x E G, there exists an element y E nX such that G = (x, y). Our interest in this type of generation is motivated by a conjecture (Brenner-Guralnick-Wiegold [18]) that every finite simple group can be generated by an arbitrary non-trivial element together with another suitable element. It was recently proved by Woldar [181] that every sporadic simple group G is pAcomplementary generated, where p is the largest prime divisor of IGI. In an attempt to further the theory of X-complementary generations of the finite simple groups, we pose the following problem. Which conjugacy classes nX of the sporadic simple groups are nX-complementary generated conjugacy classes. In this thesis we provide a complete solution to this problem for the sporadic simple groups HS, McL, C03, Co2 , Jt , J2 , J3 , J4 and Fi 22 · We partially answer the question on (l, rn, n)-generation for the said sporadic groups. A finite non-abelian group G is said to have spread r iffor every set {Xl, X2, ' , "xr } of r non-trivial distinct elements, thpre is an element y E G such that G = (Xi, y), for all i. Our interest in this type of 2-generation comes from a problem by BrennerWiegold [19] to find all finite non-abelian groups with spread 1, but not spread 2. Every sporadic simple group has spread 1 (Woldar [181]) and we show that every sporadic simple group has spread 2.Item Exact solutions for perfect fluids conformal to a Petrov type D spacetime.(2011) Mewalal, Narenee.; Hansraj, Sudan.Abstract is available from the print copy.Item 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.Item Developing an integrated decision support system for an oil refinery.(1998) Azizi, Abbas.; Hearne, John W.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.Item 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.Item 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.Item 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.Item 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.Item 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.Item 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.Item 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.