Service delivery in post-apartheid South Africa has become a topical issue both in the academia and the political arena . The rise of social movements, the xenophobic tensions of May 2008 and protest actions could be noted as the major traits of post-apartheid South Africa. Though there are divergent views on the underlying causes of these protests, lack of service delivery has most significantly been at the centre stage. In this thesis we investigate the relationship between household/population changes and the demand for piped-water connection in South Africa. There is an ample, albeit at times of questionable accuracy, supply of statistics from official and other sources. These statistics are both the source of inspiration of particular societal measures to be investigated and a gauge of the accuracy of the mathematical/statistical modelling which is the central feature of this project. We construct mathematical/statistical models which take into account demographic constituents of the problem using differential equations for modelling household dynamics and we also investigate the interaction of demographic parameters and the demand for piped-water connection using multivariate statistical techniques. The results show that with a boost in delivery the rich provinces seem to be in better standing of meeting targets and that the increasing demand in household-based services could be most significantly attributed to the fragmentation of households against other demographic processes like natural increase in population and net migration. The results imply that in as much as service delivery policies and programmes should focus on formerly disadvantaged poor communities, adequate provisions for increasing service demands in urban centres should also be a priority in view of the increasing in-migration from rural areas as households fragment. Most of the findings/results are in tabular and graphical forms for easy understanding of the reader. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2010.

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. / Thesis (M.Sc.)-University of KwaZulu-Natal, 1992.

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. / Thesis (M.Sc.)-University of Natal, Durban, 1992.

Chapter 0 In this introductory chapter, certain notational and terminological conventions are established and a summary given of background results that are needed in subsequent chapters. Chapter 1 In this chapter, the notion of a "weak conguence formula" [Tay72], [BB75] is introduced and used to characterize both subdirectly irreducible algebras and essential extensions. Special attention is paid to the role they play in varieties with definable principal congruences. The chapter focuses on residually small varieties; several of its results take their motivation from the so-called "Quackenbush Problem" and the "RS Conjecture". One of the main results presented gives nine equivalent characterizations of a residually small variety; it is largely due to W. Taylor. It is followed by several illustrative examples of residually small varieties. The connections between residual smallness and several other (mostly categorical) properties are also considered, e.g., absolute retracts, injectivity, congruence extensibility, transferability of injections and the existence of injective hulls. A result of Taylor that establishes a bound on the size of an injective hull is included. Chapter 2 Beginning with a proof of A. Day's Mal'cev-style characterization of congruence modular varieties [Day69] (incorporating H.-P. Gumm's "Shifting Lemma"), this chapter is a self-contained development of commutator theory in such varieties. We adopt the purely algebraic approach of R. Freese and R. McKenzie [FM87] but show that, in modular varieties, their notion of the commutator [α,β] of two congruences α and β of an algebra coincides with that introduced earlier by J. Hagemann and C. Herrmann [HH79] as well as with the geometric approach proposed by Gumm [Gum80a],[Gum83]. Basic properties of the commutator are established, such as that it behaves very well with respect to homomorphisms and sufficiently well in products and subalgebras. Various characterizations of the condition "(x, y) Є [α,β]” are proved. These results will be applied in the following chapters. We show how the theory manifests itself in groups (where it gives the familiar group theoretic commutator), rings, modules and congruence distributive varieties. Chapter 3 We define Abelian congruences, and Abelian and affine algebras. Abelian algebras are algebras A in which [A2, A2] = idA (where A2 and idA are the greatest and least congruences of A). We show that an affine algebra is polynomially equivalent to a module over a ring (and is Abelian). We give a proof that an Abelian algebra in a modular variety is affine; this is Herrmann's Funda- mental Theorem of Abelian Algebras [Her79]. Herrmann and Gumm [Gum78], [Gum80a] established that any modular variety has a so-called ternary "difference term" (a key ingredient of the Fundamental Theorem's proof). We derive some properties of such a term, the most significant being that its existence characterizes modular varieties. Chapter 4 An important result in this chapter (which is due to several authors) is the description of subdirectly irreducible algebras in a congruence modular variety. In the case of congruence distributive varieties, this theorem specializes to Jόnsson's Theorem. We consider some properties of a commutator identity (Cl) which is a necessary condition for a modular variety to be residually small. In the main result of the chapter we see that for a finite algebra A in a modular variety, the variety V(A) is residually small if and only if the subalgebras of A satisfy (Cl). This theorem of Freese and McKenzie also proves that a finitely generated congruence modular residually small variety has a finite residual bound, and it describes such a bound. Thus, within modular varieties, it proves the RS Conjecture. Conclusion The conclusion is a brief survey of further important results about residually small varieties, and includes mention of the recently disproved (general) RS Conjecture. / Thesis (M.Sc.)-University of Natal, Durban, 2000.

The study of second order ordinary differential equations is vital given their proliferation in mechanics. The group theoretic approach devised by Lie is one of the most successful techniques available for solving these equations. However, many second order equations cannot be reduced to quadratures due to the lack of a sufficient number of point symmetries. We observe that increasing the order will result in a third order differential equation which, when reduced via an alternate symmetry, may result in a solvable second order equation. Thus the original second order equation can be solved. In this dissertation we will attempt to classify second order differential equations that can be solved in this manner. We also provide the nonlocal transformations between the original second order equations and the new solvable second order equations. Our starting point is third order differential equations. Here we concentrate on those invariant under two- and three-dimensional Lie algebras. / Thesis (M.Sc.)-University of Natal, Durban, 2000.

In this thesis we consider spherically symmetric cosmological models when the shear is nonzero and also cases when the shear is vanishing. We investigate the role of the Emden-Fowler equation which governs the behaviour of the gravitational field. The Einstein field equations are derived in comoving coordinates for a spherically symmetric line element and a perfect fluid source for charged and uncharged matter. It is possible to reduce the system of field equations under different assumptions to the solution of a particular Emden-Fowler equation. The situations in which the Emden-Fowler equation arises are identified and studied. We analyse the Emden-Fowler equation via the method of Lie point symmetries. The conditions under which this equation is reduced to quadratures are obtained. The Lie analysis is applied to the particular models of Herlt (1996), Govender (1996) and Maharaj et al (1996) and the role of the Emden-Fowler equation is highlighted. We establish the uniqueness of the solutions of Maharaj et al (1996). Some physical features of the Einstein-Maxwell system are noted which distinguishes charged solutions. A charged analogue of the Maharaj et al (1993) spherically symmetric solution is obtained. The Gutman-Bespal'ko (1967) solution is recovered as a special case within this class of solutions by fixing the parameters and setting the charge to zero. It is also demonstrated that, under the assumptions of vanishing acceleration and proper charge density, the Emden-Fowler equation arises as a governing equation in charged spherically symmetric models. / Thesis (M.Sc.)-University of Natal, Durban, 1998.

Abstract available in pdf file.

In this thesis we study classes of static spherically symmetric spacetimes admitting a perfect fluid source, electromagnetic fields and anisotropic pressures. Our intention is to generate exact solutions that model the interior of dense, relativistic stars. We find a sufficient condition for the existence of series solutions to the condition of pressure isotropy for neutral isolated spheres. The existence of a series solution is demonstrated by the method of Frobenius. With the help of MATHEMATICA (Wolfram 1991) we recovered the Tolman VII model for a quadratic gravitational potential, but failed to obtain other known classes of solution. This establishes the weakness, in certain instances, of symbolic manipulation software to extract series solutions from differential equations. For a cubic potential, we obtained a new series solution to the Einstein field equations describing neutral stars. The gravitational and thermodynamic variables are non-singular and continuous. This model also satisfies the important barotropic equation of state p = p(p). Two new exact solutions to the Einstein-Maxwell system, that generalise previous results for uncharged stars, were also found. The first of these generalises the solution of Maharaj and Mkhwanazi (1996), and has well-behaved matter and curvature variables. The second solution reduces to the Durgapal and Bannerji (1983) model in the uncharged limit; this new result may only serve as a toy model for quark stars because of negative energy densities. In both examples we observe that the solutions may be expressed in terms of hypergeometric and elementary functions; this indicates the possibility of unifying isolated solutions under the hypergeometric equation. We also briefly study compact stars with spheroidal geometry, that may be charged or admit anisotropic pressure distributions. The adapted forms of the pressure isotropy condition can be written as a harmonic oscillator equation. Two simple examples are presented. / Thesis (M.Sc.)-University of Natal, Durban, 2002.

In order to understand how bunchgrasses achieve dominance over other plant growth forms and how they achieve dominance over one another in different environments, it is first necessary to develop a detailed understanding of how their growth strategy interacts with the resource limits of their environment. Two properties which have been studied separately in limited detail are architecture and disproportionate resource allocation. Architecture is the structural layout of organs and objects at different hierarchical levels. Disproportionate resource allocation is the manner in which resources are allocated across objects at each level of hierarchy. Clonal architecture and disproportionate resource allocation may interact significantly to determine the growth ability of clonal plants. These interactions have not been researched in bunchgrasses. This thesis employs a novel simulation technique, functional-structural plant modelling, to investigate how bunchgrasses interact with the resource constraints imposed in humid grasslands. An appropriate functional-structural plant model, the TILLERTREE model, is developed that integrates the architectural growth of bunchgrasses with environmental resource capture and disproportionate resource allocation. Simulations are conducted using a chosen model species Themeda triandra, and the environment is parameterised using characteristics of the Southern Tall Grassveld, a humid grassland type found in South Africa. Behaviour is considered at two levels, namely growth of single ramets and growth of multiple ramets on single bunchgrass clones. In environments with distinct growing and non-growing seasons, bunchgrasses are subjected to severe light depletion during regrowth at the start of each growing season because of the accumulation of dead material in canopy caused by the upright, densely packed manner in which they grow. Simulations conducted here indicate that bunchgrass tillers overcome this resource bottleneck through structural adaptations (etiolation, nonlinear blade mass accretion, residual live photosynthetic surface) and disproportionate resource allocation between roots and shoots of individual ramets that together increase the temporal resource efficiency of ramets by directing more resources to shoot growth and promoting extension of new leaves through the overlying dead canopy. The architectural arrangement of bunchgrasses as collections of tillers and ramets directly leads to consideration of a critical property of clonal bunchgrasses: tiller recruitment. Tiller recruitment is a fundamental discrete process limiting the vegetative growth of bunchgrass clones. Tiller recruitment occurs when lateral buds on parent tillers are activated to grow. The mechanism that controls bud outgrowth has not been elucidated. Based on a literature review, it is here proposed that lateral bud outgrowth requires suitable signals for both carbohydrate and nitrogen sufficiency. Subsequent simulations with the model provide corroborative evidence, in that greatest clonal productivity is achieved when both signals are present. Resource allocation between live structures on clones may be distributed proportionately in response to sink demand or disproportionately in response to relative photosynthetic productivity. Model simulations indicate that there is a trade-off between total clonal growth and individual tiller growth as the level of disproportionate allocation between ramets on ramet groups and between tillers on ramets increases, because disproportionate allocation reduces tiller population size and clonal biomass, but increases individual tiller performance. Consequently it is proposed that different life strategies employed by bunchgrasses, especially annual versus perennial life strategies, may follow more proportionate and less proportionate allocation strategies respectively, because the former favours maximal resource capture and seed production while the latter favours individual competitive ability. Structural disintegration of clones into smaller physiologically integrated units (here termed ramet groups) that compete with one another for resources is a documented property of bunchgrasses. Model simulations in which complete clonal integration is enforced are unable to survive for long periods because resource bottlenecks compromise all structures equally, preventing them from effectively overcoming resource deficits during periods when light is restrictive to growth. Productivity during the period of survival is also reduced on bunchgrass clones with full integration relative to clones that disintegrate because of the inefficient allocation of resources that arises from clonal integration. This evidence indicates that clonal disintegration allows bunchgrass clones both to increase growth efficiency and pre-empt potential death, by promoting the survival of larger ramet groups and removing smaller ramet groups from the system. The discrete nature of growth in bunchgrasses and the complex population dynamics that arise from the architectural growth and the temporal resource dynamics of the environment, may explain why different bunchgrass species dominate under different environments. In the final section this idea is explored by manipulating two species tiller traits that have been shown to be associated with species distributions across non-selective in defoliation regimes, namely leaf organ growth rate and tiller size (mass or height). Simulations with these properties indicate that organ growth rate affects daily nutrient demands and therefore the rate at which tillers are terminated, but had only a small effect on seasonal resource capture. Tiller mass size affects the size of the live tiller population where smaller tiller clones maintain greater numbers of live tillers, which allows them to them to sustain greater biomass over winter and therefore to store more reserves for spring regrowth, suggesting that size may affect seasonal nitrogen capture. The greatest differences in clonal behaviour are caused by tiller height, where clones with shorter tillers accumulate substantially more resources than clones with taller tillers. This provides strong evidence there is trade-off for bunchgrasses between the ability to compete for light and the ability to compete for nitrogen, which arises from their growth architecture. Using this evidence it is proposed that bunchgrass species will be distributed across environments in response to the nitrogen productivity. Shorter species will dominate at low nitrogen productivity, while taller species dominate at high nitrogen productivity. Empirical evidence is provided in support of this proposal. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2005.

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / In this thesis we study a standard example of refinable functions, that is, functions which can be reproduced by the integer shifts of their own dilations. Using the cardinal B-spline as an introductory example, we prove some of its properties, thereby building a basis for a later extension to the vector setting. Defining a subdivision scheme associated to the B-spline refinement mask, we then present the proof of a well-known convergence result. Subdivision is a powerful tool used in computer-aided geometric design (CAGD) for the generation of curves and surfaces. The basic step of a subdivision algorithm consists of starting with a given set of points, called the initial control points, and creating new points as a linear combination of the previous ones, thereby generating new control points. Under certain conditions, repeated applications of this procedure yields a continuous limit curve. One important goal of this thesis is to study a particular extension of scalar subdivision to matrix subdivision ...