Aspects of the fundamental concept of distance are investigated in this dissertation. Two major topics are discussed; the first considers metrics which give a measure of the extent to which two given graphs are removed from being isomorphic, while the second deals with Steiner distance in graphs which is a generalization of the standard definition of distance in graphs. Chapter 1 is an introduction to the chapters that follow. In Chapter 2, the edge slide and edge rotation distance metrics are defined. The edge slide distance gives a measure of distance between connected graphs of the same order and size, while the edge rotation distance gives a measure of distance between graphs of the same order and size. The edge slide and edge rotation distance graphs for a set S of graphs are defined and investigated. Chapter 3 deals with metrics which yield distances between graphs or certain classes of graphs which utilise the concept of greatest common subgraphs. Then follows a discussion on the effects of certain graph operations on some of the metrics discussed in Chapters 2 and 3. This chapter also considers bounds and relations between the metrics defined in Chapters 2 and 3 as well as a partial ordering of these metrics. Chapter 4 deals with Steiner distance in a graph. The Steiner distance in trees is studied separately from the Steiner distance in graphs in general. The concepts of eccentricity, radius, diameter, centre and periphery are generalised under Steiner distance. This final chapter closes with an algorithm which solves the Steiner problem and a Heuristic which approximates the solution to the Steiner problem. / Thesis (M.Sc.)-University of Natal, 1991.

In this thesis we investigate the effect of shear on radiating stars undergoing gravitational collapse. The interior spacetime is described by the most general spherically symmetric line element in the absence of rotation. The energy momentum tensor for the stellar interior is taken to be an anisotropic fluid with heat flux. The thermodynamics of a relativistic fluid is reviewed for the Eckart and causal theories. Since the star is radiating energy to the exterior in the form of a radial heat flux, the atmosphere is described by Vaidya's outgoing solution. We provide the matching conditions required for the continuity of the momentum flux across the boundary, which determines the temporal evolution junction conditions for the metric functions. We provide a general method to obtain shearing solutions of the Einstein field equations describing a radiating, collapsing sphere. A particular exact solution satisfying the boundary condition and field equations is found. The validity of this specific model is investigated by employing a causal heat transport equation which yields the temperature profile within the stellar core. The energy conditions are studied and yield interesting features of this particular model which are absent in the shear-free case. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2011.

In 1956, E. Michael proved his famous convex-valued selection theorems for l.s.c. mappings de ned on spaces with higher separation axioms (paracompact, collectionwise normal, normal and countably paracompact, normal, and perfectly normal), [39]. In 1959, he generalized the convex-valued selection theorem for mappings de ned on paracompact spaces by replacing \convexity" with \ -paraconvexity", for some xed constant 0 < 1 (see, [42]). In 1993, P.V. Semenov generalized this result by replacing with some continuous function f : (0;1) ! [0; 1) (functional paraconvexity) satisfying a certain property called (PS), [63]. In this thesis, we demonstrate that the classical Michael selection theorem for l.s.c. mappings with a collectionwise normal domain can be reduced only to compact-valued mappings modulo Dowker's extension theorem for such spaces. The idea used to achieve this reduction is also applied to get a simple direct proof of that selection theorem of Michael's. Some other possible applications are demonstrated as well. We also demonstrate that the -paraconvex-valued and the functionally-paraconvex valued selection theorems remain true for C 0 (Y )-valued mappings de ned on -collectionwise normal spaces, where is an in nite cardinal number. Finally, we prove that these theorems remain true for C (Y )-valued mappings de ned on -PF-normal spaces; and we provide a general approach to such selection theorems. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2012.

We develop a mathematical model which seeks to assess the impact of HIV Pre-Exposure Prophylaxis (PrEP) on the prevalence and incidence of HIV infection. Mathematical analysis of the model is carried out to establish the threshold conditions that determine the stability of the steady states. Numerical simulations are performed to gain insight into the use and e cacy of PrEP. Results from our model reveal that the basic reproduction number is a function of the rate at which individuals use PrEP and the rate at which PrEP protects individuals from HIV infection. Furthermore, strategies where either PrEP awareness or PrEP e cacy was low show potential loop-holes that can lead to more complications than bene ts. The best strategies revealed by our results is that a high level of awareness and high PrEP e cacy are needed. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2012.

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. / Thesis (M.Sc.)-University of Durban-Westville, 1989.

The purpose of this thesis is to investigate how "spect ralness" properties of a torsion theory T on R - Mod are reflected by properties of the ring R and its ring of quotients R,.. The development of "spectral" torsion theory owes much to Zelmanowitz [50] and Gomez-Pardo [23] . Gomez-Pardo proved that there exists a bijective correspondence between the set of spectral torsion theories on R - Modand rings of quotients of R that are Von Neumann regular and left self-injective. Chapter 1 is concerning with the notation used in the thesis and a summary of main results which are needed for understanding the sequel. Chapter 2 is concerned with the construction of a maximal ring of quotients of an arbitrary ring R by using the notion of denseness and relative injective hull. In Chapter 3, we survey the three equivalent ways of formulating Torsion Theory: by means of preradical functors on the category R- Mod, pairs of torsion / torsion-free classes and topologizing filters on rings. We shall show that Golan's approach to Torsion Theory via equivalence classes of injectives; and Dickson's one (as presented by Stenstrom) are equivalent. With a torsion theory T defined on R-Mod we associate R,. a ring of quotients of R. The full subcategory (R, T) - Mod of R- Mod whose objects are the T-torsion-free r-injective left R-modules is a Grothendieck category called the quotient category of R - Mod with respect to T. A left R,.-module that is r-torsion-free T-injective as a left R-module is injective if and only if it is injective as a left R-module (Proposition 3.6.4). Because of its use in the sequel , particular attention is paid to the lattice isomorphism that exists between the lattice of .r-pure submodules of a left Rmodule M and the lattice of subobjects of the quotient module M; in the category (R , T) - Mod. Chapter 4 introduces the definition of a spectral torsion theory: a Vll torsion theory r on R - Mod is said to be spectral if the Grothendieck category (R, r) - Mod is spectral. Using the notion of relative essential submodule, one can construct a spectral torsion theory from an arbitrary torsion theory on R - Mod. We shall show how an investigation of a general spectral torsion theory on R - Mod reduces to the Goldie torsion theory on R/tT (R) - Mod. Moreover, we shall exhibit necessary and sufficient conditions for R; to be a regular left self-injective ring (Theorem 4.2.10). In Chapter 5, after constructing the torsion functor Soce(-) which is associated with the pseudocomplement r.l of r in R - tors, we show how semiartinian rings can be characterized by means of spectral torsion theories: if a spectral torsion theory r on R - Mod is generated by the class of r-torsion simple left R-modules or, equivalently, cogenerated by the class of r-torsion-free simple left R-modules, then R is a left semiartinian ring (Proposition 5.3.2). Chapter 6 gives Zelmanowitz' important result [50]: R; is a semisimple artinian ring if and only if the torsion theory r is spectral and the associated left Gabriel topology has a basis of finitely generated left ideals. We also exhibit results due to M.J. Arroyo and J. Rios ([4] and [5]) which illustrate how spectral torsion theories can be used to describe when R; is (1) prime regular and left self-injective, (2) a left full linear ring, and (3) a direct product of left full linear rings. We also study the relationship between the flatness of the ring of quotients R; and the r- coherence of the ring R when r is a spectral torsion theory. It is proved that if r is a spectral torsion theory on R - Mod then the following conditions are equivalent: (1) R is left r-coherent; (2) (Rr)R is flat; (3) every right Rr-module is flat as a right R-module (Proposition 6.3.9). This result is an extension of Cateforis' results. / Thesis (M.Sc) - University of Natal, Pietermaritzburg, 2003

Non-negative matrices arise naturally in population models. In this thesis, we look at the theory of such matrices and we study the Perron-Frobenius type theorems regarding their spectral properties. We use these theorems to investigate the asymptotic behaviour of solutions to continuous time problems arising in population biology. In particular, we provide a description of long-time behaviour of populations depending on the nature of the associated matrix. Finally, we describe a few applications to population biology. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2010.

The valuation of real options has been of interest for some time. Recently, the model has been revised to include more than one source of randomness, e.g. Paxson and Pinto (2005). In this dissertation, we present a model with more than one diffusion process to analyze strategic interaction in a duopolistic framework. We consider a complete market where the profit per unit and the number of units sold are assumed to evolve according to distinct, but possibly correlated, geometric Brownian motions, and aim to extend Paxson and Pinto’s research to a wider context by adjusting the model to include the effect of the covariance between the stochastic factors. In particular, we present results in both the pre-emptive and non pre-emptive equilibrium case pertaining to the follower’s and leader’s value function. We also investigate the consequences for the model in relation to traditional net present value theory, and include an analysis of the comparative static relationships that exist between the parameters. We then conclude with a chapter that extends our two-variable model to three sources of randomness - first by allowing the investment cost to be modelled as a random once-off payment, and then by considering it to be a stochastically variable ongoing cost. Keywords Real options, complete markets, more than one stochastic process, competitive games, duopoly. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2009.

The concept of a frame was introduced in the mid-sixties by Dowker and Papert. Since then frames have been extensively studied by several authors, including Banaschewski, Pultr and Baboolal to mention a few. The idea of a nearness was first introduced by H. Herrlich in 1972 and that of a nearness frame by Banaschewski in the late eighties. T. Dube made a fairly detailed study of the latter concept. The purpose of this thesis is to study the property of local finiteness and metacompactness in the setting of nearness frames. J. W. Carlson studied these ideas (including Lindelof and Pervin nearness structures) in the realm of nearness spaces. The first four chapters are a brief overview of frame theory culminating in results concerning regular, completely regular, normal and compact frames. In chapter five we provide the definitions for various nearness frames: Pervin, Lindelof , Locally Finite and Metacompact to mention a few. A particular locally finite nearness structure, denoted by µLF, is studied in detail. It is defined to be the nearness structure on a regular frame L generated by the family of all locally finite covers on the frame L. Also, a particular metacompact nearness structure, denoted by µPF, is studied in detail. It is defined to be the nearness structure on a regular frame L generated by the family of all point-finite covers of the frame L. Various theorems related the above nearness frames and these nearness structures are obtained. / Thesis (M.Sc.)-University of Durban-Westvile, 1998.

The term "frame" was introduced by C H Dowker, who studied them in a long series of joint papers with D Papert Strauss. J R Isbell , in a path breaking paper [1972] pointed out the need to introduce separate terminology for the opposite of the category of Frames and coined the term "locale". He was the progenitor of the idea that the category of Locales is actually more convenient in many ways than the category of Frames. In fact, this proves to be the case in one of the approaches adopted in this thesis. Sublocales (quotient frames) have been studied by several authors, notably Dowker and Papert [1966] and Isbell [1972]. The term "sublocale" is due to Isbell, who also used "part " to mean approximately the same thing. The use of nuclei as a tool for studying sublocales (as is used in this thesis) and the term "nucleus" itself was initiated by H Simmons [1978] and his student D Macnab [1981]. Uniform spaces were introduced by Weil [1937]. Isbell [1958] studied algebras of uniformly continuous functions on uniform spaces. In this thesis, we introduce the concept of a uniform frame (locale) which has attracted much interest recently and here too Isbell [1972] has some results of interest. The notion of a metric frame was introduced by A Pultr [1984]. The main aim of his paper [11] was to prove metrization theorems for pointless uniformities. This thesis focuses on the construction of completions in Uniform Frames and Metric Frames. Isbell [6] showed the existence of completions using a frame of certain filters. We describe the completion of a frame L as a quotient of the uniformly regular ideals of L, as expounded by Banaschewski and Pultr[3]. Then we give a substantially more elegant construction of the completion of a uniform frame (locale) as a suitable quotient of the frame of all downsets of L. This approach is attributable to Kriz[9]. Finally, we show that every metric frame has a unique completion, as outlined by Banaschewski and Pultr[4]. In the main, this thesis is a standard exposition of known, but scattered material. Throughout the thesis, choice principles such as C.D.C (Countable Dependent Choice) are used and generally without mention. The treatment of category theory (which is used freely throughout this thesis) is not self-contained. Numbers in brackets refer to the bibliography at the end of the thesis. We will use 0 to indicate the end of proofs of lemmas, theorems and propositions. Chapter 1 covers some basic definitions on frames , which will be utilized in subsequent chapters. We will verify whatever we need in an endeavour to enhance clarity. We define the categories, Frm of frames and frame homomorphisms, and Lac the category of locales and frame morphisms. Then we explicate the adjoint situation that exists between Frm and Top , the category of topological spaces and continuous functions. This is followed by an introduction to the categories, RegFrm of all regular frames and frame homomorphisms, and KRegFrm the category of compact regular frames and their homomorphisms. We then present the proofs of two very important lemmas in these categories. Finally, we define the compactification of and a congruence on a frame. In Chapter 2 we recall some basic definitions of covers, refinements and star refinements of covers. We introduce the notion of a uniform frame and define certain mappings (morphisms) between uniform frames (locales) . In the terminology of Banaschewski and Kriz [9] we define a complete uniform frame and the completion of a uniform frame. The aim of Chapter 3 is twofold : first, to construct the compact regular coreflection of uniform frames , that is, the frame counterpart of the Samuel Compactification of uniform spaces [12] , and then to use it for a description of the completion of a uniform frame as an alternative to that previously given by Isbell[6]. The main purpose of Chapter 4 is to provide another description of uniform completion in frames (locales), which is in fact even more straightforward than the original topological construction. It simply consists of writing down generators and defining relations. We provide a detailed examination of the main result in this section, that is, a uniform frame L is complete of each uniform embedding f : (M,UM) -t (L,UL) is closed, where UM and UL denote the uniformities on the frames M and L respectively. Finally, in Chapter 5, we introduce the notions of a metric diameter and a metric frame. Using the fact that every metric frame is a uniform frame and hence has a uniform completion, we show that every metric frame L has a unique completion : CL - L. / Thesis (M.Sc.)-University of Durban-Westville, 1996.