There are basically two types of variables in population modelling, global and local variables. The former describes the behavior of the entire population while the latter describes the behavior of individuals within this population. The description of the population using local variables is more detailed, but it is also computationally costly. In many cases to study the dynamics of this population, it is sufficient to focus only on global variables. In applied sciences, to achieve this, the method of aggregation of variables is used. One of methods used to mathematically justify variables aggregation is the centre manifold theory. In this dissertation we provide detailed proofs of basic results of the centre manifold theory and discuss some examples of applications in population modelling. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2009.

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. / Thesis (Ph.D.)-University of Durban-Westville, 2001.

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. / Thesis (Ph.D.)-University of Natal, 1994.

The first half of this thesis deals with an aspect of domination; more specifically, we investigate the vertex integrity of n-distance-domination in a graph, i.e., the extent to which n-distance-domination properties of a graph are preserved by the deletion of vertices, as well as the following: Let G be a connected graph of order p and let oi- S s;:; V(G). An S-n-distance-dominating set in G is a set D s;:; V(G) such that each vertex in S is n-distance-dominated by a vertex in D. The size of a smallest S-n-dominating set in G is denoted by I'n(S, G). If S satisfies I'n(S, G) = I'n(G), then S is called an n-distance-domination-forcing set of G, and the cardinality of a smallest n-distance-domination-forcing set of G is denoted by On(G). We investigate the value of On(G) for various graphs G, and we characterize graphs G for which On(G) achieves its lowest value, namely, I'n(G), and, for n = 1, its highest value, namely, p(G). A corresponding parameter, 1](G), defined by replacing the concept of n-distance-domination of vertices (above) by the concept of the covering of edges is also investigated. For k E {a, 1, ... ,rad(G)}, the set S is said to be a k-radius-forcing set if, for each v E V(G), there exists Vi E S with dG(v, Vi) ~ k. The cardinality of a smallest k-radius-forcing set of G is called the k-radius-forcing number of G and is denoted by Pk(G). We investigate the value of Prad(G) for various classes of graphs G, and we characterize graphs G for which Prad(G) and Pk(G) achieve specified values. We show that the problem of determining Pk(G) is NP-complete, study the sequences (Po(G),Pl(G),P2(G), ... ,Prad(G)(G)), and we investigate the relationship between Prad(G)(G) and Prad(G)(G + e), and between Prad(G)(G + e) and the connectivity of G, for an edge e of the complement of G. Finally, we characterize integral triples representing realizable values of the triples b,i,p), b,l't,i), b,l'c,p), b,l't,p) and b,l't,l'c) for a graph. / Thesis (Ph.D.-Mathematics and Applied Mathematics)-University of Natal, 1995.

This thesis is a study of the symmetry of graphs and digraphs by considering certain homogeneous embedding requirements. Chapter 1 is an introduction to the chapters that follow. In Chapter 2 we present a brief survey of the main results and some new results in framing number theory. In Chapter 3, the notions of frames and framing numbers is adapted to digraphs. A digraph D is homogeneously embedded in a digraph H if for each vertex x of D and each vertex y of H, there exists an embedding of D in H as an induced subdigraph with x at y. A digraph F of minimum order in which D can be homogeneously embedded is called a frame of D and the order of F is called the framing number of D. We show that that every digraph has at least one frame and, consequently, that the framing number of a digraph is a well defined concept. Several results involving the framing number of graphs and digraphs then follow. Analogous problems to those considered for graphs are considered for digraphs. In Chapter 4, the notions of edge frames and edge framing numbers are studied. A nonempty graph G is said to be edge homogeneously embedded in a graph H if for each edge e of G and each edge f of H, there is an edge isomorphism between G and a vertex induced subgraph of H which sends e to f. A graph F of minimum size in which G can be edge homogeneously embedded is called an edge frame of G and the size of F is called the edge framing number efr(G) of G. We also say that G is edge framed by F. Several results involving edge frames and edge framing numbers of graphs are presented. For graphs G1 and G2 , the framing number fr(G1 , G2 ) (edge framing number ef r(GI, G2 )) of G1 and G2 is defined as the minimum order (size, respectively) of a graph F such that Gj (i = 1,2) can be homogeneously embedded in F. In Chapter 5 we study edge framing numbers and framing number for pairs of cycles. We also investigate the framing number of pairs of directed cycles. / Thesis (Ph.D.)-University of Natal, Pietermaritzburg, 1996.

In this thesis we investigate bounds on distance measures, namely, Steiner diameter and radius, in terms of other graph parameters. The thesis consists of four chapters. In Chapter 1, we define the most significant terms used throughout the thesis, provide an underlying motivation for our research and give background in relevant results. Let G be a connected graph of order p and S a nonempty set of vertices of G. Then the Steiner distance d(S) of S is the minimum size of a connected subgraph of G whose vertex set contains S. If n is an integer, 2 ≤ n ≤ p, the Steiner n-diameter, diamn(G), of G is the maximum Steiner distance of any n-subset of vertices of G. In Chapter 2, we give a bound on diamn(G) for a graph G in terms of the order of G and the minimum degree of G. Our result implies a bound on the ordinary diameter by Erdös, Pach, Pollack and Tuza. We obtain improved bounds on diamn(G) for K3-free graphs and C4-free graphs. In Chapter 3, we prove that, if G is a 3-connected plane graph of order p and maximum face length l then the radius of G does not exceed p/6 + 5l/6 + 5/6. For constant l, our bound improves on a bound by Harant. Furthermore we extend these results to 4- and 5-connected planar graphs. Finally, we complete our study in Chapter 4 by providing an upper bound on diamn(G) for a maximal planar graph G. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2011.

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. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2009.

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. / Thesis (M.Sc.)-University of Natal, 1996.

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. / Thesis (M.Sc.)-University of Natal, 1993.

We study the class of partial differential equations Utt = f(x, ux)uxx + g(x, u x), with arbitrary functions f(x, u x) and g(x, u x), from the point of view of group classification. The principal Lie algebra of infinitesimal symmetries admitted by the whole class is three-dimensional. We use the method of preliminary group classification to obtain a classification of these equations with respect to a one-dimesional extension of the principal Lie algebra and then a countable-dimensional subalgebra of their equivalence algebra. Each of these equations admits an additional infinitesimal symmetry. L.V. Ovsiannikov [9] has proposed an algorithm to construct efficiently the optimal system of an arbitrary decomposable Lie algebra. We use this algorithm to construct an optimal system of subalgebras of all dimensionalities (from one-dimensional to six- dimensional) of a seven-dimensional solvable Lie algebra. / Thesis (M.Sc)-University of Durban-Westville, 1995.