Doctoral Degrees (Mathematics and Applied Mathematics)
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Item Open Access A structural approach to the endomorphisms of certain abelian groups(University of the Free State, 2016-09) De Klerk, Ben-Eben; Meyer, J. H.English: Given a set S, and any selfmap ƒ: S→S, the functional graph associated with ƒ can be described as a graph with vertex set S and directed edge set E = {(u; v) ϵ S2 : ƒ (u) = v}. A classification of all functional graphs induced by lattice endomorphisms has recently been done by J. Szigeti ([12]). In this dissertation, we aim to achieve a similar type of classi_cation with respect to functional graphs induced by endomorphisms on certain abelian groups. A method for finding all functional graphs that can be induced by endomorphisms of a group has been developed for all groups of the form Znp with p any prime, n ϵ N, and Zn for any n ϵ N, as well as all cyclic groups. A deep connection between the functional graphs corresponding to group endomorphisms and the minimal polynomial of the matrix representation of the group endomorphism has been found.Item Open Access Analysis of numerical approximation algorithms for nonlinear differential equations using a discrete multiple scales technique(University of the Free State, 2002-12) Maré, Eben; Schoombie, S. W.English: Perturbation techniques for the solution of differential equations form an essential ingredient of the tools of mathematics as applied to physics, engineering, finance and other areas of applied mathematics. A natural extension would be to seek perturbation-type solutions for discrete approximations of differential equations. The main objective of the research project is to develop a perturbation technique for approximation. The spurious behavior, predicted theoretically, is shown to be present experimentally, independent of temporal discretization. We also detail some comparisons of central difference solutions of different orders of approximation to the KdV equation. The results show a clear benefit of higher order central differences relative to lower order methods. The benefit of the central difference methodology would also extend to more general regions over which we would solve partial differential equations. We also show that the method of multiple scales can provide an adequate explanation for spurious behavior in a difference scheme for the Van der Pal equation. the solution of discrete equations. We discuss the well-known method of multiple scales and show its use for the solution of the Korteweg-de Vries (KdV), Regularized Long Wave (RLW) and Van der Pol equations. In particular, for the KdV and RLW equations the analysis shows that the envelopes of modulated waves are governed by the nonlinear Schrödinger equation. We present a variation of the multiple scales technique which presents an ideal framework from which we devise a discrete multiple scales analysis methodology. We discuss a discrete multiple scales methodology derived by Schoombie [111], as applied to the Zabusky-Kruskal approximation of the KdV equation. This discrete multiple scales analysis methodology is generalized and applied to the solution of a generalized finite difference approximation of the KdV equation. We show the consistency of the method with the continuous analysis as the discretization parameters tend to zero. The discrete multiple scales technique is a powerful tool for the examination of modulational properties of the KdV equation. In the case of certain modes of the carrier wave, the discrete multiple scales analysis breaks down, indicating that the numerical solution deviates in behavior from that of the KdV equation. Several numerical experiments are performed to examine the spurious behavior for different orders of approximation. The spurious behavior, predicted theoretically, is shown to be present experimentally, independent of temporal discretization. We also detail some comparisons of central difference solutions of different orders of approximation to the KdV equation. The results show a clear benefit of higher order central differences relative to lower order methods. The benefit of the central difference methodology would also extend to more general regions over which we would solve partial differential equations. We also show that the method of multiple scales can provide an adequate explanation for spurious behavior in a difference scheme for the Van der Pal equation.Item Open Access Contributions to the theory of near vector spaces(University of the Free State, 2007-09) Howell, Karin-Therese; Meyer, J. H.The main purpose of this thesis is to give an exposition of and expand the theory of near vector spaces, as first introduced by Andr´e [1]. The notion of a vector space is well known. For this reason the material in this thesis is presented in such a way that the parallels between near vector spaces and vector spaces are apparent. In Chapter 1 several elementary definitions and properties are given. In addition, some important examples that will be referred to throughout this paper are cited. In Chapter 2 the theory of near vector spaces is presented. We start off with some preliminary results in 2.1 and build up to the definition of a regular near vector space in 2.5. In addition, we show how a near vector space can be decomposed into maximal regular subspaces. We conclude this chapter by showing when a near vector space will in fact be a vector space. We follow the format of De Bruyn’s thesis; however, both De Bruyn and Andr´e make use of left nearfields to define the near vector spaces. In light of the material we want to present in Chapter 4, it is more standard to use the notation as in the papers by van der Walt, [12], [13]. Thus we develop the material using right nearfields with scalar multiplication on the right of vectors. The third chapter contains some examples of near vector spaces and serves as an illustration of much of the work of Chapter 2. Examples 1, 2 and 3 were used in De Bruyn’s thesis. However, on closer inspection, it was revealed that in Example 2, the element (a, 0, 0, d) is omitted as an element of Q(V ). This error is corrected. And in keeping with our use of right nearfields, the necessary changes are made to Example 1 and 3. In particular, the definition of ◦ in Example 3 is adapted and the necessary adjustments are made. We conclude this chapter by developing a theory that allows us to characterise all finite dimensional near vector spaces over Zp, for p a prime. In Chapter 4 we turn our attention to the work done by van der Walt in [12] and [13]. In Section 4.1 we consider the effects that ‘perturbations’ in the action of a (right) nearfield F has on the well known structures, the ring of linear transformations of V and the nearring of homogeneous functions of V into itself. This first section sets the scene for the more generalised situation described in 4.2 and leads to the introduction of the nearring of matrices determined by n multiplicatively isomorphic nearfields and a matrix of isomorphisms. We conclude this chapter by summarising some properties of this nearring in 4.3 and 4.4. Note that throughout this paper, ⊂ will be used to convey a proper subset, whereas ⊆ will convey the possibility of equality.Item Open Access Adaptive dynamics for an age-structured population model with a Shepherd recruitment function(University of the Free State, 2013-06-07) Ellis, Michelle Heidi; Schoombie, S. W.English: In this study the evolution of the genetic composition of certain species will be replaced by the evolution of the traits that represent these genetic compositions. Depending on the nature of the trait of interest, a scalar valued parameter called the strategy parameter will be assigned to this trait making the simulation of strategy evolution possible. The trait of interest, and therefore the strategy associated, will be the ability of a population to keep its densities within the carrying capacity of the environment they find themselves in. The Shepherd function, on account of its wide use in population simulations as well as composing of exactly such a density parameter, will be the density curbing mechanism of choice in the age-structured population model designed here. An algorithm will be designed to simulate strategy evolution towards an evolutionary stable strategy or ESS that will ensure not only an optimal fit for this environment but also render the population immune against future invasion by other members of the population practising slight variations of this strategy. There are two ways to come by such an optimal strategy without directly involving genetics. The first is game theory, allowing strategists to compete for this position, and the second is with the use of adaptive dynamics, converting winning and loosing instead into tangible mathematics. Combining these two classics will show that the quest is an exercise in strategy optimization, not only from the point of view of an already established population but also from the point of view of an initially small one. It will be interesting!