Thesis (PhD)–Stellenbosch University, 2014. / ENGLISH ABSTRACT: See pdf for abstract / AFRIKAANSE OPSOMMING: Sien PDF vir die opsomming

Ever since Maynard-Smith and Price first introduced the concept of an evolutionary stable strategy (ESS) in 1973, there has been a growing amount of work in and around this field. Many new concepts have been introduced, quite often several times over, with different acronyms by different authors. This led to other authors trying to collect and collate the various terms (for example Lessard, 1990 & Eshel, 1996) in order to promote better understanding ofthe topic. It has been noticed that dynamic selection did not always lead to the establishment of an ESS. This led to the development ofthe concept ofa continuously stable strategy (CSS), and the claim that dynamic selection leads to the establishment of an ESSif it is a CSS. It has since been proved that this is not always the case, as a CSS may not be able to displace its near neighbours in pairwise ecological competitions. The concept of a neighbourhood invader strategy (NIS) was introduced, and when used in conjunction with the concept of an ESS, produced the evolutionary stable neighbourhood invader strategy (ESNIS) which is an unbeatable strategy. This work has tried to extend what has already been done in this field by investigating the dynamics of coexisting species, concentrating on systems whose dynamics are governed by Lotka-Volterra competition models. It is proved that an ESNIS coalition is an optimal strategy which will displace any size and composition of incumbent populations, and which will be immune to invasions by any other mutant populations, because the ESNIS coalition, when it exists, is unique. It has also been shown that an ESNIS coalition cannot exist in an ecologically stable state with any finite number of strategies in its neighbourhood. The equilibrium population when the ESNIS coalition is the only population present is globally stable in a n-dimensional system (for finite n), where the ESNIS coalition interacts with n - 2 other strategies in its neighbourhood. The dynamical behaviour of coexisting species was examined when the incumbent species interacted with various invading species. The different behaviour ofthe incumbent population when invaded by a coalition using either an ESNIS or an NIS phenotype underlines the difference in the various strategies. Similar simulations were intended for invaders who were using an ESS phenotype, but unfortunately the ESS coalition could not be found. If the invading coalition use NIS phenotypes then the outcome is not certain. Some, but not all of the incumbents might become extinct, and the degree to which the invaders flourish is very dependent on the nature ofthe incumbents. However, if the invading species form an ESNIS coalition, one is certain of the outcome. The invaders will eliminate the incumbents, and stabilise at their equilibrium populations. This will occur regardless of the composition and number of incumbent species, as the ESNIS coalition forms a globally stable equilibrium point when it is at its equilibrium populations, with no other species present. The only unknown fact about the outcome in this case is the number ofgenerations that will pass before the system reaches the globally stable equilibrium consisting ofjust the ESNIS. For systems whose dynamics are not given by Lotka-Volterra equations, the existence ofa unique, globally stable ESNIS coalition has not been proved. Moreover, simulations of a non Lotka-Volterra system designed to determine the applicability ofthe proof were inconclusive, due to the ESS coalition not having unique population sizes. Whether or not the proof presented in this work can be extended to non Lotka-Volterra systems remains to be determined. / Thesis (M.Sc.)-University of Natal, Pietermaritzburg, 2000.

An overview of the historical developments of the paradigms of classical mechanics, the free particle, oscillator and the Kepler problem, is given ito (in terms of) their conserved quantities. Next, the orbits of the three paradigms are found from quadratic forms. The quadratic forms are constructed using first integrals found by the application of Poisson's theorem. The orbits are presented ito expanding surfaces defined by the quadratic forms. The Lie and Noether symmetries of the paradigms are investigated. The free particle is discussed in detail and an overview of the work done on the oscillator and Kepler problem is given. The Lie and Noether theories are compared from various aspects. A technical description of Lie groups and algebras is given. This provides a basis for a discussion of the historical development of the paradigms of mechanics ito their group properties. Lastly the paradigms are discussed ito of Quantum Mechanics. / Thesis (M.Sc.)-University of Natal, 1996.

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. / Thesis (M.Sc.)-University of Natal, Pietermaritzburg, 1999.

We wish to apply the newly developed multigrid method [14] to the solution of ODEs resulting from the semi-discretization of time dependent PDEs by the method of lines. In particular, we consider the general form of two important PDE equations occuring in practice, viz. the nonlinear diffusion equation and the telegraph equation. Furthermore, we briefly examine a practical area, viz. atmospheric physics where we feel this method might be of significance. In order to offer the method to a wider range of PC users we present a computer program, called PDEMGS. The purpose of this program is to relieve the user of much of the expensive and time consuming effort involved in the solution of nonlinear PDEs. A wide variety of examples are given to demonstrate the usefulness of the multigrid method and the versatility of PDEMGS. / Thesis (M.Sc.)-University of Natal, Durban, 1992.

Thesis (MComm (Mathematics))--University of Stellenbosch, 2007. / The word numéraire refers to the unit of measurement used to value a portfolio of assets. The change of numéraire technique involves converting from one measurement to another. The foreign exchange markets are natural settings for interpreting this technique (but are by no means the only examples). This dissertation includes elementary facts about the change of numeraire technique. It also discusses the mathematical soundness of the technique in the abstract setting of Delbaen and Schachermayer’s Mathematics of Arbitrage. The technique is then applied to financial pricing problems. The right choice of numéraire could be an elegant approach to solving a pricing problem or could simplify computation and modelling.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2006. / The main topic of this thesis is the construction of the algebraic geometric codes (Goppa codes), and their decoding by the list-decoding, which allows one to correct beyond half of the minimum distance. We also consider the list-decoding of the Reed–Solomon codes as they are subclass of the Goppa codes, and the determination of the parameters of the non primitive BCH codes. AMS Subject Classification: 4B05, 94B15, 94B35, 94B27, 11T71, 94B65,B70. Keywords: Linear codes, cyclic codes, BCH codes, Reed–Solomon codes, list-decoding, Algebraic Geometric codes, decoding, bound on codes, error probability.

Thesis (MSc (Mathemathical Sciences))--Stellenbosch University, 2009. / We have the familiar Kuratowski-Mr owka theorem in topology, where compactness is characterised by a closure and a projection-map (X is compact i p : X Y ! Y is a closed mapping, for any space Y , i.e. p(A) = p(A) A X Y ). Using this as our starting point, we generalise compactness to a categorical setting. We then generalise even further to "asymmetric" compactness. Then we discuss a functional approach to compactness, where we do not explicitly mention closure operators. All this provides economical proofs as well as applications in di erent areas of mathematics.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2008. / Let T be a non-empty set, A a u-algebra of subsets of T and u : .A -+ Rn a bounded, countably additive measure. A set E E A is called an atom with respect to u if u(E)=/F 0 and, if F E A, FeE, then u(F) = u(E) or u(F) = 0; the measure u is atomic if there exists at least one atom (with respect to u) in A. If no such atom (with respect to u) exists in A, then u is called non-atomic. In 1940 the Russian mathematician A. A. Lyapunov published the Convexity Theorem. According to this theorem the range 'R.{u) of a bounded, finite-dimensional measure u is compact and, in the non-atomic case, convex. Since 1940 much has been published on different aspects of the range of a vector-measure. These aspects range from new and shorter proofs of the Convexity Theorem and the usefulness of it in diverse fields, to research about the geometrical characteristics of the range by using other familiar theorems, like Krein-Milman and Radon-Nikodym. In the survey at hand the Convexity Theorem in itself is studied. Applications in different fields will be looked at as well as pieces about the history of the people and the ideas involved in the development of the theorem.

Thesis (MSc (Mathematical Sciences))-- University of Stellenbosch, 2010. / ENGLISH ABSTRACT: We investigate the suitability of applying some of the probabilistic and automata theoretic ideas, that have been extremely successful in the areas of speech and natural language processing, to the area of musical style imitation. By using music written in a certain style as training data, parameters are calculated for (visible and hidden) Markov models (of mixed, higher or first order), in order to capture the musical style of the training data in terms of mathematical models. These models are then used to imitate two instrument music in the trained style. / AFRIKAANSE OPSOMMING: Hierdie tesis ondersoek die toepasbaarheid van probabilitiese en outomaatteoretiese konsepte, wat uiters suksesvol toegepas word in die gebied van spraak en natuurlike taal-verwerking, op die gebied van musiekstyl nabootsing. Deur gebruik te maak van musiek wat geskryf is in ’n gegewe styl as aanleer data, word parameters vir (sigbare en onsigbare) Markov modelle (van gemengde, hoër- of eerste- orde) bereken, ten einde die musiekstyl van die data waarvan geleer is, in terme van wiskundige modelle te beskryf. Hierdie modelle word gebruik om musiek vir twee instrumente te genereer, wat die musiek waaruit geleer is, naboots.