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.

Thesis (MComm)--University of Stellenbosch, 2003. / Please refer to full text for abstract.

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: In this thesis we describe the creation of an embodied conversational agent which exhibits the behavioural traits of a child who has Asperger Syndrome. The agent is rule-based, rather than arti cially intelligent, for which we give justi cation. We then describe the design and implementation of the agent, and pay particular attention to the interaction between emotion, personality and social context. A 3D demonstration program shows the typical output to conform to Asperger-like answers, with corresponding emotional responses. / AFRIKAANSE OPSOMMING: In hierdie tesis beskryf ons die ontwerp en implementasie van 'n gestaltegespreksagent wat die gedrag van 'n kind met Asperger se sindroom uitbeeld. Ons regverdig die besluit dat die agent reël-gebaseerd is, eerder as 'n ware skynintelligensie implementasie. Volgende beskryf ons die wisselwerking tussen emosies, persoonlikheid en sosiale konteks en hoe dit inskakel by die ontwerp en implementasie van die agent. 'n 3D demonstrasieprogram toon tipiese ooreenstemmende Asperger-agtige antwoorde op vrae, met gepaardgaande emosionele reaksies.

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2011. / Includes bibliography / ENGLISH ABSTRACT: The production and abuse of methamphetamine has increased dramatically in South Africa, especially in the Western Cape province. A typical methamphetamine use cycle consists of concealed use after initiation, addiction, treatment and recovery. The model by Nyabadza and Musekwa in [32], is extended to include a core group, fast and slow progression to addiction. The model is analysed analytically and numerically using mass action incidence function and non-linear incidence function. The analysis of the model with mass action incidence is presented in terms of the methamphetamine epidemic threshold R0. The analysis shows that the drug free equilibrium is locally asymptotically stable when R0 < 1 and drug persistent equilibrium is locally asymptotically stable when R0 > 1. The model also exhibits a backward bifurcation. Sensitivity analysis of the model on R0 is performed. The most sensitive parameters are transmission rate and recruitment rate of individuals into the core group. The non-linear incidence incorporates innovators and behaviour change. Analytically, the model is analysed in the absence of behaviour change. With behaviour change two cases were considered. Firstly without innovators and secondly with innovators. In the absence of innovators the non-linear incidence reduced to standard incidence and similar results to the ones in the first model were obtained. With the presence of innovators there is no drug free equilibrium. Numerically we fit the model to data on the number of patients who enter into treatment centers for rehabilitation. Using the fitted model, we determine the prevalence and incidence of methamphetamine abuse. We investigate the impact of behaviour change, ‘reinfection’ rate as well as uptake rate into treatment on prevalence. Our results suggest that intervention and prevention programs focusing on behaviour change and uptake rate into treatment would reduce the prevalence. Projections are made to determine the possible long term trends of the prevalence of methamphetamine abuse in the Western Cape. We give suggestions related to data that should be collected from a modelling perspective. / AFRIKAANSE OPSOMMING: Die vervaardiging en misbruik van metamfetamien het dramaties in Suid-Afrika toegeneem, veral in die Wes-Kaap provinsie. ’n Tipiese metamfetamien gebruiksiklus bestaan uit heimlike gebruik na aanvang, verslawing, behandeling en herstel. Die model deur Nyabadza en Musekwa in [32], is uitgebrei om ’n kerngroep in te sluit, vinnige en stadige verloop tot verslawing. Die model is analities en numeries ontleed deur van massa-aksie insidensie funksie en ’n nie-liniêre insidensie funksie gebruik te maak. Die ontleding van die model met massa-aksie insidensie word voorgestel in terme van die metamfetamien epidemiese drempel R0. Die ontleding toon dat die dwelmvrye ewewig lokaal asimptoties stabiel is as R0 < 1 en die dwelmblydende ewewig is lokaal asimptoties stabiel as R0 > 1. Die model beeld ook ’n terugwaartse bifurkasie uit. Sensitiwiteitsontleding van die model ten opsigte van R0 is uitgevoer. Die mees sensitiewe parameters is die oordraagbaarheidskoers en die rekrute koers van individue in die kerngroep in. Die nuwelinge en gedragsverandering word deur die nie-liniêre insidensie opgeneem. Analities, is die model ontleed in die afwesigheid van gedragsverandering. Met gedragsverandering is twee gevalle beskou. Eerstens sonder nuwelinge en tweedens met nuwelinge. In die afwesigheid van nuwelinge is die nie-liniêre insidensie herlei tot standaard insidensie en soortgelyke resultate is verkry, as dié wat in die eerste model verkry is. Met die aanwesigheid van nuwelinge is daar geen dwelmvrye ewewig nie. Numeries pas ons die model aan die data wat betrekking het met die aantal pasiënte wat in rehabilitasie sentra opgeneem word vir behandeling. Deur die gepaste model te gebruik, het ons die voorkoms en insidensie van metamfetamien misbruik bepaal. Ons ondersoek die impak van gedragsverandering, “re-infeksie” koers sowel as die koers van opname in behandeling op voorkoms. Ons resultate toon dat intervensie- en voorkomingsprogramme sal voorkoms verlaag, wat op die gedragsverandering en die koers van opname in behandeling konsentreer. Die model is ook gebruik om die aantal metamfetamien gebruikers te projekteer. Ons maak voorstelle verwant aan die data, wat vanuit ‘n modellerings-oogpunt ingesamel moet word.

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: See full text / AFRIKAANSE OPSOMMING: Sien volteks

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: In modern theory of finance, the so-called First and Second Fundamental Theorems of Asset Pricing play an important role in pricing options with no-arbitrage. These theorems gives a necessary and sufficient conditions for a market to have no-arbitrage and for a market to be complete. An early version of the First Fundamental Theorem of Asset Pricing was proven by Harrison and Kreps [30] in the case of a finite probability space. A more general version was proven by Harrison and Pliska [31] in the case of a finite probability space and discrete time. In the case of continuous time, Delbaen and Schachermayer [19] introduced a more general concept of no-arbitrage called "No-Free Lunch With Vanishing Risk" (NFLVR), and showed that for a locally-bounded semimartingale price process NFLVR is essentially equivalent to the existence of an equivalent local martingale measure. The goal of this thesis is to review the theory of arbitrage pricing and the extension of this theory to include liquidity risk. At the current time, liquidity risk is a key challenge faced by investors. Consequently there is a need to develop more realistic pricing models that include liquidity risk. We present an approach to liquidity risk by Çetin, Jarrow and Protter [10]. In to this approach the liquidity risk is embedded into the classical theory of arbitrage pricing by having investors act as price takers, and assuming the existence of a supply curve where prices depend on trade size. This framework assumes that the quantity impact on the price transacted is momentary. Using trading strategies that are both continuous and of finite variation allows one to avoid liquidity costs. Therefore, the First and Second Fundamental Theorems of Asset Pricing and the Black-Scholes model can be extended. / AFRIKAANSE OPSOMMING: In moderne finansiële teorie speel die sogenaamde Eerste en Tweede Fundamentele Stellings van Bateprysbepaling ’n belangrike rol in die prysbepaling van opsies in arbitrage-vrye markte. Hierdie stellings gee nodig en voldoende voorwaardes vir ’n mark om vry van arbitrage te wees, en om volledig te wees. ’n Vroeë weergawe van die Eerste Fundamentele Stelling was deur Harrison en Kreps [30] bewys in die geval van ’n eindige waarskynlikheidsruimte. ’n Meer algemene weergawe was daarna gepubliseer deur Harrison en Pliska [31] in die geval van ’n eindige waarskynlikheidsruimte en diskrete tyd. In die geval van kontinue tyd het Delbaen en Schachermayer [19] ’n meer algemene konsep van arbitragevryheid ingelei, naamlik “No–Free–Lunch–With–Vanishing–Risk" (NFLVR), en aangetoon dat vir lokaalbegrensde semimartingaalprysprosesse NFLVR min of meer ekwivalent is aan die bestaan van ’n lokaal martingaalmaat. Die doel van hierdie tesis is om ’n oorsig te gee van beide klassieke arbitrageprysteorie, en ’n uitbreiding daarvan wat likideit in ag neem. Hedendaags is likiditeitsrisiko ’n vooraanstaande uitdaging wat beleggers die hoof moet bied. Gevolglik is dit noodsaaklik om meer realistiese modelle van prysbepaling wat ook likiditeitsrisiko insluit te ontwikkel. Ons bespreek die benadering van Çetin, Jarrow en Protter [10], waar likiditeitsrisiko in die klassieke arbitrageprysteorie ingesluit word deur die bestaan van ’n aanbodkromme aan te neem, waar pryse afhanklik is van handelsgrootte. In hierdie raamwerk word aangeneem dat die impak op die transaksieprys slegs tydelik is. Deur gebruik te maak van handelingsstrategië wat beide kontinu en van eindige variasie is, is dit dan moontlik om likiditeitskoste te vermy. Die Eerste en Tweede Fundamentele Stellings van Bateprysbepaling en die Black–Scholes model kan dus uitgebrei word om likiditeitsrisiko in te sluit.