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.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: We study Monte Carlo methods for pricing American options where the stock price dynamics follow exponential pure jump L évy models. Only stock price dynamics for a single underlying are considered. The thesis begins with a general introduction to American Monte Carlo methods. We then consider two classes of these methods. The fi rst class involves regression - we briefly consider the regression method of Tsitsiklis and Van Roy [2001] and analyse in detail the least squares Monte Carlo method of Longsta and Schwartz [2001]. The variance reduction techniques of Rasmussen [2005] applicable to the least squares Monte Carlo method, are also considered. The stochastic mesh method of Broadie and Glasserman [2004] falls into the second class we study. Furthermore, we consider the dual method, independently studied by Andersen and Broadie [2004], Rogers [2002] and Haugh and Kogan [March 2004] which generates a high bias estimate from a stopping rule. The rules we consider are estimates of the boundary between the continuation and exercise regions of the option. We analyse in detail how to obtain such an estimate in the least squares Monte Carlo and stochastic mesh methods. These models are implemented using both a pseudo-random number generator, and the preferred choice of a quasi-random number generator with bridge sampling. As a base case, these methods are implemented where the stock price process follows geometric Brownian motion. However the focus of the thesis is to implement the Monte Carlo methods for two pure jump L évy models, namely the variance gamma and the normal inverse Gaussian models. We first provide a broad discussion on some of the properties of L évy processes, followed by a study of the variance gamma model of Madan et al. [1998] and the normal inverse Gaussian model of Barndor -Nielsen [1995]. We also provide an implementation of a variation of the calibration procedure of Cont and Tankov [2004b] for these models. We conclude with an analysis of results obtained from pricing American options using these models. / AFRIKAANSE OPSOMMING: Ons bestudeer Monte Carlo metodes wat Amerikaanse opsies, waar die aandeleprys dinamika die patroon van die eksponensiële suiwer sprong L évy modelle volg, prys. Ons neem slegs aandeleprys dinamika vir 'n enkele aandeel in ag. Die tesis begin met 'n algemene inleiding tot Amerikaanse Monte Carlo metodes. Daarna bestudeer ons twee klasse metodes. Die eerste behels regressie - ons bestudeer die regressiemetode van Tsitsiklis and Van Roy [2001] vlugtig en analiseer die least squares Monte Carlo metode van Longsta and Schwartz [2001] in detail. Ons gee ook aandag aan die variansie reduksie tegnieke van Rasmussen [2005] wat van toepassing is op die least squares Monte Carlo metodes. Die stochastic mesh metode van Broadie and Glasserman [2004] val in die tweede klas wat ons onder oë neem. Ons sal ook aandag gee aan die dual metode, wat 'n hoë bias skatting van 'n stop reël skep, en afsonderlik deur Andersen and Broadie [2004], Rogers [2002] and Haugh and Kogan [March 2004] bestudeer is. Die reëls wat ons bestudeer is skattings van die grense tussen die voortsettings- en oefenareas van die opsie. Ons analiseer in detail hoe om so 'n benadering in die least squares Monte Carlo en stochastic mesh metodes te verkry. Hierdie modelle word geï mplementeer deur beide die pseudo kansgetalgenerator en die verkose beste quasi kansgetalgenerator met brug steekproefneming te gebruik. As 'n basisgeval word hierdie metodes geï mplimenteer wanneer die aandeleprysproses 'n geometriese Browniese beweging volg. Die fokus van die tesis is om die Monte Carlo metodes vir twee suiwer sprong L évy modelle, naamlik die variance gamma en die normal inverse Gaussian modelle, te implimenteer. Eers bespreek ons in breë trekke sommige van die eienskappe van L évy prossesse en vervolgens bestudeer ons die variance gamma model soos in Madan et al. [1998] en die normal inverse Gaussian model soos in Barndor -Nielsen [1995]. Ons gee ook 'n implimentering van 'n variasie van die kalibreringsprosedure deur Cont and Tankov [2004b] vir hierdie modelle. Ons sluit af met die resultate wat verkry is, deur Amerikaanse opsies met behulp van hierdie modelle te prys.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Adolescents are regarded as a high risk group in South Africa with the highest human immunodeficiency virus (HIV) incidence occurring in this group. Prevention among adolescents is therefore a key in decreasing the HIV burden. This thesis aims to assist in the design of trials by simulating the potential outcomes of a combination prevention trial in adolescents. We develop a stochastic individual-based model stratified by sex and age. We then use this model to determine the impact of various prevention packages on HIV incidence among adolescents participating in a hypothetical trial over a three year period. The trial that is simulated involves an intervention arm, in which adolescents are offered a choice of a prevention methods (including medical male circumcision (MMC), oral pre-exposure prophylaxis (PrEP) and antiretroviral-based vaginal microbicides (ARV-VM)), and a control arm. We predict that the impact of a full prevention package on HIV incidence would be a 46% per personyear( PPY) (95% CI 45–47%) risk reduction. The combination of MMC and PrEP has a substantial impact on HIV incidence in males, with a 51% PPY (95% CI 49–53%) relative risk of HIV infection. Offering women the choice of PrEP, a microbicide gel or a microbicide in the form of a vaginal ring would be less effective, with a 57% PPY (95% CI 56–58%) relative risk of HIV acquisition. This is not substantially different from the relative risk estimated when the vaginal ring alone is offered, as the ring is assumed to be the most accept able of the three prevention methods. We determine a sample size requirement of approximately 1013 in each arm of a trial would achieve 80% power to detect a statistically significant reduction in HIV risk. We find that the relative risk is sensitive to the assumed degree of correlation between condom use and the acceptability of the prevention method. We also find that the most efficient trial design may be to offer both MMC and PrEP to males but to offer only a microbicide ring to females. Further work is required to better understand the processes by which adolescent prevention method choices are made. / AFRIKAANSE OPSOMMING: Adolessente word beskou as ‘n hoe risiko groep in Suid Afrika, met die hoogste menslike immuniteitsgebrekvirus (MIV) insidensie in hierdie groep. Voorkoming van MIV onder adolessente is daarom noodsaaklik om die MIV las te verminder. Die doel van hierdie tesis is om te help met die ontwerp van studies deur die moontlike uitkomste van ‘n kombinasie-voorkoming studie in adolessente te simuleer. Ons het ‘n stogastiese individu-gebaseerde model, gestratifiseer met betrekking tot seks en ouderdom, ontwikkel. Ons het toe die model gebruik om die impak van ‘n verskeinheid van voorkomingspakette op MIV insidensie onder adolessente wat deelneem aan ‘n hipotetiese proef oor ‘n drie jaar periode, te bepaal. Die proef wat gesimuleer word behels a intervensie groep, waarin die jong volwassenes ‘n keuse van voorbehoedings metodes (insluitende mediese manlike besnydenis (MMB), pre-blootstelling profilakse (PrBP) en anti-retrovirale vaginale mikrobisiedes (ARV-VM)) aangebied word, en ‘n kontrole groep. Ons voorspel dat die impak van ‘n volle voorkomingspaket op MIV insidensie ‘n 46% per persoon-jaar (PPJ) (95% VI 47–47%) risiko vermindering sal wees. Die kombinasie van MMB en PrBP het ‘n substansiele impak op MIV insidensie onder mans, met ‘n relatiewe risiko van MIV infeksie van 51% PPJ (95% VI 49–53%). Om die keuse van PrBP, ‘n mikrobisiede gel of ‘n mikrobisiede in die vorm van ‘n vaginale ring aan vrouens te bied, is minder effektief, met ‘n relatiewe risiko van MIV infeksie van 57% PPJ (95% VI 56%–58%). Hierdie verskil nie substansieel van die beraamde relatiewe risiko in die geval waar slegs die vaginale ring gebied word nie, aangesien daar aanvaar word dat die ring die mees aanvaarde van die drie voorkomingsmetodes is. Ons het bepaal dat ‘n steekproef van ongeveer 1013 individue in elke arm van die proef nodig is om ‘n 80% kans te he om ‘n statisties betekenisvolle afname in MIV-risiko te bespeur. Ons vind dat die relatiewe risiko sensitief is tot die aanvaarde graad van die korrelasies tussen kondoom-gebruik en die aanvaarding van die voorkomings metodes. Ons het ook gevind dat dit mag wees dat die mees doeltreffende proef ontwerp is om beide MMB en PrBP vir mans en slegs ‘n mikrobisiede ring vir vrouens te bied. Verdere werk word benodig om die prosesse waarby jong volwassenes keuses maak oor voorkomingsmetodes te verstaan.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: In recent years, HIV-associated cancers have proven to be the bane of our time, since HIV is decimating humanity across the globe, even in the twilight of the last century. Cancer rates continue to rise in developing countries, where 95% of the world’s HIV-infected population lives, yet less than 1% have access to antiretroviral therapy. HIV-infected individuals have a higher proclivity to develop cancers, mainly from immunosuppression. An understanding of the immunopathogenesis of HIV-related cancers (HRC) is therefore a major prerequisite for rationally developing and/or improving therapeutic strategies, developing immunotherapeutics and proplylatic vaccines. In this study, we explore the pathology of HIV-related cancer malignancies, taking into account the pathogenic mechanisms and their potential for improving the treatment of management of these malignancies especially in developing countries. We mathematically model the dynamics of malignant tumors in an HIV-free environment, investigate the impact of cancer malignancies on HIV-positive patients and explore the benefits of various therapeutic intervention strategies in the management of HIV-related cancers. We present two deterministic models of infectious diseases to implement these, and they were analysed. We use HIV-related lymphomas in the Western Cape of South Africa as a case study. We validated the proposed models using lymphoma incidence data from the Tygerberg Lymphoma Study Group (TLSG), Tygerberg Hospital, Western Cape, South Africa. We show that the increasing prevalence of HIV increases lymphoma cases, and thus, other HIV-related cancers. Our models also suggests that an increase in the roll-out of the HAART program can reduce the number of lymphoma cases in the nearest future, while it averts many deaths. Furthermore, the results indicate that a highly crucial factor to consider in the prognosis of the incidence of lymphoma (and other cancer types) in HIV-infected patients is their CD4 cell count, irrespective of whether the patient has developed an HRC or not.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: In this thesis we develop a self-dual categorical approach to some topics in non-abelian algebra, which is based on replacing the framework of a category with that of a category equipped with a functor to it. We also make some first steps towards a possible link between this theory and the theory of databases in computer science. Both of these theories are based around the study of Grothendieck bifibrations and their generalisations. The main results in this thesis concern correspondences between certain structures on a category which are relevant to the study of categories of non-abelian group-like structures, and functors over that category. An investigation of these correspondences leads to a system of dual axioms on a functor, which can be considered as a solution to the proposal of Mac Lane in his 1950 paper "Duality for Groups" that a self-dual setting for formulating and proving results for groups be found. The part of the thesis concerned with the theory of databases is based on a recent approach by Johnson and Rosebrugh to views of databases and the view update problem. / AFRIKAANSE OPSOMMING: In hierdie tesis word ’n self-duale kategoriese benadering tot verskeie onderwerpe in nie-abelse algebra ontwikkel, wat gebaseer is op die vervanging van die raamwerk van ’n kategorie met dié van ’n kategorie saam met ’n funktor tot die kategorie. Ons neem ook enkele eerste stappe in die rigting van ’n skakel tussen hierdie teorie and die teorie van databasisse in rekenaarwetenskap. Beide hierdie teorieë is gebaseer op die studie van Grothendieck bifibrasies en hul veralgemenings. Die hoof resultate in hierdie tesis het betrekking tot ooreenkomste tussen sekere strukture op ’n kategorie wat relevant tot die studie van nie-abelse groep-agtige strukture is, en funktore oor daardie kategorie. ’n Verdere ondersoek van hierdie ooreemkomste lei tot ’n sisteem van duale aksiomas op ’n funktor, wat beskou kan word as ’n oplossing tot die voorstel van Mac Lane in sy 1950 artikel “Duality for Groups” dat ’n self-duale konteks gevind word waarin resultate vir groepe geformuleer en bewys kan word. Die deel van hierdie tesis wat met die teorie van databasisse te doen het is gebaseer op ’n onlangse benadering deur Johnson en Rosebrugh tot aansigte van databasisse en die opdatering van hierdie aansigte.

Thesis (MSc (Mathematical Sciences))--Stellenbosch University, 2008. / Class field theory describes the abelian extensions of a given field K in terms of various class groups of K, and can be viewed as one of the great successes of 20th century number theory. However, the main results in class field theory are pure existence results, and do not give explicit constructions of these abelian extensions. Such explicit constructions are possible for a variety of special cases, such as for the field Q of rational numbers, or for quadratic imaginary fields. When K is a global function field, however, there is a completely explicit description of the abelian extensions of K, utilising the theory of sign-normalised Drinfeld modules of rank one. In this thesis we give detailed survey of explicit class field theory for rational function fields over finite fields, and of the fundamental results needed to master this topic.