The role of the scientist in the courtroom has come under more scrutiny this century than ever before. As a consequence, scientists must constantly look for ways to improve the validity of the evidence they deliver. It is here that the professional statistician can provide assistance. The use of statistics in the courtroom and in forensic science is not new, but until recently has not been common either. Statistics can provide objectivity to subjective assessments and strengthen a case for the prosecution or the defence, but only if is used correctly. The aim of this thesis is to enhance and replace the existing technology used in statistical analysis and presentation of trace evidence, i.e. all non-genetic evidence (hairs, fibres, glass, paint, etc.) and transfer problems.

Given a linear regression model y = Xβ + e, where e has a multivariate normal distribution N(0, Σ) consequences of the erroneous assumption that e is distributed as N(0, I) are considered. For a general linear hypothesis concerning the parameters β, in a general model the distribution of the statistic to test the hypothesis, derived under the erroneous assumption is studied. Particular linear hypotheses concerning particular linear models are investigated so as to describe the effects of various patterns of serial correlation on the test statistics arising from these hypotheses. Attention is specially paid to the models of one- and two- way analysis of variance.

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2011. / Thesis presented in fulfilment of the adacemic requirements for the degree of Master of Science at the University of Stellenbosch. / Please refer to full text to view abstract.

Thesis (MSc.)--University of Stellenbosch, 2000. / ENGLISH ABSTRACT: The notion of a classical graph theoretic Ramsey number is generalized by assuming that both the original graph whose edges are arbitrarily bicoloured and the monochromatic subgraphs to be forced are complete, balanced, multipartite graphs, instead of complete graphs as in the standard definition. Some small multipartite Ramsey numbers are found, while upper- and lower bounds are established for others. Analytic arguments as well as computer searches are used. / AFRIKAANSE OPSOMMING: Die klassieke grafiek-teoretiese definisie van ’n Ramsey getal word veralgemeen deur te aanvaar dat beide die oorspronklike grafiek, waarvan die lyne willekeurig met twee kleure gekleur word en die gesogte subgrafieke almal volledige, gebalanseerde, veelledige grafieke is, anders as in die standaard definisie. Klein veelledige Ramsey getalle word gevind, terwyl bo- en ondergrense vir ander daargestel word. Analitiese argumente en rekenaarsoektogte word gebruik.

Thesis (MComm)--Stellenbosch University, 2001. / ENGLISH ABSTRACT: This thesis describes the design and implementation of a data storage service. Many clients can be served simultaneously in an environment where processes execute on different physical machines and communicate via message passing primitives. The service is provided by two separate servers: one that functions at the disk block level and another that maintains files. A prototype system was developed first in the form of a simple file store. The prototype served two purposes: (1) it extended the single-user Oberon system to create a multiuser system suitable to support group work in laboratories, and (2) it provided a system that could be measured to obtain useful data to design the final system. Clients access the service from Oberon workstations. The Oberon file system (known as the Ceres file system) normally stores files on a local disk. This system was modified to store files on a remote Unix machine. Heavily used files are cached to improve the efficiency of the system. In the final version of the system disk blocks are cached, not entire files. In this way the disks used to store the data are unified and presented as a separate virtual block service to be used by file systems running on client workstations. The virtual block server runs on a separate machine and is accessed via a network. The simplicity of the block server is appealing and should in itself improve reliability. The main concern is efficiency and the goal of the project was to determine whether such a design can be made efficient enough to serve its purpose. / AFRIKAANSE OPSOMMING:Hierdie tesis omskryf die ontwerp en implementasie van 'n data stoor diens. Verskeie gebruikers word bedien deur die diens wat funksioneer in 'n verspreide omgewing: 'n omgewing waar prosesse uitvoer op verskillende masjiene en met mekaar kommunikeer met behulp van boodskappe wat rondgestuur word. Die diens word verskaf deur twee bedieners: die eerste wat funksioneer op 'n blok vlak en die ander wat lers onderhou. 'n Prototipe leer diens is ontwikkel deur middel van 'n basiese leer stoor. Die prototipe het twee funksies verrig: (1) die enkel gebruiker Oberon stelsel is uitgebrei na 'n veelvoudige gebruiker stelsel bruikbaar vir groepwerk in 'n laboratorium omgewing, en (2) 'n stelsel is verskaf wat betroubare en akkurate data kon verskaf vir die ontwerp van die finale stelsel. Oberon werkstasies word gebruik met die leer diens. Die Oberon leer stelsel (ook bekend as die Ceres leer stelsel) stoor normaalweg leers op 'n lokale skyf. Hierdie bestaande stelsel is verander om leers te stoor op 'n eksterne Unix masjien. Leers wat die meeste in gebruik is word in geheue aangehou vir effektiwiteits redes. Die finale weergawe van die stelsel berg skyf blokke in geheue, nie leers nie. Hierdie metode laat dit toe om data te stoor op 'n standaard metode, bruikbaar deur verskillende tipes leer stelsels wat uitvoer op verskeie gebruikers se werkstasies. Die virtuele blok stoor voer uit op 'n aparte masjien en is bereikbaar via 'n netwerk. Die eenvoudige ontwerp van die diens is opsigself aanloklik en behoort betroubaarheid te verbeter. Die hoof bekommernis is effektiwiteit en die hoofdoel van die projek was om te bepaal of hierdie ontwerp effektief genoeg gemaak kon word.

Thesis (MSc)--Stellenbosch University , 1993. / ENGLISH ABSTRACT: Reactive systems are event driven state machines which usually do not terminate, but remain in perpetual interaction with their environment. Such systems usually interact 'With devices which introduce a high degree of concurrency and some real time constraints to the system. Because of the concurrent nature of reactive systems they are commonly implemented as communicating concurrent processes on one or more processors. Jeffay introduces a design paradigm which requires consumer processes to consume messages faster than they are produced by producer processes. If this is guaranteed, the real time constraints of such .. system are always met, and the correctness of the process interaction is guaranteed in terms of the message passing semantics. I developed the ESE kernel, which supports Jeffay systems by providing lightweight processes which communicate over asynchronous channels. Processes are scheduled non-preemptively according to the earliest deadline first policy when they have messages pending on their input channels. The Jeffay design method and the ESE kernel have been found to be highly suitable to implement embedded reactive systems. The general requirements of embedded reactive systems, and kernel support required by such systems, are discussed. / AFRIKAANSE OPSOMMING: Reaktiewe stelsels is toeatandsoutomate wat aangedryf word deur gebeure in hul omgewins. So 'n stelsel termineer gewoonlik nie, maar bly in 'n voortdurende wisselwerking met toestelle in sy omgewing. Toestelle in die omgewing van 'n reaktiewe stelsel veroorsaak in die algemeen 'n hoë mate van gelyklopendheid in die stelsel, en plaas gewoonlik sekere intydse beperkings op die stelsel. Gelyklopende stelsels word gewoonlik as stelsel. van kommunikerende prosesse geïmplementeer op een of meer prosessors. Jeffay beskryf 'n ontwerpsmetodologie waarvolgens die ontvanger van boodskappe hulle vinniger moet verwerk as wat die sender hulle kan stuur. Indien hierdie gedrag tussen alle pare kommunikerende prosesse gewaarborg kan word, sal die stelsel altyd sy intydse beperkings gehoorsaam, en word die korrektheid van interaksies tussen prosesse deur die semantiek van die boodskapwisseling gewaarborg. Die "ESE" bedryfstelselkern wat ek ontwikkel het, ondersteun stelsels wat ontwerp en geïmplementeer word volgens Jeffay se metode. Prosesse kommunikeer oor asinkrone kanale, en die ontvanger van die boodskap met die vroegste keertyd word altyd eerste geskeduleer. Jeffay se ontwerpsmetode en die "ESE" kern blyk in die praktyk baie geskik te wees vir reaktiewe stelsels wat as substelsels van groter stelsels uitvoer. Die vereistes van reaktiewe substelsels, en die kemondersteuning wat daarvoor nodig is, word bespreek.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: The central objective of our study focuses on torsion points on elliptic curves. The case of elliptic curves over finite fields is explored up to giving explicit formulae for the cardinality of the set of points on such curves. For finitely generated fields of characteristic zero, a presentation and discussion of some known results is made. Some applications of elliptic curves are provided. In one particular case of applications, we implement an integer factorization algorithm in a computer algebra system SAGE based on Lenstra’s elliptic curve factorisation method. / AFRIKAANSE OPSOMMING: Die hoofdoel van ons studie is torsiepunte op elliptiese krommes. Ons ondersoek die geval van elliptiese krommes oor ‘n eindige liggaam met die doel om eksplisiete formules vir die aantal punte op sulke krommes te gee. Vir ‘n eindig-voortgebringde liggaam met karakteristiek nul bespreek ons sekere bekende resultate. Sommige toepassings van elliptiese krommes word gegee. In een van hierdie toepassings implementeer ons ‘n heeltallige faktoriseringalgoritme in die rekenaar-algebrastelsel SAGE gebaseer op Lenstra se elliptiese krommefaktoriseeringmetode.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: See the full text for the abstract / AFRIKAANSE OPSOMMING: Sien volteks vir die opsomming

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: See the full text for the abstract / AFRIKAANSE OPSOMMING: Sien die volteks vir die opsomming

This thesis presents mathematical models for the dynamics of vaccine preventable diseases, specifically looking at the New Zealand situation. Through the use of integral and differential equations, we develop models and compare the results of these to known data. Using game theory analysis we determine and compare the proportion of the population that needs to be vaccinated in order to minimise the expected costs to the individuals in the population and to the community. Two different scenarios and methods are considered, where the effects of vaccination last only one epidemic cycle (using an integral equation method) and where vaccination is effective over an entire lifetime (using a differential equation method). For both scenarios, we find that the minimum cost for the individuals is reached when a lower proportion of the population is vaccinated than needed for the minimum cost to the community. We then elaborate on the integral equation method to produce a model for repeated epidemics of measles in a population, where a discrete mapping is used to include the year to year demographics of the population. The results of this model show a different epidemic pattern then that produced from a differential equation model, with numerical problems encountered. From here on, we use differential equation models in our analysis. A critique and extension to an existing model for the dynamics of the hepatitis B virus is presented, with discussion on the appropriateness of the model’s construct for predicting the incidence of infection. Alternative differential equation models for hepatitis B virus and immunisation that include splitting the population into age groups with nonhomogeneous mixing are presented. The results of these models are compared with the known data on incidence of infection and carriage in New Zealand, showing how affective different immunisation schedules may have been. Differential equation models are then presented for meningococcal B virus epidemiology in New Zealand, with the models incorporating different features of the virus until the best model is found that fits the New Zealand data. Each model is compared with the known incidence of infection, with the population being either treated as a whole or split into age groups with non-homogeneous mixing. The effect of vaccination is included in this model so that we can explore the future of the infection in the population, and how best to tackle any future epidemics. The model shows that the current vaccination campaign was the best solution for controlling the epidemic, but there will be epidemics in the future that will need subsequent vaccination campaigns to limit the number of infections.