Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: The aim of this thesis is to study some methods used in solving exponential Diophan- tine equations. There is no generic method or algorithm that can be used in solving all Diophantine equations. The main focus for our study will be solving the exponential Dio- phantine equations using the modular approach and the linear forms in two logarithms approach. / AFRIKAANSE OPSOMMING: Die doel van hierdie tesis is om sommige metodes te bestudeer om sekere Diophantiese vergelykings op te los. Daar is geen metode wat alle Diophantiese vergelykings kan oplos nie. Die fokus van os studie is hoofsaaklik om eksponensiele Diophantiese vergelykings op te los met die modul^ere metode en met die metode van line^ere vorms in twee logaritmes.

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

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Symbolic execution is a well-established technique for automated test generation and for nding errors in complex code. Most of the focus has however been on programs that manipulate integers, booleans, and even, references in object-oriented programs. Recently researchers have started looking at programs that do lots of string processing, motivated, in part, by the popularity of the web and the risk that errors in web servers may lead to security violations. Attempts to extend symbolic execution to the domain of strings are mainly divided into one of two camps: automata-based approaches and approaches based on bitvector analysis. Here we investigate these two approaches in a uni ed setting, namely the symbolic execution framework of Java PathFinder. We describe the implementations of both approaches and then do an evaluation to show under what circumstances each approach performs well (or not so well). We also illustrate the usefulness of the symbolic execution of strings by nding errors in real-world examples. / AFRIKAANSE OPSOMMING: Simboliese uitvoering is 'n bekende tegniek vir automatiese genereering van toetse en om foute te vind in ingewikkelde bronkode. Die fokus sover was grotendeels op programme wat gebruik maak van heelgetalle, boolse waardes en selfs verwysings in objek geörienteerde programme. Navorsers het onlangs begin kyk na programme wat baie gebruik maak van string prosessering, deelteliks gemotiveerd deur die populariteit van die web en die gepaardgaande risiko's daarvan. Vorige implementasies van simboliese string uitvoering word binne twee kampe verdeel: die automata gebaseerde benadering en bitvektoor gebaseerde benadering. Binne hierdie tesis word die twee benaderings onder een dak gebring, naamliks Java PathFinder. Die implentasie van beide benaderings word bespreek en ge-evalueer om die omstandighede uit te wys waarbinne elk beter sou vaar. Die nut van simboliese string uitvoering word geïllustreer deur dit toe te pas in foutiewe regte wêreld voorbeelde.

Thesis (MSc)--Stellenbosch University, 2002. / ENGLISH ABSTRACT: Multiprotocol Label Switching (MPLS) is a routing technology which can manage Quality of Service (QoS) in scalable connectionless networks using relatively simple packet forwarding mechanisms. This thesis considers the optimisation of the QoS offered by an MPLS network. The QoS measure used is the expected packet delay which is minimised by switching packets along optimal label switched paths (LSPs). Two mathematical models of MPLS networks are presented together with appropriate algorithms for optimally dividing the network traffic into forwarding equivalence classes (FECs), finding optimal LSPs which minimise the expected packet delay and switching these FECs along the optimal LSPs. These algorithms are applied to compute optimal LSPs for several test networks. The mathematics on which these algorithms are based is also reviewed. This thesis provides the MPLS network operator with efficient packet routing algorithms for optimising the network's QoS. / AFRIKAANSE OPSOMMING: Multiprotocol Label Switching (MPLS) is 'n roeteringsmetode om die diensvlak (QoS) van 'n skaleerbare, verbindinglose netwerk te bestuur deur middel van relatief eenvoudige versendingsmeganismes. Hierdie tesis beskou die optimering van die QoS van 'n MPLS-netwerk. Die QoS-maatstaf is die verwagte vert raging van 'n netwerk-pakkie. Dit word geminimeer deur pakkies langs optimale "label switched paths" (LSPs) te stuur. Twee wiskundige modelle van MPLS-netwerke word ondersoek. Toepaslike algoritmes word verskaf vir die optimale verdeling van die netwerkverkeer in "forwarding equivalence classes" (FECs), die soektog na optimale LSPs (wat die verwagte pakkie-vertraging minimeer) en die stuur van die FECs langs die optimale LSPs. Hierdie algoritmes word ingespan om optimale LSPs vir verskeie toetsnetwerke op te stel. Die wiskundige teorie waarop hierdie algoritmes gegrond is, word ook hersien. Hierdie tesis verskaf doeltreffende roeteringsalgoritmes waarmee 'n MPLS-netwerkbestuurderj-es die netwerk se QoS kan optimeer.

Thesis (MSc)--Stellenbosch University , 2002. / ENGLISH ABSTRACT: Grammars with regulated rewriting are used to restrict the application of contextfree productions in order to avoid certain derivations. This enables these grammars to generate both context-free and non-context-free languages using only production rules with a context-free format. These grammars are more powerful than contextfree grammars, but usually not as powerful as context-sensitive grammars. Various grammars with regulated rewriting have been developed and some will be discussed in this thesis. Propositional linear temporal logic is a formal system used to describe truth values of propositions over time. This is done by defining a timeline together with a set of propositions. It is then possible to construct temporal logic formulae, consisting of these propositions and temporal operators, to specify the truth values of the propositions for every step in the timeline. In this thesis we define and discuss temporal grammars that combine grammars with propositionallinear temporal logic. Since a derivation can be associated with a timeline, a regulating device can be constructed from temporal logic formulae, that will control the application of productions within the derivation. The discussion on temporal grammars includes some of the properties of these grammars, while many ideas are illustrated by examples. / AFRIKAANSE OPSOMMING: Grammatikas met gereguleerde herskrywing word gebruik om 'n beperking te plaas op die toepassing van konteksvrye produksies en verhoed sodoende sekere afleidings. Hierdie grammatikas beskik oor die vermoe om beide konteksvrye en nie-konteksvrye tale te genereer deur slegs produksiereels van 'n konteksvrye formaat te gebruik. Grammatikas met gereguleerde herskrywing is dus sterker as konteksvrye grammatikas, alhoewel dit soms swakker as konteks-sensitiewe grammatikas is. 'n Verskeidenheid sulke grammatikas is al ontwikkel en sommige sal in hierdie tesis bespreek word. Proposisionele lineere temporale logika is 'n formele stelsel wat gebruik kan word om die waarheidswaardes van proposisies oor tyd te beskryf. Dit word gedoen deur 'n tydlyn, asook 'n versameling proposisies te definieer. Dit is clan moontlik om temporale operatore tesame met die proposisies te gebruik om temporale logika-formules te konstrueer wat in staat is om waarheidswaardes van die proposisies te spesifiseer vir elke oomblik in die tydlyn. In hierdie tesis word temporale grammatikas, wat grammatikas met proposisionele lineere temporale logika kombineer, gedefinieer en bespreek. Aangesien 'n afleiding met 'n tydlyn geassosieer kan word, is dit moontlik om 'n regulerende meganisme uit temporale logika-formules te konstrueer wat die toepassing van produksiereels in die afleiding kontroleer. Die bespreking van temporale grammatikas sluit 'n verskeidenheid eienskappe van die grammatikas in, asook 'n aantal voorbeelde wat ter illustrasie gebruik word.