Thesis (MComm)--Stellenbosch University, 2003. / ENGLISH ABSTRACT: In this dissertation I investigate the class number one problem in function fields. More precisely I give a survey of the current state of research into extensions of a rational function field over a finite field with principal ring of integers. I focus particularly on the quadratic case and throughout draw analogies and motivations from the classical number field situation. It was the "Prince of Mathematicians" C.F. Gauss who first undertook an in depth study of quadratic extensions of the rational numbers and the corresponding rings of integers. More recently however work has been done in the situation of function fields in which the arithmetic is very similar. I begin with an introduction into the arithmetic in function fields over a finite field and prove the analogies of many of the classical results. I then proceed to demonstrate how the algebra and arithmetic in function fields can be interpreted geometrically in terms of curves and introduce the associated geometric language. After presenting some conjectures, I proceed to give a survey of known results in the situation of quadratic function fields. I present also a few results of my own in this section. Lastly I state some recent results regarding arbitrary extensions of a rational function field with principal ring of integers and give some heuristic results regarding class groups in function fields. / AFRIKAANSE OPSOMMING: In hierdie tesis ondersoek ek die klasgetal een probleem in funksieliggame. Meer spesifiek ondersoek ek die huidige staat van navorsing aangaande uitbreidings van 'n rasionale funksieliggaam oor 'n eindige liggaam sodat die ring van heelgetalle 'n hoofidealgebied is. Ek kyk in besonder na die kwadratiese geval, en deurgaans verwys ek na die analoog in die klassieke getalleliggaam situasie. Dit was die beroemde wiskundige C.F. Gauss wat eerste kwadratiese uitbreidings van die rasionale getalle en die ooreenstemende ring van heelgetalle in diepte ondersoek het. Onlangs het wiskundiges hierdie probleme ook ondersoek in die situasie van funksieliggame oor 'n eindige liggaam waar die algebraïese struktuur baie soortgelyk is. Ek begin met 'n inleiding tot die rekenkunde in funksieliggame oor 'n eindige liggaam en bewys die analogie van baie van die klassieke resultate. Dan verduidelik ek hoe die algebra in funksieliggame geometries beskou kan word in terme van kurwes en gee 'n kort inleiding tot die geometriese taal. Nadat ek 'n paar vermoedes bespreek, gee ek 'n oorsig van wat alreeds vir quadratiese funksieliggame bewys is. In hierdie afdeling word 'n paar resultate van my eie ook bewys. Dan vermeld ek 'n paar resultate aangaande algemene uitbreidings van 'n rasionale funksieliggaam oor 'n eindige liggaam waar die van ring heelgetalle 'n hoofidealgebied is. Laastens verwys ek na 'n paar heurisitiese resultate aangaande klasgroepe in funksieliggame.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: Age and temperature are two factors that affect mortality in adult tsetse flies. Both are found to be very important, but the simultaneous effect of these factors on the mortality rate have not been studied. This study seeks to address this, with an application to a population of female tsetse, using a model based on partial differential equations. Adult mortality is agedependent and is modelled as the sum of two exponentials, with four parameters (coefficients of each exponential): numerical analysis of a population model with this mortality structure predicts exponential growth. Analysis of each of the parameters through parameter variation shows that two of these parameters control the mortality of the nulliparous (ages 0 − 10 days) flies only while the other two only take care of flies of mature ages. Measurement of the impact of these parameters on the mortality of tsetse of different ages by the normalized forward sensitivity index method is also carried out. This is followed by fitting the model based on the age-dependent mortality along with a constant tsetse birth rate to data representing the catches of female Glossina pallidipes at Rekomitjie Research station, Zimbabwe. Considering a three parameter adult tsetse mortality, parameter analysis shows the effect of one of the parameters to affect the mortality of flies of all ages while a second controls only the mature tsetse flies of reproductive ages. A further analysis resulted in the estimate of these parameters as functions of temperature, thereby leading to the establishment of an age and temperature-dependent adult tsetse mortality. Using data for the daily average temperature records obtained in 1981 on Antelope Island, Lake Kariba, Zimbabwe, daily changes in the pupal duration (adult tsetse birth rate) changes negatively with temperature change. Incorporating this (temperature-dependent ) birth rate into the model, together with the established age and temperature-dependent adult mortality, the adult tsetse population dynamics is explored numerically. The latter model is then fitted to population data of female Glossina morsitans morsitans obtained from the same Island and for the same period as used for the temperature data. The data suggests peak tsetse population to be in the month of July and lowest in the month of December. The first quarter of the year is predicted to be most favorable for breeding tsetse while the second, showed a period of stable growth rate and a time of tsetse abundance. In addition, the dynamics with both age and temperature showed a non-uniform daily population growth contrary to that with age effect only. This study has enhanced our understanding of tsetse population dynamics for age and temperature-dependent adult mortality with temperature-dependent pupal duration and suggests the period of tsetse abundance on Antelope Island. / AFRIKAANSE OPSOMMING: Geen opsomming in Afrikaans.

Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007. / Explicit towers of algebraic function fields over finite fields are studied by considering their ramification behaviour and complete splitting. While the majority of towers in the literature are recursively defined by a single defining equation in variable separated form at each step, we consider towers which may have different defining equations at each step and with arbitrary defining polynomials. The ramification and completely splitting loci are analysed by directed graphs with irreducible polynomials as vertices. Algorithms are exhibited to construct these graphs in the case of n-step and -finite towers. These techniques are applied to find new tamely ramified n-step towers for 1 n 3. Various new tame towers are found, including a family of towers of cubic extensions for which numerical evidence suggests that it is asymptotically optimal over the finite field with p2 elements for each prime p 5. Families of wildly ramified Artin-Schreier towers over small finite fields which are candidates to be asymptotically good are also considered using our method.

Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2005. / A great deal of work has been done in the area of off-line signature verification over the past two decades. Off-line systems are of interest in scenarios where only hard copies of signatures are available, especially where a large number of documents need to be authenticated. This dissertation is inspired by, amongst other things, the potential financial benefits that the automatic clearing of cheques will have for the banking industry.

Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007. / We study refinable functions where the dilation factor is not always assumed to be 2. In our investigation, the role of convolutions and refinable step functions is emphasized as a framework for understanding various previously published results. Of particular importance is a class of polynomial factors, which was first introduced for dilation factor 2 by Berg and Plonka and which we generalise to general integer dilation factors. We obtain results on the existence of refinable functions corresponding to certain reduced masks which generalise similar results for dilation factor 2, where our proofs do not rely on Fourier methods as those in the existing literature do. We also consider subdivision for general integer dilation factors. In this regard, we extend previous results of De Villiers on refinable function existence and subdivision convergence in the case of positive masks from dilation factor 2 to general integer dilation factors. We also obtain results on the preservation of subdivision convergence, as well as on the convergence rate of the subdivision algorithm, when generalised Berg-Plonka polynomial factors are added to the mask symbol. We obtain sufficient conditions for the occurrence of polynomial sections in refinable functions and construct families of related refinable functions. We also obtain results on the regularity of a refinable function in terms of the mask symbol factorisation. In this regard, we obtain much more general sufficient conditions than those previously published, while for dilation factor 2, we obtain a characterisation of refinable functions with a given number of continuous derivatives. We also study the phenomenon of subsequence convergence in subdivision, which explains some of the behaviour that we observed in non-convergent subdivision processes during numerical experimentation. Here we are able to establish different sets of sufficient conditions for this to occur, with some results similar to standard subdivision convergence, e.g. that the limit function is refinable. These results provide generalisations of the corresponding results for subdivision, since subsequence convergence is a generalisation of subdivision convergence. The nature of this phenomenon is such that the standard subdivision algorithm can be extended in a trivial manner to allow it to work in instances where it previously failed. Lastly, we show how, for masks of length 3, explicit formulas for refinable functions can be used to calculate the exact values of the refinable function at rational points. Various examples with accompanying figures are given throughout the text to illustrate our results.

Dissertation (PhD)--University of Stellenbosch, 2007. / ENGLISH ABSTRACT: See full text for abstract / AFRIKAANSE OPSOMMING: Sien volteks vir opsomming

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

Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Various properties of integer partitions are studied in this work, in particular the number of summands, the number of ascents and the multiplicities of parts. We work on random partitions, where all partitions from a certain family are equally likely, and determine moments and limiting distributions of the different parameters. The thesis focuses on three main problems: the first of these problems is concerned with the length of prime partitions (i.e., partitions whose parts are all prime numbers), in particular restricted partitions (i.e., partitions where all parts are distinct). We prove a central limit theorem for this parameter and obtain very precise asymptotic formulas for the mean and variance. The second main focus is on the distribution of the number of parts of a given multiplicity, where we obtain a very interesting phase transition from a Gaussian distribution to a Poisson distribution and further to a degenerate distribution, not only in the classical case, but in the more general context of ⋋-partitions: partitions where all the summands have to be elements of a given sequence ⋋ of integers. Finally, we look into another phase transition from restricted to unrestricted partitions (and from Gaussian to Gumbel-distribution) as we study the number of summands in partitions with bounded multiplicities. / AFRIKAANSE OPSOMMING: Verskillende eienskappe van heelgetal-partisies word in hierdie tesis bestudeer, in die besonder die aantal terme, die aantal stygings en die veelvoudighede van terme. Ons werk met stogastiese partisies, waar al die partisies in ’n sekere familie ewekansig is, en ons bepaal momente en limietverdelings van die verskillende parameters. Die teses fokusseer op drie hoofprobleme: die eerste van hierdie probleme gaan oor die lengte van priemgetal-partisies (d.w.s., partisies waar al die terme priemgetalle is), in die besonder beperkte partisies (d.w.s., partisies waar al die terme verskillend is). Ons bewys ’n sentrale limietstelling vir hierdie parameter en verkry baie presiese asimptotiese formules vir die gemiddelde en die variansie. Die tweede hooffokus is op die verdeling van die aantal terme van ’n gegewe veelvoudigheid, waar ons ’n baie interessante fase-oorgang van ’n normaalverdeling na ’n Poisson-verdeling en verder na ’n ontaarde verdeling verkry, nie net in die klassieke geval nie, maar ook in die meer algemene konteks van sogenaamde ⋋-partities: partisies waar al die terme elemente van ’n gegewe ry ⋋ van heelgetalle moet wees.

Thesis (PhD (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Classical formulations model belief revision as a deterministic process. Under certain circumstances, the process may have more than one outcome, which suggests that belief revision is non-deterministic instead. Representations exist that model belief revision in either format, and for both formats there are axiom schemes that determine whether the representation is in fact a belief revision process. Although the axiom scheme for the non-deterministic case generalises that of the deterministic case, both schemes entail that all of the beliefs held by an agent are affected by new information, which is perhaps unintuitive. Rather, one may consider that belief revision should be local, with beliefs only affected if the new information is pertinent to them. We approach the problem of belief revision from the standpoint that it is local and non-deterministic, and the purpose and contribution of this dissertation is the development of a topological framework with which to model belief revision in this manner. / AFRIKAANSE OPSOMMING: Geloofshersiening word gewoonlik as ’n deterministiese proses voorgestel. Meer as een uitkoms mag bestaan vir verskeie omstandighede, wat aandui dat die proses liewer nie-deterministies van aard is. Beide die gevalle word deur aksiomaskemas gereguleer, en die aksiomas vir die nie-deterministiese geval veralgemeen dié van die deterministiese geval. Albei aksiomaskemas stipuleer, miskien onintuïtief, dat alle gelowe van ’n agent deur die nuwe informasie geaffekteer word. ’n Beter metode is dat net daardie gelowe waarvoor die nuwe informasie toepaslik is geaffekteer word. Ons benader die probleem van geloofshersiening uit die standpunt dat dit lokaal en nie-deterministies is, en die doel en bydrae van hierdie proefskrif is dus die ontwikkeling van ’n topologiese raamwerk waarmee ons geloofshersiening op hierdie manier kan voorstel.

Prof. Marcel Wild completed his PhD with Zurick University and this is a copy of the original works / The original works can be found at http://www.hbz.uzh.ch/ / ABSTRACT: A linear representation of a modular lattice L is a homomorphism from L into the lattice Sub(V) of all subspaces of a vector space V. The representation theory of lattices was initiated by the Darmstadt school (Wille, Herrmann, Poguntke, et al), to large extent triggered by the linear representations of posets (Gabriel, Gelfand-Ponomarev, Nazarova, Roiter, Brenner, et al). Even though posets are more general than lattices, none of the two theories encompasses the other. In my thesis a natural type of finite lattice is identified, i.e. triangle lattices, and their linear representation theory is advanced. All of this was triggered by a more intricate setting of quadratic spaces (as opposed to mere vector spaces) and the question of how Witt’s Theorem on the congruence of finite-dimensional quadratic spaces lifts to spaces of uncountable dimensions. That issue is dealt with in the second half of the thesis.