Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2011. / Includes bibliography. / ENGLISH ABSTRACT: Latimer & MacDuffee showed in 1933 that there is a one-to-one correspondence between equivalence classes of matrices with a given minimum polynomial and equivalence classes of ideals of a certain ring. In the case where the matrices are taken over the integers, Behn and Van der Merwe developed an algorithm in 2002 to produce a representative in each equivalence class. We extend this algorithm to matrices taken over the ring Fq[T] of polynomials over a finite field and prove a modified version of the Latimer-MacDuffee theorem which holds for proper equivalence classes of matrices. / AFRIKAANSE OPSOMMING: Latimer & MacDuffee het in 1933 bewys dat daar 'n een-tot-een korrespondensie is tussen ekwivalensieklasse van matrikse met 'n gegewe minimumpolinoom en ekwivalensieklasse van ideale van 'n sekere ring. In die geval waar die matrikse heeltallige inskrywings het, het Behn en Van der Merwe in 2002 'n algoritme ontwikkel om verteenwoordigers in elke ekwivalensieklas voort te bring. Ons brei hierdie algoritme uit na die geval van matrikse met inskrywings in die ring Fq[T] van polinome oor 'n eindige liggaam en ons bewys 'n gewysigde weergawe van die Latimer-MacDuffee stelling wat geld vir klasse van streng ekwivalente matrikse.

Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Water-borne infections have been a menace in many countries around the globe, claiming millions of lives. Cholera in particular has spread to all continents and now on its seventh epidemic. Although control measures have been continually developed through sanitation, vaccination and rehydration, the infection still devastates populations whenever there is an outbreak. In this research work, mathematical models for cholera transmission dynamics with focus on the impact of sanitation and hygiene, metapopulation spread, optimal control and biological control using a bacteriophage specific for pathogenic Vibrio cholerae are constructed and analysed. Vital analyses for the models are precisely given as well as numerical results depicting long term behaviour and the evolution of populations over time. The results of our analysis indicate that; improved sanitation and hand-hygiene are vital in reducing cholera infections; the spread of disease across metapopulations characterised by exchange of individuals and no cross community infection is associated with synchronous fluctuation of populations in both adjacent communities; during control of cholera, the control measures/efforts ought to be optimal especially at the beginning of the epidemic where the outbreak is often explosive in nature; and biological control if well implemented would avert many potential infections by lowering the concentration of pathogenic vibrios in the aquatic environment to values lower than the infectious dose. / AFRIKAANSE OPSOMMING: Water-infeksies is ’n bedreiging in baie lande regoor die wêreld en eis miljoene lewens. Cholera in die besonder, het op sy sewende epidemie na alle kontinente versprei. Hoewel beheermaatreëls voortdurend ontwikkel word deur middel van higiëne, inentings en rehidrasie, vernietig die infeksie steeds bevolkings wanneer daar ’n uitbraak voorkom. In hierdie navorsingswerk, word wiskundige modelle vir cholera-oordrag dinamika met die fokus op die impak van higiëne, metabevolking verspreiding, optimale beheer en biologiese beheer met behulp van ’n bakteriofaag spesifiek vir patogene Vibrio cholerae gebou en ontleed. Noodsaaklike ontledings vir die modelle is gegee sowel as numeriese resultate wat die langtermyn gedrag uitbeeld en die ontwikkeling van die bevolking oor tyd. Die resultate van ons ontleding dui daarop dat; verbeterde higiëne is noodsaaklik in die vermindering van cholera infeksies; die verspreiding van die siekte oor metapopulaties gekenmerk deur die uitruil van individue en geen kruis gemeenskap infeksie wat verband houmet sinchrone skommeling van bevolkings in beide aangrensende gemeenskappe; tydens die beheer van cholera,behoort die beheermaatreëls/pogings optimaal te wees veral aan die begin van die epidemie waar die uitbreking dikwels plofbaar in die natuur is; en biologiese beheer, indien dit goed geïmplementeer word, kan baie potensiële infeksies voorkom deur ’n vermindering in die konsentrasie van patogene vibrio in die water tot waardes laer as die aansteeklike dosis.

The control of the African stalk borer Eldana saccharina Walker (Lepidoptera: Pyralidae) in sugarcane fields of KwaZulu-Natal, South Africa has proved problematical. Researchers at the South African Sugarcane Research Institute (SASRI) have since 1974 been intensively investigating various means of controlling the pest. Among the methods of control currently being investigated are biological control, chemical control, production of more resistant varieties and crop management. These investigations, however, require many years of experimentation before any conclusions can be made. In order to aid the research currently being carried out in the Entomology Department at SASRI (to investigate biological control strategies, insecticide application strategies and the carry-over decision), a simulation model of E. saccharina growth in sugarcane has been formulated. The model is cohort-based and includes the effect of temperature on the physiological development of individuals in each life-stage of the insect. It also takes into account the effect of the condition of sugarcane on the rate of E. saccharina infestation, by making use of output from the sugarcane growth model CANEGRO. Further, a crop damage index is defined that gives an indication of the history of E. saccharina infestation levels during the sugarcane’s growth period. It is linked to the physiological activity of the borer during the period spent feeding on the stalk tissue. The damage index can further be translated into length of stalks bored and hence the percentage of the stalk length bored can be calculated at each point in the simulation using the total length of stalks calculated in the CANEGRO model. Using an industry accepted relationship between percent stalks damaged and reduction in sucrose content of the crop, reductions in losses in the relative value of the crop when the various control measures are implemented can be compared. Relationships between the reduction in percent stalk length bored (and hence gains in the relative value of the crop) and the various control strategies are obtained. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermartzburg, 2008.

Increased abstraction of water in the catchment results in a reduced or altered pattern of river flow and this holds serious consequences for the downstream estuarine ecosystem. In South Africa this is a serious concern because freshwater is in limited supply and the demand for freshwater can be expected to increase in the future. A large multi-disciplinary consortium of South African scientists are working on projects to determine the freshwater requirements of estuarine ecosystems. As part of this, this thesis reports on research undertaken to develop mathematical models to determine the freshwater requirements of estuarine macrophytes. Three key macrophytes are selected. The macrophytes are Zostera capensis Setchell, Ruppia cirrhosa Grande, and Phragmites australis. They are common macrophytes in South African estuaries. Zostera and Ruppia are submerged macrophytes and Phragmites is an emergent macrophyte. They have different freshwater environments and therefore respond differently to alterations in freshwater flow. A first order differential equation model is used to determine the effect of different combinations of open and closed mouth conditions of the estuary on Zostera and Ruppia. The scenarios are selected to determine whether achieving a switch in states from a Zostera-dominated estuary to a Ruppia-dominated estuary is possible. To predict encroachment rates and colonisation patterns, a cellular automaton of the vegetative spread of existing Zostera beds is developed. After analysing various scenarios accounting for both an increase and a decrease in freshwater supply, the cellular automaton is extended to include interactions between Ruppia and Phragmites. The multi-species model is applied to the Kromme estuary, South Africa and the Great Brak estuary, South Africa. Various freshwater scenarios are examined from the natural runoff condition to the situation of no freshwater inflow. A sensitivity analysis of the spatial model with Zostera, Ruppia and Phragmites is conducted. / Thesis (Ph.D.)-University of Natal, Pietermaritzburg, 1998.

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2008. / Context-free grammars are well-studied and well-behaved in terms of decidability, but many real-world problems cannot be described with context-free grammars. Grammars with regulated rewriting are grammars with mechanisms to regulate the applications of rules, so that certain derivations are avoided. Thus, with context-free rules and regulated rewriting mechanisms, one can often generate languages that are not context-free. In this thesis we study grammars with regulated rewriting mechanisms. We consider problems in which context-free grammars are insufficient and in which more descriptive grammars are required. We compare bag context grammars with other well-known classes of grammars with regulated rewriting mechanisms. We also discuss the relation between bag context grammars and recognizing devices such as counter automata and Petri net automata. We show that regular bag context grammars can generate any recursively enumerable language. We reformulate the pumping lemma for random permitting context languages with context-free rules, as introduced by Ewert and Van der Walt, by using the concept of a string homomorphism. We conclude the thesis with decidability and complexity properties of grammars with regulated rewriting.

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2007. / The ever-increasing reliance of society on computer systems has led to a need for highly reliable systems. There are a number of areas where computer systems perform critical functions and the development of such systems requires a higher level of attention than any other type of system. The appropriate approach in this situation is known as formal methods. Formal methods refer to the use of mathematical techniques for the specification, development and verification of software and hardware systems. The two main goals of this thesis are: 1. The design of mathematical models as a basis for the implementation of error-free software for the safety interlock system at iThemba LABS (http://www.tlabs.ac.za/). 2. The comparison of formal method techniques that addresses the lack of much-needed empirical studies in the field of formal methods. Mathematical models are developed using model checkers: Spin, Uppaal, Smv and a theorem prover Pvs. The criteria used for the selection of the tools was based on the popularity of the tools, support of the tools, representation of properties, representativeness of verification techniques, and ease of use. The procedure for comparing these methods is divided into two phases. Phase one involves the time logging of activities followed by a novice modeler to model check and theorem prove software systems. The results show that it takes more time to learn and use a theorem prover than a model checker. Phase two involves the performance of the tools in relation to the time taken to verify a property, memory used, number of states and transitions generated. In spite of the differences between models, the results are in favor of Smv and this maybe attributed to the nature of the safety interlock system, as it involves a lot of hard-wired lines.

Thesis (MScEng (Mathematical Sciences)--University of Stellenbosch, 2009. / ENGLISH ABSTRACT: To reliably classify parts of an image sequence as foreground or background is an important part of many computer vision systems, such as video surveillance, tracking and robotics. It can also be important in applications where bandwidth is the limiting factor, such as video conferencing. Independent foreground motion is an attractive source of information for this task, and with static cameras, background subtraction is a particularly popular type of approach. The idea behind background subtraction is to compare the current image with a reference image of the background, and from there decide on a pixel by pixel basis, what is foreground and what is background by observing the changes in the pixel sequence. The problem is to get the useful reference image, especially when large parts of the background are occluded by moving/stationary foreground objects; i.e. some parts of the background are never seen. In this thesis four algorithms are reviewed that segment an image sequence into foreground and background components with varying degrees of success that can be measured on speed, comparative accuracy and/or memory requirements. These measures can be then effectively used to decide the application scope of the individual algorithms. / AFRIKAANSE OPSOMMING: Om betroubaar dele van ’n beeld reeks te klassifiseer as voorgrond of agtergrond is ’n belangrike deel van baie rekenaarvisie sisteme, byvoorbeeld video bewaking, volging en robotika. Dit kan ook belangrik wees in toepassings waar bandwydte die beperkende faktor is, byvoorbeeld video konferensie gesprekke. Onafhanklik voorgrond beweging is ’n aantreklike bron van informasie vir hierdie taak, en met statiese kameras, is agtergrond aftrekking ’n populêre benadering. Die idee agter agtergrond aftrekking is om die huidige beeld met ’n naslaan beeld van die agtergrond te vergelyk, en daarvandaan besluit op ’n piksel-na-piksel basis, wat is voorgrond en wat is agtergrond deur die observasies van die veranderinge in die piksel-reeks. Die probleem is om die naslaan beeld te kry om mee te werk, veral wanneer groot dele van die agtergrond onsigbaar bly as gevolg van bewegende of stilstaande voorgrond objekte en sommige dele van die agtergrond word dalk nooit gesien nie. In hierdie tesis word vier algorithms ondersoek wat ’n beeld reeks segmenteer in respektiewe voorgrond en agtergrond komponente met wisselende grade van sukses wat gemeet kan word deur spoed, vergelykbare akkuraatheid en/of geheu gebruik. Hierdie metings kan dan effektief gebruik word om die applikasie veld van die individuele algoritmes the bepaal.

Thesis (MSc (Mathematical Sciences)--University of Stellenbosch, 2005. / Image compression using weighted finite automata (WFA) is studied and implemented in Matlab. Other more prominent image compression techniques, namely JPEG, vector quantization, EZW wavelet image compression and fractal image compression are also presented. The performance of WFA image compression is then compared to those of some of the abovementioned techniques.

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / The computation of the greeks of an option is an important aspect of financial mathematics. The information gained from knowing the value of a greek of an option can help investors decide whether or not to hold on to or to sell their options to avoid losses or gain a profit. However, there are technical difficulties that arise from having to do this. Among them is the fact that the mathematical formula for the value some options is complex in nature and evaluating their greeks may be cumber- some. On the other hand the greek might have to be numerically estimated if the option does not posses an explicit evaluation formula. This could be a computationally expensive undertaking. Malliavin calculus offers us a solution to these problems. We can find formula that can be used in combination with Monte Carlo simulations to give results quickly and which are not computationally expensive to obtain and hence give us an degree of accuracy higher that non Malliavin calculus techniques. This thesis will develop the Malliavin calculus tools that will enable us to develop the tools which we will then use to compute the greeks of some known options.

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / LULU smoothers is a class of nonlinear smoothers and they are compositions of the maximum and minimum operators. By analogy to the discrete Fourier transform and the discrete wavelet transform, one can use LULU smoothers to create a nonlinear multiresolution analysis of a sequence with pulses. This tool is known as the Discrete Pulse Transform (DPT). Some research have been done into the distributional properties of the LULU smoothers. There exist results on the distribution transfers of the basic LULU smoothers, which are the building blocks of the discrete pulse transform. The output distributions of further smoothers used in the DPT, in terms of input distributions, has been a challenging problem. We motivate the use of these smoothers by first considering linear filters as well as the median smoother, which has been very popular in signal and image processing. We give an overview of the attractive properties of the LULU smoothers after which we tackle their output distributions. The main result is the proof of a recursive formula for the output distribution of compositions of LULU smoothers in terms of a given input distribution.