A credit default swap is a contract that ensures protection against losses occurring due to a default event of an certain entity. It is crucial to know how default should be modelled for valuation or estimating of credit derivatives. In this dissertation, we first review the structural approach for modelling credit risk. The model is an approach for assessing the credit risk of a firm by typifying the firms equity as a European call option on its assets, with the strike price (or exercise price) being the promised debt repayment at the maturity. The model can be used to determine the probability that the firm will default (default probability) and the Credit Spread. We second concentrate on the valuation of credit derivatives, in particular the Credit Default Swap (CDS) when the hazard rate (or even of default) is modelled as the Vasicek-type model. The other objective is, by using South African credit spread data on defaultable bonds to estimate parameters on CIR and Vasicek-type Hazard rate models such as stochastic differential equation models of term structure. The parameters are estimated numerically by the Moment Method. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2013.

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.

A large proportion of the world's land surface is covered by semi-arid grasslands, and they provide an important source of income as a grazing resource. A more comprehensive understanding of these complex ecosystems is vital for the effective management of rangelands, as it will lead to an increased and more sustainable economic output. Herbivores modify the spatial pattern of vegetation distribution and their response to spatially heterogeneous forage resources affects their performance. The spatial aspect of herbivory is often ignored although it is a necessary component of understanding grazing dynamics and the factors affecting herbivore condition. A spatial model is developed which incorporates vegetation and animal dynamics and the interactions between these two components. The effect of different spatial foraging strategies on animal performance and vegetation was investigated. Model results were compared with the output of a non-spatial model to assess the importance of spatially explicit modelling in the context of monitoring animal performance. The relative significance of a number of aspects relating to spatial grazing and animal condition was explored. The results from this research show that significant differences in output are obtained from spatial versus non-spatial models. While the purpose of a model will determine its nature, the results imply that in certain contexts, a spatial model is essential for accurate results and insight. The results also indicated that foraging strategies have a large affect on herbivore condition and that spatially explicit models are necessary in the context of investigating the effect of foraging strategies on animal performance. Various aspects that significantly affected animal condition were highlighted and are useful in directing future investigations into grazing dynamics. It is difficult to conduct field studies under spatially and temporally variable conditions where the interactions between vegetation and herbivores are so complex. In the light of this, modelling was found to be an effective tool that can be used in investigating and revealing important dynamics of semi-arid grazing systems. / Thesis (M.Sc.)-University of Natal, Pietermaritzburg, 1999.

Thesis (MScEng)--University of Stellenbosch, 2002 / ENGLISH ABSTRACT: A Representative Unit Cell (RUC) model for flow through two-dimensional porous media is presented and applied to two industrial related problems. The first application is to that of cross-flow in tube banks. Both staggered and square (inline) configurations are investigated and the model results are compared to experimental data. The second application is to flow through a stack in a timber-drying kiln. The RUC model is applied to the anisotropic timber stack ends and the centre part is modelled with a standard duct flow solution. The results of the models applied to a timber stack are compared to experimental data obtained from model tests undertaken in a wind tunnel. The results of the RUC and duct flow models are found to be in excellent agreement with the data of the experimental models. These models may be used to optimize kiln designs. / AFRIKAANSE OPSOMMING: 'n Verteenwoordigende Eenheid Sel (VES) model vir vloei deur twee-dimensionele poreuse media word weergegee en toegepas op twee industriële toepassings. Die eerste toepassing is op dwarsvloei deur banke van buise. Beide gestapelde en inlyn konfigurasies word ondersoek waarvan die model resulte met eksperimentele data vergelyk word. Die tweede toepassing is op vloei deur 'n stapel in 'n hout-droogoond. Die VES model word toegepas op die anisotropiese ente van houtstapels en die middelste seksie word gemodelleer deur 'n standaard kanaalvloei oplossing. Die resultate van die modelle toegepas op n 'houtstapel word vergelyk met eksperimentele data verkry uit model toetse wat in 'n wind-tonnel uitgevoer is. Die VES en kanaalvloei modelle se resultate stem uitstekend ooreen met die data van die eksperimentele modelle. Hierdie modelle kan gebruik word om die ontwerp van droogoonde te optimeer.

Thesis (MScEng (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2006. / When considering flow through porous media, different flow regimes may be identified. At very small Reynolds numbers the relation between the pressure gradient and the velocity of the fluid is linear. This flow regime ...

Thesis (MScEng (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2005. / Tracking manoeuvring military airborne targets with radar is problematic due to the low scan rates and the high levels of measurement noise. Surveillance systems using multiple radars have the benefit of an increased rate of observation and noise reduction but also have the problem of correlating observations from multiple sensors. Mehtods are discussed to correlate single observations from multiple radar sensors as well as assigning observations to existing tracks. Filtering methods to reduce measurement noise of the target tracks and methods to extrapolate the predicted position of targets are also explored.

Thesis (MScEng (Applied Mathematics))--University of Stellenbosch, 2006. / This thesis investigates the various components required for the development of a patient position verification system to replace the existing system used by the proton facilities of iThemba LABS1. The existing system is based on the visual comparison of a portal radiograph (PR) of the patient in the current treatment position and a digitally reconstructed radiograph (DRR) of the patient in the correct treatment position. This system is not only of limited accuracy, but labour intensive and time-consuming. Inaccuracies in patient position are detrimental to the effectiveness of proton therapy, and elongated treatment times add to patient trauma. A new system is needed that is accurate, fast, robust and automatic. Automatic verification is achieved by using image registration techniques to compare the PR and DRRs. The registration process finds a rigid body transformation which estimates the difference between the current position and the correct position by minimizing the measure which compares the two images. The image registration process therefore consists of four main components: the DRR, the PR, the measure for comparing the two images and the minimization method. The ray-tracing algorithm by Jacobs was implemented to generate the DRRs, with the option to use X-ray attenuation calibration curves and beam hardening correction curves to generate DRRs that approximate the PRs acquired with iThemba LABS’s digital portal radiographic system (DPRS) better. Investigations were performed mostly on simulated PRs generated from DRRs, but also on real PRs acquired with iThemba LABS’s DPRS. The use of the Correlation Coefficient (CC) and Mutual Information (MI) similarity measures to compare the two images was investigated. Similarity curves were constructed using simulated PRs to investigate how the various components of the registration process influence the performance. These included the use of the appropriate XACC and BHCC, the sizes of the DRRs and the PRs, the slice thickness of the CT data, the amount of noise contained by the PR and the focal spot size of the DPRS’s X-ray tube. It was found that the Mutual Information similarity measure used to compare 10242 pixel PRs with 2562 pixel DRRs interpolated to 10242 pixels performed the best. It was also found that the CT data with the smallest slice thickness available should be used. If only CT data with thick slices is available, the CT data should be interpolated to have thinner slices. Five minimization algorithms were implemented and investigated. It was found that the unit vector direction set minimization method can be used to register the simulated PRs robustly and very accurately in a respectable amount of time. Investigations with limited real PRs showed that the behaviour of the registration process is not significantly different than for simulated PRs.

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))--Stellenbosch University, 2008. / A novel implementation exploiting modern hardware is explored and found to be a significant improvement over current methods used. A 50 times performance increase in the computation time of DRRs is achieved over the conventional ray casting approach and image registration is performed in under a minute.

Thesis (MScIng)--University of Stellenbosch, 2006. / ENGLISH ABSTRACT: We designed a generic avatar animator for use in sign language related projects. The animator is capable of animating any given avatar that is compliant with the H-Anim standard for humanoid animation. The system was designed with the South African Sign Language Machine Translation (SASL-MT) project in mind, but can easily be adapted to other sign language projects due to its generic design. An avatar that is capable of accurately performing sign language gestures is a special kind of avatar and is referred to as a signing avatar. In this thesis we investigate the special characteristics of signing avatars and address the issue of finding a generic design for the animation of such an avatar. / AFRIKAANSE OPSOMMING: Ons het ’n generiese karakteranimasiestelsel ontwikkel vir gebruik in gebaretaal verwante projekte. Die animasiestelsel het die vermo¨e om enige karaktermodel wat met die H-Anim standaard versoenbaar is, te animeer. Die animasiestelsel is ontwerp met die oog op gebruik in die South African Sign Language Machine Translation (SASL-MT) projek, maar kan maklik aangepas word vir ander gebaretaalprojekte te danke aan die generiese ontwerp. ’n Karaktermodel wat in staat is om gebare akkuraat te maak is ’n spesiale tipe karaktermodel wat bekend staan as ’n gebaretaal avatar (Engels : signing avatar). In hierdie tesis ondersoek ons die spesiale eienskappe van ’n gebaretaal avatar en beskou die soektog na ’n generiese ontwerp vir die animering van so ’n karaktermodel.