Philosophiae Doctor - PhD (Mathematics)

Permanent URI for this collectionhttps://hdl.handle.net/10566/19489

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Prestasiemotivering by studente aan die Universiteit van Wes-Kaapland.
(University of the Western Cape, 1991) Brown, Alexander
The major objective of this study was to investigate the nature of the relationship between achievement motivation, autonomous and social achievement values, study habits and attitudes, locus of control and socio-economic status (SES) as independent variables on the one hand and the level of achievement as dependent variable on the other. The subjects were 548 second and third year social science students who were studying in seven different directions at the University of the Western Cape during 1990. The following measuring instruments were used in the investigation: The Ray-Lynn (1980) Achievement Orientation questionnaire; Strumpfer's (1975) questionnaire for the measuring of autonomous and social achievement values; Rotter's (1966) internal/external locus of control scale, as adapted by Collins (1974); The study habits and attitudes subscales of the Brown and Holtzman (1955) Survey of Study Habits and Attitudes (SSHA) questionnaire, as adapted for South African conditions; A brief biographical questionnaire The achievement criterion consisted of the average achievement point, which is constituted of a proportion of achievement obtained in continuous evaluation, and a proportion of achievement obtained in the final examination. The following findings were made: Achievement motivation plays a much smaller role in achievement than can be expected and its influence is gender specific. It explains only about 5% of the variance in the achievement of males, and non in the case of females. Academically successful and unsuccessful students could also not be distinguished from each other in terms of level of achievement motivation. The measuring instrument for achievement motivation, although valid and reliable, probably does not succeed in measuring aspects of achievement motivation which are related to a specific situation such as the academic. While social achievement value is not related to achievement, autonomous achievement value explains 4,8% of the variance in achievement of males but none in the case of females. Successful and unsuccessful students also do not differ from each other with regard to their achievement value orientation. Study habit and attitude do not differ in their ability to predict the achievement criterion and explain 4,1% and 5,3% of the variance in achievement of males respectively, but none in the case of females. Successful and unsuccessful students can be distinguished in terms of their study habits and attitudes. Socio-economic status has a differential influence on achievement. While higher SES females achieve at a higher level than low SES females, males do not differ in this regard. The subjects are predominantly internally orientated as far as locus of control characteristic is concerned. Although internal individuals display more "positive" characteristics compared to external individuals, the two groups do not, however, differ as far as level of achievement is concerned, irrespective of gender or socio-economic status. African students have a more positive attitude towards study compared to English and Afrikaans speaking, as well as bilingual (English and Afrikaans speaking) students. Females in this study are generally more homogenous than males. It is recommended that: The suitability of the average achievement point as a criterion of achievement be studied; A broad investigation be launched into practices and problems which might centre around the system of continuous evaluation at uwc, with specific reference to possible problems that students, lecturers and big departments may experience; The nature of differences which might exist between higher and low SES female, and low SES female and low SES male students be investigated; The nature of debilitating factors which affect the achievement of low SES female students be investigated; The tendency towards greater homogeneity among female influence thereof on university study; The adjustment of African students at uwc be studied with the objective of identifying factors that obstruct their academic progress
Codes from uniform subset graphs and cycle products
(University of the Western Cape, 2007) Fish, Washiela
In this thesis only Binary codes are studied. Firstly, the codes overs the field GF(2) by the adjacency matrix of the complement T(n), ofthe triangular graph, are examined. It is shown that the code obtained is the full space F2 s(n/2) when n= 0 (mod 4) and the dual code of the space generated by the j-vector when n= 2(mod 4). The codes from the other two cases are less trivial: when n=1 (mod 4) the code is [(n 2), (n 2 ) - n + 1, 3] code, and when n = 3 (mod 4) it is an [(n 2), (n 2) - n, 4 ] code.
Higher order numerical methods for singular perturbation problems
(University of the Western Cape, 2009) Munyakazi, Justin Bazimaziki
In recent years, there has been a great interest towards the higher order numerical methods for singularly perturbed problems. As compared to their lower order counterparts, they provide better accuracy with fewer mesh points. Construction and/or implementation of direct higher order methods is usually very complicated. Thus a natural choice is to use some convergence acceleration techniques, e.g., Richardson extrapolation, defect correction, etc. In this thesis, we will consider various classes of problems described by singularly perturbed ordinary and partial differential equations. For these problems, we design some novel numerical methods and attempt to increase their accuracy as well as the order of convergence. We also do the same for existing numerical methods in some instances. We find that, even though the Richardson extrapolation technique always improves the accuracy, it does not perform equally well when applied to different methods for certain classes of problems. Moreover, while in some cases it improves the order of convergence, in other cases it does not. These issues are discussed in this thesis for linear and nonlinear singularly perturbed ODEs as well as PDEs. Extrapolation techniques are analyzed thoroughly in all the cases, whereas the limitations of the defect correction approach for certain problems is indicated at the end of the thesis.
Mathematical epidemiology of malaria disease transmission and its optimal control analyses
(University of the Western Cape, 2010) Oare, Okosun Kazeem
In this thesis, we present and analyse an SEIR (susceptible-exposed infectious-recovered) model for malaria disease transmission. The model consists treatment and control strategies such as the use of bed nets and spray of insecticides with the costs associated with each control measure. Firstly, we analyse the model without treatment and investigate its stability and bifurcation behaviour. Then, we incorporate treatment and investigated the effects of different control strategies on the spread of malaria. Further, we use optimal control methods to determine the necessary conditions for the optimality of the disease eradication or control. We determined the most cost-effective strategies in fighting malaria disease by carrying out a cost-effectiveness study. We found that mass action model exhibited transcritical bifurcation. The disease-free equilibrium (DFE) is globally stable whenever, basic reproductive number is less than unity, while the models with standard incidence form exhibited backward bifurcation. In examining the cost-effectiveness analysis, we found that the most cost-effective strategy is the combination of insecticides spray and treatment of infective individuals. Furthermore, we modified the SEIR model to incorporate treatment and vaccination with waning immunity and an appropriate cost function. We analyse the model and investigated its stability and bifurcation property. Also, we use optimal control theory to determine the necessary optimal conditions for the disease eradication, and when eradication of the disease is unachievable, we derived the necessary conditions for its control. Further, we carried out a cost-effectiveness analysis of the control strategies. In our findings, the mass action model exhibits a backward bifurcation phenomenon, while the standard incidence model exhibited a phenomenon of multiple endemic equilibria. We also found that the most cost-effective strategy to eliminate malaria is the combination of treatment of infective individuals and vaccination. From the analysis, we found that eradication will be possible and optimal when the community marginal cost is less than the community marginal benefits.
Analysis and implementation of robust numerical methods to solve mathematical models of HIV and Malaria co-infection
(University of the Western Cape, 2011) Elsheikh, Sara Mohamed Ahmed Suleiman
There is a growing interest in the dynamics of the co-infection of these two diseases. In this thesis, firstly we focus on studying the effect of a distributed delay representing the incubation period for the malaria parasite in the mosquito vector to possibly reduce the initial transmission and prevalence of malaria. This model can be regarded as a generalization of SEI models (with a class for the latently infected mosquitoes) and SI models with a discrete delay for the incubation period in mosquitoes. We study the possibility of occurrence of backward bifurcation. We then extend these ideas to study a full model of HIV and malaria co-infection. To get further inside into the dynamics of the model, we use the geometric singular perturbation theory to couple the fast and slow models from the full model. Finally, since the governing models are very complex, they cannot be solved analytically and hence we develop and analyze a special class of numerical methods to solve them.
Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance
(University of the Western Cape, 2011) Khabir, Mohmed Hassan Mohmed
Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.
Mesh Free Methods for Differential Models In Financial Mathematics
(University of the Western Cape, 2011) Sidahmed, Abdelmgid Osman Mohammed
Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston's volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.
Robust Spectral Methods for Solving Option Pricing Problems
(University of the Western Cape, 2012) Pindza, Edson
Robust Spectral Methods for Solving Option Pricing Problems by Edson Pindza PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape Ever since the invention of the classical Black-Scholes formula to price the financial derivatives, a number of mathematical models have been proposed by numerous researchers in this direction. Many of these models are in general very complex, thus closed form analytical solutions are rarely obtainable. In view of this, we present a class of efficient spectral methods to numerically solve several mathematical models of pricing options. We begin with solving European options. Then we move to solve their American counterparts which involve a free boundary and therefore normally difficult to price by other conventional numerical methods. We obtain very promising results for the above two types of options and therefore we extend this approach to solve some more difficult problems for pricing options, viz., jump-diffusion models and local volatility models. The numerical methods involve solving partial differential equations, partial integro-differential equations and associated complementary problems which are used to model the financial derivatives. In order to retain their exponential accuracy, we discuss the necessary modification of the spectral methods. Finally, we present several comparative numerical results showing the superiority of our spectral methods.
Fischer-clifford matrices and character tables of inertia groups of maximal subgroups of finite simple groups of extension type
(University of the Western Cape, 2011) Prins, A.L.
The aim of this dissertation is to calculate character tables of group extensions. There are several well–developed methods for calculating the character tables of group extensions. In this dissertation we study the method developed by Bernd Fischer, the so–called Fischer–Clifford matrices method, which derives its fundamentals from the Clifford theory. We consider only extensions G of the normal subgroup K by the subgroup Q with the property that every irreducible character of K can be extended to an irreducible character of its inertia group in G, if K is abelian. This is indeed the case if G is a split extension, by a well-known theorem of Mackey. A brief outline of the classical theory of characters pertinent to this study, is followed by a discussion on the calculation of the conjugacy classes of extension groups by the method of coset analysis. The Clifford theory which provide the basis for the theory of Fischer-Clifford matrices is discussed in detail. Some of the properties of these Fischer-Clifford matrices which make their calculation much easier are also given. As mentioned earlier we restrict ourselves to split extension groups G in which K is always elementary abelian. In this thesis we are concerned with the construction of the character tables of certain groups which are associated with Fi₂₂ and Sp₈ (2). Both of these groups have a maximal subgroup of the form 2⁷: Sp₆ (2) but they are not isomorphic to each other. In particular we are interested in the inertia groups of these maximal subgroups, which are split extensions. We use the technique of the Fischer-Clifford matrices to construct the character tables of these inertia groups. These inertia groups of 2⁷ : Sp₆(2), the maximal subgroup of Fi₂₂, are 2⁷ : S₈, 2⁷ : Ο⁻₆(2) and 2⁷ : (2⁵ : S₆). The inertia group of 2⁷ : Sp₆(2), the affine subgroup of Sp₈(2), is 2⁷ : (2⁵ : S₆) which is not isomorphic to the group with the same form which was mentioned earlier.
Efficient numerical methods based on integral transforms to solve option pricing problems
(University of the Western Cape, 2012) Ngounda, Edgard
In this thesis, we design and implement a class of numerical methods (based on integral transforms) to solve PDEs for pricing a variety of financial derivatives. Our approach is based on spectral discretization of the spatial (asset) derivatives and the use of inverse Laplace transforms to solve the resulting problem in time. The conventional spectral methods are further modified by using piecewise high order rational interpolants on the Chebyshev mesh within each sub-domain with the boundary domain placed at the strike price where the discontinuity is located. The resulting system is then solved by applying Laplace transform method through deformation of a contour integral. Firstly, we use this approach to price plain vanilla options and then extend it to price options described by a jump-diffusion model, barrier options and the Heston’s volatility model. To approximate the integral part in the jump-diffusion model, we use the Gauss-Legendre quadrature method. Finally, we carry out extensive numerical simulations to value these options and associated Greeks (the measures of sensitivity). The results presented in this thesis demonstrate the spectral accuracy and efficiency of our approach, which can therefore be considered as an alternative approach to price these class of options.
Efficient numerical methods to solve some reaction-diffusion problems arising in biology
(University of the Western Cape, 2013) Matthew, Owolabi Kolade
In this thesis, we solve some time-dependent partial differential equations, and systems of such equations, that governs reaction-diffusion models in biology. we design and implement some novel exponential time differencing schemes to integrate stiff systems of ordinary differential equations which arise from semi-discretization of the associated partial differential equations. We split the semi-linear PDE(s) into a linear, which contains the highly stiff part of the problem, and a nonlinear part, that is expected to vary more slowly than the linear part. Then we introduce higher-order finite difference approximations for the spatial discretization. Resulting systems of stiff ODEs are then solved by using exponential time differencing methods. We present stability properties of these methods along with extensive numerical simulations for a number of different reaction-diffusion models, including single and multi-species models. When the diffusivity is small many of the models considered in this work are found to exhibit a form of localized spatiotemporal patterns. Such patterns are correctly captured by our proposed numerical schemes. Hence, the schemes that we have designed in this thesis are dynamically consistent. Finally, in many cases, we have compared our results with those obtained by other researchers.
Analysis of errors in derivatives of trigonometric functions: a case study in an extended curriculum programme
(University of the Western Cape, 2012) Siyepu, Sibawu Witness
The purpose of this study was to explore errors that are displayed by students when learning derivatives of trigonometric functions in an extended curriculum programme. The first aim was to identify errors that are displayed by students in their solutions through the lens of the APOS theory. The second aim was to address students' errors by using the two principles of Vygotsky's socio-cultural theory of learning, namely the zone of proximal development and more knowledgeable others. The research presented in this thesis is a case study located in the interpretive paradigm of qualitative research. The participants in this study comprised a group of students who registered for mathematics in the ECP at Cape Peninsula University of Technology, Cape Town, South Africa. The study was piloted in 2008 with a group of twenty students who registered for mathematics in the ECP for Chemical Engineering. In 2009 thirty students from the ECP registered for mathematics in Chemical Engineering were selected to participate in the main study. This study was conducted over a period of four and a half years. Data collection was done through students' written tasks; classroom audio and video recordings and indepth interviews. Data were analysed through categorising errors from students' written work, and finding common themes and patterns in audio and video recordings and from the in-depth interviews. The findings of this study revealed that students committed interpretation, arbitrary, procedural, linear extrapolation and conceptual errors. Interpretation errors arise when students fail to interpret the nature of the problem correctly owing to over-generalisation of certain mathematical rules. Arbitrary errors arise when students behave arbitrarily and fail to take account of the constraints laid down in what is given. Procedural errors occur when students fail to carry out manipulations or algorithms although they understand concepts in problem. Linear extrapolation errors happen through an overgeneralisation of the property f (a + b) = f (a) + f (b) , which applies only when f is a linear function Conceptual errors occur owing to failure to grasp the concepts involved in the problem or failure to appreciate the relationships involved in the problem. The findings were consistent with literature indicated that errors are based on students’ prior knowledge, as they over-generalise certain mathematical procedures, algorithms and rules of differentiation in their solutions. The use of learning activities in the form of written tasks; as well as classroom audio and video recordings assisted the lecturer to identify and address errors that were displayed by students when they learned derivatives of trigonometric functions. The students claimed in their interviews that they benefited from class discussions as they obtained immediate feedback from their fellow students and the lecturer. They also claimed that their performances improved as they continued to practice with the assistance of more knowledgeable students, as well as the lecturer. This study supports the view from the literature that identification of errors has immense potential to address students’ poor understanding of derivatives of trigonometric functions. This thesis recommends further research on errors in various sections of Differential Calculus, which is studied in an extended curriculum programme at Universities of Technology in South Africa.
The role of visualization in the teaching and learning of multivariate calculus and systems of ordinary differential equations
(University of the Western Cape, 2015) Sheikh, T.O
The purpose of this study was to investigate the role of visualization in the conceptualisation and solution of problems in multivariate calculus and dynamical systems. The theoretical basis, and the visual and analytical aspects of evaluating multiple integrals, and the stability analysis of dynamical systems, were established. To address the research questions, a teaching experiment with activities to facilitate visualization of 3D objects and phase portraits of non-linear dynamic systems was conducted with an experimental class (n = 24) which received six activity sessions in the computer Laboratory in addition to traditional lectures. The control class (n = 26) received traditional lectures and tutorial instruction. Both groups were lectured by the researcher using the same set of class notes, assignments, worksheets and tutorials. Additional support materials were posted on the Blackboard on Web-City. The activities included tasks in the computer laboratory that reinforced visualization and spatial ability factors such as surface features, nets, projections, cross-sections and rotation of 3D objects as well as phase portraits of systems of differential equations. The students were tested at several time points, and over both the short and long term to assess the impact on their visual and analytical solutions to problems in the two study domains. The pre-test on prior knowledge indicated no significant differences between the means of the experimental and control groups. Results indicate that there were no significant differences between the achievement of the two groups in Test 1 and Test 2 while the activities were ongoing, but towards the end of the semester significant differences in favour of the experimental group were recorded. A multiple linear regression analysis confirmed that in addition to prior knowledge as measured by the pretest, two of the spatial factors were significant predictors of achievement for the domains under investigation. Students had difficulties in visualising 3D regions of integration and in switching the order of triple integrals. Very few (18%) recognised the need for split integrals to span the required area or volume. While students could find analytical solutions to systems of differential equations and describe the stability of individual equilibrium points using eigenvalues, they struggled with translating rates of change into slopes on the phase portraits, with the interpretation of the solutions and in describing the global behaviour of the system. Students had difficulties in visualizing the region of integration in R³, the stability of equilibrium points in the phase portraits, and in coordinating the treatments and conversions between the geometric, numerical, symbolic and algebraic registers. The tendency to work in the algebraic register to determine the limits of the integral was noted, and students opted to use analytic methods in conducting a stability analysis of the given dynamic system rather than the geometric method. This study adds to research on visualization in mathematics by examining how exposure to technologically enhanced representations complement and promote the conceptualisation of solutions to problems involving multiple integrals and systems of differential equations.
Optimal asset allocation and capital adequacy management strategies for Basel III compliant banks
(University of the Western Cape, 2015) Muller, Grant Envar
In this thesis we study a range of related commercial banking problems in discrete and continuous time settings. The ﬁrst problem is about a capital allocation strategy that optimizes the expected future value of a commercial bank’s total non-risk-weighted assets (TNRWAs) in terms of terminal time utility maximization. This entails ﬁnding optimal amounts of Total capital for investment in different bank assets. Based on the optimal capital allocation strategy derived for the ﬁrst problem, we derive stochastic models for respectively the bank’s capital adequacy and liquidity ratios in the second and third problems. The Basel Committee on Banking Supervision (BCBS) introduced these ratios in an attempt to improve the regulation of the international banking industry in terms of capital adequacy and liquidity management. As a fourth problem we derive a multi-period deposit insurance pricing model which incorporates the optimal capital allocation strategy, the BCBS’ latest capital standard, capital forbearance and moral hazard. In the ﬁfth and ﬁnal problem we show how the values of LIBOR-in-arrears and vanilla interest rate swaps, typically used by commercial banks and other ﬁnancial institutions to reduce risk, can be derived under a specialized version of the affine interest rate model originally considered by the bank in question. More speciﬁcally, in the ﬁrst problem we assume that the bank invests its Total capital in a stochastic interest rate ﬁnancial market consisting of three assets, viz., a treasury security, a marketable security and a loan. We assume that the interest rate in the market is described by an affine model, and that the value of the loan follows a jump-diffusion process. We wish to ﬁnd the optimal capital allocation strategy that maximizes an expected logarithmic utility of the bank’s TNRWAs at a future date. Generally, analytical solutions to stochastic optimal control problems in the jump setting are very difficult to obtain. We propose an approximation method that exploits a similarity between the forms of the control problems of the jump-diffusion model and the diffusion model obtained by removing the jump. With the jump assumed sufficiently small, the analytical solution of the diffusion model then serves as a proxy to the solution of the control problem with the jump. In the second problem we construct models for the bank’s capital adequacy ratios in terms of the proxy. We present numerical simulations to characterize the behaviour of the capital adequacy ratios. Furthermore, in this chapter, we consider the approximate optimal capital allocation strategy subject to a constant Leverage Ratio, which is a speciﬁc non-risk-based capital adequacy ratio, at the minimum prescribed level. We derive a formula for the bank’s TNRWAs at constant (minimum) Leverage Ratio value and present numerical simulations based on the modiﬁed TNRWAs formula. In the third problem we model the bank’s liquidity ratios and we monitor the levels of the liquidity ratios under the proxy numerically. In the fourth problem we derive a multi-period deposit insurance pricing model, the latest capital standard a la Basel III, capital forbearance and moral hazard behaviour. The deposit insurance pricing method utilizes an asset value reset rule comparable to the typical practice of insolvency resolution by insuring agencies. We perform numerical computations with our model to study its implications. In the ﬁnal problem, we specialize the affine interest rate model considered previously to the Cox-Ingersoll-Ross (CIR) interest rate dynamic. We consider ﬁxed-for-ﬂoating interest rate swaps under the CIR model. We show how analytical expressions for the values of both a LIBOR-in-arrears swap and a vanilla swap can be derived using a Green’s function approach. We employ Monte Carlo simulation methods to compute the values of the swaps for different scenarios. We wish to make explicit the contributions of this project to the literature. A research article titled “An Optimal Portfolio and Capital Management Strategy for Basel III Compliant Commercial Banks” by Grant E. Muller and Peter J. Witbooi [1] has been published in an accredited scientiﬁc journal. In the aforementioned paper we solve an optimal capital allocation problem for diffusion banking models. We propose using the solution of the Brownian motions control problem of [1] as the proxy in problems two to four of this thesis. Furthermore, we wish to note that the methodology employed on the ﬁnal problem of this study is actually from the paper [2] of Mallier and Alobaidi. In the paper [2] the authors did not present simulation studies to characterize their pricing models. We contribute a simulation study in which the values of the swaps are computed via Monte Carlo simulation methods.
Codes, graphs and designs from maximal subgroups of alternating groups
(University of the Western Cape, 2018) Mumba, Nephtale Bvalamanja
The main theme of this thesis is the construction of linear codes from adjacency matrices or sub-matrices of adjacency matrices of regular graphs. We first examine the binary codes from the row span of biadjacency matrices and their transposes for some classes of bipartite graphs. In this case we consider a sub-matrix of an adjacency matrix of a graph as the generator of the code. We then shift our attention to uniform subset graphs by exploring the automorphism groups of graph covers and some classes of uniform subset graphs. In the sequel, we explore equal codes from adjacency matrices of non-isomorphic uniform subset graphs and finally consider codes generated by an adjacency matrix formed by adding adjacency matrices of two classes of uniform subset graphs.
On the primarity of some block intersection graphs
(University of the Western Cape, 2018) Vodah, Sunday
A tactical con guration consists of a nite set V of points, a nite set B of blocks and an incidence relation between them, so that all blocks are incident with the same number k points, and all points are incident with the same number r of blocks (See [14] for example ). If v := jV j and b := jBj; then v; k; b; r are known as the parameters of the con guration. Counting incident point-block pairs, one sees that vr = bk: In this thesis, we generalize tactical con gurations on Steiner triple systems obtained from projective geometry. Our objects are subgeometries as blocks. These subgeometries are collected into systems and we study them as designs and graphs. Considered recursively is a further tactical con guration on some of the designs obtained and in what follows, we obtain similar structures as the Steiner triple systems from projective geometry. We also study these subgeometries as factorizations and examine the automorphism group of the new structures. These tactical con gurations at rst sight do not form interesting structures. However, as will be shown, they o er some level of intriguing symmetries. It will be shown that they inherit the automorphism group of the parent geometry.
Mathematical models for optimal management of bank capital, reserves and liquidity
(University of the Western Cape, 2019) Van Schalkwyk, Garth
The aim of this study is to construct and propose continuous-time mathematical models for optimal management of bank capital, reserves and liquidity. This aim emanates from the global financial crisis of 2007 − 2009. In this regard and as a first task, our objective is to determine an optimal investment strategy for a commercial bank subject to capital requirements as prescribed by the Basel III Accord. In particular, the objective of the aforementioned problem is to maximize the expected return on the bank capital portfolio and minimize the variance of the terminal wealth. We apply classical tools from stochastic analysis to achieve the optimal strategy of a benchmark portfolio selection problem which minimizes the expected quadratic distance of the terminal risk capital reserves from a predefined benchmark. Secondly, the Basel Committee on Banking Supervision (BCBS) introduced strategies to protect banks from running out of liquidity. These measures included an increase of the minimum reserves that the bank ought to hold, in response to the global financial crisis. We propose a model to minimize risk for a bank by finding an appropriate mix of diversification, balanced against return on the portfolio. Thirdly and finally, in response to the financial crises, the Basel Committee on Banking Supervision (BCBS) designed a set of precautionary measures (known as Basel III) for liquidity imposed on banks and one of its purposes is to protect the economy from deteriorating. Recently, bank regulators wanted banks to depend on sources such as core deposits and long-term funding from small businesses and less on short-term wholesale funding.
Stochastic modeling of an HIV/AIDS epidemic with treatment
(University of the Western Cape, 2019) Nsuami, Mozart Umba
The HIV/AIDS epidemic continues to be among the most devastating diseases in human history despite the new scientific advances and serious public health interventions. The greatest burden of HIV/AIDS is still in sub-Saharan Africa, and within this specific region, women are severely affected. Despite an increase in prevention interventions, including such as ARV treatment and pre-exposure prophylaxis (PrEP), behavioural change remains a key role in the transmission of HIV/AIDS. In this thesis, we investigate several related models for the population dynamics of HIV/AIDS epidemic model with treatment. We start off with a four compartmental HIV deterministic model with stages of HIV infection and with inflow of HIV infectives. Thereafter, we impose stochastic perturbations on the underlying HIV/AIDS deterministic model without inflow of infectives. For this version of HIV stochastic model, we prove global existence and positivity of solutions to the HIV/AIDS-perturbed model. Some useful properties such as boundedness property, stochastic permanence property and asymptotic stability have been derived.
Interior operators and their applications
(University of the Western Cape, 2019) Assfaw, Fikreyohans Solomon
Categorical closure operators were introduced by Dikranjan and Giuli in [DG87] and then developed by these authors and Tholen in [DGT89]. These operators have played an important role in the development of Categorical Topology by introducing topological concepts, such as connectedness, separatedness and compactness, in an arbitrary category and they provide a uni ed approach to various mathematical notions. Motivated by the theory of these operators, the categorical notion of interior operators was introduced by Vorster in [Vor00]. While there is a notational symmetry between categorical closure and interior operators, a detailed analysis shows that the two operators are not categorically dual to each other, that is: it is not true in general that whatever one does with respect to closure operators may be done relative to interior operators. Indeed, the continuity condition of categorical closure operators can be expressed in terms of images or equivalently, preimages, in the same way as the usual topological closure describes continuity in terms of images or preimages along continuous maps. However, unlike the case of categorical closure operators, the continuity condition of categorical interior operators can not be described in terms of images. Consequently, the general theory of categorical interior operators is not equivalent to the one of closure operators. Moreover, the categorical dual closure operator introduced in [DT15] does not lead to interior operators. As a consequence, the study of categorical interior operators in their own right is interesting.
Quasi-uniform and syntopogenous structures on categories
(University of the Western Cape, 2019) Iragi, Minani
In a category C with a proper (E; M)-factorization system for morphisms, we further investigate categorical topogenous structures and demonstrate their prominent role played in providing a uni ed approach to the theory of closure, interior and neighbourhood operators. We then introduce and study an abstract notion of C asz ar's syntopogenous structure which provides a convenient setting to investigate a quasi-uniformity on a category. We demonstrate that a quasi-uniformity is a family of categorical closure operators. In particular, it is shown that every idempotent closure operator is a base for a quasi-uniformity. This leads us to prove that for any idempotent closure operator c (interior i) on C there is at least a transitive quasi-uniformity U on C compatible with c (i). Various notions of completeness of objects and precompactness with respect to the quasi-uniformity de ned in a natural way are studied. The great relationship between quasi-uniformities and closure operators in a category inspires the investigation of categorical quasi-uniform structures induced by functors. We introduce the continuity of a C-morphism with respect to two syntopogenous structures (in particular with respect to two quasi-uniformities) and utilize it to investigate the quasiuniformities induced by pointed and copointed endofunctors. Amongst other things, it is shown that every quasi-uniformity on a re ective subcategory of C can be lifted to a coarsest quasi-uniformity on C for which every re ection morphism is continuous. The notion of continuity of functors between categories endowed with xed quasi-uniform structures is also introduced and used to describe the quasi-uniform structures induced by an M- bration and a functor having a right adjoint.

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