Philosophiae Doctor - PhD (Mathematics)

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

Browse

Now showing 1 - 20 of 23

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.
Robust computational methods to simulate slow-fast dynamical systems governed by predator-prey models
(University of the Western Cape, 2019) Mergia, Woinshet D.
Numerical approximations of multiscale problems of important applications in ecology are investigated. One of the class of models considered in this work are singularly perturbed (slow-fast) predator-prey systems which are characterized by the presence of a very small positive parameter representing the separation of time-scales between the fast and slow dynamics. Solution of such problems involve multiple scale phenomenon characterized by repeated switching of slow and fast motions, referred to as relaxationoscillations, which are typically challenging to approximate numerically. Granted with a priori knowledge, various time-stepping methods are developed within the framework of partitioning the full problem into fast and slow components, and then numerically treating each component differently according to their time-scales. Nonlinearities that arise as a result of the application of the implicit parts of such schemes are treated by using iterative algorithms, which are known for their superlinear convergence, such as the Jacobian-Free Newton-Krylov (JFNK) and the Anderson’s Acceleration (AA) fixed point methods.
Mathematical modeling of the transmission dynamics of malaria in South Sudan
(University of the Western Cape, 2019) Mukhtar, Abdulaziz Yagoub Abdelrahman
Malaria is a common infection in tropical areas, transmitted between humans through female anopheles mosquito bites as it seeks blood meal to carry out egg production. The infection forms a direct threat to the lives of many people in South Sudan. Reports show that malaria caused a large proportion of morbidity and mortality in the fledgling nation, accounting for 20% to 40% morbidity and 20% to 25% mortality, with the majority of the affected people being children and pregnant mothers. In this thesis, we construct and analyze mathematical models for malaria transmission in South Sudan context incorporating national malaria control strategic plan. In addition, we investigate important factors such as climatic conditions and population mobility that may drive malaria in South Sudan. Furthermore, we study a stochastic version of the deterministic model by introducing a white noise.
An analytical model for assessing the knowledge of statistical procedures amongst postgraduate students in a higher educational environment
(University of the Western Cape, 2019) Kamleu, Germaine
Over the past decades, the use and application of statistical concepts for university students have been a big challenge learned from their previous courses. Aftermath of democracy, South African higher education focused on redressing issues of reparation and social imbalances inherited from Apartheid with the commitment to reconstruct a comprehensive educational quality framework. Growing activities lead to new models emphasised to support students and universities in their attempts to demonstrate evidence of enthusiastic statistics learning, with an acceptable degree of accuracy. This study combines quantitative and qualitative research approaches to assess the knowledge of postgraduate students in applying suitable statistical procedures in higher education (HE). The quantitative data were randomly collected from the postgraduate students (n1=307) while the qualitative data were collected through semi-structured interviews (n2=19) from two institutions (University of Cape Town [UCT] and University of the Western Cape [UWC]) in the Western Cape, South Africa. The SPSS V24 statistical package was used for quantitative data analysis and the explorative design was selected as a theoretical framework to guide the investigation, analysis and interpretation of the qualitative findings. UCT model achieved for all combined categories 73% high prediction accuracy. The UWC model revealed similar results, with ask for help, worth of statistics, fear of statistics monitors, affect, cognitive competence, support from significant others, marital status, ethnic groups and type of study as significant predictors with a high prediction accuracy of 75.49%. Additionally, the ethnic groups, marital status, postgraduate programmes, experiences in statistics and effort were significant contributed factors of SELS beliefs while findings of the combined data of UCT and UWC significantly explained the variation observed in SELS beliefs with only 60% model accuracy. Nevertheless, the qualitative data outcomes indicated that the comments of the participants provided a rich understanding of the perceived failure to choose a relevant statistical test. The results further indicated that confusion and frustration characterised the attitude of students during the selection of a suitable statistical test. The original value of this current study is bridging the inequity gap, in terms of statistics learning, and building a substantial input to the achievement of the objectives of UNESCO, the World Education Forum and the White Paper 3, while ultimately, contributing to the sustainable development of learning statistics at universities in the Western Cape, South Africa. By logical extrapolation, this current study proffers significant insights to the rest of the universities in Africa, and beyond.
Robust numerical methods to solve differential equations arising in cancer modeling
(University of the Western Cape, 2020) Shikongo, Albert
Cancer is a complex disease that involves a sequence of gene-environment interactions in a progressive process that cannot occur without dysfunction in multiple systems. From a mathematical point of view, the sequence of gene-environment interactions often leads to mathematical models which are hard to solve analytically. Therefore, this thesis focuses on the design and implementation of reliable numerical methods for nonlinear, first order delay differential equations, second order non-linear time-dependent parabolic partial (integro) differential problems and optimal control problems arising in cancer modeling. The development of cancer modeling is necessitated by the lack of reliable numerical methods, to solve the models arising in the dynamics of this dreadful disease. Our focus is on chemotherapy, biological stoichometry, double infections, micro-environment, vascular and angiogenic signalling dynamics. Therefore, because the existing standard numerical methods fail to capture the solution due to the behaviors of the underlying dynamics. Analysis of the qualitative features of the models with mathematical tools gives clear qualitative descriptions of the dynamics of models which gives a deeper insight of the problems. Hence, enabling us to derive robust numerical methods to solve such models.
High Accuracy Fitted Operator Methods for Solving Interior Layer Problems
(University of the Western Cape, 2020) Sayi, Mbani T
Fitted operator finite difference methods (FOFDMs) for singularly perturbed problems have been explored for the last three decades. The construction of these numerical schemes is based on introducing a fitting factor along with the diffusion coefficient or by using principles of the non-standard finite difference methods. The FOFDMs based on the latter idea, are easy to construct and they are extendible to solve partial differential equations (PDEs) and their systems. Noting this flexible feature of the FOFDMs, this thesis deals with extension of these methods to solve interior layer problems, something that was still outstanding. The idea is then extended to solve singularly perturbed time-dependent PDEs whose solutions possess interior layers. The second aspect of this work is to improve accuracy of these approximation methods via methods like Richardson extrapolation. Having met these three objectives, we then extended our approach to solve singularly perturbed two-point boundary value problems with variable diffusion coefficients and analogous time-dependent PDEs. Careful analyses followed by extensive numerical simulations supporting theoretical findings are presented where necessary.
Mathematical modeling of TB disease dynamics in a crowded population
(University of the Western Cape, 2020) Maku Vyambwera, Sibaliwe
Tuberculosis is a bacterial infection which is a major cause of death worldwide. TB is a curable disease, however the bacterium can become resistant to the first line treatment against the disease. This leads to a disease called drug resistant TB that is difficult and expensive to treat. It is well-known that TB disease thrives in communities in overcrowded environments with poor ventilation, weak nutrition, inadequate or inaccessible medical care, etc, such as in some prisons or some refugee camps. In particular, the World Health Organization discovered that a number of prisoners come from socio-economic disadvantaged population where the burden of TB disease may be already high and access to medical care may be limited. In this dissertation we propose compartmental models of systems of differential equations to describe the population dynamics of TB disease under conditions of crowding. Such models can be used to make quantitative projections of TB prevalence and to measure the effect of interventions. Indeed we apply these models to specific regions and for specific purposes. The models are more widely applicable, however in this dissertation we calibrate and apply the models to prison populations.
Efficient variable mesh techniques to solve interior layer problems
(University of the Western Cape, 2020) Mbayi, Charles K.
Singularly perturbed problems have been studied extensively over the past few years from different perspectives. The recent research has focussed on the problems whose solutions possess interior layers. These interior layers appear in the interior of the domain, location of which is difficult to determine a-priori and hence making it difficult to investigate these problems analytically. This explains the need for approximation methods to gain some insight into the behaviour of the solution of such problems. Keeping this in mind, in this thesis we would like to explore a special class of numerical methods, namely, fitted finite difference methods to determine reliable solutions. As far as the fitted finite difference methods are concerned, they are grouped into two categories: fitted mesh finite difference methods (FMFDMs) and the fitted operator finite difference methods (FOFDMs). The aim of this thesis is to focus on the former. To this end, we note that FMFDMs have extensively been used for singularly perturbed two-point boundary value problems (TPBVPs) whose solutions possess boundary layers. However, they are not fully explored for problems whose solutions have interior layers. Hence, in this thesis, we intend firstly to design robust FMFDMs for singularly perturbed TPBVPs whose solutions possess interior layers and to improve accuracy of these approximation methods via methods like Richardson extrapolation. Then we extend these two ideas to solve such singularly perturbed TPBVPs with variable diffusion coefficients. The overall approach is further extended to parabolic singularly perturbed problems having constant as well as variable diffusion coefficients.
Measurements of edge uncolourability in cubic graphs
(University of the Western Cape, 2020) Allie, Imran
The history of the pursuit of uncolourable cubic graphs dates back more than a century. This pursuit has evolved from the slow discovery of individual uncolourable cubic graphs such as the famous Petersen graph and the Blanusa snarks, to discovering in nite classes of uncolourable cubic graphs such as the Louphekine and Goldberg snarks, to investigating parameters which measure the uncolourability of cubic graphs. These parameters include resistance, oddness and weak oddness, ow resistance, among others. In this thesis, we consider current ideas and problems regarding the uncolourability of cubic graphs, centering around these parameters. We introduce new ideas regarding the structural complexity of these graphs in question. In particular, we consider their 3-critical subgraphs, speci cally in relation to resistance. We further introduce new parameters which measure the uncolourability of cubic graphs, speci cally relating to their 3-critical subgraphs and various types of cubic graph reductions. This is also done with a view to identifying further problems of interest. This thesis also presents solutions and partial solutions to long-standing open conjectures relating in particular to oddness, weak oddness and resistance.
Fractional Black-Scholes equations and their robust numerical simulations
(University of the Western Cape, 2020) Nuugulu, Samuel Megameno
Conventional partial differential equations under the classical Black-Scholes approach have been extensively explored over the past few decades in solving option pricing problems. However, the underlying Efficient Market Hypothesis (EMH) of classical economic theory neglects the effects of memory in asset return series, though memory has long been observed in a number financial data. With advancements in computational methodologies, it has now become possible to model different real life physical phenomenons using complex approaches such as, fractional differential equations (FDEs). Fractional models are generalised models which based on literature have been found appropriate for explaining memory effects observed in a number of financial markets including the stock market. The use of fractional model has thus recently taken over the context of academic literatures and debates on financial modelling. Fractional models are usually of a non-linear and complex nature, which pose a considerable amount of computational and theoretical difficulties in deriving their analytical solutions. To the best of our knowledge, currently, there exist no tractable exact/analytical solution methods for solving fractional Black-Scholes equations, and as such, numerical solution methods become of a vital importance in understanding nature of solutions to such models. This thesis therefore, serves to derive some Generalised (fractional) Black-Scholes Partial Differential Equations (fBS-PDEs), as well as, propose their respective tractable, efficient and robust numerical simulation methods.
Graphs of integral distance and their properties
(University of the Western Cape, 2021) Habineza, Olivier
Understanding the geometries of points in space has been attractive to mathematicians for ages. As a model, twelve years ago, Kurz and Meyer [32] considered point sets in the m-dimensional a ne space Fmq over a nite eld Fq with q = pr elements, p prime, where each squared Euclidean distance of two points is a square in Fq: The latter points are said to be at integral distance in Fmq , and the sets above are called integral point sets.
Reliable numerical simulations of problems for pricing real estate derivatives
(University of the Western Cape, 2022) Dube, Mbakisi
The globalisation of nancial systems has presented new challenges to investing in real estate assets. For example, any crisis occurring in one real estate market will have an adverse e ect on other markets regardless of them being vastly geographically distant from each other. This interconnectedness is due to the ease of acquiring property portfolios using capital from investors coming from di erent jurisdictions who would have pooled their capital to acquire those properties. This illustrates how capital can ow internationally in the process creating linkages between various markets. The globalisation of real estate markets and the amount of capital invested in them results in a need for innovative mechanisms to manage and contain the risk of having shocks in the real estate market destabilising and a ecting international nancial markets. Risk management for real estate portfolios can be e ectively done through the use of real estate index based derivatives.
Analysis and simulation of some ecological models describing weed-water hyacinth dynamics in Lake Tana, Ethiopia
(University of the Western Cape, 2022) Belay, Yeshambel Azene
The water hyacinth plant is notorious for being the world’s worst aquatic weed. In Ethiopia, it invades lake Tana which is the largest lake in Ethiopia. This is attracting attention of many researchers including mathematical modellers. To this end, in this thesis, we consider systems modelling water hyacinth effect in lake ecosystem. Among the classes of models considered in this work includes, the refuge effect of water hyacinth in lake ecosystem, the effect of water hyacinth in a lake eutrophication dynamics and singularly perturbed (slow-fast) phytoplankton-zooplankton, predator-prey systems with water hyacinth’s refuge effect.
Analysis and robust simulation of mathematical models of HIV and Leishmaniasis co-infection
(University of the Western Cape, 2022) Adamu, Elias M
The purpose of this thesis is the theoretical and numerical study of epidemiological models of Visceral Leishmaniasis (VL) and human immunode ciency virus (HIV) coinfection. We study these models theoretically and design and analyze robust numerical methods to solve them. The rst sub-model describes the transmission dynamics of VL and incorporates three populations: the human, the reservoir, and the vector host populations. Then we study the Anthroponotic Visceral Leishmaniasis (AVL) between the human and the vector host population. The second sub-model describes a deterministic SIAT (Susceptible-Infected-AIDS-Treated) HIV/AIDS model that incorporates the treatments of HIV-infected and AIDS-infected individuals.
Codes, graphs and designs related to iterated line graphs of complete graphs
(University of the Western Cape, 2011) Kumwenda, Khumbo
In this thesis, we describe linear codes over prime fields obtained from incidence designs of iterated line graphs of complete graphs Li(Kn) where i = 1,2. In the binary case, results are extended to codes from neighbourhood designs of the line graphs Li+l(Kn) using certain elementary relations. Codes from incidence designs of complete graphs, Kn' and neighbourhood designs of their line graphs, £1(Kn) (the so-called triangular graphs), have been considered elsewhere by others. We consider codes from incidence designs of Ll(Kn) and L2(Kn), and neighbourhood designs of L2(Kn) and L3(Kn). In each case, the basic parameters of the codes are determined. Further, we introduce a family of vertex-transitive graphs Rn that are embeddable into the strong product Ll(Kn) ~ K2' of triangular graphs and K2' a class that at first sight may seem unnatural but, on closer look, is a repository of graphs rich with combinatorial structures. For instance, unlike most regular graphs considered here and elsewhere that only come with incidence and neighbourhood designs, Rn also has what we have termed as 6-cycle designs. These are designs in which the point set contains vertices of the graph and every block contains vertices of a 6-cycle in the graph. Also, binary codes from incidence matrices of these graphs have other minimum words in addition to incidence vectors of the blocks. In addition, these graphs have induced subgraphs isomorphic to the family Hn of complete porcupines (see Definition 4.11). We describe codes from incidence matrices of Rn and Hn and determine their parameters. The discussion is concluded with a look at complements of Rn and Hn, respectively denoted by Rn and Hn. Among others, the complements rn are contained in the union of the categorical product Ll(Kn) x Kn' and the categorical product £1(Kn) x Kn (where £1(Kn) is the complement of the iii triangular graph £1(Kn)). As with the other graphs, we have also considered codes from the span of incidence matrices of Rn and Hn and determined some of their properties. In each case, automorphisms of the graphs, designs and codes have been determined. For the codes from incidence designs of triangular graphs, embeddings of Ll(Kn) x K2 and complements of complete porcupines, we have exhibited permutation decoding sets (PD-sets) for correcting up to terrors where t is the full error-correcting capacity of the codes. For the remaining codes, we have only been able to determine PD-sets for which it is possible to correct a fraction of t-errors (partial permutation decoding). For these codes, we have also determined the number of errors that can be corrected by permutation decoding in the worst-case.

Browse

Recent Submissions