Masters Degrees (Applied Mathematics)
Permanent URI for this collectionhttps://hdl.handle.net/10413/7108
Browse
Browsing Masters Degrees (Applied Mathematics) by Title
Now showing 1 - 20 of 74
- Results Per Page
- Sort Options
Item Algebraizing deductive systems.(1995) Van Alten, Clint Johann.; Raftery, James Gordon.; Sturm, Teo.Abstract available in PDF.Item Analysis and numerical solutions of fragmentation equation with transport.(2012) Wetsi, Poka David.; Banasiak, Jacek.; Shindin, Sergey Konstantinovich.Fragmentation equations occur naturally in many real world problems, see [ZM85, ZM86, HEL91, CEH91, HGEL96, SLLM00, Ban02, BL03, Ban04, BA06] and references therein. Mathematical study of these equations is mostly concentrated on building existence and uniqueness theories and on qualitative analysis of solutions (shattering), some effort has be done in finding solutions analytically. In this project, we deal with numerical analysis of fragmentation equation with transport. First, we provide some existence results in Banach and Hilbert settings, then we turn to numerical analysis. For this approximation and interpolation theory for generalized Laguerre functions is derived. Using these results we formulate Laguerre pseudospectral method and provide its stability and convergence analysis. The project is concluded with several numerical experiments.Item Analysis of models arising from heat conduction through fins using Lie symmetries and Tanh method.(2021) Bulunga, Vusi Andile.; Mhlongo, Mfanafikile Don.Abstract available in PDF.Item Application of the wavelet transform for sparse matrix systems and PDEs.(2009) Karambal, Issa.; Paramasur, Nabendra.; Singh, Pravin.We consider the application of the wavelet transform for solving sparse matrix systems and partial differential equations. The first part is devoted to the theory and algorithms of wavelets. The second part is concerned with the sparse representation of matrices and well-known operators. The third part is directed to the application of wavelets to partial differential equations, and to sparse linear systems resulting from differential equations. We present several numerical examples and simulations for the above cases.Item Assessment of variability in on-farm trials : a Uganda case.(2002) Lapaka, Odong Thomas; Njuho, Peter Mungai.On-farm trials techniques have become an integral part of research aimed at improving agricultural production especially in subsistence farming. The poor performance of certain technologies on the farmers' fields known to have performed well on stations have been of concern. Traditionally, on-farm trials are meant to address such discrepancies. The main problems associated with on-farm trials in most developing countries are high variability and inappropriate application of statistical knowledge known to work on station to on-farm situation. Characterisation of various on-farm variability and orientation of existing statistical methods may lead to improved agricultural research. Characterization of the various forms of variability in on-farm trials was conducted. Based on these forms of variability, estimation procedures and their strength have been assessed. Special analytical tools for handling non-replicated experiments known to be common to on-farm trials are presented. The above stated procedures have been illustrated through a review of Uganda case. To understand on-farm variability require grouping of sources of variability into agronomic, animal and socioeconomic components. This led to a deeper understanding of levels of variability and appropriate estimation procedures. The mixed model, modified stability analysis and additive main effects and multiplicative interaction methods have been found to play a role in on-farm trials. Proper approach to on-farm trials and application of appropriate statistical tools will lead to efficient results that will subsequently enhance agricultural production especially under subsistence farming.Item Bayesian analysis of cosmological models.(2010) Moodley, Darell.; Moodley, Kavilan.; Sealfon, C.In this thesis, we utilise the framework of Bayesian statistics to discriminate between models of the cosmological mass function. We first review the cosmological model and the formation and distribution of galaxy clusters before formulating a statistic within the Bayesian framework, namely the Bayesian razor, that allows model testing of probability distributions. The Bayesian razor is used to discriminate between three popular mass functions, namely the Press-Schechter, Sheth-Tormen and normalisable Tinker models. With a small number of particles in the simulation, we find that the simpler model is preferred due to the Occam’s razor effect, but as the size of the simulation increases the more complex model, if taken to be the true model, is preferred. We establish criteria on the size of the simulation that is required to decisively favour a given model and investigate the dependence of the simulation size on the threshold mass for clusters, and prior probability distributions. Finally we outline how our method can be extended to consider more realistic N-body simulations or be applied to observational data.Item Bounds on distances for spanning trees of graphs.(2018) Ntuli, Mthobisi Luca.; Morgan, Megan Jane.; Mukwembi, Simon.In graph theory, there are several techniques known in literature for constructing spanning trees. Some of these techniques yield spanning trees with many leaves. We will use these constructed spanning trees to bound several distance parameters. The cardinality of the vertex set of graph G is called the order, n(G) or n. The cardinality of the edge set of graph G is called the size, m(G) or m. The minimum degree of G, (G) or , is the minimum degree among the degrees of the vertices of G: A spanning tree T of a graph G is a subgraph that is a tree which includes all the vertices of G. The distance d(u; v) between two vertices u and v is the length of a shortest u-v path of G. The eccentricity, ec (v), of a vertex v 2 V (G) is the maximum distance from it to any other vertex in G. The diameter, diam(G) or d, is the maximum eccentricity amongst all vertices of G. The radius, rad(G), is the minimum eccentricity among all vertices of G. The average distance of a graph G, (G), is the expected distance between a randomly chosen pair of distinct vertices. We investigate how each constructed spanning tree can be used to bound diam- eter, radius or average distance in terms of order, size and minimum degree. The techniques to be considered include the radius-preserving spanning trees by Erd}os et al, the Ding et al technique, and the Dankelmann and Entringer technique. Finally, we use the Kleitman and West dead leaves technique to construct spanning trees with many leaves for various values of the minimum degree k (for k = 3; 4 and k > 4) and order n. We then use the leaf number to bound diameter.Item Bounds on the extremal eigenvalues of positive definite matrices.(2018) Jele, Thokozani Cyprian Martin.; Singh, Virath Sewnath.; Singh, Pravin.The minimum and maximum eigenvalues of a positive de nite matrix are crucial to determining the condition number of linear systems. These can be bounded below and above respectively using the Gershgorin circle theorem. Here we seek upper bounds for the minimum eigenvalue and lower bounds for the maximum eigenvalue. Intervals containing the extremal eigenvalues are obtained for the special case of Toeplitz matrices. The theory of quadratic forms is discussed in detail as it is fundamental in obtaining these bounds.Item Categorical systems biology : an appreciation of categorical arguments in cellular modelling.(2012) Songa, Maurine Atieno.; Banasiak, Jacek.; Amery, Gareth.With big science projects like the human genome project, [2], and preliminary attempts to seriously study brain activity, e.g. [9], mathematical biology has come of age, employing formalisms and tools from most branches of mathematics. Recent results, [51] and [53], have extended the relational (or categorical) approach of Rosen [44], to demonstrate that (in a very general class of systems) cellular self-organization/self-replication is implicit in metabolism and repair/stability. This is a powerful philosophical statement and removes the need of teleological argument. However, the result carries a technical limitation to Cartesian closed categories, which excludes many mathematical languages. We review the relevant literature on metabolic-repair pathways, category theory and systems theory, before performing a critique of this work. We find that the restriction to Cartesian closed categories is purely for simplicity, and describe how equivalent arguments may be built for monoidal closed categories. Moreover, any symmetric monoidal category may be "embedded" in a closed one. We discuss how these constructions/techniques provide the formal structure to treat self-organization/self-replication in most contemporary mathematical (modelling) languages. These results signicantly soften the impact on current modelling paradigms while extending the philosophical implications.Item The characteristic approach in determining first integrals of a predator-prey system.(2016) Mahomed, Abu Bakr.; Narain, Rivendra Basanth.Predator-Prey systems are an intriguing symbiosis of living species that interplay during the fluctuations of birth, growth and death during any period. In the light of understanding the behavioural patterns of the species, models are constructed via differential equations. These differential equations can be solved through a variety of techniques. We focus on applying the characteristic method via the multiplier approach. The multiplier is applied to the differential equation. This leads to a first integral which can be used to obtain a solution for the system under certain initial conditions. We then look at the comparison of first integrals by using two different approaches for various biological models. The method of the Jacobi Last Multiplier is used to obtain a Lagrangian. The Lagrangian can be used via Noether’s Theorem to obtain a first integral for the system.Item Charge distribution in neutron stars.(2017) Ndebele, Sakhile Mpilo.; Ray, Subharthi.In previous studies, people have shown that compact stars, like the neutron stars and quark stars, can hold a lot of charge during their formation resulting in a large mass and radius. It was also argued that when the charges leave the system due to repulsion from the self created field, these might render a secondary collapse to a charged black hole. In the present work, we have taken a particular type of charge distribution, with varying parameters, such that changing these parameters mimic the situation when the charge particles leaving the system. We have made a systematic study of each stage of the charge distributions. Our results reveal that when the charge distribution deviates slightly from the scenario where the charge density is proportional to the mass density, then the system is no longer able to retain the large mass and radius, and quickly attains a lower mass and radius.Item Chebyshev spectral and pseudo-spectral methods in unbounded domains.(2015) Govinder, Saieshan.; Shindin, Sergey Konstantinovich.; Parumasur, Nabendra.Chebyshev type spectral methods are widely used in numerical simulations of PDEs posed in unbounded domains. Such methods have a number of important computational advantages. In particular, they admit very efficient practical implementation. However, the stability and convergence analysis of these methods require deep understanding of approximation properties of the underlying functional basis. In this project, we deal with Chebyshev spectral and pseudo-spectral methods in unbounded domains. The first part of the project deals with theoretical analysis of Chebyshev-type spectral projection and interpolation operators in Bessel potential spaces. In the second part, we provide rigorous analyses of Chebyshev-type pseudo-spectral (collocation) scheme applied to the nonlinear Schrodinger equation. The project is concluded with several numerical experiments.Item Commissioning and characterisation of the C-Band All-Sky Survey Southern telescope.(2015) Allotey, Johannes Adotey.; Chiang, Hsin Cynthia.Abstract available in PDF file.Item A comparison study of Chebyshev spectral collocation based methods for solving nonlinear second order evolution equations.(2015) Kheswa, Khayelihle Allen.; Motsa, Sandile Sydney.In this study Spectral Quasilinearisation Method (SQLM) coupled with finite differ- ence and Bivariate Spectral Quasilinearisation Method (BSQLM) in solving second order nonlinear evolution partial differential equations are compared. Both meth- ods use Newton-Raphson quasilinearisation method (QLM) and Chebyshev spectral collocation based on Lagrange interpolation to solve the governing equations. The Spectral Quasilinearisation Method coupled with finite difference is obtained by ap- plying the spectral collocation method on space derivatives and finite difference of time derivatives while the BSQLM is a Bivariate Lagrange interpolation based scheme in which the spectral collocation method is applied independently to both time and space derivatives. The applicability of these methods is shown by solving a class of second order nonlinear evolution partial differential equations (NPDEs), namely Burgers equation, Burgers-Fisher, Fisher's equation, Newell-Whitehead-Segel equa- tion and Zeldovich equation that arise in some fields of science and engineering. The numerical approximation results are validated for accuracy by comparing them with exact solutions. Tables for Explicit, Implicit and Crank-Nicolson SQLM and BSQLM with their computational times were generated for comparison; the order of accuracy for each method and error graphs are presented.Item Conformal motions in Bianchi I spacetime.(1992) Lortan, Darren Brendan.; Maharaj, Sunil Dutt.In this thesis we study the physical properties of the manifold in general relativity that admits a conformal motion. The results obtained are general as the metric tensor field is not specified. We obtain the Lie derivative along a conformal Killing vector of the kinematical and dynamical quantities for the general energy-momentum tensor of neutral matter. Equations obtained previously are regained as special cases from our results. We also find the Lie derivative of the energy-momentum tensor for the electromagnetic field. In particular we comprehensively study conformal symmetries in the Bianchi I spacetime. The conformal Killing vector equation is integrated to obtain the general conformal Killing vector and the conformal factor subject to integrability conditions. These conditions place restrictions on the metric functions. A particular solution is exhibited which demonstrates that these conditions have a nonempty solution set. The solution obtained is a generalisation of the results of Moodley (1991) who considered locally rotationally symmetric spacetimes. The Killing vectors are regained as special cases of the conformal solution. There do not exist any proper special conformal Killing vectors in the Bianchi I spacetime. The homothetic vector is found for a nonvanishing constant conformal factor. We establish that the vacuum Kasner solution is the only Bianchi I spacetime that admits a homothetic vector. Furthermore we isolate a class of vectors from the solution which causes the Bianchi I model to degenerate into a spacetime of higher symmetry.Item Conformal symmetries : solutions in two classes of cosmological models.(1991) Moodley, Manikam.; Maharaj, Sunil Dutt.In this thesis we study the conformal symmetries in two locally rotationally symmetric spacetimes and the homothetic symmetries of a Bianchi I spacetime. The conformal Killing equation in a class AIa spacetime (MacCallum 1980), with a G4 of motions, is integrated to obtain the general solution subject to integrability conditions. These conditions are comprehensively analysed to determine the restrictions on the metric functions. The Killing vectors are contained in the general conformal solution. The homothetic vector is obtained and the explicit functional dependence of the metric functions determined. The class AIa spacetime does not admit a nontrivial special conformal factor. We also integrate the conformal Killing equation in the anisotropic locally rotationally symmetric spacetime of class A3 (MacCallum 1980), with a G4 of motions, to obtain the general conformal Killing vector and the conformal factor subject to integrability conditions. The Killing vectors are obtained as a special case from the general conformal solution. The homothetic vector is found for a nonzero constant conformal factor. The explicit functional form of the metric functions is determined for the existence of this homothetic vector. The spatially homogeneous and anisotropic A3 spacetime also does not admit a nontrivial special conformal vector. In the Bianchi I spacetime, with a G3 of motions, the conformal Killing equation is integrated for a constant conformal factor to generate the homothetic symmetries. The integrability conditions are solved to determine the functional dependence of the three time-dependent metric functions.Item Conformal symmetries and classification in shear-free spherically symmetric spacetimes.(2014) Manjonjo, Addial Mackingtosh.; Maharaj, Sunil Dutt.; Moopanar, Selvandren.In this thesis we study the conformal geometry of static and non-static spherically symmetric spacetimes. We analyse the general solution of the conformal Killing vector equation subject to integrability conditions which place restrictions on the metric func- tions. TheWeyl tensor is used to characterise the conformal geometry, and we calculate the Weyl tensor components for the spherically symmetric line element. The accuracy of our results is veri ed using Mathematica (Wolfram 2010) and Maple (2009). We show that the standard result in the conformal motions for static spacetimes is in- correct. This mistake is identi ed and corrected. Two nonlinear ordinary differential equations are derived in the classi cation of static spacetimes. Both equations are solved in general. Two nonlinear partial differential equations are derived in the classi- cation of non-static spacetimes. The rst equation is solved in general and the second equation admits a particular solution. Our treatment is the rst complete classi cation of conformal motions in static and non-static spherically symmetric spacetimes using the Weyl tensor.Item A covariant approach to LRS-II spacetime matching.(2017) Paul, Erwin Roderic.; Goswami, Rituparno.; Maharaj, Sunil Dutt.In this thesis we examine the spacetime matching conditions covariantly for Locally Rotationally Symmetric class II (LRS-II) spacetimes, of which spherical symmetry is a special case. We use the semi-tetrad 1+1+2 covariant formalism and look at two general spacetime regions in LRS-II and match them across a timelike hypersurface using the Israel-Darmois matching conditions. This gives a new and unique result which is transparently presented in terms of the matching of various geometrical quantities (e.g. the expansion, shear, acceleration). Thereafter we apply the new result to the case involving a general spherically symmetric spacetime, representing for instance the interior of a star, and the Schwarzschild spacetime, which could represent the exterior. It is shown that the matching conditions make the Misner-Sharp and Schwarzschild masses exactly the same at the boundary, and the pressure is zero on the boundary.Item Credit derivative valuation and parameter estimation for CIR and Vasicek-type models.(2013) Maboulou, Alma Prell Bimbabou.; Arunakirinathar, Kanagaratnam.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.Item Degree theory in nonlinear functional analysis.(1989) Pillay, Paranjothi.; Hill, C. K.The objective of this dissertation is to expand on the proofs and concepts of Degree Theory, dealt with in chapters 1 and 2 of Deimling [28], to make it more readable and accessible to anyone who is interested in the field. Chapter 1 is an introduction and contains the basic requirements for the subsequent chapters. The remaining chapters aim at defining a ll-valued map D (the degree) on the set M = {(F, Ω, y) / Ω C X open, F : Ὠ → X, y ɇ F(∂Ω)} (each time, the elements of M satisfying extra conditions) that satisfies : (D1) D(I, Ω, y) = 1 if y Є Ω. (D2) D(F, Ω, y) = D(F, Ω1 , y) + D(F, Ω2, y) if Ω1 and Ω2 are disjoint open subsets of Ω o such that y ɇ F(Ὠ \ Ω1 U Ω2 ). (D3) D(I - H(t, .), Ω, y(t)) is independent of t if H : J x Ὠ →X and y : J → X. An important property that follows from these three properties is (D4) F-1(y) ≠ Ø if D(F, Ω, y) ≠ 0. This property ensures that equations of the form Fx = y have solutions if D(F, Ω, y) ≠ 0. Another property that features in these chapters is the Borsuk property which gives us conditions under which the degree is odd and hence nonzero.