In this study we generate the matching conditions for a spherically symmetric radiating star in the presence of shear. Two new exact solutions to the Einstein held equations are presented which model a relativistic radiating sphere. We examine the role of anisotropy in the thermal evolution of a radiating star undergoing continued dissipative gravitational collapse in the presence of shear. Our model was the first study to incorporate both shear and pressure anisotropy, and these results were published in 2006. The physical viability of a recently proposed model of a shear-free spherically symmetric star undergoing gravitational collapse without the formation of a horizon is investigated. These original results were published in 2007. The temperature profiles of both models are studied within the framework of extended irreversible thermodynamics. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2008.

The epidemiological study of tuberculosis (TB) has been ongoing for several decades, but the most effective control strategy is yet to be completely understood. The basic reproduction number, R₀, has been found to be plausible indicator for TB control rate. The R₀ value is the average number of secondary TB cases produced by a typical infective individual in a completely susceptible population during its entire infectious period. In this study we develop two SEIR models for TB transmission; one involving treatment of active TB only, with the second incorporating both active TB treatment and post-exposure prophylaxis (PEP) treatment for latent TB. Using the next generation matrix method we obtain R₀. We determine the disease free equilibrium (DFE) point and the endemic equilibrium (EE) point. Global stability conditions of DFE are determined using the Castillo-Chavez theorem. Through model analysis of the reproduction number, R₀, we find that for R₀ < 1, the infection will die out. The value of R₀ > 1 implies that the disease will spread within the population. Through stability analysis, we show that the model exhibits backward bifurcation, a phenomenon allowing multiple stable states for fixed model parameter value. MATLAB ode45 solver was used to simulate the model numerically. Using the Latin Hypercube Sampling technique the model is sensitive to treatment and disease transmission parameters, suggesting that to control the disease, more emphasis should be placed on treatment and on reducing TB transmission. For the second model, which incorporated treatment with post-exposure prophylaxis for latently infected individuals, by means of simulations, we found that treatment of latently infected individuals may reduce R₀. Numerical simulations on the latter model also showed that it may be better to introduce a hybrid of active treatment and post-exposure treatment of the latent class. The force of infection was found to reduce when this hybrid control strategy is present. Contour plots and PRCC values highlighted the important parameters that influence the size of the Infective class. The implications of these findings are that TB control measures should emphasise on treatment. Our simplified models assume that there is homogeneous mixing. The model used have not been validated against empirical data. / M.Sc. University of KwaZulu-Natal, Pietermaritzburg 2014.

Let F be a finite field of q elements and G be a primitive group on a finite set . Then there is a G-action on , namely a map G ! , (g; !) 7! !g = g!; satisfying !gg0 = (gg0)! = g(g0!) for all g; g0 2 G and all ! 2 , and that !1 = 1! = ! for all ! 2 : Let F = ff j f : ! Fg, be the vector space over F with basis . Extending the G-action on linearly, F becomes an FG-module called an FG- permutation module. We are interested in finding all G-invariant FG-submodules, i.e., codes in F . The elements f 2 F are written in the form f = P !2 a! ! where ! is a characteristic function. The natural action of an element g 2 G is given by g P !2 a! ! = P !2 a! g(!): This action of G preserves the natural bilinear form defined by * X a! !; X b! ! + = X a!b!: In this thesis a program is proposed on how to determine codes with given primitive permutation group. The approach is modular representation theoretic and based on a study of maximal submodules of permutation modules F defined by the action of a finite group G on G-sets = G=Gx. This approach provides the advantage of an explicit basis for the code. There appear slightly different concepts of (linear) codes in the literature. Following Knapp and Schmid [83] a code over some finite field F will be a triple (V; ; F), where V = F is a free FG-module of finite rank with basis and a submodule C. By convention we call C a code having ambient space V and ambient basis . F is the alphabet of the code C, the degree n of V its length, and C is an [n; k]-code if C is a free module of dimension k. In this thesis we have surveyed some known methods of constructing codes from primitive permutation representations of finite groups. Generally, our program is more inclusive than these methods as the codes obtained using our approach include the codes obtained using these other methods. The designs obtained by other authors (see for example [40]) are found using our method, and these are in general defined by the support of the codewords of given weight in the codes. Moreover, this method allows for a geometric interpretation of many classes of codewords, and helps establish links with other combinatorial structures, such as designs and graphs. To illustrate the program we determine all 2-modular codes that admit the two known non-isomorphic simple linear groups of order 20160, namely L3(4) and L4(2) = A8. In the process we enumerate and classify all codes preserved by such groups, and provide the lattice of submodules for the corresponding permutation modules. It turns out that there are no self-orthogonal or self-dual codes invariant under these groups, and also that the automorphism groups of their respective codes are in most cases not the prescribed groups. We make use of the Assmus Matson Theorem and the Mac Williams identities in the study of the dual codes. We observe that in all cases the sets of several classes of non-trivial codewords are stabilized by maximal subgroups of the automorphism groups of the codes. The study of the codes invariant under the simple linear group L4(2) leads as a by-product to a unique flag-transitive, point primitive symmetric 2-(64; 28; 12) design preserved by the affi ne group of type 26:S6(2). This has consequently prompted the study of binary codes from the row span of the adjacency matrices of a class of 46 non-isomorphic symmetric 2-(64; 28; 12) designs invariant under the Frobenius group of order 21. Codes obtained from the orbit matrices of these designs have also been studied. The thesis concludes with a discussion of codes that are left invariant by the simple symplectic group S6(2) in all its 2-modular primitive permutation representations. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2012.

In this dissertation we study radiating stars in strong gravitational elds. We generate new classes of exact solutions to the Einstein eld equations and the boundary condition applicable to radiating relativistic stars. The model of a radiating star in general relativity, matching to the Vaidya exterior spacetime, is reviewed. The boundary condition is converted to a Riccati equation and we consider both cases involving geodesic and non-geodesic particle trajectories. We present the metrics found previously. We rst solve the boundary condition for the geodesic case and nd the gravitational potentials which are expanding and shearing. This is a new result. Secondly the boundary condition is analysed for the non-geodesic case and we seek new gravitational potentials which are accelerating, expanding and shearing. We are able to identify only geodesic solutions for this second case; this appears to be a new class of models. The solutions found are presented in terms of elementary functions which are helpful in studying the physical properties. The new solutions found cannot be categorised in existing classes of known solutions; they are examples of a new generic class di erent from previous studies. The matter variables of the model are generated . / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2012.

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2007. / In this thesis, we surveyed the most important methods for factorization of polynomials over a global field, focusing on their strengths and showing their most striking disadvantages. The algorithms we have selected are all modular algorithms. They rely on the Hensel factorization technique, which can be applied to all global fields giving an output in a local field that can be computed to a large enough precision. The crucial phase of the reconstruction of the irreducible global factors from the local ones, determines the difference between these algorithms. For different fields and cases, different techniques have been used such as residue class computations, ideal calculus, lattice techniques. The tendency to combine ideas from different methods has been of interest as it improves the running time. This appears for instance in the latest method due to van Hoeij, concerning the factorization over a number field. The ideas here can be used over a global function field in the form given by Belabas et al. using the logarithmic derivative instead of Newton sums. Complexity analysis was not our objective, nevertheless it was important to mention certain results as part of the properties of these algorithms.

Thesis (MSc (Mathematical Sciences. Physical and Mathematical Analysis))--University of Stellenbosch, 2006. / This work studies the regularity (or smoothness) of continuous finitely supported refinable functions which are mainly encountered in multiresolution analysis, iterative interpolation processes, signal analysis, etc. Here, we present various kinds of sufficient conditions on a given mask to guarantee the regularity class of the corresponding refinable function. First, we introduce and analyze the cardinal B-splines Nm, m ∈ N. In particular, we show that these functions are refinable and belong to the smoothness class Cm−2(R). As a generalization of the cardinal B-splines, we proceed to discuss refinable functions with positive mask coefficients. A standard result on the existence of a refinable function in the case of positive masks is quoted. Following [13], we extend the regularity result in [25], and we provide an example which illustrates the fact that the associated symbol to a given positive mask need not be a Hurwitz polynomial for its corresponding refinable function to be in a specified smoothness class. Furthermore, we apply our regularity result to an integral equation. An important tool for our work is Fourier analysis, from which we state some standard results and give the proof of a non-standard result. Next, we study the H¨older regularity of refinable functions, whose associated mask coefficients are not necessarily positive, by estimating the rate of decay of their Fourier transforms. After showing the embedding of certain Sobolev spaces into a H¨older regularity space, we proceed to discuss sufficient conditions for a given refinable function to be in such a H¨older space. We specifically express the minimum H¨older regularity of refinable functions as a function of the spectral radius of an associated transfer operator acting on a finite dimensional space of trigonometric polynomials. We apply our Fourier-based regularity results to the Daubechies and Dubuc-Deslauriers refinable functions, as well as to a one-parameter family of refinable functions, and then compare our regularity estimates with those obtained by means of a subdivision-based result from [28]. Moreover, we provide graphical examples to illustrate the theory developed.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2007. / In this thesis, we study the numerical approximation of indefinite integrals with algebraic or logarithmic end-point singularities. We show the derivation of the two quadrature formulas proposed by Haber based on the sinc method, as well as, on the basis of error analysis, by means of variable transformations (Single and Double Exponential), the derivation of two other formulas: Stenger’s Single Exponential (SE) formula and Tanaka et al.’s Double Exponential (DE) sinc method. Important tools for our work are residue calculus, functional analysis and Fourier analysis from which we state some standard results, and give the proof of some of them. Next, we introduce the Paley-Wiener class of functions, define the sinc function, cardinal function, when a function decays single and double exponentially, and prove some of their interesting properties. Since the four formulas involve a conformal transformation, we show how to transform from the interval (−¥,¥) to (−1, 1). In addition, we show how to implement the four formulas on two computational examples which are our test problems, and illustrate our numerical results by means of tables and figures. Furthermore, from an application of the four quadrature formulas on two test problems, a plot of the maximum absolute error against the number of function evaluations, reveals a faster convergence to the exact solution by Tanaka et al.’s DE sinc method than by the other three formulas. Next, we convert the indefinite integrals (our test problems) into ordinary differential equations (ODE) with suitable initial values, in the hope that ODE solvers such as Matlabr ode45 or Mathematicar NDSolve will be able to solve the resulting IVPs. But they all failed because of singularities in the initial value. In summary, of the four quadrature formulas, Tanaka et al.’s DE sinc method gives more accurate results than the others and it will be noted that all the formulas are applicable to both singular and non-singular integrals.

Thesis (MSc (Mathematical Sciences))-- University of Stellenbosch, 2005. / It is an open question whether or not one can always lift Galois extensions of smooth algebraic curves in characteristic p to Galois extensions of smooth relative curves in characteristic 0. In this thesis we study some of the available techniques and partial solutions to this problem. Our studies include the techniques of Oort, Sekiguchi and Suwa where the lifting problem is approached via a connection with lifting group schemes. We then move to the topic of singular liftings and for this we study the approach of Garuti. Thereafter, we move to the wild smooth setting again where we study the crucial local − global principle, and apply it by illustrating how Green and Matignon solved the p2-lifting problem.

Thesis (MSc (Mathematics))--University of Stellenbsoch, 2009. / Please refer to full text for abstract

Thesis (MSc (Mathematics))--University of Stellenbosch, 2009. / ENGLISH ABSTRACT: In this thesis, a mathematical model describing the interaction between HIV and TB in the presence of TB superinfection is presented. The model takes into account two strains of Mycobacterium tuberculosis (MTB), where one strain is drug-sensitive and the other is resistant to at least one of the first-line anti-tuberculosis drugs. The impact of TB superinfection on the incidence and prevalence of TB in HIV-negative and HIVTB coinfected individuals is evaluated. Various control measures such as condom use, antiretroviral therapy, isoniazid preventive therapy and increased TB detection are studied using this model. Numerical results show that TB superinfection increases the prevalence and incidence of TB and its impact is more in HIV-negative than HIV-TB coinfected individuals. The results also show that TB superinfection promotes strain coexistence and increases the associated HIV mortality. Increased condom use was found to have a high positive impact towards the control of the two epidemics. Antiretroviral therapy decreases the TB notification rate and its impact on HIV prevalence increases with the coverage and efficacy. Isoniazid preventive therapy has a clear effect on the TB prevalence. Finally, increased TB detection was found to have a less impact on the TB incidence in HIV-TB coinfected individuals / AFRIKAANSE OPSOMMING: In hierdie verhandeling word ´n wiskundige model vir die interaksie tussen MIV en TB, in ´n situasie met TB superinfeksie voorgelˆe. Die model neem twee variante van TB in ag. Een van die variante is sensitief vir MTB behandeling, terwyl die ander weerstandig is vir ten minste een van die eerste-linie TB behandenings. Die impak van TB superinfeksie op die insidensie and prevalensie van TB in MIV negatiewe en MIV-TB ko-ge˜ınfekteerde individu word ondersoek. Veskeie beheer maatreels soos kondoom gebruik, anti-retrovirale behandeling (vir MIV) en isonazid voorkomende behandeling en verhoodge TB deteksie (vir TB) word ondersoek. Numeriese resultate wys TB superinfeksie verhoog die prevalense en insidensie van TB en dat dit ´n groter bydrae maak by MIV negatief as by MIV-TB ko-geinfekteerde individu. Die resultate wys veder TB superinfeksie promofeer variant kohabitasie en verhoog MIV verwante mortalitieit. Verhoogde kondoom gebruik is gevind om ´n positiewe bydrae te maak tot die beheer van beide epidemies. Anti-retrovirale terapie verlaag die TB aanmeldings koers en die impak van ART verhoog saam met ´n verhoging in die dekking en effektiwiteit daarvan. Voorkomende behandeling het ´n beduidende impak op TB prevalensie. Ons vind dat TB deteksie ´n beperkte impak maak op TB insidensie by MIV-TB ko-geinfekteerde individu