Masters Degrees (Mathematics and Computer Science Education)

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Dynamics of dissipative gravitational collapse.
(2008) Naidu, Nolene Ferrari.; Maharaj, Sunil Dutt.; Govender, Megandren.
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.
The pricing theory of Asian options.
(2007) Mkhize, Ngenisile Grace Zanele; Xu, Hongjun.
An Asian option is an example of exotic options. Its payoff depends on the average of the underlying asset prices. The average may be over the entire time period between initiation and expiration or may be over some period of time that begins later than the initiation of the option and ends with the options expiration. The average may be from continuous sampling or may be from discrete sampling. The primary reason to base an option payoff on an average asset price is to make it more difficult for anyone to significantly affect the payoff by manipulation of the underlying asset price. The price of Asian options is not known in closed form, in general, if the arithmetic average is taken into effect. In this dissertation, we shall investigate the pricing theory for Asian options. After a brief introduction to the Black-Scholes theory, we derive the partial differential equations for the value process of an Asian option to satisfy. We do this in several approaches, including the usual extension to Asian options of the Black-Scholes, and the sophisticated martingale approach. Both fixed and floating strike are considered. In the case of the geometric average, we derive a closed form solution for the Asian option. Moreover, we investigate the Asian option price theory under stochastic volatility which is a recent trend in the study of path-dependent option theory.
Global embeddings of pseudo-Riemannian spaces.
(2007) Moodley, Jothi.; Amery, Gareth.
Motivated by various higher dimensional theories in high-energy-physics and cosmology, we consider the local and global isometric embeddings of pseudo-Riemannian manifolds into manifolds of higher dimensions. We provide the necessary background in general relativity, topology and differential geometry, and present the technique for local isometric embeddings. Since an understanding of the local results is key to the development of global embeddings, we review some local existence theorems for general pseudo-Riemannian embedding spaces. In order to gain insight we recapitulate the formalism required to embed static spherically symmetric space-times into fivedimensional Einstein spaces, and explicitly treat some special cases, obtaining local and isometric embeddings for the Reissner-Nordstr¨om space-time, as well as the null geometry of the global monopole metric. We also comment on existence theorems for Euclidean embedding spaces. In a recent result, it is claimed (Katzourakis 2005a) that any analytic n-dimensional space M may be globally embedded into an Einstein space M × F (F an analytic real-valued one-dimensional field). As a corollary, it is claimed that all product spaces are Einsteinian. We demonstrate that this construction for the embedding space is in fact limited to particular types of embedded spaces. We analyze this particular construction for global embeddings into Einstein spaces, uncovering a crucial misunderstanding with regard to the form of the local embedding. We elucidate the impact of this misapprehension on the subsequent proof, and amend the given construction so that it applies to all embedded spaces as well as to embedding spaces of arbitrary curvature. This study is presented as new theorems.
A comparative study of collocation methods for the numerical solution of differential equations.
(2008) Kajotoni, Margaret Modupe.; Parumasur, Nabendra.; Singh, Pravin.
The collocation method for solving ordinary differential equations is examined. A detailed comparison with other weighted residual methods is made. The orthogonal collocation method is compared to the collocation method and the advantage of the former is illustrated. The sensitivity of the orthogonal collocation method to different parameters is studied. Orthogonal collocation on finite elements is used to solve an ordinary differential equation and its superiority over the orthogonal collocation method is shown. The orthogonal collocation on finite elements is also used to solve a partial differential equation from chemical kinetics. The results agree remarkably with those from the literature.
Inhomogeneous solutions to the Einstein equations.
(2007) Govender, Gabriel.; Maharaj, Sunil Dutt.
In this dissertation we consider spherically symmetric gravitational fields that arise in relativistic astrophysics and cosmology. We first present a general review of static spherically symmetric spacetimes. aand highlight a particular class of exact solutions of the Einstein-Maxwell system for charged spheres. In the case of shear-free spacetimes with heat flow, the integration of the system is reduced to solving the condition of pressure isotropy. This condition is a second order linear differential equation with variable coefficients. By choosing particular forms for the gravitational potentials, sev-eral classes of new solutions are generated. We regain known solutions corresponding to coniformal flatness when tidal forces are absent. We also consider expanding, accelerating and shearing models when the heat flux is not present. A new general class of models is found. This new class of shearing solutions contains the model of Maharaj et al (1993) when a parameter is set to zero. Our new solution does not contain a singularity at the stellar centre, and it is therefore useful in modelling the interior of stars. Finally, we demonstrate that the shearing models obtained by Markund and Bradley (1999) do not satisfy the Einstein field equations.
Multi-parameter perturbation analysis of a second grade fluid flow past an oscillating infinite plate.
(2009) Habyarimana, Faustin.; Sibanda, Precious.
In this dissertation we consider the two dimensional flow of an incompressible and electrically conducting second grade fluid past a vertical porous plate with constant suction. The flow is permeated by a uniform transverse magnetic field. The aim of this study is to use the multi-parameter perturbation technique to study the effects of Eckert numbers on the flow of a pulsatile second grade fluid along a vertical plate. We further aim to investigate the effects of other fluid and physical parameters such as the Prandtl numbers, Hartmann numbers, viscoelastic parameter, angular frequency and suction velocity on boundary layer velocity, temperature, skin friction and the rate of heat transfer. Similarity transformations are used to reduce the governing partial differential equations to ordinary differential equations. We used perturbation methods to solve the coupled ordinary differential equations for zero Eckert number and the multiparameter perturbation technique to solve the coupled ordinary differential equations for small viscoelastic parameters and Eckert numbers. It is found that increasing the Eckert number or the viscoelastic parameter enhances the boundary layer velocity while reducing the temperature, the rate of heat transfer and the skin-friction. The results for the boundary layer velocity and the temperature are presented graphically and discussed. The results for the rate of heat transfer in terms of the Nusselt number and the skin friction are tabulated and discussed. A good agreement is found between these results and other published research. The comparison between the results for zero Eckert numbers and small Eckert numbers is also presented graphically and discussed.
Remediation of first-year mathematics students' algebra difficulties.
(2009) Campbell, Anita.; Anderson, Trevor Ryan.; Christiansen, Iben Maj.; Ewer, John Patrick Graham.
The pass rate of first-year university mathematics students at the University of KwaZulu-Natal (Pietermaritzburg Campus) has been low for many years. One cause may be weak algebra skills. At the time of this study, revision of high school algebra was not part of the major first year mathematics course. This study set out to investigate if it would be worthwhile to spend tutorial time on basic algebra when there is already an overcrowded calculus syllabus, or if students refresh their algebra skills sufficiently as they study first year mathematics. Since it was expected that remediation of algebra skills would be found to be worthwhile, two other questions were also investigated: Which remediation strategy is best? Which errors are the hardest to remediate? Five tutorial groups for Math 130 were randomly assigned one of four remediation strategies, or no remediation. Three variations of using cognitive conflict to change students’ misconceptions were used, as well as the strategy of practice. Pre- and post-tests in the form of multiple choice questionnaires with spaces for free responses were analysed. Comparisons between the remediated and non-remediated groups were made based on pre- and post-test results and Math 130 results. The most persistent errors were determined using an 8-category error classification developed for this purpose. The best improvement from pre- to post-test was 12.1% for the group remediated with cognitive conflict over 5 weeks with explanations from the tutor. Drill and practice gave the next-best improvement of 8.1%, followed by self-guided cognitive conflict over 5 weeks (7.8% improvement). A once-off intervention using cognitive conflict gave a 5.9% improvement. The group with no remediation improved by 2.3%. The results showed that the use of tutorintensive interventions more than doubled the improvement between pre-and post-tests but even after remediation, the highest group average was 80%, an unsatisfactory level for basic skills. The three most persistent errors were those involving technical or careless errors, errors from over-generalising and errors from applying a distorted algorithm, definition or theorem.
Character tables of the general linear group and some of its subgroups
(2008) Basheer, Ayoub Basheer Mohammed.; Moori, Jamshid.
The aim of this dissertation is to describe the conjugacy classes and some of the ordinary irreducible characters of the nite general linear group GL(n, q); together with character tables of some of its subgroups. We study the structure of GL(n, q) and some of its important subgroups such as SL(n, q); UT(n, q); SUT(n, q); Z(GL(n, q)); Z(SL(n, q)); GL(n, q)0 ; SL(n, q)0 ; the Weyl group W and parabolic subgroups P : In addition, we also discuss the groups PGL(n, q); PSL(n, q) and the a ne group A (n, q); which are related to GL(n, q): The character tables of GL(2; q); SL(2; q); SUT(2; q) and UT(2; q) are constructed in this dissertation and examples in each case for q = 3 and q = 4 are supplied. A complete description for the conjugacy classes of GL(n, q) is given, where the theories of irreducible polynomials and partitions of i 2 f1; 2; ; ng form the atoms from where each conjugacy class of GL(n, q) is constructed. We give a special attention to some elements of GL(n, q); known as regular semisimple, where we count the number and orders of these elements. As an example we compute the conjugacy classes of GL(3; q): Characters of GL(n, q) appear in two series namely, principal and discrete series characters. The process of the parabolic induction is used to construct a large number of irreducible characters of GL(n, q) from characters of GL(n, q) for m < n: We study some particular characters such as Steinberg characters and cuspidal characters (characters of the discrete series). The latter ones are of particular interest since they form the atoms from where each character of GL(n, q) is constructed. These characters are parameterized in terms of the Galois orbits of non-decomposable characters of F q n: The values of the cuspidal characters on classes of GL(n, q) will be computed. We describe and list the full character table of GL(n, q): There exists a duality between the irreducible characters and conjugacy classes of GL(n, q); that is to each irreducible character, one can associate a conjugacy class of GL(n, q): Some aspects of this duality will be mentioned.
Applications of symmetry analysis to physically relevant differential equations.
(2005) Kweyama, Mandelenkosi Christopher.; Govinder, Keshlan Sathasiva.; Maharaj, Sunil Dutt.
We investigate the role of Lie symmetries in generating solutions to differential equations that arise in particular physical systems. We first provide an overview of the Lie analysis and review the relevant symmetry analysis of differential equations in general. The Lie symmetries of some simple ordinary differential equations are found t. illustrate the general method. Then we study the properties of particular ordinary differential equations that arise in astrophysics and cosmology using the Lie analysis of differential equations. Firstly, a system of differential equations arising in the model of a relativistic star is generated and a governing nonlinear equation is obtained for a linear equation of state. A comprehensive symmetry analysis is performed on this equation. Secondly, a second order nonlinear ordinary differential equation arising in the model of the early universe is described and a detailed symmetry analysis of this equation is undertaken. Our objective in each case is to find explicit Lie symmetry generators that may help in analysing the model.
A classical approach for the analysis of generalized linear mixed models.
(2004) Hammujuddy, Mohammad Jahvaid. ; Matthews, Glenda Beverley.
Generalized linear mixed models (GLMMs) accommodate the study of overdispersion and correlation inherent in hierarchically structured data. These models are an extension of generalized linear models (GLMs) and linear mixed models (LMMs). The linear predictor of a GLM is extended to include an unobserved, albeit realized, vector of Gaussian distributed random effects. Conditional on these random effects, responses are assumed to be independent. The objective function for parameter estimation is an integrated quasi-likelihood (IQL) function which is often intractable since it may consist of high-dimensional integrals. Therefore, an exact maximum likelihood analysis is not feasible. The penalized quasi-likelihood (PQL) function, derived from a first-order Laplace expansion to the IQL about the optimum value of the random effects and under the assumption of slowly varying weights, is an approximate technique for statistical inference in GLMMs. Replacing the conditional weighted quasi-deviance function in the Laplace-approximated IQL by the generalized chi-squared statistic leads to a corrected profile quasilikelihood function for the restricted maximum likelihood (REML) estimation of dispersion components by Fisher scoring. Evaluation of mean parameters, for fixed dispersion components, by iterative weighted least squares (IWLS) yields joint estimates of fixed effects and random effects. Thus, the PQL criterion involves repeated fitting of a Gaussian LMM with a linked response vector and a conditional iterated weight matrix. In some instances, PQL estimates fail to converge to a neighbourhood of their true values. Bias-corrected PQL estimators (CPQL) have hence been proposed, using asymptotic analysis and simulation. The pseudo-likelihood algorithm is an alternative estimation procedure for GLMMs. Global score statistics for hypothesis testing of overdispersion, correlation and heterogeneity in GLMMs has been developed as well as individual score statistics for testing null dispersion components separately. A conditional mean squared error of prediction (CMSEP) has also been considered as a general measure of predictive uncertainty. Local influence measures for testing the robustness of parameter estimates, by inducing minor perturbations into GLMMs, are recent advances in the study of these models. Commercial statistical software is available for the analysis of GLMMs.
The construction and use of an evaluation instrument to measure attainment of objectives in mathematics learning at senior secondary level.
(1975) Moodley, Moonsamy.; Behr, A. Leslie.
This research aimed at measuring the extent to which a group of senior secondary pupils were attaining desirable cognitive objectives in mathematics. The summary of the design and procedures adopted in this study and the major findings which emerged is presented here. A scheme of objectives for mathematics learning at the senior secondary level was suggested in accordance with Bloom's Taxonomy of Educational Objectives and recent research relating to the Taxonomy and other classifications used in mathematics education. Multiple choice-type test items were constructed with reference to the above scheme of objectives and to content areas selected from the standard grade senior secondary mathematics syllabus. A pilot test was administered and analysed. The selection of items for the final form of the test was based on a consideration of item analysis data, distractors, reliability, validity, rating of items according to objectives and length of test. The final forms of the test and questionnaire were administered to a selected sample of 769 standard nine pupils from 14 Indian high schools in the Durban and District Area. The test was manually scored and the scores were subjected to statistical analyses by computerization. The findings suggest that: (i) it is possible to devise a reasonably reliable and valid test instrument to test at least two different levels of objectives in mathematics learning at senior secondary school level; (ii) the lower level objectives in mathematics are significantly easier to attain than the higher level objectives, which tends to support - in at least two levels - the assumption of hierarchical structure of a taxonomic classification of objectives; (iii) the performance in mathematics of the higher grade pupils tends to be adversely affected by being taught mathematics in mixed higher and standard grade classes.
Knowledge-directed intelligent information retrieval for research funding.
(2001) Hansraj, Sanjith.; Warren, Peter R.
Researchers have always found difficulty in attaining funding from the National Research Foundation (NRF) for new research interests. The field of Artificial Intelligence (AI) holds the promise of improving the matching of research proposals to funding sources in the area of Intelligent Information Retrieval (IIR). IIR is a fairly new AI technique that has evolved from the traditional IR systems to solve real-world problems. Typically, an IIR system contains three main components, namely, a knowledge base, an inference engine and a user-interface. Due to its inferential capabilities. IIR has been found to be applicable to domains for which traditional techniques, such as the use of databases, have not been well suited. This applicability has led it to become a viable AI technique from both, a research and an application perspective. This dissertation concentrates on researching and implementing an IIR system in LPA Prolog, that we call FUND, to assist in the matching of research proposals of prospective researchers to funding sources within the National Research Foundation (NRF). FUND'S reasoning strategy for its inference engine is backward chaining that carries out a depth-first search over its knowledge representation structure, namely, a semantic network. The distance constraint of the Constrained Spreading Activation (CSA) technique is incorporated within the search strategy to help prune non-relevant returns by FUND. The evolution of IIR from IR was covered in detail. Various reasoning strategies and knowledge representation schemes were reviewed to find the combination that best suited the problem domain and programming language chosen. FUND accommodated a depth 4, depth 5 and an exhaustive search algorithm. FUND'S effectiveness was tested, in relation to the different searches with respect to their precision and recall ability and in comparison to other similar systems. FUND'S performance in providing researchers with better funding advice in the South African situation proved to be favourably comparable to other similar systems elsewhere.
On spectral torsion theories.
(2003) Uworwabayeho, Alphonse.; Van den Berg, John Eric.
The purpose of this thesis is to investigate how "spect ralness" properties of a torsion theory T on R - Mod are reflected by properties of the ring R and its ring of quotients R,.. The development of "spectral" torsion theory owes much to Zelmanowitz [50] and Gomez-Pardo [23] . Gomez-Pardo proved that there exists a bijective correspondence between the set of spectral torsion theories on R - Modand rings of quotients of R that are Von Neumann regular and left self-injective. Chapter 1 is concerning with the notation used in the thesis and a summary of main results which are needed for understanding the sequel. Chapter 2 is concerned with the construction of a maximal ring of quotients of an arbitrary ring R by using the notion of denseness and relative injective hull. In Chapter 3, we survey the three equivalent ways of formulating Torsion Theory: by means of preradical functors on the category R- Mod, pairs of torsion / torsion-free classes and topologizing filters on rings. We shall show that Golan's approach to Torsion Theory via equivalence classes of injectives; and Dickson's one (as presented by Stenstrom) are equivalent. With a torsion theory T defined on R-Mod we associate R,. a ring of quotients of R. The full subcategory (R, T) - Mod of R- Mod whose objects are the T-torsion-free r-injective left R-modules is a Grothendieck category called the quotient category of R - Mod with respect to T. A left R,.-module that is r-torsion-free T-injective as a left R-module is injective if and only if it is injective as a left R-module (Proposition 3.6.4). Because of its use in the sequel , particular attention is paid to the lattice isomorphism that exists between the lattice of .r-pure submodules of a left Rmodule M and the lattice of subobjects of the quotient module M; in the category (R , T) - Mod. Chapter 4 introduces the definition of a spectral torsion theory: a Vll torsion theory r on R - Mod is said to be spectral if the Grothendieck category (R, r) - Mod is spectral. Using the notion of relative essential submodule, one can construct a spectral torsion theory from an arbitrary torsion theory on R - Mod. We shall show how an investigation of a general spectral torsion theory on R - Mod reduces to the Goldie torsion theory on R/tT (R) - Mod. Moreover, we shall exhibit necessary and sufficient conditions for R; to be a regular left self-injective ring (Theorem 4.2.10). In Chapter 5, after constructing the torsion functor Soce(-) which is associated with the pseudocomplement r.l of r in R - tors, we show how semiartinian rings can be characterized by means of spectral torsion theories: if a spectral torsion theory r on R - Mod is generated by the class of r-torsion simple left R-modules or, equivalently, cogenerated by the class of r-torsion-free simple left R-modules, then R is a left semiartinian ring (Proposition 5.3.2). Chapter 6 gives Zelmanowitz' important result [50]: R; is a semisimple artinian ring if and only if the torsion theory r is spectral and the associated left Gabriel topology has a basis of finitely generated left ideals. We also exhibit results due to M.J. Arroyo and J. Rios ([4] and [5]) which illustrate how spectral torsion theories can be used to describe when R; is (1) prime regular and left self-injective, (2) a left full linear ring, and (3) a direct product of left full linear rings. We also study the relationship between the flatness of the ring of quotients R; and the r- coherence of the ring R when r is a spectral torsion theory. It is proved that if r is a spectral torsion theory on R - Mod then the following conditions are equivalent: (1) R is left r-coherent; (2) (Rr)R is flat; (3) every right Rr-module is flat as a right R-module (Proposition 6.3.9). This result is an extension of Cateforis' results.
Economic evaluation of management strategies for cattle ranching in semi-arid regions.
(2001) Tarr, Heather Lucy.; Hearne, John W.
Arid and semi-arid regions have increasingly become the subject of much research and debate by scientists. By their very nature, these regions characteristically exhibit extremes which complicate the implementation of effective management strategies that ensure sustainable productivity and economic output. Namibia is one such region where low and highly variable rainfall conditions and fluctuating productivity pose a challenge to managers of commercial livestock enterprises, / who seek to optimise economic benefits while controlling the negative effect on herd production and income of unpredictable and unfavourable climatic events. Various management approaches are proposed as a means of exploiting periods of abundant productivity and so optimising income from herd production, while controlling for the effects of drought conditions. To analyse the effects of these various offtake strategies, a rainfall-driven plant-herbivore simulation model is used. The model comprises components simul~tihg vegetation and herbivore dynamics. The vegetation component incorporates soil moisture and nutrient allocation to plant parts. The herbivore dynamics sub-model comprises age and sex classes, population dynamics and animal energy requirements which govern accumulated fat reserves. The model is adapted to account for climatic and vegetation attributes specific to Namibia. An economic component including a seasonal monthly price structure is developed, and a dynamic feedback governing management decisions is incorporated. The much debated issue of whether to maintain a constant stocking rate or to track climatic variation by employing a variable stocking level is investigated, with the performance of management strategies incorporating these approaches ranked according to various factors, including annual returns, associated risk and annual stock mortality. The economic consequences of the timing of offtake are investigated, with the simulation of management strategies that implement destocking in the face of anticipated drought conditions. A dynamic projection of expected income allows the impact of forecasting potential economic gains on decision-making to be analysed. Results indicate that the performance of management strategies is not as dependent on climatic and seasonal price variability as was originally expected, with the application of a constant stocking level proving to be the most favourable strategy in terms of economic gain and variability of income. Tracking climatic variation by adapting stocking levels does not provide the improvement in economic returns from a livestock production system that was anticipated, although this approach is successful in effecting a significant reduction in annual stock mortality. Further results show the sensitivity of income to the long-term average stocking level characterising the management strategies investigated, as well as to the elasticity of the underlying price structure. The results of this study indicate that the implementation of management strategies designed to track climatic variation does not offer significant economic advantages over the application of a constant stocking approach.
A measure-theoretic approach to chaotic dynamical systems.
(2001) Singh, Pranitha.; Banasiak, Jacek.
The past few years have witnessed a growth in the study of the long-time behaviour of physical, biological and economic systems using measure-theoretic and probabilistic methods. In this dissertation we present a study of the evolution of dynamical systems that display various types of irregular behaviour for large times. Large systems, containing many elements, like e.g. bacteria populations or ensembles of gas particles, are very difficult to analyse and contain elements of uncertainty. Also, in general, it is not necessary to know the evolution of each bacteria or each gas particle. Therefore we replace the "pointwise" description of the evolution of the system with that of the evolution of suitable averages of the population like e.g. the gas or the bacteria spatial density. In particular cases, when the quantity in the evolution that we analyse has the probabilistic interpretation, say, the probability of finding the particle in certain state at certain time, we will be talking about the evolution of (probability) densities. We begin with the establishment of results for discrete time systems and this is later followed with analogous results for continuous time systems. We observe that in many cases the system has two important properties: at each step it is determined by a non-negative function (for example the spatial density or the probability density) and the overall quantity of the elements remains preserved. Because of these properties the most suitable framework to investigate such systems is the theory of Markov operators. We shall discuss three levels of "chaotic" behaviour that are known as ergodicity, mixing and exactness. They can be described as follows: ergodicity means that the only invariant sets are trivial, mixing means that for any set A the sequence of sets S-n(A) becomes, asymptotically, independent of any other set B, and exactness implies that if we start with any set of positive measure, then, after a long time the points of this set will spread and completely fill the state space. In this dissertation we describe an application of two operators related to the generating Markov operator to study and characterize the abovementioned properties of the evolution system. However, a system may also display regular behaviour. We refer to this as the asymptotic stability of the Markov operator generating this system and we provide some criteria characterizing this property. Finally, we demonstrate the use of the above theory by applying it to a system that is modeled by the linear Boltzmann equation.
Residually small varieties and commutator theory.
(2000) Swart, Istine Rodseth.; Raftery, James Gordon.
Chapter 0 In this introductory chapter, certain notational and terminological conventions are established and a summary given of background results that are needed in subsequent chapters. Chapter 1 In this chapter, the notion of a "weak conguence formula" [Tay72], [BB75] is introduced and used to characterize both subdirectly irreducible algebras and essential extensions. Special attention is paid to the role they play in varieties with definable principal congruences. The chapter focuses on residually small varieties; several of its results take their motivation from the so-called "Quackenbush Problem" and the "RS Conjecture". One of the main results presented gives nine equivalent characterizations of a residually small variety; it is largely due to W. Taylor. It is followed by several illustrative examples of residually small varieties. The connections between residual smallness and several other (mostly categorical) properties are also considered, e.g., absolute retracts, injectivity, congruence extensibility, transferability of injections and the existence of injective hulls. A result of Taylor that establishes a bound on the size of an injective hull is included. Chapter 2 Beginning with a proof of A. Day's Mal'cev-style characterization of congruence modular varieties [Day69] (incorporating H.-P. Gumm's "Shifting Lemma"), this chapter is a self-contained development of commutator theory in such varieties. We adopt the purely algebraic approach of R. Freese and R. McKenzie [FM87] but show that, in modular varieties, their notion of the commutator [α,β] of two congruences α and β of an algebra coincides with that introduced earlier by J. Hagemann and C. Herrmann [HH79] as well as with the geometric approach proposed by Gumm [Gum80a],[Gum83]. Basic properties of the commutator are established, such as that it behaves very well with respect to homomorphisms and sufficiently well in products and subalgebras. Various characterizations of the condition "(x, y) Є [α,β]” are proved. These results will be applied in the following chapters. We show how the theory manifests itself in groups (where it gives the familiar group theoretic commutator), rings, modules and congruence distributive varieties. Chapter 3 We define Abelian congruences, and Abelian and affine algebras. Abelian algebras are algebras A in which [A2, A2] = idA (where A2 and idA are the greatest and least congruences of A). We show that an affine algebra is polynomially equivalent to a module over a ring (and is Abelian). We give a proof that an Abelian algebra in a modular variety is affine; this is Herrmann's Funda- mental Theorem of Abelian Algebras [Her79]. Herrmann and Gumm [Gum78], [Gum80a] established that any modular variety has a so-called ternary "difference term" (a key ingredient of the Fundamental Theorem's proof). We derive some properties of such a term, the most significant being that its existence characterizes modular varieties. Chapter 4 An important result in this chapter (which is due to several authors) is the description of subdirectly irreducible algebras in a congruence modular variety. In the case of congruence distributive varieties, this theorem specializes to Jόnsson's Theorem. We consider some properties of a commutator identity (Cl) which is a necessary condition for a modular variety to be residually small. In the main result of the chapter we see that for a finite algebra A in a modular variety, the variety V(A) is residually small if and only if the subalgebras of A satisfy (Cl). This theorem of Freese and McKenzie also proves that a finitely generated congruence modular residually small variety has a finite residual bound, and it describes such a bound. Thus, within modular varieties, it proves the RS Conjecture. Conclusion The conclusion is a brief survey of further important results about residually small varieties, and includes mention of the recently disproved (general) RS Conjecture.
The effects of using visual literacy and visualization in the teaching and learning of mathematics problem solving on grade 6 and grade 7.
(2009) Budram, Rajesh.; Mudaly, Vimolan.
In this study I examine the effects of visualization in the teaching of problem solving in grades 6 and 7 in a school south of Durban in KwaZulu Natal. One of the goals of mathematics instruction according to the Department of Education is to prepare learners to become proficient in solving problems (DoE, 2003). Whilst many studies have been conducted in the field of problem solving, using visualization as a strategy to solve problems has been a neglected area in mathematics teaching in some schools. A literature survey shows that the link between solving problems and visualization strategies is making finding solutions easier for learners. The literature suggests that visualization assists learners to develop their problem solving skills as it allows them an opportunity to show their interpretation of the problem and the understanding of mathematical concepts. Through the use of problem centred mathematics, problem centred learning, growth of mathematical understanding and realistic mathematics education, learners see the connection and employ appropriate strategies to solve problems. This study examines the strategies employed by educators in the teaching and learning of problem solving and the strategies used by learners when solving problems. Data was collected from educators using a questionnaire, observation of grade 6 and 7 learners in the classroom and semi structured interviews. The conclusions from the data analysis have shown that problem solving is been neglected and that visualization does assist learners in solving problems.
The design and implementation of a classroom-based support programme in trignometry for use by underqualified educators.
(2005) Mkhize, Sabelo Andrias.; Brijlall, Deonarain.; Maharaj, A.
The main purpose of the study was to show the necessity of a classroom-based educator in-service support programme. Educators have unique problems being derived from the uniqueness of their school situations. Thus, the feeling that this kind of support could improve the quality of mathematics teaching and learning.
A classification of second order equations via nonlocal transformations.
(2000) Edelstein, R. M.; Govinder, Keshlan Sathasiva.
The study of second order ordinary differential equations is vital given their proliferation in mechanics. The group theoretic approach devised by Lie is one of the most successful techniques available for solving these equations. However, many second order equations cannot be reduced to quadratures due to the lack of a sufficient number of point symmetries. We observe that increasing the order will result in a third order differential equation which, when reduced via an alternate symmetry, may result in a solvable second order equation. Thus the original second order equation can be solved. In this dissertation we will attempt to classify second order differential equations that can be solved in this manner. We also provide the nonlocal transformations between the original second order equations and the new solvable second order equations. Our starting point is third order differential equations. Here we concentrate on those invariant under two- and three-dimensional Lie algebras.
Spherically symmetric solutions in relativistic astrophysics.
(2002) John, Anslyn James.; Maharaj, Sunil Dutt.
In this thesis we study classes of static spherically symmetric spacetimes admitting a perfect fluid source, electromagnetic fields and anisotropic pressures. Our intention is to generate exact solutions that model the interior of dense, relativistic stars. We find a sufficient condition for the existence of series solutions to the condition of pressure isotropy for neutral isolated spheres. The existence of a series solution is demonstrated by the method of Frobenius. With the help of MATHEMATICA (Wolfram 1991) we recovered the Tolman VII model for a quadratic gravitational potential, but failed to obtain other known classes of solution. This establishes the weakness, in certain instances, of symbolic manipulation software to extract series solutions from differential equations. For a cubic potential, we obtained a new series solution to the Einstein field equations describing neutral stars. The gravitational and thermodynamic variables are non-singular and continuous. This model also satisfies the important barotropic equation of state p = p(p). Two new exact solutions to the Einstein-Maxwell system, that generalise previous results for uncharged stars, were also found. The first of these generalises the solution of Maharaj and Mkhwanazi (1996), and has well-behaved matter and curvature variables. The second solution reduces to the Durgapal and Bannerji (1983) model in the uncharged limit; this new result may only serve as a toy model for quark stars because of negative energy densities. In both examples we observe that the solutions may be expressed in terms of hypergeometric and elementary functions; this indicates the possibility of unifying isolated solutions under the hypergeometric equation. We also briefly study compact stars with spheroidal geometry, that may be charged or admit anisotropic pressure distributions. The adapted forms of the pressure isotropy condition can be written as a harmonic oscillator equation. Two simple examples are presented.

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