# Masters Degrees (Mathematics and Computer Science Education)

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Item 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.Item Semiperfect CFPF rings.(1987) Francis, Donald Nicholas.; Pillay, Poobhalan.The Wedderburn-Artin Theorem (1927) characterised semisimple Artinian rings as finite direct products of matrix rings over division rings. In attempting to generalise Wedderburn's theorem, the natural starting point will be to assume R/RadR is semisimple Artinian. Such rings are called semilocal. They have not been completely characterised to date. If additional conditions are imposed on the radical then more is known about the structure of R. Semiprimary and perfect rings are those rings in which the radical is nilpotent and T-nilpotent respectively. In both these cases the radical is nil, and in rings in which the radical is nil, idempotents lift modulo the radical. Rings which have the latter property are called semiperfect. The characterisation problem of such rings has received much attention in the last few decades. We study semiperfect rings with a somewhat strong condition arising out of the status of generators in the module categories. More specifically, a ring R is CFPF iff every homomorphic image of R has the property that every finitely generated faithful module over it generates the corresponding module category. The objective of this thesis is to develop the theory that leads to the complete characterisation of semiperfect right CFPF rings. It will be shown (Theorem 6.3.17) that these rings are precisely finite products of full matrix rings over right duo right VR right a-cyclic right CFPF rings. As far as possible theorems proved in Lambek [16] or Fuller and Anderson [12] have not been reproved in this thesis and these texts will serve as basic reference texts. The basis for this thesis was inspired by results contained in the first two chapters of the excellent LMS publication "FPF Ring Theory" by Carl Faith and Stanley Page [11]. Its results can be traced to the works of G. Azumaya [23], K. Morita [18], Nakayama [20;21], H. Bass [4;5], Carl Faith [8;9;10], S. Page [24;25] and B. Osofsky [22]. Our task is to bring the researcher to the frontiers of FPF ring theory, not so much to present anything new.Item Population dynamics based on the McKendrick-von Foerster model.(1988) Seillier, Robyn.; Swart, John H.The current state of information concerning the classical model of deterministic, age-dependent population dynamics - the McKendrick von Foerster equation - is overviewed. This model and the related Renewal equation are derived and the parameters involved in both are elaborated upon. Fundamental theorems concerning existence, uniqueness and boundedness of solutions are outlined. A necessary and sufficient condition concerning the stability of equilibrium age-distributions is rederived along different lines. Attention is then given to generalizations of the McKendrick-von Foerster model that have arisen from the inclusion of density- dependence into the parameters of the system; the inclusion of harvesting terms; and the extension of the model to describe the dynamics of a two-sex population. A technique which reduces the model, under certain conditions on the mortality and fertility functions, to a system of ordinary differential equations is discussed and applied to specific biochemical population models. Emphasis here is on the possible existence of stable limit cycles.The Kolmogorov system of ordinary differential equations and its use in describing the dynamics of predator-prey systems is examined. The Kolmogorov theorem is applied as a simple alternative to a lengthy algorithm for determining whether limit cycles are stable. Age-dependence is incorporated into this system by means of a McKendrick - von Foerster equation and the effects on the system of different patterns of age-selective predation are demonstrated. Finally, brief mention is made of recent work concerning the use of the McKendrick - von Foerster equation to describe the dynamics of both predator and prey. A synthesis of the theory and results of a large number of papers is sought and areas valuable to further research are pointed out.Item Distances in and between graphs.(1991) Bean, Timothy Jackson.; Swart, Hendrika Cornelia Scott.Aspects of the fundamental concept of distance are investigated in this dissertation. Two major topics are discussed; the first considers metrics which give a measure of the extent to which two given graphs are removed from being isomorphic, while the second deals with Steiner distance in graphs which is a generalization of the standard definition of distance in graphs. Chapter 1 is an introduction to the chapters that follow. In Chapter 2, the edge slide and edge rotation distance metrics are defined. The edge slide distance gives a measure of distance between connected graphs of the same order and size, while the edge rotation distance gives a measure of distance between graphs of the same order and size. The edge slide and edge rotation distance graphs for a set S of graphs are defined and investigated. Chapter 3 deals with metrics which yield distances between graphs or certain classes of graphs which utilise the concept of greatest common subgraphs. Then follows a discussion on the effects of certain graph operations on some of the metrics discussed in Chapters 2 and 3. This chapter also considers bounds and relations between the metrics defined in Chapters 2 and 3 as well as a partial ordering of these metrics. Chapter 4 deals with Steiner distance in a graph. The Steiner distance in trees is studied separately from the Steiner distance in graphs in general. The concepts of eccentricity, radius, diameter, centre and periphery are generalised under Steiner distance. This final chapter closes with an algorithm which solves the Steiner problem and a Heuristic which approximates the solution to the Steiner problem.Item Teachers' perceptions of the effectiveness of in-service education and training (inset) for senior secondary school mathematics teachers in the greater Durban area.(1991) Ntenza, Philemon Simangele.; Winter, Paul.; Graham-Jolly, Mike.The project of the Shell Science Centre (SSC) started in 1985, in response to the high failure rate in mathematics amongst black pupils, the perceived inadequacy of college mathematics curricula for prospective mathematics teachers, and generally because of the destructive policies of apartheid and the inferior system of education for black pupils. One of the programmes is to organise and run in-service education and training (INSET) courses for senior mathematics teachers, in collaboration with teachers' Action Committees, with the hope of effecting curriculum change and teacher behaviour in the classroom. It is important, therefore, for the SSC to know whether the INSET programmes meet the needs of the teachers, especially those who graduated from colleges of education in KwaZulu with a Secondary Teachers' Diploma, since they form the majority of the INSET participants. Hence, this investigaton aimed to survey and analyse: (i) mathematics teachers' and college mathematics lecturers' perceptions of the college mathematics curriculum; and (ii) mathematics teachers' perception of the effectiveness of INSET courses. Initial data, about the effectiveness of INSET courses and the pre-service training of prospective mathematics teachers, was gathered through informal talks with mathematics teachers during INSET courses. Issues and themes that emerged were "fleshed out" using unstructured interviews with five (5) mathematics teachers. From these it was possible to draw up a detailed structured interview schedule, which was administered to a further seventeen (17) mathematics teachers in senior secondary schools in Umlazi and KwaMashu townships. Data, about the college mathematics curriculum, was also gathered by means of structured interviews with college mathematics lecturers, in the two colleges of education in KwaZulu, and with graduate teachers with a Secondary Teachers' Diploma in mathematics. Among the significant findings were: o Limitations in the college mathematics curriculum in as far as the mathematics content and the methodology courses were concerned; o Problems with SSC INSET courses such as teaching methods suggested by INSET tutors, timing of INSET courses, group work, etc.; o Problems with teaching mathematics at school such as the shortage of mathematics textbooks, large classes, inadequate resources, etc.; and o Problems with incorrect 'overlearnt' rules from inadequate college mathematics textbooks. The implications of these findings for the SSC were considered. It is suggested inter alia that the SSC should adopt strategies which would emphasize direct contact with the pupils of INSET participants. It is hoped that these strategies will help correct the various problems experienced by mathematics teachers in the schools.Item Cosmological models and the deceleration parameter.(1992) Naidoo, Ramsamy.; Maharaj, Sunil Dutt.In this thesis we utilise a form for the Hubble constant first proposed by Berman (1983) to study a variety of cosmological models. In particular we investigate the Robertson-Walker spacetimes, the Bianchi I spacetime, and the scalar-tensor theory of gravitation of Lau and Prokhovnik (1986). The Einstein field equations with variable cosmological constant and gravitational constant are discussed and the Friedmann models are reviewed. The relationship between observation and the Friedmann models is reviewed. We present a number of new solutions to the Einstein field equations with variable cosmological constant and gravitational constant in the Robertson-Walker spacetimes for the assumed form of the Hubble parameter. We explicitly find forms for the scale factor, cosmological constant, gravitational constant, energy density and pressure in each case. Some of the models have an equation of state for an ideal gas. The gravitational constant may be increasing in certain regions of spacetime. The Bianchi I spacetime, which is homogeneous and anisotropic, is shown to be consistent with the Berman (1983) law by defining a function which reduces to the scale factor of Robertson-Walker. We illustrate that the scalar-tensor theory of Lau and Prokhovnik (1986) also admits solutions consistent with the Hubble variation proposed by Berman. This demonstrates that this approach is useful in seeking solutions to the Einstein field equations in general relativity and alternate theories of gravity.Item The application of the multigrid algorithm to the solution of stiff ordinary differential equations resulting from partial differential equations.(1992) Parumasur, Nabendra.; Mika, Janusz R.We wish to apply the newly developed multigrid method [14] to the solution of ODEs resulting from the semi-discretization of time dependent PDEs by the method of lines. In particular, we consider the general form of two important PDE equations occuring in practice, viz. the nonlinear diffusion equation and the telegraph equation. Furthermore, we briefly examine a practical area, viz. atmospheric physics where we feel this method might be of significance. In order to offer the method to a wider range of PC users we present a computer program, called PDEMGS. The purpose of this program is to relieve the user of much of the expensive and time consuming effort involved in the solution of nonlinear PDEs. A wide variety of examples are given to demonstrate the usefulness of the multigrid method and the versatility of PDEMGS.Item Queueing and communication networks governed by generalised Lindley-Loynes equations.(1993) Rose, David Michael.; Berezner, S. A.Several decades after A.K. Erlang originated the theory of queues and queueing networks, D.V. Lindley added impetus to the development of this field by determining a recursive relation for waiting times. Part I of this thesis provides a theoretical treatment of single-server and multiserver queues described by the basic Lindley relation and its extensions, which are referred to collectively as Lindley-Loynes equations. The concepts of stability, and minimal and maximal solutions are investigated. The interdependence of theory and practice becomes evident in Part II, where the results of recent and current research are highlighted. While the main aim of the first part of the thesis is to provide a firm theoretical framework for the sequel, the objective in Part II is to derive generalised forms of the Lindley-Loynes equations from different network protocols. Such protocols are regulated by different switching rules and synchronization constraints. Parts I and II of the thesis are preceded by Chapter 0 in which several fundamental ideas (including those on notation and probability) are described. It is in this chapter too that a more detailed overview of the concept of the thesis is provided.Item Pupils' perceptions of study of Mathematics as a subject for the Senior Certificate examination: two case studies.(1995) Appanna, Sandras.; Harley, Keneth Lee.This study was conducted at two Secondary schools in the Pietermaritzburg area which is in the province of Kwazulu - Natal, South Africa. Of the 182 pupils who participated in this investigation, 97 were from a Black High school and 85 from an Indian Secondary school. The aim of this study was to gain insights into pupils; perceptions of Mathematics. The motivation was that such an exploratory investigation could contribute significantly to the understanding of some of the principal underlying factors that have contributed to the current crisis in mathematics education. The knowledge gained could inform future research in Mathematics education and educational strategies aimed at increasing the number of pupils studying Mathematics at matriculation level. Since there exists a significant racial skewing in favour of White, Coloured and Indian pupils in the percentages of matriculants studying Mathematics for the Senior Certificate Examination, the research focused on the perceptions of Black and Indian pupils. The prevention of further disruptions to the studies of matriculants and the need for a manageable sample necessitated the use of two groups of Standard 9 pupils. The study therefore acquired the characteristics of the case study method of investigation. Open - ended questionnaires, interviews and written essays were used for the purposes of data collection. In examining pupils' perceptions, factors such as biographical details, future aspirations, pupils' explanations for studying/ not studying Mathematics, their preference for the subject, pupils' views on whether more pupils should study the subject, as well as the status of the examination subjects, were considered. Findings suggested that all pupils - even those not studying Mathematics - had similar perceptions of the importance Mathematics, although their learning experiences had been significantly different. The curricula experiences of pupils appeared to have been influenced by past apartheid policies. However, the classroom experiences on which pupils' perceptions of Mathematics were based appeared to have been directly responsible for the low numbers of pupils studying Mathematics for examination purposes. Critical theory played an important role in the interpretation of the major findings. These interpretations suggest that the classroom experiences of pupils were crucial in that they influenced pupils' decisions to select or not to select Mathematics as an examination subject. The study concluded with recommendations for classroom practice and research areas in Mathematics education which would improve the existing educational experiences of disadvantaged pupils.Item The preliminary group classification of the equation utt = f(x,ux)uxx + g(x, ux)(1995) Narain, Ojen Kumar.; Kambule, M. T.We study the class of partial differential equations Utt = f(x, ux)uxx + g(x, u x), with arbitrary functions f(x, u x) and g(x, u x), from the point of view of group classification. The principal Lie algebra of infinitesimal symmetries admitted by the whole class is three-dimensional. We use the method of preliminary group classification to obtain a classification of these equations with respect to a one-dimesional extension of the principal Lie algebra and then a countable-dimensional subalgebra of their equivalence algebra. Each of these equations admits an additional infinitesimal symmetry. L.V. Ovsiannikov [9] has proposed an algorithm to construct efficiently the optimal system of an arbitrary decomposable Lie algebra. We use this algorithm to construct an optimal system of subalgebras of all dimensionalities (from one-dimensional to six- dimensional) of a seven-dimensional solvable Lie algebra.Item Closure operators on complete lattices with application to compactness.(1995) Brijlall, Deonarain.; Sturm, Teo.; Jordens, Olav.No abstract available.Item Cosmological attractors and no-hair theorems.(1996) Miritzis, John.; Cotsakis, S.Interest in the attracting property of de Sitter space-time has grown during the 'inflationary era' of cosmology. In this dissertation we discuss the more important attempts to prove the so called 'cosmic no-hair conjecture' ie the proposition that all expanding universes with a positive cosmological constant asymptotically approach de Sitter space-time. After reviewing briefly the standard FRW cosmology and the success of the inflationary scenario in resolving most of the problems of standard cosmology, we carefully formulate the cosmic no-hair conjecture and discuss its limitations. We present a proof of the cosmic no-hair theorem for homogeneous space-times in the context of general relativity assuming a positive cosmological constant and discuss its generalisations. Since, in inflationary cosmology, the universe does not have a true cosmological constant but rather a vacuum energy density which behaves like a cosmological term, we take into account the dynamical role of the inflaton field in the no-hair hypothesis and examine the no-hair conjecture for the three main inflationary models, namely new inflation, chaotic inflation and power-law inflation. A generalisation of a well-known result of Collins and Hawking [21] in the presence of a scalar field matter source, regarding Bianchi models which can approach isotropy is given. In the context of higher order gravity theories, inflation emerges quite naturally without artificially imposing an inflaton field. The conformal equivalence theorem relating the solution space of these theories to that of general relativity is reviewed and the applicability of the no-hair theorems in the general framework of f (R) theories is developed. We present our comments and conclusions about the present status of the cosmic no-hair theorem and suggest possible paths of future research in the field.Item Completion of uniform and metric frames.(1996) Murugan, Umesperan Goonaselan.; Baboolal, Deeva Lata.The term "frame" was introduced by C H Dowker, who studied them in a long series of joint papers with D Papert Strauss. J R Isbell , in a path breaking paper [1972] pointed out the need to introduce separate terminology for the opposite of the category of Frames and coined the term "locale". He was the progenitor of the idea that the category of Locales is actually more convenient in many ways than the category of Frames. In fact, this proves to be the case in one of the approaches adopted in this thesis. Sublocales (quotient frames) have been studied by several authors, notably Dowker and Papert [1966] and Isbell [1972]. The term "sublocale" is due to Isbell, who also used "part " to mean approximately the same thing. The use of nuclei as a tool for studying sublocales (as is used in this thesis) and the term "nucleus" itself was initiated by H Simmons [1978] and his student D Macnab [1981]. Uniform spaces were introduced by Weil [1937]. Isbell [1958] studied algebras of uniformly continuous functions on uniform spaces. In this thesis, we introduce the concept of a uniform frame (locale) which has attracted much interest recently and here too Isbell [1972] has some results of interest. The notion of a metric frame was introduced by A Pultr [1984]. The main aim of his paper [11] was to prove metrization theorems for pointless uniformities. This thesis focuses on the construction of completions in Uniform Frames and Metric Frames. Isbell [6] showed the existence of completions using a frame of certain filters. We describe the completion of a frame L as a quotient of the uniformly regular ideals of L, as expounded by Banaschewski and Pultr[3]. Then we give a substantially more elegant construction of the completion of a uniform frame (locale) as a suitable quotient of the frame of all downsets of L. This approach is attributable to Kriz[9]. Finally, we show that every metric frame has a unique completion, as outlined by Banaschewski and Pultr[4]. In the main, this thesis is a standard exposition of known, but scattered material. Throughout the thesis, choice principles such as C.D.C (Countable Dependent Choice) are used and generally without mention. The treatment of category theory (which is used freely throughout this thesis) is not self-contained. Numbers in brackets refer to the bibliography at the end of the thesis. We will use 0 to indicate the end of proofs of lemmas, theorems and propositions. Chapter 1 covers some basic definitions on frames , which will be utilized in subsequent chapters. We will verify whatever we need in an endeavour to enhance clarity. We define the categories, Frm of frames and frame homomorphisms, and Lac the category of locales and frame morphisms. Then we explicate the adjoint situation that exists between Frm and Top , the category of topological spaces and continuous functions. This is followed by an introduction to the categories, RegFrm of all regular frames and frame homomorphisms, and KRegFrm the category of compact regular frames and their homomorphisms. We then present the proofs of two very important lemmas in these categories. Finally, we define the compactification of and a congruence on a frame. In Chapter 2 we recall some basic definitions of covers, refinements and star refinements of covers. We introduce the notion of a uniform frame and define certain mappings (morphisms) between uniform frames (locales) . In the terminology of Banaschewski and Kriz [9] we define a complete uniform frame and the completion of a uniform frame. The aim of Chapter 3 is twofold : first, to construct the compact regular coreflection of uniform frames , that is, the frame counterpart of the Samuel Compactification of uniform spaces [12] , and then to use it for a description of the completion of a uniform frame as an alternative to that previously given by Isbell[6]. The main purpose of Chapter 4 is to provide another description of uniform completion in frames (locales), which is in fact even more straightforward than the original topological construction. It simply consists of writing down generators and defining relations. We provide a detailed examination of the main result in this section, that is, a uniform frame L is complete of each uniform embedding f : (M,UM) -t (L,UL) is closed, where UM and UL denote the uniformities on the frames M and L respectively. Finally, in Chapter 5, we introduce the notions of a metric diameter and a metric frame. Using the fact that every metric frame is a uniform frame and hence has a uniform completion, we show that every metric frame L has a unique completion : CL - L.Item Aspects of spherically symmetric cosmological models.(1998) Moodley, Kavilan.; Maharaj, Sunil Dutt.; Govinder, Keshlan Sathasiva.In this thesis we consider spherically symmetric cosmological models when the shear is nonzero and also cases when the shear is vanishing. We investigate the role of the Emden-Fowler equation which governs the behaviour of the gravitational field. The Einstein field equations are derived in comoving coordinates for a spherically symmetric line element and a perfect fluid source for charged and uncharged matter. It is possible to reduce the system of field equations under different assumptions to the solution of a particular Emden-Fowler equation. The situations in which the Emden-Fowler equation arises are identified and studied. We analyse the Emden-Fowler equation via the method of Lie point symmetries. The conditions under which this equation is reduced to quadratures are obtained. The Lie analysis is applied to the particular models of Herlt (1996), Govender (1996) and Maharaj et al (1996) and the role of the Emden-Fowler equation is highlighted. We establish the uniqueness of the solutions of Maharaj et al (1996). Some physical features of the Einstein-Maxwell system are noted which distinguishes charged solutions. A charged analogue of the Maharaj et al (1993) spherically symmetric solution is obtained. The Gutman-Bespal'ko (1967) solution is recovered as a special case within this class of solutions by fixing the parameters and setting the charge to zero. It is also demonstrated that, under the assumptions of vanishing acceleration and proper charge density, the Emden-Fowler equation arises as a governing equation in charged spherically symmetric models.Item Locally finite nearness frames.(1998) Naidoo, Inderasan.; Baboolal, Deeva Lata.; Ori, Ramesh G.The concept of a frame was introduced in the mid-sixties by Dowker and Papert. Since then frames have been extensively studied by several authors, including Banaschewski, Pultr and Baboolal to mention a few. The idea of a nearness was first introduced by H. Herrlich in 1972 and that of a nearness frame by Banaschewski in the late eighties. T. Dube made a fairly detailed study of the latter concept. The purpose of this thesis is to study the property of local finiteness and metacompactness in the setting of nearness frames. J. W. Carlson studied these ideas (including Lindelof and Pervin nearness structures) in the realm of nearness spaces. The first four chapters are a brief overview of frame theory culminating in results concerning regular, completely regular, normal and compact frames. In chapter five we provide the definitions for various nearness frames: Pervin, Lindelof , Locally Finite and Metacompact to mention a few. A particular locally finite nearness structure, denoted by µLF, is studied in detail. It is defined to be the nearness structure on a regular frame L generated by the family of all locally finite covers on the frame L. Also, a particular metacompact nearness structure, denoted by µPF, is studied in detail. It is defined to be the nearness structure on a regular frame L generated by the family of all point-finite covers of the frame L. Various theorems related the above nearness frames and these nearness structures are obtained.Item Modelling the spatial dynamics of a semi-arid grazing system.(1999) Koch, Kathryn Jane.; Hearne, John W.; O' Connor, Timothy Gordon.A large proportion of the world's land surface is covered by semi-arid grasslands, and they provide an important source of income as a grazing resource. A more comprehensive understanding of these complex ecosystems is vital for the effective management of rangelands, as it will lead to an increased and more sustainable economic output. Herbivores modify the spatial pattern of vegetation distribution and their response to spatially heterogeneous forage resources affects their performance. The spatial aspect of herbivory is often ignored although it is a necessary component of understanding grazing dynamics and the factors affecting herbivore condition. A spatial model is developed which incorporates vegetation and animal dynamics and the interactions between these two components. The effect of different spatial foraging strategies on animal performance and vegetation was investigated. Model results were compared with the output of a non-spatial model to assess the importance of spatially explicit modelling in the context of monitoring animal performance. The relative significance of a number of aspects relating to spatial grazing and animal condition was explored. The results from this research show that significant differences in output are obtained from spatial versus non-spatial models. While the purpose of a model will determine its nature, the results imply that in certain contexts, a spatial model is essential for accurate results and insight. The results also indicated that foraging strategies have a large affect on herbivore condition and that spatially explicit models are necessary in the context of investigating the effect of foraging strategies on animal performance. Various aspects that significantly affected animal condition were highlighted and are useful in directing future investigations into grazing dynamics. It is difficult to conduct field studies under spatially and temporally variable conditions where the interactions between vegetation and herbivores are so complex. In the light of this, modelling was found to be an effective tool that can be used in investigating and revealing important dynamics of semi-arid grazing systems.Item 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.Item 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.Item Computer analysis of equations using Mathematica.(2001) Jugoo, Vikash Ramanand.; Govinder, Keshlan Sathasiva.; Maharaj, Sunil Dutt.In this thesis we analyse particular differential equations that arise in physical situations. This is achieved with the aid of the computer software package called Mathematica. We first describe the basic features of Mathematica highlighting its capabilities in performing calculations in mathematics. Then we consider a first order Newtonian equation representing the trajectory of a particle around a spherical object. Mathematica is used to solve the Newtonian equation both analytically and numerically. Graphical plots of the trajectories of the planetary bodies Mercury, Earth and Jupiter are presented. We attempt a similar analysis for the corresponding relativistic equation governing the orbits of gravitational objects. Only numerical results are possible in this case. We also perform a perturbative analysis of the relativistic equation and determine the amount of perihelion shift. The second equation considered is the Emden-Fowler equation of order two which arises in many physical problems, including certain inhomogeneous cosmological applications. The analytical features of this equation are investigated using Mathematica and the Lie analysis of differential equations. Different cases of the related autonomous form of the Emden-Fowler equation are investigated and graphically represented. Thereafter, we generate a number of profiles of the energy density and the pressure for a particular solution which demonstrates that a numerical approach for studying inhomogeneity, in cosmological models in general relativity, is feasible.Item 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.