Mathematics and Computer Science Education

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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.
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.
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.
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.
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.
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.
On purity relative to an hereditary torsion theory.
(1992) Gray, Derek Johanathan.; Meijer, A. R.
The thesis is mainly concerned with properties of the concept "σ-purity" introduced by J. Lambek in "Torsion Theories, Additive Semantics and Rings of Quotients", (Springer-Verlag, 1971). In particular we are interested in modul es M for which every exact sequence of the form O→M→K→L→O (or O→K→M→L→O or O→K→L→M→O) is σ-pure exact. Modules of the first type turn out to be precisely the σ- injective modules of O. Goldman (J. Algebra 13, (1969), 10-47). This characterization allows us to study σ- injectivity from the perspective of purity. Similarly the demand that every short exact sequence of modules of the form O→K→M→L→O or O→K→L→M→O be σ-pure exact leads to concepts which generalize regularity and flatness respectively. The questions of which properties of regularity and flatness extend to these more general concepts of σ- regularity and σ-flatness are investigated. For various classes of rings R and torsion radicals σ on R-mod, certain conditions equivalent to the σ-regularity and the σ-injectivity of R are found. We also introduce some new dimensions and study semi-σ-flat and semi-σ-injective modules (defined by suitably restricting conditions on σ-flat and σ-injective modules). We further characterize those rings R for which every R-module is semi- σ-flat. The related concepts of a projective cover and a perfect ring (introduced by H. Bass in Trans. Amer. Math. Soc. 95, (1960), 466-488) are extended in a 'natural way and, inter alia , we obtain a generalization of a famous theorem of Bass. Lastly, we develop a relativized version of the Jacobson Radical which is shown to have properties analogous to both the classical Jacobson Radical and a radical due to J.S. Golan.
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.
Conformally invariant relativistic solutions.
(1993) Maharaj, M. S.; Maharaj, Sunil Dutt.; Maartens, Roy.
The study of exact solutions to the Einstein and Einstein-Maxwell field equations, by imposing a symmetry requirement on the manifold, has been the subject of much recent research. In this thesis we consider specifically conformal symmetries in static and nonstatic spherically symmetric spacetimes. We find conformally invariant solutions, for spherically symmetric vectors, to the Einstein-Maxwell field equations for static spacetimes. These solutions generalise results found previously and have the advantage of being regular in the interior of the sphere. The general solution to the conformal Killing vector equation for static spherically symmetric spacetimes is found. This solution is subject to integrability conditions that place restrictions on the metric functions. From the general solution we regain the special cases of Killing vectors, homothetic vectors and spherically symmetric vectors with a static conformal factor. Inheriting conformal vectors in static spacetimes are also identified. We find a new class of accelerating, expanding and shearing cosmological solutions in nonstatic spherically symmetric spacetimes. These solutions satisfy an equation of state which is a generalisation of the stiff equation of state. We also show that this solution admits a conformal Killing vector which is explicitly obtained.
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.
Polynomial approximations to functions of operators.
(1994) Singh, Pravin.; Mika, Janusz R.
To solve the linear equation Ax = f, where f is an element of Hilbert space H and A is a positive definite operator such that the spectrum (T (A) ( [m,M] , we approximate -1 the inverse operator A by an operator V which is a polynomial in A. Using the spectral theory of bounded normal operators the problem is reduced to that of approximating a function of the real variable by polynomials of best uniform approximation. We apply two different techniques of evaluating A-1 so that the operator V is chosen either as a polynomial P (A) when P (A) approximates the n n function 1/A on the interval [m,M] or a polynomial Qn (A) when 1 - A Qn (A) approximates the function zero on [m,M]. The polynomials Pn (A) and Qn (A) satisfy three point recurrence relations, thus the approximate solution vectors P (A)f n and Q (A)f can be evaluated iteratively. We compare the procedures involving n Pn (A)f and Qn (A)f by solving matrix vector systems where A is positive definite. We also show that the technique can be applied to an operator which is not selfadjoint, but close, in the sense of operator norm, to a selfadjoint operator. The iterative techniques we develop are used to solve linear systems arising from the discretization of Freedholm integral equations of the second kind. Both smooth and weakly singular kernels are considered. We show that earlier work done on the approximation of linear functionals < x,g > , where 9 EH, involve a zero order approximation to the inverse operator and are thus special cases of a general result involving an approximation of arbitrary degree to A -1 .
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.
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.
Aspects of distance and domination in graphs.
(1995) Smithdorf, Vivienne.; Swart, Hendrika Cornelia Scott.; Dankelmann, Peter A.
The first half of this thesis deals with an aspect of domination; more specifically, we investigate the vertex integrity of n-distance-domination in a graph, i.e., the extent to which n-distance-domination properties of a graph are preserved by the deletion of vertices, as well as the following: Let G be a connected graph of order p and let oi- S s;:; V(G). An S-n-distance-dominating set in G is a set D s;:; V(G) such that each vertex in S is n-distance-dominated by a vertex in D. The size of a smallest S-n-dominating set in G is denoted by I'n(S, G). If S satisfies I'n(S, G) = I'n(G), then S is called an n-distance-domination-forcing set of G, and the cardinality of a smallest n-distance-domination-forcing set of G is denoted by On(G). We investigate the value of On(G) for various graphs G, and we characterize graphs G for which On(G) achieves its lowest value, namely, I'n(G), and, for n = 1, its highest value, namely, p(G). A corresponding parameter, 1](G), defined by replacing the concept of n-distance-domination of vertices (above) by the concept of the covering of edges is also investigated. For k E {a, 1, ... ,rad(G)}, the set S is said to be a k-radius-forcing set if, for each v E V(G), there exists Vi E S with dG(v, Vi) ~ k. The cardinality of a smallest k-radius-forcing set of G is called the k-radius-forcing number of G and is denoted by Pk(G). We investigate the value of Prad(G) for various classes of graphs G, and we characterize graphs G for which Prad(G) and Pk(G) achieve specified values. We show that the problem of determining Pk(G) is NP-complete, study the sequences (Po(G),Pl(G),P2(G), ... ,Prad(G)(G)), and we investigate the relationship between Prad(G)(G) and Prad(G)(G + e), and between Prad(G)(G + e) and the connectivity of G, for an edge e of the complement of G. Finally, we characterize integral triples representing realizable values of the triples b,i,p), b,l't,i), b,l'c,p), b,l't,p) and b,l't,l'c) for a graph.
Closure operators on complete lattices with application to compactness.
(1995) Brijlall, Deonarain.; Sturm, Teo.; Jordens, Olav.
No abstract available.
On chain domains, prime rings and torsion preradicals.
(1995) Van den Berg, John Eric.; Raftery, James Gordon.
Abstract available in pdf file.
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.
Graph and digraph embedding problems.
(1996) Maharaj, Hiren.; Henning, Michael Anthony.
This thesis is a study of the symmetry of graphs and digraphs by considering certain homogeneous embedding requirements. Chapter 1 is an introduction to the chapters that follow. In Chapter 2 we present a brief survey of the main results and some new results in framing number theory. In Chapter 3, the notions of frames and framing numbers is adapted to digraphs. A digraph D is homogeneously embedded in a digraph H if for each vertex x of D and each vertex y of H, there exists an embedding of D in H as an induced subdigraph with x at y. A digraph F of minimum order in which D can be homogeneously embedded is called a frame of D and the order of F is called the framing number of D. We show that that every digraph has at least one frame and, consequently, that the framing number of a digraph is a well defined concept. Several results involving the framing number of graphs and digraphs then follow. Analogous problems to those considered for graphs are considered for digraphs. In Chapter 4, the notions of edge frames and edge framing numbers are studied. A nonempty graph G is said to be edge homogeneously embedded in a graph H if for each edge e of G and each edge f of H, there is an edge isomorphism between G and a vertex induced subgraph of H which sends e to f. A graph F of minimum size in which G can be edge homogeneously embedded is called an edge frame of G and the size of F is called the edge framing number efr(G) of G. We also say that G is edge framed by F. Several results involving edge frames and edge framing numbers of graphs are presented. For graphs G1 and G2 , the framing number fr(G1 , G2 ) (edge framing number ef r(GI, G2 )) of G1 and G2 is defined as the minimum order (size, respectively) of a graph F such that Gj (i = 1,2) can be homogeneously embedded in F. In Chapter 5 we study edge framing numbers and framing number for pairs of cycles. We also investigate the framing number of pairs of directed cycles.
Spherically symmetric cosmological solutions.
(1996) Govender, Jagathesan.; Maharaj, Sunil Dutt.
This thesis examines the role of shear in inhomogeneous spherically symmetric spacetimes in the field of general relativity. The Einstein field equations are derived for a perfect fluid source in comoving coordinates. By assuming a barotropic equation of state, two classes of nonaccelerating solutions are obtained for the Einstein field equations. The first class has equation of state p = ⅓µ and the second class, with equation of state p = µ, generalises the models of Van den Bergh and Wils (1985). For a particular choice of a metric potential a new class of solutions is found which is expressible in terms of elliptic functions of the first and third kind in general. A class of nonexpanding cosmological models is briefly studied. The method of Lie symmetries of differential equations generates a self-similar variable which reduces the field and conservation equations to a system of ordinary differential equations. The behaviour of the gravitational field in this case is governed by a Riccati equation which is solved in general. Another class of solutions is obtained by making an ad hoc choice for one of the gravitational potentials. It is demonstrated that for a stiff fluid a particular case of the generalised Emden-Fowler equation arises.
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.

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