This thesis provides a new conceptualisation of ethnomathematics which avoids some of the difficulties which emerge in the literature. In particular, work has been started on a philosophic basis for the field. There is no consistent view of ethnomathematics in the literature. The relationship with mathematics itself has been ignored, and the philosophical and theoretical background is missing. The literature also reveals the ethnocentricity implied by ethnomathematics as a field of study based in a culture which has mathematics as a knowledge category. Two strategies to over come this problem are identified: universalising the referent of ‘mathematics’ so that it is the same as “knowledge-making”; or using methodological techniques to minimise it. The position of ethnomathematics in relationship to anthropology, sociology, history, and politics is characterised on a matrix. A place for ethnomathematics is found close the anthropology of mathematics, but the aim of anthropology is to better understand culture in general, while ethnomathematics aims to better understand mathematics. Anthropology, however, contributes its well-established methodologies for overcoming ethnocentricity. The search for a philosophical base finds a Wittgensteinian orientation which enables culturally based ‘systems of meaning’ to gain credibility in mathematics. A definition is proposed for ethnomathematics as the study of mathematical practices within context. Four types of ethnomathematical activity are identified: descriptive, archaeological, mathematising, and analytical activity. The definition also gives rise to a categorisation of ethnomathematical work along three dimensions: the closeness to conventional mathematics; the historical time; and the type of host culture. The mechanisms of interaction between mathematical practices are identified, and the imperialistic growth of mathematics is explained. Particular features of ethnomathematical theory are brought out in a four examples. By admitting the legitimacy of other viewpoints, ethnomathematics opens mathematics to new creative forces. Within education, ethnomathematics provides new choices, and turns cultural conflict into a useful tool for teaching. Mathematical activity exists in a variety of contexts. Learning mathematics involves being aware of, and integrating, diverse concepts. Ethnomathematics expands mathematical horizons, so that cultural diversity becomes a richer contributor to the cultural structures which humans use to understand their world.

Given a linear regression model y = Xβ + e, where e has a multivariate normal distribution N(0, Σ) consequences of the erroneous assumption that e is distributed as N(0, I) are considered. For a general linear hypothesis concerning the parameters β, in a general model the distribution of the statistic to test the hypothesis, derived under the erroneous assumption is studied. Particular linear hypotheses concerning particular linear models are investigated so as to describe the effects of various patterns of serial correlation on the test statistics arising from these hypotheses. Attention is specially paid to the models of one- and two- way analysis of variance.

Linear algebra is one of the first advanced mathematics courses that students encounter at university level. The transfer from a primarily procedural or algorithmic school approach to an abstract and formal presentation of concepts through concrete definitions, seems to be creating difficulty for many students who are barely coping with procedural aspects of the subject. This research proposes applying APOS theory, in conjunction with Tall’s three worlds of embodied, symbolic and formal mathematics, to create a framework in order to examine the learning of a variety of linear algebra concepts by groups of first and second year university students. The aim is to investigate the difficulties in understanding some linear algebra concepts and to propose potential paths for preventing them. As part of this research project several case studies were conducted where groups of first and second year students were exposed to teaching and learning some introductory linear algebra concepts based on the framework and expressed their thinking through their involvements in tests, interviews and concept maps. The results suggest that the students had limited understanding of the concepts, they struggled to recognise the concepts in different registers, and their lack of ability in linking the major concepts became apparent. However, they also revealed that those with more representational diversity had more overall understanding of the concepts. In particular the embodied introduction of the concept proved a valuable adjunct to their thinking. Since difficulties with learning linear algebra by average students are universally acknowledged, it is anticipated that this study may provide suggestions with the potential for widespread positive consequences for learning.

Automatic Interaction Detection (A.I.D.) is the name of a computer program, first used in the social sciences, to find the interaction between a set of predictor variables and a single dependent variable. The program proceeds in stages, and at each stage the categories of a predictor variable induce a split of the dependent variable into two groups, so that the between groups sum of squares ( BSS ) is a maximum. In this way, the optimum split defines the interaction between predictor and dependent variable, and the criterion BSS is taken as a measure of the explanatory power of the split. One of the strengths of A.I.D. is that this interaction is established without any reference to a specific model, and for this reason it is widely used in practice. However this strength is also its weakness; with no model there is no measure of its significance. Barnard (1974) has said: “… nowadays with more and more apparently sophisticated computer programs for social science, failure to take account of possible sampling fluctuations is leading to a glut of unsound analyses … I have in mind procedures such as A.I.D., the automatic interaction detector, which guarantees to get significance out of any data whatsoever. Methods of this kind require validation …” The aim of this thesis is to supply Part of that validation by investigating the null distribution of the optimum BSS for a single predictor at a single stage of A.I.D., so that the significance of any particular split can be judged. The problem of the overall significance of a complete A.I.D. analysis, combining many stages, still remains to be solved. In Chapter 1 the A.I.D. method is described in more detail and an example is presented to illustrate its use. A null hypothesis that the dependent variable observations have independent and identical normal distributions is proposed as a model for no interaction. In Chapters 2 and 3 the null distributions of the optimum BSS for a single predictor are derived and tables of percentage points are given. In Chapter 4 the normal assumption is dropped and non-parametric A.I.D. criteria, based on ranks, are proposed. Tables of percentage points, found by direct enumeration and by Monte Carlo methods, are given. In Chapter 5 the example presented in Chapter 1 is used to illustrate the application of the theory and tables in Chapters 2, 3 and 4 and some final conclusions are drawn.

This thesis investigates structures that are presentable by finite automata working synchronously on tuples of finite words. The emphasis is on understanding the expressiveness and limitations of automata in this setting. In particular, the thesis studies the classification of classes of automatic structures, the complexity of the isomorphism problem, and the relationship between definability and recognisability.

The n-dimensional pseudospheres are the surfaces in Rn+l given by the equations x12+x22+...+xk2-xk+12-...-xn+12=1(1 ≤ k ≤ n+1). The cases k=l, n+1 give, respectively a pair of hyperboloids, and the ordinary n-sphere. In the first chapter we consider the pseudospheres as surfaces h En+1,k, where Em,k=Rk x (iR)m-k, and investigate their geometry in terms of the linear algebra of these spaces. The main objects of investigation are finite sequences of hyperplanes in a pseudosphere. To each such sequence we associate a square symmetric matrix, the Gram matrix, which gives information about angle and incidence properties of the hyperplanes. We find when a given matrix is the Gram matrix of some sequence of hyperplanes, and when a sequence is determined up to isometry by its Gram matrix. We also consider subspaces of pseudospheres and projections onto them. This leads to an n-dimensional cosine rule for spherical and hyperbolic simplices. In the second chapter we derive integral formulae for the volume of an n-dimensional spherical or hyperbolic simplex, both in terms of its dihedral angles and its edge lengths. For the regular simplex with common edge length γ we then derive power series for the volume, both in u = sinγ/2, and in γ itself, and discuss some of the properties of the coefficients. In obtaining these series we encounter an interesting family of entire functions, Rn(p) (n a nonnegative integer and pεC). We derive a functional equation relating Rn(p) and Rn-1(p). Finally we classify, up to isometry, all tetrahedra with one or more vertices truncated, for which the dihedral angles along the edges formed by the truncatons. are all π/2, and the remaining dihedral angles are all sub-multiples of π. We show how to find the volumes of these polyhedra, and find presentations and small generating sets for the orientation-preserving subgroups of their reflection groups. For particular families of these groups, we find low index torsion free subgroups, and construct associated manifolds and manifolds with boundary In particular, we find a sequence of manifolds with totally geodesic boundary of genus, g≥2, which we conjecture to be of least volume among such manifolds.

This research is a collective case study that investigates the constitutive relationship between students' representational competences and mathematical understanding of derivative. Its goal was to describe the representational abilities characterising different ways of knowing, and these were categorised as procedure-oriented, process-oriented, object-oriented, concept-oriented and versatile. The study was conducted in four Form 7 classrooms, in their Mathematics with Calculus classes, where graphic calculators were used in the teaching and learning of derivative. The choice of the context was based on the belief that the use of graphic calculators might encourage a multi-representational approach to teaching, and support the development of students' multi-representational way of thinking. The research data were collected both from teachers and their students. These data comprise teacher and student interviews, with the first interviews conducted before their lessons on derivative and the second after the lesson. The students were also given pre-lesson and post-lesson tests in order to triangulate student data. A Representational Framework of Knowing Derivative was constructed as an analysis tool, and used to explore students' representational abilities and ways of knowing. From the analysis, the students' cognitive processes were construed, together with the nature of their representational, cognitive and conceptual schemas. The representational framework of knowing was later refined to present an empirically-based theoretical framework that bridges the gap between what was theorised and what was observed. The results of the study suggest that the relationship between students' ways of knowing and their representational abilities is mediated by the following factors: (i) the students' interpretation of the mathematical notion; (ii) the representational nature of their interpretations of derivative, and the representational aspects in their problem solving activities; and (iii) the nature of the representational links that they have formed between procedures, processes, objects and sub-concepts that were construed to constitute their conceptual and cognitive schemas of derivative. With regard to the use of the graphic calculator, this research has noted a possible contribution of the graphic calculator in the development of students' multi-representational ways of thinking and learning.

This thesis is divided into two sections. The first section examines certain stability properties of implicit Runge-Kutta methods. In particular, a new stability property is defined, which is a modification to non-autonomous problems of A-stability, and its relation to B-stability is considered. A Runge-Kutta method is written as [see 01front.pdf for graphic] and classes of methods are constructed based on the property ∑sj=1aijck-1j = cki/k, i = 1,...,s and k = 1,...,s-1, where c1,...,cs are assumed to be distinct. Under this assumption a transformation is made, such that A = VsAsV-1s, where Vs is the Vandermonde matrix whose (i,j) element is cj-1i, and As has a special structure. These methods are examined in the light of the various stability criteria. It is also shown that the growth of errors can be estimated by an extension of this new stability theory and a number of examples are given. In the second section we consider the solution of stiff differential equations by implicit Runge-Kutta methods. In particular, we examine a procedure suggested by Butcher [6] which enables an efficient implementation of Runge-Kutta methods. He has shown that the most efficient methods when using this implementation are those whose characteristic polynomial of the Runge-Kutta matrix has a single real s-fold zero. Based on this criterion a family of methods, called singly-implicit methods, is constructed, and results concerning their maximum attainable order and stability properties are given. Some consideration is also given to showing how local error estimates can be obtained, by the use of embedding techniques, for both singly-implicit methods and the more general family of implicit Runge-Kutta methods. Finally, an algorithm based on these singly-implicit methods is presented. It is tested on a number of stiff differential equations, and comparisons are made between this algorithm and others currently in use.

Ordinary differential equations arise frequently in the study of the physical world. Unfortunately many cannot be solved exactly. This is why the ability to solve these equations numerically is important. Traditionally mathematicians have used one of two classes of methods for numerically solving ordinary differential equations. These are linear multistep methods and Runge–Kutta methods. General linear methods were introduced as a unifying framework for these traditional methods. They have both the multi-stage nature of Runge–Kutta methods as well as the multi-value nature of linear multistep methods. This extremely broad class of methods, besides containing Runge–Kutta and linear multistep methods as special cases, also contains hybrid methods, cyclic composite linear multistep methods and pseudo Runge–Kutta methods. In this thesis we present a class of methods known as Almost Runge–Kutta methods. This is a special class of general linear methods which retains many of the properties of traditional Runge–Kutta methods, but with some advantages. Most of this thesis concentrates on explicit methods for non-stiff differential equations, paying particular attention to a special fourth order method which, when implemented in the correct way, behaves like order five. We will also introduce low order diagonally implicit methods for solving stiff differential equations.

In this thesis we study a class of symmetric forced oscillators modeled by non-linear ordinary differential equations. Solutions for this class of systems can be symmetric or non-symmetric. When a symmetric periodic solution loses its stability as a physical parameter is varied, and two non-symmetric periodic solutions appear, this is called a symmetry breaking bifurcation. In a symmetry increasing bifurcation two conjugate chaotic attractors (i.e.,attractors which are related to each other by the symmetry) collide and form a larger symmetric chaotic attractor. Symmetry can also be restored via explosions where, as a physical parameter is varied, two conjugate attractors (chaotic or periodic) which do not intersect are suddenly embedded in one symmetric attractor. In this thesis we show that all these apparently distinct bifurcations can be realized by a single mechanism in which two conjugate attractors collide with a symmetric limit set. The same mechanism seems to operate for at least some bifurcations involving non-attracting limit sets. We illustrate this point with examples of symmetry restoration in attracting and non-attracting sets found in the forced Duffing oscillator and in a power system. Symmetry restoration in the power system is associated with a phenomenon known as ferroresonance. The study of the ferroresonance phenomenon motivated this thesis. Part of this thesis is devoted to studying one aspect of the ferroresonance phenomenon the appearance of a strange attractor with a band-like structure. This attractor was called previously a 'pseudo-periodic' attractor. Some methods for analyzing the non-autonomous systems under study are shown. We construct three different maps which highlight different features of symmetry restoring bifurcations. One map in particular captures the symmetry of a solution by sampling it every half the period of the forcing. We describe a numerical method to construct a bifurcation diagram of periodic solutions and present a non-standard approach for converting the forced oscillator to an autonomous system.