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 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.

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.

The role of the scientist in the courtroom has come under more scrutiny this century than ever before. As a consequence, scientists must constantly look for ways to improve the validity of the evidence they deliver. It is here that the professional statistician can provide assistance. The use of statistics in the courtroom and in forensic science is not new, but until recently has not been common either. Statistics can provide objectivity to subjective assessments and strengthen a case for the prosecution or the defence, but only if is used correctly. The aim of this thesis is to enhance and replace the existing technology used in statistical analysis and presentation of trace evidence, i.e. all non-genetic evidence (hairs, fibres, glass, paint, etc.) and transfer problems.

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.

This work addresses the development and application of gene set analysis methods to problems in microarray-based data sets. The work consists of three parts. In the first part a gene set analysis method (PCOT2) is developed. It utilizes inter-gene correlation to detect significant alteration in gene sets across experimental conditions. The second part is focused on the exploration of correlation-based gene sets in conjunction with the application of the PCOT2 testing method in the investigation of biological mechanisms underlying breast cancer recurrence. In the third part, statistical models for analyzing combined microarray-based expression and genomic copy number data are developed. In addition, an analysis which incorporates tumour subgroups is shown to provide more accurate prognosis assessment for breast cancer patients.

The effective integration of technology into the teaching and learning of mathematics remains one of the critical challenges facing tertiary mathematics, which has traditionally been slow to respond to technological innovation. This thesis reveals that the term integration is widely used in the literature with respect to technology and the curriculum, although its meaning can vary substantially, and furthermore, the term is seldom well defined. A review of the literature provides the basis for a survey of undergraduate mathematics educators, to determine their use of technology, their views of what an Integrated Technology Mathematics Curriculum (ITMC) may resemble, and how it may be achieved. Responses to this survey, and factors identified in the literature, are used to construct a taxonomy of integrated technology. The taxonomy identifies six defining characteristics of an ITMC, each with a number of associated elements. A visual model using radar diagrams is developed to compare courses against the taxonomy, and to identify aspects needing attention in individual courses. T Evidence from an observational study of initiatives to introduce Computer Algebra Systems into undergraduate mathematics courses at The University of Auckland, firstly using CAS-calculators and latterly computer software, is examined against the taxonomy. A number of critical issues influencing the integration of these technologies are identified. These include mandating technology use in official departmental policy, attention to congruency and fairness in assessment, re-evaluating the value of topics in the curriculum, re-establishing the goals of undergraduate courses, and developing the pedagogical technical knowledge of teaching staff. The thesis concludes that effective integration of technology in undergraduate mathematics requires a recognition of, and comprehensive attention to, the interdependence of the taxonomy components. An integrated, holistic approach, which aims for curricular congruency across all elements of the taxonomy, provides the basis for a more consistent, effective and sustainable ITMC.