The introductory chapter in this thesis examines briefly the nature of initial value problems and surveys the main types of methods used for their numerical solution. The general linear formulation of methods, first proposed by J. C. Butcher, is introduced, together with the definition of order for this class of methods. Finally in this chapter, the problem of stiffness and its effect on numerical procedures is considered. Following a review of Butcher's algebraic approach to the theory of Runge-Kutta and general linear methods in Chapter 2, the theory is applied in Chapter 3 to the search for general linear methods of various orders. As in the case of Runge-Kutta methods, the use of so-called simplifying assumptions plays a significant role in the practical determination of general linear methods. From amongst the range of possible numbers of simplifying assumptions, two cases are chosen and investigated in detail. The important question of stability is considered in the final section of Chapter 3. When a general linear method is used to approximate the solution of an initial value problem, special procedures are required to start and finish the integration. Whilst a major part of Chapter 4 is devoted to the determination of these procedures, the problems of the estimation of local truncation error and the implementation of general linear methods are also discussed. Finally, the Appendices contain Algol 60 procedures for the most important of the algorithms developed in the main body of the thesis.

Autonomous deterministic epidemiological models are known to be asymptotically stable. Asymptotic stability of these models contradicts observations. In this thesis we consider some factors which were suggested as able to destabilise the system. We consider discrete-time and continuous-time autonomous epidemiological models. We try to keep our models as simple as possible and investigate the impact of different factors on the system behaviour. Global methods of dynamical systems theory, especially the theory of bifurcations and the direct Lyapunov method are the main tools of our analysis. Lyapunov functions for a range of classical epidemiological models are introduced. The direct Lyapunov method allows us to establish their boundedness and asymptotic stability. It also helps investigate the impact of such factors as susceptibles' mortality, horizontal and vertical transmission and immunity failure on the global behaviour of the system. The Lyapunov functions appear to be useful for more complicated epidemiological models as well. The impact of mass vaccination on the system is also considered. The discrete-time model introduced here enables us to solve a practical problem-to estimate the rate of immunity failure for pertussis in New Zealand. It has been suggested by a number of authors that a non-linear dependence of disease transmission on the numbers of infectives and susceptibles can reverse the stability of the system. However it is shown in this thesis that under biologically plausible constraints the non-linear transmission is unable to destabilise the system. The main constraint is a condition that disease transmission must be a concave function with respect to the number of infectives. This result is valid for both the discrete-time and the continuous-time models. We also consider the impact of mortality associated with a disease. This factor has never before been considered systematically. We indicate mechanisms through which the disease-induced mortality can affect the system and show that the disease-induced mortality is a destabilising factor and is able to reverse the system stability. However the critical level of mortality which is necessary to reverse the system stability exceeds the mortality expectation for the majority of human infections. Nevertheless the disease-induced mortality is an important factor for understanding animal diseases. It appears that in the case of autonomous systems there is no single factor able to cause the recurrent outbreaks of epidemics of such magnitudes as have been observed. It is most likely that in reality they are caused by a combination of factors. / Subscription resource available via Digital Dissertations

This study attempts to develop the theory surrounding a controlled convergence theorem in the setting of AP integration. First, after a brief review, we spend some time on developing concepts and definitions. We are able to use these concepts to successfully present an initial control convergent theorem. Then we study our conditions more profoundly and find a number of equivalences and inequivalences between them. We are able to weaken some of the standard hypotheses significantly. The Riesz type approach to our integral has been included in our theory. Appropriately enough, we are able to use some of our approaches to extend the theory of the Henstock-Kurweil integral.

The introductory chapter in this thesis examines briefly the nature of initial value problems and surveys the main types of methods used for their numerical solution. The general linear formulation of methods, first proposed by J. C. Butcher, is introduced, together with the definition of order for this class of methods. Finally in this chapter, the problem of stiffness and its effect on numerical procedures is considered. Following a review of Butcher's algebraic approach to the theory of Runge-Kutta and general linear methods in Chapter 2, the theory is applied in Chapter 3 to the search for general linear methods of various orders. As in the case of Runge-Kutta methods, the use of so-called simplifying assumptions plays a significant role in the practical determination of general linear methods. From amongst the range of possible numbers of simplifying assumptions, two cases are chosen and investigated in detail. The important question of stability is considered in the final section of Chapter 3. When a general linear method is used to approximate the solution of an initial value problem, special procedures are required to start and finish the integration. Whilst a major part of Chapter 4 is devoted to the determination of these procedures, the problems of the estimation of local truncation error and the implementation of general linear methods are also discussed. Finally, the Appendices contain Algol 60 procedures for the most important of the algorithms developed in the main body of the thesis.

This study attempts to develop the theory surrounding a controlled convergence theorem in the setting of AP integration. First, after a brief review, we spend some time on developing concepts and definitions. We are able to use these concepts to successfully present an initial control convergent theorem. Then we study our conditions more profoundly and find a number of equivalences and inequivalences between them. We are able to weaken some of the standard hypotheses significantly. The Riesz type approach to our integral has been included in our theory. Appropriately enough, we are able to use some of our approaches to extend the theory of the Henstock-Kurweil integral.

The introductory chapter in this thesis examines briefly the nature of initial value problems and surveys the main types of methods used for their numerical solution. The general linear formulation of methods, first proposed by J. C. Butcher, is introduced, together with the definition of order for this class of methods. Finally in this chapter, the problem of stiffness and its effect on numerical procedures is considered. Following a review of Butcher's algebraic approach to the theory of Runge-Kutta and general linear methods in Chapter 2, the theory is applied in Chapter 3 to the search for general linear methods of various orders. As in the case of Runge-Kutta methods, the use of so-called simplifying assumptions plays a significant role in the practical determination of general linear methods. From amongst the range of possible numbers of simplifying assumptions, two cases are chosen and investigated in detail. The important question of stability is considered in the final section of Chapter 3. When a general linear method is used to approximate the solution of an initial value problem, special procedures are required to start and finish the integration. Whilst a major part of Chapter 4 is devoted to the determination of these procedures, the problems of the estimation of local truncation error and the implementation of general linear methods are also discussed. Finally, the Appendices contain Algol 60 procedures for the most important of the algorithms developed in the main body of the thesis.

Autonomous deterministic epidemiological models are known to be asymptotically stable. Asymptotic stability of these models contradicts observations. In this thesis we consider some factors which were suggested as able to destabilise the system. We consider discrete-time and continuous-time autonomous epidemiological models. We try to keep our models as simple as possible and investigate the impact of different factors on the system behaviour. Global methods of dynamical systems theory, especially the theory of bifurcations and the direct Lyapunov method are the main tools of our analysis. Lyapunov functions for a range of classical epidemiological models are introduced. The direct Lyapunov method allows us to establish their boundedness and asymptotic stability. It also helps investigate the impact of such factors as susceptibles' mortality, horizontal and vertical transmission and immunity failure on the global behaviour of the system. The Lyapunov functions appear to be useful for more complicated epidemiological models as well. The impact of mass vaccination on the system is also considered. The discrete-time model introduced here enables us to solve a practical problem-to estimate the rate of immunity failure for pertussis in New Zealand. It has been suggested by a number of authors that a non-linear dependence of disease transmission on the numbers of infectives and susceptibles can reverse the stability of the system. However it is shown in this thesis that under biologically plausible constraints the non-linear transmission is unable to destabilise the system. The main constraint is a condition that disease transmission must be a concave function with respect to the number of infectives. This result is valid for both the discrete-time and the continuous-time models. We also consider the impact of mortality associated with a disease. This factor has never before been considered systematically. We indicate mechanisms through which the disease-induced mortality can affect the system and show that the disease-induced mortality is a destabilising factor and is able to reverse the system stability. However the critical level of mortality which is necessary to reverse the system stability exceeds the mortality expectation for the majority of human infections. Nevertheless the disease-induced mortality is an important factor for understanding animal diseases. It appears that in the case of autonomous systems there is no single factor able to cause the recurrent outbreaks of epidemics of such magnitudes as have been observed. It is most likely that in reality they are caused by a combination of factors. / Subscription resource available via Digital Dissertations

The accuracy and efficiency of computing multiple integrals is a very important problem that arises in many scientific, financial and engineering applications. The research conducted in this thesis is designed to build on past work and develop and analyze new numerical methods to evaluate double integrals efficiently. The fundamental aim is to develop and assess techniques for (numerically) evaluating double integrals with high accuracy. The general approach presented in this thesis involves the development of new multivariate approximations from a generalaised Taylor perspective in terms of Appell type polynomials and to study their use in multi-dimensional integration. The expectation is that the new methods will provide polynomial and polynomial-like approximations that can be used for application in a straight forward manner with better accuracy. That is, we aim to devise and investigate new multiple integration formulae and as well as provide information on a priori error bounds. A further major contribution of the work builds on the research conducted in the field of Grüss type inequalities and leads to a new approximation of the one and two dimensional finite Fourier transform. The approximations are in terms of the complex exponential mean and estimate of the error of approximation for different classes of functions of bounded variation defined on finite intervals. It is believed that this work will also have an impact in the area of numerical multidimensional integral evaluation for other integral operators.

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.