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    The quotient between length and multiplicity : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University
    (Massey University, 2000) Allsop, Nicholas Frederick
    This dissertation examines the finiteness of the algebraic invariants nA(M) and θA(M). These invariants, based on the ratio of length and multiplicity and the ratio of Loewy length and multiplicity respectively, are studied in general and under certain conditions. The finiteness of θA(M) is established for a large class of algebraic structures. nA(M) is shown to be finite in the low dimensional case as well as when we restrict our attention to special sets of ideals. Also considered in this dissertation are equivalent conditions for the local case to be bounded by the graded case when evaluating nA(M).
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    A model for phenotype change in a stochastic framework
    (American Institute of Mathematical Sciences, 2008) Wake GC
    In some species, an inducible secondary phenotype will develop some time after the environmental change that evokes it. Nishimura (2006) [4] showed how an individual organism should optimize the time it takes to respond to an environmental change ("waiting time''). If the optimal waiting time is considered to act over the population, there are implications for the expected value of the mean fitness in that population. A stochastic predator-prey model is proposed in which the prey have a fixed initial energy budget. Fitness is the product of survival probability and the energy remaining for non-defensive purposes. The model is placed in the stochastic domain by assuming that the waiting time in the population is a normally distributed random variable because of biological variance inherent in mounting the response. It is found that the value of the mean waiting time that maximises fitness depends linearly on the variance of the waiting time.
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    Evolutionary analyses of large data sets : trees and beyond : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University
    (Massey University, 2001) Holland, Barbara Ruth
    The increasing amount of molecular data available for phylogenetic studies means that larger, often intra-species, data sets are being analysed. Treating such data sets with methods designed for small interspecies data may not be useful. This thesis comprises four projects within the field of phylogenetics that focus on cases where the application of current tree estimation methods is not sufficient to answer the biological questions of interest. A simulation study contrasts the accuracy of several tree estimation methods for a particular class of five-taxon, equal-rate, trees. This study highlights several difficulties with tree estimation, including the fact that some tree topologies produce “misleading" patterns that are incorrectly interpreted; that correction for multiple changes does not always increase accuracy, because of increased variance; and the difficulty of correctly placing outgroup taxa. A mitochondrial DNA data set, containing over 400 modern and ancient Adélie penguin samples, is used to estimate the rate of evolution. Straightforward tree-estimation is unhelpful because the amount of homoplasy in the data makes the construction of a single reliable tree impossible. Instead the data is represented by a network. A method, that extends statistical geometry, assesses whether or not a data set can be well-represented by a tree. The "tree-likeness" of each quartet in the data is evaluated and displayed visually, either for the entire data set or by taxon. This aids in identifying reticulate (or simply noisy) data sets, and also particular taxa that confound tree-like signal. Novel methods are developed that use pairwise dissimilarities between isolates in intra-species microbial data sets, to identify strains that are good representatives of their species or subspecies.
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    Intercensal updating of small area estimates : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University, Palmerston North, New Zealand
    (Massey University, 2010) Isidro, Marissa Cinco
    Small area estimation (SAE) involves tting statistical models to generate statistics for areas where the sample size of the survey data is insu cient for generating precise estimates. A recent application of SAE techniques is in estimating local level poverty measures in Third World countries necessary for aid allocation and monitoring of the Millennium Development Goals (MDGs). The SAE technique commonly known as ELL method (Elbers et al., 2003) is extensively implemented by the World Bank in collaboration with national statistical agencies in most Third World countries. This technique generates estimates by tting a linear mixed model to household level income or consumption using the survey and census data. The ELL method di ers in various ways from the mainstream SAE techniques, two of which are emphasized in this thesis: (1) the ELL model does not include area level e ects and (2) the model tting technique follows a non-standard weighted generalized least squares (GLS). Under the ELL method the survey and the census data are assumed to have been conducted at the same time period, hence generating updated estimates of poverty measures during non-census years is a problem. The method for SAE updating developed in this thesis is called the Extended Structure Preserving Estimation (ESPREE) method, an extension of the classical SAE technique called the structure preserving estimation (SPREE) method - an approach to SAE based on a categorical data analysis framework. The ESPREE method is structured within a generalized linear model (GLM) framework and uses information from the most recent survey and pseudocensus (census replicates) data to generate updated small area estimates under a superpopulation. The World Bank in collaboration with the National Statistical Coordination Board in the Philippines has conducted an intercensal updating project using an ELL-based method requiring time invariant variables. Comparison of the estimates generated from the ELL-based and ESPREE updating method revealed substantial di erences. The ESPREE method but not the ELL updating method generated unbiased estimates. An in-country validation exercise conducted in the Philippines supported the view that ESPREE based estimates, besides having theoretical advantages, also conformed better to local experts' opinion on current poverty levels.
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    Two elliptic generator Kleinian groups : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand
    (Massey University, 2010) Zhang, Qingxiang
    This thesis studies the discreteness of Kleinian groups and the geometry of their associated orbit spaces: hyperbolic 3-manifolds and 3-orbifolds. Thurston's geometrization theorem states that the interior of every compact 3-manifold can be decomposed into pieces which have geometric structures. Most of these pieces have hyperbolic structures. Every hyperbolic 3-manifold can be described as H3=G, where H3 is the 3-dimensional hyperbolic space and G is a torsion free Kleinian group. The compact 3-manifolds other than hyperbolic 3-manifolds have been classified. Therefore the study of Kleinian groups holds the key to understanding 3-manifolds in general. We investigate various conditions for determining the discreteness of a Kleinian group. Such a group is discrete if and only if all its two generator subgroups are discrete, thus we study the question of discreteness through two generator groups. Here we consider Kleinian groups generated by two elements of finite order. Once the order of these generators is fixed, these groups lie in a one complex-dimensional parameter space with a highly fractal boundary. Computational investigations of various types into the size and structure of this parameter space form an integral part of this thesis. In particular an analysis of Dehn surgery on 2-bridge knots and links gives us data indicating the boundary of the parameter space "internally" analogous to the boundary in the Riley slice (the parabolic case). We then investigate this boundary "externally" and numerically using the rational pleating rays and rational words, which were developed by Keen and Series to study the Riley slice boundary. We are then able to identify some boundary points via the interplay between algebraic and geometric convergence of sequences of Kleinian groups. We find an interesting connection between Conway's notation for 2-bridge knots and links and rational words. We then apply these descriptions of parameter spaces to advance the identification problem for two generator arithmetic Kleinian groups, in fact we give a conjecturally complete list of certain families of such groups that were previously identified as being the most difficult cases.
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    Monotone iterates for nonlinear singularly perturbed convection-diffusion problems : a thesis submitted in partial fulfilment of the requirements of the degree for Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand
    (Massey University, 2010) Pack, Sophie
    We are interested in monotone iterative algorithms for solving nonlinear singularly perturbed convection-diffusion problems. These problems arise in many physical phenomena. One of the most common sources of these problems is the linearization of Navier-Stokes equations with large Reynolds numbers, other sources include drift-diffusion equations of semi-conductor device modelling, financial modelling, modelling in mathematical biology, fluid dynamics and heat transport problems. Singularly perturbed convection-diffusion problems are characterized by thin areas of rapid change of solutions. Many of these problems can not be solved analytically but must instead be solved numerically. Classical numerical approaches for solving these problems do not always work and may show unsatisfactory behaviours. In this thesis, we focus on constructing monotone iterative methods for solving nonlinear singularly perturbed convection-diffusion problems. Monotone difference schemes have significant advantages: they guarantee that systems of algebraic equations based on such schemes are well-posed; the finite difference operators satisfy the discrete maximum principle. We construct a uniform convergent difference scheme for solving a nonlinear singularly perturbed two-point boundary value problem of the convection-diffusion type with discontinuous data. The uniform convergence of this scheme is proven on arbitrary meshes. A monotone iterative method is applied to computing the nonlinear difference scheme. In the past fteen years, much interest has been shown in domain decomposition techniques for solving singularly perturbed convection-diffusion problems. In this thesis, we construct one- and two-level monotone domain decomposition algorithms based on the multiplicative and additive Schwarz algorithms. These algorithms are proven to converge to the exact solution of the problem. We construct monotone relaxation methods by modifying the point and block w-Jacobi and successive underrelaxation methods. We prove that the point and block monotone relaxation methods converge to the exact solution of the problem. We combine the monotone domain decomposition algorithms and relaxation methods to construct composite monotone domain decomposition algorithms. These algorithms are proven to converge to the exact solution of the problem. Multigrid methods are generally accepted as fast efficient solvers. The standard multigrid method has been shown to be unsatisfactory when applied to singularly perturbed problems. We construct monotone multigrid methods for solving nonlinear singularly perturbed convection-diffusion problems. We prove that these methods converge to the exact solution of the problem.
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    Small area estimation via generalized linear models : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University, Palmerston North, New Zealand
    (Massey University, 2003) Noble, Alasdair D. L.
    Survey information is commonly collected to yield estimates of quantities for large geographic areas, for example, complete countries. However the estimates of those quantities at much smaller geographic areas are often of interest and the sample sizes in these areas are generally too small to give useful results. Small area estimation is used to make inference about those small areas with greater precision than the direct estimates, either by exploiting similarities between different small areas or by accessing additional information often from administrative records. The majority of the traditional small area estimation methods are examples of a simple linear model Marker (1999) and this work begins by extending the model to a generalized linear model (GLM) Nelder and Wedderburn (1972) and then including structure preserving estimation (SPREE) in the classification. This had not been done previously. SPREE had previously been fitted using the iterative proportional fitting algorithm Deming and Stephan (1940) which could be described as a "black box" approach. By expressing SPREE in terms of a GLM an alternative algorithm for fitting the method is developed which elucidates the underlying concepts. This new approach allows the method to be extended from the contingency table with categorical variables which the IPF could fit, to continuous variables and random effects models. An example including a continuous variable is given. SPREE is a method which uses auxiliary information as well as survey data. In the past assumptions about appropriate auxiliary information have been made with little theoretical support. The new approach allows these assumptions to be considered and they are found to be wanting in some cases. An example based on a national survey in New Zealand for unemployment statistics, is used extensively throughout the thesis. These data have characteristics that make analysis in the Bayesian paradigm appropriate. This paradigm has been applied and a conditional autoregressive error structure is considered. Finally relative risk models are considered. It is shown that these could have been fitted using the IPF algorithm but the new approach allows combinations of other modeling techniques which are not available using IPF.
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    Differential geometry of projectively flat Finsler spaces : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand
    (Massey University, 2003) Senerath, Padma
    The aim of this thesis is to study the theory of Finsler spaces by considering the following main objectives. (i) To present the basic concepts of Finsler geometry including connections, flag curvature, projective changes, Randers spaces and Finsler spaces with other types of (α,β)-metric, where α is a Riemannian metric and β is a one-form. (ii) To introduce a Riemannian space of non-zero constant sectional curvature by considering a locally projectively flat Finsler space. The requirement for the Riemannian connection to be metric compatible gives a system of partial differential equations. Further, we compute two standard Riemannian metrics of non-zero constant sectional curvature by choosing two solutions of this system of partial differential equations. (iii) To give two examples of locally projectively flat Randers metrics of scalar curvature by using a Riemannian metric computed in (ii) to illustrate the fact that some locally projectively flat Randers metrics of scalar curvature do not have isotropic S-curvature. We also prove that the scalar curvature of a Randers metric is not necessarily a constant if the metric has isotropic S-curvature and closed one-form by using an example. (iv) To find necessary and sufficient conditions for Finsler spaces with various types of (α,β)-metric to be locally projectively flat and determine whether the conditions, a Riemannian metric (α) is locally projectively flat and a one-form (β) is closed, can occur at the same time in the locally projectively flat Finsler spaces with various types of (α,β)-metric.
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    Contact systems and contact integrators : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand
    (Massey University, 2003) Joo, Seung-Hee
    This thesis is concerned with the study of contact systems, which are ordinary differential equations whose flow preserves a contact structure. We study contact systems from both an analytical and numerical point of view. The traditional point of view is to study the Reeb vector field of a contact form. However, if the contact Hamiltonian vanishes then its contact vector field is not the Reeb vector field of any contact form equivalent to the given one. In this thesis we study exactly this case, when the contact Hamiltonian vanishes on some submanifold of phase space. This submanifold is invariant under the flow and we study the flow on it, including its stability and fixed points. The natural numerical method for a contact system is a 'contact integrator', a map that preserves the contact structure, which is suitable for exploring the long-time dynamics of contact systems. These have not been studied very much in geometric integration. In order to formulate our results and some consequences for contact integrators, we give a thorough development of the symplectification of a contact system and have found the integrable contact systems related to integrable homogeneous Hamiltonian systems via symplectification. We develop contact integrators by the splitting method, leading to an explicit contact integrator for any polynomial contact vector field. We also study how symplectic integrators for Hamiltonian systems and volume-preserving integrators for divergence-free systems are related to contact integrators for contact systems.
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    Quantification of individual rugby player performance through multivariate analysis and data mining : a thesis presented for the fulfilment of the requirements for the degree of Doctor of Philosophy at Massey University, Albany, New Zealand
    (Massey University, 2003) Bracewell, Paul J
    This doctoral thesis examines the multivariate nature of performance to develop a contextual rating system for individual rugby players on a match-by-match basis. The data, provided by Eagle Sports, is a summary of the physical tasks completed by the individual in a match, such as the number of tackles, metres run and number of kicks made. More than 130 variables were available for analysis. Assuming that the successful completion of observed tasks are an expression of ability enables the extraction of the latent dimensionality of the data, or key performance indicators (KPI), which are the core components of an individual's skill-set. Multivariate techniques (factor analysis) and data mining techniques (self-organising maps and self-supervising feed-forward neural networks) are employed to reduce the dimensionality of match performance data and create KPI's. For this rating system to be meaningful, the underlying model must use suitable data, and the end model itself must be transparent, contextual and robust. The half-moon statistic was developed to promote transparency, understanding and interpretation of dimension reduction neural networks. This novel non-parametric multivariate method is a tool for determining the strength of a relationship between input variables and a single output variable, whilst not requiring prior knowledge of the relationship between the input and output variables. This resolves the issue of transparency, which is necessary to ensure the rating system is contextual. A hybrid methodology is developed to combine the most appropriate KPI's into a contextual, robust and transparent univariate measure for individual performance. The KPI's are collapsed to a single performance measure using an adaptation of quality control ideology where observations are compared with perfection rather than the average to suit the circumstances presented in sport. The use of this performance rating and the underlying key performance indicators is demonstrated in a coaching setting. Individual performance is monitored with the use of control charts enabling changes in form to be identified. This enables the detection of strengths/weakness in the individual's underlying skill-set (KPI's) and skills. This process is not restricted to rugby or sports data and is applicable in any field where a summary of multivariate data is required to understand performance.