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.

Quandles were introduced to Knot Theory in the 1980s as an almost complete algebraic invariant for knots and links. Like their more basic siblings, groups, they are difficult to distinguish so a major challenge is to devise means for determining when two quandles having different presentations are really different. This thesis addresses this point by studying algebraic aspects of quandles. Following what is mainly a recapitulation of existing work on quandles, we firstly investigate how a link quandle is related to the quandles of the individual components of the link. Next we investigate coset quandles. These are motivated by the transitive action of the operator, associated and automorphism group actions on a given quandle, allowing techniques of permutation group theory to be used. We will show that the class of all coset quandles includes the class of all Alexander quandles; indeed all group quandles. Coset quandles are used in two ways: to give representations of connected quandles, which include knot quandles; and to provide target quandles for homomorphism invariants which may be useful in enabling one to distinguish quandles by counting homomorphisms onto target quandles. Following an investigation of the information loss in going from the fundamental quandle of a link to the fundamental group, we apply our techniques to calculations for the figure eight knot and braid index two knots and involving lower triangular matrix groups. The thesis is rounded out by two appendices, one giving a short table of knot quandles for knots up to six crossings and the other a computer program for computing the homomorphism invariants.

No description available.

No description available.

This thesis is concerned with the well-posedness, dynamical properties and numerical treatment of the generalised Euler equations on the Bott-Virasoro group with respect to the general H[superscript]k metric , k[is greater than or equal to]2. The term “generalised Euler equations” is used to describe geodesic equations on Lie groups, which unifies many differential equations and has found many applications in such as hydrodynamics, medical imaging in the computational anatomy, and many other fields. The generalised Euler equations on the Bott-Virasoro group for k = 0, 1 are well-known and intensively studied— the Korteweg-de Vries equation for k = 0 and the Camassa-Holm equation for k = 1. Unlike these, the equations for k[is greater than or equal to]2, which we call the modified Camassa-Holm (mCH) equation, is not known to be integrable. This distinction motivates the study of the mCH equation. In this thesis, we derive the mCH equation and establish the short time existence of solutions, the well-posedness of the mCH equation, long time existence, the existence of the weak solutions, both on the circle S and [blackboard bold] R, and three conservation laws, show some quite interesting properties, for example, they do not lead to the blowup in finite time, unlike the Camassa-Holm equation. We then consider two numerical methods for the modified Camassa-Holm equation: the particle method and the box scheme. We prove the convergence result of the particle method. The numerical simulations indicate another interesting phenomenon: although mCH does not admit blowup in finite time, it admits solutions that blow up (which means their maximum value becomes infinity) at infinite time, which we call weak blowup. We study this novel phenomenon using the method of matched asymptotic expansion. A whole family of self-consistent blowup profiles is obtained. We propose a mechanism by which the actual profile is selected that is consistent with the simulations, but the mechanism is only partly supported by the analysis. We study the four particle systems for the mCH equation finding numerical evidence both for the non-integrability of the mCH equations and for the existence of the fourth integral. We also study the higher dimensional case and obtain the short time existence and well-posedness for the generalised Euler equation in the two dimension case.

The estimation of the distribution of usual intake of nutrients is important for developing nutrition policies as well as for etiological research and educational purposes. In most nutrition surveys only a small number of repeated intake observations per individual are collected. Of main interest is the longterm usual intake which is defined as long-term daily average intake of a dietary component. However, dietary intake on a single day is a poor estimate of the individual’s long-term usual intake. Furthermore, the distribution of individual intake means is also a poor estimator of the distribution of usual intake since usually there is large within-individual compared to between-individual variability in the dietary intake data. Hence, the variance of the mean intakes is larger than the variance of the usual intake distribution. Essentially, the estimation of the distribution of long-term intake is equivalent to the estimation of a distribution of a random variable observed with measurement error. Some of the methods for estimating the distributions of usual dietary intake are reviewed in detail and applied to nutrient intake data in order to evaluate their properties. The results indicate that there are a number of robust methods which could be used to derive the distribution of long-term dietary intake. The methods share a common framework but differ in terms of complexity and assumptions about the properties of the dietary consumption data. Hence, the choice of the most appropriate method depends on the specific characteristics of the data, research purposes as well as availability of analytical tools and statistical expertise.

Agrichemical spray drift is an issue of concern for the orcharding industry. Shelterbelts surrounding orchard blocks can significantly reduce spray drift by intercepting droplets from the airflow. At present, there is little information available with which to predict drift deposits downwind, particularly in the case of a fully-sheltered orchard block. In this thesis, we develop a simple mathematical model for the transport of airborne drifting spray droplets, including the effects of droplet evaporation and interception by a shelterbelt. The object is for the model to capture the major features of the droplet transport, yet be simple enough to determine an analytic solution, so that the deposit on the ground may be easily calculated and the effect of parameter variations observed. We model the droplet transport using an advection-dispersion equation, with a trapping term added to represent the shelterbelt. In order to proceed analytically, we discretise the shelterbelt by dividing it into a three-dimensional array of blocks, with the trapping in each block concentrated to the point at its centre. First, we consider the more straightforward case where the droplets do not evaporate; solutions are presented in one, two and three dimensions, along with explicit expressions for the total amount trapped and the deposit on the ground. With evaporation, the model is more difficult to solve analytically, and the solutions obtained are nestled in integral equations which are evaluated numerically. In both cases, examples are presented to show the deposition profile on the ground downwind of the shelterbelt, and the corresponding reduction in deposit from the same scenario without the shelterbelt.

Let M Φ and Mψ be elements of PSL(2,C) representing orientation preserving isometries on the upper half-space model of hyperbolic 3-space Φ and ψ respectively. The parameters β = tr2(M Φ) - 4, β1 = tr2(Mψ) - 4, γ = tr[M Φ,Mψ] - 2, determine the discrete group (Φ ,ψ) uniquely up to conjugacy whenever γ ≠ 0. This thesis is concerned with explicitly lifting this parameterisation of (Φ , ψ) to PSO(1, 3) realised as a discrete 2 generator subgroup of orientation preserving isometries on the hyperboloid model of hyperbolic 3-space. We particularly focus on the case where both Φ and ψ are elliptic.

This thesis is concerned with the well-posedness, dynamical properties and numerical treatment of the generalised Euler equations on the Bott-Virasoro group with respect to the general Hk metric , k 2. The term “generalised Euler equations” is used to describe geodesic equations on Lie groups, which unifies many differential equations and has found many applications in such as hydrodynamics, medical imaging in the computational anatomy, and many other fields. The generalised Euler equations on the Bott-Virasoro group for k = 0, 1 are well-known and intensively studied— the Korteweg-de Vries equation for k = 0 and the Camassa-Holm equation for k = 1. Unlike these, the equations for k 2, which we call the modified Camassa-Holm (mCH) equation, is not known to be integrable. This distinction motivates the study of the mCH equation. In this thesis, we derive the mCH equation and establish the short time existence of solutions, the well-posedness of the mCH equation, long time existence, the existence of the weak solutions, both on the circle S and R, and three conservation laws, show some quite interesting properties, for example, they do not lead to the blowup in finite time, unlike the Camassa-Holm equation. We then consider two numerical methods for the modified Camassa-Holm equation: the particle method and the box scheme. We prove the convergence result of the particle method. The numerical simulations indicate another interesting phenomenon: although mCH does not admit blowup in finite time, it admits solutions that blow up (which means their maximum value becomes infinity) at infinite time, which we call weak blowup. We study this novel phenomenon using the method of matched asymptotic expansion. A whole family of self-consistent blowup profiles is obtained. We propose a mechanism by which the actual profile is selected that is consistent with the simulations, but the mechanism is only partly supported by the analysis. We study the four particle systems for the mCH equation finding numerical evidence both for the non-integrability of the mCH equations and for the existence of the fourth integral. We also study the higher dimensional case and obtain the short time existence and well-posedness for the generalised Euler equation in the two dimension case.