This thesis concerns the study of geometric numerical integrators and how they preserve phase space structures of Hamiltonian ordinary differential equations. We examine the invariant sets of differential equations and investigate which numerical integrators preserve these sets, and under what conditions. We prove that when periodic orbits of Hamiltonian differential equations are discretized by a symplectic integrator they are preserved in the numerical solution when the integrator step size is not resonant with the frequency of the periodic orbit. The preservation of periodic orbits is the result of a more general theorem which proves preservation of lower dimensional invariant tori from dimension zero (fixed points) up to full dimension (the same as the number of degrees of freedom for the differential equation). The proof involves first embedding the numerical trajectory in a non-autonomous flow and then applying a KAM type theorem for flows to achieve the result. This avoids having to prove a KAM type theorem directly for the symplectic map which is generally difficult to do. We also numerically investigate the break up of periodic orbits when the integrator's step size is resonant with the frequency of the orbit. We study the performance of trigonometric integrators applied to highly oscillatory Hamiltonian differential equations with constant frequency. We show that such integrators may not be as practical as was first thought since they suffer from higher order resonances and can perform poorly at preserving various properties of the di fferential equation. We show that, despite not being intended for such systems, the midpoint rule performs no worse than many of the trigonometric integrators, and indeed, better than some. Lastly, we present a numerical study of a Hamiltonian system consisting of two magnetic moments in an applied magnetic field. We investigate the effect of both the choice of integrator and the choice of coordinate system on the numerical solutions of the system. We show that by a good choice of integrator (in this case the generalised leapfrog method) one can preserve phase space structures of the system without having to resort to a change of coordinates that introduce a coordinate singularity.

The research presented in this thesis investigates the performance of some well-known heuristic algorithms on graph decomposition problems. First, a genetic algorithm is introduced and some modifications are trialled on finding Steiner triple systems (STS) of small orders. The results show that traditional genetic algorithms are not well suited to finding graph decompositions. Then a hill climbing optimisation technique is presented and investigated in the context of cycle decompositions. Such searches have previously proved to be effective at finding STSs. However, the general hill climbing approach is not immediately applicable to decompositions into cycles of length larger than 3. A modification of the hill climbing algorithm for cycles, called slippery hill climbing, is introduced and tested on decompositions of graphs into cycles of small lengths larger than 3. Slippery hill climbing successfully decomposed complete and dense non-complete graphs of considerable sizes into cycles of small lengths. In addition, we applied the slippery hill climbing approach to completing partial latin squares. It is reasonably expected that the algorithms developed in this study will also be applicable to other related problems in combinatorics and graph theory.

Optical tweezers using highly focussed laser beams can be used to exert forces and torques and thus drive micromachines. This opens up a new field of microengineering, whose potential has yet to be fully realized. Until now, methods that have been used for modelling optical tweezers are limited to scatterers that are homogeneous or that have simple geometry. To aid in designing more general micromachines, I developed and implemented two main methods for modelling the micromachines that we use. These methods can be used for further proposed structures to be fabricated. The first is a FDFD/T-matrix hybrid method that incorporates the finite difference frequency domain (FDFD) method, which is used for inhomogeneous and anisotropic media, with vector spherical wave functions (VSWF) to formulate the T-matrix. The T-matrix is then used to calculate the torque of the trapped vaterite sphere, which is apparently composed of birefringent unit crystals but the bulk structure appears to be arranged in a sheaf-of-wheat fashion. The second method is formulating the T-matrix via discrete dipole approximation (DDA) of complex arbitrarily shaped mesoscale objects and implementing symmetry optimizations to allow calculations to be performed on high-end desktop PCs that are otherwise impractical due to memory requirements and calculation time. This method was applied to modelling microrotors. The T-matrix represents the scattering properties of an object for a given wavelength. Once it is calculated, subsequent calculations with different illumination conditions can be performed rapidly. This thesis also deals with studies of other light scattering phenomena including the modelling of scattered fields from protein molecules subsequently used to model FRET resonance, determining the limits of trappability, interferometric Brownian motion and the comparison between integral transforms by direct numerical integration and overdetermined point-matching.

No description available.

No description available.

As transnational education is becoming firmly embedded as a part of the distance education landscape, governments and universities are calling for meaningful research on transnational education. This study involved the development and validation of a model for effective transnational education programs. The study used student experience as a key indicator of program effectiveness and, following a holistic approach, took into consideration various dimensions of the transnational education context including student, instructor, curriculum and instruction design, interaction, evaluation and assessment, technology, and program management and organisational support. This selection of dimensions, together with their attributes, formed the proposed model for transnational education programs. The model was applied for validation against three transnational computing education programs currently offered by Australian universities in Hong Kong. Two methods of data collection - a survey, and group interviews with students - were used to validate the model; data was obtained from approximately three hundred subjects. The model was evaluated in terms of the perceived importance, to the students, of the various attributes of each program dimension on program effectiveness. The results of the validation indicated that the students in all the programs participating in the evaluation were in agreement as to the factors they consider most important to the effectiveness of transnational programs. The validation of the model led to its refinement; first, the least important attributes were removed from dimensions; second, a new dimension, pre-enrolment considerations, was introduced to the model; and finally, the attributes within each of the dimensions were ordered in terms of their perceived importance.

There has been considerable recent interest in geometric function theory, nonlinear partial differential equations, harmonic mappings, and the connection of these to minimal energy phenomena. This work explores Nitsche's 1962 conjecture concerning the nonexistence of harmonic mappings between planar annuli, cast in terms of distortion functionals. The connection between the Nitsche problem and the famous Grötzsch problem is established by means of a weight function. Traditionally, these kinds of problems are investigated in the class of quasiconformal mappings, and the assumption is usually made a priori that solutions preserve various symmetries. Here the conjecture is solved in the much wider class of mappings of finite distortion, symmetry-preservation is proved, and ellipticity of the variational equations concerning these sorts of general problems is established. Furthermore, various alternative interpretations of the weight function introduced herein lead to an interesting analysis of a much wider variety of critical phenomena -- when the weight function is interpreted as a thickness, density or metric, the results lead to a possible model for tearing or breaking phenomena in material science. These physically relevant critical phenomena arise, surprisingly, out of purely theoretical considerations.

The analysis of presence/absence data of plant species by regression analysis is the subject of this thesis. A nonparametric approach is emphasized, and methods which take into account correlations between species are also considered. In particular, generalized additive models (GAMs) are used, and these are applied to species’ responses to greenhouse scenarios and to examine multispecies interactions. Parametric models are used to estimate optimal conditions for the presence of species and to test several niche theory hypotheses. An extension of GAMs called vector GAMs is proposed, and they provide a means for proposing nonparametric versions of the following models: multivariate regression, the proportional and nonproportional odds model, the multiple logistic regression model, and bivariate binary regression models such as bivariate probit model and the bivariate logistic model. Some theoretical properties of vector GAMs are deduced from those pertaining to ordinary GAMs, and its relationship with the generalized estimating equations (GEE) approach elucidated. / Whole document restricted, but available by request, use the feedback form to request access.

This thesis describes the Operational Harvest Scheduling (OHS) problem and develops an algorithm that solves instances of the problem. The solution to an OHS problem is an Operational Harvest Schedule (OHS). An OHS: ² assigns forest harvesting crews to locations within a forest in the short-term (4-8 weeks); ² instructs crews to harvest specific log-types and allocates these log-types to customers; ² maximises profitability while meeting customer demand. The OHS problem is modelled as a Mixed Integer Linear Program (MILP). The formulation given in this thesis differs significantly from previous literature, especially with regard to the construction of the problem variables. With this novel formulation, the problem can be solved using techniques developed in previous work on aircraft crew scheduling optimisation (Ryan 1992). These techniques include constraint branching and column generation. The concept of relaxed integer solutions is introduced. A traditional integer solution to the OHS problem will require harvesting crews to move between harvesting locations at the end of a week. However, a relaxed integer solution allows crews to move at any time during a week. This concept allows my OHS model to more effectively model the practical problem. The OHS model is formulated for New Zealand and Australian commercial forestry operations,though the model could be applied to other intensively managed production forests. Three case studies are developed for two companies. These case studies show improvements in profitability over manual solution methods and a significant improvement in the ability to meet demand restrictions. The optimised solutions increased profit (revenue less harvesting and transportation costs) by between 3-7%, while decreasing the total value of excess or shortfall logs by between 15-86%.

This thesis describes the Operational Harvest Scheduling (OHS) problem and develops an algorithm that solves instances of the problem. The solution to an OHS problem is an Operational Harvest Schedule (OHS). An OHS: ² assigns forest harvesting crews to locations within a forest in the short-term (4-8 weeks); ² instructs crews to harvest specific log-types and allocates these log-types to customers; ² maximises profitability while meeting customer demand. The OHS problem is modelled as a Mixed Integer Linear Program (MILP). The formulation given in this thesis differs significantly from previous literature, especially with regard to the construction of the problem variables. With this novel formulation, the problem can be solved using techniques developed in previous work on aircraft crew scheduling optimisation (Ryan 1992). These techniques include constraint branching and column generation. The concept of relaxed integer solutions is introduced. A traditional integer solution to the OHS problem will require harvesting crews to move between harvesting locations at the end of a week. However, a relaxed integer solution allows crews to move at any time during a week. This concept allows my OHS model to more effectively model the practical problem. The OHS model is formulated for New Zealand and Australian commercial forestry operations,though the model could be applied to other intensively managed production forests. Three case studies are developed for two companies. These case studies show improvements in profitability over manual solution methods and a significant improvement in the ability to meet demand restrictions. The optimised solutions increased profit (revenue less harvesting and transportation costs) by between 3-7%, while decreasing the total value of excess or shortfall logs by between 15-86%.