School of Mathematics and Statistics - Theses
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ItemMathematical approaches to pattern formation in dermatology(University of Melbourne, 2005)
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ItemAspects of mixed longitudinal growth analysis(University of Melbourne, 2010)This thesis presents practical approaches to the analysis of mixed longitudinal growth data. Longitudinal studies of the human population are specifically designed to investigate changes over a limited age range in a characteristic which is measured repeatedly for each study participant. This type of data poses several methodological challenges. First, models for the analysis of longitudinal data must recognize the relationship between the observations taken from each study participant. The mixed nature of the data calls for the use of random effects and variance and correlation structures for the within group errors. Secondly, the models must be flexible enough so that they can be easily differentiated for the timing of the population growth spurts. And thirdly, longitudinal growth data of human subjects is more often than not affected by the missing data problem. In practice, the missing data mechanism needs to be understood and taken into consideration when fitting the models. These aspects of mixed longitudinal growth analysis are covered in detail in this thesis using a comprehensive data set of repeated measures of human height of hundreds of Melbourne school children ranging form the ages of 5 to 18 years.
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ItemPointwise axiomatic spectral theory in Banach algebras(University of Melbourne, 2008)
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ItemOn sample size determination for discrete data(University of Melbourne, 1993)
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ItemNew data generation and solution techniques for the Sequential Ordering Problem with Time Windows( 2014)The motivation for this thesis resulted from a study of semi-automated rail mounted gantry cranes at an Australian seaport container terminal. The Sequential Ordering Problem with Time Windows (SOP-TW) can arise as a subproblem of these crane operations. The SOP-TW corresponds to the task of scheduling a single crane to move a set of containers, where each container has: a fixed pick-up location; a fixed drop-off location; a time window during which the container must be picked up and moved to its destination; and some container movements must be performed before other container movements. A crane schedule is a list showing the pick-up time of each container. If time permits, a crane is allowed to sit idle between moving containers; however, once a container is collected from its pick-up location it must be delivered to its drop-off location. Consequently, the order of the container movements and a path for the crane can be determined from the schedule. For this application the aim is to minimise the total time the crane spends moving while not carrying a container, that is, the total time moving from the drop-off location of one container movement to the pick-up location of the next container movement. The SOP-TW can be viewed in more general terms as the problem of scheduling n jobs on a single machine with sequence-dependent setup times, generalised precedence constraints between some pairs of jobs and start time windows, where the objective is to minimise the sum of all setup times; the jobs are container moves and the setup time is the time needed for a crane to move from the drop-off location for one job to the pick-up location for the next. A detailed critique of existing SOP-TW data generators and their resulting data sets is undertaken and reveals a lack of available, feasible, difficult SOP-TW problem instances for testing methods for solving the SOP-TW. This motivates the development of a comprehensive port-specific data generator for creating feasible SOP-TW problem instances, and a variety of interesting SOP-TW instances are constructed. Importantly, the underlying methodology of this new generator is sufficiently general to be easily adapted for non-port settings. Three existing formulations for the Asymmetric Travelling Salesman Problem with Time Windows (ATSP-TW) are extended to model the SOP-TW. An efficient preprocessing routine that simultaneously addresses both time windows and precedence relations is given that ultimately reduces the size of the mixed integer programming (MIP) models implemented. A computational study of the two compact MIPs is performed to provide a baseline for comparison. Three techniques for solving the SOP-TW are identified and investigated: * a construction heuristic for generating feasible solutions; * an IP-based heuristic for generating feasible solutions; and * a scheduling approach embedded within a branch-and-bound framework. A series of comprehensive computational studies are performed to investigate the effectiveness of these techniques against each other and the baseline MIPs. While a crane scheduling application provided the initial motivation and the foundation of the data generator, it is important to note that the techniques presented in this thesis are applicable to other contexts where the SOP-TW arises.
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ItemOne-dimensional stochastic models with open boundaries: integrability, applications, and q-deformed Knizhnik–Zamolodchikov equations( 2014)This thesis contains work on three separate topics, but with common themes running throughout. These themes are drawn together in the asymmetric exclusion process (ASEP) – a stochastic process describing particles hopping on a one-dimensional lattice. The open boundary ASEP is set on a finite length lattice, with particles entering and exiting at both boundaries. The transition matrix of the open boundary ASEP provides a representation of the two boundary Temperley–Lieb algebra, and the integrability of the system allows the diagonalisation of the transition matrix through the Bethe ansatz method. We study the ASEP in the reverse bias regime, where the boundary injection and extraction rates oppose the preferred direction of flow in the bulk. We find the exact asymptotic relaxation rate along the coexistence line by analysing solutions of the Bethe equations. The Bethe equations are first solved numerically, then the form of the resulting root distribution is used as the basis for an asymptotic analysis. The reverse bias induces the appearance of isolated roots, which introduces a modified length scale in the system. We describe the careful treatment of the isolated roots that is required in both the numerical procedure, and in the asymptotic analysis. The second topic of this thesis is the study of a priority queueing system, modelled as an exclusion process with moving boundaries. We call this model the prioritising exclusion process (PEP). In the PEP, the hopping of particles corresponds to high priority customers overtaking low priority customers in order to gain service sooner. Although the PEP is not integrable, techniques from the ASEP allow calculation of exact density profiles in certain phases, and the calculation of approximate average waiting times when the expected queue length is finite. The final topic of this thesis is a study of polynomial solutions of a q-deformed Knizhnik–Zamolodchikov (qKZ) equation with mixed boundaries. The qKZ equation studied here is given in terms of the one boundary Temperley–Lieb algebra, and its solutions have a factorised form in terms of Baxterized elements of the type B Hecke algebra. We find an integral form for certain components of the qKZ solution, along with a factorised expression for a generalised sum rule. The representation of the Temperley– Lieb algebra that we study is related to the O(n) Temperley–Lieb loop model, and a specialization of the sum rule gives the normalisation of the ground state vector for the O(n = 1) model.
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ItemDiscretely holomorphic observables in statistical mechanics( 2014)In this thesis we investigate applications of discretely holomorphic observables in two-dimensional statistical mechanical lattice models. Discretely holomorphic observables or parafermionic ob- servables are functions defined on a graph embedding which depend on a real-valued parameter known as the parafermionic spin. Such observables were introduced in the context of lattice models in order to rigorously prove that the scaling limit of a given model is conformally invariant. This approach has been successfully carried out for the dimer model, percolation and the Ising model. However, these observables also have important applications at the lattice level. For example, Cardy and Ikhlef showed that they are naturally related to the Yang-Baxter integrability of the model, providing a straightforward method for obtaining the integrable weights. We further this connection by defining discretely holomorphic observables for loop models in domains with a boundary, showing that for a simple set of boundary conditions, the integrable boundary weights are obtained. This is true for models with diagonal as well as off-diagonal boundary weights, in which case the observables of Cardy and Ikhlef must be further generalised. Duminil-Copin and Smirnov made use of the discrete holomorphicity property in order to prove Nienhuis’ conjectured value for the connective constant of self-avoiding walks on the honeycomb lattice. We show that by relaxing the discrete holomorphicity condition we obtain an off-critical condition which allows us to relate certain critical exponents of different classes of self-avoiding walks in the dilute O(n) model. We are also able to derive an exponent equality and several exponent inequalities related to the winding angle distribution of the O(n) model, whose full distribution was first predicted using Conformal Field Theory techniques by Duplantier and Saleur. Finally we define parafermionic observables for the Andrews-Baxter-Forrester heights mod- els at the critical point. These models remain integrable away from the critical point. We therefore expect that the off-critical integrable weights should arise from an off-critical discrete holomorphicity condition, similar to that which we defined in the context of the dilute O(n) model.
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ItemMethods for estimating occupancy( 2014)The estimation of the probability of occupancy of a site by a species is used to monitor the distribution of that species. Occupancy models have been widely applied and several limitations have been identified. In this thesis we resolve some of these. In particular we focus on limitations of maximum likelihood estimators and the associated interval estimators, and the difficulties associated with the extension from linear to generalised additive models for the relationship between occupancy and covariates. Initially we consider in detail the basic occupancy model which includes two parameters: $\psi$ and $p$. Our primary concern is the probability that the species occupies a particular site, $\psi$. The other parameter, the detection probability $p$, is a nuisance parameter. We first derive the joint probability mass function for the sufficient statistics of occupancy which allows the exact evaluation of its mean and variance, and hence its bias. We show that estimation near the boundaries of the parameter space is difficult. For small values of detection, we show that estimation of occupancy is not possible and specify the region of the parameter space where maximum likelihood estimators exist, and give the equations for the MLEs in this region. We next demonstrate that the asymptotic variance of the estimated occupancy is underestimated, yielding interval estimators that are too narrow. Methods for constructing interval estimators are then explored. We evaluate several bootstrap-based interval estimators for occupancy. Finally, instead of the full likelihood we consider a partial likelihood approach. This gives simple closed form estimators in a basic model with only a small loss of efficiency. It greatly simplifies the inclusion of linear and nonlinear covariates by allowing the use of standard statistical software for GLM and GAM frameworks and in our simulation study there is little loss of efficiency compared to the full likelihood.
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ItemThe Steiner ratio conjecture for eight points( 2014)Given a set X of n points in the euclidean plane R2, a Steiner minimal tree is a shortest network interconnecting X. Note that a Steiner minimal tree may contain extra vertices, referred to as Steiner points, not already in X. The problem of finding a Steiner minimal tree is NP-hard. A Steiner minimal tree can, however, be approximated by a minimum spanning tree, which can be found in polynomial time in n. A measure for the performance of a Steiner minimal tree compared with that of a minimum spanning tree is given by the Steiner ratio ρ, the ratio of lengths between a Steiner minimal tree and a minimum spanning tree. In 1968, Gilbert and Pollak conjectured that ρ ≥ √3/2 for any X. The conjecture was subsequently verified for several small values of n. For example, Rubinstein and Thomas applied some ideas from the calculus of variations to verify the conjecture for n = 6. Then, in 1992, Du and Hwang published a paper concerning the general problem, with convexity and flexibility as the major ideas. It has since been pointed out, however, that the argument of Du and Hwang contains a gap. Nevertheless, their argument contains many elegant ideas which may still be retained. In this thesis, we use a combination of techniques drawn from both the work of Rubinstein and Thomas as well as that of Du and Hwang to verify the conjecture for n = 8. Whereas Du and Hwang chose to work with a restricted set of spanning trees, we opt to work with the full set of spanning trees. We introduce the notion of symmetry breaking into the conception of the problem in order to fully exploit the idea of flexibility. We also adapt some material from rigidity theory, the study of bar-and-joint frameworks, in order to ensure that our arguments apply to both generic and nongeneric situations.