A numerical study is made of the wintertime circulation within the Southeast Indian Ocean (SEIO). The downwelling favourable winds result in a continuous eastward Coastal Current (CC) extending from Cape Leeuwin to the eastern coast of Tasmania, where it forms a confluence with the south branch of the East Australian Current. An additional forcing mechanism for the CC is the Leeuwin Current in the western part of the domain. The study here is divided in two parts: (1) available data and the wintertime averaged results from the Ocean Circulation and Climate Advanced Model (OCCAM) are analysed to provide a first order description of the large-scale circulation; (2) a high resolution model (Princeton Ocean Model) is nested within OCCAM to examine the shelf-slope circulation within the eastern SEIO. The nested model is forced with climatological monthly average winds and several experiments were run to simulate the effects of surface fluxes of density, enhanced bottom friction and stronger winds. In summary, the shelf-slope circulation is governed by a surface south-eastward CC that carries around 2 Sv and reaches velocities of up to 50 cm/s, where the shelf is narrowest. The core of the current is generally constrained to the shelf-break region. Zonal winds and geostrophic control of the CC lead to a transport of 1 Sv through Bass Strait and a north-eastward jet that is directed into the strait between King Is. and Tasmania. Further south, the CC is poleward and known as the Zeehan Current (ZC). Between Cape Leeuwin and Tasmania and over the slope region, a westward current (the Flinders Current) is found at depths of 500-1000 m and has an associated transport of 5-7 Sv. The current is shown to result from a northward Sverdrup transport in the deep ocean. Meso-scale eddies are shown to result from baroclinic instability and have wavelengths of around 250 km and transports of 3-4 Sv, and can dominate the slope circulation. A comparison of the numerical results is also made with two current meter data sets and results show an interannual variability in the ZC strength, that is probably related to ENSO.

A notion of entropy is defined for the non-singular action of finite co-ordinate changes on X - the infinite product of two- point spaces. This quantity - average co-ordinate or AC entropy - is calculated for product measures and G-measures on X, and an equivalence relation is established for which AC entropy is an invariant. The Inverse Vitali Lemma is discussed in a measure preserving context, and it is shown that for a certain class of measures on X known as odometer bounded, the result will still hold for odometer actions. The foundations for a non-singular version of Rudolph's restricted orbit equivalence are established, and a size for non-singular orbit equivalence is introduced. It is shown that provided the Inverse Vitali Lemma still holds, the non-singular orbit equivalence classes can be described using this new size.

This thesis is a study of the air-sea fluxes of momentum, sensible heat and latent heat. Fluxes are estimated using the covariance, COARE2.6b bulk flux algorithm, and inertial dissipation methods. The bulk algorithm is validated against the covariance fluxes for the first time in a light-wind, shallow tropical sea, with strong atmospheric instability and low sea state conditions. The removal of ship motion contamination is investigated. This is the first study to quantify the errors associated with corrections for ship motion contamination, and the effects of motion contamination on the covariance calculated heat fluxes. Flow distortion is investigated. Bulk transfer coefficients and roughness lengths are computed and related to the sea state. Ship motion contamination is successfully removed in 86% of the runs. Error analysis of the motion removal algorithm indicates maximum uncertainties of 15% in the wind fluctuations, and 0.002 N/m/m for the wind stress. Motion correction changes the stress by more than 15% in half of the runs analysed. The ship is found to accelerate the mean air flow and deflect it above the horizontal. A correction is developed for the air flow acceleration. The scalar fluxes show good agreement on average for all the methods. As wind speed approaches zero, covariance wind stress is significantly larger than the bulk and inertial dissipation derived wind stress. The non-zero covariance wind stress is reflected in the drag coefficient, CdN10, and momentum roughness length, z0, which are much larger than the parameterisations used in the bulk algorithm. The MCTEX CdN10, wind speed (u10N) relation is 1000 x Cd10N = 1.03 + 7.88/(u10N)^2 0.8 &lt u10N &lt 7.5 m/s z0 is primarily a function of wind speed rather than sea state, with largest roughness lengths occurring as wind speed approaches zero. This relation is used in the bulk algorithm, yielding good agreement between covariance and bulk derived wind stress. A new parameterisation for the effects of gustiness, based on wind variance is developed. This brings the bulk wind stress into agreement with the covariance derived wind stress.

This thesis is concerned with the mathematical modelling for human immunodeficiency virus (HIV) and hepatitis C virus (HCV) epidemics in Australia. There are two parts to this thesis. Part I is aimed at modelling the transmission of HIV and HCV via needle sharing among injecting drug users (IDUs). The dynamical model of an epidemic through needle sharing among IDUs is derived. This model reveals the correlation between needle sharing and the epidemic prevalence among IDUs. The simulations of HIV and HCV prevalence and incidence among IDUs in Australia are made with this model. The comparison of simulated results with literature estimates shows that the modelled results are consistent with the literature estimates. The effects of needle sharing and cleaning on HIV and HCV prevalence and incidence among IDUs in Australia are evaluated. Part II is devoted to modelling the spread of HIV in the general community in Australia. A mathematical model is formulated to assess the epidemiological consequences of injecting drug use and sexual transmission in Australia. The effects of highly active antiretroviral therapies (HAART) on the HIV epidemic are included. The modelled results are in broad agreement with the literature estimates and observed data. The long-term effects of HAART are also discussed.

Bayesian approaches to prediction and the assessment of predictive uncertainty in generalized linear models are often based on averaging predictions over different models, and this requires methods for accounting for model uncertainty. In this thesis we describe computational methods for Bayesian inference and model selection for generalized linear models, which improve on existing techniques. These methods are applied to the building of flexible models for gamma ray count data (data measuring the natural radioactivity of rocks) at the Castlereagh Waste Management Centre, which served as a hazardous waste disposal facility for the Sydney region between March 1978 and August 1998. Bayesian model selection methods for generalized linear models enable us to approach problems of smoothing, change point detection and spatial prediction for these data within a common methodological and computational framework, by considering appropriate basis expansions of a mean function. The data at Castlereagh were collected in the following way. A number of boreholes were drilled at the site, and for each borehole a gamma ray detector recorded gamma ray emissions at different depths as the detector was raised gradually from the bottom of the borehole to ground level. The profile of intensity of gamma counts can be informative about the geology at each location, and estimation of intensity profiles raises problems of smoothing and change point detection for count data. The gamma count profiles can also be modelled spatially, to inform the geological profile across the site. Understanding the geological structure of the site is important for modelling the transport of chemical contaminants beneath the waste disposal area. The structure of the thesis is as follows. Chapter 1 describes the Castlereagh hazardous waste site and the geophysical data, which motivated the methodology developed in this research. We summarise the principles of Gamma Ray (GR) logging, a method routinely employed by geophysicists and environmental engineers in the detailed evaluation of hazardous site geology, and detail the use of the Castlereagh data in this research. In Chapter 2 we review some fundamental ideas of Bayesian inference and computation and discuss them in the context of generalised linear models. Chapter 3 details the theoretical basis of our work. Here we give a new Markov chain Monte Carlo sampling scheme for Bayesian variable selection in generalized linear models, which is analogous to the well-known Swendsen-Wang algorithm for the Ising model. Special cases of this sampling scheme are used throughout the rest of the thesis. In Chapter 4 we discuss the use of methods for Bayesian model selection in generalized linear models in two specific applications, which we implement on the Castlereagh data. First, we consider smoothing problems where we flexibly estimate the dependence of a response variable on one or more predictors, and we apply these ideas to locally adaptive smoothing of gamma ray count data. Second, we discuss how the problem of multiple change point detection can be cast as one of model selection in a generalized linear model, and consider application to change point detection for gamma ray count data. In Chapter 5 we consider spatial models based on partitioning a spatial region of interest into cells via a Voronoi tessellation, where the number of cells and the positions of their centres is unknown, and show how these models can be formulated in the framework of established methods for Bayesian model selection in generalized linear models. We implement the spatial partition modelling approach to the spatial analysis of gamma ray data, showing how the posterior distribution of the number of cells, cell centres and cell means provides us with an estimate of the mean response function describing spatial variability across the site. Chapter 6 presents some conclusions and suggests directions for future research. A paper based on the work of Chapter 3 has been accepted for publication in the Journal of Computational and Graphical Statistics, and a paper based on the work in Chapter 4 has been accepted for publication in Mathematical Geology. A paper based on the spatial modelling of Chapter 5 is in preparation and will be submitted for publication shortly. The work in this thesis was collaborative, to a smaller or larger extent in its various components. I authored Chapters 1 and 2 entirely, including definition of the problem in the context of the CWMC site, data gathering and preparation for analysis, review of the literature on computational methods for Bayesian inference and model selection for generalized linear models. I also authored Chapters 4 and 5 and benefited from some of Dr Nott's assistance in developing the algorithms. In Chapter 3, Dr Nott led the development of sampling scheme B (corresponding to having non-zero interaction parameters in our Swendsen-Wang type algorithm). I developed the algorithm for sampling scheme A (corresponding to setting all algorithm interaction parameters to zero in our Swendsen-Wang type algorithm), and performed the comparison of the performance of the two sampling schemes. The final discussion in Chapter 6 and the direction for further research in the case study context is also my work.

This thesis demonstrates how the geometric connection between the integrable Heisenberg spin equation, the nonlinear Schr??dinger equation and fluid flows with constant velocity magnitude along individual streamlines may be exploited. Specifically, we are able to construct explicitly the complete class of constant speed flows where the constant pressure surfaces constitute surfaces of revolution. This class is undoubtedly important as it contains many of the specific cases discussed earlier by other authors.

This thesis explains to statisticians what graphical models are and how to use them for statistical inference; in particular, how to use decomposable graphical models for efficient inference in covariance selection and multivariate regression problems. The first aim of the thesis is to show that decomposable graphical models are worth using within a Bayesian framework. The second aim is to make the techniques of graphical models fully accessible to statisticians. To achieve these aims the thesis makes a number of statistical contributions. First, it proposes a new prior for decomposable graphs and a simulation methodology for estimating this prior. Second, it proposes a number of Markov chain Monte Carlo sampling schemes based on graphical techniques. The thesis also presents some new graphical results, and some existing results are reproved to make them more readily understood. Appendix 8.1 contains all the programs written to carry out the inference discussed in the thesis, together with both a summary of the theory on which they are based and a line by line description of how each routine works.

We consider conditions under which a continuous functional calculus for a Banach space operator T ?? L(X) may be extended to a bounded Borel functional calculus, and under which a functional calculus for absolutely continuous (AC) functions may be extended to one of for functions of bounded variation (BV). The natural setting for investigating the former case is finitely spectral operators, and for the latter, well-bounded operators. Some such conditions are well-established. If X is a reflexive space, both type of Extensions are assured; in fact if X contains an isomorphic copy of co, then every Operator T ?? L(X) that has a continuous functional calculus necessarily admits a Borel one. We show that if a space X has a predual, then also every operator T ?? L(X) with a continuous functional calculus admits a bounded Borel functional Calculus. In case a Banach space X either contains an isomorphic copy of co, or has a Predual, and T ?? L(X) is an operator with an AC functional calculus, we find that the existence of a decomposition of the identity of bounded variation for T is sufficient to ensure that the AC functional calculus may be extended to a BV functional calculus. We also consider operators defined by a linear map on interpolation families of Banach spaces [Xr, X???] (r???1), where for example Xp = lp, Lp[0,1] or Cp. We show that under certain uniform boundedness conditions, the possession of a BV functional calculus by operators on the spaces Xp, p ?? (r, ???), may be extrapolated to the corresponding operators on the spaces Xr and X???.

This thesis examines finite dimensional representability of Forward Rate and LIBOR models. A new approach is examined. This approach is more general, elementary, and relevant to finance when compared with existing approaches. This new approach is applied to the following infinite dimensional equations used in finance: ?Gaussian Heath, Jarrow and Morton model; ?Free 1 Heath, Jarrow and Morton model; ?Brace, G?atarek and Musiela???s LIBOR model. Stronger results have been achieved using this approach. The results are as follows: ?The Gaussian HJM model can be represented in finite dimensions if and only if the volatility satisfies a particular differential equation. In which case the finite dimensional representation can be explicitly written; ?The Brace, G?atarek and Musiela???s LIBOR model with one dimensional Wiener process cannot be represented in finite dimensions (other than in a trivial case); ?The Brace, G?atarek and Musiela???s LIBOR model with multidimen-sional Wiener process, and Free HJM have a finite dimensional repre-sentation only if the initial yield curves satisfy a restrictive differential equation. This thesis is arranged as follows ?Chapter 1 is an introduction to this thesis and derivative pricing in general. The reader is referred to section 1.4 titled ???This Thesis?for a more detailed description of the approach of this thesis and its results. ?Chapter 2 contains a brief summary of results from the theory of stochastic processes, stochastic calculus and stochastic equations in infinite dimensions ?Chapter 3 contains an overview of spot market pricing models including the Cox, Ross and Rubinstein and Black and Scholes models. ?Chapter 4 contains an overview of the fixed income market pricing models including the Heath, Jarrow and Morton model; Musiela???s re-formulation of the HJM model; the Goldys, Musiela and Sondermann model; and the Brace, G?atarek and Musiela LIBOR model. ?Chapter 5 contains the primary results of this thesis. Finite Dimen-sional Representability is defined formally and applied to the Musiela reformulated Gaussian HJM model; Musiela reformulated free HJM model; and the Brace, G?atarek and Musiela LIBOR model. This ap-proach and results are compared with the literature.

This thesis demonstrates how the geometric connection between the integrable Heisenberg spin equation, the nonlinear Schr??dinger equation and fluid flows with constant velocity magnitude along individual streamlines may be exploited. Specifically, we are able to construct explicitly the complete class of constant speed flows where the constant pressure surfaces constitute surfaces of revolution. This class is undoubtedly important as it contains many of the specific cases discussed earlier by other authors.