The 4-manifold S1 x S3, when endowed with the structure of a certain complex Hopf surface, is an example of a principal elliptic fibration. We use this structure to study the moduli spaces of anti-self-dual connections (instantons) on SU(2) bundles over S1 x S3. Chapter 1 is introductory. We define Buchdahl's notion of stability and outline the correspondence between instantons and stable holomorphic SL(2,C) bundles over S1 x S3. In Chapter 2 we study holomorphic line and SL(2, C) bundles over a general principal elliptic surface using an extension of the ‘graph’ invariant introduced by Braam and Hurtubise. We prove some auxiliary results needed in later chapters and introduce a stratification of the moduli space. In Chapter 3 we construct elements of one of the strata using the ‘Serre construction’ of algebraic geometry and deduce a structure result for the charge 1 case. Chapter 4 applies the results of the previous chapters in the construction of monopoles on the solid torus with a hyperbolic metric. We recover easily a result of Braam and Hurtubise. In Chapter 5 we adapt a construction of Friedman to describe a method of construction for elements of the remaining strata of the moduli spaces over the Hopf surface. In the charge 1 case we again determine the diffeomorphism type of the stratum completely. Combined with the results of Chapter 3 we deduce the natural action of S1 x S3 on the charge 1 moduli space is free. In Chapter 6 we study the charge 1 instanton moduli spaces over secondary Hopf surfaces diffeomorphic to the product of S1 and a Lens space. Chapter 7 considers twistorial methods and their application in the construction of explicit solutions. We define an invariant of an instanton, the spectral surface, which is a 2-dimensional analogue of Hitchin’s spectral curve. We use it to deduce that methods of Atiyah and Ward fail to generate a full family of charge 1 solutions. Finally we show how the spectral surface can be used in a sheaf theoretic construction of the ‘missing’ solutions.

The present thesis considers the space of connections modulo based gauge equivalence on a principal SU(2) bundle over a closed simply-connected smooth four-dimensional manifold M. Up to homotopy equivalence, this is the space of basepoint-preserving maps from M to BSU('2), the classifying space of SU(2). It depends only on the homotopy type of M which is characterized by the intersection form. The Z/pZ-homology of the mapping space for p a prime not equal to 3 is computed and given in terms of the data associated to the intersection form. For the prime 3, partial results are obtained. The main method is to consider a fibration associated to a CW decomposition of M and to show that the corresponding Eilenberg- Moore spectral sequence collapses. These results generalize from manifolds to spaces homotopy equivalent to a bouquet of 2-spheres with a single 4-cell attached. For the possible homotopy types the space of connections modulo gauge equivalence ran attain, a classification is obtained in the following sense. The homotopy type of this space is uniquely determined by the rank, type and signature modulo eight of the intersection form. On the other hand, the homotopy type determines the rank, type and signature modulo four of the intersection form. Both results together give a complete classification for the case of spin manifolds. The homotopy types of the spaces of connections modulo gauge equivalence over two spin manifolds agree if and only if the intersection forms are of the same rank. These results use a classification of unimodular bilinear forms over the ring Z/4Z. In a final part, a map is constructed from the labelled configuration spaces of points in the manifold to the mapping space. This map is shown to be asymptotically surjective in homology with Z/2Z-coefficients. For homology with general coefficients, classes are constructed which are not approximated by this map.

Monodromy fields on I3 are a family of lattice fields in two dimensions which are a natural generalisation of the two dimensional Ising field occurring in the C*-algebra approach to Statistical Mechanics. A criterion for the critical limit one point correlation of the monodromy field tra(M) at a 6 l3, Um(#.(M)). is deduced for matrices M € GL(p, C) having non-negative eigenvalues. Using this criterion a non-identity 2x2 matrix is found with a finite critical limit one point correlation. The general set of p x p matrices with finite critical limit one point correlations is also considered and a conjecture for the critical limit n point correlations postulated. The boson-fermion correspondence for the representation of the CAR algebra over L3(Sl, C) defined by the (t,B) KMS state with chemical potential p is considered and the non-bijectivity shown. Using an alternative formulation the correlations are recalculated leading to a determinant identity reminiscent of Saego’s Theorem.

This work attempts to utilise perturbation theory to derive discrete mappings which describe the dynamical behaviour of a continuous, and a discrete, chaotic system. The first three chapters introduce some background to the theory of chaotic behaviour In discrete and continuous systems. Chapter 4 considers the dynamical behaviour of Duffings equation. Perturbation theory is applied to Hamiltonian solutions of the system, and a 1-D mapping is derived which models the bifurcation of the system to chaos. Chapter 5 introduces a 2-D chaotic difference map. The qualitative dynamics of the system are investigated and a form of perturbation theory is applied to a parameterised version of the map. The perturbative solutions are shown to exhibit dynamical behaviour very like the original system.

The objective of this research is to develop a design framework for virtual prototyping of electronic packaging. This framework couples computational mechanics and fluid dynamics, based on finite volume method with integrated finite element routines, with numerical optimisation and statistical methods. This integrated approach is intended as a modelling tool for calculating optimal design solutions for electronic packaging and component assembly with a focus on die reliability and the thermal management. The motivation is to introduce numerical optimisation theory as an approach for a fast, systematic and automated design approach for wide range microelectronics applications. The proposed methodology will also benefit from multi-physics numerical analysis to predict complex behaviour of electronic packages, systems and processes subject to different operational or environmental conditions. This thesis demonstrates multi-physics modelling (i.e. integrated solutions for fluid flow, heat transfer and stress) coupled with gradient/non-gradient based numerical optimisation techniques and associated statistical methods. An explanation and comparison of the two approaches to numerical optimisation — (1) Response Surface Methodology (RSM) based on Design of Experiments (DoE) and (2) direct gradient based and non-gradient methods - are given. Both the advantages and limitations of these virtual design strategies, with respect to their integration with multi-physics modelling, are discussed and demonstrated. This integrated multiphysics/optimisation design approach is demonstrated on a variety of problems from the area of microelectronics design and packaging. The thesis demonstrates this for three industrial examples. These are: The software packages used to develop the design tool and to undertake the outlined studies are PHYSICA and VisualDOC. PHYSICA is a multiphysics finite volume based simulation tool with integrated modules for finite element solid mechanics analysis. The software framework is detailed in Chapter 2, Section 2.4 and further in Chapter 4. The VisualDOC tool offers a collection of numerical optimisation routines and modules for statistical analysis (Design of Experiments) and approximate Response Surface modelling. VisualDOC framework is discussed in Chapter 4, Section 4.8.

Impeller-stirred mixing is one of the most important processes employed throughout the chemical, metallurgical and allied industries. The research reported in this thesis is focused on impeller stirred mixing associated with the refining of lead bullion. The aim of this process is to sequentially remove contained impurities such as copper, antimony, silver and bismuth. This occurs in hemispherical vessels, called kettles, where reagents are initially added to the lead bath to form surface dross that contains both the required impurity and a large amount of lead oxide. This dross is then continuously mixed back into the bath to remove the lead oxide and capture more of the required impurity. A key requirement for this process is to obtain and remove dross that contains a high concentration of the impurity. Although this process has been in operation for many years, there is very little known on how the fluid dynamics associated with the mixing process affects final dross content. The aim of this research is to fully investigate the lead refining process using scientific analysis methods that help understand the mixing process and provide design tools which can be used to optimise process conditions. The three methods of analysis used are: (1) Direct readings from a real kettle, (2) Physical modelling (using water), and finally (3) Computational Fluid Dynamics (CFD). The use of physical modelling, exploiting the techniques of similitude, to predict vortex was also validated. An Acoustic Doppler Velocimeter (ADV) probe was used for the velocity measurement at various locations inside the water model and this gave valuable insight about the flow phenomena occurring inside the refining kettle. A particular important finding was that when fluid is stirred above certain rotational speed the vortex depth becomes independent of the Reynolds number of the operation. With regards CFD technology, the Volume of Fluid (VOF) method was used to capture the free surface and the Lagrangian Particle Tracking (LPT) and Algebraic Slip Model (ASM) to simulate the dross phase. Appropriate methods were also used to represent the moving impeller region. Validation of simulation results against experimental data was very encouraging. Computed vortex depth showed the similar trend as observed during the experiments on the physical model. A design strategy was developed that integrates results from both physical and computational modelling to allow optimal process conditions to be predicted at the kettle design stage. The use of this integrated physical and computational modelling methodology successfully helped eliminate surface swirl by introducing baffles to the kettle. The design and introduction of these flow controllers was also validated to ensure that it optimised the dross mixing process and final impurity content in the dross.

In this work we use the Zakharov-Kuznetsov equation to study the evolution of a plane soliton subjected to a two-dimensional perturbation. The first part of the thesis is concerned with determining the growth rate of such a perturbation. We present two closely related methods which allow us to obtain rigorously the growth rates much more directly and simply than previous approaches. Both methods are general and hence applicable to other problems. If the perturbation is of a large enough wavelength, the plane soliton will evolve into more stable coherent structures of the form of two-dimensional solitons. This process is the subject of the remainder of the thesis. A weakly nonlinear analysis which fully describes the preliminary stages of the process is developed. We have studied how the eventual fate of a plane soliton is affected by the wavelength of the perturbation and obtained a simple formula for the variation of the number of cylindrical solitons formed with this wavelength. The methods developed in this thesis have been used to obtain an analytical description of a soliton state that occurs in coupled optical fibres.

Bosonic string theory and superstring theory are briefly overviewed. Three dimensional field theories are similarly discussed, with a focus on effective $N=2$ supersymmetric theories. It is shown how to induced contributions to the Chern-Simons level of the low energy theory, by integrating out massive matter. Such effective field theories are then shown to arise from type IIB brane configurations based on the Hanany-Witten brane configuration. Strong-weak dualities are overviewed, leading to a discussion of the three dimensional strong-weak dualities: Aharony duality for theories with zero Chern-Simons level, and Giveon-Kutasov duality for theories with non-zero Chern-Simons level. In the results section, brane configurations corresponding to three-dimensional $N=2$ $U(N_{c})$ field theories with various numbers of flavour of massive matter are investigated. The resulting low energy field theories are explained, and the flows between Aharony and Giveon-Kutasov dualities are catalogued. Three dimensional $N=2$ effective field theories obtained through the inclusion of massive adjoint matter are also examined, with the flows between Aharony and Giveon-Kutasov dualities, again, catalogued. Finally, the significance of the results and the possibilities for future research, are discussed.

This study provides an understanding of various aspects of hydrogeological systems modelling and the use of computational techniques to predict and optimise hydrological parameter assessment, anisotropic scaling, macrodispersion and solute flux measurements under unsteady, uniform/non-uniform flow conditions. The incorporated models are structured around multi-physics continuum mechanics analysis to investigate fluid flow and solute transport in hydrogeological systems. The control-volume unstructured mesh configuration, based on cell-centred or vertex-based FV algorithms for CFD and CSM problems is employed. The non-linear material behaviour exhibited by porous soils and the fluid flow evaluation under system stresses is described by elasto-visco-plastic constitutive relationships and the coupling between CFD and CSM processes. The designed simulation models are used to calibrate the flow problems associated with regional groundwater levels estimation, determination of soil hydraulic properties and moisture distribution in dry soils in response to infiltration of compressible or incompressible fluids. For solute transport problems, investigations of spatial distribution of solute species in homogeneous/layered heterogeneous systems are undertaken by accounting for chemical, geochemical and biological reactions caused by particle deposition processes and liquid-solid interactions in natural subsurface systems. The simulated shape and spread of contaminant plume are effectively influenced by the governing transport mechanism for solutes. The attention in leachate is predicted to have a significant role in reducing the level of contaminant concentration and its potential impact on the attainable groundwater resources.

This thesis describes the development of a methodology to determine large-scale and meso-scale atmospheric dispersion patterns. The research is only concerned with outdoor exposure to atmospheric pollutants and aims to identify pollution sources using dispersion modelling with the assistance of ground level measurements from British, French and other monitoring stations and remote sensing technology. Lagrangian Particle Dispersion (LPD) models compute trajectories of a large number of notional particles and can be used to numerically simulate the dispersion of a pollutant (passive tracer) in the planetary boundary layer. Two widely used atmospheric dispersion models were employed: the Hybrid Single Particle Lagrangian Integrated Trajectory (HYSPLIT) model by R. Draxler, and the model FLEXPART by Stohl et al. Both models possess forward tracking and inverse (or receptor-based) modes. Meteorological data output from the PSU/NCAR Mesoscale model (known as MM5), or datasets from the European Centre of Medium-range Weather Forecast (ECMWF) are used to drive the dispersion models. Linkage routines were developed to interpret the LPD codes with the required meteorological information. This study aims to determine whether current approaches and practice for atmospheric dispersion modelling are reliable, consistent and up-to-date. An intercomparison of the models FLEXPART and HYSPLIT is performed for known episodes to determine their accuracy, ease of use, effect of source specification and to investigate their sensitivity to input data and mesh resolution, and in particular the effect of different model formulations and assumptions followed by the models. The possibility of identifying emission sources in the near and far field is investigated, by modelling dispersion backwards in time, in particular the discrimination of multiple sources from receptor data is discussed. The effect of meteorological data resolution on the output of LPD models was evaluated and the most suitable methodology for better source definition was determined for different modelling scales, ranging from the intercontinental transport of airborne pollutants to simulating pollution episodes caused by local sources.