The research work included in this thesis concerns the study of wave propagation in elastic materials which are stressed in their initial state. This research is based on the non-linear theory of elasticity. Using the theory of invariants, the general constitutive equation for an isotropic hyperelastic material in the presence of initial stress is derived. These invariants depend on the finite deformation as well as the initial stress. In general, this derivation involves 10 invariants for a compressible material and 9 for an incompressible material. Making use of these invariants, the elasticity tensor is given in its most general form for both the deformed and the undeformed (i.e., the initially stressed reference) configurations. The equations governing infinitesimal motions superimposed on a finite deformations are then used to study the effects of initial stress and finite deformation on wave propagation. For each of the problems carried out in this thesis, the results are specialized for a prototype strain energy function which depends on the initial stress as well as the deformation. The basic theory in each of the problems is formed for the material in the deformed configuration and is later specialized for the undeformed reference configuration. Considering the special case when initial stress is zero, the results are compared with those from the linear theory of elasticity. The problem of homogeneous plane waves in an initially stressed incompressible half-space is considered. The basic theory of the problem is later used to study the reflection of plane waves from the boundary of such a half-space. The reflection coefficients of waves are calculated and graphical representations are given to study the behaviour with reference to the magnitude of initial stress and finite deformation. The study of Rayleigh and Love waves follows thereafter and the basic theory already developed in this thesis is used to study the effect of initial stress on the wave speed of these surface waves. In both cases, the secular equation is analysed in deformed and undeformed configurations and graphs are presented. The problem of wave propagation in a residually stressed inhomogeneous thick-walled incompressible tube which is axially stretched and inflated due to internal pressure, is considered. On the basis of known experimental behaviour, a simple expression for the residual stress is chosen to calculate the internal pressure used to inflate the tube and the axial load to stretch it. The effect of initial stress and stretch on pressure and axial load is studied and graphs are presented. The general theory developed for the deformed configuration for the special model is specialized to the reference configuration and the dispersion relation is analysed numerically.

We study certain non-symmetric wavefunctions associated to the quantum nonlinear Schrödinger (QNLS) model, introduced by Komori and Hikami using representations of the degenerate affine Hecke algebra. In particular, they can be generated using a vertex operator formalism analogous to the recursion that defines the symmetric QNLS wavefunction in the quantum inverse scattering method. Furthermore, some of the commutation relations encoded in the Yang-Baxter equation are generalized to the non-symmetric case.

In this thesis we consider some boundary value problems concerning nonlinear deformations and incremental motions in magnetoelastic solids. Three main problems have been addressed relating to waves propagating on the surface of a finitely deformed half-space and waves propagating along the axis of a thick-walled tube. First, the equations and boundary conditions governing linearized incremental motions superimposed on an initial motion and underlying electromagnetic field are derived and then specialized to the quasimagnetostatic approximation. The magnetoelastic material properties are characterized in terms of a ``total'' isotropic energy density function that depends on both the deformation and a Lagrangian measure of the magnetic field. In the first problem, we analyze the propagation of Rayleigh-type surface waves for different directions of the initial magnetic field and for a simple constitutive model of a magnetoelastic material in order to evaluate the combined effect of the finite deformation and magnetic field on the surface wave speed. Numerical results for a Mooney--Rivlin type magnetoelastic material show that a magnetic field in the considered (sagittal) plane in general destabilizes the material compared with the situation in the absence of a magnetic field. A magnetic field applied in the direction of wave propagation is more destabilizing than that applied perpendicular to it. In the second problem, the propagation of Love-type waves in a homogeneously and finitely deformed layered half-space is analyzed for a Mooney--Rivlin type and a neo-Hookean type magnetoelastic energy function. The resulting wave speed characteristics in general depend significantly on the initial magnetic field as well as on the initial finite deformation, and the results are illustrated graphically for different combinations of these parameters. In the absence of a layer, shear horizontal surface waves do not exist in a purely elastic material, but the presence of a magnetic field normal to the sagittal plane makes such waves possible, these being analogous to Bleustein--Gulyaev waves in piezoelectric materials. Then, we consider nonlinear axisymmetric deformations and incremental motions of a cylindrical magnetoelastic tube. The effects of internal pressure, axial stretch, and magnetic field are studied for two different kinds of energy density functions. It is found that in general an underlying azimuthal magnetic field increases the total internal pressure, affects the axial load, and induces stability in the tube. Dependence of the incremental motion on internal pressure, axial stretch, thickness of tube, and the applied magnetic field is illustrated graphically. Finally, we consider the general equations of Electrodynamics and Thermodynamics in continua. In particular, we write the equations governing mechanical waves, electromagnetic fields and temperature changes in a magnetoelastic conductor with a motivation to describe the electromagnetic acoustic transduction (EMAT) process. This is a work in progress and an open research problem for the future.

Quasideterminants are a relatively new addition to the field of integrable systems. Their simple structure disguises a wealth of interesting and useful properties, enabling solutions of noncommutative integrable equations to be expressed in a straightforward and aesthetically pleasing manner. This thesis investigates the derivation and quasideterminant solutions of two noncommutative integrable equations - the Davey-Stewartson (DS) and Sasa-Satsuma nonlinear Schrodinger (SSNLS) equations. Chapter 1 provides a brief overview of the various concepts to which we will refer during the course of the thesis. We begin by explaining the notion of an integrable system, although no concrete definition has ever been explicitly stated. We then move on to discuss Lax pairs, and also introduce the Hirota bilinear form of an integrable equation, looking at the Kadomtsev-Petviashvili (KP) equation as an example. Wronskian and Grammian determinants will play an important role in later chapters, albeit in a noncommutative setting, and, as such, we give an account of their widespread use in integrable systems. Chapter 2 provides further background information, now focusing on noncommutativity. We explain how noncommutativity can be defined and implemented, both specifically using a star product formalism, and also in a more general manner. It is this general definition to which we will allude in the remainder of the thesis. We then give the definition of a quasideterminant, introduced by Gel'fand and Retakh in 1991, and provide some examples and properties of these noncommutative determinantal analogues. We also explain how to calculate the derivative of a quasideterminant. The chapter concludes by outlining the motivation for studying our particular choice of noncommutative integrable equations and their quasideterminant solutions. We begin with the DS equations in Chapter 3, and derive a noncommutative version of this integrable system using a Lax pair approach. Quasideterminant solutions arise in a natural way by the implementation of Darboux and binary Darboux transformations, and, after describing these transformations in detail, we obtain two types of quasideterminant solution to our system of noncommutative DS equations - a quasi-Wronskian solution from the application of the ordinary Darboux transformation, and a quasi-Grammian solution by applying the binary transformation. After verification of these solutions, in Chapter 4 we select the quasi-Grammian solution to allow us to determine a particular class of solution to our noncommutative DS equations. These solutions, termed dromions, are lump-like objects decaying exponentially in all directions, and are found at the intersection of two perpendicular plane waves. We extend earlier work of Gilson and Nimmo by obtaining plots of these dromion solutions in a noncommutative setting. The work on the noncommutative DS equations and their dromion solutions constitutes our paper published in 2009. Chapter 5 describes how the well-known Darboux and binary Darboux transformations in (2+1)-dimensions discussed in the previous chapter can be dimensionally-reduced to enable their application to (1+1)-dimensional integrable equations. This reduction was discussed briefly by Gilson, Nimmo and Ohta in reference to the self-dual Yang-Mills (SDYM) equations, however we explain these results in more detail, using a reduction from the DS to the nonlinear Schrodinger (NLS) equation as a specific example. Results stated here are utilised in Chapter 6, where we consider higher-order NLS equations in (1+1)-dimension. We choose to focus on one particular equation, the SSNLS equation, and, after deriving a noncommutative version of this equation in a similar manner to the derivation of our noncommutative DS system in Chapter 3, we apply the dimensionally-reduced Darboux transformation to the noncommutative SSNLS equation. We see that this ordinary Darboux transformation does not preserve the properties of the equation and its Lax pair, and we must therefore look to the dimensionally-reduced binary Darboux transformation to obtain a quasi-Grammian solution. After calculating some essential conditions on various terms appearing in our solution, we are then able to determine and obtain plots of soliton solutions in a noncommutative setting. Chapter 7 seeks to bring together the various results obtained in earlier chapters, and also discusses some open questions arising from our work.

This thesis presents novel Markov chain Monte Carlo methodology that exploits the natural representation of a statistical model as a Riemannian manifold. The methods developed provide generalisations of the Metropolis-adjusted Langevin algorithm and the Hybrid Monte Carlo algorithm for Bayesian statistical inference, and resolve many shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlation structure. The performance of these Riemannian manifold Markov chain Monte Carlo algorithms is rigorously assessed by performing Bayesian inference on logistic regression models, log-Gaussian Cox point process models, stochastic volatility models, and both parameter and model level inference of dynamical systems described by nonlinear differential equations.