This thesis considers the statistical analysis of diffusion tensor imaging (DTI). DTI is an advanced magnetic resonance imaging (MRI) method that provides a unique insight into biological microstructure \textit{in vivo} by directionally describing the water molecular diffusion. We firstly develop a Bayesian multi-tensor model with reparameterisation for capturing water diffusion at voxels with one or more distinct fibre orientations. Our model substantially alleviates the non-identifiability issue present in the standard multi-tensor model. A Markov chain Monte Carlo (MCMC) algorithm is then developed to study the uncertainty of the model parameters based on the posterior distribution. We apply the Bayesian method to Monte Carlo (MC) simulated datasets as well as a healthy human brain dataset. A region containing crossing fibre bundles is investigated using our multi-tensor model with automatic model selection. A diffusion tensor, a covariance matrix related to the molecular displacement at a particular voxel in the brain, is in the non-Euclidean space of 3x3 positive semidefinite symmetric matrices. We define the sample mean of tensor data to be the Fréchet mean. We carry out the non-Euclidean statistical analysis of diffusion tensor data. The primary focus is on the use of Procrustes size-and-shape space. Comparisons are made with other non-Euclidean techniques, including the log-Euclidean, Riemannian, Cholesky, root Euclidean and power Euclidean methods. The weighted generalised Procrustes analysis has been developed to efficiently interpolate and smooth an arbitrary number of tensors with the flexibility of controlling individual contributions. A new anisotropy measure, Procrustes Anisotropy is defined and compared with other widely used anisotropy measures. All methods are illustrated through synthetic examples as well as white matter tractography of a healthy human brain. Finally, we use Giné’s statistic to design uniformly distributed diffusion gradient direction schemes with different numbers of directions. MC simulation studies are carried out to compare effects of Giné’s and widely used Jones' schemes on tensor estimation. We conclude by discussing potential areas for further research.

In this thesis, we present Kohn variational calculations of scattering and annihilation parameters for very low energy interactions of positrons with molecular hydrogen. Our analysis includes the first application of the Kohn method for this system in which the interelectronic potential in the molecular target is treated explicitly. All previous Kohn calculations on positron hydrogen molecule scattering have avoided this complication by the use of the method of models. The advantage of the explicit treatment over the method of models is that it allows approximate target wavefunctions of a very high accuracy to be admitted more easily to the Kohn calculations. We find that the accuracy of the approximate target wavefunction is an extremely important factor in obtaining reliable results from the calculations. We carry out an extensive investigation of anomalous, nonphysical behaviour in the results of our Kohn calculations. Our explanations of how these anomalies arise and how they may be avoided significantly improves upon the discussions of these phenomena given in earlier accounts of positron hydrogen molecule scattering calculations by other authors. As with all previous models of positron hydrogen molecule scattering, we find discrepancies between the experimental value of the annihilation parameter, Z effective, and the theoretical value of this quantity as determined from our Kohn calculations. Limitations of the model that could explain these discrepancies are discussed and suggestions for future improvements are proposed.

The Nikolaevskiy equation is considered as a simple model exhibiting spatiotemporal chaos due to the coupling of finite-wavelength patterns to a long-wavelength mode (Goldstone mode). It was originally proposed as a model for seismic waves and is also considered as a model for various physical phenomena, including electroconvection, reaction-diffusion systems and transverse instabilities of travelling fronts for chemical reactions. This equation has attracted the attention of several researchers due to its rich dynamical properties and physical applications. We are interested in studying this equation closely by means of numerical computations and asymptotic analysis. In this thesis we reinstate the dispersive terms, in contrast to most research regarding the Nikolaevskiy equation, and study the effect on the stability of spatially periodic solutions, which take the form of travelling waves. It is shown that dispersion can stabilise the travelling wave solutions, which emerge at the onset of instability of the spatially uniform state. The secondary stability plots exhibit high sensitivity on the degree of dispersion and can sometimes be remarkably complicated. Dispersive amplitude equations are derived: numerical simulations manifest behaviour similar to the non-dispersive case but there is a drift of the pattern with a certain speed. Another aspect of this thesis is analysing systems similar to the Nikolaevskiy equation, where they incorporate a Goldstone mode and possess the same symmetries. We conclude that such systems share with the Nikolaevskiy equation the fact that roll solutions are unstable at the onset of instability. We also study the amplitude equations of these systems numerically and deduce that statistical measures of their solutions depend on the ratio of the curvatures of the dispersion relation near the finite-wavelength and long-wavelength modes. Finally, we consider a system coupling a Swift-Hohenberg equation to a large-scale mode. The result of this study shows that there can be stable stationary wave solutions, in contrast to the Nikolaevskiy equation.

UNIFORM algebras have been extensively investigated because of their importance in the theory of uniform approximation and as examples of complex Banach algebras. An interesting question is whether analogous algebras exist when a complete valued field other than the complex numbers is used as the underlying field of the algebra. In the Archimedean setting, this generalisation is given by the theory of real function algebras introduced by S. H. Kulkarni and B. V. Limaye in the 1980s. This thesis establishes a broader theory accommodating any complete valued field as the underlying field by involving Galois automorphisms and using non-Archimedean analysis. The approach taken keeps close to the original definitions from the Archimedean setting. Basic function algebras are defined and generalise real function algebras to all complete valued fields. Several examples are provided. Each basic function algebra is shown to have a lattice of basic extensions related to the field structure. In the non-Archimedean setting it is shown that certain basic function algebras have residue algebras that are also basic function algebras. A representation theorem is established. Commutative unital Banach F-algebras with square preserving norm and finite basic dimension are shown to be isometrically F-isomorphic to some subalgebra of a Basic function algebra. The theory of non-commutative real function algebras was established by K. Jarosz in 2008. The possibility of their generalisation to the non-Archimedean setting is established in this thesis. In the context of complex uniform algebras, a new proof is given using transfinite induction of the Feinstein-Heath Swiss cheese “Classicalisation” theorem.

There is much current interest in systems exhibiting homoclinic snaking, in which solution curves of localised patterns snake back and forth within a narrow region of parameter space. Such solutions comprise superimposed, back-to-back stationary fronts, each front connecting a homogeneous and a patterned state. These fronts are pinned to the underlying pattern within the snaking region; elsewhere, they become travelling waves and cannot form localised solutions. Application of standard asymptotic techniques near bifurcation can only produce a stationary front at the centre of the snaking region; this is the Maxwell point, where patterned and homogeneous states are equally energetically favourable. Such methods fail to capture the pinning mechanism because it is an exponentially small effect, and must be studied using exponential asymptotics. Deriving the late terms in the asymptotic expansion and observing that it is divergent, we truncate optimally after the least term. The resultant remainder is exponentially small and governed by an inhomogeneous differential equation. Rescaling this equation near Stokes lines---lines in the complex plane at which forcing is maximal---we observe a smooth but rapid increase from zero to exponentially small in the coefficient of an exponentially growing complementary function as Stokes lines are crossed. Requiring that unbounded terms vanish fixes the phase of the underlying pattern relative to the leading-order front. Furthermore, matching two fronts together produces a set of formulae describing the snaking bifurcation diagram. We successfully apply this method to continuous and discrete systems. In the former, we also show how symmetric solutions comprising two localised patches form figure-of-eight isolas in the bifurcation diagram. In the latter, we investigate snaking behaviour of a one-dimensional localised solution rotated into a square lattice, and find that the snaking region vanishes when the tangent of the angle of orientation is irrational.

Distributive laws in Computer Science are rules governing the transformation of one programming structure into another. In programming, they are programs satisfying certain formal conditions. Their importance has been to date documented in several isolated cases by diverse formal approaches. These applications have always meant leaps in understanding the nature of the subject. However, distributive laws have not yet been given the attention they deserve. One of the reasons for this omission is certainly the lack of a formal notion of distributive laws in their full generality. This hinders the discovery and formal description of occurrences of distributive laws, which is the precursor of any formal manipulation. In this thesis, an approach to formalisation of distributive laws is presented based on the functorial approach to formal Category Theory pioneered by Lawvere and others, notably Gray. The proposed formalism discloses a rather simple nature of distributive laws of the kind found in programming structures based on lax 2-naturality and Gray's tensor product of 2-categories. It generalises the existing more specific notions of distributive laws. General notions of products, coproducts and composition of distributive laws are studied and conditions for their construction given. Finally, the proposed formalism is put to work in establishing a semantical equivalence between a large class of functional and object-based programs.

No description available.

In this thesis we study the chemical reactions and transport phenomena which occur in a microwave power assisted chemical vapour deposition (MPA-CVD) reactor which facilitates diamond growth. First we introduce a model of an underlying binary gas flow and its chemistry for a hydrogen gas mixture. This system is heated by incorporating a microwave frequency electric field, operating in a resonant mode in the CVD chamber. This heating facilitates the dissociation of hydrogen and the generation of a gas discharge plasma, a key component of carbon deposition in industrial diamond manufacture. We then proceed to summarise the discontinuous Galerkin (DG) finite element discretisation of the standard hyperbolic and elliptic partial differential operators which typically occur in conservation laws of continuum models. Additionally, we summarise the non-stabilised discontinuous Galerkin formulation of the time harmonic Maxwell operator. These schemes are then used as the basis for the discretisation method employed for the numerical approximation of the MPA-CVD model equations. The practical implementation of the resulting DG MPA-CVD model is an extremely challenging task, which is prone to human error. Thereby, we introduce a mathematical approach for the symbolic formulation and computation of the underlying finite element method, based on automatic code generation. We extend this idea further such that the DG finite element formulation is automatically computed following the user's specification of the convective and viscous flux terms of the underlying PDE system in this symbolic framework. We then devise a method for writing a library of automatically generated DG finite element formulations for a hierarchy of partial differential equations with automatic treatment of prescribed boundary conditions. This toolbox for automatically computing DG finite element solutions is then applied to the DG MPA-CVD model. In particular, we consider reactor designs inspired by the AIXTRON and LIMHP reactors which are analysed extensively in the literature.

In Chapter 1 we introduce many of the concepts and techniques, including Nevanlinna theory, referred to throughout the rest of the thesis. In Chapter 2 we extend a result of Langley and Shea concerning the distribution of zeros of the logarithmic derivative of meromorphic functions to higher order logarithmic derivatives. Chapter 3 details an alternative formulation, avoiding reference to the multiplicity of poles, of a result due to Chuang concerning differential polynomials. In Chapter 4 we generalise a theorem of Bergweiler and Eremenko concerning transcendental singularities of the the inverse of a meromorphic function. In Chapter 5 we generalise a result of Gordon to show that an unbounded analytic function on a quasidisk has a strong form of unboundedness there. Chapter 6 contains a proof of a result concerning the normality of families of analytic functions such that the composition of any of these functions with a fixed (meromorphic) outer factor has no fixpoints in a given domain.

Transformation methods are perhaps the most powerful analytic tool currently available in the study of nonlinear partial differential equations. Transformations may be classified into two categories: category I includes transformations of the dependent and independent variables of a given partial differential equation and category II additionally includes transformations of the derivatives of the dependent variables. In part I of this thesis our principal attention is focused on transformations of the category I, namely point transformations. We mainly deal with groups of transformations. These groups enable us to derive similarity transformations which reduce the number of independent variables of a certain partial differential equation. Firstly, we introduce the concept of transformation groups and in the analysis which follows three methods for determining transformation groups are presented and consequently the corresponding similarity transformations are derived. We also present a direct method for determining similarity transformations. Finally, we classify all point transformations for a particular class of equations, namely the generalised Burgers equation. Bäcklund transformations belong to category II and they are investigated in part II. The first chapter is an introduction to the theory of Bäcklund transformations. Here two different classes of Bäcklund transformations are defined and appropriate example are given. These two classes are considered in the proceeding analysis, where we search for Bäcklund transformations for specific classes of partial differential equations.