Dimension reduction is an important tool used to describe the structure of complex data (explicitly or implicitly) through a small but sufficient number of variables, and thereby make data analysis more efficient. It is also useful for visualization purposes. Dimension reduction helps statisticians to overcome the ‘curse of dimensionality’. However, most dimension reduction techniques require the intrinsic dimension of the low-dimensional subspace to be fixed in advance. The availability of reliable intrinsic dimension (ID) estimation techniques is of major importance. The main goal of this thesis is to develop algorithms for determining the intrinsic dimensions of recorded data sets in a nonlinear context. Whilst this is a well-researched topic for linear planes, based mainly on principal components analysis, relatively little attention has been paid to ways of estimating this number for non–linear variable interrelationships. The proposed algorithms here are based on existing concepts that can be categorized into local methods, relying on randomly selected subsets of a recorded variable set, and global methods, utilizing the entire data set. This thesis provides an overview of ID estimation techniques, with special consideration given to recent developments in non–linear techniques, such as charting manifold and fractal–based methods. Despite their nominal existence, the practical implementation of these techniques is far from straightforward. The intrinsic dimension is estimated via Brand’s algorithm by examining the growth point process, which counts the number of points in hyper-spheres. The estimation needs to determine the starting point for each hyper-sphere. In this thesis we provide settings for selecting starting points which work well for most data sets. Additionally we propose approaches for estimating dimensionality via Brand’s algorithm, the Dip method and the Regression method. Other approaches are proposed for estimating the intrinsic dimension by fractal dimension estimation methods, which exploit the intrinsic geometry of a data set. The most popular concept from this family of methods is the correlation dimension, which requires the estimation of the correlation integral for a ball of radius tending to 0. In this thesis we propose new approaches to approximate the correlation integral in this limit. The new approaches are the Intercept method, the Slop method and the Polynomial method. In addition we propose a new approach, a localized global method, which could be defined as a local version of global ID methods. The objective of the localized global approach is to improve the algorithm based on a local ID method, which could significantly reduce the negative bias. Experimental results on real world and simulated data are used to demonstrate the algorithms and compare them to other methodology. A simulation study which verifies the effectiveness of the proposed methods is also provided. Finally, these algorithms are contrasted using a recorded data set from an industrial melter process.

This thesis is concerned with structuring the statistical model with which we relate physical systems and computer simulators. The novelty of the work lies in the fact that we relate them via imagined smoothed versions of themselves, reflecting the belief that they are similar on large scales but discrepant when in comes to small scale details. Our central, paradigmatic example involves relating the planet’s climate to a climate simulator. Here the simulator is suspected to be incapable of faithfully reproducing changes in the system as time or certain physical parameters are changed by a small amount, but is still considered informative for the changes in the system over long time scales and large parameter changes.

We give a generalization of the concept of near-symplectic structures to 2n dimensions. According to our definition, a closed 2-form on a 2n-manifold M is near-symplectic, if it is symplectic outside a submanifold Z of codimension 3, where the (n-1)-th power of the 2-form vanishes. We depict how this notion relates to near-symplectic 4-manifolds and broken Lefschetz fibrations via some examples. We define a generalized broken Lefschetz fibration, or BLF, as a singular map with indefinite folds and Lefschetz-type singularities. We show that given such a map on a 2n-manifold over a symplectic base of codimension 2, then the total space carries such a near-symplectic structure, whose singular locus corresponds precisely to the singularity set of the fibration. A second part studies the geometry around the codimension--3 singular locus Z . We describe a splitting property of the normal bundle N_Z that is also present in dimension four. A tubular neighbourhood for Z is provided, which has as a corollary a Darboux-type theorem for near-symplectic forms.

This thesis presents the use of signatures within nonparametric predictive inference (NPI) for the failure time of a coherent system with a single type of components, given failure times of tested components that are exchangeable with those in the system. NPI is based on few modelling assumptions and here leads to lower and upper survival functions. We also illustrate comparison of reliability of two systems, by directly considering the random failure times of the systems. This includes explicit consideration of the difference between failure times of two systems. In this method we assume that the signature is precisely known. In addition, we show how bounds for these lower and upper survival functions can be derived based on limited information about the system structure, which can reduce computational effort substantially for specific inferential questions. It is illustrated how one can base reliability inferences on a partially known signature, assuming that bounds for the probabilities in the signature are available. As a further step in the development of NPI, we present the use of survival signatures within NPI for the failure time of a coherent system which consists of different types of components. It is assumed that, for each type of component, additional components which are exchangeable with those in the system have been tested and their failure times are available. Throughout this thesis we assume that the system is coherent, we start with a system consisting of a single type of components, then we extend for a system consisting of different types of components.

We consider the properties and behaviour of 2 U(2) noncommutative instantons: solutions of the NC-deformed ADHM equations which arise from U(2) 5d Yang-Mills theory. The ADHM construction allows us to find all such solutions, which form a moduli space of allowed configurations. We derive the metric for such a space, and consider the dynamics of the instantons on this space using the Manton approximation. We examine the reduction of this system to lower-dimensional soliton theories, and finally consider the effect of adding a Higgs field to the SYM theory, resulting in a potential on the instanton moduli space.

We study the symmetry and dynamics of M5-branes as well as chiral p-forms in this thesis. In the first part, we propose a model describing the gauge sector of multiple M5-branes. The model has modified six-dimensional Lorentz symmetry and its double dimensional reduction gives 5D Yang-Mills theory with higher derivative corrections. The non-abelian self-dual string solutions to this model are presented. In the second part of the thesis, we propose an alternative new action for the single M5-brane. The six-dimentional worldvolume space is covariantly split into 3+3. The relation of the new action to the conventional PST action as well as to the M2- brane action are studied. Finally, we briefly discuss the attempt to formulate the M5-brane action in a 2+4 splitting of worldvolume space and some duality properties and issues of chiral p-form actions.

We investigate delta-function contact interactions for theories of point particles and of strings. These interactions are introduced to reformulate the conventional theory of classical electrodynamics in terms of particles and strings which interact when they intersect. Upon quantisation we find that the tensionless limit of the spinning string theory generates well-known gauge invariant quantities in the worldline formulation of quantum field theory. Despite the off-shell nature of the interaction we find that the string theory does not encounter the expected break-down of conformal invariance. We further develop worldline techniques for non-Abelian theories and consider first quantised versions of some grand unified theories. This work can be seen as initiating the construction of a first-quantised version of the quantum field theory describing the standard model.

In this thesis the research presented relates to topological solitons in (2+1) and (3+1)-dimensional Skyrme theories. Solutions in these theories have topologically invariant quantities which results in stable solutions which are topologically distinct from a vacuum. In Chapter 2 we discuss the broken baby Skyrme model, a theory which breaks symmetry to the dihedral group D_N. It has been shown that the unit soliton solution of the theory is formed of N distinct peaks, called partons. The multi-soliton solutions have already been numerically simulated for N = 3 and were found to be related to polyiamonds. We extend this for higher values of N and demonstrate that a polyform structure continues. We discuss our numerical simulations studying the dynamics of this model and show that the time dependent behaviour of solutions in the model can be understood by considering the interactions of individual pairs of partons. Results of these dynamics are then compared with those of the standard baby Skyrme model. Recently it has been demonstrated that Skyrmions of a fixed size are able to exist in theories without a Skyrme term so long as the Skyrmion is located on a domain wall. In Chapter 3 we present a (2+1)-dimensional O(3) sigma model, with a potential term of a particular form, in which such Skyrmions exist. We numerically compute domain wall Skyrmions of this type. We also investigate Skyrmion dynamics so that we can study Skyrmion stability and the scattering of multi-Skyrmions. We consider scattering events in which Skyrmions remain on the same domain wall and find they are effectively one-dimensional. At low speeds these scatterings are well-approximated by kinks in the integrable sine-Gordon model. We also present more exotic fully two-dimensional scatterings in which Skyrmions initially on different domain walls emerge on the same domain wall. The Skyrme-Faddeev model is a (3+1)-dimensional non-linear field theory that has topological soliton solutions, called hopfions. Solutions of this theory are unusual in that that they are string-like and take the form of knots and links. Solutions found to date take the form of torus knots and links of these. In Chapter 4 we show results which address the question of whether any non-torus knot hopfions exist. We present a construction of fields which are knotted in the form of cable knots to which an energy minimisation scheme can be applied. We find static hopfions of the theory which do not have the form of torus knots, but instead take the form of cable and hyperbolic knots. In Chapter 5 we consider an approximation to the Skyrme-Faddeev model in which the soliton is modelled by elastic rods. We use this as a mechanism to study examples of particular knots to attempt to gain an understanding of why such knots have not been found in the Skyrme-Faddeev model. The aim of this study is to focus the search for appropriate rational maps which can then be applied in the Skyrme-Faddeev model. The material presented in this thesis relates to two published papers and corresponding to Chapters 2 and 3 respectively, which were done as part of a collaboration. In this thesis my own results are presented. Chapter 4 concerns material which relates to the preprint which is all my own work. Chapter 5 discusses my own ongoing work.

A mathematical and numerical analysis has been undertaken for three cross diffusion systems which arise in the modelling of biological systems. The first system appears in modelling the movement of multiple interacting cell populations whose kinetics are of competition type. The second model is the mechanical tumor-growth model of Jackson and Byrne that consists of nonlinear parabolic cross-diffusion equations in one space dimension for the volume fractions of tumor cells and an extracellular matrix (ECM), and describes tumor encapsulation influenced by a cell-induced pressure coefficient. The third system is the Keller-Segel model in multiple-space dimensions with an additional cross-diffusion term in the elliptic equation for the chemical signal. A fully practical piecewise linear finite element approximation for each system is proposed and studied. With the aid of a fixed point theorem, existence of fully discrete solution is shown. By using entropy type inequalities and compactness arguments, the convergence of each approximation is proved and hence existence of a global weak solution is obtained. In the case of the Keller-Segel model, we were able to obtain additional regularity to provide an improved weak formulation. Further, for the Keller-Segel model we established uniqueness results and error estimates. Finally, a practical algorithm for computing the numerical solutions of each system is described and some numerical experiments are performed to illustrate and verify the theoretical results.

Following the work of Bruinier and Funke in the orthogonal setting, we consider a regularised theta lift from weight 0 harmonic weak Maass forms on non-compact quotients of SU(1,1) to meromorphic modular forms of weight 2, and realise the result of the lift as a generating series of modular traces of those Maass forms on CM points. We also lift the non-holomorphic Eisenstein series of weight 0 and realise the derivative of a suitably normalised weight 2 Eisenstein series as the lift of the logarithm of the modular Delta function.