We present a novel shape classification method which is embedded in the Bayesian paradigm. We focus on the statistical classification of planar shapes by using methods which replace some previous approximate results by analytic calculations in a closed form. This gives rise to a new Bayesian shape classification algorithm and we evaluate its efficiency and efficacy on available shape databases. In addition we apply our results to the statistical classification of geological sand bodies. We suggest that our proposed classification method, that utilises the unique geometrical information of the sand bodies, is more substantial and can replace ad-hoc and simplistic methods that have been used in the past. Finally, we conclude this work by extending the proposed classification algorithm for shapes in three-dimensions.

This dissertation is made up of two independent parts. In Part I we consider the Pestov Identity, an identity stated for smooth functions on the tangent bundle of a manifold and linking the Riemannian curvature tensor to the generators of the geodesic flow, and we lift it to the bundle of k-tuples of tangent vectors over a compact manifold M of dimension n. We also derive an integrated version over the bundle of orthonormal k-frames of M as well as a restriction to smooth functions on such a bundle. Finally, we present a dynamical application for the parallel transport of the Grassmannian of oriented k-planes of M. In Part II we consider a family of compact and connected n-dimensional manifolds, called graph-like manifold, shrinking to a metric graph in the appropriate limit. We describe the asymptotic behaviour of the eigenvalues of the Hodge Laplacian acting on differential forms on those manifolds in the appropriate limit. As an application, we produce manifolds and families of manifolds with arbitrarily large spectral gaps in the spectrum of the Hodge Laplacian.

In this thesis we investigate unusual and non-trivial interplays between gravity and field theory. We concentrate on two examples, one related to holography and the other to the physics of false vacuum decay. In the first chapter we overview basic concepts and techniques from both these examples. In chapter 2 we construct solutions describing flows between AdS and Lifshitz spacetimes in IIB supergravity. We find that flows from AdS5 can approach either AdS3 or Lifshitz3 in the IR depending on the values of the deformation from AdS5. Surprisingly, the choice between AdS and Lifshitz in the IR depends only on the value of the deformation, not on its character; the breaking of the Lorentz symmetry in the flows with a Lifshitz IR is spontaneous. We find that the values of the deformation which lead to flows to Lifshitz make the UV field theory dual to the AdS5 geometry unstable, so that these flows do not offer an approach to defining the field theory dual to the Lifshitz spacetime. In chapter 3 we consider the possibility that small black holes can act as nu- cleation seeds for the decay of a metastable vacuum. Using a thin-wall bubble approximation for the nucleation process, we show that black holes can stimulate vacuum decay. In chapter 4 we apply this technique to the particular example of the Higgs potential with generic quantum gravity corrections. We show how small black holes can act as seeds for vacuum decay, spontaneously nucleating a new Higgs phase centred on the black hole with a lifetime measured in millions of Planck times rather than billions of years. The constraints on the parameter space of corrections to the Higgs potential are outlined. We demonstrate that for suitable parameter ranges, the vacuum decay process dominates over the Hawking evaporation process. We also comment on the application of these results to vacuum decay seeded by black holes produced in particle collisions. By relaxing the conditions for the thin-wall approximation and proceeding to the numerical calculations an expansion of the range of the parameter space is proposed.

This thesis discusses branes in string theory and M-theory. In chapter 1, we present background materials. In chapters 2 and 3, we discuss an application of a D3/D7 model in the framework of gauge/gravity duality, which uses the results of a gravity theory calculation to obtain informations about the corresponding gauge theory. Gauge/gravity duality is usually studied in a non-compact space. This thesis, however, focuses on the duality in a compact space. In chapter 2, we study a strongly coupled gauge theory in a compact space. We find that a homogeneous ground state is unstable at a sufficiently large isospin chemical potential. We then construct a new ground state which corresponds to a scalar meson condensate charged under a global SO(4) symmetry. In chapter 3, we discuss an on-going work. We study a strongly-coupled gauge theory living in a compact space with a generic value of quark mass at zero temperature in the presence of an external magnetic field. We are investigating the phase diagram of the external magnetic field and quark mass. In chapter 4 and 5, we move to M-theory. In chapter 4, we construct a complete non-linear theory of a supersymmetric single M5-brane with a 3+3 splitting. The M5-brane lives in a general background of M-theory. The idea of our construction is to get equations of motion that agree with that obtained from the superembedding approach. In chapter 5, we consider a model which attempts to describe a theory on multiple M5-branes. Using this model, we construct self-dual string solutions which describes M2/M5 branes intersection. These solutions are supported by magnetic monopoles. For self-dual strings with large number of M2 and M5-branes, we obtain the required radius-distance relationship and energy.

This thesis discusses BPS monopoles in hyperbolic space and BPS vortices in the presence of magnetic impurities. We prove explicit formulae for the spectral curve and rational map of a JNR-type hyperbolic monopole, and we use these to study some Platonic examples as well as some new 1-parameter families analogous to Euclidean monopole scattering. Explicit fields and Braam-Austin data for axial hyperbolic monopoles of a particular mass are derived using a correspondence to 1-monopoles, and this data is deformed to give new 1-parameter families. Numerical techniques are used to study the effect of magnetic impurities on vortices on a flat background. Analytic results for vortices with magnetic impurities are found by adapting previous results on vortices on the hyperbolic plane and the 2-sphere.

This thesis investigates a new bootstrap method, this method is called Nonparametric Predictive Inference Bootstrap (NPI-B). Nonparametric predictive inference (NPI) is a frequentist statistics approach that makes few assumptions, enabled by using lower and upper probabilities to quantify uncertainty, and explicitly focuses on future observations. In the NPI-B method, we use a sample of n observations to create n + 1 intervals and draw one future value uniformly from one interval. Then this value is added to the data and the process is repeated, now with n+1 observations. Repetition of this process leads to the NPI-B sample, which therefore is not taken from the actual sample, but consists of values in the whole range of possible observations, also going beyond the range of the actual sample. We explore NPI-B for data on finite intervals, real line and non negative observations, and compare it to other bootstrap methods via simulation studies which show that the NPI-B method works well as a prediction method. The NPI method is presented for the reproducibility probability (RP) of some nonparametric tests. Recently, there has been substantial interest in the reproducibility probability, where not only its estimation but also its actual definition and interpretation are not uniquely determined in the classical frequentist statistics framework. The explicitly predictive nature of NPI provides a natural formulation of inferences on RP. It is used to derive lower and upper bounds of RP values (known as the NPI-RP method) but if we consider large sample sizes, the computation of these bounds is difficult. We explore the NPI-B method to predict the RP values (they are called NPI-B-RP values) of some nonparametric tests. Reproducibility of tests is an important characteristic of the practical relevance of test outcomes.

In this thesis, we are concerned with the molecular biology processes of ionising radiation (IR) damage and its repair. In particular, we develop a mechanistic model of high dose irradiation damage to DNA in mammalian cells by considering a population of cells structured by the number of double strand breaks and mis-repairs. This framework allows us to construct a mechanistic explanation for the Linear-Quadratic (LQ) formalism. Other contributions of this thesis are a model of ataxia telangiectasia mutated (ATM) dynamics in DNA damage repair, and a discussion of bistability, an important ingredient in the construction of a general theory of non-targeted radiation effects.

The squeeze-film flow of a thin layer of Newtonian fluid filling the gap between either a flat or a curved rigid impermeable solid coated with a thin rigid or elastic layer and moving under a prescribed constant load towards a thin rigid porous bed of uniform thickness coating a stationary flat impermeable solid is considered. Three related problems are considered separately, namely the case of a flat surface coated with a rigid layer, the case of a curved surface coated with a rigid layer and the general case of a curved surface coated with an elastic layer. Unlike the case of an impermeable bed, in which an infinite time is required for the two surfaces to come into contact, for a porous bed contact occurs in a finite time. Lubrication approximations are used and numerical and (where possible) analytical solutions for the fluid layer thickness, the fluid pressure and the contact time are obtained and analysed. Furthermore, in the rigid cases the fluid particle paths are calculated, and the penetration depths of fluid particles into the porous bed are determined. In particular, the behaviour in the asymptotic limit of small permeability, in which the contact time is large but finite, is investigated. Finally, the results are interpreted in the context of lubrication in the human knee joint, and some conclusions are drawn about the contact time of the cartilage-coated femoral condyles and tibial plateau and the penetration of nutrients into the cartilage.

Outbreaks of widely spread infectious diseases, such as Human Immunodefficiency Virus (HIV), Severe Acute Respiratory Syndrome (SARS) and Swineflu (H1N1) and hospital acquired infections, such as Meticillin-resistant Staphylococcus Aureus (MRSA) and Clostridium Diffcile, are serious health problems which have been tackled by the World Health Organization and international health protection agencies. Various statistical analyses have contributed a remarkable effect on providing scientific evidence on which to base political decisions and infection control strategies. In this project, we focused on two infectious diseases: HIV and MRSA and the research project is divided into two separate parts. One is the quantification of replication in HIV anonymous test reports and the other is the effect of patient movement between wards on the acquisition of MRSA. The first research project is concerned with the analysis of an anonymous HIV test dataset. The data is collected as a set of birthdays and it is possible that there is repeated sampling of the same person. The aim is to quantify the amount of replication in the HIV data using a maximum likelihood technique and then give the confidence intervals for the estimated amount of replication using the bootstrap method. The data were provided by the Public Health Laboratory Service (PHLS), Colindale, London in 1994, who were interested in a statistical method to estimate multiple counting that possibly existed in the database. The data consists of individual records of the number of AIDS cases diagnosed, with birthdates from 1901 to 1973. There were two datasets provided by the PHLS, one of which contained 1,134 records and was provided in 1991. The other dataset was provided in 1994 with the sample size 17,137. An estimate of the true number of distinct individuals as well as the percentage of replication was obtained by programming the maximum likelihood calculation in the languages R and C. This technique is based upon evaluation of the probability that two records with the same birthdate represent two separate individuals as opposed to the same person reported twice. The results for the 1991 dataset showed that there were five out of sixteen birth years (i.e. 31.25% of the observed records in the 1991 dataset) with replication in the true number of distinct individuals. In the results of the 1994 dataset, the majority of the birth years (57/73) recorded the correct number of distinct individuals in the observations. The 95% confidence intervals for the estimated amount of replication were calculated by applying a parametric bootstrap method. The results show that the birth years in the 1991 dataset with non-zero estimated amount of replication (the birth years of 1931, 1934, 1935, 1943 and 1944) have comparatively wide 95% bootstrap confidence intervals, which implies higher uncertainty of the true amount of replication. A similar conclusion was obtained from the results of 95% bootstrap con dence intervals for the 1994 dataset. Comparing the results within the same birth years recorded in the 1991 dataset and the 1994 dataset, the data indicate that the confidence intervals for the 1994 dataset are mainly narrower than the corresponding ones in the 1991 dataset. The conclusion of this study illustrates the drawback of recording the HIV patients only with date of birth, which has now been improved by combining with 'Soundex' codes for the surname and gender. The second part of the project aims to estimate the impact of patient movement within a hospital on the risk of MRSA acquisition by using data from the MRSA screening admission and discharge studies in Scotland which took place in two hospitals in 2010. The data consist of an admission-only database (7,181 patients), a discharge-only database (2,432 patients) and a combined admission-discharge cohort (2,792 patients). The third database has complete information on MRSA status on admission, on discharge, as well as data on the wards the patient was in while in hospital. In order to understand the effect of potential risk factors on MRSA acquisition, a multivariate logistic regression model was constructed to analyse the effects of the number of wards a patient was in on MRSA acquisition as well as other risk factors. Receiver Operating Characteristic (ROC) curves were plotted and the individual area under the curve (AUC) was also calculated for indicating the reliability and the accuracy of the prediction of the models. Furthermore, we modelled the dynamic patient movement and assessed the effect of being in a ward with MRSA by imputing the unknown date of transfer, simulating the missing length of stay (along with the simulation envelope). The timelines of MRSA infection and carriage pressure in each ward of the two hospitals were then mapped for all patients in the three databases, imputing where necessary. Patient movement was measured as a volume indicator in terms of the frequency of ward to ward transfer and as cohabiting in the same ward. By using logistic regression within a bootstrap simulation, we estimated the odds ratio of acquisition of MRSA association with being in a ward with MRSA present, which was given by averaging the estimated effects from the fitted models, and generating the 95% confidence intervals. The results indicate that the number of wards that patients had moved through and patients being in a ward with MRSA present do not affect the risk of acquiring MRSA significantly over and above the patient level risk factors such as age and the presence of open wounds or catheters. Some further work which can be done in an MRSA screening programme is suggested as an implementation study.

There are many nonlinear econometric models which are useful in analysis of financial time series. In this thesis, we consider two kinds of nonlinear autoregressive models for nonnegative integer-valued time series: threshold autoregressive models and Markov switching models, in which the conditional distribution given historical information is the Poisson distribution. The link between the conditional variance (i.e. the conditional mean for the Poisson distribution) and its past values as well as the observed values of the Poisson process may be different according to the threshold variable in threshold autoregressive models, and to an unobservable state variable in Markov switching models in different regimes. We give a condition on parameters under which the Poisson generalized threshold autoregressive heteroscedastic (PTGARCH) process can be approximated by a geometrically ergodic process. Under this condition, we discuss statistical inference (estimation and tests) for PTGARCH models, and give the asymptotic theory on the inference. The complete structure of the threshold autoregressive model is not exactly specific in economic theory for the most financial applications of the model. In particular, the number of regimes, the value of threshold and the delay parameter are often unknown and cannot be assumed known. Therefore, in this research, the performance of various information criteria for choosing the number of regimes, the threshold value and the delay parameters for different sample sizes is investigated. Tests for threshold nonlinearity are applied. The characteristics of Markovian switching Poisson generalized autoregressive hetero-scedastic (MS-PGARCH) models are given, and the maximum likelihood estimation of parameters is discussed. Simulation studies and applications to modelling financial counting time series are presented to support our methodology for both the PTGARCH model and the MS-PGARCH model.