In this thesis we introduce a model for informative censoring. We assume that the joint distribution of the failure and the censored times depends on a parameter δ, which is actually a measure of the possible dependence, and a bias function B(t,θ). Knowledge of δ means that the joint distribution is fully specified, while B(t,θ) can be any function of the failure times. Being unable to draw inferences about δ, we perform a sensitivity analysis on the parameters of interest for small values of δ, based on a first order approximation. This will give us an idea of how robust our estimates are in the presence of small dependencies, and whether the ignorability assumption can lead to misleading results. Initially we propose the model for the general parametric case. This is the simplest possible case and we explore the different choices for the standardized bias function. After choosing a suitable function for B(t,θ) we explore the potential interpretation of δ through it's relation to the correlation between quantities of the failure and the censoring processes. Generalizing our parametric model we propose a proportional hazards structure, allowing the presence of covariates. At this stage we present a data set from a leukemia study in which the knowledge, under some certain assumptions, of the censored and the death times of a number of patients allows us to explore the impact of informative censoring to our estimates. Following the analysis of the above data we introduce an extension to Cox's partial likelihood, which will call "modified Cox's partial likelihood", based on the assumptions that censored times do contribute information about the parameters of interest. Finally we perform parametric bootstraps to assess the validity of our model and to explore up to what values of parameter δ our approximation holds.

This thesis examines the asymptotic behaviour of solution flows of certain stochastic differential equations utilising the theory of Lyapunov exponents. The approach is taken on two fronts. Initially flows are considered on compact manifolds that arise from embedding the manifold in some Euclidean space - the Gradient Brownian flow. In this case the existence of the Lyapunov exponents is known. Results are obtained for the sum of the exponents - which has the geometrical interpretation as the exponential rate of change of volume under the action of the flow - and for the largest exponent on generalised Clifford Tori and convex hypersurfaces. The situation on non-compact manifolds is then considered - where the existence of the exponents is uncertain due to the fact that the existence of a finite invariant measure is not guaranteed. However, by considering a stochastic mechanical system this problem is overcome and conditions for existence are obtained for both the Lyapunov spectrum and the sum' of the exponents. Some examples are then considered in the noncompact case. Finally in the Appendix a computational method for calculating the largest Lyapunov exponent on a hypersurface is considered.

This thesis, is concerned with the determination of the connected centralizers of semi-simple elements in a Chevalley group. To deal with this problem we shall use the recent work of R. Carter [6] and a new tool for the study of algebraic groups - the so called Brauer complex. This complex has been first defined by J. Humphreys [11] in the context of the modular representation theory of the finite Chevalley groups of universal type. Now, in our version, the Brauer complex can be also used for the ordinary representation theory of the finite Chevalley groups of adjoint type. For, Deligne and Lusztig in their fundamental work [9] have constructed for these groups certain families of irreducible complex representations whose degrees can be obtained if we know what subgroups of the finite Chevalley groups are the connected centralizers of semi-simple elements.

This thesis consists of two unconnected parts. In the first part we study the Cr-conjugacy classes of flows on two dimensional manifolds whose flow lines near a fixed point are diffeomorphic to the level surfaces of a Morse function near a critical point and which have no holonomy. We show how these can be decomposed into those in which every flow line is closed and those in which no flow line is closed. In the remainder of the thesis we consider the latter case and show that then the number of limit sets is finite. He describe their geometry and use the techniques of ergodic theory to show that the number of asymptotic cycles is finite in certain cases. We show that the asymptotic cycles are classifying for flow of this type on a manifold of genus 2 with exactly two non- trivial limit sets. Finally we give some new examples on manifolds of higher genus both of flows in which every flow line is [] and of flows in which each limit set is a closed, nowhere dense set which meets any transverse interval in a perfect set. In the second part we consider differential operators which are functionally associated to Hiemannian manifolds and which satisfy a regularity condition that arise in the proof of the index theorem via the heat equation. These are classified in terms of the On-equivariant representations of the general linear group.

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The likelihood for the parameters of a generalised linear mixed model involves an integral which may be of very high dimension. Because of this apparent intractability, many alternative methods have been proposed for inference in these models, but it is shown that all can fail when the model is sparse, in that there is only a small amount of information available on each random effect. The sequential reduction method developed in this thesis seeks to fill in this gap, by exploiting the dependence structure of the posterior distribution of the random effects to reduce dramatically the cost of approximating the likelihood in models with sparse structure. Examples are given to demonstrate the high quality of the new approximation relative to the available alternatives. Finally, robustness of various estimators to misspecification of the random effect distribution is considered. It is found that certain marginal composite likelihood estimators are not robust to such misspecification in situations in which the full maximum likelihood estimator is robust, providing a counterexample to the notion that composite likelihood estimators will always be at least as robust as the maximum likelihood estimator under model misspecification.

Fano varieties are one of important classes in the classi cation of algebraic varieties. In this thesis, we mainly study problems on deformations of Fano varieties motivated by the classi cation problems. In particular, we study Fano 3-folds with terminal singularities and weak Fano manifolds. In Chapter 2, we prepare necessary notions on deformation theory and singularities. We also explain about the orbifold Riemann-Roch formula and computation of numerical data of a K3 surface with Du Val singularities and a Q-Fano 3-fold. In Chapter 3, we study the deformation theory of a Q-Fano 3-fold with only terminal singularities. First, we show that the Kuranishi space of a Q-Fano 3-fold is smooth. Second, we show that every Q-Fano 3-fold with only "ordinary" terminal singularities is Q-smoothable, that is, it can be deformed to a Q-Fano 3-fold with only quotient singularities. Finally, we prove Q-smoothability of a Q-Fano 3-fold assuming the existence of a Du Val anticanonical element. As an application, we get the genus bound for primary Q-Fano 3-folds with Du Val anticanonical elements. In Chapter 4, we prove that a weak Fano manifold has unobstructed deformations. For a general variety, we investigate conditions under which a variety is necessarily obstructed. In Chapter 5, we investigate a certain coboundary map associated to a 3-fold terminal singularity which is important in the study of deformations of singular 3-folds. We determine when this map vanishes. As an application, we prove that almost all Q-Fano 3-folds have Q-smoothing. We also treat the Q-smoothability problem on Q-Calabi-Yau 3-folds. In Chapter 6, we study deformations of a pair of a Q-Fano 3-fold X with its elephant D E |-KXZ. We prove that, if X has only quotient singularities and there exists D with only isolated singularities, there is a deformation X -> [triangle]1 of X over a unit disc such that |-KXt| has a Du Val element for t E [triangle]1\0. We also give several examples of Q-Fano 3-folds without Du Val elephants.

Surface active agents (surfactants) reduce the surface tension of fluid interfaces and, via surface tension gradients, can lead to tangential forces resulting in the Marangoni effect. Biological systems take advantage of their impact on fluids with interfaces, but surfactants are also important for industrial applications such as processes of emulsification or mixing. Surfactants can be soluble in at least one of the fluid phases and the exchange of surfactants between the bulk phases and the fluid interfaces is governed by the process of adsorption and desorption. One can compute the interfacial surfactant density from the bulk surfactant density by assuming that the interface is in equilibrium with the adjacent bulk phase and imposing a closure relation (known as adsorption isotherm) between the two quantities. The assumption (known as instantaneous adsorption) is valid when the process of adsorption to the interface is fast compared to the kinetics in the bulk phases. However, it is not valid in the context of ionic surfactant systems, or when the diffusion is not limited to a thin layer. In this thesis, we derive two types of mathematical models for two-phase flow with a soluble surfactant that can account for both instantaneous and non-instantaneous adsorption. The first type is a sharp interface model, in which the interface is modelled by moving hypersurfaces. While the second type is a phase field model, in which the interface is a region of small, nonzero thickness where there is some microscopic mixing of the two fluids. Both types of models are shown to satisfy energy inequalities which guarantee thermodynamical consistency. Via a formal asymptotic analysis, we show the phase field models are related to sharp interface models in the limit that the interfacial width tends to zero. Flexibility with respect to the choice of bulk and surface free energies allows us to realise various isotherms and relations of state between surface tension and surfactant. We present some numerical simulations to support the asymptotic analysis and display the effectiveness of the our approach. As a first step towards an analysis of our models, we consider sharp interface and phase field models for soluble surfactants in a static situation. The surfactant equations become a linear elliptic coupled bulk-surface partial differential equation, and our main result is the rigorous convergence of the weak solution of the phase field models to the weak solution of the sharp interface models.

In the Seel-Strack contest (Seel and Strack [2013]), n agents each privately observe an independent copy of a drifting Brownian motion which starts above zero. Each agent chooses when to stop the process she observes, and the winner of the contest is the agent who stops her Brownian motion at the highest value amongst the set of agents. The objective of each agent is to maximise her probability of winning the contest. We will give a new derivation of the results of Seel and Strack [2013] based on a Lagrangian approach. This approach facilitates our analysis of the variants of the Seel-Strack problem. We will consider a generalisation of the Seel-Strack contest in which the observed processes are independent copies of some time-homogeneous diffusion. We will use a change of scale to reduce this contest to a contest in which the observed processes are diffusions in natural scale. It turns out that, unlike in the Seel-Strack problem, the way of breaking ties becomes important. Moreover, we will discuss an extension of the Seel-Strack contest to one in which an agent is penalised when her strategy is suboptimal, in the sense that her chosen strategy does not win the contest, but there existed an alternative strategy which would have resulted in victory. We will see that different types of penalty have different effects. Seel and Strack [2013] studied the asymmetric 2-player contest in which the observed processes start from different constants. We will redrive their results using the Lagrangian method and then study a general asymmetric n-player contest. We will find that some results in the 2-player contest do not hold for the general n-player contest. In a symmetric 2-player contest, the Seel-Strack model assumes that the observed processes start from the same positive constant. We will extend the results to the case where the starting values of the processes are independent non-negative random variables that have the same distribution.

This thesis covers theoretical and practical aspects of Bayesian inference and survival analysis, which is a powerful tool for the analysis of the time until a certain event of interest occurs. This dissertation focuses on non-standard models inspired by features of real datasets that are not accommodated by conventional models. Materials are divided in two parts. The first and more extended part relates to the development of flexible parametric lifetime distributions motivated by the presence of anomalous observations and other forms of unobserved heterogeneity. Chapter 2 presents the use of mixture families of lifetime distributions for this purpose. This idea can be interpreted as the introduction of an observation-specific random effect on the survival distribution. Two families generated via this mechanism are studied in Chapter 3. Covariates are introduced through an accelerated failure times representation, for which the interpretation of the regression coefficients is invariant to the distribution of the random effect. The Bayesian model is completed using reasonable (improper) priors that require a minimum input from practitioners. Under mild conditions, these priors induce a well-defined posterior distribution. In addition, the mixture structure is exploited in order to propose a novel method for outlier detection where anomalous observations are identified via the posterior distribution of the individual-specific random effects. The analysis is illustrated in Chapter 4 using three real medical applications. Chapter 5 comprises the second part of this thesis, which is motivated in the context of university outcomes. The aim of the study is to identify determinants of the length of stay at university and its associated academic outcome for undergraduate students of the Pontificia Universidad Católica de Chile. In this setting, survival times are defined as the time until the end of the enrollment period, which can relate to different reasons - graduation or dropout - that are driven by different processes. Hence, a competing risks model is employed for the analysis. Model uncertainty is handled through Bayesian model averaging, which leads to a better predictive performance than choosing a unique model. The output of this analysis does not account for all features of this complex dataset yet it provides a better understanding of the problem and a starting point for future research. Finally, Chapter 6 summarizes the main findings of this work and suggests future extensions.