The research presented in this thesis is on the topic of the Bayesian approach to statistical inference. In particular it focuses on the analysis of mixture models. Mixture models are a useful tool for representing complex data and are widely applied in many areas of statistics (see, for example, Titterington et al. (1985)). The representation of mixture models as missing data models is often useful as it makes more techniques of inference available to us. In addition, it allows us to introduce further dependencies within the mixture model hierarchy leading to the definition of the hidden Markov model and the hidden Markov random field model (see Titterington (1990)). Chapter 1 introduces the main themes of the thesis. It provides an overview of variational methods for approximate Bayesian inference and describes the Deviance Information Criterion for Bayesian model selection. Chapter 2 reviews the theory of finite mixture models and extends the variational approach and the Deviance Information Criterion to mixtures of Gaussians. Chapter 3 examines the use of the variational approximation for general mixtures of exponential family models and considers the specific application to mixtures of Poisson and Exponential densities. Chapter 4 describes how the variational approach can be used in the context of hidden Markov models. It also describes how the Deviance Information Criterion can be used for model selection with this class of model. Chapter 5 explores the use of variational Bayes and the Deviance Information Criterion in hidden Markov random field analysis. In particular, the focus is on the application to image analysis. Chapter 6 summarises the research presented in this thesis and suggests some possible avenues of future development. The material in chapter 2 was presented at the ISBA 2004 world conference in Viña del Mar, Chile and was awarded a prize for best student presentation.

This thesis investigates problems in a number of different areas of graph theory. These problems are related in the sense that they mostly concern the colouring or structure of the underlying graph. The first problem we consider is in Ramsey Theory, a branch of graph theory stemming from the eponymous theorem which, in its simplest form, states that any sufficiently large graph will contain a clique or anti-clique of a specified size. The problem of finding the minimum size of underlying graph which will guarantee such a clique or anti-clique is an interesting problem in its own right, which has received much interest over the last eighty years but which is notoriously intractable. We consider a generalisation of this problem. Rather than edges being present or not present in the underlying graph, each is assigned one of three possible colours and, rather than considering cliques, we consider cycles. Combining regularity and stability methods, we prove an exact result for a triple of long cycles. We then move on to consider removal lemmas. The classic Removal Lemma states that, for n sufficiently large, any graph on n vertices containing o(n^3) triangles can be made triangle-free by the removal of o(n^2) edges. Utilising a coloured hypergraph generalisation of this result, we prove removal lemmas for two classes of multinomials. Next, we consider a problem in fractional colouring. Since finding the chromatic number of a given graph can be viewed as an integer programming problem, it is natural to consider the solution to the corresponding linear programming problem. The solution to this LP-relaxation is called the fractional chromatic number. By a probabilistic method, we improve on the best previously known bound for the fractional chromatic number of a triangle-free graph with maximum degree at most three. Finally, we prove a weak version of Vizing's Theorem for hypergraphs. We prove that, if H is an intersecting 3-uniform hypergraph with maximum degree D and maximum multiplicity m, then H has at most 2D+m edges. Furthermore, we prove that the unique structure achieving this maximum is m copies of the Fano Plane.

This thesis examines the completability of an incomplete set of m-row orthogonal Latin rectangles (MOLRm) from a set theoretical viewpoint. We focus on the case of two rows, i.e. MOLR2, and define its independence system (IS) and the associated clutter of bases, which is the collection of all MOLR2. Any such clutter gives rise to a unique clutter of circuits which is the collection of all minimal dependent sets. To decide whether an incomplete set of MOLR2 is completable, it suffices to show that it does not contain a circuit therefore full knowledge of the clutter of circuits is needed. For the IS associated with 2-row orthogonal Latin rectangles (OLR2) we establish a methodology based on the notion of an availability matrix to fully characterise the corresponding clutter of circuits.

Graph theory is the study of networks of objects (called vertices) joined by links (called edges). Since many real world problems can be represented by a graph, graph theory has applications in areas such as sociology, chemistry, and computing. In this thesis, a number of open problems in graph theory are studied.

The nonequilibrium Keldysh formalism has been used to study the spin transport effects found in magnetic multi-layered nanostructures. We formulate a new methodology based on Landauer and show it to be in very good qualitative agreement with Keldysh. However, our theory provides more information regarding the physics of these effects because it allows us to calculate the contributions of individual electrons incident from either side of a junction as well as the contributions within a single layer that are incident on and reflected from an adjacent interface. Chapter 1 provides a consolidated introduction to spintronics in magnetic multilayer nanostructures (the key focus of this thesis) including phenomena such as giant magnetoresistance (GMR), tunneling magnetoresistance (TMR) and current induced switching of magnetization. We then describe how to calculate the local charge and spin current in the direction perpendicular to the layers of an arbitrary magnetic layer structure using the nonnonequilibrium Keldysh formalism before introducing our Landauer approach to investigating the transport of charge and spin current in these magnetic multilayers using the simplest paraboilic band model for electrons in each layer. In Chapter 2 we formulate our approach by defining the general solution to the wave equation for a given layer in a system in terms of the angle by which the spin polarization is rotated in-plane in that layer and the generalized wave vectors for each electron spin band. We determine a general transfer matrix that enables us to solve explicitly the coefficients of the wave functions in each layer of any general multi-layered system before defining an expression for the in-plane and out-of-plane spin current components in terms of these wave functions before detailing our Landauer formalism to calculating the local spin current in a realistic system consisting of ferromagnets with a finite exchange splitting and appropriate boundary conditions. We apply our formulated approach in Chapter 3 to a set of collinear spin problems whereby the two magnetic layers in our general multilayer junction (consisting of two ferromagnets separated by a non-magnetic spacer layer) have their rotated magnetizations either ferromagnetically or anti-ferromagnetically aligned (parallel/anti-parallel to the net magnetization). Our analytical results provide the necessary conditions for optimising tunneling magnetoresistance (TMR) and show how a `switching' effect can be used to control it. We achieve this by calculating analytically in the ferromagnetic configuration the necessary conditions to support a 100% transmission success rate in one spin channel whilst making it very difficult for transmission to occur in the other spin channel. However, we show conclusively that re-aligning the magnetization to the anti-ferromagnetic configuration under the same conditions will make it very difficult for transmission to occur in either spin channel. In Chapter 4 we investigate the spin current in a general five layer junction and show that a zero out-of-plane spin current in the nonmagnetic spacer exists only when perfect symmetry is introduced because the contribution from the left cancels exactly the contribution from the right. We identify a number of properties within the nonmagnetic layers and observe the effect of varying the angle of rotated magnetization and width of the polarizing magnet on the spin current components in the nonmagnetic layers. In Chapter 5 we define the appropriate boundary conditions for our Landauer approach before investigating analytically the origin of out-of-plane spin current in the nonmagnetic spacer in Chapter 6 by looking specifically at an interface between a semi-infinite magnet and a semi-infinite nonmagnet and obtaining qualitative insight into the out-of-plane spin current found in a non-magnetic spacer sandwiched between two finite ferromagnets. In this final chapter we also calculate numerically the effect of an additional insulating barrier on our classical junction consisting of a nonmagnetic spacer sandwiched between two ferromagnets. We compare our results for the charge and spin current in the nonmagnetic spacer to those obtained previously using the Kelydysh formalism before showing for the first time the physical dependence on multiple magnetic interfaces of the out-of-plane spin current in a nonmagnetic spacer and how the out-of-plane spin current in the spacer can be large even when the charge current and the in-plane spin current are both negligibly small.

Intuitive arguments involving standard quantum mechanical uncertainty relations suggest that at length scales close to the Planck length, strong gravity effects limit the spatial as well as temporal resolution smaller than fundamental length scale, leading to space-space as well as spacetime uncertainties. Space-time cannot be probed with a resolution beyond this scale i.e. space-time becomes "fuzzy" below this scale, resulting into noncommutative spacetime. Hence it becomes important and interesting to study in detail the structure of such noncommutative spacetimes and their properties, because it not only helps us to improve our understanding of the Planck scale physics but also helps in bridging standard particle physics with physics at Planck scale. Our main focus in this thesis is to explore different methods of constructing models in these kind of spaces in higher dimensions. In particular, we provide a systematic procedure to relate a three dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing non-commutative spaces. The representations for the corresponding operators obey algebras whose uncertainty relations lead to minimal length, areas and volumes in phase space, which are in principle natural candidates of many different approaches of quantum gravity. We study some explicit models on these types of non-commutative spaces, in particular, we provide solutions of three dimensional harmonic oscillator as well as its decomposed versions into lower dimensions. Because the solutions are computed in these cases by utilising the standard Rayleigh-Schrodinger perturbation theory, we investigate a method afterwards to construct models in an exact manner. We demonstrate three characteristically different solvable models on these spaces, the harmonic oscillator, the manifestly non-Hermitian Swanson model and an intrinsically non-commutative model with Poschl-Teller type potential. In many cases the operators are not Hermitian with regard to the standard inner products and that is the reason why we use PT -symmetry and pseudo-Hermiticity property, wherever applicable, to make them self-consistent well designed physical observables. We construct an exact form of the metric operator, which is rare in the literature, and provide Hermitian versions of the non-Hermitian Euclidean Lie algebraic type Hamiltonian systems. We also indicate the region of broken and unbroken PT -symmetry and provide a theoretical treatment of the gain loss behaviour of these types of systems in the unbroken PT -regime, which draws more attention to the experimental physicists in recent days. Apart from building mathematical models, we focus on the physical implications of noncommutative theories too. We construct Klauder coherent states for the perturbative and nonperturbative noncommutative harmonic oscillator associated with uncertainty relations implying minimal lengths. In both cases, the uncertainty relations for the constructed states are shown to be saturated and thus imply to the squeezed coherent states. They are also shown to satisfy the Ehrenfest theorem dictating the classical like nature of the coherent wavepacket. The quality of those states are further underpinned by the fractional revival structure which compares the quality of the coherent states with that of the classical particle directly. More investigations into the comparison are carried out by a qualitative comparison between the dynamics of the classical particle and that of the coherent states based on numerical techniques. We find the qualitative behaviour to be governed by the Mandel parameter determining the regime in which the wavefunctions evolve as soliton like structures. We demonstrate these features explicitly for the harmonic oscillator, the Poschl-Teller potential and a Calogero type potential having singularity at the origin, we argue on the fact that the effects are less visible from the mathematical analysis and stress that the method is quite useful for the precession measurement required for the experimental purpose. In the context of complex classical mechanics we also find the claim that "the trajectories of classical particles in complex potential are always closed and periodic when its energy is real, and open when the energy is complex", which is demanded in the literature, is not in general true and we show that particles with complex energies can possess a closed and periodic orbit and particles with real energies can produce open trajectories.

This thesis is based on the work done during my PhD studies and is roughly divided in two independent parts. The first part consists of Chapters 1 and 2 and is based on the two papers Cavaglià et al. [2011] and Cavaglià & Fring [2012], concerning the complex PT-symmetric deformations of the KdV equation and of the inviscid Burgers equation, respectively. The second part of the thesis, comprising Chapters 3 and 4, contains a review and original numerical studies on the properties of certain quasilinear dispersive PDEs in one dimension with compacton solutions. The subjects treated in the two parts of this work are quite different, however a common theme, emphasised in the title of the thesis, is the occurrence of nonsemilinear PDEs. Such equations are characterised by the fact that the highest derivative enters the equation in a nonlinear fashion, and arise in the modeling of strongly nonlinear natural phenomena such as the breaking of waves, the formation of shocks and crests or the creation of liquid drops. Typically, nonsemilinear equations are associated to the development of singularities and non-analytic solutions. Many of the complex deformations considered in the first two chapters are nonsemilinear as a result of the PT deformation. This is also a crucial feature of the compacton-supporting equations considered in the second part of this work. This thesis is organized as follows. Chapter 1 contains an introduction to the field of PT-symmetric quantum and classical mechanics, motivating the study of PT-symmetric deformations of classical systems. Then, we review the contents of Cavaglià et al. [2011] where we explore travelling waves in two family of complex models obtained as PT-symmetric deformations of the KdV equation. We also illustrate with many examples the connection between the periodicity of orbits and their invariance under PTsymmetry. Chapter 2 is based on the paper Cavaglià & Fring [2012] on the PTsymmetric deformation of the inviscid Burgers equation introduced in Bender & Feinberg [2008]. The main original contribution of this chapter is to characterise precisely how the deformation affects the gradient catastrophe. We also point out some incorrect conclusions of the paper Bender & Feinberg [2008]. Chapter 3 contains a review on the properties of nonsemilinear dispersive PDEs in one space dimension, concentrating on the compacton solutions discovered in Rosenau & Hyman [1993]. After an introduction, we present some original numerical studies on the K(2, 2) and K(4, 4) equations. The emphasis is on illustrating the different type of phenomena exhibited by the solutions to these models. These numerical experiments confirm previous results on the properties of compacton-compacton collisions. Besides, we make some original observations, showing the development of a singularity in an initially smooth solution. In Chapter 4 , we consider an integrable compacton equation introduced by Rosenau in Rosenau [1996]. This equation has been previously studied numerically in an unpublished work by Hyman and Rosenau cited in Rosenau [2006]. We present an independent numerical study, confirming the claim of Rosenau [2006] that travelling compacton equations to this equation do not contribute to the initial value problem. Besides, we analyse the local conservation laws of this equation and show that most of them are violated by any solution having a compact, dynamically evolving support. We confirm numerically that such solutions, which had not been described before, do indeed exist. Finally, in Chapter 5 we present our conclusions and discuss open problems related to this work.

The majority of species are under predatory risk in their natural habitat and targeted by predators as part of the food web. Through the process of evolution by natural selection manifold mechanisms have emerged to avoid predation. As Fisher argued, it is the ubiquitous presence of anti-predator adaptations which shows that predation plays a significant role in the ecology and evolution of ecosystems. These ecosystems are intrinsically complex which derives from the high entanglement of organisms interacting in competitive relationships: the prey is part of the predator’s environment and vice versa. As a result, the evolution of predator and prey is best described as a co-evolutionary process of predator-prey systems. It is common to classify anti-predator adaptations into ‘primary defences’ and ‘secondary defences’. Primary defences operate before an attack by reducing the frequency of detection or encounter with predators. Secondary defences, which are used after a predator has initiated prey-catching behaviour, commonly involve the expression of toxins or deterrent substances which are not observable by the predator. Hence, the possession of such secondary defence in many prey species comes with a specific signal of that defence. This pairing of a toxic secondary defence and a conspicuous primary defence is known as aposematism. Previous models mainly focused on questions of the initial evolution of aposematism in ancestrally cryptic populations. However, the field has a renewed interest in questions beyond the initial evolution of aposematism such as: how conspicuous should a signal be, and how much should be invested into secondary defence? Moreover, which factors influence evolutionary stability of aposematic solutions. Within this context, the role of co-evolution and the mechanisms of aversive learning are at the heart of the current research. On the one hand, to explain stability and persistence of aposematic signals requires a theory of co-evolution of defence and signals. On the other hand, the role of the predator and details of the predator’s aversive learning process gained renewed interest of the field. As the selective agent, aversive learning is an important aspect of predator avoidance and of the co-evolution of predator-prey systems. In the first chapter, this thesis will review the literature on aposematism and introduce the different selective pressures acting on aposematic prey. The thesis will then identify open questions of interest around aposematism. In the second chapter the thesis will focus on the perspective of the prey. The introduction of a game theoretical model of co-evolution of defence and signal will be followed by an adaptation of the model for finite populations. In finite populations, investigating the co-evolution of defence and signalling requires an understanding of natural selection as well as an assessment of the effects of drift as an additional force acting on stability. In the third chapter the thesis will adopt the perspective of the predator. It will introduce reinforcement learning as an normative framework of rational decision making in a changing environment. An analysis of the consequences of aposematism in combination with aversive learning on the predator’s diet and energy intake will be followed by a lifetime model of optimal foraging behaviour in the presence of aposematic prey in the fourth chapter. In the last chapter I will conclude that the predator’s aversive learning process plays a crucial role in the form and stability of aposematism. The introduction of temporal difference learning allows for a better understanding of the specific details of the predator’s role in aposematism and presents a way to take the discipline forward.

Let F_n denote the free group of rank n with free basis X. The palindromic automorphism group PiA_n of F_n consists of automorphisms taking each member of X to a palindrome: that is, a word on X that reads the same backwards as forwards. We obtain finite generating sets for certain stabiliser subgroups of PiA_n. We use these generating sets to find an infinite generating set for the so-called palindromic Torelli group PI_n, the subgroup of PiA_n consisting of palindromic automorphisms inducing the identity on the abelianisation of F_n. Two crucial tools for finding this generating set are a new simplicial complex, the so-called complex of partial pi-bases, on which PiA_n acts, and a Birman exact sequence for PiA_n, which allows us to induct on n. We also obtain a rigidity result for automorphism groups of right-angled Artin groups. Let G be a finite simplicial graph, defining the right-angled Artin group A_G. We show that as A_G ranges over all right-angled Artin groups, the order of Out(Aut(A_G)) does not have a uniform upper bound. This is in contrast with extremal cases when A_G is free or free abelian: in these cases, |Out(Aut(A_G))| < 5. We prove that no uniform upper bound exists in general by placing constraints on the graph G that yield tractable decompositions of Aut(A_G). These decompositions allow us to construct explicit members of Out(Aut(A_G)).

In finite dimensions, Hopf algebras have a very nice duality theory, as the vector space dual of a finite-dimensional Hopf algebra is also a Hopf algebra in a canonical way. This breaks down in the infinite-dimensional setting, as here the dual need not be a Hopf algebra. Instead, one chooses a subalgebra of the vector space dual called the finite dual. This subalgebra is always canonically a Hopf algebra. In this thesis, we aim to better understand the finite dual by trying to understand how the finite dual of a crossed product relates to the finite duals of its components. We start by investigating what the assignment sending a Hopf algebra to its finite dual does to functions. Unlike in the finite-dimensional case, this is no longer a contravariant exact monoidal functor and might not even be a functor at all. However, many of the results true thanks to this in finite dimensions still always hold, while we can find necessary and sufficient conditions for others to hold as well as specific situations in which they are always true. Crossed products generalise the notion of a smash product, which can be viewed as the Hopf algebra equivalent of the semidirect product. Many Hopf algebras of interest can be written as crossed products. We study the finite dual of such a product and find numerous results when assuming conditions such as one of the components being finite-dimensional or the crossed product being a smash product. These can be combined for strong statements about the finite dual under certain assumptions. Finally, we consider Noetherian Hopf algebras which are finite modules over central Hopf subalgebras. Many of these algebras decompose as crossed products, so that we can use our previous results to study them. However, we also find results that are true without assuming such a decomposition. This allows us to calculate the finite duals of numerous examples, including a quantised enveloping algebra at a root of unity and all the prime affine regular Hopf algebras of Gelfand-Kirillov dimension one with prime PI degree.