Quantum entanglement is the key property that makes quantum information theory different from its classical counterpart and is also a valuable physical resource with massive potential for technological applications. However, our understanding of entanglement is still far from com plete despite intense research activities. Like other physical resources, the first step towards exploiting them fully is to know how to quantify. There are many reasons to focus on the en tanglement of continuous-variable states since the underlying degrees of freedom of physical systems carrying quantum information are frequently continuous, rather than discrete. Much of the effort has been concentrated on Gaussian states, because these are common as the ground or thermal states of optical modes. Within this framework, many interesting topics have been stud ied and some significant progress made. Nevertheless, non-Gaussian states are also extremely important this is especially so in condensed-phase systems, where harmonic behaviour in any degree of freedom is likely to be only an approximation. So far, there is little knowledge about the quantification of entanglement in non-Gaussian states. This thesis aims to contribute to the active field of research in quantum entanglement by introducing a new approach to the analysis of entanglement, especially in continuous-variable states, and shows that it leads to the first systematic quantification of the (local) entanglement in arbitrary bipartite non-Gaussian states. By applying this local approach, many new insights can be gained. Notably, local entanglements of systems with smooth wavefunctions are fully characterised by the derived simple expressions, provided the wavefunction is known. The local (logarithmic) negativity of any two-mode mixed states can be directly computed from the closed-form formulae given. For multi-mode mixed states, this approach provides a scheme that permits much simpler numerical computation for quantifying entanglement than is generally possible from directly computing the full entanglement of the system.

I introduce and define Quantum Chromodynamics. I describe various well-known nonperturbative techniques for calculating quantities from the theory and discuss their merits and deficiencies. I then motivate and define a non-relativistic formulation (NRQCD) of the theory. I discuss the mechanics of the extraction of numbers from numerical simulations, and present general arguments as to the expected form of these data. I present results and details of their extraction from simulations of heavy-heavy and heavy-light mesons using NRQCD. I compare these results with those from other calculations and with experimental data, where they exist. I make suggestions for further work. An appendix contains details of the code used in the simulation together with the input parameters of the simulation.

In this thesis we study Fourier-Mukai transforms for complex projective surfaces. Extending work of A.I. Bondal and D.O. Orlov, we prove a theorem giving necessary and sufficient conditions for a functor between the derived categories of sheaves on two smooth projective varieties to be an equivalence of categories, and use it to construct examples of Fourier-Mukai transforms for surfaces. In particular we construct new transforms for elliptic surfaces and quotient surfaces. This enables us to identify all pairs of complex projective surfaces having equivalent derived categories of sheaves. We also derive some general properties of Fourier-Mukai transforms, and gives examples of their use. The main applications are to the study of moduli spaces of stable sheaves. In particular we identify many such moduli spaces on elliptic surfaces, generalising results of R. Friedman.

In 1957 Chern, Hirzebruch and Serre proved that the signature of the total space of a fibration of manifolds is equal to the product of the signatures of the base space and the fibre space if the action of the fundamental group of the base space on the fibre is trivial. In the late 1960s Kodaira, Atiyah and Hirzebruch independently discovered examples of fibrations of manifolds with non-multiplicative signature. These examples are in the lowest possible dimension where the base and fibre spaces are both surfaces. W. Meyer investigated this phenomenon further and in 1973 proved that every multiple of four occurs as the signature of the total space of a fibration of manifolds with base and fibre both surfaces. Then in 1998 H. Endo showed that the simplest example of such a fibration with non-multiplicative signature occurs when the genus of the base space is 111. We will prove two results about the signature of fibrations of Poincaré spaces. Firstly we show that the signature is always multiplicative modulo four, extending joint work with A. Ranicki and I. Hambleton on the modulo four multiplicativity of the signature in a <i>PL</i>-manifold fibre bundle. Secondly we show that if the action of the fundamental group of the base space on the middle-dimensional homology of the fibre with coefficients in Z₂ is trivial, then the signature is multiplicative modulo eight. The main ingredient of the first result is the development of absolute Whitehead torsion; this is a refinement of the usual Whitehead torsion which takes values in the absolute group <i>K</i>₁(<i>R</i>) of a ring <i>R,</i> rather than the reduced group ?₁(<i>R</i>). When applied to the algebraic Poincaré complexes of Ranicki the “sign” term (the part which vanishes in ?₁(<i>R</i>)) will be identified with the signature modulo four. We prove a formula for the absolute Whitehead torsion of the total space of a fibration and a simple calculation yields the first result. The second result is proved by means of an equivalent Pontrjagin square, a refinement of the usual one. We make use of the Theorem of Morita which states that the signature modulo eight is equal to the Arf invariant of the Pontrjagin square. The Pontrjagin square of the total space of the bundles concerned is expressed in terms of the equivalent Pontrjagin square on the base space and this allows us to compute the Arf invariant.

The thesis has two main themes: some homological results on the Conley index are put into a more natural homotopical context; and degenerate isolated critical points are studied from the point of view of the Conley index theory. A critical point of a smooth function is a rest point of the induced gradient flow so, if isolated, has a Conley index; this is the <I>k</I>-sphere, <I>S<SUP>k</SUP></I>, if the point is non-degenerate with Morse index <I>k</I>. The question as to which spaces can occur as the Conley index of a (degenerate) critical point is addressed. It is shown that the Lusternik Schnirelmann category of an invariant set (in general) is at least that of its Conley index less one. Consequently, the Conley index of a critical point can have Lusternik Schnirelmann category at most two. Conversely, the suspension of any finite <I>CW</I>-complex is shown to be the Conley index of a critical point of some function. A degenerate critical point may be broken up into a collection of non-degenerate points by perturbing the function in a neighbourhood of the point. The Conley index of the degenerate point is used to study this collection - homotopy invariants are introduced that give lower bounds on the number of critical values obtained in this manner. Despite its homotopical definition, much of the previous work using methods of algebraic topology with the Conley index concentrates on the homological properties of the index. This thesis, exploiting the definition of the Conley index as the homotopy type of a pointed space, studies the implications a flow on a space has on the homotopy of that space. It is shown that <I>S</I>-duality relates the forward and reverse flow Conley indices, generalising and clarifying a known Poincaré duality theorem on the homology of the indices.

In this thesis we study plane-wave limits and M-theory vacua. We consider several hereditary properties of the plane-wave limit but focus on that of homogeneity. We show that a sufficient condition for a plane-wave limit along a particular geodesic of any spacetime to be homogeneous is that the geodesic be homogeneous. On reductive homogeneous spacetimes we reduce the calculation to a set of algebraic formulae by two different methods; the first uses the covariant description of the plane-wave limit [Blau, O’Loughlin, Papadopoulos. JHEP,01:047,2002] and the second employs a non-adapted coordinate description of the plane-wave limit. We study how the homogeneous structure on a reductive homogeneous spacetime behaves under the plane-wave limit and apply our formulae to many relevant examples. We then consider supersymmetric M-theory vacua and the Lie supersymmetry superalgebra on these backgrounds. We show that those backgrounds which preserve more than 24 of the supersymmetries are necessarily homogeneous and provide some evidence that this boundary is sharp. The symmetric square of the spinor bundle of an 11-dimensional spacetime is isomorphic to a particular bundle of differential forms, this can be used to interpret Killing spinors as differential forms satisfying a system of first order equations [Gauntlett, Gutowski, Pakis. JHEP,12:049,2003]. We use this technique to investigate both the geometric and algebraic nature of the 24+ supergravity solutions, in particular those which are plane-waves. Finally we consider some more general homogeneous supergravity solutions, including homogeneous 5-dimensional supergravity.

The Fisher-Bulmer infinitesimal model is the classical mathematical model of phenotypic evolution in quantitative genetics. I show that it arises from certain population genetic models in the limit as the number of genes contributing to the phenotypic trait tends to infinity. The conditions which these population genetic models must satisfy are discussed, in particular, the restrictions which are placed on the strength and the form of linkage disequilibrium (statistical associations between variation in different genes) in the population. Other situations, where the Fisher-Bulmer model does not arise in the limit of infinitely many genes, are also considered. Alternative limiting models are investigated. One of these, here referred to as the 'rare alleles model', applies when each gene mostly occurs in only one form, with the alternative forms occurring much more rarely. A method is developed for analysing the behaviour of the rare alleles model. This is used to investigate the balance between mutation and selection against deleterious alleles, and the selection pressure which this generates on the outcrossing rate.

James Ax showed that, in each characteristic, there is a natural bijection from the space of complete theories of pseudo-finite fields, in first order logic, to the set of conjugacy classes of procyclic subgroups of the absolute Galois group of the prime field. I show that when the set of subgroups of a profinite group is considered to have the Vietoris (a.k.a. hyperspace, finite, exponential, neighbourhood) topology the aforementioned bijection is a homeomorphism. Thus we can think of the space of complete theories of pseudo-finite fields of a given characteristic as being encoded in the absolute Galois group of the prime field. I go on to show that there is a natural way of encoding the whole space of complete theories of pseudo-finite fields (i.e. without dependence on characteristic) in the absolute Galois group of the rationals. To do this I use: the theory of the algebraic <i>p</i>-adics; the relationship between the absolute Galois group of the <i>p</i>-adics and the absolute Galois group of the field with <i>p</i> elements; the structure of the absolute Galois group of the <i>p</i>-adics given by Iwasawa; Krasner’s lemma for henselian fields; and the Vietoris topology. At the same time, we consider the theory of algebraically closed fields with a generic automorphism (<i>ACFA</i>). By taking the theory of the fixed field, there is a surjective (but not injective) map from the space of complete theories of <i>ACFA</i> to the space of complete theories of pseudo-finite fields. For the space of complete theories of <i>ACFA,</i> there is also a bijective Galois correspondence, in each characteristic, given by restricting the automorphism to the algebraic closure of the prime field. I show that this correspondence is a homeomorphism and that there is an analogous way of encoding the whole space in the absolute Galois group of the rationals.

This work studies the modulation and kinematics of short waves riding on non-uniform velocity fields (solitary waves and long waves) and achieves theoretical and experimental conclusions. For the interactions between short waves and solitary waves, short waves and long waves, this research shows that the wavenumber, frequency and amplitude of short waves riding on solitary waves and long waves are strongly modulated. It also demonstrates that the maximum values of the modulated short wavenumber, frequency, and amplitude always occur at the crests of solitary waves and long waves. By increasing either the amplitude of solitary waves or the steepnesses of long waves the main conclusion--- that the modulated short wavenumber, frequency, and amplitude increase on the crest of solitary waves and long waves--- is achieved. The kinematics of two component waves (short waves and long waves) has been measured by PIV (Particle Image Velocimetry). Comparison of the results with Fourier theory and various stretching methods is also carried out. The mechanisms of the modulation of short waves riding on solitary waves or long waves, as studied in this thesis, provides a useful base line for work on more general and complex local water wave breaking.

This thesis is mainly concerned with the problem of generating gauge configurations for use in Monte-Carlo lattice QCD calculations that include the effects of dynamical fermions. Although algorithms to do this have been in existence for some time the computing power necessary for their application at a scale where physically relevant results can be obtained has only recently become available, so these large scale dynamical simulations are still a new feature of lattice QCD. The emphasis here is on the new experiences gained from the design, implementation and development of a particular dynamical gauge configuration algorithm, and from its initial use in production. The intention is that this will facilitate future computations where the effects of dynamical fermions in QCD can be systematically explored, and the further development of better algorithmic techniques. The first chapter outlines those features of lattice gauge theory computations that are salient to this work, concentrating particularly on the motivations for and consequences of going beyond the quenched approximation and on the properties of Markov processes used in the generation of gauge configurations. The second chapter introduces the main types of algorithm employed for dynamical gauge configuration production, viz. the multiboson algorithm and Hybrid Monte-Carlo, and explains how they work. In chapter 3 the implementation of the chosen algorithm (Generalised Hybrid Monte-Carlo) is described, along with various algorithmic investigations, coding developments, performance evaluation, and a description of the procedures used in the verification of the code. Finally, some results are presented from the first large scale production runs on the Cray-TBE which attempt to put the algorithm work into a physical context.