The equations required to determine the motion of atoms and molecules in the molecular dynamics simulations are discussed. Atom-atom pairwise additive potentials are used to describe the van der Waals type forces between atoms and molecules. The Parrinello-Rahman description of the equations of motion of particles in a periodically repeating cell are formulated. Quaternions are used to describe the orientations of molecules. The melting transition in a two-dimensional cluster of krypton atoms is simulated. The structural changes on heating are examined in some detail and the possibility of a second-order melting transition and the existence of a hexatic phase between the solid and liquid phases is discussed. The formation of a solid from a rapidly cooled liquid-like phase is investigated. Simulations of krypton atoms in a periodically repeating cell and in a cluster are performed to examine the effects of boundary conditions. Simulations on mixtures of krypton and argon atoms, in equal proportion, in both a periodically repeating cell and in a cluster are performed. The solid structures formed are discussed in terms of glass and crystalline ordering. The orientational order-disorder transition and plastic phase in adamantane is studied. The molecular reorientation in the plastic phase is examined. Molecules occupy two possible orientations which are related by inversion. By restricting molecules to these two orientations an Ising-like model for adamantane is produced and the order-disorder phase transition is examined in a Monte Carlo simulation. Comparison of the molecular dynamics and Monte Carlo models is made. The disordered phase and reverse phase transition is discussed in terms of orientational frustration and domain formation. The suitability of using pairwise additive van der Waals type potential to represent the interaction between C<SUB>60</SUB> molecules is assessed. The plastic phase of C<SUB>60</SUB> is successfully modelled, however, the unit cell at low temperature predicted in the simulation is not consistent with experimental results. A discussion of other possible models of intermolecular potential is given.

This thesis is concerned with the efficient simulation of lattice QCD with dynamical fermions. We discuss two aspects of this theme, the tuning of existing algorithms and the investigation of novel algorithms. We begin with an introduction to lattice QCD and Monte Carlo Methods for its simulation. Particular emphasis is placed on the difficulties of the lattice formulation of fermion fields. We then continue with a description of the Hybrid Monte Carlo (HMC) algorithm, focusing on the conditions the algorithm must obey for correctness and on some of the numerical methods required for its implementation. We then discuss issues of reversibility and instability for the Molecular Dynamics part of HMC algorithm. After considering the source of instabilities in the context of free field theory we adopt a working hypothesis by which we can relate this instability to the case of lattice QCD. Our tuning studies of HMC attempt to investigate the behaviour of reversibility violations and simulation cost in the molecular dynamics with varying solver target residue <i>r.</i> We also investigate the onset of instabilities in the molecular dynamics while varying the solver residue <i>r</i> and the stepsize dt. Our second subject is the investigation of novel simulation algorithms. We consider the Parallel Tempering (PT) algorithm and its application to lattice QCD. We give an introduction to the algorithm and discuss the use of action matching technologies to tune the simulation parameters for maximal swap acceptance rates. We then discuss issues of cost for PT simulations by considering the CPU time needed by the algorithm for the estimation of the expectation value of an observable of interest and comparing this with the cost of reference HMC simulations. Finally we present some numerical results which indicate that we have a reasonable understanding of the algorithm but that we have not managed to maximise the acceptance rate through action matching. Due to large errors on our measured autocorrelation time we reserve judgements on the question of cost efficiency of the algorithm.

There has been much recent interest in obtaining analytic results for rule learning using a neural network. In this thesis the performance of a simple neural network model learning a rule from noisy examples is calculated using methods of statistical mechanics. The free energy for the model is defined and order parameters that capture the statistical behaviour of the system are evaluated analytically. A weight decay term is used to regularise the effect of the noise added to the examples. The network's performance is estimated in terms of its ability to generalise to examples from outside the data set. The performance is studied for a linear network learning both linear and nonlinear rules. The analysis shows that a linear network learning a nonlinear rule is equivalent to a linear network learning a linear rule, with effective noise added to the training data and an effective gain on the linear rule. Examining the dependence of the performance measures on the number of examples, the noise added to the data and the weight decay parameter, it is possible to optimise the generalisation error by setting the weight decay parameter to be proportional to the noise level on the data. Hence, a weight decay is not only useful for reducing the effect of noisy data, but can also be used to improve the performance of a linear network learning a nonlinear rule. A generalisation of the standard weight decay term in the form of a general quadratic penalty term or regulariser, which is equivalent to a general Gaussian prior on the network's weight vector, is considered. In this case an average over a distribution of rule weight vectors is included in the calculation to remove any dependence on the exact realisation of the rule.

This thesis investigates various aspects of the relation between the lattice and continuum formulations of quantum field theories, in particular QCD. The aim of this is to gain a better insight into the theory of QCD, and to be able to relate more accurately the numbers obtained from lattice simulations to experimental values for physical quantities. The first part of this thesis (chapters 1 and 2) gives a general introduction to quantum field theory, with emphasis on the lattice formulation of QCD. The first chapter describes the functional integral formulation of gauge theories and how it can be used to study these theories non-perturbatively by discretising the space-time variables. The second chapter discusses the principles behind the renormalisation of these theories. The Ward and Slavnov-Taylor identities that are preserved non-perturbatively, and can be invoked when renormalising the theory, are derived. The final part of this chapter discusses the renormalisation of composite operators, using both perturbative and non-perturbative methods. In particular, it is shown how the chiral Ward identities can be used to extract renormalisation constants for the axial and vector currents and the ratio of the scalar to the pseudoscalar density. In chapter 3, results for <I>Z<SUB>A</SUB>, Z<SUB>V</SUB> </I>and <I>Z<SUB>P</SUB>/Z<SUB>s</SUB> </I>at β = 6.2 are presented and their effects on calculations of physical quantities like decay constants are discussed. The final chapter investigates the quark-gluon vertex. The form factors of the off-shell vertex function, and the symmetries and Slavnov-Taylor identities that may be used to reduce these form factors, are discussed. I then outline a method for extracting the running coupling from the vertex function. This also includes a discussion of the quark and gluon field renormalisation.

We consider a complete, connected, non-compact Riemannian manifold <i>M</i> without boundary. We are principally interested in the case where the Laplacian has a spectral gap (i.e.σ(-Δ)).

The thesis begins with a summary of the classical theory of Banach function spaces, including the notions of saturation and associate norms along with the various well-known ideas of completeness. We then go on to establish some rather more "practical" results. In particular we look at the problem of establishing when certain subspaces, such as the simple or continuous functions, are dense in a Banach function space <i>L_p. </i>We shall see that our intention fails us slightly when considering continuous functions, requiring us to approach that idea of saturation from a slightly different angle. This interplay between topology, measure and norm is studied in more depth when we look at function norms over locally compact abelian groups, and results will be illuminated by reviewing well-known functions spaces such as Lorentz spaces and weighted <i>L^p</i> spaces. The chapter finishes with the idea of vector-valued function spaces. In the second chapter we motivate and develop the idea of mixed (or iterated) norms, as introduced for <i>L^p</i> spaces by Benedek and Panzone, before going on to identify dense subspaces and some other elementary results. We shall see that there are certain interesting measurability problems to address here which are not evident when considering <i>L^p</i> spaces. One rather technical highlight of this measure theory will be to make rigorous the canonical identification between most mixed norm spaces and vector-valued Banach function spaces. Motivated by a trivial application of Fubini's theorem which allows us to interchange two <i>L^p </i>norms, i.e. ||||<i>f(x, y)</i>||<i>_Lp(dy)</i>||<i>_Lp(dx)</i> = ||||<i>f(x, y)</i>||<i>_Lp(dx)</i>||<i>_Lp(dy)</i>, we then consider when interchanging two general mixed norms is bounded. Although there are some positive results we shall see that this idea fails in many cases. In particular we shall show that two iterated Lorentz <i>L^pq</i> norms can be interchanged if and only if <i>p = q.</i> In chapter three we study how the classical transference theorem of Coifman and Weiss can be generalised from <i>L^p</i> spaces to arbitrary rearrangement invariant spaces.

In contrast to the equilibrium case, there is no general theoretical framework for the treatment of many-body nonequilibrium systems. Therefore, simple model systems, amenable to detailed analytical or numerical treatment, are important in the understanding of such systems. Phase transitions and ordering are fundamental phenomena which have been extensively studied in equilibrium statistical physics. In this work, we investigate these phenomena in several model driven diffusive systems. We introduce the 'bus route model', a simple microscopic model in which jamming of a conserved driven species is mediated by the presence of a non-conserved quantity. Jamming proceeds via a strict phase transition only in a prescribed limit; outside this limit, we find sharp crossovers and transient coarsening. Next, we study flocking, the collective motion of many self-driven entities, in a one-dimensional lattice model. We find the existence of an ordered phase characterized by the presence of a single large 'flock' which exhibits stochastic reversals in direction. Using numerical finite-size scaling, we analyse the continuous phase transition from this ordered phase to a homogeneous phase and we calculate critical exponents. Finally, we study a model of shear-induced clustering; we find evidence for a discontinuous jamming transition with hysteresis. We also study the kinetics of jamming.

Combinatorial optimization problems arise throughout science, industry, and commerce. The demonstration that analogue neural networks could, in principle, rapidly find near-optimal solutions to such problems - many of which appear computationally intractable - was important both for the novelty of the approach and because these networks are potentially implementable in parallel hardware. However, subsequent research, conducted largely on the travelling salesman problem, revealed problems regarding the original network's parameter sensitivity and tendency to give invalid states. Although this has led to improvements and new network designs which at least partly overcome the above problems, many issues concerning the performance of optimization networks remain unresolved. This thesis explores how to optimize the performance of two neural networks current in the literature: the elastic net, and the mean field Potts network, both of which are designed for the travelling salesman problem. Analytical methods elucidate issues of parameter sensitivty and enable parameter values to be chosen in a rational manner. Systematic numerical experiments on realistic size problems complement and support the theoretical analyses throughout. An existing analysis of how the elastic net algorithm may generate invalid solutions is reviewed and extended. A new analysis locates the parameter regime in which the net may converge to a second type of invalid solution. Combining the two analyses yields a prescription for setting the value of a key parameter optimally with respect to avoiding invalid solutions. The elastic net operates by minimizing a computational energy function. Several new forms of dynamics using locally adaptive step-sizes are developed, and shown to increase greatly the efficiency of the minimization process. Analytical work constraining the range of safe adaptation rates is presented. A new form of dynamics, with a user defined step-size, is introduced for the mean field Potts network. An analysis of the network's critical temperature under these dynamics is given, by generalizing a previous analysis valid for a special case of the dynamics.

This thesis describes theoretical and numerical investigations into the resummation of soft singularities in some perturbative QCD cross sections. The form of the quark <i>MS</i> splitting functions and coefficient function that resums all soft singularities in DIS and Drell-Yan is obtained. After obtaining the DIS and Drell-Yan quark coefficient functions that resum leading logarithms and next to leading logarithms, PDF's and higher twist are fitted with and without resummation to the DIS CCFR, BCDMS, SLAC and H1 data sets. These PDF's are then used to determine the effect of resummation on predictions of Tevatron and future LHC Drell-Yan cross sections. Variation of the renormalization and factorization scales is performed to determine if the dependence on these two quantities is reduced in the case that resummation is used, both in the fits to DIS data and in the Drell-Yan predictions. Independently, an improved treatment of heavy quarks in the calculation of <i>F</i>₂ is investigated. A major simplification of the VFNS is described, and shown theoretically to be as perturbatively good as the VFNS. A fit of PDF's and higher twist to BCDMS and SLAC data is performed with variations of the threshold scales, to determine whether there is less threshold scale dependence in the VFNS than in the ZM-VFNS.

This thesis details a study of the vacuum structure of QCD using the tool of lattice gauge theory. Chapter 1 gives an introduction to path integrals, semi-classical approximations to path integrals, instantons, topological charge and instanton phenomenology. Chapter 2 introduces lattice gauge theory and the problems of studying topological charge on the lattice. The cooling method and its pitfalls are discussed and details are given of a study undertaken of under-relaxed cooling. In Chapter 3 the algorithms that were developed to study the instantons on the cooled configurations are discussed. Chapter 4 gives the results for the structure of the vacuum: size distributions, spatial distributions, correlations between charges, and scaling of distributions with the lattice spacing. Chapter 5 discusses an exploratory study of the low-lying eigenmodes of the Wilson-Dirac operator. The zero-modes of both the unimproved and improved operators on cold and heated instantons are calculated and the lattice artefacts investigated. Chapter 6 contains my conclusions and suggestions for future work.