In the mathematical study of general relativity it is valuable to consider the classification and properties of the metric connection and Riemannian curvature derived from the spacetime metric. The aim of this thesis is to present various results obtained regarding (i) the holonomy groups of simply-connected spacetimes (ii) the problem of deciding if a connection is a metric connection (iii) a sub-class of curvature preserving symmetries and (iv) geodesic preserving symmetries. The holonomy group associated with a spacetime provides a global classification scheme for the metric connection. After the necessary background and definitions, chapter three provides an analysis of the holonomy groups possible in many of the commonly studied classes of spacetime. While every metric is uniquely associated with its metric connection, the converse is not true in general. The question of when a symmetric connection and the curvature derived from it can be associated with a Lorentz signature metric can be addressed via the existence of integrability conditions involving a metric candidate. In chapter four such an analysis is performed in terms of a classification scheme based on the rank and eigenbivector structure of the curvature. The problem is resolved in that the maximum number of such integrability conditions required to guarantee that the connection is a metric connection is given for each (non-trivial) curvature type. Transformations of a spacetime that preserve some geometric quantity or structure are particularly important in general relativity. In chapter five some results regarding transformations that preserve the curvature and its covariant derivatives are presented for spacetimes in which the preservation of the curvature alone is not enough to completely specify the nature of the curvature symmetry.

In a large class of space-times, the specification of the curvature tensor components Rabcd in some coordinate domain of the space-time uniquely determines the metric up to a constant conformal factor. The purpose of this thesis is to investigate the spaces where the metric is not so determined, and to look at the determination of the metric when the components of the derivatives of the Riemann tensor (one index up) are also specified, with special reference to the role of the infinitesimal holonomy group (ihg). In chapter one we set up the mathematical background, describing the Weyl and Ricci tensor classifications, and defining holonomy. In chapter two we look at spaces with Riemann tensors of low rank. This leads us on to decomposable spaces and the connection between decomposable spaces and relativity in three dimensions. We examine the connection between decomposability and the ihg, and relate this to the Weyl and Ricci tensor classifications. In chapter three we discuss the problem of determination of the metric by the Riemann tensor alone, and give a brief review of the history of the problem. In chapter four we go on to look at the determination of the metric by the curvature and its derivatives. It is shown that, with the exception of the generalised pp-waves, we only need look as far as the first derivatives of the Riemann tensor to obtain the best determination of the metric, unless the Riemann tensor is rank 1, when the second derivatives may also be required. The form of the metric ambiguity, the ihg and Petrov types are determined in each case. These results are then reviewed in the final chapter.

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This thesis is concerned with various aspects of perturbed conformal field theory and the methods used to calculate finite-size effects of integrable quantum field theories. Nonlinear integral equations are the main tools to find the exact ground-state energy of a quantum field theory. The thermodyamic Bethe ansatz (TBA) equations are a set of examples and are known for a large number of models. However, it is also an interesting question to find exact equations describing the excited states of integrable models. The first part of this thesis uses analytical continuation in a continuous parameter to find TBA like equations describing the spin-zero excited states of the sine-Gordon model at coupling β(^2) = 16π/3. Comparisons are then made with a further type of nonlinear integral equation which also predicts the excited state energies. Relations between the two types of equation are studied using a set of functional relations recently introduced in integrable quantum field theory. A relevant perturbation of a conformal field theory results in either a massive quantum field theory such as the sine-Gordon model, or a different massless conformal field theory. The second part of this thesis investigates flows between conformal field theories using a nonlinear integral equation. New families of flows are found which exhibit a rather unexpected behaviour. The final part of this thesis begins with a review of a connection between integrable quantum field theory and properties of certain ordinary differential equations of second- and third-order. The connection is based on functional relations which appear on both sides of the correspondence; for the second-order case these are exactly the functional relations mentioned above. The results are extended to include a correspondence between n(^th) order differential equations and Bethe ansatz system of SU(n) type. A set of nonlinear integral equations are derived to check the results.

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In examining some exact solutions of Einstein's field equations, the main approach used here is to study the geodesic motion of light, and sometimes test particles. Difficulties in solving the geodesic equations are avoided by using computer algebra to solve the equations numerically and to plot them in two- or three-dimensional diagrams. Interesting features revealed by these diagrams may then be investigated analytically. Application of this technique to the van Stockum solution for a rotating dust cylinder and to Bonnor's rotating dust cloud seems to reveal different constraints on the spatial distribution of geodesics with different parameters. Analysis then continns that, in the highest mass van Stockum case, null geodesics in the vacuum exterior are radially confined according to their initial conditions. Null geodesics plotted in Bonnor's dust cloud seem to be repelled before they can reach the centre. Although there is no event horizon, analysis reveals a central region which cannot be penetrated by light from spatial infinity and from which light cannot escape to spatial infinity. The gravitomagnetic clock effect is studied in van Stockum spacetime. The effect is found to be frame dependent and can be reduced to zero by a suitable coordinate transformation.

A theory of superfluidity (S.F.) is developed from first principles using two novel concepts, (1) that of a 'superfluid Ensemble' (S.E.) i.e. a 'Restricted Ensemble' constructed from a 'Separable Phase Space' (S.P.S.) admitting independent configurations, at least one set of which are statistically equivalent. (2) The notion of 'Dynamical Equivalence' (D.E.), satisfied if and only if (i) all dynamical symmetries are rearranged (not broken) for two Lagrangian formulations of the same problem and if (ii) the expectation values of all the constants of motion are the same, even if their functional expressions are not. The dynamical variables (d.v.) of the S.P.S. are defined from the ('q' and 'c' number) fields of the most general 'Linear Coherent State Representation', more general than those of Glauber and Bogoljubov-Valatin combined. Three independent pairs of d.v. are obtained. D.E. is proven for the Ideal Bose Gas and for a non-linear, interacting zero order Bose problem (I.Z.O.P.). An exact relation is obtained from the action principle, ensuring the cancellation of 'low and high order dangerous diagrams'. From this it follows that D.E. for the exact interacting problem must be demonstrable at infinite order of perturbation, in the finite volume limit. The I.Z.O.P. is posed in the Random Phase Approximation (R.P.A.), free from 'anomalous averages' and solved for the three branches of the excitation spectrum in a pure state description; the lowest branch is gapless, whilst the upper two coincide and show a gap. The standard strategy of linearization is found to be faulty. The partition functions for both superfluid and non-superfluid ensembles are obtained for the I.Z.O.P. in the R.P.A. The coincident upper two branches (in a pure state description) split into a band in thermal equilibrium for the superfluid ensemble, in agreement with an upper band recently observed experimentally. O.D.L.R.O. is found in the second reduced density matrix, but ruled out in the first. Integral equations are obtained in thermal equilibrium - for the I.Z.O.P. in the R.P.A.; which differ, however, from those of existing approaches for the same problem. Most existing theories of S.F. are in fact shown not to predict superfluid behaviour. The present theory is applicable to arbitrary Bose or Fermi systems, whether superfluid or not. O.D.L.R.O. is found to be sufficient for SF. No a priori assumption is made as to the occurrence of Bose-Einstein condensation, its existence being here contingent on the solution of the integral equation; in any case, it is not to be associated with O.D.L.R.O. or S.F.

We study the measure-theoretic and topological entropies of diffeo- morphisms of surfaces. In the measure theoretic case we look for relations between Lyapunov exponents, Hausdorff dimension and the entropy of ergodic invariant measures. First we describe the concept of measure-theoretic entropy in topological terms and discuss a general method of relating it with the Hausdorff dimension of ergodic invariant measures. This is done in a general setting, namely Lipschitz maps of compact metric spaces. The rest of the thesis is mainly directed to the study of diffeomorphisms of surfaces. To apply a refinement of this general method to C2 diffeomorphisms of surfaces we need Pesin's theory of non-uniform hyperbolicity, which we review in Chapter 2. Also in this chapter, we prove that the topological pressure of certain functions can be approximated by its restriction to the hyperbolic sets of the diffeomorphisms. This result is used in Chapter 3 to study the size of sets of generic points of ergodic measures supported on hyperbolic sets. The main result of Chapter 3 is that if µ is an ergodic Borel f- invariant measure for a diffeomorphism f:M -► M of a surface M . Then, provided the entropy hµ(f) > 0 , the Hausdorff dimension of the set of generic points of µ is at least 1 + hµ (f)xtµ , where xtµ is the positive Lyapunov exponent of µ. In Chapter 4 we prove that if the family of local stable manifolds is Lipschitz, then for an ergodic measure µ, hµ (f)=HD(µϩx)X+ µ for almost every X ε M . Here f is as 1n Chapter 3 and µϩx is a quotient measure defined by the family of local stable manifolds. Finally, Chapter 5 is devoted to study the topological entropy of homoclinic closures by 'counting1 homoclinic orbits.

A technique based on the Wigner distribution phase space interpretation of quantum mechanics is developed to obtain a transport theory capable of describing the high-field, inhomogeneous electronic transport in collision dominated sub-micron semiconductor devices. The problems associated with the general construction of quantum phase space distributions are considered using the Marcinkiewicz theorem which suggests that any defined quantum distribution which is both real and bounded must also, in general, be allowed to assume negative values. A pair of exact coupled transport equations for the one-electron and one-phonon Wigner distribution functions is obtained using multiple imaginary time Greens function techniques starting from a model Hamiltonian incorporating electron-electron and electron-phonon interactions as well as a coupling to externally applied space-and time-dependent electric and pressure fields. These exact equations exhibit a non-locality which may be interpreted in terms of the uncertainty relations of quantum mechanics. Assumptions are made (the many body correlation effects being approximated using functional derivative techniques) which restrict the resulting equations to the transition regime between the bulk scale device adequately modelled by Boltzmann transport and the microscopic region driven exclusively by boundary influences. These approximate equations maintain a non-locality due to the finite extent of a collision process thus allowing the collision integrals to become explicitly dependent on the driving field. Derived self-consistently within the collision integrals is a dynamical nonequilibrium screening of the interaction potentials which is also explicitly field-dependent. The field-dependence of the effective interactions is analysed for several model screening functions in two-and three-dimensional electron assemblies for a range of system parameters in GaAs. The results indicate that conventional screening overestimates the efficiency of the shielding process and that the action of a strong field within a collision event descreens the effective interaction potentials.