Non-abelian target space duality and its effect on low energy string backgrounds is investigated, with particular emphasis on Taub-NUT space. The dual solutions are analysed and the physical effects of the duality are discussed, along with the interrelationship between successive applications of target space duality. A full quantum mechanical analysis of non-abelian duality is then considered. Conditions for the transformation to be anomaly free are presented, along with the quantum corrections to the standard non-abelian duality formulae. A low energy analysis is also performed. The study of <I>D</I>-branes creates an interest in massive sigma models. The supersymmetric abelian target space duals of such models are constructed. From the dual models the correct form of the dual potentials may be inferred. Finally the effects of the duality on the solitons of the massive model are discussed. The general problem of branes in twelve dimensions is studied. A supersymmetric action principle is formulated, but in order to do this the notions of supersymmetric must generalised. The resulting (2 + 2)- brane lives in a new type of superspace with two timelike directions. It is then shown that the traditional spacetime strings and membranes may be obtained by dimensional reduction. Once one has adapted to the possibility that there are many different supersymmetry theories, it is natural to attempt a classification. Details of such a procedure are given, and the corresponding superspaces constructed. In addition, complete brane scans are given for branes in any spacetime dimension and signature, and the algebras for which they are possible. The (2+2)-brane requires there to be two time directions. The concrete properties which any supersymmetric theory with two times must have are thus discussed. The BPS states of the two-time algebras are investigated in an analysis similar to that used in supergravity. Consistency conditions are discussed and the intersection properties of the branes detailed.

This thesis is about the quantum cosmology of supergravity theories, in particular <I>N</I> = 1 supergravity. In chapter 2, the Bianchi-IX model in <I>N</I> = 1 supergravity with a cosmological constant is investigated. This example is also restricted to the case of the <I>k</I> = +1 Friedmann universe. In chapter 3, the most general solution of the Lorentz constraints is found. Using this solution, <I>N </I>= 1 supergravity in the diagonal Bianchi-IX model is studied. The Hamilton-Jacobi<I> </I>equation is derived and completely solved. The Hartie-Hawking and wormhole actions are both found among the solutions. In chapter 4, the relation between the Chern-Simons functional and the no-boundary proposal is considered. The exponential of the Chern-Simons functional is the first known exact solution of quantum general relativity with a cosmological constant, being defined in the Ashtekar variables. However, it has turned out to be possible to show that the Chern-Simons functional and the no-boundary proposal are not related to each other in general relativity and <I>N</I> = 1 supergravity by considering perturbations around the <I>k</I> = +1 Friedmann universe. In chapter 5, the canonical formulation of <I>N</I> = 1 supergravity coupled to supermatter is presented. The supersymmetry and gauge constraints are derived. This model is then studied in the <I>k</I> = +1 Friedmann universe. It is found there are solutions in the case of a scalar multiplet. However, no non-trivial solution exists for a Yang-Mills multiplet, and it is explained why there is no physical state. Chapter 6 describes a brief investigation of the quantum cosmology of <I>N</I> = 2 and <I>N</I> = 4 supergravity theories. The thesis ends with concluding remarks and an indication of directions of future development.

The work described in this dissertation concerns the development of new methods for performing computer simulations of real materials from first principles or <I>ab initio </I>i.e. using the fundamental equations of quantum mechanics and only well-controlled approximations. In particular, these methods have been developed within the framework of density-functional theory and therefore lie in the realms of both quantum chemistry and computational condensed matter physics. The work is particularly concerned with methods which are efficient in the sense that the computational effort required scales only linearly with system-size (i.e. the volume or number of electrons) whereas traditional methods have scaled with the cube of the system-size which has restricted their range of applicability. The aim of this work is therefore to extend the scope of <I>ab initio </I>quantum-mechanical calculations beyond what is currently possible. Density-functional theory has traditionally been applied by making use of a mapping from the system of interacting electrons to a fictitious system of non-interacting particles. However, the need to maintain the mutual orthogonality of the wave functions of the fictitious system leads to the cubic scaling mentioned above, and is ultimately responsible for limiting the maximum size of systems which can be treated. Making use of a reformulation of the problem in terms of the single-particle density-matrix eliminates the need to work with the wave functions directly. Moreover, exploiting the short-ranged nature of the density-matrix leads in principle to a linear-scaling method. The dissertation tackles two issues which are relevant to obtaining practical schemes for performing linear-scaling calculations. Firstly a localised basis set is developed which is used to describe the density-matrix computationally. Analytic results for several key quantities required by the calculation are derived, namely the overlap, kinetic energy and non-local pseudopotential matrix elements. These results allow accurate calculation of the total energy of the system and have been implemented and tested computationally. Secondly, the dissertation discusses several methods for imposing the difficult non-linear idempotency constraint on the density-matrix.

This thesis is concerned with the development of a new <i>ab initio </i>Monte Carlo method for the evaluation of exact, basis set correlation energies. A simple set of rules acting on signed walkers allow for the simulation of the underlying imaginary-time Schrödinger equation in a finite space of Slater determinants. These rules return probabilistic events which are stochastically realised in each step of the algorithm. The antisymmetric space in which the dynamic operates precludes the emergence of Bosonic solutions, and the Fermion sign problem is countered without approximation by inclusion of annihilation events between walkers of different sign. The method is applied to many molecular systems described by common Gaussian basis sets. Single point calculations, binding curves, basis set expansions and detailed studies of ionisation potentials are included. In these investigations, the method is compared to several alternative quantum chemical methods as well as exact full configuration interaction results to asses its qualities. The method is found to significantly reduce the memory and CPU requirements compared to exact diagonalisation methods, and includes an effective parallelisation scheme which scales almost linearly up to thousands of processors. This extends the scope of exact multireference calculations and allows for larger systems than previously possible to be treated.

Percolation models are of interest both for the wide range of their physical applications and for the mathematical challenges which they present. The basic model is as follows. Starting from an infinite connected graph such as the hypercubic lattice, each edge is declared 'open' with probability <I>p</I> or 'closed' otherwise, independently of all others. The standard theory is primarily concerned with the existence (or not) of infinite connected components of the graph of open edges, [2]. Various extensions of the basic model have been studied in detail, [1, 2, 5]. In this work we extend the model in a direction which has received less attention: rather than studying <I>connected</I> components, we consider other graph properties analogous to connectivity. We explore this idea with particular reference to two such properties which have important physical applications, [3, 4]: <I>entanglement</I> and <I>rigidity</I>. Roughly speaking, the meaning of these terms is as follows. A graph in three-dimensional space is entangled if it cannot be 'pulled apart' when the edges are regarded as physical connections made of elastic. A graph is rigid if it cannot be 'deformed' when the edges are regarded as solid rods which can pivot at the vertices. We formalise these intuitive notions for both finite and infinite graphs. In the case of infinite graphs this involves overcoming interesting challenges which are related to the issue of boundary conditions. Having defined entanglement and rigidity formally, we consider entangled and rigid graphs in the percolation model. We prove that (under suitable conditions) there is a genuine phase transition for each, occurring at critical probabilities which differ from the usual critical probability for connectivity percolation. For <I>p</I> below the appropriate critical probability, we explore the size of finite entangled or rigid components. For <I>p</I> greater than the appropriate critical probability we study the question of uniqueness of the infinite entangled or rigid component. We prove several relevant theorems including uniqueness for entanglement for large <I>p</I>, and uniqueness for rigidity for almost all <I>p. </I>

In this thesis I consider bifurcations in lattice dynamical systems, primarily coupled map lattices and lattice differential equations. First I explain how certain coupled map lattices can be considered as cellular automata, on all or part of each of state- or parameter-space, and outline what can be gained from such a consideration. Then I consider bifurcations of fixed points in locally bistable coupled map lattices, introducing various analytical techniques for the study of piecewise-linear lattice dynamical systems, and showing how numerical techniques allow smooth systems to be investigated also. In particular, bifurcation diagrams and bifurcation sets for "kink" and "bump" fixed points are constructed, and these reveal that there will be small regions of parameter space in which no stable bump fixed points exist, though many such fixed points exist on each side of these regions. Strange behaviour occurs in such regions, with very long transients being observed. Bifurcations from homogenous fixed points of lattice dynamical systems are then investigated using the methods of equivalent bifurcation theory. As well as locating such bifurcations, I obtain information regarding branching directions, numbers of branches, and branch stabilities. The effect of the local dynamical units having odd symmetry is discussed. Some aspects of the theory of infinite-dimensional lattice dynamical systems are then considered. I discuss the connection between fixed points of these systems and orbits of a certain area-preserving map of the plane, and extend some earlier results regarding the symbolic dynamics of this map. I then introduce a shadowing-based technique which allows us to infer the existence of certain fixed points on the infinite lattice using our knowledge of similar fixed points on finite lattices.

The development of a fully unsupervised algorithm to achieve image segmentation is the central theme of this dissertation. Existing literature falls short of such a goal providing many algorithms capable of solving a subset of this highly challenging problem. Unsupervised segmentation is the process of identifying and locating the constituent regions of an observed image, while having no prior knowledge of the number of regions. The problem can be formulated in a Bayesian framework and through the use of an assumed model unsupervised segmentation can be posed as a problem of optimisation. This is the approach pursued throughout this dissertation. Throughout the literature, the commonly adopted model is an hierarchical image model whose underlying components are various forms of Markov Random Fields. Gaussian Markov Random Field models are used to model the textural content of the observed image's regions, while a Potts model provides a regularisation function for the segmentation. The optimisation of such highly complicated models is a topic that has challenged researchers for several decades. The contribution of this thesis is the introduction of new techniques allowing unsupervised segmentation to be carried using a single optimisation process. It is hoped that these algorithms will facilitate the future study of hierarchical image models and in particular the discovery of further models capable of more closely fitting real world data. The extensive literature surrounding Markov Random Field models and their optimisation is reviewed early in this dissertation, as is the literature concerning the selection of features to identify the textural content of an observed image. In the light of these reviews new algorithms are proposed that achieve a fusion between concepts originating in both these areas. Algorithms previously applied in statistical mechanics form an important part of this work. The use of various Markov Chain Monte Carlo algorithms is prevalent and in particular, the reversible jump sampling algorithm is of great significance. It is the combination of several of these algorithms to form a single optimisation framework that lies at the heart of the most successful algorithms presented here.

The simplest examples, braided lines and planes, yield the full supersymmetry transformation and super-Poincaré algebra in one and two dimensions, as well as the generalisation of this to the case of fractional supersymmetry. In terms of this deformed geometry these algebras/transformations have straightforward interpretations. For example, in one dimension, (fractional) supersymmetry transformations arise as translations along the braided line in the limit as its deformation parameter goes to a root of unity, so that a theory is (fractionally) supersymmetric if it is invariant under such translations. The various structures of which one makes use when working with supersymmetry, for example the supercharge, covariant derivative, Berezin integral and superspace, arise naturally in the context of this deformed geometry, as do their fractional analogues. The properties of the <I>q</I>-deformed bosonic oscillator algebra in the limit when <I>q</I> goes to a root of unity are also discussed, and this algebra is found to decompose into the direct product of an ordinary bosonic oscillator algebra and an anyonic oscillator algebra (fermionic when <I>q</I> = -1). The corresponding Fock space decomposition is also studied. Using these results and the Schwinger realisation of <I>U<SUB>q</SUB></I>(<I>sl</I>(2)), we obtain a similar decomposition for this algebra. Motivated by this results we study the complete <I>U<SUB>q</SUB></I>(<I>sl</I>(2)) Hopf algebra in the limit as its deformation parameter goes to a root of unity. This leads to new Hopf algebras which are (fractionally) supersymmetric analogues of <I>U<SUB>q</SUB></I>(<I>sl</I>(2)), and also to a novel point of view on the origin of intrinsic spin. The properties of the quantum hermitian matrices<I> L<SUB>q</SUB></I>(2) when <I>q</I> is a root of unity are also discussed, as are those for the braided form of <I>Lq</I>(2). These also have novel structure, and can be interpreted as providing (fractionally) supersymmetric analogues of <I>GL</I><SUB>q</SUB>(2). The physical interpretation of these results suggests that both four dimensional supersymmetry and the four dimensional Dirac equation have their origin in deformed geometry.

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