This thesis examines N = 2 Super-Yang-Mills theory where the low-energy effective action of the theory is governed by a holomorphic function called the prepotential. The Seiberg-Witten solution of the theory determines the prepotential in terms of an complex curve and, once we compactify the theory on a circle, we will examine the identification of this complex curve with the spectral curve of the Calogero-Moser integrable system. Since the supersymmetry restricts the perturbative contributions to the prepotential, the results we gain are exact. Further, they are independent of the compactification radius. The generalization to the quiver models, with gauge group SU(N)k, is introduced along with the spin generalization of the integrable system. The massive vacua of these theories have been determined previously, here we examine the case of a specific gauge group in order to determine the complete phase structure, including the massless vacua. We then move on to determining contributions coming from instantons to the prepotential of the theory with gauge group SU(N). We see how by lifting the theory onto 5 dimensions the functional integral on the instanton moduli space is realized as a quantum mechanical sigma-model with the moduli space as a target. However, just such a model is shown to calculate a particular index of the manifold, in this case a particular equivariant index since the space has isometries. We account for the non-compact nature of the moduli space by removing boundary terms and then calculate explicit results in the case of SU(2).

This thesis presents results for spectral functions extracted from imaginary-time correlation functions obtained from Monte Carlo simulations using the Maximum Entropy Method (MEM). The advantages this method are (i) no a priori assumptions or parametrisations of the spectral function are needed, (ii) a unique solution exists and (iii) the statistical significance of the resulting image can be quantitatively analysed. The Gross Neveu model in d = 3 spacetime dimensions (GNM3) is a particularly interesting model to study with the MEM because at T = 0 it has a broken phase with a rich spectrum of mesonic bound states and a symmetric phase where there are resonances. Results for the elementary fermion, the Goldstone boson (pion), the sigma, the massive pseudoscalar meson and the symmetric phase resonances are presented. UKQCD Nf = 2 dynamical QCD data is also studied with MEM. Results are compared to those found from the quenched approximation, where the effects of quark loops in the QCD vacuum are neglected, to search for sea-quark effects in the extracted spectral functions. Information has been extract from the difficult axial spatial and scalar as well as the pseudoscalar, vector and axial temporal channels. An estimate for the non-singlet scalar mass in the chiral limit is given which is in agreement with the experimental value of Mao = 985 MeV.

In this thesis, we examine the construction and characteristics of generalised reflection matrices, within the a_1^(1), a_2^(1) and a_2^(2) integrable affine Toda field theories. In doing so, we generalise the existing finite-dimensional reflection matrices because our construction involves the dressing of an integrable boundary with a defect. Within this framework, an integrable defect's ability to store an unlimited amount of topological charge is exploited, therefore all generalised solutions are intrinsically infinite-dimensional and exhibit interesting features. Overall, further evidence of the rich interplay between integrable defects and boundaries is provided. It is hoped that the generalised solutions presented in this thesis are potential quantum analogues of more general classical integrable boundary conditions.

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The main aim of this thesis is to gain insights into the decidability problem for hhpb, and to analyse when it coincides with hpb; less technically, we might say, to analyse the power of the interplay between concurrency, causality, and conflict. We first examine the backtracking condition, and see that it has to dimensions: the number of transitions over which one may backtrack, and the number of backtracking moves. These dimensions translate into two hierarchies of bisimilarities; we find that both of them are strict, and that each of their levels is decidable. Our second approach is to analyse which behavioural properties of concurrent systems are crucial to the increased power of hhpb. After establishing a minimum of behavioural situations necessary to keep hpb and hhpb distinct, we study two aspects of the interplay of causality, concurrency, and conflict: a kind of synchronization, and the notion of confusion. With the help of a composition and decomposition result we prove that synchronization is essential for non-coincidence (for finite-state systems). However, we show that this is not so for confusion, which disproves the long-standing conjecture that hpb and hhpb coincide for confusion-free systems. We continue by studying two structural system classes with promising behavioural properties. First we consider basic parallel processes (BPP), with a suitable partial order semantics. These systems are infinite-state, but they restrict synchronization. Using the tableau technique, we prove the decidability and coincidence of hpb and hhpb for ‘full standard form’ BPP. The two bisimilarities do not coincide for the complete BPP class, but we separately achieve decidability of both (a known result for hpb, but not for hhpb). The second structural class is (safe) free-choice systems, an important class in Petri net theory. These systems have a controlled interplay of concurrency and conflict, and thereby exclude confusion. Having shown that hpb and hhpb do not coincide here, we identify another interesting candidate: live and strict free-choice systems. For this class, we prove that an auxiliary bisimilarity coincides with a ‘half’-hereditary version. As a consequence we put forward the coincidence of hpb and hhpb for live buffered and strict free choice systems: the only known positive result for a class with a reasonable amount of interplay between concurrency, causality and conflict.

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The major topic of this thesis is the interpretation (as a three-dimensional velocity field) of the changing intensity pattern induced by a smoothly deforming Lambertian surface of uniform albedo illuminated by a distant point light source. A constraint is derived which shows how the changing intensity pattern induced by such a deforming surface is locally constrained by the three-dimensional motion of that surface. This constraint, the 'Intensity Rate Constraint', a partial differential equation in the normal component of surface velocity, contains no terms relating to the tangential components of surface velocity, hence the problem of determining the three-dimensional motion is <i>ill-posed</i>. The application of an additional constraint on the motion, (implemented in the form of a stretch-based regulariser) is proposed. This enables certain psychologically significant classes of three-dimensional velocity field over the surface to be estimated veridically from the image intensity rate, the velocity field along the boundary and static information. This technique is successfully tested on synthetic data in experiments requiring at least ten times greater accuracy in intensity measurement than is commonly available. The thesis concludes with a suggested technique for the interpretation of smoothly deforming space-curve motion.

In this thesis we investigate four mechanisms for encoding hierarchical structures in distributed representations that are suitable for processing in connectionist systems: Tensor Product Representation, Recursive Auto-Associative Memory (RAAM). Holographic Reduced Representation (HRR), and Binary Spatter Code (BSC). IN these four schemes representations of hierarchical structures are either learned in a connectionist network or constructed by means of various mathematical operations from binary or real-value vectors. It is argued that the resulting representations carry structural information without being themselves syntactically structured. The structural information about a represented object is encoded in the position of its representation in a high-dimensional representational space. We use Principal Component Analysis and constructivist networks to show that well-separated clusters consisting of representations for structurally hierarchical objects are formed in the representational spaces of RAAMs and HRRs. The spatial structure of HRRs and RAAM representations supports the holistic yet structure-sensitive processing them. Holistic operations on RAAM representations can be learned by back-propagation networks. However, holistic operators over HRRs, Tensor Products, and BSCs have to be constructed by hand, which is not a desirable situation. We propose two new algorithms for learning holistic transformations of HRRs from examples. These algorithms are able to generalise the acquired knowledge to hierarchical objects of higher complexity than the training examples. Such generalisations exhibit systematicity of a degree which, to our best knowledge, has not yet been achieved by any other comparable learning method.

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