This thesis investigates the effect of time interval omission on properties and distributions of stochastic processes obtained by aggregating the state space of the original process into a finite number of subsets. Time interval omission is a phenomenon of missing sojourns of the aggregated process that are shorter than a certain duration, which results in an apparent, rather than the ideal, sequence of sojourns. This problem arises in the stochastic modelling of ion channels, where the underlying Markov process is not observable and only the apparent aggregated process can be reconstructed from experimental data. The methodology employed in the thesis includes embedding a Markovian sequence that contains all information about the apparent aggregated process <I>Z (t), </I>into the original process. Distributions of <I>Z (t)</I> are deduced from characteristics of an intermediary semi-Markov apparent gateway process <I>Y (t)</I>, determined by this sequence. We also construct several fundamental matrix functions of time such that probability distributions of <I>Y (t)</I> can be expressed via these functions and a unified efficient algorithm for their calculation exists for all values 0 < <I>t</I> < ∞. Three particular problems are analysed in greater detail. The multi-level model of time interval omission leads to a new enhanced definition of apparent sojourns and to the introduction of the concept of indeterminacy. Investigation of the piece-wise homogeneous model of the underlying process provides new practical tools for the analysis of pulse and single jump experiments. Explicit probability distributions are obtained for the developed time interval omission model of structural bursts. Along with the theoretical results for probability distributions of <I>Z (t)</I>, software is developed as a part of the research that provides a flexible tool for calculating various characteristics of <I>Y (t)</I> and <I>Z (t)</I>. Numerical examples considered in the thesis show the applicability of the developed methods and demonstrate non-trivial effects of time interval omission.

The Ashkin-Teller and q-state Potts models, on the square and triangular lattices, are studied by approximate methods. A series analysis is used to investigate the variation, with q, of the internal energy and specific heat of the Potts model. Estimates of the specific heat exponents are given. A number of real-space renormalization groups are then used to study both models. The decimation and cluster expansion values, for the critical temperature and exponents, imply that larger clusters are needed in order to obtain reliable quantitative results, although even the simplest of these transformations gave an accurate qualitative description of the critical surface of the Ashkin-Teller model. The lower bound variational renormalization group is modified to allow an investigation of the two-layer using representation of the square lattice Ashkin-Teller model. Wherever exact results are available, the transformation proves to be remarkably accurate. A line of fixed points is found, giving exponents which vary continuously with interaction strength. A comparison with the corresponding results for the eight vertex model indicates that, for the same ratio, of four to two-spin couplings, the models have equal exponents.

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This thesis proposes a framework, based on formal methods, for the design, specification, and analysis of 1. neurocomputers as seen from the perspective of hardware, and 2. real world tasks as seen from the perspective of a user, in order to give useful definitions of correctness. We apply the theory of synchronous concurrent algorithms to a number of well known neural network models. This gives us a detailed, and precise description of the spatial and temporal behaviour of the network's operation. Such an operational semantics is necessary for defining a notion of correctness for such sytems. In order to formally specify a task and define its correctness we extend the standard treatment of synchronous concurrent algorithm correctness to the case of approximate correctness, and using mathematical tools from universal algebra and topology, examine the semantics of a number of natural 'approximate specifications'. We apply this framework to two case studies: the requirements analysis of a hospital waiting list management system, and the implementation of a hypothetical anti-ballistic missile firing system.

The finite element method has played an important role in helping the understanding of physics; from material mechanics to plasma flow, and is an extremely versatile tool for faster and better prototyping of today's industrial products, ranging from sub-micron semiconductor devices to large scale flight vehicles and reservoir dams. The outstanding power of the finite element method lies in its capability to solve geometrically complicated problems. However, this capability can only be fulfilled by an appropriately constructed mesh. With the recent emergence of the adaptive finite element method, users are relieved from the difficulties involved in appropriate/optimal mesh design and an automatic adaptive finite element analysis seems within reach. However, the realisation of adaptive finite element methods requires extensive theoretical and numerical development, together with, in order to properly integrate them into a smoothly operating system, the redesign of system philosophy and infrastructures. it is this aspect of the finite element method that makes a modern finite element system drastically different from the more traditional mesh-based ones. This thesis is on the design and development of such an automated, adaptive finite element simulation system. The emphasis is on its automation and adaptivity. Central to the system is the geometry-based philosophy. The system comprises two crucial procedures, namely, model discretisation and accuracy assessment. Mesh generation and mesh adaptation techniques are systematically reviewed. A geometry-based automatic 3D mesh generator, based on the 2-stage scheme of the unstructured approach exploiting the novel Delaunay simplexification algorithm has been researched and successfully developed. A mesh adaptator has also been developed to assume the responsibility of mesh adaptation. The mesh adaptator is a combination of the regeneration-based and node-based schemes of the <i>h</i>-adaptation approach. Other supporting modules such as the discretisation controller, automatic attribute assigner and solution mapper are also developed to form the complete model discretisation procedure.

This work is concerned with the development of reliable and versatile computational tools for the numerical simulation of two-dimensional heat conduction and incompressible and compressible laminar fluid flow problems. Issues related to adaptive techniques, discretisation methodologies (upwind or centred type) and the design of high-resolution shock-capturing schemes are investigated in this thesis. Three distinct research works have been pursued here. In the first work, attention is focused on the construction of an adaptive finite element procedure with mesh refinement, by mesh enrichment, in time and space, and with automatic time stepping for the heat conduction problem in a stationary medium. The Galerkin finite element method and the Euler-backward time marching scheme are used as the basis to obtain the steady-state and transient approximate solutions. Particular emphasis concentrates on the design of the adaptive strategy and the combined influence of time and spatial adaptation. The second task is concerned with the derivation of adaptive remeshing strategies for both steady and unsteady solution of the incompressible Navier-Stokes equations in primitive variables. A Petrov-Galerkin formulation, which automatically introduces streamline upwinding and allows equal order interpolation for all variables, combined with either an explicit or implicit time integration represents the general discretisation methodologies adopted. The adaptive redefinition of the mesh, the error estimate and specific features, such as the presence of singularities on the solution and accumulation of interpolation errors inherent to a transient remeshing, are carefully analysed with some remedies proposed to deal with such difficulties. In the final part of the thesis, the most relevant mathematical-physical properties of the first order hyperbolic model equations are discussed. The utilisation of upstream or centred discretisation and several ways to produce high-resolution schemes to deal with this class of problems are described and compared for one-dimensional test cases. With regard to upwind discretisation techniques, the most popular flux difference splitting, flux vector splitting and some recently proposed hybrid splitting methodologies are considered.

In this dissertation we argue the case for Jaynes' Information Theory approach to the foundations of Statistical Mechanics. In Chapter 1 a brief revue of the relevant aspects of classical and quantum mechanics is given. In particular, the methods in which probability theory is applied to mechanics are discussed the emphasis being on statistical indices, i.e., probability distribution functions (classical case) and density matrices (quantum case.). Also, Jaynes' 'Maximum Entropy Principle' is introduced; this principle advocates the use of the statistical index with the maximum microscopic entropy in Statistical Mechanics. The microscopic expressions for the classical and quantum entropies are, derived, in Chapter 2, from a few basic axioms. The method of proof is identical in both cases and two new axiomatic characterisation theorems (one classical, one quantum) for the microscopic entropies are proved. Chapter 3 is mostly an exposition of the literature on Jaynes' approach to Statistical Mechanics, or 'Information Thermodynamics' as it is also known. Also, a few of the mathematical gaps are either filled or examined from a different point of view. Chapter 3 is concluded by a discussion of the Second Law of Thermodynamics. In Chapter 4 Jaynes' approach is generalised to the case of nonequilibrium Statistical Mechanics. This net theory is based on the idea of expanding the statistical mean values evaluated using the solution of the Liouville equation in terms of those mean values evaluated using the statistical index with the maximum entropy via a 'mean value perturbation theory' actually developed by Jaynes. The Laws of 'Irreversible Thermodynamics' are then derived from a microscopic basis. Chapter 5 gives a critical analysis of the Information Theory approach. Also, possible directions which future work on this subject may take are indicated.

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In this thesis some solutions to classical Euclidean Yang-Mills theory are considered and presented with emphasis on self-dual SU(2) solutions. Chapter one is a brief introduction to the subject. In chapter two the possibility of solution via inversion of the dynamic equation to obtain in terms or Fuv and then imposing a self-consistency requirement is considered. Chapter three deals with the extension of Witten's method of solution through cylindrically symmetric ansatze to self- dual SU(3) theories. In Chapter four the invariances and Backlund-type transformations inherent in a self-dual SU(n) theory are investigated and these methods are used in Chapter five to present an analytic method for constructing all self-dual SU(2) solutions.

The approach of calculating non-perturbative effects in Quantum Chromodynamics by expanding about non-trivial classical solutions of the equation of motion is described. Some of the techniques required for this are developed in references [l,2,3] on which this thesis is predominantly based. The general self-dual solutions are discussed. With these as background fields the Green and massless Dirac functions are solved for arbitrary group representation. Then with the help of these the determinants and collective coordinate zero modes required for the first order quantum corrections ar'3 calculated.