In this thesis an investigation is carried out into the uses and applications of orthogonal matrices in digital logic design. All but the first chapter are primarily concerned with the application of the Rademacher-Walsh matrix in this area. In the first chapter, it is shown how other matrices can be formed from the Walsh-Hadamard matrix, and comparisons are made between the properties of these matrices. Particular reference is made to where these other matrices have their major advantages and setbacks in logic design. The second chapter shows how the summation of certain coefficients of a given function, taken from the Rademacher-Walsh and other such complete spectra, to a specified maximum value, directly corresponds to a realisation in terms of MD, OR and majority gates. By the consideration of only a few coefficients, it is possible to define large factors of a logic function. The third chapter is devoted to the consideration of how to deal with the spectrum once the procedure described in chapter two has been carried out. Two design algorithms are described; one recursive and the other non-recursive, both of which incorporate spectral addition techniques. In chapter four, two associated design techniques are proposed, which make more optimal use of the information that the spectrum offers. The first of these considers the incorporation of don't care terms in the function specification, and in so doing overcomes some of the disadvantages of the techniques described in chapter three. The second technique is concerned with the design of multiplexer universal-logic- module circuits. In the final chapter, it is shown how spectral techniques can be applied to the diagnosis of faults in two-level combinational logic circuits. Throughout this thesis, an emphasis is placed on the practical rather than mathematical implications of the techniques.

This thesis is concerned initially with a stochastic logistic, (SL), analogue of VERHULST's equation to describe the growth of an isolated population in a limited environment. An essential feature of the SL process is that extinction will occur with probability 1, and so to begin with the analysis is concerned with the distribution of extinction times in terms of the basic parameters, which are the birth and death rates and the upper limit, N, to the integer state variable, j. The analysis then continues with an investigation into the form of the distribution of j at time t, and for small t an approximate analytic result for the related PGF is obtained. For larger t a LAPLACE Transform solution, inverted numerically, is found to be an effective method of evaluating the moments of this distributon, and this may be extended to very large t , if necessary, by considering the process conditional on non - extinction. A diffusion approximation to the SL process is then derived, and this not only extends the numerical methodology developed so far to the large N situation, but also provides a simple overall view of the SL process itself. The model is then modified so as to incorporate age and/or time dependent birth and death parameters, and theoretically it is shown that extinction will occur with probability 1 for a large class of such processes. The way in which the distribution of extinction times, in particular, and the overall behavior of the process, in general, depend on the additional parameters is then investigated by simulation, and in this respect it found useful to consider also the corresponding deterministic processes for which numerical solutions can be found. Finally the model is developed futher so as to describe the sheep populations of the islands of St. Kilda.

We consider problems of splitting and connectivity augmentation in hypergraphs. In a hypergraph G = (V +s, E), to split two edges su, sv, is to replace them with a single edge uv. We are interested in doing this in such a way as to preserve a defined level of connectivity in V . The splitting technique is often used as a way of adding new edges into a graph or hypergraph, so as to augment the connectivity to some prescribed level. We begin by providing a short history of work done in this area. Then several preliminary results are given in a general form so that they may be used to tackle several problems. We then analyse the hypergraphs G = (V + s, E) for which there is no split preserving the local-edge-connectivity present in V. We provide two structural theorems, one of which implies a slight extension to Mader’s classical splitting theorem. We also provide a characterisation of the hypergraphs for which there is no such “good” split and a splitting result concerned with a specialisation of the local-connectivity function. We then use our splitting results to provide an upper bound on the smallest number of size-two edges we must add to any given hypergraph to ensure that in the resulting hypergraph we have λ(x, y) ≥ r(x, y) for all x, y in V, where r is an integer valued, symmetric requirement function on V*V. This is the so called “local-edge-connectivity augmentation problem” for hypergraphs. We also provide an extension to a Theorem of Szigeti, about augmenting to satisfy a requirement r, but using hyperedges. Next, in a result born of collaborative work with Zoltán Király from Budapest, we show that the local-connectivity augmentation problem is NP-complete for hypergraphs. Lastly we concern ourselves with an augmentation problem that includes a locational constraint. The premise is that we are given a hypergraph H = (V,E) with a bipartition P = {P1, P2} of V and asked to augment it with size-two edges, so that the result is k-edge-connected, and has no new edge contained in some P(i). We consider the splitting technique and describe the obstacles that prevent us forming “good” splits. From this we deduce results about which hypergraphs have a complete Pk-split. This leads to a minimax result on the optimal number of edges required and a polynomial algorithm to provide an optimal augmentation.

This thesis is concerned with some practical and theoretical problems in the field of spatial stochastic point processes. The work described falls into two main areas: Chapters 1-3 describe some techniques for analysing spatial patterns, while, in Chapters 4-7, properties of certain stochastic models for spatial patterns are investigated, theoretically and empirically. We consider simple point processes in the plane, strictly stationary under rigid motions of the plane; many of the techniques and results can be generalised to more than two dimensions. Chapter 1 discusses the use of empty space in analysing spatial patterns; the method described estimates the probability that a randomly placed set of a given shape and size is empty of points of the pattern. Chapter 2 describes the use of histograms to estimate g(t)/2 for a spatial pattern; this technique is of use during the earlier stages of the analysis of a pattern. In Chapter 3, empty space techniques are generalised for the analysis of 'multitype' point patterns, and are compared with second-order methods for analysing such patterns. All the techniques of Chapters 1-3 are illustrated by examples. Chapters 4-7 deal mainly with models for 'hard core' point patterns. Chapter 4 contains a review of various models for hard core processes. Chapters 5 and 6 are concerned with spatial birth and death processes which are used to simulate realisations of certain models for point patterns, notably Kelly-Ripley models. In Chapter 5, coupling techniques from Markov Chain theory are used to obtain theoretical convergence results for such processes. Chapter 6 describes a computer algorithm for simulating hard core birth and death processes at high packing densities. In Chapter 7, this algorithm is modified to simulate the 'SSI' process and to investigate the complete packing of non-overlapping discs in a rectangular container.

The theory of finite automata provides a formal approach to the design of sequential circuits, assuming the sequential aspect of the realisation to be in the form of bistables. No formal approach has been developed, however, to take advantage of the various sequential units available in MSI (Medium Scale Integration) form. The problem can be viewed as that of "decomposing" the objective automaton into an interconnection of MSI sequential units, and this is the approach adopted in the present study. However the study of such ''composite realisations'' raises fundamental problems, for example what does an objective automaton represent? Moreover, how is an objective automaton to be formulated? It is also essential to clarify what is meant by a "realisation" of an objective automaton, so that in forming a "composite realisation" the basic aim is clearly understood. The initial aim in the present study, however, is to consider even more fundamental problems. It would seem that finite-automata theory can be developed from just a few essential concepts, furthermore these concepts are closely interrelated so a unified appreciation can be gained. By adopting this approach, the theory of finite automata can be developed in close association with more general abstract algebra, and can be developed with regard to axiomatic set theory and universal algebra.

Variable structure systems are examined from both an adaptive and an optimal control point of view. A theoretical procedure is developed which allows the optimal switching curve for a class of bang-bang variable structure systems to be directly generated. It is shown that the optimal and adaptive control strategies are highly compatible in certain cases and that the advantages of both philosophies may be obtained by the application of a correctly chosen variable structure control law. The theoretical procedures developed are applied to the practical problem of the speed control of a 5 H.P. D.C. machine. The variable structure controller was realised in the form of a Motorola 6800 Microprocessor based system. The performance of the machine to step speed demands and to step loading was investigated with both a variable structure control law and a conventional linear control law. In both cases, the respective control law is applied after an initial period where the machine is accelerated at the maximum safe value. It is seen that substantial improvements in the dynamic performance and the consistency of performance over a range of operating conditions are obtained with the variable structure control law, when compared to the performance with the conventional linear control law.

This thesis is mainly concerned with the improvement of approximate solutions of the eigenvalue problem for Fredholm integral equations with continuous kernels. It is assumed that an approximate solution has been obtained by the Nystrom (quadrature) method and more accurate solutions are required. Particular attention is paid to the application to the integral equation eigenvalue problem of the well known deferred correction technique for the improvement of numerical solutions. To illustrate the ideas the trapezoidal rule is chosen as the basic quadrature rule with the Gregory formula being used to estimate the local truncation error. In chapter 1, the main ideas of this thesis are introduced. The approximation of simple eigenvalues is discussed in chapter 2. An equation for the corrections of eigenvalues and eigenvectors is derived and gives rise to a stable iteration process for the computation of the corrections. For the case of multiple eigenvalues, the equations of chapter 2 are generalised and a more complicated correction process is obtained. Chapter 3 gives some necessary results for the matrix perturbation problem. The results for the approximations of multiple eigenvalues (defective and non-defective) of integral equations are presented in chapter 4. In chapter 5 the ideas are extended to deal with the approximation of eigenvalues for the Sturm-Liouville ordinary differential equation eigenvalue problem. A suite of programs for the numerical solution of the integral equation eigenvalue problem with continuous kernel has been written using the ideas described in chapters 2 to 4. These are described in chapter 6 and listings of the main programs are given in Appendix VI. Numerical results which illustrate the theoretical results of chapters 2, 4 and 5 are also presented.

This thesis is concerned with the developments firstly in combinational logic, where the problems involved in multi-threshold realisations are examined, and secondly in sequential logic, where the specific type of system known as serial input logic is considered. Chapter 1 is an introduction which discusses the areas covered by this thesis and indicates their relationship to one another within the general framework of logic design. Chapter 2 sets out to extend the number of functions that can be realised by a multi-threshold logic gate with near optimal weights and thresholds. Use is made of the already existing spectral translation method of obtaining a single threshold solution with additional exclusive-or gates, a multi-threshold solution being obtained by algebraic manipulation of the weights and threshold. The mathematical basis that enables this to be done is derived and examples given. Chapter 3 discusses the possible advantages of using a multi-threshold logic gate within a charge-coupled device over the alternative Boolean AND/OR and quaternary logic gates. The fundamental operations of a charge-coupled device are reviewed, and the tolerancing problems that result from the charge transfer inefficiency and voltage fluctuations are considered as limiting factors on the logical complexity of the gate. Chapter 4 is concerned with the subject of serial input logic. Initially it sets out to define serial input logic in terms of a general sequential system, and then goes on to show that with regard to state reduction using compatibles it is unique since it only requires the derivation of the implied maximal compatibles. Furthermore, a modular realisation is given, where the design procedure consists of the use of reverse response trees. Various labelling schemes are considered and finally one is considered that guarantees an optimal solution. Throughout the thesis emphasis is placed on finding general solutions whenever possible, so that not only do they apply to the situations described herein, but also may prove useful in as yet undeveloped areas of research.

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Human lung tryptase (HLT), a mast cell derived trypsin-like enzyme, was isolated from whole human lung tissue obtained at autopsy. Increased yields from this purification process allowed extensive characterization of the enzyme. One of the critical steps in the purification scheme was the use of a linear heparin gradient to elute active material from cellulose phosphate. Gel filtration studies in 1.0 M NaCl yielded an apparent M(,r) of 135,000, and subsequent electrophoresis on sodium dodecyl sulfate-polyacrylamide gels demonstrated the presence of two active species with apparent M(,r) = 30,900 and 31,600. Enzymatic activity was sensitive to NaCl concentrations above 0.05 M and was only 50% in 0.15 M NaCl, decreasing to 18% in 0.6 M NaCl. The effects of synthetic and natural inhibitors were studied, confirming the enzyme's trypsin-like characteristics and demonstrating that naturally occurring serum inhibitors are incapable of diminishing its activity. A complete amino acid analysis showed a high tryptophan content. Antisera to human lung tryptase was generated, and the immunological identity of active fractions was investigated. The stability of HLT in various buffer systems was extensively studied, and 10 mM MES buffer, pH 6.1 appeared to provide the best conditions during extended storage and purification. The effect of heparin on the enzyme's activity using the synthetic substrates Z-Lys-SBzl was studied, and heparin concentrations of 10 micromolar stabilized HLT and allowed full expression of activity even at low ionic strengths. In the presence of heparin the enzyme retained full activity after 24 hours at 37(DEGREES)C, whereas in the absence of heparin, activity was lost after 30 min at this temperature. Heparin had a similar effect on HLT's ability to cleave natural substrates such as fibronectin. Assays comparing the activity of HLT on the substrates Z-Lys-SBzl and Z-Arg-SBzl were performed. The K(,m) and V(,max) of HLT for the above substrates were determined. The substrate 4-methylumbelliferyl-p-guanidinobenzoate (MUG-B) has been used to perform an active site titration on HLT, and a k(,cat) of 610/sec was calculated for the substrate Z-Arg-SBzl.