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The thesis reviews the history and literature of reliability theory. The implicit assumptions of the basic reliability model are identified and their potential for generalisation investigated. A generalised model of reliability is constructed, in which components and systems can take any values in an ordered discrete or continuous state-space representing various levels of partial operation. For the discrete state-space case, the enumeration of suitable system structure functions is discussed, and related to the problem posed by Dedekind in 1897 on the cardinality of the free distributive lattice. Some numerical enumerations are evaluated, and several recursive bounds are derived. In the special case of the usual dichotomic reliability model, a new upper bound is shown to be superior to the best explicit and non-asymptotic upper bound previously derived. The relationship of structure functions to event networks is also examined. Some specific results for the state probabilities of components with small numbers of states are derived. Discrete and continuous examples of the generalised model of reliability are investigated, and properties of the model are derived. Various forms of independence between components are shown to be equivalent, but this equivalence does not completely generalise to the property of zero-covariance. Alternative forms of series and parallel connections are compared, together with the effects of replacement. Multiple time scales are incorporated into the formulation. The above generalised reliability model is subsequently specialised and extended so as to study the optimal tuning of partially operating components. Simple drift and catastrophic failure mechanisms are considered. Explicit and graphical solutions are derived, together with several bounds. The optimal retuning of such units is also studied and bounds are again obtained, together with some explicit solutions.

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This thesis is concerned with the extension of Zubov's method for the determination of domains of attraction. The basic definitions and theorems of Liapunov and Zubov as well as a numerical algorithm (due to White) are given in the introductory chapter. The application of the method of Zubov to some practical situations like power systems and control systems of order two is the subject of chapter two. Chapter three describes the determination of the domains of attraction for scalar time varying systems. The series solution has a similar problem of non-uniform convergence that occurs in autonomous systems. Extension of the method to third order non-linear autonomous systems is included in Chapter four so that it can be applied to second order time varying system which is described in Chapter five. Results in the form of slices or cross-sections of the stability boundaries in the various principal planes are obtained. Systems which have periodic solutions are examined and the domain of attraction of the stable limit cycle is determined in Chapter six. Approximate solutions are also used in trying to determine the domain of attraction of the periodic solutions. In Chapter seven a technique for solving global optimization problems is presented. Several one-dimensional and two-dimensional minimization problems are solved and the results indicate the accuracy of this technique.

The purpose of the thesis is to assess the mathematical achievements of Bernard Bolzano on the basis of the five early published works. The material is divided into the areas of the foundations of mathematics, geometry and analysis. In making this assessment there have been two principal considerations. Firstly, any judgement of the significance of Bolzano's work should be made in the light of the historical context, so considerable space is devoted to the relevant 18th century sources. Secondly, as a general framework to the thesis there is the question of how Bolzano's general views about mathematical proofs and concepts are related to his achievements. The main claim and conclusion of the thesis is that this relationship was unusually clear and significant in the case of Bolzano's work. There is an Appendix containing the first English translation of all five of Bolzano's works as well as the German texts of their first editions.

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