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We investigate various models of thermal convection in a fluid saturated porous medium of both Darcy and Brinkman types. The linear instability and global (unconditional) nonlinear stability thresholds are analysed. Analytical solutions and numerical solutions are obtained by employing the $D^2$ Chebyshev tau and compound matrix techniques, and we investigate the effect that the inertia term and other physical parameters have on the stability of the system. The thesis is split into two parts. In PartI we consider a coupled model of thermal convection in a fluid saturated porous material and theories of viscous fluid motion which allows heat to travel as a wave. This is discussed in the first three chapters. In Chapter 2 the instability mechanism is investigated in complete detail and it is shown that stationary convection is likely to prevail under normal terrestrial conditions, but if the thermal relaxation time is sufficiently large there is a possible parameter range which allows for oscillatory convection. However, the presence of the Guyer-Krumhansl terms has the effect of damping the oscillatory convection and returning the instability mechanism to one of stationary convection. In Chapter 3 the constitutive equation for the heat flux is governed by a couple of the Guyer-Krumhansl equations and the Cattaneo-Fox law. In particular, we study the effects of the Guyer-Krumhansl terms on oscillatory convection. It is found that for a certain range of the Guyer-Krumhansl coefficient stationary convection occurs while changing the range results in oscillatory convection. Numerical results quantify this effect. The thermal instability in a Brinkman porous medium incorporating fluid inertia for both free--free and fixed--fixed boundaries is considered in Chapter 4. We have incorporated the Cattaneo--Christov theory in the onstitutive equation for the heat flux. For fixed surfaces, the results are generated by using the $D^2$ Chebyshev tau method. The results reveal that employing the Cattaneo--Christov theory has a pronounced effect in determining the convection instability threshold. Part II concerns the effect of an anisotropic permeability on thermal instability in the modelling problems of thermal convection of Darcy type with and without the inclusion of an inertia term, which represented the last three chapters. In Chapter 5 we allow a non-zero inertia term and also allow the permeability to be an anisotropic tensor. For particular numerical results we consider the case when the vertical component of the permeability tensor is variable. Linear instability results are calculated numerically and it is proved that the nonlinear energy stability bound is the same as the linear one. We perform the linear instability and nonlinear stability analysis, in the case where the inertial term vanishes, to investigate the effect of anisotropy with rotation on the stability thresholds in Chapter 6, showing that the nonlinear critical Rayleigh numbers coincide with those of the linear analysis. The results reveal that the inclusion of the inertial term for this model can play an important role on the onset of convection in Chapter 7.

This thesis considers the inverse problem of detecting and characterising flaws within heterogeneous materials using ultrasonic phased array transducers. Many imaging techniques include subjective measurements and the aim of this thesis is to develop objective mathematical model based methods which alleviate such subjectivity. Within the first method, the Kirchhoff model is used to derive an explicit expression which relates the maximum eigenvalue from a scattering matrix to the length of a crack in a homogeneous medium. It is shown that there exists a one to one relationship between this maximum eigenvalue and the crack length. The advantage of deriving this analytical approximation is that it can then be analysed to assess the crack sizing capabilities of the method given some scattering matrices from experimental data (the inverse problem). The procedure for using this method is then demonstrated by applying it to finite element simulated data from a homogeneous medium with a 5 mm long crack inclusion, the crack length recovered using this method is 4.4 mm. A second method is then presented which exploits another feature of the scattering matrix. An analytical expression which is an approximation to the first minimum in the pulse echo response of a scattering matrix is derived from the Kirchhoff model. This approach is also illustrated by sizing a 5 mm long crack within a homogeneous medium from finite element simulated data, the crack length recovered using this method is 5.8 mm. The method is then extended to form a multi-frequency technique which enables it to be applied to finite element simulated data from a 5mm long crack inclusion in a heterogeneous medium. The method is enhanced by using a convolution method to reduce the noise prior to the multi-frequency method being used. The recovered crack length using this method once the noise has been reduced is 4 mm. Finally, a detection technique based on the first stage of a time reversal is presented, within which a detection threshold specific to steel welds is proposed. This method is applied to both finite element simulated data and experimental data. Having detected a flaw the time reversal algorithm (DORT) is then used to create images which are then compared to those obtained using the Total Focusing Method.

It is well known that in primitive recursive arithmetic with a single successor the number of parameters in a definition by recursion may be successively reduced. In this thesis I examine the possibility of effecting a similar reduction in the number of parameters in a definition by recursion in a multi-successor arithmetic. The reduction process involves the discovery in multi-successor arithmetic of analogues of pairing functions and of functions which select the elements of an ordered pair. One of the difficulties in finding such functions is the construction within multi-successor arithmetic of suitable product and square foot functions and establishing the properties of these functions, and the pairing functions, within a formalisation of multi-successor arithmetic. The reduction process involves of course an examination of what functions, if any, need to be adjoined to the initial functions to secure the generality of the reduction.

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This thesis sets out to examine the possibility of devising a theory which will give a unified account of prepositional calculi and algebraic systems. Starting from a historical account of the principal ideas tributary to the main stream of theory from Boole to the present day, it presents a technical- language framework within which it is possible to develop in a uniform format substantial portions of the theories of both sorts of system. The idea of an Interpretation then leads to a discussion of Functional Completeness, and the use of Galois fields in the algebraic representation of functions. Two particular families of systems, the Protomodules and Protorings, are selected for more detailed study. Their principal decision problems are considered, their structure examined, and their relationship to familiar systems of algebra and prepositional calculus displayed. The discussion then specialises again to the use of Galois fields in the solution of computational problems arising in connection with an important class of protorings, the so- called Galois Logics. One of these problems is of sufficient complexity to warrant the use of an automatic digital computer, and details of the computer program are presented in an appendix. Three other appendices are devoted to the presentation of material which evolved as by-products during the contemplation of the main issues; they are concerned with closely related topics, and are given here in support of the thesis rather than as part of the theory.

The thesis is in two parts. In the first, I give a method for constructing decidable classes of number theoretic sentences, and in the second, I apply this method in the construction of particular decidable classes. Let B0,B1,.. be an increasing sequence of finite Boolean algebras of subsets, of the natural numbers, N, such that {e-1} B limits for all e > 0. We call the Be limiters. We say that e limits a predicate P if the extension of each component of P belongs to We say that a function p(i) limits a function f if the inverse of each component of f maps Bi, into Bpi. By a constituent, we mean either a predicate or a function. We call a set of constituents B effectively limited if there is an effective procedure for obtaining limits, which are recursive in the case of functions, for each constituent in B. By a sentence with constituents B we mean a sentence generated from B and the equations X = 0,X=1, by substitution, the prepositional operations, bounded and unbounded quantification, and bounded and unbounded u-operations Our main result is that the class of sentences, with a given effectively limited set of recursive constituents, is decidable if b2 is recursive where b2 denotes max{min B : B Be}. In the second part, we consider several possible sequences of limiters. In each case, we show that be is recursive, and we find as large an effectively limited class B of recursive constituents as we can, so that the class of sentences with constituents B is decidable. In the case of the simplest possible limiters, our work reveals the connection between the apparently unrelated methods, used by Goodstein and Lee and by Rousseau, to determine decidable classes of equations of the form f = 0, and provides a considerable extension of these classes.

In what was almost certainly an attempt to find a new axiom system for Boolean algebra based on distributivity and the existence of complements MHA Newman discovered a remarkable set of independent postulates defining an algebra which may be regarded as a generalization of Boolean algebra and now bears his name. Shortly after publication of his paper Newman extended his discussion to a wider class of relatively complemented algebras which we call Generalized Newman algebra. Recently K.Roy investigated the properties of an algebra closely related to Newman algebra, called Dual Newman algebra, and found that it has similar properties to its progenitor the opening chapters of the thesis are devoted to a discussion of the properties of the lattices of ideals, congruence relations and filters in Newman algebra and the relationships between them. The concepts of inverse and sub-inverse limits of Newman algebras are introduced, some general properties proved, and a sub-inverse limit representation established for a particular class of Newman algebras together with an inverse limit representation for the class of infinite, couplet, Boolean algebras. Furthermore, it is proved that a Newman algebra can be represented as a direct product of simple algebras if and only if its ideal lattice is a finite Boolean algebra. In the following chapter we investigate, within the framework of Newman algebras, the analogues of the auto and ideal topologies on Boolean algebra discovered by P.S. Rema. It is shown that the set of all ideal topologies L1 on a Newman algebra N is a complete, Brouwerian, dually atomic lattice containing the set L0 of all auto topologies as a complete sub-lattice and that L0 is completely isomorphic to the lattice of filters of N. Some important types of filters in N are characterized in terms of properties of the associated auto topologies on N and the auto topology associated with a given filter characterized within the lattice of ideal topologies on N. Amongst the more general properties proved we mention that the property of a topology, compatible with the fundamental operations on N, being an ideal (auto) topology is, in the algebraic and topological sense, hereditary, productive and divisible. The various connectedness properties of ideal topologized Newman algebra N;J are considered in some detail; the components being exhibited as certain congruence classes of N and necessary and sufficient conditions found for N;J to be connected, locally connected and totally disconnected. Some results are obtained concerning complete ideal uniformities and compact ideal uniformities. The properties of a particular class of ideal uniformities, called chain uniformities, are investigated and a clear cut family of metrizable chain uniformities are exhibited. Necessary and sufficient conditions are then established for a Newman algebra endowed with a separated ideal uniformity to the metrizable. In the closing chapters of the thesis we are concerned with the axiomatics of Dual and Generalized Newman algebras. Two new sets of axioms for Dual Newman algebra are exhibited each containing one less axiom than the system due to K. Roy. A new set of axioms for Generalized Boolean algebra is found containing one less axiom than the system discovered by lawmen together with a new set of independent postulates, characterizing the direct product of an arbitrary Generalized Boolean algebra and Boolean ring, which contains two fewer axioms than the system discovered by Newman.

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Throughout this thesis we deal with the scaling properties of resonance (or <I>Arnol'd</I>) tongues of circle maps. The motivation comes from the work of A.M. Davie, which we summarise as necessary.