Includes abstract. / Includes bibliographical references. / The principal aim of this thesis is to investigate the existence of an injective hull in the categories of T-quasi-metric spaces and of T-ultra-quasi-metric spaces with nonexpansive maps.

Includes abstract. Includes bibliographical references.

Bibliography: leaves 158-169. / Secondary settling tanks (SSTs) form a crucial part of wastewater treatment plants. Besides having to produce the separation of suspended solids and clarified effluent the secondary settling tank is used to concentrate and recycle the settled sludge to the biological reactor. The efficiency of the biological reactor in the waste water treatment system is determined by the efficiency of this final clarifying process. Hydrodynamic models have been developed for simulating secondary settling tanks in order to gain a better understanding of the complex flow patterns in these tanks, and to make design and optimization of the SST internal features possible. These models use mainly the finite volume method. This thesis is concerned with the development and implementation of a finite element approach to the simulation of flows in SSTs. Although it is nowadays also possible to realise an unstructured grid within the FVM, the power of the finite element method (FEM) lies in its higher flexibility in fitting irregular domains and in providing local grid refinement. Generally, unstructured mesh procedures with the FVM require essential, additional orthogonality corrections, which affect the accuracy of the solution, and these corrections increase the computational cost due to the additional computations and increased iteration requirements. Structured mesh discretization may offer significantly shorter computation time. The FEM is therefore convenient for handling arbitrarily shaped domains and adaptation of complex internal features of SSTs, such as inlet and outlet arrangements.

Includes bibliographical references.

Bibliography: pages 102-107. / This thesis deals with further investigations of the enhanced strain finite element method, with particular attention given to the analysis of the method for isoparametric elements. It is shown that the results established earlier by B D Reddy and J C Simo for affine-equivalent meshes carry over to the case of isoparameric elements. That is, the method is stable and convergent provided that a set of three conditions are met, and convergence is at the same rate as in the standard method. The three conditions differ in some respects, though, from their counterparts for the affine case. A procedure for recovering the stress is shown to lead to an approximate stress which converges at the optimal rate to the actual stress. The concept of the equivalent parallelogram associated with a quadrilateral is introduced. The quadrilateral may be regarded as a perturbation of this parallelogram, which is most conveniently described by making use of properties of the isoparametric map which defines the quadrilateral. The equivalent parallelogram generates a natural means of defining a regular family of quadrilaterals; this definition is used together with other properties to obtain in a relatively simple manner estimates, in appropriate seminorms or norms, of the isoparametric map and it's Jacobian, for use in the determination of finite element interpolation error estimates, with regard to computations, a new basis for enhanced strains is introduced, and various examples have been tested. The results obtained are compared with those obtained using other bases, and with those found from an assumed stress approach. Favourable comparisons are obtained in most cases, with the present basis exhibiting an improvement over existing bases. Convergence of the finite element results are verified; it is observed numerically that the improvement of results due to enhancement is as a result of a smaller constant appearing in the error estimates.

Bibliography: pages 127-130. / The compact operators on a Hilbert space are those operators for which the image of the unit ball is relatively compact in the norm topology. These operators form an ideal, in the algebra of all continuous linear operators on the Hilbert space, which is closed in the uniform norm. In the case that the underlying Hilbert space is separable this is the only such ideal, while for non-separable Hilbert spaces the norm-closed ideals are easily characterised by means of cardinal numbers. The algebra of all continuous linear operators on a Hilbert space is a specific example of a van Neumann algebra, and the theory of compact operators and the ideal they form admit certain generalisations to van Neumann algebras. One of the characterisations of the ideal of compact operators is that it is the closure of the ideal of finite rank operators, and hence the closed ideal generated by the finite dimensional projections. Kaftal has considered the ideal of so called algebraically compact operators, which is defined to be the closed ideal generated by the algebraically finite projections in the von Neumann algebra, and has shown that this ideal consists of those operators which map the unit ball to sets which have compact-like properties. This characterisation was generalised to arbitrary norm-closed ideals by Stroh. In this thesis we explore the extent to which norm-closed ideals in van Neumann algebras resemble the ideal of compact operators on a Hilbert space. We extend the theory developed by Kaftal and Stroh, and show that arbitrary ideals in van Neumann algebras can be characterised in terms of homologies and topologies. We also consider continuity characterisations of norm-closed ideals in von Neumann algebras, generalising the characterisation of the compact operators as being those that are continuous from the unit ball equipped with the weak topology, to the Hilbert space equipped with the norm topology. Furthermore we briefly consider sequential continuity characterisations as first analysed by Kaftal in the case of the algebraically compact ideal. Finally, in the case of a semifinite von Neumann algebra equipped with a faithful semifinite normal trace T, we generalise the characterisation of the compact operators given in terms of the singular value sequence, by showing that the ideal of T-measurable operators whose generalised singular function decreases to 0 possess many of the same properties as the ideal of compact operators.

Bibliography: p. 221-233. / This thesis is concerned with the problem of how to effectively address the complex issue of poverty in the context of limited resources. Poverty is a multi-dimensional problem that affects different communities in different ways. In order to use the available resources in such a way as to most effectively tackle poverty, a means of measuring and benchmarking outcomes as well as evaluating choices of intervention is required. However, smart methods of allocating scarce resources are not in themselves sufficient, if they are not regarded as legitimate by the participants of the process. The imperative of legitimacy demands that we both address the issue of quantitative rigour in resource allocation methods and that we look beyond and explore too the mechanics of effective participatory methods. The approach of developmental local government adopted by the new South African government post apartheid, places this complex problem in the sphere of local government. The primary tool available to local administrators for addressing poverty, amongst other issues, is that of integrated development planning. This process draws together the stakeholders who fall broadly into three groups of participants, namely the communities that live in the municipality, the municipal officials and the elected politicians, and allocates them the task of identifying and prioritising community and municipal issues, and developing appropriate plans to address them. This package of plans or projects is compiled into a municipal budget that targets priority issues for the area, in an integrated and coherent manner. This thesis proposes a new method for tackling this specific group decision making problem, namely Participative Multi-Criteria Decision Analysis. This method was developed in an action research setting in the municipality of Stellenbosch, South Africa, and applied to their 2001/2002 integrated development planning process. The method is grounded in the principles of participative action research in which the participation of all interested and affected patties is valued, and in which there is a commitment to work for change to the fundamental fabric of knowledge and power, leading to a greater empowerment of ordinary people. This participative framework strengthens the legitimacy of the approach by promoting a stronger sense of ownership of process and products by all participants. Within this participative framework, tools of multi-criteria decision analysis are used to support the decision making process by quantifying difficult decisions that need to be addressed. It is the synthesis of these two approaches (action research and multi-criteria decision analysis) that provides both legitimacy and rigour for this method within a highly contested and complex public decision making arena. In the spirit of action research, the method is developed by drawing on theory about developmental local government and poverty, as well as multi-criteria decision analysis. In the process of the research, over forty community workshops were held throughout the Stellenbosch municipal area. Community representatives identified and prioritised the issues of their areas; and in conjunction with municipal officials, developed and evaluated projects in response to these issues. These evaluations assisted the local council to compile the final budget for 2001/2002 in Stellenbosch. In this process, the communities (divided into nine development areas) also developed community development measurement scales which formed the basis for the project evaluations and an ongoing basis for monitoring progress in these communities. It unfolded during the course of this research that a fundamental component of this proposed participative public decision making approach is the role of a central co-ordinating person, not connected to or answerable to any of the constituent groups, who can manage the process of participation, promote an awareness of effective and informative data; ensure the appropriate use of quantification tools and maintain a focus on sustainable poverty alleviation. The method developed in this thesis was successfully applied to the process of identifying, prioritising and making choices about community issues in Stellenbosch, under conditions of significant polarisation of the constituent decision making groups, conclude that this method can be used to implement key aspects of integrated development planning as it addresses the issues of legitimacy and rigour in participative public decision making.

The boundary value problems for the Helmholtz equation give rise to boundary integral equations for the unknown surface field or its normal derivative. These integral equations involve the Helmholtz surface potentials in the form of weakly singular surface integrals. This thesis is based on a method of parameterisation of the surface integrals which removes the weak singularities provided that the surface satisfies certain convexity conditions. Firstly this method of parameterisation is applied to investigate the properties of the Helmholtz surface potentials on convex surface elements, and some new proofs are given. The theory is then applied to the boundary integral equations which arise when a scalar field is incident on a bounded scatterer. The surface integrals in these integral equations are Helmholtz potentials and can be regularised by suitable parameterisation. It is assumed that the unknoWn density function is an analytical function on the boundary of the scatterer, and can therefore be expanded as a Taylor series at any point of the surface. If this expansion is substituted into the regularised integral equation and if the operations of integration and summation are formally interchanged, then the end result is a partial differential equation of infinite order involving only the field coordinates and having analytical coefficients. However, if the Taylor expansions are truncated then partial differential equations of finite orders result. The view is taken that analytical solutions of such differential equations of finite orders can serve as _approximations for the surface field or its normal derivative provided that suitable initial conditions are imposed to ensure uniqueness. On the other hand the general solution of such a differential equation can serve as a local approximation at any point on the surface. Some basic properties of the differential equations and their solutions, called analytical approximations, are discussed and the theory is then applied to the problem of acoustic scattering from a sound hard sphere.

Includes bibliographical references (leaves 204-221).

Includes bibliographical references.