This thesis follows the approach of papers (1, 2) by exploring signature changes in other metrics. The metrics we chose to investigate are the Schwarzschild metric and the Tolman metric. The Schwarzschild metric was originally chosen in order to investigate whether the neighborhood of the singularity inside a black hole can be replaced with a Euclidean region, and also to see whether this Euclidean region can lead to new universes by providing "wormholes" through to other Lorentzian universes. By this we mean that, if one follows "time-like" geodesic paths from a Lorentzian region into a Euclidean region, they bounce (instead of hitting a singularity) and can then pass through a second signature change into another Lorentzian region. Consideration of how geodesics pass through a signature change naturally leads to the Tolman metric, whose vacuum cases cover the Schwarzschild/Kruskal-Szekeres manifold with all possible sets of radial geodesic coordinates. We take the opportunity to explore several cases of signature change in other Tolman models.

This thesis has as its aim the analysis of a possible manifold structure on V, a join of two individual manifolds V⁺ and V⁻, and analysing the physics across the join, as implied by Einstein's theory of General Relativity. There are several reasons why one might want to study such a situation. Firstly, the joining of manifolds is useful in the study of shock waves, be they of gravitational or other origin - we will be able to characterise the propagation of energy in the join. Secondly piecing together manifolds is a potentially fruitful way of obtaining exact solutions of Einstein's equations which do not exhibit any symmetries in the large, and are yet sufficiently homogeneous (in some sense) to enable one to model the apparent Universe - the prototype of this is the Swiss-Cheese model, used to study light transmission in an inhomogeneous Universe. Thirdly, discontinuities in the fundamental quantities in Relativity are of prime importance in the study of singularities and in particular, it is of prime importance to single out the contributions of the differential geometry and metric structure of the Universe to the existence and nature of such singularities. Closely linked to these problems is the problem of linking the small scale structure of the Universe (which is manifestly complicated and potentially full of singularities) and the large scale structure, which seems so well modelled by assumptions of homogeneity and isotropy. In this regard, the techniques of Regge (1961), originally proposed to provide approximate solutions to the Einstein equations, assume a new theoretical importance, for the delta-type singularities in the curvature he used, in a smoothing process, to represent the (assumed) continuous curvature of space, could in themselves play a distinguished role representing the small scale structure of the Universe. Furthermore, the matching together of blocks of space-time with sharp edges and corners may enable to develop a manifold like structure in which the tangent spaces of some points had a surfeit or deficiency of vectors, so that the differential geometry of the resulting space-time forced discontinuities and singularities in the metric structure of the Universe. Although this may be aphysical, it may be a reasonable way of seeking further understanding of the Universe.

Includes bibliographical references. / In the first part of this dissertation, we present a detailed description of the eigenface technique first proposed by Sirovich and Kirby and subsequently developed by several groups, most notably the Media Lab at MIT. Other significant contributions have been made by Rockefeller University, whose ideas have culminated in a commercial system known as Faceit. For a different techniques (i.e. not eigenfaces) and a detailed comparison of some other techniques, the reader is referred to [5]. Although we followed ideas in the open literature (we believe there that there is a large body of advanced proprietary knowledge, which remains inaccessible), the implementation is our own. In addition, we believe that the method for updating the eigenfaces to deal with badly represented images presented in section 2. 7 is our own. The next stage in this section would be to develop an experimental system that can be extensively tested. At this point however, another, nonscientific difficulty arises, that of developing an adequately large data base. The basic problem is that one needs a training set representative of all faces to be encountered in future. Note that this does not mean that one can only deal with faces in the database, the whole idea is to be able to work with any facial image. However, a data base is only representative if it contains images similar to anything that can be encountered in future. For this reason a representative database may be very large and is not easy to build. In addition for testing purposes one needs multiple images of a large number of people, acquired over a period of time under different physical conditions representing the typical variations encountered in practice. Obviously this is a very slow process. Potentially the variation between the faces in the database can be large suggesting that the representation of all these different images in terms of eigenfaces may not be particularly efficient. One idea is to separate all the facial images into different, more or less homogeneous classes. Again this can only be done with access to a sufficiently large database, probably consisting of several thousand faces.

Bibliography: leaves 147-153. / This thesis contains two distinct parts: the first part introduces and explains the relevant theory and background necessary for the analytic work done on magnetized cosmological perturbations at the end of the first part. The second part discusses some issues related to observational cosmology. After an introductory chapter including an overview of the thesis, PART I starts with a discussion of the covariant approach to cosmology, introducing notation needed in the thesis. The covariant approach to perturbations is then discussed, and the basic inhomogeneity variables describing energy density, pressure and expansion pertubations are introduced. Their exact evolution equations are presented before being linearized about an FRW background.

Bibliography: leaves 77-82. / This thesis is concerned with properties of equations governing fibre suspensions. Of particular interest is the extent to which solutions, and their properties, depend on the type of closure used. For this purpose two closure rules are investigated: the linear and the quadratic closures. We show that the equations are consistent with the second law of thermodynamics, or dissipation inequality, when the quadratic closure is used. When the linear closure is used, a sufficient condition for consistency is that the particle number Np satisfies Np ≤ 35/2. Likewise, flows are found to be monotonically stable for the quadratic closure, and for the linear closure with Np ≤ 35/2. The second part of the thesis is concerned with one-dimensional problems, and their solution by finite element. The hyperbolic nature of the evolution equation for the orientation tensor necessitates a modification of the standard Galerkin-based approach. We investigate the conditions under which convergence is obtained, for unidirectional flows, with the use of the Streamline Upwind (SU) method, and the Streamline upwind Petrov/Galerkin (SUPG) method.

Includes bibliography. / The purpose of the procedures described is to assign “objects” or "observations" in some optimum fashion to one of two or more populations. In routine banking a bank manager may wish to classify clients who wish to make loans as low or high credit risks on the basis of the elements of certain accounting statements. In such a case there are two definite distinct classes. Another investigation may be initiated to determine whether buying habits are different with respect to the categories: urban, sub-urban and rural clients. Note that in the first example the classes are determined before any sample of observations is investigated, i.e. the sample results do not influence the choice of groups. In the latter case one is trespassing on the terrain of cluster analysis.In the first case we have two types of problems, namely that of devising a classification rule from samples of already classified objects and that of imposing the classification scheme on the objects. The term "discrimination" refers to the process of deriving classification rules from samples of classified objects and the term "classification" refers to applying the rules to knew objects of unknown class. Although it is possible to convert raw data into more easily grasped forms like cartoon faces (Chernoff, 1973) this still represents the problem that any grouping or classification based on these diagrams is subjective.

Bibliography: pages 64-68. / This thesis is concerned with the development of a variational formulation for the problem of viscous incompressible free surface flows, and with the development and implementation of algorithms for the solution of this problem by finite elements. The study is restricted to two-dimensional steady problems. The approach differs from those in current use, in that it is based on a two-stage strategy suggested by theoretical (existence) studies of the problem. In the first stage the free surface is kept fixed and the resulting so-called auxiliary problem is solved. In the second stage the equation for the normal stress on the free surface is used to update the free surface. Both the auxiliary problem and the normal stress equation are formulated variationally; in the case of the latter problem the unknown variable is actually the slope of the free surface. Finite element approximations are used in both problems. Algorithms are developed for determining solutions at the two stages, and for the overall problem. The key example treated is the dieswell problem, for the plane and axisymmetric cases. Solutions obtained by the present method are presented, and compared with the solutions of others where available.

We describe the Galois theory of commutative semirings as a Boolean Galois theory in the sense of Carboni and Janelidze. Such a Galois structure then naturally suggests an extension to commutative semirings of the classical theory of quadratic equations over commutative rings. We show, however, that our proposed generalization is impossible for connected commutative semirings which are not rings, leading to the conclusion that for the theory of quadratic equations, “minus is needed”. Finally, by considering semirings B which have no non-trivial additive inverses and no non-trivial zero divisors, we present an example of a normal extension of commutative semirings which has an underlying B-semimodule structure isomorphic to B×B.

We define and explore the concepts underpinning the Ryu-Takayanagi prescription for entanglement entropy in a holographic theory. We begin by constructing entanglement entropy in finite-dimensional quantum systems, and defining the boundary at infinity of a bulk spacetime. This is sufficient for a naïve application of the Ryu-Takayanagi prescription to some simple examples; nonetheless, we review the general theory of minimal submanifolds in Riemannian ambient manifolds in order to better characterise the objects involved in the prescription. Finally, we explore the symmetries of the boundary theory to which the prescription applies, and thereby extend the aforementioned examples. Throughout, emphasis is placed on making explicit the mathematical structures that are taken for granted in the research literature.

Bibliography: pages 118-121. / In the field of computational mechanics, increases in computing power and enhancements in material and kinematic models have enhanced the feasibility of performing structural design optimisation for a wide range of applications. The work presented here was motivated by the current groundswell of research effort in computational optimisation. Design Sensitivity Analysis (DSA) crucially underpins much of structural optimisation and, as such, is focussed on more intently than the optimisation theory itself: various approaches to the Direct Differentiation Method (DDM) DSA procedure are investigated and computationally implemented. The procedures implemented were chosen so as to involve a range of important issues in computational sensitivity analysis, particularly * Shape and non-shape sensitivity analysis, * Total and Updated Lagrange-based DSA, * DSA of displacement and non-displacement based response functionals, * Multiparameter DSA. * DSA for large strain behaviour The primary objectives of this thesis are: I. Development of a design sensitivity formulation which, when discretised, resembles the standard displacement based kinematic element formulation, thus enabling the implementation of design sensitivity analysis in established Finite Element Analysis (FEA) codes as a 'pseudo'-element routine. II. lmplemention of several design sensitivity formulations and structural optimisation into the FEA code ABAQUS as a verification of the first objective. Numerical results provided in this work demonstrate the successful completion of the above-mentioned objectives. The discretised DSA formulations presented, as well as the 'pseudo'-element approach adopted, particularly in the case of shape DSA are entirely original. To the best of the author's knowledge, DSA and DSA-based structural optimisation had never before been attempted with ABAQUS. The research conducted here lays the foundation for potentially very fruitful future work.