We consider here two classes of non-linear systems giving different degrees of non-linearity. In both cases the systems arise from finite difference discretisations of non-linear elliptic partial differential equations. Our solution methods can also fit into two categories - linearisation and non-linearisation techniques - and in our studies we have pursued three main objectives. 1) For mildly non-linear systems we generalise certain iterative techniques from the solution of linear systems to the solution of nonlinear systems with symmetric Jacobians. We are especially concerned with the effect of preconditioning of the equations here. 2) We consider the use of bidiagonalisation on non-linear systems, using preconditioning in two ways and considering both classes of nonlinear problem. 3) We solve the laminar flow problem and assess the effects of multigrid acceleration on non-linear S.I.P. techniques.

This thesis has three sections. Section one contains two chapters, the first describing those techniques used later, principally multidimensional scaling, procrustes fitting and cluster analysis. Least squares scaling, preprocessing the dissimilarity matrix and clustering by maximum likelihood partition are less known. The second chapter reviews simulation studies previously published in multidimensional scaling literature. Section two contains one chapter detailing four simulation studies in multidimensional scaling. The first considers the robustness of classical scaling in the presence of error in the dissimilarity matrix. Four probabilistic models generating euclidean-distance-like dissimilarity functions are proposed, which reflect some of the ways dissimilarities actually arise, and allow dependence between dissimilarities to be studied. Next we compare how well various scaling methods reconstruct specific configurations, given the same dissimilarity matrix. Properties of preprocessing the matrix and least squares scaling are demonstrated. Thirdly we describe a study, designed to measure the redundancy in a dissimilarity matrix, which justifies subsequent use of scaling with missing data. Finally we determine the robustness of approximations to procrustes statistics obtained from perturbational analysis of classical scaling by Sibson (1979). Section three contains four applications chapters. Firstly multidimensional scaling is applied to data concerning the voting behaviour of M.P.s in 1861. This large data set requires special handling, some dissimilarity values being best treated as unknown. The results identify both unusual and regular voting behaviour. The second application is in ethnology. Dissimilarity values derived from phonetic differences between languages are used to derive their genetic origin. The techniques, especially clustering by maximum likelihood partition, reproduce known relationships satisfactorily and suggest others. The third example uses morphological and meristic parameters to generate dissimilarities between specimens of the fish species Colisa. Here the aim is taxonomic. Finally we consider dietary changes across Britain through time to identify regional and temporal differences.

Conformal methods have proven to be very useful in the analysis global properties and stability of vacuum spacetimes in general relativity. These methods transform the physical spacetime into a different Lorentzian manifold known as the unphysical spacetime where the ideal points at infinity are located at a finite position. This thesis makes use of conformal methods and applies them to various problems involving trace-free matter models. In particular, it makes progress towards the understanding of the evolution of unphysical spacetimes perturbed by trace-free matter as well as the behaviour of the the matter itself. To this end, evolution equations (wave equations) are derived and analyzed for both the unphysical spacetime and the matter. To investigate the relation between solutions of these wave equations to the Einstein field equations, a suitable system of subsidiary evolution equations is also derived. Furthermore, this thesis looks in detail at the behaviour of an unphysical spacetime coupled to the simplest matter trace free model: the confomally invariant scalar field. Finally, the system of conformal wave equations is used to show that the deSitter spacetime is non-linearly stable under perturbations by trace-free matter.

Understanding the macroscopic properties of matter, based on the microscopic interactions of the single particles requires to bring together the areas of statistical physics and dynamical systems. For deterministic diffusion one of the most prominent models is the Lorentz gas in which a point particle performs specular reflections with hard disks distributed in the plane. This model generates deterministic chaos and has led to many mathematical results revealing the origin of diffusion starting from chaotic dynamics. For the periodic Lorentz gas on a triangular lattice, it is possible to understand the diffusion coefficient, in the limit of high scatterer densities, in terms of random walk approximations. The key question addressed in this thesis is: What happens to the diffusion coefficient, as a function of control parameters, if the hard potential walls of the Lorentz gas scatterers are replaced by a soft potential? In this study we use a repulsive Fermi potential from which the hard limit can be recovered by varying a control parameter. We then performed computer simulations and analytical random walk approximations to understand the functional form of the diffusion coefficient as a function of the following parameters: the minimal distance between two scatters, the softness of the potential and the energy of a moving particle. Our main results is that the diffusion coefficient is a highly irregular function of each of these control parameters. Under certain assumptions one can construct analytical approximations that describe the coarse shape of the diffusion coeffi- cient when it exists: For high densities of scatterers we develop suitable random walk approximations, in the low density regime we apply a more elaborate argument that tests for memory effects. We find that diffusion in our soft Lorentz gas exhibits different random walk regimes, where either randomization characterizes the evolution of diffusion or spatio-temporal correlations take place. Via Poincare surfaces of section we show that the irregularities appearing in the diffusion coef- cient, as a function of parameters, which strongly deviate from simple random walk dynamics, come from non-trivial quasi-ballistic trajectories in con guration space.

The hypercube, Qd, is a natural and much studied combinatorial object, and we discuss various extremal problems related to it. A subgraph of the hypercube is said to be (Qd; F)-saturated if it contains no copies of F, but adding any edge forms a copy of F. We write sat(Qd; F) for the saturation number, that is, the least number of edges a (Qd; F)-saturated graph may have. We prove the upper bound sat(Qd;Q2) < 10 2d, which strongly disproves a conjecture of Santolupo that sat(Qd;Q2) = 1 4 + o(1) d2d 1. We also prove upper bounds on sat(Qd;Qm) for general m. Given a down-set A and an up-set B in the hypercube, Bollobás and Leader conjectured a lower bound on the number of edge-disjoint paths between A and B in the directed hypercube. Using an unusual form of the compression argument, we confirm the conjecture by reducing the problem to a the case of the undirected hypercube. We also prove an analogous conjecture for vertex-disjoint paths using the same techniques, and extend both results to the grid. Additionally, we deal with subcube intersection graphs, answering a question of Johnson and Markström of the least r = r(n) for which all graphs on n vertices may be represented as subcube intersection graph where each subcube has dimension exactly r. We also contribute to the related area of biclique covers and partitions, and study relationships between various parameters linked to such covers and partitions. Finally, we study topological properties of uniformly random simplicial complexes, employing a characterisation due to Korshunov of almost all down-sets in the hypercube as a key tool.

Network structures can encode information from datasets that have a natural representation in terms of networks, for example datasets describing collaborations or social relations among individuals in science or society, as well as from data that can be mapped into graphs due to their intrinsic correlations, such as time series or images. Developing models and algorithms to characterise the structure of complex networks at the micro and mesoscale is thus of fundamental importance to extract relevant information from and to understand real world complex data and systems. In this thesis we will investigate how modularity, a mesoscopic feature observed almost universally in real world complex networks can emerge, and how this phenomenon is related to the appearance of a particular type of network motif, the triad. We will shed light on the role that motifs play in shaping the mesoscale structure of complex networks by considering two special classes of networks, multiplex networks, that describe complex systems where interactions of different nature are involved, and visibility graphs, a family of graphs that can be extracted from the time series of dynamical processes.

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A critical review of some current malaria models is given in which a new model of superinfection is presented. An alternative malaria model, partly stochastic and party deterministic, is then proposed and results of the simulation of the model are discussed. Simplified versions of the model are used to analyse longitudinal survey data from a World Health Organization malaria project, carried out in Northern Nigeria.

Many differential geometric concepts such as (isometric) immersion, stability, etc., realized in Euclidean spaces proved to be also realized in manifolds without conjugate points while other concepts are found to be strictly associated with Euclidean spaces. In fact, this thesis may be considered as a trial for finding out to what extent geometric phenomena in Euclidean spaces are still l valid in manifolds without conjugate points. In the introduction, we have quoted the necessary background material for the following chapters. Specially, we have concentrated on the geometry of submanifolds. The interesting problem of rigidity of submanifolds lies in three different categories : finite rigidity, continuous rigidity and infinitesimal rigidity. These three types of rigidity have been studied in hyperbolic spaces in chapter I, sections 1 and 2. K. Nomizu, B. Snmyth (1969) and S. Braidi, C.C. Hsuing (1970) studied some geometric properties of immersed submanifolds in Euclidean sphere essentially the behaviour of the second fundamental form and the Gauss map. In chapter II (sections 1, 2) we have carried out similar study for immersed submanifolds in hyperbolic spaces which shows some deviations from the corresponding one in Euclidean sphere. Since B.Y. Chen's paper (1973) which established the geometric concept of stability of submanifolds in Euclidean spaces, other geometers tried to extend this concept to non-Euclidean spaces. In chapter II (section 3) we share this development through studying stability of surfaces in hyperbolic 3-dimensional space. The most interesting part of our thesis is the last chapter which deals with tight and taut (convex-minimal) immersions in manifolds without conjugate points. Some geometric concepts such as (spherical) two-piece property, h-two-piece property, total (absolute) curvature,... e t c . , have been introduced. Relations between the above concepts have been adopted. We expect for this part to receive more attention in the future to discover more results and to generalize other Euclidean concepts which we did not touch.

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