In this thesis, we develop a deeper and much more extensive theory of synchronisation of trajectories of random dynamical systems (RDS) than currently exists. In particular, focusing on random dynamical systems with memoryless noise, we achieve two main goals: Firstly, we demonstrate that the notion of "statistical equilibria" is purely a property of the measurable dynamics of a RDS on a standard Borel space; and yet, within such statistical equilibria is "encoded" the phenomenon of noise-induced synchronisation (which may then be observed in *any* compatible metric on the phase space). Secondly, we provide new, widely applicable criteria for synchronisation in RDS, considerably improving upon some of the existing criteria for synchronisation.

This thesis considers the numerical solution to elliptic boundary value problems (BVPs) in convex domains. Specifically we look at the two-dimensional problem in a polygon, and the three dimensional problem in a polyhedron. The nature of elliptic equations means that, knowing the values of a solution on the boundary, one can reconstruct this function inside the domain. This amounts to finding a Dirichlet-to-Neumann (D2N) map, which reconstructs the unknown (Neumann) boundary data from the known (Dirichlet) boundary data. Much is known about the solution to elliptic equations, both theoretically and numerically, but we shall pursue a newer development called the unified approach of [Fok08], or “Fokas method”. It is hoped that the positive results we present here will motivate further inclusion of the Fokas method in numerical packages.

The dimensional reduction method for solving Laplace's equation on a flat plate, an arch and a spherical shell is investigated, extending previous work on laminated plates by Vogelius and Babuska (1981). Convergence rates for the error in energy are obtained, extending previous results by deriving explicit values for the constant of approximation and its dependence on the thickness of the domain and the model order. The framework for laminated plates is shown to easily extend to other geometries. Numerical results are given which verify the convergence rates in terms of the thickness. Details are given as to how to implement the dimensional reduction technique and in particular, for a spherical shell, a method is given which reduces the problem to that of inverting relatively small matrices. A posteriori error estimators are given for each of the geometries under consideration. Error estimators are already known for flat plates. It is shown how the estimators for flat plates can be modified for use in the arch case, and for shells, techniques for estimating the discretization error and modelling error are presented. The a posteriori estimators are then used to derive a refinement algorithm for adaptively constructing hierarchic models for representative problems on each of the geometries under consideration.

The k-ary n-cube interconnection network Qkn, for k 3 and n 2, is n-dimensional network with k processors in each dimension. A k-ary n-cube parallel computer consists of kn identical processors, each provided with its own sizeable memory and interconnected with 2n other processors. The k-ary n-cube has some attractive features like symmetry, high level of concurrency and efficiency, regularity and high potential for the parallel execution of various algorithms. It can efficiently simulate other network topologies. The k-ary n-cube has a smaller degree than that of its equivalent hypercube (the one with at least as many nodes) and it has a smaller diameter than its equivalent mesh of processors. In this thesis, we review some topological properties of the k-ary n-cube Qkn and show how a Hamiltonian cycle can be embedded in Qkn using the Gray codes strategy. We also completely classify when a Qkn contains a cycle of some given length. The problem of embedding a large cycle in a Qkn with both faulty nodes and faulty links is considered. We describe a technique for embedding a large cycle in a k-ary n-cube Qkn with at most n faults and show how this result can be extended to obtain embeddings of meshes and tori in such a faulty k-ary n-cube. Embeddings of Hamiltonian cycles in faulty k-ary n-cubes is also studied. We develop a technique for embedding a Hamiltonian cycle in a k-ary n-cube with at most 4n-5 faulty links where every node is incident with at least two healthy links. Our result is optimal as there exist k-ary n-cubes with 4n - 4 faults (and where every node is incident with at least two healthy links) not containing a Hamiltonian cycle. We show that the same technique can be easily applied to the hypercube. We also show that the general problem of deciding whether a faulty k-ary n-cube contains a Hamiltonian cycle is NP-complete, for all (fixed) k 3.

In 1950, Laurent Schwartz marked a convenient starting point for the theory of generalized functions as a subject in its own right. He developed and unified much of the earlier work by Hadamard, Bochner, Sobolev and others. Since then an enormous literature dealing with both theory and applications has grown up, and the subject has undergone extensive further development. The original Schwartz treatment defined a distribution as a linear continuous functional on a space of test functions. This thesis can be considered a part of the development going in that direction. It is partly an extension of earlier contributions by Fisher, Kuribayashi, Itano and others. After introducing the background and basic definitions in Chapter One, we developed some basic results concerning the cosine integral Ci(lambda x) and its associated functions Ci+(lambda x) and Ci-(lambdax) as well as the neutrix convolution products of the cosine integral. Chapter Three is devoted to similar results concerning the sine integral Si(lambdax). In Chapter Four, we generalize some earlier results by Fisher and Kuribayashi concerning the product of the two dimensions xl+ and x-l-r+ . Moreover, other results are obtained concerning the neutrix product of |x|lambda-r lnp |x| and sgn x| x|lambda-r. Other theorems are proved about the matrix product of some other distributions such as xl+ ln x+ and x-l-r- . Chapter Five contains new results about the composition of distributions. It involves the applications of the neutrix limit to establish such relationships between different distributions.

In this PhD thesis we explore the concept of a multidimensional asymptotic class. This is a new notion in model theory, arising as a generalisation of the Elwes–Macpherson–Steinhorn notion of an N-dimensional asymptotic class [22] and thus ultimately as a development of the Lang–Weil estimates of the number of points of a variety in a finite field [47]. We provide the history and motivation behind the topic before developing its basic theory, paying particular attention to multidimensional exact classes, a special kind of multidimensional asymptotic class where the measuring functions provide the precise sizes of the definable sets, rather than only approximations. We describe a number of examples and non-examples and then show that multidimensional asymptotic classes are closed under bi-interpretability. We use results about smoothly approximable structures [35] and Lie coordinatisable structures [18] to prove the following result, as conjectured by Macpherson: For any countable language L and any positive integer d the class C(L,d) of all finite L-structures with at most d 4-types is a polynomial exact class in L; here a polynomial exact class is a multidimensional exact class with polynomial measuring functions. We finish the thesis by posing some open questions, indicating potential further lines of research.

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