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This thesis studies the theory of Riemannian 4-symmetric spaces. It follows the methods first introduced by E. Cartan to study ordinary symmetric spaces, and extended by J. Wolf and A. Gray and by Kac. The theory of generalized n-symmetric spaces was initiated by A. Ledger in 1967, and 2- and 3-symmetric spaces have already been classified. The theory of generalized n-symmetric spaces is completely new. The thesis naturally divides into two chapters. The first chapter treats the geometry of the spaces. Their homogeneous structure and their invariant connections are studied. The existence of a canonical invariant almost product structure is pointed out. A fibration over 2-symmetric spaces with 2-symmetric fibers is obtained. Root systems are used to obtain geometric invariants. Finally a local characterization in terms of curvature is obtained. Chapter II centres on the problems of classification. A local classification is given for the compact spaces in terms of simple Lie algebras. A global formulation is given for the compact classical simple Lie algebras. A final section is devoted to invariant almost complex structures. A characterization is given in terms of their homogeneous structure. It is shown that they can bear both Hodge and non-Kahler structures.

Recent developments in computing technology have opened new prospects for computationally intensive numerical methods such as the finite element method. More complex and refined problems can be solved, for example increased number and order of the elements improving accuracy. The power of Computer Algebra systems and parallel processing techniques is expected to bring significant improvement in such methods. The main objective of this work has been to assess the use of these techniques in the finite element method. The generation of interpolation functions and element matrices has been investigated using Computer Algebra. Symbolic expressions were obtained automatically and efficiently converted into FORTRAN routines. Shape functions based on Lagrange polynomials and mapping functions for infinite elements were considered. One and two dimensional element matrices for bending problems based on Hermite polynomials were also derived. Parallel solvers for systems of linear equations have been developed since such systems often arise in numerical methods. Both symmetric and asymmetric solvers have been considered. The implementation was on Transputer-based machines. The speed-ups obtained are good. An analysis by finite element method of a free surface flow over a spillway has been carried out. Computer Algebra was used to derive the integrand of the element matrices and their numerical evaluation was done in parallel on a Transputer-based machine. A graphical interface was developed to enable the visualisation of the free surface and the influence of the parameters. The speed- ups obtained were good. Convergence of the iterative solution method used was good for gated spillways. Some problems experienced with the non-gated spillways have lead to a discussion and tests of the potential factors of instability.

Multi-instanton solutions of four dimensional HP(^1) models are sought, and a singular two instant solution in flat Euclidean space time is constructed. Non-singular multi-instanton solution can be constructed if a gravitational field is introduced, as first pointed out by Gursey et al. Their method is developed, and in the process a formalism for the construction of an (anti) self-dual SU(2) Yang- Mills field tensor in curved space-times is exhibited. Demanding that a potential for the SU(2) field exists implies that, for a space of non-zero scalar curvature, Einstein's field equations must be satisfied, and conditions on the Weyl tensor are found. It is shown how the formalism relates to the work of Charap and Duff Finally the method is applied to the four dimensional complex projective space and the four dimensional manifold consisting of the outer product of two two spheres.

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This thesis gives some asymptotic formulae associated with some non-negative multiplicative functions, and introduce a new method for proof of some classical formulae in number theory using the Laplace transform.

Matrix equations have been studied by Mathematicians for many years. Interest in them has grown due to the fact that these equations arise in many different fields such as vibration analysis, optimal control, stability theory etc. This thesis is concerned with methods of solution of various matrix equations with particular emphasis on quadratic matrix equations. Large scale numerical techniques are not investigated but algebraic aspects of matrix equations are considered. Many established methods are described and the solution of a matrix equation by consideration of an equivalent system of multivariable polynomial equations is investigated. Matrix equations are also solved by a method which combines the given equation with the characteristic equation of the unknown matrix. Several iterative processes used for the solution of scalar equations are applied directly to the matrix equation. A new iterative process based on elimination methods is also described and examples given. The solutions of the equation x2 = P are obtained by a method which derives a set of polynomial equations connecting the characteristic coefficients of X and P. It is also shown that the equation X2 = P has an infinite number of solutions if P is a derogatory matrix.

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Methods for computing an efficient and accurate numerical solution of the real monic unilateral quadratic matrix equation, are few. They are not guaranteed to work on all problems. One of the methods performs a sequence of Newton iterations until convergence occurs whilst another is a matrix analogy of the scalar polynomial algorithm. The former fails from a poor starting point and the latter fails if no dominant solution exists. A recent approach, the Elimination method, is analysed and shown to work on problems for which other methods fail. . The method requires the coefficients of the characteristic polynomial of a matrix to be computed and to this end a comparative numerical analysis of a number of methods for computing the coefficients is performed. A new minimisation approach for solving the quadratic matrix equation is proposed and shown to compare very favourably with existing methods . . A special case of the quadratic matrix equation is the matrix square root problem, where P = o. There have been a number of method proposed for it's solution, the more successful ones being based upon Newton iterations or the Schur factorisation. The Elimination method is used as a basis for generating three methods for solving the matrix square root problem. By means of a numerical analysis and results it is shown that for small order problems the Elimination methods compare favourably with the existing methods. The algebraic Riccati equation of stochastic and optimal control is, where the solution of interest is the symmetric non-negative definite one. The current methods are based on Newton iterations or the determination of the invariant subspace of the associated Hamiltonian matrix. A new method based on a reformulation of Newton's method is presented. The method reduces the work involved at each iteration by introducing a Schur factorisation and a sparse linear system solver. Numerical results suggest that it may compare favourably with well-established methods. Central to the numerical issues are the discussions on conditioning, stability and accuracy. For a method to yield accurate results, the problem must be well-conditioned and the method that solves the problem must be stable-consequently discussions on conditioning and stability feature heavily in this thesis. The units of measure we use to compare the speed of the methods are the operations count and the Central Processor Unit (CPU) time. We show how the CPU time accurately reflects the amount of work done by an algorithm and that the operations counts of the algorithms correspond with the respective CPU times.

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