Using the method of dimensional reduction of N=l supersymmetric Yang-Mills theories from higher dimensions down to four dimensions, all possible supersymmetric Yang-Mills theories in four dimensions are obtained. The conserved currents associated with the symmetries of these models are then developed using Noether's theorem in ordinary space-time. By the variation of these conserved currents under supersymmetry transformations the supercurrent multiplets for the different models are obtained. Supersymmetric gauge theories are then discussed in superspace where differential geometry can be used to obtain Bianchi identities for the supersymmetric field strengths. The constraints on the field strengths that give rise to off-shell representations for each of the different supersymmetric gauge theories are then obtained and off-shell Lagrangians written down. The connection with supersymmetric gauge theories in ordinary space-time is made. The supercurrents in superspace are then derived using the generalization of Noether's theorem to superspace.

This thesis deals with nonlocal models for solid-solid phase transitions, such as ferromagnetic phase transition or phase separation in binary alloys. We discuss here, among others, nonlocal versions of the Allen-Cahn and Cahn-Hilliard equations, as well as a nonlocal version of the viscous Cahn-Hilliard equation. The analysis of these models can be motivated by the fact that their local analogues fail to be applicable when the wavelength of microstructure is very small, e. g. at the nanometre scale. Though the solutions of these nonlocal equations and those of the local versions share some common properties, we find many differences between them, which are mainly due to the lack of compactness of the semigroups generated by nonlocal equations. Directly from microscopic considerations, we derive and analyse two new types of equations. One of the equations approximately represents the dynamic Ising model with vacancy-driven dynamics, and the other one is the vacancy-driven model obtained using the Vineyard formalism. These new equations are being put forward as possible improvements of the local and nonlocal Cahn-Hilliard models, as well as of the mean-field model for the Ising model with Kawasaki dynamics.

A comparison of boundary approximations used in numerical solution of one-dimensional hyperbolic systems of partial differential equations is undertaken. Stability and accuracy studies of the boundary approximations are conducted for a variety of interior schemes. A new fourth order accurate finite-element scheme is proposed.

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Let A be a C*-algebra and B be a C*-subalgebra of A. B is said to have the extension property in A if every pure state of B has unique state extension to A. Such pairs (A, B) are considered where B is a maximal abelian self-adjoint subalgebra (masa) of A with the extension property. For A unital and n-homogeneous, results are obtained for the structure of (A, B) as fibre bundles over the primitive ideal space of A (2.2). For A a Type I C -algebra, necessary and sufficient conditions are given for B" to be maximal abelian in A" for all representations of A (2.3). The problem of whether an atomic masa of JL(H) has the extension property in i(H) is related to questions about other pairs (A,B) (3.4.2). A subalgebra of 3t(H) is constructed in which the atomic masa does have the extension property (3.2). Let Fn denote the free group on n generators, A = C*(Fn) and B be the masa of A generated by the image of one of the group generators. It is shown that states of A which restrict to pure states of B are pairwise non-equivalent (3.4.3). A similar result is obtained for states of the Calkin algebra Q which restrict to pure states of a masa of Q constructed by Anderson (3.4.4). In particular, Anderson's masa is not the image of an atomic masa of (H). Conditions of a dynamical nature are given for B to have the extension property in A for certain cases where B is a masa of a C*-crossed product algebra A (4.2). Let A be the W*-crossed product algebra for a W*-dynamical system (M,G,?). For G an amenable group, certain G-invariant states of M are shown to induce conditional expectations from A onto the group von Neumann algebra (G) (4.3, 4.4). This result generalises that of Anderson concerning the existence of many conditional expectations from (H) onto a continuous masa of (H). In conclusion it must be said that the results of this thesis throw little light on the question of whether pure states of an atomic masa of have unique state extensions to (H). The limitations of an analytic approach are illustrated by the failure of Chapter 4 to provide results for actions by compact groups. In view of the remarks of 3.3.5 it seems likely that a solution of the atomic masa problem will require hard results of a geometrical or combinatorial nature for finite dimensional algebras to settle the question of uniform compressibility in ? ? Mn.

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No description available.

This thesis is a study of the integration of proximity functions over certain compact groups. Mean values are found of the ultrametric valuation of certain rational functions associated with a divisor on an abelian variety, and it is shown how these may be expressed in terms of an integral, thus finding the analogue, for an abelian variety, of Mahler's definition of the measure of a polynomial. These integrals are shown to arise in a manner which mimics classical Riemann sums, and their relation with the global canonical height is investigated. It is shown that the measure is a rational multiple of log p. Similar results are given for elliptic curves, taking the divisor to be the identity of the group law, and somewhat stronger mean value theorems proven in this more specific case by working directly with local canonical heights rather than approaching them through related functions. Effective asymptotic formulae for the local height are derived, first for the kernel of reduction of a curve and then, via a detailed analysis of the local reduction of the curve, for the group of rational points. The theory of uniform distribution is used to show that the mean value also takes an integral form in the case of an archimedean valuations, and recent inequalities for elliptic forms in logarithms are used to give error terms for the convergence towards the measure. This is undertaken first for the local height on an elliptic curve, and then, in terms of general theta-functions, on an abelian variety. We then seek to exploit these generalisations of the Mahler measure to yield an alternative method to that of Silverman and Tate for the determining of the global height. The integration over a cyclic group of the laws satisfied locally by the height allows us to reformulate our theorems in a manner conducive to practical application. It is demonstrated how our asymptotic formulae may be used together with an appropriate computer software package, PARI in our case, to calculate the mean value of heights, and, more generally, of rational functions, on an elliptic curve and on abehan varieties of higher genus. Some such calculations are displayed, with comments on their efficacy and their possible future development.

Sliding mode control is a well-known approach to the problem of the control of uncertain systems, since it is invariant to a class of parameter variations. Well-established investigations have shown that the sliding mode controller/ observer is a good approach from the point of view of robustness, implementation, numerical stability, applicability, ease of design tuning and overall evaluation. In the sliding mode control approach, the controller and/ or observer is designed so that the state trajectory converges to a surface named the sliding surface. It is desired to design the sliding surface so that the system stability is achieved. Many new methods and design techniques for the sliding controller/ observer are presented in this thesis. LQ frequency shaping sliding mode is a way of designing the sliding surface and control. By using this method, corresponding to the weighting functions in conventional quadratic performance, a compensator can be designed. The intention of observer design is to find an estimate for the state and, if the input is unknown, estimate a suitable input. Using the sliding control input form, a suitable estimated input can be obtained. The significance of the observer design method in this thesis is that a discontinuous observer for full order systems, including disturbance inputs, is designed. The system may not be ideally in the sliding mode and the uncertainty may not satisfy the matching condition. In discrete-time systems instead of having a hyperplane as in the continuous case, there is a countable set of points comprising a so-called lattice; and the surface on which these sliding points lie is named the latticewise hyperplane. Control and observer design using the discrete-time sliding mode, the robust stability of the sliding mode dynamics and the problem of stabilization of discrete-time systems are also studied. The sliding mode control of time-delay systems is also considered. Time-delay sliding system stability is studied for the cases of full information about the delay and also lack of information. The sliding surface is delay-independent as for the traditional sliding surface, and the reaching condition is achieved by applying conventional discontinuous control. A well-known method of control design is to specify eigenvalues in a region in the left-hand half-plane, and to design the gain feedback matrix to yield these eigenvalues. This method can also be used to design the sliding gain matrix. The regions considered in this thesis are, a sector, an infinite vertical strip, a disc, a hyperbola and the intersection ii of two sectors. Previous erroneous results are rectified and new theory developed. The complex Riccati equation, positivity of a complex matrix and the control of complex systems are significant problems which arise in many control theory problems and are discussed in this thesis.

This thesis is concerned with Electrical Impedance Tomography (EIT), an imaging technique in which pictures of the electrical impedance within a volume are formed from current and voltage measurements made on the surface of the volume. The focus of the thesis is the mathematical and numerical aspects of reconstructing the impedance image from the measured data (the reconstruction problem). The reconstruction problem is mathematically difficult and most reconstruction algorithms are computationally intensive. Many of the potential applications of EIT in medical diagnosis and industrial process control depend upon rapid reconstruction of images. The aim of this investigation is to find algorithms and numerical techniques that lead to fast reconstruction while respecting the real mathematical difficulties involved. A general framework for Newton based reconstruction algorithms is developed which describes a large number of the reconstruction algorithms used by other investigators. Optimal experiments are defined in terms of current drive and voltage measurement patterns and it is shown that adaptive current reconstruction algorithms are a special case of their use. This leads to a new reconstruction algorithm using optimal experiments which is considerably faster than other methods of the Newton type. A tomograph is tested to measure the magnitude of the major sources of error in the data used for image reconstruction. An investigation into the numerical stability of reconstruction algorithms identifies the resulting uncertainty in the impedance image. A new data collection strategy and a numerical forward model are developed which minimise the effects of, previously, major sources of error. A reconstruction program is written for a range of Multiple Instruction Multiple Data, (MIMD), distributed memory, parallel computers. These machines promise high computational power for low cost and so look promising as components in medical tomographs. The performance of several reconstruction algorithms on these computers is analysed in detail.