Thesis (Ph. D.)--University of Hong Kong, 1982.

published_or_final_version / Civil and Structural Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Civil and Structural Engineering / Doctoral / Doctor of Philosophy

A nonlinear wave equation invariant with respect to unitary representations of the Lorentz group is considered in an attempt to describe extended particles with spin and positive definite energy by means of a self-confined classical field. The wave function has an infinite number of components and, in the specific representations used, the corresponding internal degree of freedom is identified with the spin. A fractional power of the scalar bilinear invariant appears as an appropriate choice for the nonlinearity in order that all the stationary states be localized. Two approximation methods are proposed and both lead to results that bear a resemblance to the results of the MIT bag model.

Most real world phenomena is modeled by ordinary and/or partial differential equations. Most of these equations are highly nonlinear and exact solutions are not always possible. Exact solutions always give a good account of the physical nature of the phenomena modeled. However, existing analytical methods can only handle a limited range of these equations. Semi-numerical and numerical methods give approximate solutions where exact solutions are impossible to find. However, some common numerical methods give low accuracy and may lack stability. In general, the character and qualitative behaviour of the solutions may not always be fully revealed by numerical approximations, hence the need for improved semi-numerical methods that are accurate, computational efficient and robust. In this study we introduce innovative techniques for finding solutions of highly nonlinear coupled boundary value problems. These techniques aim to combine the strengths of both analytical and numerical methods to produce efficient hybrid algorithms. In this work, the homotopy analysis method is blended with spectral methods to improve its accuracy. Spectral methods are well known for their high levels of accuracy. The new spectral homotopy analysis method is further improved by using a more accurate initial approximation to accelerate convergence. Furthermore, a quasi-linearisation technique is introduced in which spectral methods are used to solve the linearised equations. The new techniques were used to solve mathematical models in fluid dynamics. The thesis comprises of an introductory Chapter that gives an overview of common numerical methods currently in use. In Chapter 2 we give an overview of the methods used in this work. The methods are used in Chapter 3 to solve the nonlinear equation governing two-dimensional squeezing flow of a viscous fluid between two approaching parallel plates and the steady laminar flow of a third grade fluid with heat transfer through a flat channel. In Chapter 4 the methods were used to find solutions of the laminar heat transfer problem in a rotating disk, the steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation and the classical von Kάrmάn equations for boundary layer flow induced by a rotating disk. In Chapter 5 solutions of steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain bounded by two permeable surfaces and the MHD viscous flow problem due to a shrinking sheet with a chemical reaction, were solved using the new methods. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.

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Much of the low energy phenomenology which can be extracted from the field theory limit of the intrinsically ten dimensional E 8 ® E 8 heterotic superstring depends upon the topological and geometrical properties of the six dimensional compactified component of spacetime. After briefly reviewing the topological constraints on the latter manifold which ensure the survival of N=l four dimensional supersymmetry, we present and apply the mathematics necessary for the rigorous construction of vacuum solutions and the determination of the four dimensional massless field content. Two phenomenologically attractive classes of solutions, with unbroken E₈ ⨂ SU(5) and E₈ ⨂ SO(10) gauge groups, arise if the vacuum configuration contains a Ricci flat Kahler manifold with SU(3) holonomy (Calabi-Yau manifold), which admits certain SU(5) or SU(4) vector bundles. Further reduction of the gauge group and emergence of naturally light weak Higgs doublets may also occur by flux breaking if the Calabi Yau manifold is multiply connected. We analyse the feasibility of such scenarios for Calabi Yau manifolds with any possible fundamental group. Phenomenological considerations place severe constraints on the dimensions and transformation properties of certain cohomology groups and thereby lead to a highly restricted class of acceptable models. We then present the mathematical analysis of a three generation heterotic superstring inspired model, with E₈ ⨂ E₆ gauge symmetry. A detailed description of the manifold of compactification is given, along with a determination of its Hodge numbers and of the associated light supermultiplet structure. For a particular choice of vacuum moduli we derive this manifold's symmetry group, and determine its action on the massless fields in the theory. Preliminary investigation indicates that these transformation properties give rise to a remarkably realistic model. In the second volume we derive cosmological constraints on a supersymmetric extension of the standard model in which weak gauge symmery breaking is triggered at the tree level by a Higgs singlet superfield. The fermionic component of this gauge singlet (the "nino") is shown to be the lightest supersymmetric particle with a relic abundance near the critical closure density for a surprisingly wide range of the unconstrained parameters. The previously favoured photino dark matter scenario has been eliminated by the non observation of high energy solar neutrinos. After briefly reviewing this argument, we extend the analysis to eliminate Higgsino dark matter scenarios with and#60H₁°and#62 ≠ and#60H₂°and#62. We show that the nino produces an acceptably low level of solar neutrinos and that it may also account for the anomalously high level of cosmic ray antiproton flux.

We investigate the dynamical generation of fermion mass in quantum electrodynamics (QED). This non-perturbative study is performed using a truncated set of Schwinger-Dyson equations for the fermion and the photon propagator. First, we study dynamical fermion mass generation in quenched QED with the Curtis-Pennington vertex, which satisfies the Ward-Takahashi identity and moreover ensures the multiplicative renormalizability of the fermion propagator. We apply bifurcation analysis to determine the critical point for a general covariant gauge. In the second part of this work we investigate the dynamical generation of fermion mass in full, unquenched QED. We develop a numerical method to solve the system of three coupled non-linear equations for the dynamical fermion mass, the fermion wavefunction renormalization and the photon renormalization function. Much care is taken to ensure the high accuracy of the solutions. Moreover, we discuss in detail the proper numerical cancellation of the quadratic divergence in the vacuum polarization integral and the requirement of using smooth approximations to the solutions. To achieve this, we improve the numerical method by introducing the Chebyshev expansion method. We apply this method to the bare vertex approximation to unquenched QED to determine the critical coupling for a variety of approximations. This culminates in the detailed, highly accurate, solution of the Schwinger-Dyson equations for dynamical fermion mass generation in QED including both, the photon renormalization function and the fermion wavefunction renormalization in a consistent way, in the bare vertex approximation and, for the first time, using improved vertices. We introduce new improvements to the numerical method, to achieve the accuracy necessary to avoid unphysical quadratic divergences in the vacuum polarization with the Ball-Chiu vertex.

This thesis addresses the problem of how to examine a metabolic pathway and identify what are the key elements, specifically with respect to rate-limitation. The aim is to be able to analyze a pathway, identify the bottlenecks and implement genetic modifications to remove these bottlenecks. This is done by defining the system of interest and developing a predictive model using kinetic data. The model predictions can then be verified using fermentation data and genetic techniques to make the appropriate changes for improved performance. The test system chosen for this study was the TOL meta-cleavage pathway for the degradation of benzoate. This system was chosen on the basis of the application of pathway engineering principles to other systems. The modelling strategy and software was developed using principles from metabolic control theory and biochemical systems theory. By applying this to the TOL pathway using kinetic data, the control coefficients for the pathway were obtained as well as the system parameters required for the optimization of the pathway. The simulated results obtained from this model must be validated by experiment. Errors can arise both from incorrect assumptions in the model and from the fact that the kinetic data taken from individual in vitro experiments may not be applicable to the in vivo system. The effect of the presence of the TOL pathway on the behaviour of E.coli JM107 during fermentation was investigated and the transient concentration data necessary to identify the bottlenecks in the pathway measured. This data is then used to calculate the flux control coefficients for the TOL pathway. The predictive results were verified by the fermentation data which identified the first two enzymes in the pathway as having significant flux control coefficients. This final chapter also addresses the issue of flux analysis, that is, the calculation of the fluxes in the system to determine where fluxes to unwanted by-products occur and to indicate points of control. A graphical user interface is used to provide a user-friendly and intuitive means of building and customising metabolic pathways which can then be interfaced with instrumentation to provide on-line flux analysis.

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