A class of perfect fluids in general relativity

This thesis is concerned with exact solutions of Einstein's field equations of general relativity, in particular, when the source of the gravitational field is a perfect fluid with a purely electric Weyl tensor. General relativity, cosmology and computer algebra are discussed briefly. A mathematical introduction to Riemannian geometry and the tetrad formalism is then given. This is followed by a review of some previous results and known solutions concerning purely electric perfect fluids. In addition, some orthonormal and null tetrad equations of the Ricci and Bianchi identities are displayed in a form suitable for investigating these space-times. Conformally flat perfect fluids are characterised by the vanishing of the Weyl tensor and form a sub-class of the purely electric fields in which all solutions are known (Stephani 1967). The number of Killing vectors in these space-times is investigated and results presented for the non-expanding space-times. The existence of stationary fields that may also admit 0, 1 or 3 spacelike Killing vectors is demonstrated. Shear-free fluids in the class under consideration are shown to be either non-expanding or irrotational (Collins 1984) using both orthonormal and null tetrads. A discrepancy between Collins (1984) and Wolf (1986) is resolved by explicitly solving the field equations to prove that the only purely electric, shear-free, geodesic but rotating perfect fluid is the Godel (1949) solution. The irrotational fluids with shear are then studied and solutions due to Szafron (1977) and Allnutt (1982) are characterised. The metric is simplified in several cases where new solutions may be found. The geodesic space-times in this class and all Bianchi type 1 perfect fluid metrics are shown to have a metric expressible in a diagonal form. The position of spherically symmetric and Bianchi type 1 space-times in relation to the general case is also illustrated.

Variational mean-field algorithm for efficient inference in large systems of stochastic differential equations

This work introduces a Gaussian variational mean-field approximation for inference in dynamical systems which can be modeled by ordinary stochastic differential equations. This new approach allows one to express the variational free energy as a functional of the marginal moments of the approximating Gaussian process. A restriction of the moment equations to piecewise polynomial functions, over time, dramatically reduces the complexity of approximate inference for stochastic differential equation models and makes it comparable to that of discrete time hidden Markov models. The algorithm is demonstrated on state and parameter estimation for nonlinear problems with up to 1000 dimensional state vectors and compares the results empirically with various well-known inference methodologies.

A mean field theory of coded CDMA systems

We present a mean field theory of code-division multiple access (CDMA) systems with error-control coding. On the basis of the relation between the free energy and mutual information, we obtain an analytical expression of the maximum spectral efficiency of the coded CDMA system, from which a mean field description of the coded CDMA system is provided in terms of a bank of scalar Gaussian channels whose variances in general vary at different code symbol positions. Regular low-density parity-check (LDPC)-coded CDMA systems are also discussed as an example of the coded CDMA systems.

Developments and extensions of Roth's method of double Fourier series with applications to the solution of electromagnetic field problems

The first part of the thesis compares Roth's method with other methods, in particular the method of separation of variables and the finite cosine transform method, for solving certain elliptic partial differential equations arising in practice. In particular we consider the solution of steady state problems associated with insulated conductors in rectangular slots. Roth's method has two main disadvantages namely the slow rate of con­vergence of the double Fourier series and the restrictive form of the allowable boundary conditions. A combined Roth-separation of variables method is derived to remove the restrictions on the form of the boundary conditions and various Chebyshev approximations are used to try to improve the rate of convergence of the series. All the techniques are then applied to the Neumann problem arising from balanced rectangular windings in a transformer window. Roth's method is then extended to deal with problems other than those resulting from static fields. First we consider a rectangular insulated conductor in a rectangular slot when the current is varying sinusoidally with time. An approximate method is also developed and compared with the exact method.The approximation is then used to consider the problem of an insulated conductor in a slot facing an air gap. We also consider the exact method applied to the determination of the eddy-current loss produced in an isolated rectangular conductor by a transverse magnetic field varying sinusoidally with time. The results obtained using Roth's method are critically compared with those obtained by other authors using different methods. The final part of the thesis investigates further the application of Chebyshdev methods to the solution of elliptic partial differential equations; an area where Chebyshev approximations have rarely been used. A poisson equation with a polynomial term is treated first followed by a slot problem in cylindrical geometry.

Resonant gradient force on atom interacting with laser field

The gradient force, as a function of position and velocity, is derived for a two-level atom interacting with a standing-wave laser field. Basing on optical Bloch equations, the numerical solutions for the gradient force f_(|_;n) (n = 0, 1, 2, 3, 4, ...) pointing in the direction of the transverse of the laser beam are given. It is shown the higher order gradient force plays important role at strong intensity (G = 64), the contribution of them can not be neglected.

Gravity field recovery using two low satellites in different orbital planes

This study is concerned with gravity field recovery from low-low satellite to satellite range rate data. An improvement over a coplanar mission is predicted in the errors associated with certain parts of the geopotential by the separation of the orbital planes of the two satellites. Using Hill's equations an analytical scheme to model the range rate residuals is developed. It is flexible enough to model equally well the residuals between pairs of satellites in the same orbital plane or whose planes are separated in right ascension. The possible benefits of such an orientation to gravity field recovery from range rate data can therefore be analysed, and this is done by means of an extensive error analysis. The results of this analysis show that for an optimal planar mission improvements can be made by separating the satellites in right ascension. Gravity field recoveries are performed in order to verify and gauge the limitations of the analytical model, and to support the results of the error analysis. Finally the possible problem of the differential decay rates of two satellites due to the diurnal bulge are evaluated.

Boundary integral equations for acoustical inverse sound-soft scattering

The shape of a plane acoustical sound-soft obstacle is detected from knowledge of the far field pattern for one time-harmonic incident field. Two methods based on solving a system of integral equations for the incoming wave and the far field pattern are investigated. Properties of the integral operators required in order to apply regularization, i.e. injectivity and denseness of the range, are proved.

The single-phase induction motor:a critical appraisal of the rotating-field and cross-field theories with particular reference to skin effect

The purpose of this thesis is twofold: to examine the validity of the rotating-field and cross-field theories of the single-phase induction motor when applied to a cage rotor machine; and to examine the extent to which skin effect is likely to modify the characteristics of a cage rotor machine. A mathematical analysis is presented for a single-phase induction motor in which the rotor parameters are modified by skin effect. Although this is based on the usual type of ideal machine, a new form of model rotor allows approximations for skin effect phenomena to be included as an integral part of the analysis. Performance equations appropriate to the rotating-field and cross-field theories are deduced, and the corresponding explanations for the steady-state mode of operation are critically examined. The evaluation of the winding currents and developed torque is simplified by the introduction of new dimensionless factors which are functions of the resistance/reactance ratios of the rotor and the speed. Tables of the factors are included for selected numerical values of the parameter ratios, and these are used to deduce typical operating characteristics for both cage and wound rotor machines. It is shown that a qualitative explanation of the mode of operation of a cage rotor machine is obtained from either theory; but the operating characteristics must be deduced from the performance equations of the rotating-field theory, because of the restrictions on the values of the rotor parameters imposed by skin effect.

Using overlay test to evaluate fracture properties of field-aged asphalt concrete

Field material testing provides firsthand information on pavement conditions which are most helpful in evaluating performance and identifying preventive maintenance or overlay strategies. High variability of field asphalt concrete due to construction raises the demand for accuracy of the test. Accordingly, the objective of this study is to propose a reliable and repeatable methodology to evaluate the fracture properties of field-aged asphalt concrete using the overlay test (OT). The OT is selected because of its efficiency and feasibility for asphalt field cores with diverse dimensions. The fracture properties refer to the Paris’ law parameters based on the pseudo J-integral (A and n) because of the sound physical significance of the pseudo J-integral with respect to characterizing the cracking process. In order to determine A and n, a two-step OT protocol is designed to characterize the undamaged and damaged behaviors of asphalt field cores. To ensure the accuracy of determined undamaged and fracture properties, a new analysis method is then developed for data processing, which combines the finite element simulations and mechanical analysis of viscoelastic force equilibrium and evolution of pseudo displacement work in the OT specimen. Finally, theoretical equations are derived to calculate A and n directly from the OT test data. The accuracy of the determined fracture properties is verified. The proposed methodology is applied to a total of 27 asphalt field cores obtained from a field project in Texas, including the control Hot Mix Asphalt (HMA) and two types of warm mix asphalt (WMA). The results demonstrate a high linear correlation between n and −log A for all the tested field cores. Investigations of the effect of field aging on the fracture properties confirm that n is a good indicator to quantify the cracking resistance of asphalt concrete. It is also indicated that summer climatic condition clearly accelerates the rate of aging. The impact of the WMA technologies on fracture properties of asphalt concrete is visualized by comparing the n-values. It shows that the Evotherm WMA technology slightly improves the cracking resistance, while the foaming WMA technology provides the comparable fracture properties with the HMA. After 15 months aging in the field, the cracking resistance does not exhibit significant difference between HMA and WMAs, which is confirmed by the observations of field distresses.