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.

Looking under the bonnet:factors affecting student adoption of e-learning systems in Jordan

The primary questions addressed in this paper are the following: what are the factors that affect students’ adoption of an e-learning system and what are the relationships among these factors? This paper investigates and identifies some of the major factors affecting students’ adoption of an e-learning system in a university in Jordan. E-learning adoption is approached from the information systems acceptance point of view. This suggests that a prior condition for learning effectively using e-learning systems is that students must actually use them. Thus, a greater knowledge of the factors that affect IT adoption and their interrelationships is a pre-cursor to a better understanding of student acceptance of e-learning systems. In turn, this will help and guide those who develop, implement, and deliver e-learning systems. In this study, an extended version of the Technology Acceptance Model (TAM) was developed to investigate the underlying factors that influence students’ decisions to use an e-learning system. The TAM was populated using data gathered from a survey of 486 undergraduate students using the Moodle based e-learning system at the Arab Open University. The model was estimated using Structural Equation Modelling (SEM). A path model was developed to analyze the relationships between the factors to explain students’ adoption of the e-learning system. Whilst findings support existing literature about prior experience affecting perceptions, they also point to surprising group effects, which may merit future exploration.

Hanchett-Stamford V (1) HM Attorney-General and (2) Dr William Johnston Jordan [2008] Ewhc 330 (CH), 2008 WL 1968855

Casenote considers meaning and impact of ruling of High Court in Hanchett Stamford v HM A.G. The decision of Mr Justice Lewison in Hanchett-Stamford v HM Attorney General and Dr William Johnston Jordan1 provides us with a useful analysis of the legal principles relating to the thorny issues of: (i) how unincorporated associations hold property; (ii) the applicability of the law of charities to unincorporated associations and (iii) the property rights of a declining membership upon the dissolution of such associations.

A Computer Algebra Application to Determination of Lie Symmetries of Partial Differential Equations

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006

The Automorphism Group of the Free Algebra of Rank Two

The theorem of Czerniakiewicz and Makar-Limanov, that all the automorphisms of a free algebra of rank two are tame is proved here by showing that the group of these automorphisms is the free product of two groups (amalgamating their intersection), the group of all affine automorphisms and the group of all triangular automorphisms. The method consists in finding a bipolar structure. As a consequence every finite subgroup of automorphisms (in characteristic zero) is shown to be conjugate to a group of linear automorphisms.

Operational Methods in the Environment of a Computer Algebra System

This article presents the principal results of the doctoral thesis “Direct Operational Methods in the Environment of a Computer Algebra System” by Margarita Spiridonova (Institute of mathematics and Informatics, BAS), successfully defended before the Specialised Academic Council for Informatics and Mathematical Modelling on 23 March, 2009.

Quasi-Monte Carlo Methods for some Linear Algebra Problems. Convergence and Complexity

We present quasi-Monte Carlo analogs of Monte Carlo methods for some linear algebra problems: solving systems of linear equations, computing extreme eigenvalues, and matrix inversion. Reformulating the problems as solving integral equations with a special kernels and domains permits us to analyze the quasi-Monte Carlo methods with bounds from numerical integration. Standard Monte Carlo methods for integration provide a convergence rate of O(N^(−1/2)) using N samples. Quasi-Monte Carlo methods use quasirandom sequences with the resulting convergence rate for numerical integration as good as O((logN)^k)N^(−1)). We have shown theoretically and through numerical tests that the use of quasirandom sequences improves both the magnitude of the error and the convergence rate of the considered Monte Carlo methods. We also analyze the complexity of considered quasi-Monte Carlo algorithms and compare them to the complexity of the analogous Monte Carlo and deterministic algorithms.

Automorphisms of the Planar Tree Power Series Algebra and the Non-Associative Logarithm

2000 Mathematics Subject Classification: 17A50, 05C05.

Microlocal Approach to Tensor Tomography and Boundary and Lens Rigidity

2000 Mathematics Subject Classification: 53C24, 53C65, 53C21.

On Generalized Models and Singular Products of Distributions in Colombeau Algebra G(R)

MSC 2010: 46F30, 46F10

Convexity around the Unit of a Banach Algebra

2000 Mathematics Subject Classification: Primary: 46B20. Secondary: 46H99, 47A12.