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41 
Conformal symmetries : solutions in two classes of cosmological models.Moodley, Manikam. January 1991 (has links)
In this thesis we study the conformal symmetries in two locally rotationally symmetric
spacetimes and the homothetic symmetries of a Bianchi I spacetime. The
conformal Killing equation in a class AIa spacetime (MacCallum 1980), with a G4
of motions, is integrated to obtain the general solution subject to integrability conditions.
These conditions are comprehensively analysed to determine the restrictions
on the metric functions. The Killing vectors are contained in the general conformal
solution. The homothetic vector is obtained and the explicit functional dependence
of the metric functions determined. The class AIa spacetime does not admit a nontrivial
special conformal factor. We also integrate the conformal Killing equation
in the anisotropic locally rotationally symmetric spacetime of class A3 (MacCallum
1980), with a G4 of motions, to obtain the general conformal Killing vector and the
conformal factor subject to integrability conditions. The Killing vectors are obtained
as a special case from the general conformal solution. The homothetic vector is found
for a nonzero constant conformal factor. The explicit functional form of the metric
functions is determined for the existence of this homothetic vector. The spatially
homogeneous and anisotropic A3 spacetime also does not admit a nontrivial special
conformal vector. In the Bianchi I spacetime, with a G3 of motions, the conformal
Killing equation is integrated for a constant conformal factor to generate the homothetic
symmetries. The integrability conditions are solved to determine the functional
dependence of the three timedependent metric functions. / Thesis (M.Sc.)University of Natal, Durban, 1991.

42 
Applications of symmetry analysis to physically relevant differential equations.January 2005 (has links)
We investigate the role of Lie symmetries in generating solutions to differential equations that arise in particular physical systems. We first provide an overview of the Lie analysis and review the relevant symmetry analysis of differential equations in general. The Lie symmetries of some simple ordinary differential equations are found t. illustrate the general method. Then we study the properties of particular ordinary differential equations that arise in astrophysics and cosmology using the Lie analysis of differential equations. Firstly, a system of differential equations arising in the model of a relativistic star is generated and a governing nonlinear equation is obtained for a linear equation of state. A comprehensive symmetry analysis is performed on this equation. Secondly, a second order nonlinear ordinary differential equation arising in the model of the early universe is described and a detailed symmetry analysis of this equation is undertaken. Our objective in each case is to find explicit Lie symmetry generators that may help in analysing the model. / Thesis (M.Sc.)University of KwaZuluNatal, Durban, 2005.

43 
The pricing theory of Asian options.January 2007 (has links)
An Asian option is an example of exotic options. Its payoff depends on the average of the underlying asset prices. The average may be over the entire time period between initiation and expiration or may be over some period of time that begins later than the initiation of the option and ends with the options expiration. The average may be from continuous sampling or may be from discrete sampling. The primary reason to base an option payoff on an average asset price is to make it more difficult for anyone to significantly affect the payoff by manipulation of the underlying asset price. The price of Asian options is not known in closed form, in general, if the arithmetic average is taken into effect. In this dissertation, we shall investigate the pricing theory for Asian options. After a brief introduction to the BlackScholes theory, we derive the partial differential equations for the value process of an Asian option to satisfy. We do this in several approaches, including the usual extension to Asian options of the BlackScholes, and the sophisticated martingale approach. Both fixed and floating strike are considered. In the case of the geometric average, we derive a closed form solution for the Asian option. Moreover, we investigate the Asian option price theory under stochastic volatility which is a recent trend in the study of pathdependent option theory. / Thesis (M.Sc.)University of KwaZuluNatal, Westville, 2007.

44 
Global embeddings of pseudoRiemannian spaces.Moodley, Jothi. January 2007 (has links)
Motivated by various higher dimensional theories in highenergyphysics and cosmology, we consider the local and global isometric embeddings of pseudoRiemannian manifolds into manifolds of higher dimensions. We provide the necessary background in general relativity, topology and differential geometry, and present the technique for local isometric embeddings. Since an understanding of the local results is key to the development of global embeddings, we review some local existence theorems for general pseudoRiemannian embedding spaces. In order to gain insight we recapitulate the formalism required to embed static spherically symmetric spacetimes into fivedimensional Einstein spaces, and explicitly treat some special cases, obtaining local and isometric embeddings for the ReissnerNordstr¨om spacetime, as well as the null geometry of the global monopole metric. We also comment on existence theorems for Euclidean embedding spaces. In a recent result, it is claimed (Katzourakis 2005a) that any analytic ndimensional space M may be globally embedded into an Einstein space M × F (F an analytic realvalued onedimensional field). As a corollary, it is claimed that all product spaces are Einsteinian. We demonstrate that this construction for the embedding space is in fact limited to particular types of embedded spaces. We analyze this particular construction for global embeddings into Einstein spaces, uncovering a crucial misunderstanding with regard to the form of the local embedding. We elucidate the impact of this misapprehension on the subsequent proof, and amend the given construction so that it applies to all embedded spaces as well as to embedding spaces of arbitrary curvature. This study is presented as new theorems. / Thesis (M.Sc.)University of KwaZuluNatal, Westville, 2007.

45 
Inhomogeneous solutions to the Einstein equations.Govender, Gabriel. January 2007 (has links)
In this dissertation we consider spherically symmetric gravitational fields that arise in relativistic astrophysics and cosmology. We first present a general review of static spherically symmetric spacetimes. aand highlight a particular class of exact solutions of the EinsteinMaxwell system for charged spheres. In the case of shearfree spacetimes with heat flow, the integration of the system is reduced to solving the condition of pressure isotropy. This condition is a second order linear differential equation with variable coefficients. By choosing particular forms for the gravitational potentials, several classes of new solutions are generated. We regain known solutions corresponding to coniformal flatness when tidal forces are absent. We also consider expanding, accelerating and shearing models when the heat flux is not present. A new general class of models is found. This new class of shearing solutions contains the model of Maharaj et al (1993) when a parameter is set to zero. Our new solution does not contain a singularity at the stellar centre, and it is therefore useful in modelling the interior of stars. Finally, we demonstrate that the shearing models obtained by Markund and Bradley (1999) do not satisfy the Einstein field equations. / Thesis (M.Sc.)University of KwaZuluNatal, Durban, 2007.

46 
Closure operators on complete lattices with application to compactness.Brijlall, Deonarain. January 1995 (has links)
No abstract available. / Thesis (M.Sc.)University of Natal, Westville, 1995

47 
Multiparameter perturbation analysis of a second grade fluid flow past an oscillating infinite plate.Habyarimana, Faustin. January 2009 (has links)
In this dissertation we consider the two dimensional flow of an incompressible and electrically conducting second grade fluid past a vertical porous plate with constant suction. The flow is permeated by a uniform transverse magnetic field. The aim of this study is to use the multiparameter perturbation technique to study the effects of Eckert numbers on the flow of a pulsatile second grade fluid along a vertical plate. We further aim to investigate the effects of other fluid and physical parameters such as the Prandtl numbers, Hartmann numbers, viscoelastic parameter, angular frequency and suction velocity on boundary layer velocity, temperature, skin friction and the rate of heat transfer. Similarity transformations are used to reduce the governing partial differential equations to ordinary differential equations. We used perturbation methods to solve the coupled ordinary differential equations for zero Eckert number and the multiparameter perturbation technique to solve the coupled ordinary differential equations for small viscoelastic parameters and Eckert numbers. It is found that increasing the Eckert number or the viscoelastic parameter enhances the boundary layer velocity while reducing the temperature, the rate of heat transfer and the skinfriction. The results for the boundary layer velocity and the temperature are presented graphically and discussed. The results for the rate of heat transfer in terms of the Nusselt number and the skin friction are tabulated and discussed. A good agreement is found between these results and other published research. The comparison between the results for zero Eckert numbers and small Eckert numbers is also presented graphically and discussed. / Thesis (M.Sc.)  University of KwaZuluNatal, Pietermaritzburg, 2009.

48 
Population dynamics based on the McKendrickvon Foerster model.Seillier, Robyn. January 1988 (has links)
The current state of information concerning the classical model of deterministic, agedependent population dynamics  the McKendrick von Foerster equation  is overviewed. This model and the related Renewal equation are derived and the parameters involved in both are elaborated upon. Fundamental theorems concerning existence, uniqueness and boundedness of solutions are outlined. A necessary and sufficient condition concerning the stability of equilibrium agedistributions is rederived along different lines. Attention is then given to generalizations of the McKendrickvon Foerster model that have arisen from the inclusion of density dependence into the parameters of the system; the inclusion of harvesting terms; and the extension of the model to describe the dynamics of a twosex population. A technique which reduces the model, under certain conditions on the mortality and fertility functions, to a system of ordinary differential equations is discussed and applied to specific biochemical population models. Emphasis here is on the possible existence of stable limit cycles.The Kolmogorov system of ordinary differential equations and its use in describing the dynamics of predatorprey systems is examined. The Kolmogorov theorem is applied as a simple alternative to a lengthy algorithm for determining whether limit cycles are stable. Agedependence is incorporated into this system by means of a McKendrick  von Foerster equation and the effects on the system of different patterns of ageselective predation are demonstrated. Finally, brief mention is made of recent work concerning the use of the McKendrick  von Foerster equation to describe the dynamics of both predator and prey. A synthesis of the theory and results of a large number of papers is sought
and areas valuable to further research are pointed out. / Thesis (M.Sc.)University of Natal, Durban, 1988.

49 
Cluster mass reconstruction via gravitational lensing.Musonda, Ededias. January 2009 (has links)
The presence of massive objects is detectable in observations via the gravitational lensing effect on light from more distant sources. From this effect it is possible to reconstruct the masses of clusters, and the distribution of matter
within the cluster. However, further theoretical work needs to be done to properly contextualize any proposed projects involving, for instance, SALT data sets. Observational lensing studies use one of two techniques to recover the lens mass distribution: parametric (model dependent) techniques; and, a more recent innovation, nonparametric methods. The latter deserves further study as a tool for cluster surveys. To this end, we provide a comprehensive analysis of existing nonparametric algorithms and software, as well as estimates on the likely errors to be expected when used as an astronomical tool. / Thesis (M.Sc.)University of KwaZuluNatal, Westville, 2009.

50 
Semiperfect CFPF rings.Francis, Donald Nicholas. January 1987 (has links)
The WedderburnArtin Theorem (1927) characterised
semisimple Artinian rings as finite direct products of
matrix rings over division rings. In attempting to
generalise Wedderburn's theorem, the natural starting point
will be to assume R/RadR is semisimple Artinian. Such
rings are called semilocal. They have not been completely
characterised to date. If additional conditions are
imposed on the radical then more is known about the
structure of R. Semiprimary and perfect rings are those
rings in which the radical is nilpotent and Tnilpotent
respectively. In both these cases the radical is nil, and
in rings in which the radical is nil, idempotents lift
modulo the radical. Rings which have the latter property
are called semiperfect. The characterisation problem of
such rings has received much attention in the last few
decades.
We study semiperfect rings with a somewhat strong condition
arising out of the status of generators in the module
categories. More specifically, a ring R is CFPF iff every
homomorphic image of R has the property that every finitely
generated faithful module over it generates the
corresponding module category.
The objective of this thesis is to develop the theory that
leads to the complete characterisation of semiperfect right
CFPF rings. It will be shown (Theorem 6.3.17) that these
rings are precisely finite products of full matrix rings
over right duo right VR right acyclic right CFPF rings.
As far as possible theorems proved in Lambek [16] or Fuller
and Anderson [12] have not been reproved in this thesis and
these texts will serve as basic reference texts.
The basis for this thesis was inspired by results contained
in the first two chapters of the excellent LMS publication
"FPF Ring Theory" by Carl Faith and Stanley Page [11]. Its
results can be traced to the works of G. Azumaya [23], K.
Morita [18], Nakayama [20;21], H. Bass [4;5],
Carl Faith [8;9;10], S. Page [24;25] and B. Osofsky [22].
Our task is to bring the researcher to the frontiers of FPF
ring theory, not so much to present anything new. / Thesis (M.Sc.)University of DurbanWestville, 1987.

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