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Analysis of shear-free spherically symmetric charged relativistic fluids.Kweyama, Mandlenkosi Christopher. January 2011 (has links)
We study the evolution of shear-free spherically symmetric charged fluids
in general relativity. This requires the analysis of the coupled Einstein-Maxwell
system of equations. Within this framework, the master field equation to be
integrated is
yxx = f(x)y2 + g(x)y3
We undertake a comprehensive study of this equation using a variety of ap-
proaches. Initially, we find a first integral using elementary techniques (subject
to integrability conditions on the arbitrary functions f(x) and g(x)). As a re-
sult, we are able to generate a class of new solutions containing, as special
cases, the models of Maharaj et al (1996), Stephani (1983) and Srivastava
(1987). The integrability conditions on f(x) and g(x) are investigated in detail
for the purposes of reduction to quadratures in terms of elliptic integrals. We
also obtain a Noether first integral by performing a Noether symmetry analy-
sis of the master field equation. This provides a partial group theoretic basis
for the first integral found earlier. In addition, a comprehensive Lie symmetry
analysis is performed on the field equation. Here we show that the first integral
approach (and hence the Noether approach) is limited { more general results
are possible when the full Lie theory is used. We transform the field equation
to an autonomous equation and investigate the conditions for it to be reduced
to quadrature. For each case we recover particular results that were found pre-
viously for neutral fluids. Finally we show (for the first time) that the pivotal
equation, governing the existence of a Lie symmetry, is actually a fifth order
purely differential equation, the solution of which generates solutions to the
master field equation. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2011.
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Applications of Lie symmetries to gravitating fluids.Msomi, Alfred Mvunyelwa. January 2011 (has links)
This thesis is concerned with the application of Lie's group theoretic method
to the Einstein field equations in order to find new exact solutions. We analyse
the nonlinear partial differential equation which arises in the study of non-
static, non-conformally flat fluid plates of embedding class one. In order to find
the group invariant solutions to the partial differential equation in a systematic and comprehensive manner we apply the method of optimal subgroups.
We demonstrate that the model admits linear barotropic equations of state in
several special cases. Secondly, we study a shear-free spherically symmetric
cosmological model with heat flow. We review and extend a method of generating solutions developed by Deng. We use the method of Lie analysis as a
systematic approach to generate new solutions to the master equation. Also,
general classes of solution are found in which there is an explicit relationship
between the gravitational potentials which is not present in earlier models.
Using our systematic approach, we can recover known solutions. Thirdly, we
study generalised shear-free spherically symmetric models with heat flow in
higher dimensions. The method of Lie generates new solutions to the master equation. We obtain an implicit solution or we can reduce the governing equation to a Riccati equation. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2011.
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Impact of exogenous reinfection on TB infection in a genetically susceptible population.Mwangi, Wangari Isaac. 17 December 2013 (has links)
In this study we investigated the impact of exogenous reinfection on genetically resistant and genetically sensitive sub populations. We qualitatively analysed the dynamics of TB by assuming that TB is transmitted in two ways namely homogeneous and heterogeneous modes of transmission. Analytically, we computed the fundamental thresholds used to measure disease persistence; the basic reproduction number R₀; and found that the exogenous reinfection parameters do not appear in the basic reproduction number. Hence, basic reproduction number derived in presence of exogenous reinfection does not adequately predict the course of a TB epidemic. We obtained the exogenous reinfection threshold which indicated that exogenous reinfection complicates TB dynamics. Both analytical and simulation results disclosed that when exogenous reinfection is above exogenous reinfection threshold TB dynamics were governed by
a backward bifurcation implying TB may continue to invade the population despite basic reproduction number being less than one. We computed critical value of basic reproduction numbers Rᴄ and found that TB can only be eradicated if basic reproduction number is reduced below critical value Rc. Furthermore, we incorporated TB therapy in heterogeneous model among individuals with clinically active TB and performed sensitivity and uncertainty analysis using Latin Hypercube Sampling. The sensitivity and uncertainty results showed that transmission rates, reactivation rates and proportion that is genetically resistant greatly infuenced outcome
variables of our TB model. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.
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Bayesian analysis of cosmological models.Moodley, Darell. January 2010 (has links)
In this thesis, we utilise the framework of Bayesian statistics to discriminate between models
of the cosmological mass function. We first review the cosmological model and the formation
and distribution of galaxy clusters before formulating a statistic within the Bayesian framework,
namely the Bayesian razor, that allows model testing of probability distributions. The Bayesian
razor is used to discriminate between three popular mass functions, namely the Press-Schechter,
Sheth-Tormen and normalisable Tinker models. With a small number of particles in the simulation,
we find that the simpler model is preferred due to the Occam’s razor effect, but as the size of
the simulation increases the more complex model, if taken to be the true model, is preferred. We
establish criteria on the size of the simulation that is required to decisively favour a given model
and investigate the dependence of the simulation size on the threshold mass for clusters, and
prior probability distributions. Finally we outline how our method can be extended to consider
more realistic N-body simulations or be applied to observational data. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2010.
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Exact solutions for perfect fluids conformal to a Petrov type D spacetime.Mewalal, Narenee. January 2011 (has links)
Abstract is available from the print copy. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.
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Group analysis of equations arising in embedding theory.Okelola, Michael. January 2010 (has links)
Embedding theories are concerned with the embedding of a lower dimensional manifold (dim = n, say) into a higher dimensional one (usually dim = n+1, but not necessarily so). We are concerned with the particular case of embedding 4D spherically symmetric equations into 5D Einstein spaces. This scenario is of particular relevance to contemporary cosmology and astrophysics. Essentially, they are 5D vacuum field equations with initial data given on a 4D spacetime hypersurface. The equations that arise in this framework are highly nonlinear systems of ordinary differential equations and they have been particularly resistant to solution techniques over the past few years. As a matter of fact, to date, despite theoretical results for the existence of solutions for embedding classes of 4D space times, no general solutions to the local embedding equations are known. The Lie theory of extended groups applied to differential equations has proved to be very successful since its inception in the nineteenth century. More recently, it has been successfully utilized in relativity and has provided solutions where none were previously found, as well as explaining the existence of ad hoc methods. In our work, we utilize this method in an attempt to find solutions to the embedding equations. It is hoped that we can place the analysis of these equations onto a firm theoretical basis and thus provide valuable insight into embedding theories. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2010.
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The theory of option valuation.Sewambar, Soraya. January 1992 (has links)
Although options have been traded for many centuries, it has remained a relatively
thinly traded financial instrument. Paradoxically, the theory of option
pricing has been studied extensively. This is due to the fact that many of the
financial instruments that are traded in the market place have an option-like
structure, and thus the development of a methodology for option-pricing may
lead to a general methodology for the pricing of these derivative-assets.
This thesis will focus on the development of the theory of option pricing.
Initially, a fundamental principle that underlies the theory of option valuation
will be given. This will be followed by a discussion of the different types
of option pricing models that are prevalent in the literature.
Special attention will then be given to a detailed derivation of both the
Black-Scholes and the Binomial Option pricing models, which will be followed
by a proof of the convergence of the Binomial pricing model to the
Black-Scholes model.
The Black-Scholes model will be adapted to take into account the payment
of dividends, the possibility of a changing inter est rate and the possibility of
a stochastic variance for the rate of return on the underlying as set. Several
applications of the Black-Scholes model will finally be presented. / Thesis (M.Sc.)-University of Natal, 1992.
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Queueing and communication networks governed by generalised Lindley-Loynes equations.Rose, David Michael. January 1993 (has links)
Several decades after A.K. Erlang originated the theory of queues and queueing
networks, D.V. Lindley added impetus to the development of this field
by determining a recursive relation for waiting times.
Part I of this thesis provides a theoretical treatment of single-server and
multiserver queues described by the basic Lindley relation and its extensions,
which are referred to collectively as Lindley-Loynes equations. The
concepts of stability, and minimal and maximal solutions are investigated.
The interdependence of theory and practice becomes evident in Part II, where
the results of recent and current research are highlighted. While the main
aim of the first part of the thesis is to provide a firm theoretical framework
for the sequel, the objective in Part II is to derive generalised forms of the
Lindley-Loynes equations from different network protocols. Such protocols
are regulated by different switching rules and synchronization constraints.
Parts I and II of the thesis are preceded by Chapter 0 in which several fundamental
ideas (including those on notation and probability) are described.
It is in this chapter too that a more detailed overview of the concept of the
thesis is provided. / Thesis (M.Sc.)-University of Natal, Durban, 1993.
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Identifying and modeling the dynamics of a core cancer sub-network.Bayleyegn, Yibeltal Negussie. January 2011 (has links)
Many recent studies have shown that the initiation of human cancer is due to the malfunction
of some genes at the R-checkpoint during the G1-to-S transition of the cell cycle. Identifying
and modeling the dynamics of these genes has a paramount advantage in controlling and, possibly,
treating human cancer. In this study, a new mathematical model for the dynamics of a
cancer sub-network concentration is developed. Positive equilibrium points are determined and
rigorously analyzed. We have found a condition for the existence of the positive equilibrium
points from the activation, inhibition and degradation parameter values of the dynamical system.
Numerical simulations have also been carried out. These results confirm analyses in the
literature. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.
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Categorical systems biology : an appreciation of categorical arguments in cellular modelling.Songa, Maurine Atieno. 23 April 2014 (has links)
With big science projects like the human genome project, [2], and preliminary
attempts to seriously study brain activity, e.g. [9], mathematical biology
has come of age, employing formalisms and tools from most branches of
mathematics.
Recent results, [51] and [53], have extended the relational (or categorical)
approach of Rosen [44], to demonstrate that (in a very general class of systems)
cellular self-organization/self-replication is implicit in metabolism and
repair/stability. This is a powerful philosophical statement and removes the
need of teleological argument. However, the result carries a technical limitation
to Cartesian closed categories, which excludes many mathematical
languages.
We review the relevant literature on metabolic-repair pathways, category
theory and systems theory, before performing a critique of this work. We
find that the restriction to Cartesian closed categories is purely for simplicity,
and describe how equivalent arguments may be built for monoidal closed
categories. Moreover, any symmetric monoidal category may be "embedded"
in a closed one. We discuss how these constructions/techniques provide the
formal structure to treat self-organization/self-replication in most contemporary
mathematical (modelling) languages. These results signicantly soften
the impact on current modelling paradigms while extending the philosophical
implications. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2012.
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