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Many physics and engineering problems are modeled by differential equations. In many instances these equations are nonlinear and exact solutions are difficult to obtain. Numerical schemes are often used to find approximate solutions. However, numerical solutions do not describe the qualitative behaviour of mechanical systems and are insufficient in determining the general properties of certain systems of equations. The need for analytical methods is self-evident and major developments were seen in the 1990’s. With the aid of faster processing equipment today, we are able to compute analytical solutions to highly nonlinear equations that are more accurate than numerical solutions. In this study we discuss solutions to nonlinear partial differential equations with focus on non-perturbation analytical methods. The non-perturbation methods of choice are the homotopy analysis method (HAM) developed by Shijun Liao and the variational iteration method (VIM) developed by Ji-Huan He. The aim is to compare the solutions obtained by these modern day analytical methods against each other focusing on accuracy, convergence and computational efficiency. The methods were applied to three test problems, namely, the heat equation, Burgers equation and the Bratu equation. The solutions were compared against both the exact results as well as solutions generated using the finite difference method, in some cases. The results obtained show that the HAM successfully produces solutions which are accurate, faster converging and requires less computational resources than the VIM. However, the VIM still provides accurate solutions that are also in good agreement with the closed form solutions of the test problems. The FDM also produced good results which were used as a further comparison to the analytical solutions. The findings of this study is in agreement with those published in the literature. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

In this thesis we study classes of static spherically symmetric solutions to the Einstein and Einstein–Maxwell equations that may be used to model the interior of compact stars. We also study the spherical accretion of fluids on to bodies in both general relativity and the Newtonian theory of gravity. The condition for pressure isotropy is obtained upon specifying one of the gravitational potentials and the electric field intensity. A series solution was found after specifying a cubic form for the potential. The pressure and energy density appear to be non–singular and continuous inside the star. This solution admits an explicit equation of state that, in regions close to the stellar centre, may be approximated by a polytrope. Another class of exact solutions to the Einstein–Maxwell solutions was found with charge. These solutions are in the form of hypergeometric functions with two free parameters. For particular parameter values we recovered two previously known exact solutions that are reasonable models for the interior of compact stars. We demonstrated two new solutions for other choices of the parameters. One of these has well behaved pressure, energy density and electric field intensity variables within the star. The other was rejected as unphysical on the grounds that it has a negative energy density. This violates the energy conditions. We obtained the mass accretion rate and critical radius of a polytrope accreting onto a D– dimensional Schwarzschild black hole. The accretion rate, ˙M , is an explicit function of the black hole mass, M, as well as the gas boundary conditions and the dimensionality, D, of the spacetime. We also found the asymptotic compression ratios and temperature profiles below the accretion radius and at the event horizon. This generalises the Newtonian expressions of Giddings and Mangano (2008) which examined the accretion of TeV black holes. We obtained the critical radius and accretion rates of a generalised Chaplygin gas accreting on to body under a Newtonian potential. The accretion rate is about 2 - 4 times greater than that for neutral hydrogen. The Rankine–Hugoniot relations for shocked GCG flow were also found. We found general expressions for the pressure and density compression ratios. Some post shock states imply negative volumes. We suspect that these may be thermodynamically forbidden. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2012.

Individuals have different implicit theories. They may hold an incremental theory and believe traits are malleable or an entity theory and believe traits are fixed. In past research implicit theories have been linked to achievement goals. The present study extends this research by investigating the relationship between implicit theories and personal goals. An experimental design was used to examine if priming individuals with a specific implicit theory impacts their goal prioritization. It was expected that individuals primed with an entity theory would place greater importance on intrinsic goals while individuals primed with an incremental theory would place greater importance on extrinsic goals. It was found that female participants rated the importance of financial success higher in the Entity condition compared to the Incremental condition and rated the importance of community feeling higher in the Incremental condition compared to the Entity condition. No significant differences were found with male participants. / Graduate

This thesis investigates the algebraic structure of certain quantum field theories in one space and one time dimension. These theories are integrable - essentially, highly constrained and therefore soluble. Thus, instead of having to use perturbative techniques, it is possible to conjecture their exact 5-matrices, which have the property that they are factorized into two-particle 5-matrices. In particular, there are two types of such theory: in one, scattering is purely elastic, whilst in the other, there is additional structure dictated by the Yang-Baxter equation. This thesis explores the algebraic structure of the latter and its links with the former. We begin, in chapter one, with an informal summary of the development of the subject, followed by a more mathematical exposition in chapter two. Chapter three constructs explicitly some exact factorized 5-matrices with Yang-Baxter structure, and comments on their features, both intrinsic and in relation to purely elastic 5-matrices. In particular, there is an unexplained close correspondence between the mass spectra and particle fusings in the two types of theory. The next three chapters attempt to shed some light on these features. Chapter four constructs similar 5-matrices, but based on quantum-deformed algebras rather than classical algebras. In chapter five we describe the structure of the 5-matrices when the particles they describe transform in irreducible representations of classical algebras. This leads us to consider the Yangian algebra, the representation theory of which underlies Yang-Baxter dependent 5-matrices, and which we therefore review briefly. We begin chapter six by reviewing the work which shows that the Yangian is also the charge algebra of the integrable quantum field theory, and subsequently show that the Yangian is also to a great extent present in the corresponding classical theory. We conclude with a brief seventh chapter describing the outlook for further research, followed by appendices containing respectively details of the Lagrangians of some integrable quantum field theories, a continuum formulation of the quantum inverse problem, explicit expressions for some of the R-matrices computed in the text, and a summary of known solutions of the Yang-Baxter equation.

This thesis is concerned with the semantic function of linguistic elements which do not seem to contribute to the truth conditions of an utterance, that is, with 'non-truth-conditional' linguistic devices. The first part of the thesis is devoted to theoretical considerations, while the second part concentrates on 'concessive' linguistic devices, which form a sub-class of 'non-truth-conditional' expressions. The first chapter outlines the way in which traditional semantic theories have employed the notion of truth conditions to capture linguistic meaning and a series of problems with this approach are pointed out. The chapter ends with an overview of 'non-truth-conditional' linguistic devices. Chapter 2 is concerned with ways in which fundamentally truth-conditional theories of linguistic semantics have attempted to accommodate such expressions in their frameworks. In chapter 3, the discussion focuses on Argumentation Theory, which does not just accommodate non-truth-conditional meaning but, ultimately, treats all linguistic meaning in non-truth-conditional terms and leads to the untenable conclusion that the general intuition that utterances can give information about the world is an illusion. This is followed by a chapter devoted to Sperber & Wilson's cognitive Relevance Theory. It is argued that this theory offers an ideal framework for a semantic analysis of 'truth-conditional' and 'non-truth-conditional' expressions alike, while avoiding the problems encountered by other theories. The next three chapters investigate the nature of linguistic 'concessivity' and provide a critical survey of existing analyses of three specific 'concessive' devices: but, although, and even if. In each case, an original relevance-theoretic analysis in procedural terms is proposed. In the last chapter, the possibility of purely pragmatic (that is, unencoded) 'concessive' interpretations is explored, and, finally, the role of the concept of 'truth-conditional content' in a theory of utterance interpretation is reassessed.

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A simple two scale compactification scheme for the E(8) x E(8) heterotic string is studied. The internal space used is a direct product of two compact spaces, each with its own length scale. Compactification on the smaller 4-dimensional (4d) manifold is carried out to obtain 6d theories with simple supersymmetry (SUSY). Assuming the background torsion vanishes, we show that this manifold must be K3. Compactification on K3 is studied in detail. Also analyzed are the two possible torsion-free compactifications on the orbifold K3$ sp prime$ (the limit of the manifold K3). The compactification from 6d to 4d on the larger scale 2d manifold results in Grand Unified Theories (GUT's) with broken SUSY. We show that it is not possible to generate a realistic theory using our scheme. Strings exclude what is conceivable from the perspective of point field theories: getting a realistic GUT from a 6d theory with simple SUSY.

Cultural differences in compliments were examined across five studies. The results are consistent with cultural differences in self-enhancement and self-criticism and suggest that compliment responses may reflect underlying differences in self-views. Asian golfers were less accepting and more rejecting of compliments about a tournament win than European golfers (Study 1). Cultural differences in responses to compliments about close others were found to mirror those about the self. Asian Canadian mothers were less accepting and more rejecting of compliments about their children than were European Canadian mothers (Study 2). Study 3 examined cultural differences in response to compliments that focus on natural ability (person-praise) versus those that focus on effort (process-praise). European Canadians were more accepting and less rejecting of person-praise compliments about their basketball shooting ability than Asian Canadians, whereas no differences were found in responses to process-praise compliments. Cultural differences in giving compliments were examined using both cultural artifacts (Study 4) and self-report (Study 5). The results are consistent with previous research on differences in implicit theories of ability. Chinese graduation cards contained more process- than person-praise compliments, whereas the reverse was true of American cards (Study 4). Chinese parents indicated that they would be more likely to select and Chinese students indicated that they would be more likely to receive graduation card messages containing process- versus person-praise compliments (Study 5). American parents and students showed no effects of type of compliment.