Includes bibliographical references. / Finite element models of the aortic heart valve have been successfully used in the past to gain insight into the mechanics of the valve and to aid in understanding of valve failure. Moreover such models are indispensable tools for further developments in heart valve prosthetic design. In previous stress analyses linear elastic constitutive models have predominantly been used to model aortic valve leaflets, despite aortic valve tissue showing highly non-linear behaviour in tension tests. In view of recent developments towards tissue engineering of heart valves, these linear constitutive models of aortic valve leaflets are not likely to produce results sufficiently accurate to correlate cell behaviour with mechanical stimuli. To study how non-linear material behaviour affects the results of stress analyses of the aortic valve, static finite element analyses of the valve including the aortic root and leaflets have been carried out. An isotropic linear elastic material model was assigned to the aortic root with Young's modulus adjusted for the simulated compliance to match physiological values. Linear elastic models for the aortic valve leaflets with parameters used in previous studies were then compared with hyperelastic materials. The parameters used for the exponential strain energy function of the latter were obtained from fits of uniaxial tension test results of fresh porcine aortic valve leaflets. As natural leaflets show anisotropy with a pronounced stiff direction along the circumference of the valve, isotropic models of the leaflets were extended to account for this behaviour by incorporating transverse isotropy. The results display a stark impact of a transversely isotropic hyperelastic material on leaflet mechanics, Le. increased coaptation with peak values of stress and strain in the elastic limit. Interestingly, the alignment of maximum principal stress of all models seems to approximately follow the coarse collagen fibre distribution found ill aortic valve leaflets.

In this thesis an approximation method, the method of spherical currents, is advanced for the determination of scattered fields (acoustic and electromagnetic) from convex bodies for plane harmonic incident waves. The method is worked out for the scalar (acoustic) case and applied to sound soft and sound hard obstacles of a given shape. The surface fields and backscattering cross sections of these obstacles are calculated according to the method and the results are compared with those of other methods. For the vector (electromagnetic) case only the basic theory is worked out for a perfectly conducting convex body.

Includes abstract. / Includes bibliographical references (leaves 153-169). / This study pursued a two-fold objective: firstly, to broaden our knowledge about the rainfall-plant-animal system dynamics; and secondly, to re-examine the rotational grazing system with modelling as a substitute to field trials.

We investigate various aspects of noncommutative geometry and fuzzy field theory and their relations to string theory. In particular, we study the BPS and non-BPS solutions of the CJPN nonlinear sigma model on the noncommutative plane in some detail and show among other things that a class of its solitonic excitations may be built from bound states of noncommutative scalar solitons. We then go on to construct a fuzzy extension of the semilocal SU(N)a x U(l)L Yang-Mills-Riggs model. We find that not only does this noncommutative model support a large class of BPS vortex solutions but, unlike in the commutative model, these are exact solutions of the BPS equations. We also study the large coupling limit of the semilocal model and demonstrate conclusively the metamorphosis of the semilocal vortex to an appropriate degree instanton of the fuzzy CJPN model. In the second part of this work, we study the perpendicular intersection of Dl- and D7-branes in type liB string theory and the fuzzy 6-sphere that resolves the singularity of the intersection. We demonstrate the equivalence of the D7 and dual D-string descriptions by computing the energy, charge and radial profiles of the solution in each description. We conclude the thesis with a foray into cosmology by constructing a realisation of a recently proposed singularity-free inflating universe. We discuss the basic characteristics of this model and show that none are at odds with current observations.

Bibliography: p. 228-249. / Research into student understanding of university level physics has been extremely extensive over the past decade with many international studies affirming the inherent complexity and difficulty that undergraduate students typically experience in learning physics. An important and informative element of this research has focused attention on describing the variation in ways which students make sense of concepts which they experienced as being fundamentally counter-intuitive. One of the areas of most prolific research has been Newtonian dynamics. However, one important aspect from the teaching perspective which has not been examined at all is the variety of the ways in which multi-conceptual links are understood. This study has begun the examination of this aspect of Newtonian physics understanding using a group of first-year physics students enrolled at a typically good South African university. Since the study was primarily aimed at characterising the variance in understanding of conceptual links in Newtonian dynamics the fundamental theoretical framework chosen for the study was drawn from phenomenography. The data consisted of a set of concept maps created by the students involved in the study and in-depth interviews with these students about the understanding they were attempting to represent on a multi-conceptual level on these maps. Since an integral part of the study included exploring the role of counter- intuitiveness, the method involved creating ideal data-generating contexts for thematising drawn from an everyday problem with varying degrees of abstraction. In brief, these thematised scenarios incorporated the following: a familiar everyday experience with normal friction conditions; a familiar everyday experience with reduced friction conditions; and, an unfamiliar everyday experience with greatly reduced friction conditions. The set of interviews formed what is known as a 'pool of meaning' in phenomenography for its associated analytic process. The analysis had two components. The first component involved analysing the interview data across individuals to develop what is known phenomenographically as 'categories of description' to characterise the nature of the understanding in terms of conceptualisation or experience. The second component focused on learning as a function of the students' everyday and educational experiences. Here critical educational aspects emerged and their variance was identified. For example, intuition and context emerged as deeply influential factors in the ways Newtonian links are understood. Following contemporary thrusts in phenomenography this component of the analysis also looked at intra-contextual and inter-contextual shifts. The analysis produced four distinct qualitative ways of understanding or conceptualising Newtonian links and showed the critical influence of intuition and context in the development of understanding of Newtonian dynamics at the introductory level. The analysis also contributed to phenomenographic theoretical concerns about a way of experiencing or understanding a phenomenon and the evolvement of such understanding. Collectively, these results are used to suggest important pedagogical implications for informing the improvement of physics teaching at this level.

Bibliography: pages 151-162. / A new internal variable theory of plasticity is presented. This theory is developed within a framework of non-smooth convex analysis; a unification of ideas concerning the postulates of plasticity is achieved by using the powerful tools provided by results in this branch of mathematics. A firm mathematical foundation for the study of qualitative aspects of problems involving plastic deformations is provided. Among the features of the theory is the establishment of a clear relationship between conventional formulations, which make use of yield functions, and those formulated in terms of a dissipation function. The role of the principle of maximum plastic work is also made precise. Attention is focussed on application of the theory to finite-strain plasticity. Quasi-static initial-boundary-value problems involving large plastic deformations are considered. An incremental form of such problems arises from a discretisation in time. A variational form of the incremental boundary-value problem is derived using the new theory. This incremental formulation is based on a generalised midpoint rule, evolution equations for plastic variables are defined in terms of a dissipation function, and an assumption of isochoric plastic deformation is imposed explicitly. A spatially discrete form of the incremental problem is obtained by application of the finite element method. An algorithm for solving this discrete problem, based on the Newton-Raphson procedure and having the typical predictor-corrector structure used in computational plasticity, is proposed and investigated. This algorithm is implemented in NOSTRUM, the in-house finite element code of The FRD/UCT Centre for Research in Computational and Applied Mechanics, at the University of Cape Town. A number of standard example problems are analysed using this code and results are compared with those obtained by others. It is shown that a corrector algorithm based on use of a dissipation function is a viable alternative to the conventional return mapping algorithms. While this alternative approach is not necessarily better than the conventional one for simple models of plasticity, it may prove valuable when considering more complex models for materials which exhibit dissipative behaviour. The manner in which an assumption of isochoric plastic deformation is incorporated into the incremental form of the problem is shown to play an important role.

Includes abstract. / Includes bibliographical references (p. 112-122). / One of the distinct trends in modern cosmology is in testing its theoretical aspects against the high precision data available in recent years. Although the standard cosmological model has already shown certain satisfactory results in matching the data from observations of the Cosmic Microwave Background and Large Scale Structure, we are by far unable to be convinced of its ability in describing the complete nature of the universe. Therefore, detailed study has to be done in the theory of higher order cosmological perturbation to enable us the ability in describing the non-linear aspects of the universe.

Bibliography: leaves 158-167. / We conduct three investigations in Relativistic Cosmology that is the Einstein Field Equations applied to the largest scales with source field typically taken to be a perfect fluid and fundamental observers comoving with the preferred fluid four-velocity. We show using a tetrad analysis of the evolution equations for the dynamical variables and all the constraints these satisfy in classical General Relativity, that there are no new consistent perfect fluid cosmologies with the kinematic variables and the electric and/or magnetic parts of the Weyl curvature all rotationally symmetric about a common axis in an open neighbourhood Ա of an event. The consistent solutions of this kind are either locally rotationally symmetric, or most generally are subcases of the Szekeres model-an inhomogeneous dust model with no Killing symmetries. This result and its obvious future generalisations provides an input into the equivalence problem in cosmology necessary for a mathematically consistent understanding of probability and a measure set for universes required in quantum cosmology, for instance. We investigate such generalisations and find that similar results hold under some further assumptions dependent on the level of generalisation. In particular, we examine situations where either the electric part or the magnetic part of the free gravitational field are not rotationally symmetric, and also make a brief comment on the most general case where only the shear is rotationally symmetric. We use a tetrad analysis to show that the well-known result that holds for relativistic shear-free dust cosmologies in Einstein's classical theory either the expansion vanishes or the flow is irrotational - has an analogue in the Kaluza-Klein universe model, which has its roots presumably in string theory (or M-theory), recently proposed by Randall and Sundrum. The Big Bang singularity of General Relativity can not be avoided in these so-called brane universes in the situation where we neglect non-local tidal effects on the dynamics by allowing the vorticity to spin up as the singularity is approached in shear-free cases. Moreover, we show that in the general case of a shearing perfect fluid, the singularity at the start of the universe is approached even more strongly than in classical General Relativity in the case of no tidal interaction. Finally, we reconsider the issue of proving large scale spatial homogeneity of the universe in classical General Relativity, given isotropic observations about us and the possibility of source evolution both in numbers and luminosities. We use a spherically symmetric dust universe model (compatible with observations) for our investigation and we solve the field equations on the null cone analytically for the first time. Two theorems make precise the freedom available in constructing cosmological models that will fit the observations. They make quite clear that homogeneity cannot be proven without either a fully determinate theory of source evolution, or availability of distance measures that are independent of source evolution. We contrast this goal with the standard approach that assumes spatial homogeneity a priori, and determines source evolution functions on the basis of this assumption.

Word processed copy. Includes bibliographical references (p. [171]-182).

Includes bibliographical references. / Over the last few decades, astronomy and cosmology have become data-driven fields. The parallel increase in computational power has naturally lead to the adoption of more sophisticated statistical techniques for data analysis in these fields, and in particular, Bayesian methods. As the next generation of instruments comes online, this trend should be continued since previously ignored effects must be considered rigorously in order to avoid biases and incorrect scientific conclusions being drawn from the ever-improving data. In the context of supernova cosmology, an example of this is the challenge from contamination as supernova datasets will become too large to spectroscopically confirm the types of all objects. The technique known as BEAMS (Bayesian Estimation Applied to Multiple Species) handles this contamination with a fully Bayesian mixture model approach, which allows unbiased estimates of the cosmological parameters. Here, we extend the original BEAMS formalism to deal with correlated systematics in supernovae data, which we test extensively on thousands of simulated datasets using numerical marginalization and Markov Chain Monte Carlo (MCMC) sampling over the unknown type of the supernova, showing that it recovers unbiased cosmological parameters with good coverage. We then apply Bayesian statistics to the field of radio interferometry. This is particularly relevant in light of the SKA telescope, where the data will be of such high quantity and quality that current techniques will not be adequate to fully exploit it. We show that the current approach to deconvolution of radio interferometric data is susceptible to biases induced by ignored and unknown instrumental effects such as pointing errors, which in general are correlated with the science parameters. We develop an alternative approach - Bayesian Inference for Radio Observations (BIRO) - which is able to determine the joint posterior for all scientific and instrumental parameters. We test BIRO on several simulated datasets and show that it is superior to the standard CLEAN and source extraction algorithms. BIRO fits all parameters simultaneously while providing unbiased estimates - and errors - for the noise, beam width, pointing errors and the fluxes and shapes of the sources.