Postural distortion of human bite marks on skin occurs when photographing a bite mark in a body position other than the position of the body at the time of biting. Postural distortion in the bite mark may introduce significant changes in both the shape and size of the recorded marks. As a result, the analysis of the marks may be hindered and a proper comparison between the bite mark and the causal dentition may be precluded. Therefore, a method by which postural distortion in a bite mark photographic record can be evaluated and eliminated with minimal operator subjectivity is required. This study describes the development of an objective technique for evaluating postural distortion in bite mark photographic records and for minimising postural distortion during photography of bite marks. The source for developing these techniques was provided by digitally analysing the symmetry of dental arches in a defined population including males and females (236 subjects) whose ages ranged between 20 and 30 years. The analysis resulted in quantifying a mathematical relationship between the biting edges of each homologous pair of the anterior teeth and specified reference lines. The validity of the analytical method of dental arch symmetry is discussed. The developed techniques were applied to posturally distorted (test) bite marks. The results demonstrate the validity of the developed techniques in determining postural distortion and recording correct images (shown to resemble the biters dentition) of the test bites. Suggestions for further work are proposed.

During the twentieth century, particle physics developed into a cornerstone of modern physics, culminating in the Standard Model. Even though this theory has proved to be of extraordinary power, it is still incomplete in several respects. It is our aim in this bachelor thesis to discuss some possible theories beyond the Standard Model, the main focus being on Grand Unified Theories, while also taking a look at attempts of further unication via discrete family symmetry. At the heart of all these theories lies the concept of local gauge invariance, which is introduced as a fundamental principle, followed by an overview of the Standard Model itself. No theory has so far managed to unify all elementary particles and their interactions, but some interesting features are highlighted. We also give a hint at some possible paths to go in the future in the quest for a unication in particle physics. / Under 1900-talet utvecklades partikelfysiken till en av de fundamentala teorierna inom fysiken, och kom att sammanfattas i den s.k. Standardmodellen. Även om denna modell rönt exceptionella framgånger vad gäller beskrivningen av elementarpartiklar och deras växelverkan, är den fortfarande ofullständig på flera sätt. Syftet med denna kandidatuppsats är att diskutera möjliga teorier bortom Standardmodellen såsom Storförenande Teorier och diskreta familjesymmetrier vars avsikt är att koppla samman de tre familjerna av fermioner i Standardmodellen. Men först introduceras idén om lokal gaugeinvarians, vilken ligger till grund for dessa teorier, varpå en översikt av Standardmodellen följer. Ingen teori har ännu lyckats ge en helt tillfredsställande bild av elementarpartiklar och deras interaktion, men en del intressanta egenskaper hos föreslagna teorier belyses i denna uppsats. Slutligen ges en del spekulativa förslag på väger att gå i framtida försök till föreningar inom partikelfysiken.

Numerical solutions of quantum mechanical problems have witnessed tremendous advances over the past years. In this thesis, we develop an iterative approach to problems of double-well potentials and their variants with parity-time-reversal symmetry- breaking perturbations. We show that the method provides an efficient scheme for obtaining accurate energies and wave functions. We discuss in this thesis potential applications to a variety of related topics such as phase transitions, symmetry breaking, and external field-induced effects.

This dissertation has two areas of interest with regard to the four dimensional Kerr black hole; the rst being its conformal nature in its near region and second it characteristic frequencies. With it now known that the scalar solution space of the four dimensional Kerr black hole has a two dimensional conformal symmetry in its near region, it was the rst focus of this dissertation to see if this conformal symmetry is unique to the near region scalar solution space or if it is also present in the spin-half solution space. The second focus of this dissertation was to explore techniques which can be used to calculate these quasinormal mode (characteristic) frequencies, such as the WKB(J) approximation which has been improved from third order to sixth order recently and applied to the perturbations of a Schwarzschild black hole. The additional correction terms show a signi cant increase of accuracy when comparing to numerical methods. This dissertation shall use the sixth order WKB(J) method to calculate the quasinormal mode frequencies for both the scalar and spin-half perturbations of a four dimensional Kerr black hole. An additional method used was the asymptotic iteration method, a relatively new technique being used to calculate the quasinormal mode frequencies of black holes that have been perturbed. Prior to this dissertation it had only been used on a variety of Schwarzschild black holes and their possible perturbations. For this dissertation the asymptotic iteration method has been used to calculate the quasinormal frequencies for both the scalar and spin-half perturbations of the four dimensional Kerr black hole. The quasinormal mode frequencies calculated using both the sixth order WKB(J) method and the asymptotic iteration method were compared to previously published values and each other. For the most part, they both compare favourably with the numerical values, with di erences that are near negligible. The di erences did become more apparent when the mode number (or angular momentum per unit mass increased), but less so when the angular number increased. The only factor that separates these two methods signi cantly, was that the computational time for the sixth order WKB(J) method is less than than that of the asymptotic iteration method.

In this research report we explore some applications of symmetry methods for boundary value problems in the pricing of barrier options. Various nancial instruments satisfy the Black-Scholes partial di erential equation (pde) but with di erent domain, maturity date and boundary conditions. We nd Lie symmetries that leave the Black-Scholes (pde) invariant and will guarantee that the relevant solutions satisfy the boundary conditions. Using these sym- metries, we can thus generate group-invariant solutions to the boundary value problem.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy. Johannesburg, August 2015. / We construct, using rst principles, a number of non-trivial conservation laws of some partial di erence equations, viz, the discrete Liouville equation and the discrete Sine-Gordon equation. Symmetries and the more recent ideas and notions of characteristics (multipliers) for di erence equations are also discussed. We then determine the symmetry generators of some ordinary di erence equations and proceed to nd the rst integral and reduce the order of the di erence equations. We show that, in some cases, the symmetry generator and rst integral are associated via the `invariance condition'. That is, the rst integral may be invariant under the symmetry of the original di erence equation. We proceed to carry out double reduction of the di erence equation in these cases. We then consider discrete versions of the Painlev e equations. We assume that the characteristics depend on n and un only and we obtain a number of symmetries. These symmetries are used to construct exact solutions in some cases. Finally, we discuss symmetries of linear iterative equations and their transformation properties. We characterize coe cients of linear iterative equations for order less than or equal to ten, although our approach of characterization is valid for any order. Furthermore, a list of coe cients of linear iterative equations of order up to 10, in normal reduced form is given.

ABSTRACT In jet problems the conserved quantity plays a central role in the solution process. The conserved quantities for laminar jets have been established either from physical arguments or by integrating Prandtl's momentum boundary layer equation across the jet and using the boundary conditions and the continuity equation. This method of deriving conserved quantities is not entirely systematic and in problems such as the wall jet requires considerable mathematical and physical insight. A systematic way to derive the conserved quantities for jet °ows using conservation laws is presented in this dissertation. Two-dimensional, ra- dial and axisymmetric °ows are considered and conserved quantities for liquid, free and wall jets for each type of °ow are derived. The jet °ows are described by Prandtl's momentum boundary layer equation and the continuity equation. The stream function transforms Prandtl's momentum boundary layer equation and the continuity equation into a single third- order partial di®erential equation for the stream function. The multiplier approach is used to derive conserved vectors for the system as well as for the third-order partial di®erential equation for the stream function for each jet °ow. The liquid jet, the free jet and the wall jet satisfy the same partial di®erential equations but the boundary conditions for each jet are di®erent. The conserved vectors depend only on the partial di®erential equations. The derivation of the conserved quantity depends on the boundary conditions as well as on the di®erential equations. The boundary condi- tions therefore determine which conserved vector is associated with which jet. By integrating the corresponding conservation laws across the jet and imposing the boundary conditions, conserved quantities are derived. This approach gives a uni¯ed treatment to the derivation of conserved quantities for jet °ows and may lead to a new classi¯cation of jets through conserved vectors. The conservation laws for second order scalar partial di®erential equations and systems of partial di®erential equations which occur in °uid mechanics are constructed using di®erent approaches. The direct method, Noether's theorem, the characteristic method, the variational derivative method (mul- tiplier approach) for arbitrary functions as well as on the solution space, symmetry conditions on the conserved quantities, the direct construction formula approach, the partial Noether approach and the Noether approach for the equation and its adjoint are discussed and explained with the help of an illustrative example. The conservation laws for the non-linear di®usion equa- tion for the spreading of an axisymmetric thin liquid drop, the system of two partial di®erential equations governing °ow in the laminar two-dimensional jet and the system of two partial di®erential equations governing °ow in the laminar radial jet are discussed via these approaches. The group invariant solutions for the system of equations governing °ow in two-dimensional and radial free jets are derived. It is shown that the group invariant solution and similarity solution are the same. The similarity solution to Prandtl's boundary layer equations for two- dimensional and radial °ows with vanishing or constant mainstream velocity gives rise to a third-order ordinary di®erential equation which depends on a parameter. For speci¯c values of the parameter the symmetry solutions for the third-order ordinary di®erential equation are constructed. The invariant solutions of the third-order ordinary di®erential equation are also derived.

There is a well established connection between one parameter Lie groups of transformations and conservation laws for differential equations. In this thesis, we construct conservation laws via the invariance and multiplier approach based on the wellknown result that the Euler-Lagrange operator annihilates total divergences. This technique will be applied to some plasma physics models. We show that the recently developed notion of the association between Lie point symmetry generators and conservation laws lead to double reductions of the underlying equation and ultimately to exact/invariant solutions for higher-order nonlinear partial di erential equations viz., some classes of Schr odinger and KdV equations.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2012. / Unable to load abstract.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, in requirement for the degree Doctor of Philosophy, Johannesburg, 2013. / The (1+3) dimensional wave and Klein-Gordon equations are constructed using the covariant d'Alembertian operator on several spacetimes of interest. Equations on curved geometry inherit the nonlinearities of the geometry. These equations display interesting properties in a number of ways. In particular, the number of symmetries and therefore, the conservation laws reduce depending on how curved the manifold is. We study the symmetry properties and conservation laws of wave equations on Freidmann-Robertson-Walker, Milne, Bianchi, and de Sitter universes. Symmetry structures are used to reduce the number of unknown functions, and hence contribute to nding exact solutions of the equations. As expected, properties of reduction procedures using symmetries, variational structures and conservation laws are more involved than on the well known at (Minkowski) manifold.