A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. May 30, 2016. / In this dissertation, we consider a number of modi ed Black-Scholes equations being either non-linear or given in higher dimensions. In particular we focus on the non-linear Black-Scholes equation describing option pricing with hedging strategies in one case, and two dimensional models in the other. Classical Lie point symmetry techniques are employed in an attempt to construct exact solutions. Some large symmetry algebras are admitted. We proceeded by determining the one dimensional optimal systems of sub-algebras for the admitted Lie algebras. The elements of the optimal systems are used to reduce the number of variables by one. In some cases, exact solutions are constructed. For the cases for which exact solutions are di cult to construct, we employed the numerical solutions. Some simulations are observed and interpreted / MT2016

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulflment of the requirements for the degree of Master of science. Johannesburg, December 2012. / Solutions of linear iterative equations and expressions for these solutions in terms of the parameters of the source equation are obtained. Based on certain properties of iterative equations, nding the solutions is reduced to nding group-invariant solutions of the second-order source equation. We have therefore found classes of solutions to the source equations. Regarding the expressions of the solutions in terms of the parameters of the source equation, an ansatz is made on the original parameters r and s, by letting them be functions of a speci c type such as monomials, functions of exponential and logarithmic type. We have also obtained an expression for the source parameters of the transformed equation under equivalence transformations and we have looked for the conservation laws of the source equation. We conducted this work with a special emphasis on second-, third- and fourth-order equations, although some of our results are valid for equations of a general order.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2015. / This work deals with conditional symmetries of ordinary di erential equations (ODEs). We re ne the de nition of conditional symmetries of systems of ODEs in general and provide an algorithmic viewpoint to compute such symmetries subject to root di erential equations. We prove a proposition which gives important and precise criteria as to when the derived higher-order system inherits the symmetries of the root system of ODEs. We rstly study the conditional symmetry properties of linear nth-order (n 3) equations subject to root linear second-order equations. We consider these symmetries for simple scalar higherorder linear equations and then for arbitrary linear systems. We prove criteria when the derived scalar linear ODEs and even order linear system of ODEs inherit the symmetries of the root linear ODEs. There are special symmetries such as the homogeneity and solution symmetries which are inherited symmetries. We mention here the constant coe cient case as well which has translations of the independent variable symmetry inherited. Further we show that if a system of ODEs has exact solutions, then it admits a conditional symmetry subject to the rst-order ODEs related to the invariant curve conditions which arises from the known solution curves. This is even true if the system has no Lie point sym

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa, in fulfilment of the requirements for the degree of Master of Science. 21 March 2016. / The problem of turbulent hydraulic fracturing is considered. Despite it being a known phenomenon, limited mathematical literature exists in this field. Prandtl’s mixing length model is utilised to describe the eddy viscosity and a mathematical model is developed for two distinct cases: turbulence where the kinematic viscosity is sufficiently small to be neglected and the case where it is not. These models allow for the examination of the fluid’s behaviour and its effect on the fracture’s evolution through time. The Lie point symmetries of both cases are obtained, and a wide range of analytical and numerical solutions are explored. Solutions of physical significance are calculated and discussed, and approximate solutions are constructed for ease of fracture estimation. The non-classical symmetries of these equations are also investigated. It was found that the incorporation of the kinematic viscosity within the modelling process was important and necessary. / MT2016

An MSc dissertation by Shivani Singh. University of Witwatersrand Faculty of Science, School of Mathematics. August 2016. / These symmetries have interesting applications to enumerative combinatorics and to Ramsey theory. The aim of this thesis will be to present some important results in these fields. In particular, we shall enumerate the r-ary symmetric bracelets of length n. / LG2017

A research report submitted to the Faculty of Science, University of the Witwatersrand, in ful lment of the requirements for the degree of Master of Science. School of Mathematics Johannesburg February 15, 2016 / We obtain a solution of the Black-Scholes equation with a non-smooth bound- ary condition using symmetry methods. The Black-Scholes equation along with its boundary condition are rst transformed into the one dimensional heat equation and an initial condition respectively. We then nd an appro- priate general symmetry generator of the heat equation using symmetries of the heat equation and the fundamental solution of the heat equation. The method we use to nd the symmetry generator is such that the boundary condition is left invariant and yet the symmetry can still be used to solve the heat equation. We then use the help of Mathematica to nd the solution to the heat equation. Then the solution is then transformed backwards to a solution of the Black-Scholes equation using the same change of variables that were used for the forward transformations. The solution is then nally checked if it satis es the boundary condition of the Black-Scholes equation.

Many real life problems can naturally be modeled as constraint satisfaction problems (CSPs), which can sometimes contain both variable symmetries and value symmetries. Tree search based CSP solving algorithms often suffer from symmetries, which creates symmetrically equivalent states in the search tree. Exploring more than one of the symmetrically equivalent states is a waste of search efforts. Adding symmetry breaking constraints to a CSP can force the search to visit only one of the symmetrical regions and helps reduce search space. While variable symmetry breaking constraints can be expressed relatively easily and executed efficiently by enforcing lexicographic ordering, value symmetry breaking constraints are often difficult to formulate. In this thesis, we propose two methods of using symmetry breaking constraints to tackle value symmetries. In the first method, we show theoretically when value symmetries in one CSP model correspond to variable symmetries in another CSP model of the same problem. We also show when variable symmetry breaking constraints in the two models, combined using channeling constraints, are consistent. Such results allow tackling value symmetries efficiently using additional CSP variables and channeling constraints. In the second method, we identify a common and important class of value symmetries, namely symmetries of indistinguishable values, and introduce value precedence to break such symmetries. Although value precedence can be expressed straightforwardly using if-then constraints in existing constraint programming systems, the resulting formulation is inefficient both in terms of size and runtime. We present two propagation algorithms for implementing global constraints on value precedence for integer and set variables respectively. We also characterize the propagation levels attained by various usages of the global constraints and the conditions when the constraints are consistent with variable symmetry breaking constraints. Extensive experiments are conducted to verify the feasibility and efficiency of our two proposals. / Law Yat Chiu. / "September 2005." / Adviser: Jimmy Ho Man Lee. / Source: Dissertation Abstracts International, Volume: 67-07, Section: B, page: 3905. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 112-123). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract in English and Chinese. / School code: 1307.

Thesis (Ph. D.)--University of Alberta, 2010. / Title from pdf file main screen (viewed on July 30, 2010). A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics, Department of Mathematical and Statistical Sciences, University of Alberta. Includes bibliographical references.

In this thesis we study left-invariant control offine systems on the symmetry group of a. model of hyperbolic plane geometry, the matrix Lie group SO(1, 2)₀. We determine that there are 10 distinct classes of such control systems and for typical elements of two of these classes we provide solutions of the left-invariant optimal wntrol problem with quauratic costs. Under the identification of the Lie allgebra .so(l, 2) with Minkowski spacetime R¹̕'², we construct a controllabilility criterion for all left-invariant control affine systems on 50(1. 2)₀ which in the inhomogeneous case depends only on the presence or absence of an element in the image of the system's trace in R¹̕ ²which is identifiable using the inner product. For the solutions of both the optimal control problems, we provide explicit expressions in terms of Jacobi elliptic functions for the solutions of the reduced extremal equations and determine the nonlinear stability of the equilibrium points.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. September 26, 2014. / The aim of the dissertation is to extend on the work done by Hydon in [17]. We only consider second order ordinary difference equations and calculate their symmetry generators, first integrals and reduce their order, that is, find a general solution. We investigate the association between a symmetry generator and a first integral. Furthermore, we investigate when a reduced equation may be further reduced and lead to a double reduction. The examples considered are obtained from [17]. ii