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Symmetry reductions of some non-linear 1+1 D and 2+1 D black-scholes modelsSeoka, Nonhlanhla 19 September 2016 (has links)
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
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Symmetry and transformation properties of linear iterative ordinary differential equationFolly-Gbetoula, Mensah Kekeli 06 August 2013 (has links)
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.
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Conditional symmetry properties for ordinary differential equationsFatima, Aeeman 07 May 2015 (has links)
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
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Turbulent hydraulic fracturing described by Prandtl's mixing lengthNewman, Despina 19 September 2016 (has links)
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
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Combinatorial aspects of symmetries on groupsSingh, Shivani January 2016 (has links)
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
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Smoothness conditions and symmetries of partial differential equationsMamba, Siphamandla 10 May 2016 (has links)
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.
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Using constraints to break value symmetries in constraint satisfaction problems. / CUHK electronic theses & dissertations collection / Digital dissertation consortiumJanuary 2005 (has links)
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.
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Interpolating refinable function vectors and matrix extension with symmetryZhuang, Xiaosheng. January 2010 (has links)
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.
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The symmetry group of a model of hyperbolic plane geometry and some associated invariant optimal control problemsHenninger, Helen Clare January 2012 (has links)
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.
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Difference equations and their symmetriesNdlovu, Lungelo Keith 29 January 2015 (has links)
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].
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