Thesis (Ph. D.)--University of Oregon, 2006. / Typescript. Includes vita and abstract. "In this dissertation I investigate Poisson structures on symmetric and exterior algebras of modules over complex reductive Lie algebras. I use the results to study the braided symmetric and exterior algebras"--P. 1. Includes bibliographical references (leaves 150-152). Also available for download via the World Wide Web; free to University of Oregon users.

Made available in DSpace on 2014-06-11T19:24:52Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-11-16Bitstream added on 2014-06-13T20:13:17Z : No. of bitstreams: 1 neves_prv_me_rcla.pdf: 1296931 bytes, checksum: b45ea16895bf9e20db063325be68a349 (MD5) / Este trabalho teve o objetivo de produzir um conjunto de atividades para analisar como o uso do caleidoscópio associado ao estudo dos ornamentos planos pode contribuir no ensino de grupos de simetria e transformações geométricas em um curso de graduação em Matemática. Esta pesquisa tem caráter qualitativo e foi desenvolvida segundo a proposta metodológica de Romberg. Elaborou-se uma proposta de ensino baseada na metodologia de Resolução de Problemas que foi aplicada a um grupo de professores (alguns em fase de formação) de matemática. As atividades tiveram a finalidade de fazer com que os alunos usassem o caleidoscópio para reproduzir ornamentos planos e, a partir de então, discutissem, com base em argumentos geométricos e algébricos, quais as possibilidades (e impossibilidades) que esse instrumento oferece para obtenção desses ornamentos e suas respectivas justificativas. A coleta de dados ocorreu, essencialmente, por observação participante em sala de aula por meio do uso de questionários, anotações e registros fotográficos. Após a coleta de dados, foi feita uma análise das possibilidades e limitações do material desenvolvido para o ensino de grupos de simetria e transformações geométricas, bem como o uso do caleidoscópio enquanto recurso didático / The purpose of this work was to develop a set of activities to analyze how the use of kaleidoscope associated to the study of ornaments can contribute to the teaching of symmetry groups and geometric transformations on a undergraduate course in Mathematics. This is a qualitative research and it was developed according to the methodological proposal of Romberg. A teaching proposal was drafted and was applied to a group of mathematics teachers. Activities were designed following the methodology of problem-solving and intended to make students to use the kaleidoscope to reproduce some ornaments and thereafter, discuss, based on geometric and algebraic arguments, the possibilities and impossibilities that this tool provides to obtain ornaments and their respective justifications. Data collection occurred primarily by participant observation in the classroom through the use of questionnaires, notes and photographic records. After the end of the course a viability analysis of the activities was done (possibilities and limitations) for teaching symmetry groups and geometric transformations as well as the use of Kaleidoscope as a didactic tool

The purpose of the thesis is to give an alternative and more efficient method for working with finite groups by constructing finite groups as homomorphic images of progenitors. The method introduced can be applied to all finite groups that possess symmetric generating sets of involutions. Such groups include all finite non-abelian simple groups, which can then be constructed by the technique of manual double coset enumeration.

M.Sc. / The innovative work of Black and Scholes [1, 2] extended the mathematical understanding of the options pricing model, beginning the deliberate study of the theory of option pricing. Its impact on the nancial markets was immediate and unprecedented and is arguably one of the most important discoveries within nance theory to date. By just inserting a few variables, which include the stock price, risk-free rate of return, option's strike price, expiration date, and an estimate of the volatility of the stock's price, the option-pricing formula is easily used by nancial investors. It allows them to price various derivatives ( nancial instrument whose price and value are derived from the value of assets underlying them), including options on commodities, nancial assets and even pricing of employee stock options. Hence, European1 and American2 call or put options on a non-dividend-paying stock can be valued using the Black-Scholes model. All further advances in option pricing since the Black-Scholes analysis have been re nements, generalisations and expansions of the original idea presented by them.

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, 2017 / Killing vectors are widely used to study conservation laws admitted by spacetime metrics or to determine exact solutions of Einstein field equations (EFE) via Killing’s equation. Its solutions on a manifold are in one-to-one correspondence with continuous symmetries of the metric on that manifold. Two well known spherically symmetric static spacetime metrics in Relativity that admit maximal symmetry are given by Minkowski and de-Sitter metrics. Some other spherically symmetric metrics forming interesting solutions of the EFE are known as Schwarzschild, Kerr, Bertotti-Robinson and Einstein metrics. We study the symmetry properties and conservation laws of the geodesic equations following these metrics as well as the wave and Klein-Gordon (KG) type equations constructed using the covariant d’Alembertian operator on these manifolds. As expected, properties of reduction procedures using symmetries are more involved than on the well known flat (Minkowski) manifold. / XL2017

A thesis submitted to the Faculty of Science, University of the Witwatersrand in fulfillment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 3 April 2017. / The effect of suction and blowing at the base on the horizontal spreading under gravity of a two-dimensional thin fluid film and an axisymmetric liquid drop is in- vestigated. The velocity vn which describes the suction/injection of fluid at the base is not specified initially. The height of the thin film satisfies a nonlinear diffusion equation with vn as a source term. The Lie group method for the solution of partial differential equations is used to reduce the partial differential equations to ordinary differential equations and to construct group invariant solutions. For a group invari- ant solution to exist, vn must satisfy a first order linear partial differential equation. The two-dimensional spreading of a thin fluid film is first investigated. Two models for vn which give analytical solutions are analysed. In the first model vn is propor- tional to the height of the thin film at that point. The constant of proportionality is β (−∞ < β < ∞). The half-width always increases to infinity as time increases even for suction at the base. The range of β for the thin fluid film approximation to be valid is determined. For all values of suction and a small range of blowing the maximum height of the film tends to zero as time t → ∞. There is a value of β corresponding to blowing for which the maximum height remains constant with the blowing balancing the effect of gravity. For stronger blowing the maximum height tends to infinity algebraically, there is a value of β for which the maximum height tends to infinity exponentially and for stronger blowing, still in the range for which the thin film approximation is valid, the maximum height tends to infinity in a finite time. For blowing the location of a stagnation point on the centre line is determined by solving a cubic equation approximately by a singular perturbation method and then exactly using a trigonometric solution. A dividing streamline passes through the stagnation point which separates the flow into two regions, an upper region consisting of fluid descending due to gravity and a lower region consisting of fluid rising due to blowing. For sufficiently strong blowing the lower region fills the whole of the film. In the second model vn is proportional to the spatial gradient of the height with constant of proportionality β∗ (−∞ < β∗ < ∞). The maximum height always decreases to zero as time increases even for blowing. The range of β∗ for the thin fluid film approximation to be valid is determined. The half-width tends to infinity algebraically for all blowing and a small range of weak suction. There is a value of β∗ corresponding to suction for which the half-width remains constant with the suction balancing the spreading due to gravity. For stronger suction the half-width tends to zero as t → ∞. For even stronger suction there is a value of β∗ for which the half-width tends to zero exponentially and a range of β∗ for which it tends to zero in a finite time but these values lie outside the range for which the thin fluid film approximation is valid. For blowing there is a stagnation point on the centre line at the base. Two dividing streamlines passes through the stagnation point which separate fluid descending due to gravity from fluid rising due to blowing. An approximate analytical solution is derived for the two dividing streamlines. A similar analysis is performed for the axisymmetric spreading of a liquid drop and the results are compared with the two-dimensional spreading of a thin fluid film. Since the two models for vn are still quite general it can be expected that general results found will apply to other models. These include the existence of a divid- ing streamline separating descending and rising fluid for blowing, the existence of a strength of blowing which balances the effect of gravity so the maximum height remains constant and the existence of a strength of suction which balances spreading due to gravity so that the half-width/radius remains constant. / MT 2017

We consider second and third order limit theorems--namely central-limit theorems, Berry-Esseen bounds and concentration inequalities-- and extend them for "symmetric" random objects, and general estimators of exchangeable structures. At first, we consider random processes whose distribution satisfies a symmetry property. Examples include exchangeability, stationarity, and various others. We show that, under a suitable mixing condition, estimates computed as ergodic averages of such processes satisfy a central limit theorem, a Berry-Esseen bound, and a concentration inequality. These are generalized further to triangular arrays, to a class of generalized U-statistics, and to a form of random censoring. As applications, we obtain new results on exchangeability, and on estimation in random fields and certain network model; extend results on graphon models; give a simpler proof of a recent central limit theorem for marked point processes; and establish asymptotic normality of the empirical entropy of a large class of processes. In certain special cases, we recover well-known properties, which can hence be interpreted as a direct consequence of symmetry. The proofs adapt Stein's method. Subsequently, we consider a sequence of-potentially random-functions evaluated along a sequence of exchangeable structures. We show that, under general stability conditions, those values are asymptotically normal. Those conditions are vaguely reminiscent of those familiar from concentration results, however not identical. We require that the output of the functions does not vary significantly when an entry is disturbed; and the size of this variation should not depend markedly on the other entries. Our result generalizes a number of known results, and as corollaries, we obtain new results for several applications: For randomly sub-sampled subgraphs; for risk estimates obtained by K-fold cross validation; and for the empirical risk of double bagging algorithms. The proof adapts the martingale central-limit theorem.

Children develop the ability to perceive symmetry very early in life; symmetry is abundant in the world around us, and it is a naturally occurring theme in children’s play and creative endeavors. Symmetry is a type of pattern structure and organization of visual information that has been found by psychologists to aid adults in the processing and recall of visual information. Symmetry plays an important role across branches of mathematics and at all levels, and it provides a link between mathematics and a variety of fields and areas of study. Despite this, symmetry does not figure prominently in early childhood mathematics curriculum in the United States. The purpose of this study is to develop, implement, and evaluate a computer program that expands young children’s innate perception and understanding of symmetry and its subtopics—reflection, translation, and rotation. Eighty-six first and second grade children were randomly assigned to one of two conditions: nine sessions using the symmetry computer program designed for this study, or nine sessions using a non-geometry-related computer program. Results showed that children assigned to the experimental condition were better able to identify symmetry subtypes, accurately complete translation tasks and symmetry tasks overall, and explain symmetric transformations. These findings suggest that children are capable of learning about symmetry and its subtypes, and the symmetry software program designed for this study has the potential to improve children’s understanding of symmetry beyond what is currently taught in the early elementary mathematics curriculum. Recommendations for other researchers, educators, and future research are discussed

We consider the study of symmetry analysis of difference equations. The original work done by Lie about a century ago is known to be one of the best methods of solving differential equations. Lie's theory of difference equations on the contrary, was only first explored about twenty years ago. In 1984, Maeda [42] constructed the similarity methods for difference equations. Some work has been done in the field of symmetries of difference equations for the past years. Given an ordinary or partial differential equation (PDE), one can apply Lie algebra techniques to analyze the problem. It is commonly known that the number of independent variables can be reduced after the symmetries of the equation are obtained. One can determine the optimal system of the equation in order to get a reduction of the independent variables. In addition, using the method, one can obtain new solutions from known ones. This feature is interesting because some differential equations have apparently useless trivial solutions, but applying Lie symmetries to them, more interesting solutions are obtained. The question arises when it happens that our equation contains a discrete quantity. In other words, we aim at investigating steps to be performed when we have a difference equation. Doing so, we find symmetries of difference equations and use them to linearize and reduce the order of difference equations. In this work, we analyze the work done by some researchers in the field and apply their results to some examples. This work will focus on the topical review of symmetries of difference equations and going through that will enable us to make some contribution to the field in the near future. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2008. / Subdivision techniques have, over the last two decades, developed into a powerful tool in computer-aided geometric design (CAGD). In some applications it is required that data be preserved exactly; hence the need for interpolatory subdivision schemes. In this thesis,we consider the fundamentals of themathematical analysis of symmetric interpolatory subdivision schemes for curves, also with the property of polynomial filling up to a given odd degree, in the sense that, if the initial control point sequence is situated on such a polynomial curve, all the subsequent subdivision iterates fills up this curve, for it to eventually also become also the limit curve. A subdivision scheme is determined by its mask coefficients, which we find convenient to mathematically describe as a bi-infinite sequnce a with finite support. This sequence is in one-to-one correspondence with a corresponding Laurent polynomial A with coefficients given by the mask sequence a. After an introductory Chapter 1 on notation, basic definitions, and an overview of the thesis, we proceed in Chapter 2 to separately consider the issues of interpolation, symmetry and polynomial filling with respect to a subdivision scheme, eventually leading to a definition of the class Am,n of mask symbols in which all of the above desired properties are combined. We proceed in Chapter 3 to deduce an explicit characterization formula for the classAm,n, in the process also showing that its optimally local member is the well-known Dubuc–Deslauriers (DD) mask symbol Dm of order m. In fact, an alternative explicit characterization result appears in recent work by De Villiers and Hunter, in which the authors characterized mask symbols A ∈Am,n as arbitrary convex combinations of DD mask symbols. It turns out that Am,m = {Dm}, whereas the class Am,m+1 has one degree of freedom, which we interpret here in the formof a shape parameter t ∈ R for the resulting subdivision scheme. In order to investigate the convergence of subdivision schemes associated with mask symbols in Am,n, we first introduce in Chapter 4 the concept of a refinement pair (a,φ), consisting of a finitely-supported sequence a and a finitelysupported function φ, where φ is a refinable function in the sense that it can be expressed as a finite linear combination, as determined by a, of the integer shifts of its own dilation by factor 2. After presenting proofs of a variety of properties satisfied by a given refinement pair (a,φ), we next introduce the concept of an interpolatory refinement pair as one for which the refinable function φ interpolates the delta sequence at the integers. A fundamental result is then that the existence of an interpolatory refinement pair (a,φ) guarantees the convergence of the interpolatory subdivision scheme with subdivision mask a, with limit function © expressible as a linear combination of the integer shifts of φ, and with all the subdivision iterates lying on ©. In Chapter 5, we first present a fundamental result byMicchelli, according to which interpolatory refinable function existence is obtained for mask symbols in Am,n if the mask symbol A is strictly positive on the unit circle in complex plane. After showing that the DD mask symbol Dm satisfies this sufficient property, we proceed to compute the precise t -interval for such positivity on the unit circle to occur for the mask symbols A = Am(t |·) ∈Am,m+1. Also, we compare our numerical results with analogous ones in the literature. Finally, in Chapter 6, we investigate the regularity of refinable functions φ = φm(t |·) corresponding to mask symbols Am(t |·). Using a standard result fromthe literature in which a lower bound on the Hölder continuity exponent of a refinable function φ is given explicitly in terms of the spectral radius of a matrix obtained from the corresponding mask sequence a, we compute this lower bound for selected values of m.