ETD Collection

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Now showing 1 - 9 of 9
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    An archaeological study of the two-hundred-year-old raised structures in southern Gauteng, South Africa
    (2017) Croll, Kathryn D.
    This study was first and foremost a descriptive study of the so-called enigmatic raised structures found within the Suikerbosrand Nature Reserve and the neighbouring farm. The study also aimed to determine whether the raised structures could possibly be called ‘grain bin bases’, ‘grave markers’, ‘cairns’ or ‘monuments’. The study followed a quantitative methodology which involved field surveys and an exploratory GIS study of the LiDAR data by using the following variables: slope, aspect, ruggedness, elevation, viewshed area and the number of other raised structures visible from each raised structure. It is difficult to ascertain the precise function of these structures using the methods applied in this study, however, the structures do fulfil the characteristics of monumental architecture. This ties in with other research which detailed that the Group II stone walling within the Suikerbosrand area was built by the most politically stratified groups which occupied the region during the Late Iron Age. Keywords: GIS, LiDAR, Suikerbosrand, Group II stonewalling, Raised structures, monumental architecture
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    Graphs and graph polynomials
    (2017) Kriel, Christo
    In this work we study the k-defect polynomials of a graph G. The k defect polynomial is a function in λ that gives the number of improper colourings of a graph using λ colours. The k-defect polynomials generate the bad colouring polynomial which is equivalent to the Tutte polynomial, hence their importance in a more general graph theoretic setting. By setting up a one-to-one correspondence between triangular numbers and complete graphs, we use number theoretical methods to study certain characteristics of the k-defect polynomials of complete graphs. Specifically we are able to generate an expression for any k-defect polynomial of a complete graph, determine integer intervals for k on which the k-defect polynomials for complete graphs are equal to zero and also determine a formula to calculate the minimum number of k-defect polynomials that are equal to zero for any complete graph.
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    Orthogonal polynomials and three-term recurrence relations
    (1991) Engelbrecht, Kevin Peter
    Orthogonal polynomials have had a long history. They have featured in the work of Legendre on planetary motion, continued fractions of Stieltjes, mechanical quadrature of Gauss etc. After the publication of 'Orthogonal Polynomials' by Gabor Szego in 1938 relatively little was published on orthogonal polynomials. This changed in the 1970's when increased interest in approximation theory brought about by the incredible upsurge in the use of the computer in the sciences occurred. [Abbreviated Abstract. Open document to view full version]
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    Exact global symmetry generators for restricted Schur polynomials
    (2016) Bornman, Nicholas
    The six scalar fields in N = 4 super Yang-Mills theory enjoy a global SO(6) symmetry, and large N but non-planar limits of this theory are well-described by adopting a group representation approach. Studies have shown that the one-loop dilatation operator is highly determined by the action of the su(2)=su(3) subalgebras on restricted Schur polynomials. These actions involve the traces of products of projection operators. In this dissertation, exact analytical formulae for these traces are found which in turn are used to find the exact action of these algebras on restricted Schur polynomials. The potential of the su(2) algebra to determine the one-loop dilatation operator is also explored. This is done by exploiting necessary symmetry conditions and moving to a continuum limit in order to derive a number of partial differential equations which determine the dilatation operator. The ultimate goal of this work is to provide tools to find the exact one-loop dilatation operator in the non-planar limit.
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    Weighted approximation for Erdos weight
    (1995) Damelin, Steven Benjamin
    We investigate Mean Convergence of Lagrange Interpolation and Rates of Approximation for Erdo's Weights on the Real line. An Erdos Weight is of the form, W = exp[-Q], where typically Q is even, continous and is of faster than polynomial growth at infinity. Concerning Lagrange Interpolation, we first investigate the problem of formulating and proving the correct Jackson Theorems for Erdos Weights. [ Abbreviated abstract : Open document to view full version]
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    The distinguished guests of giants
    (2016) Mathwin, Christopher Richard
    The convenient pictorial descriptions of the half-BPS and near-BPS sectors of the AdS=CFT equivalent theories of N = 4, D = 4 super Yang-Mills and D = 10 Type IIB superstring theory on AdS5 S5 are exploited in this thesis by using Schur polynomials labelled by Young diagrams as a basis for the gauge invariant operators in the eld theory. We use a \Fourier transform" on these operators to construct asymptotic eigenstates of the dilatation operator, the spectrum of which agrees precisely with the rst two leading order terms in the smallcoupling expansion of the exact result determined by symmetry. Motivated by the geometric description of the systems of open strings with magnon excitations to which the operators are dual, we propose a simple and minimal all-loop expression that interpolates between anomalous dimensions computed in the gauge theory and energies computed in the string theory. The connection to the string theory result provides the insight necessary to understand the interpretation of our Gauss graphs in the magnon language. Symmetry determines the two-body scattering matrix for the magnons up to a phase, and it is demonstrated that integrability is spoiled by the boundary conditions on the open strings. The Schur polynomial construction is then applied to the study of closed strings on a class of half- BPS excitations of the AdS5 S5 background. The string theory predictions for the magnon energies are again reproduced by calculating the anomalous dimensions of particular linear combinations of our operators. Group theoretic quantities which can be read o the Young diagram labels provide the correct modi cation of terms in the dilatation action to account for the energies of magnons at di erent radii on the LLM plane. The representation theory implies a natural splitting of the full symmetry group - the distinction between what is the background and what is the excitation is accomplished in the choice of the subgroup and representations used to construct the operator. Connecting the descriptions utilised in obtaining these results is expected to allow the construction of operators dual to general open string con gurations on the class of backgrounds considered.
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    The chromatic polynomial of a graph
    (2016) Adam, A A
    Firstly we express the chromatic polynomials of some graphs in tree form. We then Study a special product that comes natural and is useful in the calculation of some Chromatic polynomials. Next we use the tree form to study the chromatic polynomial Of a graph obtained from a forest (tree) by "blowing up" or "replacing" the vertices Of the forest (tree) by a graph. Then we give explicit expressions, in terms of induced Subgraphs, for the first five coefficients of the chromatic polynomial of a connected Graph. In the case of higher order graphs we develop some useful computational Techniques to obtain some higher order coefficients. In the process we obtain some Useful combinatorial identities, some of which are new. We discuss in detail the Application of these combinatorial identities to some families of graphs. We also discuss Pairs of graphs that are chromatically equivalent and graph that are chromatically Unique with special emphasis on wheels. In conclusion,
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    The transition across the cognitive gap - the case for long division - : Cognitive architecture for division : base ten decomposition as an algorithm for long division
    (2008-11-04T12:53:01Z) Du Plessis, Jacques Desmond
    This is an action research study which focuses on a didactical model founded on base ten decomposition as an algorithm for performing division on naturals. Base ten decomposition is used to enhance the algebraic structure of division on naturals in an attempt to cross the cognitive divide that currently exists between arithmetic long division on naturals and algebraic long division on polynomials. The didactical model that is proposed and implemented comprises three different phases and was implemented over five one hour lessons. Learners’ work and responses which were monitored over a fiveday period is discussed in this report. The structure of the arithmetic long division on naturals formed the conceptual basis from which shorter methods of algebraic long division on polynomials were introduced. These methods were discussed in class and reported on in this study.