3. Electronic Theses and Dissertations (ETDs) - All submissions
Permanent URI for this communityhttps://wiredspace.wits.ac.za/handle/10539/45
Browse
4 results
Search Results
Item Investigation into competent teachers’ choice and use of examples in teaching algebraic functions in Grade 11 in South African context: a case of two teachers.(2016) Moeti, Makhalanyane Phillipiii ABSTRACT The study focused on two competent, qualified, experienced secondary Mathematics teachers working in contrasting South African school contexts (fee-paying and no fee schools). The study investigated: on how teachers chose and used examples and how they explained their choices and usage; and what considerations were in play when these teachers chose and used examples. These teachers were purposely selected because we can learn more from their experiences as Mathematics teachers especially when they teach quadratic functions. Quadratic functions were used as unit of analysis to illuminate their choice and use of examples.Item The algebra and geometry of continued fractions with integer quaternion coefficients(2015-05-06) Mennen, Carminda MargarethaWe consider continued fractions with coe cients that are in K, the quaternions. In particular we consider coe cients in the Hurwitz integers H in K. These continued fractions are expressed as compositions of M¨obius maps in M R4 1 that act, by Poincar´e extension, as isometries on H5. This dissertation explores groups of 2 2 matrices over K and two particular determinant type functions acting on these groups. On the one hand we find M R4 1 , the group of orientation preserving M¨obius transformations acting on R4 1 in terms of a determinant D [19],[38]. On the other hand K may be considered as a Cli ord algebra C3 based on two generators i and j, or more generally i1 and i2, where i j = k or i1i2 = k. It is shown this group of matrices over C4 defined in terms of a pseudo-determinant [1],[37] can also be used to establish M R4 1 . Through this relationship we are able to connect the determinant D to the pseudo-determinant when acting on the matrices that generate M R4 1 . We explore and build on the results of Schmidt [30] on the subdivision of a Farey simplex into 31 Farey simplices. These results are reinterpreted in H5 with boundary K1 using the group of M¨obius transformations on R4 1 [19], [38]. We investigate the unimodular group G = PS DL(2;K) with its generators and derive a fundamental domain for this group in H5. We relate this domain to the 24-cells PU and r that tessellate K. We define the concepts of Farey neighbours, Farey geodesics and Farey simplices in the Farey tessellation of H5. This tessellation of H5 by a Farey pentacross under a discrete subgroup G of M R4 1 is analogous to the Farey tessellation by Farey triangles of H2 under the modular group [31]. The result in Schmidt [30], that for each quaternion there is a chain of Farey simplices that converge to , is reinterpreted as a continued fraction, with entries from H, that converges to . We conclude with a review of Pringsheim’s theorem on convergence of continued fractions in higher dimensions [5].Item Minimum L∞ norm solutions to finite dimensional algebraic underdetermined linear systems(2015-02-04) Earle, Adam ChristopherA new method of solution to the problem of nding the minimum `1 norm solution to an algebraic underdetermined linear system is developed. The new method is a geometrically clear, primal method. Like some existing methods, the new method can be logically divided into two parts. A number of new techniques are suggested in this part of the algorithm, including an iterative ascent procedure. In the second part of the solution process, the particular solution obtained in the rst part is iteratively improved. We have developed a number of new techniques here corresponding to both single and multi-element exchange procedures. Central to the new method is the development of descent criteria for a direction vector, and the stopping condition.The performance of our algorithm is also compared with two well-known methods from the literature. Our method is shown to be much superior to these well known-methods with respect to both the number of iterations and the wall-clock time required. The iterative computational complexity of the new method also compares favourably with most well-known methods. A geometric heuristic is developed for initial active constraint set selection and a number of theoretical results are given. The heuristic stands to be much more valuable if the results presented herein can be generalisedItem Preservation theorems for algebraic and relational models of logic(2013-07-30) Morton, WilmariIn this thesis a number of different constructions on ordered algebraic structures are studied. In particular, two types of constructions are considered: completions and finite embeddability property constructions. A main theme of this thesis is to determine, for each construction under consideration, whether or not a class of ordered algebraic structures is closed under the construction. Another main focus of this thesis is, for a particular construction, to give a syntactical description of properties preserved by the construction. A property is said to be preserved by a construction if, whenever an ordered algebraic structure satisfies it, then the structure obtained through the construction also satisfies the property. The first four constructions investigated in this thesis are types of completions. A completion of an ordered algebraic structure consists of a completely lattice ordered algebraic structure and an embedding that embeds the former into the latter. Firstly, different types of filters (dually, ideals) of partially ordered sets are investigated. These are then used to form the filter (dually, ideal) completions of partially ordered sets. The other completions of ordered algebraic structures studied here include the MacNeille completion, the canonical extension (also called the completion with respect to a polarization) and finally a prime filter completion. A class of algebras has the finite embeddability property if every finite partial subalgebra of some algebra in the class can be embedded into some finite algebra in the class. Firstly, two constructions that establish the finite embeddability property for residuated ordered structures are investigated. Both of these constructions are based on completion constructions: the first on the Mac- Neille completion and the second on the canonical extension. Finally, algebraic filtrations on modal algebras are considered and a duality between algebraic and relational versions of filtrations is established.