Magister Scientiae - MSc (Mathematics and Applied Mathematics)

Permanent URI for this collectionhttps://hdl.handle.net/10566/19488

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Generalization of a theorem of fitting on the product of two normal nilpotent subgroups of a group
(University of the Western Cape, 1977) Fray, R.L
Fitting proved that if H and K are normal nilpotent subgroups of G, then so is HK (t1 l'Hilfssatz 10, p. 100). The question arises if this result could be generalized to other group theoretical properties
The determination of diffracted wave fields by an annulus according to Braunbek’s method
(University of the Western Cape, 1981) Van Staden, P.W.J
In this thesis a short wave approximation, the method oI W Braunbek, is used to determine the diffracted fields (acoustic and electromagnetic) of plane harmonic waves by an annular aperture. Integral representations of the rigorous diffracted field in terms of the surface field and its normal derivative are derived. Babinet's theorem is proved for acoustic as well as electromagnetic plane harmonic incident waves. A derivation of Sommerfeld's solution for the diffraction of plane harmonic waves by a half-plane is included
Maximal left ideals and idealizers in matrix rings
(University of the Western Cape, 1984) Fransman, Andrew
In this chapter we supply all the necessary definitions as well as the required results needed in this work. All the notation and terminology will also be explained carefully. §1 DEFINITIONS AND NOTATION R will always denote a ring with identity and Mn(R) will denote the ring of nxn matrices over R. As usual the ring of integers, the ring of integers modulo n and the field of rational numbers will be denoted by z, Zn and Q respectively. ng of polynomials in the indeterminate x. The constant term of any polynomial fER[x] will be denoted by const(f). Ideal (or module) will always mean left ideal (or module). In order to simplify notation we shall adopt the convention M,N, M/N, etc. in stead of RM' RN, RM/N, etc., for left Rmodules. It will however always be evident from the context, to which ring R we are referring. Max(R) will denote the collection of all maximal left ideals of R. Mand N will be generic symbols for maximal left ideals. Normally mappings will be written on the left except in the cases of Proposition 1.12 and 1.15. R will be considered as a subring of Mn(R) via the natural embedding r + diag(r, ••• ,r).
Conjugacy classes of some projective linear groups
(University of the Western Cape, 1992) Dietrich, Ernest Athur
Given a finite set X of distinct symbols the symmetric group S* and the alternating group A* are obtained without further constructions. More interesting groups are contrived, however, by imposing a certain structure on the set X and observing the subgroups formed by those elements of S* that preserve this structure. In this thesis, we concern ourselves with one such imposition viz. that defining the notion of a finite projective plane. We look at the different subgroups of S* arising in this manner, with particular emphasis on the projective linear groups and their action on the projective plane. We conclude this work with a detailed study of the structure of the projective linear groups of orders 168 and 5616, respectively. Of particular interest to us are the distinct conjugacy classes of these groups, and the manner in which they relate to one another, within each particular group.
In-service female teachers' anxieties about mathematics: a reflective study on mathematics classroom practice at a college of education
(University of the Western Cape, 1994) Williams, E
The incidence of mathematics anxiety manifesting itself in in-service college students is generally on the increase. Such anxiety does not only affect the mathematical performance of students but also their teaching of the subject. Thus a need exists to investigate measures to alleviate mathematics anxiety as displayed by practicing teachers. It is with these factors in mind that I have embarked upon a study to analyze the role of my teaching practice in the context of mathematics anxiety and learning theories.
Variances of purity within torsion-free Abelian groups
(University of the Western Cape, 1994) Van Ster, Lynnette
In 1921, Prufer introduced the concept of a pure subgroup of an abelian group. This concept, which is applicable only to abelian groups, proved to be a very useful one. Subsequently, this concept has sparked off numerous definitions of subgroups of abelian groups which are either generalizations or refinements of the pure subgroup. We look firstly, at how these ideas have developed since Prufer's time. This picture has been gleaned by the perusal of the Mathematical Reviews to see which papers have been published regarding this topic and then, where available, by studying these papers to try to understand the rationale of the author. Secondly, we group certain concepts which are comparable and then study the interrelation between these concepts. In chapter 3.1, it is shown that, for a torsion-free abelian groupG, the following conditions are equivalent: (i) G is a finite rank completely decomposable group, (ii) all pure subgroups of G are summands, (iii) all pure subgroups of G are balanced in G. One of the interesting results of section 3.2 is the theorem that states that a subgroup of a finite rank completely decomposable group is *-purely generated if and only if it is strongly regular pure and that of 3.3 is that any finite rank *-pure subgroup of a separable group is a completely decomposable summand. Section 3.4 uses for a basis, the theorem proved by P. Hill and C. Megibben which states that a l-pure subgroup of a k-group is itself a k-group. What is so interesting about this theorem is that one of its corollaries states that a -pure subgroup of a separable group is also strongly pure. The last section of the dissertation discusses the relationship between knice subgroups and balanced subgroups. A pure subgroup is knice if and only if it is balanced and its quotient group is a k-group. This result looks as though it could be helpful when trying to look at alternative definitions of balancedness.
Left versus right canonical wiener-hopf factorization for rational matrix functions: an alternative version
(University of the Western Cape, 1995) Petersen, Mark Adam
It is an established fact that both canonical and non-canonical Wiener-hopf factorizations of matrix functions play an important role in various aspects of mathematical analysis and its applications. Indeed, for instance, the Fredholm properties of a block Toeplitz operator 7, with symbol W from the rn x rn matrix Wiener algebra Wnaxm over the unit circle T, may be read off from a (right) Wiener- hopf factorization W(^)-t\/-())r())14l+()) , )€1r, (0.1) where lVa and W- are in W*'*, the funct\on Wa has an analytic extension to the open unit disc D such that det Wa@) f 0 for z e D, th" function W- has an analytic extension to 0 U {-}\ D, such that det W-(z) I 0 for z € Q u {-}\ D, and D()) : diag ()K,)7, , (0.2) with rc1 t. . .t K^ integers. In particular, 7 is invertible if and only if the factorization is canonocal i.e., the indices Ktt . . . ) Krn are all equal to zero, and in this case the inverse of 7 may be constructed
Singular integral equations and realization: A survey of the state space method
(University of the Western Cape, 1996) Gantsho, Yolanda Vuyokazi
Different methods for solving singular integral equations exist. One of the most recent methods is the so-called state space method. This method is based on the fact that a rational matrix function VV(^) which is analytic and invertible at infinity can be represented by vv(^): D * C(AI - A)-'B, (0.1) where A is a square matrix whose order may be larger than that of I,7()), and .8. C and D are matrices of appropriate sizes. The representation (0.1) allows one to reduce analytic problems about rational matrix functions to linear algebra ones involving constant matrices, and often it provides explicit and readily computable formulas for the solutions. In the last fifteen years the state space approach has proved to be effective in solving various problems of mathematical analysis (see the survey paper [BGK3]). In this mini-thesis we employ the state space method to solve singular integral equations. These equations serve as a tool to solve problems in numerous fields of application. For the general theory and examples of applications (see, for instance, [GKr], [M] and [V]).
An examination of a didactical procedure to engage first year university students in meaningful mathematical activity
(University of the Western Cape, 1996) Kannemeyer, Larry Dickson
The challenge to empower learners whose mathematical powers have been underdeveloped has lead the author to search for and implement teaching innovations designed to enhance students learning of mathematics. Undergraduate students receive much of their exposure to mathematics during lectures which are characterised by frontal teaching, a demonstration only by the lecturer. Students are passive recipients of the lecturer's knowledge and end up coming to lectures to gather information to learn at a later stage. The low level activity inherent in traditional lectures results in most students not developing the skills to grapple meaningfully with mathematical concepts and ideas. This has prompted the author to investigate changing the format of lectures in the quest to provide an environment in which students can engage more meaningfully with mathematical concepts. Adopting the position of education theorists who advocate that mathematics as a meaningful activity is engendered by "doing" mathematics and that students must be allowed to enter the culture of the mathematical enterprise, the author has designed a teaching procedure called the "workshop-lecture". This mini-thesis reports on an examination of the design and implementation of the "workshoplecture" which affords first year university students the opportunity to be involved in meaningful mathematical activity. This examination provides evidence that the format of the "workshop-lecture" is conducive to more meaningful interaction amongst students and between lecturer and students than would be the case in a traditional lecture, even with constraints such as venues with fixed, tiered seating and a relatively large class size of 56 students. It also highlights issues such as lecturer intervention, and the learning materials and aids that facilitate student interest and involvement in meaningful mathematical activity. A way of expanding the notion of the "workshop-lecture" to create opportunities for students to recognize more their own responsibility for their learning is proposed, as well as a strong recommendation for changing the curriculum to allow for meaningful involvement by students.
Computation of the character tables of certain group extensions
(University of the Western Cape, 1997) James, C.
The work done in this mini-thesis deals mainly with different methods of calculating character tables of split extensions of finite groups. Three of the six character tables that are calculated are done with the use of Fischer matrices. In this work, the method of Fischer is applied on groups of the form N.G where N is an elementary abelian group. In fact, only one of the six groups of which the character tables are calculated is not of this form and so Fischer matrices could easily have been used to calculate five of the character tables. The aim of the work done here however is to exhibit a variety of methods to calculate the character tables of split extensions. In Chapter One a review of basic definitions and results on group extensions and a description of a method for finding the conjugacy tables of group extensions is given. An example of the application of this method is also given. Chapter two deals with basic concepts and results on representation and character theory as well as the application of some of these results in calculating the character tables of some group extensions. In Chapter Three we discuss Fischer matrices and how it is used to calculate the character tables of group extensions of the form N.G where N is an elementary abelian group.
Block Toeplitz operators with rational symbols and discrete singular systems
(University of the Western Cape, 2001) Konegerie, Abraham
This thesis concerns block Toeplitz operators (equations). Consider the block Toeplitz operator T = [ k-j ]k,j=O' where the k are complex m x m matrices such that 00 (0.1) v=-oo The norm in (0.1) is the usual operator norm on an m x m matrix. The condition (0.1) means that the symbol 00 ( 0.2) v=-oo belongs t.o the Wiener class w mxm of all absolutely convergent sequences of complex m x m matrices. Let 1 ~ p ~ oo be fixed. The block Toeplitz operator T induces a bounded linear operator (also denoted by T) on l'';, namely, 00 (0.3) Yk = (Tx )k = L k-vXv , k = 0, 1, 2, ... ' v=O where x = (xo, x 1,X2, ... ) Et;.
Character tables of some groups of extension type
(University of the Western Cape, 2002) Prins, Abraham L.
The main aim of this mini-thesis is to give a description of some of the basic methods and techniques that have been developed to calculate the character tables of groups of extension type. We restrict our attention to split extensions G of the normal subgroup N of G by the subgroup G with the property that every irreducible character of N can be extended to an irreducible character of its inertia group in G. This is particularly true when N is abelian. We are therefore interested in this special case for which Bernd Fischer developed the theory of Fischer matrices based on the Clifford Theory, to calculate the character tables for both split and non-split extensions.
A critical analysis of the pre-calculus course at U.W.C.
(University of the Western Cape, 2002) Myburgh, Namari
Mathematical knowledge and skills have, over the last few decades, become very important in terms of study and work opportunities. Unfortunately, schools in South Africa have not met the expectations of their pupils or the country in terms of delivering the appropriate number of students mathematically equipped for scientific study at universities. To redress this imbalance many tertiary institutions offer foundation courses to accommodate students who did not attain a matriculation exemption to gain entry to tertiary study. The Pre-Calculus course at the University of the Western Cape forms part of the foundation course of the university. This study analyzes the Pre-Calculus course for the identification of its attributes that would act as substitute for a matriculation exemption in terms of higher grade mathematics and for the empowering of its students to study other subjects in the science faculty. To this end questions from higher grade mathematics matriculation examination papers were analyzed to identify the mathematical thinking and algorithmic skills that are tested at this level. Subsequently it is shown that the Pre-Calculus course does indeed have the content that can facilitate the development of these skills. The students in the Pre-Calculus course were given the opportunity via an extensive questionnaire to give their opinion of the course, the problems they had while studying Pre-Calculus, their motivation for studying mathematics and the effect the course had on them emotionally. They also had the opportunity to criticize and make recommendations about the course. The information gained from the questionnaire supplemented by the observations of the author gives good insight into the problems and ideals of these students. Recommendations to improve the effectiveness of the Pre-Calculus course and recommendations for further research conclude this study.
Computing Mislin genera of certain groups with non-abelian torsion radicals
(University of the Western Cape, 2004) Hess, Victor George
In this mini-thesis we present some generalities of non-cancellation and localization and we compute non-cancellation groups. We consider groups belonging to the class X0 of all finitely generated groups that have finite commutator subgroups. For a X0-group H, we study the non - cancellation set, x(H), which is defined to be the set of all isomorphism classes of groups K such that H x Z ~ K x Z. In particular, we prove some basic facts such as that for a group G which is either finite or finitely generated abelian, we have H x Z ~ G x Z only if G"' H. For a finitely generated nilpotent group N , the Mislin genus, Q(N), is defined to be the set of all isomorphism classes of finitely generated nilpotent groups M such that for every prime p, the groups M and N have isomorphic p-localizations. It was shown by Warfield that if N is a nilpotent X0-group, then x(N) = Q(N). Various calculations of such Hilton-Mislin genus groups can be found in the literature, for example, in an article of Hilton and Scevenels. Most of these calculations are for a special subclass of nilpotent X0-groups, in particular, groups with abelian torsion radicals.
Throughput of UWC students who did at least one semester of third-year Statistics
(University of the Western Cape, 2005) Latief, Abduraghiem
The study explores the completion rates (the number of years a student takes to complete a degree) of graduates at the University of the Western Cape (UWC) in South Africa. The graduates in the study all did at least one semester of statistics in their final year of study. The students' completion will be described with respect to school results and socio-demographics. Differences between students who finished their studies in the prescribed time of three years and those who took longer than the prescribed time will be highlighted. Factors that aid or hinder students from successfully completing their studies in the prescribed time will be analyzed. An entry selection model will be developed to screen the students. This will assist with an enrolment strategy. The most significant result found was that the political environment played the most significant role in throughput. The next significant result from the study showed that the grade 12 aggregate played a significant role in throughput. It is suggested that UWC be proactive in developing alternative methods of selecting students, since the new Further Education Training (FET) school system, which will be implemented in 2006, will omit the grade 12 aggregate.
Numerical methods for the valuation of financial derivatives
(University of the Western Cape, 2005) Ntwiga, Davis Bundi
Numerical methods form an important part of the pricing of financial derivatives and especially in cases where there is no closed form analytical formula. We begin our work with an introduction of the mathematical tools needed in the pricing of financial derivatives. Then, we discuss the assumption of the log-normal returns on stock prices and the stochastic differential equations. These lay the foundation for the derivation of the Black Scholes differential equation, and various Black Scholes formulas are thus obtained. Then, the model is modified to cater for dividend paying stock and for the pricing of options on futures. Multi-period binomial model is very flexible even for the valuation of options that do not have a closed form analytical formula. We consider the pricing of vanilla options both on non dividend and dividend paying stocks. Then show that the model converges to the Black-Scholes value as we increase the number of steps. We discuss the Finite difference methods quite extensively with a focus on the Implicit and Crank-Nicolson methods, and apply these numerical techniques to the pricing of vanilla options. Finally, we compare the convergence of the multi-period binomial model, the Implicit and Crank Nicolson methods to the analytical Black Scholes price of the option. We conclude with the pricing of exotic options with special emphasis on path dependent options. Monte Carlo simulation technique is applied as this method is very versatile in cases where there is no closed form analytical formula. The method is slow and time consuming but very flexible even for multi dimensional problems.
A survey of the computer enhanced services of the Outreach Project of UWC developed for grade 12 mathematic learners and a critical appraisal of the MICSEC2000 program.
(University of the Western Cape, 2005) Isaacs, Brian Ernest Leonard
The Outreach Project of the University of the Western Cape has since 1982 through various computer supported services been assisting grade 12 mathematics learners and educators at previously disadvantaged Western Cape high schools. This thesis described and appraised the MICSES2000 program of the Outreach Program, the latest innovative computer enhancing service to schools, with respect to its implementation, perceived and achieved programs by participating educators.
Investigating an integrated teaching methodology as a means to prepare students for university studies in mathematics
(University of the Western Cape, 2005) Ceasar, Reginald Raymon
A key issue for the success of students entering a first year mathematics course at tertiary level is whether or not they have an integrated understanding and view of the mathematical concepts acquired at school. Various integrated applications from first year mathematics suggest that a compartmentalised view of mathematics would be detrimental to any student's chances of passing mathematics at this level. This study tried to assess whether learners do have an integrated understanding of mathematics at grade 12 level.
Transport modelling in the Cape Town Metropolitan Area
(University of the Western Cape, 2005) Munyakazi, Justin Bazimaziki
The use of MEPLAN by the Metropolitan Transport Planning Branch of the Cape Town City Council since 1984 was not successful due to apartheid anomalies. EMME/2 was then introduced in 1991 in replacement of MEPLAN. In this thesis we first introduce some aspects of transport modelling. Secondly we summarize the above-mentioned models before we undertake their comparative study in a post-apartheid situation. A mathematical proof of why MEPLAN was discarded is provided. The strengths and weaknesses of both MEPLAN and EMME/2 are recorded.
Analysis and implementation of a positivity preserving numerical method for an HIV model
(University of the Western Cape, 2007) Wyngaardt, Jo-Anne
This thesis deals with analysis and implementation of a positivity preserving numerical method for a vaccination model for the transmission dynamics of two HIVsubtypesnin a given community. The continuous model is analyzed for stability and equilibria. The qualitative information thus obtained is used while designing numerical method(s). Three numerical methods, namely, Implicit Finite Difference Method (IFDM), Non-standard Finite Difference Method (NSFDM) and the Runge-Kutta method of order four (RK4), are designed and implemented. Extensive numerical simulation are carried out to justify theoretical outcomes

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