Faculty of Natural and Agricultural Sciences
Department of Mathematics and Applied Mathematics
Selected Highlights from Research Findings
A financial market becomes incomplete when some 'friction' in the form transaction costs, taxes etc., is introduced to buyers and sellers. The result is that it becomes difficult to be protected in the opposition direction each time one takes a position with a financial instrument, that is, perfect hedging becomes impossible. Hence, there exists a continuum of equivalent martingale pricing measures, Q and correspondingly arbitrage-free prices which are not unique. The question of choosing this Q to give the best price becomes a crucial one. The Randon-Nykodym density process gives a way of transforming and obtaining the best pricing measure.
In this paper we chose the density of the so called minimal entropy martingale measure and managed to express it in terms of the solution of a semilinear partial differential equation (PDE). By doing so we managed to prove the existence and uniqueness of the solution of the PDE in the case of a time-dependent general stochastic volatility model. We applied our results to two particular cases of the stochastic volatility markets: the Stein-Stein and the Heston models and obtained the best candidate density processes of the minimum entropy martingale measure.
Contact person: Dr R Kufakunesu.
The supplemental instruction (SI) programme has been well-established worldwide and the resulting success of the programme is indisputable. The University of Pretoria has decided on SI as the model to be used for addressing the underpreparedness of students entering the university, largely brought about by the changes in the curricula at secondary school level. The SI model was piloted in two courses, one in mathematics and another in chemistry, each consisting of more than a thousand students.
This article addresses implementation issues of SI for such a large group of students in mathematics. It cautions would-be implementers to pitfalls and shortcomings of the SI model and suggests how the model could be adapted
to answer the current needs. This article also shows that despite problems in strictly adhering to SI principles in the implementation of the programme, participants showed increased performance. (Co-authors: Prof JC Engelbrecht and Me A Verwey)
Contact person: Prof AF Harding.
This article describes the experiences and mathematics performance of Grade 12 learners selected to participate in a mathematics intervention project using digital video disk (DVD) technology within a blended learning context. Blended learning in the context of this study is defined as employing a variety of appropriate methods of delivery to enhance the teaching and learning process. DVD technology was used as an ingredient in this blended learning
approach, since it is easily available and accessible to the majority of learners and the schools they attend. The study reported on here forms part of a larger study using action research methodology. This article reports on a single stage of the action research: implementing a change to improve the situation and observing the consequences of this action. Mathematics Incubator School Project (ISP) learners completed questionnaires with open-ended questions which pertained to their experiences of the blended learning approach. The observations of the facilitators were also recorded. A single school was used as a case study and the mathematics performance of learners who participated in the ISP was compared with that of those who did not. The findings suggest that use of DVD technology in this blended learning approach impacted on mathematics learning and enhanced the mathematics performance of learners. (Co-authors: P Padayachee, H Boshoff and W Olivier)
Contact person: Prof AF Harding.
We provide and analyse a model for the growth of bacterial biofilms based on the concept of an extracellular polymeric substance as a polymer solution, whose viscoelastic rheology is described by the classical Flory-Huggins theory. The proposed model describes biofilm growing on an impermeable substratum based on the material properties of a polymer solution. The biofilm cellular volume fraction is assumed to be small. Estimates of typical scales lead to a different mathematical description from earlier models. We show that one-dimensional solutions exist, which take the form at large times of travelling waves, and we characterize their form and speed in terms of the describing parameters of the problem. Numerical solutions of the time-dependent problem converge to the travelling wave solutions. (Co-authors: HF Winstanley, MJ McGuinness and AC Fowler)
Contact person: Dr M Chapwanya.
We develop numerical solutions of a theoretical model which has been proposed to explain the formation of subglacial bedforms. The model has been shown to have the capability of producing bedforms in two dimensions, when they may be interpreted as ribbed moraine. However, these investigations have left unanswered the question of whether the theory is capable of producing fully three-dimensional bedforms such as drumlins. Three-dimensional calculations of two versions of the instability theory of drumlin formation suggest that the segmented ridge pattern which is typically observed can be well simulated by the model. However, no genuine three-dimensional bedforms (drumlins) are produced. In both versions of the model presented here, (a static one, where water is everywhere in hydraulic equilibrium, and a dynamic one, where transient effects due to groundwater flow are included) it is essentially assumed that hydraulic drainage occurs through a passive stream system which does not interact with the ice flow. We show that, while the three-dimensional calculations show realistic quasi-three-dimensional features such as dislocations in the ribbing pattern, they do not produce genuine three-dimensional drumlins. We suggest that this inadequacy is due to the treatment of subglacial drainage in the theory as a passive variable, and thus that the three-dimensional forms may be associated with conditions of sufficient subglacial water flux. (Co-authors: CD Clark and AC Fowler)
Contact person: Dr M Chapwanya.
This paper considers the problem of constructing finite-difference methods that are qualitatively consistent with the original continuous-time model they approximate. To achieve this goal, a deterministic continuous-time model for the transmission dynamics of two strains of an arbitrary disease, in the presence of an imperfect vaccine, is considered. The model is rigorously analysed, first of all, to gain insights into its dynamical features. The analysis reveal that it undergoes a vaccine-induced backward bifurcation when the associated reproduction threshold is less than unity. For the case where the vaccine is 100% effective, the disease-free equilibrium of the model is shown to be globally-asymptotically stable if the reproduction number is less than unity. The model also exhibits the phenomenon of competitive exclusion, where the strain with the higher reproduction number dominates (and drives out) the other. Two finite-difference methods are presented for numerically solving the model. The central objective is to determine which of the two methods gives solutions that are dynamically consistent with those of the continuous-time model. The first method, an implicitly-derived explicit finite-difference method, is considered for its computational simplicity, being a Gauss-Seidel-like algorithm. However, this method is shown to suffer numerous scheme-dependent numerical instabilities and spurious behaviour (such as convergence to the wrong steady-state solutions and failing to preserve many of the main essential dynamical features of the model), particularly when relatively large step-sizes are used in the simulations. On the other hand, the second numerical method, constructed based on Mickens’ non-standard finite-difference discretization framework, is shown to be free of any numerical instabilities and contrived behaviour regardless of the size of the step-size used in the numerical simulations. In other words, unlike the first method, the non-standard method is shown to be dynamically consistent with the original continuous-time model, and, therefore, it is more suited for use to study the asymptotic dynamics of the disease transmission model being considered. (Co-authors: SM Garba and AB Gumel)
Contact person: Prof JM-S Lubuma.
This paper considers the problem of constructing finite-difference methods that are qualitatively consistent with the original continuous-time model they approximate. To achieve this goal, a deterministic continuous-time model for the transmission dynamics of two strains of an arbitrary disease, in the presence of an imperfect vaccine, is considered. The model is rigorously analysed, first of all, to gain insights into its dynamical features. The analysis reveal that it undergoes a vaccine-induced backward bifurcation when the associated reproduction threshold is less than unity. For the case where the vaccine is 100% effective, the disease-free equilibrium of the model is shown to be globally-asymptotically stable if the reproduction number is less than unity. The model also exhibits the phenomenon of competitive exclusion, where the strain with the higher reproduction number dominates (and drives out) the other. Two finite-difference methods are presented for numerically solving the model. The central objective is to determine which of the two methods gives solutions that are dynamically consistent with those of the continuous-time model. The first method, an implicitly-derived explicit finite-difference method, is considered for its computational simplicity, being a Gauss-Seidel-like algorithm. However, this method is shown to suffer numerous scheme-dependent numerical instabilities and spurious behaviour (such as convergence to the wrong steady-state solutions and failing to preserve many of the main essential dynamical features of the model), particularly when relatively large step-sizes are used in the simulations. On the other hand, the second numerical method, constructed based on Mickens’ non-standard finite-difference discretization framework, is shown to be free of any numerical instabilities and contrived behaviour regardless of the size of the step-size used in the numerical simulations. In other words, unlike the first method, the non-standard method is shown to be dynamically consistent with the original continuous-time model, and, therefore, it is more suited for use to study the asymptotic dynamics of the disease transmission model being considered. (Co-authors: JM-s Lubuma and AB Gumel)
Contact person: Dr SM Garba.
The supplemental instruction (SI) programme has been well-established worldwide and the resulting success of the programme is indisputable. The University of Pretoria has decided on SI as the model to be used for addressing the underpreparedness of students entering the university, largely brought about by the changes in the curricula at secondary school level. The SI model was piloted in two courses, one in mathematics and another in chemistry, each consisting of more than a thousand students.
This article addresses implementation issues of SI for such a large group of students in mathematics. It cautions would-be implementers to pitfalls and shortcomings of the SI model and suggests how the model could be adapted
to answer the current needs. This article also shows that despite problems in strictly adhering to SI principles in the implementation of the programme, participants showed increased performance. (Co-authors: Prof AF Harding and Me A Verwey)
Contact person: Prof JC Engelbrecht.
The 2009 intake of university students were the first to have received complete school education within the recently implemented Outcomes-Based Education (OBE) system. A feature of the matriculation examination results of these students was the exceptionally high Grade 12 marks for Mathematics. This paper addresses the question of how the 2009-intake of students performed at university with respect to general performance, general
attributes, mathematical attributes and content related attributes. It appears that these students are better prepared with respect to personal attributes such as confidence.
However, in many instances they are weaker than their predecessors with respect to mathematical and content related attributes. Yet, there are positive indications that these students adapt and improve over a semester. Some suggestions are made on how to make the transition from secondary to university mathematics somewhat smoother. (Co-Authors: Prof AF Harding and Dr P Phiri)
Contact person: Prof JC Engelbrecht.
This paper reports on an international study about mathematics students’ ideas of how they will use mathematics in their future study and careers. This builds on a previous research into students’ conceptions of mathematics. In this paper, data is used from two groups of students studying mathematics: those who participated in an in-depth interview and those who completed an open-ended questionnaire. It was found that their responses could be grouped into four categories: don’t know; procedural skills; conceptual skills; and professional skills. Although some students held clear ideas about the role of mathematics, many were not able to articulate how it would be used in their future. This has implications for their approach to learning and our approach to teaching. (Co-authors: AF Harding, LN Wood, G Mather, P Petocz, A Reid, K Houston, GH Smith and G Perrett)
Contact person: Prof JC Engelbrecht.
Abstract. A deterministic model is designed and used to theoretically assess the impact of antiviral drugs in controlling the spread of the 2009 swine influenza pandemic. In particular, the model considers the administration of the antivirals both as a preventive as well as a therapeutic agent. Rigorous analysis of the model reveals that its disease-free equilibrium is globally-asymptotically stable under certain conditions involving having the associated reproduction number less than unity. Furthermore, the model has a unique endemic equilibrium if the reproduction threshold exceeds unity. The model provides a reasonable fit to the observed H1N1 pandemic data for the Canadian province of Manitoba. Numerical simulations of the model suggest that the singular use of antivirals as preventive agents only makes a limited population-level impact in reducing the burden of the disease in the population (except if the effectiveness level of this “prevention-only” strategy is high). On the other hand, the combined use of the antivirals (both as preventive and therapeutic agents) resulted in a dramatic reduction in disease burden. Based on the parameter values used in these simulations, even a moderately-effective combined
treatment-prevention antiviral. (Co-authors: M Imran and MT Malik)
Contact person: Dr SM Garba.
We investigate the zeros of polynomial solutions to a differential-difference equation. In particular we address the question of when the zeros are real and simple and whether the zeros of polynomials of adjacent degree are interlacing. Our result holds for general classes of polynomials including sequences of classical orthogonal polynomials as well as Euler-Frobenius, Bell and other polynomials. (Co-authors: Diego Dominici and Kathy Driver)
Contact person: Prof KH Jordaan.
Classical orthogonal polynomials have several special properties. The number of zeros of a polynomial is the same as the degree and, if the polynomial is orthogonal, these zeros are all real, distinct and lie in the interval of orthogonality. A classical result on interlacing of zeros of orthogonal polynomials is due to Stieltjes who proved that if m < n, then between any two successive zeros of the polynomial of degree m there is at least one zero of the polynomial of degree n. Interlacing properties such as Stieltjes interlacing provide more information on the location of the zeros and are useful in many other contexts, most notably in numerical quadrature. Collaborative research with Prof K Driver (UCT) proved that Stieltjes interlacing extends, under certain conditions, to the zeros of Laguerre polynomials from different orthogonal sequences. Furthermore, in collaboration with Prof Driver and Mrs Jooste (UP) it was shown that the same property also holds for two polynomials, in a two parameter family of orthogonal polynomials known as Jacobi polynomials, with different parameter values.
Contact person: Prof KH Jordaan.
This work expands the mathematical theory which connects continuous dynamical systems and the discrete dynamical systems obtained from the associated numerical schemes. The problem is considered within the setting of Topological Dynamics. The topological dynamic consistency of a family of DDSs and the associated continuous system is defined as topological equivalence between the evolution operator of the continuous system and the set of maps defining the respective DDSs, for all positive time-step sizes. The one-dimensional theory is developed and a few important representative examples are studied in detail. It is found that the design of non-standard topologically dynamically consistent schemes requires some care. (Co-authors: R Anguelov and M Shillor)
Contact person: Prof JM-S Lubuma.
This work expands the mathematical theory which connects continuous dynamical systems and the discrete dynamical systems obtained from the associated numerical schemes. The problem is considered within the setting of Topological Dynamics. The topological dynamic consistency of a family of DDSs and the associated continuous system is defined as topological equivalence between the evolution operator of the continuous system and the set of maps defining the respective DDSs, for all positive time-step sizes. The one-dimensional theory is developed and a few important representative examples are studied in detail. It is found that the design of non-standard topologically dynamically consistent schemes requires some care. (Co-authors: JM-S Lubuma and M Shillor)
Contact person: Prof R Anguelov.
In 2011 Dr SM Maepa classified all 14 natural tensor norms of Grothendieck according to the possession or the lack of the Radon Nikodym property (RNP) and the Lewis Radon Nikodym property (Lewis RNP). These are the properties of tensor norms he first defined and discussed in an article of 2007 and the possession or the lack of which was limited to the smallest, the largest and the Hilbertian tensor norms. There are, up to equivalence, 14 natural tensor norms of Grothendieck.
Contact person: Dr SM Maepa.
The 2009 intake of university students were the first to have received complete school education within the recently implemented Outcomes-Based Education (OBE) system. A feature of the matriculation examination results of these students was the exceptionally high Grade 12 marks for Mathematics. This paper addresses the question of how the 2009-intake of students performed at university with respect to general performance, general
attributes, mathematical attributes and content related attributes. It appears that these students are better prepared with respect to personal attributes such as confidence.
However, in many instances they are weaker than their predecessors with respect to mathematical and content related attributes. Yet, there are positive indications that these students adapt and improve over a semester. Some suggestions are made on how to make the transition from secondary to university mathematics somewhat smoother. (Co-Authors: Prof JC Engelbrecht and Dr P Phiri)
Contact person: Prof AF Harding.
Trees are important set-theoretical structures which are used in mathematics in the study of set theory, model theory, graph theory and temporal logic and which also find application in the theory of databases and linguistics. This paper examines the formal theory of trees from the perspective of first-order logic. First-order logic, considered to be the mathematically most natural formal logical language, allows for the negation, disjunction, conjunction and implication of statements as well as for existential and universal quantification over individual nodes of a tree. Quantification over sets of nodes is not allowed. (Co-author: V Goranko)
The first-order theories of some naturally important classes of trees, such as finite trees, well-founded trees, and finitely-branching trees, have been studied in detail, but general first-order theories of arbitrary classes of trees have not yet been comprehensively studied. For every class of linear orders there naturally arise eight classes of trees determined by how the paths of those trees relate to the given linear orders. This paper examines the relationship between the first-order theories of these classes of trees and the first-order theory of the underlying class of linear orders. The set-theoretical relationships between these eight classes of trees, as well as between their corresponding first-order theories, is completely determined. We also establish the non-trivial facts that (i) if a path in a tree is at all parametrically definable then that path can be defined using a single node from the path itself as parameter, and (ii) the expressive power of nodes improves with their height, in the sense that every parametrically definable set of nodes in a tree can be defined using a set of parameters which lie higher up in the tree than the original parameters.
Contact person: Dr R Kellerman.
This paper reports on an international study about mathematics students’ ideas of how they will use mathematics in their future study and careers. This builds on a previous research into students’ conceptions of mathematics. In this paper, data is used from two groups of students studying mathematics: those who participated in an in-depth interview and those who completed an open-ended questionnaire. It was found that their responses could be grouped into four categories: don’t know; procedural skills; conceptual skills; and professional skills. Although some students held clear ideas about the role of mathematics, many were not able to articulate how it would be used in their future. This has implications for their approach to learning and our approach to teaching. (Co-authors: JC Engelbrecht (UP), LN Wood, G Mather, P Petocz, A Reid, K Houston, GH Smith and G Perrett)
Contact person: Prof AF Harding.
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