Faculty of Natural and Agricultural Sciences
Department of Mathematics and Applied Mathematics
Selected Highlights from Research Findings
Classical numerical methods for the linear reaction-diffusion equation on a rough geometry perform poorly due to the lack of sufficient smoothness of the solution. We design reliable numerical schemes by incorporating the intrinsic feature of the continuous problem into the discretizations.
Contact person: Prof JM-S Lubuma.
Examples of signals in one dimension are observations from the world around us such as radio waves, voice and sound to name a few. Signals are presented as discrete sequences of values of, say, length . The LULU operators, and , were initially developed for signals in the 1980’s by Carl Rohwer from Stellenbosch University. He developed a vast theory on them and their compositions. This includes structure preserving properties, results on how they smooth and filter a signal corrupted with noise, and the Discrete Pulse Transform – the repeated application of the operators and with increasing from 1 up to the length of the signal - and how it ‘slices’ a signal horizontally to obtain a decomposition of the said signal.
In this paper we have provided the extension theory, together with the proofs, of this work initiated by Rohwer and other collaborators in one dimension into two dimensions and higher. All the properties and results developed in one dimension mentioned above have been defined in higher dimensions and proved to hold for our higher dimensional LULU operators.
The immediate setting for the two dimensional operators and and the resulting Discrete Pulse Transform, is image analysis and results in a multiresolution decomposition of an image into scale levels which each represent the details in the image which have respective scale of the level. An image is represented as a matrix with, say, rows and columns. Applications in image analysis, specifically making use of the Discrete Pulse Transform to attain improved feature extraction, segmentation, noise removal and other analyses, are now possible. In addition, analysis in three dimensions now allows for similar work in videos.
Contact person: Prof R Anguelov.
Traditional parametric Value at Risk (VaR) estimates assume normality in financial returns data. However, it is well known that this distribution, while convenient and simple to implement, underestimates the kurtosis demonstrated in most financial returns. Huisman, Koedijk and Pownall (1998) replace the normal distribution with the Student’s t distribution in modelling financial returns for the calculation of VaR. In this paper, we extend their approach to the Monte Carlo simulation of VaR on both linear and non linear instruments with application to the South African equity market. We show, via backtesting, that the t distribution produces superior results to the normal one.
Contact person: Prof E Maré.
Motivated by statistical mechanics there is a need to develop mathematical results, tools and theories to successfully describe the long term behaviour of quantum systems. Although much work has been done on this topic, it still is not as well developed as for classical systems (where the e ects of quantum physics are ignored). This paper deals with the extension of a famous result for classical systems to the quantum domain, namely Furstenberg's
recurrence theorem. This result describes a certain highly nontrivial pattern of how dynamical systems tend to return to their initial state, called the Szemeredi property. The classical result was motivated by problems in
number theory and combinatorics.
In this paper a partial extension of the result is obtained. The basic mathematical framework is that of operator algebras, and the approach is roughly to decompose a given system into simpler parts or subsystems, and
then to prove the result for these simpler systems. This requires first building a theory for identifying simpler subsystems in a given system. The dfferent types of simpler systems then each require radically different techniques in their analysis. The relevant theories and techniques are developed in the paper.
In subsequent work by Tim Austin, Tanja Eisner and Terence Tao, a further extension is obtained, again using the idea of identifying suitable subsystems. Along with a previous paper by Niculescu, Stroh and Zsido, the
Austin-Eisner-Tao paper also shows that there are limits on how far one can extend the result. It is clear that a full extension to the quantum domain is not possible, showing yet again that quantum behaviour is much more
complicated than classical behaviour.
Contact person: Prof A Ströh.
Many problems in Mathematics, Engineering, Economics and other fields of study can be reduced to or involve the solution of systems of nonlinear equations. Most currently available methods of solution are based on Newton's method. These methods encounter problems of convergence when either the initial trial solution is far removed from the true solution or the derivatives are too small during iteration. Much research in this field is concerned with ways of circumventing these difficulties. This paper presents a new method for solving nonlinear equations. The method is free of derivatives and so it circumvents the difficulties encountered by currently available Newton- based methods. It is also easy to implement. Although its convergence is linear, numerical experiments conducted indicate that its performance compares well not only with Newton-based methods of higher order but with other derivative-free methods as well.
Contact person: Dr PA Phiri.
A usual way of approximating Hamilton-Jacobi equations is to couple space finite element discretization with time finite difference discretization. This classical approach leads to a severe restriction on the time step size for the scheme to be monotone. In this paper, we couple the finite element method with the nonstandard finite difference method, which is based on the Mickens' rule of nonlocal approximation. The scheme obtained in this way is unconditionally monotone. The convergence of the new method is discussed and numerical results that support the theory are provided.
Contact person: Prof R Anguelov.
The diminishing total variation of the solution is an essential property of conservation laws. Preserving it by a numerical scheme is important from both theoretical (e.g. for proving convergence) and from practical (correctly replicating the physical phenomenon) points of view. The paper proposes nonstandard finite difference schemes which preserve this property. Computationally simple implicit schemes are derived by using nonlocal approximation of nonlinear terms. Renormalization of the denominator of the discrete derivative is used for deriving explicit schemes of first or higher order. Unlike the standard explicit methods, the solutions of these schemes have diminishing total variation for any time step size.
Contact person: Prof R Anguelov.
This paper is concerned with the derivation of the partial differential equations that govern the propagation of sonic
disturbances in an ideal gas under isentropic conditions. The result is a quasilinear hyperbolic system of first order equations and an inequality constraint. The speed of propagation is pressure dependent. It is shown how to deal with the equations and the constraint and how to calculate characteristics and solutions. It is also shown that shock discontinuities can develop which distinguishes the equations from the traditional linear wave equation.
Contact person: Prof N Sauer.
Recently, the author introduced the Space-Time Foam Differential Algebras of Generalized Functions whose interest is in their ability to deal with, so far, the largest family of singularities. These singularities can be on arbitrary subsets, with the only condition that their complementaries should be dense. Thus the singularities can have larger cardinal than the nonsingular points. Furthermore, no conditions on the generalized functions are required in the neigbourhood of singularities. The Cuachy-Kovalevskaia theorem is a paradigmatic classical result regarding its nonlinear generality, as well as the regularity of the solutions obtained. The limitation of that theorem is in the local nature of the result it obtains. The paper, with the use of the mentioned algebras, gives a global version of that theorem, a version which improves regarding the regularity of solutions. This result is an improvement on the first global version of the Cauchy-Kovalevskaia theorem given in an earlier paper of the author.
Short description of my research activity.
Since the early 1960s I pursued the first general solution method in the mathematical literature for large classes of nonlinear partial differential equations, or PDEs. This was done by constructing infinitely many differential algebras of generalized functions in which the generalized solutions of such nonlinear PDEs can be found. The respective branch of mathematics has been listed by the American Mathematical Society under 46F30. Several important related additional results obtained are the first time complete solutions of the 1954 Schwartz "impossibility", as well as of the celebrated 1900 Fifth Problem of Hilbert. Also, global actions of Lie groups were defined for the first time on usual and generalized functions and solution. As a further extension, the theory of genuine Lie semigroups was established. A second general solution method for large classes of nonlinear PDEs was introduced in the early 1990s, based on the order completion method. The power of that theory is shown, among others, by the fact that it gives the first time solution of the 1957 Lewy "impossibility". He also pursued a number of other research directions, among them in the numerical solution of large classes of nonlinear PDEs, optimization, multiple criteria decision making, or MCDM, relativity, quantum theory and quantum computation. In MCDM he discovered the Principle of Increasing Irrelevance of Preference Type Information, or PIIPTI, which has a significant applicative relevance when the number of conflicting criteria is larger than half a dozen. In quantum theory he pursued the deeper analysis of the fundamental phenomenon of entanglement, by defining the supporting structure of tensor products within a far more general setup than the usual algebraic one.
Contact person: Prof EE Rosinger.
In this paper a model (called the QI model) is developed to measure how good a mathematics question is, in other words to decide whether a mathematics question is of good or poor quality. This model is used both to quantify and visualise the quality and nature of a mathematics question. The QI model provides assessors and educators with a useful and comprehensive measuring tool to improve the quality of the mathematics items in their mathematics assessment programmes. In particular, what distinguishes this measurement tool from existing measures is that it includes the human element, taking issues such as the intention of the item writer, his judgement of the difficulty level of the item, the experience of the student and the confidence with which the student answers the item into consideration.
Contact person: Prof AF Harding.
This is a computational study of gravity-driven fingering instabilities in unsaturated porous media. The governing equations and corresponding numerical scheme are based on the work of Nieber et al. (2003) in which non-monotonic saturation profiles are obtained by supplementing the Richards equation with a non-equilibrium capillary pressure-saturation relationship, as well as including hysteretic effects. The first part of the study takes an extensive look at the sensitivity of the finger solutions to certain key parameters in the model such as capillary shape parameter, initial saturation, and capillary relaxation coefficient. The second part is a comparison to published experimental results that demonstrates the ability of the model to capture realistic fingering behaviour.
Contact person: Dr M Chapwanya.
This paper follows in a series concerning university student conceptions of mathematics. An international team has been collecting data from 1200 undergraduate students of mathematics in Australia, the UK, Canada, South Africa and Brunei. Responses to a questionnaire were classified leading to an outcome space of four level of conceptions about mathematics. This paper focuses on the finding that for many students modeling is fundamental to their conception of what mathematics is. Modelling gives applications of mathematics a structure and purpose in the same way that abstract concepts give algebra and geometry a structure and purpose. The overarching view is that, since the vast majority of students will work outside university and schools, it is vital that they come to appreciate not only the beauty of mathematics but also its usefulness and it is the task of the educator to guide students towards this.
Contact person: Prof AF Harding.
|
