Faculty of Natural and Agricultural Sciences
School of Mathematical Sciences
Department of Mathematics and Applied Mathematics
Selected Highlights from Research Findings
In 1835 Plücker wrote about a set of nine points in the complex plane with the property that every line through two of them contained a third point of the set.
In 1893 Sylvester asked whether such a set could be found in the real plane. Let us call a finite set S of points a Sylvester-Gallai (SG) configuration if the line through any two points in S contains a third point in S.
The question of Sylvester was again asked independently in 1933 by Paul Erdös and in the same year Gallai proved that an SG configuration in any real projective space is collinear, that is one-dimensional.
In the light of the 9-point SG configuration of Plücker in the complex plane, Serre asked in 1966 whether an SG configuration in a complex projective space must necessarily be coplanar. In 1986 it was proved by Kelly to be the case by using a deep inequality of Hirzebruch.
In this article we give an elementary proof of Kelly’s result and extend it to show that an SG configuration in projective space over the quaternions must be contained in a three-dimensional flat
Contact person: Prof LM Pretorius.
In computational mechanics, the problem of using low order finite elements for the computation of displacement, strain and stress in nearly incompressible regime is a well known. The classical formulation exhibits bad behaviour and low convergence rate in bending dominate problems. In these work, we first analyze the existing formulation, in fact analyze the reasons of their bad behaviour when bending is important. We also introduced a new three field formulation based on the elastic-viscous-split-stress already known in the context of fluid mechanic. The new formulation has the potential to overcome the locking phenomena observe for the classical formulation
Contact person: Dr J Djoko Kamdem.
Hausdorff continuous (H-continuous) functions are special interval-valued functions, which are commonly used in practice, e.g. histograms, are such functions.
However, in order to avoid arithmetic operations with intervals, such functions are traditionally treated by means of corresponding semi-continuous functions, which are real-valued functions.
One difficulty in using H-continuous functions is that, if we add two H- continuous functions that have interval values at same argument using point-wise interval arithmetic, then we may obtain as a result an interval function, which is not H-continuous.
In this work the operation addition is defined in such a way that the set of H-continuous functions is closed under this operation. Moreover, the set of H-continuous functions is turned into a linear space. It is also proved that this space is the largest linear space of interval functions.
These results make H-continuous functions an attractive tool in real analysis and provides a bridge between real and interval analysis
Contact person: Prof R Anguelov.
The notion of recurrence plays a central role in the study of dynamical systems (systems that evolve with time, movement of planets, the heart’s beat, pendulums). It was discovered by Henri Poincare and states that a dynamical system having a finite amount of energy and confined to a finite spatial volume will always returns to a position close to its initial state, if a sufficient amount of time elapses.
H. Furstenberg obtained recurrence results concerning weakly mixing and almost periodic measure preserving transformations. That result was later extended by Prof Anton Ströh and his collaborators to C*-algebras.
The present work of de Beers, Duvenhage and Stroh extend previous results on noncommutative recurrence in unital *-algebras over the integers, to the case of locally compact Hausdorff groups.
They derive a generalization of Khintchine's recurrence theorem, as well as a form of multiple recurrence. This is done using the mean ergodic theorem in Hilbert space, via the GNS construction
Contact person: Dr RdV Duvenhage.
The problem of noise control in aircrafts and submarines has given rise to the mathematical investigation of so-called structural acoustic models in which an acoustic medium (which may be a gas), interacts with an elastic medium (which may be a plate).
The composite dynamics of the structure is described by systems of partial differential equations, which model the dynamics of the gas and the plate. However each system of partial differential equations is augmented by additional terms, which describe the interaction between the two media.
An important addition to the existing literature is a model, recently developed by M. Grobbelaar-Van Dalsen, in which the dynamics of the elastic medium, here a plate, takes account, not only of the transverse deflections of the plate and rotational inertia effects, but also of transverse shear effects.
This yields an even more complex model, which contains two additional variables representing the shear angles of filaments of the plate. The model which uses the Reissner-Mindlin plate equations to describe the plate dynamics, is not only more accurate over the whole frequency range, at least as far as the vibrations of the elastic medium are concerned, but is also valid at high frequencies when the Euler-Bernoulli equation ceases to be valid.
The existence of unique solutions both for two- and three-dimensional linear and non-linear models is established and in the two-dimensional case, carefully chosen multipliers is shown to yield uniform stability of the energy associated with both linear and non-linear models.
Only feedback controls at the walls of the acoustic medium, as well as at the edges of the elastic medium, a Timoshenko beam in the two-dimensional case, are required
Contact person: Dr M Grobbelaar-van Dalsen.
Singularly perturbed differential equations, i.e. equations in which a small parameter is multiplied to the highest derivative, arise in various fields of science and engineering to model the physical phenomenon of boundary layers. Classical numerical methods are not reliable in the solution of such problems.
For the class of problems investigated in this paper, we design non-standard finite difference schemes, which converge uniformly with respect to the perturbation parameter and which replicate essential physical properties of the exact solutions
Contact person: Prof JM-S Lubuma.
This work investigates the flow down an inclined plane of a non-Newtonian fluid (generalized second grade fluid type). Non-Newtonian fluids are those that do not obey the Newton’s law of viscosity and have a wide range of industrial applications and examples include consumer goods such as plastics, paints, toothpaste, foodstuffs such as tomato sauce and biological fluids such as blood.
The simplest model for non-Newtonian fluids is the second grade Rivlin-Ericksen model for fluids with constant viscosity, which cannot be used to model fluids with shear dependent viscosity. Another popular model is the power law model where viscosity is proportional to a power of the velocity gradient and successfully models shear-thinning and shear thickening fluids.
The model we used in this study is one that combines the second grade and the power-law fluids. In the study both the exact and the numerical solutions of a fully developed flow of a generalized second grade fluid with power law temperature dependent viscosity are obtained. Closed form analytical solutions are possible only for the linear case of the governing equation. For the non-linear case only numerical solutions were possible
Contact person: Dr EW Mureithi.
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