Faculty of Natural and Agricultural Sciences
School of Mathematical Sciences
Department of Mathematics and Applied Mathematics
Selected Highlights from Research Findings
With the rapid development of 'smart material technology' there has, during the past two decades, been an increasing interest in the mathematical analysis of models for hybrid elastic structures. From the Department of Mathematics and Applied Mathematics there appeared two papers in 2004 dealing with such problems. M van Dalsen-Grobbelaar studied dynamic boundary stabilization of a rectangular Reisser-Mindlin plate with a Timoshenko beam implanted in a free edge, and proved that for transversal vibrations in the plate the system of partial differential equations can in principle be solved and is stable.
NF Janse van Rensburg and co-workers considered the vibrations of a cantilevered Timoshenko beam with a rigid body attached to the free end. This lead to good procedures for the numerical solution of the system of partial differential equations and the boundary conditions that describe the situation. The numerical calculations in turn, lead to the discovery of interesting phenomena concerning the basic modes of vibration.
Contact person: Dr M Grobbelaar-van Dalsen.
With the rapid development of 'smart material technology' there has, during the past two decades, been an increasing interest in the mathematical analysis of models for hybrid elastic structures. From the Department of Mathematics and Applied Mathematics there appeared two papers in 2004 dealing with such problems. M van Dalsen-Grobbelaar studied dynamic boundary stabilization of a rectangular Reisser-Mindlin plate with a Timoshenko beam implanted in a free edge, and proved that for transversal vibrations in the plate the system of partial differential equations can in principle be solved and is stable.
NF Janse van Rensburg and co-workers considered the vibrations of a cantilevered Timoshenko beam with a rigid body attached to the free end. This lead to good procedures for the numerical solution of the system of partial differential equations and the boundary conditions that describe the situation. The numerical calculations in turn, lead to the discovery of interesting phenomena concerning the basic modes of vibration.
Contact person: Prof NF Janse van Rensburg.
In the traditional study of dynamic systems cause and effect are placed in the same descriptive framework. There are, however, situations which occur in science where the observable effects are less complex or different from the events that cause them. For situations such as these cause and effect should be described in different frameworks. Such a theory in which the causal relationship is extremely simple was published in 1997 by N Sauer. In a paper published in 2004 TJ Brown and N Sauer develop a theory in which the relationship is of an extremely complex nature and show that the new backdrop can handle dynamic boundary conditions for problems in heat transfer and wave phenomena described in terms of symmetric hyperbolic systems of partial differential equations.
Contact person: Prof N Sauer.
In nonlinear partial differential equations the development of 'shock' discontinuities in solutions is a well-known phenomenon. These 'discontinuous' solutions cannot satisfy a differential equation in the traditional sense. If one resorts to so-called interval-valued functions, the differential equation may again make sense. This approach, pioneered by prof. Rosinger of the department, is now established, but certain fundamental mathematical questions in the theory remain open. One such question is to determine the place of continuous functions amongst true interval-valued functions. In a paper on Dedekind Order Completion by R Anguelov this question and related ones are answered and leads to a deeper understanding of the objects we would like to call solutions of nonlinear partial differential equations.
Contact person: Dr R Anguelov.
Three papers not explicitly related to applications have appeared in 2004. J Wentzel studied the UMD (Unconditional Martingale Difference) property for summation operators. PP Ntumba (Mamelodi campus) continued his work on paracompact Sikorsky complexes, and Ströh and Gopalraj published results on essential lower bounds in von Neumann algebras.
Contact person: Prof A Ströh.
Three papers not explicitly related to applications have appeared in 2004. J Wentzel studied the UMD (Unconditional Martingale Difference) property for summation operators. PP Ntumba (Mamelodi campus) continued his work on paracompact Sikorsky complexes, and Ströh and Gopalraj published results on essential lower bounds in von Neumann algebras.
Contact person: Dr PP Ntumba.
A famous theorem of Motzkin and Rabin is the following: Take any finite set of points in the Euclidean plane and colour the points either blue or red. If the points do not lie on a straight line there is a line containing at least two points of one colour and no points of the other. Known proofs of this theorem are indirect. In a paper which appeared in 2004, LM Pretorius and KJ Swanepoel give a constructive proof. They construct an algorithm for finding such lines and prove that the algorithm will always work. This remarkable approach forges a link between the methods of theoretical computer science and combinatorial mathematics.
Contact person: Prof LM Pretorius.
In a 2004 paper JC Engelbrecht and A Harding report on a study in which student preferences for online and a combination of paper and online assessment are determined. The finding is that students lean towards online assessment, but prefer a paper component in the major tests. It is also reported that students perform equally well on the paper and online components of semester tests. The findings are related to a first year calculus course only.
Contact person: Dr AF Harding.
In a 2004 paper JC Engelbrecht and A Harding report on a study in which student preferences for online and a combination of paper and online assessment are determined. The finding is that students lean towards online assessment, but prefer a paper component in the major tests. It is also reported that students perform equally well on the paper and online components of semester tests. The findings are related to a first year calculus course only.
Contact person: Prof JC Engelbrecht.
|
