Faculty of Natural and Agricultural Sciences
School of Mathematical Sciences
Department of Mathematics and Applied Mathematics
Selected Highlights from Research Findings
It is an ongoing process to find better models for plates and beams, the aim being to determine a sufficiently accurate model, which is not too complicated. To model a plate-beam system, one is faced by the additional complication of the interaction between the plate and the beam. Once a plate-beam model is designed, the next step is to solve it numerically, a task which is not trivial even for the simplest of plate models. Joint research by Zietsman, van der Merwe, Geldenhuys and van Rensburg have provided a modified model for a system consisting of a rectangular plate supported at opposing sides by two beams. The new model is reliable from the numerical point of view having extremely high rate of convergence of the finite element approximations.
Contact person: Prof NF Janse van Rensburg.
The famous mathematician Henri Poincaré introduced the notion of recurrence to classical mechanics which states that, if at a certain time a physical system is in some observable state, then almost surely the system will eventually return to this observable state. In a series of papers Duvenhage, Niculescu (Romania), Ströh and Zsido (Rome) extended the notion of recurrence to quantum mechanics by showing that a sharper, quantitative version of Poincaré recurrence is valid for quantum systems. The mathematical theory of operator algebras, their states and automorphisms were used to obtain this and a number of other important conclusions on multiple recurrence. This work leads to the creation of a noncommutative ergodic theory and it is forseen that it will have an extensive impact on further developments in quantum statistical mechanics
Contact person: Prof A Ströh.
At the turn of the century and after a decade of changes in South Africa, Engelbrecht and Harding considered it timely to conduct a survey to investigate the trends in numbers of students majoring in mathematics at South African universities and to compare these to international trends. The suspected general trend of decline in these numbers over the past decade is in line with what has been happening internationally and is cause for concern. Observed local trends include:
· Traditionally Afrikaans universities show a steeper decline than the traditionally English universities but also seem to be recovering lately.
· Rural universities show a greater decline than urban universities.
· Historically disadvantaged institutions declined even more in numbers of mathematics majors than the other universities.
· The number of residential students is declining whereas the numbers of distance students seems to be increasing slightly.
· Recoveries are taking place at certain institutions where new, more pragmatic mathematics courses have been introduced recently.
Possible reasons for the decline, from within and from outside the mathematics structure are discussed. This is the first time that an investigation of this kind has been done in South Africa.
Contact person: Dr AF Harding.
With the rapid development of 'smart material technology', the mathematical analysis of hybrid structures has become a subject of extensive research. One of the burning issues is the stability of such structures. For a one-dimensional hybrid structure consisting of a vibrating thermo-elastic beam with a rigid body attached to one end, uniform stability of the energy associated with the complex interactive system of partial differential equations was accomplished by making use of abstract functional analytic methods. This research is a significant contribution towards the topical field of multi-linked structures.
Contact person: Dr M Grobbelaar-van Dalsen.
Advection-reaction equations and reaction-diffusion equations arise in many fields in science and engineering to model systems on which reaction processes lead to diffusion in time of some quantity. Often, the solutions enjoy a number of physical properties such as positivity, boundedness, conservation of energy. Unfortunately, these equations cannot be solved explicitly by analytical methods. Thus, the need of numerical methods. In a series of joint papers that appeared in 2003, Anguelov, Lubuma and their team designed several remarkable non-standard finite difference schemes which, unlike the classical ones, replicate the physical properties. The study is systematic and constitutes a unique contribution to the mathematical justification for the success of the widely used empirical procedures in R.E. Mickens' non-standard approach.
Contact person: Prof JM-S Lubuma.
Orthonormal systems of function spaces play an enormously important role in Banach space geometry. It is of specific importance to compare different orthonormal systems for a given function space. It is a long standing open question whether the trigonometric system is equivalent to some rearrangement of the Walsh system in a given function space. In a paper published in Studia Mathematica, Wenzel and Hinrichs (Jena) make use of algebraic combinatorics to 'cleverly' construct a number of orderings for which non-equivalence holds. Previously proved in 1976, this was known for the Walsh-Paley order only.
Contact person: Prof J Wenzel.
See Research Report Vol 1
Contact person: Prof J Swart.
See Research Report Vol 1
Contact person: Mr AJ van Zyl.
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