Faculty of Natural and Agricultural Sciences
School of Mathematical Sciences
Department of Mathematics and Applied Mathematics
Selected Highlights from Research Findings
The Riesz Theorem. The classical Riesz Theorem establishes a unique correspondence between the continuous linear functionals on spaces of continuous functions and spaces of measures. This fundamental result has been extended to a number of other situations. In the compiled volume Handbook of Measure Theory, (Elsevier, North Holland, 2002) Diestel and Swart contributed a comprehensive survey of the results concerning the Riesz Theorem, its applications and the nature of the difficulties associated with its extensions.
Riesz se Stelling. Riesz se klassieke stelling bevestig 'n eenduidige ooreenkoms tussen die kontinue lineêre funksionale op ruimtes van kontinue funksies en sekere maatruimtes. Hierdie fundamentele resultaat is na verskeie ander situasies uitgebrei. In die saamgestelde volume Handbook of Measure Theory (Elsevier, North Holland, 2002) gee Diestel en Swart 'n volledige en oorsigtelike bydrae oor die resultate rondom Riesz se Stelling, sy toepassings en die aard van die probleme wat met die veralgemenings gepaard gaan.
Contact person: Prof J Swart.
Grothendieck's resumé. In a set of some 80 pages of mimeographed lecture notes, Grothendieck announced and sketched without clear and comprehensive mathematical proofs the most profound results on tensor products in Banach spaces. In a remarkable series of papers, the first of which appeared in Quaestiones Mathematicae in 2002, Diestel, Fourie and Swart provide for the first time the rich setting for and the precise proofs of Grothendieck's announcements. This is a deep and lasting contribution to the mathematical understanding of a fundamental topic.
Grothendieck se resumé. In 'n stel van 80 afgerolde bladsye van lesingnotas het Grothendieck sonder duidelike en volledige wiskundige bewyse, die diepste resultate oor tensorprodukte in Banachruimtes aangekondig. In 'n merkwaardige reeks artikels waarvan die eerste in Quaestiones Mathematicae, 2002, verskyn het, verskaf Diestel, Fourie en Swart vir die eerste keer die ryk agtergrond vir en volledige bewyse van Grothendieck se aankondigings. Dit is 'n diep en blywende bydrae tot die wiskundige begrippe van 'n fundamentele onderwerp.
Contact person: Prof J Swart.
The Navier-Stokes equations and boundary permeation. The model for permeation is based on the simple assumption that it is driven by differences of normal stresses at the fluid-boundary interface. Sauer and Hlomuka showed that sufficiently small perturbations of the rest states decay back to the rest state in a polynomial manner. The same model for non-Newtonian fluids has earlier led to the result that the decay is exponential, thus showing one distinction between the two kinds of fluids, namely that the nonlinear fluid exhibits stress-dissipation at the boundary which the linear fluid does not.
Die Navier-Stokes vergelykings en wandsypeling. Die sypelingsmodel is gegrond op die eenvoudige aanname dat dit deur verskille in normaalspannings aangedryf word. Sauer en Hlomuka het aangetoon dat voldoende klein versteurings van die rustoestand op 'n polinoom-manier na die rustoestand terugkeer. Dieselfde model vir nie-Newtonse vloeistowwe het vantevore tot eksponensiële terugkeer gelei. Hierdeur word 'n onderskeid tussen die twee soorte vloeistowwe aangetoon, naamlik dat daar by nie-Newtonse vloeistowwe dissipasie van spanning is, terwyl dit by lineêre vloeistowwe afwesig is.
Contact person: Prof N Sauer.
Quantum mechanics. The notion of Poincaré recurrence in classical mechanics, namely that a system bounded to a finite volume will almost certainly return to its initial state, is also valid in quantum theory. This is the conclusion of a paper by Duvenhage (International Journal of Theoretical Physics). The author concludes in another paper (Foundations of Physics) that classical and quantum mechanics differ in the nature of information. In classical mechanics an observer's information may be described in terms of standard probability theory, whilst in quantum theory the notion of conditional probability has to be changed in a drastic manner.
Kwantummeganika. Die Poincaré rekurrensie in klassieke meganika, naamlik dat 'n stelsel wat tot 'n eindige volume beperk is, byna seker na die begintoestand sal terugkeer, is ook geldig in kwantum teorie. Dit is die resultaat van 'n artikel deur Duvenhage (International Journal of Theoretical Physics). Die skrywer toon in 'n ander artikel (Foundations of Physics) aan dat klassieke- en kwantummeganika verskil ten opsigte van die aarde van inligting. In klassieke meganika kan 'n waarnemer se inligting met behulp van standaard waarskynlikheidsteorie beskryf word, terwyl die begrip van voorwaardelike waarskynlikheid in kwantumteorie drasties gewysig moet word.
Contact person: Dr RdV Duvenhage.
Thermo-elasticity. By using concepts and techniques developed by Sauer and Grobbelaar-van Dalsen for the study of dynamic systems associated with implicit evolution equations, it was possible to prove that intricate systems fulfill all the requirements of a mathematically well-posed problem.
Termo-elastisiteit. Deur van begrippe en tegnieke wat deur Sauer en Grobbelaar-van Dalsen vir die studie van dinamiese stelsels geassosieer met implisiete evolusievergelykings ontwikkel is, was dit moontlik om te bewys dat komplekse stelsels van vergelykings vir die vibrasie van termo-elastiese plate met on-starre randstrukture aan al die vereistes vir 'n wiskundig goed-gestelde probleem voldoen.
Contact person: Dr M Grobbelaar-van Dalsen.
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