Research Projects in which various members the Department of Mathematics are involved are listed below. Follow the link to a short description of the research project. This is followed by a list of research outputs - scientific publications and presentations - that appeared in 1998. The personal home pages of some staff members contain more information about their personal research interests and publications.
Certain operators on limit vector spaces
Properties of asymptotically quasi-compact operators, Ruston operators and Riesz operators on Hausdorff limit vector spaces are established.
GFC de Bruyn.
Spline wavelets with respect to uniform knots on the whole real line have been constructed and thoroughly analysed in the literature. This project consists of a study of the more general case where the knots of the underlying space are allowed to have arbitrary positions. For this purpose we restrict attention to the bounded interval case. A further related topic under investigation is the possibility of constructing a multiresolutional analysis, and corresponding wavelets, based on local spline interpolation. Two joint papers have already appeared.
CK Chui (Texas A&M University, USA) and JM de Villiers.
Numerical integration based on nodal spline interpolation
Using the known approximation properties of a nodal spline interpolation operator, as established in a sequence of papers by JM de Villiers and CH Rohwer, the aim of this project is to develop an error analysis for the quadrature rule which results from integration of the nodal spline interpolant. By employing Peano-kernel techniques, error bounds are obtained in terms of the mesh parameters corresponding to the spline knot sequence. In particular, for the quadratic case, explicit calculations are shown to yield sharp error estimates. The nodal spline interpolant is also applied in the context of product integration, and its efficiency is compared to those of existing methods in the literature. During two joint research visits in Stellenbosch (January) and Pretoria (July) between SA de Swardt and JM de Villiers, two joint papers were prepared, both of which have been accepted for publication in, respectively, Journal of Approximation Theory and Numerische Mathematik.
SA de Swardt and JM de Villiers.
Wavelet application in analysis of trajectories
The potential advantages of wavelet bases in quadratic spline spaces for handling test data at the Overberg Test Range was previously investigated under contract. Simulations were done, together with postgraduate students, to demonstrate this convincingly. Further advantages were investigated. A lecture was given at the test range and co-operation is continued.
JM de Villiers, K Goosen, B Rohwer, CH Rohwer.
Subdivision techniques and wavelet construction by means of interpolation
In a joint research project which started in November during a research visit by JM de Villiers to CA Micchelli in New York, two aspects were investigated:
(a) the family of masks yielding a refinable function with prescribed values at the integers; (b) a method based on local interpolation for the construction of an interpolatory subdivision scheme;
and in both cases studying the resulting wavelet spaces. A joint research paper on (a) has been submitted for publication, whereas one on (b) is in preparation.
JM de Villiers, CA Micchelli (TJ Watson Research Center, Yorktown Heights, NY, USA), T Sauer (University Erlangen-Nuremberg, Erlangen, Germany).
Geometry of curves and fundamental groups
In joint research with M Matignon and M Saidi we have continued to focus on problems from this area, and in particular the question of the existence of Galois liftings of smooth curves from positive characteristic to characteristic zero. This work builds on results of Grothendieck, Oort, Suwa and Sekigushi. It is well known that there are obstructions to the lifting problem in general. One of the main results we prove has been to give local criteria for the lifting problem to be solvable. Using this criterion we have been able to prove the conjecture for ep2-cyclic covers, (e,p) = 1. The study of automorphisms of formal power series over valuation rings is crucial for this investigation, and this is the area we are currently working in. This project is partially supported by an FRD grant, as well as the Deutsche Forschungsgemeinschaft Project: Arithmetic of Fields and Fundamental Groups.
BW Green, M Matignon (Laboratory for Pure Mathematics, University of Bordeaux, France), F Pop (Bonn), P Roquette (Heidelberg), M Saidi (Bonn).
Arithmetic of curves and applications to coding theory
During the 1970s the Russian mathematician Goppa discovered an important link between coding theory and algebraic geometry. This discovery led to the construction of powerful new error correcting codes using techniques from algebraic geometry, and in particular the theory of curves over finite fields. This also led to enthusiastic work in this area in many schools with specialists in coding theory and algebraic / arithmetic geometry, in particular those of Stichtenoth and Frey from Essen. In a recently initiated project we have begun working together in this area, focusing on the connection between the arithmetic properties of special curves, diophantine equations and their application to coding theory and cryptography. This project is supported by the Volkswagen Foundation.
G Frey (Institute for Experimental Mathematics, University GH Essen, Germany), BW Green, R Omar (University of the Western Cape), H Stichtenoth (Mathematics Institute, University GH Essen, Germany) and AB van der Merwe.
Perfect maps in uniform spaces
Recent category theoretic approaches to generalising the topological notion of a perfect map produce unsatisfactory results in the category of uniform spaces and uniformly continuous maps. This is the starting point for this ongoing research project which investigates other approaches to perfectness. In particular the role of the Samuel compactification and the spin-offs in related categories (e.g. quasi-uniform spaces, pseudo-metric spaces) are of interest.
GCL Brümmer (University of Cape Town), E Giuli (University of L’Aquila, Italy) and D Holgate.
Closure operators with respect to a functor
The first two collaborators developed a theory of closure operators that act on "non-subobjects" (e.g. subsets of a topological group are subspaces, but not necessarily subgroups). This project continues this work, specifically studying initial and final maps within this theory.
G Castellini (University of Puerto Rico), E Giuli (University of L’Aquila, Italy) and D Holgate.
Completion and compactification
When can a mathematical object (metric space, topological group etc.) be considered to be complete or compact? When can a modification procedure be considered to be a completion or compactification of an object? This project uses the methods of categorical closure operators to investigate these questions. The essential difference between compactification and completion is investigated and specific examples in a range of applications are studied.
D Holgate.
This project is in its final stages. It involves a survey of the convexity theorem of AA Lyapunov as well as a survey of the wide effect that J Lindenstrauss’s proof of the convexity theorem has on the functional analytic aspects of measure theory.
P Maritz.
The extension of a finite Baire measure to a regular Borel measure on a normal paracompact topological space is being investigated.
P Maritz.
Domination properties in ordered Banach algebras
An interesting problem in the spectral theory of ordered Banach algebras is that of finding conditions under which properties of a positive element b will be inherited by any positive element smaller than, or dominated by, b. This problem was originally investigated for bounded linear operators on a Banach lattice, and results have been obtained by Aliprantis, Burkinshaw, Dodds, Fremlin and others. Let A be an ordered Banach algebra. We are specifically interested in finding conditions under which it follows from 0 £ a £ b in A and b being in the radical of A, that a is in the radical of A. Interesting answers can be obtained by the use of subharmonic analysis.
HduT Mouton (Electric and Electronic Engineering) and S Mouton.
Approximation with locally monotone sequences
The concept of local monotoncity is complementary to that of impulsive noise in some strict sense. It is known that best approximations to a given sequence exist in any of the usual p-norms, but that these are computationally inappropriate. The search is on for simple, stable, local near-optimal approximation methods.
CH Rohwer.
Results of research into in-line vibrations of a beam were presented at the 1998 SANUM conference. Derivation of simple methods for determining approximations to square roots by elementary mathematical methods is investigated. Several optimal properties can be argued, and the process is pedagogically illuminating.
CH Rohwer.
Decomposition of LULU operators for analysis and computational efficiency
Analysis of smoothing methods with LULU-theory and Mathematical Morphology is possible by decomposing such operators into simple minimum- and maximum operators. Computational advantages can be expected as well. Ongoing research is done. Progress was reported at a workshop in Umtata and in a publication that was submitted.
CH Rohwer and M Wild.
Formal methods for the generation of pictures
The field of picture generation by formal methods such as grammars and automata is large and multifaceted. An overview of the field was obtained and several papers on original research have been completed. The work on the project continues.
S Ewert and APJ van der Walt.
Bi-ideals of quasi-reflexive rings
This project is merely a continuation of ‘Some results concerning n-socles of quasi-reflexive rings’. We are looking for specific results concerning non-nilpotent / nilpotent minimal bi-ideals in quasi-reflexive rings.
GWS van Rooyen.
Rings (semigroups) containing strongly quasi-ideals
This work is in its final stages. Our purpose is to give a much simpler characterisation of completely (0-)simple (semigroups with 0) rings than the condition given by O Steinfeld.
GWS van Rooyen.
Centralizer- and matrix nearrings
The classes of centralizer nearrings and matrix nearrings are active research fields in nearring theory. We are investigating isomorphisms between nearrings in these two classes.
KC Smith (Texas A&M University, USA) and L van Wyk.
Idempotent generators of matrix algebras
What is the minimum number of idempotents needed to generate a matrix algebra? Interesting answers to this question are known for full and upper triangular matrix algebras. We investigate this problem in blocked triangular matrix algebras.
AB van der Merwe and L van Wyk.
A characterisation of upper triangular matrix rings
The purpose of this project is to find an internal characterization of upper triangular matrix rings in terms of nilpotent elements (and possibly invertible elements) which is easier to handle than the existing internal characterization in terms of matrix units.
L van Wyk.
F.Breuer, J.M.Robson. 1998. Strategy and Complexity of the Game
of Squares.
Bull.London.Math.Soc 30: 274-282.
F.Breuer. 1998. A Note on a Paper by Glaser and Schoffl. The
Fibonacci Quarterly 36(5): 463-466.
Chui CK, De Villiers JM. 1998. Spline-wavelets with arbitrary
knots on a bounded interval: orthogonal decomposition and computational
algorithms. Communications in Applied Analysis 2(4): 457-486.
Conradie AE, Buys JD, Kröger DG. 1998. Performance optimization of dry-cooling systems for power plants through SQP methods. Applied Thermal Engineering 18(1-2): 25-45.
Dascalescu S, Van Wyk L. 1998. Complete blocked triangular matrix rings over a Noetherian ring. Journal of Pure and Applied Algebra 133: 65-68.
Ewert S, Van der Walt APJ. 1998. Generating pictures using random forbidding context. International Journal of Pattern Recognition and Artificial Intelligence 12(7): 939-950.
Green BW, Matignon M. 1998. Liftings of galois covers of smooth curves. Compositio Mathematica 113: 237-272.
Green BW. 1998. Geometric Families of Constant Reductions and the Skolem Property. Transactions of the American Mathematical Society 350(4): 1379-1393.
Holgate D. 1998. Linking the closure and orthogonality properties of perfect morphisms in a category. Commentationes Mathematicae Universitatis Carolinae 39(3): 587-607.
Kelarev AV, Van Der Merwe AB, Van Wyk L. 1998. The Minimum Number of Idempotent Generation of an Upper Triangular Matrix Algebra. Journal of Algebra 205: 605-616.
Loubser MM, Muller MA. 1998. Bank overdrafts: Limitation of interest by the in duplum rule; and prescription. The South African Law Journal 115(4): 598-612.
Maritz P. 1998. Multifunctions as approximation operations in Generalized Approximation Spaces. Proceedings First International Conference, RSCTC ‘98 Warsaw, Poland, 1998: Lecture Notes in Artificial Intelligence, 1424, Springer, Berlin: 131-138.
Maxson CJ, Van der Merwe AB. 1998. Rings of homogeneous functions. Journal of Pure and Applied Algebra 124: 211-226.
Mouton S, Raubenheimer H. 1998. More spectral theory in ordered Banach algebras. Positivity 1: 305-317.
Rohwer CH. 1998. One-sided quadratic spline approximation. Rendiconti del Circolo Matematico di Palermo, Serie II 52: 757-764.
Smith KC, Van Wyk L. 1998. Generalized blocked triangular matrix rings associated with finite abelian centralizer near-rings. Proceedings of the Edinburgh Mathematical Society 41: 177-195.
Smith KC, Van Wyk L. 1998. Solution of the J2-radical problem in structural matrix near-rings. Proceedings of the Royal Society of Edinburgh (Section A-Mathematics) 128A: 137-145.
Van der Merwe AB. 1998. Homogeneous Maps and Divisorial Ideals. Journal of Algebra 207: 547-556.
Van der Merwe AB. 1998. Near-rings of homogeneous maps. Quaestiones Mathematicae 21: 1-9.
Van Wyk L. 1997. Good ideals in blocked triangular matrix rings over division rings. Annales Universitatis Scientiarum Budapestinensis (de Rolando Eotvos Nominate-Sectio Mathematica) 40: 75-83.
Wild M. 1998. Three Notes on Distributive Lattices. Advances in applied Mathematics 20: 44-49.
Keller HA, Künzi UM, Wild M. 1998. (reds) Orthogonal Geometry in Infinite Dimensional Vector Spaces. Bayreuther Mathematische Schriften, No 53. Mathematisches Institut der Universität Bayreuth: 326pp.
Herrmann C, Moresi R, Schuppli R, Wild M. 1998. Classification of Subspaces. In: Keller HA, Künzi UM, Wild M, (reds) Orthogonal Geometry in Infinite dimensional Vector Spaces. Bayreuther Mathematische Schriften No 53, Mathematisches Institut der Universität Bayreuth: 55-170.
JM de Villiers
Interpolatory spline wavelets. 22nd South African Symposium on Numerical Mathematics, Cape Town.
On the construction of a multiresolutional analysis based on spline interpolation. Ninth International Conference on Approximation Theory, Nashville, Tenessee, USA.
BW Green
1) A characterisation of order p automorphisms of the p-adic open disc. 2) Reduction theory of Drinfeld modules over finite fields. Workshop on Function Fields of one variable over Finite Fields, Drinfeld modules and Coding Theory, Institute for Experimental Math., Essen, Germany.
Geometric Henselian Fields and the Skolem Property. 41st SAMS Conference, RAU, Johannesburg.
On the geometry of order p automorphisms of the open disc of a p-adic field: Plenary Talk, Oberwolfach Modelltheorie Tagung.
D Holgate
Completion, Compactification and Closure. International Symposium on Applications of Categorical Methods, Antwerpen, Belgium.
G Brümmer and D Holgate
Closure operators, direct reflections and perfect morphisms in uniform spaces. International Symposium on Applications of Categorical Methods, Antwerpen, Belgium.
P Maritz
Multifunctions as approximation operations in Generalized Approximation Spaces. First International Conference on Rough Sets and Current Trends in Computing (RSCTC ‘98), Warsaw, Poland.
S Mouton
Some Spectral Properties in ordered Banach algebras. International Congress of Mathematicians, Berlin, Germany.
CH Rohwer
Splines Camshafts and Vibrations. 22nd South African Symposium on Numerical Mathematics, Cape Town.
Nonlinear Analysis and Impulsive Noise Separation. Workshop in "Nonlinear Analysis and its applications," Umtata.
L van Wyk
A natural centralizer and the smallest essential ideal in matrix rings. 41st SAMS Conference, RAU, Johannesburg.
Ewert, S. Random Context Picture Grammars. Ph.D. (Computer Science). (Promoter: prof. APJ van der Walt).