Collaborators and Vistors

My fields of current research interest include cryptology, graph theory, operations research and also previously differential equations. During my career I have had the opportunity to meet and work with a number of excellent people in each of these fields. I shall discuss the contacts that I was fortunate to forge in these fields separately.

	Cryptology
	Graph Theory
	Operations Research
	Differential Equations

Cryptology

My interest in cryptology is relatively new. I only became involved in the field in 1999, through my friendship with Niël van Greunen (National Communication Centre), with whom I developed two new courses on the subject at the Department of Applied Mathematics: one undergraduate (a third year course) and the other postgraduate (a fourth year course). Besides Niël, the other people who have sparked my interest in and shaped my views on cryptology, include Prof Gerhard Geldenhuys (formerly from the Department of Applied Mathematics here at Stellenbosch University), Prof Walter Penzhorn (Department of Electrical and Electronic Engineering, University of Pretoria) and Prof Alko Meijer (GENNAN Systems).

This picture (above) shows Niël van Greunen and myself. Niël works for the National Communication Centre in Pretoria and is a past student of the Department of Applied Mathematics at Stellenbosch University. In 1999-2000 Niël and I developed two new courses on applications of discrete mathematics within the field of cryptology. These courses have been running as Applied Mathematics 314 and Applied Mathematics 785/885 since 1999 and 2000 respectively, and Niël's input into the courses as co-author of class notes and general advisor has ensured that students get a flavour of practical relevance, while studying such beautiful mathematical topics as number theory and algebraic group theory.

Here (above) I appear with Johan Botha, manager of the Cryptology Department at the National Communication Centre. Johan is Niël's direct boss, and has shown a tremendous amount of good will towards our efforts to establish applied mathematics at Stellenbosch as a centre for training young mathematical cryptologists, by allowing Niël to devote a considerable amount of his work time to student supervision and course development matters. Johan also visits our department regularly to speak to students about job opportunities in the information security sector.

This picture shows Prof Alko Meijer, chief cryptologist at GENNAN Systems, and myself. Before occupying his current position, Alko was head cryptologist at the South African Communication Security Agency (SACSA), and before that he was Dean of Sciences at the University of Natal, Durban. He is a mathematician and his field of specialisation is algebraic group theory. Alko has acted as external thesis examiner for some of our graduate students and has visited the Department of Applied Mathematics at Stellenbosch University to address our final year students on the role of mathematics within cryptology. He has also taken part in our colloquium series at the department, and I am a regular visitor of his in Pretoria. During the period 2002-2005 I served on Working Group 2 of SABS standing committee 27 under chairmanship of Alko in Pretoria. The working group (consisting of advisors in academia, the banking industry, the IT sector, various government sectors and the military) had the task of advising the South African Bureu of Standards (SABS) with respect to the adoption of cryptological cipher and protocol standards on a national level, and I learned an immense amount of practical cryptology during sessions of this work group.

On January 27th, 2003 Dr Kamil Kulesza of the Cryptography & Data Security Group at the Polish Academy of Sciences (and then visiting professor at Rhodes University) visited our department to forge links between the Department of Applied Mathematics at Stellenbosch University and the Polish Academy of Sciences. He gave a very interesting talk on the use of graph colouring to develop a check-digit error detection coding scheme - thereby linking two of my favourate fields of research: data security and graph theory. We continue to correspond.

The picture above shows myself with Stephen Berjak, who obtained his PhD in Applied Mathematics (Cryptology) under my supervision in December 2003. I was fortunate to work with Stephen on a fascinating project (in collaboration with the National Communication Centre (NCC), in Pretoria) in which the aim was to break an advanced, internationally operational block-oriented cryptographic stream cipher. Stephen was able to break the system after years of hard work and unfailing dedication, and we learnt a lot about the intricasies of practical cryptology during this project. Stephen now works for the NCC as a result of the success of his PhD project.

Another interesting graduate project in cryptology was Hendri Botha's master's thesis, in which he set out to analyse and exploit weaknesses in another advanced, internationally operational stream cipher. In this project close collaboration was again achieved between the NCC and the Department of Applied Mathematics at Stellenbosch University, in the sense that Dr Stephen Berjak (of the NCC) and I acted as co-supervisors for Hendri Botha. Hendri analysed the stream cipher in question in terms of its diffusion properties and investigated the possibility of using diffusion weaknesses to design a real-time computerised attack against the system. He submitted his thesis early in 2005.

Graph Theory

Graph theory is my true passion. In this field I have been fortunate to have collaborated with a number of very fine mathematicians from Durban and Pretoria, as well as from abroad (Canada & France). Of these people I have probably worked most closely together with Dr Alewyn Burger (currently a post-doctoral fellow in Applied Mathematics here at the University of Stellenbosch, formerly a post-doctoral fellow in Mathematics at the University of Victoria in Canada (2003-2004), and before that from the Department of Mathematics, Applied Mathematics & Astronomy at UNISA) over the period 1998-2005, on topics such as graph theoretic Ramsey numbers and various problems in graph domination theory. We work particularly well together, bringing together very different viewpoints on and approaches to the problems we study. I have also worked particularly well with Dr Paul Grobler (formerly from the School of Mathematical Sciences at the University of Natal, Durban) on two occasions, before he joined our faculty here at the Department of Applied Mathematics at Stellenbosch in 2003: the first occasion, in 1999, was when I embarked on a study of graph theoretic Ramsey numbers together with Eugene Stipp (a former master's student of mine), Alewyn Burger and Paul Grobler. This collaboration, in hindsight, was a turning point in my research career, when my field of interest gradually started to shift from the continuous end of the mathematical spectrum (differential equations) more towards the discrete end. The second occasion was in 2002, when Paul and I worked on a new kind of domination, called Roman domination, a notion introduced to us by Prof Ernie Cockayne (Department of Mathematics and Statistics, University of Victoria, Canada) during a "Graph Theory Concentration Camp" organised by Prof Kieka Mynhardt (then from the Department of Mathematics, Applied Mathematics & Astronomy, UNISA). Since Paul Grobler's arrival here at our department, we have collaborated on numerous occasions on such facinating topics as graph protection strategies and progressive sequences within graph edge orderings, to name but a few.

This picture (above) shows, from left to right, Paul Grobler, Eugene Stipp, myself and Alewyn Burger. Eugene and I held a one-day workshop on multipartite Ramsey numbers together with these two excellent graph theoreticians in 1999. The workshop was a very productive one, and sparked an excellent piece of work by Eugene for his master's thesis. We four also later wrote a paper on diagonal multipartite Ramsey numbers, which is due to appear in Utilitas Mathematica [an abstract and postscript copy of the full paper may be obtained here]. Alewyn and I later followed this up by writing two more papers, this time on off-diagonal multipartite Ramsey numbers - these two papers have appeared in Discrete Mathematics [an abstract and electronic copy of the first of these papers may be downloaded here, while those of the second paper may be obtained here].

Here I appear with Prof Gary MacGillivray, from the Department of Mathematics & Statistics at the University of Victoria in Canada. I have never had the opportunity to work with Gary, but he visited our department in 2001 and gave a very interesting colloquium. During the course of his visit we also talked extensively about design theory and two problems in combinatorial theory in particular, known as the covering problem and the packing problem. As a result of our discussions Alewyn Burger and I were helped onto the right track with a combinatorial problem on lotteries, which we were considering at that time. Gary, as head of department at Victoria, was also so kind as to host Werner Gründlingh, one of my PhD students when Werner won the prestigeous Wilhelm Frank Stipendium, enabling him to spend the year 2004 abroad during his PhD studies on the lottery problem.

The graph theory community in South Africa is a close-knit one, and delightful local workshops and specialised conferences are often organised by its members. An example of such a specialised conference was the "International Graph Theory Conference" organised by Michael Henning (School of Mathematical Sciences at the University of Natal, Pietermaritzburg), which took place in May 2001 at the Ithala Game Reserve just south of the Swaziland border. This event was organised as a tribute to the life-long contributions of the "mother" and "father" of graph theory in South Africa: Henda Swart (School of Mathematical Sciences, University of Natal, Durban) and Izak Broere (Mathematics Department, Rand Afrikaans University) respectively - reportedly the first two graph theorists in South Africa. Henda Swart was lured to graph theory by its shear beauty, while working in geometry during her graduate studies in the United States, and Izak Broere was introduced to the field during his undergraduate years, while working on a joint paper with Izak Bouwer (who subsequently moved to Ottawa, where he has now retired). The picture below shows the participants at this tribute conference - posing while a partial solar eclipse was in progress!

Another example of a special graph theory event was the so-called "Graph Theory Concentration Camp" organised by Kieka Mynhardt in January 2002 at the Sunnyside Campus of UNISA. On the first day of this week-long workshop a number of delegates made presentations of ongoing research projects on open problems. At the start of the second day the rest of the delegates then divided into groups to tackle these problems. Werner Gründlingh (a masters student of mine, at that time) and I joined a group led by Ernie Cockayne on a new kind of graph domination problem, called Roman domination. Other members of the group included Paul Grobler, Justin Munganga (Department of Mathematics, Applied Mathematics & Astronomy at UNISA) and Ken Halland (Department of Computer Science at UNISA). The work done during the group session gave rise to a joint paper on graph protection, which is due to appear in Utilitas Mathematica [an abstract and electronic copy of the full paper may be obtained here]. The following picture shows a delightful picnic at the Pretoria Botanical Gardens during this "concentration camp."

Starting with Prof Jean Dunbar (Converse College) in the wheel chair, and moving around the circle in an anti-clockwise direction, we find Ms Joy Singleton (UNISA), Dr Paul Grobler (then from the University of Natal, Durban), Mr Werner Gründlingh (University of Stellenbosch), Dr Sheng Bau (then from the University of Natal, Pietermaritzburg), Prof Ernie Cockayne (University of Victoria, Canada), Prof Kieka Mynhardt (currently of the University of Victoria, Canada, but then from UNISA), Prof Henda Swart (University of Natal, Durban; and the mother of graph theory in South Africa), Dr Alewyn Burger (then post-doctoral fellow at the University of Victoria, Canada) & Prof Marietjie Frick (UNISA).

Still at the "graph theory concentration camp," during a tea time discussion, we have from left to right, Mr Olof de Wet (UNISA), Prof Marietjie Frick (UNISA), Prof Kieka Mynhardt (then from UNISA; and organiser of the concentration camp), Dr Elna Ungerer (then from Rand Afrikaans University) and also from the Rand Afrikaans University, Prof Izak Broere (father of graph theory in South Africa).

A direct consequence of this "graph theory concentration camp" was a visit by Alewyn Burger, Ernie Cockayne, Odile Favaron (then from Laboratoire de Recherche en Informatique, Université Paris Sud, France) and Kieka Mynhardt to our department in April 2002. We had an enjoyable day during which Ernie and three of my then master's students in graph theory, George Groves, Werner Gründlingh and Wynand Winterbach (shown below), made short presentations on the problems they were researching at the time. The three visitors provided us with valuable information and contacts on ongoing research projects elsewhere in the world bearing a resemblance to the problems described by the students. Odile also give a fascinating colloquium about known bounds on the upper irredundance parameter, IR(G).

As a result of this visit by Ernie Cockayne, Odile Favaron and Kieka Mynhardt, a research weekend on the West Coast took place in May 2002, during which Alewyn Burger, Werner Gründlingh, Wynand Winterbach and I secluded ourselves in a holiday flat in Yzerfontein to work on a problem about higher order domination in graphs, presented by Ernie during his visit in April. The weekend, with many walks along the beach, turned out to be very productive, and culminated in two joint papers in the end, which have appeared in the Journal of Combinatorial Mathematics and Combinatorial Computing [an abstract and electronic copy of the first paper, on finite order graph protection strategies may be obtained here, whilst those of the second paper, on infinite order graph protection strategies may be downloaded here]. The picture below shows, from left to right, Alewyn, Wynand and Werner tackling a tantalizing conjecture in higher order graph domination, which we believed to be true for quite a while, but just could not prove. The conjecture was, in fact, disproved in November 2002 by Alewyn and Kieka Mynhardt, using an exhaustive computer search for a counter-example.

Stephen Benecke completed our project on higher order graph domination or protection strategies for graphs, by developing a general framework for higher order domination in his master's thesis, for which Paul Grobler and I acted as co-supervisors. This thesis was submitted in September 2004 [an electronic copy of his thesis may be downloaded here]. Stephen did excentional work for his thesis, which culminated in a joint paper by Stephen, Paul and myself, which is due to appear in Utilitas Mathematica, and in which the optimal strategies for the protection of complete multipartite graphs is fully established [an abstract and electronic copy of this paper may be downloaded here]. The picture below shows, from left to right, myself, Stephen Benecke and Paul Grobler towards the end of the project.

During the period 2000-2005 I was involved in a fascinating joint project with Alewyn Burger on a combinatorial problem involving lotteries. This work gave rise to the master's thesis topic of one of my students, Werner Gründlingh. The project was initiated by an enquiry of Prof Dirk Laurie (Department of Mathematics, Stellenbosch University), who writes a regular column on lotteries for News 24. The picture below shows Werner Gründlingh, Alewyn Burger and myself at the start of the project.

The problem concerns the minimal number of tickets that a lottery player should buy in order to be guaranteed a lesser prize at the game Lotto. The project turned out to be much more challenging and rewarding than we thought possible at the outset. Alewyn, Werner and I wrote a total of four papers on this topic. In the first paper, due to appear in the Journal of Combinatorial Mathematics and Combinatorial Computing, we proved the optimality of 192 lottery design bounds listed at the lottery repository website of R Belic and improved upon 429 of the designs listed at the site, in the process establishing a total of 204 new lottery numbers [an abstract and electronic copy of this paper may be obtained here]. In the second paper, due to appear in the Utilitas Mathematica, we formulated and studied (in algorithmic fashion) a generalisation of the classical lottery problem, in which the player does not require a guarantee of winning a lesser lottery prize, but instead only requires a 100a% assurance of doing so [an abstract and electronic copy of this paper may be downloaded here]. In the third paper, submitted in 2003, we focussed on counting the number of structurally different optimal solutions to the above generlised lottery problem for values of the parameter a in the range (0,1] [an abstract and electronic copy of this paper may be downloaded here]. In the fourth paper, submitted in 2004, we designed a computerised search technique capable of characterising structurally different optimal solutions to small instances of the classical lottery problem [an abstract and electronic copy of this paper may be obtained here].

The project is still ongoing - in January 2005 Basie Kok and Francois du Toit designed and implemented a database-driven repository website for lottery-related results, where researchers are able to submit improvements to lottery number bounds online. The picture above shows the full "lottery team" at the University of Stellenbosch.

The breadth and scope of Werner Gründlingh's master's thesis on the lottery problem was such that his external examiners unanimously recommended that the thesis be upgraded to a PhD dissertation after limited additional work. During 2003-2004 Werner did the additional work required, of which he spent 2004 finalising his dissertation under the supervision of Alewyn Burger at the Department of Mathematics and Statistics at the University of Victoria in Canada, where Alewyn was completing a two-year term as post-doctoral fellow in Mathematics. Werner's visit to Canada was made financially possible by the prestigious Wilhelm Stipendium, which he won for the original work in his master's thesis. He submitted his PhD dissertation in December 2004 [an electronic copy of his dissertation may be downloaded here] and the picture above shows Werner, Alewyn Burger (Werner's co-promoter) and myself at Werner's degree ceremony.

Another very interesting graph theory project in which Wynand Winterbach and I were involved concerned crossing numbers of graphs embedded in the plane. Wynand completed an honours degree in computer science in 2001 and this presented him with an ideal platform from which to launch his masters's studies on crossing numbers - a topic that requires skills in both analytic graph theoretic argument and in computer search implementations. In this study we focussed on multipartite graphs: we were specifically interested in forbidden sub-graph structures for crossing number 1, analogous to K(3,3) and K(5) for crossing number zero, by Kuratowski's famous theorem. Wynand did excellent work for his thesis, developing a first heuristic algorithmic approach toward establishing lower bounds for the plane crossing number of a graph (various heuristics and exact algorithms are known for upper bounds on the crossing number). Wynand submitted his thesis in December 2004 [an electronic copy of his thesis may be downloaded here]. The picture above shows Wynand and myself at the start of the project.

Ernie Cockayne, Kieka Mynhardt and Gerhard Geldenhuys (formerly from this department) visited our department again on 14-16 April 2003 and on 17-19 December 2003 to work with us on a fascinating new problems concerning edge-orderings in graphs. The picture below shows, from left to right, Ernie Cockayne, Isabelle Nieuwoudt, Kieka Mynhardt, Paul Grobler and myself in the back row, and Stephen Benecke, Wynand Winterbach & Werner Gründlingh in the front row, during the April visit.

During these two visits Ernie, Kieka, Gerhard, Paul and I worked on an interesting problem in which one seeks minima of maximal increasing label sequence in edge-labeleld graphs. This study gave rise to a new parameter, called the depression of a graph (closely related to a dual parameter, called altitude, on which Kieka and Ernie had previously been working). We wrote a paper on the subject, which was submitted in December 2004 [an abstract and electronic copy of this paper may be downloaded here].

Operations Research

Although my primary field of research interest remains graph theory, I have a very soft spot for operations research (OR). And as a bonus, these two fields (graph theory and OR) often intersect. I am a proud member of the Operations Research Society of South Africa (ORSSA). Since 1999 I have had the opportunity to serve on the national executive of ORSSA, first as additional member, during 2002-2004 as chair of the Western Cape Chapter of the Society (charged with the responsibility of organising chapter events each year, ranging from regular style seminars or colloquia, to one-day workshops, to hosting full conferences) and for my sins I have been appointed editor of ORiON, the official journal of the Society, since 2004.

During one of my first projects in operations research (in 1998), I worked with Werner Gründlingh, a former honours student of mine (in cryptology) and also a former master's and PhD student (in graph theory). He appears with me on the picture above during this OR project, in which we sought optimal release strategies for large-scale open-air irrigation reservoirs. Werner implemented our solution to this problem in the form of an automated interactive decision support system, called ORMADSS. This system is currently used by a group of 15 farmers near Worcester in the South African Western Cape to determine real-time release strategies for Keerom Dam, the second largest privately owned reservoir in South Africa. ORMADSS settled a very long and intense dispute about what seemed to be a good way to manage the reservoir.

Werner and I took part in the 1999 OR in Development Paper Competition, with our work on Keerom Dam. This competition is organised by the International Federation of OR Societies (IFORS) every three years, and draws entries from developing countries across the world. The papers are adjudicated by a panel of international OR experts, resulting in the invitation of 8 finalists to present their papers at the triennial IFORS conference. Werner & I made the list of finalists and went on to present our work in Beijing, China. We won the silver medal in the end, and on the picture above Prof Goutam Dutta (competition chair) awards our prize to Werner and myself, while Elise del Rosario (then chair of the developing countries portfolio of IFORS)looks on. [An abstract of our winning paper may be obtained here].

I have indeed been fortunate to work with a number of outstanding students in OR over the years. On the picture below I appear with Dr Philip Fourie (President of ORSSA during 2001-2002) on the left, and Elmien Wagenaar (a former master's student of mine) on the right. Elmien did an extremely interesting piece of work for her honours year project on the evaluation of investment portfolio managers. She won the annual ORSSA prize for the best student OR-project in the country in 2000. We wrote up Elmien's work in a paper [an abstract and electronic copy of the full paper may be downloaded here]. Philip Fourie presents Elmien with her winning certificate and prize money, while I look on (clearly very pleased) in the picture below.

In September 2000 our department was honoured with a visit by Elise del Rosario, vice-president of IFORS. She flew in from the Philippines to appear as keynote speaker at the 2000 annual conference of ORSSA. Elise is a past-recipient of the prestigious Franz Edelman award, which is awarded annually by the American OR Society to an individual or team who, as OR-practitioner(s), made a marked contribution in their working environments. Elise won the award for her 20 year long leadership of the OR team of the Philippine San Miguel Company, which started out as a bear manufacturer, but which has subsequently expanded into just about every economic sector and has emerged as a world leader in its field as a direct consequence of Elise's team's work. She had a lot of wisdom to impart to our graduate students in applied mathematics and OR during her visit to the department. In the picture below she appears (in the middle) with three of my OR colleagues and myself, from left to right, Dr Philip Fourie, Dr Elmari Roos (then du Toit) and Ms Isabelle Nieuwoudt.

To me the annual ORSSA-conference is one of the highlights of the year. The conference venue is nearly always some interesting or exotic spot in South Africa, and the conference is usually graced with papers by leading OR practitioners in the country. I always try and take along as many of my graduate (and even undergraduate) students as possible - and they invariably enjoy the experience and exposure to the application of mathematical techniques in real-world situations. Here four graduate students appear with a colleague of mine, Isabelle Nieuwoudt (wearing the white top), during a boat trip on the Vaal River at the 2001 Conference in Vanderbijlpark. The students are, from left to right, David Coleman, Margarete Louw, Stephen Berjak and Stephen Benecke.

David and Margarete have gone on to pursue their master's studies in OR (on a somewhat daunting production scheduling bottleneck-type problem experienced at Loubser Woods Components, Inc, a factory in the Western Cape industria producing wooden panels for products such as doors and furniture). Stephen Berjak was elected to the national executive of ORSSA, with the society's newsletter as portfolio for the period on 2001-2002, whilst Stephen Benecke was elected to the national executive of ORSSA, with the society's website as portfolio for the period on 2003-2004. Isabelle Nieuwoudt and I offer a number of OR-related courses at the Department of Applied Mathematics at Stellenbosch University (both on undergraduate and graduate levels) - details on one of our project driven honours courses may be downloaded here.

In 2001 Margarete Louw, a former honours student of mine and of a colleague of mine, Isabelle Nieuwoudt, won the annual chapter student competition organised by the Western Cape Chapter of ORSSA. She was awarded the prize for work done during her honours year project on the determination of efficient duty rosters for nursing staff at large hospitals. She implemented a tabu-search type heuristic as part of a decision support system, called NURODSS, which takes as input work pattern preferences of individual staff members, as well as their annual leave and remuneration levels and gives as output a suggested duty roster which contrasts cost efficiency against fairness of duty shifts allocated. This system is currently in use at a large hospital in the Western Cape. In the picture above Margarete is congratulated and handed her prize (of a year's free student membership to ORSSA) by Prof Wim Gevers, one of the adjudicators of the competition, which took place at the US Graduate School of Business in Bellville. A part of Margarete's prize was that her project was selected as the Western Cape's official entry for the annual student competition on national level during the following year, 2002. She went on to win the national student competition as well, this time being awarded a substantial cash prize. In the (somewhat dark) picture on the left below Johann Heymann, from SAS (who sponsored the competition), hands over Margarete's cheque to her during the Annual Conference of ORSSA in September 2002, at Goudini Spa.

Isabelle Nieuwoudt, Margarete Louw and I wrote up Margarete's nurse scheduling work in a paper, for which Basie Kok, an undergraduate student in applied mathematics shown in the picture on the right above, helped us expertly with the computer programming of a host of experimental work. This paper was submitted to Journal od Scheduling in April 2004, and an abstract and electronic copy of the paper may be obtained here].

An unexpected honour was also bestowed on Grant Huddlestone (a former honours student of mine) and myself, shown in the picture below, in 2001. We jointly won the prestigious Tom Rozwadowski medal (the highest honour bestowed by ORSSA for the best publication by a member of ORSSA) for work we did on the development of optimal pulping schedules at a large fruit juice producer in the Western Cape [an abstract of our winning paper may be obtained here]. This honour was unexpected, because the work had already been completed by the start of 1998 - the reason for the delay was of course due to the refereeing and publishing process of our paper.

In a very interesting OR-project I was fortunate to work with George Groves (a master's student from the Department of Industrial Engineering at Stellenbosch) and Jeanne le Roux (Department of Quantitative Management at UNISA) on a new kind of vehicle routing problem, which we dubbed the Scheduled Multiple Traversal Postman Problem (SMTPP). The problem is to find a cost-efficient route for a service vehicle through a transportation network in which different network links require a potentially different number of services per given time interval. In addition to being cost-efficient, the routing must also be scheduled such that consecutive traversals of the same link are spread out as much as possible over the routing time window. This problem is NP-complete and we followed a tabu-search heuristic together with a number of graph theoretic algorithms to solve the problem approximately. We also applied our solution methodology to a large-scale real-world problem of SPOORNET (the semi-privatised South African railways authority), who sought a service route for their track testing vehicle, the IM2000. The picture below shows George Groves and myself towards the start of George's masters studies on the SMTPP.

George Groves did exceptional work during in his master's thesis on the SMTPP - so much so that his thesis was upgraded to PhD status. We were also selected as finalists in the international triennial OR in development Competition organised by IFORS and went on to present our work during the IFORS 2002 Conference in Edinburgh, Scotland, during a special competition stream. We won the silver medal, and appear on the picture below with (from left to right) Elize del Rosiario (chair of the Development Committee of IFORS), Goutam Dutta (chair of the Prize Competition Committee) and Paulo Toth (president of IFORS) during the officail prize awards ceremony at Murray Field Stadium in Scotland.

To our surprise George Groves, Jeanne le Roux and I were awarded the Tom Rozwadowski model again in 2002 - this time for our work on the SMTPP. The medal was awarded during the Annual Conference of ORSSA in September 2002, at Goudini Spa. The two pictures below show Jeanne and myself receiving our medals from the president of the Society, Hans Ittmann. Unfortunately George was not at the conference to receive his medal, but it was presented to him at a later stage. Our winning paper has appeared in Internatonal Transactions in Operationsl Research, and its abstract as well as an electronic copy of the full paper may be obtained here.

George Groves submitted his upgraded PhD dissertation on the SMTPP in May 2004 [an electronic copy of his dissertation may be downloaded here], and he passed with high praise from his external exminers. The picture below shows George and myself, just after George reveived his doctorate. We submitted another paper on work on the SMTPP emanating from George's dissertation to the journal Networks in 2004 - this time testing the solution methodology on a number of benchmark problems [an abstract an electronic copy of the paper may be obtained here]. It turns out that a famous routing problem in graph theory, called the rural postman problem, is a special case of the SMTPP, and we submitted a final paper focussing on the rural postman aspect of the SMTPP and comparing the qualities of our solutions on a number of benchmark problems to those of other authors over the years. This paper is due to appear in ORiON, and an abstract and electronic copy of the paper may be downloaded here.

The picture below shows two master's students of mine, Maragerete Louw and David Coleman (seated in the middle). We worked together in a team to try and resolve an interesting bottleneck problem on the production floor of Loubser Woods Components, Inc, a wood product production plant in the Western Cape. Margarete tackled the job shop problem from a network flow (heuristic) point of view for her master's thesis in OR, which was co-supervised by Isabelle Nieuwoudt (on the left in the picture below) and myself (middle). David considered the same problem from a computer simulation angle, and was co-spuervised by Isabelle Nieuwoudt, Jeanne le Roux (on the right in the picture below) and myself. The project involved numerous visits to the factory site and some interesting OR models emerged during the course of our work. David and Margarete both submitted their theses in 2005, and electronic copies may be downloaded here.

I have also been involved in a very interesting OR project in collaboration with the Institute of Maritime Technology in Simon's Town, through Nicky Pantland who was a cadet there during 2001-2003. Nicky enrolled for a masters thesis in applied mathematics in which the objective was to implement an automatic, computerised procedure whereby the foot of a continental slope could be traced, as the ridge of maximum curvature at the base of the continental rise. Many coastal states (including South Africa) are signatories to United Nations Convention on The Law of the Sea (UNCLOS), are thus bound by the articles contained therein and have a time limit of 10 years from signing of the agreement for submitting a once-off professional and scientifically substantiated claim for extended maritime estate, bringing with it increased mineral rights. In the case of South Africa, the deadline for submission is 2009 and a significant area of seafloor estate has been identified as potential claim material, and this masters study formed part of the realisation of this potential. Nicky did very interesting computational work during this project, based on real satellite data of the seafloor around the South African coast, and appears in the picture on the right towards the end of the project. An electronic copy of her thesis may be downloaded here.

During the period 2003-2005 I was involved in a large project coordinated for the South African fresh fruit export industry by Dr Esbeth van Dyk from the Transportek Division of the CSIR in Stellenbosch. In this project I supervised a masters student, Frank Ortmann, who modelled the export infrastructure of the industry (comprising a large number of packhouses and cold stores, sea and airports and a number of modes of transportation of fruit between these components) with the aim of estimating (a) the maximum volume of fruit that may flow through the infrastructure as whole during any harvesting season, and (b) a minimum cost strategy with respect to exporting a given volume of fruit from South Africa at any given time. During the course of this study some interesting results emerged, such as that feelings by role players in the fruit industry that the capacity of the export infrastructure is reached at South African ports during the peak summer and winter export weeks is merely a perception and that bottlenecks that are observed are, in fact, as a result of poor scheduling practices, worker strikes and adverse weather conditions. Frank submitted his thesis in June 2005 and an electronic copy of his thesis may be downloaded here. The picture below shows Frank Ortmann, Esbeth van Dyk and myself towards the end of the project.

In 2005 I supervised an honours student, Rickus Jooste, during his year project on a facinating topic, involving the use of self-orthogonal latin squares in the construction of optimal spouse avoiding mixed doubles tennis tournaments. The topic was originally suggested by Prof Dirk Laurie of the Department of Mathematics at Stellenbosch, and during the course of his work Rickus used a wide variety of discrete mathemematical tools (including groups, rings and fields and their "addition" tables) to design and implement a computerised decision support system capable of aiding a sports manager in designing a clash-free spouse avoiding mixed doubles tennis tournament that utilises as few tennis cours as possible in the minimum number of matches such that (a) every person opposes every person of the same sex exactly one, (b) every person teams up with and opposes each person of the opposite sex exactly once and (c) no person teams up with or opposes his/her spouse. The picture below shows Rickus and myself during the project.

Differential Equations

My master's and doctoral thesis topics were on ordinary and partial differential equations respectively. During my 1991-1992 master's studies at the University of Stellenbosch I was fortunate to have Prof Tom Dreyer as supervisor on a topic involving the modelling of 2-dimensional shapes of and tension forces in elastic cables with bending stiffness exhibiting no torsion. The study included the numerical solution of the resulting differential system model, which was a most challenging task in those days, before computing packages were readily available and computers became as fast and powerful as they are nowadays. Tom and I wrote a paper together on the work emanating from my thesis, which has appeared in Applied Mathematical Modelling [an abstract and an electronic copy of the full paper may be obtained here].

I continued my studies by enrolling for a DPhil at the University of Oxford in 1992. Dr John Norbury acted as my promoter on a dissertation topic involving the analytic determination of simple criteria by which non-mathematicians would be able to verify the permanence (or otherwise) of solutions to existing models of biological competition of a number of cohabiting species in a given geographic region, without actually having to solve the models. I learned a lot from John, who accepted me as I am, with all my eccentricities, and gave me the space to develop my thesis as I saw fit, yet always cautioning well in advance of a potential disaster, when my avenues of investigation seemed to take strange turns. I cherish our weekly meetings in his office off chapel quad and in the Lincoln bursary during the mid-nineties. John and I have written two papers together on permanence in biological reaction-diffusion theory and I continue to visit him at Oxford regularly. Our first paper laid the theoretical foundation and has appeared in the Proceedings of the Royal Society of Edinburgh [an abstract and an electronic copy of the full paper may be downloaded here]. In our second paper, which has appeared in the Australian & New Zealand Industrial and Applied Mathematics Journal, we made the work in the first paper more accessible, by applying the theory to sixteen well-known models of biological competition, deriving conditions for permanence for each in terms of the model parameters [an abstract of the full paper may be downloaded here]. We were also invited to contribute a chapter to the book Advances in Mathematical Population Dynamics - Molecules, Cells & Man (published by World Scientific in Singapore). In this chapter we summarised our work of the two papers mentioned above [an abstract of the chapter may be downloaded here].

The picture above shows John Norbury and myself in Main Quad, Lincoln College, University of Oxford. Only fellows and pigeons are allowed on the beautifully kept lawn behind us - but strictly no students! The college dates from 1427, and the "Oxford experience" made a very deep impression on me as an immature graduate student. I shall always remember the feeling of awe when walking in college, working in the library or attending a concert in the Sheldonian: I often felt like a little, naïve mortal being tolerated by a wise, ancient and enduring institution. Many a value encountered at Oxford have accompanied me since my departure there in November 1995, and I try to instil some of these values in my own students here at Stellenbosch.

I wrote another two papers on the work for my doctoral dissertation. In the first, which appeared in the IMA Journal of Applied Mathematics, I demonstrated the existence of a family of travelling wave front solutions to a fairly general set of partial differential equations (under zero-flux Neumann boundary conditions), typically used to model biological competition. An abstract and an electronic copy of the full paper may be downloaded here. In the second paper, which also appeared in the IMA Journal of Applied Mathematics, I considered the resilience of solutions to the same partial differential model, after pertubations away from equilibrium densities. An abstract and an electronic copy of the full paper may be downloaded here.

In 1996 I was asked by Gerhard Geldenhuys (then head of department) to return to the Department of Applied Mathematics at Stellenbosch University - this time as a staff member. One of my first tasks was to develop and present a course on partial differential equations to honours students at our department. I spent most of December 1995 and January 1996 developing a set of lecture notes, based on the scope of a course I took and on a course I tutored at Oxford. I offered this course as part of our department's honours programme during the period 1996-2001. Dr Milton Maritz now offers this course at our department. During the years 1996-2000 I also enjoyed presenting a second year course on ordinary differential equations, which I inherited from Tom Dreyer, my master's supervisor. In 1998-2000 I shared presentation of this course with Dr Francois Smit, who was able to impart a considerable amount of practical experience in differential equation modelling (which he brought with him from industry) to the students. Prof Andre Weideman and Ms Karin Hunter now present this course at our department ... I no longer lecture any courses in differential equations, as a result of other commitments - something I regret.

I have been involved in a number of graduate and other research projects involving differential equations. For example, in 1997 Mariza Goosen (a former honours student of mine) and I worked on a differential equation model from which optimal release strategies could be calculated for open-air reservoirs via the theory of optimal control. This gave rise to the Keerom Dam project discussed above. But one of the high points of my work in differential equations was certainly joint work in 2001 with Stephen Benecke (then an honours student). This project entailed generalising my master's work with Tom Dreyer to the 3-dimensional case where the cable is not only elastic and imperfectly flexible, but also exhibits torsion as a result of axial twist applied at its endpoints. Stephen did exceptionally good work during this project; his model was able to capture first points of twisting bifurcation (where so much twist is applied to the end points of the cable that it jumps from an unstable three-dimensional configuration to a stable one). We wrote a paper on the subject together, which has appeared in Applied Mathematical Modelling [an abstract and an electronic copy of the full paper may be downloaded here. The picture to the right shows Stephen at his graduation ceremony: he received the Dean of Science's medal for the highest aggregate over all four BSc & (Hons)BSc years within his class group. Stephen went on to enrol for a masters degree on a topic in graph theory, as described above.

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