Discrete Mathematics
Chapter
Abstract

The wombat with the broad hairy nose.

Identification number ICME 5 1234123413.

The peculiar animal peculiar to South Australia.

What has Vombatus Platyrhinus got to do with my talk?

That is a question for you to try to answer.
Keywords
Period Solution Equilibrium Point Difference Equation Minimal Span Tree Network Theory
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
 The references listed below, although confined to the English mathematical literature, should be readily accessible and should prove useful for those wishing to find out more about the material in my talk.Google Scholar
 1.May, R.M. (1976) `Simple mathematical models with very complicated dynamics’. Nature 261, pp. 459–467.Google Scholar
 Quoted from the abstract: “Firstorder difference equations arise in many contexts in the biological, economic and social sciences. Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points to a bifurcating hierarchy of stable cycles, to apparently random fluctuations. There are consequently many problems, some concerned with delicate mathematical aspects of the fine structure of the trajectories, and some concerned with the practical implications and applications”.Google Scholar
 This is an excellent review article by one of the world’s leading mathematical biologists. I wholeheartedly agree with the author’s conclusion that “the most important applications may be pedagogical”. As he states: “The elegant body of mathematical theory pertaining to linear systems tends to dominate even moderately advanced University courses. The mathematical intuition so developed ill equips the student to confront the bizarre behaviour exhibited by the simplest of discrete nonlinear systems [such as (5)]. Yet such nonlinear systems are surely the rule, not the exception, outside the physical sciences. I would therefore urge that people be introduced to, say, equation (5) early in their mathematical education”.Google Scholar
 2.Hofstadter, D.R. (1981) `Strange attractors: mathematical patterns delicately poised between order and chaos’. Scientific American 245, pp. 16–29.Google Scholar
 The author is a regular contributor of metamagical themas to Scientific American, and is wellknown for his fascinating Pulitzer Prize winning book “Gödel, Escher, Bach”. This popular article on strange attractors, with a touch of romance, provides excellent background material for a study of nonlinear difference equations.Google Scholar
 3.Thompson, J.M.T., and Thompson, R.J. (1980) `Numerical experiments with a strange attractor’. Bull. Inst. Maths and its Appl. 16, pp. 150–154.Google Scholar
 An interesting father and son paper analysing the system of nonlinear difference equations (11). The computer programming was done by the son during school holidays.Google Scholar
 4.Boole, G. Calculus of finite differences. 1st edition (1860); 4th edition Chelsea (1970).Google Scholar
 Difference equations is old mathematics!Google Scholar
 5.Lighthill, James (ed.) (1980) Newer uses of mathematics. Penguin. The editor of this book is one of the world’s leading applied mathematicians, one who has taken a keen interest in mathematical education, and is a former ICMI President.Google Scholar
 Chapter 4 of the book, written by the present author, gives an introduction to networks with applications to a wide variety of problems in transportation, telecommunications and industry.Google Scholar
 6.School Mathematics Project,New Book 3, Part 2. C 1982.Google Scholar
 An interesting snippet on route matrices of networks (p.76).Google Scholar
 7.Penny, D., Foulds, L.R., and Hendy, M.D. (1982) `Testing the theory of evolution by comparing phylogenetic trees constructed from five different protein sequences.’ Nature 297, pp. 197–207.Google Scholar
 This paper gives a detailed analysis of the minimal spanning tree problem for 11 species. The example described in the talk for just 4 species is a simplified version. With more species one becomes aware of the rapid growth in the size of the problem and begins to realise that it could become too large for any computer to handle. The authors claim that their results are consistent with the theory of evolution.Google Scholar
 8.Zweng, M.J., et al. (ed.) (1983) Proceedings of the Fourth International Congress on Mathematical Education (ICME IV). Birkhäuser. Is calculus essential? (p. 50 )Google Scholar
 What should be dropped from the secondary school mathematics curriculum to make room for new topics? (p. 390)Google Scholar
 Where do we go from here? (p. 434)Google Scholar
 What should have been accomplished in four years, to the next ICME? (p. 435)Google Scholar
Copyright information
© Springer Science+Business Media New York 1986