The Lyness equation (1)

\begin{equation}{X_{n + 1}} = \frac{{{X_n} + a}}{{{X_{n - 1}}}},\,(a,{x_1},{x_2} > 0)\end{equation}

THE first Mathematical Colloquium held in Edinburgh met during the first week of August, and proved a great success. It was organised by the office-bearers of the Edinburgh Mathematical Society, A. G. Burgess, M.A., F.R.S.E., and Peter Comrie, M.A., B.Sc, F.R.S.E., respectively the President and Secretary of that Society, being also President and Secretary of the Colloquium. The idea of holding such a colloquium was an outcome of Professor Whittaker’s announcement that he purposed organising, as part of the Mathematical Honours curriculum in the University of Edinburgh, a mathematical laboratory for systematic numerical discussion of functions and methods of calculation. Several correspondents had expressed the hope that vacation courses in this line of study might be established; and it was decided to make a first experiment. It was resolved, however, not to limit the colloquium to a discussion of one branch of mathematics, but to enlarge its scope by the inclusion of two other domains of mathematical thought. The broad features of the programme we owe to Professor Whittaker; and its variety was such as to appeal to all types of mathematical mind.

Family history research has become a very popular hobby during the last few years due to the ready access to records of births and marriages. Many people have rigorously traced their ancestors back to 1700 and earlier, corresponding to some 12 generations. The principal motivation of most amateur genealogists is to learn about the lives of their forebears and they trust these ‘legal’ records without question and do not usually acknowledge that cuckoldry may have occurred and that their true ancestors may not have been the ones recorded.

Recent notes (1, 2) suggest the problem. Find necessary and sufficient conditions on a set of n(>3) real constants a1 …, anfor the inequality

The present age is witnessing the rapid growth of a wide variety of communication and control systems using electrical signals. Examples are radio and commercial communications, television and facsimile transmission, radar and aircraft control systems, computers and industrial control mechanisms. All these have in common the fact that information has to be transmitted and processed through a channel, and they can all be represented by a block schematic such as in fig. 1. The efficiency of these systems must be assessed by comparing their information handling capacity with their cost in time, transmission channel bandwidth, equipment complexity and money. It is therefore essential to define a quantitative measure of information and to derive relations between this quantity and the parameters of a given communication system and also to determine criteria for a maximum information flow under given conditions. Information theory concerns itself with these problems, based on simplified and somewhat idealized mathematical models of actual systems. The origins of information theory go back to the 1920’s, but consistent logical foundations for the theory were laid by Shannon in the late 1940’s and the modern theory dates from then. It might appear that information is too loosely defined as a concept to be submitted to quantitative analysis; and indeed, before we can do so, we must clearly differentiate between the amount of information in a message, its meaning (semantic aspects), and its value (philosophical aspects).

In “The music of the spheres”, Jacob Bronowski takes us on a mathematical tour of the Alhambra in Grenada. This includes a visit to the Sala de las Camas, that most private inner sanctum, in order to study the tessellations on the walls. We have used this film with teachers in training at Leeds University for some years now. It provides a stimulating introduction to the history of mathematics and enables consideration of the place of historical references in the teaching of mathematics in schools. We have also compiled a worksheet, based on the film, which includes the task of constructing one of the tessellations, illustrated overleaf.

At the Cambridge Conference, 1969, I proposed the following problem to several members:

Find a binary operation on the reals which is associative but not commutative.

The following two theorems are extensions of the Cayley-Hamilton theorem. The first gives the explicit expression for AN as a polynomial in A, where A is an arbitrary square matrix. The second expands the first result to the case of a matrix series.

This paper was originally planned as an exposition of finite geometry in a form in which it could be used to illustrate all the work covered in the normal school geometry course. The beginning of it, however, was anticipated by an article for the Gazette by Mr. H. Martyn Cundy (1), which he has kindly allowed me to read before publication. This present paper shows how a coordinate system can be constructed for the geometry described in his article by using the field of residue classes (mod. 5); that is, the Galois field GF(5). A definition of distance is introduced which extends that adopted by Cundy to a form suited to the discussion of a wider range of theorems.