Showing papers in "The Mathematical Gazette in 1985"

Cayley graphs for some well-known groups

Abstract: First it may be necessary to remind some readers of what is meant by the representation of a group by a Cayley graph. Each element of the group is represented by a point, the points being connected by lines (or edges) each of which represents a generating operation of the group. Suppose, for example, that x and y are two elements of a group, while another element;? is taken as a generator, and that these are connected: xp = y . (Throughout this article we shall take the identity element of a group to be 1, inverses, to be indicated x -1 , while xp means “first x, thenp” .) This would be graphed:

Complex analysis and applications

Abstract: Analytic functions. Complex integration. Infinite series. Contour integration. Conformal mappings. Boundary-value and initial value problems.

The absolute correlation coefficient

Abstract: The two most common measures of central tendency and dispersion in statistics are the mean and standard deviation on the one hand, and the median and absolute deviation on the other. For most purposes the former measure is preferred for two very good reasons; the first is that the squares of quantities are easier to handle analytically than their moduli; and secondly for all the common symmetrical distributions, such as the normal, uniform and binomial distributions, for which the mean and median coincide, if a sample is taken to estimate the central value, then the mean of that sample has a smaller variance than the median, and is therefore relatively more efficient as an estimator of the central value of the parent population.

The twenty-one card trick

Abstract: 1. Introduction Nearly forty years ago (and I barely knowing the three times table), my father taught me the twenty-one card trick, amid background murmuring from mother as to its mysterious mathematical nature. Now, with a rather more questioning six year old, I have been made to look at how it works, and see what variations a budding Paul Daniels might introduce.

A-level mathematics in an information age

Abstract: as is often the case in an economic context, the methods of calculus, analysis and even topology are appropriate [6]. Returning to \"pure\" Arrow's theorem, there are alternative approaches and proofs: Roberts' treatment in [7] is graph theoretic, and in [8] he points to connections with graph theory, semi-orders and indifference relations. In [9] the individual rankings are points in a metric space and the election is decided by minimising a certain distance in this space. [2] contains a version of Arrow's theorem in terms of Boolean matrices. Finally, a totally elementary and entertaining account of the shortcomings of several \"reasonable\" ways of conducting elections is given in [10], using nothing more sophisticated than arithmetic. Most of these books contain references to more detailed papers.

Investigational work at A-level

Abstract: I read the first page of John Hersee’s article in the June 1984 Gazette and sat down immediately, “stimulated to contribute comments” on his question. “What do we mean by ‘investigationary work’… in the context of A-level?”

69.30 A remarkable approximation to π

Abstract: In the first book of Kings, we find a description of the temple built by King Solomon in which the measurements of the various parts are stated. In Chapter 7 v. 23 the ‘molten sea’, a large basin containing water in which the priests washed their hands and feet before performing the rites, is described. The verse states (in the A.V.)