The material sketched here is mostly well known and concerns the geometrical inter-relations of vectors in a Hilbert (that is, complete inner product) space. The discussion and references are (obviously) not exhaustive but I hope the reader will find here some interesting problems.

We shall permit the scalars to be real, complex or quaternionic numbers. The inner product of vectors x and y will be denoted (x|y).

A sequence {un}, n=0, 1, 2, 3,… is said to be an integral recurrence of order r if the terms satisfy the equation

for n=r+1, r+2,…, and a1, a2,…, ar are integers, ar≠0. In this case we will say that {un} satisfies the relation [a1, a2,…, ar]. The sequence {un} is uniquely determined when u1, u2,…, ur are given specified values. If u1, u2,…, ur are integers all the terms of {un} are integers. The generating function f(t)=u1t + u2t2+… takes on the form where Q(t) depends on the values of u1, u2,…, ur and R(t)=tr-a1tr-1-a2tr-2-…-ar. We will refer to R(t) as the characteristic polynomial of the recurrence.

Let there be given n(≥l) distinct sets of points Pi=(Pi1,…, Pini), with ni≥1, for i=1,…, n. If joining each pair of points not in the same set is a line oriented towards one, and only one, point of the pair the resulting configuration will be called an n-partite tournament. If the line joining Pij and Pkl is oriented towards the latter point we shall indicate this by Pij →Pkl, and similarly if the orientation is in the opposite sense.

A recent publication entitled "Magic Squares and Cubes", by W. S. Andrews, published by the Dover Publications, New York, is an excellent authoritative book on magic squares, and magic cubes. In fact, chapter XIV entitled "Magic Octrahedroids", has shown examples of the extension into four-dimensional space. Andrews' method seems to be that of extending symmetrical considerations, which he calls reversions, in forming higher dimensional forms of magic squares. Neither he, nor any other author, to the best of my knowledge, has extended magic squares to higher than four dimensions.

La Hire must be given full credit for his method of breaking down magic squares into component squares, and conversely constructing magic squares from component squares. The approach of this author is to extend La Hire's method for n-dimensional space by means of a relation:

This note concerns a countably additive measure on a Boolean ring of subsets of an abstract set, this measure being real-valued, admitting ∞ as a possible value. We are interested only in unique extensions, so we suppose the measure to be σ- finite. The following well known result will be referred to as the "extension theorem": "Every σ-finite measure on a ring extends uniquely to a σ-finite measure on the generated σ-ring. "Besides the familiar proof using outer measure, there is a Borel-type proof using transfinite induction [4]. We attempt here to reduce the Borel-type proof to its ultimate simplicity, reducing the problem to the bounded case.

In a recent paper [1], L. Moser and E.L. Whitney have proved the following.

Theorem: The number of compositions of n into parts=1, 2, 4 or 5 (mod 6) and involving an even number of parts=4 or 5 (mod 6) exceeds by n the number of compositions of n into parts=1, 2, 4 or 5 (mod 6) and involving an odd number of parts=4 or 5 (mod 6).

Their method of proof utilizes the notion of weighted compositions and the method of generating series. They remark that they have not been able to find a direct combinatorial proof. The purpose of this note is to give a direct proof of a more general result.

Given a set of n points, with each pair of distinct points joined by a line that is oriented towards exactly one of the points, then the resulting configuration is called a (roun-drobin) tournament. A tournament is reducible if the points can be separated into two non-empty subsets, A and B, such that every line that joins a point in A to a point in B is oriented towards the point in B. If a tournament is not reducible it is called irreducible. The object of this note is to derive an approximation for P(n), the probability that a tournament on n point, chosen at random from the set of possible ones, will be irreducible. p(1)=1, by definition.

We shall obtain a little more general formula than the one given in the problem P44.

Put

1

First of all we shall prove the identity

2

where Cmi are constants whose values will also be determined