As usual we denote the number of integers not exceeding n and relatively prime to n by Euler's ϕ function ϕ(n). Lehmer calls the ϕ(n) integers

the totatives of n.

Denote by ϕ(k,l,n) the number of a's satisfying

If nl ≡ 0 (mod k) or n(l + 1) ≡ 0 (mod k) then, since n > k, (n - l/k, n) > 1 and (n(l + 1)/k, n)>1 respectively. Thus ϕ(k, l, n) is the number of totatives of n satisfying

According to a well-known theorem of algebra [3, p.47], an integral domain can be embedded in a field, called its field of quotients. Every freshman is familiar with the simplest form of this theorem concerning the integers and the rational numbers. Many generalisations have been given in which a "ring of quotients" is constructed for a given ring [e.g. 6, 7, 11, 12, 13]. Of course, we cannot expect a ring of quotients to be a field, or even a skew field. As Malcev [9] has shown, there even exist rings without proper divisors of zero which cannot be embedded isomorphically in a skew field. The most recent construction, by Utumi [12], gives a ring of quotients for any ring with zero left annihilator. We show in this paper that this construction can be extended to arbitrary rings, in fact, to arbitrary modules. The method used is more abstract: a fundamental relation between rings, defined by Utumi in terms of their elements, is here replaced by a corresponding relation between modules, defined by means of homomorphisms.

In this paper we consider the following combinatorial problem. In how many ways can n distinguishable objects be placed into an unrestricted number of indistinguishable boxes, if each box can hold at most r objects? Let us denote this number by Gn, r

Special cases of this problem have been the object of considerable study. In the case r = 2 we have the numbers Gn, 2 = Tn which have been treated by Rothe [12] as early as 1800. Tn is also the number of solutions of x2 = 1 in the symmetric group on n letters , and in this and related guises has been studied by Touchard [13], Chowla, Herstein and Moore [3] and two of the present authors [7].

There are so many proofs of this theorem in the literature, that an excuse is necessary before exhibiting another. Such may be found by examining the proof given below for the following: it uses no matrices, almost no topology and makes little use of the geometry of convex sets; it applies equally well to the case where only one of the pure strategy spaces is finite; also there is no assumption that the payoff function is bounded. Thus it can provide a short route to the more general forms of the theorem.

In this section we continue to study the indecomposable matrix pairs adopting the same notation as in part I of this paper.

If the matrix A is regular and if it is symmetric or anti-symmetric such that

3.1

and if the matrix pair (X, A) is indecomposable then the corresponding representation space either is indecomposable or it is the direct sum of two indecomposable invariant subspaces. These are operator isomorphic if and only if the minimal polynomial mX of X is equal to (x-δ)μ where

3.2

at any rate there is even a decomposition of the representation space into the direct sum of two isotropic indecomposable invariant subspaces provided the characteristic of F is not 2.

It is the purpose of the calculus described in the preceding section (i) to yield as theorems all tautologies which can be formulated in terms of the given variables, and (ii) to yield no other formulae as theorems. We shall establish presently that these two conditions are indeed satisfied. However, there are questions of a more general character which can be solved both with regard to the present calculus and with regard to many other calculi, including some with a more comprehensive vocabulary.

For the past four years we have given, at the University of Alberta, a course entitled "The Nature of Mathematics". This course is open to first and second year students in arts and science and in education, and is designed primarily for those who will be taking only a single course in mathematics at the university. The only prerequisite for the course is high school mathematics and the course is not prerequisite for any other course.

1. Explain briefly what is meant by each of the following:

• (i) Euler path

• (ii) fixed point theorem

• (iii) four color problem

• (iv) unit braid of order n

• (v) MBbius band.

The new Bulletin has started off most successfully and congratulations to Peter Scherk, Jim Lambek, John Peck and all the others who have worked together to launch it so successfully. The reports on the Seminar at Edmonton and the Congress at Banff, which appeared in the first number of the Bulletin, happily make it unnecessary for me to do more than to say that these seemed to be as successful meetings as any we have held. The settings both at Edmonton and Banff were ideal, but without our hosts1 careful planning and continued attention to detail, these would have been of no avail.

The Montreal Mathematical Colloquium is an organization of mathematicians in the Montreal area formed for the purpose of holding regular mathematical meetings and for inviting outside speakers. This year (1957-1958) the following talks were given by the invited speakers.

University of Alberta: Professor J.G. Kemeny spent four days here, March 12 to 15, as a visiting lecturer for the Mathematical Association of America. He gave two colloquium talks on "Markov chains in probability theory" and public lectures on "New developments in the teaching of mathematics" and on "Mathematics in the Social Sciences". He was guest speaker at a dinner meeting of the Mathematics and Physics club, at which he gave a delightful popular talk on the way group theory might have originated, and he also addressed several groups of mathematics students. His visit was greatly appreciated.