The writer of the following lines is aware of the possibility that the property discussed below is not a new discovery. He will, of course, be grateful to any reader who will provide him with bibliographical references.

It is well known-that the nine point circle of a triangle is tangent to the 4 contact (i.e. the inscribed and the escribed) circles. Given two tangent circles and we propose to find the locus of vertices of the triangles (Δ) which admit as their nine point circle and as one of their contact circles.

Let f(x) be an increasing function. Recently there have been several papers which proved that under fairly general conditions on f(x) the density of integers n for which (n, f(n)) = 1 is 6/π2 and that (d(n) denotes the number of divisors of n)

In particular both of these results hold if f(x) = xα, 0 < α < 1 and the first holds if f(x) = [α x], α irrational.

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

A question which Chalk and L. Moser asked me several years ago led me to the following problem: Let 0 < x ≤ y. Estimate the smallest f(x) so that there should exist integers u and v satisfying

1

I am going to prove that for every ∊ > 0 there exist arbitrarily large values of x satisfying

2

but that for a certain c > 0 and all x

3

A sharp estimation of f(x) seems to be a difficult problem. It is clear that f(p) = 2 for all primes p. I can prove that f(x)→ ∞ and f(x)/loglog x → 0 if we neglect a sequence of integers of density 0, but I will not give the proof here.

The purpose of this note is to give an elementary proof of a special case of a theorem suggested by Th. Bang (2; 3) and proved by Lee et al (5; see also 1; 4; 6; 7; 8).

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].

Let R denote the rectangle: |t-t0| ≤ a, | x-x0| ≤ b (a,b > 0) in the (t,x) plane and let f(t, x) be a function of two real variables t and x, defined and continuous on R. If I is the interval |t—t0| ≤ d with d = min(a,b/M), where M = max(|f(t, x)|, (t, x) ϵ R), then every solution x = x (t) of the differential equation x' = f(t, x) defined on I and which satisfies the initial condition x(t0) = x0, satisfies the integral equation

1.1

and conversely. In some cases, in order to prove the existence and uniqueness of the solutions of (1.1) on I, one forms the successive approximations

1.2

A partial homomorphism ϕ from B into A will be called a fractional homomorphism if dom ϕ ≤ B (A). Every extension of a fractional homomorphism is again fractional, and each fractional homomorphism has a unique irreducible extension.

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.

The relationship between the representation theory of the full linear group GL(d) of all non-singular linear transformations of degree d over a field of characteristic zero and that of the symmetric group Sn goes back to Schur and has been expounded by Weyl in his classical groups, [4; cf also 2 and 3]. More and more, the significance of continuous groups for modern physics is being pressed on the attention of mathematicians, and it seems worth recording a remark made to the author by Philip Hall in Edmonton.

As is well known, the irreducible representations of Sn are obtainable from the Young diagrams [λ]=[λ1, λ2 ,..., λr] consisting of λ1 nodes in the first row, λ2 in the second row, etc., where λ1≥λ2≥ ... ≥λr and Σ λi = n. If we denote the jth node in the ith row of [λ] by (i,j) then those nodes to the right of and below (i,j), constitute, along with the (i,j) node itself, the (i,j)-hook of length hij.

Given an integer n, we define an r-composition of n as follows:

An r-composition of n, (t1, ... , tr), is a set of ti where ti ≥ 1 is an integer for i = 1, ... , r such that

If r is an integer such that 1 ≤ r ≤ n, we have, obviously, distinct r - compositions of n.

The space Hp, 1 ≤ p ≤ ∞ consists of those analytic functions f(z) regular in the unit circle, for which Mp (f;r) is bounded for O ≤ r ≤ 1, where

These spaces have been extensively studied.

One well known result concerning these spaces is that if f(z) = Σ ∞n=0 anzn and {an} ɛ lp for some p, 1 ≤ p ≤ 2, then f ɛ Hq, where p-1+q-1 = 1, and conversely if f ɛ Hp, 1 ≤ p ≤ 2, then {an} ɛ lq. We propose to generalize this result to deal with functions f(z) = Σ ∞n=0 anzn with {n-λ an; n = 1, 2,...} ɛ lp, where λ ≥ 0. The resulting generalization is contained in the theorems below.

However, in order to make these generalizations we must first generalize the spaces Hp. To this end we make the following definition.

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.

Let x, y be column matrices of n real homogeneous coordinates xj, yj (j = 1,..., n) representing points in (n - 1) - dimensional real projective space Pn - 1. Let A be an n × n real symmetric matrix. The equation x' Ax = 0 represents a quadric in Pn - 1 and the equation

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.

Given a vector space V = {x, y, ...} over an arbitrary field. In V a symmetric bilinear form (x,y) i s given. A subspace W is called totally isotropic [t.i.] if (x,y) = 0 for every pair x W, y W.

Let Vn and Vm be two t.i. subspaces of V; n < m. Lower indices always indicate dimensions. It is a well known and fundamental fact of analytic geometry that there exists a t.i. subspace Wm of V containing Vn [cf. Dieudonné: Les Groupes classiques , P. 18]. As no simple direct proof seems to be available, we propose to supply one.

At the Edmonton Meeting of the Canadian Mathematical Congress E. Wigner asked me whether one knew something about the distribution of the characteristic roots of the linear transformations that leave invariant the quadratic form t2+x2-y2-z2, just as one knows that a Lorentz transformation has two complex conjugate characteristic roots and two real characteristic roots that are either inverse to one another or the numbers 1 and -1.

In this paper an answer to E. Wigner’s question will be obtained.

We are concerned with the pairs of matrices (X,A) with coefficients in a field of reference F such that the condition

0.1

is satisfied, where XT = (ξki) is the transpose of the matrix X = (ξki). It follows that both matrices are quadratic of the same degree d.

Let x, y denote real column vectors with n elements. Let A be a regular symmetric real n×n matrix. Dashes indicate transposition.

If x is fixed, x' Ax > 0, the discriminant of A at x is the quadratic form

y' where S= S(x)= x'x'Ax ¨ Axx'A.

In Can. Math. Bull. 1, pp. 175-179, Dr. Schwerdtfeger proved the equivalence of the following properties of A:

1. (i) A is of the congruence type [+, -,..., -].

2. (ii) y' Sy≤ 0 for all y, equality holding if and only if y is a multiple of x. His note is of particular interest because he also discusses the eigen-values of S. If only the quoted result is aimed at, the following procedure may be shorter.

University of Alberta: Dr. I.N. Baker, a graduate of the University of Adelaide, has been appointed Assistant Professor. His thesis on entire functions has been accepted at the University of Tubingen (Germany) for the degree of* Dr. rer. nat. Dr. G.C. Crée, a graduate of McGill and Washington University (St. Louis), has joined the staff as Assistant Professor. He was formerly at the University of Nebraska, in Lincoln. Dr. Shanti S. Gupta, a graduate in mathematical statistics of the University of North Carolina, and latterly employed with the Bell Telephone Laboratories at Allentown, Pa., has been appointed Associate Professor. Dr. H.F.J. Lowig of the University of Tasmania (Hobart) and formerly from Czechoslovakia, has been appointed Associate Professor; his special field is algebra.