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.

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.

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

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.

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.

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.

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.

This is the first of (presumably) three articles on the subject mentioned in the title. The exposition is based on a course of fifteen lectures which formed part of the Edmonton (1957) Seminar of the Canadian Mathematical Congress. Limitations of space (and, originally, of time) compel us to be selective in two directions. First, while we shall refer to other branches of logic in passing, we shall be concerned principally with the two fundamental calculi - of propositions and of predicates (of the first order). Thus, except for a number of modern developments which are included here, our exposition will be similar in scope, though not in detail, to the first and third chapters of the well-known "Principles of Mathematical Logic" by D. Hilbert and W. Ackermann (English translation, Chelsea, New York, 1950) and this was in fact the recommended text for the Edmonton course. However, there exists a growing number of other good introductions to the field and some of these will be listed later.

The sixth mathematics seminar was held on the University of Alberta campus in Edmonton, Alberta, in conjunction with the first seminar of the Canadian Association of Physicists (Theoretical Physics Division). The following courses were given.

Immediately following the seminar in Edmonton the fourth quadrennial Mathematical Congress was held in Banff, Alberta. The Canadian Association of Physicists (Theoretical Physics Division) participated in this, as they did in the seminar.

Finance. Grants to support the Journal were received as follows

In addition, 35,530 was received from subscriptions, advertising, reprints, etc. Total cost of printing (at the University of Totonto Press) was about 312,500, leaving, after other expenses, a balance of $5,700.

The Institute was in session from May 15 to August 15, 1957. There were 32 Fellows and visitors from the staffs of 19 Canadian universities and from the University of London and Rutgers University. Fellowship holders.