In [5, p. 105] attention has been called to a set of propositions, due to H. Cox [3, p. 67], which are related to another set, due to Clifford [2, p. 145; 4, p. 447], concerning points and circles in the plane or on the sphere. One may state Cox's chain of theorems as follows:

In a projective 3-space, S3, let (1), (2), (3), (4) be four points lying in a plane α such that no three of them are collinear. Every two determine a line; let one plane such as [12], pass through each line. There are six such planes. The planes [12], [23], [13] determine a point (123); there are four such points. The first theorem of the chain states that they all lie in one plane [1234], It is not difficult to see that this is, in fact, a rewording of Möbius's theorem on mutually inscribed pairs of tetrahedra [4, p. 444].

Write

1

2

where the maximum is over all real θ, and the lower bound is over all sets of positive integers a1 ≤ a2 ≤ … ≤ an. The problem of the order of magnitude of f(n) was posed by Erdös and Szekeres [1], side by side with a number of other interesting questions. Writing g(n) = log f(n), it is obvious that g(n) is sub-additive, in the sense that g(m+n) ≤ g(m) + g(n), and also that g(1) = log 2, so that g(n) ≤ n log 2.

On an orientable surface of genus p, a set of 2p fundamental circuits can be selected which all pass through a single point A. After cutting along the 2p circuits, the surface can be unfolded into a flat region bounded by a 4p-gon so that: the set of vertices corresponds to the one point A; and the 2p pairs of edges to the 2p fundamental circuits; and the interior of the polygon to the remainder of the surface. If the edges of the polygon are directed, the 2 edges which correspond to one fundamental circuit will be directed in opposite sense, since the surface is orientable [1]. The sequence and direction of the edges is the same as the sequence of the fundamental circuits.

In 1954 A.J. Hoffman and O. Taussky [1] showed that if A is an n-square complex matrix with eigenvalues λ = (λ1, …, λn ) and P is a permutation matrix for which αA + βA* has eigenvalues for some αβ ≠ 0 then A is normal. Here is the conjugate vector of λ. As a companion result they also proved that if the eigenvalues of AA* are , i = 1, …, n then A is normal.

A fundamental problem in combinatorial analysis is the classification of the permutations of 1, 2, …, n which satisfy a system of constraints. Thus one may ask such questions as how many permutations are there which have exactly r k-cycles; how many have at least s cycles regardless of cycle length. Again, one may ask how many permutations are there in which k ascending sequences appear; or how many permutations are there in which specified numbers may not appear in specified places or at specified distances from other numbers. The literature on these problems is quite extensive. References [1,2,5,7,10,14,17] give an indication of the present, status of these problems.

By a composition of a positive integer n is meant a representation of n as a sum of one or more positive integers where the order of the summands is taken into account. Thus for example 4 has the eight compositions 4 = 3 + 1 = 1 +3 = 2 + 2 = 2 + 1 + 1 = 1 + 2 + 1 = 1 + 1 + 2 = 1 + 1 + 1 + 1. Now n can be written in the form 1 + 1 + … + 1 with n - 1 plus signs. Deletion of any subset of these plus signs breaks n into parts which form a composition of n. Conversely, any composition of n corresponds to a subset of plus signs, so that the number of compositions of n is the number of subsets of a set with n - 1 elements, namely 2n - 1. In this note we obtain a number of generalizations of this rather obvious remark by making use of the notion of a weighted composition and the method of generating series.

A parallelohedron is a convex polyhedron, in real affine three-dimensional space, which can be repeated by translation to fill the whole space without interstices. It has centrally symmetrical faces [4, p. 120] and hence is centrally symmetrical.

Let Fi denote the number of faces each having exactly i edges, Vi denote the number of vertices each incident with exactly i edges, E denote the number of edges, n denote the number of sets of parallel edges, F denote the total number of faces, V denote the total number of vertices.

A complex number λ will be said to be free if the multiplicative group Fλ generated by the two matrices

is a free group, and non-free, otherwise. Very little is known about the distribution of free and non-free numbers [1]. It is, for instance, unknown whether the domain

contains an open set which consists of only free points.

This note is to complement the earlier paper on the same subject (Canad. Math. Bull. 3 (1960), 41–57) in two points. The first part presents a simpler proof of the minimum property (cf. 1. c. section 3) of the orthogonal functions Ψv(x) (cf. 1. c. p. 46). In the second part we introduce another orthogonal system of sectionally linear functions χo(x), …, χn(x) which leads to a particularly simple interpolation formula. These functions appeared, mutatis mutandis, in the author's study on sectionally linear functions over an infinite range about which a report will be given elsewhere.

Let H = (Hi, j) (1 ≦ i, j ≦ n) be an nk × nk matrix with complex coefficients, where each Hi, j is itself a k × k matrix (n, k ≧ 2). Let |H| denote the determinant of H and let ∥H∥ = |(|H i, j|)| (1 ≦ i, j ≦ n ). The purpose of this note is to prove the following theorem.

Theorem. If H is positive definite Hermitian then |H| ≦∥H∥. Moreover, |H| = ∥H∥ if and only if Hi, j = 0 whenever i ≠ j.

The case n = 2 of this theorem is contained in [1].

A subset K of a lattice is said to be directed if for any a, b∊K there is c∊K with c ≥ a, b. A complete lattice L is called upper continuous if for every directed subset (aα) and every element b.

The following is a slight improvement of [4; Anmerkung 1. 11, p. 11].

On a recent visit to Toronto, Professor Riesz made an interesting remark on the invariance of a certain cross-ratio, I wish to present a simple proof of his result.

Let x = ( x1, x2, x3, x4), … denote homogeneous coordinates in real projective three-space; x(t), … are differentiable curves in that space.