This note proves two propositions on infinite doubly stochastic matrices, both of which already appear in the literature: one with an unnecessarily sophisticated proof (Kendall [2]) and the other with the incorrect assertion that the proof is trivial (Isbell [l]). Both are purely algebraic; so we are, if you like, in the linear space of all real doubly infinite matrices A = (aij).

Proposition 1. Every extreme point of the convex set of ail doubly stochastic matrices is a permutation matrix.

Kendall's proof of this depends on an ingenious choice of a topology and the Krein-Milman theorem for general locally convex spaces [2]. The following proof depends on practically nothing: for example, not on the axiom of choice.

Let there be given two sets of points, P = {P1,…, Pm}1 m and Q={Q1,…, Qn}, such that joining each pair of points (Pi, Qk), for i=1,…, m and k=1…, n, is a line oriented towards one, and only one, of the pair. Such a configuration will be called an m×n bipartite tournament. If the line joining Pi to Qk is oriented towards Qk we may indicate this by pi→Qk, and similarly if the line is oriented in the other sense. The points Pi, Pj, Qk, and Ql. will be said to form a 4-cycle if either Pi→Qk→Pj→Ql→Pi or Pi→Ql→Pj→Qk→Pi. C(m, n), the number of 4-cycles in a given m×n bipartite tournament, provides, in some sense, a measure of the degree of transitivity of the relationship indicated by the orientation of the lines, and the complete configuration may be thought of as representing the outcome of comparing each member of one population with each member of a second population, and making a decision, upon some basis, as to which component of each pair is the preferred one.

In (4), pp. 25−27, Coxeter divided the regular maps on a surface of genus 1 into three infinite families. They are:

1. (i) Maps of type {4, 4}.

2. (ii) Maps of type {6, 3}.

3. (iii) Maps of type {3, 6} (the duals of (ii)).

We consider the family (iii). By adjoining an element to the group of any map in (iii) we shall derive the group of a regular map of type {6, 6}. Thus we produce a 1−1 correspondence between the members of the family (iii) and of the new family. Corresponding members in the two families have certain properties in common, the most interesting of which is the property of reflexibility. Our results are summarized in Theorems 1 and 2.

S. M. Nikolski [4, Theorem 1; cf. 3, esp. pp. 144 and 148] considered the remainder term in the approximation by the n-th Fejér mean, σn(x), to a function, f(x), of period 2π satisfying a Lipschitz condition of order α, 0<α≤1. In this connection, he introduced the quantity

1

where the maximum is taken over all x and the supremum is taken over all functions of period 2π, bounded by 1 (a notational convenience only) and satisfying a Laps chitz condition of order α.

Soient un anneau A et un sous-ensemble (non vide) B de A. L'anneau A est dit:

Extension radicale de B, si pour tout a ∊ A, il existe un entier n(a)>0, dépendant de a, tel que an(a)∊B.

Extension quasi-radicale de B, si pour tout a ∊ A, il existe un entier n(a)>1, dépendant de a, tel que an(a)-a∊B.

En imposant certaines conditions à A ou B, on peut alors obtenir certains renseignements sur la structure de A. Dans le cas où B est un sous-anneau de A, plusieurs travaux ont été consacrés récemment à ce sujet. Notamment, on a cherché quelles conditions, imposées à A ou B, entraînent la commutativité de A. (Voir en particulier [2], [3], [4], [5] et [6].)

Let B be a Boolean algebra and let ℳ and n be two systems of subsets of B, both containing all finite subsets of B. Let us assume further that the join ∨M of every set M∊ℳ and the meet ∧N of every set N∊n exist. Several authors have treated the question under which conditions there exists an isomorphism φ between B and a field δ of sets, satisfying the conditions:

Let xi, yi (i=1,2,…, t) and n be non-negative integers. A function (n; x1, …, xt) may be defined recursively as follows: let(0;0,…, 0)=1 and

1

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.

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.