This paper is concerned with undirected graphs which may be infinite and may contain multiple edges. The Axiom of Choice is assumed. The terms path, infinite path and circuit are used in the same sense as Weg, unendlicher Weg and Kreis, respectively, are used in D. Konig's book [1]. The valency of a vertex is the number of edges incident with it.The length of a path is the number of edges in it. The following theorem is a generalization of the well known fact that if a vertex of a graph is not a cut-vertex (Artikulation [2]) and has valency ≧2, then the graph contains at least one circuit to which the vertex belongs.

Let L be a complemented, χ-complete modular lattice. A theorem of Amemiya and Halperin (see [l], Theorem 4.3) asserts that if the intervals [O, a] and [O, b], a, bεL, are upper χ-continuous then [O, a∪b] is also upper χ-continuous. Roughly speaking, in L upper χ-continuity is additive. The following question arises naturally: is χ-completeness an additive property of complemented modular lattices? It follows from Corollary 1 to Theorem 1 below that the answer to this question is in the negative.

A complemented modular lattice is called a Von Neumann geometry or continuous geometry if it is complete and continuous. In particular a complete Boolean algebra is a Von Neumann geometry. In any case in a Von Neumann geometry the set of elements which possess a unique complement form a complete Boolean algebra. This Boolean algebra is called the centre of the Von Neumann geometry. Theorem 2 shows that any complete Boolean algebra can be the centre of a Von Neumann geometry with a homogeneous basis of order n (see [3] Part II, definition 3.2 for the definition of a homogeneous basis), n being any fixed natural integer.

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.

Un anneau partiellement ordonné A est un anneau sur lequel est définie une relation d'ordre partiel telle que:

1. 1. a≥b entraîne a+c≥+c pour tout c∈A.

2. 2. a≥o et b≥o entraînent ab≥o.

Si la relation d'ordre est une relation d1 ordre total, l'anneau A est dit un anneau totalement ordonné.

Il est bien connu (Théorème d'artin-Schreier [l]) que, pour qu'un corps commutatif K puisse être totalement ordonné, il faut et il suffit que la relation entraîne a1=…=an=o. Ce théorème a été généralisé par T. Szele [4] qui a montré que, pour qu'un corps quelconque K puisse être totalement ordonné, il faut et il suffit que le demi-groupe additif et multiplicatif S, engendré par les éléments de K qui sont des carrés non nuls, ne contienne pas I'élément zéro de K. Ce résultat a été étendu au cas d1 un anneau d'intégrité par R.E. Johnson [2] de la manière suivante.

By a result due to Tricomi(1), it is known that the La guerre polynomials have an expansion in terms of the Bessel functions, Jv(z), of the form

1.1

where the coefficients Am are determined by

1.2

and

1.3

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.

Let A be an n by n matrix with non-negative entries. If a permutation matrix P exists such that P-1 is of the form

where A and C are square matrices and O is a zero-matrix then A is said to be reducible. Otherwise, A is irreducible. If A is irreducible, then A is said to be primitive if there is an integer k such that Ak0, i.e., Ak has no zero entries. If A is primitive, the least integer m for which Am0 is called the exponent of A. In (4), Wielandt has shown that for n by n primitive matrices the exponent is at most (n-1)2+1. In this paper, the theory of directed graphs is used to determine conditions under which the exponent is less than (n-1)2+1. The following results are obtained. If A contains r non-zero entries along its main diagonal then its exponent is at most 2n-r-1, and this result is the best possible. If all the diagonal entries of A are zero but its graph KA (see next section for definition) contains a cycle of length d, then the exponent of A is at most d(2n-d-1). For the case where this improves Wielandt's result.

P. Scherk a posé le problème suivant (Canadian Mathematical Bulletin, vol.4, no. 3, 1961, p. 306, problème 49):

Let x(t) and y(t) be two differentiable functions defined in some interval. Does the function have the Darboux property?

La réponse est négative. En effet, Witold Wilkosz (Some properties of derivative functions, Fundam. Math. vol.2, 1921, p. 145−154) a montré (p. 146−147 loc. cit.) que la fonction

The object of this paper is to give simple criteria for the existence of direct limits in categories and for the permuting of a functor with direct limits.

The notion of direct limit of a diagram that we shall use here is essentially that of Kan (4), which is more general than the usual notion of direct limit of a directed diagram.

Our treatment is based on the fact (Lemma 2) that the usual process for constructing the direct limit of a diagram of modules, which consists in taking a direct sum of the modules in the diagram and then considering a certain homomorphic image of this direct sum (3, p. 220), is essentially, once certain notions have been properly generalized, the only process for constructing the direct limit of any diagram in any category.

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.

It is well known that ℳn, the multiplicative group n consisting of n-rowed square matrices with integer entries and determinant equal to ±1 can be generated by:

It is also known that U4 can be generated by U1, U2, and U3 (cf.[2] p.85).

However, by a construction which is much simpler than the one just mentioned for U4, it is possible to generate U3 by just U2 and U4. Since U2 and U4 affect only the first

In a recent interesting paper [1] on the stability of linear mechanical systems S.P. Diliberto discusses certain reduction theorems for symmetric and skew-symmetric Hamiltonian matrices with respect to symplectic orthogonal similarity. In what follows it is shown that a unified simple argument will handle both of these problems together and then a s an example it is indicated how the argument can be also used to obtain the reduction theorem for symplectic orthogonal matrices.

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].)

Utumi has shown [3, Claim 5.1] that for a certain class of rings the associated maximal rings of right quotients are isomorphic to the endomorphism rings of modules over division rings. We shall prove a generalization of this theorem and then show how it is obtained as a corollary. The following proofs do not depend on Utumi's paper; instead, they make extensive use of results proved in [1]. The terminology and notations employed here are the same as in [1].

I wish to thank Dr. B. Banaschewski for his suggestions and helpful criticism.

LEMMA: If J is a left ideal with zero left annihilator in a ring R then a maximal ring of right quotients of R is also a maximal ring of right quotients of J.

In a recent issue of this Bulletin a problem equivalent to the following is proposed by Moser and Pounder [1]:

If ax2+bx+c is a polynomial with real coefficients and real roots then a+b+c ≤9/4 max (a, b, c).

The object of this note is to prove the following theorems which generalise this result.

Theorem 1. Let αn be the smallest constant such that n for all polynomials

1

In this note we obtain some simple criteria which show that in certain algebraic number fields factorization of elements is not unique. All the arguments depend only on the most elementary ideas (except in §6) so that probably many of the results are not new. However the proofs are short and direct and therefore should be of some interest.

We now list some basic facts, whose proofs can be found in any of the books listed in the bibliography, at the same time fixing our notation.

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

Marshall Hall [l] shows how projective planes of very general structure may be constructed and at the same time exhibits an extensive class which are non-Desarguesian. Here we shall indicate how his method of free extension can be generalized to yield a class of planes which seem to be distinct from those which he obtains.

A partial plane is a system consisting of two distinct sets of elements, a set of "points" P, Q,… and a set of "lines" l, m,…, and a relation between these two sets, called "incidence", such that for any two distinct points, there is at most one line incident with both (or, equivalently, for any two distinct lines, there is at most one point incident with both). A partial plane is complete if every two distinct points are joined by a line and every two distinct lines intersect in a point.