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.

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

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.

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.

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

The material sketched here is mostly well known and concerns the geometrical inter-relations of vectors in a Hilbert (that is, complete inner product) space. The discussion and references are (obviously) not exhaustive but I hope the reader will find here some interesting problems.

We shall permit the scalars to be real, complex or quaternionic numbers. The inner product of vectors x and y will be denoted (x|y).

A sequence {un}, n=0, 1, 2, 3,… is said to be an integral recurrence of order r if the terms satisfy the equation

for n=r+1, r+2,…, and a1, a2,…, ar are integers, ar≠0. In this case we will say that {un} satisfies the relation [a1, a2,…, ar]. The sequence {un} is uniquely determined when u1, u2,…, ur are given specified values. If u1, u2,…, ur are integers all the terms of {un} are integers. The generating function f(t)=u1t + u2t2+… takes on the form where Q(t) depends on the values of u1, u2,…, ur and R(t)=tr-a1tr-1-a2tr-2-…-ar. We will refer to R(t) as the characteristic polynomial of the recurrence.

This note concerns a countably additive measure on a Boolean ring of subsets of an abstract set, this measure being real-valued, admitting ∞ as a possible value. We are interested only in unique extensions, so we suppose the measure to be σ- finite. The following well known result will be referred to as the "extension theorem": "Every σ-finite measure on a ring extends uniquely to a σ-finite measure on the generated σ-ring. "Besides the familiar proof using outer measure, there is a Borel-type proof using transfinite induction [4]. We attempt here to reduce the Borel-type proof to its ultimate simplicity, reducing the problem to the bounded case.

We shall obtain a little more general formula than the one given in the problem P44.

Put

1

First of all we shall prove the identity

2

where Cmi are constants whose values will also be determined