It is well known that a module M has finite length if and only if it is semi-artinian and Noetherian or, equivalently, semi-noetherian and artinian. Our main result shows that finite length is often achieved by just assuming that M is semi-artinian, semi-noetherian and has finitely generated socle.

In a paper read before the Research Branch of the Royal Statistical Society (Ref. 1, p. 150) the following case was considered:

Let the expression be given; introduce, for c, a linear form in and obtain

If the yi are sample values from a normal population with unit variance, then it is known (Ref. 2) that (1) is distributed as where zi varies as chi-squared with one degree of freedom and the li are the latent roots of the matrix of the quadratic form. If these latent roots are f times unity and n—f times zero, then this reduces to a chi-squared distribution with f degrees of freedom.

We prove inequalities for convex functions, Lp norms, and sums of powers. Our results sharpen recently published inequalities of C. E. M. Pearce and J. E. Pečarić.

This curious device is figured at p. 149 of De Morgan's Budget of Paradoxes, where it is described as a “hollow semi-cylinder, but not with a circular curve,” revolving on pivots. The form of the cylinder is such that, whatever quantity of oil it may contain, it turns itself till the oil is flush with the wick, which is placed at the edge.

In a recent investigation of a conjecture on an upper bound for permanents of (0, 1)-matrices (2) we obtained some inequalities involving the function (r!)1/r which are of interest in themselves. Probably the most interesting of them, and certainly the hardest to prove, is the inequality

where ø(r = (r!)1/r. In the present paper we prove (1) and other inequalities involving the function ø(r.

Let f(x) and F(x) be polynomials which are supposed to be decomposed into a series of n terms as

Further let θ be a distributive operation and φ(x) be a given function. Then the functional equation

has a solution of the form

where yν (x) (ν = 1, 2, … n) is the solution of the equation

We consider a subclass of the Dirichlet series studied by Chandrasekharan and Narasimhan in (1). Our objective is to generalize some identities due to Landau (3) concerning r2(n), the number of representations of the positive integer n as the sum of 2 squares. We shall also give a slight extension of Theorem III in (1).

A graph in which each line is designated as either positive or negative is called a signed graph S. The sign of a cycle in S is a product of the signs of its lines. A signed graph in which every cycle is positive is called balanced. This concept was introduced by Harary in (3) and the following characterisation of balanced signed graphs was given.

It is well known that the real, skew-symmetric, non-singular, bilinear forms of n + n variables have no invariants. In fact, each of these forms may be transformed into one and the same form, for instance into the one which occurs in the usual representation of the complex group. The standard proofs of this theorem break down in case of infinite forms which are bounded in the sense of Hilbert, one of the impediments being the possibility of a continuous spectrum. The object of this note is to show that, while the usual proofs break down, the theorem itself is true in Hubert's case also.

The concept of “root vectors” is investigated for a class of multiparameter eigenvalue problems

where operate in Hilbert spaces Hm and . Previous work on this “uniformly elliptic” class has demonstrated completeness of the decomposable tensors x1 ⊗…⊗ xk in a subspace G of finite codimension in H=H1 ⊗…⊗ Hk, but questions remain about extending this to a basis of H. In this work, bases of elements ym, in general nondecomposable but satisfying recursive equations of the type are constructed for the “root subspaces” corresponding to λ∈ℝk.