In a paper recently published in this journal [1], R. E. Graves proved a closure criterion for orthonormal sets of functions. A refined form of it may be stated as follows :

The boundary value problems considered in this paper relate to harmonic p-tensors on Riemannian manifolds with boundary. We study the equation of Beltrami-Laplace

Let A be a closed subset of a topological space. We show that the following three conditions are equivalent.

1.1 Any countable neighbourhood-finite open covering of A (the topological terms referring to the relative topology of A) has a refinement which can be extended to be a countable neighbourhood-finite open covering of X.

The present paper is a sequel to one I published lately (3) on regular surfaces of genus 3, and like it, is intended to fill up some of the gaps in our detailed knowledge of the regular surfaces of moderately low genus p = pg = pa and linear genus p(1) = n + 1 > 1. (The surfaces for which p(1) = 1 form a rather separate field of study on which a good deal of work has been done, and I shall not consider them.)

In a recent paper, Kaplansky [2] proved the following theorem: Let R be a ring with centre Z, and such that xn(x) ∈ Z for every x∈ R. If R, in addition, is semi-simple then it is also commutative

The existence of non-commutative rings in which every element is nilpotent rules out the possibility of extending this result to all rings. One might hope, however, that if R is such that xn(x) ∈ Z for all x ∈ R and the nilpotent elements of R are reasonably “well-behaved,” then Kaplansky's theorem should be true without the restriction of semi-simplicity.

Formerly Hua [1] proved that if A is a division ring with centre Z and if there exists a natural number n such that an ∈ Z for every a ∈ A, then A is commutative; this generalizes Wedderburn's theorem on finite division rings. Another generalization of Wedderburn's theorem, due to Jacobson [3], asserts that every algebraic division algebra over a finite field is commutative. On the other hand, a theorem of Noether and Jacboson [3] states that every noncommutative algebraic division algebra contains an element which is not contained in the centre Z and is separable over Z.

Let 5 be a set of positive integers no three of which are in arithmetical progression, i.e., if A, B, C are distinct elements of S, A + B ≠ 2C. We call such a set a non-averaging set. Let v(n) denote the maximum number of elements not exceeding n in any non-averaging set. The problem of finding bounds for v(n) has been treated by several authors [1, 3, 5, 6, 7]. The question first arose in connection with a theorem of van der Waerden [8].

The question of the irrationality of functions defined by power series, for rational values of the variable, has attracted much attention for over a hundred years. Legendre, in generalizing Lambert's proof of the irrationality of tan x for rational x, proved an important theorem on the irrationality of continued fractions with integer elements. Here we use Legendre's theorem (Lemma 3) to prove that at least one of a certain pair of power series is irrational whenever the variable is rational and satisfies a further condition. We prove the following:

The consideration of relative extrema to correspond to the absolute extremum which is the critical lattice has been going on for some time. As far back as 1873, Korkine and Zolotareff [6] worked with the ellipsoid in hyperspace (i.e., with quadratic forms), and later Minkowski [8] worked with a general convex body in two or three dimensions. They showed how to find critical lattices by selection from among a finite number of relative extrema. They were aided by the long-recognized premise that only a finite number of lattice points can enter into consideration [1] when one deals with lattices “admissible to convex bodies.”

This paper is the first in a series dealing with Banach spaces L whose elements are functions on a measure space S. If W is a family of non-negative weight functions wα we sometimes write LWp when the norm is given as

Let f(x) be real valued, bounded and, integrable in the sense of Riemann on the interval X = (0 < x < 1), with the value of its integral over X equal to one. For brevity we call such a function admissible.The symbol Xnk will always denote the interval an arbitrarily chosen point of Xnk and δ any specified set of intermediate points

Einstein, in his Generalized Theory of Gravitation [I], deals with a tensor

whose ninety-six independent components are complex-valued functions of four real variables xα. Schouten [4, p. 261 (89)] has decomposed the general relative tensor, antisymmetric in α, β, γ, and in ρ,σ, into five irreducible parts. This decomposition can be applied to the R of Einstein if we define the tensor v by

The Jacobi polynomials may be defined by

where (a)n = a (a + 1) … (a + n — 1). Putting β = α gives the ultraspherical polynomials which have as a special case the Legendre polynomials .

Let

Then if p is a prime > 3, Glaisher [4] proved

Using some known results on Gauss sums in a finite field, it is shown that the sum (1.3) defined below can either be evaluated explicitly or expressed in terms of a Kloosterman sum. The same result applies to the more general sum S(α, λ, Q) defined in (5.1). The latter sum also satisfies the reciprocity formula (5.5). Some related sums are discussed in §§6, 7.

The fact that the symmetric difference is a group operation in a Boolean algebra is, of course, well known. Not so well known is the fact observed by Ellis [3] that it possesses some of the desirable properties of a metric distance function. Specifically, if * denotes this operation, it is easy to verify that

McCoy, following Frobenius, studied a problem which can be described as follows. Let k be an arbitrary field, kc its algebraic closure, and any algebra of n ⨯ n matrices over k which contains the identity I . Define a canonical ordering to be a set of n mappings λ of , or of a subset of , into kc such that the sequence λ1(A),λ2(A), …, λn(A), for each A ∈ , consists of the characteristic values (roots of det(A — xI) = 0) of A, each with the right multiplicity. Define a canonical ordering to be a Frobenius ordering if, for all non-commutative polynomials f(x1, x2, … , xm) and all finite subsets A1, A2, …, Am of ,

In [2], we derived some character relations of the symmetric group Sn. These relations were also obtained in [1] independently. In the present paper, we shall study the properties of these character relations in some detail. In the last section, using a result obtained in [3], we shall further determine the number of modular irreducible representations in a p-block of Sn.

It has been shown by Diederichsen [2] that for integral representations of a finite group, the irreducible constituents in any complete reduction are not necessarily unique up to order and unimodular equivalence. In this same article, it is shown that for certain finite groups, such as the cyclic group of order 4, there are infinitely many classes of indecomposable representations under unimodular equivalence.