1. Introduction. Several recent papers have dealt with the number of solutions of xd = 1 in Sn, the symmetric group of degree n. Let us denote this number by An,d and let An,2 = Tn.

Introduction. In this paper we study the (discrete) group ring Γ of a finitely generated torsion free nilpotent group over a field of characteristic zero. We show that if Δ is the ideal of Γ spanned by all elements of the form G − 1, where G ∈ , then

and the only element belonging to Δw for all w is the zero element (cf. (4.3) below).

The algebras considered here arose in the investigation of an algebra connected with the orthogonal group. We consider an algebra of dimension mn over a field K of characteristic zero, and possessing a basis {eij} (1 ≤ i ≤ m; 1 ≤ j ≤ n) with the multiplication property

1,

The field elements ϕij form a matrix Φ = (ϕij) of order n × m. It will be called the multiplication matrix of the algebra relative to the basis {ϕij}.

Matrices with real quaternion elements have been dealt with in earlier papers by Wolf (10) and Lee (4). In the former, an elementary divisor theory was developed for such matrices by using an isomorphism between n×n real quaternion matrices and 2n×2n matrices with complex elements. In the latter, further results were obtained (including, mainly, the transforming of a quaternion matrix into a triangular form under a unitary similarity transformation) by using a different isomorphism.

Several authors (1; 2; 3; 4; 5; 6) have recently studied the existence and non-existence of certain types of extensions of a given field. In this note we prove a theorem closely related to these results which, in a sense, contains essential portions of each of these.

In algebraic geometry it is of interest to examine polynomial surfaces F which transform into themselves under the collineation T defined by:

where Ep = 1, and p is a prime (2). One of the most obvious ways to ensure invariance of a surface is for each term X1ax2bX3cX4d of F to go into itself. We present initially, therefore, a theorem which will be useful in the study of such termwise invariance for polynomials of composite degree.

1. Introduction. Let T be an orientation-preserving homeomorphism of period two of the 3-sphere S3 onto itself; further let T be different from the identity and have at least one fixed point. It has been shown by Smith (8, p. 162) that the set F of all fixed points of T is a simple closed curve. However, very little is known about the position of F in S3.

Introduction. In his article (22), de Possel laid the foundations for an abstract theory of differentiation of set functions, the term “abstract” being meant in the sense of Fréchet-Nikodym, that is, without reference to a euclidean, metric, or topological background. In 1.1, we adopt, substantially, his notion of derivation basis. De Possel considered two Vitali properties for a derivation basis.

1. Introduction. An interesting feature of recent topological developments is the increasingly important role played by locally finite coverings. Point-finite coverings, on the other hand, even though conceptually simpler, have received very little attention. And deservedly so, since they are much less useful. Nevertheless, it sometimes happens (as it did to the author in (5)) that one is confronted by a covering which is known to be point-finite, but not necessarily locally finite.

1. Introduction. Consider the (vector, n-component) system of differential equations

1,

where f(x) is of class C1. Let Ω denote a set of points, x, consisting of unrestricted solution paths x(t), so that the x(t) exist and lie in Ω for — ∞ < t < ∞.

1. Introduction. In a previous paper (3) the study of one-dimensional, unsteady flows, isentropic or anisentropic, was reduced to the integration of a Monge-Ampère partial differential equation

(1)

For a polytropic gas, the specific volume

(2) ,

takes the form

,

once the entropy distribution function S = S(Ψ) is specified.

Introduction. It is the purpose of this paper to develop a Lebesgue theory of integration of scalar functions with respect to a countably additive measure whose values lie in a Banach space. The class of integrable functions reduces to the ordinary space of Lebesgue integrable functions if the measure is scalar valued. Convergence theorems of the Vitali and Lebesgue type are valid in the general situation. The desirability of such a theory is indicated by recent developments in spectral theory.

1. Introduction. Our problem is related to the construction of geographical maps as follows. The reason for using conformai geographical maps is that the scale (viz., the ratio of two corresponding line-elements) does not depend on the direction of these line-elements. In an ideal map the scale, being responsible for the preservation of shape, would also be independent of the points where the line-elements begin.

The spaces Δ (α) and M(α) were defined by Lorentz (2) as follows. Let 0 < α < l , 0 < l ≤ ∞ ; let ϕ be measurable on (0,l), and, in case l = ∞, let the set where |ϕ(x)| > ∊ have finite measure for each positive ∊.

1. Introduction. One of the striking results of the work done by Murray and von Neumann (9) in the analysis of rings of operators on a Hilbert space is the reduction of the unitary equivalence problem for certain types of factors to the problem of algebraic equivalence. Roughly speaking, they associate with each concrete representation of a factor a number (which measures the relative size of the factor and its commutant)—the so-called “coupling constant.”

1. Reducible diophantine equations. The present paper will provide a general method for obtaining the complete parametric representation for the rational integer solutions of the multiplicative diophantine equation

1.1

for some specified range of k, where the aijk,bijk are non-negative integers and the fki, hki are decomposable forms, that is to say they are integral irreducible homogeneous polynomials over the rational field R of degree k in k variables which can be written as the product of k linear forms.

1. Introduction. Several authors have proved theorems of the following type:

Let x0, y0 be any real numbers. Then for certain functions f(x, y), there exist numbers x, y such that

1.1 x ≡ x0, y ≡ y0 (mod 1),

and

1.2 .

The first result of this type, but with replaced by min , was given by Barnes (3) for the case when the function is an indefinite binary quadratic form. A generalisation of this was proved by elementary geometry by K. Rogers (6).

This paper is concerned with the numbers which are relatively prime to a given positive integer

where the p's are the distinct prime factors of n. Since these numbers recur periodically with period n, it suffices to study the ϕ(n) numbers ≤n and relatively prime to n.

1. Introduction. On recent occasions papers have been presented concerned with the problem of solving a system of linear congruences. Apparently the authors were not aware that this problem was solved very neatly and completely a long time ago by H. J. S. Smith (5; 6).

1. Introduction. The use of incomplete block designs for estimating and judging the significance of the difference of treatment effects is now a standard statistical technique. A special kind of incomplete block design is the Partially Balanced Incomplete Block Design (PBIBD) introduced in (3).