Nous appellerons groupoïde un ensemble non vide, G, muni d'une loi (×) faisant correspondre à tout couple ordonné x, y ∊ G, au plus un élément z de G, appelé produit de x par y, et satisfaisant à la loi d'homogénéité (2).

A pyramid clearly has all its projections closed, even when the line segments from vertex to base are extended to infinite half-lines. Not so a circular cone. For if the cone is on its side and supported by the (x, y) plane in such a way that its infinite half-line of support coincides with the positive x axis, then its horizontal projection on the (y, z) plane is the open upper half-plane y > 0, together with the single point (0, 0).

The purpose of this paper is to establish the following theorems, which were obtained by Hopf and Voss in their joint paper (2) for the case where n = 2.

THEOREM 1. Let Vn, V*n be two closed orientable hyper surfaces twice differ entiably imbedded in a Euclidean space En+l of dimension n + 1 ≥ 3.

The principal result of this paper is the representation of the Mathieu group M 23 as a group of 11 × 11 matrices over the Galois Field GF(2). This is a new representation of M 23 and in §5 an indication of how the techniques of this result might be extended to the Mathieu group M 11 is given.

It is our purpose in this paper to present certain aspects of a cohomology theory of a ring R relative to a subring S, basing the theory on the notions of induced and produced pairs of our earlier paper (2), but making the paper self-contained except for references to a few specific results of (2). The cohomology groups introduced occur in dual pairs. Generic cocycles are defined,and the groups are related to the protractions and retractions of R-modules.

In his lecture at the University of Kyoto on September 23, 1955, Professor Artin gave an important theorem on Noetherian rings, which seems to have not a few interesting consequences. It is the purpose of our present note to point out one of them. We begin by quoting a special case of the theorem.

1. Introduction. In a recent paper (1) it was remarked that the theory of zero-dimensional spaces is exactly that part of general topology which can be described in terms of equivalence relations. Here, it will be shown how this idea can be used to obtain the following characterizations of certain types of zero-dimensional spaces:

Any compact zero-dimensional space which has a denumerable basis for its open sets and is dense in itself is homeomorphic to the space of 2-adic integers.

1. Introduction. Let be a Boolean ring of at least two elements containinga unit 1. Form the set of matrices A, B, … of order n havingentries aiJ , bij , … (i, j = 1, 2, …, n), which are members of . A matrix U of is called unimodular if there exists a matrix V of such that VU= I, the identity matrix. Two matrices A and B are said to be left-associates if there exists a unimodular matrix U satisfying UA = B.

Let F be a field and let V be a finite dimensional vector space over F which is also a module over the ring F[a]. Here a may lie in any extension ring of F. We do not assume, as yet, that V is a faithful module, so that a need not be a linear transformation on V. It is known that by means of a decomposition of V into cyclic F[a]-modules we may obtain a definition of the characteristic polynomial of a on V which does not involve determinants.

If n is a non-negative integer, define pr (n) as the coefficient of xn in

;

otherwise define pr (n) as 0. In a recent paper (1) the author has proved that if r has any of the values 2, 4, 6, 8, 10, 14, 26 and p is a prime > 3 such that r(p + 1) ≡ 0 (mod 24), then

1 , ,

where n is an arbitrary integer.

All matrices considered here have rational integral elements. In particular some circulants of this nature are investigated. An n × n circulant is of the form

The following result concerning positive definite unimodular circulants was obtained recently (3 ; 4 ):

Let C be a unimodular n × n circulant and assume that C = AA' where A is an n × n matrix and A' its transpose. Then it follows that C = C1C1', where C1 is again a circulant.

Let F denote the Galois field GF(pr) with pr elements, where p is an odd prime and r is a positive integer. Suppose further that m and n are arbitrary elements of F and that αi, βi pt (i = 1, …, s) are nonzero elements of F. The purpose of this paper is to evaluate the function N s(m, n), defined, for an arbitrary positive integer s, to be the number of simultaneous solutions in F of the equations

1.1 .

Let M be the normed linear space whose general element, x, is a bounded sequence

of real numbers, and ‖x‖ = l.u.b. |ξn |. Let T denote the linear operation (of norm 1) defined by Tx = (ξ2, ξ3, … , ξn+1,…). A generalized limit is a linear functional ϕ on M which satisfies the conditions

.

We extend some observations of Popken (2) on the algebraic foundations of the theory of asymptotic series. The main result is the theorem in §5 which characterizes, for a particular function space, a class of linear functionals defined in §4. In §3 we discuss another class of linear functionals related to asymptotic series. In the first two paragraphs we give definitions which render this note self-contained.

In the paper (5), Ward defines an integral of Perron type of a finite function f with respect to another finite function g, where g need not be of bounded variation. There arise two problems, (a) and (b) below, that have not been dealt with in (5).

If f = j at a countable number of points everywhere dense in (a, b), where f and j are both integrable with respect to g, then f — j can be nonzero on a large set of points of (a, b).

If a < u < b and n > 0 then

(1)

is a so-called improper integral owing to the infinity in the integrand at x = u. When n = 0 we have associated with (1) the well-known Cauchy principal value, namely

(2) .

Hadamard (1, p. 117 et seq.) derives from an improper integral an expression which he calls its finite part and which, as he shows, possesses many important properties.

This paper is concerned with the Fresnel integrals

1.1

in the complex domain.

Recent research work in different fields of physical and technical applications of mathematics shows that an increasing number of problems require a detailed knowledge of elementary and higher functions for complex values of the argument.

In the study of approximate methods for solving ordinary differential equations, an interesting question arises. To state it roughly for a single first order expression, let y 0(t) be the solution of the equation

(1.1)

which satisfies the initial condition y(a) = n a. Let n b be an approximation to the value of y 0 at a later time, t = b.

In the theory of hyperbolic differential equations a mixed boundary value problem involves two types of auxiliary conditions which may be described as initial and boundary conditions respectively. The problem of Cauchy, in which only initial conditions are present, has been studied in great detail, starting with the early work of Riemann and Volterra, and the well-known monograph of Hadamard (4). A modern treatment of great generality has been given by Leray (7).

This paper may be regarded as a sequel to (1), where the initial value or Cauchy problem for harmonic tensors on a normal hyperbolic Riemann space was treated. The mixed problems to be studied here involve boundary conditions on a timelike boundary surface in addition to the Cauchy data on a spacelike initial manifold. The components of a harmonic tensor satisfy a system of wave equations with similar principal part, and we assign two initial conditions and one boundary condition for each component.