Let E be a given field and G some (not necessarily the total) group of automorphisms of E. We introduce a topology in G by saying that a net of elements Tα in G converges to T in G if for each x∈E, Tα(x) coincides with Tα(x) ultimately. We shall call this the Krull topology on G.

The Fourier transform F(y) of a function f(t) in L1(Ek) where Ek is the k-dimensional cartesian space will be defined by .

We prove that factor groups of cartesian powers of finite non-abelian simple groups cannot be countably infinite. Thisis not our main result, but it had been our original aim. The proof is based on a similar fact concerning σ-complete Boolean algebras, and on a representation of certain subcartesian powers of a group in its group ring over a Boolean ring. This representation, to which we give the name “Boolean power”, will be our central theme, and we begin with it.

The object of this note is to show that under suitable restrictions some results on the wreath product of groups can be carried over to topological groups. We prove in particular the following analogue of the well-known theorem of Krasner and Kaloujnine (see for example [2] Theorem 3.5): Theorem. Let A and B be two locally compact topological groups, and let (C, ε) be an extension of A by B. If there exists a continuous left inverseof ε, that is to say a continuous mapping τ: B → C such that re is the identity on B, then there exists a continuous monomorphism of C into the topological standard wreath product of A by B.

It is shown algebraically that the full defining relation for the σκ(img)ν is effectively contained in its symmetric part.

In 1927 Schreier [8] proved the Nielsen-Schreier Theorem that a subgroup H of a free group F is a free group by selecting a left transversal for H in F possessing a certain cancellation property. Hall and Rado [5] call a subset T of a free group F a Schreier system in F if it possesses this cancellation property, and consider the existence of a subgroup H of F such that a given Schreier system T is a left transversal for H in F.

Let ℒ V denote the algebra of all linear transformations on an n-dimensional vector space V over a field Φ. A subsemigroup S of the multiplicative semigroup of ℒ V will be said to be an affine semigroup over Φ if S is a linear variety, i.e., a translate of a linear subspace of ℒ V.

This concept in a somewhat different form was introduced and studied by Haskell Cohen and H. S. Collins [1]. In an appendix we give their definition and outline a method of describing possibly infinite dimensional affine semigroups in terms of algebras and supplemented algebras.

1. P. Heywood [3] proved the following theorems:

Theorem A. If 0 ≦ λ < 2, if xy−1g(x) ε L(0, π), and if

for n = 1, 2, 3,…, then the seriesis convergent.

Theorem B. If 0 ≦ λ < 1, if xy−1f(x)ε L(0, π), and if

for n = 1, 2, 3,…, then the seriesn−yanis convergent.

In this paper we shall be concerned with the set Mn(B) of n x n matrices whose elements belong to a given Boolean algebra B(≤, ∪,∩,').

It is well known that Mn(B) also forms a Boolean algebra with respect to the partial ordering ≼ defined by

in which union (⋎), intersection (⋏) and complementation (*) are given by

A solution is obtained of the problem of diffusion from an elevated point source into a turbulent atmospheric flow over a horizontal ground z = 0. The mechanism of the turbulence is the one considered by D. R. Davies [1[ when he obtained a solution of the same problem in the case when the source is at ground level, the specifications for the mean wind velocity V, and for the across-wind diffusivity Ky, and for the vertical diffusivity Kz, being V αzm, Ky αzm, Kz αz1−m, where m is a constant. Predictions of the solution are that the maximum concentration at ground level of the diffusing matter varies inversely as h1·8 in adiabatic conditions, where h is source height, and that the distance downwind from the source to the point where this maximum concentration is attained varies, in these conditions, as h1·3.

Green [8] has shown that a constitutive relation of the form

arises as a special case of an incompressible anisotropic simple fluid, where S is the stress tensor or matrix,

and V is the velocity gradient matrix at time t, all measured in a fixed rectangular cartesian coordinate system. Also, if F is the displacement gradient measured with respect to some curvilinear reference system θi, then

where R is a proper orthogonal matrix, and M and K are positive definite symmetric matrices. In addition

and

1. Let p be a prime, t a positive integer, and ϕ a cubic polynomial with integral coefficients, and an integral constant term. We study the congruence

because of its obvious connection with the equation

No systematic study seems to have been made of so natural a question as the analogue for matrices of quadratic residues. One generalization of x2 (x an integer) is X2 (X an integral matrix). Another is X′ X, where the prime means “transpose”. We study here the solvability for X of the congruence

where p is a prime, r ≥ 1; I (the identity matrix) and X are n-by-n; and a is an integer not divisible by p2.

Let N be a natural number, and let be a non-empty subset of the set of integers

Let Z be the number of elements of . We denote by Z(p, h) the number of elements of falling into the congruence class h modulo p, so that

The two examples of fluid motion in a container which are described in this paper can be easily demonstrated in any kitchen. The first motion was noticed by Professor C. A. Rogers while attempting to dissolve chlorine tablets in water to improve its drinkability. The water nearly filled a cylindrical jar and he had shaken it, with the axis of the jar horizontal, in such a way that the water had a considerable angular momentum about the axis. When the axis of the jar was suddenly moved into the vertical position, he noticed that the water was now rotating about the vertical, which prompted the question of the source of this vertical component of angular momentum. A simplified version of this motion is determined mathematically in §2, and the observations are found to be in general agreement with the theoretical prediction.

For a subset S of a real linear space, let ck S denote the set of all points from which S is starshaped; that is p∈ ck S if and only if S contains the segment [p, s] for all s∈ S. The set ck S, which is necessarily convex, was introduced by H. Brunn [2[ in 1913 as the Kerneigebeit or convex kernel of the set S. Of course ck S = S if and only if the set S itself is convex. L. Fejes Tóth asked for a characterization of those plane convex bodies which can be realized as the convex kernels of nonconvex plane domains, and it was proved by N. G. de Bruijn and K. Post that every plane convex body can be so realized. Here we establish a stronger result.