Let (respectively, ) denote the set of all complex-valued infinitely differentiable functions defined on the reals whose supports are bounded on the left (respectively, right). Under addition and convolution is a commutative algebra.

The purpose of this note is to discuss the relationship between the concepts of approximate continuity and almost continuity of real functions of a real variable.

Contrary to the papers of T. Husain [1], [2], [3], we shall show that there is little relationship between these concepts, that approximate continuity does not imply almost continuity without additional restrictions and we shall give one such restriction.

Let R be a ring and let Lγ(R) be the lattice of right ideals. We define that I ∊ Lγ(R) is a prime right ideal provided that if AB⊆I for some A, B in Lγ(R) such that AI⊆I then either A⊆I or B⊆I. Any prime ideal of a ring R is a prime right ideal and if R is commutative then an ideal is prime if and only if it is a prime right ideal. If R is a ring and a∊R, let aR={x ∊ R | x=ar for some r∊R} and aR1={x ∊ R | x = na+ar for some integer n and r ∊ R}.

Let L denote a finite-dimensional simple Lie algebra over an algebraically closed field K of characteristic zero. It is well known that every finite-dimension 1, irreducible representation of L admits a weight space decomposition; moreover every irreducible representation of L having at least one weight space admits a weight space decomposition.

J. C. Burkill [1] has defined the AP-integral whose indefinite integral is approximately continuous. An (approximately continuous) function which is approximately derivable at all points of an interval is necessarily an indefinite AP-integral of its approximate derivative.

Let L be a solvable Lie algebra and A be an abelian ideal of L. For a ∊ A, let da be the (right) inner derivation of L generated by a and let exp da = 1 + da. Since A is abelian, exp da is an automorphism and exp da exp db = exp da+b for all a, b ∊ A. Let I(L, A) be the subgroup of the automorphism group of L generated by exp da for all a ∊ A.

A recent issue of the Bulletin contained a paper by Robillard [13] in which results from the theory of confounded factorial designs were used to obtain some lower bounds for Ramsey numbers. We shall derive, by more elementary methods, bounds which are much better than Robillard's in every example which he considered.

In this note we prove the following theorem, which does not seem to appear explicitly in the literature.

Theorem. Let A be a finite abelian group and p the smallest prime which divides |A|. Then there are p—1 mutually orthogonal quasigroups of order |A|, one of which is A.

A well-known embedding theorem of Banach and Mazur [1, p. 185] states that every separable Banach space is isometrically isomorphic to a subspace of C[0, 1], establishing C[0, 1] as a universal separable Banach space. The embedding theorem one encounters in a course in topological vector spaces states that every Hausdorff locally convex space (l.c.s.) is topologically isomorphic to a subspace of a product of Banach spaces.

In this paper, an elementary proof is given of the following proposition:

Theorem 1. If Qp is an arbitrary field of p-adic numbers, then it contains normal subfields Ln(2 ≤ n ≤ p) which have symmetric groups Sn as their respective Galois groups over Q, the field of rational numbers. Furthermore, each Ln may be chosen to be totally real.

In this note we prove the following

Theorem. If G is a nonunimodular locally compact group and 1<p<∞, then there is an open set, U, in G and there are functions, f simultaneously in every Lr(G), p≤r ≤∞, and g simultaneously in every Lq(G), 1 ≤q≤∞, such that the convolution, f * g(y), is not defined for any y in U.

In this paper we state and prove three theorems on positive solutions of a Volterra equation of the second kind. The equation considered is

1

where K(x, t) is a Volterra type kernel, that is, K(x, t) = 0 for t > x.

Let {Xa, a ∊ A} be a family of completely regular Hausdorff spaces, {βXa} the corresponding family of their Stone-Čech compactifications and ΠaXa the usual topological product. The following theorem was proved by Glicksberg [2] and subsequently by Frolík [1].

It has been conjectured that if p(z) is a polynomial of degree n then the product P(z)=p(z)p'(z)p"(z)…p(n-1)(z) has at least n+1 distinct zeros unless p(z)=c(z—a)n. Professor P. Erdös who mentioned this problem in a lecture at the University of Montreal attributed it to Tiberiu Popoviciu.

If I is a right ideal of a ring R, I is said to be an annihilator right ideal provided that there is a subset S in R such that

I is said to be injective if it is injective as a submodule of the right regular R-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective.

The following result is proved in [1, p. 6].

Theorem 1. Let X be a complete metric space, and let T and Tn(n = 1, 2,…)be contraction mappings of X into itself with the same Lipschitz constant k<1, and with fixed points u and un respectively. Suppose that limn → ∞ Tn(x) = T(x) for every x ∊ X. Then limn → ∞ un = u.

The purpose of this note is to give a direct proof of Koh's main theorem in [2, p. 31] applying well-known theorems of Faith and Utumi. While doing so, we shall mention a general theorem from which Koh's theorem may be obtained as a corollary.

Let A be a closed linear operator with domain DA dense in a Banach space B; DA is also a Banach space under the graph norm. By we represent the space of continuous linear mappings from B to DA and R(λ; A) = (λI—A)-1 denotes the resolvent of A; λ ∊ C (complex plane).

Let E be a Hausdorff topological vector space. A subset A of E is a polytope iff A is the convex hull of a finite number of points. In this note a necessary condition for every maximal convex subset of a subset B of E to be a polytope is given. This is related to a problem first posed by Klee [1] for compact three-cells in Euclidean 3 space.

All spaces in this note are regular, Hausdorff topological spaces.

At the topology conference in Pullman, Washington, in March 1970, E. Michael posed the following question: if X is Lindelôf Y is separable and metrizable and I × Y is paracompact, must X × Y be Lindelöf?