Let E1 and E2 be real Banach spaces. We denote by E12 the Banach space of all bounded linear operators mapping E1 into E2 with the usual operator norm. By Dr we denote the open ball of radius r, centred at the origin.

We record here two further remarks about the systems, studied in [1] and [2], consisting of a vector space U and a set K of subspaces of U. In § 1, we show that such a system may be viewed as a module over a suitable artinian ring; the results of [1] and [2] thus serve to illustrate the complexity of structure of these modules. The main idea, a little wider than one introduced by Mitchell in Chapter IX of [3], is to view a diagram of vector spaces, with a small category as the scheme of the diagram, as a module over the ‘category ring’ of the category.

Using the notion of spherical modification and results from Morse theory a general technique is described for constructing manifolds whose strong category is small (≦ 3) but whose homological structure is complex.

In a series of papers K. D. Magill, Jr. (see [1] and its references) has proved that, in various semigroups of mappings on topological spaces, every automorphism is inner, where an automorphism φ of a semigroup A is a bijection of A such that for all ƒ and g in A, and it is said to be inner if there exists a bijection h ∈ A such that h−1 (the inverse of h) belongs to A and for every ƒ ∈ A.

Problem 24 of Hanna Neumann's book [3] reads: Does there exist, for a given integer n > 0, a Cross variety that is generated by its k-generator groups and contains (k+n)-generator critical groups? In such a variety, is every critical group that needs more than k generators a factor of a k-generator critical group, or at least of the free group of rank k? In a recent paper [1], R. G. Burns pointed out that the answer to the first question is an easy affirmative, and asked instead the question which presumably was intended: Given two positive integers k, l, does there exist a variety 23 generated by k-generator groups and also by a set S of critical groups such that S contains a group G minimally generated by k+l elements and S/{G} does not generate B? The purpose of this note is to record a simple example which shows that the answer to the question of Burns is affirmative at least for k = 2, l = 1, and also that the answer to the second question of Hanna Neumann's Problem 24 is negative.

Let G denote a finite group with a fixed-point-free automorphism of prime order p. Then it is known (see [3] and [8]) that G is nilpotent of class bounded by an integer k(p). From this it follows that the length of the derived series of G is also bounded. Let l(p) denote the least upper bound of the length of the derived series of a group with a fixed-point-free automorphism of order p. The results to be proved here may now be stated: Theorem 1. Let G denote a soluble group of finite order and A an abelian group of automorphisms of G. Suppose that (a) ∣G∣ is relatively prime to ∣A∣; (b) GAis nilpotent and normal inGω, for all ω ∈ A#; (c) the Sylow 2-subgroup of G is abelian; and (d) if q is a prime number andqk+ 1 divides the exponent of A for some integer k then the Sylow q-subgroup of G is abelian.

This note is concerned with a translation of some concepts and results about characteristic subgroups of a group into the language of categories. As an example, consider strictly characteristic and hypercharacteristic subgroups of a group: the subgroup H of the group G is called strictly characteristic in G if it admits all ependomorphisms of G; that is all homomorphic mappings of G onto G; and H is called hypercharacteristic2 in G if it is the least normal subgroup with factor group isomorphic to G/H, that is if H is contained in every normal subgroup K of G with G/K ≅ G/H.

In a recent paper (see [2]), Orrin Frink introduced a method to provide Hausdorff compactifications for Tychonoff or completely regular T1 spaces X. His method utilized the notion of a normal base. A normal base ℒ for the closed sets of a space X is a base which is a disjunctive ring of sets, disjoint members of which may be separated by disjoint complements of members of ℒ.

Consider the following second order nonlinear differential equation: where a(t) ∈ C3[0, ∞) and f(x) is a continuous function of x. We are here concerned with establishing sufficient conditions such that all solutions of (1) satisfy (2) Since a(t) is differentiable and f(x) is continuous, it is easy to see that all solutions of (1) are continuable throughout the entire non-negative real axis. It will be assumed throughout that the following conditions hold: Our main results are the following two theorems: Theorem 1. Let 0 < α < 1. If a(t) satisfieswhere a(t) > 0, t ≧ t0 and = max (−a′(t), 0), andthen every solution of (1) satisfies (2).

Consider a non-degenerate convex body K in a Euclidean (n + 1)-dimensional space of points (x, z) = (x1,…, xn, z) where n ≧2. Denote by μ the maximum length of segments in K which are parallel to the z-axis, and let Aj, signify the area (two dimensional volume) of the orthogonal projection of K onto the linear subspace spanned by the z- and xj,-axes. We shall prove that the volume V(K) of K satisfies After this, some applications of (1) are discussed.