Let G be a simply connected, semi-simple, compact Lie group, let K* denote Z/2-graded, representable K-theory, and K* the corresponding homology theory. The K-theory of G and of its classifying space BG are well known, (8),(1). In contrast with ordinary cohomology, K*(G) and K*(BG) are torsion-free and have simple multiplicative structures. If ΩG denotes the space of loops on G, it seems natural to conjecture that K*(ΩG) should have, in some sense, a more simple structure than H*(ΩG).

Some time ago, Dr I. B. S. Passi pointed out to me the connexion between the structure of the terms of the α-central series of a wreath product and that of polynomial groups which he had defined and studied. In discussing this problem with him, we looked closely at a paper (1) which I had written on nilpotent wreath products. In the process of this, I found a mistake in Theorem 3·7, the main result of (1). The results in (2) depend onthis theorem. Fortunately, it is possible to obtain a correct version of Theorem 3·7 and to modify the proof in (2) which depends on this theorem in such a way that all the results of (2) remain valid.

The detailed study of n-dimensional crystallography appears to have been initiated by Hermann but unfortunately only a small fragment of his work was ever published(4). Recently, interest in the subject has been revived by the work of Bülow, Neubüser and Wondratschek(2), who not only obtained definitive results on 4-dimensional lattices and space groups but also proposed general definitions and procedures for the n-dimensional case. In particular they introduced the concept of crystal family and so resolved satisfactorily the controversy on whether there are six or seven 3-dimensional crystal systems (see(1)). Their point of view is arithmetic, that is, they use a coordinate system adapted to the particular lattice under consideration and represent elements of its symmetry group by integer matrices.

In this paper comparisons are made between the class of continuous functions of generalized bounded variation and the class of continuous functions with graphs having σ-finite length i.e. linear measure. An investigationof the differentiability and approximate differentiability of such functions discloses the fact that the latter class is considerably more extensivethan the former one. The following definitions will be needed:

(1) A function f is said to be of bounded variation (VB) on a set E if

where the supremum is taken over all sequences {[ai, bi]} of non-overlapping intervals with endpoints in E.

In (3) the second author has shown that all sub-Hopf-algebras of the mod 2 Steenrod algebra have a certain form; this resulthas been used in (1) and (2). The analogous result for the mod p Steenrod algebra, where p is an odd prime, is contained in (6). The object of this paper is to give a shorter proof of a slightly sharper result; we construct all the subalgebras in question. In the mod 2 case this sharper result has also been obtained by Anderson and Davis (2).

In (5), a generalization of a Morse-Smale diffeomorphism known as a Smale diffeomorphism is given and all such diffeomorphisms are shown to be C° dense on a C∞ compact manifold and structurally stable. Also Morse inequalities for Smale diffeomorphisms are given which supersede the classical Morse-Smale inequalities (3).

In a finite projective plane S2, q of order q, a (k, n)-arc is a set of k points, of which some n, but no n + 1 are collinear. Barlotti (1) proved that if (n, q) = 1 then

All the sets with which we are concerned are subsets of the real Euclidean plane E2. By Lm we denote those subsets X of E2 for which, if pl, p2, …, Pm are any m points of X, then at least one segment pipj, i ≠ j consists entirely of points of X. L2 is the class of convex subsets of E2. We shall show that if X is closed and X ∈ Lm. then X is the union of finitely many convex sets. This extends a result of Valentine (4). See also (1),(2),(3).

The importance of the spectral theory of operators in Functional Analysis cannot be over emphasized. Spectral theory is in its best form when one considers normal operators in Hilbert spaces. However, for te dimensional spaces one has the reduction to Jordan's Canonical form. In an attempt to generalize this reduction for arbitrary Banach spaces, Dunford introduced the concept of spectral operators. Considerable work has been done in recent times in the study of spectral theory in Banach spaces, almost all of which stems from the pioneering work of Dunford.

The notion of orthogonality is axiomatically defined on a poset. Various notions of orthocomplementation are distinguished and conditions are given in order to induce an orthocomplementation from an orthogonality and vice versa. Subsequently ⊥-modular ⊥-poset are defined and the set of morphisms between two posets with orthogonality is briefly discussed. Given the notions of additive monoid and of positive semi-ring, an orthogonality relation is introduced on the set of idempotent elements of a positive semi-ring. Finally, the obtained results are applied to the set of idempotent and absorbent endomorphisms of an additive monoid.

In this note, we state general conditions which imply that the numerical range of a function mapping a set S into a normed linear space Y is the closed convex hull of the spatial numerical range of the function. This conclusion is of interest since, for example, it is equivalent to an extension to non-compact spaces of Kolmogoroff's characterization of functions of best approximation.

Nina M. Roy has proved (3) that every separable G-space is square, thus exposing an error in my paper (1), in which I proposed a construction of non-square G-spaces which purports to include separable examples. The mistake lies in Lemma 5, whose proof confuses the weak-* topology with the bounded weak-* topology. This note is to salvage the construction to the extent possible, which is in the nonseparable case. The context and notations of (1) are used without further comment.

Let E[τ] be a locally convex Hausdorif topological vector space, with a Schauder basis {xi, x′j where

for each x ∈ E. The partial summation operator Sn, defined by

is a linear operator on E, whose definition extends at once to a linear operator mapping (E′)* into E, where (E′)* is the algebraic dual of E′. The dual of Sn is the operator S′n, mapping E* into E′, defined by

and 〈Snx, x′〉 = 〈x, S′nx′〉 for each x ∈ (E′)*. It is easy to see that S′nx′ → x′ with respect to the weak topology σ(E′, E) for each x′ ∈ E′.

In this paper, we use the results of (3) to show that a condition which is formally much weaker than that given by Sherman in (11) is necessary and sufficient for the commutativity of a C*-algebra. The basic idea behind the proof is then used again to obtain a characterization in order-theoretic terms of the self-adjoint elements in the centre of a C*-algebra. We shall use the notation and terminology of (3).

In the structure theory of Banach spaces as developed in (1), an important role is played by subspaces which are the ranges of projections having norm properties akin to those of the classical Banach spaces. A linear projection e on a Banach space V is called an M-projection if

and an L-projection if, instead

A closed subspace J of V is called an M-Summand if it is the range of an M-projection and an M-Ideal if J0 is the range of an L-projection in V′. Every M-Summand is an M-Ideal but the reverse is false.

Approximating sequences have been extensively studied in many branches of mathematics, for example, in number theory (approximating real numbers by rationals) and in numerical analysis (approximations to functions by polynomials). In (1), A. Hyllengren introduced a type of approximating sequence ‘majorizing sequences’ which he used in solving a problem in complex analysis. In this note we study a very similar concept, which is general enough to be applicable to any separable metric space, and which turns out to have strong connexions with the theory of Hausdorff measures (as did Hyllengren's majorizing sequences).

A typical Riesz product on the circle is the weak* limit

where – 1 ≤ rk ≤ 1, øk ∈ R, λT is Haar measure, and the positive integers nk satisfy nk+1/nk ≥ 3. A classical result of Zygmund (11) implies that either µ is absolutely continuous with respect to λT (when ) or µ is purely singular (when ).

For t > 0, let

where xn+r = xr ≥ 0 and xr+1 + xr+2 > 0 for each r.

Let f(x, y) be a bounded real function on the unit square. Consider the Riemann integrals (if they exist)

Certain infinite products occur in potential problems involving plane boundaries and elements, such as line charges, which produce logarithmic potentials. In this paper, the author derives a theorem which is used to evaluate a particular product which arose in the calculation of the electric potential distribution in a flat, square sheet of conducting material when an electric current enters and leaves respectively at line-contact electrodes attached to adjacent corners of the square.