We axiomatize the theory of real and complex numbers in Boolean-valued models of set theory, and prove that every Horn sentence true in the complex numbers is true in any complete Stonean algebra, and provable from its axioms.

A theorem concerning M-summands in dual spaces is used to prove that certain known M-ideals are not M-summands. In some cases where this information was already known, our procedure greatly simplifies the earlier proofs. Finally, we give a condition to determine which M-ideals in dual spaces are M-summands and which are not.

Let R be a 2-torsion free associative ring with involution. It is shown that if the set S of symmetric elements is nilpotent as a Jordan ring then R is nilpotent.

The concept of H-semidirect product structure on a grouplike space is introduced. It is shown that the loop space ΩX of any based CW-complex X is the H-semidirect product of the identity path-component of ΩX with π1,X. The set of free homotopy classes of maps into a Hsemidirect product inherits the structure of a semidirect product. This leads to new results concerning the nilpotency of homotopy classes of maps into a group-like space.

Let G(n) denote the number of non-isomorphic groups of order n. We prove that for squarefree integers n, there is a constant A such that

where denotes the Euler function. This upper bound is essentially best possible, apart from the constant A.

In this paper we introduce several properties closely related to the fixed point property of a partially ordered set P: the comparability property, the fixed point property for cones, and the fixed point extension property. We apply these properties to the sets of common bounds of the minimal (maximal) elements of the partially ordered set P in order to derive fixed point theorems for P.

Plumbing a Hopf band on the fibre-surface of a simple fibred knot is a geometric operation that produces another such knot. We show by algebraic methods that every high odd-dimensional simple fibred knot is obtained from the unknot by using this operation and its inverse.

Let P - {p1,p2,. . . ,pn} be an independent point-set in ℝd (i.e., there are no d + 1 on a hyperplane). A simplex determined by d + 1 different points of P is called empty if it contains no point of P in its interior. Denote the number of empty simplices in P by fd(P). Katchalski and Meir pointed out that . Here a random construction Pn is given with , where K(d) is a constant depending only on d. Several related questions are investigated.

Recently J. B. Reade determined the spectrum of C, the Cesaro matrix of order 1, considered as an operator on cO, the space of null sequences. Previously F. P. A. Cass and the author had determined the spectra for a large class of weighted mean operators on c, the space of convergent sequences. Subsequently the author determined the fine spectra of these operators over c. This paper examines the spectra and fine spectra of weighted mean operators on cO, obtaining the result of Reade as a special case.

The purpose of this paper is to give an explicit formula for the action of the n dimensional Dickson invariants on the admissible mod 2 Steenrod operations of length n.

We give a dual characterization of the following uniformization of the Mazur's intersection property of balls in a Banach space X: for every ∊ > 0 there is a K > 0 such that whenever a closed convex set C ⊂ X and a point p ∊ X are such that diam C ≤ 1/∊ and dist(p, C) ≤ ∊, then there is a closed ball B of radius ≤ K with B ⊃ C and dist(p,B) ≥ ∊/2.

Let X be a metric space such that every separable subspace of X has size less than the continuum. We answer a question of D. H. Fremlin by showing that MA + ┐CH does not necessarily imply that every subset of X is analytic.

Let P be a polynomial of degree n having all its zeros in the closed unit disk. Given that a is a zero (of P) of multiplicity k we seek to determine the radius ρ(n; k; a) of the smallest disk centred at a containing at least k zeros of the derivative P'. In the case k = 1 the answer has been conjectured to be 1 and is known to be true for n ≦ 5. We prove that ρ(n; k; a) ≦ 2k/(k + 1) for arbitrary k ∊ N and n ≦ k + 4.

We construct a uniformly asymptotically regular, Lipschitzian mapping acting on a weakly compact convex subset of l2 which has no fixed points.