There are many concepts which arise naturally in a discussion of injectivity in an equational class; for example, weak injective algebras, absolute subretracts, essential extensions, the congruence extension property, and the amalgamation property (see [3; 9; 17; 18]). It has already been demonstrated in several papers, notably [9; 17; 26; 27; 28], that the study of these concepts is greatly enriched by the assumption that the algebras under consideration have distributive congruence lattices. In this work attention is focused on weak injective algebras (Section 2) and the congruence extension property (Section 3).

Let A be a compact operator on a separable Hilbert space . The aim of this paper is to investigate the relationship between the weak closure of the algebra of polynomials in A (denoted by U(A)) and its invariant subspace lattice Lat A.

We are concerned with the extent to which the structure of a Boolean algebra (or BA, for brevity) is reflected in its group of automorphisms, Aut . In particular, for which algebras can one conclude that if Aut Aut , then Monk has conjectured [3] that this implication holds for denumerable BA's with at least one atom. We shall refute his conjecture, but show that the implication does hold if and are denumerable, if each has at least one atom, and if the sum of the atoms exists in . In fact, under those assumptions the algebra 21 can be rather neatly recovered from its abstract automorphism group.

A much-studied equation in recent years has been the second order nonlinear ordinary differential equation

where q and f are continuous on the real line and, in addition, f is monotone increasing with yf(y) > 0 for y ≠ 0. Although the original interest in (1) lay largely with the case that q﹛t) ≧ 0 for all large values of t, a number of papers have recently appeared in which this sign restriction is removed.

A number of results are given concerning the character and cardinality of symmetrizable and related spaces. An example is given of a symmetrizable Hausdorff space containing a point that is not a regular Gδ , and a proof is given that if a point p of a symmetrizable Hausdorff space has a neighborhood base of cardinality , then p is a Gδ . It is shown that for each cardinal number m there exists a locally compact, pseudocompact, Hausdorff -space X with |X| ≧ m. Several questions of A. V. Arhangel'skii and E. Michael are partially answered.

Garten and Knopp [7] introduced the notion of infinite iteration of Césaro (C1 ) averages, which they called H∞ summability. Flehinger [6] (apparently unaware of [7]) produced the first nontrivial example of an H∞ summable sequence: the sequence ﹛ai ﹜ ∞i=1 where at is 1 or 0 as the lead digit of the integer i is one or not. Duran [2] has provided an elegant treatment of H∞ summability as a special case of summability with respect to an ergodic semigroup of transformations.

The algebraic structure of a topological algebra influences its topological structure in a way which is profound but not well understood. (See § 7 below for various examples.) Here we examine this influence rather generally, and give a fairly complete analysis of one of the many forms it can take, namely, the influence of the identities of on the group identities obeyed by the homotopy group (or groups of the components) of .

The classical multiplier rule. The purpose of this section is to review the multiplier rule in order to place the results of this report in perspective. Let us begin by considering the following problem of Mayer in the calculus of variations: we seek to minimize

Let G and F be groups. A G-defining subgroup of F is a normal subgroup N of F such that F/N is isomorphic to G. The automorphism group Aut (F) acts on the set of G-defining subgroups of F. If G is finite and F is finitely generated, one obtains a finite permutation representation of Out (F), the outer automorphism group of F. We study these representations in the case that F is a free group.

Let D be the unit disk, |z| < 1, and H(D) the Fréchet space of holomorphic functions on D, provided with the topology of uniform convergence on compact subsets of D. If f is meromorphic in D, we denote by

The present work stems from the following classical result, due to G. H. Hardy, J. E. Littlewood, G. Pólya [7], and R. Rado [10].

THEOREM 1. Concerning a pair of n-tuples x, y ϵ Rn, the following four statementsare equivalent:

(a) for every continuous, convex function f : R → R

Recently, a great deal of attention has been paid to the concept of quasipure injectivity introduced by L. Fuchs as Problem 17 in [5]. An abelian group G is said to be quasi-pure-injective (q.p.i.) if every homomorphism from a pure subgroup of G to G can be lifted to an endomorphism of G. D. M. Arnold, B. O'Brien and J. D. Reid have succeeded in [1] to characterize torsion free q.p.i. of finite rank, whereas in [2] we solved the torsion case and in [3] we studied certain classes of infinite rank torsion free q.p.i. groups.

The Grunsky inequalities [6] and their generalizations (e.g., [5; 14; 17]) have become an increasingly important tool for the study of the coefficients of normalized univalent functions defined on the unit disc. In particular, proofs based upon the Grunsky inequalities have now settled the Bieberbach conjecture for the fifth [15] and sixth [13] coefficients. For bounded univalent functions the situation is similar, although the Grunsky inequalities go over to those of Nehari [11].

Let F be a formally real field, and let A be a preordering of F; that is, a subset of F satisfying Δ + Δ = Δ, Δ Δ = Δ, F2 ⊆ Δ. Denote by X Δ the set of all orderings P of F satisfying P ⊇ Δ. Thus Δ = ⋂ p ∈xΔP. This result is well known. It was first proved by Artin [3, Satz 1] in the case Δ = ∑ F2.

S(X) is the semigroup under composition of all continuous selfmaps of the topological space X. For certain spaces X and Y we classify completely the homomorphisms from S(X) into S(Y). An application of the main result to S(I) the semigroup of all continuous selfmaps of the closed unit interval I results in the solution of a problem which was suggested in the closing paragraph of [6].

For a locally compact group G, let LP(G) be the usual Lebesgue space with respect to left Haar measure m on G. For x ϵ G define the left and right translation operators Lx and Rx by Lx f(y) = f(xy), Rx f(y) = f(yx)(f ϵ Lp(G),y ϵ G). The purpose of this paper is to prove the following theorem.

Dans cette note on se propose d'étendre certains résultats de L. Mirsky concernant des matrices doublement sous-stochastiques [5; 6; 7].

On commence par rappeler des définitions et notations qu'on aura à utiliser par la suite.

K. Mahler [8] has proposed the following problem. Let Ωr for r ≧ 1 be a sequence of n X n non-negative rational integer matrices. Each Ωr — (ωrij) defines a map Ωr : Cn —⟶ Cn by

A group g is called right-orderable (or an ro-group) if there exists an order relation ≦ on g such that a ≦ b implies ac ≦ be for all a, b, c in g. this is equivalent to the existence of a subsemigroup p of g such that p ⋂ p-1 = ﹛e﹜ and p ⋃ p-1 = g. given the order relation ≦, p can be taken to be the set of positive elements and conversely, given p, define a ≦ b if and only if ba-1ϵ p. a group g together with a given right-order relation on g is called right-ordered.

Following [10] an ergodic measure-preserving transformation is called rank one if it admits a sequence of approximating stacks. Rank one transformations have been studied in [1] and [2] where it was shown that any rank one transformation has simple spectrum. More generally it has been shown by Chacon [4] that a transformation of rank n has spectral multiplicity at most n. M. A. Akcoglu and J. R. Baxter have asked whether the converse is true. In particular: does simple spectrum imply rank one? In this paper we give a negative answer to this question.