We show that the ring of complex-valued regular functions on an affine irreducible nonsingular real algebraic variety X is factorial if dim X = 1 or dim X = 2 and X has no compact connected components or X is compact and the second cohomology group of X with integral coefficients vanishes.

Let be a separable Hilbert space, the algebra of all bounded operators on and Ai, Bi, ∊ , i = 1,. . . , r. It is shown that if no nontrivial linear combination of the operators Ai, is compact, then there exist X, Y ∊ such that X Ai, Y = Bi, for all i. A related (but much milder) result is discussed in other algebras with the unique maximal ideal and an application to elementary operators is given.

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.

Recently, Balder proved a version of FatoiTs lemma in several dimensions which, inter alia, generalizes a version of this lemma due to Artstein. Here we show how the latter result can be used to derive the former, by using Chacon's biting lemma.

It is a classical conjecture of E. Artin that any integer a > 1 which is not a perfect square generates the co-prime residue classes (mod ρ) for infinitely many primes ρ. Let E be the set of a > 1, a not a perfect square, for which Artin's conjecture is false. Set E(x) = card(e ∊ E: e ≤ x). We prove that E(x) = 0(log6 x) and that the number of prime numbers in E is at most 6.

In this paper the ergodic Hilbert transform is investigated at the operator theoretic level. Let T be an invertible positive operator on Lp = Lp(X, , μ) for some fixed p, 1 < p < ∞, such that sup{||Tn||p: — ∞ < n < ∞} < ∞. It is proved that the limit

exists almost everywhere and in the strong operator topology, where the prime denotes that the term with zero denominator is omitted. Related results are also proved.

We explicitly describe an infinite collection of pairs of Azumaya algebras over the ring of integers of real quadratic number fields K which are maximal orders in the usual quaternion algebra over K, hence Brauer equivalent, but are not isomorphic. The result follows from an identification of the groups of norm one units, using the classification of Coxeter.

L'objet de cet article est de construire un modèle pour l'élément β1, de Toda (premier élément non nul de la composante p - primaire de l'homotopie stable des sphères qui n'est pas dans l'image du J-homomorphisme, p ≠ 2). Le modèle construit possède en outre la propriété de se plonger, comme variété différentiable, en codimension 1.

La construction, basée sur la J-théorie et la chirurgie, exhibe en outre des complexes cellulaires satisfaisant à certaines conditions de plongement, répondant partiellement à un problème posé par le 2e et le 3e auteur.

Let R be a prime ring with involution satisfying x/2 ∊ R whenever x ∊ R. Assume that R has two nontrivial symmetric idempotents e1, e2 whose sum is not 1, and that the subrings determined by e1, e2, 1 — (e1 + e2) are not orders in simple rings of dimension at most 4 over their centers. Then if L is a Lie derivation of the skew elements K into R there exists a subring A of R, A ⊆ , a derivation D :A → RC, the central closure of R, and a mapping T:R → C, satisfying L = D + T on K and = 0.

A word W in a group G is a geodesic (weighted geodesic) if W has minimum length (minimum weight with respect to a generator weight function α) among all words equal to W. For finitely generated groups, the word problem is equivalent to the geodesic problem. We prove: (i) There exists a group G with solvable word problem, but unsolvable geodesic problem, (ii) There exists a group G with a solvable weighted geodesic problem with respect to one weight function α1, but unsolvable with respect to a second weight function α2. (iii) The (ordinary) geodesic problem and the free-product geodesic problem are independent.

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 R be a ring with center Z, and S a nonempty subset of R. A mapping F from R to R is called centralizing on S if [x, F(x)] ∊ Z for all x ∊ S. We show that a semiprime ring R must have a nontrivial central ideal if it admits an appropriate endomorphism or derivation which is centralizing on some nontrivial one-sided ideal. Under similar hypotheses, we prove commutativity in prime rings.

Let P be a partially ordered set. For an element x ∊ P, a subset C of P is called a cutset for x in P if every element of C is noncomparable to x and every maximal chain in P meets {x} ∪ C. The following result is established: if every element of P has a cutset having n or fewer elements, then P has at most 2n maximal elements. It follows that, if some element of P covers k elements of P then there is an element x ∊ P such that every cutset for x in P has at least log2k elements.

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.

In this paper we extend and greatly generalize, with some new information, the well known results that an approximately continuous function is Darboux, and that a finite approximate derivative has the mean value property and is Darboux. Our theorems on Darboux and mean value properties of derivatives include also those of selective derivatives and I-approximate derivatives.

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

Let p be odd prime and suppose that G = 〈a, b〉 where ap-1 = bp = 1, a-1 ba = br, and r is a generator of the multiplicative group of integers mod p. An explicit characterization of the group of normalized units V of the group ring ZG is given in terms of a subgroup of GL(p - 1, Z). This characterization is used to exhibit a normal complement for G in V.

We determine an algebraic condition necessary and sufficient for a group G to act freely on the nth Cartesian power of an even sphere, and characterize the abelian groups that satisfy this condition.

A new look at Manià's classical example of the Lavrentiev Phenomenon leads to several pertinent observations.