In the isomorphism theory of Bernoulli shifts, the or Hamming distance plays an important role. Two finite state stationary processes are close in the metric if, on the average, strings from one process can be matched with strings from the other process so that their outputs agree with high frequency. Since such agreement must hold in the arbitrarily distant future, it might at first seem difficult to control the pairing.

It was in the year 1931 that Dieudonné [4] proved the following necessary and sufficient condition for a polynomial to be univalent in the unit disk.

THEOREM A (Dieudonné criterion). The polynomial

(1)

is univalent in |z| < 1 if and only if for every θ in [0, π/2] the associated polynomial

(2)

does not vanish in |z| < 1. For θ = 0, ϕ(z, θ) is to be interpreted as Pn'(z).

Let be a complex Banach space and the algebra of bounded linear operators on . In this paper we study functions from the complex numbers to of the form

In this paper we consider a generalization of the discrete Poisson kernel for the Hahn polynomials obtained recently by Gasper [6]. The Hahn polynomials of degree n are defined by

(1.1)

and are known to be orthogonal on the set of non-negative integers x = 0, 1, . . . , N provided Re α, β > - l or Re α, β < -N [7; 8].

We investigate the extent to which knot groups determine knot manifolds and knot types. Let Ki(i = 1, 2) denote a tame knot in S3, let Ci denote a Ki-knot manifold, and assume that Π1(C1) ≈ Π1(C2). The first named author recently showed (in [6]) that, if C1 has no essential annulus, then C1 ≅ C2, and so K1 and K2 are equivalent, if K1 has property P.

Let Λ be an artin algebra, that is, an artin ring that is a finitely generated module over its center C which is also an artin ring. We denote by mod Λ the category of finitely generated left Λ-modules. We recall that the category of finitely generated modules modulo projectives is the category given by the following data: the objects are the finitely generated Λ-modules.

In [1], a number of “multilinear” functors Lpq, denned for finitely generated free modules, were introduced. They arose as cycles in a generic Koszul complex, and in turn gave rise to a large family of other generic complexes. One of the things we will do in this paper is study some of these new complexes in order to obtain new multilinear functors on free modules which appear as their cycles.

The first example of a nuclear Fréchet space without a basis was given by B. S. Mitiagin and N. M. Zobin [9; 10]. The question of existence of subspaces without bases in nuclear Fréchet spaces was recently settled in papers by P. Djakov and B. S. Mitiagin [2] and Ed Dubinsky [5]. In this paper we consider the analogous question for quotient spaces. As in the case of subspaces we obtain a complete solution to the problem.

A Baer ring is a ring in which every right (and left) annihilator ideal is generated by an idempotent. Generalizing quite naturally from the fact that the endomorphism ring of a vector space is a Baer ring, Wolfson [5; 6] investigated questions such as when the endomorphism ring of a free module is a Baer ring, and when the ring of continuous linear transformations on a pair of dual vector spaces is a Baer ring. A further generalization was made in [7], where the question of when the endomorphism ring of a torsion-free module over a semiprime left Goldie ring is a Baer ring was treated.

Let R be a ring with involution*. In this paper, we study additive subgroups A of R which are invariant under all mappings of the form ϕx : a → xax*. That is, xAx* ⊆ A for all x ∈ R. Obvious examples of such subgroups A are ideals of R, the set of symmetric elements, and the set of skew-symmetric elements. We will prove that when R is *-prime, these examples are essentially the only ones.

This paper completes a study, begun in [7], of conditions under which a generalized Hankel conjugate transformation Hƛ is bounded between a pair of μα -rearrangement invariant function spaces, the measure μα being defined by dμα(t) = tα∼1dt. Examples of such spaces are the Lp(μα) of Lebesgue and generalizations of them due respectively to Orlicz and Lorentz.

Throughout this paper the word “Artinian (Noetherian) ring” means an associative ring with minimum (maximum) condition on left ideals. According to C. Hopkins, an Artinian ring is Noetherian if it contains a left or right identity [3, p. 728]. However we shall consider Artinian rings without the assumption of existence of such an identity, and the theorem of Hopkins will be reproved.

The following problem was the starting point for this investigation: Can every desarguesian affine Hjelmslev plane be embedded into a desarguesian projective Hjelmslev plane [8]? An affine Hjelmslev plane is called desarguesian if it can be coordinatized by a right chain ring R with a maximal ideal J(R) consisting of two-sided zero divisors. A projective Hjemslev plane is called desarguesian if the coordinate ring is in addition a left chain ring, i.e. a chain ring. This leads to the algebraic version of the above problem, namely the embedding of right chain rings into suitable chain rings. We can prove the following result.

σ-ideals (collections of sets which are closed under subset and countable union) are certainly important mathematically—consider first category sets, sets of measure zero, nonstationary sets, etc.—but aside from the observation that in certain spaces the first category σ-ideal is proper, cr-ideals have not been extensively studied by topologists. In this note we study a natural topology determined by a d-ideal, exploiting the interplay between the set-theoretic properties of the σ-ideal and the topological properties of the associated space.

Dans ce travail, nous étudions l'approximation polynomiale pondérée usuelle et aussi unilatérale dans l'espace L. Le poids considéré est w(x) = exp (—|x|) ce qui constitue un cas extremal; rappelons à ce propos quelques résultats de G. Freud. A chaque poids de la forme wQ(x) = exp ( — (Q(#)) correspond une suite caractéristique {qn} de nombres positifs définis par la relation qnQ’ (qn) = n. Dans le sixième chapitre de [1] (voir aussi [14]), Freud démontre que toute fonction/ pour laquelle ƒwQ ∊ L peut être approchée par un polynôme pn de degré ≤n en commettant une erreur (∫|fƒ -Pn\wQ) d'au plus Ω(ƒ, qn/n) où Ω est un module de continuité adéquat et obtient des résultats précisés lorsque la fonction ƒ est supposée differentiate. Dans [2], Freud montre que si la r ième dérivée f(r) est à support compact et à variation bornée et si |ƒ| admet une majoration polynomiale, alors/ peut être approchée unilatéralement par des polynômes de degré ≦n (pn ≦ ƒ ≦ Pn) avec une erreur (∫(Pn — pn)wQ) inférieure à K(ƒ)(qn/n)r+1. Lorsque Q(x) = |x|p, on a qn ~ n1/p de telle sorte que, si ρ > 1, qn/n tend vers zéro avec \/n mais, si ρ = 1, le prolongement formel direct des résultats de Freud fournit une majoration de l'erreur ne tendant pas vers zéro avec 1/n. Cependant, il suit des recherches de M. Riesz [3] (voir aussi [4]) que les fonctions à croissance polynomiale peuvent être approchées unilatéralement dans l'espace L avec poids w et, en particulier, que les polynômes sont denses dans cet espace.

Main results. Let G be a group generated by two subnormal subgroups H and K. Denoting the class of nilpotent groups by 𝔑, and the limit of the lower central series by G𝔑, Wielandt showed in [14], for groups with a composition series that

(*)

Let the group be presented on three generators a, b, c with three defining relations

where p, q, r, s, t, u are integers, the “parameters” of the presentation.

L. L. Herrington recently characterized completely Hausdorff-closed topological spaces in terms of arbitrary filterbases and a type of convergence for filterbases called f-convergence [2], Before this, characterizations of these spaces via filterbases was by open filterbases [1]. In this article, we employ Herrington's characterizations to obtain characterizations of completely Hausdorff-closed spaces in terms of projections and in terms of graphs of functions into the spaces; both of these—projections and graphs—are utilized in conjunction with a class S of spaces containing as a subclass the Hausdorff completely normal fully normal spaces to effect the characterizations.

Let D be a division algebra and let D* denote the multiplicative group of nonzero elements of D. In [3] Herstein and Scott asked whether any subnormal subgroup of D* must be normal in D*. Our purpose here is to show that division algebras over certain p-local fields do not satisfy such a “subnormal property”.

If a locally compact group G acts as a measure preserving transformation group on a Lebesgue space X, then there is a naturally induced unitary representation of G on L2(X), and one can study the action on X by means of this representation. The situation in which the representation has discrete spectrum (i.e., is the direct sum of finite dimensional representations) and the action is ergodic was examined by von Neumann and Halmos when G is the integers or the real line [7], and by Mackey for general non-abelian G [10].