1. Let

and denote by Aδ the class of functions f analytic in the strip Sδ = {z = x + iy| |y| < δ}, real on the real axis, and satisfying |Ref(z)| ≤ 1,z∊Sδ. Then N.I. Achieser ([1], pp. 214–219; [8], pp. 137–8, 149) proved that each f∊Aδ can be uniformly approximated on the whole real axis by an entire function fc of exponential type at most c with an error

where ∥·∥∞ is the sup norm on ℝ. Furthermore ([7], pp. 196–201), if f∊Aδ is 2π-periodic, then the uniform approximation Ẽn (Aδ) of the class Aδ by trigonometric polynomials of degree at most n is given by

In this paper we present a proof of the mean-value theorem required by Levinson to show that at least one-third of the zeros of ζ(s) are on the critical line. As in Levinson [3], let

where Χ(s)=ζ(s)/ζ(1−s) is the usual factor from the functional equation, and let

where

and

where

In [1], Julian Cole asked whether every topos is definable over a Boolean one. Peter Johnstone partially answered this in [2] (corollary 3·7) by giving an example of a topos admitting no bounded morphism to any Boolean topos. Here we give a simple example where boundedness is unnecessary.

Given two homomorphisms of rings j1:R1 → R' and j2: R2 → R' we may construct a new ring R called the pullback of R1 and R2 over R', together with homomorphisms i1:R →R1 and i2:R→R2 such that j1i1= j2i2. There are many instances where special cases of pullbacks have been studied, either to construct new examples of rings or to reduce various problems to those concerning much simpler rings. A few of these instances are given in the references. Thus a general study of the properties of pullbacks in terms of those of the component rings would seem to be useful.

The aim of this paper is to prove:

Theorem. Let R be an affine (finitely generated) algebra over a field k and of Gelfand-Kirillov dimension one. Then R satisfies a polynomial identity. Consequently, if N is the prime radical of R, then N is nilpotent and R/N is a finite module over its Noetherian centre.

In 1955 [1], Amitsur determined all the finite groups G that can be embedded in the multiplicative group T* = GL(1, T) of some division ring T of characteristic zero. If G can be so embedded, then the rational span of G in T is a division ring of finite dimension over ℚ, and G acts on it by right multiplication in such a way that every non-trivial element operates fixed point freely. The finite groups admitting such a representation had earlier been determined by Zassenhaus[24; 4, XII. 8], and Amitsur begins by quoting Zassenhaus' results, which show in particular that the only perfect group that can be embedded in the multiplicative group of a division ring of characteristic zero is SL(2,5). The more difficult part of Amitsur's paper is the determination of the possible soluble groups. Here the main tool is Hasse's theory of cyclic algebras over number fields.

Let G be a group. A subset D of G will be called a set of 3-transpositions if |x| =2 for all xεD and |xy| = 3 whenever x, yεD do not commute. We will call the set D closed if xDx = D for each xεD. For each xεD, let

In this paper we complete the work begun in [2] on the subgroup structure of the Lyons simple group Ly of order

For V a complex algebraic threefold with k( V) = 0 and having at least one minimal model (for definition, see below), we show that (except possibly in one rather special case) the components of an effective divisor mKv are birationally ruled surfaces.

Naturally reductive homogeneous spaces have been studied by a number of authors as a natural generalization of Riemannian symmetric spaces. A general theory with many examples was well-developed by D'Atri and Ziller[3]. D'Atri and Nickerson have proved that all naturally reductive spaces are spaces with volume-preserving local geodesic symmetries (see [1] and [2]).

For a topologist, the fundamental group G of a space is usually the most important non-abelian algebraic object of study. However, under many equivalence relationships G is not invariant, so topologists have been led to examine other algebraic objects. In particular, for questions of concordance the lower central series of G seems to play the crucial role. Recall that the lower central series Gn(n = 1,2,...) of G is defined by G1 = G, Gn = [G, Gn_1] for n > 1, and the lower central sequence of G is the sequence of quotients G/Gn.

Stone-Čech compactifications of semigroups have aroused a good deal of interest recently. Several authors, for example Milnes [6], Marcri [5] and Baker and Butcher [1], have concentrated on problems of the existence of continuous extensions to βS of the operation in a topological semigroup S. For a discrete semigroup a separately continuous extension always exists, and others such as Pym and Vasudeva[8] have studied the compactifications of particular classes of semigroups. Further interest has centred on the algebraic structure of these compactifications; see for example Hindman[3].

1. Introduction. A linear operator T on a vector lattice is band preserving if x⊥ y implies Tx ⊥ y. Much is known about the order bounded band preserving operators on an Archimedean vector lattice. The collection of all of these forms an Abelian algebra under composition and a vector lattice for the operator order (see [7], [8] and [13] amongst others). Very little appears to be known about band preserving operators which are not order bounded apart from some isolated examples ([11], [13], [1] and [17]) and some non-existence results ([11] and [1]).

In this short note we prove that Istrǎƫescu's notion of k-uniform (k-locally uniform) convexity of a Banach space is actually equivalent to the notion of uniform (locally uniform) convexity. Thus theorem 2 in [3] and theorem 2·6·28 in [2] are trivially true.

One of the most important concepts in the theory of approximation by analytic functions is that of analytic continuation. In a typical problem, for example, there is generally a region Ω, a Banach space B of functions analytic in Ω and a subfamily ℱ ⊂ B, each member of which is analytic in some larger open set, and one might be asked to decide whether or not ℱ is dense in B. It often happens, however, that either ℱ is dense or the only functions which can be so approximated have a natural analytic continuation across ∂Ω. A similar phenomenon is also known to occur even for approximation on sets without interior. In this article we shall describe a method for proving such theorems which can be applied in a variety of settings and, in particular, to: (1)  the Bernštein problem for weighted polynomial approximation on the real line; (2)  the completeness problem for weighted polynomial approximation on bounded simply connected regions; (3) the Shapiro overconvergence problem for sequences of rational functions with sparse poles; (4) the Akutowicz-Carleson minimum problem for interpolating functions. Although we shall present no new results, the method of proof, which is based on an argument of the author [6], seems sufficiently versatile to warrant exposition.

Recently Stadje [9] gave asymptotic formulas for some Bessel functions by using a probabilistic argument: some Bessel functions can be represented by the probability distribution of partial sums of symmetrized independent and identically distributed random variables (Feller [4], p. 149). Stadje used this idea and the local limit theorem, which is well known in probability theory.

It is shown that P, depending on x, u and derivatives of u, satisfies Div P ≥ 0 for all such x, u, if and only if Div p = Φ(x) ≥ 0 where Φ is independent of u. Applications to theories of continuum thermomechanics are discussed.

The joint distribution of the n(n− l)/2 distances between n normally distributed points in d dimensions is studied. Moment generating functions and probability density functions are obtained. It is shown that when n = d the squared distances are jointly exponentially distributed subject only to the constraint that a valid n point configuration is prescribed. In the case n = d = 3 the distributions of the ordered distance are obtained explicitly.

We propose a smooth correction for skewness in an asymptotically normal statistic. Unlike the case of approximation by Edgeworth expansion, our correction applies uniformly in all values of the level. The correction is based on a ' comparison statistic', and, in the case of the Studentized mean, it enables removal of all effects of skewness up to terms of order .

In this paper we study the asymptotic behaviour of large axisymmetric deformations of closed axisymmetric nonlinearly elastic shells under internal hydrostatic pressure. These shells can suffer flexure, extension, and shear. Since there are spherical shells that can enclose an arbitrarily large volume at a finite pressure (cf. [1]), we take the volume rather than the pressure as the large parameter.