1. Let A be an associative algebra over a commutative field K; A is then a vector space over K, and multiplication is defined in A in such a way that

for x, y, z in A and λ in K. We suppose in addition that A does not contain a unit, and we denote by Ae the algebra obtained by adjoining a unit e to A.

Let L be a finite-dimensional Lie algebra and U(L) its universal envelope. It is known that U(L) is a Noetherian domain (see (5), theorem v. 3·4) and therefore U(L) has a field of fractions. (Throughout the paper we use the term ‘field’ in the sense of skew field.) We prove in this article the following theorem.

We show that under quite general circumstances a Banach algebra R need not be closed in its tilda algebra . In answer to a query of Katznelson we show that there is a very strongly homogeneous algebra B ≠ C0(R) on R with R as carrier space such that non-analytic functions operate on B.

A simple solution of a general problem involving a bifurcated wave guide is presented. The purpose of the work is to explain a new and simple method of solving such problems and to exhibit an organic connexion between Sturm–Liouville theory and the theory of two-part boundary-value problems.

Soit $f$ une fonction arithmétique. Posons $$m(n;f) \,{\coleq}\, \min_{d|n}\!{\Vert}f(d){\Vert}\qquad(n\,{\geq}\,1),$$ oú ${\Vert} u {\Vert}$ désigne la distance du nombre réel $u$ à l'ensemble des entiers. Le comportement normal de $m$($n$;$f$) est une mesure de la nature probabiliste de l'ensemble des diviseurs d'un entier [Lt ]statistique[Gt ]. Une hypothèse standard d'équirépartition conduit à l'évaluation $$m(n;f) = 1/\tau(n)^{1+o(1)}\quad {\rm pp}$$, oú $\tau$($n$) désigne le nombre total des diviseurs d'un entier naturel $n$, et oú, ici et dans la suite, nous utilisons la mention pp (presque partout) pour indiquer qu'une propriété relative à un entier générique est valable sur une suite de densité naturelle unité.

In this paper, we will investigate discrete coactions on Hilbert C*-modules. In particular, we obtain a one-to-one correspondence between Hilbert C*-modules with discrete coactions and Hilbert C*-modules over the crossed products of the original C*-algebras which satisfies some nice properties (see 3·6 and 3·7). Then we will give some applications of this correspondence in the last three sections.

In Section 2, we give some results about discrete coactions on Hilbert C*-modules which mainly correspond to those about discrete coactions on C*-algebras (see [7]).

A theorem of Besicovitch, namely that, assuming the continuum hypothesis, there exists in any uncountable complete separable metric space a set of cardinality the continuum all of whose Hausdorif h-measures are zero, is here deduced by appeal to Martin's Axiom. It is also shown that for measures λ of Hausdorff type the union of fewer than 2ℵ0 sets of λ-measure zero is also of λ-measure zero; furthermore, the union of fewer than 2ℵ0 λ-measurable sets is λ-measurable.

A generalized triangle group is a group with a presentation

GT(l, m, n; w) = 〈a, b | al = bm = wn = 1〉;

where l, m, n [ges ] 2, and w is a cyclically reduced word in the free product on {a, b}. In the setting of one-relator products of cyclic groups, Fine et al. [3] calculated their Euler characteristics for n [ges ] 3 under certain conditions on the defining word w.

For t > 0, let

where xn+r = xr ≥ 0 and xr+1 + xr+2 > 0 for each r.

J. R. Paktington. ‘On nearly uniformly convex Banach spaces.’

Theorem 1 was printed incorrectly. It should read

If X is (UKK) then so is lp(x) (1 ≤ p < ∞).

A similar replacement of L by l should also be made on p. 127, 1. 4* and p. 128, 1. 13, while p. 128, ll. 11*–10* should read

Also the l2 direct sum of lp, p = 2, 3, … is not even isomorphic to any (UKK) space but is (BS).

Let k be a positive integer and F(x, k) denote the number of integers n < x which have a divisor in every residue class prime to k. Erdös (1) proved that for every fixed ε > 0, we have F(x, k) ˜ x when

and conjectured the following result, which we prove in this paper.

It was first proved by R. Lashof in [4], using the work of S. Cappell and J. Shaneson on four-dimensional surgeryu (see [1]), that there exist locally flat topological knots S3 ∪ S5 which are not smoothable. In [2] (compare also [6]) S. Cappell and J. Shaneson have constructed infinitely many non-smoothable locally fat topological knots as the fixed points of locally nice (= locally smoothable) Zp actions on S5, therefore giving non-trivial examples of locally smoothable but equivariantly non-smoothable actions of Zp on S5.

If H(ξ, η) is a harmonic function which is defined and positive in η > 0, then there is a non-negative number D and a bounded non-decreasing function G(x) such that

(For a proof, see Loomis and Widder, Duke Math. J. 9 (1942), 643–5.) If we write

where λ > 1, then the equation

defines a harmonic function h which is positive in υ > 0. Hence there is a non-negative number d and a bounded non-decreasing function g(x) such that

The problem of finding the connexion between the functions G(x) and g(x) has been mentioned by Loomis (Trans. American Math. Soc. 53 (1943), 239–50, 244).

In this paper we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $\pi$ we consider a special small $\mathrm{GL}_2(\mathbb{Z}_p)$-type V in $\pi$ and prove global sup-norm bounds for an average over an orthonormal basis of V. We achieve a non-trivial saving when the dimension of V grows.

In Eagleson and Weber [2] a central limit theorem for weakly exchangeable arrays is given as a consequence of a reverse martingale central limit theorem. As noted in their remarks, a direct application of this is a central limit theorem for the classical U-statistics. Here we give a corollary to the functional law of the iterated logarithm of Scott and Huggins [4] and use this to obtain laws of the iterated logarithm for weakly exchangeable arrays and hence for U-statistics under a finite (2 + δ)th moment condition.

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.

Let H(t, θ) be the hyperplane in Rn (n ≥ 2) which is perpendicular to the unit vector θ, and distant t from the origin; that is H(t, θ) = {x ε Rn: x.θ = t}. (Note that H(t, θ) and H(−t, − θ) are the same hyperplane.) If f(x) εℒ1(Rn), we will denote the integral of f with respect to (n − 1)-dimensional Lebesgue measure over H(t, θ) by F(t, θ), termed the projection or sectional integral of f over H(t,θ). By Fubini's theorem, F(t, θ) exists for almost all t for any θ. Throughout this paper we will assume that f(x) has support in X, a compact convex subset of Rn. In Section 2 we examine some of the topologies that may be defined on functions on X in terms of the F(t, θ), and in the remainder of the paper we examine the extremal problem suggested by Croft (4), that of maximizing the integral of f over the set X with the constraint that the F(t, θ) are uniformly bounded above. We examine in particular how the extremal values depend on the convex set X. In the final section the extremal problem is related to a generalization of Bang's plank theorem and the theory of capacities, and several conjectures are proposed.

We determine the rate of growth of the expected number of real zeros of a class of random polynomials whose coefficients form a stationary uniformly mixing sequence.

In this paper we consider complex projective surfaces V, defined by an equation of the form fn–1 (x, y, z) w + fn (x, y, z) = 0, where fi is homogeneous of degree i, and relate the geometry of the intersections of the piane projective curves fn–1 = 0 and fn = 0 with the singularities of V. The results we obtain clarify and generalize some of those presented by Bruce and Wall (3).