Let F denote a family of analytic functions in the unit disk Δ. Suppose that one has a “sharp” estimate on the almost everywhere radial variation of functions in the class Δ. We prove that if Δ is contained in the Nevanlinna class N then the estimate will be “sharp” in the algebra A of functions analytic in Δ and continuous in Δ.

Let X be a reflexive Banach space. This article presents a number of new characterizations of the topology of Mosco convergence TM for convex sets and functions in terms of natural geometric operators and functional. In addition, necessary and sufficient conditions are given for TM to agree with the weak topology generated by {d(x, C): x є X}, where each distance functional is viewed as a function of the set argument.

We determine infinite products in the field of Laurent series with the property that the truncations of the product yield every second continued fraction convergent of the product. We mention some related examples and specialize to obtain numerical results.

Let K be an algebraic number field, [K: Q] = κ є N; only the case κ > 1 is of interest in this paper. Let f be any non-zero ideal in ZK, the ring of integers of K, and let b be any ray-class (modx f) of K. In this paper we answer a question of P. Erdös (private communication) about the “maximum-growth-rate” of the functions

and

the sum here taken over all ray-classes (modx f), while N(a) is the absolute norm of a. Let

and

where, as usual, for x є R, log+ x - log max {1, x}. We prove

THEOREM 1. For every f, b we have

Soit Y un sous-ensemble algébrique de codimension 1 de R'. Une distribution globale (resp. partielle) de signes sur Rn est une application qui associe un signe a chaque composante connexe (resp. à certaines parmi les composantes connexes) de Rn – Y.

This paper is concerned with convex subsets of finite dimensional vtctor spaces, over the field of real scalars. As in [10, p. 244] and [20] we say that a compact convex set A, symmetric about the origin, is reducible, if there is a nonsymmetric closed convex set B for which A = B - B. The latter term denotes the set of all differences, {x-y: x, y є B}. Equivalently, A is reducible if, and only if, A =1/2 (B - B) for some B which is not a translate of A. If the identity A = B - B is only possible when B is centrally symmetric, then A is irreducible. If A is symmetric about a point other than the origin, we can say that it is (ir)reducible when it is the translate of an (ir)reducible set which is symmetric about the origin. It is well known that a parallelotope of any dimension is irreducible [6, Hilfssatz 3], that any 2-dimensional convex body other than a parallelogram is reducible [9, p. 217], and that euclidean balls of any dimension (other than one) are reducible [4, Ch. 7]. For more general convex bodies, the determination of reducibility is not a simple problem. Shephard [20] showed that a set is reducible if, and only if, it has an asymmetric summand, and he used this to study reducibility of polytopes. The main purpose of this paper is to give a new condition, necessary and sufficient for a symmetric polytope to be reducible. This condition may be expressed in the form: does a certain finite family of linear equations have a nontrivial solution? Thus, to determine the reducibility of a given polytope, it suffices to find the rank of some matrix. Using our criterion, we are able to describe some large families of irreducible polytopes. For example, every n- dimensional symmetric polytope with 4n – 2 or fewer vertices is irreducible (unless n = 2). We also establish the existence of irreducible, smooth, strictly convex bodies.

In the part (16-3) of his extensive study on measurability in Banach spaces, Talagrand [12] considered the Banach space C(K) of continuous functions on a dyadic topological space K. He proved that C(K) is realcompact in its weak topology, if, and only if, the topological weight of K is not a twomeasurable cardinal (Theorem 16-3-1). Then he asked for an alternative to a rather complicated proof presented there (p. 214) and posed the problem whether C(K) is measure-compact whenever the weight of K is not a realmeasurable cardinal (Problem 16-3-2).

We improve W. Schmidt's lower bound for the slice (intersection of two halfspheres) discrepancy of point distributions on spheres and show that this estimate is up to a logarithmic factor best possible. It is shown that the slice and spherical cap discrepancies are equivalent for the definition of uniformly distributed sequences on spheres.

Almost seventy years ago Jeffery [1] showed that a finite velocity can result at infinity when the biharmonic equation is solved for the titled problem. Here, we extend his calculations to show that finite vorticity is the more general conclusion, and then indicate a resolution of the apparent paradox.

One may perhaps doubt whether in the geometry of numbers any particular family of lattices deserves such an attention as, for example, BCH codes receive in coding theory. However, only recently a quite interesting family has emerged. The general case of these lattices considered by Rosenbloom and Tsfasman [5, Section 2] parallels Goppa's construction of codes from algebraic curves. Here we shall take a closer look at the case of genus zero where some special features of Goppa's early codes will show up again: There is a lattice Λ (L, g) in n-dimensional euclidean space associated with a subset L of the field , and a polynomial g satisfying g(є) ≠ for all λ є L. For g = zd previously known sphere packings are recovered and generalized. A nonconstructive argument shows that for n → ∞ and some irreducible polynomials g Minkowski's lower packing bound is met (this being not achieved in [5] where q is fixed, but the genus grows; cf. also [4]).

A right S-system over a monoid S is a set A on which S acts unitarily on the right. That is, there is a function A such that (φ,1)φ and (a, st)φ = ((a, s) t)φ for all a є A and for all s, t є S. We shall refer to right S-systems simply as S-systems. It is clear what is meant by S-homomorphism, S-subsystem etc.; further details of the terms used in this Introduction are given in Section 2.

For satellite knots there is a well-known formula which relates the Alexander polynomial of the satellite to those of a companion knot and the corresponding pattern. If &s, &C and &P are the Alexander polynomials of a satellite, companion and pattern respectively then

where is the linking number of P with a meridian of the companion torus (see [BZ], p. 118). Analogous relationships do not exist for other knot polynomials [MS]. This suggests that the existence of the above formula depends more on the geometry underlying the polynomial than on the geometry of the satellite construction.

In this paper two expansions are obtained by contour integration methods for the velocity potential describing two-dimensional time-harmonic surface waves due to a free-surface wave source on water of infinite depth in the presence of surface tension. First the series expansion at r = 0 is found and then the asymptotic expansion as Kr®¥, where K is the wave number for progressive waves and r the radial distance from the source. The corresponding expansions for the more important submerged wave source in terms of the radial distance from the image source in the free surface may then easily be deduced. The latter are required in a number of surface wave problems, particularly those of a short-wave asymptotic nature, and are also relevant in obtaining expansions for finite constant depth.

This article considers the effect of more than one quotient and improves a theorem of Tong which is a generalization of a theorem of Segre on asymmetric approximation.

The purpose of this work is to investigate the relationship between Radon transforms and centrally symmetric convex bodies. Because of the injectivity properties of the Radon transform it is natural to consider transforms on the sphere separately from those on the higher order Grassmannians. Here we shall concentrate on the latter, whilst the former will be the subject of another article presently in preparation, Goodey and Weil [1991].

The problem concerning the distribution of the fractional parts of the sequence ank (k an integer exceeding one) was first considered by Hardy and Littlewood [6] and Weyl [20] earlier this century. This work was developed, with the focus on small fractional parts of the sequence, by Vinogradov [17], Heilbronn [13] and Danicic [2] (see [1]). Recently Heath-Brown [12] has improved the unlocalized versions of these results for k ≥ 6 (a slightly stronger result than Heath-Brown's for K = 8 is given on page 24 of [8]. The method mentioned there can, after some numerical calculation, improve Heath-Brown's result for 8 ≤ k ≤ 20, but still stronger results have recently been obtained by Dr. T. D. Wooley). The cognate problem regarding the sequence apk, where p denotes a prime, has also received some attention. In this situation even the case k = 1 proves to be difficult (see [9] and [14]). The first results in this field were given by Vinogradov (see Chapter 11 of [19] for the case k = 1, [18] for k ≥ 2). For k = 2 the best result to date has been supplied by Ghosh [5], and for ≥, by Harman (Theorem 3 in [9], building on the work in [7] and [8]). In this paper we shall improve the known results for 2 ≤ k ≤ 12. For larger k, Theorem 3 in [8] is more efficient. The theorem we prove is as follows.

Using several transformation formulae from Ramanujan's second Notebook we achieve distribution results on random variables related to dynamic data structures (so-called “tries”). This continues research of Knuth, Flajolet and others via an approach that is completely new in this subject.