For a class of functions containing polynomials over ℤm, we give an inequality relating the cardinality of the value set to the additive order of differences of elements in that set. To do this, we find some inequalities concerning the combinatorics of substrings of sequences on finite sets which are related to an interesting matrix inequality.

Group actions on ℝ-trees may be split into different types, and in Section 1 of this paper five distinct types are defined, with one type splitting into two sub-types. For a group G acting as a group of isometries on an ℝ-tree, conditions are considered under which a subgroup or a factor group may inherit the same type of action as G. In Section 2 subgroups of finite index are considered, and in Section 3 normal subgroups and also factor groups are considered. The results obtained here, Theorems 2.1 and 3.4, allow restrictions on possible types of actions for hypercentral, hypercyclic and hyperabelian groups to be given in Theorem 3.6. In Section 4 finitely generated subgroups are considered, and this gives rise to restrictions on possible actions for groups with certain local properties. The results throughout are stated in terms of group actions on trees. Using Chiswell's construction in [3], they could equally be stated in terms of restrictions on possible types of Lyndon length functions.

The results developed by Watson [1] are interpreted to indicate how the slow viscous flow due to the rotation of a small circular cylinder in the presence of a stationary cylinder can be calculated. It is shown how the stream function is given as a combination of the force-free representations corresponding to a line rotlet and a line stokeslet outside the stationary body, plus the streaming flow past the body. The coefficients which multiply these representations are calculated by techniques already described by Watson.

Let R⊂S be two orders in a number field, and let ER and ES be the respective groups of units in each ring. Then ES/ER and S/R are both finite. We consider the problem of bounding the order of ES/ER in terms of the index of R in S. In this paper we solve this problem in the special case that S/R is cyclic as a module over Z.

Let F:Z→X be a minimal usco map from the Baire space Z into the compact space X. Then a complete metric space P and a minimal usco G:P→X can be constructed so that for every dense Gδ-subset P1 of P there exist a dense Gδ Z1 of Z and a (single-valued) continuous map f: Z1→P1 such that F(Z)⊂G(f(z)) for every z∈Z1. In particular, if G is single valued on a dense Gδ-subset of P, then F is also single-valued on a dense Gδ-subset of its domain. The above theorem remains valid if Z is Čech complete space and X is an arbitrary completely regular space.

These factorization theorems show that some generalizations of a theorem of Namioka concerning generic single-valuedness and generic continuity of mappings defined in more general spaces can be derived from similar results for mappings with complete metric domains.

The theorems can be used also as a tool to establish that certain topological spaces contain dense completely metrizable subspaces.

In earlier treatments of the title problem it was found that it is impossible, in general, to obtain solutions of Stokes's equations of slow viscous flow in which the fluid velocity vanishes at infinity. It is shown here that the paradox can be resolved by the introduction of a resultant force on the cylinders. This enables the solution to be matched to an outer solution of the full Navier-Stokes equations.

Packing measures have been introduced to complement the theory of Hausdorff measures in [13,14]. (For a new treatment see also [10, Chapter 5]. While Hausdorff measures are intimately connected to upper density estimates (see, e.g., [5,2.10.18]), the importance of packing measures stems from their connection to lower density estimates.

A new type of convergence (called uniformly pointwise convergence) for a sequence of scalar valued functions is introduced. If (fn) is a uniformly bounded sequence of functions in l∞(Γ), it is proved that:

(i) (fn) converges uniformly pointwise on Γ to some function f if, and only if, every subsequence of (fn) is Cesaro summable in l∞(Γ); and

(ii) there exists a subsequence (f′n) of (fn) such that either (f′n) converges uniformly pointwise on Γ to some f or no subsequence of (f′n) is Cesaro-summable in l∞(Γ).

Applications of the above results in Banach space theory are given.

This communication concerns our paper which appeared, without proper proofreading, in Mathematika, 41 (1994), pp. 239–250. Listed below are corrections to those of the misprints/omissions which, in our judgement, most significantly interfere with efficient reading of the paper. We apologize for any inconvenience that may have resulted, and we thank the editors of Mathematika for an opportunity to make up the mistake.

This paper is concerned with the geometry of a measure μ, and in particular with the relationship between various .s-dimensional densities of μ, the geometry of the support of μ and the question of whether s is an integer.

In this short paper, we shall give a new estimate for the exponential sum S(H, M, N), where

e( ξ,) = exp (2πiξ;) for a real number ξ, am and bn are complex numbers with |am| ≤ 1 and |bn| ≤ l, H, M, N ≤1, , x is a large number, ε is a sufficiently small positive number, and Y ≤ x(½)−ε (h ∼ H means 1≤h/H < 2 and so on). In making application of the Rosser-Iwaniec linear sieve of Iwaniec [6] to find almost primes in short intervals of the type (x − y, x], Halberstam, Heath-Brown and Richert [4] first considered an estimate for S(H, M, N) to the effect that

with MN as large as possible. Later, better estimates were given in Iwaniec and Laborde [7], and Fouvry and Iwaniec [3]. Of course, the most interesting case would be finding P2 numbers in a short interval (x − y, x]. The related estimate of [7] implies that (1) holds provided that

Let X be a compact metric space and Y a separable metric space. Any Baire class one function from X to Y can be recovered from its values on a certain countable set via a simple algorithm.

Let r, k, h be positive integers. For any Dirichlet character χ modulo k write