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

Let

Asymptotic formulae for Ik(T) have been established for the cases k=1 (Hardy-Littlewood, see [13]) and k = 2 (Ingham, see [13]). However, the asymptotic behaviour of Ik(T) remains unknown for any other value of k (except the trivial k = 0, of course). Heath-Brown, [6], and Ramachandra, [10], [11], independently established that, assuming the Riemann Hypothesis, when 0≤K≤2, Ik(T) is of the order T(log T)k2 One believes that this is the right order of magnitude for Ik(T) even when k = 2 and indeed expects an asymptotic formula of the form

where Ck is a suitable positive constant. It is not clear what the value of Ck should be.

It is shown that in all dimensions n ≥ 11 there exists a lattice which is generated by its minimal vectors but in which no set of n minimal vectors forms a basis.

Many positive results are known to hold for the class of Banach spaces known as Asplund spaces and it was for a time conjectured that Asplund spaces should admit equivalent norms with good smoothness and strict convexity properties. A counterexample to these conjectures, in the form of a space of continuous real-valued functions on a suitably chosen tree, was presented in [5]. In this paper we show that the bad behaviour of that example is shared by a wider class of Banach spaces, associated with a wider class of trees. The immediate aim of this extension of the original result is to answer a question posed by Deville and Godefroy [3]. They introduced and studied a subclass of Asplund spaces, those with Corson compact bidual balls, and asked whether this additional assumption is enough to guarantee the existence of nice renormings. We show that it is not.

We study infima of families of topologies on the hyperspace of a metrizable space. We prove that Kuratowski convergence is the infimum, in the lattice of convergences, of all Wijsman topologies and that the cocompact topology on a metric space which is complete for a metric d is the infimum of the upper Wijsman topologies arising from metrics that are uniformly equivalent to d.

Wielandt [4] has shown that a common subnormal subgroup of two permutable subgroups of finite group is subnormal in their product. When G is infinite it seems unlikely that Wielandt's theorem will still be true, but an example illustrating this appears to be difficult, even if G is an FC-group (that is groups in which each element has only finitely many conjugates; see [2]). However, if we replace subnormality by ascendancy we have the following.

In [7] S. Pride gave a family of examples of finitely presented groups of cohomological dimension 2 having no non-trivial action on a simplicial tree. We show here that his examples have no non-trivial action on a Λ-tree, for any ordered abelian group Λ. This provides further slight evidence for an affirmative answer to Question A in §3.1 of [8]. We also give another similar family of examples.

Let I be an ideal of a Noetherian ring R. The purpose of this paper is to study the relationship between the vanishing of the local cohomology modules , and the comparison of the topologies defined by the I-adic {In}n≥0, the symbolic {I(n)}n≥0 and the integral filtration

Given a topological space X, we denote by Cp(X) the space of real-valued continuous functions on X, equipped with the topology of pointwise convergence.

Every homogeneous convex body in ℝd (d≥2) put to sit on a horizontal hyperplane finds a position of stable equilibrium. A cube has 2d such positions and an ellipsoid with pairwise distinct axis-lengths has 2. How many positions of stable equilibrium have most convex bodies?

The term “most” is understood in the Baire category sense. For various other results on most convex bodies, see [2], [4].

Let g(n) be a complex valued multiplicative function such that |g(n)| ≤ 1. In this paper we shall be concerned with the validity of the inequality

under the weak condition g(p)∈ for all primes p, where is a fixed subset of the closed unit disc Thus our point of view is similar to that of Halász [Hz 2] in that we seek a general inequality in terms of simple quantities, albeit g(p) may have a quite irregular distribution. We are not concerned here with the problem of asymptotic formulae for the sum on the left of (1) studied by (among others) Delange [D], Halász [Hz 1] and Wirsing [W].