Let Ω(n) denote the number of prime factors of n, counted according to multiplicity. We shall consider the following question. Are there infinitely many natural numbers n for which Ω(n) = Ω(n + 1)? Erdős and Mirsky [4] have asked a closely related question concerning the divisor function d(n)—are there infinitely many n for which d(n) = d(n + 1)? The fact that Ω(n)is completely additive makes our problem slightly easier.

It is a familiar fact that the sequence of K-free numbers, i.e. those having no K-th power divisor greater than 1, has asymptotic density 1/ζ(K). Let χ(K) denote the characteristic function of this sequence, χ(2) = χ. Roth [5] proved that, if k/(n3/13(log n)4/13) → ∞, then

In this paper n always denotes an arbitrary but fixed positive integer. Let S be a subset of n-dimensional euclidean space En and (Si) = (S1, S2, …) a finite or infinite sequence of subsets of En. The sequence (Si) is called a covering of S if S ⊂ ⋃Si, and a packing in S if ⋃Si ⊂ S and int Si, ⋂ int Sj = Ø (for all i ≠ j). We say that (Si) permits an isometric covering of S or packing in S if there are rigid motions σi so that (σiSi) is a covering of S or a packing in S, respectively. If there are not only rigid motions but translations τi so that (τiSi) is a covering or packing, we express this by saying that (Si) permits a translative covering or packing. We consider sequences (Si) rather than sets {Si} not because the ordering is of any importance but because some of the sets Si may appear repeatedly.

The construction, for a module M over a commutative ring A (with identity) and a multiplicatively closed subset S of A, of the module of fractions S-1M is, of course, one of the most basic ideas in commutative algebra. The purpose of this note is to present a generalization which constructs, for a positive integer n and what is called a triangular subset U of An = A × A × … × A (n factors), a module U-n M of generalized fractions, a typical element of which has the form

where m∈ M and (u1, … un)∈U.

This paper is concerned with two aspects of the theory of measures on compact totally ordered spaces (the topology is to be the order topology). In Section 2, we clarify a recent construction of Sapounakis [11, 12] and, in so doing, we are able to say a little more about it. It should be added here that Sapounakis had other ends in view. To be precise, let I be the closed unit interval [0, 1] and let λ be Lebesgue measure on I. We shall construct another totally ordered set Ĩ which is compact in its order topology, a continuous increasing surjection τ : Ĩ → I with the property that card τ−1(t) = 2 for all t ∈ ]0,1[ (these brackets denote the open interval), and a measure on Ĩ such that τ() = λ. Then the following theorem holds.

Hyperplane mean values of non-negative subharmonic functions have been studied in many papers, of which [2] and [3] are examples. Recently, Armitage [1] began a study of hyperplane means of non-negative superharmonic functions. One of his results [1, Theorem 2] shows that, if w is a positive superharmonic function on

and n/(n + 1) p < 1, then

as t → ∞, while another [1, Theorem 1] shows that, if 0 < p ≤ n/(n + l), then the integral in (1) is always infinite. However, he did not present a complete analogue of the result of Flett and Brawn [2, 3], which states that, if Ф: [0, ∞ [ → [0, ∞ [ is a non-decreasing convex function such that Ф(u)/u → 0 as u → 0 +, and w is a nonnegative subharmonic function on Rn × ]0, ∞ [, then under certain conditions on the size of w, the integral mean

tends to zero as t → ∞. In this note we present an analogue for superharmonic functions of the above result, in which the mean M(Ф(w); t) is shown to tend to infinity with t, provided that Ф(u)/u → ∞ as u → 0 +, and which therefore generalizes (1). It might be expected that, in dealing with the superharmonic case, the function Ф would have to be concave, so that Ф(w) would also be superharmonic. It turns out that this condition is unnecessary.

Two spheres of different radii are approaching each other with equal and opposite velocities, the fluid flow around them being at low Reynolds number. The forces on the spheres can be calculated when they are very close by applying an asymptotic analysis — usually called lubrication theory — to the flow in the gap between the spheres. If the non-dimensional gap width is ε, the force is calculated here correct to O(ε In ε) for all ratios of the two spheres' radii. The analysis can be combined with earlier numerical calculations to find all the constants in the asymptotic expansion correct to O(ε).

Given a second-order, linear, partial differential equation, it is sometimes the case that an arbitrary non-negative solution on a strip or half-space ℝn × ]0, c[, where 0 < c ≤ ∞, can be represented by the integral of a kernel function with respect to a non-negative measure on ℝn. The solution is thus, at least theoretically, determined by the measure. This paper is concerned with the determination of the measure, given the solution.

Let λ1, …, λ8 be any non-zero real numbers not all of the same sign and not all in rational ratios. According to a theorem of Davenport and Roth [3], given a real number κ, the inequality

has infinitely many solutions in positive integers for any ε > 0. Recently Liu, Ng and Tsang [5] gave a refinement of this result: for any δ > 0, the inequality

has infinitely many solutions in positive integers. In the present note we obtain a better exponent.

Let K be an algebraic number field of degree n = rl + 2r2 (in the usual notation) over the rationals with discriminant d. Let ZK denote the ring of integers in K. It is usual to speak of an integer Πi ∈ Zk as an almost-prime of order l, if the principal ideal (Πi) has at most l prime ideal factors, counted according to multiplicity. Let P1, …, Pn be positive real numbers with Pk = Pk+r2, k = r1 + l, …, r1 + r2 and P = P1 … Pn ≥ 1.

If П is a k-dimensional vector subspace of Rn and E is a subset of Rn, let projп(E) denote the orthogonal projection of E onto П. Marstrand [8] and Kaufman [6] have developed results on the Hausdorff dimension and measure of projп(E) in terms of the dimension of E, leading to the very general theory of Mattila [11]. In particular, Mattila shows that if the Hausdorff dimension dim E of the Souslin set E is greater than k, then projп(E) has positive k-dimensional Lebesgue measure for almost all П ∈ Gn, k (in the sense of the usual normalized invariant measure on the Grassmann manifold Gn, k of k-dimensional subspaces of Rn).

Let G be a locally compact abelian group. Then there is a finitely additive regular set function m defined on an algebra A of Borel sets in G, m(G) = 1, such that m(T-1F) = m(F) for all F ∈ A and all surjective group endomorphisms T of G onto G.

For a convex polytope P, let s(P) and σ(P) denote, respectively, the largest t such that P has a t-dimensional face (section) which is a simplex. Using Gale diagrams, lower bounds are obtained for these numbers and formulae are developed for the behaviour of s and σ over Cartesian and other products.

Let ℱ denote a set of subsets of X = {1, 2,…, n). Let deg(i) be the number of members of ℱ containing i and val(ℱ) = min {deg (i): i ∈ X). Suppose no k members of ℱ have union X. We conjecture val(ℱ) ≤ 2n-k-1 for k ≥ 3. This is known for n ≤ 2k and we prove it for k ≥ 25. For k = 2 an example has val(ℱ) > 2n-2(l−n-0·651) and we prove val(ℱ) ≤ 2n-2(1–n-1). We also prove that if the union of k sets one from each of ℱ1,…, ℱk has cardinality at most n – t then min {cardinality ℱj} < 2nαt where αk = 2α − 1 and ½ < α < 1.

The aim of this paper is to give a clear statement, and, I hope, a reasonably clear proof, of a theorem of Thorn, which occurs in his important and difficult paper “Ensembles et morphismes stratifiés” [10]. The theorem to which I refer is Théorème 1.D.1 of [10]. “Tout espace stratifié compact admet une présentation associée aux applications kYX données”. At least, I think that the theorem herein described is equivalent to the above, but I could not swear to it. The main difficulty is that, despite strenuous efforts on my part, I have always found it easier to rig up my own system of definitions than to work within the framework suggested by Thorn. However, the two accounts clearly say the same sort of thing. In particular, §1 of the present paper is closely related to, and heavily influenced by, the material on page 250 of [10].