An innovation by D. R. Heath-Brown on square-full numbers in short intervals is applied to establish an asymptotic formula for the number of cube-full numbers in the interval

Let ordq (a) be the exponent with which a prime q occurs in the factorization of a rational number a ≠ 0 and, if ordq (a) = 0, let Mq(a) be the multiplicative group generated by a modulo q. In the course of a group-theoretical investigation J. S. Wilson found he needed some results about integers a, b such that Mq(a) = Mq(b), indeed also for algebraic integers, and he proved some of what he needed. J. W. S. Cassels observed that Wilson's argument naturally proved the existence of infinitely many primes q with Mq(a) = Mq(b) for rational integers a, b with ab > 0, |a| > 1, |b| > 1. J. G. Thompson found a proof for the case of integers a, b with ab < 0, |a| > 1, |b| > 1. He also posed the problem for rational a, b (all this is unpublished). The aim of this paper is to prove that the answer to Thompson's question is affirmative. We also include the case ab > 0 settled by Thompson himself. We use the same technique devised by Wilson which has been elaborated by Cassels and Thompson. We thank Professor Cassels for the simplification of our original exposition and the referee for his suggestions.

The aim of this paper is to prove an inequality of Golod-Shafarevich type for metabelian Lie algebras and to show that this inequality is best possible up to a constant factor. Investigations of this kind were started in [4] in connection with the solution of the class field tower problem. It was shown that if there is a finite p-group which may be presented as a pro-p-group with d ≥ 2 generators and r relations then the inequality r > ¼(d − 1)2 holds; and Vinberg [8] improved this result by showing

The inequality (1) also holds for presentations of nilpotent Lie algebras (see [6]) (with the exception (d, r) = (2, 1)) and nilpotent associative algebras (see [4, 8, 6]). If the relations have of degree at least m ≥ 3 then more relations are needed. More precisely, Koch [5] has shown that the inequality

holds for finite p-groups, and the corresponding results hold for associative algebras and Lie algebras.

The purpose of this paper is to study the structure of locally free modules over the ring of differential operators on projective space. Let be a non-singular, complex, algebraic variety. Denote by the sheaf of rings of differential operators over and by its ring of global sections. A -module M is called locally free if the associated sheaf ⊗ M is locally free as a sheaf of -modules. Locally free modules arise naturally in -module theory as inverse images of determined modules; see [1] for definitions and examples.

An indefinite binary quadratic form ƒ gives rise to a certain function M on the torus. The properties of M, especially those related to its maximum – the so-called inhomogeneous minimum of ƒ – are the subject of numerous papers. Here we continue this study, putting more emphasis on the general behaviour of M.

Throughout this paper ∈ denotes an arbitrary positive number. For real α, ‖α‖ denotes the distance from α to the nearest integer. For natural numbers k we write K = 2k−1. In 1948 Heilbronn [8] showed that for any real α and N > C1(∈)

This theorem has since been generalized in many ways. In particular, results of the following type have been proved for natural numbers k ≥ 2, h = 1,2 and s.

Let G be a hyperfinite locally soluble group and let A be a noetherian ℤZ;G-module. In [2], we proved that A is the direct sum of a ℤZ;G-submodule Af each of whose irreducible ℤZ;G-module sections is finite and a ℤZ;G-submodule each of whose irreducible ℤZ;G-module sections is infinite. In this paper we study the structure of the ℤZ;G-submodules Af and . Our main result gives a complete description of Af.

Suppose that K = {K0, K1} is a partition of some finite combinatorial power [S]r of a set S. Let X(K) be the compact subspace of the Tychonoff cube [0, 1]S consisting of those functions ƒ whose supports Sƒ = {ƒ > 0} are 0-homogeneous i.e., [Sƒ]r ⊆ K0. It can be said that almost every example of a space witnessing the distinction between various chain conditions is a variation of X(K) (see [2]). This comes from the fact that the subject is closely related to the subject of Partition Calculus. (See [11] for an explanation of this point.) To have examples which are even more topologically interesting one usually tries to make them small, i.e., as closely related to the unit interval as possible. The operation X(K) has a considerable drawback in that respect. For example, if we want X(K) to be ccc this becomes equivalent to the fact that the poset of all finite 0-homogeneous sets is ccc which amounts to the fact that K1 must be very small. Hence K0 is big in the sense that there exist large 0-homogeneous sets. This usually results in X(K) having large size and having points of large character. One attempt to solve this problem was given by van Douwen in [5] by going to the subspace Xm(K) of X(K) consisting of those ƒ for which Sƒ are maximal 0-homogeneous subsets of S. Unfortunately, while Xm(K) usually does have small character it is almost never compact. This might have been the reason for his question ([2], p. 207) whether the Continuum Hypothesis implies that the class of all first countable compacta distinguishes the standard chain conditions that lie between ‘ccc’ and ‘separable’. In this paper we solve this problem completely. Moreover, we shall not go beyond the usual axioms of set theory in constructing the examples. The sequence of examples will start with a compact space of small character whose chain condition is not productive and it will end with a compact space of size c and small character which is ccc in a strong sense but which fails to have calibre θ for some regular uncountable cardinal θ, i.e., it fails to have the property of Shanin. Note that one cannot go further and show that, for example, compact spaces of small character distinguish between ‘the property of Shanin’ and ‘separable’. This follows from an old result of Efimov [3] that, under CH, first-countable spaces of calibre ℵ1 are separable. The combinatorics behind our examples have been developed in a series of papers that deal with the subject of forcing axioms in general and Martin's axiom in particular ([10, 11, 12, 13, 14]). Martin's axiom was originally invented in connection with the Souslin Problem, i.e., to show that certain compact ccc spaces must be separable (see [8]). A result of the aforementioned study of MA showed that this axiom is nothing more than the statement that all compact ccc spaces of π-weight < c must be separable (see [12]). An analysis of the fact that MA implies SH, due to Hajnal and Juhasz[6] (see also [4], §43 for a definite result in that direction due to Shapirovskii), revealed that MA implies that every compact ccc space X with the property χ(X)+ < c must be separable. This result explains why the compact ccc non-separable spaces X that we construct in this paper have the property that χ(x, X) < c for all x m X rather than the stronger property χ(X) < c or even χ(X) = ℵ0. Note that our examples show, answering a question from [6], that the Hajnal–Juhasz result is sharp in the sense that the assumption χ(X)+ < c cannot be weakened to χ(X) < c. The first example to show this was constructed by Bell [1] using a consequence of MA rather than just ZFC for its construction. Another feature of our examples is that they all are remainders of certain compactifications of the integers. This is of independent interest in certain constructions of weak P-points in compact F-spaces. An explanation of this can be found in [1] and [9] where the first examples of ccc non-separable remainders were constructed and used.

Let denote a sequence of integers exceeding 1, and let τ(n, ) be the number of those divisors of n which belong to . We say that is a Behrend sequence if

where, here and in the sequel, we use the notation p.p. to indicate that a relation holds on a set of asymptotic density one.

If g is an element of torsion of a group G, its order o(g) is the least positive integer such that go(g) = e. Let OG = {o(g);g ∈ G}. If G is a group of homeomorphisms of a space X, and if g ∈ G, x ∈ X let q(g, x) denote the cardinality of the g-orbit {gnx}n∈Z of x.

An important problem in the topology of 3-manifolds is to classify manifolds obtained by Dehn surgeries on a knot in a closed 3-manifold, or equivalently, Dehn fillings of a 3-manifold M with boundary a torus.

Let ℚp and ℤp respectively denote the p-adic numbers and the p-adic integers for a fixed prime number p. For any subset Y of and n ∈ ℕ we denote by Nn(Y) the cardinality of the image of Y under the natural map . (Thus we have Nn(Y)≤ pnr.)

Let R be a ring (all rings considered here are commutative with identity and Noetherian), M a finitely generated R-module, and I an ideal of R. The jth local cohomology module of M with support in I is defined by

In this paper, we prove a uniform version of a theorem of Brodmann about annihilation of local cohomology modules. As a corollary of this, we deduce a generalization of a theorem of Hochster and Huneke about uniform annihilation of Koszul homology.

In [5], we studied the behaviour of the Kähler cone of Calabi–Yau threefolds under deformations. We saw that the Kähler cone is locally constant in a smooth family of Calabi–Yau threefolds, unless some of the threefolds Xb contain elliptic ruled surfaces. Moreover, if X is a Calabi–Yau threefold containing an elliptic ruled surface, then the Kähler cone is not invariant under a generic small deformation.

All manifolds in this paper are piecewise linear (or smooth if one wishes).

In the paper [12] Shcherbak studied some duality properties of projective curves and applied them to obtain information concerning central projections of surfaces in projective three space. He also states some interesting results relating the contact of a generic surface with lines and the contact of its dual with lines in the dual space. In this paper we extend this duality to cover non-generic surfaces. Our proof is geometric, and uses deformation theory. The basic idea is the following. Given a surface X in projective 3-space we can consider the lines tangent to X, and measure their contact. The points on the surface with a line yielding at least 4-point contact are classically known as the flecnodal. (The reason is that the tangent plane meets the surface in a nodal curve, one branch of which has an inflexion at the point in question; see Proposition 7 below. The line in question is the inflexional tangent, which is clearly asymptotic.)

Let M be a smooth n-manifold, X be a smooth (2n − 1)-manifold, and g:M → X be a map. It was proved in [6] that g is always homotopic to an immersion. The set of homotopy classes of monomorphisms from TM into g*TX, which is denoted by Sg, may be enumerated either by the method of I. M. James and E. Thomas or by the singularity method of U. Koschorke (see [1] and references therein). When the natural action of π1(XM, g) on Sg is trivial, for example, if X is euclidean, the set Sg is in one-to-one correspondence with the set of regular homotopy classes of immersions homotopic to g (see e.g. [4]).

A special monoid is a monoid presented by generators and defining relations of the form w = e, where w is a non-empty word on generators and e is the empty word. Groups are special monoids. But there exist special monoids that are not groups. Special monoids have been extensively studied by Adjan[1] and Makanin[7] (see also [2]).

The classification of all finite primitive distance-transitive graphs is basically divided into two cases. In the one case, known as the almost simple case, we have an almost simple group acting primitively as a group of automorphisms on the graph. In the other case, known as the affine case, the vertices of the graph can be identified with the vectors of a finite-dimensional vector space over some finite field. In this case the automorphism group G of the graph Γ contains a normal p-subgroup N which is elementary Abelian and acts regularly on the set of vertices of Γ. Let G0 be the subgroup of G that stabilizes a vertex. Identifying the vertices of Γ with G0-cosets in G, one obtains a vector space V on which N acts as a group of translations, G0, stabilizes 0 and, as Γ is primitive, G0 acts irreducibly on V.

Throughout this paper p denotes an odd prime. The groups we shall consider are central extensions of a cyclic subgroup by Cp ⊕ Cp, and may be presented as

The group P(n) is defined for each n ≥ 3, and has order pn. The group P(3) is the non-abelian group of order p3 and exponent p. The mod-p cohomology rings of the other groups of order p3 have been known for some time (see [14] or [5] for that of the non-abelian metacyclic group), and so this paper completes the calculation of the mod-p cohomology rings of the groups of order p3.