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

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.

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.

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.

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.

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.)

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.

In this note we prove universal coefficient theorems for Borel cohomology and related theories. Whatever other merit this may have the comment of Borel [5] applies ‘ …elle a au moms l'utilité de bien mettre en évidence le rôle fondamental joué dans cette question par la cohomologie des groupes’.

Indeed the purpose of the enterprise is to use homological properties of the group cohomology ring H*(BG+) to study properties of G-spaces. Because of the relative simplicity of ordinary cohomology much attention in the proofs and applications is concentrated on change of groups, and on changes in the way the group action is exploited. Nonetheless we are able to adapt the non-equivariant approach of Adams ([1, 2]; see also [3]). Thus the existence of universal coefficient theorems automatically gives Kiinneth theorems as special cases.

Let G be a connected Lie group and let {λi} be a sequence of probability measures on G converging (in the usual weak topology) to a probability measure λ. Suppose that {αi} is a sequence of affine automorphisms of G such that the sequence {αi,(λi)} also converges, say to a probability measure μ. What does this imply about the sequence {αi}? It is a classical observation that if G = ℝn for some n, and neither of λ and μ is supported on a proper affine subspace of ℝn, then under the above condition, {αi} is relatively compact in the group of all affine automorphisms of ℝn.

For a compact irreducible 3-manifold containing 2-sided projective planes and admitting a ℤ2-action, we consider the problem of when there exists a ℤ2-equivariant hierarchy. We show that a ℤ2-equivariant hierarchy always exists if either every irreducible summand of the orientable double cover has infinite first homology or every 2-sided projective plane is boundary parallel.

Casson–Gordon invariants were first used to prove that certain algebraically slice knots in S3 are not slice knots [2, 3]. Since then they have been applied to a wide range of problems, including embedding problems and questions relating to boundary links [2, 10, 21, 25]. The most general Casson–Gordon invariant takes its value in L0(ℚ(ζd)(t)) ⊗ ℚ; here ζd denotes a primitive dth root of unity. Litherland [20] observed that one could usually tensor with ℤ(2) instead of ℚ, and in this way preserve the 2-torsion in the Witt group. He then constructed new examples of non-slice genus two knots which were detected with torsion classes in L0(ℚ(ζd)) ⊗ ℤ(2) modulo the image of L0(ℚ(ζd)) ⊗ ℤ(2).

We prove that the series

are irrational and not Liouville whenever q is an integer (q ╪ 0, ±1) and r is a nonzero rational (r ╪ −qn).

For a bounded open set U ⊂ ℂ, we denote by H∞(U) the collection of all bounded analytic functions on U. We let X denote bdy (U), the boundary of U, Y denote the polynomial hull of U (the complement of the unbounded component of ℂ / X), and U* denote mt (Y), the interior of Y. We denote the sup norm of a function f: A → ℂ by ∥f∥A:

We denote the space of all analytic polynomials by ℂ[z], and we denote the open unit disc by D and the unit circle by S1.

Compact homomorphisms between Banach algebras are not particularly unusual – for example if A is the algebra of complex sequences {an} with nan → 0 as n → ∞ and ∥{an}∥ = supn(nan) and B = c0 then the embedding map ø:A → B is a compact homomorphism. Nevertheless for really well behaved Banach algebras it is often the case that the only weakly compact homomorphisms are those with a finite-dimensional range. A number of interesting results in this area appear in [1] and elsewhere. The purpose of this paper is to extend those results.

Let Λ be a perfect Köthe function space in the sense of Dieudonné, and Λ× its Köthe-dual. Let E be a normed space. Then the topological dual of the space Λ(E) of Λ-Bochner integrable functions equals the corresponding Λ×(E′) if and only if E′ has the Radon–Nikodým property. We also give some results concerning barrelledness for spaces of this kind.

For certain Fréchet convolution algebras associated with a weight w on the half-line [0, ∞), we are interested in the question of which Radon measures on [0, ∞) determine the derivations on the algebra. For particular weights, we show that the derivations are determined by those measures in the multiplier algebra of the original algebra. However, we also give an example of a weight for which that characterization fails. The results show a connection between the space of derivations and the behaviour for large y of the ratio w(x + y)/w(x) w(y).

Subspaces of Lp spanned by symmetric independent identically distributed random variables were identified as Orlicz spaces by Bretagnolle and Dacunha-Castelle[1], who showed that, conversely, in the case p ≤ 2, every p-convex, 2-concave Orlicz space is isomorphic to a subspace of Lp. This was extended by Dacunha-Castelle [3] to subspaces of Lp with symmetric basis, which appear as ‘p-means’ of Orlicz spaces (see [9] for the corresponding finite-dimensional result, and [12] for the case of rearrangement invariant function spaces). On the contrary the only subspaces with symmetric basis of Lp for p ≥ 2 are lp and l2 (if one does not care about isomorphy constants).

We obtain weak and strong Gaussian approximations for logarithmic averages of indicators of normalized partial sums. The proofs are based on invariance principles for integrals of an Ornstein–Uhlenbeck process and on strong approximations of normalized partial sums by Orstein–Uhlenbeck processes.

In this paper, we consider the diffraction problem for wave propagation in inhomogeneous, penetrable media with an unbounded interface. The low frequency behaviour of the resolvent for the reduced wave operator is studied, and the validity of the limiting-amplitude principle for such an operator is proved.