In 1982, Glyn Harman [2] proved that for almost all n, the interval [n, n+n(1/10)+ε] contains a prime number. By this we mean that the set of n[les ]N for which the interval does not contain a prime has measure o(N) as n→+∞. It follows from Huxley's work [6] that if θ>1/6 then there will almost always be asymptotically nθ(log n)−1 primes in the interval [n, n+nθ]. In 1983, Glyn Harman [3] pointed that for almost all n, the interval [n, n+n(1/12)+ε] contains a prime number, and meantime Heath-Brown gave the outline of this result in [5]. The exponent was reduced to 1/13 by Jia [10], 2/27 by Li [12] and 1/14 by Jia [11], and meantime N. Watt [16] got the same result. In this paper we shall prove the following result.

THEOREM. For almost all n, the interval

formula here

contains a prime number.

Results of Bunge and Funk and of Johnstone, providing constructively sound descriptions of the global points of the lower and upper powerlocales, are extended here to describe the generalized points and proved in a way that displays in a symmetric fashion two complementary treatments of frames: as suplattices and as preframes. Also described here are the points of the Vietoris powerlocale.

In each of two special cases, an exponential $D ($ being the Sierpiński locale) is shown to be homeomorphic to a powerlocale: to the lower powerlocale when D is discrete, and to the upper powerlocale when D is compact regular.

It is known that, for a transcendental entire function f, the Hausdorff dimension of the Julia set of f satisfies 1[les ]dim J(f)[les ]2. In this paper we introduce a family of transcendental entire functions fp, K for which the set {dim J(fp, K)[ratio ]0<p, K<∞} has infemum 1 and supremum 2.

Given a Lie group, it is often useful to have a parametrization of the set of its lattices. In Euclidean space ℝn, for example, each lattice corresponds to a basis, and any lattice is equivalent to the standard integer lattice under an automorphism in GL(n, ℝ). In the nilpotent case, the lattices of the Heisenberg groups are classified, up to automorphisms, by certain sequences of positive integers with divisibility conditions (see [1]). In this paper we will study the set of lattices in a class of simply connected, solvable, but not nilpotent groups G. The construction of G depends on a diagonal n×n matrix Δ with distinct non-zero eigenvalues, of trace 0; we will write

formula here

In this paper, we obtain some new lower bounds for the degree of commutativity of a p-group of maximal class. These bounds show the relationship between a new invariant l(G) associated to the normal structure of G (which we use in the calculation of the defining relations of G) and the degree of commutativity.

Let Sylp(G) denote the set of Sylow p-subgroups of a finite group G. We say that G has Property Ip if, for any S∈Sylp(G) and U, V[les ]G, S∩U∈Sylp(U) and S∩V∈Sylp(V) implies S∩U∩V∈Sylp (U∩V). Below, every group is assumed to be finite.

In this paper the unitary equivalence of unbounded *-representations of *-algebras is investigated. It is shown that if closed *-representations π1 and π2 of a *-algebra [Ascr ] satisfy a certain density condition for the intertwining spaces [Jscr ](π1, π2) and [Jscr ](π2, π1), then a *-isomorphism Φ between the O*-algebras π1([Ascr ]) and π2([Ascr ]) is defined by Φ(π1(x))=π2(x), x∈[Ascr ] and it induces a *-isomorphism Φ¯, between the von Neumann algebras (π1([Ascr ])′w)′ and (π2([Ascr ])′w)′, and further if Φ¯, is spatial (that is, it is unitarily implemented), then π1 and π2 are unitarily equivalent.

For knots in S3 we obtain new criteria for periodicity. We show that the Casson–Gordon invariants of a periodic knot are preserved under the periodic action lifted to the cyclic covers. As an application, we consider a family of knots with the Seifert form of a period 3 knot, and using Casson–Gordon invariants show that knots in this family do not have period 3. We also obtain periodicity criteria in terms of the homology groups of cyclic branched covers of S3.

The present paper is a continuation of [Ga], [GL1] and [GO]. Using a key lemma we compare two currently existing definitions of finite type invariants of oriented integral homology spheres and show that type 3m invariants in the sense of Ohtsuki [Oh] are included in type m invariants in the sense of the first author [Ga]. This partially answers question 1 of [Ga]. We show that type 3m invariants of integral homology spheres in the sense of Ohtsuki map to type 2m invariants of knots in S3, thus answering question 2 from [Ga].

Examples are given of legendrian links in the manifold of cooriented contact elements of the plane, or equivalently, in the 1-jet space of the circle which are not equivalent via an isotopy of contact diffeomorphisms. These examples have generalizations to linked legendrian spheres in contact manifolds diffeomorphic to ℝn×Sn−1. These links are distinguished by applying the theory of generating functions to contact manifolds.

We measure the distance of a general shape from the set of shapes having a certain sequence of their vertices collinear and also describe the behaviour of shapes as they move along a geodesic towards this set. This models problems which arise in computer vision and other applications of shape theory.

This paper investigates the qualitative behaviour of solutions of the second order equation x¨+f(x, x¯) x¯+g(x)=0. The sufficient conditions for all solutions to be oscillatory are established. The theorems on the existence of a nontrivial periodic solution are proved and the criteria for the origin to be a global centre and globally asymptotically stable are given.

We consider the transmission problem for second-order differential equations with unbounded interfaces. The spaces B and B* are used to prove the limiting absorption principle for the steady-state problem and to establish the resolvent estimate at low frequencies for the steady-state operator. These are then applied in showing the validity of the limiting amplitude principle for such an operator.

We generalize the Novikov inequalities for 1-forms in two different directions: first, we allow non-isolated critical points (assuming that they are non-degenerate in the sense of R. Bott) and, secondly, we strengthen the inequalities by means of twisting by an arbitrary flat bundle. The proof uses Bismut's modification of the Witten deformation of the de Rham complex; it is based on an explicit estimate on the lower part of the spectrum of the corresponding Laplacian.

In particular, we obtain a new analytic proof of the degenerate Morse inequalities of Bott.

It is shown that the spectrum of a one-dimensional Dirac operator with a potential q tending to infinity at infinity, and such that the positive variation of 1/q is bounded, covers the whole real line and is purely absolutely continuous. An example is given to show that in general, pure absolute continuity is lost if the condition on the positive variation is dropped. The appendix contains a direct proof for the special case of subordinacy theory used.