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.

Let X be a curve with an involution T which fixes r points. We show that the Weierstrass weight of a fixed point is at least (r−2)(r−4)/8. Our proof is independent of the recent result of Torres.

We consider the case where X=Fn, the nth Fermat curve, and T is any of the involutions of Fn. We find that our bound is equal to the actual weight in all known cases (n[les ]7) and compute then n=8 case to demonstrate that the equality continues to hold.

Let K=ℚ(√−3). Assume E is an elliptic curve defined over K, with complex multiplication by OK, the ring of integers of K. Ω∈[Copf]X is a minimal period of E. For each prime v of K, let cv denote the Tamagawa number [E(Kv)[ratio ]E0(Kv)]. Let L(E/K, s) denote the Hasse–Weil L-function of E/K. [lfloor][mid ][rfloor](E/K)=ker{H1(K, E)→[oplus ]vH1(Kv, E)} denotes the Tate-Shafarevic group of E/K.

The purpose of this paper is to prove irrationality results from Jacobi's triple product identity, which can be written, for x∈[Copf]*, y∈[Copf], [mid ]y[mid ]<1:

formula here

There are various proofs of this identity; the classical one rests on the theory of theta functions ([3], theorem 6, p. 69). An alternative proof uses Heine's summation formula ([10], p. 12). An elementary, self-contained proof, can be found in [9], p. 227.

In this paper, we will use the same elementary methods as in [5] and [6], and prove the following theorems.

In recent years there has been an interest in using elliptic logarithms to find integral points on elliptic curves defined over the rationals, see [23], [17], [6] and [12]. This has been partly due to work of David [5], who gave an explicit lower bound for linear forms in elliptic logarithms. Previously, integral points on elliptic curves had been found by Siegel's method; that is, a reduction to a set of Thue equations which could be solved, in principle, by the methods in [19]. For examples of this method see [3], [7], [16], [18], [21], [22] and [8]. Other techniques can be used to find all integral points in some special cases, see, for instance, [14].

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.

We prove Du's positivity conjecture for the canonical basis of the q-Schur algebra, using elementary arguments and the positivity result for Lusztig's canonical basis for U+q(sln). We also describe a family of subalgebras of the q-Schur algebra, each of which is spanned by the canonical basis elements it contains.

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.

Triquotient maps of topological spaces were introduced by E. Michael as a natural generalization of both open and proper surjections. We introduce the notion of localic triquotient map. Our main result is that localic triquotient maps are effective descent maps. This generalizes the corresponding results for proper surjections (J. Vermeulen) and open surjections (A. Joyal and M. Tierney). Further results concern stability of triquotiency under various operations, for instance, arbitrary products and filtered (inverse) limits. Among the applications are a new constructive proof of Tychonoff's theorem, and a new result on stability of open surjections under filtered limits.

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

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Let [Sfr ]n and [Afr ]n denote the symmetric and alternating groups of degree n∈ℕ respectively. Let p be a prime number and let F be an arbitrary field of characteristic p. We say that a partition of n is p-regular if no p (non-zero) parts of it are equal; otherwise we call it p-singular. Let SλF denote the Specht module corresponding to λ. For λ a p-regular partition of n let DλF denote the unique irreducible top factor of SλF. Denote by ΔλF=DλF↓[Afr ]n its restriction to [Afr ]n. Recall also that, over F, the ordinary quiver of the modular group algebra FG is a finite directed graph defined as follows: the vertices are labelled by the set of all simple FG-modules, L1, ..., Lr, and the number of arrows from Li to Lj equals dimFExtFG(Li, Lj). The quiver gives important information about the block structure of G.

Let PI (respectively CPI) be the convex compact set of all unital positive (respectively completely positive) linear maps from the matrix algebra Mm([Copf]) into Mn([Copf]). We show that maximal faces of CPI correspond to one dimensional subspaces of the vector space Mm, n([Copf]). Furthermore, a maximal face of CPI lies on the boundary of PI if and only if the corresponding subspace is generated by a rank one matrix.

The theory of reductive monoids has been well developed by Renner and the author, cf. [8, 13]. This paper concerns the finite reductive monoids MF where M is a reductive monoid and F[ratio ]M→M is a Frobenius map. We show that the Deligne–Lusztig characters RT, θ of finite reductive groups GF have natural extensions to MF. We accomplish this by first showing that the action of a finite monoid (by partial transformations) on an algebraic variety gives rise to a virtual character of the monoid via étale cohomology. We then find the correct analogue of the algebraic set L−1(U) (where L is the Lang map) for M. The action of MF on this algebraic variety gives rise to virtual characters RT¯, θ where T is an F-stable maximal torus of G, T¯ the Zariski closure of T in M and θ an irreducible character of the finite commutative monoid T¯F. We go on to show that any irreducible character of MF is a component of some RT¯, θ.

Let R be a commutative ring, Γ a finite group acting on R, and let k∈ℕ be invertible in R. Generalizing a definition of Kervaire, we construct an Adams operation ψk on the Grothendieck group and on the higher K-theory of projective modules over the twisted group ring R#Γ. For this, we generalize Atiyah's cyclic power operations and use shuffle products in higher K-theory. For the Grothendieck group, we show that ψk is multiplicative and that it commutes with base change, with the Cartan homomorphism, and with ψl for any other l which is invertible in R.

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.

We study the question of whether the Morava K-theory of the classifying space of an elementary abelian group V is a permutation module (in either of two distinct senses, defined below) for the automorphism group of V. We use Brauer characters and computer calculations. Our algorithm for finding permutation submodules of modules for p-groups may be of independent interest.

A classical fact is that Seifert manifolds with non-empty boundary are covered by surface bundles over the circle S1 and closed Seifert manifolds may or may not be covered by surface bundles over S1. Some closed graph manifolds are not covered by surface bundles over S1 ([LW] and [N]). Thurston asked if complete hyperbolic 3-manifolds of finite volume are covered by surface bundles [T]. J. Luecke and Y. Wu asked if graph manifolds with non-empty boundary are covered by surface bundles over S1 ([LW]). In this paper we prove:

THEOREM 0·1. Each graph manifold with non-empty boundary is finitely covered by a surface bundle over the circle S1.

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.

In [7], the second author proved that there is an integer k such that every element of a finite non-abelian simple group S is a product of k commutators in S. The motivation for proving this result came from a model-theoretic question about simple groups. The proof depended on the classification of the finite simple groups, a theorem of Malle, Saxl and Weigel [5] which shows that in many finite simple classical groups S there is a real conjugacy class R such that S=R3∪{1}, and an ultraproduct argument. Here we shall use a similar combination of ideas to prove the following result.