We write ‖x‖ to denote the least distance from x to an integer, and write p for a prime variable. Duffin and Schaeffer [l] showed that for almost all real α the inequality

has infinitely many solutions if and only if

diverges. Thus f(x) = (x log log (10x))−1 is a suitable choice to obtain infinitely many solutions for almost all α. It has been shown [2] that for all real irrational α there are infinitely many solutions to (1) with f(p) = p−/13. We will show elsewhere that the exponent can be increased to 7/22. A very strong result on primes in arithmetic progressions (far stronger than anything within reach at the present time) would lead to an improvement on this result. On the other hand, it is very easy to find irrational a such that no convergent to its continued fraction expansion has prime denominator (for example (45– √10)/186 does not even have a square-free denominator in its continued fraction expansion, since the denominators are alternately divisible by 4 and 9).

A polynomial system is a real autonomous system of ordinary differential equations on the plane with polynomial nonlinearities:

with aij, bij ∈ ℝ and where x = x(t) and y = y(t) are real-valued functions.

The problem of analysing limit cycles (isolated periodic solutions) in polynomial systems was first discussed by Poincaré[16]. Then, in the famous list of 23 mathematical problems stated in 1900, Hilbert[9] asked in the second part of the 16th problem for an upper bound for the number of limit cycles for nth degree polynomial systems, in terms of n. Recently, it has been proved that, given a particular choice of coefficients for a system of form (1·1), the number of limit cycles is finite. This result is known as Dulac's theorem, see Ecalle[8] or Il'yashenko[10]. However, it is unknown whether or not there exists an upper bound for the number of limit cycles in system (1·1) in terms of n. Even for quadratic systems (i.e. polynomial systems with quadratic nonlinearities) this remains an open question.

Estimates for rational trigonometric sums are of great importance in analysing the local aspects of many additive problems. Indeed, bounds for the sums

in which e(α) denotes exp (2πiα), play a fundamental rôle in the application of the Hardy–Littlewood method to Waring's problem (see [11]), and also in the analysis of the local solubility of systems of additive equations (see, for example, [2]). When k ≥ 2 is an integer, and p is a prime number it is well known (see [5] or [11, lemma 4·3]) that

We define a notion of group functor G on categories of graded modules, which unifies previous concepts of a group functor G possessing a notion of elementary subfunctor E. We show under a general condition which is easily checked in practice that the elementary subgroup E(M) of G(M) is normal for all quasi-weak Noetherian objects M in the source category of G. This result includes all previous ones on Chevalley and classical groups G of rank ≥ 2 over a commutative or module finite ring M (since such rings are quasi-weak Noetherian) and settles positively unanswered cases of normality for these group functors.

A class of groups that is closed with respect to subnormal subgroups and normal products is called a Fitting class. Given a finite soluble group G, one may ask for the Fitting class (G) generated by G, that is the intersection of all Fitting classes containing G. For simple or nilpotent groups G it is easy to compute (G), but in other cases the determination of (G) seems to be surprisingly difficult, and there is no general method of solving this problem. In recent years there has been a lot of work in this area, see for instance Bryce and Cossey[l], [2], Hawkes[6] (or [5], IX. 9. Var. II), Heineken[7] and McCann[10].

In this paper we prove a conjecture of Smith's on the number of simple modules of algebras which are similar to the enveloping algebra of sl(2).

In this note, we construct a sequence of l.t. surfaces (Xn)n ∈ ℕ such that KXn is ample for all n and such that (K2Xn)n ∈ ℕ is a strictly increasing series with limit equal to 1. This answers (in the affirmative) a question by Alexeev, cf. [Al], 11·1. Here, an l.t. surface is a normal complex projective surface with at most quotient singularities (which is the same as ‘at most log terminal singularities’). A main result of [Al] implies that it is impossible to find a sequence (Xn)n ∈ ℕ of l.t. surfaces with KXn ample for all n such that K2Xn is strictly decreasing. Although our construction is not too difficult, the example is new and has several interesting implications, see Section 4.

Without further explanation, we use some fundamental tools concerning l.t. surfaces like Mumford's intersection theory or the notion of minimality; the reader should consult [Blb] and the references quoted there.

In 1962 Kodaira[11] proved that ifX ↪ Y is a compact complex submanifold with normal bundle N such that H1(X, N) = 0, then X belongs to a locally complete family {Xt: t ∈ M} of complex submanifolds Xt of Y with the moduli space M being a (dimcH0(X, N))-dimensional complex manifold, and there exists a canonical isomorphism

between the tangent space of M at a point t ∈ M and the space of all global sections of the normal bundle Nt of the embedding Xt ↪ Y.

For each integer n ≥ 0, we give a distinct closed model category structure to the categories of spaces and of simplicial sets. Recall that a non-empty map is said to be a weak equivalence if it induces isomorphisms on the homotopy groups for any choice of base point. Putting the condition on dimensions ≥ n, we have the notion of a weak n-equivalence which is at the base of the nth closed model category structure given here.

For any topological space X and integer n ≥ 1, denote by Cn(X) the configuration space

The symmetric group Sn acts by permuting coordinates on Cn(X) and we are concerned in this note with the induced graded representation of Sn on the cohomology space H*(Cn(X)) = ⊕iHi (Cn(X), ℂ), where Hi denotes (singular or de Rham) cohomology. When X = ℂ, Cn(X) is a K(π, 1) space, where π is the n-string pure braid group (cf. [3]). The corresponding representation of Sn in this case was determined in [5], using the fact that Cn(C) is a hyperplane complement and a presentation of its cohomology ring appears in [1] and in a more general setting, in [8] (see also [2]).

For a function f ∈ Lp(ℝd) we define the spherical means

where dσ is the rotationally invariant measure on Sd−1, normalized such that σ(Sd−1) = 1. We consider the problem of pointwise convergence of the means , for any particular sequence tj → 0.

This paper studies the homology of the homotopy quotient of a G-manifold. We start by formulating a complex based on differential forms to compute this homology. This leads to a topological version of integration formulas over fibred products of G-manifolds. In particular the TQFT formulas studied by Witten and Atiyah–Jeffrey now appear as a pairing of homology and cohomology classes. We compare our construction with both the double complex of Axelrod and Witten and with the distributional complex of Duflo and Vergne.

A compact, orientable 3-manifold M is called hyperbolic if int M admits a complete hyperbolic structure (Riemannian metric of constant curvature − 1) of finite volume. Any hyperbolic 3-manifold M is irreducible, and each component of ∂M is an incompressible torus. Let f: M → N be a proper, continuous map between hyperbolic 3-manifolds. By Mostow's Rigidity Theorem [8], if f is π1-isomorphic then f is properly homotopic to a diffeomorphism g: M → N such that g | int M: int M → int N is isometric. In particular, the topological type of int M determines uniquely the hyperbolic structure on int M up to isometry, so the volume vol (int M) of int M is well-defined. This Rigidity Theorem is generalized by Thurston[11, theorem 6·4] so that any proper, continuous map f:M → N between hyperbolic 3-manifolds with vol(int M) = deg(f) vol(int N) is properly homotopic to a deg(f)-fold covering g:M → N such that g | int M: int M → int N is locally isometric.

One of the most useful theorems for proving the existence of fixed points, or more generally, periodic points of a continuous self-map f of a compact manifold, is the Lefschetz fixed point theorem. When studying the periodic points of f it is convenient to use the Lefschetz zeta function Zf(t) of f, which is a generating function for the Lefschetz numbers of all iterates of f. The function Zf(t) is rational in t and can be computed from the homological invariants of f. See Section 2 for a precise definition. Thus there exists a relation, based on the Lefschetz fixed point theorem, between the periodic points of a self-map of a manifold f:M → M and the properties of the induced homomorphism f*i on the homology groups of M. This relation has been used in several papers, namely [F1], [F2], [F3] and [M]. In these papers, sufficient conditions are given for the existence of infinitely many periodic points in the case when all the zeros and poles of the associated Lefschetz zeta function are roots of unit. Here we restrict ourselves to maps defined on manifolds with a certain homology type. For transversal maps f defined on this class of manifolds, it is possible to extend the techniques introduced in [F1], [F3] and [M] in order to obtain information on the set of periods of f. We recover the above mentioned results of J. Franks and T. Matsuoka, and derive new results on the set of periods of f when the associated Lefschetz zeta function has zeros or poles outside the unit circle.

Let X be a smooth projective variety of dimension n over the field of complex numbers. We denote by Kx the canonical bundle of X. By Mori's theory, if Kx is not numerically effective (i.e. if there exists a curve on X which has negative intersection number with Kx), then there exists an extremal ray ℝ+[C] on X and an elementary contraction fR: X → Y associated with ℝ+[C].fR is called a small contraction if it is bi-rational and an isomorphism in co-dimension one.

Curves of genus 2 with good reduction away from 2 with a rational Weierstrass point

By J. R. Merriman and N. P. Smart

(Volume 114 (1993) 203–214)

A case had been omitted in the calculation of Table 1 (p. 212). This we now correct. Table 1 should contain the following additional 3 lines.