Let q and N be integers, let a be an integer coprime to q, and let zN be defined implicitly by q = (log N)log22−ZN √(log2N). We show that for large N, an integer n has at least one divisor d with q [les ] d [les ] N and d ≡ a(mod q) with probability approximately Φ(zN), where Φ denotes the distribution function of the Gaussian Law. This solves a conjecture of Hall.

Infinitesimal conformal transformations of ℝn are always polynomial and finitely generated when n > 2. Here we prove that the Lie algebra of infinitesimal conformal polynomial transformations over ℝn, n > 2, is maximal in the Lie algebra of polynomial vector fields. When n is greater than 2 and p, q are such that p + q = n, this implies the maximality of an embedding of so(p + 1, q + 1, ℝ) into polynomial vector fields that was revisited in recent works about equivariant quantizations. It also refines a similar but weaker theorem by V. I. Ogievetsky.

The purpose of this note is initially to present an elementary but surprising connectedness principle pertaining to the intersection of a fixed subvariety X of some ambient space Z with another subvariety Y which is ‘mobile’ (in the sense of being movable, rather than actually moving). It is via this mobility that monodromy enters the picture, permitting the crucial passage from ‘relative’ or total-space irreducibility to ‘absolute’ or fibrewise connectedness (and sometimes irreducibility). A general form of this principle is given in Theorem 2 below.

Suppose that M is a finite module under the Galois group of a local or global field. Ever since Tate's papers [17, 18], we have had a simple and explicit formula for the Euler–Poincaré characteristic of the cohomology of M. In this note we are interested in a refinement of this formula when M also carries an action of some algebra [Ascr ], commuting with the Galois action (see Proposition 5.2 and Theorem 5.1 below). This refinement naturally takes the shape of an identity in a relative K-group attached to [Ascr ] (see Section 2). We shall deduce such an identity whenever we have a formula for the ordinary Euler characteristic, the key step in the proof being the representability of certain functors by perfect complexes (see Section 3). This representability may be of independent interest in other contexts.

Our formula for the equivariant Euler characteristic over [Ascr ] implies the ‘isogeny invariance’ of the equivariant conjectures on special values of the L-function put forward in [3], and this was our motivation to write this note. Incidentally, isogeny invariance (of the conjectures of Birch and Swinnerton-Dyer) was also a motivation for Tate's original paper [18]. I am very grateful to J-P. Serre for illuminating discussions on the subject of this note, in particular for suggesting that I consider representability. I should also like to thank D. Burns for insisting on a most general version of the results in this paper.

Let p be a prime number, and let G be a p-group which is not elementary abelian. For every mod-p cohomology class ξ of G which restricts trivially to all proper subgroups, we show that ξp = 0. This gives upper bounds for nilpotency degrees of such classes of G and of nilpotent mod-p cohomology classes of finite groups.

Let F be a field, and let Σn be the symmetric group on n letters. In this paper we address the following question: given two irreducible FΣn-modules D1 and D2 of dimensions greater than 1, can it happen that D1 [otimes ] D2 is irreducible? The answer is known to be ‘no’ if char F = 0 [12] (see also [2] for some generalizations). So we assume from now on that F has positive characteristic p. The following conjecture was made in [4].

CONJECTURE. Let D1and D2be two irreducible FΣn-modules of dimensions greater than 1. Then D1 [otimes ] D2 is irreducible if and only if p = 2, n = 2 + 4l for some positive integer l; one of the modules corresponds to the partition (2l + 2, 2l) and the other corresponds to a partition of the form (n − 2j − 1, 2j + 1), 0 [les ] j < l. Moreover, in the exceptional cases, one has

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The main result of this paper is the following theorem, which establishes a big part of the conjecture.

In his book Abelian groups, L. Fuchs raised the question as to whether, in general, in the factorization of a finite (cyclic) abelian group one factor may always be replaced by some subgroup. The answer turned out to be negative in general, but positive in certain cases. In this paper the complete answer for cyclic groups is given. In all previously unsolved cases, the answer turns out to be positive. It is shown that a cyclic group has the property that in every factorization, one factor may be replaced by a subgroup if and only if the group has order equal to the product of a prime and a square-free integer. Certain results are also given in non-cyclic cases.

Let μ be a real number. The Möbius group Gμ is the matrix group generated by

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It is known that Gμ is free if [mid ]μ[mid ] [ges ] 2 (see [1]) or if μ is transcendental (see [3, 8]). Moreover, there is a set of irrational algebraic numbers μ which is dense in (−2,2) and for which Gμ is non-free [2, p. 528]. We may assume that μ > 0, and in this paper we consider rational μ in (0, 2). The following problem is difficult.

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Let [Gscr ] denote the set of all rational numbers μ in (0, 2) for which Gμ is non-free. In 1969 Lyndon and Ullman [8] proved that [Gscr ]nf contains the elements of the forms p/(p2 + 1) and 1/(p + 1), where p = 1, 2, …, and that if μ0 ∈ [Gscr ]nf then μ0/p ∈ [Gscr ]nf for p = 1, 2, …. In 1993 Beardon [2] studied problem (P) by means of the words of the form ArBsAt and ArBsAtBuAv, and he obtained a sufficient condition for solvability of (P), included implicitly in [2, pp. 530–531], by means of the following Diophantine equations:

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Using properties of the Steinberg character, we obtain a congruence modulo p for the number of ways in which a p-regular element may be expressed as a commutator in a finite simple group G of Lie type of characteristic p. This congruence shows that such an element is a commutator in G. We also show that if K and L are conjugacy classes of G consisting of elements whose centralizers have order relatively prime to p, then any p-regular element of G is expressible as the product of an element of K and an element of L.

Functions with bounded variation and with a (total) variation are examined within Bishop's constructive mathematics. It is shown that the property of having a variation is hereditary downward on compact intervals, and hence that a real-valued function f with a variation on a compact interval can be expressed as a difference of two increasing functions. Moreover, if f is sequentially continuous, then the corresponding variation function, and hence f itself, is uniformly continuous.

For f a meromorphic function on the plane domain D and a ∈ [Copf ], let Ēf(a) = {z ∈ D[ratio ]f(z) = a}. Let [Fscr ] be a family of meromorphic functions on D, all of whose zeros are of multiplicity at least k. If there exist b ≠ 0 and h > 0 such that for every f ∈ [Fscr ], Ēf(0) = Ēf(k)(b) and 0 < [mid ]f(k+1)(z)[mid ] [les ] h whenever z ∈ Ēf(0), then [Fscr ] is a normal family on D. The case Ēf(0) = Ø is a celebrated result of Gu [5].

We consider the Abel equation

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where p and q are polynomials, and a is a fixed constant. We denote the solution of (1) by y(x, c), where y(a, c) = c. Standard existence theorems ensure that y(x, c) is well-defined and analytic in both its arguments, for c sufficiently small. If y(b, c) = c, then we call y(x, c) a periodic solution. Likewise, if y(b, c) ≡ c for all c close to 0, then we say that the system has a centre between a and b. The numbers a and b are not important; by a simple transformation, we can always choose a = 0 and b = 1, and we shall usually do so from now on.

Abel equations arise in several circumstances, but perhaps the main reason for their recent study is connected to the family of systems

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where M and N are homogeneous polynomials of the same degree n. A transformation due to Cerkas allows us to bring these systems to the form (1), where p and q are now trigonometric polynomials. It is not hard to show that, setting a = 0 and b = 2π, the definitions of periodic solution and centre for (1) coincide with their usual definitions in the planar system (2). There are also transformations to Abel-type equations for more general systems; see [8, 10].

This trigonometric Abel equation has been used in a large number of works in order to estimate the number of limit cycles or obtain centre conditions, as well as in more general investigations relating the derivatives of the return map with iterated integrals. However, studying system (2) in whatever form is by no means easy, and a natural question is to ask whether we can still capture the essence of this problem if we take p and q to be polynomials, in the hope that the calculations will become easier. The recent series of investigations by Briskin, Françoise and Yomdin [3, 4, 5] seems to indicate that this could be the case.

Our interest here is to see what conditions the existence of a centre in (1) imposes on the defining equations. For ease of reference, we shall always denote the antiderivative of the polynomials p and q as P and Q; that is,

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Let Ω be a pseudoconvex domain in C2 with smooth boundary, and let A ⊆ C2 [setmn ] Ω be a smooth embedded analytic disc intersecting ∂Ω transversally along the curve ∂A. Then ∂A is knotted in ∂Ω.

It is shown that Voiculescu's topological entropy for the canonical endomorphism of a simple Cuntz–Krieger algebra [Oscr ]A equals the logarithm of the spectral radius of A.

We study the behaviour of non-negative solutions to the equation

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This equation is an ‘explicitly’ soluble special case of a class of non-local evolution equations, of which the behaviour of solutions has been studied by Uchiyama [9]. In this paper, certain fine properties of solutions such as a local relaxation within their supports are obtained for the present special case. The asymptotic forms (for large time) of the solutions whose initial measures have compact supports are identified within the error of the order O(t−3/2). It is also shown that the comparison principle does not hold. With a simple change of variables, the equation ([midast ]) is transformed to the equation arising in Dyson's approach to the Wigner semi-circle law of eigenvalues of random matrices, and the present results have immediate consequences on the latter.

Given an oriented manifold and its immersion in a euclidean space, we compute the oriented cobordism class of the manifold of [sum ]1r singular points of the projection of the immersion to a hyperplane. For immersions of non-oriented manifolds, we show that the cobordism class of the domain manifold determines those of all [sum ]1r singularity manifolds of the hyperplane projection. Finally, we investigate the possible (algebraic) number of cusps (that is, [sum ]1,1 singular points) of generic maps of oriented 4t-manifolds in R6t−1.