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].

Let G be the group given by the following presentation:

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The subgroup generated by ab is infinite-cyclic and normal, with quotient the dihedral group of order 6, so G is cyclic-by-finite. The subgroups H = 〈a, c〉 and K = 〈b, c〉 are both dihedral of order 6, and G is isomorphic to the free product of H and K amalgamating L = H∩K. We study K0(kG), the Grothendieck group of isomorphism classes of finitely generated projective kG-modules, and, in particular, the dependence of K0(kG) on the choice of field k. As usual, let ℚ, ℝ and [Copf ] stand for the rationals, reals and complex numbers, respectively. We prove the following.

We show that the Suzuki group Sz(32) is a subgroup of E8(5), and so is its automorphism group. Both are unique up to conjugacy in E8([ ]) for any field [ ] of characteristic 5, and the automorphism group Sz(32)[ratio ]5 is maximal in E8(5).

Mather has shown that almost all linear projections of a smooth submanifold of a vector space to a subspace yield a generic mapping (so in the nice dimensions, most linear projections give rise to ‘stable’ mappings). In this paper we show that the same is true if we replace the submanifold by the image of a stable mapping.

We prove weak laws of large numbers for freely independent, uniformly bounded and non-identically distributed random variables belonging to non-commutative Ciach and, in particular, Lorentz Lp-spaces.

Given a principal ideal domain R of characteristic zero, containing 1/2, and a two-cone X of appropriate connectedness and dimension, we present sufficient algebraic conditions for the Hopf algebra FH(ΩX; R) to be isomorphic with the universal enveloping algebra of an R-free graded Lie algebra; here, F stands for free part (that is, quotient by the R-torsion), H for homology, and Ω for the Moore loop space functor.

We say that a Hopf algebra is copolynomial if its dual is polynomial as an algebra. We re-derive Milnor's result that the mod 2 Steenrod algebra is copolynomial by means of a more general result that is also applicable to a number of other related Hopf algebras.

In this note, we show that a homotopy class of [[ ]3, [ ]2] admits an [ ]1-invariant harmonic map if and only if its Hopf invariant is ±k2 for some integer k, where [ ]3 is the unit 3-sphere and [ ]2 is a 2-sphere with an arbitrary metric.

We prove upper and lower heat kernel bounds for the Laplacian on weighted graphs which include the case that the weights have no strictly positive lower bound. Our estimates give rise to a very explicit probabilistic interpretation, and can be formulated in terms of a weighted metric. Interestingly, this metric is not equivalent to the intrinsic metric.

Answering a question of Gromov [7], we shall present an example of a finitely generated group Γ and two non-principal ultrafilters [Ascr ], [Bscr ] such that the asymptotic cones Con[Ascr ] Γ and Con[Bscr ] Γ are not homeomorphic.

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 G be either a free product with amalgamation A[midast ]CB or an HNN group A[midast ]C, where all normal subgroups of C are finitely generated. Suppose that both A and B have no non-trivial finitely generated normal subgroups of infinite indices. We show that if G contains a finitely generated normal subgroup N which intersects A or B non-trivially but is not contained in C, then the index of N in G is finite.

Philip Holgate was born in Chesterfield on 8 December 1934. His family moved from Derbyshire to Devon in 1945, and he was educated at Newton Abbot Grammar School from 1945 to 1952, and the University College of the South-West (now Exeter University, but which then awarded London degrees) from 1952 to 1955. He qualified as both a teacher (King's College, London, 1955–56) and a statistician (University College, London, 1956–57).

After teaching mathematics and physics at Burgess Hill School, in Hampstead and then in Borehamwood, Philip joined the Statistics Section at Rothamsted Experimental Station (1961–62). He then spent five years in the Biometrics Section of the Nature Conservancy. His first publications date from this period, and the interests he acquired then were to develop into what became his most enduring, and distinctive, scientific interests.

Philip joined the Department of Statistics, Birkbeck College, University of London, in 1967, and remained at Birkbeck for the rest of his career, until his death from a heart attack on 13 April 1993.

In a further generalisation of the classical Ito formula, we show that if B = (At + A[dagger]t [ratio ] t ∈ [0, 1]) is the Brownian quantum martingale and J = (Jt [ratio ] t ∈ [0, 1]) is a bounded quantum semimartingale, then M = (Bt + Jt [ratio ] t ∈ [0, 1]) satisfies the functional quantum Ito formula [10, Section 6]

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The proof, which relies on the classical theory of unbounded self-adjoint operators, may be adapted to the case where B is replaced by a classical Brownian martingale.

Let Γ be a group having finite virtual cohomological dimension. Let BΓ be its classifying space, and let C(Γ) be its Quillen category, whose objects are elementary abelian 2-subgroups V of Γ and whose morphisms are restrictions of conjugacy of Γ. The group Γ is called a Quillen group if the natural map

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is an isomorphism. The symmetric groups (see [8]) and the Coxeter groups of finite order (see [7]) are examples of Quillen groups.

J. H. Gunawardena, J. Lannes and S. Zarati obtained the characterization of Quillen groups in terms of unstable modules over the mod 2 Steenrod algebra [8] (see Proposition 2 below). By making use of this result, M. Errokh and F. Grazzini obtained the classification of Coxeter groups on three generators which are Quillen groups [8]. The purpose of the present paper is to obtain a necessary and sufficient condition for aspherical Coxeter groups to be Quillen groups (Theorem 1). Our result is very simple, and the previously mentioned result by Errokh and Grazzini follows easily from ours (see Example 1).

The paper is organized as follows. In Section 2, after giving relevant definitions concerning Coxeter groups, we shall state our main result. The next two sections are devoted to preliminaries. In Section 3, we shall review necessary definitions and facts concerning unstable modules over the mod 2 Steenrod algebra. In Section 4, we shall discuss the Leray spectral sequences converging to the cohomology of Coxeter groups. The proof of Theorem 1 will be given in Section 5. In Section 6, we shall give examples illustrating Theorem 1.

Notation. The cardinality of a finite set X is denoted by [mid ]X[mid ].

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.

Let Λ and Γ be finite dimensional algebras. It is shown that any stable equivalence f [ratio ] mod Λ → mod Γ between the categories of finitely generated modules induces a bijection M [map ] Mf between the sets of isomorphism classes of generic modules over Λ and Γ such that the endolength of Mf is bounded by the endolength of M up to a scalar which depends only on f. Using Crawley-Boevey's characterization of tame representation type in terms of generic modules, one obtains as a consequence a new proof for the fact that a stable equivalence preserves tameness. This proof also shows that polynomial growth is preserved.

If u is a superharmonic function on ℝ2, then

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for all (x, y) ∈ ℝ2. This follows from the fact that a line segment in ℝ2 is non-thin at each of its constituent points. (See Doob [1, 1.XI] or Helms [7, Chapter 10] for an account of thin sets and the fine topology.) The situation is different in higher dimensions. For example, if u is the Newtonian potential on ℝ3 defined by

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then

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Corollary 2 below will show that, nevertheless, for nearly every vertical line L, the value of a superharmonic function at any point X of L is determined by its lower limit along L at X.

Throughout this paper, we let n [ges ] 3. A typical point of ℝn will be denoted by X or (X′, x), where X′ ∈ ℝn−1 and x ∈ ℝ. Given any function f[ratio ]ℝn → [−∞, +∞] and any point X, we define the vertical cluster set of f at X by

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and the fine cluster set of f at X by

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