Let G be a finite group, let V be an [ ]G-module of finite dimension d, and denote by β(V, G) the minimal number m such that the invariant ring S(V)G is generated by finitely many elements of degree at most m. A classical result of E. Noether says that β(V, G)[les ][mid ]G[mid ] provided that char [ ] is coprime to [mid ]G[mid ]!. If char [ ] divides [mid ]G[mid ], then no bounds for β(V, G) are known except for very special choices of G. In this paper we present a constructive proof of the following. If H[les ]G with [G[ratio ]H]∈[ ]*, and if the restriction V[mid ]H is a permutation module (for example, if V is a projective [ ]G-module and H∈Sylp(G)), then β(V, G)[les ]max{[mid ]G[mid ], d([mid ]G[mid ]−1)} regardless of char [ ].

Suppose that K is a linear space of functions analytic in some domain D in the complex plane. A sequence Λ = (λk) of distinct points from D is said to be a set of uniqueness for K if ƒ∈K and ƒ(λk) = 0 for all k imply ƒ≡0. Depending on the dispersion and the density of Λ on the one hand, and the growth of the functions in K on the other, one may often require only |ƒ(λk)| [les ]ak for some sequence of positive numbers ak, and still conclude that ƒ≡0 for ƒ∈K. Of particular interest are sharp conditions on the decay of ak, which reflect the interplay between growth and decay of analytic functions.

If p is a prime, then a finite group is p-soluble if each of its composition factors is either a p-group or has order coprime to p. For example, soluble groups are p-soluble. However, there are many insoluble groups that are p-soluble. We shall prove the following result.

We give a formula for the Betti numbers of 3-Sasakian manifolds or orbifolds which can be obtained as 3-Sasakian quotients of a sphere by a torus. This answers a question of Galicki and Salamon about the topology of 3-Sasakian manifolds.

We deal with weighted inequalities of the type

formula here

where:

T is either the Hardy–Littlewood maximal operator or a singular integral operator; G is any measurable subset of ℝn; ƒ is any measurable function, vanishing outside G, such that Tƒ is well-defined; v and w are weights, that is, nonnegative locally integrable functions on G;

p, q ∈(1, ∞).

In [1], Beardon introduced the Apollonian metric α defined for any domain D in ℝ by

formula here

This metric is Möbius invariant, and for simply connected plane domains it satisfies the inequality αD[les ]2ρD, where ρD denotes the hyperbolic distance in D, and so gives a lower bound on the hyperbolic distance. Furthermore, it is shown in [1, Theorem 6.1] that for convex plane domains, the Apollonian metric satisfies ρD[les ]4 sinh [½αD], and, by considering the example of the infinite strip {x+iy[ratio ][mid ]y[mid ]<1}, that the best possible constant in this inequality is at least π. In this paper we make the following improvements.

We give an explicit example of an exotic (non-simplicial), geometric free action of the free group [ ]3 on an ℝ-tree T. We begin by associating an interval translation mapping of the unit interval to an automorphism ψ of [ ]3. We use a result of D. Gaboriau and G. Levitt to obtain an [ ]3-action on an ℝ-tree T. We show that for our particular choice of ψ, the resulting [ ]3-action is minimal, free and exotic.

We introduce several successively stronger conditions which are all necessary for a sequence in the unit ball to be interpolating for the space of bounded holomorphic functions, and which all imply Varopoulos' classical necessary condition.

Belyi˘'s Theorem implies that a Riemann surface X represents a curve defined over a number field if and only if it can be expressed as U/Γ, where U is simply-connected and Γ is a subgroup of finite index in a triangle group. We consider the case when X has genus 1, and ask for which curves and number fields Γ can be chosen to be a lattice. As an application, we give examples of Galois actions on Grothendieck dessins.

We precisely evaluate the operator norm of the uncentred Hardy–Littlewood maximal function on Lp(ℝ1). Consequently, we compute the operator norm of the ‘strong’ maximal function on Lp(ℝn), and we observe that the operator norm of the uncentred Hardy–Littlewood maximal function over balls on Lp(ℝn) grows exponentially as n[xrarr ]∞.

Let K. denote the graded Koszul complex associated to the regular sequence (x0, …, xn) in the graded polynomial ring A = k[x0, …, xn], [mid ]xi[mid ] = 1 for all i, over an arbitrary field k. Let K′. denote the Koszul complex associated to another regular sequence of homogeneous elements (p0, …, pn) in A. In [5] we have studied ranks of graded chain complex morphisms f.[ratio ]K′.→K′. with the property f0 = id. Let Ωk (respectively, Ω′k) denote the kernel of the Koszul differential d[ratio ]Kk→ Kk−1 (respectively, d′[ratio ] K′k→ K′k−1), and let fk[ratio ] Ω′k→Ωk denote the restriction of fk. The main result was that Rank (fk)>n−k.

We give an elementary proof for the fact that the Julia set of a rational or entire function is the closure of the repelling periodic points.

We show that the zero curvature limit of the space of hyperbolic monopoles gives the Euclidean monopoles, settling a conjecture of Atiyah. We also study the infinite curvature limit of the space of hyperbolic monopoles, and show that the associated rational maps appear explicitly here.

We consider the functional equation F∘T−F=f, where T is a measure-preserving transformation and f is a continuous function. We show that if there is an L∞ function F which satisfies this equation, then F is constrained to satisfy a number of regularity conditions, and, in particular, if T is a one-sided Bernoulli shift, then we show that there is a continuous function F satisfying this equation. We show that this is not the case for the two-sided shift.

The purpose of this paper is to answer some questions posed by Doob [2] in 1965 concerning the boundary cluster sets of harmonic and superharmonic functions on the half-space D given by D = ℝn−1 × (0, +∞), where n[ges ]2. Let f[ratio ]D→[−∞, + ∞] and let Z∈δD. Following Doob, we write BZ (respectively CZ) for the non-tangential (respectively minimal fine) cluster set of f at Z. Thus l∈BZ if and only if there is a sequence (Xm) of points in D which approaches Z non-tangentially and satisfies f(Xm)→l. Also, l∈CZ if and only if there is a subset E of D which is not minimally thin at Z with respect to D, and which satisfies f(X)→l as X→Z along E. (We refer to the book by Doob [3, 1.XII] for an account of the minimal fine topology. In particular, the latter equivalence may be found in [3, 1.XII.16].) If f is superharmonic on D, then (see [2, §6]) both sets BZ and CZ are subintervals of [−∞, + ∞]. Let λ denote (n−1)-dimensional measure on δD. The following results are due to Doob [2, Theorem 6.1 and p. 123].

With the death of Brian Kuttner on 2 January 1992, near Birmingham, England, summability theory has lost one of its leading exponents. During the period 1934–1991 he published over 100 research papers, including work on Fourier series, strong summability, Riesz means, Nörlund methods and Tauberian theory. Since his death, further papers with joint authorship have continued to appear; his influence persists.

Let [Ascr ] be a complex algebra (with unit e), X a left [Ascr ] module and a = (a1, . . ., am) a commuting tuple of elements in [Ascr ]. Following Taylor [8], we have to consider the Koszul complex K·(a, X) in order to define the joint spectrum of the tuple a.

1. Introduction

Let k be an algebraically closed field of any characteristic. A toric variety over k is a normal variety X containing the algebraic group T=(k*)n as an open dense subset, with a group action T × X→X extending the group law of T.

On any smooth variety X over a field k, we can define the sheaf of differential operators [Dscr ], which carries a natural structure as an [Oscr ]X-bisubalgebra of Endk([Oscr ]X). A [Dscr ]-module on X is a sheaf [Fscr ] of abelian groups having a structure as a left [Dscr ]-module, such that [Fscr ] is quasi-coherent as an [Oscr ]X-module. A smooth variety X is called [Dscr ]-affine if for every [Dscr ]-module [Fscr ] we have

[bull ] [Fscr ] is generated by global sections over [Dscr ],

[bull ] Hi(X, [Fscr ])=0, i>0.

Beilinson and Bernstein have shown [1] that every flag variety over a field of characteristic zero is [Dscr ]-affine, from which they deduced a conjecture of Kazhdan and Lusztig. In fact, flag varieties are the only known examples of [Dscr ]-affine projective varieties. In this paper we prove that the [Dscr ]-affinity of a smooth complete toric variety implies that it is a product of projective spaces. Part of the method will be to translate a proof of the non [Dscr ]-affinity of a 2-dimensional Schubert variety, given by Haastert in [4], into the language of toric varieties.

We show in this note the following statement which is an improvement over a result of R. M. Dudley and which is also of independent interest. Let X be a set of a Hilbert space with the property that there are constants ρ, σ>0, and for each n∈ℕ, the set X can be covered by at most n balls of radius ρn−σ. Then, for each n∈ℕ, the convex hull of X can be covered by 2n balls of radius cn−½−σ. The estimate is best possible for all n∈ℕ, apart from the value c=c(ρ, σ, X). In other words, let N(ε, X), ε>0, be the minimal number of balls of radius ε covering the set X. Then the above result is equivalent to saying that if N(ε, X)=O(ε−1/σ) as ε↓0, then for the convex hull conv (X) of X, N(ε, conv (X)) =O(exp(ε−2/(1+2σ))). Moreover, we give an interplay between several covering parameters based on coverings by balls (entropy numbers) and coverings by cylindrical sets (Kolmogorov numbers).

It is proved that every commutative, nilpotent extension of a commutative C*-algebra splits topologically.