The preceding paper ‘Strong statements of analysis’ by A. R. D. Mathias defends a so-called full-blooded set theory without full detail [3]. He again objects to a weak set theory which he calls ‘Mac’, in which the usual Zermelo–Fraenkel separation scheme is required only for formulas with suitably ‘restricted’ quantifiers. I had proposed that such separation is adequate for all standard uses of set theory in mathematics. But Mathias has not produced any counter examples of actual mathematics which requires the use of a stronger separation.

Let $p$ be a monic complex polynomial of degree $n$, and let $K$ be a measurable subset of the complex plane. Then the area of $p(K)$, counted with multiplicity, is at least $\pi n ( \Area (K)/\pi )^n$, and the area of the pre-image of $K$ under $p$ is at most $\pi^{1-1/n} (\Area (K) )^{1/n}$. Both bounds are sharp. The special case of the pre-image result in which $K$ is a disc is a classical result, due to Pólya. The proof is based on Carleman's classical isoperimetric inequality for plane condensers.

Let [ ]P2,p (where 2 < p) be the class of all finite nilpotent groups of class 2 and of exponent p with an additional predicate for a subgroup of the centre that contains the commutator subgroup. The Fraïssé limit D of this class exists. Non-forking is described for Th(D).

The context of this note is as follows. One considers a connected reductive group G and a Frobenius endomorphism F[ratio ]G→G defining G over a finite field of order q. One denotes by GF the associated (finite) group of fixed points.

Let [lscr ] be a prime not dividing q. We are interested in the [lscr ]-blocks of the finite group GF. Such a block is called unipotent if there is a unipotent character (see, for instance, [6, Definition 12.1]) among its representations in characteristic zero. Roughly speaking, it is believed that the study of arbitrary blocks of GF might be reduced to unipotent blocks (see [2, Théorème 2.3], [5, Remark 3.6]). In view of certain conjectures about blocks (see, for instance, [9]), it would be interesting to further reduce the study of unipotent blocks to the study of principal blocks (blocks containing the trivial character). Our Theorem 7 is a step in that direction: we show that the local structure of any unipotent block of GF is very close to that of a principal block of a group of related type (notion of ‘control of fusion’, see [13, §49]).

Let G be a transitive permutation group on a set Ω such that, for ω∈Ω, the stabiliser Gω induces on each of its orbits in Ω\{ω} a primitive permutation group (possibly of degree 1). Let N be the normal closure of Gω in G. Then (Theorem 1) either N factorises as N=GωGδ for some ω, δ∈Ω, or all unfaithful Gω-orbits, if any exist, are infinite. This result generalises a theorem of I. M. Isaacs which deals with the case where there is a finite upper bound on the lengths of the Gω-orbits. Several further results are proved about the structure of G as a permutation group, focussing in particular on the nature of certain G-invariant partitions of Ω.

Using basic harmonic analysis on the n-torus, it is shown that the non-commutative n-tori are the only primitive C*-algebras (up to isomorphism) on which the n-torus [ ]n acts freely and ergodically. An explicit construction recovers the generating unitaries in a natural way from the action.

There is a Weil cohomology theory with coefficients in a pseudofinite field of characteristic zero.

For all smooth nondecreasing functions f on [0, 1], the local time Lf spent by standard Brownian motion on the graph of f satisfies P(Lf<x) [ges ]P(2L0<x), for x>0. The constant 2 in this inequality may not be replaced with a smaller one.

A simple proof that no subset of the plane that meets every line in precisely two points is an Fσ-set in the plane is presented. It was claimed that this result can be generalized for sets that meet every line in either one point or two points. No proof of this assertion is known, however. The main results in this paper form a partial answer to the question of whether the claim is valid. In fact, it is shown that a set that meets every line in the plane in at least one but at most two points must be zero-dimensional, and that if it is σ-compact then it must be a nowhere dense Gδ-set in the plane. Generalizations for similar sets in higher-dimensional Euclidean spaces are also presented.

Let $G$ be a group that has a presentation with $n$ generators and $m$ relators, where $m\,{<}\,n$, and let $S$ be an arbitrary generating set for $G$. Then some subset of $n-m$ elements of $S$ freely generates a free subgroup of $G$. This leads to the proof of a conjecture of Gromov on growth of groups.

A well-known formula of Bendixson states that solutions of first-order differential equations, as functions of their initial conditions, satisfy a certain partial differential equation. A consequence is Alekseev's nonlinear variation of parameters formula. In this paper, corresponding results are proved for difference equations. To achieve this, use is made of the recently introduced concept of alpha derivatives, rather than of differences or of the usual derivatives. This technique allows the results to be generalized to alpha dynamic equations, which include among others ordinary differential and difference equations.

Let v be a henselian valuation of a field K. In this paper it is proved that any finite extension (K′, v′) of (K, v) is tame if and only if there exists α ≠ 0 in K′ such that v′(α) = v(TrK′/K(α)) using elementary results of valuation theory. A special case of this result, when the characteristic of the residue field of v is p > 0 and (K′, v′)/(K, v) is an extension of degree p, was proved in 1990 by J. P. Tignol (J. Reine Angew. Math. 404 (1990) 1–38).

This paper investigates how to relate the syzygy periodicity of a self-injective algebra $A$ to its Auslander–Reiten periodicity. Moreover, a characterization is provided of the Auslander–Reiten bounded $A$–$A$-bimodules that are periodic.

It is proved that Vaught's conjecture is true for modules over an arbitrary countable Dedekind prime ring. It follows from the structural result that every module with few types over a countable Dedekind prime ring is ω-stable.

Let V be a vector space over some division ring D, and G a finitary subgroup of GL(V). If G is locally completely reducible, then the D-G modules V, [V, G] and V/CV(G) need not be completely reducible, even if dimDV is finite. Moreover, if F is a field, then V and V/CV(G) need not be completely reducible. We prove here that if D is a finite-dimensional division algebra and G is locally completely reducible, then [V, G] is always a completely reducible D-G module.

The omega limit sets of a nonlinear operator $T$ which is defined on a positive cone and satisfies certain ray-contractive type conditions are discussed. Under the assumption that the restriction of $T$ to a compact subset is surjective, the following alternatives are proved: the omega limit set of a point in the cone either consists of a fixed point or forms a 2-cycle. In addition, new proofs and extensions to relevant results are given.

We determine the intersection theory on the Igusa compactification and the second Voronoi compactification of ${\cal A}_4$.

The author proves in this paper that every profinite group G with polynomial subgroup growth is boundedly generated; that is, it is a product of finitely many procyclic subgroups. This answers a question of P. Zalesskii. By contrast, if G is a boundedly generated group, then the subgroup growth of G is at most nclogn . As a byproduct, a short, elementary proof demonstrates that Aut(Fr) (for r [ges ] 2) and many other related groups are not boundedly generated.

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.

Requirements for boundary conditions are found which imply sharp by order estimates in L∞-metrics of eigenfunctions of the ordinary differential operator

formula here

It is shown that the inequalities

formula here

for the eigenfunctions {yλ} yield certain restrictions on the spectrum.