The paper presents an improvement of the author's earlier estimate for the average number of limit cycles of a planar polynomial vector field situated in a neighbourhood of the origin, provided that the field is close enough to a linear centre in a larger neighbourhood. The result follows from a new distributional inequality for the number of zeros of a family of univariate holomorphic functions depending holomorphically on a parameter.

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.

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.

Let V be a commutative valuation domain of arbitrary Krull-dimension (rank), with quotient field F, and let K be a finite Galois extension of F with group G, and S the integral closure of V in K. If, in the crossed product algebra K [midast ] G, the 2-cocycle takes values in the group of units of S, then one can form, in a natural way, a ‘crossed product order’ S [midast ] G ⊆ K [midast ] G. In the light of recent results by H. Marubayashi and Z. Yi on the homological dimension of crossed products, this paper discusses necessary and/or sufficient valuation-theoretic conditions, on the extension K/F, for the V-order S [midast ] G to be semihereditary, maximal or Azumaya over V.

We prove tight closure analogues of results of Watanabe about chains and families of integrally closed ideals.

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.

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.

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.

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

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.

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.

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.

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 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).

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.

We prove that for an arbitrary measurable set A ⊂ ℝ2 and a σ-finite Borel measure μ on the plane, there is a Borel set of lines L such that for each point in A, the set of directions of those lines from L containing the point is a residual set, and, moreover, μ(A) = μ({∪[lscr ] : [lscr ] ∈ L}). We show how this result may be used to characterise the sets of the plane from which an invisible set is visible. We also characterise the rectifiable sets C1, C2 for which there is a set which is visible from C1 and invisible from C2.

We show that any gauge-equivalence class of solutions (∇, Φ) of the monopole equation F∇ = *d∇Φ, defined on an open set Ω of S3 (equipped with its standard metric), induces canonically a holomorphic function on the complex 2-dimensional space of all geodesics on S3 entirely contained in Ω.

A flip can be thought of as a diagram X− → X ← X+ of complex threefolds satisfying some conditions. One often thinks of a flip as being in two parts: the first part, X− → X, is the given, while the second, X ← X+, is the unknown. I calculate cohomological properties of the canonical classes, K− = KX− and so on, and in particular properties of the function

formula here

In the case of the toric flips of Danilov [3] and Reid [7], this function can be expressed in terms of lattice points of multiplies of a polyhedron giving sharpness for the more general result.

Sufficient conditions are given on a Banach space $X$ which ensure that $\ell_{\infty}$ embeds in ${\mathcal L}\, (X)$, the space of all bounded linear operators on $X$. A basic sequence $(e_n)$ is said to be quasisubsymmetric if for any two increasing sequences $(k_n)$ and $(\ell_n)$ of positive integers with $k_n \leq \ell_n$ for all $n$, $(e_{k_n})$ dominates $(e_{\ell_n})$. If a Banach space $X$ has a seminormalized quasisubsymmetric basis then $\ell_{\infty}$ embeds in ${\mathcal L}\, (X)$.

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.