Let R be a commutative domain, and let us denote by (R) the set of non-zero fractional ideals of R, which is a commutative semigroup under multiplication.

Let G be a finite soluble group and let Σ be a Hall system of G. A subgroup U of G is said to be Σ-permutable if U permutes with every member of Σ. In [1; I, 4·29] it is proved that if U and V are Σ-permutable subgroups of G then so also are U ∩ V and 〈U, V〉.

We show that, over suitable rings, q-Young modules for the Hecke algebra of type A have a filtration by q-Specht modules. The multiplicities are also determined.

Throughout this paper (A, , k) denotes a Noetherian local ring of dimension d and stands for the ith local cohomology functor with respect to . We refer the reader to [16] for any unexplained terminology.

Roughly speaking, a link of an ideal of a Noetherian ring R is an ideal of the form I = (z): , where z = z1, …, zg is a regular sequence and g is the codimension of . This is a very common operation in commutative algebra, particularly in duality theory, and plays an important role in current methods to effect primary decomposition of polynomial ideals (see [2]).

We show that the seif-affine sets considered by McMullen [15] and Bedford [2] have infinite packing measure in their packing dimension θ except when all non-empty rows of the initial pattern have the same number of rectangles. More precisely, the packing measure is infinite in the gauge tθ|logt|−1 and zero in the gauge tθ|logt|−1−δ for any δ > 0.

An orbit of the shift σ: t ↦ 2t on the circle = ℝ/ℤ is ordered if and only if it is contained in a semi-circle Cμ = [μ, μ+½]. We investigate the ‘devil's staircase’ associating to each μ ε the rotation number ν of the unique minimal closed σ-invariant set contained in Cμ; we present algorithms for μ in terms of ν, and we prove (after Douady) that if ν is irrational then μ is transcendental. We apply some of this analysis to questions concerning the square root map, and mode-locking for families of circle maps, we generalize our algorithms to orbits of the shift having ‘sequences of rotation numbers’, and we conclude with a characterization of all orders of points around realizable by orbits of σ.

For p > 2, is the first positive even-dimensional element in the stable homotopy groups of spheres. A classical theorem of Nishida[1] states that all elements of positive dimension in the stable homotopy groups of spheres are nilpotent. In fact, Toda [4] proved . For p = 3 he showed that while . In [2] the second author computed the first thousand stems of the stable homotopy groups of spheres at the prime 5. One of the consequences of this computation is that while .

Over the last thirty years, the study of C*-algebras has proceeded in a number of directions. On one hand, much effort has been devoted to understanding the structure of particular classes of algebras, such as the approximately finite (AF) algebras. On the other, general structure theorems have been sought. Classes of algebras defined by certain abstract properties have been investigated with a view to obtaining more concrete descriptions of the algebras. One of the earliest results of this type was the theorem of Glimm [13], later extended by Sakai [20] to the inseparable case, characterizing the non-type I C*-algebras as those algebras which contain a subalgebra with a quotient *-isomorphic to the CAR algebra.

The complex hyperbolic version of Shimizu's lemma gives an upper bound on the radii of isometric spheres of maps in a discrete subgroup of PU(n, 1) containing a vertical Heisenberg translation. The purpose of this paper is to show that in a neighbourhood of this bound radii of isometric spheres only take values in a particular discrete set. When the group contains certain ellipto-parabolic maps this upper bound can be improved and the set of values of the radii is more restricted. Examples are given that show that these results cannot be improved.

The aim of this paper is to analyse Riemannian Kähler–Einstein metrics g in real dimension four admitting an isometric action of SU(2) with generically three-dimensional orbits. The Kähler condition means that there is a complex structure I, with respect to which the metric is hermitian, such that the two-form Ωdefined by

is closed. It is well-known that if this condition holds then Ω is in fact covariant constant.

Borel measures in ℝd are called fractal if locally at a.e. point their Hausdorff and packing dimensions are identical. It is shown that the product of two fractal measures is fractal and almost all projections of a fractal measure into a lower dimensional subspace are fractal. The results rely on corresponding properties of Borel subsets of ℝd which we summarize and develop.

In this paper, we consider transmission problems for wave propagation in two inhomogeneous half-spaces with a locally perturbed hyperplane interface. A radiation condition is obtained for the problem in the framework of the spaces B and B*. It is then used, together with the spaces B, B* and the limiting absorption method to prove the existence of the unique solution to the problem.

We consider the Vlasov—Einstein system in a spherically symmetric setting and prove the existence of static solutions which are asymptotically flat and have finite total mass and finite extension of the matter. Among these there are smooth, singularity-free solutions, which have a regular centre and have isotropic or anisotropic pressure, and solutions which have a Schwarzschild-singularity at the centre.