We give a combinatorial description of the homotopy groups of the suspension of a K(π, 1) and of wedges of 2-spheres. In particular, all of the homotopy groups of the 2-sphere are given as the centres of certain combinatorially described groups.

We prove that if k, F are fields of different characteristics then the permutation module kFn for the affine group AGL(n, F) has simple augmentation submodule. This proves a conjecture of A. R. Camina.

The functional relations between the coordinates of points on a manifold make the study of Diophantine approximation on manifolds much harder than the classical theory in which the variables are independent. Nevertheless there has been considerable progress in the metric theory of Diophantine approximation on smooth manifolds. To describe this, some notation and terminology are needed.

Let [sum ]∞an denote the set of cluster points of the sequence of partial sums of a series [sum ]an with terms in ℝd. For any permutation f of the set ℕ of positive integers, [Cscr ]f (ℝd) denotes the set of all sets [sum ]∞af(n) arising from series [sum ]an with terms in ℝd and sum 0. For each f, we use the Max-Flow Min-Cut Theorem to determine all convex sets in [Cscr ]f(ℝd) which are symmetric about a point. These sets depend only on a parameter w(f) ∈ ℕ ∪ {0, ∞}, called the width of f. We show that w(f), when it is a positive integer, falls far short of completely determining [Cscr ]f(ℝd) but, for each q ∈ ℕ, we find the largest of the sets [Cscr ]f(ℝd) arising from permutations f of width q. We also describe the smallest of these sets when q = 1.

The main result of the paper is that the real characters of a group G of type FPm (Fm respectively) that do not vanish on a normal locally polycyclic-by-finite subgroup represent elements of the geometric invariant [sum ]m(G, ℤ) ([sum ]m(G) respectively). In the case m = 2 a stronger result is proved. Some consequences of the main result are considered.

We prove that for an arbitrary Borel measurable function f on the space of all planar lines there exists a set which intersects almost every line [lscr ] in a set of packing dimension f([lscr ]).

For E a subset of ℝn and s ∈ [0, n] we define upper and lower box dimension profiles, B-dimsE and B-dimsE respectively, that are closely related to the box dimensions of the orthogonal projections of E onto subspaces of ℝn. In particular, the projection of E onto almost all m-dimensional subspaces has upper box dimension B-dimmE and lower box dimension B-dimmE. By defining a packing type measure with respect to s-dimensional kernels we are able to establish the connection to an analogous packing dimension theory.

In this paper, various Euler characteristic formulas for simplicial maps are obtained, which generalize the Izumiya–Marar formula [14], the Banchoff triple point formula [3] and the formula due to Szücs for maps of surfaces into 3-space [27]. Moreover, we obtain new results about the Euler characteristics of the multiple point sets and the images of generic smooth maps and the numbers of their singularities.

It is proved that, for a complex minimal smooth projective surface S of general type with a pencil of genus g = 3 or 4, any Abelian automorphism group of S is of order [les ] 12K2S + 96(g − 1), provided K2S > 8(g − 1)2, where KS is the canonical divisor of S.

An algebra of Colombeau generalized ultradistributions in which the corresponding space of ultradistributions is embedded via regularizations and in which the multiplication of ultradifferentiable functions of an appropriate class is the ordinary multiplication is constructed. This enables the development of the local and micro-local analysis in the framework of a new scale of Colombeau algebras. Sufficient conditions for a generalized ultradistribution being an ultradifferentiable function are given.

In the classical Lagrangian approach to conservation laws of gauge-natural field theories a suitable vector density is known to generate the so-called conserved Noether currents. It turns out that along any section of the relevant gauge-natural bundle this density is the divergence of a skew-symmetric tensor density, which is called a superpotential for the conserved currents.

We describe gauge-natural superpotentials in the framework of finite order variational sequences according to Krupka. We refer to previous results of ours on variational Lie derivatives concerning abstract versions of Noether's theorems, which are here interpreted in terms of ‘horizontal’ and ‘vertical’ conserved currents. The gauge-natural lift of principal automorphisms implies suitable linearity properties of the Lie derivative operator. Thus abstract results due to Kolář, concerning the integration by parts procedure, can be applied to prove the existence and globality of superpotentials in a very general setting.

In this note we first show, using results of McDuff, that for a Cr-manifold S, diffeomorphic to the interior of a compact manifold with boundary, the class of all Cr-diffeomorphisms lying in a one parameter group of S generates the connected component of 1S in Diffr (S, S). Then we use this result to obtain two extension theorems for stratified maps defined on some strata of a stratified space [Xscr ]. Our extension theorems hold for Mather's abstract stratified sets, for Whitney (b)-regular, Bekka (c)-regular, Verdier (w)-regular and Lipschitz-regular stratified spaces.

For bond percolation on the two-dimensional triangular lattice with arbitrary retention parameter p ∈ [0, 1], we show that the number of infinite rigid components is a.s. at most 1. This proves a conjecture by Holroyd. Further results, concerning simultaneous uniqueness, and continuity (in p) of the probability that a given edge is in an infinite rigid component, are also obtained.

(Volume 125 (1999), 463–509)

Case (x) in Theorem 6 is missing. The correct statement of Theorem 6 in the paper should read as follows.

THEOREM 6. Let x: Mn → Hn+1 (−1) ⊂ [ ]n+21 (n > 2) be an isometric immersion of a conformally flat n-manifold. Then it satisfies equality (1·2) if and only if one of the following ten cases occurs.

We determine first the structure of certain exponential reduced subsemigroups of the Lie group ℝ × G, where G is a compact and connected Lie group. Next, we construct the Bohr compactification of these subsemigroups. The main result is a structure theorem of this Bohr compactification. In the final section we make some links to the theory of one-parameter semigroups of subsets of Lie groups (due to Rådström).

We generalize the Carathéodory version of the Schwarz reflection principle for pseudo-metrics

formula here

where w is analytic in some domain of the complex plane, [mid ] w [mid ] < 1 and use a unit disk version of the new result to study certain homeomorphisms of the unit ball in [Hscr ]([ ]) onto itself.

The random record distribution ν associated with a probability distribution μ can be written as a convolution series, ν = [sum ]∞n=1n−1 (n + 1)−1μ*n. Various authors have obtained results on the behaviour of the tails ν((x, ∞)) as x → ∞, using Laplace transforms and the associated Abelian and Tauberian theorems. Here we use Gelfand transforms and the Wiener–Lévy–Gelfand Theorem to obtain expansions of the tails under moment conditions on μ. The results differ notably from those known for other convolution series.