The paper is concerned with rings of polynomial invariants of finite groups. In particular, it will be shown that these rings are isomorphic as modules over the Steenrod algebra [Pscr ]* if and only if the group representations are pointwise conjugate. An application to cohomology is the construction of classifying spaces of finite groups which are not homotopy equivalent, but where the cohomology rings are isomorphic as unstable modules over the (topological) Steenrod algebra.

It is shown that D. Cohen's inequality bounding the isoperimetric function of a group by the double exponential of its isodiametric function is valid in the more general context of locally finite simply connected complexes. It is shown that in this context this bound is ‘best possible’. Also studied are second-dimensional isoperimetric functions for groups and complexes. It is shown that the second-dimensional isoperimetric function of a group is bounded by a recursive function. By a similar argument it is shown that the area distortion of a finitely presented subgroup of a finitely presented group is recursive. Cohen's inequality is extended to second-dimensional isoperimetric and isodiametric functions of 2-connected simplicial complexes.

It is proved that an excellent ring which is the homomorphic image of a regular ring of finite Krull dimension and having a resolution of singularities satisfies the uniform Artin–Rees property. Extensions of some results of Lipman, Sathaye and Teissier are also obtained.

All finite groups G are determined that admit a subgroup K of index pa, p a prime, under the assumption that G has an irreducible and faithful GF(q)-module in characteristic p whose dimension over GF(q) is at most a. As an application to the theory of permutation groups, the maximal transitive subgroups of the primitive affine permutation groups are determined. The above-mentioned classification is generalized by dropping the assumption that p|q. In both cases surprisingly nice results are obtained.

Let k[x](x) be the polynomial ring k[x] localized in the maximal ideal (x) ⊂ k[x]. The paper studies the Hilbert functor parametrizing ideals of colength n in this ring having support at the origin. The main result is that this functor is not representable. A complete description of the functor as a limit of representable functors is also given.

We are interested in Weil representations of Sp(2n, R), where R is the ring Z/plZ, p is an odd prime and l is a positive integer, or, more generally, R = [Oscr ]/[pfr ]l, where [Oscr ] is the ring of integers of a local field, [pfr ] is the maximal ideal of [Oscr ] and [Oscr ]/[pfr ] has odd characteristic. One reason for this interest is that a continuous finite-dimensional complex representation of Sp(2n, [Oscr ]) has to factor through a representation of Sp(2n, [Oscr ]/[pfr ]l) for some l.

It is shown that if a locally compact group acts isometrically on a Banach space X leaving a closed subspace M invariant, and if the induced actions on M and X/M are strongly continuous, then the action on X is strongly continuous. Since this may be of interest for one-parameter semigroups, similar results are proved for actions of suitable topological semigroups. Other generalizations are given for (suitable) non-isometric actions, non-locally compact groups, and non-Banach spaces; corollaries concerning 1-cocycles and uniformly continuous actions are given.

Let M be a smooth, compact, oriented, odd-dimensional Riemannian manifold and let Γ → Mˆ → M be a normal covering of M. It is proved that the relative von Neumann eta-invariant ρ(2)(Mˆ) of Cheeger and Gromov is a homotopy invariant when Γ is torsion-free, discrete and the Baum–Connes assembly map μmax[ratio ]K0(BΓ) → K0(C*Γ) is an isomorphism.

We study here some foundational aspects of the classification problem for torsion-free abelian groups of finite rank. These are, up to isomorphism, the subgroups of the additive groups (ℚn, +), for some n = 1, 2, 3,…. The torsion-free abelian groups of rank [les ] n are the subgroups of (ℚn, +).

In this paper we provide a minimal constructive integration process of Riemann type which includes the Lebesgue integral and also integrates the derivatives of differentiable functions. We provide a new solution to the classical problem of recovering a function from its derivative by integration, which, unlike the solution provided by Denjoy, Perron and many others, does not possess the generality which is not needed for this purpose.

The descriptive version of the problem was treated by A. M. Bruckner, R. J. Fleissner and J. Foran in [2]. Their approach was based on the trivial observation that for the required minimal integral, a function F is the indefinite integral of f if and only if F' = f almost everywhere and there exists a differentiable function H such that F – H is absolutely continuous. They strengthen this definition by proving that F – H can have arbitrary small variation. Nevertheless, their definition still needs a choice of a differentiable function.

It is proved that a Tychonoff topological group is locally compact if and only if it is of pointwise countable type and its left uniformity is cofinally complete. From this result a characterization is derived of those T0 paratopological groups (X, τ) of pointwise countable type for which (X, τ ∨ τ−1) is locally compact and also a characterization is deduced of locally pseudocompact topological groups in terms of cofinal completeness. Also characterized are the Tychonoff topological groups of pointwise countable type for which their left uniformity has property U. Finally, cofinal completeness of the Hausdorff–Bourbaki uniformity of a topological group is studied.

Ritt has shown that any complex polynomial p can be written as the composition of polynomials p1,…,pm, where each pj is prime in the sense that it cannot be written as a non-trivial composition of polynomials. The factors pj are not unique but the number m of them is, as is the set of the degrees of the pj. The paper extends Ritt's theory and, in particular, a third invariant of the decomposition is introduced.

The paper gives an explicit classification of the finite groups which are homogeneous in the sense of Fraïssé.

A point λ in the complex plane is said to be free if the group generated by the matrices

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is free. The paper gives an infinite family of polynomials whose roots are the non-free points. The main idea in the paper is to employ a symmetry relation.

Let [Gscr ] be a family of functions analytic in a domain D in the complex plane. It is proved that [Gscr ] is a normal family, provided that for each f∈[Gscr ], there exists k = k(f) > 1 such that the kth iterate fk has no repulsive fixpoint in D. A new proof of a result of Bergweiler and Terglane concerning the dynamics of entire functions is also given.

A theorem due to Hardy states that, if f is a function on R such that |f(x)| [les ] C e−α|x|2 for all x in R and |fˆ(ξ)| [les ] C e−β|ξ|2 for all ξ in R, where α > 0, β > 0, and αβ > 1/4, then f = 0. A version of this celebrated theorem is proved for two classes of Lie groups: two-step nilpotent Lie groups and harmonic NA groups, the latter being a generalisation of noncompact rank-1 symmetric spaces. In the first case the group Fourier transformation is considered; in the second case an analogue of the Helgason–Fourier transformation for symmetric spaces is considered.

The definiteness of the Peano kernel is proved for a functional associated with the mean-value property of Picone and Bramble and Payne for polyharmonic functions in the ball. An important corollary of this is that if a function f satisfying (−1)pΔpf>0 vanishes on p concentric spheres centered at 0, then f(0)>0. This generalizes a well-known property of subharmonic functions (which arise in the special case p = 1).

The aim of this paper is to determine which (finite) 3-dimensional classical groups are completions of the Goldschmidt G3-amalgam. We recall, first, that an amalgam (of rank 2) consists of three groups P1, P2, B and two group monomorphisms ϕ1, ϕ2 such that

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Usually, when ϕ1 and ϕ2 are understood, this amalgam is denoted by [Ascr ](P1, P2, B). Now a group G is a completion of [Ascr ](P1, P2, B) if there exist group homomorphisms ψi[ratio ]Pi → G (i = 1, 2) satisfying G = 〈im ψ1, im ψ2〉 and ψ1ϕ1 = ψ2ϕ2[ratio ]B→G. The Goldschmidt G3-amalgam, which appears in [6], is defined as follows: P1 ≅ Sym(4) ≅ P2, B ≅ Dih(8) (Dih(n) denotes the dihedral group of order n) with ϕ−11(O2(P1)) ≠ ϕ−12(O2(P2)).

The paper is concerned with the uniqueness and existence problem for a multidimensional reacting and convection system with the vanishing viscosity method. The uniqueness theorem is obtained from the stability with respect to the initial data. To solve the existence problem, the uniqueness and existence of solutions to a viscous system are first proved, and then the L1-modulus of continuity of solutions independent of the small viscosity is obtained.

For a given integer n, all zero-mean cosine polynomials of order at most n which are non-negative on [0,(n/(n+1))π] are found, and it is shown that this is the longest interval [0,θ] on which such cosine polynomials exist. Also, the longest interval [0,θ] on which there is a non-negative zero-mean cosine polynomial with non-negative coefficients is found.

As an immediate consequence of these results, the corresponding problems of the longest intervals [θ,π] on which there are non-positive cosine polynomials of degree n are solved.

For both of these problems, all extremal polynomials are found. Applications of these polynomials to Diophantine approximation are suggested.