Volume 21, Issue 2

Let 𝒟 be a large category which is cocomplete. We construct a model structure (in the sense of Quillen) on the category of small functors from 𝒟 to simplicial sets. As an application we construct homotopy localization functors on the category of simplicial sets which satisfy a stronger universal property than the customary homotopy localization functors do.

Many of the known ovoids and spreads of finite polar spaces admit a transitive group of collineations, and in 1988, P. Kleidman classified the ovoids admitting a 2-transitive group. A. Gunawardena has recently extended this classification by determining the ovoids of the seven-dimensional hyperbolic quadric which admit a primitive group. In this paper we classify the ovoids and spreads of finite polar spaces which are stabilised by an insoluble transitive group of collineations, as a corollary of a more general classification of m -systems admitting such groups.

We show that every group of exponent 2 m · 3 n ( m, n ∈ ℕ, n ≤ 2) that acts freely on some abelian group is finite.

The aim of this paper is to prove the asymptotic behaviour of class numbers h ( d ) in the case that the class numbers are ordered by the size of the fundamental solution of Pell's equation t 2 – du 2 = 4 and the considered discriminants d belong to an arithmetic progression. As an application of this result we prove a result for the asymptotic behaviour of class numbers for fundamental discriminants.

An RD-space 𝒳 is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in 𝒳, or equivalently, that there exists a constant a 0 > 1 such that for all x ∈ 𝒳 and 0 < r < diam(𝒳)/ a 0 , the annulus B ( x , a 0 r ) \ B ( x,r ) is nonempty, where diam(𝒳) denotes the diameter of the metric space (𝒳, d ). An important class of RD-spaces is provided by Carnot-Carathéodory spaces with a doubling measure. In this paper, the authors introduce some spaces of Lipschitz type on RD-spaces, and discuss their relations with known Besov and Triebel-Lizorkin spaces and various Sobolev spaces. As an application, a difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces is obtained.

A topological group G is defined to have property (OB) if any G -action by isometries on a metric space, which is separately continuous, has bounded orbits. We study this topological analogue of strong uncountable cofinality in the context of Polish groups, where we show it to have several interesting reformulations and consequences. We subsequently apply the results obtained in order to verify property (OB) for a number of groups of isometries and homeomorphism groups of compact metric spaces. We also give a proof that the isometry group of the rational Urysohn metric space of diameter 1 has strong uncountable cofinality.

In this paper we present a unified treatment of certain class of majorization results on eigenvalues and singular values of matrices.

We construct an example of conjugacy separable group possessing a subgroup of finite index which is not conjugacy separable. We give also a sufficient condition for a conjugacy separable group to preserve this property when passing to subgroups of finite index. We establish also conjugacy separability of finitely presented residually free groups using impressive results of Bridson and Wilton [Subgroup separability in residually free groups].

In this paper the strong limit power functions are extended from ℝ + to ℝ. A group Q is found whose dual group is shown to be a Bohr-like compactification of ℝ. Some characterizations of the compactification are established. The compactification is applied to investigate properties of strong limit power functions. The normality of the functions is proven. A converse problem is investigated. The summability of the Fourier series is set up.