The spherical functions of triangle buildings can be described in terms of certain two-dimensional orthogonal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynimial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups. We present strong laws of large numbers, a central limit theorem, and a local limit theorem; all these results are well-known for homogeneous trees. Proofs are based on moment functions on hypergroups and on explicit expansions of the hypergroup characters in terms of certain two-dimensional Tchebychev polynimials.

This paper is a contiunation of the study of the rings for which every principal right ideal (respectively, every right ideal) is a direct summand of a right annihilator initiated by Stanley S. Page and the author in [20, 21].

We define equivariant completion of a G-complex and define residually finite G-spaces. We show that the group of G-homotopy classes of G-homotopy self equivalences of a finite, residually finite G-complex, in residually finite. This generalizes some results of Roitberg.

The first purpose of this paper is to give a tensor product formula of the characteristic invariant and modular invariant for a tensor product action of a discrete group G on AFD factors. The second purpose is to describe a characteristic invariant and modular invariant of the extended action to a crossed product in terms of the original invariants.

Davey and Quackenbush proved a strong duality for each dihedral group Dm with m odd. In this paper we extend this to a strong duality for each finite group with cyclic Sylow subgroups (such groups are known to be metacyclic).

Let and denote respectively the variety of groups of exponent dividing e, the variety of nilpotent groups of class at most c, the class of nilpotent groups and the class of finite groups. It follows from a result due to Kargapolov and Čurkin and independently to Groves that in a variety not containing all metabelian groups, each polycyclic group G belongs to . We show that G is in fact in , where c is an integer depending only on the variety. On the other hand, it is not always possible to find an integer e (depending only on the variety) such that G belongs to but we characterize the varieties in which that is possible. In this case, there exists a function f such that, if G is d-generated, then G ∈ So, when e = 1, we obtain an extension of Zel'manov's result about the restricted Burnside problem (as one might expect, this result is used in our proof). Finally, we show that the class of locally nilpotent groups of a variety forms a variety if and only if for some integers c′, e′.

In this paper we prove that if a Cantor set has ratios of dissection bounded away from zero, then there is a natural number N, such that its N-fold sum is an interval. Moreover, for each element z of this interval, we explicitly construct the N elements of C whose sum yields z. We also extend a result of Mendes and Oliveria showing that when s is irrational is an interval if and only if a /(1−2a) as/(1−2as) ≥ 1.

Extremal partitions of domains into configurations of certain to pological form are studied. The extremal value of the weighted sum of reduced moduli of circular domains and digons is obtained. These results are applied to some problems about distortion under bounded conformal maps of the unit disk with two preassigned values.

In 1986, Higgins proved that T(X), the semigroup (under composition) of all total transformations of a set X, has a proper dense subsemigroup if and only if X is infinite, and he obtained similar results for partial and partial one-to-one transformations. We consider the generalised transformation semigroup T(X, Y) consisting of all total transformations from X into Y under the operation α * β = αθβ, where θ is any fixed element of T(Y, X). We show that this semigroup has a proper dense subsemigroup if and only if X and Y are infinite and | Yθ| = min{|X|,|Y|}, and we obtain similar results for partial and partial one-to-one transformations. The results of Higgins then become special cases.