In this paper we generalize techniques used by Klyachko and the authors to prove some tessellation results about S2. These results are applied to prove the solvability of certain equations with torsion-free coefficients.

The six quaternary quartic forms

were first obtained by Maschke; it has recently been explained that the quartic surfaces obtained by equating these forms to zero are important constituents of Klein's famous configuration derived from six linear complexes that are mutually in involution. The quartic surface Φi = 0 will be denoted, for each of the six suffixes i, by Mi.

The problem discussed is that of dissecting two given triangles into triangular parts which shall consist of mutually similar pairs of triangles, so that the first given triangle A being dissected into the triangles a1, a2, a3 …, and the second given triangle B being dissected into the triangles b1, b2, b3 …, we shall have a1 similar to b1, a2 to b2, and so on.

The integral equations we consider are

and

where 0 < α < 1. The asymptotic behaviour of the eigen-values of the latter equation is already known (see (1) and (4)). The former equation has been studied by many authors but as yet no explicit statement seems to have been made about the behaviour of its eigen-values.

In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P.

Given a partial automorphism of a group G, i.e. an isomorphic mapping μ of a subgroup A of G onto a second subgroup B of G, it is known (2, Theorem I) that there always exists a group H containing G and an inner automorphism of H which extends µ; i.e. there exists an element t of H, such that the transform by t of any element of A is its image under µ.

The object of this paper is to investigate some properties of series which satisfy conditions of the form

where 0 < ρ ≦ p. denotes, as usual, the n-th Cesàro sum of order p for the series ∑an and the binomial coefficient . It is convenient to state here some properties of and to which we must constantly refer in the sequel.

Recently Passman (attributing the origin of the idea to Zalesskii) has defined the following ideal in a ring, (2).

Definition. For a unitary ring R,

N * R = {α ∈ R | αS is nilpotent for all finitely generated subrings S of R}.

For a group algebra KG over a field K of characteristic p ≠ 0, he has proved the radical property:

We present an example of a $C^*$-subalgebra $A$ of $\mathbb{B}(H)$ and a bounded linear map from $A$ to $\mathbb{B}(K)$ which does not admit any bounded linear extension. This generalizes the result of Robertson and gives the answer to a problem raised by Pisier. Using the same idea, we compute the exactness constants of some Q-spaces. This solves a problem raised by Oikhberg. We also construct a Q-space which is not locally reflexive.

AMS 2000 Mathematics subject classification: Primary 46L05. Secondary 46L07

Let X be an infinite set, be the group of all bijections of X and S be a semigroup of total transformations of X with the composition of transformations f and g in S defined by the formula

We say that S is a -normal semigroup if