In this paper we give an elementary method for classifying conjugacy classes of involutions in a Coxeter group (W, S). The classification is in terms of (W-equivalence classes of certain subsets of S).

Let G be a compact Lie group and M be a compact G-manifold. We investigate the class of equivariant diffeomorphisms of M covering the identity map on the orbit space M/G.

The following result has been established by Stadje and Gross:

THEOREM. If 〈X, d〉 is a compact connected metric space, then there is a unique positive real number a(〈X, d〉) with the property that for each natural number n, and for all {x1, x2, …, xn} ⊂ X, there exists y ∈ X for which

Previous work concerning boundary regularity of solutions of the non-parametric least area problem leaves open the question of regularity of solutions at points where the mean curvature of the boundary of the domain vanishes. We here prove that the solutions may be discontinuous at such points, even when the given boundary data is smooth. We also give a sufficient condition which will ensure continuity at such points.

We give a simple proof of the fact that compact, connected topological spaces have the “average distance property”. For a metric space (X, d), this asserts the existence of a unique number a = a(X) such that, given finitely many points x1, …, xn ∈ X, then there is some y ∈ X with

We examine the possible values of a(X) , for subsets of finite dimensional normed spaces. For example, if diam(X) denotes the diameter of some compact, convex set in a euclidean space, then a(X) ≤ diam(X)/√2 . On the other hand, a(X)/diam(X) can be arbitrarily close to 1 , for non-convex sets in euclidean spaces of sufficiently large dimension.

We give a simple proof of the result that if the total space of a holomorphic fiber bundle is (complete) hyperbolic then both the fiber and the base manifold must be (complete) hyperbolic. Shoshichi Kobayashi tried to set up examples where the total space is hyperbolic but the base is not; our theorem shows that any such example is bound to fail.

We describe a class of weight functions, which admit Tchebycheff quadrature.

Let G be a graph. With every connected subgraph of G, let us associate a weight (Wα and with every spanning subgrapha or cover) C of G, a weight

where αi (i = 1, 2, …, n) are the components of C. The subgraph polynomial of G is

where the summation is taken over all the spanning subgraphs in G.

The basic properties of subgraph polynomials are given, expressions for the subgraph polynomials of multigraphs and pseudographs are derived, and the chromatic and dichromatic polynomials are shown to be special cases of the subgraph polynomial.

Nearlattices, or lower semllattices in which any two elements have a supremum whenever they are bounded above, provide an interesting generalization of lattices. In this context, we study standard, neutral, and central elements, as well as standard ideals. A new perspective is obtained in the well established case of lattices.

We call a group G cyclically separated if for any given cyclic subgroup B in G and subgroup A of finite index in B, there exists a normal subgroup N of G of finite index such that N ∩ B = A. This is equivalent to saying that for each element x ∈ G and integer n ≥ 1 dividing the order o(x) of x, there exists a normal subgroup N of G of finite index such that Nx has order n in G/N. As usual, if x has infinite order then all integers n ≥ 1 are considered to divide o(x). Cyclically separated groups, which are termed “potent groups” by some authors, form a natural subclass of residually finite groups and finite cyclically separated groups also form an interesting class whose structure we are able to describe reasonably well. Construction of finite soluble cyclically separated groups is given explicitly. In the discussion of infinite soluble cyclically separated groups we meet the interesting class of Fitting isolated groups, which is considered in some detail. A soluble group G of finite rank is Fitting isolated if, whenever H = K/L (L ⊲ K ≤ G) is a torsion-free section of G and F(H) is the Fitting subgroup of H then H/F(H) is torsion-free abelian. Every torsion-free soluble group of finite rank contains a Fitting isolated subgroup of finite index.

In 1978 Almkvist and Fossum examined the decomposition of the exterior powers of basis modules in the modular representation algebra of a cyclic group of prime order. In particular they developed an isomorphism between these exterior powers and terms of binomial coefficient type in the algebra.

We derive several recurrence relations for these terms.

Each continuous positive linear form on a commutative locally m-convex *-algebra E with a bounded approximate identity, accepts an integral representation on the hermitian spectrum (hermitian characters) of E. An alternative form of the latter is also obtained. The presented results constitute an abstract form of the Bochner-Weil-Raikov theorem within the frame of topological *-algebras.

Since the author's article “A unified treatment of transfinite constructions …”, in Volume 22 (198O) of this Bulletin, had an encyclopaedic goal, he now takes the opportunity to answer two further questions raised since that article was submitted. The lesser of these asks whether the only pointed endofunctors for which every action is an isomorphism are the well-pointed ones, at least when the endofunctor is cocontinuous; a counter-example provides a negative answer. The more important question concerns the reflexion from the comma-category T/A into the category of algebras for the pointed endofunctor T of A, and the algebra-reflexion sequence which converges to this reflexion; and asks for simplified descriptions in the special case where T is cocontinuous. We give closed formulas in this case, both for the reflexion and for the sequence which converges to it. The reader may wonder why we care about the approximating sequence when we have a closed formula for the reflexion; the answer is that, in certain applications, we need to separate the roles of finite colimits and filtered ones.

By a sketch we here mean a small category S together with a small set φ of projective cones in S, each cone φ ∈ φ being indexed by a small category Lφ. A model of S in any category B is a functor G: S → B such that each Gφ is a limit-cone. Let F be any small set of small categories containing all the Lφ. A small category T admitting all F-limits (that is, an F-complete small T ) is called an F-theory; it is considered as a sketch in which the distinguished cones are all the F-limit-cones. It is an important result of modern universal algebra, due originally to Ehresmann, that each sketch S = (S, φ) with every Lφ ∈ F determines an F-theory T, with a generic model M: S → T of S, such that composition with M induces an equivalence M* between the category of T-models in B and that of S-models in B, whenever B is F-complete. We give a simple proof of this result – one which generalizes directly to the case of enriched categories and indexed limits; and we make the new observation that the inverse to M* is given by (pointwise) right Kan extension along M.

A simple change of the operation in a commutative semigroup leads to a groupoid. It is shown that it has no finite basis for its laws.

This paper provides an example of a non-abelian 2-group with a non-abelian automorphism group in which every automorphism is central.

An LCA group G is called largely open if every large subgroup (that is, subgroup of finite index) of G is open. G is power-open if the continuous endomorphism, g → ng, g ∈ G, takes open sets of G onto open sets of G for every integer n > 1.