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).

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

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.

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.