The basic properties of polytopes and their faces are derived from a set of axioms which are satisfied, in particular, by polytopes in Euclidean, hyperbolic or (hemi-)spherical space. The underlying space is not assumed to be either dense or unbounded.

In this note we answer a question of W. G. Bade by showing that if a normal, unital Banach function algebra A is strongly regular at one of its characters φ has a bounded approximate identity, then A has bounded relative units at φ. In particular, every strong Ditkin algebra has bounded relative units at all points of its character space. There need not, however, be a global bound available for the relative units.

Relative Property (T) for a subgroup H of a group G and its connection with positive definite functions are studied. A relation with the Haagerup approximation property is established. We show that if H is a non-normal subgroup of a group G which has Property (T) and G/H is amenable as a graph then H has finite index in G.

We find topological conditions on a space X under which the left (right, or two-sided) uniformity of the free topological group F(X) induces the universal uniformity or the product uniformity ux × ux on the square of X. Special attention is given to kω metrisable spaces. The main technical tool in the paper is an extension of certain continuous pseudometrics from X2 to F(X) considered by the author in the previous volume of this journal.

A group G is called a CSA-group if all its maximal Abelian subgroups are malnormal; that is, Mx ∩ M = 1 for every maximal Abelian subgroup M and x ∈ G − M. The class of CSA-groups contains all torsion-free hyperbolic groups and all groups freely acting on λ-trees. We describe conditions under which HNN-extensions and amalgamated products of CSA-groups are again CSA. One-relator CSA-groups are characterised as follows: a torsion-free one-relator group is CSA if and only if it does not contain F2 × Z or one of the nonabelian metabelian Baumslag-Solitar groups B1, n = 〈x, y | yxy−1 = xn〉, n ∈ Z ∂ {0, 1}; a one-relator group with torsion is CSA if and only if it does not contain the infinite dihedral group.

Approximation by smooth convex functions and questions on the Smooth Variational Principle for a given convex function f on a Banach space are studied in connection with majorising f by C1-smooth functions.

We extend the recent work of Dummit and Hayes on rank one Drinfeld modules on elliptic curves and hyperelliptic curves in the case that the infinite place is ramified to the case that the infinite place is inert.

Constructed are strictly increasing smooth families Σt ⊆ ∂D × C2, t ∈ [0, 1], of fibrations over the unit circle with strongly pseudoconvex fibers all diffeomorphic to the ball such that there is no analytic selection of the polynomial hull of Σ0 and which end at the product fibration . In particular these examples show that the continuity method for describing the polynomial hull of a fibration over ∂D fails even if the complex geometry of the fibers is relatively simple.

We classify, modulo the kernel of the action, finite groups G that act 2-transitively on Sylr(G) for some prime r dividing |G|. We furthermore prove that any finite group that acts 2-transitively on Sylr(G) for each prime r is solvable and of nilpotent length at most 3.

A module M is called a CS-module if every submodule of M is essential in a direct summand of M. It is shown that a ring R is semilocal if and only if every semiprimitive right R-module is CS. Furthermore, it is also shown that the following statements are equivalent for a ring R: (i) R is semiprimary and every right (or left) R-module is injective; (ii) every countably generated semiprimitive right R-module is a direct sum of a projective module and an injective module.

We obtain a result about the maximal circumradius of a planar compact convex set having circumcentre O and containing no non-zero lattice points in its interior. In addition, we show that under certain conditions, the set with maximal circumradius is a triangle with an edge containing two lattice points.

In this paper we consider the situation that a group G has a subgroup H which is a dihedral 2-group and with connected transversals A and B in G. We show that G is then solvable and moreover, if G is generated by the set A ∪ B, then H is subnormal in G. We apply these results to loop theory and it follows that if the inner mapping group of a loop Q is a dihedral 2-group then Q is centrally nilpotent.

For a continuous convex function on an open convex subset of any Banach space a separability condition on its image under the subdifferential mapping is sufficient to guarantee the generic Fréchet differentiability of the function. This gives a direct insight into the characterisation of Asplund spaces.

For years it has remained a problem to find suitable functionals satisfying certain maximum principles for solutions of the equation Δu + f(x, u) = 0. In this paper, maximum principles for certain functionals which are defined on solutions of semilinear elliptic equations subject to mixed or Robin boundary conditions are obtained. The principles derived may be used to deduce bounds on important quantities in physical problems of interest.