A number of papers have been written concerning the properties of triangles which circumscribe convex sets, see for example (1), (2), (3).

In this note we shall characterise two set functionals, one of which has already been introduced in (6). This will enable us to produce some new results in the spirit of those obtained in the papers mentioned above. These two functionals are interesting in that although they are defined in a rather abstruse manner, they turn out to have an intuitive geometric meaning, namely that they are the minimal widths of certain circumscribing triangles.

In (8) Todd and Coxeter described an algorithm for enumerating the cosets of a finitely generated subgroup of finite index in a finitely presented group. Several authors ((1), (2), (5), (6), (7)) have discussed a modification of the algorithm to give also a presentation of the subgroup in terms of the given generators.

In their recently published book Bonsall and Duncan ask the following question ((1), p. 65): if A is a Banach star algebra with an identity, is it true that

In this paper we continue our study of the tensor product of distributive lattices which was begun in (2). We obtain a representation of the tensor product as a ring of sets and we describe a simple way to construct the tensor product of finite distributive lattices.

A bounded linear operator T on a complex reflexive Banach space is said to be well-bounded if it is possible to choose a compact interval J = [a, b] and a positive constant M such that

for every complex polynomial p, where ‖p‖J denotes sup {|p(t)|:t ∈ J}. Such operators were introduced and first studied by Smart (4). They are of interest principally because they admit (and in fact are characterised by) an integral representation similar to, but in general weaker than, the integral representation of a self-adjoint operator on a Hilbert space. (See (2) and (4) for details.) It is easily seen, by verifying (1) directly, that T is well-bounded if it is a scalar-type spectral operator with real spectrum.

The inverse problem we will consider in this paper has its origins in the following problem connected with the scattering of acoustic waves in a nonhomogeneous medium. Let an incoming plane acoustic wave of frequency ω moving in the direction of the z axis be scattered off a “soft” sphere Ω of radius one which is surrounded by a pocket of rarefied or condensed air in which the local speed of sound is given by c(r) where r = |x| for x ∈ R3. Let us(x)eiωt be the velocity potential of the scattered wave and let r, θ, φ be spherical coordinates in R3. Then from a knowledge of the far field pattern f(θ, φ, λ) for λ = ω/c0 contained in some finite interval 0 < λ0 ≤ λ ≤ λ1,, we would like to determine the unknown function c(r).

Foguel (8) and Fixman (7) independently proved that an invertible spectral operator, which is power-bounded, is of scalar type. Their proofs rely heavily on a result of Dunford on spectral operators whose resolvents satisfy a growth condition. (See Lemma 3.16 of (6, p. 609).) Observe that the resolvent of an invertible power-bounded operator T satisfies an inequality of the form