A theory of positive definite kernels in the context of Hilbert C*-modules is presented. Applications are given, including a representation of a Hilbert C*-module as a concrete space of operators and a construction of the exterior tensor product of two Hilbert C*-modules.

The possibility of the steady motion of a spherical vortex of constant vorticity in an infinite homogeneous liquid was first pointed out by Hill in the Phil. Trans., 1894, pp. 213–245. He had already discussed a case of motion which had for the surfaces always containing the same particles those given by the equation

a particular surface being

It is known, for each 1<p<∞, p≠2, that there exist differential operators in LP(ℝN) which are not (unbounded) decomposable operators in the sense of C. Foiaş. In this note we exhibit large classes of differential (and unbounded multiplier operators which are decomposable in LP(ℝN) and hence have good spectral mapping properties; the arguments are based on the existence of a sufficiently rich functional calculus. The basic idea is to take advantage of existing classical results on p-multipliers and use them to generate appropriate functional calculi.

It is shown that if G is a group and Aut G is a countable periodic CC-group then Aut G is FC.

Minkowski proved the following (for a proof see (4)): if A and B are n × n positive semi-definite hermitian matrices then

It is known (4) that if both A and B are non-singular, then the equality holds in (1) if and only if B = cA where c is a positive number.

This paper contains an extension of a result obtained by H. Bart, M. A. Kaashoek and D. C. Lay in (2). These authors studied the reduced algebraic multiplicity RM(A; λ0) of a meromorphic operator function at a point λ0 ∈ C. They proved that under certain conditions this quantity has logarithmic behaviour, i.e.,

For more restricted cases such results had been proved by others, notably I. C. Gohberg and E. I. Sigal (see (4) and (5)). Here we shall prove that such a result also holds for a larger class of operator functions than the diagonable functions considered in (2).

Let G be a locally compact topological group, with left-invariant Haar measure. If L1(G) is the usual class of complex functions which are integrable with respect to this measure, and μ is any bounded Borel measure on G, then the convolution-product μ⋆f, defined for any f in Li by

is again in L1, and

We determine all functions f(z) meromorphic in the plane such that f′(z)/f(z) has finite order and f(z) and F(z) have only finitely many zeros, where F(z) = f″(z) + Af(z) for some constant A.

An elegant symbolic method of solving differential equations was developed by Heaviside in his “Electrical Papers” and “Electromagnetic Theory,” chiefly in connexion with problems concerning electric currents in net-works of wires. Attention has recently been called to the method by Bromwich, who applied it to a wider range of problems and gave an extension of Heaviside's formula; another generalisation of the formula has been obtained by Carson.

In the present paper a formula is obtained which contains the formulae of Heaviside, Bromwich and Carson as particular cases, and whose form is such that it may be readily applied to physical problems.

The well-known algebraic concept of tensor product exists for any variety of algebras.The tensor product of groups and of rings have been studied extensively. For other varieties, such as the variety of semigroups, the tensor product has been investigated more recently (5). In this paper we investigate the tensor product of distributive lattices.

In this paper we consider the quartic diophantine equation 3(y2 – 1) = 2x2(x2 – 1) in integers x and y. We show that this equation does not have any other solutions (x, y) with x≧0 than those given by x = 0,1,2,3,6,91. Two approaches are emphasized, one based on diophantine approximation techniques, the other depends on the structure of certain quartic number fields.