Recently Scheiblich (7) and Munn (3), amongst others, have given explicit constructions for FIA, the free inverse semigroup on a non-empty set A. Further, Reilly (5) has investigated the free inverse subsemigroups of FIA. In this note we generalise two of Reilly's lesser results, and also characterise the surjective endomorphisms of FIA. The latter enables us to determine the group of automorphisms of FIA, and to show that if A is finite then FIA is Hopfian (a result proved independently by Munn (3)). Finally, we give an alternative proof of Reilly's main theorem, which uses Munn's theory of birooted trees.

We study the eigenproblem

where

and Tm, Vmn are self-adjoint operators on separable Hilbert spaces Hm. We assume the Tm to be bounded below with compact resolvents, and the Vmn to be bounded and to satisfy an “ellipticity” condition. If k = 1 then ellipticity is automatic, and if each Tm is positive definite then the problem is “left definite”.

Keller (6) considered a generalisation of a problem of Minkowski (7) concerning the filling of Rn by congruent cubes. Hajós (4) reduced Minkowski's conjecture to a problem concerning the factorization of finite abelian groups and then solved this problem. In a similar manner Hajós (5) reduced Keller's conjecture to a problem in the factorization of finite abelian groups, but this problem remains unsolved, in general. It occurs also as Problem 80 in Fuchs (3). Seitz (10) has obtained a solution for cyclic groups of prime power order. In this paper we present a solution for cyclic groups whose order is the product of two prime powers.

In a highly interesting critical account of the mathematical work of James Gregory (1638–1675), written for the Proceedings of the Edinburgh Mathematical Society, (1) 41 (1923), 2–25, by the late Professor G. A. Gibson, there occurs at p. 8 something of a mathematical puzzle. On that page a pair of formulae are quoted, which certainly are striking examples of the analytical power of Gregory, and which run as follows:—

Gibson adds that “there is another formula (Rigaud, p. 207), but it is of a very complicated character and I do not reproduce it.” It will be convenient to refer to the above pair as formulae A′ and B′, and to the more complicated but analogous series as formula C, and to the original series, from which the above were transcribed, as formulse A and B. I am indebted to Mr A. Inglis for drawing my attention to the problem.

A triple (or monad) in a category K is a triple = (T, μ, η) where, T: K → K is a functor and μ: TT →, T, η: 1k → T are natural transformations for which (1.1) and (1.2) commute:

In these diagrams the component of a natural transformation α at an object x is denoted xα. Thus for example (kη)T is the value of the functor T applied to the component of η at k, whereas (kT)η is the component of η at the object kT. I write functions and functors on the right and composition from left to right.

Let R be an associative, commutative ring with identity, and let A be a (unitary) R-module. It is well known that if A is a Noetherian R-module then every submodule of A has a primary decomposition in A. The object of the present paper is to dualise this result; that is, to show that if A is an Artinian R-module then every submodule of A can be expressed as the sum of a finite number of coprimary submodules of A.

§ 1. The device here described has been found to simplify greatly the “somewhat laborious discussion” of the different musical intervals as given, say, in Sedley Taylor's Sound and Music (chap, viii.) or in Helmholtz's Sensations of Tone. It has been found particularly helpful in giving an account, necessarily rapid, of the nature of harmony to classes studying sound as a part of physics, from whom much familiarity with musical terms and notation is not to be expected.

We present some applications of monotonicity methods to the solution of certain nonscalar reaction-diffusion problems. In particular we prove existence under appropriate conditions and we introduce a convergent algorithm.