In 1966, J. M. Howie characterised the transformations of an arbitrary set that can be written as a product (under composition) of idempotent transformations of the same set. In 1967, J. A. Erdos considered the analogous problem for linear transformations of a finite-dimensional vector space and in 1983, R. J. Dawlings investigated the corresponding idea for bounded operators on a separable Hilbert space. In this paper we study the case of arbitrary vector spaces.

A nonlinear boundary value problem (P) having positive parameters L and a is considered. We associate with it a family of perturbed problems () affected by the presence of a barrier parameter γ related to L and a. There is a critical value L*(a) of the parameter L such that for L >L*(a), (P) has no regular solution. Then some natural extensions of (P), solutions of a free boundary value problem, arise as singular limits of ().

We use a modified form of the Stieltjes inversion formula due to Atkinson together with some recent results on the asymptotic form of Titchmarch–Weyl m-functions to obtain an asymptotic expression for the spectral functions τ(t) associated with the differential equation

We determine which subvarieties of the class of MS-algebras enjoy the amalgamation property, the embedding property, the strong amalgamation property, or the special amalgamation property.

We obtain lower bounds for the number of common systems of distinct representatives of two families of sets and the number of symmetric chain decompositions of certain ranked partially ordered sets.

We obtain the best constants for multipoint extensions of the Rayleigh–Ritz inequality, and use our results to find good upper bounds for the constants involved in the standard spline approximation and interpolation error bound theory.

Let S11× V1 be a direct product of a cyclic monoid S11with S1 not a group, and a semigroup V1 with an adjoined identity. We prove thatboth the lattice of congruences on (S11 × V1)/{(1,1)} and the lattice of congruences on (S11 × V1) are neither lower semimodular nor uppersemimodular. We then prove that the lattice of congruences on a free finitely generated commutative semigroup with more than one generator is neither lower semimodular nor upper semimodular.