We study the semilinear equation –Δu + β(u) = f in ℝ2, where β is a continuous increasing real function with β(0) = 0 and f is a bounded Radon measure. We show the existence of a solution, which is unique in the appropriate class, provided that each of the point masses contained in f does not exceed some critical value denned in terms of the growth of (β at ∞ This condition is shown to be necessary for the existence of solutions, even locally. The one-dimensional situation is also discussed.

Let W be a semigroup with W\W2 non-empty, such that if ρ is a congruence on W with xpy for all x, y= W\W2, then zpw for all z, w= W2. We prove that the lattice of congruences on W is directly indecomposable, and conclude that a direct product of cyclic semigroups, with at least two non-group direct factors, has a directly indecomposable lattice of congruences. We find that the lattice of congruences on a direct product S1×S2×V of two non-trivial cyclic semigroups S1 and S2, one not being a group, and any other semigroup V, is not lower semimodular, and hence, not modular. We then prove that any finite ideal extension of a group by a nil semigroup has an upper semimodular lattice of congruences, and conclude that a finite direct product of finite cyclic semigroups has an upper semimodular lattice of congruences.

The following conjecture of I. D. Macdonald is confirmed in this paper. If G is a finite ρ-group generated by elements of breadth at most n, then the nilpotency class of G is bounded in terms of nonly.

The Weyl theory assets that there exist circles, possibly degenerate, in the complex plane which are associated with Sturm–Liouville equations. Closely related to these circles is the Titchmarch–Weyl m-function which plays a major role in the theory of Sturm-Liouville equations. In this paper, we investigate the asymptotic behaviour of the m-function and derive results of the form

for λ in a certain subset of the compledx plane.

An E-unitary inverse semigroup, S, has the property that, if x=S, and e2 = e=S, then (xe)2 = xe implies that x2 = x. As a consequence of this, we can see that S is an extension of its semilattice of idempotents, E, by its maximal group morphic image, G. Thus, following McAlister (1974), we attempt to describe S in terms of E and G. If we extend the semilattice E to a larger semilattice F, we are able to describe S in terms of a semi-direct product of F and G, giving a new interpretation to the approach of Schein (1975).

The following conjecture of Race will be proved: if τ is a formally J-symmetric quasi-differential expression on a real interval I, such that for some λ = ℂ all solutions of τy = τy belong to L2(I), then λ belongs to the regularity field of the minimal operator To generated by τ in L2(l).

A class of evolution problems is investigated [see (1.5), (1.6), (1.7) of the present paper] which includes, as a particular case, an evolution problem previously considered by a different author. Existence and uniqueness theorems are given in several function spaces. It is shown that, when the solution is required to belong to spaces of smooth functions, the problem becomes overdetermined. The necessary and sufficient integro-differential equations, to be satisfied by the datum, for the existence of a smooth solution, are given.

Explicit formulae and numerical values for upper and lower bounds for the best constant in Landau/s inequality on the real line are given. For p > 3, the value of the upper bound is less than the value of the best constant conjectured by Gindler and Goldstein (J. Analyse Math. 28 (1975), 213–238).

The present paper is concerned with developing the existence and asymptotic properties of the state density N(λ) associated with certain higher order random ordinary differential operators A of the form

where Ao has homogeneous and ergodic coefficients with respect to the σ-algebra generated by the Wiener process q(ω, x). The analysis uses the Weyl min-max principle to determine rough upper and lower bounds for N(λ).

In an earlier paper (1981), the present authors made a conjecture about the number of solutions of a semilinear elliptic boundary value problem which has been investigated extensively in the past decade. The conjecture is proved in the one-dimensional case.

A closed summation operator, whose spectrum lies within a certain region, generates a derivation and antiderivation, and an Euler–Maclaurin sum formula among these three operators.

Generic singularities occurring in dispersion relations are discussed within the framework of imperfect bifurcation theory and classified up to codimension four. Wave numbers are considered as bifurcation variables x =(x1,…, xn) and the frequency is regarded as a distinguished bifurcation parameter λ. The list of normal forms contains, as special cases, germs of the form ±λ +f(x), where f is a standard singularity in the sense of catastrophe theory. Since many dispersion relations are ℤ(2)-equivariant with respect to the frequency, bifurcation equations which are ℤ(2)-equivariant with respect to the bifurcation parameter are introduced and classified up to codimension four in order to describe generic singularities which occur at zero frequency. Physical implications of the theory are outlined.

The problem of classifying homogeneous null Lagrangians satisfying an nth order divergence identity is completely solved. All such differential polynomials are affine combinations of higher order Jacobian determinants, called hyperjacobians, which can be expressed as higher dimensional determinants of higher order Jacobian matrices. Special cases, called transvectants, are of importance in classical invariant theory. Transform techniques reduce this question to the characterization of the symbolic powers of certain determinantal ideals. Applications to the proof of existence of minimizers of certain quasi-convex variational problems with weakened growth conditions are discussed.

We develop a systematic method for calculating real numbers Λ1, and Λ2 such that there are no non-trivial solutions of the equation

which belong to L2(a, ∞) for λ> Λ1, or λ < Λ2. We also give conditions under which this equation has no solutions in L2(a,∞) for any real number λ.