Using the method of generalized characteristics, we discuss the regularity and large time behaviour of admissible weak solutions of a single conservation law, in one space variable, with one inflection point.

We show that when the initial data are C∞ then, generically, the solution is C∞ except: (a) on a finite set of C∞ arcs across which it experiences jump discontinuities (genuine shocks or left contact discontinuities); (b) on a finite set of straight line characteristic segments across which its derivatives of order m, m = 1, 2,…, experience jump discontinuities (weak waves of order m); and (c) on the finite set of points of interaction of shocks and weak waves. Weak waves of order 1 are triggered by rays grazing upon contact discontinuities. Weak waves of order m, m ≥ 2, are generated by the collision of a weak wave of order m − 1 with a left contact discontinuity.

We establish sharp decay rates for solutions with initial data of the following types: (a) with bounded primitive; (b) with primitive having sublinear growth; (c) in L1; (d) of compact support; and (e) periodic.

In this paper, we prove that a positive perturbation T = T0 + q (q ≧ 0 and in ) of an essentially self-adjoint Schrödinger operator T0 = −Δ + q0 on is again essentially self-adjoint if T is relatively bounded with respect to T0. An application of the method of the proof to positive approximations of elements u ≧ 0 in D(T) by a positive sequence in is given.

We describe five different sheaf representations of a ring, all of which are full and four of which are faithful. We give a characterization of strongly harmonic rings, and show that for such rings, the four faithful representations agree.

It is shown that all travelling or standing plane-wave solutions to certain radially symmetric parabolic systems may be found by solving a related scalar ordinary differential equation (ODE). The radially symmetric systems considered here are those whose reaction term is radially directed and points inward near infinity. The stability of these waves is also discussed. Many systems arising in the physical sciences are included in the class studied and so the classification and stability of the travelling waves has physical significance.

A necessary condition is obtained for a complete graph to have a decomposition as the line-disjoint union of three isomorphic strongly regular subgraphs. The condition is used to determine the number of non-trivial solutions of the equation x3+y3 = z3 in a finite field of characteristic p ≡ 2 mod 3.

A local homeomorphism from a compact, connected manifold with boundary to a simply connected manifold without boundary is shown to be one-to-one if it is one-to-one on each component of the boundary.

A scattering theory is developed for transmission problems associated with the plate equation. Asymptotic methods of solution for large time are examined as are questions concerning regularity of solution, nature of the associated spectrum and existence of appropriate wave operators. It is shown that in contrast to solutions of the wave equation, signals can propagate with an infinite dispersion velocity.

An idempotent-separating congruence μ is studied further in this paper. It is shown to satisfy special properties with respect to regular elements and to group-bound elements. It is shown that for any semigroup S, μ is the identity congruence on S/μ. From this, it can be shown that S/μ is fundamental for any semigroup S. Some alternative characterizations of μ are given and applied to yield sufficient conditions for a subsemigroup T of S to satisfy μ (T) = μ (S) ∩ (T × T), whence T is fundamental if S is fundamental.

Topological ideas based on the notion of flows and Wazewski sets are used to establish the existence of homoclinic orbits to a class of Hamiltonian systems. The results, as indicated, are applicable to a variety of reaction diffusion equations including models of bundles of unmyelinated nerve axons.

The well-known theorem of Bernstein is extended to generalized polynomial maps of closed balanced convex sets in complex Hausdorff topological vector spaces.

In this paper, we consider the equation

where A is a linear operator, N = ψ′ with ψ ∈ C1(E, R), and E is an Hilbert space. We suppose that N has a derivative at infinity N′(∞) and that 0 belongs to the spectrum of A–N′(∞). We prove an abstract theorem for multiplicity of solutions for the above equation. We then apply this theorem to the study of periodic solutions of Hamiltonian systems and of semilinear wave equations when the period is prescribed.

In this paper, a theory of fractional calculus is developed for certain spaces D′p,μ of generalised functions. The theory is based on the construction of fractionalpowers of certain simple differential and integral operators. With the parameter μ suitably restricted, these fractional powers are shown to coincide with the Riemann-Liouville and Weyl operators of fractional integration and differentiation. Standard properties associated with fractional integrals and derivatives follow immediately from results obtained previously by the author on fractional powers of operators; see [6], [7]. Some spectral properties are also obtained.

We study the sign of solutions to a class of semilinear Dirichlet problems when the nonlinearity is, for instance, a concave-convex function which interacts with the spectrum of the linear part. We are able to prove, in some cases, exact multiplicity results for positive and negative solutions. For the proofs we employ a device which splits the given problem into two others satisfying a suitable version of the Ambrosetti–Prodi result.

Results are obtained on the existence, multiplicity, blow-up and asymptotic behaviour of the solutions to the integro-differential equation

which arises in the linearised, quasi-static theory of buckling for a viscoelastic rod.

A new description is provided for the nil radical of the algebra RS of a commutative semigroup S over a commutative ring R with a 1. It is shown that the Jacobson radical of RS is nil if the Jacobson radical of R is nil and that the converse holds in the case where S is periodic.

Page 277, line 14: Read “measure” instead of “distribution”.

Page 287, line 1: Read “(Ω)” instead of “′(Ω)”,

line 13: Read “explanation” instead of “generalization”,

line 23: Read “In fact, Corollary 1” instead of “In fact, Corollary 2”.

We thank Professor L. Véron for directing our attention to these mistakes.