General convergence theorems are established for the lower part of the intermediate operator spectra arising in lower bound methods of Aronszajn type.Convergence of the T*T method is considered in detail, as are constructions that allow perturbations that are nonclosable in the underlying Hilbert space. Problems with essential spectra are admitted, as is the use of infinite rank projections capable of displacing essential spectra. These last features are necessary for a theory applicable to Schrodinger operators corresponding to atomic systems.

Conditions, generalizing the usual Liouville–Green conditions, are obtained under which the equation {r(x)y′}′ + q(x)y = 0, where q and r are positive and have derivatives of a sufficiently high order, has solutions of the form y ∼ (rq)–¼e10. 0 ∼ f (q/r)½ dx as x →∞. A related result is obtained for a system of two first-order equations y′ = A(x)y, where the eigenvalues of A are pure imaginary.

The ring of a finite projective plane, introduced by the author in a previous paper, is shown to be a Gorenstein ring. Its integral closure and conductor are also determined.

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.

Decay estimates for the energy are derived for the initial boundary value problem of the wave equation with a degenerate dissipative term:

where Ω is a bounded domain in Rn, a(×) is a nonnegative function such that a1 ∊Lp(Ω) for some p > 0 and f is a function tending to 0 rapidly as t → ∞

By assuming that a linear scalar functional differential equation (FDE) has only the zero eigenvalue on the imaginary axis, it is shown that the flows on the centre manifolds of all Cr-perturbations of this equation coincide with the flows obtained from scalar ordinary differential equations (ODEs) of order m, where m is the multiplicity of the zero eigenvalue. Furthermore, it is shown that the above situation can be realized through differential difference equations with m – 1 fixed distinct delays.

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.

In this paper some eigenvalue problems for elliptic as well as hyperbolic equations are solved. The main tool used is an abstract critical point theorem on an unbounded manifold of the form {u | (Lu, u) = constant} (where L is a nonpositive selfadjoint operator), which makes use of a linking type argument on a manifold.

We study the possibility of perturbing a matrix A by a diagonal matrix so that an eigenvalue problem with leading matrix A has specifiedeigenvalues when A is replaced by A+D. The particular cases presented are the one-parameter generalized eigenvalue problem (A× = λB μ×, a two-parameter eigenvalue problem (A + λB + μC)× = 0, a linked system ofsuch two-parameter problems and a quadratic eigenvalue problem (A + λB + λ2C)× = 0. The work extends results of Hadeler for the classical problem A× = λ×.

We give a direct and simple proof of a central result of the above paper.

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.

Estimates on the gradient of solutions to the Dirichlet problem for a semilinear elliptic equation are given when the nonlinearity in the equation is quadratic with respect to the gradient of the solution. These estimates extend results of F. Tomi to less smooth boundary data and results of the author to the full quadratic growth.

For a smooth manifold M ⊆ ℝn, the symmetry set S(M) is defined to be the closure of the set of points u∈ℝn which are centres of spheres tangent to M at two or more distinct points. (The idea has its origin in the theory of shape recognition.) The connexion with singularities is that S(M) can be described alternatively as the levels bifurcation set of the family of distance-squared functions on M. In this paper a multi-germ version of the standard uniqueness result for versal unfoldings of potential functions is used to obtain a complete list of local normal forms (up to diffeomorphism) for the symmetry sets of generic plane curves, generic space curves, and generic surfaces in 3-space. For these cases the authors verify that M can be recovered as the envelope of a family of spheres centred at smooth points of S(M).

The existence and certain qualitative properties of travelling-wave solutions to the Korteweg-de Vries-Burgers equation,

are established. The limiting behaviour of these waves, when ε tends to zero and when δ tends to zero is examined together with a singular limit wherein both ε and δ tend to zero.

The deficiency index of each power of the differential expression M[y] = w−1(−(py′)′ + qy), defined on [a, ∞), is calculated exactly in terms of the behaviour of a simple function of p and w for a large class of expressions satisfying a hypothesis which requires that p be large compared with w and q. In general, not all powers of M are limit-point.

We give local results related to Hopf bifurcation for parabolic equations. The linear part about the equilibrium point can have zero eigenvalues. In our results the information about the perturbation is essential and it is possible to obtain bifurcation even if some ‘i’ or zero remains on the imaginary axis for all values of the parameter.

We prove the existence of the thermodynamic limit of the free energy per particle for a twocomponent plasma in one space dimension and with a logarithmic pair interaction.

In catalysis theory there is interest in the number of solutions to the equation

with the boundary conditions

the parameters λ,β, γ being all positive and p a non-negative integer. The paper answers this question when γ is large, which is the interesting situation physically. Although the treatment is somewhat different in the cases p = 0 and p ≠ 0, the final answer is the same, that is, given β, there exist two positive functions λ1(γ) and λ2(γ) such that the problem has one solution if λ<λ1(γ), or λ>λ2(γ), three solutions if λ1(γ)<λ <λ2(γ), and two solutions if λ=λ1(γ) or λ=λ2(γ).