The Hopf bifurcation theorem describes the creation of a limit cycle from an isolated singular point of a system of first-order differential equations depending on a parameter. This paper describes a method for determining explicitly a range of values of the parameter throughout which the Hopf configuration continues to exist; only the three-dimensional case is described in this paper, but the method can be generalized.

Rollo Davidson conjectured in 1968 that every stationary second order line process in the plane which has a.s. no parallel lines is necessarily a Cox (or doubly stochastic Poisson) process. This conjecture is disproved here. An affirmative answer is further given to the question whether there exists a lattice type point process in the plane which is stationary under arbitrary area preserving affine transformations.

A deeper mathematical insight into physical problems is gained when a relatively elementary problem is generalized mathematically beyond the bounds of physical reality, particularly when the results are independent of the generalization. See, for example, Heading (3).

The onset of turbulence in many systems appears disorganized and unpredictable. However, the present detailed study of a well-known model reveals a series of well-defined transitions from steady motion to highly non-periodic behaviour. The bifurcation structure of this simple model, which describes both a reversing disc dynamo and Benard convection in thin fluid loops, is characteristic of a class of systems with subcritical instabilities. It is found that the bifurcation curve for this instability does not have a stable branch and that non-periodic and linearly stable steady solutions coexist. The boundary between transient and non-periodic behaviour is marked by a non-uniformity in the number of oscillations between reversals. This non-uniformity, not previously observed, is a striking corroboration of the model's relationship to the geodynamo. Other features of reversal in the disc dynamo such as the presence of two time scales in the oscillations are also exhibited by geomagnetic fields. In addition to a geometric description of the transitions, a one-dimensional mapping first constructed by Lorenz for the system beyond marginal stability, is extended to the subcritical regime. This type of mapping, which can be interpreted in terms of invariant surfaces of the system, may be of value in dissecting more general systems with subcritical Hopf bifurcations.

Atkinson and Reuter(1) consider travelling wave solutions for the deterministic epidemic, with or without removals, spreading along the line. In the case where there are no removals, they reformulate the problem in terms of the solutions X(·) to the integral equation

which satisfy X(− ∞) = − ∞, X( + ∞) = 0, X(u) < 0 for u ∈ (− ∞, ∞), where

is the left hand tail of the contact distribution, and where ʗ > 0 is the velocity of the wave corresponding to X. They show that no solution is possible unless

converges for λ > 0 sufficiently small, and that any solution X must satisfy

for some C > 0. They then prove the following existence theorems.

We prove the existence and completeness of the wave operators for quantum-mechanical scattering by a potential which does not decrease to zero at infinity in two of the three space directions. We also obtain a new abstract result concerning the continuous dependence of the wave operators on the potential.

Bernstein's theorem states that the only complete minimal graphs in R3 are the hyperplanes. We shall produce evidence in favour of some conjectural generalizations of this theorem for the cases of spacelike hypersurfaces of constant mean curvature in Minkowski space and in de Sitter space. The results suggest that the class of asymptotically simple space-times admitting a complete spacelike hypersurface of constant mean curvature may well be considerably smaller than the general class of asymptotically simple space-times.

A number of papers have appeared over the past decade or so which study questions of the existence and stability of positive steady-state solutions for parabolic initial-boundary value problems of the general form

The ‘static-geometric analogy’ in thin shell structures is a formal correspondence between equilibrium equations on the one hand and geometric compatibility equations on the other. It is well known as a fact, but no satisfactory explanation of its basis has been given. The paper gives an explanation for the analogy, within the framework of shallow-shell theory. The explanation is facilitated by two innovations: (i) separation of the shell surface conceptually into separate stretching (S) and bending (B) surfaces; (ii) use of change of Gaussian curvature as a prime variable. Various limitations of the analogy are pointed out, and a scheme for numerical calculation which embodies the most useful features of the analogy is outlined.

The purpose of these remarks is to use elementary mathematics to describe some simple connexions between multi-valued Legendre transformations and certain elementary catastrophes. Multi-valued Legendre transforms appear in subjects such as non-linear elasticity and non-convex optimization.

Zeeman has drawn attention to the sequences in which catastrophes, or modes of instability, can be linked, and it is a common observation that sequences of catastrophes of low order are always found in the environment of catastrophes of higher order. In this paper, a simple buckling model is presented that generates in an elegant manner a complete sequence of the umbilic catastrophes as represented by the semi-symmetric branching points. The scan of a single fundamental parameter of this model is shown to trace a route through all regimes of the umbilic bracelet, giving in turn the hyperbolic, symbolic, elliptic, parabolic and hyperbolic umbilic catastrophes.

Elastic work functions may be restricted by material symmetry requirements and/ or by the Cauchy symmetry of the first order moduli with respect to the Green strain. This investigation enables the construction of any work function so constrained. The crystallographic point groups are considered in detail, and examples of such work functions are given.