In the Session of 1876, Mr John Murray communicated to this Society a paper on the distribution of volcanic debris over the floor of the ocean, and in it announced the discovery of cosmic dust in deep sea deposits. It was shown that at points, where neither the action of waves, rivers, or currents can transport the debris of continents, volcanic materials play the most important role in the formation of the mineral constituents of the deep sea deposits. It was pointed out that pumice, on account of its structure, was able-to float to great distances, but in time became waterlogged and sank to the bottom, there to decompose. On the other hand, incoherent volcanic matters, ejected in the form of lapilli, sand, and ashes, into the higher regions of the atmosphere, may, ceteris paribus, be conveyed, in consequence of their small dimensions and structure, to greater distances than other mineral particles derived from the continents. The possibility was also admitted that submarine volcanic eruptions might also contribute to the accumulation of those silicates and pyrogenous minerals and rocks, whose microscopic characters and distribution at the bottom of the sea we shall presently point out.

Let B1 be a ball of radius R1 in RN with centre at the origin and let B0 be a smaller ball of radius R0 contained inside it. Let u be the solution of the problem −Δu = 1 in B1\B0 vanishing on the boundary. It is shown that is minimal if and only if the balls are concentric. It is also shown that the first (Dirichlet) eigenvalue of the Laplacian in B1\B0 is maximal if and only if the balls are concentric. Generalizations are indicated.

Subcritical fronts are shown to exist in a quintic version of the well-known complex Ginzburg–Landau equation, which has a subcritical pitchfork as well as a supercritical saddle-node bifurcation. The fronts connect a finite amplitude plane wave state to a stable zero solution. The unstable manifold at finite amplitude and stable manifold of vanishing amplitude solutions are shown to intersect transversely on an invariant zero-wavenumber manifold with parameters set to be real. By the persistence of transverse intersection, frontal connections exist for a continuum of nearly real fronts parametrised by appropriate variables that exhibit some interesting changes in dimension.

We consider a semilinear elliptic system with both concave—convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

Starting from an age-structured model, we derive a partial differential equation satisfied by the total number of mature adult members of a population, on an infinite one-dimensional domain. The formulation involves a distribution of possible ages of maturation and uses a probability density function on which ecologically realistic assumptions are made. It is found that the existence and value of a positive equilibrium solution depends on the mean maturation delay. When no positive equilibrium exists, we prove global attractivity of the zero solution. For a particular ecologically reasonable choice of the distribution function, we show that travelling fronts exist connecting the zero equilibrium with the positive one provided the mean maturation delay is sufficiently small, and the dependence of the front's propagation speed on the mean delay is discussed.

In order to examine the hypothesis, so ably enunciated by the Steinach school, that senility is a phase of life that is attended by, perhaps even caused by, the inefficient functioning of one or more of the components of the endocrine system, the writer set out upon a series of attempts to evoke rejuvenation in the aged fowl. In a previous communication (Crew, 1925) an account was given of an unsuccessful attempt to produce this result in the aged cock by means of unilateral vasoligation. In this paper the results of thyroid administration to aged birds of both sexes are reviewed.

In previous papers we have studied oscillation properties of Sturm–Liouville problems (−Py′)′ + qy = λry, with λ-dependent boundary conditions, under various ‘definiteness’ conditions. Here we present a new, unified, approach which also covers cases previously untreated, e.g. of semidefinite weight, and also the fully indefinite problem.

This paper is concerned with the existence of minimizers for functionals having a double-well integrand with affine boundary conditions. Such functionals are related to the so-called Kohn–Strang functional, which arises in optimal shape design problems in electrostatics or elasticity. They are known to be not quasiconvex, and therefore existence of minimizers is, in general, guaranteed only for their quasiconvex envelopes. We generalize the previous results of Allaire and Francfort, and give necessary and sufficient conditions on the affine boundary conditions for existence of minimizers. Our method relies on the computation of the quasiconvexification of these functionals by using homogenization theory. We also prove by a general argument that their rank-one convexifications coincide with their quasiconvexifications.

Components in the function space of maps from a space X to the classifying space BG of a topological group G can sometimes be distinguished up to homotopy type by a Samelson product method. When X is a closed Riemann surface and G is a unitary group, this method is nearly sufficient to classify the components up to homotopy type.

This paper is devoted to the study of time-periodic solutions to the nonlinear one-dimensional wave equation with x-dependent coefficients u(x)ytt – (u(x)yx)x + g(x,t,y) = f(x,t) on (0,π) × ℝ under the periodic boundary conditions y(0,t) = y(π,t), yx(0,t) = yx(π,t) or anti-periodic boundary conditions y(0, t) = –y(π,t), yx[0,t) = – yx(π,t). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. Our main concept is that of the ‘weak solution’. For T, the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.

Using the theory of indicator measures, a lower semicontinuity result for quasiconvex functions in W1,1 and assuming only L1 convergence is obtained.

Oscillation, comparison and asymptotic theory for the Sturm-Liouville problem

with 1/p, q, r ε L1 ([0, 1]), p, r > 0, are studied subject to eigenvalue-dependent boundary conditions

This continues previous work on cases with (− 1)j δj ≦ 0 where δj = ajdj − bjcj. We now consider the remaining sign conditions for δj, exploiting the interplay between the graph of cot θ− (λ, 1), for a modified Prüfer angle θ−, and the eigencurves of a related two-parameter problem.

Reaction-diffusion systems on the real line are considered. Localized travelling waves become unstable when the essential spectrum of the linearization about them crosses the imaginary axis. In this article, it is shown that this transition to instability is accompanied by the bifurcation of a family of large patterns that are a superposition of the primary travelling wave with steady spatially periodic patterns of small amplitude. The bifurcating patterns can be parametrized by the wavelength of the steady patterns; they are time-periodic in a moving frame. A major difficulty in analysing this bifurcation is its genuinely infinite-dimensional nature. In particular, finite-dimensional Lyapunov–Schmidt reductions or centre-manifold theory do not seem to be applicable to pulses having their essential spectrum touching the imaginary axis.

The equation considered in this paper is tp(φp(x′))′ + g(x) = 0, where φp(x′) = |x′|p−2x′ with p > 1, and g(x) satisfies the signum condition xg(x) > 0 if x ≠ 0 but is not assumed to be monotone. Our main objective is to establish a criterion on g(x) for all non-trivial solutions to be non-oscillatory. The criterion is the best possible. The method used here is the phase-plane analysis of a system equivalent to this differential equation. The asymptotic behaviour is also examined in detail for eventually positive solutions of a certain half-linear differential equation.

The abstract non-linear non-autonomous functional differential equation

is considered. An evolution operator is associated with the solutions of this equation and existence and stability results are obtained.

We study the blow-up of positive solutions of the Cauchy problem for the semilinear parabolic equation

where u is a scalar function of the spatial variable x ∈ ℝN and time t > 0, a ∈ ℝV, a ≠ 0, 1 < p and 1 ≦ q. We show that: (a) if p > 1 and 1 ≦ q ≦ p, there always exist solutions which blow up in finite time; (b) if 1 < q ≦ p ≦ min {1 + 2/N, 1 + 2q/(N + 1)} or if q = 1 and 1 < p ≦ l + 2/N, then all positive solutions blow up in finite time; (c) if q > 1 and p > min {1 + 2/N, 1 + 2q/N + 1)}, then global solutions exist; (d) if q = 1 and p > 1 + 2/N, then global solutions exist.

It is shown that, generically, scalar one-dimensional parabolic equations ut = (a2(x)ux)x + f(u), x ∈ [0, 1], with Neumann boundary conditions, have all the equilibrium solutions hyperbolic.

Moreover, the bifurcations of these equilibria are generically of the saddle-node type.

This paper proves the existence of (at least) one solution of the following equation:

Here, is an elliptic operator of Leray-Lions type acting from into Lp′(0, T; W−1.p′ (Ω)), (1/p + 1/p′ = 1) and |F(u, ∇u)| ≧b(|u|)(l + |≧u|P). There are no smoothness assumptions on the bounded open set Ω; the operator and the nonlinearity F(u, ∇u) are denned in terms of Carathéodory functions. These points are the most characteristic features of this paper.

Assuming the existence of upper and lower solutions allows us to obtain L∞(Q)-estimates. An estimate is then proved. The final step is to prove the strong convergence in of the approximations. This proof relies on the method introduced by L. Boccardo, F. Murat and J. P. Puel for elliptic and parabolic problems of this type.

We give some results on existence and uniqueness for the solution of elliptic boundary value problems of type

when the βi are not necessarily smooth.

The purpose of the present paper is to give as simple an account as possible of the general-relativity theory of two-component spinors, and to investigate its geometrical and analytical consequences. The work was suggested by courses of lectures given at Edinburgh in 1932 and 1935 by Professor E. T. Whittaker, who, on the basis of the special-relativity spinor theory of van der Waerden (1929), obtained the completely tensorized form of Dirac's equations given by him in a recent paper (1937).