The problem discussed is that of determining the sequence {an} such that

where (λn) is the sequence of positive zeros of the function λJν(λ) + HJν(λ), arranged in order of increasing magnitude, þ, ν and H are real constants (−I <þ < I, ν > −½) and f1(ρ), f2(ρ) are prescribed. By expressing the sequence {cn} in terms of a sequence of integrals involving a function g(t) the problem is reduced to the solution of a non-singular Fredholm integral of the second kind for g(t).

Uniqueness of positive solutions of the following equation:

has been obtained, where B ⊂ ℝn is the ball of unit radius and λ > 0. Moreover, if n = 2, then the solutions are nondegenerate.

We show that there exists a function f, meromorphic in the plane C, such that the family of all functions g holomorphic in the unit disc D for which f ∘ g has no fixed point in D is not normal. This answers a question of Hinchliffe, who had shown that this family is normal if Ĉ\f(C) does not consist of exactly one point in D. We also investigate the normality of the family of all holomorphic functions g such that f(g(z)) ≠ h(z) for some non-constant meromorphic function h.

In this paper we study the asymptotic behaviour of reaction–diffusion systems with a small parameter by using the n-dimensional Feynman–Kac formula and large deviation theory. The generalised solutions are introduced in Section 2. We obtain the travelling wave joining an unstable steady state and an asymptotically stable steady state of a diffusionless dynamical system in a reaction–diffusion system with nonlinear ergodic interactions, and a special case with nonlinear reducible interactions.

In this paper we focus on the existence of a weak solution to a system describing a self-propelled motion of a single deformable body in a viscous compressible fluid that occupies a bounded domain in the three-dimensional Euclidean space. The governing system considered for the fluid is the isentropic compressible Navier–Stokes equation. We prove the existence of a weak solution up to a collision.

Let G be a simple, compact Lie group and let be the gauge group of the principal G-bundle over S4 with second Chern class k. McGibbon classified the groups G that are homotopy commutative when localized at a prime p. We show that in many cases the homotopy commutativity of G, or its failure, determines that of .

We obtain infinitely many non-radial rupture solutions of the equation

with

by constructing infinitely many radially symmetric regular solutions of the equation on SN−1:

We prove a Lusin type theorem for a certain class of linear partial differential operators G(D), reducing to [1, Theorem 1] when G(D) is the gradient. Moreover, we describe the structure of the set {G(D)f = F}, under assumptions of non-integrability on F, in terms of lower dimensional rectifiability and superdensity. Applications to Maxwell type system and to multivariable Cauchy–Riemann system are provided.

L'hiver passé je reçus une invitation de Sir John Murray à venir, en qualité d'assistant, au Challenger Office étudier sous sa direction une nombreuse collection de concrétions phosphatées et nodules provenant de dépôts dragués par les bateaux du “Department of Agriculture” du Cap de Bonne Espérance et envoyée par Dr J. D. F. Gilchrist, “Government Biologist of the Cape of Good Hope.”

Le Sage, born at Geneva in 1724, devoted the last sixty-three years of a life of eighty to the investigation of a mechanical theory of gravitation. The probable existence of a gravific mechanism is admitted and the importance of the object to which Le Sage devoted his life pointed out, by Newton and Rumford in the following statements:—

“It is inconceivable that inanimate brute matter should, without the mediation of something else, which is not material, operate upon, and affect other matter without mutual contact; as it must do, if gravitation, in the sense of Epicurus, be essential and “inherent in it.

The paper which I have the honour of presenting to you to-night is similar in construction to those I have previously brought under your notice on the For, Madi, and Waganda tribes.

As in former papers, I have written from notes of my own observations made when in the country and from information supplied me by people of the tribe. This time, however, in order to complete the description of the people, I have had recourse to the works published by Sir Samuel Baker, Dr Emin Pasha, Capt. Casati, and the late Dr W. Junker, and I have to acknowledge my indebtedness to these authors for information gleaned from their pages.

In 1916, in a paper on the size of the particles in deep-sea deposits, Odén showed that the distribution by mass of a suspension into classes of particles of different size could be inferred from a study of the course of sedimentation of such a suspension from a state of uniform dispersion in water. The experimental data required constitute the “sedimentation curve ” of the suspension, showing the total weight deposited, at a fixed depth, at the end of any assigned length of time.

In the course of the detailed investigations which, during the past six or seven years, have been carried on by the officers of the Geological Survey into the history of the glacial period in Britain, the desire naturally arose to compare the phenomena of glaciation so familiar in this country with those of some other region where they might be linked on to the action of still existing glaciers. No other part of Europe offered so many facilities for such a comparison as were to be found in Scandinavia. It was accordingly planned by my colleague, Dr John Young, and myself, to visit Norway in the summer of 1863. Unforeseen circumstances delayed the journey, and ultimately deprived me of the companionship of my friend.

Dr Pettersson has recently published a paper on “Gezeitenaehnliche Bewegungen des Tiefenwassers” (Publications de Circonstance, No. 47), in which he describes certain movements in the deep waters of the Skagerak observed in January, February, and March 1909 at the Swedish station Bornö, in the Gullmarfjord, lat. 58° 24′ N., long. 11° 33′E. The observations were taken daily at 12.30 p.m. by means of soundings made through the ice. The diagram accompanying this paper is a reproduction of a diagram prepared by Dr Pettersson, and shows the observed variations in the position of the isotherms and isohalines. No observations appear to have been made on 7th, 8th, 10th, and 11th February, nor from 15th to 18th March, so that there are large interpolations on the diagram. Fig. 1 is a map of the Skagerak, the point of observation being marked by a cross.

Uniform asymptotic expansions are obtained for the associated Legendre functions and , and the Ferrers functions and , as the order μ → ∞. The approximations are uniformly valid for 0 ≤ ν + ½ ≤ μ(1 − δ), where δ ∈ (0, 1) is fixed, x ∈ (−1, 1) in the real-variable case and Re z ≥ 0 in the complex-variable case. Explicit error bounds are available for all approximations. In the complex-variable case, expansions are obtained by an application of two existing general asymptotic theories to the associated Legendre differential equation: the first case (in which ν is fixed) applies to regions containing an isolated simple pole; and the second case (in which 0 ≤ ν + ½ ≤ μ(1 − δ)) applies to regions containing a coalescing turning point and double pole. In both cases, the expansions involve modified Bessel functions. In the real-variable case (in which 0 ≤ ν + ½ ≤ μ(1 − δ)), asymptotic expansions of Liouville–Green type are obtained, which involve elementary functions.

In [6] and [9] two different methods are given for comparing solutions of Dirichlet problems for second order quasilinear elliptic equations on convex regions. In this paper a general comparison technique is outlined—one which contains the methods of [6] and [9] as special cases. This technique is then applied to a number of special examples, comparisons with known results are given and a number of possible extensions are indicated.

The asymptotic expansions of solutions of a class of linear ordinary differential equations of arbitrary order n are investigated for large values of the independent variable z in the complex plane. Solutions are expressed in terms of Mellin-Barnes integrals and their asymptotic expansions are subsequently determined by means of the asymptotic theory of integral functions of the hypergeometric type. Three classes of solutions are considered: (i) solutions whose behaviour is either exponentially large or algebraic for |z|→∞ in different sectors of the z-plane, (ii) solutions which are even and odd functions of z when the order n of the differential equation is even and (iii) solutions which are exponentially damped as |z|→∞ in a certain sector of the z-plane.

The semilinear damped wave equations

subject to homogeneous Neumann boundary conditions, admit spatially homogeneous solutions (i.e. u(x, t) = u(t)). In order that every solution tends to a spatially homogeneous one, we look for conditions on the coefficients a and d, and on the Lipschitz constant of f with respect to u.

We prove that a finite semigroup which is decomposable into a direct product of cyclic semigroups which are not groups is uniquely so decomposable, and show how the non-group cyclic direct factors of a finite semigroup, if they exist, can be found.