This paper discusses the existence, under fairly general conditions, of solutions of the two-parameter eigenvalue problem denned by the differential equation,

and three point Sturm-Liouville boundary conditions.

The attempt is made to exhibit the necessary correlations between fragment excitation (or prompt neutron number), fragment kinetic energy, and a-particle energy, in α-particle-accompanied ternary fission. Treating a single mode of mass and charge division on the assumption that the ternary process develops directly out of an intermediate binary phase, and using the mutual electrostatic potential energy of the nascent binary fragments of this phase and the additional kinetic energy developed at the moment of α-particle emission as independent variables, various formal results are obtained and discussed in the light of the experimental evidence.

In terms of a classical description, it appears likely that (for a given mode of division) the nuclear configuration at a-particle release in ternary fission is subject to much smaller variations than is the nuclear configuration at scission in binary fission in the corresponding mode. Possible inadequacies of this classical description are very briefly discussed.

This paper is concerned with high-frequency scattering in a medium, the square of whose refractive index varies linearly with height from a plane boundary. Two asymptotic methods are examined, namely the method of stationary phase and evaluation by residue series. The first of these corresponds to geometric optics and gives the high-frequency field in the illuminated region, while the second complements the first in the sense that if thsre is no point of stationary phase, the residue series is an asymptotic expansion of the field. The Airy functions in the residue series can be replaced by their asymptotic developments in terms of exponentials, and when this is done only the first term or first creeping wave is of genuine significance.

The differential equation

in N dimensions is considered, where q(x) is periodic. When N = 1, it is known that the conditional stability set coincides with the spectrum and that these also coincide with two other sets involving eigenvalues of associated eigenvalue problems. These results have been proved by means of the Floquet theory and the discriminant. Here an alternative proof is given which avoids the Floquet theory and which applies to the general case of N dimensions.

Closed form solutions of some triple equations involving trigonometric series have been obtained, in each case, by reducing them to a single integral equation. The results have been applied to determine the distribution of stress in the interior of an infinitely long elastic strip containing a pair of Griffith cracks situated on a line perpendicular to the bounding lines of the strip.

In the present paper we consider the uniqueness of the solutions of the equations governing the motion of an incompressible second-order fluid in a bounded region. For such a fluid model the stress tensor S in a rectangular Cartesian coordinate system xi at a point xi of the fluid is determined from

where I is the unit tensor, p the hydrostatic pressure and μ, β, γ are material constants. The tensors A1 and A2 are the first and second Rivlin-Ericksen tensors which are defined in terms of the velocity field vi by

Let p be an odd prime and denote by [p] the field of p elements. Let C = C(n, p) be the set of points x = (x1 …, xn) of Zn (n ≥ 1) satisfying

A general convex body in Euclidean space can be approximated by smooth convex bodies, and many results, arising first in the differential geometry of smooth convex bodies, have been extended to yield corresponding results for general convex bodies. Although convex bodies can be approximated by convex polyhedra, very little of the rich theory of convex polyhedra has been extended to general convex bodies. In this paper, we extend the concept of the one-skeleton of a convex polytope to yield the concept of the one-skeleton of a general convex body. We investigate the connectivity properties of this one-skeleton, and we extend a result of Balinski [1], on paths in the one-skeleton of a convex polytope to the class of convex bodies.

A PV-number is defined to be an algebraic integer θ, of modulus greater than one, all of whose conjugates (excluding θ itself) lie inside the unit circle. Salem [1] has shown that the set S of PV-numbers forms a closed subset of the real line.

The stability equation of the asymptotic suction boundary layer profile, using a linear stability analysis, is transformed into a generalised hypergeometric equation. The solution of the stability problem may thus be written formally in terms of the relevant generalised hypergeometric functions. An asymptotic analysis is carried out on these functions for large values of the Reynolds number, and the asymptotic representation of the solutions shown to agree with that given by the usual Orr–Sommerfeld analysis.

1. If w(mod 2ω1, 2ω2) is an elliptic parameter for points of a normal elliptic curve C = 1Cn[n − 1], then it is well known that the sets of n points in which C is met by primes have a constant parameter sum k (mod 2ωl, 2ω2), and we may express this for convenience by saying that k is the prime parameter sum for the parametrisation of C by w. If we take the origin of w (the point for which w ≡ 0) to be one of the points of hyperosculation of C, then k ≡ 0, and we may say that w is a normal parameter for C. In the same way, if Γ is the Grassmannian image curve of the generators of a normal elliptic scroll 1R2n[n − 1], then a normal parametrisation of Γ defines a normal parameter w for the generators of 1R2n, such that n of the generators have parameter sum zero if and only if they belong to a linear line-complex not containing all the generators of 1R2n; or, in particular, if they all meet a space [n – 3] that is not met by every generator of the scroll. In this paper we are concerned in the first instance with the type of normal elliptic scroll 1R22m+1[2m] whose points can be represented by the unordered pairs (u1; u2) of values of an elliptic parameter u(mod 2ω1, 2ω2); and we establish a significant connection between any normal parametrisation of the generators of 1R2m+1 and an associated parametric representation (u1u2) of its points. We also add a brief note to indicate the lines along which this kind of connection can be extended to apply to a general normal elliptic scrollar variety 1Rkmk+1[mk] whose points can be represented by the unordered sets (u1, …, uk) of values of an elliptic parameter u.

Let G be a locally compact Abelian non-discrete group. Let M(G) be the convolution algebra of Radon measures on G. Let µ be an element of M(G) with its Lebesgue decomposition [1]

into absolutely continuous, purely discontinuous and continuous singular parts. The chief problem one encounters in the study of the invertibility of µ is with the case µs ≠ 0. As observed by Wiener and Pitt [2], the problem can be handled provided µs be “not too large”. In fact, Wiener and Pitt (loc. cit.) proved the following:

Let µ be a Radon measure on R (the real line) such that

whereare the Fourier transforms of µ, µd, and ≑µs≑ is the variational norm of µs. Then, µ has an inverse in M (R).

A transversely isotropic elastic material can transmit three body waves in each direction, a quasi-longitudinal (QL) wave, a quasi-transverse (QT) wave, and a purely transverse wave. When the material is able to conduct heat the properties of small amplitude QL and QT waves are modified and we consider here the analysis of such thermo-elastic interactions in plane harmonic disturbances. The modified QL and QT waves are both found to exhibit frequency-dependent dispersion and damping of the kind known to affect dilatational waves in isotropic heat-conducting elastic materials, and in addition we show that the particle paths in the associated motions are ellipses with their axes inclined to the wave normal. This latter effect is peculiar to body waves travelling in anisotropic heat-conducting elastic materials and seems not to have been studied in detail hitherto. Numerical results referring to the propagation of plane harmonic body waves in a single crystal of zinc are presented and discussed.

Erdös and Straus have conjectured that for every integer n > 1,

is soluble in positive integers x, y, z. Schinzel has conjectured that for every a > 0 if n > no(a),

is soluble in positive integers x, y, z.

The fluid mechanical problem with which we are concerned is the behaviour of fluid occupying the half-space x > 0 above a rotating disc which is coincident with the horizontal plane x = 0 and rotating about its axis which remains fixed. Studying rotationally symmetric solutions of this problem, von Kármán [1] (see also [2; p. 93]; [3; p. 133]) reduced it to the solution of two simultaneous equations in functions f(x), g(x) which may, with suitable normalisation, be written in the form

with the boundary conditions

Generating functions for the number of linearly independent invariants of a set of tensors under a given group of transformations are given by the theory of group representations. For the full and proper orthogonal groups these generating functions are in the form of definite integrals. The classical theory of algebraic invariants gives generating functions for the number of invariants of tensors under two-dimensional unimodular transformations, these generating functions being algebraic expressions. Because of a correspondence between the two-dimensional unimodular group and the three-dimensional proper orthogonal group, the corresponding generating functions are equivalent. The main result of this paper is an explicit demonstration of this equivalence. In addition, algebraic generating functions for the three-dimensional full orthogonal group are obtained and the use of the algebraic generating functions illustrated by applying them to a third order symmetric tensor.