Let x1, x2,…, xn, be n consecutive observations generated by a stationary time series {x}, t = 0, ±1, ±2,…, with E(xt2) < ∈. The periodogram of the set of observations, which may be defined as a function In of angular frequency with range [0, π] that is proportional to , plays an important part in methods of making inferences about the structure of {xt}, particularly its spectral distribution function or spectral density.

A “roughness parameter”, first used by the author in 1953 (Feather 1953 b) has been re-calculated for 348 points on the mass surface. Systematic features are identified in relation to the variation of this parameter with charge number (Z) and isotopie number (D). In the region of small Z these regularities provide evidence for the persistence of some degree of alpha-unit structure at least as far as Ca. In the region of greater Z (20 < Z < 50) they provide evidence for neutron-proton interactions among the last-added nucleons. Overall, they indicate that the “residuals” characterizing the various semi-empirical mass equations currently in use very probably arise in large part from sub-shell effects which it would be impracticable to attempt to include in the equations.

The paper describes an investigation of the terminal velocity of uniformly dispersed particles of various shapes, sizes and densities falling through water.

It is concluded that for concentrations above 0–5 per cent by weight, the suspension as a whole behaves as though it were viscous even though the individual particles lie well outside the Stokes range. The shape of the particles has a significant effect only when the concentration is less than 0·5 per cent, and for concentrations between 0·5 and 7·0 per cent, the relative changes in velocity of descent are adequately described for a range of particle shapes from highly angular to spherical and for sizes at least up to 0·65 mm. nominal diameter, by the power series

in which U is the velocity of the suspension, U0 that of a single particle, d the nominal diameter (i.e. that of a sphere having the same volume) and s the mean spacing of the particles.

If the concentration is lower than 4 per cent, the equation may be assumed linear in (d/s) without serious error.

In this note we consider the problem of determining the stress on the boundary y = 0 of the elastic half-plane y ≡ 0 when there are prescribed body forces acting in the interior and the boundary is free from applied stress. Expressions for the components of stress at a general point of the half-plane when the imposed body force is concentrated at a single point have been derived by Melan [1], Sneddon [2] and Green [3], each author making use of a different method.

Many results concerning real orthogonal matrices have their counterparts in the theory of orthogonal Boolean matrices. In particular, the analogues for Boolean matrices of certain theorems due to Kronecker are established. The structure of the group of orthogonal Boolean matrices of order n is determined in the case where the underlying Boolean algebra is finite.

where be an entire

function represented by a Dirichlet series whose order (R) and proximate order (R) are respectively ρ (0 < ρ < ∞) and ρ(σ). For proximate order (R) and its properties, see the paper of Balaguer [4, p. 28].

The distribution of stress in the neighbourhood of an infinite row of collinear cracks in an elastic body is calculated for general distributions of internal pressure. The case in which each of the cracks is opened out by the same constant pressure is considered in detail; the variation with the ratio of the width of the crack to the distance between two cracks of quantities of physical interest is shown in a series of diagrams.

Let and Hn(x) be the nth Laguerre and Hermite polynomials, respectively. Two well-known bilinear generating formulas are the Hardy-Hille formula [1, p. 101]

and the mehler formula [1, p. 377]

This suggests the following problem. Consider the equation

where fa(x) is a polynomial in x of degree n with highest coefficient equal to 1,

A0 = B0 = 1. We shall also assume that ak = 1 and y0y1y2 … yk–1 ≠ 0. We seek all sets of polynomials {fn(x)} which satisfy (1.3), (1.4) and (1.5).

Let G be a graph with p points and q lines, and genus γ. The thicknesst(G) has been introduced as the minimum number of planar subgraphs whose union is G. This topological invariant of a graph has been studied by Battle, Harary and Kodama [1], Tutte [7], Beineke, Harary and Moon [3], and Beineke and Harary [2].

It is natural to generalise this concept of the thickness of a graph to the union of graphs with a specified genus. We say that the n-thickness of G is the minimum number of subgraphs of genus at most n whose union is G. Denoting the n-thickness of G by tn, we write in particular t0 the 0-thickness, i.e., the thickness.

In this note, we first establish an integral transform pair where the kernel of each integral involves the Gaussian hypergeometric function. Special cases of Theorem 1 have been studied by several authors [1, 2, 5, 6]. In Theorem 2 a similar integral transform pair involving a confluent hypergeometric function is given.

The Lucas numbers υn and the Fibonacci numbers υn are defined by υ1 = 1, υ2= 3, υn+2 = υn+1 + υn and u1 = u2 = 1, un+2 = un+1 + un for all integers n. The elementary properties of these numbers are easily established; see for example [2].However, despite the ease with which many such properties are proved, there are a number of more difficult questions connected with these numbers, of which some are as yet unanswered. Among these there is the well-known conjecture that un is a perfect square only if n = 0, ± 1, 2 or 12. This conjecture was proved correct in [1]. The object of this paper is to prove similar results for υn, ½un and ½υn, and incidentally to simplify considerably the proof for un. Secondly, we shall use these results to solve certain Diophantine equations.

The ellipsoidal wave equation is the name given to the ordinary differential equation which arises when the wave equation (Helmholtz equation) is separated in ellipsoidal co-ordinates. In this paper, solutions of the equation are expressed as Neumann series (series of Bessel functions of increasing order).

A matrix analysis is derived, using the concept of finite elements, for beam-columns of continuously varying cross-section and “small” initial curvature subject to tangential and normal forces acting at discrete points along the centroidal axis.

The relationships between loads applied to the ends of the member and their corresponding deformations are established in the form of stiffness matrices, which are the basis of the Equilibrium Method of Analysis of linear or non-linear elastic plane frames composed of such membeis. In addition, the end loads induced by the tangential and normal forces for various types of boundary conditions are obtained.

Finally, it is shown that the buckling load may be calculated by the determination of the lowest eigenvalue of a certain matrix, provided there is no lateial translation of the ends of the member.

In the case of Boolean matrices a given eigenvector may have a variety of eigenvalues. These eigenvalues form a sublattice of the basic Boolean algebra and the structure of this sublattice is investigated. Likewise a given eigenvalue has a variety of eigenvectors which form a module of the Boolean vector space. The structure of this module is examined. It is also shown that if a vector has a unique eigenvalue λ, then λ satisfies the characteristic equation of the matrix.

In [2] a condition, originally due to Olagunju, was given for the spectra of certain compact operators to be on the real axis of the complex plane. Here, by using conformal mappings, this result is extended to more general curves. The problem divides naturally into two cases depending on whether or not the curve under consideration passes through the origin. Discussion is confined to the prototype curves C0 and C1. The case of C0, the unit circle of centre the origin, is considered in § 3; this problem is a simple one as the spectrum is a finite set. In § 4 results are given for C1 the unit circle of centre the point 1, and some results on ideals of compact operators, given in § 2, are needed. No attempt has been made to state results in complete generality (see [2]); this paper is kept within the framework of Hilbert space, and particularly simple conditions may be given if the operators are normal.

In recent years interest in the mixed boundary value problems of mathematical physics has increased appreciably because of various applications. The mixed boundary value problems for simply-connected regions have been investigated widely and it can reasonably be hoped that within a short time the theory will reach a satisfactory stage. It appears, however, that very few problems for multiply-connected domains have been solved. Recently Srivastav [2] has considered the problem of rinding an axisymmetric potential function for a half space with a cylindrical cavity subject to mixed type boundary conditions. In a subsequent paper [1], Srivastav extends the analysis to the asymmetric problem and formulates the problem in terms of dual integral equations involving Bessel functions of the first and second kinds whose solution leads to the solution of the potential problem. The latter paper, however, involves heavy manipulations and complicated contour integrals.

The traditional method of solution to problems in linear viscoelasticity theory involves the direct application of the Laplace transform to the relevant field equations and boundary conditions. If the shape of the body under consideration or the type of boundary condition specified at a point or both vary with time then this method no longer works. In this paper we investigate the applicability of stress function solutions to this situation. It is shown that for time-dependent ablating regions a generalization of the Papkovich Neuber stress function solution of elasticity holds. As an example the stress and displacement fields are calculated for the problem of an infinite viscoelastic body with a spherical ablating stress free cavity and prescribed time-dependent stresses at infinity.