1. Given a series

we will throughout write {sn} for the sequence of its partial sums; that is to say,

We will write p, q to denote the sequences {pn}, {qn}; we will write throughout

and will take P−1 = p−1 = 0.

In the contour integral

f and g are regular functions of z in a neighbourhood of the contour C and of the complex parameters (α1, α2, …, αp) = α in a domain of . N is a positive parameter and asymptotic expansions are considered as N → + ∞. The method of steepest descents provides an asymptotic expansion for each fixed value of α, but this is non-uniform with respect to α near critical values α0 at which certain saddle points tend to coincidence. A more complicated expansion, involving Airy functions or generalizations thereof, is valid and uniform for α near α0. This expansion is known to be valid in a neighbourhood of α0 which does not depend on N, in contrast to certain other expansions having regions of validity which contract to a point or surface as N → ∞. It is thereby suggested that the actual region of validity has a definite geometrical shape determined by the functions f and g and the contour C, just as steepest descents expansions are valid in regions bounded by Stokes' lines. In this paper, procedures are derived for the determination of such regions, subject to assumptions concerning f, g and C. The validity of the expansion is established in regions so determined.

Let denote the group of transformations of the upper-half complex plane U onto itself of the form

If, on U, we introduce the Riemannian metric ds = |dz| y−1 (z = x + iy), then U becomes a model of the hyperbolic plane and its group of isometries. The set of elements of type I, the orientation-preserving isometries form a subgroup of index two in , which we denote by .

Let , where pk are complex numbers, have 0 < ρ ≤ ∞ for radius of convergence and assume that P(x) ≠ 0 for α ≤ x < ρ (α < ρ is some real constant). Assuming that is convergent for all (x ∈ [0, ρ), we define the P-limit of the sequence s = {sk} by

This, so called, power method of limitation (see (3), Definition 9 and (1) Definition 6) will be denoted by P. The best known power methods are Abel's (P(x) = 1/(1 – x), α = 0, ρ = 1) and Borel's (P(x) = ex, α = 0, ρ = ∞). By Cp we denote the set of all sequences, P-limitable to a finite limit and by the set of all sequences, P-limitable to zero.

In this paper the classical Hardy transformation (5) has been extended to distributions. The celebrated Hankel transformation, Y-transformation and H-transformation are particular cases of the Hardy transformation. The Hankel transformation has been extended to distributions by Zemanian(12). The distributional theory has been explored by Zemanian(13), Koh and Zemanian(6), and Dubey and Pandey(2). The famous Y- and its reciprocal H-transformations have not been extended to distributions. The inversion and some of the properties of these distributional transformations can be deduced as particular cases of the results contained in this paper.

Consider a differential game of survival governed by the differential equation

in , with pay-off

where tF is the entry time of the trajectory (t, x(t)) into a given terminal set F. Under suitable conditions on f, g, h and the terminal set F, it was shown in (3) that the question of existence of value of such a game can be approached by considering a certain pair of partial differential equations called the Isaacs-Bellman equations.

In this paper, we improve on the results of two previous papers (8, 9) by establishing a general existence theorem (section 1·3, below) for a class of periodic oscillations of a wide class of non-linear differential equations of the second order in the real domain which are perturbations of the autonomous differential equation

where g(x) is strictly non-linear. We then, by way of illustrating the power of the theorem, apply it to the problems which Morris (section 2·2 below), Shimuzu (section 2·3 below) and Loud (section 2·5 below) set themselves on the existence of periodic oscillations of certain differential equations which are perturbations of equations of the form (1·1·1).

An investigation of the random or stochastic integral equations of the form

and

is presented, where ω ∈ Ω, the supporting set of the probability measure space (Ω, A, P). The existence and uniqueness of a random solution, a second-order stochastic process, of the equations is considered. Several theorems utilizing fixed point theorems and successive stochastic approximations give sufficient conditions for the existence of a random solution.

Let Zn = n−½(X1 + X2 + … + Xn), where {Xn} is a sequence of independent and identically distributed random variables with EX1 = 0, and a common distribution function F and characteristic function ω. Suppose |ω|r is integrable for some integer r ≥ 1. For all n ≥ r, then Zn has a probability density function fn obtained by using the inversion formula.

In electrostatic, elastostatic and hydrodynamic problems involving a plane annulus in space, two mixed boundary value potential problems are frequently encountered. For reasons given below, we consider here the axisymmetric cases of them; and we describe them using cylindrical polar coordinates (r, θ, z) so chosen that the annulus is a < r < b, z = 0.

Whitham's exact averaged variational principle is found by a new method. Comparisons are drawn between this method and the method advanced by Whitham. Further results about the Euler equations from this variational principle are obtained.

In these notes we give proofs of some properties of surface solitary waves. Assuming the existence of solitary-wave solutions to the nonlinear boundary-value problem (P) defined below, it is shown (i) that the wave is a wave of elevation alone, and (ii) that at large distances it is asymptotic to a uniform supercritical stream (i.e. the Froude number , where c is the speed of the stream, h is its depth, and g is the gravity constant).

We also deduce a number of inequalities relating F2 to a/h, where a is the maximum displacement of the free surface from its value at infinity. In particular, it is shown for the wave of greatest height that 1·480 < F2 < 2.

In this paper we derive the Lorentz-Dirac equation of motion for a charged particle moving in an external electromagnetic field. We use Maxwell's electromagnetic field equations together with the assumptions (1) that all fields are retarded and (2) that the 4-force acting on the charged particle is a Lorentz 4-force. To define the self-field on the world-line of the charge we utilize a contour integral representation for the field due to A. W. Conway. This by-passes the need to define an ‘average field’. In an appendix the case of a scalar field is briefly discussed.

A solution which is exact within the framework of the classical theory of elasticity is obtained for the title problem assuming that the half space is homogeneous and isotropic, and that the die indents at a constant rate. If the shape of the die and the elastic medium are specified, the rate of indentation uniquely determines the outward speed of the edge of the expanding contact zone. The magnitude of this speed, relative to the speeds of the dilatational, rotational and Rayleigh waves in the elastic medium, determines which of four possible characteristic transient stress states will occur. Each of the four ranges of contact speed is solved by the method of rotational superposition of self-similar potentials which is briefly described in the Appendix.

The simply supported rhombic plate under transverse load has received extensive attention from elasticians, applied mathematicians and engineers. All known solutions are based on approximate procedures. Now, an exact solution in a fast converging explicit series form is derived for this problem, by applying Stevenson's tentative approach with complex variables. Numerical values for the central deflexion and moments are obtained for various corner angles. The present solution provides a basis for assessing the accuracy of approximate methods for analysing problems of skew plates or domains.

The percolation probabilities of the face-centred cubic and close-packed hexagonal lattices are found not to be identical; the identity of their critical percolation probabilities remains an open question. Other pairs of closely similar lattices are discussed.

(1) Proc. Cambridge Philos. Soc. 73 (1973), page 173

Dr E. Leibowitz has called my attention to a mistake in my paper on isotropic tensors. (1) On p. 173, if 12 is annihilated as stated and then we annihilate l13 by the same method, a new contribution to l12 appears. The correct method is to choose α so as to annihilate l21. Then a suitable rotation on the 13 plane removes l31 without re-introducing l21. The rest of the argument stands.