Let T2 = {(eix1, eix2):0 ≦ xj<2π, j=1,2} be a two dimensional torus and r, s, t and k be positive integers with k>r+s+t–2. Our main object is to study the approximation and interpolation properties of a class of smooth functions whose restrictions to each triangle of a three direction mesh lie in the linear span of or 0≦μ≦r–1, r+s–l≦μ+ν≦r+s+t–2, or 0≦ν≦s–1, r+s–1≦μ+ν≦r+s+t–2} Where (z1, z2) ∈ T2.

Many problems in mathematical analysis require a knowledge of the asymptotic behaviour of Γ(z + α)/Γ(z + β) for large values of |z|, where α and β are bounded quantities. Tricomi and Erdélyi in (1), gave the asymptotic expansion

where the are the generalised Bernoulli polynomials, see (2), defined by

In this note, we show that if, instead of considering z to be the large variable, we consider a related large variable, (1) can be improved from a computational viewpoint.

In the following paper I give a complete list of the types of covariants belonging to the concomitant system of three quaternary quadrics, where covariant is used in its restricted sense and refers solely to a concomitant involving the variable x alone. A complete list of the types, 62 in number, is given in §1. In §§(6–10) the covariants are determined, and in §§(11–12) a list of the identities used in the reduction of the covariants is given, along with typical examples of the process.

Dual integral equations of the form

where f(x) and g(x) are given functions, ψ(x) is unknown, k≧0, μ, v and α are real constants, have applications to diffraction theory and also to dynamical problems in elasticity. The special cases v = −μ, α = 0 and v = μ = 0, 0<α2<1 were treated by Ahiezer (1). More recently, equations equivalent to the above were solved by Peters (2) who adapted a method used earlier by Gordon (3) for treating the (extensively studied) case μ = v, k = 0.

The inverse scattering problem for acoustic waves in shallow oceans are different from that in the spaces of R2 and R3 in the way that the “propagating” far-field pattern can only carry the information from the N +1 propagating modes. This loss of information leads to the fact that the far-field pattern operator is not injective. In this paper, we will present some properties of the far-field pattern operator and use this information to construct an injective far-field pattern operator in a suitable subspace of L2(∂Ω). Based on this construction an optimal scheme for solving the inverse scattering problem is presented using the minimizing Tikhonov functional.

At the close of the preceding meeting of the Society a discussion arose concerning the effect of a uniform rise in prices upon the amount of small money necessary for the transaction of business. It was clear that the total amount of money must be increased, but in the case of small money, which is used only when fractional parts of the larger unit are involved, the effect was less obvious. It is the fact, however, that more small money required. The present paper attempts an explanation of that fact.

The basic reciprocity of j-differential and LM-integral

for bounded functions f(x) with simple discontinuities but continuous on the left at each point and for g(x) in the somewhat restricted class B of functions of bounded variation and also left-continuous, was established in (2) and (3); the dot here indicates the lower product of and (jg, g+ (x+)dx), with , and the integral indicated is the RJDS-integral, equivalent to (LM) .

The Kontorovich-Lebedev Transform of a function f(r), 0<r<∞, may be written in a general form as fg(μ) where,

A suitable function f(x) 0 ≤ x < ∞, can be expanded into a Fourier series of Laguerre polynomials , whose interval of orthogonality is 0 ≤ x < ∞. The usual problems as to convergence and, lacking convergence, summability, and also the asymptotic behaviour of Lebesgue constants, arise for such developments. A summary of work on these convergence and summability problems, together with extensive references to the literature, can be found in the standard treatise by G. Szegö (5, especially Chapter IX) to whom many of these results are due.