It is a classical result in the theory of special functions that Bessel functions are limits in an appropriate sense of Legendre polynomials. For example in (11), § 17.4, the following result is attributed to Heine:

such limiting formulae are also known for certain other special functions (cf. (1), (5)). Apart from their intrinsic interest, these formulae have been used in the theory of special functions to obtain product formulae, etc. for the limit function from those of the approximating sequence.

A simple sufficient condition is given for a stochastic process x(t), 0 ≤ t ≤ 1, to have the following property: There is an integer m ≥ 2 such that for any non-degenerate subinterval J ⊂ [0, 1], there exist m disjoint subintervals I1, …, Im ⊂ J such that the intersection of the images of I1,…, Im under the mapping by x(·) has positive Lebesgue measure, almost surely. There is also a version for vector random fields; and the main result is shown to apply to large classes of processes.

In the classical theory of plane deformations in isotropic plastic media, the field equations are hyperbolic and the orthogonal families of characteristics are known as Hencky-Prandtl nets. Their distinctive geometry has been given symbolic expression by Collins (1968), in an algebra of infinite matrices associated with canonical series representations of the general solution. This has become the standard technique when investigating boundary-value problems, both analytically and numerically. The basic framework of the algebra is here reorganized and developed. A systematic approach then leads to new identities which are shown to be fundamental in the algebraic hierarchy.

It is known, (see [7], theorem 4 or [8], corollary II, 10.1), that a positive generated, ordered Banach space X is order-isomorphic to l1 iff X has a Schauder basis which generates its positive cone X+ and X+ has a bounded base.

J. C. Wood. ‘An extension theorem for holomorphic mappings.’

As pointed out by Y.-T. Siu [MR82d:32021] the theorems and corollary require the additional hypothesis that the boundary ∂X, is hyper-(m–1)-convex where m = dim X, i.e. for all p ∈ ∂X, the sum of the eigenvalues of the restriction of the Levi form of ∂X to the complex tangent space Tp(∂X) ∩ JTp(∂X) is non-negative.

J. R. Paktington. ‘On nearly uniformly convex Banach spaces.’

Theorem 1 was printed incorrectly. It should read

If X is (UKK) then so is lp(x) (1 ≤ p < ∞).

A similar replacement of L by l should also be made on p. 127, 1. 4* and p. 128, 1. 13, while p. 128, ll. 11*–10* should read

Also the l2 direct sum of lp, p = 2, 3, … is not even isomorphic to any (UKK) space but is (BS).

In this note we use the notion of the quasiasymptotic behaviour of distributions (introduced in [2]) in order to obtain a final value Abelian theorem for the distributional Stieltjes transformation. At the end of the note we give a few examples in which two different concepts of the asymptotic behaviour of distributions are compared.

A theory is developed relating to the stability of solitary-wave solutions of the so-called Benjamin-Ono equation. This equation was derived by Benjamin (5) as a model for the propagation of internal waves in an incompressible non-diffusive heterogeneous fluid for which the density is non-constant only within a layer whose thickness is much smaller than the total depth. In his article, Benjamin wrote in closed form the one-parameter family of solitary-wave solutions of his model equation whose stability will be the focus of attention presently.

For a system of partial differential equations, the existence of appropriate conservation laws is often a key ingredient in the investigation of its solutions and their properties. Conservation laws can be used in proving existence of solutions, decay and scattering properties, investigation of singularities, analysis of integrability properties of the system and so on. Representative applications, and more complete bibliographies on conservation laws, can be found in references [7], [8], [12], [19]. The more conservation laws known for a given system, the more tools available for the above investigations. Thus a complete classification of all conservation laws of a given system is of great interest. Not many physical systems have been subjected to such a complete analysis, but two examples can be found in [11] and [14]. The present paper arose from investigations ([15], [16]) into the conservation laws of the equations of elasticity.

In § 7 of [1] we described how a stochastic integral of non-commuting process cesan be constructed. This was achieved via a Doob-Meyer decomposition for the square of a self-adjoint L2-martingale. We introduced a ‘condition D'’ derived from the lsquo;class D’ of stochastic process theory, and showed that if the square of a self-adjoint L2-martingale satisfies this condition then it has a decomposition of the Doob-Meyer type. The purpose of this paper is to extend the results of § 7 of [1] and to introduce another construction of the stochastic integral that does not employ condition D.

The method to be described provides an alternative means of dealing with certain non-standard linear boundary-value problems. It is developed in several applications to the theory of gravity-capillary waves. The analysis is based on a variational formulation of the hydrodynamic problem, being motivated by and extending the original study by Benjamin and Scott [3].