Let N be an arbitrary near-ring. Each element a ∈ N determines in a natural way a new multiplication on the elements of N which results in a near-ring Na whose additive group coincides with that of N but whose multiplicative semigroup generally differs. Specifically, we define the product x * y of two elements in Na by x * y = x a y where a product in the original near-ring is denoted by juxtaposition. One easily checks that Na is a near-ring with addition identical to that of N. The original near-ring N will be referred to as the base near-ring, Na will be referred to as a laminated near-ring of N and a will be referred to as the laminating element or sometimes more simply as the laminator.

It can be shown as follows that Newton's Theorem can be derived from elementary considerations without making use of the idea of an equation and its roots.

The equations considered are Fredholm integral equations of the second kind with regular kernels, whose argument depends only on the difference of the variables. Approximate solutions are sought for a given finite range of the eigenvalues, and for large values of the range of integration. Certain special conditions are imposed on the general form of the Fourier transforms of the kernel. Then it is shown that approximate solutions may be obtained in terms of the solutions of the corresponding (singular) Wiener-Hopf equations. Approximations to the eigenvalues are also found. It is shown that the eigenfunctions are unique, and that except possibly near the end points of the range, the solutions are of trigonometric type with the zeros of successive solutions interlacing.

Throughout this paper D denotes a division ring with centre F and n a positive integer. A subgroup G of GL(n,D) is absolutely irreducible if the F-subalgebra F[G] enerated by G is the full matrix ring Dn ×n. It is completely reducible (resp. irreducible) if row n-space Dn over D is completely reducible (resp. irreducible), as D–G bimodule in the obvious way. Absolutely irreducible skew linear groups have a more restricted structure than irreducible skew linear groups, see for example [7],[8], [8] and [10]. Here we make a start on elucidating the structure of locally nilpotent suchgroups.

The subject of the Singular Solutions of Differential Equations of higher orders than the first is not touched in the ordinary textbooks. Their existence, for instance, is not mentioned by Forsyth in his Treatise. This is probably due to the fact that, while in the case of equations of the first order a theory has been developed by Cayley and others which connects the singular solution in a geometrical manner with the ordinary solutions (the singular solution being, of course, the envelope of the ordinary solutions), in the case of equations of, say, the second order no corresponding theory exists—at any rate, no corresponding theory has yet been developed. Our only guide in the subject at present is Cauchy's Existence Theorem, which points out where we are to look for singular solutions.

In 1640, when only 18 years of age, Pascal published a tract of a few pages with the above title. It contains only a few enunciations, and concludes with the statement that the author has several other theorems and problems, but that his inexperience, and the distrust he has of his own powers, do not allow him to publish them till they have been examined by competent judges. He afterwards wrote a complete work (opus completum) on the Conics, which was submitted to Leibnitz by M. Périer, Pascal's brother-in-law. Leibnitz recommended that it should be published; but this was not done, and we know its contents only from the analysis which Leibnitz sent back to M. Périer.

In this paper, we shall obtain two results on the class of far field patterns corresponding to the scattering of time harmonic acoustic plane waves by an inhomogeneous medium of compact support. Although the problem of characterizing the class of far field patterns is of basic importance in inverse scattering theory, very little is known about this class other than the fact that the far field patterns are entire functions of their independent (complex) variables for each positive fixed value of the wave number. In particular, the class of far field patterns is not all of L2(∂Ω) where ∂Ω is the unit sphere and this implies that the inverse scattering problem is improperly posed since the far field patterns are, in practice, determined from inexact measurements. The purpose of this paper is to show that while the class of far field patterns corresponding to the scattering of time harmonic plane waves by an inhomogeneous medium is not all of L2(∂Ω), it is dense in L2(∂Ω) for sufficiently small values of the wave number. In addition, a related result will be obtained for a special translation of the class of far field patterns. Analogous results for the scattering of time harmonic acoustic waves by a homogeneous scattering obstacle have recently been obtained by Colton [1], Colton and Kirsch [2], Colton and Monk [3, 4] and Kirsch [8].

This paper is concerned with two notions of cohomological triviality for Banach algebras, weak amenability and cyclic amenability. The first is defined within Hochschild cohomology and the latter within cyclic cohomology. Our main result is that where ℱ is a Banach algebraic free product of two Banach algebras and ℬ. It follows that cyclic amenability is preserved under the formation of free products.

We consider a medium in which the equation satisfied by a disturbance ø(x,t) is capable of solutions of the form sin(vx—wt), with v and ω real but not necessarily of the same sign. If the phase velocity U = ω/v is not of constant magnitude the medium is said to be dispersive, and the group velocity V may conveniently be defined as dω/v, an expression easily re-written in other familiar forms. From this definition the two physical interpretations of V may easily be seen. In one interpretation we consider a superposition of two harmonic waves with slightly different v (and ω); the velocity of advance of the group form, being the velocity of advance of a point at which the two waves have a constant phase difference, is dω/v.