The potential barrier governed by a second-order ordinary differential equation has been studied for decades, both exactly and approximately. These concepts are generalized so as to be applicable to differential equations of arbitrary even order. The barrier is defined, and the approximate solutions both inside and outside are derived. Transmission through this barrier is investigated by deriving connexion formulae through the two transition points of arbitrary odd order. The transmitted solution is linked through the barrier to the incident and reflected solutions in a unique manner, enabling an approximate transmission coefficient to be calculated and its structure to be ascertained. Every concept in the investigation is a suitable generalization of the elementary case, the generalization possible being a product of the approximate processes adopted and how they limit the nature of the generalization. An inspection of the elementary case and its transmission coefficient gives no hint as to the nature of the generalization, the limitations necessarily imposed, and the structure of the final transmission coefficient, since the elementary case has special features not present in the generalization.

The method of Part I(3) is applied to problems involving boundary conditions on an equatorial plane.

We obtain upper and lower bounds of the same order of magnitude for the error between the distribution of a sum of independent and identically distributed random variables, and a normal approximation by a portion of a Chebychev-Cramér series. Our results are sufficiently general to contain the familiar characterizations by Ibragimov(4), Heyde and Leslie (3) and Lifshits(5), and complement some of those obtained earlier by the author (2).

Let be a very ample vector bundle of rank 2 on with c1() = 4 and c2() = 6. Then it is proved that is the cokernel of a bundle monomorphism , where is the tangent bundle of . This gives a new example of a threefold containing a Bordiga surface as a hyperplane section.

Two collections of locally convex space topologies are shown to have the intertwined completeness property. This is done by relating their completions with sets of sequentially continuous functionals on the dual.

A proof by means of graphical techniques is given of an inequality conjectured by Nakanishi concerning the Feynman integrand for any term in the perturbation expansion of the scattering amplitude.

We characterize those countable rooted trees with non-trivial components whose full automorphism group has uncountable strong cofinality, and those whose full automorphism group contains an open subgroup with ample generics.

A problem of fundamental importance in Spectral Theory consists of finding criteria for an operator to be of scalar-type in the sense of N. Dunford [1]. One relatively general approach in determining such criteria is based on the method of integral transforms (see for example [4], [5], [6], [11], [12]). For example, if X is a Banach space and T is a continuous linear operator on X, then the group {eitT; t real} exists. As noted by several authors (e.g. [4], [6]), this group can then be effectively used for analysing the operator T.

We determine the stable decompositions of the classifying spaces of metacyclic 2-groups into wedges of indecomposable spectra. The stable decompositions of classifying spaces of finite groups with metacyclic 2-Sylow subgroups are also determined.

We obtain a simple new proof of Bonsall's characterization theorem for type 1 semi-algebras of continuous functions. An extension of our methods produces several new results on stability properties of wedges.

This paper contains an attempt, begun several years ago and only partially successful, to do for the canonical curve of genus five something similar to what W. P. Milne had done for that of genus four. A fundamental feature of Milne's work was the use of a rational normal curve of order three drawn through the six points of contact of a quadric with the sextic curve; here it is not in general possible to pass a rational curve of order four through the eight points of contact. (Exceptionally it may be possible to do so and then the development follows much the same lines as in Milne's paper: only the general case is discussed in what follows.)

It is proved that every locally convex space is linearly homeomorphic to a subspace of a space which admits no non-trivial continuous linear functionals.

In recent years Cantor manifolds have assumed great importance in topology. This is due chiefly to the work of Urysohn and to the later work of Alexandroff. Urysohn, in the paper in which he introduced the theory of dimensions, showed that a 2-dimensional closed surface is a 2-dimensional Cantor manifold. Alexandroff has not only extended this result to n dimensions but has also established the existence on the cutting set Z of the surface F of generalized cycles homologous to zero on the various parts of F − Z. These results show that the deepest structural characteristics of the closed surface come to light when one dissects it.

The Gliding of a Plate on a Stream of Finite Depth*

Page 597. The line after equation (44) should read

Proc. Cambridge Phil. Soc. 31 (1935), 589–603.

It was proved by Littlewood that, for every large positive T, ζ (s) has a zero β + iγ satisfying

where A is an absolute constant.

Let us say that a set points in a Euclidean space has property (2) if no 2 points X, Y ∈ S have X Y > 1. Then an easy observation is the following:

The maximum area (Lebesgue measure) of a plane set S with property(2) is ¼π, attained only when S is (essentially) a disc.

A group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later, in 1921, he published a proof that every abelian group of composite degree is a B-group. Both proofs are character-theoretic and both have serious flaws. Indeed, the second result is false. In this paper we explain these flaws and prove that every cyclic group of composite order is a B-group, using only Burnside’s character-theoretic methods. We also survey the related literature, prove some new results on B-groups of prime-power order, state two related open problems and present some new computational data.

It is well known that a regular neighbourhood of a polyhedron in a piecewise linear manifold may be regarded as a simplicial mapping cylinder. The aim of this paper is to show that if the polyhedron is a locally unknotted submanifold of the interior then the class of maps giving rise to such regular neighbourhoods has a simple characterization. At the same time, it is possible to answer the question: Given a simplicial map f defined on a combinatorial manifold, when is the image of f also a combinatorial manifold? Marshall Cohen has answered this question when the image is required to be isomorphic to the domain; the methods used here are those developed in (1), to which the reader is referred for definitions and notation.

If a receiving aerial is placed in the field of a wave produced by a distant transmitter, currents are induced in the receiving aerial and these dissipate energy in the ohmic resistances of the aerial. This energy may be thought of as being absorbed directly from the incident wave, but this viewpoint gives no insight into the mechanism by which the absorption takes place. In this paper the following alternative way of regarding the phenomenon is considered. The currents induced in the aerial cause it to re-radiate a secondary wave. The electric and magnetic fields of this secondary wave combine with those of the unabsorbed incident wave to give resultant fields which have, in general, a direction different from those of the incident wave. These resultant fields give rise to energy flows which are different from those in the incident wave, both in magnitude and direction. In this way a flow of energy into the aerial is produced, by the superposition of the incident and the re-radiated fields. It is important to notice that if this viewpoint is adopted we must add the secondary re-radiated wave to the unabsorbed main wave, the addition, in itself, producing the requisite absorption.

Let p be an odd prime, and h*( − ) a multiplicative mod p cohomology theory with a complex orientation satisfying the 2(p − 1)-sparseness condition. A formula concerning the formal group law for such h*( − ) is used to investigate the structure of h*(BG), G being a finite group whose p-Sylow subgroups are cyclic.