A bilinear map ø Ra x Rb → Rc is non-singular if ø (x, y) = 0 implies x = 0 or y = 0. For background information on such maps see (4, 5, 6, 14). If we apply the ‘Hopf construction’ to ø, we get a map

defined by 2ø(x, y)) for all x ∈ Ra, y ∈ Rb satisfying ∥x∥2 + ∥y∥2 = 1. Homotopically, Jø coincides with the map obtained by first restricting and normalizing ø to , and then applying the standard Hopf construction ((13), p. 112). In any case, one gets an element [Jø] in , which in turn determines a stable homotopy class of spheres {Jø} in the d-stem , where d = a + b − c −1. An element in which equals {Jø} for some non-singular bilinear map ø will be called bilinearly representable. The first purpose of this paper is to prove

The aim of this paper is to give some results on the determination of Hamiltonian structure for a set of evolutionary equations, either differential or non-local. There are several reasons why the recognition of Hamiltonian structure is important. Firstly, as in the case of Lagrangian systems, conservation laws are often connected with symmetry groups of the equations. This point is addressed in §4. Secondly, variational principles for steady solutions can often be found by restricting attention to the steady equations. Connected to the second point, methods have been developed by Arnol'd[2] and more recently by Marsden, Weinstein etc. (see, e.g. [24]) to study the nonlinear stability of steady solutions, on lines similar to the proof of nonlinear stability for the solitary-wave solution of the Korteweg–de Vries equation by Benjamin [4] and Bona[9].

Locally convex spaces which are generated by a weakly compact subset (or a sequence of weakly compact subsets) and their subspaces are studied. Various characterizations and the permanence properties of these spaces are obtained. Certain results valid for weakly compactly generated Banach spaces are extended. These spaces are shown to have sequential properties which extend well-known properties of separable locally convex spaces.

Let $D\,{=}\,\pi_1\cdots \pi_n$, where $\pi_1,\dots,\pi_n$ are distinct Gaussian odd primes and $n$ is any positive integer. Let $E_{4D^2}$ be the elliptic curve $y^2\,{=}\,x^3-(2D)^2x$. We prove that the value at $s\,{=}\,1$ of the complex $L$-function of $E_{4D^2}$, divided by its natural period(which is $\omega/ \surd{(2D)}$, where $\omega\,{=}\,2{\cdot}6220575\dots$), is always divisible by $2^{n-1}$, and we give a simple combinatorial criterion for it to be exactly divisible by $2^{n-1}$. As a corollary, we construct a series of even non-congruent numbers having an arbitrarily large number of prime factors. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton–Dyer.

We introduce the concepts of local spectral and walk dimension for fractals. For a class of finitely ramified fractals we show that, if the Laplace operator on the fractal is defined with respect to a multifractal measure, then both the local spectral and walk dimensions will have associated non-trivial multifractal spectra. The multifractal spectra for both dimensions can be calculated and are shown to be transformations of the original underlying multifractal spectrum for the measure, but with respect to the effective resistance metric.

This result does not appear to be in the literature, though the method of proof is related to those used in (1) and, particularly, (2) to prove A. H. Stone's theorem that metrics have paracompact topologies.

We construct a ‘structure invariant’ of a one-ended, finitely presented group that describes the way in which the factors of its JSJ decomposition over two-ended subgroups fit together. For hyperbolic groups satisfying a very general condition, these invariants completely reduce the problem of classifying such groups up to quasi-isometry to a relative quasi-isometry classification of the factors of their JSJ decomposition. Under some additional assumption, our results extend to more general finitely presented groups, yielding a far-reaching generalisation of the quasi-isometry classification of some 3–manifolds obtained by Behrstock and Neumann.

The same approach also allows us to obtain such a reduction for the problem of determining when two hyperbolic groups have homeomorphic Gromov boundaries.

Copson(1) has shown that the problem of determining the surface density of electric charge induced on a thin circular disk maintained at a constant potential in an external electrostatic field can be solved by two applications of the known solution of Abel's integral equation. This note shows that Copson's method can be extended to determine the surface charge density induced on the disk when situated inside an earthed coaxial infinitely long hollow cylinder. Whilst this problem is not solved in closed form, it is shown to be governed by a Fredholm integral equation of the second kind, which can be solved approximately by iteration when the radius of the cylinder is large compared with that of the disk.

In order to be able to identify the type to which an adsorbed film belongs from its experimentally determined behaviour, it is necessary to know theoretically how various simple and general types, which are likely to occur frequently, behave. Films formed on a rectangular solid lattice with another atom at the centre of each rectangle are considered. Each adsorbed molecule is assumed sufficiently large to exclude the possibility of adsorption on the four neighbouring sites and, owing to the rectangular shape of the lattice, reacts appreciably with two and not with four neighbouring adsorbed molecules. The statistics and kinetics of the film are considered. It is shown that the variation of heat of adsorption with fraction of sites occupied is quite different from that of other types of film which have so far been treated. Immobile films of the same type are briefly discussed.

Let us assume throughout that (pn) denotes a sequence of reals which satisfies

For real sequences (sn) with increments an = sn – sn−1 for n ≥ 0,(where s−1 = 0), we consider the power seriesmethod of summability (P), where we say

The power series methods (P) containthe so-called (Jp)-methods (R = 1)and the Borel-type methods (Bp)(R = ∞). We consider only regular (P)-methods, i.e. sn → s implies sn → s(P). By theorem 5 in [5], p.49, we have regularity if and only if

Here we are interested in the converse conclusion, namely sn → s(P) implies sn → s, which can only be validiffurther conditions, so-called Tauberian conditions are satisfied by (sn). These so-called Tauberian theorems for power series methods have a long history; see e.g. the books [5, 14, 23], and they found new attentionrecently in the papers [6, 18, 19, 20] and [8, 9, 10, 11, 12]. The latter papers contain certain o- Tauberian theorems for all power series methods in question and O-Tauberian theorems, if the weight sequence (pn) can be interpolated by alogarithmico-exponential function g(·)(see e.g. [4]), i.e.

Let Ω be a Riemann matrix whose 2n columns are vectors of Cn. It is well-known (e.g. (10)) that the field of meromorphic functions on Cn with these vectors among their periods is of transcendence degree n over C. More precisely, this field can be written as C(A, B) where A = (A1, …, An) is a vector of algebraically independent functions of the variable z = (z1, …, zn) and B is algebraic over C(A). We shall assume that B is in fact integral of degree d over the ring C[A]. Since the derivatives ∂f/∂zi of a periodic function f(z) are also periodic, the field is mapped into itself by the differential operators ∂/∂zi. Thus there exists a function C(z) in C[A] such that these operators map the ring C[A, B, C−1] into itself.

A method is described for measuring the heat of adsorption of hydrogen on a clean tungsten filament. The value obtained is 2·8 × 104 calories per mol.

A celebrated result of Halász describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions providing similar asymptotics, thus verifying a two dimensional variant of a conjecture of Elliott. As a consequence, we get several convergence results for such multilinear expressions, one of which generalises a well known convergence result of Wirsing. The key ingredients are a recent structural result for multiplicative functions with values on the unit disc proved by the authors and the mean value theorem of Halász.

It is known that if every prime power branched cyclic cover of a knot in $S^3$ is a homology sphere, then the knot has vanishing Casson–Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in $S^3$ whose prime power branched cyclic covers are homology spheres. We show that these knots generate an infinite rank subgroup of $\scrf_{(1.0)}/\scrf_{(1.5)}$ for which Casson–Gordon invariants vanish in Cochran–Orr–Teichner's filtration of the classical knot concordance group. As a corollary, it follows that Casson–Gordon invariants are not a complete set of obstructions to a second layer of Whitney disks.

Matroid theory was first studied by Whitney (1) as an abstract theory of linear independence in vector spaces. Recently its importance in graph theory has been noticed by Tutte (2), Edmonds (3) and Nash-Williams (4,5). Less interest has been shown in the extremely close relationship between matroids and incidence geometries. In this note we develop the more geometrical aspects of matroid theory, paying particular attention to the fundamental role of the hyperplanes of a matroid in this theory.

1. This paper investigates the boundedness, as t → ∞, of the solutions of the real differential equation

where α is a constant and φ2, φ3, ψ depend on the arguments shown with φ2, φ′3, ψ continuous.

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.

James Clerk Maxwell, in his days of early development, made a practice of communicating his progress in ideas by informal letters to his scientific friends G. G. Stokes and W. Thomson, who were in the habit of preserving their correspondence. The record, so far as revealed in the letters to Stokes, has been published in volume 2 of Prof. Stokes' Scientific Correspondence. The letters which are here printed have emerged among Lord Kelvin's manuscript remains. They had been arranged apparently by Prof. S. P. Thompson when he was preparing his biography of Lord Kelvin's practical activities. I find that they had been examined by myself when a project of publishing Lord Kelvin's scientific correspondence was contemplated, after the manner of that of Stokes; which afterwards proved to be impracticable, as the material had largely been skimmed over.

A simple method for constructing irreducible representations of the symmetric group is given. It is particularly suitable for use in quantum mechanical calculation of interactions in many particle systems, using electronic digital computers.

The decay period of the α-activity produced when boron is bombarded with neutrons from a (Li + D) source has been measured, and found to agree with the known decay period of 8Li. It has been shown that no activity is produced when (Be + D) neutrons (maximum energy 4·5 × 106 e.V.) are used. The results are explained by postulating the formation of radioactive 8Li according to the process

A short investigation of the number-range distribution for the α-particles emitted by 8Li indicated that a maximum occurred in the distribution at a range of about 6 mm.