Let r be any fixed integer with, r≥ 2; then, given any positive integer n, we can find* integers αk(r, n) (k = 0, 1, 2, …) such that

where, subject to the conditions

the integers αk(r, n) are uniquely determined, and, in fact, clearly

αk(r, n) = [n/rk] − r[n/rk+1]

(square brackets denoting integral parts, according to the usual convention).

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].

We consider invariant positive sesquilinear forms on a (partial) *-algebra A without unit. First we investigate the relationship between extendability and representability for such a form ø; in particular we discuss under which conditions the two concepts are equivalent. Then we introduce the notions of weak representability and strict unrepresentability, and we show that every fully invariant positive sesquilinear form on A × A is uniquely decomposed into a weakly representable part and a strictly unrepresentable part.

Let K(k) and E(k) denote respectively the complete elliptic integrals of the first and second kind with modulus k, as defined by Byrd and Friedman ([5] 110·06 and 110·07), and let k′ = √(1 − k2), the complementary modulus.

The following Conjecture was made in (6).

CONJECTURE A. Let Xn be a closed manifold of type K(π, 1). Then X˜n is homeornorphic to.

In section 4, we prove the conjecture when n ≥ 5 and π is a non-trivial direct product. More general than Conjecture A would be

CONJECTURE B. Let Xn be a manifold of type K(π, 1) with ∂X = Ø. Then X˜n is homeomorphic to.

In section 4 we give an example to show that Conjecture B is false in each dimension n ≥ 4.

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 formal conditions of Lorentz-invariance apply to coordinates x1, x2, x3, x4 which form a 4-vector, and to expressions associated with them. In this paper attention is called to the frequent disregard in quantum investigations of the proviso that the coordinates form a 4-vector.

In most quantum investigations with a practical application the coordinates employed are relative space coordinates ξ, η, ζ coupled with a progressive time coordinate, so that (ξ, η, ζ, it) is not a 4-vector. Nevertheless, conditions of Lorentz-invariance are applied by many authors. Alternatively, they attempt to base the investigation on wave functions of non-relative coordinates x, y, z, t. It is here pointed out that such wave functions give no information about the eigenstates, and that there is no means of deriving wave functions of ξ, η, ζ from those of x, y, z.

Special attention is directed to the misapplication of Lorentz-invariance in Dirac's theory of the hydrogen atom, and in the calculation of degeneracy energy of ionized material.

Absorption curves in paper have been obtained for the β-rays of actinium (B + C) and actinium C′. Some inferences regarding the shape of their energy distribution curves have been drawn, and the ranges of the β-rays of actinium B and actinium C′ fix their maximum β-ray velocities at Hρ 3430 and 6140. The potency of the method is emphasized for elements which cannot be easily investigated directly in a magnetic spectrograph.

The γ-ray ionizations of actinium B and actinium C′ have been compared under the same conditions with the β-ray ionization of actinium C′.

In conclusion, the writer wishes to thank Professor Sir Ernest Rutherford for his kind interest in the work. Thanks are also due to Dr J. Chadwick and Dr C. D. Ellis for many valuable suggestions and discussions.

In a previous paper (2) by one of the authors some types of periodic potential functions were obtained and were applied to problems in which the boundaries were cylinders or spheres arranged at regular intervals in a row or ring. It is now our object to extend the methods of that paper to a number of other cases.

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.

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.

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.

For nilpotent spaces p-completion is well behaved and reasonably well understood. By p–completion we mean Bousfield–Kan completion with respect to the field Fp [BK]. For non-nilpotent spaces the completion process often has a chaotic effect, this is true even for small spaces. One knows, however, that the classifying space of a compact Lie group is Fp-good even though it is usually non-nilpotent.

The paper deals with the problem of investigating the effect of magneto-thermo-elastic interactions on the cooling process of an infinite circular cylinder when its boundary is subjected to a time-decaying temperature. The Hooke–Duhamel law of elasticity, the equation of heat conduction together with the electromagnetic equations of Maxwell supplemented by Ohm's law have been used to solve the problem. It is found that the Laplace transformation serves as an important tool for the solution of the problem.

We show that any nonminimal bridge decomposition of a torus knot is stabilized and that n-bridge decompositions of a torus knot are unique for any integer n. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphere in two essential loops.

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.

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.

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.

Let X be an affine real algebraic variety, i.e., up to biregular isomorphism an algebraic subset of ℝn. (For definitions and notions of real algebraic geometry we refer the reader to the book [6].) Let denote the ring of regular functions on X ([6], chapter 3). (If X is an algebraic subset of ℝn then is comprised of all functions of the form f/g, where g, f: X → ℝ are polynomial functions with g−1(O) = Ø.) In this paper, assuming that X is compact, non-singular, and that dim X ≤ 3, we compute the Grothendieck group of projective modules over (cf. Section 1), and the Grothendieck group and the Witt group of symplectic spaces over (cf. Section 2), in terms of the algebraic cohomology groups and generated by the cohomology classes associated with the algebraic subvarieties of X. We also relate the group to the Grothendieck group KO(X) of continuous real vector bundles over X, and the groups and to the Grothendieck group K(X) of continuous complex vector bundles over X.