The theory of the evaporation of adsorbed atoms and ions of the alkali metals from a hot metal surface has been developed from somewhat different viewpoints by Langmuir and Kingdon and by Becker. These two treatments, which yield essentially the same results, are applicable only when equilibrium conditions obtain on the surface, so that the surface concentration is not changing. It is the purpose of the present note to extend Becker's treatment to take account of the rate of attainment of equilibrium on the surface, in order to lead to an interpretation of some experiments on the rate of evaporation of the alkali metals adsorbed on hot tungsten.

The aim of this paper is to make some remarks about the theory of parallel immersions in ℝn. We show that the existence of parallels to a curve in ℝn depends on the values of the integrals of the n − 1 curvature functions along the curve. We then give examples of immersions of S1 in ℝ3 and of immersions of codimension 2 with local parallel rank zero, thereby answering certain questions in [1].

It is concluded that the “overshoot phenomenon” in superconductivity described by F. B. Silsbee was due to his method of mounting the specimen.

The relevant results from the writer's previous paper on the relation between spectra of atoms of different atomic structure are summarised.

For non-penetrating orbits no new theoretical results are obtained, and there are few known spectra (other than those of lithium-like or sodium-like atoms already treated in the first paper) on which to test the relations previously obtained. Values of the polarisability for the A1+ and Si++ ions are calculated from terms of Al I and Si II respectively corresponding to non-penetrating orbits, and are shown to be very much greater than the values of the polarisability of the neon-like ions Al+++ and Si++++.

The main new results are those for penetrating orbits. Assuming a central field of force, it is shown that the quantum defect q for such an orbit can be expressed as the sum of contributions from the electrons in groups of core orbits of different principal quantum number n, and further that if for a given atom in different states of ionisation corresponding orbits of the series electron are compared, the contribution to q from a set of core orbits of given n is very nearly independent of the degree of ionisation so long as the number of electrons in core orbits in the group remains the same. It follows that, if q is the quantum defect for a term of the spectrum of an atom core charge C, the core of which contains SM orbits of principal quantum number M and none of higher quantum number, and q″ is the quantum defect for the corresponding term in the spectrum of the atom of the same element with core charge C + sM, which differs from the atom of core charge C only in lacking the core orbits of principal quantum number M, then q −; q″ is approximately the contribution from the core orbits of principal quantum number M to the quantum defect for the term of the atom core charge C.

Further, it is shown that if corresponding terms of different atoms of the same electronic structure are compared, then for large values of C the contribution to q from any group of core orbits should tend asymptotically to be proportional to the average time mean radius of these orbits, and its reciprocal should tend asymptotically to be linear in C.

Somewhat similar relations are obtained for the quantity Q = dq/d (ν/R) which measures the variation of quantum defect within a sequence.

These theoretical results, and in particular the result that l/(q − q″) should tend asymptotically to be proportional to C, are compared with the values of q deduced from the terms of such observed spectra of aluminium-like and copper-like atoms as are available, and it is found that though the theoretical relations are only established as asymptotically true for large C, there is a considerable measure of agreement with spectra for small values of C, which are the only ones which can be observed.

Let R be an arbitrary associative ring, and X a set of generators of R. The elements of X generate a Lie ring, [X], say, with respect to the addition and subtraction in R, and the multiplication [a, b] = ab − ba. In this note we shall be concerned with the following question: if [X] is given to be nilpotent as a Lie ring, what does this imply about R?

In a previous paper [1] we gave examples of positive integral operators on L2( − 1, 1) and estimates of their eigenvalues. In theorem 1 we treated operators with kernels of the form Σan sn tn, where (an) is a sequence of non-negative real numbers satisfying an ≃ α2n and 0 < α < 1 (here and throughout the notation an ≃ bn shall mean that an = O(bn) and bn = O(an)). In this paper we prove the more comprehensive Theorem 1 a below; theorem 1 of [1] is just the case b = 0. The term q(2α/(1 + α2)) will be explained immediately after the statement of the result.

We prove a generalisation of Roth's theorem for arithmetic progressions to d-configurations, which are sets of the form {ni+nj+a}1 ≤ i ≤ j ≤ d with a, n1,. . .,nd ∈ $\mathbb{N}$, using Roth's original density increment strategy and Gowers uniformity norms. Then we use this generalisation to improve a result of Sudakov, Szemerédi and Vu about sum-free subsets [10] and prove that any set of n integers contains a sum-free subset of size at least logn(log(3)n)1/32772 − o(1).

1. The simple group of order 6048 has a permutation representation of degree 28 and two of degree 63. These are conspicuous in SU(3,32), the representation as a group of 3-rowed matrices, unitary and of determinant + 1, whose elements all belong to GF(32), or to J as we name this field. When these matrices as linear transformations in the projective plane ϖ over J they leave invariant the unit Hermitian form

and the single capital U will henceforth signify the group so acting. There are 28 isotropic points n in ϖ satisfying H3 = 0; they furnish the representation of degree 28. The other 63 points p in ϖ are non-isotropic and furnish one of the representations of degree 63. The other is furnished by 63 triangles T; the vertices of a T are all p and every pair of them is linked in the antipolarity Π set up by H3; if their coordinate vectors are y and z then y′z = 0. Each p is a vertex of 3 T. These matters are fully explained in (7). Those points linked by II to a given point A are collinear, on the polar of A; no p lies on its polar, every n does. The polars t of points n are called tangents of H3 = 0, and the only isotropic point on t is its pole. The polars s of points p are secants of H3 = 0: each consists of 4 n with 3 pairs of linked p and, given s, it is these 3 pairs that complete those T having a vertex at the pole of s.

The object of this note is to direct attention to an entirely false inference which was drawn from the analysis given in my previous paper with the same title.

The transonic, two-dimensional flow past a thin airfoil is treated by approximating the transonic small perturbation equation by a linear equation valid in the neighbourhood of the airfoil. The solution is derived by representing the body by a line of suitable sources. Pressure distributions are given for airfoils with maximum thickness both forward and aft, and agreement with experiment improves as the position of maximum thickness moves aft.

Bagnera(1) shows that when a certain primitive finite group of order 576, G576, is represented as a group of collineations in three-dimensional space, it contains twenty-four harmonic inversions with respect to points and planes, and he shows how to calculate the coordinates of the points and planes of the inversions. The purpose of this note is to discuss the configuration of these points, which we shall call vertices, to show that the whole group is generated by the twenty-four harmonic inversions, termed projections after Prof. Baker (2) and Dr J. A. Todd (4), and to enumerate the conjugate sets of the group.

If a particle counter is subject to the radiation from a pure radioactive source of sensibly constant activity, it is axiomatic to suppose that the series of events constituted of the traversals of the counter by ionizing particles is a random series in relation to distribution in time. There are various reasons why the related series of events formed by the occasions on which the counter responds to these ionizing particles is not a random one—and there may also be cases in which the primary source does not, in fact, give rise to particles traversing the counter in a random sequence. The counter responses will necessarily depart from strict randomness in time because of the finite recovery time of the counter; they may also depart from strict randomness through a not uncommon counter defect, the occurrence of ‘spurious’ discharges correlated in some way with the ‘true’ discharges brought about as a result of the ionization produced by the particles. On the other hand, the original ionizing events will not have the effect of a random sequence, even with a pure radioactive source, if secondary radiations are emitted—even in a fraction of the disintegrations—with a time delay appreciable in relation to the counter resolving time, and, clearly, they may depart considerably from true randomness in time if the source is not a pure source, and, in particular, if there is produced in the source material a ‘daughter’ radioelement of lifetime comparable with this resolving time, or with the mean time between counter discharges.

If G is a locally compact group such that G/G0 contains a uniform compactly generated nilpotent subgroup, then G has a maximal compact normal subgroup K such that G/G is a Lie group. A topological group G is an N-group if, for each neighbourhood U of the identity and each compact set C ⊂G, there is a neighbourhood V of the identity such that for each g∈G. Several results on N-groups are obtained and it is shown that a related weaker condition is equivalent to local finiteness for certain totally disconnected groups.

For a random walk on a finite group, the distribution after n steps will converge, as n→∞, to the uniform distribution (under mild conditions). The asymptotic behaviour of such walks can be studied easily using standard Markov chain theory. But there are many natural problems about the non-asymptotic behaviour which have no simple solution in general. For example, what is

the time until a specified element is first visited?

the time until all element have been visited?

the time until the distribution approaches the uniform distribution?

Stone-Čech compactifications of semigroups have aroused a good deal of interest recently. Several authors, for example Milnes [6], Marcri [5] and Baker and Butcher [1], have concentrated on problems of the existence of continuous extensions to βS of the operation in a topological semigroup S. For a discrete semigroup a separately continuous extension always exists, and others such as Pym and Vasudeva[8] have studied the compactifications of particular classes of semigroups. Further interest has centred on the algebraic structure of these compactifications; see for example Hindman[3].

In section 3 we shall prove the following results: Let G be a separable locally compact abelian group, R a von Neumann algebra acting on a separable Hilbert space, and α a weakly continuous representation of G by inner *-automorphisms of R, say α(g) = ad Wg with Wg ∈ U(R). Then there is a weakly continuous unitary representation of G, by unitaries in R, implementing α if and only if the Wg's commute with each other. The result was motivated by the proof of (7), theorem 1. Suppose now Gis a discrete amenable group of *-automorphisms of a countably decomposable von Neumann algebra R. In section 3 we give a necessary and sufficient condition for the existence of a faithful normal G-invariant state on R. This generalizes a result of Hajian and Kakutani on invariant measures (2).

Two spatial embeddings of a graph are said to be delta edge-homotopic if they are transformed into each other by self delta moves and ambient isotopies. In this paper we classify theta curves up to delta edge-homotopy in terms of the third coefficient of the Conway polynomial of an associated 2-component link. In particular, we show that every boundary theta curve is delta edge-homotopically trivial, and two cobordant theta curves are delta edge-homotopic.

A Kolmogorov-type inequality is derived from a Jackson-type inequality and a Bernstein-type inequality. Several new results follow as corollaries.

Two perpendicular intersecting straight lines will be said to be normal to each other. Then the Petersen-Morley theorem is: Given arbitrarily three lines a, b, c, let a′ be the unique normal to b and c, b′ that to c and a, and c′ that to a and b; then the lines x, y, z, normal respectively to the pairs of lines a and a′, b and b′, c and c′, have a common normal.

For a hyperelliptic curve ${\hbox{\ax C}}$ of genus $g$ with a divisor class of order $n = g + 1$, we shall consider an associated covering collection of curves ${\hbox{\ax D}}_\delta$, each of genus $g^2$. We describe, up to isogeny, the Jacobian of each ${\hbox{\ax D}}_\delta$ via a map from ${\hbox{\ax D}}_\delta$ to ${\hbox{\ax C}}$, and two independent maps from ${\hbox{\ax D}}_\delta$ to a curve of genus $g(g-1)/2$. For some curves, this allows covering techniques that depend on arithmetic data of number fields of smaller degree than standard 2-coverings; we illustrate this by using 3-coverings to find all ${\Bbb Q}$-rational points on a curve of genus 2 for which 2-covering techniques would be impractical.