A simple explanation of Pauli's principle was first given with the wave mechanics. Its interpretation in the new theory was that the wave functions of Schrödinger were antisymmetrical in all the electrons concerned. Thus when the interactions of the electrons may be neglected, the wave function (for a system of n electrons) can never be of the form

in nature, but only of the form

Hence σ, τ,…ω must be all distinct.

The Abel sum of the series can be written in the form

where

This suggests that we should define the “Abel limit”, as u → ∞, of any function A(u) as being given by the expression (1) whenever this exists. We shall, however, in this paper, restrict ourselves to functions A(u) which are bounded in any finite interval. Since we are concerned only with the behaviour of A(u) as u → ∞, this does not involve any serious loss of generality, while we avoid difficulties arising from the divergence of integrals at finite points. We note that the expression (1) can be written in the form

where

A1(u) may conveniently be described as the “Abel transform” of A(u).

In these Proceedings, Vol. xx (1920), pp. 198–204, the writer ventured (l.c. p. 203) on some remarks as to the reason why Sir G. Darwin and Mr Jeans had obtained discordant results for the stability of the pear-shaped figure of equilibrium, suggesting that this was due to a different mode of expansion of the functions involved. Sir G. Darwin used an infinite series of Lamé functions; Mr Jeans' method was equivalent to using the early parts from an expansion, of which every term, when expressed as an integral series of Lamé functions, would be an infinite series. At that time there was difficulty in obtaining Liapounoff's papers; since then, by the kindness of M. Belopolsky, of Pulkova, the whole of the four parts of Liapounoff's publication “Sur les figures d'équilibre peu différentes des ellipsoïdes d'une masse liquide homogène douèe d'un mouvement de rotation,” in all over 750 large folio pages, have become available; and these are now in the market. It is particularly interesting to see that the fourth part (1914) is devoted precisely to that change in the method of development which would arise in passing from Sir G. Darwin's expansion to the other expansion referred to above. And it is only by this change that Liapounoff is able to give the general proof of a form of expression of his results—in terms of polynomials and not infinite series—upon which his theorem of instability is made to depend. The careful consideration of the convergence of his expansions, which adds so greatly to the length of Liapounoff's papers, supplies materials for the proof that the expansion used in Mr Jeans' paper can be placed on a sure foundation, while Sir G. Darwin's expansion requires an estimation of the remainder.

In (1) A. H. Boers has studied the properties of n-PA rings, i.e. rings where any n-product (any product of n factors) is independent of its bracketing; a private communication is quoted in that paper to the effect that the n-PA rings were defined by Y. Ilamed starting from problems in mathematical physics.

In this note the author has derived three Fourier series expansions for H-function which is more general than Meijer's G-function. Some integrals and a recurrence relation for H-function have been derived as corollaries.

In a paper by Slater a new theory of radiation is discussed and the following explanation of the breadths of spectral lines is given: “An atom in the ith. state has a probability Pi of suffering in unit time a transition. Thus there is a probability Pi that the vibrations of each of the oscillators will simultaneously cease. But we shall assume that, in addition to this probability Pi of ceasing its oscillation altogether, each oscillator has also an independent probability Pj of suffering an interruption in which it ceases its oscillation as if it were leaving the state, but immediately begins again as if it were entering the same state” (i.e. with an arbitrary phase difference). “This term Pj is the same as the probability that an atom in the jth state will leave that state.” The total probability of interruption of vibration is thus (Pi + Pj) which is symmetrical with regard to the two end states. This makes the breadths of absorption and emission lines equal and so satisfies Kirchhoff's law.

The most outstanding discrepancy between experiment and Dirac's theory of the electron is at present that shown by Dymond's experiments on the polarisation of beams of electrons by double scattering. It has been shown by Mott, using Dirac's relativistic wave equation, that one would expect a 15% asymmetry of beams of 140 k.v. electrons scattered twice at 90° by gold. The experiments of Dymond referred to above show that there cannot be more than 1% asymmetry. Several attempts have been made to explain this discrepancy, but without success. (For an account of these see Dymond's paper II.) One possibility which has not yet been investigated is that of exchange between the incident electron and the electrons of the struck atom. It is evident that only elastic exchange need be considered, since the probability of excitation of any form is known to be small. Only exchange with the K and L shells may be expected to give any appreciable effect.

In paper(1) it has been proved that

where the Jacobi polynomial is denned as ((3), p. 255)

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.

It is familiar that if on an algebraic surface there is an exceptional curve, that is an irreducible rational curve of virtual grade − 1 when no points of it are assigned as base points, and if there is on the surface a canonical system containing some actual curves, so that pg ≥ 1, then the exceptional curve is a fixed constituent of every curve of the canonical system, generally a simple constituent, and in that case has no intersections with the residual constituent. More generally, if there is on the surface a reducible exceptional curve, i.e. a set of curves which can be transformed into the neighbourhoods of a family of simple points (some of which are in the neighbourhoods of others) on a surface birationally equivalent to the given one, then the canonical system has as a fixed constituent of all its curves at least that combination of the curves which corresponds to the sum of the total neighbourhoods of the points, and generally just this combination, in which case this fixed part has no intersection with the residual variable part.

Kampé de Fériet (l) has defined a generalized hypergeometric function of two variables as

where ∏(σp)s stands for the product (σ1)s (σ2)s … (σp)s.

We prove that every finitely-generated right-angled Artin group embeds into some Brin–Thompson group nV. It follows that any virtually special group can be embedded into some nV, a class that includes surface groups, all finitely-generated Coxeter groups, and many one-ended hyperbolic groups.

Let C be a soluble smooth genus one curve over a Henselian discrete valuation field. There is a unique minimal Weierstrass equation defining C up to isomorphism. In this paper we consider genus one equations of degree n defining C, namely a (generalised) binary quartic when n = 2, a ternary cubic when n = 3 and a pair of quaternary quadrics when n = 4. In general, minimal genus one equations of degree n are not unique up to isomorphism. We explain how the number of these equations varies according to the Kodaira symbol of the Jacobian of C. Then we count these equations up to isomorphism over a number field of class number 1.

The properties p-convexity and q-concavity are fundamental in the study of Banach sequence spaces (see [L-TzII]), and in recent years have been shown to be of great significance in the theory of the corresponding Schatten ideals ([G-TJ], [LP-P] and many other papers). In particular, the notions 2-convex and 2-concave are meaningful in Schatten ideals. It seems to have been noted only recently [LP-P] that a Schatten ideal has either of these properties if the underlying sequence space has. One way of establishing this is to use the fact that if (E, ‖ ‖E) is 2-convex, then there is another Banach sequence space (F, ‖ ‖F) such that ‖x;‖ = ‖x2‖F for all x ε E. The 2-concave case can then be deduced using duality, though this raises some difficulties, for example when E is inseparable.

Let

be a series of type Ω of the group G. By this we mean that Ω is an ordered set (we shall always understand ‘ordered’ to mean ‘totally ordered’) and Λσ and Vσ are subgroups of G satisfying

(i) Vσ ◃ Λσ for all σεΩ,

(ii) Λσ ≤ Vτ if σ < τ

(iii) , where G−1 denotes the set of elements ≠ 1 of G and Λσ−Vσ the set of elements of Λσ not belonging to Vσ,

(iv) Vσ ≠ Λσ for all σ ε Ω.

The rate of sputtering of magnesium in oxygen was observed with an apparatus which permitted the field immediately in front of the target to be reversed. Change in the sign of the field did not affect the rate of sputtering which remained abnormally low. It is therefore concluded that, while the low rate is in some way due to the presence of an oxide film, it is not due to charged particles being returned to the cathode by the electric field, as had been suggested by other workers.

A random environment is modelled by an arbitrary stochastic process, the future of which is described by a σ-algebra. Renewal processes and alternating renewal processes are defined in this environment by considering the conditional distributions of random variables generated by the processes with respect to the σ-algebra. Generalizations of several of the standard limit theorems of renewal theory are derived.

The aim of this note is to find out some self-reciprocal functions and kernels for Fourier-Bessel integral transforms. Following Hardy and Titchmarsh(i), we shall denote by Rp the class of functions which satisfy the homogeneous integral equation

where Jp(x) is a Bessel function of order p ≥ − ½. For particular values of p = ½, − ½, we write Rs and Rc irrespectively.

We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch’s characterisation of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvature, and which holds for large classes of complexes.

As an application, we study the hyperbolicity of groups obtained by small cancellation over a graph of hyperbolic groups.

Ever since the solution of the -Neumann problem by Kohn [4], refinements, extensions and estimates of the solution have been made by a lot of people up to the present day. Kohn discovered that, where the -Neumann problem was solvable, the space of harmonic forms is finite dimensional and it followed from Hörmander's solution [3] that on bounded pseudoconvex domains in ℂn the appropriate spaces of harmonic forms are zero dimensional.