In this note we utilize the atomic decomposition of weighted Hardy spaces to prove weighted versions of Hardy's inequality for the Fourier transform with Muckenhoupt weight. The result extends to certain integral operators with homogeneous kernels of degree −1.

In a previous paper it has been shown that the corrosion of zinc, cadmium, iron and lead is connected with electric currents set up through unequal distribution of oxygen; the currents have been measured and compared with the weight of metal corroded away. The “unaerated” areas are anodic, and thus the attack tends to become concentrated upon the points relatively inaccessible to oxygen. When a drop of salt water is placed on an iron sheet, the central part of the drop suffers anodic corrosion, giving rise to ferrous salts, whilst the marginal ring over which oxygen has ready access to the metal becomes cathodic, alkali being produced there. Where the ferrous salt and alkali come into contact, ferrous hydroxide is precipitated, which oxidises to ferric hydroxide; after a few hours we get a membrane of brown ferric hydroxide extending right over the drop, but surrounded by a ring of clear alkaline liquid. All the actual corrosion of the metal is confined to the area within the membrane; the metal below the external ring of clear liquid, to which oxygen has access, suffers no attack.

If f: X × Y → Z is a map, then the classical Hopf construction associates to f a map hf: X * Y → ΣZ, where X * Y is the join of X and Y and ΣZ the suspension of Z. Since X * Y has the homotopy type of Σ(X Λ Y), the suspension of the smash product of X and Y, the homotopy class of hf can be regarded as an element Hf ↦ [Σ(X Λ Y), ΣZ]. Now elements of [Σ(X Λ Y), ] are in one to one correspondence with homotopy classes in the group [σ(X Λ Y), ΣZ] which are trivial on the suspension of the wedge Σ(X ≷ Y).

Let (M, g) be an arbitrary space-time of dimension ≥ 2 and let d = d(g): M × M → ℝ ∪ {∞} (where d(p, q) = 0 for q∉J+(p)) denote the Lorentzian distance function of (M, g). Also let C(M, g) denote the space of Lorentzian metrics for M globally con-formal to g. Here g1 is said to be globally conformal to g if there exists a smooth function Ω: M → (0, ∞) such that g1 = Ωg.

In a recent paper, Porter [9] devised two generalized Volterra operators which convert integral equations with the Hankel function kernel into Cauchy singular equations. The transformations were exploited in [9], and in a subsequent paper (Porter and Chu [10]), in relation to certain wave diffraction problems.

We give an elegant simple proof of a result concerning the first normal bundle of ideal Lagrangian immersions. This enables us to simplify and clarify several of the results in an earlier paper.

We study the spectrum containment of almost periodic solutions to differential equations with piecewise constant argument (EPCA, for short). We find an important property, which is different from that given by Cartwright for Ordinary Differential Equations (ODE). Some known (periodic, quasi-periodic) results may be expanded. As a corollary, it is shown that EPCA with periodic perturbations possess a quasi-periodic solution and no periodic solution. As a consequence, a new class of quasi-periodic functions is described. This new phenomenon is due to the piecewise constant argument and illustrates a crucial difference between ODE and EPCA.

Summary. The main result of the present paper is contained in the following: Theorem ℜ α ≠ −1, −2, −3,…

where2F1[.,.;.;.] and 1f1[.;.;.] (l).

We give a precise description of combed trees in terms of Kelly–Mac Lane graphs. We show that any combed tree is uniquely expressed as an allowable Kelly–Mac Lane graph of a certain shape. Conversely, we show that any such Kelly–Mac Lane graph uniquely defines a combed tree. We show how to apply this to the construction of opetopes.

By studying dressing orbits a complete account is given of the (full) super-horizontal holomorphic maps of finite type from the 2-plane into a full flag manifold. It is shown that precisely one dressing orbit contains all 2-spheres of finite type. From this follows a simple description of all harmonic 2-spheres in CPn of finite type.

The proportion ρk of gaps with length k between square-free numbers is shown to satisfy logρk=−(1+o(1))(6/π2)klogk as k→∞. Such asymptotics are consistent with Erdős's challenge to prove that the gap following the square-free number t is smaller than clogt/log logt, for all t and some constant c satisfying c>π2/12. The results of this paper are achieved by studying the probabilities of large deviations in a certain ‘random sieve’, for which the proportions ρk have representations as probabilities. The asymptotic form of ρk may be obtained in situations of greater generality, when the squared primes are replaced by an arbitrary sequence (sr) of relatively prime integers satisfying [sum ]r1/sr<∞, subject to two further conditions of regularity on this sequence.

1. Introduction. It is known (6) that the connectivity conditions on the manifolds in Irwin's embedding theorem (7) can be replaced by a connectivity condition on the map. This has led to the suggestion that in Hudson's analogue of Irwin's theorem for bounded manifolds (5), (6), the connectivity conditions on the manifolds modulo boundary can be replaced by the corresponding connectivity condition on the map modulo boundary. The main purpose of the present note is to show that this is in fact false, as is the formal analogue of Irwin's and Hudson's theorems for manifold triads. Its other purpose is to announce a related embedding theorem for manifolds with boundary which appeared in (3).

On a (pseudo-)Riemannian manifold (${\mathcal M}$, g), some fields of endomorphisms i.e. sections of End(T${\mathcal M}$) may be parallel for g. They form an associative algebra $\mathfrak e$, which is also the commutant of the holonomy group of g. As any associative algebra, $\mathfrak e$ is the sum of its radical and of a semi-simple algebra $\mathfrak s$. Here we study $\mathfrak s$: it may be of eight different types, including the generic type $\mathfrak s$ = ${\mathbb R}$ Id, and the Kähler and hyperkähler types $\mathfrak s$ ≃ ${\mathbb C}$ and $\mathfrak s$ ≃ ${\mathbb H}$. This is a result on real, semi-simple algebras with involution. For each type, the corresponding set of germs of metrics is non-empty; we parametrize it. We give the constraints imposed to the Ricci curvature by parallel endomorphism fields.

1. Polarisation measurements are described on the Hg lines λ 4358, 5461, 5770, 5791 emitted from a low-pressure electron-maintained arc to which a longitudinal field of up to 3000 gauss may be applied.

2. Two types of polarisation effect have been found. The first is typified by the line λ 5461. It is a polarisation of the light with the electric vector perpendicular to the lines of force. Its maximum value is 9% but it decreases to zero (a) with decreasing field strength and (b) with decreasing current in the tube. It is suggested that this effect is due to a self-reversal of λ 5461 owing to the metastability of the 2 3p2 state and that the polarisation may be caused through a slight non-uniformity of the magnetic field.

The second type of effect is found for λ5770 and 5791. The polarisation is in a plane parallel to the discharge, and is found in a zero magnetic field. This effect is further discussed elsewhere. The line λ 4358 is unpolarised under all conditions tried.

3. Neither of these effects implies a polarisation of the total light emitted directly from atoms in the Transverse Zeeman Effect when the atoms are excited under isotropic conditions and the experiments give definite evidence against the existence of such a polarisation effect.

Gyroscopic effects furnish important instances of dynamical actions, but their general discussion is outside the range of most students reading for Part I of the Natural Sciences Tripos. If, however, we limit ourselves to the case in which the axis of the wheel makes a constant angle with the vertical, the theory becomes elementary and simple devices suffice for recording the movements of the revolving wheel. The recording gyroscope was designed to give students at the Cavendish Laboratory, Cambridge, an opportunity of making practical measurements to verify the theoretical results.

The elastic scattering of Yukawa's particles, or mesons, by a static electric field is recalculated by considering the scattering as a process in which the incident meson is absorbed and another meson, i.e. the scattered meson, is created, and Laporte's result is confirmed. This method promises an easier handling of the second approximation.

A similar dependence of the angular distribution of the emitted quanta upon the spin of the incident mesons in a process like Y− + P → N + hν is found. Thus, the angular distribution contains a dependence upon the azimuthal angle in general. For the reverse process, the angular distribution of the emitted mesons, both longitudinal and transverse, contains also a dependence upon the azimuthal angle, which disappears only when the incident light is circularly polarized.

The writer wishes to thank Prof. W. Pauli for calling his attention to the azimuthal dependence in the problem of elastic scattering and Dr C. Møller for help given during the preparation of the paper.

Let G be a locally compact group and L1(G) be the group algebra of G. We show that G is abelian or compact if every continuous automorphism of L1(G)** maps L1(G) onto L1(G) This characterizes all groups with this property and answers a question raised by F. Ghahramani and A. T. Lau in [7]. We also show that if G is a compact group and θ is any (algebra) isomorphism from L1(G)** onto L1(H)**, then H is compact and θ maps L1(G) onto L1(H).

The problem of the double integral in the calculus of variations, when expressed in the parametric notation, was first fully discussed by Kobb. In this paper Kobb finds conditions for the minimising of the double integral

taken over the surface

We discuss a new method to compute the canonical height of an algebraic point on a hyperelliptic jacobian over a number field. The method does not require any geometrical models, neither p-adic nor complex analytic ones. In the case of genus 2 we also present a version that requires no factorisation at all. The method is based on a recurrence relation for the ‘division polynomials’ associated to hyperelliptic jacobians, and a diophantine approximation result due to Faltings.

In various communications published in the Journal of the Chemical Society the author has discussed the effect of different nuclei on the absorption spectra of various compounds. The problem has been further investigated with a number of substances derived from aniline, o-and m-cresols, and α-and β-naphthylamines, and this communication is to give an account of the results. A condensed Cd-spark was used as the source of radiant energy, and alcoholic solutions of the substances were examined in the usual way.