We give a combinatorial characterization of the known examples of intervals of ℤ having the Littlewood-Paley (LP) property. This leaves an interesting open problem. We also note that certain previously known necessary conditions are equivalent, and give two examples of intervals which are not LP but whose endpoints form thin sets (e.g. Sidon or Λ(p) for all p < ∞).

For a ε R, we define a subset V of Rn to be a-convex if x, y ε V implies

Clearly V is (1 — a)-convex iff it is a-convex: V is convex iff it is a-convex for all a ε [0, 1], and any set is 1-convex. We define the a-convex hull of V to be the intersection of all a-convex subsets of Rn containing V. Put D = 0,1} and D(a) the a-convex hull of D in R1. If V is a-convex and x, y ε V, then V must contain the points of the line through x, y given by {tx + (1 — t) y: t ε D(a)} and so in this paper we concentrate on the study of D(a).

It is possible to obtain approximate theoretical relations between the spectra of different atoms ionised to such an extent that they possess the same electron structure. In this paper these relations are used to extrapolate values of the terms of the spectra of various lithium-like and sodium-like atoms.

From the terms so estimated lines can be calculated, and in many cases correlated with lines observed by Millikan and Bowen in the hot spark spectrum of the corresponding element. Revised term values can then be calculated from the lines so identified.

The spectra of lithium-like atoms treated in this way include those from Be II up to O VI, and those of sodium-like atoms, the spectra P V and S VI.

About twenty years ago a number of results were given expressing the sum of n terms of an ordinary hypergeometric series with unit argument in terms of an infinite series of the type 3F2. The interest in the subject was aroused by a theorem due to Ramanujan, who stated that

Consider a sphere of radius $\sqrt{n}$ in $n$ dimensions, and consider $\vc{X}$, a random variable uniformly distributed on its surface. Poincaré's Observation states that for large $n$, the distribution of the first $k$ coordinates of $\vc{X}$ is close in total variation distance to the standard normal $N(\vc{0},I_k)$. In this paper we consider a larger family of manifolds, and $\vc{X}$ taking a more general distribution on the surfaces. We establish a bound in the stronger Kullback–Leibler sense of relative entropy, and discuss its sharpness, providing a necessary condition for convergence in this sense. We show how our results imply the equivalence of ensembles for a wider class of test functions than is standard. We also deduce results of de Finetti type, concerning a generalisation of the idea of orthogonal invariance.

Some years ago Evelyn and Linfoot found an asymptotic formula, with estimation of remainder, for the number of representations of a large integer n as the sum of s r-free integers. For s ≥ 4 their formula was subsequently sharpened by Barham and Estermann, and for s = 3 recently by me.

This paper considers an invariant of modules over a finite-dimensional Hopf algebra, called the critical group. This generalises the critical groups of complex finite group representations studied in [1, 11]. A formula is given for the cardinality of the critical group generally, and the critical group for the regular representation is described completely. A key role in the formulas is played by the greatest common divisor of the dimensions of the indecomposable projective representations.

The polarization constant of a Banach space X is defined as

\[{\text{c}}(X){\text{ }}{\text{ }}\mathop {\lim }\limits_{k \to \infty } {\text{ }}\sup {\text{c}}{(k,X)^{\frac{1}{k}}},\]

where \[{\text{c}}(k,X)\] stands for the best constant \[C > 0\] such that \[\mathop P\limits^ \vee \leqslant CP\] for every k-homogeneous polynomial \[P \in \mathcal{P}{(^k}X)\]. We show that if X is a finite dimensional complex space then \[{\text{c}}(X) = 1\]. We derive some consequences of this fact regarding the convergence of analytic functions on such spaces.

The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak’s complexification procedure.

We also study some other properties connected with polarization. Namely, we provide necessary conditions related to the geometry of X for \[c(2,X) = 1\] to hold. Additionally we link polarization constants with certain estimates of the nuclear norm of the product of polynomials.

We explicitly find lower bounds on the volume of threefolds and fourfolds of general type in order to have non-vanishing of pluricanonical systems and birationality of pluricanonical maps. In the case of threefolds of large volume, we also give necessary and sufficient conditions for the fourth canonical map to be birational.

1. The object of this paper is to prove some formulae of Jacobi polynomials including a generating function. The results (2·l)–(2·4), (2·6)–(2·9), (3·l)–(3·4), and (4·1) are believed to be new.

An amenable semigroup of positive linear unital mappings on a W*-algebra is considered. Two main questions are dealt with: the existence of a normal faithful state invariant with respect to this semigroup and the description of ergodicity conditions. An explicit form of the ergodic projection, useful in treating the ergodicity problems, is also derived.

Let A and G be locally compact groups and α a continuous action of G on A, and let denote the semi-direct product of A and G. Then we prove that the left Hilbert algebra of continuous functions with compact support, has the same achieved left Hilbert algebra, as the crossed product of K(A)" by the associated action α̃ of G on . As a consequence we obtain that the canonical weight on is the dual weight of the canonical weight on K(A)".

The idea of gauge-invariance in general relativity was first introduced by Weyl(1) who proposed that the field equations of gravitation should be invariant, not only under the general group of coordinate transformations, but also under the gauge-transformation

where is the symmetric metric tensor, is the symmetric affine connexion and λ(x8) is an arbitrary scalar function of the coordinates. In this way it was possible to introduce into the theory a four-vector Ak which in consequence of (1·1) transformed as

such that the six-vector remained an invariant quantity under the gauge-transformation. It was Weyl's hope that by widening the invariance properties gauge-transformation. It was Weyl's hope that by widening the invariance properties of general relativity in this way the vector Ak and its associated six-vector Fik could be interpreted as representing the electromagnetic field. However, no obvious or unique way of doing this was found. More recently (see Stephenson (2,3) and Higgs (4)) gaugeinvariant variational principles formed from Lagrangians quadratic in the Riemann—Christoffel curvature tensor and its contractions have been discussed by performing the variations with respect to the symetric and symetric independently (following the palatini method).

In a paper published in these Proceedings I proved that there are only a finite number of quadratic fields in which Euclid's Algorithm (E.A.) holds. Recently Davenport has found a new proof of this theorem based on the theory of the minima of the product of linear inhomogeneous forms.

A general theory of ‘strength’ for Hausdorff methods T of summability was given in the first part of this paper, to which we shall refer in the following as I. Those methods T which are regular (that is, at least as strong as ordinary convergence C), are defined by the linear transforms

of a given sequence (sk). Here ø(t) (that is, each of its real and imaginary parts) is of bounded variation (b.v.) in the closed interval 〈0, 1〉 [see I, (4)]. The additional condition for consistency (with C) is that ø(t) is continuous at t = 0, and that ø(1) − ø(0) = 1 [I, H. 3]. We write also T ˜ ø(t).

Actinium emanation was introduced into a cloud chamber for the purpose of making a study of the secondary β-rays. The energies were in agreement with those given by Surugue using the magnetic spectrum method, except that his strong line at 105 kV. was absent. By comparing the numbers of β-rays with the relative intensities of the groups of α-particles, measured by Rosenblum, it was found that the internal conversion coefficients were in general agreement with previous experimental values in the same region of energy. Tracks of Auger electrons associated with the secondary β-rays were found in the photographs. In general, the results from the cloud chamber method agreed with accepted theories and confirmed the results obtained by other methods.

The theory of unsteady two-dimensional compressible boundary layers is studied in two parts. Part I deals with the solutions when the main stream is steady and the body is in an arbitrary motion while Part II deals with the solutions when the main stream is steady and the body is at rest with an arbitrary unsteady suction or injection perturbing about a zero mean. General methods of obtaining solutions are given when the coefficient of viscosity is proportional to temperature and the Prandtl number is unity. The main feature of the work in this paper lies in subjecting the equations to a transformation similar to that of Howarth and in linearization similar to that of Lighthill. Following the work of Sarma in incompressible flows two types of solutions are developed, one for large times and the other for small times. General expressions for the skin friction, the pressure ratio, the heat transfer and the difference between the normal coordinates of compressible and incompressible flows are obtained.

The effect of surface layers in modifying the thermionic emission has been explained by Nordheim on the assumption that the atoms in the surface contamination create an electrical double layer on the surface. If the layer is electro-positive with respect to the metal the work of final extraction of a given electron is reduced. This reduces the value of the work function χ, but at the same time requires the escaping electrons to pass through a region in which the potential energy is greater than their total energy. It follows that the emission coefficient D (W) is decreased at the same time. The details of the connection between χ and D (W) have been examined further by Fowler.

A doubly-stochastic matrix is an n × n matrix with non-negative elements such that each row and each column sums to 1. A permutation matrix is the result of permuting the rows of the unit n × n matrix. Birkhoff's theorem states that the doubly-stochastic matrices constitute the convex hull of the permutation matrices. Using Birkhoff's theorem, Farahat and Mirsky (1) showed that each doubly-stochastic matrix could be expressed as a convex combination of n2 − 2n + 2 permutation matrices, though not in general of fewer. Given Birkhoff's theorem, the Farahat-Mirsky refinement can also be proved quite shortly as follows.

Let X be a complex normed space with dual space X′ and let T be a bounded linear operator on X. The numerical range of T is defined as

and the numerical radius is v(T) = sup {|ν: νε V(T)}. Most known results and problems concerning numerical range can be found in the notes by Bonsall and Duncan (5).