In this paper we discuss the structure of a partially ordered set X by studying in detail certain subsets of X.

Let f = f(x1, …, xn) be a positive definite quadratic form in n ( ≤ 8) variables; then we consider the old problem of estimating the minimum of f (its least value for integers xi not all 0) in terms of the determinant Δ(f). Normalizing by supposing the minimum to be 1, the known results may be stated as

each inequality being best possible. Further, for each n, all forms for which equality holds in (1) have integral coefficients and are equivalent to each other.

Let b ⩾ 2 be an integer. Among other results we establish, in a quantitative form, that any sufficiently large integer which is not a multiple of b cannot simultaneously be divisible only by very small primes and have very few nonzero digits in its representation in base b.

The theory is given of the structure and behaviour of immobile adsorbed films on a simple quadratic lattice in which each particle is so large that it occupies more than one site and makes it impossible for adsorption to occur on sites which are neighbours to that on which it is adsorbed. The results are obtained partly by statistical methods and partly by the use of a model of the film.

The final or complete film of this type is not homogeneous and has gaps in it. The sites on the surface can be divided into two groups, the gaps occurring where regions of the surface occupied by particles adsorbed on one group and those occupied by particles on the other group meet. The number of these gaps or “boundary sites” is determined, and the possible configurations of particles in the boundary regions are discussed in detail.

The results are considered in connexion with some experiments on the adsorption of oxygen on tungsten. In this case a second layer is formed over the boundary sites and the amount that can be so adsorbed or the number of “second layer sites” to which the boundary sites give rise is obtained. It is shown that numerically the experimental results for oxygen are in better agreement with the view that the first or very stable layer is a molecular layer of the above type than with the view that it is atomic.

The classical result of Gohberg, Markus and Feldman states that, when $E$ is one of the classical Banach sequence spaces $E=l^p$ for $1\leq p<\infty$ or $E=c_0$, the only closed, two-sided, non-trivial ideal in $\cal B(E)$, the Banach algebra of operators on a Banach space $E$, is $\cal K(E)$, the ideal of compact operators. Gramsch and Luft completely classified the closed, two-sided ideals in $\cal B(H)$ for an arbitrary Hilbert space $H$ through the idea of $\kappa$-compact operators, for infinite cardinals $\kappa$. This paper presents an extension of this result to the non-separable versions of $l^p$ and $c_0$.

Let $p$ be a prime and $\vartheta$ an integer of order $t$ in the multiplicative group modulo $p$. In this paper, we continue the study of the distribution of Diffie–Hellman triples$(\vartheta^x, \vartheta^y, \vartheta^{xy})$ by considering the closely related problem of estimating exponential sums formed from linear combinations of the entries in such triples. We show that the techniques developed earlier for complete sums can be combined, modified and developed further to treat incomplete sums as well. Our bounds imply uniformity of distribution results for Diffie–Hellman triples as the pair $(x,y)$ varies over small boxes.

1. Introduction: In (4) Moran considers two independent random variables X and Y taking non-negative integral values to give a characterization of the Poisson distribution. He establishes that the conditional distribution of X, given the total X + Y, is binomial for all given values of X + Y and there exists at least one i so that P(x = i) > 0, P( Y = i) > 0 if and only if X and Y have Poisson distributions. A slightly improved version of this result is given by Chatterji (1). For a comprehensive bibliography on the Poisson distribution the reader is referred to (3).

The optical properties of reflecting echelette gratings, used in the infra-red region of the spectrum, are discussed theoretically. Expressions are derived for various important characteristics including the wave-length relationship, the dispersion and the resolving power of spectrometers using such gratings. The intensity distribution is discussed and illustrated by reference to a particular spectrometer fitted with a grating of 1200 lines to the inch. The relative importance of other factors such as the variation of available energy with wave-length, the dispersion of the preliminary optical system, etc., is considered and tabulated. Particular consideration is given to the properties of such gratings when used in much higher orders of interference than are common at present. With large aperture spectrometers, this possibility of using comparatively coarse gratings with a high order of interference is of great practical importance.

1. Since many writers, among them G. I. Taylor (1) and W. L. Bragg (2), have based theories of the strength of metals upon the assumption that there is some type of dislocation in the metal, it has seemed desirable to give an account of a dislocation in which displacement and stress are given, according to the classical theory of elasticity, by simple closed formulae.

In this paper we are concerned with curves (C) of the following types:

where k1, k2, …, kh−1, kh = 0 (h ≤ n) are the curvatures of (C) relative to the space Vn in which (C) lie. Hayden proved that a curve in a Vn is an (A)2m, h = 2m + 1, if and only if it admits an auto-parallel vector along it which lies in the osculating space of the curve and makes constant angles with the tangent and the principal normals. Independently, Sypták∥ stated without proof that a curve in an Rn is a (B)n if and only if it admits a certain number of fixed R2's having the same angle properties; he also gave to such a curve a set of canonical equations from which many interesting properties follow as immediate consequences.

The group of order 168 discovered by Klein, which it is now known can be generated by two operations E, of order 7, and ϑ, of order 2, which satisfy the relations E7 = 1, ϑ2 = 1, (Eϑ)3 = 1, (E4ϑ)4 = 1, has a vast literature. But for the most part each author pursues the matter from his own point of view; and it seems it may be useful to present a simplified approach to the theory which takes account of various possible aspects, in particular the geometrical. This is the object of the present note; for most of its contents I have found it necessary to do fresh work, so that the paper is by no means a transcript of what is already available.

Two alternative forms of the CO2 molecule have been suggested by various authors who have discussed the band spectrum data. The specific heat curves based on these models are considered here. It is found that neither is quite satisfactory over the whole range of temperature and we discuss the difficulties for the low temperature and high temperature portions separately. In order to get agreement for low temperatures we find it necessary to introduce a further hypothesis about the molecular model which also seems to explain one or two outstanding difficulties in interpreting the fine structure of the bands. This assumption does not make any difference at higher temperatures where we show the error in one of the curves to be of the order we should expect to be accounted for by a centrifugal stretching of the molecule.

A variety satisfies the amalgamation property if given G, H1, and embeddings τ1: G → H1 and τ2: G → H2, there exist and embeddings σ1: H1 → K and σ2: H2 → K such that σ1τ1 = σ2τ2.

1. Let f(n) be a real-valued additive arithmetic function. Let α(x) and β(x) > 0 be real valued functions, defined for x ≥ 2. Define the frequencies

Reade[11] has shown that positive definite kernels K(x, t) which satisfy a Lipschitz condition of order α on a bounded region have eigenvalues which are asymptotically O(1/n1+α). In this paper we extend this result to positive definite kernels whose symmetric derivative Krr(x, t) = ∂2rK(x, t)/∂xτ ∂tτ is in Lipα and establish λn(K) = O(1/n2r+1+α). If ∂Krr/∂t is in Lipα, the anticipated asymptotic estimate is also derived.

The proofs use a well-known result of Chang [2], recently rederived by Ha [5], and estimates based upon finite rank approximations to the kernels in question. In these latter estimates we employ the familiar piecewise linear ‘hat’ basis functions of approximation theory.

Recently, the present author together with M. Crampin proved a structure theorem for a certain subclass of geometric objects known as almost tangent structures (Crampin and Thompson [8]). As the name suggests, an almost tangent structure is obtained by abstracting one of the tangent bundle's most important geometrical ingredients, namely its canonical 1–1 tensor, and using it to define a certain class of G-structures. Roughly speaking, the structure theorem referred to above may be paraphrased by saying that, if an almost tangent structure is integrable as a G-structure and satisfies some natural global hypotheses, then it is essentially the tangent bundle of some differentiable manifold. (I shall have a further remark to make about the conclusion of that theorem at the end of Section 2.)

An alternative to Fock's proof of the uniqueness of the electromagnetic energy momentum tensor is presented. The proof is four-dimensional and is applicable in the curved space-time of general relativity.

The Bourbaki–Witt principle states that any progressive map on a chain-complete poset has a fixed point above every point. It is provable classically, but not intuitionistically.

We study this and related principles in an intuitionistic setting. Among other things, we show that Bourbaki–Witt fails exactly when the trichotomous ordinals form a set, but does not imply that fixed points can always be found by transfinite iteration. Meanwhile, on the side of models, we see that the principle fails in realisability toposes, and does not hold in the free topos, but does hold in all cocomplete toposes.

In what follows, G will denote a compact, connected Lie group.

Definition. A complex-valued function f ε L2(G) lies in the space A(G) if it can be written in the form

where

The A(G) norm of f is the infimum of all sums (2) with respect to all decompositions (1).

This paper describes a repetition of J. B. Johnson's determination of Boltzmann's constant by observing the thermal agitation of electrons in a conductor. The principle of the measurement is described and the various sources of error are examined separately. The attenuator system of the calibrating circuit is discussed in detail and mutual agreement established between the three different attenuating systems. The valve amplifier had a maximum response which could be arranged to occur at any frequency between 1000 and 4000 cycles/sec. The mean value of 52 determinations of Boltzmann's constant (with the amplifier set to respond to one of four frequencies within the range 1 to 4 k.c./s.) was found to be 1·361 × 10−16 ergs per degree, with a probable error of 1·42%: this value is 0·8% less than the accepted value of 1·372 × 10−16 ergs per degree. The investigation developed out of a research initiated with a directly practical objective; we desire to express our thanks to the Hebdomadal Council of the University of Oxford, who by a grant to one of us have enabled this section of the work to be completed.