Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11, 12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspired by the groupoid model of (intensional) Martin–Löf type theory [14] due to Hofmann and Streicher [9]. In particular, we show that a form of Martin–Löf type theory can be soundly modelled in any model category. This result indicates moreover that any model category has an associated “internal language” which is itself a form of Martin-Löf type theory. This suggests applications both to type theory and to homotopy theory. Because Martin–Löf type theory is, in one form or another, the theoretical basis for many of the computer proof assistants currently in use, such as Coq and Agda (cf. [3] and [5]), this promise of applications is of a practical, as well as theoretical, nature.

The notion of fractional dimensions is one which is now well known. The object of the present paper is the investigation of the dimensional numbers of sets of points which, when expressed as continued fractions, obey some simple restriction as to their partial quotients. The sets considered are naturally of linear measure zero. Those properties of the partial quotients which hold for almost all continued fractions make up the subject called by Khintchine ‘the measure theory of continued fractions’.

In treating plane wave scattering by a periodic surface, Lord Rayleigh (10) assumed that the discrete, outgoing and evanescent plane wave representation for the scattered field was valid on the surface itself. Recently, this Rayleigh assumption has been questioned and criticized. For the surface y = b cos kx on which the total field vanishes, Petit and Cadilhac(8) have demonstrated its invalidity when Kb > 0·448. The present paper discusses scattering of a wave, incident from y > 0, by an analytic periodic surface y = f(x) ( – ∞ < x < ∞), and shows that the Rayleigh assumption is valid if and only if the solution can be continued analytically across the boundary at least to the line y = minf(x). Conformal mapping and results relating to the analytic continuation of solutions to elliptic partial differential equations reduce the problem to one involving the location of singularities and critical points of a potential Green's function. Provided that the perturbation of the surface from a plane is sufficiently gentle, the validity of the Rayleigh assumption is established. For the surface y = b cos kx it is shown that the assumption is valid if Kb < γ, where γ is a positive number no greater than 0·448, the precise value of which is unknown. Possible extensions of the analysis to different or more general situations are suggested.

The problem was suggested by Professor G. I. Taylor as being of interest and importance in the phenomena of rupture in a solid, due to the presence of a crack, and the slip in a crystal. In experiments on the distortion of a crystal of aluminium under a tensile stress the conclusion is reached that “…as far as these experiments go, the distortion of a crystal of aluminium under compression is of the same nature as the distortion which occurs when a uniform single-crystal bar is stretched. The distortion is due to slipping parallel to a certain crystal plane and in a certain crystallographic direction, and the choice of which of twelve possible crystallographically similar types of slipping actually occurs depends only on the components of shear stress in the material and not at all on whether the stress normal to the slip plane is a pressure or a tension.”

The purpose of this paper is to obtain some necessary conditions for a link in Euclidean 3-space to be spanned by a locally unknotted surface of given type in one half of 4-space. In particular necessary conditions for two links to be cobordant are proved.

In order to explain the phenomena of melting, tensile strength, etc., we have to investigate the stability of crystals for finite deformations, for which deviations from Hooke's law occur. Although these deviations are in most cases of an irreversible character, it is necessary, for a systematic study, to develop mathematical methods for treating the mechanical (reversible) case of a highly strained crystal lattice, where terms of higher order than the second in the deformation energy must be taken into account.

The NUT metric is interpreted as the field of (a) a mass around the origin of coordinates, and (b) a semi-infinite massless source of angular momentum.

This paper describes, in general terms, a Monte Carlo method for estimating statistical parameters of quantum-mechanical systems. In this method, we construct a Markov chain of transitions between finite sequences of indices, and obtain these parameters in terms of parameters of the limit distribution. This is an extension of the method of Metropolis for classical systems, and may have equally wide application.

This paper describes a method for computing the coefficients in the Chebyshev expansion of a solution of an ordinary linear differential equation. The method is valid when the solution required is bounded and possesses a finite number of maxima and minima in the finite range of integration. The essence of the method is that an expansion in Chebyshev polynomials is assumed for the highest derivative occurring in the equation; the coefficients are then determined by integrating this series, substituting in the original equation and equating coefficients.

Comparison is made with the Fourier series method of Dennis and Foots, and with the polynomial approximation method of Lanczos. Examples are given of the application of the method to some first and second order equations, including one eigenvalue problem.

When we treat an atom containing a number n of electrons by the method of the self-consistent field, we assume that each electron has its own particular “orbit,” specified by a wave function (q|r) in four variables q. These four variables are usually taken to be the three coordinates of the electron together with a variable describing the spin, but according to the transformation theory of quantum mechanics, they may be any four independent commuting functions of the coordinates, momenta and spin variables.

It is shown that all the classical thermodynamical properties and the equation of state of a rectilinear assembly of spherical molecules with short-range attractive forces acting only between neighbours can be derived from a single partition function which is an analytic function of the pressure and the temperature. The thermodynamical functions are simply related to the Laplace transform of the function e−E(x)/(kT), where E(x) is the interaction energy between two neighbouring molecules. Although this linear model behaves like a perfect gas at high temperatures and like a crystal near the absolute zero, there are no critical phenomena, owing to the analytic character of the partition function. It is, however, pointed out that at very low temperatures the slope of the isothermal curves can suffer extremely sharp changes which may be interpreted as changes of phase, and this point is illustrated by an example.

Finally, I should like to express my sincere gratitude to Prof. H. Jones for his guidance of this work and for his invaluable advice throughout. My thanks are also due to the Turkish Ministry of Education for the award of a scholarship.

The rheological equations of state for an isotropic Bingham solid may be written in tensor form:

where

The notation is explained fully in a previous paper (1). The frame of reference is essentially Eulerian. Primes denote deviatoric components of the stress, strain and rate of strain tensors, pik, εik and eik (for example, ). The dilatation is denoted by Δ. The rigidity and bulk moduli μ and κ are, if not constants, essentially scalar functions of the stress tensor. The variable viscosity η involves two constants, the yield value ϑ and the reciprocal mobility η1.

This investigation owes its origin to a remark by Professor Lennard-Jones that it should be possible to calculate the approximate force between two atomic systems at large distances apart by a fairly direct method. The inter-atomic force is known to be of the form kR−7 at a large distance R, and an attempt to obtain the value of the constant k for two hydrogen atoms was made by Wang. The method suggested to the writer of this paper by Lennard-Jones was used by London and Eisenschitz for this case, and they obtained a value ofk differing from that given by Wang. The value given by London and Eisenschitz has been confirmed by following the method given in this paper, but further applications to the calculation of the inter-atomic force at large distances were postponed as the problem of calculating the polarizability of helium provided, on Lennard-Jones' suggestion, a simpler case on which the method could be tested.

The notion of statistical convergence was introduced by Fast[1] and has been investigated in a number of papers[2, 5, 6]. Recently, Fridy [2] has shown that k(xk–xk+l) = O(1) is a Tauberian condition for the statistical convergence of (xk). Existing work on statistical convergence appears to have been restricted to real or complex sequences, but in the present note we extend the idea to apply to sequences in any locally convex Hausdorif topological linear space. Also we obtain a representation of statistical convergence in terms of strong summability given by a modulus function, an idea recently introduced in Maddox [3, 4]. Moreover Fridy's Tauberian result is extended so as to apply to sequences of slow oscillation in a locally convex space, and we also examine the local convexity of w(f) spaces.

The method of the self-consistent field is applied to a discussion of the energy and wave-functions of the ground state of the hydrogen molecule; an analytical expansion of the wave-function is given in terms of spheroidal coordinates, and the distribution of charge is determined. Two simpler, though less accurate, wave-functions for the molecule are also included, and the possibility of using the method for more complex molecules is discussed.

1. The reflexion of waves on the surface of water by a thin plane vertical barrier is considered and the coefficient of reflexion (the ratio of the amplitudes, at a great distance from the barrier, of the reflected and incident waves) is calculated. If the top edge is at a depth a below the surface, it is found that the coefficient of reflexion is about ¼ when where T is the period of the incident waves, so that the condition that the coefficient may exceed ¼ is a .

When a vessel of liquid has been emptied and put aside, a thin film of liquid clings to the inside and gradually drains down to the bottom under the action of gravity. The layer being thin, the motion is very nearly laminar flow, and the curvature of the surface in a horizontal direction may be ignored. Thus the problem for a cylindrical vessel is reducible to that of a wet plate standing vertically.

The stability of a charged raindrop has been discussed mathematically by Lord Rayleigh. The case of an uncharged drop in a uniform electric field is perhaps of more meteorological importance but a mathematical discussion of the conditions for stability turns out to be very much more difficult in this case, owing to the fact that the drop ceases to be spherical before it bursts. Moreover it does not seem possible to express its geometrical shape by means of any simple mathematical expressions. On the other hand, by using a soap bubble instead of a water drop it was found possible to carry out experiments under well-defined conditions in this case, whereas experiments with Rayleigh's charged drop would be difficult.

The concept of ideal strength of perfect crystals as an instability phenomenon is critically reviewed in the context of generalized moduli associated with arbitrary measures of stress and strain. Further aspects of elastic response to finite strain are discussed in relation to the classical model of an atomic lattice.

In a recent paper (3) Mordell proposed for solution

PROBLEM 1. To find the constants a1, …, an so that

where xn+i = xi ≥ 0 for all i.