We prove that the order of any subloop of a finite Moufang loop is a factor of the order of the loop, thus obtaining an analog of Lagrange's theorem for finite Moufang loops.

This paper presents a general theory of storage for dams subject to a steady release, and with continuous inputs having infinitely divisible distributions. The paper extends some earlier work by Downton, Lindley, Smith and Kendall ((10), see discussion), and gives a detailed account of results previously sketched by the authors (6).

1. The concept of a unique factorization domain (UFD) has been defined, for commutative (e.g. (4) page 21) and non-commutative (1) integral domains. We take the theory a stage further here by defining a unique factorization ring (UFR), where throughout, a ring is understood to mean a commutative ring with identity, possibly containing proper zero-divisors.

It was shown by Sas (1) that, if K is a plane convex body, then it is possible to inscribe in K a convex n-gon occupying no less a fraction of its area than the regular n-gon occupies in its circumscribing circle. It is the object of this note to establish the n-dimensional analogue of Sas's result, giving incidentally an independent proof of the plane case. The proof is a simple application of the Steiner method of symmetrization.

Some features of the linear theory of an elastic Cosserat plate are discussed and special attention is given to a bending theory which corresponds to the bending of a transversely isotropic three-dimensional plate. A useful representation for the solution of the system of differential equations characterizing the bending of an elastic Cosserat plate is deduced and the results are applied to the problem of an infinite Cosserat plate with a circular hole, the plate being in a state of uniform bending at infinity.

A method is presented for the accurate numerical treatment of molecular vibration problems in which the potential energy function is of the form

The treatment is carried through in detail for the case V(q) = Aq2 + Bq4, which, in most instances affords an adequate description of the potential. There is no restriction on the relative magnitudes of quadratic and quartic terms so that the method is equally applicable to the purely anharmonic oscillations occurring in types of ring bending (1), where V(q) = aq4.

An approximate formula is derived for the energy levels in the general case. The case V = aq4 affords an illustration of the accurate treatment; the first five eigen-values and eigen-functions are computed and from them the associated transitionprobabilities. Numerical results are presented in a form convenient for further application. On comparison the approximate formula is found to yield results in error by approximately 1 %, the error decreasing for the higher levels for which the formula tends to a B.W.K. expression. In conclusion a ground-state eigen-function of simple analytical form is found to approximate remarkably well to the case of purely quartic vibrations.

The method to be described provides an alternative means of dealing with certain non-standard linear boundary-value problems. It is developed in several applications to the theory of gravity-capillary waves. The analysis is based on a variational formulation of the hydrodynamic problem, being motivated by and extending the original study by Benjamin and Scott [3].

1. Making use of the familiar abbreviation

let us adopt a contracted notation for the generalized hypergeometric function AFB[x] and write

where (a) denotes the sequence of parameters

A long column of viscous liquid of radius a and uniform density p is rotating about its axis with angular velocity ω. It is shown that this motion is stable to plane perturbations of wave number 8 provided that the surface tension T satisfies . This critical value is higher than that required for stability of the similar motion of a nonviscous liquid, but is otherwise independent of the coefficient of viscosity. The rate of development of instability when T is less than the critical value is also studied. Some numerical results are given.

The condition has recently been obtained by Hocking (2) for the special cases of very high and very low Reynolds numbers.

The dynamical problem of the “diatomic molecule” is solved on the new mechanics. The terms of the rotational energy are , where ; the weights of the corresponding states are 2m; the frequencies differ a little from the classical ones. Finally the intensities are slightly different from those computed by Kemble; the main term agrees with that of Fowler, but the positive branch is only slightly stronger than the negative. The central line vanishes. The intensities are valid only for the fundamental band.

Proof of convergence of the Crank-Nicolson procedure, an ‘implicit’ numerical method for solving parabolic partial differential equations, is given for the case of the classical ‘problem of limits’ for one-dimensional diffusion with zero boundary conditions. Orders of convergence are also given for different classes of initial functions. Results do not support the validity of so-called h2-extrapolation in some cases.

A number of methods are formulated for solving the Schrödinger equation for systems of finite extent. The methods are developed in detail for the particular case of an electron moving in a Coulomb field (e.g. hydrogen-like atom), with a boundary consisting of a sphere of given radius. In the second part of the paper these results are transformed into those for an electron moving in a uniform magnetic field, the boundary of the system being cylindrical.

The answers to many important questions in the harmonic analysis of orthogonal polynomials are known to depend on the determination of when formulas of the types

and their duals

hold, where pn(x) and qn(x) are suitably normalized orthogonal polynomials or orthogonal polynomials multiplied by certain functions; e.g. e−pxLn(x).

Let

formula here

be the complete elliptic integral of the first kind. Recently, S.-L. Qiu and M. K. Vamanamurthy have proved for all r∈(0, 1):

formula here

with r′=√1−r2. In this paper we establish that the converse inequality

formula here

holds for all r∈(0, 1). Moreover, we show that the constant factors ¼ and π/(4 log (2))−1 in (a 1) and (a 2) are best possible.

All spaces are assumed to be locally path connected and semi-locally 1-connected. Let X be a connected topological group with identity e, and let be the universal cover of the underlying space of X. It follows easily from classical properties of lifting maps to covering spaces that for any point ẽ in with pẽ = e there is a unique structure of topological group on such that ẽ is the identity and is a morphism of groups. We say that the structure of topological group on X lifts to .

The grand partition function of any statistical assembly may be defined by the equation

where E denotes any value of the energy of the assembly, k is Boltzmann's constant, T the thermodynamic absolute temperature and λi a parameter which is later to be connected with the chemical potential, μi, of the ith species in the assembly by the simple formula

In a recent joint paper (1) with Prof. Besicovitch we announced the conjecture that for almost all one-dimensional Brownian paths, the set of zeros has dimensional number ½, and zero A½-measure. It is the purpose of this paper to give a proof of this result. In doing so we consider the graph C(ω) of a Brownian path ω as a point set in the plane, and prove that, with probability 1, C(ω) has dimensional number ¾ and zero Λ¾-measure.

The comparison theorem is an algebraic theorem, which corresponds to the following topological situation. A map between two fibre spaces induces a homomorphism between the two corresponding spectral sequences relating the homologies of the base, fibre and fibre space; if the map induces an isomorphism on any two of these quantities, then it also does on the third. J. C. Moore (l) proved the theorem using a mapping cylinder, which requires that the spectral sequences arise from filtered differential groups. The present proof assumes only the existence of the spectral sequences from the E2 terms onwards. Moreover, we generalize the theorem, assuming that the homomorphisms on the two given quantities are isomorphic only up to given dimensions, and deducing the same for the third quantity. For algebraic translucence we use a simpler hypothesis (property (iii)) than Moore's concerning the E2 terms. However, if his hypothesis is assumed, together with the existence of E1 terms, then his theorem can be deduced as a corollary. The analogous result for cohomology is similar, and is stated at the end.

In this note I prove the following theorem:

Theorem 1. Given 3N points in a plane, we can divide them into N triads such that, when we form a triangle with the points of each triad, the N triangles will all have a common point.

The proof depends on three lemmas; the first two are well known, and are quoted without proof; the third I believe to be new, and leads directly to the proof of the theorem.