Littlewood posed the following problem: for each pair of real numbers, θ and φ, and each ε > 0, is there a positive integer n such that

where ‖α‖ denotes the difference between a and the nearest integer.

A hemisphere, resting on a horizontal plane, is initially at rest relative to an incompressible, inviscid, non-diffusive fluid whose density is vertically stratified. The hemisphere is then given, impulsively, a small constant horizontal velocity which is maintained thereafter. Assuming that the Froude number is small, and using the Boussinesq approximation, the equations of motion are linearised and solved using a Laplace transform. The disturbance in the fluid is analysed for large times and is found to contain a steady component of purely horizontal flow, an internal wave field and internal oscillations at the Brunt-Väisälä frequency, together with their various interactions. The effects of viscosity and diffusivity are discussed qualitatively by considering their effects on an internal wave.

We investigate the steady pressure driven flow of conducting fluid through an insulating circular pipe in the presence of a transverse applied magnetic field. It is well known that an exact solution is available, and the series expressions for the fluid velocity and induced magnetic field have often appeared in the literature, Uflyand (1960), Uhlenbusch and Fischer (1961), Gold (1962); however, no progress has been made in deriving asymptotic expansions for large Hartmann number directly from these series. Boundary layer methods have been used by Shercliff (1953. 1962) to determine the core structure, the Hartmann layer structure, the current distribution and the flow rate; however, these analyses do not describe the tangential layers (called obscure layers by Shercliff), which are located in the vicinity of the points at which the applied magnetic field is tangential to the pipe.

Consider an incompressible fluid of density p and kinematic viscosity v in an infinite two-dimensional domain. We assume that the fluid has a uniform velocity U∞, in the direction of the positive x*-axis of a rectangular Cartesian coordinate system Ox* y*, at large distances from a fixed flat plate of zero thickness which occupies the interval −l < x* < 0 of the line Ox*. Of special interest here is the structure of the flow when ε ≪ 1 where

Re being the Reynolds number of the flow. A first examination was made by Blasius (1908), using Prandtl's theory of the boundary layer, who found inter alia that the leading term of the drag D on one side of the plate is given by

the numerical factor being determined by Goldstein (1930).

This paper presents an analysis of two canonical problems for the steady supersonic inviscid flow past a wing-body combination with the body nonlifting and the wings at a very small angle of attack. Pressure distributions in the vicinity of the lines separating the interaction regions from the remainder of the wing and body are determined from asymptotic expansions by modifying a method due to Friedlander, who considered the diffraction of sound pulses by a cylinder. In a manner similar to Keller's geometrical theory of diffraction the results of the canonical problem are directly applicable to the case where the cross section of the body is any closed convex curve which meets the wings at right angles. In the first canonical problem, the leading edge of the wings has no sweepback, that is, it is normal to the body surface. Here results are compared with those of an approach using boundary layer methods by Stewartson and of a Green's function method by Jones. In the second problem which is even more important in practice and has been neglected in the literature, the leading edge of the wings has positive sweepback; our results are valid provided that the leading edge of the wings is supersonic.

The model of the complete system of triangles lying in a plane is a sixfold, whose nature of course depends on the chosen definition of the geometric variable. An earlier paper [1] concentrated on the variety associated with the Schubert triangle, although it contained brief descriptions of the corresponding models for other types. We now study the relation between these sixfolds, paying particular attention to the partial dilatation which transforms the model of ordered triangles into the Schubert variety.

Let X be a finite set of points in Ed. Then a partition of X into two non-empty subsets X1 and X2 (X1 ∪ X2 = X, X1 ∩ X2 = ∅) will be called a Radon partition if

.

The object of the present note is to prove the following theorem. †

Theorem. Let P be a convex d-polytope in d-dimensional real vector space Rd (d ≥ 2). If each d-polytope which is the intersection of two translates of P has the same number of vertices as P, then P is a direct linear sum ‡ of simplices.

Weber proves in §§114-124 of his Algebra [19] that if w is complex quadratic and ℤ[ω] is the ring of integers of the field ℚ(ω) then the absolute class field of ℚ(ω) is generated by the modular invariant j(w); he calls j(w) a class invariant. He goes on in §§125-144 to consider the values f(w) of other modular functions f(z); he shows that in certain cases the degree of the extension ℚ(ω, f(ω)) of ℚ(ω, j(ω)) is much less than that of ℚ(f(z)) over ℚ(f(z)); indeed, f(ω) is often in ℚ(ω, j(ω)), and in such circumstances Weber calls f(ω) a class invariant too. Using such results, Weber computes many class invariants—an end in itself, since the numbers are so beautiful. More recently, results of this type have been applied to determine all the complex quadratic fields with class number 1, and to prove that elliptic curves of certain families always have infinitely many rational points-see [9, 2, 5].

In an earlier paper [Shail, 1] one of the present authors considered the effect of a magnetic field on the frictional couple experienced by an axisymmetric solid insulator which rotates slowly in a bounded viscous conducting fluid, the applied magnetic field being parallel to the axis of rotation. Results for the couple in various geometrical configurations were obtained in the form of power series expansions in the Hartmann number M, and hence are valid only for M ≪ 1. However, because of the increased rigidity given to the fluid by the applied field, it is physically evident that the magnetic field will have a more dramatic effect on the flow pattern for large values of M. Thus one object of this paper is to investigate the frictional couple on a solid insulator rotating in an unbounded fluid for values of M ≫ 1.

In this note we shall show that the assumptions made by Murray [1] on the initial state of the gas and the initial magnetic field are inconsistent with the steady state momentum equation and the energy balance equation (20). The form of second term in equation (20) requires

Irrotational gravity waves of finite height: a numerical study. The sentence on page 146 in which equation (30) occurs should read:

The method consists of expanding the equation as

An analogue of a problem of Littlewood. The arguments of this paper are not wholly correct; a detailed corrigendum will be published in due course.