Martin(1) has shown that if the pressure p and the Lagrange variable ψ (ψ = constant on a trajectory) are taken as independent variables, then problems in one-dimensional unsteady flow of an ideal gas reduce to the question of finding solutions of the Monge-Ampère equation

where τ = 1/ρ, ρ being the density, and S = S(ψ) is the specific entropy. Thus the right-hand term is a function of p and ψ depending upon the equation of state and the function S(ψ).

A graph G is a pair (V{G), E(G)); V(G) is the set of vertices of G, and E(G) the set of edges of G; E(G) is a subset of the Cartesian product V(G) × V(G) (all pairs (υi, υj) where υi∈ V(G), υj ∈ V(G)). If υ is a vertex of G we shall write υ ∈ G. The graph-theoretic terminology used in this paper is that of Berge (1). We shall consider only graphs which are finite, undirected, and without loops.

The compact–open topology for function spaces is usually attributed to R. H. Fox in 1945 [16]; and indeed, there is no earlier publication to attribute it to. But it is clear from Fox's paper that the idea of the compact–open topology, and its notable success in locally compact spaces, were already familiar. The topology of course goes back to Riemann; and to generalize to locally compact spaces needs only a definition or two. The actual contributions of Fox were (1) to formulate the partial result, and the problem of extending it, clearly and categorically; (2) to show that in separable metric spaces there is no extension beyond locally compact spaces; (3) to anticipate, partially and somewhat awkwardly, the idea of changing the category so as to save the functorial equation. (Scholarly reservations: Fox attributes the question to Hurewicz, and doubtless Hurewicz had a share in (1). As for (2), when Fox's paper was published R. Arens was completing a dissertation which gave a more general result [1] – though worse formulated.)

The magnetic susceptibility of antimony both parallel and perpendicular to the trigonal axis is independent of field down to 4° K. The numerical value of the susceptibility parallel to the trigonal axis decreases with increasing temperature, similarly to that of bismuth, but perpendicular to the trigonal axis there is no temperature dependence. The results at higher temperatures are compared with earlier measurements and the comparison suggests that the susceptibility of antimony, like that of bismuth, is very sensitive to addition of foreign elements.

1. In this paper, a continuation of an earlier paper(1), we consider the two-dimensional motion of incompressible viscous liquid past a projection, the motion being one of uniform shear apart from the disturbance caused by the projection. A special form is assumed for the boundary, so that the area in the z-plane (Fig. 1) can be represented conformally on a circle in the ζ-plane by a rational function of ζ; this function contains a parameter a (0 < a ≤ 1), and by varying a the shape of the projection can be varied. Since a rational function is concerned in the conformal transformation a method lately developed by N. Muschelišvili(2) can be used in solving the biharmonic equation for the stream function, though the method actually used differs in some points of detail from that originally proposed by Muschelišvili and appears to be somewhat simpler.

The main purpose is to characterise continuous maps that are n-branched coverings in terms of induced maps on the rings of functions. The special properties of Frobenius n-homomorphisms between two function spaces that correspond to n-branched coverings are determined completely. Several equivalent definitions of a Frobenius n-homomorphism are compared and some of their properties are proved. An axiomatic treatment of n-transfers is given in general and properties of n-branched coverings are studied and compared with those of regular coverings.

Let S be a smooth hypersurface in ℝn with surface area measure ds and Gaussian curvature κ(s). Define the convolution operator T by

formula here

for suitable functions f on ℝn. We are interested in the Lp − Lq mapping properties of T. Write [Sscr ] for the type set of T, the set

formula here

It is well known (see, e.g. [O1]) that [Sscr ] is contained in the closed triangle [Tscr ] with vertices (0, 0), (1, 1) and (n/(n+1), 1/(n+1)). This paper is concerned with estimates of the form

formula here

The estimate (1) is interesting because it serves as a weak substitute for the L(n+1)/n − Ln+1 boundedness of T. For example, if S is compact and (1) holds, then well-known arguments show that [Sscr ] differs from the full triangle [Tscr ] by at most the point (n/(n + 1), 1/(n + 1)). Our main result is a condition sufficient to imply (1). Its statement requires the following definition.

The object of the paper was to find the forces acting in bubble rafts and to use them for a description of the mechanical properties of these rafts. The first section is based entirely on Nicolson's work and gives expressions for the cohesive forces in the lattice. The second and third sections deal with the repulsive forces arising when a bubble is compressed uniaxially, and are really the basis of the paper. The fourth section introduces the idea of ‘interaction’ between compressions in the three close-packed directions—a conception which explains the breakdown of Cauchy's relations. The conditions of equilibrium in the lattice are investigated in § 5, and in § 6 all these forces are combined to give a complete account of the elastic properties of the rafts.

The remaining section gives a brief account of the experimental checks of the theory; it appears to be a reasonable approximation for small bubbles, and, indeed, we could certainly not expect it to apply to large ones, since for them not only are all the calculations of the attractive potentials invalid because of the large local slopes of the surface, but also the ‘flats’ and ‘caps’ become so large as to interfere with one another.

The forces are considered well enough supported to justify their further use in an investigation of the plastic behaviour of the bubble rafts—an investigation which has explained several interesting experimental facts but brought us little nearer an understanding of the low yield strains of metallic single crystals (see (2)).

In important work some thirty years ago, G. Birkhoff[2, 3] and E. Hopf [16, 17] showed that large classes of positive linear operators behave like contraction mappings with respect to certain ‘almost’ metrics. Hopf worked in a space of measurable functions and took as his ‘almost’ metric the oscillation ω(y/x) of functions y and x with x(t) > 0 almost everywhere, defined by

For 0 < α < 1, let for 0 ≤ x < 1, where is the sequence of Rademacher functions. We give a class of fα so that their graphs have Hausdorff dimension 2 − α. The result is closely related to the corresponding unsolved question for the Weierstrass functions.

In a recent paper (1) the authors considered some generalizations of Cauchy's inequality. The method of approach was by the construction of certain convex functions (where by a convex function we mean a function f(x) satisfying

for every pair of unequal values x1 and x2). For example, it was shown that if (a), (b) are the sets of non-negative real numbers a1, …,an; bl, …, bn, the function

which had been used earlier by Callebaut(2), is a convex function with minimum at x = 0, except if the sets (a) and (b) are proportional, in which case the function is a constant. This function takes the value (σab)2 when x = 0 and (σa2) (Σb2) when x = 1, and the property of convexity gives Cauchy's inequality.

A discussion is given of the convergence of the eigenvalues Λ (N) of two-dimensional finite-difference equations towards the eigenvalues Λ of the corresponding second-order differential equation, and it is shown that

where h = N−1 and ν2 is a constant. As in our previous paper (4), this can be used to make an accurate estimate of λ by extrapolating to h = 0. After an account of the relaxation method used for computing Λ(N) and a discussion of the residual vector, results are presented for an approximation to the lowest spatially symmetric and antisymmetric states of two electrons in a sphere, interacting through their Coulomb potential.

This paper represents a new method for obtaining an asymptotic representation for integrals of the form when p is large. It is shown that if f(x) satisfies certain conditions this representation is also convergent. Numerical calculations seem to show that the first term of the representation gives a close approximation to the value of the integral for a wide range of values of p.

1.1. There are two familiar methods of summation of divergent series usually called the methods (R, 1) and (R, 2). If sn = u0 + u1 + … + un and, as it will be convenient to suppose throughout, u0 = 0, then

when h → + 0§: the convergence of the series for small positive h is presupposed.

In this paper we are concerned with the two-dimensional, unsteady flow of an inviscid, polytropic gas whose adiabatic index γ lies between 1 and 3. We recall that comparatively early in the study of gas dynamics we encounter two exact solutions of gas dynamic problems. One, in one-dimensional unsteady flow, is the expansion of a semi-infinite column of gas which is initially at rest behind a piston which, at time t = 0, begins to move with constant speed away from the gas. The second, in two-dimensional, steady, supersonic flow, is the Prandtl–Meyer flow round a sharp convex corner. Both of those flows may be regarded as special cases of more general exact solutions which are obtained by the method of characteristics (see, for example, Courant and Friedrichs(1)). On the other hand, each may be obtained directly from the appropriate equations by making use of the fact that, in so far as neither problem contains any characteristic length parameter in its formulation, the principle of dynamic similarity can be used to reduce the system of partial differential equations to one of ordinary differential equations. In the first case the independent variables x and t occur only in the combination x/t and in the second the independent variables x and y occur only in the combination x/y. Interesting and instructive as the derivation of these solutions from such principles may be, it is an unfortunate fact that they are the only non-trivial solutions of the respective equations. This is not altogether surprising as the equations are ordinary with (in this case) a limited number of non-trivially distinct solutions.

1. General remarks. The theorem that has come to be known as Cochran's theorem in works on mathematical statistics was published in these Proceedings in 1934(1). If x1, …, xn are independently distributed standard normal deviates, and q1, …, qk are k real quadratic forms in the xi with ranks n1, …, nk respectively, and such that then Cochran's Theorem II states that a necessary and sufficient condition that q1 …, qk are independently distributed in χ2 forms with n1, …, nk degrees of freedom is that Σnj = n. The necessity of the condition is obvious. Cochxan proves its sufficiency by expressing each qj as a sum, involving nj squares of real linear forms in the xi; it follows easily that the coefficients ci are in fact + 1, and that the transformation is orthogonal. The theorem then follows immediately from the properties of orthogonal transformations in relation to independent normal deviates.

We study generalized symbolic powers and form ideals of powers and compare their growth with the growth of ordinary powers, and we discuss the question of when the graded rings attached to symbolic powers or to form ideals of powers are finitely generated.

The general linear difference-differential equation takes the form

where x is a real variable, ν(x) and Aμν(x) are known functions and

Mr Denis Lichtman has kindly pointed out to us that an added hypothesis is needed in Theorem (4·1). The revised theorem follows.