In a recent paper [1], we have given some account of theories of equivalence and intersection on a singular algebraic surface and have shown that such theories share many of the simple properties enjoyed by corresponding theories on non-singular surfaces. Another paper [2], now in preparation, will extend this work to singular varieties of arbitrary dimension. In the meantime, Zobel [3] has drawn attention to some suspect arguments of Samuel [4] concerning the specialization of intersections on a singular variety.

1. Introduction. Let I0 be a closed rectangle in Euclidean n-space, and let ℬ be the field of Borel subsets of I0. Let ℱ be the space of completely additive set functions F, having a finite real value F(E) for each E of ℬ, and left undefined for sets E not in ℬ. In recent work, we used Hausdorff measures in an attempt to analyze the set functions F of ℱ. If h(t) is a monotonic increasing continuous function of t with h(0) = 0, a measure h-m(E) is generated by the method first defined by Hausdorff [2].

The main object of this paper is to give a self-contained elementary proof of a result (Theorem 1, below), which could be deduced from a theorem of Siegel ([1], Satz 2). It seems worth while to do so, because Siegel's proof is long and difficult, though his result is deeper and more precise than mine.

A method is developed for finding the distribution of velocity, density, pressure and magnetic field behind an expanding strong cylindrical shock wave in an infinitely conducting fluid in the presence of a magnetic field.

In the flow of such a fluid there are two distinct methods for producing a strong shock:

(i) by imposing the usual density ratio across the shock, as in the non-magnetic cnse, and

(ii) by imposing a large magnetic field, such that the Alfvén velocity is very much larger than the speed of sound. The distribution of the various physical quantities and the velocity of propagation of the shock are discussed for both cases, and numerical results given.

The purpose of this paper is to call attention to a very simple example of intersections on a singular variety, and to its effect on intersection theories relating to ambient varieties with singularities; in particular it will be shown how the example—to which we refer throughout as Example A—invalidates an aspect of the theory put forward by P. Samuel ([1], Ch. VI).

Let p be a prime and let F be a polynomial in one variable with coefficients in GF(p), the field of p elements. Let d be the degree of F, and let r+1 denote the number of distinct values F(µ) as µ. ranges over GF(p). A generalization of the Waring problem modulo p leads to the question the determination of a lower bound for r.

Let S be a point set of the Euclidean plane, such that

(i) S is bounded,

(ii) the closure of S has unit Lebesgue measure.

Let P be an arbitrary set of n points contained in S, and let l(P) denote the total length of the shortest system of lines connecting the points of P together. Define ln to be the supremum of l(P), taken over all sets P of n points in S. Beardwood, Halton, and Hammersley [1[ proved that there exists an absolute constant α, independent of S, such that

In an earlier paper [2] by the present writer, a solution of the equations of elasticity in complete aeolotropy was found under the assumption that the stresses and therefore the strains are linear in the third cartesian coordinate z. In the present paper, the solution is extended to the case where the stresses and therefore the strains are polynomials in z. This provides a wider scope of applications to the problem of elastic equilibrium of a completely aeolotropic cylinder under resultant end forces and couples and a general distribution of tractions on the lateral surface of the cylinder. These tractions may take any form consistent with the elastic equilibrium of the cylinder, provided they are polynomials of any degree in z. Of the many applications of the present theory, Luxenberg [1] considered the very particular case of torsion of a cylinder made up of a material having a plane of elastic symmetry, under a constant lateral loading.

In previous papers [1, 2] the author has considered the stability of a current-vortex sheet in a non-diffusing incompressible fluid, the magnetic field being parallel to the plane of discontinuity. In [1] a criterion was given for a parallel magnetic field to stabilise a vortex sheet, and in [2] the energy balance of this system was considered and it was shown how the magnetic energy level is increased at the expense of mechanical kinetic energy when the system becomes unstable. In this paper the oscillations on a plane interface are considered when the magnetic field is not parallel to the interface.

For any (real or complex) transcendental number ξ and any integer n > 0 let ϑn(ξ) be the least upper bound of the set of all positive numbers σ for which there exist infinitely many polynomials p1(x), p2(x), … of degree n, with integer coefficients, satisfying

where ‖pi‖ denotes the “height” of pi(x), i.e. the maximum modulus of the coefficients. Plainly ϑn(ξ) serves as a measure of how well (or how badly) the number zero can be approximated by values of nth degree integral polynomials at the point ξ. It can be shown by means of the “Schubfachprinzip” that, at worst,

if the transcendental number ξ is real, and

if it is complex, i.e.ϑn(ξ) is never smaller than these bounds. Furthermore, a conjecture of K. Mahler may be interpreted as stating that for almost all real and for almost all complex numbers the equations (2) and (3), respectively, are actually true; in other words, almost all transcendental numbers have the worst possible approximation property for any degree n.

Let D(f) denote the discriminant of a binary cubic form

having integral coefficients. We restrict ourselves to forms with D(f) ≠ 0, and further (as a matter of convenience) to forms which are irreducible in the rational field. It is a problem of some interest, in connection with the approximation properties of transcendental numbers, to estimate the sum

as H→∞, where ‖f‖ = max(|a|, |b|, |c|, |d|). Since |D(f)| ≪ H4 when ‖f‖ < H, there is the trivial lower bound

In ring theory there is the following theorem (cf. [1], p. 39):

If R is a ring satisfying the descending chain condition for left ideals, then the following three conditions are equivalent: (i) R is primitive, (ii) R is simple, and (iii) R is isomorphic to the complete ring of linear transformations in a finite dimensional vector space over a division ring.

Let be a plane bounded convex set whose width in the direction φ is . It has been shown by L. A. Santaló that

is invariant under unimodular affine transformations. Santaló [1] established a number of properties of this invariant and conjectured that if the area of is A() then and that equality characterizes triangles. In this note Santaló's conjecture is shown to be true.

An expression is found for the biharmonic Green's function, G(x, y; x1, y1), for an infinite area in the x, y-plane bounded internally by a single curve; at all points of the boundary G has a zero of the second order.

A solution is obtained by real variable methods for a Cauchy problem for the generalised radially symmetric wave equation. A solution of this problem has been given by Mackie [1] employing the contour integral methods developed by Copson [2] and Mackie [3] for a class of problems occurring in gas dynamics. The present approach employs a simple definite integral representation for the solution and reduces the problem to solving an Abel integral equation. The real variable approach avoids the unnecessary restriction of the initial data to be analytic and also avoids the difficulty encountered in the complex variable approach in continuing the solution across a characteristic. The solution is in fact obtained in a form valid everywhere in the region of interest.

In an earlier paper (cf. [1]) I had given a generalization of the concept of an absolute discriminant to arbitrary finite number fields K as base fields. In a second paper (cf. [2] 2. 3, see also [3] 1. 3) it was shown that the discriminant δ(Λ/K) of a finite extension Λ of K determines the structure of the ring ς of algebraic integers in Λ qua module over the ring ο of algebraic integers in K. The purpose of the present note is to establish a corresponding result for an arbitrary Dedekind domain ο, and finite separable extensions Λ of its quotient field K. The general theory of discriminants and module invariants developed in [1] and [2] for algebraic integers applies in principle to arbitrary Dedekind domains, as already pointed out in the earlier papers. It is usually evident what further hypotheses—if any—have to be imposed to ensure the validity of any particular theorem. For the quoted result of [2] this is, however, not at all clear. In fact the proof involves the proposition:

I. If an element of K is a square everywhere locally then it is a square in K.

In a recent paper (Shail [1]) the present author considered the problem of finding a two-centred expansion of the retarded Helmholtz Green's function This work formed an extension to that of Carlson and Rushbrooke [2] (also Buehler and Hirschfelder [3]) on the Coulomb Green's function and arose out of considerations of the interaction energy of two charge distributions taking account of electromagnetic retardation. The two-centred expansion obtained in [1] took the form of a double Taylor series, each term being interpreted as a Maxwell multipole—multipole interaction energy between two “charge” distributions coupled through a retarded scalar field. The first few terms in the expansion were also given in spherical polar coordinates.

It is shown that the greatest value of the resultant shear in the Saint-Venant torsion problem for an aeolotropic material possessing digonal elastic symmetry occurs on the boundary of the cross-section.