In the present paper a simple technique will be developed for the arithmetical determination of certain class group components and class number factors in finite number fields. This technique is based on classical theories (Hilbert's work on inertia groups, the theory of absolutely Abelian fields as class fields of congruence groups, absolute class fields of number fields). In keeping with the traditional approach to the subject we shall use here the language of ideal theory. The only non-classical concepts to be used (which, however, are of fundamental importance) are those of the inertia groups and the congruence groups associated to p-adic fields. We shall also give some illustrations of the use of our technique in some special cases. Further applications will follow in subsequent papers.

The indentation produced by an axially symmetrical punch bearing on the plane surface of an elastic half-space has been considered by Harding and Sneddon [1], who used Hankel transforms and a well-known pair of dual integral equations, and for the case of a spherical punch they took the indenting surface to be part of the approximating paraboloid of revolution. Chong [2], also using these dual integral equations has treated the case of a symmetrical punch of polynomial form and considers a two-termed expansion for a spherical punch. More recently, Payne [3] has given the exact solution for a spherical punch using either oblate spheroidal coordinates or toroidal coordinates.

Let X be a locally compact space, C(X) the algebra (with point-wise operations) of continuous numerical functions on X. On C(X) we introduce the topology of compact convergence. If f ε C(X), Zf denotes the set of zeros of f; and if I is a subset of C(X), we define

It is well known that an indefinite quadratic form with integral coefficients in 5 or more variables always represents zero properly, and this has raised the problem of proving a similar result for forms of higher degree, namely that such a form, of degree r, represents zero properly if the number of variables exceeds some number depending only on r. For a form of odd degree, no condition corresponding to indefiniteness is needed, but for a form of even degree (4 or more) some even stronger condition must be required.

About twenty years ago, in a note of the same title [2], I obtained the following result.

THEOREM 1. Let u and v be relatively prime integers satisfying u> v ≥ 2 and let ε be an arbitrarily small positive number. Suppose the inequality

is satisfied by an infinite sequence of positive integers n1 n2, … Then

J. R. M. Radok [1] has applied complex variable methods to problems of dynamic plane elasticity. The object of this paper is to show that his results may be obtained in a somewhat simpler way by a more systematic use of complex variable analysis.

In fact it is shown that the problems may be reduced to a form similar to that of the static aelotropic plane strain problems considered by Green and Zerna [2].

Let λ be a random variable with the distribution function F(λ). A transform of F which has, in effect, been used in several recent papers ([1], [2], [3], [4]; see also [6]) is

defined formally by the equation

It is the main purpose of this paper to prove the inversion formulae given in the two theorems below.

If p is a prime other than 2, half of the numbers

1, 2, … p—1

are quadratic residues (mod p) and the other half are quadratic non-residues. Various questions have been proposed concerning the distribution of the quadratic residues and non-residues for large p, but as yet only very incomplete answers to these questions are known. Many of the known results are deductions from the inequality

found independently by Pólya and Vinogradov, the symbol being Legendre's symbol of quadratic character.

Let S be an ordered set, i.e. a set with a transitive irreflexive binary relation “<” such that, for any a, bεS, either a = b or a < b or b < a. By an order automorphism of S we mean a one-one mapping α of S onto itself such that

This paper deals with problems of transverse displacements of thin anisotropie plates with the most general type of digonal symmetry [1]. Proofs of uniqueness of solution under certain conditions are given for problems of plates occupying both finite and infinite regions. This is a generalization to anisotropy of the uniqueness theorems given by Tiff en [2] for isotropic plates.

The problem considered here is the determination of the stresses and displacements in a semi-infinite elastic plate which contains a thin notch perpendicular to its edge, and is in a state of plane strain or generalized plane stress under the action of given loads. The axes of x and y are taken along the infinite edge and along the notch, and the scale is chosen so that the depth of the notch is unity (Fig. 1).

Let K be a bounded n-dimensional convex body, with its centroid at the origin o. Let ϑ denote the density of the most economical lattice covering of the whole of space by K (i.e. the lower bound of the asymptotic densities of the coverings of the whole space by a system of bodies congruent and similarly situated to K, their centroids forming the points of a lattice); and let ϑ* denote the density of the most economical covering of the whole space by K (i.e. the lower bound of the asymptotic lower densities of the coverings of the whole space by a system of bodies congruent and similarly situated to K).

Let Q be a local ring and let q be an m-primary ideal of Q, where m is the maximal ideal of Q. With q we may associate a ring F(Q, q), termed the form ring of Q relative to the ideal q. If u1, …, um is a basis of q, and if B denotes the quotient ring Q/q, there is a homomorphism of the ring B[X1, …, Xm] of polynomials over B in indeterminates X1 …, Xm onto F(Q, q). The kernel of this homomorphism is a homogeneous ideal of B[X1 …, Xm]. Finally, if a is an ideal of Q there is a homomorphism of F(Q, q) onto F(Q/a, q+a/a). The kernel of this latter homomorphism will be termed the form ideal relative to q of a and denoted by ā.

Let ƒ = ƒ(x1, …, xk) be a quadratic form in k variables, which has integral coefficients and is not degenerate. Let n ≠ 0 be any integer representable by ƒ, that is, such that the equation

is soluble in integers x1, …, xk. We shall call a solution of (1) a bounded representation of n by ƒ if it satisfies

It is well known that the thinnest covering of the plane by equal circles (of radius 1, say) occurs when the centres of the circles are at the points of an equilateral lattice, i.e. a lattice whose fundamental cell consists of two equilateral triangles. The density of thinnest covering is

Our main object in this note is to establish (Theorem 1) a necessary and sufficient condition to be satisfied by a sequence {εn} so that a series Σ an εnmay be summable | A |whenever the series Σanis summable (C, — 1). We suppose that an and εn are complex numbers. The condition is unchanged if the an are restricted to be real, but our proof is adapted to the case where they may be complex. Theorem 1 has been quoted by Bosanquet and Chow [12] in order to fill a gap in the theory of summability factors. We also obtain some related results, which are discussed in the Appendix.

An expression is found here for the small transverse displacement of a thin elliptic plate due to a force applied at an arbitrary point of the plate. The plate is in the form of a complete ellipse and is clamped along the boundary. The displacement is expressed in terms of infinite series in §§2–4. The convergence of the series is rapid unless the eccentricity of the ellipse is nearly unity. The simplest case in which the force is applied at the centre of the plate is considered in §5; the displacement of the centre due to this force is compared in §6 with the corresponding displacements of a circular plate and of an infinite strip.