In the equation

dis any positive integer which is not a perfect square. For convenience we shall consider only those solutions of (1) for which x and yare both positive. All the others can be obtained from these. In fact, it is well known that if (x0, y0) is the minimum positive integer solution of (1), then all integer solutions (x, y) are given by

and, in particular, all positive integer solutions are given by

Marshall Hall has proved that every real number is representable as the sum of two continued fractions with partial quotients at most 4. This implies that for any real β1, β2 there exists a real α such that

for all integers x > 0 and y, where C is a positive constant. In this note I prove a generalization to r numbers β2, …, βr. The case r = 2 implies a result similar to Marshall Hall's but with a larger number (71) in place of 4.

Let (x1, y1), …, (xN, yN) be N points in the square 0 ≤ x < 1, 0 ≤ y < 1. For any point (ξ, η) in this square, let S(ξ, η) denote the number of points of the set satisfying

A complex-valued function ƒ is said by W. Maak [1] to be almost periodic (a.p.) on Rn if for every positive number ε there is a decomposition of Rn into a finite number of sets S such that

for all h in Rn and all pairs x, y belonging to the same S. This definition is equivalent to that of Bohr when ƒ is continuous.

In what follows, groups are written additively, and commutators are denoted by brackets:

A group is metabelian if it satisfies the law

It is conceivable, though not plausible, that this law is equivalent to a law, or a set of laws, in only three variables, or even two. The present note shows that this is not the case.

The recurrence formulae for the Bessel, Legendre, hypergeometric and other such functions can all be related to each other by means of the E-functions. In this paper it will be shown how, starting from known recurrence formulae for the hypergeometric function, others can be derived. The E-function formulae are deduced in § 2, and the others in § 3.

It has long been conjectured that any indefinite quadratic form, with real coefficients, in 5 or more variables assumes values arbitrarily near to 0 for suitable integral values of the variables, not all 0. The basis for this conjecture is the fact, proved by Meyer in 1883, that any such form with rational coefficients actually represents 0.

In a recent paper [1] we showed that there is a (1,) -correspondence between the homomorphisms of an inverse semigroup S and its normal subsemigroups. The normal subsemigroup of S corresponding to and determining the homomorphism μ of S is the inverse image under μ of the set of idempotents of Sμ and is called the kernel of the homomorphism μ. The inverse image of each idempotent of Sμ is itself an inverse semigroup [1], and each such inverse semigroup is said to be a component of the normal subsemigroup determined by μ.

Following, for example, Kurošs [8], we define the (transfinite) upper central series of a group G to be the series

such that Zα + 1/Za is the centre of G/Zα, and if β is a limit ordinal, then If α is the least ordinal for which Zα =Zα+1=…, then we say that the upper central series has length α, and that Zα= His the hypercentre of G. As usual, we call G nilpotent if Zn= Gfor some finite n.

In the following pages there will be found an account of the properties of a certain class of local rings which are here termed semi-regular local rings. As this name will suggest, these rings share many properties in common with the more familiar regular local rings, but they form a larger class and the characteristic properties are preserved under a greater variety of transformations. The first occasion on which these rings were studied by the author was in connection with a problem concerning the irreducibility of certain ideals, but about the same time they were investigated in much greater detail by Rees [7] and in quite a different connection. In his discussion, Rees made considerable use of the ideas and techniques of homological algebra. Here a number of the same results, as well as some additional ones, are established by quite different methods. The essential tools used on this occasion are the results obtained by Lech [3] in his important researches concerning the associativity formula for multiplicities. Before describing these, we shall first introduce some notation which will be used consistently throughout the rest of the paper.

In an earlier paper [1] on groups which are the products of two finite cyclic groups with trivial intersection, certain permutations, called “semi-special”, played a certain role. The permutation π of the numbers 1, 2,…, n is semi-special if† πn=n, and if, for every y ε [n],

is again a permutation, namely a power (depending on y) of π.

The problem of a concentrated normal force at any point of a thin clamped circular plate was treated in terms of infinite series by Clebsch [1], who gave the general solution of the biharmonic equation D∇4w = p. Using the method of inversion Michell [2] found a solution for the same problem in finite terms. The method of complex potentials was used by Dawoud [3] to solve the problem of an isolated load on a circular plate under certain boundary conditions. Applying Muskhelishvili's method Washizu [4] obtained the same results for clamped and hinged boundaries. The complex variable method was applied by the authors [5] to obtain solutions for a thin circular plate having an eccentric circular patch symmetrically loaded with respect to its centre under a particular form of boundary condition defining certain types of boundary constraints which include the usual clamped and hinged boundaries as well as other special cases. Flügge [6] gave the solution for a linearly varying load over the complete simply supported circular plate. Using complex variable methods Bassali [7] found the solution for the same load distributed over the area of an eccentric circle under the boundary conditions mentioned before [5], and the authors [8] obtained the solutions for general loads of the type cos nϑ(or sinnϑ), spread over the area of a circle concentric with the plate. In this paper the solutions for a circular plate subjected to the same boundary conditions are obtained when the plate is acted upon by the following types of loading: (a) a concentrated load at an arbitrary point; (b) a line load spread on any part of a diameter; (c) a load distributed over the area of a sector of the plate; (d) a concentrated couple at an arbitrary point of the plate. As a limiting case we find the deflexion at any point of a thin elastic plate having the form of a half plane clamped along the straight edge and subject to an isolated couple at any point.

In 1955 a programme of study of the first 10000 zeros of the Riemann Zeta-function

was completed. Use was made of the high-speed digital computer SWAC and a report of this programme has appeared recently [1]. More recently still, the programme has been extended to the first 25000 zeros. All these zeros have σ= ½ The purpose of this paper is to summarize the methods needed for this (and possibly future) work from the highspeed computer point of view.

The closed graph theorem is one of the deeper results in the theory of Banach spaces and one of the richest in its applications to functional analysis. This note contains an extension of the theorem to certain classes of topological vector spaces. For the most part, we use the terminology and notation of N. Bourbaki [1], contracting “locally convex topological vector space over the real or complex field” to “convex space”; here we confine ourselves to convex spaces.

Let

denote an indefinite quadratic form in n variables with real coefficients and with determinant Δn≠0. Blaney ([1], Theorem 2) proved that for any γ ≥0 there is a number Γ = Γ(γ, n) such that the inequalities

are soluble in integers x1, …, xn for any real α1, …, αn The object of this note is to establish an estimate for Γ as a function of γ. The result obtained, which is naturally only significant if γ is large, is as follows.

Let Λ be a lattice in three-dimensional space with the property that the spheres of radius 1 centred at the points of Λ cover the whole of space. In other words, every point of space is at a distance not more than 1 from some point of Λ. It was proved by Bambah that then

equality occurring if and only if Λ is a body-centred cubic lattice with the side of the cube equal to 4/√5. Another way of stating the result is to say that the least density of covering of three-dimensional space by equal spheres, subject to the condition that the centres of the spheres form a lattice, is . Another proof of Bambah's result was given recently by Barnes. Both proofs depend on the theory of reduction of ternary quadratic forms.