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.

Let A be an arbitrary set of positive integers (finite or infinite) other than the empty set or the set consisting of the single element unity. Let p(n) = pA(n) denote the number of partitions of the integer n into parts taken from the set A, repetitions being allowed. In other words, p(n) is the number of ways n can be expressed in the form n1a1 + n2a2 + …, where a1, a2, … are the distinct elements of A and n1, n2, … are arbitrary non-negative integers. In this paper we shall prove that p(n) is a strictly increasing function of n for sufficiently large n if and only if A has the following property (which we shall subsequently call property P1): A contains more than one element, and if we remove any single element from A, the remaining elements have greatest common divisor unity.

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.

Let be a positive definite quadratic form in n variables. Here x denotes a column vector with components x1, x2, …, xn, and we write A for the symmetric matrix {ajk} so that

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.

The following well-known conjecture is generally attributed to Minkowski:

Let L1, …, Ln be n real homogeneous linear forms of determinant Δ ≠ 0 in the n variables x1, …, xn; and let (x1′, …, xn′) be any point. Then there exists a point (x1, xn) congruent to (x1′, …, xn′) (mod 1) at which

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.

In [3] Pontrjagin proved the following form of the Alexander duality theorem:

Theorem A. Let K be a sub-polyhedron of the n–dimensional sphere, Sn. Let G, G* be orthogonal topological groups, G being compact. Then Hr(K; G) and Hn–r–1(Sn–K; G*) are orthogonal with the product of αεHr(K; G) and αεHn−r−1(SnK; G) determined as the linking coefficientof some cycle of class a with some cycle of class α

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.

If a group G is presented in terms of generators and relations, then the classical Reidemeister-Schreier Theorem [1] gives a presentation for any subgroup of G. If G is a free product of groups Gα each of which is presented in terms of generators and relations, then the main result of this paper is a presentation for any subgroup H of G, which shows the nature of H as a free product of certain subgroups of G. This result is a generalization of the celebrated Kuroš Theorem [2]. It also includes the Reidemeister–Schreier Theorem and the Schreier Theorem [1] which states that any subgroup of a free group is free.

Although there is an extensive literature dealing with the location of characteristic roots of matrices, the problem of estimating the maximum distance between two characteristic roots of a given matrix does not appear to have attracted much attention. In the present note we shall be concerned with this problem.

Various attempts have been made to identify the slip lines or Lüder lines which are observed in solids with surfaces of discontinuity or characteristic surfaces associated with solutions of equations of plasticity. Results such as those obtained in [1], together with the observed fact that such lines occur in a variety of types of experiments, indicate that, for two well-known theories of plasticity, characteristic surfaces fail to exist in situations in which such lines are observed. This can come about in two ways, one being that real characteristic directions do not exist, the other being that they do, but that the characteristic surface elements do not unite to form surfaces. The latter situation seems to arise from the fact that, even in truly three-dimensional problems, the equations considered admit only a finite number of characteristic directions. Results such as those given in [2] indicate that, if real characteristic directions do not always exist, there is some doubt as to whether one can identify such lines with surfaces of discontinuity. Another point to be considered is the ease with which solutions may be obtained. For equations possessing real characteristics, the method of characteristics is a powerful tool to use in solving two-dimensional problems. It is noted in [3], Ch. X, that, in axially symmetric problems, one cannot use this method to obtain solutions of the von Mises equations. In treating such problems, it may be easier to use equations which appear to be more complicated, but which possess real characteristics. These facts suggest that plasticity equations which always possess real characteristic directions are to be preferred to those which do not. Some workers in plasticity appreciate this, as is indicated by remarks made in [4]. However, no one has taken a rather general theory of plasticity, such as the theory of perfectly plastic solids, and attempted to determine which of the equations included in it have this property. We made an unsuccessful attempt to do so for a theory which is roughly equivalent. The purpose of this paper is to present this theory, to indicate the basic mathematical problem involved, and to record a partial solution of it.

The theory of two-dimensional anisotropic dielectrics is developed using complex variable methods, and the problems of an elliptic cylinder in a uniform electric field and of a line charge before a dielectric plane and circular cylinder are then discussed. The method is believed to be more general than that given by Netushil [1].

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

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.

In this paper we shall set out the generalization, for n-dimensional space Sn, of some recent results about complete quadrics and complete collineations in S2, S3 and S4. For the results about complete conies in S2, originally introduced by Study [1], we refer the reader to papers by Severi ([2], [3]), van der Waerden [4], Semple [5]; for those about complete quadrics in S3, to Semple ([6], [7]); for the extension to S4 to Alguneid [8]; for the general concept of complete collineations in Sn, and for results in S2 and S3, to Semple [9].

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.

It is well known that the star domain

in the plane is boundedly reducible, that is, it contains a bounded star body with the same lattice determinant, namely √5. Hence the bounded star domain

has the same lattice determinant as K has if r is sufficiently large. The following result is therefore perhaps a little surprising.

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.