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

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.

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].

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].

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

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.

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 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

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.

Several writers (4), (6), (7), (9) have used orthogonal expansions in discussing properties of Fourier transformations, and Kober (3) has used such expansions to derive fractional Fourier and Hankel transformations. In 1950 Barrucand (1) noted a reciprocity holding between the coefficients in the expansions in Laguerre polynomials of pairs of functions which are transforms with respect to the kernel J0(2x½).

Let R be a ring and let Rn be the complete ring of n × n matrices with coefficients from R.

The development of the theory of local rings has been greatly stimulated by the importance of the applications to algebraic geometry, but it is none the less true that this stimulus has produced a theory which, on aesthetic grounds, is somewhat unsatisfactory. In the first place, if a local ring Q arises in the ordinary way from a geometric problem, then Qwill have the same characteristic as its residue field. It is partly for this reason that our knowledge of equicharacteristic local rings is much more extensive than it is of those local rings which present the case of unequal characteristics. Again, in the geometric case, the integral closure of Q in its quotient field will be a finite Q-module. Here, once more, we have a special situation which it would be desirable to abandon from the point of view of a general abstract theory.

The basic formula to be proved is

where p≧q + 1, z ≠0; | amp z | < π, R(n)>0, r = 1, 2,…,p. For other values of pand qthe result holds if the integral converges. From this formula some results, involving Bessel functions and Confluent Hypergeometric functions, will be deduced.

In this paper we prove a theorem in Operational Calculus and use it to evaluate a few infinite integrals involving Legendre, Bessel and E-functions. We write

when

and

when

(2) is a generalisation of (1) as given by Meijer [2] and it reduces to (1) when v = ±½ by virtue of the relation

The object of this paper is to evaluate a few infinite integrals involving E-functions by applying the Parseval-Goldstein [1] theorem of Operational Calculus; that, if

and

then

when the integrals are convergent.

The first formula to be proved is

where R(z)>0.

The main result in this paper, contained in Theorem 1, is a generalisation of the inequality of the arithmetic-geometric means. A result of a similar character has been proved by Siegel (2). The present result gives an improvement in the inequality in the case when the variables involved are not all distinct, whereas Siegel's result does not. The theorem is used in § 3 to obtain a result in connection with totally real and positive algebraic integers.