Let X denote the general point with coordinates (x1, x2, x3) in 3-dimensional space; and let P(X) be the function defined by

1. Introduction. Necessary and sufficient conditions are established for a real quadratic form to be positive definite on a linear manifold, in a real vector space, explicit in terms of the dual Grassmann coordinates for the manifold.

In this paper we formalize the two-valued propositional calculus with implication and the logical constant 0 (denoting a false statement), as primitives. We can define the logical constant 1 (denoting a true statement), by

and we can define negation by NP = df. CP0.

1. Let f(x1, x2, …, xn) be a homogeneous form with real coefficients in n variables x1, x2, …, xn. Let a1, a2, …, an be n real numbers. Define mf(a1, …, an) to be the lower bound of | f(x1 + a1, …, xn + an) | for integers x1, …, xn. Let mf be the upper bound of mf(a1, …, an) for all choices of a1, …, an. For many forms f it is known that there exist estimates for mf in terms of the invariants alone of f. On the other hand, it follows from a theorem of Macbeath* that no such estimates exist if the region

has a finite volume. However, for such forms there may be simple estimates for mf dependent on the coefficients of f; for example, Chalk has conjectured that:

If f(x,y) is reduced binary cubic form with negative discriminant, then for any real a, b there exist integers x, y such that

1. In this paper we consider the following problems, for continuous linear transformations in the space ω, whose points are arbitrary sequences of complex numbers, with the usual topology:

(a) the establishment of a spectral representation theorem; and

(b) the general description of the spectrum.

Let ξ, η, ζ be linear forms in u, v, w with real coefficients and determinant Δ ≠ 0. Then there exists a number ℳ such that, corresponding to any real numbers a, b, c, there exist rational integers u, v, w for which

1. It is well known that, if two n × n matrices A, B commute, then there is a non-singular matrix P such that P−1AP, P−2BP are both triangular (i.e. have all their subdiagonal elements zero). This result has been generalized, and, in particular, has been shown to hold even if the commutator K = AB − BA is not zero, provided that K is properly nilpotent in the polynomial ring generated by A, B.

It is the purpose of this paper to study the properties of multiplicative systems, for which the associative law is not assumed, when these systems are given in terms of generators and relations. We confine ourselves mainly to loop theory, although the general theory holds also for groupoids, groupoids with division on one side, and quasigroups. Throughout the paper we are guided by two main considerations, to discover how far the concepts and results of group theory carry over to the non-associative case, and to exhibit a specific example of some of the fundamental concepts of abstract algebra. In many ways, in the general theory, we are able to obtain more complete results than in group theory. There remain, however, many interesting analogues of group theoretical concepts. It is hoped to deal with some of these in a later paper.

1. Introductory. The present paper is mainly concerned with some forms for the reciprocal of the square matrix u of which the typical element is

and k is not an integer within the range ± (n − 1) inclusive but is otherwise arbitrary; thus

The problem of evaluating the non-homogeneous critical determinant of the plane region |y − x2| ≤ 1 is solved as a particular case of a result concerning the non-homogeneous properties of convex regions proved in the author's thesis (not yet published). The search for an alternative method of proof led to a proof of the existence of an infinite sequence of minima analogous to the Markoff chain. This is the subject of the present paper.

1. Mazur(1) has shown that any normed algebra A over the real field in which the norm is multiplicative in the sense that

is equivalent (i.e. algebraically isomorphic and isometric under one and the same mapping) to one of the following algebras: (i) the real numbers, (ii) the complex numbers, (iii) the real quaternions, each of these sets being regarded as normed algebras over the real field. Completeness of A is not assumed by Mazur. A relevant discussion is given also in Lorch (2).

The groups of symmetries of three-dimensional lattices have been known for some time. They consist of finite rotation groups, the crystal classes, and infinite discrete motion groups, which include both rotations and translations. The general theory of the corresponding groups in higher dimensional Euclidean spaces has also been developed. This theory includes a demonstration that in Euclidean space of n dimensions the number of motion groups is finite, and leads to a method† for calculating the motion groups, the first step being to determine the crystal classes. The explicit calculation of the various groups by the general method is not simple, and has so far been confined to the case of two and three dimensions. In the special case of the crystal classes in four dimensions, however, we may make use of the results of a paper by Goursat‡. In this paper Goursat sets up a correspondence between the finite rotation groups in four-dimensional Euclidean space, and a set of groups each of which is formed by associating two of Klein's groups of linear non-homogeneous substitutions in one variable. Using this result he is able to evaluate explicitly all the proper and improper finite four-dimensional rotation groups which include the element −I, where I is the four-rowed unit matrix.

The purpose of this note is to show how a considerable part of the local theory of the prime divisors of a field of algebraic functions can be extended to analytically irreducible geometric quotient rings. In doing this we shall make frequent use of Chevalley's paper on local and semi-local rings (2). For brevity this paper will be referred to as ‘L.R.’. We begin the discussion by recalling for the reader's convenience the definition of the Kronecker product of two fields over a common subfield, since this concept will play an important role later.

Zygmund (1) proved in 1936 that if f(x) ∈Lp(−∞, ∞) (1 ≤ p ≤ 2), the Fourier transform

exists for almost every λ, in the sense that

exists for almost every λ. The purpose of this note is to provide a short proof of this theorem.

Suppose that dυ and dυ′ are two volume elements situated at points P and P′ respectively in a three-dimensional right circular cylinder, that y is the distance PP′, that z(y) is a given function of y, and that we wish to evaluate the sixfold integral

taken over all pairs of points P, P′ within the cylinder. We observe that z(y) is a function of y only; so that the sixfold integral can be expressed as a single integral

that is to say a weighted mean of z(y) over the relevant values of y, where the weight function is evidently given by

Introduction. If ξ is a real number we denote by ∥ ξ ∥ the difference between ξ and the nearest integer, i.e.

It is well known (e.g. Koksma (3), I, Satz 4) that if θ1, θ2, …, θn are any real numbers, the inequality

has infinitely many integer solutions q > 0. In particular, if α is any real number, the inequality

has infinitely many solutions.

When two functions are given, each with a finite radius of convergence, a theorem due independently to Hurwitz and Pincherle (1, 2) provides information about the position of the singularities of the function

in terms of the positions of the singularities of f(z) and g(z).

If V is an irreducible variety and W is an irreducible simple subvariety of V, then one of the properties of the quotient ring of W in V is that it is a unique factorization domain. A proof of this theorem has been given by Zariski ((2), Theorem 5, p. 22), based on the structure theorems for complete local rings, and the fact that the local rings which arise geometrically are always analytically unramified. Here the theorem is deduced from certain properties of functions and their divisors which will be established by entirely different considerations. The terminology which will be employed is that proposed by A. Weil in his book(1), and we shall use, for instance, F-viii, Th. 3, Cor. 1, when referring to Corollary 1 of the third theorem in Chapter 8. Before proceeding to details it should be noted that Weil and Zariski differ in then-definitions, and that in particular the terms ‘variety’ and ‘simple point’ do not mean quite the same in the two theories. The effect of this is to make Zariski's result somewhat stronger than Theorem 3 of this paper.

1. In his paper ‘On certain arithmetical functions' Ramanujan (23) considers the function τ(n) defined by the expansion

This function appears in the discussion of an asymptotic formula for the function

and also in Ramanujan's formula for the number of representations of an integer as the sum of 24 squares. It is also of interest as the coefficient in the expansion of g(z), which plays an important part in the theory of modular functions.

1. Introduction. In the course of an important memoir on integral functions of finite order†, G. Valiron discusses ‘fonctions orientées’, that is, functions with zeros an such that arg an tends to a limit as n → ∞. He obtains results that include the following two theorems, in which Vρ(x) denotes a function of the form , where α1, …, αν are positive integers, and n(r) has its usual significance in the theory of integral functions:.