The object of this paper is to show how a well-known method of demonstrating the independence of one member of a system of axioms can be incorporated into a systematic method of testing for independence, suitable for solution on a logical computer of the type now being built at the University of Nottingham.

Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, and

where γ is Euler's constant. Then P(x) is the error term in the problem of the lattice points of the circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of the rectangular hyperbola.

The purpose of this paper is twofold. In order to state our first aim, let U denote the ‘infinite’ unitary group, and let BU be a classifying space for U. Then Bott (2),(3) has shown that we propose to investigate the ‘Postnikov system’ of BU.

Let Q be a local domain of dimension d with maximal ideal m and let q be an m-primary ideal. Then we define the degree function dq(x) to be the multiplicity of the ideal , where x; is a non-zero element of m. The degree function was introduced by Samuel (5) in the case where q = m. The function dq(x) satisfies the simple identity

The main purpose of this paper is to obtain a formula

where vi(x) denotes a discrete valuation centred on m (i.e. vi(x) ≥ 0 if x ∈ Q, vi(x) > 0 if x ∈ m) of the field of fractions K of Q. The valuations vi(x) are assumed to have the further property that their residue fields Ki have transcendence degree d − 1 over k = Q/m. The symbol di(q) denotes a non-negative integer associated with vi(x) and q which for fixed q is zero for all save a finite set of valuations vi(x).

Let K be a finite algebraic number field, of degree R. Then those integers of K which may be expressed as a sum of dth powers generate a subring JK, d of the integers of K (JK, d need not be an ideal of K, as the simplest example K = Q(i), d = 2 shows. JK, d is an order, it in fact contains all integer multiples of d!; it also contains all rational integers). Siegel(12) showed that every sufficiently large totally positive integer of JK, d is the sum of at most (2d−1 + R) Rd totally positive dth powers; and he conjectured that the number of dth powers necessary should be independent of the field K—for instance, he had proved(11) that five squares are enough for every K. In this paper, we will show that, as far as the analytic part of the argument is concerned, Siegel's conjecture is correct. I have not been able to deal properly with the problem of proving that the singular series is positive; but since Siegel wrote, a good deal of extra information about singular series has been obtained, in particular by Stemmler(14) and Gray(4). The most spectacular consequence of all this is that if p is prime, then every large enough totally positive integer of JK, p is a sum of (2p + 1) totally positive pth. powers.

In (10), (11) Wall determined the structure of the cobordism ring introduced by Thom in (9). Among Wall's results is a certain exact sequence relating the oriented and unoriented cobordism groups. There is also another exact sequence, due to Rohlin(5), (6) and Dold(3) which is closely connected with that of Wall. These exact sequences are established by ad hoc methods. The purpose of this paper is to show that both these sequences are ‘cohomology-type’ exact sequences arising in the well-known way from mappings into a universal space. The appropriate ‘cohomology’ theory is constructed by taking as universal space the Thom complex MSO(n), for n large. This gives rise to (oriented) cobordism groups MSO*(X) of a space X.

Recently Cunningham and Valentine gave in (3) an axiomatic description of the one-dimensional real affine space in terms of its order structure and the (abstract) group of affine transformations It is the purpose of the present note to show that the system of axioms in (3) (cf. (L. 1)–(L. 5) of this note) leads in a natural way to a model of the real number field. Our method is suggested by a result of Hall ((4), p. 382), namely, that an infinite doubly transitive Frobenius group is isomorphic to the group of affine transformations in a near-field, provided that there is at most one transformation displacing all points and taking a given point a into a given point b. The salient point of our investigation is the redundancy of the latter condition in the case where the underlying space is endowed with a certain linear order structure which is invariant under the transformations of the given group.

In two papers, (5) and (6), D. G. Northcott and the author considered the notion of the reductions of an ideal a of a Noether ring A. A reduction of a is an ideal b contained in a which satisfies ar+1 = arb for all sufficiently large r. This notion was inspired by the following elementary property of a reduction. Suppose that A is a local ring with maximal ideal m, and that a is m-primary. It is well known (Samuel (10)) that the length of the ideal an is, for large values of n equal to Pa(n) where Pa(n) is a polynomial in n whose degree d is equal to the dimension of A. If we write the coefficient of nd in Pa(n) in the form e(a)/d!, e(a) is a positive integer termed the multiplicity of a. If now b is a reduction of a, then b is also m-primary, and e(b) = e(a).

Tensor operators under any group are defined and the theory is developed for semi-simple continuous groups. Coupled tensor operators are introduced and the matrix elements of tensor operators are expressed in terms of the coupling coefficients. The structure of generalized Casimir operators is investigated.

In a previous paper (2) it was shown how the work of finding the units of a totally real cubic field could be facilitated by consideration of the sum of squares of differences between a number and its conjugates. In the present paper it is shown that the same ideas can be helpful in the calculation of class-numbers, and a list of the fields with class-number greater than unity and discriminant less than 20,000 is given.

In any valuated ring R, more specifically, a ring with a degree-function d(a) as defined in §2, the notion of right R-dependence may be denned as follows: (i) The elements a1, …, ar of R are right R-dependent if there exist b1, …, br ∈ R such that . (ii) An element a of R is said to be right R-dependent on the elements a1, …, ar of R if a = 0 or if there exist c1, …, cr ∈ R such that .

Ω will denote the space of all plane paths ω, so that ω is a short way of denoting the curve . we assume that there is a probability measure μ defined on a Borel field of (measurable) subsets of Ω, so that the system (Ω, , μ) forms a mathematical model for Brownian paths in the plane. [For details of the definition of μ, see for example (9).] let L(a, b; μ) be the plane set of points z(t, ω) for a≤t≤b. Then with probability 1, L(a, b; μ) is a continuous curve in the plane. The object of the present note is to consider the measure of this point set L(a, b; ω).

The commutators of the infinitesimal operators of the n-dimensional rotation group Rn with vector operators under Rn are expressed in a vectorial notation. The infinitesimal operators for the representations (l 0…0) are treated in detail. Shift operators for l are constructed and are used to derive the branching rule for these representations. The energy levels and degeneracy of bound states of a particle under an inverse square force in n dimensions are found by wave mechanics and by expressing the Hamiltonian in terms of Casimir's operator for Rn+1. Differential operators which transform one radial wave function into another are obtained.

The theorems of Bott (4), (5) on the stable homotopy of the classical groups imply that the sphere Sn is not parallelizable for n ≠ 1, 3, 7. This was shown independently by Kervaire(8) and Milnor(7), (9). Another proof can be found in (3), §26·11. The work of J. F. Adams (on the non-existence of elements of Hopf invariant one) implies more strongly that Sn with any (perhaps extraordinary) differentiable structure is not parallelizable if n ≠ 1, 3, 7. Thus there exist already four proofs for the non-parallelizability of the spheres, the first three mentioned relying on the Bott theory, as given in (4), (5). The purpose of this note is to show how the refined form of Bott's results given in (6) leads to a very simple proof of the non-parallelizability (only for the usual differentiable structures of the spheres). We shall prove in fact the following theorem due to Milnor (9) which implies the non-parallelizability.

In (1), (2), (3) and (4) it is shown that homological algebra (5) can be applied to stable homotopy-theory. In this application, we deal with A-modules, where A is the mod p Steenrod algebra. In the present paper, we shall prove a finiteness theorem for the cohomology of the Steenrod algebra. This theorem is stated as Corollary 2 below. It is purely algebraic, but it is not claimed that it has any algebraic interest; it is inspired solely by the application mentioned above. Here it has the following uses.

Recent work of Atiyah (1) on the Grothendieck rings of classifying spaces of finite groups has yielded, among many other interesting results, a spectral sequence relating the simple integer cohomology of a finite group to its representation ring. He has determined the first differential operator of the spectral sequence which affects the p-part of the cohomology, and the question naturally arises, when is it the only non-zero one. My object in this note is to show that this is so for a large class of groups.

Because of the symmetry of the problems encountered in the applications of matrices to physics, electrical engineering and numerical analysis, it frequently turns out that the matrix to be considered is a composite matrix, that is, a matrix whose elements are matrices. For example, the matrix may be where A1 and A2 are square matrices of order n which need not commute. It is easy to prove that the eigenvalues of this matrix of order 2n are the eigenvalues of the two matrices, A1+A2 and A1−A2 of order n.

We consider H-spaces which are Abelian ‘up to homotopy’ (cf. (9)). We analyse this condition in terms of the homotopy of an associated space called ‘the projective plane’. The analysis is applied to show that ΩCP(3) is homotopy Abelian.

A group G will be called measurable if there exists a real-valued, non-negative set function μ(A) defined on all subsets A ⊂ G which satisfies .

Let G be an Abelian group. A binary relation ≥ denned in G is called an order of G if for each x, y, z ε G,

(i) x ≥ y or y ≥ x (and hence x ≥ x);

(ii) x ≥ y and y ≥ x ⇒ x = y,

(if x ≥ y and x ≠ y, we write x > y);

(iii) x ≥ y and y ≥ z ⇒ x = z;

(iv) z ≥ y ⇒ x + z ≥ y + z.