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).

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).

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 .

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.

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.

If X is a space with base point, let ΩX denote the space of loops of X based at its base point. We will denote the homology and cohomology suspensions by σ* and σ*, respectively, , . Let K(π, n) be the Eilenberg-MacLane complex with only one non-vanishing homotopy group πn(K(π, n))= π, πi(K(π, n)) = 0 for i ≠ n. If π is abelian K(π, n) = ΩK(π, n+1). We will assume that π is a finitely generated abelian group throughout this note. Let K(π, n) × K(π, n) → K(π, n) denote the multiplication in K(π, n).

Several relations between the Cesàro and the Riemann methods of summation are known. For instance, Verblunsky(3) has shown that a series summable (C, k − δ), where k is a positive integer, is also summable (R,k+ 1) and Kuttner (2) has proved that, for k = 1, 2, summability (R, k) implies summability (C, k + δ). In this paper we consider Riesz's typical means and a generalized Riemann summability both of which are intimately connected with almost periodic functions. The result we establish is similar to Verblunsky's, except that we start from a Riesz mean of integral order k*.

Let {λn} be a sequence of non-negative numbers increasing to infinity. Let be any series. Following the usual terminology, we say that the series is summable (R, λn, k) to s if as u → ∞, where . If, further, u−kA(k)(u) is of bounded variation in (0, ∞), we say that the series is summable s. We say that the series is summable (A, λn) to s if converges for all σ > 0, and tends to s as σ →0+. If A, B are two summability methods, we write A → B (‘A implies B’) if they have the property that any series summable A is necessarily summable B to the same sum.

Ramanujan's Continued Fraction may be stated as follows: Let where there are eight gamma functions in each product and the ambiguous signs are so chosen that the argument of each gamma function contains one of the specified number of minus signs. Then where the products and the sums on the right range over the numbers α, β, γ, δ, ε: provided that one of the numbers β, γ, δ, ε is equal to ± ±n, where n is a positive integer. In 1935, Watson (3) proved the theorem by induction and also gave a basic analogue. In this paper we give a new proof of Ramanujan's Continued Fraction by using the transformation of Bauer and Muir in the theory of continued fractions (Perron (1), §7;(2), §2).

If the components x1, x2,…, xk of a vector X have a non-singular multivariate normal distribution having a null vector of means and variance-covariance matrix Σ= σ2, the matrix R=[ρij] (where ρii = 1) is known in certain cases but σ2 is unknown. If s2 is an estimate of σ2 based on ƒ degrees of freedom and is distributed independently of X, the distribution of the vector t=x/s is known as the multivariate t-distribution. This distribution was first obtained by Dunnett and Sobel (6) and independently by Cornish (3). Dunnett, Sobel and Bechhofer(2) have discussed some practical applications of this distribution. Cornish (3) obtained this distribution while considering the pre-treatment to be given to certain types of replicated experiments. This distribution possesses some useful properties and makes it suitable as a basis for exact tests of significance in various problems, and Dunnett and Sobel (6), by providing tables of the probability integral, have taken the first step towards its use in practice. Cornish, in a later paper (4) considered the sampling distribution of statistics derived from the multivariate t-distribution and using this he obtained the well-known ((7), (8)) distribution of the sample regression coefficient of one variate with respect to another, when both have a bivariate normal distribution.

A theoretical treatment is given of the relaxation time for the absorption of longitudinally polarized vibrations in an isotropie medium. Employing standard phonon interaction theory, the relaxation time is obtained in terms of certain integrals, depending parametrically on the ratio of transverse to longitudinal sound velocity. These integrals are evaluated numerically over the range which this ratio takes in practice. The results are applied to a consideration of solid argon, sodium chloride and germanium.

It was shown by Baber and Hassé(1) that the Schrödinger equation for the single electron in H+2 with fixed nuclei is separable in spheroidal coordinates. The separated equations were integrated by Bates, Ledsham and Stewart (2) and their solutions may be regarded as exact. However, several attempts have been made to derive approximate solutions for this, which is the simplest of all molecular problems, generally with the hope that it might be possible to extend the analysis to larger diatomic systems. Recently a single-centre expansion has been much favoured, since it avoids the difficult two-centre integrals; but unfortunately this advantage has been offset by the fact that the rate of convergence for the lowest states has been rather slow since the deviations of diatomic H+2 from the united-atom H+e are serious. But successful approximations to some excited states suggest that a single-centre approach may still be useful, provided that the trial functions are suitably chosen.

Domb and Salter (3) have postulated that the Debye characteristic temperature for the rare gas solids should be given by where C involves atomic constants and the atomic mass of the rare gas, r is the mean interatomic distance and Φ(r) is the interatomic potential. This formula provides a good approximation for the heavier rare gas solids but breaks down for the lighter inert elements. On the basis of Born's argument (1) that the dynamic rather than the static equilibrium positions should be considered, Domb (2) suggested that (1) might be replaced by , where the additional term is effectively the zero-point energy contribution to the lattice equilibrium. (R is the gas constant.) From experimental measurements of the Gruneisen parameter γ Domb obtained numerical values which, when substituted in the right-hand side of (2), yielded values of vD that appeared to compare favourably with direct experiments. More recently Hooton (6) considered the problem of lattice dynamics with anharmonic forces (which are the dominant factor in the lighter rare gas solids) and developed a theoretical argument which also led to Domb's equation (2).

The effects of the spin-other-orbit and spin-spin interactions on the energy levels of the 2p3 3s configuration of oxygen are investigated, and they are found to make an appreciable contribution to the fine structure. The interactions are shown to have a negligible effect on the strengths of all the stronger lines of the 2p4-2p3 3s array, and the more important of the weaker transitions are likewise little affected. The value to be adopted for the radial integral involved in the absolute oscillator strengths is discussed, and numerical values of the oscillator strengths are given.

An investigation is made of the effect of a non-linear addition to Schrödinger's equation for s-wave scattering of a beam of particles by a finite region of potential energy. The non-linear region is bounded and of an especially simple form so that an analytic solution is possible. The R-function for the problem is explicitly constructed and some related formulae such as the resonant energies and the scattering length are given. The analytical nature of R as a function of the momentum considered as a complex variable is discussed in relation to causality.

The equations of motion of a biased bowl rolling on a plane horizontal grass green are derived by Lagrange's method, using an appropriate representation of the effect of the green. The equations are solved approximately under general initial conditions which include the presence of a perturbing wobble.

An expression is derived for the total angle of precession of the bowl, and the equation of the path is found in parametric form. The theory predicts that (contrary to universal belief) the effect of an initial wobble on the bowl path is negligible under normal playing conditions.

A detailed comparison is made between the predictions of the theory and experimental results obtained on several greens of different speeds.

A long column of viscous liquid of radius a and uniform density p is rotating about its axis with angular velocity ω. It is shown that this motion is stable to plane perturbations of wave number 8 provided that the surface tension T satisfies . This critical value is higher than that required for stability of the similar motion of a nonviscous liquid, but is otherwise independent of the coefficient of viscosity. The rate of development of instability when T is less than the critical value is also studied. Some numerical results are given.

The condition has recently been obtained by Hocking (2) for the special cases of very high and very low Reynolds numbers.

A method employed by the author for a class of boundary-value problems arising in diffraction theory is applied to the problem of the oscillations of water waves on a sloping beach. The two cases of long and short waves are considered. The approach is valid for all values of the parameters and particular cases do not have to be treated separately. It is shown that the present approach enables certain solutions to be obtained which cannot arise in earlier solutions of the problem.

Complex variable methods are applied to obtain exact solutions for the complex potentials and deflexions of thin isotropic slabs bounded by regular curvilinear polygonal contours with n sides and subject to symmetrical loading distributed over a concentric circle. The supported boundary is either clamped or has equal boundary cross-couples. The plates taken in the z-plane are conformally mapped on the unit circle in the ζ-plane by the mapping function . Polynomial approximations to the Schwarz—Christoffel transformations are then used to discuss the bending of clamped and simply supported rectilinear plates symmetrically loaded over a concentric circle or acted upon by a central point load.

The Jacobi identity for Whitehead products,

where α ɛ πl(X), β ɛ πm(M), γ ɛ πn (X), l, m, n ≥ 2 is now well known. In this note we extend the identity to the case l, m, n > 1; the extension is easy and we are merely filling a small gap in the literature. However, the extension does enable us to improve a result in (2).