Let R denote a commutative Noetherian ring with an identity element. Hilbert's basis theorem states that the polynomial ring in a finite number of indeterminates over R is also Noetherian. (See Northcott ], theorem 8, p. 26; Zariski and Samuel [4], theorem 1, p. 201). Hilbert's original theorem in [2] is stated for the case when R is a field or the ring of integers. The standard proofs of this fundamental theorem are essentially of a direct type. The analogue of Hilbert's basis theorem in the ring of formal power series in a finite number of indeterminates over R is also true (Chevalley [1]; see also Northcott [3], theorem 3, p. 89; Zariski and Samuel [5], theorem 4, p. 138). In the present note we bring together concise proofs of Hilbert's and Chevalley's theorems by using a single indirect (i.e. reductio ad absurdum) argument. By induction it is sufficient to consider the case of a polynomial ring S and a formal power series ring T in a single indeterminate X over R.

Starting from their defining presentation by generators and relations, we develop the basic structure and representation theory of generalised q-Schur algebras of finite type.

1. When, as will be presumed henceforward, no two of a0, a1, …, an are equal the n + 1 equations

are linearly independent; x0, x1, …, xn are homogeneous coordinates in [n] projective space of n dimensions—and the simplex of reference S is self-polar for all the quadrics.

A simple sufficient condition is given for a stochastic process x(t), 0 ≤ t ≤ 1, to have the following property: There is an integer m ≥ 2 such that for any non-degenerate subinterval J ⊂ [0, 1], there exist m disjoint subintervals I1, …, Im ⊂ J such that the intersection of the images of I1,…, Im under the mapping by x(·) has positive Lebesgue measure, almost surely. There is also a version for vector random fields; and the main result is shown to apply to large classes of processes.

The problem of the diffusion and convection of heat from an arbitrarily placed heat source in the potential convection field due to uniform streaming past an infinite circular cylinder is considered. Solutions are obtained for the regions inside and outside the cylinder and are given in terms of Mathieu Functions. An appendix giving the required properties of these functions is included.

A general method is given for the determination of diagonal matrix elements of relative state operators in shell model states of a given supermultiplet and SU3 symmetry, for states in s, p and sd shells, and for cross-terms between these shells, enabling the excitation energies of particle–hole configurations to be evaluated.

This paper contains all those formulae which are necessary for the numerical evaluation of the basic confluent hypergeometric functions, including asymptotic formulae, and three skeleton tables which can be used as a basis for the study of the numerical behaviour of these functions over a wide field.

Second-order perturbation theory, in conjunction with the usual treatment of electron-phonon interaction, allows us to calculate the correction to the first-order result for the scattering of phonons by electrons. It is shown that the second-order term is much smaller, and therefore negligible. This justifies the use of the first-order theory in the treatment of the interaction in metals and semi-conductors.

We have called a linear partial differential operator P almost hypoelliptic in some of the variables if the distribution u is strongly regular in those variables in every open subset where this is true of Pu. In an earlier paper (l) we have characterised almost hypoelliptic operators with constant coefficients. Following techniques of Hörmander (2) and Yoshikawa(5) we obtain in this paper some a priori inequalities for almost hypoelliptic operators and derive results about the solvability of the adjoint operator P*. Our notation for various standard distribution spaces and multi-indices follows Hörmander (2).

1. Introduction. Bernoulli trials. Consider a sequence of Bernoulli trials. Let p, assumed to satisfy 0<p < 1, be the probability of success at any given trial and let q = 1–p. If Nn is the number of successes in the first n trials, it is well known that Nn/n→p almost surely as n→∞ so that for every ∈> 0,

as n→∞, and it is clearly of great interest to know quantitatively how this probability depends upon n and ∈.

It is well known (Watson, A treatise on Bessel functions) that if Jν(z) is a Bessel function of the first kind, and A is a real constant, then all roots of the equation are either real or purely imaginary.

The compression of fairly short solid cylinders under axial load is considered. Radial expansion is prevented over a central region of the outer surface by a rigid constraint concentric with the cylinder. Physically the situation arises when a fairly soft rivet is expanded into a relatively rigid plate. Two types of elastic material are considered; firstly, rubber-like materials governed by a strain energy function of the Mooney form, and secondly, metals which have a quadratic strain energy function. In the former case a finite axial compression is permitted prior to contact between the cylinder and the constraint. In both cases the irregularities introduced by the constraint are sufficiently small that they can be described by infinitesimal elasticity theory. The analysis utilizes displacement potential functions and is reduced to solving a set of dual cosine series. The particular case in which the cylinder height and diameter are equal and the contact height is equal to the radius is examined in detail and the contact stresses are given graphically.

A linear vector m-space Rm defined along a curve (C) in a Vn and lying in the complete osculating space of (C) will be called a characteristic Rm of (C) if it is auto-parallel along the curve and makes constant angles with its tangent and the principal normals. Curves admitting a characteristic R1 have been studied by Hayden under the name of generalized helices and generalized by me§. In this paper we give a complete determination of the curves with a characteristic R2. Curves whose curvatures are proportional to a set of constants, which have been considered by Syptak for the particular case when Vn is an Rn, form one of the classes of curves of this type. As a consequence, the existence of the curves admitting a characteristic Rm (m > 2) is partly established, but the problem has not been completely solved. At the end we prove two theorems in connexion with two particular types of characteristic Rm's.

It is shown that if we know the Love's strain function due to the existence of any arbitrary axisymmetric singularity in the elastic whole space, then the corresponding Love's function due to the existence of the same singularity in the interior of one of two dissimilar semi-infinite solids separated by a dissimilar thick layer can be obtained by differentiation and integration of that for the elastic whole space. This discovery rests on the principle that to every singularity in the layered space there will correspond an infinite system of images. Remarkably, the number of images is finite when the phases have equal rigidities but different compressibilities. The analysis thus brings under one general theorem the solutions due to such influencing singularities as a normal point force, a normal point force doublet, an infinitesimal prismatic dislocation loop, a centre of dilatation and a dilatation doublet. From the theorem, we readily infer that the elastic field at large distances from any arbitrary axisymmetric influencing singularity becomes nearly that for two perfectly bonded semi-infinite solids without an interface layer.

Volume 98 (1985), 403–412

‘On the mobility of bodies in ℝn’

The proof of the following theorem which I gave in the above paper is incorrect. A correct proof is given here.

Theorem 2·1. Let ℬ be an intersection regular, finite collection of bodies, and ℱ a manoeuvre on ℬ. ℱ is strongly separating if and only if every component of S(ℬ, ℱ) is positive; and, if any S(i, j, p) (ℬ, ℱ) is negative, ℱ is colliding.

In this short note we prove that Istrǎƫescu's notion of k-uniform (k-locally uniform) convexity of a Banach space is actually equivalent to the notion of uniform (locally uniform) convexity. Thus theorem 2 in [3] and theorem 2·6·28 in [2] are trivially true.

A general form of the covering principle and relative differentiation of additive functions. II*.

By A. S. Besicovitch

P. 5, line 7 from the bottom, after ‘F-singular’ add ‘in the class of B-sets’.

P. 5, the bottom line, after ‘function Θ(X)’ add ‘in the class of B-sets’.

P. 7, line 9 from the bottom, forG⊃Xnread.

P. 8, line 4, for.

P. 8, line 5, for.

* Vol. 42 (1946), pp. 1–10.