1. Let A be an associative algebra over a commutative field K; A is then a vector space over K, and multiplication is defined in A in such a way that

for x, y, z in A and λ in K. We suppose in addition that A does not contain a unit, and we denote by Ae the algebra obtained by adjoining a unit e to A.

Introduction. By the content of a polynomial with coefficients in a ring R, one understands either the R-ideal or (better) the additive group which the coefficients generate. There are important connexions between the contents of two polynomials and the content of their product, and results of this kind are to be found in the work of Gauss. Here we shall give a wide generalization of a theorem of this type not involving polynomials at all; however, as the reader will observe, the proof is essentially a reduction to the polynomial case.

In this note I prove the following theorem:

Theorem 1. Given 3N points in a plane, we can divide them into N triads such that, when we form a triangle with the points of each triad, the N triangles will all have a common point.

The proof depends on three lemmas; the first two are well known, and are quoted without proof; the third I believe to be new, and leads directly to the proof of the theorem.

1. Introduction. We say that a smooth compact manifold is embedded in q–space when there is given a regular one-one map of the manifold into q–dimensional Euclidean space. Whitney (5) has shown that embeddings always exist when q is not less than twice the dimension of the manifold.

We prove that the length of the shortest closed path through n points in a bounded plane region of area v is ‘almost always’ asymptotically proportional to √(nv) for large n; and we extend this result to bounded Lebesgue sets in k–dimensional Euclidean space. The constants of proportionality depend only upon the dimensionality of the space, and are independent of the shape of the region. We give numerical bounds for these constants for various values of k; and we estimate the constant in the particular case k = 2. The results are relevant to the travelling-salesman problem, Steiner's street network problem, and the Loberman—Weinberger wiring problem. They have possible generalizations in the direction of Plateau's problem and Douglas' problem.

1. Introduction. Since the first proofs by Kolmogorov (13) and Smirnov ((14), (15)) of their well-known results on the limit distribution of the deviations of the sample distribution function, many alternative proofs of these results have been given. For example, we may cite the various approaches of Feller (4), Doob (3), Kac (8), Gnedenko and Korolyuk(7), and Anderson and Darling (1). The approaches of (3), (8) and (1) rest on a probabilistic computation regarding the Wiener process, and are justified by the paper of Donsker (2) (see also (11)). Of all these approaches, only those of (8) and (1) can be extended to obtain the limit distributions of the ‘k–sample’ generalizations of the Kolmogorov-Smirnov statistics suggested in (9), and the author ((9), (10)) and Gihman(6) carried out such proofs.

1. Introductory discussion and summary. Consider a sequence {ui} of independent real or complex-valued random variables such that E(ui) = 1, and a sequence of mutually exclusive events S1, S2,…, such that Si depends only on u1, u2, …,ui, with ΣP(Sj) = 1. Define the random variable n = n(u1, u2,…) = m when Sm occurs. We shall obtain the necessary and sufficient conditions under which

referred to as the product theorem.

The Wigner 6j–symbol, written in the form

is shown to be invariant under separate permutations of A, B, C alone, separate permutations of α, β, γ alone and separate change in sign of any pair of α, β, γ; results equivalent to the new symmetry relations of Regge. Alternatively, written in the form

with J0 + J1 + J2 + J3 = K1 + K2 + K3, Jr(r = 0, 1, 2, 3) and K3 (s = 1, 2, 3) integral, it is invariant for separate permutations of the Jr and of the Ks. If Jm = max (J0, J1, J2, J3), then each 6j-symbol with distinct value may be associated with an ordered partition of Jm into 6 integral parts: Jm = n1 + n2 + n3 + p1 + p2 + p3, n1 ≥ n2 ≥ n3; p1 ≥ p2 ≥ n3. The 6j-symbol is proportional to the Saalschützian

of unit argument.

1. Introduction. The subject of this paper is the propagation of small disturbances in a compressible fluid which is also a conductor of electricity, in the presence of a magnetic field. The characteristic feature of the problem is that it combines sound waves and magneto-hydrodynamic waves. The theory is based on linearized equations which generalize the equations of motion of the theory of sound. The disturbances with which it deals might therefore be called magneto-acoustic waves. Energy dissipation by viscosity, heat conduction and Joule heat is neglected, and so is the displacement current. Even so, a hyperbolic system of seven partial differential equations of the first order is obtained, whose solution requires analysis of considerable complexity.