Let R be a commutative ring. We say that R satisfies the ascending chain condition for principal ideals, or that R has property (M), if each ascending sequence (a1) ⊆ (a2) ⊆ … of principal ideals of R terminates. Property (M) is equivalent to the maximum condition on principal ideals; that is, under the partial order of set containment, each collection of principal ideals of R has a maximum element. Noetherian rings, of course, have property (M), but the converse is not true; for if R has property (M) and if {Xλ} is a set of indeterminates over R, then the polynomial ring R[{Xλ}] has property (M). Krull domains, and hence unique factorization domains, have property (M).

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.

The following extension problem is of considerable geometrical interest. Given a compact manifold X = ∂Z the boundary of a compact manifold Z, does the normal bundle of X in Z extend to a line bundle on Z? A related question involving foliations is if X is a compact boundary, is X a leaf of a codimension one foliation on any compact manifold bounded by X? Obviously if it were such a leaf, its normal bundle in the bounded manifold would extend. Reinhart posed both these problems in the language of cobordism theories(1), (2) defining a vector cobordism theory to a deal with questions of the first genre and a foliated-cobordism theory to deal with the second. He was able to obtain complete results in his vector-cobordism theory, whilst the foliated-cobordism theory remains an unsolved problem. Our aim in this article is to extend Reinhart's results on vector-cobordism theory to unoriented line bundles. Whereas in vector-cobordism the Euler number is a cobordism invariant, in the cobordism theory that we shall describe, the modulus of the Euler number becomes a cobordism invariant.

Introduction. The space Cp is the class of operators on a Hilbert space for which the norm ∥K∥p = [trace (KK*)P/2]/p is finite. Equivalently, a compact operator is in Cp if

where the μn are the so-called ‘singular values’ of K (characteristic values of the non-negative compact operator [K] ≡ (KK*)½). The case p = 2 gives the familiar class of Hilbert–Schmidt operators, while C1 is the collection of trace-class or nuclear operators considered by Schatten(12), Lidskii(11), and Gohberg and Krein(7), among others.

In (1) (where references and terminology may be found), C. A. Rogers has proved the following surprising theorem.

Letbe a class of sets, all Souslin-in a topological space X. If A is a Souslin-set and B is an analytic set that does not meet A, then there is a Borelian-set C with A ⊂ C and C ∩ B = Ø

Let X1, X2, … be a sequence of independent random variables such that, for each n ≥ 1, EXn = 0 and and assume that then converges almost surely as N → ∞. Let and let Fn(x) denote the distribution function of Xn. Loynes (2) observed that the sequence {Sn} is a reversed martingale, and applied his central limit theorem to it: however, stronger results are obtainable, in precise duality with the classical theory of partial sums of independent random variables. These results describe the fluctuations of the sequence {Sn}, and hence the way in which converges to its limit.

1. Introduction. We shall employ the customary abbreviations: p.d.f. for probability density function; d.f. for distribution function; c.f. for characteristic function.

A very general exact solution of Einstein's equation Rik = σξiξk, is given in terms of the Kerr-Schild metric, gik = ηij + Hξiξk, where ξi is a shearfree geodetic null congruence. The Kerr-Schild solution for Rik = 0 is derived as a particular case.

In this paper, the problem of diffraction of an impulsive P wave by a fluid circular cylinder has been considered. The cylinder is embedded in an unbounded isotropic homogeneous elastic medium and it is filled with inviscid fluid material. The line source, giving rise to the incident front, is situated outside the cylinder parallel to its axis.

The exact solution of the problem is obtained by using the method of dual integral transformations. The solution is evaluated approximately to obtain the motion on the wave front in the shadow zone of the elastic medium. Further, we interpret the approxi mate solutions in terms of Keller's geometrical theory of diffraction. Our result also gives a correction to an earlier investigation of the similar problem by Knopoff and Gilbert(s).

In this paper wave-maker theory including the effect of surface tension is determined for three-dimensional motion of water in a semi-infinite rectangular channel with outgoing surface wave modes allowed for at infinity; the motion is generated by a harmonically oscillating vertical plane wave-maker at the end of the channel and the cases of both infinite and finite constant depth are treated. The solution of the boundary-value problem for the velocity potential is more complicated in the presence of surface tension due mainly to the additional effect of the channel walls at which the normal free surface slopes are prescribed—as also is the slope at the wave-maker—to ensure uniqueness. The simpler three-dimensional solution for a semi-infinite region—obtained long ago by Sir Thomas Havelock in the absence of surface tension for the case of infinite depth—is also noted.

The use of even-order isotropic tensors in non-linear elasticity theory is discussed in this paper. A notation is adopted through which these tensors can be represented conveniently so that their interdependence is clearly shown. Information about the number of independent elastic constants required is then readily available for use in an expansion of the stress to various orders in the strain relative to the undistorted configuration of the elastic material in question.

For an incompressible isotropic hyperelastic solid, it is shown that each principal component of the distortional part of the stress is expressible as a function only of the corresponding principal component of strain to the fourth order. Under certain conditions, which are not too restrictive, this result can be extended to higher orders.