Let R be a (commutative Noetherian) local ring (with identity) having maximal ideal and dimension d≧l. It is shown in [5,3.6rsqb; that the local cohomology module may be described as a module of generalized fractions: if x1…,xd is a system of parameters for R, then , where U(x)d+1 is the triangular subset [4,2.1] of Rd+1 given by

Throughout this paper all spaces are T1 and N will denote the set of all positive integer numbers.

A quasi-metric on a set X is a non-negative real-valued function d on X × X such that, for all x, y, z ∈ X, (i) d(x, y) = 0 if, and only if, x = ysemicolon (ii) d(x, y)≦d(x, z) + d(z, y).

On reading Dr Peddie's paper, the following modification of the proof, which avoids summation, occurred to me:—

If in Figure 7 we take a point S on the circle BQD such that PQ + PS = 2a, where a is the radius, and a corresponding point S′ such that PQ′ + PS′ = 2a, then it is clear by Dr Peddie's construction that the potential at P due to the zone of the spherical surface lying between planes through Q and Q′ perpendicular to BD is given by 2πσ(PQ′−PQ) · a/CP, and is therefore equal to that due to the corresponding zone between S and S′, since

Throughout this paper all near-rings considered will be zero-symmetric and left distributive. All groups will be written additively, but this does not imply commutativity. The near-ring of all zero-fixing maps of a group V into itself will be denoted by Mo(V). If N is a near-ring withan identity and α ≠ 1 is an element of N such that α2 = 1, then α will be called an involution of N. Let V be a group. An involution a of Mo(V) will be called an involution on V.

We prove that for F: [0,∞) × ℝn → K (ℝn) a Lipschitzian multifunction with compact values, the set of derivatives of solutions of the Cauchy problem

is a retract of

A problem in descriptive set theory, in which the objects of interest are compact convex sets in linear metric spaces, primarily those having extreme points.

§ 1. To inscribe in a triangle ABC a triangle similar to the triangle DEF, and having its sides parallel to those of DEF.

In order to inscribe in the triangle ABO (fig. 21), a triangle having its sides parallel to those of DEF, through D, E, F, draw lines parallel to the sides of ABC, and then reduce the figure A′B′C′ in the ratio BC : B′C′.

We shall concern ourselves with the class of continuous, four-parameter, one-sided probability distributions which can be characterized by the probability density function (pdf) class

It depends on the four parameters: shift c ∈ R, scale b > 0, initial shape p < 1, and terminal shape β > 0. For p ≦ 0, the definition of f(x) can be completed by setting f(c) = β/bΓ(β−1)>0 if p = 0, and f(c) = 0 if p < 0. For 0 < p < 1, f(x) remains undefined at x = c; f(x)↑ + ∞ as x↓c.

In his book [1] Divinsky refers to eight radicals as classical. In [6] radicals were considered such that the radical of each one-sided ideal of a ring may be expressed as the intersection of a left ideal and a right ideal of the ring. From results obtained there it was deduced that seven of these eight radicals have this property. The purpose of this note is to give a proof that this property also holds for the remaining one of these classical radicals.

In 1901, when I was drawing up notes on Mental Arithmetic, I looked into many text-books in search of a simple method for dividing by such numbers as 19, 29, 99, 87, etc., but found none. The following method, viz., that of using a multiple of ten as divisor instead of a given divisor, was then discovered by me, and I think it simple enough to be learned and practised by any one.

If it be required to divide A by D, let the quotient at any step be Q, then the product at that step will be DQ, and the remainder, R = A - DQ, where R cannot exceed D nor be less than zero.

The purpose of this brief note is to draw attention to a type of inverse factorial series which, so far as the writer can judge, has not been intensively studied. The central difference formulae of interpolation and the corresponding infinite series in central factorial polynomials have in the past received much attention, and so also has the ordinary inverse factorial series

which from the time of Stirling (Methodus Differentialis, 1730) has been known to be of value in transforming slowly convergent series into series of more rapid convergence.

In my paper on the continuous (α + β) – theorem (these Proceedings 12 (1961), 209–211) the proof of Lemma 6 assumes that if A and B are closed sets in the relative topology as subsets of the space P of positive reals then C = A + B is closed.

Certain cases of motion in a gas contained in a spherical or in a cylindrical envelope or surrounding a sphere or a cylinder have been considered by Professor Stokes and Lord Rayleigh, of which a clear idea may be obtained from chapters xvii. and xviii. of Rayleigh's Sound. Their chief object is to determine the motion in the gas when the motion in the bounding surface is given, or to determine the modification in the nature of a wave being propagated through the gas by the presence of an obstacle.