In this note we show that the variety of near-rings generated by d.g. near-rings is the class of all near-rings R satisfying

This extends a result of J. J. Malone (3) on the embedding of near-rings.

In this paper is considered the form taken in Polar co-ordinates by the equivalents of certain well known expressions in Cartesian co-ordinates which occur in Elastic Solids and in Hydrodynamics.

The accompanying figure (fig. 29) will enable us to transform from Cartesians to Polars by simple geometry.

It is shown that the lower radical construction of Tangeman and Kreiling need not terminate at any ordinal.

Let the points A, B, C be the centres of three given circles, whose radii are a, b, c; and let d, e, f be the distances BC, CA, AB between the centres (Fig. 41). It is required to find the radii of the circles which touch the circles A, B, C.

The aim of the paper is to prove the Theorem: Let M be a surface in the euclidean space E3 which is diffeomorphic to the sphere and suppose that all geodesies of M are congruent. Then M is a euclidean sphere.

We study the structure of inductive limits of weighted spaces of harmonic and holomorphic functions defined on the open unit disk of ℂ, and of the associated weighted locally convex spaces. Using a result of Lusky we prove, for certain radial weights on the open unit disk D of ℂ, that the spaces of harmonic and holomorphic functions are isomorphic to complemented subspaces of the corresponding Köthe sequence spaces. We also study the spaces of harmonic functions for certain non-radial weights on D. We show, under a natural sufficient condition for the weights, that the spaces of harmonic functions on D are isomorphic to corresponding spaces of continuous or bounded functions on ∂D.

Let R be a ring and X a right R-module (all rings have identities and all modulesare unitary). The intersection of all non-zero submodules of X is denoted by μ(X). The module X is called monolithic if and only if μ(X)≠0 and in this case μ(X) is anessential simple submodule of X. (Recall that a submodule Y of X is essential if and only if Y ⊂ A ≠ 0 for every non-zero submodule A of X.) It is well known that a module X is monolithic if and only if there is a simple right R-module U such that X is a submodule of the injective hull E(U) of U. If x is a non-zero element of an arbitrary right. R-module X then by Zorn's Lemma there is a submodule Yx of X maximal with the property x ∉ Yx. It can easily be checked that X/Yx is monolithic and ⊂ Yx = 0, where the intersection is taken over all non-zero elements x of X.

1. It is convenient to begin with a brief statement of the notation which will be used throughout this paper.

Let k be any positive number and let

where is the coefficient of xn in the formal expansion of (1 – x )–k–1, and let

Then the series Σαn is said to be summable(C, k) if is convergent, that is, if tends to a limit, and absolutely summable (C, k), or summable |C, k|, if is absolutely convergent.

Since Brownian motion is point recurrent in R1, recurrent in R2 and transient in Rn, n ≧ 3 (see (7)), it follows that the total time spent in a bounded open set in R1 or R2 is unbounded. With the following ergodic theorems for Brownian motion in R1 and R2 as motivation, we examine the rate of convergence in these theorems. Note that there is no ergodic property in Rn for n ≧ 3 since Brownian motion is not dense there.

Interpolation is one of the most frequent processes in calculation, and yet it is the process in which most computers find the ordinary methods least satisfactory and most troublesome. Indeed, whenever linear interpolation is not practicable, it is usually worth while to find out a method depending on the nature the functions involved in the calculation, and use it in preference to the ordinary difference or Lagrangian formulae. In interpolation by differences there is the want of adequate tables of the coefficients, and worse than that, the necessity for watching the signs and the decimal points, a necessity which in these days calculating machines is relatively a great trouble. There is usually, moreover, a lack of system about interpolation by differences that makes it peculiarly susceptible to slips of working. In this connection I might mention a useful and not too well-known arrangement of the work for Newton's formula which Legendre gives in his Traité des fonctions elliptiqueg.

In a recent paper Dr J. M. Whittaker has shown that the Fourier series

of a function f(θ) which has a Lebesgue integral in (— π, π), is absolutely summable (A) to sum l, if

exists, where

.

Let X be a set and the semigroup (under composition) of all total transformations from X into itself. In ([6], Theorem 3) Howie characterised those elements of that can be written as a product of idempotents in different from the identity. We gather from review articles that his work was later extended by Evseev and Podran [3, 4] (and independently for finite X by Sullivan [15]) to the semigroup of all partial transformations of X into itself. Howie's result was generalized in a different direction by Kim [8], and it has also been considered in both a topological and a totally ordered setting (see [11] and [14] for brief summaries of this latter work). In addition, Magill [10] investigated the corresponding idea for endomorphisms of a Boolean ring, while J. A. Erdos [2] resolved the analogous problem for linear transformations of a finite–dimensional vector space.

Let B be a complex Banach algebra with an identity 1 and an involution x→x*. Kadison (1) has shown that, if B is a B*-algebra, [the set of extreme points of its unit ball coincides with the set of elements x of B for which