In this paper we will show that large classes of near-rings are embeddable into (or even isomorphic to) near-rings Gν[x] of polynomials over G in a suitably chosen variety ν of Ω-groups.

For near-rings see [2], for polynomials [1]; for near-rings of polynomials over Ω-groups see [3] [4], [5].

Let L be a positive invertible self-adjoint operator in L2(X;C). Using transference methods for locally bounded groups of operators we obtain multipliers for the group of complex powers Liu on vector-valued Lebesgue spaces. Using a Mellin inversion formula, we derive a sufficient condition for a function to be a multiplier of the semigroup e-tL on Lp(X;E) where E is a UMD Banach space and 1<p<∞.

In a recent paper Einstein has proposed a new type of geometry in which it is possible to give an arbitrary vector in the field a displacement which conserves parallelism at a distance. It may be briefly stated as follows:

Let the metric ofthe space, supposed to be of n dimensions, be given by

At each point of the space we consider a set of n unit vectors, denoted by the n2 quantities phq, which form an orthogonal ennuple such that

The object of this paper is to bring together the principal properties of figures described on the surface of the sphere that can be established without the use of Solid Geometry or of Trigonometry.

The following properties of the spherical surface, which correspond to the definitions and axioms in Plane Geometry, are assumed. They may be considered as derived from one's general notion of the sphere.

The chief purpose of the paper was to indicate the rise of transformations of the type where the integral in the first member of the equation is taken throughout a closed surface, and that in the second member over the surface, a being the angle made with the axis of x by the normal to the element dS drawn outwards. It is on this transformation the analytical proof of Green's theorem depends, and it was shown to have been employed in various forms by Poisson, Duhamel, Gauss, and others, before Green's essay was generally known on the Continent. It may be observed that the essay was published at Nottingham in 1828, and seems to have been unknown to continental mathematicians till its reprint in Crelle's Journal, vols. 39 (1850), 44 (1852), and 47 (1854).

§1. In a recent paper with this title Prof. W. P. Milne has discussed the properties of the conics which pass through two fixed points of a plane quartic curve and touch the curve at three other points. In dealing with a numerous family of curves such as this it is very desirable to have a scheme of marks or labels to distinguish the different members of the family; Hesse's notation for the double tangents of a C4 illustrates this. By using another line of approach to the subject, by projecting the curve of intersection of a quadric and a cubic surface from a point at which (under exceptional circumstances) the surfaces touch, I find that a fairly simple notation for the 64 conics, in harmony with that for the bitangents, can be obtained. This paper, let it be said, from start to finish is no more than an adaptation of results known for the sextic space-curve referred to; it will be sufficient therefore to state results with short explanations.

It is well known that a ring R is semiprime Artinian if and only if every right ideal is an injective right R-module. In this paper we shall be concerned with the following general question: given a ring R all of whose right ideals have a certain property, what implications does this have for the ring R itself? In practice, it is not necessary to insist that all right ideals have the property, usually the maximal or essential right ideals will suffice. On the other hand, Osofsky proved that a ring R is semiprime Artinian if and only if every cyclic right R-module is injective. This leads to the second general question: given a ring R all of whose cyclic right R-modules have a certain property, what can one say about R itself?

1. In deriving an expression for the number of representations of a sufficiently large integer N in the form

where k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the type

where ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sum

where α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the form

where s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri prime factors.

We study the semicentre of a group algebra K[G] where K is a field of characteristic zero and G is a polycyclic-by-finite group suchthat Δ(G) is torsion-free abelian. Several properties about the structure of this ring are proved, in particular as to when is the semicentre a UFD. Examples are constructed when this is not the case. We also prove necessary and sufficient conditions for every normal element of K[G] which belongs to K[Δ(G)] to be the product of a unit and a semi-invariant.

A q-analogue of the integral ∣f(t)dt is defined by means of

which is an inverse of the q–derivative

The present author (2) has recently obtained a q–nalogue of a formula of Cauchy, namely,

where, for real or complex α and N a positive integer,

When a perfectly conducting uniform thin circular disc is kept at a potential V0 in an external electrostatic field of potential Φ, electric charge is induced on the surface of the disc; the problem is to find the surface-density σ of this induced charge and its potential V so that the total potential V + Φ has the constant value V0 on the surface of the disc. This problem was first discussed by Green in 1832, and the solution in the case when there is no external field was deduced by Lord Kelvin from the known formula for the gravitational potential of an elliptic homoeoid. The problem is still of interest since similar ideas occur in the theory of diffraction by a circular disc and in the theory of the generation of sound waves by a vibrating disc when the wave-length is large compared with the radius of the disc.