For a valued field (K, v), let Kv denote the residue field of v and Gv its value group. One way of extending a valuation v defined on a field K to a simple transcendental extension K(x) is to choose any α in K and any μ in a totally ordered Abelian group containing Gv, and define a valuation w on K[x] by w(Σici(x – α)i) = mini (v(ci) + iμ). Clearly either Gv is a subgroup of finite index in Gw = Gv + ℤμ or Gw/Gv is not a torsion group. It can be easily shown that K(x)w is a simple transcendental extension of Kv in the former case. Conversely it is well known that for an algebraically closed field K with a valuation v, if w is an extension of v to K(x) such that either K(x)w is not algebraic over Kv or Gw/Gv is not a torsion group, then w is of the type described above. The present paper deals with the converse problem for any field K. It determines explicitly all such valuations w together with their residue fields and value groups.

Let R be a commutative integral domain and let S be its quotient field. The group GL2(R) acts on Ŝ = S ∪ {∞} as a group of linear fractional transformations in the usual way. Let F2(R, z) be the stabilizer of z ∈ Ŝ in GL2(R) and let F2(R) be the subgroup generated by all F2(R, z). Among the subgroups contained in F2(R) are U2(R), the subgroup generated by all unipotent matrices, and NE2(R), the normal subgroup generated by all elementary matrices.

We prove a structure theorem for F2(R, z), when R is a Krull domain. A more precise version holds when R is a Dedekind domain. For a large class of arithmetic Dedekind domains it is known that the groups NE2(R),U2(R) and SL2(R) coincide. An example is given for which all these subgroups are distinct.

By a ring we shall mean an associative ring not necessarily containing an identity element. The fundamental definitions and properties of radicals may be found in Divinsky [2]. Similarly we refer to Howie [3] for the semigroup concepts.

If R is a ring Mn(R) will denote the ring of n × n matrices with entries from R. For many important radicals α it has been shown that α(Mn(R)) = Mn(α(R)) for all rings R and all positive integers n. However this is not the case for all radicals α. Associated with each radical α we define a set of positive integers S(α) by

In the short sketch which I propose to give of the History of Mathematics in Scotland up to the end of the 18th century I must limit myself mainly to the work of the Universities. An adequate treatment of the subject would involve considerations of a general educational character that would range over the relations of the school to the University, the distribution of the various subjects of study and the place of mathematics in the educational system; but it is, of course, impossible to undertake such an extensive investigation at present, though it seems to me that an investigation, with special reference to mathematics, is greatly needed and might form the subject of a research that would be of real value as a contribution to the development of educational ideas. It would be improper, however, to omit all reference to school mathematics, since the school conditions determine, to a considerable extent, those of the University, as current discussions in Scotland clearly show, even though a sound appreciation of the relations between school and University may at times be lacking.

On p. 403 of Greenhill's Calculus (2nd Ed.) the following sentence occurs:— “By differentiation of the integral

with respect to A, B, or C we can deduce the results of

For the evaluation of tlie typical form in which f(x) is a linear function, especially when A, B, etc., are given numbers, the method of differentiation does not seem very suitable; be that as it may, it may perhaps be of some interest to investigate a formula of reduction analogous to those in use for the integrals in which Ax2 + 2Bx + C is replaced by a linear function and f(x) is a constant.

Einstein has studied a universe in which the time coordinate t is uncurved and the spatial section is the surface of a sphere

in four dimensions. Some interest attaches to the case where this surface is replaced by

where n is a positive integer.

I begin by recalling the well known definitions for summability by the methods of Cesaro and Riesz.

The series Σan is said to be summable (C, K), k> – 1, to the sum s if, as n → ∞

where

and is defined formally by the relation

An equation is derived for the strains of an arbitrary elastic field in an infinite matrix perturbed by several inclusions. The equation is solved exactly when the shear moduli of the inclusions and matrix are identical, and also when only a single ellipsoidal inclusion perturbs a field uniform at infinity. Mean-values of the strains are then calculated for non-uniform fields perturbed either by an ellipsoid or by a system of weakly-interacting spheres.

Over a field of characteristic p>0 the group algebra of a finite group has a unique maximal nilpotent ideal, the Jacobson radical of the algebra. The powers of the radical form a decreasing and ultimately vanishing series of ideals and it would be of interest to determine the least vanishing power. Apart from the work of Jennings (3) on p-groups little is known in general (cf. (5)) about this particular power of the radical (cf. Remarks of Brauer in (4), p. 144. Problem 15). In this paper we give non-trivial lower bounds for the index of the least vanishing power of the radical when the group is p-soluble. Of the lower bounds we give we show that that lower bound, which is dependent solely on the order of the group, is the best possible such bound and that this bound is invalid if the assumption of p-solubility is omitted.

Banach modules over C*-algebras (von Neumann algebras) that can be represented isometrically as operator modules (normal operator modules, respectively) are characterised.

In this note I prove the following result:—

Theorem. A compact, orientable, Riemannian manifold Mn, with positive definite metric and zero Ricci curvature, is flat if the first Betti number R1 exceeds n — 4.

In this statement of the theorem it is assumed that the dimensions of Mn are not less than four. If this is not the case, the result is still valid but appears as a purely local result and is true for a metric of arbitrary signature.

Two n-planes Γ and Δ in real Euclidean r-space Rr are called isoclinic with parameter λ if the angle θ between any x in Γ and its orthogonal projection Px on Δ is unique, with cos2 θ = λ. Let vλ(n, r) denote the maximum number of equi-isoclinic (i.e. pairwise isoclinic) n-planes in Rr with parameter λ.