The complex Stiefel manifold Wn,k, where n≦k≦1, is a space whose points are k-frames in Cn. By using the formula of McCarty [4], we will make the calculations of the Whitehead products in the groups π*(Wn,k). The case of real and quaternionic will be treated by Nomura and Furukawa [7]. The product [[η],j1l] appears as generator of the isotropy group of the identity map of Stiefel manifolds. In this note we use freely the results of the 2-components of the homotopy groups of real and complex Stiefel manifolds such as Paechter [8], Hoo-Mahowald [1], Nomura [5], Sigrist [9] and Nomura-Furukawa [6].

The rational treatment of Geometry has this important disadvantage, that for want of suitable demonstrations it seems impossible to preserve the natural grouping of the facts developed. The study of Rational Geometry, in fact, should always be supplemented by a systematic attempt to array the facts demonstrated according to their subject-matter; for it will hardly be denied that a direct and systematic knowledge of the Properties of Geometrical Figures has an intrinsic value apart from the knowledge of their demonstrations. In pursuing such a retrospective scheme as this in connection with the Second Book of Euclid, I have found that a very comprehensive view of the subject-matter is obtained by adding a Third Mode of Section of a straight line to the two which are already recognised. This third mode of section, for which I have not been able to find a more suitable name than “Circuitous Section,” along with the other two known as Internal and External Section respectively, exhausts the possible modes of section of a line—for three-dimensional space at any rate. From this point of view, the elementary treatment of the subject may be arranged as follows. It will be observed that several important properties of triangles and polygons acquire a new-interpretation as cases of circuitous section.

Let {Ai; i ∈ Ω} be a family of C*-algebras acting on a Hubert space H and let A be the C*-algebra that they generate. We shall assume throughout that C*-algebras always contain the identity operator. Let M(A) denote the space of characters, that is, multiplicative linear functionals, acting on A, with the weak star topology. We obtain here a natural characterisation of M(A) as a subset of the product space determined by {M(Ai); i ∈ Ω}. In the case of singly generated C*-algebras this characterisation is related to the joint normal spectrum (5) of a family of operators.

In this article some properties on subseries convergent sequences in locally convex topological vector spaces are studied and some open questions of (2) are answered.

In (1), § 6.2, a multiplying factor method has been used to solve certain dual integral equations. The results are then used to solve a single integral equation of the Wiener-Hopf type. In this note we indicate how a related technique can be used to solve Wiener-Hopf integral equations directly. Consider

where

Define

where α = σ+iτ, and F+(α) is regular for τ>q; K(α) is regular and non-zero in −p < τ < p. For simplicity we restrict ourselves to the case where

The purpose of this note is to extend the results of Reilly and Scheiblich (6) (see also Scheiblich (7) and Hall (2)) on the θ-class decomposition of the congruence lattice of a regular semigroup and, at the same time, to provide an alternative proof of these results.

Etant donne un triangle OAB, on demande de mener par le sommet O une droite OM telle qu' en abaissant les perpendiculaires Aa, Bb sur cette droite, les surfaces des triangles OAa, OBb soient entre elles dans un rapport r donné, c'est-à-dire qu'on ait la relation

We consider the following problem: A potential function φ satisfies Laplace's equation ∇2φ = φxx + φyy = 0 in a region R bounded by a closed curve C on which mixed boundary conditions are specified, i.e. φ = f(s) on a part A of the boundary and ∂φ/∂n = g(s) on a part B, where C = A + B and distance along C is denoted by s. Electrostatic problems of this type have been solved approximately in (1) and (2) by formulating them in terms of integral equations and then applying variational principles to the integral equations. In that approach, attention is concentrated on integrals over the boundary of the region R. The most common type of variational principle for potential problems involves integrals over the region R rather than integrals over the boundary of R. An example is given by the Rayleigh-Ritz method which depends on the stationary character of Dirichlet's integral

In this paper we show that the variational principles used in (1), (2), are closely connected with the more usual type of variational principles, by deriving the principles used in (1), (2) from inequalities deduced by considering integrals of type (1) over the region R.

Consider the dual equations

where

A new finite integral transformation (an extension of those given by Sneddon (1)), whose kernel is given by cylindrical functions, is used to solve the problem of finding the temperature at any point of a hollow cylinder of any height, with boundary conditions of radiation type on the outside and inside surfaces, with independent radiation constants. It is to be noticed that all possible problems on boundary conditions in hollow cylinders can be solved by particularising the method described here.

We describe for every natural n the class of rings R such that if R is an accessible (left accessible) subring of a ring then R is an n-accessible (n-left-accessible) subring of the ring. This is connected with the problem of the termination of Kurosh's construction of the lower (lower strong) radical. The result for n = 2 was obtained by Sands in a connection with some other questions.

§ 1. Copson and Ferrar have proved the following theorem:

Ifis a divergent series of positive terms, and (pn + qn) un−pnun−1 = ynalways, where yn → 0, then un → 0.

The present paper describes briefly a notation for representing continued fractions in many dimensions, which has the advantage providing a direct method of attack and of rendering intuitive, results which are usually proved by induction. The notation is the outcome of a generalisation which I previously made [1] in connection with the solution of certain difference equations. Only formal theorems are considered here. For a discussion of convergence reference may be made to the works [2, 3, 4, 5] cited at the end. The paper by Paley and XJrsell is particularly important since these authors discuss very fully the non-cyclic simple continued fraction

In a recent paper some general properties of γ-matrices were proved and Dienes' theorem on regular γ-matrices extended to semiregular γ-matrices and the binomial series. In section 2 of this paper the previous results will be extended to certain classes of Taylor series. Section 3 gives some new results on Borel's exponential summation, and section 4 introduces matrices efficient for Taylor series on the circle of convergence and others efficient for Dirichlet series on the line of convergence. A knowledge of the definitions and results of the paper mentioned above is assumed.