This paper is a continuation of our project on “inverse interpolation”, begun in [6]. In brief, the task of inverse interpolation is to deduce some property of a function f from some given property of the set L of its Lagrange interpolants. In the present work, the property of L is that it be a uniformly bounded set of functions when restricted to the domain of f. In particular (see Section 3), when the domain is a disc, we deduce sharp bounds on the successive derivatives of f. As a result, f must extend to be an analytic function (of restricted growth) in the concentric disc of thrice the original radius.

An expression for the sum of a series of terms which takes account of the different rates of oscillation of the functions involved is of importance for series such as occur in the problems of diffraction, and a relation of this kind has been given by Poisson. This relation can be written in the form

where Sm denotes the sum of m terms of the series Σun, beginning with the term un.

1. Let AB be the half-side of any n-gon, OB its in-radius (r), and OA its circum-radius (R). Draw OA11 to bisect ∠ AOB and AA1C ⊥ to it meeting OB in C. Then A1B2 to AB is the half-side of a 2n-gon having the same perimeter as the n-gon, OB1 its inradius (r1), and OA1 its circum-radius (R1).

If G is a one-relator group on at least 3 generators, or is a one-relator group with torsion on at least 2 generators, then it follows from results in [1] and [6] that G has a subgroup of finite index which can be mapped homomorphically onto F2, the free group of rank 2. In the language of [2], G is equally as large as F2, written G⋍F2.

In the Bulletin of the American Mathematical Society Whittaker defines the confluent hypergeometric function Wk,m(x) by the equation

where the path of integration begins at t = + ∞, and after encircling the point t = Q in the counter-clockwise direction, returns to t = + ∞ again.

In a recent paper (1), Jones extended the definition of the convolution of two distributions to cover certain pairs of distributions which could not be convolved in the sense of the previous definition. The convolution ω1 * ω2 of two distributions ω1 and ω2 was defined as the limit of the sequence ω1n * ω2n, provided the limit ω exists in the sense that

for all fine functions φ in the terminology of Jones (2) where

and τ is an infinitely differentiate function satisfying the following conditions:

The definition of a suitable Jordan analogue of C*-algebras (which we call JB*-algebras in this paper) was recently suggested by Kaplansky (see (26)). The theory of unital JB*-algebras is now comparatively well understood due to the work of Alfsen, Shultz and Størmer (1) from which a Gelfand-Neumark theorem for unital JB*-algebras can be obtained (26). Independently, from work on simply connected symmetric complex Banach manifolds with base point, Kaup introduced the definition of C*-triple systems in (14) and subsequently in (7) it was shown that every unital JB*-algebra is a C*-triple system. In this paper, we wish to extend this result to show that every JB*-algebra is a C*-triple system.

Several textbooks of Elementary Geometry have recently been put on the market, and in nearly all that I have examined (and I have gone carefully through many of them) the treatment of tangents is based on what the writers call the Method of Limits. The usual form given to the proof that the tangent at any point of a circle is at right angles to the radius to the point of contact is somewhat as follows.

For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

In a very remarkable work on the operational Calculus, Dr Balth. van der Pol has introduced a new function, playing with respect to Bessel function of order zero the same part as the cosine- or sine-integral with respect to the ordinary cosine or sine. He showed that this function—which he called Bessel-integral junction—can be used to express the differential coefficient of any Bessel function with respect to its index. But he did not investigate the further properties of his new function. I propose to give here some of them, which appear to be interesting, and to introduce and study the functions connected, in the same way, with Bessel functions of any order.

The following notation is here used for the quantities occurring in the discussion of tortuous curves.

Length = s; curvature = ; torsion = ; direction cosines of tangent, principal normal and binomial:—α, β, γ; l, m, n; λ, μ, ν.

Frenet's formulae are therefore,

with two corresponding sets.

The determinant

each row of wliich contains the same n elements in the same cyclic order, with ′a1 always in the leading diagonal, is the product of n linear factors, which we shall write as follows

where ρ is any primitive nth root of unity.