In an earlier paper (Aitken, 1932) the author put forward a method of polynomial interpolation which was designed to make full use of the speed and power of modern calculating machines in forming and summing products of two factors. The present paper is devoted to two further applications of the method. It is necessary to give a brief review and explanation of the process used, in order that these applications may be properly understood.

Till within recent years the epiphysis cerebri has been generally regarded as a mesial outgrowth from the roof of the thalamencephalon in Vertebrates. The researches of Béraneck (5), Dendy (11), Hill (17), and Locy (19), however, tend to demonstrate the fact that this structure arises in the form of two bilateral outgrowths; while Gaskell (12) has drawn attention to its bilateral nature in Ammocœtes. Some observations which the author made on the development of the epiphysis in Amphibia (8 and 9) were found to agree in the main with those of the above-mentioned workers. The present research was therefore undertaken with the view of corroborating the results which had been obtained in the Amphibia, and it was found that these received support in the case of the chick.

The shape of radicals of semigroups algebras of commutative and cancellative semigroups is studied. The questions asto when a radical of those algebras is homogeneous and if homogeneous radicals have more regular form are examined.

We study the evaluation of multilinear forms under arbitrary rearrangements of the entries of increasing n-tuples x(1), x(2),…, x(m), and we show that the difference of two such multilinear forms under certain circumstances can be written as a sum of obviously definite forms.

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.

We define spaces of functions on an interval naturally related to functional differential equations of retarded type in the sense that exponential solutions of such equations form a uniform basis (or Riesz basis) in these spaces. Embedding theorems relating these spaces to the Sobolev spaces are established using a result on Carleson's measure defined on a half plane.

The deficiency indices (mean deficiency index) and the essential spectrum for a class of odd order ordinary differential expressions are determined. The considered expressions are relatively bounded or relatively compact perturbations of symmetric expressions with odd order terms having as coefficients real powers of the independent variable.

In 1975 K. W. Brodie and W. N. Everitt dealt with integral inequalities of the form

in the two cases T = ℝ, T = ℝ+, and obtained the best possible constants KT(μ) for all μ ε ℝ. The proof was not elementary, but an elementary proof was given in 1977 by E. T. Copson. This note shows how Copson's method can be greatly simplified so as to obtain the results in a more straightforward manner.

A Ljapunov equation XL − BX = C appears in stabilisation studies of linear systems. Here, the operators L, B, and C are given linear operators working in infinite-dimensional Hilbert spaces, which are derived from a specific control system. We have so far considered the case where L is a general elliptic operator of order 2 in a bounded domain of an Euclidean space. When L is instead a self-adjoint elliptic operator working in an interval of ℝ1, we derive here a stronger geometrical character of the solution X to the Ljapunov equation. The result is applied to stabilisation of one-dimensional diffusion equations.

We prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed m there are at most ${\cal O}_{\epsilon }(n^{{3}/{5}+\epsilon })$ solutions of ${m}/{n} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$. This improves upon a result of Browning and Elsholtz (2011) and extends a result of Elsholtz and Tao (2013) who proved this when m=4 and n is a prime. Moreover, there exists an algorithm finding all solutions in expected running time ${\cal O}_{\epsilon }(n^{\epsilon }({n^3}/{m^2})^{{1}/{5}})$, for any $\epsilon \gt 0$. We also improve a bound on the maximum number of representations of a rational number as a sum of k unit fractions. Furthermore, we also improve lower bounds. In particular, we prove that for given $m \in {\open N}$ in every reduced residue class e mod f there exist infinitely many primes p such that the number of solutions of the equation ${m}/{p} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$ is $\gg _{f,m} \exp (({5\log 2}/({12\,{\rm lcm} (m,f)}) + o_{f,m}(1)) {\log p}/{\log \log p})$. Previously, the best known lower bound of this type was of order $(\log p)^{0.549}$.

The probability that a neutron leads to a divergent chain reaction in a nuclear reactor is governed by a nonlinear integro-partial-differential equation [1]. A model case of this equation was completely analysed by Pazy and Rabinowitz [2,3]. The purpose of this paper is to extend their results to the general case and to tackle some related topics.

The author stated, as the results of some experiments which he had made on the phosphorescence of a species of Beröe (Idyia) which abounded during the last summer in the Firth of Forth, that the light is never emitted as long as the animal is exposed to daylight; and that withdrawal from the daylight for some time is necessary, in order to impart to the Beröe the property of phosphorescence.

In December 1908 I received from Mr A. Earland a fragment of a rock, believed to be chalk, which had been dredged from the bottom of the North Sea. Mr Earland also informed me that he believed boulders of a similar rock occurred in some profusion near what is known as the Kinnaird Deep, off the northern coast of Aberdeenshire. As a result of our correspondence a little later, Professor D'Arcy Thompson asked me to investigate such boulders as might presumably be chalk, dredged from the northern parts of the North Sea during the operations of the s.s. Goldseeker, a vessel employed by the North Sea Fisheries Commission, and he has kindly permitted me to include a description of the boulders found by the Goldseeker in the details of this paper.

In the course of a study of fecundity in Drosophila funebris (Donald and Lamy, 1937), it was noted that a particular female was laying very few eggs, and that the shape of these was peculiar. Among her descendants other females of similarly low fecundity and laying eggs of the same abnormal shape were encountered, and it was assumed therefore that a mutation affecting fecundity and egg shape had been recognised. In this paper evidence concerning the type of heredity of these characteristics is presented, and a statistical description of the morphological aspect of the new mutation, the size and shape of the egg, as compared with those of the normal D. funebris egg, is given.

We consider a nonlinear eigenvalue problem in the form F(x, λ) = Ax – T(λ)x – R(x, λ) = 0 with F:X × ℝ →Y where X, Y are Banach spaces. We assume that F(0, λ) = 0 for all λ ∈ ℝ and seek bifurcation points; that is, values λ0 ∈ ℝ for which there are solutions to F(x, λ) = 0 with x ≠ 0 in any neighbourhood of (0, λ0). Corresponding to these bifurcation points we obtain global properties of the maximal connected subset of solutions to F(x, λ) = 0 containing (0, λ0).

Generalised topological degree techniques are employed in the proofs of our results without requiring a transversality condition. The operators involved belong to the general class of A-proper mappings which include compact and k-ball contractive perturbations of the identity operator, accretive mappings, and many more.

The injection of adrenaline hydrochloride into mature female mice is able to inhibit the occurrence of oestrus.

The injection of adrenaline hydrochloride into immature female mice prevents the onset of sexual maturity.

Adrenaline does not interfere with the action of α-hormone (œstrin) on the uterus and vagina; it has an inhibitory effect on the action of gonadotropic hormones (prepared from pregnancy urine) on the ovary.

I should like to thank Mr P. G. Marshall for the supply of gonadotropic preparation from pregnancy urine used in these experiments.

General formulae are obtained for the reflection and transmission of harmonic acoustic waves by a curved interface between two media when the frequency is high. In addition to refracted rays there turn up tunnelling rays, if the surface is concave to the source, which are emitted from an evanescent region when the phenomenon of total internal reflection would be anticipated. Uniformly valid formulae dealing with the transition from refraction to tunnelling in both transmission and reflection are derived.

The theory is applied to the circular cylinder and to the top-hat circular jet. In the latter case it is suggested that radiation may tend to be more significant at inclinations of 50°-65° (downstream) and 25°-40° (upstream) to the axis of the cylinder. The augmentation due to tunnelling rays in propagation upstream is mentioned.

This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem The function f is allowed to change sign and has an asymptotically linear or a superlinear behaviour.

Let a, c ≥ 0 and let B be a compact set of scalars. We show that if X is a Banach space such that the canonical projection π from X*** onto X* satisfies the inequality

and 1 ≤ λ < max |B| + c, then every λ-commuting bounded compact approximation of the identity of X is shrinking. This generalizes a theorem by Godefroy and Saphar from 1988. As an application, we show that under the conditions described above both X and X* have the metric compact approximation property (MCAP). Relying on the Willis construction, we show that the commuting MCAP does not imply the approximation property.