Semilinear elliptic partial differential systems of second order with weak coupling are considered in exterior domains Ω ⊆ ℝN, N≧3. Conditions on the nonlinearities are given which guarantee the existence of solutions u with positive components in Ω such that u|∂Ω = 0 and u(x)→0 uniformly as |x|→∞. Asymptotic decay estimates for the solutions are established, including an exponential decay law under extra hypotheses.

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 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.

We show that the property of linear operators to be in the surjective hull (injective hull) of the ideal of strictly singular (strictly cosingular) operators between Banach spaces is an interpolation property with respect to the real interpolation method with parameters 0 < ủ < 1 and < p < ℞.

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.

The methods developed in I, II of this series of papers are applied to a solution of a variety of dual series relations involving trigonometric series. In general the problem is reduced to one of solving (usually by numerical methods) a Fredholm integral equation of the second kind for an auxiliary function g(t), but for certain values of the parameters it is possible to obtain analytical solutions of the integral equations and these cases are considered in detail.

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.

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.

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.

The volume marked “Logarithms 0” now placed on the Society's table contains, to 28 places, the Logarithms of all numbers up to Ten Thousand. In preparing it, care was taken that each prime should be placed in connection with two distinct sets of other prime numbers, in order that all likelihood of mistake should be avoided. This amounts to the computation of each logarithm by two independent processes. Opposite each prime, reference is made to the page of the record of calculations; and the accompanying volumes marked “Construction, I., II.” contain minutes of each articulate step, so that the genesis of any logarithm may readily be traced and examined.

According to a usage which has been adopted from the German of Helmholtz by the best English scientific writers on sound, a sound is called a “simple tone,” or without qualification a “tone,” when the variation of pressure of the air in the neighbourhood of the ear which is the immediate excitant of the sense is according to a simple harmonic function of the time; that is to say, when the whole pressure of the air varies in simple proportion to the distance, from a fixed plane, of a point moving uniformly in a circle.

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.

According to Percy's definition, “clays are termed fireclays or refractory clays when they resist exposure to a high temperature without melting or becoming in a sensible degree soft and pasty.” The refractoriness of fireclay is due to its low proportions of fluxes. The best known British fireclays occur beneath the coal seams of the coal measures, and are therefore known as “under-clays” or “seat-clays.” The explanation of their paucity of fluxes usually offered is that the alkalies have been withdrawn as food by the vegetation which formed the coal. Thus Professor Tarr remarks that fireclays “are particularly abundant in the Carboniferous rocks associated with coal beds, the plants having been instrumental in the withdrawal of the alkalies.” This view has been widely accepted, and the death of the successive coal forests has been attributed to the exhaustion of the plant foods in the underlying soils. Caution is, however, necessary in the application of this theory; for plants can only withdraw soluble alkalies from the soil, and the alkalies are usually present in the form of silicates, which decompose slowly. Hence the total removal of alkalies from a soil by vegetation would be a very lengthy process. Moreover, alkalies are not the only, or indeed the most common of fluxes, for iron is usually the most potent. Further, some good fireclays are not associated with coal seams, while well-washed river muds may be poorer in alkali than an underclay, the alkalies having been washed out of the material. Thus the river muds from the Rhine near Bonn, analysed by Bischoff, contain only ·89 per cent. of potash and ·39 per cent. of soda; or after deducting the water and organic material the percentages of potash and soda are 1·02 and 0·45 respectively. Another analysis by Bischoff of river mud from the Rhine collected above Lake Constance contained potash ·55 per cent. and soda ·54 per cent., or after deducting the water and carbonates the percentage of these two constituents only rose to ·91 per cent. of potash and ·90 per cent. of soda.

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.

The Glossopharyngeal.—The glossopharyngeal is usually looked upon as the most typical of the cranial nerves. In Amia, according to Allis, it consists of post- and præ-branchial branches, a visceral or pharyngeal branch, and a dorsal branch which takes part in innervating the lateral line system—supplying by a single twig one of the sense organs of the lateral canal, and in addition a row of pit organs.

In this paper we consider ‘slowly’ oscillating perturbations of almost periodic Duffing-like systems, i.e. systems of the form ü = u − (a(t) + α(wt))W′(u), t ∈ ℝ, u ∈ ℝN, where W ∈ C2N(ℝN, ℝ) is superquadratic and a and α are positive and almost periodic. By variational methods, we prove that if w > 0 is small enough, then the system admits a multibump dynamics. As a consequence we get that the system ü = u − a(t)W′(u), t ∈ ℝ, u ∈ ℝN, admits multibump solutions whenever a belongs to an open dense subset of the set of positive almost periodic continuous functions.

Using the constitutive relationship for a Maxwell body, as an example of a viscoelastic fluid, the equations of motion are derived in the Fourier wavenumber-time domain, and specialised to the case of isotropic turbulence. It is shown that, for grid-generated turbulence, the model predicts increased spectral intensity levels, reduced decay rates and steepening of the spectrum in wavenumber, relative to the Newtonian case. These forms of behaviour have been observed in dilute solutions of drag reducing polymers. The key factor in this is found to be the presence of an ‘elastic’ non-linear term in the equations of motion: this term reverses the normal direction of turbulent energy transfer in wavenumber.

A perturbation theorem for symmetric differential expressions is given and applied to even-order expressions. We assume the coefficients p0(t), pn(t) to be eventually positive and to have real powers of t as dominating terms. Then we are able to admit for the absolute values of the other coefficients a rate of growth depending on the growth of both of the coefficients p0, pn in order to obtain minimal deficiency indices.

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.