Formulas are determined for the deficiency numbers of a formally symmetric ordinary differential operator with complex coefficients which have asymptotic expansions of a prescribed type on a half-axis. An implication of these formulas is that for any given positive integer there exists a formally symmetric ordinary differential operator whose deficiency numbers differ by that positive integer.

The self-adjointness of Δ2 + κ|x|−4 (κ>κ0 = κ0(N)) in L2(ℝN) is established as an application of the perturbation theorem in terms of Re(Au, Bεu), u ∈ D(A), for two non-negative self-adjoint operators A, B in a Hilbert space, where the family {Bε}ε>0 is the Yosida approximation of B. A key to the proof lies in a new inequality for the functions ν ∈ L2(ℝN) with |x|2Δν ∈ L2(ℝN) derived by using two real parameters.

We discuss the existence, nonexistence and multiplicity of solutions for a class of elliptic equations in the unit ball with zero Dirichlet boundary conditions involving nonlinearities with supercritical growth. By using Pohozaev type identity we prove a nonexistence result for a class of supercritical problems with variable exponent which allow us to complement the analysis developed in (Calc. Var. (2016) 55:83). Moreover, we establish existence results of positive solutions for semilinear elliptic equations involving nonlinearities which are subcritical at infinity just in a part of the domain, and can be supercritical in a suitable sense.

Bivariational principles are constructed that yield upper and lower bounds to the quantity 〈g0, f〉, where f is the solution of the equation f0−Tf = 0, g0 is a given function and T is a non-self-adjoint linear operator from a Hilbert space into itself. The theory is illustrated by an integral equation of Fredholm type.

In this paper we obtain necessary and sufficient conditions for the existence of Lipschitz minimisers of a functional of the type

where h is a convex function converging to infinity at zero and u is subjected to displacement boundary conditions. We provide examples of body forces f for which the infimum of J(.) is not attained.

We study non-totally geodesic Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so-called angle functions and that such Lagrangian submanifolds are closely related to minimal surfaces in 𝕊3. Indeed, starting from an arbitrary minimal surface, we can construct locally a large family of such Lagrangian immersions, including one exceptional example. We also show that locally all such Lagrangian submanifolds can be obtained in this way.

This paper considers an extension of the following inequality given in the book Inequalities by Hardy, Littlewood and Polya; let f be real-valued, twice differentiable on [0, ∞) and such that f and f are both in the space fn, ∞), then f′ is in L,2(0, ∞) and

The extension consists in replacing f′ by M[f] where

choosing f so that f and M[f] are in L2(0, ∞) and then seeking to determine if there is an inequality of the form

where K is a positive number independent of f.

The analysis involves a fourth-order differential equation and the second-order equation associated with M.

A number of examples are discussed to illustrate the theorems obtained and to show that the extended inequality (*) may or may not hold.

Using the Leray–Schauder degree, we study the existence of solutions for the following periodic differential equation with relativistic acceleration and singular nonlinearity:

where μ > 1 and the weight h: [0, T] → ℝ is a continuous sign-changing function. There are no a priori estimates on the set of positive solutions (a condition used in general to apply the Leray–Schauder degree), and we prove that no solution of the equation appears on the boundary of an unbounded open set during the deformation to an autonomous problem.

Since the discovery by Heusler in 1903 of a magnetic alloy of copper, manganese, and aluminium, considerable interest has been aroused by the peculiarities of its properties and the difficulty of their explanation. In the summer of 1905 a preliminary investigation of a sample of the alloy was made in the Physical Laboratory of Glasgow University, and the results obtained have been published in a paper by Professor A. Gray.

Numerous authors have been concerned with phenomena and problems of senescence, duration of life, order of dying-out, and related subjects. With the exception of studies on Drosophila (Pearl and Parker, 1921; Pearl, 1922), other insects (for literature see Korschelt, 1924) and, in some respects, human subjects (see actuarial literature), most of the research of previous authors was embarrassed by the lack of sufficiently large and sufficiently homogeneous material; many conclusions rest on an insecure basis, since they refer only to isolated specimens or to experiments carried out on small numbers of animals.

The genus Branchiomaldane, represented by a single species vincenti, was founded by Langerhans to contain certain small Polychætes, which he discovered living in sand-covered tubes, among algæ, on the northern shore of Teneriffe. Professor Mesnil has since found specimens, inhabiting mucous tubes situated on the lower side of the calcareous alga Lithothamnion, in rock-pools at St Martin, near Cape la Hague, and has given an account of some of the structural features of these worms, which he showed to be adult and hermaphrodite. Professor Fauvel considered B. vincenti to be a dwarf Arenicola, arrested in development; he therefore merged Branchiomaldane with Arenicola, and designated the worm Arenicola vincenti. The writer has investigated the anatomy of several specimens of this worm, kindly sent to him by Professor Mesnil, and has concluded that the genus Branchiomaldane should be retained; the original name of the worm—B. vincenti—is therefore employed in the following account.

Of the more recent researches on the subject of this paper, mention may be made of the work of H. Zahn* on the galvanomagnetic and thermomagnetic effects in various metals. Zahn has measured these effects in many different metals, and has used his results to test the electron theory of the properties of metals as developed by P. Drude. He has also determined in some cases the temperature variation of the effects.

The author of the present paper has confined his attention to the thermomagnetic transverse effects and the Hall effect. These have been measured in magnetic fields of various strengths and at temperatures varying over a range of about 100 Centigrade degrees.

The experiments were carried out with a view to obtaining some light on the electron theory, and the ratios of the effects are discussed in relation to this theory.

Mr Clerk Maxwell ingeniously suggests the extension of the common theory of the foci of the conic sections to curves of a higher degree of complication in the following manner:—

(1.) As in the ellipse and hyperbola, any point in the curve has the sum or difference of two lines drawn from two points or foci = a constant quantity, so the author infers, that curves to a certain degree analogous, may be described and determined by the condition that the simple distance from one focus plus a multiple distance from the other, may be = a constant quantity; or more generally, m times the one distance + n times the other = constant.

In this paper it is proposed to study the problem of the vibrations of a uniformly-stretched membrane whose superficial density σ depends upon the distance r from a fixed point in the membrane in accordance with the law

in which C and a are constants. The constant a will be known as the critical radius and the circle r = a as the critical circle.

Monthly observations, extending over one year, were made on the rectal temperature of 114 domestic fowls (Gallus gallus, ♀) and records from forty-one of these were obtained for two years. Six different breeds were used, each located in a separate pen, all under similar conditions, and the mean temperatures for each group were plotted out to form an annual temperature curve. It was found that—

1. The lowest temperatures occur in December, January, and February, and the highest in June, July, and August, corresponding in a general way with the temperature of the external air.

2. Barometric pressure does not appear to have any influence on the body temperature of the hen.

3. The curve of egg-production does not coincide with the annual temperature curve, the former reaching its highest level in April and May, the latter in June, July, and August.

If we compare the mean rectal temperature at two periods of the year when the external or weather conditions are approximately the same (April-May and September-October), but when the vitality of the birds, as indicated by the curve of egg-production, moulting, etc., is at a maximum and minimum respectively, we find that the figures are practically identical. This would seem to show that cyclical bodily changes have little effect on body temperature as compared with outside influences.

When thio-urea is treated with suitable oxidising agents in presence of acids, salts are formed corresponding to the general formula (CSN2H4)2X2:—

Of these salts the di-nitrate is very sparingly soluble, and is precipitated on the addition of nitric acid or a nitrate to solutions of the other salts. The salts, as a class, are not very stable, and their solutions decompose, especially on warming, with formation of sulphur, thio-urea, cyanamide, and free acid. A corresponding decomposition results immediately on the addition of alkali, and this constitutes a very characteristic reaction for these salts.

The fundamental impulse determining the migration of birds is connected with the reproductive cycle; but when that has been recognised it must also be admitted that there are secondary conditions of environment which regulate the actual movements of which a migration is compounded. Of these weather has long been looked upon as one of the most important. There is still, however, much divergence of opinion as to the extent and nature of the influence of weather, and indeed in recent years statistical analysis has led workers, such as Bretscher (1915), to doubt whether any satisfactory connection between meteorological conditions and migration can be demonstrated, and Lincoln to announce boldly that “the state of the weather at any point has little if anything to do with the time of arrival of migratory birds” (1935, p. 59).

Among many geological questions which wait solution, there is probably none more interesting or perplexing than the agency by which Boulders or “blocs erratiques,” as the French term them, have come to their present sites. I allude, of course, not to blocks lying at the foot of some mountain crag from which they have fallen by the decay or weathering of the overhanging rocks, but to blocks which have manifestly been transported great distances, after being detached from the rocks of which they originally formed part.

In August 1901 Mr Thomas Anderson, merchant, Hillswick, to whom and to other members of whose family I have on several occasions been indebted for specimens to illustrate the Zoology of Shetland, wrote to tell me that a dead whale had been found floating near Hillswick and had been towed into Roeness Voe. It was within three miles of land when seen by the fishermen and was claimed by the Crown. From the appearance of the animal Mr Anderson thought that it was a sperm whale, and he sent me the following description, from which it was clear that he had correctly recognised it.

We prove that the two-dimensional Brown–Ravenhall operator is bounded from below when the coupling constant is below a specified critical value—a property also referred to as stability. As a consequence, the operator is then self-adjoint. The proof is based on the strategy followed by Evans et al. and Lieb and Yau, with some relevant changes characteristic of the dimension. Our analysis also yields a sharp Kato inequality.