Cases of absence or defect of the corpus callosum are of interest not only because of their great rarity, but because of the light which they throw on the distribution and functions of this commissure, and on the development of the mesial aspects of the cerebral hemispheres.

Let n ⩾ 3 and 0 < m < (n − 2)/n. We extend the results of Vazquez and Winkler (2011, J. Evol. Equ. 11, no. 3, 725–742) and prove the uniqueness of finite points blow-up solutions of the fast diffusion equation ut = Δum in both bounded domains and ℝn × (0, ∞). We also construct initial data such that the corresponding solution of the fast diffusion equation in bounded domain oscillates between infinity and some positive constant as t → ∞.

In this paper, we study a symmetry-breaking phenomenon of a least-energy solution to a nonlinear Schrödinger equation under suitable assumptions on V(x), where λ > 1, p > 2 and χA is the characteristic function of the set A = [−(l + 2), −l] ∪ [l,l + 2] with l > 0. We also study asymptotic profiles of least-energy solutions for the singularly perturbed problem for small ε > 0.

An explicit solution is derived for the dual integral equations

when A(ξ) takes the form

It is then shown how the basic technique can be adapted to derive explicit solutions for more general forms of A(ξ) such as

and

and linear combinations of such sums.

Any solution to the problem

is shown to decay to zero in the strong topology of the energy space as t→ ∞. The function β is allowed to be nonmonotone and d is a nonnegative function strictly positive on a nonvoid subset of (0, л).

For a family of elliptic operators with periodically oscillating coefficients, $-{\rm div}(A(\cdot /\varepsilon )\nabla )$ with tiny ε > 0, we comprehensively study the first-order expansions of eigenvalues and eigenfunctions (eigenspaces) for both the Dirichlet and Neumann problems in bounded, smooth and strictly convex domains (or more general domains of finite type). A new first-order correction term is introduced to derive the expansion of eigenfunctions in L2 or $H^1_{\rm loc}$. Our results rely on the recent progress on the homogenization of boundary layer problems.

L'Expédition Antarctique Nationale Ecossaise, sous le commandement de M. W. S. Bruce, a recueilli une importance collection de Poissons au cours de son exploration de l'Atlantique austral et de la Mer de Weddell (1902–1904).

A la suite de la publication de mon mémoire sur les Poissons de la Belgica,* M. Bruce m'ayant fait l'honneur de me prier d'étudier les matériaux ichthyologiqucs qu'il avait rapportés, je me propose de soumettre à la Société Royale d'Edimbourg, si elle veut bien me le permettre, les résultats de mes recherches, au fur et à mesure de leurs progrèss, afin d'assurer à la Scotia la priorité de ses découvertes, en attendant que je sois assez avancé pour faire paraître mon travail définitif.

§ 1. In 1880 Stringham proved that in space of four dimensions there exist six and no more regular rectilinear figures, whose boundaries are regular polyhedra. The same result was arrived at independently by Hoppe in the following year. In 1883 Schlegel gave an extensive investigation of the same problem, and constructed projection-models of the six regular figures, which were exhibited at the Magdeburg meeting of the Society of German Naturalists in 1884. This series of models was published by the firm L. Brill of Darmstadt and is obtainable from their successors, Martin Schilling in Leipzig.

That liquids, if finitely compressible, must (at any one temperature) become steadily less compressible as the pressure is raised, seems to be obvious without any attempt at proof. Yet the assertion is even now generally made, mainly in consequence of an erroneous statement of Örsted, which has been supported by some comparatively recent investigations of Cailletet and others, that the compressibility of water (at any one temperature) is practically the same at all pressures not exceeding a few hundred atmospheres.

In the early part of 1937, during the course of excavations for a sewage extension tunnel at Queen's Park, on the south side of Glasgow, some pieces of bone were obtained by the workmen from a bed of gravel at a depth of 70 feet from the surface. Through the kindness of Mr T. Somers, Master of Works and City Engineer to the Corporation of Glasgow, these remains were loaned to the Geological Survey and, again with his sanction, submitted for examination to Professor James Ritchie of Edinburgh University, one of the collaborators in this note. Professor Ritchie reported that the fragments of bone fitted together to form the greater part of the right radius and ulna of a reindeer. At a later stage of the excavations a portion of an antler was disinterred from the same bed. It seems desirable to record these occurrences in some detail, and for convenience this communication has been divided into two parts. In the first of these the geological position and age of the fossiliferous deposit are briefly discussed, while the second part is devoted to a description of the remains themselves and to a comparison with similar remains elsewhere.

The object of the present communication is to give the purport of the remainder of the letter referred to in the title, as well as that of other letters bearing on the same subject written by the late Mr Froude shortly afterwards, which were not at hand at the time the extract referred to was printed.

The following paper is a description of experiments carried out to determine the coefficient of diffusion of alkali salt vapours in a bunsen flame.

The colouration of the flame when a bead of salt is heated in it seems to be due to the presence of neutral atoms of the metallic vapour, and the intensity of the colour at any part of the flame depends on the concentration of the metallic vapour at the particular point.

The fossil cuticles of the paper-coal from the Moscow Basin in Central Russia were first described and figured by Auerbach and Trautschold in 1860. Since then these cuticles have been the subject of much discussion, especially with regard to the interpretation of their numerous regularly disposed perforations. According to the various views of previous investigators, several generic and specific names have been assigned to these cuticles. Auerbach and Trautschold, regarding the perforations as the result of the fall of the leaf-cushions as well as the leaves, considered the cuticles to belong to the genus Lepidodendron, and named the plant to which they belonged Lepidodendron tenerrimum.

Investigations into the nature of the organic matter in oil-shales began at the time of the famous Torbanehill case in 1854, when experts attempted to settle the question as to whether the substance known as “ Torbanite ” or “ Boghead Mineral” was a coal or an oil-shale. Several witnesses at the trial (Gillespie v. Russel, Session Papers, 1854) maintained that the oilproducing material in the Mineral was of organic origin, while others pronounced it to be bituminous and produced by subaqueous eruptions. T. S. Traill, M.D., proposed for the Boghead Mineral the name “ Bitumenite,” as it seemed to him to “ consist of much bitumen, mingled with earthy matter” (Trans. Roy. 8oc. Edin., 1857, xxi. p. 7). Dr Redfern (Quart. Jonrn. Micros. Soc, 1855, x. pp. 118-119), on the other hand, supposed the round orange-yellow bodies which occur in torbanite to have had their origin in "“ a mass of vegetable cells and tissues which have been disintegrated and otherwise changed by maceration, pressure, and chemical action, and subsequently solidified."” C. E. Bertrand and B. Renault (Bull. Soc. Hist. Nat. Autun, 1892-3) on microscopic examination have classed these bodies as the remains of gelatinous algae which have been altered by bacterial action.

In 1903 I communicated to the Society a paper on the relation between magnetization and resistance of nickel at high temperatures (1). In this paper the magnetization was along the direction in which the resistance was measured. A second paper, in which the magnetization was transverse to the direction in which the resistance was measured, was communicated in 1906 (2). In these later experiments a flat coil of nickel wire was used; and it was necessary to use very high fields before an appreciable change of resistance was obtained.

For comfort the total future world population faces a limit. Three or more educated generations seem likely before the population growth will be checked. It seems prudent to envisage a world population of about 15 thousand million in a century or two. Their energy needs for food, water, warmth, clean air, locomotion and management of wastes may be assessed at 5 to 50 thermal kilowatts per capita. That amounts to 0·05 per cent, to 0·5 per cent, of the energy supply from the sun. Generating this amount by nuclear fission would cause no severe pressure on world resources of uranium and thorium for hundreds of centuries, if near-breeders such as CANDU reactors are used exploiting the uranium-thorium fuel cycle. Neither the cost of fuel supply nor of waste management to keep the environment unburdened by extra radioactivity need be as high as prices now paid for coal and oil. The addition of breeders or other means such as nuclear fusion or spallation for increasing the neutron supply would significantly lower the use of uranium and thorium but would not necessarily reduce the cost of power.

Using a fixed point theorem on operators expanding a cone in a Banach space we prove the existence of positive solutions of superlinear boundary value problems

At the same time we get bounds (or even inclusions) of positive solutions.

We prove that, for every non-empty open subset of a rectangular plate, there exists a positive number T such that no vibration of this plate with fixed edges can remain strictly above the rest position at all points of the subset during a period T. Moreover, the result remains valid for every non-empty segment parallel to one of the sides of the plate. Our proof is based on some results of non-harmonic analysis.

This paper, extending the work of Stratford [6] considers a boundary layer with uniform pressure when x < x0, and with the pressure in x > x0 so chosen that the layer is just on the point of separation for all x >x0. The required pressure distribution is shown to be

The displacement and momentum thicknesses are also derived as series in powers of ξ (and log ξ), and the shape parameter H then obtained as a similar series. The continuous change in H from the Blasius value (when ξ = 0) towards the Falkner-Skan [3] separation value is convincingly demonstrated, with the aid of the leading terms of an asymptomatic expansion for large ξ.

We study an implicit second-order ordinary differential equation with complete integral. The class of second-order Clairaut-type equations is an important class of completely integrable equations. Indeed, the notion of second-order Clairaut-type equations is the generalization of second-order classical Clairaut equations. We give generic classifications of second-order Clairaut-type equations under the weak equivalence and the one-parameter strict equivalence relations.