An attempt is made to provide a sound basis for the method of singular eigenfunction expansions which has been in vogue in linear transport theory for some decades. The procedure is exemplified by a treatment of the one-dimensional neutron transport equation with a degenerate scattering function. Full-range as well as half-range results are derived. At the end of the paper the implications for a certain matrix factorization problem are given.

The four transverse galvanomagnetic and thermomagnetic effects were determined in the case of iron, nickel, cobalt, silver, copper, zinc, cadmium, aluminium.

The results were considered in the light of the electron theory of conduction in metals as formulated by G. H. Livens, and were found to be in fair accord with the theory as regards the ratios of the effects, but they could not be reconciled with the expressions given for the effects themselves.

Present knowledge of the oogenesis of caddis-flies is scanty. Marshall's paper (17) on the ovary of Platyphylax designatus is the only detailed account of the growth and history of the ovarian elements during the larval and pupal stages, but he does not deal with the cytoplasmic inclusions.

We study the Helmholtz equation with electromagnetic-type perturbations, in the exterior of a domain, in dimension n ≥ 3. Using multiplier techniques in the style of Morawetz, we prove a family of a priori estimates from which the limiting absorption principle follows. Moreover, we give some standard applications to cases with an absence of embedded eigenvalues and zero resonances, under explicit conditions on the potentials.

Hyperasymptotic expansions were recently introduced by Berry and Howls, and yield refined information by expanding remainders in asymptotic expansions. In a recent paper of Olde Daalhuis, a method was given for obtaining hyperasymptotic expansions of integrals that represent the confluent hypergeometric U-function. This paper gives an extension of that method to neighbourhoods of the so-called Stokes lines. At each level, the remainder is exponentially small compared with the previous remainders. Two numerical illustrations confirm these exponential improvements.

Transition probabilities are calculated which make the construction of diffusions on finitely ramified fractals straightforward. In contrast to former approaches using Brouwer's Fixed Point Theorem, we consider an approximation procedure based on the iteration of a nonlinear map L. Physically, this is done by ‘coarse-graining-renormalisation of finite electric resistor networks’. Mathematically, it is a convergence problem for quotients of Dirichlet forms on finite graphs. These graphs approximate finitely ramified fractals. The basic tool is a contraction theorem for the renormalisation map L which allows the use of known results about nested fractals for non-nested (p.c.f. self-similar) ones. In general, the above contraction is not strict because several linear independent fixed points occur.

The so–called rectal gland of Elasmobranch fishes claimed notice early in the history of scientific research, as might indeed be expected from the obviousness of its appearance and the invariability of its occurrence. But, in spite of this fact, the rectal gland remains one of those organs the knowledge of the structure of which is unsatisfactory, and the conjecture as to the function of which is consequently hazy.

Our knowledge of the anatomy of Dicksonia is mainly based on the description of D. Barometz given by Gwynne-Vaughan. The stem of this species possesses a radial dictyostele, which, on account of the shortness of the leaf gaps, is not far removed from the solenostelic condition. The leaf trace arises as a broad ribbon of tissue with inwardly folding margins, which soon, however, breaks up to form a horse-shoe of isolated strands.

The article is devoted to Hardy type inequalities on closed manifolds. By means of various weighted Ricci curvatures, we establish several sharp Hardy type inequalities on closed weighted Riemannian manifolds. Our results complement in several aspects those obtained recently in the non-compact Riemannian setting.

We study the two-phase Stokes flow driven by surface tension with two fluids of equal viscosity, separated by an asymptotically flat interface with graph geometry. The flow is assumed to be two-dimensional with the fluids filling the entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces up to critical regularity. The main technical tools are an analysis of nonlinear singular integral operators arising from the hydrodynamic single-layer potential and abstract results on nonlinear parabolic evolution equations.

Let Ω be a bounded Lipschitz domain in R2, let H be a 2 × 2 diagonal matrix with det(H) = 1. Let ε > 0 and consider the functional over AF ∩ W2,1(Ω), where AF is the class of functions from Ω satisfying affine boundary condition F. It can be shown by convex integration that there exists F ∉ SO(2) ∪ SO(2)H and u ∈ AF with I0(u) = 0. Let 0 < ζ1 < 1 < ζ2 < ∞, .

In this paper we begin the study of the asymptotics of mε ≔ infBF∩W2,1Iε for such F. This is one of the simplest minimization problems involving surface energy for which we can hope to see the effects of convex integration solutions. The only known lower bounds are lim infε→0mε/ε = ∞.

We link the behaviour of mε to the minimum of I0 over a suitable class of piecewise affine functions. Let {τi} be a triangulation of Ω by triangles of diameter less than h and let denote the class of continuous functions that are piecewise affine on a triangulation {τi}. For the function u ∈ BF let be the interpolant, i.e. the function we obtain by defining ũ⌊τi to be the affine interpolation of u on the corners of τi. We show that if for some small ω > 0 there exists u ∈ BF ∩ W2,1 with then, for h = ε(1+6399ω)/3201, the interpolant satisfies I0(ũ) ≤ h1−cω.

Note that it is trivial that , so we reduce the problem of non-trivial (scaling) lower bounds on mε/ε to the problem of non-trivial lower bounds on .

Of the explanations advanced in respect of the observed inequality in the viability and longevity of the sexes in man, the sex‐linked lethal theory advocated by Lenz (1923), Gunther (1923), Geiser (1924–5), Huxley (1924), Schirmer (1929), amongst others, has attracted most support. One reason for this is undoubtedly that it is a genetic theory promulgated at a time when biological thought was being profoundly influenced by the many spectacular discoveries which had been made in this particular science.

Observations were made on the body temperature of a large number of diving and swimming birds of eighteen different species in the Orkney Islands and Firth of Forth, Scotland, and on and around Cayuga Lake, N.Y., U.S.A., immediately after they were killed by shooting.

1. In all the species examined, where the sex was determined, it was found that the rectal temperature of the male was slightly below that of the female.

2. Of the orders examined the highest temperatures were found in the Longipennes and the lowest in the Tubinares. When arranged according to body temperature the series does not run parallel with the zoological series.

In conclusion, I wish to express my indebtedness to Dr H. D. Reed and Dr A. H. Wright for help in identifying the specimens obtained from Cayuga Lake and the surrounding district.

The chemical composition of submarine glauconite is of considerable interest in view of the fact that glauconite is the only silicate which is synthesized at the bottom of the sea, and apparently nowhere else. Numerous analyses of this mineral have been published from time to time, but the results are far from uniform, because the material almost always—certainly in most of the older analyses—was anything but pure. The analysis which inspires most confidence on this score is one which was recently carried out in the Challenger laboratory by Collet and Lee on a purified granular glauconite dredged off the Californian coast by U.S.S. Tuscarora (1879).

In nearly every narrative of exploration in uncivilised tropical regions, accounts are given, often no doubt somewhat fanciful, of poisonous substances which are said to possess the most remarkable properties. Usually these poisons are of vegetable origin; and the great majority may be included in the two divisions of ordeal and of arrow poisons, according as they are applied to one or other of these purposes. Among the most remarkable of the ordeal poisons are the Tanghinia venifera of Madagascar, the Physostigma venenosum of Old Calabar, and the Akazga poison of the Gaboon; and of the arrow-poisons, the famous Curara or Wourali of South America, and the Antiaris toxicaria of Java.

Professor Allman described a new genus of Polyzoa, obtained by the Rev. A. M. Norman and Mr J. Gwyn Jeffreys, from deep-sea dredgings in Shetland.

Its cœnœcium consists of a branched tube, partly adherent and partly free, the free portion forming tubes of egress, through which the polypides move in the acts of exsertion and retraction. In the walls of the adherent portion a rigid chitinous rod is developed along their attached side, and to this rod the polypides are connected from distance to distance each by a flexible cord or funiculus.

The processes of long-wave low-temperature radiation in the atmosphere have been studied with a view to investigating the extent to which they may be regarded as diffusion-like; the decrease of water-vapour with height is taken to be exponential, that of temperature to be constant up to the tropopause, and then to be zero. Full allowance is made for the radiation being diffuse, involving the functions ; such properties of these functions as are required in the further work are studied. The expression for the upward flow of radiation is obtained in the form of a power series in the temperature gradient in the troposphere; the coefficient of the first power of the temperature gradient is defined as the “coefficient of radiative diffusion.” There is shown to be a certain amount of ambiguity about the coefficient so defined.

A model atmosphere resembling fairly well the conditions in N.W. Europe is considered: the upward flow of radiation and the coefficient of radiative diffusion are calculated; for this purpose we require Hettner's measurements of water-vapour absorption as modified by Simpson.

The effect of the heating of the ground during the day to a temperature above that of the air near by is then considered; it is shown that the study of the results of this factor which are of importance in the interpretation of continuous temperature records near the ground requires measurements of the absorption of water-vapour for the wave-lengths for which it is particularly opaque; these do not appear to exist.

We study the asymptotic behaviour of solutions of the fast diffusion equation near extinction. For a class of initial data, the asymptotic behaviour is described by a singular Barenblatt profile. We complete previous results on rates of convergence to the singular Barenblatt profile by describing a new phenomenon concerning the difference between the rates in time and space.

Using a regular Borel measure μ ⩾ 0 we derive a proper subspace of the commonly used Sobolev space D1(ℝN) when N ⩾ 3. The space resembles the standard Sobolev space H1(Ω) when Ω is a bounded region with a compact Lipschitz boundary ∂Ω. An equivalence characterization and an example are provided that guarantee that is compactly embedded into L1(RN). In addition, as an application we prove an existence result of positive solutions to an elliptic equation in ℝN that involves the Laplace operator with the critical Sobolev nonlinearity, or with a general nonlinear term that has a subcritical and superlinear growth. We also briefly discuss the compact embedding of to Lp(ℝN) when N ⩾ 2 and 2 ⩽ p ⩽ N.

A representation theory for the Hankel transformation Hv, defined for suitable f and for v > −1, by

on the spaces ℒμ, p defined by the norm

is developed. Using this, representation theories are developed for some related transformations.