Special operator-ideal approximation properties (APs) of Banach spaces are employed to solve the problem of whether the distance functions S ↦ dist(S*, I(F*, E*)) and S ↦ dist(S, I*(E, F)) are uniformly comparable in each space L(E, F) of bounded linear operators. Here, I*(E, F) = {S ∈ L(E, F) : S* ∈ I(F*, E*)} stands for the adjoint ideal of the closed operator ideal I for Banach spaces E and F. Counterexamples are obtained for many classical surjective or injective Banach operator ideals I by solving two resulting ‘asymmetry’ problems for these operator-ideal APs.

In the following paper, the horizontal line method (the Rothe method) is applied to Maxwell's initial boundary value problem. By means of results from abstract perturbation theory, convergence results and error estimates are established.

Let P be an operator of Dirac type on a compact Riemannian manifold with smooth boundary. We impose spectral boundary conditions and study the asymptotics of the heat trace of the associated operator of Laplace type.

We discuss convex l-subgroups of an l-group G in their role as direct summands, not so much of G as of each other. This is done by writing A ≥dB for subgroups A, B to mean that A is a direct summand of B, and studying the properties of the resulting poset. It is shown to be a hypolattice, that is, to have local lattice properties in a certain sense. However it need not be a lattice; and there may exist meets of pairs of elements, outside the hypolattice structure. It need not be conditionally complete even when G is conditionally complete. We look also at the map which sends a subgroup to its lattice-closure; the lattice-closed subgroups also form a hypolattice. Our main result asserts that this hypolattice is conditionally complete if G is. The paper ends with some examples and counter examples in C(X).

Given and similar , modelled on radial Laplace-Beltrami operators (ρp = , in this paper we begin the study of transmutations which leads to elliptic equations Working with and transmutations Qm → −D2 for m > −½ and −D2 → for m < −½, we obtain a transmutation formulation and derivation of many results of generalized axially symmetric potential theory in the first case and in both cases generalized Hilbert transforms (different). Canonical generalizations are then automatic using general transmutation theory.

We give a concise approach to generalising the inequalities of Wirtinger, Hardy, Weyl and Opial by using the well-known inequality: if X and Y are non-negative, then

for p > 1 (0 < p < 1), respectively.

This is the second part of our study on the spatially heterogeneous predator–prey model where the interaction is governed by a Crowley–Martin type functional response. In part I, we have proved that when the predator competition is strong (i.e. k is large), the model has at most one positive steady-state solution for any $\mu \in \mathbb {R}$, moreover it is globally asymptotically stable for any $\mu >0$. This part is denoted to study the effect of saturation. Our result shows that the large saturation coefficient (i.e. large m) can not only lead to the uniqueness of positive solutions, but also lead to the multiplicity of positive solutions, moreover the stability of the corresponding positive solutions is also completely obtained. This work can be regarded as a supplement of Ref. [10].

We study the $\Gamma$-convergence of nonconvex vectorial integral functionals whose integrands satisfy possibly degenerate growth and coercivity conditions. The latter involve suitable scale-dependent weight functions. We prove that under appropriate uniform integrability conditions on the weight functions, which shall belong to a Muckenhoupt class, the corresponding functionals $\Gamma$-converge, up to subsequences, to a degenerate integral functional defined on a limit weighted Sobolev space. The general analysis is then applied to the case of random stationary integrands and weights to prove a stochastic homogenization result for the corresponding functionals.

In this paper, we investigate the $H^{p}(G) \rightarrow L^{p}(G)$, $0< p \leq 1$, boundedness of multiplier operators defined via group Fourier transform on a graded Lie group $G$, where $H^{p}(G)$ is the Hardy space on $G$. Our main result extends those obtained in [Colloq. Math. 165 (2021), 1–30], where the $L^{1}(G)\rightarrow L^{1,\infty }(G)$ and $L^{p}(G) \rightarrow L^{p}(G)$, $1< p <\infty$, boundedness of such Fourier multiplier operators were proved.

Little is known regarding the integration of Legendre Functions with respect to their degrees. In this paper several such integrals are evaluated, three different methods being employed. In § 2 proofs are given of a number of formulae which are required later. In § 3 an example is given of the evaluation of an integral by contour integration. The following section contains the proof of a formula of the Fourier Integral type, a special case of which was given in a previous paper (Proc. Roy. Soc. Edin., vol. li, 1931, p. 123). In § 5 an integral is evaluated by employing Fourier's Integral Theorem; while in § 6 other integrals are evaluated by. means of expansions in series.

The object of the present paper is to introduce certain surfaces, which will be shown to play the same part in the general electromagnetic field as Faraday's tubes of force do in electrostatic and magnetostatic fields.

The value of Faraday tubes in electrostatics and magnetism has led many investigators to seek for them a wider application, in connection with variable electromagnetic fields.

In this paper chemical substances are represented as “operands,” and chemical processes as “operators”—thus, φ·x is the substance produced by acting on the substance x by the process φ. When φ, χ, ψ, &c., act independently on x, this is indicated by the notation . When they act one on another to produce a complex operator which acts on x, the notation φ·χ·ψ·x is used. It is shown that in the latter case the commutative law does not apply to operators.

But a few years ago the diverse phenomena of optical activity, such as the changes of rotation which occur with alteration of temperature, of colour of light, of solvent, or of concentration in a solvent, appeared, in spite of the great accumulation of relative data, to be practically independent of each other, and gave very little hope of satisfactory generalisation. Quite recently, however, the possibility of bringing into one scheme all these different branches of the subject has become more than a mere aspiration, and the progress which has been made by several investigators reveals clearly the existence of a deep-seated and farreaching regularity underlying the remarkable sensitiveness to external conditions of the phenomena in question.

The ultra-violet absorption spectra of C2N2, CNCl, CNBr and CNI have been photographed.

Measurements are given of band edges of the C2N2 absorption spectrum in the region 2380 A–1850 A. The influence of temperature on the relative intensities of the bands has been studied.

The long wave-length limits of the regions of continuous absorption due to the cyanogen halides are: CNCl, 2240 A; CNBr, 2540 A; CNI, 3100 A and 2150 A.

We desire to thank Dr Baker, of the Royal Observatory, Edinburgh, for making photometer records of several spectra, Dr Ludlam for his helpful interest in our work, and Imperial Chemical Industries, Ltd., for a grant towards the cost of quartz apparatus.

1. Reference is made to experiments conducted by Flaskämper as a result of which he stated that the removal of the cotyledons at an early stage from seedlings of Phaseolus sp. and of Vicia Faba resulted in an elimination of the pith from the subsequently formed part of the root in both species, and further induced a reduction in the number of poles in the root of the second species.

2. The significance of such observations in a consideration of the theory of the Size Factor is pointed out.

3. The experiments have been repeated. The malnutrition resulting from the decotylation affected the varieties employed in the present work more adversely than with Flaskämper, and usually caused the death of the seedling after a shorter or longer period.

4. Sufficient evidence has been obtained, however, to throw very serious doubt on Flaskämper's conclusions. The two reduction phenomena he described are now shown to occur in the normal development of the roots of these species, and there is no indication that their occurrence is affected by decotylation.

5. A teleological explanation, based on the theory of the Size Factor recently advanced by Bower, is made of this reduction in the number of protoxylem poles which occurs in the normal development of the roots of these plants.

6. The significance of the appearance of a pith in the basal parts only of these roots is discussed.

In conclusion, I desire to express my indebtedness to Professor F. O. Bower, F.R.S., for his constant interest in this work, and to thank Dr S. Williams and other members of the Botany Department for much valuable help. Also my best thanks are due to the Executive Committee of the Carnegie Trust for a grant towards the expense of the illustrations both of the present work and those of a recent paper on “The Stem-Endodermis in the Genus Piper,” published in the Transactions of the Royal Society of Edinburgh.

That the electrical resistance of palladium is increased by hydrogenisation has long been known, but the lack of any very definite information induced me to investigate the matter last winter.

To illustrate my lectures on Molecular Forces, I sometimes use piles of marbles of equal size. Each of these is taken to represent the sphere of action of a symmetrical integrant molecule, in the sense that the attractive and repulsive forces necessary to a theory of molecules are balanced, when the distance of two molecules is equal to the diameter of the sphere. If the integrant molecules be not symmetrical, their spheres of action will be spheroids or ellipsoids.

In arranging such piles, there are two obvious ways of constructing the layers, and two of applying layer to layer.

In the present paper I propose to give the result of my examination of the specimens of fishes which were obtained by Mr John Murray on the West Coast of Scotland, whilst dredging during the last eleven months on board of the “Medusa.” Exact observations as to the bathymetrical distribution of British fishes at certain seasons and localities, such as have been obtained during the cruises of the “Medusa,” are much needed, and if methodically carried out for some years, will prove a most valuable contribution to the British fauna, especially if they are supplemented by similar reports on the invertebrates which were collected simultaneously with, and form part of the food of, the fishe.

Heisenberg and Pauli (1929) developed a general scheme for the quantization of a field, if the field equations can be derived from a variation principle,

Here za denotes the field variables, L is the Lagrangian, and

The scheme of Heisenberg and Pauli is known to be Lorentz invariant. It is the purpose of this paper to show that it is also invariant with regard to all co-ordinate transformations allowed by the general theory of relativity. The method adopted to prove this is that used by Infeld (1937) and Pryce (1937) to prove the invariance of the “New Field Equations” against Lorentz transformations.

In contributions to the theory of circulants this twenty-year period is found to have been still more fruitful than its fruitful predecessor, the number of writings belonging to it being, indeed, a fourth more, and the percentage of unoriginal communications much about the same. The language which appears oftenest is English, the next in order of frequency being Italian, French, German,—an unusual sequence: the American share of the first of these is considerable, and shows a tendency to increase. In our treatment of the subject papers on block circulants have not been kept separate from those on ordinary circulants: strict segregation has been found to necessitate considerable repetition, and the gain got from it in facility of reference is not so great as to overweigh this disadvantage.