A new method is developed for identifying real-valued coefficients r(x), p(x), and q(x) for which all solutions of the fourth-order differential equation

are L2(0, ∞). The results are compared with those derived from the asymptotic theory of Devinatz, Walker, Kogan and Rofe-Beketov.

We consider a system of integrodifferential equations

where f(t, x) and F(t, s, x, y) are almost periodic in t uniformly for parameters, and we assume that the system has a bounded solution u(t). To discuss the existence of an almost periodic solution, we consider the relationship between the total stability of u(t) with respect to a certain metric ρ and the separation condition with respect to ρ. Moreover, we discuss a sufficient condition for the existence of a positive almost periodic solution of a model of the dynamics of an n-species system.

The problems of the evolution and ancestry of the numerous breeds of domestic animals have not yet been finally solved. In many cases craniological methods of investigation have been used, and the results obtained have formed the basis of far-reaching conclusions. Such data are, however, open to serious criticism because the influence of environmental agencies during the life of the individual may be responsible for great variability of characterisation of the plastic animal body. The difficulty is increased by innumerable migrations of the ancestral stocks, by extensive hybridisation between distinct though closely related types, and the subsequent mixing of the various types in the modern breeds.

1. Advantage has been taken of the large size of Glyptonotus to study certain structural features, especially skeletal, which cannot be so readily investigated in smaller isopods. Where possible, an attempt has been made to correlate structural peculiarities with functional use.

2. The Legs.—The peculiar articulation, analogous to a spheroidal bony joint, between the coxopodites and basipodites of the thoracic limbs, is described in detail. A comparison is instituted between the full flexion-complex of the isopodan and of the brachyuran walking limb. The peræopods are contrasted with the gnathopods of Glyptonotus, and functional peculiarities pertaining to each group of limbs are discussed.

3. The Peræon.—The medial split in the thoracic sternites is interpreted as a device for allowing of distension of the body, say, after a meal. The arched thoracic somites articulate with each other in a special way, not by a crossed articulation, like that in a pair of scissors, but by a rocker articulation, like that in a spring clothes-pin; this combination of arch and rocker articulation is interpreted as a means of preventing change of internal volume in body flexion. The phenomena pertaining to the moulting process throw light upon the numerical grouping of the successive fusions of coxæ with somites which have occurred in isopods.

4. The Pleon.—The pleon consists of four movable portions, the last four of the seven constituent pieces being welded together. Of the four fused segments, only the first has a complete sternite, like those of somites 1, 2, and 3. In the skeleton of the pleonic floor a wide opening is thus left posteriorly, which is closed by a diaphragm of soft cuticle; part of this diaphragm is elevated into two long parallel folds or valves, one on each side of the anus.

5. The Pleonic Appendages.—The protopodite of each of the anterior three pleopods is composed of two complete pieces; a third piece more proximally situated is interpreted, not as evidence of an additional primitive segment, but as a secondary development in the articular membrane. The uropods have acquired their present position by a forward rotation of the sternite of the sixth primitive somite, and the gill-directed surface of the uropod corresponds to the posterior surface of the pleopods. It follows that what has commonly been called the exopodite of the uropod is a real exopodite. In addition to the pleopods, the internal walls of the whole uropodal chamber subserve a respiratory function.

6. The Cephalosome.—Two thoracic somites are fused with the head to form a cephalosome. By welded inturnings of the anterior border of these two and of the posterior border of the mandibular cephalic somite a strong internal bracing is formed. The tergites of the maxillary somites have apparently disappeared from the dorsal aspect of the cephalon; the endo-skeletal structures described by Lloyd in Bathynomus, and by him named “sternal alæ,” functionally correspond to these tergites in so far as they serve for attachment of muscles for the maxillæ. These “sternal alæ” (which also appear to form a covering for the maxillary excretory gland) with other ventral endoskeletal structures are described in detail. All are rigidly fixed to the maxillo-sternal framework, which is capable of independent movement.

7. The Alimentary Canal.—The foregut of Glyptonotus is not a gastric mill; the muscle-provided involutions of its wall (exactly analogous to those in Bathynomus) are concerned simply with onward propulsion of the food, i.e. with swallowing. Between the midgut and the hindgut, the two opposed ends of which are not in exact alignment, is a strongly contractile part of the gut, which acts as a sphincter. The hepatic cæca could not be examined. Glyptonotus is carnivorous.

8. The Eyes and the Chromatophore-System.—By experiment on colour change of Idotea it has been shown that the eye of this animal acts as a double mechanism, the ventral half being functionally separable from the dorsal. Glyptonotus has retractile chromatophores, and probably undergoes colour change in accordance with différent modes of eye illumination. Its eye is divided into two entirely separate parts, one lying on the dorsal the other on the ventral aspect of the cephalon.

The cost of providing the illustrations which accompany this paper was defrayed by a grant from the Earl of Moray Endowment for the promotion of research in the University of Edinburgh; and the expense of preparing the blocks was defrayed by a grant from the Carnegie Trust.

I have to record my thanks to Dr W. S. Bruce, not only for the material provided, but for information relating to the quarters inhabited by Glyptonotus.

The intertidal and shallow-water sands of the coasts of the British Isles contain, in many cases as the dominant amphipod fauna, species of the three genera Haustorius, Urothoë, and Bathyporeia of the family Haustoriidæ, often coexistent in the same habitat and obtainable in the same samples. The swimming and burrowing mechanisms of Haustorius arenarius have been described by Dennell (1933), and I have described (1939) the mechanisms in some species of the genus Bathyporeia. This paper deals mainly with Urothoë marina; no reference is made to the other species of the genus, although U. brevicornis has been examined and found to be similar to U. marina. Since all species of the genus Urothoë are very similar morphologically and live in similar habitats, it may be assumed that the following description of U. marina will apply to them also, but I have had no opportunity of examining the more southern species U. grimaldii or U. pulchella. Crawford (1937) states that the burrowing habits of U. brevicornis and U. grimaldii var. poseidonis are very similar to those of Haustorius.

The following investigation was made in consequence of certain peculiarities in the earlier results of some recent measurements of ripples by Prof. Michie Smith, in my Laboratory, which will, I hope, soon be communicated to the Society. These seemed to suggest that viscosity might have some influence on the results, as might also the film of oxide, &c, which soon gathers on a free surface of mercury. I therefore took account of the density, as well as of possible rigidity, of this surface layer, in addition to the surface tension which, was the object of Prof. Smith's work. The later part of the paper, where Cartesian coordinates are employed, runs somewhat on the lines of an analogous investigation in Basset's Hydrodynamics. My original object, however, was different from his, as I sought the effects of viscosity on waves steadily maintained by means of a tuning-fork used as a current interruptor; not on waves once started and then left to themselves. Besides obtaining his boundary conditions in a singular manner, I think that in his § 521 Mr Basset has made an erroneous investigation of the effects of very great viscosity.

Let (Tn) be a non-decreasing sequence of self-adjoint operators which are semi-bounded from below and converge to some self-adjoint operator T in the sense of strong resolvent convergence. For every λ which is eventually below the essential spectrum of Tn it is shown that ‖En(λ)−E(λ)‖→0 for n→∞, where En(·) and E(·) are the spectral resolution of Tn and T, respectively.

We study the existence of left approximate units, left approximate identities and bounded left approximate identities in the algebras (X)of all compact operators on a Banach space X and ℱ(X)− of all operators uniformly approximable by finite rank operators. In the case of bounded left approximate identities, necessary and sufficient conditions on X are obtained. In the other cases, sufficient conditions are obtained, together with an example of non-existence using a space constructed by Szankowski. The possibility of the sufficient conditions being also necessary depends on the question of whether every compact set is contained in the closure of the image of the unit ball under an operator in (X)(or ℱ(X)−). Sufficient conditions on X are obtained for this to be true, but it is conjectured that the answer for general X is negative.

Suppose that f is meromorphic of finite order in the plane, and that f″ has only finitely many zeros. We prove a strong estimate for the frequency of distinct poles of f. In particular, if the poles of f have bounded multiplicities, then f has only finitely many poles.

We show that shooting methods for homoclinic or heteroclinic orbits in dynamical systems may automatically guarantee the topological transversality of the stable and unstable manifolds. The interest of such results is twofold. First, these orbits persist under perturbations which destroy the structure allowing the shooting method and, second, topological transversality is often sufficient when some kind of transversality is required to obtain chaotic dynamics. We shall focus on heteroclinic solutions in the extended Fisher–Kolmogorov equation.

We study the motion of a rigid body with a cavity filled with a viscous liquid. The main objective is to investigate the well-posedness of the coupled system formed by the Navier–Stokes equations describing the motion of the fluid and the ordinary differential equations for the motion of the rigid part. To this end, appropriate function spaces and operators are introduced and analysed by considering a completely general three-dimensional bounded domain. We prove the existence of weak solutions using the Galerkin method. In particular, we show that if the initial velocity is orthogonal, in a certain sense, to all rigid velocities, then the velocity of the system decays exponentially to zero as time goes to infinity. Then, following a functional analytic approach inspired by Kato's scheme, we prove the existence and uniqueness of mild solutions. Finally, the functional analytic approach is extended to obtain the existence and uniqueness of strong solutions for regular data.

We determine a set of conditions that are necessary and sufficient for an ordered regular semigroup to be a perfect Dubreil-Jacotin semigroup; and a necessary and sufficient condition for a perfect Dubreil-Jacotin semigroup to be orthodox.

We generalize to the anisotropic case some classical and recent results on the (n – 1)-Minkowski content of rectifiable sets in ℝn, and on the outer Minkowski content of subsets of ℝn. In particular, a general formula for the anisotropic outer Minkowski content is provided; it applies to a wide class of sets that are stable under finite unions.

Groups having two generators and two relations are studied. The Reidemeister-Schreier algorithm is used to determine presentations for their derived groups. This enables the orders of the groups to be found. Necessary and sufficient conditions are given for the groups to be metabelian. Certain classes closely related to the class are also discussed.

The rigorous definition of time-ordered exponentials, solving quantum linear stochastic differential equations, is extended to Boson and Fermion stochastic calculi with infinitely many degrees of freedom. The relation to the classicalmultiplicative stochastic integral, solving the Doleans exponential equation, is discussed.

A new construction of derived equivalences is given, which relates different endomorphism rings and, more generally, cohomological endomorphism rings, including higher extensions, of objects in triangulated categories. These objects need to be connected by certain universal maps that are cohomological approximations and that exist in very general circumstances. The construction turns out to be applicable to a wide variety of situations, covering finite-dimensional algebras as well as certain infinite-dimensional algebras, Frobenius categories and n-Calabi–Yau categories.

We define a quantity called the reduced C* exponential rank rcel (A) of a C*-algebra A, which satisfies rcel (A) ≦ cel (A). We show that rcel (A) = ∞ whenever A has two distinct normalised traces which agree on K0(A), and we prove a partial converse. This gives some understanding of why cel (A) = π cer (A) for some C*-algebras A but not for others. We also characterise rcel (A) as the supremum of the rectifiable distances from unitaries in the identity component of the unitary group to the commutator subgroup of this component.

A degenerate parabolic partial differential equation with a time derivative and first- and second-order derivatives with respect to one spatial variable is studied. The coefficients in the equation depend nonlinearly on both the unknown and the first spatial derivative of a function of the unknown. The equation is said to display finite speed of propagation if a non-negative weak solution which has bounded support with respect to the spatial variable at some initial time, also possesses this property at later times. A criterion on the coefficients in the equation which is both necessary and sufficient for the occurrence of this phenomenon is established. According to whether or not the criterion holds, weak travelling-wave solutions or weak travelling-wave strict subsolutions of the equation are constructed and used to prove the main theorem via a comparison principle. Applications to special cases are provided.

Let p(n) denote the partition function. In this paper, we will prove that for $n\ges 222$,

$$\left| {\matrix{ {p(n)} & {p(n + 1)} & {p(n + 2)} \cr {p(n-1)} & {p(n)} & {p(n + 1)} \cr {p(n-2)} & {p(n-1)} & {p(n)} \cr } } \right| > 0.{\rm }$$

$n\ges 222$

$$(p(n)^2-p(n-1)p(n+1))^2-(p(n-1)^2-p(n-2)p(n))(p(n+1)^2-p(n)p(n+2))>0.$$

In this paper, we obtain the variational characterization of Hardy space Hp for $p\in (((n)/({n+1})),1]$, and get estimates for the oscillation operator and the λ-jump operator associated with approximate identities acting on Hp for $p\in (((n)/({n+1})),1]$. Moreover, we give counterexamples to show that the oscillation and λ-jump associated with some approximate identity cannot be used to characterize Hp for $p\in (((n)/({n+1})),1]$.