In a recent paper embodying the results of my work upon Actinotrocha, I was led to suggest a theory of the segmentation of cœlomate animals, the essential features of which were as follows:—

1. There can be demonstrated, in the morphology of the Cœlomata, two distinct types of segmentation—(a) a primitive or archimeric type, having in its constitution certain evidences of a radial origin, and (b) a secondary or metameric segmentation, superposed upon the former and bearing evidence of a bilateral origin.

2. A certain number of the Cœlomata retain, under varied disguise, the archimeric segmentation throughout life, together with a number of other primitive characters in common. These groups, being genetically allied, may be conveniently classified in one division, for which the name Archi-cœlomata is proposed.

3. In the truly segmented animals, such as the Annelida, Arthropoda, and Eu-chorda, the bilateral or metameric segmentation ⟨of the posterior archimeric segment⟩ tends to completely replace the archimeric. In at least two of these groups there has been an independent evolution of metameric segmentation.

In attempting to bring forward facts in support of this theory, one may approach the subject from three stand-points.

(A.) Firstly, an attempt may be made, by general reasoning, to construct a hypothetical ancestor of the Cœlomata as a group. In this way may be obtained a schematic outline of an organism possessing each system of organs in its most archaic condition.

The South Orkneys are a small group of islands situated in the Southern Ocean, in about 62° S. lat. and 45° W. long., roughly 800 miles S.E. of Cape Horn. A single landing was made from the “Scotia “on Saddle Island, a small island on the north side of the group, and another on Coronation Island, the largest and most westerly. With these two exceptions all the rock specimens were obtained on Laurie Island, the most easterly of the group.

In this paper, a formally J-symmetric, linear differential expression of 2nth order, with complex-valued coefficients, is considered. A number of results concerning the location of the essential spectrum of associated operators are obtained. These are extensions of earlier work dealing with complex Strum-Liouville operators, and include results which, in the real case, are due to Birman, Glazman and others. They lead to criteria, for the non-emptiness of the regularity field, of the corresponding minimal operator-a condition which is needed in the theory of J-selfadjoint extensions.

We introduce algorithms for calculating minimum length factorisations of order-preserving mappings on a finite chain into products of idempotents, and into products of idempotents of defect one. The least upper bounds for these lengths are given.

We construct rank-one convex functions which are not convex and have linear growth at infinity. We show that these functions could be useful in some problems concerning weak convergence of gradients if we were able to prove that they are quasiconvex. This question, however, seems to be open.

Necessary and sufficient conditions on weights u and v are given for the inequality

to hold for all g ∊ .

The problem of classifying homogeneous null Lagrangians satisfying an nth order divergence identity is completely solved. All such differential polynomials are affine combinations of higher order Jacobian determinants, called hyperjacobians, which can be expressed as higher dimensional determinants of higher order Jacobian matrices. Special cases, called transvectants, are of importance in classical invariant theory. Transform techniques reduce this question to the characterization of the symbolic powers of certain determinantal ideals. Applications to the proof of existence of minimizers of certain quasi-convex variational problems with weakened growth conditions are discussed.

A Volterra integro-partial differential equation of parabolic type, which describes the time evolution of a population in a bounded habitat, subject both to past history and space diffusion effects, is investigated; general homogeneous boundary conditions are admissible. Under suitable conditions, the unique nontrivial nonnegative equilibrium is shown to be globally attractive in the supremum norm. Monotone methods are the main tool of the proof.

Some characterisations of commutatitivity for C*-algebras are given in terms of inequalities involving sums and products of self-adjoint elements, and optimal constants are obtained for the corresponding inequalities for non-commutative C*-algebras.

In this paper, we consider the existence and multiplicity of periodic solutions for the Duffing equation $x''+g(x)=p(t)$ with a singularity. When the time map has oscillating properties, $g(x)$ possesses a singularity at the origin and tends to $+\infty$ as $x\to+\infty$ and other conditions hold. We obtain the existence of harmonic solutions and the multiplicity of subharmonic solutions of the given equation by using the phase-plane analysis methods and the generalized Poincaré–Birkhoff twist theorem.

A subalgebra M of a Lie algebra L is called modular in L if M is a modular element in the lattice of the subalgebras of L. Our aim is to study the finite-dimensional Lie algebras all of whose maximal subalgebras are modular. We characterize these algebras over any field of characteristic zero.

Let ψ ∈ C2[0,1] be a positive function on (0, 1]. Under certain assumptions on ψ, the set

is a pseudoconvex domain with C2-boundary, for which it is possible to construct a Henkin-type operator Hψ = Kψ + Bψ solving in Dψ. The operator Bψ, is L∞-continuous because it has a Riesz potential type kernel, while the L∞-continuity of Kψ depends on the flatness of ψ at 0. Our main result states that Kψ is continuous from L∞(∂Dψ) into L∞(Dψ) if and only if

The tow-nettings on which this survey is based were taken mostly from Keppel Pier. The tide as a rule runs down the channel strongly for several hours after high water, and the current is sufficiently strong to hold out the tow-nets more or less horizontal. Owing to some peculiarity in the configuration of the sea-floor at Keppel, the downward current is much the stronger and often begins an hour or several hours before high water. The flood-tide is never strong enough nor constant enough to hold out a tow-net for any length of time, and so most of the observations are based on tow-nettings taken on the ebb. Tow-nettings were taken almost every day and the tow-nets were usually left out for an hour.

The problem of fitting a polynomial to data by Least Squares has engaged the attention of many writers. The methods of approach have been many and various. Continued fractions, determinants, the calculus of finite differences and sums, the method of moments, the linear combination of data, the use of the orthogonal polynomials of Legendre and Tchebychef, these and doubtless other instruments of analysis have been pressed into service. At the end of the present paper is given a selective bibliography, which we hope on a future occasion to complete and to supplement by adding brief indications of the standpoint and achievement of each investigator.

Let T be the formally self-adjoint second-order elliptic differential expression

in ℝn, where the coefficients bj, ajk, q are real-valued and ajk ≈ akj. In this paper sufficient conditions for all positive integer powers Tm of T to be essentially self-adjoint on are obtained.

1. When a suspension of a large number of small rigid spheres in a viscous fluid is allowed to sediment under the action of gravity the rate of fall of any of the particles in the body of the suspension is

U = U0(1 – κΦ), where U0 is the rate of fall of an isolated single particle calculated according to Stokes' law , where α is the radius of the particle, η the viscosity of the fluid, ρ the density of the particle, σ the density of the fluid, and g the constant of gravity. Φ is the volume of suspended material per unit volume of the suspension, and κ is a constant whose value according to theory is 5·5 + c, where c is probably in the neighbourhood of 1·6.

We show that the pseudovariety generated by all semigroups of order-preserving transformations of a finitechain has no finite pseudoidentity basis.

We prove the existence of a weak solution for the system of partial differential equations describing the shearing of stratified thermoviscoplastic materials with temperature-dependent non-homogeneous viscosity.

This paper is devoted to a study of differential equations and inequalities of the form

and

The results are mainly concerned with the existence of positive solutions, their uniqueness in the case of (*), and bounds for these solutions.

We investigate critical points of the free energy Eε(u) of the Cahn–Hilliard model over the unit square under the constraint of a mean value ü. We show that for any fixed value ü in the so-called spinodal region and to any mode of an infinite class, there are critical points of Eε(u) having the characteristic symmetries of that mode provided ε > 0 is small enough. As ε tends to zero, these critical points have singular limits forming characteristic patterns for each mode. Furthermore, any singular limit is a stable critical point of E0(u)). Our method consists of a global bifurcation analysis of critical points of the energy Eε(u) where the bifurcation parameter is the mean value ü.