The solutions of the differential equation Lny + p(x)y = 0, where Lny = ρn(ρn−1 … (ρ1(ρ0y)′)′ …)′ and p(x) is of one sign, are classified according to their behaviour as x → ∞. The solution space is decomposed into disjoint, non-empty sets Sk, 0≦K≦n, such that (−1)n−kp(x)≦0. We study the growth properties and the density of the zeros of the solutions which belong to the different sets Sk, the structure of the sets and its connection with (k, n − k)-disfocality.

The space group for paraelectric KH2PO4 is described and tables of irreducible representations are presented. These are used to derive the symmetry of the vibrational modes for both the heavy atoms and the hydrogen atoms, the latter being treated by a simple pseudo-spin model. Recent experiments on KH2PO4 and related materials are analysed in terms of the mode symmetries, and the light these experiments throw upon the nature of the phase transitions is briefly discussed.

The aim of this paper is to establish some new integral inequalities of the Sobolev type involving functions of several independent variables. The analysis used in the proofs is elementary and the results established provide new estimates for this type of inequality.

The nonlinear evolution problem [Bu(t)]′ = A(t, Bu)u + f(t, Bu) with B a constant linear operator and A = A(t, Bu) a time-dependent nonlinear operator from one Banach space to another, is studied. Existence and uniqueness results are obtained by making use of the theory of B-evolutions and the fractional powers of A and B. Two examples are presented in which the theory is applied to nonlinear equations with dynamic boundary conditions.

From the Proceedings of the Royal Society of Edinburgh, No. 106, p. 501, it appears the colouring of maps is receiving attention. This note bears chiefly upon the history of the matter.

Some thirty years ago, when I was attending Professor De Morgan's class, my brother, Francis Guthrie, who had recently ceased to attend them (and who is now professor of mathematics at the South African University, Cape Town), showed me the fact that the greatest necessary number of colours to be used in colouring a map so as to avoid identity of colour in lineally contiguous districts is four. I should not be justified, after this lapse of time, in trying to give his proof, but the critical diagram was as in the margin. With my brother's permission I submitted the theorem to Professor De Morgan, who expressed himself very pleased with it; accepted it as new; and, as I am informed by those who subsequently attended his classes, was in the habit of acknowledging whence he had got his information.

In this paper we examine the influence of capillarity on existence and uniqueness of travelling wave solutions in an isothermal system of van der Waals fluids. Existence and non-uniqueness theorems are proved using phase-space analysis and topological methods.

We establish the uniqueness of positive radial solutions of −Δu = up − u in B(R1, R2), u = 0 on ∂B(R1, R2), where B(R1, R2) is an annulus and 0 < R1 < R2 ≤ ∞, in the following cases.

(a) n ∈ {3, 4} and 1 < p ≤ n/(n − 2).

(b) n ∈ {5, 6, 7, 8} and 1 < p ≤ p0(n) for some p0(n) < n/(n − 2).

Earlier to this result, the uniqueness has been obtained by Coffman for n = 3 and 1 < p ≤ 3 and by Yadava for p ≥ (n + 2)/(n − 2) and n ≥ 3.

In the course of a search for criteria which would assist in deciding whether a particular series of numbers was randomly arranged, it was found necessary to investigate some of the more obvious characters of a random series of numbers. If the series is an infinite one and all the numbers are unequal (the series of decimals between o and 1 selected in some random fashion may be considered) certain of the numbers will be maximal, that is greater than their immediate neighbours, whilst others will be minimal, that is less than their immediate neighbours. Every maximal number is succeeded by a run down, that is by a series of descending numbers ending in the succeeding minimal number. The length of a run down is the number of numbers it contains, including the maximal number which begins it and the minimal number which ends it. The length of a run up is similarly defined. Clearly the minimum length of a run is 2.

We give a representation formula for the convex, polyconvex and rank one convex hulls of a set of n × n matrices with prescribed singular values.

For linear systems with a multi-symplectic structure, arising from the linearization of Hamiltonian partial differential equations about a solitary wave, the Evans function can be characterized as the determinant of a matrix, and each entry of this matrix is a restricted symplectic form. This variant of the Evans function is useful for a geometric analysis of the linear stability problem. But, in general, this matrix of two-forms may have branch points at isolated points, shrinking the natural region of analyticity. In this paper, a new construction of the symplectic Evans matrix is presented, which is based on individual vectors but is analytic at the branch points—indeed, maximally analytic. In fact, this result has greater generality than just the symplectic case; it solves the following open problem in the literature: can the Evans function be constructed in a maximally analytic way when individual vectors are used? Although the non-symplectic case will be discussed in passing, the paper will concentrate on the symplectic case, where there are geometric reasons for evaluating the Evans function on individual vectors. This result simplifies and generalizes the multi-symplectic framework for the stability analysis of solitary waves, and some of the implications are discussed.

Dual extremum principles characterising the solution of initial value problems for the heat equation are obtained by imbedding the problem in a two-point boundary-value problem for a system in which the original equation is coupled with its adjoint. Bounds on quantities of interest in the original initial value problem are obtained. Such principles are examples of ones which can be obtained for a general class of linear operators on a Hilbert space.

In 1987, Weidmann proved that, for a symmetric differential operator τ and a real λ, if there exist fewer square-integrable solutions of (τ−λ)y = 0 than needed and if there is a self-adjoint extension of τ such that λ is not its eigenvalue, then λ belongs to the essential spectrum of τ. However, he posed an open problem of whether the second condition is necessary and it has been conjectured that the second condition can be removed. In this paper, we first set up a formula of the dimensions of null spaces for a closed symmetric operator and its closed symmetric extension at a point outside the essential spectrum. We then establish a formula of the numbers of linearly independent square-integrable solutions on the left and the right subintervals, and on the entire interval for nth-order differential operators. The latter formula ascertains the above conjecture. These results are crucial in criteria of essential spectra in terms of the numbers of square-integrable solutions for real values of the spectral parameter.

We are interested in standing waves of a modified Schrödinger equation coupled with the Chern–Simons gauge theory. By applying a constraint minimization of Nehari-Pohozaev type, we prove the existence of radial ground state solutions. We also investigate the nonexistence for nontrivial solutions.

In this paper we study the problem of the axial symmetry of solutions of some semilinear elliptic equations in unbounded domains. Assuming that the solutions have Morse index one and that the nonlinearity is strictly convex in the second variable, we are able to prove several symmetry results in Rn and in the exterior of a ball. The case of some bounded domains is also discussed.

We study a special class of linear differential operators well-behaved with respect to weakconvergence. Questions related to weak lower semicontinuity, associated Young measures, weak continuity and quasi-convexity are addressed. Specifically, it is shown that the well-known necessary conditions for weak lower semicontinuity are also sufficient in this case. Some examples are given, including a discussion on how well the operator curl fits inthis context.

We show that some free discontinuity problems can be obtained as a limit of nonconvex local functionals with a singular perturbation of higher order.

During the last five years the greater part of my time has been taken up by the study of the plankton in the fresh-water lakes of Denmark, and in the spring of 1904 the first part of an extensive paper upon this subject was published by me. In that paper I have compared the results arrived at by me with the results of plankton explorations in other countries, and have tried to collect together everything known at the present moment regarding the periodicity and geographical distribution of the European freshwater plankton.

The region known as the Faroe Channel is that portion of the Atlantic Ocean to the north of the mainland of Scotland, which is bounded on the N.W. by the Faroe Islands, with their south-west-ward extending fishing-banks, and on the S.E. and S. by the Shetland and Orkney Islands, the shores of Caithness and Sutherland, and the Hebrides.

The Faroe Islands are composed of basaltic rocks, while the north of Scotland, and the Scottish Islands, are chiefly made up of Laurentian gneiss, Silurian, and Devonian rocks.

Nettles of the two species Urtica dioica and Urtica urens are commonly infested with the aphid Macrosiphum urticæ, and amid the living aphids and a collection of moulted skins dead examples are almost certain to be found. These dead aphids are almost spherical in shape and are of a brown colour, quite different from the colour of a normal aphid. They are attached to the nettle-leaf, often to the upper surface, by their ventral side, the leaf generally being discoloured at the point of attachment. If a number of these dead aphids are collected it will be found that, after a time, hymenopterous parasites, a species of the family Braconidæ, issue from the swollen body of the dead aphids. This parasite has been identified as Aphidius avence (Hal.), and the present paper deals with the life-history and habits of this species.

By Cauchy the word “symmetric,” as applied to functions, was used in a more general sense than that in which we ordinarily use it now, or than that in which it was used before his time. He spoke of “fonctions symétriques permanentes” and “fonctions symétriques alternées:” we apply the word symmetric only to certain of the first of these; the second we call simply “alternating functions.” Thus the expression

was called by him a permanent symmetric function, since if a1, a2, a3 are interchanged in order with b1, b2, b3, the function remains unaltered. It would be more definite to say that it is a permanent symmetric function with respect to a1, a2, a3 and b1, b2, b3.