We introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces…which show that, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollobás property for the numerical radius is given.

We introduce a class of essentially self-adjoint Schrödinger operators, where essential self-adjointness is stable under positive potential perturbations. We show that this class is “stable” under certain perturbations and contains operators discussed by Simon and Kato. Finally, an extended essentially self-adjointness criterion is given.

The Lochs of Stenness and Harray on the mainland of Orkney are the only known stations in Scotland for Neritina fluviatilis L. The Loch of Stenness has a salinity varying between 9 and 26·8 parts per thousand; 24 species of invertebrates live in the loch, as well as several kinds of fish. The distribution of H. jenkinsi Smith and Peringia ulvœ Pennant is peculiar and only partly explicable by the salinity. The Loch of Harray is almost fresh and only slightly affected by spring tides, the salinity varying between 0·6 and 4·3 parts per thousand. Twenty-seven species of invertebrates are to be found in it, most of which are fresh-water species. The calcium content of the lochs varies from 56 to 139 mg. per litre.

Using Leray–Schauder degree theory, a theorem of upper and lower solutions and a strong maximum principle for the telegraph equation we prove an Ambrosetti–Prodi-type result for periodic solutions of the telegraph equation.

In this paper we study the global existence and nonexistence of spatially periodic solutions to the initial value problem of the Ginzburg–Landau equation.

We consider nonlinear singular Volterra integral equations of the second kind. We generalise the transformation method introduced in Part I of this paper [6] to cope with both the nonlinearity and slightly more general singular kernels. We also consider a particular class of nonlinear equation for which the solution behaviour is known. Using this a priori knowledge, we propose a modification of the transformation technique which results in a numerical method with good asymptotic stability properties. Applying the general theory of Part I of this paper, we prove convergence of this scheme.

We classify group-theoretically all separable coordinate systems for the eigenvalue equation of the Laplace-Beltrami operator on the hyperboloid = 1, finding 71 orthogonal and 3 non-orthogonal systems. For a number of cases the explicit spectral resolutions are worked out. We show that our results have application to the problem of separation of variables for the wave equation and to harmonic analysis on the hyperboloid and the group manifold SL(2, R). In particular, most past studies of SL(2, R) have employed only 6 of the 74 coordinate systems in which the Casimir eigenvalue equation separates.

In a previous paper, the authors gave a complete description of the number of even harmonic solutions of Duffing's equation without damping for the parameters varying in a full neighbourhood of the origin in the parameter space. In this paper, the analysis is extended to the case of an independent small damping term. It is also shown that all solutions of the undamped equation are even functions of time.

Consider the following heat conduction problem: let Hm−1 be a hemisphere of Sm−1 and suppose that Hm−1 has temperature 1 at time t = 0, while the boundary of Hm−1 is kept at temperature 0 for all time t>0. We obtain a combinatorial formula and a uniform estimate for the amount of heat in Hm−1 at time t.

Linear semi-group theory can be used to prove the existence of solutions to the equations of linear elasticity when the elasticity tensor is positive definite. Here, it is shown that this condition is also necessary for the existence of a semi-group. The method is also applied to linear dissipative equations.

Plena descriptio Botanices Socotrensis quantum nobis cognitæ, brevi tempore subjicietur Societati Regiæ Edinensis ut in suis Actis in lucem emittatur et divulgetur. Quoniam verum spatio certo temporis opus erit ut singulæ partes et minutiæ Floræ tam abundantis et copiosæ plene elaborentur et perlustrentur, et quoniam perplures formæ endemicæ tarn generum quam specierum in ea Flora comprehenduntur, oportet et congruit opportunis temporis intervallis, opere progrediente, breves diagnoses novarum plantarum in medium proponere, ut prioritas vindicetur nominibus quæ his novis tribuantur, et quæ in volumine Actorum Societatis Regiæ publice emittentur.

One of the main objects of the study of animal distribution is to determine the probable areas of origin and dispersal—Osborn's Centres of Radiation—of the different families and genera of the various classes of animals. This is not an easy matter, and, as reference to any standard work on molluscs would show, the questions of the origin, in time and space, of the various families of molluscs, as also their relationships, are often severely left alone. The classification is, as a result, often quite arbitrary and does not show the genetic relationships of the different members. In dealing with families with a world-wide and, more particularly, a discontinuous distribution, it is a matter of grave doubt whether several of the families, as defined, are homogeneous units which originated in any one area of the globe and spread to other regions.

This peculiar iridescent pinkish Annelid was dredged in 1871 in Bressay Sound, in four or five fathoms, on a bottom of coarse sand and gravel, which abounded with finely-branched Melobesia calcarea, Ell. and Soland.

We give a simple explicit description of the norm in the complex interpolation space (Cp[Lp(M)], Rp[Lp(M)])θ for any von Neumann algebra M and any 1 ≤ p ≤ ∞.

It is known that the parametric equation u'∊+ a∊u∊ = f, u∊ (0)= 0,with α ≦ a∊ ≦ β, for all ∊ > 0 and almost everywhere in a bounded domain Ω of ℝN, and f in L∞((0, T) × Ω), shows, at the limit, a memory effect. In this work the associated minimisation problem is considered and we describe how the memory effect appears in the Γ-limit, for the weak topology H1:(0, T; L2(Ω)) of the corresponding functional. The sequence a∊ has no dependence in time.

If a Sturm—Liouville problem is given in an open interval of the real line, then regular boundary value problems can be considered on compact sub-intervals. For these regular problems, all with necessarily discrete spectra, the eigenvalues depend on both the end-points of the compact intervals, and upon the choice of the real separated boundary conditions at these end-points. These eigenvalues are not, in general, continuous functionsof the end-points and boundary conditions. The paper shows the surprising form of these discontinuities. The results have applications to the approximations of singular Sturm—Liouville problems by regular problems, and to the theoretical aspects of the Sleign2 Computer program.

Two alternative characterizations of semidirect products of semigroups are given. Characterizations are provided of such products that are groups, regular semigroups, and inverse semigroups, respectively.

Based on earlier results on existence, we study the asymptotic behaviour of solutions to the coalescence-breakage equations, including the volume-scattering phenomenon and high-energy collisions. The solutions are shown to converge towards one particular equilibrium, provided the kernels satisfy a kind of reversibility. We also derive stability of these equilibria in a suitable topology.

In this paper we present a simpler proof of a result of Lewis concerning the continuity of weak solutions to the two-dimensional thermistor problem in the case where the temperature can blow up in a region with non-empty interior. Some other regularity properties are also discussed.

We use the notations and results of Part I of this paper [see 7]. Sections and formulae of that paper will be referred to as section I.4, formula (1.5), etc. In particular, T, A, TN0, K, σ, F and Λ(λ) will have the same meaning as in [7].