In a Preliminary Note, read before this Society on the 31st of May 1869 (see Proceedings), a number of experiments were described, which proved that the lethal action of certain doses of physostigma can be prevented by the administration of atropia. Further, it was pointed out, that antagonism between any two substances, in the sense of the lethal action of the one being preventible by the physiological action of the other, had not previously been shown to exist by any certain and satisfactory evidence. In the various instances where experiment seemed to indicate the existence of such an antagonism, sufficient proof was not given that the dose of the substance whose action appeared to be antagonised was certainly a lethal one. The conflicting opinions and doubts this fallacy has given origin to, have induced the author to follow a plan whereby it may be completely avoided.

To understand the heterogeneous spatial effect on predator–prey models, we study the behaviour of the positive steady states of a predator–prey model as certain parameters are small or large. We compare the case when the model has a spatial degeneracy with the case when it does not have such a degeneracy. Our results show that the effect of the degeneracy can be clearly observed in one limiting case, but not in the others.

We consider a smooth curve with singular points in the Euclidean space. As a smooth curve with singular points, we have introduced a framed curve or a framed immersion. A framed immersion is a smooth curve with a moving frame and the pair is an immersion. We define an evolute and a focal surface of a framed immersion in the Euclidean space. The evolutes and focal surfaces of framed immersions are generalizations of each object of regular space curves. We give relationships between singularities of the evolutes and of the focal surfaces. Moreover, we consider properties of the evolutes, focal surfaces and repeated evolutes.

The Riemann initial value problem is studied for scalar conservation laws whose fluxes have a single inflection point. For a regularization consisting of balanced diffusive and dispersive terms, the travelling wave criterion is used to select admissible shocks. In some cases, the Riemann problem solution contains an undercompressive shock. The analysis is illustrated by exploring parameter space for the Buckley–Leverett flux. The boundary of the set of parameters for which there is a physical solution of the Riemann problem for all data is computed. Within the region of acceptable parameters, the solution hasseveral different forms, depending on the initial data; the different forms are illustrated by numerical computations. Qualitatively similar behaviour is observed in Lax–Wendroff approximations of solutions of the Buckley–Leverett equation with no dissipation or dispersion.

There is a residual subset of embeddings of an m-manifold, M in Rm+1 (m ≦ 6), for which the induced Maxwell subset on the sphere Sm is a stratified subset. We define and study two different stratifications of this subset and their extensions to the whole Sm: the Gauss stratification and the core stratification. We also find relations between the Euler numbers of the strata of the core stratification and the “exposed” singularities of the Gauss map on M.

As far back as 1678 Martin Lister, in his Historiæ Animalium Tractatus, has noted that, in the case of certain Molluscs, the fæces are of a regular form, and that their shape differs according to the species; he deduces from this that the intestine must show a corresponding difference in the various species.

It is well known that many taxes do not fall ultimately on the person from whom they are in the first instance levied. The merchant advances the duties imposed on goods, but the tax ultimately falls on the consumer. The problem of discovering the ultimate or true incidence of each tax is one of great importance, and of considerable complexity. The following paper contains an investigation of the methods by which this incidence may in some cases be experimentally determined, and of the principles regulating the incidence in all cases, these principles being stated in a mathematical form.

For a class of non-conservative hyperbolic systems of partial differential equations endowed with a strictly convex mathematical entropy, we formulate the initial-value problem by supplementing the equations with a kinetic relation prescribing the rate of entropy dissipation across shock waves. Our condition can be regarded as a generalization to non-conservative systems of a similar concept introduced by Abeyaratne, Knowles and Truskinovsky for subsonic phase transitions and by LeFloch for non-classical undercompressive shocks to nonlinear hyperbolic systems. The proposed kinetic relation for non-conservative systems turns out to be equivalent, for the class of systems under consideration at least, to Dal Maso, LeFloch and Murat's definition based on a prescribed family of Lipschitz continuous paths. In agreement with previous theories, the kinetic relation should be derived from a phase-plane analysis of travelling-wave solutions associated with an augmented version of the non-conservative system. We illustrate with several examples that non-conservative systems arising in the applications fit in our framework, and for a typical model of turbulent fluid dynamics we provide a detailed analysis of the existence and properties of travelling waves which yields the corresponding kinetic function.

We obtain weighted norm inequalities for maximal truncated operators of multi-linear singular integrals with non-smooth kernels in the sense of Duong et al. This class of operators extends the class of multi-linear Calderón-Zygmund operators introduced by Coifman and Meyer and includes the higher-order commutators of Calderón. The weighted norm inequalities obtained in this work are with respect to the new class of multiple weights of Lerner et al. The key ingredient in the proof is the introduction of a new multi-sublinear maximal operator that plays the role of the Hardy-Littlewood maximal function in a version of Cotlar's inequality. As applications of these results, new weighted estimates for the mth order Calderón commutators and their maximal counterparts are deduced.

This contribution is a continuation of a former paper on the same subject, references to which in this paper will, for the sake of brevity, be referred to as ante, p.…, or ante, fig.… The present pages deal especially with the following districts of Scotland:—

1. North-west Kirkcudbrightshire.

2. South-east Kirkcudbrightshire.

3. Wigtownshire.

4. Fife and Kinross.

The present paper is a continuation of my monographs already published by the Royal Microscopical Society, London, in their Journals for June and December 1888. It has been carried out under similar conditions in the Botanical Department of the British Museum (Natural History), South Kensington, but I am especially indebted to Edmund Grove, Esq., F.R.M.S., for much valuable cooperation, and to Julien Deby, Esq., for the readiness with which he has placed at my command the resources alike of his library and cabinet. Herr E. Weissflog and Dr James Rae, R.N., have also furnished me with many excellent preparations.

A model of a predator–prey system showing group defence on the part of the prey is formulated, and reduced to a three-parameter family of quartic polynomial systems of equations. Mathematically, this system contains the Volterra–Lotka system, and yields numerous kinds of bifurcation phenomena, including a codimension-two singularity of cusp type, in a neighbourhood of which the quartic system realises every phase portrait possible under small smooth perturbation. Biologically, the nonmonotonic behaviour of the predator response function allows existence of a second singularity in the first quadrant, so that the system exhibits an enrichment paradox, and, for certain choices of parameters, coexistence of stable oscillation and a stable equilibrium.

In this paper, we provide sufficient conditions which guarantee the uniform stability as well as asymptotic stability of the positive equilibrium for a food limited population model with time delay.

We study the first initial–boundary-value problem for the three-dimensional non-autonomous Navier–Stokes–Voigt equations in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality. The existence of a weak solution to the problem is proved by using the Faedo–Galerkin method. We then show the existence of a unique minimal finite-dimensional pull-back -attractor for the process associated with the problem, with respect to a large class of non-autonomous forcing terms. We also discuss relationships between the pull-back attractor, the uniform attractor and the global attractor.

We propose a generalisation of the Mullins–Sekerka problem to model phase separation in multi-component systems. The model includes equilibrium equations in bulk, the Gibbs–Thomson relation on the interfaces, Young's law at triple junctions, together with a dynamic law of Stefan type. Using formal asymptotic expansions, we establish the relationship to a transition layer model known as the Cahn-Hilliard system. We introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds, together with measure theoretical tools. Through an implicit time discretisation, we construct approximate solutions by stepwise minimisation. Under the assumption that there is no loss of area as the time step tends to zero, we show the existence of a weak solution.

Asymptotic estimates are established for nontrivial positive radial eigenfunctions of the nonlinear eigenvalue problem −Δu = λ(up − uq) in the unit ball B in ℝN (N > 2) with Neumann boundary conditions, as the supremum norm tends to infinity. Here p is the critical Sobolev exponent (N + 2)/(N − 2) and 0 < q < p − 1 = 4/(N − 2).

As the first step towards a classification of simple weight modules with finite dimensional weight spaces over Witt algebras Wn, we explicitly describe the supports of such modules. We also obtain some descriptions of the support of an arbitrary simple weight module over a ℤn-graded Lie algebra having a root space decomposition with respect to the abelian subalgebra , with the property for all α, β ∈ ℤn, α ≠ β (this class contains the algebra Wn).

The diffraction of a line source of sound by an absorbing semi-infinite half plane in the presence of a fluid flow is examined. It is found that the radiated sound intensity, in the half space in which the source is located, can be considerably reduced by a suitable choice of the absorption parameter. For subsonic flow the system exhibits no acoustic instabilities.

This paper is concerned with the existence, non-existence and qualitative properties of cylindrically symmetric travelling fronts for time-periodic reaction–diffusion equations with bistable nonlinearity in ℝm with m ≥ 2. It should be mentioned that the existence and stability of two-dimensional time-periodic V-shaped travelling fronts and three-dimensional time-periodic pyramidal travelling fronts have been studied previously. In this paper we consider two cases: the first is that the wave speed of a one-dimensional travelling front is positive and the second is that the one-dimensional wave speed is zero. For both cases we establish the existence, non-existence and qualitative properties of cylindrically symmetric travelling fronts. In particular, for the first case we furthermore show the asymptotic behaviours of level sets of the cylindrically symmetric travelling fronts.

About the year 1695 Martin Martin visited the Hebrides, and of the Island of North Uist he writes, “There is such a number of fresh water lakes here as can hardly be believed. … They are generally well stocked with trout and eels and some of them with salmon, and, which is yet more strange, cod, ling, mackerel, etc., are taken in these lakes into which they are brought by the spring tides.” This old reference suggested that the brackish-water fauna of the Hebrides might be of considerable interest and extent. Consequently I spent part of the summers of 1933 and 1935 in North Uist in order to study the fauna of the lochs and the conditions under which the animals were living.