We obtain upper and lower bounds for tr (e−th−etΔ), where H = −Δ + V is a Schrödinger operator on L2 (ℝm), and ℝ is the Laplace operator for ℝm. The bounds are obtained for a class of negative valued Borel measurable potentials with compact support and in L∞(ℝm).

Using asymptotic estimates for the Green's function of irregular multipoint eigenvalue problems, we state sufficient conditions for the uniform convergence of the expansion of functions into a series of eigenfunctions.

A new proof is given that all regular biordered sets come from regular semigroups.

We study Lotka–Volterra models with fractional Laplacian. To do this we study in detail the logistic problem and show that the sub–supersolution method works for both the scalar problem and for systems. We apply this method to show the existence and non-existence of positive solutions in terms of the system parameters.

This paper is concerned with solutions of the complex Korteweg–de Vries (KdV) equation. We achieve two goals. First, we prove local well-posedness results for the complex KdV equation on a line, in a periodic domain and in a finite domain. These results are in line with the local well-posedness theory for the real KdV equation. Second, we establish a rigorous connection between the local (in space) regularity of the real part and that of the imaginary part of any solution to the complex KdV equation. This result partly validates the numerical observation that the real and imaginary parts of a singular solution of the complex KdV equation blow up at the same point and at the same time.

A previous attempt to systematize various conditions from multiparameter spectral theory is extended to weaker forms of definiteness. The latter not only occur in the established literature but are also under active investigation at present. Several algebraic and geometrical formulations exist, and questions concerning their equivalence are approached in a unified fashion where possible.

We construct nontrivial, non-negative quasiconvex functions denned on M2×2 with p-th order growth such that the zero sets of the functions are Lipschitz graphs of mappings from subsets of a fixed two-dimensional subspace to its orthogonal complement. We assume that the graphs do not have rank-one connections with the Lipschitz constants sufficiently small. In particular, we are able to construct quasiconvex functions which are homogeneous of degree p (p > 1) and ‘conjugating’ invariant.

An inequality of C. R. Putnam involving a Dirichlet functional in the singular theory of the Sturm-Liouville differential equation is generalised. The corresponding result for the Sturm-Liouville theory over a compact interval is established and this is extended to the singular case by means of a Tauberian argument.

The position taken by Mr Darwin is, that all species have arisen by the natural process of ordinary generation. That the differences which we now see in them have arisen from slight variations in individuals having from time to time occurred, which have been perpetuated by inheritance, by successive stages and slow degrees, through unlimited spaces of time. Some of these slight variations he considers to originate in causes beyond our power of explanation, and which, although not the work of chance, we may call chance, for want of a better appellation—others to arise from habit, or from the excessive use or disuse of certain organs; but that when such a variation has once appeared, it is preserved by hereditary descent through a principle which he calls “natural selection,” and which he deduces as a corollary from the struggle for existence which we see constantly going on around us.

The apparatus used in the following determinations is that of Victor Meyer, modified as described in a paper published in 1879, “On the Vapour Densities of Potassium and Sodium,”* by Professor Dewar and the author. In the experiments here described, the platinum vessel was further protected by a casing of iron, and the intervening space filled with magnesia, the iron casing being embedded in sand enclosed between two crucibles.

During the deep sea dredging expedition of H.M. ships ‘Lightning’ and ‘Porcupine,’ in the summers of 1868–69 and 1870, two or three nearly perfect specimens, and a number of fragments were procured of three species of regular echinideans, which were referred by the author to a new family, the Echinothuridæ, intermediate in their more essential characters between the Cidaridæ and the Diadematidæ.

In these urchins the test is circular and greatly depressed. The plates of the perisom are long and strap-shaped, and the interambulacral plates overlap one another regularly from the apical towards the oral poll, while the ambulacral plates overlap in a similar way in the opposite direction.

From M. Lefort's opening sentence it appears that he had only recently received the copy of my remarks which had been posted to him on 22d December. Perhaps on this account M. Lefort has been hurried in the perusal of my paper, and so has fallen into several mistakes as to my meaning. These will be apparent to any one who peruses the writings, and I shall pass them over entirely, confining myself to the very few points which are essential to the subject in hand. The only extraneous matter to which I shall allude is this, that while M. Lefort has obviously and justly been desirous of upholding the dignity of the Grandes Tables du Cadastre, he has, in the true spirit of an inquirer after truth, clearly and faithfully exhibited even those points which press most sorely on his own position.

Zeleny (Physical Review, xii., p. 321, 1901) has investigated the variation with temperature of the photo-electric discharge from a platinum wire in air at ordinary atmospheric pressure.

The fact that the temperature of the earth increases with the depth below the surface, implies a continual loss of heat from the interior by conduction outwards, through or into the upper crust. Since the upper crust does not become hotter from year to year, there must therefore be a secular loss of heat from the whole earth. It is possible that no cooling may result from this loss of heat, but only exhaustion of potential energy, which in this case could scarcely be other than chemical affinity between substances forming part of the earth's mass. But it is certain that either the earth is becoming, on the whole, cooler from age to age, or that the heat conducted out is generated in the interior by temporary dynamical action (such as chemical combination).

In Kayser's Spectroscopie, vol. iii. pp. 45–49, an account is given of all the different methods that have hitherto been employed for photometry in the ultra-violet part of the spectrum. Photography, phosphorescent plates, selenium cells, and the ionising effect of ultra-violet light have all been used, but with limited success, and Professor Kayser considers that the only really practical method is that which has been recently introduced by Pfluger. Pfliiger has discovered that there is relatively an enormous amount of energy in the ultra-violet spectrum of the electric spark produced by the discharge of a condenser between metal electrodes. He therefore uses the electric spark as a source, and takes the deflections with a thermopile and galvanometer, just as in the infra-red. He finds the deflections to be wonderfully steady, considering the inconstant nature of the spark.

In this paper, we study existence of solutions with boundary layer and peaks for an elliptic problem. We prove that the problem has a mountain-pass-type solution, which has a boundary layer and a single peak near the boundary. Moreover, we also study the existence of solutions with interior peak.

A seven-dimensional Euclidean space considered as the space of purely imaginary Cayley numbers is called a Cayley space. The six-dimensional sphere in a Cayley space admits an almost complex structure which is not integrable. Moreover the algebraic properties of the imaginary Cayley numbers induce an almost complex structure on any oriented differentiable hypersurface in the Cayley space. The Riemannian metric induced on the hypersurface from the metric of the Cayley space is Hermitian with respect to the almost complex structure.

It is proved that the induced Hermitian structure of an oriented hypersurface in the Cayley space is almost Kaehlerian if and only if it is Kaehlerian, that a necessary and sufficient condition for a hypersurface in a Cayley space to be an almost Tachibana space is that the hypersurface be totally umbilical, and that a totally umbilical hypersurface in a Cayley space admits a complex structure when and only when it is totally geodesic.

For a hypersurface in the Cayley space with the induced Hermitian structure which is an *O-space it is proved that all the principal curvatures of the hypersurface are constant, and from this is deduced a classification of such *O-spaces.

We show that the Schur multiplier of a Noetherian group need not be finitely generated. We prove that the non-abelian tensor product of a polycyclic (resp. polycyclic-by-finite) group and a Noetherian group is a polycyclic (resp. polycyclic-by-finite) group. We also prove new versions of Schur's theorem.

In this paper, we study the Cauchy problem of a cubic autocatalytic chemical reaction system

$$ u_{1,t}=u_{1,xx}-u_1u^{2}_2,\qquad u_{2,t}=du_{2,xx}+u_1u^{2}_2 $$

with non-negative initial data, where the constant $d>0$ is the Lewis number. Our purpose is to study the global dynamics of solutions under mild decay of initial data as $|x|\rightarrow\infty$. In particular, we show that, for a substantial class of $L^1$ initial data, the exact large-time behaviour of solutions is characterized by a universal, non-Gaussian spatio-temporal profile, subject to the apparent conservation of total mass.

We describe completely the link invariants constructed using Markov traces on the Yokonuma–Hecke algebras in terms of the linking matrix and the Hoste–Ocneanu–Millett–Freyd–Lickorish–Yetter–Przytycki–Traczyk (HOMFLY-PT) polynomials of sublinks.