In this paper, we consider the semilinear elliptic equation −Δu = λf(x,u) with the Dirichlet boundary value, and under suitable assumptions on the nonlinear term f with a more general growth condition. Some existence results of solutions are given for all λ > 0 via the variational method and some analysis techniques.

In a previous communication (Proc. Roy. Soc. Edin., vol. lii, 1932, p. 337) a visual method of measuring the diffusion coefficients of Br2 : H2, Br2 : N2, Br2 : O2, and Br2 : CO2 was described. The investigation has been extended to include similar measurements for bromine diffusing into argon, nitrous oxide, methane, and hydrogen chloride. Preliminary experiments in which a photoelectric cell replaces the human eye have also been made and confirmatory results have been obtained.

Depuis Murray et Eenard deux travaux importants sont venus enrichir la bibliographie de la Glauconie: le mémoire de MM. Calderon et Cliaves, “Contribuciones al estudio de la Glauconita,” et l'œuvre magistrale de M. L. Cayeux, Etude micrographique des terrains sédimentaires.

Murray et Eenard avaient étudié spécialement la Glauconie actuelle; MM. Calderon et Cliaves baserent leur synthese de la Glauconie actuelle sur une analyse de Glauconie des roches sédimentaires; M. L. Cayeux limita strictement ses belles recherches à la Glauconie des roches sédimentaires et n'attaquapas le problème de sa genèse.

We consider the homogenization of parabolic systems with Dirichlet boundary conditions when the operators and the domains in which the problems are posed vary simultaneously. We assume the operators do not depend on t. Then we show that the corrector obtained in a previous paper for the elliptic problem still gives a corrector for the parabolic one. From this result, we easily obtain the limit problem in the parabolic case.

Let n, $N \in {\open N}$ with $\Omega \subseteq {\open R}^n$ open. Given ${\rm H} \in C^2(\Omega \times {\open R}^N \times {\open R}^{Nn})$, we consider the functional 1

$${\rm E}_\infty (u,{\rm {\cal O}})\, : = \, \mathop {{\rm ess}\,\sup}\limits_{\rm {\cal O}} {\rm H}(\cdot, u,{\rm D}u),\quad u\in W_{{\rm loc}}^{1,\infty} (\Omega, {\open R}^N),\quad {\rm {\cal O}}{\Subset}\Omega.$$

$L^\infty $

2

$$\left\{\matrix{{\rm H}_{P}(\cdot, u, {\rm D}u)\, {\rm D}\left({\rm H}(\cdot, u, {\rm D} u)\right) = \, 0, \hfill \cr {\rm H}(\cdot, u, {\rm D} u) \, [\![{\rm H}_{P}(\cdot, u, {\rm D} u)]\!]^\bot \left({\rm Div}\left({\rm H}_{P}(\cdot, u, {\rm D} u)\right)- {\rm H}_{\eta}(\cdot, u, {\rm D} u)\right) = 0,\hfill}\right.$$

$[\![A]\!]^\bot := {\rm Proj}_{R(A)^\bot }$

${\cal D}$

Liouville theorems are obtained for fourth order elliptic systems of the form Δ2U + B Δ U + AU = 0 and for fourth order nonlinear equations of the form Δ2u−q(x) Δu + p(x)f(u) = 0 as a consequence of two related subharmonic functions, the mean value property of subharmonic functions, and a basic Green identity.

The nonlinear operator equation (N), , describes both semilinear elliptic boundary value problems (P) and their natural discretisations (Ph). (Here is a positive compact linear operator and [t]+ ≡ (t + |t|)/2 for all t ∊ ℝ.) It is proved that, for q ≧ 0 (q ≢ 0), in a Banach lattice E the equation (N) has an unbounded continuum ℱ of nontrivial solutions (λ, ψ) ∊ ℝ × E bifurcating from infinity at (λ1, ∞) (here is the first (positive) eigenvalue of ). All nontrivial solutions (λ, ψ) have λ ≧ λ1, and if maps a smaller cone Ks, into itself then there is a ℱs ⊂ℝ×Ks with similar properties to ℱ. The existence theory for (N) is applied to problems (P) and (Ph) which are defined on the simply-connected domain Ω. It is shown that the projection of ℱ on the λ-axis is either unbounded if the continuous function q > 0 except on a set of measure zero in Ω, or is bounded if q ≡ 0 on a subdomain of Ω. If is the second eigenvalue for in (Ph) then there is at most one nontrivial solution for each λ satisfying λ1 < λ < λ2; in the corresponding uniqueness result for (P) q is restricted to being strictly positive somewhere on ∂Ω Additional properties for the solutions of (P) and (Ph) are also proved.

We construct the Conley index for maps. We do not assume any compactness of map or space. We prove the Ważewski property, additivity property and continuation property.

We consider a cylinder with arbitrary cross section moving in a viscous incompressible fluid parallel to a plane wall. Formal asymptotic expansions of the solution for small Reynolds numbers are constructed by using boundary integral equations of the first kind. In contrast to the problem without a wall, we show that there exists a unique solution to the zeroth order problem. However, the problem considered here is still singular in the sense that we find the Stokes paradox in the next higher order problem. A justification of the formal asymptotic expansion for the first two terms is established rigorously.

This paper describes the characterization of optimal constants for some coercivity inequalities in W1,p(Ω), 1 < p < ∞. A general result involving inequalities of p-homogeneous forms on a reflexive Banach space is first proved. The constants are shown to be the least eigenvalues of certain eigenproblems with equality holding for the corresponding eigenfunctions. This result is applied to three different classes of coercivity results on W1,p(Ω). The inequalities include very general versions of the Friedrichs and Poincaré inequalities. Scaling laws for the inequalities are also described.

Conjectures are given for Hilbert series related to polynomial invariants of finite general linear groups: one for invariants mod Frobenius powers of the irrelevant ideal and one for cofixed spaces of polynomials.

It is assumed that the glaciation of Northern Europe during the great Ice Age was due to a shift of the pole associated with appreciable variations of solar radiation.

The shift of the pole brought Europe into sufficiently high latitudes to permit of the formation of an ice-sheet; but the large variations of climate during the Ice Age, as shown by the interglacial epochs, were due to the oscillations of the solar energy.

If two complete cycles of solar radiation occurred during the Pleistocene Period, it is possible to account for four advances of the ice in the Alps as demonstrated by Penck and Brückner, but the interglacial epochs were not all warm. The Günz-Mindel and the Riss-Würm interglacial epochs occurred at the maximum of the solar radiation and were, therefore, warm interglacial epochs; but the Mindel-Riss interglacial epoch occurred at a minimum of solar radiation and was, therefore, a cold interglacial epoch.

At a maximum of solar radiation—that is, during a warm interglacial epoch—the climate of North-West Europe was warm and very wet, with a relatively small annual variation of temperature. As the intensity of solar radiation decreased, the mean temperature fell and the annual variation of temperature increased. At the same time the amount of precipitation decreased. The fall of temperature occurred sufficiently rapidly compared with the decrease in precipitation to cause the glaciers of the Alps to advance and for an ice-sheet to form over Scandinavia. As the solar radiation still further decreased, the lack of precipitation caused the glaciers of the Alps to retreat. At the minimum of solar radiation there was a cold interglacial epoch with low mean temperature, a large annual variation of temperature and very low precipitation; in fact, a truly continental climate.

With these changes of climate went a corresponding change in the flora, the sequence being: park land, forest, tundra, grass with sparse trees, and steppe. In this way it has been found possible to determine a sequence of climates and of fauna and flora for the whole Pleistocene Period, which is supported by the geological and archæological evidence available. In particular it is possible to arrange the sequence of human culture, the geological strata of East Anglia, and the history of the ice in the Alps into the scheme of climate change.

The two maxima of solar radiation were accompanied by increased precipitation in all parts of the world, so accounting for the two pluvial periods which are known to have occurred during the Pleistocene Period.

Korn's inequalities are proved for star-shaped domains and it is shown how the constants in these inequalities depend on the dimensions of the domain. These inequalities are then used to prove a generalisation of Saint-Venant's Principle for nonlinear elasticity and additionally to establish the asymptotic behaviour of solutions to the traction boundary value problem for a non-prismatic cylinder.

When we consider that the Phlogistic Theory formed, as it were, the central point round which the facts of chemistry first crystallized into regular scientific form, and that for more than a hundred years it was recognised by all as the foundation of the science, we might reasonably suppose that it should contain at least some germ of truth.

I think I shall be able to show in the following note that not only is this the case, but that the theory itself, as stated by its founders, Beccher and Stahl, is, if not strictly true, a very close approximation to what we now recognise as truth.

Given a positive, decreasing sequence a, whose sum is L, we consider all the closed subsets of [0, L] such that the lengths of their complementary open intervals are in one-to-one correspondence with the sequence a. The aim of this paper is to investigate the possible values that Assouad-type dimensions can attain for this class of sets. In many cases, the set of attainable values is a closed interval whose endpoints we determine.

Large time behaviour of solutions to a damped quasi-linear wave equation are studied. Conditions are obtained which guarantee the global existence of a classical solution. The asymptotic behaviour of this solution is studied in the case of a unique equilibrium solution and in the case of multiple equilibria. The results are applied to various special examples.

The purpose of this paper is to give universal bounds for the order of the Postnikov k-invariants of infinite loop spaces. This is done by giving universal bounds for the order of the k-invariants of m-connected r-fold loop spaces in dimensions ≦ r + 2m. An application of the result provides information on the Hurewicz homomorphism between the algebraic K-theory of aring and the homology of its general linear group.

A local homeomorphism from a compact, connected manifold with boundary to a simply connected manifold without boundary is shown to be one-to-one if it is one-to-one on each component of the boundary.

We compute the heat content asymptotics of odd dimensional hemispheres in closed form.