Making use of the Karamata regular variation theory and the López-Gómez localization method, we establish the uniqueness and asymptotic behaviour near the boundary ∂Ω for the large solutions of the singular boundary-value problem

where Ω is a smooth bounded domain in ℝN. The weight function b(x) is a non-negative continuous function in the domain, which can vanish on the boundary ∂Ω at different rates according to the point x0 ∊ ∂Ω. f(u) is locally Lipschitz continuous such that f(u)/u is increasing on (0, ∞) and f(u)/up = H(u) for sufficiently large u and p > 1, here H(u) is slowly varying at infinity. Our main result provides a sharp extension of a recent result of Xie with f satisfying limu→f(u)/up = H for some positive constants H > 0 and p > 1.

Let D(A) be the domain of an m-accretive operator A on a Banach space E. We provide sufficient conditions for the closure of D(A) to be convex and for D(A) to coincide with E itself. Several related results and pertinent examples are also included.

For a model of a driven interface in an elastic medium with random obstacles we prove the existence of a stationary positive supersolution at non-vanishing driving force. This shows the emergence of a rate-independent hysteresis through the interaction of the interface with the obstacles despite a linear (force = velocity) microscopic kinetic relation. We also prove a percolation result, namely, the possibility to embed the graph of an only logarithmically growing function in a next-nearest neighbour site percolation cluster at a non-trivial percolation threshold.

We study the elasticity domain for an antiplane deformation of a perfect elastoplastic medium, which is described by the Prandtl-Reuss equations. We prove that a boundary of this domain can be found by solving a system of nonlinear functional equations. In the simplest case of simple shear deformations, this system of equations is studied in detail.

The object of the present paper is to put upon record for the guinea-pig data which I have collected regarding (1) the relation between the number of young in a litter and the number of corpora lutea of pregnancy in the ovaries, (2) the distribution of corpora lutea between the two ovaries in two successive ovulations, and (3) the utilisation of the uterine horns in successive pregnancies. The data were obtained in the course of other researches and are arranged in tabular form (Tables I and II). The records from which they have been collected refer to 40 female guinea-pigs killed post partum, of which 16 were pregnant and 24 had not been reimpregnated. The size of the previous litter is known for 38 of these females; both ovaries were serially sectioned at 10 microns in the 40 animals, but in 4 the records of only one ovary were preserved.

We consider, for m ∈ (0, 1) and q > 1, the porous media equation with absorption We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in Rn × [0, ∞)\{(0, 0)}, and satisfy u(x, 0) = 0 for all x ≠ 0. We prove the following results. When q ≥ m + 2/n, there does not exist any such singular solution. When q < m + 2/n, there exists, for every c > 0, a unique singular solution u = u(c), called the fundamental solution with initial mass c, which satisfies ∫Rnu(·, t) → c as t ↘ 0. Also, there exists a unique singular solution u = u∞, called the very singular solution, which satisfies ∫Rnu∞(·, t) → ∞ as t ↘ 0.

In addition, any singular solution is either u∞ or u(c) for some finite positive c, u(c1) < u(c2) when c1 < c2, and u(c) ↗ u∞ as c ↗ ∞.

Furthermore, u∞ is self-similar in the sense that u∞(x, t) = t−αw(|x| t−αβ) for α = 1/(q − 1), β = ½(q − m), and some smooth function w defined on [0, ∞), so that is a finite positive constant independent of t > 0.

Let

be, for each λ∈ℝk, a compact symmetric operator on a complex Hilbert space. Let the“fundamental” eigenset Z be denned by the relation λ∈Z if and only if W(λ) has maximal eigenvalue one. Conditions are given for Z to be the boundary of an open convex set P. A detailed investigation is given of the structure of P, including its recession cone and its representations as intersections of half-spaces.

Results are obtained on the existence of positive solutions to the following elliptic system:

in a bounded region Ω in Rn with a smooth boundary, where the diffusion terms φ ψ are non-negative functions and the system could be degenerate, β γ are strictly increasing functions, k,σ ≧ 0 are constants. We assume also that the growth rates f, g satisfy certain monotonicities. Applications to biological interactions with density-dependent diffusions are given.

Let Cf, Pf, Qf and Rf be respectively the convex, polyconvex, quasi-convex and rank-one-convex envelopes of a given function f. If fp: RNxM→R and fq(ξ) behaves as |ξ|q at infinity q ∈ (1, ∞), we show that . This is the case for (Pfp)p provided that q ≠1,…, min (N, M), otherwise . In the last part of this work, we show that f(ξ) = g(|ξ|) does not imply in general Pf = Qf.

We consider two model reaction-diffusion systems of bistable type arising in the theory of phase transition; they appear in various physical contexts, such as thin magnetic films and diblock copolymers. We prove the convergence of the solution of these systems to the solution of free-boundary problems involving modified motion by mean curvature.

We investigate two kinds of Fricke families, those consisting of Fricke functions and those consisting of Siegel functions. In terms of their special values we then generate ray class fields of imaginary quadratic fields over the Hilbert class fields, which are related to the Lang–Schertz conjecture.

We prove that a class of A-stable symplectic Runge–Kutta time semi-discretizations (including the Gauss–Legendre methods) applied to a class of semilinear Hamiltonian partial differential equations (PDEs) that are well posed on spaces of analytic functions with analytic initial data can be embedded into a modified Hamiltonian flow up to an exponentially small error. Consequently, such time semi-discretizations conserve the modified Hamiltonian up to an exponentially small error. The modified Hamiltonian is O(hp )-close to the original energy, where p is the order of the method and h is the time-step size. Examples of such systems are the semilinear wave equation, and the nonlinear Schrödinger equation with analytic nonlinearity and periodic boundary conditions. Standard Hamiltonian interpolation results do not apply here because of the occurrence of unbounded operators in the construction of the modified vector field. This loss of regularity in the construction can be taken care of by projecting the PDE to a subspace in which the operators occurring in the evolution equation are bounded, and by coupling the number of excited modes and the number of terms in the expansion of the modified vector field with the step size. This way we obtain exponential estimates of the form O(exp(–c/h 1/(1+q))) with c > 0 and q ⩾ 0; for the semilinear wave equation, q = 1, and for the nonlinear Schrödinger equation, q = 2. We give an example which shows that analyticity of the initial data is necessary to obtain exponential estimates.

Among optical phenomena which originate over the surface of water there is one so ill-defined and ill-observed as to be still mysterious; till now it has received no valid explanation. The Italians call it the Fata Morgana. Under conditions still lacking precise description, there appear on the far side of the Straits of Messina certain fantastic visions, fortresses and castles of unknown cities, which seem to emerge from the sea, soon to vanish again. These are the “palaces” of the “fairy Morgana,” which appear and disappear at the capricious stroke of the magician's wand.

The “Challenger” Expedition had no opportunity of visiting Barbadoes, and this I regretted greatly, as Sir Rawson Rawson, who was at that time governor of the island, had paid great attention to the marine fauna, and was anxious to introduce us to his fine collection, which included many specimens of the rare and remarkable forms for which the sea of the Antilles is famous.

In a series of papers dating from 1897, resumés of winch are given in the Bulletin international de I'académie des Sciences de Cracovie, Professor Browicz gives the results of observations on the liver cells, both in the normal state and in a pathologically altered condition, and draws from these observations certain important deductions. The most prominent fact which is recorded is the appearance (under normal conditions in the liver of the newborn child, and under pathological conditions in the adult human liver; under normal conditions, and after the intravenous injection of hæmoglobin some hours prior to death, in the liver of the dog) of erythrocytes, singly and in groups, and of free hæmoglobin and hæmoglobin crystals, as well as brown pigment, in the form both of granules and crystalline clumps, within the hepatic cells, both in the nucleus and in the cytoplasm. After affirming the undoubted existence of intracellular biliary passages communicating with intercellular bile ducts,—which intracellular passages he regards, not as accidental and temporary vacuoles, but as preformed channels within the cells, — Professor Browicz states that his observations, especially the presence within the liver cells of erythrocytes, demonstrate that the cells must be in communication with the hepatic capillaries.

In a paper presented to the Society in 1867–8 I deduced from certain hypothetical considerations regarding Dissipation of Energy results connected with the thermal and electric conductivity of bodies, the electric convection of heat, &c. As these were all of a confessedly somewhat speculative character, I printed at the time only that connected with thermal conductivity, which I had the means of comparing with experiment, and which seemed to accord fairly with Forbes' experimental results. But the assumption on which this was based was essentially involved in all the other portions of the paper.

The problem of the simultaneous transport of solute and water through a cell membrane is examined from a kinetic standpoint. It is shown that the results obtained by this approach are in accord with those obtained conventionally by irreversible thermodynamics. Stress is laid on the fact that the kinetic theory has certain advantages over the conventional approach in that it provides, among other things, a form of membrane model and at the same time its range of validity is indicated.

This paper deals with the existence of bounded plane wave fronts of reaction-diffusion systems. The main result ensures the partial invariance of a certain region, under kinetic conditions commonly used in the literature. This allows us to construct bounded plane wave fronts taking their values in that domain. We also give an estimate of the minimum permissible value of the propagation velocity of those plane wave fronts. Some examples are given.

Given a set valued mapping ∑: ℝ → ∑n, we prove a closure property with respect to -convergence for the differential inclusion

under very mild assumptions on ∑.