We show that, if f : M2×2 → R is rank-one convex on the hyperboloid is the set of 2 × 2 real symmetric matrices, then f can be approximated by quasi-convex functions on M2×2 uniformly on compact subsets of . Equivalently, every gradient Young measure supported on a compact subset of is a laminate.

This paper is concerned with the blowup of positive solutions of the semilinear heat equation

with zero boundary conditions. The domain Ω is supposed to be smooth, convex and bounded. We first show that, under the assumption that the initial data are uniformly monotone near the boundary, solutions that exist on the time interval (0, T form a compact family in a suitable topology. We then derive some localisation properties of these solutions. In particular, we discuss a general criterion, independent of the initial data, which in some cases ensures single-point blowup.

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.

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.

We study the differentiability properties of the topological equivalence between a uniformly asymptotically stable linear nonautonomous system and a perturbed system with suitable nonlinearities. For this purpose, we construct a homeomorphism inspired in the Palmer's one restricted to the positive half line, studying additional continuity properties and providing sufficient conditions ensuring its Cr–smoothness.

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

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

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

${\cal D}$

Let τ denote the second-order elliptic expression

where the coefficients bj and q are complex-valued, and let Ω be a spherical shell Ω = {x:x ∈ ℝn, l <|x|<m} with l≧0, m≦∞. Under the conditions assumed on the coefficients of τ and with either Dirichlet or Neumann conditions on the boundary of Ω, τ generates a quasi-m-sectorial operator T in the weighted space L2(Ω;w). The main objective is to locate the spectrum and essential spectrum of T. Best possible results are obtained.

A theorem of Rudin states that if B is the open unit ball in ℂN, N > 1, if 0<ρ < 1, if is the family of all complex lines in ℂN at a distance ρ from the origin and if f ∈ C(∂B) is such that for every Λ∈ the function f|Λ∂B has a continuous extension to Λ ∩ B which is holomorphic in Λ ∩ B, then f has a continuous extension to B which is holomorphic in B. In this paper we show that when N = 2, the theorem still holds if is replaced by a considerably smaller family.

We show that a gap phenomenon occurs for general variational integrals for mappings from a domain Rn into a Riemannian manifold if has a non-trivial topology.

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.

This paper considers a model for oncolytic virotherapy given by the doubly haptotactic cross-diffusion system

\[ \left\{\begin{array}{@{}ll} u_t=D_u\Delta u-\xi_u\nabla\cdot(u\nabla v)+\mu_u u(1-u)-\rho uz,\\ v_t={-} (\alpha_u u+\alpha_w w)v,\\ w_t=D_w\Delta w-\xi_w\nabla\cdot(w\nabla v)- w+\rho uz,\\ z_t=D_z\Delta z-\delta_z z- \rho uz+\beta w, \end{array}\right. \]

$D_u,D_w,D_z,\xi _u,\xi _w,\delta _z,\rho$

$\alpha _u,\alpha _w,\mu _u,\beta$

$\Omega \subset {\mathbb {R}}^{2}$

$\beta <1$

$(u,v,w,z)$

$(1, 0, 0, 0)$

$(L^{\infty }(\Omega ))^{4}$

$t\rightarrow \infty$

In this paper, we study a dissipative systems modelling electrohydrodynamics in incompressible viscous fluids. The system consists of the Navier–Stokes equations coupled with a classical Poisson–Nernst–Planck equations. In the three-dimensional case, we establish a global regularity criteria in terms of the middle eigenvalue of the strain tensor in the framework of the anisotropic Lorentz spaces for local smooth solution. The proof relies on the identity for entropy growth introduced by Miller in the Arch. Ration. Mech. Anal. [16].

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.

Balanus nubilis, one of the sessile Cirripedes, grows to a considerable size, a large specimen measuring some 10 to 11 cm. across the base and an equal distance in height. During a visit to Nanaimo Biological Station an opportunity was taken to study in this giant form a few features pertaining to the mechanism of movement of the operculum, which is attached within the cup of the shell by the opercular membrane. Peripherally the membrane is attached close to the downwardly directed sharp edge of an overhanging shelf formed by the inner wall of the shell; centrally it is attached to the free edge of the operculum on its basal aspect.

This paper commences with the following extract from the author's private journal, of date January 6, 1858:—

“Let be rectangular components of an impulsive force and an impulsive couple applied to a solid of invariable shape, with or without inertia of its own, in a perfect liquid, and let u, ν, ω, ѽ, ρ, σ, be the components of linear and angular velocity generated.

We consider the positive solutions to singular boundary-value problems of the formwhere λ > 0, β ∈ (0,1) and Ω is a bounded domain in ℝN, N ≥ 1, with smooth boundary ∂Ω. Here, we assume that f: [0, ∞) → (0, ∞) is a C1 non-decreasing function and f(s)/sβ is decreasing for s large. We establish the uniqueness of the positive solution when λ is large.

Given any countably infinite set of isloated points on the ℷ -axis, it is shown that there is a continuous q(x) such that these points constitute exactly the point-continuous spectrum for the equation yn″(x) + (ℷ —q(x))y(x) = 0(0≦x<∞) with some homogenous boundary condition at x = 0. This extends a result given by Eastham and McLeod for countably infinite sets of isolated points on the positive ℷ-axis.

This paper is concerned with the existence results for generalized transition waves of space periodic and time heterogeneous lattice Fisher-KPP equations. By constructing appropriate subsolutions and supersolutions, we show that there is a critical wave speed such that a transition wave solution exists as soon as the least mean of wave speed is above this critical speed. Moreover, the critical speed we construct is proved to be minimal in some particular cases, such as space-time periodic or space independent.

The ventricular rhythm differs from the auricular, firstly, on the occurrence of a ventricular extra-systole, and, secondly, in cases where there is allorhythmia due to depression of conductivity of the heart muscle. When conductivity is depressed, the contraction stimulus at regular or irregular intervals may be blocked at the auriculo-ventricular muscle bridge; so that whereas some stimuli induce both auricular and ventricular systole, others induce auricular systole alone, the ventricular systole being missed. That condition, incomplete heart-block, is of not infrequent occurrence in the human heart, and has been graphically recorded by Mackenzie (1), Finkelnburg (2), and other writers.