The purpose of this paper is to prove strong-type inequalities with one-sided weights for commutators (with symbol b ∈ BMO) of several one-sided operators, such as the one-sided discrete square function, the one-sided fractional operators, or one-sided maximal operators given by the convolution with a smooth function. We also prove that b ∈ BMO is a necessary condition for the boundedness of commutators of these one-sided operators.

It is shown that four-dimensional generalized symmetric spaces can be naturally equipped with some additional structures defined by means of their curvature operators. As an application, those structures are used to characterize generalized symmetric spaces.

We study the asymptotic behaviour of solutions of the fast diffusion equation near extinction. For a class of initial data, the asymptotic behaviour is described by a singular Barenblatt profile. We complete previous results on rates of convergence to the singular Barenblatt profile by describing a new phenomenon concerning the difference between the rates in time and space.

The chemical composition of submarine glauconite is of considerable interest in view of the fact that glauconite is the only silicate which is synthesized at the bottom of the sea, and apparently nowhere else. Numerous analyses of this mineral have been published from time to time, but the results are far from uniform, because the material almost always—certainly in most of the older analyses—was anything but pure. The analysis which inspires most confidence on this score is one which was recently carried out in the Challenger laboratory by Collet and Lee on a purified granular glauconite dredged off the Californian coast by U.S.S. Tuscarora (1879).

Under appropriate conditions on b and a function g with 2k +1 simple zeros, the equation

has a maximal compact invariant set Ab,g in C([−1,0]R), consisting of the zeros of g and the one-dimensional unstable manifolds of these zeros. For k =2, it is shown that there may be a saddle connection in the flow on Ab,g for some g. This implies that the zeros of g as elements of the flow on Ab,g cannot be given the natural order of the reals.

We first prove existence and uniqueness of non-negative solutions of the equation

in in the range 1 < p < 1 + 2/N, when initial data u(x, 0) = a|x|−2(p−1), x ≠ 0, for a > 0. It is proved that the maximal and minimal solutions are self-similar with the form

where g = ga satisfies

After uniqueness is proved, the asymptotic behaviour of solutions of

is studied. In particular, we show that

The case for a = 0 is also considered and a sharp decay rate of the above equation is derived. In the final, we reveal existence of solutions of the first and third equations above, which change sign.

It has long been known that yellow phosphorus is slowly oxidised in air at the ordinary temperature and that the oxidation is accompanied by a glow easily visible in a darkened room. If oxygen at atmospheric pressure is substituted for air no glow is perceived, and the oxidation, if it occurs at all, is incomparably slower than in air. If the pressure of oxygen is reduced either directly or by admixture with an indifferent gas the glow may be made to reappear. The seeming contradiction of the law of mass action involved in these phenomena has been the subject of many experiments, and the main object of the present investigation is to attempt to throw some light on this anomaly.

The mapping properties of the finite Hilbert-transform (respectively the Hilbert transform on the half axis) are studied. Invertibility, surjectivity, injectivity and bounded ness from below of the transform are characterised in general weighted spaces. The results are applied to the restriction of the operator with logarithmic kernel.

Using a dual variational approach, we obtain nontrivial real-valued solutions of the critical nonlinear Helmholtz equation

$$-\Delta u-k^2u = Q(x) \vert u \vert ^{2^*-2}u,\quad u\in W^{2,2^*}({\open R}^{N})$$

$Q \in L^{\infty }({\open R}^{N}){\setminus }\{0\}$

Using a regular Borel measure μ ⩾ 0 we derive a proper subspace of the commonly used Sobolev space D1(ℝN) when N ⩾ 3. The space resembles the standard Sobolev space H1(Ω) when Ω is a bounded region with a compact Lipschitz boundary ∂Ω. An equivalence characterization and an example are provided that guarantee that is compactly embedded into L1(RN). In addition, as an application we prove an existence result of positive solutions to an elliptic equation in ℝN that involves the Laplace operator with the critical Sobolev nonlinearity, or with a general nonlinear term that has a subcritical and superlinear growth. We also briefly discuss the compact embedding of to Lp(ℝN) when N ⩾ 2 and 2 ⩽ p ⩽ N.

This paper is concerned with the existence of a global attractor for a semi-flowgoverned by the weak solutions to a nonlinear one-dimensional thermoviscoelasticsystem with clamped boundary conditions in shape memory alloys. The constitutiveassumptions for the Helmholtz free energy include the model for the study ofmartensitic phase transitions in shape memory alloys. To describe physicallyphase transitions between different configurations of crystal lattices, we workin a framework in which the strain u belongsto L∞. New approachesare introduced and more delicate estimates are derived to establish the crucialL∞ -estimate ofstrain u in the course of showing thecompactness of the orbit of the semi-flow and existence of an absorbing set.

We consider an emulsion of two Stokes fluids, one of which is periodically distributed in the form of small spherical bubbles. The effects of surface tension on the bubble boundaries are modelled mathematically, as in the work of G. I. Taylor, by a jump only in the normal component of the traction. For a given volume fraction of bubbles, we consider the two-scale convergence, and in the fine phase limit we find that the bulk flow is described by an anisotropic Stokes fluid. The effective viscosity tensor is consistent with the bulk stress formula obtained by Batchelor [2].

We study the effect of algebraically localized impurities on striped phases in one spatial dimension. We therefore develop a functional-analytic framework that allows us to cast the perturbation problem as a regular Fredholm problem despite the presence of the essential spectrum, caused by the soft translational mode. Our results establish the selection of jumps in wavenumber and phase, depending on the location of the impurity and the average wavenumber in the system. We also show that, for select locations, the jump in the wavenumber vanishes.

In nearly every narrative of exploration in uncivilised tropical regions, accounts are given, often no doubt somewhat fanciful, of poisonous substances which are said to possess the most remarkable properties. Usually these poisons are of vegetable origin; and the great majority may be included in the two divisions of ordeal and of arrow poisons, according as they are applied to one or other of these purposes. Among the most remarkable of the ordeal poisons are the Tanghinia venifera of Madagascar, the Physostigma venenosum of Old Calabar, and the Akazga poison of the Gaboon; and of the arrow-poisons, the famous Curara or Wourali of South America, and the Antiaris toxicaria of Java.

Professor Allman described a new genus of Polyzoa, obtained by the Rev. A. M. Norman and Mr J. Gwyn Jeffreys, from deep-sea dredgings in Shetland.

Its cœnœcium consists of a branched tube, partly adherent and partly free, the free portion forming tubes of egress, through which the polypides move in the acts of exsertion and retraction. In the walls of the adherent portion a rigid chitinous rod is developed along their attached side, and to this rod the polypides are connected from distance to distance each by a flexible cord or funiculus.

The processes of long-wave low-temperature radiation in the atmosphere have been studied with a view to investigating the extent to which they may be regarded as diffusion-like; the decrease of water-vapour with height is taken to be exponential, that of temperature to be constant up to the tropopause, and then to be zero. Full allowance is made for the radiation being diffuse, involving the functions ; such properties of these functions as are required in the further work are studied. The expression for the upward flow of radiation is obtained in the form of a power series in the temperature gradient in the troposphere; the coefficient of the first power of the temperature gradient is defined as the “coefficient of radiative diffusion.” There is shown to be a certain amount of ambiguity about the coefficient so defined.

A model atmosphere resembling fairly well the conditions in N.W. Europe is considered: the upward flow of radiation and the coefficient of radiative diffusion are calculated; for this purpose we require Hettner's measurements of water-vapour absorption as modified by Simpson.

The effect of the heating of the ground during the day to a temperature above that of the air near by is then considered; it is shown that the study of the results of this factor which are of importance in the interpretation of continuous temperature records near the ground requires measurements of the absorption of water-vapour for the wave-lengths for which it is particularly opaque; these do not appear to exist.

This paper is concerned with the asymptotic behaviour of solutions to quasilinear hyperbolic equations with nonlinear damping on the quarter-plane (x, t) ∈ ℝ+ x ∈ ℝ+. We obtain the Lp (1 ≤ p ≤ +∞) convergence rates of the solution to the quasilinear hyperbolic equations without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton.

We define and study twisted Alexander-type invariants of complex hypersurface complements. We investigate torsion properties for the twisted Alexander modules and extend the local-to-global divisibility results of Maxim and of Dimca and Libgober to the twisted setting. In the process, we also study the splitting fields containing the roots of the corresponding twisted Alexander polynomials.

We consider the eigenvalue gap for Schrödinger operators on an interval with Dirichlet or Neumann boundary conditions. For a class of symmetric potentials, we prove that the gap between the two lowest eigenvalues is maximized when the potential is constant. We also give some related results for doubly symmetric potentials.

This corrigendum corrects a result in [1].