A theorem is proved characterising representable, multiplicative commutative cohomology theories that split as sums of singular cohomologies after localisation at 2. This theorem is shown to be equivalent to one proved by Würgler and Pazhitnov and Rudyak, for which we provide a simplified proof. We also provide a simple proof of a related theorem of Boardman.

A large class of first-order partial nonlinear differential equations in two independent variables which possess an infinite set of polynomial conservation laws derived from an explicit generating function is constructed. The conserved charge densities are all homogeneous polynomials in the unknown functions which satisfy the differential equations in question. The simplest member of the class of equations is related to the Born–Infeld Equation in two dimensions. It is observed that some members of this class possess identical charge densities. This enables the construction of a set of multivariable equations with an infinite number of conservation laws.

This paper addresses the computation of normal forms for periodic retarded functional differential equations (FDEs) with autonomous linear part. The analysis is based on the theory previously developed for autonomous retarded FDEs. Adequate nonresonance conditions are derived. As an illustration, the Bogdanov–Takens and the Hopf singularities are considered.

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.

We prove the existence of an arbitrarily large number of periodic solutions for a class of nonlinear differential equations generalising the dynamics of a forced pendulum with small length.

This is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains, and it concentrates on exploring the structure of these spaces, such as the embeddings of weighted Sobolev spaces, the relation between weighted Sobolev spaces and weighted continuous function spaces, and the relations between the weighted Sobolev spaces and countably weighted Sobolev spaces in Cartesian coordinates and in the spherical and cylindrical coordinates. These well-defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in ℝ3, which are essential for the design of effective computation and the analysis of the h – p version of the finite element method for solving elliptic problems in three-dimensional nonsmooth domains arising from mechanics and engineering.

We derive necessary and sufficient conditions for a unit regular monoid to have a uniquely unit regular, idempotent separating cover.

We define the rational de Rham cohomology associated with the generalised confluent hypergeometric functions. Purity of the cohomology is proved and an explicit ℂ-basis of the nontrivial cohomology is computed.

We consider certain n-dimensional operators of Hardy type and we study their boundedness in These spaces were introduced by M. J. Carro and J. Soria and include weighted Lp, q spaces and classical Lorentz spaces. As an application, we obtain mixed weak-type inequalities for Calderón—Zygmund singular integrals, improving results due to K. Andersen and B. Muckenhoupt.

We present a simple probability approach for establishing a gradient estimate for a solution of an elliptic equation on a compact Riemannian manifold with convex boundary, or on a noncompact complete manifold. Our method can also be applied to derive a similar gradient estimate for a nonlinear parabolic equation, and an abstract gradient estimate for a Markov semigroup.

In this paper, we prove that the Hele–Shaw problem with kinetic condition and surface tension is the limit case of the supercooled Stefan problem in the classical sense when specific heat ε goes to zero. The method is the use of a fixed-point theorem; the key is to construct a workable function space. The main feature is to obtain the existence and the uniform estimates with respect to ε > 0 at the same time for the solutions of the supercooled Stefan problem. For the sake of simplicity, we only consider the case of one phase, although the method used here is also applicable in the case of two phases.

Consider the unique continuation problem for the nonlinear Schrödinger (NLS) equation

By using the inverse scattering transform and some results from the Hardy function theory, we prove that if u ∈ C(R; H1(R)) is a solution of the NLS equation, then it cannot have compact support at two different moments unless it vanishes identically. In addition, it is shown under certain conditions that if u is a solution of the NLS equation, then u vanishes identically if it vanishes on two horizontal half lines in the x–t space. This implies that the solution u must vanish everywhere if it vanishes in an open subset in the x–t space.

We prove that connected subsets of M2×2 without rank-one connections are Lipschitz graphs of mappings from subsets of a fixed two-dimensional subspace to its orthogonal complement. Under a weaker condition that the set does not have rank-one connections locally, we are able to establish some global results on the set. We also establish some results on Lipschitz extensions of the functions thus obtained.