We show that no infinite-dimensional Banach space provided with a strictly convex norm satisfies Lindenstrauss's property B. This is a generalization of previous results by Lindenstrauss for rotund spaces isomorphic to C0 and by Gowers for ℓp (1 < p < ∞). Also, there is an appropriate complex version of the announced result that works for all the C-strictly convex spaces. As a consequence, the Hardy space H1, any infinite-dimensional complex L1(μ), and, in general, any infinite-dimensional predual of a von Neumann algebra lacks Lindenstrauss's property B.

We introduce the concept of an elementary operator between two algebras, and thereby extend the study of operators of the form

where ai, bi are fixed elements in an algebra. Elementary operators on some classes of algebras are considered and their general form is described in some special cases.

We show that shooting methods for homoclinic or heteroclinic orbits in dynamical systems may automatically guarantee the topological transversality of the stable and unstable manifolds. The interest of such results is twofold. First, these orbits persist under perturbations which destroy the structure allowing the shooting method and, second, topological transversality is often sufficient when some kind of transversality is required to obtain chaotic dynamics. We shall focus on heteroclinic solutions in the extended Fisher–Kolmogorov equation.

We consider the Ginzburg–Landau type equation

where G is a smooth bounded domain in ℝ2, g ∊ C∞(∂G;ℝ2 / {0}), and ε > 0 is a small parameter. We prove the uniqueness of solutions to this equation under some non-vanishing assumptions on uε, or under conditions on the boundary function g.

Using the viscosity vanishing method, we obtain the existence of a smooth solution to the inhomogeneous Heisenberg chain equations in one dimension. The uniqueness of the solution to the viscosity equation is also given.

A class of Noetherian semigroup algebras K[S] is described. In particular, we show that, for any submonoid S of the semigroup Mn of all monomial n × n matrices over a polycyclic-by-finite group G, K[S] is right Noetherian if and only if S satisfies the ascending chain condition on right ideals. This is then used to prove that every prime homomorphic image of a semigroup algebra of a finitely generated Malcev nilpotent semigroup S satisfying the ascending chain condition on right ideals is left and right Noetherian.

In this paper, we study the blow-up behaviour of the radially symmetric non-negative solutions u of the semilinear heat equation with supercritical power nonlinearity up (that is, (N – 2)p> N + 2). We prove the existence of non-trivial self-similar blow-up patterns of u around the blow-up point x = 0. This result follows from a convergence theorem for a nonlinear parabolic equation associated to the initial one after rescaling by similarity variables.

In this paper we establish a non-global Cα estimation on the boundary for the gradient of the solution of a Neumann exterior problem in ℝ3. The boundary data of this problem are with small compact support and with C0 norm tending to infinity when the diameter of support data goes to zero. This problem appears, for instance, in the numerical shape optimization problems when the Newton method is used. It turns out that the gradient decreases when the distance to the boundary data supports increases (Saint Venant's principle), which, when used in applications, leads to a strong reduction of the gradient computational cost.

Let X be a Banach space and Y a closed subspace. We introduce an intrinsic geometric property of Y—the k-ball sequence property—which is a weakening of the famous k-ball property due to Alfsen & Effros. We prove that Y satisfies the 2-ball sequence property if and only if Y has the Phelps uniqueness property U (i.e. every continuous linear functional g ∈Y* has a unique norm-preserving extension f ∈X*). We prove that Y is an ideal having property U if and only if Y satisfies the 3-ball sequence property, and in this case, Y satisfies the k-ball sequence property for all k.

Reaction-diffusion systems on the real line are considered. Localized travelling waves become unstable when the essential spectrum of the linearization about them crosses the imaginary axis. In this article, it is shown that this transition to instability is accompanied by the bifurcation of a family of large patterns that are a superposition of the primary travelling wave with steady spatially periodic patterns of small amplitude. The bifurcating patterns can be parametrized by the wavelength of the steady patterns; they are time-periodic in a moving frame. A major difficulty in analysing this bifurcation is its genuinely infinite-dimensional nature. In particular, finite-dimensional Lyapunov–Schmidt reductions or centre-manifold theory do not seem to be applicable to pulses having their essential spectrum touching the imaginary axis.

We study the existence and the asymptotic behaviour of global solutions of the semilinear parabolic equation u(0) = ϧwhere a, b ∈ℝ, q > 1, p > 1. Forq=2p/(p+1) and ½ 1(p-1)>1 (equivalently, q > (n + 2)/(n + 1)), we prove the existence of mild global solutions for small initial data with respect to some norm. Some of those solutions are asymptotically self-similar.

For the Euclidean squared-distance function f(·) = dist2(·, K), with K ⊂ MN×n, we show that K is convex if and only if f(·) equals either its rank-one convex, quasiconvex or polyconvex relaxations. We also establish that if (i) K is compact and contractible or (ii) dim C(K) = k < Nn, K is convex if and only if f equals one of the semiconvex relaxations when dist2(P, K) is sufficiently large, and for case (i), P ∈MNxn; for case (ii), P ∈ Ek—a k-dimensional plane containing C(K). We also give some estimates of the difference between dist2(P, K) and its semiconvex relaxations. Some possible extensions to more general p-distance functions are also considered.