The main objects of study are the homotopically stratified metric spaces introduced by Quinn. Closed unions of strata are shown to be stratified forward tame. Stratified fibrations between spaces with stratifications are introduced. Paths that lie in a single stratum, except possibly at their initial points, form a space with a natural stratification, and the evaluation map from that space of paths is shown to be a stratified fibration. Applications to mapping cylinders and to the geometry of manifold stratified spaces are expected in future papers.

Let f be a rotationally invariant function defined on the set Lin+ of all tensors with positive determinant on a vector space of arbitrary dimension. Necessary and sufficient conditions are given for the rank 1 convexity of f in terms of its representation through the singular values. For the global rank 1 convexity on Lin+, the result is a generalization of a two-dimensional result of Aubert. Generally, the inequality on contains products of singular values of the type encountered in the definition of polyconvexity, but is weaker. It is also shown that the rank 1 convexity is equivalent to a restricted ordinary convexity when f is expressed in terms of signed invariants of the deformation.

We introduce the spectral heat function H associated with the Dirichlet-Laplace–Beltrami operator ΔM on a compact smooth Riemannian manifold M with a non-empty smooth boundary. We obtain two-term asymptotics for H without assuming any billiard conditions on M. As a corollary, we obtain estimates for the integral of the k′th Dirichlet eigenfunction of ΔM over M for k→∞.

We consider the Stokes problem with a small viscosity. When the viscosity goes to zero, the boundary-layer phenomenon can appear. In this case, the solution of the given perturbed Stokes equation cannot be properly approximated by the solution of its limiting equation ‘near’ the boundary Γ of the domain of study, say Ω To overcome this problem, we need to construct a corrector term in the neighbourhood of Γ Lions has studied this problem and has constructed a corrector for the case where Ω is a half space in ℝ2. The case where Ω is an open and bounded domain of ℝ2 or ℝ3, which remained unsolved, is the concern of this paper. The construction of the corrector to the perturbed Stokes equation depends heavily on the geometry of Ω In two dimensions, we construct the corrector in the form of a stream function, while in ℝ3 we construct it in the form of a potential vector. The corrector acts effectively in a neighbourhood of Γ that is the boundary layer. Using similar methods to those of Baranger and Tartar, we define the thickness of the boundary layer in a natural way. In addition, in this paper we study the behaviour of the corrected solution in some Hölder spaces.

Local models are given for the singularities that can appear on the trajectories ofgeneral motions of the plane with more than two degrees of freedom. Versal unfoldings of these model singularities give rise to computer-generated pictures describing the family of trajectories arising from small deformations of the tracing point, and determine the local structure of the bifurcation curves.

We consider a class of inhomogeneous systems of reaction-diffusion equations that includes a model for cavity dynamics in the semiconductor Fabry–Pérot interferometer. By adapting topological and geometrical methods, we prove that a standing pulse solution to this system is stable in a certain parameter regime, under the simplification of homogeneous illumination. Moreover, we explain two bifurcation mechanisms which can cause a loss of stability, yielding travelling and standing pulses, respectively. We compute conditions for these bifurcations to persist when inhomogeneity is restored through a certain general perturbation. Under certain of these conditions, a Hopf bifurcation results, producing periodic solutions called edge oscillations. These inhomogeneous bifurcation mechanisms represent new means for the generation of solutions displaying edge oscillations in a reaction-diffusion system. The oscillations produced by each inhomogeneous bifurcation are expected to depend qualitatively on the properties of the corresponding homogeneous bifurcation.

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 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.

Let X be a smooth projective curve of genus g ≥ 2. Here we construct stable vector bundles on X equipped with a filtration with suitable numerical properties and with stable graded subquotients.

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.

In this note, I will present some new problems, and also some old ones which, in my opinion, have been undeservedly forgotten or neglected.

Let An and Gn (respectively, A′n and G′n) be the weighted arithmetic and geometric means of x1, …, xn (respectively, 1 – x1, …, 1 – xn). We present sharp upper and lower bounds for the differences and . And we determine the best possible constants r and s such that

holds for all xi ∈ [a, b] (i = 1, …, n; 0 < a < b < 1). Our theorems extend and sharpen results of Fan, Cartwright and Field, McGregor and the author.

A vector equation which gives the velocity of a vortex filament embedded in an inviscid incompressible flow is considered. It comprises terms representing effects from the localized self-induction and from the external flow. The initial value problem is proved to have at least a solution for a suitable external flow term.

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.

It is commonly believed that, as far as stabilities are concerned, ‘small delays are negligible in some modelling processes’. However, to have an affirmative answerfor this common belief is still an open problem for many nonlinear equations. In this paper, the classical Lotka–Volterra prey–predator equation with discrete delays

is considered, and, by using degenerate Lyapunov functionals method, an affirmative answer to this open problem on both local and global stabilities of the prey–predator delay equations is given. It is shown that degenerate Lyapunov functional method is a powerful tool for studying the stability of such nonlinear delay systems. A detailed and explicit procedure of constructing such functionals is provided. Furthermore, some explicit estimates on the allowable sizes of the delays are obtained.

Using the ‘monotonicity trick’ introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais–Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the form

We assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity f satisfies (i) f(x, s)s−1 → a ∈)0, ∞) as s →+∞; and (ii) f(x, s)s–1 is non decreasing as a function of s ≥ 0, a.e. x → ℝN.