The acid-mediation hypothesis, that is, the hypothesis that acid produced by tumours, as a result of aerobic glycolysis, provides a mechanism for invasion, has so far been considered as a relatively closed system. The focus has mainly been on the dynamics of the tumour, normaltissue, acid and possibly some other bodily components, without considering the effect of an external intervention such ...

If there exists a diffeomorphism f on a closed, orientable n-manifold M such that the non-wandering set Ω(f) consists of finitely many orientable (±) attractors derived from expanding maps, then M is a rational homology sphere; moreover all those attractors are of topological dimension n − 2. Expanding maps are expanding on (co)homologies.

The asymptotical limit of the trajectory deened by the continuous steepest descent method for a proper closed convex function f on a Hilbert space is characterized in the set of minimizers of f via an asymp-totical variational principle of Brezis-Ekeland type. The implicit discrete analogue (prox method) is also considered.

Let G = {etA : t ∈ R} be a closed one-parameter subgroup of the general linear group of matrices of order n acting on Rn by matrix-vector multiplication. We assume that all eigenvalues of A are rationally related. We study conditions for which the set { f ( et1A· ) , · · · , f ( etmA· )} is linearly dependent in Lp (Rn) with 1 ≤ p <∞. AMS Subject Classification: 39B52