Flat currents modulo p in metric spaces and filling radius inequalities

We adapt the theory of currents in metric spaces, as developed by the first-mentioned author in collaboration with B. Kirchheim, to currents with coefficients in Z_p. Building on S. Wenger's work in the orientable case, we obtain isoperimetric inequalities mod(p) in Banach spaces and we apply these inequalities to provide a proof of Gromov's filling radius inequality (and therefore also the systolic inequality) which applies to nonorientable manifolds, as well. With this goal in mind, we use the Ekeland principle to provide quasi-minimizers of the mass mod(p) in the homology class, and use the isoperimetric inequality to give lower bounds on the growth of their mass in balls.Comment: 31 pages, to appear in Commentarii Mathematici Helvetic

Minimal Curvature Trajectories: Riemannian Geometry Concepts for Model Reduction in Chemical Kinetics

In dissipative ordinary differential equation systems different time scales cause anisotropic phase volume contraction along solution trajectories. Model reduction methods exploit this for simplifying chemical kinetics via a time scale separation into fast and slow modes. The aim is to approximate the system dynamics with a dimension-reduced model after eliminating the fast modes by enslaving them to the slow ones via computation of a slow attracting manifold. We present a novel method for computing approximations of such manifolds using trajectory-based optimization. We discuss Riemannian geometry concepts as a basis for suitable optimization criteria characterizing trajectories near slow attracting manifolds and thus provide insight into fundamental geometric properties of multiple time scale chemical kinetics. The optimization criteria correspond to a suitable mathematical formulation of "minimal relaxation" of chemical forces along reaction trajectories under given constraints. We present various geometrically motivated criteria and the results of their application to three test case reaction mechanisms serving as examples. We demonstrate that accurate numerical approximations of slow invariant manifolds can be obtained.Comment: 22 pages, 18 figure