Abstract We introduce a new metric homology theory, which we call Moderately Discontinuous Homology, designed to capture Lipschitz properties of singular subanalytic germs. The main novelty our approach is allow “moderately discontinuous” chains, are specially advantageous for capturing the subtleties outer phenomena. Our invariant finitely generated graded abelian group any and homomorphisms ....

In this article, we prove convergence of the weakly penalized adaptive discontinuous Galerkin methods. Unlike other works, we derive the contraction property for various discontinuous Galerkin methods only assuming the stabilizing parameters are large enough to stabilize the method. A central idea in the analysis is to construct an auxiliary solution from the discontinuous Galerkin solution by ...

We perform the nonlinear stability analysis of a one-dimensional chemotaxis model bacterial self-organization, assuming that bacteria respond sharply to chemical signals. The resulting discontinuous advection speed represents key challenge for analysis. follow perturbative approach, where shape cellular profile is clearly separated from its global motion, allowing us circumvent discontinuity is...