On the superlinear convergence of PCG algorithms: Numerical experiments for convection-diffusion equations

AbstractThe CGM is studied for nonsymmetric elliptic problems with both Dirichlet and mixed boundary conditions. The mesh independence of the convergence is an important property when symmetric part preconditioning is applied to the FEM discretizations of the boundary value problem. Computations in two dimensions are presented to illustrate the mesh independent superlinear convergence for convection-diffusion equations with both types of boundary conditions. Preconditioning by the leading term plus a zeroth-order term is also investigated in the aspect of superlinear convergence through numerical computations

Política social e racismo como desafios para historiadores da educação Social policy and racism as challenges for educational historians

Turing mechanisms can yield a large variety of patterns from noisy, homogenous initial conditions and have been proposed as patterning mechanism for many developmental processes. However, the molecular components that give rise to Turing patterns have remained elusive, and the small size of the parameter space that permits Turing patterns to emerge makes it difficult to explain how Turing patterns could evolve.We have recently shown that Turing patterns can be obtained with a single ligand if the ligand-receptor interaction is taken into account. Here we show that the general properties of ligand-receptor systems result in very large Turing spaces. Thus, the restriction of receptors to single cells, negative feedbacks, regulatory interactions among different ligand-receptor systems, and the clustering of receptors on the cell surface all greatly enlarge the Turing space. We further show that the feedbacks that occur in the FGF10-SHH network that controls lung branching morphogenesis are sufficient to result in large Turing spaces. We conclude that the cellular restriction of receptors provides a mechanism to sufficiently increase the size of the Turing space to make the evolution of Turing patterns likely. Additional feedbacks may then have further enlarged the Turing space. Given their robustness and flexibility, we propose that receptor-ligand-based Turing mechanisms present a general mechanism for patterning in biology