A geometric view of Krylov subspace methods on singular systems

We give a geometric framework for analysing iterative methods on singular linear systems A x = b and apply them to Krylov subspace methods. The idea is to decompose the method into the ?(A) component and its orthogonal complement ?(A)^?, where ?(A) is the range of A. We apply the framework to GMRES, GMRES(k) and GCR(k), and derive conditions for convergence without breakdown for inconsistent and consistent singular systems. The approach also gives a geometric interpretation and different proofs of the conditions obtained by Brown and Walker for GMRES. We also give examples arising in the finite difference discretization of two‐point boundary value problems of an ordinary differential equation. Copyright © 2010 John Wiley & Sons, Ltd.     

Electro‐conductive double‐network hydrogels

Novel electro‐conductive and mechanically‐tough double network polymer hydrogels (E‐DN gels) were synthesized by polymerization of 3, 4‐ethylenedioxythiophene in the presence of a double network hydrogel (DN gel) matrix. The E‐DN gels showed not only excellent mechanical performance, having a fracture stress of 1.4–2.1 MPa, but also electrical conductivity as high as 10^?3 S cm^?1, both under dry and water‐swollen states. The fracture stress and fracture energy of the E‐DN gel was increased by 1.7 and 3.4 times, respectively, as compared with the DN gel. From scanning electron microscope and AFM observations, it was found that electro‐conductive poly(3,4‐ethylenedioxythiophene) (PEDOT) was incorporated into DN gel matrix, apparently due to the formation of a poly‐ion complex with sulfonic acid group of the DN gel network. Thus, PEDOT incorporated into the DN gel matrix greatly improves not only electronic conductivity, but also mechanical properties, reinforcing the double network gel matrix. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

Spontaneous and Widespread Electricity Generation in Natural Deep-Sea Hydrothermal Fields

Deep-sea hydrothermal vents discharge abundant reductive energy into oxidative seawater. Herein, we demonstrated that in situ measurements of redox potentials on the surfaces of active hydrothermal mineral deposits were more negative than the surrounding seawater potential, driving electrical current generation. We also demonstrated that negative potentials in the surface of minerals were widespread in the hydrothermal fields, regardless of the proximity to hydrothermal fluid discharges. Lab experiments verified that the negative potential of the mineral surface was induced by a distant electron transfer from the hydrothermal fluid through the metallic and catalytic properties of minerals. These results indicate that electric current is spontaneously and widely generated in natural mineral deposits in deep-sea hydrothermal fields. Our discovery provides important insights into the microbial communities that are supported by extracellular electron transfer and the prebiotic chemical and metabolic evolution of the ocean hydrothermal systems.