Algebraic characteristic classes for idempotent matrices

This paper contains the algebraic analog for idempotent matrices of the Chern-Weil theory of characteristic classes. This is used to show, algebraically, that the canonical line bundle on the complex projective space is not stably trivial. Also a theorem is proved saying that for any smooth manifold there is a canonical epimorphism from the even dimensional algebraic de Rham cohomology of its algebra of smooth functions onto the standard even dimensional de Rham cohomology of the manifold

Cation‐dependent stabilization of electrogenerated naphthalene diimide dianions in porous polymer thin films and their application to electrical energy storage

Porous polymer networks (PPNs) are attractive materials for capacitive energy storage because they offer high surface areas for increased double-layer capacitance, open structures for rapid ion transport, and redox-active moieties that enable faradaic (pseudocapacitive) energy storage. Here we demonstrate a new attractive feature of PPNs--the ability of their reduced forms (radical anions and dianions) to interact with small radii cations through synergistic interactions arising from densely packed redox-active groups, only when prepared as thin films. When naphthalene diimides (NDIs) are incorporated into PPN films, the carbonyl groups of adjacent, electrochemically generated, NDI radical anions and dianions bind strongly to K(+), Li(+), and Mg(2+), shifting the formal potentials of NDI's second reduction by 120 and 460 mV for K(+) and Li(+)-based electrolytes, respectively. In the case of Mg(2+), NDI's two redox waves coalesce into a single two-electron process with shifts of 240 and 710 mV, for the first and second reductions, respectively, increasing the energy density by over 20 % without changing the polymer backbone. In contrast, the formal reduction potentials of NDI derivatives in solution are identical for each electrolyte, and this effect has not been reported for NDI previously. This study illustrates the profound influence of the solid-state structure of a polymer on its electrochemical response, which does not simply reflect the solution-phase redox behavior of its monomers.Porous polymer networks (PPNs) are attractive materials for capacitive energy storage because they offer high surface areas for increased double‐layer capacitance, open structures for rapid ion transport, and redox‐active moieties that enable faradaic (ps54451322513229FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO2013/25527‐1This research was supported by an NSF GRFP (DGE‐1144153) award to C.R.D. W.R.D. acknowledges support from the Alfred P. Sloan and Camille and Henry Dreyfus Foundations. This work was supported in part (K.H.B., H.D.A.) through grant DE‐FG02‐87ER45298, by