Rechargeable aqueous zinc-ion batteries are promising energy storage devices due to their high safety and low cost. However, they remain in their infancy because of the limited choice of positive electrodes with high capacity and satisfactory cycling performance. Furthermore, their energy storage mechanisms are not well established yet. Here we report a highly reversible zinc/sodium vanadate system, where sodium vanadate hydrate nanobelts serve as positive electrode and zinc sulfate aqueous solution with sodium sulfate additive is used as electrolyte. Different from conventional energy release/storage in zinc-ion batteries with only zinc-ion insertion/extraction, zinc/sodium vanadate hydrate batteries possess a simultaneous proton, and zinc-ion insertion/extraction process that is mainly responsible for their excellent performance, such as a high reversible capacity of 380 mAh g–1 and capacity retention of 82% over 1000 cycles. Moreover, the quasi-solid-state zinc/sodium vanadate hydrate battery is also a good candidate for flexible energy storage device.

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The η-invariant of an odd dimensional manifold with boundary is investigated. The natural boundary condition for this problem requires a trivialization of the kernel of the Dirac operator on the boundary. The dependence of the η-invariant on this trivialization is best encoded by the statement that the exponential of the η-invariant lives in the determinant line of the boundary. Our main results are a variational formula and a gluing law for this invariant. These results are applied to reprove the formula for the holonomy of the natural connection on the determinant line bundle of a family of Dirac operators, also known as the ‘‘global anomaly formula.’’ The ideas developed here fit naturally with recent work in topological quantum field theory, in which gluing (which is a characteristic formal property of the path integral and the classical action) is used to compute global invariants on closed manifolds from local invariants on manifolds with boundary.

We study the asymptotic of the Bergman kernel of the spin c Dirac operator on high tensor powers of a line bundle.

Civita connection of gM. Our first main result is Theorem 0.1 (Adiabatic limit formula). Assume that ker Dy is a vector bundle on B. Assume further that dim(ker D x) becomes constant when 0 < x :$ Xo for some small xO. Then limx-+o 17(DJ exists in lR and (0.1) lim 17(Dx) =2 [A(R 2 B)/\ij+17(D B ®kerDy)+lim I: sgnA x ' x-+O