We present a fairly general method for nding deterministic constructions obeying what we call krestrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)-universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2 k con gurations appear) and families of perfect hash functions. The near-optimal constructions of these objects imply the very e cient derandomization of algorithms in learning, of xed-subgraph nding algorithms, and of near optimal threshold formulae. In addition, they derandomize the reduction showing the hardness of approximation of set cover. They also yield deterministic constructions for a local-coloring protocol, and for exhaustive testing of circuits.

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The rapid development of flexible and wearable electronics proposes the persistent requirements of high-performance flexible batteries. Much progress has been achieved recently, but how to obtain remarkable flexibility and high energy density simultaneously remains a great challenge. Here, a facile and scalable approach to fabricate spine-like flexible lithium-ion batteries is reported. A thick, rigid segment to store energy through winding the electrodes corresponds to the vertebra of animals, while a thin, unwound, and flexible part acts as marrow to interconnect all vertebra-like stacks together, providing excellent flexibility for the whole battery. As the volume of the rigid electrode part is significantly larger than the flexible interconnection, the energy density of such a flexible battery can be over 85% of that in conventional packing. A nonoptimized flexible cell with an energy density of 242 Wh L is demonstrated with packaging considered, which is 86.1% of a standard prismatic cell using the same components. The cell also successfully survives a harsh dynamic mechanical load test due to this rational bioinspired design. Mechanical simulation results uncover the underlying mechanism: the maximum strain in the reported design (≈0.08%) is markedly smaller than traditional stacked cells (≈1.1%). This new approach offers great promise for applications in flexible devices.

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