Potassium‐ion batteries (KIBs) are very promising alternatives to lithium‐ion batteries (LIBs) for large‐scale energy storage. However, traditional carbon anode materials usually show poor performance in KIBs due to the large size of K ions. Herein, a carbonization‐etching strategy is reported for making a class of sulfur (S) and oxygen (O) codoped porous hard carbon microspheres (PCMs) material as a novel anode for KIBs through pyrolysis of the polymer microspheres (PMs) composed of a liquid crystal/epoxy monomer/thiol hardener system. The as‐made PCMs possess a porous architecture with a large Brunauer–Emmett–Teller surface area (983.2 m2 g−1), an enlarged interlayer distance (0.393 nm), structural defects induced by the S/O codoping and also amorphous carbon nature. These new features are important for boosting potassium ion storage, allowing the PCMs to deliver a high potassiation capacity of 226.6 mA h g−1 at 50 mA g−1 over 100 cycles and be displaying high stability by showing a potassiation capacity of 108.4 mA h g−1 over 2000 cycles at 1000 mA g−1. The density functional theory calculations demonstrate that S/O codoping not only favors the adsorption of K to the PCMs electrode but also reduces its structural deformation during the potassiation/depotassiation. The present work highlights the important role of hierarchical porosity and S/O codoping in potassium storage.

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Piezoelectric and triboelectric nanogenerators (NGs) have been proposed in the past few years to effectively harvest mechanical energy from the environment. Here, a polydimethylsiloxane (PDMS) layer is placed under the aluminum electrode of polyvinylidene fluoride (PVDF), thus forming an r-shaped hybrid NG. Micro/nanostructures have been fabricated on the PDMS surface and the aluminum electrodes of PVDF to enhance the output performance. Power densities of the piezoelectric part and the triboelectric part are 10.95 and 2.04 mW/cm(3), respectively. Moreover, influence of the triboelectric charges on the piezoelectric output voltage is investigated. Both finite element method simulations and experimental measurements are conducted to verify this phenomenon. The novel hybrid NG is also demonstrated as a power source for consumer electronics. Through one cycle of electric generation, 10 light-emitting diodes are lighted up instantaneously, and a 4-bit liquid crystal display can display continuously for more than 15 s. Besides, the device is integrated into a keyboard to harvest energy in the typing process.

In this paper, we first generalize the common index jump theorem for symplectic matrix paths proved in 2002 by Long and Zhu in [LoZ], and get an enhanced version of it. As its applications, we further prove that for a compact simply-connected manifold (M,

By constructing normal coordinates on a quaternionic contact manifold M, we can osculate the quaternionic contact structure at each point by the standard quaternionic contact structure on the quaternionic Heisenberg group. By using this property, we can do harmonic analysis on general quaternionic contact manifolds, and solve the quaternionic contact Yamabe problem on M if its Yamabe invariant satisfies λ(M) < λ(H n ).