On reduction complexity of Heegaard splittings

AbstractLet ∪F W be a Heegaard splitting of a closed, connected, orientable 3-manifold M with genus n. We introduce a reduction complexity, δ, for the Heegaard splitting, and conclude that: 1.(1) δ ⩾ max{2, n} if and only if V ∪F W is reducible;2.(2) δ ⩾ 2 if and only if V ∪F W is weakly reducible,3.(3) δ > n if and only if M has exactly δ − n connected sum factors which are homeomorphic to S1 × S2

Taylor Genetic Programming for Symbolic Regression

Genetic programming (GP) is a commonly used approach to solve symbolic regression (SR) problems. Compared with the machine learning or deep learning methods that depend on the pre-defined model and the training dataset for solving SR problems, GP is more focused on finding the solution in a search space. Although GP has good performance on large-scale benchmarks, it randomly transforms individuals to search results without taking advantage of the characteristics of the dataset. So, the search process of GP is usually slow, and the final results could be unstable. To guide GP by these characteristics, we propose a new method for SR, called Taylor genetic programming (TaylorGP). TaylorGP leverages a Taylor polynomial to approximate the symbolic equation that fits the dataset. It also utilizes the Taylor polynomial to extract the features of the symbolic equation: low order polynomial discrimination, variable separability, boundary, monotonic, and parity. GP is enhanced by these Taylor polynomial techniques. Experiments are conducted on three kinds of benchmarks: classical SR, machine learning, and physics. The experimental results show that TaylorGP not only has higher accuracy than the nine baseline methods, but also is faster in finding stable results

Polypyrrole-assisted nitrogen doping strategy to boost vanadium dioxide performance for wearable nonpolarity supercapacitor and aqueous zinc-ion battery

Fiber-shaped energy-storage devices with high energy and power density are crucial for powering wearable electronics. However, the improvement of their energy and power density is limited by the low mass loading of active materials and slow diffusion of ions, which further hinders the application as flexible energy-storage devices. Herein, a facile and cost-effective strategy is proposed to fabricate polypyrrole (PPy)-assisted nitrogen-doped vanadium dioxide/nitrogen-doped carbon (N–VO2@NC) heterostructures by the pyrolyzation of vanadium oxide (VOx)/PPy supported on carbon nanotube fiber (CNTF). The carbonization of PPy nanowire not only forms nitrogen-doped carbon 3D conductive scaffold for enhancing ion transport pathways and mass loading of N–VO2 but also provides source of nitrogen in situ doping into VOx to produce N–VO2 for improving electronic conductivity and energy-storage capacity. Consequently, the well-designed N–VO2@NC@CNTF electrode delivers impressive electrochemical performance and extraordinary mechanical flexibility both applied in all-solid-state fiber-shaped nonpolarity supercapacitors and aqueous zinc-ion batteries. Furthermore, the results of theoretical calculations discovered that the band gap of PPy-assisted N–VO2 can be significantly reduced from 0.55 to 0.23 eV and thus its conductivity is greatly enhanced. This work sheds light on the construction of high-performance free-standing electrodes for next-generation wearable aqueous energy-storage devices.J.G., L.L., and J.L. contributed equally to this work. This work was supported by the Suzhou Institute of Nano-Tech and Nano-Bionics, the Chinese Academy of Sciences (Start-up grant E1552102), and the Natural Science Foundation of China (22075043 and 21875034)