Newton’s iteration is a fundamental tool for numerical solutions of systems of equations. The well-known iteration ?rapidly refines a crude initial approximation X0?to the inverse of a general nonsingular matrix. In ...Newton’s iteration is a fundamental tool for numerical solutions of systems of equations. The well-known iteration ?rapidly refines a crude initial approximation X0?to the inverse of a general nonsingular matrix. In this paper, we will extend and apply this method to n× n?structured matrices M?, in which matrix multiplication has a lower computational cost. These matrices can be represented by their short generators which allow faster computations based on the displacement operators tool. However, the length of the generators is tend to grow and the iterations do not preserve matrix structure. So, the main goal is to control the growth of the length of the short displacement generators so that we can operate with matrices of low rank and carry out the computations much faster. In order to achieve our goal, we will compress the computed approximations to the inverse to yield a superfast algorithm. We will describe two different compression techniques based on the SVD and substitution and we will analyze these approaches. Our main algorithm can be applied to more general classes of structured matrices.展开更多
Hard carbon(HC)is widely regarded as one of the most promising anode materials for commercial sodium-ion batteries due to its excellent electrochemical performance and cost-effectiveness.Although organic polymers offe...Hard carbon(HC)is widely regarded as one of the most promising anode materials for commercial sodium-ion batteries due to its excellent electrochemical performance and cost-effectiveness.Although organic polymers offer compositional homogeneity and structural tunability as HC precursors,their high raw material costs and uncontrollable carbonization processes limit large-scale applications.Here,we introduce a liquid-phase carbonization strategy to recycle waste polyethylene terephthalate(PET)into porous micro/nanostructured HC enriched with intrinsic carbon defects(LHC-3,LHC=liquid-phase-prepared hard carbon).These carbon defects and the morphological structures were modulated by bubbles generated from the decomposition of PET in the presence of N,N’-dimethylformamide and zinc acetate.The synergistic effects between intrinsic carbon defects and micro/nanostructure endow LHC-3 anode with high specific capacity(355 mAh·g^(-1) at 0.1 A·g^(-1)),superfast charging capability(132.6 mAh·g^(-1) input within 13 s of charging),and ultralong cycling stability(100,000 stable cycles at 50 A·g^(-1)).The sodium storage mechanism of LHC-3 anode was investigated by ex-situ Raman spectroscopy,X-ray photoelectron spectroscopy,and ion diffusion kinetics analysis.Theoretical calculations indicate that intrinsic carbon defects with non-zero curvature structure in LHC-3 enhance its ability to accommodate more Na+.These findings are expected to have broader applications in energy storage and waste management.展开更多
文摘Newton’s iteration is a fundamental tool for numerical solutions of systems of equations. The well-known iteration ?rapidly refines a crude initial approximation X0?to the inverse of a general nonsingular matrix. In this paper, we will extend and apply this method to n× n?structured matrices M?, in which matrix multiplication has a lower computational cost. These matrices can be represented by their short generators which allow faster computations based on the displacement operators tool. However, the length of the generators is tend to grow and the iterations do not preserve matrix structure. So, the main goal is to control the growth of the length of the short displacement generators so that we can operate with matrices of low rank and carry out the computations much faster. In order to achieve our goal, we will compress the computed approximations to the inverse to yield a superfast algorithm. We will describe two different compression techniques based on the SVD and substitution and we will analyze these approaches. Our main algorithm can be applied to more general classes of structured matrices.
基金the National Natural Science Foundation of China(Nos.U24A20507,22271203,and 22478152)the State Key Laboratory of Organometallic Chemistry of Shanghai Institute of Organic Chemistry(No.2024KF005)+1 种基金Open Research Fund of State Key Laboratory of Coordination Chemistry,School of Chemistry and Chemical Engineering,Nanjing University,the Collaborative Innovation Center of Suzhou Nano Science and Technologythe Project of Scientific and Technologic Infrastructure of Suzhou(No.SZS201905).
文摘Hard carbon(HC)is widely regarded as one of the most promising anode materials for commercial sodium-ion batteries due to its excellent electrochemical performance and cost-effectiveness.Although organic polymers offer compositional homogeneity and structural tunability as HC precursors,their high raw material costs and uncontrollable carbonization processes limit large-scale applications.Here,we introduce a liquid-phase carbonization strategy to recycle waste polyethylene terephthalate(PET)into porous micro/nanostructured HC enriched with intrinsic carbon defects(LHC-3,LHC=liquid-phase-prepared hard carbon).These carbon defects and the morphological structures were modulated by bubbles generated from the decomposition of PET in the presence of N,N’-dimethylformamide and zinc acetate.The synergistic effects between intrinsic carbon defects and micro/nanostructure endow LHC-3 anode with high specific capacity(355 mAh·g^(-1) at 0.1 A·g^(-1)),superfast charging capability(132.6 mAh·g^(-1) input within 13 s of charging),and ultralong cycling stability(100,000 stable cycles at 50 A·g^(-1)).The sodium storage mechanism of LHC-3 anode was investigated by ex-situ Raman spectroscopy,X-ray photoelectron spectroscopy,and ion diffusion kinetics analysis.Theoretical calculations indicate that intrinsic carbon defects with non-zero curvature structure in LHC-3 enhance its ability to accommodate more Na+.These findings are expected to have broader applications in energy storage and waste management.
