A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations.Let S_(n)(σ_(1),σ_(2),...,σ_(r))denote the set of permutations in the symmetric group S_(n)...A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations.Let S_(n)(σ_(1),σ_(2),...,σ_(r))denote the set of permutations in the symmetric group S_(n)which avoid patternsσ_(1),σ_(2),...,σ_(r).The aim of this paper is to investigate when,for certain patternsσ_(1),σ_(2),...,σ_(r),S_(n)(σ_(1),σ_(2),...,σ_(r))is sign-balanced for every integer n>1.We prove that for any{σ_(1),σ_(2),...,σ_(r)}?S_3,if{σ_(1),σ_(2),...,σ_(r)}is sign-balanced except for{132,213,231,312},then S_(n)(σ_(1),σ_(2),...,σ_(r))is sign-balanced for every integer n>1.In addition,we give some results in the case of avoiding some patterns of length 4.展开更多
Electrode materials with strong desalting ability is an important research direction of capacitive deionization.In this study,HKUST-1 was successfully synthesized by the solvothermal method,and MOFsderived porous carb...Electrode materials with strong desalting ability is an important research direction of capacitive deionization.In this study,HKUST-1 was successfully synthesized by the solvothermal method,and MOFsderived porous carbon/Cu@Cu_(2)O composites were prepared by simple pyrolysis as cathode materials for CDI.After high-temperature pyrolysis,the Cu^(+) site with unsaturated coordination is generated,and the structure changes from micropores to the coexistence of mesoporous and micropores.The complex pore structure is conducive to strengthening ion migration and diffusion.The results show that the porous carbon/Cu@Cu_(2)O materials derived from MOFs depend on the pseudocapacitance behavior for capacitive deionization and desalination.At a voltage window of-1.2V~1.2V,a current density of 40mA/g.and 5 mmol/L NaCl,the HDC-1100 exhibited the best desalting capacity of 30.9 mg/g.HDC-1100 also has good cycle stability.After 20 cycles of adsorption and desorption,the desalting capacity almost does not decrease.Therefore,MOFs derived porous carbon/Cu@Cu_(2)O composites are expected to be an excellent choice for CDI cathode materials.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.12061030)the Natural Science Foundation of Hainan Province(Grant No.122RC652)2023 Excellent Science and Technology Innovation Team of Jiangsu Province Universities(Real-Time Industrial Internet of Things).
文摘A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations.Let S_(n)(σ_(1),σ_(2),...,σ_(r))denote the set of permutations in the symmetric group S_(n)which avoid patternsσ_(1),σ_(2),...,σ_(r).The aim of this paper is to investigate when,for certain patternsσ_(1),σ_(2),...,σ_(r),S_(n)(σ_(1),σ_(2),...,σ_(r))is sign-balanced for every integer n>1.We prove that for any{σ_(1),σ_(2),...,σ_(r)}?S_3,if{σ_(1),σ_(2),...,σ_(r)}is sign-balanced except for{132,213,231,312},then S_(n)(σ_(1),σ_(2),...,σ_(r))is sign-balanced for every integer n>1.In addition,we give some results in the case of avoiding some patterns of length 4.
基金supported by The National Natural Science Foundation of China(Nos.22276137,52170087).
文摘Electrode materials with strong desalting ability is an important research direction of capacitive deionization.In this study,HKUST-1 was successfully synthesized by the solvothermal method,and MOFsderived porous carbon/Cu@Cu_(2)O composites were prepared by simple pyrolysis as cathode materials for CDI.After high-temperature pyrolysis,the Cu^(+) site with unsaturated coordination is generated,and the structure changes from micropores to the coexistence of mesoporous and micropores.The complex pore structure is conducive to strengthening ion migration and diffusion.The results show that the porous carbon/Cu@Cu_(2)O materials derived from MOFs depend on the pseudocapacitance behavior for capacitive deionization and desalination.At a voltage window of-1.2V~1.2V,a current density of 40mA/g.and 5 mmol/L NaCl,the HDC-1100 exhibited the best desalting capacity of 30.9 mg/g.HDC-1100 also has good cycle stability.After 20 cycles of adsorption and desorption,the desalting capacity almost does not decrease.Therefore,MOFs derived porous carbon/Cu@Cu_(2)O composites are expected to be an excellent choice for CDI cathode materials.
