This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An...This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.展开更多
A simple calcination method was employed to prepare a Z-scheme N-doped K4Nb6O17/g-C3N4(KCN)heterojunction photocatalyst,in which the electronic structure of K4Nb6O17 was regulated by N-doping,and g-C3N4 was formed bot...A simple calcination method was employed to prepare a Z-scheme N-doped K4Nb6O17/g-C3N4(KCN)heterojunction photocatalyst,in which the electronic structure of K4Nb6O17 was regulated by N-doping,and g-C3N4 was formed both on the surface and within the interlayer spaces of K4Nb6O17.The KCN composite showed profoundly improved photocatalytic activity for both H2 generation and RhB degradation compared to its counterparts.This improved performance was attributed to the synergistic effects of N-doping,which broadened its light harvesting ability,and heterojunction formation,which increased the charge separation rate.The relatively low BET specific surface area of the KCN composite had little effect on its photocatalytic activity.Based on ESR spectroscopy studies,•O2^−,•OH,and h^+are the main active species in the photocatalytic degradation of RhB.Thus,it is reasonable to propose a Z-scheme photocatalytic mechanism over the KCN composite,which exhibits the dual advantages of efficient charge separation and high redox ability.Our work provides a simple approach for constructing large-scale Z-scheme heterojunction photocatalysts with high photocatalytic performance.展开更多
{[K.18-Crown-6]Br3}n,a unique tribromide-type catalyst,was utilized for the N-boc protection of amines and trimethylsilylation(TMS)and tetrahydropyranylation(THP)of alcohols.The method is general for the preparation o...{[K.18-Crown-6]Br3}n,a unique tribromide-type catalyst,was utilized for the N-boc protection of amines and trimethylsilylation(TMS)and tetrahydropyranylation(THP)of alcohols.The method is general for the preparation of N-boc derivatives of aliphatic(acyclic and cyclic)and aromatic,and primary and secondary amines and also various TMS-ethers and THP-ethers.The simple separation of the catalyst from the product is one of the many advantages of this method.展开更多
由公式C<sub>n</sub><sup>k</sup>+C<sub>n</sub><sup>k+1</sup>=C<sub>n+1</sub><sup>k+1</sup>,可得:C<sub>2</sub><sup>2</sup>+...由公式C<sub>n</sub><sup>k</sup>+C<sub>n</sub><sup>k+1</sup>=C<sub>n+1</sub><sup>k+1</sup>,可得:C<sub>2</sub><sup>2</sup>+C<sub>3</sub><sup>2</sup>+…+C<sub>n</sub><sup>2</sup>=C<sub>n+1</sub><sup>3</sup>,sum from k=2 to nC<sub>k</sub><sup>2</sup>=C<sub>n+1</sub><sup>3</sup>,展开更多
文摘This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.
文摘A simple calcination method was employed to prepare a Z-scheme N-doped K4Nb6O17/g-C3N4(KCN)heterojunction photocatalyst,in which the electronic structure of K4Nb6O17 was regulated by N-doping,and g-C3N4 was formed both on the surface and within the interlayer spaces of K4Nb6O17.The KCN composite showed profoundly improved photocatalytic activity for both H2 generation and RhB degradation compared to its counterparts.This improved performance was attributed to the synergistic effects of N-doping,which broadened its light harvesting ability,and heterojunction formation,which increased the charge separation rate.The relatively low BET specific surface area of the KCN composite had little effect on its photocatalytic activity.Based on ESR spectroscopy studies,•O2^−,•OH,and h^+are the main active species in the photocatalytic degradation of RhB.Thus,it is reasonable to propose a Z-scheme photocatalytic mechanism over the KCN composite,which exhibits the dual advantages of efficient charge separation and high redox ability.Our work provides a simple approach for constructing large-scale Z-scheme heterojunction photocatalysts with high photocatalytic performance.
基金support for this work from the research affairs of Hamedan University of Medical Sciences,Hamedan,I.R.Iranpartial support of this work by the Research Affairs Office of Bu-Ali Sina UniversityCenter of Excellence in Development of Chemical Method(CEDCM)Hamedan,I.R.Iran
文摘{[K.18-Crown-6]Br3}n,a unique tribromide-type catalyst,was utilized for the N-boc protection of amines and trimethylsilylation(TMS)and tetrahydropyranylation(THP)of alcohols.The method is general for the preparation of N-boc derivatives of aliphatic(acyclic and cyclic)and aromatic,and primary and secondary amines and also various TMS-ethers and THP-ethers.The simple separation of the catalyst from the product is one of the many advantages of this method.
文摘由公式C<sub>n</sub><sup>k</sup>+C<sub>n</sub><sup>k+1</sup>=C<sub>n+1</sub><sup>k+1</sup>,可得:C<sub>2</sub><sup>2</sup>+C<sub>3</sub><sup>2</sup>+…+C<sub>n</sub><sup>2</sup>=C<sub>n+1</sub><sup>3</sup>,sum from k=2 to nC<sub>k</sub><sup>2</sup>=C<sub>n+1</sub><sup>3</sup>,
