We calculate the Hormander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system ...We calculate the Hormander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.展开更多
Polymeric carbon nitride(CN)as a metal-free photocatalyst holds great promise to produce high-value chemicals and H_(2) fuel utilizing clean solar energy.However,the wider deployment of pristine CN is critically hampe...Polymeric carbon nitride(CN)as a metal-free photocatalyst holds great promise to produce high-value chemicals and H_(2) fuel utilizing clean solar energy.However,the wider deployment of pristine CN is critically hampered by the poor charge carrier transport and high recombination.Herein,we develop a facile salt template-assisted interfacial polymerization strategy that insitu introduces alkali ions(Na+,K+)and nitrogen defects in CN(denoted as v-CN-KNa)to simultaneously promote charge separation and transportation and steer photoexcited holes and electrons to their oxidation and reduction sites.The photocatalyst exhibits an impressive photocatalytic H_(2) evolution rate of 8641.5μmol·g^(−1)·h^(−1)(33-fold higher than pristine CN)and also works readily in real seawater(10752.0μmol·g^(−1)·h^(−1))with a high apparent quantum efficiency up to 18.5%at 420 nm.In addition,we further demonstrate that the v-CN-KNa can simultaneously produce H_(2) and N-benzylidenebenzylamine without using any other sacrificial reagent.In situ characterizations and DFT calculations reveal that the alkali ions notably promote charge transport,while the nitrogen defects generate abundant edge active sites,which further contribute to efficient electron excitation to trigger photoredox reactions.展开更多
The authors prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space.Then a direct proof of the iteration formulae for the Masl...The authors prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space.Then a direct proof of the iteration formulae for the Maslov-type indices of symplectic paths is given.展开更多
We study the pseudoholomorphic curves with brake symmetry in symplectization of a closed contact manifold.We introduce the pseudoholomorphic curves with brake symmetry and the corresponding moduli space.Then we get th...We study the pseudoholomorphic curves with brake symmetry in symplectization of a closed contact manifold.We introduce the pseudoholomorphic curves with brake symmetry and the corresponding moduli space.Then we get the virtual dimension of the moduli space.展开更多
基金Acknowledgements The authors would like to thank the referees for their critical reading and very helpful comments and suggestions. This work was partially supported by the National Natural Science Foundation of China （Grant Nos. 11221091, 11471169） and LPMC of MOE of China.
文摘We calculate the Hormander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.
基金supported by the National Key Research and Development Program of the MOST(Nos.2021YFA1500400 and 2018YFA0208603)the National Natural Science Foundation of China(NSFC,Nos.21571167,51502282,22075266,and 21890751)the Fundamental Research Funds for the Central Universities(Nos.WK2060190053 and WK2060190100).
文摘Polymeric carbon nitride(CN)as a metal-free photocatalyst holds great promise to produce high-value chemicals and H_(2) fuel utilizing clean solar energy.However,the wider deployment of pristine CN is critically hampered by the poor charge carrier transport and high recombination.Herein,we develop a facile salt template-assisted interfacial polymerization strategy that insitu introduces alkali ions(Na+,K+)and nitrogen defects in CN(denoted as v-CN-KNa)to simultaneously promote charge separation and transportation and steer photoexcited holes and electrons to their oxidation and reduction sites.The photocatalyst exhibits an impressive photocatalytic H_(2) evolution rate of 8641.5μmol·g^(−1)·h^(−1)(33-fold higher than pristine CN)and also works readily in real seawater(10752.0μmol·g^(−1)·h^(−1))with a high apparent quantum efficiency up to 18.5%at 420 nm.In addition,we further demonstrate that the v-CN-KNa can simultaneously produce H_(2) and N-benzylidenebenzylamine without using any other sacrificial reagent.In situ characterizations and DFT calculations reveal that the alkali ions notably promote charge transport,while the nitrogen defects generate abundant edge active sites,which further contribute to efficient electron excitation to trigger photoredox reactions.
基金supported by the National Natural Science Foundation of China(Nos.11221091,11471169)the Key Laboratory of Pure Mathematics and Combinatorics,the Ministry of Education of China
文摘The authors prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space.Then a direct proof of the iteration formulae for the Maslov-type indices of symplectic paths is given.
基金The first author was supported by the China Scholarship Council(CSC)(No.201806200130),the LPMC of Ministry of Education of China,and Nankai Zhide Foundation and Nankai UniversityThe second author was supported by the National Natural Science Foundation of China(Grant Nos.11971245,11771331),the LPMC of Ministry of Education of China,the Nankai Zhide Foundation and Nankai University.
文摘We study the pseudoholomorphic curves with brake symmetry in symplectization of a closed contact manifold.We introduce the pseudoholomorphic curves with brake symmetry and the corresponding moduli space.Then we get the virtual dimension of the moduli space.
