In this article, we discuss several properties of the basic contact process on hexagonal lattice H, showing that it behaves quite similar to the process on d-dimensional lattice Zd in many aspects. Firstly, we constru...In this article, we discuss several properties of the basic contact process on hexagonal lattice H, showing that it behaves quite similar to the process on d-dimensional lattice Zd in many aspects. Firstly, we construct a coupling between the contact process on hexagonal lattice and the oriented percolation, and prove an equivalent finite space-time condition for the survival of the process. Secondly, we show the complete convergence theorem and the polynomial growth hold for the contact process on hexagonal lattice. Finally, we prove exponential bounds in the supercritical case and exponential decay rates in the subcritical case of the process.展开更多
For a graph G and two positive integers j and k, an m-L（j, k）-edge-labeling of G is an assignment on the edges to the set {0, 1, 2,..., m}, such that adjacent edges which receive labels differ at least by j, and edg...For a graph G and two positive integers j and k, an m-L（j, k）-edge-labeling of G is an assignment on the edges to the set {0, 1, 2,..., m}, such that adjacent edges which receive labels differ at least by j, and edges which are distance two apart receive labels differ at least by k. The λj,k-number of G is the minimum m such that an m-L（j, k）-edge-labeling is admitted by G. In this article, the L（1, 2）-edge-labeling for the hexagonal lattice, the square lattice and the triangular lattice are studied, and the bounds for λj,k-numbers of these graphs are obtained.展开更多
Promoting the intrinsic catalytic activity of transition metal dichalcogenides(TMDs)for the hydrogen evolution reaction(HER)is very beneficial for developing Pt-free single-component electrocatalysts.Here we report a ...Promoting the intrinsic catalytic activity of transition metal dichalcogenides(TMDs)for the hydrogen evolution reaction(HER)is very beneficial for developing Pt-free single-component electrocatalysts.Here we report a single-component TMD-based electrocatalyst,Cu_(0.4)NbS_(2−δ),with chemical intercalation and sulfur vacancies.Cu_(0.4)NbS_(2−δ)was synthesized conveniently using a solid-phase reaction and it exhibits a hexagonal lattice with intercalated Cu atoms in a four-coordinated tetrahedral configuration.Comprehensive experiments and theoretical calculations indicate that the intercalation and S vacancies redistributed the electronic structure of pristine NbS_(2),leading to multi-functional catalytic sites including defect hollow sites in the basal plane and different Cu sites in between the layers.Therefore,hydrogen spillover can proceed along pathways at the atomic level in Cu_(0.4)NbS_(2−δ)from proton adsorption at defect hollow sites,through interlayer sites for readily occurring hydrogen migration,and finally to Cu sites for fast H_(2)desorption.Based on this atomic-scale hydrogen spillover effect,the polycrystalline Cu_(0.4)NbS_(2−δ)exhibits a low overpotential of 153 mV at 10 mA cm^(-2)and great durability after 3000 cycles.Our work provides an effective strategy to build hydrogen spillover pathways in TMDs,which is instructive for exploring economical single-component electrocatalysts with high intrinsic activity.展开更多
基金Supported in part by the NNSF of China (10531070,10625101)the National Basic Research Program of China (2006CB805900)
文摘In this article, we discuss several properties of the basic contact process on hexagonal lattice H, showing that it behaves quite similar to the process on d-dimensional lattice Zd in many aspects. Firstly, we construct a coupling between the contact process on hexagonal lattice and the oriented percolation, and prove an equivalent finite space-time condition for the survival of the process. Secondly, we show the complete convergence theorem and the polynomial growth hold for the contact process on hexagonal lattice. Finally, we prove exponential bounds in the supercritical case and exponential decay rates in the subcritical case of the process.
基金Supported by the National Natural Science Foundation of China(10971025 and 10901035)
文摘For a graph G and two positive integers j and k, an m-L（j, k）-edge-labeling of G is an assignment on the edges to the set {0, 1, 2,..., m}, such that adjacent edges which receive labels differ at least by j, and edges which are distance two apart receive labels differ at least by k. The λj,k-number of G is the minimum m such that an m-L（j, k）-edge-labeling is admitted by G. In this article, the L（1, 2）-edge-labeling for the hexagonal lattice, the square lattice and the triangular lattice are studied, and the bounds for λj,k-numbers of these graphs are obtained.
基金National Natural Science Foundation of China(no.52271199 and 51832010)National Key R&D Program of China(no.2020YFA0406202)+1 种基金Fundamental Research Funds for the Central Universities(FRF-EYIT-23-04)Open Fund of State Key Laboratory of Advanced Metallurgy(KF21-01)。
文摘Promoting the intrinsic catalytic activity of transition metal dichalcogenides(TMDs)for the hydrogen evolution reaction(HER)is very beneficial for developing Pt-free single-component electrocatalysts.Here we report a single-component TMD-based electrocatalyst,Cu_(0.4)NbS_(2−δ),with chemical intercalation and sulfur vacancies.Cu_(0.4)NbS_(2−δ)was synthesized conveniently using a solid-phase reaction and it exhibits a hexagonal lattice with intercalated Cu atoms in a four-coordinated tetrahedral configuration.Comprehensive experiments and theoretical calculations indicate that the intercalation and S vacancies redistributed the electronic structure of pristine NbS_(2),leading to multi-functional catalytic sites including defect hollow sites in the basal plane and different Cu sites in between the layers.Therefore,hydrogen spillover can proceed along pathways at the atomic level in Cu_(0.4)NbS_(2−δ)from proton adsorption at defect hollow sites,through interlayer sites for readily occurring hydrogen migration,and finally to Cu sites for fast H_(2)desorption.Based on this atomic-scale hydrogen spillover effect,the polycrystalline Cu_(0.4)NbS_(2−δ)exhibits a low overpotential of 153 mV at 10 mA cm^(-2)and great durability after 3000 cycles.Our work provides an effective strategy to build hydrogen spillover pathways in TMDs,which is instructive for exploring economical single-component electrocatalysts with high intrinsic activity.
