This paper is mainly concerned with corank-2 and corank-3 symmetrybreaking bifurcation point in Z2×Z2-symmetric nonlinear problems. Regular extended systems are used to compute corank-2 and corank-3 symmetry--bre...This paper is mainly concerned with corank-2 and corank-3 symmetrybreaking bifurcation point in Z2×Z2-symmetric nonlinear problems. Regular extended systems are used to compute corank-2 and corank-3 symmetry--breaking bifurcation points. Two numerical examples are given. In addition, we show that there exist three quadratic pitchfork bifurcation point curves passing through corank-2 symmetry breaking bifurcation point.展开更多
Let J_(*,k)~r 2. denote the ideal in MO_* of cobordism classes containing arepresentative that admits (Z_2)~k-actions with a fixed point set of constant codimension r. Inthis paper we determine J_(*,k)^(2^k+2) and J_(...Let J_(*,k)~r 2. denote the ideal in MO_* of cobordism classes containing arepresentative that admits (Z_2)~k-actions with a fixed point set of constant codimension r. Inthis paper we determine J_(*,k)^(2^k+2) and J_(*,3)^(2^3+1).展开更多
This paper is concerned with the computation of Hopf branches emanating from a Hopf/Pitchfork point in a two-parametor nonlinear problem satisfying a Z2symnletry condition. Our aim is to present a new al,proach to ...This paper is concerned with the computation of Hopf branches emanating from a Hopf/Pitchfork point in a two-parametor nonlinear problem satisfying a Z2symnletry condition. Our aim is to present a new al,proach to the theoretical and computational analysis of the bifurcating Hopf branches at this singular point by using the system designed to calculate Hopf points and exploring its symmetry. It is shown that a Hopf/Pitchfork point is a pitchfork bifurcation point in the system.Hence standard continuation and branch-switching can be used to compute these Hopf branches. In addition, an effect method based on the extended system of the singular points is developed for the computation of branch of secondary (nonsymmetric) Hopf points. The implementation of Newton's method for solving the extended system is also discussed. A numerical example is given.Keyworks: Hopf/Pitchfork point, Z2-symmetry, Hopf point, bifurcation, Extended system展开更多
Machine learning(ML)integrated with density functional theory(DFT)calculations have recently been used to accelerate the design and discovery of single-atom catalysts(SACs)by establishing deep structure–activity rela...Machine learning(ML)integrated with density functional theory(DFT)calculations have recently been used to accelerate the design and discovery of single-atom catalysts(SACs)by establishing deep structure–activity relationships.The traditional ML models are always difficult to identify the structural differences among the single-atom systems with different modification methods,leading to the limitation of the potential application range.Aiming to the structural properties of several typical two-dimensional MA_(2)Z_(4)-based single-atom systems(bare MA_(2)Z_(4) and metal single-atom doped/supported MA_(2)Z_(4)),an improved crystal graph convolutional neural network(CGCNN)classification model was employed,instead of the traditional machine learning regression model,to address the challenge of incompatibility in the studied systems.The CGCNN model was optimized using crystal graph representation in which the geometric configuration was divided into active layer,surface layer,and bulk layer(ASB-GCNN).Through ML and DFT calculations,five potential single-atom hydrogen evolution reaction(HER)catalysts were screened from chemical space of 600 MA_(2)Z_(4)-based materials,especially V_(1)/HfSn_(2)N_(4)(S)with high stability and activity(Δ_(GH*)is 0.06 eV).Further projected density of states(pDOS)analysis in combination with the wave function analysis of the SAC-H bond revealed that the SAC-dz^(2)orbital coincided with the H-s orbital around the energy level of−2.50 eV,and orbital analysis confirmed the formation ofσbonds.This study provides an efficient multistep screening design framework of metal single-atom catalyst for HER systems with similar two-dimensional supports but different geometric configurations.展开更多
This paper deals with the classification of the simple higher-order symmetry-breaking bifurcation in multiparameter nonlinear problems with Z2-symmetry. The regular extended systems for computing the simple higher-ord...This paper deals with the classification of the simple higher-order symmetry-breaking bifurcation in multiparameter nonlinear problems with Z2-symmetry. The regular extended systems for computing the simple higher-order symmetry-breaking bifurcation points with different singularities are proposed. An etficient algorithm for solving the extended systems is given. Finally, some numerical examples are shown to demonstrate the efficiency of the algorithm.展开更多
We study entanglement in dimerized Heisenberg systems. In particular, we give exact results of groundstate pairwise entanglement for the four-qubit model by identifying a Z2 symmetry. Although the entanglements cannot...We study entanglement in dimerized Heisenberg systems. In particular, we give exact results of groundstate pairwise entanglement for the four-qubit model by identifying a Z2 symmetry. Although the entanglements cannot identify the critical point of the system, the mean entanglement of the nearest-neighbor qubits really does, namely, it reaches a maximum at the critical point.展开更多
We consider double high order S-breaking bifurcation points of two-Parameter dependent nonlinear problems with Z_2×Z_2-symmetry. Because of the underlying symmetry we could propose some regular extended systems...We consider double high order S-breaking bifurcation points of two-Parameter dependent nonlinear problems with Z_2×Z_2-symmetry. Because of the underlying symmetry we could propose some regular extended systems to determine double high order S-breaking bifurcation points. and we could also show that there exist two quadratic pitchfork bifurcation point paths passing through the point being considered.展开更多
文摘This paper is mainly concerned with corank-2 and corank-3 symmetrybreaking bifurcation point in Z2×Z2-symmetric nonlinear problems. Regular extended systems are used to compute corank-2 and corank-3 symmetry--breaking bifurcation points. Two numerical examples are given. In addition, we show that there exist three quadratic pitchfork bifurcation point curves passing through corank-2 symmetry breaking bifurcation point.
基金Supported by the National Natural Sciences Foundation of P.R.China(No.10371029)the Natural Sciences Foundation of Hebei Province(No.103144)the Doctoral Foundation of Hebei Normal University(No.103257)
文摘Let J_(*,k)~r 2. denote the ideal in MO_* of cobordism classes containing arepresentative that admits (Z_2)~k-actions with a fixed point set of constant codimension r. Inthis paper we determine J_(*,k)^(2^k+2) and J_(*,3)^(2^3+1).
文摘This paper is concerned with the computation of Hopf branches emanating from a Hopf/Pitchfork point in a two-parametor nonlinear problem satisfying a Z2symnletry condition. Our aim is to present a new al,proach to the theoretical and computational analysis of the bifurcating Hopf branches at this singular point by using the system designed to calculate Hopf points and exploring its symmetry. It is shown that a Hopf/Pitchfork point is a pitchfork bifurcation point in the system.Hence standard continuation and branch-switching can be used to compute these Hopf branches. In addition, an effect method based on the extended system of the singular points is developed for the computation of branch of secondary (nonsymmetric) Hopf points. The implementation of Newton's method for solving the extended system is also discussed. A numerical example is given.Keyworks: Hopf/Pitchfork point, Z2-symmetry, Hopf point, bifurcation, Extended system
基金supported by the National Key R&D Program of China(2021YFA1500900)National Natural Science Foundation of China(U21A20298,22141001).
文摘Machine learning(ML)integrated with density functional theory(DFT)calculations have recently been used to accelerate the design and discovery of single-atom catalysts(SACs)by establishing deep structure–activity relationships.The traditional ML models are always difficult to identify the structural differences among the single-atom systems with different modification methods,leading to the limitation of the potential application range.Aiming to the structural properties of several typical two-dimensional MA_(2)Z_(4)-based single-atom systems(bare MA_(2)Z_(4) and metal single-atom doped/supported MA_(2)Z_(4)),an improved crystal graph convolutional neural network(CGCNN)classification model was employed,instead of the traditional machine learning regression model,to address the challenge of incompatibility in the studied systems.The CGCNN model was optimized using crystal graph representation in which the geometric configuration was divided into active layer,surface layer,and bulk layer(ASB-GCNN).Through ML and DFT calculations,five potential single-atom hydrogen evolution reaction(HER)catalysts were screened from chemical space of 600 MA_(2)Z_(4)-based materials,especially V_(1)/HfSn_(2)N_(4)(S)with high stability and activity(Δ_(GH*)is 0.06 eV).Further projected density of states(pDOS)analysis in combination with the wave function analysis of the SAC-H bond revealed that the SAC-dz^(2)orbital coincided with the H-s orbital around the energy level of−2.50 eV,and orbital analysis confirmed the formation ofσbonds.This study provides an efficient multistep screening design framework of metal single-atom catalyst for HER systems with similar two-dimensional supports but different geometric configurations.
文摘This paper deals with the classification of the simple higher-order symmetry-breaking bifurcation in multiparameter nonlinear problems with Z2-symmetry. The regular extended systems for computing the simple higher-order symmetry-breaking bifurcation points with different singularities are proposed. An etficient algorithm for solving the extended systems is given. Finally, some numerical examples are shown to demonstrate the efficiency of the algorithm.
基金The project supported by National Natural Science Foundation of China under Grant No. 10405019
文摘We study entanglement in dimerized Heisenberg systems. In particular, we give exact results of groundstate pairwise entanglement for the four-qubit model by identifying a Z2 symmetry. Although the entanglements cannot identify the critical point of the system, the mean entanglement of the nearest-neighbor qubits really does, namely, it reaches a maximum at the critical point.
文摘We consider double high order S-breaking bifurcation points of two-Parameter dependent nonlinear problems with Z_2×Z_2-symmetry. Because of the underlying symmetry we could propose some regular extended systems to determine double high order S-breaking bifurcation points. and we could also show that there exist two quadratic pitchfork bifurcation point paths passing through the point being considered.
