Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and ...Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.展开更多
Key points:Throughout the ages,rule of law has been an indispensable means of governing a state.However,as General Secretary Xi Jinping has pointed out,it is not enough to govern a state by the rule of law
The structure and dynamic succession law of natural secondary forest after severe fire interference in recent 20 years were studied by adopting the method of deducing time series from the spatial sequence of vegetatio...The structure and dynamic succession law of natural secondary forest after severe fire interference in recent 20 years were studied by adopting the method of deducing time series from the spatial sequence of vegetation in Heihe region, Heilongjiang, China. Two typical and widely distributed forest types in the study area, namely forest type A and forest type B, were selected as study subjects. Forest type A is pure broadleaf forest or broadleaf mixed forest mainly composing of superior Betula platyphylla and Populus davidiana in the area with gradient 〈25°, while forest type B is pure forest or mixed forest composing of superior Quercus mongolica and Betula davurica in the area with gradient 〉25°. Species richness, vegetation coverage, important value, and similarity index of commtmity in different layers （Herb, shrub, small tree, and arbor layers） were investigated and analyzed for the two typical forests. The results show that after fire interference, the species richness and coverage in each layer in forest type A were higher than that in forest type B. Both for forest type A and B, with elapse of post-fire years, the species richness and coverage of herbs and shrubs showed a decline tendency, while those of arbor layer present a rising tendency. Through comparison of the important values of species in each layer and analysis of community structure changes, the dynamic process of post-fire vegetation succession for forest type A and B was separately determined. Post-fire 80 years＇ succession tendency of forest type A is B. platyphylla and Larix gmelinii mixed forest. Its shrub layer is mainly composed of Corylus heterophylla and Vaccinium uliginosum, and herb layer is dominated by Carex tristachya, Athyrium multidentatum, and Pyrola incarnate; whereas, the post-fire 80 years＇ succession of forest type B is Q. mongolica and B. davurica mixed forest. Its shrub layer is mainly composed of lespedeza bicolar and corylus heterophylla and herb layer is dominated by Carex tristachya, Asparagus densiflorus, and Hemerocallis minor.展开更多
In this article,we explore the famous Selkov–Schnakenberg(SS)system of coupled nonlinear partial differential equations(PDEs)for Lie symmetry analysis,self-adjointness,and conservation laws.Moreover,miscellaneous sol...In this article,we explore the famous Selkov–Schnakenberg(SS)system of coupled nonlinear partial differential equations(PDEs)for Lie symmetry analysis,self-adjointness,and conservation laws.Moreover,miscellaneous soliton solutions like dark,bright,periodic,rational,Jacobian elliptic function,Weierstrass elliptic function,and hyperbolic solutions of the SS system will be achieved by a well-known technique called sub-ordinary differential equations.All these results are displayed graphically by 3D,2D,and contour plots.展开更多
The structure and dynamic succession law of natural secondary forest after severe fire interference in recent 20 years were studied by adopting the method of deducing time series from the spatial sequence of vegetatio...The structure and dynamic succession law of natural secondary forest after severe fire interference in recent 20 years were studied by adopting the method of deducing time series from the spatial sequence of vegetation in Heihe region, Heilongjiang, China.Two typical and widely distributed forest types in the study area, namely forest type A and forest type B, were selected as study subjects.Forest type A is pure broadleaf forest or broadleaf mixed forest mainly composing of superior Betula platyphylla and Populus davidiana in the area with gradient <25°, while forest type B is pure forest or mixed forest composing of superior Quercus mongolica and Betula davurica in the area with gradient >25°.Species richness, vegetation coverage, important value, and similarity index of community in different layers(Herb, shrub, small tree, and arbor layers) were investigated and analyzed for the two typical forests.The results show that after fire interference, the species richness and coverage in each layer in forest type A were higher than that in forest type B.Both for forest type A and B, with elapse of post-fire years, the species richness and coverage of herbs and shrubs showed a decline tendency, while those of arbor layer present a rising tendency.Through comparison of the important values of species in each layer and analysis of community structure changes, the dynamic process of post-fire vegetation succession for forest type A and B was separately determined.Post-fire 80 years' succession tendency of forest type A is B.platyphylla and Larix gmelinii mixed forest.Its shrub layer is mainly composed of Corylus heterophylla and Vaccinium uliginosum, and herb layer is dominated by Carex tristachya, Athyrium multidentatum, and Pyrola incarnate;whereas, the post-fire 80 years' succession of forest type B is Q.mongolica and B.davurica mixed forest.Its shrub layer is mainly composed of lespedeza bicolar and corylus heterophylla and herb layer is dominated by Carex tristachya, Asparagus densiflorus, and Hemerocallis minor.展开更多
This paper extends Hojman＇s conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The gener...This paper extends Hojman＇s conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results.展开更多
Lie symmetry and Mei conservation law of continuum Lagrange system are studied in this paper. The equation of motion of continuum system is established by using variational principle of continuous coordinates. The inv...Lie symmetry and Mei conservation law of continuum Lagrange system are studied in this paper. The equation of motion of continuum system is established by using variational principle of continuous coordinates. The invariance of the equation of motion under an infinitesimal transformation group is determined to be Lie-symmetric. The condition of obtaining Mei conservation theorem from Lie symmetry is also presented. An example is discussed for applications of the results.展开更多
The nonlinear evolution equations have a wide range of applications,more precisely in physics,biology,chemistry and engineering fields.This domain serves as a point of interest to a large extent in the world’s mathem...The nonlinear evolution equations have a wide range of applications,more precisely in physics,biology,chemistry and engineering fields.This domain serves as a point of interest to a large extent in the world’s mathematical community.Thus,this paper purveys an analytical study of a generalized extended(2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering.The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model.We further reduce the equation using the subalgebras obtained.Besides,more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique.Consequently,we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright,dark,periodic(cnoidal and snoidal),compact-type as well as singular solitons.The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time,are outlined with suitable diagrams.This can be of interest to oceanographers and ocean engineers for future analysis.Furthermore,to visualize the dynamics of the results found,we present the graphic display of each of the solutions.Conclusively,we construct conservation laws of the understudy equation via the application of Noether’s theorem.展开更多
In this paper, a detailed Lie symmetry analysis of the(2+1)-dimensional coupled nonlinear extension of the reaction-diffusion equation is presented. The general finite transformation group is derived via a simple dire...In this paper, a detailed Lie symmetry analysis of the(2+1)-dimensional coupled nonlinear extension of the reaction-diffusion equation is presented. The general finite transformation group is derived via a simple direct method,which is equivalent to Lie point symmetry group actually. Similarity reduction and some exact solutions of the original equation are obtained based on the optimal system of one-dimensional subalgebras. In addition, conservation laws are constructed by employing the new conservation theorem.展开更多
文摘Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.
文摘Key points:Throughout the ages,rule of law has been an indispensable means of governing a state.However,as General Secretary Xi Jinping has pointed out,it is not enough to govern a state by the rule of law
文摘The structure and dynamic succession law of natural secondary forest after severe fire interference in recent 20 years were studied by adopting the method of deducing time series from the spatial sequence of vegetation in Heihe region, Heilongjiang, China. Two typical and widely distributed forest types in the study area, namely forest type A and forest type B, were selected as study subjects. Forest type A is pure broadleaf forest or broadleaf mixed forest mainly composing of superior Betula platyphylla and Populus davidiana in the area with gradient 〈25°, while forest type B is pure forest or mixed forest composing of superior Quercus mongolica and Betula davurica in the area with gradient 〉25°. Species richness, vegetation coverage, important value, and similarity index of commtmity in different layers （Herb, shrub, small tree, and arbor layers） were investigated and analyzed for the two typical forests. The results show that after fire interference, the species richness and coverage in each layer in forest type A were higher than that in forest type B. Both for forest type A and B, with elapse of post-fire years, the species richness and coverage of herbs and shrubs showed a decline tendency, while those of arbor layer present a rising tendency. Through comparison of the important values of species in each layer and analysis of community structure changes, the dynamic process of post-fire vegetation succession for forest type A and B was separately determined. Post-fire 80 years＇ succession tendency of forest type A is B. platyphylla and Larix gmelinii mixed forest. Its shrub layer is mainly composed of Corylus heterophylla and Vaccinium uliginosum, and herb layer is dominated by Carex tristachya, Athyrium multidentatum, and Pyrola incarnate; whereas, the post-fire 80 years＇ succession of forest type B is Q. mongolica and B. davurica mixed forest. Its shrub layer is mainly composed of lespedeza bicolar and corylus heterophylla and herb layer is dominated by Carex tristachya, Asparagus densiflorus, and Hemerocallis minor.
文摘In this article,we explore the famous Selkov–Schnakenberg(SS)system of coupled nonlinear partial differential equations(PDEs)for Lie symmetry analysis,self-adjointness,and conservation laws.Moreover,miscellaneous soliton solutions like dark,bright,periodic,rational,Jacobian elliptic function,Weierstrass elliptic function,and hyperbolic solutions of the SS system will be achieved by a well-known technique called sub-ordinary differential equations.All these results are displayed graphically by 3D,2D,and contour plots.
基金supported by Heilongjiang Natural Foundation (C200625)Forestry Science and Technology Sup-porting Program (2006BAD03A0805)
文摘The structure and dynamic succession law of natural secondary forest after severe fire interference in recent 20 years were studied by adopting the method of deducing time series from the spatial sequence of vegetation in Heihe region, Heilongjiang, China.Two typical and widely distributed forest types in the study area, namely forest type A and forest type B, were selected as study subjects.Forest type A is pure broadleaf forest or broadleaf mixed forest mainly composing of superior Betula platyphylla and Populus davidiana in the area with gradient <25°, while forest type B is pure forest or mixed forest composing of superior Quercus mongolica and Betula davurica in the area with gradient >25°.Species richness, vegetation coverage, important value, and similarity index of community in different layers(Herb, shrub, small tree, and arbor layers) were investigated and analyzed for the two typical forests.The results show that after fire interference, the species richness and coverage in each layer in forest type A were higher than that in forest type B.Both for forest type A and B, with elapse of post-fire years, the species richness and coverage of herbs and shrubs showed a decline tendency, while those of arbor layer present a rising tendency.Through comparison of the important values of species in each layer and analysis of community structure changes, the dynamic process of post-fire vegetation succession for forest type A and B was separately determined.Post-fire 80 years' succession tendency of forest type A is B.platyphylla and Larix gmelinii mixed forest.Its shrub layer is mainly composed of Corylus heterophylla and Vaccinium uliginosum, and herb layer is dominated by Carex tristachya, Athyrium multidentatum, and Pyrola incarnate;whereas, the post-fire 80 years' succession of forest type B is Q.mongolica and B.davurica mixed forest.Its shrub layer is mainly composed of lespedeza bicolar and corylus heterophylla and herb layer is dominated by Carex tristachya, Asparagus densiflorus, and Hemerocallis minor.
基金supported by the National Natural Science Foundation of China (Grant No 10872037)the Natural Science Foundation of Anhui Province of China (Grant No 070416226)
文摘This paper extends Hojman＇s conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results.
基金Project supported by the National Natural Science Foundation of China （Grant No. 11072218） and the Natural Science Foundation of Zhejiang Province of China （Grant No. Y6100337）.
文摘Lie symmetry and Mei conservation law of continuum Lagrange system are studied in this paper. The equation of motion of continuum system is established by using variational principle of continuous coordinates. The invariance of the equation of motion under an infinitesimal transformation group is determined to be Lie-symmetric. The condition of obtaining Mei conservation theorem from Lie symmetry is also presented. An example is discussed for applications of the results.
基金Supported by National Natural Science Foundation of China(11371293)the Natural Science Foundation of Shaanxi Province(2014JM2-1009)and the Science and Technology Innovation Foundation of Xi’an(CYX1531WL41,CYX1531WL40)
文摘The nonlinear evolution equations have a wide range of applications,more precisely in physics,biology,chemistry and engineering fields.This domain serves as a point of interest to a large extent in the world’s mathematical community.Thus,this paper purveys an analytical study of a generalized extended(2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering.The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model.We further reduce the equation using the subalgebras obtained.Besides,more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique.Consequently,we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright,dark,periodic(cnoidal and snoidal),compact-type as well as singular solitons.The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time,are outlined with suitable diagrams.This can be of interest to oceanographers and ocean engineers for future analysis.Furthermore,to visualize the dynamics of the results found,we present the graphic display of each of the solutions.Conclusively,we construct conservation laws of the understudy equation via the application of Noether’s theorem.
基金Supported by the National Natural Science Foundation of China under Grant No.11275072Research Fund for the Doctoral Program of Higher Education of China under Grant No.20120076110024+3 种基金Innovative Research Team Program of the National Natural Science Foundation of China under Grant No.61321064Shanghai Knowledge Service Platform Project under Grant No.ZF1213Shanghai Minhang District Talents of High Level Scientific Research ProjectTalent Fund and K.C.Wong Magna Fund in Ningbo University
文摘In this paper, a detailed Lie symmetry analysis of the(2+1)-dimensional coupled nonlinear extension of the reaction-diffusion equation is presented. The general finite transformation group is derived via a simple direct method,which is equivalent to Lie point symmetry group actually. Similarity reduction and some exact solutions of the original equation are obtained based on the optimal system of one-dimensional subalgebras. In addition, conservation laws are constructed by employing the new conservation theorem.
