The compatible left-symmetric algebra structures on the twisted Heisenberg-Virasoro algebra with some natural grading conditions are completely determined. The results of the earlier work on left-symmetric algebra str...The compatible left-symmetric algebra structures on the twisted Heisenberg-Virasoro algebra with some natural grading conditions are completely determined. The results of the earlier work on left-symmetric algebra structures on the Virasoro algebra play an essential role in determining these compatible structures. As a corollary, any such left-symmetric algebra contains an infinite-dimensional nontrivial subalgebra that is also a submodu]e of the regular module.展开更多
The finite-dimensional indecomposable solvable Lie algebras s with Q2n+1as their nilradical are studied and classified, it turns out that the dimension of s is dim Q2n+1+1.Then the Hom-Lie algebra structures on solvab...The finite-dimensional indecomposable solvable Lie algebras s with Q2n+1as their nilradical are studied and classified, it turns out that the dimension of s is dim Q2n+1+1.Then the Hom-Lie algebra structures on solvable Lie algebras s are calculated.展开更多
The prolongation structure methodologies of Wahlquist-Estabrook [H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 （1975） 1] for nonlinear differential equations are applied to a more general set of coupled integ...The prolongation structure methodologies of Wahlquist-Estabrook [H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 （1975） 1] for nonlinear differential equations are applied to a more general set of coupled integrable dispersionless system. Based on the obtained prolongation structure, a Lie-Algebra valued connection of a closed ideal of exterior differential forms related to the above system is constructed. A Lie-Algebra representation of some hidden structural symmetries of the previous system, its Biicklund transformation using the Riccati form of the linear eigenvalue problem and their general corresponding Lax-representation are derived. In the wake of the previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new （2 ＋ 1）-dimensional coupled integrable dispersionless system is unveiled along with its inverse scattering formulation, which applications are straightforward in nonlinear optics where additional propagating dimension deserves some attention.展开更多
The algebraic structures of the dynamical equations for the rotational relativistic systems are studied. It is found that the dynamical equations of holonomic conservative rotational relativistic systems and the speci...The algebraic structures of the dynamical equations for the rotational relativistic systems are studied. It is found that the dynamical equations of holonomic conservative rotational relativistic systems and the special nonholonomic rotational relativistic systems have Lie's algebraic structure, and the dynamical equations of the general holonomic rotational relativistic systems and the general nonholonomic rotational relativistic systems have Lie admitted algebraic structure. At last the Poisson integrals of the dynamical equations for the rotational relativistic systems are given.展开更多
The construction of the biproduct of Hopf algebras, which consists of smash product and the dual notion of smash coproduct, was first formulated by Radford. In this paper we study the quasitriangular structures over b...The construction of the biproduct of Hopf algebras, which consists of smash product and the dual notion of smash coproduct, was first formulated by Radford. In this paper we study the quasitriangular structures over biproduct Hopf algebras B*H. We show the necessary and sufficient conditions for biproduct Hopf algebras to be quasitriangular. For the case when they are, we determine completely the unique formula of the quasitriangular structures. And so we find a way to construct solutions of the Yang-Baxter equation over biproduct Hopf algebras in the sense of (Majid, 1990).展开更多
The algebraic structure and Poisson＇s integral theory of mechanico-electrical systems are studied. The Hamilton canonical equations and generalized Hamilton canonical equations and their the contravariant algebraic f...The algebraic structure and Poisson＇s integral theory of mechanico-electrical systems are studied. The Hamilton canonical equations and generalized Hamilton canonical equations and their the contravariant algebraic forms for mechanico-electrical systems are obtained. The Lie algebraic structure and the Poisson＇s integral theory of Lagrange mechanico-electrical systems are derived. The Lie algebraic structure admitted and Poisson＇s integral theory of the Lagrange-Maxwell mechanico-electrical systems are presented. Two examples are presented to illustrate these results.展开更多
A new Lie algebra G of the Lie algebra sl（2） is constructed with complex entries whose structure constants are real and imaginary numbers. A loop algebra G corresponding to the Lie algebra G is constructed, for whic...A new Lie algebra G of the Lie algebra sl（2） is constructed with complex entries whose structure constants are real and imaginary numbers. A loop algebra G corresponding to the Lie algebra G is constructed, for which it is devoted to generating a soliton hierarchy of evolution equations under the framework of generalized zero curvature equation which is derived from the compatibility of the isospectral problems expressed by Hirota operators. Finally, we decompose the Lie algebra G to obtain the subalgebras G1 and G2. Using the G2 and its one type of loop algebra G2, a Liouville integrable soliton hierarchy is obtained, furthermore, we obtain its bi-Hamiltonian structure by employing the quadratic-form identity.展开更多
Granular computing is a very hot research field in recent years. In our previous work an algebraic quotient space model was proposed,where the quotient structure could not be deduced if the granulation was based on an...Granular computing is a very hot research field in recent years. In our previous work an algebraic quotient space model was proposed,where the quotient structure could not be deduced if the granulation was based on an equivalence relation. In this paper,definitions were given and formulas of the lower quotient congruence and upper quotient congruence were calculated to roughly represent the quotient structure. Then the accuracy and roughness were defined to measure the quotient structure in quantification. Finally,a numerical example was given to demonstrate that the rough representation and measuring methods are efficient and applicable. The work has greatly enriched the algebraic quotient space model and granular computing theory.展开更多
A new algorithm of structure random response numerical characteristics, namedas matrix algebra algorithm of structure analysis is presented. Using the algorithm, structurerandom response numerical characteristics can ...A new algorithm of structure random response numerical characteristics, namedas matrix algebra algorithm of structure analysis is presented. Using the algorithm, structurerandom response numerical characteristics can easily be got by directly solving linear matrixequations rather than structure motion differential equations. Moreover, in order to solve thecorresponding linear matrix equations, the numerical integration fast algorithm is presented. Thenaccording to the results, dynamic design and life-span estimation can be done. Besides, the newalgorithm can solve non-proportion damp structure response.展开更多
When the parameters of structures are uncertain, the structural natural frequencies become uncertain. The interval parameters description, i.e. the unknown-but bounded parameters description is one of the methods to s...When the parameters of structures are uncertain, the structural natural frequencies become uncertain. The interval parameters description, i.e. the unknown-but bounded parameters description is one of the methods to study the uncertainty. In this paper, the vibration problem of structure with interval parameters was studied, and the eigenvalue problem of the structures with interval parameters was transferred into two different eigenvalue problems. The perturbation method was applied to the vibration problem of the structures with interval parameters. The numerical results show that the proposed method is sufficiently accurate and needs less computational efforts.展开更多
Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identi...Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity, and the conserved functionals were proved to be in involution in pairs under the defined Poisson bracket. As its reduction,special cases of this nonlinear super integrable couplings were obtained.展开更多
Nonlinear super integrable couplings of the super Yang hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity. ...Nonlinear super integrable couplings of the super Yang hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity. As its reduction, nonlinear integrable couplings of Yang hierarchy were obtained.展开更多
The algebraic structure and the Poisson method for a weakly nonholonomic system are studied.The differential equations of motion of the system can be written in a contravariant algebra form and its algebraic structure...The algebraic structure and the Poisson method for a weakly nonholonomic system are studied.The differential equations of motion of the system can be written in a contravariant algebra form and its algebraic structure is discussed.The Poisson theory for the systems which possess Lie algebra structure is generalized to the weakly nonholonomic system.An example is given to illustrate the application of the result.展开更多
In this paper, based on a discrete spectral problem and the corresponding zero curvature representation,the isospectral and nonisospectral lattice hierarchies are proposed. An algebraic structure of discrete zero curv...In this paper, based on a discrete spectral problem and the corresponding zero curvature representation,the isospectral and nonisospectral lattice hierarchies are proposed. An algebraic structure of discrete zero curvature equations is then established for such integrable systems. the commutation relations of Lax operators corresponding to the isospectral and non-isospectral lattice flows are worked out, the master symmetries of each lattice equation in the isospectral hierarchyand are generated, thus a τ-symmetry algebra for the lattice integrable systems is engendered from this theory.展开更多
In this note, the compatible left-symmetric superalgebra structures on an infinite-dimensional Lie superalgebra with some natural grading conditions are completely determined. The results of earlier work on left-symme...In this note, the compatible left-symmetric superalgebra structures on an infinite-dimensional Lie superalgebra with some natural grading conditions are completely determined. The results of earlier work on left-symmetric algebra structures on the Virasoro algebra play an essential role in determining these compatible structures.展开更多
A compatible left-symmetric algebra is a pair of left-symmetric algebras satisfying that any linear combination of the two left-symmetric products is a left-symmetric product.We construct a bialgebra theory of compati...A compatible left-symmetric algebra is a pair of left-symmetric algebras satisfying that any linear combination of the two left-symmetric products is a left-symmetric product.We construct a bialgebra theory of compatible left-symmetric algebras as an analogue of a Lie bialgebra.They can also be regarded as a"compatible version"of leftsymmetric bialgebras,that is,a pair of left-symmetric bialgebras satisfying that any linear combination of the two left-symmetric bialgebras is still a left-symmetric bialgebra.Many properties of compatible left-symmetric bialgebras as the"compatible version"of the corresponding properties of left-symmetric bialgebras are presented.In particular,there is a coboundary compatible left-symmetric bialgebra theory and an S-equation,which is an analogue of the classical Yang-Baxter equations in Lie algebras.展开更多
Two new explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Giachetti- Johnson （G J） hierarchy and the Yang hierarchy are obtained, respectively. By employing the variational ide...Two new explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Giachetti- Johnson （G J） hierarchy and the Yang hierarchy are obtained, respectively. By employing the variational identity their ttamiltonian structures are also generated. The approach presented in the paper can also provide nonlinear integrable couplings of other soliton hierarchies of evolution equations.展开更多
Structural controllability is critical for operating and controlling large-scale complex networks. In real applications, for a given network, it is always desirable to have more selections for driver nodes which make ...Structural controllability is critical for operating and controlling large-scale complex networks. In real applications, for a given network, it is always desirable to have more selections for driver nodes which make the network structurally controllable. Different from the works in complex network field where structural controllability is often used to explore the emergence properties of complex networks at a macro level,in this paper, we investigate it for control design purpose at the application level and focus on describing and obtaining the solution space for all selections of driver nodes to guarantee structural controllability. In accord with practical applications,we define the complete selection rule set as the solution space which is composed of a series of selection rules expressed by intuitive algebraic forms. It explicitly indicates which nodes must be controlled and how many nodes need to be controlled in a node set and thus is particularly helpful for freely selecting driver nodes. Based on two algebraic criteria of structural controllability, we separately develop an input-connectivity algorithm and a relevancy algorithm to deduce selection rules for driver nodes. In order to reduce the computational complexity,we propose a pretreatment algorithm to reduce the scale of network's structural matrix efficiently, and a rearrangement algorithm to partition the matrix into several smaller ones. A general procedure is proposed to get the complete selection rule set for driver nodes which guarantee network's structural controllability. Simulation tests with efficiency analysis of the proposed algorithms are given and the result of applying the proposed procedure to some real networks is also shown, and these all indicate the validity of the proposed procedure.展开更多
基金supported by Postdoctoral Science Foundation of China(Grant No.201003326)National Natural Science Foundation of China(Grant Nos.11101056 and 11271056)
文摘The compatible left-symmetric algebra structures on the twisted Heisenberg-Virasoro algebra with some natural grading conditions are completely determined. The results of the earlier work on left-symmetric algebra structures on the Virasoro algebra play an essential role in determining these compatible structures. As a corollary, any such left-symmetric algebra contains an infinite-dimensional nontrivial subalgebra that is also a submodu]e of the regular module.
基金Foundation item： Supported by the National Natural Science Foundation of China（11071187） Supported by the Natural Science Foundation of Henan Province（13A110785）
文摘The finite-dimensional indecomposable solvable Lie algebras s with Q2n+1as their nilradical are studied and classified, it turns out that the dimension of s is dim Q2n+1+1.Then the Hom-Lie algebra structures on solvable Lie algebras s are calculated.
文摘The prolongation structure methodologies of Wahlquist-Estabrook [H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 （1975） 1] for nonlinear differential equations are applied to a more general set of coupled integrable dispersionless system. Based on the obtained prolongation structure, a Lie-Algebra valued connection of a closed ideal of exterior differential forms related to the above system is constructed. A Lie-Algebra representation of some hidden structural symmetries of the previous system, its Biicklund transformation using the Riccati form of the linear eigenvalue problem and their general corresponding Lax-representation are derived. In the wake of the previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new （2 ＋ 1）-dimensional coupled integrable dispersionless system is unveiled along with its inverse scattering formulation, which applications are straightforward in nonlinear optics where additional propagating dimension deserves some attention.
文摘The algebraic structures of the dynamical equations for the rotational relativistic systems are studied. It is found that the dynamical equations of holonomic conservative rotational relativistic systems and the special nonholonomic rotational relativistic systems have Lie's algebraic structure, and the dynamical equations of the general holonomic rotational relativistic systems and the general nonholonomic rotational relativistic systems have Lie admitted algebraic structure. At last the Poisson integrals of the dynamical equations for the rotational relativistic systems are given.
文摘The construction of the biproduct of Hopf algebras, which consists of smash product and the dual notion of smash coproduct, was first formulated by Radford. In this paper we study the quasitriangular structures over biproduct Hopf algebras B*H. We show the necessary and sufficient conditions for biproduct Hopf algebras to be quasitriangular. For the case when they are, we determine completely the unique formula of the quasitriangular structures. And so we find a way to construct solutions of the Yang-Baxter equation over biproduct Hopf algebras in the sense of (Majid, 1990).
基金Project supported by the State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences and the National Natural Science Foundation of China （Grant Nos 10471145 and 10372053） and the Natural Science Foundation of Henan Provincial Government of China （Grant Nos 0311011400 and 0511022200）.
文摘The algebraic structure and Poisson＇s integral theory of mechanico-electrical systems are studied. The Hamilton canonical equations and generalized Hamilton canonical equations and their the contravariant algebraic forms for mechanico-electrical systems are obtained. The Lie algebraic structure and the Poisson＇s integral theory of Lagrange mechanico-electrical systems are derived. The Lie algebraic structure admitted and Poisson＇s integral theory of the Lagrange-Maxwell mechanico-electrical systems are presented. Two examples are presented to illustrate these results.
基金supported by the National Natural Science Foundation of China under Grant No.10471139
文摘A new Lie algebra G of the Lie algebra sl（2） is constructed with complex entries whose structure constants are real and imaginary numbers. A loop algebra G corresponding to the Lie algebra G is constructed, for which it is devoted to generating a soliton hierarchy of evolution equations under the framework of generalized zero curvature equation which is derived from the compatibility of the isospectral problems expressed by Hirota operators. Finally, we decompose the Lie algebra G to obtain the subalgebras G1 and G2. Using the G2 and its one type of loop algebra G2, a Liouville integrable soliton hierarchy is obtained, furthermore, we obtain its bi-Hamiltonian structure by employing the quadratic-form identity.
基金Supported by the National Natural Science Foundation of China(No.61772031)the Special Energy Saving Foundation of Changsha,Hunan Province in 2017
文摘Granular computing is a very hot research field in recent years. In our previous work an algebraic quotient space model was proposed,where the quotient structure could not be deduced if the granulation was based on an equivalence relation. In this paper,definitions were given and formulas of the lower quotient congruence and upper quotient congruence were calculated to roughly represent the quotient structure. Then the accuracy and roughness were defined to measure the quotient structure in quantification. Finally,a numerical example was given to demonstrate that the rough representation and measuring methods are efficient and applicable. The work has greatly enriched the algebraic quotient space model and granular computing theory.
基金This project is supported by National Natural Science Foundation of China (No.59805001)
文摘A new algorithm of structure random response numerical characteristics, namedas matrix algebra algorithm of structure analysis is presented. Using the algorithm, structurerandom response numerical characteristics can easily be got by directly solving linear matrixequations rather than structure motion differential equations. Moreover, in order to solve thecorresponding linear matrix equations, the numerical integration fast algorithm is presented. Thenaccording to the results, dynamic design and life-span estimation can be done. Besides, the newalgorithm can solve non-proportion damp structure response.
文摘When the parameters of structures are uncertain, the structural natural frequencies become uncertain. The interval parameters description, i.e. the unknown-but bounded parameters description is one of the methods to study the uncertainty. In this paper, the vibration problem of structure with interval parameters was studied, and the eigenvalue problem of the structures with interval parameters was transferred into two different eigenvalue problems. The perturbation method was applied to the vibration problem of the structures with interval parameters. The numerical results show that the proposed method is sufficiently accurate and needs less computational efforts.
基金Supported by the Natural Science Foundation of Henan Province（162300410075） the Science and Technology Key Research Foundation of the Education Department of Henan Province（14A110010） the Youth Backbone Teacher Foundationof Shangqiu Normal University（2013GGJS02）
文摘Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity, and the conserved functionals were proved to be in involution in pairs under the defined Poisson bracket. As its reduction,special cases of this nonlinear super integrable couplings were obtained.
文摘Nonlinear super integrable couplings of the super Yang hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity. As its reduction, nonlinear integrable couplings of Yang hierarchy were obtained.
基金supported by the National Natural Science Foundation of China(10772025,10932002,10972031)the Beijing Municipal Key Disciplines Fund for General Mechanics and Foundation of Mechanics
文摘The algebraic structure and the Poisson method for a weakly nonholonomic system are studied.The differential equations of motion of the system can be written in a contravariant algebra form and its algebraic structure is discussed.The Poisson theory for the systems which possess Lie algebra structure is generalized to the weakly nonholonomic system.An example is given to illustrate the application of the result.
基金Supported by the National Natural Science Foundation of China(Grant No.11301144,11771122,11801141).
文摘We give a complete description of the Batalin-Vilkovisky algebra structure on Hochschild cohomology of the self-injective quadratic monomial algebras.
基金Supported by the National Science Foundation of China under Grant No.11371244the Applied Mathematical Subject of SSPU under Grant No.XXKPY1604
文摘In this paper, based on a discrete spectral problem and the corresponding zero curvature representation,the isospectral and nonisospectral lattice hierarchies are proposed. An algebraic structure of discrete zero curvature equations is then established for such integrable systems. the commutation relations of Lax operators corresponding to the isospectral and non-isospectral lattice flows are worked out, the master symmetries of each lattice equation in the isospectral hierarchyand are generated, thus a τ-symmetry algebra for the lattice integrable systems is engendered from this theory.
文摘In this note, the compatible left-symmetric superalgebra structures on an infinite-dimensional Lie superalgebra with some natural grading conditions are completely determined. The results of earlier work on left-symmetric algebra structures on the Virasoro algebra play an essential role in determining these compatible structures.
基金Supported by the Meritocracy Research Funds of China West Normal University and the Fundamental Research Funds of China West Normal University(17C048,17YC392)by Scientific Research Fund of Sichuan Provincial Education Department(17AZ0378).
文摘A compatible left-symmetric algebra is a pair of left-symmetric algebras satisfying that any linear combination of the two left-symmetric products is a left-symmetric product.We construct a bialgebra theory of compatible left-symmetric algebras as an analogue of a Lie bialgebra.They can also be regarded as a"compatible version"of leftsymmetric bialgebras,that is,a pair of left-symmetric bialgebras satisfying that any linear combination of the two left-symmetric bialgebras is still a left-symmetric bialgebra.Many properties of compatible left-symmetric bialgebras as the"compatible version"of the corresponding properties of left-symmetric bialgebras are presented.In particular,there is a coboundary compatible left-symmetric bialgebra theory and an S-equation,which is an analogue of the classical Yang-Baxter equations in Lie algebras.
基金Supported by the Fundamental Research Funds of the Central University under Grant No. 2010LKS808the Natural Science Foundation of Shandong Province under Grant No. ZR2009AL021
文摘Two new explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Giachetti- Johnson （G J） hierarchy and the Yang hierarchy are obtained, respectively. By employing the variational identity their ttamiltonian structures are also generated. The approach presented in the paper can also provide nonlinear integrable couplings of other soliton hierarchies of evolution equations.
基金supported by the National Science Foundation of China(61333009,61473317,61433002,61521063,61590924,61673366)the National High Technology Research and Development Program of China(2015AA043102)
文摘Structural controllability is critical for operating and controlling large-scale complex networks. In real applications, for a given network, it is always desirable to have more selections for driver nodes which make the network structurally controllable. Different from the works in complex network field where structural controllability is often used to explore the emergence properties of complex networks at a macro level,in this paper, we investigate it for control design purpose at the application level and focus on describing and obtaining the solution space for all selections of driver nodes to guarantee structural controllability. In accord with practical applications,we define the complete selection rule set as the solution space which is composed of a series of selection rules expressed by intuitive algebraic forms. It explicitly indicates which nodes must be controlled and how many nodes need to be controlled in a node set and thus is particularly helpful for freely selecting driver nodes. Based on two algebraic criteria of structural controllability, we separately develop an input-connectivity algorithm and a relevancy algorithm to deduce selection rules for driver nodes. In order to reduce the computational complexity,we propose a pretreatment algorithm to reduce the scale of network's structural matrix efficiently, and a rearrangement algorithm to partition the matrix into several smaller ones. A general procedure is proposed to get the complete selection rule set for driver nodes which guarantee network's structural controllability. Simulation tests with efficiency analysis of the proposed algorithms are given and the result of applying the proposed procedure to some real networks is also shown, and these all indicate the validity of the proposed procedure.
