In this paper, we characterize all of the Darboux polynomials of the L system and prove that the system is not algebraically integrable, using the weight homogeneous polynomials and the method of characteristic curves...In this paper, we characterize all of the Darboux polynomials of the L system and prove that the system is not algebraically integrable, using the weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations.展开更多
A kind of integrable coupling of soliton equations hierarchy with self-consistent sources associated with s/（4） has been presented （Yu F J and Li L 2009 Appl. Math. Comput. 207 171; Yu F J 2008 Phys. Lett. A 372 6...A kind of integrable coupling of soliton equations hierarchy with self-consistent sources associated with s/（4） has been presented （Yu F J and Li L 2009 Appl. Math. Comput. 207 171; Yu F J 2008 Phys. Lett. A 372 6613）. Based on this method, we construct two integrable couplings of the soliton hierarchy with self-consistent sources by using the loop algebra sl（4）. In this paper, we also point out that there are some errors in these references and we have corrected these errors and set up new formula. The method can be generalized to other soliton hierarchy with self-consistent sources.展开更多
In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras？H and？T, respectively. By using the loop algebra？H and the Tu scheme, ...In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras？H and？T, respectively. By using the loop algebra？H and the Tu scheme, we obtain an integrable hierarchy from which we derive a new Darboux transformation to produce a set of exact periodic solutions. With the loop algebra？T, a new integrable-coupling hierarchy is obtained and reduced to some variable-coefficient nonlinear equations, whose Hamiltonian structure is derived by using the variational identity. Furthermore, we construct a higher-dimensional loop algebraˉH of the Lie algebra H from which a new Liouville-integrable hierarchy with 5-potential functions is produced and reduced to a complex m Kd V equation, whose 3-Hamiltonian structure can be obtained by using the trace identity. A new approach is then given for deriving multiHamiltonian structures of integrable hierarchies. Finally, we extend the loop algebra？H to obtain an integrable hierarchy with variable coefficients.展开更多
This paper outlines the necessity of the knowledge representation for the geometrical shapes （KRGS）. We advocate that KRGS for being powerful must contain at least three major components, namely （1） fu...This paper outlines the necessity of the knowledge representation for the geometrical shapes （KRGS）. We advocate that KRGS for being powerful must contain at least three major components, namely （1） fuzzy logic scheme; （2） the machine learning technique; and （3） an integrated algebraic and logical reasoning. After arguing the need for using fuzzy expressions in spatial reasoning, then inducing the spatial graph generalized and maximal common part of the expressions is discussed. Finally, the integration of approximate references into spatial reasoning using absolute measurements is outlined. The integration here means that the satisfiability of a fuzzy spatial expression is conducted by both logical and algebraic reasoning.展开更多
In this paper, we study the invariant algebraic surfaces of a system, which generalizes the Lorenz system. Using the weight homogeneous polynomials and the method of characteristic curves for solving linear partial di...In this paper, we study the invariant algebraic surfaces of a system, which generalizes the Lorenz system. Using the weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations, we characterize all the Darboux invariants, the irreducible Darboux polynomials, the rational first integrals and the algebraic integrability of this system.展开更多
We consider an integrable three-dimensional system of ordinary differential equations introduced by S. V. Kovalevskaya in a letter to G. Mittag- Leffler. We prove its isomorphism with the three-dimensional Euler top, ...We consider an integrable three-dimensional system of ordinary differential equations introduced by S. V. Kovalevskaya in a letter to G. Mittag- Leffler. We prove its isomorphism with the three-dimensional Euler top, and propose two integrable discretizations for it. Then we present an integrable generalization of the Kovalevskaya system, and study the problem of integrable discretization for this generalized system.展开更多
This paper deals with the analytic Feynman integral of functionals on a Wiener space. First the authors establish the existence of the analytic Feynman integrals of functionals in a Banach algebra S_α. The authors th...This paper deals with the analytic Feynman integral of functionals on a Wiener space. First the authors establish the existence of the analytic Feynman integrals of functionals in a Banach algebra S_α. The authors then obtain a formula for the first variation of integrals. Finally, various analytic Feynman integration formulas involving the first variation are established.展开更多
基金supported by the NNSF of China grant No.10231020
文摘In this paper, we characterize all of the Darboux polynomials of the L system and prove that the system is not algebraically integrable, using the weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations.
基金Project supported by the Natural Science Foundation of Shanghai (Grant No. 09ZR1410800)the Science Foundation of Key Laboratory of Mathematics Mechanization (Grant No. KLMM0806)+2 种基金the Shanghai Leading Academic Discipline Project (Grant No. J50101)the Key Disciplines of Shanghai Municipality (Grant No. S30104)the National Natural Science Foundation of China (Grant Nos. 61072147 and 11071159)
文摘A kind of integrable coupling of soliton equations hierarchy with self-consistent sources associated with s/（4） has been presented （Yu F J and Li L 2009 Appl. Math. Comput. 207 171; Yu F J 2008 Phys. Lett. A 372 6613）. Based on this method, we construct two integrable couplings of the soliton hierarchy with self-consistent sources by using the loop algebra sl（4）. In this paper, we also point out that there are some errors in these references and we have corrected these errors and set up new formula. The method can be generalized to other soliton hierarchy with self-consistent sources.
基金Supported by the Research Grant Council of the Hong Kong Special Administrative Region(Grant No.City U101211)the National Natural Science Foundation of China(Grant No.11371361)the Natural Science Foundation of Shandong Province(Grant No.ZR2013AL016)
文摘In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras？H and？T, respectively. By using the loop algebra？H and the Tu scheme, we obtain an integrable hierarchy from which we derive a new Darboux transformation to produce a set of exact periodic solutions. With the loop algebra？T, a new integrable-coupling hierarchy is obtained and reduced to some variable-coefficient nonlinear equations, whose Hamiltonian structure is derived by using the variational identity. Furthermore, we construct a higher-dimensional loop algebraˉH of the Lie algebra H from which a new Liouville-integrable hierarchy with 5-potential functions is produced and reduced to a complex m Kd V equation, whose 3-Hamiltonian structure can be obtained by using the trace identity. A new approach is then given for deriving multiHamiltonian structures of integrable hierarchies. Finally, we extend the loop algebra？H to obtain an integrable hierarchy with variable coefficients.
文摘This paper outlines the necessity of the knowledge representation for the geometrical shapes （KRGS）. We advocate that KRGS for being powerful must contain at least three major components, namely （1） fuzzy logic scheme; （2） the machine learning technique; and （3） an integrated algebraic and logical reasoning. After arguing the need for using fuzzy expressions in spatial reasoning, then inducing the spatial graph generalized and maximal common part of the expressions is discussed. Finally, the integration of approximate references into spatial reasoning using absolute measurements is outlined. The integration here means that the satisfiability of a fuzzy spatial expression is conducted by both logical and algebraic reasoning.
基金supported by the NNSF of China (11171191 and 11201266)
文摘In this paper, we study the invariant algebraic surfaces of a system, which generalizes the Lorenz system. Using the weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations, we characterize all the Darboux invariants, the irreducible Darboux polynomials, the rational first integrals and the algebraic integrability of this system.
文摘We consider an integrable three-dimensional system of ordinary differential equations introduced by S. V. Kovalevskaya in a letter to G. Mittag- Leffler. We prove its isomorphism with the three-dimensional Euler top, and propose two integrable discretizations for it. Then we present an integrable generalization of the Kovalevskaya system, and study the problem of integrable discretization for this generalized system.
基金supported by the research fund of Dankook University in 2015
文摘This paper deals with the analytic Feynman integral of functionals on a Wiener space. First the authors establish the existence of the analytic Feynman integrals of functionals in a Banach algebra S_α. The authors then obtain a formula for the first variation of integrals. Finally, various analytic Feynman integration formulas involving the first variation are established.
