We prove an estimate for Donaldson's Q-operator on a prequantized compact symplectic manifold.This estimate is an ingredient in the recent result of Keller and Lejmi(2017) about a symplectic generalization of Dona...We prove an estimate for Donaldson's Q-operator on a prequantized compact symplectic manifold.This estimate is an ingredient in the recent result of Keller and Lejmi(2017) about a symplectic generalization of Donaldson's lower bound for the L^2-norm of the Hermitian scalar curvature.展开更多
For a compact symplectic manifold which is s-Lefschetz which is weaker than the decomposition for de hard Lefschetz property, we prove that the Lefschetz Rham cohomology also holds.
In this paper,the modern geometrical structure of analytical mechanics,the exterior differential forms and the geometrical meaning of dynamic equations are briefly discussed.
By using the discrete variational method,we study the numerical method of the general nonholonomic system in the generalized Birkhoffian framework,and construct a numerical method of generalized Birkhoffian equations ...By using the discrete variational method,we study the numerical method of the general nonholonomic system in the generalized Birkhoffian framework,and construct a numerical method of generalized Birkhoffian equations called a self-adjoint-preserving algorithm.Numerical results show that it is reasonable to study the nonholonomic system by the structure-preserving algorithm in the generalized Birkhoffian framework.展开更多
The important notions and results of the integral invariants of Poincard and Cartan-Poincard and the relationship between integral invariant and invariant form established first by E. Cartan in the classical mechanics...The important notions and results of the integral invariants of Poincard and Cartan-Poincard and the relationship between integral invariant and invariant form established first by E. Cartan in the classical mechanics are generalized to Hamilton mechanics on Kiihler manifold, by the theory of modern geometry and advanced calculus, to get the corresponding wider arid deeper results. differential展开更多
In this paper, a new completely integrable system related to the complex spectral problem -φ xx+(i/4)wpx+(i/4)(wp)x+(1/4)vφ=iλφxand the constrained flows of the Boussinesq equations axe generated. Accor...In this paper, a new completely integrable system related to the complex spectral problem -φ xx+(i/4)wpx+(i/4)(wp)x+(1/4)vφ=iλφxand the constrained flows of the Boussinesq equations axe generated. According to the viewpoint of Hamiltonian mechanics, the Euler-Lagrange equations and the Legendre transformations, a reasonable Jacobi-Ostrogradsky coordinate system is obtained. Moreover, by means of the constrained conditions between the potentiaJ u, v and the eigenfunction φ, the involutive representations of the solutions for the Boussinesq equation hieraxchy axe given.展开更多
This paper is divided in two parts: in Section 2, we define recursively a privileged basis of the primitive forms in a symplectic space(V^(2n), ω). Successively, in Section 3, we apply our construction in the se...This paper is divided in two parts: in Section 2, we define recursively a privileged basis of the primitive forms in a symplectic space(V^(2n), ω). Successively, in Section 3, we apply our construction in the setting of Heisenberg groups H^n, n ≥ 1, to write in coordinates the exterior differential of the so-called Rumin's complex of differential forms in H^n.展开更多
The complex surface X obtained by 8 points blown up on CP2 and Barlow’s surface Y are homeomorphic,but not diffeomorphic.Using the Gromov-Witten invariant Ruan showed that the stabilized manifolds X×S2and Y×...The complex surface X obtained by 8 points blown up on CP2 and Barlow’s surface Y are homeomorphic,but not diffeomorphic.Using the Gromov-Witten invariant Ruan showed that the stabilized manifolds X×S2and Y×S2are not deformation equivalent.In this note,we show that the stabilized manifolds X×S1and Y×S1are diffeomorphic and non-deformation equivalent in cosymplectic sense.展开更多
We give some new genus-3 universal equations for Gromov-Witten invariants of compact symplectic manifolds. These equations were obtained by studying relations in the tautological ring of the moduli space of2-pointed g...We give some new genus-3 universal equations for Gromov-Witten invariants of compact symplectic manifolds. These equations were obtained by studying relations in the tautological ring of the moduli space of2-pointed genus-3 stable curves. A byproduct of our search for genus-3 equations is a new genus-2 universal equation for Gromov-Witten invariants.展开更多
A real Liouville domain is a Liouville domain with an exact anti-symplectic involution. The authors call a real Liouville domain uniruled if there exists an invariant finite energy plane through every real point. Asym...A real Liouville domain is a Liouville domain with an exact anti-symplectic involution. The authors call a real Liouville domain uniruled if there exists an invariant finite energy plane through every real point. Asymptotically, an invariant finite energy plane converges to a symmetric periodic orbit. In this note, they work out a criterion which guarantees uniruledness for real Liouville domains.展开更多
This paper investigates the relationship between state feedback and Hamiltonian realization. First, it is proved that a completely controllable linear system always has a state feedback state equation Hamiltonian real...This paper investigates the relationship between state feedback and Hamiltonian realization. First, it is proved that a completely controllable linear system always has a state feedback state equation Hamiltonian realization. Necessary and sufficient conditions are obtained for it to have a Hamiltonian realization with natural output. Then some conditions for an affine nonlinear system to have a Hamiltonian realization are given. For generalized outputs, the conditions of the feedback, keeping Hamiltonian, are discussed. Finally, the admissible feedback controls for generalized Hamiltonian systems are considered.展开更多
Let (M, ω) be a closed symplectic 2n-dimensional manifold. Donaldson in his paper showed that there exist 2m-dimensional symplectie submanifolds (V^2m,ω) of (M,ω), 1 ≤m ≤ n - 1, with (m - 1)-equivalent in...Let (M, ω) be a closed symplectic 2n-dimensional manifold. Donaldson in his paper showed that there exist 2m-dimensional symplectie submanifolds (V^2m,ω) of (M,ω), 1 ≤m ≤ n - 1, with (m - 1)-equivalent inclusions. On the basis of this fact we obtain isomorphic relations between kernel of Lefschetz map of M and kernels of Lefschetz maps of Donaldson submanifolds V^2m, 2 ≤ m ≤ n - 1. Then, using this relation, we show that the flux group of M is discrete if the action of π1 (M) on π2(M) is trivial and there exists a retraction r : M→ V, where V is a 4-dimensional Donaldson submanifold. And, in the symplectically aspherical case, we investigate the flux groups of the manifolds.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.11401232 and 11528103)Agence nationale de la recherche(Grant No.ANR-14-CE25-0012-01)funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative and Deutsche Forschungsgemeinschaft Funded Project Sonderforschungsbereich Transregio 191
文摘We prove an estimate for Donaldson's Q-operator on a prequantized compact symplectic manifold.This estimate is an ingredient in the recent result of Keller and Lejmi(2017) about a symplectic generalization of Donaldson's lower bound for the L^2-norm of the Hermitian scalar curvature.
基金Acknowledgements The authors were very grateful to their advisor Prof. Hongyu Wang for discussion and suggestions. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11371309, 11471145, 11401514), the University Science Research Project of Jiangsu Province (14KJB110027), and the Foundation of Yangzhou University (2014CXJ004).
文摘For a compact symplectic manifold which is s-Lefschetz which is weaker than the decomposition for de hard Lefschetz property, we prove that the Lefschetz Rham cohomology also holds.
基金Work supported by NSF of Henan Education Commission
文摘In this paper,the modern geometrical structure of analytical mechanics,the exterior differential forms and the geometrical meaning of dynamic equations are briefly discussed.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11472124,11572145,11202090,and 11301350)the Doctor Research Start-up Fund of Liaoning Province,China(Grant No.20141050)+1 种基金the China Postdoctoral Science Foundation(Grant No.2014M560203)the General Science and Technology Research Plans of Liaoning Educational Bureau,China(Grant No.L2013005)
文摘By using the discrete variational method,we study the numerical method of the general nonholonomic system in the generalized Birkhoffian framework,and construct a numerical method of generalized Birkhoffian equations called a self-adjoint-preserving algorithm.Numerical results show that it is reasonable to study the nonholonomic system by the structure-preserving algorithm in the generalized Birkhoffian framework.
文摘The important notions and results of the integral invariants of Poincard and Cartan-Poincard and the relationship between integral invariant and invariant form established first by E. Cartan in the classical mechanics are generalized to Hamilton mechanics on Kiihler manifold, by the theory of modern geometry and advanced calculus, to get the corresponding wider arid deeper results. differential
文摘In this paper, a new completely integrable system related to the complex spectral problem -φ xx+(i/4)wpx+(i/4)(wp)x+(1/4)vφ=iλφxand the constrained flows of the Boussinesq equations axe generated. According to the viewpoint of Hamiltonian mechanics, the Euler-Lagrange equations and the Legendre transformations, a reasonable Jacobi-Ostrogradsky coordinate system is obtained. Moreover, by means of the constrained conditions between the potentiaJ u, v and the eigenfunction φ, the involutive representations of the solutions for the Boussinesq equation hieraxchy axe given.
基金Supported by University of Bolognafunds for selected research topics+1 种基金supported by the Gruppo Nazionale per l’Analisi Matematica,la Probabilita e le loro Applicazioni(GNAMPA)of the Istituto Nazionale di Alta Matematica(INdA M)supported by P.R.I.N.of M.I.U.R.,Italy
文摘This paper is divided in two parts: in Section 2, we define recursively a privileged basis of the primitive forms in a symplectic space(V^(2n), ω). Successively, in Section 3, we apply our construction in the setting of Heisenberg groups H^n, n ≥ 1, to write in coordinates the exterior differential of the so-called Rumin's complex of differential forms in H^n.
基金supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education(Grant No.2013004848)
文摘The complex surface X obtained by 8 points blown up on CP2 and Barlow’s surface Y are homeomorphic,but not diffeomorphic.Using the Gromov-Witten invariant Ruan showed that the stabilized manifolds X×S2and Y×S2are not deformation equivalent.In this note,we show that the stabilized manifolds X×S1and Y×S1are diffeomorphic and non-deformation equivalent in cosymplectic sense.
基金supported by National Security Agency(Grant No.H98230-10-1-0179)the National Science Foundation of USA(Grant No.DMS-0905227)+2 种基金a Tian-Yuan Special Fund of National Natural Science Foundation of China(Grant No.11326023)Specialized Research Fund for the Doctoral Program of Ministry of Higher Education(Grant No.20120001110051)Peking University 985 Fund
文摘We give some new genus-3 universal equations for Gromov-Witten invariants of compact symplectic manifolds. These equations were obtained by studying relations in the tautological ring of the moduli space of2-pointed genus-3 stable curves. A byproduct of our search for genus-3 equations is a new genus-2 universal equation for Gromov-Witten invariants.
基金supported by National Research Foundation of Korea(No.2012-011755)a stipend from the Humboldt foundation
文摘A real Liouville domain is a Liouville domain with an exact anti-symplectic involution. The authors call a real Liouville domain uniruled if there exists an invariant finite energy plane through every real point. Asymptotically, an invariant finite energy plane converges to a symmetric periodic orbit. In this note, they work out a criterion which guarantees uniruledness for real Liouville domains.
基金This research is supported partly by the National Natural Science Foundation of China(No.G59837270)and National 973 Project(No.G
文摘This paper investigates the relationship between state feedback and Hamiltonian realization. First, it is proved that a completely controllable linear system always has a state feedback state equation Hamiltonian realization. Necessary and sufficient conditions are obtained for it to have a Hamiltonian realization with natural output. Then some conditions for an affine nonlinear system to have a Hamiltonian realization are given. For generalized outputs, the conditions of the feedback, keeping Hamiltonian, are discussed. Finally, the admissible feedback controls for generalized Hamiltonian systems are considered.
文摘Let (M, ω) be a closed symplectic 2n-dimensional manifold. Donaldson in his paper showed that there exist 2m-dimensional symplectie submanifolds (V^2m,ω) of (M,ω), 1 ≤m ≤ n - 1, with (m - 1)-equivalent inclusions. On the basis of this fact we obtain isomorphic relations between kernel of Lefschetz map of M and kernels of Lefschetz maps of Donaldson submanifolds V^2m, 2 ≤ m ≤ n - 1. Then, using this relation, we show that the flux group of M is discrete if the action of π1 (M) on π2(M) is trivial and there exists a retraction r : M→ V, where V is a 4-dimensional Donaldson submanifold. And, in the symplectically aspherical case, we investigate the flux groups of the manifolds.