Aim To study singular points, closed orbits, stable manifolds and unstable manifolds of a second order autonomous Birkhoff system. Methods Qualitative methods of ordinary differential equation were used. Results and ...Aim To study singular points, closed orbits, stable manifolds and unstable manifolds of a second order autonomous Birkhoff system. Methods Qualitative methods of ordinary differential equation were used. Results and Conclusion The criteria for singular points, closed orbits and hyperbolic equilibrium points of a second order autonomous Birkhoff system are given. Moreover the stability of equilibria, stable manifolds and unstable manifolds are obtained.展开更多
We determine the set of degrees between some classes of oriented closed （n - 1）-connected 2n-manifolds by using the arithmetic theory of quadratic forms.
Let （M,g（t））, 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative Ricci c for some constant C 〉 0 and g（t） evolving by the Ricci flow curvature, │Rc│ ≤C/t for some constant C 〉 0 and g（t） e...Let （M,g（t））, 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative Ricci c for some constant C 〉 0 and g（t） evolving by the Ricci flow curvature, │Rc│ ≤C/t for some constant C 〉 0 and g（t） evolving by the Ricci flow gij/ t=-2Rij.In this paper, we derive a differential Harnack estimate for positive solutions to parabolic equations of the type u~ = /△u - aulogu - bu on M x （0,T], where a 〉 0 and b ∈ R.展开更多
文摘Aim To study singular points, closed orbits, stable manifolds and unstable manifolds of a second order autonomous Birkhoff system. Methods Qualitative methods of ordinary differential equation were used. Results and Conclusion The criteria for singular points, closed orbits and hyperbolic equilibrium points of a second order autonomous Birkhoff system are given. Moreover the stability of equilibria, stable manifolds and unstable manifolds are obtained.
基金Supported by Morningside Center of Mathematics, National Natural Science Foundation of China (Grant Nos. 10325105 and 10531060)KRF (2003-070-C00001)
文摘We determine the set of degrees between some classes of oriented closed （n - 1）-connected 2n-manifolds by using the arithmetic theory of quadratic forms.
基金Supported by National Natural Science Foundation of China （Grant N0s. 10926109 and 11001268） and Chinese Universities Scientific Fund （2009JS32 and 2009-2-05）
文摘Let （M,g（t））, 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative Ricci c for some constant C 〉 0 and g（t） evolving by the Ricci flow curvature, │Rc│ ≤C/t for some constant C 〉 0 and g（t） evolving by the Ricci flow gij/ t=-2Rij.In this paper, we derive a differential Harnack estimate for positive solutions to parabolic equations of the type u~ = /△u - aulogu - bu on M x （0,T], where a 〉 0 and b ∈ R.
