The authors generalize the works in [5] and [6] to prove a Hopf index theorem associated to a smooth section of a real vector bundle with non-isolated zero points.
A mod 2 index theorem for the twisted Signature operator on 4 q + 1 dimensional manifolds is established. This result generalizes a result of Farber and Turaev, which was proved for the case of orthogonal flat bundles...A mod 2 index theorem for the twisted Signature operator on 4 q + 1 dimensional manifolds is established. This result generalizes a result of Farber and Turaev, which was proved for the case of orthogonal flat bundles, to arbitrary real vector bundles. It also provides an analytic interpretation of the sign of the Poincare-Reidemeister scalar product defined by Farber and Turaev.展开更多
In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed＇s odd dimensional index formula. Then we extend this theorem to the noncommuta- tive geometry framework. As a c...In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed＇s odd dimensional index formula. Then we extend this theorem to the noncommuta- tive geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.展开更多
In this article, we study the nonexistence of solution with finite Morse index for the following Choquaxd type equation -△u=∫Rn｜u（y）｜p/｜x-y｜αdy｜u（x）｜p-2u（x） in RN,where N≥3,0〈α〈min {4,N}.Suppose tha...In this article, we study the nonexistence of solution with finite Morse index for the following Choquaxd type equation -△u=∫Rn｜u（y）｜p/｜x-y｜αdy｜u（x）｜p-2u（x） in RN,where N≥3,0〈α〈min {4,N}.Suppose that 2 〈 p 〈2N-α/N-2,we will show that this problem does not possess nontrivial solution with finite Morse index. While for p =2N-α/N-2,if i（u） 〈∞, then we have ∫RN∫RN｜u（x）｜p｜u（y）｜p dxdy 〈∞ and ∫RN｜△u｜2 dx=｜∫RN∫RN｜u（x）｜p/｜x-y｜a dxdy.展开更多
A two-state system was taken as an example to show the method that we derived for analyzing the instantaneous reliability index,which was an important issue in the field of reliability.The paper first described the sy...A two-state system was taken as an example to show the method that we derived for analyzing the instantaneous reliability index,which was an important issue in the field of reliability.The paper first described the system as an abstract Cauchy problem by choosing suitable operators and state spaces.Then,with the TrotterKato theorem and strong continuous semi-group theorem,the method of solving the instantaneous reliability index of the repairable system was constructed.The convergence of the method was also proved in theory in this paper.To show the effectiveness of this method,some numerical examples were given at the end of the paper.展开更多
In this paper, the class of uniform limit mappings of set-valued, strick set-contractive mappings is discussed. Furthermore, the fixed point index theory for the uniform limit mappings is established. Using the fixed ...In this paper, the class of uniform limit mappings of set-valued, strick set-contractive mappings is discussed. Furthermore, the fixed point index theory for the uniform limit mappings is established. Using the fixed point index theory, some positive fixed point theorems are proved. Our theorems generalize some results in [1,4,5,7].展开更多
Simple formulas for the number of different cyclic and dihedral necklaces containing nj beads of the j-th color, and , are derived, using the Pólya enumeration theorem.
文摘The authors generalize the works in [5] and [6] to prove a Hopf index theorem associated to a smooth section of a real vector bundle with non-isolated zero points.
基金Project partially supported by the National Natural Science Foundation of China (Grant No. 19525102)the Fok Ying-Tung Foundation and the Qiu Shi Foundation
文摘A mod 2 index theorem for the twisted Signature operator on 4 q + 1 dimensional manifolds is established. This result generalizes a result of Farber and Turaev, which was proved for the case of orthogonal flat bundles, to arbitrary real vector bundles. It also provides an analytic interpretation of the sign of the Poincare-Reidemeister scalar product defined by Farber and Turaev.
基金Supported by National Natural Science Foundation of China(Grant No.11271062)Program for New Century Excellent Talents in University(Grant No.13-0721)
文摘In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed＇s odd dimensional index formula. Then we extend this theorem to the noncommuta- tive geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.
文摘In this article, we study the nonexistence of solution with finite Morse index for the following Choquaxd type equation -△u=∫Rn｜u（y）｜p/｜x-y｜αdy｜u（x）｜p-2u（x） in RN,where N≥3,0〈α〈min {4,N}.Suppose that 2 〈 p 〈2N-α/N-2,we will show that this problem does not possess nontrivial solution with finite Morse index. While for p =2N-α/N-2,if i（u） 〈∞, then we have ∫RN∫RN｜u（x）｜p｜u（y）｜p dxdy 〈∞ and ∫RN｜△u｜2 dx=｜∫RN∫RN｜u（x）｜p/｜x-y｜a dxdy.
基金Joint Funds of the National Natural Science Foundation of China(NSAF)(No.U1430125)
文摘A two-state system was taken as an example to show the method that we derived for analyzing the instantaneous reliability index,which was an important issue in the field of reliability.The paper first described the system as an abstract Cauchy problem by choosing suitable operators and state spaces.Then,with the TrotterKato theorem and strong continuous semi-group theorem,the method of solving the instantaneous reliability index of the repairable system was constructed.The convergence of the method was also proved in theory in this paper.To show the effectiveness of this method,some numerical examples were given at the end of the paper.
文摘In this paper, the class of uniform limit mappings of set-valued, strick set-contractive mappings is discussed. Furthermore, the fixed point index theory for the uniform limit mappings is established. Using the fixed point index theory, some positive fixed point theorems are proved. Our theorems generalize some results in [1,4,5,7].
文摘Simple formulas for the number of different cyclic and dihedral necklaces containing nj beads of the j-th color, and , are derived, using the Pólya enumeration theorem.
