This paper deals with a Dirac operator with periodic and finite-bands potentials.Taking advantage of the commutativity of the monodromy operator and the Dirac operator, we define the Bloch functions and multiplicator ...This paper deals with a Dirac operator with periodic and finite-bands potentials.Taking advantage of the commutativity of the monodromy operator and the Dirac operator, we define the Bloch functions and multiplicator curve, which leads to the formula of DubrovinNovikov＇s type. Further, by calculation of residues on the complex sphere and via gauge transformation, we get the trace formulae of eigenfunctions corresponding to the left endpoints and right end-points of the spectral bands, respectively. As an application, we obtain a completely integrable Hamiltonian system in Liouville sense through nonlinearization of the Dirac spectral problem.展开更多
In order to sustain and increase students＇ interest in learning English, we tried to design several motivational approaches based on teaching psychology and pedagogy so as to improve our English teaching. The three o...In order to sustain and increase students＇ interest in learning English, we tried to design several motivational approaches based on teaching psychology and pedagogy so as to improve our English teaching. The three orientations were identified to be the principles for the study： 1.Inspiration out of tracing the retrogression to its source; 2.Stimulating learning through integrative internal and external drives; 3.Providing chances to get more inspiration of creativity. The study of integrative motivation was made for two years, which turned out to be efficacious in our English teaching when it was finished.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.61473332)the Natural Science Foundation of Zhejiang Province(Grant No.LQ14A010009)the Natural Science Foundation of Huzhou City(Grant No.2013YZ06)
文摘This paper deals with a Dirac operator with periodic and finite-bands potentials.Taking advantage of the commutativity of the monodromy operator and the Dirac operator, we define the Bloch functions and multiplicator curve, which leads to the formula of DubrovinNovikov＇s type. Further, by calculation of residues on the complex sphere and via gauge transformation, we get the trace formulae of eigenfunctions corresponding to the left endpoints and right end-points of the spectral bands, respectively. As an application, we obtain a completely integrable Hamiltonian system in Liouville sense through nonlinearization of the Dirac spectral problem.
文摘In order to sustain and increase students＇ interest in learning English, we tried to design several motivational approaches based on teaching psychology and pedagogy so as to improve our English teaching. The three orientations were identified to be the principles for the study： 1.Inspiration out of tracing the retrogression to its source; 2.Stimulating learning through integrative internal and external drives; 3.Providing chances to get more inspiration of creativity. The study of integrative motivation was made for two years, which turned out to be efficacious in our English teaching when it was finished.
