Matrix structuring is a very beautiful way to place Bernoulli numbers, by which a new view to the numbers is opened. Natural Numbers are mathematics seeds and Natural Number System （NNS） breeds the whole world mathe...Matrix structuring is a very beautiful way to place Bernoulli numbers, by which a new view to the numbers is opened. Natural Numbers are mathematics seeds and Natural Number System （NNS） breeds the whole world mathematically.展开更多
Within the framework of compact density matrix approach and iterative procedure, a detailed procedure for the calculation of the second-harmonic generation (SHG)susceptibility tensor is given in the electric-field-bia...Within the framework of compact density matrix approach and iterative procedure, a detailed procedure for the calculation of the second-harmonic generation (SHG)susceptibility tensor is given in the electric-field-biased parabolic and semi-parabolic quantum wells (QWs). The simple analytical formula for the SHG susceptibility in the systems is also deduced. Numerical results on typical AlGaAs/GaAs materials show that, for the same effective width,the SHG susceptibility in semi-parabolic QW is larger than that in parabolic QW due to the self-asymmetry of the semiparabolic QW, and the applied electric field can make the SHG susceptibilities in both systems enhance remarkably.Moreover, the SHG susceptibility is also related to the parabolic confinement frequency and the relaxation rate of the systems.展开更多
We extend two known dynamical systems obtained by Blaszak, et al. via choosing Casimir functions and utilizing Novikov-Lax equation so that a series of novel dynamical systems including generalized Burgers dynamical s...We extend two known dynamical systems obtained by Blaszak, et al. via choosing Casimir functions and utilizing Novikov-Lax equation so that a series of novel dynamical systems including generalized Burgers dynamical system, heat equation, and so on, are followed to be generated. Then we expand some differential operators presented in the paper to deduce two types of expanding dynamical models. By taking the generalized Burgers dynamical system as an example, we deform its expanding model to get a half-expanding system, whose recurrence operator is derived from Lax representation, and its Hamiltonian structure is also obtained by adopting a new way. Finally, we expand the generalized Burgers dynamical system to the （29-1）-dimensional case whose Hamiltonian structure is derived by Poisson tensor and gradient of the Casimir function. Besides, a kind of （29-1）-dimensional expanding dynamical model of the （29-1）-dimensionaJ dynamical system is generated as well.展开更多
文摘Matrix structuring is a very beautiful way to place Bernoulli numbers, by which a new view to the numbers is opened. Natural Numbers are mathematics seeds and Natural Number System （NNS） breeds the whole world mathematically.
文摘Within the framework of compact density matrix approach and iterative procedure, a detailed procedure for the calculation of the second-harmonic generation (SHG)susceptibility tensor is given in the electric-field-biased parabolic and semi-parabolic quantum wells (QWs). The simple analytical formula for the SHG susceptibility in the systems is also deduced. Numerical results on typical AlGaAs/GaAs materials show that, for the same effective width,the SHG susceptibility in semi-parabolic QW is larger than that in parabolic QW due to the self-asymmetry of the semiparabolic QW, and the applied electric field can make the SHG susceptibilities in both systems enhance remarkably.Moreover, the SHG susceptibility is also related to the parabolic confinement frequency and the relaxation rate of the systems.
基金Supported by the Fundamental Research Funds for the Central University under Grant No.2017XKZD11
文摘We extend two known dynamical systems obtained by Blaszak, et al. via choosing Casimir functions and utilizing Novikov-Lax equation so that a series of novel dynamical systems including generalized Burgers dynamical system, heat equation, and so on, are followed to be generated. Then we expand some differential operators presented in the paper to deduce two types of expanding dynamical models. By taking the generalized Burgers dynamical system as an example, we deform its expanding model to get a half-expanding system, whose recurrence operator is derived from Lax representation, and its Hamiltonian structure is also obtained by adopting a new way. Finally, we expand the generalized Burgers dynamical system to the （29-1）-dimensional case whose Hamiltonian structure is derived by Poisson tensor and gradient of the Casimir function. Besides, a kind of （29-1）-dimensional expanding dynamical model of the （29-1）-dimensionaJ dynamical system is generated as well.
