The theory of velocity-dependent symmetries(or Lie symmetry) and non-Noether conserved quantities are presented corresponding to both the continuous and discrete electromechanical systems.Firstly,based on the invarian...The theory of velocity-dependent symmetries(or Lie symmetry) and non-Noether conserved quantities are presented corresponding to both the continuous and discrete electromechanical systems.Firstly,based on the invariance of Lagrange-Maxwell equations under infinitesimal transformations with respect to generalized coordinates and generalized charge quantities,the definition and the determining equations of velocity-dependent symmetry are obtained for continuous electromechanical systems;the Lie's theorem and the non-Noether conserved quantity of this symmetry are produced associated with continuous electromechanical systems.Secondly,the operators of transformation and the operators of differentiation are introduced in the space of discrete variables;a series of commuting relations of discrete vector operators are defined.Thirdly,based on the invariance of discrete Lagrange-Maxwell equations under infinitesimal transformations with respect to generalized coordinates and generalized charge quantities,the definition and the determining equations of velocity-dependent symmetry are obtained associated with discrete electromechanical systems;the Lie's theorem and the non-Noether conserved quantity are proved for the discrete electromechanical systems.This paper has shown that the discrete analogue of conserved quantity can be directly demonstrated by the commuting relation of discrete vector operators.Finally,an example is discussed to illustrate the results.展开更多
Porous tetragonal BaTiO 3 ceramic was successfully prepared by a combination of hydrothermal and low-temperature-sintering method.The hollow TiO2@BaCO 3 as the sintering precursor was synthesized via a simple hydrothe...Porous tetragonal BaTiO 3 ceramic was successfully prepared by a combination of hydrothermal and low-temperature-sintering method.The hollow TiO2@BaCO 3 as the sintering precursor was synthesized via a simple hydrothermal method,and then porous BaTiO 3 was generated by calcining the hollow TiO2@BaCO 3 precursor at 900 ℃ without additive.The hollow TiO2@BaCO 3 structure plays two important roles in the preparing of the porous BaTiO 3 ceramic.First,the TiO2@BaCO 3 hollow structure provides high surface areas and increases the contact points between BaCO 3 and TiO2,which can reduce the sintering temperature of the BaTiO 3 ceramic.Second,the cavity of the ordered arranged TiO2@BaCO 3 hollow sphere shows important influence on the porous structure,and the pore size of the as-prepared porous BaTiO 3 ceramic can be tuned from several nanometers to hundreds nanomters by changing the sintering temperature.The formation mechanism of the porous BaTiO 3 ceramic was proposed.展开更多
基金supported by the National Natural Science Foundation of China (Grant Nos.10672143 and 11072218)
文摘The theory of velocity-dependent symmetries(or Lie symmetry) and non-Noether conserved quantities are presented corresponding to both the continuous and discrete electromechanical systems.Firstly,based on the invariance of Lagrange-Maxwell equations under infinitesimal transformations with respect to generalized coordinates and generalized charge quantities,the definition and the determining equations of velocity-dependent symmetry are obtained for continuous electromechanical systems;the Lie's theorem and the non-Noether conserved quantity of this symmetry are produced associated with continuous electromechanical systems.Secondly,the operators of transformation and the operators of differentiation are introduced in the space of discrete variables;a series of commuting relations of discrete vector operators are defined.Thirdly,based on the invariance of discrete Lagrange-Maxwell equations under infinitesimal transformations with respect to generalized coordinates and generalized charge quantities,the definition and the determining equations of velocity-dependent symmetry are obtained associated with discrete electromechanical systems;the Lie's theorem and the non-Noether conserved quantity are proved for the discrete electromechanical systems.This paper has shown that the discrete analogue of conserved quantity can be directly demonstrated by the commuting relation of discrete vector operators.Finally,an example is discussed to illustrate the results.
基金supported by the National Natural Science Foundation of China (50972130,20701033,51172209,91122022)Zhejiang Environmental Protection Bureau Foundation (2011B19)China Postdoctoral Science Foundation (201003048)
文摘Porous tetragonal BaTiO 3 ceramic was successfully prepared by a combination of hydrothermal and low-temperature-sintering method.The hollow TiO2@BaCO 3 as the sintering precursor was synthesized via a simple hydrothermal method,and then porous BaTiO 3 was generated by calcining the hollow TiO2@BaCO 3 precursor at 900 ℃ without additive.The hollow TiO2@BaCO 3 structure plays two important roles in the preparing of the porous BaTiO 3 ceramic.First,the TiO2@BaCO 3 hollow structure provides high surface areas and increases the contact points between BaCO 3 and TiO2,which can reduce the sintering temperature of the BaTiO 3 ceramic.Second,the cavity of the ordered arranged TiO2@BaCO 3 hollow sphere shows important influence on the porous structure,and the pore size of the as-prepared porous BaTiO 3 ceramic can be tuned from several nanometers to hundreds nanomters by changing the sintering temperature.The formation mechanism of the porous BaTiO 3 ceramic was proposed.
