Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and ...Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.展开更多
In this paper,a novel"composite reaching law"was explained in details:1)the equation of discrete motion for a control system;2)the design of discrete-time variable structure control.In addition,the model of ...In this paper,a novel"composite reaching law"was explained in details:1)the equation of discrete motion for a control system;2)the design of discrete-time variable structure control.In addition,the model of a three-storey shear-type building structure was used to verify the effectiveness of the discrete variable structure control method.The results of numerical example analysis of the model show that the control law can effectively reduce the peak value of seismic response of the building structure and the chattering effect of the control system.展开更多
文摘Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.
文摘In this paper,a novel"composite reaching law"was explained in details:1)the equation of discrete motion for a control system;2)the design of discrete-time variable structure control.In addition,the model of a three-storey shear-type building structure was used to verify the effectiveness of the discrete variable structure control method.The results of numerical example analysis of the model show that the control law can effectively reduce the peak value of seismic response of the building structure and the chattering effect of the control system.
