In this article,we give a further survey of some progress of the applications of group actions in the complex geometry after my earlier survey around 2020,mostly related to my own interests.
An exact three-dimensional solution for stochastic chaos of I wave groups of M random internal waves governed by the Navier-Stokes equations is developed. The Helmholtz decomposition is used to expand the Dirichlet pr...An exact three-dimensional solution for stochastic chaos of I wave groups of M random internal waves governed by the Navier-Stokes equations is developed. The Helmholtz decomposition is used to expand the Dirichlet problem for the Navier-Stokes equations into the Archimedean, Stokes, and Navier problems. The exact solution is obtained with the help of the method of decomposition in invariant structures. Differential algebra is constructed for six families of random invariant structures: random scalar kinematic structures, time-complementary random scalar kinematic structures, random vector kinematic structures, time-complementary random vector kinematic structures, random scalar dynamic structures, and random vector dynamic structures. Tedious computations are performed using the experimental and theoretical programming in Maple. The random scalar and vector kinematic structures and the time-complementary random scalar and vector kinematic structures are applied to solve the Stokes problem. The random scalar and vector dynamic structures are employed to expand scalar and vector variables of the Navier problem. Potentialization of the Navier field becomes available since vortex forces, which are expressed via the vector potentials of the Helmholtz decomposition, counterbalance each other. On the contrary, potential forces, which are described by the scalar potentials of the Helmholtz decomposition, superimpose to generate the gradient of a dynamic random pressure. Various constituents of the kinetic energy are ascribed to diverse interactions of random, three-dimensional, nonlinear, internal waves with a two-fold topology, which are termed random exponential oscillons and pulsons. Quantization of the kinetic energy of stochastic chaos is developed in terms of wave structures of random elementary oscillons, random elementary pulsons, random internal, diagonal, and external elementary oscillons, random wave pulsons, random internal, diagonal, and external wave oscillons, random group pulsons, random internal, diagonal, and external group oscillons, a random energy pulson, random internal, diagonal, and external energy oscillons, and a random cumulative energy pulson.展开更多
In this article,we give a survey of some progress of the complex geometry,mostly related to the Lie group actions on compact complex manifolds and complex homogeneous spaces in the last thirty years.In particular,we e...In this article,we give a survey of some progress of the complex geometry,mostly related to the Lie group actions on compact complex manifolds and complex homogeneous spaces in the last thirty years.In particular,we explore some works in the special area in Di erential Geometry,Lie Group and Complex Homogeneous Space.Together with the special area in nonlinear analysis on complex manifolds,they are the two major aspects of my research interests.展开更多
We present in this paper a structural decomposition for linear multivariable singular systems. Such a decomposition has a distinct feature of capturing and displaying all the structural properties, such as the finite ...We present in this paper a structural decomposition for linear multivariable singular systems. Such a decomposition has a distinct feature of capturing and displaying all the structural properties, such as the finite and infinite zero structures, invertibility structures, and redundant dynamics of the given system. As its counterpart for non-singular systems, we believe that the technique is a powerful tool in solving control problems for singular systems.展开更多
In this paper, we use the theory of generalized Poisson bracket (GPB) to build the Poisson structure of three-dimensional 'frozen' systems Of Hamiltonian systems with slow time variable,and show that under pro...In this paper, we use the theory of generalized Poisson bracket (GPB) to build the Poisson structure of three-dimensional 'frozen' systems Of Hamiltonian systems with slow time variable,and show that under proper conditions, there exists an adiabatic invariant on every closed simply connected symplectic leaf for the time-dependent Hamiltonian systems. If the Hamiltonian H(p,q,τ) on these symplectic leaves are periodic with respect to τ and the frozen systems are in some sense strictly nonisochronous, then there are perpetual adiabatic invariants. To illustrate these results, we discuss the classical Lotka-Volterra equation with slowly periodic time-dependent coefficients modeling the interactions of three species.展开更多
文摘In this article,we give a further survey of some progress of the applications of group actions in the complex geometry after my earlier survey around 2020,mostly related to my own interests.
文摘An exact three-dimensional solution for stochastic chaos of I wave groups of M random internal waves governed by the Navier-Stokes equations is developed. The Helmholtz decomposition is used to expand the Dirichlet problem for the Navier-Stokes equations into the Archimedean, Stokes, and Navier problems. The exact solution is obtained with the help of the method of decomposition in invariant structures. Differential algebra is constructed for six families of random invariant structures: random scalar kinematic structures, time-complementary random scalar kinematic structures, random vector kinematic structures, time-complementary random vector kinematic structures, random scalar dynamic structures, and random vector dynamic structures. Tedious computations are performed using the experimental and theoretical programming in Maple. The random scalar and vector kinematic structures and the time-complementary random scalar and vector kinematic structures are applied to solve the Stokes problem. The random scalar and vector dynamic structures are employed to expand scalar and vector variables of the Navier problem. Potentialization of the Navier field becomes available since vortex forces, which are expressed via the vector potentials of the Helmholtz decomposition, counterbalance each other. On the contrary, potential forces, which are described by the scalar potentials of the Helmholtz decomposition, superimpose to generate the gradient of a dynamic random pressure. Various constituents of the kinetic energy are ascribed to diverse interactions of random, three-dimensional, nonlinear, internal waves with a two-fold topology, which are termed random exponential oscillons and pulsons. Quantization of the kinetic energy of stochastic chaos is developed in terms of wave structures of random elementary oscillons, random elementary pulsons, random internal, diagonal, and external elementary oscillons, random wave pulsons, random internal, diagonal, and external wave oscillons, random group pulsons, random internal, diagonal, and external group oscillons, a random energy pulson, random internal, diagonal, and external energy oscillons, and a random cumulative energy pulson.
基金the Natural Science Foundation of Henan University。
文摘In this article,we give a survey of some progress of the complex geometry,mostly related to the Lie group actions on compact complex manifolds and complex homogeneous spaces in the last thirty years.In particular,we explore some works in the special area in Di erential Geometry,Lie Group and Complex Homogeneous Space.Together with the special area in nonlinear analysis on complex manifolds,they are the two major aspects of my research interests.
文摘We present in this paper a structural decomposition for linear multivariable singular systems. Such a decomposition has a distinct feature of capturing and displaying all the structural properties, such as the finite and infinite zero structures, invertibility structures, and redundant dynamics of the given system. As its counterpart for non-singular systems, we believe that the technique is a powerful tool in solving control problems for singular systems.
文摘In this paper, we use the theory of generalized Poisson bracket (GPB) to build the Poisson structure of three-dimensional 'frozen' systems Of Hamiltonian systems with slow time variable,and show that under proper conditions, there exists an adiabatic invariant on every closed simply connected symplectic leaf for the time-dependent Hamiltonian systems. If the Hamiltonian H(p,q,τ) on these symplectic leaves are periodic with respect to τ and the frozen systems are in some sense strictly nonisochronous, then there are perpetual adiabatic invariants. To illustrate these results, we discuss the classical Lotka-Volterra equation with slowly periodic time-dependent coefficients modeling the interactions of three species.
