In the process of eliminating variables in a symbolic polynomial system,the extraneous factors are referred to the unwanted parameters of resulting polynomial.This paper aims at reducing the number of these factors vi...In the process of eliminating variables in a symbolic polynomial system,the extraneous factors are referred to the unwanted parameters of resulting polynomial.This paper aims at reducing the number of these factors via optimizing the size of Dixon matrix.An optimal configuration of Dixon matrix would lead to the enhancement of the process of computing the resultant which uses for solving polynomial systems.To do so,an optimization algorithm along with a number of new polynomials is introduced to replace the polynomials and implement a complexity analysis.Moreover,the monomial multipliers are optimally positioned to multiply each of the polynomials.Furthermore,through practical implementation and considering standard and mechanical examples the efficiency of the method is evaluated.展开更多
It is shown in this paper that any state space realization （A, b, c） of a given transfer function T（s） =β（s）/α（s）with α（s）monic and dim（A）=deg（α（s））,satisfies the identity β（A）=Qe（A,b）Sα Qo（...It is shown in this paper that any state space realization （A, b, c） of a given transfer function T（s） =β（s）/α（s）with α（s）monic and dim（A）=deg（α（s））,satisfies the identity β（A）=Qe（A,b）Sα Qo（A,c）where Qc （A,b）and Qo（A, c） are the controllability matrix and observability matrix of the matrix triple （A, b, c）, respectively, and S,~ is a nonsingular symmetric matrix. Such an identity gives a deep relationship between the state space description and the transfer function description of single-input single-output （SISO） linear systems. As a direct conclusion, we arrive at the well-known result that a realization of any transfer function is minimal if and only if the numerator and the denominator of the transfer function is coprime. Such a result is also extended to the SISO descriptor linear system case. As an applications, a complete solution to the commuting matrix equation AX --- XA is proposed and the minimal realization of multi-input multi-output （MIMO） linear system is considered.展开更多
According to quantum mechanics, the commutation property of the energy Hamiltonian with the momentum operator should give the definite values not only for energy but also for the momentum quantum levels. A difficulty ...According to quantum mechanics, the commutation property of the energy Hamiltonian with the momentum operator should give the definite values not only for energy but also for the momentum quantum levels. A difficulty provided by the standing-like boundary conditions of the electron gas is that the Hamiltonian eigenfunctions are different than eigenfunctions of the momentum operator. In results the electron momenta are obtained from the correspondence rule between the classical and quantum mechanics given by Landau and Lifshits. As a consequence the statistics of solutions representing not only the energy values but also the electron momenta should be taken into account. In the Heisenberg picture of quantum mechanics, the momenta are easily obtained because the electron oscillators are there directly considered. In fact, the Hamiltonian entering the Heisenberg method can be defined in two different ways each giving the set of the electron energies known from the Schr?dinger’s approach.展开更多
文摘In the process of eliminating variables in a symbolic polynomial system,the extraneous factors are referred to the unwanted parameters of resulting polynomial.This paper aims at reducing the number of these factors via optimizing the size of Dixon matrix.An optimal configuration of Dixon matrix would lead to the enhancement of the process of computing the resultant which uses for solving polynomial systems.To do so,an optimization algorithm along with a number of new polynomials is introduced to replace the polynomials and implement a complexity analysis.Moreover,the monomial multipliers are optimally positioned to multiply each of the polynomials.Furthermore,through practical implementation and considering standard and mechanical examples the efficiency of the method is evaluated.
基金the Chinese Outstanding Youth Foundation(No. 69925308)Program for Changjiang Scholars and Innovative Research Team in University.
文摘It is shown in this paper that any state space realization （A, b, c） of a given transfer function T（s） =β（s）/α（s）with α（s）monic and dim（A）=deg（α（s））,satisfies the identity β（A）=Qe（A,b）Sα Qo（A,c）where Qc （A,b）and Qo（A, c） are the controllability matrix and observability matrix of the matrix triple （A, b, c）, respectively, and S,~ is a nonsingular symmetric matrix. Such an identity gives a deep relationship between the state space description and the transfer function description of single-input single-output （SISO） linear systems. As a direct conclusion, we arrive at the well-known result that a realization of any transfer function is minimal if and only if the numerator and the denominator of the transfer function is coprime. Such a result is also extended to the SISO descriptor linear system case. As an applications, a complete solution to the commuting matrix equation AX --- XA is proposed and the minimal realization of multi-input multi-output （MIMO） linear system is considered.
文摘According to quantum mechanics, the commutation property of the energy Hamiltonian with the momentum operator should give the definite values not only for energy but also for the momentum quantum levels. A difficulty provided by the standing-like boundary conditions of the electron gas is that the Hamiltonian eigenfunctions are different than eigenfunctions of the momentum operator. In results the electron momenta are obtained from the correspondence rule between the classical and quantum mechanics given by Landau and Lifshits. As a consequence the statistics of solutions representing not only the energy values but also the electron momenta should be taken into account. In the Heisenberg picture of quantum mechanics, the momenta are easily obtained because the electron oscillators are there directly considered. In fact, the Hamiltonian entering the Heisenberg method can be defined in two different ways each giving the set of the electron energies known from the Schr?dinger’s approach.
