This note is a contribution to the application of generalized inverse of homomorphisms of modules in ring(module)theory.Using the{1}-and{2}-inverses of homomorphisms of modules,we characterize a class of rings and an ...This note is a contribution to the application of generalized inverse of homomorphisms of modules in ring(module)theory.Using the{1}-and{2}-inverses of homomorphisms of modules,we characterize a class of rings and an important class of modules respectively.展开更多
We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenva...We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is (N1,Nn) , where Ni is the i–th iprincipal minor of N=M–In , where In is the identity matrix of dimension n. In the noncommutative case, this eigenvector is (P1-1,Pn-1) , where Pi is the sum in Q《αij》 of the corresponding labels of nonempty paths starting from i and not passing through i in the complete directed graph associated to M .展开更多
This note contains three main results.Firstly,a complete solution of the Linear Non-Homogeneous Matrix Differential Equations(LNHMDEs)is presented that takes into account both the non-zero initial conditions of the ps...This note contains three main results.Firstly,a complete solution of the Linear Non-Homogeneous Matrix Differential Equations(LNHMDEs)is presented that takes into account both the non-zero initial conditions of the pseudo state and the nonzero initial conditions of the input.Secondly,in order to characterise the dynamics of the LNHMDEs correctly,some important concepts such as the state,slow state(smooth state)and fast state(impulsive state)are generalized to the LNHMDE case and the solution of the LNHMDEs is separated into the smooth(slow)response and the fast(implusive)response.As a third result,a new characterization of the impulsive free initial conditions of the LNHMDEs is given.展开更多
Let K be the Calderón-Zygmund convolution kernel on R^d(d≥2).Christ and Journé defined the commutator associated with K and a∈L~∞(R^d)by T_af(x)=p.v.∫_(R^d)K(x-y)m_x,y^a·f(y)dy,which is an extension...Let K be the Calderón-Zygmund convolution kernel on R^d(d≥2).Christ and Journé defined the commutator associated with K and a∈L~∞(R^d)by T_af(x)=p.v.∫_(R^d)K(x-y)m_x,y^a·f(y)dy,which is an extension of the classical Calderón commutator. In this paper, we show that T_a is weighted weak type(1,1) bounded with A,1 weight for d≥2.展开更多
文摘This note is a contribution to the application of generalized inverse of homomorphisms of modules in ring(module)theory.Using the{1}-and{2}-inverses of homomorphisms of modules,we characterize a class of rings and an important class of modules respectively.
文摘We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is (N1,Nn) , where Ni is the i–th iprincipal minor of N=M–In , where In is the identity matrix of dimension n. In the noncommutative case, this eigenvector is (P1-1,Pn-1) , where Pi is the sum in Q《αij》 of the corresponding labels of nonempty paths starting from i and not passing through i in the complete directed graph associated to M .
文摘This note contains three main results.Firstly,a complete solution of the Linear Non-Homogeneous Matrix Differential Equations(LNHMDEs)is presented that takes into account both the non-zero initial conditions of the pseudo state and the nonzero initial conditions of the input.Secondly,in order to characterise the dynamics of the LNHMDEs correctly,some important concepts such as the state,slow state(smooth state)and fast state(impulsive state)are generalized to the LNHMDE case and the solution of the LNHMDEs is separated into the smooth(slow)response and the fast(implusive)response.As a third result,a new characterization of the impulsive free initial conditions of the LNHMDEs is given.
基金supported by National Natural Science Foundation of China (Grant Nos. 11371057,11471033 and 11571160)Specialized Research Fund for the Doctoral Program of Higher Education (Grant No.20130003110003)the Fundamental Research Funds for the Central Universities (Grant No. 2014KJJCA10)
文摘Let K be the Calderón-Zygmund convolution kernel on R^d(d≥2).Christ and Journé defined the commutator associated with K and a∈L~∞(R^d)by T_af(x)=p.v.∫_(R^d)K(x-y)m_x,y^a·f(y)dy,which is an extension of the classical Calderón commutator. In this paper, we show that T_a is weighted weak type(1,1) bounded with A,1 weight for d≥2.
