Almost all of the existing results on the explicit solutions of the matrix equationAX-XB= C are obtained under the condition that A and B have no eigenvalues incommon For both symmetric or skewsymmetric matrices A and...Almost all of the existing results on the explicit solutions of the matrix equationAX-XB= C are obtained under the condition that A and B have no eigenvalues incommon For both symmetric or skewsymmetric matrices A and B. we shall give outthe explicit general solutions of this equation by using the notions of eigenprojectionsThe results we obtained are applicable not only to any cases of eigenvalues regardlessof their multiplicities but also to the discussion of the general case of this equation展开更多
In this paper we study a matrix equation AX+BX=C(I)over an arbitrary skew field,and give a consistency criterion of(I)and an explicit expression of general solutions of(I).A convenient,simple and practical method of s...In this paper we study a matrix equation AX+BX=C(I)over an arbitrary skew field,and give a consistency criterion of(I)and an explicit expression of general solutions of(I).A convenient,simple and practical method of solving(I)is also given.As a particular case,we also give a simple method of finding a system of fundamental solutions of a homogeneous system of right linear equations over a skew field.展开更多
In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic (MHD) equations. Such solutions can be explicitly expr...In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic (MHD) equations. Such solutions can be explicitly expressed by appropriate formulae. Once the required matrices are chosen, solutions to the MHD equations axe directly constructed.展开更多
In this paper, we discuss a discrete time repairable queuing system with Markovian arrival process, where lifetime of server, service time and repair time of server are all discrete phase type random variables. Using...In this paper, we discuss a discrete time repairable queuing system with Markovian arrival process, where lifetime of server, service time and repair time of server are all discrete phase type random variables. Using the theory of matrix geometric solution, we give the steady state distribution of queue length and waiting time. In addition, the stable availability of the system is also provided.展开更多
It is known that the solution to a Cauchy problem of linear differential equations:x'(t)=A(t)x(t),with x(t0)=x0,can be presented by the matrix exponential as exp(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition...It is known that the solution to a Cauchy problem of linear differential equations:x'(t)=A(t)x(t),with x(t0)=x0,can be presented by the matrix exponential as exp(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition for the coefficient matrix A(t)holds:[∫_(t0)^(t)A(s)ds,A(t)]=0.A natural question is whether this is true without the commutativity condition.To give a definite answer to this question,we present two classes of illustrative examples of coefficient matrices,which satisfy the chain rule d/dt exp(∫_(t0)^(t)A(s)ds)=A(t)exp(∫_(t0)^(t)A(s)ds),but do not possess the commutativity condition.The presented matrices consist of finite-times continuously differentiable entries or smooth entries.展开更多
In this paper,we consider a GI/M/1 queue operating in a multi-phase service environment with working vacations and Bernoulli vacation interruption.Whenever the queue becomes empty,the server begins a working vacation ...In this paper,we consider a GI/M/1 queue operating in a multi-phase service environment with working vacations and Bernoulli vacation interruption.Whenever the queue becomes empty,the server begins a working vacation of random length,causing the system to move to vacation phase 0.During phase 0,the server takes service for the customers at a lower rate rather than stopping completely.When a vacation ends,if the queue is non-empty,the system switches from the phase 0 to some normal service phase i with probability qi,i=1,2,⋯,N.Moreover,we assume Bernoulli vacation interruption can happen.At a service completion instant,if there are customers in a working vacation period,vacation interruption happens with probability p,then the system switches from the phase 0 to some normal service phase i with probability qi,i=1,2,⋯,N,or the server continues the vacation with probability 1−p.Using the matrix geometric solution method,we obtain the stationary distributions for queue length at both arrival epochs and arbitrary epochs.The waiting time of an arbitrary customer is also derived.Finally,several numerical examples are presented.展开更多
In this paper, we consider a nonlinear hybrid dynamic (NHD) system to describe fedbatch culture where there is no analytical solutions and no equilibrium points. Our goal is to prove the strong stability with respec...In this paper, we consider a nonlinear hybrid dynamic (NHD) system to describe fedbatch culture where there is no analytical solutions and no equilibrium points. Our goal is to prove the strong stability with respect to initial state for the NHD system. To this end, we construct corresponding linear variational system (LVS) for the solution of the NHD system, also prove the boundedness of fundamental matrix solutions for the LVS. On this basis, the strong stability is proved by such boundedness.展开更多
Quasi-birth and death processes with block tridiagonal matrices find many applications in various areas. Neuts gave the necessary and sufficient conditions for the ordinary ergodicity and found an expression of the st...Quasi-birth and death processes with block tridiagonal matrices find many applications in various areas. Neuts gave the necessary and sufficient conditions for the ordinary ergodicity and found an expression of the stationary distribution for a class of quasi-birth and death processes. In this paper we obtain the explicit necessary and sufficient conditions for/-ergodicity and geometric ergodicity for the class of quasi-birth and death processes, and prove that they are not strongly ergodic. Keywords ergodicity, quasi-birth and death process.展开更多
We report results of a large computational 'alloy by design' study, in which the 'chemical composition-mechanical strength' space is explored for austenitic, ferritic and martensitic creep resistant steels. The ap...We report results of a large computational 'alloy by design' study, in which the 'chemical composition-mechanical strength' space is explored for austenitic, ferritic and martensitic creep resistant steels. The approach used allows simultaneously optimization of alloy composition and processing parameters based on the integration of thermodynamic, thermo-kinetics and a genetic algorithm optimization route. The nature of the optimisation depends on both the intended matrix(ferritic, martensitic or austenitic) and the desired precipitation family. The models are validated by analysing reported strengths of existing steels. All newly designed alloys are predicted to outperform existing high end reference grades.展开更多
文摘Almost all of the existing results on the explicit solutions of the matrix equationAX-XB= C are obtained under the condition that A and B have no eigenvalues incommon For both symmetric or skewsymmetric matrices A and B. we shall give outthe explicit general solutions of this equation by using the notions of eigenprojectionsThe results we obtained are applicable not only to any cases of eigenvalues regardlessof their multiplicities but also to the discussion of the general case of this equation
文摘In this paper we study a matrix equation AX+BX=C(I)over an arbitrary skew field,and give a consistency criterion of(I)and an explicit expression of general solutions of(I).A convenient,simple and practical method of solving(I)is also given.As a particular case,we also give a simple method of finding a system of fundamental solutions of a homogeneous system of right linear equations over a skew field.
文摘In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic (MHD) equations. Such solutions can be explicitly expressed by appropriate formulae. Once the required matrices are chosen, solutions to the MHD equations axe directly constructed.
文摘In this paper, we discuss a discrete time repairable queuing system with Markovian arrival process, where lifetime of server, service time and repair time of server are all discrete phase type random variables. Using the theory of matrix geometric solution, we give the steady state distribution of queue length and waiting time. In addition, the stable availability of the system is also provided.
基金supported in part by the Established Researcher Grant and the CAS Faculty Development Grant of the University of South Florida,Chunhui Plan of the Ministry of Education of China,Wang Kuancheng Foundation,the National Natural Science Foundation of China(Grant Nos.10332030,10472091 and 10502042)the Doctorate Foundation of Northwestern Polytechnical University(Grant No.CX200616).
文摘It is known that the solution to a Cauchy problem of linear differential equations:x'(t)=A(t)x(t),with x(t0)=x0,can be presented by the matrix exponential as exp(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition for the coefficient matrix A(t)holds:[∫_(t0)^(t)A(s)ds,A(t)]=0.A natural question is whether this is true without the commutativity condition.To give a definite answer to this question,we present two classes of illustrative examples of coefficient matrices,which satisfy the chain rule d/dt exp(∫_(t0)^(t)A(s)ds)=A(t)exp(∫_(t0)^(t)A(s)ds),but do not possess the commutativity condition.The presented matrices consist of finite-times continuously differentiable entries or smooth entries.
基金the National Natural Science Foundation of China(No.61773014)。
文摘In this paper,we consider a GI/M/1 queue operating in a multi-phase service environment with working vacations and Bernoulli vacation interruption.Whenever the queue becomes empty,the server begins a working vacation of random length,causing the system to move to vacation phase 0.During phase 0,the server takes service for the customers at a lower rate rather than stopping completely.When a vacation ends,if the queue is non-empty,the system switches from the phase 0 to some normal service phase i with probability qi,i=1,2,⋯,N.Moreover,we assume Bernoulli vacation interruption can happen.At a service completion instant,if there are customers in a working vacation period,vacation interruption happens with probability p,then the system switches from the phase 0 to some normal service phase i with probability qi,i=1,2,⋯,N,or the server continues the vacation with probability 1−p.Using the matrix geometric solution method,we obtain the stationary distributions for queue length at both arrival epochs and arbitrary epochs.The waiting time of an arbitrary customer is also derived.Finally,several numerical examples are presented.
基金This work was supported by the National Science Foundation for the Youth of China (Grant Nos. 11501574, 11401073 and 11701063), the National Natural Science Foundation of China (Grant Nos. 11771008, 61673083 and 61773086), the National Science Foundation for the Tianyuan of China (Grant No. 11626053), the Natural Science Foundation of Shandong Province in China (Grant No.: ZR2015FM014, ZR2015AL010 and ZR2017MA005), the Fundamental Research Funds for the Cen- tral Universities in China (Grant No. DUT16LK07) and the Project funded by China Postdoctoral Science Foundation (Grant No. 2016M601296).
文摘In this paper, we consider a nonlinear hybrid dynamic (NHD) system to describe fedbatch culture where there is no analytical solutions and no equilibrium points. Our goal is to prove the strong stability with respect to initial state for the NHD system. To this end, we construct corresponding linear variational system (LVS) for the solution of the NHD system, also prove the boundedness of fundamental matrix solutions for the LVS. On this basis, the strong stability is proved by such boundedness.
基金partially supported by NSFC(No.10171009)Research Fund for PhD Programs of MOE of China(No.20010533001)Research Fund for Educational Innovation for Doctorates of CSU(No.030602)
文摘Quasi-birth and death processes with block tridiagonal matrices find many applications in various areas. Neuts gave the necessary and sufficient conditions for the ordinary ergodicity and found an expression of the stationary distribution for a class of quasi-birth and death processes. In this paper we obtain the explicit necessary and sufficient conditions for/-ergodicity and geometric ergodicity for the class of quasi-birth and death processes, and prove that they are not strongly ergodic. Keywords ergodicity, quasi-birth and death process.
基金financial support from the Chinese Scholarship Council(CSC)and internal funding of TU Delft
文摘We report results of a large computational 'alloy by design' study, in which the 'chemical composition-mechanical strength' space is explored for austenitic, ferritic and martensitic creep resistant steels. The approach used allows simultaneously optimization of alloy composition and processing parameters based on the integration of thermodynamic, thermo-kinetics and a genetic algorithm optimization route. The nature of the optimisation depends on both the intended matrix(ferritic, martensitic or austenitic) and the desired precipitation family. The models are validated by analysing reported strengths of existing steels. All newly designed alloys are predicted to outperform existing high end reference grades.