For discrete spectrum of 1D second-order differential/difference operators(with or without potential(killing),with the maximal/minimal domain),a pair of unified dual criteria are presented in terms of two explicit mea...For discrete spectrum of 1D second-order differential/difference operators(with or without potential(killing),with the maximal/minimal domain),a pair of unified dual criteria are presented in terms of two explicit measures and the harmonic function of the operators.Interestingly,these criteria can be read out from the ones for the exponential convergence of four types of stability studied earlier,simply replacing the‘finite supremum’by‘vanishing at infinity’.Except a dual technique,the main tool used here is a transform in terms of the harmonic function,to which two new practical algorithms are introduced in the discrete context and two successive approximation schemes are reviewed in the continuous context.All of them are illustrated by examples.The main body of the paper is devoted to the hard part of the story,the easier part but powerful one is delayed to the end of the paper.展开更多
Based on a series of recent papers, a powerful algorithm is reformulated for computing the maximal eigenpair of self-adjoint complex tridiagonal matrices. In parallel, the same problem in a particular case for computi...Based on a series of recent papers, a powerful algorithm is reformulated for computing the maximal eigenpair of self-adjoint complex tridiagonal matrices. In parallel, the same problem in a particular case for computing the sub-maximal eigenpair is also introduced. The key ideas for each critical improvement are explained. To illustrate the present algorithm and compare it with the related algorithms, more than 10 examples are included.展开更多
This paper is a continuation of our previous work [Front. Math. China, 2016, 11（6）： 1379-1418] where an efficient algorithm for computing the maximal eigenpalr was introduced first for tridiagonal matrices and then...This paper is a continuation of our previous work [Front. Math. China, 2016, 11（6）： 1379-1418] where an efficient algorithm for computing the maximal eigenpalr was introduced first for tridiagonal matrices and then extended to the irreducible matrices with nonnegative off-diagonal elements. This paper introduces mainly two global algorithms for computing the maximal eigenpair in a rather general setup, including even a class of real （with some negative off-diagonal elements） or complex matrices.展开更多
This paper introduces some efficient initials for a well-known algorithm （an inverse iteration） for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algori...This paper introduces some efficient initials for a well-known algorithm （an inverse iteration） for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algorithm but are also unexpectedly efficient. The initials presented here are based on our analytic estimates of the maximal eigenvalue and a mimic of its eigenvector for many years of accumulation in the study of stochastic stability speed. In parallel, the same problem for computing the next to the maximal eigenpair is also studied.展开更多
For a large class of integral operators or second-order differential operators,their isospectral(or cospectral)operators are constructed explicitly in terms of htransform(duality).This provides us a simple way to exte...For a large class of integral operators or second-order differential operators,their isospectral(or cospectral)operators are constructed explicitly in terms of htransform(duality).This provides us a simple way to extend the known knowledge on the spectrum(or the estimation of the principal eigenvalue)from a smaller class of operators to a much larger one.In particular,an open problem about the positivity of the principal eigenvalue for birth–death processes is solved in the paper.展开更多
This paper is a continuation of our previous paper[Front.Math.China,2017,12(5):10231043]where global algorithms for computing the maximal cigcnpair were introduced in a rather general setup.The efficiency of the globa...This paper is a continuation of our previous paper[Front.Math.China,2017,12(5):10231043]where global algorithms for computing the maximal cigcnpair were introduced in a rather general setup.The efficiency of the global algorithms is improved in this paper in terms of a good use of power iteration and two quasi-symmetric techniques.Finally,the new algorithms are applied to Hua’s economic optimization model.展开更多
For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case （i.e., the ergodic case） is even harder, and wa...For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case （i.e., the ergodic case） is even harder, and was deduced from the Dirichlet one plus a use of duality and the coupling method. In this paper, an alternative and more direct proof for the basic estimates is presented. The estimates in the Dirichlet case are then improved by a typical application of a recent variational formula. As a dual of the Dirichlet case, the refine problem for bilateral Neumann boundary condition is also treated. The paper starts with the continuous case （one-dimensional diffusions） and ends at the discrete one （birth-death processes）. Possible generalization of the results studied here is discussed at the end of the paper展开更多
This paper is devoted to the study on the spectrum of Hermitizable tridiagonal matrices.As an illustration of the application of the author’s recent results on Hermitizable matrices,an explicit criterion for discrete...This paper is devoted to the study on the spectrum of Hermitizable tridiagonal matrices.As an illustration of the application of the author’s recent results on Hermitizable matrices,an explicit criterion for discrete spectrum of the matrices is presented,with a slight and technical restriction.The problem is well known,but from the author’s knowledge,it has been largely opened for quite a long time.It is important in various application,in quantum mechanics for instance.The main tool to solve the problem is the isospectral technique developed a few years ago.Two alternative constructions of the isospectral operator are presented;they are helpful in theoretical analysis and in numerical computations,respectively.Some illustrated examples are included.展开更多
This paper deals with the principal eigenvalue of discrete p-Laplacian on the set of nonnegative integers. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequalities. The main goal is...This paper deals with the principal eigenvalue of discrete p-Laplacian on the set of nonnegative integers. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequalities. The main goal is the quantitative estimates of the eigenvalue. The paper begins with the case having reflecting boundary at origin and absorbing boundary at infinity. Several variational formulas are presented in different formulation： the difference form, the single summation form, and the double summation form. As their applications, some explicit lower and upper estimates, a criterion for positivity （which was known years ago）, as well as an approximating procedure for the eigenvalue are obtained. Similarly, the dual case having absorbing boundary at origin and reflecting boundary at presented at the end of Section 2 to infinity is also studied. Two examples are illustrate the value of the investigation.展开更多
The top eigenpairs at the title mean the maximal, the submaximal, or a few of the subsequent eigenpairs of an Hermitizable matrix. Restricting on top ones is to handle with the matrices having large scale, for which o...The top eigenpairs at the title mean the maximal, the submaximal, or a few of the subsequent eigenpairs of an Hermitizable matrix. Restricting on top ones is to handle with the matrices having large scale, for which only little is known up to now. This is different from some mature algorithms, that are clearly limited only to medium-sized matrix for calculating full spectrum. It is hoped that a combination of this paper with the earlier works, to be seen soon, may provide some effective algorithms for computing the spectrum in practice, especially for matrix mechanics.展开更多
The mixed principal eigenvalue of p-Laplacian （equivalently, the optimal constant of weighted Hardy inequality in Lp space） is studied in this paper. Several variational formulas for the eigenvalue are presented. As...The mixed principal eigenvalue of p-Laplacian （equivalently, the optimal constant of weighted Hardy inequality in Lp space） is studied in this paper. Several variational formulas for the eigenvalue are presented. As applications of the formulas, a criterion for the positivity of the eigenvalue is obtained. Furthermore, an approximating procedure and some explicit estimates are presented case by case. An example is included to illustrate the power of the results of the paper.展开更多
To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by t...To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is shown that the sharp factor is meaningful for each finite interval and a classical sharp model is re-examined.展开更多
The first aim of the paper is to study the Hermitizability of secondorder differential operators,and then the corresponding isospectral operators.The explicit criteria for the Hermitizable or isospectral properties ar...The first aim of the paper is to study the Hermitizability of secondorder differential operators,and then the corresponding isospectral operators.The explicit criteria for the Hermitizable or isospectral properties are presented.The second aim of the paper is to study a non-Hermitian model,which is now well known.In a regular sense,the model does not belong to the class of Hermitizable operators studied in this paper,but we will use the theory developed in the past years,to present an alternative and illustrated proof of the discreteness of its spectrum.The harmonic function plays a critical role in the study of spectrum.Two constructions of the function are presented.The required conclusion for the discrete spectrum is proved by some comparison technique.展开更多
This paper is a continuation of the author＇s previous papers [Front. Math. China, 2016, 11（6）： 1379-1418; 2017, 12（5）： 1023-1043], where the linear case was studied. A shifted inverse iteration algorithm is int...This paper is a continuation of the author＇s previous papers [Front. Math. China, 2016, 11（6）： 1379-1418; 2017, 12（5）： 1023-1043], where the linear case was studied. A shifted inverse iteration algorithm is introduced, as an acceleration of the inverse iteration which is often used in the non-linear context （the p-Laplacian operators for instance）. Even though the algorithm is formally similar to the Rayleigh quotient iteration which is well-known in the linear situation, but they are essentially different. The point is that the standard Rayleigh quotient cannot be used as a shift in the non-linear setup. We have to employ a different quantity which has been obtained only recently. As a surprised gift, the explicit formulas for the algorithm restricted to the linear case （p = 2） is obtained, which improves the author＇s approximating procedure for the leading eigenvalues in different context, appeared in a group of publications. The paper begins with p-Laplacian, and is closed by the non-linear operators corresponding to the well-known Hardy-type inequalities.展开更多
The ＂Workshop on Markov Processes and Related Topics＂ were held nine times since 2002, organized mainly by the Probability Group in the Research Center for Stochastics, Beijing Normal University. The tenth will be h...The ＂Workshop on Markov Processes and Related Topics＂ were held nine times since 2002, organized mainly by the Probability Group in the Research Center for Stochastics, Beijing Normal University. The tenth will be held in the coming August, 2014. This workshop happens almost in each year except 2003 （cancelled because of SARS）, and 2009 （cancelled because too many related conferences were held in that year）. In 2005, the ard workshop is merged into the Joint Meeting of CSPS/IMS. The research group was mainly supported by the National Natural Science Foundation of China （NSFC） in the period of 2002-2010, and by the Ministry of Education （985 project） during 2012-2014. Among the ten workshops, five of them were or will be co-organized with some universities out of Beijing.展开更多
基金The author thanks S.Kotani for introducing[7]and[9]to him and R.O˘ınarov for sending him the original version of[12].Thanks are also given to H.J.Zhang and Z.W.Liao for their corrections of an earlier version of the paper.Research supported in part by the National Natural Science Foundation of China(No.11131003)the“985”project from the Ministry of Education in China,and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions。
文摘For discrete spectrum of 1D second-order differential/difference operators(with or without potential(killing),with the maximal/minimal domain),a pair of unified dual criteria are presented in terms of two explicit measures and the harmonic function of the operators.Interestingly,these criteria can be read out from the ones for the exponential convergence of four types of stability studied earlier,simply replacing the‘finite supremum’by‘vanishing at infinity’.Except a dual technique,the main tool used here is a transform in terms of the harmonic function,to which two new practical algorithms are introduced in the discrete context and two successive approximation schemes are reviewed in the continuous context.All of them are illustrated by examples.The main body of the paper is devoted to the hard part of the story,the easier part but powerful one is delayed to the end of the paper.
基金supported in part the National Natural Science Foundation of China (Grant No. 11771046)the Project from the Ministry of Education in Chinathe Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘Based on a series of recent papers, a powerful algorithm is reformulated for computing the maximal eigenpair of self-adjoint complex tridiagonal matrices. In parallel, the same problem in a particular case for computing the sub-maximal eigenpair is also introduced. The key ideas for each critical improvement are explained. To illustrate the present algorithm and compare it with the related algorithms, more than 10 examples are included.
文摘This paper is a continuation of our previous work [Front. Math. China, 2016, 11（6）： 1379-1418] where an efficient algorithm for computing the maximal eigenpalr was introduced first for tridiagonal matrices and then extended to the irreducible matrices with nonnegative off-diagonal elements. This paper introduces mainly two global algorithms for computing the maximal eigenpair in a rather general setup, including even a class of real （with some negative off-diagonal elements） or complex matrices.
基金Acknowledgements The main results of the paper have been reported at Anhui Normal University, Jiangsu Normal University, the International Workshop on SDEs and Numerical Methods at Shanghai Normal University, Workshop on Markov Processes and Their Applications at Hunan University of Arts and Science, and Workshop of Probability Theory with Applications at University of Macao. The author acknowledges Professors Dong-Jin Zhu, Wan-Ding Ding, Ying-Chao Xie, Xue-Rong Mao, Xiang-Qun Yang, Xu-Yan Xiang, Jie Xiong, Li-Hu Xu, and their teams for very warm hospitality and financial support. The author also thanks Ms. Yue-Shuang Li for her assistance in computing large matrices. This work was supported in part by the National Natural Science Foundation of China （Grant No. 11131003）, the ＂985＂ project from the Ministry of Education in China, and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘This paper introduces some efficient initials for a well-known algorithm （an inverse iteration） for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algorithm but are also unexpectedly efficient. The initials presented here are based on our analytic estimates of the maximal eigenvalue and a mimic of its eigenvector for many years of accumulation in the study of stochastic stability speed. In parallel, the same problem for computing the next to the maximal eigenpair is also studied.
基金The results of the paper were presented several times in our seminar,from which the authors are benefited a lot from the discussions and suggestions.Research supported in part by the National Natural Science Foundation of China(No.11131003)the“985”project from the Ministry of Education in China,and the Project Funded by the Priority Academic Program Development of JiangsuHigher Education Institutions.
文摘For a large class of integral operators or second-order differential operators,their isospectral(or cospectral)operators are constructed explicitly in terms of htransform(duality).This provides us a simple way to extend the known knowledge on the spectrum(or the estimation of the principal eigenvalue)from a smaller class of operators to a much larger one.In particular,an open problem about the positivity of the principal eigenvalue for birth–death processes is solved in the paper.
基金This work was supported in part by the National Natural Science Foundation of China(Grant No.11771046)the Project from the Ministry of Education in China,and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘This paper is a continuation of our previous paper[Front.Math.China,2017,12(5):10231043]where global algorithms for computing the maximal cigcnpair were introduced in a rather general setup.The efficiency of the global algorithms is improved in this paper in terms of a good use of power iteration and two quasi-symmetric techniques.Finally,the new algorithms are applied to Hua’s economic optimization model.
基金This work was supported in part by the National Natural Science Foundation of China （Grant No. 11131003） and the ＂985＂ Project from the Ministry of Education in China.
文摘For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case （i.e., the ergodic case） is even harder, and was deduced from the Dirichlet one plus a use of duality and the coupling method. In this paper, an alternative and more direct proof for the basic estimates is presented. The estimates in the Dirichlet case are then improved by a typical application of a recent variational formula. As a dual of the Dirichlet case, the refine problem for bilateral Neumann boundary condition is also treated. The paper starts with the continuous case （one-dimensional diffusions） and ends at the discrete one （birth-death processes）. Possible generalization of the results studied here is discussed at the end of the paper
基金This work was supported in part by the National Natural Science Foundation of China(Grant No.11771046),the project from the Ministry of Education in China,and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘This paper is devoted to the study on the spectrum of Hermitizable tridiagonal matrices.As an illustration of the application of the author’s recent results on Hermitizable matrices,an explicit criterion for discrete spectrum of the matrices is presented,with a slight and technical restriction.The problem is well known,but from the author’s knowledge,it has been largely opened for quite a long time.It is important in various application,in quantum mechanics for instance.The main tool to solve the problem is the isospectral technique developed a few years ago.Two alternative constructions of the isospectral operator are presented;they are helpful in theoretical analysis and in numerical computations,respectively.Some illustrated examples are included.
基金Acknowledgements The authors would like to thank Professors Yonghua Mao and Yutao Ma for their helpful comments and suggestions. This work was supported in part by the National Natural Science Foundation of China （Grant No. 11131003）, the Specialized Research Fund for the Doctoral Program of Higher Education （Grant No. 20100003110005）, the ＇985＇ project from the Ministry of Education in China, and the Fundamental Research Funds for the Central Universities.
文摘This paper deals with the principal eigenvalue of discrete p-Laplacian on the set of nonnegative integers. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequalities. The main goal is the quantitative estimates of the eigenvalue. The paper begins with the case having reflecting boundary at origin and absorbing boundary at infinity. Several variational formulas are presented in different formulation： the difference form, the single summation form, and the double summation form. As their applications, some explicit lower and upper estimates, a criterion for positivity （which was known years ago）, as well as an approximating procedure for the eigenvalue are obtained. Similarly, the dual case having absorbing boundary at origin and reflecting boundary at presented at the end of Section 2 to infinity is also studied. Two examples are illustrate the value of the investigation.
基金This work was supported in part by the National Natural Science Foundation of China(Grant Nos.12090011,11771046,11771188,11771189)the National Key R&D Program of China(No.2020YFA0712900)+1 种基金the Natural Science Foundation of Jiangsu Province(Grant No.BK20171162)the project from the Ministry of Education in China,and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘The top eigenpairs at the title mean the maximal, the submaximal, or a few of the subsequent eigenpairs of an Hermitizable matrix. Restricting on top ones is to handle with the matrices having large scale, for which only little is known up to now. This is different from some mature algorithms, that are clearly limited only to medium-sized matrix for calculating full spectrum. It is hoped that a combination of this paper with the earlier works, to be seen soon, may provide some effective algorithms for computing the spectrum in practice, especially for matrix mechanics.
基金Acknowledgements The authors would like to thank Professor Yonghua Mao for his helpful comments and suggestions. This work was supported in part by the National Natural Science Foundation of China （Grant No. 11131003）, the Specialized Research Fund for the Doctoral Program of Higher Education （Grant No. 20100003110005）, the ＇985＇ project from the Ministry of Education in China, and the Fundamental Research Funds for the Central Universities.
文摘The mixed principal eigenvalue of p-Laplacian （equivalently, the optimal constant of weighted Hardy inequality in Lp space） is studied in this paper. Several variational formulas for the eigenvalue are presented. As applications of the formulas, a criterion for the positivity of the eigenvalue is obtained. Furthermore, an approximating procedure and some explicit estimates are presented case by case. An example is included to illustrate the power of the results of the paper.
基金Supported by National Natural Science Foundation of China(Grant No.11131003)the "985" Project from the Ministry of Education in Chinathe Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions
文摘To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is shown that the sharp factor is meaningful for each finite interval and a classical sharp model is re-examined.
基金supported in part by the National Natural Science Foundation of China(Grant No.11771046)the project from the Ministry of Education in China,and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘The first aim of the paper is to study the Hermitizability of secondorder differential operators,and then the corresponding isospectral operators.The explicit criteria for the Hermitizable or isospectral properties are presented.The second aim of the paper is to study a non-Hermitian model,which is now well known.In a regular sense,the model does not belong to the class of Hermitizable operators studied in this paper,but we will use the theory developed in the past years,to present an alternative and illustrated proof of the discreteness of its spectrum.The harmonic function plays a critical role in the study of spectrum.Two constructions of the function are presented.The required conclusion for the discrete spectrum is proved by some comparison technique.
基金supported in part by the National Natural Science Foundation of China (GrantNo.11771046)the project from the Ministry of Education in China,and the Project Fundedby the Priority Academic Program Development of Jiangsu Higher Education Institutions.
基金Acknowledgements The author thanks Yue-Shuang Li＇s contribution in the earlier stage of looking for the new algorithm, especially a lot of work on computer checking. Thanks are also given to Zhong-Wei Liao for his corrections on the earlier version of the paper. The author acknowledges the referees for their careful comments and corrections. This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11626245, 11771046）, the Project from the Ministry of Education in China, and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘This paper is a continuation of the author＇s previous papers [Front. Math. China, 2016, 11（6）： 1379-1418; 2017, 12（5）： 1023-1043], where the linear case was studied. A shifted inverse iteration algorithm is introduced, as an acceleration of the inverse iteration which is often used in the non-linear context （the p-Laplacian operators for instance）. Even though the algorithm is formally similar to the Rayleigh quotient iteration which is well-known in the linear situation, but they are essentially different. The point is that the standard Rayleigh quotient cannot be used as a shift in the non-linear setup. We have to employ a different quantity which has been obtained only recently. As a surprised gift, the explicit formulas for the algorithm restricted to the linear case （p = 2） is obtained, which improves the author＇s approximating procedure for the leading eigenvalues in different context, appeared in a group of publications. The paper begins with p-Laplacian, and is closed by the non-linear operators corresponding to the well-known Hardy-type inequalities.
文摘The ＂Workshop on Markov Processes and Related Topics＂ were held nine times since 2002, organized mainly by the Probability Group in the Research Center for Stochastics, Beijing Normal University. The tenth will be held in the coming August, 2014. This workshop happens almost in each year except 2003 （cancelled because of SARS）, and 2009 （cancelled because too many related conferences were held in that year）. In 2005, the ard workshop is merged into the Joint Meeting of CSPS/IMS. The research group was mainly supported by the National Natural Science Foundation of China （NSFC） in the period of 2002-2010, and by the Ministry of Education （985 project） during 2012-2014. Among the ten workshops, five of them were or will be co-organized with some universities out of Beijing.
