Some complete variational formulas and approximation theorems for the first eigenvalue of elliptic operators in dimension one or a class of Markov chains are presented.
We investigate Noether symmetries and conservation laws of the discrete nonconserved systems with nonregular lattices. The operators of discrete transformation and discrete differentiation to the right and left are in...We investigate Noether symmetries and conservation laws of the discrete nonconserved systems with nonregular lattices. The operators of discrete transformation and discrete differentiation to the right and left are introduced for the systems. Based on the invariance of discrete Hamilton action on nonregular lattices of the systems with the nonconserved forces under the infinitesimal transformations with respect to the time and generalized coordinates, we give the discrete analog of generalized variational formula. From this formula we derive the discrete analog of generalized Noether-type identity, and then we present the generalized quasi-extremal equations and properties of these equations for the systems. We also obtain the discrete analog of Noether-type conserved laws and the discrete analog of generalized Noether theorems for the systems. We discuss an example to illustrate these results.展开更多
We investigate Noether symmetries and conservation laws of the discrete mechanico-electrical systems with nonregular lattices.The operators of discrete transformation and discrete differentiation to the right and left...We investigate Noether symmetries and conservation laws of the discrete mechanico-electrical systems with nonregular lattices.The operators of discrete transformation and discrete differentiation to the right and left are introduced for the systems.Based on the invariance of discrete Hamilton action on nonregular lattices of the systems with the dissipation forces under the infinitesimal transformations with respect to the time,generalized coordinates and generalized charge quantities,we work out the discrete analog of the generalized variational formula.From this formula we derive the discrete analog of generalized Noether-type identity,and then we present the generalized quasi-extremal equations and properties of these equations for the systems.We also obtain the discrete analog of Noether-type conserved laws and the discrete analog of generalized Noether theorems for the systems.Finally we use an example to illustrate these results.展开更多
In author's one previous paper, the same topic was studied for onedimensional diffusions. As a continuation, this paper studies the discrete case, that is thebirth-death processes. The explicit criteria for the in...In author's one previous paper, the same topic was studied for onedimensional diffusions. As a continuation, this paper studies the discrete case, that is thebirth-death processes. The explicit criteria for the inequalities, the variational formulas andexplicit bounds of the corresponding constants in the inequalities are presented. As typicalapplications, the Nash inequalities and logarithmic Sobolev inequalities are examined.展开更多
The aim of the paper is to establish two dual variational formulas for the first Dirichlet eigenvalue of the second order elliptic operators on half-line. Some explicit bounds of the eigenvalue depending only on the c...The aim of the paper is to establish two dual variational formulas for the first Dirichlet eigenvalue of the second order elliptic operators on half-line. Some explicit bounds of the eigenvalue depending only on the coefficients of the operators are presented. Moreover, the corresponding problems in the discrete case and the higher-order eigenvalues in the continuous case are also studied.展开更多
Motivated from the study on logarithmic Sobolev. Nash and other functional inequalities, the variational formulas for Poincare inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the l...Motivated from the study on logarithmic Sobolev. Nash and other functional inequalities, the variational formulas for Poincare inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examined.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 is one of a series of papers exploring the stability speed of one-dimensional stochastic processes. The present paper emphasizes on the principal eigenvalues of elliptic operators. The eigenvalue is just the best...This is one of a series of papers exploring the stability speed of one-dimensional stochastic processes. The present paper emphasizes on the principal eigenvalues of elliptic operators. The eigenvalue is just the best constant in the L2-Poincare inequality and describes the decay rate of the corresponding diffusion process. We present some variational formulas for the mixed principal eigenvalues of the operators. As applications of these formulas, we obtain case by case explicit estimates, a criterion for positivity, and an approximating procedure for the eigenvalue.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
This paper describes a theoretical method for reconstruction of the skin friction topology in complex separated flows,which is developed based on the exact relation between skin friction and surface pressure through t...This paper describes a theoretical method for reconstruction of the skin friction topology in complex separated flows,which is developed based on the exact relation between skin friction and surface pressure through the boundary enstrophy flux(BEF).The key of this method is that a skin friction field is reconstructed from a surface pressure field as an inverse problem by applying a variational method.For applications,the approximate method is proposed,where the composite surface pressure field is given by a linear superposition of the base-flow surface pressure field and the surface pressure variation field and the base-flow BEF field is used as the first-order approximation.This approximate method is constructive in a mathematical sense since a complex skin friction field in separated flows can be reconstructed from some elemental skin friction structures(skin friction source/sink,vortex and their combinations)by a linear superposition of some simple surface pressure structures.The distinct topological features,such as critical points,separation lines and attachment lines,naturally occur as a result of such reconstruction.As examples,some elemental skin friction structures in separated flows are reconstructed in simulations,and the skin friction fields in shock-wave/boundary-layer interactions(SWBLIs)are reconstructed from pressure sensitive paint(PSP)images obtained in wind tunnel experiments.展开更多
A variational formula for the asymptotic variance of general Markov processes is obtained.As application,we get an upper bound of the mean exit time of reversible Markov processes,and some comparison theorems between ...A variational formula for the asymptotic variance of general Markov processes is obtained.As application,we get an upper bound of the mean exit time of reversible Markov processes,and some comparison theorems between the reversible and non-reversible diffusion processes.展开更多
By the generalized variational principle of two kinds of variables in general me-chanics,it was demonstrated that two Lagrangian classical relationships can be applied to both holonomic systems and nonholonomic system...By the generalized variational principle of two kinds of variables in general me-chanics,it was demonstrated that two Lagrangian classical relationships can be applied to both holonomic systems and nonholonomic systems. And the restriction that two Lagrangian classical relationships cannot be applied to nonholonomic systems for a long time was overcome. Then,one important formula of similar La-grangian classical relationship called the popularized Lagrangian classical rela-tionship was derived. From Vakonomic model,by two Lagrangian classical rela-tionships and the popularized Lagrangian classical relationship,the result is the same with Chetaev's model,and thus Chetaev's model and Vakonomic model were unified. Simultaneously,the Lagrangian theoretical framework of dynamics of nonholonomic system was established. By some representative examples,it was validated that the Lagrangian theoretical framework of dynamics of nonholonomic systems is right.展开更多
An optimal control problem for a controlled backward stochastic partial differential equation in the abstract evolution form with a Bolza type performance functional is considered. The control domain is not assumed to...An optimal control problem for a controlled backward stochastic partial differential equation in the abstract evolution form with a Bolza type performance functional is considered. The control domain is not assumed to be convex, and all coefficients of the system are allowed to be random. A variational formula for the functional in a given control process direction is derived, by the Hamiltonian and associated adjoint system. As an application, a global stochastic maximum principle of Pontraygins type for the optimal controls is established.展开更多
Constructing some proper functional spaces, we obtain the corresponding norm for the operator (-.L)^-1, and then, via spectral theory, we revisit two variational formulas of the spectral gap, given by M. F. Chen [Fr...Constructing some proper functional spaces, we obtain the corresponding norm for the operator (-.L)^-1, and then, via spectral theory, we revisit two variational formulas of the spectral gap, given by M. F. Chen [Front. Math. China, 2010, 5(3): 379-515], for transient birth-death processes.展开更多
In this paper,we introduce a kind of submanifold called translating solitons with density,and obtain two variational formulas for it,and show some geometric quantities of it.
基金This work was supported in part bythe National Natural Science Foundation of China (Grant No. 19631060) Mathematical Tian Yuan Foundation, Qiu Shi Science & Technology Foundation, RFDP and MCEC.
文摘Some complete variational formulas and approximation theorems for the first eigenvalue of elliptic operators in dimension one or a class of Markov chains are presented.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10672143 and 60575055)
文摘We investigate Noether symmetries and conservation laws of the discrete nonconserved systems with nonregular lattices. The operators of discrete transformation and discrete differentiation to the right and left are introduced for the systems. Based on the invariance of discrete Hamilton action on nonregular lattices of the systems with the nonconserved forces under the infinitesimal transformations with respect to the time and generalized coordinates, we give the discrete analog of generalized variational formula. From this formula we derive the discrete analog of generalized Noether-type identity, and then we present the generalized quasi-extremal equations and properties of these equations for the systems. We also obtain the discrete analog of Noether-type conserved laws and the discrete analog of generalized Noether theorems for the systems. We discuss an example to illustrate these results.
基金supported by the National Natural Science Foundation of China (Grant Nos 10672143 and 60575055)
文摘We investigate Noether symmetries and conservation laws of the discrete mechanico-electrical systems with nonregular lattices.The operators of discrete transformation and discrete differentiation to the right and left are introduced for the systems.Based on the invariance of discrete Hamilton action on nonregular lattices of the systems with the dissipation forces under the infinitesimal transformations with respect to the time,generalized coordinates and generalized charge quantities,we work out the discrete analog of the generalized variational formula.From this formula we derive the discrete analog of generalized Noether-type identity,and then we present the generalized quasi-extremal equations and properties of these equations for the systems.We also obtain the discrete analog of Noether-type conserved laws and the discrete analog of generalized Noether theorems for the systems.Finally we use an example to illustrate these results.
文摘In author's one previous paper, the same topic was studied for onedimensional diffusions. As a continuation, this paper studies the discrete case, that is thebirth-death processes. The explicit criteria for the inequalities, the variational formulas andexplicit bounds of the corresponding constants in the inequalities are presented. As typicalapplications, the Nash inequalities and logarithmic Sobolev inequalities are examined.
基金supported in part by 973 Projectthe National Natural Science Foundation of China(Grant Nos.10121101,10101003)and RFDP.
文摘The aim of the paper is to establish two dual variational formulas for the first Dirichlet eigenvalue of the second order elliptic operators on half-line. Some explicit bounds of the eigenvalue depending only on the coefficients of the operators are presented. Moreover, the corresponding problems in the discrete case and the higher-order eigenvalues in the continuous case are also studied.
基金Research supported in part by NSFC (No. 10121101)973 ProjectRFDP
文摘Motivated from the study on logarithmic Sobolev. Nash and other functional inequalities, the variational formulas for Poincare inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examined.
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Acknowledgements 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 is one of a series of papers exploring the stability speed of one-dimensional stochastic processes. The present paper emphasizes on the principal eigenvalues of elliptic operators. The eigenvalue is just the best constant in the L2-Poincare inequality and describes the decay rate of the corresponding diffusion process. We present some variational formulas for the mixed principal eigenvalues of the operators. As applications of these formulas, we obtain case by case explicit estimates, a criterion for positivity, and an approximating procedure for the eigenvalue.
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.
文摘This paper describes a theoretical method for reconstruction of the skin friction topology in complex separated flows,which is developed based on the exact relation between skin friction and surface pressure through the boundary enstrophy flux(BEF).The key of this method is that a skin friction field is reconstructed from a surface pressure field as an inverse problem by applying a variational method.For applications,the approximate method is proposed,where the composite surface pressure field is given by a linear superposition of the base-flow surface pressure field and the surface pressure variation field and the base-flow BEF field is used as the first-order approximation.This approximate method is constructive in a mathematical sense since a complex skin friction field in separated flows can be reconstructed from some elemental skin friction structures(skin friction source/sink,vortex and their combinations)by a linear superposition of some simple surface pressure structures.The distinct topological features,such as critical points,separation lines and attachment lines,naturally occur as a result of such reconstruction.As examples,some elemental skin friction structures in separated flows are reconstructed in simulations,and the skin friction fields in shock-wave/boundary-layer interactions(SWBLIs)are reconstructed from pressure sensitive paint(PSP)images obtained in wind tunnel experiments.
基金Supported by NSFC(Grant No.11901096)NSF-Fujian(Grant No.2020J05036)+3 种基金the Program for Probability and Statistics:Theory and Application(Grant No.IRTL1704)the Program for Innovative Research Team in Science and Technology in Fujian Province University(IRTSTFJ)the National Key R&D Program of China(2020YFA0712900,2020YFA0712901)the National Natural Science Foundation of China(Grant No.11771047)。
文摘A variational formula for the asymptotic variance of general Markov processes is obtained.As application,we get an upper bound of the mean exit time of reversible Markov processes,and some comparison theorems between the reversible and non-reversible diffusion processes.
基金Supported by the National Natural Science Foundation of China (Grant No. 10272034)the Research Fund for the Doctoral Program of Higher Education of Chinathe Basic Research Foundation of Harbin Engineering University (Grant No. 20060217020)
文摘By the generalized variational principle of two kinds of variables in general me-chanics,it was demonstrated that two Lagrangian classical relationships can be applied to both holonomic systems and nonholonomic systems. And the restriction that two Lagrangian classical relationships cannot be applied to nonholonomic systems for a long time was overcome. Then,one important formula of similar La-grangian classical relationship called the popularized Lagrangian classical rela-tionship was derived. From Vakonomic model,by two Lagrangian classical rela-tionships and the popularized Lagrangian classical relationship,the result is the same with Chetaev's model,and thus Chetaev's model and Vakonomic model were unified. Simultaneously,the Lagrangian theoretical framework of dynamics of nonholonomic system was established. By some representative examples,it was validated that the Lagrangian theoretical framework of dynamics of nonholonomic systems is right.
基金Supported by the National Natural Science Foundation of China(11101140,11301177)the China Postdoctoral Science Foundation(2011M500721,2012T50391)the Zhejiang Natural Science Foundation of China(Y6110775,Y6110789)
文摘An optimal control problem for a controlled backward stochastic partial differential equation in the abstract evolution form with a Bolza type performance functional is considered. The control domain is not assumed to be convex, and all coefficients of the system are allowed to be random. A variational formula for the functional in a given control process direction is derived, by the Hamiltonian and associated adjoint system. As an application, a global stochastic maximum principle of Pontraygins type for the optimal controls is established.
基金Acknowledgements This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11101040, 11131003), the 985 Project, the 973 Project (No. 2011CB808000), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20100003110005), and the Fundamental Research Funds for the Central Universities.
文摘Constructing some proper functional spaces, we obtain the corresponding norm for the operator (-.L)^-1, and then, via spectral theory, we revisit two variational formulas of the spectral gap, given by M. F. Chen [Front. Math. China, 2010, 5(3): 379-515], for transient birth-death processes.
基金Supported partially by the key project foundation of Henan Province(Grant No.18A110014)。
文摘In this paper,we introduce a kind of submanifold called translating solitons with density,and obtain two variational formulas for it,and show some geometric quantities of it.