Glaucoma is a group of eye diseases characterized by progressive loss of retinal ganglion cells(RGCs)and optic nerve degeneration.During this process,the visual field is reduced,and blindness may ultimately occur.Worl...Glaucoma is a group of eye diseases characterized by progressive loss of retinal ganglion cells(RGCs)and optic nerve degeneration.During this process,the visual field is reduced,and blindness may ultimately occur.Worldwide,glaucoma is the second leading cause of blindness,with about 80 million people affected.Glaucoma is a multifactorial disease and due to its complexity,the exact pathomechanisms are not fully understood yet.However,different risk factors,such as elevated intraocular pressure(IOP),age,or myopia,have been identified to date(EGS,2021).展开更多
Due to the novel applications of flexible pipes conveying fluid in the field of soft robotics and biomedicine,the investigations on the mechanical responses of the pipes have attracted considerable attention.The fluid...Due to the novel applications of flexible pipes conveying fluid in the field of soft robotics and biomedicine,the investigations on the mechanical responses of the pipes have attracted considerable attention.The fluid-structure interaction(FSI)between the pipe with a curved shape and the time-varying internal fluid flow brings a great challenge to the revelation of the dynamical behaviors of flexible pipes,especially when the pipe is highly flexible and usually undergoes large deformations.In this work,the geometrically exact model(GEM)for a curved cantilevered pipe conveying pulsating fluid is developed based on the extended Hamilton's principle.The stability of the curved pipe with three different subtended angles is examined with the consideration of steady fluid flow.Specific attention is concentrated on the large-deformation resonance of circular pipes conveying pulsating fluid,which is often encountered in practical engineering.By constructing bifurcation diagrams,oscillating shapes,phase portraits,time traces,and Poincarémaps,the dynamic responses of the curved pipe under various system parameters are revealed.The mean flow velocity of the pulsating fluid is chosen to be either subcritical or supercritical.The numerical results show that the curved pipe conveying pulsating fluid can exhibit rich dynamical behaviors,including periodic and quasi-periodic motions.It is also found that the preferred instability type of a cantilevered curved pipe conveying steady fluid is mainly in the flutter of the second mode.For a moderate value of the mass ratio,however,a third-mode flutter may occur,which is quite different from that of a straight pipe system.展开更多
This paper considers the rational expectations model with multiplicative noise and input delay,where the system dynamics rely on the conditional expectations of future states.The main contribution is to obtain a suffi...This paper considers the rational expectations model with multiplicative noise and input delay,where the system dynamics rely on the conditional expectations of future states.The main contribution is to obtain a sufficient condition for the exact controllability of the rational expectations model.In particular,we derive a sufficient Gramian matrix condition and a rank condition for the delay-free case.The key is the solvability of the backward stochastic difference equations with input delay which is derived from the forward and backward stochastic system.展开更多
This paper discusses a queueing system with a retrial orbit and batch service, in which the quantity of customers’ rooms in the queue is finite and the space of retrial orbit is infinite. When the server starts servi...This paper discusses a queueing system with a retrial orbit and batch service, in which the quantity of customers’ rooms in the queue is finite and the space of retrial orbit is infinite. When the server starts serving, it serves all customers in the queue in a single batch, which is the so-called batch service. If a new customer or a retrial customer finds all the customers’ rooms are occupied, he will decide whether or not to join the retrial orbit. By using the censoring technique and the matrix analysis method, we first obtain the decay function of the stationary distribution for the quantity of customers in the retrial orbit and the quantity of customers in the queue. Then based on the form of decay rate function and the Karamata Tauberian theorem, we finally get the exact tail asymptotics of the stationary distribution.展开更多
Sample size determination typically relies on a power analysis based on a frequentist conditional approach. This latter can be seen as a particular case of the two-priors approach, which allows to build four distinct ...Sample size determination typically relies on a power analysis based on a frequentist conditional approach. This latter can be seen as a particular case of the two-priors approach, which allows to build four distinct power functions to select the optimal sample size. We revise this approach when the focus is on testing a single binomial proportion. We consider exact methods and introduce a conservative criterion to account for the typical non-monotonic behavior of the power functions, when dealing with discrete data. The main purpose of this paper is to present a Shiny App providing a user-friendly, interactive tool to apply these criteria. The app also provides specific tools to elicit the analysis and the design prior distributions, which are the core of the two-priors approach.展开更多
The objectives of this paper are to demonstrate the algorithms employed by three statistical software programs (R, Real Statistics using Excel, and SPSS) for calculating the exact two-tailed probability of the Wald-Wo...The objectives of this paper are to demonstrate the algorithms employed by three statistical software programs (R, Real Statistics using Excel, and SPSS) for calculating the exact two-tailed probability of the Wald-Wolfowitz one-sample runs test for randomness, to present a novel approach for computing this probability, and to compare the four procedures by generating samples of 10 and 11 data points, varying the parameters n<sub>0</sub> (number of zeros) and n<sub>1</sub> (number of ones), as well as the number of runs. Fifty-nine samples are created to replicate the behavior of the distribution of the number of runs with 10 and 11 data points. The exact two-tailed probabilities for the four procedures were compared using Friedman’s test. Given the significant difference in central tendency, post-hoc comparisons were conducted using Conover’s test with Benjamini-Yekutielli correction. It is concluded that the procedures of Real Statistics using Excel and R exhibit some inadequacies in the calculation of the exact two-tailed probability, whereas the new proposal and the SPSS procedure are deemed more suitable. The proposed robust algorithm has a more transparent rationale than the SPSS one, albeit being somewhat more conservative. We recommend its implementation for this test and its application to others, such as the binomial and sign test.展开更多
Editor's Note:In early spring,22 students from the border counties of Xigaze,Xizang,visited Beijing for a seven-day study tour.It is the first time for the children to travel by train and also their first time to ...Editor's Note:In early spring,22 students from the border counties of Xigaze,Xizang,visited Beijing for a seven-day study tour.It is the first time for the children to travel by train and also their first time to Beijing.In just a few days'study activities,the children have a lot of novel experience.In this age of curiosity and desire for knowledge,having such a new experience in life is exactly the significance of the study tour.展开更多
In dynamic problems the electric and magnetic fields are inseparable. At the same time, a multitude of electrostatic and magnetostatic effects permit mutually independent description. This separation appears to be pos...In dynamic problems the electric and magnetic fields are inseparable. At the same time, a multitude of electrostatic and magnetostatic effects permit mutually independent description. This separation appears to be possible and thermodynamically consistent when the bulk energy density depends only on the polarization density or, alternatively, on the magnetization density. However, when the bulk energy density depends simultaneously on the both densities, then, the electrostatic and magnetostatic effects should be studied together. There appear interesting cross-effects;among those are the change of the internal electrostatic field inside a specimen under the influence of the external magnetic fields, and vice versa. Below, in the framework of thermodynamic approach the boundary value problem for magnetoelectric plate is formulated and analyzed. The exact solution is established for the isotropic pyroelectric plate.展开更多
This new work aims to develop a full coupled thermomechanical method including both the temperature profile and displacements as primary unknowns of the model.This generic full coupled 3D exact shell model permits the...This new work aims to develop a full coupled thermomechanical method including both the temperature profile and displacements as primary unknowns of the model.This generic full coupled 3D exact shell model permits the thermal stress investigation of laminated isotropic,composite and sandwich structures.Cylindrical and spherical panels,cylinders and plates are analyzed in orthogonal mixed curved reference coordinates.The 3D equilibrium relations and the 3D Fourier heat conduction equation for spherical shells are coupled and they trivially can be simplified in those for plates and cylindrical panels.The exponential matrix methodology is used to find the solutions of a full coupled model based on coupled differential relations with respect to the thickness coordinate.The analytical solution is based on theories of simply supported edges and harmonic relations for displacement components and sovra-temperature.The sovra-temperature magnitudes are directly applied at the outer faces through static state hypotheses.As a consequence,the sovra-temperature description is assumed to be an unknown variable of themodel and it is calculated in the sameway as the three displacements.The final systemis based on a set of coupled homogeneous differential relations of second order in the thickness coordinate.This system is reduced in a first order differential relation system by redoubling the number of unknowns.Therefore,the exponential matrix methodology is applied to calculate the solution.The temperature field effects are evaluated in the static investigation of shells and plates in terms of displacement and stress components.After an appropriate preliminary validation,new benchmarks are discussed for several thickness ratios,geometrical data,lamination sequences,materials and sovra-temperature values imposed at the outer faces.Results make evident the accordance between the uncoupled thermo-mechanical model and this new full coupled thermo-mechanical model without the need to separately solve the Fourier heat conduction relation.Both effects connected with the thickness layer and the related embedded materials are included in the conducted thermal stress analysis.展开更多
The recently developed hard-magnetic soft(HMS)materials manufactured by embedding high-coercivity micro-particles into soft matrices have received considerable attention from researchers in diverse fields,e.g.,soft ro...The recently developed hard-magnetic soft(HMS)materials manufactured by embedding high-coercivity micro-particles into soft matrices have received considerable attention from researchers in diverse fields,e.g.,soft robotics,flexible electronics,and biomedicine.Theoretical investigations on large deformations of HMS structures are significant foundations of their applications.This work is devoted to developing a powerful theoretical tool for modeling and computing the complicated nonplanar deformations of flexible beams.A so-called quaternion beam model is proposed to break the singularity limitation of the existing geometrically exact(GE)beam model.The singularity-free governing equations for the three-dimensional(3D)large deformations of an HMS beam are first derived,and then solved with the Galerkin discretization method and the trustregion-dogleg iterative algorithm.The correctness of this new model and the utilized algorithms is verified by comparing the present results with the previous ones.The superiority of a quaternion beam model in calculating the complicated large deformations of a flexible beam is shown through several benchmark examples.It is found that the purpose of the HMS beam deformation is to eliminate the direction deviation between the residual magnetization and the applied magnetic field.The proposed new model and the revealed mechanism are supposed to be useful for guiding the engineering applications of flexible structures.展开更多
In this paper, the problem of finding exact solutions to the magnetohydrodynamic(MHD) equations in the presence of incompressible mass flows with helical symmetry is considered. For ideal flows, a similarity reduction...In this paper, the problem of finding exact solutions to the magnetohydrodynamic(MHD) equations in the presence of incompressible mass flows with helical symmetry is considered. For ideal flows, a similarity reduction method is used to obtain exact solutions for several MHD flows with nonlinear variable Mach number. For resistive flows parallel to a magnetic field, the governing equilibrium equation is derived. The MHD equilibrium state of a helically symmetric incompressible flow is governed by a second-order elliptic partial differential equation(PDE) for the helical magnetic flux function. Exact solutions for the latter equation are obtained. Also, the equilibrium equations of a gravitating plasma with incompressible flow are derived.展开更多
In this paper,we present a new modification of the newly developed semi-analytical method named the Optimal Auxilary Function Method(OAFM)for fractional-order equations using the Caputo operator,which is named FOAFM.T...In this paper,we present a new modification of the newly developed semi-analytical method named the Optimal Auxilary Function Method(OAFM)for fractional-order equations using the Caputo operator,which is named FOAFM.The mathematical theory of FOAFM is presented and the effectiveness of this method is proven by using it with well-known Fornberg-Whitham Equations(FWE).The FOAFM results are compared with other method results along with their exact solutions with the help of tables and plots to prove the validity of FOAFM.A rapidly convergent series solution is obtained from FOAFMand is validated by comparison with other results.The analysis proves that ourmethod is simply applicable,contains less computationalwork,and is rapidly convergent to the exact solution at the first iteration.A series solution to the problem is obtained with the help of FOAFM.The validity of FOAFM results is validated by comparing its results with the results available in the literature.It is observed that FOAFM is simply applicable,contains less computational work,and is fastly convergent.The convergence and stability are obtained with the help of optimal constants.FOAFM is very easy in applicability and provides excellent results at the first iteration for complex nonlinear initial/boundary value problems.FOAFM contains the optimal auxiliary constants through which we can control the convergence as FOAFM contains the auxiliary functions D_(1),D_(2),D_(3)...in which the optimal constants G_(1),G_(2),...and the control convergence parameters exist to play an important role in getting the convergent solution which is obtained rigorously.The computational work in FOAFM is less when compared to other methods and even a low-specification computer can do the computational work easily.展开更多
Efficient numerical algorithm for stochastic differential equation has been an important object in the research of statistical physics and mathematics for a long time.In this work we study the highly accurate numerica...Efficient numerical algorithm for stochastic differential equation has been an important object in the research of statistical physics and mathematics for a long time.In this work we study the highly accurate numerical algorithm for the overdamped Langevin equation.In particular,our interest is in the behaviour of the numerical schemes for solving the overdamped Langevin equation in the harmonic system.Based on the large friction limit of the underdamped Langevin dynamic scheme,three algorithms for overdamped Langevin equation are obtained.We derive the explicit expression of the stationary distribution of each algorithm by analysing the discrete time trajectory for both one-dimensional case and multi-dimensional case.The accuracy of the stationary distribution of each algorithm is illustrated by comparing with the exact Boltzmann distribution.Our results demonstrate that the“BAOA-limit”algorithm generates an accurate distribution of the harmonic system in a canonical ensemble,within a stable range of time interval.The other algorithms do not produce the exact distribution of the harmonic system.展开更多
基金supported by the Deutsche Forschungsgemeinschaft(Germany,RE-4543/1-1 to SR).
文摘Glaucoma is a group of eye diseases characterized by progressive loss of retinal ganglion cells(RGCs)and optic nerve degeneration.During this process,the visual field is reduced,and blindness may ultimately occur.Worldwide,glaucoma is the second leading cause of blindness,with about 80 million people affected.Glaucoma is a multifactorial disease and due to its complexity,the exact pathomechanisms are not fully understood yet.However,different risk factors,such as elevated intraocular pressure(IOP),age,or myopia,have been identified to date(EGS,2021).
基金Project supported by the National Natural Science Foundation of China (Nos.12072119,12325201,and 52205594)the China National Postdoctoral Program for Innovative Talents (No.BX20220118)。
文摘Due to the novel applications of flexible pipes conveying fluid in the field of soft robotics and biomedicine,the investigations on the mechanical responses of the pipes have attracted considerable attention.The fluid-structure interaction(FSI)between the pipe with a curved shape and the time-varying internal fluid flow brings a great challenge to the revelation of the dynamical behaviors of flexible pipes,especially when the pipe is highly flexible and usually undergoes large deformations.In this work,the geometrically exact model(GEM)for a curved cantilevered pipe conveying pulsating fluid is developed based on the extended Hamilton's principle.The stability of the curved pipe with three different subtended angles is examined with the consideration of steady fluid flow.Specific attention is concentrated on the large-deformation resonance of circular pipes conveying pulsating fluid,which is often encountered in practical engineering.By constructing bifurcation diagrams,oscillating shapes,phase portraits,time traces,and Poincarémaps,the dynamic responses of the curved pipe under various system parameters are revealed.The mean flow velocity of the pulsating fluid is chosen to be either subcritical or supercritical.The numerical results show that the curved pipe conveying pulsating fluid can exhibit rich dynamical behaviors,including periodic and quasi-periodic motions.It is also found that the preferred instability type of a cantilevered curved pipe conveying steady fluid is mainly in the flutter of the second mode.For a moderate value of the mass ratio,however,a third-mode flutter may occur,which is quite different from that of a straight pipe system.
基金supported by the National Natural Science Foundation of China under Grants 61821004,62250056,62350710214,U23A20325,62350055the Natural Science Foundation of Shandong Province,China(ZR2021ZD14,ZR2021JQ24)+2 种基金High-level Talent Team Project of Qingdao West Coast New Area,China(RCTD-JC-2019-05)Key Research and Development Program of Shandong Province,China(2020CXGC01208)Science and Technology Project of Qingdao West Coast New Area,China(2019-32,2020-20,2020-1-4).
文摘This paper considers the rational expectations model with multiplicative noise and input delay,where the system dynamics rely on the conditional expectations of future states.The main contribution is to obtain a sufficient condition for the exact controllability of the rational expectations model.In particular,we derive a sufficient Gramian matrix condition and a rank condition for the delay-free case.The key is the solvability of the backward stochastic difference equations with input delay which is derived from the forward and backward stochastic system.
文摘This paper discusses a queueing system with a retrial orbit and batch service, in which the quantity of customers’ rooms in the queue is finite and the space of retrial orbit is infinite. When the server starts serving, it serves all customers in the queue in a single batch, which is the so-called batch service. If a new customer or a retrial customer finds all the customers’ rooms are occupied, he will decide whether or not to join the retrial orbit. By using the censoring technique and the matrix analysis method, we first obtain the decay function of the stationary distribution for the quantity of customers in the retrial orbit and the quantity of customers in the queue. Then based on the form of decay rate function and the Karamata Tauberian theorem, we finally get the exact tail asymptotics of the stationary distribution.
文摘Sample size determination typically relies on a power analysis based on a frequentist conditional approach. This latter can be seen as a particular case of the two-priors approach, which allows to build four distinct power functions to select the optimal sample size. We revise this approach when the focus is on testing a single binomial proportion. We consider exact methods and introduce a conservative criterion to account for the typical non-monotonic behavior of the power functions, when dealing with discrete data. The main purpose of this paper is to present a Shiny App providing a user-friendly, interactive tool to apply these criteria. The app also provides specific tools to elicit the analysis and the design prior distributions, which are the core of the two-priors approach.
文摘The objectives of this paper are to demonstrate the algorithms employed by three statistical software programs (R, Real Statistics using Excel, and SPSS) for calculating the exact two-tailed probability of the Wald-Wolfowitz one-sample runs test for randomness, to present a novel approach for computing this probability, and to compare the four procedures by generating samples of 10 and 11 data points, varying the parameters n<sub>0</sub> (number of zeros) and n<sub>1</sub> (number of ones), as well as the number of runs. Fifty-nine samples are created to replicate the behavior of the distribution of the number of runs with 10 and 11 data points. The exact two-tailed probabilities for the four procedures were compared using Friedman’s test. Given the significant difference in central tendency, post-hoc comparisons were conducted using Conover’s test with Benjamini-Yekutielli correction. It is concluded that the procedures of Real Statistics using Excel and R exhibit some inadequacies in the calculation of the exact two-tailed probability, whereas the new proposal and the SPSS procedure are deemed more suitable. The proposed robust algorithm has a more transparent rationale than the SPSS one, albeit being somewhat more conservative. We recommend its implementation for this test and its application to others, such as the binomial and sign test.
文摘Editor's Note:In early spring,22 students from the border counties of Xigaze,Xizang,visited Beijing for a seven-day study tour.It is the first time for the children to travel by train and also their first time to Beijing.In just a few days'study activities,the children have a lot of novel experience.In this age of curiosity and desire for knowledge,having such a new experience in life is exactly the significance of the study tour.
文摘In dynamic problems the electric and magnetic fields are inseparable. At the same time, a multitude of electrostatic and magnetostatic effects permit mutually independent description. This separation appears to be possible and thermodynamically consistent when the bulk energy density depends only on the polarization density or, alternatively, on the magnetization density. However, when the bulk energy density depends simultaneously on the both densities, then, the electrostatic and magnetostatic effects should be studied together. There appear interesting cross-effects;among those are the change of the internal electrostatic field inside a specimen under the influence of the external magnetic fields, and vice versa. Below, in the framework of thermodynamic approach the boundary value problem for magnetoelectric plate is formulated and analyzed. The exact solution is established for the isotropic pyroelectric plate.
文摘This new work aims to develop a full coupled thermomechanical method including both the temperature profile and displacements as primary unknowns of the model.This generic full coupled 3D exact shell model permits the thermal stress investigation of laminated isotropic,composite and sandwich structures.Cylindrical and spherical panels,cylinders and plates are analyzed in orthogonal mixed curved reference coordinates.The 3D equilibrium relations and the 3D Fourier heat conduction equation for spherical shells are coupled and they trivially can be simplified in those for plates and cylindrical panels.The exponential matrix methodology is used to find the solutions of a full coupled model based on coupled differential relations with respect to the thickness coordinate.The analytical solution is based on theories of simply supported edges and harmonic relations for displacement components and sovra-temperature.The sovra-temperature magnitudes are directly applied at the outer faces through static state hypotheses.As a consequence,the sovra-temperature description is assumed to be an unknown variable of themodel and it is calculated in the sameway as the three displacements.The final systemis based on a set of coupled homogeneous differential relations of second order in the thickness coordinate.This system is reduced in a first order differential relation system by redoubling the number of unknowns.Therefore,the exponential matrix methodology is applied to calculate the solution.The temperature field effects are evaluated in the static investigation of shells and plates in terms of displacement and stress components.After an appropriate preliminary validation,new benchmarks are discussed for several thickness ratios,geometrical data,lamination sequences,materials and sovra-temperature values imposed at the outer faces.Results make evident the accordance between the uncoupled thermo-mechanical model and this new full coupled thermo-mechanical model without the need to separately solve the Fourier heat conduction relation.Both effects connected with the thickness layer and the related embedded materials are included in the conducted thermal stress analysis.
基金Project supported by the National Key Research and Development Program of China(No.2018YFA0703200)the National Natural Science Foundation of China(Nos.52205594 and51820105008)+1 种基金the China National Postdoctoral Program for Innovative Talents(No.BX20220118)the China Postdoctoral Science Foundation(No.2021M701306)。
文摘The recently developed hard-magnetic soft(HMS)materials manufactured by embedding high-coercivity micro-particles into soft matrices have received considerable attention from researchers in diverse fields,e.g.,soft robotics,flexible electronics,and biomedicine.Theoretical investigations on large deformations of HMS structures are significant foundations of their applications.This work is devoted to developing a powerful theoretical tool for modeling and computing the complicated nonplanar deformations of flexible beams.A so-called quaternion beam model is proposed to break the singularity limitation of the existing geometrically exact(GE)beam model.The singularity-free governing equations for the three-dimensional(3D)large deformations of an HMS beam are first derived,and then solved with the Galerkin discretization method and the trustregion-dogleg iterative algorithm.The correctness of this new model and the utilized algorithms is verified by comparing the present results with the previous ones.The superiority of a quaternion beam model in calculating the complicated large deformations of a flexible beam is shown through several benchmark examples.It is found that the purpose of the HMS beam deformation is to eliminate the direction deviation between the residual magnetization and the applied magnetic field.The proposed new model and the revealed mechanism are supposed to be useful for guiding the engineering applications of flexible structures.
文摘In this paper, the problem of finding exact solutions to the magnetohydrodynamic(MHD) equations in the presence of incompressible mass flows with helical symmetry is considered. For ideal flows, a similarity reduction method is used to obtain exact solutions for several MHD flows with nonlinear variable Mach number. For resistive flows parallel to a magnetic field, the governing equilibrium equation is derived. The MHD equilibrium state of a helically symmetric incompressible flow is governed by a second-order elliptic partial differential equation(PDE) for the helical magnetic flux function. Exact solutions for the latter equation are obtained. Also, the equilibrium equations of a gravitating plasma with incompressible flow are derived.
文摘In this paper,we present a new modification of the newly developed semi-analytical method named the Optimal Auxilary Function Method(OAFM)for fractional-order equations using the Caputo operator,which is named FOAFM.The mathematical theory of FOAFM is presented and the effectiveness of this method is proven by using it with well-known Fornberg-Whitham Equations(FWE).The FOAFM results are compared with other method results along with their exact solutions with the help of tables and plots to prove the validity of FOAFM.A rapidly convergent series solution is obtained from FOAFMand is validated by comparison with other results.The analysis proves that ourmethod is simply applicable,contains less computationalwork,and is rapidly convergent to the exact solution at the first iteration.A series solution to the problem is obtained with the help of FOAFM.The validity of FOAFM results is validated by comparing its results with the results available in the literature.It is observed that FOAFM is simply applicable,contains less computational work,and is fastly convergent.The convergence and stability are obtained with the help of optimal constants.FOAFM is very easy in applicability and provides excellent results at the first iteration for complex nonlinear initial/boundary value problems.FOAFM contains the optimal auxiliary constants through which we can control the convergence as FOAFM contains the auxiliary functions D_(1),D_(2),D_(3)...in which the optimal constants G_(1),G_(2),...and the control convergence parameters exist to play an important role in getting the convergent solution which is obtained rigorously.The computational work in FOAFM is less when compared to other methods and even a low-specification computer can do the computational work easily.
基金Project supported by the Basic and Applied Basic Research Foundation of Guangdong Province,China(Grant No.2021A1515010328)the Key-Area Research and Development Program of Guangdong Province,China(Grant No.2020B010183001)the National Natural Science Foundation of China(Grant No.12074126)。
文摘Efficient numerical algorithm for stochastic differential equation has been an important object in the research of statistical physics and mathematics for a long time.In this work we study the highly accurate numerical algorithm for the overdamped Langevin equation.In particular,our interest is in the behaviour of the numerical schemes for solving the overdamped Langevin equation in the harmonic system.Based on the large friction limit of the underdamped Langevin dynamic scheme,three algorithms for overdamped Langevin equation are obtained.We derive the explicit expression of the stationary distribution of each algorithm by analysing the discrete time trajectory for both one-dimensional case and multi-dimensional case.The accuracy of the stationary distribution of each algorithm is illustrated by comparing with the exact Boltzmann distribution.Our results demonstrate that the“BAOA-limit”algorithm generates an accurate distribution of the harmonic system in a canonical ensemble,within a stable range of time interval.The other algorithms do not produce the exact distribution of the harmonic system.