To address the issue that static densest subgraph mining algorithms often exhibit low efficiency when handling large scale dynamic graphs,this paper proposes a heuristic approximation algorithm.The algorithm approxima...To address the issue that static densest subgraph mining algorithms often exhibit low efficiency when handling large scale dynamic graphs,this paper proposes a heuristic approximation algorithm.The algorithm approximates the densest k-subgraphs of the entire graph through four steps:partitioning the large-scale dynamic graph,constructing a partial set of the densest k-subgraphs,heuristically merging the subgraph sets,and finally extracting the densest k-subgraphs.This approach significantly reduces the computational time for large-scale dynamic graphs while simultaneously improving the quality of the resulting subgraphs.This algorithm is applicable to various definitions of“density”and can accommodate diverse requirements on the number of edges.When integrated with existing static densest subgraph detection algorithms,it achieves scalability and computational efficiency.Theoretical analysis demonstrates that the optimal density of the densest k-subgraphs extracted by the proposed algorithm reaches 0.9.To evaluate the performance of the algorithm,experiments were conducted on four billion-scale datasets:Friendster,Orkut,YouTube,and DBLP.The results indicate that the proposed algorithm outperforms static methods in both runtime efficiency and subgraph quality on large-scale dynamic graphs.展开更多
The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential pr...The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing-van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.展开更多
In this paper, we establish a new type of alternation theory for more general restricted ranges Chebyshev approximation with equalities. The uniqueness and strong uniqueness theorems are given. Applying the results, w...In this paper, we establish a new type of alternation theory for more general restricted ranges Chebyshev approximation with equalities. The uniqueness and strong uniqueness theorems are given. Applying the results, we obtain the alternation theorem and uniqueness theorem for best coposilive approximation.展开更多
In this note,we develop,without assuming the Haar condition,a generalized simultaneous Chebyshev approximation theory which is similar to the classical Chebyshev theory and con- rains it as a special case.Our results ...In this note,we develop,without assuming the Haar condition,a generalized simultaneous Chebyshev approximation theory which is similar to the classical Chebyshev theory and con- rains it as a special case.Our results also contain those in[1]and[3]as a special case,and the two conjectures proposed by C.B.Dunham in[2]are proved to be true in the case of simulta- neous approximation.展开更多
In this paper, Remes algorithm is applied to compute the numerical solution of the best chebyshev approximation from varisolvent family. Feasibility and convergence of the algorithm are discussed carefully.
In this paper, we construct Chebyshev biorthogonal multiwavelets, and use this multiwavelets to approximate signals (functions). The convergence rate for signal approximation is derived. The fast signal decomposition ...In this paper, we construct Chebyshev biorthogonal multiwavelets, and use this multiwavelets to approximate signals (functions). The convergence rate for signal approximation is derived. The fast signal decomposition and reconstruction algorithms are presented. The numerical examples validate the theoretical analysis.展开更多
The critical point set plays a central role in the theory of Tchebyshev approximation. Generally, in multivariate Tchebyshev approximation, it is not a trivial task to determine whether a set is critical or not. In th...The critical point set plays a central role in the theory of Tchebyshev approximation. Generally, in multivariate Tchebyshev approximation, it is not a trivial task to determine whether a set is critical or not. In this paper, we study the characterization of the critical point set of S^01（△） in geometry, where A is restricted to some special triangulations （bitriangular, single road and star triangulations）. Such geometrical characterization is convenient to use in the determination of a critical point set.展开更多
The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J.P.Boyd's rational...The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J.P.Boyd's rational Chebyshev basis.展开更多
双曲偏微分方程是重要的偏微分方程之一。提出求解电报方程的Chebyshev谱法,采用Chebyshev-Gauss-Lobatto配点,利用Chebyshev多项式构造导数矩阵,将电报方程近似为常微分方程,证明了电报方程的离散Chebyshev谱法的误差估计,采用Runge-Ku...双曲偏微分方程是重要的偏微分方程之一。提出求解电报方程的Chebyshev谱法,采用Chebyshev-Gauss-Lobatto配点,利用Chebyshev多项式构造导数矩阵,将电报方程近似为常微分方程,证明了电报方程的离散Chebyshev谱法的误差估计,采用Runge-Kutta进行求解。将该法得到的数值结果与精确解进行比较,验证了方法的有效性,数据结果的误差与其他方法相比有较高的精确度。Hyperbolic partial differential equation is one of the important partial differential equations. The Chebyshev spectral method is proposed to solve the telegraph equation. Chebyshev-gauss-lobatto is used to assign points, the derivative matrix is constructed by Chebyshev polynomial, and the telegraph equation is approximated as an ordinary differential equation. The error estimation of the discrete Chebyshev spectral method for the telegraph equation was proved. Runge-Kutta was used to solve the problem. The numerical results obtained by the method are compared with the exact solution, and the effectiveness of the method is verified. The error of the data results is more accurate than that of other methods.展开更多
在求解奇异摄动两点边值问题时,本文构造了基于Chebyshev点的B样条配置法。该方法采用三次B样条函数作为基函数,利用Chebyshev点作为配置点直接对方程进行求解。文中探讨了该方法在实施时的具体步骤及需要注意的若干细节。通过奇异摄动...在求解奇异摄动两点边值问题时,本文构造了基于Chebyshev点的B样条配置法。该方法采用三次B样条函数作为基函数,利用Chebyshev点作为配置点直接对方程进行求解。文中探讨了该方法在实施时的具体步骤及需要注意的若干细节。通过奇异摄动扩散反应问题、奇异摄动对流扩散反应问题这两个算例的研究,表明基于Chebyshev点的B样条配置法与等距节点下的B样条配置法相比,前者具有高精度和高效率的优势。In solving the singular perturbation two-point boundary value problems, this paper constructs a Chebyshev B-spline collocation method. This method uses cubic B-spline functions as basis functions and utilizes the Chebyshev point as the configuration point to solve the equation directly. The specific steps in the implementation of the method and several details that need to be noted are discussed in the paper. Through the study of two arithmetic cases, namely, the singular regent diffusion response problem and the singular regent convection diffusion response problem, it is shown that the Chebyshev B-spline collocation method has the advantages of high accuracy and high efficiency as compared with the B-spline configuration method under equidistant nodes.展开更多
In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to...In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to the concentration level of singular Trudinger-Moser functional on the unit ball.展开更多
Theoretical analysis has demonstrated that the dispersion relation of chorus waves plays an essential role in the resonant interaction and energy transformation between the waves and magnetospheric electrons.Previous ...Theoretical analysis has demonstrated that the dispersion relation of chorus waves plays an essential role in the resonant interaction and energy transformation between the waves and magnetospheric electrons.Previous quantitative analyses often simplified the chorus dispersion relation by using the cold plasma assumption.However,the applicability of the cold plasma assumption is doubtful,especially during geomagnetic disturbances.We here present a systematic statistical analysis on the validity of the cold plasma dispersion relation of chorus waves based on observations from the Van Allen Probes over the period from 2012 to 2018.The statistical results show that the observed magnetic field intensities deviate substantially from those calculated from the cold plasma dispersion relation and that they become more pronounced with an increase in geomagnetic activity or a decrease in background plasma density.The region with large deviations is mainly concentrated in the nightside and expands in both the radial and azimuthal directions as the geomagnetic activity increases or the background plasma density decreases.In addition,the bounce-averaged electron scattering rates are computed by using the observed and cold plasma dispersion relation of chorus waves.Compared with usage of the cold plasma dispersion relation,usage of the observed dispersion relation considerably lowers the minimum resonant energy of electrons and lowers the scattering rates of electrons above tens of kiloelectronvolts but enhances those below.Furthermore,these differences are more pronounced with the enhancement of geomagnetic activity or the decrease in background plasma density.展开更多
Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for ...Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).展开更多
This paper studies the Smoluchowski–Kramers approximation for a discrete-time dynamical system modeled as the motion of a particle in a force field.We show that the approximation holds for the drift-implicit Euler–M...This paper studies the Smoluchowski–Kramers approximation for a discrete-time dynamical system modeled as the motion of a particle in a force field.We show that the approximation holds for the drift-implicit Euler–Maruyama discretization and derive its convergence rate.In particular,the solution of the discretized system converges to the solution of the first-order limit equation in the mean-square sense,and this convergence is independent of the order in which the mass parameterμand the step size h tend to zero.展开更多
基金The National Natural Science Foundation of China(No.62172443).
文摘To address the issue that static densest subgraph mining algorithms often exhibit low efficiency when handling large scale dynamic graphs,this paper proposes a heuristic approximation algorithm.The algorithm approximates the densest k-subgraphs of the entire graph through four steps:partitioning the large-scale dynamic graph,constructing a partial set of the densest k-subgraphs,heuristically merging the subgraph sets,and finally extracting the densest k-subgraphs.This approach significantly reduces the computational time for large-scale dynamic graphs while simultaneously improving the quality of the resulting subgraphs.This algorithm is applicable to various definitions of“density”and can accommodate diverse requirements on the number of edges.When integrated with existing static densest subgraph detection algorithms,it achieves scalability and computational efficiency.Theoretical analysis demonstrates that the optimal density of the densest k-subgraphs extracted by the proposed algorithm reaches 0.9.To evaluate the performance of the algorithm,experiments were conducted on four billion-scale datasets:Friendster,Orkut,YouTube,and DBLP.The results indicate that the proposed algorithm outperforms static methods in both runtime efficiency and subgraph quality on large-scale dynamic graphs.
基金Project supported by the National Natural Science Foundation of China （Grants Nos 10472091 and 10332030）.
文摘The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing-van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.
文摘In this paper, we establish a new type of alternation theory for more general restricted ranges Chebyshev approximation with equalities. The uniqueness and strong uniqueness theorems are given. Applying the results, we obtain the alternation theorem and uniqueness theorem for best coposilive approximation.
文摘In this note,we develop,without assuming the Haar condition,a generalized simultaneous Chebyshev approximation theory which is similar to the classical Chebyshev theory and con- rains it as a special case.Our results also contain those in[1]and[3]as a special case,and the two conjectures proposed by C.B.Dunham in[2]are proved to be true in the case of simulta- neous approximation.
文摘In this paper, Remes algorithm is applied to compute the numerical solution of the best chebyshev approximation from varisolvent family. Feasibility and convergence of the algorithm are discussed carefully.
文摘In this paper, we construct Chebyshev biorthogonal multiwavelets, and use this multiwavelets to approximate signals (functions). The convergence rate for signal approximation is derived. The fast signal decomposition and reconstruction algorithms are presented. The numerical examples validate the theoretical analysis.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 1027102260373093+3 种基金605330601110136661100130)the Innovation Foundation of the Key Laboratory of High-Temperature Gasdynamics of Chinese Academy of Sciences
文摘The critical point set plays a central role in the theory of Tchebyshev approximation. Generally, in multivariate Tchebyshev approximation, it is not a trivial task to determine whether a set is critical or not. In this paper, we study the characterization of the critical point set of S^01（△） in geometry, where A is restricted to some special triangulations （bitriangular, single road and star triangulations）. Such geometrical characterization is convenient to use in the determination of a critical point set.
文摘The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J.P.Boyd's rational Chebyshev basis.
文摘双曲偏微分方程是重要的偏微分方程之一。提出求解电报方程的Chebyshev谱法,采用Chebyshev-Gauss-Lobatto配点,利用Chebyshev多项式构造导数矩阵,将电报方程近似为常微分方程,证明了电报方程的离散Chebyshev谱法的误差估计,采用Runge-Kutta进行求解。将该法得到的数值结果与精确解进行比较,验证了方法的有效性,数据结果的误差与其他方法相比有较高的精确度。Hyperbolic partial differential equation is one of the important partial differential equations. The Chebyshev spectral method is proposed to solve the telegraph equation. Chebyshev-gauss-lobatto is used to assign points, the derivative matrix is constructed by Chebyshev polynomial, and the telegraph equation is approximated as an ordinary differential equation. The error estimation of the discrete Chebyshev spectral method for the telegraph equation was proved. Runge-Kutta was used to solve the problem. The numerical results obtained by the method are compared with the exact solution, and the effectiveness of the method is verified. The error of the data results is more accurate than that of other methods.
文摘在求解奇异摄动两点边值问题时,本文构造了基于Chebyshev点的B样条配置法。该方法采用三次B样条函数作为基函数,利用Chebyshev点作为配置点直接对方程进行求解。文中探讨了该方法在实施时的具体步骤及需要注意的若干细节。通过奇异摄动扩散反应问题、奇异摄动对流扩散反应问题这两个算例的研究,表明基于Chebyshev点的B样条配置法与等距节点下的B样条配置法相比,前者具有高精度和高效率的优势。In solving the singular perturbation two-point boundary value problems, this paper constructs a Chebyshev B-spline collocation method. This method uses cubic B-spline functions as basis functions and utilizes the Chebyshev point as the configuration point to solve the equation directly. The specific steps in the implementation of the method and several details that need to be noted are discussed in the paper. Through the study of two arithmetic cases, namely, the singular regent diffusion response problem and the singular regent convection diffusion response problem, it is shown that the Chebyshev B-spline collocation method has the advantages of high accuracy and high efficiency as compared with the B-spline configuration method under equidistant nodes.
文摘In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to the concentration level of singular Trudinger-Moser functional on the unit ball.
基金supported by the National Natural Science Foundation of China (NSFC) through Grant Number 42074193
文摘Theoretical analysis has demonstrated that the dispersion relation of chorus waves plays an essential role in the resonant interaction and energy transformation between the waves and magnetospheric electrons.Previous quantitative analyses often simplified the chorus dispersion relation by using the cold plasma assumption.However,the applicability of the cold plasma assumption is doubtful,especially during geomagnetic disturbances.We here present a systematic statistical analysis on the validity of the cold plasma dispersion relation of chorus waves based on observations from the Van Allen Probes over the period from 2012 to 2018.The statistical results show that the observed magnetic field intensities deviate substantially from those calculated from the cold plasma dispersion relation and that they become more pronounced with an increase in geomagnetic activity or a decrease in background plasma density.The region with large deviations is mainly concentrated in the nightside and expands in both the radial and azimuthal directions as the geomagnetic activity increases or the background plasma density decreases.In addition,the bounce-averaged electron scattering rates are computed by using the observed and cold plasma dispersion relation of chorus waves.Compared with usage of the cold plasma dispersion relation,usage of the observed dispersion relation considerably lowers the minimum resonant energy of electrons and lowers the scattering rates of electrons above tens of kiloelectronvolts but enhances those below.Furthermore,these differences are more pronounced with the enhancement of geomagnetic activity or the decrease in background plasma density.
基金Supported by NSFC(Nos.12301006,12471009,12071238,11901566,12001047,11971476)Beijing Natural Science Foundation(No.1242003)。
文摘Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).
基金supported by the PhD Research Startup Foundation of Hubei University of Economics(Grand No.XJ23BS42).
文摘This paper studies the Smoluchowski–Kramers approximation for a discrete-time dynamical system modeled as the motion of a particle in a force field.We show that the approximation holds for the drift-implicit Euler–Maruyama discretization and derive its convergence rate.In particular,the solution of the discretized system converges to the solution of the first-order limit equation in the mean-square sense,and this convergence is independent of the order in which the mass parameterμand the step size h tend to zero.
