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Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity
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作者 Zheng Sun Chi-Wang Shu 《Communications on Applied Mathematics and Computation》 2021年第4期671-700,共30页
A time discretization method is called strongly stable(or monotone),if the norm of its numerical solution is nonincreasing.Although this property is desirable in various of contexts,many explicit Runge-Kutta(RK)method... A time discretization method is called strongly stable(or monotone),if the norm of its numerical solution is nonincreasing.Although this property is desirable in various of contexts,many explicit Runge-Kutta(RK)methods may fail to preserve it.In this paper,we enforce strong stability by modifying the method with superviscosity,which is a numerical technique commonly used in spectral methods.Our main focus is on strong stability under the inner-product norm for linear problems with possibly non-normal operators.We propose two approaches for stabilization:the modified method and the filtering method.The modified method is achieved by modifying the semi-negative operator with a high order superviscosity term;the filtering method is to post-process the solution by solving a diffusive or dispersive problem with small superviscosity.For linear problems,most explicit RK methods can be stabilized with either approach without accuracy degeneration.Furthermore,we prove a sharp bound(up to an equal sign)on diffusive superviscosity for ensuring strong stability.For nonlinear problems,a filtering method is investigated.Numerical examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance. 展开更多
关键词 Runge-Kutta(RK)methods strong stability Superviscosity Hyperbolic conservation laws Discontinuous Galerkin methods
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Almost Sure Convergence Theorem and Strong Stability for Weighted Sums of NSD Random Variables 被引量:14
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作者 Yan SHEN Xue Jun WANG +1 位作者 Wen Zhi YANG Shu He HU 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2013年第4期743-756,共14页
In this paper, Kolmogorov-type inequality for negatively superadditive dependent (NSD) random variables is established. By using this inequality, we obtain the almost sure convergence for NSD sequences, which extend... In this paper, Kolmogorov-type inequality for negatively superadditive dependent (NSD) random variables is established. By using this inequality, we obtain the almost sure convergence for NSD sequences, which extends the corresponding results for independent sequences and negatively associated (NA) sequences. In addition, the strong stability for weighted sums of NSD random variables is studied. 展开更多
关键词 Almost sure convergence negatively superadditive dependent strong stability
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Strong Stability of Linear Forms in φ-Mixing Random Variables 被引量:2
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作者 GAN Shixin 《Wuhan University Journal of Natural Sciences》 CAS 2009年第1期6-10,共5页
In this paper some new results of strong stability of linear forms in φ-mixing random variables are given. It is mainly proved that for a sequence of φ-mixing random variables {xn,n≥1} and two sequences of positive... In this paper some new results of strong stability of linear forms in φ-mixing random variables are given. It is mainly proved that for a sequence of φ-mixing random variables {xn,n≥1} and two sequences of positive numbers {an,n≥1} and {bn,n≥1} there exist d dn∈R,n = 1,2,..., such that bn^-1∑i=1^naixi-dn→0 a.s.under some suitable conditions. The results extend and improve the corresponding theorems for independent identically distributed random variables. 展开更多
关键词 strong stability linear form φ-mixing random variable sequence strong law of large numbers
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STRONG STABILITY PRESERVING PROPERTY OF THE DEFERRED CORRECTION TIME DISCRETIZATION 被引量:2
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作者 Yuan Liu Chi-Wang Shu Mengping Zhang 《Journal of Computational Mathematics》 SCIE CSCD 2008年第5期633-656,共24页
In this paper, we study the strong stability preserving (SSP) property of a class of deferred correction time discretization methods, for solving the method-of-lines schemes approximating hyperbolic partial differen... In this paper, we study the strong stability preserving (SSP) property of a class of deferred correction time discretization methods, for solving the method-of-lines schemes approximating hyperbolic partial differential equations. 展开更多
关键词 strong stability preserving Deferred correction time discretization.
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HERMITE WENO SCHEMES WITH STRONG STABILITY PRESERVING MULTI-STEP TEMPORAL DISCRETIZATION METHODS FOR CONSERVATION LAWS
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作者 Xiaofeng Cai Jun Zhu Jianxian Qiu 《Journal of Computational Mathematics》 SCIE CSCD 2017年第1期52-73,共22页
Based on the work of Shu [SIAM J. Sci. Stat. Comput, 9 (1988), pp.1073-1084], we construct a class of high order multi-step temporal discretization procedure for finite volume Hermite weighted essential non-oscillat... Based on the work of Shu [SIAM J. Sci. Stat. Comput, 9 (1988), pp.1073-1084], we construct a class of high order multi-step temporal discretization procedure for finite volume Hermite weighted essential non-oscillatory (HWENO) methods to solve hyperbolic conservation laws. The key feature of the multi-step temporal discretization procedure is to use variable time step with strong stability preserving (SSP). The multi-step tem- poral discretization methods can make full use of computed information with HWENO spatial discretization by holding the former computational values. Extensive numerical experiments are presented to demonstrate that the finite volume HWENO schemes with multi-step diseretization can achieve high order accuracy and maintain non-oscillatory properties near discontinuous region of the solution. 展开更多
关键词 Key words: Multi-step temporal discretization Hermite weighted essentially non-oscillatoryscheme Uniformly high order accuracy strong stability preserving Finite volume scheme.
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Strong stability of an optimal control hybrid system in fed-batch fermentation
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作者 Zhenyu Dong Bing Tan +4 位作者 Yuduo Zhang Jinlong Yuan Enmin Feng Hongchao Yin Zhilong Xiu 《International Journal of Biomathematics》 SCIE 2018年第4期1-17,共17页
In this paper, we consider a nonlinear hybrid dynamic (NHD) system to describe fedbatch culture where there is no analytical solutions and no equilibrium points. Our goal is to prove the strong stability with respec... In this paper, we consider a nonlinear hybrid dynamic (NHD) system to describe fedbatch culture where there is no analytical solutions and no equilibrium points. Our goal is to prove the strong stability with respect to initial state for the NHD system. To this end, we construct corresponding linear variational system (LVS) for the solution of the NHD system, also prove the boundedness of fundamental matrix solutions for the LVS. On this basis, the strong stability is proved by such boundedness. 展开更多
关键词 Nonlinear hybrid system linear variational system fundamental matrix solution strong stability.
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Strong stability of optimal design to dynamic system for the fed-batch culture
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作者 Jinxing Zhang Jinlong Yuan +3 位作者 Zhenyu Dong Enmin Feng Hongchao Yin Zhilong Xiu 《International Journal of Biomathematics》 2017年第2期65-82,共18页
Most economic and industrial processes are governed by inherently nonlinear dynamic system in which mathematical analysis (with few exceptions) is unable to provide general solutions; even the conditions to the exis... Most economic and industrial processes are governed by inherently nonlinear dynamic system in which mathematical analysis (with few exceptions) is unable to provide general solutions; even the conditions to the existence of equilibrium point for the nonlinear dynamic system are simply not established in some special cases. In this paper, based on numerical solution of a nonlinear multi-stage automatic control dynamic (NMACD) in fed-batch culture of glycerol bioconversion to 1,3-propanediol (1,3-PD) induced by KlebsieUa pneumoniae (K. pneumoniae), we consider an optimal design of the NMACD system. For convenience, the NMACD system is reconstructed together with the existence, uniqueness and continuity of solutions are discussed. Our goal is to prove the strong stability with respect to the perturbation of initial state for the solution to the NMACD system. To this end, we construct corresponding linear variational system for the solution to the NMACD system, and also prove the boundedness of fundamental matrix solutions to the linear variational system. On this basis, we prove the strong stability appearing above through the application of this boundedness. 展开更多
关键词 Nonlinear dynamic system linear variational system strong stability.
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SOME LIMIT THEOREMS FOR SEQUENCES OF PAIRWISE NQD RANDOM VARIABLES 被引量:8
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作者 甘师信 陈平炎 《Acta Mathematica Scientia》 SCIE CSCD 2008年第2期269-281,共13页
In this article, the authors study some limit properties for sequences of pairwise NQD random variables, which are not necessarily identically distributed. They obtain Baum and Katz complete convergence and the strong... In this article, the authors study some limit properties for sequences of pairwise NQD random variables, which are not necessarily identically distributed. They obtain Baum and Katz complete convergence and the strong stability of Jamison's weighted sums for pairwise NQD random variables, which may have different distributions. Some wellknown results are improved and extended. 展开更多
关键词 Pairwise NQD random variable sequence convergence in probability almost sure convergence complete convergence strong stability
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Direct discontinuous Galerkin method for the generalized Burgers-Fisher equation 被引量:3
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作者 张荣培 张立伟 《Chinese Physics B》 SCIE EI CAS CSCD 2012年第9期72-75,共4页
In this study, we use the direct discontinuous Galerkin method to solve the generalized Burgers-Fisher equation. The method is based on the direct weak formulation of the Burgers-Fisher equation. The two adjacent cell... In this study, we use the direct discontinuous Galerkin method to solve the generalized Burgers-Fisher equation. The method is based on the direct weak formulation of the Burgers-Fisher equation. The two adjacent cells are jointed by a numerical flux that includes the convection numerical flux and the diffusion numerical flux. We solve the ordinary differential equations arising in the direct Galerkin method by using the strong stability preserving Runge^Kutta method. Numerical results are compared with the exact solution and the other results to show the accuracy and reliability of the method. 展开更多
关键词 direct discontinuous Galerkin method Burgers Fisher equation strong stability pre-serving Runge-Kutta method
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Solvability Condition for Robust Stabilization Problem of Control Systems with Parameter Uncertainties 被引量:1
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作者 伍清河 《Journal of Beijing Institute of Technology》 EI CAS 1999年第3期258-263,共6页
Aim The solvability condition for robust stabilization problem associated with a plant family P(s,δ) having parameter uncertainty δ was considered. Methods Using Youla parameterization of the stabilizers this pro... Aim The solvability condition for robust stabilization problem associated with a plant family P(s,δ) having parameter uncertainty δ was considered. Methods Using Youla parameterization of the stabilizers this problem was transformed into a strong stabilization problem associated with a related plant family G (s, δ). Results A necessary solvability condition was established in terms of the parity interlacing property of each element in G(s,δ). Another apparently necessary solvability condition is that every element in P(s,δ) must be stabilizable. Conclusion The two conditions will be compared with each other and it will be shown that every element in G(s,δ) possesses parity interlacing property if P(s,δ) is stabilizable. 展开更多
关键词 robust stabilization parameter uncertainties strong stabilization parity interlacing property
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A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Diffusion Problems 被引量:1
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作者 Qiang Du Lili Ju +1 位作者 Jianfang Lu Xiaochuan Tian 《Communications on Applied Mathematics and Computation》 2020年第1期31-55,共25页
There have been many theoretical studies and numerical investigations of nonlocal diffusion(ND)problems in recent years.In this paper,we propose and analyze a new discontinuous Galerkin method for solving one-dimensio... There have been many theoretical studies and numerical investigations of nonlocal diffusion(ND)problems in recent years.In this paper,we propose and analyze a new discontinuous Galerkin method for solving one-dimensional steady-state and time-dependent ND problems,based on a formulation that directly penalizes the jumps across the element interfaces in the nonlocal sense.We show that the proposed discontinuous Galerkin scheme is stable and convergent.Moreover,the local limit of such DG scheme recovers classical DG scheme for the corresponding local diff usion problem,which is a distinct feature of the new formulation and assures the asymptotic compatibility of the discretization.Numerical tests are also presented to demonstrate the eff ectiveness and the robustness of the proposed method. 展开更多
关键词 Nonlocal diff usion Discontinuous Galerkin method Interior penalty Asymptotic compatibility strong stability preserving
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High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations
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作者 Giacomo Albi Lorenzo Pareschi 《Communications on Applied Mathematics and Computation》 2021年第4期701-718,共18页
We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form,typically used in implicit-explicit(IMEX)methods,is not... We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form,typically used in implicit-explicit(IMEX)methods,is not possible.As shown in Boscarino et al.(J.Sci.Comput.68:975-1001,2016)for Runge-Kutta methods,these semi-implicit techniques give a great flexibility,and allow,in many cases,the construction of simple linearly implicit schemes with no need of iterative solvers.In this work,we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype lineal'advection-diffusion equation and in the setting of strong stability preserving(SSP)methods.Our findings are demonstrated on several examples,including nonlinear reaction-diffusion and convection-diffusion problems. 展开更多
关键词 Semi-implicit methods Implicit-explicit methods Multistep methods strong stability preserving High order accuracy
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On the order of stable compensators for a class of time-delay system 被引量:3
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作者 HanlinHE ZhongshengWANG XiaoXinLIAO 《控制理论与应用(英文版)》 EI 2004年第1期85-88,共4页
The stabilization using a stable compensator does not introduce additional unstable zeros into the closed-loop transfer function beyond those of the original plant, so it is a desirable compensator, the price is that ... The stabilization using a stable compensator does not introduce additional unstable zeros into the closed-loop transfer function beyond those of the original plant, so it is a desirable compensator, the price is that the compensator’s order will go up. This note considered the order of stable compensators for a class of time-delay systems. First, it is shown that for single-loop plants with at most one real right-half plane zero, a special upper bound for the minimal order of a strongly stabilizing compensator can be obtained in terms of the plant order; Second, it is shown that approximate unstable pole-zero cancellation does not occur, and the distances between distinct unstable zeroes are bounded below by a positive constant, then it is possible to find an upper bound for the minimal order of a strongly stabilizing compensator. 展开更多
关键词 STABILIZATION strong stabilization Time-delay system INTERPOLATION
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Stability of Multiplier Ideal Sheaves
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作者 Qi'an GUAN Zhenqian LI Xiangyu ZHOU 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 2022年第5期819-832,共14页
In the present article, the authors find and establish stability of multiplier ideal sheaves, which is more general than strong openness.
关键词 Plurisubharmonic function Multiplier ideal sheaf strong openness and stability Coherent analytic sheaf L^(2)estimate
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Incompressible limit and stability of all-time solutions to 3-D full Navier-Stokes equations for perfect gases
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作者 REN Dan Dan OU Yao Bin 《Science China Mathematics》 SCIE CSCD 2016年第7期1395-1416,共22页
This paper studies the incompressible limit and stability of global strong solutions to the threedimensional full compressible Navier-Stokes equations, where the initial data satisfy the "well-prepared" cond... This paper studies the incompressible limit and stability of global strong solutions to the threedimensional full compressible Navier-Stokes equations, where the initial data satisfy the "well-prepared" conditions and the velocity field and temperature enjoy the slip boundary condition and convective boundary condition, respectively. The uniform estimates with respect to both the Mach number ∈(0, ∈] and time t ∈ [0, ∞) are established by deriving a differential inequality with decay property, where ∈∈(0, 1] is a constant.As the Mach number vanishes, the global solution to full compressible Navier-Stokes equations converges to the one of isentropic incompressible Navier-Stokes equations in t ∈ [0, +∞). Moreover, we prove the exponentially asymptotic stability for the global solutions of both the compressible system and its limiting incompressible system. 展开更多
关键词 incompressible limit full Navier-Stokes equations global strong solution asymptotic stability
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ODE-Based Multistep Schemes for Backward Stochastic Differential Equations
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作者 Shuixin Fang Weidong Zhao 《Numerical Mathematics(Theory,Methods and Applications)》 SCIE CSCD 2023年第4期1053-1086,共34页
In this paper,we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations(BSDEs).By the nonlinear Feynman-Kac formula,we reformulate the BSDE into a pair of referen... In this paper,we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations(BSDEs).By the nonlinear Feynman-Kac formula,we reformulate the BSDE into a pair of reference ordinary differential equations(ODEs),which can be directly discretized by many standard ODE solvers,yielding the corresponding numerical schemes for BSDEs.In particular,by applying strong stability preserving(SSP)time discretizations to the reference ODEs,we can propose new SSP multistep schemes for BSDEs.Theoretical analyses are rigorously performed to prove the consistency,stability and convergency of the proposed SSP multistep schemes.Numerical experiments are further carried out to verify our theoretical results and the capacity of the proposed SSP multistep schemes for solving complex associated problems. 展开更多
关键词 Backward stochastic differential equation parabolic partial differential equation strong stability preserving linear multistep scheme high order discretization
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A Second-Order Implicit-Explicit Scheme for the Baroclinic-Barotropic Split System of Primitive Equations
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作者 Rihui Lan Lili Ju +1 位作者 Zhu Wang Max Gunzburger 《Communications in Computational Physics》 SCIE 2023年第10期1306-1331,共26页
The baroclinic-barotropic mode splitting technique is commonly employed in numerical solutions of the primitive equations for ocean modeling to deal with the multiple time scales of ocean dynamics.In this paper,a seco... The baroclinic-barotropic mode splitting technique is commonly employed in numerical solutions of the primitive equations for ocean modeling to deal with the multiple time scales of ocean dynamics.In this paper,a second-order implicit-explicit(IMEX)scheme is proposed to advance the baroclinic-barotropic split system.Specifically,the baroclinic mode and the layer thickness of fluid are evolved explicitly via the second-order strong stability preserving Runge-Kutta scheme,while the barotropic mode is advanced implicitly using the linearized Crank-Nicolson scheme.At each time step,the baroclinic velocity is first computed using an intermediate barotropic velocity.The barotropic velocity is then corrected by re-advancing the barotropic mode with an improved barotropic forcing.Finally,the layer thickness is updated by coupling the baroclinic and barotropic velocities together.In addition,numerical inconsistencies on the discretized sea surface height caused by the mode splitting are alleviated via a reconciliation process with carefully calculated flux deficits.Temporal truncation error is also analyzed to validate the second-order accuracy of the scheme.Finally,two benchmark tests from the MPAS-Ocean platform are conducted to numerically demonstrate the performance of the proposed IMEX scheme. 展开更多
关键词 Primitive equations baroclinic-barotropic splitting implicit-explicit strong stability preserving RK SSH reconciliation
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On Complete Hypersurfaces with Constant Mean Curvature and Finite L^p-norm Curvature in R^(n+1)
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作者 Yi Bing SHEN Xiao Hua ZHU 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2005年第3期631-642,共12页
By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1+1 must be minimal, so that M is a hyperplane if it is strongly stabl... By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1+1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n+1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1+1 are considered. 展开更多
关键词 Constant mean curvature strong stability L^p-norm curvature
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A Continuous State Feedback Controller Design for High-order Nonlinear Systems with Polynomial Growth Nonlinearities 被引量:2
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作者 Guo-Dong Zhao Na Duan 《International Journal of Automation and computing》 EI CSCD 2013年第4期267-274,共8页
In this paper, we investigate the problem of global stabilization for a general class of high-order and non-smoothly stabilizable nonlinear systems with both lower-order and higher-order growth conditions. The designe... In this paper, we investigate the problem of global stabilization for a general class of high-order and non-smoothly stabilizable nonlinear systems with both lower-order and higher-order growth conditions. The designed continuous state feedback controller is recursively constructed to guarantee the global strong stabilization of the closed-loop system. 展开更多
关键词 Nonlinear systems lower-order and high-order nonlinearities state feedback adding a power integrator global strong stabilization.
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Strategically supported cooperation in dynamic games with coalition structures 被引量:5
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作者 WANG Lei GAO HongWei +2 位作者 PETROSYAN Leon QIAO Han SEDAKOV Artem 《Science China Mathematics》 SCIE CSCD 2016年第5期1015-1028,共14页
The problem of strategic stability of long-range cooperative agreements in dynamic games with coalition structures is investigated. Based on imputation distribution procedures, a general theoretical framework of the d... The problem of strategic stability of long-range cooperative agreements in dynamic games with coalition structures is investigated. Based on imputation distribution procedures, a general theoretical framework of the differential game with a coalition structure is proposed. A few assumptions about the deviation instant for a coalition are made concerning the behavior of a group of many individuals in certain dynamic environments.From these, the time-consistent cooperative agreement can be strategically supported by ε-Nash or strong ε-Nash equilibria. While in games in the extensive form with perfect information, it is somewhat surprising that without the assumptions of deviation instant for a coalition, Nash or strong Nash equilibria can be constructed. 展开更多
关键词 cooperative game theory coalition structure strategic stability imputation distribution procedure deviation instant ε-Nash equilibrium strong ε-Nash equilibrium
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