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On equivalence of state-based potential games

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science China(Information sciences) 2024年第6期67卷 313-327页

作者： Han WU Jinhuan WANG institute of mathematics School of ScienceHebei University of Technology

In this paper, we explore state-based potential games using the semi-tensor product of matrices.First, applying the potential equation, we derive both a necessary and sufficient condition as well as a sufficient condition to verify whether a state-based game qualifies as a potential game. Next, we present two static equivalence conditions of state-based potential games. We then delve into dynamic equivalence. We propose a criterion that allows us to identify state-based games that are dynamically equivalent to state-based potential games and share similar dynamic properties. Ultimately, we introduce the concept of state-based networked evolutionary games. We provide a necessary and sufficient condition to ensure that a state-based networked evolutionary game can be classified as a state-based potential game.

关键词： semi-tensor product of matrices state-based potential game verification equivalence state-based networked evolutionary game

C^(α)regularity of weak solutions of non-homogeneous ultraparabolic equations with drift terms

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science China mathematics 2024年第1期67卷 23-44页

作者： Wendong Wang Liqun Zhang school of mathematical sciences Dalian University of TechnologyDalian 116024China institute of mathematics Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

We consider a class of non-homogeneous ultraparabolic differential equations with singular drift terms arising from some physical models,and prove that weak solutions are H?lder continuous,which are sharp in some sense and also generalize the well-known De Giorgi-Nash-Moser theory to degenerate parabolic equations satisfying the H?rmander hypoellipticity condition.The new ingredients are manifested in two aspects:on the one hand,for lower-order terms,we exploit a new Sobolev inequality suitable for the Moser iteration by improving the result of Pascucci and Polidoro(2004);on the other hand,we explore the G-function from an early idea of Kruzhkov(1964)and an approximate weak Poincaréinequality for non-negative weak sub-solutions to prove the H?lder regularity.

关键词： ultraparabolic equations Moser iteration Poincare inequality C^a regularity

2-local unstable homotopy groups of indecomposable A32-complexes

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science China mathematics 2024年第3期67卷 607-626页

作者： Zhongjian Zhu Jianzhong Pan College of mathematics and Physics Wenzhou University Hua Loo-Keng Key Laboratory of mathematics Institute of MathematicsAcademy of Mathematics and Systems ScienceChinese Academy of Sciences school of mathematical sciences University of Chinese Academy of Sciences

In this paper,we calculate the 2-local unstable homotopy groups of indecomposable A32-complexes.The main technique used is analysing the homotopy property of J(A,X),defined by Gray(1973) for a CW-pair(X,A),which is ho... 详细信息

关键词： homotopy groups A 2 n -complexes fibration sequence relative James construction 55P15

Boundary points, minimal L^(2) integrals and concavity property Ⅱ: Weakly pseudoconvex Kähler manifolds

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science China mathematics 2024年第7期67卷 1665-1718页

作者： Qi’an Guan Zhitong Mi Zheng Yuan school of mathematical sciences Peking UniversityBeijing100871China school of mathematics and Statistics Beijing Jiaotong UniversityBeijing100044China institute of mathematics Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing100190China

In this paper, we consider minimal L^(2) integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex K?hler manifolds with Lebesgue measurable gain related to modules at boundary points of the sublevel sets, and establish a concavity property of the minimal L^(2) integrals. As applications, we present a necessary condition for the concavity degenerating to linearity, a concavity property related to modules at inner points of the sublevel sets, an optimal support function related to modules, a strong openness property of modules and a twisted version, and an effectiveness result of the strong openness property of modules.

关键词： minimal L^(2)integrals plurisubharmonic functions boundary points weakly pseudoconvex K?hler manifold

High-order Soliton Matrix for the Third-order Flow Equation of the Gerdjikov-Ivanov Hierarchy Through the Riemann-Hilbert Method

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Acta Mathematicae Applicatae Sinica 2024年第2期40卷 358-378页

作者： Jin-yan ZHU Yong CHEN school of mathematical sciences Shanghai Key Laboratory of Pure Mathematics and Mathematical PracticeEast China Normal UniversityShanghai 200062China College of mathematics and Systems science Shandong University of Science and TechnologyQingdao 266590China

The Gerdjikov-Ivanov(GI)hierarchy is derived via recursion operator,in this article,we mainly investigate the third-order flow GI equation.In the framework of the Riemann-Hilbert method,the soliton matrices of the third-order flow GI equation with simple zeros and elementary high-order zeros of Riemann-Hilbert problem are constructed through the standard dressing process.Taking advantage of this result,some properties and asymptotic analysis of single soliton solution and two soliton solution are discussed,and the simple elastic interaction of two soliton are proved.Compared with soliton solution of the classical second-order flow,we find that the higher-order dispersion term affects the propagation velocity,propagation direction and amplitude of the soliton.Finally,by means of a certain limit technique,the high-order soliton solution matrix for the third-order flow GI equation is derived.

关键词： Gerdjikov-Ivanov hierarchy third-order flow GI equation Riemann-Hilbert method high-order soliton

A Deep Learning Method for Computing Eigenvalues of the Fractional Schrödinger Operator

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Journal of Systems science & Complexity 2024年第2期37卷 391-412页

作者： GUO Yixiao MING Pingbing LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China school of mathematical sciences University of Chinese Academy of SciencesBeijing 100049China

The authors present a novel deep learning method for computing eigenvalues of the fractional Schrödinger operator.The proposed approach combines a newly developed loss function with an innovative neural network architecture that incorporates prior knowledge of the problem.These improvements enable the proposed method to handle both high-dimensional problems and problems posed on irregular bounded domains.The authors successfully compute up to the first 30 eigenvalues for various fractional Schrödinger operators.As an application,the authors share a conjecture to the fractional order isospectral problem that has not yet been studied.

关键词： Eigenvalue problem deep learning fractional Schrödinger operator isospectral problem

Revealing excited states of rotational Bose-Einstein condensates

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The Innovation 2024年第1期5卷 41-48页

作者： Jianyuan Yin Zhen Huang Yongyong Cai Qiang Du Lei Zhang school of mathematical sciences Laboratory of Mathematics and Applied MathematicsPeking UniversityBeijing 100871China Department of mathematics National University of SingaporeSingapore 119076Singapore Department of mathematics University of CaliforniaBerkeleyBerkeleyCA 94720USA school of mathematical sciences Beijing Normal UniversityBeijing 100875China Department of Applied Physics and Applied mathematics and Data science institute Columbia UniversityNew YorkNY 10027USA Beijing International Center for mathematical Research Center for Quantitative BiologyCenter for Machine Learning ResearchPeking UniversityBeijing 100871China

Rotational Bose-Einstein condensates can exhibit quantized vortices as topological excitations.In this study,the ground and excited states of the rotational Bose-Einstein condensates are systematically studied by calculating the stationary points of the Gross-Pitaevskii energy functional.Various excited states and their connections at different rotational frequencies are revealed in solution landscapes constructed with the constrained high-index saddle dynamics method.Four excitation mechanisms are identified:vortex addition,rearrangement,merging,and splitting.We demonstrate changes in the ground state with increasing rotational frequencies and decipher the evolution of the stability of ground states.

关键词： method rotational excited

Threshold Selection and Resource Allocation for Quantized Identification

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Journal of Systems science & Complexity 2024年第1期37卷 204-229页

作者： WANG Ying LI Xin ZHAO Yanlong ZHANG Ji-Feng Key Laboratory of Systems and Control Institute of Systems ScienceAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China school of mathematical sciences University of Chinese Academy of SciencesBeijing 100049China

This paper is concerned with the optimal threshold selection and resource allocation problems of quantized identification, whose aims are improving identification efficiency under limited resources. Firstly, the first-order asymptotically optimal quantized identification theory is extended to the weak persistent excitation condition. Secondly, the characteristics of time and space complexities are established based on the Cramér-Rao lower bound of quantized systems. On these basis, the optimal selection methods of fixed thresholds and adaptive thresholds are established under aperiodic signals, which answer how to achieve the best efficiency of quantized identification under the same time and space complexity. In addition, based on the principle of maximizing the identification efficiency under a given resource, the optimal resource allocation methods of quantized identification are given for the cases of fixed thresholds and adaptive thresholds, respectively, which show how to balance time and space complexity to realize the best identification efficiency of quantized identification.

关键词： Quantized output resource allocation system identification threshold selection.

Uniqueness of the Reissner-Nordstr?m metric near the null infinity

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science China mathematics 2024年第8期67卷 1933-1956页

作者： Xiaokai He Xiaoning Wu Naqing Xie school of mathematics and Statistics Hunan First Normal University institute of mathematics Academy of Mathematics and Systems Science and Hua Loo-Keng Key Laboratory of MathematicsChinese Academy of Sciences school of mathematical sciences Fudan University

In the Newman-Penrose formalism, we prove that the Reissner-Nordstr?m metric is the only asymptotically fat, static, axially symmetric, Petrov type-D analytic solution of the electro-vacuum Einstein-Maxwell equations ... 详细信息

关键词： uniqueness theorem Reissner-Nordström metric Newman-Penrose formalism