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Stabilities of delay stochastic McKean-Vlasov equations in the G-framework

Science China(Information Sciences) 2025年第1期68卷 255-277页

作者： Haiyan YUAN Quanxin ZHU Department of Mathematics Heilongjiang Institute of Technology Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education) School of Mathematics and Statistics Hunan Normal University

This article focuses on a type of stochastic Mc Kean-Vlasov equation in the G-framework(G-SMVE) and addresses the stability issue for delay stochastic Mc Kean-Vlasov equations in the G-framework(G-SMVDEs). Distribution dependence and uncertainty prevent us from directly applying the stochastic analysis method for stochastic Mc Kean-Vlasov equations(SMVEs)to G-SMVEs directly. To overcome this difficulty, we introduce definitions, including the derivative of a function with a law under the G-expectation and the Lions derivatives. Then we construct a new G-It? formula for G-SMVEs according to the G-It? formula for stochastic differential equations in the G-framework(G-SDEs) and the It? formula for SMVEs. Using the new G-It? formula and the Lyapunov functional method, we investigate the moment exponential stability and almost sure asymptotic stability of G-SMVDEs. To overcome the difficulty in obtaining the distribution dependence of the exact solution to the G-SMVDE, we introduce the empirical measure and the corresponding interacting particle system, and then prove the stability equivalence between the underlying G-SMVDE and the corresponding interacting particle system. Two examples are used to confirm our theoretical results.

关键词： delay stochastic McKean-Vlasov equations Wasserstein distance G-Brownian motion Lions derivative stability

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A CELL-CENTERED MULTIGRID SOLVER FOR THE FINITE VOLUME DISCRETIZATION OF ANISOTROPIC ELLIPTIC INTERFACE PROBLEMS ON IRREGULAR DOMAINS

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Journal of Computational mathematics 2025年第1期43卷 18-42页

作者： Kejia Pan Xiaoxin Wu Hongling Hu Zhilin Li School of Mathematics and Statistics HNP-LAMACentral South UniversityChangsha 410083China Key Laboratory of Computing and Stochastic Mathematics(Ministry of Education) School of Mathematics and StatisticsHunan Normal UniversityChangsha 410081China Department of Mathematics North Carolina State UniversityRaleighNC 27695-8205USA

The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a union of blocks. We assume that the curved interface falls exactly on the boundaries of blocks. The transfinite interpolation technique is applied to generate block-wise distorted quadrilateral meshes, which can resolve the interface with fine geometric details. By an extensive study of the harmonic average point method, an interpolation-free nine-point FV scheme is then derived on such multi-block grids for anisotropic elliptic interface problems with non-homogeneous jump conditions. Moreover, for the resulting linear algebraic systems from cell-centered FV discretization, a high-order prolongation operator based fast cascadic multigrid solver is developed and shown to be robust with respect to both the problem size and the jump of the diffusion coefficients. Various non-trivial examples including four interface problems and an elliptic problem in complex domain without interface, all with tens of millions of unknowns, are provided to show that the proposed multigrid solver is dozens of times faster than the classical algebraic multigrid method as implemented in the code AMG1R5 by Stüben.

关键词： Elliptic interface problem Discontinuous coefficients Anisotropic coefficients Cascadic multigrid method Richardson extrapolation

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The Hilbert expansion of the Boltzmann equation in the incompressible Euler level in a channel

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Science China mathematics 2025年第1期68卷 39-88页

作者： Feimin Huang Weiqiang Wang Yong Wang Feng Xiao Academy of Mathematics and Systems Science Chinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China Key Laboratory of Computing and Stochastic Mathematics(Ministry of Education) School of Mathematics and StatisticsHunan Normal UniversityChangsha 410081China

The study of the hydrodynamic limit of the Boltzmann equation with physical boundary is a challenging problem due to the appearance of the viscous and Knudsen boundary *** this paper,the hydrodynamic limit from the Boltzmann equation with the specular reflection boundary condition to the incompressible Euler equations in a channel is *** on the multi-scaled Hilbert expansion,the equations with boundary conditions and compatibility conditions for interior solutions,and viscous and Knudsen boundary layers are derived under different scaling,***,some uniform estimates for the interior solutions,and viscous and Knudsen boundary layers are *** the help of the L^(2)-L^(∞) framework and the uniform estimates obtained above,the solutions to the Boltzmann equation are constructed by the truncated Hilbert expansion with multiscales,and hence the hydrodynamic limit in the incompressible Euler level is justified.

关键词： Boltzmann equation incompressible Euler equations hydrodynamic limit Hilbert expansion specular reflection boundary condition viscous boundary layer Knudsen boundary layer

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Liouville-type theorem for the stationary inhomogeneous Navier-Stokes equations

arXiv

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arXiv 2025年

作者： Ding, Huiting Tan, Wenke Key Laboratory of Computing and Stochastic Mathematics Ministry of Education School of Mathematics and Statistics Hunan Normal University Changsha Hunan410081 China

In this manuscript, a new Liouville-type theorem for the three-dimensional stationary inhomogeneous Navier-Stokes equations is established. We first localize the Dirichlet energy into the region near the origin in frequency spaces by two times localizations. The first localization is to eliminate the non-zero frequency part coming from the interaction between ρu and u, the second one is to eliminate the non-zero frequency part coming from the interaction between ρ and u. Based on the local formula of Dirichlet energy, we can establish suitable estimates on different frequency parts of u and ρ, then show our new Liouville-type theorem. Copyright © 2025, The Authors. All rights reserved.

关键词： Liouville equation

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Spectrality of a class of Cantor-dust type self-affine tiles

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Bulletin of the Malaysian Mathematical Sciences Society 2025年第4期48卷 1-28页

作者： Liu, Jing-cheng Wang, Jia-jie Zheng, Jia Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education) School of Mathematics and Statistics Hunan Normal University Changsha P.R. China School of Mathematics and Statistics Wuhan University Wuhan P.R. China

We say that a set $$\Omega \subseteq {\mathbb {R}}^n$$ with positive Lebesgue measure is a spectral set if $$L^{2}(\Omega )$$ admits an orthogonal basis of exponentials. In this paper, we study the spectrality of self-affine tiles. We prove a class of sets $$T:=T(M,D)$$ with positive Lebesgue measure satisfying $$MT=\bigcup _{d\in D}(T+d)$$ are spectral sets, where $$M\in M_{2}({\mathbb {Z}})$$ is an expanding matrix with $$|\det {(M)}|=4$$ and $$D=\{{\textbf{0}},\mathbf {\alpha },\beta ,-(\alpha +\beta )\}\subset {\mathbb {Z}}^2$$ is a non-collinear digit set. As an application, we conclude that T is a spectral set if and only if it tiles $${\mathbb {R}}^2$$ by translations.

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Hopf and double Hopf bifurcations in a delayed lateral vibration model of footbridges induced by pedestrians

arXiv

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arXiv 2025年

作者： Li, Xuemei Liu, Yechi Key Laboratory of High Performance Computing and Stochastic Information Processing Department of Mathematics Hunan Normal University Hunan Changsha410081 China

In this paper, we investigate the dynamical behaviors of a delayed lateral vibration model of footbridges proposed based on the facts that pedestrians will reduce their walking speed or stop walking when the response of the footbridge becomes sufficiently large, and that the bridge velocity can not be changed at once when the pedestrians begin to walk on the bridge. By analyzing the distribution of roots of the associated characteristic equation, we find that there are only two types of bifurcations in this model: Hopf bifurcation and double Hopf bifurcation, and give the condition on the stability of the trivial solution. By using the center manifold theorem and bifurcation theory of delayed differential equations, we obtain the dynamical behavior in these bifurcations, specially including the stability of periodic solutions and invariant tori bifurcating from the trivial solution in these bifurcations. Finally, we prove that this model exhibits quasi-periodic vibrations by KAM theorems, besides periodic vibrations. © 2025, CC BY-NC-ND.

关键词： Hopf bifurcation

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Unified Feature Selection Approach for Complex Data Based on Fuzzy β-Covering Reduction via Information Granulation

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IEEE Transactions on Neural Networks and Learning Systems 2025年 PP卷 PP页

作者： Zou, Xiongtao Dai, Jianhua Hunan Normal University Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education) Hunan Provincial Key Laboratory of Intelligent Computing and Language Information Processing College of Information Science and Engineering School of Mathematics and Statistics Changsha 410081 China

Feature selection, as an important step of data analysis, is widely used in the fields of data mining, machine learning, and artificial intelligence. It can not only effectively alleviate the curse of dimensionality and improve model performance but also enhance model interpretability. In the real world, data is usually complex such as different feature types, the presence of missing values, and so on. However, most existing feature selection approaches are only capable of handling data with a single feature type. To address the issue of feature selection under the environment of complex data, this article proposes a unified feature selection (UFS) approach for complex data based on fuzzy β-covering reduction via information granulation. To begin with, several monotonic uncertainty measures for fuzzy β-covering are constructed from the viewpoints of algebra and information theory. Based on the proposed measures, two forward heuristic algorithms are designed for fuzzy β-covering reduction. Meanwhile, the complex data with multiple features is represented by fuzzy β-covering via information granulation. On this basis, a UFS approach is put forward for complex data. Finally, the effectiveness and superiority of the proposed approach are verified through a series of experiments compared with 12 state-of-the-art feature selection approaches. © 2012 IEEE.

关键词： Complex data feature selection fuzzy conditional entropy (FCE) fuzzy rough set (FRS) fuzzy β-covering granular computing

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Correction to: Optimal Error Estimates for Gegenbauer Approximations in Fractional Spaces

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Journal of Scientific computing 2025年第2期103卷 1-2页

作者： Xie, Ruiyi Liu, Wenjie Wang, Haiyong Wu, Boying School of Mathematics Harbin Institute of Technology Harbin People’s Republic of China Key Laboratory of Computing and Stochastic Mathematics Ministry of Education (LCSM) Changsha People’s Republic of China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan People’s Republic of China China and Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan People’s Republic of China

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CPST-GAN: Conditional Probabilistic State Transition Generative Adversarial Network With the Biomedical Large Foundation Models

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IEEE Transactions on Neural Networks and Learning Systems 2025年 PP卷 PP页

作者： Wang, Qiong Xu, Luyun Shan, Yinglu Shen, Wenzhuo Li, Lou Bi, Xia-An Liu, Zhonghua Hunan Normal University Key Laboratory of Computing and Stochastic Mathematics Ministry of Education School of Mathematics and Statistics College of Information Science and Engineering Changsha410081 China Hunan Normal University College of Business Changsha410081 China Hunan Normal University College of Information Science and Engineering Changsha410081 China

The risk prediction of Alzheimer's disease (AD) is crucial for its early prevention and treatment. However, current risk prediction methods face challenges in effectively extracting and fusing multiomics features, particularly overlooking the multilevel evolutionary mechanisms of AD. This article combines biomedical large foundation models with the conditional generative adversarial network (GAN) to mine the evolutionary patterns of AD by considering the regulatory effect of genes on brain lesions. Specifically, we first use biomedical large foundation models to effectively construct high-quality imaging genetic features. Next, a conditional probabilistic state transition mathematical model is constructed to describe AD progression as state transitions of brain regions under genetic regulations. Based on the mathematical model, a conditional probabilistic state transition GAN (CPST-GAN) is proposed. This algorithm can mine the dynamic evolutionary patterns of AD by fusing brain imaging and genetic features to achieve risk prediction of AD. Finally, experiments on the public imaging genetics datasets validate the effectiveness and superiority of CPST-GAN in evolutionary pattern mining and risk prediction of AD. This article not only provides a reliable intelligence algorithm for early intervention of AD but also offers new insights for future research on AD pathogenesis. The code has been published at ***/fmri123456/CPST-GAN. © 2025 IEEE.

关键词： Generative adversarial networks

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Brain-Inspired Fuzzy Graph Convolution Network for Alzheimer's Disease Diagnosis Based on Imaging Genetics Data

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IEEE Transactions on Fuzzy Systems 2025年

作者： Bi, Xia-An Huang, Yangjun Shen, Wenzhuo Yang, Zicheng Mao, Yuhua Xu, Luyun Liu, Zhonghua Hunan Normal University College of Information Science and Engineering Changsha China Hunan Normal University Key Laboratory of Computing and Stochastic Mathematics Ministry of Education School of Mathematics and Statistics Hunan Changsha410081 China Hunan Normal University College of Business Changsha410081 China Hunan Normal University Hunan Changsha410081 China

The analysis of multi-omics biomedical data has become increasingly critical in clinical decision-making for brain diseases like Alzheimer's Disease (AD). However, the inherent fuzziness of biomedical information limits the classification performance of existing methods, and current disease models struggle to explore pathogenetic mechanisms. Facing with these issues, this paper develops a fuzzy graph-based deep learning method to achieve accurate diagnosis and pathogeny identification for brain diseases. Firstly, fuzzy graphs are constructed to describe the associations between pathogenies using fuzzy memberships. Secondly, a mathematical model inspired by the fuzzy mechanisms of brain is established, effectively capturing the fuzzy congregation patterns of feature information across brain regions and genes. Finally, a Brain-Inspired Fuzzy Graph Convolutional Network (BI-FGCN) is proposed. In BI-FGCN, white-boxed convolutional operations are designed based on the mathematical model. Experimental results across multiple brain disease datasets demonstrate the superiority of BI-FGCN in AD diagnosis and pathogeny identification. We provide a reliable supporting method for the diagnosis and treatment of brain diseases. The code of BI-FGCN is available at: ***/fmri123456/BI-FGCN. © 1993-2012 IEEE.

关键词： Convolution

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