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Incomplete crossing and semi-topological horseshoes

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arXiv 2025年

作者： Cheng, Junfeng Yang, Xiao-Song School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

This paper enriches the topological horseshoe theory using finite subshift theory in symbolic dynamical systems, and develops an elementary framework addressing incomplete crossing and semi-horseshoes. Two illustrative examples are provided: one from the perturbed Duffing system and another from a polynomial system proposed by Chen, demonstrating the prevalence of semi-horseshoes in chaotic systems. Moreover, the semi-topological horseshoe theory enhances the detection of chaos and improves the accuracy of topological entropy estimation. Copyright © 2025, The Authors. All rights reserved.

关键词： Topology

Complex generalized Gauss-Radau quadrature rules for Hankel transforms of integer order

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arXiv 2024年

作者： Wang, Haiyong Wu, Menghan School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

Complex Gaussian quadrature rules for oscillatory integral transforms have the advantage that they can achieve optimal asymptotic order. However, their existence for Hankel transform can only be guaranteed when the order of the transform belongs to [0, 1/2]. In this paper we introduce a new family of Gaussian quadrature rules for Hankel transforms of integer order. We show that, if adding certain value and derivative information at the left endpoint, then complex generalized Gauss-Radau quadrature rules that guarantee existence can be constructed and their nodes and weights can be calculated from a half-size Gaussian quadrature rule with respect to the generalized Prudnikov weight function. Orthogonal polynomials that are closely related to such quadrature rules are investigated and their existence for even degrees is proved. Numerical experiments are presented to show the performance of the proposed *** Codes 65R10, 65D32 © 2024, CC BY-NC-ND.

关键词： Orthogonal functions

Stochastic homogenization of nonlinear evolution equations with space-time nonlocality

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arXiv 2023年

作者： Chen, Junlong Tang, Yanbin School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China

In this paper we consider the homogenization problem of nonlinear evolution equations with space-time non-locality, the problems are given by Beltritti and Rossi [JMAA, 2017, 455: 1470-1504]. When the integral kernel J(x,t;y,s) is re-scaled in a suitable way and the oscillation coefficient ν(x,t;y,s) possesses periodic and stationary structure, we show that the solutions uΕ(x,t) to the perturbed equations converge to u0(x,t), the solution of corresponding local nonlinear parabolic equation as scale parameter Ε → 0+. Then for the nonlocal linear index p = 2 we give the convergence rate such that (formula presented). Furthermore, we obtain that the normalized difference (formula presented) converges to a solution of an SPDE with additive noise and constant coefficients. Finally, we give some numerical formats for solving non-local space-time homogenization. Copyright © 2023, The Authors. All rights reserved.

关键词： Additive noise

Density functions for the overdamped generalized Langevin equation and its Euler-Maruyama method: smoothness and convergence

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arXiv 2024年

作者： Dai, Xinjie Jin, Diancong School of Mathematics and Statistics Yunnan University Kunming650504 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

This paper focuses on studying the convergence rate of the density function of the Euler- Maruyama (EM) method, when applied to the overdamped generalized Langevin equation with fractional noise which serves as an important model in many fields. Firstly, we give an improved upper bound estimate for the total variation distance between random variables by their Malliavin-Sobolev norms. Secondly, we establish the existence and smoothness of the density function for both the exact solution and the numerical one. Based on the above results, the convergence rate of the density function of the numerical solution is obtained, which relies on the regularity of the noise and kernel. This convergence result provides a powerful support for numerically capturing the statistical information of the exact solution through the EM method. Copyright © 2024, The Authors. All rights reserved.

关键词： Differential equations

Open-loop and closed-loop solvabilities for discrete-time mean-field stochastic linear quadratic optimal control problems

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arXiv 2023年

作者： Song, Teng Liu, Bin Department of Mathematics Wuhan University of Technology China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China

This paper discusses the discrete-time mean-field stochastic linear quadratic optimal control problems, whose weighting matrices in the cost functional are not assumed to be definite. The open-loop solvability is characterized by the existence of the solution to a mean-field forward-backward stochastic difference equations with a convexity condition and a stationary condition. The closed-loop solvability is presented by virtue of the existences of the regular solution to the generalized Riccati equations and the solution to the linear recursive equation, which is also shown by the uniform convexity of the cost functional. Moreover, based on a family of uniformly convex cost functionals, the finiteness of the problem is characterized. Also, it turns out that a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem. Finally, some examples are given to illustrate the theory *** Codes 93E20, 93C55, 49N10 Copyright © 2023, The Authors. All rights reserved.

关键词： Riccati equations

Multiple-relaxation-time finite-difference lattice Boltzmann model for the nonlinear convection-diffusion equation

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Multiple-relaxation-time finite-difference lattice Boltzmann...

第十二届全国流体力学学术会议

作者： Xinmeng Chen Zhenhua Chai Jinlong Shang Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Key Laboratory of Engineering Modeling and Scientific Computing

In this paper, a multiple-relaxation-time finite-difference lattice Boltzmann method(MRT-FDLBM) is developed for the nonlinear convection-diffusion equation(NCDE). Through designing the equilibrium distribution function and the source term properly, the NCDE can be recovered exactly from MRT-FDLBM. We also conduct the von Neumann stability analysis on the present MRT-FDLBM and its special case, i.e., single-relaxationtime finite-difference lattice Boltzmann method(SRT-FDLBM). Then, a simplified version of MRT-FDLBM(SMRT-FDLBM) is also proposed, which can save about 15% computational cost. In addition, a series of real and complex-value NCDEs, including the isotropic convection-diffusion equation, Burgers-Fisher equation, sine-Gordon equation, heat-conduction equation, and Schr?dinger equation, are used to test the performance of MRT-FDLBM. The results show that both MRT-FDLBM and SMRT-FDLBM have second-order convergence rates in space and time. Finally, the stability and accuracy of five different models are compared, including the MRT-FDLBM, SMRT-FDLBM,SRT-FDLBM, the previous finite-difference lattice Boltzmann method [H. Wang, B. Shi et al., ***. Comput. 309, 334(2017)], and the lattice Boltzmann method(LBM). The stability tests show that the sequence of stability from high to low is as follows: MRT-FDLBM, SMRT-FDLBM,SRT-FDLBM, the previous finite-difference lattice Boltzmann method, and LBM. In most of the precision test results, it is found that the order from high to low of precision is MRT-FDLBM,SMRT-FDLBM, SRT-FDLBM, and the previous finite-difference lattice Boltzmann method.

关键词： lattice Boltzmann method finite-difference multiple-relaxation-time

Bridging Fairness Gaps: A (Conditional) Distance Covariance Perspective in Fairness Learning

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arXiv 2024年

作者： Huang, Ruifan Liu, Haixia School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan Hubei China School of Mathematics and Statistics Institute of Interdisciplinary Research for Mathematics and Applied Science Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan China

We bridge fairness gaps from a statistical perspective by selectively utilizing either conditional distance covariance or distance covariance statistics as measures to assess the independence between predictions and sensitive attributes. We enhance fairness by incorporating sample (conditional) distance covariance as a manageable penalty term into the machine learning process. Additionally, we present the matrix form of empirical (conditional) distance covariance for parallel calculations to enhance computational efficiency. Theoretically, we provide a proof for the convergence between empirical and population (conditional) distance covariance, establishing necessary guarantees for batch computations. Through experiments conducted on a range of real-world datasets, we have demonstrated that our method effectively bridges the fairness gap in machine learning. Copyright © 2024, The Authors. All rights reserved.

关键词： Contrastive Learning

Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations

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Science China Mathematics 2022年第8期65卷 1731-1748页

作者： Waixiang Cao Dongfang Li Zhimin Zhang School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Beijing Computational Science Research Center Beijing 100193China Department of Mathematics Wayne State UniversityDetroitMI 48202USA

In this paper,we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave *** error estimates in time and superconvergent error estimates in space are established without certain time-step *** key is to estimate directly the solution bounds in the H-norm for both the nonlinear wave equation and the corresponding fully discrete scheme,while the previous investigations rely on the temporal-spatial error splitting *** examples are presented to confirm energy-conserving properties,unconditional convergence and optimal error estimates,respectively,of the proposed fully discrete schemes.

关键词： scalar auxiliary variable wave equations stability error estimate superconvergence

ON THE KELLER-SEGEL MODELS INTERACTING WITH A STOCHASTICALLY FORCED INCOMPRESSIBLE VISCOUS FLOW IN R2

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arXiv 2022年

作者： Zhang, Lei Liu, Bin School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China

This paper considers the Keller-Segel model coupled to stochastic Navier-Stokes equations (KS-SNS, for short), which describes the dynamics of oxygen and bacteria densities evolving within a stochastically forced 2D incompressible viscous flow. Our main goal is to investigate the existence and uniqueness of global solutions (strong in the probabilistic sense and weak in the PDE sense) to the KS-SNS system. A novel approximate KS-SNS system with proper regularization and cut-off operators in Hs(R2) is introduced, and the existence of approximate solution is proved by some a priori uniform bounds and a careful analysis on the approximation scheme. Under appropriate assumptions, two types of stochastic entropy-energy inequalities that seem to be new in their forms are derived, which together with the Prohorov theorem and Jakubowski-Skorokhod theorem enables us to show that the sequence of approximate solutions converges to a global martingale weak solution. In addition, when χ(·) ≡ const. > 0, we prove that the solution is pathwise unique, and hence by the Yamada-Wantanabe theorem that the KS-SNS system admits a unique global pathwise weak solution. Copyright © 2022, The Authors. All rights reserved.

关键词： Navier Stokes equations