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An accurate and efficient space-time Galerkin spectral method for the subdiffusion equation

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Science China Mathematics 2024年第10期67卷 2387-2408页

作者： Wei Zeng Chuanju Xu school of {1. sciences and Fujian Provincial Key Laboratory of {1. Modeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper, we design and analyze a space-time spectral method for the subdiffusion ***, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the initial time. Consequently, traditional high-order numerical methods in time are inefficient. The second obstacle is that the resulting system of the space-time spectral approach is usually large and time-consuming to solve. We aim at overcoming the first difficulty by proposing a novel approach in time, which is based on variable transformation techniques. Suitable ψ-fractional Sobolev spaces and a new variational framework are introduced to establish the well-posedness of the associated variational problem. This allows us to construct our space-time spectral method using a combination of temporal generalized Jacobi polynomials(GJPs) and spatial Legendre polynomials. For the second difficulty, we propose a fast algorithm to effectively solve the resulting linear system. The fast algorithm makes use of a matrix diagonalization in space and QZ decomposition in time. Our analysis and numerical experiments show that the proposed method is exponentially convergent with respect to the polynomial degrees in both space and time directions, even though the exact solution has very limited regularity.

关键词： subdiffusion equations variable transformation Ψ-Sobolev spaces well-posedness space-time Galerkin spectral method error estimate fast algorithm

Optimal regularity of positive solutions of the Hénon-Hardy equation and related equations

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Science China Mathematics 2024年第10期67卷 2283-2302页

作者： Zongming Guo Fangshu Wan College of Mathematics and Information Science Henan Normal UniversityXinxiang 453007China school of {1. sciences University of Science and Technology of ChinaHefei 230026China school of {1. sciences Anhui UniversityHefei 230601China

We present a new method to determine the optimal regularity of positive solutions u∈C^(4)(Ω\1.)∩C^(0)(Ω) of the Hénon-Hardy equation,i.e.,Δ^(2)u=|x|^(α)u^(p)inΩ,(0,1. where Ω■RN(N≥4) is a bounded smooth domain with 0∈Ω,α>-4,and p∈*** is clear that 0 is an isolated singular point of solutions of(0.1. and the optimal regularity of u in Ω relies on the parameter α.It is also important to see that the regularity of u at x=0 determines the regularity of u in Ω.We first establish asymptotic expansions up to arbitrary orders at x=0 of prescribed positive solutions u ∈C^(4)(Ω1.) ∩ C^(0)(Ω)of(0.1..Then we show that the regularity at x=0 of each positive solution u of(0.1. can be determined by some terms in asymptotic expansions of the related positive radial solution of the equation(0.1. with Ω=B,where B is the unit ball of R^(N).The main idea works for more general equations with singular weights.

关键词： H´enon-Hardy equation positive solutions optimal regularity singular point asymptotic expansions

MODIFIED STOCHASTIC EXTRAGRADIENT METHODS FOR STOCHASTIC VARIATIONAL INEQUALITY

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Journal of Computational Mathematics 2024年第2期42卷 390-414页

作者： Ling Zhang Lingling Xu school of Management and Economic Beijing Institute of TechnologyBeijing 100081China school of {1. sciences Nanjing Normal UniversityNanjing 210023China school of {1. sciences and Jiangsu Key Laboratory for NSLSCS Nanjing Normal UniversityNanjing 210023China

In this paper,we consider two kinds of extragradient methods to solve the pseudo-monotone stochastic variational inequality ***,we present the modified stochastic extragradient method with constant step-size(MSEGMC)and prove the convergence of *** the strong pseudo-monotone operator and the exponentially growing sample sequences,we establish the R-linear convergence rate in terms of the mean natural residual and the oracle complexity O(1.ǫ).Second,we propose a modified stochastic extragradient method with adaptive step-size(MSEGMA).In addition,the step-size of MSEGMA does not depend on the Lipschitz constant and without any line-search ***,we use some numerical experiments to verify the effectiveness of the two algorithms.

关键词： Stochastic variational inequality Pseudo-monotone Modified stochastic ex-tragradient methods Adaptive step-size

Stochastic Maximum Principle for Square-Integrable Optimal Control of Linear Stochastic Systems

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Chinese Annals of Mathematics,Series B 2024年第5期45卷 661-676页

作者： Shanjian TANG Xueqi WANG Department of Finance and Control sciences School of Mathematical Sciences Fudan University school of {1. sciences Fudan University

The authors give a stochastic maximum principle for square-integrable optimal control of linear stochastic systems. The control domain is not necessarily convex and the cost functional can have a quadratic growth. In ... 详细信息

关键词： 60H1. 93E20 Linear stochastic system Optimal control Square integrability Stochastic maximum principle

Aubry-Mather theory for contact Hamiltonian systems Ⅲ

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Science China Mathematics 2024年第11期67卷 2541-2570页

作者： Panrui Ni Lin Wang Shanghai Center for mathematical {1. Fudan University school of Mathematics and Statistics Beijing Institute of Technology

By exploiting the contact Hamiltonian dynamics（T*M × R,Φ_t） around the Aubry set of contact Hamiltonian systems,we provide a relation among the Mather set,the Φ_t-recurrent set,the strongly static set,the Aubry set,the Ma?é set,and the Φ_t-non-wandering ***,we consider the strongly static set,as a new flow-invariant set between the Mather set and the Aubry set in the strictly increasing *** show that this set plays an essential role in the representation of certain minimal forward weak Kolmogorov-Arnold-Moser（KAM） solutions and the existence of transitive orbits around the Aubry set.

关键词： Aubry-Mather theory weak KAM theory contact Hamiltonian systems Hamilton-Jacobi equations

Diophantine equations and Fermat's last theorem for multivariate(skew-)polynomials

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Applied Mathematics(A Journal of Chinese Universities) 2024年第1期39卷 159-173页

作者： PAN Jie JIA Yu-ming LI Fang school of {1. sciences Zhejiang UniversityHangzhou 310058China Henan Experimental High school Zhengzhou 450003China

Fermat’s Last Theorem is a famous theorem in number theory which is diﬃcult to ***,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very *** aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.

关键词： Fermat's last theorem polynomial ring skew polynomial ring

On Stability Discrimination of Limit Cycles for Piecewise Smooth Systems

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Acta Mathematica Sinica,English Series 2024年第7期40卷 1785-1803页

作者： Mao An HAN Xia Yu ZHOU school of {1. sciences Zhejiang Normal UniversityJinhua321004P.R.China

This paper is concerned with the problem of stability discrimination of limit cycles for piecewise smooth *** first establish the Poincaré map near a periodic orbit,and deduce the first order derivative of the map for general piecewise smooth systems on the ***,we obtain a sufficient condition for determining the stability of limit cycles for these systems.

关键词： Piecewise smooth system limit cycle stability

Quantitative Green's function estimates for lattice quasi-periodic Schrodinger operators

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Science China Mathematics 2024年第5期67卷 1011-1058页

作者： Hongyi Cao Yunfeng Shi Zhifei Zhang school of {1. sciences Peking UniversityBeijing 100871China school of Mathematics Sichuan UniversityChengdu 610064China

In this paper,we establish quantitative Green’s function estimates for some higher-dimensional lattice quasi-periodic(QP)Schrodinger *** resonances in the estimates can be described via a pair of symmetric zeros of certain functions,and the estimates apply to the sub-exponential-type non-resonance *** the application of quantitative Green’s function estimates,we prove both the arithmetic version of Anderson localization and the finite volume version of(1.2-)-Holder continuity of the integrated density of states(IDS)for such QP Schrodinger *** gives an affirmative answer to Bourgain’s problem in Bourgain(2000).

关键词： Holder continuity of the IDS quantitative Green's function estimates quasi-periodic Schrodinger operators arithmetic Anderson localization multi-scale analysis

On hypersurfaces of H2×H2

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Science China Mathematics 2024年第2期67卷 339-366页

作者： Dong Gao Hui Ma Zeke Yao school of Science Beijing University of Civil Engineering and Architecture Department of mathematical {1. Tsinghua University

In this paper, we study hypersurfaces of H2× H2. We first classify the hypersurfaces with constant principal curvatures and constant product angle functions. Then we classify homogeneous hypersurfaces and isoparametric hypersurfaces, respectively. Finally, we classify the hypersurfaces with at most two distinct constant principal curvatures, as well as those with three distinct constant principal curvatures under some additional conditions.

关键词： constant principal curvature homogeneous hypersurface isoparametric hypersurface 53C42 53B25 53C40