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NSNO:Neumann Series Neural Operator for Solving Helmholtz Equations in Inhomogeneous Medium

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Journal of Systems Science & Complexity 2024年第2期37卷 413-440页

作者： CHEN Fukai LIU Ziyang LIN Guochang CHEN Junqing SHI Zuoqiang Department of mathematical sciences Tsinghua UniversityBeijing 100084China Yau mathematical sciences center Tsinghua UniversityBeijing 100084China Yanqi Lake Beijing Institute of mathematical sciences and Applications Beijing 101408China

In this paper,the authors propose Neumann series neural operator(NSNO)to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to *** equation is a crucial partial differential equation(PDE)with applications in various scientific and engineering ***,efficient solver of Helmholtz equation is still a big challenge especially in the case of high ***,deep learning has shown great potential in solving PDEs especially in learning solution *** by Neumann series in Helmholtz equation,the authors design a novel network architecture in which U-Net is embedded inside to capture the multiscale *** experiments show that the proposed NSNO significantly outperforms the state-of-the-art FNO with at least 60%lower relative L^(2)-error,especially in the large wavenumber case,and has 50%lower computational cost and less data ***,NSNO can be used as the surrogate model in inverse scattering *** tests show that NSNO is able to give comparable results with traditional finite difference forward solver while the computational cost is reduced tremendously.

关键词： Helmholtz equation inverse problem neumann series neural network solution operator

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Markovian Quadratic BSDEs with an Unbounded Sub-quadratic Growth

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Chinese Annals of Mathematics,Series B 2024年第3期45卷 441-462页

作者： Jingnan JU Shanjian TANG Shanghai center for mathematical sciences Fudan UniversityShanghai 200433China Department of Finance and Control sciences School of Mathematical SciencesFudan UniversityShanghai 200433China

This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations(BSDEs for short)with bounded terminal *** generator is allowed to have an unbounded sub-quadratic growth in the second unknown variable *** existence and uniqueness results are given to these *** an application,an existence result is given to a system of coupled forward-backward stochastic differential equations with measurable coefficients.

关键词： Markovian BSDE Quadratic growth Unbounded sub-quadratic term coeficients Coupled FBSDE

Trace inequalities, isocapacitary inequalities, and regularity of the complex Hessian equations

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Science China Mathematics 2024年第3期67卷 557-576页

作者： Jiaxiang Wang Bin Zhou Beijing International center for mathematical Research Peking UniversityBeijing 100871China School of mathematical sciences Peking UniversityBeijing 100871China

In this paper, we study the relations between trace inequalities(Sobolev and Moser-Trudinger types), isocapacitary inequalities, and the regularity of the complex Hessian and Monge-Amp`ere equations with respect to a general nonnegative Borel measure. We obtain a quantitative characterization for these relations through the properties of the capacity-minimizing functions.

关键词： complex Monge-Ampère equations plurisubharmonic functions Sobolev inequality Moser-Trudinger inequality

On the Bicanonical Map of Algebraic Threefolds of General Type

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Acta Mathematica Sinica,English Series 2024年

作者： Meng CHEN Chen JIANG Jian Shi YAN School of mathematical sciences & Shanghai center for mathematical sciences Fudan University Shanghai center for mathematical sciences & School of mathematical sciences Fudan University Department of Mathematics Northeastern University

We answer an open problem of the first author and Zhang(see Open Problem 6.4(3) in Math. Z. 258(2008), 565–585) and prove that, for any nonsingular projective 3-fold of general type with the geometric genus greater t...

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Remarks on the Global Existence for Incompressible Navier-Stokes Equations

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Chinese Annals of Mathematics,Series B 2024年第4期45卷 529-536页

作者： Sheng WANG Zexian ZHANG Yi ZHOU Shanghai center for mathematical sciences Fudan University School of Mathematics Science Fudan University

In this article, the authors use the special structure of helicity for the threedimensional incompressible Navier-Stokes equations to construct a family of finite energy smooth solutions to the Navier-Stokes equations... 详细信息

关键词： 17B40 17B50 Critical norm Global existence Helicity Navier-Stokes euqations

Oscillatory Dynamics of Heterogeneous Stem Cell Regeneration

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Communications on Applied Mathematics and Computation 2024年第1期6卷 431-453页

作者： Xiyin Liang Jinzhi Lei School of mathematical sciences Center for Applied MathematicsTiangong UniversityTianjin300387China

Stem cell regeneration is an essential biological process in the maintenance of tissue homeostasis;dysregulation of stem cell regeneration may result in dynamic diseases that show oscillations in cell *** heterogeneity and plasticity are necessary for the dynamic equilibrium of tissue homeostasis;however,how these features may affect the oscillatory dynamics of the stem cell regeneration process remains poorly ***,based on a mathematical model of heterogeneous stem cell regeneration that includes cell heterogeneity and random transition of epigenetic states,we study the conditions to have oscillation solutions through bifurcation analysis and numerical *** results show various model system dynamics with changes in different parameters associated with kinetic rates,cellular heterogeneity,and *** show that introducing heterogeneity and plasticity to cells can result in oscillation dynamics,as we have seen in the homogeneous stem cell regeneration ***,increasing the cell heterogeneity and plasticity intends to maintain tissue homeostasis under certain *** current study is an initiatory investigation of how cell heterogeneity and plasticity may affect stem cell regeneration dynamics,and many questions remain to be further studied both biologically and mathematically.

关键词： Stem cell regeneration Heterogenous Hopf bifurcation Hematopoietic dynamical disease

Q-BioLiP:A Comprehensive Resource for Quaternary Structure-based Protein-ligand Interactions

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Genomics, Proteomics & Bioinformatics 2024年第1期22卷 151-159页

作者： Hong Wei Wenkai Wang Zhenling Peng Jianyi Yang School of mathematical sciences Nankai UniversityTianjin 300071China MOE Frontiers Science center for Nonlinear Expectations Research Center for Mathematics and Interdisciplinary SciencesShandong UniversityQingdao 266237China

Since its establishment in 2013,BioLiP has become one of the widely used resources for protein-ligand ***,several known issues occurred with it over the past *** example,the protein-ligand interactions are represented in the form of single chain-based tertiary structures,which may be inappropriate as many interactions involve multiple protein chains(known as quaternary structures).We sought to address these issues,resulting in Q-BioLiP,a comprehensive resource for quaternary structure-based protein-ligand *** major features of Q-BioLiP include:(1)representing protein structures in the form of quaternary structures rather than single chain-based tertiary structures;(2)pairing DNA/RNA chains properly rather than separation;(3)providing both experimental and predicted binding affinities;(4)retaining both biologically relevant and irrelevant interactions to alleviate the wrong justification of ligands’biological relevance;and(5)developing a new quaternary structure-based algorithm for the modelling of protein-ligand complex *** these new features,Q-BioLiP is expected to be a valuable resource for studying biomolecule interactions,including protein-small molecule interaction,protein-metal ion interaction,protein-peptide interaction,protein-protein interaction,protein-DNA/RNA interaction,and RNA-small molecule interaction.Q-BioLiP is freely available at https://***/Q-BioLiP/.

关键词： Protein-ligand interaction Quaternary structure Protein-ligand binding site Protein-protein interaction Binding affinity

Weak type(1, 1) of the Riesz transform on some direct product manifolds with exponential volume growth

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Science China Mathematics 2024年第7期67卷 1599-1622页

作者： Hong-Quan Li Jie-Xiang Zhu School of mathematical sciences Fudan UniversityShanghai200433China center for Applied Mathematics Tianjin UniversityTianjin300072China

In this paper, we are concerned with the Riesz transform on the direct product manifold H^(n)× M,where H^(n) is the n-dimensional real hyperbolic space, and M is a connected complete non-compact Riemannian manifold satisfying the volume doubling property and generalized Gaussian or sub-Gaussian upper estimates for the heat kernel. We establish its weak type(1, 1) property. In addition, we obtain the weak type(1, 1) of the heat maximal operator in the same setting. Our arguments also work for a large class of direct product manifolds with exponential volume growth. Particularly, we provide a simpler proof of weak type(1, 1) boundedness of some operators considered in the work of Li et al.(2016).

关键词： Riesz transform heat kernel direct product real hyperbolic space

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Holder continuous weak solutions of the 2D Boussinesq equation with thermal diffusion

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Science China Mathematics 2024年第8期67卷 1777-1806页

作者： Tianwen Luo Tao Tao Liqun Zhang Yau mathematical sciences center Tsinghua UniversityBeijing 100084China School of Mathematics Shandong UniversityJinan 250100China Institute of Mathematics Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of mathematical sciences University of Chinese Academy of SciencesBeijing 100049China

We show the existence of Holder continuous periodic weak solutions of the 2D Boussinesq equation with thermal diffusion which satisfy the prescribed kinetic *** precisely,for any smooth e(t):[0,1]→R+andε∈(0,110),there exist v∈C 110−ε([0,1]×T2)andθ∈C 1,120−εt 2 C 2,1 x 10−ε([0,1]×T2),which satisfy(1.1)in the sense of distribution and e(t)=ˆT2|v(t,x)|2 dx,∀t∈[0,1].

关键词： 2D Boussinesq equation with thermal diffusion Holder continuous periodic weak solutions prescribed kinetic energy