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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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GLOBAL SOLUTIONS IN THE CRITICAL SOBOLEV SPACE FOR THE LANDAU EQUATION

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Acta Mathematica Scientia 2024年第4期44卷 1347-1372页

作者： Hao WANG department of mathematical sciences Tsinghua UniversityBeijing 100084China

The Landau equation is studied for hard potential with-2≤γ≤*** a perturbation setting,a unique global solution of the Cauchy problem to the Landau equation is established in a critical Sobolev space H_(x)^(d)L_(v)^(2)(d>3/2),which extends the results of[11]in the torus domain to the whole space R_(x)^(3).Here we utilize the pseudo-differential calculus to derive our desired result.

关键词： Landau equation nonlinear energy method global existence pseudo-differential calculus

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

Pseudo-Differential Operators of Homogeneous Symbol Class Associated with the Weinstein Transform

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Acta Mathematica Sinica,English Series 2024年第6期40卷 1533-1562页

作者： Santosh Kumar UPADHYAY Mohd SARTAJ department of mathematical sciences Indian Institute of Technology(BHU)VaranasiIndia

In this paper,pseudo-differential operators with homogeneous symbol classes associated with the Weinstein transform are *** boundedness of pseudo-differential operators and commutator between two pseudo-differential o... 详细信息

关键词： Pseudo-Differential Operators Homogeneous Symbol Class Weinstein Operator Sobolev space Weinstein transform

Equivariant Tautological Integrals on Flag Varieties

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Chinese Annals of Mathematics,Series B 2024年第3期45卷 463-486页

作者： Xiaobo ZHUANG department of mathematical sciences Zhejiang Sci-Tech UniversityHangzhou 310018China

The author apples the Atiyah-Bott-Berline-Vergne formula to the equivariant tautological integrals over flag varieties of types A,B,C,D,and recovers the formulas expressing the integrals as iterated residues at infini... 详细信息

关键词： Localization Flag variety Residue

Integralinput-to-statestability ystemswithsmalldelays Integral input-to-state stability of systems with small delays

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Control Theory and Technology 2024年第3期22卷 455-467页

作者： R.H.Harsha Nawarathna Yuandan Lin Yuan Wang department of mathematical sciences Florida Atlantic UniversityBoca RatonFL 33431USA

We consider how small delays affect the integral-input-to-state stability(iss)property for a *** result is similar to the input-to-state stability(Iss)result obtained in[1]:the iss property will be preserved in a practical and semi-global manner if the delay interval is small ***,since the iss quantifies the robust stability in terms of a generalized Li norm of the inputs instead of a generalized Loo norm of the inputs for the Iss case,the techniques and proofs for the Iss case do not apply to the iss case *** the proofs in[1]are based on the Lyapunov-Razumikhin approach,our proofs are based on the iss-Lyapunov functions for the zero-delay *** addition to the interest by its own in showing how the iss property is affected by small delays,the result also serves to the study of the iss property for singularly perturbed systems.

关键词： Integral-input-to-state stability Nonlinear systems Small delays Lyapunov method

Dilation,discrimination and Uhlmann's theorem of link products of quantum channels

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Chinese Physics B 2024年第3期33卷 201-206页

作者：雷强操刘桁 Asutosh Kumar 武俊德 School of mathematical sciences Harbin Institute of TechnologyHarbin 150001China department of Physics Gaya CollegeMagadh UniversityRampurGaya 823001India School of mathematical sciences Zhejiang UniversityHangzhou 310027China

We establish the Stinespring dilation theorem of the link product of quantum channels in two different ways,discuss the discrimination of quantum channels,and show that the distinguishability can be improved by self-linking each quantum channel n times as n *** also find that the maximum value of Uhlmann's theorem can be achieved for diagonal channels.

关键词： quantum channels link products Stinespring dilation theorem Uhlmann's theorem

Emergent Geometry of KP Hierarchy

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Acta Mathematica Sinica,English Series 2024年第1期40卷 3-25页

作者： Jian ZHOU department of mathematical sciences Tsinghua UniversityBeijng 100084P.R.China

We explain how to construct a quantum deformation of a spectral curve associated to a tau-function of the KP *** construction is applied to Witten-Kontsevich tau-function to give a natural explanation of some earlier ... 详细信息

关键词： KP hierarchy tau-functions spectral curves

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 sciences 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