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Science China mathematics 2024年第11期67卷 2433-2452页

作者： Feng Shao institute of mathematics Academy of Mathematics and Systems ScienceChinese Academy of Sciences School of mathematical sciences University of Chinese Academy of Sciences

Let X be an irreducible Hermitian symmetric space of compact type(IHSS for short).In this paper,we give the irreducible decomposition of *** a by-product,we give a cohomological characterization of the rank of ***,we introduce pseudoeffective thresholds to measure the bigness of tangent bundles of smooth complex projective varieties precisely and calculate them for irreducible Hermitian symmetric spaces of compact type.

关键词： tangent bundle pseudoeffective threshold irreducible Hermitian symmetric space of compact type (IHSS) multiplicity free action

Jackson-type inequalities on high-dimensional spheres

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Science China mathematics 2024年第11期67卷 2571-2586页

作者： Yeli Niu Feng Dai department of mathematics Shanghai Normal University department of mathematical and Statistical sciences University of Alberta

We study Jackson's inequality on high-dimensional spheres with respect to the modulus of smoothness defined via the rotation *** obtain a version of Jackson's inequality with a dimensionfree constant,extending Newman and Shapiro's well-known results in 1964 from the case of r = 1 and p = ∞ to more general *** results partially overcome the curse of *** also establish similar results on the equivalence of the K-functional and modulus of smoothness.

关键词： modulus of smoothness $K$-functional Jackson's inequality high-dimensional approximation

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 sciences 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

Tighter sum uncertainty relations via(α,β,γ)weighted Wigner–Yanase–Dyson skew information

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Communications in Theoretical Physics 2024年第3期76卷 91-99页

作者： Cong Xu Zhaoqi Wu Shao-Ming Fei School of mathematical sciences Capital Normal UniversityBeijing 100048China department of mathematics Nanchang UniversityNanchang 330031China Max-Planck-institute for mathematics in the sciences 04103 LeipzigGermany

We establish tighter uncertainty relations for arbitrary finite observables via(α,β,γ)weighted Wigner–Yanase–Dyson((α,β,γ)WWYD)skew *** results are also applicable to the(α,γ)weighted Wigner–Yanase–Dyson((α,γ)WWYD)skew information and the weighted Wigner–Yanase–Dyson(WWYD)skew *** also present tighter lower bounds for quantum channels and unitary channels via(α,β,γ)modified weighted Wigner–Yanase–Dyson((α,β,γ)MWWYD)skew *** examples are provided to illustrate the tightness of our uncertainty relations.

关键词： quantum channels (α,β,γ)WWYD skew information (α,β,γ)MWWYD skew information uncertainty relations

2-local unstable homotopy groups of indecomposable A3～2-complexes

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Science China mathematics 2024年第3期67卷 607-626页

作者： Zhongjian Zhu Jianzhong Pan College of mathematics and Physics Wenzhou University Hua Loo-Keng Key Laboratory of mathematics Institute of MathematicsAcademy of Mathematics and Systems ScienceChinese Academy of Sciences School of mathematical sciences University of Chinese Academy of Sciences

In this paper,we calculate the 2-local unstable homotopy groups of indecomposable A3～*** main technique used is analysing the homotopy property of J（A,X）,defined by Gray（1973） for a CW-pair（X,A）,which is homoto... 详细信息

关键词： homotopy groups A 2 n -complexes fibration sequence relative James construction 55P15

On Differential Equations of Integrable Billiard Tables

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Acta Mathematica Sinica,English Series 2024年第1期40卷 417-424页

作者： Vladimir DRAGOVIC Andrey E.MIRONOV department of mathematical sciences The University of Texas at DallasRichardson TXUSA mathematical institute SANU BelgradeSerbia Novosibirsk State University NovosibirskRussia Sobolev institute of mathematics of the Siberian Branch of the Russian Academy of sciences NovosibirskRussia

We introduce a method to find differential equations for functions which define tables,such that associated billiard systems admit a local first *** illustrate this method in three situations:the case of(locally)integrable wire billiards,for finding surfaces in R^(3)with a first integral of degree one in velocities,and for finding a piece-wise smooth surface in R^(3)homeomorphic to a torus,being a table of a billiard admitting two additional first integrals.

关键词： Polynomial first integrals wire billiards piece-wise smooth surfaces

ON THE UNIVALENCE OF CERTAIN POLYHARMONIC MAPPINGS WITH BOUNDED LENGTH DISTORTIONS

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Acta Mathematica Scientia 2024年第6期44卷 2125-2138页

作者：汪欣 Saminathan PONNUSAMY 刘名生 department of mathematics Shenzhen Polytechnic University department of mathematics Indian Institute of Technology Madras School of mathematical sciences South China Normal University

The main purpose of this paper is to investigate the univalence of normalized polyharmonic mappings with bounded length distortions in the unit *** first establish the coefficient estimates for polyharmonic mappings with bounded length ***,using these results,we establish five Landau-type theorems for subclasses of polyharmonic mappings F and L(F), where F has bounded length distortion and L is a differential operator.

关键词： biharmonic mappings coefficients estimates harmonic mappings Landau-type theorems polyharmonic mappings

C^(α)regularity of weak solutions of non-homogeneous ultraparabolic equations with drift terms

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Science China mathematics 2024年第1期67卷 23-44页

作者： Wendong Wang Liqun Zhang School of mathematical sciences Dalian University of TechnologyDalian 116024China institute of mathematics Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

We consider a class of non-homogeneous ultraparabolic differential equations with singular drift terms arising from some physical models,and prove that weak solutions are H?lder continuous,which are sharp in some sense and also generalize the well-known De Giorgi-Nash-Moser theory to degenerate parabolic equations satisfying the H?rmander hypoellipticity *** new ingredients are manifested in two aspects:on the one hand,for lower-order terms,we exploit a new Sobolev inequality suitable for the Moser iteration by improving the result of Pascucci and Polidoro(2004);on the other hand,we explore the G-function from an early idea of Kruzhkov(1964)and an approximate weak Poincaréinequality for non-negative weak sub-solutions to prove the H?lder regularity.

关键词： ultraparabolic equations Moser iteration Poincare inequality C^a regularity

A Deep Learning Method for Computing Eigenvalues of the Fractional Schrödinger Operator

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Journal of Systems Science & Complexity 2024年第2期37卷 391-412页

作者： GUO Yixiao MING Pingbing LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of mathematical sciences University of Chinese Academy of SciencesBeijing 100049China

The authors present a novel deep learning method for computing eigenvalues of the fractional Schrödinger *** proposed approach combines a newly developed loss function with an innovative neural network architecture that incorporates prior knowledge of the *** improvements enable the proposed method to handle both high-dimensional problems and problems posed on irregular bounded *** authors successfully compute up to the first 30 eigenvalues for various fractional Schrödinger *** an application,the authors share a conjecture to the fractional order isospectral problem that has not yet been studied.

关键词： Eigenvalue problem deep learning fractional Schrödinger operator isospectral problem

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