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Optimal Quadrature Problem on n-Information for Hardy-Sobolev Classes

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Acta Mathematica Sinica,English Series 2011年第12期27卷 2371-2378页

作者： Xue Hua LI gen sun fang College of Science China Agricultural University Beijing 100083 P. R. China School of Mathematical Sciences Beijing Normal University Beijing 100875 P. R. China

For β 〉 0 and an integer r 〉 2, denote by H∞,β those 2π-periodic, real-valued functions f on R, which are analytic in Sβ = {z ∈ C： [ImzI 〈β} and satisfy the restriction If（r）（z）[ ≤ 1, z ∈ Sβ. The optimal quadrature formulae about information composed of the values of a function and its kth （k ： 1,..., r - 1） derivatives on free knots for the classes H∞,β are obtained, and the error estimates of the optimal quadrature formulae are exactly determined.

关键词： Hardy-Sobolev class analytic function optimal quadrature formula n-information

Bernstein n-Widths of Besov Embeddings on Lipschitz Domains

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Acta Mathematica Sinica,English Series 2013年第12期29卷 2283-2294页

作者： Yue Wu LI gen sun fang School of Mathematical Sciences Hulunbeier University School of Mathematical Sciences Beijing Normal University

In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bq0s＋t（Lp0（Ω））→Bq1s（Lp1（Ω））, t〉max{d（1/p0-1/p1）, 0}, 1 ≤ p0, p1, q0, q1 ≤∞,where Bq0s＋t（Lp0（Ω）） is a Besov space defined on the bounded Lipschitz domain Ω ？ Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.

关键词： Bernstein widths Besov spaces Lipschitz domains

Optimal Recovery on the Classes of Functions with Bounded Mixed Derivative

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Acta Mathematica Sinica,English Series 2009年第2期25卷 279-286页

作者： gen sun fang Li Qin DUAN School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education Beijing 100875 P. R. China School of Mathematical Sciences Beijing Normal University Beijing 100875 P. R. China Institute of Mathematics Hangzhou Dianzi University Hangzhou 310018 P. R. China

Temlyakov considered the optimal recovery on the classes of functions with bounded mixed derivative in the Lp metrics and gave the upper estimates of the optimal recovery errors. In this paper, we determine the asymptotic orders of the optimal recovery in Sobolev spaces by standard information, i.e., function values, and give the nearly optimal algorithms which attain the asymptotic orders of the optimal recovery.

关键词： optimal recovery standard information class of functions with bounded mixed derivative

Optimization on class of operator equations in the probabilistic case setting

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Science China Mathematics 2007年第1期50卷 100-104页

作者： gen-sun fang Li-xin QIAN School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China College of Mathematics and Physics Zhejiang Normal UniversityJinhua 321004China

In this paper, we introduce a problem of the optimization of approximate solutions of operator equations in the probabilistic case setting, and prove a general result which connects the relation between the optimal approximation order of operator equations with the asymptotic order of the probabilistic width. Moreover, using this result, we determine the exact orders on the optimal approximate solutions of multivariate Freldholm integral equations of the second kind with the kernels belonging to the multivariate Sobolev class with the mixed derivative in the probabilistic case setting.

关键词： probabilistic width Gauss measure multivariate Sobolev space operator equation 41A46 41A63 45L05 60G15

Non-Linear Approximation of Functions with Mixed Smoothness by Sets of Finite Pseudo-Dimension

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Acta Mathematica Sinica,English Series 2007年第4期23卷 671-676页

作者： Guang Gui CHEN gen sun fang Yuan Long RUAN School of Mathematics and Computer Engineering Xihua University Chengdu 611930 P. R. China School of Mathematical Science Beijing Normal University Beijing 100875 P. R. China

In this paper, the non-linear approximation on the class of multivariate functions with bounded mixed derivatives is investigated, and the asymptotic degree of the non-linear width on this class is determined.

关键词： pseudo-dimension non-linear width mixed smoothness

Trigonometric Widths and Best N-Term Approximations of the generalized Periodic Besov Classes B_(p,θ)~Ω

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Journal of Mathematical Research and Exposition 2011年第1期31卷 129-141页

作者： Li Qin DUAN gen sun fang Institute of Mathematics Hangzhou Dianzi University Zhejiang 310018 P. R. China School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education Beijing 100875 P. R. China

In this paper, we determine the estimates exact in order for the trigonometric widths and the best n-term trigonometric approximations of the generalized classes of periodic functions B^Ωpθ in the space Lq for some ... 详细信息

关键词： trigonometric width best n-term approximation generalized Besov classes.

The ε-Complexity of the Approximate Solution of Integral Equations with Isotropic Kernels

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Acta Mathematica Sinica,English Series 2002年第4期18卷 679-686页

作者： Wan MA gen sun fang College of Mathematics & Information Science of Wenzhou University, Wenzhou 325027, P. R. China Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China College of Mathematics & Information Science of Wenzhou University Wenzhou 325027 P. R. China Department of Mathematics Beijing Normal University Beijing 100875 P. R. China

The exact order of s-complexity is determined in L_p (2≤p≤∞) spaces for the second kind of Fredholm integral equations with kernels belonging to an isotropic Sobolev class.

关键词： Fredholm equation ε-complexity Sobolev classes

Zooplankton community composition in summers of 2021and 2022 in the northern East China Sea and Yellow Sea,and their linkage with abiotic and biotic factors

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Journal of Oceanology and Limnology 2025年第1期43卷 159-176页

作者： Dongjie GUO fang ZHANG Shuangyan ZHANG Song sun CAS Key Laboratory of Marine Ecology and Environmental Sciences Institute of OceanologyChinese Academy of SciencesQingdao 266071China College of Marine Science University of Chinese Academy of SciencesQingdao 266071China Jiaozhou Bay National Marine Ecosystem Research Station Institute of OceanologyChinese Academy of SciencesQingdao 266071China

The northern East China Sea and Yellow Sea are highly dynamic marginal seas,serving as important fishing grounds and primary areas for *** zooplankton community structure,including abundance,biovolume,and normalized biovolume size spectra(NBSS)within this ecosystem in summers 2021 and 2022 was assessed through ZooScan digital imaging *** abundances of zooplankton in summers 2021 and 2022 were 3364.22±2190.53 and 4435.52±2520.06 inds./m^(3),respectively,whereas the corresponding biovolumes were 649.10±519.63 and 1064.86±1254.87 mm^(3)/m^(3),*** copepods as the dominant zooplankton functional group in terms of abundance in both years,while chaetognath and medusae were the dominant groups in terms of *** dominance of medusae and chaetognath in terms of biovolume,along with the flatter slopes of NBSS compared to a stable community,indicated that more energy accumulated in gelatinous organisms within the study *** were significant positive relationships between temperature and tunicates or medusae,with a significantly higher population size observed in summer 2022 than in *** lower abundance and biovolume of zooplankton,mainly of small copepods,in summer 2021 may be related to the larger population size of the large jellyfish and the proliferation of green *** investigation provided basic data and fundamental insights for achieving a comprehensive understanding of zooplankton community structure and their long-term dynamics.

关键词： zooplankton community spatial distribution global warming Nemopilema nomurai Ulva prolifera

赋以混合范数的各向异性Besov类在不同度量下的嵌入定理

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数学学报（中文版） 2006年第2期49卷 381-390页

作者：钱李新房艮孙北京师范大学数学科学学院北京100875

本文给出了赋以混合范数的各向异性多元Besov类BPθr(Rd)的一个表现定理, 利用此表现定理,证明了它在不同度量下的一个嵌入定理BPθr(Rd)→Bqθr'(Rd),其中 1≤P≤q≤∞,r'=(1-∑jd=1(1/pj-1/qj)1/rj)r．

关键词：表现定理嵌入定理各向异性Besov类