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Uniqueness and local stability for the inverse scattering problem of determining the cavity

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Science china Mathematics 2005年第8期48卷 1113-1123页

作者： FENG Lixin & MA Fuming LMAM, school of mathematical sciences, Peking university, Beijing 100871, china school of mathematical sciences, Heilongjiang university, Harbin 150080, china school of mathematical sciences, Jilin university, Changchun 130012, china LMAM School of Mathematical Sciences Peking University Beijing China school of mathematical sciences Heilongjiang University Harbin China school of mathematical sciences Jilin University Changchun China

Considering a time-harmonic electromagnetic plane wave incident on an arbitrarily shaped open cavity embedded in infinite ground plane, the physical process is modelled by Maxwell's equations. We investigate the inverse problem of determining the shape of the open cavity from the information of the measured scattered field. Results on the uniqueness and the local stability of the inverse problem in the 2-dimensional TM (transverse magnetic) polarization are proved in this paper.

关键词： uniqueness, stability, Maxwell's equations, inverse scattering problem.

INFINITE ELEMENT METHOD FOR THE EXTERIOR PROBLEMS OF THE HELMHOLTZ EQUATIONS

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Journal of Computational Mathematics 2000年第6期18卷 657-672页

作者： Lung-an Ying (school of mathematical sciences, Peking university, Beijing 100871, china) 北京大学北京 100871

There are two cases of the exterior problems of the Helmholtz equation. If λ ≥ 0 the bilinear form is coercive, and if λ < 0 it is the scattering problem. We give a new approach of the infinite element method, which enables us to solve these exterior problems as well as corner problems. A numerical example of the scattering problem is given. [ABSTRACT FROM AUTHOR]

关键词： Helmholtz equation exterior problem infinite element method

Conformal isoparametric hypersurfaces with two distinct conformal principal curvatures in conformal space

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Science china Mathematics 2010年第4期53卷 953-965页

作者： NIE ChangXiong 1,2,LI TongZhu 3,HE YiJun 4 & WU ChuanXi 5 1 Faculty of Mathematics and Computer sciences,Hubei university,Wuhan 430062,china 2 school of mathematical sciences,Peking university,Beijing 100871,china 3 Faculty of sciences,Beijing Institute of Technology,Beijing 100081,china 4 school of mathematical sciences,shanxi university,taiyuan 03.006,china 5 Institute of Mathematics,Hubei university,Wuhan 430062,china Faculty of Mathematics and Computer sciences Hubei University school of mathematical sciences Peking University Faculty of sciences Beijing Institute of Technology school of mathematical sciences Shanxi University

The conformal geometry of regular hypersurfaces in the conformal space is *** classify all the conformal isoparametric hypersurfaces with two distinct conformal principal curvatures in the conformal space up to confor... 详细信息

关键词： Lorentz conformal geometry conformal invariants conformal isoparametric hypersurfaces

Sharpness on the Lower Bound of the Lifespan of Solutions to Nonlinear Wave Equations

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Chinese Annals of Mathematics,Series B 2011年第4期32卷 521-526页

作者： Yi ZHOU 1 Wei HAN 2 1 Nonlinear mathematical Modeling and Methods Laboratory Shanghai Key Laboratory for Contem- porary Applied Mathematics school of mathematical sciences, Fudan university, Shanghai 200433, china. 2 school of mathematical sciences, Fudan university, Shanghai 200433, china Department of Mathematics, North university of china, taiyuan 030051, china. Nonlinear mathematical Modeling and Methods Laboratory Shanghai Key Laboratory for Contemporary Applied Mathematics School of Mathematical Sciences Fudan University Shanghai 200433 China

This paper is devoted to proving the sharpness on the lower bound of the lifespan of classical solutions to general nonlinear wave equations with small initial data in the case n = 2 and cubic nonlinearity （see the results of T. T. Li and Y. M. Chen in 1992）. For this purpose, the authors consider the following Cauchy problem： { where □=δt^2-∑i=1n δx^^2 is the wave operator, g（x）（≡/） 0 is a smooth non-negative function with compact support, and ε 〉 0 is a small parameter. It is shown that the solution blows up in a finite time, and the lifespan T（ε） of solutions has an upper bound T（ε） ≤ exp（Aε-2） with a positive constant A independent of ε, which belongs to the same kind of the lower bound of the lifespan.

关键词： Nonlinear wave equation Cauchy problem Lifespan

TWO POSITIVE SOLUTIONS TO THREE-POINT SINGULAR BOUNDARY VALUE PROBLEMS

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Acta Mathematica Scientia 2011年第1期31卷 29-38页

作者：李宇华梁占平 school of mathematical sciences Shanxi University

In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem： -u′′（t） = λf（t, u（t）） for all t ∈ （0, 1） subjecting to u（0） = 0 and αu（η） = u（1）, where η ∈ （0, 1）, α ∈ [0, 1）, and λ is a positive parameter. The nonlinear term f（t, u） is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ＊ ∈ （0, ＋∞], such that the above problem has at least two positive solutions for any λ ∈ （0, λ＊） under certain conditions on the nonlinear term f.

关键词： Three-point singular boundary value problem positive solutions fixed point index

On Riemann-Finsler geometry

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Chinese Science Bulletin 1998年第6期43卷 447-450页

作者： MO Xiaohuan school of mathematical sciences, Peking university, Beijing 100871, china school of mathematical sciences Peking University Beijing China

The history of Finsler geometry is reviewed and briefly recent development in Finsler geometry and its application is completed systematically. Furthermore, an interesting open problem has been proposed in this field.

关键词： Finsler space Chern connection Hilbert form Minkowski potential projective sphere bundle.

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灵光独耀:沈宗灵与中国法理学

中国法律评论 2024年第4期 149-154页

作者： Peking university Law school 北京大学法学院

在其诸多弟子的追忆中,沈宗灵先生一生专心治学、淡泊名利,由此常常对世事保持一种有意的疏离。“灵光独耀,迥脱根尘”,或许便是这位法理学大家的灵魂写照。壹“经国济民”:沈宗灵的早年岁月1923年2月24日,沈宗灵出生于浙江杭州的一个... 详细信息

在其诸多弟子的追忆中,沈宗灵先生一生专心治学、淡泊名利,由此常常对世事保持一种有意的疏离。“灵光独耀,迥脱根尘”,或许便是这位法理学大家的灵魂写照。壹“经国济民”:沈宗灵的早年岁月1923年2月24日,沈宗灵出生于浙江杭州的一个商人家庭,后随家人迁居上海。

关键词：沈宗灵淡泊名利法理学浙江杭州灵光

Hausdorff dimension of Julia set of a polynomial with a Siegel disk

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Science china Mathematics 2006年第9期49卷 1284-1296页

作者： SHEN Liang LMAM, school of mathematical sciences, Peking university, Beijing 100871, china LMAM School of Mathematical Sciences Peking University Beijing China

Let f(z) = e2πiθz(1+z/d)d,θ∈R\Q be a polynomial. Ifθis an irrational number of bounded type, it is easy to see that f(z) has a Siegel disk centered at 0. In this paper, we will show that the Hausdorff dimension o... 详细信息

关键词： Hausdorff dimension Siegel disk porous set

A note on periodic solutions for semi-ratio-dependent predator-prey systems

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Applied Mathematics(A Journal of Chinese Universities) 2010年第1期25卷 1-8页

作者： LIU Xiu-xiang school of mathematical sciences, South china Normal university, Guangdong 510631, china school of mathematical sciences South China Normal University Guangdong China

This note investigates the existence of periodic solutions for semi-ratio-dependent predator-prey models with functional response. New sharp criteria without any other nontrivial assumptions are presented by the invariance property of homotopy and analysis technique, which improve and extend many previous work. Some interesting numerical examples are given to illustrate our results.

关键词： Periodic solutions predator-prey system functional response.