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The isometry of Riemannian Manifold to a Sphere

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Chinese Quarterly Journal of Mathematics 1996年第2期11卷 56-59页

作者：赵培标安徽财贸学院数学系安徽合肥 233041

In this paper the quasi-constant curvature space and the Riemannian manifold contained the totally umbilical hypersurface family are studied,and two theorems are given at the same time.

关键词： quasi-constant curvature spaces sphere isometry totally umbilical hypersurface

isometry Between Two Indefinite Metrics

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Tsinghua Science and Technology 2001年第4期6卷 347-349,354页

作者：吴岚李海中 Department of Mathematics People University of China Beijing 100872 China Department of Mathematical Sciences Tsinghua University Beijing 100084 China

This paper solves the isometry problem between two indefinite metrics by using exterior differential systems. We prove that an indefinite metric is either rigid, admits a one parameter group of isometries (rotation... 详细信息

关键词： isometry indefinite metric principal bundle

The Extension of isometry between Unit Spheres of Normed Space E and l^1

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Journal of Mathematical Research and Exposition 2009年第5期29卷 901-906页

作者： ZHAN Hua Ying College of Science Tianjin University of Technology Tianjin 300384 China

The main result of this paper is to prove Fang and Wang＇s result by another method： Let E be any normed linear space and Vo ： S（E）→ S（l^1） be a surjective isometry. Then V0 can be linearly isometrically extend... 详细信息

关键词： isometry surjective linearly isometric extension.

Small into-isomorphism from L_∞ (A,μ) into L_∞ (B,v)

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Science China Mathematics 2001年第3期44卷 273-279页

作者：定光桂 1. Department of Mathematics Nankai University 300071 Tianjin China

In this paper we shall assert that if T is an isomorphism of L∞(Ω1, A, μ) into L∞(Ω2, B, υ) satisfying the condition ‖T‖·‖T ?1‖?1+? for ?∈ $\left( {0,\frac{1}{5}} \right)$ , then $\frac{T}{{\parallel T\par... 详细信息

关键词： ε-isomorphism isometry small into-isomorphism

The Riemannian Manifolds with Boundary and Large Symmetry

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Chinese Annals of Mathematics,Series B 2010年第3期31卷 347-360页

作者： Zhi CHEN Yiqian SHI Bin XU Department of Mathematics University of Science and Technology of China Hefei 230026 China. Department of Mathematics University of Science and Technology of China Hefei 230026 China.

Seventy years ago, Myers and Steenrod showed that the isometry group of a Riemannian manifold without boundary has a structure of Lie group. In 2007, Bagaev and Zhukova proved the same result for a Riemannian orbifold. In this paper, the authors first show that the isometry group of a Riemannian manifold M with boundary has dimension at most 1/2 dim M（dim M - 1）. Then such Riemannian manifolds with boundary that their isometry groups attain the preceding maximal dimension are completely classified.

关键词： Riemannian manifold with boundary isometry Rotationally symmetric metric Principal orbit

Extension of Isometries on the Unit Sphere of L^p Spaces

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Acta Mathematica Sinica,English Series 2012年第6期28卷 1197-1208页

作者： Dong Ni TAN Department of Mathematics Tianjin University of Technology School of Mathematical Science Nankai University

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of L^p（μ）（1 〈 p 〈∞, p ≠ 2） and a Banach space E can be extended to a linear isometry from L^p（μ） onto E, which means that if the unit sphere of E is （metrically） isometric to the unit sphere of L^P（μ）, then E is linearly isometric to L^p（μ）. We also prove that every surjective 1-Lipschitz or anti-l-Lipschitz map between the unit spheres of L^p（μ1, H1） and L^p（μ2, H2） must be an isometry and can be extended to a linear isometry from L^p（μ2, H2） to L^p（μ2, H2）, where H1 and H2 are Hilbert spaces.

关键词： Tingley's problem 1-Lipschitz anti-l-Lipschitz isometry isometric extension

THE ALEKSANDROV PROBLEM FOR UNIT DISTANCE PRESERVING MAPPING

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Acta Mathematica Scientia 2000年第3期20卷 359-364页

作者：马玉梅 Dalian Univ Dept Math Dalian 116622 Peoples R China

In this paper, one of the Aleksandrov problem was resolved, the proof that a mapping f which preserve unit distance between two real p-normed spaces X and Y is an isometry if Y is a p-strictly convex space and f satisfies locally Lipschitz condition was shown, and a same result in normed spaces was given. In addition, a proof which there doesn't exist any isometry between some spaces was obtained.

关键词： isometry p-normed space Dopp

Regularized Principal Component Analysis

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Chinese Annals of Mathematics,Series B 2017年第1期38卷 1-12页

作者： Yonathan AFLALO Ron KIMMEL Samsung Research Labs Ramat GanIsrael Computer Science Department TechnionIsrael Institute of TechnologyHaifaIsrael.

Given a set of signals, a classical construction of an optimal truncatable basis for optimally representing the signals, is the principal component analysis(PCA for short)approach. When the information about the signals one would like to represent is a more general property, like smoothness, a different basis should be considered. One example is the Fourier basis which is optimal for representation smooth functions sampled on regular grid. It is derived as the eigenfunctions of the circulant Laplacian operator. In this paper,based on the optimality of the eigenfunctions of the Laplace-Beltrami operator(LBO for short), the construction of PCA for geometric structures is regularized. By assuming smoothness of a given data, one could exploit the intrinsic geometric structure to regularize the construction of a basis by which the observed data is represented. The LBO can be decomposed to provide a representation space optimized for both internal structure and external observations. The proposed model takes the best from both the intrinsic and the extrinsic structures of the data and provides an optimal smooth representation of shapes and forms.

关键词： Laplace-Beltrami operator Principal component analysis isometry

Truncated sparse approximation property and truncated q-norm minimization

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Applied Mathematics(A Journal of Chinese Universities) 2019年第3期34卷 261-283页

作者： CHEN Wen-gu LI Peng Institute of Applied Physics and Computational Mathematics Beijing100088China Graduate School China Academy of Engineering PhysicsBeijing 100088China

This paper considers approximately sparse signal and low-rank matrix’s recovery via truncated norm minimization minx∥xT∥q and minX∥XT∥Sq from noisy *** first introduce truncated sparse approximation property,a more general robust null space property,and establish the stable recovery of signals and matrices under the truncated sparse approximation *** also explore the relationship between the restricted isometry property and truncated sparse approximation *** we also prove that if a measurement matrix A or linear map A satisfies truncated sparse approximation property of order k,then the first inequality in restricted isometry property of order k and of order 2k can hold for certain different constantsδk andδ2k,***,we show that ifδs(k+|T^c|)isometry property of order sk.

关键词： truncated norm minimization truncated sparse approximation property restricted isometry property sparse signal recovery low-rank matrix recovery Dantzig selector