This paper studies two isometric problems between unit spheres of Banach spaces.In the first part,we introduce and study the Figiel type problem of isometric embeddings between unit spheres.However,the classical Figie...This paper studies two isometric problems between unit spheres of Banach spaces.In the first part,we introduce and study the Figiel type problem of isometric embeddings between unit spheres.However,the classical Figiel theorem on the whole space cannot be trivially generalized to this case,and this is pointed out by a counterexample.After establishing this,we find a natural necessary condition required by the existence of the Figiel operator.Furthermore,we prove that when X is a space with the T-property,this condition is also sufficient for an isometric embedding T:S_(X)→S_(Y) to admit the Figiel operator.This answers the Figiel type problem on unit spheres for a large class of spaces.In the second part,we consider the extension of bijectiveε-isometries between unit spheres of two Banach spaces.It is shown that every bijectiveε-isometry between unit spheres of a local GL-space and another Banach space can be extended to be a bijective 5ε-isometry between the corresponding unit balls.In particular,whenε=0,this recovers the MUP for local GL-spaces obtained in[40].展开更多
The present paper is concerned with the existence of golbal smooth solutions for the homogeneous Dirichlet boundary value problem of the Darboux equation and the case degenerate onthe boundary is contained As some app...The present paper is concerned with the existence of golbal smooth solutions for the homogeneous Dirichlet boundary value problem of the Darboux equation and the case degenerate onthe boundary is contained As some applications the smooth isometric embeddings of positivelyand nonnegatively curved disks into R^3 are constructed.展开更多
A new method to solve the Gauss-Codazzi system is given in which we transform the linearized system to a partial differential equation of second order, and by the method we solve the problem of semi-global isometric e...A new method to solve the Gauss-Codazzi system is given in which we transform the linearized system to a partial differential equation of second order, and by the method we solve the problem of semi-global isometric embedding of surfaces with Gaussian curvature changing sign cleanly.展开更多
In the paper, the authors show that any abstract smooth surface can be locally isometricallyembedded into a class of 3-dimensional spaces Nρ0 (ρ0>0) with the non-positively sectionalcurvature being fixed sufficie...In the paper, the authors show that any abstract smooth surface can be locally isometricallyembedded into a class of 3-dimensional spaces Nρ0 (ρ0>0) with the non-positively sectionalcurvature being fixed sufficiently small.展开更多
详细介绍了一种新的机器学习的方法——流形学习。流形学习是一种新的非监督学习方法,可以有效地发现高维非线性数据集的内在维数并进行维数约简,近年来越来越受到机器学习和认知科学领域的研究者的重视。目前已经出现了很多有效的流形...详细介绍了一种新的机器学习的方法——流形学习。流形学习是一种新的非监督学习方法,可以有效地发现高维非线性数据集的内在维数并进行维数约简,近年来越来越受到机器学习和认知科学领域的研究者的重视。目前已经出现了很多有效的流形学习算法,如等度规映射(ISOMAP)、局部线性嵌套(Locally Linear Embedding,LLE)等。详细讲述了当前常用的几种流形学习算法以及在流形方面已经取得的研究成果,并对流形学习目前在各方面的应用作了较为细致的阐述。最后展望了流形学习的研究发展趋势,且提出了流形学习中仍需解决的关键问题。展开更多
基金the National Nature Science Foundation of China(11671214,11971348,12071230)the Hundred Young Academia Leaders Program of Nankai University(63223027,ZB22000105)+1 种基金the Undergraduate Education and Teaching Project of Nankai University(NKJG2022053)the National College Students’Innovation and Entrepreneurship Training Program of Nankai University(202210055048)。
文摘This paper studies two isometric problems between unit spheres of Banach spaces.In the first part,we introduce and study the Figiel type problem of isometric embeddings between unit spheres.However,the classical Figiel theorem on the whole space cannot be trivially generalized to this case,and this is pointed out by a counterexample.After establishing this,we find a natural necessary condition required by the existence of the Figiel operator.Furthermore,we prove that when X is a space with the T-property,this condition is also sufficient for an isometric embedding T:S_(X)→S_(Y) to admit the Figiel operator.This answers the Figiel type problem on unit spheres for a large class of spaces.In the second part,we consider the extension of bijectiveε-isometries between unit spheres of two Banach spaces.It is shown that every bijectiveε-isometry between unit spheres of a local GL-space and another Banach space can be extended to be a bijective 5ε-isometry between the corresponding unit balls.In particular,whenε=0,this recovers the MUP for local GL-spaces obtained in[40].
文摘The present paper is concerned with the existence of golbal smooth solutions for the homogeneous Dirichlet boundary value problem of the Darboux equation and the case degenerate onthe boundary is contained As some applications the smooth isometric embeddings of positivelyand nonnegatively curved disks into R^3 are constructed.
基金supported by the Fundamental Research Funds for the Central Universities(Grant No. ZYGX2010J109)National Natural Science Foundation of China (Grant No. 11101068)the Sichuan Youth Science and Technology Foundation (Grant No. 2011JQ0003)
文摘A new method to solve the Gauss-Codazzi system is given in which we transform the linearized system to a partial differential equation of second order, and by the method we solve the problem of semi-global isometric embedding of surfaces with Gaussian curvature changing sign cleanly.
文摘In the paper, the authors show that any abstract smooth surface can be locally isometricallyembedded into a class of 3-dimensional spaces Nρ0 (ρ0>0) with the non-positively sectionalcurvature being fixed sufficiently small.
文摘详细介绍了一种新的机器学习的方法——流形学习。流形学习是一种新的非监督学习方法,可以有效地发现高维非线性数据集的内在维数并进行维数约简,近年来越来越受到机器学习和认知科学领域的研究者的重视。目前已经出现了很多有效的流形学习算法,如等度规映射(ISOMAP)、局部线性嵌套(Locally Linear Embedding,LLE)等。详细讲述了当前常用的几种流形学习算法以及在流形方面已经取得的研究成果,并对流形学习目前在各方面的应用作了较为细致的阐述。最后展望了流形学习的研究发展趋势,且提出了流形学习中仍需解决的关键问题。
