Banach空间中一类度量投影的判据及表达式

X为自反、严格凸Ｂａｎａｃｈ 空间,L为X中闭子空间,P：X→L为单值算子,该文给出P成为L上度量投影P_L的判据及P_L为线性算子的充分必要条件．在自反Ｂａｎａｃｈ空间中,利用对偶映射,给出超平面上（值）度量投影的表达式．对于自反、严格凸、光滑的Ｂａｎａｃｈ 空间中线性流形上的（单值）度量投影,利用广义右逆的表示,求出其表达式．在后继文章中将给出此表达式的应用．     

Kähler流形上的Schwarz导数(Ⅱ)

证明了当目标流形是复投影空间CPⁿ和复超球Bⁿ(具有典则度量)时,Kahler流形的全纯映照的单射定理.至此,连同已经证明的关于复Euclid空间Cⁿ(具有平坦度量)的同样结果,对于全纯截曲率为一1,0,+1的目标流形建立了单射定理.     

自反严格凸Banach空间中约束极值解问题的扰动分析

借助于代数度量广义逆方面的扰动结论,同时利用一般的约束极值解问题和无约束极值问题的一个等价转化,该文在自反严格凸Banach空间中获得了具有等式约束的极值解问题的扰动估计.最后,作为主要结论的推论,该文分别考虑了不适定算子方程的极值解、最佳逼近解和点投影到线性流形等问题的扰动分析.     

Banach空间中拟线性投影算子

证明了 :在自反 Banach空间＄X＄中 ,每个闭子空间 L,都存在 X到 L上的拟线性投影算子 SL.一般说来 ,SL 既非度量投影算子 ,又非线性算子 .     

满旗流形SO(8)/T上不变爱因斯坦度量（英文）

本文研究了迷向表示分为12个不可约子空间的满旗流形SO(8)/T上不变爱因斯坦度量的问题.利用计算机计算满旗流形SO(8)/T爱因斯坦方程组的方法,得到了满旗流形SO(8)/T上有160个不变爱因斯坦度量(up to a scale)的结果,在等距情况下考虑这160个不变爱因斯坦度量,其中1个是凯莱爱因斯坦度量,4个是非凯莱爱因斯坦度量.推广了只对迷向表示分为小于等于6个不可约子空间的满旗流形上不变爱因斯坦度量的研究.     

Banach空间中线性流形的单值度量投影算子部分

为了研究Banach空间集值线性映射包含y∈M(x)的最小范数极值解,其中X,Y为Banach空间,M X×Y为线性流形,本文引入Banach空间X×Y中线性流形的单值度量算子部分,并给出了该算子部分的结构的刻划.为在另文将Lee S J与NashedM Z所引进并研究的Hilbert空间中集值线性映射包含的最小二乘解推广到Banach空间奠定了理论基础.     

Banach空间闭超平面上度量投影的连续性

该文给出Banach空间X的对偶空间X~*中闭超平面上度量投影的表达式,并在Banach空间中研究了闭超平面上度量投影的连续性.     

矢量丛的微分几何

Sasaki,S.曾经在黎曼流形的切丛和切球面丛上引进了典型的黎曼度量,並研究了这种度量的微分几何。本文把Sasaki.S.的工作推广到黎曼流形上带连络的任意矢量丛,研究了丛空间上的黎曼连络、测地线和全测地子流形等问题。     

满旗流形SO(8)=T上不变爱因斯坦度量

本文研究了迷向表示分为12个不可约子空间的满旗流形SO(8)=T上不变爱因斯坦度量的问题.利用计算机计算满旗流形SO(8)=T爱因斯坦方程组的方法, 得到了满旗流形SO(8)=T上有160 个不变爱因斯坦度量(up to a scale)的结果, 在等距情况下考虑这160个不变爱因斯坦度量, 其中1个是凯莱爱因斯坦度量, 4 个是非凯莱爱因斯坦度量. 推广了只对迷向表示分为小于等于6个不可约子空间的满旗流形上不变爱因斯坦度量的研究.     

Perelman熵和Khler-Ricci流的稳定性

本文研究Perelman熵在一个Fano流形上Khler度量空间中的第二变分.特别地,本文证明了Perelman熵在一个Khler-Einstein流形上是稳定的.     

广义正交分解定理与Tseng度量广义逆

本文将Banach空间中广义正交分解定理从线性子空间拓广至非线性集—太阳集,分别给出了一算子为度量投影算子和一度量投影算子为有界线性算子的充要条件;得到了判别Banach空间中子空间广义正交可补的充要条件;建立了王玉文和季大琴(2000年)新近引入的Banach空间中的线性算子的Tseng度量广义逆存在的特征刻划条件;这些工作本质地把王玉文等人的新近结果从自反空间拓广至非自反空间的情形.     

Geometric Gibbs theory

We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space, which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero.

     

Geometric Gibbs theory In Memory of Professor Shantao Liao

We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space,which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero.     

Perturbation Analysis of Moore–Penrose Quasi-linear Projection Generalized Inverse of Closed Linear Operators in Banach Spaces

In this paper, we investigate the perturbation problem for the Moore–Penrose bounded quasi-linear projection generalized inverses of a closed linear operaters in Banach space. By the method of the perturbation analysis of bounded quasi-linear operators, we obtain an explicit perturbation theorem and error estimates for the Moore–Penrose bounded quasi-linear generalized inverse of closed linear operator under the T-bounded perturbation, which not only extend some known results on the perturbation of the oblique projection generalized inverse of closed linear operators, but also extend some known results on the perturbation of the Moore–Penrose metric generalized inverse of bounded linear operators in Banach spaces.     

非自反Banach空间中的度量投影

该文给出非自反Banach空间中一类超平面上度量投影的表达式．在近严格凸Banach空间中,研究了它们的连续性．对于对偶Banach空间X*,给出弱*闭子集上度量投影的一些连续性结果．     

Manifolds of linear involutions

The set of bounded linear involutions on a complex Banach space $\text{X}$ is equipped with a Banach manifold structure and an affine connection compatible with its embedding into $\text{B}$($\text{X}$). Geodesic lines are characterized. Moreover, if $\text{X}$ is a Hilbert space and the topology of the self-adjoint part of the manifold is strengthened to be compatible with the Hilbert-Schmidt metric, these geodesics are identified as minimal arcs between pairs of self-adjoint involutions whose straight line distance is less than 2.     

The best approximation theorems and variational inequalities for discontinuous mappings in Banach spaces

We discuss Ky Fan's theorem and the variational inequality problem for discontinuous mappings f in a Banach space X. The main tools of analysis are the variational characterizations of the metric projection operator and the order-theoretic fixed point theory. Moreover, we derive some properties of the metric projection operator in Banach spaces. As applications of our best approximation theorems, three fixed point theorems for non-self maps are established and proved under some conditions. Our results are generalizations and improvements of various recent results obtained by many authors.     

Continuity of metric projections in uniformly convex and uniformly smooth Banach spaces

The continuity of the metric projection onto an approximatively compact set in a uniformly convex and uniformly smooth Banach space is investigated. An explicit modulus of continuity for the metric projection which depends on the directional radius of curvature at a certain point of the set is obtained. The results generalize and improve those obtained by B. O. Björnest l.     

The Maslov-Morse index of indefinite metrics

We compute the Maslov-Morse index of geodesics on a manifold with indefinite metric. It is shown that the multiplicity of conjugate points in the sense of Maslov is equal to the signature of a quadratic form obtained by restricting the metric to the space of degeneracy for the projection of the tangent space onto the Lagrangian manifold, if this latter spans the principal axes of the metric.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 703–708, May, 1973.