ε-近似保正交映射的稳定性与扰动

对于Hilbert空间之间的ε-近似保正交线性映射T,给出了|〈T(x),T(y)〉-||T||~2〈x,y〉|的一个估计,得到了ε-近似保正交线性映射的充分条件,研究了ε-近似保正交线性映射的稳定性,并获得了ε-近似保正交线性映射的扰动定理.     

关于近似Birkhoff正交的注记

在复赋范空间中,给出了近似Birkhoff正交的一个等价刻画,推广了已有的结论,并且证明了近似保ρ+-正交和近似保ρ--正交线性映射都是近似保Birkhoff正交线性映射.     

保正交性或与|·|~k交换的可加映射

设H是复Hilbert空间,B（H）表示H上所有有界线性算子构成的代数.本文刻划了B（H）上保正交性的可加映射和von Neumann代数上与运算|·|k交换的可加映射.     

保正交性或与|·|~k交换的可加映射

设H是复Hilbert空间,B（H）表示H上所有有界线性算子构成的代数.本文刻划了B（H）上保正交性的可加映射和von Neumann代数上与运算|·|k交换的可加映射.     

B(X)上的保秩线性映射

记B(X)为复Banach空间X上有界线性算子全体所成的Banach代数,本文讨论B(X)上把一秩算子映为最多一秩的算子的弱连续线性映射,给出了这种映射所具有的形式,并由此得到B(X)上保秩线性映射,保谱线性映射以及保正线性映射的一些表示定理。     

二维Minkowski空间到完备Riemannσ流形Dirac波映射的Cauchy问题

本文证明了当目标流形是曲率有界的完备流形时, 在Cauchy条件下, 存在唯一的Dirac波映射满足 的Euler-Lagrange方程     

保正交性或与|·|κ交换的可加映射

设H是复Hilbert空间,B(H)表示H上所有有界线性算子构成的代数.本文刻划了B(H)上保正交性的可加映射和von Neumann代数上与运算|·|κ交换的可加映射.     

球面间λ2-特征映射的一些新结果

研究了球面间的λ2-特征映射和欧氏空间上的正交乘,利用Tang等人的思想方法,构造了一些λ2-特征映射,并找到了正交乘Rm×Rn到Rn的具体表达式。正交乘在λ2-特征映射的构造中是很有用的。     

相似不变子空间和保相似线性映射

本文给出了可分无限维Hilbert空间H上有界线性算子全体B(H)中的相似不变子空间的构造,同时给出了B(H)上双边保相似线性映射的表示.     

由向量细分方程生成的双正交多重小波

双正交多重小波是用多分辩率分析由向量细分函数生成的, 文中通过如下形式的向量细分方程的解给出紧支撑双正交多重小波一般性的构造方法: 其中向量函数j=(j1,…, j r)T属于 中, 是一个具有有限长的矩阵值序列, 称为细分面具, M 是一个满足limn→∞M-n=0的s×s整数矩阵. 我们的刻画是在一般的情形下. 文中的主要结果是一些已知重要结果的实质性延拓     

Stability of the orthogonality preserving property in finite-dimensional inner product spaces

We establish a stability property for inner product preserving (not necessarily linear) mappings. Then, as a consequence, we show that a linear mapping, defined on a finite-dimensional inner product space, which approximately preserves orthogonality can be approximated by a linear, orthogonality preserving one.     

On mappings approximately preserving orthogonality

We answer a question posed by Chmieliński, whether a linear map which approximately preserves orthogonality must be close to an orthogonality preserving one. Furthermore, we give a short proof of the stability of the orthogonality equation on finite dimensional Hilbert spaces.     

Stability of disjointness preserving mappings

Let and be compact Hausdorff spaces and let . A linear mapping is called -disjointness preserving if implies that . If is a continuous or surjective -disjointness preserving linear mapping, we prove that there exists a disjointness preserving linear mapping satisfying . We also prove that every unbounded -disjointness preserving linear functional on is disjointness preserving.

     

Quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality

In this paper we introduce a new geometry constant D(X) to give a quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality. We show that 1 and is the upper and lower bound for D(X), respectively, and characterize the spaces of which D(X) attains the upper and lower bounds. We calculate D(X) when X=(R²,‖⋅p_‖) and when X is a symmetric Minkowski plane respectively, we show that when X is a symmetric Minkowski plane D(X)=D(X^∗).     

Diameter preserving surjections in the geometry of matrices

We consider a class of graphs subject to certain restrictions, including the finiteness of diameters. Any surjective mapping φ:Γ→Γ^′ between graphs from this class is shown to be an isomorphism provided that the following holds: Any two points of Γ are at a distance equal to the diameter of Γ if, and only if, their images are at a distance equal to the diameter of Γ^′. This result is then applied to the graphs arising from the adjacency relations of spaces of rectangular matrices, spaces of Hermitian matrices, and Grassmann spaces (projective spaces of rectangular matrices).     

Approximate range searching using binary space partitions

We show how any BSP tree for the endpoints of a set of n disjoint segments in the plane can be used to obtain a BSP tree of size for the segments themselves, such that the range-searching efficiency remains almost the same. We apply this technique to obtain a BSP tree of size O(nlogn) such that -approximate range searching queries with any constant-complexity convex query range can be answered in O(min_>0{(1/)+k}logn) time, where k is the number of segments intersecting the -extended range. The same result can be obtained for disjoint constant-complexity curves, if we allow the BSP to use splitting curves along the given curves.We also describe how to construct a linear-size BSP tree for low-density scenes consisting of n objects in such that -approximate range searching with any constant-complexity convex query range can be done in O(logn+min_>0{(1/^d−1)+k}) time.     

Second-Order Optimality and Beyond: Characterization and Evaluation Complexity in Convexly Constrained Nonlinear Optimization

High-order optimality conditions for convexly constrained nonlinear optimization problems are analysed. A corresponding (expensive) measure of criticality for arbitrary order is proposed and extended to define high-order \(\epsilon \)-approximate critical points. This new measure is then used within a conceptual trust-region algorithm to show that if derivatives of the objective function up to order \(q \ge 1\) can be evaluated and are Lipschitz continuous, then this algorithm applied to the convexly constrained problem needs at most \(O(\epsilon ^{-(q+1)})\) evaluations of f and its derivatives to compute an \(\epsilon \)-approximate qth-order critical point. This provides the first evaluation complexity result for critical points of arbitrary order in nonlinear optimization. An example is discussed, showing that the obtained evaluation complexity bounds are essentially sharp.