On the generation of updates for quasi-Newton methods

We present a unified technique for updating approximations to Jacobian or Hessian matrices when any linear structure can be imposed. The updates are derived by variational means, where an operator-weighted Frobenius norm is used, and are finally expressed as solutions of linear equations and/or unconstrained extrema. A certain behavior of the solutions is discussed for certain perturbations of the operator and the constraints. Multiple secant relations are then considered. For the nonsparse case, an explicit family of updates is obtained including Broyden, DFP, and BFGS. For the case where some of the matrix elements are prescribed, explicit solutions are obtained if certain conditions are satisfied. When symmetry is assumed, we show, in addition, the connection with the DFP and BFGS updates.This work was partially supported by a grant from Control Data     

Banach空间中的拟范均值

在算术平均值和几何平均值的基础上,给出了一种新的平均值—拟算数平均值以及拟几何平均值,并将它们引入到Banach空间进行了讨论.给出了一些定义并在其基础上得到了这几类平均值在Banach空间中的几个有用的性质.     

Strong convergence of iterative sequences for nonexpansive mappings

An iterative algorithm is proposed for finding a fixed point of a nonexpansive self-mapping of a closed convex subset of a Banach space with a uniformly Gâteaux differentiable norm, and an estimate of the convergence speed also is given.     

Strong convergence of an iterative method for non‐expansive mappings

Let E be a real reflexive Banach space having a weakly continuous duality mapping J_φ with a gauge function φ, and let K be a nonempty closed convex subset of E. Suppose that T is a non‐expansive mapping from K into itself such that F (T) ≠ ??. For an arbitrary initial value x₀ ∈ K and fixed anchor u ∈ K, define iteratively a sequence {x_n } as follows: x_{n +1} = α_n u + β_n x_n + γ_n Tx_n , n ≥ 0, where {α_n }, {β_n }, {γ_n } ? (0, 1) satisfies α_n +β_n + γ_n = 1, (C 1) lim_{n →∞} α_n = 0, (C 2) ∑^∞_{n =1} α_n = ∞ and (B) 0 < lim inf_{n →∞} β_n ≤ lim sup_{n →∞} β_n < 1. We prove that {x_n } converges strongly to Pu as n → ∞, where P is the unique sunny non‐expansive retraction of K onto F (T). We also prove that the same conclusions still hold in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm or in a uniformly smooth Banach space. Our results extend and improve the corresponding ones by C. E. Chidume and C. O. Chidume [Iterative approximation of fixed points of non‐expansive mappings, J. Math. Anal. Appl. 318 , 288–295 (2006)], and develop and complement Theorem 1 of T. H. Kim and H. K. Xu [Strong convergence of modified Mann iterations, Nonlinear Anal. 61 , 51–60 (2005)]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Operator trigonometry of Hotelling correlation, Frobenius condition, Penrose twistor

Three new applications of the author’s operator trigonometry are presented. These take place in three diverse mathematical domains: matrix statistics, numerical analysis, and theoretical physics. The goal is to continue to develop and expand the scope of the operator trigonometry.     

ON L^P-MATRICIALLY NORMED SPACES

1Introduction and PreliminariesThroughout this paper,we consider linear spaces over the complex numbers C.Given alinear space X and integers m and n,let Mm,n(X)=X?Mm,n denote the linear space ofm×n matricesx=?????x.x..m1,1,1.x..xm1,n,n?????with entries x…     

Spherical Minimax Location Problem

This paper presents an algorithm for determining a minimax location to service demand points that are equally weighted and distributed over a sphere. The norm under consideration is geodesic. The algorithm presented here is based on enumeration and has a polynomial time complexity.     

Anisotropic norms of dirichlet kernels and some other trigonometric polynomials

In the paper new estimates of anisotropic norms for trigonometric polynomials are obtained. Translated fromMatematischeskie Zametki, Vol. 67, No. 5, pp. 686–701, May, 2000.     

基于灰色关联分析的财产保险公司偿付能力的模糊综合评判

为解决财产保险公司的偿付能力的评价问题,建立了基于灰色关联分析的模糊综合评判模型.选取了2003-2005年我国8家财产保险公司的相关数据,用灰色关联分析中的点关联系数求解隶属度得出评判矩阵,用范数灰关联度法确定因素集中各因素的权重,从而对其偿付能力进行模糊综合评判,得出各公司的偿付能力所属的类别.并对评语集中各评语赋予适当的权重,对所研究的8家财产保险公司依据其偿付能力的情况进行了排序.     

Counting the number of indecomposable representations and a q-analogue of the Weyl-Kac denominator identity of type _n

By using Frobenius maps and F-stable representations,we count the number of isomor- phism classes of indecomposable representations with the fixed dimension vector of a species of type _n over a finite field,first,and then,as an application,give a q-analogue of the Weyl-Kac denominator identity of type _n.