Perturbation bounds for weighted polar decomposition in the weighted unitarily invariant norm

In this paper, by generalizing the ideas of the (generalized) polar decomposition to the weighted polar decomposition and the unitarily invariant norm to the weighted unitarily invariant norm, we present some perturbation bounds for the generalized positive polar factor, generalized nonnegative polar factor, and weighted unitary polar factor of the weighted polar decomposition in the weighted unitarily invariant norm. These bounds extend the corresponding recent results for the (generalized) polar decomposition. In addition, we also give the comparison between the two perturbation bounds for the generalized positive polar factor obtained from two different methods. Copyright © 2008 John Wiley & Sons, Ltd.     

赋权图中的路和圈

本文研究了赋权图中的最长路和最长圈,将关于非赋权图中最长路和最长圈的一些结果推广到赋权图上.     

带权Paneitz算子和带权圆盘振动问题的特征值不等式

本文研究光滑度量测度空间上带权Paneitz算子的闭特征值问题和带权圆盘振动问题,给出Euclid空间、单位球面、射影空间和一般Riemann流形的n维紧子流形的权重Paneitz算子和带权圆盘振动问题的前n个特征值上界估计.进一步地,本文给出带权Ricci曲率有界的紧致度量测度空间上带权圆盘振动问题的第一特征值的下界估计.     

加权幂平均不等式的加强

对加权幂平均不等式的加强作了全面的研究，推广了一个加强的加权算术-几何不等式．     

Products of weighted composition and differentiation operators into weighted Zygmund and Bloch spaces

We characterize boundedness and compactness of products of differentiation operators and weighted composition operators between weighted Banach spaces of analytic functions and weighted Zygmund spaces or weighted Bloch spaces with general weights.     

Dimensions of weighted Besicovitch sets

We consider weighted Besicovitch sets which are defined in terms of the weighted frequency of digit 1 in the dyadic expansion of real numbers. Explicit formulas for their Hausdorff dimensions are given.     

A note on weighted composition operators on the weighted Bergman space

This paper gives a note on weighted composition operators on the weighted Bergman space, which shows that for a fixed composition symbol, the weighted composition operators are bounded on the weighted Bergman space only with bounded weighted symbols if and only if the composition symbol is a finite Blaschke product.     

广义分数次积分算子在加权Herz型Hardy空间的有界性

利用加权Herz型Hardy空间的原子分解理论,讨论了广义分数次积分算子Tl从加权Lp空间到加权Lq空间,以及从加权Herz型Hardy空间到加权Herz空间的有界性问题.将已有的分数次积分算子的结论推广到广义分数次积分算子的情形.     

Expansions and polynomial limit representations of weighted pseudoinverses

Expansions of weighted pseudoinverses with positive definite or singular weights in matrix power series or power products with negative exponents and arbitrary positive parameters are proposed and analyzed. Based on these expansions, polynomial limit representations of weighted pseudoinverses are obtained. Issues related to the construction of direct and iterative methods for calculating weighted pseudoinverses and weighted normal pseudosolutions, as well as solving constrained least squares problems, are examined.     

Bochner—Riesz算子交换子在加权Morrey空间上的有界性

运用了Sharp极大函数估计的方法证明了当权函数满足一定条件时,Bochner～Riesz算子与加权BMO函数生成的交换子在加权Morrey空间上的有界性．     

Convergence of two-dimensional weighted integrals

A two-dimensional weighted integral in is proposed as a tool for analyzing higher-dimensional unweighted integrals, and a necessary and sufficient condition for the finiteness of the weighted integral is obtained.

     

Further spectral properties of the weighted finite Fourier transform operator and approximation in weighted Sobolev spaces

In this work, we first give some mathematical preliminaries concerning the generalized prolate spheroidal wave function (GPSWF). This set of special functions have been defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator has a super‐exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As an application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a weighted Sobolev space. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.     

Error estimation for a weighted minimum-norm least squares solution with positive definite weights

The weighted least squares problem is considered. Given a generally inconsistent system of linear algebraic equations, error estimates are obtained for its weighted minimum-norm least squares solution under perturbations of the matrix and the right-hand side, including the case of rank modifications of the perturbed matrix.     

Pointwise weighted approximation by Bernstein operators

We consider the pointwise weighted approximation by Bernstein operators. The related weight functions are Jacobi weights . We obtain approximation equivalence theorems, using a weighted modulus of smoothness , which is an extension of~[5].     

Heavy cycles passing through some specified vertices in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex υ is called the weighted degree of υ. The weight of a cycle is defined as the sum of the weights of its edges. In this paper, we prove that: (1) if G is a 2‐connected weighted graph such that the minimum weighted degree of G is at least d, then for every given vertices x and y, either G contains a cycle of weight at least 2d passing through both of x and y or every heaviest cycle in G is a hamiltonian cycle, and (2) if G is a 2‐connected weighted graph such that the weighted degree sum of every pair of nonadjacent vertices is at least s, then for every vertex y, G contains either a cycle of weight at least s passing through y or a hamiltonian cycle. AMS classification: 05C45 05C38 05C35. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Weighted pseudoinverses and weighted normal pseudosolutions with singular weights

Weighted pseudoinverses with singular weights can be defined by a system of matrix equations. For one of such definitions, necessary and sufficient conditions are given for the corresponding system to have a unique solution. Representations of the pseudoinverses in terms of the characteristic polynomials of symmetrizable and symmetric matrices, as well as their expansions in matrix power series or power products, are obtained. A relationship is found between the weighted pseudoinverses and the weighted normal pseudosolutions, and iterative methods for calculating both pseudoinverses and pseudosolutions are constructed. The properties of the weighted pseudoinverses with singular weights are shown to extend the corresponding properties of weighted pseudoinverses with positive definite weights.     

Heavy cycles in 2-connected triangle-free weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v, denoted by dw(v). The weight of a cycle is defined as the sum of the weights of its edges. Fujisawa proved that if G is a 2-connected triangle-free weighted graph such that the minimum weighted degree of G is at least d, then G contains a cycle of weight at least 2d. In this paper, we proved that if G is a2-connected triangle-free weighted graph of even size such that dw(u) + dw(v) ≥ 2d holds for any pair of nonadjacent vertices u, v ∈ V(G), then G contains a cycle of weight at least 2d.     

The finite Hilbert transform and weighted Sobolev spaces

The boundedness of the finite Hilbert transform operator on certain weighted L_p spaces is well known. We extend this result to give the boundedness of that operator on certain weighted Sobolev spaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixing weighted values of non-atomic games

Weighted values of non-atomic games were introduced by Hart and Monderer (1997). They study these values by using two approaches: the potential approach and the asymptotic approach. In this study we develop the random order approach (the mixing value, Aumann and Shapley, 1974) to weighted values and prove that these values coincide with the asymptotic weighted values of Hart and Monderer in pNA. Received April 1995/Revised version May 1996/ Final version October 1997