Weighted Variational Inequalities

In this paper, we introduce weighted variational inequalities over product of sets and system of weighted variational inequalities. It is noted that the weighted variational inequality problem over product of sets and the problem of system of weighted variational inequalities are equivalent. We give a relationship between system of weighted variational inequalities and systems of vector variational inequalities. We define several kinds of weighted monotonicities and establish several existence results for the solution of the above-mentioned problems under these weighted monotonicities. We introduce also the weighted generalized variational inequalities over product of sets, that is, weighted variational inequalities for multivalued maps and systems of weighted generalized variational inequalities. Extensions of weighted monotonicities for multivalued maps are also considered. The existence of a solution of weighted generalized variational inequalities over product of sets is also studied. The existence results for a solution of weighted generalized variational inequality problem give also the existence of solutions of systems of generalized vector variational inequalities. The first and third author express their thanks to the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The authors are also grateful to the referees for comments and suggestions improving the final draft of this paper.     

Existence and uniqueness of weighted pseudoinverse matrices and weighted normal pseudosolutions with singular weights

For one definition of weighted pseudoinversion with singular weights, we establish necessary and sufficient conditions for the existence and uniqueness of a solution of a system of matrix equations. Expansions of weighted pseudoinverse matrices in matrix power series and matrix power products are obtained. A relationship between weighted pseudoinverse matrices the weighted normal pseudosolutions is established. Iterative methods for the calculation of weighted pseudoinverse matrices and weighted normal pseudosolutions are constructed.     

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.     

Computational Tools in Weighted Persistent Homology

In this paper, the authors study further properties and applications of weighted homology and persistent homology. The Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology are introduced. For applications, the authors show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. They also prove a theorem to calculate the mod p² weighted persistent homology provided with some information on the mod p weighted persistent homology.     

Branch-and-bound algorithms for scheduling in permutation flowshops to minimize the sum of weighted flowtime/sum of weighted tardiness/sum of weighted flowtime and weighted tardiness/sum of weighted flowtime,weighted tardiness and weighted earliness of jobs

The problem of scheduling in permutation flowshops is considered in this paper with the objectives of minimizing the sum of weighted flowtime/sum of weighted tardiness/sum of weighted flowtime and weighted tardiness/sum of weighted flowtime, weighted tardiness and weighted earliness of jobs, with each objective considered separately. Lower bounds on the given objective (corresponding to a node generated in the scheduling tree) are developed by solving an assignment problem. Branch-and-bound algorithms are developed to obtain the best permutation sequence in each case. Our algorithm incorporates a job-based lower bound (integrated with machine-based bounds) with respect to the weighted flowtime/weighted tardiness/weighted flowtime and weighted tardiness, and a machine-based lower bound with respect to the weighted earliness of jobs. The proposed algorithms are evaluated by solving many randomly generated problems of different problem sizes. The results of an extensive computational investigation for various problem sizes are presented. In addition, one of the proposed branch-and-bound algorithms is compared with a related existing branch-and-bound algorithm.     

独立随机序列加权和的强收敛性

本文研究独立随机变量序列加权和的强收敛性,利用截尾法和Borel-Cantelli引理,证明了加权系数ank为列阵情形的强收敛性,在一般双下标加权系数的加权部分和的强收敛性,并对Jamison型加权部分和情形证明了其强收敛的充要条件,推广了Chow与Teicher(1971)[3]的相应结果.     

加权极分解

In this paper, a new matrix decomposition called the weighted polar decomposition is considered. Two uniqueness theorems of weighted polar decomposition are presented, and the best approximation property of weighted unitary polar factor and perturbation bounds for weighted polar decomposition are also studied.     

A weighted pseudoinverse,generalized singular values,and constrained least squares problems

The weighted pseudoinverse providing the minimum semi-norm solution of the weighted linear least squares problem is studied. It is shown that it has properties analogous to those of the Moore-Penrose pseudoinverse. The relation between the weighted pseudoinverse and generalized singular values is explained. The weighted pseudoinverse theory is used to analyse least squares problems with linear and quadratic constraints. A numerical algorithm for the computation of the weighted pseudoinverse is briefly described.This work was supported in part by the Swedish Institute for Applied Mathematics.     

On the modified Gram-Schmidt algorithm for weighted and constrained linear least squares problems

A framework and an algorithm for using modified Gram-Schmidt for constrained and weighted linear least squares problems is presented. It is shown that a direct implementation of a weighted modified Gram-Schmidt algorithm is unstable for heavily weighted problems. It is shown that, in most cases it is possible to get a stable algorithm by a simple modification free from any extra computational costs. In particular, it is not necessary to perform reorthogonalization.Solving the weighted and constrained linear least squares problem with the presented weighted modified Gram-Schmidt algorithm is seen to be numerically equivalent to an algorithm based on a weighted Householder-likeQR factorization applied to a slightly larger problem. This equivalence is used to explain the instability of the weighted modified Gram-Schmidt algorithm. If orthogonality, with respect to a weighted inner product, of the columns inQ is important then reorthogonalization can be used. One way of performing such reorthogonalization is described.Computational tests are given to show the main features of the algorithm.     

Weighted limits in an $$(\infty ,1)$$ ( ∞ , 1 ) -category

We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal’s approach: we identify a meaningful construction for the quasi-category of weighted cones over a diagram in a quasi-category, whose terminal object is the weighted limit of the considered diagram. We then show that each weighted limit can be expressed as an ordinary limit. When the quasi-category arises as the homotopy coherent nerve of a category enriched over Kan complexes, we generalize an argument by Riehl-Verity to show that the weighted limit agrees with the homotopy weighted limit in the sense of enriched category theory, for which explicit constructions are available. When the quasi-category is complete, tensored and cotensored over the quasi-category of spaces, we discuss a possible comparison of our definition of weighted limit with the approach by Gepner-Haugseng-Nikolaus.

     

Weighted allocation rules for standard fixed tree games

In this paper we consider standard fixed tree games, for which each vertex unequal to the root is inhabited by exactly one player. We present two weighted allocation rules, the weighted down-home allocation and the weighted neighbour-home allocation, both inspired by the painting story in Maschler et al. (1995) . We show, in a constructive way, that the core equals both the set of weighted down-home allocations and the set of weighted neighbour allocations. Since every weighted down-home allocation specifies a weighted Shapley value (Kalai and Samet (1988)) in a natural way, and vice versa, our results provide an alternative proof of the fact that the core of a standard fixed tree game equals the set of weighted Shapley values. The class of weighted neighbour allocations is a generalization of the nucleolus, in the sense that the latter is in this class as the special member where players have all equal weights.     

α-Admissibility of Observation Operators in Discrete and Continuous Time

In this paper a weighted form of the Weiss conjecture is studied. For certain weights, the conjecture is shown to hold for normal contraction operators related to discrete time linear systems. This is proved by an application of the Carleson measure theorem for weighted Dirichlet spaces. The result for discrete time systems is used to show that a weighted form of the Weiss conjecture holds for normal operators generating bounded C₀-semigroups. Previously, weighted admissibility has been characterised for generators of analytic semigroups. No such assumption of analyticity is made here. Additionally, results are presented regarding weighted Carleson measures, fractional powers of normal operators and weighted composition operators.     

Weighted Shapley hierarchy levels values

A new class of values for cooperative games with level structure is introduced. We apply a multi-step proceeding to the weighted Shapley values. For characterization, two well-known axiomatizations of the weighted Shapley values are extended, the first one by efficiency and weighted balanced contributions and the second one by weighted standardness for two-player games and consistency. We get a new axiomatization of the Shapley levels value too.     

Basic inequalities for weighted entropies

The concept of weighted entropy takes into account values of different outcomes, i.e., makes entropy context-dependent, through the weight function. In this paper, we establish a number of simple inequalities for the weighted entropies (general as well as specific), mirroring similar bounds on standard (Shannon) entropies and related quantities. The required assumptions are written in terms of various expectations of weight functions. Examples are weighted Ky Fan and weighted Hadamard inequalities involving determinants of positive-definite matrices, and weighted Cramér-Rao inequalities involving the weighted Fisher information matrix.     

Hermitian Weighted Composition Operators and Bergman Extremal Functions

Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are ${(1 - \overline{w}z)^{-\kappa}}$ for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces.     

Essential spectrum of the weighted Laplacian on noncompact manifolds and applications

We obtain upper estimates for the bottom (that is, greatest lower bound) of the essential spectrum of weighted Laplacian operator of a noncompact weighted manifold under assumptions of the volume growth of their geodesic balls and spheres. Furthermore, we find examples where the equality occurs in the estimates obtained. As a consequence, we give estimates for the weighted mean curvature of complete noncompact hypersurfaces into weighted manifolds.     

二元Boolean矩阵的加权Moore-Penrose逆

主要研究了二元Boolean矩阵A的加权Moore-Penrose逆的存在性问题,给出了二元Boolean矩阵A的加权Moore-Penrose逆存在的一些充分必要条件,并讨论了加权Moore-Penrose逆存在时的若干等价刻画及惟一性问题.     

Normal-mode attenuation and group velocity in shallow water

The computational methods of the mode attenuation and group,velocity in shallow water are discussed and two novel modified weighted integral formulae are presented. The accuracy of the modified weighted integral formulae, cycle distance formulae and weighted integral formulae in the Pekeris channel is compared theoretically and numerically. Transmission losses are calculated by those three kinds of formulae in a shallow water. Theoretical analyses and numerical results show that the modified weighted integral formulae provide higher accuracy than the weighted integral formulae. So the modified weighted integral formulae can be widely used for computing the shallow water acoustic field. Project supported by the National Natural Science Foundation of China (Grant No. 19574061).     

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.     

Approximation algorithms for the weighted independent set problem in sparse graphs

The approximability of the unweighted independent set problem has been analyzed in terms of sparseness parameters such as the average degree and inductiveness. In the weighted case, no corresponding results are possible for average degree, since adding vertices of small weight can decrease the average degree arbitrarily without significantly changing the approximation ratio. In this paper, we introduce two weighted measures, namely weighted average degree and weighted inductiveness, and analyze algorithms for the weighted independent set problem in terms of these parameters.