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On Nonnegative Solution of Multi-Linear System with Strong M_(z)-Tensors

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Numerical Mathematics(Theory,Methods and Applications) 2021年第1期14卷 176-193页

作者： Changxin Mo yimin wei School of Mathematical Sciences Fudan UniversityShanghai 200433P.R.of China School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics Fudan UniversityShanghai 200433P.R.of China

A class of structured multi-linear system defined by strong M_(z)-tensors is *** prove that the multi-linear system with strong M_(z)-tensors always has a nonnegative solution under certain condition by the fixed point *** also prove that the zero solution is the only solution of the homogeneous multi-linear system for some structured tensors,such as strong M-tensors,H^(+)-tensors,strictly diagonally dominant tensors with positive diagonal *** examples are presented to illustrate our theoretical results.

关键词： M_(z)-tensor multi-linear system nonnegative solution M-tensor tensor equation fixed point theory

weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product

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Frontiers of Mathematics in China 2017年第6期12卷 1319-1337页

作者： Jun JI yimin wei Department of Mathematics Kennesaw State University Marietta GA 30060 USA School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics ~-hldan University Shanghai 200433 China

We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the mill spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares （M） solutions to a multilineax system and establish the relationship between the minimum-norm （N） leastsquares （M）solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.

关键词： Fundamental theorem weighted Moore-Penrose inverse multi- linear system null space and range tensor equation

H-tensors and nonsingular H-tensors

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Frontiers of Mathematics in China 2016年第3期11卷 557-575页

作者： Xuezhong WANG yimin wei School of Mathematical Sciences Fudan University Shanghai 200433 China School of Mathematics and Statistics Hexi University Zhangye 734000 China

The H-matrices are an important class in the matrix theory, and have many applications. Recently, this concept has been extended to higher order H-tensors. In this paper, we establish important properties of diagonally dominant tensors and H-tensors. Distributions of eigenvalues of nonsingular symmetricH-tensors are given. An J（t%-tensor is semi-positive, which enlarges the area of semi-positive tensor from H-tensor to H＋-tensor. The spectral radius of Jacobi tensor of a nonsingular （resp. singular） H-tensor is less than （resp. equal to） one. In particular, we show that a quasi-diagonally dominant tensor is a nonsingular H-tensor if and only if all of its principal sub-tensors are nonsingular H-tensors. An irreducible tensor H is an H-tensor if and only if it is quasi-diagonally dominant.

关键词： Diagonally dominant irreducible diagonally dominant H-tensor,nonsingular

CONDITION NUMBER FOR weiGHTED LINEAR LEAST SQUARES PROBLEM

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Journal of Computational Mathematics 2007年第5期25卷 561-572页

作者： yimin wei Huaian Diao Sanzheng Qiao School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Sciences Fudan University Shanghai 200433 China Department of Mathematics The City University of Hong Kong 83 Tat Chee Avenue Kowloon Hong Kong Department of Computing and Software McMaster University Hamilton Ontario L8S 4LT Canada

In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem： minx ｜｜Ax- b｜｜2, where A is an m-by-n （m ≥ n） rank deficient matrix. We first derive an explicit expression for the condition number in the weighted Frobenius norm ｜｜ [AT,βb] ｜｜F of the data A and b, where T is a positive diagonal matrix and β is a positive scalar. We then discuss the sensitivity of the standard 2-norm condition numbers for the generalized matrix inversion and rank deficient least squares and establish relations between the condition numbers and their condition numbers called level-2 condition numbers.

关键词： Moore-Penrose inverse Condition number Linear least squares.

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Splitting Method for Support Vector Machine in Reproducing Kernel Banach Space with a Lower Semi-continuous Loss Function

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Chinese Annals of Mathematics,Series B 2024年第6期45卷 823-854页

作者： Mingyu MO yimin wei Qi YE Academy for Advanced Interdisciplinary Studies Peking University School of Mathematical Science South China Normal University School of Mathematical Science Key Laboratory of Mathematics for Nonlinear Sciences Fudan University Corresponding author.School of Mathematical Science South China Normal University

In this paper, the authors employ the splitting method to address support vector machine within a reproducing kernel Banach space framework, where a lower semicontinuous loss function is utilized. They translate support vector machine in reproducing kernel Banach space with such a loss function to a finite-dimensional tensor optimization problem and propose a splitting method based on the alternating direction method of multipliers. Leveraging Kurdyka-Lojasiewicz property of the augmented Lagrangian function,the authors demonstrate that the sequence derived from this splitting method is globally convergent to a stationary point if the loss function is lower semi-continuous and subanalytic. Through several numerical examples, they illustrate the effectiveness of the proposed splitting algorithm.

关键词： 46E22 68P01 68Q32 68T05 Lower semi-continuous loss function Reproducing kernel Banach space Splitting method Support vector machine Tensor optimization problem

The Generalized Order Tensor Complementarity Problems

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Numerical Mathematics(Theory,Methods and Applications) 2020年第1期13卷 131-149页

作者： Maolin Che Liqun Qi yimin wei School of Economic Mathematics Southwestern University of Finance and EconomicsChengdu 611130SichuanChina Department of Applied Mathematics The Hong Kong Polytechnic UniversityHong Kong School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics Fudan UniversityShanghai 200433China

The main propose of this paper is devoted to study the solvability of the generalized order tensor complementarity *** define two problems:the generalized order tensor complementarity problem and the vertical tensor comple-mentarity problem and show that the former is equivalent to the *** the degree theory,we present a comprehensive analysis of existence,uniqueness and stability of the solution set of a given generalized order tensor complementarity problem.

关键词： Generalized order tensor complementarity problems tensor complementarity prob-lems structured tensors degree theory

Influence of Axial Loads to Propagation Characteristics of the Elastic Wave in a Non-Uniform Shaft

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Chinese Journal of Mechanical Engineering 2019年第4期32卷 140-152页

作者： yimin wei Zhiwei Zhao Wenhua Chen Qi Liu National and Local Joint Engineering Research Center of Reliability Analysis and Testing for Mechanical and Electrical Products Zhejiang SciTech University

The vibration propagates through the shaft in the form of elastic waves. The propagation characteristics of the elastic waves are a ected by the axial loads. The influence of the axial loads to the propagation characteristics of the elastic waves is studied in this paper. Firstly, the transfer matrix of the elastic waves for the non-uniform shaft with axial loads is deduced by combining the transfer matrix without axial load and the additional equation caused by the axial load. And then, a numerical method is used to study the influence of the axial load, non-uniformity and the rotating speed to the propagation characteristics of the elastic waves. It’s found that a new Stop Band will appear due to the axial force, and the central frequency of which will decrease as the increase of the force, while the band width of which remains the same. The central frequency of the new Stop Band will also increase as the increase of the cross-section area ratio;however, the rotating speed of the shaft doesn’t a ect the propagation characteristics of the elastic waves obviously. Finally, an experimental rig is built up for further study, even though there are some small local errors, the results of experiments match well with the numerical ones, which indicates the validation of the theoretical results. The result can help to study the influence of the axial load to the dynamics of a non-uniform shaft and help to reveal the vibration propagating mechanism in such a shaft.

关键词： Axial force Propagation characteristic Non-uniform shaft Elastic wave Transfer matrix

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T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

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Communications on Applied Mathematics and Computation 2021年第2期3卷 201-220页

作者： Yun Miao Liqun Qi yimin wei School of Mathematical Sciences Fudan UniversityShanghai 200433China Department of Applied Mathematics The Hong Kong Polytechnic UniversityHong KongChina School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics Fudan UniversityShanghai 200433China

In this paper,we investigate the tensor similarity and propose the T-Jordan canonical form and its *** concepts of the T-minimal polynomial and the T-characteristic polynomial are *** a special case,we present properties when two tensors commute based on the tensor *** prove that the Cayley-Hamilton theorem also holds for tensor ***,we focus on the tensor decompositions:T-polar,T-LU,T-QR and T-Schur decompositions of tensors are *** an F-square tensor is not invertible with the T-product,we study the T-group inverse and the T-Drazin inverse which can be viewed as the extension of matrix *** expressions of the T-group and T-Drazin inverses are given by the T-Jordan canonical *** polynomial form of the T-Drazin inverse is also *** the last part,we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverses can be given by a limit process.

关键词： T-Jordan canonical form T-function T-index Tensor decomposition T-Drazin inverse T-group inverse T-core-nilpotent decomposition

Targeted energy transfer of a parallel nonlinear energy sink

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Applied Mathematics and Mechanics(English Edition) 2019年第5期40卷 621-630页

作者： yimin wei Sha wei Qianlong ZHANG Xinjian DONG Zhike PENG Wenming ZHANG State Key Laboratory of Mechanical System and Vibration Shanghai Jiao Tong University

A parallel nonlinear energy sink(NES) is proposed and analyzed. The parallel NES is composed of a vibro-impact(VI) NES and a cubic NES. The dynamical equation is given, and the essential analytical investigation is carried out to deal with the cubic nonlinearity and impact nonlinearity. Multiple time-scale expansion is introduced, and the zeroth order is derived to give a rough outline of the system. The underlying Hamilton dynamic equation is given, and then the optimal stiffness is expressed. The clearance is regarded as a critical factor for the VI. Based on the periodical impact treatment by analytical investigation, the relationships of the cubic stiffness, the clearance, and the zeroth-order attenuation amplitude of the linear primary oscillator(LPO) are obtained.A cubic NES under the optimal condition is compared with the parallel NES. Harmonic signals, harmonic signals with noises, and the excitation generated by a second-order?lter are considered as the potential excitation forces on the system. The targeted energy transfer(TET) in the designed parallel NES is shown to be more e?cient.

关键词： targeted energy transfer (TET) nonlinear energy sink (NES) nonlinear vibration absorber impact vibration reduction

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