On weakly irreducible nonnegative tensors and interval hull of some classes of tensors

Based on the S-type eigenvalue localization set developed by Li et al. (Linear Algebra Appl. 493 (2016) 469-483) for tensors, a modified S-type eigenvalue localization set for tensors is established in this paper by excluding some sets from the existing S-type eigenvalue localization set developed by Huang et al. (arXiv: 1602.07568v1, 2016). The proposed set containing all eigenvalues of tensors is much sharper compared with that employed by Li et al. and Huang et al. As its applications, a criteria, which can be utilized for identifying the nonsingularity of tensors, is developed. In addition, we provide new upper and lower bounds for the spectral radius of nonnegative tensors and the minimum H-eigenvalue of weakly irreducible strong M-tensors. These bounds are superior to some previous results, which is illustrated by some numerical examples.

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Some inequalities on the spectral radius of nonnegative tensors

Open Mathematics ◽

10.1515/math-2020-0143 ◽

2020 ◽

Vol 18 (1) ◽

pp. 262-269

Author(s):

Chao Ma ◽

Hao Liang ◽

Qimiao Xie ◽

Pengcheng Wang

Keyword(s):

Spectral Radius ◽

Nonnegative Tensors

Abstract The eigenvalues and the spectral radius of nonnegative tensors have been extensively studied in recent years. In this paper, we investigate the analytic properties of nonnegative tensors and give some inequalities on the spectral radius.

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Vibration-based damage detection with uncertainty quantification by structural identification using nonlinear constraint satisfaction with interval arithmetic

Structural Health Monitoring ◽

10.1177/1475921718806476 ◽

2018 ◽

Vol 18 (5-6) ◽

pp. 1569-1589 ◽

Cited By ~ 3

Author(s):

Timothy Kernicky ◽

Matthew Whelan ◽

Ehab Al-Shaer

Keyword(s):

Constraint Satisfaction ◽

Interval Arithmetic ◽

Model Updating ◽

Structural Identification ◽

Inverse Eigenvalue Problem ◽

Degree Of Freedom ◽

Nonlinear Constraint ◽

Complete Set ◽

Feasible Solutions ◽

Interval Hull

Structural identification has received increased attention over recent years for performance-based structural assessment and health monitoring. Recently, an approach for formulating the finite element model updating problem as a constraint satisfaction problem has been developed. In contrast to widely used probabilistic model updating through Bayesian inference methods, the technique naturally accounts for measurement and modeling errors through the use of interval arithmetic to determine the set of all feasible solutions to the partially described and incompletely measured inverse eigenvalue problem. This article presents extensions of the constraint satisfaction approach permitting the application to larger multiple degree-of-freedom system models. To accommodate for the drastic increase in the dimensionality of the inverse problem, the extended methodology replaces computation of the complete set of solutions with an approach that contracts the initial search space to the interval hull, which encompasses the complete set of feasible solutions with a single interval vector solution. The capabilities are demonstrated using vibration data acquired through hybrid simulation of a 45-degree-of-freedom planar truss, where a two-bar specimen with bolted connections representing a single member of the truss serves as the experimental substructure. Structural identification is performed using data acquired with the undamaged experimental member as well as over a number of damage scenarios with progressively increased severity developed by exceeding a limit-state capacity of the member. Interval hull solutions obtained through application of the nonlinear constraint satisfaction methodology demonstrate the capability to correctly identify and quantify the extent of the damage in the truss while incorporating measurement uncertainties in the parameter identification.

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