scholarly journals Identifying the Positive Definiteness of Even-Order Weakly Symmetric Tensors via Z-Eigenvalue Inclusion Sets

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1239
Author(s):  
Feichao Shen ◽  
Ying Zhang ◽  
Gang Wang
Keyword(s):  
Asymptotic Stability ◽  
Sufficient Conditions ◽  
Positive Definiteness ◽  
Polynomial Systems ◽  
Invariant Polynomial ◽  
Symmetric Tensors ◽  
Even Order ◽  
Weakly Symmetric ◽  
Time Invariant ◽  
The Given

The positive definiteness of even-order weakly symmetric tensors plays important roles in asymptotic stability of time-invariant polynomial systems. In this paper, we establish two Brauer-type Z-eigenvalue inclusion sets with parameters by Z-identity tensors, and show that these inclusion sets are sharper than existing results. Based on the new Z-eigenvalue inclusion sets, we propose some sufficient conditions for testing the positive definiteness of even-order weakly symmetric tensors, as well as the asymptotic stability of time-invariant polynomial systems. The given numerical experiments are reported to show the efficiency of our results.

scholarly journals Sharp Z-eigenvalue inclusion set-based method for testing the positive definiteness of multivariate homogeneous forms

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 3131-3139
Author(s):  
Gang Wang ◽  
Linxuan Sun ◽  
Yiju Wang
Keyword(s):  
Convergence Rate ◽  
Numerical Experiments ◽  
Sufficient Conditions ◽  
Positive Definiteness ◽  
Upper Bounds ◽  
Rank One ◽  
Inclusion Set ◽  
Even Order ◽  
Weakly Symmetric ◽  
The Given

In this paper, we establish a sharp Z-eigenvalue inclusion set for even-order real tensors by Z-identity tensor and prove that new Z-eigenvalue inclusion set is sharper than existing results. We propose some sufficient conditions for testing the positive definiteness of multivariate homogeneous forms via new Z-eigenvalue inclusion set. Further, we establish upper bounds on the Z-spectral radius of weakly symmetric nonnegative tensors and estimate the convergence rate of the greedy rank-one algorithms. The given numerical experiments show the validity of our results.

scholarly journals An optimal $ Z $-eigenvalue inclusion interval for a sixth-order tensor and its an application

2021 ◽  
Vol 7 (1) ◽  
pp. 967-985
Author(s):  
Tinglan Yao ◽  
Keyword(s):  
Homogeneous Polynomial ◽  
Symmetric Tensor ◽  
Positive Definiteness ◽  
Polynomial Systems ◽  
Sixth Order ◽  
Invariant Polynomial ◽  
Sufficient Condition ◽  
Order Tensor ◽  
Inclusion Interval ◽  
Time Invariant

<abstract><p>An optimal $ Z $-eigenvalue inclusion interval for a sixth-order tensor is presented. As an application, a sufficient condition for the positive definiteness of a sixth-order real symmetric tensor (also a homogeneous polynomial form) is obtained, which is used to judge the asymptotically stability of time-invariant polynomial systems.</p></abstract>

scholarly journals Practical Criteria for H-Tensors and Their Application

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 155
Author(s):  
Min Li ◽  
Haifeng Sang ◽  
Panpan Liu ◽  
Guorui Huang
Keyword(s):  
Sufficient Conditions ◽  
Symmetric Tensor ◽  
Positive Definiteness ◽  
Tensor Analysis ◽  
Symmetric Tensors ◽  
Numerical Examples ◽  
Even Order

Identifying the positive definiteness of even-order real symmetric tensors is an important component in tensor analysis. H-tensors have been utilized in identifying the positive definiteness of this kind of tensor. Some new practical criteria for identifying H-tensors are given in the literature. As an application, several sufficient conditions of the positive definiteness for an even-order real symmetric tensor were obtained. Numerical examples are given to illustrate the effectiveness of the proposed method.

scholarly journals M-Eigenvalues-Based Sufficient Conditions for the Positive Definiteness of Fourth-Order Partially Symmetric Tensors

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
Gang Wang ◽  
Linxuan Sun ◽  
Lixia Liu
Keyword(s):  
Spectral Radius ◽  
Fourth Order ◽  
Quantum Physics ◽  
Sufficient Conditions ◽  
Positive Definiteness ◽  
Symmetric Tensors ◽  
Nonlinear Elastic Material ◽  
Numerical Examples ◽  
Nonlinear Elastic ◽  
Partially Symmetric

M-eigenvalues of fourth-order partially symmetric tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we introduce M-identity tensor and establish two M-eigenvalue inclusion intervals with n parameters for fourth-order partially symmetric tensors, which are sharper than some existing results. Numerical examples are proposed to verify the efficiency of the obtained results. As applications, we provide some checkable sufficient conditions for the positive definiteness and establish bound estimations for the M-spectral radius of fourth-order partially symmetric nonnegative tensors.

scholarly journals Analytical expressions of copositivity for fourth-order symmetric tensors

2020 ◽  
pp. 1-22 ◽  
Author(s):  
Yisheng Song ◽  
Liqun Qi
Keyword(s):  
Lower Bound ◽  
Fourth Order ◽  
Scalar Potential ◽  
Sufficient Conditions ◽  
Scalar Fields ◽  
Positive Definiteness ◽  
Necessary Condition ◽  
Symmetric Tensors ◽  
Scalar Potentials ◽  
Sufficient And Necessary Condition

In particle physics, scalar potentials have to be bounded from below in order for the physics to make sense. The precise expressions of checking lower bound of scalar potentials are essential, which is an analytical expression of checking copositivity and positive definiteness of tensors given by such scalar potentials. Because the tensors given by general scalar potential are fourth-order and symmetric, our work mainly focuses on finding precise expressions to test copositivity and positive definiteness of fourth-order tensors in this paper. First of all, an analytically sufficient and necessary condition of positive definiteness is provided for fourth-order 2-dimensional symmetric tensors. For fourth-order 3-dimensional symmetric tensors, we give two analytically sufficient conditions of (strictly) copositivity by using proof technique of reducing orders or dimensions of such a tensor. Furthermore, an analytically sufficient and necessary condition of copositivity is showed for fourth-order 2-dimensional symmetric tensors. We also give several distinctly analytically sufficient conditions of (strict) copositivity for fourth-order 2-dimensional symmetric tensors. Finally, these results may be applied to check lower bound of scalar potentials, and to present analytical vacuum stability conditions for potentials of two real scalar fields and the Higgs boson.

scholarly journals POSITIVE SEMIDEFINITENESS OF DISCRETE QUADRATIC FUNCTIONALS

2003 ◽  
Vol 46 (3) ◽  
pp. 627-636 ◽  
Author(s):  
Martin Bohner ◽  
Ondřej Došlý ◽  
Werner Kratz
Keyword(s):  
Sufficient Conditions ◽  
Positive Definiteness ◽  
Quadratic Functional ◽  
Quadratic Functionals ◽  
Positive Semidefiniteness ◽  
Difference Systems ◽  
Special Cases ◽  
Even Order ◽  
Sturm Liouville ◽  
Necessary And Sufficient

AbstractWe consider symplectic difference systems, which contain as special cases linear Hamiltonian difference systems and Sturm–Liouville difference equations of any even order. An associated discrete quadratic functional is important in discrete variational analysis, and while its positive definiteness has been characterized and is well understood, a characterization of its positive semidefiniteness remained an open problem. In this paper we present the solution to this problem and offer necessary and sufficient conditions for such discrete quadratic functionals to be non-negative definite.AMS 2000 Mathematics subject classification: Primary 39A12; 39A13. Secondary 34B24; 49K99

scholarly journals New iterative codes for 𝓗-tensors and an application

2016 ◽  
Vol 14 (1) ◽  
pp. 212-220 ◽  
Author(s):  
Feng Wang ◽  
Deshu Sun
Keyword(s):  
Homogeneous Polynomial ◽  
Sufficient Conditions ◽  
Symmetric Tensor ◽  
Positive Definiteness ◽  
Polynomial Form ◽  
Numerical Examples ◽  
Even Order

AbstractNew iterative codes for identifying 𝓗 -tensor are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor, i.e., an even-degree homogeneous polynomial form are given. Advantages of results obtained are illustrated by numerical examples.

scholarly journals New criteria for H-tensors and an application

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Feng Wang ◽  
Deshu Sun
Keyword(s):  
Sufficient Conditions ◽  
Symmetric Tensor ◽  
Positive Definiteness ◽  
Numerical Examples ◽  
Even Order ◽  
New Criteria

AbstractSome new criteria for identifying H-tensors are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor are given. Advantages of results obtained are illustrated by numerical examples.

scholarly journals On the Asymptotic Stability of Linear Discrete Time Delay Systems

Volume 1 ◽  
2004 ◽  
Author(s):  
S. B. Stojanovic ◽  
D. Lj. Debeljkovic
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Numerical Example ◽  
Discrete Time Delay ◽  
Time Invariant ◽  
Working Out

This paper extends some of the basic results in the area of Lyapunov (asymptotic) to linear, discrete, time invariant time-delay systems. These results are given in the form of only sufficient conditions and represent other generalisation of some previous ones or completely new results. In the latter case it is easy to show that, in the most cases, these results are less conservative then those in existing literature. A numerical example has been working out to show the applicability of results derived. To the best knowledge of the authors, such results have not yet been reported.

scholarly journals Stability of Volterra system with impulsive effect

1991 ◽  
Vol 4 (1) ◽  
pp. 83-93 ◽  
Author(s):  
M. Ramamohana Rao ◽  
S. Sivasundaram
Keyword(s):  
Asymptotic Stability ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Uniform Stability ◽  
Impulsive Effect ◽  
Property A ◽  
Uniform Asymptotic Stability ◽  
Volterra System ◽  
Piecewise Continuous ◽  
The Given

Sufficient conditions for uniform stability and uniform asymptotic stability of impulsive integrodifferential equations are investigated by constructing a suitable piecewise continuous Lyapunov-like functionals without the decresent property. A result which establishes no pulse phenomena in the given system is also discussed.

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