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

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.

Download Full-text

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

Filomat ◽

10.2298/fil2009131w ◽

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.

Download Full-text

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

AIMS Mathematics ◽

10.3934/math.2022058 ◽

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>

Download Full-text

Practical Criteria for H-Tensors and Their Application

Symmetry ◽

10.3390/sym14010155 ◽

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.

Download Full-text

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

Complexity ◽

10.1155/2020/2474278 ◽

2020 ◽

Vol 2020 ◽

pp. 1-8 ◽

Cited By ~ 1

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.

Download Full-text

Analytical expressions of copositivity for fourth-order symmetric tensors

Analysis and Applications ◽

10.1142/s0219530520500049 ◽

2020 ◽

pp. 1-22 ◽

Cited By ~ 1

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.

Download Full-text

POSITIVE SEMIDEFINITENESS OF DISCRETE QUADRATIC FUNCTIONALS

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091502001086 ◽

2003 ◽

Vol 46 (3) ◽

pp. 627-636 ◽

Cited By ~ 15

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

Download Full-text

New iterative codes for 𝓗-tensors and an application

Open Mathematics ◽

10.1515/math-2016-0022 ◽

2016 ◽

Vol 14 (1) ◽

pp. 212-220 ◽

Cited By ~ 1

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.

Download Full-text

New criteria for H-tensors and an application

Open Mathematics ◽

10.1515/math-2015-0058 ◽

2015 ◽

Vol 13 (1) ◽

Cited By ~ 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.

Download Full-text

On the Asymptotic Stability of Linear Discrete Time Delay Systems

Volume 1 ◽

10.1115/esda2004-58291 ◽

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.

Download Full-text

Stability of Volterra system with impulsive effect

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953391000059 ◽

1991 ◽

Vol 4 (1) ◽

pp. 83-93 ◽

Cited By ~ 3

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.

Download Full-text