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 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 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 An Upper Bound for the Symmetric Tensor Rank of a Low Degree Polynomial in a Large Number of Variables

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-2 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Upper Bound ◽  
Homogeneous Polynomial ◽  
Symmetric Tensor ◽  
Tensor Rank ◽  
Degree Polynomial ◽  
Linear Forms ◽  
Symmetric Tensor Rank ◽  
Low Degree ◽  
Powers Of Linear Forms

Fix integers m≥5 and d≥3. Let f be a degree d homogeneous polynomial in m+1 variables. Here, we prove that f is the sum of at most d·⌈(m+dm)/(m+1)⌉d-powers of linear forms (of course, this inequality is nontrivial only if m≫d.)

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