Strong ${\mathcal {H}}$-tensors play a significant role in identifying the positive definiteness of homogeneous polynomials. Motivated by the definition of $\mathcal {N}$-scal matrices, this paper first introduces two new subclasses of strong ${\mathcal {H}}$-tensors, namely ${{\mathcal {N}}}$-scal tensors and strong ${{\mathcal {N}}}$-scal tensors. Second, it analyzes the relationships among ${{\mathcal {N}}}$-scal tensors, strong ${{\mathcal {N}}}$-scal tensors, and Nekrasov tensors. Finally, as applications, two new methods are proposed to determine the positive definiteness of even-order real symmetric tensors based on ${{\mathcal {N}}}$-scal tensors and strong ${{\mathcal {N}}}$-scal tensors, with numerical examples provided to illustrate the results.
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Funding
National Natural Science Foundations of China(31600299)
RIGHTS & PERMISSIONS
Shanghai University
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