<p>Strong <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathcal {H}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>-tensors play a significant role in identifying the positive definiteness of homogeneous polynomials. Motivated by the definition of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation>-scal matrices, this paper first introduces two new subclasses of strong <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\mathcal {H}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>-tensors, namely <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${{\mathcal {N}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation>-scal tensors and strong <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${{\mathcal {N}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation>-scal tensors. Second, it analyzes the relationships among <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${{\mathcal {N}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation>-scal tensors, strong <InlineEquation ID="IEq9"> <EquationSource Format="TEX">${{\mathcal {N}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation>-scal tensors, and <i>Nekrasov</i> tensors. Finally, as applications, two new methods are proposed to determine the positive definiteness of even-order real symmetric tensors based on <InlineEquation ID="IEq10"> <EquationSource Format="TEX">${{\mathcal {N}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation>-scal tensors and strong <InlineEquation ID="IEq11"> <EquationSource Format="TEX">${{\mathcal {N}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation>-scal tensors, with numerical examples provided to illustrate the results.</p>

Two New Subclasses of ${\mathcal {H}}$-Tensors and Their Applications

期刊论文
发表日期：

2026年6月19日

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Keru Wen,
Min Hui,
Yaqiang Wang

摘要

Strong ${\mathcal {H}}$ $H$ -tensors play a significant role in identifying the positive definiteness of homogeneous polynomials. Motivated by the definition of $\mathcal {N}$ $N$ -scal matrices, this paper first introduces two new subclasses of strong ${\mathcal {H}}$ $H$ -tensors, namely ${{\mathcal {N}}}$ $N$ -scal tensors and strong ${{\mathcal {N}}}$ $N$ -scal tensors. Second, it analyzes the relationships among ${{\mathcal {N}}}$ $N$ -scal tensors, strong ${{\mathcal {N}}}$ $N$ -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 ${{\mathcal {N}}}$ $N$ -scal tensors and strong ${{\mathcal {N}}}$ $N$ -scal tensors, with numerical examples provided to illustrate the results.

Two New Subclasses of \({\mathcal {H}}\)-Tensors and Their Applications

摘要