Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Further Results on Cauchy Tensors and Hankel Tensors   In this article, we present various new results on Cauchy tensors and Hankel
tensors. { We first introduce the concept of generalized Cauchy tensors which
extends Cauchy tensors in the current literature, and provide several
conditions characterizing positive semi-definiteness of generalized Cauchy
tensors with nonzero entries.} As a consequence, we show that Cauchy tensors
are positive semi-definite if and only if they are SOS (Sum-of-squares)
tensors.} Furthermore, we prove that all positive semi-definite Cauchy tensors
are completely positive tensors, which means every positive semi-definite
Cauchy tensor can be decomposed { as} the sum of nonnegative rank-1 tensors. We
also establish that all the H-eigenvalues of nonnegative Cauchy tensors are
nonnegative. Secondly, we present new mathematical properties of Hankel
tensors. { We prove that an even order Hankel tensor is Vandermonde positive
semi-definite if and only if its associated plane tensor is positive
semi-definite. We also show that, if the Vandermonde rank of a Hankel tensor
$\mathcal{A}$ is less than the dimension of the underlying space, then positive
semi-definiteness of $\mathcal{A}$ is equivalent to the fact that $\mathcal{A}$
is a complete Hankel tensor, and so, is further equivalent to the SOS property
of $\mathcal{A}$. Lastly, we introduce a new structured tensor called
Cauchy-Hankel tensors, which is a special case of Cauchy tensors and Hankel
tensors simultaneously.} Sufficient and necessary conditions are established
for an even order Cauchy-Hankel tensor to be positive definite. Final remarks
are listed at the end of the paper.

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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc