Polynomial stability and potentially stable patterns

Cavers, Michael S.

doi:10.1016/j.laa.2020.12.015

Linear Algebra and its Applications

2021

DOI: 10.1016/j.laa.2020.12.015

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Polynomial stability and potentially stable patterns

Michael S. Cavers

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Cited by 3 publications

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“…Note that, trivially, m n ≤ S n for any n since A is assumed to be stable. The following has been proved in [5] (for n ≤ 6 and n ≥ 9), [6] (for n = 7) and [3]…”

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confidence: 83%

Minimum number of non-zero-entries in a stable matrix exhibiting Turing instability

Hambric

Pelejo

et al. 2022

DCDS-S

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<p style='text-indent:20px;'>It is shown that for any positive integer <inline-formula><tex-math id="M1">\begin{document}$ n \ge 3 $\end{document}</tex-math></inline-formula>, there is a stable irreducible <inline-formula><tex-math id="M2">\begin{document}$ n\times n $\end{document}</tex-math></inline-formula> matrix <inline-formula><tex-math id="M3">\begin{document}$ A $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ 2n+1-\lfloor\frac{n}{3}\rfloor $\end{document}</tex-math></inline-formula> nonzero entries exhibiting Turing instability. Moreover, when <inline-formula><tex-math id="M5">\begin{document}$ n = 3 $\end{document}</tex-math></inline-formula>, the result is best possible, i.e., every <inline-formula><tex-math id="M6">\begin{document}$ 3\times 3 $\end{document}</tex-math></inline-formula> stable matrix with five or fewer nonzero entries will not exhibit Turing instability. Furthermore, we determine all possible <inline-formula><tex-math id="M7">\begin{document}$ 3\times 3 $\end{document}</tex-math></inline-formula> irreducible sign pattern matrices with 6 nonzero entries which can be realized by a matrix <inline-formula><tex-math id="M8">\begin{document}$ A $\end{document}</tex-math></inline-formula> that exhibits Turing instability.</p>

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“…Note that, trivially, m n ≤ S n for any n since A is assumed to be stable. The following has been proved in [5] (for n ≤ 6 and n ≥ 9), [6] (for n = 7) and [3]…”

mentioning

confidence: 83%

Minimum number of non-zero-entries in a stable matrix exhibiting Turing instability

Hambric

Pelejo

et al. 2022

DCDS-S

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“…Few methods to construct higher-order potentially stable sign pattern matrices from lower-order potentially stable sign pattern matrices are given in [2,8]. Cavers [5] gave few methods to identify some sign pattern matrices which are not potentially stable.…”

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confidence: 99%

Potentially stable and 5-by-5 spectrally arbitrary tree sign pattern matrices with all edges negative

Das

2021

ELA

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Characterization of potentially stable sign pattern matrices has been a long-standing open problem. In this paper, we give some sufficient conditions for tree sign pattern matrices with all edges negative to allow a properly signed nest. We also characterize potentially stable star and path sign pattern matrices with all edges negative. We give a conjecture on characterizing potentially stable tree sign pattern matrices with all edges negative in terms of allowing a properly signed nest which is verified to be true for sign pattern matrices up to order 6. Finally, we characterize all 5-by-5 spectrally arbitrary tree sign pattern matrices with all edges negative.

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On allowable properties and spectrally arbitrary sign pattern matrices

Jadhav

Deore

2022

Research in Mathematics

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Polynomial stability and potentially stable patterns

Cited by 3 publications

References 18 publications

Minimum number of non-zero-entries in a stable matrix exhibiting Turing instability

Minimum number of non-zero-entries in a stable matrix exhibiting Turing instability

Potentially stable and 5-by-5 spectrally arbitrary tree sign pattern matrices with all edges negative

On allowable properties and spectrally arbitrary sign pattern matrices

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