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17 pages, 334 KB

Open AccessArticle

Eventual Positivity of a Class of Double Star-like Sign Patterns

by Ber-Lin Yu, Zhongshan Li, Gu-Fang Mou and Sanzhang Xu

Symmetry 2022, 14(3), 512; https://doi.org/10.3390/sym14030512 - 2 Mar 2022

Viewed by 2194

Identifying and classifying the potentially eventually positive sign patterns and the potentially eventually exponentially positive sign patterns of orders greater than 3 have been raised as two open problems since 2010. In this article, we investigate the potential eventual positivity of the class [...] Read more.

S_{(n, m, 1)}

whose underlying graph

G (S_{(n, m, 1)})

is obtained from the underlying graph

G (S_{(n, m)})

of the

(n + m)

-by-

(n + m)

double star sign patterns

S_{(n, m)}

by adding an additional vertex adjacent to the two center vertices and removing the edge between the center vertices. We firstly establish some necessary conditions for a double star-like sign pattern to be potentially eventually positive, and then identify all the minimal potentially eventually positive double star-like sign patterns. Secondly, we classify all the potentially eventually positive sign patterns in the class of double star-like sign patterns

S_{(n, m, 1)}

. Finally, as an application of our results about the potentially eventually positive double star-like sign patterns, we identify all the minimal potentially eventually exponentially positive sign patterns and characterize all the potentially eventually exponentially positive sign patterns in the class of double star-like sign patterns

S_{(n, m, 1)}

. Full article

(This article belongs to the Section Mathematics)

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9 pages, 249 KB

Open AccessArticle

On the Eventual Exponential Positivity of Some Tree Sign Patterns

by Ber-Lin Yu, Zhongshan Li and Sanzhang Xu

Symmetry 2021, 13(9), 1669; https://doi.org/10.3390/sym13091669 - 10 Sep 2021

Viewed by 1757

Abstract

n \times n

matrix A is called eventually exponentially positive (EEP) if

e^{t A} = \sum_{k = 0}^{\infty} \frac{t^{k} A^{k}}{k!} > 0

for all

t \geq t_{0}

, where

t_{0} \geq 0

. [...] Read more.

n \times n

matrix A is called eventually exponentially positive (EEP) if

e^{t A} = \sum_{k = 0}^{\infty} \frac{t^{k} A^{k}}{k!} > 0

for all

t \geq t_{0}

, where

t_{0} \geq 0

. A matrix whose entries belong to the set

{+, -, 0}

is called a sign pattern. An

n \times n

sign pattern

A

is called potentially eventually exponentially positive (PEEP) if there exists some real matrix realization A of

A

that is EEP. Characterizing the PEEP sign patterns is a longstanding open problem. In this article,

A

is called minimally potentially eventually exponentially positive (MPEEP), if

A

is PEEP and no proper subpattern of

A

is PEEP. Some preliminary results about MPEEP sign patterns and PEEP sign patterns are established. All MPEEP sign patterns of orders

n \leq 3

are identified. For the

n \times n

tridiagonal sign patterns

T_{n}

, we show that there exists exactly one MPEEP tridiagonal sign pattern

T_{n}^{o}

. Consequently, we classify all PEEP tridiagonal sign patterns as the superpatterns of

T_{n}^{o}

. We also classify all PEEP star sign patterns

S_{n}

and double star sign patterns

DS (n, m)

by identifying all the MPEEP star sign patterns and the MPEEP double star sign patterns, respectively. Full article

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