scholarly journals On the Eventual Exponential Positivity of Some Tree Sign Patterns

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1669
Author(s):  
Ber-Lin Yu ◽  
Zhongshan Li ◽  
Sanzhang Xu
Keyword(s):  
Open Problem ◽  
Double Star ◽  
Real Matrix ◽  
Sign Pattern ◽  
Preliminary Results ◽  
Sign Patterns

An n×n matrix A is called eventually exponentially positive (EEP) if etA=∑k=0∞tkAkk!>0 for all t≥t0, where t0≥0. A matrix whose entries belong to the set {+,−,0} is called a sign pattern. An n×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≤3 are identified. For the n×n tridiagonal sign patterns Tn, we show that there exists exactly one MPEEP tridiagonal sign pattern Tno. Consequently, we classify all PEEP tridiagonal sign patterns as the superpatterns of Tno. We also classify all PEEP star sign patterns Sn and double star sign patterns DS(n,m) by identifying all the MPEEP star sign patterns and the MPEEP double star sign patterns, respectively.

Potentially eventually positive star sign patterns

2016 ◽  
Vol 31 ◽  
pp. 541-548
Author(s):  
Yu Ber-Lin ◽  
Huang Ting-Zhu ◽  
Jie Cui ◽  
Deng Chunhua
Keyword(s):  
Positive Integer ◽  
Necessary Conditions ◽  
Positive Sign ◽  
Real Matrix ◽  
Sign Pattern ◽  
Positive Real ◽  
Sign Patterns

An $n$-by-$n$ real matrix $A$ is eventually positive if there exists a positive integer $k_{0}$ such that $A^{k}>0$ for all $k\geq k_{0}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is potentially eventually positive (PEP) if there exists an eventually positive real matrix $A$ with the same sign pattern as $\mathcal{A}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is a minimal potentially eventually positive sign pattern (MPEP sign pattern) if $\mathcal{A}$ is PEP and no proper subpattern of $\mathcal{A}$ is PEP. Berman, Catral, Dealba, et al. [Sign patterns that allow eventual positivity, {\it ELA}, 19(2010): 108-120] established some sufficient and some necessary conditions for an $n$-by-$n$ sign pattern to allow eventual positivity and classified the potentially eventually positive sign patterns of order $n\leq 3$. However, the identification and classification of PEP signpatterns of order $n\geq 4$ remain open. In this paper, all the $n$-by-$n$ PEP star sign patterns are classified by identifying all the MPEP star sign patterns.

scholarly journals Potentially Eventually Positive 2-generalized Star Sign Patterns

2019 ◽  
Vol 35 ◽  
pp. 100-115
Author(s):  
Yu Ber-Lin ◽  
Ting-Zhu Huang ◽  
Xu Sanzhang
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Real Matrix ◽  
Sign Pattern ◽  
Open Problems ◽  
Sign Patterns ◽  
Necessary And Sufficient

A sign pattern is a matrix whose entries belong to the set $\{+, -, 0\}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is said to be potentially eventually positive if there exists at least one real matrix $A$ with the same sign pattern as $\mathcal{A}$ and a positive integer $k_{0}$ such that $A^{k}>0$ for all $k\geq k_{0}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is said to be potentially eventually exponentially positive if there exists at least one real matrix $A$ with the same sign pattern as $\mathcal{A}$ and a nonnegative integer $t_{0}$ such that $e^{tA}=\sum_{k=0}^{\infty}\frac{t^{k}A^{k}}{k!}>0$ for all $t\geq t_{0}$. Identifying necessary and sufficient conditions for an $n$-by-$n$ sign pattern to be potentially eventually positive (respectively, potentially eventually exponentially positive), and classifying these sign patterns are open problems. In this article, the potential eventual positivity of the $2$-generalized star sign patterns is investigated. All the minimal potentially eventually positive $2$-generalized star sign patterns are identified. Consequently, all the potentially eventually positive $2$-generalized star sign patterns are classified. As an application, all the minimal potentially eventually exponentially positive $2$-generalized star sign patterns are identified. Consequently, all the potentially eventually exponentially positive $2$-generalized star sign patterns are classified.

Potentially nilpotent tridiagonal sign patterns of order 4

2016 ◽  
Vol 31 ◽  
pp. 706-721
Author(s):  
Yubin Gao ◽  
Yanling Shao
Keyword(s):  
Linear Algebra ◽  
Positive Answer ◽  
Electronic Journal ◽  
Real Matrix ◽  
Sign Pattern ◽  
Open Questions ◽  
Sign Patterns ◽  
Arbitrary Tree

An $n\times n$ sign pattern ${\cal A}$ is said to be potentially nilpotent (PN) if there exists some nilpotent real matrix $B$ with sign pattern ${\cal A}$. In [M.~Arav, F.~Hall, Z.~Li, K.~Kaphle, and N.~Manzagol.Spectrally arbitrary tree sign patterns of order $4$, {\em Electronic Journal of Linear Algebra}, 20:180--197, 2010.], the authors gave some open questions, and one of them is the following: {\em For the class of $4 \times 4$ tridiagonal sign patterns, is PN (together with positive and negative diagonal entries) equivalent to being SAP?}\ In this paper, a positive answer for this question is given.

scholarly journals Sign patterns that require eventual exponential nonnegativity

2015 ◽  
Vol 30 ◽  
Author(s):  
Craig Erickson
Keyword(s):  
Sign Pattern ◽  
Sign Patterns ◽  
Necessary And Sufficient

Sign patterns that require exponential nonnegativity are characterized. A set of conditions necessary for a sign pattern to require eventual exponential nonnegativity are established. It is shown that these conditions are also sufficient for an upper triangular sign pattern to require eventual exponential nonnegativity and it is conjectured that these conditions are both necessary and sufficient for any sign pattern to require eventual exponential nonnegativity. It is also shown that the maximum number of negative entries in a sign pattern that requires eventual exponential nonnegativity is (n−1)(n−2)/2 + 2

scholarly journals Photometric Observations of Three Double Stars by the Occultation Method — Comparison with Interferometric Results and with Preliminary Results of HIPPARCOS

1992 ◽  
Vol 135 ◽  
pp. 546-548
Author(s):  
M. Froeschlé ◽  
C. Meyer
Keyword(s):  
Diffraction Pattern ◽  
Method Comparison ◽  
Short Description ◽  
Double Star ◽  
Preliminary Results ◽  
Double Stars ◽  
Photometric Observations

AbstractWe first briefly recall the geometry of the occultation of a double star by the Moon’s edge. Then we give a short description of the principle of the formation of the diffraction pattern. We present the results for three double stars and compare them with those obtained by other methods.

scholarly journals On the infinite Borwein product raised to a positive real power

2021 ◽  
Author(s):  
Michael J. Schlosser ◽  
Nian Hong Zhou
Keyword(s):  
Asymptotic Formula ◽  
Circle Method ◽  
Sign Pattern ◽  
Real Numbers ◽  
Positive Real ◽  
Infinite Products ◽  
Real Power ◽  
Sign Patterns ◽  
Divisibility Properties ◽  
Made In

AbstractIn this paper, we study properties of the coefficients appearing in the q-series expansion of $$\prod _{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta $$ ∏ n ≥ 1 [ ( 1 - q n ) / ( 1 - q pn ) ] δ , the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power $$\delta $$ δ . We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients. For $$p=3$$ p = 3 we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $$\delta $$ δ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $$p=3$$ p = 3 case.

Inertia sets allowed by matrix patterns

2018 ◽  
Vol 34 ◽  
pp. 343-355 ◽  
Author(s):  
Adam Berliner ◽  
Dale Olesky ◽  
Pauline Van den Driessche
Keyword(s):  
Dynamical Systems ◽  
Sufficient Condition ◽  
Sign Pattern ◽  
Zero Eigenvalue ◽  
Sign Patterns

Motivated by the possible onset of instability in dynamical systems associated with a zero eigenvalue, sets of inertias $\sn_n$ and $\SN{n}$ for sign and zero-nonzero patterns, respectively, are introduced. For an $n\times n$ sign pattern $\mc{A}$ that allows inertia $(0,n-1,1)$, a sufficient condition is given for $\mc{A}$ and every superpattern of $\mc{A}$ to allow $\sn_n$, and a family of such irreducible sign patterns for all $n\geq 3$ is specified. All zero-nonzero patterns (up to equivalence) that allow $\SN{3}$ and $\SN{4}$ are determined, and are described by their associated digraphs.

Robust separations in inductive inference

2011 ◽  
Vol 76 (2) ◽  
pp. 368-376 ◽  
Author(s):  
Mark Fulk
Keyword(s):  
Language Learning ◽  
Open Problem ◽  
Formal Language ◽  
Inductive Inference ◽  
Function Learning ◽  
Preliminary Results ◽  
Original Idea ◽  
Important Open Problem ◽  
Inference Theory ◽  
Formal Language Learning

AbstractResults in recursion-theoretic inductive inference have been criticized as depending on unrealistic self-referential examples. J. M. Bārzdiņš proposed a way of ruling out such examples, and conjectured that one of the earliest results of inductive inference theory would fall if his method were used. In this paper we refute Bārzdiņš' conjecture.We propose a new line of research examining robust separations; these are defined using a strengthening of Bārzdiņš' original idea. The preliminary results of the new line of research are presented, and the most important open problem is stated as a conjecture. Finally, we discuss the extension of this work from function learning to formal language learning.

scholarly journals Epis are onto for finite regular semigroups

1983 ◽  
Vol 26 (2) ◽  
pp. 151-162 ◽  
Author(s):  
T. E. Hall ◽  
P. R. Jones
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Open Problem ◽  
Inverse Semigroups ◽  
Regular Semigroups ◽  
Preliminary Results ◽  
Absolutely Closed ◽  
Finite Regular Semigroup

After preliminary results and definitions in Section 1, we show in Section 2 that any finite regular semigroup is saturated, in the sense of Howie and Isbell [8] (that is, the dominion of a finite regular semigroup U in a strictly containing semigroup S is never S). This is equivalent of course to showing that in the category of semigroups any epi from a finite regular semigroup is in fact onto. Note for inverse semigroups the stronger result, that any inverse semigroup is absolutely closed [11, Theorem VII. 2.14] or [8, Theorem 2.3]. Further, any inverse semigroup is in fact an amalgamation base in the class of semigroups [10], in the sense of [5]. These stronger results are known to be false for finite regular semigroups [8, Theorem 2.9] and [5, Theorem 25]. Whether or not every regular semigroup is saturated is an open problem.

scholarly journals Spectral Properties of Sign Patterns

2020 ◽  
Vol 36 (36) ◽  
pp. 183-197
Author(s):  
Michael Cavers ◽  
Jonathan Fischer ◽  
Kevin N. Vander Meulen
Keyword(s):  
Spectral Properties ◽  
Infinite Family ◽  
Sign Pattern ◽  
Sign Patterns ◽  
Spectrally Arbitrary

In this paper, an infinite family of irreducible sign patterns that are spectrally arbitrary, for which the nilpotent-Jacobian method does not apply, is given. It is demonstrated that it is possible for an irreducible sign pattern to be refined inertially arbitrary and not spectrally arbitrary. It is observed that not every nonzero spectrally arbitrary pattern has a signing which is spectrally arbitrary. It is also shown that every superpattern of the reducible pattern $\T_2 \oplus \T_2$ is spectrally arbitrary.

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