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

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Christopher Logan Hambric ◽  
Chi-Kwong Li ◽  
Diane Christine Pelejo ◽  
Junping Shi
Keyword(s):  
Positive Integer ◽  
Turing Instability ◽  
Sign Pattern ◽  
Minimum Number ◽  
Stable Matrix

<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>

On the metric dimension and diameter of circulant graphs with three jumps

2018 ◽  
Vol 10 (01) ◽  
pp. 1850008
Author(s):  
Muhammad Imran ◽  
A. Q. Baig ◽  
Saima Rashid ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Metric Spaces ◽  
Metric Dimension ◽  
Resolving Set ◽  
Circulant Graphs ◽  
Infinite Class ◽  
Metric Properties ◽  
Minimum Number ◽  
Metric Basis

Let [Formula: see text] be a connected graph and [Formula: see text] be the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The diameter of [Formula: see text] is defined as [Formula: see text] and is denoted by [Formula: see text]. A subset of vertices [Formula: see text] is called a resolving set for [Formula: see text] if for every two distinct vertices [Formula: see text], there is a vertex [Formula: see text], [Formula: see text], such that [Formula: see text]. A resolving set containing the minimum number of vertices is called a metric basis for [Formula: see text] and the number of vertices in a metric basis is its metric dimension, denoted by [Formula: see text]. Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let [Formula: see text] be a family of connected graphs [Formula: see text] depending on [Formula: see text] as follows: the order [Formula: see text] and [Formula: see text]. If there exists a constant [Formula: see text] such that [Formula: see text] for every [Formula: see text] then we shall say that [Formula: see text] has bounded metric dimension, otherwise [Formula: see text] has unbounded metric dimension. If all graphs in [Formula: see text] have the same metric dimension, then [Formula: see text] is called a family of graphs with constant metric dimension. In this paper, we study the metric properties of an infinite class of circulant graphs with three generators denoted by [Formula: see text] for any positive integer [Formula: see text] and when [Formula: see text]. We compute the diameter and determine the exact value of the metric dimension of these circulant graphs.

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 Region Coloring in Minahasa Regency Using Sequential Color Algorithm

d'CARTESIAN ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 86
Author(s):  
Yevie Ingamita ◽  
Nelson Nainggolan ◽  
Benny Pinontoan
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Chromatic Number ◽  
Human Life ◽  
Minimum Number ◽  
Mathematical Sciences ◽  
Map Coloring ◽  
Graph Application

Graph Theory is one of the mathematical sciences whose application is very wide in human life. One of theory graph application is Map Coloring. This research discusses how to color the map of Minahasa Regency by using the minimum color that possible. The algorithm used to determine the minimum color in coloring the region of Minahasa Regency that is Sequential Color Algorithm. The Sequential Color Algorithm is an algorithm used in coloring a graph with k-color, where k is a positive integer. Based on the results of this research was found that the Sequential Color Algorithm can be used to color the map of Minahasa Regency with the minimum number of colors or chromatic number χ(G) obtained in the coloring of 25 sub-districts on the map of Minahasa Regency are 3 colors (χ(G) = 3).

Distance Vertex Identification in Graphs

2021 ◽  
pp. 2150005
Author(s):  
Gary Chartrand ◽  
Yuya Kono ◽  
Ping Zhang
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Identification Number ◽  
Minimum Number ◽  
Positive Integers ◽  
Distance Formula

A red-white coloring of a nontrivial connected graph [Formula: see text] is an assignment of red and white colors to the vertices of [Formula: see text] where at least one vertex is colored red. Associated with each vertex [Formula: see text] of [Formula: see text] is a [Formula: see text]-vector, called the code of [Formula: see text], where [Formula: see text] is the diameter of [Formula: see text] and the [Formula: see text]th coordinate of the code is the number of red vertices at distance [Formula: see text] from [Formula: see text]. A red-white coloring of [Formula: see text] for which distinct vertices have distinct codes is called an identification coloring or ID-coloring of [Formula: see text]. A graph [Formula: see text] possessing an ID-coloring is an ID-graph. The problem of determining those graphs that are ID-graphs is investigated. The minimum number of red vertices among all ID-colorings of an ID-graph [Formula: see text] is the identification number or ID-number of [Formula: see text] and is denoted by [Formula: see text]. It is shown that (1) a nontrivial connected graph [Formula: see text] has ID-number 1 if and only if [Formula: see text] is a path, (2) the path of order 3 is the only connected graph of diameter 2 that is an ID-graph, and (3) every positive integer [Formula: see text] different from 2 can be realized as the ID-number of some connected graph. The identification spectrum of an ID-graph [Formula: see text] is the set of all positive integers [Formula: see text] such that [Formula: see text] has an ID-coloring with exactly [Formula: see text] red vertices. Identification spectra are determined for paths and cycles.

scholarly journals On the Minimum Number of Monochromatic Generalized Schur Triples

10.37236/6490 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Thotsaporn Thanatipanonda ◽  
Elaine Wong
Keyword(s):  
Positive Integer ◽  
General Problem ◽  
Original Proof ◽  
Minimum Number ◽  
Fixed Positive Integer

The solution to the problem of finding the minimum number of monochromatic triples $(x,y,x+ay)$ with $a\geq 2$ being a fixed positive integer over any 2-coloring of $[1,n]$ was conjectured by Butler, Costello, and Graham (2010) and Thanathipanonda (2009). We solve this problem using a method based on Datskovsky's proof (2003) on the minimum number of monochromatic Schur triples $(x,y,x+y)$. We do this by exploiting the combinatorial nature of the original proof and adapting it to the general problem.

scholarly journals MINIMUM NUMBER OF FOX COLORS FOR SMALL PRIMES

2012 ◽  
Vol 21 (03) ◽  
pp. 1250025 ◽  
Author(s):  
PEDRO LOPES ◽  
JOÃO MATIAS
Keyword(s):  
Positive Integer ◽  
Prime Divisor ◽  
Exact Results ◽  
Minimum Number

This article concerns exact results on the minimum number of colors of a Fox coloring over the integers modulo r, of a link with non-null determinant.Specifically, we prove that whenever the least prime divisor of the determinant of such a link and the modulus r is 2, 3, 5, or 7, then the minimum number of colors is 2, 3, 4, or 4 (respectively) and conversely.We are thus led to conjecture that for each prime p there exists a unique positive integer, mp, with the following property. For any link L of non-null determinant and any modulus r such that p is the least prime divisor of the determinant of L and the modulus r, the minimum number of colors of L modulo r is mp.

The 3-Irreducible Partially Ordered Sets

1977 ◽  
Vol 29 (2) ◽  
pp. 367-383 ◽  
Author(s):  
David Kelly
Keyword(s):  
Positive Integer ◽  
Finite Dimension ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Partial Ordering ◽  
Ordered Sets ◽  
Linear Orders ◽  
Compactness Property ◽  
Partially Ordered ◽  
Minimum Number

The dimension [4] of a partially ordered set (poset) is the minimum number of linear orders whose intersection is the partial ordering of the poset. For a positive integer m, a poset is m-irreducible[10] if it has dimension m and removal of any element lowers its dimension. By the compactness property of finite dimension, every m-irreducible poset is finite and every poset of dimension ≧ m contains an m-irreducible subposet.

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.

scholarly journals Minimum Number of Colours to Avoid k-Term Monochromatic Arithmetic Progressions

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 247
Author(s):  
Kai An Sim ◽  
Kok Bin Wong
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Upper Bound ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Arithmetic Progressions ◽  
Minimum Number ◽  
Necessary And Sufficient

By recalling van der Waerden theorem, there exists a least a positive integer w=w(k;r) such that for any n≥w, every r-colouring of [1,n] admits a monochromatic k-term arithmetic progression. Let k≥2 and rk(n) denote the minimum number of colour required so that there exists a rk(n)-colouring of [1,n] that avoids any monochromatic k-term arithmetic progression. In this paper, we give necessary and sufficient conditions for rk(n+1)=rk(n). We also show that rk(n)=2 for all k≤n≤2(k−1)2 and give an upper bound for rp(pm) for any prime p≥3 and integer m≥2.

scholarly journals Minimum number of non-zero-entries in a 7 × 7 stable matrix

2019 ◽  
Vol 572 ◽  
pp. 135-152
Author(s):  
Christopher Logan Hambric ◽  
Chi-Kwong Li ◽  
Diane Christine Pelejo ◽  
Junping Shi
Keyword(s):  
Minimum Number ◽  
Stable Matrix
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