Rational realization of the minimum ranks of nonnegative sign pattern matrices

Wei Fang; Wei Gao; Yubin Gao; Fei Gong; Guangming Jing; Zhongshan Li; Yan Ling Shao; Lihua Zhang

Displaying similar documents to “Rational realization of the minimum ranks of nonnegative sign pattern matrices”

Factorization of CP-rank- $3$ completely positive matrices

Jan Brandts, Michal Křížek (2016)

Czechoslovak Mathematical Journal

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A symmetric positive semi-definite matrix $A$ is called completely positive if there exists a matrix $B$ with nonnegative entries such that $A = B B^{⊤}$ . If $B$ is such a matrix with a minimal number $p$ of columns, then $p$ is called the cp-rank of $A$ . In this paper we develop a finite and exact algorithm to factorize any matrix $A$ of cp-rank $3$ . Failure of this algorithm implies that $A$ does not have cp-rank $3$ . Our motivation stems from the question if there exist three nonnegative polynomials of degree at...

Possible isolation number of a matrix over nonnegative integers

LeRoy B. Beasley, Young Bae Jun, Seok-Zun Song (2018)

Czechoslovak Mathematical Journal

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Let $ℤ_{+}$ be the semiring of all nonnegative integers and $A$ an $m \times n$ matrix over $ℤ_{+}$ . The rank of $A$ is the smallest $k$ such that $A$ can be factored as an $m \times k$ matrix times a $k \times n$ matrix. The isolation number of $A$ is the maximum number of nonzero entries in $A$ such that no two are in any row or any column, and no two are in a $2 \times 2$ submatrix of all nonzero entries. We have that the isolation number of $A$ is a lower bound of the rank of $A$ . For $A$ with isolation number $k$ , we investigate the possible values of the...

Bounds for the characteristic rank and cup-length of oriented Grassmann manifolds

Tomáš Rusin (2018)

Archivum Mathematicum

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We estimate the characteristic rank of the canonical $k$ –plane bundle over the oriented Grassmann manifold ${\tilde{G}}_{n, k}$ . We then use it to compute uniform upper bounds for the $ℤ_{2}$ –cup-length of ${\tilde{G}}_{n, k}$ for $n$ belonging to certain intervals.

Cardinalities of DCCC normal spaces with a rank 2-diagonal

Wei-Feng Xuan, Wei-Xue Shi (2016)

Mathematica Bohemica

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A topological space $X$ has a rank 2-diagonal if there exists a diagonal sequence on $X$ of rank $2$ , that is, there is a countable family ${𝒰_{n} : n \in ω}$ of open covers of $X$ such that for each $x \in X$ , ${x} = ⋂ {{St}^{2} (x, 𝒰_{n}) : n \in ω}$ . We say that a space $X$ satisfies the Discrete Countable Chain Condition (DCCC for short) if every discrete family of nonempty open subsets of $X$ is countable. We mainly prove that if $X$ is a DCCC normal space with a rank 2-diagonal, then the cardinality of $X$ is at most $𝔠$ . Moreover, we prove that if $X$ is a first...

On the real $X$ -ranks of points of $ℙ^{n} (ℝ)$ with respect to a real variety $X \subset ℙ^{n}$

Edoardo Ballico (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $X \subset ℙ^{n}$ be an integral and non-degenerate $m$ -dimensional variety defined over $ℝ$ . For any $P \in ℙ^{n} (ℝ)$ the real $X$ -rank $r_{X, ℝ} (P)$ is the minimal cardinality of $S \subset X (ℝ)$ such that $P \in 〈 S 〉$ . Here we extend to the real case an upper bound for the $X$ -rank due to Landsberg and Teitler.

On soluble groups of module automorphisms of finite rank

Bertram A. F. Wehrfritz (2017)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring, $M$ an $R$ -module and $G$ a group of $R$ -automorphisms of $M$ , usually with some sort of rank restriction on $G$ . We study the transfer of hypotheses between $M / C_{M} (G)$ and $[M, G]$ such as Noetherian or having finite composition length. In this we extend recent work of Dixon, Kurdachenko and Otal and of Kurdachenko, Subbotin and Chupordia. For example, suppose $[M, G]$ is $R$ -Noetherian. If $G$ has finite rank, then $M / C_{M} (G)$ also is $R$ -Noetherian. Further, if $[M, G]$ is $R$ -Noetherian and if only certain abelian...

Simultaneous stabilization in $A_{ℝ} ()$

Raymond Mortini, Brett D. Wick (2009)

Studia Mathematica

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We study the problem of simultaneous stabilization for the algebra $A_{ℝ} ()$ . Invertible pairs $(f_{j}, g_{j})$ , j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that $α f_{j} + β g_{j}$ is invertible in this algebra for j = 1,..., n. For n = 2, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_{ℝ} ()$ has stable rank two, we are faced here with a different...

G-matrices, $J$ -orthogonal matrices, and their sign patterns

Frank J. Hall, Miroslav Rozložník (2016)

Czechoslovak Mathematical Journal

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A real matrix $A$ is a G-matrix if $A$ is nonsingular and there exist nonsingular diagonal matrices $D_{1}$ and $D_{2}$ such that $A^{- T} = D_{1} A D_{2}$ , where $A^{- T}$ denotes the transpose of the inverse of $A$ . Denote by $J = diag (\pm 1)$ a diagonal (signature) matrix, each of whose diagonal entries is $+ 1$ or $- 1$ . A nonsingular real matrix $Q$ is called $J$ -orthogonal if $Q^{T} J Q = J$ . Many connections are established between these matrices. In particular, a matrix $A$ is a G-matrix if and only if $A$ is diagonally (with positive diagonals) equivalent to a column permutation...

Linear preservers of rc-majorization on matrices

Mohammad Soleymani (2024)

Czechoslovak Mathematical Journal

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Let $A$ , $B$ be $n \times m$ matrices. The concept of matrix majorization means the $j$ th column of $A$ is majorized by the $j$ th column of $B$ and this is done for all $j$ by a doubly stochastic matrix $D$ . We define rc-majorization that extended matrix majorization to columns and rows of matrices. Also, the linear preservers of rc-majorization will be characterized.

Class groups of large ranks in biquadratic fields

Mahesh Kumar Ram (2024)

Czechoslovak Mathematical Journal

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For any integer $n > 1$ , we provide a parametric family of biquadratic fields with class groups having $n$ -rank at least 2. Moreover, in some cases, the $n$ -rank is bigger than 4.

The 4-string braid group $B_{4}$ has property RD and exponential mesoscopic rank

Sylvain Barré, Mikaël Pichot (2011)

Bulletin de la Société Mathématique de France

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We prove that the braid group $B_{4}$ on 4 strings, its central quotient $B_{4} / 〈 z 〉$ , and the automorphism group $Aut (F_{2})$ of the free group $F_{2}$ on 2 generators, have the property RD of Haagerup–Jolissaint. We also prove that the braid group $B_{4}$ is a group of intermediate mesoscopic rank (of dimension 3). More precisely, we show that the above three groups have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.

Composite rational functions expressible with few terms

Clemens Fuchs, Umberto Zannier (2012)

Journal of the European Mathematical Society

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We consider a rational function $f$ which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number $ℓ$ of terms. Then we look at the possible decompositions $f (x) = g (h (x))$ , where $g, h$ are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of $g$ is bounded only in terms of $ℓ$ (and we provide explicit bounds). This supports and quantifies...

Localization of dominant eigenpairs and planted communities by means of Frobenius inner products

Dario Fasino, Francesco Tudisco (2016)

Czechoslovak Mathematical Journal

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We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix $A$ . The result exploits the Frobenius inner product between $A$ and a given rank-one landmark matrix $X$ . Different choices for $X$ may be used, depending on the problem under investigation. In particular, we show that the choice where $X$ is the all-ones matrix allows to estimate the signature of the leading eigenvector of $A$ , generalizing previous results on Perron-Frobenius properties of matrices...

-simplicity of interval max-min matrices

Ján Plavka, Štefan Berežný (2018)

Kybernetika

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A matrix $A$ is said to have $𝐗$ -simple image eigenspace if any eigenvector $x$ belonging to the interval $𝐗 = {x : \underset{̲}{x} \leq x \leq \bar{x}}$ containing a constant vector is the unique solution of the system $A \otimes y = x$ in $𝐗$ . The main result of this paper is an extension of $𝐗$ -simplicity to interval max-min matrix $𝐀 = {A : \underset{̲}{A} \leq A \leq \bar{A}}$ distinguishing two possibilities, that at least one matrix or all matrices from a given interval have $𝐗$ -simple image eigenspace. $𝐗$ -simplicity of interval matrices in max-min algebra are studied and equivalent conditions for...