Monotone substochastic operators and a new Calderón couple

Karol Leśnik

Displaying similar documents to “Monotone substochastic operators and a new Calderón couple”

Row Hadamard majorization on $𝐌_{m, n}$

Abbas Askarizadeh, Ali Armandnejad (2021)

Czechoslovak Mathematical Journal

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An $m \times n$ matrix $R$ with nonnegative entries is called row stochastic if the sum of entries on every row of $R$ is 1. Let $𝐌_{m, n}$ be the set of all $m \times n$ real matrices. For $A, B \in 𝐌_{m, n}$ , we say that $A$ is row Hadamard majorized by $B$ (denoted by $A ≺_{R H} B)$ if there exists an $m \times n$ row stochastic matrix $R$ such that $A = R \circ B$ , where $X \circ Y$ is the Hadamard product (entrywise product) of matrices $X, Y \in 𝐌_{m, n}$ . In this paper, we consider the concept of row Hadamard majorization as a relation on $𝐌_{m, n}$ and characterize the structure of all linear operators $T : 𝐌_{m, n} \to 𝐌_{m, n}$ preserving...

On the combinatorial structure of $0 / 1$ -matrices representing nonobtuse simplices

Jan Brandts, Abdullah Cihangir (2019)

Applications of Mathematics

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A $0 / 1$ -simplex is the convex hull of $n + 1$ affinely independent vertices of the unit $n$ -cube $I^{n}$ . It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute $0 / 1$ -simplices in $I^{n}$ can be represented by $0 / 1$ -matrices $P$ of size $n \times n$ whose Gramians $G = P^{⊤} P$ have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part $D$ of the transposed inverse $P^{- ⊤}$ of $P$ is doubly stochastic and has the...

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.

On reliability analysis of consecutive $k$ -out-of- $n$ systems with arbitrarily dependent components

Ebrahim Salehi (2016)

Applications of Mathematics

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In this paper, we consider the linear and circular consecutive $k$ -out-of- $n$ systems consisting of arbitrarily dependent components. Under the condition that at least $n - r + 1$ components ( $r \leq n$ ) of the system are working at time $t$ , we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive $k$ -out-of- $n$ systems. In the following, we investigate the inactivity time of the component with lifetime...

G-tridiagonal majorization on $𝐌_{n, m}$

Ahmad Mohammadhasani, Yamin Sayyari, Mahdi Sabzvari (2021)

Communications in Mathematics

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For $X, Y \in 𝐌_{n, m}$ , it is said that $X$ is majorized by $Y$ (and it is denoted by $X ≺_{g t} Y$ ) if there exists a tridiagonal g-doubly stochastic matrix $A$ such that $X = A Y$ . In this paper, the linear preservers and strong linear preservers of $≺_{g t}$ are characterized on $𝐌_{n, m}$ .

On row-sum majorization

Farzaneh Akbarzadeh, Ali Armandnejad (2019)

Czechoslovak Mathematical Journal

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Let $𝕄_{n, m}$ be the set of all $n \times m$ real or complex matrices. For $A, B \in 𝕄_{n, m}$ , we say that $A$ is row-sum majorized by $B$ (written as $A ≺^{rs} B$ ) if $R (A) ≺ R (B)$ , where $R (A)$ is the row sum vector of $A$ and $≺$ is the classical majorization on $ℝ^{n}$ . In the present paper, the structure of all linear operators $T : 𝕄_{n, m} \to 𝕄_{n, m}$ preserving or strongly preserving row-sum majorization is characterized. Also we consider the concepts of even and circulant majorization on $ℝ^{n}$ and then find the linear preservers of row-sum majorization of these relations on $𝕄_{n, m}$ . ...

On linear preservers of two-sided gut-majorization on $𝐌_{n, m}$

Asma Ilkhanizadeh Manesh, Ahmad Mohammadhasani (2018)

Czechoslovak Mathematical Journal

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For $X, Y \in 𝐌_{n, m}$ it is said that $X$ is gut-majorized by $Y$ , and we write $X ≺_{gut} Y$ , if there exists an $n$ -by- $n$ upper triangular g-row stochastic matrix $R$ such that $X = R Y$ . Define the relation $\sim_{gut}$ as follows. $X \sim_{gut} Y$ if $X$ is gut-majorized by $Y$ and $Y$ is gut-majorized by $X$ . The (strong) linear preservers of $≺_{gut}$ on $ℝ^{n}$ and strong linear preservers of this relation on $𝐌_{n, m}$ have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of $\sim_{gut}$ on $ℝ^{n}$ and $𝐌_{n, m}$ .

Maps on upper triangular matrices preserving zero products

Roksana Słowik (2017)

Czechoslovak Mathematical Journal

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Consider $𝒯_{n} (F)$ —the ring of all $n \times n$ upper triangular matrices defined over some field $F$ . A map $φ$ is called a zero product preserver on $𝒯_{n} (F)$ in both directions if for all $x, y \in 𝒯_{n} (F)$ the condition $x y = 0$ is satisfied if and only if $φ (x) φ (y) = 0$ . In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map $φ$ may act in any bijective way, whereas for the zero divisors and zero matrix one can write $φ$ as a...

Computing the greatest $𝐗$ -eigenvector of a matrix in max-min algebra

Ján Plavka (2016)

Kybernetika

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A vector $x$ is said to be an eigenvector of a square max-min matrix $A$ if $A \otimes x = x$ . An eigenvector $x$ of $A$ is called the greatest $𝐗$ -eigenvector of $A$ if $x \in 𝐗 = {x; \underset{̲}{x} \leq x \leq \bar{x}}$ and $y \leq x$ for each eigenvector $y \in 𝐗$ . A max-min matrix $A$ is called strongly $𝐗$ -robust if the orbit $x, A \otimes x, A^{2} \otimes x, \dots$ reaches the greatest $𝐗$ -eigenvector with any starting vector of $𝐗$ . We suggest an $O (n^{3})$ algorithm for computing the greatest $𝐗$ -eigenvector of $A$ and study the strong $𝐗$ -robustness. The necessary and sufficient conditions for strong $𝐗$ -robustness are introduced...

On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

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Given an integer $n \geq 3$ , let $u_{1}, ..., u_{n}$ be pairwise coprime integers $\geq 2$ , $𝒟$ a family of nonempty proper subsets of ${1, ..., n}$ with “enough” elements, and $ε$ a function $𝒟 \to {\pm 1}$ . Does there exist at least one prime $q$ such that $q$ divides $\prod_{i \in I} u_{i} - ε (I)$ for some $I \in 𝒟$ , but it does not divide $u_{1} \dots u_{n}$ ? We answer this question in the positive when the $u_{i}$ are prime powers and $ε$ and $𝒟$ are subjected to certain restrictions. We use the result to prove that, if $ε_{0} \in {\pm 1}$ and $A$ is a set of three or more primes that contains all prime divisors of any...

A new characterization of symmetric group by NSE

Azam Babai, Zeinab Akhlaghi (2017)

Czechoslovak Mathematical Journal

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Let $G$ be a group and $ω (G)$ be the set of element orders of $G$ . Let $k \in ω (G)$ and $m_{k} (G)$ be the number of elements of order $k$ in $G$ . Let nse $(G) = {m_{k} (G) : k \in ω (G)}$ . Assume $r$ is a prime number and let $G$ be a group such that nse $(G) =$ nse $(S_{r})$ , where $S_{r}$ is the symmetric group of degree $r$ . In this paper we prove that $G ≅ S_{r}$ , if $r$ divides the order of $G$ and $r^{2}$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.