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 $A{\prec }^{\mathrm{rs}}B$) if $R\left(A\right)\prec R\left(B\right)$, where $R\left(A\right)$ is the row sum vector of $A$ and $\prec$ 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 ${𝕄}_{n,m}$. ...

For $X,Y\in {𝐌}_{n,m}$ it is said that $X$ is gut-majorized by $Y$, and we write $X{\prec }_{\mathrm{gut}}Y$, if there exists an $n$-by-$n$ upper triangular g-row stochastic matrix $R$ such that $X=RY$. Define the relation ${\sim }_{\mathrm{gut}}$ as follows. $X{\sim }_{\mathrm{gut}}Y$ if $X$ is gut-majorized by $Y$ and $Y$ is gut-majorized by $X$. The (strong) linear preservers of ${\prec }_{\mathrm{gut}}$ 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 ${\sim }_{\mathrm{gut}}$ on ${ℝ}^n$ and ${𝐌}_{n,m}$.

Let $m$ and $n$ be positive integers, and let $R=(r_1,...,r_m)$ and $S=(s_1,...,s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m\times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F\left(R\right)$ be the $m\times n$ $(0,1)$-matrix, where for each $i$, the $i$th row of $F(R,S)$ consists of $r_i$ 1’s followed by $(n-r_i)$ 0’s. Let $A\in A(R,S)$. The discrepancy of A, $\mathrm{disc}\left(A\right)$, is the number of positions in which $F\left(R\right)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m\times n$ matrices over...

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 $G=P^{\top }P$ 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 $P^{-\top }$ of $P$ is doubly stochastic and has the...

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.

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 $x\in 𝐗=\{x;\underline{x}\le x\le \overline{x}\}$ and $y\le x$ for each eigenvector $y\in 𝐗$. A max-min matrix $A$ is called strongly $𝐗$-robust if the orbit $x,A\otimes x,A^2\otimes x,\cdots$ reaches the greatest $𝐗$-eigenvector with any starting vector of $𝐗$. We suggest an $O\left(n^3\right)$ 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...

Let $T$ be a tree with $n$ vertices. To each edge of $T$ we assign a weight which is a positive definite matrix of some fixed order, say, $s$. Let $D_{ij}$ denote the sum of all the weights lying in the path connecting the vertices $i$ and $j$ of $T$. We now say that $D_{ij}$ is the distance between $i$ and $j$. Define $D:=\left[D_{ij}\right]$, where $D_{ii}$ is the $s\times s$ null matrix and for $i\ne j$, $D_{ij}$ is the distance between $i$ and $j$. Let $G$ be an arbitrary connected weighted graph with $n$ vertices, where each weight is a positive definite matrix of order...

Suppose that $A$ is a real symmetric matrix of order $n$. Denote by $m_A\left(0\right)$ the nullity of $A$. For a nonempty subset $\alpha$ of $\{1,2,...,n\}$, let $A\left(\alpha \right)$ be the principal submatrix of $A$ obtained from $A$ by deleting the rows and columns indexed by $\alpha$. When $m_{A\left(\alpha \right)}\left(0\right)=m_A\left(0\right)+\left|\alpha \right|$, we call $\alpha$ a P-set of $A$. It is known that every P-set of $A$ contains at most $\lfloor n/2\rfloor$ elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...