6 problems

Let $n\ge 2$, and let $F_n$ be the $n\times n$ matrix whose $(i,j)$ entry is $1$ when $j\le i+1$ and $0$ otherwise. The matrix $F_n$ is a union of permutation matrices correspondin…

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices, and let $PM_n$ be the union of the convex hulls of the $k$-th roots of…

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices, and let a convex combination of permutation matrices mean a matrix of t…

Let $D$ be a doubly stochastic matrix of order $n$, meaning that it has nonnegative real entries and every row and column sums to $1$. Its binary diagonal X-ray is the vector…

Spectral determination conjecture. For $0<a\leq n$, the only elements of $\Delta_n^s(a)$ that are DS in $\Delta_n^s$ are points on the line segments $[I_n,C_n]$, $[I_n,P]$ and…

Let $\phi$ be a doubly stochastic $n\times n$ matrix, so that its entries are non-negative and every row and column sums to $1$. The permanent is denoted by…