26 problems

Let $\Omega_n$ be the set of doubly-stochastic $n\times n$ matrices, and define … The authors exhibit doubly-stochastic matrices $A$ for which…

Let $\Omega_n$ be the set of doubly stochastic matrices of order $n$, and let $\omega_n$ denote its eigenvalue region. For $j\geq 2$, let $\Pi_j$ be the convex hull of the $j$th ro…

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…

Let $B_n$ be the polytope of $n\times n$ doubly stochastic matrices, and let $e(B_n,t)$ denote the number of lattice points in $tB_n$. The dimension of $B_n$ is $(n-1)^2$. For a la…

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 $A$ be an $n\times n$ doubly stochastic matrix, and let $l$ be a positive integer. Its $l\times l$ permanental minors are the permanents of the submatrices obtained by choosing…

Let ${\bf A}=(A_1,\ldots,A_n)$ be an $n$-tuple of complex $n\times n$ matrices. Let $D({\bf A})$ be their mixed discriminant, and let $D_n$ be the set of tuples satisfying … and ……

Limiting permanent conjecture. The liminf is in fact a limit, namely

Let $\Omega_n$ denote the set of all doubly stochastic matrices of order $n$. For a matrix $A$, let $\sigma_k(A)$ denote its $k$th-order subpermanent. Holens–Doković conjecture. If…

Let $\Omega_n$ denote the set of all doubly stochastic matrices of order $n$, and let $J_n$ be the $n\times n$ matrix with every entry equal to $\frac{1}{n}$. Here…

Let $\Omega_n$ denote the set of all doubly stochastic matrices of order $n$, and let $J_n$ be the $n\times n$ matrix with every entry equal to $\frac{1}{n}$. Here…

Let $\c2$ be a doubly stochastic matrix, and let $\mu$ and $\lambda$ be the parameters defining its induced norm. Write $\sigma_2$ for the second largest singular value of $\c2$. E…

Let ${\mathrm {PDS}}_4$ be the set of permutative doubly stochastic matrices of order $4$, and let $\Pi_3$ and $\Pi_4$ be the convex hulls of the third and fourth roots of unity, r…

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. Let $PM_n$ be the union of the convex hulls of the $k$-th roots of unit…

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices. For permutations $P,Q$, let $I$ denote the identity permutation matrix,…

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…

For $n\in\mathbb{N}$, let $\Theta_n$ denote the set of possible eigenvalues of $n\times n$ stochastic matrices. Let $\Pi_n$ be the convex hull of the $n$th roots of unity, … Let…

Let $\Omega_n$ be the set of all single eigenvalues of all $n\times n$ doubly stochastic matrices, and let $\Pi_n$ be the convex hull of the $n$th roots of unity. Let…

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 $X$ be a random doubly stochastic matrix of size $n$, let ${\mathbf{E}}X$ denote its expectation, and consider the empirical spectral distribution of…

Strong Conjecture. For every $P\in\Omega_n$,

Let $\Omega_n$ be the set of $n\times n$ doubly stochastic matrices, and let $\mathrm{perm}_k(C)$ denote the sum of the permanents of all $k\times k$ submatrices of $C$. Tverberg's…

Let $X=(X_{ij})_{i,j=1,\ldots,n}$ be a uniformly chosen doubly stochastic matrix, and set … The empirical spectral distribution of $\bar{X}$ is the probability measure assigning ma…