15 problems

Large-$N$ trace conjecture. The reduced traces satisfy

Kirkland–Šmigoc's conjecture. Then $A$ is permutation similar to a matrix whose directed graph $\Gamma$ satisfies the conditions of Proposition 7.1 in Kirkland and Šmigoc. Kirkland…

Let $P$ be an $n\times n$ tridiagonal stochastic matrix. Its eigenvalues are real when $P$ is reversible, and the second largest eigenvalue means the largest algebraic eigenvalue a…

Let $r_\text{conv}(s)$ denote the convergence rate as a function of the packet size $s$, and let $r_\text{conv}(r_\text{drop})$ denote the convergence rate as a function of the pac…

Let $A$ be an irreducible $4\times4$ tridiagonal stochastic matrix of the form … where $0\leq\alpha,\beta,\gamma,\delta,\phi,\kappa\leq1$, $\beta+\gamma\leq1$, and…

Ran and Teng's conjecture. The nonreal spectral region is determined by the Karpelevich constraint together with the algebraic boundary

Let $s(n)$ be the set of $n\times n$ stochastic matrices, and let $G$ be a generating set of a similar form to the one considered for three-dimensional stochastic matrices. Let…

Let $s(n)$ denote the set of $n\times n$ stochastic matrices, and let $N_G(S)$ be the factorization length associated with a generating set $G$ for a stochastic matrix $S$. In thre…

Generic 2n-support conjecture. Generically, the support set $\Omega$ needs at least $2n$ elements: one per column to satisfy…

Product conjecture. The product of the $u_i$-tilted matrices

Let $A$ be a stochastic matrix with distinct eigenvalues. Say that $A$ has the strong anti-diagonal eigenvalue property when it satisfies the stronger anti-diagonal eigenvalue cond…

Let $n$ be a positive integer and let $\mathcal{S}$ be an SIA set of $n \times n$ stochastic matrices. Its SIA-index is the length of the shortest SIA product formed from matrices…

For each positive integer $n$, let $\operatorname{sia}(n)$ denote the largest SIA-index among all SIA sets of $n \times n$ stochastic matrices. Linear-growth conjecture. The actual…

Let $n$ be a positive integer and let $\mathcal{S}$ be a set of stochastic $n \times n$ matrices. A positive-column product is a product of matrices from $\mathcal{S}$ having a col…

Uniqueness and convergence conjecture. There is only one nontrivial equilibrium, it lies in the interior of $\Delta$, and for every $x(0)\in\Delta\setminus\{e_1,\dots,e_n\}$, the t…