16 problems

Let $A$ be a nonnegative substochastic matrix, meaning … for all $i$ and $j$. Suppose that above the diagonal, $A$ has nonzero entries only at distance $1$ from the diagonal, while…

Let $A$ be a nonnegative substochastic matrix, meaning … for all $i$ and $j$. Suppose that above the diagonal, $A$ has nonzero entries only at distance $1$ from the diagonal, while…

Basic semialgebraicity conjecture. The set of spectra realizable by symmetric nonnegative matrices forms a basic semi-algebraic set.

Let $\mathcal{P}_{n,2n}$ be the cone of polynomials of degree at most $2n$ that preserve nonnegative matrices of order $n$, and let $\mathbf{v}$ denote the coefficient vector of su…

Normalized-volume decay conjecture.

Continuity conjecture. The largest $t\in\mathbb{R}^+$ such that $p_t\in\mathcal{P}_n$ for all such $m,s$ is a continuous function of $m$ and $s$, bounded between $1$ and $2$. The c…

Let $A$ be a nonnegative substochastic matrix, meaning … for all $i$ and $j$. Suppose that above the diagonal, $A$ has nonzero entries only at distance $1$ from the diagonal, while…

Consider the Markov chain on $\mathbb{N}$ in which every vertex $i$ has an edge of weight $r$ to $i-1$ and edges of weight $c_l$ to $i+l$ for $0\leq l\leq k$, with … Let $p_l$ be t…

Let $C_n$ denote the $n\times n$ Chet matrix, and let $C=\{n:C_n\text{ is entry-wise nonnegative}\}$. Chet Conjecture. One has … More strongly, … The paper presents this as its mai…

For $n\geq 3$, let $\mathscr{P}_n$ denote the set of polynomials that preserve entrywise nonnegative $n\times n$ matrices under polynomial evaluation. Hierarchy conjecture. … The p…

Let $A$ be a square matrix over a subring $\mathcal R$ of $\mathbb R$. Its nonzero spectrum is the multiset of its nonzero eigenvalues. Boyle–Handelman's strong generalized spectra…

Let $\mathcal R$ be a subring of $\mathbb R$, and let $\Lambda=(\lambda_1,\dots,\lambda_k)$ be a $k$-tuple of complex numbers. The necessary conditions are the Perron condition, th…

Let $N$ be a non-negative real and $k$ a positive integer. For a square matrix $A$ with non-negative real entries, define its weight by … Assume that $A$ is nilpotent and has weigh…

Let $CE_n$ denote the critical exponent for $n$-by-$n$ generalized doubly nonnegative matrices, and let $MIP_n$ be the maximum index of primitivity among primitive $n$-by-$n$ gener…

Let $H\foral R^{n\times n}$ be symmetric and entrywise nonnegative, with geometrically simple spectral-radius eigenvalue $\rho(H)$. Let $v,w\in\mathbb R^n$ satisfy $v>0$ and $w>0$.…

Non-negative least-squares column-exclusion conjecture. There exists at least one column ${\bf g}_*$ of $G$ such that $\hat{\bf x}$ has non-negative entries.