3 problems

Bigelow–Gonçalves–Kamada–Nakamura conjecture. The image $C(P_m^+)$ is the set of matrices $M=(M_{ij})\in {\rm Mat}_m^0$ satisfying: (I) $M$ is symmetric; (II) every $M_{ij}$ is a n…

Positive pure braid crossing-matrix conjecture. An $n \times n$ integer matrix $M$ is the crossing matrix of some positive pure $n$-braid if and only if $M$ is a non-negative integ…

Let $A=(a_{ij})$ be a positively realizable symmetric matrix, and call a position $(i,j)$ fully supported if, for every $k$ between $i$ and $j$, at least one of $a_{ik}$ and…