3 problems

Let $\mathcal{L}_d$ be the set of bracelet matrices of dimension $d$. A monoid is a set closed under matrix multiplication and containing an identity element $\mathbb{I}_d$. Monoid…

Let ${\mathcal B}_n$ denote the set of bistochastic $n\times n$ matrices, and let $d_{\min}(n)$ be the smallest value of $d$ such that every matrix in ${\mathcal B}_n$ is $d$-ortho…

Let $\lambda_n$ be the smallest nonnegative number such that … and let $\rho_n$ be the smallest nonnegative number such that … Here $\mathcal{B}_n^{sym}$ is the set of $n$ by $n$ s…