10 problems

Let $T(n,\tilde{k})$ be the family of matrices with real row-weight parameter $\tilde{k}>1$, and let $M_T(n,\tilde{k})$ be its maximum determinant. Define … and … Let $M_R(m,n,k)$…

Bruhn et al.'s conjecture. The true exponential growth rate is

Let $S(n,k)$ denote the family of $n\times n$ zero-one matrices with every row and column sum equal to $k$, and let $M(n,k)$ be the maximum absolute determinant of a matrix in…

Let $A$ be an $n\times n$ circulant matrix whose entries are real numbers in the interval $[0,1]$. The first row determines $A$ through parameters $a_0,\ldots,a_{n-1}$, so these ma…

Let $D_{01}(n)$ denote the maximum determinant of an $n\times n$ circulant matrix with entries in $\{0,1\}$, and let $U_{01}(n)$ be the upper bound defined by … Here…

Maximal-determinant conjecture. Maximal determinants of 0/1-matrices are attained by 0/1-matrices that represent acute 0/1-simplices.

Universal lower-bound conjecture. The condition that $n$ be sufficiently large can be omitted: the inequality

Rokicki et al.'s conjecture. Based on computational results for $n\leq 120$, one has

Let $A$ be a maxdet $\\{\pm 1\\}$ matrix of order $n$, and let $M$ range over all $m\times m$ submatrices of $A$, with $n\geq m$. Write $E(\det(M)^2)$ for the mean value of…

For each positive integer $k$, let $d_n=D_n/2^{n-1}$, where $D_n$ is the maximum absolute determinant of an $n\times n$ matrix with entries in $\{+1,-1\}$. Divisibility conjecture.…