9 problems

Let $G$ be a simple graph with vertex set $V(G)=\{v_1,\ldots,v_n\}$. Let $Z^o_G$ be the scheme of nondegenerate symmetric $n\times n$ matrices $M$ satisfying … Equivalently, for a…

Fix an integer $d\geq 2$ and a constant $C>0$. Let $r(k),n_k\in\mathbb{N}$ satisfy … For each $k$, let $M^{(k)}\in\{-1,0,1\}^{k\times n_k}$ be a deterministic matrix whose every co…

A matrix is sparse when its relevant rows and columns have uniformly boundedly many nonzero entries. A sparse LU decomposition is a factorization … where $P$ and $Q$ are permutatio…

A sparse $2$-complex over $\mathbb{F}_2$ is a chain complex in degrees $0$, $1$, and $2$ whose boundary operators are sparse. A stable lift is obtained by adding finitely many cont…

A chain complex over $\mathbb{F}_2$ is sparse when every row and column of every boundary operator has $O(1)$ nonzero entries; a sparse lift to $\mathbb{Z}$ has uniformly bounded r…

Let $A\in\{0,1\}^{n\times n}$ have at most $3n$ non-zero entries. Bruhn–Rautenbach's conjecture. The determinant of $A$ is at most … The bound is motivated by the incidence matrix…

Let $A$ be the matrix, let $B$ be the output of incomplete selected inversion, and let $F$ denote the error matrix in the notation of the paper's selected-inversion setup. Write…

Let $H$ be the matrix and let $E$ denote the error matrix in the notation of the paper's selected-inversion setup. Write $\mathrm{level}(i,j)$ for the level of fill between indices…

Let $\hat L^{k_1,k_2}$ be a block of the matrix $\hat L$ arising in the covariance-based cryo-electron microscopy algorithm, with $k_1,k_2$ its associated frequency indices. Let…