25 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…

Let $A$ be an $m\times n$ matrix of rank $m$ whose displayed nonzero entries are algebraically independent over $\mathbb{Q}$. Let … be elementary vectors in the row space of $A$. F…

Sparse elliptic limiting-measure conjecture. For every $c>0$, there exists a deterministic measure $\mu_c$ that is the weak limit of the empirical spectral measures of the model wi…

The implementations use orderings for star and acyclic bicoloring, and these orderings are applied to an augmented symmetric matrix rather than to the graph representation for whic…

The sparse Cholesky methods MF, LL, and RL are compared through their factorization performance on large matrices. Method MF uses a stack whose data movement is described in line18…

The methods LL, RL, and MF are compared on large sparse matrices using serial and multithreaded BLAS. Their assembly operations are performed serially, whereas RLB performs every f…

Failure conjecture. Even if the slra code could tackle such a large input, its approach is likely to fail on this example.

Consider a slightly non-symmetric matrix and an $LDU$ factorization whose triangular factors include an upper-triangular factor $U$. Let a reordering be a permutation of the matrix…

Let ${\bf B}_n$ be a Bernoulli matrix in the regime where $pn$ is of order one, and consider its eigenvectors and the limiting spectral measure, whose atomic part consists of Dirac…

Let $H$ be the sparse random matrix and let $q$ denote its sparsity parameter. Theorem 1 and Theorem 2 concern the noise sensitivity results established in the paper for the top ei…

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 $\mathcal{C}$ be a family of low-density parity-check codes whose associated chain complexes over $\mathbb{F}_2$ are sparse. A lift is a chain complex over $\mathbb{Z}$ whose b…

Let $A\in\{0,1\}^{n\times n}$ be a $C_4$-free $0$-$1$ matrix with at most $3n$ non-zero entries. Fano-plane permanent conjecture. Its permanent satisfies … The conjecture is motiva…

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 $A\in\{0,1\}^{n\times n}$ have at most $2n$ non-zero entries. Bruhn–Rautenbach's conjecture. The determinant of $A$ is at most … The conjecture improves the earlier bound of Br…

Let a matrix approximation be specified by the powers of two used for its nonzero entries and by the locations of those entries. For a $K\times K$ matrix with $NK$ nonzero entries,…

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 $A\in\{0,1\}^{n\times n}$ be a $0/1$-matrix with at most $2n$ non-zero entries. Sparse maximal determinant conjecture. Then … The bound is motivated by block-diagonal matrices…

Consider sparse random matrices with average degree parameter $pN$, where $p\ll 1$. Let $p_c>1$ be a critical value. Critical-threshold conjecture. For $pN>p_c$, the bulk eigenvalu…

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…

Let $n$ be the blocklength, let $\rho(n)=1/n^\gamma$ for a fixed $\gamma$ with $0<\gamma<1$, and let the generating matrices be drawn according to the Bernoulli$(n,k,\rho(n))$ dist…