13 problems

Monotonicity conjecture. For all $m\in\mathbb N$, one has

The Discrete ALIF inner-loop convergence conjecture asserts that if

Let ${\bf B}_{m+\ell}$ be partitioned as … and define … Here $\widetilde{{\bf B}}_\ell$ is the corresponding projected bidiagonal block produced by the recycled hybrid projection m…

Expanding orthogonal similarity variant conjecture. For almost all matrices of dimension at most $7$, repeatedly applying this eigenvector construction and random orthogonal or uni…

Let $A$ be a matrix of dimension at most $7$, and repeatedly construct eigenvector matrices $X_i$ whose columns have unit $\mathbb{L}_2$ norm. A matrix property holds for almost al…

Let the roots of the corresponding univariate polynomials be all distinct, and let $\mathbf{a}$ be a vector whose entries are points in the complex plane. Let…

Let $A$ be nonsingular. Define the non-orthogonal coarse-grid correction using transfer operators … The correction is stable in the sense that … for some constant $C$ independent o…

Let $\zeta_1,\dots,\zeta_d\in\mathbb{C}$ be the spectral points used in the finite-dimensional Orthomin analysis, and let $\operatorname{Hull}(\zeta_1,\dots,\zeta_d)$ denote their…

Let $\mathbb{S}^1$ be the unit circle with Haar probability measure $d\mu_0$, and let $U:L^2(\mathbb{S}^1,d\mu_0)\to L^2(\mathbb{S}^1,d\mu_0)$ be the multiplication operator … Cons…

Let $k\in\mathbb{N}$. For each $d\in\mathbb{N}$, let $U=\operatorname{diag}(1,\zeta_d,\zeta_d^2,\dots,\zeta_d^{d-1})$, where $\zeta_d=\exp(2\pi \mathbf{i}/d)$ is a primitive root o…

Laszkiewicz et al.'s convergence conjecture. If $k\ge m-1$ and $x_0$ lies in $L^{(\operatorname{Pad\acute{e}})}_p$, then

Let $V$ denote the matrix of eigenvectors for the force-based QC operator $L^{{\rm qcf}}_F$. Assume $A_F>0$. Eigenvector conditioning conjecture. As $N\to\infty$, … The conjecture…

Let $N\geq4$, let $1\leq K\leq N-2$, and let $F>0$. Denote by $L^{{\rm qcf}}_F$ and $L^{{\rm qnl}}_F$ the force-based and quasinonlocal QC operators, respectively. Spectral equival…