6 problems

Let $l,g$ be two even functions with $g>0$ on $(0,\pi)$, and suppose that $f\equiv l/g$ is monotone increasing over $(0,\pi)$. Set $X_n\equiv T_n^{-1}(g)T_n(l)$ for all $n$. Let…

Let $A$ be the discretized integral operator in the referenced two- or three-dimensional models, and let $T^{-1}_{\text{Schwarz}}$ be the Schwarz preconditioner associated with the…

Let ${\rm dom}$ be a convex domain, let $s$ satisfy $0<s<1$, and let ${\mathcal H}_h^s$ and ${\mathcal G}_h^s$ denote the fine-grid and two-grid operators, respectively, with mesh…

Uniform condition-number conjecture. There exist $s<0$ and a constant $C$ such that

Let ${\widetilde W}$ denote the matrix of eigenvectors for the symmetrically preconditioned force-based QC operator ${L}^{-1/2}L^{{\rm qcf}}_F{L}^{-1/2}$. Assume $A_F>0$. Symmetric…

Let $N\geq4$, let $1\leq K\leq N-2$, and let $F>0$. Let ${L}^{-1}$ be the inverse of the discrete Laplacian operator $L$, and consider the preconditioned operators…