6 problems

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…

Let the force-based quasicontinuum (QCF) operator for the homogeneous periodic chain be represented by a matrix, and let a matrix of its eigenvectors be chosen. Denote the conditio…

Let the linearized force-based quasicontinuum (QCF) operator act on the homogeneous periodic chain, and consider its spectrum of $\ell^2$-eigenvalues. Let the quasinonlocal QC meth…

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…

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…