In this paper, we consider three already known perturbation estimates of the solution to symmetric matrix Riccati differential equation. The aim paper is analyse experimentally effectiveness bounds considered, compare their sharpness for problems with increasing conditioning, and specify area application analysed. analytical scalar equation one experimental models proved in theorem. results sug...

We investigate the noise sensitivity of top eigenvector a sparse random symmetric matrix. Let v be an N×N matrix with average d non-zero centered entries per row. resample k randomly chosen and obtain another realization v[k]. Building on recent results matrices analysis previously developed for Wigner matrices, we prove that, if d?N2?9, high probability, when k?N5?3, vectors v[k] are almost co...

The molecular ruby analogue [Cr(ddpd)2]3+ (ddpd=N,N’-dimethyl-N,N’-dipyridine-2-ylpyridine-2,6-diamine) exhibits near infrared (NIR) emission with a high photoluminescence (PL) quantum yield ?PL of 11 % and lifetime 898 ?s in deaerated water at room temperature. While ligand-based control the photophysical properties has received much attention, influences counter anions microenvironment are st...