a ring $r$ is a strongly clean ring if every element in $r$ is the sum of an idempotent and a unit that commutate. we construct some classes of strongly clean rings which have stable range one. it is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.

Abstract The Parker Solar Probe mission (PSP) has completed seven orbits around the Sun. Wide-field Imager for (WISPR) on PSP consists of two visible light heliospheric imagers, which together image interplanetary medium between 13.°5 and 108° elongation. PSP/WISPR nominal science observing window occurs during solar encounters, take place when spacecraft (S/C) is within 0.25 au from During Orb...

The black hole photon ring is a prime target for upcoming space-based VLBI missions seeking to image the fine structure of astrophysical holes. classical Lyapunov exponents corresponding nearly bound null geodesics control quasinormal ringing perturbed as it settles back down equilibrium, and they admit holographic interpretation in terms quantum Ruelle resonances microstate dual Kerr hole. Rec...