  We present optically thin solutions for magnetized, advective disc-outflow
symbiosis around black holes (BHs). The main objective is to explain the
bright, hard state observations of accreting systems with stellar-mass to
supermassive BHs. We include the effects of magnetic fields and radiation
counterpart in entropy gradient based on the first law of thermodynamics to
represent energy advection. The cooling process includes bremsstrahlung,
synchrotron radiation, and inverse Comptonization process. One of our main
ventures is to explain some long-standing issues of ultra-luminous X-ray
sources (ULXs). The existing physical scenarios to explain their unusual high
luminosity are either the existence of the missing class of intermediate-mass
BH (IMBH), or super-Eddington accretion around a stellar-mass BH. However, most
ULXs with steep power-law spectrum can be well explained through
super-Eddington accretion, while the existence of IMBH is indeed disputed
extensively. Nevertheless, the interpretation of ULXs with a hard power-law
dominated state remains mysterious. Here we show that our magnetically
dominated disc-outflow symbiosis around rapidly spinning stellar-mass BHs can
achieve such large luminosity even for sub-Eddington accretion rate. The
magnetic field at the outer zone of the advective flow is more than the
corresponding Eddington limit. Such field becomes dynamically dominant near the
BH through continuous accretion process due to flux freezing, but maintaining
its Eddington limit. This unique field configurations enhance the synchrotron,
and synchrotron self-Comptonization process to achieve very large luminosity.
Through the same mechanism, our solutions for supermassive BHs can explain the
unusual large luminosity of ultra-luminous quasars.
