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IC 10 X-1 is one of close X-ray binaries containing a Wolf-Rayet donor, which can provide an evolutionary link between high-mass X-ray binaries and gravitational wave sources. It is still unclear about the precise nature of the accreting compact object in IC 10 X-1, although it looks more like a black hole than a neutron star. In this work, we use a binary population synthesis method to simulate the formation of IC 10 X-1 like binaries by assuming different common-envelope ejection efficiencies. This work represents a big step forward over previous studies since we adopt new criteria of mass-transfer stability. These criteria allow the formation of IC 10 X-1 like systems without experiencing common envelope evolution. Based on our calculations, we propose that the compact object in IC 10 X-1 is a black hole with mass of $\sim 10-30M_\odot$ and the progenitor evolution of this binary probably just experienced stable mass transfer., Comment: 17 pages, 10 figures, accepted by ApJ

Neutron star$-$white dwarf (NSWD) binaries are one of the most abundant sources of gravitational waves (GW) in the Milky Way. These GW sources are the evolutionary products of primordial binaries that experienced many processes of binary interaction. We employ a binary population synthesis method to investigate the properties of Galactic NSWD binaries detectable by the Laser Interferometer Space Antenna (LISA). In this paper, only the NSWD systems with a COWD or ONeWD component are included. We consider various models related to mass transfer efficiencies during primordial binary evolution, supernova explosion mechanisms at NS formation, common envelope ejection efficiencies, and critical WD masses that determining the stability of mass transfer between WDs and NSs. Based on our calculations, we estimate that tens to hundreds of LISA NSWD binaries exist in the Milky Way. We find that the detection of LISA NSWD binaries is able to provide profound insights into mass transfer efficiencies during the evolution of primordial binaries and critical WD masses during mass transfer from a WD to an NS., Comment: 11 pages, 5 figures, 1 table, accepted to MNRAS