  We report the discovery of milli-hertz (mHz) quasi-periodic oscillations
(QPOs) and QPO-modulated pulsations during large X-ray flares from the
high-mass X-ray binary pulsar LMC X-4 using data from the Rossi X-Ray Timing
Explorer (RXTE). The lightcurves of flares show that, in addition to ~74 mHz
coherent pulsations, there exist two more time-varying temporal structures at
frequencies of ~0.65-1.35 and ~2-20 mHz. These relatively long-term structures
appear in the power density spectra as mHz QPOs and as well-developed sidebands
around the coherent pulse frequency as well, indicating that the amplitudes of
the coherent pulsation is modulated by those of the mHz QPOs. One interesting
feature is that, while the first flare shows symmetric sidebands around the
coherent pulse frequency, the second flare shows significant excess emission in
the lower-frequency sidebands due to the ~2-20 mHz QPOs. We discuss the origin
of the QPOs using a combination of the beat-frequency model and a modified
version of the Keplerian-frequency model. According to our discussion, it seems
to be possible to attribute the origin of the ~0.65-1.35 and ~2-20 mHz QPOs to
the beating between the rotational frequency of the neutron star and the
Keplerian frequency of large accreting clumps near the corotation radius and to
the orbital motion of clumps at Keplerian radii of 2-10 times 10^9 cm,
respectively.
