We present GALEX near ultraviolet (NUV:1750 - 2750A) and far ultraviolet (FUV: 1350 - 1750A) imaging observations of two 1.2 degree diameter fields in the Hyades and Pleiades open clusters in order to detect possible UV variability of the member stars. We have performed a detailed software search for short-term UV flux variability during these observations of the approx 400 sources detected in each of the Hyades and Pleiades fields to identify flare-like (dMe) stellar objects. This search resulted in the detection of 16 UV variable sources, of which 13 can be directly associated with probable M-type stars. The other UV sources are G-type stars and one newly discovered RR Lyrae star, USNOB1.0 1069-0046050, of period 0.624 day and distance 4.5-7.0 kpc. Light curves of photon flux versus time are shown for 7 flare events recorded on six probable dMe stars. UV energies for these flares span the range 2E27 to 5E29 erg, with a corresponding NUV variability change of 1.82 mag. Only one of these flare events (on the star Cl* Melotte 25 LH129) can definitely be associated with an origin on a member the Hyades cluster itself. Finally, many of our M-type candidates show long periods of enhanced UV activity but without the associated rapid increase in flux that is normally associated with a flare event. However, the total UV energy output during such periods of increased activity is greater than that of many short-term UV flares. These intervals of enhanced low-level UV activity concur with the idea that, even in quiescence, the UV emission from dMe stars may be related to a superposition of many small flare events possessing a wide range of energies.
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