************************************************************************* 82062 We propose deep follow-up observations of two LMC remnants, N23 and DEM 71, in order to complete a program to characterize their hard spectral emission and measure or set limits on their nonthermal X-ray luminosities. Existing observations are too brief (30 ks) to achieve these goals. The presence of a nonthermal spectral component could be the signature of acceleration of electrons within the shock to highly relativistic energy, and hence the creation of cosmic rays. *************************************************************************