  In this paper, we investigate the ground-state properties of a bosonic
Tonks-Girardeau gas confined in a one-dimensional periodic potential. The
single-particle reduced density matrix is computed numerically for systems up
to $N=265$ bosons. Scaling analysis of the occupation number of the lowest
orbital shows that there are no Bose-Einstein Condensation(BEC) for the
periodically trapped TG gas in both commensurate and incommensurate cases. We
find that, in the commensurate case, the scaling exponents of the occupation
number of the lowest orbital, the amplitude of the lowest orbital and the
zero-momentum peak height with the particle numbers are 0, -0.5 and 1,
respectively, while in the incommensurate case, they are 0.5, -0.5 and 1.5,
respectively. These exponents are related to each other in a universal
relation.

Statistical Mechanics;Mesoscale and Nanoscale Physics