  We show that in the translation invariant case and in the antiferromagnetic
phase, the reduced density matrix $\rho _2$ has no off-diagonal long-range
order of on-site electron pairs for the single-band Hubbard model on a cubic
lattice away from half filling at finite temperatures both for the positive
coupling and for the negative coupling. In these cases the model can not give a
mechanism for the superconductivity caused by the condensation of on-site
electron pairs and the nearest-neighbor electron pairs.
