  The Hubbard model has been investigated widely by many authors, while this
work may be new in two aspects. One, we focus on the possible effects of the
positions of the gaps associated with the pairing and the spin density wave.
Two, we suggest that the models with different parameters are appropriate for
different materials (or a material in different doped regions). This will lead
to some new insights into the high temperature superconductors. It is shown
that the SDW can appear at some temperature region when the on-site Coulomb
interaction is larger, while the SC requires a decreased U at a lower
temperature. This can qualitatively explain the relationship between
superconducting and pseudogap states of Cu-based superconductors in underdoped
and optimally doped regions. The superinsulator is also discussed.
