  On the basis of the entropy of incomplete statistics (IS) and the joint
probability factorization condition, two controversial problems existing in IS
are investigated, where one is what the correct expression of the internal
energy for a composite system is and the other is whether the zeroth law of
thermodynamics is true or not. Some new equivalent expressions of the internal
energy of a composite system are derived through a precise mathematical
calculation. Moreover, a self-consistent calculation is used to expound that
the zeroth law of thermodynamics is also suitable for IS, but it can't be
proven from theory. Finally, it is pointed out that the generalized zeroth law
of thermodynamics and the expressions of the internal energy of a composite
system derived in literature are incorrect, because two irrational assumptions
have been implicitly introduced.
