  There is a very important problem that has not attracted sufficient attention
in academia, i.e., nonlinear field normalization citation counts at the paper
level obtained using nonlinear field normalization methods cannot be added or
averaged. Unfortunately, there are many cases adding or averaging the nonlinear
normalized citation counts of individual papers that can be found in the
academic literature, indicating that nonlinear field normalization methods have
long been misused in academia. In this paper, we performed the following two
research works. First, we analyzed why the nonlinear normalized citation counts
of individual papers cannot be added or averaged from the perspective of
theoretical analysis in mathematics: we provide mathematical proofs for the
crucial steps of the analysis. Second, we systematically classified the
existing main field normalization methods into linear and nonlinear field
normalization methods. The above two research works provide a theoretical basis
for the proper use of field normalization methods in the future, avoiding the
continued misuse of nonlinear data. Furthermore, because our mathematical proof
is applicable to all nonlinear data in the entire real number domain, our
research works are also meaningful for the whole field of data and information
science.
