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.

翻译：学术界存在一个尚未引起足够重视的重要问题，即通过非线性领域归一化方法获得的论文层面引用次数不能相加或平均。遗憾的是，学术文献中存在大量将单篇论文的非线性归一化引用次数进行相加或平均的案例，表明非线性领域归一化方法在学术界长期被误用。本文完成了以下两项研究工作：首先，从数学理论分析的角度揭示了单篇论文的非线性归一化引用次数为何不能相加或平均，并对分析的关键步骤提供了数学证明；其次，系统地梳理现有主流领域归一化方法，将其划分为线性与非线性两类。上述两项研究为未来正确使用领域归一化方法提供了理论基础，避免非线性数据的持续误用。此外，由于我们的数学证明适用于整个实数域的所有非线性数据，本研究对数据与信息科学全域亦具有重要参考价值。