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In this study, we obtain an exact time-dependent solution of the chemical master equation (CME) of an extension of the two-state telegraph model describing bursty or non-bursty protein expression in the presence of positive or negative autoregulation. Using the method of spectral decomposition, we show that the eigenfunctions of the generating function solution of the CME are Heun functions, while the eigenvalues can be determined by solving a continued fraction equation. Our solution generalizes and corrects a previous time-dependent solution for the CME of a gene circuit describing non-bursty protein expression in the presence of negative autoregulation [Ramos et al., Phys. Rev. E 83, 062902 (2011)]. In particular, we clarify that the eigenvalues are generally not real as previously claimed. We also investigate the relationship between different types of dynamic behavior and the type of feedback, the protein burst size, and the gene switching rate. 

Humans acquire and accumulate knowledge through language usage and eagerly exchange their knowledge for advancement. Although geographical barriers had previously limited communication, the emergence of information technology has opened new avenues for knowledge exchange. However, it is unclear which communication pathway is dominant in the 21st century. Here, we explore the dominant path of knowledge diffusion in the 21st century using Wikipedia, the largest communal dataset. We evaluate the similarity of shared knowledge between population groups, distinguished based on their language usage. When population groups are more engaged with each other, their knowledge structure is more similar, where engagement is indicated by socio-economic connections, such as cultural, linguistic, and historical features. Moreover, geographical proximity is no longer a critical requirement for knowledge dissemination. Furthermore, we integrate our data into a mechanistic model to better understand the underlying mechanism and suggest that the main channel of information distribution in the 21st century is based online.