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Abstract This paper investigates the problem of recovering source terms in abstract initial value problems (IVP) commonly used to model various scientific phenomena in physics, chemistry, economics, and other fields. We consider source terms of the form$$F=h+\eta $$ $F=h+\eta$, where$$\eta $$ $\eta$is a Lipschitz continuous background source. The primary objective is to estimate the unknown parameters of non-instantaneous sources$$h(t)=\sum \limits _{j=0}^M h_je^{-\rho _j(t-t_j)}\chi _{[t_j,\infty )}(t)$$ $h\left(t\right)=\sum _{j=0}^M{h}_j{e}^{-{\rho }_j(t-t_j)}{\chi }_{[t_j,\infty )}\left(t\right)$, such as the decay rates, initial intensities and activation times. We present two novel recovery algorithms that employ distinct sampling methods of the solution of the IVP. Algorithm 1 combines discrete and weighted average measurements, whereas Algorithm 2 uses a different variant of weighted average measurements. We analyze the performance of these algorithms, providing upper bounds on the recovery errors of the model parameters. Our focus is on the structure of the dynamical samples used by the algorithms and on the error guarantees they yield. 

Well-preserved coprolites (fossil faecal pellets) were found from lignite seams of the Lower Cretaceous Huolinhe Formation at the Huolinhe Basin in eastern Inner Mongolia, Northeast China. These coprolites provide a combination of following features: oval to cylindrical shaped with six longitudinal ridges, hexagonal to elliptical cross-sections, and one blunt end and the other pointed end. According to these distinct features and their size range, the producers of these coprolites are attributed to termites. Termites were estimated to have originated in the earliest Cretaceous with an evolutionary radiation in the Early Cretaceous. The presence of wood debris in the coprolites indicate that the Early Cretaceous termites from the Huolinhe Basin had wood-feeding habits; and anatomical features displaying on the wood debris further suggest their feeding preference was coniferous wood. Besides, the results of a k-means clustering analysis performed for these coprolites indicate that three clusters with different proportion were present, suggesting the division of labor in termites’ sociality existed as early as the Early Cretaceous.