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Binary search trees with costs α and β on the left and right edges (lopsided binary search trees) are important in the construction of optimum prefix codes. In this paper we derive lower bounds for the external pathlengths of lopsided binary trees. It is found that the lower bound is tight if the cost difference (the difference in maximum cost and the minimum cost) is small but quite sharp when the cost difference is large. We suggest alternative ways to construct the lopsided binary tree when the cost difference is high to improve the lower bound.

The origin of the knee in the energy spectrum of cosmic rays is one of the central questions of high-energy astrophysics. One possible explanation is the energy dependent leakage of nuclei from the Galaxy due to their propagation. The latter is investigated in a combined method using numerical calculations of trajectories and the diffusion approximation. The life time of cosmic rays in the Galaxy and the corresponding pathlength are presented. The resulting energy spectra as observed at Earth are discussed and compared to experimental data.

Near infrared spectroscopy (NIRS) is based on molecular overtone and combination vibrations. It is difficult to assign specific features under complicated system. So it is necessary to find the relevance between NIRS and target compound. For this purpose, the chondroitin sulfate (CS) ethanol precipitation process was selected as the research model, and 90 samples of 5 different batches were collected and the content of CS was determined by modified carbazole method. The relevance between NIRS and CS was studied throughout optical pathlength, pretreatment methods and variables selection methods. In conclusion, the first derivative with Savitzky–Golay (SG) smoothing was selected as the best pretreatment, and the best spectral region was selected using interval partial least squares (iPLS) method under 1 mm optical cell. A multivariate calibration model was established using PLS algorithm for determining the content of CS, and the root mean square error of prediction (RMSEP) is 3.934 g⋅L^-1. This method will have great potential in process analytical technology in the future.