Abstract. Many high and steep slopes are comprised of special topographic and geomorphic types and formed through mining activities during the construction of mountain expressways. Severe soil erosion may also occur under heavy rainfall conditions. Therefore, predicting soil loss on highway slopes is important in protecting infrastructure and human life. In this study, we investigate Xinhe Expressway located at the southern edge of the Yunnan–Guizhou Plateau. The Revised Universal Soil Loss Equation (RUSLE) is used as the prediction model for soil and water loss on slopes. Geographic information systems, remote sensing technology, field surveys, run-off plot observation testing, cluster analysis and co-kriging calculations are also utilised. The partition of the prediction units of soil loss on the expressway slope in the mountainous area and the spatial distribution of rainfall on a linear highway are studied. Given the particularity of the expressway slope in the mountainous area, the model parameter is modified, and the risk of soil loss along the mountain expressway is simulated and predicted under 20- and 1-year rainfall return periods. The following results are obtained. (1) Natural watersheds can be considered for the prediction of slope soil erosion to represent the actual situation of soil loss on each slope. Then, the spatial location of the soil erosion unit can be determined. (2) Analysis of actual observation data shows that the overall average absolute error of the monitoring area is 0.39 t ha−1, the overall average relative error is 33.96 % and the overall root mean square error is between 0.21 and 0.66, all of which are within acceptable limits. The Nash efficiency coefficient is 0.67, indicating that the prediction accuracy of the model satisfies the requirements. (3) Under the 1-year rainfall return period condition, we find through risk classification that the percentage of prediction units with no risk of erosion is 78 %. The soil erosion risk is low and does not affect road traffic safety. Under the 20-year return period rainfall condition, the percentage of units with high and extremely high risks is 7.11 %. The prediction results can help adjust the design of water and soil conservation measures for these units.

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In the present paper, we continue our research on numerical ranges of operators. With newly developed techniques, we show thatLet N be a nest on a Hilbert space H and T ∈ T (N ), where T (N ) denotes the nest algebra associated with N . Then for given ε > 0, there exists a compact operator K with K < ε such that T + K ∈ T (N ) and the numerical range of T + K is closed.As applications, we show that the statement of the above type holds for the class of Cowen-Douglas operators, the class of nilpotent operators and the class of quasinilpotent operators.