In this study, we define the unrestricted Pell and Pell-Lucas quaternions. We give generating functions, Binet formulas and some generalizations of well-known identities such as Vajda’s, Catalan’s, Cassini’s d’Ocagne’s identities.

Lianghuai Mining Area is one of the 13 large coal bases in China. It is an important coal and coal production base in China. Mine water inrush accidents occur frequently, resulting in economic and human resource losses, reflecting the importance of the study of hydrogeology in mining areas. In this paper, the hydrogeological conditions of Bozhou and Huainan Panxie mine are analyzed, and the similarities and differences between the hydrogeological conditions of the two mines are summarized. The shallow pore water group in the Bozhou area is composed of the Quaternary system of the Quaternary system (Q4d) and the upper part of the upper part of the Mao Tong group (Q3m). The lithology of the aquifer is silt, silt and fine sand. The shallow pore water group of the Panxian Pancho Formation in Huainan is composed of the Upper Pleistocene of the Quaternary system and the Holocene strata. The lithology is mainly composed of fine sand. The main sources of shallow pore water supply in the two areas are precipitation infiltration, mainly for evaporation, lateral runoff, artificial mining and deep flow and discharge to the river.

Heat recovery is one of the measures proposed for the appropriate use of ammonia in tropical countries. This article analyzes a heat recovery system installed in an industrial refrigeration plant. Based on comparative readings of operating parameters of the installation, determined the effectiveness of the heat exchange, the increase in the efficiency of the refrigeration system, as well as the fuel saved by heating water in the industry. The results obtained reported that the thermal design based on heat exchange in annular spaces allows a significant saving of resources and a high rate of thermal utilization.

To address the problem that the imaging inversion method based on a single model in integrated aperture imaging is difficult to effectively correct model errors and perform accurate image reconstruction, a dual-model (DM)-based integrated aperture imaging inversion method is proposed for correcting the parametric errors of the inversion model and performing highly accurate millimeter-wave image reconstruction of the target scene. In view of the different parameter sensitivities of the Fourier transform (MFFT) model and the G-matrix (GM) model, the proposed DM method first corrects the imaging parameters with errors accurately by comparing the reconstruction errors of the two models; then recon-structs a high-precision target image based on the accurate GM model with the help of an improved regularization method. It is proved by simulation experiments that the proposed DM method can effectively correct the parameter errors of the imaging model and reconstruct the target scene with high accuracy in millimeter wave images compared with the traditional single-model imaging method.