Consider the one dimensional nonlinear beam equation u_tt + u_xxxx + mu + u³ = 0 under Dirichlet boundary conditions. We show that for any m > 0 but a set of small Lebesgue measure, the above equation admits a family of small-amplitude quasi-periodic solutions with n-dimensional Diophantine frequencies. These Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proofs are based on an infinite dimensional Kolmogorov-Arnold-Moser iteration procedure and a partial Birkhoff normal form.

Accurate prediction of Air Quality Index (AQI) is of significant importance for environmental governance. In this paper, the CNN-LSTM prediction model based on Pelican Optimization Algorithm (POA) was proposed to study the air quality index (AQI) and its influencing factors in Changchun. The model initially employs LightGBM and SHAP methods for feature engineering, constructs feature and label data, and increases the data dimensionality. The Pelican Optimization Algorithm (POA) is utilized to identify optimal performance parameters, ensuring the model achieves peak efficiency in parameter selection. The model evaluation showed that the mean absolute error was 4.2767, the root mean squared error is 6.7421, the coefficient of determination R-squared was 0.9871 and the explained variance score was 0.9877. The results of our study indicate the effectiveness of the POA-optimized CNN-LSTM prediction method in air quality forecasting. This model demonstrates the capacity to learn long-term dependencies and is well-suited for processing time series data.

Accurate soybean yield estimation is critically imperative for modern agricultural systems amidst escalating global food security pressures, yet conventional methodologies are constrained for large-scale, high-frequency monitoring. To address this, an innovative deep learning framework, TransBiHGRU-PSO, is proposed for precise large-scale soybean yield estimation via effective fusion of multi-source, multitemporal remote sensing data, emphasizing robust and accurate estimation even with anomalous yield data. This framework synergistically integrates an optimized Bidirectional Hierarchical Gated Recurrent Unit (BiHGRU), a Transformer encoder, and a novel Greenness and Water Content Composite Index (GWCCI), with critical parameters optimized by Particle Swarm Optimization (PSO). County-level yield data from 12 U.S. states were used, supplemented by multi-temporal remote sensing datasets (MODIS surface reflectance, vegetation indices, environmental variables). Empirical analyses showed TransBiHGRU-PSO demonstrated improved estimation capability and generalizability compared to multiple benchmark models. Notably, with anomalous yield data retained, the model achieved solid test set performance (coefficient of determination (R²) of 0.71, root mean square error (RMSE) of 4.2812 bushels/acre). Compared to the best traditional machine learning model (SVR), R² increased by 52.96% and RMSE decreased by 26.05%; and relative to the best deep learning baseline model (LSTM), R² and RMSE improved by 7.04% and 7.04%, respectively. Furthermore, validation of interannual stability (2008–2018, anomalies retained) revealed a mean R² of 0.70 and mean RMSE of 4.4701 bushels/acre, affirming its consistency under complex real-world conditions. This TransBiHGRU-PSO algorithmic framework, combined with multi-source and multitemporal remote sensing data, offers a valuable exploration for large-scale soybean yield estimation.

Automatically measuring the Vertebral Heart Scale (VHS) from canine thoracic X-ray images is an effective approach for the early screening and prevention of cardiomegaly. Establishing automated and precise measurement tools is crucial for advancing population-based research on canine cardiac health. In this study, an integrated framework based on an improved U-Net architecture (SWA-UNet) and YOLOv11 is proposed for precise heart segmentation and key point detection to enable automated and accurate VHS calculation. It provides an efficient, objective, and repeatable auxiliary diagnostic tool for clinical practice, significantly enhancing the standardization of cardiac assessment and diagnostic efficiency. The method achieved a mean Intersection over Union of 95.34 % and a Dice coefficient of 98.20 % for heart segmentation. For key point detection, YOLOv11 attained a precision rate of 96.50 % and a recall rate of 95.20 % for the T3-T10 vertebrae, demonstrating high detection accuracy and reliability. Ultimately, the predicted VHS values showed a strong linear association with manual measurements, with a Pearson correlation coefficient of 0.8985. Furthermore, Bland-Altman analysis indicated that the automated system maintains high consistency with expert assessments, characterized by a minimal mean difference of -0.0696 cm. This result fully validates the effectiveness of this method in addressing challenges such as low-contrast images, blurred boundaries, and structural overlaps. As an auxiliary tool for veterinary radiologists, this method demonstrates great potential and can contribute to efficient and standardized assessment of canine heart diseases.

In the present paper, the series solutions and the approximate solutions of the time–space fractional differential equations are obtained using two different analytical methods. One is the homotopy perturbation Sumudu transform method (HPSTM), and another is the variational iteration Laplace transform method (VILTM). It is observed that the approximate solutions are very close to the exact solutions. The solutions obtained are very useful and significant to analyze many phenomena, and the solutions have not been reported in previous literature. The salient feature of this work is the graphical presentations of the third approximate solutions for different values of order $\alpha$.

To address issues such as significant scale differences, complex pose variations, strong background interference, and similar category characteristics of pests in the images obtained from field traps, this study proposes a pest recognition method based on a two-stage “segmentation–detection” approach to improve the accuracy of field pest situation monitoring. In the first stage, an improved segmentation model, BAE-UNet (Background-Aware and Edge-Enhanced U-Net), is adopted. Based on the classic U-Net framework, a Background-Aware Contextual Module (BACM), a Spatial-Channel Refinement and Attention Module (SCRA), and a Multi-Scale Edge-Aware Spatial Attention Module (MESA) are introduced. These modules respectively optimize multi-scale feature extraction, background suppression, and boundary refinement, effectively removing complex background information and accurately extracting pest body regions. In the second stage, the segmented pest body images are input into the YOLOv8 model to achieve precise pest detection and classification. Experimental results show that BAE-UNet performs excellently in the segmentation task, achieving an mIoU of 0.930, a Dice coefficient of 0.951, and a Boundary F1 of 0.943, significantly outperforming both the baseline U-Net and mainstream models such as DeepLabV3+. After segmentation preprocessing, the detection performance of YOLOv8 is also significantly improved. The precision, recall, mAP50, and mAP50–95 increase from 0.748, 0.796, 0.818, and 0.525 to 0.958, 0.971, 0.977, and 0.882, respectively. The results verify that the proposed two-stage recognition method can effectively suppress background interference, enhance the stability and generalization ability of the model in complex natural scenes, and provide an efficient and feasible technical approach for intelligent pest trap image recognition and pest situation monitoring.