In this note, we establish six Gallai theorems involving twelve minority and majority parameters. Accordingly, the complexity problems corresponding to some of these parameters are obtained.

The promotion of industrial digital transformation is a crucial breakthrough in the evolution of economic structures and the physical layout of spaces. It has the potential to elevate the entire industrial chain to a high-end value chain, creating more profit opportunities and enhancing the influence of domestic industries in the international cycle. This study uses the cities in the Yangtze River Delta Economic Belt as a case study to explore the spatial effects of digital transformation on the healthy transformation of traditional industrial structures. It constructs relevant spatial coupling models and empirically verifies them by testing specific assumptions. The experimental results indicate that the model is significant at a level greater than 5%, making it suitable for selecting spatial measurement models. The mean square error of its network simulation output is 0.1333, confirming the expected hypothesis and demonstrating that digital transformation has a significant spatial driving effect on industrial upgrading.

A \(k\)-tree is a graph that can be formed by starting with \(K_{k+1}\) and iterating the operation of making a new vertex adjacent to all the vertices of a \(k\)-clique of the existing graph. A structural characterization of 3-trees with diameter at most 2 is proven. This implies a corollary for planar 3-trees which leads to a description of their degree sequences.

Electric shock accidents remain a major safety concern for distribution workers. Recent advancements in video AI applications allow for detecting when workers cross safety lines, but determining their height and the spatial distance between them and live equipment is still a challenge. This article proposes a pre-control system using LiDAR, an edge processing module, and a warning module to ensure safe operations in power distribution scenarios. The system scans the area in real time, uses deep learning to identify objects like distribution stations, human bodies, high-voltage equipment, and transmission lines in point clouds, and calculates the distance between operators and high-voltage equipment. When this distance approaches or exceeds safety limits, the warning module issues voice alerts. Experimental results show that this system significantly reduces false alarms compared to video-based methods, accurately measures distances, and provides timely warnings, making it a practical solution for enhancing worker safety in power distribution operations.

In this paper, we present a new combinatorial characterization of Hermitian cones in \(\mathrm{PG}(3,q^2)\).

Let \(K_n\), \(P_n\), and \(Y_n\) respectively denote a complete graph, a path, and a \(Y\)-tree on \(n\) vertices, and let \(K_{m,n}\) denote a complete bipartite graph with \(m\) and \(n\) vertices in its parts. Graph decomposition is the process of breaking down a graph into a collection of edge-disjoint subgraphs. A graph \(G\) has a \((H_1, H_2)\)-multi-decomposition if it can be decomposed into \(\alpha \geq 0\) copies of \(H_1\) and \(\beta \geq 0\) copies of \(H_2\), where \(H_1\) and \(H_2\) are subgraphs of \(G\). In this paper, we derive the necessary and sufficient conditions for the \((P_5, Y_5)\)-multi-decomposition of \(K_n\) and \(K_{m,n}\).

Criminal evidence serves as the foundation for criminal proceedings, with evidence used to ascertain the facts of cases being critical to achieving fairness and justice. This study explores the application of digital information technology in building a data resource base for criminal cases, formulating standard evidence guideline rules, and optimizing evidence verification procedures. A named entity recognition model based on the SVM-BiLSTM-CRF framework is proposed, coupled with an evidence relationship extraction model using the Transformer framework to improve evidence information extraction through sequential features and global feature capturing. Results show that the F1 value for entity recognition in criminal cases reaches 94.19%, and the evidence extraction model achieves an F1 value of 81.83% on the CAIL-A dataset. These results are utilized to construct evidence guidelines, helping case handlers increase case resolution rates to approximately 99%. The application of digital technology enhances evidence collection efficiency, accelerates case closures, and offers a pathway to improving judicial credibility.

In this paper we consider some new weighted and alternating weighted generalized Fibonomial sums and the corresponding \(q-\)forms. A generalized form of weight sequences which contains squares in subscripts is discussed for the first time in the literature. The main key to get success in sums is an ability to change one sum into another that is simpler in some way. Thus, in order to prove these sums by doing some manipulations and tricks, our approach is to use classical \(q-\)analysis, in particular a formula of Rothe, a version of Cauchy binomial theorem and Gauss identity.