We use a representation for the spanning tree where a parent function maps non-root vertices to vertices. Two spanning trees are defined to be adjacent if their function representations differ at exactly one vertex. Given a graph \(G\), we show that the graph $H$ with all spanning trees of \(G\) as vertices and any two vertices being adjacent iff their parent functions differ at exactly one vertex is connected.
The application of technologies such as big data, mobile Internet, artificial intelligence and so on has triggered a major change in the field of education and promoted the classroom reform in colleges and universities. Taking deep learning theory as the research perspective, this paper constructs a college English teaching model based on deep learning, and applies the model to actual teaching practice, with a view to promoting students’ English learning level and enhancing their intercultural communication ability. Among them, the K-means algorithm improved by the whale optimization algorithm is also used to cluster and stratify the English proficiency of students in a class to illustrate the specific application of deep learning in English teaching. The results classified the sample students into four categories, A, B, C and D. The English level of students in category A is the highest and the largest, accounting for 35.56%, and teachers can design differentiated teaching based on the results of student stratification. After carrying out the experiment of the teaching model, the practicing students’ English scores improved by 4.01%, and at the same time, they gained 18.87%~28.45% and 18.82%~39.01% of competence in the personal domain and the communicative domain, respectively, which confirms the effect of the constructed English teaching model on the enhancement of the students’ English learning level and cross-cultural communicative competence.
The wires and ground wires on transmission towers cannot be straight lines, but present different sizes of arcs, which directly affect the safety and transmission quality of the line. In response to this, a research proposes an online monitoring system for transmission towers based on computer video algorithms. The system collects environment and mechanism parameters of transmission lines by installing sensors on transmission towers, monitors them through computer video algorithms, and combines grey wolf algorithm and deep learning models to predict sag, thereby achieving crisis warning of the power grid around transmission towers. The outcomes denoted that during the field testing process, the warning accuracy of the system reaches over 98.57%, and the response time is only 0.5 seconds. The false negative rate and false positive rates are 2% and 0.5%, respectively. Based on the above content, it can be concluded that the proposed online monitoring system for transmission towers can effectively achieve line anomaly warning and maintain stable line operation.
A double Italian dominating function on a digraph \( D \) with vertex set \( V(D) \) is defined as a function \( f: V(D) \to \{0,1,2,3\} \) such that each vertex \( u \in V(D) \) with \( f(u) \in \{0,1\} \) has the property that \(\sum_{x \in N^{-}[u]} f(x) \geq 3,\) where \( N^{-}[u] \) is the closed in-neighborhood of \( u \). The weight of a double Italian dominating function is the sum \(\sum_{v \in V(D)} f(v),\) and the minimum weight of a double Italian dominating function \( f \) is the double Italian domination number, denoted by \( \gamma_{dI}(D) \). We initiate the study of the double Italian domination number for digraphs, and we present different sharp bounds on \( \gamma_{dI}(D) \). In addition, several relations between the double Italian domination number and other domination parameters such as double Roman domination number, Italian domination number, and domination number are established.
1970-2025 CP (Manitoba, Canada) unless otherwise stated.