An (unrooted) binary tree is a tree in which every internal vertex has degree \(3\). In this paper, we determine the minimum and maximum number of total dominating sets in binary trees of a given order. The corresponding extremal binary trees are characterized as well. The minimum is always attained by the binary caterpillar, while the binary trees that attain the maximum are only unique when the number of vertices is not divisible by~\(4\). Moreover, we obtain a lower bound on the number of total dominating sets for \(d\)-ary trees and characterize the extremal trees as well.

This paper proposes an optimized Backpropagation (BP) neural network for improving intelligent elderly care talent training. To address BP’s limitations, including noise sensitivity and slow convergence, we introduce Particle Swarm Optimization (PSO) to refine network weights and thresholds. The model integrates course quality, teacher effectiveness, platform support, and market demand, aiming to optimize elderly care service talent cultivation. Experimental results demonstrate a significant improvement in prediction accuracy, with average error reduced from 9.94% to 6.3%. This enhanced model offers a more efficient and accurate solution for aligning educational outcomes with industry needs.

Amnesty international is recognized as a key force in promoting social development, with higher education also facing the need for innovation. This paper explores new opportunities in educational theory and policy proposed in a recent initiative. The proposal emphasizes filtering ideology, political education, and public opinion to enhance the accuracy of ideological and political teaching. By incorporating personal suggestions through interviews, the model recommends learning materials tailored to student characteristics. System implementation and testing demonstrate its potential as a core tool for ideological education in colleges, supporting the integration of knowledge, politics, and technology to meet students’ educational needs.

Networks with smaller strong diameters generally have better fault tolerance because they enable closer connections between vertices, leading to shorter information paths. This allows the network to maintain communication and functionality more effectively during attacks or failures. In contrast, larger strong diameters mean vertices are connected over longer distances, increasing vulnerability to disruptions. Thus, the strong diameter is a key metric for assessing and optimizing network fault tolerance. This paper determines the optimal orientations for the Cartesian and strong products of even cycles, provides the minimum strong diameters and their bounds under specific conditions, and establishes a lower bound for the maximum strong diameter. A conjecture about the exact value of the maximum strong diameter is also proposed.

For a graph \(F\) and a positive integer \(t\), the edge-disjoint Ramsey number \(ER_t(F)\) is the minimum positive integer \(n\) such that every red-blue coloring of the edges of the complete graph \(K_n\) of order \(n\) results in \(t\) pairwise edge-disjoint monochromatic copies of a subgraph isomorphic to \(F\). Since \(ER_1(F)\) is in fact the Ramsey number of \(F\), this concept extends the standard concept of Ramsey number. We investigate the edge-disjoint Ramsey numbers \(ER_t(K_{1, n})\) of the stars \(K_{1, n}\) of size \(n\). Formulas are established for \(ER_t(K_{1, n})\) for all positive integers \(n\) and \(t = 2, 3, 4\) and bounds are presented for \(ER_t(K_{1, n})\) for all positive integers \(n\) and \(t \ge 5\). Furthermore, exact values of \(ER_t(K_{1, n})\) are determined for \(n = 3, 4\) and several integers \(t \ge 5\).

The development of artificial intelligence enables computers to not only simulate human artistic creations, but also synthesize fine art works with deeper meanings based on natural images. This study digitally parses the fusion of fine art and philosophy visual expressions, and develops a visual expression system based on the fusion of fine art and philosophy by utilizing a variety of key big data algorithms for visual expressions such as adversarial networks. Research on pattern recognition of this system in art creation is carried out through model training, recommendation performance evaluation, pattern recognition strategy application and regression analysis. The model in this paper works best when the number of nearest neighbors k=15, and the recommendation model in this paper can provide a personalized list of artwork recommendations for different people. The recognition of the system in this paper in the five dimensions of “spiritual level”, “value level”, “philosophical level”, “aesthetic level” and “technical level” is distributed between 4.24\(\mathrm{\sim}\)4.55. The results of regression analysis indicated that the system in this paper can improve the artistic creation as well as pattern recognition.

With development of Internet of Things, big data and artificial intelligence, cell phone signaling data, point-of-interest data and machine learning methods have been widely used in research of various fields of transportation. The use of big data processing techniques and machine learning methods to mine intercity travel data collected by various types of traffic detectors provides a new way of thinking to study travel mode selection behavior. In this paper, we pre-processed cell phone signaling data, geospatial data and interest point data around three aspects: personal attributes, travel attributes and travel mode attributes, and designed intercity travel target group extraction, travel chain extraction, travel mode extraction and travel purpose extraction algorithms, which provide basis for travel feature analysis and travel mode choice behavior prediction modeling.

Conventional techniques to electric power network (EPN) design and management are insufficient to handle extreme weather events like hurricanes due to the growing complexity and fragility of power systems. As a sophisticated simulation and optimization tool, digital twin (DT) technology may offer real-time power infrastructure monitoring and prediction. This study aims to investigate the possible application of digital twin technology in enhancing power system resilience and streamlining the design process, as well as to use it for the 3D design of the full substation engineering infrastructure process. A digital twin-based EPN model that incorporates all of the main components of the power system—power plants, substations, transmission and distribution networks, and customers—is proposed in this paper. Every component of the power system undergoes vulnerability analysis, and the chance of the system failing is calculated using a Bayesian network (BN) model and a parametric vulnerability function. According to modeling projections, Hurricane Ike will cause the majority of consumers’ power supplies to be interrupted. The model predicts that power consumption for residential, commercial, and industrial buildings will be 96.4%, 96.0%, and 94.2%, respectively, depending on the kind of building.

Radar ranging and speed measurement are common applications in daily life, with performance largely dependent on the radar signal processor. However, existing civilian radar signal processors struggle with weak signal reception and low analysis efficiency. This study designs a high-speed radar signal processor based on FPGA architecture, incorporating a fusion processing algorithm to integrate different radar signal bands, enhancing processing efficiency and accuracy. The design includes data feature analysis, storage, and fusion modules. Tests showed that the processor achieved real-time performance with a processing time under 1ms, a ranging error below 1m, and speed measurement accuracy within 5m/s, meeting practical requirements.

Intriguing symmetries are uncovered regarding all magic squares of orders 3, 4, and 5, with 1, 880, and 275,305,224 distinct configurations, respectively. In analogy with the travelling salesman problem, the distributions of the total topological distances of the paths travelled by passing through all the vertices (matrix elements) only once and spanning all elements of the matrix are analyzed. Symmetries are found to characterize the distributions of the total topological distances in these instances. These results raise open questions about the symmetries found in higher-order magic squares and the formulation of their minimum and maximum total path lengths.