A node in the \(n\)-dimensional hypercube \(Q_n\) is called an odd node (resp. an even node) if the sum of all digits of the node is odd (resp. even). Let \(F\subset E(Q_n)\) and let \(L\) be a linear forest in \(Q_n-F\) such that \(|E(L)|+|F|\leq n-2\) for \(n\geq 2\). Let \(x\) be an odd node and \(y\) an even node in \(Q_n\) such that none of the paths in \(L\) has \(x\) or \(y\) as internal node or both of them as end nodes. In this note, we prove that there is a Hamiltonian path between \(x\) and \(y\) passing through \(L\) in \(Q_n-F\). The upper bound \(n-2\) on \(|E(L)|+|F|\) is sharp.

As a product of the revolutionary war years, red culture, with its strong vitality, strong cohesion and extraordinary charm, with its incomparable positive energy, resists vulgar and flattering culture, promotes people to rebuild their faith, purify their minds, stimulate their motivation, and promote the process of cultural power. Yan’an, represented by red culture, is rich in resources. This is the holy land of Chinese revolution, the first batch of famous historical and cultural cities named by the State Council, and the three major education bases of patriotism, revolutionary tradition, and Yan’an spirit. The development and utilization of such resources have great political, cultural, educational and economic values. This research is based on the development of red culture, and uses the distributed machine learning system to realize in the system architecture of parameter server. In the distributed system set in this study, node downtime and network interruption are random. When the parameter server system adopts static scheduling, it leads to poor scalability and robustness. The experimental results show that under the intelligent simulation of machine learning system, the development of red culture resources meets the expected assumptions, and the accuracy of the model is relatively high.

In this paper, we introduce a class of restricted symmetric permutations, called half-exceeded symmetric permutations. We deduce the enumerative formula of the permutations of \(\{1,2,\ldots,2n\}\) and give it a refinement according to the distribution of the inverse pairs. As a consequence, we obtain new combinatorial interpretations of some well-known sequences, such as Stirling numbers of the second kind and ordered Bell numbers. Moreover, we introduce the ordered Stirling number of the second kind and establish a combinatorial proof of the recursive relation of the sequence.

The total labeling of a graph \(G=(V,E)\) is a bijection from the union of the vertex set and the edge set of \(G\) to the set \(\{1,2,\dots,|V(G)|+|E(G)|\}\). The edge-weight of an edge under a total labeling is the sum of the label of the edge and the labels of the end vertices of that edge. The vertex-weight of a vertex under a total labeling is the sum of the label of the vertex and the labels of all the edges incident with that vertex. A total labeling is called edge-magic or vertex-magic when all the edge-weights or all the vertex-weights are the same, respectively. When all the edge-weights or all the vertex-weights are different then a total labeling is called edge-antimagic or vertex-antimagic total, respectively.

In this paper we deal with the problem of finding a~total labeling of some classes of graphs that is simultaneously vertex-magic and edge-antimagic or simultaneously vertex-antimagic and edge-magic, respectively.
We show several results for stars, paths and cycles.

This study presents a pioneering federated multi-modal data classification model tailored for smart optical cable monitoring systems. By harnessing federated learning techniques, the model ensures data privacy while achieving performance on par with centralized models. Through comprehensive experiments spanning various modalities, including vision and auditory data, our approach showcases promising outcomes, as evidenced by accuracy and precision metrics. Furthermore, comparative analyses with centralized models highlight the superior data security and reduced network strain offered by federated learning. Moreover, we delineate the design and deployment of a smart optical cable monitoring system leveraging edge computing, accentuating the pivotal role of information technology in elevating operational efficiency within the cable monitoring domain. Through meticulous analysis and simulations, our proposed system adeptly monitors environmental variables, thereby bolstering safety and efficiency in smart optical cable monitoring applications.

The created public art sculpture is a material form that expresses the public spirit of the city. This paper proposes a deep model capable of enhancing the aesthetic quality of public art sculptures. The model uses the inverse mapping network of the augmented network to weaken the restriction of paired data sets required for training, and at the same time designs an effective loss function, that is, constructs the color and texture losses that are actively learned in training through generative adversarial rules, and enhances generative sculpture. The total variational loss of smoothness that improves the aesthetic quality of the sculpture to some extent. On this basis, this paper improves the design idea of content consistency loss. Experiments on the interaction between public art sculptures and the urban environment and the enhancement of aesthetics.

With the increasing scale of college enrollment and the increasing complexity of college teaching management, college finance department should innovate the traditional financial management mode while adapting to the reform of teaching management, and make use of the openness and real-time characteristics of Internet to improve the quality of college financial management and reduce the risk of college financial management. To this end, this paper designs a university financial system based on multi-scale deep learning. In the hardware design, the system adds multiple sensors and scans all the information in the financial database using a coordinator. In the software design, the weights that can connect the financial information of the same attribute are set by establishing a database form; according to the multilayer perceptual network topology, a full interconnection model based on multi-scale deep learning is designed to realize the system’s deep extraction of data. The experimental results show that the financial risk is based on the risk warning capability for university finance, and compared with the system under the traditional design, the university finance system designed in this time has the most categories of financial information parameters extracted.

This work suggests predicting student performance using a Gaussian process model classification in order to address the issue that the prediction approach is too complex and the data set involved is too huge in the process of predicting students’ performance. In order to prevent overfitting, a sample set consisting of the three typical test outcomes from 465 undergraduate College English students is divided into training and test sets. The cross-validation technique is used in this study. According to the findings, Gaussian process model classification can accurately predict 92% of the test set with a prediction model, and it can also forecast students’ final exam marks based on their typical quiz scores. Furthermore, it is discovered that the prediction accuracy increases with the sample set’s distance from the normal distribution; this prediction accuracy rises to 96% when test scores with less than 60 points are taken out of the analysis.

Fix integers \(k, b, q\) with \(k \ge 2\), \(b \ge 0\), \(q \ge 2\). Define the function \(p\) to be: \(p(x) = kx + b\). We call a set \(S\) of integers \emph{\((k, b, q)\)-linear-free} if \(x \in S\) implies \(p^i(x) \notin S\) for all \(i = 1, 2, \dots, q-1\), where \(p^i(x) = p(p^{i-1}(x))\) and \(p^0(x) = x\). Such a set \(S\) is maximal in \([n] := \{1, 2, \dots, n\}\) if \(S \cup \{t\}, \forall t \in [n] \setminus S\) is not \((k, b, q)\)-linear-free. Let \(M_{k, b, q}(n)\) be the set of all maximal \((k, b, q)\)-linear-free subsets of \([n]\), and define \(g_{k, b, q}(n) = \min_{S \in M_{k, b, q}(n)} |S|\) and \(f_{k, b, q}(n) = \max_{S \in M_{k, b, q}(n)} |S|\). In this paper, formulae for \(g_{k, b, q}(n)\) and \(f_{k, b, q}(n)\) are proposed. Also, it is proven that there is at least one maximal \((k, b, q)\)-linear-free subset of \([n]\) with exactly \(x\) elements for any integer \(x\) between \(g_{k, b, q}(n)\) and \(f_{k, b, q}(n)\), inclusively.

Nanoparticles have potential applications in a wide range of fields, including electronics, medicine and material research, because of their remarkable and exceptional attributes. Carbon nanocones are planar carbon networks with mostly hexagonal faces and a few non-hexagonal faces (mostly pentagons) in the core. Two types of nanocone configurations are possible: symmetric and asymmetric, depending on where the pentagons are positioned within the structure. In addition to being a good substitute for carbon nanotubes, carbon nanocones have made an identity for themselves in a number of fields, including biosensing, electrochemical sensing, biofuel cells, supercapacitors, gas storage devices, and biomedical applications. Their astonishing chemical and physical attributes have made them well-known and widely accepted in the fields of condensed matter physics, chemistry, material science, and nanotechnology. Mathematical and chemical breakthroughs were made possible by the concept of modeling a chemical structure as a chemical graph and quantitatively analyzing the related graph using molecular descriptors. Molecular descriptors are useful in many areas of chemistry, biology, computer science, and other sciences because they allow for the analysis of chemical structures without the need for experiments. In this work, the quotient graph approach is used to establish the distance based descriptors of symmetrically configured two-pentagonal and three-pentagonal carbon nanocones.