Stanley considered Dyck paths where each maximal run of down-steps to the \(x\)-axis has odd length; they are also enumerated by (shifted) Catalan numbers. Prefixes of these combinatorial objects are enumerated using the kernel method. A more challenging version of skew Dyck paths combined with Stanley’s restriction is also considered.

For \(r=1,2,…, 6\), we obtain generating functions \(F^{(r)}_{k}(y)\) for words over the alphabet \([k]\), where \(y\) tracks the number of parts and \([y^n]\) is the total number of distinct adjacent \(r\)-tuples in words with \(n\) parts. In order to develop these generating functions for \(1\le r\le 3\), we make use of intuitive decompositions but for larger values of \(r\), we switch to the cluster analysis method for decorated texts that was introduced by Bassino et al. Finally, we account for the coefficients of these generating functions in terms of Stirling set numbers. This is done by putting forward the full triangle of coefficients for all the sub-cases where \(r=5\) and 6. This latter is shown to depend on both periodicity and number of letters used in the \(r\)-tuples.

We consider the following variant of the round-robin scheduling problem: \(2n\) people play a number of rounds in which two opposing teams of \(n\) players are reassembled in each round. Each two players should play at least once in the same team, and each two players should play at least once in opposing teams. We provide an explicit formula for calculating the minimal numbers of rounds needed to satisfy both conditions. Moreover, we also show how one can construct the corresponding playing schedules.

Two binary structures \(\mathfrak{R}\) and \(\mathfrak{R’}\) on the same vertex set \(V\) are \((\leq k)\)-hypomorphic for a positive integer \(k\) if, for every set \(K\) of at most \(k\) vertices, the two binary structures induced by \(\mathfrak{R}\) and \(\mathfrak{R’}\) on \(K\) are isomorphic. A binary structure \(\mathfrak{R}\) is \((\leq k)\)-reconstructible if every binary
structure \(\mathfrak{R’}\) that is \((\leq k)\)-hypomorphic to \(\mathfrak{R}\) is isomorphic to \(\mathfrak{R}\). In this paper, we describe the pairs of \((\leq 3)\)-hypomorphic posets and the pairs of \((\leq 3)\)-hypomorphic bichains. As a consequence, we characterize the \((\leq 3)\)-reconstructible posets and the \((\leq 3)\)-reconstructible bichains. This answers a question suggested by Y. Boudabbous and C. Delhommé during a personal communication.

A tremendous amount of drug experiments revealed that there exists a strong inherent relation between the molecular structures of drugs and their biomedical and pharmacology characteristics. Due to the effectiveness for pharmaceutical and medical scientists of their ability to grasp the biological and chemical characteristics of new drugs, analysis of the bond incident degree (BID) indices is significant of testing the chemical and pharmacological characteristics of drug molecular structures that can make up the defects of chemical and medicine experiments and can provide the theoretical basis for the manufacturing of drugs in pharmaceutical engineering. Such tricks are widely welcomed in developing areas where enough money is lacked to afford sufficient equipment, relevant chemical reagents, and human resources which are required to investigate the performance and the side effects of existing new drugs. This work is devoted to establishing a general expression for calculating the bond incident degree (BID) indices of the line graphs of various well-known chemical structures in drugs, based on the drug molecular structure analysis and edge dividing technique, which is quite common in drug molecular graphs.

In this paper, we introduce a graph structure, called component intersection graph, on a finite dimensional vector space \(\mathbb{V}\). The connectivity, diameter, maximal independent sets, clique number, chromatic number of component intersection graph have been studied.

A linear system is a pair \((P,\mathcal{L})\) where \(\mathcal{L}\) is a finite family of subsets on a finite ground set \(P\) such that any two subsets of \(\mathcal{L}\) share at most one element. Furthermore, if for every two subsets of \(\mathcal{L}\) share exactly one element, the linear system is called intersecting. A linear system \((P,\mathcal{L})\) has rank \(r\) if the maximum size of any element of \(\mathcal{L}\) is \(r\). By \(\gamma(P,\mathcal{L})\) and \(\nu_2(P,\mathcal{L})\) we denote the size of the minimum dominating set and the maximum 2-packing of a linear system \((P,\mathcal{L})\), respectively. It is known that any intersecting linear system \((P,\mathcal{L})\) of rank \(r\) is such that \(\gamma(P,\mathcal{L})\leq r-1\). Li et al. in [S. Li, L. Kang, E. Shan and Y. Dong, The finite projective plane and the 5-Uniform linear intersecting hypergraphs with domination number four, Graphs and 34 Combinatorics (2018) , no.~5, 931–945.] proved that every intersecting linear system of rank 5 satisfying \(\gamma(P,\mathcal{L})=4\) can be constructed from a 4-uniform intersecting linear subsystem \((P^\prime,\mathcal{L}^\prime)\) of the projective plane of order 3 satisfying \(\tau(P^\prime,\mathcal{L}^\prime)=\nu_2(P^\prime,\mathcal{L}^\prime)=4\), where \(\tau(P^\prime,\mathcal{L}^\prime)\) is the transversal number of \((P^\prime,\mathcal{L}^\prime)\). In this paper, we give an alternative proof of this result given by Li et al., giving a complete characterization of these 4-uniform intersecting linear subsystems. Moreover, we prove a general case, that is, we prove if $q$ is an odd prime power and \((P,\mathcal{L})\) is an intersecting linear system of rank \((q+2)\) satisfying \(\gamma(P,\mathcal{L})=q+1\), then this linear system can be constructed from a spanning \((q+1)\)-uniform intersecting linear subsystem \((P^\prime,\mathcal{L}^\prime)\) of the projective plane of order \(q\) satisfying \(\tau(P^\prime,\mathcal{L}^\prime)=\nu_2(P^\prime,\mathcal{L}^\prime)=q+1\).

We classify all near hexagons of order \((3,t)\) that contain a big quad. We show that, up to isomorphism, there are ten such near hexagons.

Let \(G=(V,E)\) be a simple graph. A vertex \(v\in V(G)\) ve-dominates every edge \(uv\) incident to \(v\), as well as every edge adjacent to these incident edges. A set
\(D\subseteq V(G)\) is a vertex-edge dominating set if every edge of \(E(G)\) is ve-dominated by a vertex of \(D.\) The MINIMUM VERTEX-EDGE DOMINATION problem is to find a vertex-edge dominating set of minimum cardinality. A linear time algorithm to find the minimum vertex-edge dominating set for proper interval graphs is proposed. The vertex-edge domination problem is proved to be APX-complete for bounded-free graphs and NP-Complete for Chordal bipartite and Undirected Path graphs.

In this paper, we investigate the \((d,1)\)-total labelling of generalized Petersen graphs \(P(n,k)\) for \(d\geq 3\). We find that the \((d,1)\)-total number of \(P(n,k)\) with \(d\geq 3\) is \(d+3\) for even \(n\) and odd \(k\) or even \(n\) and \(k=\frac{n}{2}\), and \(d+4\) for all other cases.

By employing Kummer and Thomae transformations, we examine four classes of nonterminating \(_3F_2\)(1)-series with five integer parameters. Several new summation formulae are established in closed form.

Let \(c\) be a proper \(k\)-coloring of a connected graph \(G\) and \(\pi=\{S_{1},S_{2},\ldots,S_{k}\}\) be an ordered partition of the vertex set \(V(G)\) into the resulting color classes, where \(S_{i}\) is the set of all vertices that receive the color \(i\). For a vertex \(v\) of \(G\), the color code \(c_{\pi}(v)\) of \(v\) with respect to \(\pi\) is the ordered \(k\)-tuple \(c_{\pi}(v)=(d(v,S_{1}),d(v,S_{2}),\ldots,d(v,S_{k}))\), where \(d(v,S_{i})=min\{d(v,u):\textit{ } u\in S_{i}\}\) for \(1\leqslant i \leqslant k\). If all distinct vertices of \(G\) have different color codes, then \(c\) is called a locating coloring of \(G\). The locating chromatic number is the minimum number of colors needed in a locating coloring. In this paper, we determine the locating-chromatic number for the middle graphs of Path, Cycle, Wheel, Star, Gear and Helm graphs.

There has been significant research dedicated towards computing the crossing numbers of families of graphs resulting from the Cartesian products of small graphs with arbitrarily large paths, cycles and stars. For graphs with four or fewer vertices, these have all been computed, but there are still various gaps for graphs with five or more vertices. We contribute to this field by determining the crossing numbers for fifteen such families.

A graph \(G\) with vertex set \(V = V(G)\) and edge set \(E = E(G)\) is harmonious if there exists a harmonious labeling of \(G\); which is an injective function \(f:V(G) \rightarrow \mathbf{Z}_m\) provided that whenever \(e_1, e_2 \in E\) are distinct with endpoints \(u_1,v_1\) and \(u_2,v_2\), respectively, then \(f(u_1) + f(v_1) \not\equiv f(u_2) + f(v_2) (\hbox{mod } m )\). Using basic group theory, we prove in a different manner an already established result that a disjoint union of an odd cycle and a path is harmonious provided their lengths satisfy certain conditions. We apply the same basic idea to establish that, under the same conditions, a disjoint union of an odd cycle with a certain starlike tree is harmonious (where a starlike tree consists of a central vertex that is adjacent to an endpoint of a certain number of fixed length paths). Finally, we extend the latter result to include specifying that the central vertex in the tree be adjacent to different vertices in each of the \(t\)-many \(s\)-paths.

Under the background of my country’s new rural construction and the implementation of the rural revitalization strategy, the rural economy has ushered in an unprecedented opportunity for development. Agricultural economic management plays a catalytic role in providing direction guidance for rural economic development, promoting sustainable rural economic development, and providing a good environment for rural economic development. However, there are some drawbacks in agricultural economic management, which are mainly reflected in the imperfect agricultural economic management system and the lag in information infrastructure construction. In view of this, the author puts forward corresponding the advanced VAR model from the aspects of improving the agricultural economic management system, improving the application level of information technology, and improving the quality of the agricultural economic management team. Research shows that: through effective management of agricultural economy, the obstacles to agricultural economic development can be reduced from the source, thereby promoting the healthy and sustainable development of rural economy.

This paper mainly focuses on the algorithms related to local path planning and path tracking control of unmanned vehicles in the process of obstacle avoidance. By introducing the temporal dimension as a reference, the perceptual results are projected onto the 3D spatio-temporal navigation map by combining the multi-target behavior prediction and other means; by increasing the temporal dimension, the static obstacles and dynamic obstacles are unified into the same parameter space. Under this parameter space, the front-end A* path search initializes the unified B spline curve control points, designs the trajectory cost function and performs nonlinear optimization to generate a spatio-temporal trajectory that satisfies the safety collision-free and vehicle motion constraints (speed and acceleration limits), thus transforming the decision and planning problem under the two-dimensional fence dynamic physical space into a static scene decision and planning problem under the three-dimensional spatio-temporal space. Through simulation verification, the whole process of the proposed trajectory planning method takes 51.27ms on average, which meets the driving requirements of driverless cars. In addition, by adjusting the search conditions of the A* algorithm, its overall planning efficiency is improved by 27.86% compared with the search speed of the traditional algorithm. The actual feeling and data results from the real vehicle experiments show its good tracking effect, which verifies the effectiveness and practicality of the algorithm proposed in this paper.

Based on the visual servo technology, this paper focuses on the visual tracking algorithm of moving objects and the dynamic grasping control method of robots, and realizes the automatic loading and unloading of moving workpieces to improve production efficiency. Firstly, aiming at the difficulties in the selection of high-dimensional features extracted by visual servo, this paper proposes a training method of generation countermeasure network based on heuristic algorithm by using the efficient search ability of heuristic algorithm. Secondly, we use image processing technology to realize real-time recognition and location of workpieces under complex background. According to the positioning results, an adaptive dual rate unscented Kalman filter visual tracking algorithm is proposed to solve the problem of delay and multi sampling rate in visual servo, and realize visual tracking of moving objects. The experimental results show that the proposed visual tracking algorithm has better stability and real-time performance.

For any positive integer \(h\), a graph \(G=(V,E)\) is said to be \(h\)-magic if there exists a labeling \(l:E(G)\to \mathbb{Z}_h -\{0\}\) such that the induced vertex set labeling \(\ l^+ : V(G) \to \mathbb{Z}_h \) defined by
\[
l^+ (v)=\sum_{uv \in E(G)} \ l(uv)
\]
is a constant map. The integer-magic spectrum of a graph \(G\), denoted by \(IM(G)\), is the set of all \(h \in \mathbb{N}\) for which \(G\) is \(h\)-magic. So far, only the integer-magic spectra of trees of diameter at most five have been determined. In this paper, we determine the integer-magic spectra of trees of diameter six and higher.

A total Roman \(\{2\}\)-dominating function on a graph \(G = (V,E)\) is a function \(f:V\rightarrow\{0,1,2\}\) with the properties that (i) for every vertex \({v}\in V\) with \(f({v})=0\), \(f(N({v}))\ge2\) and (ii) the set of vertices with \(f({v})>0\) induces a subgraph with no isolated vertices. The weight of a total Roman \(\{2\}\)-dominating function is the value \(f(V)=\sum_{{v}\in V}f({v})\), and the minimum weight of a total Roman \(\{2\}\)-dominating function is called the total Roman \(\{2\}\)-domination number and denoted by \(\gamma_{tR2}(G)\). In this paper, we prove that for every graph \(G\) of order \(n\) with minimum degree at least two, \(\gamma_{tR2}({G})\leq \frac{5n}{6}\).

The penetration of virtual classroom teaching into German teaching is the presentation of teaching innovation in the information age. In this work, we explore the necessity of applying virtual classroom teaching in German classrooms and the effective strategies of German teaching innovation in virtual classrooms, to provide some suggestions for the reform of German teaching. First, the GPS trajectories are transformed into a sequence of hotspot regions using the spatiotemporal properties of GPS points. Then, a sequential pattern mining algorithm of asynchronous cycles with multiple minimum supports based on pattern growth is adopted, and the sequential patterns of asynchronous cycles are deeply recursively mined according to the multiple minimum supports. Experiments show that the proposed IoT-assisted teaching scheme can effectively integrate equipment resources, mine spatiotemporal information, and help students and teachers establish a new educational method of integrating space and land. Compared with the baseline, it can fully exploit the characteristics of German.

The limitations of existing procedures make it difficult to locate and identify old subterranean culverts in urban infrastructure management. In order to effectively manage urban infrastructure, subterranean pipe culverts must be accurately located and detected. In this research, we investigate the method of computing the shortest distance from the point to the ellipse and propose a pipeline collision detection method based on the projection of the direction of the common perpendicular. In the positioning accuracy test, we simulate the detection of straight and curved paths and obtain satisfactory results; the experimental results show that the detection errors are within acceptable limits for different azimuth and bending angles; in the correctness test, we compared with AutoCAD and ArcGIS, and found that the algorithm in this paper shows superiority in collision detection, especially when dealing with complex spatial relationships and large amounts of data, with evident efficiency advantages. Through theoretical analysis and experimental verification, we demonstrate the effectiveness and reliability of the method.

We identified, via a computer search, 143 excluded minors of the spindle surface, the space formed by the identification of two points of the sphere. Per our search, any additional excluded minors must have at least 12 vertices and 28 edges. We also identified 847 topological obstructions for the spindle surface. We conjecture that our lists of excluded minors and topological obstructions are complete.

This paper analyzes the prediction model of enterprise human resource demand based on Internet of Things (IoT) technology and data mining technology. It also analyzes the impact of the company’s growth scale and other key factors on the demand for human resources, tries to establish a coupling factor model of enterprise development and economic benefits, and then analyzes and forecasts the enterprise’s personnel structure and quality structure. The experimental results show that the optimized human resource demand forecasting model integrates the advantages of the grey system model in data processing, can mine the inherent laws of unorganized data information, and provides a certain convenience for forecasting. Through the linear mapping and processing of sample data, the input and output reflect a kind of correlation, thus changing the fault tolerance of information, making the prediction in the calculation process more accurate, and its comprehensive accuracy can exceed 92.5%.

Low efficiency and poor accuracy are caused by missing data in traditional 3D reconstruction methods. This study suggests a new 3D point cloud recognition technique for substation equipment based on 3D laser scanning point clouds, which combines the k-nearest neighbour (KNN) classification algorithm and particle swarm optimisation (PSO) algorithm, to address these issues. The particle swarm optimisation algorithm optimises the coefficient weights of each subspace feature. The k-nearest neighbour classification algorithm is then used to finish the classification. To confirm the superiority and accuracy of the suggested approach, the impact of the point cloud subspace’s size and loss rate on the recognition effect is examined experimentally and contrasted with the enhanced iterative nearest point algorithm. With an average recognition time of 0.19 seconds and a recognition accuracy of over 95\%, the experimental results demonstrate the method’s good performance in terms of efficiency and accuracy, opening up a wide range of potential applications.

Users may receive personalised information services and decision support from personalised recommendations. In this paper, a hybrid algorithm-based personalised recommendation approach for learning English is proposed. The user model is created by merging user interest tags, and the Person Rank algorithm is then recommended based on user information. Second, the question-and-answer model is created once the question-and-answer data has been labelled, and the Problem Rank algorithm is suggested in accordance with the question-and-answer data. Then, the approach of tag-based recommendation, comparable user recommendation, and multi-dimensional sliding window are used to construct the recommendation algorithm model. The experimental findings demonstrate that, following the model’s training with the gradient descent technique, the recommendation accuracy is steady at around 0.78, the suggested information can accommodate users who are learning English, and the personalised recommendation effect is enhanced.

In this paper, we introduce the edge version of doubly resolving set of a graph which is based on the edge distances of the graph. As a main result, we computed the minimum cardinality \(\psi_E\) of edge version of doubly resolving sets of family of \(n\)-sunlet graph \(S_n\) and prism graph \(Y_n\).

Consider the simple connected graph G with vertex set V(G) and edge set E(G). A graph \(G\) can be resolved by \(R\) if each vertex’s representation of distances to the other vertices in \(R\) uniquely identifies it. The minimum cardinality of the set \(R\) is the metric dimension of \(G\). The length of the shortest path between any two vertices, x, y in V(G), is signified by the distance symbol d(x, y). An ordered k-tuple \(r(x/R)=(d(x,z_1),d(x,\ z_2),…,d(x,z_k))\) represents representation of \(x\) with respect to \(R\) for an ordered subset \(R={\{z}_1,z_2,z_3…,z_k\}\) of vertices and vertex \(x\) in a connected graph. Metric dimension is used in a wide range of contexts where connection, distance, and connectedness are essential factors. It facilitates understanding the structure and dynamics of complex networks and problems relating to robotics network design, navigation, optimization, and facility location. Robots can optimize their localization and navigation methods using a small number of reference sites due to the pertinent idea of metric dimension. As a result, many robotic applications, such as collaborative robotics, autonomous navigation, and environment mapping, are more accurate, efficient, and resilient. A claw-free cubic graph (CCG) is one in which no induced subgraph is a claw. CCG proves helpful in various fields, including optimization, network design, and algorithm development. They offer intriguing structural and algorithmic properties. Developing algorithms and results for claw-free graphs frequently has applications in solving of challenging real-world situations. The metric dimension of a couple of claw-free cubic graphs (CCG), a string of diamonds (SOD), and a ring of diamonds (ROD) will be determined in this work.

Using blockchain technology to handle the entire chain of digital copyrights in digital libraries not only helps to improve the economy, validity, and fairness of the libraries’ digital resource offerings, but it also increases the revenue of digital copyright subjects in a sustainable manner. In this work, a decentralized, secure, and traceable digital copyright transaction system is designed and implemented using blockchain technology. The system serves creators, administrators, and subscribers through its user layer, business model layer, and Fabric network layer. To guarantee the accuracy and integrity of transaction data, smart contracts are used for the registration of digital works, transaction supervision, and smart contract execution. Fabric Composer is used in the development of the system and offers good scalability. The system still has issues with privacy protection, increasing performance, and complying with laws and regulations. It is anticipated that the digital copyright transaction system will advance in the area of digital copyright protection as blockchain technology develops.

As computer and mechanical automation technologies advance, machine vision-based non-destructive testing technology finds use in a multitude of domains. Non-destructive testing technologies can be used on apple sorting equipment to decrease apple damage while simultaneously increasing sorting efficiency. As a result, the apple sorting machine’s image identification system now incorporates machine vision technology. The automatic classification of apple grades is accomplished by gathering, processing, extracting, and computing the contour features of apple photographs using preset sorting levels. The automatic control system then sorts apples of different grades to designated locations, thus achieving the automation of apple sorting. Tests were run on the sorting machine’s image recognition system to confirm the solution’s viability. The outcomes demonstrate that the sorting machine can effectively classify fruit automatically based on their perimeter, which is important for fruit sorting automation.

The gearbox gearbox transmission system, which is the foundation of a new energy vehicle, is responsible for the crucial duty of power transmission. In reality, the reducer gearbox system is the primary source of noise inside cars because of the design of the system, mistakes made during manufacturing and assembly, and gear engagement impulses. The research target is the second-stage retarder gearbox system of a new energy vehicle. A three-dimensional model of the retarder gearbox system is created using the Romax software.Static and dynamic analyses were carried out in Romax software based on the five typical conditions of start, acceleration, equal speed, deceleration, and stop in order to derive performance data such as maximum contact and bending stresses of the gears, single-position length load distribution, gearbox error, etc. In the NVH analysis, the system’s vibration acceleration was ascertained using the findings of the gearbox error analysis. In order to provide comparative data for vibration and noise reduction of gear modification, the comparative study analyses the data output results under various working conditions and analyses the relationship between gear engagement force and gear vibration.

Let \(G\) be a connected graph. A pebbling move is defined as taking two pebbles from one vertex and the placing one pebble to an adjacent vertex and throwing away the another pebble. A dominating set \(D\) of a graph \(G=(V,E)\) is a non-split dominating set if the induced graph \(\) is connected. The Non-split Domination Cover(NDC) pebbling number, \(\psi_{ns}(G)\), of a graph $G$ is the minimum of pebbles that must be placed on \(V(G)\) such that after a sequence of pebbling moves, the set of vertices with a pebble forms a non-split dominating set of \(G\), regardless of the initial configuration of pebbles. We discuss some basic results and determine \(\psi_{ns}\) for some families of standard graphs.

Graph theory is playing vital role in almost every field of our routine life. You make a conference call with your friends by using vertices (yourself and your friends) and edges (network connection). You construct a printed grid floor with different faces in your home by the help of graph theory. Authors in this study are using labelling of graphs and applying it in choosing best friends around you. The helping graphs in this article will be plane graphs which will be labelling under \(\Bbbk-\)labelling \(\mathrm{M}\) of kind \((\lambda,\mu,\nu)\). This study can be applied in many fields of everyday life.

A complex Hadamard matrix is a matrix \(H_n \in {\{\omega^i | 1\leq i \leq m \}}^{n\times n}\) of order \(n\), where \(\omega\) is a primitive \(m^{th}\) root of unity, that satisfies \(H_n{H}^{*}_n=n{I_{n}}\), where \(H_n^{*}\) denotes the complex conjugate transpose of \(H_n\). We show that the Scarpis technique for constructing classic Hadamard matrices generalizes to Butson-type complex Hadamard matrices.

With the rapid development of the country’s economy, politics, and culture, China has swiftly ascended to the ranks of global powers. Its participation in international organizations, including the WTO, has significantly bolstered its global standing and diplomatic ties, making it an indispensable player in international politics. Meanwhile, domestically, China has implemented numerous initiatives aimed at improving the lives of its citizens, such as anti-corruption campaigns, efforts to uphold integrity, crackdowns on criminal organizations, and poverty alleviation programs. As a result, the well-being of the populace has seen a steady increase. Furthermore, China has embarked on a new era of education characterized by its unique attributes, with civic education platforms experiencing comprehensive development. This paper examines these developments through text and knowledge mapping, assessing the efficacy of this approach within the framework of course ideology and politics.

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.

Let \(\varepsilon_{0}\), \(\varepsilon_{1}\) be two linear homogenous equations, each with at least three variables and coefficients not all the same sign. Define the \(2\)-color off-diagonal Rado number \(R_2(\varepsilon_{0},\varepsilon_{1})\) to be the smallest \(N\) such that for any 2-coloring of \([1,N]\), it must admit a monochromatic solution to \(\varepsilon_{0}\) of the first color or a monochromatic solution to \(\varepsilon_{1}\) of the second color. Mayers and Robertson gave the exact \(2\)-color off-diagonal Rado numbers \(R_2(x+qy=z,x+sy=z). \) Xia and Yao established the formulas for \(R_2(3x+3y=z,3x+qy=z) \) and \(R_2(2x+3y=z,2x+2qy=z) \). In this paper, we determine the exact numbers \(R_2(2x+qy=2z,2x+sy=2z)\), where \(q, s\) are odd integers with \(q>s\geq1\).

Let \(X\) be bipartite mixed graph and for a unit complex number \(\alpha\), \(H_\alpha\) be its \(\alpha\)-hermitian adjacency matrix. If \(X\) has a unique perfect matching, then \(H_\alpha\) has a hermitian inverse \(H_\alpha^{-1}\). In this paper we give a full description of the entries of \(H_\alpha^{-1}\) in terms of the paths between the vertices. Furthermore, for \(\alpha\) equals the primitive third root of unity \(\gamma\) and for a unicyclic bipartite graph \(X\) with unique perfect matching, we characterize when \(H_\gamma^{-1}\) is \(\pm 1\) diagonally similar to \(\gamma\)-hermitian adjacency matrix of a mixed graph. Through our work, we have provided a new construction for the \(\pm 1\) diagonal matrix.

Student-centeredness is a teaching theory proposed by British and American scholars in linguistics, psycholinguistics, applied linguistics, and second language acquisition theory. “The student-centered approach is different from the traditional teacher-centered approach, but it is implemented in a teacher-led environment. In this study, word2vec, paragraph2vec, pos2vec and LDA (latent dirichlet allocation) are combined to form a semantic representation vector for college business English translation. The key point of the college business English translation reform is to update the concept and theoretical understanding, so as to improve the teachers’ business English teaching theory and teaching practice, and to do a good job of college business English translation reform. Finally, it is shown that the proposed intelligent evaluation framework is more accurate than the traditional method in terms of automatic grading and rubric generation for college business English translation.

We study real algebras admitting reflections which commute. In dimension two, we show that two commuting reflections coincide and we specify the two and four-dimensional real algebras cases. We characterize real algebras of division of two-dimensional to third power-associative having a reflection. Finally We give a characterization in four-dimensional, the unitary real algebras of division at third power-associative having two reflections that commute. In eight-dimensional, we give an example of algebra so the group of automorphisms contains a subgroup isomorphic to \(\mathbb{Z}_2\times\mathbb{Z}_2\).

Let \(G(V,E)\) be a simple graph of order \(n\) with vertex set \(V\) and edge set \(E\). Let \((u, v)\) denote an unordered vertex pair of distinct vertices of \(G\). For a vertex \(u \in G,\) let \(N(u)\) be the set of all vertices of \(G\) which are adjacent to \(u\) in \(G.\) Then for \(0\leq i \leq n-1\), the \(i\)-equi neighbor set of \(G\) is defined as: \(N_{e}(G,i)=\{(u,v):u, v\in V, u\neq v\) and \(|N(u)|=|N(v)|=i\}.\) The equi-neighbor polynomial \(N_{e}[G;x]\) of \(G\) is defined as \(N_{e}[G;x]=\sum_{i=0}^{(n-1)} |N_{e}(G,i)| x^{i}.\) In this paper we discuss the equi-neighbor polynomial of graphs obtained by some binary graph operations.

The presence of unknown synchronization characteristics, unclear instability mechanism, and various fault mode evolution laws, lacking corresponding theoretical support and analysis methods and instability criteria, are defined with clear physical concepts. It is still impossible to systematically understand the transient synchronization mechanism of the wind power grid-connected system from the perspective of the whole fault stage. Therefore, this study uniformly reveals its temporary synchronous stability problem and proposes a large/small disturbance adaptive synchronous stability control method, which improves the dynamic characteristics of the wind turbine through the control of the inverter itself to improve the system stability—using different scenarios, such as single doubly-fed wind turbines. The experimental results show that the small disturbance on the AC side significantly impacts the system characteristics, followed by a bit of annoyance on the DC side. The DC side fault will cause a change in system frequency characteristics, especially at the receiving end. However, compared with the Voltage Source Converters-High Voltage Direct Current (VSC-HVDC)system, Modular Multilevel Converters-High Voltage Direct Current (MMC-HVDC) systems operate at a much higher frequency and produce less low-frequency harmonics. This makes them less likely to induce subsynchronous oscillations in the system.

With China’s educational reform, college physical education teaching mode has also made some innovations. Sports club is a new modern education model developed on the basis of traditional physical education courses. It provides students with more choices and is convenient for autonomous learning, thus forming a student-centered engineering education model. With the background of sports reform, this paper investigates and analyzes the reform of sports club system in universities, puts forward specific implementation means to stimulate the development process of sports reform in universities in China, puts forward data analysis schemes, and analyzes and guides the reform of sports club system. The specific research results show that our reform plan has been recognized by 92% of students and 94% of teachers.

Paths that consist of up-steps of one unit and down-steps of \(k\) units, being bounded below by a horizontal line \(-t\), behave like \(t+1\) ordered tuples of \(k\)-Dyck paths, provided that \(t\le k\). We describe the general case, allowing \(t\) also to be larger. Arguments are bijective and/or analytic.

The lens array is a multi-functional optical element, which can modulate the incident light such as diffusion, beam shaping, light splitting, and optical focusing, thereby achieving large viewing angle, low aberration, small distortion, high temporal resolution, and infinite depth of field. Meanwhile, it has important application potential in the form, intelligence and integration of op-to electronic devices and optical systems. In this paper, the optical principle and development history of lens arrays are introduced, and the lens array fabrication technologies such as ink jet printing, laser direct writing, screen printing, photo lithography, photo polymerization, hot melt reflow and chemical vapor deposition are reviewed. The application progress of lens arrays in imaging sensing, illumination light source, display and photovoltaic fields is presented. And this paper prospected the development direction of lens arrays, and discussed the development trends and future challenges of new directions such as curved lenses, superimposed compound eye systems, and the combination of lenses and new op-to electronic materials.

Innovation and entrepreneurship and education have become an important topic in China’s higher education. Based on pedagogy theory, this paper divides innovation and entrepreneurship education in universities into three levels: ideological education, innovative education and entrepreneurial education. Innovation is the content of higher education, and it is also the ability that contemporary college students must have. Only with education can there be innovation, only with innovation can there be entrepreneurship, and only with entrepreneurship can there be innovation. This is of great significance to the development of multi-level education, universal education, innovation and entrepreneurship education, and the improvement of education, teaching and child-rearing levels. In order to promote the optimization practice of college students’ innovation and entrepreneurship education, this paper designs a software system which is convenient for college students’ project application, project implementation, data verification and progress report. At the same time, it can help people review and select team members, thus greatly improving management efficiency.

People’s aesthetic requirements for landscape environment are improving, and we can also see very beautiful as well as characteristic urban parks, street side green areas and scenic spots with certain aesthetic value around us, and we can find that people’s demand for the living environment they live in regarding beauty is also strengthening. The synergistic development of edge computing and cloud computing is an important development trend in the future, and integrating them into landscape design is an inevitable choice and requirement for developing gardens and building a beautiful China. Based on this, this study first proposes a methodological framework based on machine learning to model and predict GSS, and then proposes a data-driven multi-style terrain synthesis method. The experimental results prove that the optimized landscape perception model optimizes the landscape path aesthetics according to the relevant theories and actual cases of landscape planning and construction.

Multimode fibre optic communication systems, employing mode/mode group multiplexing, present challenges in accurately identifying numerous modes and mode groups for improved performance. In this study, we propose an intelligent identification model utilizing a fully convolutional neural network (CNN) to precisely identify multimode fibre modes and their clusters. The model is simulated and experimentally validated, considering noise influences on linear polarisation modes. Using a platform with OM2 multimode fibre and a multiplane optical conversion mode multiplexer, we capture optical field information for 10 modes and their corresponding mode groups. Extensive data are employed for training and validation, achieving a 100% recognition rate for all modes and mode groups in experiments. Notably, when employing a 44-photodetector array, an impressive 98.3% recognition efficiency is attained, showcasing the potential of deep learning in advancing multimode fibre optic communication systems.