The modern concept of the matrix was gradually formed in the 19th century. In 1801, German mathematician Gauss took all the coefficients of a linear transformation. In 1844, German mathematician Eisenstein discussed ‘transformation’ (matrix) and its product. In 1850, the British mathematician Sylvester first used the term matrix. In 1858, British mathematician Gloria published ‘Research Report on Matrix Theory’. He first studied matrices as an independent mathematical object and first published a series of articles on this subject. Therefore, he was considered the founder of matrix theory. He gave a series of definitions that are commonly used now, such as two matrices, equality, zero matrix, sum of two matrices, quantity product of a number and a matrix, product of two matrices, matrix inverse, transposed matrix, etc [1]. Kelley also noticed that matrix multiplications are combinable but generally not commutative and

The maximum order

The ranks of the transposed matrices are equal, that is, ^{T}

Suppose that the _{ij}_{m×n} are exchanged to obtain the matrix

Suppose that row

Let's multiply the

If

If

If

Since _{1} and _{2} are a _{1} = 0, _{2} = 0. Thus,

From the above three cases, we can see that all the sub-forms of matrix

In fact, performing a row or column elementary transformation is equivalent to multiplying this matrix to the left or right by an invertible matrix. Let _{ij}_{ij}

We only need to further subtract the appropriate multiples of line

In fact, the matrix is transformed into a step shape by elementary transformation, and the rank of the number of non-zero rows in the step shape matrix is the rank of the matrix. This results in another equivalent definition of matrix rank:

The number of non-zero rows in the staircase formed by matrix _{ij}_{m×n} after elementary transformation becomes the rank of the matrix. The rank of matrix

The elementary transformation is used to transform the matrix

Any matrix

If the matrix ^{−1} ^{1} ^{−1}

If _{A}_{A}_{B}_{B}_{A}_{B}

Set
_{1})^{n1},(_{2})^{n2},⋯(_{s}^{ns}, so

If the rank of matrix

The matrix of contracts has the same rank. That is, if

If ^{T} AC^{T}^{T}

0 ≤ _{m}_{×n}) ≤ min{^{T}

Let _{1},_{2},⋯, _{p}_{p}_{+1},_{p}_{+2},⋯, _{q}

So, have:
_{A}B_{1}. Obviously, the remaining rows are zero, and so, _{A}B_{1}) ≤ _{1} _{q}AB^{−1} (

This property is extended to the case of the product of any

First prove

(Sylvester inequality) Let

(Proven using a block matrix) Since

If the matrices

Prove: because
_{n}_{n}_{n}

Because
_{n}_{n}_{n}

If

Because
^{n×n}

Among them,

If one of _{1} (_{1} (_{1} (_{1} (

This results in the conditions for the equality of Sylvester and Frobenius inequality:

Assume ^{n×n}

If (

With Theorem 1.4.4

Assume ^{n×n}

If

If

Assume ^{n×n}

due to (

Assume ^{n×n}_{i}_{j}_{i}_{j}

due to (_{i}_{j}

Analysis of typical examples:

^{2} =

^{2} = ^{2} −

^{2} =

^{n×n}

^{l}^{m}^{k} = ^{m}^{+1} =

^{l} + ^{m}^{−1} + ^{m}^{−2} + ⋯ + ^{k} = ^{m}

prove:

w ^{l}^{m}^{k}, then (

make ^{l}, ^{m}^{−1} + ^{m}^{+2} + ⋯ + ^{k}, then (

For any matrix

Let

Use the elementary transformation indicated by Lemma 1.1.1 above to turn _{r}_{+1} = _{r}_{+2} = ⋯ = _{m}

The rank of a matrix is a basic concept and one of the most important quantitative characteristics of a matrix. It is an invariant under elementary transformation. The rank of a matrix is an important concept that reflects the inherent characteristics of a matrix. It has an important role in linear algebra, analytic geometry and even probability theory.

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Hand-Drawn art Food Packaging Design Based on Deep Neural Network Model Innovative Development Model of Ideological and Political Education Under “Self-media” Based on Big Data Thoughts and Design of Ideological and Political Construction Based on the ADDIE Model VR-based computer maintenance practical training platform development design and application research Design and implementation of an integrated management system for discipline construction and postgraduate education in universities Research on innovative human capital for China’s economic development based on STI model In-depth analysis of the artistic expression of paper-cut elements in the design of boat space Construction and intelligent analysis of power grid physical data knowledge graph based on Internet of Things for power system Research on the effects of financial literacy on rural household credit constraint Research on educational resource recommendation system based on MRLG Rec Visual analysis of the trend of mental health changes among Chinese college students in the context of the normalisation of the new crown epidemic A study on the phenomenon of anaphoric correction in college students’ English conversation Design of fitness walker for the elderly based on ergonomic SAPAD model Research on the innovation of collaborative education model of scientific research and teaching in colleges and universities based on the theory of process reengineering The innovative path of diversified vocal music education reform in colleges and universities under the advanced educational concepts of China and the West Rhino application in ceramic packaging design teaching Analysis and countermeasures of cultivating independent learning ability in colleges teaching English based on OBE theory Influencing Factors of Ideological and Political Construction Work based on VSBPC Model Projection of Early Warning Identification of Hazardous Sources of Gas Explosion Accidents in Coal Mines Based on NTM Deep Learning Network Coupling Coordination of Long Triangle Digitization and High Quality Development of Manufacturing Industry based on Differential Equation The application of graph theory teaching method in ideological and political education of discrete mathematics Construction of a personalized learning platform based on genetic algorithm for specialized education in industrial colleges in the context of industry-education integration Analysis of the application of music curriculum education integrating modern network technology Evaluation of capability maturity of digital transformation of manufacturing industry in Yangtze River Delta based on SDAF model Online and Offline Audio-visual-oral English Teaching Model Construction Based on ARCS Research on new ways of ideological and political education in new era universities Application of ecological architecture in architectural design based on information technology integration Strategies for the implementation of aesthetic education in music teaching in colleges and universities under the threshold of big data RETRACTION NOTE Analysis of tennis techniques and tactics based on multiple linear regression model Automatic Detection of Transformer Health Based on Bayesian Network Model Application and value of painting technique in art education based on improved ARMA model 3D Virtual Art Design Method Based on ARCS Model Construction of intelligent terminal equipment for Electric Internet of Things based on edge computing model Analysis of bridge foundation pile detection based on convolutional neural network model Application of deep learning model in computer data mining intrusion detection Defect prediction of big data computer network based on deep learning model Analysis of News Communication Law Based on PSR Model Formative Assessment of College English Teaching Based on the CLA Model Application of animation design in the digital media art industry based on artificial intelligence technology An applied mathematical model based on discrete regression algorithm for computational prediction of blood collection volume Fractional Differential Equation in the Cost Control System of Cross-Border E-Commerce Optimization of innovative information-based teaching paths in college sports psychology based on principal component analysis Data Information Security Algorithm Based on Chaos and Hash Function Research on the innovative inheritance application of Guangxi minority elements in clothing design under the perspective of rural revitalization A study of English learning anxiety regulation strategies based on the deep learning model Application of power electronic circuit design based on BP artificial neural network in environmental monitoring Research on Curriculum Civic Education and Innovation and Entrepreneurship Education Reform in Undergraduate Universities under the Web-based Teaching Collaboration Platform Optimized reform strategy for quality education of Chinese language and literature education for college students based on information fusion technology Research on the innovation path of university library management and service under the background of big data Construction and practice of a standardized training system for outpatient clinics based on the cultivation of TCM thinking Genetic algorithm-based analysis of heat production prediction in electronic devices Analysis of the Implementation Approach of University Budget Performance Project Based on Equity Realization Perspective A study of multimodal education and discursive reading model in English based on multidimensional cognitive information aggregation theory Teaching Effect Evaluation of Music Education Based on Partial Differential Equation Model Analysis of the effect of music on personal emotional experience and identification of psychophysiological signals The application of body symbols in contemporary art from the perspective of Da Vinci’s theory A Hybrid Stochastic/Robust Planning Model for Integrated Energy System Considering Multiple Uncertainties Railroad freight service quality evaluation system based on a logistic regression model The Design and Development of Network Multimedia Music Teaching Based on Multiple Linear Regression Algorithm Research on the relationship between investment in preschool education and economic growth in Chongqing Deep Learning-based Fuzzy Translation Problem in Chinese-English Epidemic News Reporting Innovation and application of digital reading model for college students based on deep learning model Study of trip card data analysis system under big data Construction of a text complexity grading model for English textbooks in the context of globalization Design and research of big data technology based on e-commerce platform A study of cultural differences in Sino-Thai translation based on big data analysis On the Application of Artificial Intelligence in Local Legislation Big data analysis based on the correlation between live-streaming with goods, perceived value and consumer repurchase Big data methods and news communication innovation from theoretical definition to operational route Design and Practice of Human Geography Field Teaching in the Age of Big Data The innovation path of virtual practice teaching in college Civics class based on the Ridge regression model Application of CAE for mold flow analysis in plastic molding mold design Intelligent control system of port turning line based on WITNESS software Symmetry Analysis and Exact Solutions of Extended Kadomtsev-Petviashvili Equation in Fluids The reform of university English education inspired by cultural differences between China and foreign countries New approach to regularise the perturbed two-body problem Analysis of the factors influencing the development of college students’ Civic Science course based on ISO model The development trend of music art based on content analysis method The construction of cross-regional cooperation innovation network governance system based on data fusion technology Analysis of the innovative development path of university civic education based on the era of big data On new aspects of Chebyshev polynomials for space-time fractional diffusion process Rural landscape design strategy based on deep learning model Innovative Development Path of E-Commerce and Civil and Commercial Law in the Information Age Based on Discrete Regression Algorithm Stability of equilibrium price-production of dynamic duopoly Bertrand–Cournot game in a transnational supply chain with asymmetric information Mathematical model on Type 1 diabetes and Healthy state: Mechanisms of β cells on the interaction between M1 and M2 MacrophagesScientific Training Model of Sports Tourism Based on Nonlinear Differential Equation The dilemma and path of rural college students returning to their hometowns for employment and entrepreneurship in the post-epidemic period under the background of big data Financial big data control and intelligent analysis method for investment decision of renewable energy projects Analysis of the dilemma of higher vocational thinking education in China under the background of “Internet+” Analysis of the current situation and development countermeasures of college sports training and management based on the background of artificial intelligence A Review of the Design and Enhancement of Pastoral Culture Creativity in Chinese Rural Ecotourism Research on Computer Network Security Protection Technology Incorporating Full Convolutional Networks Research on the Logical Turn and Practical Path of China’s High-Quality Economic Development in the Context of Digitalization Influencing factors of modern and contemporary decorative pattern design on users’ visual perception characteristics Construction machinery hydraulic transmission system fault analysis and elimination methods Optimization strategy of ideological and political education in colleges and universities based on modern information technology A Review of the Research on Labor Supply Effect of Active Aging Application of traditional cultural elements in landscape design Driving Model of Electronic Information System Based on Agent Modeling and Simulation Research on the path of high-quality development of the rural tourism industry driven by big data technology The application of cloud accounting in enterprise financial decision making in the era of big data Variable combinatorial gap-filling method for single-cell RNA-seq data Research on the development path of integration of rural sports culture and sports tourism in the context of big data Risk Assessment of Financial Markets in the Era of Big Data Based on Eviews Econometric Model Cost Control System of Cross-Border E-Commerce Based on Fractional Differential Equation Fourier transform-based electronic logic information processing architecture design The Development of Enterprise Management Innovation in the Digital Economy Era Based on the Schumpeter Model Neural Networks and Migration Learning Models in Sports Video Course Applications Research on performance evaluation and optimization of college budget management under the background of big data Risks and preventive measures for manufacturing enterprises in the Context of cloud accounting Research and application of Augmented Reality Technology and digital media Technology in mobile visual media Research on Information Visualization of Chinese Cultural Heritage Patterns in the Digital Perspective The modernization strategy of constructing teacher-student relationship in the new era of universities Research on optimal path planning technology for vehicle positioning and navigation system Construction and practice of training system for dance professionals based on OBE teaching model combined with Kinect motion capture technology The construction of environmental cost accounting system - based on ecological perspective Interactive experience of information visualization in mechanical design Analysis of tea culture communication path based on the principal component analysis method Programming application of contemporary jewelry design forms based on Grasshopper software Construction of a multimodal poetry translation corpus based on AdaBoost model Analysis of the practical education model and its value in the context of big data in university civic education Information Fusion Technology-based Approach to Mining Civic Political Elements in College Curriculum Design of a decentralized Internet-based control system for industrial robot robotic arms General Architecture of Sustainable Urban Spatial Development Model Based on Smart Cities Design of a mechanical automatic transmission system for automobiles based on torsional vibration damping Optimization of the quality management path of physical education teaching in colleges and universities integrating modern network technology Construction of a mathematical modeling teaching quality assessment system for universities based on Eviews model Research on Information Literacy Education Model in the Big Data Environment - A Case Study of six Hainan high school Libraries Innovative model of teaching in law experiment center based on LETS software Fuzzy PID control theory in multi-objective optimal longitudinal following of intelligent trolley A Study on the Implications of Critical Discourse Analysis Theory for College English Teaching in the Context of Big Data The Construction of “Party Building + Curriculum Civics” Education Model for Chinese Grassroots Party Organizations in the Context of “Internet+” The artistic aesthetic and cultural value orientation of 3D printing A multidimensional interpretation of Marxist theory based on a decentralized Internet perspective Research on key technologies for connected vehicle autonomous driving based on 5G big data Student management model of college student associations based on ant colony Algorithm Research on practical teaching of Ideological and political education based on VR technology in the information age The Construction of Cultural Translation Research System under Big Data Technology Analysis of factors influencing the development of Civic Education for senior students of Chinese party culture based on ISO model Research on the application of big data technology in enterprise project management An investigation into the practice of blended teaching in English courses based on the BOPPPS model Analysis of student management path optimization in higher education institutions with ISM Strategies for optimising chemical instrumental analysis methods based on the ADDIE model The improvement path of international Chinese education management in universities based on big data technology Analysis of the coupling correlation between environmental protection efforts and economic development level based on artificial intelligence technology Seismic finite element analysis of underground integrated pipe corridor based on generalized reaction displacement method Research on the construction and operation mode of smart city information service system based on IOT technology Innovative reform strategy of artistic practice of university folk dance education based on information fusion technology Biosafety management strategy based on multiple linear regression model Thinking about the strategy and practice path of modern agricultural industry development in the context of big data New ideas of precision teaching development based on the background of big data Design and research of news recommendation system based on perceptron model in big data era Combined application of employment education and big data internet technology based on the context of vocational education reform Research on structural damage identification and localization based on artificial neural network College students' innovation and entrepreneurship model based on probability theory statistics Research on the design and application of intelligent teaching management system of intelligent training room in the background of internet Innovation and entrepreneurship model of higher vocational college students based on probability theory statistics Analysis of distribution network planning with distributed power supply based on ELM algorithm A fuzzy mathematical model-based approach to customer-side energy management for power companies Research on the evaluation of the employment effect of Jiangsu artificial intelligence based on neural network on colleges and universities Rural Tourism Development Planning Based on Nonlinear Differential Equation Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Strategies for Marine remote sensing target detection based on deep learning model and internal wave simulation imaging technology Construction of a smart grid load forecasting platform based on clustering algorithm Variation mechanisms of metadiscourse verb patterns in English journal articles from intra- and interdisciplinary perspectives Construction of a refined people management model for SMEs based on discrete regression algorithm Color and composition under big data technology: the art of visual communication in film art FPGA-based Intelligent Logistics Management for Universities Study on the design of old residential district landscape space renewal based on Cite Space from the perspective of big data Digital design and experimental testing of a compressor's suction muffler transmission loss A Study on the Impact of the Internet+ Era on Korean Film Consumer Culture SCP model based on the new energy vehicle collaborative detection configuration method research Research on the innovation path of teaching interior design majors in universities based on big data analysis The cardiovascular promotion of college taekwondo based on fractional differential equation A Systematic Study of Intercultural Listening Strategies in Higher Education Institutions in the Context of Big Data Analysis and research on the economic benefits of employee labor in enterprises in the context of the Internet Design and application of an intelligent communication platform for aesthetics education for internationalization The boundedness of functional differential equations of hospital financial software under information Multi-UAV Aerial Mission Intelligent Assignment Algorithm Based on LightGBM Model Deep learning model-based brand design 3D image construction The Evolution of Marxist Theory of Financial Capital in the ACC Payment Model The character image setting of 3D animation works based on Poisson equation The Impact of College English Teaching on Improving Students’ Writing Effectiveness in the Context of Big Data Research on the interaction between university physical culture education and college students’ physical exercise in the context of big data Innovative research on the construction of experimental teaching platform of economics and management in colleges and universities based on the background of big data The optimization path of core literacy of innovative talents in universities based on the improved Markov model Analysis of the innovation path of university education management informatization in the era of big data Research on visual recognition and positioning of industrial robots based on big data technology Coupled coordination analysis of green finance on economic growth based on big data Information technology system to promote drug production command supervision Exploring English-Chinese Translation Skills Based on Big Data Technology Decentralized Internet-based calligraphy education leads the internal development of aesthetic education in schools Research on the construction and effect evaluation of OMO mode of professional practical training teaching of landscape engineering technology based on virtual simulation technology Theoretical and practical research on mathematical modeling of economy and finance based on artificial intelligence Calculation and evaluation of similarity in sports information resources based on combined algorithm Research on personalized Japanese language teaching model based on big data technology A multidimensional interpretation of Marxist globalization theory based on decentralized internet Analysis of the improvement path of English listening and speaking ability based on big data technology Study on the absorption technology of distribution network with high-proportion distributed power supply Application of BOPPPS teaching model in online and offline blended teaching of college Public Basic Mathematics Information Development and Student Talent Cultivation of Modern Chinese Language and Literature in the Context of Big Data Grouting reinforcement strategy for tunnel sand layer based on BP neural network Multimodal discourse analysis of automotive websites Research on university laboratory network security based on Cloud Computing Optimization of journalism communication digital education ecology based on SEM model Títul Mobile visual media design based on digital media technology and augmented reality Spatial planning layout design of national land in the context of rural revitalization based on MOLA model Influence of 3D characters image transfer on animation drawing based on detail feature preservation Títul Sports consumption behavior and consumption demand of urban residents in the context of “Internet+” Theoretical Innovation of English Teaching Methodology Based on Continuous Discretization Method Study on the influencing factors of the development of art education in colleges and universities based on the deep learning model Evaluation mechanism of local universities’ faculty construction based on SOAP technology An analysis of the cultural heritage of traditional Chinese festivals based on the perspective of big data Research on the mechanism of cultivating the English listening ability of college students based on the background of big data Reform and innovation of saxophone teaching mode in colleges and universities based on Bayesian model Media-based information display in university libraries based on a generalized differential equation numerical method Research on detection of parking space based on the combination of magnetoresistance and microwave radar sensor under IoT platform Research on sustainable development mechanism of regional ‘Artisan Workshop’ under big data environment Innovative Teaching Strategies for Aesthetic Arts in Higher Education Institutions Based on Big Data Technology Application of hybrid kernel function in economic benefit analysis and evaluation of enterprises The analysis method of the impact of missing audit information on the securities investment decisions based on Logistic regression models Research on the close-range photogrammetry accuracy of 3D digital interactive platform of Hunan intangible cultural heritage Research and implementation of visual question and answer system based on deep learning Títul Research on Wireless Network Coverage for Transformation and Upgrading of Exhibition Management with Artificial Intelligence Technology Study on the application of multi-angle imaging related technology in the construction process The combination application of Civic Education and Big Data Internet Technology based on the background of college education reform Economic statistical strategies for the transportation industry based on time-varying parametric state space models The digital economy is an important path to boosting common prosperity Public Management Model Based on Mathematical Equation of Statistics of Probability Theory Optimization path of corporate ESG performance and compensation performance management under carbon neutral target New anergy tide control strategy based on Eviews econometric model Optimization of vocal singing training method based on intelligent big data technology The construction of blockchain platform for engineering museums based on vectorized image processing technology Green supply chain innovation management strategy based on the combination of low carbon economy and e-commerce with big data technology Numerical simulation analysis of load-bearing capacity at joints of symmetrical cantilever-assembled bridges based on the logistic regression model Research on the current situation and countermeasures of cultivating talents in recreational sports under the perspective of artificial intelligence The Construction of a Coordinated Integration System of Rural Tourism and Habitat Environment in China Based on Coupled Coordination Degree Model Multi-objective Evolutionary Algorithm Supported Construction of Architectural Window Design Model Based on Visual Comfort - A Case Study of Office Buildings in Cold Regions Optimization analysis of cross-border e-commerce marketing strategy based on the SCOR model Construction and implementation of teacher support services model under “5G+Smart Education” Probabilistic neural network-based diagnosis of blockage faults in air conditioning and refrigeration units The Meaning, Value, and Realisation of Internet-based Culturally Responsive Teaching Analysis of digital humanistic knowledge production based on natural language processing Design of language assisted learning model and online learning system under the background of artificial intelligence An experimental study of the carbohydrate exchange method combined with aerobic exercise to intervene in the physical health of obese college students Design of a text evaluation and interaction system for university English textbooks based on a perceptual machine model in the context of globalization Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering Research on loyalty prediction of e-commerce customer based on data mining The use of red cultural and creative design in tourism cultural and creative design based on the background of big data Innovative measures of student management by counselors in higher education institutions under a big data environment Digital terrain interactive editing technology for landscape planning and design based on deep learning LightGBM model applied in the teaching course of Civic Education integrating red culture Research on personalized recommendation method of preschool e-learning resources Key technology research on decentralized Internet-based smart agriculture platform Optimization of cross-border e-commerce marketing strategy based on deep learning model An innovative study of art teaching strategies for preschool children based on big data analysis The function of ideological and political education of campus culture in the context of big data and its realization path The use of new media technology in the creation of media animation under the threshold of big data Research on the construction of early warning model of customer churn on e-commerce platform Research and Development of “Gathering Green” Digital Cultural and Creative Collection Creative Platform under the Technical Support of Big Data and Advocacy of Rural Revitalization: A Typical Practice Based on Qingdao Coastal Villages Evaluation and prediction of regional human capital based on optimised BP neural network A Regionalization Study of Market-Based Allocation of Data Factors in China A study of local smoothness-informed convolutional neural network models for image inpainting IoT-based interaction design of smart home products for elderly families A multidimensional interpretation analysis of Marxism globalization theory based on distributed Internet Strategy for financing mode optimization in international trade supply chain based on deep learning model Early warning and detection of geological disasters based on intelligent genetic algorithm Exploring the Application of Modern Fine Arts Artistic Modeling and Color in Modern Art Design Based on Information Fusion Technology Decisions of competing supply chain with altruistic retailer under risk aversion Research on China’s tourism marketing model in the data perspective: based on ecological perspective Research on human resource management performance evaluation based on BP algorithm Problem identification and solution of voice change phenomenon in vocal singing based on the AHP method A study on the effect of music on college students’ depression and anxiety based on big data analysis Inferring the role of fiscal decentralization and carbon taxes through the ARDL approach Some Necessary Conditions for Feedback Functions of de Bruijn Sequences A new model of tax governance in the context of the integration of traditional industries and the digital economy Improvement of the financial management supervision system of universities in the context of big data Analysis of the current situation and teaching strategies of university table tennis teaching in the context of big data Fractional Differential Equations in Sports Training in Universities Current situation and training strategies of preschool students’ skills in the Internet era Construction of sustainable development professional cluster in private universities based on data mining model Law of interest rate changes in financial markets based on the differential equation model of liquidity Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Innovative cultivation path of great craftsmanship based on logistic regression model Research on Fault Prevention and Maintenance System of Automatic Substation Primary Equipment Based on Decision Tree Algorithm The construction of innovation system of college physical education reform based on Bp neural network Analysis of the impact of college students’ innovation and entrepreneurship on human resource index of science and technology enterprises based on logistic regression model Noise Reduction of Cable Partial Discharge Vibration Signal based on EEMD-CIIT method Application of image processing technology based on hierarchical analysis in visual communication system Research on the cultivation mechanism of “double-creative” talents in ethnic regions with the integration of industry and education in the context of the Internet Research on the organization of public space relations in commercial complex buildings in the context of big data The use of traditional Chinese elements in modern interior design in the context of multimedia Application of biofilm-membrane bioreactor in municipal wastewater treatment Regional Logistics Routing Optimization System Based on Fractional Differential Equation Modeling Orientation and triage of preschool students’ interest development based on deep learning model AHM-FC Model in the Evolution Analysis of Ideological and Political Subjects System Construction of Intercity Culture and Xinjiang Music Integration in the Context of “One Belt, One Road” Based on SCP Model 3D Scene Simulation System of Japanese Teaching Based on Finite Element Equation Assessment of tax-related risks in corporate betting agreements based on vector autoregressive model Multi-Governance of Basketball Event Organization in Ethnic Areas Based on PESTEL Analysis Model Design of a linear regression model-based Internet exit anomaly detection method Building Tilt Scanning Measurement Based on Binary Linear Regression Analysis Exploring the application of composite materials in ceramic design based on fuzzy numerical analysis Construction of law and economics litigation service platform based on multimedia retrieval Application of motion capture technology based on dance big data in dance retrieval The Innovation and Application of Computer Multimedia-assisted Teaching Methods in College English Teaching Deep learning-based fault diagnosis and localization method for fiber optic cables in communication networks A Study on Innovative Teaching Strategies of English in Universities Based on VAR Model Research on the integration and innovation of red culture and ideological and political education in universities in the new media era Analysis of the impact of ADDIE education model based on logistic regression model on teaching contemporary cultural and creative product design Research on the optimization of prefabricated building preset scheme integrating BIM technology and BiLSTM model Research on practical education model of mental health development of college students in higher education institutions in the context of big data Badminton players’ trajectory under numerical calculation method Evaluation of the Integration Path of Ideological and Political Elements in English Major Courses Based on the ADDIE Model Application of multiple regression models in the analysis of kinematic parameters in competitive gymnastics Analysis of the influencing factors of college students’ mental health literacy based on multiple linear regression model Scale and efficiency of China’s radiation industry based on LightGBM model The construction of a mathematical model for college English interactive distance teaching Application of Data Mining Algorithms in Power Marketing Predictive Analytics Analysis of music teaching technology based on a data mining model Research on the Development of Young Teachers’ Informatization Teaching Ability in Universities Based on Big Data Technology Interactive texture replacement of cartoon characters based on deep learning model Application of Multiple Linear Regression Model in the Sustainable Development of National Traditional Sports Exploring Innovative Strategies for Teaching English in Higher Education with the Support of Big Data Analysis of media content recommendation in the new media era considering scene clustering algorithm The generalized matrix equation for systematic evaluation of oral Russian self-study The application of computer VR technology in physical education Evaluation of Corporate Social Responsibility and Core Competitiveness Based on GA-SVM Model Gymnastic movement recognition based on support vector machine classification model Extraction of Design Factors of Chinese Dance Culture based on Analytic Hierarchy Model Cultural and artistic dissemination of traditional Chinese silk patterns in the 17th-18th centuries in the context of big data Theory of physical education ecosystem based on SFIC model Research on the integration of online teaching resources in higher education institutions under the perspective of industry-education integration The Analysis of the Expression Form of Ideological and Political Elements in Philosophy Courses Based on Multimedia Information Fusion Strategy Diversified Integration of Civics Teaching Resources in Universities Based on Discrete Regression Model Study on the rehabilitation training effect of exoskeletal diseases based on the skeletal muscle geometry modeling method Exploring the teaching practice of visual communication design in colleges and universities under the background of big data Dance Rehearsal System Based on Linear Partial Differential Mathematical Equation Big data-based output-oriented teaching competency development for mathematics teacher trainees Research on dynamic detection method of bridge vehicle load based on optimization algorithm Deep Learning Based Sports Video Classification Research Design of energy big data and carbon emission monitoring system based on perceptron model in the context of carbon neutral and carbon peaking A model of online opinion dissemination based on vector autoregressive model of emotional information entropy Application of decision tree algorithm mining model in career planning goal simulation Multimodal construction of human movement dataset under the development of physical dance teaching in colleges and universities Visual Communication Design Method Based on Multimedia Information Processing Technology and Its Application Optimization of Teaching Path of Artificial Intelligence Programming Course in the Context of New Engineering Construction Application of short video recommendation technology in classical music teaching Reading promotion mode of university libraries in the context of information development The application of video text generation technology in assessing the effectiveness of teaching ethnic traditional sports Design of online art education system based on 3D virtual simulation technology Using Multimedia Information Retrieval Technology to Optimize the Strategies of College English Teaching Research on the current situation and development countermeasures of Chinese sports industry companies in the context of big data Research on digital transformation innovation of the exhibition industry in the era of big data A structural model of teachers’ teaching competencies based on multimodal affective features of support vector machines Research on information visualization for digital media design methodology based on big data technology Research on the physical training of athletes in ice and snow sports based on big data Rural tourism service design based on collaborative filtering algorithm after epidemic normalization Innovation of China’s international trade methods based on the perspective of cross-border e-commerce The theory of innovation and entrepreneurship education practice with the integration of modern information technology and new engineering The effect of fitness qigong exercise on college students’ physical fitness based on big data technology A study on the management of big data technology in financial decision-making of enterprise cloud accounting Empirical evidence of coupling and coordination between circular economy and foreign trade based on big data A strategy study based on artificial intelligence data analysis of rural governance into tourism industry development research Study on the Flipped Classroom Teaching Model of “Business English Translation” in the Context of Big Data Research on brand operation strategy of integrating traditional culture into the modern tourism industry based on big data analysis The Fit of Constructivist Teaching Theory and Foreign Language Teaching Reform in the Background of the Internet Era Heritage development of traditional culture in folk art education based on the decentralized Internet Intelligent algorithm-based analysis of corporate financial decisions in the era of cloud accounting The use of management accounting and financial accounting in financial management based on multiple linear regression algorithm Construction of intelligent practice system of college Civics teaching based on VR technology in the information age Application of nonlinear regression theory based on edge computing to mathematical modeling in universities Coupling research of Confucianism and Chinese philosophy based on data mining model The construction of a localized teaching model of the Orff music teaching method based on big data analysis A study on the application of non-fiction writing in writing courses in local universities based on a decentralized Internet model An Exploration of Chinese Language International Education Exchange and Cooperation Model Based on Random Forest Model A Study on Risk Management of Commercial Banks’ Home Mortgage Loans Based on Risk Management Framework A study on the application of creative writing in literature teaching based on multiple linear regression algorithm The positive impact of increased government bailout on weakening corporate default risk Teaching reform and innovation of vocational development and employment guidance courses in colleges and universities based on random forest model Modeling the connection between oil painting creation and university students’ inspiration in the context of “Internet+” The cultivation system of cross-media narrative ability of scriptwriter talent based on cluster model The path of optimizing and improving the teaching innovation ability of visual communication based on the SWOT model The Structure and Influencing Factors of Students’ Innovation and Entrepreneurship Ability Based on the Structural Equation Model The Influence of Green Entrepreneurship Orientation on Enterprise Performance Based on the TPB Model Computer Security Active Defense Technology Based on Bayesian Model Visual Operation and Maintenance Monitoring of Power System Data Network Based on Neural Network Model Design Method and Application of Intelligent Sports Equipment Based on Stochastic Differential Equation Color Processing System of Art Color Based on Fractional Differential Equation Global efficiency estimation of complex networks and regional technological and economic development based on fractal network model DMAIC Model’s Relative Efficiency Analysis of Input and Output in Cultural Construction of University Library and Mental Health Education of College Students Integrated Operation and Maintenance of Power System Communication Based on the Markov Model Oscillation and Asymptotic Behavior of a Special Delay Third Order Nonlinear Neutral Functional Differential Equation Statistical Analysis of Export Waybill Data of Cross-Border E-Commerce Based on ISM Model Zagreb Connection Indices for Metal Trihalides Adaptation of the COVASIM model to incorporate non-pharmaceutical interventions: Application to the Dominican Republic during the second wave of COVID-19 Karnauph Classifier: A Hybrid Mathematical Model for Data Classification An Improvement of Parameter Estimation Accuracy of Structural Equation Modeling using Hybridization of Artificial Neural Network in the Entrepreneurship Structural Model Mathematical model for portraits of members in large-scale department stores Analysis of the Innovation of University Students’ Education Management Based on the APT Model Construction of College Vocal Music Innovation Teaching Mode Based on ARCS Model The Clustering Effect Analysis of College Marxist Theory Network Propagation Based on Ant Colony Intelligent Algorithm Effects of Aerobic Training on Body Function Based on Multiple Linear Regression Analysis Limit cycles of a generalised Mathieu differential system The nonlinear effects of ageing on national savings rate – An Empirical Study based on threshold model Virtual Museum Scene Design Based on VRAR Realistic Interaction under PMC Artificial Intelligence Model Karnauph Classifier: A Hybrid Mathematical Model for Data Classification Construction Cost Control and Management Based on Mapping Model and BIM Research on innovative cultivation mode of English and American literature translation talents based on internet environment Improved K-SVD Model in Higher Vocational Physics Teaching Dance Emotion Characteristic Parameters Based on Deep Learning Model BIM Model Design of Deep Foundation Pit Engineering Based on BP Network Analysis of quantitative management of online intelligent monitoring of tailing ponds based on the perspective of safety prevention and control The Innovative Development Model of Ideological and Political Education Based on Big Data and We Media Analysis of rural tourism culture advertising content based on LSTM-CNN model Construction of network mathematics model for college English mixed teaching Optimization of college physical education curriculum based on G-DINA Strategic Policy Innovation of Regional Development Strategy Transformation in the New Era Based on Progressive Differential Model Optimization of local ethnic music teaching transmission path based on logistic regression model Vocational Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Exploring the Construction of Curriculum System of Theater and Film Studies Based on ADDIE Model Innovative research on propagation modes of short video works based on digital technology Analysis of the effect of compound training for explosive power on improving the psychological resilience of football athletes Construction of feature extraction model for machine foreign language translation evaluation system Exploration on the path of ideological and political education and cultural education in colleges and universities from the perspective of red culture Effect evaluation of ideological and political guidance in university party construction based on hierarchical analysis model Can policy coordination facilitate unimpeded trade? An empirical study on factors influencing smooth trade along the Belt and Road Research on design of customer portrait system for E-commerce Research on rule extraction method based on concept lattice of intuitionistic fuzzy language Application of data mining model in English text writing training system Design and evaluation of intelligent teaching system on basic movements in PE A Hierarchical Analysis Structure Model for the Integration and Optimization of Sports Industry Digitalization and Tourism Industry Construction of an economic environment scheduling system based on the multi-objective fuzzy optimization model Statistical Analysis of Advanced Mathematics Teaching Quality Evaluation Based on Quantile Regression Model Analysis of music teaching technology based on a data mining model The curriculum optimization exercise system of college physical aerobics based on partial differential equation Prediction of Cyberspace Security Data Based on the Markov Chain Model Scheduling Optimization of hotel supply chain based on heterogeneous multipopulation ant colony algorithm The application of multiple regression model in interactive English classroom teaching Sports Talent Training System of Sports Aerobics Based on Fractional Differential Equation Research on evaluation system of cross-border E-commerce platform based on the combined model Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing New Green’s Function Method for Solving Underground Oil Seepage Problems The zero-Hopf bifurcations of a new hyperchaotic system Retraction: Wei, Wei and Lu, Liu. “The influence of X fuzzy mathematics method in basketball tactics scoring” Applied Mathematics and Nonlinear Sciences, vol.7, no.1, 2022, pp.677-684 Retraction of: Pan, Liurong, Alshalabi, Riyad, Li, Pingfen, Naminse, Eric Yaw and Tan, Fuqiang. “Analysis of digital humanistic knowledge production based on natural language processing” Applied Mathematics and Nonlinear Sciences, vol.0, no.0, 2023, pp.-. https://doi.org/10.2478/amns.2022.1.00029 Retraction Retraction: Retraction of: Mundewadi, R. A. and S, Kumbinarasaiah. “Numerical Solution of Abel's Integral Equations using Hermite Wavelet” Applied Mathematics and Nonlinear Sciences, vol.4, no.2, 2019, pp.395-406 A new expansion auxiliary equation method for traveling wave solutions of the simplified MCH equation and the ZKBBM equation Geometric Properties of a Linear Complex Operator on a Subclass of Meromorphic Functions: An Analysis of Hurwitz-Lerch-Zeta Functions Study of Enterprise Resource Optimization Scheme from the Perspective of Knapsack Problems