Due to the rapid development of the Internet, music data is growing explosively. The need for people to retrieve audio data is growing. The traditional retrieval method is to manually annotate the music into text, and then people use the traditional text retrieval method to retrieve the database of the annotated information. This mainly has the following shortcomings. First, the number of songs is increasing day by day, and the cost of manual labeling is huge; second, the information of manual labeling is mostly concentrated in song name, song style, song author and other information, but does not contain song melody information; third, in human natural thinking Songs are mainly identified by melody, and retrieval of songs by text is not in line with the natural habits of human beings [1]. Therefore, it is necessary to propose a new characteristic theme to automatically label songs. There are two explanations for the word main melody. One explanation is that it is the melody line composed of the fundamental frequency that occupies the main position at each moment in the song. Another explanation is that the melody line formed by the fundamental frequency of the vocals in the song is the main melody. Because the melody of the human voice is more widely spread and used in general songs, the main melody in this topic refers to the second interpretation method, that is, the fundamental frequency curve of the human voice in the song. Selecting it as a new feature is mainly based on the following three aspects. First, the main melody is an intuitive description that can be perceived by people without professional musical training, which means that the main melody is a concise and natural description of a song by humans [2]. Second, the main melody plays a significant role in identifying song fragments in popular songs, which can bring good results in many applications and promote the development of various related applications. Third, even in very complex song signals, the main melody can maintain its simple and highly recognizable characteristics, which shows that the main melody description method has better adaptability to different types of songs. Theme extraction has been widely used in many fields [3]. The most common application is to use the extracted main melody as an index to generate a humming search database or an acoustic search database. Humming search is a song retrieval method that is most in line with the natural expression habits of human beings. The searcher hums a tune of the song they are looking for. The retrieval system automatically returns the song information corresponding to the tune, such as song name, author, download link and other information. In addition to humming retrieval, vocal melody extraction can also be applied in a wide range of other fields. For example, music vocal separation, automatic karaoke soundtrack generation, song retrieval, song structure analysis and song classification, etc. [4].

Murray E and others proposed a main melody extraction method based on speech separation. Experiments show that the performance of the main melody extraction algorithm with speech separation is higher than that of the algorithm without [5]. Zhu M X and others used a data-driven approach to train a deep neural network model for each fundamental frequency label with sufficient labeled data. Due to the simple singing melody localization algorithm, the overall accuracy of the algorithm is slightly lower than other algorithms. However, the original pitch accuracy of the algorithm has been improved compared with other algorithms, so as long as a more effective singing melody localization algorithm is added, the overall accuracy will be significantly improved [6]. Combining two saliency function calculation methods, source filter model and harmonic weighting, multiple candidate melody lines are extracted according to the continuity of the melody, and finally singing melody localization and main melody extraction are carried out according to a series of features of the candidate melody lines. Through the above research status, it is found that the research on the pitch saliency function algorithm runs through the entire research process of the automatic main melody extraction algorithm, while the research on the singing melody localization algorithm has only attracted the attention of scholars in recent years, and has not yet reached the upper limit of performance. Experiments show that the introduction of a simple singing melody localization algorithm is helpful to improve the overall accuracy, and further improving the accuracy of the singing melody localization algorithm will help to improve the overall accuracy of the automatic main melody extraction algorithm [7].

Summarizing the advantages and disadvantages of previous studies, this article introduces the patterns of note segmentation and basic frequency differentiation in terms of tone differentiation based on the volume tightening function. Adding note-based time stability to note segmentation based on spectral distance and dividing the sound into paragraphs with a relatively stable frequency spectrum helps to control the melody and determine the melody to be sung. In the localization of the melody, a basic model of frequency differentiation using a neural network based on different timbre and accompaniment of the vocal cords is added, and the probability that the predominant basic frequency trajectory belongs to the singing melody is based on segmented statistics, it can effectively reduce the false alarm speed of tone localization and improve the overall accuracy.

The overall framework of the algorithm is shown in Figure 1.

According to the overall framework, the main melody extraction steps of vocal music are as follows: The first step is to preprocess the input music, mainly including signal down sampling, normalization, framing, windowing and time-frequency domain transformation; The second step is to segment notes and detect voiced segments through spectrum analysis; The third step is pitch estimation. The pitch saliency is calculated on each frame of the voiced segment, and three candidate fundamental frequencies are selected according to the fundamental frequency characteristics and pitch saliency function; The fourth step is to track the dominant fundamental frequency, and extract a main melody line by Viterbi algorithm in each voiced segment; The fifth step is to distinguish the main melody of the song. Since the extracted main melody is not dominated by the song, a song discrimination model is used to judge whether the dominant fundamental frequency belongs to the song and connect all the pitch sequences belonging to the song to form the main melody of the vocal music [8].

Singing is a time-varying non-stationary speech signal. In order to better analyze the information it carries, a series of measures are usually taken to preprocess it first, including down sampling and normalization of the input audio signal, framing, windowing and STFT.

Considering that the frequency range of voice signal is concentrated in 300 ~ 3400 Hz, in order to reduce the amount of calculation of subsequent signal processing, the audio signal used is uniformly down sampled to 8 kHz. In addition, the input signal is normalized to ensure the consistency of further processing. The speech signal is almost stable in a short time. Based on this, the speech signal can be processed by frame windowing. In this paper, Hamming window is adopted, and its formula is:

Where, N is the window length, the frame length is 40 ms, and the frame shift is 20 ms. As required, STFT (short-time Fourier transform) can be used to time-frequency transform the signal after framing.

Music consists of musical notes. Each note has a relatively stable spectral character. By segmenting the music, we can obtain a more reliable and accurate sound frequency based on the calculated note frequency. The notes were segmented using a distance-based audio segmentation algorithm (DIS algorithm). One feature used to segment notes is the short-term amplitude spectrum. Slide the data window along the frame, and calculate the disk measurement distance of the sound function of the two information windows before and after the L frame of the audio as follows.

Where, _{t}_{,1} and _{t}_{,2} represent the mean vector of two adjacent segments of audio features, and

After obtaining the DIS measurement distance curve, find the maximum point of DIS(t) and screen the note segmentation points according to certain screening rules. The screening rule is: compare the standard deviation threshold with the maximum point. If the maximum point is greater than the standard deviation threshold, it is the note segmentation point, otherwise it is rounded off [9].

In addition, there is a non voiced segment in the music signal, and there is no fundamental frequency in the non voiced segment. Therefore, these segments should be excluded before fundamental frequency estimation. In this paper, the spectral variance method is used to judge the voiced and non voiced segments in the sound segment.

In polyphonic music, there are often multiple fundamental frequencies at the same time. The system uses the pitch significance function to extract the quadratic wave equation. Specifically, the extraction method of fundamental quadratic wave equation based on pitch estimation filter with amplitude compression (PEFAC) is adopted. If the fundamental frequency of a voiced signal is, the signal can be expressed as:

Where _{k}

Where _{0}, convolute it with a comb filter, and the convolution junction _{0} = log _{0}. The position in the frequency domain is the fundamental frequency of the voiced segment of the required signal.

The comb filter actually used in this paper is:

Parameter

The significance function of each frame is defined as:

For the multi fundamental frequency signal, find the frequency set {_{1}} corresponding to all peak positions of S (q) as the candidate fundamental frequency. In addition, the fundamental frequency range of singing is usually 70 ~ 1000 Hz, and calculating S(q) according to formula (7) at each frequency point requires a large amount of computation. In order to reduce the computational complexity, according to the characteristics of singing spectrum, only some frequency points with large signal spectrum amplitude are selected to calculate the significance value, which can greatly reduce the amount of calculation.

The improved pitch saliency calculation is realized as follows: set a threshold T and judge whether the amplitude of the logarithmic amplitude spectrum of the signal is greater than t for the frequency point of 70 ~ 1000 Hz. If it is greater than t, calculate the significance value according to formula (8); Otherwise, set the significance value corresponding to the frequency point directly to 0 [11].

Where q is the frequency point in log domain, Y (q) is the spectrum in log domain, and h (q) is the comb filter function in log domain; S (q) is the pitch saliency function.

Since only the significance values of a limited number of frequency points with large energy of 70 ~ 1000 Hz are calculated, the computational complexity is significantly reduced. Finally, from all frequency points, find the largest three frequencies that do not become double or half frequency to each other as the final candidate fundamental frequency. Where, _{n}_{n}

A similar method is used to track the dominant fundamental frequency trajectory. According to the peak value of pitch saliency function, multiple candidate fundamental frequencies are extracted, and then Viterbi algorithm is used to track the dominant fundamental frequency in the voiced segment (in each note).

For each voiced segment, Viterbi algorithm based on pitch likelihood and pitch transition probability is used to extract the optimal pitch sequence of the segment, that is, the dominant pitch trajectory, so that it meets the formula:

Where, _{1} represents the segment length of voiced segment, _{t}_{t}_{t}

The main melody of vocal music refers to the melody of the song sung by people, and the dominant fundamental frequency track of each note segment extracted by the main melody extraction method may belong to the song or the accompaniment instrument. Therefore, it is necessary to introduce the fundamental frequency discrimination model to judge whether the dominant fundamental frequency track of each audio belongs to the song melody.

The original algorithm uses the shallow BP neural network as the fundamental frequency discrimination model. At present, the deep neural network shows better performance in the classification task and has been widely used in the field of speech signal processing. Therefore, sparse self coding neural network (SAENN) and softmax classifier will be used to establish the discrimination model of training fundamental frequency, so as to achieve better recognition effect of vocal fundamental frequency.

Sparse self coding (SAE) neural network adds sparse constraint on the basis of self coding (AE) neural network. An unsupervised feature learning algorithm is adopted. The algorithm limits the sparsity to the neurons in the middle hidden layer. The trained sparse noise reduction self coding neural network is equivalent to a nonlinear filter. A small number of hidden layer activation units are used to characterize the original signal and automatically screen the signal significance atoms, which is of great significance to improve the recognition rate of vocal fundamental frequency.

The front end of the fundamental frequency discrimination model is a sparse self coding neural network, and its structure is 12-200-50-16, that is, the input is a 12 dimensional MFCC feature that can reflect the envelope characteristics of the sound spectrum. The hidden layer neurons used in the middle are 200, 50 and 16 respectively. The back end of the last hidden layer neuron is connected to the softmax classifier, which contains two outputs. They represent the fundamental frequency of singing and instrumental music respectively [13].

In addition, the super parameters are set as follows: the weight penalty parameter

During the training, a square wave equation algorithm is used to optimize the weight. The algorithm stores only the last m iteration curve information and uses it to construct a matrix close to the Hessian matrix. The cost of each iteration is low, and each iteration can provide positive clarity of the matrix. Based on the above basic frequency discrimination model. Determine if the dominant frequency of each sound segment is the main melody of the song. The specific steps are as follows.

Firstly, a comb filter with rectangular wave

Find the harmonic spectrum corresponding to the fundamental frequency _{0}: filter the amplitude spectrum of the signal with the comb filter constructed in the above step, and extract the Mel cepstrum parameter MFCC of the harmonic spectrum (the product of the combination of sound generation mechanism and human auditory perception characteristics, which is one of the important characteristic parameters of audio signal analysis).

The MFCC parameters are sent into the trained fundamental frequency discrimination model for recognition, and the judgment result of whether _{0} is the fundamental frequency of singing is obtained.

If the number of frames judged as the fundamental frequency of the song in each voiced segment is less than 1 / 2 of the total number of frames in a segment, the track of _{0} is not the main melody of the song, on the contrary, it is the main melody of the song.

The music data sets used in the experiment are all taken from the mir-1k data set provided by Hsu. It is composed of 110 Chinese karaoke songs. The singing part is 1000 music clips recorded by amateur song lovers with a sampling rate of 16 kHz. The background music and song are input into the left and right channels respectively, including the song fundamental frequency tag with a time interval of 10 ms. 500 pieces of music are randomly selected to train the neural network of the dominant fundamental frequency discrimination model, and the other 500 pieces of music are used as the main melody extraction test set.

In order to compare with the fundamental frequency discriminant model of the original algorithm. The same experimental conditions are used to train and test the SAE-based fundamental frequency discriminant model. The data used is shown in Table 1.

Test data of model training data

Number of dominant fundamental frequency frames | Style dominant fundamental frame number | Total frames | |
---|---|---|---|

Training data | 13789 | 14576 | 28365 |

test data | 12475 | 17543 | 30018 |

Use the above training data. The quadratic wave equation algorithm is used to train the fundamental frequency discrimination model based on sparse self coding neural network. After completing the model training, the performance test is carried out by using the test set data. Firstly, extract the MFCC parameters of each frame of the test data signal and its corresponding label, then send the MFCC parameters of each frame of the signal into the fundamental frequency discrimination model, obtain the judgment result of whether it is the fundamental frequency of human voice, compare it with the label result, and finally count the recognition accuracy

Where _{1} is the number of frames judged to be correct in the test data set, and N is the number of frames of the total test data.

Experiments show that the recognition accuracy of the model is 85.1%, and the performance index is about 5% higher than that of the original algorithm, which is better than the fundamental frequency discrimination model of the original algorithm. Subsequent experiments show that the application of this model can reduce the false alarm rate of melody location and improve the overall accuracy of the algorithm [14].

There are two main tasks in extracting the main melody of a song: one is to judge whether the melody really exists, and the other is to accurately calculate the melody of the main melody (there should be no difference between the calculated sound level and the reference value. More than half a tone). Around the main task, the algorithm's performance is also assessed using five performance criteria: melody call speed (VR), pitch false alarm level (false signal level, VFAR), and original sound level (raw frequency accuracy, RPA), raw chromium accuracy (raw chromium accuracy, RCA), general accuracy (total accuracy, 0A). Based on the first five commonly used evaluation indices, the “average extraction time” of the sixth evaluation index is added to compare the calculated size of the improved algorithm and the original algorithm.

The experimental data used were 500 randomly selected music. The following experiments were performed under SIR 0 dB and SIR 5 dB. The specific results are shown in Figure 2 and Figure 3.

As shown in Figure 2, Figure 3, all performance parameters of the enhanced algorithm are higher than the original algorithm, tone localization speed (VRR), initial sound frequency accuracy (RPA), initial chroma accuracy level (RCA), and the General Accuracy Level (OA) of the Four Parameters The false alarm level (VFAR) of the tone localization is reduced by about 2% if the signal and interference ratios are improved under different conditions, indicating that the improved algorithm is able to more accurately identify the accessory. and singing melody., because the accuracy of recognizing the basic frequency differential model in an enhanced algorithm is higher than in the original algorithm. In addition, the AET of the improved algorithm is reduced by approximately 0.12 seconds compared to the original algorithm, which ensures that the process calculation and complexity of the frequency function function of the improved algorithm is lower than that of the original algorithm melody of music [15].

This paper proposes an algorithm for extracting the quadratic wave equation and extracting the sound theme based on the basic frequency differentiation of the singing. Compared to traditional methods, this paper introduces a segmentation algorithm based on the stability properties of notes and a basic model for distinguishing frequencies based on the timbre characteristics of singing instruments. The note segmentation algorithm helps to control the predominant base frequency trajectory and the localization of the melody. Experiments have shown that the basic tone differentiation algorithm in this article can effectively reduce octave error when the signal interference is 5 dB and 0 dB, and when the false signal level of tone localization is much lower than other algorithms. The overall level of accuracy is higher than other algorithms, which can effectively produce the basic melody of the song.

#### Test data of model training data

Number of dominant fundamental frequency frames | Style dominant fundamental frame number | Total frames | |
---|---|---|---|

Training data | 13789 | 14576 | 28365 |

test data | 12475 | 17543 | 30018 |

Law of interest rate changes in financial markets based on the differential equation model of liquidity Research and implementation of smart city public information mining analysis system based on mobile edge model of game theory Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Design of college education evaluation based on accompanying data acquisition and mathematical analysis Enterprise financial strategy and performance management analysis based on principal component analysis Study on the Dynamic Change of Regional Water Level and Climate Based on Forecast Equation Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Municipal Civil Engineering Construction Based on Finite Element Differential Equations Financial Risk Prevention Model of Financial Institutions Based on Linear Partial Differential Equation Influence of heterogeneity of local officials on the economy of resource-based cities developed with high quality Analysis of influencing factors of SPOC course teaching effect using structural equation modelling Training Model of Basketball Offensive Route Based on Nonlinear Differential Equation Research on the construction of rural interface style based on aesthetic rules Optimization Algorithm of New Media Hot Event Push Based on Nonlinear Differential Equation Mathematical Differential Equation in Calculating the Strength of Building Beam Structure Impact Resistance Stability of Building Structural Engineering Based on Fractional Differential Equations The Technical Research on the Assessment of Network Security Situation Based on D-S Evidence Theory Computer big data modeling system based on finite element mathematical equation simulation Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation The Composition System of Pop Music Movement Based on Finite Element Differential Equations The Structure and Influencing Factors of Innovation and Entrepreneurship Ability of Higher Vocational Students Based on Structural Equation Model Model System Study of Accordion Score Based on Fractional Differential Equations Data mining of Chain convenience stores location Parameter Id of Metal Hi-pressure State Equation P-Matrix Reasoning and Information Intelligent Mining Research on Detection Model of Abnormal Data in Engineering Cost List Research on deformation monitoring of tunnel engineering based on 3D laser scanning Solidification treatment effect of wellsite waste mud based on physical adsorption of a composite curing agent A study of immersive technology for product usability improvement design based on comprehensive value evaluation Intelligent Recommendation System for English Vocabulary Learning – Based on Crowdsensing Application of Nonlinear Fractional Differential Equations in Computer Artificial Intelligence Algorithms Application of calculus function and value chain analysis method in the quantification of financial flexibility management indicators Optimal Research in Piano Shape Sound and Sound Propagation Model Based on Nonlinear Differential Equations Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations Badminton players’ trajectory under numerical calculation method Mathematical methodology in the seismic resilience evaluation of the water supply system Prediction of surface quality in end milling based on modified convolutional recurrent neural network Analysis of IPO pricing efficiency under the registration system Case application data research of traditional ink art elements in packaging design Research on an early warning model of effectiveness evaluation in ideological and political teaching based on big data Application of intelligent teaching resource organisation model and construction of performance evaluation model Secure transmission of simultaneous wireless information and power transfer system for Internet of things Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Numerical simulation of vortex vibration in main girder of cable-stayed bridge based on bidirectional fluid–structure coupling Research on location algorithm of new energy vehicle charging station based on multi-objective decision Optimisation of construction mode of residential houses based on the genetic algorithm under BIM technology A study on the reform of college English education inspired by the cultural differences between China and the United States Research on innovative strategies of college students’ English teaching under the blessing of big data Research on multi-dimensional optimisation design of user interface under Rhino/GH platform Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption Attitude control for the rigid spacecraft with the improved extended state observer A long command subsequence algorithm for manufacturing industry recommendation systems with similarity connection technology Application of Higher-Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Evaluation of spoken English self-study system considering speech knowledge recognition algorithm Application research on piano teaching in colleges and universities based on remote wireless network communication Construction of Financial Risk Evaluation Index System for Biomass Graphene Fibre Industrialisation Project System dynamics model of output of ball mill Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Design of Morlet wavelet neural network to solve the non-linear influenza disease system An online statistical analysis of the hotel management and operation performance model Research on the post-purchase behaviour tendency of the product quality to customers in fast-selling marketing Motion about equilibrium points in the Jupiter-Europa system with oblateness Ultra-short-term power forecast of photovoltaic power station based on VMD–LSTM model optimised by SSA Optimal preview repetitive control for impulse-free continuous-time descriptor systems Design of information management system based on wireless communication under the background of Internet of Things The impact of global warming on the small Scottish Fishing Company Development of main functional modules for MVB and its application in rail transit Analysis of agricultural economic development and optimisation measures under the strategy of rural revitalisation Study on the impact of forest fire prevention policy on the health of forest resources Value Creation of Real Estate Company Spin-off Property Service Company Listing Selection by differential mortality rates Research on the relationship between government subsidies, R&D investment and high-quality development of manufacturing industry Research on the protection and inheritance of intangible cultural heritage under the background of rural revitalisation Research on behavioural differences in the processing of tenant listing information: An eye-movement experiment Innovation of Economic Management Risk Control in Retailer Supply Chain Based on Differential Equation Model Computer vision recognition and tracking algorithm based on convolutional neural network A review of the treatment techniques of VOC Study on structural parameter design and life analysis of large four-point contact ball bearing channel Some classes of complete permutation polynomials in the form of ( x ^{pm}−x +δ )^{s}+ax ^{pm}+bx overF _{p2m}Digital marketing solutions based on consumer data and homomorphic encryption The consistency method of linguistic information and other four preference information in group decision-making Statistical Model of College Ideological and Political Learning Based on Fractional Differential Equations Research on the driving principle and guiding strategy of the public's collaborative supervision of the sharing economy in my country Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development Nonlinear Differential Equation in Anti-aging Test of Polymer Nanomaterials Application research of bel canto performance based on artificial intelligence technology Fractal structure of magnetic island in tokamak plasma Mechanics of Building Structural Materials Based on Lagrangian Mathematical Model Analysis The Mental Health Education Management of Higher Vocational Students Based on Fractional Differential Equations Application of regression function model based on panel data in financial risk management of bank resource allocation Application of knowledge graph in smart grid fault diagnosis University Ideological and Political Learning Model Based on Statistical Memory Curve Mathematical Equation Research on the optimisation of logistics parcel intelligent sorting and conveying chain combined with variable clustering mathematical method Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Research on Resonance Properties of Semantic Wave Fractal Fractals Based on Quantitative Analysis of English Corpus Research on urban landscape big data information processing system based on ordinary differential equations Modeling of fractional differential equation in cloud computing image fusion algorithm Application of Discriminative Training Algorithm Based on Intelligent Computing in English Translation Evaluation Research on the application of GLE teaching mode in English-medium colleges The application of directional derivative in the design of animation characters and background elements Research on product process design and optimisation model based on IoT intelligent computing Conventional Algorithms in Sports Training Based on Fractional Differential Equations Dynamic Nonlinear System Based on Complex System Theory in the Development of Vocational Education LTE wireless network coverage optimisation based on corrected propagation model The Algorithm Accuracy of Mathematical Model to Improve the Transmission Speed of E-commerce Platform Study on tourism development income index calculation of finite element ordinary differential mathematical equation The Security of Database Network Model Based on Fractional Differential Equations Electric Vehicle Mechanical Transmission System Based on Fractional Differential Equations The Practice System of Physics and Electronics Courses in Higher Vocational Colleges Based on Fractional Differential Equations The Economic Model of Rural Supply and Demand Under the Data Analysis Function Based on Ordered Probit 3D Modeling System of Indoor Environment Art Landscape Design under Statistical Equation Electronic Information Security Model of Nonlinear Differential Equations The Optimization Model of College Students' Physical Exercise Motivation and Self-control Ability Based on the Mathematical Model of Probability Theory Paris panxiensis , a new combination of Trilliaceae from Sichuan, ChinaImpact of ASEAN-China free trade area on fishery value chain based on difference-in-difference method Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas Criminal law imputation path for biometric information Research on composite dynamic disaster prevention and control system of mine earthquake and shock in thick and hard rock mines Research on innovative strategies of college students’ English teaching under the background of artificial intelligence Electromagnetic interference prediction technology of new energy motor drive system Research on the application of PLC technology in electrical automation engineering Research on indoor environment design of comprehensive commercial shopping center based on numerical simulation Application of matrix multiplication in signal sensor image perception Empirical analysis of the economic absolute income hypothesis based on mathematical statistics Analysing the variation of metadiscourse verb patterns in English academic papers from within and between disciplines Impact of COVID-19 policy on stock prices of listed property companies Realization of Book Collaborative Filtering Personalized Recommendation System Based on Linear Regression Equation Research on the experimental principle of deep integration of LETS software and criminal procedure under the background of artificial intelligence Study on Interactive Relations between Enterprise Social Media and Decision Style Based on a vector Autoregressive Model Research on Dynamics of Flexible Multibody System with Deployable Antenna Based on Static Lagrangian Function The Optimization of Mathematics Teaching Models in Colleges and Universities Based on Nonlinear Differential Equations Study on spatial planning and design of learning commons in university libraries based on fuzzy matrix model The Stability Model of Piano Tone Tuning Based on Ordinary Differential Equations Construction and application of automobile user portrait based on k-mean clustering model The Homework Model of Screening English Teaching Courses Based on Fractional Differential Equations VR-based computer maintenance practical training platform development design and application research 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 Knowledge graph construction and Internet of Things optimisation for power grid data knowledge extraction Modeling the pathway of breast cancer in the Middle East Construction and intelligent analysis of power grid physical data knowledge graph based on Internet of Things for power system Research on industrial Internet of Things and power grid technology application based on knowledge graph and data asset relationship model Research on the effects of financial literacy on rural household credit constraint Calculus Logic Function in Integrated Manufacturing Automation of Single Chip Microcomputer Football Offense Training Strategy Based on Fractional Differential Mathematical Modeling Research on educational resource recommendation system based on MRLG Rec The Mathematical Analysis Model of Educational System in Music Courses in Colleges and Universities Continuing Education Network Data Center Model Based on Fractional Differential Mathematical Equations A study on the phenomenon of anaphoric correction in college students’ English conversation Computer Art Design Model Based on Nonlinear Fractional Differential Equations The Optimization Model of Public Space Design Teaching Reform Based on Fractional Differential Equations The Approximate Solution of Nonlinear Vibration of Tennis Based on Nonlinear Vibration Differential Equation Graphical Modular Power Technology of Distribution Network Based on Machine Learning Statistical Mathematical Equation Employment and Professional Education Training System of College Graduates Based on the Law of Large Numbers Economic Research on Multiple Linear Regression in Fruit Market inspection and Management Nonlinear Differential Equations in Preventing Financial Risks Lagrange’s Mathematical Equations in the Sports Training of College Students Simulation Research of Electrostatic Precipitator Power Supply Voltage Control System Based on Finite Element Differential Equation Research on the effect of generative adversarial network based on wavelet transform hidden Markov model on face creation and classification Research on Lightweight Injection Molding (CAE) and Numerical Simulation Calculate of New Energy Vehicle Power Flow Calculation in Smart Distribution Network Based on Power Machine Learning Based on Fractional Differential Equations Demonstration of application program of logistics public information management platform based on fuzzy constrained programming mathematical model Basketball Shooting Rate Based on Multiple Regression Logical-Mathematical Algorithm The Optimal Application of Lagrangian Mathematical Equations in Computer Data Analysis Similarity Solutions of the Surface Waves Equation in (2+1) Dimensions and Bifurcation Optimal decisions and channel coordination of a green supply chain with marketing effort and fairness concerns Game theoretic model for low carbon supply chain under carbon emissions reduction sensitive random demand Limit cycles of a generalised Mathieu differential system Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Application of data mining in basketball statistics The nonlinear effects of ageing on national savings rate – An Empirical Study based on threshold model Design of fitness walker for the elderly based on ergonomic SAPAD model AtanK-A New SVM Kernel for Classification Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Study of a linear-physical-programming-based approach for web service selection under uncertain service quality The Relationship Between College Students’ Taekwondo Courses and College Health Based on Mathematical Statistics Equations Analysis and countermeasures of cultivating independent learning ability in colleges teaching English based on OBE theory A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Fractional Differential Equations in the Exploration of Geological and Mineral Construction AdaBoost Algorithm in Trustworthy Network for Anomaly Intrusion Detection Projection of Early Warning Identification of Hazardous Sources of Gas Explosion Accidents in Coal Mines Based on NTM Deep Learning Network Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China Enterprise Financial Risk Early Warning System Based on Structural Equation Model A Study on the Application of Quantile Regression Equation in Forecasting Financial Value at Risk in Financial Markets Fractional Differential Equations in the Model of Vocational Education and Teaching Practice Environment Information transmission simulation of Internet of things communication nodes under collision free probability equation Image denoising model based on improved fractional calculus mathematical equation Random Fourier Approximation of the Kernel Function in Programmable Networks The Complexity of Virtual Reality Technology in the Simulation and Modeling of Civil Mathematical Models University Library Lending System Model Based on Fractional Differential Equations Calculation and Performance Evaluation of Text Similarity Based on Strong Classification Features Intelligent Matching System of Clauses in International Investment Arbitration Cases Based on Big Data Statistical Model Evaluation and Verification of Patent Value Based on Combination Forecasting Model Financial Institution Prevention Financial Risk Monitoring System Under the Fusion of Partial Differential Equations Prediction and Analysis of ChiNext Stock Price Based on Linear and Non-linear Composite Model Calculus Logic Function in Tax Risk Avoidance in Different Stages of Enterprises The Psychological Memory Forgetting Model Based on the Analysis of Linear Differential Equations Optimization Simulation System of University Science Education Based on Finite Differential Equations The Law of Large Numbers in Children's Education Optimization System of Strength and Flexibility Training in Aerobics Course Based on Lagrangian Mathematical Equation Data structure simulation for the reform of the teaching process of university computer courses RETRACTION NOTE Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Research on non-linear visual matching model under inherent constraints of images Good congruences on weakly U-abundant semigroups Can policy coordination facilitate unimpeded trade? An empirical study on factors influencing smooth trade along the Belt and Road Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Research on design of customer portrait system for E-commerce Research on rule extraction method based on concept lattice of intuitionistic fuzzy language Fed-UserPro: A user profile construction method based on federated learning A multi-factor Regression Equation-based Test of Fitness Maximal Aerobic Capacity in Athletes Design and evaluation of intelligent teaching system on basic movements in PE Garment Image Retrieval based on Grab Cut Auto Segmentation and Dominate Color Method Financial Risk Prediction and Analysis Based on Nonlinear Differential Equations Constructivist Learning Method of Ordinary Differential Equations in College Mathematics Teaching Multiple Effects Analysis of Hangzhou Issuing Digital Consumer Coupons Based on Simultaneous Equations of CDM Model Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations A Hybrid Computational Intelligence Method of Newton's Method and Genetic Algorithm for Solving Compatible Nonlinear Equations Pressure Image Recognition of Lying Positions Based on Multi-feature value Regularized Extreme Learning Algorithm English Intelligent Question Answering System Based on elliptic fitting equation Precision Machining Technology of Jewelry on CNC Machine Tool Based on Mathematical Modeling Application Research of Mathematica Software in Calculus Teaching Computer Vision Communication Technology in Mathematical Modeling Skills of Music Creation Based on Homogeneous First-Order Linear Partial Differential Equations Mathematical Statistics Technology in the Educational Grading System of Preschool Students Music Recommendation Index Evaluation Based on Logistic Distribution Fitting Transition Probability Function Children's Educational Curriculum Evaluation Management System in Mathematical Equation Model Query Translation Optimization and Mathematical Modeling for English-Chinese Cross-Language Information Retrieval The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization research on prefabricated concrete frame buildings based on the dynamic equation of eccentric structure and horizontal-torsional coupling Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Green building considering image processing technology combined with CFD numerical simulation Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching Research on evaluation system of cross-border E-commerce platform based on the combined model A mathematical model of PCNN for image fusion with non-sampled contourlet transform Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Design of Tennis Mobile Teaching Assistant System Based on Ordinary Differential Equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Animation VR scene mosaic modeling based on generalized Laplacian equation Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations The influence of accounting computer information processing technology on enterprise internal control under panel data simultaneous equation Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Construction of comprehensive evaluation index system of water-saving irrigation project integrating penman Montei the quation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression Analysis of technical statistical indexes of college tennis players under the win-lose regression function equation Automatic extraction and discrimination of vocal main melody based on quadratic wave equation Analysis of wireless English multimedia communication based on spatial state model equation Optimization of Linear Algebra Core Function Framework on Multicore Processors Application of hybrid kernel function in economic benefit analysis and evaluation of enterprises Research on classification of e-commerce customers based on BP neural network The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Fractional Differential Equations in Electronic Information Models BIM Engineering Management Oriented to Curve Equation Model Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Design of language assisted learning model and online learning system under the background of artificial intelligence Study on the influence of adolescent smoking on physical training vital capacity in eastern coastal areas Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering Action of Aut( G ) on the set of maximal subgroups ofp -groupsThe internal mechanism of corporate social responsibility fulfillment affecting debt risk in China: analysis of intermediary transmission effect based on degree of debt concentration and product market competitive advantage Study on transmission characteristics in three kinds of deformed finlines based on edge-based finite element method Asymptotic stability problem of predator–prey system with linear diffusion Research on loyalty prediction of e-commerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Evaluation and prediction of regional human capital based on optimised BP neural network Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long short-term memory networks A study of local smoothness-informed convolutional neural network models for image inpainting Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of vertical video creation under the background of mobile communication Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithm-based congestion control optimisation for mobile data network