In recent years, multimodal audio and video signal processing has been a research hotspot in speech recognition research. The fusion of speech auditory and visual speech information during speaking can improve speech signals' recognition rate and robustness in noisy environments. However, the experimental results of speech recognition show that although the visual movement of the lips is related to the speech signal, it is not synchronized [1]. Lip visual motion precedes speech signal by about 120ms on average. A complete audio-visual speech recognition system should reflect this fact as accurately as possible.

There are two main methods for extracting visual speech features: video image pixel and mouth contour feature extraction. The former directly performs some transformation on the image area of the mouth. The algorithm can then extract the transform domain parameters as features [2]. The advantage is that it can retain information such as teeth and tongue when speaking. But the algorithm is susceptible to visual image background and different speakers. The latter extracts the contour features of the mouth shape when speaking.

The algorithm is not easily affected by the experience of the visual image. At present, the optical elements based on the mouth contour mainly directly use the coordinates of the mouth contour feature points or the geometric features (perimeter, height, width, etc.) of the mouth contour points. Some scholars use genetic algorithms to obtain contour coordinate points and then perform discrete Fourier transform on the coordinate points and the contour coordinate points of adjacent frames. Then we can get the dynamic characteristics of mouth shape. However, the genetic algorithm only uses larger eigenvalues and corresponding eigenvectors in model reconstruction and does not consider the residual part. Significantly when the lips change drastically, the algorithm is not very accurate for the calibration of the mouth contour points. Therefore, if we directly use the coordinate points of the contour to perform discrete Fourier transform, the visual features obtained are not robust.

This paper adopts the Bayesian Tangent Shape Model (BTSM) algorithm. This paper obtains a more accurate mouth contour by establishing a distribution model for the residual part of the principal component analysis [3]. Then we extract the geometric features of lips such as the width, height, and angle of lips as the geometric features of visual speech according to the contour feature points. Finally, we transform these visual geometric features using discrete Fourier transforms. This eliminates the redundancy of simple geometric elements and reduces the feature dimension. At this time, we can obtain more robust lip-dynamic speech features.

Audio and video fusion methods include feature fusion, model fusion, and decision fusion. Model fusion has been proven to be a better fusion method. Some scholars have conducted analysis and identification experiments on the Multi-Stream Hidden Markov Model (MSHMM) and its various extended models [4]. These models describe, to some extent, the asynchrony between audio and video streams. However, these models can only model synchronous and asynchronous relations at the phoneme level in the continuous speech recognition of audio and video with large and medium vocabularies. However, in many cases, the asynchronous degree of audio and video data flow has exceeded the boundary of the phoneme. Describing the asynchrony of audio and video streams over more extended periods should yield better recognition results.

Because HMM has certain limitations on complex multimodal speech recognition expression ability, more related research has been opened in the academic community. Some scholars use single-stream and multi-stream dynamic Bayesian network (DBN) models to conduct continuous speech recognition research. HMM is a particular case of DBN, and DBN is a generalization of HMM. Compared with HMM, the DBN model has better scalability and interpretability [5]. Some scholars have established a multi-stream DBN model. The algorithm forces the audio and video streams to synchronize through word transition probabilities at word nodes. But this model is only applied to speech recognition of multiple audio features. Some scholars have extended this model and used it for audio and video speech recognition. The study found an improvement in recognition rates. Some scholars have found that the influence of audio and video streams on word transition probability nodes is very controversial. The state transition probabilities of video streams are simultaneously affected by the corresponding nodes in both streams. This limits the description of the asynchrony of audio and video streams to some extent.

This paper introduces a multi-stream asynchronous DBN model. We call it the MS-ADBN model for short, in which the audio and video streams are synchronized at word nodes. The transition probability of the word node is jointly determined by the nodes of the audio stream and the video stream [6]. This model truly reflects the asynchrony of audio and video streams at the word level. We use the MS-ADBN model to conduct joint speech recognition experiments on audio and video streams. At the same time, the model is compared with the recognition results of the MSHMM model under the same experimental conditions. We got better results here.

Visual features based on lip contours have been a better visual speech feature. Since the mouth is a flexible target, we use the traditional genetic algorithm model to analyze the labeled lip training data. At this point, we build a statistical parametric model. Then we use this model to match the new lip image to get the mouth contour. However, the genetic algorithm only uses larger shape space eigenvalues to describe the model, which does not consider the reconstruction error. The BTSM model establishes a distribution model to describe the reconstruction error using principal component analysis (PCA) to describe the model [7]. The algorithm can tell the mouth contour more accurately than the genetic algorithm model. At the same time, we use the EM (Expectation-maximization) algorithm to train and estimate the matching parameters of the model. The main steps are as follows.

First, we manually annotate the training samples of mouth images. And we normalize it to the same reference frame.

Principal component analysis and prior shape model. We use _{t}_{1}, _{1}, ⋯, _{t}

Where

Where _{i}

Establish a local point, morphological model. The algorithm performs grayscale sampling in the average direction of each contour feature point based on the training samples. We build the local contour point model. We sample

Search the new image mouth contour.

We compute the model parameters within a Bayesian framework for the image _{1}, ⋯ _{N}_{1}, ⋯, _{N}

In this paper, a total of 20 points (shown in Figure 1) of the lips' inner and outer contour features obtained by the BTSM algorithm are used to represent the mouth shape. In this paper, the geometric features of the mouth shape are extracted directly according to the contour feature points [8]. The content includes 20-dimensional features of the opening degree (8 horizontal and eight vertical distances) and the angle of opening (_{1}, _{2}, ⋯, _{N}

Although this paper uses BTSM to obtain more accurate contour feature points, the recognition effect is general when we directly use static geometric features as visual speech features. It is more important for speech signals to reflect the dynamic characteristics of mouth shape changes. The dimension is relatively large if we take differential features for geometric components [9]. The existence of information redundancy between features increases the intermediate parameters and training parameters required by the model. In this paper, the method of the document is used to perform discrete Fourier transform on multi-frame features to obtain the dynamic characteristics of mouth shape based on Visemic discrete Fourier. Later, this paper directly performs discrete Fourier transform on the contour coordinate points of multiple frames. However, the mouth shape coordinates obtained by the parametric model are not very accurate, significantly when the lips change drastically. In this paper, discrete Fourier transform is performed based on geometric features such as the mouth shape's height, width, and angle. In this way, we can obtain more stable visual speech dynamic features.

The voice data sampling rate is 100Hz, and the corresponding video data sampling rate is 25Hz. We first perform linear interpolation to make the audio and video feature sampling rates consistent [10]. The article reflects the dynamic information of visual geometric features. This paper takes the feature information of 5 frames before and after this data frame and connects them in sequence. This is equivalent to moving an observation window with a length of 11 frames in the lip-shaped geometric feature sequence to obtain multi-frame connected lip-shaped features. It will be the input to the discrete Fourier transform.

The number of classifications needs to be determined before the discrete Fourier transform is performed. Since a viseme is the basic speech unit in the visual domain, there is a particular mapping relationship between phonemes and visemes. Phonemes can be clustered into several graphic viseme classes in the visual environment [11]. At this point, we can think that vise retains the basic discriminative information of speech-visual features. In this paper, the number of categories of visemes is regarded as the number of classes of discrete Fourier. The digital audio and video database of this paper have 13 viseme units.

Earlier, we determined the input and output dimensions of the discrete Fourier. The input is 220 dimensions, and the output is 13 dimensions. We first need to train the discrete Fourier. The input training data is _{ci},

Where _{B}_{w}

Pr(_{c}_{c}_{W}_{B}_{W}_{lda}_{lda}X_{lda}

The audio and video database adopts the English digital continuous voice audio and video database jointly recorded by the laboratory. The database was recorded in a quiet environment. The video image is a frontal head video image. The database contains 11 words with numbers 0-10. This involves a total of 22 phonemes. The database scripts are recorded in sentence order of the Aurora 2.0 database. This paper uses 100 sentences of pure audio and video data as the training set. Another 50 sentences and the corresponding audio and video data of the noise-added speech are used as the test set.

The frame rate of audio data is 100 frames/s. We adopt HTK to extract 13-dimensional MFCC features and energy features of audio data. We obtain 13-dimensional visual features from the video data according to the BTSM mouth contour feature extraction algorithm and discrete Fourier projection [12]. At this point, we call it discrete Fourier features. We used the 20-dimensional static geometric features of the mouth shape as a comparison.

There are 22 phonemes in both audio and video streams in the experiment. At the same time, we add mute and pause, a total of 50 Gaussian model parameters need to be trained. The training set has more than 60 samples per word. The above model parameters can be appropriately trained. This article uses the single-stream DBN model, conventional HMM, and multi-stream asynchronous HMM (MSHMM) for comparative simulation [13]. The author conducts speech recognition experiments using MFCC-D-A audio features, discrete Fourier visual features, and geometric visual features. For the multi-stream DBN (MS-ADBN) model and the multi-stream HMM, the authors also conducted speech recognition experiments under the same experimental conditions using a combination of MFCC-D-A audio features and discrete Fourier or geometric visual features, respectively. The results are shown in Table 1.

Comparison of word recognition rates under various HMM and DBN models

System | SNR/% | |||
---|---|---|---|---|

0dB | 5dB | 10dB | 15dB | |

SS-DBN (MFCC-D-A) | 42.94 | 66.1 | 71.75 | 77.97 |

HMM (MFCC-D-A) | 27.55 | 42.76 | 62.67 | 74.62 |

SS-DBN (Static Geometric Features) | 54.58 | 54.58 | 54.58 | 54.58 |

SS-DBN (Discrete Fourier) | 58.67 | 58.67 | 58.67 | 58.67 |

HMM (Static Geometric Features) | 46.54 | 46.54 | 46.54 | 46.54 |

HMM (Discrete Fourier) | 54.21 | 54.21 | 54.21 | 54.21 |

MSHMM (MFCC-D-A and static geometry features) | 29.28 | 49.72 | 62.84 | 77.97 |

MSHMM (MFCC-D-A and Discrete Fourier) | 25.76 | 55.76 | 70 | 82.84 |

MS-ADBN (MFCC-D-A and static geometric features) | 41.18 | 56.21 | 72.2 | 82.66 |

MS-ADBN (MFCC-D-A and Discrete Fourier) | 42.79 | 67.86 | 75.82 | 82.01 |

System | SNR/% | |||

20dB | 30dB | Clean | 0-15dB | |

SS-DBN (MFCC-D-A) | 82.46 | 96.62 | 97.74 | 64.69 |

HMM (MFCC-D-A) | 85.67 | 98.2 | 98.44 | 52.88 |

SS-DBN (Static Geometric Features) | 54.58 | 54.58 | 54.58 | 54.58 |

SS-DBN (Discrete Fourier) | 58.67 | 58.67 | 58.67 | 58.67 |

HMM (Static Geometric Features) | 46.54 | 46.54 | 46.54 | 46.54 |

HMM (Discrete Fourier) | 54.42 | 54.42 | 54.42 | 54.42 |

MSHMM (MFCC-D-A and static geometry features) | 84.75 | 92.42 | 94.24 | 55.24 |

MSHMM (MFCC-D-A and Discrete Fourier) | 88.28 | 92.66 | 96.05 | 62.09 |

MS-ADBN (MFCC-D-A and static geometric features) | 88.89 | 94.22 | 96.08 | 64.56 |

MS-ADBN (MFCC-D-A and Discrete Fourier) | 89.72 | 94.56 | 96.74 | 67.62 |

When the signal-to-noise ratio of the speech signal is relatively high, the recognition rate of the audio feature data of the DBN model is slightly lower than that based on the HMM. The most likely reason is that this DBN model uses a monotone model, while the HMM uses a triphone model. When the signal-to-noise ratio is low, especially when it is less than or equal to 15dB, the recognition rate of the DBN model is 12.81% higher than that of HMM on average. It has better robustness to noise.

Regardless of HMM or SSDBN model, the recognition rate of discrete Fourier features is 7.71% and 4.09% higher than that of geometric elements, respectively. This shows that discrete Fourier features are more robust. When discrete Fourier video features are used in this paper, the recognition rate of the SS-DBN model is 4.36% higher than that of HMM.

When the signal-to-noise ratio is low, the performance of the multi-stream model is better than the recognition rate of the corresponding single-stream model due to the combination of video and speech features [14]. The recognition performance of the multi-stream model that fuses audio elements and discrete Fourier features is better than that of the multi-stream model that connects audio features and geometric features. The MSHMM and MS-ADBN models, which fuse audio features and discrete Fourier features, improve performance by 9.21% and 2.93%, respectively, over the corresponding single-stream models. This shows that the discrete Fourier visual speech features used are relatively robust.

The performance of the MS-ADBN model is significantly better than that of MSHMM. Because the MS-ADBN model describes the synchronous and asynchronous relationship of the audio and video streams within the word, the MSHMM model restricts the audio and video to be synchronized at the phoneme nodes. The average recognition rate of the MSADBN model is 4.92% and 6.53% higher than MSHMM in the test environment with a signal-to-noise ratio of 0–30dB.

The highest recognition rate of the multi-stream model in Table 1 under pure speech is only 96.73%. This is because the training data is not yet sufficient. The flow index problem involving complex audio and video streams is not discussed in more detail in this paper. Model details require more in-depth study.

This paper constructs a multi-stream asynchronous dynamic Bayesian network (MS-ADBN) model. The word transition probabilities of this model are jointly determined by the audio stream and the video stream. It reflects the synchronous and asynchronous nature of audio and video streams at the word level. At the same time, we use the Bayesian Tangent Shape Model (BTSM) algorithm to extract the mouth contour and introduce a novel visual feature. The model performs discrete Fourier transform on multi-frame mouth geometric features to obtain a robust discrete Fourier visual speech dynamic quality. We adopt the mouth geometry feature and discrete Fourier feature, and the average recognition rate (0~30dB) of the MS-ADBN model is 6.53% and 4.92% higher than MSHMM, respectively. This is because MSHMM enforces audio and video synchronization at the phoneme level, while the MS-DBN model relaxes this asynchrony limit to the word level. For the single-stream SS-DBN model, the speech recognition rate using discrete Fourier visual speech features is higher than that using mouth geometry features. This indicates that the extracted discrete Fourier visual features are more robust.

#### Comparison of word recognition rates under various HMM and DBN models

System | SNR/% | |||
---|---|---|---|---|

0dB | 5dB | 10dB | 15dB | |

SS-DBN (MFCC-D-A) | 42.94 | 66.1 | 71.75 | 77.97 |

HMM (MFCC-D-A) | 27.55 | 42.76 | 62.67 | 74.62 |

SS-DBN (Static Geometric Features) | 54.58 | 54.58 | 54.58 | 54.58 |

SS-DBN (Discrete Fourier) | 58.67 | 58.67 | 58.67 | 58.67 |

HMM (Static Geometric Features) | 46.54 | 46.54 | 46.54 | 46.54 |

HMM (Discrete Fourier) | 54.21 | 54.21 | 54.21 | 54.21 |

MSHMM (MFCC-D-A and static geometry features) | 29.28 | 49.72 | 62.84 | 77.97 |

MSHMM (MFCC-D-A and Discrete Fourier) | 25.76 | 55.76 | 70 | 82.84 |

MS-ADBN (MFCC-D-A and static geometric features) | 41.18 | 56.21 | 72.2 | 82.66 |

MS-ADBN (MFCC-D-A and Discrete Fourier) | 42.79 | 67.86 | 75.82 | 82.01 |

System | SNR/% | |||

20dB | 30dB | Clean | 0-15dB | |

SS-DBN (MFCC-D-A) | 82.46 | 96.62 | 97.74 | 64.69 |

HMM (MFCC-D-A) | 85.67 | 98.2 | 98.44 | 52.88 |

SS-DBN (Static Geometric Features) | 54.58 | 54.58 | 54.58 | 54.58 |

SS-DBN (Discrete Fourier) | 58.67 | 58.67 | 58.67 | 58.67 |

HMM (Static Geometric Features) | 46.54 | 46.54 | 46.54 | 46.54 |

HMM (Discrete Fourier) | 54.42 | 54.42 | 54.42 | 54.42 |

MSHMM (MFCC-D-A and static geometry features) | 84.75 | 92.42 | 94.24 | 55.24 |

MSHMM (MFCC-D-A and Discrete Fourier) | 88.28 | 92.66 | 96.05 | 62.09 |

MS-ADBN (MFCC-D-A and static geometric features) | 88.89 | 94.22 | 96.08 | 64.56 |

MS-ADBN (MFCC-D-A and Discrete Fourier) | 89.72 | 94.56 | 96.74 | 67.62 |

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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 Application of improved RBF neural network algorithm in hierarchical management of enterprise 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