As the main content of animation creation, animation characters are an important factor to attract the audience, which affects the creation quality of the whole animation work [1, 2]. The rise of the animation industry has gradually become clear, so the animation character design market has also been promoted, involving animation derivatives, games, advertisements, posters, mascots, etc., and the commercial value of animation characters has gradually increased [3, 4]. The characters in animation works, like the actors in film and television dramas, undertake the role of interpreting the content of the story, promoting the development of the plot, highlighting the character of the characters and forming the style of the film [5–7]. Excellent animated images can attract audiences of different ages. With time, audiences may gradually forget the plots of animation works, but animated characters with unique personality charm and modelling characteristics often leave a deep impression on audiences [8–10]. Excellent animation characters convey a correct outlook on life, world outlook and values by their pleasing modelling design, smart, humorous and optimistic personality characteristics [11, 12]. To a certain extent, it has the role of ideological inspiration for the audience [13–15]. There are many problems in animation creation. The application of directional derivatives and visual restoration can well solve the problems caused by animation characters during construction and can perform theoretical calculations and modelling more efficiently [16].

Video repair is the process of repairing the incomplete area of picture information, mainly to restore the information contained in the primitive itself. In today’s society where data technology is becoming more and more advanced, picture information has also become one of the important means of signal transmission for human beings. Video inpainting technology can be based on computer-aided geometric feature inpainting and texture-based video inpainting (completion) in two directions [17, 18]. Video enhancement is one of the key technologies in animation design and background element design. Due to the instability of the scene, inaccurate internal parameter values of the computer, equipment damage, etc., uneven lighting, colour distortion, and overlapping areas may appear in the creation caused by visual degradation. Under the higher requirements of human beings for visual quality, the contrast of the video occupies an important position. The greater the contrast, the better the visual effect of the video, and the higher the clarity of details [19, 20].

Compared with the above-mentioned simulation application of video texture and primitive repair, this paper will apply the theory of directional derivative to the design of animation characters and background elements. According to the information positioning of the directional derivative, the stable field model and equation of the local texture of the static video can be established, and the definition and information entropy can be obtained by calculating each known pixel point [21,22]. This can be now be compared with known algorithms, and then the realistic degree and effect of animation background elements under the directional derivative theory are evaluated [23,24]. In addition, the directional derivative will be calculated in the background based on the restoration time and the peak noise reduction ratio, and the accurate image restoration parameters will be obtained through research, which will confirm the feasibility of the directional derivative in the design of animation characters and background elements from the side [25, 26].

The knowledge of partial derivatives lets us know that the partial derivatives actually reflect the rate of change of the multivariate function at a point along the coordinate axis. Many problems in life involve solving the rate of change of a multivariate function at a point in other directions. Under this phenomenon, the concept of the directional derivative of a multivariate function at a point along a given direction is proposed.

The most widely admired is ‘Advanced Mathematics’ printed by Tongji University Press. The book takes the binary function

When the moving length of the moving point

This article will study the connection between it and the design of animation elements according to the modified part.

Animation character design is defined when the character designer analyses the character’s shape and character’s characteristics by refining the character information in the script and related materials, and then draws the character’s appearance, expression, siblings, movements, clothing, etc. The production activity lays the groundwork. In the production process of animation works, character design is an important link, and the character image directly affects the style of the work and the expression of the plot. It can transmit visual information to the audience for the first time and promote the development of the story. In animation works, there are usually several animation characters. When designing, we should fully consider the character’s characteristics of different characters and their effects on the development of the storyline, to produce animation characters with different characteristics and improve the audio-visual effect of animation works. Excellent animation character design should follow certain principles, which also play an important role in the development of animation derivatives and stand guarantee for commercial benefits.

The design of animation characters should have the following ideas:

Visual effects are one of the important criteria to measure the quality of animation works. To enjoy the visual enjoyment of animation works, audiences need to have beautiful pictures, which include pleasing animation characters.

The concise and lively character image helps to deepen the audience’s memory, and is also conducive to the development of animation derivative products.

The character design of animation should have individuality, and it will have more appeal and vitality.

The animation character design cannot be separated from the storyline and scene design, and the three must be consistent.

When designing character actions, first of all, there must be a clear purpose. The design is based on the overall style and character’s characteristics of animation works, follows the laws of motion, and uses the methods of elasticity, inertia and curve motion to make moderate exaggeration. Scientific and reasonable character pose design, which can be the key to shaping the character and unfolding the plot. Second, attention to details is imperative and needs to be paid. When designing a character’s action pose, a look, a smile, a frown, a widened nostril, clenched teeth, etc., are all detailed actions that express the character’s emotion and personality. The detailed action and posture designed based on the character’s character can make the character fresher, more infectious, and easy to resonate with the audience. Character designers should observe carefully in life, pay attention to accumulation, create creations according to the plot and character characteristics, and pursue better design effects.

The character pedigree of animation work is drawn after the design work of all the appearing characters is completed, and all characters are placed on the same horizon on the design drawing, reflecting the high degree of change and contrast between each character. For original artists and animators, character lineage is to define the relative proportions between different characters to ensure that when multiple characters appear in the same scene, the correct height ratio and body shape can be maintained between characters. This puts forward a stricter requirement for portraits, especially in terms of the clarity and fidelity of imaging, and in close-up shots, it is not allowed to have confusion in the structure and proportions between characters. Therefore, this requires original artists and animators to draw characters strictly under the character pedigree settings to ensure the accuracy and rigour of animation works.

There are many types of software for animation character creation. The software includes two categories: plane animation production software and 3D animation production software. For details, see Figure 1:

In recent years, researchers have extended the application field of directional derivatives into the design and processing of animation characters and scene elements so that the application research of directional derivatives has a new development direction and space. Mathematical variables are used to monitor and control the key parameters of the image in real time, to enhance the video of animations and scenes so that the edge of the image is easier for everyone to identify. At the same time, the texture information of the smooth area of the image is preserved. The G–L (Grümwald–Letnikov) fractional image enhancement method is used in the literature to solve the problem that the effect of integer order differentiation on image texture enhancement is not obvious, and the edge colour of RGB colour images will be distorted after differentiation. Combining the G–L (Grümwald–Letnikov) fractional differential with the directional derivative, the adjacent pixel information of the processed pixels can be fully utilised, and the edge information and texture detail information of the image can be significantly enhanced.

Caputo and G–L definitions are equivalent under the condition that function

In the directional derivative of this part, redefine the directional function mentioned in the second part, then there is, let

According to the correlation characteristics of directional derivatives,

Similarly, if

Among them,

The G–L fractional derivative is defined as:

Let’s analyse it here. The meaning of the left superscript in

Let H be a variable function, which is defined to represent the change of function

To make the pixels on the image boundary to be processed, it is necessary to calculate the processed image correspondingly in the computer-aided measurement system, and explore the enhancement index that can realise the image. Since the differential order of the fractional differential calculation can be adjusted, different fractional orders are used to process the image so that the better order of enhancement effect can be selected. To verify the superiority of the directional derivative, we compare the Laplacian method, the G–L fractional differential method and the directional derivative itself. Laplacian is the mainstream practical application method of image enhancement, but in most cases, the enhanced image is under micro-vision. It will still look rough, not only lost more detail information, but also has a lot of noise, the enhancement effect is not good. And the algorithm processing of the dark part of the image is not very effective, especially the texture part. However, the G–L fractional differential methods have enhanced it, and the enhancement amplitude is slightly smaller, and there is no sharpening phenomenon. However, the enhancement effect is still not obvious; the enhancement effect of the directional derivative is better, but there is a slight blurring phenomenon at the edge of the image; when the directional derivative is used as a parameter aid, the contrast of the overall grey level of the animated character and the background image is significantly increased, and the local details Obviously enlarged, the texture is clearly visible. The experimental results show that the directional derivative has the best enhancement effect when k = 2–4, and k 3 is selected in this paper. Therefore, in practical applications, the method in this paper can appropriately select the value of k according to the degree of image enhancement, to achieve the optimal situation of the image under the assistance of directional derivative parameterisation.

The quality of image enhancement in this paper is evaluated in a quantitative direction. Quantitative methods are used to evaluate the enhancement effect of the image to reflect the change of the grey value of the image and finally, conduct a comprehensive evaluation. Let _{in}_{out}

Among them,

Clarity is the characteristic of the contrast unit of the tiny details of the video.

Let the video grayscale set be

The larger the

The differential order of the G–L fractional differential operator can be adjusted, and the image is processed accordingly, and a better order of enhancement effect is reserved. To verify the superiority of the method in this paper, we analyse and compare the Laplacian method, G–L fractional derivative method and directional derivative (k = 2) method, the test results are shown in Table 1.

Information entropy and clarity of various enhancement methods

To enhance the image | Method | |||
---|---|---|---|---|

Laplacian | G–L | Directional derivative (k=2) | ||

Image 1 | S | 0.2305 | 6.5574 | 12.0354 |

H | 1.0455 | 7.5638 | 7.8928 | |

Image 2 | S | 0.1788 | 5.4325 | 5.9807 |

H | 1.0022 | 6.8725 | 7.0901 | |

Image 3 | S | 0.4482 | 10.1075 | 14.8263 |

H | 1.0228 | 6.5826 | 7.0304 |

It can be seen from Table 1 that the sharpness and information entropy of the enhanced image obtained by the Laplacian method are relatively low, resulting in a poor visual effect of the enhanced image; the G–L method has a small enhancement range, resulting in low image sharpness. The derivative (application of the method discussed in this paper) enhances the image with the highest clarity and the largest information entropy, indicating that this method has the best enhancement effect on the image. To sum up, the directional derivative proposed in this paper makes full use of the information of the surrounding pixels. It can be seen from both subjective evaluation and objective evaluation methods that the directional derivative has a significant effect on image enhancement. It is deduced from this that in the design that requires a higher quality of video processing technology for animation characters and background elements, the directional derivative can bring a better effect.

At present, the research of image inpainting technology is more popular in texture special effects and set feature video inpainting, which is mainly used for the reconstruction of small defect areas, and most of them use the calculation mode of the partial differential equation (PDE) solution. The edge information of the defect region is diffused from the region boundary to the boundary, and the diffusion information and diffusion direction are determined by the edge information. Its representative algorithms are: the BSCB model, and the curvature diffusion repair algorithm (CDD model). In addition to the different mathematical models used in the above algorithms, the main factors affecting the restoration effect are the selection of diffusion information and the control of diffusion direction. In general, these methods aim to make the overall effect of the restored image imperceptible to the human eye and focus on the overall visual effect, rather than on the reconstruction of the accurate value of a certain point, which also makes them difficult to detect. The clarity of edge and texture detail repair is not deliberately pursued. In addition, because this type of repair model requires repeated iterations and high-order partial derivative calculations, it will affect its repair efficiency. Through the image texture itself and the image texture formation mechanism. The video inpainting based on directional derivatives is proposed first.

The directional derivative of a scalar field describes the magnitude of the rate at which the scalar function changes in a certain direction in the field. As shown in Figure 2, for the image scalar field with defects, let _{i}_{i}_{i}_{i}

Therefore, when

The ray drawn from point _{i}_{i}

Since the influence of points far away is too small to be ignored, when actually reconstructing a defective pixel point, the area with a distance not greater than

Let A represent the gradient of the pixel at point B, and C represent the directional derivative of point D in the direction of E. Figure 3 is a randomly selected point, and these quantities are marked. On the premise that F satisfies the appropriate order of magnitude (take the normalised value) available

Let _{i}_{i}

In the formula, _{i}

In the formula,

After the above parameters are calculated and processed mathematically, Table 1 lists the repair or reconstruction time and peak signal-to-noise ratio (PSNR) of the above three algorithms. It is convenient to compare the impact of image size on the repair or reconstruction time of several algorithms; at the same time, Image 1 and Image 4 are destroyed respectively, and the number of defective pixels in each image is quite different, so as to compare the impact of the size of the defect area on the repair and reconstruction time of each algorithm.

It can be seen from Table 2 that the repair time of the BSCB algorithm is mainly affected by the size of the image itself; the repair time of the CDD algorithm is greatly affected by the size of the image itself and the size of the defect area; while the repair time of the algorithm in this paper is almost completely determined by the size of the defect area. A horizontal comparison of the repair time in the table shows that the image scalar directional derivative repair efficiency is higher than the previous two algorithms. The larger the image size, the more obvious the advantage of this algorithm in terms of efficiency. From the perspective of PSNR, the scalar directional derivative algorithm is obviously better than the other ones. The algorithm in this paper and the BSCB and CDD methods have their own advantages for different types of images and different types of damage, but they are not as good as the image scalar directional derivative. To sum up, the image scalar directional derivative and the first two algorithms have better overall visual effects on image reconstruction or restoration; since the reconstruction model in this paper focuses on the reconstruction of each defective pixel point, in the reconstruction process, each defective point is reconstructed. All calculations are performed, so that the algorithm in this paper can reconstruct the edge and texture details more clearly and accurately; but because the calculation accuracy of the gradient and the influence function is not enough, the reconstruction accuracy of the edge details by the algorithm in this paper is not high enough, and the edge cannot be completely and accurately reconstructed. Since the reconstruction model in this paper avoids repeated iterations and calculation of high-order partial derivatives, it has high efficiency, especially for large-size images, the reconstruction time advantage of image scalar directional derivatives is obvious.

Comparison of repair time and PSNR of several algorithms. PSNR, peak signal-to-noise ratio

Pattern | The total number of pixels of the image | Defect pixels | Repair time/s | PSNR/dB | ||||
---|---|---|---|---|---|---|---|---|

BSCB | CDD | directional derivative | BSCB | CDD | directional derivative | |||

Image 1 | 128*128 | 366 | 5 | 38 | 2 | 36.8 | 38.2 | 46.9 |

Image 2 | 512*512 | 2405 | 58 | 287 | 6 | 45 | 47.3 | 36.8 |

Image 3 | 686*1024*3 | 5933 | 389 | 1167 | 15 | 45.5 | 46.8 | 42.5 |

Through the specific application research of the directional derivative in video enhancement and video restoration, it can be known that its directional derivative is also suitable for the design of animation characters and background elements, and can be more prominent and better than the traditional algorithm. The directional derivative can improve the clarity of the video and the amount of information carried by a higher level, which is based on video enhancement; the directional derivative can also be repaired in the case of video damage, and the repair time and the PSNR should be high-quality and outstanding compared with the comparison object algorithm. Based on the above description, the conclusions can be summarised as follows:

In the aspect of video enhancement, traditional Laplacian, G–L fractional derivative and directional derivative can achieve better video visualisation effect, but in terms of quantitative analysis, whether it is the result of clarity and entropy, the research value of the directional derivative is the same. It is higher than the other two methods, indicating that the directional derivative has an excellent ability to enhance the image.

In terms of image restoration, when dealing with higher pixel images, the BSCB model, the curvature diffusion restoration algorithm CDD model, and the directional derivative model show that the restoration time and PSNR values of the two index values show that the directional derivative has the shortest restoration time among the three. The PSRN value is the smallest, and it maintains a good repair and shaping force for the texture details and edge construction of the video.

Directional derivatives have practical applications in both areas of visual enhancement and restoration in the design of animated characters and background elements. And play an important guiding role in the design of the rising national comics and the design of scene background elements. Especially in the function of texture filling and realistic characterisation and self-healing of edge details, the algorithm principle of directional derivative can bring higher efficiency.

#### Information entropy and clarity of various enhancement methods

To enhance the image | Method | |||
---|---|---|---|---|

Laplacian | G–L | Directional derivative (k=2) | ||

Image 1 | S | 0.2305 | 6.5574 | 12.0354 |

H | 1.0455 | 7.5638 | 7.8928 | |

Image 2 | S | 0.1788 | 5.4325 | 5.9807 |

H | 1.0022 | 6.8725 | 7.0901 | |

Image 3 | S | 0.4482 | 10.1075 | 14.8263 |

H | 1.0228 | 6.5826 | 7.0304 |

#### Comparison of repair time and PSNR of several algorithms. PSNR, peak signal-to-noise ratio

Pattern | The total number of pixels of the image | Defect pixels | Repair time/s | PSNR/dB | ||||
---|---|---|---|---|---|---|---|---|

BSCB | CDD | directional derivative | BSCB | CDD | directional derivative | |||

Image 1 | 128*128 | 366 | 5 | 38 | 2 | 36.8 | 38.2 | 46.9 |

Image 2 | 512*512 | 2405 | 58 | 287 | 6 | 45 | 47.3 | 36.8 |

Image 3 | 686*1024*3 | 5933 | 389 | 1167 | 15 | 45.5 | 46.8 | 42.5 |

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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 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 -groupsResearch 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 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