Transportation is the foundation and the key to rejuvenating the country. In the construction of transportation infrastructure, special geographical conditions and geological environment are often encountered, and thus tunnel excavation plays a significant role. After decades of development, China's tunnel construction technology has reached a level that is on par with those of most other countries [1]. At the same time, the problems of tunnel engineering are gradually exposed where the tunnel will deform under the load, but when the deformation is too large to exceed the safety limit, there will be a hidden danger of major safety accidents [2]. Therefore, it is necessary to monitor the deformation of tunnel structures.

Three-dimensional (3D) laser scanning technology was born in the 1990s. It is a data acquisition technology that has set off another revolution in technology application after GPS. It has many advantages such as increased automation, high precision, large amount of data, holography and so on, and has been widely used for surveying and mapping in the science circles [3]. 3D laser scanning technology is based on the principle of laser-ranging, high-density, high-precision and rapid acquisition of spatial 3D coordinate information of the surface, which realises the ‘real-scene reproduction’ of the measured entity [4]. At the beginning of this century, 3D laser scanning technology has developed rapidly and has found application in most industries, including engineering monitoring, reverse modelling of structures, protection of ancient buildings and historical relics, scene restoration of traffic accidents, natural disaster monitoring, 3D map making and many other fields [5]. Point cloud data is obtained by 3D laser scanning technology, that is, the collection of spatial 3D coordinate data of high-density points, which is a collection of massive coordinate points on the surface of an entity. It contains 3D coordinate information of solid space, surface laser reflectivity, RGB colour and other information. The data structure of point cloud data is concise and flexible. In recent years, the field of data processing and application for point cloud data has also developed rapidly [6].

Therefore, the 3D laser scanning technology is used to obtain the tunnel point cloud data, which implements the reproduction of the tunnel spatial information under the 3D real scene. By using computer technology to develop processing software based on tunnel point cloud data, fitting the central axis of the tunnel, extracting continuous sections, fitting the section curve of a single section and extracting monitoring point, 3D laser scanning technology realises the high efficiency, high precision, visual analysis and measurement of tunnel structure, and then achieves the monitoring of tunnel deformation, thus improving the efficiency of deformation monitoring, which in turn improves the construction quality and retrenches the project investment.

3D laser scanning technology records the 3D coordinates, reflectivity and other information of the scanned object at the rate of tens of thousands or even millions of points per second. In the data processing of the office, the high-precision data processing system is used to efficiently model the measured object, and the practical engineering problems are solved by analysing the point cloud model [7]. There are many kinds of scanners that differ in structure, but all 3D laser scanners are mainly composed of laser transmitter, receiver, camera, timer, power control system and other components [8]. The laser transmitter actively emits the laser in front of the rotating lens, and the laser will be reflected back once it touches the surface of the measured object. Due to the fast propagation rate of light, the reflected laser is collected by the laser receiver along almost the same path. Then the timer calculates the distance by calculating the corresponding changes of laser before and after transmission and reception. At the same time, the encoder simultaneously measures the current attitude of the lens, namely the horizontal rotation angle and vertical rotation angle [9]. In this way, the X, Y and Z coordinates of the illuminated point can be obtained based on the coordinate system. After the number of scanning points per second is set before scanning, automatic and adaptive transmission devices can be set inside the scanner to ensure that the all-round and high-precision 3D position of the target object can be captured at different distances [10,11,12,13,14,15]. The 3D laser scanner is based on the instrument coordinate system to save the spatial position information of scanning points. The principle of the coordinate system is that the coordinate origin is in the centre of the instrument, that is, the location of laser beam emission, where the Z-axis is vertically upward in the vertical scanning plane, while the X-axis and the Z-axis are vertically in the horizontal scanning plane. The Y-axis is the scanning direction of laser pulse and it represents the right-hand rectangular coordinate system with X-axis and Z-axis. Figure 1 is a schematic diagram of a common scanner, where the instrument scans the surface of an object to capture the oblique distance S between the target point P and the laser beam emission place, the transverse scanning angle

The traditional measurement means that it obtains the 3D position information of the target point by using a single point. The 3D laser scanning technology has the advantages of rapidity, continuity and automation when collecting the surface of the target object. In the protection of cultural relics, the on-site records can be quickly completed, and the real restoration can be carried out in the computer through 3D information. 3D laser scanning is a way to obtain information pertaining to the surfaces of objects at a certain resolution, and it has the following characteristics:

High precision. Due to the influence of scanning distance and scanning angle, the single-point accuracy of the instrument may not reach the accuracy of the total station when acquiring the scanning point position in the middle and long distance, but modelling by the point cloud data has great advantages over the total station data single-point modelling [16].

High rate of data collection. With the continuous improvement of 3D laser scanning technology, the rate at which the instrument collects points can reach tens of thousands to millions of points per second, which has great benefits in large-area data collection [17].

Real-time. 3D laser actively emits light source signals, and it is thus not restricted by external conditions such as illumination and climate, and achieves all-weather observation, with long working time, high efficiency and convenient data update [18].

Non-contact. 3D laser scanning transmits information of scanning points through laser signals. Without erecting prisms, the 3D data of the target surface will be collected directly, while the data of inaccessible places and the information of dangerous areas are received, which effectively reduce the risk of field measurement [19].

Full automatic collection. 3D laser scanning technology can simply and effectively obtain the surface information of objects, and it contains data processing and modelling capabilities that facilitate the output of data; moreover, this technology can manage point cloud files of various formats compatibly [20].

External digital camera used with GPS system. The application of digital cameras can enable the instrument to obtain more detailed and accurate information when collecting data in complex spatial conditions and bad environments, and provide a certain basis in the later point cloud data processing. The cooperation of GPS system further improves the accuracy of measurement [21].

Panoramic scanning. Scanning at 360° in the horizontal direction and 320° in the vertical direction advances the scanning efficiency, and allows direct generation of the 3D model by processing the scanned point cloud.

According to the principle of 3D laser scanning technology and point cloud registration, the workflow of applying 3D laser scanning technology in tunnels can be divided into four stages: pre-preparation, point cloud data collection, point cloud data processing and deformation analysis.

Pre-preparation. First, the monitoring method is established through the design data and the construction scheme, which is further improved through on-site reconnaissance, so as to determine the location of the station and the monitoring time. Depending upon the construction environment and arrangement in the tunnel, it might be necessary to arrange targets or prisms rationally, and thus ensure the protection of registration instruments. Generally, before measurement, a field scanning test is required to ensure that the instrument works effectively in the tunnel, the target is identified clearly and the prism is unobstructed [22].

Point cloud data collection. In this stage, tunnel point cloud data is collected according to the monitoring scheme, usually once or twice a day. During scanning, it is necessary to guarantee that there is no interference noise and vibration in the tunnel. In case of accidents, it is essential to make timely records and take remedial measures. Meanwhile, point cloud data should be filed in a unified format to facilitate post-processing [23].

Point cloud data processing. Point cloud data processing usually includes point cloud registration, preprocessing, axis fitting, tunnel surface fitting and other steps. The target registration method needs to complete the registration of point cloud before pre-processing, and backward registration method can directly perform pre-processing [24]. In tunnel deformation monitoring, data pre-processing includes the steps of pre-processing by way of eliminating irrelevant data, data reduction and noise reduction to reduce the number of point clouds and improve the processing efficiency. The original point cloud data can be transformed into a data format capable of deformation analysis through operations such as axis fitting and tunnel surface fitting [25].

Deformation analysis. Tunnel deformation analysis based on point cloud data includes two parts: section convergence value and full-space deformation field. The full-space deformation field directly reflects the distribution of deformation for tunnel lining, and the section convergence value can be selected as the data support of the longitudinal distribution curve, which further guides the parameters of tunnel support design and support timing [26].

Using a 3D laser scanner to carry out measurement can improve the efficiency of measurement, but the amount of data collected is large. Compared with traditional measurement methods, the point cloud data processing scanned by 3D laser scanner is a complicated process, in which point cloud data pre-processing is the first and most important step. The result of pre-processing directly affects the difficulty of later data application and the accuracy of application results. Each 3D laser scanner uses special pre-processing software, and there are also many commercial software, which can be used to pre-process point cloud data, or the corresponding point cloud pre-processing algorithm can be compiled to fulfil the purpose of data processing. Point cloud data pre-processing mainly includes point cloud registration, point cloud denoising, point cloud compression and so on.

Usually, the 3D laser scanner cannot scan the measured object completely through one scan and the complexity of the measured object or the range of the measured area determines the number of times. As a result of multi-station scanning, there are many coordinate systems in the point cloud data, and it is thus necessary to splice the point cloud data of two adjacent stations to present complete information. Point cloud registration means converting the point cloud data in multiple coordinate systems to the same coordinate system to form a complete scanning. There are two forms of point cloud registration: automatic registration and manual registration. In this paper, automatic registration is mainly used.

Automatic registration is to eliminate the displacement error between two point clouds by some specific algorithms or statistical rules. Iterative nearest point (ICP) algorithm is the most classic algorithm in automatic registration. Its principle is to determine an objective function, and then determine the corresponding nearest point from the point set to iterate until the objective function is satisfied. The basic ideas are as follows: 1P and 2P are two groups of point sets in point cloud data, 1P is transformed into 2P by space to generate a functional mapping relationship, and the relationship between single points in point sets is as follows:

The transformation parameter vector x can be expressed as:
_{0} of the parameter vector

Search the nearest point set _{2K} according to the coordinates in _{1K}.

Get the parameter vector _{k}_{+1} through the registration process between point sets, and then calculate the sum of distance squares _{k}_{+1}.

The rotated point set
_{1k}. This is done by calculating the distance d between _{2k} and

Stop iteration when the change of the sum of squares of distances is less than the threshold

Measuring instruments will produce a series of errors in the process of collecting data, and 3D laser scanner is no exception. The collected point cloud data contains unnecessary redundant points and noise points mixed with effective point clouds. There are many reasons for this, including scanning to points outside the measurement area, points not belonging to the research object, influence of external environment, etc. Redundant point and noise will not only multiply the amount of point cloud data but also affect the application of point cloud data. Therefore, it is necessary to denoise the point cloud to facilitate further research [27]. Owing to the different arrangements of point cloud data, the denoising methods are different. According to the spatial distribution of noise points, noise points can be roughly divided into the following four categories:

Drift points: Those sparse and scattered points that are perceptibly far away from the main body of the point cloud and floating above the point cloud.

Isolated points: Those small and dense point clouds that are far away from the centre of the point cloud.

Redundant points: Those redundant scanning points that are beyond the predetermined scanning area.

Mixed points: Those noise points that are confused with the correct point cloud.

Denoising methods of ordered and partially ordered point clouds include: least square filtering, Kalman filtering, median filtering, mean filtering, Gaussian filtering and Wiener filtering. Denoising of unordered and scattered point clouds cannot directly apply the denoising method of ordered point clouds. It is necessary to sort out the scattered point clouds or establish the logical relationship between data points beforehand; once this is accomplished, thereafter the ordered point cloud denoising algorithm can be used for the denoising exercise. This sorting of scattered point clouds is a very complicated task, and the denoising method is inefficient and difficult to apply. Therefore, there are other algorithms that can be directly applied to scattered point cloud data. The most classic ones are: Laplacian algorithm, mean-shift algorithm, bilateral filtering algorithm and mean curvature flow filtering algorithm [28].

In order to adapt to the practical application and scientific research, it is necessary to simplify the massive point cloud data, so as to not only ensure the data accuracy but also improve the processing efficiency, that is, extract enough useful information to express the characteristics of the model from the original sampling point cloud according to different application requirements, which is referred to as point cloud compression. According to the different forms of point cloud data, scholars have proposed some methods for point cloud compression, and these commonly include: random sampling method, uniform segmentation method, octree algorithm and so on.

The ideal point cloud compression algorithm can express the most complete information with the least amount of data. In practical application, the following points are taken as the evaluation criteria of point cloud compression algorithm: (1) high compression ratio, that is, maximum compression of point cloud data on the premise of ensuring the minimum distortion; (2) meet the accuracy requirement, that is, the simplified point cloud data meets the accuracy requirement of application; (3) the algorithm is concise and the execution rate is high; and (4) no specificity [29].

The purpose of fitting the tunnel axis is to accurately intercept the tunnel section and to draw the longitudinal deformation curve of the tunnel according to the axis.

Van Gosligaf fitted the tunnel cylinder by the least square method, and defined the shortest distance between any measured point in the point cloud and the axis of the cylinder to be determined as follows:
_{i}_{i}, y_{i}, z_{i}_{0}, _{0}, _{0}) is a fixed point on the axis of the cylindrical surface; and

The tunnel axis fitting problem can be transformed into an optimisation problem:

During the construction, the structural shape of the tunnel only satisfies the horizontal symmetry, and the preprocessed original point cloud data of the tunnel is irregularly distributed in spatial coordinates. The longitudinal slope of the tunnel is usually between 0.3% and 3%, and thus the tunnel axis is not a completely horizontal curve in space. In this paper, based on the projection method, a fitting method for the central axis of point cloud data is proposed.

The projection equation of the central axis on the XOY and XOZ planes can be calculated by the projection boundary of point cloud data. Along the axial direction, the left and right boundaries of the XOY plane projection are parallel curves. Therefore, the boundaries of the XOY plane projection with polynomials are fitted as follows:
_{0}, _{0}) on one of the boundaries, the tangent equation is:
_{0}, _{0}) exist on the central axis, and the curve with quadratic function can be fitted to obtain the projection of the central axis of the tunnel on the XOY plane:

Based on a cross-section coordinate system, for any cross-section plane _{0}, setting the intercept thickness as

The polar coordinate system is constructed as follows:

Import a point cloud matrix in 3D coordinates

For any section _{0}, the 2D coordinates (

The radial displacement Δ

The underground utility tunnel of Tianhe Smart City has a total length of about 19.39 km, and the main routes are arranged along the existing Keyun Road, Kexiang Road, Huaguan Road, Gaotang Road, Ruan Road, Gaopu Road, Daguan Road, Guangcan Road, Kemulang South Road and Mubei West Road, Lingcen Road. The section of shield construction is 7.6 km, and the pipe gallery section is circular; further, it is mainly divided into Keyun Road and Huaguan Road. The open cut construction section is 10.8 km, and the section of the pipe gallery is rectangular. The road section studied in this paper is the sum of Huaguan Road and the pipe gallery. In order to get the deformation of the pipe section, a comparative analysis of the same section of the pipe section is made by three periods.

In this paper, the total station measured data is used to measure the control points, the geodetic coordinates of two fixed points outside the tunnel are measured by the total station and the coordinates of the fixed points are discussed in the forthcoming content. This coordinate of fixed point is the common point of scanning data. Through these two fixed points, the multi-phase scanning data can be transformed in the same coordinate system to analyse the segment deformation.

Geodetic coordinates of fixed points

Fixed points | X | Y | H |
---|---|---|---|

1 | 33756.7338 | 52120.5613 | 9.1703 |

2 | 33742.4468 | 52249.2831 | 9.1259 |

As an ultra-long linear structure, the scanning interval is 1.6 km in total. In this paper, a scheme is made before scanning, which is divided into 15 stations for scanning and splicing. The stations are named as Station 1 to Station 11. The station is arranged in the middle of the tunnel, which is scanned in all directions using 360° scanning.

By measuring the geodetic coordinates of three stations, a complete control network system can be formed. Among them, the geodetic coordinates of Station 1, Station 5 and Station 11 are measured by the total station. The layout scheme and geodetic coordinate values are shown in Table 2.

Geodetic coordinates of stations

Fixed points | X | Y | H |
---|---|---|---|

1 | 33763.6312 | 51987.3442 | 8.8299 |

5 | 33799.6582 | 51345.2493 | 8.3854 |

11 | 33760.9238 | 51345.2493 | 8.9284 |

The 3D laser scanner on the ground collects point cloud data through the arrangement of the previous control network, and there will be a large number of point clouds irrelevant to the collected point cloud data, which may even become the data needed to interfere with the collection. These point cloud are called noise, and the existence of noise will affect the fitting of the subsequent tunnel section, resulting in the decrease of the accuracy of segment deformation analysis [30]. Moreover, the scanning distance of the ground 3D laser scanning instrument is limited. When the control network is laid out in the early stage, a total of 15 measuring stations are arranged. The coordinate system of each measuring station is independent, and the coordinates of each measuring station need to be unified under the same coordinate system.

When the point cloud registration is completed, data processing is needed and the noise that interferes with the effective point cloud is eliminated, which is the key step in extracting the model of point cloud and further comparing the segment deformation using 3DReshaper software [31]. In this paper, the segment point cloud of shield tunnel is the extraction point cloud, while the railings, air ducts, tracks and water pipes inside the tunnel are noises. Under the environment of interactive editing, the redundant point clouds irrelevant to the extracted information and the compensation of systematic missing information can be realised interactively. The figure of point cloud before and after denoising can be clearly distinguished, as shown in Figures 3 and 4.

After the scheme of 3D laser scanning is made, the first part of deformation analysis needs to be conducted, and the same is described in the previous section. The application of Leica P50, a ground 3D laser scanning instrument, and cyclone, its supporting software, is used to collect and splice point cloud data to complete the preliminary work of the model. In this section, we need to fit and transform the spliced point cloud data that is transformed into a curved surface model for overall section analysis, which includes three parts: the unification of point cloud coordinates, the transformation of point cloud data, the establishment of fitting surface and the extraction of cross-section.

By inputting the coordinate data of each phase into Leica cyclone software, the point cloud data after scanning and splicing can be unified into the same coordinate system, which is convenient for subsequent data transformation and comparison model establishment.

Geodetic coordinates of stations in different periods

Times | Station 1 | Station 2 | Station 3 |
---|---|---|---|

1 | (33763.6312, 51987.3442, 8.8299) | (33799.6582, 51345.2493, 8.3854) | (33760.9238, 51345.2493, 8.9284) |

2 | (33764.8412, 51946.8742, 8.4865) | (33796.8549, 51545.3584, 8.8746) | (33765.8429, 51654.3467, 8.7873) |

3 | (33765.6597, 51966.2433, 8.6596) | (33799.8496, 51898.3672, 8.8379) | (33799.6364, 51996.7442, 8.7633) |

Based on the unified point cloud data in the previous section, the output is imported into 3DReshaper (3D point cloud data modelling) software to extract the central axis and create the cross-section along the central axis.

The content of this paper is the length range of segment from 500 ring to 670 ring, the width of segment is 1.5 m, and the total length of research area is 255 m. A part of the tunnel section is selected, and cross-sectional interception is carried out with every 10 m as a truncated surface, which is formed into 25 truncated surfaces, as shown in Figure 5.

As shown in Figure 6, there are four models near the same position; one is the basic circle model, and the other three are the cubic measurement models. By extracting the cross-section information, a comparison chart of 25 cross-sections is obtained, as shown in Figure 7.

In order to study the distribution law of each characteristic monitoring point with time, the changes of arch crown, arch bottom and arch waist of lining segment measured by scanning in three periods are sorted out. It can be seen that the four monitoring points of vault, arch bottom and arch waist on both sides have similar changes with time. Mainly, as time goes by, the deformation of each monitoring point becomes larger [32].

The speed of deformation from ring 500 (section 0) to ring 670 (section 24) is also gradually increasing, which is due to the fact that ring 670 (section 24) is closer to the excavated face than ring 500 (section 0). In the first scanning survey in the early stage, the lining segment was installed soon, and with the continuous excavation of shield, the surrounding rock around the lining segment changed more. The lining segment is deformed by the secondary stress of surrounding rock. In addition, ring 500 (section 0) is farther away from the excavated face, and the disturbance of surrounding rock at this section is small. The surrounding rock is supported by the lining segment and stable grouting effect, and the deformation is small and tends to converge. For the multi-stage monitoring of a large number of sections, the whole deformation monitoring of shield tunnel is formed, which can search the deformation law and the deformation characteristic points that need to be controlled. Compared with the limited number of monitoring points in traditional methods, the ground 3D laser scanning monitoring method can monitor the whole section. It can also extract the deformation monitoring feature points of the whole section, with a larger amount of data, which has a greater guiding role for the safety and stability of the lining tunnel.

In this paper, based on 3D laser scanning technology, the monitoring scheme of deformation about shield tunnel segment lining is implemented. The point cloud data are spliced and denoised, and the point cloud data at three time points of shield excavation, shield shutdown and shield excavation are selected to fit the curved surface model. Then, the cross-section is selected for comparative analysis, which indicates that four monitoring points of vault, arch bottom and arch waist on both sides have similar change rules with time. This can be explained by the observation that after the later spliced lining segments are installed, the surrounding rock around the lining segment changes greatly with the continuous excavation of the shield, and the lining segment is deformed by the secondary stress of surrounding rock. Moreover, the surrounding rock is supported by the lining segments whose deformation is small and tends to converge. Therefore, in the process of tunnel excavation, in order to reduce the increasing speed of deformation of spliced segments in the later period, the timeliness of synchronous grouting should be ensured, and the appropriate amount of grouting should be selected to make the mortar behind the segments filled with saturated mortar.

#### Geodetic coordinates of stations in different periods

Times | Station 1 | Station 2 | Station 3 |
---|---|---|---|

1 | (33763.6312, 51987.3442, 8.8299) | (33799.6582, 51345.2493, 8.3854) | (33760.9238, 51345.2493, 8.9284) |

2 | (33764.8412, 51946.8742, 8.4865) | (33796.8549, 51545.3584, 8.8746) | (33765.8429, 51654.3467, 8.7873) |

3 | (33765.6597, 51966.2433, 8.6596) | (33799.8496, 51898.3672, 8.8379) | (33799.6364, 51996.7442, 8.7633) |

#### Geodetic coordinates of fixed points

Fixed points | X | Y | H |
---|---|---|---|

1 | 33756.7338 | 52120.5613 | 9.1703 |

2 | 33742.4468 | 52249.2831 | 9.1259 |

#### Geodetic coordinates of stations

Fixed points | X | Y | H |
---|---|---|---|

1 | 33763.6312 | 51987.3442 | 8.8299 |

5 | 33799.6582 | 51345.2493 | 8.3854 |

11 | 33760.9238 | 51345.2493 | 8.9284 |

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sustainable development and renewal of Macao inner harbour under the background of digitisation Support design of main retracement passage in fully mechanised coal mining face based on numerical simulation Study on the crushing mechanism and parameters of the two-flow crusher Interaction design of financial insurance products under the Era of AIoT Modeling the pathway of breast cancer in the Middle East ARMA analysis of the green innovation technology of core enterprises under the ecosystem – Time series data Image design and interaction technology based on Fourier inverse transform Calculus Logic Function in Integrated Manufacturing Automation of Single Chip Microcomputer Football Offense Training Strategy Based on Fractional Differential Mathematical Modeling Research on China interregional industrial transformation slowdown and influencing factors of industrial transformation based on numerical simulation The medical health venture capital network community structure, information dissemination and the cognitive proximity 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 generative design method of building layout generated by path Analysis of the causes of the influence of the industrial economy on the social economy based on multiple linear regression equation Research of neural network for weld penetration control 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 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projection and finite element numerical analysis of integral differential equations in modern art design Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Research on motion capture of dance training pose based on statistical analysis of mathematical similarity matching Application of data mining in basketball statistics Application of B-theory for numerical method of functional differential equations in the analysis of fair value in financial accounting Research on the influence of fuzzy mathematics simulation model in the development of Wushu market Study on audio-visual family restoration of children with mental disorders based on the mathematical model of fuzzy comprehensive evaluation of differential equation Difference-in-differences test for micro effect of technological finance cooperation pilot in China Application of multi-attribute decision-making methods based on normal random variables in supply chain risk management Exploration on the collaborative relationship between government, industry, and university from the perspective of collaborative innovation The impact of financial repression on manufacturing upgrade based on fractional Fourier transform and probability AtanK-A New SVM Kernel for Classification Validity and reliability analysis of the Chinese version of planned happenstance career inventory based on mathematical statistics Visual positioning system for marine industrial robot assembly based on complex variable function 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 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 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 Calculating university education model based on finite element fractional differential equations and macro-control analysis Educational research on mathematics differential equation to simulate the model of children's mental health prevention and control system Analysis of enterprise management technology and innovation based on multilinear regression model Verifying the validity of the whole person model of mental health education activities in colleges based on differential equation 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 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 Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets 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 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 MCM of Student’s Physical Health Based on Mathematical Cone Sports health quantification method and system implementation based on multiple thermal physiology simulation Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations 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 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 Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on short-term human movement Human gait modelling and tracking based on motion functionalisation The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model 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 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 Research on threat assessment problems of island air defence system based on the leader-follower model 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 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 Evaluation of ecosystem health in Futian mangrove wetland based on the PSR-AHP model A study of local smoothness-informed convolutional neural network models for image inpainting Towards more efficient control of the ironmaking blast furnace: modelling gaseous reduction of iron ores in H _{2}-N_{2}atmosphereAlgorithm of overfitting avoidance in CNN based on maximum pooled and weight decay 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 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 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 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 Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model