This paper proposes a mathematical function movement model based on the gait movement of the human body and, in particular, on the trajectory of the limbs during human movement. The article systematically measures and experimentally deals with the trajectory of the limbs of 40 students in the walking movement. The linear high-order polynomial fitting method eliminates the motion error. Simultaneously, the linear relationship least square method is used to obtain the expression of the limb motion function. Finally, the mathematical model of the limb motion trajectory is obtained. It is verified through experiments that the model proposed in the thesis can calculate the law of limb movement and movement parameters of any person under normal walking movement. This research has high research value for human movement rehabilitation and the design of wearable equipment.

#### Keywords

- Human motion
- motion functionalisation
- gait modelling
- least square method
- polynomial fitting method

#### MSC 2010

- 62J12

With the rapid development of biomedical engineering technology at home and abroad, higher and higher requirements have been put forward for obtaining human motion parameters. Since the 19th century, various research methods for extracting gait parameters have emerged at home and abroad [1], involving visual tracking of human motion. To avoid the shortcomings associated with these methods, the techniques used in this paper realise the detection, positioning and tracking of motion by analysing the video sequence shot by the camera without human intervention or with minimal intervention. On this basis, the human body posture and motion parameters are obtained for further semantic analysis and behavioural understanding. In short, the paper aims to realise the automation of motion analysis without infringing on the analysed person, to liberate people from tedious labour.

In the field of functional modelling of gait movement, some scholars fit the gait movement of the human body with b-spline. Some scholars assume that the human body movement obeys the characteristics of a sinusoidal signal and only estimates the signal amplitude, frequency and phase. The former is currently mainly used in robotics. Because this method ignores the periodicity of human motion, it limits its modelling accuracy and the application value of the model. At the same time, the latter is due to the simplicity of the model, making it unable to reflect the differences between people accurately. Therefore, the question arises as to the optimal means to increase the parameters of the model based on consideration of the periodicity of human motion, while ensuring the accuracy of the modelling and its results; the difference between one human motion and another needs to be accurately reflected by adjusting the model parameters to perform human motion modelling, thus facilitating actual human motion to serve as the basis for tracking applications.

In the process of upright walking of the human body, the limbs play an important balancing role. When the limbs produce abnormal movement trajectories, it will directly affect the movement characteristics of the gait [2]. For this reason, the movement trajectory characteristics of the limbs need to be judged, including standard features, which are of great significance in the fields of rehabilitation medicine and sports training.

Gait movement is a natural human movement that can be seen everywhere. Through careful observation, we can find that the gait has prominent periodic characteristics. The two strides during normal walking can be regarded as a gait cycle, and the human gait movement is a repeated process of such a gait cycle. We need to extract the four critical states in the gait cycle to get a complete gait cycle. In the gait movement model, the uncertain changes are the eight joint angle parameters of the upper arm and the longitudinal body axis, the upper arm and the forearm, the thigh and the longitudinal body axis, the upper leg and the lower leg and the changes of the two ankle joints and the lower leg [3]. The changing relationship is approximately unchanged. We use ‘maker’ to locate the absolute joint position, calculate the rotation angle of the human body's degree of freedom through the joint position and then model the respective degree of movement through curve fitting.

The three-dimensional motion trajectory measurement system of the limbs is composed of two parts: a trajectory detection system and a plantar force detection system, which mainly includes a treadmill, a support frame, a pull-wire displacement sensor, a PC, a Dspace system and a force measurement insole and interface card. In the experimental measurement process, the subject only needs to walk naturally on the treadmill at a given speed. The operator observes the subject and starts to record the data after the subject is observed to be walking smoothly. The displacement sensor is measured by the interface card, Dspace system and Matlab software. The output signal of the load cell is collected in real-time, and matrix operation is performed. The movement trajectory curve of the limb in the coronal, sagittal and transverse planes is displayed on the computer screen through the ControlDesk interface. All experimental data are recorded after the measurement is completed [4]. The Matlab software processes the data stored in the computer for secondary data processing to obtain the characteristic parameters such as the amplitude, phase, frequency and other related gait parameters of the limbs in the coronal, sagittal and transverse planes, as well as the gait cycle and other related gait parameters.

When the subject walks naturally on the treadmill, subjective factors and the influence of the environment will cause the subject to randomly experience irregular gait on the track, such as when the walking line swings from side to side or back and forth, which results in three-dimensional movement of the limbs. The curve has obvious, irregular up and down fluctuations. This process can be observed in real-time on the computer screen. To eliminate the measurement error caused by irregular gait, we obtain the standard limb movement trajectory and various characteristic parameters, and analyse the same using secondary offline data processing. In the process, a high-order polynomial fitting method is first used to fit the actual measurement data to obtain the error curve and the error data at each sampling point [5]. This curve reflects the macroscopic position of the limbs during walking. This article aims to study the microscopic movement of the limbs based on this macroscopic movement. Then, the actual measurement data at each time point and the error value are reversely superimposed. The obtained data is the relative movement trajectory of the final limb, and thereafter we draw each set of curves.

The paper selects 40 students of different heights as the measurement objects, and each subject walks naturally on a treadmill at a given speed. Figure 1 shows the trajectory of the limbs when one of the subjects walks at different speeds. Figure 2 shows the limb movement trajectories of nine randomly selected subjects of different heights walking at a fixed speed [6]. The abscissa is the percentage of the gait cycle, and the ordinate is the displacement _{p}

This result shows that the walking speed and the height of the subject have a particular influence on the trajectory of the limbs and show a regular trend of changes in parameters such as the motion amplitude _{y}_{y}_{y}

Table 1 shows the experimental measurement results of the limb motion amplitude of the above nine subjects walking at different speeds [7]. We use height as the abscissa and motion amplitude as the ordinate and plot the measurement results at each speed on the same graph. As shown in Figure 3, the ‘*’ point corresponds to the data in Table 1, h is the height of the subject and _{y}

In the formula,

The amplitude of limb movement when subjects of different heights walk at different speeds.

^{−1} |
|||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 1.86 | 44.03 | 42.25 | 38.04 | 34.35 | 32.52 | 29.45 | 24.43 | 22.03 |

2 | 1.84 | 40.64 | 35.24 | 33.48 | 31.33 | 28.82 | 25.13 | 24.26 | 21.26 |

3 | 1.68 | 34.34 | 31.25 | 28.85 | 24.58 | 23.12 | 22.55 | 19.4 | 19.43 |

4 | 1.64 | 31.88 | 30.5 | 25.24 | 24.85 | 22.45 | 20.95 | 16.24 | 16.11 |

5 | 1.624 | 30.28 | 25.44 | 24.23 | 22.05 | 18.32 | 15.99 | 14.66 | 14.03 |

6 | 1.6 | 26.84 | 24.88 | 22.55 | 18.45 | 18.32 | 15.23 | 16.41 | 14.19 |

7 | 1.684 | 24.18 | 22.85 | 20.55 | 15.88 | 14.44 | 14.31 | 14.1 | 11.4 |

8 | 1.64 | 22.16 | 16.86 | 16.84 | 14.48 | 14.82 | 13.34 | 11.94 | 10.99 |

9 | 1.614 | 16.68 | 14.1 | 14.22 | 13.44 | 12.26 | 11.42 | 10.44 | 9.94 |

It can be seen from Figure 3 that there is a fitting error between the fitted curve and the measured value. The expression for calculating the fitting error Δ_{y}

The formula
_{y}^{*}) is the fitting function value obtained when the height is taken from the measured value. The expression for calculating the variance _{y}

In the above equation, m is the number of subjects, i.e. the number of measurement points at each speed. We take the average of the fitting variance of each speed, and the expression of the average variance Δ_{y}

In the above equation,

Table 2 shows the fitting coefficients _{y}_{y}

Fitting coefficients and errors at different speeds.

^{−1} |
|||||
---|---|---|---|---|---|

0.4 | 104.16 | −160.77 | 1.0683 | 1.0316 | 2.31% |

0.5 | 88.81 | −144.08 | 0.8648 | ||

0.6 | 88.36 | −118.31 | 1.1636 | ||

0.7 | 83.85 | −111.87 | 1.45 | ||

0.8 | 77.67 | −114.01 | 0.8586 | ||

0.9 | 69.9 | −101.71 | 1.135 | ||

1 | 59.09 | −95.39 | 1.1549 | ||

1.1 | 53.91 | −77.55 | 1.63 |

Similarly, the relative fitting error of coefficient a is 1.50%, and the relative fitting error of coefficient b is 1.86%. Substituting Eqs (7) and (8) into Eq. (1), it can be inferred that when healthy young people walk naturally on the treadmill, the mathematical model of the amplitude of the limbs in the left and right directions is as follows:

From the experimental data, it can be concluded that during the walking process of the subject, the limb swings once in one gait cycle, which indicates that the limb movement cycle is consistent with the gait cycle [8]. Table 3 shows the same subject in the gait cycle, with the limb movement cycles at different walking speeds. Table 4 shows the limb movement cycles when subjects with different heights walk at a speed of 0.8 m/s.

Exercise cycles of the same subject at different walking speeds.

^{−1} |
^{−1} |
||
---|---|---|---|

0.4 | 1.94 | 0.8 | 1.35 |

0.5 | 1.6 | 0.9 | 1.21 |

0.6 | 1.45 | 1 | 1.18 |

0.7 | 1.35 | 1.1 | 1.13 |

Exercise cycle of subjects with different heights when walking at a speed of 0.8 m/s.

1.97 | 1.17 | 1.684 | 1.15 |

1.94 | 1.14 | 1.67 | 1.13 |

1.735 | 1.13 | 1.644 | 1.16 |

1.72 | 1.14 | 1.614 | 1.11 |

1.705 | 1.15 |

We draw the experimental data with walking speed and height as the abscissa and the limb movement period as the ordinate. As shown in Figure 5, the ‘*’ points are the measurement data points. The least-squares method is also used for data fitting, and the period and walking are obtained. The fitting curve of the relationship between speed and height is shown in the Figure 5. From this, it can be seen that during the natural walking of healthy young people, the limb movement cycle is mainly affected by walking speed, and height has no effect on it. It is derived by the polynomial fitting method. The mathematical model of the relationship between the cycle and walking speed is as follows [Eq. (10)]. Similarly, the relative fitting error of the fitting function is 4.3%.

The walking speed is taken as 0.8 m/s and substituted into the above formula, and the gait period is 1.25 s. The average value of the measurement data in Table 3 is 1.24 s, which shows that the measured value and the fitted value are the same.

_{y}

It can be seen from Figures 1 and 2 that at the beginning of the gait cycle, the limb is located on the left side of the intermediate balance position (taking the left heel landing time as the gait cycle dividing point), and then the first wave peak is reached through the balance position; so is the initial phase of the sine curve [10]. Table 5 shows the initial phase _{y}_{y}

The initial phase of the same subject at different walking speeds.

^{−1} |
_{y} |
^{−1} |
_{y} |
---|---|---|---|

0.4 | −0.566 | 0.8 | −0.442 |

0.5 | −0.505 | 0.9 | −0.442 |

0.6 | −0.46 | 1 | −0.552 |

0.7 | −0.442 | 1.1 | −0.552 |

The initial phases of subjects with different heights when walking at a speed of 0.8 m/s.

1.87 | −0.471 | 1.685 | −0.6 |

1.84 | −0.451 | 1.67 | −0.566 |

1.735 | −0.417 | 1.655 | −0.614 |

1.71 | −0.518 | 1.615 | −0.487 |

1.705 | −0.534 |

Similarly, the relative error of this fitting function is 1.8%.

The paper uses a sine function to establish a mathematical model of the limb movement trajectory, and the expression is

The paper substitutes the models in Eqs (9), (10) and (11) of the amplitude, period and initial phase obtained in the above sections into Eq. (12), respectively, to obtain the trajectory of the limbs in the left and right directions during the natural walking of healthy young people. The mathematical model is as follows:

An important factor that affects the accuracy of human motion is whether the model can reflect the absolute motion of the human body [11]. In this part of the research, we use the previously established model as a tool for particle state prediction. The algorithm is as follows:

When tracking the first frame, we don’t have any tracking history information to guide the estimation of the state. If the blind search is used, it will inevitably involve a heavy computational cost and affect the tracking efficiency. Therefore, we need a specific method to roughly estimate the state of the human body in the first frame [12]. The prior knowledge we can use is the periodic function model of joint angles in gait motion obtained through model learning. We select a particle set of 500 particles to be distributed according to Gaussian at the initialisation time.

The dynamic model uses continuity assumptions and prior knowledge (motion model and motion constraints) to predict the human pose represented by the particle state in the current frame according to the motion function of each joint [13]. A particle set of 500 particles is selected for error correction, and the corrected particle set has a

We match the motion regions extracted from the video with the motion regions represented by the particles to detect the degree of matching for all particles. Regional matching of the head, upper body and arm parts is carried out. The movement of the head and the upper body torso is relatively simple, and the movement of the arm is more complicated, but the phenomenon of occlusion between the torso and the torso is serious when walking, and the functionalisation of human movement can be in the arm. We can predict the movement of the arm more accurately during occlusion [14]. Therefore, in the feature matching stage, we only use region matching for the head, upper body and arms. Subsequently, regional matching and edge matching of the legs is carried out. Among the changes of various parts of the human body during walking, the movement of the legs is the part that best reflects the characteristics of human walking. Therefore, in the field of human motion tracking, especially gait tracking, the accuracy of leg tracking is higher. In the research carried out in this paper, the feature matching of the legs uses both edge features and regional features, as well as the final region matching error Er, and edge matching error and feature error, which determine the proportion of regional matching error and edge matching error in the feature error.

The particle update process in this paper is different from the re-sampling update of the ordinary particle filter [15]. In the tracking study of this part, the particle state represents the error; so, the difference between the particle states at adjacent moments is not significant; thus, in the process of particle updating, the re-sampling operation is no longer needed. The generation of the new particle set is also sampled according to the Gaussian distribution. The mean of this Gaussian distribution is the output of the best particle state at the last moment, and the variance remains unchanged. The tracking effect is shown in Figure 6.

Based on the experimental data, this article establishes a mathematical model of the trajectory of the limbs of healthy young people when they walk naturally. This model has a particular general significance and application value. In the case of there being no actual measurement, as long as the height of a certain young man is given together with the walking speed, this model can be used to obtain the physical characteristics of the limbs that the person should have under the condition of physical health. In rehabilitation training for patients with walking dysfunction, these parameters can provide a basis for rehabilitation evaluation. The parameters obtained from the model are compared with the actual morbid parameters to evaluate the rehabilitation effect. The research on the mathematical model of the human limb movement trajectory has particular scientific significance and application value in many fields such as sports, anthropology, aerospace and ergonomics.

#### Fitting coefficients and errors at different speeds.

^{−1} |
|||||
---|---|---|---|---|---|

0.4 | 104.16 | −160.77 | 1.0683 | 1.0316 | 2.31% |

0.5 | 88.81 | −144.08 | 0.8648 | ||

0.6 | 88.36 | −118.31 | 1.1636 | ||

0.7 | 83.85 | −111.87 | 1.45 | ||

0.8 | 77.67 | −114.01 | 0.8586 | ||

0.9 | 69.9 | −101.71 | 1.135 | ||

1 | 59.09 | −95.39 | 1.1549 | ||

1.1 | 53.91 | −77.55 | 1.63 |

#### Exercise cycles of the same subject at different walking speeds.

^{−1} |
^{−1} |
||
---|---|---|---|

0.4 | 1.94 | 0.8 | 1.35 |

0.5 | 1.6 | 0.9 | 1.21 |

0.6 | 1.45 | 1 | 1.18 |

0.7 | 1.35 | 1.1 | 1.13 |

#### Exercise cycle of subjects with different heights when walking at a speed of 0.8 m/s.

1.97 | 1.17 | 1.684 | 1.15 |

1.94 | 1.14 | 1.67 | 1.13 |

1.735 | 1.13 | 1.644 | 1.16 |

1.72 | 1.14 | 1.614 | 1.11 |

1.705 | 1.15 |

#### The amplitude of limb movement when subjects of different heights walk at different speeds.

^{−1} |
|||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 1.86 | 44.03 | 42.25 | 38.04 | 34.35 | 32.52 | 29.45 | 24.43 | 22.03 |

2 | 1.84 | 40.64 | 35.24 | 33.48 | 31.33 | 28.82 | 25.13 | 24.26 | 21.26 |

3 | 1.68 | 34.34 | 31.25 | 28.85 | 24.58 | 23.12 | 22.55 | 19.4 | 19.43 |

4 | 1.64 | 31.88 | 30.5 | 25.24 | 24.85 | 22.45 | 20.95 | 16.24 | 16.11 |

5 | 1.624 | 30.28 | 25.44 | 24.23 | 22.05 | 18.32 | 15.99 | 14.66 | 14.03 |

6 | 1.6 | 26.84 | 24.88 | 22.55 | 18.45 | 18.32 | 15.23 | 16.41 | 14.19 |

7 | 1.684 | 24.18 | 22.85 | 20.55 | 15.88 | 14.44 | 14.31 | 14.1 | 11.4 |

8 | 1.64 | 22.16 | 16.86 | 16.84 | 14.48 | 14.82 | 13.34 | 11.94 | 10.99 |

9 | 1.614 | 16.68 | 14.1 | 14.22 | 13.44 | 12.26 | 11.42 | 10.44 | 9.94 |

#### The initial phases of subjects with different heights when walking at a speed of 0.8 m/s.

1.87 | −0.471 | 1.685 | −0.6 |

1.84 | −0.451 | 1.67 | −0.566 |

1.735 | −0.417 | 1.655 | −0.614 |

1.71 | −0.518 | 1.615 | −0.487 |

1.705 | −0.534 |

#### The initial phase of the same subject at different walking speeds.

^{−1} |
_{y} |
^{−1} |
_{y} |
---|---|---|---|

0.4 | −0.566 | 0.8 | −0.442 |

0.5 | −0.505 | 0.9 | −0.442 |

0.6 | −0.46 | 1 | −0.552 |

0.7 | −0.442 | 1.1 | −0.552 |

Law of interest rate changes in financial markets based on the differential equation model of liquidity Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Industrial transfer and regional economy coordination based on multiple regression model Response model for the psychological education of college students based on non-linear finite element equations Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Analysis of the relationship between industrial agglomeration and regional economic growth based on the multi-objective optimisation model Constraint effect of enterprise productivity based on constrained form variational computing The impact of urban expansion in Beijing and Metropolitan Area urban heat Island from 1999 to 2019 Ultrasonic wave promoting ice melt in ice storage tank based on polynomial fitting calculation model Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations Badminton players’ trajectory under numerical calculation method Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Nonlinear Differential Equations in the Teaching Model of Educational Informatisation The evaluation of college students’ innovation and entrepreneurship ability based on nonlinear model Smart Communities to Reduce Earthquake Damage: A Case Study in Xinheyuan, China Institutional investor company social responsibility report and company performance Mathematical analysis of China's birth rate and research on the urgency of deepening the reform of art education First-principles calculations of magnetic and mechanical properties of Fe-based nanocrystalline alloy Fe _{80}Si_{10}Nb_{6}B_{2}Cu_{2}Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption Attitude control for the rigid spacecraft with the improved extended state observer Cognitive Computational Model Using Machine Learning Algorithm in Artificial Intelligence Environment Research on tourism income index based on ordinary differential mathematical equation Application of Higher-Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Sports Science Teaching of Athletics Based on Nonlinear Mathematical Equation Informatisation of educational reform based on fractional differential equations Research on the control of quantitative economic management variables under the numerical method based on stochastic ordinary differential equations Network monitoring and processing accuracy of big data acquisition based on mathematical model of fractional differential equation System dynamics model of output of ball mill Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Design of Morlet wavelet neural network to solve the non-linear influenza disease system Motion about equilibrium points in the Jupiter-Europa system with oblateness Badminton players’ trajectory under numerical calculation method Optimal preview repetitive control for impulse-free continuous-time descriptor systems Development of main functional modules for MVB and its application in rail transit Study on the impact of forest fire prevention policy on the health of forest resources Value Creation of Real Estate Company Spin-off Property Service Company Listing Selection by differential mortality rates Digital model creation and image meticulous processing based on variational partial differential equation The modelling and implementation of the virtual 3D animation scene based on the geometric centre-of-mass algorithm The policy efficiency evaluation of the Beijing–Tianjin–Hebei regional government guidance fund based on the entropy method The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations Research on behavioural differences in the processing of tenant listing information: An eye-movement experiment A review of the treatment techniques of VOC Some classes of complete permutation polynomials in the form of ( x ^{pm}−x +δ )^{s}+ax ^{pm}+bx overF _{p2m}Deformation and stress theory of surrounding rock of shallow circular tunnel based on complex variable function method The consistency method of linguistic information and other four preference information in group decision-making Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development A mathematical model of the fractional differential method for structural design dynamics simulation of lower limb force movement step structure based on Sanda movement Numerical calculation and study of differential equations of muscle movement velocity based on martial articulation body ligament tension Study on the maximum value of flight distance based on the fractional differential equation for calculating the best path of shot put Sports intensity and energy consumption based on fractional linear regression equation Translog function in government development of low-carbon economy Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Research on the Psychological Distribution Delay of Artificial Neural Network Based on the Analysis of Differential Equation by Inequality Expansion and Contraction Method Study on Establishment and Improvement Strategy of Aviation Equipment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode The Model of Sugar Metabolism and Exercise Energy Expenditure Based on Fractional Linear Regression Equation Constructing Artistic Surface Modeling Design Based on Nonlinear Over-limit Interpolation Equation Numerical Simulation Analysis Mathematics of Fluid Mechanics for Semiconductor Circuit Breaker Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Human Body Movement Coupling Model in Physical Education Class in the Educational Mathematical Equation of Reasonable Exercise Course The contribution of structural equation model analysis to higher education agglomeration and innovation and entrepreneurship Study on the evolutionary game theory of the psychological choice for online purchase of fresh produce under replicator dynamics formula The influence of X fuzzy mathematics method in basketball tactics scoring Mathematical statistics algorithm in the bending performance test of corroded reinforced concrete beams under fatigue load Nonlinear strategic human resource management based on organisational mathematical model Back propagation mathematical model for stock price prediction Evolutionary game research on the psychological choice of online shopping of fresh agricultural products based on dynamic simulation model Differential equation model of financial market stability based on big data Multi-attribute decision-making methods based on normal random variables in supply chain risk management Linear fractional differential equations in bank resource allocation and financial risk management model Construction and reform of art design teaching mode under the background of the integration of non-linear equations and the internet Spatial–temporal graph neural network based on node attention A contrastive study on the production of double vowels in Mandarin Financial accounting measurement model based on numerical analysis of rigid normal differential equation and rigid generalised functional equation Research of cascade averaging control in hydraulic equilibrium regulation of heating pipe network Mathematical analysis of civil litigation and empirical research of corporate governance Health monitoring of Bridges based on multifractal theory College students’ innovation and entrepreneurship ability based on nonlinear model Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Mathematical simulation analysis of optimal testing of shot puter's throwing path Performance evaluation of college laboratories based on fusion of decision tree and BP neural network Application and risk assessment of the energy performance contracting model in energy conservation of public buildings The term structure of economic management rate under the parameter analysis of the estimation model based on common differential equation Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas The Spatial Form of Digital Nonlinear Landscape Architecture Design Based on Computer Big Data The improvement of museum information flow based on paste functional mapping method The art design of industrialised manufacturing furniture products based on the simulation of mathematical curves TOPSIS missile target selection method supported by the posterior probability of target recognition Research on Evaluation of Intercultural Competence of Civil Aviation College Students Based on Language Operator The incentive contract of subject librarians in university library under the non-linear task importance Modelling and Simulation of Collaborative Innovation System in Colleges and Universities Based on Interpreted Structural Equation Model Small amplitude periodic solution of Hopf Bifurcation Theorem for fractional differential equations of balance point in group competitive martial arts The Optimal Solution of Feature Decomposition Based on the Mathematical Model of Nonlinear Landscape Garden Features Composite mechanical performance of prefabricated concrete based on hysteresis curve equation Higher education innovation and reform model based on hierarchical probit Application of Fuzzy Mathematics Calculation in Quantitative Evaluation of Students’ Performance of Basketball Jump Shot The teaching of sports science of track and field-based on nonlinear mathematical equations Visual error correction of continuous aerobics action images based on graph difference function Ecological balance model of effective utilization of agricultural water resources based on fractional differential equations Application of Higher Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Precision algorithms in second-order fractional differential equations Application of Forced Modulation Function Mathematical Model in the Characteristic Research of Reflective Intensity Fibre Sensors Fractional differential equations in National Sports Training in Colleges and Universities Radioactive source search problem and optimisation model based on meta-heuristic algorithm Visualized calculation of regional power grid power data based on multiple linear regression equation Application of mathematical probabilistic statistical model of base – FFCA financial data processing Least-squares method and deep learning in the identification and analysis of name-plates of power equipment Research on a method of completeness index based on complex model Distribution network monitoring and management system based on intelligent recognition and judgement Fake online review recognition algorithm and optimisation research based on deep learning Research on the 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 Topological optimisation technology of gravity dam section structure based on ANSYS partial differential equation operation Interaction design of financial insurance products under the Era of AIoT Modeling the pathway of breast cancer in the Middle East Corporate social responsibility fulfilment, product-market competition and debt risk: Evidence from China ARMA analysis of the green innovation technology of core enterprises under the ecosystem – Time series data Reconstruction of multimodal aesthetic critical discourse analysis framework Image design and interaction technology based on Fourier inverse transform What does students’ experience of e-portfolios suggest 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 optimal model of employment and entrepreneurship models in colleges and universities based on probability theory and statistics 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 Analysing the action techniques of basketball players’ shooting training using calculus method Engineering project management based on multiple regression equation and building information modelling technology Research on predictive control of students’ performance in PE classes based on the mathematical model of multiple linear regression equation Beam control method for multi-array antennas based on improved genetic algorithm The influence of X fuzzy mathematical method on basketball tactics scoring Mathematical model of back propagation for stock price forecasting Application of regression function model based on panel data in bank resource allocation financial risk management Application of Logical Regression Function Model in Credit Business of Commercial Banks Research on aerobics training posture motion capture based on mathematical similarity matching statistical analysis Application of Sobolev-Volterra projection and finite element numerical analysis of integral differential equations in modern art design Research on motion capture of dance training pose based on statistical analysis of mathematical similarity matching 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 Application of Lane-Emden differential equation numerical method in fair value analysis of financial accounting Regression function model in risk management of bank resource allocation Application of numerical method of functional differential equations in fair value of financial accounting Optimal solution of fractional differential equations in solving the relief of college students’ mental obstacles Risk contagion in financial markets based on copula model Calculating university education model based on finite element fractional differential equations and macro-control analysis Automatic parameter selection ZVD shaping algorithm for crane vibration suppression based on particle swarm optimisation 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 Calculation of tourism development income index based on finite element ordinary differential mathematical equation Adoption of deep learning Markov model combined with copula function in portfolio risk measurement Radar system simulation and non-Gaussian mathematical model under virtual reality technology Comparison of compression estimations under the penalty functions of different violent crimes on campus through deep learning and linear spatial autoregressive models Research and application of constructing football training linear programming based on multiple linear regression equation Research on management evaluation of enterprise sales cash flow percentage method based on the application of quadratic linear regression equations Mathematical simulation analysis of optimal detection of shot-putters’ best path Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Mathematical modelling of enterprise financial risk assessment based on risk conduction model Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets Mathematical simulation experiment based on optimisation of heat treatment process of aluminium alloy materials Mathematical model of transforming image elements to structured data based on BP neural network Educational reform informatisation based on fractional differential equation 3D Mathematical Modelling Technology in Visual Rehearsal System of Sports Dance 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 Informationisation of teaching model for track and field education based on finite element higher-order fractional differential equation Information technology of preschool education reform of fine arts based on fractional differential equation Information Teaching Model of Preschool Art Education in Colleges and Universities Based on Finite Element Higher-Order Fractional Differential Equation Application of artificial intelligence algorithm in mathematical modelling and solving College Students’ Mental Health Climbing Consumption Model Based on Nonlinear Differential Equations Communication architecture of power monitoring system based on incidence matrix model Differential equation to verify the validity of the model of the whole-person mental health education activity in Universities Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Analysis and Prediction of College Students’ Mental Health Based on K-means Clustering Algorithm Mathematical function data model analysis and synthesis system based on short-term human movement Human gait modelling and tracking based on motion functionalisation Analysis and synthesis of function data of human movement Energy-saving technology of BIM green buildings using fractional differential equation Study on the training model of football movement trajectory drop point based on fractional differential equation Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation User online consumption behaviour based on fractional differential equation Differential equation model of financial market stability based on Internet big data Multi-attribute Decision Method Based on Normal Random Variable in Economic Management Risk Control Children’s cognitive function and mental health based on finite element nonlinear mathematical model Dichotomy model based on the finite element differential equation in the educational informatisation teaching reform model Nonlinear Dissipative System Mathematical Equations in the Multi-regression Model of Information-based Teaching Stock price analysis based on the research of multiple linear regression macroeconomic variables Fractional Linear Regression Equation in Agricultural Disaster Assessment Model Based on Geographic Information System Analysis Technology