With the rapid development of digital media technology and the popularisation of mobile electronic devices, we have entered an era of information explosion . At least 80% of the information obtained by humans is obtained through vision. A picture is worth a thousand words. Images occupy a very large proportion because of their rich content and vivid expressions. Image and video data play an important role in many fields, such as surveillance security, virtual reality, 3D film and television, ultra-high-definition video, mobile terminals, etc. [2, 3]. In the information age, image data is very easy to obtain, and everyone can become a self-media, so massive amounts of image data are generated and transmitted on the network every day [4, 5]. On 30 August, 2019, the 44th ‘Statistical Report on Internet Development in China’ released by China Internet Network Information Centre (CNNIC) pointed out that as of June 2019, the number of Internet users in China reached 854 million, and the number of online video users reached 759 million. In the first half of 2019, mobile Internet access traffic reached 55.39 billion GB .
The traditional signal sampling technology needs to follow the Nyquist sampling theorem, that is, the sampling frequency needs to be greater than twice the highest frequency of the signal. It can realise signal sampling and compression at the same time, reduce the signal sampling frequency, reduce the loss of hardware resources, and can realise the encryption of information, so it is very suitable for wireless sensor networks . The current research mainly focuses on the sparse representation of the signal, the construction of the measurement matrix and the design of the reconstruction algorithm. Image perception is a major branch of perceptual hashing, and its feature extraction and encoding stages need to contain more visual perception information to meet its robustness requirements. However, the current research on image-aware hashing algorithms lacks the consideration of human visual characteristics and focuses on general image features such as image greyscale and feature points . Literature  proposes that image perception technology breaks the limitation of the traditional sampling theorem and provides people with a new technology that integrates signal acquisition, which has attracted widespread attention from scholars and research institutions around the world. Literature [9, 10] proposed that the current image perception technology has been widely used in signal processing, medical imaging, wireless communication, target positioning, radar detection and other fields. Choi et al.  proposed a binary block diagonal matrix, which is a simple and convenient construction method. It is easy to implement in hardware, and the effect of sensor signal reconstruction is also improved. Abonyi and Honti  proposes a fast structured random measurement matrix. This measurement matrix adopts block processing and has the characteristics of fast calculation speed. It is suitable for large-scale information transmission and image perception application scenarios with high real-time requirements. Literature  proposed two sparse measurement matrix construction methods, which not only ensured the reconstruction accuracy of the sensor signal but also reduced the energy consumption of the device and the cost of hardware fabrication. The current reconstruction and recovery algorithms are mainly combined algorithms, convex optimisation algorithms and greedy algorithms. Commonly used algorithms include the matching pursuit algorithm and the orthogonal matching pursuit algorithm. Zhao et al.  proposes a fast sensor signal reconstruction and reconstruction algorithm. This reconstruction algorithm not only has high reconstruction accuracy of the sensor signal but also has a small amount of calculation. Mallat and Zhang  proposes a reconstruction algorithm based on dictionary learning, which improves the reconstruction accuracy of the sensor signal at a low sampling rate. structure effect. On the one hand, this feature information reflects the perceptual characteristics of the image, and on the other hand, the algorithms based on them also maintain operation and transformation robustness to various contents to varying degrees. However, these image-aware hashing algorithms have not done special research on the process of human perception of images and the perceptual characteristics of human vision. Although signal sampling, compression and encryption can be achieved, the encryption implemented by compressed sensing is weak and cannot effectively resist various attacks. (The first paragraph which introduces compressed sensing is a bit redundant, and the title is the application of matrix multiplication in signal sensor image perception)
Phuttharak and Loke  and Silva et al.  have made groundbreaking work on compressed sensing theory and established a theoretical framework. Literature [17, 18] established the Restricted Isometry Principle (RIP). Although this theoretical characteristic is excellent, it cannot be used to guide the design of the measurement matrix, because it is only a sufficient condition for the measurement matrix. Pal et al.  and Hewa et al.  mentioned that the sparseness of perceptual data is prevalent in personalised filtering recommendation models for cross-border e-commerce. To change the user similarity calculation based on compressed sensing and use particle distance (PD) to realise the similarity calculation of cross-item mobile users, an e-commerce personalised filtering recommendation algorithm based on the fusion of particle-level compressed sensing and e-commerce customer trust relationship is proposed. The experimental results show that, compared with other methods, the user similarity calculation method fused with granular compressed sensing can alleviate the impact of sparse sensing data on the personalised filtering algorithm of e-commerce [21–23]. The advantages and disadvantages of wireless sensor networks are analysed, and the basic knowledge for the application of compressed sensing in this environment is provided. The three important parts of the theory, the sparse representation of the signal, the construction of the measurement matrix and the design of the reduction algorithm are studied, and the existing problems are summarised respectively.
Based on the theory of tensor compressed sensing, this paper introduces the concept of the auxiliary matrix to realize flexible and safe observation of information. Aiming at the high-security requirements of wireless sensor networks for image information transmission, an efficient image encryption algorithm based on DNA coding operation and a chaotic system is proposed. Furthermore, the combination of the new P-tensor compressed sensing model and the DNA encoding encryption algorithm is applied to the image transmission in wireless sensor networks, which realises the efficient utilisation of resources, low time consumption and high security of image transmission.
It is an important prerequisite for realising compressed sensing that the signal can be sparsely represented. It is precisely the signal in nature that can be compressed, that is, it can be converted into a sparse signal in a certain transform domain. Assuming that
Common signal sparse representations are divided into two categories. One is to create sparse signals through sparse transform bases to achieve the effect of sparseness. Common sparse transform bases include discrete wavelet transform bases, discrete cosine transform bases, and discrete Fourier transform bases. Transformation bases, etc.; the other is to build a sparse dictionary library through dictionary learning, and then use the sparse dictionary library to represent the signal.
The compressed observation process uses the measurement matrix Φ to linearly project the signal
The measurement matrix plays an important role in compressive sensing to achieve accurate signal recovery, and it is an important link between data compression sampling and reconstruction recovery. For the uniqueness and accurate reconstruction of the signal, the construction of the measurement matrix usually needs to satisfy three constraints, namely the Spark property, the incoherence and the finite isometric principle.
Satisfying the Spark property is a necessary condition to ensure the success of reconstruction. The Spark property represents the minimum number of linear correlation vector groups in the column vector of the matrix, which can be expressed as:
There is at most one signal
The finite isometric principle is defined as the existence of a constant
As long as the Spark property and the finite isometric principle are satisfied, the original signal can be guaranteed to be successfully reconstructed, but in practical use, the above two constraints are not easy to verify. Therefore, this paper proposes an easily verifiable constraint, the incoherence.
The smaller the value of
The reconstruction and recovery of the signal is the process of recovering the original signal
This is a non-convex optimisation problem, and Eq. (6) can be transformed into a convex optimisation problem, namely:
At this time,
Common reconstruction and recovery algorithms are mainly divided into greedy algorithms and convex optimisation algorithms [24, 25]. Although the number of observations in the convex optimisation algorithm is small, the computational complexity is higher than other algorithms: the greedy algorithm is easy to operate, has a fast recovery speed, and has a wide range of applications, but The sparsity of the reconstructed signal needs to be analysed before using the greedy algorithm. Different reconstruction algorithms have different characteristics. Therefore, it is necessary to select a suitable reconstruction recovery algorithm in combination with specific application scenarios.
First, the matrix-tensor product, semi-tensor product and P-tensor product are introduced, and then the semi-tensor compressed sensing (STP-CS) theory and the P-tensor compressed sensing theory are combined.
The traditional matrix multiplication needs to satisfy the dimension matching, but the tensor product can realise the multiplication of two arbitrarily sized matrices. The tensor integral is the left tensor product and the right tensor product. This chapter takes the right tensor product as an example. For example, if the matrix
The tensor product has the following operational properties:
Associativity: Distributive law: Transpose: Inverse of a matrix: Rank of the matrix:
Inverse of a matrix:
Rank of the matrix:
If the matrices
Among them, Associativity: Distributive law: If If
It can be seen from the definition that the semi-tensor product operation can realize the operation of two matrices with mismatched dimensions, so the semi-tensor product has been widely used once it was proposed. It has strong practicality in nonlinear problems and multi-linear problems [26, 27]. At present, the semi-tensor product operation has been widely used in Boolean networks, graph theory, linear algebra, control and other fields.
Now, the restoration effect is analysed for the greyscale image, the qualitative analysis of the reconstructed image is performed from the visual point of view, and the peak signal-to-noise ratio (PSNR) of the reconstructed image is quantitatively analysed from the mathematical point of view. The PSNR can measure the restoration effect of the reconstructed image. Before calculating the PSNR, the mean square error (MSE) needs to be calculated. The MSE refers to the mean of the squared error between the reconstructed image and the original image. The specific calculation equation is as follows:
Associativity: Distributive Law: If If the matrices
If the matrices
The semi-tensor product operation of matrices not only breaks the limitation that the traditional matrix multiplication must satisfy the dimension matching but also retains the characteristics of the traditional matrix multiplication operation . It is known that the number of columns of the left factor matrix in the semi-tensor product operation does not have to be equal to the number of rows of the right factor matrix, which greatly improves the flexibility of matrix multiplication and has been widely used in many fields. Since compressed sensing involves both one-dimensional information processing and two-dimensional information processing, compressed sensing is closely related to matrix multiplication. Based on this, the literature proposes a STP-CS model, which can be expressed as:
Except for the first row and column, every other element is the same as its upper-left corner element. In addition, let:
Then a special form of the Toeplitz matrix is obtained, a circulant matrix commonly used in coding.
The product of the Toeplitz matrix and the vector corresponds to the convolution of the signal with the impulse response of the channel parameterised by the first column of the Toeplitz matrix, and the product of the further circulant matrix and the signal vector corresponds to the channel impulse response and the signal Circular convolution. For example, a signal B of length A passes through a channel with C tap coefficients, and the impulse response of the channel is denoted, then the Toeplitz matrix representation of the convolution of the signal and the channel is:
After completing the construction of the above sparse Toeplitz/circular block matrix, according to the general proof method proposed, it is theoretically proved that the sparse Toeplitz/circular matrix constructed in this chapter satisfies the RIP property. To facilitate the proof, the observation matrix discussed in this section is based on the Toeplitz matrix shown in the equation column unitisation, and the proof of the circulant matrix is its direct extension, so it will not be repeated.
For the block matrix, the properties of the Gram matrix can be further obtained by discussing the properties of the Gram matrix block. Let
Then the diagonal block of the Grammatic matrix can be easily calculated from the properties of the diagonal shift matrix
Then further, off-diagonal blocks can be written as:
Going back to each element of the Gram matrix, from the previous proof, we get
Combining the probabilities of the diagonal and off-diagonal elements of the Gram matrix, we get:
The random Gaussian matrix and two improved sparse matrices, which are commonly used in compressed sensing, are used as comparison objects for simulation comparison. The operating environment is shown in Table 1. All simulations involved in this article are in the same operating environment.
Time complexity comparison
|Method||Space complexity||Time complexity|
|Sparse two-dimensional matrix2 D matrices||990,080||0.0079s|
|Improved sparse two-dimensional matrix||32,768||0.0027s|
|Improved sparse random matrix||32,768||0.0112s|
|The space of the random Gaussian matrices||2,990,080||0.0142s|
The first thing to do is to verify the parameter values of the two measurement matrices. The representation in the sparse two-dimensional matrix divides the sampled values
The simulation experiment takes the original random data length
In the simulation process, in order to ensure the reliability of the algorithm in real-world applications, the actual data of the St. Helens volcano is used for testing. The raw data is taken from a segment when the seismic wave arrives, and the severe shaking of the image perception data can be clearly seen. Take the test data length,
It can be seen from the restoration effect in Figure 3 that the information reconstruction has reached an ideal state. The improved sparse random matrix is also simulated, and the above data is also used, but to see the restoration effect more clearly, only the part with large jitter at the back end of the above number is selected. In the experiment, the data length A and the measured value
As can be seen from Figure 3, the sparse random matrix is the same as the sparse two-dimensional matrix, and finally has a very ideal effect, which does not affect the experiment of the reduction algorithm. However, since the generation period of the sparse random matrix is slightly longer than that of the sparse two-dimensional matrix, the following experiments all use the sparse two-dimensional matrix as the measurement matrix to test the reduction algorithm. For the analysis of actual completion time and storage volume, the minimum completion time for wavelet transform is 1.29 for sparse estimation for preliminary P-wave selection 0.07 for compressed sensing 0.20. The maximum completion time for wavelet transform was 1.32 for sparse estimation 0.62 for preliminary P-wave selection 0.17 for compressed sensing 0.88. It can be seen from this that the processing time of the overall operation in the sensor will not exceed 3 s, and the running time is only 0.22-0.88 s, which meets the requirements of real-time performance.
Next, to better represent the effectiveness of the overall algorithm, we selected four periods of original observation data in different periods, and only intercepted a part with obvious changes, data length
It can be seen from Figure 4 that the overall algorithm is stable enough to be suitable for various signals. The improved restoration algorithm can basically complete accurate reconstruction, and the quality of the reconstructed signal is stable. The results of the first three experiments can fully verify that the overall algorithm guarantees high-precision reducibility.
Then proceed to simulate the specific advantages of the improved algorithm. Select several popular algorithms as comparison objects, such as orthogonal matching algorithm (OMP, compressive sampling matching pursuit (CoSaMP), model-based algorithm (Model-based), an iterative algorithm with a hard threshold (IHT), orthogonalised hard threshold Iterative algorithm (OSIHT), etc. Two indicators are mainly used to measure the properties of the algorithm: RMSE and calculation time. In the simulation, a large number of repeated experiments need to be set to obtain the average value as the result.
After comparing several restoration algorithms that are popular in the field of compressed sensing today, it can be seen that the improved algorithm converges faster than other algorithms, and the overall reconstruction time is much less than other algorithms.
Next, to more intuitively show the excellent performance of the improved algorithm, several restoration algorithms are used to restore the same signal, and the results are shown in Figure 5.
As can be clearly seen in Figure 6, the restoration effect of the improved algorithm is far better than that of the comparison object, and a high-precision restoration is ensured. It can be concluded from Figures 5 and 6 that the improved hard-threshold iterative reconstruction algorithm based on the wavelet tree model has better performance than the existing algorithm, and can basically and accurately restore the original signal.
The next step is to compare the restoration effects of the four restoration algorithms. Different measurement values will be selected to see the reconstruction error results of the algorithms, and the RMSE will be used to uniformly represent the error. The comparison results are shown in Figure 7. It can be seen from Figure 7 that the proposed algorithm has the lowest error rate.
This paper makes a comprehensive analysis of the theory of compressed sensing and applies the theory to the actual situation to solve the problems in the application. First, the introduction of wireless sensor network and the elaboration of compressive sensing theory, analyses the advantages and disadvantages of wireless sensor networks, and provide a basic knowledge for the application of compressed sensing in this environment; second, it studies the three important parts of the theory, which are the sparse representation of the signal, the construction of the measurement matrix and the design of the restoration algorithm, and respectively summarize the existing problems, and finally get the following conclusions through simulation experiments:
Improvement of the measurement matrix. Due to the small memory and limited energy of wireless sensor nodes, if a complete NM × measurement matrix is generated in practical applications, it will cause storage overflow and take too long to form. Improvement of the restoration algorithm. The improved restoration algorithm is based on the wavelet tree modelling algorithm and applies the hard threshold iterative algorithm, and then adds an adaptive step size in the iterative process. The step size is also simpler and faster than the step size calculation proposed by the existing algorithm speed while ensuring the accuracy of the restoration. Apply the improved algorithm to distributed sensor networks. By comparing the calculation time of the reduction algorithm, the convergence time CoSaMp is 0.043, the IHT is 1.92*e-3, the Model-based is 2.86*e-4, and the improved algorithm is 1.65*e-4. The total time CoSaMp is 1.984, the IHT is 0.014, the Model-based is 6.82*e-4, and the improved algorithm is 2.58*e-4. Through experiments, it is concluded that the two improved measurement matrices have great advantages in terms of time and storage space required for forming; the improved restoration algorithm has better effects than the existing algorithms in terms of time and restoration accuracy. Therefore, it can be concluded that the improved algorithm can meet the problem of small memory of general wireless sensor nodes, solve the problem of large errors after back-end restoration processing, reduce the number of observation samples required, and speed up the overall operation speed. It is suitable for wireless sensor real-time monitoring network.
Improvement of the measurement matrix. Due to the small memory and limited energy of wireless sensor nodes, if a complete NM × measurement matrix is generated in practical applications, it will cause storage overflow and take too long to form.
Improvement of the restoration algorithm. The improved restoration algorithm is based on the wavelet tree modelling algorithm and applies the hard threshold iterative algorithm, and then adds an adaptive step size in the iterative process. The step size is also simpler and faster than the step size calculation proposed by the existing algorithm speed while ensuring the accuracy of the restoration.
Apply the improved algorithm to distributed sensor networks. By comparing the calculation time of the reduction algorithm, the convergence time CoSaMp is 0.043, the IHT is 1.92*e-3, the Model-based is 2.86*e-4, and the improved algorithm is 1.65*e-4. The total time CoSaMp is 1.984, the IHT is 0.014, the Model-based is 6.82*e-4, and the improved algorithm is 2.58*e-4. Through experiments, it is concluded that the two improved measurement matrices have great advantages in terms of time and storage space required for forming; the improved restoration algorithm has better effects than the existing algorithms in terms of time and restoration accuracy. Therefore, it can be concluded that the improved algorithm can meet the problem of small memory of general wireless sensor nodes, solve the problem of large errors after back-end restoration processing, reduce the number of observation samples required, and speed up the overall operation speed. It is suitable for wireless sensor real-time monitoring network.
Time complexity comparison
|Method||Space complexity||Time complexity|
|Sparse two-dimensional matrix2 D matrices||990,080||0.0079s|
|Improved sparse two-dimensional matrix||32,768||0.0027s|
|Improved sparse random matrix||32,768||0.0112s|
|The space of the random Gaussian matrices||2,990,080||0.0142s|Law of interest rate changes in financial markets based on the differential equation model of liquidity Research and implementation of smart city public information mining analysis system based on mobile edge model of game theory Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Design of college education evaluation based on accompanying data acquisition and mathematical analysis Enterprise financial strategy and performance management analysis based on principal component analysis Study on the Dynamic Change of Regional Water Level and Climate Based on Forecast Equation Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Municipal Civil Engineering Construction Based on Finite Element Differential Equations Financial Risk Prevention Model of Financial Institutions Based on Linear Partial Differential Equation Influence of heterogeneity of local officials on the economy of resource-based cities developed with high quality Analysis of influencing factors of SPOC course teaching effect using structural equation modelling Training Model of Basketball Offensive Route Based on Nonlinear Differential Equation Research on the construction of rural interface style based on aesthetic rules Optimization Algorithm of New Media Hot Event Push Based on Nonlinear Differential Equation Mathematical Differential Equation in Calculating the Strength of Building Beam Structure Impact Resistance Stability of Building Structural Engineering Based on Fractional Differential Equations The Technical Research on the Assessment of Network Security Situation Based on D-S Evidence Theory Computer big data modeling system based on finite element mathematical equation simulation Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation The Composition System of Pop Music Movement Based on Finite Element Differential Equations The Structure and Influencing Factors of Innovation and Entrepreneurship Ability of Higher Vocational Students Based on Structural Equation Model Model System Study of Accordion Score Based on Fractional Differential Equations Data mining of Chain convenience stores location Parameter Id of Metal Hi-pressure State Equation P-Matrix Reasoning and Information Intelligent Mining Symmetry Analysis and Exact Solutions of Extended Kadomtsev-Petviashvili Equation in Fluids Research on Detection Model of Abnormal Data in Engineering Cost List Research on deformation monitoring of tunnel engineering based on 3D laser scanning Solidification treatment effect of wellsite waste mud based on physical adsorption of a composite curing agent A study of immersive technology for product usability improvement design based on comprehensive value evaluation Intelligent Recommendation System for English Vocabulary Learning – Based on Crowdsensing Application of Nonlinear Fractional Differential Equations in Computer Artificial Intelligence Algorithms Application of calculus function and value chain analysis method in the quantification of financial flexibility management indicators Optimal Research in Piano Shape Sound and Sound Propagation Model Based on Nonlinear Differential Equations Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations Badminton players’ trajectory under numerical calculation method Prediction of surface quality in end milling based on modified convolutional recurrent neural network Analysis of IPO pricing efficiency under the registration system Case application data research of traditional ink art elements in packaging design Research on an early warning model of effectiveness evaluation in ideological and political teaching based on big data Application of intelligent teaching resource organisation model and construction of performance evaluation model Secure transmission of simultaneous wireless information and power transfer system for Internet of things Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Numerical simulation of vortex vibration in main girder of cable-stayed bridge based on bidirectional fluid–structure coupling Research on location algorithm of new energy vehicle charging station based on multi-objective decision Optimisation of construction mode of residential houses based on the genetic algorithm under BIM technology A study on the reform of college English education inspired by the cultural differences between China and the United States Research on innovative strategies of college students’ English teaching under the blessing of big data Research on multi-dimensional optimisation design of user interface under Rhino/GH platform Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption Attitude control for the rigid spacecraft with the improved extended state observer A long command subsequence algorithm for manufacturing industry recommendation systems with similarity connection technology Application of Higher-Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Evaluation of spoken English self-study system considering speech knowledge recognition algorithm Application research on piano teaching in colleges and universities based on remote wireless network communication Construction of Financial Risk Evaluation Index System for Biomass Graphene Fibre Industrialisation Project System dynamics model of output of ball mill Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Design of Morlet wavelet neural network to solve the non-linear influenza disease system An online statistical analysis of the hotel management and operation performance model Research on the post-purchase behaviour tendency of the product quality to customers in fast-selling marketing Motion about equilibrium points in the Jupiter-Europa system with oblateness Ultra-short-term power forecast of photovoltaic power station based on VMD–LSTM model optimised by SSA Optimal preview repetitive control for impulse-free continuous-time descriptor systems Design of information management system based on wireless communication under the background of Internet of Things The impact of global warming on the small Scottish Fishing Company Development of main functional modules for MVB and its application in rail transit Analysis of agricultural economic development and optimisation measures under the strategy of rural revitalisation Study on the impact of forest fire prevention policy on the health of forest resources Value Creation of Real Estate Company Spin-off Property Service Company Listing Selection by differential mortality rates Research on the relationship between government subsidies, R&D investment and high-quality development of manufacturing industry Research on the protection and inheritance of intangible cultural heritage under the background of rural revitalisation Research on behavioural differences in the processing of tenant listing information: An eye-movement experiment Innovation of Economic Management Risk Control in Retailer Supply Chain Based on Differential Equation Model Computer vision recognition and tracking algorithm based on convolutional neural network A review of the treatment techniques of VOC Study on structural parameter design and life analysis of large four-point contact ball bearing channel Some classes of complete permutation polynomials in the form of ( x p m− x+ δ) s+ ax p m+ bxover F p2 m Digital marketing solutions based on consumer data and homomorphic encryption The consistency method of linguistic information and other four preference information in group decision-making Statistical Model of College Ideological and Political Learning Based on Fractional Differential Equations Research on the driving principle and guiding strategy of the public's collaborative supervision of the sharing economy in my country Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development Nonlinear Differential Equation in Anti-aging Test of Polymer Nanomaterials Application research of bel canto performance based on artificial intelligence technology Fractal structure of magnetic island in tokamak plasma Mechanics of Building Structural Materials Based on Lagrangian Mathematical Model Analysis The Mental Health Education Management of Higher Vocational Students Based on Fractional Differential Equations Application of regression function model based on panel data in financial risk management of bank resource allocation Application of knowledge graph in smart grid fault diagnosis University Ideological and Political Learning Model Based on Statistical Memory Curve Mathematical Equation Research on the optimisation of logistics parcel intelligent sorting and conveying chain combined with variable clustering mathematical method Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Research on Resonance Properties of Semantic Wave Fractal Fractals Based on Quantitative Analysis of English Corpus Research on urban landscape big data information processing system based on ordinary differential equations Modeling of fractional differential equation in cloud computing image fusion algorithm Application of Discriminative Training Algorithm Based on Intelligent Computing in English Translation Evaluation Research on the application of GLE teaching mode in English-medium colleges The application of directional derivative in the design of animation characters and background elements Research on product process design and optimisation model based on IoT intelligent computing Conventional Algorithms in Sports Training Based on Fractional Differential Equations Dynamic Nonlinear System Based on Complex System Theory in the Development of Vocational Education LTE wireless network coverage optimisation based on corrected propagation model The Algorithm Accuracy of Mathematical Model to Improve the Transmission Speed of E-commerce Platform Study on tourism development income index calculation of finite element ordinary differential mathematical equation The Security of Database Network Model Based on Fractional Differential Equations Electric Vehicle Mechanical Transmission System Based on Fractional Differential Equations The Practice System of Physics and Electronics Courses in Higher Vocational Colleges Based on Fractional Differential Equations The Economic Model of Rural Supply and Demand Under the Data Analysis Function Based on Ordered Probit 3D Modeling System of Indoor Environment Art Landscape Design under Statistical Equation Electronic Information Security Model of Nonlinear Differential Equations The Optimization Model of College Students' Physical Exercise Motivation and Self-control Ability Based on the Mathematical Model of Probability Theory Impact of ASEAN-China free trade area on fishery value chain based on difference-in-difference method Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas Criminal law imputation path for biometric information Research on composite dynamic disaster prevention and control system of mine earthquake and shock in thick and hard rock mines Research on innovative strategies of college students’ English teaching under the background of artificial intelligence Electromagnetic interference prediction technology of new energy motor drive system Research on the application of PLC technology in electrical automation engineering Research on indoor environment design of comprehensive commercial shopping center based on numerical simulation Application of matrix multiplication in signal sensor image perception Empirical analysis of the economic absolute income hypothesis based on mathematical statistics Analysing the variation of metadiscourse verb patterns in English academic papers from within and between disciplines Impact of COVID-19 policy on stock prices of listed property companies Realization of Book Collaborative Filtering Personalized Recommendation System Based on Linear Regression Equation Research on the experimental principle of deep integration of LETS software and criminal procedure under the background of artificial intelligence Study on Interactive Relations between Enterprise Social Media and Decision Style Based on a vector Autoregressive Model Research on Dynamics of Flexible Multibody System with Deployable Antenna Based on Static Lagrangian Function The Optimization of Mathematics Teaching Models in Colleges and Universities Based on Nonlinear Differential Equations Study on spatial planning and design of learning commons in university libraries based on fuzzy matrix model The Stability Model of Piano Tone Tuning Based on Ordinary Differential Equations Construction and application of automobile user portrait based on k-mean clustering model The Homework Model of Screening English Teaching Courses Based on Fractional Differential Equations VR-based computer maintenance practical training platform development design and application research Research on innovative human capital for China’s economic development based on STI model In-depth analysis of the artistic expression of paper-cut elements in the design of boat space Knowledge graph construction and Internet of Things optimisation for power grid data knowledge extraction Modeling the pathway of breast cancer in the Middle East Construction and intelligent analysis of power grid physical data knowledge graph based on Internet of Things for power system Research on industrial Internet of Things and power grid technology application based on knowledge graph and data asset relationship model Research on the effects of financial literacy on rural household credit constraint Calculus Logic Function in Integrated Manufacturing Automation of Single Chip Microcomputer Football Offense Training Strategy Based on Fractional Differential Mathematical Modeling Research on educational resource recommendation system based on MRLG Rec The Mathematical Analysis Model of Educational System in Music Courses in Colleges and Universities Continuing Education Network Data Center Model Based on Fractional Differential Mathematical Equations A study on the phenomenon of anaphoric correction in college students’ English conversation Computer Art Design Model Based on Nonlinear Fractional Differential Equations The Optimization Model of Public Space Design Teaching Reform Based on Fractional Differential Equations The Approximate Solution of Nonlinear Vibration of Tennis Based on Nonlinear Vibration Differential Equation Graphical Modular Power Technology of Distribution Network Based on Machine Learning Statistical Mathematical Equation Employment and Professional Education Training System of College Graduates Based on the Law of Large Numbers Economic Research on Multiple Linear Regression in Fruit Market inspection and Management Nonlinear Differential Equations in Preventing Financial Risks Lagrange’s Mathematical Equations in the Sports Training of College Students Simulation Research of Electrostatic Precipitator Power Supply Voltage Control System Based on Finite Element Differential Equation Research on the effect of generative adversarial network based on wavelet transform hidden Markov model on face creation and classification Research on Lightweight Injection Molding (CAE) and Numerical Simulation Calculate of New Energy Vehicle Power Flow Calculation in Smart Distribution Network Based on Power Machine Learning Based on Fractional Differential Equations Demonstration of application program of logistics public information management platform based on fuzzy constrained programming mathematical model Basketball Shooting Rate Based on Multiple Regression Logical-Mathematical Algorithm The Optimal Application of Lagrangian Mathematical Equations in Computer Data Analysis Similarity Solutions of the Surface Waves Equation in (2+1) Dimensions and Bifurcation Optimal decisions and channel coordination of a green supply chain with marketing effort and fairness concerns Game theoretic model for low carbon supply chain under carbon emissions reduction sensitive random demand Limit cycles of a generalised Mathieu differential system Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Application of data mining in basketball statistics The nonlinear effects of ageing on national savings rate – An Empirical Study based on threshold model Design of fitness walker for the elderly based on ergonomic SAPAD model AtanK-A New SVM Kernel for Classification Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Study of a linear-physical-programming-based approach for web service selection under uncertain service quality The Relationship Between College Students’ Taekwondo Courses and College Health Based on Mathematical Statistics Equations Analysis and countermeasures of cultivating independent learning ability in colleges teaching English based on OBE theory A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Fractional Differential Equations in the Exploration of Geological and Mineral Construction AdaBoost Algorithm in Trustworthy Network for Anomaly Intrusion Detection Projection of Early Warning Identification of Hazardous Sources of Gas Explosion Accidents in Coal Mines Based on NTM Deep Learning Network Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China Enterprise Financial Risk Early Warning System Based on Structural Equation Model A Study on the Application of Quantile Regression Equation in Forecasting Financial Value at Risk in Financial Markets Fractional Differential Equations in the Model of Vocational Education and Teaching Practice Environment Information transmission simulation of Internet of things communication nodes under collision free probability equation Image denoising model based on improved fractional calculus mathematical equation Random Fourier Approximation of the Kernel Function in Programmable Networks The Complexity of Virtual Reality Technology in the Simulation and Modeling of Civil Mathematical Models University Library Lending System Model Based on Fractional Differential Equations Calculation and Performance Evaluation of Text Similarity Based on Strong Classification Features Intelligent Matching System of Clauses in International Investment Arbitration Cases Based on Big Data Statistical Model Evaluation and Verification of Patent Value Based on Combination Forecasting Model Financial Institution Prevention Financial Risk Monitoring System Under the Fusion of Partial Differential Equations Prediction and Analysis of ChiNext Stock Price Based on Linear and Non-linear Composite Model Calculus Logic Function in Tax Risk Avoidance in Different Stages of Enterprises The Psychological Memory Forgetting Model Based on the Analysis of Linear Differential Equations Optimization Simulation System of University Science Education Based on Finite Differential Equations The Law of Large Numbers in Children's Education Optimization System of Strength and Flexibility Training in Aerobics Course Based on Lagrangian Mathematical Equation Data structure simulation for the reform of the teaching process of university computer courses RETRACTION NOTE Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Research on non-linear visual matching model under inherent constraints of images Good congruences on weakly U-abundant semigroups Can policy coordination facilitate unimpeded trade? An empirical study on factors influencing smooth trade along the Belt and Road Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Research on design of customer portrait system for E-commerce Research on rule extraction method based on concept lattice of intuitionistic fuzzy language Fed-UserPro: A user profile construction method based on federated learning A multi-factor Regression Equation-based Test of Fitness Maximal Aerobic Capacity in Athletes Design and evaluation of intelligent teaching system on basic movements in PE Garment Image Retrieval based on Grab Cut Auto Segmentation and Dominate Color Method Financial Risk Prediction and Analysis Based on Nonlinear Differential Equations Constructivist Learning Method of Ordinary Differential Equations in College Mathematics Teaching Multiple Effects Analysis of Hangzhou Issuing Digital Consumer Coupons Based on Simultaneous Equations of CDM Model Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations A Hybrid Computational Intelligence Method of Newton's Method and Genetic Algorithm for Solving Compatible Nonlinear Equations Pressure Image Recognition of Lying Positions Based on Multi-feature value Regularized Extreme Learning Algorithm English Intelligent Question Answering System Based on elliptic fitting equation Precision Machining Technology of Jewelry on CNC Machine Tool Based on Mathematical Modeling Application Research of Mathematica Software in Calculus Teaching Computer Vision Communication Technology in Mathematical Modeling Skills of Music Creation Based on Homogeneous First-Order Linear Partial Differential Equations Mathematical Statistics Technology in the Educational Grading System of Preschool Students Music Recommendation Index Evaluation Based on Logistic Distribution Fitting Transition Probability Function Children's Educational Curriculum Evaluation Management System in Mathematical Equation Model Query Translation Optimization and Mathematical Modeling for English-Chinese Cross-Language Information Retrieval The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization research on prefabricated concrete frame buildings based on the dynamic equation of eccentric structure and horizontal-torsional coupling Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Green building considering image processing technology combined with CFD numerical simulation Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching Research on evaluation system of cross-border E-commerce platform based on the combined model A mathematical model of PCNN for image fusion with non-sampled contourlet transform Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Design of Tennis Mobile Teaching Assistant System Based on Ordinary Differential Equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Animation VR scene mosaic modeling based on generalized Laplacian equation Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations The influence of accounting computer information processing technology on enterprise internal control under panel data simultaneous equation Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Construction of comprehensive evaluation index system of water-saving irrigation project integrating penman Montei the quation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression Analysis of technical statistical indexes of college tennis players under the win-lose regression function equation Automatic extraction and discrimination of vocal main melody based on quadratic wave equation Analysis of wireless English multimedia communication based on spatial state model equation Optimization of Linear Algebra Core Function Framework on Multicore Processors Application of hybrid kernel function in economic benefit analysis and evaluation of enterprises Research on classification of e-commerce customers based on BP neural network The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Fractional Differential Equations in Electronic Information Models BIM Engineering Management Oriented to Curve Equation Model Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Design of language assisted learning model and online learning system under the background of artificial intelligence Study on the influence of adolescent smoking on physical training vital capacity in eastern coastal areas Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering Action of Aut( G) on the set of maximal subgroups of p-groups Research on loyalty prediction of e-commerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Evaluation and prediction of regional human capital based on optimised BP neural network Study on inefficient land use determination method for cities and towns from a city examination perspective A 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