Support vector machine (SVM) is a classical kernel method. It has grown substantially due to its good generalization capability even in cases of low quantity training setting over the last decades. It comes from Vapnik’s theory, and develops on the structural risk minimization principle [1]. Illuminated by SVM, many kernel based methods, such as kernel principle component analysis (KPCA) [2], kernel clustering (KC) [3] kernel Fisher discriminant analysis (KFDA) [4], kernel canonical correlation analysis (KCCA) [5], and kernel linear discriminant analysis (GDA) [6] are proposed to solve various different problems in machine learning community.

In most practical problems the data samples are linear non-separable, kernel methods map these points into a high dimensional space and even infinite-dimensional space (named feature space) to gain a better linear separability. The mapping is defined implicitly by the chosen kernel function which represents the inner product of each pair of data points in the feature space without computing their mapping functions directly. Thus the linear algorithm which has a poor performance in the default space produces a very impressive performance in a suitable feature space. This is the well-known “kernel trick”, and it is the key to the success of kernel methods. The importation of kernel functions makes the SVM more robust against the curse of dimensionality. Based on this technique of “kernel trick”, kernel methods have been numerous algorithms developed due to their attractive advantages in terms of theoretical guarantees, efficient training time, and delicate high-dimensional data handle ability.

Regardless of the prosper of kernel methods, a poorly selected kernel can lead to a worse prediction performance than that of their linear counterparts in the original input space. The reason lies in the nature of data is hard to catch. So the estimation of the underlying relationship of data is now converted into the choice of kernel function and its parameters. As we all know, selecting an appropriate kernel or an optional kernel parameter, namely the appropriate feature space is a crucial target. [1, 7, 8]

There are several commonly used kernel functions, such as the linear kernel (^{n},^{2}/2^{2}),^{2}),

In the literature, there are three approaches to choose kernel function or kernel parameters. First, the kernel selection can be reduced to real-valued parameter optimization. Especially, the Gaussian kernel has shown to be an optimal choice of kernel function. Many operations are taken to select its hyperparameters for different tasks. The most generally used method is grid search technique and cross-validation [1, 7, 8]. This approach works well in practice, however it is quite time-consuming because it need train SVMs with all desired combinations of parameters and screens them according to the training accuracy. Kernel alignment, Fisher discriminant criterion, Bayes predictive method and independence criterion are proposed subsequently [9–12]. And these operations are all based on kernel matrix and do not require the information of the trained SVMs, so they can save running time compared with cross-validation strategy. However, these methods based on target function optimization have their own disadvantages such as the minimum is not global, and their preferred accuracy depending on the type of data to be processed. Some dynamic evolutionary algorithms have been adapted to optimize the SVM parameters [13] [14]. But they are still time-consuming because of the training of SVMs from the population evolved generation by generation to converge. So far, there is no uniform standard or paradigm to solve the kernel parameter optimization problem of the Gaussian kernel function.

The second approach is combing several base kernels instead of using only one specific kernel to achieve better results. In the recent years, multiple kernel learning method (MKL) are proposed to overcome the difficulty of the Gaussian kernel optimization [15–17]. MKL adopts a set of predefined based kernels instead of selecting a single kernel function. The combination function of kernel functions has several forms, such as linear function, nonlinear function, or a data-dependent combination. Sincerely, choosing a suitable combination form, and further constructing the specific combination function is a challenging work. Many approaches are proposed to pursuit the combination coefficients by a joint optimization algorithm, or by using a two-stage alternate procedure. Testing results show that using multiple kernels can improve classification accuracy in practice compared with the single kernel approach, but the computational time of MKL is higher for determining the kernel function combination form.

The third way is creating new kernel functions to evade the kernel selection in the practice setting. A. Jacot et al. prove that the evolution of an ANN during training can also be described by a kernel, and proposed a new kernel-the Neural Tangent Kernel [18]. Built on the Cauchy distribution, Basak proposed the Cauchy kernel and observed the new proposed kernel has a long-tailed property which triggers its special performance sometime [19]. N. E. Ayat et al. propose a new kernel with moderate decreasing for pattern recognition [20]. Subsequently, Zhang and Wang presented a kernel function modifying the kernel proposed above, and the new proposed kernel can obtain better generalization performance compared with the old one [21]. Recently, some sets of generalized kernels based on orthogonal functions are proposed to simplify the kernel parameter optimization [22, 23]. M. Tian et al. analyzed the construction modes and the properties of several kinds of kernels based on orthogonal polynomials, and highlighted the similarities and differences among them [24]. But the specific construction of these kernels makes the calculation of kernel function time-consuming.

A new kernel function based on the arctangent function is presented in this paper. The obtained kernel function has concentrated on a high similarity in the neighborhood of the inspection sample and a low similarity toward infinity. Meanwhile, this kernel has a quick similarity descent rate near to the inspection sample and a slow similarity descent rate far away from the inspection sample. We provide experimental results showing their good performances on synthetic and UCI benchmark data. The paper focuses on the classification tasks and considers the SVM algorithm is popular in various applications. The applicability and robustness of the proposed kernel will be investigated here.

The remaining part of this paper is organized as follows. Section 2 shows a clear description of the SVM for classification. Section 3 provides a detailed introduction of the new presented kernel. We give experimental results and discussions on performance in section 4. Section 5 concludes this article.

The support vector machine is a classical machine learning approach. In recent years, SVM has become a hot topic in pattern recognition, and SVM has been successfully applied in many practice fields. In particular, a classical SVM classifier is capable of producing a linear decision boundary between two different classes, and maximizing the margin in the feature space between them at the same time. Introductions to SVM for binary classification task are summarized in the following.

Consider a set of training labeled sample represented by _{i} ∈ ^{n}, ^{n} denotes _{i}_{i}. The SVM classifier training process which builds the separating hyperplane between these two classes can be formulated by that:
_{1},…,ξ_{l})^{T} is the vector of slack variables and

The Lagrange dual operation converts the optimization problem above into the dual formulation, which can be rewritten as:
_{i}_{i∈{1,…,l}} are Lagrangian multipliers obtained by computing the optimization problems corresponding to each separation constraint. The samples with _{i} > 0 are named as support vectors and they are integral to define the separating boundary.

To solve real-world classification problems, the dot product < _{i},x_{j} > in the former algorithm can be replaced with a kernel function _{i}, _{j}). The kinds of kernel K satisfying some defined characterization of dot product are all samples _{i} and _{j}, which can be formulated as:
_{i} and _{j} in the augmented space

In practice, some theorems have been proved and provided to verify whether a function can be a kernel function. In these theorems, the original and most common condition for a symmetric function _{i}, _{j}) to be a kernel was raised by Mercer in 1909, namely a kernel is the function must be symmetric, continue and positive definite.

Based on unimodality and asymptotic of kernel functions, we proposed a new kind of kernel function. This kernel function is formulated as:

To prove the AtanK kernel is a Mercer kernel. We give the following theorem which is helpful to prove positive definite of it.

^{n},f^{2}).

^{n},

^{∞}. Let ^{2}, and Eq. (1) can be simplified as in the following form:

If 0 < _{1} < _{2}, we have
_{1} < ξ < _{2}. So

Fig.1 shows the output of AtanK kernel function for different

N. E. Ayat et al. pointed that a good kernel function should have different decrease speed in the neighborhood and the infinity of the inspection sample. That is to say, a quick and a moderate decrease speed should behold in the neighborhood and the infinity of the inspection sample, respectively [20]. And the authors pointed that the Gaussian kernel satisfies the requirement in the neighborhood of the inspection sample but not the requirement in the infinity of the inspection sample. Based on this, the authors proposed a kernel with moderate decreasing (denoted as KMOD), whose analytic expression is as:

Where ^{x}

Note that these two kinds of kernel functions both need a pair of parameters. Obviously, optimizing these two parameters is a very time-consuming target. For simplicity, Ref. [20] set the parameters chosen from some predetermined sets and Ref. [21] set L = ^{2} and

Fig. 2 shows the kernel outputs of the Gaussian kernel, the ERBF kernel, the UKF kernel, and the AtanK kernel for the parameters 0.2, 1, and 2, respectively. In order to give a detail comparison, the curves of the Gaussian kernel, the ERBF kernel, the UKF kernel, and the AtanK kernel are all given here. We discarded the comparison between the UKF kernel and the KMOD kernel due to the reason that Ref. [21] have shown the UKF can learn more separation information compared with the KMOD kernel. For each parameter, the kernel curves near to and far away from the neighborhood of the inspection sample zero in two-dimension plane are both shown.

From Fig.2, we can see that the AtanK kernel satisfies the quick decrease in the neighborhood of the inspection sample and the moderate decrease far away from the inspection sample. When

Seen from the kernel function discussed above, we find that when describing the change of the kernel function value with respect to the inspection sample in two-dimensional space, the curve of the kernel function presents a unimodal function curve whose maximum point is exactly the inspection sample, whether the kernel function constructed based on the inner product form of <

This paper discusses the different trends of similarity change near to and far away from the inspection sample, which in fact provides intuitive guidance for the similarity measurement requirements around the inspection sample. Various similarity measurement methods provide different inner product, or similarity, between mapping vectors in different feature spaces, which makes it possible to find the optimal feature spaces with better classification performance. However, because of the diversity and particularity of the real problems, we cannot determine which kernel function is the optimal kernel function for the specific problem. Therefore, we all choose the Gaussian kernel function as the universal kernel. As we all know, in the field of mathematical statistics, the normal distribution is a key type of data distribution. The existence of the maximum likelihood probability theorem makes the normal distribution be the highest peak in the classical statistical field. However, with the development of statistical technology, people have found that some specific data distribution cannot be characterized by the normal distribution in real life. Gradually, some new distributions such as the Chi-square distribution and the student distribution, are springing up. It can be imagined that the construction of kernel function is also faced with the same problem. The universal kernel function represented by the Gaussian kernel cannot describe all the similarity measurement. It is necessary to construct new kernel functions constantly to provide more possibilities for different practical problems and diverse similarity measurements.

The effectiveness of the proposed kernel function is assessed on 2 artificial datasets and 11 UCI benchmark datasets compared with the polynomial kernel, the Gaussian kernel, the UKF kernel, and the exponential radial basis kernel [26]. For simplicity, the polynomial kernel, the Gaussian kernel and the exponential radial basis kernel, are denoted as Poly, Gauss and ERBF, respectively.

To evaluate the performance and robustness of the AtanK kernel, we have performed a simple classification task in two dimensions. The results of the AtanK kernel in relation to the four kernel functions mentioned above (Poly, Gauss, UKF and ERBF) are performed on the checkerboard dataset.

The original training population is a set of 592 points, uniformly distributed in the unit square. We divide the square into 9 grids, and set these points of adjacent grids into two classes alternately. In the experiment, we select randomly 496 training samples and the last 96 samples as testing samples.

For simplicity, we set the coefficient

Table 1 lists the chosen parameter, the training accuracy and the minimal number of support vectors corresponding to several different kernels when they obtain their highest testing accuracy. Fig.3 shows the default checkerboard data set and the best classification resulting boundaries with different kernels.

Results on the Checkerboard dataset.

Kernel | Poly | Gauss | ERBF | UKF | AtanK |
---|---|---|---|---|---|

Parameter | σ=0.5 | σ=0.7 | σ=1.1 | σ=0.7 | |

Training accuracy (%) | 67.14 | 98.18 | 97.18 | 94.15 | |

Testing accuracy (%) | 71.87 | 92.71 | 89.58 | 92.71 | |

Number of SV | 381 | 129 | 199 | 183 |

Results on the Cosexp dataset.

Kernel | Poly | Gauss | ERBF | UKF | AtanK |
---|---|---|---|---|---|

Parameter | σ=1.1 | σ=0.7 | σ=0.5 | σ=1.3 | |

Trainging accuracy (%) | 75.20 | 88.20 | 98.00 | 78.20 | |

Testing accuracy (%) | 71.00 | 71.00 | 70.00 | 72.00 | |

Number of SV | 281 | 193 | 134 | 128 |

From the results of Table 1, it can be seen that the AtanK kernel have both the highest testing accuracy and the lowest support vector numbers in spite of its ordinary training accuracy in these 5 kernel functions. As we all know the support vector ratio can be seen as an index reflecting the generalization error rate, so we take the number of SV into consideration here. On this index, the AtanK kernel is pretty much the same as the Gaussian kernel. In all, the performance of Poly is the worst. The Gaussian kernel is the best in the training accuracy, while its generalization is not so amazing. Seen from the testing accuracy performance, the Gaussian kernel and the UKF kernel have the same results, and both fall behind the AtanK kernel. And the AtanK kernel has the best classification performance in these kernels discussed above.

Seen from Fig. 3, the polynomial kernel has the worst performance that it does not find the adjacent grids structure at all. Though the ERBF kernel has the highest training accuracy, but its classification boundary is not smooth enough. Part of the reason lies in the quick decrease in the neighborhood of

We give another artifical data set-the Cosexp dataset to evaluate the performance of the new proposed kernel [25]. Let the function

The results of the AtanK kernel compared with the four kernel functions are investigated on the Cosexp dataset in Table 2. Table 2 shows the chosen parameter, the training accuracy and the minimal number of support vectors obtained with these kernels when they each obtain their highest testing accuracy. From the results of Table 2, it can be seen that the AtanK kernel and the polynomial kernel receive the better test accuracy, in spite of their naive training accuracy. And the UKF, ERBF and AtanK all have the good performance in terms of support vector numbers.

Fig. 4 shows the best classification resulting boundaries with different kernels on the Cosexp dataset. Seen from Fig. 4, we can see that the classification boundaries obtained by the ERBF kernel and the UKF kernel are more identical to the default boundary. They both have good local structure preserving ability, and it may be responsible for the result. At the same time, the difference of boundries obtained by the Gaussian kernel and the AtanK kernel shows that these two kernel functions have an obvious difference in classification, although they both have symmetry and unimodality.

Combined the results of Table 2 and Fig. 4, we found the Poly kernel has the best classification accuracy despite it fails to find the structure of default separating boundary, while the UKF kernel has the worst classification accuracy despite it catch the more vivid structure of default separating boundary. It can be seen as a reflection of the difficulty of realizing the structural risk minimization in kernel selection. From this point of view, the Atank kernel has a naive training accuracy performance, a better testing classification result, and a rougher separating boundary structure, so it is a well-balanced choice.

We evaluate the classification performance improvement of our proposed kernel over the four commonly used kernels mentioned above on 11 data sets from the UCI benchmark repository [26]. The specifications of these data sets are listed in Table 3. The regularization parameter C in SVM is chosen from the set of {2^{–4},2^{–3},2^{–2},2^{–1},2^{0},2^{1}, 2^{2},2^{3},2^{4}}. And the description of the hyperparameter values of kernels are summarized in Table 4.

In the experiment, the experimental methodology is set as follows: For a given data set, we throw randomly two-thirds of samples into the training set and the remaining one-third into the testing set, the stratified sampling is exceeded in this program. For single dataset, five kernel functions are adopted separately. To ensure the objectivity of experimental results, we run 10 randomly independent performances to evaluate the classification performance by the best classification accuracy rate by average. The average classification accuracies with standard deviations over 10 trials using the optimal parameter obtained by cross validation are summarized in Table 5. In Table 5, the bold font denotes the best classification performance across the kernel functions compared. The t-test results are summarized in Win-tie-loss (W-T-L), and they are attached at the bottoms of Table 5.

According to the prediction accuracy in Table 5, we note that the AtanK kernel obtains the best accuracy on 6 out of 11 datasets. The ERBF kernel and the UKF kernel fall behind, give better performance on 2 datasets, and the Gaussian kernel provides the best accuracy on only one dataset. Of these five kernel functions, the polynomial kernel has the worst behavior in the experiment. The W-T-L summarization shows that the AtanK kernel has an obvious advantage compared with the Poly kernel, the Gauss kernel, the ERBF kernel and the UKF kernel on 10 datasets, 7 datasets, 9 datasets and 7 datasets out of 11 datasets in Table 5.

The specification of data sets for experiments.

Data set | Number of features | Number of samples | Number of classes |
---|---|---|---|

Heart | 13 | 270 | 2 |

Ringnorm | 20 | 1000 | 2 |

German | 24 | 1000 | 2 |

Wdbc | 30 | 569 | 2 |

Ionosphere | 34 | 351 | 2 |

Sonar | 60 | 208 | 2 |

Splice | 60 | 1000 | 2 |

Australian | 14 | 690 | 2 |

Twonorm | 20 | 1000 | 2 |

Yeast | 8 | 1136 | 2 |

IJK | 16 | 2241 | 2 |

The values corresponding to kernel parameters in experiments.

Kernel | Parameters | Values |
---|---|---|

Poly | 1,2,3,4,5,6,7,8 | |

Gauss | 2^{–4},2^{–3},2^{–2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

ERBF | 2^{–4},2^{–3},2 ^{2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

AtanK | 2^{–4},2^{–3},2^{–2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

UKF | 2^{–4},2^{–3},2^{–2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

1 |

The classification accuracy of different kernels on benchmark data sets (mean(%)± std).

Data set | Poly | Gauss | ERBF | AtanK | UKF |
---|---|---|---|---|---|

Heart | 76.00±0.0457 | 78.44±0.0393 | 78.59±0.0149 | 77.22±0.0417 | |

Ringnorm | 95.09±0.0102 | 96.29±0.0049 | 97.37±0.0457 | 97.84±0.0070 | |

German | 72.72±0.0184 | 73.38±0.0103 | 73.05±0.0252 | 73.26±0.0192 | |

Wdbc | 94.74±0.0096 | 97.00±0.0145 | 97.21±0.0050 | 96.84±0.0253 | |

Ionosphere | 86.15±0.0248 | 88.37±0.0092 | 92.48±0.0184 | 91.28± 0.0212 | |

Sonar | 91.43±0.0316 | 90.43±0.0415 | 92.43±0.0234 | 91.29±0.0549 | |

Splice | 80.36±0.0401 | 82.99±0.0229 | 81.95±0.0183 | 83.83±0.0268 | |

Australian | 82.74±0.0334 | 81.57±0.0148 | 83.48± 0.0319 | 82.87±0.0340 | |

Twonorm | 96.41± 0.0101 | 96.32±0.0047 | 96.26± 0.0046 | 96.38±0.0065 | |

Yeast | 66.78± 0.0144 | 65.80±0.0091 | 67.28± 0.0270 | 66.33±0.0210 | |

IJK | 96.80± 0.0045 | 98.11±0.0091 | 98.14± 0.0087 | 98.30±0.0054 | |

W-T-L | 10-0-1 | 7-3-1 | 9-0-2 | 7-2-2 | — |

Besides comparing the testing accuracy of these kernels for each data set, we take an overall comparison on 11 UCI data sets by combing the nonparametric Friedman’s test and the Tukey’s honestly significant difference criterion. The Friedman’s test is an extension of the sign test and is better known, it is widely useful for a much larger range of treatments [27]. The overall evaluation of the classification accuracy of these kernels is displayed in Fig. 5.

Seen from Fig. 5, the performance of the polynomial kernel is outperformed by other kernel functions. The performance of the proposed AtanK has the best rank of these five kernel functions, and it has a statistically significantly accuracy on 11 datasets. It means the AtanK kernel is a better and robust kernel compared with the polynomial kernel, the ERBF kernel, the Gaussian kernel, and the UKF kernel.

Kernel selection is fundamental to kernel based methods. Driven by the similarity measure defined by the arctangent function, we described a new universal kernel in this paper. Comprehensive experimental shows the new proposed AtanK kernel can properly learn separating boundary for many classification application. Schematic diagrams show the AtanK kernel is a new set of SVM kernels that allows a quick decreasing around the sample and a moderate decreasing toward infinity.

Experiments done on the checkerboard dataset and the Cosexp dataset show the ability of the AtanK kernel in separating the patterns. Thus we find the new kernel can preserve the all data on closeness information while still penalize the far neighborhood more reliably. Moreover, a comparison is done with previously constructed SVM models based on the UCI benchmark datasets, which was obtained by using the polynomial kernel, the Gaussian kernel, the ERBF kernel, and the UKF kernel. The result revealed that the AtanK kernel can improve classification accuracy.

Further numerical experiments are required in order to assess the power of our evolved kernel. And an automatic algorithm of the AtanK kernel parameters optimization will be valuable. Many mature kernel parameter optimization tricks can extend to the new kernel easily.

#### The classification accuracy of different kernels on benchmark data sets (mean(%)± std).

Data set | Poly | Gauss | ERBF | AtanK | UKF |
---|---|---|---|---|---|

Heart | 76.00±0.0457 | 78.44±0.0393 | 78.59±0.0149 | 77.22±0.0417 | |

Ringnorm | 95.09±0.0102 | 96.29±0.0049 | 97.37±0.0457 | 97.84±0.0070 | |

German | 72.72±0.0184 | 73.38±0.0103 | 73.05±0.0252 | 73.26±0.0192 | |

Wdbc | 94.74±0.0096 | 97.00±0.0145 | 97.21±0.0050 | 96.84±0.0253 | |

Ionosphere | 86.15±0.0248 | 88.37±0.0092 | 92.48±0.0184 | 91.28± 0.0212 | |

Sonar | 91.43±0.0316 | 90.43±0.0415 | 92.43±0.0234 | 91.29±0.0549 | |

Splice | 80.36±0.0401 | 82.99±0.0229 | 81.95±0.0183 | 83.83±0.0268 | |

Australian | 82.74±0.0334 | 81.57±0.0148 | 83.48± 0.0319 | 82.87±0.0340 | |

Twonorm | 96.41± 0.0101 | 96.32±0.0047 | 96.26± 0.0046 | 96.38±0.0065 | |

Yeast | 66.78± 0.0144 | 65.80±0.0091 | 67.28± 0.0270 | 66.33±0.0210 | |

IJK | 96.80± 0.0045 | 98.11±0.0091 | 98.14± 0.0087 | 98.30±0.0054 | |

W-T-L | 10-0-1 | 7-3-1 | 9-0-2 | 7-2-2 | — |

#### Results on the Checkerboard dataset.

Kernel | Poly | Gauss | ERBF | UKF | AtanK |
---|---|---|---|---|---|

Parameter | σ=0.5 | σ=0.7 | σ=1.1 | σ=0.7 | |

Training accuracy (%) | 67.14 | 98.18 | 97.18 | 94.15 | |

Testing accuracy (%) | 71.87 | 92.71 | 89.58 | 92.71 | |

Number of SV | 381 | 129 | 199 | 183 |

#### The values corresponding to kernel parameters in experiments.

Kernel | Parameters | Values |
---|---|---|

Poly | 1,2,3,4,5,6,7,8 | |

Gauss | 2^{–4},2^{–3},2^{–2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

ERBF | 2^{–4},2^{–3},2 ^{2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

AtanK | 2^{–4},2^{–3},2^{–2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

UKF | 2^{–4},2^{–3},2^{–2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

1 |

#### The specification of data sets for experiments.

Data set | Number of features | Number of samples | Number of classes |
---|---|---|---|

Heart | 13 | 270 | 2 |

Ringnorm | 20 | 1000 | 2 |

German | 24 | 1000 | 2 |

Wdbc | 30 | 569 | 2 |

Ionosphere | 34 | 351 | 2 |

Sonar | 60 | 208 | 2 |

Splice | 60 | 1000 | 2 |

Australian | 14 | 690 | 2 |

Twonorm | 20 | 1000 | 2 |

Yeast | 8 | 1136 | 2 |

IJK | 16 | 2241 | 2 |

#### Results on the Cosexp dataset.

Kernel | Poly | Gauss | ERBF | UKF | AtanK |
---|---|---|---|---|---|

Parameter | σ=1.1 | σ=0.7 | σ=0.5 | σ=1.3 | |

Trainging accuracy (%) | 75.20 | 88.20 | 98.00 | 78.20 | |

Testing accuracy (%) | 71.00 | 71.00 | 70.00 | 72.00 | |

Number of SV | 281 | 193 | 134 | 128 |

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Information Course Selection System Influencing factors of modern and contemporary decorative pattern design on users’ visual perception characteristics Big Data Analytics Technology and Applications in Cloud Computing Perspective Construction machinery hydraulic transmission system fault analysis and elimination methods Cost management of power grid transmission and substation project based on BIM technology Optimization strategy of ideological and political education in colleges and universities based on modern information technology Analysis of IPO pricing efficiency under the registration system Case application data research of traditional ink art elements in packaging design Research on the identification of common faults of agricultural machinery based on vibration characteristics A Review of the Research on Labor Supply Effect of Active Aging Research on an early warning model of effectiveness evaluation in ideological and political teaching based on big data Design and implementation of computer network security protection system based on artificial intelligence technology 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 Application of traditional cultural elements in landscape design Research on the application of traditional culture in landscape design Driving Model of Electronic Information System Based on Agent Modeling and Simulation The Evaluation Model of College Students’ Mental Health Under the Fractional Differential Equation The application of cloud accounting in enterprise financial decision making in the era of big data Research on the development path of integration of rural sports culture and sports tourism in the context of big data Numerical simulation of vortex vibration in main girder of cable-stayed bridge based on bidirectional fluid–structure coupling Deep learning-based foreign object detection method for aviation runways Cost Control System of Cross-Border E-Commerce Based on Fractional Differential Equation Research on location algorithm of new energy vehicle charging station based on multi-objective decision Ancient Building Surveying and Mapping Modeling Protection System Based on Lagrangian Mathematical Equation The Development of Enterprise Management Innovation in the Digital Economy Era Based on the Schumpeter Model Sports Science Movement model based on fractional differential equation Research on performance evaluation and optimization of college budget management under the background of big data Training fatigue and recovery of Wushu Sanda athletes based on comprehensive environmental testing Application of grey correlation degree algorithm in the evaluation system of college students’ innovation and entrepreneurship in applied undergraduate institutions Research and application of Augmented Reality Technology and digital media Technology in mobile visual media Optimisation of construction mode of residential houses based on the genetic algorithm under BIM technology The development of the integration of red gene and curriculum thinking based on BP neural network mathematical algorithm The modernization strategy of constructing teacher-student relationship in the new era of universities Construction of a social audit platform based on big data for “industry price, quality and credit” 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 Structure innovation of agricultural supply chain in china: from farmers’ perspective Construction and practice of training system for dance professionals based on OBE teaching model combined with Kinect motion capture technology Research on multi-dimensional optimisation design of user interface under Rhino/GH platform Quantitative Analysis of the Influence of Coal Mining on the Underflow Zone of Rivers by Using Double Integro-differential Equations The construction of environmental cost accounting system - based on ecological perspective Optimisation analysis of distribution network planning based on the IVIF-AHP method Exploration of “English Plus” teaching mode under the background of blockchain construction A long command subsequence algorithm for manufacturing industry recommendation systems with similarity connection technology Analysis of tea culture communication path based on the principal component analysis method Research on the strategy of using graphic creativity in visual communication design in the context of Internet+ Supply chain network model of cost accounting based on improved particle swarm optimisation algorithm Construction of a multimodal poetry translation corpus based on AdaBoost model Civil and commercial law improves the construction of social credit system in the context of “Internet +” Research on personalised recommendation algorithm for college students’ employment Combined application of ideological and political education and big data Internet technology in the context of education reform 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 Design of a decentralized Internet-based control system for industrial robot robotic arms Promotion of VR technology in taijiquan multimedia technology in the context of artificial intelligence Design of a mechanical automatic transmission system for automobiles based on torsional vibration damping Application research on piano teaching in colleges and universities based on remote wireless network communication Construction of a mathematical modeling teaching quality assessment system for universities based on Eviews model Construction of Financial Risk Evaluation Index System for Biomass Graphene Fibre Industrialisation Project Innovative model of teaching in law experiment center based on LETS software Correlation analysis between talent training quality and regional economic development based on multivariate statistical analysis model A Study on the Implications of Critical Discourse Analysis Theory for College English Teaching in the Context of Big Data Computational research on flat image design and style based on perceptual feature quantification The artistic aesthetic and cultural value orientation of 3D printing A multidimensional interpretation of Marxist theory based on a decentralized Internet perspective Student management model of college student associations based on ant colony Algorithm Analysis of the evolution of China’s energy legal system under the background of discrete model theory and dual carbon 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 relationship between the construction of school teachers’ team literacy and the educational environment Exploration of the impact of rural road maintenance and management on rural economy based on VAR model Research on the post-purchase behaviour tendency of the product quality to customers in fast-selling marketing Engineering archive management model based on big data analysis and deep learning model Motion about equilibrium points in the Jupiter-Europa system with oblateness Design of an embedded machine vision system for smart cameras Ultra-short-term power forecast of photovoltaic power station based on VMD–LSTM model optimised by SSA The development model of marine science and technology innovation under the background of regional integration in the Yangtze River Delta in the new era 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 Application of digital design technology in the design of intelligent agricultural machinery and equipment The impact of global warming on the small Scottish Fishing Company Research on heat transfer properties of graphene composites Financial risk assessment management of cloud accounting SOEs based on spillover index 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 An empirical study on the economic linkage characteristics of Yangtze River port cities based on discrete second-order derivatives Study on the impact of forest fire prevention policy on the health of forest resources Analysis of the development direction of the history of American war photography in the context of big data Tourism architecture design strategy in tourism area planning based on G-DINA model Practice and research on non-foreign heritage culture inheritance education in universities based on information fusion technology Research on the innovation of teaching model of college board tennis based on GDINA model 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 Computer vision recognition and tracking algorithm based on convolutional neural network Innovative reform strategy of artistic practice of university folk dance education based on information fusion technology 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 ^{pm}−x +δ )^{s}+ax ^{pm}+bx overF _{p2m}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 Research on the driving principle and guiding strategy of the public's collaborative supervision of the sharing economy in my country Influence of innovative human capital on economic development of China through the STI model New ideas of precision teaching development based on the background of big data Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development Application research of bel canto performance based on artificial intelligence technology Research on the path of integrating blockchain technology into the teaching of film and television courses in higher education institutions Combined application of employment education and big data internet technology based on the context of vocational education reform Fractal structure of magnetic island in tokamak plasma College students' innovation and entrepreneurship model based on probability theory statistics Exploring the path of network security and student privacy protection in smart campus based on Markov model Innovation and entrepreneurship model of higher vocational college students based on probability theory statistics Application of knowledge graph in smart grid fault diagnosis Research on the innovation of youth network media literacy education in the context of Internet + education A fuzzy mathematical model-based approach to customer-side energy management for power companies Research on the optimisation of logistics parcel intelligent sorting and conveying chain combined with variable clustering mathematical method Research on power quality assessment based on ubiquitous power IoT Rural Tourism Development Planning Based on Nonlinear Differential Equation Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Analysis of network information security issues under the background of big data Strategies for Marine remote sensing target detection based on deep learning model and internal wave simulation imaging technology Research on SOC estimation of residual power of lithium-ion batteries for electric vehicles based on extended Kalman filtering algorithm Variation mechanisms of metadiscourse verb patterns in English journal articles from intra- and interdisciplinary perspectives Harmonic current variation characteristics of shunt capacitors based on full-phase FFT algorithm Color and composition under big data technology: the art of visual communication in film art Urban Public Epidemic Prevention and Control Model Based on Nonlinear Differential Equations Research on computer vision technology based on BP-LSTM hybrid network Research progress of computer vision tasks based on deep learning and SAE network 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 Simulation research on music feature recognition based on mobile big data and smart sensors Digital design and experimental testing of a compressor's suction muffler transmission loss LTE wireless network coverage optimisation based on corrected propagation model SCP model based on the new energy vehicle collaborative detection configuration method research Study on tourism development income index calculation of finite element ordinary differential mathematical equation The cardiovascular promotion of college taekwondo based on fractional differential equation Analysis and research on the economic benefits of employee labor in enterprises in the context of the Internet Application of renewable energy decorative materials in modern architectural design under low-carbon concept The boundedness of functional differential equations of hospital financial software under information Deep learning model-based brand design 3D image construction The character image setting of 3D animation works based on Poisson equation Research on the interaction between university physical culture education and college students’ physical exercise in the context of big data The optimization path of core literacy of innovative talents in universities based on the improved Markov model Analysis of the innovation path of university education management informatization in the era of big data Impact of ASEAN-China free trade area on fishery value chain based on difference-in-difference method Research on indoor environment design of comprehensive commercial shopping center based on numerical simulation Empirical analysis of the economic absolute income hypothesis based on mathematical statistics Realization of Book Collaborative Filtering Personalized Recommendation System Based on Linear Regression Equation 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 MATLAB-based simulation of industrial robots in water environment monitoring An innovative teaching model of automotive manufacturing process based on ADDIE model The train of thought and design of university talents training based on ADDIE model from the perspective of ideological and political education Research on the innovation of model based on the cultivation of talents in higher vocational art and design Calculus Logic Function in Integrated Manufacturing Automation of Single Chip Microcomputer Football Offense Training Strategy Based on Fractional Differential Mathematical Modeling 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 Study on the factors influencing the evolution mechanism of cobalt trade network from the perspective of national competition 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 Reliability analysis of reinforced concrete beams based on the stochastic model of resistance calculation 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 The Relationship Between College Students’ Taekwondo Courses and College Health Based on Mathematical Statistics Equations Fractional Differential Equations in the Exploration of Geological and Mineral Construction AdaBoost Algorithm in Trustworthy Network for Anomaly Intrusion Detection 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 A multi-factor Regression Equation-based Test of Fitness Maximal Aerobic Capacity in Athletes Garment Image Retrieval based on Grab Cut Auto Segmentation and Dominate Color Method Financial Risk Prediction and Analysis Based on Nonlinear Differential Equations Constructivist Learning Method of Ordinary Differential Equations in College Mathematics Teaching Multiple Effects Analysis of Hangzhou Issuing Digital Consumer Coupons Based on Simultaneous Equations of CDM Model Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations A Hybrid Computational Intelligence Method of Newton's Method and Genetic Algorithm for Solving Compatible Nonlinear Equations Pressure Image Recognition of Lying Positions Based on Multi-feature value Regularized Extreme Learning Algorithm English Intelligent Question Answering System Based on elliptic fitting equation Precision Machining Technology of Jewelry on CNC Machine Tool Based on Mathematical Modeling Application Research of Mathematica Software in Calculus Teaching Computer Vision Communication Technology in Mathematical Modeling Skills of Music Creation Based on Homogeneous First-Order Linear Partial Differential Equations Mathematical Statistics Technology in the Educational Grading System of Preschool Students Music Recommendation Index Evaluation Based on Logistic Distribution Fitting Transition Probability Function Children's Educational Curriculum Evaluation Management System in Mathematical Equation Model Query Translation Optimization and Mathematical Modeling for English-Chinese Cross-Language Information Retrieval The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization research on prefabricated concrete frame buildings based on the dynamic equation of eccentric structure and horizontal-torsional coupling Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Green building considering image processing technology combined with CFD numerical simulation Construction of an enterprise financial risk management system based on F-score model Research on personalized Japanese language teaching model based on big data technology Research on education management system based on machine learning and multidimensional data modeling A multidimensional interpretation of Marxist globalization theory based on decentralized internet Analysis of the improvement path of English listening and speaking ability based on big data technology Multi-featured multi-scale combination of high-resolution remote sensing images for building extraction Study on the absorption technology of distribution network with high-proportion distributed power supply Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Electronic properties of diamond semiconductor materials: based on response surface model Application of BOPPPS teaching model in online and offline blended teaching of college Public Basic Mathematics Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching Deep learning techniques in modern women’s smart clothing design Grouting reinforcement strategy for tunnel sand layer based on BP neural network Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology A study on the aesthetic characteristics of ethnic music based on radial basis neural network model Research on university laboratory network security based on Cloud Computing The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Analysis of the application methods of film and television media and images in the era of big data cloud Títul Mobile visual media design based on digital media technology and augmented reality Air data computer simulation and design credibility assessment considering ADC calculation model Spatial planning layout design of national land in the context of rural revitalization based on MOLA model Analysis of news communication strategies in the era of full media based on big data mining On the improvement, innovation and inheritance of stage makeup styling in opera under the background of big data Influence of 3D characters image transfer on animation drawing based on detail feature preservation Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Evaluation Mechanism of Teaching Staff’s Building in Local Vocational Colleges Based on Big Data Technology Títul Sports consumption behavior and consumption demand of urban residents in the context of “Internet+” 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 Research on detection of parking space based on the combination of magnetoresistance and microwave radar sensor under IoT platform Research on sustainable development mechanism of regional ‘Artisan Workshop’ under big data environment Application of hybrid kernel function in economic benefit analysis and evaluation of enterprises The analysis method of the impact of missing audit information on the securities investment decisions based on Logistic regression models Research on the innovation theory of college students’ ideological and political education under the background of big data Research on the close-range photogrammetry accuracy of 3D digital interactive platform of Hunan intangible cultural heritage Research on classification of e-commerce customers based on BP neural network Research on the problems and reform strategies of government statistics based on big data technology Research and implementation of visual question and answer system based on deep learning Títul Research on Wireless Network Coverage for Transformation and Upgrading of Exhibition Management with Artificial Intelligence Technology The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Adaptive fusion of multi-exposure images based on perceptron model Study on the application of multi-angle imaging related technology in the construction process Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Exploring the Path of Integrating Excellent Traditional Culture into Civic Education in Universities in the Internet+ Era Economic statistical strategies for the transportation industry based on time-varying parametric state space models Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Design and research of environmentally friendly field sewage purification mechanical device Public Management Model Based on Mathematical Equation of Statistics of Probability Theory 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System An analysis of time commitment for college students to online English writing and peer assessment learning Optimization path of corporate ESG performance and compensation performance management under carbon neutral target Research on network security management monitoring platform based on Internet of things technology New anergy tide control strategy based on Eviews econometric model Study of frictional wear properties of materials for mechanical seals Optimization of vocal singing training method based on intelligent big data technology Fractional Differential Equations in Electronic Information Models Exploring the construction of an all-round cost management system for investment control in university engineering construction The construction of blockchain platform for engineering museums based on vectorized image processing technology BIM Engineering Management Oriented to Curve Equation Model Relationship Between Physical Activity Sedentary Behavior and Body Mass Index in Children and Adolescents Based on FIT Modelo Green supply chain innovation management strategy based on the combination of low carbon economy and e-commerce with big data technology 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 Grid data asset relationship and intelligent classification integrating knowledge graph and Internet of Things 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 Analysis of digital humanistic knowledge production based on natural language processing 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 The Construction of Judgment Model for Cultivating Intelligent Accounting Professionals in Universities Based on Logarithmic Periodic Power Law Model Action of Aut( G ) on the set of maximal subgroups ofp -groupsDigital Regeneration and Database Construction of Hunan Embroidery Needlework under the Perspective of Artificial Intelligence Research on the practice of innovation and entrepreneurship education in universities under the environment of big data Application of digital technology in environmental art design Research on loyalty prediction of e-commerce customer based on data mining Research on the evaluation system of tourism management classroom teaching based on multimedia big data analysis The application of ARCS motivation model in college oral English teaching Research on Innovation of Translation Teaching and Translation Strategies for College Students in Multimedia Background A multi-channel convolutional neural network based on attention mechanism fusion for facial expression recognition Exploring the development of cultural and creative tourism under the perspective of ecological theory of tourism destination management Modeling Analysis of User Consumption Data of Internet Culture Industry in the Context of Emerging Technologies Reference: An algorithm for recognizing the main melody of orchestral music based on artificial intelligence of music melody contour 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 The construction of a practical platform for party building and ideological and political education based on information fusion A study on the influence of the spread of Yangming Studies in Japan on the psychology of the Japanese people based on big data analysis Research on personalized recommendation algorithm for micromechanical sensors based on cloud model Research on the value of modern vernacular literature in contemporary rural construction based on ADDIE model Robust speech recognition based on deep learning for sports game review IoT-based interaction design of smart home products for elderly families Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Model Construction of Influencing Factors of Physical Education Core Quality Cultivation Based on Multiple Linear Regression A multidimensional interpretation analysis of Marxism globalization theory based on distributed Internet Application of Nonlinear Differential Equation in Electric Automation Control System Research on the operation and management mechanism of cross-border e-commerce shared practice training base Strategy for financing mode optimization in international trade supply chain based on deep learning model Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Strategies for improving Chinese language proficiency based on artificial intelligence technology Early warning and detection of geological disasters based on intelligent genetic algorithm The success rate model of college students’ new entrepreneurship based on nonlinear differential equations Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Research on the application of marketing strategy of national fitness exercise and dance events in the construction of sports culture based on big data technology Exploring the Application of Modern Fine Arts Artistic Modeling and Color in Modern Art Design Based on Information Fusion Technology Decisions of competing supply chain with altruistic retailer under risk aversion Strategies for constructing mathematical models of nonlinear systems based on multiple linear regression models Research on China’s tourism marketing model in the data perspective: based on ecological perspective Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations Research on human resource management performance evaluation based on BP algorithm 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 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 Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Badminton players’ trajectory under numerical calculation method Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Innovation of Economic Management Risk Control in Retailer Supply Chain Based on Differential Equation Model Statistical Model of College Ideological and Political Learning Based on Fractional Differential Equations Nonlinear Differential Equation in Anti-aging Test of Polymer Nanomaterials 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 University Ideological and Political Learning Model Based on Statistical Memory Curve Mathematical Equation 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 Conventional Algorithms in Sports Training Based on Fractional Differential Equations Dynamic Nonlinear System Based on Complex System Theory in the Development of Vocational Education The Algorithm Accuracy of Mathematical Model to Improve the Transmission Speed of E-commerce Platform 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 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 urban energy Internet planning based on fuzzy comprehensive method 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 intelligent path of college students’ network ideological and political education based on big data mining technology Research on the application of PLC technology in electrical automation engineering Application of matrix multiplication in signal sensor image perception 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 Research on the experimental principle of deep integration of LETS software and criminal procedure under the background of artificial intelligence VR-based computer maintenance practical training platform development design and application research Design and implementation of an integrated management system for discipline construction and postgraduate education in universities 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 Research on educational resource recommendation system based on MRLG Rec Visual analysis of the trend of mental health changes among Chinese college students in the context of the normalisation of the new crown epidemic A study on the phenomenon of anaphoric correction in college students’ English conversation 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 Research on the innovation of collaborative education model of scientific research and teaching in colleges and universities based on the theory of process reengineering AtanK-A New SVM Kernel for Classification The innovative path of diversified vocal music education reform in colleges and universities under the advanced educational concepts of China and the West Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Rhino application in ceramic packaging design teaching Study of a linear-physical-programming-based approach for web service selection under uncertain service quality Analysis and countermeasures of cultivating independent learning ability in colleges teaching English based on OBE theory Virtual Museum Scene Design Based on VRAR Realistic Interaction under PMC Artificial Intelligence Model A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Projection of Early Warning Identification of Hazardous Sources of Gas Explosion Accidents in Coal Mines Based on NTM Deep Learning Network Research on innovative cultivation mode of English and American literature translation talents based on internet environment Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China Analysis of quantitative management of online intelligent monitoring of tailing ponds based on the perspective of safety prevention and control Analysis of rural tourism culture advertising content based on LSTM-CNN model Optimization of college physical education curriculum based on G-DINA Optimization of local ethnic music teaching transmission path based on logistic regression model Exploring the Construction of Curriculum System of Theater and Film Studies Based on ADDIE Model Innovative research on propagation modes of short video works based on digital technology Analysis of the effect of compound training for explosive power on improving the psychological resilience of football athletes Exploration on the path of ideological and political education and cultural education in colleges and universities from the perspective of red culture 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 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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 Design and evaluation of intelligent teaching system on basic movements in PE Research on identifying psychological health problems of college students by logistic regression model based on data mining 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 Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Retraction: Wei, Wei and Lu, Liu. “The influence of X fuzzy mathematics method in basketball tactics scoring” Applied Mathematics and Nonlinear Sciences, vol.7, no.1, 2022, pp.677-684 Retraction Retraction: Retraction of: Mundewadi, R. A. and S, Kumbinarasaiah. “Numerical Solution of Abel's Integral Equations using Hermite Wavelet” Applied Mathematics and Nonlinear Sciences, vol.4, no.2, 2019, pp.395-406