English for college students is one of the compulsory courses in university teaching, and its importance is reflected in all aspects, especially in conversation, and due to the limited English level of college students, various problems often occur in English conversation, which requires learning conversation repair; conversation self-repair is a self-corrective behaviour taken by the speaker after monitoring his or her own discourse errors, which is the main form of conversation repair [1–3]. Since 1977, when American sociologists Shegloff and Sacks [4] conducted a systematic study of conversational repair and first proposed the internal structure and judgment criteria of conversational self-repair, conversational self-repair has received the attention of many researchers at home and abroad. Over the past two decades, self-repair has achieved many results in the field of second language acquisition and other research areas. It is believed that self-repair can lead to comprehensible output, which is an important factor for successful language learning and a measure of success in second language learning.

Referring back is a common linguistic phenomenon that is frequently used in everyday conversation and writing. In order to meet the needs of communicative activities, language has developed a complete and complex system of reflexive language. We need to study not only the use of reflexive language in communication but also the structural features of the reflexive system itself. Specifically, the study of reflexives involves the following questions: Why do reflexives exist in their current structure? What are the roots of the differences between different reflexive structures and each other? How are reflexives used in discourse and what functions do they serve? How is the meaning of the reflexive interpreted in the process of communication? And so on. After the first mention of a person, thing or event in a discourse, we repeat the person, thing or event a second or third time in subsequent lines. This ‘‘‘repeated mention’’’ is a reflexive reference. The person, thing or event first mentioned exists in memory as a conceptual entity and becomes the object of the subsequent reflexive reference, which we call a prior concept. It is called a prior concept because it is already present in the context as opposed to the ‘‘‘repeated mention’’’ that follows. The word or phrase used in the first mention of the concept is called the ‘‘‘antecedent’’’, and it is the word that needs to be referred to in order to correctly interpret the reflexive that follows the repeated mention of the concept.

A reflexive is a word or phrase that is used to refer to a prior concept repeatedly in a discourse. The object referred to by the reflexive should be the same as the object referred to by the antecedent, i.e., it should be in a ‘‘‘homonymous’’’ relationship with the antecedent [5–10].

However, the questions of why there are structural and distributional differences between different reflexives and how they are interpreted in the communicative process are central to the study of reflexives. Different linguistic theories have attempted to answer these questions. As a typical linguistic phenomenon, it has become the best experimental object for linguistic schools to test the validity of their theoretical explanations. The explanation of the phenomenon of reflexivity depends fundamentally on our understanding of the nature of linguistic structure. Different linguistic theories have provided different explanations for the phenomenon from their own perspectives. Broadly speaking, there are three basic positions on the understanding of the nature of the reflexive phenomenon: formal, pragmatic and cognitive, and the corresponding representative theories are generative grammar, neo-Griceanism and accessibility theory, respectively [11–16].

Generative grammar has long dominated the field of language studies. The main researchers who have used the theory of generative grammar to study reflexivity are Reinhart (1983) and Chomsky (1981a, 1982). Chomsky’s theory of constraint attempts to find a set of explanatory rules for the internal referents of sentences. His study focuses on three types of referential phenomena: referring expression, anaphor and pronoun. Referring expressions include proper names, noun phrases, etc. (equivalent to what we call name words); corresponding expressions include reflexive pronouns, mutual pronouns, etc.; pronouns refer mainly to what is traditionally called personal pronouns in grammar. Like all other theories, Chomsky’s theory is based on a set of basic categories, and Chomsky’s syntactic theoretical system consists mainly of a series of conversion rules for the generation of small sentences. These rules are mainly based on a system of categories of sentence constituents and their relations. His study of reflexivity is also based on this purely syntactic system of categories.

Among the many studies, Pillai [17] proposed a model of the internal structure of self-repair that has greatly influenced the subsequent related studies. The model he developed is as follows (Figure 1):

The above model shows that the talker self-monitors his cognitive speech behaviour, and when he detects a speech error after crossing ‘‘‘at the’’’, he stops the flow and enters the second phase of self-repair, i.e., the editing phase. In order to fill the gaps in this stage, the speaker used the filler word ‘‘‘no’’’. In this stage, he tries to re-plan the speech and speech repair strategies in order to repair the speech errors (hereinafter referred to as speech errors). Finally, in the third stage of repair, a replacement strategy was used, replacing ‘‘‘left’’’ with ‘‘‘right’’’. In the third stage of repair, the speaker did not repair the verbal error directly but redirected it to the beginning of the verb phrase, and the process went from self-repair to self-replanning to self-repair, where ‘‘‘turn’’’ was the ‘‘‘initial word of retraced constituent’’’ (hereinafter referred to as the initial word of retraced).

It was found that in the error correction stage of self-repair, the phenomenon of retracing is more common. The reflexive indicates that the speaker tends to go back to the beginning of the repaired constituent when performing the repair, rather than directly after the wrong word. This terminology was introduced by Levelt in 1983 in his model of the internal structure of self-repair described above and has been used ever since. A corpus study by Nootboom [18] found that lexical errors were more likely to cause reflexivity than phonological errors, but there is not much literature on the prominent phenomenon of verbal reflexive repair. Later, in a colour naming experiment with native speakers, the self-repair of speaker errors was divided into three types, with instant repair (starting the repair directly with the word at error) accounting for 42% and reflexive repair (anticipatory retracing: tracing to and repeating some word prior to the trouble element) accounting for 35%, and starting with fresh material that was not part of, Levelt (henceforth L1) argues that retracing indicates that the speaker is trying to help the listener with the problem of speech continuation because the listener is faced with the problem of distinguishing and identifying the word that the speaker is trying to express from the interrupted word, and thus, retracing helps the listener to relate the corrected word to the original word. Nagano [19] drew on Levelt’s experimental model to compare the difference in the use of verbal self-repair between L1 and second language learners (hereafter L2, Japanese learners of English in this study) and found that L2 used more reflexive repair than L1 under proper repair and wrong repair, with a frequency of 33%:25% and 41%:63%, respectively. He explained this phenomenon by the fact that L2 used finger-back repair to enable listeners to keep up with the communication; the speakers bought time to re-plan the language and used it as a delaying strategy.

A domestic researcher, Yaojianpeng [20], classified the strategies used by speakers for finger retrieval as retracing without overt correction (hereafter referred to as self-repetition), retracing with correction (which can be subdivided into replacement and insertion retracing) and retracing with reformulation (retracing with reformulation). The study found that function words were more likely to cause repetition than real words in self-repetition and that 70% of the retracing was caused by pronouns, i.e., retracing was most likely to be caused by function words. The study also pointed out that one of the motivations of the speaker for reflexive repair was to make the repaired speech accurate and syntactically and semantically acceptable. In addition, by referring back, the speaker expects the hearer to follow and understand his or her communicative intention. However, this study investigated the L1’s reflexive repair. The present study aims to investigate this common linguistic phenomenon in order to fill the gap. This study aims to answer the following two questions: (1) What are the characteristics of Chinese college students’ use of English conversational reflexive repair strategies and (2) What is the distribution of the refer-back words among Chinese college students in English conversational reflexive repair?

The corpus of this study was obtained from the College Learners Spoken English Corpus (hereinafter referred to as COLSEC) [21]. The source of the COLSEC corpus was the oral test of the National College English Test (CET-SET). The corpus is based on the transcribed materials of the CET-SET. The corpus was selected using a random proportional sampling method, proportional to the candidates’ regional origin, major, test scores and topics of conversation [22]. The types of discourse in the candidate corpus include three types: teacher-student interview, student discussion and teacher-student discussion, and all three parts of the examination sessions revolve around certain topics. A simple random sample of 31 texts was selected from 2000 to 2002, involving 29 male and 65 female examinees. There were 14 topics in the test, including mass media, the use of computer, examination, staying healthy, spending, big or small families and caring for others. The topics are divided into three main categories: campus life, social and natural concerns and personal and interpersonal issues, which have a strong focus and can easily reflect the characteristics of vocabulary use and the use of communication strategies.

In classifying the reflexive repair strategies of the COLSEC corpus, we drew on previous classification models and adjusted the classification model with the present corpus to classify the reflexive repair strategies into three major categories: the first category is reflexive without apparent error correction (i.e., self-repetition, hereafter referred to as repetition), which can be subdivided as follows: two-word repetition, multi-word repetition and continuous repetition (excluding word fragment repetition and one-word repetition which are common in self-repair because they do not have the characteristic of referring back). Two-word repetition is the repetition of two lexical items, which is the most common in this corpus. Multi-word repetition refers to repetitions involving more than two lexical items. There is a particular type of repetition in this corpus, where the speaker consistently makes a two-word or multi-word repetition. Researchers have interpreted repetition differently, and Hieke [23] states that repetition performs two main functions, namely, delay and repair. In addition, it acts as a pause, indicating that the speaker is assembling speech, during which the speaker’s memory is temporarily blocked, lexical extraction becomes difficult and time is needed to extract the exact word to overcome the current lexical selection problem [24]. The second category is error correction back to the finger, which includes substitution, insertion and deletion. Substitution is mainly accomplished through the speaker’s self-monitoring of lexical items, which suggests that the speaker’s motivation for this repair is to use more appropriate lexical and grammatical expressions to convey his or her communicative intentions. The insertion not only indicates a high level of self-monitoring ability and awareness of natural speech communication but also clearly indicates that the discourse is made more complete and accurate by inserting information to express the discourse’s communicative intentions. Deletion is a new type found in this corpus. In this type of reflexive, the discourse participant undergoes a reflexive-repair process when performing substitution, insertion or deletion repair. The third category is reorganisation: in this type of repair, the speaker refers back to the beginning of the constituent, but after referring back to the word, he or she re-plans the discourse in terms of syntactic structure or semantic composition. We manually annotated the corpus according to the above classification model and then counted the frequency of each type of repair using the search function of corpus retrieval software AntConc 3.2.2.

However, the classification of reflexives can be multifaceted: we can classify them by their formal structural features, by their syntactic function or by the conceptual relationship between the reflexive and its antecedent. In a broad sense, all repeated references to people, things, features, behaviours, etc. in a discourse can be regarded as reflexive. In sentences, these different referred objects often belong to different semantic roles or event elements of the sentence proposition and correspond to different sentence constituents. For example, people and things often correspond to nouns, features often correspond to adjectives or adverbs and actions often correspond to verbs. Therefore, some scholars classify reflexives according to the syntactic features of the antecedent.

Classification methods are based on context-sensitive contexts and can be traced back to coronal and prepositional error correction studies in the early days. In the classification approach, for a specific syntactic error type, the classification model eventually outputs a label representing the correct form of the target word (e.g., on the singular-plural type of a noun, 0 for the singular form and 1 for the plural form). Classification models are divided into two main categories: statistical classification models and deep classification models [25].

Statistical classification models are more common in GEC research before the recovery of deep learning and involve models such as average perceptual machines, plain Bayesian classifiers and N-gram-based language models [26, 27].

The perceptron is a linear classification model that works by classifying an input feature vector [28] into two classes (+1 or –1) by means of a linear discriminant function, specifically the linear discriminant function has the following mathematical Eq. (1):

Linear Eq. (3) represents a hyperplane on the feature space

There is also the plain Bayesian classifier, a classification model based on Bayes’ theorem [29], which takes the category with the highest probability as the category of the item to be classified by finding the probability of occurrence of each category given the conditions of occurrence of the item to be classified. Bayes’ theorem is able to calculate the conditional probability P(B∣A) of event B occurring on the basis of knowing the probability P(A) of event A occurring, the probability P(B) of event B occurring and the conditional probability P(A∣B) of event A occurring on the basis of B occurring, as in Eq. (5):

Suppose Eq. (6) is a term to be classified, where _{i}_{1} ∣x), P(y_{2} ∣x), …, P(y_{n}mid_{k} is as follows:

N-gram is a natural language processing model based on a statistical approach [30], and its core principle is to process text using a sliding window of length N bytes to produce a sequence of byte fragments of length N. Each byte fragment is called a gram. The algorithm counts the self-count of all the grams that appear in the text, and it then filters out the grams with low occurrences according to a certain threshold, thus obtaining a list of key grams representing the feature space of the text vector, where each gram in the list represents one feature vector dimension of the space.

The N-gram assumes that the occurrence probability of the Nth word is related to the previous N-1 words only, and the occurrence probability of the whole sentence is the product of the occurrence probabilities of individual words. At present, the most commonly used N-grams are binary gram and ternary gram.

N-grams can be used to evaluate whether an utterance is reasonable or not. Suppose there is a sequence of m words and we want to get the probability

Using the Markov chain assumption that the current word is only related to a limited number of words before it, the computational effort can be significantly reduced. The specific formula is as follows (10):

The common binary and ternary models are defined as Eqs (11) and (12), respectively:

Therefore, given the training corpus, the above conditional probabilities can be computed using Bayes’ theorem. When selecting the more correct sentence among the two corrected sentences, the N-gram can be used to calculate the probabilities of the two sentences separately, and the one with the higher probability is selected as the final correct sentence.

This paper presents important research results in the field of English grammar error correction in recent years and research works that are closely related to the algorithms in this paper. These research studies mainly include word embedding and Seq2Seq models.

Word embedding method is to map each word or phrase from a high-dimensional space with dimensionality of all the number of words to a low-dimensional real domain space, where each word or phrase corresponds to a vector in this low-dimensional space. Word2Vec and GloVe are two excellent algorithms for generating word embedding vectors.

Word2Vec learns the vector representation of words from a corpus in an unsupervised way, including two models, CBOW and Skip-gram, assuming a window size of c. A word

GloVe is an unsupervised machine learning algorithm that trains word vectors based on global word frequency statistics. Similar to the Word2Vec method, GloVe word vectors can also obtain some semantic properties between words, such as similarity and analogy. GloVe is implemented in three steps as follows:

Construct a co-occurrence matrix X based on the corpus, where each element _{ij}_{i}_{j}

Construct the approximate relationship between the word vector and the co-occurrence matrix, and the formula is Eq. (14):_{i}_{i}

Construct a loss function based on the mean variance as in Eq. (15):

Here, the _{max} value of 100 is taken.

Compared with Word2Vec, GloVe is easier to parallelise during the training process and therefore faster to train on large data sets. In practice, GloVec word vectors perform better on test sets for some tasks, and word2Vec word vectors perform better on test sets for other tasks. Overall, there are no clear advantages or disadvantages of these two methods, and both are the preferred word embedding methods today.

The Seq2Seq model was first applied to neural machine translation research. The Seq2Seq model first obtains the vector representation of each word in the input sentence by word embedding and then inputs the vector sequence

The RNN unit in the Seq2Seq model is designed to evaluate the conditional probability, as in Eq. (17):_{1} is represented by Eq. (18) and the output sequence _{2} is represented by Eq. (19); the lengths _{1} and _{2} may not be equal, and

The Seq2Seq model in English grammar error correction is basically the same as that in machine translation, except that the parallel corpus is no longer one country’s language corresponding to another country’s language, but sentences with grammatical errors corresponding to correct sentences. Various improvement and optimisation methods for the Seq2Seq model are applicable to both scenarios.

We counted the refer-back words (same method as above) and compared them with the distribution of refer-back words in the Santa Barbara American Spoken English Corpus Linguistic Data Consortium (LDC) 2003S06 from Yaojianpeng [20] (Table 1).

Distribution of anaphora in corpus COLSEC

Function word | Pronoun | 120 | 45.2 |

Preposition | 20 | 9.1 | |

Conjunction | 25 | 11.2 | |

In total | 165 | 65.5 | |

Notional words | Noun | 4 | 2.1 |

Verb | 40 | 19.2 | |

Adjective | 9 | 4.3 | |

Adverb | 5 | 2.2 | |

In total | 58 | 27.8 |

From the previous discussion, it can be seen that Chinese college students used the most repetition among the finger-back repair strategies, among which two-word repetition, multi-word repetition and continuous repetition were dominant, which is basically consistent with the related studies at home and abroad. The majority of error-correcting reflexes were substitution, insertion, and deletion, and students showed a more flexible level of substitution, while only a small proportion of reorganisation reflexes were used. We suggest that these characteristics of students’ use of finger-repair strategies are influenced by their self-monitoring ability, information processing ability and attention allocation in verbal communication.

Self-monitoring is essential for self-repair of verbal errors, and in natural speech monitoring theory, perceptual circuit theory is involved in this process. In this context, the speech production model consists of four modules (conceptualiser, formulator, articulator and self-monitoring), two loops (internal loop and external loop perceptual loops) and a speech comprehension system (see Figure 2), in which the speaker self-monitors his or her speech through two perceptual loops. The inner loop monitors the information between the syntactic formers and the concept formers through the speech understanding system, while the outer loop monitors the information between the articulators and the concept formers through the listening and speech understanding system. In this way, not only implicit speech errors are monitored but also explicit speech errors are monitored so that the communication between the speaker and the listener can be carried out smoothly and accurately.

Self-monitoring is closely related to controlled processing and attention, and since human attention is limited, if attention is not sufficient to perform the task, language use will be affected, as well as the efficiency of the monitoring process and the number and type of errors noticed by the speaker. Also, according to our research, the speaker’s attention allocated is related to the context and the task, and the speaker may only pay attention to certain errors or disfluencies and ignore others. In cognitive psychology, attention is seen as an internal mechanism that enables control of stimulus selection and regulates behaviour, i.e., discarding some information in order to process important information efficiently. Then there are two modes of information processing in attention theory: controlled processing and automatic processing, which are able to control the correction of college students’ English fingerspelling. However, controlled processing is a kind of processing that requires the application of attention and has a limited capacity, which is also called attentional processing; automatic processing is a kind of processing that is not controlled by human beings, which does not require the application of attention and does not have a certain capacity to control, and once formed, it is difficult to change. When applied to second language learning, it can be understood that learners have a limited capacity to process information. Therefore, to master a new language, one can only learn its sub-skills gradually and then connect each sub-skill in series and finally reach the level of proficiency in the whole skill, which is the process from controlled processing to automatic processing, and finally to the gradual internalisation of the learner’s language knowledge.

Our research shows that the most important reason why L2 monitoring is different from that of the native language is that L2 lexical, syntactic and phonological processing requires attentional capacity, which is limited. In the process of bilingual production, speakers have limited knowledge of the target language and encounter difficulties in extracting appropriate lexical items and syntactic structures, and the encoding of phonological plan components is far from automatic; so, they cannot express their thoughts coherently, which inevitably leads to many speech errors. Therefore, the limited attentional resources of the speakers can only focus on monitoring and self-repairing those linguistic components that may affect the accuracy of information, while preserving those errors that do not affect the accuracy of information. Take the substitutions ‘‘‘And you know, many big companies er make big deals through the Internet. There is a very fashion, is a very fashionable word(s) called commercial business.’’’ for example that occur more frequently in this corpus. This example is part of the candidate’s answer to the examiner’s question about the negative impact of computers. The candidate clearly had difficulty in extracting the word ‘‘‘fashionable’’’ and the concept ‘‘‘commercial business’’’, and his limited attention was focused on monitoring the language components that might have affected the accuracy of the information. The limited attention was focused on the monitoring of linguistic components that might affect the accuracy of the information and then stalled for time by repeating ‘‘‘is a very’’’ to restore the memory to its cognitive resources, finally replacing the wrong word ‘‘‘fashion’’’ with ‘‘‘fashionable’’’, while those errors that did not affect the accuracy of the information, such as ‘‘‘a very fashionable words’’’, were retained. The above example shows that when fixing a complete error of speech, the speaker tends to keep the syntax of the original speech in the repair, repeating a part of the original speech and fixing the part that needs to be replaced. However, when the error is merely incomplete, the speaker often reconstructs it, either by inserting new materials into the original discourse, using an insertion strategy, or by choosing to suspend the flow of discourse online and re-plan the discourse, using a restructuring strategy. These two repair strategies require a high level of information processing and online language processing skills.

English education for college students is of great developmental importance; this study analysed the patterns and characteristics of the finger-back repair and the characteristics of the use of finger-back words among Chinese college students and came to the following conclusions: (1) The frequency of finger-back repair is high, reflecting that college students have a strong sense of communicative interaction and prompt the listener to solve the problem of topic connection through finger-back repair; (2) Students’ excessive use of certain double-word or multi-word repetition gives a ‘‘‘self-centered’’’ (I perspective) perspective and often gives listeners a monotonous and inauthentic feeling, which may be due to the influence of their native language; and (3) The proportion of self-repetition is the highest among the college students, reaching about 75%, and the proportion of reorganisation and insertion strategies is very small, about 23%, which reflects the characteristics of college students’ ability in linguistic information processing and online processing. The present study aims to make an indicative role for the subsequent in-depth investigation of the function of verbal finger-referential repair.

#### Distribution of anaphora in corpus COLSEC

Function word | Pronoun | 120 | 45.2 |

Preposition | 20 | 9.1 | |

Conjunction | 25 | 11.2 | |

In total | 165 | 65.5 | |

Notional words | Noun | 4 | 2.1 |

Verb | 40 | 19.2 | |

Adjective | 9 | 4.3 | |

Adverb | 5 | 2.2 | |

In total | 58 | 27.8 |

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 ^{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 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 ofp -groupsResearch on loyalty prediction of e-commerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Evaluation and prediction of regional human capital based on optimised BP neural network Study on inefficient land use determination method for cities and towns from a city examination perspective A study of local smoothness-informed convolutional neural network models for image inpainting Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of vertical video creation under the background of mobile communication Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithm-based congestion control optimisation for mobile data network