In this paper, two iron salts, ferrous chloride (FeCl_{2}) and ferric chloride (FeCl_{3}), are directly added into an aeration tank for phosphorus removal, and their effects on the biochemical system are studied; the water quality parameters such as pH and alkalinity are also investigated. The extent of influence of the added iron salts on the pH and alkalinity of aerated solutions is demonstrated to be FeCl_{3} > FeCl_{2}. When the dosage of iron ions is 20 mg/L, the decrease in pH and alkalinity caused by FeCl_{3} is 0.5 and 65 mg/L, which is higher than FeCl_{2} by 2% and 26%. The initial phosphorus removal effect of FeCl_{2} is worse than that of FeCl_{3}, but after continued aeration and oxidation, the phosphorus removal effect of FeCl_{2} can be improved; however, the final phosphorus removal effect is basically the same as that of FeCl_{3} added directly. The results show that FeCl_{2} is preferred when iron salt is added directly into the aeration tank to remove phosphorus. The proposed scheme can reduce the effect of iron salts on the alkalinity of the biochemical system on the premise of ensuring the phosphorus removal effect of the system, and is conducive to ensuring the stable operation of the biochemical system.

#### Keywords

- iron salt
- aeration tank
- pH
- alkalinity
- total phosphorus

As the urbanisation process is expedited and people's living standards have improved in China, the quantity of wastewater discharge is also rising year by year [1], and the black odour water body and eutrophication are increasingly becoming a serious concern [2,3,4]. In order to reduce the total wastewater and pollutant discharge, the level-I A discharge standard of GB 18918–2002 has been fixed: The Discharge Standard of Pollutants for Municipal Wastewater Treatment Plant is being implemented comprehensively and may become even more stringent than usual [5,6,7]. This has caused most wastewater treatment plants to upgrade their standard of water quality indexes, such as the total phosphorus and nitrogen, mainly by process adjustment or extension to enhance the nitrogen and phosphorus removal effect [8,9]. In terms of phosphorus removal, there are mainly two approaches: (1) directly adding coagulated phosphorus removing agent into aeration tanks, and (2) extending the coagulative precipitation tank [10,11]. Extending an aeration tank requires large investment, takes a long period of construction, and requires increasing the floor area. On the other hand, adding a phosphorus removing agent directly into aeration tanks requires only minimal investment, and does not increase the land area along with fast reconstruction, and so is widely applied, although its impact on the biochemical system cannot be ignored.

Coagulated phosphorus removing agents mainly include aluminium and ferric salts. Since aluminium ions may have certain effects on microorganisms [12], and the phosphorus removing effect of ferric salt coagulant is better than aluminium salt [13], ferric salt is more widely used in actual phosphorus removing projects. The common ferric salt coagulants are FeCl_{3}, Fe_{2}(SO_{4})_{3}, FeCl_{2}, FeSO_{4} and so on [14], among which FeCl_{3} is frequently used in phosphorus removing projects. In projects that involve removal of phosphorus from wastewater, the FeCl_{3} used is primarily prepared on the basis of FeCl_{2}. By adding oxidising agent, catalyst and hydrochloric acid, the FeCl_{2} is oxidised to FeCl_{3}. The added hydrochloric acid accounts for 20% [15,16], so the FeCl_{3} added will influence the pH and alkalinity of the biochemical system to a certain extent, and will further produce adverse effects on the performance of the biochemical system, especially the sludge nitration system. If FeCl_{2} is directly added to aeration tanks, it will be transformed into FeCl_{3} under the oxidisation effect of aeration oxygenation. This not only reduces the hydrochloric acid added and the agent preparation cost but also prevents the effects of FeCl_{3} on the biochemical system. Recently, there are no reports related to the research on the effects of FeCl_{2} and FeCl_{3} added to the biochemical system.

This paper mainly investigates the changes in pH, alkalinity and phosphorus removing effect after adding FeCl_{2} and FeCl_{3} into a solution, and studies the effects of FeCl_{2} and FeCl_{3} on the biochemical system of aeration tanks, so as to provide a reference for wastewater plants, which directly add coagulated phosphorus removing agents; this will enable us to select the appropriate phosphorus removing agent in upgrading standards and reconstruction projects.

Experimental drugs used were sodium bicarbonate (AR), sodium carbonate (AR), sodium phosphate (AR), ferric chloride (AR), ferrous chloride (AR), and hydrochloric acid (AR).

Coagulants used were solid ferrous chloride and ferric trichloride; these were placed in a beaker to prepare the solution with iron ionic concentration of 10%.

The experimental device is shown in Table 1.

Main instrument table for testing.

Experimental device | Type |
---|---|

Digital electronic analytical balance | AUY220 |

Precise PH meter | PHS-3C |

Vacuum extractor | MC-600D |

Stirrer used in coagulation test | ZR4-6 |

Magnetic stirrer | MS7-H550-Pro |

Silent air pump | AP9903A |

UV spectrophotometer | UVmini-1240 |

The test water is the simulation solution and the effluent is taken from the aeration tank of a wastewater treatment plant in Qingdao. In studying the effects of two ferric salts, FeCl_{2} and FeCl_{3}, added to the pH, alkalinity and phosphorus removing effect of aeration tanks, in view of the complex quality of wastewater in wastewater plant, to prevent the disturbance of other elements in the solution to test results, the simulation solution is used for study, and the test results are verified in real wastewater.

Prepare a simulation solution with alkalinity of 120 mg/L (calculated with CaCO_{3}, same as below), and measure its initial pH value. Add FeCl_{2} and FeCl_{3} (with iron ion content of 10 mg/L, 15 mg/L, 20 mg/L, 25 mg/L, mg/L 30 mg/L and 35 mg/L, respectively) into two groups of solutions, stir them with magnetic stirrer (at a speed of 200 r/min) and aerate (for 30 min), then let it stand still. Next, measure the pH value of the solution, take the supernatant and measure the alkalinity of the solution. Add ferric salt into two groups of solution, respectively, stir them with stirrer (at speeds of 200 r/min 1 min; 100 r/min 5 min), let them stand still and then measure the pH value and alkalinity of the solution.

Prepare a simulation solution with alkalinity of 120 mg/L and total phosphorus content of 4 mg/L. Mix the solution with HCl to make its pH the same as the initial pH of solution in step (1). Operate according to the addition quantity of ferric salt in step (1) and the test steps, filter it and measure the remaining phosphorus content of the filtrate.

The effluent from the aeration tank of a wastewater plant in Qingdao is taken and we measured its total phosphorus content as 4 mg/L. Based on the test results of simulation solution, the optimal dosage of ferric ion is determined. The test steps are the same as in Section 1.2.2, and the changes in pH value and alkalinity of the solution and its phosphorus removing effect are observed.

Potentiometric titration in the Water and Wastewater Analysis and Test Method (fourth version) is used to measure the alkalinity content.

The standards specified in the Water quality-Determination of total phosphorus-Ammonium molybdate spectrophotometric method (GB 118993-89) are used to measure the total phosphorus content.

Activated sludge is the key to the effect of wastewater disposal in aeration tanks, while the pH value and alkalinity of aeration tanks are closely related to the survival, propagation and metabolic function of microorganisms [17,18,19]. Therefore, in order to enhance the phosphorus removing effect without influencing the biological removal of nitrogen, this paper studies the effects of adding FeCl_{2} and FeCl_{3} on the biochemical system of aeration tanks through three indexes: the pH value, alkalinity and phosphorus removing effect.

It can be seen from Figure 1 that adding FeCl_{2} and FeCl_{3} to the solution and conducting aeration treatment will reduce the pH value of the solution; as the dosage increases, the decreasing trend of pH value becomes more obvious. When the initial pH value of the solution is 8.5, and the ferric ion dosage is 10 mg/L and 35 mg/L, FeCl_{2} makes the pH value to reduce by 0.2 and 0.9, and FeCl_{3} makes the pH value to reduce by 0.2 and 1.8, which results in a greater impact on pH. The reason why addition of ferric salt can lead to a decrease in pH is that ferric ions need to consume OH^{−} during hydrolysis. The reaction equation of ferric ion and OH^{−} is as follows:

Under neural or alkalescence condition, ferrous ions can easily be oxidised to ferric ions. Its chain reaction with OH^{−} is as follows:

That is, adding 1 mol FeCl_{2} to oxidise to FeCl_{3} can consume 1 mol OH^{−} less than directly adding 1 mol FeCl_{3}. So, directly addition of FeCl_{2} into aeration tanks has lesser impact on pH than FeCl_{3}.

The pH value is an important factor that affects biological nitrogen and phosphorus removal. Li Nan [20] et al. found that in acidic conditions, the glycogen-accumulating organisms fight with phosphorus-accumulating bacteria on the substrate, leading to a decrease in biological phosphorus removal performance. In neural or alkalescene conditions, however, the phosphorus-accumulating bacteria account for 60% in activated sludge, which is a dominant bacteria with stable and efficient biological phosphorus removal performance. Nitrifying bacteria react very sensitively to pH, and Li Li et al. [21] discover from research that in neural and alkalescene conditions, it is more conducive for microorganism floccules to assimilate the ammonia nitrogen in the water body. Hence, the neural and alkalescene conditions are the optimal pH range of biological nitrogen and phosphorus removal. When the FeCl_{2} dosage is 35 mg/L, the pH of solution is 7.6, which is still within the optimal pH range of biological phosphorus removal and ammonia nitrogen assimilation; but when the FeCl_{3} dosage is 35 mg/L, the pH of solution is 6.7, showing faintly acidic nature, which has a certain effect on biological nitrogen and phosphorus removal.

It can be seen from Figure 1 that adding FeCl_{2} and FeCl_{3} to the solution will reduce its pH value, and the pH value of aerated solution is one unit higher than that of non-aerated solution. This indicates that aeration is helpful to increase the pH value of a biochemical system, and to bring down the effects of ferric salt added to the pH value of the biochemical system. It is inferred that the causes of raised pH value of solution after aeration treatment may be that (1) the pH value is significantly positively correlated with the dissolved oxygen concentration in the water, with the correlation coefficient >0.9 [22]. Aeration can elevate the concentration of dissolved oxygen in the water, resulting in a rising pH value of solution; and (2) aeration can release CO_{2} in the water, leading to a decrease in carbonate and an increase in pH value by one unit [23].

In order to verify the specific reason for rising pH value of aerated solution, the following experiment was carried out. Test I: conduct aeration treatment on pure water, observe the change in pH value of pure water before and after aeration; Test II: fill CO_{2} in pure water, conduct aeration treatment and observe the change in pH value of pure water before and after aeration. See test results in Table 2.

Pure water test.

Test I | Initial pH | The pH after aeration | ||

Pure water | 6.4 | 6.3 | ||

Tap water | 7.7 | 8.3 | ||

Test II | Initial pH | The pH after filling in CO_{2} |
The pH after aeration | |

Pure water | 6.2 | 5.3 | 6.4 |

From the data of Test I, it is observed that the pH value of pure water is basically the same before and after aeration, demonstrating that the dissolved oxygen is not the reason for the rising pH value of solution after aeration treatment. The pH value of tap water rises by 0.6 after aeration treatment. Compared to the change in pH value of pure water, it is inferred that aeration brings some substances out of tap water and leads to the rising pH value of solution. From the results of Test II, it is observed that the initial pH value of pure water is 6.2, the pH value of solution decreases by 0.9 after filling CO_{2} in the solution and the pH value of solution after aeration treatment has little difference with the initial pH value. These observations indicate that the reason for the raised pH value of solution after aeration treatment is that aeration brings down the CO_{2} content in the solution.

Alkalinity is a vital factor that affects biological nitrogen removal. In the nitration reaction, oxidising ammonia nitrogen into nitrate needs to consume 7.1 g alkalinity, so as to balance the acidity generated in nitration. At the same time, nitrifying bacteria and nitrococcus also consume alkalinity to propagate. The denitrifying bacteria can produce 3.5 g alkalinity every time it reduces ammonia nitrogen [24]. If we want the biological nitration reaction to proceed normally, it is essential to satisfy the remaining alkalinity to be >100 mg/L, and if the alkalinity is insufficient, it is necessary to add alkaline matter to supplement the alkalinity. So, when adding ferric salt to improve the phosphorus removal effect, it is required to consider the consumption of alkalinity by ferric salt. To guarantee the biological nitrogen removal effect, it is necessary to ensure the alkalinity. If the alkalinity is insufficient, it is required to add alkaline matter to supplement the alkalinity.

From Figure 2, it is observed that the initial alkalinity of the solution is 123 mg/L. After adding FeCl_{2} and FeCl_{3} to the solution and conducting aeration treatment, the dosage of ferric ion is 10~35 mg/L, and the FeCl_{2} added makes the solution's alkalinity to reduce by 20 mg/L, 26 mg/L, 33 mg/L, 44 mg/L, 51 mg/L and 59 mg/L. Under the same dosage, FeCl_{3} consumes twice as much alkalinity as compared to FeCl_{2}. When the dosage of ferric iron is 20 mg/L, FeCl_{3} consumes >50% alkalinity. From the above data, it is shown that ferric salt added will consume the solution's alkalinity, and the more the dosage is, the more the alkalinity will be consumed; under the same dosage, FeCl_{3} has a greater impact on the alkalinity. The reasons why addition of ferric salt can bring down a solution's alkalinity are that: (1) ferric ion consumes OH^{−} during hydrolysation (see reaction equation in Section 2.1) and (2) ferric iron consumes HCO_{3}^{−}, and the reaction equation of ferric ion is as below:

In the process by which FeCl_{2} reacts with O_{2} to generate FeCl_{3}, partial ferrous ions reform to Fe(OH)_{3} sediment; therefore, the free ferric ions are fewer in comparison with those obtained pursuant to direct addition of FeCl_{3}, and HCO_{3}^{−} is consumed less. Thus, directly adding FeCl_{2} into aeration tanks consumes less alkalinity of biochemical system than that of FeCl_{3}.

From Figure 2, we can see that there is little change between the alkalinity of aerated and non-aerated solutions, which indicates that aeration has less effect on the alkalinity change of solution after adding FeCl_{2} and FeCl_{3}.

In Figure 3, the initial total phosphorus content of the solution is 4 mg/L, and FeCl_{2} and FeCl_{3} added can reduce the total phosphorus content of the solution; as the dosage increases, the total phosphorus content of the solution tends to be stable after a decline. Without aeration treatment, the phosphorus removing effect of adding FeCl_{2} is worse than that of FeCl_{3}; after aeration treatment, the phosphorus removing effect of FeCl_{2} and FeCl_{3} does not differ much. When ferric ion dosage reaches to 20 mg/L, the total phosphorus contents of their supernatant are 0.48 mg/L and 0.49 mg/L (<0.5 mg/L), respectively, which meets the level-I. A discharge standard of the Discharge Standard of Pollutants for Municipal Wastewater Treatment Plant (GB18918-2002) (TP≤0.5 mg/L). Therefore, adding ferric salt into aeration tanks directly, whether FeCl_{2} or FeCl_{3}, can achieve the anticipatory phosphorus removing effect. But FeCl_{2} added has less impact on the pH value and alkalinity of the solution, with more advantage of cost performance.

To avoid the effects of organophosphorus content in wastewater, the effluent from the aeration tank of a wastewater treatment plant in Qingdao was taken to verify the effects of adding FeCl_{2} and FeCl_{3} on the aeration tank. The initial pH and alkalinity of the solution are 7.1 mg/L and 126 mg/L, and the prepared total phosphorus content is 4 mg/L (the total phosphorus content of raw water is 0.27 mg/L). The dosage of ferric ion is 20 mg/L (see test results in Figure 4).

It is observed from Figure 4 that, after adding FeCl_{2} and FeCl_{3} into the solution, the pH value of the solution reduces by 0.2 and 0.5, and the alkalinity reduces by 19.8% and 42.0%. After conducting aeration treatment on the solution, the aerated pH values rise by one unit more than the non-aerated solution, and the alkalinity remains unchanged, which conforms to the impact of simulation solution on pH value and alkalinity. After adding FeCl_{2} in the solution, the solution is also treated by aeration, and the phosphorus removing effect can reach to 99.7%. At this time, the total phosphorus content of the supernatant is 0.01 mg/L<0.50 mg/L, which meets the level-I A discharge standard of the Discharge Standard of Pollutants for Municipal Wastewater Treatment Plant (GB18918-2002), and the phosphorus removing effect is better than that of non-aerated solution (with removing rate of 60.7%), which approximates to that of directly adding FeCl_{3} in the solution (with removing rate of 97.2). So, with regard to the premise of guaranteeing the phosphorus removing effect, directly adding FeCl_{2} has less impact on the biochemical system of an aeration tank than FeCl_{3}.

Adding FeCl_{2} and FeCl_{3} may reduce the pH value of an aeration tank. Under the same dosage, FeCl_{3} has greater effects on pH value than FeCl_{2}. But aeration can improve the pH value of the biochemical system, and lessen the effects of adding ferric salt on the pH value of a biochemical system to a certain extent, so as to lessen the effects of pH value on the biological nitrogen and phosphorus removal performance.

Adding ferric salt will consume the alkalinity of aeration tank and affect the biological nitrogen removal effect. Under the same dosage, FeCl_{3} consumes more alkalinity than FeCl_{2}, which tendency conforms to the required impact on pH value; though aeration can affect the pH value, it basically does not affect the consumption of alkalinity by FeCl_{2} and FeCl_{3}.

FeCl_{3} has a better phosphorus removal effect than FeCl_{2}, so it is a priority selection in coagulated phosphorus removal process. When adding ferric salt in aeration tanks, FeCl_{2} has a better phosphorus removal effect than FeCl_{3}, so it is a priority selection in directly adding ferric salt in aeration tanks for phosphorus removal, with the ferric ion dosage of 20 mg/L.

Directly adding ferric ions into aeration tanks for phosphorus removal has a certain effect on the pH value and alkalinity of the biochemical system, while its specific impact on the nitrogen removal by nitration and denitrification needs to be further investigated.

#### Main instrument table for testing.

Experimental device | Type |
---|---|

Digital electronic analytical balance | AUY220 |

Precise PH meter | PHS-3C |

Vacuum extractor | MC-600D |

Stirrer used in coagulation test | ZR4-6 |

Magnetic stirrer | MS7-H550-Pro |

Silent air pump | AP9903A |

UV spectrophotometer | UVmini-1240 |

#### Pure water test.

Test I | Initial pH | The pH after aeration | ||

Pure water | 6.4 | 6.3 | ||

Tap water | 7.7 | 8.3 | ||

Test II | Initial pH | The pH after filling in CO_{2} |
The pH after aeration | |

Pure water | 6.2 | 5.3 | 6.4 |

Financial accounting measurement model based on numerical analysis of rigid normal differential equation and rigid generalised functional equation Health monitoring of Bridges based on multifractal theory College students’ innovation and entrepreneurship ability based on nonlinear model Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Educational reform informatisation based on fractional differential equation Mathematical simulation analysis of optimal testing of shot puter's throwing path Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption Application and risk assessment of the energy performance contracting model in energy conservation of public buildings 3D Mathematical Modelling Technology in Visual Rehearsal System of Sports Dance The term structure of economic management rate under the parameter analysis of the estimation model based on common differential equation Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas The Spatial Form of Digital Nonlinear Landscape Architecture Design Based on Computer Big Data Attitude control for the rigid spacecraft with the improved extended state observer Sports health quantification method and system implementation based on multiple thermal physiology simulation The improvement of museum information flow based on paste functional mapping method The art design of industrialised manufacturing furniture products based on the simulation of mathematical curves Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations Research on the normalisation method of logging curves: taking XJ Oilfield as an example Research on Evaluation of Intercultural Competence of Civil Aviation College Students Based on Language Operator Information technology of preschool education reform of fine arts based on fractional differential equation Information Teaching Model of Preschool Art Education in Colleges and Universities Based on Finite Element Higher-Order Fractional Differential Equation College Students’ Mental Health Climbing Consumption Model Based on Nonlinear Differential Equations Visual error correction of continuous aerobics action images based on graph difference function Application of fuzzy mathematics calculation in quantitative evaluation of students’ performance of basketball jump shot Application of Forced Modulation Function Mathematical Model in the Characteristic Research of Reflective Intensity Fibre Sensors System dynamics model of output of ball mill Application of mathematical probabilistic statistical model of base – FFCA financial data processing Dynamics of infectious diseases: A review of the main biological aspects and their mathematical translation Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Least-squares method and deep learning in the identification and analysis of name-plates of power equipment Mathematical function data model analysis and synthesis system based on short-term human movement Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Human gait modelling and tracking based on motion functionalisation Analysis and synthesis of function data of human movement Support design of main retracement passage in fully mechanised coal mining face based on numerical simulation Energy-saving technology of BIM green buildings using fractional differential equation Topological optimisation technology of gravity dam section structure based on ANSYS partial differential equation operation Study on the training model of football movement trajectory drop point based on fractional differential equation Modeling the pathway of breast cancer in the Middle East Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation User online consumption behaviour based on fractional differential equation Differential equation model of financial market stability based on Internet big data Multi-attribute Decision Method Based on Normal Random Variable in Economic Management Risk Control Image design and interaction technology based on Fourier inverse transform Children’s cognitive function and mental health based on finite element nonlinear mathematical model Research on China interregional industrial transformation slowdown and influencing factors of industrial transformation based on numerical simulation The optimal model of employment and entrepreneurship models in colleges and universities based on probability theory and statistics Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations Value Creation of Real Estate Company Spin-off Property Service Company Listing Analysing the action techniques of basketball players’ shooting training using calculus method Research on predictive control of students’ performance in PE classes based on the mathematical model of multiple linear regression equation The influence of X fuzzy mathematical method on basketball tactics scoring Application of regression function model based on panel data in bank resource allocation financial risk management Application of Logical Regression Function Model in Credit Business of Commercial Banks Digital model creation and image meticulous processing based on variational partial differential equation Dichotomy model based on the finite element differential equation in the educational informatisation teaching reform model Nonlinear Dissipative System Mathematical Equations in the Multi-regression Model of Information-based Teaching The policy efficiency evaluation of the Beijing–Tianjin–Hebei regional government guidance fund based on the entropy method Stock price analysis based on the research of multiple linear regression macroeconomic variables Application of B-theory for numerical method of functional differential equations in the analysis of fair value in financial accounting Fractional Linear Regression Equation in Agricultural Disaster Assessment Model Based on Geographic Information System Analysis Technology The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations Research on the influence of fuzzy mathematics simulation model in the development of Wushu market Application of multi-attribute decision-making methods based on normal random variables in supply chain risk management The impact of financial repression on manufacturing upgrade based on fractional Fourier transform and probability Deformation and stress theory of surrounding rock of shallow circular tunnel based on complex variable function method A mathematical model of the fractional differential method for structural design dynamics simulation of lower limb force movement step structure based on Sanda movement Numerical calculation and study of differential equations of muscle movement velocity based on martial articulation body ligament tension Research on the Psychological Distribution Delay of Artificial Neural Network Based on the Analysis of Differential Equation by Inequality Expansion and Contraction Method Study on Establishment and Improvement Strategy of Aviation Equipment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode The Model of Sugar Metabolism and Exercise Energy Expenditure Based on Fractional Linear Regression Equation Constructing Artistic Surface Modeling Design Based on Nonlinear Over-limit Interpolation Equation Calculating university education model based on finite element fractional differential equations and macro-control analysis Optimal allocation of microgrid using a differential multi-agent multi-objective evolution algorithm Numerical Simulation Analysis Mathematics of Fluid Mechanics for Semiconductor Circuit Breaker Educational research on mathematics differential equation to simulate the model of children's mental health prevention and control system Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Verifying the validity of the whole person model of mental health education activities in colleges based on differential equation RETRACTION NOTE Human Body Movement Coupling Model in Physical Education Class in the Educational Mathematical Equation of Reasonable Exercise Course Calculation of tourism development income index based on finite element ordinary differential mathematical equation Adoption of deep learning Markov model combined with copula function in portfolio risk measurement Radar system simulation and non-Gaussian mathematical model under virtual reality technology Comparison of compression estimations under the penalty functions of different violent crimes on campus through deep learning and linear spatial autoregressive models Research and application of constructing football training linear programming based on multiple linear regression equation Research on management evaluation of enterprise sales cash flow percentage method based on the application of quadratic linear regression equations Mathematical simulation analysis of optimal detection of shot-putters’ best path Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Mathematical modelling of enterprise financial risk assessment based on risk conduction model Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets Nonlinear Differential Equations in the Teaching Model of Educational Informatisation The evaluation of college students’ innovation and entrepreneurship ability based on nonlinear model Institutional investor company social responsibility report and company performance Mathematical analysis of China's birth rate and research on the urgency of deepening the reform of art education First-principles calculations of magnetic and mechanical properties of Fe-based nanocrystalline alloy Fe _{80}Si_{10}Nb_{6}B_{2}Cu_{2}