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  《兰州理工大学学报》为自然科学类综合性学术期刊,由兰州理工大学主办。创刊于1975年,国内外公开发行,辟有材料科学与工程、机械工程与动力工程、化工与轻工、自动化技术与计算机技术、建筑科学、数理科学等栏目。本刊为中国科技论文统计源期刊、中国科学引文数据库来源期刊、中国学术期刊综合评价数据库来源期刊。自1987年以来,本刊陆续被美国《化学文摘》(CA)、俄罗斯《文摘杂志》(PЖ)、美国《数学评论》(MR)、 德国《数学文摘》(Zbl)、《中国化学文献数据库》、《中国数学文摘》、《中国机械工程文摘》、《机械制造文摘》等国内外权威数据库和文摘刊物收录。 更多...
28 June 2026, Volume 52 Issue 3 Previous Issue   
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Materials Science and Engincering
Composition optimization and deformation mechanism of CoCrMo alloy by molecular dynamics simulation
LIU Fen-xia, JIA Jin-feng, LU Su-jun, REN Jun-qiang, LIU Jun-zhao
2026, 52 (3):  1-7. 
Abstract ( 47 )   PDF (5271KB) ( 194 )  
Molecular dynamics simulations were employed to construct CoCrMo alloy models with varying compositions, systematically investigating the effects of composition on stacking fault energy and lattice constant.Based on compositional optimization, criteria for selecting the optimal alloy composition were established.The results indicated that the lattice constant of the CoCrMo alloy increases monotonically with Mo content.When Mo content is held constant, variations in Cr content show no significant effect on the lattice constant.Furthermore, increasing Mo content markedly enhances the shear modulus of the alloy while reducing its intrinsic stacking fault energy, thereby improving the overall mechanical performance.The optimized alloy composition, Co70Cr26Mo4, exhibits a plastic deformation mechanism dominated by intrinsic stacking faults, extrinsic stacking faults, and phase transformation during tensile loading.
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The wide temperature range tribological behavior of plasma sprayed NiCoCrAlY-Cr2O3-Ag-Mo-Ta composite coating
XU Fu-rong, CHENG Jun
2026, 52 (3):  8-14. 
Abstract ( 37 )   PDF (11716KB) ( 30 )  
High temperature represents one of the most extreme and demanding environments for mechanical systems in service. High-performance lubricating and wear-resistant coating with a wide temperature range capabilities serve as an effective solution to address high-temperature friction and wear in mechanical systems, enabling reliable long-term service.A NiCoCrAlY-Cr2O3-Ag-Mo-Ta composite coating designed for wide-temperature-range lubricating and wear-resistance was fabricated using advanced atmospheric plasma spraying technology. The coating primarily consists of (Ni, Cr), Cr2O3, Ag, and a small amount of Al8(CoCr)3.6Y, indicating that during the thermal spraying process, the raw materials do not undergo significant oxidation or decomposition, and the lubricating phases do not participate in chemical reactions. The hardness of the coating is 405.70 HV0.2. Tribological studies reveal that the coating exhibits poor lubrication performance from room temperature to 200 ℃. In contrast, optimal lubrication and wear resistance are achieved at 400 ℃ and 600 ℃, with the friction coefficient and wear rate of 0.44 and 5.23×10-5 mm3/(N·m) at 400 ℃, and 0.48 and 4.88×10-5 mm3/(N·m) at 600 ℃. At the high temperature of 800 ℃, the coating maintains comparatively good lubricity but suffers severe wear, primarily because excessive oxidation of the coating produces a loose surface oxides that hinder the formation of a dense glazed oxide layer.
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Effects of fine clay content on the properties of microbial solidified coastalsands
JIANG Ming-qiang, LIU Xiao-yan, WANG Teng, FENG Qiong
2026, 52 (3):  15-22. 
Abstract ( 24 )   PDF (6471KB) ( 29 )  
Against the background of coastal erosion problems, Bacillus pasteurii that was acclimatized to seawater and low-temperature was used to solidify coastal sands with different fine clay mass-fraction (0%~28%). Through physical and mechanical property tests of solidified soil bodies, including unconfined compressive strength, triaxial shear characteristics, and calcium carbonate generation amount, combined with microstructure analysis, the solidification effect of microbial solidification technology on coastal sands was comprehensively evaluated, and the influence mechanism of fine clay content on the solidification effect was explored. The results showed that microbial acclimatization effectively enhances bacterial activity in the seawater environment, increasing the strength of treated specimens by 7.7% compared with the unacclimatized group. The mass-fraction of fine clay has a significant impact on the solidification effect. When its content is 12%, both calcium carbonate generation and unconfined compressive strength reach their peak values. An appropriate amount of fine clay can serve as nucleation sites to promote cementation, while excessive fine clay (>12%) tends to block pores and hinder slurry migration, leading to a significant decrease in strength and calcium carbonate generation.
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The influence of compaction method on the fine mechanics of asphalt mixture structure
WANG Xiu-shan, LI Zhen-nan, HU Yi-qi, LIU Qi, YU Ye-wei
2026, 52 (3):  23-29. 
Abstract ( 24 )   PDF (3071KB) ( 31 )  
The equal-size 3D asphalt mixtures were reconstructed using laser scanning equipment and image processing techniques. Three-dimensional asphalt mixture models with identical gradation and size were then developed via the discrete element method (DEM) to simulate the compaction processes of Marshall impact compaction and gyratory compaction. A novel intelligent granular sensor is embedded to capture mechanical response data and validate the DEM simulations. The contact force Gini index, the probability of force chain distribution for different strengths P(f), the characteristics of force chain direction distribution, and the constitutive anisotropy of asphalt mixtures were analysed under different loading methods. The results showed that the Gini index G of contact force between aggregate-aggregate, aggregate-mortar, and mortar-mortar are 0.178, 0.129, and 0.107, respectively, for the specimens moulded by percussive action. The distribution probability P(f) of the strong force chain (f≥1) reaches 32.4%, and is predominantly concentrated near the angle of 0° of the force chain, leading to a higher anisotropy of the group structure. In contrast, under kneading-dominant compaction, the contact force between particles tends to be uniform, weakening the formation of highly resistant strong-compaction regions. The distribution probabilities P(f) of strong force chain, sub-strong force chain (0.5<f<1), and weak force chain (f≤0.5) are 26.1%, 30.7%, and 43.2%, respectively, and the distribution orientation of the force chain develops toward anisotropic uniformity.
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Mechanical Engineering and Power Engineering
Analysis of tooth surface error of non-circular gear based on mesh constraint optimization method
LIU Yong-ping, XING Meng-yi, DONG Chang-bin, LI Da-wei, HUANG Chuan-hong
2026, 52 (3):  30-38. 
Abstract ( 26 )   PDF (3681KB) ( 28 )  
When measuring non-circular gears, it will encounter problems such as mutual coupling of coordinate system error and processing error, many noise points, low measurement efficiency, etc. In order to detect the true condition of the tooth surface, it is necessary to plan measurements of the theoretical tooth surface and to establish a measuring coordinate system. Accordingly, a mathematical model is established based on the relationship between the measurement coordinate system and design coordinate system, which is optimized and solved. At the same time, a mathematical model is established for the relationship between the measurement point and the theoretical point. It also analyses the effect of measurement error on the detection results and compensates the measurement error. Further, the accuracy of this error detection method is verified by coordinate measuring machine error detection experiments. The results show that the degree of mesh distortion of this method is reduced by 15.6% and 6.6% compared to the conventional and unconstrained methods, respectively, thus indicating its correctness and effectiveness.
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High-precision adjustment strategy of rudder for hypersonic vehicles
TANG Wei-qiang, SHI Wen-ke, JIA Cheng-chao
2026, 52 (3):  39-46. 
Abstract ( 21 )   PDF (4753KB) ( 29 )  
A high precision rudder adjustment strategy based on linear-nonlinear switching active disturbance rejection control is proposed for hypersonic vehicles with strong uncertainty. Firstly, the angle of attack system model for attitude control is derived. Secondly, the piecewise continuous function in active disturbance rejection control is replaced by a continuous smooth function , and the associated gain characteristics are analyzed. Finally, a linear-nonlinear switching extended state observer is designed based on the function , and a smooth switching rule of linear and nonlinear control law is proposed. In addition, control parameters are estimated in real time to provide better accuracy of disturbance observation and compensation. The results show that the proposed adjustment strategy has stronger anti-disturbance ability than single linear and nonlinear active disturbance rejection control, and achieves the goal of high precision rudder adjustment.
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Polishing characteristics of magnetic compound fluid based on an annular magnetic field
WANG You-liang, ZHANG Tong, SHI Xiao-feng
2026, 52 (3):  47-52. 
Abstract ( 23 )   PDF (3631KB) ( 26 )  
In this paper, in order to solve the problems such as the reduction of topography accuracy in the polishing process of optical planar elements, a polishing process based on circular Magnetic compound fluid (MCF) was proposed. Optical camera and stylus roughness meter were used to observe the forming characteristics of magnetic composite fluid and the surface roughness changes of workpiece before and after polishing, and MAXWELL was used to simulate the surface magnetic field intensity of workpiece. At the same time, the influence rules of magnetic composite fluid components, liquid carrier plate speed nc, magnet speed nm, processing gap Δ on workpiece surface roughness Ra were analyzed, and the influence mechanism of different parameters on surface quality was explored. The results show that a ring polishing tool is formed based on the ring magnetic field MCF, which is suitable for polishing optical planar elements. The smoothest surface was obtained after 20 min polishing when the MCF containing carbonyl-iron powder 40%, abrasive particle 12%, α-fiber 3%, magnetic fluid 45% and nc =300 r·m-1, nm =400 r·m-1, Δ =1 mm. The workpiece surface roughness decreases from 0.592 μm to 0.035 μm, with a decrease rate of 94.1%. It is proved that MCF polishing tool under annular magnetic field show good polishing performance.
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Accurate calculation of instantaneous geometric flow characteristics of involute external meshing spur gear pump
WANG Jian-sen, LI Wen-xuan, SI Guo-lei, LI Jin-xuan, YAN Hong-liang, HE Chun-yang
2026, 52 (3):  53-60. 
Abstract ( 19 )   PDF (2692KB) ( 29 )  
Accurate, efficient, and high-precision modeling and calculation methods are the foundation of digital twin technology for hydraulic components and systems, taking the involute external meshing spur gear pump as an example to explore an accurate calculation method for its instantaneous geometric flow rate. The axial projection areas of the pump’s discharge chamber and the trapped oil chamber are viewed respectively as a simple connected domain enclosed by piecewise smooth curves, and a parameterized mathematical model of the piecewise curve of the calculation domain is established by the principle of coordinate transformation, using Green's formula, the variation law of the area of the domain with the gear rotation angle can be accurately calculated, and then the geometric instantaneous flow characteristics of the gear pump can be obtained. Through case verification, the results obtained by this method are completely consistent with those based on CAD assisted measurement, compared with traditional approximate calculation methods, the instantaneous geometric flow pulsation rate of the pump obtained by this method is higher than the approximate calculation results, and the displacement of the pump obtained is lower than the approximate calculation results.This method has a certain degree of universality and can be extended to calculate and analyze the instantaneous geometric flow characteristics of gear pumps with other tooth profiles.
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Chemical Industry and Light Industry
Analysis of viscous thermal effect of liquid film of high speed annular groove and spiral groove mechanical seal
ZHANG Wei-zheng, HAN Dong-min, ZHAO Ji-jun, LIN Hua, LIU Jing-wei, HUANG Wen-bin
2026, 52 (3):  61-71. 
Abstract ( 14 )   PDF (6319KB) ( 28 )  
At high speeds, substantial viscous heat is generated in the lubricating fluid of end clearance by the strong shearing action of the rotor. The hydrodynamic lubrication (HD) and thermal hydrodynamic lubrication (THD) characteristics of the annular fluid-spiral groove composite mechanical seal (ASG) model are numerically analyzed. The influence mechanism of liquid film viscous thermal effects on mechanical seal performance is clarified, and the influence laws of ASG geometric parameters on seal performance are investigated. The results indicate that similar pressure distributions are presented by HD and THD models, whereas the hydrodynamic pressure effect of the ASG model is weakened when viscous thermal effects are considered. A significantly higher temperature is observed in the non-grooved zone of the stator end face compared with the grooved zone, while a slightly lower temperature is detected on the rotor end face than on the stator end face. The temperatures of the liquid film, rotor and stator are all gradually increased from the outer diameter to the inner diameter. Under various geometric parameters and working conditions, lower predicted liquid film opening force and friction coefficient values are obtained by the THD model than by the HD model. Rotor temperature is raised with the increase of rotational speed. Larger groove width ratio, groove dam ratio and sealing gap are conducive to temperature reduction but result in an increased leakage rate. Effective temperature reduction cannot be achieved by simply increasing groove depth, and an optimal groove depth is available for thermal management optimization.
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Automation Technique and Computer Technology
Hierarchical coordination optimization of microgrid group based on the consistency principle and ADMM
WANG Xiao-lan, GUAN Ya-hao, GONG Wei-li
2026, 52 (3):  72-80. 
Abstract ( 32 )   PDF (3403KB) ( 33 )  
In order to solve the problem of wind and solar power consumption and power imbalance in the microgrid, a method of hierarchical coordination and optimization of the microgrid group combined with ADMM was proposed. In the microgrid, considering the uncertainties of wind and solar power generation and load, Gaussian regression is used to predict wind, solar, and load. The power fluctuation caused by wind and solar power generation and load consumption is accommodated by optimizing the purchase power, diesel engine power generation, and energy storage charge and discharge power of the grid. Among microgrids, the consensus algorithm is employed to exchange expected interconnection power information between neighboring microgrids and compute their average values. According to the global information provided by the consensus algorithm, ADMM optimizes the distributed power of each microgrid to obtain the optimal distribution of the interactive power between the microgrids. Case studies demonstrate that the proposed optimization algorithm has a faster convergence speed than the traditional ADMM. While realizing the lowest cost operation of microgrids, the total operating cost of microgrid clusters is the lowest through the power interaction between microgrids.
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Event-triggered security control of cyber-physical systems using an adaptive sliding mode control strategy
ZHU Chao-qun, LI Tian-cheng
2026, 52 (3):  81-90. 
Abstract ( 26 )   PDF (2511KB) ( 32 )  
In this paper, the event-triggered adaptive sliding mode control problem is investigated for a class of discrete-time nonlinear cyber-physical systems under hybrid cyber attacks. Firstly, the adaptive event-triggered communication mechanism is introduced in the measurement channels to reduce the communication load. By considering the mode characteristics of hybrid cyber attacks, an augmented closed-loop system model is established based on sliding mode and error dynamics. Secondly, in terms of Lyapunov stability theory and linear inequality technique, sufficient conditions are derived to ensure the asymptotic stability of the augmented closed-loop system. Furthermore, a new adaptive sliding mode control law is proposed to compensate for the negative impact of hybrid cyber attacks on system performance, ensuring that the system state is driven into the quasi sliding mode domain and guaranteeing the accessibility of the sliding region. Finally, the effectiveness and superiority of the proposed control method are verified by a simulation example.
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Traffic signal priority control based on deep reinforcement learning
WANG Zhi-wen, YU Yu-ling, YANG Kang-kang, WANG Hao-xu, MIAO Wei
2026, 52 (3):  91-101. 
Abstract ( 28 )   PDF (3733KB) ( 25 )  
The existing traffic signal priority control system cannot effectively adapt to the increasingly complex and changing urban traffic environment, and is prone to cause unnecessary delays to non-priority vehicles when implementing traffic signal priority control. To address this shortcoming, this study proposes improvements from both the control strategy and optimization objectives. Based on the real-time traffic information of intersections and combining the characteristics of deep reinforcement learning to efficiently process high-dimensional continuous data and self-learning, a priority control strategy based on deep reinforcement learning is proposed to improve the deep double-Q network based on a dueling architecture to solve the signal priority control strategy. At the same time, a dual-objective optimization incentive function is constructed to balance the access needs of both public transport vehicles and general traffic,aiming to reduce delays of public transport vehicles while maximizing the capacity of intersections. Finally, a simulation environment is built using the SUMO to compare the improved algorithm with the three base class algorithms of D3QN,DDQN and Q-Learning. The experimental results show that the improved algorithm can effectively improve the access efficiency of buses and social vehicles in a coordinated manner.
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Multi-scale attention residual network for fault diagnosisof rolling bearings
ZHAO Xiao-qiang, LIU Tong-guo
2026, 52 (3):  102-110. 
Abstract ( 29 )   PDF (3830KB) ( 24 )  
Aiming at the problem of noise interference and working condition differences leading to a significant reduction in the accuracy of rolling bearing fault diagnosis, this study proposes a method for rolling bearing fault diagnosis. By adopting the technical means of hierarchical multi-scale feature extraction and fusion, and exploring the design principles of the multi-scale channel coupling learning, the work advances the design methodology in the multi-scale channel coupling feature extraction module and feature screening separable convolutional residual module design method. This leads to the construction of an Inception attention residual module, culminating in a multi-scale attention residual network for rolling bearing fault diagnosis. Noise experiments and load experiments are carried out on the Case Western Reserve bearing dataset, demonstrating that the proposed method has better anti-noise performance and better generalization performance for different loads. The average diagnostic accuracy of gearbox bearings under different working conditions reaches 92.19%. Furthermore, it shows better generalization performance under different working conditions. The confusion matrix of gearbox bearing diagnosis indicates a a classification accuracy of 98% across five fault states, each with 200 samples, underscoring the high precision of the proposed approach.
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Architectural Sciences
Comparative analysis of new energy machinery and fuel machinery in high-altitude tunnel construction
ZHAO Yong-wei, WANG Zheng-zhen, HU Bin, HE Qing-wen, DAI Guo-liang
2026, 52 (3):  111-119. 
Abstract ( 28 )   PDF (2720KB) ( 22 )  
A highway tunnel section under construction in Gansu was taken as the background by this paper. The multiple values of new energy machinery and fuel machinery under the same working conditions (noise and vibration under different working conditions in the tunnel) were monitored and analyzed, and the full life cycle use costs of the two were calculated and compared. The entropy weight evaluation method was employed to analyze the aforementioned data, and their comprehensive scores were utilized to evaluate and compare the strengths and weaknesses of the two types of machinery, providing a reference for the future use of new energy machinery in tunnel projects. The main conclusions are as follows: the noise of new energy machinery is generally lower than that of fuel machinery under all working conditions, with the difference being particularly noticeable during full-load driving and stationary waiting periods. New energy machinery demonstrate superior performance in overall noise control. The body vibration values and amplitude ranges of new energy machinery under various working conditions are consistently smaller than those of traditional fuel machinery. Although new energy machinery require a larger initial investment, significant advantages in maintenance costs, energy expenses (electricity or fuel), and environmental friendliness during the operational phase are exhibited. When the total life cycle costs of bothmachinery are calculated, new energy machinery demonstrate higher cost-effectiveness compared to fuel machinery.
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Study on the mechanical properties of recycled concrete composite column reinforced with thin-wall steel tube and fiber reinforced plastic
CHEN Hua, GUAN Ji-wen, CHEN Yao-jia, YANG Jing-xiong
2026, 52 (3):  120-130. 
Abstract ( 23 )   PDF (7406KB) ( 16 )  
To study the mechanical properties of fiber-reinforced polymer (FRP) reinforced thin-walled steel tube recycled concrete composite columns, axial compression tests were conducted on one reinforced concrete (RC) column and five FRP reinforced composite columns to investigate the effects of initial axial compression ratio, newly added recycled concrete strength, and FRP type on the failure mode and stress mechanism of the specimens.Important mechanical indicators such as bearing capacity, displacement, thin-walled steel tube strain, and FRP strain were measured.Based on the limit equilibrium theory, a calculation method for bearing capacity and ductility coefficient was proposed.The results indicate that increasing the initial axial compression ratio would continuously reduce the bearing capacity of the reinforced specimen.Improving the strength of recycled concrete can enhance the bearing capacity of reinforced specimens.The finite element simulation results are in good agreement with the experimental results, and the variation pattern of mechanical parameters is basically consistent.The calculation methods for bearing capacity and ductility coefficient proposed are in good agreement with experimental results, finite element calculation results, and relevant literature fitting results.The calculation model meets the relevant mechanical indicators of FRP reinforced thin-walled steel tube recycled concrete composite columns and meets the requirements of engineering structural reinforcement design.
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Study on the shear performance of assembled RC bridge piers with mortise-tenon and cast-in-situ UHPC hybrid connection
LIN Wen, WANG Wen-ping, LIN Yong-jie, LIN Shang-shun
2026, 52 (3):  131-139. 
Abstract ( 21 )   PDF (4940KB) ( 19 )  
To investigate the shear behavior and influencing factors of assembled reinforced concrete (RC) bridge piers with mortise and tenon-cast-in-situ UHPC hybrid connection, quasi-static tests were conducted. Based on the experimental results, a refined finite element model was established to systematically analyze the effects of key parameters, including axial compression ratio, longitudinal reinforcement ratio, stirrup ratio, and UHPC thickness, on the shear capacity of the specimens. Meanwhile, existing calculation methods for the shear strength of precast assembled bridge piers were reviewed. Using the finite element results as the basis, the applicability and accuracy of the shear strength formulas in relevant domestic and international design codes were evaluated, and a modified shear strength calculation formula applicable to precast assembled bridge piers with hybrid connection joints was further proposed. The results indicated that the mortise-tenon and cast-in-situ UHPC hybrid-connected assembled RC bridge piers exhibit favorable shear performance, the modified calculation method proposed demonstrates higher accuracy and better applicability.
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Research on selection method of equilibrium point during self-balanced static load test for super-long pile under loess area
GUO Xiang-jun, GAO Jun
2026, 52 (3):  140-146. 
Abstract ( 23 )   PDF (3327KB) ( 19 )  
In order to reduce the influences of side friction resistance for pile body during the determination process of equilibrium point of self-balanced method, the finite element simulation was used, the finite element model of pile-soil was established and the equilibrium point positions of pile under the different conditions of foundation bearing capacity, pile lengths and pile diameters were analyzed. When the bearing capacity of loess is 120 kPa, the pile length is 75m and the pile diameter is 800mm, the 1/3, 1/7, 7/25, 21/100 and 17/100 points of pile body are selected as the calculation nodes of equilibrium point according to the site test results and the principle of bisection method. The displacements of the upper and lower piles are 4 mm and 4.1 mm respectively when the 17/100 point of pile body is selected as the calculation node of equilibrium point, and the result is reasonable. According to the established finite element model, the equilibrium points were obtained under the different working conditions such as when the foundation bearing capacity increases from 110 to 250 kPa, the pile length increases from 30 to 110 m, and the pile diameter increases from 600 to 1 500 mm, a series of equilibrium diagrams were drawn. Equilibrium diagrams can provide a reference for the determination equilibrium point in the self-balancing method.
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Prediction of CO concentration in extra-long tunnels based on time series
LU Jian-qiang, HE Xing, LIU Wei, LIN Hai-cheng, SUN Cheng-hui
2026, 52 (3):  147-154. 
Abstract ( 26 )   PDF (2781KB) ( 17 )  
The time series analysis method has the advantage of modeling and predicting the time series data, which can more accurately reveal the law and trend of CO concentration in tunnels. By collecting the historical data of CO concentration, vehicle flow, vehicle speed and wind speed in Liupanshan tunnel for three consecutive years, a prediction model of CO concentration was constructed by time series analysis method. At the same time, the factors affecting the change of CO concentration (vehicle flow, vehicle speed, wind speed) were analyzed. The results show that the increase of vehicle flow, the decrease of vehicle speed and the lower wind speed may lead to the increase of CO concentration. The average difference between the predicted value of the prediction model and the measured value is small, indicating that the prediction model has certain accuracy and prediction ability. The application of the prediction model can provide a scientific basis for traffic management and environmental protection, reduce potential safety risks, and provide early warning and decision-making for tunnel managers.
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Scientific
Weak injective modules and weak flat modules over formal triangular matrix rings
WU De-jun, MA Jie
2026, 52 (3):  155-160. 
Abstract ( 23 )   PDF (471KB) ( 19 )  
Let $T=\left(\begin{array}{ll} A & 0 \\ U & B \end{array}\right)$ be a formal triangular matrix ring, where A and B are rings and U is a (B,A)-bimodule. Characterizations of Gorenstein weak injective left T-modules and Gorenstein weak flat right T-modules over the formal triangular matrix ring T are given. Moreover, the weak injective dimensions of left T-modules and weak flat dimensions of right T-modules, Gorenstein weak injective dimensions of left T-modules and Gorenstein weak flat dimensions of right T-modules are discussed.
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Steiner (n-1)-Wiener and Steiner 3-Wiener indices of cacti
YIN Yue-qin, WU Fan, AN Xin-hui
2026, 52 (3):  161-167. 
Abstract ( 19 )   PDF (820KB) ( 16 )  
For 2≤k≤n-1, the Steiner k-Wiener index of a connected graphG is the sum of Steiner distances of all k-subsets S of V(G). The bounds of the Steiner (n-1)-Wiener index and the bound of the Steiner 3-Wiener index of the cactus are studied and the corresponding Extremal graphs are given.
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Direct and inverse problems in the application of Green’s theorem
GONG Xing-tian, YANG Shu-wei
2026, 52 (3):  168-172. 
Abstract ( 20 )   PDF (849KB) ( 20 )  
Using Green’s theorem, a new theorem and formula are proposed and proved to convert the double integral into the curve integral with the help of a homogeneous function, which not only reflects the advantage of dimensionality reduction to a certain extent, but also provides a new way of solving double integrals. The typical examples demonstrate the ingenious use of the theory in simplifying the integral calculation.
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