Table of Content

15 December 2025, Volume 42 Issue 6 Previous Issue   

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An Improved Accelerate Proximal Gradient for Low-rank Matrix Completion YAN Xihong, LAN Ze, LU Jingyu 2026, 42 (6):  991-1004.  doi: 10.3969/j.issn.1005-3085.2025.06.001 Abstract ( 51 )   Save This paper proposes a variable step-size accelerated proximal gradient algorithm for low-rank matrix completion. The method employs an adaptive step-size selection strategy based on the Armijo rule, which ensures a monotonic decrease of the objective function per iteration and enhances computational efficiency. The convergence of the algorithm is rigorously established under standard assumptions. Numerical experiments demonstrate the algorithm's superiority over existing methods in terms of both convergence speed and overall efficacy. Related Articles | Metrics

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Global Analysis of Discrete SIR Model with Saturated Incidence HU Xinli, LI Hanghang, WU Hang 2026, 42 (6):  1005-1013.  doi: 10.3969/j.issn.1005-3085.2025.06.002 Abstract ( 38 )   Save This paper  discretizes the SIR model with a saturated incidence rate using a nonstandard finite difference method, then solves and analyzes the properties of the  discrete SIR model. The existence positivity and boundedness of the model's solution are obtained. It is shown that the disease-free equilibrium $E_0$ is globally asymptotically stable if the basic reproductive number $R_0<1$, the endemic equilibrium $E^*$ is globally asymptotically stable if $R_0>1$ by constructing a proper V-function. Finally, the numerical simulation supports the proposed theoretical results. According to the simulation, the step length $h$ of the nonstandard finite difference method, rather than that of the forward euler method, has no effect on the global dynamics of the discrete model. Related Articles | Metrics

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Stability and Simulation of Nonlinear Multivariable Dynamical Systems with Time-delay WANG Chunsheng, LI Yongming, FENG Weitong, CHEN Han 2026, 42 (6):  1014-1028.  doi: 10.3969/j.issn.1005-3085.2025.06.003 Abstract ( 39 )   Save In this paper, asymptotic stability theorems for zero solutions of a class of nonlinear multivariate time-lagged generalized Halanay inequalities are given using the fixed point method. This study is an attempt of the fixed point method in the study of the stability of Halanay inequalities, and the results show that the use of the fixed point method to study the stability of Halanay inequalities has certain superiority compared with other methods in certain situations. According to the different characteristics of time lag system coefficients, the proof process of this paper introduces the corresponding functions to construct the operator, which makes the conclusion more flexible and feasible. After that, the obtained results are applied to study the stability of neural network systems in this paper, and better conclusions are obtained. The numerical simulation results combined with the cases verify the accuracy and superiority of the conclusions of this paper. This study adds new methods and new ideas to the study of the stability of Halanay inequalities and enriches the application area of the fixed point method. Related Articles | Metrics

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Statistical Inference of Partially Linear Spatial Autoregressive Model under Constraint Conditions#br# and Heteroscedasticity#br# LI Tizheng, TAN Yundi, CHENG Yaoyao 2026, 42 (6):  1029-1046.  doi: 10.3969/j.issn.1005-3085.2025.06.004 Abstract ( 34 )   Save In many applications of regression analysis, regression coefficients usually satisfy constraint conditions and error terms are heteroscedastic. This paper considers statistical inference of partially linear spatial autoregressive model with constraint conditions and heteroscedasticity. The constrained estimators of unknown parameters and function are established by combining local polynomial smoothing method, generalized method of moments and Lagrange multiplier method, and their asymptotic properties are investigated under appropriate regularity conditions. Furthermore, a Wald test statistic is constructed to test appropriateness of a linear constraint condition on the regression coefficients. It is shown that the proposed test statistic asymptotically follows Chi-squared distribution under both null and alternative hypotheses. Simulation studies and real data analysis are conducted to demonstrate finite sample property of the proposed estimation and testing methods. Related Articles | Metrics

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Chaos Mechanism Analysis and Energy Evolution Simulation of Flow System WANG Heyuan 2026, 42 (6):  1047-1062.  doi: 10.3969/j.issn.1005-3085.2025.06.005 Abstract ( 29 )   Save In order to study the generation mechanism of chaos in flow system, the dynamical mechanism and energy conversion of plane five-mode Lorenz-like equations are studied by moment analysis method. The five-mode system was transformed into a Kolmogorov type system, which is decomposed into three types of torques: inertial torque, dissipation and external torque (drive torque). Through the coupling of different torques, the mechanism of chaos and the relation between energy and external torques were analyzed. Dynamic behavior evolution and energy conversion of the system are analyzed by Casimir function, and the bound of chaotic attractor is estimated. The physical mechanism of flow instability and turbulence is preliminarily explained. Related Articles | Metrics

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Analysis of a Complex Four-dimensional Chaotic System and\\ Its Application ZHANG Yong, ZHANG Fuchen, XIAO Min 2026, 42 (6):  1063-1072.  doi: 10.3969/j.issn.1005-3085.2025.06.006 Abstract ( 69 )   Save In this paper, the complex dynamical behaviors of a four-dimensional chaotic system, including boundedness, global attractive domain and Hamiltonian energy function, are studied. We can judge whether the equilibrium point of chaotic system is stable or not according to the result of the globally exponential attractive set. According to the result of the globally exponential attractive set of this system, the equilibrium point of this system is globally exponential stable. As far as the author knows, the study of boundedness and global attraction domain of chaotic systems is mostly limited to low-dimensional chaotic systems, but due to the structural complexity of high-dimensional chaotic systems, there are few studies on the boundedness and the global attraction domain of high-dimensional chaotic systems. The innovation of this paper is that not only the boundedness of a high-dimensional chaotic system is obtained, but also the rate estimation of the attractive domain of the orbit is obtained. The mathematical expression of a cluster of global attractive sets for the chaotic system is obtained. The research results of this paper provide theoretical basis and support for secure communication based on chaos synchronization. Related Articles | Metrics

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Extrapolation Jacobi Methods for Multiparameter Eigenvalue Problems WU Jiayan, CHEN Xiaoshan 2026, 42 (6):  1073-1088.  doi: 10.3969/j.issn.1005-3085.2025.06.007 Abstract ( 40 )   Save The multiparameter eigenvalue problem originates from the theory of canonical correlation analysis, which is used to analyze the correlation between multiple sets of variables. The canonical correlation analysis is one of the important methods of multivariate statistics, and has a wide range of applications in econometrics, biostatistics and signal processing. The extrapolation Jacobi method is used to solve two kinds of multiparameter eigenvalue problems and the convergence of this method is proved. The effectiveness of the extrapolated Jacobi method is illustrated by numerical examples, and it is compared with the classical Jacobi method, Gauss-Seidel method and SOR method. Related Articles | Metrics

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Research on Signal Reconstruction Theory and Frequency Division Channel Technology Based on Linear Equations SHAN Guangjun 2026, 42 (6):  1089-1097.  doi: 10.3969/j.issn.1005-3085.2025.06.008 Abstract ( 106 )   Save Based on the signal reconstruction theory of linear equation groups, a novel multi-carrier technology called frequency division channel utilization (FDCU) is proposed, aimed at effectively enhancing channel capacity. Unlike other multi-carrier signals such as OFDM, the harmonics of FDCU signals are not fully orthogonal mathematically, which renders traditional Fourier signal processing methods unsuitable for their demodulation. An FDCU signal comprising three frequency components was successfully demodulated using a back-propagation (BP) neural network, demonstrating the effectiveness of BP neural networks in processing non-orthogonal signals. This approach provides a new perspective on signal application and processing methods in fields such as electronic communications, image processing, and remote sensing. Related Articles | Metrics

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A Study on the Forecast of Import and Export Scale Based on Grey Markov Model ZHANG Wenjie, LU Yuyan 2026, 42 (6):  1098-1114.  doi: 10.3969/j.issn.1005-3085.2025.06.009 Abstract ( 36 )   Save The scale of import and export of service trade is closely related to the transformation and upgrading of economy, and the accurate prediction of it can realize the effective grasp of the overall situation of service trade. However, how to realize accurate prediction under the condition of `little data and poor information' has become a difficult problem in the research at domestic and abroad. Based on the data of import and export of Chinese service trade from 2005 to 2018, a grey Markov model is used to simulate and to forecast the import and export scale and industry scale of Chinese service trade from 2019 to 2021. The prediction results show that the grey Markov model has good applicability and high prediction accuracy. At the same time, ${\rm GM}(1,1)$ model and exponential smoothing method are used to simulate and to forecast the same set of data. After comparing the prediction results of the three methods, it is found that the grey Markov model has more advantages in the accuracy and applicability of prediction. In order to further improve the prediction accuracy of the grey Markov model, this paper proposes two kinds of improved methods based on data accumulation and data rolling. The data results show that the two kinds of improved methods can further improve the prediction accuracy of the grey Markov model. This paper provides an effective solution for the forecast of import and export scale of service trade under the condition of `little data and poor information'. Related Articles | Metrics

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Entropy-based Optimal Scale Selection for Wu-Leung Information Systems with a Multi-scale Decision WU Weizhi, SONG Maolin, WANG Jinbo 2026, 42 (6):  1115-1130.  doi: 10.3969/j.issn.1005-3085.2025.06.010 Abstract ( 33 )   Save We investigate the optimal scale selection and the attribute reduction problems in Wu-Leung information systems with a multi-scale decision. Firstly, the concept of entropy-based optimal scale selection is defined in a consistent Wu-Leung information system with a multi-scale decision, which can be used to characterize the optimal scale selection in the same system. Secondly, the concept of an entropy-based optimal scale selection for a single object is introduced, and an attribute reduction method based on the optimal scale selection is further presented. Finally, entropy-based optimal scale selection and attribute reduction methods for inconsistent Wu-Leung information systems with a multi-scale decision are explored. Related Articles | Metrics

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Inventory System with Modified Delayed Vacation Strategy XU Qingzhe, LI Jianjun, LIU Liwei 2026, 42 (6):  1131-1148.  doi: 10.3969/j.issn.1005-3085.2025.06.011 Abstract ( 35 )   Save We introduce a modified delayed vacation strategy. We consider an inventory system with lost sales and $(s,S)$ strategy under the vacation strategy. When the inventory level is zero, the server goes to a random time to prepare for vacation in which if the replenishment is successful, the server still serves the customer normally. The random time is called a modified delayed vacation time. When the delayed vacation is over, the server immediately takes multiple vacations. We analyze the stationary distributions of the inventory system under the vacation strategy by using the Markov theory. On the basis, some performance measures and the average cost function are obtained. Finally, we get the optimal inventory policy by a genetic algorithms. On the basis, some performance measures and expected cost also are discussed by numerical examples. Related Articles | Metrics

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Variable Selection for High-dimensional Partially Linear Models with Longitudinal Data Based on Orthogonal Projection#br# YANG Yiping, QIN Renyu, ZHAO Peixin 2026, 42 (6):  1149-1170.  doi: 10.3969/j.issn.1005-3085.2025.06.012 Abstract ( 50 )   Save The variable selection problem of a partially linear model is considered when the number of the parameters diverges for longitudinal data. The nonparametric components are first eliminated by B-spline and QR decomposition, and the penalized objective function of the regression coefficient in the partial linear model is constructed by combining SCAD penalty and quadratic inference function, the variable selection and estimation of regression coefficients are obtained at the same time. Further, the estimates of the nonparametric components are obtained by combination with the quadratic inference function. The asymptotic properties of the regression coefficient estimates and nonparametric component estimates are demonstrated under some regular conditions. The simulation results show that the proposed method has a good estimation effect regardless of whether the correlation structure of longitudinal data is specified correctly. Finally, the method proposed is used to analyze the factors affecting the operating performance of real estate listed companies. Related Articles | Metrics

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Analysis to a Repairable Retrial Queue with Setup Times and Reserved Idle Times TIAN Ruiling, WU Xinyu 2026, 42 (6):  1171-1188.  doi: 10.3969/j.issn.1005-3085.2025.06.013 Abstract ( 28 )   Save A repairable $M/M/1$ retrial queue with setup times and reserved idle time is studied. The server holds a period of idle time after finishing the service, and customers who arrive at this time can receive the service immediately. Otherwise, the server will be closed down to reduce operating costs until a new customer arrives to activate the system. The server may break down during the busy and setup period. Firstly, the steady-state probability and stationary performance measures of the system are obtained by using the probability generating function method. Secondly, based on the reward-cost structure, the customer's equilibrium strategy and the socially optimal strategy that maximizes social benefits are obtained. Finally, we use the genetic algorithm to find the best combination of parameters that minimize the cost of the system. In addition, we construct a bi-objective optimal model and makes it possible to achieve an appropriate balance between the system cost and the service quality. The regression equation between the minimum cost and the waiting time is proposed, which is helpful to determine the minimum cost that meets service quality. Related Articles | Metrics