中国科学院数学与系统科学研究院期刊网

28 March 2024, Volume 47 Issue 2
    

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  • A Modified Primal-dual Algorithm for Matrix Completion Problems
    YAN Xihongy ZHANG Ning
    Acta Mathematicae Applicatae Sinica. 2024, 47(2): 175-192. https://doi.org/10.20142/j.cnki.amas.202401052
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    As an important problem in the field of information science such as machine learning and image processing, low rank matrix completion has been widely studied. The first-order primal-dual algorithm is one of the classical algorithms for solving this problem. However, the data processed in practical applications is often large-scale. Therefore, based on the framework of the primal-dual algorithm, this paper proposes a modified primal-dual algorithm for large-scale matrix completion problems by exploring correction strategy with the variable step size. In each iteration of the new algorithm, the primal and dual variables are firstly updated by the primal-dual algorithm, and then the correction strategy with the variable step size is used to further correct the two variables. Under certain assumptions, the global convergence of the new algorithm is proved. Finally, the new algorithm is verified to be efficient by solving some random low rank matrix completion problems and examples of image restoration.
  • Uniformity of Combined Two-level U-type Designs Under Projection Weighted Symmetric $L_2$-discrepancy
    LEI Yiju, OU Zujun
    Acta Mathematicae Applicatae Sinica. 2024, 47(2): 193-203. https://doi.org/10.20142/j.cnki.amas.202401056
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    The uniform designs are accepted widely because of its robust and easy to use, flexible characteristics. In order to distribute the points evenly in the experimental domain, many criteria ($L_2$-discrepancy) have been forwarded to measure the uniformity of the design array. At present, centered $L_2$-discrepancy, wrap-around $L_2$-discrepancy, mixed discrepancy and so on are widely used. Symmetric $L_2$-discrepancy has better geometric sense, but the poor performance at projection uniformity limits the use of SD. To refine the projection properties of SD, a projection weighted SD is proposed. The SD was exponentially weighted. The projection weighted SD can retain the excellent properties of the original discrepancy, and overcome the original defects effectively, and has better performance. The foldover is a useful technique in construction of factorial designs. In this paper, the projection weighted symmetric $L_2$-discrepancy is used as the optimality criterion to evaluate the quality of the foldover scheme. Lower bounds for projection weighted symmetric $L_2$-discrepancy on combined two-level U-type designs under a general foldover plan are obtained, which can be used as a benchmark for searching optimal foldover plans.
  • B-spline Estimations of Partially Functional Linear Multiplicative Model
    DING Jianhua, YU Ping, DING Yanping
    Acta Mathematicae Applicatae Sinica. 2024, 47(2): 204-225. https://doi.org/10.20142/j.cnki.amas.202401047
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    In this paper, the partial functional linear multiplicative model is considered. This model, which becomes partial functional linear regression after taking logarithmic transformation, is useful in analyzing data with positive responses. Based on B-splines, two estimation methods are proposed by minimizing the least absolute relative error (LARE) and the least product relative error (LPRE), respectively. The dimension of the B-spline bases is selected using the Schwarz information criteria. Consistency and asymptotic normality of the two methods are investigated. For the slope function, we prove that its convergence rate achieves the optimal rate of nonparametric function. Monte Carlo simulations are conducted to evaluate and compare the finite sample performance of the proposed estimators with the least squares (LS) estimator and the least absolute deviation (LAD) estimator under the different random error settings. Simulation results show that the proposed methods are comparable to other methods. Finally, an example of real data analysis is given to illustrate the application of the model.
  • Blow-up of Solutions to a Dissipative Generalized Tricomi Equation with Variable Coefficients and Nonlinearity of Derivative Type
    OUYANG Baiping
    Acta Mathematicae Applicatae Sinica. 2024, 47(2): 226-237. https://doi.org/10.20142/j.cnki.amas.202401055
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    Blow up phenomena of solutions to a dissipative generalized Tricomi equation with variable coefficients and nonlinearity of derivative type in the subcritical case are considered. By constructing some time-dependent functionals associated with test function methods and Bessel equations, an iterative frame and the first lower bound of the time-dependent functional are obtained. Then, blow-up of solutions and upper bound estimate for the lifespan to the Cauchy problem are proved via iteration arguments.
  • Randomized Test of Mean Function for High-frequency Functional Data
    ZHAO Fanrong, YUE Lili, ZHANG Baoxue
    Acta Mathematicae Applicatae Sinica. 2024, 47(2): 238-254. https://doi.org/10.20142/j.cnki.amas.202401029
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    This paper studies the problem of testing the mean of high-frequency functional data. For functional data with infinite number of principal components and spiked eigenvalues of covariance operators, the classical Chi-square or mixed Chi-square test constructed based on the dimension reduction method using functional principal components will become invalid due to insufficient sample size and strong conditions of covariance operators. Therefore, this paper proposes a randomized test to solve this problem, and proves the large sample properties. Further, the numerical simulation of limited samples is used to verify the effectiveness of the proposed test. Finally, this method is applied to the phoneme data.
  • Admitted Lie Groups and Group Invariant Solutions of Two Kinds of Population Balance Equations with Fragmentation Processes
    LIN Fubiao, ZHANG Qianhong
    Acta Mathematicae Applicatae Sinica. 2024, 47(2): 255-268. https://doi.org/10.20142/j.cnki.amas.202401020
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    In this work, aiming at the problem of lacking analytical methods and it is typically difficult to find explicit exact solutions to the population balance equations with fragmentation processes. We consider the admitted Lie groups, group invariant solutions, reduced integro-ordinary differential equations and explicit exact solutions of two classes of integro-partial differential equations (population balance equations with fragmentation processes) by use of the method of scaling transformation group analysis, the method of classical Lie group analysis and the method of developed Lie group analysis. Firstly, the admitted scaling transformation Lie groups of the integro-partial differential equations are explored by using the method of scaling transformation group analysis. Secondly, the integro-partial differential equations are transformed into pure partial differential equations, the admitted Lie groups of the pure partial differential equations are calculated by use of the methods of classical Lie group analysis. Thirdly, the admitted Lie groups of the original integro-partial differential equations are determined by use of the methods of developed Lie group analysis combining with the related results obtained by method of scaling transformation group and the method of classical Lie group analysis. Finally, the admitted Lie groups of the original integro-partial differential equations are successfully found. All group invariant solutions, reduced integro-ordinary differential equations and explicit exact solutions are given. The related analysis for dynamic behavior characteristics of a solution with evolution of the size distribution are also presented.
  • The Pointwise Estimates to Solutions for 1-dimensional Linear Navier-Stokes-Fourier Equations
    AN Zhengda, ZHANG Qi
    Acta Mathematicae Applicatae Sinica. 2024, 47(2): 269-283. https://doi.org/10.20142/j.cnki.amas.202401011
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    In this paper, we study the 1-dimensional linear Navier-Stokes-Fourier equations and obtain the pointwise estimates of the decay properties of the solution under the appropriate initial value conditions, and describe the decay direction of the solution, and verify that the generalized Huygens’ principle holds. To this end, we divide the Fourier transform of the Green function of the equations into low-frequency, mid-frequency and high-frequency parts by means of Fourier transform, and prove the decay properties of the Green function in the corresponding frequency parts, and then obtain the decay estimates of the solutions by means of the Fourier inverse transform and the properties of the fundamental solutions.
  • Strategies Research on the Discrete-time Queueing System with Risk-sensitive Customers
    CAO Can, LIU Zaiming, GAO Shan, WU Yifan
    Acta Mathematicae Applicatae Sinica. 2024, 47(2): 284-311. https://doi.org/10.20142/j.cnki.amas.202401022
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    The research on customer strategy behavior in queueing system combined with game theory is a hot topic in the current queueing theory. This paper studies the strategic behavior of risk-sensitive customers in discrete-time queueing systems. Different from the classical economics of queues, the utility function in this paper is an expectation-variance quadratic utility function. Based on the Nash equilibrium and Markov process theory, we study the game behavior of Geo/Geo/1 queueing system with risk-sentitive customers under fully observable case and fully unobservable case, respectively. The individual optimal joining strategy, the joining strategy for the social net welfare and the server’s profit optimization are obtained. It is found that the smaller the risk sensitivity coefficient is, the more customers like to take risks and the stronger the willingness to join the system. Some numerical experiments are provided to illustrate the effect of the risk sensitivity coefficient on the customer strategic behavior.
  • Higher-Order Generalized Weak Studniarski Epiderivatives and Applications to Composite Set-Valued Optimization Problems
    HE Liu, WANG Qilin, ZHANG Xiaoyan, TANG Tian
    Acta Mathematicae Applicatae Sinica. 2024, 47(2): 312-332. https://doi.org/10.20142/j.cnki.amas.202401049
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    In this paper, we introduce the notion of higher-order generalized weak Studniarski epiderivatives for a set-valued map without lower-order approximating directions and obtain chain and sum operation rules of the epiderivative. Then by applying the higher-order epiderivative, we establish the optimality conditions of the weakly efficient solution for unconstrained composite set-valued optimization problems. Some illustrative examples are provided as well.
  • European Option Pricing under the Log Mean-Reverting Jump Diffusion Stochastic Volatility Model
    MA Aiqin, GUO Jingjun, WANG Yubing, ZHANG Cuiyun
    Acta Mathematicae Applicatae Sinica. 2024, 47(2): 333-354. https://doi.org/10.20142/j.cnki.amas.202401028
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    Considering the uncertainty of financial market data volatility, a new logmean reversion jump diffusion 4/2 random volatility (LMRJ-4/2-SV) model was proposed in this paper. Firstly, the LMRJ-4/2-SV model was constructed, and the European option pricing formula based on LMRJ-4/2-SV model was obtained by using FFT and other methods. Secondly, descriptive statistical analysis of the actual market data was carried out to discuss the price change characteristics of the underlying asset and the applicability of the LMRJ-4/2-SV model, and the model parameters were estimated by particle swarm optimization algorithm. Finally, European options were priced based on the option pricing formula and parameter estimates under the LMRJ-4/2-SV model, and the pricing results were compared with the 4/2, 3/2, Heston model estimates and market prices. The results show that the pricing error of European option based on LMRJ-4/2-SV model is minimal, and the pricing results have obvious advantages over other stochastic volatility models.
  • $\mathscr{M}_{\alpha}$-shadowing Property on the Measure Center
    WU Xinxing
    Acta Mathematicae Applicatae Sinica. 2024, 47(2): 355-368. https://doi.org/10.20142/j.cnki.amas.202401019
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    In this paper, it is proved that if a dynamical system has the periodic $\mathscr{M}_{\alpha}$-shadowing property or periodic $\mathscr{M}^{\alpha}$-shadowing property, then the dynamical system restricted on its measure center has the same shadowing property. Conversely, if a dynamical system restricted on its measure center has the periodic $\mathscr{M}_{\alpha}$-shadowing property (resp., periodic $\mathscr{M}^{\alpha}$-shadowing property), then the dynamical system has the periodic $\mathscr{M}_{\beta}$-shadowing property (resp., periodic $\mathscr{M}^{\beta}$-shadowing property) for any $\beta\in [0, \alpha)$. Moreover, it is obtained that for an equicontinuous system, many shadowing properties are equivalent to the condition that it has a trivial measure center.
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