Table of Content

    25 January 2023, Volume 36 Issue 1
    Preface to the Special Topic on Computer Mathematics
    CHEN Shaoshi, MOU Chenqi
    2023, 36(1):  1-2.  DOI: 10.1007/s11424-023-3000-4
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    Improve Robustness and Accuracy of Deep Neural Network with $L_{2,\infty}$ Normalization
    YU Lijia, GAO Xiao-Shan
    2023, 36(1):  3-28.  DOI: 10.1007/s11424-022-1326-y
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    In this paper, the $L_{2,\infty}$ normalization of the weight matrices is used to enhance the robustness and accuracy of the deep neural network (DNN) with Relu as activation functions. It is shown that the $L_{2,\infty}$ normalization leads to large dihedral angles between two adjacent faces of the DNN function graph and hence smoother DNN functions, which reduces over-fitting of the DNN. A global measure is proposed for the robustness of a classification DNN, which is the average radius of the maximal robust spheres with the training samples as centers. A lower bound for the robustness measure in terms of the $L_{2,\infty}$ norm is given. Finally, an upper bound for the Rademacher complexity of DNNs with $L_{2,\infty}$ normalization is given. An algorithm is given to train DNNs with the $L_{2,\infty}$ normalization and numerical experimental results are used to show that the $L_{2,\infty}$ normalization is effective in terms of improving the robustness and accuracy.
    Isogeometric Analysis-Based Topological Optimization for Heterogeneous Parametric Porous Structures
    HU Chuanfeng, HU Hui, LIN Hongwei, YAN Jiacong
    2023, 36(1):  29-52.  DOI: 10.1007/s11424-022-1290-6
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    Porous structures widely exist in nature and artifacts, which can be exploited to reduce structural weight and material usage or improve damage tolerance and energy absorption. In this study, the authors develop an approach to design optimized porous structures with Triply Periodic Minimal Surfaces (TPMSs) in the framework of isogeometric analysis (IGA)-based topological optimization. In the developed method, by controlling the density distribution, the designed porous structures can achieve the optimal mechanical performance without increasing the usage of materials. First, the implicit functions of the TPMSs are adopted to design several types of porous elements parametrically. Second, to reduce the cost of computation, the authors propose an equivalent method to forecast the elastic modulus of these porous elements with different densities. Subsequently, the relationships of different porous elements between the elastic modulus and the relative density are constructed. Third, the IGA-based porous topological optimization is developed to obtain an optimal density distribution, which solves a volume constrained compliance minimization problem based on IGA. Finally, an optimum heterogeneous porous structure is generated based on the optimized density distribution. Experimental results demonstrate the effectiveness and efficiency of the proposed method.
    Curvature-Based $r$-Adaptive Isogeometric Analysis with Injectivity-Preserving Multi-Sided Domain Parameterization
    JI Ye, WANG Mengyun, YU Yingying, ZHU Chungang
    2023, 36(1):  53-76.  DOI: 10.1007/s11424-022-1293-3
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    Inspired by the $r$-refinement method in isogeometric analysis, in this paper, the authors propose a curvature-based $r$-adaptive isogeometric method for planar multi-sided computational domains parameterized by toric surface patches. The authors construct three absolute curvature metrics of isogeometric solution surface to characterize its gradient information, which is more straightforward and effective. The proposed method takes the internal weights as optimization variables and the resulting parameterization is analysis-suitable and injectivity-preserving with a theoretical guarantee. Several PDEs are solved over multi-sided computational domains parameterized by toric surface patches to demonstrate the effectiveness and efficiency of the proposed method
    New Results on the Equivalence of Bivariate Polynomial Matrices
    ZHENG Xiaopeng, LU Dong, WANG Dingkang, XIAO Fanghui
    2023, 36(1):  77-95.  DOI: 10.1007/s11424-023-1304-z
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    This paper investigates the equivalence problem of bivariate polynomial matrices. A necessary and sufficient condition for the equivalence of a square matrix with the determinant being some power of a univariate irreducible polynomial and its Smith form is proposed. Meanwhile, the authors present an algorithm that reduces this class of bivariate polynomial matrices to their Smith forms, and an example is given to illustrate the effectiveness of the algorithm. In addition, the authors generalize the main result to the non-square case.
    Nonlinear Inverse Relations of the Bell Polynomials via the Lagrange Inversion Formula (II)
    MA Xinrong, WANG Jin
    2023, 36(1):  96-116.  DOI: 10.1007/s11424-022-1300-8
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    In this paper, by means of the classical Lagrange inversion formula, the authors establish a general nonlinear inverse relation as the solution to the problem proposed in the paper [J. Wang, Nonlinear inverse relations for the Bell polynomials via the Lagrange inversion formula, J. Integer Seq., Vol. 22 (2019), Article 19.3.8]. As applications of this inverse relation, the authors not only find a short proof of another nonlinear inverse relation due to Birmajer, et al. (2012), but also set up a few convolution identities concerning the Mina polynomials.
    The Log-Concavity of Kazhdan-Lusztig Polynomials of Uniform Matroids
    XIE Matthew H Y, ZHANG Philip B
    2023, 36(1):  117-128.  DOI: 10.1007/s11424-022-1296-0
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    Elias, et al. (2016) conjectured that the Kazhdan-Lusztig polynomial of any matroid is logconcave. Inspired by a computer proof of Moll’s log-concavity conjecture given by Kauers and Paule, the authors use a computer algebra system to prove the conjecture for arbitrary uniform matroids.
    Ramp Scheme Based on CRT for Polynomial Ring over Finite Field
    DING Jian, KE Pinhui, LIN Changlu, WANG Huaxiong
    2023, 36(1):  129-150.  DOI: 10.1007/s11424-022-1292-4
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    Chinese Reminder Theorem (CRT) for integers has been widely used to construct secret sharing schemes for different scenarios, but these schemes have lower information rates than that of Lagrange interpolation-based schemes. In ASIACRYPT 2018, Ning, et al. constructed a perfect $(r,n)$-threshold scheme based on CRT for polynomial ring over finite field, and the corresponding information rate is one which is the greatest case for a $(r,n)$-threshold scheme. However, for many practical purposes, the information rate of Ning, et al. scheme is low and perfect security is too much security. In this work, the authors generalize the Ning, et al. $(r,n)$-threshold scheme to a $(t,r,n)$-ramp scheme based on CRT for polynomial ring over finite field, which attains the greatest information rate $(r-t)$ for a $(t,r,n)$-ramp scheme. Moreover, for any given $2\leq r_1 < r_2\leq n$, the ramp scheme can be used to construct a $(r_1,n)$-threshold scheme that is threshold changeable to $(r',n)$-threshold scheme for all $r'\in \{r_1+1,r_1+2,\cdots,r_2\}$. The threshold changeable secret sharing (TCSS) scheme has a greater information rate than other existing TCSS schemes of this type.
    Smith Form of Triangular Multivariate Polynomial Matrix
    LIU Jinwang, WU Tao, LI Dongmei
    2023, 36(1):  151-164.  DOI: 10.1007/s11424-022-1289-z
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    The Smith form of a matrix plays an important role in the equivalence of matrix. It is known that some multivariate polynomial matrices are not equivalent to their Smith forms. In this paper, the authors investigate mainly the Smith forms of multivariate polynomial triangular matrices and testify two upper multivariate polynomial triangular matrices are equivalent to their Smith forms respectively.
    On PID Control Theory for Nonaffine Uncertain Stochastic Systems
    ZHANG Jinke, ZHAO Cheng, GUO Lei
    2023, 36(1):  165-186.  DOI: 10.1007/s11424-022-1486-9
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    PID (proportional-integral-derivative) control is recognized to be the most widely and successfully employed control strategy by far. However, there are limited theoretical investigations explaining the rationale why PID can work so well when dealing with nonlinear uncertain systems. This paper continues the previous researches towards establishing a theoretical foundation of PID control, by studying the regulation problem of PID control for nonaffine uncertain nonlinear stochastic systems. To be specific, a three dimensional parameter set will be constructed explicitly based on some prior knowledge on bounds of partial derivatives of both the drift and diffusion terms. It will be shown that the closed-loop control system will achieve exponential stability in the mean square sense under PID control, whenever the controller parameters are chosen from the constructed parameter set. Moreover, similar results can also be obtained for PD (PI) control in some special cases. A numerical example will be provided to illustrate the theoretical results.
    Differentially Private Distributed Parameter Estimation
    WANG Jimin, TAN Jianwei, ZHANG Ji-Feng
    2023, 36(1):  187-204.  DOI: 10.1007/s11424-022-2012-9
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    Data privacy is an important issue in control systems, especially when datasets contain sensitive information about individuals. In this paper, the authors are concerned with the differentially private distributed parameter estimation problem, that is, we estimate an unknown parameter while protecting the sensitive information of each agent. First, the authors propose a distributed stochastic approximation estimation algorithm in the form of the differentially private consensus+innovations (DP-CI), and establish the privacy and convergence property of the proposed algorithm. Specifically, it is shown that the proposed algorithm asymptotically unbiased converges in mean-square to the unknown parameter while differential privacy-preserving holds for finite number of iterations. Then, the exponentially damping step-size and privacy noise for DP-CI algorithm is given. The estimate approximately converges to the unknown parameter with an error proportional to the step-size parameter while differential privacy-preserving holds for all iterations. The tradeoff between accuracy and privacy of the algorithm is effectively shown. Finally, a simulation example is provided to verify the effectiveness of the proposed algorithm.
    Consensus Analysis of Fractional Multi-Agent Systems with Delayed Distributed PI Controller
    XIE Yingkang, MA Qian
    2023, 36(1):  205-221.  DOI: 10.1007/s11424-022-1256-8
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    The consensus problem for fractional multi-agent systems (MASs) with time delay is considered. The distributed fractional proportional-integral (PI)-type controller is designed so that the consensus of the proposed systems is achieved. Moreover, explicit condition to determine the crossing directions is developed. The results show that with the increase of time delay, the closed-loop system has two different dynamic characteristics: From consensus to nonconsensus and consensus switching. Furthermore, delay margin within which consensus of MASs will always hold is determined. The results should provide useful guidelines in the consensus analysis and in the analytical design of the distributed controllers.
    Event-Triggered Optimal Nonlinear Systems Control Based on State Observer and Neural Network
    CHENG Songsong, LI Haoyun, GUO Yuchao, PAN Tianhong, FAN Yuan
    2023, 36(1):  222-238.  DOI: 10.1007/s11424-022-1146-0
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    This paper develops a novel event-triggered optimal control approach based on state observer and neural network (NN) for nonlinear continuous-time systems. Firstly, the authors propose an online algorithm with critic and actor NNs to solve the optimal control problem and provide an event-triggered method to reduce communication and computation burdens. Moreover, the authors design weight estimation for critic and actor NNs based on gradient descent method and achieve uniformly ultimate boundednesss (UUB) estimation results. Furthermore, by using bounded NN weight estimation and dead-zone operator, the authors propose a triggering condition, prove the asymptotic stability of closed-loop system from Lyapunov stability perspective, and exclude the Zeno behavior. Finally, the authors provide a numerical example to illustrate the effectiveness of the proposed method.
    Sampled-Data $H_{\infty }$ Dynamic Output Feedback Controller Design for Fuzzy Markovian Jump Systems
    LIN Yuqian, ZHUANG Guangming, XIA Jianwei, CHEN Guoliang, LU Junwei
    2023, 36(1):  239-256.  DOI: 10.1007/s11424-022-1196-3
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    This paper considers the issue of $H_{\infty }$ dynamic output feedback controller design for T-S fuzzy Markovian jump systems under time-varying sampling with known upper bound on the sampling intervals. The main aim is to realize sampled-data fuzzy dynamic output feedback control so as to demonstrate the stochastic stability and $H_{\infty }$ performance index of the closed-loop sampled-data fuzzy Markovian jump systems. Then, by making the most of the information within the sampling interval, a suitable closed-loop function is constructed and the integral terms are handled by using free weighted matrix method and improved integral inequality technique. Numerical example and single-link robot arm are presented to show the effectiveness of the developed method.
    Transformer-Based Deep Learning Network for Tooth Segmentation on Panoramic Radiographs
    SHENG Chen, WANG Lin, HUANG Zhenhuan, WANG Tian, GUO Yalin, HOU Wenjie, XU Laiqing, WANG Jiazhu, YAN Xue
    2023, 36(1):  257-272.  DOI: 10.1007/s11424-022-2057-9
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    Panoramic radiographs can assist dentist to quickly evaluate patients’ overall oral health status. The accurate detection and localization of tooth tissue on panoramic radiographs is the first step to identify pathology, and also plays a key role in an automatic diagnosis system. However, the evaluation of panoramic radiographs depends on the clinical experience and knowledge of dentist, while the interpretation of panoramic radiographs might lead misdiagnosis. Therefore, it is of great significance to use artificial intelligence to segment teeth on panoramic radiographs. In this study, SWinUnet, the transformer-based Ushaped encoder-decoder architecture with skip-connections, is introduced to perform panoramic radiograph segmentation. To well evaluate the tooth segmentation performance of SWin-Unet, the PLAGH-BH dataset is introduced for the research purpose. The performance is evaluated by F1 score, mean intersection and Union (IoU) and Acc, Compared with U-Net, LinkNet and FPN baselines, SWin-Unet performs much better in PLAGH-BH tooth segmentation dataset. These results indicate that SWin-Unet is more feasible on panoramic radiograph segmentation, and is valuable for the potential clinical application.
    Boundary Control of Coupled Non-Constant Parameter Systems of Time Fractional PDEs with Different-Type Boundary Conditions
    CHEN Juan, ZHUANG Bo
    2023, 36(1):  273-293.  DOI: 10.1007/s11424-023-0204-6
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    This paper addresses a boundary state feedback control problem for a coupled system of time fractional partial differential equations (PDEs) with non-constant (space-dependent) coefficients and different-type boundary conditions (BCs). The BCs could be heterogeneous-type or mixed-type. Specifically, this coupled system has different BCs at the uncontrolled side for heterogeneous-type and the same BCs at the uncontrolled side for mixed-type. The main contribution is to extend PDE backstepping to the boundary control problem of time fractional PDEs with space-dependent parameters and different-type BCs. With the backstepping transformation and the fractional Lyapunov method, the Mittag-Leffler stability of the closed-loop system is obtained. A numerical scheme is proposed to simulate the fractional case when kernel equations have not an explicit solution.
    Business Environment and Resource Allocation Based on the Perspective of the National Value Chain
    ZOU Wei, LEI Hao
    2023, 36(1):  294-327.  DOI: 10.1007/s11424-023-2357-8
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    China is actively upgrading its industrial structure through industries transferring between developed and undeveloped areas; however, the overall level of the national value chain is still not high, and the asymmetric competition pattern between the upstream and the downstream has not been broken. Therefore, this paper establishes a competitive equilibrium model for the production of manufacturing enterprises, with factor price distortion, under the condition of constant returns to scale. The authors derive the relative distortion coefficients of each factor price, calculate the misallocation indices of capital and labor, and construct an industry resource misallocation measure. Furthermore, this paper applies the regional value-added decomposition model to calculate the national value chain index and matches the market index of the China Market Index Database with the Chinese Industrial Enterprises Database and the Inter-Regional Input-Output Tables through quantitative analysis. From the perspective of the national value chain, the authors study the improvement effect and mechanism of the business environment on the resource allocation in industry. The study shows that industry resource allocation will be improved by 17.89% if the business environment level is improved by one standard deviation. This effect is most prevalent in the eastern and central regions, not so much in the west; the effect of downstream industries in the national value chain is higher than that of upstream industries; the improvement effect on capital allocation is higher in downstream industries than in the upstream industries; and the improvement effect on labor misallocation is basically the same in both the upstream and the downstream. Compared with labor intensive industries, capital intensive industries are more influenced by the national value chain, while the effect of upstream industries is weaker. At the same time, it is well documented that participation in the global value chain can improve the efficiency of regional resource allocation, and the construction of high-tech zones can improve resource allocation for both upstream and downstream industries. Based on the results of study, the authors propose suggestions for optimizing business environments, suiting the national value chain construction, and improving resource allocation in the future.
    Vulnerable European Call Option Pricing Based on Uncertain Fractional Differential Equation
    LEI Ziqi, ZHOU Qing, WU Weixing, WANG Zengwu
    2023, 36(1):  328-359.  DOI: 10.1007/s11424-023-1140-1
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    This paper presents two new versions of uncertain market models for valuing vulnerable European call option. The dynamics of underlying asset, counterparty asset, and corporate liability are formulated on the basis of uncertain differential equations and uncertain fractional differential equations of Caputo type, respectively, and the solution to an uncertain fractional differential equation of Caputo type is presented by employing the Mittag-Leffler function and α-path. Then, the pricing formulas of vulnerable European call option based on the proposed models are investigated as well as some algorithms. Some numerical experiments are performed to verify the effectiveness of the results.
    The Coupling Relationships and Influence Mechanisms of Green Credit and Energy-Environment-Economy Under China’s Goal of Carbon Neutrality
    CHAI Jian, WANG Yabo, HU Yi, ZHANG Xuejun, ZHANG Xiaokong
    2023, 36(1):  360-374.  DOI: 10.1007/s11424-023-1253-6
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    Under the goal of carbon neutrality, it is critical for China to give full play to the role of green credit, and promote the coordinated development of energy-environment-economy (3E) system. Based on the data of China from 2000 to 2020, the authors build the environmental pollution index, energy transformation index and high-quality economic development index. By using Bayesian network model (BN), the authors investigate the coupling relationships and influence mechanisms of green credit and 3E system. The results show that the main cause of environmental pollution is the annual increase of carbon dioxide emissions. Green credit can reduce carbon emissions to a certain extent, and alleviate environmental pollution through energy structure, technological progress and per capita GDP. Clean energy utilization and per capita GDP growth also help to control environmental pollution. Green credit can stimulate technological progress and accelerate energy transformation together with technological progress. Clean energy utilization can facilitate the upgrading of industrial structure, industrial structure upgrading and green credit can restrict the level of opening up. Technological progress promotes per capita GDP growth. Per capita GDP growth can reduce energy intensity and improve urbanization and per capita energy consumption.
    The General Compromise Value for Cooperative Games With Transferable Utility
    SUN Panfei, HOU Dongshuang, SUN Hao
    2023, 36(1):  375-392.  DOI: 10.1007/s11424-023-1159-3
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    The authors introduce the general compromise value for cooperative games with transferable utility. With respect to a set of potential payoffs of which the maximal and minimal potential payoff vectors are regarded as the upper and lower bounds for players, the unique pre-imputation lying on the straight line segment with these two vectors as the extreme points is defined as the general compromise value. Potential-consistency and maximal proportional property are introduced to characterize the general compromise value.
    Distance-Based Regression Analysis for Measuring Associations
    SHI Yuke, ZHANG Wei, LIU Aiyi, LI Qizhai
    2023, 36(1):  393-411.  DOI: 10.1007/s11424-023-2070-7
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    Distance-based regression model, as a nonparametric multivariate method, has been widely used to detect the association between variations in a distance or dissimilarity matrix for outcomes and predictor variables of interest in genetic association studies, genomic analyses, and many other research areas. Based on it, a pseudo-$F$ statistic which partitions the variation in distance matrices is often constructed to achieve the aim. To the best of our knowledge, the statistical properties of the pseudo-$F$ statistic has not yet been well established in the literature. To fill this gap, the authors study the asymptotic null distribution of the pseudo-$F$ statistic and show that it is asymptotically equivalent to a mixture of chi-squared random variables. Given that the pseudo-$F$ test statistic has unsatisfactory power when the correlations of the response variables are large, the authors propose a square-root $F$-type test statistic which replaces the similarity matrix with its square root. The asymptotic null distribution of the new test statistic and power of both tests are also investigated. Simulation studies are conducted to validate the asymptotic distributions of the tests and demonstrate that the proposed test has more robust power than the pseudo-$F$ test. Both test statistics are exemplified with a gene expression dataset for a prostate cancer pathway.
    Least Squares Model Averaging for Two Non-Nested Linear Models
    GAO Yan, XIE Tianfa, ZOU Guohua
    2023, 36(1):  412-432.  DOI: 10.1007/s11424-023-1172-6
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    This paper studies the least squares model averaging methods for two non-nested linear models. It is proved that the Mallows model averaging weight of the true model is root-$n$ consistent. Then the authors develop a penalized Mallows criterion which ensures that the weight of the true model equals 1 with probability tending to 1 and thus the averaging estimator is asymptotically normal. If neither candidate model is true, the penalized Mallows averaging estimator is asymptotically optimal. Simulation results show the selection consistency of the penalized Mallows method and the superiority of the model averaging approach compared with the model selection estimation.
    An Adjusted Gray Map Technique for Constructing Large Four-Level Uniform Designs
    2023, 36(1):  433-456.  DOI: 10.1007/s11424-023-1144-x
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    A uniform experimental design (UED) is an extremely used powerful and efficient methodology for designing experiments with high-dimensional inputs, limited resources and unknown underlying models. A UED enjoys the following two significant advantages: (i) It is a robust design, since it does not require to specify a model before experimenters conduct their experiments; and (ii) it provides uniformly scatter design points in the experimental domain, thus it gives a good representation of this domain with fewer experimental trials (runs). Many real-life experiments involve hundreds or thousands of active factors and thus large UEDs are needed. Constructing large UEDs using the existing techniques is an NP-hard problem, an extremely time-consuming heuristic search process and a satisfactory result is not guaranteed. This paper presents a new effective and easy technique, adjusted Gray map technique (AGMT), for constructing (nearly) UEDs with large numbers of four-level factors and runs by converting designs with $s$ two-level factors and $n$ runs to (nearly) UEDs with $2^{t-1}s$ four-level factors and $2^tn$ runs for any $t\geq0$ using two simple transformation functions. Theoretical justifications for the uniformity of the resulting four-level designs are given, which provide some necessary and/or sufficient conditions for obtaining (nearly) uniform four-level designs. The results show that the AGMT is much easier and better than the existing widely used techniques and it can be effectively used to simply generate new recommended large (nearly) UEDs with four-level factors.