15 March 2023, Volume 66 Issue 2
    

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  • Schur-type Theorems about Angles and Heights
    Xiao Le SU, Yi TAN, Yu Sheng WANG
    Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 199-208.
    DOI:10.12386/A20210060
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    In differential geometry, there is a classical result, named Schur's Theorem, which is about the comparison of chords of two curves in $\mathbb E^3$. Inspired by it, this paper presents Schur-type theorems about the comparison of chord tangent angles of two curves, and the comparison of heights of two curves relative to their chords.
  • Closed Strongly Irreducible Operators on Banach Spaces
    Li Qiong LIN, Jia Hua QUE, Yun Nan ZHANG
    Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 209-222.
    DOI:10.12386/A20210126
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    This paper firstly gives the definition of closed strongly irreducible operators on Banach spaces and gives an example of unbounded strongly irreducible operator. It obtains some properties of closed strongly irreducible operators. In particular, it obtains some equivalent descriptions of closed strongly irreducible operators. It also demonstrates some sufficient conditions for the strongly irreducibility of closed operators which have the forms of upper triangular operator matrices.
  • A Simplified Dynamical Systems Method for Solving Nonlinear Equation
    Jing Yue HUANG, Xing Jun LUO, Rong ZHANG
    Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 223-238.
    DOI:10.12386/A20210092
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    A simplified dynamical systems method for solving the nonlinear equation $F(u)=f$ is studied in this paper. Under certain conditions of the operator $F$ and the exact solution $y$, the error estimate of the solution of the dynamical systems equation is given, and the discrepancy principle of the posterior selection of regularized parameter is proposed to ensure the optimal rate of convergence of the solution of the dynamical systems equation. Compared with the traditional dynamical systems method, the simplified dynamical systems method reduces the computation amount of derivatives.
  • Hypothesis Testing in Partial Functional Linear Spatial Autoregressive Model
    Gao Sheng LIU, Yang BAI, Ping YU
    Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 239-252.
    DOI:10.12386/A20210176
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    The paper proposes a hypothesis testing of spatial autoregressive and parametric component in partial functional linear spatial autoregressive model. The functional principle component analysis is employed to approximate the slope function. And generalized method of moments (GMM) is used to estimate parameters. Basis on consistent estimators, we construct a test statistic of the residual sums of squares under null and alternative hypothesis. In addition, we establish the asymptotic properties of the proposed test. Simulation studies show the proposed test has good size and power with finite sample size. Finally, a real data analysis of growth data is conducted to investigate the significance of spatial autoregressive and parametric coefficients with partial functional linear spatial autoregressive model.
  • On Two Dimensional Riemann Problem with Three Constant States for Chaplygin Gas
    Jie CHENG, Fang Qi CHEN, Ze Jun WANG
    Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 253-262.
    DOI:10.12386/B20210196
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    In this paper, the two dimensional Riemann problem of the Euler system for Chaplygin gas with three pieces of constant states is studied. The three states are divided by the $x$-axis and the positive semi-axis of the $y$-axis. Based on the assumption that each jump in initial data outside of the origin projects exactly one planar wave of shocks, centered rarefaction waves, or slip planes, and by using of the method of generalized characteristic analysis, we give the structures of the solution in detail. In fact, we divide the analysis into ten cases and among them, only four subcases are reasonable.
  • Distributed Statistical Inference for Linear Models with Multi-source Massive Heterogeneous Data
    Xin YANG, Mi Xia WU
    Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 263-276.
    DOI:10.12386/A20210120
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    We focus on the distributed statistical inference for linear models with multi-source massive heterogeneous data. First, a communication-efficient distributed aggregation method is proposed to estimate the unknown parameter vector, and the derived estimator is proved to be best linear unbiased and asymptotically normal under some regularity conditions. Then, a distributed test method is proposed to test the heterogeneity among a large number of data sources. Finally, the simulations are conducted to illustrate the effectiveness of the proposed method.
  • Spacelike Hypersurfaces of Constant rth F-mean Curvature with Light-like Boundary
    Yuan Zheng ZHANG
    Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 277-292.
    DOI:10.12386/A20210147
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    Let $\overline{W}_{F(t)}$, corresponding to a rotation invariant function $F(t(\nu))$ with a convexity condition on the upper hyperboloid $\mathbb{H}_+^n$, be a compact space-like Wulff shape bounded by a light-like $(n-1)$-round sphere. By applying perturbation metric and some integral formulae, we show that the only spacelike hypersurface with constant $r$th $F$-mean curvature in $\mathbb{L}^{n+1}$, which is tangent to $\overline{W}_{F(t)}$ on the boundary, is the Wulff shape.
  • Softplus Beta Negative Binomial Integer-valued GARCH Model
    Le Le QI, Fu Kang ZHU
    Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 293-308.
    DOI:10.12386/A20210063
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    INGARCH models are often constructed based on Poisson distribution, negative binomial distribution and so on. Beta negative binomial (BNB) distribution is a flexible distribution. Recently, the related BNB-INGARCH model was proposed, whose conditional mean is linear, the parameters are restricted to non-negative and negative autocorrelation cannot be modeled. In this paper, we first propose the log-linear BNB-INGARCH model to solve the above problems, but the simple form of linear mean and ARMA-like structures are lost. So we further construct softplus BNB-INGARCH$(p,q)$ model by using the softplus function, which is the main research object. When $p$ and $q$ are equal to 1, the stationarity and ergodicity of the model are proved and the conditions for the existence of the second moment are given. In addition, the strong consistency and the asymptotic normality of the maximum likehood estimator are shown. Finally, the analysis of real-data examples show the usefulness of the proposed model.
  • Gröbner-Shirshov Bases for the Cyclotomic Hecke Algebra of Type A
    Munayim DILXAT, Abdukadir OBUL, Dong LIU
    Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 309-316.
    DOI:10.12386/A20210074
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    In this paper, we discuss the Gröbner--Shirshov bases and a linear bases of the cyclotomic Hecke algebra of type $A$. First, by computing the compositions, we construct a Gröbner--Shirshov bases of the cyclotomic Hecke algebra of type $A$. Then using this Gröbner--Shirshov bases and the composition-diamond lemma we give a linear bases of the cyclotomic Hecke algebra of type $A$.
  • An Inexact Newton-Lanczos Method for Solving a System of Nonlinear Equations
    Chao GU, Jue Yu WANG, De Tong ZHU
    Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 317-338.
    DOI:10.12386/A20210026
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    We present an inexact Newton-Krylov subspace method with incomplete line search technique for solving symmetric nonlinear equations, in which the Krylov subspace method uses the Lanczos-type decomposition technique. The iterative direction is obtained by approximately solving the Newton’s equations of the nonlinear equations using the Lanczos method. The global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions. Finally, the numerical results show the effectiveness of the proposed algorithm.
  • Linear Arboricity of 1-planar Graphs
    Dan Jun HUANG, Yan JIANG
    Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 339-352.
    DOI:10.12386/A20210093
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    An edge-partition of a graph $G$ is a decomposition of $G$ into subgraphs $G_1, G_2,\ldots,G_m$ such that $E(G)=E(G_1)\cup\cdots\cup E(G_m)$ and $E(G_i)\cap E(G_j)=\emptyset$ for any $i\neq j$. A linear forest is forest in which each connected component is a path. The linear arboricity ${\rm la}(G)$ is the least integer $m$ such that $G$ can be edge-partitioned into $m$ linear forests. In this paper, we use the discharging method to study the linear arboricity ${\rm la}(G)$ of 1-planar graphs, and prove that ${\rm la}(G)=\lceil\frac{\Delta(G)}{2}\rceil$ for each 1-planar graph $G$ with $\Delta(G)\ge25$, where $\Delta(G)$ is the maximum degree of $G$.
  • Geometric Construction of Pythagorean Hodograph C-curves
    Yu Jun LI, Lin Cong FANG
    Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 353-362.
    DOI:10.12386/A20210145
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    We study the geometric characteristics of C-Bézier curves that possess the Pythagorean Hodograph (PH) property. Based on the algebraic necessary and sufficient conditions for PH C-curves, we prove that a C-Bézier curve is a PH C-curve if and only if the interior angles of its control polygon are equal, and the second leg length of the control polygon is the geometric mean of the first and the last ones. Our main idea is to represent a planar parametric curve in complex form. We claim that the geometric characteristics of PH C-curves are quite similar to polynomial PH curves, which can be used to identify PH C-curves and their constructions. As an application, we give some examples of $G^1$ Hermite interpolation using PH C-curves. We point out that there are no more than two PH C-curves for any given $G^1$ Hermite conditions.
  • Analysis of M/G/1 Queue with Bi-level Threshold (m,N)-Policy and Uninterrupted Single Vacation
    Wen Ping GAO, Ying Hui TANG
    Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 363-388.
    DOI:10.12386/A20200026
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    This paper studies the $M/G/1$ queueing system with startup time, bi-level threshold $(m,N)$-policy and single server vacation without interruption. In this system, when the server is transferred on vacation, the server starts the system immediately if the number of waiting customers is no less than a given positive integer threshold $m\ (m\ge 1)$, and when the system startup is complete, the server begins service immediately if the number of waiting customers is no less than another given positive integer threshold $N\ (N\ge m)$. Assume that the server's vacation time and the startup time of the system follow general distributions, both the transient queue-length distribution and the steady-state queue-length distribution of the system are discussed by using the renewal process theory, the total probability decomposition technique and Laplace transform tool. The expressions of the Laplace transformation of the transient queue-length distribution with respect to time $t$ are obtained. Furthermore, the recursive expressions of the steady-state queue-length distribution are derived by a direct calculation. Meanwhile, the stochastic decomposition structure of the steady-state queue-length and the explicit expression of the additional queue-length distribution are presented. Finally, the explicit expression of the long-run expected cost per unit time is derived under a given cost model. And the numerical example is given to determine the optimal control policy $({{m}^{*}},{{N}^{*}})$ for minimizing the long-run expected cost per unit time.
  • Distribution Properties and Applications of Consecutive Quadratic Residues
    Xiao WANG, Ai Hua LI
    Acta Mathematica Sinica, Chinese Series. 2023, 66(2): 389-398.
    DOI:10.12386/A20210160
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    Consider any prime number $p$. In this paper, we use analytic methods, properties of Legendre's symbol modulo $p$, and estimation for character sums to study distribution properties of triples of consecutive quadratic residues (named 3-CQR) and consecutive quadratic non-residues (3-CQN) modulo $p$. We provide exact formulas for the numbers $S_1(p)$ and $S_2(p)$ of 3-CQRs and 3-CQNs when $p\equiv 3$ or $7\pmod{8}$. Asymptotic formulas are given for $p\equiv 1$ or $5\pmod{8}$. Similarly, triples of quadratic residues with equal distance 2 are investigated and corresponding enumeration formulas are given. As an application, we further apply 3-CQRs to construct magic squares of squares of full degree over $\mathbb F_p$.
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