中国科学院数学与系统科学研究院期刊网

2020年, 第36卷, 第4期　刊出日期：2020-10-15
  

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  • Construction of Normal Bimagic Squares of Order 2u
    Wen LI, Feng-chu PAN, Guang-zhou CHEN
    应用数学学报(英文版). 2020, 36(4): 771-789. https://doi.org/10.1007/s10255-020-0977-5
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    An n×n matrix A consisting of nonnegative integers is a general magic square of order n if the sum of elements in each row, column, and main diagonal is the same. A general magic square A of order n is called a magic square, denoted by MS(n), if the entries of A are distinct. A magic square A of order n is normal if the entries of A are n2 consecutive integers. Let A*d denote the matrix obtained by raising each element of A to the d-th power. The matrix A is a d-multimagic square, denoted by MS(n, d), if A*e is an MS(n) for 1 ≤ e ≤ d. In this paper we investigate the existence of normal bimagic squares of order 2u and prove that there exists a normal bimagic square of order 2u, where u and 6 are coprime and u ≥ 5.
  • Quantile Regression under Local Misspecification
    Xiao-gang DUAN, Qi-hua WANG
    应用数学学报(英文版). 2020, 36(4): 790-802. https://doi.org/10.1007/s10255-020-0973-9
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    The frequentist model averaging (FMA) and the focus information criterion (FIC) under a local framework have been extensively studied in the likelihood and regression setting since the seminal work of Hjort and Claeskens in 2003. One inconvenience, however, of the existing works is that they usually require the involved criterion function to be twice differentiable which thus prevents a direct application to the case of quantile regression (QR). This as well as some other intrinsic merits of QR motivate us to study the FIC and FMA in a locally misspecified linear QR model. Specifically, we derive in this paper the explicit asymptotic risk expression for a general submodel-based QR estimator of a focus parameter. Then based on this asymptotic result, we develop the FIC and FMA in the current setting. Our theoretical development depends crucially on the convexity of the objective function, which makes possible to establish the asymptotics based on the existing convex stochastic process theory. Simulation studies are presented to illustrate the finite sample performance of the proposed method. The low birth weight data set is analyzed.
  • New Constructions of Orbit Codes Based on the Operations of Orbit Codes
    Shang-di CHEN, Jun-ying LIANG
    应用数学学报(英文版). 2020, 36(4): 803-815. https://doi.org/10.1007/s10255-020-0974-8
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    An orbit code is a special constant dimension subspace code, which is an orbit of a subgroup of a general linear group acting on the set of all subspaces in the given ambient space. This paper presents some methods of constructing new orbit codes from known orbit codes. Firstly, we introduce the sum operation, intersection operation and union operation of subspace codes, and then we give some methods to obtain new orbit codes from known orbit codes by fully applying the sub-orbits of permutation groups and the direct product operation of the groups. Finally, as a special application, partial spread codes are researched and a condition of orbit codes with constant distance is given.
  • Identifying Significant Effects in Unreplicated Regular Two-level Factorial Experiments
    Hui LI, Sheng-li ZHAO
    应用数学学报(英文版). 2020, 36(4): 816-824. https://doi.org/10.1007/s10255-020-0981-9
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    The paper gives a new method to identify significant effects in two-level factorial experiments, and compares the new method with the exiting methods using Monte Carlo simulation. The existing methods only perform well under the assumption of effect sparsity, but the new method performs well without effect sparsity.
  • The Skew-commutator Relations and Gröbner-Shirshov Bases of Quantum Group of Type C3
    Cheng-xiu QIANG, Abdukadir OBUL
    应用数学学报(英文版). 2020, 36(4): 825-835. https://doi.org/10.1007/s10255-020-0982-8
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    In this paper, we give a Gröbner-Shirshov basis of quantum group of type C3 by using the RingelHall algebra approach. For this, first we compute all skew-commutator relations between the isoclasses of indecomposable reprersentations of Ringel-Hall algebras of type C3 by using an “inductive” method. Precisely, we do not use the traditional way of computing the skew-commutative relations, that is first compute all Hall polynomials then compute the corresponding skew-commutator relations; contrarily, we compute the “easier” skew-commutator relations which corresponding to those exact sequences with middile term indecomposable or the split exact sequences first, then “inductive” others from these “easier” ones and this in turn gives Hall polynomials as a byproduct. Then we prove that the set of these relations is closed under composition. So they constitutes a minimal Gröbner-Shirshov basis of the positive part of quantum group of type C3. Dually, we get a Gröbner-Shirshov basis of the negative part of quantum group of type C3. And finally we give a Gröbner-Shirshov basis for the whole quantum group of type C3.
  • Blow up of Solutions for a Nonlinear Petrovsky Type Equation with Time-dependent Coefficients
    Xiao-xiao ZHENG, Ya-dong SHANG, Xiao-ming PENG
    应用数学学报(英文版). 2020, 36(4): 836-846. https://doi.org/10.1007/s10255-020-0984-6
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    In this paper, we study a nonlinear Petrovsky type equation with nonlinear weak damping, a superlinear source and time-dependent coefficients
    utt + △2u + k1(t)|ut|m-2ut = k2(t)|u|p-2u, x ∈ Ω, t > 0,
    where Ω is a bounded domain in Rn. Under certain conditions on k1(t), k2(t) and the initial-boundary data, the upper bound for blow-up time of the solution with negative initial energy function is given by means of an auxiliary functional and an energy estimate method if p > m. Also, a lower bound of blow-up time are obtained by using a Sobolev-type inequality and a first order differential inequality technique for n = 2, 3 and n > 4.
  • Some Exponential Inequalities for Negatively Orthant Dependent Random Variables
    Xue-jun WANG, Shu-he HU
    应用数学学报(英文版). 2020, 36(4): 847-856. https://doi.org/10.1007/s10255-020-0985-5
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    In the paper, we establish some exponential inequalities for non-identically distributed negatively orthant dependent (NOD, for short) random variables. In addition, we also establish some exponential inequalities for the partial sum and the maximal partial sum of identically distributed NOD random variables. As an application, the Kolmogorov strong law of large numbers for identically distributed NOD random variables is obtained. Our results partially generalize or improve some known results.
  • On the General Stability of a Viscoelastic Wave Equation with an Integral Condition
    FARIDA BELHANNACHE, SALIM A. MESSAOUDI
    应用数学学报(英文版). 2020, 36(4): 857-869. https://doi.org/10.1007/s10255-020-0979-3
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    We consider a nonlocal boundary value problem for a viscoelastic equation with a Bessel operator and a weighted integral condition and we prove a general decay result. We also give an example to show that our general result gives the optimal decay rate for ceratin polynomially decaying relaxation functions. This result improves some other results in the literature.
  • Existence of Mild Solutions for a Class of Fractional Non-autonomous Evolution Equations with Delay
    Bo ZHU, Bao-yan HAN, Wen-guang YU
    应用数学学报(英文版). 2020, 36(4): 870-878. https://doi.org/10.1007/s10255-020-0980-x
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    In this paper, we consider the existence and uniqueness of the mild solutions for a class of fractional non-autonomous evolution equations with delay and Caputo’s fractional derivatives. By using the measure of noncompactness, β-resolvent family, fixed point theorems and Banach contraction mapping principle, we improve and generalizes some related results on this topic. At last, we give an example to illustrate the application of the main results of this paper.
  • Rainbow k-connectivity of Random Bipartite Graphs
    Xiao-lin CHEN, Xue-liang LI, Hui-shu LIAN
    应用数学学报(英文版). 2020, 36(4): 879-890. https://doi.org/10.1007/s10255-020-0970-z
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    A path in an edge-colored graph G is called a rainbow path if no two edges of the path are colored the same color. The minimum number of colors required to color the edges of G such that every pair of vertices are connected by at least k internally vertex-disjoint rainbow paths is called the rainbow k-connectivity of the graph G, denoted by rck(G). For the random graph G(n, p), He and Liang got a sharp threshold function for the property rck(G(n, p)) ≤ d. For the random equi-bipartite graph G(n, n, p), Fujita et. al. got a sharp threshold function for the property rck(G(n, n, p)) ≤ 3. They also posed the following problem: For d ≥ 2, determine a sharp threshold function for the property rck(G) ≤ d, where G is another random graph model. This paper is to give a solution to their problem in the general random bipartite graph model G(m, n, p).
  • Optimality Conditions for Strictly Efficient Solutions in Set-valued Optimization
    Yi-hong XU
    应用数学学报(英文版). 2020, 36(4): 891-901. https://doi.org/10.1007/s10255-020-0971-y
    摘要 ( ) PDF全文 ( )   可视化   收藏
    A new kind of tangent derivative, M-derivative, for set-valued function is introduced with help of a modified Dubovitskij-Miljutin cone. Several generalized pseudoconvex set-valued functions are introduced. When both the objective function and constraint function are M-derivative, under the assumption of near conesubconvexlikeness, by applying properties of the set of strictly efficient points and a separation theorem for convex sets, Fritz John and Kuhn-Tucker necessary optimality conditions are obtained for a point pair to be a strictly efficient element of set-valued optimization problem. Under the assumption of generalized pseudoconvexity, a Kuhn-Tucker sufficient optimality condition is obtained for a point pair to be a strictly efficient element of set-valued optimization problem.
  • Nonparametric Quantile Inference for Cause-specific Residual Life Function Under Length-biased Sampling
    Fei-Peng ZHANG, Cai-Yun FAN, Yong ZHOU
    应用数学学报(英文版). 2020, 36(4): 902-916. https://doi.org/10.1007/s10255-020-0972-x
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    This paper considers a competing risks model for right-censored and length-biased survival data from prevalent sampling. We propose a nonparametric quantile inference procedure for cause-specific residual life distribution with competing risks data. We also derive the asymptotic properties of the proposed estimators of this quantile function. Simulation studies and the unemployment data demonstrate the practical utility of the methodology.
  • Projection-based Consistent Test for Linear Regression Model with Missing Response and Covariates
    Su-jin ZHENG, Si-yu GAO, Zhi-hua SUN
    应用数学学报(英文版). 2020, 36(4): 917-935. https://doi.org/10.1007/s10255-020-0976-6
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    In recent years, there has been a large amount of literature on missing data. Most of them focus on situations where there is only missingness in response or covariate.
    In this paper, we consider the adequacy check for the linear regression model with the response and covariates missing simultaneously.
    We apply model adjustment and inverse probability weighting methods to deal with the missingness of response and covariate, respectively. In order to avoid the curse of dimension, we propose an empirical process test with the linear indicator weighting function. The asymptotic properties of the proposed test under the null, local and global alternative hypothetical models are rigorously investigated. A consistent wild bootstrap method is developed to approximate the critical value. Finally, simulation studies and real data analysis are performed to show that the proposed method performed well.
  • Green's Function for Periodic Solutions in Alternately Advanced and Delayed Differential Systems
    Kuo-Shou CHIU
    应用数学学报(英文版). 2020, 36(4): 936-951. https://doi.org/10.1007/s10255-020-0975-7
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    In this paper we investigate the existence of the periodic solutions of a nonlinear differential equation with a general piecewise constant argument, in short DEPCAG, that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use the Green’s function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii’s fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results.
  • Regularity of Invariant Sets in Variable Internal Damped Wave Equations
    Gao-cheng YUE, Yu-xin LIANG, Jia-jia YANG
    应用数学学报(英文版). 2020, 36(4): 952-974. https://doi.org/10.1007/s10255-020-0978-4
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    In this paper we prove that every compact invariant subset A associated with the semigroup {Sn,k(t)}t≥0 generated by wave equations with variable damping, either in the interior or on the boundary of the domain Ω, where Ω⊂R3 is a smooth bounded domain, in H10 (Ω)×L2(Ω) is in fact bounded in D(B0)×H10(Ω). As an application of our results, we obtain the upper-semicontinuity for global attractor of the weakly damped semilinear wave equation in the norm of H1(Ω)×L2(Ω) when the interior variable damping converges to the boundary damping in the sense of distributions.
  • Solutions in the Discrete Fractional Forms of Two Differential Equations with Singular Points
    Okkes OZTURK
    应用数学学报(英文版). 2020, 36(4): 975-981. https://doi.org/10.1007/s10255-020-0954-z
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    In present note, we apply the Leibniz formula with the nabla operator in discrete fractional calculus (DFC) due to obtain the discrete fractional solutions of a class of associated Bessel equations (ABEs) and a class of associated Legendre equations (ALEs), respectively. Thus, we exhibit a new solution method for such second order linear ordinary differential equations with singular points.
  • Continuity of Hardy-Littlewood Maximal Function
    Di WU, Dun-yan YAN
    应用数学学报(英文版). 2020, 36(4): 982-990. https://doi.org/10.1007/s10235-020-0983-z
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    In the paper, we characterize a necessary and sufficient condition which ensures the continuities of the non-centered Hardy-Littlewood maximal function Mf and the centered Hardy-Littlewood maximal function Mcf on Rn. As two applications, we can easily deduce that Mcf and Mf are continuous if f is continuous, and Mf is continuous if f is of local bounded variation on R.
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