2021年, 第37卷, 第2期　刊出日期：2021-04-15

  

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  Some Kinds of Bargaining Equilibria of Multi-objective Games

  Chun WANG, Hui YANG

  应用数学学报(英文版). 2021, 37(2): 201-213. https://doi.org/10.1007/s10255-021-1003-2

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  To solve the choice of multi-objective game’s equilibria, we construct general bargaining games called self-bargaining games, and define their individual welfare functions with three appropriate axioms. According to the individual welfare functions, we transform the multi-objective game into a single-objective game and define its bargaining equilibrium, which is a Nash equilibrium of the single-objective game. And then, based on certain continuity and concavity of the multi-objective game’s payoff function, we proof the bargaining equilibrium still exists and is also a weakly Pareto-Nash equilibrium. Moreover, we analyze several special bargaining equilibria, and compare them in a few examples.
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  Sphericity and Identity Test for High-dimensional Covariance Matrix Using Random Matrix Theory

  Shou-cheng YUAN, Jie ZHOU, Jian-xin PAN, Jie-qiong SHEN

  应用数学学报(英文版). 2021, 37(2): 214-231. https://doi.org/10.1007/s10255-021-1004-1

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  This paper addresses the issue of testing sphericity and identity of high-dimensional population covariance matrix when the data dimension exceeds the sample size. The central limit theorem of the first four moments of eigenvalues of sample covariance matrix is derived using random matrix theory for generally distributed populations. Further, some desirable asymptotic properties of the proposed test statistics are provided under the null hypothesis as data dimension and sample size both tend to infinity. Simulations show that the proposed tests have a greater power than existing methods for the spiked covariance model.
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  Degree Conditions for k-Hamiltonian [a, b]-factors

  Jie WU, Si-zhong ZHOU

  应用数学学报(英文版). 2021, 37(2): 232-239. https://doi.org/10.1007/s10255-021-1005-0

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  Let a, b, k be nonnegative integers with 2 ≤ a < b. A graph G is called a k-Hamiltonian graph if G - U contains a Hamiltonian cycle for any subset U ⊆ V (G) with |U| = k. An [a, b]-factor F of G is called a Hamiltonian [a, b]-factor if F contains a Hamiltonian cycle. If G - U admits a Hamiltonian [a, b]-factor for any subset U ⊆ V (G) with |U| = k, then we say that G has a k-Hamiltonian [a, b]-factor. Suppose that G is a k-Hamiltonian graph of order n with n ≥ ((a+b-4)(2a+b+k-6))/(b-2) + k and δ(G) ≥ a + k. In this paper, it is proved that G admits a k-Hamiltonian [a, b]-factor if max{d_G(x),d_G(y)} ≥ ((a-2)n+(b-2)k)/(a+b-4) + 2 for each pair of nonadjacent vertices x and y in G.
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  A Construction of Variable Strength Covering Arrays

  Ling JIANG, Ce SHI

  应用数学学报(英文版). 2021, 37(2): 240-250. https://doi.org/10.1007/s10255-021-1006-z

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  Covering arrays (CA) of strength t, mixed level or fixed level, have been applied to software testing to aim for a minimum coverage of all t-way interactions among components. The size of CA increases with the increase of strength interaction t, which increase the cost of software testing. However, it is quite often that some certain components have strong interactions, while others may have fewer or none. Hence, a better way to test software system is to identify the subsets of components which are involved in stronger interactions and apply high strength interaction testing only on these subsets. For this, in 2003, the notion of variable strength covering arrays was proposed by Cohen et al. to satisfy the need to vary the size of t in an individual test suite. In this paper, an effective deterministic construction of variable strength covering arrays is presented. Based on the construction, some series of variable strength covering arrays are then obtained, which are all optimal in the sense of their sizes. In the procedure, two classes of new difference matrices of strength 3 are also mentioned.
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  Optimality Conditions for Minimax Optimization Problems with an Infinite Number of Constraints and Related Applications

  Li-nan ZHONG, Yuan-feng JIN

  应用数学学报(英文版). 2021, 37(2): 251-263. https://doi.org/10.1007/s10255-021-1019-7

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  This paper is concerned with the study of optimality conditions for minimax optimization problems with an infinite number of constraints, denoted by (MMOP). More precisely, we first establish necessary conditions for optimal solutions to the problem (MMOP) by means of employing some advanced tools of variational analysis and generalized differentiation. Then, sufficient conditions for the existence of such solutions to the problem (MMOP) are investigated with the help of generalized convexity functions defined in terms of the limiting subdifferential of locally Lipschitz functions. Finally, some of the obtained results are applied to formulating optimality conditions for weakly efficient solutions to a related multiobjective optimization problem with an infinite number of constraints, and a necessary optimality condition for a quasi ε-solution to problem (MMOP).
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  Quantile Regression for Thinning-based INAR(1) Models of Time Series of Counts

  Dan-shu SHENG, De-hui WANG, Kai YANG, Zi-ang WU

  应用数学学报(英文版). 2021, 37(2): 264-277. https://doi.org/10.1007/s10255-021-1014-z

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  In this paper, we develop the quantile regression (QR) estimation for the first-order integer-valued autoregressive (INAR(1)) models by defining the smoothing INAR(1) process. Jittering method is used to derive the QR estimators for the autoregressive coefficient and the quantile of innovations. The consistency and asymptotic normality of the proposed estimators are established. The performances of the proposed estimation procedures are evaluated by Monte Carlo simulations. The results show that the proposed procedures perform well for simulations and a real data application.
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  Conflict-free Connection Number and Independence Number of a Graph

  Jing WANG, Meng JI

  应用数学学报(英文版). 2021, 37(2): 278-286. https://doi.org/10.1007/s10255-021-1013-0

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  An edge-colored graph G is conflict-free connected if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is defined as the minimum number of colors that are required in order to make G conflict-free connected. In this paper, we investigate the relation between the conflict-free connection number and the independence number of a graph. We firstly show that cfc(G) ≤ α(G) for any connected graph G, and give an example to show that the bound is sharp. With this result, we prove that if T is a tree with ∆(T) ≥ (α(T)+2)/2, then cfc(T) = ∆(T).
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  Ambarzumyan Theorems for Dirac Operators

  Chuan-fu YANG, Feng WANG, Zhen-you HUANG

  应用数学学报(英文版). 2021, 37(2): 287-298. https://doi.org/10.1007/s10255-021-1007-y

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  We consider the inverse eigenvalue problems for stationary Dirac systems with differentiable selfadjoint matrix potential. The theorem of Ambarzumyan for a Sturm-Liouville problem is extended to Dirac operators, which are subject to separation boundary conditions or periodic (semi-periodic) boundary conditions, and some analogs of Ambarzumyan’s theorem are obtained. The proof is based on the existence and extremal properties of the smallest eigenvalue of corresponding vectorial Sturm-Liouville operators, which are the second power of Dirac operators.
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  The Consistency for the Estimators of Semiparametric Regression Model with Dependent Samples

  Yi WU, Xue-jun WANG, Ling CHEN, Kun JIANG

  应用数学学报(英文版). 2021, 37(2): 299-318. https://doi.org/10.1007/s10255-021-1008-x

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  For the semiparametric regression model: Y^(j)(x_in, t_in) = t_inβ+g(x_in)+e^(j)(x_in), 1 ≤ j ≤ k, 1 ≤ i ≤ n, where t_in ∈ R and x_in ∈ R^p are known to be nonrandom, g is an unknown continuous function on a compact set A in R^p, e^j(x_in) are m-extended negatively dependent random errors with mean zero, Y^(j)(x_in, t_in) represent the j-th response variables which are observable at points x_in, t_in. In this paper, we study the strong consistency, complete consistency and r-th (r > 1) mean consistency for the estimators β_k,n and g_k,n of β and g, respectively. The results obtained in this paper markedly improve and extend the corresponding ones for independent random variables, negatively associated random variables and other mixing random variables. Moreover, we carry out a numerical simulation for our main results.
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  Infinite Horizon Forward-Backward Doubly Stochastic Differential Equations and Related SPDEs

  Qing-feng ZHU, Liang-quan ZHANG, Yu-feng SHI

  应用数学学报(英文版). 2021, 37(2): 319-336. https://doi.org/10.1007/s10255-021-1009-9

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  A type of infinite horizon forward-backward doubly stochastic differential equations is studied. Under some monotonicity assumptions, the existence and uniqueness results for measurable solutions are established by means of homotopy method. A probabilistic interpretation for solutions to a class of stochastic partial differential equations combined with algebra equations is given. A significant feature of this result is that the forward component of the FBDSDEs is coupled with the backward variable.
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  Continuous Forcing Spectra of Even Polygonal Chains

  He-ping ZHANG, Xiao-yan JIANG

  应用数学学报(英文版). 2021, 37(2): 337-347. https://doi.org/10.1007/s10255-021-1010-3

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  Let G be a graph that admits a perfect matching M. A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G. The cardinality of a forcing set of M with the smallest size is called the forcing number of M, denoted by f(G, M). The forcing spectrum of G is defined as: Spec(G) = {f(G, M)|M is a perfect matching of G}. In this paper, by applying the Z-transformation graph (resonance graph) we show that for any polyomino with perfect matchings and any even polygonal chain, their forcing spectra are integral intervals. Further we obtain some sharp bounds on maximum and minimum forcing numbers of hexagonal chains with given number of kinks. Forcing spectra of two extremal chains are determined.
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  Partial Regularity of Suitable Weak Solutions to the System of the Incompressible Shear-thinning Flow

  Ya-zhou CHEN, Hai-liang LI, Xiao-ding SHI

  应用数学学报(英文版). 2021, 37(2): 348-363. https://doi.org/10.1007/s10255-021-1011-2

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  This paper is devoted to the partial regularity of suitable weak solutions to the system of the incompressible shear-thinning flow in a bounded domain Ω ⊂ Rⁿ, n ≥ 2. It is proved that there exists a suitable weak solution of the shear-thinning fluid in the n-D smooth bounded domain (for n ≥ 2). For 3D model, it is proved that the singular points are concentrated on a closed set whose 1 dimensional Hausdorff measure is zero.
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  A Direct Method of Moving Planes to Fractional Power SubLaplace Equations on the Heisenberg Group

  Xin-jing WANG, Peng-cheng NIU

  应用数学学报(英文版). 2021, 37(2): 364-379. https://doi.org/10.1007/s10255-021-1016-x

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  We give the direct method of moving planes for solutions to the conformally invariant fractional power subLaplace equation on the Heisenberg group. The method is based on four maximum principles derived here. Then symmetry and nonexistence of positive cylindrical solutions are proved.
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  L^p Decay Rate for a Nonlinear Convection Diffusion Reaction Equation in Rⁿ

  Guo-wei LIU, Hong-mei XU, Yuan-mei XIA

  应用数学学报(英文版). 2021, 37(2): 380-392. https://doi.org/10.1007/s10255-021-1018-8

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  This paper studies the asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in Rn. Firstly, the global existence and uniqueness of classical solutions for small initial data are established. Then, we obtain the L^p, 2 ≤ p ≤ +∞ decay rate of solutions. The approach is based on detailed analysis of the Green function of the linearized equation with the technique of long wave-short wave decomposition and the Fourier analysis.
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  Exact Confidence Limits for the Parameter of an Exponential Distribution in the Accelerated Life Tests under Type-I Censoring

  De-qiang ZHENG, Xiang-zhong FANG

  应用数学学报(英文版). 2021, 37(2): 393-408. https://doi.org/10.1007/s10255-021-1021-0

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  Life data frequently arise in many reliability studies, such as accelerated life tests studies. This paper considers the part of life data where failure and censoring observations may exist. To develop statistical methods and theory for the analysis of these data, a new approach was proposed to obtain the exact lower and upper confidence limits for the mean life of the exponential distribution with Type-I censoring data. It is assumed that the acceleration factor is a random variable, and that the distribution of the acceleration factor is known from some empirical information or the meta analysis. A method for constructing the lower and upper confidence limits for the parameter based on an ordering relation among the sample space was proposed. Simulation studies and analyses of two examples suggest that the proposed method performed well.
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  The Number of Circles of a Maximum State of a Plane Graph with Applications

  Xian-an JIN, Jun GE, Xiao-Sheng CHENG, Yu-qing LIN

  应用数学学报(英文版). 2021, 37(2): 409-420. https://doi.org/10.1007/s10255-021-1020-1

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  Motivated by the connection with the genus of the corresponding link and its application on DNA polyhedral links, in this paper, we introduce a parameter s_max(G), which is the maximum number of circles of states of the link diagram D(G) corresponding to a plane (positive) graph G. We show that s_max(G) does not depend on the embedding of G and if G is a 4-edge-connected plane graph then s_max(G) is equal to the number of faces of G, which cover the results of S. Y. Liu and H. P. Zhang as special cases.
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  A Worst-Case Risk Measure by G-VaR

  Zi-ting PEI, Xi-shun WANG, Yu-hong XU, Xing-ye YUE

  应用数学学报(英文版). 2021, 37(2): 421-440. https://doi.org/10.1007/s10255-021-1002-3

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  G-VaR, which is a type of worst-case value-at-risk (VaR), is defined as measuring risk incorporating model uncertainty. Compared with most extant notions of worst-case VaR, G-VaR can be computed using an explicit formula, and can be applied to large portfolios of several hundred dimensions with low computational cost. We also apply G-VaR to robust portfolio optimization, thereby providing a tractable means to facilitate optimal allocations under the condition of market ambiguity.