2019年, 第35卷, 第2期　刊出日期：2019-05-15

  

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  Parameter Estimation Based on Set-valued Signals:Theory and Application

  Ting WANG, Hang ZHANG, Yan-long ZHAO

  应用数学学报(英文版). 2019, 35(2): 255-263. https://doi.org/10.1007/s10255-019-0822-x

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  This paper summarizes the parameter estimation of systems with set-valued signals, which can be classified to three catalogs:one-time completed algorithms, iterative methods and recursive algorithms. For one-time completed algorithms, empirical measure method is one of the earliest methods to estimate parameters by using set-valued signals, which has been applied to the adaptive tracking of periodic target signals. The iterative methods seek numerical solutions of the maximum likelihood estimation, which have been applied to both complex diseases diagnosis and radar target recognition. The recursive algorithms are constructed via stochastic approximation and stochastic gradient methods, which have been applied to adaptive tracking of non-periodic signals.
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  Two-step Estimation for Longitudinal Data When the Working Correlation Matrix is a Linear Combination of Some Known Matrices

  Yu-ling LI, Wei GAO, Man-Lai TANG, Shu-rong ZHENG

  应用数学学报(英文版). 2019, 35(2): 264-273. https://doi.org/10.1007/s10255-019-0809-7

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  The generalized estimating equations(GEE) approach is perhaps one of the most widely used methods for longitudinal data analysis. While the GEE method guarantees the consistency of its estimators under working correlation structure misspecification, the corresponding efficiency can be severely affected. In this paper, we propose a new two-step estimation method in which the correlation matrix is assumed to be a linear combination of some known working matrices. Asymptotic properties of the new estimators are developed. Simulation studies are conducted to examine the performance of the proposed estimators. We illustrate the methodology with an epileptic data set.
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  Local Least Product Relative Error Estimation for Varying Coefficient Multiplicative Regression Model

  Da-hai HU

  应用数学学报(英文版). 2019, 35(2): 274-286. https://doi.org/10.1007/s10255-018-0794-2

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  In this article, we consider the varying coefficient multiplicative regression model, which is very useful to model the positive response. The criterion of least product relative error (LPRE) is extended to the varying coefficient multiplicative regression model by kernel smoothing techniques. Consistency and asymptotic normality of the proposed estimator are established. Some numerical simulations are carried out to assess the performance of the proposed estimator. As an illustration, the ethanol data is analyzed.
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  An Inverse Design Method for Non-uniform Flow Inlet with a Given Shock Wave

  Wen-you QIAO, An-yuan YU, Yu-hui WANG

  应用数学学报(英文版). 2019, 35(2): 287-304. https://doi.org/10.1007/s10255-019-0819-5

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  For the hypersonic inlet and fore-body integrated design, the non-uniform incoming flow generated by the fore-body will bring a relatively big challenge to the inward-turning inlet design. To make the inlet match the non-uniform incoming flow, this paper, based on previous studies, develops a cross-stream marching plus (CSMP) method, by which an aerodynamic surface used to generate a given shock shape can be acquired. The method can correct such solution points as may give rise to grid distortions or flow-field abnormity and overcome the shortcoming of the insufficient stability of previous methods. Numerical simulation results of the conical supersonic flowfield show that the error obtained from the proposed CSMP method drops with the reduction of the grid dimension and the marching step, being less than 0.05% for reducing the marching step to 10%; that with this method the maximum relative error of the pressure on the profile is less than 0.23%. In the design process of the inward-turning inlets that match the fuselage fore-body, it's found that in comparison with the results of the inviscid CFD results, the aerodynamic surface designed with the CSMP method can fully generate the given shock wave shape. Thus, the CSMP method provides a new direction for the inlet/fore-body integrated design.
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  Pricing Vulnerable Options with Correlated Credit Risk Under Jump-diffusion Processes When Corporate Liabilities Are Random

  Qing ZHOU, Jiao-jiao YANG, Wei-xing WU

  应用数学学报(英文版). 2019, 35(2): 305-318. https://doi.org/10.1007/s10255-019-0821-y

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  In this paper, we consider an improved model of pricing vulnerable options with credit risk. We assume that the vulnerable European options not only face default risk, but also face the rare shocks of the underlying assets and the counterparty assets. The dynamics of two correlated assets are modeled as a class of jump diffusion processes. Furthermore, we assume that the dynamic of the corporate liability is a geometric Brownian motion that is related to the underlying asset and the counterparty asset. Under this new framework, we give an explicit pricing formula of the vulnerable European options.
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  Extremal Graphs with Respect to Matching Energy for Random Six-membered Ring Spiro Chains

  Hua-mei CHEN, Yan LIU

  应用数学学报(英文版). 2019, 35(2): 319-326. https://doi.org/10.1007/s10255-019-0820-z

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  Gutman and Wagner (in the matching energy of a graph, Disc. Appl. Math., 2012) defined the matching energy of a graph and pointed out that its chemical applications go back to the 1970s. Now the research on matching energy mainly focuses on graphs with pendent vertices and only a few papers reported the progress on matching energy of graphs without pendent vertices. For a random six-membered ring spiro chain, the number of k-matchings and the matching energy are random variables. In this paper, we determine the extremal graphs with respect to the matching energy for random six-membered ring spiro chains which have no pendent vertices.
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  Modifid Interpolatory Projection Method for Weakly Singular Integral Equation Eigenvalue Problems

  Xin ZHANG, Yun-hui HE

  应用数学学报(英文版). 2019, 35(2): 327-339. https://doi.org/10.1007/s10255-019-0823-9

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  This paper deals with eigenvalue problems for linear Fredholm integral equations of the second kind with weakly singular kernels. A new discrete method is proposed for the approximation of eigenvalues. Compactness of the integral operator in L¹[0, 1] space is obtained. This method is based on the approximation of the integral operator by modified interpolatory projection. Different from traditional methods, norm convergence of operator approximation is proved theoretically. Further, convergence of eigenvalue approximation is obtained by analytical tools. Numerical examples are presented to illustrate the theoretical results and the efficiency of the method.
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  Homogenization of Linear Parabolic Equations with a Certain Resonant Matching Between Rapid Spatial and Temporal Oscillations in Periodically Perforated Domains

  Tatiana LOBKOVA

  应用数学学报(英文版). 2019, 35(2): 340-358. https://doi.org/10.1007/s10255-019-0810-1

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  In this article, we study homogenization of a parabolic linear problem governed by a coefficient matrix with rapid spatial and temporal oscillations in periodically perforated domains with homogeneous Neumann data on the boundary of the holes. We prove results adapted to the problem for characterization of multiscale limits for gradients and very weak multiscale convergence.
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  A Primal-dual Interior-point Algorithm for Symmetric Cone Convex Quadratic Programming Based on the Commutative Class Directions

  S. ASADI, H. MANSOURI, M. ZANGIABADI

  应用数学学报(英文版). 2019, 35(2): 359-373. https://doi.org/10.1007/s10255-018-0789-z

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  In this paper, we present a neighborhood following primal-dual interior-point algorithm for solving symmetric cone convex quadratic programming problems, where the objective function is a convex quadratic function and the feasible set is the intersection of an affine subspace and a symmetric cone attached to a Euclidean Jordan algebra. The algorithm is based on the[13] broad class of commutative search directions for cone of semidefinite matrices, extended by[18] to arbitrary symmetric cones. Despite the fact that the neighborhood is wider, which allows the iterates move towards optimality with longer steps, the complexity iteration bound remains as the same result of Schmieta and Alizadeh for symmetric cone optimization problems.
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  Standing Waves for Discrete Nonlinear Schrödinger Equations with Nonperiodic Bounded Potentials

  Tie-shan HE, Meng ZHANG, Kai-hao LIANG, Peng-fei GUO

  应用数学学报(英文版). 2019, 35(2): 374-385. https://doi.org/10.1007/s10255-018-0787-1

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  In this paper, we investigate standing waves in discrete nonlinear Schrödinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we prove the existence and infinitely many sign-changing solutions of the equation. The results on the exponential decay of standing waves are also provided.
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  The Well-Posedness of Solution to Semilinear Pseudo-parabolic Equation

  Wei-ke WANG, Yu-tong WANG

  应用数学学报(英文版). 2019, 35(2): 386-400. https://doi.org/10.1007/s10255-019-0817-7

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  In this paper, we use the Green's function method to get the pointwise convergence rate of the semilinear pseudo-parabolic equations. By using this precise pointwise structure and introducing negative index Sobolev space condition on the initial data, we release the critical index of the nonlinearity for blowing up. Our result shows that the global existence does not only depend on the nonlinearity but also the initial condition.
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  Testing Serial Correlation in Partially Linear Additive Models

  Jin YANG, Chuan-hua WEI

  应用数学学报(英文版). 2019, 35(2): 401-411. https://doi.org/10.1007/s10255-019-0808-8

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  As an extension of partially linear models and additive models, partially linear additive model is useful in statistical modelling. This paper proposes an empirical likelihood based approach for testing serial correlation in this semiparametric model. The proposed test method can test not only zero first-order serial correlation, but also higher-order serial correlation. Under the null hypothesis of no serial correlation, the test statistic is shown to follow asymptotically a chi-square distribution. Furthermore, a simulation study is conducted to illustrate the performance of the proposed method.
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  A Differential Oligopoly Game Under Price Competition

  Jun WANG

  应用数学学报(英文版). 2019, 35(2): 412-420. https://doi.org/10.1007/s10255-019-0811-0

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  A dynamic actual demand function was used to portray a market environment for fast moving consumer goods and to establish oligopoly differential games under price competition. We confirmed the stable point in n-player price competition as the saddle point of differential games, and acquired the optimal price and demand at equilibrium. Analysis on optimal price and demand shows that, to obtain more profits, a manufacturer should control costs, rapidly occupy the market, and improve product uniqueness.
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  Distribution Approximation of Shrinkage Estimate in Censored Regression Model via Randomly Weighting Method

  Xian-hui LIU, Zhan-feng WANG, Yao-hua WU

  应用数学学报(英文版). 2019, 35(2): 421-434. https://doi.org/10.1007/s10255-019-0812-z

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  Censored regression ("Tobit") model is a special case of limited dependent variable regression model, and plays an important role in econometrics. Based on this model, all kinds of methods for variable or group variable selection have been developed and the corresponding shrinkage parameter estimates are widely studied. However, asymptotic distributions of the shrinkage estimates involve unknown nuisance parameters, such as density function of error term. To avoid estimating nuisance parameters, this paper presents a randomly weighting method to approximate to the asymptotic distribution of the shrinkage estimate. A computation procedure of random approximation is provided and asymptotic properties of the randomly weighting estimates are also obtained. The proposed methods are evaluated with extensively numerical studies and a women labor supply example.
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  A Class of Alternating Linearization Algorithms for Nonsmooth Convex Optimization

  Dan LI, Jie SHEN, Yuan LU, Li-Ping PANG, Zun-Quan XIA

  应用数学学报(英文版). 2019, 35(2): 435-443. https://doi.org/10.1007/s10255-019-0814-x

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  We consider the problems of minimizing the sum of a continuously differentiable convex function and a nonsmooth convex function in this paper. These problems arise in many applications of practical interest. A class of alternating linearization methods is presented for solving these problems. The global convergence rate is also obtained under certain mild conditions. Numerical experiments validate the theoretical convergence analysis and verify the implementation of the proposed algorithm.
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  On Traceability of Claw-_o-1-heavy Graphs

  Bin-long LI, Sheng-gui ZHANG

  应用数学学报(英文版). 2019, 35(2): 444-451. https://doi.org/10.1007/s10255-019-0815-9

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  A graph is called traceable if it contains a Hamilton path, i.e., a path passing through all the vertices. Let G be a graph on n vertices. G is called claw-_o-1-heavy if every induced claw (K_1,3) of G has a pair of nonadjacent vertices with degree sum at least n - 1 in G. In this paper we show that a claw-_o-1-heavy graph G is traceable if we impose certain additional conditions on G involving forbidden induced subgraphs.
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  On Signed Star Domination in Graphs

  Yan-cai ZHAO, Er-fang SHAN, Lian-ying MIAO, Zuo-song LIANG

  应用数学学报(英文版). 2019, 35(2): 452-457. https://doi.org/10.1007/s10255-019-0816-8

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  For a graph G=(V, E) without isolated vertex, a function f:E(G) → {-1, 1} is said to be a signed star dominating function of G if ∑_e∈E(v) f(e) ≥ 1 for every v ∈ V (G), where E(v)={uv ∈ E(G)|u ∈ V (G)}. The minimum value of ∑_e∈E(G) f(e), taken over all signed star dominating functions f of G, is called the signed star domination number of G and is denoted by γ_ss(G). This paper studies the bounds and algorithms of signed star domination numbers in some classes of graphs. In particular, sharp bounds for the signed star domination number of a general graph and a linear-time algorithm for the signed star domination problem in a tree is presented.
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  Remarks on Fractional ID-k-factor-critical Graphs

  Si-zhong ZHOU, Lan XU, Zu-run XU

  应用数学学报(英文版). 2019, 35(2): 458-464. https://doi.org/10.1007/s10255-019-0818-6

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  Let G be a graph, and k a positive integer. A graph G is fractional independent-set-deletable k-factor-critical (in short, fractional ID-k-factor-critical) if G - I has a fractional k-factor for every independent set I of G. In this paper, we present a sufficient condition for a graph to be fractional ID-k-factor-critical, depending on the minimum degree and the neighborhoods of independent sets. Furthermore, it is shown that this result in this paper is best possible in some sense.
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  On a Problem of Potentially Pancyclic-graphic Sequences Due to S.B. Rao

  Jian-Hua YIN

  应用数学学报(英文版). 2019, 35(2): 465-474. https://doi.org/10.1007/s10255-019-0813-y

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  A non-increasing sequence π=(d₁, d₂,…, d_n) of nonnegative integers is said to be potentially hamiltonian-graphic (resp. potentially pancyclic-graphic) if it is realizable by a simple graph on n vertices containing a hamiltonian cycle (resp. containing cycles of every length from 3 to n). A.R. Rao and S.B. Rao (J. Combin. Theory Ser.B, 13(1972), 185-191) and Kundu (Discrete Math., 6(1973), 367-376) presented a characterization of π=(d₁, d₂,…, d_n) that is potentially hamiltonian-graphic. S.B. Rao (Lecture Notes in Math., No. 855, Springer Verlag, 1981, 417-440, Unsolved Problem 2) further posed the following problem:present a characterization of π=(d₁, d₂,…, d_n) that is potentially pancyclic-graphic. In this paper, we first give solution to this problem for the case of 4 ≤ n ≤ 11. Moreover, we also show that a near regular graphic sequence π=(d₁, d₂,…, d_n) with d_n ≥ 3 is potentially pancyclic-graphic.