15 April 2020, Volume 36 Issue 2
    

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  • ARTICLES
    Nash Embedding, Shape Operator and Navier-Stokes Equation on a Riemannian Manifold
    Shizan FANG
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 237-252.
    DOI:10.1007/s10255-020-0928-1
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    What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic representation formula for Navier-Stokes equations on a general compact Riemannian manifold is obtained when de Rham-Hodge Laplacian is involved.
  • ARTICLES
    Talagrand Inequality on Free Path Space and Application to Stochastic Reaction Diffusion Equations
    Feng-yu WANG, Tu-sheng ZHANG
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 253-261.
    DOI:10.1007/s10255-020-0926-3
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    By using a split argument due to[1], the transportation cost inequality is established on the free path space of Markov processes. The general result is applied to stochastic reaction diffusion equations with random initial values.
  • ARTICLES
    Controlled Stochastic Partial Differential Equations for Rabbits on a Grassland
    Suzanne LENHART, Xiao TANG, Jie XIONG, Jiong-min YONG
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 262-282.
    DOI:10.1007/s10255-020-0925-4
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    A new approach to modeling populations incorporating stochasticity, a random environment, and individual behavior is illustrated with a specific example of two interacting populations:rabbits and grass. The derivation of the system of stochastic partial differential equations (SPDEs) to show how the individual mechanisms of both populations are included. This model also has an unusual feature of a nonlocal term. The harvesting of the rabbit population is introduced as a control variable.
  • ARTICLES
    LIL for the Length of the Longest Increasing Subsequences
    Zhong-gen SU
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 283-293.
    DOI:10.1007/s10255-020-0942-3
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    Let X1, X2, …, Xn, … be a sequence of i.i.d. random variables uniformly distributed on[0; 1], and denote by Ln the length of the longest increasing subsequences of X1, X2, …, Xn. Consider the poissonized version Hn based on Hammersley's representation in the 2-dimensional space. A law of the iterated logarithm for Hn is established using the well-known subsequence method and Borel-Cantelli lemma. The key technical ingredients in the argument include superadditivity, increment independence and precise tail estimates for the Hn's. The work was motivated by recent works due to Ledoux (J. Theoret. Probab. 31, (2018)). It remains open to establish an analog for the Ln itself.
  • ARTICLES
    Self-normalized Cramér-type Moderate Deviations for Functionals of Markov Chain
    Xin-wei FENG, Qi-Man SHAO
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 294-313.
    DOI:10.1007/s10255-020-0924-5
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    Let {xn, n ≥ 0} be a Markov chain with a countable state space S and let f(·) be a measurable function from S to R and consider the functionals of the Markov chain yn:=f(xn). We construct a new type of self-normalized sums based on the random-block scheme and establish a Cramér-type moderate deviations for self-normalized sums of functionals of the Markov chain.
  • ARTICLES
    Robust Nonparametric Function Estimation for Errors-in-variables Models
    Chao-xia YUAN, Heng-jian CUI
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 314-331.
    DOI:10.1007/s10255-020-0944-1
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    This paper discusses robust nonparametric estimators of location regression function for errorsin-variables model with de-convolution kernel. The local constant smoother is used for the estimation of the nonparametric function, and the local linear smoother is proposed to deal with the boundary problem, as well as to improve the local constant smoother. We establish the asymptotic properties of the estimator, the influence function of the statistical functional and the breakdown point. A simulation study is carried out to demonstrate robust performance of the proposed estimator. The motorcycle data is presented to illustrate the application of the robust estimator further.
  • ARTICLES
    Robust Variable Selection and Estimation in Threshold Regression Model
    Bo-wen LI, Yun-qi ZHANG, Nian-sheng TANG
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 332-346.
    DOI:10.1007/s10255-020-0939-y
    Abstract ( )    Download PDF ( )   Knowledge map   Save
    We combine the robust criterion with the lasso penalty together for the high-dimensional threshold model. It estimates regression coefficients as well as the threshold parameter robustly that can be resistant to outliers or heavy-tailed noises and perform variable selection simultaneously. We illustrate our approach with the absolute loss, the Huber's loss, and the Tukey's loss, it can also be extended to any other robust losses. Simulation studies are conducted to demonstrate the usefulness of our robust approach. Finally, we use our estimators to investigate the presence of a shift in the effect of debt on future GDP growth.
  • ARTICLES
    Reflected SPDEs Driven by Fractional Noises
    Juan YANG, Qing ZHOU
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 347-360.
    DOI:10.1007/s10255-020-0938-z
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    In this paper, we study the stochastic partial differential equation with two reflecting smooth walls h1 and h2, driven by a fractional noise, which is fractional in time and white in space. The large deviation principle for the law of the solution to this equation, will be established through developing a classical method. Furthermore, we obtain the Hölder continuity of the solution.
  • ARTICLES
    Moments of Continuous-state Branching Processes with or Without Immigration
    Li-na JI, Zeng-hu LI
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 361-373.
    DOI:10.1007/s10255-020-0935-2
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    For a positive continuous function f satisfying some standard conditions, we study the f-moments of continuous-state branching processes with or without immigration. The main results give criteria for the existence of the f-moments. The characterization of the processes in terms of stochastic equations plays an essential role in the proofs.
  • ARTICLES
    Two-sample Testing for Mean Functions with Incompletely Observed Functional Data
    Yan-qiu ZHOU, Yan-ling WAN, Tao ZHANG
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 374-389.
    DOI:10.1007/s10255-020-0934-3
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    In functional data analysis, the collected data are often assumed to be fully observed on the domain. However, in dealing with real data (for example, environmental pollution data), we are often faced with the scenario that some functional data are fully observed on dense lattice while others are incompletely observed. In this paper, we propose a method for testing equivalence of mean functions of two samples under this scenario. Some asymptotic results of the proposed methods are established. The proposed test is employed to analyze an environmental pollution study in Liuzhou City of China. Simulations show that the proposed test has a good control of the type-I error, and is more powerful than the complete case test in most cases.
  • ARTICLES
    Multiple Periodic Solutions of Differential Delay Equations with 2k-1 Lags
    Lin LI, Hua-fei SUN, Wei-gao GE
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 390-400.
    DOI:10.1007/s10255-020-0946-z
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    In this paper, we study the periodic solutions to a type of differential delay equations with 2k-1 lags. The 4k-periodic solutions are obtained by using the variational method and the method of Kaplan-Yorke coupling system. This is a new type of differential delay equations compared with all the previous researches. And this paper provides a theoretical basis for the study of differential delay equations. An example is given to demonstrate our main results.
  • ARTICLES
    Solitary Wave Solutions of Some Nonlinear Physical Models Using Riccati Equation Approach
    Zaid ODIBAT, Ahmed ALSAEDI, Tasawar HAYAT
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 401-418.
    DOI:10.1007/s10255-020-0929-0
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    Riccati equation approach is used to look for exact travelling wave solutions of some nonlinear physical models. Solitary wave solutions are established for the modified KdV equation, the Boussinesq equation and the Zakharov-Kuznetsov equation. New generalized solitary wave solutions with some free parameters are derived. The obtained solutions, which includes some previously known solitary wave solutions and some new ones, are expressed by a composition of Riccati differential equation solutions followed by a polynomial. The employed approach, which is straightforward and concise, is expected to be further employed in obtaining new solitary wave solutions for nonlinear physical problems.
  • ARTICLES
    Classical and Inverse Median Location Problems under Uncertain Environment
    Akram SOLTANPOUR, Fahimeh BAROUGHI, Behrooz ALIZADEH
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 419-438.
    DOI:10.1007/s10255-020-0945-0
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    In this paper, we first consider the classical p-median location problem on a network in which the vertex weights and the distances between vertices are uncertain variables. The uncertainty distribution of the optimal objective value of the p-median problem is given and the concepts of the α-p-median, the most p-median and the expected p-median are introduced. Then, it is shown that the uncertain p-median problem is NP-hard on general networks. However, if the underlying network is a tree, an efficient algorithm for the uncertain 1-median problem with linear time complexity is proposed. Finally, we investigate the inverse 1-median problem on a tree with uncertain vertex weights and present a programming model for the problem. Then, it is shown that the proposed model can be reformulated into a deterministic programming model.
  • ARTICLES
    List Vertex-arboricity of Planar Graphs without Intersecting 5-cycles
    Wei-fan WANG, Li HUANG, Min CHEN
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 439-447.
    DOI:10.1007/s10255-020-0936-1
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    The vertex-arboricity a(G) of a graph G is the minimum number of colors required for a vertex coloring of G such that no cycle is monochromatic. The list vertex-arboricity al(G) is the list-coloring version of this concept. In this paper, we prove that every planar graph G without intersecting 5-cycles has al(G) ≤ 2. This extends a result by Raspaud and Wang[On the vertex-arboricity of planar graphs, European J. Combin. 29 (2008), 1064-1075] that every planar graph G without 5-cycles has a(G) ≤ 2.
  • ARTICLES
    Simultaneous Estimation of Multiple Conditional Regression Quantiles
    Yan-ke WU, Ya-nan HU, Jian ZHOU, Mao-zai TIAN
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 448-457.
    DOI:10.1007/s10255-020-0932-5
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    In this article, we put forward a new approach to estimate multiple conditional regression quantiles simultaneously. Unlike the double summation method in most of the literatures, our proposed model allows continuous variety for the quantile level over (0,1). As a result, all the quantile curves can be obtained via a 2-dimensional surface simultaneously. Most importantly, the proposed minimizing criterion can be readily transformed to a linear programming problem. We use tensor product bi-linear quantile smoothing B-splines to fit it. The asymptotic property of the estimator is derived and a real data set is analyzed to demonstrate the proposed method.
  • ARTICLES
    Variable Selection for Varying Coefficient Models Via Kernel Based Regularized Rank Regression
    Kang-ning WANG, Lu LIN
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 458-470.
    DOI:10.1007/s10255-020-0937-0
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    A robust and efficient shrinkage-type variable selection procedure for varying coefficient models is proposed, selection consistency and oracle properties are established. Furthermore, a BIC-type criterion is suggested for shrinkage parameter selection and theoretical property is discussed. Numerical studies and real data analysis also are included to illustrate the finite sample performance of our method.
  • ARTICLES
    Nonconforming Finite Element Methods for the Constrained Optimal Control Problems Governed by Nonsmooth Elliptic Equations
    Hong-bo GUAN, Dong-yang SHI
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 471-481.
    DOI:10.1007/s10255-020-0931-6
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    In this paper, nonconforming finite element methods (FEMs) are proposed for the constrained optimal control problems (OCPs) governed by the nonsmooth elliptic equations, in which the popular EQ1rot element is employed to approximate the state and adjoint state, and the piecewise constant element is used to approximate the control. Firstly, the convergence and superconvergence properties for the nonsmooth elliptic equation are obtained by introducing an auxiliary problem. Secondly, the goal-oriented error estimates are obtained for the objective function through establishing the negative norm error estimate. Lastly, the methods are extended to some other well-known nonconforming elements.
  • ARTICLES
    A Novel Genetic Algorithm for Global Optimization
    Chun-feng WANG, Kui LIU, Pei-ping SHEN
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 482-491.
    DOI:10.1007/s10255-020-0930-7
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    This paper presents a novel genetic algorithm for globally solving un-constraint optimization problem. In this algorithm, a new real coded crossover operator is proposed firstly. Furthermore, for improving the convergence speed and the searching ability of our algorithm, the good point set theory rather than random selection is used to generate the initial population, and the chaotic search operator is adopted in the best solution of the current iteration. The experimental results tested on numerical benchmark functions show that this algorithm has excellent solution quality and convergence characteristics, and performs better than some algorithms.
  • ARTICLES
    On Some Cycles in Wenger Graphs
    Ye WANG, Felix LAZEBNIK, Andrew THOMASON
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 492-502.
    DOI:10.1007/s10255-020-0927-2
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    Let p be a prime, q be a power of p, and let Fq be the field of q elements. For any positive integer n, the Wenger graph Wn(q) is defined as follows:it is a bipartite graph with the vertex partitions being two copies of the (n+1)-dimensional vector space Fqn+1, and two vertices p=(p(1), …, p(n+1)) and l=[l(1), …, l(n+1)] being adjacent if p(i) + l(i)=p(1)l(1)i-1, for all i=2, 3, …, n + 1.
    In 2008, Shao, He and Shan showed that for n ≥ 2, Wn(q) contains a cycle of length 2k where 4 ≤ k ≤ 2p and k≠ 5. In this paper we extend their results by showing that
    (i) for n ≥ 2 and p ≥ 3, Wn(q) contains cycles of length 2k, where 4 ≤ k ≤ 4p + 1 and k≠ 5;
    (ii) for q ≥ 5, 0 < c < 1, and every integer k, 3 ≤ k ≤ qc, if 1 ≤ n < (1-c-7/3 logq 2)k-1, then Wn(q) contains a 2k-cycle. In particular, Wn(q) contains cycles of length 2k, where n + 2 ≤ k ≤ qc, provided q is sufficiently large.
  • ARTICLES
    Dispersion Analysis of Multi-symplectic Scheme for the Nonlinear Schrödinger Equations
    Hao-chen LI, Jian-qiang SUN, Hang YE, Xue-jun HE
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 503-515.
    DOI:10.1007/s10255-020-0933-4
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    In this paper, we study the dispersive properties of multi-symplectic discretizations for the nonlinear Schrödinger equations. The numerical dispersion relation and group velocity are investigated. It is found that the numerical dispersion relation is relevant when resolving the nonlinear Schrödinger equations.
  • ARTICLES
    Periodic Solutions for a Class of n-dimensional Prescribed Mean Curvature Equations
    Zai-tao LIANG, Shi-ping LU
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 516-526.
    DOI:10.1007/s10255-020-0943-2
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    In this paper, by using an extension of Mawhin's continuation theorem and some analysis methods, we study the existence of periodic solutions for the following prescribed mean curvature system
    d/dt φ(x') + ▽W(x)=p(t),
    where x ∈ Rn, W ∈ C1(Rn, R), p ∈ C(R, Rn) is T-periodic and
    φ(x)=(x/(√1+|x|2)).
  • ARTICLES
    Modified Projective Synchronization of Fractional-order Chaotic Systems with Different Dimensions
    Xi ZHANG, Ran-chao WU
    Acta Mathematicae Applicatae Sinica(English Series). 2020, 36(2): 527-538.
    DOI:10.1007/s10255-020-0941-4
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    In this paper, the modified projective synchronization between two fractional-order chaotic systems with different dimensions is investigated. The added-order scheme and the reduced-order scheme are proposed, respectively. Based on the Laplace transformation and feedback control theory, controllers are designed such that two chaotic systems with different dimensions could be synchronized asymptotically under the presented schemes. Corresponding numerical simulations are given to show the effectiveness of the proposed schemes.
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