15 July 2013, Volume 29 Issue 3
    

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  • Improvement of Convergence to Steady State Solutions of Euler Equations with Weighted Compact Nonlinear Schemes
    Shu-hai ZHANG, Xiao-gang DENG, Mei-liang MAO, Chi-Wang SHU
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 449-464.
    DOI:10.1007/s10255-013-0230-6
    Abstract ( )      Knowledge map   Save
    The convergence to steady state solutions of the Euler equations for weighted compact nonlinear schemes (WCNS) [Deng X. and Zhang H. (2000), J. Comput. Phys. 165, 22-44 and Zhang S., Jiang S. and Shu C.-W. (2008), J. Comput. Phys. 227, 7294-7321] is studied through numerical tests. Like most other shock capturing schemes, WCNS also suffers from the problem that the residue can not settle down to machine zero for the computation of the steady state solution which contains shock waves but hangs at the truncation error level. In this paper, the techniques studied in [Zhang S. and Shu. C.-W. (2007), J. Sci. Comput. 31, 273-305 and Zhang S., Jiang S and Shu. C.-W. (2011), J. Sci. Comput. 47, 216-238], to improve the convergence to steady state solutions for WENO schemes, are generalized to the WCNS. Detailed numerical studies in one and two dimensional cases are performed. Numerical tests demonstrate the effectiveness of these techniques when applied to WCNS. The residue of various order WCNS can settle down to machine zero for typical cases while the small post-shock oscillations can be removed.
  • Asymptotic Analysis of Quantum Dynamics in Crystals：the Bloch-Wigner Transform, Bloch Dynamics and Berry Phase
    Weinan E, Jian-feng LU, Xu YANG
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 465-476.
    DOI:10.1007/s10255-011-0095-5
    Abstract ( )      Knowledge map   Save
    We study the semi-classical limit of the Schrödinger equation in a crystal in the presence of an external potential and magnetic field. We first introduce the Bloch-Wigner transform and derive the asymptotic equations governing this transform in the semi-classical setting. For the second part, we focus on the appearance of the Berry curvature terms in the asymptotic equations. These terms play a crucial role in many important physical phenomena such as the quantum Hall effect. We give a simple derivation of these terms in different settings using asymptotic analysis.
  • Generalized Profile LSE in Varying-Coefficient Partially Linear Models with Measurement Errors
    Yun-bei MA, Jin-hong YOU, Yong ZHOU
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 477-490.
    DOI:10.1007/s10255-013-0229-z
    Abstract ( )      Knowledge map   Save
    This paper is concerned with the estimating problem of a semiparametric varying-coefficient partially linear errors-in-variables model Yi = Xiτ +Ziτ (Ui)+εi, Wi = Xi+ ξi, i = 1,…, n. Due to measurement errors, the usual profile least square estimator of the parametric component, local polynomial estimator of the nonparametric component and profile least squares based estimator of the error variance are biased and inconsistent. By taking the measurement errors into account we propose a generalized profile least squares estimator for the parametric component and show it is consistent and asymptotically normal. Correspondingly, the consistent estimation of the nonparametric component and error variance are proposed as well. These results may be used to make asymptotically valid statistical inferences. Some simulation studies are conducted to illustrate the finite sample performance of these proposed estimations.
  • A Trust Region Algorithm for Solving Bilevel Programming Problems
    Guo-shan LIU, Shi-qin XU, Ji-ye HAN
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 491-498.
    DOI:10.1007/s10255-013-0231-5
    Abstract ( )      Knowledge map   Save
    In this paper, we present a new trust region algorithm for a nonlinear bilevel programming problem by solving a series of its linear or quadratic approximation subproblems. For the nonlinear bilevel programming problem in which the lower level programming problem is a strongly convex programming problem with linear constraints, we show that each accumulation point of the iterative sequence produced by this algorithm is a stationary point of the bilevel programming problem.
  • Flows Associated to Cameron-martin Type Vector Fields on Path Spaces Over a Riemannian Manifold
    Jing-xiao ZHANG
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 499-508.
    DOI:10.1007/s10255-013-0245-z
    Abstract ( )      Knowledge map   Save
    The flow on theWiener space associated to a tangent process constructed by Cipriano and Cruzeiro, as well as by Gong and Zhang does not allow to recover Driver’s Cameron-Martin theorem on Riemannian path space. The purpose of this work is to refine the method of the modified Picard iteration used in the previous work by Gong and Zhang and to try to recapture and extend the result of Driver. In this paper, we establish the existence and uniqueness of a flow associated to a Cameron-Martin type vector field on the path space over a Riemannian manifold.
  • Large Time Behavior of Entropy Solutions to Some Hyperbolic System with Dissipative Structure
    Hui-min YU
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 509-516.
    DOI:10.1007/s10255-011-0097-3
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    In this paper the large time behavior of the global L∞ entropy solutions to the hyperbolic system with dissipative structure is investigated. It is proved that as t→∞ the entropy solutions tend to a constant equilibrium state in L2 norm with exponential decay even when the initial values are arbitrarily large. As an illustration, a class of 2×2 system is studied.
  • On the Spectral Spread of Bicyclic Graphs with Given Girth
    Bing WANG, Ming-qing ZHAI, Jin-long SHU
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 517-528.
    DOI:10.1007/s10255-013-0228-0
    Abstract ( )      Knowledge map   Save
    The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. A graph G is said to be bicyclic, if G is connected and |E(G)| = |V(G)| + 1. Let B(n, g) be the set of bicyclic graphs on n vertices with girth g. In this paper some properties about the least eigenvalues of graphs are given, by which the unique graph with maximal spectral spread in B(n, g) is determined.
  • A Specification Test of Stochastic Diffusion Models
    Shu-lin ZHANG, Zheng-hong WEI, Qiu-xiang BI
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 529-540.
    DOI:10.1007/s10255-013-0226-2
    Abstract ( )      Knowledge map   Save
    In this paper, we propose a hypothesis testing approach to checking model mis-specification in continuous-time stochastic diffusion model. The key idea behind the development of our test statistic is rooted in the generalized information equality in the context of martingale estimating equations. We propose a bootstrap resampling method to implement numerically the proposed diagnostic procedure. Through intensive simulation studies, we show that our approach is well performed in the aspects of type I error control, power improvement as well as computational efficiency.
  • Existence and Nonexistence of Global Positive Solutions for a Weakly Coupled P-Laplacian System
    Xian-zhong ZENG, Zhen-hai LIU, Yong-geng GU
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 541-554.
    DOI:10.1007/s10255-013-0232-4
    Abstract ( )      Knowledge map   Save
    In this paper, we deal with a weakly coupled evolution P-Laplacian system with inhomogeneous terms. We obtain a critical criterion concerning existence and nonexistence of its global positive solutions. Such a criterion is different from that of the weakly coupled evolution P-Laplacian system with homogeneous terms. Further, we demonstrate existence and nonexistence of its global positive solutions.
  • Global Convergence of a Modified Limited Memory BFGS Method for Non-convex Minimization
    Yun-hai XIAO, Ting-feng LI, Zeng-xin WEI
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 555-566.
    DOI:10.1007/s10255-013-0233-3
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    In this paper, a modified limited memory BFGS method for solving large-scale unconstrained optimization problems is proposed. A remarkable feature of the proposed method is that it possesses global convergence property without convexity assumption on the objective function. Under some suitable conditions, the global convergence of the proposed method is proved. Some numerical results are reported which illustrate that the proposed method is efficient.
  • Non-stationary Subdivision for Exponential Polynomials Reproduction
    Bao-jun LI, Zhi-ling YU, Bo-wen YU, Zhi-xun SU, Xiu-ping LIU
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 567-578.
    DOI:10.1007/s10255-013-0234-2
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    In this paper we develop a novel approach to construct non-stationary subdivision schemes with a tension control parameter which can reproduce functions in a finite-dimensional subspace of exponential polynomials. The construction process is mainly implemented by solving linear systems for primal and dual subdivision schemes respectively, which are based on different parameterizations. We give the theoretical basis for the existence, uniqueness, and refinement rules of schemes proposed in this paper. The convergence and smoothness of the schemes are analyzed as well. Moreover, conics reproducing schemes are analyzed based on our theory, and a new idea that the tensor parameter ωk of the schemes can be adjusted for conics generation is proposed.
  • An M[X]/G/1 Retrial G-queue with Single Vacation Subject to the Server Breakdown and Repair
    Shu-ping YANG, Jin-biao WU, Zai-ming LIU
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 579-596.
    DOI:10.1007/s10255-013-0237-z
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    An M[X]/G/1 retrial G-queue with single vacation and unreliable server is investigated in this paper. Arrivals of positive customers form a compound Poisson process, and positive customers receive service immediately if the server is free upon their arrivals; Otherwise, they may enter a retrial orbit and try their luck after a random time interval. The arrivals of negative customers form a Poisson process. Negative customers not only remove the customer being in service, but also make the server under repair. The server leaves for a single vacation as soon as the system empties. In this paper, we analyze the ergodical condition of this model. By applying the supplementary variables method, we obtain the steady-state solutions for both queueing measures and reliability quantities.
  • The Fractal Dimensions of the Level Sets of the Generalized Iterated Brownian Motion
    Chang-qing TONG, Zheng-yan LIN, Jing ZHENG
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 597-602.
    DOI:10.1007/s10255-013-0236-0
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    Let {W1(t), t∈R+} and {W2(t, t∈R+} be two independent Brownian motions with W1(0) = W2(0) = 0. {H(t) = W1(|W2(t)|), t∈R+} is called a generalized iterated Brownian motion. In this paper, the Hausdorff dimension and packing dimension of the level sets
    {t ∈[0, T], H(t) = x}
    are established for any 0 < T ≤ 1.
  • Nonlinear Discrete Inequalities of Bihari-type and Applications
    Yu WU
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 603-614.
    DOI:10.1007/s10255-013-0235-1
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    Discrete Bihari-type inequalities with n nonlinear terms are discussed, which generalize some known results and may be used in the analysis of certain problems in the theory of difference equations. Examples to illustrate the boundedness of solutions of a difference equation are also given.
  • ARTICLES
    On the Adjacent Vertex-distinguishing Equitable Edge Coloring of Graphs
    Jing-wen LI, Cong WANG, Zhi-wen WANG
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 615-622.
    DOI:10.1007/s10255-013-0239-x
    Abstract ( )      Knowledge map   Save
    Let G(V,E) be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if (1) C(u) ≠C(v) for uv ∈ E(G), where C(u) = {f(uv)|uv ∈ E}, and (2) for any i, j = 1, 2,…k, we have ||Ei|-|Ej|| ≤ 1, where Ei = {e|e ∈ E(G) and f(e) = i}. Xave'(G) = min{k| there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number of G. In this paper, we obtain the Xave'(G) of some special graphs and present a conjecture.
  • Binomial Proportion Estimation in Longitudinal Data with Non-ignorable Non-response
    Xue-li WANG
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 623-630.
    DOI:10.1007/s10255-013-0238-y
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    Non-random missing data poses serious problems in longitudinal studies. The binomial distribution parameter becomes to be unidentifiable without any other auxiliary information or assumption when it suffers from ignorable missing data. Existing methods are mostly based on the log-linear regression model. In this article, a model is proposed for longitudinal data with non-ignorable non-response. It is considered to use the pre-test baseline data to improve the identifiability of the post-test parameter. Furthermore, we derive the identified estimation (IE), the maximum likelihood estimation (MLE) and its associated variance for the posttest parameter. The simulation study based on the model of this paper shows that the proposed approach gives promising results.
  • Asymptotic Properties in Semiparametric Partially Linear Regression Models for Functional Data
    Tao ZHANG
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 631-644.
    DOI:10.1007/s10255-013-0240-4
    Abstract ( )      Knowledge map   Save
    We consider the semiparametric partially linear regression models with mean function XTβ +g(z), where X and z are functional data. The new estimators of β and g(z) are presented and some asymptotic results are given. The strong convergence rates of the proposed estimators are obtained. In our estimation, the observation number of each subject will be completely flexible. Some simulation study is conducted to investigate the finite sample performance of the proposed estimators.
  • Estimation of Semi-Varying Coefficient Model with Surrogate Data and Validation Sampling
    Ya-zhao LÜ, Ri-quan ZHANG, Zhen-sheng HUANG
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 645-660.
    DOI:10.1007/s10255-013-0241-3
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    In this paper, we investigate the estimation of semi-varying coefficient models when the nonlinear covariates are prone to measurement error. With the help of validation sampling, we propose two estimators of the parameter and the coefficient functions by combining dimension reduction and the profile likelihood methods without any error structure equation specification or error distribution assumption. We establish the asymptotic normality of proposed estimators for both the parametric and nonparametric parts and show that the proposed estimators achieves the best convergence rate. Data-driven bandwidth selection methods are also discussed. Simulations are conducted to evaluate the finite sample property of the estimation methods proposed.
  • ARTICLES
    Positive Solutions to m-point Boundary Value Problem of Fractional Differential Equation
    Yuan-sheng TIAN
    Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(3): 661-672.
    DOI:10.1007/s10255-013-0242-2
    Abstract ( )      Knowledge map   Save
    In this paper, we study the multiplicity of positive solutions to the following m-point boundary value problem of nonlinear fractional differential equations:
    ,
    where q∈R, 1<q≤2, 0<ξ1<ξ2<…<ξm-2≤1/2, μi∈[0,+∞) and p =(q-1)/2, , Dq is the standard Riemann-Liouville differentiation, and f∈C([0,1]×[0,+∞),[0,+∞)). By using the Leggett-Williams fixed point theorem on a convex cone, some multiplicity results of positive solutions are obtained.
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