2013年, 第29卷, 第2期　刊出日期：2013-04-15

  

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  Parametric Duality Models for Semiinfinite Multiobjective Fractional Programming Problems Containing Generalized (α,η,ρ)-V-Invex Functions

  G. J. ZALMAI, Qing-hong ZHANG

  应用数学学报(英文版). 2013, 29(2): 225-240. https://doi.org/10.1007/s10255-013-0213-7

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  In this paper, we present several parametric duality results under various generalized (α,η,ρ)-V-invexity assumptions for a semiinfinite multiobjective fractional programming problem.
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  Analyzing Left-truncated and Right-censored Data under Cox Models with Long-term Survivors

  Fei-peng ZHANG, Yong ZHOU

  应用数学学报(英文版). 2013, 29(2): 241-252. https://doi.org/10.1007/s10255-013-0215-5

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  We analyze left-truncated and right-censored data using Cox proportional hazard models with long-term survivors. The estimators of covariate coefficients and the long-term survivor proportion are obtained by the partial likelihood method, and their asymptotic properties are also established. Simulation studies demonstrate the performance of the proposed estimators, and an application to a real dataset is provided.
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  Construction of Supersaturated Design with Large Number of Factors by the Complementary Design Method

  Yan LIU, Min-Qian LIU

  应用数学学报(英文版). 2013, 29(2): 253-262. https://doi.org/10.1007/s10255-013-0214-6

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  Supersaturated designs (SSDs) have been widely used in factor screening experiments. The present paper aims to prove that the maximal balanced designs are a kind of special optimal SSDs under the E(f_NOD) criterion. We also propose a new method, called the complementary design method, for constructing E(f_NOD) optimal SSDs. The basic principle of this method is that for any existing E(f_NOD) optimal SSD whose E(f_NOD) value reaches its lower bound, its complementary design in the corresponding maximal balanced design is also E(f_NOD) optimal. This method applies to both symmetrical and asymmetrical (mixed-level) cases. It provides a convenient and efficient way to construct many new designs with relatively large numbers of factors. Some newly constructed designs are given as examples.
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  Θ-type Calderón-Zygmund Operators with Non-doubling Measures

  Ru-long XIE, Li-sheng SHU

  应用数学学报(英文版). 2013, 29(2): 263-280. https://doi.org/10.1007/s10255-013-0217-3

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  Let μ be a Radon measure on R^d which may be non-doubling. The only condition that μ must satisfy is μ(B(x, r))≤Crⁿ for all x∈R^d, r>0 and for some fixed 0<n≤d. In this paper, under this assumption, we prove that θ-type Calderón-Zygmund operator which is bounded on L²(μ) is also bounded from L¹(μ) into RBMO(μ) and from H_atb^1,∞(μ) into L¹(μ). According to the interpolation theorem introduced by Tolsa, the L^p(μ)-boundedness (1<p<1) is established for θ-type Calderón-Zygmund operators. Via a sharp maximal operator, it is shown that commutators and multilinear commutators of θ-type Calderón-Zygmund operator with RBMO(μ) function are bounded on L^p(μ) (1<p<1).
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  Hölder Continuity for a Class of Strongly Degenerate Schrödinger Operator Formed by Vector Fields

  Yong-yang JIN, Jie-lin Lü, Rong-fei LIN

  应用数学学报(英文版). 2013, 29(2): 281-288. https://doi.org/10.1007/s10255-011-0098-2

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  In this paper we obtain the Hölder continuity property of the solutions for a class of degenerate Schrödinger equation generated by the vector fields:
   ,
  where X={X₁, ···,X_m} is a family of C^∞ vector fields satisfying the Hörmander condition, and the lower order terms belong to an appropriate Morrey type space.
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  Regularity Criteria of Axisymmetric Weak Solutions to the 3D Magnetohydrodynamic Equations

  Bao-quan YUAN, Feng-ping LI

  应用数学学报(英文版). 2013, 29(2): 289-302. https://doi.org/10.1007/s10255-013-0223-5

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  In this paper, we study the regularity criteria for axisymmetric weak solutions to the MHD equa- tions in R³. Let ω_θ, J_θ and u_θ be the azimuthal component of ω, J and u in the cylindrical coordinates, respectively. Then the axisymmetric weak solution (u, b) is regular on (0, T) if (ω_θ, J_θ)∈L^q(0, T;L^q) or (ω_θ,∇(u_θe_θ))∈L^q(0, T;L^q) with 3/p+2/q≤2, 3/2<p<∞. In the endpoint case, one needs conditions (ω_θ, J_θ)∈L¹(0, T; B_∞,∞⁰) or (ω_θ,∇∇(_uθe_θ))∈L¹(0, T; B_∞,∞⁰).
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  A Separation Theorem for Stochastic Singular Linear Quadratic Control Problem with Partial Information

  Hong-ji MA, Ting HOU

  应用数学学报(英文版). 2013, 29(2): 303-314. https://doi.org/10.1007/s10255-013-0218-2

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  In this paper, we provide a separation theorem for the singular linear quadratic (LQ) control problem of Itô-type linear systems in the case of the state being partially observable. Above all, the Kalman- Bucy filtering of the dynamics is given by means of Girsanov transformation, by which the suboptimal feedback control of the LQ problem is determined. Furthermore, it is shown that the well-posedness of the LQ problem is equivalent to the solvability of a generalized differential Riccati equation (GDRE).
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  Existence of Solutions for the Critical Elliptic System with Inverse Square Potentials

  Zhao-xia LIU, Zhao-hui LIU

  应用数学学报(英文版). 2013, 29(2): 315-328. https://doi.org/10.1007/s10255-013-0225-3

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  Let Ω∋0 be an open bounded domain in R^N (N≥3) and 2^*(s)=2(N-s)/N-2 , 0<s<2. We consider the following elliptic system of two equations in H₀¹(Ω)×H₀¹(Ω):
  ,
  where λ, μ>0 and α,β> 1 satisfy α+β=2^*(s). Using the Moser iteration, we prove the asymptotic behavior of solutions at the origin. In addition, by exploiting the Mountain-Pass theorem, we establish the existence of solutions.
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  Solutions to BSDEs Driven by Both Standard and Fractional Brownian Motions

  Wei-yin FEI, Deng-Feng XIA, Shu-guang ZHANG

  应用数学学报(英文版). 2013, 29(2): 329-354. https://doi.org/10.1007/s10255-013-0219-1

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  The backward stochastic differential equations driven by both standard and fractional Brownian motions (or, in short, SFBSDE) are studied. A Wick-Itô stochastic integral for a fractional Brownian motion is adopted. The fractional Itô formula for the standard and fractional Brownian motions is provided. Introducing the concept of the quasi-conditional expectation, we study some its properties. Using the quasi-conditional expectation, we also discuss the existence and uniqueness of solutions to general SFBSDEs, where a fixed point principle is employed. Moreover, solutions to linear SFBSDEs are investigated. Finally, an explicit solution to a class of linear SFBSDEs is found.
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  Complex Dynamics in a Discrete-time Predator-prey System without Allee Effect

  Xian-wei CHEN, Xiang-ling FU, ZHU-JUN JING

  应用数学学报(英文版). 2013, 29(2): 355-376. https://doi.org/10.1007/s10255-013-0221-7

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  In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes (invariant circle and inverse period- doubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation) to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting (2,3,4) chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.
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  Weak Centers and Local Bifurcations of Critical Periods at Infinity for a Class of Rational Systems

  Wen-tao HUANG, Valery G. ROMANOVSKI, Wei-nian ZHANG

  应用数学学报(英文版). 2013, 29(2): 377-390. https://doi.org/10.1007/s10255-013-0220-8

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  We describe an approach to studying the center problem and local bifurcations of critical periods at infinity for a class of differential systems. We then solve the problem and investigate the bifurcations for a class of rational differential systems with a cubic polynomial as its numerator.
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  Bayesian Estimation for Generalized Exponential Distribution Based on Progressive Type-Ⅰ Interval Censoring

  Xiu-yun PENG, Zai-zai YAN

  应用数学学报(英文版). 2013, 29(2): 391-402. https://doi.org/10.1007/s10255-013-0222-6

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  In this study, we consider the Bayesian estimation of unknown parameters and reliability function of the generalized exponential distribution based on progressive Type-Ⅰ interval censoring. The Bayesian estimates of parameters and reliability function cannot be obtained as explicit forms by applying squared error loss and Linex loss functions, respectively; thus, we present the Lindley's approximation to discuss these estimations. Then, the Bayesian estimates are compared with the maximum likelihood estimates by using the Monte Carlo simulations.
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  Superconvergence Analysis and Extrapolation of Quasi-Wilson Nonconforming Finite Element Method for Nonlinear Sobolev Equations

  Dong-yang SHI, Fen-ling WANG, Yan-min ZHAO

  应用数学学报(英文版). 2013, 29(2): 403-414. https://doi.org/10.1007/s10255-013-0216-4

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  Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equations is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that (∇_h(u-I_hu),∇h^vh)h may be estimated as order O(h²) when u∈h³(Ω), where I_hu denotes the bilinear interpolation of u, v_h is a polynomial belongs to quasi-Wilson finite element space and ∇h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O(h²)/O(h³) in broken H¹-norm, which is one/two order higher than its interpolation error when u∈H³(Ω)/H⁴(Ω). Then we derive the optimal order error estimate and su- perclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapolation result of order O(H³), two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.
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  On the Existence of Solutions for an Elliptic System of Equations with Arbitrary Order Growth

  Zhong-yuan LIU

  应用数学学报(英文版). 2013, 29(2): 415-424. https://doi.org/10.1007/s10255-013-0224-4

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  Let B_R be the ball centered at the origin with radius R in R^N (N≥2). In this paper we study the existence of solution for the following elliptic system
   
  where λ>0, μ>0 p≥2, q≥2, υ is the unit outward normal at the boundary ∂BR. Under certain assumptions on κ(|x|), using variational methods, we prove the existence of a positive and radially increasing solution for this problem without growth conditions on the nonlinearity.
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  Stability and Bifurcation Analysis in a System of Four Coupled Neurons with Multiple Delays

  Xiu-ling LI, Jun-jie WEI

  应用数学学报(英文版). 2013, 29(2): 425-448. https://doi.org/10.1007/s10255-013-0212-8

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  A system of delay differential equations is studied which represent a model for four neurons with time delayed connections between the neurons and time delayed feedback from each neuron to itself. The linear stability and bifurcation of the system are studied in a parameter space consisting of the sum of the time delays between the elements and the product of the strengths of the connections between the elements. Meanwhile, the bifurcation set are drawn in the parameter space.