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Quantile Regression for RightCensored and LengthBiased Data
Xuerong CHEN, Yong ZHOU
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(3): 443462.
DOI:
10.1007/s1025501201573
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Lengthbiased data arise in many important fields, including epidemiological cohort studies, cancer screening trials and labor economics. Analysis of such data has attracted much attention in the literature. In this paper we propose a quantile regression approach for analyzing rightcensored and lengthbiased data. We derive an inverse probability weighted estimating equation corresponding to the quantile regression to correct the bias due to lengthbias sampling and informative censoring. This method can easily handle informative censoring induced by lengthbiased sampling. This is an appealing feature of our proposed method since it is generally difficult to obtain unbiased estimates of risk factors in the presence of lengthbias and informative censoring. We establish the consistency and asymptotic distribution of the proposed estimator using empirical process techniques. A resampling method is adopted to estimate the variance of the estimator. We conduct simulation studies to evaluate its finite sample performance and use a real data set to illustrate the application of the proposed method.
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Efficient Algorithms for Generating Truncated Multivariate Normal Distributions
Junwu YU, Guoliang TIAN
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(4): 601612.
DOI:
10.1007/s102550110110x
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Sampling from a
truncated multivariate normal distribution
(TMVND) constitutes the core computational module in fitting many statistical and econometric models. We propose two efficient methods, an iterative
data augmentation
(DA) algorithm and a noniterative
inverse Bayes formulae
(IBF) sampler, to simulate TMVND and generalize them to multivariate normal distributions with linear inequality constraints. By creating a Bayesian incompletedata structure, the posterior step of the DA algorithm directly generates random vector draws as opposed to single element draws, resulting obvious computational advantage and easy coding with common statistical software packages such as SPLUS, MATLAB and GAUSS. Furthermore, the DA provides a ready structure for implementing a fast EM algorithm to identify the mode of TMVND, which has many potential applications in statistical inference of constrained parameter problems. In addition, utilizing this mode as an intermediate result, the IBF sampling provides a novel alternative to Gibbs sampling and eliminates problems with convergence and possible slow convergence due to the high correlation between components of a TMVND. The DA algorithm is applied to a linear regression model with constrained parameters and is illustrated with a published data set. Numerical comparisons show that the proposed DA algorithm and IBF sampler are more efficient than the Gibbs sampler and the acceptreject algorithm.
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Ranks of the Common Solution to Six Quaternion Matrix Equations
Qingwen Wang, Yan Zhou, Qin Zhang
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(3): 443462.
DOI:
10.1007/s1025501100839
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A new expression is established for the common solution to six classical linear quaternion matrix equations
A
_{1}
X
=
C
_{1}
,
XB
_{1}
=
C
_{3}
,
A
_{2}
X
=
C
_{2}
,
XB
_{2}
=
C
_{4}
,
A
_{3}
XB
_{3}
=
C
_{5}
,
A
_{4}
XB
_{4}
=
C
_{6}
which was investigated recently by Wang, Chang and Ning (Q. Wang, H. Chang, Q. Ning, The common solution to six quaternion matrix equations with applications, Appl. Math. Comput. 195: 721732 (2008)). Formulas are derived for the maximal and minimal ranks of the common solution to this system. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition is presented for the invariance of the rank of the general solution to this system. Some known results can be regarded as the special cases of the results in this paper.
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Superconvergence of a Combined Mixed Finite Element and Discontinuous Galerkin Method for a Compressible Miscible Displacement Problem
Jiming Yang, Yanping Chen
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(3): 481494.
DOI:
10.1007/s102550110081y
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A combined mixed finite element and discontinuous Galerkin method for a compressible miscible displacement problem which includes molecular diffusion and dispersion in porous media is investigated. That is to say, the mixed finite element method with RaviartThomas space is applied to the flow equation, and the transport one is solved by the symmetric interior penalty discontinuous Galerkin (SIPG) approximation. Based on projection interpolations and induction hypotheses, a superconvergence estimate is obtained. During the analysis, an extension of the Darcy velocity along the Gauss line is also used in the evaluation of the coefficients in the Galerkin procedure for the concentration.
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Stochastic Differential Games with Reflection and Related Obstacle Problems for Isaacs Equations
Rainer BUCKDAHN, Juan LI
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(4): 647678.
DOI:
10.1007/s1025501100688
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In this paper we first investigate zerosum twoplayer stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower HamiltonJacobiBellmanIsaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of El Karoui, Kapoudjian, Pardoux, Peng and Quenez (1997), which turns out to be very useful because it allows us to estimate the
L
^{p}
distance of the solutions of two different RBSDEs by the
p
th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.
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Analysis of
SE
^{τ}
IR
^{ω}
S
Epidemic Disease Models with Vertical Transmission in Complex Networks
Xia LIU, Deju XU
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(1): 6374.
DOI:
10.1007/s1025501200941
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When the role of network topology is taken into consideration, one of the objectives is to understand the possible implications of topological structure on epidemic models. As most real networks can be viewed as complex networks, we propose a new delayed
SE
^{τ}
IR
^{ω}
S
epidemic disease model with vertical transmission in complex networks. By using a delayed ODE system, in a smallworld (SW) network we prove that, under the condition
R
_{0}
≤ 1, the diseasefree equilibrium (DFE) is globally stable. When
R
_{0}
> 1, the endemic equilibrium is unique and the disease is uniformly persistent. We further obtain the condition of local stability of endemic equilibrium for
R
_{0}
> 1. In a scalefree (SF) network we obtain the condition
R
_{1}
> 1 under which the system will be of nonzero stationary prevalence.
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A Twogrid Method with Expanded Mixed Element for Nonlinear Reactiondiffusion Equations
Wei Liu, Hongxing Rui, Hui Guo
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(3): 495502.
DOI:
10.1007/s1025501100866
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Expanded mixed finite element approximation of nonlinear reactiondiffusion equations is discussed. The equations considered here are used to model the hydrologic and biogeochemical phenomena. To linearize the mixedmethod equations, we use a twogrid method involving a small nonlinear system on a coarse gird of size
H
and a linear system on a fine grid of size
h
. Error estimates are derived which demonstrate that the error is
O
(Δ
t
+
h
^{k+1}
+
H
^{2k+2d/2}
) (
k
≥ 1), where
k
is the degree of the approximating space for the primary variable and
d
is the spatial dimension. The above estimates are useful for determining an appropriate
H
for the coarse grid problems.
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A New Algorithm for Total Variation Based Image Denoising
Yiping XU
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 721730.
DOI:
10.1007/s1025501201840
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512
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We propose a new algorithm for the total variation based on image denoising problem. The split Bregman method is used to convert an unconstrained minimization denoising problem to a linear system in the outer iteration. An algebraic multigrid method is applied to solve the linear system in the inner iteration. Furthermore, Krylov subspace acceleration is adopted to improve convergence in the outer iteration. Numerical experiments demonstrate that this algorithm is efficient even for images with large signaltonoise ratio.
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A New Periodic Solution to Jacobi Elliptic Functions of MKdV Equation and BBM Equation
Hongcai MA, ZhiPing ZHANG, Aiping DENG
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(2): 409415.
DOI:
10.1007/s1025501201537
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Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations. New exact solutions to the Jacobi elliptic function of MKdV equations and BenjaminBonaMahoney (BBM) equations are obtained with the aid of computer algebraic system Maple. The method is also valid for other (1+1)dimensional and higher dimensional systems.
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On Exact Solutions to Partial Differential Equations by the Modified Homotopy Perturbation Method
Gang YANG, Ruyun CHEN, Luogen YAO
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(1): 9198.
DOI:
10.1007/s1025501201199
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Based on the modified homotopy perturbation method (MHPM), exact solutions of certain partial differential equations are constructed by separation of variables and choosing the finite terms of a series in
p
as exact solutions. Under suitable initial conditions, the PDE is transformed into an ODE. Some illustrative examples reveal the efficiency of the proposed method.
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Optimal Dividend and Dynamic Reinsurance Strategies with Capital Injections and Proportional Costs
Yidong WU, Junyi GUO
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(3): 505524.
DOI:
10.1007/s1025501201662
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We consider an optimization problem of an insurance company in the diffusion setting, which controls the dividends payout as well as the capital injections. To maximize the cumulative expected discounted dividends minus the penalized discounted capital injections until the ruin time, there is a possibility of (cheap or noncheap) proportional reinsurance. We solve the control problems by constructing two categories of suboptimal models, one without capital injections and one with no bankruptcy by capital injection. Then we derive the explicit solutions for the value function and totally characterize the optimal strategies. Particularly, for cheap reinsurance, they are the same as those in the model of no bankruptcy.
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Optimal Selling Time in Stock Market over a Finite Time Horizon
S.C.P. YAM, S.P. YUNG, W. ZHOU
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(3): 557570.
DOI:
10.1007/s102550120169z
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1005
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In this paper, we examine the best time to sell a stock at a price being as close as possible to its highest price over a finite time horizon [
0, T
], where the stock price is modelled by a geometric Brownian motion and the ‘closeness’ is measured by the relative error of the stock price to its highest price over [
0, T
]. More precisely, we want to optimize the expression:
where (
V
_{t}
)
_{t≥0}
is a geometric Brownian motion with constant drift α and constant volatility
is the running maximum of the stock price, and the supremum is taken over all possible stopping times 0 ≤ τ ≤ T adapted to the natural filtration (
F
_{t}
)
_{t≥0}
of the stock price. The above problem has been considered by Shiryaev, Xu and Zhou (2008) and Du Toit and Peskir (2009). In this paper we provide an independent proof that when α = 1/2σ
^{2}
, a selling strategy is optimal if and only if it sells the stock either at the terminal time T or at the moment when the stock price hits its maximum price so far. Besides, when α > 1/2σ
^{2}
, selling the stock at the terminal time T is the unique optimal selling strategy. Our approach to the problem is purely probabilistic and has been inspired by relating the notion of dominant stopping
ρ
_{τ}
of a stopping time τ to the optimal stopping strategy arisen in the classical “Secretary Problem”.
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Asymptotic Behaviour of Solutions to the Navierstokes Equations of a Twodimensional Compressible Flow
Yinghui ZHANG, Zhong TAN
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(4): 697712.
DOI:
10.1007/s1025501101155
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In this paper, we are concerned with the asymptotic behaviour of a weak solution to the NavierStokes equations for compressible barotropic flow in two space dimensions with the pressure function satisfying p(
ρ
) = a
ρ
log
^{d}
(
ρ
) for large
ρ
. Here
d
> 2,
a
>0. We introduce useful tools from the theory of Orlicz spaces and construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is also briefly discussed.
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Local Influence Analysis for Semiparametric Reproductive Dispersion Nonlinear Models
Xuedong CHEN, Niansheng TANG, XuerenWANG
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(1): 7590.
DOI:
10.1007/s102550120124z
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The present paper proposes a semiparametric reproductive dispersion nonlinear model (SRDNM) which is an extension of the nonlinear reproductive dispersion models and the semiparameter regression models. Maximum penalized likelihood estimates (MPLEs) of unknown parameters and nonparametric functions in SRDNM are presented. Assessment of local influence for various perturbation schemes are investigated. Some local influence diagnostics are given. A simulation study and a real example are used to illustrate the proposed methodologies.
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Multiple Positive Solutions for Semipositone
m
point Boundary Value Problems
Chengbo Zhai, Cheng Yang
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(3): 419426.
DOI:
10.1007/s1025500961803
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In this article, we establish the existence of at least two positive solutions for the semipositone
m
point boundary value problem with a parameter
where
λ
> 0 is a parameter, 0 < ξ1 < ξ2 < · · · < ξ
m
2 < 1 with 0 <
and
f
(
t, u
) ≥ 
M
with
M
is a positive constant. The method employed is the LeggettWilliams fixedpoint theorem. As an application, an example is given to demonstrate the main result.
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Adaptive Local Linear Quantile Regression
Yunan Su, Maozai Tian
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(3): 509516.
DOI:
10.1007/s1025501100875
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In this paper we propose a new method of local linear adaptive smoothing for nonparametric conditional quantile regression. Some theoretical properties of the procedure are investigated. Then we demonstrate the performance of the method on a simulated example and compare it with other methods. The simulation results demonstrate a reasonable performance of our method proposed especially in situations when the underlying image is piecewise linear or can be approximated by such images. Generally speaking, our method outperforms most other existing methods in the sense of the mean square estimation (MSE) and mean absolute estimation (MAE) criteria. The procedure is very stable with respect to increasing noise level and the algorithm can be easily applied to higher dimensional situations.
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Dynamic Concept of Returns to Scales and Its Characteristics on Production Frontier in Intersection Form
Quanling Wei, Hong Yan
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(3): 355366.
DOI:
10.1007/s1025501100759
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This paper gives a dynamic concept and a new nonparametric method for evaluating returns to scale (RTS) of economic units with multiple inputs and outputs. It is frequently noticed that when we increase the input of a decision making unit (DMU) with a certain status of RTS, different status of RTS is observed. For example, when we increase the input of a DMU with constant RTS under the traditional method, a decreasing RTS is often observed instead of the expected constant RTS. We thus define the RTS of each DMU in both input expansion and contraction regions respectively. The research starts from transferring the production possibility set into the intersection form, by giving the explicit linear inequality representation of production frontiers. The RTS structural characteristics of DMUs' on the production frontier are described. Status of RTS of those DMUs on the production frontier include increasing RTS, constant RTS, decreasing RTS, saturated RTS and evidence of congestion. Necessary and sufficient conditions for RTS evaluation are provided. The definition and evaluation method given here provide more detailed economic characteristics of DMU for policy makers.
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Histogramkernel Error and Its Application for BinWidth Selection in Histograms
Xiuxiang Wang, Jianfang Zhang
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(3): 607624.
DOI:
10.1007/s102550077081y
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Histogram and kernel estimators are usually regarded as the two main classical databased nonparametric tools to estimate the underlying density functions for some given data sets. In this paper we will integrate them and define a histogramkernel error based on the integrated square error between histogram and binned kernel density estimator, and then exploit its asymptotic properties. Just as indicated in this paper, the histogramkernel error only depends on the choice of bin width and the data for the given prior kernel densities. The asymptotic optimal bin width is derived by minimizing the mean histogramkernel error. By comparing with Scott’s optimal bin width formula for a histogram, a new method is proposed to construct the databased histogram without knowledge of the underlying density function. Monte Carlo study is used to verify the usefulness of our method for different kinds of density functions and sample sizes.
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The Hot Spots Conjecture on a Class of Domains in
R
^{n}
with
n
≥ 3
Pengfei YANG
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(4): 639646.
DOI:
10.1007/s1025501101128
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In this paper, we define a class of domains in
R
^{n}
. Using the synchronous coupling of reflecting Brownian motion, we obtain the monotonicity property of the solution of the heat equation with the Neumann boundary conditions. We then show that the hot spots conjecture holds for this class of domains.
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Dimension Splitting Method for the Three Dimensional Rotating NavierStokes Equations
Kaitai LI, Jiaping YU, Feng SHI, Aixiang HUANG
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(3): 417442.
DOI:
10.1007/s1025501201617
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In this paper, we propose a dimensional splitting method for the three dimensional (3D) rotating NavierStokes equations. Assume that the domain is a channel bounded by two surfaces
Im
and is decomposed by a series of surfaces
Im
_{i}
into several subdomains, which are called the layers of the flow. Every interface
Im
_{i}
between two subdomains shares the same geometry. After establishing a semigeodesic coordinate (Scoordinate) system based on
Im
_{i}
, NavierStoke equations in this coordinate can be expressed as the sum of two operators, of which one is called the membrane operator defined on the tangent space on
Im
_{i}
, another one is called the bending operator taking value in the normal space on
Im
_{i}
. Then the derivatives of velocity with respect to the normal direction of the surface are approximated by the Euler central difference, and an approximate form of NavierStokes equations on the surface
Im
_{i}
is obtained, which is called the twodimensional threecomponent (2D3C) NavierStokes equations on a two dimensional manifold. Solving these equations by alternate iteration, an approximate solution to the original 3D NavierStokes equations is obtained. In addition, the proof of the existence of solutions to 2D3C NavierStokes equations is provided, and some approximate methods for solving 2D3C NavierStokes equations are presented.
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Variation of Parameters Formula and Gronwall Inequality for Differential Equations with a General Piecewise Constant Argument
KuoShou CHIU, Manuel PINTO
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(4): 561568.
DOI:
10.1007/s1025501101075
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1117
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A variation of parameters formula and Gronwall type integral inequality are proved for a differential equation involving general piecewise alternately advanced and retarded argument.
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Some Numerical Quadrature Schemes of a Nonconforming Quadrilateral Finite Element
Xiaofei GUAN, Mingxia LI, Shaochun CHEN
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(1): 117126.
DOI:
10.1007/s1025501201279
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Numerical quadrature schemes of a nonconforming finite element method for general second order elliptic problems in two dimensional (2D) and three dimensional (3D) space are discussed in this paper. We present and analyze some optimal numerical quadrature schemes. One of the schemes contains only three sampling points, which greatly improves the efficiency of numerical computations. The optimal error estimates are derived by using some traditional approaches and techniques. Lastly, some numerical results are provided to verify our theoretical analysis.
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Componentwise Complementary Cycles in Multipartite Tournaments
Zhihong HE, Guojun LI, Xueqin ZHOU
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(1): 201208.
DOI:
10.1007/s1025501201359
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The problem of complementary cycles in tournaments and bipartite tournaments was completely solved. However, the problem of complementary cycles in semicomplete
n
partite digraphs with
n
≥ 3 is still open. Based on the definition of componentwise complementary cycles, we get the following result. Let
D
be a 2strong
n
partite (
n
≥ 6) tournament that is not a tournament. Let
C
be a 3cycle of
D
and
D
\
V
(
C
) be nonstrong. For the unique acyclic sequence
D
_{1}
,
D
_{2}
, … ,
D
_{α}
of
D
\
V
(
C
), where
α
≥ 2, let
D
_{c}
= {
D
_{i}

D
_{i}
contains cycles, i = 1, 2, … ,
α
},
D
_{c}
= {
D
_{1}
,
D
_{2}
, … ,
D
_{α}
} \
D
_{c}
. If
D
_{c}
≠ Ø, then
D
contains a pair of componentwise complementary cycles.
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A Remark on the BealeKatoMajda Criterion for the 3D MHD Equations with Zero Kinematic Viscosity
Sadek GALA, Xiaochun CHEN
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(2): 209214.
DOI:
10.1007/s102550120140z
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In this paper, we study the blowup criterion of smooth solutions to the 3D magnetohydrodynamic system in
B
_{∞,∞}
^{0}
. We show that a smooth solution of the 3D MHD equations with zero kinematic viscosity in the whole space
R
^{3}
breaks down if and only if certain norm of the vorticity blows up at the same time.
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Strong Convergence Theorems of Common Elements for Equilibrium Problems and Fixed Point Problems in Banach Spaces
Xinghui GAO, Haiyun ZHOU
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(2): 337350.
DOI:
10.1007/s1025501201484
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In this paper, we consider hybrid algorithms for finding common elements of the set of common fixed points of two families quasiφnonexpansive mappings and the set of solutions of an equilibrium problem. We establish strong convergence theorems of common elements in uniformly smooth and strictly convex Banach spaces with the property (K).
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Constant Barrier Strategies in a Twostate Markovmodulated Dual Risk Model
Xuemin MA, Kui LUO, Guangming WANG, Yijun HU
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(4): 679690.
DOI:
10.1007/s1025501101137
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In this paper, we consider the dividend problem in a twostate Markovmodulated dual risk model, in which the gain arrivals, gain sizes and expenses are influenced by a Markov process. A system of integrodifferential equations for the expected value of the discounted dividends until ruin is derived. In the case of exponential gain sizes, the equations are solved and the best barrier is obtained via numerical example. Finally, using numerical example, we compare the best barrier and the expected discounted dividends in the twostate Markovmodulated dual risk model with those in an associated averaged compound Poisson risk model. Numerical results suggest that one could use the results of the associated averaged compound Poisson risk model to approximate those for the twostate Markovmodulated dual risk model.
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On a BMAP/G/1 Gqueue with Setup Times and Multiple Vacations
Yi PENG, Xiangqun YANG
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(4): 625638.
DOI:
10.1007/s1025501100523
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In this paper, we consider a BMAP/G/1 Gqueue with setup times and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process (BMAP) and Markovian arrival process (MAP) respectively. The arrival of a negative customer removes all the customers in the system when the server is working. The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. By using the supplementary variables method and the censoring technique, we obtain the queue length distributions. We also obtain the mean of the busy period based on the renewal theory.
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Existence of Solutions to Nonlinear Neumann Boundary Value Problems with
p
Laplacian Operator and Iterative Construction
Li Wei, Haiyun Zhou, Ravi P. Agarwal
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(3): 463470.
DOI:
10.1007/s1025501100848
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By using some results of pseudomonotone operator, we discuss the existence and uniqueness of the solution of one kind nonlinear Neumann boundary value problems involving the
p
Laplacian operator. We also construct an iterative scheme converging strongly to this solution.
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Generalized Commutators for Marcinkiewicz Type Integrals with Variable Kernels
Huixia Mo, Shanzhen Lu
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(3): 471480.
DOI:
10.1007/s1025501100857
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Let
A
be a function with derivatives of order
m
and
D
^{ γ}
A
∈
_{β}
(0 <
β
< 1, 
γ
 =
m
). The authors in the paper prove that if Ω(
x, z
) ∈
L
^{∞}
(R
^{n}
) ×
L
^{s}
(
S
^{n1}
) (
s
≥
n
/(
nβ
)) is homogenous of degree zero and satisfies the mean value zero condition about the variable
z
, then both the generalized commutator for Marcinkiewicz type integral
μ
_{Ω}
^{A}
and its variation
are bounded from
L
^{p}
(R
^{n}
) to
L
^{q}
(R
^{n}
), where 1 <
p
<
n
/
β
and 1/
q
= 1/
pβ
/
n
. The authors also consider the boundedness of
μ
_{Ω}
^{A}
and its variation on Hardy spaces.
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Stabilization of Nonuniform EulerBernoulli Beam with Locally Distributed Feedbacks
Xianbing CAO, Qingxu YAN
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(1): 131138.
DOI:
10.1007/s1025501201297
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In this article, we study the stabilization problem of a nonuniform EulerBernoulli beam with locally distributed feedbacks. Firstly, using the semigroup theory, we establish the wellposedness of the associated closed loop system. Then by proving the uniqueness of the solution of a related ordinary differential equations, we derive the asymptotic stability of the closed loop system. Finally, by means of the piecewise frequency domain multiplier method, we prove that the corresponding closed loop system can be exponentially stabilized by only one of the two distributed feedback controls proposed in this paper.
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PainlevéKuratowski Convergences of the Solution Sets to Perturbed Generalized Systems
Zhimiao FANG, Shengjie LI
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(2): 361370.
DOI:
10.1007/s1025501201493
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In this paper, we obtain the PainlevéKuratowski Convergence of the efficient solution sets, the weak efficient solution sets and various proper efficient solution sets for the perturbed generalized system with a sequence of mappings converging in a real locally convex Hausdorff topological vector spaces.
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Regularity Criteria for the Threedimensional MHD Equations
Lan LUO, Yongye ZHAO, Qing YANG
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(4): 581594.
DOI:
10.1007/s1025501101084
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In this paper, we consider regularity criteria for solutions to the 3D MHD equations with incompressible conditions. By using some classical inequalities, we obtain the regularity of strong solutions to the threedimensional MHD equations under certain sufficient conditions in terms of one component of the velocity field and the magnetic field respectively.
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Approximation by
q
Baskakov Beta Operators
Vijay GUPTA, Ali ARAL
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(4): 569580.
DOI:
10.1007/s102550110072z
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In the present paper we introduce the
q
analogue of the Baskakov Beta operators. We establish some direct results in the polynomial weighted space of continuous functions defined on the interval [0, ∞). Then we obtain pointwise estimate, using the Lipschitz type maximal function.
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Uzawa Iteration Method for Stokes Type Variational Inequality of the Second Kind
Yuan Li, Kaitai Li
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(2): 303316.
DOI:
10.1007/s1025501100630
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In this paper, the Uzawa iteration algorithm is applied to the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind. Firstly, the multiplier in a convex set is introduced such that the variational inequality is equivalent to the variational identity. Moreover, the solution of the variational identity satisfies the saddlepoint problem of the Lagrangian functional
L
. Subsequently, the Uzawa algorithm is proposed to solve the solution of the saddlepoint problem. We show the convergence of the algorithm and obtain the convergence rate. Finally, we give the numerical results to verify the feasibility of the Uzawa algorithm.
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On the Existence of Solutions for an Elliptic System of Equations with Arbitrary Order Growth
Zhongyuan LIU
Acta Mathematicae Applicatae Sinica(English Series). 2013, 29(2): 415424.
DOI:
10.1007/s1025501302244
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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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Remarks on the Lower Bounds for the Average Genus
Yichao Chen
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(3): 517526.
DOI:
10.1007/s1025501100884
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785
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Let
G
be a graph of maximum degree at most four. By using the overlap matrix method which is introduced by B. Mohar, we show that the average genus of
G
is not less than (1/3) of its maximum genus, and the bound is best possible. Also, a new lower bound of average genus in terms of girth is derived.
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Approximate Damped Oscillatory Solutions for Compound KdVBurgers Equation and Their Error Estimates
Weiguo ZHANG, Yan ZHAO, Xiaoyan TENG
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(2): 305324.
DOI:
10.1007/s1025501201475
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In this paper, we focus on studying approximate solutions of damped oscillatory solutions of the compound KdVBurgers equation and their error estimates. We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdVBurgers equation. We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions. Furthermore, we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient r of the equation. We obtain two critical values of r, and find that a bounded traveling wave appears as a kink profile solitary wave if 
r
 is greater than or equal to some critical value, while it appears as a damped oscillatory wave if 
r
 is less than some critical value. By means of analysis and the undetermined coefficients method, we find that the compound KdVBurgers equation only has three kinds of bell profile solitary wave solutions without dissipation. Based on the above discussions and according to the evolution relations of orbits in the global phase portraits, we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method. Finally, using the homogenization principle, we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions. Moreover, we also give the error estimates for these approximate solutions.
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The Decision of Prime and Primary Ideal
Jinwang LIU, Dongmei LI
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(4): 595600.
DOI:
10.1007/s1025501101093
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1097
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We give more efficient criteria to characterise prime ideal or primary ideal. Further, we obtain the necessary and sufficient conditions that an ideal is prime or primary in real field from the Gröbner bases directly.
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Symmetry of the Point Spectrum of Infinite Dimensional Hamiltonian Operators and Its Applications
Hua WANG, Alatancang, Junjie HUANG
Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(1): 149156.
DOI:
10.1007/s1025501201301
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1070
)
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This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator
H
. Note that the point spectrum of
H
can be described as
σ
_{p}
(
H
) =
σ
_{p}
(
A
)∪
σ
_{p}
^{1}
(
A
*). Using the characteristic of the set
σ
_{p}
^{1}
(
A
*), we divide the point spectrum
σ
_{p}
(
A
) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which
σ
_{p}
^{1}
(
A
*) and one part of
σ
_{p}
(
A
) are symmetric with respect to the real axis each other. Based on this result, the symmetry of
σ
_{p}
(
H
) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.
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Periodic Solution to BAMtype CohenGrossberg Neural Network with Timevarying Delays
Anping Chen, Qunhua Gu
Acta Mathematicae Applicatae Sinica(English Series). 2011, 27(3): 427442.
DOI:
10.1007/s102550110082x
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862
)
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By using the continuation theorem of Mawhin's coincidence degree theory and the Liapunov functional method, some sufficient conditions are obtained to ensure the existence, uniqueness and the global exponential stability of the periodic solution to the BAMtype CohenGrossberg neural networks involving timevarying delays.
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