Acta Mathematicae Applicatae Sinica(English Series)

15 October 2012, Volume 28 Issue 4

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ARTICLES
Partitioning Complete Graphs by Heterochromatic Trees

Ze-min JIN, Xue-liang LI

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 625-630.

DOI:10.1007/s10255-012-0177-z
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A heterochromatic tree is an edge-colored tree in which any two edges have different colors. The heterochromatic tree partition number of an r-edge-colored graph G, denoted by t_r(G), is the minimum positive integer p such that whenever the edges of the graph G are colored with r colors, the vertices of G can be covered by at most p vertex-disjoint heterochromatic trees. In this paper we determine the heterochromatic tree partition number of r-edge-colored complete graphs. We also find at most t_r(K_n) vertex-disjoint heterochromatic trees to cover all the vertices in polynomial time for a given r-edge-coloring of K_n.
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On the Covering Number c_λ(3,W₄⁽³⁾, v)

Yan WU, Yan-xun CHANG

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 631-638.

DOI:10.1007/s10255-012-0178-y
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A t-hyperwheel (t ≥ 3) of length l (or W_l^(t) for brevity) is a t-uniform hypergraph (V,E), where E = {e₁, e₂, · · · , e_l} and v₁, v₂, · · · , v_l are distinct vertices of V =() such that for i = 1, · · · , l, v_i, v_i+1 ∈ e_i and e_i ∩ e_j = P, j () {i - 1, i, i + 1}, where the operation on the subscripts is modulo l and P is a vertex of V which is different from v_i, 1 ≤ i ≤ l. In this paper, the minimum covering problem of MC_λ(3,W₄⁽³⁾, v) is investigated. Direct and recursive constructions on MC_λ(3,W₄⁽³⁾, v) are presented. The covering number c_λ(3,W₄⁽³⁾, v) is finally determined for any positive integers v ≥ 5 and λ.
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A Note on the Blow-up Criterion of Smooth Solutions to the 3D Incompressible MHD Equations

Sadek GALA

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 639-642.

DOI:10.1007/s10255-012-0175-1
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In this note, we will give a new proof of the blow-up criterion of smooth solutions to the 3D incompressible magneto-hydrodynamic equations by a simple application of Gagliardo-Nirenberg' s inequality.
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Two Efficient Parameterized Boundaries for Ve?e?'s Asian Option Pricing PDE

Bai-min YU

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 643-652.

DOI:10.1007/s10255-012-0179-x
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In this paper, we derive two general parameterized boundaries of finite difference scheme for Ve?e?'s PDE which is used to price both fixed and floating strike Asian options. Using these two boundaries, we can deal with all kinds of situations, especially, some extreme cases, like overhigh volatility, very small volatility, etc, under which the Asian option is usually mispriced in many existing numerical methods. Numerical results show that our boundaries are pretty efficient.
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A Multiplicity Result of Positive Solutions for a Class of Multi-parameter Ordinary Differential Systems

Xi-you CHENG, Xiao-lin YAN

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 653-662.

DOI:10.1007/s10255-012-0180-4
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In this paper, we establish a result of Leray-Schauder degree on the order interval which is induced by a pair of strict lower and upper solutions for a system of second-order ordinary differential equations. As applications, we prove the global existence of positive solutions for a multi-parameter system of second-order ordinary differential equations with respect to parameters. The discussion is based on the result of Leray- Schauder degree on the order interval and the fixed point index theory in cones.
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Joint Modeling of Failure Time Data with Transformation Model and Longitudinal Data When Covariates are Measured with Errors

Xi-ming CHENG, Qi GONG

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 663-672.

DOI:10.1007/s10255-012-0192-0
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Semiparametric transformation models provide a class of flexible models for regression analysis of failure time data. Several authors have discussed them under different situations when covariates are timeindependent (Chen et al., 2002; Cheng et al., 1995; Fine et al., 1998). In this paper, we consider fitting these models to right-censored data when covariates are time-dependent longitudinal variables and, furthermore, may suffer measurement errors. For estimation, we investigate the maximum likelihood approach, and an EM algorithm is developed. Simulation results show that the proposed method is appropriate for practical application, and an illustrative example is provided.
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Dimensional Results for Cartesian Products of Homogeneous Moran Sets

Li CAO, Xing-Gang HE

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 673-680.

DOI:10.1007/s10255-012-0181-3
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M(J,{m_s * n_s}, {c_s}) be the collection of Cartesian products of two homogenous Moran sets with the same ratios {c_s} where J = [0, 1]×[0, 1]. Then the maximal and minimal values of the Hausdorff dimensions for the elements in M are obtained without any restriction on {m_sn_s} or {c_s}.
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The Existence of Solutions for the Problem in the Optical Lattice with Nonlocal Nonlinearity

Rui-feng ZHANG, Ren-tao LIU

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 681-690.

DOI:10.1007/s10255-012-0182-2
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We address the impact of nonlocality in the physical features exhibited by solitons supported by Kerr-type nonlinear media with an imprinted optical lattice. We discuss the solitons solution for a class of nonlinear Schrödinger equations in the optical lattice with nonlocal nonlinearity. We also show via a uniform priori estimate that existence and uniqueness of the global solution for the initial problem.
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A Finite Difference Scheme for Solving the Undamped Timoshenko Beam Equations with Both Ends Free

Fu-le LI, Zong-hu XIU

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 691-704.

DOI:10.1007/s10255-012-0183-1
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In this paper, a boundary feedback system of a class of non-uniform undamped Timoshenko beam with both ends free is considered. A linearized three-level difference scheme for the Timoshenko beam equations is derived by the method of reduction of order on uniform meshes. The unique solvability, unconditional stability and convergence of the difference scheme are proved by the discrete energy method. The convergence order in maximum norm is of order two in both space and time. The validity of this theoretical analysis is verified experimentally.
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On the Seidel Integral Complete Multipartite Graphs

Sheng-mei LV, LiangWEI, Hai-xing ZHAO

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 705-710.

DOI:10.1007/s10255-012-0126-x
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For a simple undirected graph G, denote by A(G) the (0, 1)-adjacency matrix of G. Let thematrix S(G) = J-I-2A(G) be its Seidel matrix, and let S_G(λ) = det(λI-S(G)) be its Seidel characteristic polynomial, where I is an identity matrix and J is a square matrix all of whose entries are equal to 1. If all eigenvalues of S_G(λ) are integral, then the graph G is called S-integral. In this paper, our main goal is to investigate the eigenvalues of S_G(λ) for the complete multipartite graphs G = K_{n₁,n₂,···,n_t} . A necessary and sufficient condition for the complete tripartite graphs K_m,n,t and the complete multipartite graphs () to be S-integral is given, respectively.
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Perturbation of Spectra for a Class of 2?2 Operator Matrices

ALATANCANG, Guo-lin HOU, Guo-jun HAI

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 711-720.

DOI:10.1007/s10255-012-0195-x
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In this paper, we study the perturbation of spectra for 2 × 2 operator matrices such as M_X = and M_Z = on the Hilbert space H ⊕ K and the (tu711-3) P_σ(M_X), R_σ(M_X) and (tu711-4) σ(M_Z), (tu711-4) P_σ(M_Z), R_σ(M_Z), (tu711-4) C_σ(M_Z), where R(C) is a closed subspace, are characterized
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A New Algorithm for Total Variation Based Image Denoising

Yi-ping XU

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 721-730.

DOI:10.1007/s10255-012-0184-0
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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 multi-grid 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 signal-to-noise ratio.
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Empirical Likelihood Analysis of Longitudinal Data Involving Within-subject Correlation

Shuang HU, Lu LIN

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 731-744.

DOI:10.1007/s10255-011-0070-1
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In this paper we use profile empirical likelihood to construct confidence regions for regression coefficients in partially linear model with longitudinal data. The main contribution is that the within-subject correlation is considered to improve estimation efficiency. We suppose a semi-parametric structure for the covariances of observation errors in each subject and employ both the first order and the second order moment conditions of the observation errors to construct the estimating equations. Although there are nonparametric estimators, the empirical log-likelihood ratio statistic still tends to a standard χ_p² variable in distribution after the nuisance parameters are profiled away. A data simulation is also conducted.
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A New Subspace Correction Method for Nonlinear Unconstrained Convex Optimization Problems

Rong-liang CHEN, Jin-ping ZENG

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 745-756.

DOI:10.1007/s10255-012-0185-z
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This paper gives a new subspace correction algorithm for nonlinear unconstrained convex opti- mization problems based on the multigrid approach proposed by S. Nash in 2000 and the subspace correction algorithm proposed by X. Tai and J. Xu in 2001. Under some reasonable assumptions, we obtain the convergence as well as a convergence rate estimate for the algorithm. Numerical results show that the algorithm is effective.
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Supports of Fourier Transforms of Refinable Frame Functions and Their Applications to FMRA

Yun-zhang LI, Chun-hua HAN

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 757-768.

DOI:10.1007/s10255-012-0186-y
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Let M be a d × d expansive matrix, and FL²(Ω) be a reducing subspace of L²(R^d). This paper characterizes bounded measurable sets in R^d which are the supports of Fourier transforms of M-refinable frame functions. As applications, we derive the characterization of bounded measurable sets as the supports of Fourier transforms of FMRA (W-type FMRA) frame scaling functions and MRA (W-type MRA) scaling functions for FL²(Ω), respectively. Some examples are also provided.
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Variable Selection in the Partially Linear Errors-in-Variables Models for Longitudinal Data

Yi-ping YANG, Liu-gen XUE, Wei-hu CHENG

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 769-780.

DOI:10.1007/s10255-012-0187-x
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This paper proposes a new approach for variable selection in partially linear errors-in-variables (EV) models for longitudinal data by penalizing appropriate estimating functions. We apply the SCAD penalty to simultaneously select significant variables and estimate unknown parameters. The rate of convergence and the asymptotic normality of the resulting estimators are established. Furthermore, with proper choice of regularization parameters, we show that the proposed estimators perform as well as the oracle procedure. A new algorithm is proposed for solving penalized estimating equation. The asymptotic results are augmented by a simulation study.
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Asymptotic Normality of Wavelet Density Estimator under Censored Dependent Observations

Si-li NIU

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 781-794.

DOI:10.1007/s10255-012-0188-9
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In this paper, we discuss the asymptotic normality of the wavelet estimator of the density function based on censored data, when the survival and the censoring times form a stationary α-mixing sequence. To simulate the distribution of estimator such that it is easy to perform statistical inference for the density function, a random weighted estimator of the density function is also constructed and investigated. Finite sample behavior of the estimator is investigated via simulations too.
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Estimates for the Finite-time Ruin Probability with Insurance and Financial Risks

Min ZHOU, Kai-yongWANG, Yue-baoWANG

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 795-806.

DOI:10.1007/s10255-012-0189-8
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The paper gives estimates for the finite-time ruin probability with insurance and financial risks. When the distribution of the insurance risk belongs to the class L(γ) for some γ > 0 or the subexponential distribution class, we abtain some asymptotic equivalent relationships for the finite-time ruin probability, respectively. When the distribution of the insurance risk belongs to the dominated varying-tailed distribution class, we obtain asymptotic upper bound and lower bound for the finite-time ruin probability, where for the asymptotic upper bound, we completely get rid of the restriction of mutual independence on insurance risks, and for the lower bound, we only need the insurance risks to have a weak positive association structure. The obtained results extend and improve some existing results.
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Heavy Traffic Limit Theorems for a Queue with Poisson ON/OFF Long-range Dependent Sources and General Service Time Distribution

Wan-yang DAI

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 807-822.

DOI:10.1007/s10255-012-0190-2
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In Internet environment, traffic flow to a link is typically modeled by superposition of ON/OFF based sources. During each ON-period for a particular source, packets arrive according to a Poisson process and packet sizes (hence service times) can be generally distributed. In this paper, we establish heavy traffic limit theorems to provide suitable approximations for the system under first-in first-out (FIFO) and work-conserving service discipline, which state that, when the lengths of both ON- and OFF-periods are lightly tailed, the sequences of the scaled queue length and workload processes converge weakly to short-range dependent reflecting Gaussian processes, and when the lengths of ON- and/or OFF-periods are heavily tailed with infinite variance, the sequences converge weakly to either reflecting fractional Brownian motions (FBMs) or certain type of longrange dependent reflecting Gaussian processes depending on the choice of scaling as the number of superposed sources tends to infinity. Moreover, the sequences exhibit a state space collapse-like property when the number of sources is large enough, which is a kind of extension of the well-known Little's law for M/M/1 queueing system. Theory to justify the approximations is based on appropriate heavy traffic conditions which essentially mean that the service rate closely approaches the arrival rate when the number of input sources tends to infinity.
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Decomposition of λK_v into Five Graphs with Six Vertices and Eight Edges

Lan-dang YUAN, Qing-de KANG

Acta Mathematicae Applicatae Sinica(English Series). 2012, 28(4): 823-832.

DOI:10.1007/s10255-012-0191-1
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In this paper, we discuss the G-decomposition of λK_v (G-GD_λ(v)) into five graphs with six vertices and eight edges. We present some recursive structures and a number of G-designs of small orders, holey Gdesigns, and incomplete G-designs are constructed. Finally, the spectrum of the existence of G-GD_λ(v) is determined.

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15 October 2012, Volume 28 Issue 4

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