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    20 January 2005, Volume 25 Issue 1 Previous Issue    Next Issue
    Articles
    TWO REMARKS ON SCHWARZ FORMULA
    DING Jia-Qi, LUO Pei-Zhu
    Acta mathematica scientia,Series B. 2005, 25 (1):  1-6. 
    Abstract ( 601 )   RICH HTML PDF (97KB) ( 1048 )   Save

    This paper discusses two problems. Firstly the authors give the Schwarz
    formula for a holomorphic function in unit disc when the boundary value of its real part
    is in the class H of generalized functions in the sense of Hua. Secondly the authors use
    the classical Schwarz formula to give a new proof of the zero free region of the Riemann
    zeta-function.

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    NECESSARY CONDITIONS FOR OPTIMAL CONTROLS OF SEMILINEAR ELLIPTIC VARIATIONAL INEQUALITIES INVOLVING STATE CONSTRAINT
    HONG Geng-Sheng
    Acta mathematica scientia,Series B. 2005, 25 (1):  7-22. 
    Abstract ( 580 )   RICH HTML PDF (160KB) ( 827 )   Save

    This paper deals with maximum principle for some optimal control problem governed by some elliptic variational inequalities. Some state constraints are discussed. The basic techniques used here are based on those in [1] and a new penalty functional defined in this paper.

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    THE CLASSIFICATION AND PERIOD OF STATES FOR MARKOV CHAIN IN RANDOM
    ENVIRONMENT
    HU Di-He
    Acta mathematica scientia,Series B. 2005, 25 (1):  23-29. 
    Abstract ( 565 )   RICH HTML PDF (112KB) ( 1251 )   Save

    This paper is a continuation of [8]. In Section 1, three kinds of communication
    are introdnced for two states and the relations among them are investigated. In Section 2,
    two kinds of period of a state are introdnced and it is obtained that the period is a “class
    property”,i.e. two states x and y belong to same class implies the period of x is equal to
    the period of y.

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    THE EXISTENCE THEOREM OF OPTIMAL GROWTH MODEL
    GONG Liu-Tang, BANG Xian-Ze
    Acta mathematica scientia,Series B. 2005, 25 (1):  30-40. 
    Abstract ( 632 )   RICH HTML PDF (137KB) ( 900 )   Save

    This paper proves a general existence theorem of optimal growth theory. This
    theorem is neither restricted to the case of a constant technology progress, nor stated
    in terms of mathematical conditions which have no direct economic interpretation and
    moreover, are difficult to apply.

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    CYCLE SPACES OF GRAPHS ON THE SPHERE AND THE PROJECTIVE PLANE
    LIN Han, LIU Pan-Pei, MA De-Ju, LEI Dun-Jie
    Acta mathematica scientia,Series B. 2005, 25 (1):  41-49. 
    Abstract ( 581 )   RICH HTML PDF (191KB) ( 866 )   Save

    Cycle base theory of a graph has been well studied in abstract mathematical
    field such matroid theory as Whitney and Tutte did and found many applications in prat-
    ical uses such as electric circuit theory and structure analysis, etc. In this paper graph
    embedding theory is used to investigate cycle base structures of a 2-(edge)-connected graph
    on the sphere and the projective plane and it is shown that short cycles do generate the
    cycle spaces in the case of “small face-embeddings”. As applications the authors find the
    exact formulae for the minimum lengthes of cycle bases of some types of graphs and present
    several known results. Infinite examples shows that the conditions in their main results are
    best possible and there are many 3-connected planar graphs whose minimum cycle bases
    can not be determined by the planar formulae but may be located by re-embedding them
    into the projective plane.

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    CONSISTENCY OF LS ESTIMATOR IN SIMPLE
    LINEAR EV REGRESSION MODELS
    LIU Ji-Hua, CHEN Xi-Ru
    Acta mathematica scientia,Series B. 2005, 25 (1):  50-58. 
    Abstract ( 567 )   RICH HTML PDF (127KB) ( 1226 )   Save

    Consistency of LS estimate of simple linear EV model is studied. It is shown
    that under some common assumptions of the model, both weak and strong consistency of
    the estimate are equivalent but it is not so for quadratic-mean consistency.

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    GLOBAL ATTRACTIVITY IN A CLASS OF HIGHER-ORDER NONLINEAR DIFFERENCE EQUATION
    LI Mo-Tong, ZHANG Yan-Gong, SU Wei-Hui
    Acta mathematica scientia,Series B. 2005, 25 (1):  59-66. 
    Abstract ( 587 )   RICH HTML PDF (117KB) ( 1336 )   Save

    In this paper the global attractivity of the nonlinear difference equation

    x_{n+1}=\frac{a+bx_n}{A+x_{n-k}},\quad n=0,1,\cdots ,

    is investigated, where $a,b,A\in (0,\infty ),$ $k$ is an positive integer

    and the initial conditions $x_{-k},\cdots,x_{-1}$ and $x_0$ are arbitrary

    positive numbers. It is shown that the unique positive equilibrium of the

    equation is global attractive. As a corollary, the result gives a positive

    confirmation on the conjecture presented by Kocic and Ladas [1,p154].

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    A SUPERLINEARLY CONVERGENT TRUST REGION ALGORITHM FOR LC1 CONSTRAINED OPTIMIZATION PROBLEMS
    OU Yi-Gui, HOU Ding-Pi
    Acta mathematica scientia,Series B. 2005, 25 (1):  67-80. 
    Abstract ( 544 )   RICH HTML PDF (155KB) ( 1221 )   Save

    In this paper, a new trust region algorithm for nonlinear equality constrained
    LC1 optimization problems is given. It obtains a search direction at each iteration not by
    solving a quadratic programming subproblem with a trust region bound, but by solving
    a system of linear equations. Since the computational complexity of a QP-Problem is in
    general much larger than that of a system of linear equations, this method proposed in
    this paper may reduce the computational complexity and hence improve computational
    efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm
    is globally and super-linearly convergent to a solution of the original problem. Some
    numerical examples are reported, showing the proposed algorithm can be beneficial from
    a computational point of vie

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    MULTIPLE POSITIVE SOLUTIONS OF SINGULAR THIRD-ORDER PERIODIC BOUNDARY VALUE
    SUN Ji-Xian, LIU Yan-Qing
    Acta mathematica scientia,Series B. 2005, 25 (1):  81-88. 
    Abstract ( 529 )   RICH HTML PDF (123KB) ( 1419 )   Save

    This paper deals with the singular
    nonlinear third-order periodic boundary value problem
    $u^{\prime\prime\prime}+\r^3u = f(t, u)$, $0\l t \l 2\pi$, with
    $u^{(i)}(0)= u^{(i)}(2\pi)$, $i=0, 1, 2$, where
     $\r\in (0, \f{1}{\s})$ and $f$ is singular at $t=0$, $t=1$
    and $u=0$. Under
      suitable weaker conditions than those of [1], it is proved by
      constructing a special cone in $C[0, 2\pi]$ and
      employing the fixed point index theory that the
      problem has at least one or at least two positive solutions.

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    INTERIOR ESTIMATES IN MORREY SPACES FOR SOLUTIONS OF ELLIPTIC EQUATIONS AND WEIGHTED BOUNDEDNESS FOR COMMUTATORS OF SINGULAR INTEGRAL OPERATORS
    LIU Lan-Zhe
    Acta mathematica scientia,Series B. 2005, 25 (1):  89-94. 
    Abstract ( 812 )   RICH HTML PDF (117KB) ( 2279 )   Save

    It is proved that, for the nondivergence elliptic equations $\sum_{i,j=1}^n$

     $a_{ij}u_{x_ix_j}=f$,  if $f$ belongs to the generalized Morrey spaces

    $L^{p,\varphi}(\omega)$, then $u_{x_ix_j}\in L^{p, \varphi}(\omega)$, where $u$ is the $W^{2,p}$-solution of the equations.  In order to obtain this, the author first establish

    the weighted boundedness for the commutators of some singular integral operators on $L^{p,\varphi}(\omega)$. \noindent\ke{\bf Key words}{\rm  Nondivergence elliptic equation, generalized Morrey space, commutator of singular integral operator, $A_p$ weight}

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    STATIONARY SOLUTION FOR A STOCHASTIC LI´ENARD EQUATION WITH MARKOVIAN SWITCHING
    XI Fu-Bao, DIAO Li-Qin
    Acta mathematica scientia,Series B. 2005, 25 (1):  95-104. 
    Abstract ( 556 )   RICH HTML PDF (135KB) ( 1181 )   Save

    This paper considers a stochastic Li´enard equation with Markovian switching.
    The Feller continuity of its solution is proved by the coupling method and a truncation
    argument. The existence of a stationary solution for the equation is also proved under the
    Foster-Lyapunov drift condition.

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    COMPOSITION OPERATORS ON HARDY-ORLICZ SPACES
    Liu-Li-Fang, CAO Guang-Fu, WANG Xiao-Feng
    Acta mathematica scientia,Series B. 2005, 25 (1):  105-111. 
    Abstract ( 602 )   RICH HTML PDF (129KB) ( 950 )   Save

    Properties of composition operators induced by analytic self-maps on the unit
    disk of the complex plane in Hardy-Orlicz spaces are discussed. Results are concerned
    about boundedness, invertibility, compactness, Fredholm operators and spectra of compo-
    sition operators.

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    REMARKS ON BRUALDI’S THEOREM FOR SPECTRUM INCLUSION REGION OF AN IRREDUCIBLE MATRIX
    CHEN Shen-Can
    Acta mathematica scientia,Series B. 2005, 25 (1):  112-118. 
    Abstract ( 534 )   RICH HTML PDF (111KB) ( 864 )   Save

    This paper obtains a necessary and sufficient condition for an irreducible com-
    plex matrix whose comparison matrix is a singular M-matrix to be singular. This is used
    to establish a necessary and sufficient condition for a boundary point of Brualdi’s inclusion
    region of the eigenvalues of an irreducible complex matrix to be an eigenvalue.

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    ASYMPTOTIC STABILITY OF RAREFACTION WAVE FOR GENERALIZED BURGERS EQUATION
    XU Yan-Ling, JIANG Mai-Na
    Acta mathematica scientia,Series B. 2005, 25 (1):  119-129. 
    Abstract ( 596 )   RICH HTML PDF (155KB) ( 1202 )   Save

    This paper is concerned with the stability of
    the rarefaction wave for the Burgers equation \\
    $$
      \left\{\begin{array}{l}
        u_t+f(u)_x=\mu t^{\alpha}u_{xx},
          \ \ \ \mu >0,\ \ x \in {\bf R},\ \ t > 0,\\
        u|_{t=0}=u_0(x) \rightarrow u_{\pm},\ \ \ x
        \rightarrow{\pm}{\infty},
    \end{array}
    \right.
    \eqno({\rm I})
    $$
    where $ 0\leq {\alpha}<{\frac{1}{4q}}$ ($q$
     is determined by $(2.2)$). Roughly
    speaking, under the assumption that
    $u_-<u_+$, the authors prove the existence of the global smooth solution
    to the Cauchy problem (I), also find
    the solution $u(x,t)$ to the Cauchy problem (I)
    satisfying $\sup\limits_{x\in {\bf R}}|u(x,t)-u^R(x/t)| \rightarrow 0$
    as $t \rightarrow \infty$, where $u^R(x/t)$ is the rarefaction wave of
    the non-viscous Burgers equation $ u_t+f(u)_x=0 $ with Riemann initial
    data
    $    u(x,0)=\left\{\begin{array}{l}
            u_-, \ \ x<0, \\
            u_+, \ \ x>0.
    \end{array}
    \right.

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    DIFFERENTIABILITY FOR THE HIGH DIMENSIONAL POLYNOMIAL-LIKE ITERATIVE EQUATION
    LI Xiao-Pei, DENG Ku-Fu
    Acta mathematica scientia,Series B. 2005, 25 (1):  130-136. 
    Abstract ( 611 )   RICH HTML PDF (108KB) ( 1275 )   Save

    This paper studies the smoothness of solutions of the higher dimensional
    polynomial-like iterative equation. The methods given by Zhang Weinian[7] and by Kul-
    czycki M, Tabor J.[3] are improved by constructing a new operator for the structure of the
    equation in order to apply fixed point theorems. Existence, uniqueness and stability of
    continuously differentiable solutions are given.

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    COMMUTATORS OF SINGULAR INTEGRAL OPERATORS WITH NON-SMOOTH KERNELS
    DENG Dong-Gao, YA Li-Xin
    Acta mathematica scientia,Series B. 2005, 25 (1):  137-144. 
    Abstract ( 568 )   RICH HTML PDF (130KB) ( 1248 )   Save

    Let $T$ be a singular integral operator
     bounded on $L^p({{\bf R}^n})$ for some $p$, $1<p<\infty$.
     The authors give
    a sufficient condition on the kernel of
    $T$ so that when $b\in $BMO,  the  commutator 
    $[b, T](f)=T(b f)-b T(f)
    $ is bounded on the space $L^p$ for all $p, 1<p<\infty$.
    The condition of this paper is weaker than the usual pointwise
    H\"ormander condition.

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    ON f-EDGE COVER-COLOURING OF SIMPLE GRAPHS
    SONG Hui-Min, LIU Gui-Zhen
    Acta mathematica scientia,Series B. 2005, 25 (1):  145-151. 
    Abstract ( 519 )   RICH HTML PDF (112KB) ( 976 )   Save

    In this paper all graphs are simple and finite, so they
    will have no loops or multiple edges. A graph $G$ with vertex set
    $V$ and edge set $E$ is denoted by $G=(V,E)$. In an ordinary
    edge cover-colouring, each colour appears at every vertex at
    least one time. The maximum number of colours needed for an edge
    cover colouring of $G$ is called the edge cover chromatic
    index of $G$, denoted by $\chi^{'}_{c}(G)$. Gupta's theorem$^{[1]}$
     tells that $\delta(G)-1\leq \chi^{'}_{c}(G)\leq \delta(G)$.

      Let $f$ be a positive integer-valued function defined on $V$ such that
      $f(v)\leq d(v)$ for every vertex $v \in V$.  An $f$-edge
    cover-colouring of $G$ is a colouring of edges of $G$  such
    that every colour appears at each vertex $v \in V$ at least $f(v)$
    times. The maximum number of colours needed for $f$-edge cover
    colour $G$ is called the $f$-edge cover chromatic index
    of $G$, denoted by $\chi^{'}_{fc}(G)$.

    If there is a vertex $x\in V(G)$ with $\frac{d(x)}{2}<f(x)\leq
    d(x)$, then $\chi^{'}_{fc}(G)\leq 1$. In general, we suppose
    $f(v)\leq \frac{d(v)}{2}$ for every vertex $v\in V$.

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    SUPER EFFICIENCY IN THE NEARLY CONE-SUBCONVEXLIKE VECTOR OPTIMIZATION
    WITH SET-VALUED FUNCTIONS
    XU Xi-Gong, LIU San-Yang
    Acta mathematica scientia,Series B. 2005, 25 (1):  152-160. 
    Abstract ( 591 )   RICH HTML PDF (118KB) ( 1331 )   Save

    Some properties for convex cones are discussed, which are used to obtain an
    equivalent condition and another important property for nearly cone-subconvexlike set-
    valued functions. Under the nearly cone-subconvexlikeness, some characterizations of the
    super efficiency are given in terms of scalarization and Lagrangian multipliers. Related
    results are generalized.

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    ROBUST GLOBAL EXPONENTIAL STABILITY OF UNCERTAIN IMPULSIVE SYSTEMS
    LIU Bin, LIU Xin-Zhi, LIAO Xiao-Cuan
    Acta mathematica scientia,Series B. 2005, 25 (1):  161-169. 
    Abstract ( 591 )   RICH HTML PDF (138KB) ( 1254 )   Save

    By using the quasi-Lyapunov function, some sufficient conditions of global
    exponential stability for impulsive systems are established, which is the basis for the fol-
    lowing discussion. Then, by employing Riccati inequality and Hamilton-Jacobi inequal-
    ity approach, some sufficient conditions of robust exponential stability for uncertain lin-
    ear/nonlinear impulsive systems are derived, respectively. Finally, some examples are given
    to illustrate the applications of the theory.

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    GLOBAL CONVERGENCE RESULTS OF A THREE TERM MEMORY GRADIENT METHOD WITH A
    NON-MONOTONE LINE SEARCH TECHNIQU
    SUN Qing-Ying
    Acta mathematica scientia,Series B. 2005, 25 (1):  170-178. 
    Abstract ( 743 )   RICH HTML PDF (114KB) ( 1183 )   Save

    In this paper, a new class of three term memory gradient method with non-monotone line search technique for unconstrained optimization is presented. Global convergence properties of the new methods are
    discussed. Combining the quasi-Newton method with the  new method, the former is modified to have global convergence property. Numerical results show that the new algorithm is efficient.

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    LOCAL INFLUENCE ASSESSMENT IN A MULTIVARIATE t-MODEL WITH RAO’S SIMPLE STRUCTURE
    JU Qing-Meng, ZHANG Fu-Xiong
    Acta mathematica scientia,Series B. 2005, 25 (1):  179-192. 
    Abstract ( 480 )   RICH HTML PDF (170KB) ( 903 )   Save

    The local influence analysis is an important problem in statistical inference
    and some models have been discussed in many literatures[1−5] . This paper deals with
    the problem of assessing local influences in a multivariate t-model with Rao’s simple struc-
    ture(RSS). Based on Cook’s likelihood displacement, the effects of some minor perturbation
    on the statistical inference is assessed. As an application, a common covariance-weighted
    perturbation is thoroughly discussed.

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