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20 January 2005, Volume 25 Issue 1 Previous Issue    Next Issue

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TWO REMARKS ON SCHWARZ FORMULA DING Jia-Qi, LUO Pei-Zhu Acta mathematica scientia,Series B. 2005, 25 (1):  1-6.  Abstract ( 602 )   RICH HTML PDF (97KB) ( 1057 )   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. References | Related Articles | Metrics

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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 ( 582 )   RICH HTML PDF (160KB) ( 833 )   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. References | Related Articles | Metrics

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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 ( 568 )   RICH HTML PDF (112KB) ( 1278 )   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. References | Related Articles | Metrics

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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 ( 633 )   RICH HTML PDF (137KB) ( 904 )   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. References | Related Articles | Metrics

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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 ( 585 )   RICH HTML PDF (191KB) ( 871 )   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. References | Related Articles | Metrics

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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 ( 570 )   RICH HTML PDF (127KB) ( 1325 )   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. References | Related Articles | Metrics

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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 ( 588 )   RICH HTML PDF (117KB) ( 1351 )   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]. References | Related Articles | Metrics

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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 ( 546 )   RICH HTML PDF (155KB) ( 1233 )   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 References | Related Articles | Metrics

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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 ( 530 )   RICH HTML PDF (123KB) ( 1439 )   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. References | Related Articles | Metrics

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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 ( 816 )   RICH HTML PDF (117KB) ( 2309 )   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} References | Related Articles | Metrics

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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) ( 1190 )   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. References | Related Articles | Metrics

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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 ( 606 )   RICH HTML PDF (129KB) ( 954 )   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. References | Related Articles | Metrics

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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 ( 541 )   RICH HTML PDF (111KB) ( 866 )   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. References | Related Articles | Metrics

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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 ( 597 )   RICH HTML PDF (155KB) ( 1209 )   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. References | Related Articles | Metrics

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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 ( 613 )   RICH HTML PDF (108KB) ( 1282 )   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. References | Related Articles | Metrics

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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 ( 573 )   RICH HTML PDF (130KB) ( 1276 )   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. References | Related Articles | Metrics

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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 ( 520 )   RICH HTML PDF (112KB) ( 991 )   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 $v\in V$. References | Related Articles | Metrics

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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) ( 1344 )   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. References | Related Articles | Metrics

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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 ( 598 )   RICH HTML PDF (138KB) ( 1263 )   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. References | Related Articles | Metrics

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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 ( 744 )   RICH HTML PDF (114KB) ( 1188 )   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. References | Related Articles | Metrics

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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 ( 484 )   RICH HTML PDF (170KB) ( 910 )   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. References | Related Articles | Metrics