15 February 2003, Volume 46 Issue 2

    

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论文
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  A Generalization of BHW Theorem on Holomorphic Extension

  Hui Ping ZHANG(1),Xiang Yu ZHO

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 209-222. https://doi.org/10.12386/A2003sxxb0029

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  BHW theorem is about holomophic extensions for invariant holomorphic functions. In this paper a generalization of the theorem is given.
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  The Comparison Theorem on Super-Cartan Domains of the Third Type

  Wei Ping YIN,Xiao Xia ZHAO

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 223-236. https://doi.org/10.12386/A2003sxxb0030

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  In this note we get the explicit formula of holomorphic sectional curvature on super-Cartan domains of the third type under the invariant Kalher metric. By using the completeness of its Bergman metric, we construct an invariant Kalher metric which is not less than the Bergman metric and prove that the holomorphic sectional curvature with this invariant Kalher metric is bounded from above by a negative constant, hence we obtain the comparison theorm for the Bergman and Kobayashi metrics on super-Cartan domains of the third type.
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  The Distribution of Zeros for Differential Polynomials f~kQ[f] + P[f]

  Xiao Ping ZHAN(1),Hai Tao CAI(

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 237-244. https://doi.org/10.12386/A2003sxxb0031

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  The distribution of zeros for differential polynomials fkf'+ a and fkQ[f] + P[f] has been studied in [4] and [1]. There are some difficulties in proving zeros distribution of differential polynomials because of the presence of poles. So the above results are only for entire functions. In this paper, we prove that for a transcendental meromorphic function f and two differential polynomials Q[f], P[f] of f, fkQ[f] + P[f] has infinitely many zeros outside the union of the disce which do not contain the poles of f,Q[f], P[f] and the zeros of P[f]. Where f satisfies δ(∞, f) >and P are weights of Q[f] and P[f]. The result improves that of [1] and [2,4,6].
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  On the Terai's Conjecture Concerning the Exponential Diophantine Equation a~x + b~y = c~z

  Mao Hua LE

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 245-296. https://doi.org/10.12386/A2003sxxb0032

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  Let a = |m(m4 - 10m2 + 5)|, b =5m4 - 10m2 + 1, c = m2 + 1, where m is an even integer. In this paper, we prove that if m ≥ 542, then the equation ax + by = cz has only one positive integer solution (x,y, z) = (2,2,5).
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  Generalized Steepness Lemma of Projected Gradient Operator and Its Applications

  Chang Yu WANG(1),Zun Quan XIA(

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 251-260. https://doi.org/10.12386/A2003sxxb0033

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  A generalized steepness lemma of projected gradient operator on a closed convex set is discussed in this paper. The global convergence and strong convergence of the projected gradient methods using several slacked step-size rules are proved.
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  A Generalization of Herstein's Theorem

  Chang Lin FU,Xin Song YANG,Gua

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 261-268. https://doi.org/10.12386/A2003sxxb0034

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  Herstein generalized Jacobson's famous theorem on the commutativity of rings in 1955 as follows. Theorem A. if for every x, y ∈ R, there is a polynomial p(t) with integer coefficients, depending possibly on x and y, such that [x - x2p(x), y] = 0, then R is commutative. In this paper, we define a property Fk of the polynomial f(x1,X2,…, xn) and prove, on the bassing is property, a theorem on the commutativity of rings. Herstein's theorem is just the one case when n = 1 in our theorem.
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  The L~p-Besov Estimates on the Dispersive Equation

  Xiao Feng LIU,Meng WANG

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 269-278. https://doi.org/10.12386/A2003sxxb0035

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  Let u(x, t) = (SΩf)(x,t) be the solution of Cauchy problem of the general dispersive equation tu-iΩ(D)u = 0, u(x,0) = f(x), (x,t) ×Rn ∈R, and SΩ*f, SΩ**f be the local and global maximal operators. This paper gives some estimates about them.
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  Hardy's Inequalities and Pohozaev's Identities on the Heisenberg Group

  Peng Cheng NIU,Hui Qing ZHANG,

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 279-290. https://doi.org/10.12386/A2003sxxb0036

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  The aim of this paper is to construct the fundamental solution of p-sub-Laplacian on the Heisenberg group and establish Hardy's inequalities by proving Pi-cone's identity on vector fields. Furthermore, Pohozaev's identities are given by using the noncommutative properties of vector fields. These results generalize those by Fol-land, Garofalo-Lanconelli, and some methods in this paper are new.
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  Some Results on Harmonic Maps from Surfaces into Complex Grassmann Manifolds

  Bing Ye WU

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 291-296. https://doi.org/10.12386/A2003sxxb0037

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  In this paper, we study some topics for harmonic maps from surfaces into complex Grassmann manifolds. We obtain an equivalent condition for harmonic maps to be '-irreducible or "-irreducible, and provide an explicit formula to calculate the isotropy order of harmonic maps.
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  Computing the 5-Order Sums of ζ(k)

  Ping SUN

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 297-302. https://doi.org/10.12386/A2003sxxb0038

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  Let u1, u2, … be independent random variables uniformly distributed on (0,1). This paper derives that for any n ≥ 1, the (n - 1)th moments of 1 - u1u2… uk are exponential complete Bell polynomials Yk(ζn(1), 1!ζn(2), 2!ζn(3), 3!ζn(4),…). Here the partial sums ζn(r) =∑j=1n/jr, r≥1, so the Riemann-Zeta function ζ(k) can be expanded as the series involving Stirling numbers of the first kind. Furthermore, all the 5-order sums of ζn(r) such as ∑n>1 Hn2/n3 have been given.
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  Area Comparison and Superharmonic Functions and Applications

  Yi Hu YANG

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 303-308. https://doi.org/10.12386/A2003sxxb0039

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  In this paper, we study the geometrical and analytic properties of manifolds with nonnegatively sectional curvature at infinity. Then, we apply these results to the study of harmonic maps.
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  On the Conditions of a Center and General Integrals of Quadratic Differential Systems

  Wei Yin YE(1),Yan Qian YE(2)

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 309-316. https://doi.org/10.12386/A2003sxxb0040

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  We give general integral of quadratic differential system with center in two cases under the Chinese classification.
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  On the Bloch Condtaut for K-Quasiconformal Mappings in Several Complex Variables

  J. Y. Gamaliel,Huai Hui CHEN

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 317-320. https://doi.org/10.12386/A2003sxxb0041

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  This paper study the Bloch constant for K-quasiconformal holomorphic mappings of the unit ball B of Cn. The final result we prove is: If f is a K-quasiconformal holomorphic mappings of the unit ball B of Cn such that det (f'(0)) = 1, then f(B)contains a schlicht ball of radius at least (CnK)1-n f01(1+t)n-1/(1-t)2 exp {-(n+1)t/1-t}dt where Cn > 1 is a constant depending on n only, and Cn→ 10 as n→∞.
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  Complete Formulas of the Volumes for Tubes about Curves in a Reimannian Manifold

  Fa En WU

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 321-324. https://doi.org/10.12386/A2003sxxb0042

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  By using of the Taylor expansions of the solutions of Jacobi equations, we obtain the complete formulas for the volumes of tubes about curves in a Riemannian manifold. This unifies the known results and simplifies the compulations involved in this direction. In the special case of surfaces, we also obtain the corresponding complete formula which generalize the known results.
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  Multipliers on Dirichlet Type Spaces

  Peng Yan HU(1),Ji Huai SHI(2)

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 325-332. https://doi.org/10.12386/A2003sxxb0043

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  In this paper, we characterize the pointwise multiplier space M(Dτ ,Dμ) of Dirichlet type spaces in the unit ball of Cn for the values of τ,μ in three cases: (i) τ< 0, μ < 0, (ii) τ< μ, (iii)τ ≥ μ, τ> n, and construct two functions to show that the relations of M(Dτ) Dτ if r≤ n and M(Dτ) M(Dμ) if τ > μ and τ > n-1 are proper.
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  K_1 Group of Finite Dimensional Path Algebra

  Xue Jun GUO

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 333-336. https://doi.org/10.12386/A2003sxxb0044

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  In this paper, by calculating the commutator subgroup of the unit group of finite path algebra kΔ and the unit group abelianized, we explicitly characterize the K1 group of finite dimensional path algebra over an arbitrary field.
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  A Characterization for Gevrey-Sobolev Wave Front Set

  Hua CHEN

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 337-340. https://doi.org/10.12386/A2003sxxb0045

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  In this note, we use the so-called microlocal energy method to give a characterization of Gevrey-Sobolev wave front set, which will be useful in the study of non-linear microlocal analysis in Gevrey classes.
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  Average Widths of Sobolev-Wiener Classes with Mixed Smoothness in L_q(R~d)

  He Ping WANG

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 341-346. https://doi.org/10.12386/A2003sxxb0046

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  In this paper, the average σ-K width of Sobolev-Wiener classes SpqrW with mixed smoothness in Lq(Rd) is studied for 1 < q ≤ p < ∞, and the weak asymptotical behavior of these quantities is obtained.
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  On a Class of Finitely Supported Scaling Functions with Properties of Interpolation and Symmetry

  Ze Yin ZHANG

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 347-350. https://doi.org/10.12386/A2003sxxb0047

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  In this paper, a class finitely supported orthogonal filters with interpolation and symmetry are constructed.
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  The Blowup of Solutions for Two Dimensional Irrotational Compressible Euler Equations

  Hui Cheng YIN,Qin ZHENG,Shu Ze

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 351-360. https://doi.org/10.12386/A2003sxxb0048

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  For two dimensional irrotational compressible Euler equations with initial data that is a small perturbation from a constant state, we prove that the first order derivatives of ρ, v blow up at the blowup time while ρ, v remain continuous. In particular, in the irrotational case we prove Alinhac's S. conjecture.
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  On Singularity Methods of the Duffing Equation

  Hong Bin CHEN(1),Kai Tai LI(2)

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 361-368. https://doi.org/10.12386/A2003sxxb0049

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  Consider the Duffing equation x + g(x) = p(t), x(0) =x(2π), x(0) = x(2π), where g ∈ C2(R) is a strictly convex function, g'(x) crosses the first resonant point asymptotically. We apply the singularity theory to obtain that Duffing operator Fx(t) = x(t) + g(x(t)) is global equivalent to the fold mapping in the sense of H. Whitney. In particular, we obtain the exact number of periodic solutions.
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  A Note on the Uniqueness of Limit Cycles of a Class of Planar Systems of Degree n + 2

  Xi Shun YUE(1),Xian Wu ZENG(2)

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 369-374. https://doi.org/10.12386/A2003sxxb0050

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  This paper continues the work of [1] and [2]. Using the generalized Lienard equation and Zhang Zhifen uniqueness theorem we give a proof of the uniqueness of limit cycles of a class of systems of biochemical reaction of degree n + 2 for the case n ≥ 3. So far the problem of the uniqueness of limit cycles of the system has been completely solved.
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  Stability of (δ,)-Isometries

  Wei PAN

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 375-378. https://doi.org/10.12386/A2003sxxb0051

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  We investigate the stability of nonlinear and non-surjective isometries between Banach spaces. Under some assumptions, we give some positive answers to such isometries. Thus we correct some results of [1-4], and in addition, we have as well generalized them.
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  Quasiconformal Mappings with Non-Decreasable Beltrami Coefficients

  Ze Min ZHOU,Ji Xiu CHEN

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 379-384. https://doi.org/10.12386/A2003sxxb0052

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  In this paper, we discuss the relationship between the local (infinitesimally) non-decreasability and the global (infinitesimally) non-decreasability of Beltrami coefficients, and give a method to construct a family of (infinitesimally) non-decreasable Beltrami coefficients which are extremal and of non-constant absolute values.
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  The w-Frchet Differentiability Property, the Radon-Nikod(?)m Property and w-Asplund Spaces

  Li Xin CHENG(1),Cong Xin WU(2)

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 385-390. https://doi.org/10.12386/A2003sxxb0053

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  We say an extended real-valued w*-lower semi-continuous convex function f on a dual Banach space E* has w*-Frechet differentiablity property (w*-FDP), if for every w*-lower semi-continuous proper convex function g on E*, g ≤f, implies that g is Frechet differentiable at each point of a dense Gδ-subset of the interior of dom f (the effective domain of f). This paper characterizes the w*-FDP of the functions f by the Radon- Nikodym property of subsets in the pre-dual E, and as a direct consequence, it gives a localized Collier's theorem.
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  Blow-Up for Degenerate Semiliuear Parabolic Equations with Nonlocal Source

  Fu Cai LI,Qi Lin LIU,Chun Hong

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 391-396. https://doi.org/10.12386/A2003sxxb0054

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  In this paper, we investigate degenerate semilinear parabolic equations xqut -uxx= f0a f(u)dx with Dirichlet boundary condition and q > 0. It is proved that the local solution is existent and unique and the solution blows up in a finite time provided that the initial data u0(x) is sufficiently large. Furthermore, it is shown that the blowup set is the entire interval [0, a], which is quite different from the results for the solution with local source.
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  Random Dirichlet Series Dealing with Small Functions

  Zong Sheng GAO(1),Dao Chun SUN

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 397-402. https://doi.org/10.12386/A2003sxxb0055

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  For certain random Dirichlet series of order ρ (0 < ρ < ∞) in the right half plane, almost surely every point on the imaginary axis is a Borel point with no exceptional small functions.
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  The Existence of Two Positive Solutions of Singular Boundary Value Problem of Fourth Order Differential Equations

  Chang Ci PANG,Zhong Li WEI

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 403-410. https://doi.org/10.12386/A2003sxxb0056

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  The existence of two C2[0,1] positive solutions as well as C3[0,1] positive solutions is established for a singular boundary value problem of fourth order differential equations by using the fixed point theorems of cone expansion and cone compression of norm type.
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  Solvability of an Operator Equation in Partial Ordered Space

  Yu Qiang FENG,San Yang LIU

  Acta Mathematica Sinica, Chinese Series. 2003, 46(2): 411-416. https://doi.org/10.12386/A2003sxxb0057

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  In this paper, techniques of partial order theory are used to study the solvability of an operator equation Lx = Nx in complete metric space and Banach space, respectively. Existence results are obtained and then applied to a two point boundary value problem of an intergro-differential equation.