15 January 2009, Volume 25 Issue 1

    

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  Irreducibility of The Igusa Tower

  Haruzo HIDA

  Acta Mathematica Sinica. 2009, 25(1): 1-20. https://doi.org/10.1007/s10114-008-6490-z

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  We shall give a simple (basically) purely in characteristic p proof of the irreducibility of the Igusa tower over Shimura varieties of PEL type. Our result covers Shimura variety of type A and type C classical groups, in particular, the Siegel modular varieties, the Hilbert–Siegel modular varieties, Picard surfaces and Shimura varieties of inner forms of unitary and symplectic groups over totally real fields.
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  Maximal Regularity of Second Order Delay Equations in Banach Spaces

  Shang Quan BU

  Acta Mathematica Sinica. 2009, 25(1): 21-28. https://doi.org/10.1007/s10114-007-1030-9

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  Using known C^α-multiplier result,we give necessary and sufficient conditions for the secondorder delay equations:?(t)=Au(t)+Fu^t+Gút+f(t),t∈Rto have maximal regularity in Holder continuous function spaces C^α(R,X),where X is a Banach space,A is a closed operator in X,F,G∈￡(C([-r,0],X),X)are delay operators for some fixed r>0.
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  On the Stability of the Generalized Sine Functional Equations

  Gwang Hui KIM

  Acta Mathematica Sinica. 2009, 25(1): 29-38. https://doi.org/10.1007/s10114-008-6621-6

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  The aim of this paper is to study the stability problem of the generalized sine functional equations as follows:Namely,we have generalized the Hyers–Ulam stability of the(pexiderized)sine functional equation.
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  Symmetric é Type Operator

  Jian WANG

  Acta Mathematica Sinica. 2009, 25(1): 39-46. https://doi.org/10.1007/s10114-008-7154-8

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  The study of symmetric property in the L²-sense for the non-positive definite operator ismotivated by the theory of probability and analysis.This paper presents some sufficient conditions forthe existence of symmetric measure for Léevy type operator.Some new examples are illustrated.The present study is an important step for considering various ergodic properties and functional inequalities of Léevy type operator.
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  A Structural Theorem on Embedded Graphs and Its Application to Colorings

  Bao Gang XU, Xiao Xu LU

  Acta Mathematica Sinica. 2009, 25(1): 47-50. https://doi.org/10.1007/s10114-008-7011-9

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  In this paper,a Lebesgue type theorem on the structure of graphs embedded in the surface of characteristicσ≤0 is given,that generalizes a result of Borodin on plane graphs.As a consequence, it is proved that every such graph without i-circuits for 4≤i≤11-3σis 3-choosable,that offiers a new upper bound to a question of Y.Zhao.
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  The Existence of Global Attractors for 2D Navier–Stokes Equationsin H^k Spaces

  Yin Di ZHANG, Ling Yu SONG

  Acta Mathematica Sinica. 2009, 25(1): 51-58. https://doi.org/10.1007/s10114-008-6594-5

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  In this paper,we prove that the 2D Navier–Stokes equations possess a global attractor inH^k(Ω,R²)for any k≥1,which attracts any bounded set of H^k(Ω,R²)in the H^k-norm.The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator,together with a classical existence theorem of global attractor.This extends Ma,Wang and Zhong's conclusion.
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  Inequalities on the Inner Radius of Univalency and the Norm of Pre-Schwarzian Derivative

  Tao CHENG, Yue Ming KANG, Ji Xiu CHEN

  Acta Mathematica Sinica. 2009, 25(1): 59-64. https://doi.org/10.1007/s10114-008-6519-3

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  In this paper,we get a lower bound of inner radius of univalency of Schwarzian derivativeby means of the norm of pre-Schwarzian derivative.Furthermore,we apply the theory of Universal Teichmuller Space to explain its geometric meaning which shows the relationship between the inner radius in Universal Teichmuller Space embedded by Schwarzian derivative and the norm defined in Universal Teichmuller Space embedded by pre-Schwarzian derivative.
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  Notes on the Borwein–Choi Conjecture of Littlewood Cyclotomic Polynomials

  Shao Fang HONG, Wei CAO

  Acta Mathematica Sinica. 2009, 25(1): 65-76. https://doi.org/10.1007/s10114-008-6444-5

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  Borwein and Choi conjectured that a polynomial P(x)with coecients±1 of degree N-1 is cyclotomiciff P(x)=±Φ_p1(±x)Φ_p2(±x^p1)···Φ_pr(±x^{p1,p2···pr-1}),where N=p1p2···pr and the pi are primes,not necessarily distinct.HereΦp(x):=(x^p-1)/(x-1)is the p-th cyclotomic polynomial.They also proved the conjecture for N odd or a power of 2.In this paper we introduce a so-called E-transformation,by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to investigate the conjecture.
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  Higher Order Willmore Hypersurfaces in Euclidean Space

  Zhen GUO

  Acta Mathematica Sinica. 2009, 25(1): 77-84. https://doi.org/10.1007/s10114-008-6422-y

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  Let x:Mⁿ→Rⁿ⁺¹ be an n(≥2)-dimensional hypersurface immersed in Euclidean spaceRⁿ⁺¹.Letσi(0≤i≤n)be the ith mean curvature and Q_n=Σⁿ_i=₀(-1)^{i+1 n}_iσ^n-i₁σ_i.Recently,theauthor showed that W_n(x)=∫_M Q_ndM is a conformal invariant under conformal group of Rⁿ⁺¹ and called it the nth Willmore functional of x.An extremal hypersurface of conformal invariant functional Wn is called an nth order Willmore hypersurface.The purpose of this paper is to construct concrete examples of the 3rd order Willmore hypersurfaces in R⁴ which have good geometric behaviors.The ordinary diffierential equation characterizing the revolutionary 3rd Willmore hypersurfaces is established and some interesting explicit examples are found in this paper.
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  An Analogue of Beurling's Theorem for the Jacobi Transform

  Ji Zheng HUANG, He Ping LIU, Jian Ming LIU

  Acta Mathematica Sinica. 2009, 25(1): 85-94. https://doi.org/10.1007/s10114-008-6307-0

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  In this paper,we prove Beurling's theorem for the Jacobi transform,from which we derive some other versions of uncertainty principles.
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  On the Structure of Gradedλ-Hopf Algebras

  Jian Hua SUN, Pu ZHANG

  Acta Mathematica Sinica. 2009, 25(1): 95-108. https://doi.org/10.1007/s10114-008-6146-z

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  Let G be an abelian group,B the G-gradedλ-Hopf algebra withλbeing a bicharacteron G.By introducing some new twisted algebras(coalgebras),we investigate the basic properties ofthe graded antipode and the structure for B.We also prove that a G-gradedλ-Hopf algebra can beembedded in a usual Hopf algebra.As an application,it is given that if G is a finite abelian group thenthe graded antipode of a finite dimensional G-gradedλ-Hopf algebra is invertible.
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  On the Global Existence and Uniqueness of Solutions to Prandtl's System

  Xin Ying XU, Jun Ning ZHAO

  Acta Mathematica Sinica. 2009, 25(1): 109-132. https://doi.org/10.1007/s10114-008-6050-6

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  In this paper,we consider the Prandtl system for the non-stationary boundary layer in thevicinity of a point where the outer flow has zero velocity.It is assumed that U(t,x,y)=x^mU₁(t,x),where 0≤x≤L and m≥1.We establish the global existence of the weak solution to this problem.Moreover the uniqueness of the weak solution is proved.
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  The Method of A-Harmonic Approximation and Boundary Regularity for Nonlinear Elliptic Systems under the Natural Growth Condition

  Shu Hong CHEN, Zhong TAN

  Acta Mathematica Sinica. 2009, 25(1): 133-156. https://doi.org/10.1007/s10114-008-6005-y

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  We consider the questions of boundary regularity for weak solutions of second-order non-linear elliptic systems under the natural growth condition.We obtain a general criterion for a weaksolution to be regular in the neighborhood of a given boundary point.The proof yields directly theoptimal regularity for the solution in this neighborhood.This result is new for the situation under thenatural growth conditions.
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  From Markov Jump Systems to Two Species Competitive Lotka–Volterra Equations with Di?usion

  Xue Qiang WANG, Xue Qiang WANG, Yong Jin WANG

  Acta Mathematica Sinica. 2009, 25(1): 157-170. https://doi.org/10.1007/s10114-008-5587-8

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  In this paper,we start at a random evolution system on biological particles,which isdescribed by a Markov jump system.Under a suitable scaling,we perform a proper approximationprocedure.Then the so-called weak convergence of Markov processes and Martingales allow us toestablish a(deterministic)two species competitive Lotka–Volterra equation.