10 June 2010, Volume 26 Issue 6

    

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  On the Maximal Operator Associated with Certain Rotational Invariant Measures

  Adrian INFANTE, Fernando SORIA

  Acta Mathematica Sinica. 2010, 26(6): 993-1004. https://doi.org/10.1007/s10114-010-9121-4

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  The aim of this work is to investigate the integrability properties of the maximal operatorM_μ,associated with a non-doubling measureμdefined on Rⁿ.We start by establishing for a wide class of radial and increasing measuresμ,that Mμis bounded on all the spaces L^p_μ(Rⁿ),p>1.Also,we show that there is a radial and increasing measureμfor which M_μdoes not map L^p_μ(Rⁿ)into weakL^p_μ(Rⁿ),1≤p<∞.
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  Linear Biseparating Maps between Vector-valued Little Lipschitz Function Spaces

  A.JIMENEZ-VARGAS, Ya-Shu WANG

  Acta Mathematica Sinica. 2010, 26(6): 1005-1018. https://doi.org/10.1007/s10114-010-9146-8

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  In this paper we provide a complete description of linear biseparating maps betweenspaces lip0(X^α,E)of Banach-valued little Lipschitz functions vanishing at infinity on locally com-pact H¨older metric spaces X^α=(X,d^α_X)with 0<α<1.Namely,it is proved that any linear bijection T:lip0(X^α,E)→lip0(Y^α,F)satisfying that ‖Tf()‖_F‖Tg(y)‖_F=0 for all y∈Y if and only if ‖f(x)‖_E‖g(x)‖_E=0 for all x ∈X,is a weighted composition operator of the form Tf(y)=h(y)(f(φ(y))),whereφis a homeomorphism from Y onto X and h is a map from Y into the set of all linear bijections from E onto F.Moreover,T is continuous if and only if h(y)is continuous for all y ∈Y.In this case,φbecomes a locally Lipschitz homeomorphism and h a locally Lipschitz map from Y^α into the space of all continuous linear bijections from E onto F with the metric induced by the operator canonical norm.This enables us to study the automatic continuity of T and the existence of discontinuous linear biseparating maps.
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  Counting Isomorphism Classes of Pointed Hyperelliptic Curves of Genus 4 over Finite Fields with Even Characteristic

  Huah CHU, Ying Pu DENG

  Acta Mathematica Sinica. 2010, 26(6): 1019-1054. https://doi.org/10.1007/s10114-010-7186-8

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  This paper is devoted to counting the number of isomorphism classes of pointed hyperellip-tic curves over finite fields.We deal with the genus 4 case and the finite fields are of even characteristics.The number of isomorphism classes is computed and the explicit formulae are given.This number canbe represented as a polynomial in q of degree 7,where q is the order of the finite field.The resultcan be used in the classification problems and it is useful for further studies of hyperelliptic curvecryptosystems,e.g.it is of interest for research on implementing the arithmetics of curves of low genusfor cryptographic purposes.It could also be of interest for point counting problems;both on modulispaces of curves,and on finding the maximal number of points that a pointed hyperelliptic curve overa given finite field may have.
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  The Stochastic Wave Equations Driven by Fractional and Colored Noises

  Dan TANG, Yong Jin WANG

  Acta Mathematica Sinica. 2010, 26(6): 1055-1070. https://doi.org/10.1007/s10114-010-8469-9

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  We investigate a wave equation in the plane with an additive noise which is fractional intime and has a non-degenerate spatial covariance.The equation is shown to admit a process-valuedsolution.Also we give a continuity modulus of the solution,and the Hölder continuity is presented.
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  Maps Preserving Numerical Radius or Cross Norms of Products of Self-adjoint Operators

  Kan HE, Jin Chuan HOU, Xiu Ling ZHANG

  Acta Mathematica Sinica. 2010, 26(6): 1071-1086. https://doi.org/10.1007/s10114-010-9001-y

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  Let H be a complex Hilbert space with dim H≥3,B_s(H)the(real)Jordan algebra of all self-adjoint operators on H.Every surjective mapΦ:B_s(H)→B_s(H)preserving numerical radius of operator products(respectively,Jordan triple products)is characterized.A characterization of surjective maps on B_s(H)preserving a cross operator norm of operator products(resp.Jordan triple products of operators)is also given.
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  On Corepresentations of Multiplier Hopf Algebras

  Shuan-hong WANG, Hai-xing ZHU

  Acta Mathematica Sinica. 2010, 26(6): 1087-1114. https://doi.org/10.1007/s10114-010-8484-x

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  We present a general construction producing a unitary corepresentation of a multiplierHopf algebra in itself and study RR-corepresentations of twisted tensor coproduct multiplier Hopfalgebra.Then we investigate some properties of functors,integrals and morphisms related to thecategories of the crossed modules and covariant modules over multiplier Hopf algebras.By doing so,we can apply our theory to the case of group-cograded multiplier Hopf algebras,in particular,the caseof Hopf group-coalgebras.
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  The Products of Three Theta Functions and the General Cubic Theta Functions

  Xiao Mei YANG

  Acta Mathematica Sinica. 2010, 26(6): 1115-1124. https://doi.org/10.1007/s10114-010-8435-6

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  In this paper we establish two theta function identities with four parameters by the theoryof theta functions.Using these identities we introduce common generalizations of Hirschhorn–Garvan–Borwein cubic theta functions,and also re-derive the quintuple product identity,one of Ramanujan'sidentities,Winquist's identity and many other interesting identities.
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  On Some Diophantine Fourier Series

  Hai Long LI, Jing MA, Wen Peng ZHANG

  Acta Mathematica Sinica. 2010, 26(6): 1125-1132. https://doi.org/10.1007/s10114-010-8387-x

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  We continue our study on arithmetical Fourier series by considering two Fourier serieswhich are related to Diophantine analysis.The first one was studied by Hardy and Littlewood inconnection with the classification of numbers and the second one was studied by Hartman and Wintnerby Lebesgue integration theory.
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  Powers of General Digital Sums

  Feng Juan CHEN

  Acta Mathematica Sinica. 2010, 26(6): 1133-1138. https://doi.org/10.1007/s10114-010-8416-9

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  Let m0,m1,m2,...be positive integers with mi≥2 for all #em/em#.It is well known that eachnonnegative integer n can be uniquely represented asn=a0+a1m0+a2m0m1+···+atm0m1m2···mt-1,where 0≤ai≤mi-1 for all #em/em# and at≠0.Let each f_i be a function defined on{0,1,2,...,mi-1} with f_i(0)=0.Write In this paper,we give the asymptotic formula. for 1/x∑_n≤_x^S(n)^k,where k is a positive integer.
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  The Join of Split Graphs Whose Half-strong Endomorphisms Form a Monoid

  Hai Long HOU, Yan Feng LUO, Rui GU

  Acta Mathematica Sinica. 2010, 26(6): 1139-1148. https://doi.org/10.1007/s10114-010-8342-x

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  In this paper,the half-strong endomorphisms of the join of split graphs are investigated.We give the conditions under which the half-strong endomorphisms of the join of split graphs form amonoid.
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  On F-Almost Split Sequences

  Xiao Jin ZHANG, Zhao Yong HUANG

  Acta Mathematica Sinica. 2010, 26(6): 1149-1164. https://doi.org/10.1007/s10114-009-7665-y

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  LetΛbe an Artinian algebra and F an additive subbifunctor of Ext¹_Λ(-,-)having enough projectives and injectives.We prove that the dualizing subvarieties of mod Λ closed under F-extensions have F-almost split sequences.Let T be an F-cotilting module in mod Λ and S a cotilting module over Γ=End(T).Then Hom(-,T)induces a duality between F-almost split sequences in ^⊥FT and almost split sequences in^⊥FS,where addS=Hom(P(F),T).LetΛbe an F-Gorenstein algebra,T a strongF-cotilting module and 0→A→B→C→0 an F-almost split sequence in^⊥FT.If the injective dimension of S as a Γ-module is equal to d,then C≌(Ω^-d_CMΩ^dDTrA^*)^*,where(-)^*=Hom(-,T).In addition,if the F-injective dimension of A is equal to d,then F≌Ω^-d_CMFDΩ^-d_Fop TrC≌Ω^-d_CMFΩ^d_FDTrC.
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  Viscosity Approximation Methods for Nonexpansive Multimaps in Banach Spaces

  Habtu ZEGEYE, Naseer SHAHZAD

  Acta Mathematica Sinica. 2010, 26(6): 1165-1176. https://doi.org/10.1007/s10114-010-7521-0

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  We prove strong convergence of the viscosity approximation process for nonexpansivenonself multimaps.Furthermore,an explicit iteration process which converges strongly to a fixed pointof multimap T is constructed.It is worth mentioning that,unlike other authors,we do not impose thecondition“Tz={z}”on the map T.
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  Gradient Estimates for the EquationΔu+cu^-α=0 on Riemannian Manifolds

  Yun Yan YANG

  Acta Mathematica Sinica. 2010, 26(6): 1177-1182. https://doi.org/10.1007/s10114-010-7531-y

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  Let(M,g)be a complete non-compact Riemannian manifold without boundary.In thispaper,we give the gradient estimates on positive solutions to the following elliptic equation withsingular nonlinearity:Δu(x)+cu^-α(x)=0 in M,whereα>0,c are two real constants.When c<0 and M is a bounded smooth domain in Rⁿ,the above equation is known as the thin film equation,which describes a steady state of the thin film(see Guo–Wei[Manuscripta Math.,120,193–209(2006)]).The results in this paper can be viewed asan supplement of that of J.Li[J.Funct.Anal.,100,233–256(1991)],where the nonlinearity is the positive power of u.
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  Center,Limit Cycles and Isochronous Center of a Z₄-equivariant Quintic System

  Chao Xiong DU

  Acta Mathematica Sinica. 2010, 26(6): 1183-1196. https://doi.org/10.1007/s10114-010-7541-9

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  In this paper,we study the limit cycles bifurcations of four fine focuses in Z₄-equivariantvector fields and the problems that its four singular points can be centers and isochronous centersat the same time.By computing the Liapunov constants and periodic constants carefully,we showthat for a certain Z₄-equivariant quintic systems,there are four fine focuses of five order and fivelimit cycles can bifurcate from each,we also find conditions of center and isochronous center for thissystem.The process of proof is algebraic and symbolic by using common computer algebra soft such asMathematica,the expressions after being simplified in this paper are simple relatively.Moreover,whatis worth mentioning is that the result of 20 small limit cycles bifurcating from several fine focuses is goodfor Z₄-equivariant quintic system and the results where multiple singular points become isochronouscenters at the same time are less in published references.