20 January 2010, Volume 26 Issue 1

    

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  Virtual Manifolds and Localization

  Bohui CHEN, Gang TIAN

  Acta Mathematica Sinica. 2010, 26(1): 1-24. https://doi.org/10.1007/s10114-010-9538-9

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  In this paper, we explore the virtual technique that is very useful in studying the moduli problem from a differential geometric point of view. We introduce a class of new objects “virtual manifolds/orbifolds”, on which we develop the integration theory. In particular, the virtual localizationformula is obtained.
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  Chern–Simons Invariant and Conformal Embedding of a 3-Manifold

  Chiakuei PENG, Zizhou TANG

  Acta Mathematica Sinica. 2010, 26(1): 25-28. https://doi.org/10.1007/s10114-010-8559-8

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  This note studies the Chern–Simons invariant of a closed oriented Riemannian 3-manifold M. The first achievement is to establish the formula CS(e) ? CS(e) = degA, where e and e are two (global) frames of M, and A : M → SO(3) is the “difference” map. An interesting phenomenon is that the “jumps” of the Chern–Simons integrals for various frames of many 3-manifolds are at least two, instead of one. The second purpose is to give an explicit representation of CS(e₊) and CS(e_?), where e₊ and e_? are the “left” and “right” quaternionic frames on M³ induced from an immersion M³ → E⁴, respectively. Consequently we find many metrics on S³ (Berger spheres) so that they can not be conformally embedded in E⁴.
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  Realizing Enveloping Algebras via Varieties of Modules

  Ming DING, Jie XIAO, Fan XU

  Acta Mathematica Sinica. 2010, 26(1): 29-48. https://doi.org/10.1007/s10114-010-9070-y

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  By using the Ringel–Hall algebra approach,we investigate the structure of the Lie alge-bra L(Λ)generated by indecomposable constructible sets in the varieties of modules for any finite-dimensional C-algebraΛ.We obtain a geometric realization of the universal enveloping algebra R(Λ)of L(Λ),this generalizes the main result of Riedtmann.We also obtain Green’s formula in a geometric form for any finite-dimensional C-algebraΛand use it to give the comultiplication formula in R(Λ).
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  Representations of Four-Derivation Lie Algebras of Block Type

  Yu Feng ZHAO, Zhi Bin LIANG

  Acta Mathematica Sinica. 2010, 26(1): 49-76. https://doi.org/10.1007/s10114-010-9170-8

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  Block introduced certain analogues of the Zassenhaus algebras over a field of character-istic 0.The nongraded infinite-dimensional simple Lie algebras of Block type constructed by Xu can be viewed as generalizations of the Block algebras.In this paper,we construct a family of irreducible modules in terms of multiplication and di?erentiation operators on“polynomials”for four-devivation nongraded Lie algebras of Block type based on the finite-dimensional irreducible weight modules with multiplicity one of general linear Lie algebras.We also find a new series of submodules from which some irreducible quotient modules are obtained.
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  The Chen–Ruan Cohomology of Almost Contact Orbifolds

  Fan DING, Yun Feng JIANG, Jian Zhong PAN

  Acta Mathematica Sinica. 2010, 26(1): 77-88. https://doi.org/10.1007/s10114-010-6623-z

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  Comparing to the Chen–Ruan cohomology theory for the almost complex orbifolds,we study the orbifold cohomology theory for almost contact orbifolds.We define the Chen–Ruan cohomol-ogy group of any almost contact orbifold.Using the methods for almost complex orbifolds,we define the obstruction bundle for any 3-multisector of the almost contact orbifolds and the Chen–Ruan cup product for the Chen–Ruan cohomology.We also prove that under this cup product the direct sum of all dimensional orbifold cohomology groups constitutes a cohomological ring.Finally we calculate two examples.
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  Large Deviation Principle for the Fourth-order Stochastic Heat Equations with Fractional Noises

  Yi Ming JIANG, Ke Hua SHI, Yong Jin WANG

  Acta Mathematica Sinica. 2010, 26(1): 89-106. https://doi.org/10.1007/s10114-010-7366-6

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  In this paper,we shall study a fourth-order stochastic heat equation driven by a fractionalnoise,which is fractional in time and white in space.We will discuss the existence and uniqueness of the solution to the equation.Furthermore,the regularity of the solution will be obtained.On the other hand,the large deviation principle for the equation with a small perturbation will be established through developing a classical method.
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  A General Law of Precise Asymptotics for Products of Sums under Dependence

  Yong ZHANG, Xiao Yun YANG, Zhi Shan DONG

  Acta Mathematica Sinica. 2010, 26(1): 107-116. https://doi.org/10.1007/s10114-010-8236-y

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  Let{X_n,n≥1}be a strictly stationary LNQD(LPQD)sequence of positive random variables with EX₁=μ>0,and VarX₁=σ²<∞.Denote by S_n=−ⁿ_i=1 X_i andγ=σ/μthe coeffcients of variation.In this paper,under some suitable conditions,we show that a general law of precise asymptotics for products of sums holds.It can describe the relations among the boundary function,weighted function,convergence rate and limit value in the study of complete nvergence.
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  Parabolic Equations with VMO Coeffcients in Generalized Morrey Spaces

  Lin ZHANG, Yin Sheng JIANG, Jiang ZHOU

  Acta Mathematica Sinica. 2010, 26(1): 117-130. https://doi.org/10.1007/s10114-010-8127-2

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  In this paper,the authors establish the regularity in generalized Morrey spaces of solutions to parabolic equations with VMO coeffcients by means of the theory of singular integrals and linear commutators.
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  Pseudo-Differential Operators on Sobolev and Lipschitz Spaces

  Yan LIN, Shan Zhen LU

  Acta Mathematica Sinica. 2010, 26(1): 131-142. https://doi.org/10.1007/s10114-010-8109-4

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  In this paper,by discovering a new fact that the Lebesgue boundedness of a class of pseudo-differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators,the authors establish the boundedness of pseudo-differential operators with symbols in ^m_ρ,δ on Sobolev spaces,where m∈R,ρ≤1 andδ≤1.As its applications,the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced.Moreover,the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.
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  Automorphism Group and Representation of a Twisted Multi-loop Algebra

  Cui CHEN, Hai Feng LIAN, Shao Bin TAN

  Acta Mathematica Sinica. 2010, 26(1): 143-154. https://doi.org/10.1007/s10114-010-8062-2

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  Let G be the complexification of the real Lie algebra so(3)and A=C[t^±1₁,t^±1₂] be the Lau-rent polynomial algebra with commuting variables.Let L(t₁,t₂,1)=G _c A be the twisted multi-loop Lie algebra.Recently we have studied the universal central extension,derivations and its vertex opera-tor representations.In the present paper we study the automorphism group and bosonic representations of L(t₁,t₂,1).
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  On the Density of Integers of the Form 2^k+p in Arithmetic Progressions

  Xue Gong SUN

  Acta Mathematica Sinica. 2010, 26(1): 155-160. https://doi.org/10.1007/s10114-010-8013-y

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  Consider all the arithmetic progressions of odd numbers,no term of which is of the form 2^k+p,where K is a positive integer and p is an odd prime.Erdös ever asked whether all these progressions can be obtained from covering congruences.In this paper,we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2^k+p,and give a quantitative form of Romanoff’s theorem on arithmetic progressions.As a corollary,we prove that the answer to the above Erdös problem is affirmative.
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  OD-characterization of Almost Simple Groups Related to U₃(5)

  Liang Cai ZHANG, Wu Jie SHI

  Acta Mathematica Sinica. 2010, 26(1): 161-168. https://doi.org/10.1007/s10114-010-7613-x

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  Let G be a finite group with order|G|=p^α1₁p^α2₂···p^αk_k,where p₁<p₂<···<p_k are prime numbers.One of the well-known simple graphs associated with G is the prime graph(or Gruenberg–Kegel graph)denoted byΓ(G)(or GK(G)).This graph is constructed as follows:The vertex set of it isπ(G)={p₁,p₂,...,p_k}and two vertices p_i,p_j with i≠j are adjacent by an edge(and we write p_i～p_j)if and only if G contains an element of order p_ip_j.The degree deg(p_i)of a vertex p_i∈π(G)is the number of edges incident on pi.We define D(G):=(deg(p₁),deg(p₂),...,deg(p_k)),which is called the degree pattern of G.A group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H such that|H|=|G|and D(H)=D(G).Moreover,a 1-fold OD-characterizable group is simply called OD-characterizable.Let L:=U₃(5) be the projective special unitary group.In this paper,we classify groups with the same order and degree pattern as an almost simple group related to L.In fact,we obtain that L and L.2 are OD-characterizable;L.3 is 3-fold OD-characterizable;L.S₃ is 6-fold OD-characterizable.
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  Reduction Theorems for Principal and Classical Connections

  Miroslav DOUPOVEC, Wlodzimierz M.MIKULSKI

  Acta Mathematica Sinica. 2010, 26(1): 169-184. https://doi.org/10.1007/s10114-010-7333-2

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  We prove general reduction theorems for gauge natural operators transforming principal connections and classical linear connections on the base manifold into sections of an arbitrary gauge natural bundle.Then we apply our results to the principal prolongation of connections.Finally we describe all such gauge natural operators for some special cases of a Lie group G.
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  On Exit Time from Balls of Jump-type Symmetric Markov Processes

  Toshihiro UEMURA

  Acta Mathematica Sinica. 2010, 26(1): 185-192. https://doi.org/10.1007/s10114-010-6173-4

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  We obtain upper and lower bounds of the exit times from balls of a jump-type symmetric Markov process.The proofs are delivered separately.The upper bounds are obtained by using the Lévy system corresponding to the process,while the precise expression of the(L²-)generator of the Dirichlet form associated with the process is used to obtain the lower bounds.
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  Modified Mann Iterations for Nonexpansive Semigroups in Banach Space

  Ru Dong CHEN, Hui Min HE, Muhammad Aslam NOOR

  Acta Mathematica Sinica. 2010, 26(1): 193-202. https://doi.org/10.1007/s10114-010-7446-7

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  Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^*,and C be a nonempty closed convex subset of E.Let{T(t):t≥0}be a nonexpansive semigroup on C such that F:=∩_t≥0 Fix(T(t))≠Ø,and f:C→C be a fixed contractive mapping.If{α_n},{β_n},{a_n},{b_n},{t_n}satisfy certain appropriate conditions,then we suggest and analyze the two modified iterative processes as: y_n=α_nx_n+(1-αn)T(tn)xn,xn=β_nf(x_n)+(1-β _n)y_n. u₀∈C,v_n=a_nu_n+(1-a_n)T(t_n)u_n,u_n+1=b_nf(u_n)+(1-b_n)v_n.We prove that the approximate solutions obtained from these methods converge strongly to q∈∩_t≥0Fix(TT(t)),which is a unique solution in F to the following variational inequality:(I-f)q,j(q-u)≤0∀u∈F.Our results extend and improve the corresponding ones of Suzuki[Proc.Amer.Math.Soc.,131,2133–2136(2002)],and Kim and XU[Nonlear Analysis,61,51–60(2005)]and Chen and He[Appl.Math.Lett.,20,751–757(2007)].
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  A Remark on Extension of Into Isometries

  Rui Dong WANG

  Acta Mathematica Sinica. 2010, 26(1): 203-208. https://doi.org/10.1007/s10114-010-6656-3

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  In this paper,we prove that an into isometry form S(l^∞(_(n)))to S(E),which under someconditions,can be extended to be a linear isometry defined on the whole space.Therefore we improve the results of[Ding,G.G.:The isometric extension of an into mapping from the unit sphere S(l^∞(₍₂₎)) to S(L¹(_μ)).Acta Mathematica Sinica,English Series,22(6),1721–1724(2006)].