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中国科学：数学（英文版）
中国科学：数学（英文版）

周光召

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中国科学：数学（英文版）/Journal Science China(Mathematics)CSCDCSTPCDSCI
查看更多>>《中国科学》是中国科学院主办、中国科学杂志社出版的自然科学专业性学术刊物。《中国科学》任务是反映中国自然科学各学科中的最新科研成果，以促进国内外的学术交流。《中国科学》以论文形式报道中国基础研究和应用研究方面具有创造性的、高水平的和有重要意义的科研成果。在国际学术界，《中国科学》作为代表中国最高水平的学术刊物也受到高度重视。国际上最具有权威的检索刊物SCI，多年来一直收录《中国科学》的论文。1999年《中国科学》夺得国家期刊奖的第一名。
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    Multiplicity of solutions for the semilinear subelliptic Dirichlet problem

    Hua ChenHong-Ge ChenJin-Ning LiXin Liao...
    475-504页
    查看更多>>摘要：In this paper,we study the semilinear subelliptic equation {-ΔXu=f(x,u)+g(x,u)in Ω,u=0 on ∂Ω,where ΔX=-∑mi=1X*iXi is the self-adjoint Hörmander operator associated with the vector fields X=(X1,X2,.…,Xm)satisfying the Hörmander's condition,f(x,u)∈ C(-Ω) ×R),g(x,u)is a Carathéodory function on Ω × R,and Ω is an open bounded domain in Rn with smooth boundary.Combining the perturbation from the symmetry method with the approaches involving the eigenvalue estimate and the Morse index in estimating the minimax values,we obtain two kinds of existence results for multiple weak solutions to the problem above.Furthermore,we discuss the difference between the eigenvalue estimate approach and the Morse index approach in degenerate situations.Compared with the classical elliptic cases,both approaches here have their own strengths in the degenerate cases.This new phenomenon implies that the results in general degenerate cases would be quite different from the situations in classical elliptic cases.
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    The stable Auslander-Reiten components of certain monomorphism categories

    Rasool HafeziYi Zhang
    505-526页
    查看更多>>摘要：Let Λ be an Artin algebra and let Gprj-Λ denote the class of all the finitely generated Gorenstein projective Λ-modules.In this paper,we study the components of the stable Auslander-Reiten quiver of a certain subcategory of the monomorphism category S(Gprj-Λ)containing boundary vertices.We describe the shape of such components.It is shown that certain components are linked to the orbits of an auto-equivalence on the stable category Gprj-Λ.In particular,for the finite components,we show that under certain mild conditions,their cardinalities are divisible by 3.We see that this three-periodicity phenomenon reoccurs several times in the paper.
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    Global well-posedness for 2D inhomogeneous asymmetric fluids with large initial data

    Chenyin QianBeibei HeTing Zhang
    527-556页
    查看更多>>摘要：In this paper,by using time-weighted global estimates and the Lagrangian approach,we first investigate the global existence and uniqueness of the solution for the 2D inhomogeneous incompressible asymmetric fluids with the initial(angular)velocity being located in sub-critical Sobolev spaces Hs(R2)(0＜s＜1)and the initial density being bounded from above and below by some positive constants.The global unique solvability of the 2D incompressible inhomogeneous asymmetric fluids with the initial data in the critical Besov space(u0,w0)∈(B)02,1(R2)and ρ-1-1∈(B)ε2/ε,1(R2)is established.In particular,the uniqueness of the solution is also obtained without any more regularity assumptions on the initial density which is an improvement on the recent result of Abidi and Gui(2021)for the 2D inhomogeneous incompressible Navier-Stokes system.
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    Trace inequalities,isocapacitary inequalities,and regularity of the complex Hessian equations

    Jiaxiang WangBin Zhou
    557-576页
    查看更多>>摘要：In this paper,we study the relations between trace inequalities(Sobolev and Moser-Trudinger types),isocapacitary inequalities,and the regularity of the complex Hessian and Monge-Ampère equations with respect to a general nonnegative Borel measure.We obtain a quantitative characterization for these relations through the properties of the capacity-minimizing functions.
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    The KAM theorem on the modulus of continuity about parameters

    Zhicheng TongJiayin DuYong Li
    577-592页
    查看更多>>摘要：In this paper,we study the Hamiltonian systems H(y,x,ξ,ε)=(ω(ξ),y)+εP(y,x,ξ,ε),where ω and P are continuous about ξ.We prove that persistent invariant tori possess the same frequency as the unperturbed tori,under a certain transversality condition and a weak convexity condition for the frequency mapping ω.As a direct application,we prove a Kolmogorov-Arnold-Moser(KAM)theorem when the perturbation P holds arbitrary Hölder continuity with respect to the parameter ξ.The infinite-dimensional case is also considered.To our knowledge,this is the first approach to the systems with the only continuity in the parameter beyond Hölder's type.
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    Cohomology groups of a new class of Kadison-Singer algebras

    Guangyu AnXing ChengJun Sheng
    593-606页
    查看更多>>摘要：Let N be a maximal discrete nest on an infinite-dimensional separable Hilbert space H,ξ=∑∞n=1en/2n be a separating vector for the commutant N',Eξ be the projection from H onto the subspace[Cξ]spanned by the vector ξ,and Q be the projection from K=H ⊕ H ⊕H onto the closed subspace{(η,η,η)T:η ∈H}.Suppose that L is the projection lattice generated by the projections(Eξ00000000),{(E00000000):E∈N,(I000I0000)and Q.We show that L is a Kadison-Singer lattice with the trivial commutant.Moreover,we prove that every n-th bounded cohomology group Hn(AlgL,B(K))with coefficients in B(K)is trivial for n≥1.
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    2-local unstable homotopy groups of indecomposable A23-complexes

    Zhongjian ZhuJianzhong Pan
    607-626页
    查看更多>>摘要：In this paper,we calculate the 2-local unstable homotopy groups of indecomposable A23-complexes.The main technique used is analysing the homotopy property of J(A,X),defined by Gray(1973)for a CW-pair(X,A),which is homotopy equivalent to the homotopy fibre of the pinch map X ∪ CA → EA.
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    Berry-Esseen bounds and moderate deviations for the norm,entries and spectral radius of products of positive random matrices

    Hui XiaoIon GramaQuansheng Liu
    627-646页
    查看更多>>摘要：Let(gn)n≥1 be a sequence of independent and identically distributed positive random d × d matrices and consider the matrix product Gn=gn… g1 Under suitable conditions,we establish the Berry-Esseen bounds on the rate of convergence in the central limit theorem and Cramér-type moderate deviation expansions,for any matrix norm ‖Gn‖ of Gn,its entries Gi,jn and its spectral radius ρ(Gn).Extended versions of their joint law with the direction of the random walk Gnx are also established,where x is a starting point in the unit sphere of Rd.
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    Data-driven slicing for dimension reduction in regressions:A likelihood-ratio approach

    Peirong XuTao WangLixing Zhu
    647-664页
    查看更多>>摘要：To efficiently estimate the central subspace in sufficient dimension reduction,response discretization via slicing its range is one of the most used methodologies when inverse regression-based methods are applied.However,existing slicing schemes are almost all ad hoc and not widely accepted.Thus,how to define data-driven schemes with certain optimal properties is a longstanding problem in this field.The research described here is then twofold.First,we introduce a likelihood-ratio-based framework for dimension reduction,subsuming the popularly used methods including the sliced inverse regression,the sliced average variance estimation and the likelihood acquired direction.Second,we propose a regularized log likelihood-ratio criterion to obtain a data-driven slicing scheme and derive the asymptotic properties of the estimators.A simulation study is carried out to examine the performance of the proposed method and that of existing methods.A data set concerning concrete compressive strength is also analyzed for illustration and comparison.
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    Nitsche-XFEM for a time fractional diffusion interface problem

    Tao WangYanping Chen
    665-682页
    查看更多>>摘要：In this paper,we propose a space-time finite element method for a time fractional diffusion interface problem.This method uses the low-order discontinuous Galerkin(DG)method and the Nitsche extended finite element method(Nitsche-XFEM)for temporal and spatial discretization,respectively.Sharp pointwise-in-time error estimates in graded temporal grids are derived,without any smoothness assumptions on the solution.Finally,three numerical examples are provided to verify the theoretical results.
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