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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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1674-7283

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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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    Reverse mathematics and local rings

    Huishan Wu
    2671-2680页
    查看更多>>摘要：In this paper,we study local rings from the perspective of reverse mathematics.We define local rings in a first-order way by using Π02 properties of invertible elements,where for a ring R possibly not commutative,R is left(resp.right)local if for any non-left(resp.non-right)invertible elements x,y ∈ R,x+y is not left(resp.right)invertible;R is local if for any non-invertible elements x,y ∈ R,x+y is not invertible.Firstly,we solve a question of Sato on characterizations of commutative local rings in his PhD thesis(Question 6.22 in Sato(2016))and prove that the statement"a commutative ring is local if and only if it has at most one maximal ideal"is equivalent to ACA0 over RCA0.We also obtain a nice corollary in computable mathematics,i.e.,there is a computable non-local ring with exactly two maximal ideals such that each of them Turing computes the Halting set K.Secondly,we study the equivalence among left local rings,right local rings,and local rings,showing that these three kinds of first-order local rings are equivalent over the weak basis theory RCA0.Finally,we extend the results of reverse mathematics on commutative local rings to noncommutative rings.
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    Splitting of the virtual class for genus one stable quasimaps

    Sanghyeon LeeMu-Lin Li
    2681-2700页
    查看更多>>摘要：We analyze the local structure of the moduli space of genus one stable quasimaps.Combining it with the p-fields theory developed by Chang and Li(2020),we prove the splitting formula for the virtual cycle of stable quasimaps to complete intersections in Pn.
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    Verdier quotients of homotopy categories of rings and Gorenstein-projective precovers

    Manuel Cortés-Izurdiaga
    2701-2712页
    查看更多>>摘要：Let R be a ring,Proj be the class of all the projective right R-modules,κ be the full subcategory of the homotopy category K(Proj)whose class of objects consists of all the totally acyclic complexes,and Morκbe the class of all the morphisms in K(Proj)whose cones belong to κ.We prove that if K(Proj)has enough Morκ-injective objects,then the Verdier quotient K(Proj)/κ has small Hom-sets,and this last condition implies the existence of Gorenstein-projective precovers in Mod-R and of totally acyclic precovers in C(Mod-R).
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    Almost global smooth solutions of the 3D quasilinear Klein-Gordon equations on the product space R2 × T

    Jun LiFei TaoHuicheng Yin
    2713-2752页
    查看更多>>摘要：In this paper,for the 3D quadratic nonlinear Klein-Gordon equation on the product space R2 × T,we focus on the lower bound of the lifespan of the smooth solution with slowly decaying initial data.When the size of initial data is bounded by ∈o＞0,it is shown that a smooth solution exists up to the time ec0/∈20 with ∈0 being sufficiently small and c0＞0 being some suitable constant.Note that the solution of the corresponding 3D linear homogeneous Klein-Gordon equation on R2 × T only admits the optimal time-decay rate(1+t)-1,from which we generally derive the lifespan of the nonlinear Klein-Gordon equation up to ec0/∈0 rather than the more precise ec0/∈20 here.
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    Global structure of spectra of periodic non-Hermitian Jacobi operators

    Hui LuJiangong You
    2753-2770页
    查看更多>>摘要：We give global structure of spectra of periodic non-Hermitian Jacobi operators by the discriminant and its stationary points.We also give necessary and sufficient conditions for real spectra and single interval spectra.
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    Coanalytic models for Hardy-type operators

    Xiangdi FuKunyu GuoFugang Yan
    2771-2788页
    查看更多>>摘要：We establish coanalytic models for a broad class of Hardy-type operators on L2[0,1].In particular,we show that the logarithmic Hardy operator is unitarily equivalent to the difference between the identity operator and the backward shift on a Bergman-type space.This result leads to several applications related to zero sets and invariant subspaces in weighted Bergman spaces.Additionally,we study logarithmic Hardy operators on Lp[0,1]and obtain results concerning their boundedness,operator norms,and spectra.
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    On uniqueness and existence of conformally compact Einstein metrics with homogeneous conformal infinity.Ⅱ

    Gang Li
    2789-2822页
    查看更多>>摘要：In this paper,we show that for an Sp(k+1)-invariant metric g on S4k+3(k≥1)close to the round metric,the conformally compact Einstein(CCE)manifold(M,g)with(S4k+3,[(g)])as its conformal infinity is unique up to isometry.Moreover,by the result in Li et al.(2017),g is the Graham-Lee metric(see Graham and Lee(1991))on the unit ball B1 ⊂ R4k+4.We also give an a priori estimate of the Einstein metric g.As a by-product of the a priori estimates,based on the estimate and Graham-Lee and Lee's seminal perturbation results(see Graham and Lee(1991)and Lee(2006)),we directly use the continuity method to obtain an existence result of the non-positively curved CCE metric with prescribed conformal infinity(S4k+3,[g])when the metric g is Sp(k+1)-invariant.We also generalize the results to the case of conformal infinity(S15,[g])with g a Spin(9)-invariant metric in the appendix.
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    Explicit results for ergodic properties of SDEs driven by cylindrical symmetric stable noises

    Lu-Jing HuangJian Wang
    2823-2842页
    查看更多>>摘要：We consider the exponentially ergodic properties of systems of SDEs in Rn driven by cylindrical stable processes,potentially with different indices across different coordinates.Our approach is based on the well-known Foster-Lyapunov criteria and a careful selection of Lyapunov functions,alongside recent advances in regularity and transition density estimates for solutions to SDEs driven by Lévy processes with independent coordinates.These results are novel,even in the one-dimensional case.Notably,our findings suggest that multiplicative cylindrical stable processes can enhance the ergodicity of the system when the stable noise indices in all directions fall within[1,2).
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    Model-averaging-based semiparametric modeling for conditional quantile prediction

    Chaohui GuoWenyang Zhang
    2843-2872页
    查看更多>>摘要：In real data analysis,the underlying model is frequently unknown.Hence,the modeling strategy plays a key role in the success of data analysis.Inspired by the idea of model averaging,we propose a novel semiparametric modeling strategy for the conditional quantile prediction,without assuming that the underlying model is any specific parametric or semiparametric model.Due to the optimality of the weights selected by leave-one-out cross-validation,the proposed modeling strategy provides a more precise prediction than those based on some commonly used semiparametric models such as the varying coefficient and additive models.Asymptotic properties are established in the proposed modeling strategy along with its estimation procedure.We conducted extensive simulations to compare our method with alternatives across various scenarios.The results show that our method provides more accurate predictions.Finally,we applied our approach to the Boston housing data,yielding more precise quantile predictions of house prices compared with commonly used methods,and thus offering a clearer picture of the Boston housing market.
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    Improved error estimates for a modified exponential Euler method for the semilinear stochastic heat equation with rough initial data

    Xinping GuiBuyang LiJilu Wang
    2873-2898页
    查看更多>>摘要：A class of stochastic Besov spaces BpL2(Ω;Hα(O)),1≤p≤∞ and α ∈[-2,2],is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation du-△udt=f(u)dt+dW(t),under the following conditions for some α ∈(0,1]:||∫toe-(t-s)AdW(s)||L2(Ω;L2(O))≤ Ctα/2 and||∫toe-(t-s)AdW(s)||B∞L2(Ω;(H)α(O))≤ C.The conditions above are shown to be satisfied by both trace-class noises(with α=1)and one-dimensional space-time white noises(with α=1/2).The latter would fail to satisfy the conditions with α=1/2 if the stochastic Besov norm||·||B∞L2(Ω;(H)α(O))isreplaced by the classical Sobolev norm||·||L2(Ω;(H)α(O)),and this often causes reduction of the convergence order in the numerical analysis of the semilinear stochastic heat equation.In this paper,the convergence of a modified exponential Euler method,with a spectral method for spatial discretization,is proved to have order α in both the time and space for possibly nonsmooth initial data in L4(Ω;(H)β(O))withβ＞-1,by utilizing the real interpolation properties of the stochastic Besov spaces and a class of locally refined stepsizes to resolve the singularity of the solution at t=0.
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