# Frontiers of Mathematics in China

ISSN 1673-3452 (Print)
ISSN 1673-3576 (Online)
2016 Impact Factor: 0.333

#### Current Issue

, Volume 12 Issue 3
RESEARCH ARTICLE
 Select Infinite-dimensional 3-Lie algebras and their connections to Harish-Chandra modules Ruipu BAI, Zhenheng LI, Weidong WANG Front. Math. China. 2017, 12 (3): 515-530.   DOI: 10.1007/s11464-017-0606-7 Abstract   PDF (186KB) We construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and contain the smallest full toroidal Lie algebras without center.
 Select Multiple weighted estimates for maximal vector-valued commutator of multilinear Calderón-Zygmund singular integrals Dongxiang CHEN, Shanzhen LU, Suzhen MAO Front. Math. China. 2017, 12 (3): 531-558.   DOI: 10.1007/s11464-017-0634-3 Abstract   PDF (274KB) This paper concerns with multiple weighted norm inequalities for maximal vector-valued multilinear singular operator and maximal commutators. The Cotlar-type inequality of maximal vector-valued multilinear singular integrals operator is obtained. On the other hand, pointwise estimates for sharp maximal function of two kinds of maximal vector-valued multilinear singular integrals and maximal vector-valued commutators are also established. By the weighted estimates of a class of new variant maximal operator, Cotlar’s inequality and the sharp maximal function estimates, multiple weighted strong estimates and weak estimates for maximal vector-valued singular integrals of multilinear operators and those for maximal vector-valued commutator of multilinear singular integrals are obtained.
 Select An occupation time related potential measure for diffusion processes Ye CHEN, Yingqiu LI, Xiaowen ZHOU Front. Math. China. 2017, 12 (3): 559-582.   DOI: 10.1007/s11464-017-0625-4 Abstract   PDF (234KB) In this paper, for homogeneous diffusion processes, the approach of Y. Li and X. Zhou [Statist. Probab. Lett., 2014, 94: 48–55] is adopted to find expressions of potential measures that are discounted by their joint occupation times over semi-infinite intervals (−∞, a) and (a,∞). The results are expressed in terms of solutions to the differential equations associated with the diffusions generator. Applying these results, we obtain more explicit expressions for Brownian motion with drift, skew Brownian motion, and Brownian motion with two-valued drift, respectively.
 Select Koszulity and Koszul modules of dual extension algebras Huanhuan LI, Yunge XU Front. Math. China. 2017, 12 (3): 583-596.   DOI: 10.1007/s11464-016-0601-4 Abstract   PDF (177KB) Let Aand Bbe algebras, and let T be the dual extension algebra of A and B. We provide a different method to prove that Tis Koszul if and only if both A and B are Koszul. Furthermore, we prove that an algebra is Koszul if and only if one of its iterated dual extension algebras is Koszul, if and only if all its iterated dual extension algebras are Koszul. Finally, we give a necessary and sufficient condition for a dual extension algebra to have the property that all linearly presented modules are Koszul modules, which provides an effective way to construct algebras with such a property.
 Select Deformation of conic negative Kähler-Einstein manifolds Yan LI Front. Math. China. 2017, 12 (3): 597-606.   DOI: 10.1007/s11464-016-0600-5 Abstract   PDF (148KB) In this note, we investigate the behavior of a smooth flat family of n-dimensional conic negative Kähler-Einstein manifolds. By H. Guenancia’s argument, a cusp negative Kähler-Einstein metric is the limit of conic negative Kähler-Einstein metric when the cone angle tends to 0. Furthermore, it establishes the behavior of a smooth flat family of n-dimensional cusp negative Kähler-Einstein manifolds.
 Select Uniform positivity of Lyapunov exponent for a class of smooth Schrödinger cocycles with weak Liouville frequencies Jinhao LIANG, Po-Jen KUNG Front. Math. China. 2017, 12 (3): 607-639.   DOI: 10.1007/s11464-017-0619-2 Abstract   PDF (407KB) We prove uniform positivity of the Lyapunov exponent for quasiperiodic Schrödinger cocycles with C2 cos-type potentials, large coupling constants, and fixed weak Liouville frequencies.
 Select Minimal P-symmetric period problem of first-order autonomous Hamiltonian systems Chungen LIU, Benxing ZHOU Front. Math. China. 2017, 12 (3): 641-654.   DOI: 10.1007/s11464-017-0627-2 Abstract   PDF (177KB) Let P ∈ Sp(2n) satisfying Pk = I2n. We consider the minimal Psymmetric period problem of the autonomous nonlinear Hamiltonian system $x˙(t)=JH′(x(t))$. For some symplectic matrices P, we show that for any τ>0, the above Hamiltonian system possesses a kτ periodic solution x with kτ being its minimal P-symmetric period provided H satisfies Rabinowitz’s conditions on the minimal period conjecture, together with that H is convex and H(Px) = H(x).
 Select Sums of Fourier coefficients of cusp forms of level D twisted by exponential functions Huan LIU Front. Math. China. 2017, 12 (3): 655-673.   DOI: 10.1007/s11464-016-0534-y Abstract   PDF (214KB) Let g be a holomorphic or Maass Hecke newform of level D and nebentypus χD, and let ag(n) be its n-th Fourier coefficient. We consider the sum $S1=∑X and prove that S1 has an asymptotic formula when β = 1/2 and αis close to $±2q/D$ for positive integer $q≤X/4$ and X sufficiently large. And when 0<β<1 and α, β fail to meet the above condition, we obtain upper bounds of S1. We also consider the sum $S2=∑n>0ag(n)e(anβ)ϕ(n/X)$ with $ϕ(x)∈Cc∞(0,+∞)$ and prove that S2 has better upper bounds than S1 at some special α and β.
 Select Weighted stationary phase of higher orders Mark MCKEE, Haiwei SUN, Yangbo YE Front. Math. China. 2017, 12 (3): 675-702.   DOI: 10.1007/s11464-016-0615-y Abstract   PDF (255KB) The subject matter of this paper is an integral with exponential oscillation of phase f(x) weighted by g(x) on a finite interval [α, β]. When the phase f(x) has a single stationary point in (α, β), an nth-order asymptotic expansion of this integral is proved for $n≥2$. This asymptotic expansion sharpens the classical result for n = 1 by M. N. Huxley. A similar asymptotic expansion was proved by V. Blomer, R. Khan and M. Young under the assumptions that f(x) and g(x) are smooth and g(x) is compactly supported on $?$. In the present paper, however, these functions are only assumed to be continuously differentiable on [α, β] 2n + 3 and 2n + 1 times, respectively. Because there are no requirements on the vanishing of g(x) and its derivatives at the endpoints α and β, the present asymptotic expansion contains explicit boundary terms in the main and error terms. The asymptotic expansion in this paper is thus applicable to a wider class of problems in analysis, analytic number theory, and other fields.
 Select Two recursive inequalities for crossing numbers of graphs Zhangdong OUYANG, Jing WANG, Yuanqiu HUANG Front. Math. China. 2017, 12 (3): 703-709.   DOI: 10.1007/s11464-016-0618-8 Abstract   PDF (119KB) In this paper, two recursive inequalities for crossing numbers of graphs are given by using basic counting method. As their applications, the crossing numbers of $K1,3,n$and $K4,n\e$ are determined, respectively.
 Select Precise large deviations for sums of random vectors with dependent components of consistently varying tails Xinmei SHEN, Yuqing NIU, Hailan TIAN Front. Math. China. 2017, 12 (3): 711-732.   DOI: 10.1007/s11464-017-0635-2 Abstract   PDF (231KB) Let be a sequence of independent and identically distributed nonnegative m-dimensional random vectors. The univariate marginal distributions of these vectors have consistently varying tails and finite means. Here, the components of $X1$ are allowed to be generally dependent. Moreover, let $N(·)$ be a nonnegative integer-valued process, independent of the sequence .Under several mild assumptions, precise large deviations for and are investigated. Meanwhile, some simulation examples are also given to illustrate the results.
 Select Generalized twistors of nonlocal vertex algebras Jiancai SUN, Minjing WANG Front. Math. China. 2017, 12 (3): 733-748.   DOI: 10.1007/s11464-016-0507-1 Abstract   PDF (154KB) We introduce and study the concept of (weak) pseudotwistor for a nonlocal vertex algebra, as a generalization of the notion of twistor. We give the relations between pseudotwistors and twisting operators. Furthermore, we study the inverse of an invertible weak pseudotwistor and the composition of two weak pseudotwistors.
 Select An equivalent characterization of BMO with Gauss measure Zhehui WANG, Dongyong YANG Front. Math. China. 2017, 12 (3): 749-768.   DOI: 10.1007/s11464-017-0624-5 Abstract   PDF (217KB) Let γbe the Gauss measure on ℝn.We establish a Calderón-Zygmund type decomposition and a John-Nirenberg type inequality in terms of the local sharp maximal function and the median value of function over cubes. As an application, we obtain an equivalent characterization of known BMO space with Gauss measure.
13 articles