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Frontiers of Mathematics in China

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, Volume 12 Issue 4 Previous Issue   
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RESEARCH ARTICLE
Riccati equations and Toeplitz-Berezin type symbols on Dirichlet space of unit ball
Jianjun CHEN, Xiaofeng WANG, Jin XIA, Guangfu CAO
Front. Math. China. 2017, 12 (4): 769-785.   DOI: 10.1007/s11464-017-0640-5
Abstract   PDF (192KB)

The present paper mainly gives some applications of Berezin type symbols on the Dirichlet space of unit ball. We study the solvability of some Riccati operator equations of the form XAX+ XBCX= Drelated to harmonic Toeplitz operators on the Dirichlet space. Especially, the invariant subspaces of Toeplitz operators are also considered.

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A class of simple Lie algebras attached to unit forms
Jinjing CHEN, Zhengxin CHEN
Front. Math. China. 2017, 12 (4): 787-803.   DOI: 10.1007/s11464-016-0616-x
Abstract   PDF (213KB)

Let n≥3.The complex Lie algebra, which is attached to a unit form q(x1,x2,,xn)=i=1nxi2+1ijn(1)jixixj and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type An,and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.

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Asymptotic behavior for log-determinants of several non-Hermitian random matrices
Lei CHEN, Shaochen WANG
Front. Math. China. 2017, 12 (4): 805-819.   DOI: 10.1007/s11464-017-0629-0
Abstract   PDF (277KB)

We study the asymptotic behavior for log-determinants of two unitary but non-Hermitian random matrices: the spherical ensembles A−1B, where A and B are independent complex Ginibre ensembles and the truncation of circular unitary ensembles. The corresponding Berry-Esseen bounds and Cramér type moderate deviations are established. Our method is based on the estimates of corresponding cumulants. Numerical simulations are also presented to illustrate the theoretical results.

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Limit theorems for functionals of Gaussian vectors
Hongshuai DAI, Guangjun SHEN, Lingtao KONG
Front. Math. China. 2017, 12 (4): 821-842.   DOI: 10.1007/s11464-016-0620-1
Abstract   PDF (224KB)

Operator self-similar processes, as an extension of self-similar processes, have been studied extensively. In this work, we study limit theorems for functionals of Gaussian vectors. Under some conditions, we determine that the limit of partial sums of functionals of a stationary Gaussian sequence of random vectors is an operator self-similar process.

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Discrete α-Yamabe flow in 3-dimension
Huabin GE, Shiguang MA
Front. Math. China. 2017, 12 (4): 843-858.   DOI: 10.1007/s11464-016-0603-2
Abstract   PDF (190KB)

We generalize the discrete Yamabe flow to αorder. This Yamabe flow deforms the α-order curvature to a constant. Using this new flow, we manage to find discrete α-quasi-Einstein metrics on the triangulations of S 3.

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Finite dimensional characteristic functions of Brownian rough path
Xi GENG, Zhongmin QIAN
Front. Math. China. 2017, 12 (4): 859-877.   DOI: 10.1007/s11464-017-0648-x
Abstract   PDF (207KB)

The Brownian rough path is the canonical lifting of Brownian motion to the free nilpotent Lie group of order 2. Equivalently, it is a process taking values in the algebra of Lie polynomials of degree 2, which is described explicitly by the Brownian motion coupled with its area process. The aim of this article is to compute the finite dimensional characteristic functions of the Brownian rough path in ?d and obtain an explicit formula for the case when d = 2.

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Some q-inequalities for Hausdorff operators
Jiuhua GUO, Fayou ZHAO
Front. Math. China. 2017, 12 (4): 879-889.   DOI: 10.1007/s11464-017-0622-7
Abstract   PDF (136KB)

We calculate the sharp bounds for some q-analysis variants of Hausdorff type inequalities of the form+0(+0ϕ(t)tfxtdqt)pdqxCϕb0fp(t)dqt.As applications, we obtain several sharp q-analysis inequalities of the classical positive integral operators, including the Hardy operator and its adjoint operator, the Hilbert operator, and the Hardy-Littlewood-Pólya operator.

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Involutions in Weyl group of type F4
Jun HU, Jing ZHANG, Yabo WU
Front. Math. China. 2017, 12 (4): 891-906.   DOI: 10.1007/s11464-017-0646-z
Abstract   PDF (317KB)

Let W be the Weyl group of type F4: We explicitly describe a nite set of basic braid I*-transformations and show that any two reduced I*-expressions for a given involution in W can be transformed into each other through a series of basic braid I*-transformations. Our main result extends the earlier work on the Weyl groups of classical types (i.e., An; Bn; and Dn).

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Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions
Ruo LI, Tiao LU, Zhangpeng SUN
Front. Math. China. 2017, 12 (4): 907-919.   DOI: 10.1007/s11464-017-0612-9
Abstract   PDF (163KB)

We study the stationary Wigner equation on a bounded, onedimensional spatial domain with inflow boundary conditions by using the parity decomposition of L. Barletti and P. F. Zweifel [Transport Theory Statist. Phys., 2001, 30(4-6): 507–520]. The decomposition reduces the half-range, two-point boundary value problem into two decoupled initial value problems of the even part and the odd part. Without using a cutoff approximation around zero velocity, we prove that the initial value problem for the even part is well-posed. For the odd part, we prove the uniqueness of the solution in the odd L2-space by analyzing the moment system. An example is provided to show that how to use the analysis to obtain the solution of the stationary Wigner equation with inflow boundary conditions.

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Graphs with small total rainbow connection number
Yingbin MA, Lily CHEN, Hengzhe LI
Front. Math. China. 2017, 12 (4): 921-936.   DOI: 10.1007/s11464-017-0651-2
Abstract   PDF (169KB)

A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we investigate graphs with small total rainbow connection number. First, for a connected graph G, we prove that trc(G)=3?if(n12)+1|E(G)|(n2)1, and trc(G)=6?if(n22)+2. Next, we investigate the total rainbow connection numbers of graphs G with |V(G)|=n, diam(G)2, and clique number ω(G)=ns?for?1?s?3. In this paper, we find Theorem 3 of [Discuss. Math. Graph Theory, 2011, 31(2): 313–320] is not completely correct, and we provide a complete result for this theorem.

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Neighbor sum distinguishing total chromatic number of K4-minor free graph
Hongjie SONG, Changqing XU
Front. Math. China. 2017, 12 (4): 937-947.   DOI: 10.1007/s11464-017-0649-9
Abstract   PDF (165KB)

A k-total coloring of a graph G is a mapping φ: V (G) ∪ E(G) →{1, 2, . . . , k} such that no two adjacent or incident elements in V (G) ∪ E(G) receive the same color. Let f(v) denote the sum of the color on the vertex v and the colors on all edges incident with v. We say that φ is a k-neighbor sum distinguishing total coloring of G if f(u) ≠ f(v) for each edge uvE(G). Denote XΣ''(G) the smallest value k in such a coloring of G. Pilśniak andWoźniak conjectured that for any simple graph with maximum degree Δ(G), XΣ''(G)Δ(G)+3. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for K4-minor free graph G with Δ(G)≥5, XΣ''(G)=Δ(G)+1 if G contains no two adjacent Δ-vertices, otherwise, XΣ''(G)=Δ(G)+2.

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Lie bialgebra structures on derivation Lie algebra over quantum tori
Xiaomin TANG, Lijuan LIU, Jinli XU
Front. Math. China. 2017, 12 (4): 949-965.   DOI: 10.1007/s11464-017-0630-7
Abstract   PDF (188KB)

We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H1(W,WW) is trivial.

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A restriction theorem for oscillatory integral operator with certain polynomial phase
Shaozhen XU, Dunyan YAN
Front. Math. China. 2017, 12 (4): 967-980.   DOI: 10.1007/s11464-017-0637-0
Abstract   PDF (180KB)

We consider the oscillatory integral operator Tα,mf(x)=?nei(x1α1y1m+?+xnαnynm)f(y)dy, where the function f is a Schwartz function. In this paper, the restriction theorem on Sn1 for this operator is obtained. Moreover, we obtain a necessary condition which ensures validity of the restriction theorem.

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Ideal counting function in cubic fields
Zhishan YANG
Front. Math. China. 2017, 12 (4): 981-992.   DOI: 10.1007/s11464-016-0570-7
Abstract   PDF (157KB)

For a cubic algebraic extension K of ℚ, the behavior of the ideal counting function is considered in this paper. More precisely, let aK(n) be the number of integral ideals of the field K with norm n, we prove an asymptotic formula for the sum n12+n22xaK(n12+n22).

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Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications
Hui ZHANG, Chunyan QI, Baode LI
Front. Math. China. 2017, 12 (4): 993-1022.   DOI: 10.1007/s11464-016-0546-7
Abstract   PDF (295KB)

Anisotropy is a common attribute of the nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathematics, can be expressed by a fairly general discrete group of dilations {Ak : k ∈ Z}, where A is a real n × n matrix with all its eigenvalues λ satisfy |λ|>1. The aim of this article is to study a general class of anisotropic function spaces, some properties and applications of these spaces. Let ϕ: Rn×[0,∞) →[0,∞) be an anisotropic p-growth function with p ∈ (0, 1]. The purpose of this article is to find an appropriate general space which includes weak Hardy space of Fefferman and Soria, weighted weak Hardy space of Quek and Yang, and anisotropic weak Hardy space of Ding and Lan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type HAφ,(?n) and obtain its atomic characterization. As applications, we further obtain an interpolation theorem adapted to HAφ,(?n) and the boundedness of the anisotropic Calderón-Zygmund operator from HAφ,(?n) to Lφ,(?n). It is worth mentioning that the superposition principle adapted to the weak Musielak-Orlicz function space, which is an extension of a result of E. M. Stein, M. Taibleson and G. Weiss, plays an important role in the proofs of the atomic decomposition of HAφ,(?n) and the interpolation theorem.

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