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Frontiers of Mathematics in China

ISSN 1673-3452 (Print)
ISSN 1673-3576 (Online)
CN 11-5739/O1
Postal Subscription Code 80-964
2018 Impact Factor: 0.565

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, Volume 14 Issue 3

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SURVEY ARTICLE
Development of powerful algorithm for maximal eigenpair
Mu-Fa CHEN, Yue-Shuang LI
Front. Math. China. 2019, 14 (3): 493-519.  https://doi.org/10.1007/s11464-019-0769-5
Abstract   PDF (438KB)

Based on a series of recent papers, a powerful algorithm is reformulated for computing the maximal eigenpair of self-adjoint complex tridiagonal matrices. In parallel, the same problem in a particular case for computing the sub-maximal eigenpair is also introduced. The key ideas for each critical improvement are explained. To illustrate the present algorithm and compare it with the related algorithms, more than 10 examples are included.

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RESEARCH ARTICLE
Non-leaving-face property for marked surfaces
Thomas BRÜSTLE, Jie ZHANG
Front. Math. China. 2019, 14 (3): 521-534.  https://doi.org/10.1007/s11464-019-0767-7
Abstract   PDF (610KB)

We consider the polytope arising from a marked surface by flips of triangulations. D. D. Sleator, R. E. Tarjan, and W. P. Thurston [J. Amer. Math. Soc., 1988, 1(3): 647{681] studied the diameter of the associahedron, which is the polytope arising from a marked disc by flips of triangulations. They showed that every shortest path between two vertices in a face does not leave that face. We give a new method, which is different from the one used by V. Disarlo and H. Parlier [arXiv: 1411.4285] to establish the same non-leaving-face property for all unpunctured marked surfaces.

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Limiting weak-type behaviors for Riesz transforms and maximal operators in Bessel setting
Xianming HOU, Huoxiong WU
Front. Math. China. 2019, 14 (3): 535-550.  https://doi.org/10.1007/s11464-019-0774-8
Abstract   PDF (288KB)

We establish the limiting weak type behaviors of Riesz transforms associated to the Bessel operators on ℝ+; which are closely related to the best constants of the weak type (1; 1) estimates for such operators. Meanwhile, the corresponding results for Hardy-Littlewood maximal operator and fractional maximal operator in Bessel setting are also obtained.

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Continuity of functors with respect to generalized inductive limits
Jiajie HUA, Xiaochun FANG, Xiao-Ming XU
Front. Math. China. 2019, 14 (3): 551-566.  https://doi.org/10.1007/s11464-019-0772-x
Abstract   PDF (287KB)

Let (Ai,|ϕi,i+1) be a generalized inductive system of a sequence (Ai) of unital separable C*-algebras, with A=limi(Ai,ϕi,i+1). Set ϕj,i=ϕi1,iooϕj+1,j+2oϕj,j+1 for all i>j: We prove that if ϕj,i are order zero completely positive contractions for all j and i>j; and L:=inf{λ|λσ(ϕj,i(1Aj))} for all j and i>j}>0; where σ(ϕj,i(1Aj)) is the spectrum of ϕj,i(1Aj) ; then limi(Cu(ai),Cu(ϕi,i+1))=Cu(A) ; where Cu(A) is a stable version of the Cuntz semigroup of C*-algebra A: Let (An,ϕm,n) be a generalized inductive system of C*-algebras, with the ϕm,n order zero completely positive contractions. We also prove that if the decomposition rank (nuclear dimension) of An is no more than some integer k for each n; then the decomposition rank (nuclear dimension) of A is also no more than k:

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Variational study of bifurcations in von Kármán equations
Rongrong JIN, Guangcun LU
Front. Math. China. 2019, 14 (3): 567-590.  https://doi.org/10.1007/s11464-019-0766-8
Abstract   PDF (379KB)

For a class of nonlinear elliptic boundary value problems including the von Kármán equations considered by D. M. Duc, N. L. Luc, L. Q. Nam, and T. T. Tuyen [Nonlinear Anal., 2003, 55: 951{968], we give a new proof of a corresponding theorem of three solutions via Morse theory instead of topological degree theory. Several bifurcation results for this class of boundary value problems are also obtained with Morse theory methods. In addition, for the von Kármán equations studied by A. Borisovich and J. Janczewska [Abstr. Appl. Anal., 2005, 8: 889{899], we prove a few of bifurcation results under Dirichlet boundary conditions based on the second named author's recent work about parameterized splitting theorems and bifurcations for potential operators.

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Convergence of truncated rough singular integrals supported by subvarieties on Triebel-Lizorkin spaces
Feng LIU, Qingying XUE, K^oz^o YABUTA
Front. Math. China. 2019, 14 (3): 591-604.  https://doi.org/10.1007/s11464-019-0765-9
Abstract   PDF (429KB)

Let be a function of homogeneous of degree zero and satisfy the cancellation condition on the unit sphere. Suppose that h is a radial function. Let Th,Ω,p be the classical singular Radon transform, and let Th,Ω,pε be its truncated operator with rough kernels associated to polynomial mapping p which is defined by Th,Ω,pεf(x)=||y|εf(xp(y))h(|y|)Ω(y)|y|ndy|. In this paper, we show that for any α(,) and (p,q) satisfying certain index condition, the operator Th,Ω,pε enjoys the following convergence properties limε0Th,Ω,pεfTh,Ω,pfF ˙αp,q(d)=0 and limε0Th,Ω,pεfTh,Ω,pfB ˙αp,q(d)=0 provided that ΩL(log+L)β(Sn1) for some β(0,1] or ΩH1(Sn1), or Ω(1qBq(0,0)(Sn1)).

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Perfect 3-colorings on 6-regular graphs of order 9
Ziqiu LIU, Yunfeng ZHAO, Yuqin ZHANG
Front. Math. China. 2019, 14 (3): 605-618.  https://doi.org/10.1007/s11464-019-0768-6
Abstract   PDF (275KB)

The concept of a perfect coloring, introduced by P. Delsarte, generalizes the concept of completely regular code. We study the perfect 3-colorings (also known as the equitable partitions into three parts) on 6-regular graphs of order 9. A perfect n-colorings of a graph is a partition of its vertex set. It splits vertices into n parts A1,A2,...,An such that for all i,j{1,2,...,n}, each vertex of Ai is adjacent to aij vertices of Aj. The matrix A=(aij)n×n is called quotient matrix or parameter matrix. In this article, we start by giving an algorithm to find all different types of 6-regular graphs of order 9. Then, we classify all the realizable parameter matrices of perfect 3-colorings on 6-regular graphs of order 9.

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Interaction solutions to Hirota-Satsuma-Ito equation in (2+ 1)-dimensions
Wen-Xiu MA
Front. Math. China. 2019, 14 (3): 619-629.  https://doi.org/10.1007/s11464-019-0771-y
Abstract   PDF (2167KB)

Abundant exact interaction solutions, including lump-soliton, lumpkink, and lump-periodic solutions, are computed for the Hirota-Satsuma-Ito equation in (2+1)-dimensions, through conducting symbolic computations with Maple. The basic starting point is a Hirota bilinear form of the Hirota-Satsuma-Ito equation. A few three-dimensional plots and contour plots of three special presented solutions are made to shed light on the characteristic of interaction solutions.

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Lump wave and hybrid solutions of a generalized (3+ 1)-dimensional nonlinear wave equation in liquid with gas bubbles
Hui WANG, Shoufu TIAN, Tiantian ZHANG, Yi CHEN
Front. Math. China. 2019, 14 (3): 631-643.  https://doi.org/10.1007/s11464-019-0775-7
Abstract   PDF (1469KB)

We investigate a generalized (3+ 1)-dimensional nonlinear wave equation, which can be used to depict many nonlinear phenomena in liquid containing gas bubbles. By employing the Hirota bilinear method, we derive its bilinear formalism and soliton solutions succinctly. Meanwhile, the first-order lump wave solution and second-order lump wave solution are well presented based on the corresponding two-soliton solution and four-soliton solution. Furthermore, two types of hybrid solutions are systematically established by using the long wave limit method. Finally, the graphical analyses of the obtained solutions are represented in order to better understand their dynamical behaviors.

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Absence of eigenvalues for quasiperiodic Schrödinger type operators
Jiahao XU, Xin ZHAO
Front. Math. China. 2019, 14 (3): 645-659.  https://doi.org/10.1007/s11464-019-0773-9
Abstract   PDF (292KB)

We obtain the matrix-valued Schrödinger-type operators [Hα,θ] with Lipschitz potentials having no eigenvalues on the set {E: L(E)<δC,d(α,θ)}, where δ is an explicit function depending on the sampling function C(θ), dimension d, phase θ, and frequency α, and L(E) is the Lyapunov exponent.

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Estimations on upper and lower bounds of solutions to a class of tensor complementarity problems
Yang XU, Weizhe GU, Zheng-Hai HUANG
Front. Math. China. 2019, 14 (3): 661-671.  https://doi.org/10.1007/s11464-019-0770-z
Abstract   PDF (260KB)

We introduce a class of structured tensors, called generalized row strictly diagonally dominant tensors, and discuss some relationships between it and several classes of structured tensors, including nonnegative tensors, B-tensors, and strictly copositive tensors. In particular, we give estimations on upper and lower bounds of solutions to the tensor complementarity problem (TCP) when the involved tensor is a generalized row strictly diagonally dominant tensor with all positive diagonal entries. The main advantage of the results obtained in this paper is that both bounds we obtained depend only on the tensor and constant vector involved in the TCP; and hence, they are very easy to calculate.

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