Jun 2019, Volume 14 Issue 3

    

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• SURVEY ARTICLE

  Development of powerful algorithm for maximal eigenpair

  Mu-Fa CHEN, Yue-Shuang LI

  2019, 14(3): 493-519. https://doi.org/10.1007/s11464-019-0769-5

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  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.

• RESEARCH ARTICLE

  Non-leaving-face property for marked surfaces

  Thomas BRÜSTLE, Jie ZHANG

  2019, 14(3): 521-534. https://doi.org/10.1007/s11464-019-0767-7

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  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.

• RESEARCH ARTICLE

  Limiting weak-type behaviors for Riesz transforms and maximal operators in Bessel setting

  Xianming HOU, Huoxiong WU

  2019, 14(3): 535-550. https://doi.org/10.1007/s11464-019-0774-8

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  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.

• RESEARCH ARTICLE

  Continuity of functors with respect to generalized inductive limits

  Jiajie HUA, Xiaochun FANG, Xiao-Ming XU

  2019, 14(3): 551-566. https://doi.org/10.1007/s11464-019-0772-x

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  Let ({{\displaystyle A}}_i,|{{\displaystyle \varphi }}_{i,i+\mathrm{1}}) be a generalized inductive system of a sequence (A_i) of unital separable C*-algebras, with A={{\displaystyle \lim }}_{i\to \infty }({{\displaystyle A}}_i,{{\displaystyle \varphi }}_{i,i+\mathrm{1}}). Set {{\displaystyle \varphi }}_{j,i}={{\displaystyle \varphi }}_{i-\mathrm{1},i}\mathrm{o}\cdots \mathrm{o}{{\displaystyle \varphi }}_{j+\mathrm{1},j+\mathrm{2}}\mathrm{o}{{\displaystyle \varphi }}_{j,j+\mathrm{1}} for all i>j: We prove that if {{\displaystyle \varphi }}_{j,i} are order zero completely positive contractions for all j and i>j; and L:=\inf \{\lambda |\lambda \in \sigma ({{\displaystyle \varphi }}_{j,i}({{\displaystyle \mathrm{1}}}_{{{\displaystyle A}}_j}))\} for all j and i>j}>0; where \sigma ({{\displaystyle \varphi }}_{j,i}({{\displaystyle \mathrm{1}}}_{{{\displaystyle A}}_j})) is the spectrum of {{\displaystyle \varphi }}_{j,i}({{\displaystyle \mathrm{1}}}_{{{\displaystyle A}}_j}) ; then {{\displaystyle \lim }}_{i\to \infty }(\mathrm{C}\mathrm{u}({{\displaystyle a}}_i),\mathrm{C}\mathrm{u}({{\displaystyle \varphi }}_{i,i+\mathrm{1}}))=\mathrm{C}\mathrm{u}(A) ; where Cu(A) is a stable version of the Cuntz semigroup of C*-algebra A: Let ({{\displaystyle A}}_n,{{\displaystyle \varphi }}_{m,n}) be a generalized inductive system of C*-algebras, with the {{\displaystyle \varphi }}_{m,n} order zero completely positive contractions. We also prove that if the decomposition rank (nuclear dimension) of A_n is no more than some integer k for each n; then the decomposition rank (nuclear dimension) of A is also no more than k:

• RESEARCH ARTICLE

  Variational study of bifurcations in von Kármán equations

  Rongrong JIN, Guangcun LU

  2019, 14(3): 567-590. https://doi.org/10.1007/s11464-019-0766-8

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  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.

• RESEARCH ARTICLE

  Convergence of truncated rough singular integrals supported by subvarieties on Triebel-Lizorkin spaces

  Feng LIU, Qingying XUE, K^oz^o YABUTA

  2019, 14(3): 591-604. https://doi.org/10.1007/s11464-019-0765-9

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  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 {{\displaystyle T}}_{h,\mathrm{\Omega },p} be the classical singular Radon transform, and let {{\displaystyle T}}_{h,\mathrm{\Omega },{p}}^{\epsilon } be its truncated operator with rough kernels associated to polynomial mapping p which is defined by {{\displaystyle T}}_{h,\mathrm{\Omega },p}^{\epsilon }f(x)=\left|{{\displaystyle \int }}_{\left|y\right|\rangle \epsilon }f(x-p(y))h(\left|y\right|)\mathrm{\Omega }(y){\left|y\right|}^{-n}\mathrm{d}y\right|. In this paper, we show that for any \alpha \in (-\infty ,\infty ) and (p,q) satisfying certain index condition, the operator {{\displaystyle T}}_{h,\mathrm{\Omega },p}^{\epsilon } enjoys the following convergence properties {{\displaystyle \lim }}_{\epsilon \to \mathrm{0}}{\Vert {{\displaystyle T}}_{h,\mathrm{\Omega },p}^{\epsilon }f-{{\displaystyle T}}_{h,\mathrm{\Omega },p}f\Vert }_{{{\displaystyle \dot{F}}}_{\alpha }^{p,q}({ℝ}^d)}=\mathrm{0} and {{\displaystyle \lim }}_{\epsilon \to \mathrm{0}}{\Vert {{\displaystyle T}}_{h,\mathrm{\Omega },p}^{\epsilon }f-{{\displaystyle T}}_{h,\mathrm{\Omega },p}f\Vert }_{{{\displaystyle \dot{B}}}_{\alpha }^{p,q}({ℝ}^d)}=\mathrm{0} provided that \mathrm{\Omega }\in {L({\log }^{+}L)}^{\beta }(S^{n-\mathrm{1}}) for some \beta \in (\mathrm{0},\mathrm{1}] or \mathrm{\Omega }\in H^{\mathrm{1}}(S^{n-\mathrm{1}}), or \mathrm{\Omega }\in ({{\displaystyle \cup }}_{\mathrm{1}\langle q\langle \infty }{{\displaystyle B}}_q^{(\mathrm{0},\mathrm{0})}(S^{n-\mathrm{1}})).

• RESEARCH ARTICLE

  Perfect 3-colorings on 6-regular graphs of order 9

  Ziqiu LIU, Yunfeng ZHAO, Yuqin ZHANG

  2019, 14(3): 605-618. https://doi.org/10.1007/s11464-019-0768-6

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  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 {{\displaystyle A}}_{\mathrm{1}},{{\displaystyle A}}_{\mathrm{2}},\mathrm{...},{{\displaystyle A}}_n such that for all i,j\in \{\mathrm{1},\mathrm{2},\mathrm{...},n\}, each vertex of A_i is adjacent to a_ij vertices of A_j. The matrix A={{\displaystyle ({{\displaystyle a}}_{ij})}}_{n\times 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.

• RESEARCH ARTICLE

  Interaction solutions to Hirota-Satsuma-Ito equation in (2+ 1)-dimensions

  Wen-Xiu MA

  2019, 14(3): 619-629. https://doi.org/10.1007/s11464-019-0771-y

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  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.

• RESEARCH ARTICLE

  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

  2019, 14(3): 631-643. https://doi.org/10.1007/s11464-019-0775-7

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  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.

• RESEARCH ARTICLE

  Absence of eigenvalues for quasiperiodic Schrödinger type operators

  Jiahao XU, Xin ZHAO

  2019, 14(3): 645-659. https://doi.org/10.1007/s11464-019-0773-9

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  We obtain the matrix-valued Schrödinger-type operators [H_α,θ] with Lipschitz potentials having no eigenvalues on the set {E: L(E)<δ_C,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.

• RESEARCH ARTICLE

  Estimations on upper and lower bounds of solutions to a class of tensor complementarity problems

  Yang XU, Weizhe GU, Zheng-Hai HUANG

  2019, 14(3): 661-671. https://doi.org/10.1007/s11464-019-0770-z

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  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.