Mar 2018, Volume 13 Issue 2
    

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  • RESEARCH ARTICLE
    Haibin CHEN, Liqun QI, Yisheng SONG

    Stimulated by the study of sufficient matrices in linear complementarity problems, we study column sufficient tensors and tensor complementarity problems. Column sufficient tensors constitute a wide range of tensors that include positive semi-definite tensors as special cases. The inheritance property and invariant property of column sufficient tensors are presented. Then, various spectral properties of symmetric column sufficient tensors are given. It is proved that all H-eigenvalues of an even-order symmetric column sufficient tensor are nonnegative, and all its Z-eigenvalues are nonnegative even in the odd order case. After that, a new subclass of column sufficient tensors and the handicap of tensors are defined. We prove that a tensor belongs to the subclass if and only if its handicap is a finite number. Moreover, several optimization models that are equivalent with the handicap of tensors are presented. Finally, as an application of column sufficient tensors, several results on tensor complementarity problems are established.

  • RESEARCH ARTICLE
    Lizhong CHU, Shisen LIU, Zhongming TANG

    Let S = K[x1, x2, . . . , xn] be the polynomial ring in n variables over a field K, and let I be a squarefree monomial ideal minimally generated by the monomials u1, u2, . . . , um. Let w be the smallest number t with the property that for all integers 1i1<i2<?<itm such that lcm (ui1,ui2, . . . , uit) =lcm(u1, u2, . . . , um). We give an upper bound for Castelnuovo-Mumford regularity of I by the bigsize of I. As a corollary, the projective dimension of I is bounded by the number w.

  • RESEARCH ARTICLE
    Mehmet GÜRDAL, Ula¸s YAMANCI, Mübariz GARAYEV

    We investigate some problems for truncated Toeplitz operators. Namely, the solvability of the Riccati operator equation on the set of all truncated Toeplitz operators on the model space Kθ = H2ΘθH2 is studied. We study in terms of Berezin symbols invertibility of model operators. We also prove some results for the Berezin number of the truncated Toeplitz operators. Moreover, we study some property for H2-functions in terms of noncyclicity of co-analytic Toeplitz operators and hypercyclicity of model operators.

  • RESEARCH ARTICLE
    Yuan HOU, An CHANG, Lei ZHANG

    The k-uniform s-hypertree G = (V,E) is an s-hypergraph, where 1≤sk 1, and there exists a host tree T with vertex set V such that each edge of G induces a connected subtree of T. In this paper, some properties of uniform s-hypertrees are establised, as well as the upper and lower bounds on the largest H-eigenvalue of the adjacency tensor of k-uniform s-hypertrees in terms of the maximal degree Δ. Moreover, we also show that the gap between the maximum and the minimum values of the largest H-eigenvalue of k-uniform s-hypertrees is just Θ(Δs/k).

  • RESEARCH ARTICLE
    Yifen KE, Changfeng MA, Zhiru REN

    Based on the special positive semidefinite splittings of the saddle point matrix, we propose a new alternating positive semidefinite splitting (APSS) iteration method for the saddle point problem arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current problem. We prove that the new APSS iteration method is unconditionally convergent for both cases of the simple topology and the general topology. The new APSS matrix can be used as a preconditioner to accelerate the convergence rate of Krylov subspace methods. Numerical results show that the new APSS preconditioner is superior to the existing preconditioners.

  • RESEARCH ARTICLE
    Lei LIU

    Based on the relationship between symplectic group Sp(2) and Λ(2), we provide an intuitive explanation (model) of the 3-dimensional Lagrangian Grassmann manifold Λ(2), the singular cycles of Λ(2), and the special Lagrangian Grassmann manifold SΛ(2). Under this model, we give a formula of the rotation paths defined by Arnold.

  • RESEARCH ARTICLE
    Xiaole SU, Hongwei SUN, Yusheng WANG

    Let Mn(n3) be a complete Riemannian manifold with sec?M1, and let Mini(i=1,2) be two complete totally geodesic submanifolds in M. We prove that if n1 + n2 = n − 2 and if the distance |M1M2|π/2, then Mi is isometric to Sni/?h,?Pni/2/?2, or ?Pni/2/?2 with the canonical metric when ni>0, and thus, M is isometric to Sn/?h,?Pn/2, or ?Pn/2/?2 except possibly when n = 3 and M1 (or M2) isoS1/?h with h2 or n = 4 and M1 (or M2) iso?P2.

  • RESEARCH ARTICLE
    Zhihua WANG, Libin LI

    Some equivalent conditions for double Frobenius algebras to be strict ones are given. Then some examples of (strict or non-strict) double Frobenius algebras are presented. Finally, a sufficient and necessary condition for the trivial extension of a double Frobenius algebra to be a (strict) double Frobenius algebra is given.

  • RESEARCH ARTICLE
    Pai YANG, Liangwen LIAO, Qiaoyu CHEN

    We take up a new method to prove a Picard type theorem. Let f be a meromorphic function in the complex plane, whose zeros are multiple, and let R be a Möbius transformation. If lim?¯rT(r,f)r2=??????then??????f'(z)=R(ez) has infinitely many solutions in the complex plane.

  • RESEARCH ARTICLE
    Songting YIN

    We obtain the Laplacian comparison theorem and the Bishop-Gromov comparison theorem on a Finsler manifold with the weighted Ricci curvature Ric∞ bounded below. As applications, we prove that if the weighted Ricci curvature Ric∞ is bounded below by a positive number, then the manifold must have finite fundamental group, and must be compact if the distortion is also bounded. Moreover, we give the Calabi-Yau linear volume growth theorem on a Finsler manifold with nonnegative weighted Ricci curvature.

  • RESEARCH ARTICLE
    Haixia YU, Junfeng LI

    We obtain the operator norms of the n-dimensional fractional Hardy operator Hα(0αN) from weighted Lebesgue spaces L|x|pp(?n) to weighted weak Lebesgue spaces L|x|βq,(?n).

  • RESEARCH ARTICLE
    Leihong ZHANG, Weihong YANG, Chungen SHEN, Jiang FENG

    Because of its vital role of the trust-region subproblem (TRS) in various applications, for example, in optimization and in ill-posed problems, there are several factorization-free algorithms for solving the large-scale sparse TRS. The truncated Lanczos approach proposed by N. I. M. Gould, S. Lucidi, M. Roma, and P. L. Toint [SIAM J. Optim., 1999, 9: 504–525] is a natural extension of the classical Lanczos method for the symmetric linear system and eigenvalue problem and, indeed follows the classical Rayleigh-Ritz procedure for eigenvalue computations. It consists of 1) projecting the original TRS to the Krylov subspaces to yield smaller size TRS’s and then 2) solving the resulted TRS’s to get the approximates of the original TRS. This paper presents a posterior error bounds for both the global optimal value and the optimal solution between the original TRS and their projected counterparts. Our error bounds mainly rely on the factors from the Lanczos process as well as the data of the original TRS and, could be helpful in designing certain stopping criteria for the truncated Lanczos approach.

  • RESEARCH ARTICLE
    Xiaobo ZHUANG

    By using the Bialynicki-Birula decomposition and holomorphic Lefschetz formula, we calculate the Poincaré polynomials of the moduli spaces in low degrees.