Feb 2013, Volume 8 Issue 1
    

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  • EDITORIAL
    Spectral Theory of Nonnegative Tensors
    Qingzhi YANG, Liping ZHANG, Tan ZHANG, Guanglu ZHOU
    2013, 8(1): 1. https://doi.org/10.1007/s11464-012-0273-7
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  • RESEARCH ARTICLE
    Best rank one approximation of real symmetric tensors can be chosen symmetric
    Shmuel FRIEDLAND
    2013, 8(1): 19-40. https://doi.org/10.1007/s11464-012-0262-x
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    We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does not lie on a certain real algebraic variety.

  • RESEARCH ARTICLE
    lk,s-Singular values and spectral radius of rectangular tensors
    Chen LING, Liqun QI
    2013, 8(1): 63-83. https://doi.org/10.1007/s11464-012-0265-7
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    The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we study the singular values/vectors problem of real nonnegative partially symmetric rectangular tensors. We first introduce the concepts of lk,s-singular values/vectors of real partially symmetric rectangular tensors. Then, based upon the presented properties of lk,s-singular values /vectors, some properties of the related lk,s-spectral radius are discussed. Furthermore, we prove two analogs of Perron-Frobenius theorem and weak Perron-Frobenius theorem for real nonnegative partially symmetric rectangular tensors.

  • RESEARCH ARTICLE
    Maximal number of distinct H-eigenpairs for a two-dimensional real tensor
    Kelly J. PEARSON, Tan ZHANG
    2013, 8(1): 85-105. https://doi.org/10.1007/s11464-012-0263-9
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    Based on the generalized characteristic polynomial introduced by J. Canny in Generalized characteristic polynomials [J. Symbolic Comput., 1990, 9(3): 241–250], it is immediate that for any m-order n-dimensional real tensor, the number of distinct H-eigenvalues is less than or equal to n(m-1)n-1. However, there is no known bounds on the maximal number of distinct Heigenvectors in general. We prove that for any m≥2, an m-order 2-dimensional tensor A exists such that A has 2(m - 1) distinct H-eigenpairs. We give examples of 4-order 2-dimensional tensors with six distinct H-eigenvalues as well as six distinct H-eigenvectors. We demonstrate the structure of eigenpairs for a higher order tensor is far more complicated than that of a matrix. Furthermore, we introduce a new class of weakly symmetric tensors, called p-symmetric tensors, and show under certain conditions, p-symmetry will effectively reduce the maximal number of distinct H-eigenvectors for a given two-dimensional tensor. Lastly, we provide a complete classification of the H-eigenvectors of a given 4-order 2-dimensional nonnegative p-symmetric tensor. Additionally, we give sufficient conditions which prevent a given 4-order 2-dimensional nonnegative irreducible weakly symmetric tensor from possessing six pairwise distinct H-eigenvectors.

  • RESEARCH ARTICLE
    Geometric simplicity of spectral radius of nonnegative irreducible tensors
    Yuning YANG, Qingzhi YANG
    2013, 8(1): 129-140. https://doi.org/10.1007/s11464-012-0272-8
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    We study the real and complex geometric simplicity of nonnegative irreducible tensors. First, we prove some basic conclusions. Based on the conclusions, the real geometric simplicity of the spectral radius of an evenorder nonnegative irreducible tensor is proved. For an odd-order nonnegative irreducible tensor, sufficient conditions are investigated to ensure the spectral radius to be real geometrically simple. Furthermore, the complex geometric simplicity of nonnegative irreducible tensors is also studied.

  • RESEARCH ARTICLE
    Linear convergence of an algorithm for largest singular value of a nonnegative rectangular tensor
    Liping ZHANG
    2013, 8(1): 141-153. https://doi.org/10.1007/s11464-012-0260-z
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    An algorithm for finding the largest singular value of a nonnegative rectangular tensor was recently proposed by Chang, Qi, and Zhou [J. Math. Anal. Appl., 2010, 370: 284–294]. In this paper, we establish a linear convergence rate of the Chang-Qi-Zhou algorithm under a reasonable assumption.

  • RESEARCH ARTICLE
    Efficient algorithms for computing the largest eigenvalue of a nonnegative tensor
    Guanglu ZHOU, Liqun QI, Soon-Yi WU
    2013, 8(1): 155-168. https://doi.org/10.1007/s11464-012-0268-4
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    Consider the problem of computing the largest eigenvalue for nonnegative tensors. In this paper, we establish the Q-linear convergence of a power type algorithm for this problem under a weak irreducibility condition. Moreover, we present a convergent algorithm for calculating the largest eigenvalue for any nonnegative tensors.

  • RESEARCH ARTICLE
    Factorization of simple modules for certain restricted two-parameter quantum groups
    Min LI, Xiuling WANG
    2013, 8(1): 169-190. https://doi.org/10.1007/s11464-012-0236-z
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    We study the representations of the restricted two-parameter quantum groups of types B and G. For these restricted two-parameter quantum groups, we give some explicit conditions which guarantee that a simple module can be factored as the tensor product of a one-dimensional module with a module that is naturally a module for the quotient by central group-like elements. That is, given θ a primitive lth root of unity, the factorization of simple ?θy,θz,( )-modules is possible, if and only if (2(y - z), l) = 1 for =??2n+1; (3(y - z), l) = 1 for g= G2.

  • RESEARCH ARTICLE
    Singular soliton solution and bifurcation analysis of Klein-Gordon equation with power law nonlinearity
    Ming SONG, Zhengrong LIU, Essaid ZERRAD, Anjan BISWAS
    2013, 8(1): 191-201. https://doi.org/10.1007/s11464-012-0252-z
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    In this paper, the Klein-Gordon equation (KGE) with power law nonlinearity will be considered. The bifurcation analysis as well as the ansatz method of integration will be applied to extract soliton and other wave solutions. In particular, the second approach to integration will lead to a singular soliton solution. However, the bifurcation analysis will reveal several other solutions that are of prime importance in relativistic quantum mechanics where this equation appears.

  • RESEARCH ARTICLE
    Filtration, automorphisms, and classification of infinite-dimensional odd contact superalgebras
    Jixia YUAN, Wende LIU
    2013, 8(1): 203-216. https://doi.org/10.1007/s11464-012-0185-6
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    The principal filtration of the infinite-dimensional odd contact Lie superalgebra over a field of characteristic p>2 is proved to be invariant under the automorphism group by investigating ad-nilpotent elements and determining certain invariants such as subalgebras generated by some ad-nilpotent elements. Then, it is proved that two automorphisms coincide if and only if they coincide on the -1 component with respect to the principal grading. Finally, all the odd contact superalgebras are classified up to isomorphisms.

  • RESEARCH ARTICLE
    Nowhere-zero 3-flows in matroid base graph
    Yinghao ZHANG, Guizhen LIU
    2013, 8(1): 217-227. https://doi.org/10.1007/s11464-012-0246-x
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    The base graph of a simple matroid M=(E,?) is the graph G such that V(G)=? and E(G)={BB′:B,B′∈?,|B\B′|=1}, where the same notation is used for the vertices of G and the bases of M. It is proved that the base graph G of connectedsimple matroid M is Z3-connected if |V (G)|≥5. We also proved that if M is not a connected simple matroid, then the base graph G of M does not admit a nowhere-zero 3-flow if and only if |V (G)| = 4. Furthermore, if for every connected component Ei (i≥2) of M, the matroid ase graph Gi of Mi = M|Ei has |V (Gi)|≥5, then G is Z3-connected which also implies that G admits nowhere-zero 3-flow immediately.

  • RESEARCH ARTICLE
    A characterization of λ-central BMO space
    Fayou ZHAO, Shanzhen LU
    2013, 8(1): 229-238. https://doi.org/10.1007/s11464-012-0251-0
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    We give a characterization of the λ-central BMO space via the boundedness of commutators of n-dimensional Hardy operators.