15 April 2015, Volume 31 Issue 4

    

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  Nonorientable Genera of Petersen Powers

  Wen Zhong LIU, Ting Ru SHEN, Yi Chao CHEN

  Acta Mathematica Sinica. 2015, 31(4): 557-564. https://doi.org/10.1007/s10114-015-4096-9

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  In the paper, we prove that for every integer n≥1, there exists a Petersen power Pⁿ with nonorientable genus and Euler genus precisely n, which improves the upper bound of Mohar and Vodopivec's result [J. Graph Theory, 67, 1-8 (2011)] that for every integer k(2≤k≤n-1), a Petersen power Pⁿ exists with nonorientable genus and Euler genus precisely k.
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  Polynomials with Palindromic and Unimodal Coefficients

  Hua SUN, Yi WANG, Hai Xia ZHANG

  Acta Mathematica Sinica. 2015, 31(4): 565-575. https://doi.org/10.1007/s10114-015-4331-4

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  Let f(q)=a_rq_r+···+ a_sq^s, with a_r=0 and a_s=0, be a real polynomial. It is a palindromic polynomial of darga n if r+s=n and a_r+i=a_s-i for all i. Polynomials of darga n form a linear subspace P_n(q) of R(q)_n+1 of dimension [n/2]+1. We give transition matrices between two bases {q^j(1+q+··· +q^n-2j)},{q^j(1+q)^n-2j and the standard basis q^j(1+q^n-2j) of P_n(q). We present some characterizations and sufficient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.
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  Inferences in Linear Mixed Models with Skew-normal Random Effects

  Ren Dao YE, Tong Hui WANG

  Acta Mathematica Sinica. 2015, 31(4): 576-594. https://doi.org/10.1007/s10114-015-3326-5

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  For the linear mixed model with skew-normal random effects, this paper gives the density function, moment generating function and independence conditions. The noncentral skew chi-square distribution is defined and its density function is shown. The necessary and sufficient conditions under which a quadratic form is distributed as noncentral skew chi-square distribution are obtained. Also, a version of Cochran's theorem is given, which modifies the result of Wang et al. (2009) and is used to set up exact tests for fixed effects and variance components of the proposed model. For illustration, our main results are applied to a real data problem.
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  Vectorial Variational Principle with Variable Set-Valued Perturbation

  Jian ZHANG, Jing Hui QIU

  Acta Mathematica Sinica. 2015, 31(4): 595-614. https://doi.org/10.1007/s10114-015-3587-z

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  We give a general vectorial Ekeland's variational principle, where the objective function is defined on an F-type topological space and taking values in a pre-ordered real linear space. Being quite different from the previous versions of vectorial Ekeland's variational principle, the perturbation in our version is no longer only dependent on a fixed positive vector or a fixed family of positive vectors. It contains a family of set-valued functions taking values in the positive cone and a family of subadditive functions of topology generating quasi-metrics. Hence, the direction of the perturbation in the new version is a family of variable subsets which are dependent on the objective function values. The general version includes and improves a number of known versions of vectorial Ekeland's variational principle. From the general Ekeland's principle, we deduce the corresponding versions of Caristi-Kirk's fixed point theorem and Takahashi's nonconvex minimization theorem. Finally, we prove that all the three theorems are equivalent to each other.
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  Nielsen Theory on 3-manifolds Covered by S²×R

  Daciberg GONÇALVES, Peter WONG, Xue Zhi ZHAO

  Acta Mathematica Sinica. 2015, 31(4): 615-636. https://doi.org/10.1007/s10114-015-3742-6

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  Let f: M → M be a self-map of a closed manifold M of dimension dim M ≥ 3. The Nielsen number N(f) of f is equal to the minimal number of fixed points of f among all self-maps f in the homotopy class of f. In this paper, we determine N(f) for all self-maps f when M is a closed 3-manifold with S²×R geometry. The calculation of N(f) relies on the induced homomorphisms of f on the fundamental group and on the second homotopy group of M.
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  A Note on Quasi-weakly Almost Periodic Point

  Qi YAN, Jian Dong YIN, Tao WANG

  Acta Mathematica Sinica. 2015, 31(4): 637-646. https://doi.org/10.1007/s10114-015-4202-z

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  Recently, He et al. [On quasi-weakly almost periodic points. Sci. China Math., 56, 597-606 (2013)] constructed two binary sub-shifts to solve an open problem posed by Zhou and Feng in [Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: A brief survey of recent results. Nonlinearity, 17, 493-502 (2004)]. In this paper, we study more dynamical properties of those two binary sub-shifts. We show that the first one has zero topological entropy and is transitive but not weakly mixing, while the second one has positive topological entropy and is strongly mixing.
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  Bounded Fatou Components of Transcendental Entire Functions with Order Less than 1/2

  Cun Ji YANG, Yu Hua LI

  Acta Mathematica Sinica. 2015, 31(4): 647-658. https://doi.org/10.1007/s10114-015-3698-6

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  Let f be a transcendental entire function with order ρ<1/2 and let ρ be a sufficiently large constant. We prove that if there exists r₀>1 such that, for all r>r₀ and any small ε>0,
  M(r^σ, f)≥M(r, f)^σ+ε,
  then every component of the Fatou set F(f) is bounded.
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  Eigenproblem for p-Laplacian and Nonlinear Elliptic Equation with Nonlinear Boundary Conditions

  Berrajaa MOHAMMED, Chakrone OMAR, Diyer FATIHA, Diyer OKACHA

  Acta Mathematica Sinica. 2015, 31(4): 659-674. https://doi.org/10.1007/s10114-015-3662-5

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  In this article, we study the solvability of nonlinear problem for p-Laplacian with nonlinear boundary conditions. We give some characterization of the first eigenvalue of an intermediary eigenvalue problem as simplicity, isolation and its strict monotonicity. Afterward, we character also the second eigenvalue and its strictly partial monotony. On the other hand, in some sense, we establish the non-resonance below the first and furthermore between the first and second eigenvalues of nonlinear Steklov-Robin.
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  Homological Dimensions of the Extension Algebras of Monomial Algebras

  Hong Bo SHI

  Acta Mathematica Sinica. 2015, 31(4): 675-694. https://doi.org/10.1007/s10114-015-3623-z

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  The main objective of this paper is to study the dimension trees and further the homological dimensions of the extension algebras—dual and trivially twisted extensions—with a unified combinatorial approach using the two combinatorial algorithms—Topdown and Bottomup. We first present a more complete and clearer picture of a dimension tree, with which we are then able, on the one hand, to sharpen some results obtained before and furthermore reveal a few more hidden subtle homological phenomenons of or connections between the involved algebras; on the other hand, to provide two more efficient combinatorial algorithms for computing dimension trees, and consequently the homological dimensions as an application. We believe that the more refined complete structural information on dimension trees will be useful to study other homological properties of this class of extension algebras.
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  Commuting Toeplitz Operators on the Hardy Space of the Polydisk

  Ling Hui KONG, Yu Feng LU

  Acta Mathematica Sinica. 2015, 31(4): 695-702. https://doi.org/10.1007/s10114-015-4174-z

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  In this paper, we show that two Toeplitz operators T_f and T_g on the Hardy space of the polydisk can commute if and only if the Berezin transform of the commutator [T_f, T_g] is n-harmonic.
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  Schatten-p Class (0<p≤∞) Toeplitz Operators on Generalized Fock Spaces

  Lian Hua XIAO, Xiao Feng WANG, Jin XIA

  Acta Mathematica Sinica. 2015, 31(4): 703-714. https://doi.org/10.1007/s10114-015-3531-2

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  In this paper, we discuss the Schatten-p class (0<p≤∞) of Toeplitz operators on generalized Fock space with the symbol in positive Borel measure. It makes a great difference from other papers by using the estimates of the kernel and the weight together instead of separately estimating each other. We also get the equivalent conditions when a Toeplitz operator is in the Schatten-p class.
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  Isomorphisms of Finite Semi-Cayley Graphs

  Majid AREZOOMAND, Bijan TAERI

  Acta Mathematica Sinica. 2015, 31(4): 715-730. https://doi.org/10.1007/s10114-015-4356-8

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  Let G be a finite group. A Cayley graph over G is a simple graph whose automorphism group has a regular subgroup isomorphic to G. A Cayley graph is called a CI-graph (Cayley isomorphism) if its isomorphic images are induced by automorphisms of G. A well-known result of Babai states that a Cayley graph Γ of G is a CI-graph if and only if all regular subgroups of Aut(Γ) isomorphic to G are conjugate in Aut(Γ). A semi-Cayley graph (also called bi-Cayley graph by some authors) over G is a simple graph whose automorphism group has a semiregular subgroup isomorphic to G with two orbits (of equal size). In this paper, we introduce the concept of SCI-graph (semi-Cayley isomorphism) and prove a Babai type theorem for semi-Cayley graphs. We prove that every semi-Cayley graph of a finite group G is an SCI-graph if and only if G is cyclic of order 3. Also, we study the isomorphism problem of a special class of semi-Cayley graphs.