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    20 October 2018
    Volume 53 Issue 10
    Semilattice-ordered completely regular periodic semigroups
    SHAO Yong
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(10):  1-5.  doi:10.6040/j.issn.1671-9352.0.2017.506
    Abstract ( 1015 )   PDF (315KB) ( 352 )   Save
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    It is proved that the multiplicative reducts of semilattice-ordered completely regular periodic semigroups are regular orthocryptogroups through the research of semilattice-ordered completely regular periodic semigroups. Semilattice-ordered completely regular periodic semigroups are equivalent characterizations of semilattice-ordered regular band and distributive lattices are obtained by using partial orders.
    Characterizing a band variety in terms of forbidden subsemigroups
    WANG Dan, WANG Zheng-pan
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(10):  6-8.  doi:10.6040/j.issn.1671-9352.0.2018.077
    Abstract ( 737 )   PDF (386KB) ( 190 )   Save
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    The left semiregular band variety defined by the equation [x3x1x2=x3x1x2x3x2x1x2] is an important proper subvariety of band variety. The variety is described in terms of two prohibited subsemigroups by the existing literature. Based on that, a new proper subvariety [x4x2x1x3=x4x2x1x3x4x2x3x2x1x3] of band varieties is characterized by relevant prohibited subsemigroups.
    Characterization of monoids by C(P')acts
    LIANG Xing-liang, WU Su-peng, REN Jun
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(10):  9-13.  doi:10.6040/j.issn.1671-9352.0.2017.512
    Abstract ( 933 )   PDF (343KB) ( 306 )   Save
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    Let S be a monoid. By using ring theory and module theory, C(P')right S-acts in the category of S-acts are introduced, and some important monoids, such as P(P')monoids, periodic monoids and so on, are characterized by C(P')property of S-acts. Further, some important results on C(P)right S-acts are extended. Meanwhile, an important significance to the application of regularity in S-acts is presented.
    The number of tilting modules over Gorenstein algebras
    CHEN Wen-qian, ZHANG Xiao-jin, ZAN Li-bo
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(10):  14-16.  doi:10.6040/j.issn.1671-9352.0.2017.654
    Abstract ( 794 )   PDF (297KB) ( 198 )   Save
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    For a Gorenstein algebra A, the number of tilting right A-modules is equal to the number of tilting left A-modules. A counter-example is given to show that for a Gorenstein algebra B with self-injective dimension no less than 2, the number of classical tilting left B-modules is not necessary to be equal to that of classical tilting right B-modules.
    Gorenstein homological dimensions of modules under exact zero-divisors
    GUO Shou-tao, WANG Zhan-ping
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(10):  17-21.  doi:10.6040/j.issn.1671-9352.0.2017.576
    Abstract ( 766 )   PDF (320KB) ( 198 )   Save
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    Let R be a commutative Noetherian ring with identity, x be an exact zero-divisor over R. Gorenstein homological dimensions of modules under exact zero-divisors are investigated. M/xM is Gorenstein projective(injective, flat)R/xR-module if M is Gorenstein projective(injective, flat)R-module, the results of corresponding dimensions are gained. The result can also be obtained for Ding projective(injective)R- modules.
    Graded version of Enochs theorem
    WU Xiao-ying, WANG Fang-gui
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(10):  22-26.  doi:10.6040/j.issn.1671-9352.0.2018.147
    Abstract ( 844 )   PDF (405KB) ( 166 )   Save
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    It is shown that graded version of Enochs theorem is proved. Let B be a finitely generated graded R-module and let A be a graded submodule of B. If every graded homomorphsim f: A→E can be extended to B for any FP-gr-injective R-module E, then A is finitely generated. It follows that a finitely generated graded R-module M is finitely presented if and only if EXT1R(M,E)=0 for any FP-gr-injective module E.
    Maple calculation of the transition polynomial of plane graph
    LI Mei-lian, DENG Qing-ying
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(10):  27-34.  doi:10.6040/j.issn.1671-9352.0.2017.218
    Abstract ( 739 )   PDF (1547KB) ( 197 )   Save
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    A new algorithm to evaluate transition polynomials of plane graph is obtained by improving the existing computational method. The algorithm uses cyclic permutations to count the number of Euler cycles to each transition system. The algorithm can be implemented easily by the computer program written in Maple environment.
    k-frugal list coloring of planar graphs without 4 and 5-cycles
    FANG Qi-ming, ZHANG Li
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(10):  35-41.  doi:10.6040/j.issn.1671-9352.0.2017.648
    Abstract ( 795 )   PDF (603KB) ( 138 )   Save
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    For a graph G, c is a proper vertex coloring of G. If every color appears at most k-1 times in the neighbors of each vertex v, then c is called a k-frugal coloring of G. There are two conclusions on the k-frugal list coloring of planar graphs without 4 and 5-cycles:(1)For each planar graph without 4 and 5-cycles, if its maximum degree is Δ and Δ≥3k+8, then the k-frugal list chromatic number is less than or equal to 「(Δ)/(k-1)+2; (2)For each planar graph without 4 and 5-cycles, its k-frugal list chromatic number is no more than 「(Δ)/(k-1)+5.
    Commutators of multilinear Calderón-Zygmund operator on Herz-type Hardy spaces with variable exponent
    ZHAO Huan, ZHOU Jiang
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(10):  42-50.  doi:10.6040/j.issn.1671-9352.0.2017.567
    Abstract ( 852 )   PDF (397KB) ( 211 )   Save
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    Using the atomic decomposition theorem, we obtain the boundedness of a commutator generated by multilinear Calderón-Zygmund operator and BMO functions in Herz-type Hardy spaces with variable exponent. The proof is based on the boundedness of multilinear Calderón-Zygmund operator on variable exponent Lebesgue spaces.
    Differential invariants and exact solutions of variable coefficients Benjamin-Bona-Mahony-Burgers equation
    LI Hui-hui, LIU Xi-qiang, XIN Xiang-peng
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(10):  51-60.  doi:10.6040/j.issn.1671-9352.0.2017.607
    Abstract ( 817 )   PDF (1544KB) ( 225 )   Save
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    The Lie symmetry method is performed for the variable coefficients Benjamin-Bona-Mahony-Burgers(BBMB)equation and the continuous equivalence transformations are obtained. Starting with the equivalent algebra, the differential invariants of order one are constructed. It is found that there is no zero-order differential invariant for this equation, but there are eight first-order invariants that are independent of each other. Using the obtained first-order differential invariants, we make the group classification. Finally, the general variable coefficient BBMB equations are mapped to the constant coefficient BBM equation or Burgers equation or BBMB equation by the given equivalent transformation. And then a series of new exact solutions of those variable coefficient equations are obtained. The images of the exact solution of the special BBM equation with variable coefficients and Burgers equation of the exact solution are made.
    Time dependent parabolic inverse source problem based on variational theory
    ZHEN Wei-wei, ZENG Jian, REN Jian-long
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(10):  61-71.  doi:10.6040/j.issn.1671-9352.0.2018.093
    Abstract ( 767 )   PDF (457KB) ( 184 )   Save
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    The problem of source term reconstruction for a class of parabolic equation with variable coefficients is studied, and the source term here is only time dependent. Different from the previous work, the additional condition in this paper is about the integration of the spatial variable, this type of additional conditions can help to eliminate the errors caused by random selection. At the same time, it will lead to many analysis methods unavailable. Based on the variational theory, firstly, the variational formula is given and the uniqueness of the solution is proved by the variational formula; Secondly, the time discrete model is given, and based on the variational form of linear discretization, a series of priori estimates are derived and the existence of weak solutions is proved.
    Preinvexity of n-dimensional fuzzy number-valued functions
    GONG Zeng-tai, GAO Han
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(10):  72-81.  doi:10.6040/j.issn.1671-9352.0.2017.631
    Abstract ( 926 )   PDF (486KB) ( 196 )   Save
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    By means of the partial order relation in n-dimensional fuzzy number space, some preinvexity of n-dimensional fuzzy number-valued functions are defined, including preinvexity, weakly preinvexity, strictly preinvexity, weakly strictly preinvexity, prequasiinvexity, weakly prequasiinvexity, strictly prequasiinvexity, weakly strictly prequasiinvexity, and so on. In addition, their interrelation of the preinvexity of n-dimensional fuzzy number-valued functions are discussed, and some examples are given to illustrate the interrelation of the preinvexity.
    Judgement of Weyls theorem for bounded linear operators
    ZHANG Ying, CAO Xiao-hong, DAI Lei
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(10):  82-87.  doi:10.6040/j.issn.1671-9352.0.2018.134
    Abstract ( 939 )   PDF (374KB) ( 198 )   Save
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    Let H be an infinite dimensional separable complex Hilbert space and B(H) be the algebra of all bounded linear operators on H. T∈B(H) satisfies Weyls theorem if σ(T)\σw(T)=π00(T), where σ(T) and σw(T) denote the spectrum and the Weyl spectrum respectively, π00(T)={λ∈iso σ(T):0<n(T-λI)<∞}. An operator T∈B(H) is said to have the single-valued extension property, if for every open set U⊆C, the only analytic solution of the equation (T-λI)f(λ)=0(for all λ∈U)is zero function on U. Using the single-valued extension property, we give a new judgement for Weyls theorem.
    Improvement on the solution of pressure equation based on alternating direction in chemical flooding model
    CAO Wei-dong, DAI Tao, YU Jin-biao, WANG Xiao-hong, SHI An-feng
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(10):  88-94.  doi:10.6040/j.issn.1671-9352.0.2018.103
    Abstract ( 819 )   PDF (626KB) ( 215 )   Save
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    Based on the general chemical flooding mathematical model for numerical simulation at home and abroad, an improved alternating direction iterative algorithm for solving phase pressure equation is proposed. An alternating direction iterative scheme for three dimensional parabolic equations is constructed, and a new calculation method for the iterative parameters is proposed with the second kind boundary condition and strong heterogeneous condition which is common in reservoir simulation model and practical geological model. The numerical algorithm is realized with SLCHEM, which is a simulation software of Shengli oilfield with independent intellectual property rights. Examples tests of actual field models show that the new structure of the alternating direction iterative algorithm improves the computational speed by more than 16% compared to the original preconditioned conjugate gradient algorithm, and the calculation accuracy could meet the application demand.