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    20 April 2021
    Volume 56 Issue 4
    Semiring varieties defined by Greens relations on a semiring
    CHENG Yan-liang, SHAO Yong
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  1-7.  doi:10.6040/j.issn.1671-9352.0.2020.453
    Abstract ( 5 )   PDF (378KB) ( 3 )   Save
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    The properties and characterizations of congruence openings determined by Greens relations of a semiring are given. We obtain that several classes of semirings by means of these congruence openings, prove that these classes of semirings are varieties of semirings, and uncover relationships between these varieties. By exploring opening operators on the lattice of all subvarieties of varieties of semirings, the order embedding theorem of the lattice of all subvarieties of the variety of multiplicatively idempotent semirings into the direct product of the lattice of open varieties is given.
    Finite groups with some m-S-supplemented subgroups
    LIU Xin, WU Zhen-feng, YANG Nan-ying
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  8-13.  doi:10.6040/j.issn.1671-9352.0.2020.399
    Abstract ( 7 )   PDF (367KB) ( 3 )   Save
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    Let G be a finite group. A subgroup S of G is said to be m-S-permutable in G, if S=〈A,B〉 for some modular subgroup A and S-permutable subgroup B of G. A subgroup H of G is said to be m-S-supplemented in G, if there are an m-S-permutable subgroup S and a subgroup T of G such that G=HT and H∩T≤S≤H. By using the m-S-supplemented subgroups to characterize the structure of finite groups. Some new characterizations of p-nilpotency and supersolubility of finite groups are obtained and generalize some known results.
    Linear deformations and Abelian extensions of Lie supertriple systems
    TENG Wen, JIN Jiu-lin, YOU Tai-jie
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  14-19.  doi:10.6040/j.issn.1671-9352.0.2020.517
    Abstract ( 10 )   PDF (398KB) ( 6 )   Save
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    Firstly, Lie supertriple systems structure is characterized by using two order linear deformations. It leads to a definition of Nijenhuis operators. Secondly, a representation and a cocycle are constructed as a tool to get the Abelian extensions of the Lie supertriple system.
    Characterizations of uniquely clean elements
    YING Zhi-ling, ZHANG Jing, HU Guo-lei, ZHOU Hua
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  20-24.  doi:10.6040/j.issn.1671-9352.0.2020.625
    Abstract ( 3 )   PDF (362KB) ( 1 )   Save
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    An element of a ring is called uniquely clean if it is uniquely the sum of an idempotent and a unit. An element a in a ring R is said to be left uniquely exchange if there exists a uniquely idempotent e∈R such that e∈Ra and 1-e∈R(1-a). If every idempotent of a ring R is in the center, then a∈R is uniquely clean iff a is left uniquely exchange. A characterization of uniquely clean elements in a commutative ring endowed with the Zariski topology is also given.
    Equivalent characterization of(strong)Kasch triangular matrix ring of order n
    CHEN Ling-qiao, TANG Gou-liang, DI Zhen-xing
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  25-30.  doi:10.6040/j.issn.1671-9352.0.2020.648
    Abstract ( 2 )   PDF (392KB) ( 1 )   Save
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    Let T be a triangular matrix ring of order n, where n≥2 is an integer. The explicit forms of maximal ideal and minimal ideal of T, and the equivalent characterizations of T being a semi-local ring and a semi-perfect ring are given. In particular, the equivalent characterizations of left Kasch and strong left Kasch triangular matrix rings of order n are given.
    Stability of modules with finite Ext-strongly Ding projective dimensions
    GUO Hui-ying, YANG Fu-xia, ZHANG Cui-ping
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  31-38.  doi:10.6040/j.issn.1671-9352.0.2020.364
    Abstract ( 3 )   PDF (540KB) ( 3 )   Save
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    Ext-strongly Ding projective dimensions are introduced, some equivalent characterizations of Ext-strongly Ding projective dimensions are presented, this type of module is stable relative to modules with finite projective dimensions and to Ext-strongly Ding projective modules are further investigated.
    Structure of semilinear spaces based on semi-tensor addition
    LI Xiao-chao
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  39-45.  doi:10.6040/j.issn.1671-9352.0.2020.422
    Abstract ( 5 )   PDF (387KB) ( 2 )   Save
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    Semi-tensor addition of matrices is a generalization of the usual matrix addition. Based on the semi-tensor addition of matrices, the matrix semilinear spaces on nonnegative real number semiring are obtained. The dimension, basis and direct sum of the matrix semilinear spaces are researched, and a necessary and sufficient condition for the sum of two subspaces to be a direct sum is given.
    Finitely embedded modules over triangular matrix rings of order n
    SUN Bo, YIN Yu-jie, DI Zhen-xing
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  46-53.  doi:10.6040/j.issn.1671-9352.0.2020.521
    Abstract ( 1 )   PDF (361KB) ( 2 )   Save
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    Let Γ be a triangular matrix ring of order n, where n≥2 is an integer. The equivalent characterization of a Γ-module whose submodule is an essential submodule, and the concrete description of the socle of a Γ-module are given. In particular, as an application of the above conclusion, a necessary and sufficient condition for a Γ-module to be a finitely embedded module is given.
    Excellent extensions and Auslander-type conditions
    ZHANG Ying-ying
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  54-56.  doi:10.6040/j.issn.1671-9352.0.2020.392
    Abstract ( 6 )   PDF (312KB) ( 2 )   Save
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    Let Γ be an excellent extension of an Artin algebra Λ. It is proved that Λ satisfies (l,n)-condition if and only if Γ satisfies (l,n)-condition.
    On zeros and uniqueness of differential-difference polynomials
    CHEN Wen-jie, SUN Gui-rong, HUANG Zhi-gang
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  57-65.  doi:10.6040/j.issn.1671-9352.0.2020.559
    Abstract ( 3 )   PDF (390KB) ( 5 )   Save
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    By Nevanlinna value distribution theory, this paper investigates the zeros distribution of a differential-difference polynomial of a transcendental entire function with zero order [f(qz+c)nmj=1f (j)(z)](k) concerning a small function α(z), where n,j,m,k are positive integers and n≥(m(m+5))/2+k+2. Furthermore, the uniqueness result of a differential-difference polynomial [f(qz+c)nmj=1f (j)(z)](k) sharing one value with the other differential-difference polynomial [g(qz+c)nmj=1g(j)(z)](k) is obtained, where n,j,m,k are positive integers and n≥(m(m+7))/2+2k+5.
    Boundedness for commutators of parameterized Marcinkiewicz integral on weighted Herz-Morrey spaces with variable exponent
    XIN Yin-ping
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  66-75.  doi:10.6040/j.issn.1671-9352.0.2020.195
    Abstract ( 4 )   PDF (441KB) ( 1 )   Save
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    The weighted boundedness of parameterized Marcinkiewicz integral on a class of generalized Lebesgue spaces with variable exponent is established. Furthermore, by using hierarchical decomposition of functions and the weighted inequalities, the boundedness of the higher order commutators generated by parameterized Marcinkiewicz integral and function of bounded mean oscillation(BMO)b on weighted Herz spaces and weighted Herz-Morrey spaces with variable exponent are obtained.
    Geometric probability and its extremes in En
    ZHAO Jiang-fu
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  76-85.  doi:10.6040/j.issn.1671-9352.0.2020.356
    Abstract ( 1 )   PDF (431KB) ( 0 )   Save
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    The probability that three planes in 3-dimensional Euclidean space intersecting a convex body K have their common point inside K has been obtained. In order to extend the above conclusion to more general n-dimensional Euclidean space, let L, G, H be three randomly chosen hyperplanes, that intersect a convex body K in En. The probability that L∩G∩H intersecting the convex body K is given by method of integral geometry. And then the Extreme value of this probabilistic sequence is obtained through isoperimetric inequalities. The maximum of probability that L∩G intersecting the convex body K is found by using of Minkowski inequality and Cauchys formula. As their application, two inequalities about hypergeometric functions are given.
    Some results on Sasakian statistical manifolds of constant φ -curvature
    WU Feng, ZHANG Liang
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  86-93.  doi:10.6040/j.issn.1671-9352.0.2020.464
    Abstract ( 6 )   PDF (404KB) ( 2 )   Save
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    The statistical immersion of codimension one from a Sasakian statistical manifold of constant φ -curvature to a statistical manifold of constant curvature and its converse are studied in this paper. It can be proved that in both cases the constant φ -curvature of the Sasakian statistical manifold must be one. Besides, several examples of Sasakian statistical manifolds are constructed.
    Commutative properties of generalized number operators
    ZHOU Yu-lan, XUE Rui, CHENG Xiu-qiang, CHEN Jia
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  94-101.  doi:10.6040/j.issn.1671-9352.0.2020.494
    Abstract ( 3 )   PDF (413KB) ( 1 )   Save
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    This paper considers the commutative ralations of the generalized number operator Nh and the quantum Bernoulli noise σ*σ:σ∈Γ} indexed by Γ, such as Lie bracket, the expressions of the composition of Nh and əσ*σ), the commutative relation of Nh and əσə*σ*σəσ). The family of bounded linear operators σ*σ:σ∈Γ} on L2(M) satisfies the canonical anticommutative relation, nilpotence and the composition are commutative if the intersection of the index is empty. Especially, σ*σ:σ∈Γ} satisfy “absorbing commutative relation”. In the following, the paper considers the commutative relations of Nh and {əσ*σ:σ∈Γ}. For any nonnegative function h on N, the Lie bracket of Nh and the σ-creation ə*σ-annihilation əσ)are just #h(σ)ə*σ(#h(σ)əσ). Especially, if the support of h is not N, then Nh is commutative with some special kind of ə*σσ). If the support of h is a finite subset of N, the composition of Nh and a special kind of ə*σσ) are just the creation type(annihilation type)operators. Moreover, the paper obtains that Nh is commutative with σə*σ*σəσ:σ∈Γ}.
    Properties for r-H-circulant matrices and polynomial algorithm of their inverse
    LEI Lin, LI Xiao-li, HE Cheng-yuan
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):  102-110.  doi:10.6040/j.issn.1671-9352.0.2020.492
    Abstract ( 6 )   PDF (391KB) ( 2 )   Save
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    The concept of r-H-circulant matrices is proposed, and five equivalent conditions and two necessary and sufficient conditions for nonsingularity of r-H-circulant matrices are gained. Furthermore, utilizing the relationship between the r-H-circulant matrices and polynomial and the sufficient and necessary conditions for nonsingular and singular of r-H-circulant matrices, a polynomial algorithm for the inverse of the r-H-circulant matrices is given. Finally, a number of examples according to the structural characteristics of r-H-circulant matrices is provided.