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    20 October 2020
    Volume 55 Issue 10
    Cut sets of outer P-fuzzy sets and extended rough sets models
    HAO Xiu-mei, LIU Ji-qin
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  1-6.  doi:10.6040/j.issn.1671-9352.0.2020.196
    Abstract ( 180 )   PDF (384KB) ( 266 )   Save
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    The P-fuzzy sets are a pair of inner and outer P-fuzzy sets. Inner and outer P-fuzzy sets are two different kinds of dynamic fuzzy sets. The cut sets method is frequently used in fuzzy sets discussed. The concepts of λ-cut sets, strong λ-cut sets, interval cut sets and cut sets granularity are proposed, and the cut sets and interval cut sets granularity theorems of outer P-fuzzy sets are discussed. Then, the interval cut sets decomposition theorems of outer P-fuzzy sets are given. By using the concept of the outer P-fuzzy rough membership function, four rough sets extended models are proposed such as FF) rough sets, (1F,0F) rough sets, FF)probability rough sets and FF) variable precision rough sets. The FF)rough sets theorems of the outer P-fuzzy sets are discussed. In the end, the quantitative characteristics and relationship theorem are given.
    Intelligent data classification and intelligent retrieval-recognition of class
    ZHANG Ling, REN Xue-fang
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  7-14.  doi:10.6040/j.issn.1671-9352.0.2020.178
    Abstract ( 198 )   PDF (525KB) ( 106 )   Save
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    Based on the P-set models with dynamic characteristics, and for the research of ∧ type big data, several new concepts such as αF-data equivalence class, α(-overF)-data equivalence class and F(-overF))-data equivalence class are defined, the data boundary characteristics and measurement are analyzed, and data reasoning and its structure are proposed. The main points of this paper are to define the measurement of data boundary contraction and expansion for data element redundancy and missing, analysis data reasoning to obtain the generation theorems of data equivalence class, design data intelligent retrieval algorithm and data intelligent retrieval and recognition criteria, and finally give the application.
    Upper(Lower)approximation based on congruence on free semilinear spaces
    LI Xiao-chao
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  15-19.  doi:10.6040/j.issn.1671-9352.0.2020.121
    Abstract ( 142 )   PDF (379KB) ( 82 )   Save
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    This paper introduces the ideas of rough sets into semilinear spaces. Firstly, based on the subspace of free semilinear space, the congruence relationship is defined, and the properties of congruence classes are given. Next, the upper(lower)approximation of congruence of non empty subsets in free semilinear space is defined, and some properties of the upper(lower)approximation are studied. Lastly, the rough subspaces and quotient spaces of free semilinear spaces are studied.
    Principal congruence on Tarski algebra and modal algebra
    CAO Fa-sheng
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  20-23.  doi:10.6040/j.issn.1671-9352.0.2019.763
    Abstract ( 119 )   PDF (325KB) ( 78 )   Save
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    The principal congruences of Tarski algebras and modal algebras are studied. Combined with the result of principal congruence of Boolean algebra, the characterization of principal congruence of Tarski algebra and modal algebra is given.
    Strongly Gorenstein C-injective module and strongly Gorenstein C-flat module
    ZHANG Cui-ping, LIU Ya-juan
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  24-30.  doi:10.6040/j.issn.1671-9352.0.2020.060
    Abstract ( 151 )   PDF (385KB) ( 112 )   Save
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    Some properities of strongly Gorenstein C-injective and strongly Gorenstein C-flat modules are studied, and give the connection between the two types of modules on the specific ring.
    Ext-strong Ding projective modules
    GUO Hui-ying, ZHANG Cui-ping
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  31-36.  doi:10.6040/j.issn.1671-9352.0.2020.001
    Abstract ( 123 )   PDF (318KB) ( 103 )   Save
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    Ext-strong Ding projective modules are introduced, the class of Ext-strong Ding projective modules is projective resolving are investigated, and the Ext-strong Ding projective precover are discussed.
    Strongly Ding projective and strongly Ding injective modules over formal triangular matrix rings
    ZHAO Yang, ZHANG Wen-hui
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  37-45.  doi:10.6040/j.issn.1671-9352.0.2020.032
    Abstract ( 125 )   PDF (381KB) ( 97 )   Save
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    Strongly Ding projective and strongly Ding injective modules are investigated over the formal triangular matrix ring T=(A 0U B). It is proved that if both UA and BU have finite flat dimensions, and left T-module (M1M2)φM is strong Ding projective, left A-module M1 is strong Ding projective, φM is a monomorphism, left B-module M2/Im φM is strong Ding projective.
    q-Cartan matrices of self-injective Nakayama algebras
    ZHAO Tiao, ZHANG Chao
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  46-51.  doi:10.6040/j.issn.1671-9352.0.2020.256
    Abstract ( 135 )   PDF (433KB) ( 78 )   Save
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    The present paper mainly proves that the q-Cartan matrix of any self-injective Nakayama algebra A is diagonalizable and the determinant of q-Cartan|CA(q)|={(1-(qn)m)/(1-qn), if(n,m)=1;((1-q[m,n])(m,n))/(1-qn),if (n,m)≠1,where n is the number of simple modules, m is the length of the shortest paths in the homogeneous ideal I, and (n,m) is the greatest common divisor of n and m, [n,m] is the least common multiple of n and m.
    A note on polynomial automorphisms
    XU Tao
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  52-54.  doi:10.6040/j.issn.1671-9352.0.2020.005
    Abstract ( 122 )   PDF (280KB) ( 45 )   Save
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    A polynomial automorphism of a group G is an automorphism of the formx|→(v-11xε1v1)(v-12xε2v2)…(v-1mxεmvm),where εi=±1 and v1,v2,…,vm are fixed elements in G. In this paper we prove that if G is a(nilpotent of class c)-by-(soluble of derived length d)group, then the group generated by the polynomial automorphism of G is(nilpotent of class≤c-1)-by-(soluble of derived length≤2d).
    Maximal subsemigroups of the finite transformation semigroup of weak Y-stabilizer
    JIN Jiu-lin, TENG Wen, ZHU Fu-yang, YOU Tai-jie, QU Yun-yun
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  55-62.  doi:10.6040/j.issn.1671-9352.0.2019.636
    Abstract ( 178 )   PDF (405KB) ( 81 )   Save
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    Let X be a non-empty set. Denote by T (X)the full transformation semigroup on X. Given a non-empty subset Y of X, letT(X,Y)={α∈T (X):Yα⊆Y}, and call it the transformation semigroup of weak Y-stabilizer. In this paper, we show that structure and complete classification of maximal subsemigroups of T(X,Y)whenever Y is a proper non-singleton subset of finite set X.
    Rank and relative rank of the semigroup PD(n,r)
    ZHANG Qian-tao, LUO Yong-gui, ZHAO Ping
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  63-70.  doi:10.6040/j.issn.1671-9352.0.2019.922
    Abstract ( 153 )   PDF (387KB) ( 36 )   Save
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    Let PDn be the semigroup of all distance-preserving partial one-to-one singular transformations on a finite-chain [n](n≥3), and let PD(n,r)={α∈PDn:|im(α)|≤r}(0≤r≤n-1)be the two-sided ideal of the semigroup PDn. By analyzing the elements with rank r of the semigroup PDn, the minimal generating set and rank of the semigroup PD(n,r)are obtained. Furthermore, the relative rank of the semigroup PD(n,r)with respect to its ideal PD(n,l)is determined for 0≤l≤r.
    Maximum matching forcing number in some special graphs
    LI Jing-jing, BIAN Hong, YU Hai-zheng
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  71-76.  doi:10.6040/j.issn.1671-9352.0.2019.613
    Abstract ( 119 )   PDF (550KB) ( 60 )   Save
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    The topological index of chemical molecule is a kind of numerical invariant, which can predict indirectly the physical and chemical properties of the corresponding chemical. The reformulated reciprocal degree distance index is a generalized of reciprocal degree distance, defined as:(-overR)t(G)=∑{u,v}⊆V(G)(dG(u)+dG(v))/(dG(u,v)+t), t≥0[1]. Phenylene chain is a kind of important aromatic chemical molecule, which have strongly chemical background. We determined extremal phenylene chains with minimum and maximum reformulated reciprocal degree distance indices in all phenylene chains with n hexagons.
    On the facility systems reliability based on network location
    WEI Zong-tian, FANG Hui, LI Yin-kui
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  77-82.  doi:10.6040/j.issn.1671-9352.0.2019.914
    Abstract ( 134 )   PDF (556KB) ( 102 )   Save
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    The structure of the facility system is represented by a network, where vertices represent service facilities or customers, and the edges represent the transmission channel of goods or information. The reliability of such a facility system is to a large degree adversely affected by the edge failures in the network. In order to measure the reliability of a network in this situation, a feasible reliability concept is proposed. Based on the three classical facility location problems, set covering location problem, p-median problem and uncapacitated fixed charge location problem, a comprehensive location model is established. A heuristic algorithm is designed for solving the discrete optimal problem. It is shown by an instance that the system reliability can be improved obviously with a little cost increase.
    Judgement of(ω)property for operators functional calculus
    JIANG Hu, CAO Xiao-hong
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  83-87.  doi:10.6040/j.issn.1671-9352.0.2020.083
    Abstract ( 145 )   PDF (365KB) ( 60 )   Save
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    A new spectrum set is defined basing on the semi-Fredholm perturbation theory, and by using this spectrum set, this paper characterizes the(ω)property for the bounded linear operators and their functional calculus.
    Judgement of a-Weyls theorem for bounded linear operators
    FENG-GAO Hui-zi, CAO Xiao-hong
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  88-94.  doi:10.6040/j.issn.1671-9352.0.2019.729
    Abstract ( 150 )   PDF (441KB) ( 62 )   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 the a-Weyls theorem if σa(T)\σea(T)=πa00(T), where σa(T) and σea(T) denote the approximate point spectrum and essential approximate point spectrum respectively, and πa00(T)={λ∈iso σa(T):0<n(T-λI)<∞}. A new judgement for the a-Weyls theorem for operators and operator functions is given. Also, the spectrum mapping theorem related to spectrum is considered.
    Existence of positive solutions for singular boundary value problems of Hadamard fractional differential equations in abstract space
    SUN Yan-yan, LIU Yan-sheng
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  95-103.  doi:10.6040/j.issn.1671-9352.0.2019.833
    Abstract ( 152 )   PDF (425KB) ( 95 )   Save
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    The existence of positive solutions of singular boundary value problems in Banach space is investigated. By constructing a new cone and using the fixed point index theory of strict set compression operator, it is established that there are at least two positive solutions for the corresponding approximation of the boundary value problem. Then, using Ascoli-Arzela theorem and the relative compactness of the sequence of solutions, a sufficient condition is obtained for the existence of multiple positive solutions to boundary value problem.
    Global structure of the set of positive solutions for a class of nonlinear third-order boundary value problems
    ZHAO Jiao
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2020, 55(10):  104-110.  doi:10.6040/j.issn.1671-9352.0.2020.179
    Abstract ( 157 )   PDF (349KB) ( 87 )   Save
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    This paper considers the global structure of the set of positive solutions of nonlinear third-order ordinary differential equations boundary value problems{-u(3)(t)=λf(t,u(t)), a.e. t∈[0,1],u(0)=u'(0)=0, u'(1)=αu'(η),where f:[0,1]×R→[0,∞)is a L1-Carathéodory function, 0<η<1, 1<α<1 are given constants. When f satisfies the linear growth condition, the paper obtains the global structure of the set of positive solutions of the problem by using Rabinowitz global bifurcation theorem.