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Table of Content

      
    20 February 2021
    Volume 56 Issue 2
    P-unknown data sets and their filtering-separation
    LIU Ji-qin, PAN Zheng-kun
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(2):  1-6.  doi:10.6040/j.issn.1671-9352.0.2020.363
    Abstract ( 589 )   PDF (370KB) ( 334 )   Save
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    When the attribute set α of the common set X changes dynamically, the unknown data information is hidden in X. According to the situation, the concepts of internal P-unknown data set, outer P-unknown data set and P-unknown data sets are proposed by using P-sets(X(-overF),XF)generated by X and the structure of ∧-type big data, and their numerical characteristics are given. The concepts of P-dependence degree and P-filtering degree of unknown data sets are defined, and the relationships between P-dependence degree and P-filtering degree are discussed. The conditions for filtering and identifying P-unknown data sets are analyzed by the filtering degree, and the filtering-separation and identification principle, the filtering surplus-separation and identification principle of P-unknown data sets are given, the filtering-separation and identification theorems and the filtering surplus-separation and identification theorems of P-unknown data sets are obtained. Finally, the application of filtering and identifying internal P-unknown data set is given.
    Hesitant fuzzy ideals in non-involutive residuated lattices
    LIU Chun-hui
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(2):  7-16.  doi:10.6040/j.issn.1671-9352.0.2020.418
    Abstract ( 734 )   PDF (486KB) ( 251 )   Save
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    In this paper, the problem of ideals in non-involutive residuated lattices is studied by using the method and principle of hesitant fuzzy sets. The notion of hesitant fuzzy ideals of non-involutive residuated lattices is introduced. Some properties of hesitant fuzzy ideals are investigated. Some equivalent characterizations of hesitant fuzzy ideals are obtained. The relation between hesitant fuzzy ideals and ideals is discussed. It is proved that the hesitant fuzzy intersection of some hesitant fuzzy ideals, homomorphism image and inverse image of a hesitant fuzzy ideal are also hesitant fuzzy ideals. At the same time, a sufficient condition which makes the hesitant fuzzy union of some hesitant fuzzy ideals to be a hesitant fuzzy ideal is given. This work further expands the way for revealing the structural characteristics of non-involutive residuated lattices.
    Optimal granularity selection based on minimum cost of extension domain change
    LI Min, YANG Ya-feng, LEI Yu, LI Li-hong
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(2):  17-27.  doi:10.6040/j.issn.1671-9352.0.2020.236
    Abstract ( 765 )   PDF (881KB) ( 257 )   Save
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    Aiming at the problem that the cost of dynamic change of decision domain is seldom involved in the current optimal granularity selection algorithm, the extension set method is introduced, and the optimal granularity selection model based on the minimum cost of change of extension domain is proposed by combining the three-way decision. Firstly, the index grade discretization data table is determined by extension evaluation method, and the weight is used as the particle to carry out granulation, and the granular space is constructed by using binary relation crossover operator. Secondly, three domains are divided by fusing three decisions, and five domains of extension set are determined based on the dynamic changes of the three domains. Then, the measurement method of extension domain change is studied, the cost matrix is constructed, and the optimal granular layer is determined by the minimum cost of extension domain change. The model considers both static and dynamic characteristics comprehensively, and provides a new way to choose the optimal granularity. Finally, taking the data of water resources carrying capacity in Heilongjiang Province as an example, the validity of the model is verified, and the sensitivity analysis is carried out by using classification and regression trees. The results show that the model has good generalization.
    Some characterizations of Markov quantum states
    LYU Xiao-le, CHEN Zheng-li, NIU Meng-fei
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(2):  28-33.  doi:10.6040/j.issn.1671-9352.0.2020.423
    Abstract ( 618 )   PDF (349KB) ( 288 )   Save
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    By using operator theory and matrix theory, the significance of studying Markov quantum states is given. According to the definition of Markov quantum states and the related properties of von Neumann entropy, two necessary and sufficient conditions for a pure state to be a Markov quantum state and two sufficient conditions for a mixed state to be a Markov quantum state are proved.
    Judgement of Browders theorem for bounded linear operators
    SUN Chen-hui, BAI Zhen-gui, CAO Xiao-hong
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(2):  34-40.  doi:10.6040/j.issn.1671-9352.0.2020.428
    Abstract ( 644 )   PDF (407KB) ( 380 )   Save
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    By using the newly defined spectrum set, Browders theorem which is a variant of Weyls theorem is studied. The sufficient and necessary conditions for a bounded linear operator defined on a Hilbert space holding Browders theorem are established. As a consequence of the main result, a new judgement of Browders theorem for operator function is discussed.
    Nonlinear maps preserving mixed Jordan triple η-product on the von Neumann algebras
    PANG Yong-feng, ZHANG Dan-li, MA Dong
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(2):  41-47.  doi:10.6040/j.issn.1671-9352.0.2020.347
    Abstract ( 779 )   PDF (341KB) ( 323 )   Save
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    Let M and N be two factor von Neumann algebras that their dimensions are larger than 1. It is proved that every bijective map preserving mixed Jordan triple η-products with η≠-1 from M into N is of the form A→εΦ(A), where ε∈{-1,1} and εΦ is a linear *-isomorphism or conjugate linear *-isomorphism when η∈R and εΦ is a linear *-isomorphism when η∈C\R.
    A class of non-global higher derivable nonlinear maps on triangular algebras
    MA Shuai-ying, ZHANG Jian-hua
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(2):  48-55.  doi:10.6040/j.issn.1671-9352.0.2020.345
    Abstract ( 696 )   PDF (359KB) ( 356 )   Save
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    Let T =Tri(A,M,B )be a triangular algebra, and {δn}n∈N:T →T be a family of maps(without the assumption of additivity and where δ0 is the identity map). If n}n∈N satisfies δn(UV)=∑i+j=nδi(U)δj(V)for any U,V∈T with at least one of them is idempotent, then {δn}n∈N is an additive higher derivation on T.
    Existence of positive solutions for periodic boundary value problems of secondorder damped difference equations
    SU Xiao-xiao, ZHANG Ya-li
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(2):  56-63.  doi:10.6040/j.issn.1671-9352.0.2020.361
    Abstract ( 921 )   PDF (374KB) ( 274 )   Save
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    This paper studies the existence of positive solutions for periodic boundary value problems of second order damped difference equations{Δ2x(t-1)+p(t)Δx(t-1)+q(t)x(t)=f(t,x(t),Δx(t-1)), t∈[1,T]Z, x(0)=x(T), Δx(0)=Δx(T)with vanishing Greens function, where T > 2 is a integer, p(·),q(·) are functions, f(t,x,y):[1,T]Z×(0,∞)×R→R is continuous with respect to (x,y)∈(0,∞)×R. The proof of main results is based on nonlinear alternative of Leray-Schauder and Schauders fixed point theorem.
    Ambrosetti-Prodi type results of the second-order discrete Neumann boundary value problem
    YANG Xiao-mei, LU Yan-qiong, WANG Rui
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(2):  64-74.  doi:10.6040/j.issn.1671-9352.0.2020.351
    Abstract ( 840 )   PDF (445KB) ( 255 )   Save
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    By using the method of the upper and lower solutions and topological degree, this paper obtains the relationship between s and the number of solutions for the second-order discrete Neumann boundary value problem{ Δ2u(t-1)+g(t,u(t))=s, t∈[1,T]Z,Δu(0)=Δu(T)=0,where s∈R is a real parameter, g:[1,T]Z×R→R is continuous,[1,TZ:={1, 2, …, T}, there exists s0∈R such that the problem has no solution if s<s0, at least one solution if s=s0 and at least two solutions if s>s0
    Existence of solutions for singular fourth-order m-point boundary value problems
    WU Ruo-fei
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(2):  75-83.  doi:10.6040/j.issn.1671-9352.0.2020.289
    Abstract ( 721 )   PDF (364KB) ( 390 )   Save
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    This article considers the existence of positive solutions for the singular fourth-order m-point boundary value problems {u(4)(t)=f(t,u(t),u'(t)), a.e. t∈(0,1),u'(0)=0, u(1)=∑m-2i=1aiu(ξi),u(0)=0, u″(1)=∑m-2i=1aiu″(ξi)where ξi∈(0,1), i=1,2,…,m-2, 0<ξ12<…<ξm-2<1, ai∈R and ∑m-2i=1ai≠1. By using the fixed point theorem of Leray-Schauder and under some suitable assumptions of f, the existence of solutions is obtained. Note that nonlinear term f has some suitable singularities at t=1.
    Multiplicity of positive solutions for fourth-order boundary value problems with nonlinear boundary conditions
    LIU Meng-xue, LI Jie-mei, YAO Yan-yan
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(2):  84-91.  doi:10.6040/j.issn.1671-9352.0.2020.486
    Abstract ( 897 )   PDF (903KB) ( 359 )   Save
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    This paper considers a class of the fourth-order boundary value problems with nonlinear boundary conditions{u(4)(t)=f(t,u(t)), t∈(0,1),u(0)=u″(0)=u(1)=0,u'(1)+C(u(1))u(1)=0,where f:[0,1]×R→[0,∞)satisfies L1-Carathéodory conditions, C:[0,∞)→[0,∞)is continuous. The existence and multiplicity of the above problems are obtained by analyzing the properties of Green function and using Leggett-Williams fixed point theorem. Finally, an example is given to verify the validity of the obtained theorem.
    Solution manifold and its C1-smoothness for differential equations with state-dependent delay
    MA Wei-feng, CHEN Peng-yu
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(2):  92-96.  doi:10.6040/j.issn.1671-9352.0.2020.084
    Abstract ( 549 )   PDF (363KB) ( 501 )   Save
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    This paper investigates the existence of solution manifolds and its C1-smoothness for differential equations with state-dependent delay in finite dimensional space under the condition that the nonlinear function is Lipschitz continuous.
    Local existence and blow-up criterion of solutions to a class of generalised incompressible Boussinesq equations
    HOU Chun-juan, LI Yuan-fei, GUO Lian-hong
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(2):  97-102.  doi:10.6040/j.issn.1671-9352.0.2020.234
    Abstract ( 720 )   PDF (396KB) ( 297 )   Save
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    A kind of adhesive, zero spread of the existence of the generalized local solution Boussinesq equations is considered. Using the regularization method, the compression mapping principle and the classical energy estimation method, the adhesive, zero spread of the local existence of the generalized Boussinesq equations are derived. And using the technique of Sobolev inequality, a blasting principles is obtained. The results of the study reveals a kind of special physical phenomenon of fluid movement.
    Continuous dependence for a class of fluid equations in porous medium
    OUYANG Bai-ping, LI Yuan-fei
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(2):  103-110.  doi:10.6040/j.issn.1671-9352.0.2020.141
    Abstract ( 515 )   PDF (387KB) ( 377 )   Save
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    The structural stability for the Brinkman-Forchheimer equations in a bounded region is considered. Firstly, some bounds for the temperature and the salt concentration are given. Then an energy expression is formulated and the expression that satisfies a differential inequality is deduced. By integrating differential inequality, the continuous dependence for the solution on the boundary coefficients is obtained.