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    20 August 2018
    Volume 53 Issue 8
    Local optimal granularity selections in generalized multi-scale decision systems
    GU Shen-ming, LU Jin-lu, WU Wei-zhi, ZHUANG Yu-bin
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(8):  1-8.  doi:10.6040/j.issn.1671-9352.4.2018.184
    Abstract ( 1312 )   PDF (489KB) ( 849 )   Save
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    People tend to choose the appropriate level of granularity to solve problems in application. In the classical multi-scale decision systems, during the process of constructing the levels of granularity, some fixed levels of granularity are selected manually for attribute values. Aiming at generalized multi-scale decision systems in this paper, the levels of granularity are constructed by using the scale combination of attribute values. Furthermore, the selection problems of local optimal granularity are studied. The concept of generalized multi-scale decision systems is introduced firstly. Then, notions of the optimal granularity and the local optimal granularity in consistent generalized multi-scale decision system are defined, and the algorithms for finding optimal granularities and the local optimal granularities are described. Finally, the generalized decisions are introduced to inconsistent generalized multi-granular decision systems, the generalized optimal granularity and the generalized local optimal granularity are defined, and the algorithms for finding the generalized optimal granularity and the generalized local optimal granularity are also investigated.
    Attribute reduction of incomplete contexts based on similarity relations
    LI Tong-jun, HUANG Jia-wen, WU Wei-zhi
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(8):  9-16.  doi:10.6040/j.issn.1671-9352.4.2018.100
    Abstract ( 1326 )   PDF (450KB) ( 273 )   Save
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    The paper focuses on the attribute reduction of incomplete contexts. First, by comparing the values of the objects on each of the attributes, one kind of similarity relations is proposed in incomplete contexts, based on which decision rules with clear meaning can be revealed via rough approximation operators. Subsequently, one type of attribute reduction of incomplete contexts is defined, under which the similarity relations keep unchanged, some judgment theorems are given for attribute reduction, and different types of attributes for attribute reduction are characterized by the similarity relations. At last, by constructing a Boolean function with discernibility attributes among objects, an approach for attribute reduction is obtained, and an illustration example is taken to show the reliability of approach.
    Composition and structure on attribute reduction of interval-set concept lattices
    ZHANG En-sheng
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(8):  17-24.  doi:10.6040/j.issn.1671-9352.4.2018.052
    Abstract ( 1574 )   PDF (491KB) ( 408 )   Save
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    Interval-set provides a research tool for processing partially known concepts and for approximating undefinable or complex concepts. Concept lattices is a powerful tool for data analysis in machine learning, data mining, knowledge discovery, information retrieval, and so on. Interval-set concept lattices is the product of the combination concept lattices and the interval-set theory,which is a powerful tool for data analysis in machine learning, data mining, knowledge discovery and information retrieval on the information systems of partially known concepts or undefinable concepts. The attribute reduction of interval-set concept lattices is a kind of the method which reveals the elementary character of interval-set concept lattices attribute. This paper reveals the composition and structure of the attribute reduction of interval-set concept lattices. The equivalence relative necessary attributes are not in the same attribute reduction; and the intersection of attribute reduction and any relative necessary attribute equivalence class is nonempty. The set of the core attributes and the relative necessary attributes chosen by taking an attribute from each relative necessary attribute equivalence class must be an attribute reduction.
    Block discernibility matrix based on decision classification and its algorithm finding the core
    ZUO Zhi-cui, ZHANG Xian-yong, MO Zhi-wen, FENG Lin
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(8):  25-33.  doi:10.6040/j.issn.1671-9352.4.2018.121
    Abstract ( 1263 )   PDF (590KB) ( 206 )   Save
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    Attribute reduction is the fundamental approach of rough set theory to implement data mining, and its relevant algorithms are mainly based on the core. For the core, both its representation of the discernibility matrix and its calculation for finding the core exhibit important significance, but the existing discernibility matrix and its core algorithm have time and space limitations. According to the sparsity and large scale of the discernibility matrix, the block discernibility matrix based on the decision classification and its algorithm finding the core are proposed, and thus the decision classification information is directly applied to the form structure and problem solving. At first, the block discernibility matrix is defined by the decision classification, and its calculation algorithm is achieved. Then, based on the block discernibility matrix, the essence and algorithm of the core are provided. Finally, the proposed methods effectiveness is verified by the example and experiment. The block discernibility matrix based on the decision classification effectively implements the information extraction and dimensionality reduction, so its relevant algorithm finding the core well decreases the time and space complexities of the corresponding algorithm based on the discernibility matrix.
    Decomposition for L2(Rn)by subspaces composed of high-dimensional tight framelet packets
    GAI Xiao-hua, GUO Xue-jun, FENG Jin-shun, CHEN Qing-jiang, CHENG Zheng-xing
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(8):  34-42.  doi:10.6040/j.issn.1671-9352.0.2018.058
    Abstract ( 1203 )   PDF (454KB) ( 245 )   Save
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    The decomposition for space L2(Rn)by subspaces composed of framelet packets are investigated. The characteristics of the high-dimensional wavelet frame packets with a quantity dilation matrix are described by using time-frequency analysis method and functional analysis method. The subspaces from the high-dimensional framelet packets are constructed. Moreover the direct decomposition for space L2(Rn)is obtained from these subspaces composed of framelet packets. The frequency-field formulas for the high-dimensional framelet packets are presented. A sufficient condition is suggested that a Parseval frame constituted from the high-dimensional tight framelet packets of space L2(Rn). These enrich the wavelet frame theory, so that they can be applied to a wider range.
    Some notes on E2-term of Adams spectral sequence
    LIU Yan-fang, WANG Yu-yu
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(8):  43-48.  doi:10.6040/j.issn.1671-9352.0.2017.383
    Abstract ( 1204 )   PDF (358KB) ( 285 )   Save
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    The E2-term of the Adams spectral sequence, which is the cohomology of mod p Steenrod algebra A, will be discussed by using the May spectral sequence. In the Adams spectral sequence, the non-triviality of (~overγ)s+3b1hn is given, and it doesnt be hitted by any element. These results are important for the detection of new family of elements in the stable homotopy groups of spheres.
    On homological classification of weakly torsion free Rees factor S-posets
    QIAO Hu-sheng, SHI Xue-qin
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(8):  49-52.  doi:10.6040/j.issn.1671-9352.0.2018.004
    Abstract ( 1138 )   PDF (298KB) ( 215 )   Save
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    The characterization of pomonoids over which all weakly torsion free Rees factor S-posets have some kind of flatness is given, and the homological classification problem of weakly torsion free Rees factor S-posets are obtained. As applications, the homological classification of torsion free and po-torsion free Rees factor S-posets are got.
    Vertex-distinguishing IE-total coloring and general-total coloring of K1,3,p and K1,4,p
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(8):  53-60.  doi:10.6040/j.issn.1671-9352.0.2017.240
    Abstract ( 1167 )   PDF (403KB) ( 167 )   Save
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    The vertex-distinguishing IE-total coloring and vertex-distinguishing general total coloring of complete tripartite graphs K1,3,p and K1,4,pare discussed by distributing the color sets in advance,constructing the colorings and proving by contradiction. The vertex-distinguishing IE-total chromatic number and vertex-distinguishing general total chromatic number of K1,3,p and K1,4,p have been determined.
    Order of matching energy and Hosoya index of Q-shape graphs
    LIU Xiao-hua, MA Hai-cheng
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(8):  61-65.  doi:10.6040/j.issn.1671-9352.0.2017.280
    Abstract ( 1161 )   PDF (336KB) ( 153 )   Save
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    The Q(s,t)(s≥2t≥1)is the graph with s+t+1 vertices. A complete order of matching energy of the Q-shape graphs is given. As a consequence,the order of Hosoya index also be obtained.
    Inverse spectral problem of Jacobi matrices and its application
    YU Qian-qian, WEI Guang-sheng
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(8):  66-76.  doi:10.6040/j.issn.1671-9352.0.2018.109
    Abstract ( 1153 )   PDF (455KB) ( 243 )   Save
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    The inverse spectral problem of Jacobi matrices is considered. A new inverse spectral theorem for Jacobi matrices with some rank 1 perturbations of the matrix Jn is proved. And the theorem is applied to the corresponding mass-spring system. The necessary and sufficient conditions for the existence and uniqueness of the solution are derived. Furthermore, the numerical algorithm and some numerical examples are given.
    Comparison principles for viscosity solution of fully nonlinear parabolic equations with superlinear gradient nonlinearities
    WANG Jun-fang, ZHAO Pei-hao
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(8):  77-83.  doi:10.6040/j.issn.1671-9352.0.2017.341
    Abstract ( 1199 )   PDF (399KB) ( 200 )   Save
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    A problem of fully nonlinear degenerate parabolic partial differential equations with a superlinear gradient nonlinearity is studied. A comparison result is proved between semicontinuous viscosity subsolutions and supersolutions having superlinear growth. We extend our result to monotone systems of parabolic equations.
    Existence of periodic solutions of a class of third order delay differential equations in Banach spaces
    CHEN Yu-jia, YANG He
    JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2018, 53(8):  84-94.  doi:10.6040/j.issn.1671-9352.0.2017.582
    Abstract ( 1094 )   PDF (510KB) ( 335 )   Save
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    Applying the monotone iterative method of upper and lower solutions, we discuss the existence of 2π-periodic solutions for the third-order differential equation with delays in ordered Banach space Eu(t)+M0u(t-τ0)=f(t,u(t), u(t-τ1), u(t-τ2)), t∈R,where f:R×E3→E is a continuous function which is 2π-periodic in t, and τ012 are positive constants. By establishing a new maximum principle, a monotone iterative procedure for the equation is constructed. Some existence and uniqueness results of 2π-periodic solutions for this equation are obtained.