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Semantics of the soft set induced by intuitionistic fuzzy set and its three-way decision GONG Zeng-tai, TA Guang-peng JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2022, 57 (8): 68-76.   DOI: 10.6040/j.issn.1671-9352.0.2021.419 Abstract （607）      PDF（pc） （540KB）（292）       Save In this paper, by means of the shadowed set, this paper transforms intuitionistic fuzzy set into pessimistic shadowed set and optimistic shadowed set by using intuitionistic fuzzy parameters, and proposes pessimistic three-way decision and optimistic three-way decision of intuitionistic fuzzy set. The method of inducting soft set by intuitionistic fuzzy set is given, and the qualitative model of three-way decision with soft set is proposed by using three-way decision theory. Combined with pessimistic three-way decision and optimistic three-way decision of intuitionistic fuzzy set, a quantitative model of three-way decision with soft sets is given by using intuitionistic fuzzy parameters. Reference | Related Articles | Metrics

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Multiple solutions of fractional p-q-Laplacian system with sign-changing weight functions LI Chun-ping, SANG Yan-bin JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2022, 57 (8): 95-102.   DOI: 10.6040/j.issn.1671-9352.0.2021.220 Abstract （596）      PDF（pc） （421KB）（213）       Save The multiple solutions of fractional p-q-Laplacian system involving concave-convex nonlinearities and sign-changing weight functions is considered. The system has at least two nontrivial solutions when the pair of the parameters(λ, μ)belongs to a certain subset of Rn are proved by making use of the Nehari manifold and Ekelands variational principle. Reference | Related Articles | Metrics

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Complementary concepts and their properties and generation in FCA HAN Pei-lei, WEI Ling, WANG Zhen, ZHAO Si-yu JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2022, 57 (8): 60-67.   DOI: 10.6040/j.issn.1671-9352.0.2022.055 Abstract （586）      PDF（pc） （471KB）（189）       Save In formal concept analysis, partial order relation of concept lattice describes generalization and specialization between concepts. There also exists some specific semantic information that is worth studying between incomparable concepts. Inspired by this, complementary concepts are studied in this paper. Firstly, the complementary degree of two object subsets is described by considering a specific attribute subset in formal context, based on which the strong and weak complementary relations are defined according to different actual needs. Further, the complementary concepts and their related properties are studied, and the optimal strong complementary concept and optimal weak complementary concept are explored under different semantics, and the corresponding algorithms are also given. Finally, examples are presented given to illustrate the rationality and applicability of the content of this paper. Reference | Related Articles | Metrics

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Multi-label feature selection with streaming and missing labels ZHANG Zhi-hao, LIN Yao-jin, LU Shun, WU Yi-lin, WANG Chen-xi JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2022, 57 (8): 39-52.   DOI: 10.6040/j.issn.1671-9352.7.2021.141 Abstract （580）      PDF（pc） （8432KB）（145）       Save In the practical tasks of supervised learning, the high dimensionality of feature space, the dynamic and missing of labels bring severe challenges to supervised learning. To address these problems, a multi-label feature selection with streaming and missing labels algorithm is proposed. Firstly, to solve the impact of missing labels, the missing matrix is completed by learning label correlations. Secondly, sparse learning is utilized to select label-specific features for each newly arrived label. Then, a representative feature subset is selected by calculating the score of each label-specific features of label. Finally, a series of experiments on 11 benchmark data sets demonstrate that the proposed algorithm can effectively select a representative feature subset with better classification performance. Reference | Related Articles | Metrics

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Bifurcation phenomena and nonlinear wave solutions of BKK equation HAN Qing-xiu, LIU Hong-xia, WU Yun JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2022, 57 (8): 88-94.   DOI: 10.6040/j.issn.1671-9352.0.2021.615 Abstract （579）      PDF（pc） （831KB）（175）       Save The bifurcation phenomena and nonlinear wave solutions of Broer-Kaup-Kupershmidt(BKK)equation are studied. Firstly, the first integral and singularities of the BKK equation are obtained by travelling wave transformation. Then the bifurcation phase portraits of the BKK equation in each region are presented by using the bifurcation method and qualitative theory of dynamical systems, and some nonlinear wave solutions of the equation are obtained. Furthermore, three bifurcation phenomena of kink wave are revealed. Finally, Maple software is used to simulate these bifurcation phenomena. Reference | Related Articles | Metrics

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Stability of the bidirectional ring networks with multiple time delays and multiple parameters PANG Yu-ting, ZHAO Dong-xia, BAO Fang-xia JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2022, 57 (8): 103-110.   DOI: 10.6040/j.issn.1671-9352.0.2021.618 Abstract （577）      PDF（pc） （586KB）（400）       Save The dynamic analysis of the four-neuron ring neural network with multiple time delays and multiple parameters is carried out. Firstly, the system is linearized near the trivial solution, and the characteristic equation of the system is obtained. Then the characteristic equation is factorized into four first-order exponential polynomials. Furthermore, by using the zero-point distribution property of exponential polynomial, sufficient conditions for the delay-dependent and delay-independent stability are established. Moreover, an exhaustive comparison analysis is made between the results obtained and those in the existing literature. Finally, some numerical simulations are presented to show the effectiveness and superiority of the results. Reference | Related Articles | Metrics

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Cognition of network concepts based on variable precision possibility operator FAN Min, LUO Shan, LI Jin-hai JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2022, 57 (8): 1-12.   DOI: 10.6040/j.issn.1671-9352.0.2022.066 Abstract （573）      PDF（pc） （650KB）（341）       Save Concept cognition in a network is an important direction in the field of network data analysis. Starting from the necessity operator and possibility operator in formal concept analysis, this paper puts forward variable precision possibility operator, and illustrates that the classical possibility operators are special cases of the variable precision possibility operators. Furthermore, some properties of the possibility operators are studied, and their significances in the study of infectious disease networks are explained. Then, based on the variable precision possibility operators, two generalized network concepts and their upper approximations, lower approximations and boundary regions are proposed, and the upper and lower approximations and their boundary regions under variable precision possibility operators are illustrated to have much richer semantics through an example. After that, combined with the network eigenvalue method in complex network analysis, the network weak concepts are defined, and a network weak concept acquisition method based on variable precision possibility operator is presented. Finally, our algorithm is used to conduct some experiments on the UCI database, and the obtained results show that variable-precision possibility operators have advantages in network concept cognition. Reference | Related Articles | Metrics

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Generalized interval-valued Pythagorean triangular fuzzy aggregation operator and application in decision making SU Xiao-yan, CHEN Jing-rong, YIN Hui-ling JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2022, 57 (8): 77-87.   DOI: 10.6040/j.issn.1671-9352.0.2021.448 Abstract （557）      PDF（pc） （639KB）（273）       Save In this article, the interval-valued Pythagorean triangular fuzzy number is proposed for the decision making problems when the sum of membership degree and non-membership degree of decision variables exceeds 1 on continuous sets, and the decision application of its generalized aggregation operator is analyzed. Firstly, the concept of interval-valued Pythagorean triangular fuzzy numbers is given, and its algorithm is obtained. Secondly, the weighted average operator, weighted geometric operator, ordered weighted average operator, ordered weighted geometric operator, generalized ordered weighted average operator and generalized ordered weighted geometric operator of interval-valued Pythagorean triangular fuzzy numbers are defined, and their related properties are introduced. Finally, a multi-attribute decision making model based on generalized interval-valued Pythagorean triangular fuzzy aggregation operator is constructed, and the stability of generalized ordered weighted average operator and generalized ordered weighted geometric operator is analyzed according to an example. The figures are used to prove intuitively that the former is superior to the latter when dealing with decision making problems. The effectiveness and feasibility of the decision making model are illustrated. Reference | Related Articles | Metrics

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Block cipher algorithm Eslice based on Feistel structure LIANG Li-fang, DU Xiao-ni, LI Kai-bin, XIE Xin, LI Xiao-dan JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (3): 85-92.   DOI: 10.6040/j.issn.1671-9352.0.2022.283 Abstract （525）      PDF（pc） （704KB）（171）       Save A family of highly secure block cipher algorithms Eslice, including three versions: Eslice-64-64, the block length and key length are both 64 bits; Eslice-64-128, the block length is 64 bits, and the key length is 128 bits; Eslice-128-128, the block length and key length are both 128 bits. Eslices design was inspired by LBlock. The Feistel structure is adopted as a whole, and the SP structure is used for the round function. The selected S-box have all the optimal cryptographic properties. The linear transformation has only two operations: cyclic shift and XOR, and the key generation algorithm and the encryption algorithm use the same S-box. Furthermore, the security of Eslice for differential, linear, integral and other cryptanalysis methods is analyzed. The search model based on Mixed Integer Linear Programming(MILP)are used, and the number of minimum active boxes in 20 rounds is 41, which is 3 less than that of LBlock in 20 rounds. The differential probability and linear deviation are estimated by the number of active boxes, and the security of the algorithm is evaluated. The results show that Eslice can effectively resist differential attack and linear attack. Reference | Related Articles | Metrics

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Meta computing: a new computing paradigm under zero trust Xiuzhen CHENG,Weifeng LYU,Minghui XU,Runyu PAN,Dongxiao YU,Chenxu WANG,Yong YU,Xue XIAO JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (9): 1-15.   DOI: 10.6040/j.issn.1671-9352.0.2023.168 Abstract （511）   HTML （11）    PDF（pc） （7010KB）（357）       Save The popularity of the Internet has had a significant impact on the development of computing paradigms. With the continuous improvement of the new generation of information technology infrastructures, academia and industry have been constantly exploring new computing paradigms to fully exploit computing powers. The huge data generated by massive IoT devices gradually exceeds the processing capability of the high-performance back-end represented by cloud servers. Edge computing alleviates this problem through cloud-edge-end coordination, but there still exist difficulties and challenges such as low computing power utilization, low computing and storage fault tolerance capability, and low integration of computing resorces. "Meta Computing" is a new computing paradigm that aims to break down the barriers of computing powers in a zero-trust environment. It can integrate all available connected computing and storage resources, provide efficient, byzantine fault-tolerant, and personalized services, while protecting the data and user privacy. Furthermere, meta computing can ensure the correctness of results, and realize "the entire network can be regarded as a giant computer for a user", that is, "Network-as-a-Computer, NaaC", which is also called "Meta Computer". The meta computer architecture includes a number of function modules including a device management module and a zero-trust computing management module, as well as the cloud-edge-end device resources. The device management module abstracts massive heterogeneous device resources into objects that can be freely manipulated, while the zero-trust computing management module can transparently allocate computing resources to user tasks according to service requirements, complete the tasks with strong fault tolerance and verifiable outputs, and finally settle accounts. Based on the analysis of the architecture and the functional characteristics of meta computing, we put forward suggestions and advice on the development path of meta computing, that is, transitioning from local to global, and analyze the most-influential future application scenarios of meta computing and provide a reasonable plan for the implementation and development of meta computing in future. Table and Figures | Reference | Related Articles | Metrics

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Bernoulli shift of decision evolution set HU Yu-wen, XU Jiu-cheng, XU Tian-he JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2022, 57 (8): 13-20.   DOI: 10.6040/j.issn.1671-9352.7.2021.146 Abstract （505）      PDF（pc） （1245KB）（199）       Save Decision evolution set is a new method to deal with evolution problem of decision rules changing in time series. Decision evolution matrix is an important concepts of decision evolution set. Through the decision evolution matrix, we can intuitively understand the performance of each conditional attribute at a single time point and a continuous time series. However, the decision evolution matrix is defined in the current system, there is still a lack of numerical quantization. In this paper, Bernoulli shift is introduced to quantify the decision evolution matrix, and an example is given to demonstrate the role of these quantized values in prediction. Reference | Related Articles | Metrics

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Numerical properties of Gödel semantics in fuzzy argumentation frameworks ZHAO Shuang-yan, WU Jia-chao JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2022, 57 (8): 53-59.   DOI: 10.6040/j.issn.1671-9352.0.2021.803 Abstract （445）      PDF（pc） （503KB）（73）       Save The Gödel semantics in fuzzy argumentation frameworks covers several basic extension semantics, but does not further study the properties of each extension semantics. This brings some difficulties to the calculation and quick recognition of the semantic system. Through the in-depth study of the numerical properties of each extension semantics, this paper provides some solutions to the above problems. With the help of the basic properties of the Gödel t-norm, the numerical properties of the conflict-free set, the admissible extension, the complete extension, the preferred extension(the stable extension), and the grounded extension in the Gödel semantics are deduced one by one, and the algorithms of the grounded extension are given in fuzzy AFs without cycles and fuzzy AFs consisting of odd cycles or even cycles. These conclusions are the theoretical generalization of the semantic system. At the same time, the method of bypassing the definition and directly judging the extension semantics based on numerical values, as well as the algorithm of calculating the grounded extension from empty sets, are the development of the semantics in terms of algorithm and quick identification. Reference | Related Articles | Metrics

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Robust clustering based on adaptive graph regularization and low-rank matrix decomposition LI Xin-yu, FAN Hui, LIU Jing-lei JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2022, 57 (8): 21-38.   DOI: 10.6040/j.issn.1671-9352.7.2021.069 Abstract （440）      PDF（pc） （15500KB）（137）       Save Clustering is an important research content in the field of data mining and machine learning. Generally, similarity graphs are constructed based on data samples, and then the samples are divided into corresponding classes based on the similarity graphs. However, the real data is often damaged, resulting in inaccurate similarity graphs, which directly affects the clustering results. In order to solve these problems, a robust clustering-oriented adaptive graph adjustment and low-rank matrix decomposition method is proposed. The core idea of the method is to decompose the original data X into pure data D and noisy data S, then construct a Laplacian matrix based on pure data and perform adaptive graph adjustment. Subsequently, a joint learning framework is given, which integrates data separation, adaptive graph regularization, noise removal and low-rank matrix decomposition into an objective function. Use augmented Lagrangian multiplier method to update variables separately. Finally, the paper theoretically proves the convergence of the algorithm and conduct experiments. The experimental results show that the proposed method has certain advantages compared with some existing methods. Reference | Related Articles | Metrics

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Milloux type inequality on holomorphic curves Fei LI,Liu YANG JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (6): 99-106.   DOI: 10.6040/j.issn.1671-9352.0.2022.214 Abstract （354）   HTML （2）    PDF（pc） （673KB）（116）       Save Based on the derived holomorphic curves, the Milloux type inequality of holomorphic curves is established, and the Picard type theorem is proved: let $f$ be a holomorphic curve from $\mathbf{C}$ to $P^{n}(\mathbf{C}), \nabla f$ is a derived holomorphic curve of $f$ and $\left\{H_{j}\right\}_{j=1}^{2 n+1}$ be hyperplanes in $P^{n}(\mathbf{C})$ located in general position, if $f$ omits the hyperplane family $\left\{H_{j}\right\}_{j=1}^{n+1}, \nabla f$ omits the hyperplane family $\left\{H_{j}\right\}_{j=n+2}^{2 n+1}$, then $f$ is constant, where $H_{1}, H_{2}, \cdots, H_{n+1}$ are $n+1$ coordinate hyperplanes. And some examples are given to show that the number of hyperplanes cannot be reduced and the coordinate hyperplanes cannot be generalized to arbitrary hyperplanes when $n \geqslant 2$. Reference | Related Articles | Metrics

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Quasi-J-clean rings Yao WANG,Jianghuan CHEN,Yanli REN JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (6): 1-8.   DOI: 10.6040/j.issn.1671-9352.0.2022.507 Abstract （350）   HTML （0）    PDF（pc） （692KB）（222）       Save The concept of quasi-J-clean rings is introduced using quasi-idempotent elements. Some examples of quasi-J-clean rings are given and their basic properties are discussed. The following results are proved. (1) If R is a quasi-J-clean ring, then the full matrix ring Mn(R) is a quasi-J-clean ring; (2) A ring R is a UJ-ring if and only if all quasi-clean elements in R are quasi-J-clean elements; (3) If R is a commutative ring, then the necessary and sufficient condition for R to be a quasi-J-clean ring is that R/I=J(R/I)∪U(R/I) if I is an ideal of R contained in J(R) and such that R/I is an indecomposable ring. Reference | Related Articles | Metrics

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Probabilistic q-rung hesitant fuzzy TODIM method and its application Yu ZHOU,Ligang ZHOU,Zhichao LIN,Xin XU JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (6): 9-17.   DOI: 10.6040/j.issn.1671-9352.0.2022.260 Abstract （349）   HTML （0）    PDF（pc） （931KB）（431）       Save To measure differences in probabilistic q-rung hesitant fuzzy information, a novel measure of the probabilistic q-rung hesitant fuzzy distance is proposed based on the score function and overall hesitancy. The properties of this measure are discussed. A probabilistic q-rung hesitant fuzzy TODIM method is proposed based on the new distance measure. Finally, a practical example is presented to illustrate the reasonableness and effectiveness of the proposed method, and a sensitivity analysis is performed. Table and Figures | Reference | Related Articles | Metrics

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The structure of proper glrac semigroups LIU Hai-jun, GUO Xiao-jiang JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (3): 14-18.   DOI: 10.6040/j.issn.1671-9352.0.2022.223 Abstract （347）      PDF（pc） （326KB）（142）       Save The concept of glrac semigroupoids is introduced and a structure theorem of proper glrac semigroups is established by means of the action of a monoid upon a proper glrac semigroupoid on the left. As an application, the construction of proper left GC-lpp semigroups is also considered. Reference | Related Articles | Metrics

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Gorenstein DG-injective complexes YIN Jun-qi, YANG Gang JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2022, 57 (10): 28-33.   DOI: 10.6040/j.issn.1671-9352.0.2021.727 Abstract （340）      PDF（pc） （450KB）（143）    PDF（mobile） （450KB）（10）    Save A notion of Gorenstein DG-injective complexes is introduced and some properties of these complexes are discussed. It is proven that a complex G is Gorenstein DG-injective if and only if Gn is Gorenstein injective in R-Mod for each n∈Z and the complex HomR(E,G) is exact whenever E is a DG-injective complex. Reference | Related Articles | Metrics

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The number of homomorphisms from dihedral groups to any finite groups Jie TAN,Liang ZHANG,Jidong GUO JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (6): 31-34.   DOI: 10.6040/j.issn.1671-9352.0.2022.305 Abstract （325）   HTML （2）    PDF（pc） （613KB）（250）       Save Automorphism groups and normal subgroups of dihedral groups are studied, and the number of homomorphisms from dihedral groups to any finite group is obtained. As an application of this result, the conjecture of Asai and Yoshida is verified for dihedral groups. Reference | Related Articles | Metrics

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The k-path vertex cover in some products graphs of star graph and bipartite graph Huiling YIN,Jingrong CHEN,Xiaoyan SU JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (6): 18-24, 39.   DOI: 10.6040/j.issn.1671-9352.0.2021.420 Abstract （310）   HTML （1）    PDF（pc） （671KB）（273）       Save For a subset $S \subset V(G)$, if any k-path contains at least one vertex from S, then it is called a k-path vertex cover set of the graph G. The minimum cardinality of k-path vertex cover set is called the k-path vertex cover number, which is denoted by ψk(G). The k-path vertex cover problem is studied for Cartesian product graphs, lexicographic product graphs, directed product graphs of star graph and bipartite graph, and obtain an upper and a lower bound by enumeration and related concepts of subgraphs. Reference | Related Articles | Metrics

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Global existence of large solutions for a class of chemotaxis-fluid model Zhongbo CAI,Jihong ZHAO JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (6): 84-91.   DOI: 10.6040/j.issn.1671-9352.0.2022.414 Abstract （308）   HTML （1）    PDF（pc） （697KB）（92）       Save In this paper, we are concerned with a class of chemotaxis-fluid models, which is a coupled system by parabolic-parabolic Keller-Segel equations and incompressible Navier-Stokes equations. Making full use of the weighted Chemin-Lerner type norm, the interpolation theory in Besov spaces and Fourier localization technique, the global existence of large solutions is obtained in critical Besov spaces. Reference | Related Articles | Metrics

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Entire solutions of differential-difference equation of Fermat type Mingxin ZHAO,Guirong SUN,Zhigang HUANG JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (6): 107-112.   DOI: 10.6040/j.issn.1671-9352.0.2022.540 Abstract （290）   HTML （1）    PDF（pc） （625KB）（155）       Save By using Nevanlinna value distribution theory, this paper investigates the existence of transcendental entire solution with finite order of Fermat type differential-difference equation, and obtains one result. Reference | Related Articles | Metrics

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R&D innovation strategy and behavior choice of competition drug supply chain Yan WEN,Xiyuan YU,Jinsong HU JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (7): 18-31.   DOI: 10.6040/j.issn.1671-9352.0.2021.417 Abstract （283）   HTML （1）    PDF（pc） （2658KB）（144）       Save Based on the short-sighted and far-sighted behaviors of members of competitive drug supply chain, four dynamic differential game models were constructed to study the influences of different behavior choices, changes in competition intensity and reimbursement ratio of medical insurance on drug manufacturers' R & D innovation investment, medical institution' cost-sharing strategies and supply chain performance. The study found that although the competition between drug manufacturers harms the profits of short-sighted manufacturer, it increases the overall performance of the drug supply chain and promotes the higher R & D innovation investment of far-sighted manufacturer. With higher reimbursement rate, both far-sighted drug manufacturer and medical institution have increased their profits, but not short-sighted drug manufacturer. In addition, medical institution sharing costs with one manufacturer would depress the profits of the other, when medical institution choose the behavior mode of sharing drug R & D innovation costs for two competing manufacturers simultaneously, they can benefit the profits of all members of the drug supply chain. Table and Figures | Reference | Related Articles | Metrics

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Channel entry and buyback strategies of heterogeneous products under mixed leasing and retail channels Youguo JING,Junling REN,Xiaoyang LIU,Hailong WANG,Jian SONG JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (7): 1-17.   DOI: 10.6040/j.issn.1671-9352.0.2021.617 Abstract （278）   HTML （0）    PDF（pc） （1615KB）（229）       Save A supply chain of mixed leasing and retail channels comprising of a manufacturer, a leaser and a retailer is considered. The manufacturer distributes two kinds of heterogeneous products through leasing and retail channels respectively, and buybacks part of the old rental products from the leasing channel in order to avoid the competition conflict between the retail channel and the leasing channel through which these old rental products may flow into the retail market. The following channel entry and buyback strategies of two heterogeneous products: high-quality and low-quality products entering the leasing (retail) and retail (leasing) channels respectively, which are referred to as strategies of H and L. Two two-cycle dynamic game models of channel entry and buyback strategies are compared and analyzed by using the dynamic game theory. The results show that, under the above two strategies, when the buyback price is high, a so-called buybacks situation will arise. In the case of buyback situation, the buyback price is always lower than the resale price, the manufacturer's profit margin through the retail channel is always higher than that of the leasing channel, and the manufacturer's profit margin of the strategy L is higher than that of the strategy H; the buyback price, buyback quantity and resale price of the strategy H are always higher than that of the strategy L. Finally, the numerical simulation analysis shows that when the product quality difference coefficient is small, the manufacturer should choose the strategy L; otherwise, the manufacturer should choose the strategy H. Table and Figures | Reference | Related Articles | Metrics

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The numerical range of the operator P+QP Chen ZHANG,Weiyan YU JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (6): 92-98.   DOI: 10.6040/j.issn.1671-9352.0.2022.485 Abstract （272）   HTML （0）    PDF（pc） （1160KB）（103）       Save In this paper, the numerical range of the combination P+QP of two orthogonal projections P and Q on a complex separable Hilbert space $\mathscr{H}$ is studied. Firstly, the support function of the numerical range of P+QP is given by using the method of operator block. Next, by using the properties of the support function, a geometric characterization of the numerical range of the operator P+QP is given. That is, the closure of its numerical range is a closed convex hull of some explicit ellipses parameterized by points in the spectrum. Table and Figures | Reference | Related Articles | Metrics

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A new method of neighbor sum distinguishing total coloring of graphs Jingzhi CHANG,Chao YANG,Bing YAO JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (6): 35-39.   DOI: 10.6040/j.issn.1671-9352.0.2022.220 Abstract （264）   HTML （0）    PDF（pc） （636KB）（187）       Save The neighbor sum distinguishing 2-total coloring of a tree is presented along with the independence of the minimum decycling set of a 3-regular graph and the acyclicity of the corresponding decycling subgraphs. This method provides a simple way to prove that the neighbor sum distinguishing total chromatic number of a 3-regular graph satisfies the 1-2 Conjecture. The independent decycling set method is used to determine the neighbor sum distinguishing total chromatic number of an r-regular graph, Halin graph, and Cartesian product graph of paths. Reference | Related Articles | Metrics

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Peak value and peak time of nonlinear heterogeneous epidemic model Shengqiang LIU,Ningjuan MA JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (10): 1-12.   DOI: 10.6040/j.issn.1671-9352.0.2023.340 Abstract （257）   HTML （15）    PDF（pc） （2578KB）（319）       Save The study focuses on analyzing the transmission mechanisms of nonlinear heterogeneous epidemic models within a short-time frame. As a result, novel criteria for determining the peak value and peak time of a heterogeneous epidemic model are achieved, and numerical fitting is applied to prove the efficient of the mold. Table and Figures | Reference | Related Articles | Metrics

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Existence of mild solutions for the nonlocal problem of second-order impulsive evolution equations Li LI,He YANG JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (6): 57-67.   DOI: 10.6040/j.issn.1671-9352.0.2022.595 Abstract （256）   HTML （0）    PDF（pc） （685KB）（212）       Save A new definition of mild solutions of the second-order impulsive evolution equations involving nonlocal condition $ u(0)=\sum\limits_{k=1}^n C_k u\left(\tau_k\right)$ is given by introducing a Green function. Then, the existence of mild solutions of the concerned problem is proved by applying the Sadovskii fixed point theorem. At last, an example is provided as an application of the obtained abstract result. Reference | Related Articles | Metrics

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Continuous dependence of solution for impulsive evolution equations with infinite delay Pan SUN,Xuping ZHANG JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (6): 77-83, 91.   DOI: 10.6040/j.issn.1671-9352.0.2022.469 Abstract （255）   HTML （0）    PDF（pc） （696KB）（124）       Save By comprehensive applying the theory of operator semigroups and method of nonlinear analysis, this paper will investigate the existence of mild solutions and the continuous dependence of mild solutions to initial values for the initial problem of impulsive evolution equations with infinite delay. Reference | Related Articles | Metrics

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Rational conjugacy of torsion units in integral group rings of nilpotent-by-Abelian groups Xian LING,Jinke HAI JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (8): 1-5.   DOI: 10.6040/j.issn.1671-9352.0.2022.555 Abstract （255）   HTML （1）    PDF（pc） （652KB）（229）       Save Let G be a semidirect product of a finite nilpotent group N by a finite Abelian group A with (|N|, |A|)=1, it is shown that under some conditions any normalized torsion unit of the integral group ring ZG of G is rationally conjugate to an element of G. Reference | Related Articles | Metrics

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Steady-state solutions of a Holling type Ⅱ competition model in heterogeneous environment Zipeng HE,Yaying DONG JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (8): 73-81.   DOI: 10.6040/j.issn.1671-9352.0.2021.782 Abstract （254）   HTML （2）    PDF（pc） （712KB）（134）       Save The steady-state solutions of a competitive model with Holling type Ⅱ functional response function in heterogeneous environment is investigated. Using the spectral theory of linear operators and comparison principle of parabolic equations, the global asymptotic stability of the trivial and semi-trivial solutions of the model is obtained, and the sufficient conditions for the competitive repulsion of two species are derived. With the help of topological degree theory, the sufficient conditions for the existence of positive steady-state solutions are established, which gives the sufficient conditions for the coexistence of the two species. Reference | Related Articles | Metrics

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θ-almost periodic solutions for stochastic differential equations in infinite dimensions with Stepanov almost periodic coefficients Yejun CHEN,Huisheng DING JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (6): 113-126.   DOI: 10.6040/j.issn.1671-9352.0.2022.486 Abstract （253）   HTML （2）    PDF（pc） （761KB）（142）       Save Based on the work of Raynaud de Fitte, in this paper, we consider almost periodicity of stochastic differential equations in infinite dimensions with Stepanov almost periodic coefficients $\mathrm{d} X(t)=A X(t)\mathrm{d} t+F(t, X(t))\mathrm{d} t+G(t, X(t))\mathrm{d} W(t).$ Under the weaker conditions (C0-semigroup generated by A is not necessarily contractive, and F, G can be Stepanov almost periodic and not necessarily almost periodic), we obtain the existence and uniqueness of θ-almost periodic solutions of these equations, and furthermore, we prove that the solution is almost periodic in path distribution. Reference | Related Articles | Metrics

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Graph N[S]-T reconstruction based on neighbor connectivity optimization Le CHANG,Zongtian WEI JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (6): 40-45, 76.   DOI: 10.6040/j.issn.1671-9352.0.2021.454 Abstract （252）   HTML （1）    PDF（pc） （2214KB）（145）       Save Considering the neighbor invulnerability recovery of a damaged network based on the optimization idea, the neighbor subversion strategy and the concept of reconstruction are proposed. The problem of graph reconstruction based on neighbor connectivity optimization is studied. An optimal reconstruction method and an algorithm of Cartesian product graph of two complete graphs under the optimal neighbor subversion strategy are presented. Table and Figures | Reference | Related Articles | Metrics

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Balancing numbers which are concatenations of three repdigits Shanshan ZHU,Yunyun QU,Jianhua ZHOU JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (6): 25-30.   DOI: 10.6040/j.issn.1671-9352.0.2022.459 Abstract （247）   HTML （0）    PDF（pc） （639KB）（333）       Save It is proved that 204 and 1 189 are the only Balancing numbers which are concatenations of three repdigits by using the linear forms in logarithms of algebraic numbers and the Baker-Davenport reduction method. Reference | Related Articles | Metrics

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Transition behavior of underdamped periodic potential system driven by Gaussian noise and Lévy noise Jiaxin DING,Yongfeng GUO,Lina MI JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (8): 111-117.   DOI: 10.6040/j.issn.1671-9352.0.2022.538 Abstract （245）   HTML （1）    PDF（pc） （1348KB）（81）       Save In the study of an underdamped periodic potential system, multiplicative Gaussian white noise and additive Lévy noise are introduced simultaneously. First, the second-order underdamped periodic potential system is equivalent to two first-order stochastic differential equations. Then, Lévy noise is generated by Janicki-Weron algorithm, the steady-state probability density function of the equation is simulated by the numerical simulation, and the dynamic characteristics of the underdamped periodic potential system are analyzed. It is found that system parameter, friction coefficient, stability index, skew parameter, multiplicative noise intensity and additive Lévy noise intensity can induce phase transition. In addition, it can be observed that the increase of system parameter and the friction coefficient is conducive to the stable state of particles. Table and Figures | Reference | Related Articles | Metrics

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Dynamic neighborhood rough sets approaches for updating knowledge while attributes generalization Chengxiang HU,Li ZHANG,Xiaoling HUANG,Huibin WANG JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (7): 37-51.   DOI: 10.6040/j.issn.1671-9352.4.2022.5896 Abstract （242）   HTML （0）    PDF（pc） （9251KB）（171）       Save We propose the matrix-based dynamic methods for updating three-way regions in neighborhood decision systems while attributes generalizing. First, we construct the matrix representation strategy of three-way regions for each decision class by means of neighborhood relation matrix and neighborhood distance matrix.Subsequently, considering the variation of the attributes, the updating strategies for related matrices are analyzed by using prior knowledge. Based on these strategies, we propose the matrix-based incremental methods for updating neighborhood three-way regions of each decision class. Finally, we evaluate the efficiency of the proposed incremental methods on public available data sets. Table and Figures | Reference | Related Articles | Metrics

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Optimal reinsurance contract design based on Stackelberg game Peng YANG,Xiaoyan ZHANG JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (11): 1-14,26.   DOI: 10.6040/j.issn.1671-9352.0.2022.265 Abstract （242）   HTML （3）    PDF（pc） （1443KB）（367）       Save For the optimal reinsurance contract design problem, that is, the reinsurance premium pricing problem, we assume that there is an insurance company and a reinsurance company on the market. Among them, the insurance company is engaged in two types of insurance business, and the reinsurance company is engaged in one type of insurance business. To reflect the competition between the insurance company and the reinsurance company, we assume that these three insurance businesses are interdependent, and the insurance company can engage in the reinsurance business. Using the relative performance, the competition between the insurance company and the reinsurance company is quantified. The aims of the insurance company and the reinsurance company are to find an optimal reinsurance contract to maximize the mean of the terminal wealth and minimize the variance of the terminal wealth. Under the Stackelberg game framework, by using stochastic calculus and stochastic control theory, the explicit solutions for the optimal reinsurance contract and the optimal value function are obtained. Finally, the influence of model parameters on the optimal reinsurance contract is analyzed through numerical experiments and the relationship between some special cases and general cases is compared. Table and Figures | Reference | Related Articles | Metrics

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On S-acts satisfying Condition (PI) Xingliang LIANG,Yun DANG,Xueying CHEN JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (8): 6-12.   DOI: 10.6040/j.issn.1671-9352.0.2022.437 Abstract （240）   HTML （0）    PDF（pc） （691KB）（149）       Save Let I be an ideal of a monoid S. Firstly, the basic properties of the S-acts satisfying Condition (PI) are investigated. The relations between coproducts, directed colimits, Rees short exact sequences, and Condition (PI) are established respectively, and a necessary and sufficient condition of Condition (PI) is provided by purity of epimorphisms for S-acts. Then, using the concept of I-(locally) cyclic S-acts, the sufficient and necessary conditions of direct products of S-acts preserving Condition (PI) are given. Finally, cover of S-acts with Condition (PI) is studied, and a new characterization of monoids over which every S-act has a Condition (PI)-cover is presented. Reference | Related Articles | Metrics

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Extension sequential three-way decision model and its application Junyu WANG,Yafeng YANG,Jingxuan XUE,Lihong LI JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (7): 67-79.   DOI: 10.6040/j.issn.1671-9352.4.2022.9768 Abstract （236）   HTML （0）    PDF（pc） （6139KB）（152）       Save The basic method of extension evaluation and the "rule by three divisions" decision-making idea are integrated, and the sequential idea is introduced to construct an extension sequential three-way decision model and realize the purpose of dynamic decision-making and mining optimization indicators. Firstly, the data is standardized and the weight of attributes is calculated. The extension evaluation method is used as the evaluation criterion of the three-way decisions, new decision rules are defined, and the rationality of dividing the three domains is explained. Then, according to the attribute weight, the sequential evaluation attributes of multiple granularity are obtained, the multi-stage three-way decision is made, and the decision results are given. Finally, according to the decision-making results, the indicators that cause changes in sample division are analyzed, and optimization suggestions are put forward. The model was applied to the analysis of water resources carrying capacity, and compared with the entropy weight matter-element extension decision-making model. The results show that the evaluation results obtained by using the extension sequential three-way decision model and the entropy weight matter-element extension decision-making model are basically consistent, the accuracy rate reaches 84.55%, which verifies the validity and practicability of the model. Table and Figures | Reference | Related Articles | Metrics

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Analysis of evaluation model on teaching effect based on cluster analysis and discrimination information Xin JIN,Feifan YU,Yutong DAI,Ziqian LI,Yongkui ZOU JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)    2023, 58 (7): 115-120.   DOI: 10.6040/j.issn.1671-9352.0.2022.523 Abstract （227）   HTML （1）    PDF（pc） （826KB）（111）       Save In this paper, a cluster analysis algorithm and discriminant information method are used to construct a correlation model for evaluating the academic level of college students and student management measures. Based on the average number, median number, pass rate, excellent rate and the degree of distinction, a coefficient is proposed to evaluate the effectiveness of teaching reform measures and the effectiveness of the coefficient is analyzed. We apply this method to 9 years of cumulative grade data including multiple courses, analyze the correlation between grade changes and related management measures, and verify the validity of the model. Table and Figures | Reference | Related Articles | Metrics