JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)

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

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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 M_n(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.

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

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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.

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

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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 ψ_k(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.

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

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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.

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

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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.

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

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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.

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

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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.

Global BMO estimations for obstacle problems of a class of non-uniformly elliptic equations

Yanmin GUO,Song ZHAO,Yuxia TONG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(6): 46-56. doi:10.6040/j.issn.1671-9352.0.2022.520

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The regularity of weak solutions to the obstacle problem for non-uniformly elliptic equations is studied. When the gradient of the solution to the asymptotic regular problem is close to infinity, the solution of the regular problem is used to obtain an approximate solution to the asymptotic regular problem, and the bounded mean oscillation (BMO) estimates of the weak solutions gradient of the obstacle problem are obtained based on the Young inequality and perturbation discussion.

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

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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.

Existence and multiplicity of positive solutions for a class of second-order periodic boundary value problems

Congcong KANG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(6): 68-76. doi:10.6040/j.issn.1671-9352.0.2022.433

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This paper considers the existence and multiplicity of positive solutions for second-order periodic boundary value problems $\left\{\begin{array}{l}u^{\prime \prime}(t)+p u^{\prime}(t)+q u(t)=\lambda f(t, u(t)), t \in(0, 2 \pi), \\u(0)=u(2 \pi), u^{\prime}(0)=u^{\prime}(2 \pi), \end{array}\right.$ where p, q>0 are constants and satisfy p²>4q, λ>0 is a parameter, f: [0, 2π]×[0, +∞)→[0, +∞) is continuous. The proof of the main results are based on the fixed point theorem of cone expansion-compression.

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

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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.

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

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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.

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

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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.

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

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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 $n \geqslant 2$ .

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

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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.

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

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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.

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