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20 August 2023
Volume 58 Issue 8

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

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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.

On S-acts satisfying Condition (P_I)

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

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Let I be an ideal of a monoid S. Firstly, the basic properties of the S-acts satisfying Condition (P_I) are investigated. The relations between coproducts, directed colimits, Rees short exact sequences, and Condition (P_I) are established respectively, and a necessary and sufficient condition of Condition (P_I) 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 (P_I) are given. Finally, cover of S-acts with Condition (P_I) is studied, and a new characterization of monoids over which every S-act has a Condition (P_I)-cover is presented.

Some equivalent characterizations on EP elements

Mengge GUAN,Hainan ZHOU,Junchao WEI

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(8):  13-17.  doi:10.6040/j.issn.1671-9352.0.2022.345

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By studying the properties of EP elements and their association with idempotents and Jacobson radicals, some equivalent characterizations of EP elements are presented. A new structure is introduced to facilitate the equivalent characterizations of EP elements through Moore-Penrose(MP) inverses and the relationship between group inverses. Finally, by constructing some equations on the ring, the equivalent characterizations of EP elements are established by finding solutions on specific sets.

Recollements of Gorenstein flat cotorsion modules over triangular matrix rings

Xianhong YANG,Guoliang TANG,Zhenxing DI

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(8):  18-25.  doi:10.6040/j.issn.1671-9352.0.2022.318

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Let $\mathit{\boldsymbol{T}} = \left( {\begin{array}{*{20}{c}} A&0\ U&B \end{array}} \right)$ be a triangular matrix ring, where A and B are rings and U is a (B, A)-bimodule. The equivalent characterization of Gorenstein flat cotorsion modules in the category of left T-modules is given. With the help of the above result, it is proved that when ring A and bimodule U satisfy some conditions, there is a left recollement of the stable category of Gorenstein flat cotorsion T-module category. Under stronger conditions, a recollement of this stable category is further obtained.

$\mathscr{X}$-strongly n-Gorenstein projective modules

Cuiping ZHANG,Wenfei ZHANG,Fuxia YANG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(8):  26-32, 42.  doi:10.6040/j.issn.1671-9352.0.2022.502

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Let $\mathscr{X}$ be a class of containing projective modules. In this paper, the concept of the $\mathscr{X}$-strongly n-Gorenstein projective modules is introduced. We give some basic properties and prove that the $\mathscr{X}$-Gorenstein projective dimension of a module M does not exceed a non-negative integer n if and only if M is the direct summand of a $\mathscr{X}$-strongly n-Gorenstein projective module.

Construction of Gorenstein projective modules over tensor rings

Guoliang TANG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(8):  33-37.  doi:10.6040/j.issn.1671-9352.0.2022.634

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Let R be any ring, M a nilpotent R-bimodule, T_R(M) is the associated tensor ring. Suppose that Ext_R¹(G, M^⊗_Ri⊗_RP)=0=Tor₁^R(M, M^⊗_Ri⊗_RG), where P is a projective R-module, G is a Gorenstein projective R-module, and i≥0. It is proved that a T_R(M)-module: (X, u) is a Gorenstein projective T_R(M)-module if u is a monomorphism and the cokernel of u is a Gorenstein projective R-module.

Gorenstein cotorsion modules of Noetherian rings

Yadong LUO,Gang YANG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(8):  38-42.  doi:10.6040/j.issn.1671-9352.0.2022.630

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Let R be a left Noetherian ring. It is proved that if the Gorenstein flat cover of any injective precover of a left R-module M is injective, then the Gorenstein flat cover of M is injective if and only if every injective f : E→M of M is surjective and Ker f is Gorenstein cotorsion.

Gorenstein weak projective comodules over a coalgebra

Longjiao YANG,Hailou YAO

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(8):  43-47, 56.  doi:10.6040/j.issn.1671-9352.0.2022.472

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The Gorenstein weak projective comodules over a coalgebra are introduced and studied. The relationships between weak projectivity and Gorenstein weak projectivity are obtained by using the push-out diagram and the short exact sequence of the left C-comodules. Meanwhile, an equivalent characterization is also obtained that a comodule is a Gorenstein weak projective comodule. Finally, n-Gorenstein weak projective comodules are intruoduced and proved that every n-Gorenstein weak projective comodule is a Gorenstein weak projective comodule, and if M is an n-Gorenstein weak projective comodule with finite projective dimension, then M is a weak projective comodule.

Gorenstein AC-representations of linear quivers

Qing SUN,Gang YANG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(8):  48-56.  doi:10.6040/j.issn.1671-9352.0.2022.558

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Let Rep(Q, R) be the category of representations of R-modules over the linear quiver $ Q=(\bullet \rightarrow \bullet \rightarrow \cdots \rightarrow \bullet)$. It is mainly studied and characterized Gorenstein AC-injective representations and Gorenstein AC-flat representations in Rep(Q, R) in the paper.

The p-edge neighbor scattering number of graphs

Yuyuan SU,Zongtian WEI,Yan WANG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(8):  57-62.  doi:10.6040/j.issn.1671-9352.0.2022.328

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In order to measure the network invulnerability more precisely, the concept of edge neighbor scattering number with edge failure probability p is combined, the concept of p-edge neighbor scattering number is proposed. The parameter calculation formulas of some basic graphs and the algorithm of p-edge neighbor scattering number of the star tree are given. The relationship between p-edge neighbor scattering number and the edge number of graphs and edge failure probability p is analyzed.

L(2, 1)- edge coloring algorithm for several kinds of composite graphs

Lina ZHU,Jingwen LI,Shuai SUN

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(8):  63-72.  doi:10.6040/j.issn.1671-9352.0.2021.513

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An L(2, 1)- edge coloring algorithm is designed to solve the L(2, 1)- edge coloring for random graphs. By analyzing the experimental results, the coloring properties of five classes of join graphs are found, and F_n↑S_m、C_n↑C_m、F_n⁽²⁾、C₃⁽ⁿ⁾↑S_m and C_n^(m) are used to describe these graphs. The related theorems and proofs are given as well.

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

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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.

Existence and Ulam stability for positive solutions of conformable fractional coupled systems

Yun NI,Xiping LIU

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(8):  82-91.  doi:10.6040/j.issn.1671-9352.0.2023.034

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A nonlocal boundary value problem for a class of conformable fractional differential equations coupled system with p-Laplacian operator are studied. First, by constructing a special Banach space and using the Schauder fixed-point theorem and Banach contraction mapping principle, several results on the existence and uniqueness for positive solutions to the system are obtained, and provide sufficient conditions for the existence and uniqueness of the solution. Then, the stability of the system is studied, and the conclusion that the system has the generalized Hyers-Ulam stability is obtained. Finally, the applicability of the main conclusions obtained is demonstrated through a specific example.

Hopf bifurcation analysis in the Lengyel-Epstein reaction diffusion system with time delay

Yadi WANG,Hailong YUAN

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(8):  92-103.  doi:10.6040/j.issn.1671-9352.0.2022.539

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The Lengyel-Epstein reaction diffusion system with time delay subject to Neumann boundary conditions is considered. By choosing the time delay as the bifurcation parameter, the stability/instability of the unique positive constant equilibrium and the existence of Hopf bifurcation are investigated. In addition, the formulae to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by applying the normal form theory and center manifold theorem for partial differential equation are derived. Finally, some numerical simulations are carried out to support the analytical results.

MPEWMA chart for detecting tandem queuing network

Qi GAO,Hongshuai DAI,Yanhua WU

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(8):  104-110, 117.  doi:10.6040/j.issn.1671-9352.0.2021.786

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We constructed a multivariate Poisson exponential weighted moving average (MPEWMA) chart, which can detect a series queuing network with intermediate inputs. The MPEWMA chart can be used to monitor and control multiple nodes in the system simultaneously. We detect the change points of the external arrival rate λ_iby the MPEWMA chart, segment the process according to the change points, and estimate the segmented parameters using the MLE method when each segment meets the stationarity of each segment. Under the constraint of the lowest cost, the estimated value of the arrival rate γ_i is calculated from external inputs λ_i, and the service process is optimized in a timely manner based on changes in the arrival rate. The service rate is adjusted reasonably to reduce customer waiting time. The random simulation results indicate that when the customer's arrival rate changes, the MPEWMA chart can timely detect changes in external arrival rate λ_i and issue alerts, playing an important role in the smooth operation of the queuing system.

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

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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.

A new modified efficient Shamanskii-like Levenberg-Marquardt method for solving systems of nonlinear equations

Minglei FANG,Defeng DING,Ming WANG,Yuting SHENG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(8):  118-126.  doi:10.6040/j.issn.1671-9352.0.2022.597

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A new Shamanskii-like Levenberg-Marquardt method(SLM) to solve systems of nonlinear equations is presented by introducing parameters in this paper. With m order nonmonotone Armijo line search, the global convergence of the new algorithm is proved and the convergence rate of whom is shown to be m+1. Numerical experiments demonstrate that the new algorithm can solve large scale problems effectively.