JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)

Srongly 2-idem-J-clean rings

WANG Yao, CHEN Jianghuan, REN Yanli

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 1-10. doi:10.6040/j.issn.1671-9352.0.2023.198

Abstract ( 100 )

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In this paper, we introduce the concept of strongly 2-idem-J-clean rings. A ring R is called a strong 2-idempotent J-clean ring, if for any a∈R there exists e, f∈Id(R), j∈J(R) such that a=e+f+j, and any two of e, f, j satisfy multiplicative commutativity. Their basic properties and the relation between them and related rings are given to further enrich the theory of clean rings.

Characterizing SEP elements based on generalized inverse elements in rings with involution

LIANG Jiahui, WANG Long, WEI Junchao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 11-18. doi:10.6040/j.issn.1671-9352.0.2023.156

Abstract ( 81 )

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In this paper, we firstly study some characterizations of SEP elements in rings with involution by the solvability of certain equation and construct the group inverses, MP inverses and invertible elements; Then we depict SEP elements by the solution of the equations, generalized inverse equations and the general solution of bivariate equations in a certain set.

Two kinds of McCoy group rings

AN Fangyu, CHENG Zhi

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 19-23. doi:10.6040/j.issn.1671-9352.0.2023.194

Abstract ( 53 )

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Two kinds of McCoy group rings are studied. We prove that R［G］ is a McCoy ring, where R is a McCoy ring and G is an infinite cyclic group, and Z［G］ is a McCoy ring if G is a finite group. We also introduce the conception of a McCoy group ring related to its annihilator ideal, and study the properties of quotient McCoy group ring.

A structure theorem of post-Hopf modules over a cocommutative post-Hopf algebra

WANG Zhongwei

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 24-30. doi:10.6040/j.issn.1671-9352.0.2023.389

Abstract ( 57 )

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In this paper, we mainly give a structure theorem of post-Hopf modules over a cocommutative post-Hopf algebra. Meanwhile, we construct a Hopf module structure of a post-Hopf modules over a cocommutative post-Hopf algebra, and construct a post-Hopf module structure of a Hopf module by the aid of relative Rota-Baxter operator with respect to a module bialgebra.

Some Kazhdan-Lusztig coefficients for an affine Weyl group of type( ~)/C ₃

WANG Yulu, WANG Liping, HE Houzhi

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 31-39. doi:10.6040/j.issn.1671-9352.0.2023.187

Abstract ( 46 )

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In the study of the nonlocal finiteness of W-graph, Lusztig introduced a system of semi-linear equations related to the Kazhdan-Lusztig coefficients μ(y,w). We analyze the basic structure of the affine Weyl group of type( ~)/C ₃, and uses the properties of the semi-linear equation system to compute the full correlation a_λ,λ'. Then it derives some b_λ,λ″ and the corresponding Kazhdan-Lusztig coefficients.

The Hopf algebraic automorphism group of quantum double of 8-dimensional Radford Hopf algebra

HU Mi, SUN Hua, LI Libin

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 40-45. doi:10.6040/j.issn.1671-9352.0.2023.390

Abstract ( 65 )

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Let R2,2(-1) be the 8-dimensional Radford Hopf algebra, and D(R2,2(-1)) the quantum double of R2,2(-1), it is proved that the Hopf algebraic automorphism group of D(R2,2(-1)) is isomorphic to Klein four-group.

Several classes of good congruences on r-wide semigroups

SHAO Lei, GONG Chunmei

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 46-53. doi:10.6040/j.issn.1671-9352.0.2023.142

Abstract ( 44 )

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Several classes of good congruences on r-wide semigroups are characterized by(*,~)-Greens relations. It is showed that for r-wide semigroups, such as the least class of(left; right)regular bands-good congruence, the least class of(left; right)regular quasi-weakly adequate semigroups-good congruence and so on.

On the rank of the semigroup P F_(n,r)

SHU Qin, GONG Heyu, ZHAO Ping

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 54-59. doi:10.6040/j.issn.1671-9352.0.2023.141

Abstract ( 52 )

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Let P_n be the partial transformation semigroup on X_n={1,2,…,n}. Let 1≤r≤n, putP F_(n,r)={α∈P_n:iα=i, ∠i∈dom(α)∩{1,2,…,r}},it is obvious that the semigroup P F_(n,r) is subsemigroup of P_n. In this paper, we shall study the rank of the semigroup P F_(n,r), where 1≤r≤n-1. For n≥3, we prove thatrank(P F_(n,r))={2+2r, n-r=1,3+2r, n-r=2,4+2r, n-r≥3.

The number of endomorphisms and automorphisms of a class of non-abelian groups with order 8p

WU Haigang, GAO Baijun

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 60-65. doi:10.6040/j.issn.1671-9352.0.2023.196

Abstract ( 63 )

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Applying the relevant theory of solutions to congruence equations, we study all endomorphisms and automorphisms of a class of ternary generated non-abelian groups with 8p order, and verify that the number of endomorphisms satisfies the conjectures of T.Asai and T.Yoshida.

Gorenstein weak injective complexes

GUO Ting, SONG Yanhui

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 66-72. doi:10.6040/j.issn.1671-9352.0.2023.040

Abstract ( 59 )

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The notion of Gorenstein weak injective complexes is introduced, and some homological characterizations of Gorenstein weak injective complexes are given. Moreover, the relationship between weak cosyzygy and Gorenstein weak cosyzygy of a complex is discussed.

Julia limiting directions of entire solutions of complex differential equations

XIANG Xuxu, LIU Jianming, WANG Qin, OUYANG Ruiqi

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 73-78. doi:10.6040/j.issn.1671-9352.0.2023.193

Abstract ( 48 )

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The Julia limiting directions of entire solutions of complex differential equations f ⁽ⁿ⁾+P_n-1(z)f ^(n-1)+… +P0(z)f=0 and f ″+A(z)f '+B(z)f=0 are studied by using Phragmén-Lindelöf indicator function and the properties of the completely regular growth functions, the lower bound of the measure of the set of common Julia limiting directions of the derivatives and primitives of infinite order entire solutions of the equations mentioned above is obtained, where n≥2, P_j(z)(j=1,2,…,n-1) are entire functions, P0(z), A(z) and B(z) are entire functions of completely regular growth.

Existence of one-signed solutions for three-point boundary value problems of second-order differential equations

LIU Huijuan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 79-86. doi:10.6040/j.issn.1671-9352.0.2023.038

Abstract ( 51 )

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In this paper, we study the existence of one-signed solutions for three-point boundary value problems of nonlinear second-order differential equations{u″+a(t)f(u)=0, t∈［0,1］,u(0)=0, u(1)=u(ε)where ε∈(0,1), a∈C(［0,1］,(0,∞)), f∈C(R,R)with sf(s)>0 for s≠0, λ1 is the principal eigenvalue of the linear eigenvalue problem: u″+λa(t)u=0, u(0)=0, u(1)=u(ε), t∈［0,1］. Assume that either(λ1)/(f_∞)<1<(λ1)/(f0) or (λ1)/(f0)<1<(λ1)/(f_∞), the problem has at least one positive solution u(t) and one negative solution v(t). The proof of main results is based on bifurcation techniques.

Flow of 2-D pseudo-steady magnetofluid around a convex corner

LI Xing, YOU Shouke

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 87-95. doi:10.6040/j.issn.1671-9352.0.2023.074

Abstract ( 39 )

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In this paper, we study the 2-D isentropic irrotational pseudo-steady supersonic flow around a convex corner for Noble-Abel gas magnetofluid. First, the properties of principal part of the centered simple wave for Noble-Abel gas magnetofluid are given. Second, by constructing the centered simple wave near convex corner, it is proved that the supersonic incoming flow can turn the convex corner by a centered rarefaction wave or a centered compression wave. Additionally, the critical angles corresponding to the arriving of vacuum and sonic state are obtained respectively.

Existence of positive solutions for a class of second-order semi-positone problems with Dirichlet boundary conditions

LI Cunli

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 96-101. doi:10.6040/j.issn.1671-9352.0.2023.003

Abstract ( 52 )

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The existence of positive solutions for second-order semi-positone problem{-u″(t)=λ(f(u(t))+a(t)), t∈(0,1),u(0)=u(1)=0is studied, where λ is a positive parameter, a∈C(［0,1］, R), f∈C(［0,∞),［0,∞)). When f has a superlinear growth, we prove that there exists a constant λ^*>0 such that the problem has a positive solution for 0<λ<λ^*. The proof of the main results is based on a fixed point theorem in cones.

Spectrum and norm of the combinations of two orthogonal projections

MA Nan, LI Yuan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 102-108. doi:10.6040/j.issn.1671-9352.0.2023.201

Abstract ( 43 )

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The paper mainly uses Halmos decomposition to investigate the spectrum and norm of the combinations of two orthogonal projections P and Q. The norm of sum and difference of(PQ)ⁿ and(QP)ⁿ is generalized,and some related results of the spectrum of αP+βQ and the norm of I+αPQ are obtained.

Resilience of dynamical systems for the integer groups Z-partial actions

CHENG Dandan, WANG Guotao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 109-113. doi:10.6040/j.issn.1671-9352.0.2023.380

Abstract ( 43 )

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Dynamical systems for integer groups Z-partial action, the notions of various partially recurrent points are introduced. Simultaneously, the relationship and properties of various partially recurrent points are studied. Finally, we investigate the relationship of various partially recurrent points of two semi-equivalent topological partial actions.

Periodic solutions of a non-autonomous piecewise bilinear system

FAN Jianhang, WU Kuilin

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 114-121. doi:10.6040/j.issn.1671-9352.0.2023.202

Abstract ( 56 )

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The existence of periodic solutions of a non-autonomous piecewise bilinear system is concerned. First, a bounded closed domain is constructed such that any orbit of the system intersecting its boundary enters the interior of the bounded closed domain in the positive direction. Then, Brouwer fixed point theorem is used to prove the existence of crossing periodic solutions and deal with the relationship between periods of the periodic solution and the periods of forced terms.

Statistical inference of step-stress partially accelerated life test model under Type-Ⅱ censoring

LONG Bing, JIANG Zaifu

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 122-129. doi:10.6040/j.issn.1671-9352.0.2023.064

Abstract ( 54 )

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Under the step-stress partially accelerated life test, based on the test data, the maximum likelihood estimation and Bayesian estimation of the distribution parameter, acceleration factor and reliability function are obtained by the classical method and Bayesian method, respectively. According to the asymptotic theory of maximum likelihood estimation, the approximate confidence intervals of the distribution parameter and the acceleration factor are constructed. The failure times of the censored units are predicted by using the observed data, including the best unbiased predictor and the conditional median predictor. The mean values and average relative errors of various estimators are calculated by Monte-Carlo method, and the influence of sample size on estimation accuracy is discussed. Finally, a numerical example is analyzed using the method presented in the paper.

A novel unsupervised feature selection method

WANG Tinghua, HU Zhenwei, ZHAN Hongxiang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(12): 130-140. doi:10.6040/j.issn.1671-9352.7.2023.5296

Abstract ( 69 )

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Most HSIC-based feature selection methods are subject to the following limitations. First, these methods are typically only applied to labeled data, which is not feasible since most of the data in real-world applications is unlabeled. Second, existing HSIC-based unsupervised feature selection methods only address the general correlation between the selected features and the output values representing the underlying clustering structure, while ignoring the redundancy between different features. To address these issues, a new unsupervised feature selection method based on HSIC(UFSHSIC)is proposed, which utilizes HSIC as a correlation criterion to explore the correlation between features and the overall sample structure, as well as the redundancy between features. Experimental comparison with other classical feature selection methods on multiple real datasets shows that the proposed method can effectively perform feature selection from unlabeled samples, and the selected feature subset produces equivalent or better performance than supervised feature selection methods.

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