JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)

Nijenhuis paired Hopf modules and their constructions

ZHANG Liangyun, LIAO Meilin, JIANG Runzi, CAI Mingchao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 1-8. doi:10.6040/j.issn.1671-9352.0.2024.336

Abstract ( 30 )

PDF (689KB) ( 31 )

Save

References | Related Articles | Metrics

The concept of Nijenhuis paired Hopf modules is introduced by combining Nijenhuis paired modules with Nijenhuis paired comodules. Nijenhuis paired Hopf modules are then constructed from the antipode and group-like element of a Hopf algebra. The structure theorem for Nijenhuis paired Hopf modules is provided.

On S-acts satisfying condition(PWPK)

QIAO Husheng, WANG Li

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 9-12. doi:10.6040/j.issn.1671-9352.0.2024.104

Abstract ( 28 )

PDF (652KB) ( 8 )

Save

References | Related Articles | Metrics

Let K be an ideal of a monoid S. The basic properties of the S-acts satisfying condition(PWPK)are investigated. The relations between coproducts, directed colimits, Rees short exact sequences, and condition(PWPK)are established respectively. A characterization of the condition(PWPK)is given by using the purity of S-epimorphism.

Universal extensions in n-exangulated categories

HE Jian, HE Jing, ZHOU Panyue

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 13-18. doi:10.6040/j.issn.1671-9352.0.2024.158

Abstract ( 24 )

PDF (670KB) ( 10 )

Save

References | Related Articles | Metrics

Let(C,E,s)be an n-exangulated category. The notions of semi-universal extensions and universal extensions are introduced. It is shown that semi-universal extensions always exist. A necessary and sufficient condition of the existence of universal extensions is given. The results generalize some conclusions in extriangulated categories.

Classification of Z₊-modules over the representation ring of group algebra kA4

SUN Hua, WANG Wei, YIN Zetao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 19-24. doi:10.6040/j.issn.1671-9352.0.2024.189

Abstract ( 30 )

PDF (362KB) ( 13 )

Save

References | Related Articles | Metrics

Let k be an algebraically closed field with char(k)12, denoted r(kA4) the representation ring of group algebra r(kA4). All irreducible Z₊-modules over r(kA4) are classified. We prove that there are 9 non-equivalent irreducible Z₊-modules.

Lower K-groups of the recollement of extriangulated categories

ZHENG Min, CHEN Qinghua

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 25-36. doi:10.6040/j.issn.1671-9352.0.2024.114

Abstract ( 33 )

PDF (779KB) ( 6 )

Save

References | Related Articles | Metrics

This paper proves that, under certain conditions, there exists the direct and isomorphic relation among the K_i-groups of three categories in the left(right)recollement of extriangulated categories, which generalizes the conclusions of K_i-groups of recollements of abelian(triangulated)categories(i=0,1). The recollement of extriangulated categories is used to describe the additivity of K_i-groups of the idempotent completion categories of its three categories(i=0,1).

The orthogonality of matrices over three kinds of semirings

CHENG Chonghua, WANG Aifa,WANG Lili

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 37-41. doi:10.6040/j.issn.1671-9352.0.2024.170

Abstract ( 32 )

PDF (671KB) ( 4 )

Save

References | Related Articles | Metrics

For n×n matrices A and B, they are considered orthogonal when both AB and BA are zero matrices, and A is deemed self-orthogonal when A² is a zero matrix. The orthogonality of a specific class of tropical(0,-1)matrices, as well as the self-orthogonality of matrices on binary Boolean algebras and chain semirings is studied. The self-orthogonality of matrices on binary Boolean algebras are studied and the zero-square matrix form on those algebras is ndirectly characterized.

Influence of the generalized normality on p-nilpotent of the finite group

LUO Xiaoliang, JIA Lei

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 42-45. doi:10.6040/j.issn.1671-9352.0.2024.312

Abstract ( 25 )

PDF (318KB) ( 8 )

Save

References | Related Articles | Metrics

The n-σ-embedded properties of the maximal subgroups of the Sylow subgroups are investigated, and some sufficient conditions for finite groups to be p-nilpotent are obtained.

Van den Bergh duality for a class of skew Calabi-Yau algebras

LI Wen, LIU Liyu

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 46-51. doi:10.6040/j.issn.1671-9352.0.2024.043

Abstract ( 22 )

PDF (670KB) ( 3 )

Save

References | Related Articles | Metrics

A class of three-dimensional skew Calabi-Yau algebras is constructed using ungraded Ore extensions of the polynomial algebra in two variables. The Nakayama automorphisms of these skew Calabi-Yau algebras are computed, and the Van den Bergh duality between their Hochschild homology and cohomology is established.

Frobenius functors and X -Gorenstein projective objects

LI Aizhu, LIANG Li

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 52-55. doi:10.6040/j.issn.1671-9352.0.2024.407

Abstract ( 33 )

PDF (647KB) ( 3 )

Save

References | Related Articles | Metrics

let A be an abelian category with enough projective objects, and let X be a class of objects in A closed under isomorphism and containing the projective objects. The main purpose of this paper is to study under what conditions the Frobenius functors preserve the X -Gorenstein projective dimension of objects, and the conclusion that the Frobenius functors preserve the Ding projective dimension of objects is proved.

Derivation extensions and Wells exact sequences of 3-Lie algebras

XU Senrong, ZHAO Jia

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 56-61. doi:10.6040/j.issn.1671-9352.0.2024.044

Abstract ( 33 )

PDF (396KB) ( 3 )

Save

References | Related Articles | Metrics

Given an extension of 3-Lie algebras of the form 0→AiLpB→ 0 with ［A,A,L］_L=0, where i:A→L is the inclusion map. In this paper, we establish the Wells exact sequence which relates 1-order cocycles, derivation pairs and the second cohomology groups of 3-Lie algebras. In particular, when the above extension is split, we show that the corresponding Wells exact sequence is reduced to a short exact sequence and is also split.

Notes on the continuity of directed spaces

WANG Wu

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 62-68. doi:10.6040/j.issn.1671-9352.0.2024.279

Abstract ( 26 )

PDF (705KB) ( 4 )

Save

References | Related Articles | Metrics

The continuity and meet continuity of directed spaces are important properties to study topological space. We give the concepts of one-step closure and weak one-step closure, and study their relationship with meet continuous spaces. An equivalent characterization of a continuous space is given using the directed probability powerspace. If the directed space has a one-step closure, then it is meet continuous space. The directed space is an meet continuous space with a weak one-step closure if and only if it has a one-step closure. If the directed probability powerspace of a directed space is continuous, then it is meet continuous. Furthermore, a directed space is continuous if and only if its directed probability powerspace is continuous.

Two kinds of character sums and their recurrence properties

WANG Xiao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 69-77. doi:10.6040/j.issn.1671-9352.0.2025.333

Abstract ( 23 )

PDF (703KB) ( 6 )

Save

References | Related Articles | Metrics

Methods of analytic number theory, together with the properties of generalized Gauss sums modulo p are used to derive some third order recurrence formulas involving character sums A_k(p) and T_k(p) under certain conditions for an odd prime p. These results can be applied to solving problems regarding the number of solutions to the general diagonal equation.

Value distribution of rational function with multiple zero

YANG Pai, LIN Huli, YANG Jinhua

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 78-83. doi:10.6040/j.issn.1671-9352.0.2024.107

Abstract ( 28 )

PDF (655KB) ( 5 )

Save

References | Related Articles | Metrics

Let n≥1 be an integer, and f(z) be a meromorphic function in complex plane, whose zeros are multiple. If f '(z)≠zⁿ for each z∈C, then f(z) must be restricted to specific forms of rational function.

A depth-based and fusion class information reconstruction method for functional data

HUANG Jiewu, CHEN Xingyue, WANG Linjie, RAO Wenkang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 84-91. doi:10.6040/j.issn.1671-9352.0.2024.438

Abstract ( 27 )

PDF (1140KB) ( 4 )

Save

References | Related Articles | Metrics

A depth-based and fusion class information reconstruction method is proposed for partially observed functional data. By applying the depth-based reconstruction method and the inter-class information of sample curves derived from K-means clustering, each partially observed sample curve is reconstructed under different classification scenarios. Then, with the weights dynamically assigned by the self-weighted ensemble learning algorithm, final reconstructed curves are obtained by combining the reconstructed curves of each class. Simulation studies and case analysis show that the proposed method outperforms the depth-based reconstruction method and the regularized regression method under the mean-square prediction error criterion when the proportion of partially observed sample curves in the sample is large. Conversely, the regularized regression method performs better when the proportion of partially observed sample curves is small.

Resistance distance and Kirchhoff index in graph operation

SHEN Yunrui, MEI Yinzhen

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 92-101. doi:10.6040/j.issn.1671-9352.0.2024.330

Abstract ( 29 )

PDF (1259KB) ( 4 )

Save

References | Related Articles | Metrics

Let G be an undirected connected graph, S_T(G),Z_T(G),H_T(G)are the operation graphs of G. By utilizing the principles of electrical networks and combinatorial methods, the Kirchhoff indices of S_T(G),Z_T(G),H_T(G) are obtained, as well as the relationships between the Kirchhoff indices of graph operations and the Kirchhoff index of G, multiplicative degree-Kirchhoff index, additive degree-Kirchhoff index, the number of edges, and the number of vertices.

Analysis of network invulnerability based on the multi-fire source burning connectivity

BAI Yuerong, WEI Zongtian, WANG Deli

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 102-108. doi:10.6040/j.issn.1671-9352.0.2024.245

Abstract ( 26 )

PDF (1838KB) ( 8 )

Save

References | Related Articles | Metrics

The multi-fire source burning connectivity is given for several types of Cartesian product graphs. The relationship between multi-fire source burning connectivity and graph structure is analyzed, and the inverse problem of multi-fire source burning connectivity is proposed, that is, given a positive number m, determine the fire source such that the graph is burned as disconnected or empty within m steps at most, and contains as few vertices as possible(the minimum fire source). Finally, an algorithm of the minimum fire source of graphs is designed.

An equivalence characterization of uniform domains in metric spaces

YAN Shasha, LIU Hongjun, YAN Fang, XIA Ling

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 109-116. doi:10.6040/j.issn.1671-9352.0.2024.140

Abstract ( 23 )

PDF (697KB) ( 9 )

Save

References | Related Articles | Metrics

Some related properties of uniform domains in metric spaces are investigated, and an equivalence result of uniform domains in metric spaces is proved.

Weak solutions and concentration of sublinear Schrödinger-Poisson system

CHENG Rong, WANG Jinshui

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 117-122. doi:10.6040/j.issn.1671-9352.0.2024.112

Abstract ( 30 )

PDF (679KB) ( 26 )

Save

References | Related Articles | Metrics

A class of sublinear Schrödinger-Poisson system with more general form is studied by using variational method. The existence of non-trivial weak solutions and the concentration of the weak solution sequence for such Schrödinger-Poisson systems are obtained under weaker conditions. The results generalize established conclusions.

Combination solution method for stochastic differential equations with Poisson jumps

DUAN Yingpeng, HU Lin

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 123-132. doi:10.6040/j.issn.1671-9352.0.2024.122

Abstract ( 35 )

PDF（mobile） (1218KB) ( 5 )

Save

References | Related Articles | Metrics

For stochastic differential equations containing only one random term(Poisson term), the Poisson-type Itô formula is given. The existence and uniqueness conditions for the solution of stochastic differential equations containing only Poisson terms are obtained. The boundedness and convergence of the compensated θ method on such equations are proved. For general stochastic differential equations with Poisson jumps, a new combined solution method is established. This combined solution method can help obtain analytical and numerical solutions for some stochastic differential equations with Poisson jumps, and can also be used to correct numerical solutions and improve the convergence of numerical solutions.

Superconvergence analysis of nonlinear Sobolev-Galpern equations

LI Suli, XIE Huazhao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(4): 133-142. doi:10.6040/j.issn.1671-9352.0.2024.006

Abstract ( 25 )

PDF (772KB) ( 6 )

Save

References | Related Articles | Metrics

The nonlinear Sobolev-Galpern equations are studied with low order mixed finite element (Q11+Q01×Q10). By utilizing the high precision results of the finite element Q11+Q01×Q10, and mean-value technique, the superconvergence results of order O(h2)are obtained for the semi-discrete scheme of the equations. For the linearized fully discrete scheme, the superconvergence results of order O(h2+τ2)are also derived, here h is the subdivision parameter, τ is the time step. Finally, a numerical example is provided to confirm our theoretical analysis.

Table of Content