JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)

Characterization of (α, β)-derivation on rings with idempotents

GAO Shi-juan, ZHANG Jian-hua

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 1-5. doi:10.6040/j.issn.1671-9352.0.2018.254

Abstract ( 1138 )

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Let ring A with the unit I contain a nontrivial idempotent P and α, β:A→A be automorphisms. We prove that if δ:A→A is a linear map satisfying δ(AB)=δ(A)β(B)+α(A)δ(B)for any A,B∈A with AB=P, then δ is a(α, β)-derivation.

Boundedness of generalized multilinear operators on variable exponent spaces

FANG Xiao-zhen, SUN Ai-wen, WANG Min, SHU Li-sheng

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 6-16. doi:10.6040/j.issn.1671-9352.0.2018.194

Abstract ( 1396 )

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Boundedness of multilinear Littlewood-Paley operators on variable exponent function spaces are studied. Based on the boundedness of the general Littlewood-Paley operator g_φ on Lebesgue space, using the boundedness of Sharp maximal operator on variable Lebesgue space L^p(·), the boundedness of multilinear Littlewood-Paley operator on variable exponent Lebesgue space and variable exponent Herz-Morrey space are obtained.

Normal criteria concerning derivative functions and shared meromorphic functions

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 17-21. doi:10.6040/j.issn.1671-9352.0.2018.180

Abstract ( 1104 )

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Aiming at normal criteria concerning derivative functions and shared meromorphic functions, the result is showed as follows. Let Ω and a(z)(≠0) be a family of functions meromorphic and a meromorphic function in D, respectively. If every μ(z)∈Ω satisfies conditions as follows:(1)μ(z)≠0,(2)for each of the same poles of μ(z) and a(z), the multiplicities in μ(z) are greater than or equal to the multiplicities in a(z),(3)for every functions {μ(z), ν(z)}⊂Ω, μ^(m)(z) and ν^(m)(z) share a(z), then Ω is normal in D. And two examples are given to verify the necessity of conditions(1)and(2).

Applications of variational method to impulsive differential systems with variable exponent

ZHANG Shen-gui

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 22-28. doi:10.6040/j.issn.1671-9352.0.2018.279

Abstract ( 1832 )

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By using variational method, a class of impulsive differential systems with variable exponent and nonlocal coefficient are investigated. When the nonlinearity has a sublinear growth near zero, some results for existence of multiple periodic solutions are obtained via a variant Clarks theorem.

Positive periodic solutions of higher-order ordinary differential equations with delayed derivative terms

ZHANG Huan, LI Yong-xiang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 29-36. doi:10.6040/j.issn.1671-9352.0.2018.258

Abstract ( 1225 )

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This paper deals with the existence of positive ω-periodic solutions for the higher-order ordinary differential equation with delayed derivative terms in nonlinearityu⁽ⁿ⁾(t)+a(t)u(t)=f(t, u(t-τ₀(t)), u'(t-τ₁(t)),…, u^(n-1)(t-τ_n-1(t))), t∈R,where n≥2, a:R→(0,∞)is a continuous function which is ω-periodic, f:R×［0,∞)×R^n-1→［0,∞)is a continuous function which is ω-periodic on t, and τ_k:R→［0,∞)is a continuous function which is ω-periodic, k=0,1,…,n-1. By using the perturbation method of positive operator and fixed point index theory in cones, we obtain the existence results of positive ω-periodic solutions for the equation.

Painlevé analysis, Lie symmetry and exact solutions to the generalized time-dependent coefficients Gardner equation

WANG Qi, LI Lian-zhong

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 37-44. doi:10.6040/j.issn.1671-9352.0.2018.110

Abstract ( 1220 )

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A generalized Gardner equation with time-dependent coefficients is investigated in this paper, which arise in fluid dynamics, nonlinear lattice and plasma physics. By applying the combination of Painlevé analysis and Lie symmetry analysis method, the integrable conditions, symmetries and corresponding geometric vector fields of the time-dependent coefficient Gardner equation are investigated. Moreover, based on Painlevé analysis and the idea of symmetry reduction, the partial differential equations are reduced to ordinary differential equations. Combined with power series method, exact solutions to the reduced equations and a series of exact solutions to the original equations are obtained.

The(~)-good congruences on normal ortho-u-monoids

WU Dan-dan, GONG Chun-mei

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 45-53. doi:10.6040/j.issn.1671-9352.0.2018.225

Abstract ( 1191 )

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(~)-good congruences on normal ortho-u-monoids are characterized by means of(~)-good congruence pairs. The results about congruences on normal orthogroups in regular semigroups are generalized to(~)-semiabundant semigroups.

n-strongly Gorenstein graded projective modules

ZHAO Yu-peng, ZHAO Ren-yu

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 54-59. doi:10.6040/j.issn.1671-9352.0.2018.411

Abstract ( 1127 )

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The notion of n-strongly Gorenstein gr-projective modules is introduced. Some properties of the n-strongly Gorenstein gr-projective modules are studied, The relationship of n-strongly Gorenstein gr-projective modules, strongly Gorenstein gr-projective modules and Gorenstein gr-projective modules are obtained, and some equivalent characterizations of n-strongly Gorenstein gr-projective modules are given.

n-Ding projective and n-Ding injective modules

ZHANG Ming, ZHANG Wen-hui

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 60-66. doi:10.6040/j.issn.1671-9352.0.2018.405

Abstract ( 1094 )

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n-Ding projective and n-Ding injective modules are introduced, the homological properties of these two kinds of modules are discussed, and the hereditary property of the n-Ding projectiveness of module under the excellent extension is investigated.

Complements of partial silting objects in triangulated categories

LIU Hong-jin, LIU Li-min

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 67-71. doi:10.6040/j.issn.1671-9352.0.2018.158

Abstract ( 1237 )

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The existence of the complements of the presilting object P in a Krull-Schmidt triangulated category with a silting object T is investigated. It is shown that if all the indecomposable direct summands of P are generalized two-term related to T, then there exist complements of P. Consequently, P is partial silting. As an application, the presilting complexes over algebras which are derived equivalent to hereditary algebras are partial silting complexes.

On homological properties of C-torsionless and C-reflexive modules

LUO Xiao-qiang, TAN Ling-ling, XING Jian-min

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 72-79. doi:10.6040/j.issn.1671-9352.0.2018.105

Abstract ( 1340 )

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The relative notions and homological properties of torsionless and reflexive modules with respect to a semidualizing module are investigated. If C is a 1-dualizing module, then each submodule of a projective module is totally C-reflexive, and then the characterizations of C-torsionless and C-reflexive modules with finite G_C-dimension are investigated. Moreover, the relations between the G_C-dimension and the left C-orthogonal dimension of a module are studied, and some sufficient conditionson the existence of C-reflexive envelope are given.

Reconstruction of the solutions of the generalized Yang-Baxter equation based on crossed structures

WU Hui, ZHANG Xiao-hui

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 80-85. doi:10.6040/j.issn.1671-9352.0.2018.409

Abstract ( 1100 )

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The generalized Drinfeld twists over Turaev group-coalgebras generalize the twisting theory over Hopf algebras. Some properties and relations among twists and the representations, and the crossed quasitriangular structures are investigated. Furthermore, a new solution of generalized Yang-Baxter equation from the given ones is obtained.

K₄-factors in random graphs

ZHANG Wei-wei, CAI Jian-sheng

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 86-90. doi:10.6040/j.issn.1671-9352.0.2017.652

Abstract ( 1168 )

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For a graph G with n vertices, where 4 divides n, a K4-factor is a subgraph of G, consisting of(n)/4 vertex disjoint K₄. The minimal probability p=p(n) for which a random graph G(n,p)almost surely contains a K₄-factor. For p=O(n^-0.44), the random graph G(n,p)almost surely contains a K4-factor.

Complete exponential sum estimates in function fields

LI Jiao, CAO Ya-meng, LI Guo-quan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 91-99. doi:10.6040/j.issn.1671-9352.0.2018.126

Abstract ( 2524 )

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Let F_q be the finite field of q elements whose characteristic is p. Let F_q［t］ be the polynomial ring over F_q. Let e(·)denote a certain non-trivial character of the field of formal Laurent series in 1/t over F_q. Let k∈N with k≥2, a,b∈F_q［t］ and m=(m1,…,m_k)∈(F_q［t］)^k. Define the complete exponential sumS_k(a/b,m)=∑_{d∈F_q［t］, deg d<deg b}e(a/b∑^k_i=1m_id ⁱ).The following result is proved. Suppose that b≠0, gcd(b,a)=1 and gcd(b,m1,…,m_k)=1. If p>k, then |S_k(a/b,m)|≤C_k|b|1-1/k, where C2=1 and C_k=(k-1)2(k-1)⁽2k)/(k-2) when k≥3.

Research on the network heterogeneity based on Gini coefficient

WU Liu-san, CHEN Li

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 100-104. doi:10.6040/j.issn.1671-9352.0.2018.356

Abstract ( 1642 )

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In order to quantify the network heterogeneity, the concepts of Lorentz curve and Gini coefficient are extend to the study of network heterogeneity, the Gini coefficient of complex network base on importance degree of connectivity sub-graph is proposed, and it can be used to characterize network heterogeneity. Then, the conditions for obtaining the heterogeneity maximum and minimum of the network with n nodes and k connected sub-graphs are given. Finally, an example is given to illustrate the validity and practicability of this paper.

Dynamic updating of object-oriented granular concepts in formal concept analysis

LI Fen-ning, FAN Min, LI Jin-hai

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 105-115. doi:10.6040/j.issn.1671-9352.0.2018.296

Abstract ( 1980 )

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Granular computing(GrC)is a new method to simulate humans thinking in the field of computational intelligence research. GrC includes all theories, methods and technologies related to granularity, and it is an effective tool to solve complex problems. Granular concept is an important notion which is defined by combining granular computing and formal concept analysis. In order to update the granular concepts in a dynamic formal context, this paper introduces the notion of an object-oriented granular concept, analyzes how the extent and intent of the object-oriented granular concept are changed, and puts forward approaches to update the object-oriented granular concepts when objects and attributes are gradually removed from a formal context. Finally, some numerical experiments are conducted to show the effectiveness of the proposed algorithms.

Duality in multiobjective programming under(V, η)-type I symmetrical invexity

WANG Xue-feng, WANG Rui-jie, GAO Xiao-yan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(4): 116-126. doi:10.6040/j.issn.1671-9352.0.2018.384

Abstract ( 2889 )

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By using symmetrical gradient, a class of new generalized invex function, (V, η)-type I symmetrical invex function,(V, η)-type I strictly symmetrical quasi invex function and(V, η)-type I strictly symmetrical quasiqseudo invex function are defined. Under the new generalized invexity, the Mond-Weir dual model in a class of multi-objective programming problem is discussed, and several weak duality results, strong duality result and strict converse duality result are obtained.

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