JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)

The concatenation of thin languages and r-disjunctive languages ——several studies on combinatorial semigroups(Ⅰ)

LIU Zu-hua, GUO Yu-qi

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 2-7. doi:10.6040/j.issn.1671-9352.0.2019.172

Abstract ( 1340 )

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The background of this paper is from ［1］ and another paper⁽¹⁾ by Guo et al., we merge some results in above two papers as Proposition 1: L1L∈D_f(D_t, D_r)implies L∈D_f(D_t, D_r), where L1,L are languages over alphabet A and L1 is finite. In this paper, a new and simple proof of Proposition 1 is given for D_r. It is proved that Proposition 1 is also true for D and D_i. Replacing “finite” with “thin”, Proposition 1 is true for D, D_f and D_t; and some examples are given which show that Proposition 1 is not true for D_i and D_r.

Some notes on infix chains ——several studies on combinatorial semigroups(Ⅱ)

LENG Jing, GUO Yu-qi

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 8-10. doi:10.6040/j.issn.1671-9352.0.2019.173

Abstract ( 1051 )

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Some notes on infix chains and a characterization of completely dense languages are given in this paper.

Anote-on-1-(k,m)-comma-codes ——several studies on combinatorial semigroups(Ⅲ)

LIU Hai-yan, GUO Yu-qi

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 11-15. doi:10.6040/j.issn.1671-9352.0.2019.174

Abstract ( 1549 )

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Let L be a nonempty language over A, k∈N⁰ and m∈N. If L satisfies (LA^k)^mL∩A⁺(LA^k)^m-1LA⁺=Ø, then it is a code called (k,m)-comma code. If every singleton of L is a (k,m)-comma code, then it is called a 1-(k,m)-comma code. It is known that the class of 1-(k,m)-comma code is 2^Xk\{Ø}, where X_k ={u∈A⁺|(∠w∈A^k)uwu∩A⁺uA⁺=Ø}. Jürgensen et al. showed that X0 is the set of primitive words. Cui et al. gave a characterization of X1 in terms of bordered words, unbordered words, and primitive words. In this paper, we discuss X_k for any k≥2.

Some properties of weakly projective acts

QIAO Hu-sheng, ZHANG Xiao-qin

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 16-20. doi:10.6040/j.issn.1671-9352.0.2018.463

Abstract ( 1127 )

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Weakly projective property of acts is investigated, the conditions that preserve weakly projective property under coproducts and products are discussed, and transitivity of weakly projective property on Rees short exact sequences is also considered.

Characterization of monoids by strong faithful S-acts

QIAO Hu-sheng, CHEN Qian

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 21-24. doi:10.6040/j.issn.1671-9352.0.2018.447

Abstract ( 1214 )

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Let S be a monoid. The new properties of strong faithful S-acts are investigated, and the homological classification problem of strong faithful property and flatness properties are discussed.

Characterization of pomonoids by inverse S-posets

QIAO Hu-sheng, FENG Le-ting

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 25-29. doi:10.6040/j.issn.1671-9352.0.2018.452

Abstract ( 1179 )

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Let S be a pomonoid. The inverse S-acts are extended, the properties and homological classification problem of inverse S-posets are investigated.

8-dimensional Manin Triple of 3-Lie algebras

BAI Rui-pu, WU Ying-li, HOU Shuai

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 30-33. doi:10.6040/j.issn.1671-9352.0.2018.196

Abstract ( 1097 )

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Dual 3-Lie algebras are constructed by involutive derivations and dual modules. It is proved that there exist involutive derivations on 4-dimensional 3-Lie algebras over an algebraically closed field of characteristic zero, and by the involutive derivations on 4-dimensional 3-Lie algebras, seven non-isomorphic 8-dimensional Manian Triples of 3-Lie algebras are constructed.

The number of homomorphisms from dihedral groups to a class of metacyclic groups

LI Hong-xia, GUO Ji-dong, HAI Jin-ke

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 34-40. doi:10.6040/j.issn.1671-9352.0.2018.362

Abstract ( 1374 )

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Based on the theory group of the structure of metacyclic group and characteristics of the group elements, by using the basic method of algebra and number theory, it is calculated that the number of homomorphism between dihedral groups and a class of metacyclic groups. As an application, it verifies that T.Asai and T.Yoshidas conjectures establish such a cyclic group.

Strongly g(x)-nil-clean rings

CHEN Yi-ning, QIN Long

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 41-46. doi:10.6040/j.issn.1671-9352.0.2018.343

Abstract ( 1117 )

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The concepts of strongly g(x)-nil-clean rings are introduced. The relations of several classes of strongly g(x)-clean rings are discussed. Some equivalent characters of such rings and strongly nil-clean rings are given. Moreover, basic properties of such rings are investigated. Properties of several kinds of strongly g(x)-nil-clean rings are studied.

The construct of Rota-Baxter algebra on the Sweedler 4-dimensional Hopf algebra

ZHANG Qian, LI Xuan, LI Xin, ZHENG Hui-hui, LI Lin-han, ZHANG Liang-yun

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 47-52. doi:10.6040/j.issn.1671-9352.0.2018.336

Abstract ( 1092 )

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The nontrivial Rota-Baxter operators with weight -1 are constructed from the Sweedler four-dimensional Hopf algebra and its subalgebras.

On the Hermitian-incidence energy of mixed graphs

WANG Wei-zhong, ZHOU Kun-qiang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 53-58. doi:10.6040/j.issn.1671-9352.0.2018.300

Abstract ( 994 )

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By introducing a new concept— Hermitian-incidence energy(HIE)of a mixed graph M, HIE(M)=∑ⁿ_i=1(q_i)^1/2(where q_i is the i-th eigenvalues of the Hermitian quasi-Laplacian matrix of M), we mainly point out some bounds to HIE using the number of vertices, edges, and the maximum degrees of M.

Interval edge-coloring of the generalized θ-chain

CHEN Xun, HUANG Qiong-xiang, CHEN Lin

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 59-70. doi:10.6040/j.issn.1671-9352.0.2017.577

Abstract ( 1195 )

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An edge-coloring of a graph G with colors 1,2,…,t is an interval t-coloring if all colors are used, and the colors of edges incident to each vertex of G are distinct and form a continuous interval of integer. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. The set of all interval colorable graphs is denoted by N. The deficiency def(G)of a graph G is the minimum number of pendant edges whose attachment to G makes it interval colorable. Obviously, G∈N if and only if def(G)=0. A generalized θ-chain, denoted by θ_m1,m2,…,m_k, is a simple graph obtained by substituting each edge v_i-1v_i of the path P=［v0,v1,…,v_k］(k≥1)for m_i≥2 pairwise internally disjoint(v_i-1,v_i)-paths, where i=1,2,…,k. In this paper, the conclusions of deficiency of generalized θ-graph are generalized, and the deficiency of the generalized θ-chain θ_m1,m2,…,m_k is determined.

The classification of f-coloring of random graphs

XIONG Ya-ping, CAI Jian-sheng

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 71-74. doi:10.6040/j.issn.1671-9352.0.2018.333

Abstract ( 1254 )

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A random graph G(n,p) is a graph on n labeled vertices, in which every pair of vertices is chosen to be an edge of G randomly and independently with probability p. In particular, when p=1/2, a probability space is given where all labeled graphs on n vertices are equiprobable. For a simple graph G=(V,E) with vertex set V and edge set E, an f-coloring c of G is a generalized edge-coloring in which each color class appears at each vertex v at most f(v) times where f(v) is a positive integer assigned to v. A sufficient condition for the random graph G(n,1/2) to be f-class 1 is given.

Uniqueness of positive solutions of singular p-biharmonic equations with Hardy terms

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 75-80. doi:10.6040/j.issn.1671-9352.0.2018.496

Abstract ( 1207 )

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We study a class of singular p-biharmonic equations with Hardy terms. The existence and uniqueness of the positive solution for above problem is obtained by minimization method.

Existence of infinitely many high energy solutions of a class of fourth-order elliptic equations with nonlocal terms

ZHANG Nian, JIA Gao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 81-87. doi:10.6040/j.issn.1671-9352.0.2018.620

Abstract ( 1128 )

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We study a class of fourth-order elliptic equations with nonlocal term,{Δ²u-(a+b∫_R^N|∇u|²dx)Δu+V(x)u-1/2Δ(u²)u=f(x,u), x∈R^N,u(x)∈H ²(R^N),Where N≤5, constants a>0, b≥0, Δ²=Δ(Δ)is the biharmonic operater, the nonlinearity f(x,u) doesnt satisfy AR condition and the potential function V(x) is also allowed to be sign-changing. We establish the existence of a sequence of high energy weak solutions for this class of elliptic equations via variational methods.

Existence of positive solutions for a class of periodic boundary value problems of nonlinear second-order systems

MA Man-tang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 88-95. doi:10.6040/j.issn.1671-9352.0.2018.344

Abstract ( 932 )

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We consider the existence of positive solutions for the periodic boundary value problems of nonlinear second-order systems{u″+A(t)u=ΛG(t)F(u), 0<t<1,u(0)=u(1), u'(0)=u'(1)where u=(u₁,…,u_n)T, A(t)=diag［a₁(t),…,a_n(t)］, a_i(t)can change the sign in ［0,1］ (i=1,…,n), G(t)=diag［g₁(t),…,g_n(t)］, F(u)=(f₁(u),…, f_n(u))T, Λ=diag(λ₁,…,λ_n), λ_i is a positive parameter(i=1,…,n). Under the assumption that the nonlinear term F satisfies superlinear, sublinear and asymptotic growth condition, the existence of positive solutions of the problem are obtained by using the fixed-point theorem of cone expansion-compression. The conclusions in this paper generalize and improve the related results.

Study on the approximation properties of Gauss-Weierstrass operator

WANG Tao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 96-98. doi:10.6040/j.issn.1671-9352.0.2018.256

Abstract ( 1152 )

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We use analytic methods and function decomposition technique to study bounded functions for Gauss-Weierstrass operators in probability type operators, and obtain the pointwise asymptotic expansion and preserving properties for bounded functions.

Wavelet estimation for anisotropic density functions

CAO Kai-kai

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 99-105. doi:10.6040/j.issn.1671-9352.0.2018.481

Abstract ( 897 )

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This paper provides an estimation for uncompactly supported density functions based on the wavelet method over an anisotropic Besov space and gives the linear wavelet estimator, then its upper bound under L^p(2≤p<+∞) risk is obtained. Furthermore, it assumes that the density function has independent structures so that the dimension disaster is reduced, and the corresponding proof is given.

Nonlinear maps preserving mixed Lie triple ξ-product on factor von Neumann algebras

ZHOU You, ZHANG Jian-hua

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 106-111. doi:10.6040/j.issn.1671-9352.0.2018.257

Abstract ( 1249 )

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In this paper, we prove that every bijective map preserving mixed Lie triple ξ-products with ξ≠1 from a factor von Neumann algebra M with dim M >1 into another factor von Neumann algebra N with dim N >1 is of the form A→εΨ(A), where ε∈{1,-1} and Ψ:M→N is a linear or conjugate linear *-isomorphism when ξ∈R and Ψ is a linear *-isomorphism when ξ∈C\R.

Consistent invertibility and the judgement of Weyls theorem

LIU Ying, CAO Xiao-hong

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 112-117. doi:10.6040/j.issn.1671-9352.0.2018.584

Abstract ( 1016 )

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H is an infinite dimensional separable complex Hilbert space and B(H) is the algebra of all bounded linear operators on H. An operator T∈B(H) is said to be “consistent in invertibility” provided that for each S∈B(H), TS and ST are both or neither invertible. Based on the property of consistency in invertibility, we give the necessary and sufficient conditions for T and its functional calculus which the Weyls theorem hold.

Error analysis of a linearized Crank-Nicolson scheme for the phase field crystal equation

LI Juan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(6): 118-126. doi:10.6040/j.issn.1671-9352.0.2018.146

Abstract ( 1670 )

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The phase field crystal model is a high order nonlinear evolutionary equation with the sixth order derivative in space. A linearized Crank-Nicolson scheme is presented. The scheme is explicit at the first-and second-time level. We just only to solve an implicit linearized scheme at the rest of the time level. In the derivation of the scheme, the nonlinear term (u3)_xx is rewritten to be (3u2 u_x)_x, and then be discretized by central difference quotient. The priori estimate of the numerical solution and unconditional convergence is proved in L2 norm. The convergence order is two in time and space. Some numerical examples are presented to demonstrate the theoretical results.

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