JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)

Attractor family and dimension for a class of high-order nonlinear Kirchhoff equations

LIN Guo-guang, LI Zhuo-xi

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 1-11. doi:10.6040/j.issn.1671-9352.0.2018.709

Abstract ( 1315 )

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The initial boundary value problem for a class of high-order Kirchhoff equations with nonlinear nonlocal source terms and strong damping terms is studied. For the nonlinear nonlocal source term and the Kirchhoff stress term, the existence and uniqueness of the global solution of the equation are firstly proved by Galerkin finite element method and a prior estimate. Then the bounded absorption set is obtained by a prior estimate, so the global attractor family of high-order nonlinear Kirchhoff equation is obtained. By linearizing the equation and proving the Frechet differentiable of the solution semigroup, it further proves the decay of the volume element of the linearization problem. Finally, the Hausdorff dimension and Fractal dimension of the global attractor family are proved to be finite.

Continuous dependence on viscosity coefficient for primitive equations

LI Yuan-fei

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 12-23. doi:10.6040/j.issn.1671-9352.0.2019.539

Abstract ( 1109 )

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The continuous dependence of the solutions of the 3D primitive equations of large scale ocean under oscillating random force in a cylindrical region is considered. Using the technique of differential inequality, the a prior bounds of the solutions of the equations are derived. By the energy analysis methods, the continuous dependence on the viscosity coefficient of the solution of the three-dimensional viscous primitive equation of large-scale ocean is obtained.

Solving explicit new travelling wave solutions of KdV-Burgers-Kuramoto equation by Riccati equation

LIN Fu-biao, ZHANG Qian-hong

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 24-31. doi:10.6040/j.issn.1671-9352.0.2018.513

Abstract ( 1198 )

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Firstly, 8 types explicit new analytical solutions of the Riccati equation are presented by the trial function method combing with the related properties of solutions for the Riccati equation. Secondly, the reduced equations and invariant solutions of KdV-Burgers-Kuramoto(KBK)equation are given by the Lie group analysis method. Finally, the extended tanh-function method and 8 types explicit new analytical solutions of the Riccati equation are used to solve the reduced equation of KBK equation, moreover, many types explicit new travelling wave solutions of KBK equation are found. In addition, periodic types, rational types of exponential function and trigonometric function of explicit new travelling wave solutions of other similar nonlinear partial differential equations can be obtained by use of the extended tanh-function method, 8 types explicit new analytical solutions of the Riccati equation and the Lie group analysis method.

Monotone positive solutions of fourth-order boundary value problems with integral boundary conditions

HE Yan-qin, HAN Xiao-ling

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 32-37. doi:10.6040/j.issn.1671-9352.0.2019.320

Abstract ( 1039 )

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By using the monotone iterative technique, this article studies the existence of monotone positive solutions for fourth order boundary value problems with integral boundary conditions{u⁽⁴⁾(t)=f(t,u(t),u'(t)), t∈(0,1),u(0)=u'(1)=u(1)=0,u″(0)=∫¹₀g(t)u″(t)dt,where f:［0,1］×［0,+∞)2→［0,+∞) is continuous, g:［0,1］→［0,+∞) is continuous, not only the existence of the positive solution of the problem is obtained, but also the initial value of the iterative sequence is a simple zero function or a primary function.

Existence of positive solutions for periodic boundary conditions of singular second-order damped difference equations

SU Xiao-xiao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 38-45. doi:10.6040/j.issn.1671-9352.0.2019.461

Abstract ( 822 )

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This paper studies the existence of positive solutions for periodic boundary value problems of second order damped difference equations{Δ²x(t-1)+αΔx(t-1)+βx(t)=f(t,x(t), Δx(t-1)), t∈［1,T］_Z,x(0)=x(T), Δx(0)=Δx(T)where T >2 is an integer, α, β are constants, f(t,x,y):［1,T］_Z×(0,∞)×R→R is continuous with respect to (x,y)∈(0,∞)×R, f may be singular at x=0, which means that lim_x→0⁺ f(t,x,y)=+∞,(t,y)∈［1,T］_Z×R. The proof of main results is based on nonlinear alternative of Leray-Schauder.

Strong new product preserving maps on *-algebras

ZHANG Fang-juan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 46-49. doi:10.6040/j.issn.1671-9352.0.2019.391

Abstract ( 1055 )

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Let R be a unital *-algebra with a nontrivial symmetric idempotent P which satisfies:(1)ARP={0} implies A=0;(2)AR(I-P)={0} implies A=0. Let φ:R→R be a surjective map. Then φ is strong new product preserving if and only if there exists an element Z∈Z_S(R)with Z²=I such that φ(X)=ZX for all X∈R. As an application, a characterization of strong new product preserving on von Neumann algebras with no central summands of type I₁ and prime *-ring are obtained.

Nonlinear Jordan higher derivable maps on triangular algebras by Lie product square zero elements

FEI Xiu-hai, DAI Lei, ZHU Guo-wei

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 50-58. doi:10.6040/j.issn.1671-9352.0.2019.100

Abstract ( 1226 )

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Let U be a 2-torsion free triangular algebra, D={d_n}_n∈N is a nonlinear Jordan higher derivable map on triangular algebra U by Lie product square zero elements. In this paper, it is shown that every nonlinear Jordan higher derivable map on triangular algebra U by Lie product square zero elements is a higher derivation. As its application, we get that every nonlinear Jordan higher derivable map on a nest algebra or a 2-torsion free block upper triangular matrix algebra U by Lie product square zero elements is a higher derivation.

Number of homomorphisms between groups with Abel direct factors and finite groups

LI Ning-ying, GUO Ji-dong, HAI Jin-ke

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 59-62. doi:10.6040/j.issn.1671-9352.0.2019.092

Abstract ( 1221 )

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Let H,G be finite group. If the subgroup A of H is an Abel direct factor of H, then the number of homomorphism from H to G is a multiple of the largest common factor of |A| and |G|. The famous T.Yoshidas theorem is generalized.

L-fuzzy ideals in Quantale

LUO Qing-jun

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 63-67. doi:10.6040/j.issn.1671-9352.0.2018.669

Abstract ( 1144 )

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In the present paper, by applying L-fuzzy theory to Quantale, we introduce the notations of L-fuzzy ideals, L-fuzzy prime ideals and L-fuzzy primary ideals in Quantale, and investigate their properties. Several characterizations of such ideals are present. The concert structure of L-fuzzy ideals generated by L-fuzzy points is also obtained.

Some extensions of T-nilpotent rings

MA Guang-lin, WANG Yao, REN Yan-li

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 68-73. doi:10.6040/j.issn.1671-9352.0.2019.135

Abstract ( 1091 )

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Some extension properties of T-nilpotent rings are investigated. It is mainly proved that(1)If R is a ring and α an automorphism of R, then R is a left T-nilpotent ring if and only if the skew polynomial ring R［x;α］ over R is a left T-nilpotent ring, if and only if the skew Laurent polynomial ring R［x,x^-1;α］ is a left T-nilpotent ring;(2)R is a left T-nilpotent ring if and only if the Nagata extension over R is a left T-nilpotent ring, if and only if the skew triangular matrix ring over R is a left T-nilpotent ring.

Ding projective modules over Frobenius extensions

WANG Zhan-ping, ZHANG Rui-jie

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 74-78. doi:10.6040/j.issn.1671-9352.0.2018.756

Abstract ( 993 )

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Ding projective modules over Frobenius extensions and Ding projective dimensions are investigated. Let R⊂A be a separable Frobenius extension, and let M be any left A-module. It is proved that M is a Ding projective left A-module if and only if M is Ding projective left R-module if and only if A_RM and HomR(A,M) are Ding projective left A-modules. Some the conclusion on Ding projective Dimensions are obtained.

Gorenstein FP-projective modules and its stability

ZHANG Yu, ZHAO Ren-yu

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 79-85. doi:10.6040/j.issn.1671-9352.0.2018.713

Abstract ( 964 )

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The concept of Gorenstein FP-projective modules is introduced as a generalization of FP-projective modules. Some properties and equivalent characterizations of Gorenstein FP-projective modules are given. The stability of the class of Gorenstein FP-projective modules is studied.

A note on intersective polynomials in function fields

QIAN Kun, LIU Bao-qing, LI Guo-quan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 86-96. doi:10.6040/j.issn.1671-9352.0.2018.710

Abstract ( 1117 )

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Let Z denote the ring of rational integers. Let L be a field, and let p be its characteristic. Let f(x)=∑ⁿ_j=0a_jx^j∈L［x］, the polynomial ring over L. Suppose that f(x)=a∏^r_i=1(x-η_i)^eⁱ over some algebraic closure of L, where a∈L, all the η_i are distinct, and r,e1,e2,…,e_r are positive integers with r≥2 and n=∑^r_j=1e_j. The semidiscriminant Δ(f) of f is defined by Δ(f)=a2n-1∏1≤i,j≤r_i≠j(η_i-η_j)^{e_i e_j}. It is proved that if n<p, then there exist a positive integer m with m|n! and a polynomial G∈Z［x₀,x₁,…,x_n］, which depend only on the vector (e1,e2,…,e_r), such that Δ(f)=1/mG(a0,a1,…,a_n). This result is applied to investigate a question on intersective polynomials over the ring L［x］, where L is a finite field.

A note on the hybrid power mean of the character sums and exponential sums

WANG Xiao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 97-101. doi:10.6040/j.issn.1671-9352.0.2019.025

Abstract ( 897 )

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The main purpose is using the properties of trigonometric sums and character sums to study the computational problem of one kind hybrid power mean involving the two-term exponential sums and polynomial character sums, and give the precise computational formulae for it.

Tate homology of modules of finite Gorenstein flat dimension with respect to a semidualizing module

PAN Xiao-ling, LIANG Li

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 102-105. doi:10.6040/j.issn.1671-9352.0.2018.719

Abstract ( 1083 )

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For a semidualing module C, Tate homology Tor^F_C of modules admitting Tate F_C-resolutions is investigated. In particular, a long exact sequence connecting Tor^F_C, Tor^F_C and Tor^GF_C is built. As applications, the vanishing and the balance of this Tate homology theory are proved.

Uniqueness of meromorphic solutions of nonlinear differential equations

WANG Ling-shuang, HUANG Zhi-gang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 106-114. doi:10.6040/j.issn.1671-9352.0.2019.457

Abstract ( 1485 )

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This paper is devoted to studying the uniqueness of meromorphic functions by Nevanlinna value distribution theory. Obtain that if f(z) is a finite order meromorphic solution of the equation f 'f=F(z) and shares 0、1、∞ with finite order meromorphic function g(z) where F(z) is an entire function and λ(F(z))<1, then f(z)=g(z). If f(z) is a finite order meromorphic solution of the equation f '+A(z)f ⁿ=F(z) and shares 0、1、∞ with finite order meromorphic function g(z), where A(z) is a nonzero polynomial and F(z) is an entire function, λ(F(z))<1, A(z)≠F(z), then f(z)=g(z).

On the fundamental group of complete noncompact Riemannian manifolds

CHEN Ai-yun, XUE Qiong, XIAO Xiao-feng

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 115-119. doi:10.6040/j.issn.1671-9352.0.2018.604

Abstract ( 1160 )

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We study the topology of complete noncompact Riemannian manifolds with Ricci curvature satisfies RicM≥-(n-1)k(k>0). By using the uniform estimates for the distance from a point to halfway point of minimal geodesics, we prove that a manifold with linear diameter growth has a finitely generated fundamental group.

Fractional Tikhonov method of a non-characteristic Cauchy problem for a parabolic equation

CHEN Ya-wen, XIONG Xiang-tuan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12): 120-126. doi:10.6040/j.issn.1671-9352.0.2018.607

Abstract ( 871 )

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The ill-posed non-characteristic Cauchy problem for a parabolic equation is considered. A fractional Tikhonov regularization method is applied to solve the the problem. Some stability error estimates under for a-poriori and a-posteriori choice rules are given.

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