Table of Content

  

20 August 2017
Volume 52 Issue 8

Representation of Choquet integral of the set-valued functions with respect to fuzzy measures and the characteristic of its primitive

GONG Zeng-tai, KOU Xu-yang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  1-9.  doi:10.6040/j.issn.1671-9352.0.2016.450

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The analytic properties of the Choquet integral of set-valued functions with respect to fuzzy measures are discussed, such as the characteristics of the primitive, representation of integral, differentiability of the primitive, and so on. Firstly, based on the previous results, the calculation of Choquet integral of set-valued function is investigated, and a representation theorem of Choquet integral for set-valued function is obtained as a Radon-Nikodym property in some sense. In addition, the characteristics of the primitive of the Choquet integral for set valued functions are given. Finally, the definition of the Choquet integral of set-valued functions with respect to fuzzy measures is improved, and the concepts of the above functions and below function of the set-valued functions are proposed, which achieved the domination of the Choquet integral of set-valued functions with respect to fuzzy measures.

Approximation of lagrange interpolation polynomials based on the fourth Chebyshev nodes

ZHANG Yan-yan, YAN Chao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  10-16.  doi:10.6040/j.issn.1671-9352.0.2016.315

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This paper first provides the exact approximation errors for an analytic function class under the maximum norm based on the Lagrange interpolation polynomials with the fourth Chebyshev nodes in the worst setting. As for the norm of L^p(p>1), the corresponding strong asymptotic order for that kind of analytic function class is obtained.

Existence and stability for multiple gabor frames

SONG Liang, FENG Jin-shun, CHENG Zheng-xing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  17-24.  doi:10.6040/j.issn.1671-9352.0.2016.189

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Since redundant frames are play very important role in signal processing and image processing, existence and perturbation for multiple Gabor frames are studied in this paper. The sufficient condition for the existence of multiple Gabor frames is proposed. It is shown that a function which is the generator of Gabor frames is also a generator of Gabor frames after it is perturbed. According to the frame stability criterion, we prove that a class of irregular Gabor frames are stable frames as well.

Expanded mixed finite element method for compressible miscible displacement in heterogeneous porous media

YU Jin-biao, REN Yong-qiang, CAO Wei-dong, LU Tong-chao, CHENG Ai-jie, DAI tao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  25-34.  doi:10.6040/j.issn.1671-9352.0.2017.040

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We consider the numerical methods for the mathematical model describing the transport and diffusion process of porous media flow. There are two kinds of fluids, one is displaced by another, which are compressible and miscible. The media is supposed to be heterogeneous, so the permeability is of full tensor form. For the pressure equation, an expanded mixed finite element method is introduced to solve the variables of pressure, gradient, and velocity. For the concentration equation, a Galerkin finite element formulation is constructed to solve the variable of concentration. This approach aims at obtaining more reliable numerical solutions for porous media flow with heterogenous permeability. By means of theoretical analysis, optimal error estimates in L²-norm for pressure and in H¹-norm for concentration are derived.

Convergence analysis for inverse problems in a degenerate parabolic equation

ZHANG Tai-nian, LI Zhao-xing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  35-42.  doi:10.6040/j.issn.1671-9352.0.2017.005

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This paper consider an inverse problem of reconstructing the principal coefficient in a second order degenerate parabolic equation of non-divergence form by using some additional observation data. It is transformed into an optimal control problem. The major problem is that the principal coefficient is unknown, but the degenerate degree of equation is determined, in general, by the principal coefficient. By introducing some weighted Sobolev spaces and some new source conditions, and adding a strong regularity condition to the admissible function set of the principal term, we prove the convergence of the optimal solution.

Some research about Wigner-Yanase-Dyson skew information

WANG Li-li, CHEN Zheng-li

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  43-47.  doi:10.6040/j.issn.1671-9352.0.2017.036

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Based on the study of some properties of the Wigner-Yanase-Dyson skew information and the Wigner-Yanase correlation, the Wigner-Yanase-Dyson skew information is given for different quantum channels. Finally, the concavity with the Wigner-Yanase-Dyson skew information is proven.

Characterizations of n-fold positive implicative filter in residuated lattice

LIU Li-jun

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  48-52.  doi:10.6040/j.issn.1671-9352.0.2016.491

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The concept of n-fold implicative filter and n-fold positive implicative filter are introduced in residuated lattice. By studying their properties, a series of characterization theorems are given. On this basis, the sufficient and necessary condition that non-empty subsets of residuated lattice become n-fold(positive)implicative filter are proposed. And the relation between n-fold implicative filter and n-fold positive implicative filter is discussed. The results of the study further extend the filter theory of the residuated lattice. It is important theoretical foundation for the application of algebraic logic and computer information processing.

Parametric estimations for renewal-geometric process

LIANG Xiao-lin, GUO Min, LI Jing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  53-57.  doi:10.6040/j.issn.1671-9352.0.2016.576

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According to the definition and the properties of renewal-geometric process, we get the parametric estimation of renewal-geometric process by Change-points detection via cross-validation and the improved least-squares method. Some simulation experiments are performed, and the simulation results show that the proposed method is effective.

A^† graded hard thresholding pursuit algorithm

SHI Zhang-lei, LI Wei-guo

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  58-64.  doi:10.6040/j.issn.1671-9352.0.2016.474

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Inspired by hard thresholding pursuit algorithm(HTP). A^† graded hard thresholding pursuit algorithm(APGHTP)was proposed for solving compressive sensing problems. The theoretical guarantees of the new algorithm were given under restricted isometry property(RIP)condition.In the numerical experiment, regardless of whether the measured value contains error, APGHTP performance is better, which proves the sparse recovery ability of the algorithm.When recovering sparse vectors, the number of iterations required for APGHTP is the same as that of sparse vectors.

A class of minimal inequalities for demimartingales

FENG De-cheng, ZHANG Xiao, ZHOU Lin

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  65-69.  doi:10.6040/j.issn.1671-9352.0.2016.532

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In the background of the demimartingale sequence {S_n, n≥1}, a class of minimal inequalities for the sequence {c_nS_n, n≥1} and {c_ng(S_n), n≥1} are given in this paper. These results generalize the corresponding results in recent papers.

On the mean value of a hybrid arithmetic function

LI Ting-ting, LAO Hui-xue

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  70-74.  doi:10.6040/j.issn.1671-9352.0.2016.505

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Let λ_f(n) be the Fourier coefficients of cusp forms of the normalized Hecke operator T_n, σ(n) be the sum-of-divisors function, and φ(n) be the Eulers function. The average order estimation of hybrid arithmetic functions of the form λ_f(n)σ^b(n)φ^c(n) is investigated by using analytic methods, and the upper bound estimate ∑_n≤xλ_f(n)σ^b(n)φ^c(n)≪x^b+c+1/2+ε,for any ε>0 is obtained.

Relative torsionless modules

XU Hui, ZHAO Zhi-bing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  75-80.  doi:10.6040/j.issn.1671-9352.0.2016.458

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Some elementary properties of relative torsionless modules with respect to a faithful and balanced bimodule are studied. Some equivalent conditions for a left or right Noether ring is a QF-ring is gotten. It is proved that a finitely generated module is an ω-torsionless module if and only if it is an ω-1-syzygy module if and only if it is an ω-1-torsionfree module when ω is a generalized tilting bimodule. Moreover, the extension closure of the class of ω-torsionless modules is investigated, and some classical results are generalized to the relative case.

Characterizations of rings relative to an ideal

WANG Yong-duo, HE Jian

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  81-84.  doi:10.6040/j.issn.1671-9352.0.2016.432

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Let I be an ideal of a ring R. The concept of pseudo semiprojective I-covers is introduced. It is shown that every left R-module has a pseudo semiprojective I-cover if and only if every left R-module has a proj-ective I-cover. It is also proved that the class of all pseudo semiprojective modules is a projectivity class, and then some well known results are generalized.

Homological dimensions with respect to semidualizing modules and excellent extensions

CHEN Xiu-li, CHEN Jian-long

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  85-89.  doi:10.6040/j.issn.1671-9352.0.2016.386

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Let C be a semidualizing R-module with R being a commutative ring. It is investigated the transfer properties of C-homological dimensions under(almost)excellent extensions, and it is discussed that the precovering and preenveloping properties of the C-projectives, C-injectives, and C-flats. As applications, it is proved that if S≥R is an excellent extension, then R is Noetherian if and only if S is Noetherian, and R is coherent if and only if S is coherent.

Prehomomorphisms and restricted products of regular semigroups with a multiplicative inverse transversal

WANG Shou-feng

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  90-93.  doi:10.6040/j.issn.1671-9352.0.2016.507

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Prehomomorphisms and restricted products of regular semigroups with a multiplicative inverse transversal are introduced. It is proved that the class of regular semigroups with a multiplicative inverse transversal together with prehomomorphisms form a category, by using restricted products some characterizations of prehomomorphisms between two regular semigroups with a multiplicative inverse transversal are obtained.

Neighbor sum distinguishing index of a kind of sparse graphs

PAN Wen-hua, XU Chang-qing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  94-99.  doi:10.6040/j.issn.1671-9352.0.2016.326

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Let φ be a proper k-edge coloring of G. For each vertex v∈V(G), set f_φ(v)=∑_uv∈E(G)φ(uv). φ is called a k-neighbor sum distinguishing edge coloring of G if f_φ(u)≠f_φ(v) for each edge uv∈E(G). The smallest k such that G has a k-neighbor sum distinguishing edge coloring is called the neighbor sum distinguishing index, denoted by χ'_Σ(G). The neighbor sum distinguishing index of a kind of sparse graphs is determined. It is proved that if G is a graph without isolated edges, Δ≥6 and mad(G)≤5/2, then χ'_Σ(G)=Δ if and only if G has no adjacent vertices of maximum degree.

Total coloring of planar graphs with maximum degree seven

WANG Xiao-li, WANG Hui-juan, LIU Bin

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  100-106.  doi:10.6040/j.issn.1671-9352.0.2016.363

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Let G be a planar graph with maximum degree Δ≥7. It is proved that if chordal 5-cycles of G are not adjacent to chordal 6-cycles, then its total chromatic number is Δ+1 by the discharging method.

The fundamental theorem forAM^H_H in Yetter-Drinfeld module categories

CHEN Hua-xi, XU Qing-bing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(8):  107-110.  doi:10.6040/j.issn.1671-9352.0.2016.582

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The definitions of weak Hopf algebra and weak comodule algebras in Yetter-Drinfeld module categories are introduced, and the fundamental theorem forAM^H_H in Yetter-Drinfeld module categories is obtained.