JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)

Gap functions and error bounds for generalized vector variational inequalities

CHEN Xia, CHEN Chun-rong

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 1-5. doi:10.6040/j.issn.1671-9352.0.2016.469

Abstract ( 1067 )

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Gap functions for generalized vector variational inequalities were established via linear scalarization approaches. By using some properties of generalized f-projection operators, the regularized gap function was verified. With the condition of the generalized strong pseudomonotonicity, error bounds were obtained.

A modified FR spectral conjugate gradient method with Wolfe line search

LIN Sui-hua

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 6-12. doi:10.6040/j.issn.1671-9352.0.2016.406

Abstract ( 1318 )

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A modified FR spectral conjugate gradient method is proposed for unconstrained optimization. This method can automatically generate descent direction at every iterations depending on no any line search. Under the conventional assumption, it is proved that the corresponding method with Wolfe line search is globally convergent. The numerical results show that the spectral conjugate gradient method is effective.

Precise asymptotics in the law of iterated logarithm for the moment convergence of ρ-mixing sequences

ZHANG Ya-yun, WU Qun-ying

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 13-20. doi:10.6040/j.issn.1671-9352.0.2016.249

Abstract ( 1118 )

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Let {X_n, n≥1}be a sequence of strictly stationary of ρ-mixing random variables with zero means and finite variances. Set S_n=∑X_i, M_n=max1≤i≤n|S_i|. Using the moment inequality and the central limit theorem of the ρ-mixing random variables.The precise asymptotics in the law of iterated logarithm for the moment convergence of ρ-mixing random variables of the partial sum and the maximum of the partial sum are obtained.

Convergence properties of the kernel-type density estimator under WOD dependent samples

HU Xue-ping, ZHANG Hong-mei

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 21-25. doi:10.6040/j.issn.1671-9352.0.2016.329

Abstract ( 1403 )

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Let {X_n,n≥1} be an identically distributed WOD random sequence with a commen density functiong f(x). Based on the Rosenthal-type inequality and Bernstein-type inequality for WOD sequence, the kernel estimator for density function f(x)was investigated under suitable conditions, and the consistency in r order mean, the pointwise strong consistency and uniform consistency in L¹ were obtained.

Comparison theorem for backward doubly stochastic differential equations driven by white noises and Poisson random measures

YANG Xu, LI Shuo

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 26-29. doi:10.6040/j.issn.1671-9352.0.2016.327

Abstract ( 1780 )

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In this paper, a comparison theorem for a class of backward doubly stochastic differential equations driven by white noises and Poisson random measures was established.

Vertex-distinguishing I-total colorings of the join of two paths

CHEN Xiang-en, MIAO Ting-ting, WANG Zhi-wen

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 30-33. doi:10.6040/j.issn.1671-9352.0.2016.242

Abstract ( 1370 )

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The problems of vertex-distinguishing I-total colorings and vertex-distinguishing VI-total colorings of the join of two paths are discussed by using the method of constructing concrete coloring. Vertex-distinguishing I-total chromatic numbers and vertex-distinguishing VI-total chromatic numbers of P_m∨P_n are determined,and the VDITC conjecture and VDVITC conjecture are valid for P_m∨P_n.

Every 1-planar graph without cycles of length 3 or 4 is 5-colorable

WANG Ye, SUN Lei

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 34-39. doi:10.6040/j.issn.1671-9352.0.2016.171

Abstract ( 1284 )

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A graph G is called 1-planar graph if it can be drawn on the plane so that each edge is crossed by at most one other edge. A propervertexcoloring of G is an assignment φ of integer to the vertex of G such that φ(u)≠φ(v)if the vertex u and v adjacent in G. A k-coloring is a proper vertex coloring using k colors at least. The theorem that every 1-planar graph without cycles of length 3 or 4 is 5-colorable was given by the Discharging Method.

The pointwise estimates of solutions to the Cauchy problem of Keller-Segel equations with cross-diffusion

DUAN Shuang-shuang, QIAN Yuan-yuan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 40-47. doi:10.6040/j.issn.1671-9352.0.2016.330

Abstract ( 1334 )

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We consider the Canchy problem for a class of Keller-Segel equations with cross-diffusion. By utilizing the method of Greens function, we obtain the pointwise estimates of solutions to the Cauchy problem for small initial data, and the W ^s,p(1≤p≤∞) decay properties of solutions.

Existence of radial positive solutions of second-order semi-positone elliptic differential equations

LI Tao-tao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 48-55. doi:10.6040/j.issn.1671-9352.0.2016.485

Abstract ( 1325 )

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We consider the existence of radial positive solutions of second-order semi-positone elliptic differential equationΔu+λg(|x|)f(u)=0, R1<|x|<R2,with Dirichlet and Robin boundary conditions, where g∈C(［R1,R2］,［0,+∞)), f∈C(［0,+∞), R). The proof of the main results are based on the fixed point theorems in cones.

Lower bounds for blow up time of two nonlinear wave equations

DONG Li

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 56-60. doi:10.6040/j.issn.1671-9352.0.2016.224

Abstract ( 1049 )

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The initial boundary value problem for the nonlinear viscoelastic euqation with strong damping term and dispersive term and the nonlinear Petrovsky equation is investigated. Under the premise of the solutions blow up of the equations, the lower bound of the blow up time is obtained by the proper perturbation.

The judgement for the small compact perturbation of SVEP for upper triangular operator matrices

SONG Jia-jia, CAO Xiao-hong, DAI Lei

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 61-67. doi:10.6040/j.issn.1671-9352.0.2016.521

Abstract ( 998 )

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When A∈B(H), B∈B(K)are given, we denote by M_C an upper triangular operator matrix, acting on the Hilbert space H⊕K, of the form M_C=(A C0 B). The properties of the component A,B are discussed in the matrix such that M_C satisfies the single-valued extension property of small compact perturbation by defining a kind of new resolvent set. The necessary and sufficient conditions for upper triangular operator matrix which satisfies the single-valued extension property under small compact perturbation are studied, and some examples are given to illustrate the essence of the conditions given in the main theorem.

Some constants in a summation formula

GUO Ru-ting, WANG Ming-qiang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 68-71. doi:10.6040/j.issn.1671-9352.0.2016.190

Abstract ( 1070 )

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Some constants are given in an asymptotic formula. The constants are about the mean value estimates of the divisor function of a quadratic form. We have evaluated two singular series and one singular integral. We also have given the analytic formula of the other singular integral.

Dynamic behavior analysis of a stochastic SIR epidemic model

BAI Bao-li, ZHANG Jian-gang, DU Wen-ju, YAN Hong-ming

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 72-82. doi:10.6040/j.issn.1671-9352.0.2016.614

Abstract ( 1802 )

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Taking the random factors into account,we introduced the randomness into the SIR epidemic model and established the nonlinear differential equation of the random SIR epidemic model with stage structure. Then by applying stochastic center manifold and stochastic average method,the stochastic differential equation was reduced order and we got the corresponding Ito differential equation. Based on the Oseledec multiplicative ergodic theorem,the conditions of local and global stability of the system were discussed by using the largest Lyapunov exponent and boundary category. Besides,we selected some of these parameters as the bifurcate parameter,and the stochastic Hopf bifurcation behavior of the system were analyzed by the stochastic averaging method of the quasi-non-integrable Hamiltonian system. Finally,the functional image of stationary probability density and jointly stationary probability density were simulated,and the bifurcate point from the probability and location was verified.

Estimates for the Hessian of f-harmonic functions and their applications to splitting theorem

DENG Yi-hua

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 83-86. doi:10.6040/j.issn.1671-9352.0.2016.364

Abstract ( 1439 )

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A new estimate for the Hessian of f-harmonic functions is obtained. Using the new estimate, we improve a splitting theorem on smooth metric measure space with weighted Poincaré inequality under the condition that the Bakry-Émery Ricci curvature is bounded from below by some negative functions.

Surplus-loss value theorems of attribute cardinal numbers and internal-outer separation of information laws

XU Feng-sheng, YU Xiu-qing, SHI Kai-quan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 87-92. doi:10.6040/j.issn.1671-9352.0.2016.509

Abstract ( 1295 )

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Function packet sets(function P-sets)is a special form of packet sets(P-sets). Based on the rasearch on the attribute sets of function P-sets, some concepts are proposed, such as the attribute cardinal number, information separation, and so on. Accordingly, internal separation theorem, outer separation theorem, and internal-outer separation theorem about information laws are derived by a series of reasoning. Finally, the new theoretical results are applied to camouflage-hiding of information images.

P-information fusion and its P-matrix reasoning intelligent generation

ZHANG Xiu-quan, LI Xiao-chao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 93-99. doi:10.6040/j.issn.1671-9352.0.2016.338

Abstract ( 1366 )

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Based on P-augmented matrix reasoning, the theory and application research of information fusion intelligent screening-discovery are presented. Information fusion intelligent screening-discovery is obtained under the condition of P-augmented matrix reasoning. P-augmented matrix is obtained by using the structure and dynamic characteristics of P-set and improving common augmented matrix. The method for generating P-augmented matrix by using P-set and the structure of P-augmented matrix are given. The reasoning model of P-augmented matrix and the intelligent generation method of information fusion are presented. The information fusion intelligent screening-discovery theorem and the application of these theoretical results are given.

The digraphs representation of finite topologies

MA Hai-cheng, LI Sheng-gang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 100-104. doi:10.6040/j.issn.1671-9352.0.2016.362

Abstract ( 1227 )

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A digraph which is called topology graph is defined for each finite topology space. First, an equivalence relation between elements is defined, and then, the equivalence classes is came into being. These digraphs is defined by using the inclusion relation between equivalence classes. Topological space and its topology graph to determine each other is proved. It is easy to calculate the closure, the derived set, the interior and border of a set by using the topology graph. The connectedness consistent of the topological space and its topology graph is proved. The number of non-homeomorphism topology with 1≤n≤4 elements is calculated by using the topology graph.

Stabilization for discrete-time stochastic systems with multiple input delay

GAO Rong, ZHANG Huan-shui

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(4): 105-110. doi:10.6040/j.issn.1671-9352.0.2016.075

Abstract ( 930 )

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This paper mainly investigates the mean square stabilization problem for discrete-time stochastic system with multiple input delay and multiplicative noises based on receding horizon control(RHC). First, a new cost was designed and the property of this cost function was studied. Then the sufficient mean square stabilization condition was obtained. This condition can be verified by solving linear matrix inequality. The explicit controller can be derived under this condition.

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